DEVELOPMENT OF LIQUID CRYSTAL INFRARED IMAGING SENSORS. A dissertation submitted. to Kent State University in partial

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1 DEVELOPMENT OF LIQUID CRYSTAL INFRARED IMAGING SENSORS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Valerie Finnemeyer August 2016 i

2 Dissertation written by Valerie Finnemeyer B.S. Allegheny College, Meadville PA 2007 M.S. Drexel University, Philadelphia, PA 2010 Ph.D., Kent State University, Kent, OH 2016 Approved by, Dr. Philip Bos Chair, Doctoral Dissertation Committee, Dr. Deng-ke Yang Members, Doctoral Dissertation Committee, Dr. Liang-Chy Chien, Dr. John West. Dr. Elizabeth Mann Accepted by, Dr. Hiroshi Yokoyama Chair, Department of Chemical Physics, Dr. James Blank Dean, College of Arts and Sciences ii

3 Table of Contents List of Figures..viii List of Tables xxi Dedication xxii Acknowledgements.xxiii Chapter 1: Introduction.1 Chapter 2: Introduction to Spectrometer Systems Introduction Wavelength Regimes Visible Spectroscopy IR Spectroscopy Spectrometers Dispersive Spectrometers Fourier-Transform Spectrometers Michelson Interferometer Birefringent Interferometers Imaging Spectrometers Michelson Imaging Spectrometers Birefringent Imaging Spectrometers The Polarization-Interference Fourier-Transform Spectrometer (PIFTIS) System Principles of Operation Device Limitations...17 Chapter 3: Wide-View Polarization Interference Fourier-Transform Spectrometer: Design and Simulation Introduction 19 iii

4 3.2 Our Device Design Principles of the Bi-Crystal Design Methods Basic System Parameters Optical Path Calculations Michelson Interferometer Polarization-Interference Spectrometer Simulated Interferogram Michelson Interferometer Polarization-Interference Spectrometer Simulated Spectra Zemax Simulations Results Off-axis OPD Change Simulated Spectra Spectral Degradation Amplitude Loss Spectral Noise Beam Walk-off Conclusion Future Work...50 Chapter 4: Consideration of Variable Elements Introduction Polarization Rotators Hybrid Designs Methods.54 iv

5 4.2.1 Simulated Liquid Crystal Cells TN Cell Thickness Director Configuration Signal Generation Optical Path Difference Scan Time Results Determining TN Cell Thickness Spectral Degradation in Hybrid Models Optical Path Difference Scan Speed Conclusions 67 Chapter 5: Introduction to the Liquid Crystal Thermal Imager Thermal Imaging Thermal Imaging Systems Liquid Crystal Thermal Imager Principles of Operation System Design Alignment in the Liquid Crystal Thermal Imager Mechanical Generation of Grooves Director Fluctuations and Noise Fluctuation Modes Isolating Fluctuation Modes Addition of Quarter-wave Plate Maximizing Signal to Noise...89 Chapter 6: Reactive-Mesogen Stabilized Azo Dye Photoalignment...93 v

6 6.1 Introduction Photoalignment Techniques Azo Dye Photoalignment Surface-Localized Polymer Layer Model for Verifying Surface-Localization Verifying Surface-Localization of the Polymer Layer Experimental Details Results Photostability Testing Experimental Details Results Without Polymer Layer With Polymer Layer Anchoring Energy Details of the Method Experimental Details Results Conclusions Chapter 7: Optimization of the Surface-Localized Polymer Stabilization Layer for Infused Photoalignment Introduction Experimental Details Surface-Localization Photostability in Infused Bulk Cells Results Phase Retardation vs. Voltage Curves.116 vi

7 Samples Polymerized at 0V Samples Polymerized at 10V Considering Polymer Layer Thickness Thickness of Fully Separated Polymer Thickness of Polymer to Generate Effective Pretilt Results Mathematical Model Development of Simulation Simulated Results Comparing Simulation and Experiment Photostability Conclusion..139 Chapter 8: Conclusion Appendix A: Fourier-Transform Simulations Appendix B: Modeling Director Fluctuations on the Poincare Sphere.175 Appendix C: Development of Infused Photoalignment Cells 186 Appendix D: Photostability and Polymer Separation in Other Experimental Conditions Appendix E: Details of Surface Localization Modeling 203 Appendix F: Experimental Procedures and Methods 217 References..232 vii

8 List of Figures Figure 2.1. Schematic diagram of a basic Fourier-transform spectrometer 14. Figure 2.2. Schematic of a Michelson interferometer based Fourier-transform spectrometer 15. Figure 2.3. Wollaston prism and the interference pattern it creates 21. Figure 2.4. Optical scheme of a static Fourier-transform spectrometer using a modified beam splitter cube 24. Figure 2.5. Generation of path difference for off-axis rays through a Michelson interferometer 28. Figure 2.6. Schematic of a Polarization-Interference Fourier-Transform Spectrometer utilizing birefringent elements 28. Figure 3.1. Diagram of a uniaxial-crystal polarization-interference Fourier-transform spectrometer Figure 3.2. Viewing angle dependence of the retardation through a biaxial film with different relative indices as defined by the factor, N z, given in Equation Figure 3.3. Proposed dual-crystal design mimicking wide-view Lyot filter design. n x=n z in Crystal 1 and n y=n z in Crystal 2. Figure 3.4. Diagrammatic representation of different azimuthal angles of incidence with varying polar incident angle of light. (Left) Variation in polar angle of incidence in this orientation causes variation of effective extraordinary index of refraction. (Right) Variation in polar angle of incidence in this orientation does not cause a variation in the extraordinary index of refraction. Figure 3.5. Off-axis change in net retardation through a 5μm thick cell with a uniaxial material in planar orientation (n e=1.75 and n o=1.5) for in-plane and out-of-plane polar variations in angle based on Figure 3.4 (with light propagating from the left-hand side). Figure 3.6. Light propagation through a Michelson interferometer 28. Figure 3.7. Geometric diagram of off-axis change in OPD in a Michelson interferometer. Figure 3.8. Geometric diagram of off-axis changes in OPD through a birefringent crystal stages. viii

9 Figure 3.9. Example interference pattern from Michelson interferometer simulation using a single input signal with =0.8 m [Michelson Simulated code included in Appendix A, flowchart on page 147. Input parameters: N=0.8, lamd=0.8, DH=14, θ=0]. Figure Example interference pattern from Polarization-Interference Spectrometer simulation using a single input signal with =0.8 m [Digital Simulated code included in Appendix A, flowchart on page 150. Input parameters: lam=0.8, lamd=0.8, DH=14, θ=0]. Figure Example of spectrum output from Michelson interferometer simulation using a single input signal with λ=0.8μm [Michelson Simulated code included in Appendix A, flowchart on page 147. Input parameters: N=0.8, lamd=0.8, DH=14, θ=0]. Figure Zemax diagram of a single stage dual crystal system with 400μm retardation and 20-degree off-axis light propagation. Figure Incident angle limit based on OPD criterion. Comparison between single-crystal and dualcrystal models as well as the equivalent Michelson interferometer dependent on device size [Code shown in Appendix A (A.7.7)]. Figure Simulated output spectra of 14-stage dual-crystal WA-PIFTS (BC) and single-crystal PIFTS (U) interferometers as well as the equivalent Michelson interferometer (M) design. Simulated for on-axis (0) and 25-degrees off-axis (25) propagation of 0.8 m light. [Simulated code in appendix A. Michelson (flowchart on page 147, input parameters: N=0.8, lamd=0.8, DH=14, θ=0,25; Digital single and dual crystal models (flowchart on page 150, input parameters: lam=0.8, lamd=0.8, DH=14, θ=0,25]. Figure Simulated output spectra of 14-stage dual-crystal WA-PIFTS for on-axis (0) and 25-degrees off-axis (25) propagation of 0.8 m light. [Simulated code in Appendix A. Digital dual crystal models (flowchart on page 147, input parameters: lam=0.8, lamd=0.8, DH=14, θ=0,25]. Figure Incident angle limit using amplitude loss criteria. Comparison between uniaxial and bicrystal models as well as the equivalent Michelson interferometer dependent on system size. ix

10 [Simulated code in Appendix A. Michelson (flowchart on page 147, input parameters: N=0.8, lamd=0.8, DH=14, θ=0-40; Digital single and dual crystal models (flowchart on page 150, input parameters: lam=0.8, lamd=0.8, DH=14, θ=0-40]. Figure Incident angle limit using spectral noise criterion. Comparison between single-crystal and dual-crystal models as well as the equivalent Michelson interferometer dependent on system size [Simulated code in Appendix A. Michelson (flowchart on page 147, input parameters: N=0.8, lamd=0.8, DH=14, θ=0-90; Digital single and dual crystal models (flowchart on page 150, input parameters: lam=0.8, lamd=0.8, DH=14, θ=0-90]. Figure Walk-off of the extraordinary ray at 5-degrees off axis or angle at which 10μm walk-off will occur at maximum path difference through a single Calcite-Quartz dual-crystal stage. Figure 4.1. Schematic diagram of the hybrid designs. P=Polarizer, R=Polarization Rotator, C=Compensation Element, E=ECB Cell, B=Birefringent Crystal (+ or indicate sign of birefringence), m, n indicate repeating units. Figure μm and 2.00μm spectral peak for 13-stage digital dual-crystal model for (a) ideal polarization rotators and various TN cell thicknesses (b) 1 st transmission minimum, (c) 3 rd transmission minimum and (d) 4 th transmission minimum. Shown for on-axis signal propagation [Digital: lam=1.99,2.00, lamd=0.8, DH=13, θ=0, (a) U=1, (b) U=2, (c) U=3, (d) U=4]. Figure µm spectral peak for 14-stage digital dual-crystal model for selected TN cell thickness for on-axis signal propagation and 10 and 25 degree off-axis signal propagation [Simulated code in Appendix A. Digital (flowchart on page 150) with TN cells (code in Section A.7.2 and A.7.3), Input parameters: lam=0.8, lamd=0.8, DH=14, θ=0,10,25, U=4]. Figure m spectral peak for a 14-stage hybrid analog design HA8 with dual-crystal design for onaxis signal propagation and 10-degree off-axis signal propagation [Simulation code Section A.7.4 and A.7.5. Input parameters: lam=0.8, lamd=0.8, DH=14, HA=8, θ=0,10]. x

11 Figure 4.5. Incident angle limit using OPD criterion. Comparison of the Hybrid systems compared with single and dual-crystal PIFTS based on total system size (effective total crystal stages) and the number of fixed stages replaced with variable elements [Simulated code in Appendix A (A.7.8), ECB Cell Thicknesses shown in Table 4.1]. Figure 4.6. Scan speeds calculated for Hybrid systems compared with digital PIFTS based on total system size (effective total crystal stages) and the number of fixed stages replaced with variable elements. Figure 5.1. Diagram of a single microbolometer 47. Figure 5.2. Temperature-dependence of the order parameter for a typical liquid crystal 50. Figure 5.3. Diagram of a thermal imaging system utilizing temperature-dependent changes in liquid crystal birefringence 48. Figure 5.4. Experimental [MITLL] (top) and simulated (bottom) transmission vs. temperature curves. 2μm thick 5CB cell with incident light of λ=505nm. Figure 5.5. (a) Cartoon thermal pixel cross-section and (b) SEM photograph of a thermal pixel 48. Figure 5.6. Poor-quality alignment generated on scratched silicon nitride substrates when fabricated into a 90-degree twist cell. Shown under the microscope at 5X between crossed polarizers. Figure 5.7. Simplified schematic of optical system. Figure 5.8. Transmission and noise in Senarmont setup measured with and without quarter-wave plate. [MITLL] Figure 5.9. Illustration of twist fluctuations in cell viewed from the front (left) and splay fluctuations in cell viewed from the side (right). Figure Poincare sphere plot of a fixed director configuration in a liquid crystal cell with no fluctuation (left), in-plane fluctuation (center), or out-of-plane fluctuation (right). [Left: σ T=0deg, σ s=0deg; Center: σ T=10deg, σ s=0deg; Right: σ T=0deg, σ s=10deg] xi

12 Figure Poincare sphere (left) and noise (right) of a moving director model with in-plane fluctuations (top) or out-of-plane fluctuations (bottom). [Top: σ T=5deg, σ s=0deg, nloops=50; Bottom: σ T=0deg, σ s=5deg, nloops=50] Figure Original experimental configuration without quarter-wave plate. Figure Modified setup with liquid crystal and polarizer perpendicular to one another. Figure Poincare sphere illustration (top) and noise (bottom) of circularly polarized light with inplane fluctuations (left), out-of-plane fluctuations (center), or both fluctuation types (right). [Left: σ T=5deg, σ s=0deg, nloops=50; Center: σ T=0deg, σ s=5deg, nloops=50; Right: σ T=5deg, σ s=5deg, nloops=50] Figure Experimentally detected noise for circularly polarized light using setup shown in Figure 5.12 [MITLL]. Figure Experimentally detected [MITLL] (left) and simulated (right) noise for setup shown in Figure Simulation included both fluctuation types. [Simulated: σ T=5deg, σ s=5deg, nloops=50] Figure Poincare sphere plot of a fixed director configuration of a liquid crystal cell (green) then a quarter-wave plate (blue) with no fluctuation (left), in-plane fluctuation (center), or out-of-plane fluctuation (right). [Left: σ T=0deg, σ s=0deg; Center: σ T=10deg, σ s=0deg; Right: σ T=0deg, σ s=10deg] Figure Poincare sphere (top) and noise (bottom) of a moving director with in-plane fluctuations (left), out-of-plane fluctuations (center) or both (right). [Left: σ T=5deg, σ s=0deg, nloops=500; Center: σ T=0deg, σ s=5deg, nloops=500; Right: σ T=5deg, σ s=5deg, nloops=500] Figure Spatial correlation between points as it depends on the distance, r, between those points and the thickness, d, of the cell. Figure Relationship between anchoring energy, cell thickness, and dominant fluctuation mode (wavenumber) for the case of weak anchoring. Figure 6.1. Molecular structure of Brilliant Yellow azo dye 61. xii

13 Figure 6.2. Absorption spectrum of BY on glass substrate spun on using 1%wt BY in DMF. Figure 6.3. (a) Monomers are present in the liquid crystal sample which exist in some distribution across the cell normal, localized near the surfaces. (b) Applied voltage will reorient the liquid crystal. (c) UV illumination will crosslink the monomers, locking in the liquid crystal orientation during this stage. (d) After UV illumination, this liquid crystal orientation is locked in (where the polymer network exists), even after voltage is removed. Figure 6.4. Molecular structure of reactive mesogen RM257. Figure 6.5. Simulated dielectric data using (a) surface concentration X 0=0.08 and decay length ξ=0.2d and (b) surface concentration X 0=0.8 and decay length ξ=0.02d 77. Figure 6.6. Simulated dielectric data in which the polymer network is assumed to be infinitesimally thin (i.e. highly localized) 77. Figure 6.7. Diagram of cells prepared for testing surface localization of reactive mesogen. Polymide alignment layers on each substrate were rubbed in opposite directions (as indicated). Figure 6.8. Phase profiles for 5µm planar cells filled with pure LC or LC with 1.5%wt RM257 which was polymerized at either 0V or 20V. Figure 6.9. Expected visual appearance of cells (when viewed between crossed and parallel polarizers) depending on the condition of the photoalignment layer following exposure to either polarized blue light at 20mW/cm 2 oriented at 45 degrees to the photoalignment axis (top) or unpolarized blue light at 120mW/cm 2 (bottom). Figure Sample filled with pure BL006 (a) before exposure or after (b) 50 minutes, (c) 100 minutes, (d) 150 minutes, and (e) 200 minutes of exposure to ~20mW/cm 2 blue light oriented at 45 degrees to the photoalignment axis. Before images shown between crossed (left) and parallel (right) polarizers. After images shown between parallel (left) and crossed (center) polarizers as well as polarized oriented at 45 degrees (right). Figure Samples filled with pure BL006 shown (a) before and (b) after exposure to unpolarized blue light at ~120mW/cm 2 for 20 minutes. Shown between parallel (left) or crossed (right) polarizers. xiii

14 Figure Samples filled with 1.5%wt RM257 in BL006 shown between parallel (left) or crossed (right) polarizers. (a) Before and (b) after exposure to polarized blue light of ~20mW/cm 2 for 21 days and (c) before and (d) after exposure to unpolarized blue light of ~120mW/cm 2 for 21 days. Figure TV curves of 5µm samples with rubbed polyimide alignment on one substrate and BY alignment on the other substrate, aligned in 90-degree twisted configuration. Curve taken between crossed polarizers with the sample aligned with the polarizers as shown in the embedded diagram. Sample either unexposure (pure BL006) or exposed to 120mW/cm 2 unpolarized blue light for 3 weeks (1.5%wt RM257 in BL006). Figure Diagram of experimental setup used to measure twist angle of TN cells for the purpose of determining anchoring energy. Figure 7.1. Monomers distribution along the cell gap direction under different mixing conditions. Bright areas mean the dye is more concentrated. 0.66% RM84, 0.08% dye 79. Figure 7.2. Monomers distribution along the cell gap direction under different mixing conditions. Intensity was measured along a random vertical cross-section of each image shown in Figure Figure 7.3. Simulated (blue) and experimental (orange) data for samples filled with 0.5%wt RM257 in E7 vortexed for either 10 seconds (top) or 6 minutes (bottom) and separated for 1hr (left) or 1 day (right) for cells polymerized at 0V. Figure 7.4. Simulated (blue) and experimental (orange) data for samples filled with 1%wt RM257 in E7 vortexed for either 10 seconds (top) or 6 minutes (bottom) and separated for 1hr (left) or 1 day (right) for cells polymerized at 0V. Figure 7.5. Simulated (blue) and experimental (orange) data for samples filled with 2%wt RM257 in E7 vortexed for either 10 seconds (top) or 6 minutes (bottom) and separated for 1hr for cells polymerized at 0V. Figure 7.6. Normalized retardation vs. voltage for all samples polymerized at 10V. xiv

15 Figure 7.7. Effective pretilt angle of 1.5μm cells filled with 0.5%, 1%, or 2%wt pre-polymer in liquid crystal vortexed for 10 seconds or 6 minutes and polymerized at 10V 1 hour or 1 day after filling. Figure 7.8. Director angle in the first 20nm near the substrate surface in a simulated cell (no polymer) with 10V applied. Red shows the area near the surface through which the polymer network must exist (to lock in the 10.2 degree pretilt in this cell). Figure 7.9. Calculated exponential decay of the coupling strength of a surface-localized polymer network in a 1.5μm cell based on Equation 1 for various decay lengths, ξ. Figure Simulated phase curves of samples with either a weaker (top) or stronger (bottom) average coupling strength, W avg, and varying values of the decay length, ξ, for cells polymerized at either 0V (left) or 10V (right). Figure Zero volt retardation vs. decay length for simulated RM-stabilized cell polymerized at 10V with W avg=6000. Figure Proportion of pre-polymer between the substrate and various points through the cell with ξ=0.025 (left) or ξ=0.25 (right). Figure Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) for a selected sample at each pre-polymer concentration. Figure Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 6 minutes and separated for 1 hr prior to polymerization. Experimental data simulated with W avg=16000 and ξ= Figure Comparison between simulated and experimental phase vs. retardation curves for samples with 2%wt RM vortexed for 10 seconds and separated for 1 day polymerized at 0V (left) or 10V (right) with W avg=40000 and ξ= Figure Infused photoaligned cells filled with pure E7 (no RM) or 0.5%wt pre-polymer in E7 vortexed for 10 seconds or 6 minutes and polymerized at 0V 1 hour or 1 day after filling. Shown xv

16 on the light table between crossed polarizers with the photoalignment axis oriented along the original photoalignment axis. Each cell is shown before and after 20 min exposure to 20mW/cm 2 blue (447nm) light polarized at 45 degrees to the original photoalignment axis. Figure Infused photoaligned samples filled with 1%wt RM257 in E7 (a) vortexed for 10 seconds and polymerized 1 hour after filling, (b) vortexed for 10 seconds and polymerized 1 day after filling, (c) vortexed for 6 minutes and polymerized 1 hour after filling, or (d) vortexed for 6 minutes and polymerized 1 day after filling. Shown on the microscope at 5X between crossed polarizers with the original photoalignment axis along one of the polarizers (i.e. the dark state). (Top) Before photostability testing; (bottom) after 5 weeks of exposure to 20mW/cm 2 blue light polarized at 45 degrees to the initial photoalignment axis. Figure B.1. Poincare Sphere Figure B.2. Azimuthal and equatorial angles on the Poincare sphere. Figure B.3. Simplified schematic of optical system. Figure B.4. Example Poincare sphere for a fixed director simulation. Figure B.5. Example Poincare sphere fixed director simulation with in-plane director fluctuations. [σ T=10deg, σ s=0deg]. Figure B.6. Example of transmitted intensity versus analyzer angle for single photon Poincare sphere model without quarter-wave plate (green) and with quarter-wave plate (blue). Figure B.7. Example Poincare sphere moving director simulation [σ T=5deg, σ s=5deg, nloops=50]. Figure B.8. Example of noise vs. analyzer angle for the moving director Poincare sphere model without quarter-wave plate (green) and with quarter-wave plate (blue). [σ T=5deg, σ S=0deg, nloops=500]. Figure C.1. Infused photoaligned samples prepared with 2-day-old (left) or fresh (right) 3%wt BY in DMF. Shown on the light table between crossed polarizers oriented with the photoalignment layer along one of the polarizers. Figure C.2. Infused photoaligned cells with identical preparation except preparation occurred when fabrication room was at 37% relative humidity (left) or 45% relative humidity (right). Shown on xvi

17 the light table between crossed polarizers oriented with the photoalignment axis along one of the polarizers. Figure C.3. (Top) Images of an infused cell at 10 seconds into the solvent evaporation stage. (Bottom) The same cell shown after complete processing and fill on the light table between crossed polarizers oriented with the photoalignment axis along one of the polarizers. Figure C.4. (a) DI water selectively wetting plain glass region and dewetting from central region where ITO-coating remains. (b) Infused photoaligned prepared using glass which exhibited this selective wetting phenomenon. Shown filled with pure E7 on light table between crossed polarizers with photoalignment axis along one of the polarizers (dark state). (c) The same cell shown on the microscope at the edge of the ITO-coated region. Also shown in the dark state between crossed polarizers. Figure C.5. Infused photoalignment cells prepared with ITO-coated glass Figure C.6. Infused photoalignment cells with a 1 hour bake in a vacuum bake at a reduced temperature of 70C. Shown on the light table between crossed polarizers oriented with the photoalignment axis along one of the polarizers. Figure C.7. Microscopic texture in poorly aligned regions of infused photoaligned liquid crystal cells. Shown on the microscope between crossed polarizers (with photoalignment axis oriented along one of the polarizers) at 5X under normal camera exposure conditions (left) or 50X with 10-times increase in camera exposure time (right). Figure C.8. Various textures of a dried dye droplet on glass. Shown on the microscope at 50X between crossed polarizers. Figure C.9. Example microcavity geometry; (a) side-view and (b) top-view. Figure C.10. Bright (a, c) and dark (b, d) state of photoaligned LC microcavities, between crossed polarizes, which operate in either reflective (a, b) or transmissive (c, d) mode. Photoalignment layer prepared using 0.5%wt BY in DMF. Microcavity diameter is ~12μm (top) or ~20μm (bottom). Bright and dark images for each taken with equal exposure/light settings. xvii

18 Figure C.11. Infused photoalignment cell which has been disassembled then reassembled into a twisted configuration. Shown on the light table between crossed (left) and parallel (right) polarizers. Figure C.12. Liquid crystal director configuration through a defect line. Figure D.1. 5µm planar samples filled with 1.5%wt RM257 in either E7 or BL006 vortexed for 6 minutes then separated for 30 minutes, 2 hours, or 1 day prior to polymerization at 50V (1kHz) using 3.5mW/cm 2 UV light for 10 minutes. Figure D.2. 5µm cells filled with 1.5%wt RM257 in BL006 vortexed for 10 seconds or 6 minutes, then separated for 30 minutes prior to polymerization at 50V (1kHz) using UV (365nm) light for 10 minutes at either 3.5mW/cm 2 or 14mW/cm 2. Figure D.3. 5µm cells filled with 1.5%wt RM257 in E7 vortexed for 10 seconds or 6 minutes, then separated for 30 minutes prior to polymerization at 50V (1kHz) using UV (365nm) light for 10 minutes at either 3.5mW/cm 2 or 14mW/cm 2. Figure D.4. 5µm or 1.4µm planar cells filled with either 1%wt or 2%wt RM257 in BL006, vortexed for 10 seconds then separated for 2 hours prior to polymerization at 50V (for 5µm) or 10V (for 1.5µm) using 3.5mW/cm 2 UV (365nm) light. Figure D.5. 5μm thick spun photoalignment samples filled with 0.1%wt (left) or 0.5%wt (right) RM257 in E7 shown before (top) or after (bottom) 20 minutes of photoexposure at 20mW/cm 2 polarized at 45-degrees to the photoalignment axis. Figure D.6. 5μm thick spun photoalignment sample filled with 1%wt RM257 in E7 shown before (a) or after 20 minutes (b), 3 hours (c), or 1 day (d) of photoexposure at 20mW/cm 2 polarized at 45- degrees to the photoalignment axis. Figure E.1. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 0.5%wt RM vortexed for 10 seconds and separated for 1 hr prior to polymerization. Experimental data simulated with W avg=4000 and ξ= xviii

19 Figure E.2. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 0.5%wt RM vortexed for 10 seconds and separated for 1 day prior to polymerization. Experimental data simulated with W avg=4000 and ξ= Figure E.3. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 0.5%wt RM vortexed for 6 minutes and separated for 1 hr prior to polymerization. Experimental data simulated with W avg=4000 and ξ= Figure E.4. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 0.5%wt RM vortexed for 6 minutes and separated for 1 day prior to polymerization. Experimental data simulated with W avg=4000 and ξ= Figure E.5. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 1%wt RM vortexed for 10 seconds and separated for 1 hr prior to polymerization. Experimental data simulated with W avg=8000 and ξ= Figure E.6. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 1%wt RM vortexed for 10 seconds and separated for 1 day prior to polymerization. Experimental data simulated with W avg=8000 and ξ= Figure E.7. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 1%wt RM vortexed for 6 minutes and separated for 1 hr prior to polymerization. Experimental data simulated with W avg=8000 and ξ= Figure E.8. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 1%wt RM vortexed for 6 minutes and xix

20 separated for 1 day prior to polymerization. Experimental data simulated with W avg=8000 and ξ= Figure E.9. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 10 seconds and separated for 1 hr prior to polymerization. Experimental data simulated with W avg=16000 and ξ= Figure E.10. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 10 seconds and separated for 1 day prior to polymerization. Experimental data simulated with W avg=16000 and ξ= Figure E.11. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 6 minutes and separated for 1 hr prior to polymerization. Experimental data simulated with W avg=16000 and ξ= Figure E.12. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 6 minutes and separated for 1 day prior to polymerization. Experimental data simulated with W avg=16000 and ξ= xx

21 List of Tables Table 2.1. Different wavelength ranges in IR Spectroscopy Table 3.1. Maximum path difference and resolution for different numbers of stages. Table 4.1. ECB cell thickness and switching time for various Hybrid Analog models. Table 6.1. Calculated anchoring energies (in J/cm 2 ) for twist cells filled with either pure E7 (photoalignment) or RM257 in E7 (RM-stabilized photoalignment). Table 7.1. Comparing theoretical RM thickness with distance from the surface required to lock-in the measured pretilt in the cell. Table 7.2. Estimated decay lengths for the various experimental cells based on the effective pretilt calculated for cells polymerized at 10V and average coupling strength, W avg, and decay length, ξ, used to match simulated and experimental data for the various experimental conditions. Table C.1. Outline of processing steps for fabrication of infused photoaligned bulk cells. Table C.2. Outline of processing steps for fabrication of infused photoaligned microcavities. Table C.3. Defect line width measured in liquid crystal microcavities with various alignment layers. Table D.1 Summary of RM-stabilized Photostability Results xxi

22 DEDICATION This dissertation is dedicated to my husband, whose constant support was integral to my ability to reach my goals, and to my two children, who made me push myself harder in order to achieve those goals. xxii

23 ACKNOWLEDGEMENTS I would like to start by thanking my advisor, Dr. Phil Bos, who helped me countless times along the way. He has provided a great deal of advice, feedback, and criticism that has been necessary for me to become a better researcher. I would also like to thank Doug Bryant for his help on a number of projects in both teaching me to use the various equipment in the cleanroom and providing fabrication support. I thank Dr, Robin Selinger for providing invaluable guidance and advice when needed, particularly when it came to navigating the start of my career with a husband and children. Additionally, I acknowledge and support of my research partners at MIT Lincoln Laboratory, namely Bob Reich, Shaun Berry, Carl Bozler, and Harry Clark, who provided valuable support and technical expertise. I learned a great deal both technically and professionally during my research collaboration with Lincoln Laboratory and believe that, without their interaction, I would not be on the path I am on today. Finally, I am extremely grateful to my husband, who not only bore the extra burden of my long work hours but also read/critiqued countless drafts of papers, reports, and this dissertation, and to my mother, who was a constant support along the way no matter what. Valerie Finnemeyer April 2016, Kent, Ohio xxiii

24 CHAPTER 1 Introduction Liquid crystals are a state of matter that exists, for some molecules, between the liquid and crystalline states. This state is possible due to the high aspect ratio of the molecules, which also gives them a number of anisotropic properties. Most commonly, liquid crystals are utilized in display applications where their optical and electrical anisotropies are leveraged to create switchable transmission when sandwiched between polarizers. There are, however, a number of other, more specialized, applications of liquid crystals including biological 1,2, photonic 3,4, beam steering 5,6, etc.. Additionally, while liquid crystals are generally utilized in visible ranges, their use has already begun to expand into other wavelength regions. This dissertation investigates applications which utilize liquid crystals in the infrared range of the electromagnetic spectrum. First, a birefringent Fourier-transform imaging spectrometer is described which would, ideally, operate in the near-infrared range from 0.8μm to 2.0μm. Chapter 2 provides an introduction to Fourier-transform spectroscopy and the use of birefringent crystals to create a polarization-interference Fourier transform spectrometer which operates electro-optically rather than with moving parts. Unfortunately, the applicability of this system is limited due to off-axis effects. 1

25 Chapter 3 describes a design modification based on a wide-angle Lyot filter design. In this case, the birefringent crystal stages of the spectrometer are replaced with two birefringent crystals of opposite birefringence and equal retardation so that the total retardation through the two crystals equals the retardation of the original stage. This device, along with the original design and an equivalent Michelson interferometer design, is simulated to investigate the improvement in off-axis performance offered by such a design modification. Chapter 4, then, considers the variable elements necessary for this type of spectrometer system. Polarization rotators must be chosen that are both achromatic and fast. Twisted nematic cells are used as an example to consider the balance of these two concerns. The smaller birefringent crystal stages can also be replaced with variable liquid crystal retarders, which may reduce the off-axis gains offered by this new device configuration. Again, the trade-offs of the use of these elements are explored, using electrically-controlled birefringence (ECB) cells as an example. In Chapter 5, an application of liquid crystals to thermal imaging is described. Here, the liquid crystals are used to measure temperature changes due to the absorption of light in the longwave infrared region, from 8-12μm. The temperature changes induce a change in the birefringence of the liquid crystals which is measured by detecting intensity changes through the liquid crystal cell sandwiched between two linear polarizers. A key issue in the development of the liquid crystal thermal imager is the creation of an alignment layer which can be applied into the confined geometry of the isolated liquid crystal pixels. An existing method for developing this alignment is described, but the anchoring energy is low, which leads to an increase in the liquid crystal director fluctuations in the cell. Chapter 5 also describes the signal noise created by these liquid crystal director fluctuations in great depth. 2

26 These issues have motivated the development of an azo dye photoalignment layer which can be infused into the liquid crystal microcavities. However, because this type of photoalignment layer is unstable against subsequent light exposure, a surface-localized polymer layer must be used to stabilize the liquid crystal alignment against such exposure conditions. This layer is introduced into the liquid crystal cell by mixing a pre-polymer into the liquid crystal and allowing it to separate to the substrate surfaces prior to polymerization. Chapter 6 demonstrates the feasibility of such a surface-localized polymer layer. This is accomplished utilizing spun-on photoalignment layers in bulk liquid crystal cells. In Chapter 7, the infused photoalignment layer is developed to test the important parameters in the generation of a surface-localized polymer layer to stabilize this alignment. The degree of surfacelocalization under a number of different liquid crystal/polymer mixture and cell preparation conditions are explored. The effect of these conditions on the resultant photostability of alignment in infused photoaligned cells is also shown. 3

27 CHAPTER 2 Introduction to Spectrometer Systems 2.1 Introduction Spectroscopy is a useful tool for chemical identification. Spectrometers can be found in the laboratory, airports, field-kits, space, and many other locations across a variety of fields and applications. A number of these applications require spectrometers which are rugged and provide real-time images of a scene. This has produced a drive to develop imaging spectrometers which can rapidly image a scene yet require no moving parts. In this section, a novel electro-optic Fourier-transform (FTS) system design is proposed. This chapter will cover the background of relevant spectroscopic techniques and spectrometer systems available today. The next two chapters will discuss the design modifications and describe a variety of techniques used to analyze the benefits of these device configurations and discuss design trade-offs. 2.2 Wavelength Regimes Visible Spectroscopy While sometimes expanding into the UV range, visible spectroscopy typically measures within the range of µm. Visible spectroscopy involves the absorption of photons to excite electrons from their ground to an excited state. Typically, it is electrons involved in double and 4

28 triple bonds (or lone pairs) which absorb in the visible range 7. This electron absorption can extend up into the near infrared (NIR) range, to about 1.1µm 8. One important disadvantage of visible spectroscopy is the presence of fluorescence. Absorbed light causes the excitation of electrons to a high energy state; these electrons will then return to the ground state, emitting photons in the process. This fluorescence causes interference and makes any chemical species exhibiting fluorescence difficult to identify IR Spectroscopy Molecules can be considered to be a system of weak harmonic oscillators. The bonds within the molecules have a specific set of frequencies at which they vibrate. Incident electromagnetic radiation at these same frequencies causes resonant vibrations within the molecule, resulting in the absorption of these frequencies 9. Because each molecule has its own unique absorption spectrum, measuring the spectrum of an unknown material can help to identify it. In order for such vibrations to result in absorption requires an inherent dipole moment. As such, homonuclear diatomic molecules are not detectable via IR spectroscopy 10. IR spectroscopy can be broken up into a number of spectral ranges: near-infrared/shortwave-infrared (NIR/SWIR), mid-wave infrared (MWIR), and long-wave infrared (LWIR). The division between these regions is due, in part, to the performance ranges of detectors. Table 2.1 outlines the rough wavelength ranges for each of these spectral regions; however, it is important to note that there is no exact maximum or minimum wavelength for each range. NIR/SWIR is commonly used for low-light security, night vision, crop health, forgery detection, and chemical imaging. MWIR and LWIR are typically used to detect radiated heat, such as in safety inspections of power plants or refineries to detect a leak 11. 5

29 Table 2.1. Different wavelength ranges in IR Spectroscopy Spectral Range Minimum λ (µm) Maximum λ (µm) NIR/SWIR MWIR LWIR IR molecular absorptions require a change in the dipole moment, µ, of a molecule; this absorbed frequency matches one of the normal vibrational modes of the molecule (any number of stretching or bending modes). Typically, these normal modes occur in the MWIR region. As such, the signals in the NIR/SWIR region tend to be overtones or combinations of these MWIR fundamental bands. The more anharmonic the bonds within the molecule, the more intense the NIR/SWIR signal will be. Therefore, CH, NH, and OH bonds have relatively strong signals in the NIR/SWIR region 12. Due to the fact that liquid crystal materials are not readily available which do not possess strong absorption in the MWIR and LWIR ranges, the discussion of this FTS system will focus on the NIR range. 2.3 Spectrometers Because all of the techniques discussed in this section involve the detection of reflected (or scattered) light, the instrumentation is, essentially, the same for all of these. The main spectrometer systems discussed here are dispersive spectrometers and Fourier-transform (FT) spectrometers. Imaging spectrometers will also be discussed, though this discussion will be limited to the FT-type systems. 6

30 2.3.1 Dispersive Spectrometers In conventional spectroscopy, scanning occurs using a dispersive element (e.g. a prism) which is slowly rotated or adjusted so that each frequency of incident/transmitted intensity within the desired range can be measured independently. Generally speaking, independent wavelengths are selected using a slit or filter, the resolution of which is controlled by the width of the slit or transmission peak of the filter 13. Due to this sequential scanning, dispersive devices have a number of limitations. For any particular step in the scan, all radiation not passing through the slit or filter is wasted, lowering the throughput/efficiency of the system. In addition, because the throughput is limited, the noise in the detected signal is also relatively high, decreasing the sensitivity. In many cases, these devices have limited reproducibility because the accuracy of the wavelength range being measured depends on correct alignment of the dispersive element; misalignment can cause spectral peaks to be interpreted at the wrong wavelength. Finally, because each wavelength must be scanned one at a time, this device has a somewhat long scan time which scales linearly with spectral range for equivalent resolutions Fourier-Transform Spectrometers In contrast to dispersive techniques, a Fourier-transform spectrometer (FTS) accomplishes a spectrum by sending all wavelengths through the system simultaneously and relying on interference between different components of the beam to create an interferogram which can then be Fourier-transformed to provide the spectrum. Mathematically speaking, this is much more complex, but this system has several advantages over the dispersive type 9. 7

31 It is well known that the two main advantages of the FTS are the so-called Fellgett and Jacquinot advantages. The Fellgett advantage of the FTS is due to the fact that the FTS detects all wavelengths simultaneously so, for the same collection time, an FTS will have collected more data for each wavelength than a dispersive device. The Jacquinot advantage of the FTS is based on the fact that the detector collects more signal at any particular time than a dispersive device, since it does not have to limit the signal. Both of these lead to a significantly higher signal-tonoise ratio. In addition, for broadband scans with the same resolution, FTS can accomplish the scan much more quickly than dispersive devices 14. Figure 2.1. Schematic diagram of a basic Fourier-transform spectrometer 14. 8

32 A diagram of a FTS is shown in Figure 2.1. In this example, the white light source and He-Ne laser source are used to determine the relative path-lengths of the beams (which results in more accurate and reproducible results). The spectral signal generates a beam of all wavelengths within a given range. For example, a mercury lamp would produce a signal with wavelengths in the far-ir region of the spectrum. All three beams travel through the interferometer, creating multiple, superimposed, interference patterns. The white light source and He-Ne laser source are then directed to the detector while the spectral signal is sent to the sample before entering the detector. While a beam of one wavelength, when passed through an interferometer, will produce a characteristic cosine wave interferogram, the spectral signal will result in the superposition of the interferograms of all wavelengths in its range. An inverse Fourier-transform can then be performed on this spectral signal after it has traveled to the sample to provide an absorption/emission spectrum of that sample Michelson Inteferometer When the first FTS was produced in , it employed a Michelson interferometer in order to create the interference pattern. The general layout of this sort of interferometer can be seen in Figure 2.2. The incoming light is split to travel along two paths, one of which having a fixed mirror at the other end and the other having a moving mirror. The moving mirror creates a variable path difference between the two beams which, when recombined, create the interference pattern which is Fourier-transformed. 9

33 Figure 2.2. Schematic of a Michelson interferometer based Fourier-transform spectrometer 15. Since then, other common interferometers have been used in FTS, such as diffractive grating and Fabry-Perot interferometers, but the Michelson interferometer remains the most commonly used due to its ease of construction and high resolving power 14. In portable applications, Michelson interferometer FTS have a number of limitations. Miniaturization limits the maximum optical path difference (OPD) achievable, but more importantly, the ruggedness of the device becomes a significant concern. Researchers have developed highly accurate MOEMS-based Michelson interferometer FTS capable of high resolution and fast scans 16,17. The engineering and precision of these parts, however, makes them an expensive choice 18. In addition, a number of these devices are so new that the true lifetime of these parts under extreme conditions is unknown. 10

34 Birefringent Interferometers A number of attempts have been made to eliminate the requirement for moving parts associated with the Michelson interferometer. Many of these involve the use of birefringent elements. One of the oldest birefringent techniques was introduced by Billings 19 in 1967; it utilizes a single Z-cut uniaxial birefringent crystal plate between two circular polarizers. Beams of different angles of incidence experience different path differences between the two eigenmodes, which creates a spatial interference pattern on the image plane. As early as , other researchers have explored using birefringent materials to construct a Wollaston prism FTS, the schematic of which is shown in Figure 2.3. In this type of prism, the principle axis of each wedge of the prism is perpendicular to the other. As the beam travels through the prism and hits the wedge interface, the two orthogonal components of the incoming beam are sent along different paths, resulting in a radial spatial interference pattern, as shown. Figure 2.3. Wollaston prism and the interference pattern it creates 21. A number of modifications to the basic Wollaston prism design have been introduced, including one modification in which the two wedges of the Wollaston prism are replaced with 11

35 wedged twisted nematic cells and another in which a flat twisted nematic cell is placed between the two wedges 21,22. Other techniques have been proposed which do not utilize a Wollaston prism to accomplish an interference pattern. One such technique, developed by Lu et al in , uses a birefringent crystal cell followed by a liquid crystal cell, each with their principal axis directed parallel to one another and at 45 degrees to a pair of crossed polarizers which sandwich the system. The thickness of the crystal is increased incrementally in a direction perpendicular to the propagation of light, requiring as many different crystal thicknesses as steps in the scan. In yet another technique, the interferometer is created using a specially designed beam splitter cube, shown in Figure 2.4, which uses angled back planes to spread the split incoming beam into an interference pattern 24. Figure 2.4. Optical scheme of a static Fourier-transform spectrometer using a modified beam splitter cube

36 2.3.3 Imaging Spectrometers In many of these static FTS cases, a spatial interference pattern is created, either in one or two dimensions. This makes it incredibly challenging to develop any of these techniques into imaging applications. One recent study has developed an imaging spectrometer based on a Wollaston prism approach by using a slit to effectively reduce the incoming signal to one spatial dimension and scanning the slit across the orthogonal direction to create a spatially scanning spectrometer 25. However, this technique only allows for imaging along one spatial dimension and, due to the use of the slit, may have limited sensitivity compared to traditional FTS techniques. For many applications, video-rate or real-time scanning is important. That is, pixels cannot be scanned one-by-one or row-by-row, as is fairly common in current imaging spectroscopic techniques such as the pushbroom or whiskbroom techniques employed for scanning from aircraft 26. For instance, explosives identification from fast-moving vehicles across rugged terrain would require a device capable of scanning the full scene quickly and simultaneously to avoid measurement/location error due to the unsteady motion of the vehicle. In addition, even with spectrometers developed to include no moving parts for non-imaging applications, a pushbroom/whiskbroom method of scanning requires at least one moving part to shift the scan area. Aside from the previously explained drawbacks of these moving parts, some applications (such as satellite spectroscopy) require no moving parts due to device inertia and limited possibility of the requirement of human correction/intervention 27. Typically, then, a video-rate imaging spectrometer requires certain trade-offs which may be application specific. 13

37 Michelson Imaging Spectrometers In theory, the simplest imaging spectrometer would be a basic Michelson interferometer device with mirrors large enough to image the entire scene. However, this type of device suffers significantly from off-axis changes in device performance, which limits the numerical aperture and, therefore, the amount of light that can be collected (decreasing signal-to-noise). That is, it works perfectly well for highly collimated beams, but when collecting light from a scene, there may be significant deviation in the polar angle of incident light. As light goes further and further off-axis, it experiences a greater and greater path length, resulting in a different path length for off-axis light than for on-axis light. The higher the desired spectral resolution of the device (and, therefore, the higher the required maximum path difference), the greater the path difference between these two beams 28. An illustration of this is shown in Figure 2.5. The expected off-axis performance losses due to this effect will be discussed in greater depth in Chapter 3. Figure 2.5. Generation of path difference for off-axis rays through a Michelson interferometer 28. One technique was developed to accomplish a snapshot imaging spectrometer utilizing a Michelson interferometer, known as a Multiple-Image Fourier-Transform Spectrometer 14

38 (MFTS) 29. In this design, the sample image is sent through an 8x8 lens array, which creates 64 miniaturized collimated images. These are sent through a modified Michelson interferometer in which the moving mirror is replaced by a fixed, tilted mirror in order to create a path difference between the images. Aside from the low spatial resolution of this device (due to the miniaturization of the scene and limitations on the detector pixel size), this system suffers some of the same downfalls as the Michelson interferometer when it comes to portability: it is relatively large and difficult to ruggedize but no longer suffers from portability issues due to the lack of moving parts Birefringent Imaging Spectrometers An improved version of the MFTS system, referred to as the Snapshot Hyperspectral Imaging Fourier-Transform (SHIFT) Spectrometer 30, replaces the Michelson interferometer with a Wollaston prism. The OPD of this system varies along the wedge axis. If the prism axis is rotated by a small angle with respect to the axis of the focal plane array used to split the image into sub-images, then each image will experience a different OPD and the scene can be imaged at all path-differences simultaneously. As with the MFTS, this device has limited spatial resolution due to the miniaturization of the images The Polarization-Interference Fourier-Transform Spectrometer (PIFTS) System Other techniques have been developed which utilize birefringent elements between parallel polarizers to create a polarization interference pattern. These devices share one distinct feature which makes them quite desirable in imaging applications; namely, all light travels along a common path so no potential for extraneous path difference due to misalignment is possible. 15

39 Principles of Operation Figure 2.6. Schematic of a Polarization-Interference Fourier-Transform Spectrometer utilizing birefringent elements 28. Figure 2.6 shows the schematic of one such spectrometer 27,28. This device consists of n birefringent stages, each stage having a fixed birefringent element (such as Quartz) and an achromatic half-wave plate. The entire system is sandwiched between two parallel polarizers oriented at 45 degrees with respect to the principle axis of the birefringent elements. Since the light is polarized at a 45 degree angle, the light is equally split between an extraordinary and orthogonal ordinary component. The extraordinary component will pick up a phase difference with respect to the ordinary component; the exact phase difference will be dependent upon the design parameters of the birefringent stages (thickness and birefringence) and the wavelength of light and will, therefore, vary across the spectral range. After the light passes through the analyzer, the two beam components are, effectively recombined. That is, the change in polarization state of the net beam affects the net transmission through the system and onto the detector. When the achromatic rotators are switched (on or off), the system is capable of 16

40 scanning through various path difference to generate the required interferogram. The device design in this case, is quite similar to that of a Lyot filter; a tunable Lyot filter is usable as a dispersive spectrometer. A number of others have proposed alternatives to this basic design. For instance, Miller et al 30 proposed a version which used only liquid crystal variable retarders between the parallel polarizers to achieve the path difference. One claimed benefit of this device over the device of Chao is the ability to step continuously through the path range, which also means that the tolerancing of the elements is less strict. Using only liquid crystal cells limits the maximum attainable path difference as large path differences would require either unrealistically thick cells or a prohibitively large number of cells. In addition, liquid crystals oriented with their directors at an intermediate angle with respect to the cell substrate (such as 45 degrees) are known to have significant viewing angle problems, which also become prohibitively large at large path differences even with viewing angle improvements such as anti-parallel cells or compensation films. Miller suggested adding fixed birefringent elements, as with Chao s design, to attain these large path differences Device Limitations One drawback of these birefringent designs which is not well discussed is their limited off-axis performance. DeHoog et al 28 provided a brief analysis of the field-of-view of the Chao system. With the maximum field-of-view defined as the point at which the OPD between the onand off-axis rays equals a quarter-wave (for the minimum wavelength of the range), the field-ofview becomes fairly prohibitive (less than 5-degrees) for larger systems. 17

41 In terms of design trade-offs, this limited field-of-view means that the numerical aperture of a system with large spectral resolution must be quite small. If a large numerical aperture is needed, then the spectral resolution must be reduced. The trade-off between these, as well as scan time (time it takes to collect a single hyperspectral image of the scene) and spatial resolution, can be quite application specific, meaning that one design can be preferable in one application but not in another. However, any improvements, such as reducing off-axis effects, can make these trade-offs less restrictive. A dispersive Lyot filter will experience these same limitations, but requires a much smaller system size to accomplish the same spectral resolution. On the other hand, this system suffers from a significant reduction in signal due to the requirement that all light outside of the wavelength of interest for each step of the scan is being thrown away a common drawback of dispersive designs. In the next chapters, the off-axis performance of the PIFTS design is explored in further depth. A design modification for improved performance is described and analyzed. Finally, the incorporation of variable elements, such as the variable retarders proposed by Miller, is considered in terms of the trade-offs such elements may require. While the specific trade-offs chosen must be made based on a given application, these chapters point to the key issues and analyses that may be used to make such considerations. 18

42 CHAPTER 3 Wide-Angle Polarization Interference Fourier-Transform Spectrometer: Design and Simulation 3.1 Introduction Using polarization-interference to create a spectrometer has been shown to have a great deal of promise for both portable and imaging applications. However, at the current state, this design has significantly reduced performance for off-axis light, which limits its usability by requiring significant trade-offs between numerical aperture and spectral resolution. In this chapter a design modification is proposed which utilizes a combination of positive and negative birefringence crystals in order to significantly widen the usable off-axis angles, allowing for this type of spectrometer system to be used for applications which require higher spectral resolution or rapid imaging of a scene (larger numerical aperture). Basic principles of this device modification are discussed and mathematical justification is provided for its improvement of the system. Computer code is generated which models the systems and allows for understanding of and comparison between the different device designs. Finally, the limitations of this modeling in determining the off-axis performance of these devices is discussed as well as potential areas important for further investigation of these systems. Particularly, the achievable spatial resolution of these devices may require in-depth analysis to fully consider its implications. 19

43 3.2 Our Device Design The starting point of the designs used in this study is similar to the device design proposed by Chao et al 27 and discussed in Chapter 2. The basic schematic of the design proposed here is shown in Figure 3.1. Each birefringent crystal in the series will have a different associated path difference, given by x m = ( x o 2 ) 2m, (3.1) where xo is the desired step size (in μm) of the scan and m is the integer representing the particular birefringent crystal (m = 0, 1, 2,, N-1). N is the number of crystals required to achieve the maximum desired path difference. Note that there are two xo stages to ensure that a net zero path difference through the system is possible. Figure 3.1. Diagram of a uniaxial-crystal polarization-interference Fourier-transform spectrometer The total number of birefringent crystal stages in the system is n=n+1. The total retardation of the system, xtot, is given by 20

44 x tot = ( x o 2 ) 2n 1. (3.2) Using this equation, one can determine the resolution of the system for a given wavelength by considering the path difference to be the maximum beat frequency 31 detectable by the system. Written in terms of the wavelength, the maximum beat length is also the total retardation of the system, xtot, and can be written as where = -. The resolution,, can then be reduced to 1 x tot = 1 λ 1 λ, (3.3) λ = λ2 x tot λ. (3.4) Note that the resolution is smaller at the low-wavelength end of the spectral range and larger at the high-wavelength end of the spectral range. The step size of the scan should be set so that the wavelength at the bottom of the spectral range, min, is sampled a sufficient number of times, which we propose to be at least 4. That is, the maximum step size is given by x o,max = λ min 4. (3.5) Next, this equation can be substituted into xo in Equation 3.2 and this result then substituted into Δxtot in Equation 3.4. This gives the resolution, Δλ, at the minimum wavelength, λmin, in terms of this wavelength and the total number of stages as λ min = λ min ( 1 2 n 4 1 ). (3.6) Note that this is only valid for a system with at least 5 stages. In this study, the NIR spectral range from 0.8 m to 2.0 m is considered. It is expected that changes in off-axis performance are greatest for the shortest wavelength in the spectral range, so focus on simulations of this 21

45 wavelength, λmin=0.8μm. However, as resolution is smallest at the upper end of the spectral range, discussions of resolution consider 2.0μm light Principles of the Dual Crystal Design Figure 3.2. Viewing angle dependence of the retardation through a biaxial film with different relative indices as defined by the factor, Nz, given in Equation The modification presented in this chapter is based on the principal of a viewing angle independent optical retarder which can be created using a biaxial crystal with an index of refraction through the plane (nz) being the average of the two in-plane indices (nx) and (ny). Fujimura et al 32 proposed a biaxial film whereby the three indices of refraction could be related by a factor, Nz, by 22

46 N z = n s n z n s n f, (3.7) where ns and nf are the indices along the slow and fast axis, respectively, and nz is the refractive index across the thickness of the cell. Figure 3.2 shows the change in retardation through the retarder film with incident angle for different Nz factors. The dependence of retardation on incident angle is seen to be least when Nz=0.5. In this case, the relationship between the three indices of refraction is given by, n z = n s+n f. (3.8) 2 While using such a biaxial retarder is infeasible, a set of uniaxial crystals can approximately achieve the same outcome. This technique is similar to the viewing-angle improvement of a Lyot filter 33. Figure 3.3 shows a basic schematic of the crystal axis orientation in this dual-crystal design. The length of each arrow represents the relative magnitude of the index along that axis with respect to the others within the same crystal. As can be seen, Crystal 1 has a negative birefringence while Crystal 2 has a positive birefringence (though this scheme can also be reversed). The front face of each of the crystals has been accentuated to better depict the indices. Figure 3.3. Proposed dual-crystal design mimicking wide-view Lyot filter design. nx=nz in Crystal 1 and ny=nz in Crystal 2. 23

47 Figure 3.4. Diagrammatic representation of different azimuthal angles of incidence with varying polar incident angle of light. (Left) Variation in polar angle of incidence in this orientation causes variation of effective extraordinary index of refraction. (Right) Variation in polar angle of incidence in this orientation does not cause a variation in the extraordinary index of refraction. In a uniaxial retarder, the (polar) angular dependence of the retardation through the cell depends on the azimuthal angle of incidence. The two key scenarios for this are shown diagrammatically in Figure 3.4. In both cases, incident light is initially perpendicular to the optic axis. At one orientation, varying the polar angle of the incident light keeps this relationship constant (right-hand diagram), so the birefringence is also kept constant. However, the path length will increase with increasing polar angle of incidence. In the perpendicular orientation (left-hand diagram), variation in polar angle causes the light to change its orientation with 24

48 respect to the optic axis. This light will experience a decrease in birefringence which can counteract and even dominate the effect of increasing path length. The optical path length change with polar angle for each of these scenarios is shown in Figure 3.5 (using ne=1.75 and no=1.5 and d=5μm). These effects are inverted in the case of the negative birefringence material (such as Calcite or Sapphire) given the relative orientations shown in Figure 3.3. In the combination however, the polar angle variation will result in an increase in retardation through one crystal and a decrease in retardation through the other (and vice versa). Figure 3.5. Off-axis change in net retardation through a 5μm thick cell with a uniaxial material in planar orientation (ne=1.75 and no=1.5) for in-plane and out-of-plane polar variations in angle based on Figure 3.4 (with light propagating from the left-hand side). 25

49 The next step, then, is to determine the optimal combination such that the relative thicknesses of the two crystals minimizes the off-axis changes in optical path length. The average index of refraction of the entire set of two crystals, along any particular axis, will be a weighted average based on the relative thickness of the two crystals. The value of this weighted average index is given by n i,ave = D 1n i,1 +D 2 n i,2 D 1 +D 2. (3.9) However, these two thicknesses are related to one another. Therefore, they can be, instead, represented as a fraction of the total thickness, whereby d1+d2=1. Put another way, the relative thickness of the first crystal can be defined as d 1 = 26 D 1 D 1 +D 2. (3.10) The thickness of the second crystal, d2, can be expressed similarly. This allows the average index to be rewritten as n i,ave = d 1 n i,1 + (1 d 1 )n i,2. (3.11) This set of equations does not, in itself, help to determine the relative thicknesses of the two crystals. However, this can be combined with the relationship given in Equation 3.8 for a viewing angle independent biaxial device. If the refractive indices in this expression are taken as the average refractive indices, each given by Equation 3.11, then the relative thickness of the two crystals can be expressed as d 1 = n x,2 +n y,2 2n z,2 2n z,1 2n z,2 n y,1 +n y,2 n x,1 +n x,2. (3.12) Based on the crystals as diagrammed in Figure 3.3, nx,1=nz,1=no,1, ny,1=ne,1, nx,2=ne,2, and ny,2=nz,2=no,2. For each of the birefringent crystals, the birefringence can be written as Δn1=ne,1- no,1 and Δn2=ne,2-no,2. Plugging these into Equation 3.12, the expression reduces to

50 d 1 = n 2 n 2 n 1. (3.13) Equations 3.10 and 3.13 can then be set equal to one another and solved for D2, giving D 2 = D 1 [( n 2 n 1 n 2 ) 1]. (3.14) The ratio of the retardations of the two crystals is then given by Г 1 Г 2 = n 1D 1 n 2 D 2 = n 1 n 2 ( n 2 n1 n2 ) n 2 = 1, (3.15) where the negative sign in the numerator is due to the fact that the principle axis of the first crystal is orthogonal to that of the second. This means that the retardations of the two crystals should be identical which can only be done if each crystal accounts for exactly half of the total retardation of the system. 3.3 Methods Basic System Parameters As previously stated, the system is tested in the NIR range from 0.8μm to 2.5μm, meaning that the design wavelength for the system is λmin=0.8μm. The maximum step size for this wavelength is xo,max=0.2μm. Systems with 7 to 14 stages are investigated. Additionally, Quartz is utilized as the positive birefringence material (ne=1.553, no=1.544) and Calcite as the negative birefringence material (ne=1.486, no=1.658) and dispersion is ignored. The maximum path length for each of the system sizes, as well as other relevant parameters for these systems, are outlined in Table

51 Table 3.1. Maximum path difference and resolution for different numbers of stages. # Stages Max Path (μm) Resolution at 0.8μm (μm) Optical Path Calculations Michelson Interferometer Figure 3.6. Light propagation through a Michelson interferometer

52 Figure 3.6 shows a diagram of the light propagation through a Michelson interferometer, including added path length due to off-axis light propagation. If the effect of the beam splitter is neglected, the change in OPD can be considered from a purely geometric viewpoint, as shown in Figure 3.7. The change in OPD for off-axis light is the difference between the off-axis OPD, r, and the on-axis OPD, x. This can be simplified to OPD = x ( 1 1). (3.16) cosθ Figure 3.7. Geometric diagram of off-axis change in OPD in a Michelson interferometer. For the off-axis calculations, x is considered to be the maximum OPD through the interferometer, Δxtot, which is given in Equation 3.2 (with xo set by Equation 3.5). Equation 3.16 is then solved for θ with the limit in incident angle is defined as the angle at which the change in OPD is equal to ¼ of the minimum wavelength in the spectral range (0.8 µm in our case). That is, the change in OPD is set as ΔOPD=λ/4=0.2μm. Beyond this angle, the changes in propagation of off-axis light are assumed to be too great (degrade the signal so much that it is no 29

53 longer usable). This creates an angle limit which is entirely independent of the design wavelength of the system, as given by θ max = cos 1 ( 2n 2 ), (3.17) 2 n 2 +1 recalling that n is the total number of birefringent stages required to achieve the maximum path of an equivalent PIFTS Polarization-Interference Spectrometer In this case, the off-axis calculations are somewhat more complicated. As previously stated, incident light is split into two polarization states: the ordinary and extraordinary rays. These rays each have a different path length through the material due to the different indices of refraction and, for off-axis light, different direction of propagation. A diagram of the propagation of the two rays in a single dual-crystal stage is shown in Figure 3.8. Here, only the effect of Snell s law on the propagation direction of the beams is considered to maintain simplicity (the differences in the direction of the Poynting vector are neglected). 30

54 Figure 3.8. Geometric diagram of off-axis changes in OPD through a birefringent crystal stages. The incident beam is initially in glass, at angle θ, striking the Calcite substrate at point S. The p-polarization experiences the ordinary index and is refracted by angle θa1, while the s- polarization experiences the effective extraordinary index and is refracted by angle θb1. These rays strike the Quartz at points X and Y, respectively. The p-polarization then experiences the 31

55 extraordinary index and is refracted by angle θa2 while the s-polarization experiences the ordinary index and is refracted by angle θb2. These rays strike the exit glass at points A and B, respectively. Both rays are then refracted into the glass by the angle θ. The path difference, considered at points A and B, between the two rays is the difference between the optical paths SXA A and SYB, which also needs to take into account the speed of each ray through the media, affected by the index experienced by each ray. For light propagated at an azimuthal angle perpendicular to that shown in Figure 3.8, the full extraordinary index of Calcite would be experienced by the p-polarized ray while the effective extraordinary index of Quartz would be experienced by the s-polarized ray. For rays propagated at any other azimuthal angle, the situation is quite a bit more complicated. However, these rays are also expected to experience a smaller off-axis change in OPD and are not considered in this investigation. In order to calculate the OPD change in a spectrometer model, the entire system is approximated as one stage with retardation equal to the total retardation of the system. The net retardation through the system for the various system sizes are as defined in Table 3.1. In the dual-crystal system, this retardation is evenly split between the Calcite and Quartz crystals. The off-axis angle limit is defined in the same way as with the Michelson interferometer calculations the angle at which the off-axis change in OPD is equal to ¼ of the design wavelength of the system. Since the off-axis change in OPD will be different depending on the azimuthal angle, both previously discussed scenarios are models (parallel and perpendicular to the page as shown in Figure 3.8). The azimuthal angle with poorer off-axis performance is used to calculate the incident angle limit of the device. 32

56 To actually determine the incident angle limits based on this criterion, a simulation was generated which calculated the off-axis OPD for gradually increasing incident angle (in 0.1 degree steps) until the change in OPD equaled or exceeded the λ/4 definition; the simulation output the incident angle limit for the various system sizes. It is important to note that this OPD calculation does not include of real polarization rotators (such as TN cells). In addition, the OPD model only takes into account the maximum OPD configurations of these systems Simulated Interferogram Michelson Interferometer For a monochromatic source, the interference pattern created is a cosine function. The intensity of the superimposed interference signals for any number of frequencies input is given by 9 I(x) = A(ν) cos(2πν x). (3.18) All amplitudes were considered to be unity to simplify investigations. This interference signal was constructed by looping over all input wavelengths and all path differences. To create a symmetric scan, the results for an individual positive path difference was stored for the negative path difference as well. An example of the interference pattern created by this simulation (with =0.8 m being the only input) is shown in Figure

57 Intensity (a.u.) Path step (#) Figure 3.9. Example interference pattern from Michelson interferometer simulation using a single input signal with =0.8 m [Michelson Simulated code included in Appendix A, flowchart on page 147. Input parameters: N=0.8, lamd=0.8, DH=14, θ=0] Polarization-Interference Spectrometer To calculate the polarization state of light as it travels through the optical interferometer system, the Extended Jones Matrix method 34 is used, representing each element with its own Jones matrix. The matrix representing a particular element in the system is given by with G n = exp (ik z1d) 0, (3.19) 0 exp (ik z2 d) k z1 k o = [n 2 o ( k x ) 2 1/2 ]. (3.20) k o 34

58 and k z2 k o = ε xx ε zz k x + [ε k zz (1 n e 2 n2 o o n2 e cos 2 θsin 2 φ) ( k x ) 2 1/2 ] n en o, (3.21) k o ε zz with θ representing the angle of the principle axis from the x-y plane and φ representing the angle of the projection of the principle axis onto the x-y plane from the x-axis. The wave vector of the light is defined by and where k x = k o sinθ k, (3.22) k z = k o cosθ k, (3.23) k o = 2π λ. (3.24) o Quartz and Calcite were used for the positive and negative birefringence crystals, respectively. The single-crystal model utilized Calcite only. The polarizers were modeled as an anisotropic material with one index having an imaginary component. It is assumed that there are no inter-element reflections or losses associated with substrates; only losses from the entrance and exit of the component stack (assumed to be glass with n=1.5) are considered. The polarization rotators were modeled as 90- degree rotation matrices for their rotating state and were neglected for their non-rotating state. 35

59 Intensity (a.u.) Path step (#) Figure Example interference pattern from Polarization-Interference Spectrometer simulation using a single input signal with =0.8 m [Digital Simulated code included in Appendix A, flowchart on page 150. Input parameters: lam=0.8, lamd=0.8, DH=14, θ=0]. A stack of elements, then, utilizes a series of Jones matrices, all of which operate on the incident polarization state (for a particular wavelength) in succession. This produces a polarization state which travels through the final polarizer; the polarization state prior to that polarizer determines the intensity of the light transmitted through the polarizer, given by I = E x 2 +cos 2 θ k E y 2 1+cos 2 θ k, (3.25) where Ex and Ey are the x- and y- components of the light incident on the exit polarizer. These can be summed up over many incident wavelengths and for each system configuration (i.e. path 36

60 difference) to generate an interference pattern. An example of the interference pattern created by this simulation (with =0.8 m being the only input) is shown in Figure Simulated Spectra Once the simulated interferogram was generated, MatLab s built-in Fast Fourier- Transform (FFT) algorithm was used to generate the transform of the interferogram into frequency information. This output was then adjusted and scaled to output data on the appropriate wavenumber (frequency) scale. Figure 3.11 shows an example of the output spectrum from the Michelson model using an input signal of λ=0.8μm. Figure Example of spectrum output from Michelson interferometer simulation using a single input signal with λ=0.8μm [Michelson Simulated code included in Appendix A, flowchart on page 147. Input parameters: N=0.8, lamd=0.8, DH=14, θ=0]. 37

61 3.3.5 Zemax Simulations As may be evident in Figure 3.7 and Figure 3.8, the change in optical path difference for off-axis light is not the only problem associated with off-axis propagation. There is also a difference in position between the two components of the signal. While the beam walk-off does not affect the output signal in a non-imaging device, it can pose a significant problem for imaging devices as it could result in the ordinary and extraordinary rays reaching different pixels of the detector. To avoid this, the pixels must be made sufficiently large as to fully capture both rays on the same pixel for the majority of the light. This, in turn, will reduce the spatial resolution of the device. However, if a specific spatial resolution is required, then the spectral resolution may need to be reduced in order to decrease beam walk-off. The Zemax software package was used to investigate ray propagation through the model systems. This software package is adept at considering light propagation through a variety of optical components and allow for easy visualization of the beam walk-off phenomenon. The birefringent crystal types were specified by name and the Zemax glass catalog, included with the software, provided data such as the wavelength-dependent indices of refraction. The principle axes of the crystals were specified shown in Figure 3.2. Polarizers were modeled as ideal linear polarizers. An ideal rotation matrix was sandwiched between the two crystals. The only way to appropriately model ray propagation through this birefringent system is in Sequential Mode using Multiple Configuration Mode whereby each configuration can propagate an individual ray through the each crystal as either the extraordinary ray or the ordinary ray (with the rotation matrix either on or off), resulting in a very large number of possible configurations, which effectively overloads Zemax. However, tests of systems with limited size (i.e. only a few stages), indicate that the case in which the ordinary ray is propagated 38

62 through all of the crystals and the case in which the extraordinary ray is propagated through all of the crystals represent the largest walk-off between the two beams. Note that the rotation matrix is on for both of these cases. As such, all investigations of ray walk-off have utilized only these two possible configurations and considered the difference in position between them on the image plane to be the system walk-off. XMZ.tseTgifnoCSTF4 Y 61 lla :noitarugifnoc tuoyal D3 2102/01/01 Z X Figure Zemax diagram of a single stage dual crystal system with 400μm retardation and 20-degree off-axis light propagation. Figure 3.12 shows an example of the system configuration plotted in Zemax. As stated, this system consists of an optical stack using two birefringent crystals an entrance and exit polarizer and a rotation matrix between the two crystals. The two rays (one ordinary, one extraordinary) are propagated through the system from a specified angle and their different optical 39

63 paths can be measured. Zemax allows for the difference in their two positions to be manually measured on the image plane. This difference is defined as the walk-off of the system for a given input angle. 3.4 Results Off-axis OPD Change Basic field-of-view analysis, such as the DeHoog analysis 28 discussed previously, considers the off-axis OPD change that occurs through the system at its maximum path difference for the design wavelength in the spectral range. The angle limit, in this analysis, was based on the criterion of the angle at which the change in OPD from on-axis OPD is equal to one-quarter of the design wavelength. To determine the usefulness of the dual-crystal model, these OPD analyses were conducted for both the single and dual-crystal systems (with Calcite and Quartz). Figure 3.13 shows the results for these two systems as well as the equivalent Michelson interferometer system. Note that this result is the same regardless of the design wavelength used in the calculations. As with the previous analysis, the single-crystal model shows poorer performance than the standard Michelson model. However, the dual-crystal model shows around 45% improvement over the uniaxial model and around 33% improvement over the Michelson model at all system sizes. It is important to note that the angle limit is also dependent on the birefringence of the crystals being used. As such, results for other system types may vary. 40

64 Figure Incident angle limit based on OPD criterion. Comparison between singlecrystal and dual-crystal models as well as the equivalent Michelson interferometer dependent on device size [Code shown in Appendix A (A.7.7)] Simulated Spectra Figure 3.14 shows the simulated output spectrum for each of the three models considered, with one input signal of =0.8 m. The peaks shown for each model are the on-axis peak (0) and the 25-degrees off-axis peak (25). The off-axis peak of the dual-crystal model is very near the original location of the on-axis peaks while the off-axis peaks of the single-crystal and Michelson models have shifted to higher wavelengths. 41

65 Figure Simulated output spectra of 14-stage dual-crystal WA-PIFTS (BC) and singlecrystal PIFTS (U) interferometers as well as the equivalent Michelson interferometer (M) design. Simulated for on-axis (0) and 25-degrees off-axis (25) propagation of 0.8 m light. [Simulated code in appendix A. Michelson (flowchart on page 147, input parameters: N=0.8, lamd=0.8, DH=14, θ=0,25; Digital single and dual crystal models (flowchart on page 150, input parameters: lam=0.8, lamd=0.8, DH=14, θ=0,25] Spectral Degradation Amplitude Loss It is evident from the previous discussion of the off-axis behavior of the spectral signals for the various models that one of the ways that the spectral begins to degrade is either a generalized decrease in the spectral peak amplitude or a shift in the location of the peak. As such, our first analysis technique involves measuring the amplitude loss at the original peak 42

66 location. Either generalized amplitude loss or a shift in the location of the spectral peak will be seen as a decrease in the amplitude at this location. The limit of the incident angle is set to be the angle at which the amplitude loss is 10%. Note that this has been arbitrarily determined only to allow comparison among models and not as an accurate determination of device angle limit. Figure 3.15 shows an enlarged close-up of the two dual-crystal curves shown in Figure The spectral signal with an incident angle of 25 degrees has shifted to the left of the initial peak location and has only decreased slightly in amplitude. The dotted line shown on the plot is a vertical line from the original spectral peak (the correct location) to the x-axis. As the spectral signal shifts with greater off-axis angles, the signal detected at the correct location decreases significantly. So, while the amplitude of the off-axis signal has decreased less than 5%, the amplitude at the original peak location (where this curve intersects with the dotted line) has decreased approximately 70%. 43

67 Figure Simulated output spectra of 14-stage dual-crystal WA-PIFTS for on-axis (0) and 25-degrees off-axis (25) propagation of 0.8 m light. [Simulated code in Appendix A. Digital dual crystal models (flowchart on page 147, input parameters: lam=0.8, lamd=0.8, DH=14, θ=0,25]. All discussed models have been analyzed based on this amplitude loss criteria; the results of this analysis are shown in Figure As seen in Figure 3.14, the performance is best for the dual-crystal design. The angular limits shown here are somewhat different than those obtained using the OPD criterion, shown in Figure 3.13, for the two crystal systems. 44

68 Figure Incident angle limit using amplitude loss criteria. Comparison between uniaxial and bi-crystal models as well as the equivalent Michelson interferometer dependent on system size. [Simulated code in Appendix A. Michelson (flowchart on page 147, input parameters: N=0.8, lamd=0.8, DH=14, θ=0-40; Digital single and dual crystal models (flowchart on page 150, input parameters: lam=0.8, lamd=0.8, DH=14, θ=0-40]. All models show a monotonous decrease in the incident angle limit with system size with the Michelson model showing significantly lower angular limits than any of the others. Both PIFTS models appear to have equivalent angular limits for very low system sizes (8 stages) Spectral Noise Another way the signals were degraded was through the introduction of noise outside of the spectral peak location. This noise is any portion of the signal which is different than the 45

69 input wavelength. For the most part, this is caused by a shift in the spectral signal away from its input value (see Figure 3.14). In considering whether there is any noise in the output, the data point representing the original intended spectral peak location, along with one data point on either side of this peak, are ignored. Any signal at any other data point, then, is interpreted as noise. The incident angle limit is defined to be the angle at which noise outside of the spectral peak location exceeds 10% of the peak amplitude. That is, ideally speaking, there should be no intensity measured outside of the input spectral peak. Noise is introduced when there is intensity detected in any region other than the spectral peak location. This analysis places the arbitrary constraint that the good system must have no false signals (intensity at any other spectral location) that are greater than 10% of the amplitude of the actual, input peak. This result is shown in Figure Again, this is an arbitrary definition to allow comparison among models and is not meant to serve as an accurate determination of device angle limit. Indeed, the incident angle limit based on this definition of spectral noise may be affected by the sampling pitch, but general trends of angular limit between the models are expected. It is important to note that, as this model considers the spectral peak location to be that of the on-axis signal, significant shift in the location of the spectral peak for off-axis light propagation will also be interpreted as the introduction of spectral noise 46

70 Figure Incident angle limit using spectral noise criterion. Comparison between single-crystal and dual-crystal models as well as the equivalent Michelson interferometer dependent on system size [Simulated code in Appendix A. Michelson (flowchart on page 147, input parameters: N=0.8, lamd=0.8, DH=14, θ=0-90; Digital single and dual crystal models (flowchart on page 150, input parameters: lam=0.8, lamd=0.8, DH=14, θ=0-90]. As with the other analysis methods, the Michelson model shows the worst performance of all models while the dual-crystal WA-PIFTS shows the best. In fact, the incident angle limit of the dual-crystal WA-PIFTS actually reaches beyond what would be physically realizable, suggesting that the dual-crystal WA-PIFTS offers significant reduction in off-axis noise. Additionally, the results for this system appear to suggest that the incident angle limit actually decreases when the system size is reduced below 11 stages. However, this effect is simply due 47

71 to the broadening of the spectral peak for these device resolutions which have a wider spectral peak Beam Walk-off Figure Walk-off of the extraordinary ray at 5-degrees off axis or angle at which 10μm walk-off will occur at maximum path difference through a single Calcite-Quartz dualcrystal stage. Figure 3.18 shows the walk-off through a dual-crystal system. Recall that, for the purposes of this investigation, walk-off is defined as the difference in output position between the e-ray and the o-ray. The blue curve, plotted against the left-hand y-axis, shows the walk-off experienced by the extraordinary ray when incoming light is input at 5-degrees from the normal. The green curve, plotted against the right-hand y-axis, shows the angle of incoming light that 48

72 results in a 10μm walk-off of the extraordinary ray for the various system sizes. The single crystal model shows approximately the same walk-off as the dual-crystal model for all system sizes and is, therefore, not plotted here. These graphs provide clear evidence that there may be a significant trade-off between spectral resolution (increased system size) and spatial resolution. If the spatial resolution required is larger, then larger pixels can be utilized in the imaging array and walk-off can be larger before the ordinary and extraordinary rays strike different pixels. 3.5 Conclusion The dual-crystal design modification proposed in this chapter offers significant improvements to the incident angle limit of the PIFTS system. The incorporation of the dualcrystal WA-PIFTS design, which equally splits the retardation of the crystal stage between a positive birefringence crystal and a negative birefringence crystal with orthogonal optic axes, reduces the off-axis change in the OPD experienced between the two polarization states. The introduction of this design modification reduces the trade-off necessary between spectral resolution, bandwidth, and numerical aperture in the PIFTS design. These trade-offs were discussed in terms of a number of different aspects of system performance. In this chapter, an analysis has been provided of this incident angle improvement through multiple means, including the standard OPD analysis conducted on this type of model by previous researchers as well as simulating output spectra to determine off-axis performance. All methods of comparison have shown significant gains with the dual-crystal WA-PIFTS over both the single-crystal PIFTS and the Michelson model. This indicates that polarization interference is a promising technique for Fourier-transform imaging spectroscopy in applications that require no moving parts. 49

73 3.6 Future Work The dual-crystal design modification has been shown to be effective in significantly improving the incident angle limit of the PIFTS system at a given spectral resolution. However, it has also been shown, using Zemax simulations, that beam walk-off is still a problem for high resolution systems. Further investigation would be required in order to improve the beam walkoff in the PIFTS design. It may be possible, then, to reduce the trade-offs required when considering walk-off in the same way as was achieved with the use of a dual-crystal design in terms of off-axis changes in OPD discussed in this chapter. 50

74 CHAPTER 4 Consideration of Variable Elements 4.1 Introduction In the previous chapter, a dual-crystal design modification to the Polarization- Interference Fourier-Transform Spectrometer (PIFTS) was described which significantly improved its incident angle limit. However, none of this analysis took into account the polarization rotators, which can affect both the scan speed of the device and its off-axis performance (as well as its achromaticity). Scan speed can be improved by utilizing some variable retarders instead of exclusively fixed birefringent retarders. However, as the fixed retarders are replaced by variable retarders, the angular limit decreases. In this chapter, the consideration of these elements is discussed, providing analysis techniques to evaluate the tradeoffs between different options Polarization Rotators In order to properly switch polarization states, real polarization rotators are required which were neglected in the previous chapter. However, the actual polarization rotators must be considered to get an accurate picture of the incident angle limit for various device configurations. These elements possess limitations with respect to both achromaticity and off- 51

75 axis performance, and these trade-offs must be weighed to determine final device design. This is particularly true for a standard twisted nematic (TN) cell, which is considered in this analysis. Historically, a number of strategies have been employed to provide fast-switching, achromatic polarization rotators. One simple strategy uses dual-frequency liquid crystal materials in thick cells to improve speed without reducing thickness 35. While this has also utilized this in the analysis provided in this chapter, the performance of the dual-frequency materials in these situations is questionable. Thick cells using these techniques have problem with backflow, though these problems may be mitigated using a polymer stabilization technique 36. However, while fast, these cells still suffer from limited viewing angle due to their thickness. A number of strategies have been employed to use a thinner TN (or FLC) cell in combination with other fixed retarder elements; most of these have been optimized for the visible range However, this technique has been developed in the NIR range, with optimal performance from 1155nm to 2437nm Hybrid Designs These designs combine the benefits of both the digital birefringent crystal model (discussed in the previous chapter) and the liquid crystal model (as proposed by Miller 30 ). In this analysis, both the incident angle limit and scan speed implications of various device designs using variable retarders are considered. In the device configuration considered in this chapter, some of the birefringent crystal stages are replaced with Electrically-Controlled-Birefringence (ECB) cells (starting with the thinnest crystal stages); these ECB cells are paired sets in which the pretilts are equal and opposite in order to minimize the angular dependence of path difference 52

76 introduced by these cells. A simple schematic of this design is shown in Figure 4.1. A Compensation Element is associated with the ECB stage in order to compensate for residual retardation of the voltage-applied state so that the stage can achieve a zero micron path difference. A polarization rotator is also associated with the ECB stage so that the contribution to the net path difference can be both positive (additive) or negative (subtractive) as with the fixed birefringent crystals. Figure 4.1. Schematic diagram of the hybrid designs. P=Polarizer, R=Polarization Rotator, C=Compensation Element, E=ECB Cell, B=Birefringent Crystal (+ or indicate sign of birefringence), m, n indicate repeating units. In this design, the ECB cells are tuned continuously from low to high voltage. The total number of ECB cells is kept constant at 4, meaning that the cells become thicker as more of the birefringent crystal stages are replaced with ECB cells. While this is not necessarily the optimum in terms of the incident angle limit, it provides a simple way to illustrate model comparison for any type of system. In order to effectively differentiate among the various Hybrid models in terms of the total retardation as well as the amount of the system made up of ECB cells versus birefringent 53

77 crystals, a coding system was developed. Each model was labeled with HA# where HA represents Hybrid Analog and the number indicates how many of the birefringent crystal stages are replaced with ECB cells. For example, HA6 is the Hybrid Analog model in which the 6 smallest stages are replaced with ECB cells. 4.2 Methods Simulated Liquid Crystal Cells Many of the methods utilized in Chapter 3 are carried over to this chapter. However, the variable liquid crystal elements utilized in the models studied in this chapter require additional methods. In particular, TN cells are modeled for the polarization rotators and ECB cells for variable retarders, though these analyses are conducted separately. The methods used to model each are described in this section TN Cell Thickness Previous efforts to analyze the performance of the PIFTS system have ignored effects due to the polarization rotators. However, the wide wavelength range over which they must operate requires that they be taken into consideration. For general purposes, how these elements might be considered is discussed in this chapter. To do this, the use of a twisted nematic cell is assumed. For a TN cell to properly rotate the polarization state of a particular wavelength of light, it must be any of a number of specific thicknesses. For thicknesses or wavelengths that deviate from these values, the polarization state of the light will rotate more or less than the intended amount. The transmission of a TN cell between parallel polarizers is given by 42 T = 1 2 sin 2 ( π 2 1+u2 ) 1+u 2, (4.1) 54

78 where u is given by u = 2 nd λ. (4.2) If the TN cell rotates the input polarization by 90 degrees as designed, the transmission will be at a minimum. The wavelength dependence of this relation is particularly problematic for the FTS since it must function properly over a large spectral range. However, as the cell becomes thicker (to the next transmission minimum), the deviation in transmission with wavelength becomes less and less. Unfortunately, the thicker the TN cell, the longer it will take to switch and, therefore, the longer the scan time. A balance must be found between achromatic performance and thickness/speed Director Configuration To tune the ECB cells through all necessary steps, the free energy associated with a particular orientation of the liquid crystal directors within the cell with a given applied voltage is considered; the free energy density 43 is given by f d = k 11 2 ( n )2 + k 33 2 (n n )2 1 2 (D z E ) 2, (4.3) where Dz is the z-component (normal to the cell surface) of the electric displacement due to the applied voltage and splay and bend elastic constants of the liquid crystal are k11=10pn and k33=18pn, respectively. Using a separate simulation, the ECB cell was tuned through small voltage steps until the appropriate retardation was found; this director configuration was utilized for a particular step in the system scan. To simplify simulations, all four ECB cells are assumed to tune simultaneously, though it should be noted that this results in a worst-case for the incident angle limit comparisons. 55

79 Signal Generation The interference pattern utilized in the FTS with variable elements is constructed identically to the ideal models described in Chapter 3. For models utilizing the polarization rotators, director configurations for an off- and on-state of the TN cell were calculated and stored. In hybrid models, the required director configurations were calculated using the voltages determined by the simulation described in Section Optical Path Difference To consider the degree of reduction of the off-axis performance of these systems, the change in optical path difference (OPD) was calculated; this was chosen as the method which was most directly comparable to available literature. This is done in the same way as it was described in Chapter 3. As before, only the configuration in which the fixed birefringent stages all have an additive effect on the net OPD of the system was considered. It is assumed that one set of antiparallel ECB cells will be switched simultaneously while the other set will be either in their voltage-off (planar) state. Due to the fact that off-axis changes in the effective extraordinary index will be largest when the director angle in the center of the cell is oriented at 45 degrees with respect to normally incident light, the first set of cells is assumed to be in this orientation. Note that the polarization rotators are not included in this analysis. The azimuthal angle is taken such that the variations in polar angle result in a change in the effective extraordinary index (variation in the plane of the principle axis of the liquid crystal). 56

80 As before, Snell s Law was used to consider the angle of refraction of both the ordinary and extraordinary rays (taking care to adjust the angle of the extraordinary ray to account for the effective extraordinary index). Considering the optical path of each ray, the retardation through each ECB cell and crystal in the system was determined. The off-axis OPD was compared with the OPD for on-axis propagation. The incident angle limit was defined as the angle at which the change in the OPD compared with the on-axis propagation is equal to one-quarter of the lowest wavelength in the spectral range ( min=0.8 m) Scan Time In some cases, the incorporation of variable retarders can decrease the scan speed of the device, particularly if the switching speed of the variable retarders is faster than the switching speed of the polarization rotators. As an example of this, a hybrid system with 20 m thick TN cells is considered. There are two factors which affect the scan speed for a particular device design: (1) the number of cell switches required to complete a full scan and (2) the switching speed of the cells. The cell switching speed should account for both the switching speed of the polarization rotators and the switching speed of any analog cell. To complete an entire scan, each possible combination of polarization rotators is required. For instance, if a system consisted of three polarization rotators, the following combinations would be required (in no particular order): (1) on, on, on; (2) on, on, off; (3) off, on, on, (4) on, off, on; (5) off, off, on; (6) off, on, off; (7) on, off, off; and (8) off, off, off. Because each cell has two possible states (on or off), the total number of switches, S#, required to complete a scan is given by 57

81 S # = 2 R #, (4.4) where R# is the total number of polarization rotators in the device (not the total number of birefringent stages). This must then be multiplied by the switching time of the polarization rotators and the total switching time for the variable elements between these switches. That is, the total scan time, Ttot, can be written as T tot = S # T PR T VR, (4.5) where TPR is the switching speed of the polarization rotators and TVR is the total switching time of the variable retarders for each time the polarization rotators are switched. To determine these times, it was first necessary to choose an appropriate approximation; using traditional LC-only cells would result in a prohibitively long switching speed and several seconds to complete a scan, even in the fastest device configurations. Instead, it is assumed that all LC cells were filled with dual-frequency material, though faster switching cells may be available in the literature. According to Golovin et al 36, the switching speed of a 20 m thick TN cell (Von and Voff are the same) was experimentally determined to be 2ms. This paper also discussed the strategy of splitting one thick TN cells with 90 degree twist into two half-thick TN cells each with 45-degree twist for faster switching speed, which is also used in these calculations. Assuming a 1/d 2 proportionality of the switching time, this data was used to determine the approximate switching speed of all variable cells (ECB and half-thick TN pairs). The use of a dual-frequency material was chosen for this analysis due to the fact that it simplified these calculations that is, the on- and off- switching times were the same so the calculation of total scan time was straightforward. However, if this material is not chosen, the exact nature of switching in the device of interest must be considered. For example, if at least one polarization 58

82 rotator is being switched off each time the polarization rotators are switched, then the switch-off time must be used for TPR. It was also assumed that the measurement (signal) could be taken while the ECB cell was actively switching (no delay in measurements); all ECB cells were assumed to be switching simultaneously. The total number of ECB cells was always kept at 4; resulting in a different ECB cell thickness and switching time for each system type. This means that, for each polarization rotator switch, the ECB cells must be tuned through their entire range of retardations. 4.3 Results Determining TN Cell Thickness In order to build a model with good performance while also being fast scanning using TN cells, it is necessary to determine the minimum possible thickness of the TN cells which allows for an output spectrum with an acceptably low noise level. This is also important to reduce losses to the FOV introduced by the TN cells. On-axis simulations of the dual-crystal digital PIFTS model were conducted in which the TN cell thickness was adjusted through various transmission minima, given by using Equations 4.1 and 4.2. In this case, a 13-stage design was used with two input wavelengths 1.99μm and 2.0μm. This particular schematic was chosen because the device, as designed (in the ideal case), should have the ability to resolve these two input signals. 59

83 Power (a.u.) Power (a.u.) (a) (b) Power (a.u.) Power (a.u.) Wavenumber (cm -1 ) Wavenumber (cm -1 ) (c) (d) Wavenumber (cm -1 ) Wavenumber (cm -1 ) Figure μm and 2.00μm spectral peak for 13-stage digital dual-crystal model for (a) ideal polarization rotators and various TN cell thicknesses (b) 1 st transmission minimum, (c) 3 rd transmission minimum and (d) 4 th transmission minimum. Shown for on-axis signal propagation [Digital: lam=1.99,2.00, lamd=0.8, DH=13, θ=0, (a) U=1, (b) U=2, (c) U=3, (d) U=4] Figure 4.2 shows the output spectrum for these two input signals in the case of ideal polarization rotators (represented as rotation matrices as in Chapter 3) and for the 1 st, 3 rd, and 4 th transmission minima TN cells. At the 1 st -transmission minimum, only 1 input signal is visible this system has no ability to resolve these two inputs. By the 3 rd transmission minimum, the 60

84 signals are resolvable, but there is a great deal of measured intensity outside of these two peaks (i.e. noise). Finally, by the 4 th transmission minimum, both peaks are easily resolved with performance near that of the ideal model and only a small amount of intensity outside of the two peaks. Therefore, the 4 th -minimum can be chosen as the thinnest TN cell which maintains relatively high on-axis performance. Figure µm spectral peak for 14-stage digital dual-crystal model for selected TN cell thickness for on-axis signal propagation and 10 and 25 degree off-axis signal propagation [Simulated code in Appendix A. Digital (flowchart on page 150) with TN cells (code in Section A.7.2 and A.7.3), Input parameters: lam=0.8, lamd=0.8, DH=14, θ=0,10,25, U=4]. Figure 4.3, then, shows the spectral output of this model using a 4 th -minimum TN cell for on-axis signal propagation as well as signal propagation at 10 degrees and 25 degrees off-axis and a 0.8μm input signal (note that the azimuthal angle is simulated at 0-degrees, which is the 61

85 entrance angle of the TN cell). At 10 degrees, the spectral peak has shown significant degradation, but there is little to no noise outside this region. However, by 25-degrees, the spectral peak has completely disappeared and there is noise across the plot, though this noise is still fairly minimal compared to the on-axis signal Spectral Degradation in Hybrid Models The next step is to understand the effect of ECB cells on the off-axis behavior of the spectrum. For this, a 14-stage equivalent HA8 model is used. Ideal polarization rotators are used in these simulations; TN cells have been excluded in this analysis. Again, the dual crystal model is used. The on-axis and 10-degree off-axis spectra for this model are shown in Figure 4.4. Here, it can be seen that, off-axis, the model suffers from the introduction of a somewhat periodic noise. The peak amplitude of the off-axis signal remains almost identical to the on-axis peak amplitude. While it is not entirely evident from this plot, the periodic noise is somewhat centralized around the spectral peak (and, therefore, higher in the plotted spectral range). This noise is likely due to the switching of the ECB cells through their entire range; this changes the field-of-view dependence of the signal periodically through the scan. 62

86 Figure m spectral peak for a 14-stage hybrid analog design HA8 with dual-crystal design for on-axis signal propagation and 10-degree off-axis signal propagation [Simulation code Section A.7.4 and A.7.5. Input parameters: lam=0.8, lamd=0.8, DH=14, HA=8, θ=0,10] Optical Path Difference The effect of replacing more and more of the fixed birefringent stages with variable ECB cells is considered, using the same limit as previously defined with the off-axis angular limit set to the angle at which the change in off-axis OPD from on-axis OPD is equal to λmin/4. These hybrid models are compared to the dual-crystal WA-PIFTS as well as the single crystal PIFTS, with Calcite, the results of which are shown in Figure 4.5. Recall that these are for the dualcrystal model with the polarization rotators neglected. For the most part, the incident angle limits of the hybrid models is effectively sandwiched between the poor-performing uniaxial PIFTS and the high-performing bi-crystal WA-PIFTS. When all stages have been replaced with 63

87 ECB cells, the performance drops below that of the uniaxial PIFTS. Additionally, when only one crystal stage remains, the angular limit of the hybrid device is approximately the same as the uniaxial PIFTS. For all other scenarios, the performance is better than the uniaxial PIFTS. Figure 4.5. Incident angle limit using OPD criterion. Comparison of the Hybrid systems compared with single and dual-crystal PIFTS based on total system size (effective total crystal stages) and the number of fixed stages replaced with variable elements [Simulated code in Appendix A (A.7.8), ECB Cell Thicknesses shown in Table 4.1]. 64

88 4.3.4 Scan Speed To determine the switching speed of the various systems, it was first necessary to determine the thickness of the ECB cells for the various Hybrid Analog models. The results of this calculation, as well as the switching time, are shown in Table 4.1. Table 4.1. ECB cell thickness and switching time for various Hybrid Analog models. HA Thickness ( m) Switch Time (ms) As previously described, the hybrid systems are referred to with an abbreviation HA (for Hybrid Analog) and a number indicating how many of the fixed birefringent stages have been replaced by the ECB cells; these stages are always the smallest. The effective total number of stages (the total number of stages which would be present in a digital device made exclusively of binary stages) is also referenced. For example, a 14-stage HA8 would refer to a system with a total path of xmax=819.2 m in which the smallest 8 binary stages, a total retardation of xecb=12.8 m, have been replaced with a set of 4 ECB cells, each of which is 14 m thick. Note that the thickness of the ECB cells, with a birefringence of n=0.25, results in a larger total retardation for the four ECB cells than xecb due to the inclusion of the compensation element, which decreases the total retardation through the analog portion of the device. 65

89 Figure 4.6. Scan speeds calculated for Hybrid systems compared with digital PIFTS based on total system size (effective total crystal stages) and the number of fixed stages replaced with variable elements. Based on these cell thicknesses and the methods described in Section 4.2.3, the resultant scan time for the various Hybrid Analog designs, as well as the digital PIFTS, are shown in Figure 4.6. All Hybrid Analog models offer a significant reduction in scan speed from the Digital device with the best gains being for Hybrid Analog 8. As more and more fixed elements are replaced by variable ones, the gains are canceled due to the thicker ECB cells required. Note 66

90 that these scan times can be significantly reduced by further dividing these ECB cells into thinner cells; the scan times calculated here can be considered a worst-case scenario. 4.4 Conclusions In this chapter, the consideration of variable elements in the PIFTS system has been discussed. One element that must be considered is the polarization rotators, which can significantly affect not only the off-axis performance, but the achromaticity and switching speed of the system. Methods have been illustrated which can be used to consider the trade-offs of polarization rotator design using TN cells. Replacing some of the fixed stages with variable retarders provides the benefit of decreasing system cost (by reducing mechanical tolerancing) and limiting any losses caused by the achromatic polarization rotators. However, by replacing bi-crystal stages with tunable uniaxial ones, some of the incident angle advantage is lost. The tunable cells add spectral noise (output peaks which are in a different spectral location from those input into the system). In this chapter, methods are illustrated for considering these elements using ECB cells as an example. Based on the methods used, as long as at least two dual-crystal stages remain, the system still has a larger incident angle limit benefit than the single crystal PIFTS system. When designing a system and selecting the number of stages to replace, balancing all of the losses against those from the polarization rotators being used is an important step in finding the optimal system design. The methods described here can be used on any combination of polarization rotators and variable retarders to consider these trade-offs. 67

91 CHAPTER 5 Introduction to the Liquid Crystal Thermal Imager 5.1 Thermal Imaging All objects with a temperature above absolute zero radiate infrared waves. The peak wavelength radiated from a body is proportional to its temperature. Typically, this radiation is detected in the long-wave infrared range from 8μm to 12μm (sometimes extended from 7μm to 14μm) 44. As this radiation is given off by bodies, it can also be absorbed by other bodies. Thermal imaging is the ability to utilize the absorption of IR radiation given off in a scene to generate an image of that scene 45. While thermal imaging can be designed to actually measure incident photons with wavelengths in the LWIR range, it is typically accomplished by simply absorbing the radiation, or heat given off by bodies in the scene to determine the temperature (or relative temperature) of these bodies 46. Thermal imaging is used in a wide variety of applications 46. Thermal imaging can be used to detect weak spots in the insulation of a building or identify hot spots in electronic circuits. It can be used to take a child s temperature without contact or to measure heat distribution in a body to monitor for illness. It can also be used for surveillance or as a motion detector

92 5.2 Thermal Imaging Systems In general, thermal imaging systems depend on a temperature-dependent change in a measureable property of a material in the imager. In some cases, this requires using an imaging system which is cooled to near absolute zero 46. However, these systems are bulky and have energy requirements that make them unusable for many applications. Figure 5.1. Diagram of a single microbolometer 47. Instead, some applications allow for the use of an uncooled system. The most common way of achieving thermal imaging in an uncooled system is through the use of a microbolometer, a diagram of which is shown in Figure 5.1. In this case, an IR absorbing material is placed on top of a pedestal the top region is the imaging pixel, the size of which defines the spatial resolution of the device 46. The absorption of IR radiation (heat) on the top of this pixel causes a change in resistance of this material. This resistance change in detected by running electrodes up the thermally insulating legs and probing with a very small current. Unfortunately, these microbolometers are difficult to fabricate, which has limited device yield and maximum achievable resolution (pixel size and/or array size). The requirement of electrical circuitry requires a large number of photolithographic steps defects introduced during 69

93 fabrication can destroy an entire array of these microbolometers 48. As such, there is a great deal of motivation to develop an alternative technology that does not have these problems. 5.3 Liquid Crystal Thermal Imager It is well known that liquid crystals possess a temperature-dependent birefringence. In fact, many years ago, researchers attempted to use liquid crystals to replace microbolometers in uncooled thermal imaging applications 49. Unfortunately, limitations in microfabrication techniques meant that liquid crystal pixels could not be thermally isolated from one another, so this technique were not pursued. However, recent developments in device processing have allowed for renewed interest in such devices Principles of Operation Figure 5.2. Temperature-dependence of the order parameter for a typical liquid crystal 50. The driving principle behind the use of liquid crystals as a thermal imaging device is the temperature-dependence of their liquid crystalline order. Figure 5.2 shows the temperature- 70

94 dependence of the order parameter, S, for a typical liquid crystal the actual values on this curve are dependent on the properties of the liquid crystal material. In general, the temperaturedependence of the order parameter can be approximated 50 by S = (1 T T ) β, (5.1) c where Tc is the clearing temperature of the liquid crystal and β is a material constant which can be experimentally determined for a given material (for 5CB, this has been experimentally determined as β= ). The temperature-dependence of the birefringence, then, can be shown to have this same dependence as n(t) = ( n) o (1 T T ) β, (5.2) c where (Δn)o is the birefringence of the liquid crystalline material in its crystalline state. Once these material constants are known, the birefringence of the liquid crystal at any temperature within its liquid crystalline range also becomes known. When the liquid crystal cell, then, is used between a pair of polarizers, the cell functions in much the same way as a planar cell based on electrically-controlled birefringence (ECB). In this case, though, changes in the net birefringence/retardation through the cell are induced by a change in temperature of the liquid crystalline material. This translates into a variable change in the polarization state of the light exiting the liquid crystal cell and, therefore, a change in transmission exiting the system. So, unlike the electrical probing required by microbolometer arrays, the change in temperature of a liquid crystal based thermal imager can be probed optically and does not require complex device processing

95 5.3.2 System Design Figure 5.3 shows an example layout of a thermal imaging system utilizing a liquid crystal transducer. In this case, thermally-isolated liquid crystal pixels are fabricated on top of a transparent substrate. From the front side, IR radiation incident from a scene is imaged onto the liquid crystal array. From the back side, polarized monochromatic light is propagated through the liquid crystal. The absorbed IR radiation causes changes in the transmitted intensity of the visible light which can then be imaged onto an off-the-shelf visible imaging array and turned into a thermal image of the scene 48. Figure 5.3. Diagram of a thermal imaging system utilizing temperature-dependent changes in liquid crystal birefringence

96 Transmission Detected signal (a.u.) Temperature ( o C) Temperature ( o C) Figure 5.4. Experimental [MITLL] (top) and simulated (bottom) transmission vs. temperature curves. 2μm thick 5CB cell with incident light of λ=505nm. Figure 5.4 shows the intensity measured through a 2μm thick 5CB cell between crossed polarizers with 505nm incident light. This is shown for measurements of an actual thermal imager 73

97 device as well as a calculated plot based on the temperature-dependent indices of refraction for 5CB that have been previously calculated 52. Figure 5.5 shows a basic schematic of one of the thermal isolated pixels, as well as an SEM photograph of a thermal pixel on an actual device. The fabrication details of these pixels is beyond the scope of this dissertation and are, therefore, not described here. However, these images show the thermal legs used to isolate each pixel from the substrate (and each other) in order to maintain high temporal resolution (fast response time). The liquid crystal pixels are fabricated as fully enclosed with a small ~2μm diameter hole in the top layer of the pixel through which the liquid crystal material must be infiltrated into the microcavity. Once the liquid crystal is inside, a capping layer must be applied to keep the material in. - Cap Layer - (a) Mask Layer (b) Figure 5.5. (a) Cartoon thermal pixel cross-section and (b) SEM photograph of a thermal pixel Alignment in the Liquid Crystal Thermal Imager As is well known, in order to generate a well-aligned liquid crystal system, the substrate surfaces must impose a preferred direction onto the liquid crystals. Unfortunately, the traditional rubbed polyimide alignment layer cannot be used because it is unable to survive the subsequent processing steps necessary to complete the microcavities. An alternative alignment layer must therefore be utilized that can either be applied while these microcavities are being fabricated and 74

98 survive subsequent processing or that can be infiltrated into the fully fabricated microcavities. The two options which were explored were mechanical rubbing of the substrates to generate surface grooves or photoalignment using azo dyes. Efforts to investigate the usability of photoalignment in these microcavities will be described in subsequent chapters Mechanical Generation of Grooves Figure 5.6. Poor-quality alignment generated on scratched silicon nitride substrates when fabricated into a 90-degree twist cell. Shown under the microscope at 5X between crossed polarizers. The first method considered by MIT Lincoln Laboratory to generate alignment in these microcavities was to use mechanical rubbing to generate grooves in the inner surfaces of the microcavity. After the bottom substrate of the microcavities is deposited, the array is essentially scratched with a diamond grit lapping paper. A layer of molybdenum which acts as a sacrificial layer (serves as the interior of the microcavity and is etched out later in fabrication) is then deposited and is also scratched. Finally, the top layer of the microcavity is applied; the groove 75

99 pattern generated in the molybdenum should then be transferred to the inner surface of the top layer of the microcavity. While this is a relatively simple way to generate liquid crystal alignment and has been shown to do so effectively, yield in this process is somewhat low. Additionally, alignment quality is somewhat poor. Figure 5.6 shows alignment generated when two silicon nitride substrates (1-inch squares) were scratched and used to assemble a 90-degree twist cell. The grooves in this case can be seen even under low microscope magnification and the alignment generated is quite uneven. Berreman s calculation of anchoring energy generated by surface topography alone has shown that anchoring energy of these layers is likely to be quite poor 53. Therefore, while mechanical scratching is usable in the short term to generate proof-of-concept of liquid crystal thermal imagers, it is not seen as a long-term solution for fieldable devices Director Fluctuations and Noise Noise is the limiting factor on the sensitivity of a liquid crystal thermal imager. Experimental and modeling efforts have been conducted to better understand the noise in this thermal imager system as well as the achievable device resolution when noise is limited. The experimental setup used to investigate the liquid crystal thermal imager system is shown in Figure

100 Figure 5.7. Simplified schematic of optical system Fluctuation Modes Initially, the optical system shown in Figure 5.7 was used by MIT Lincoln Laboratory to detect the presence of liquid crystal-based noise. Transmission and noise measurements were taken as the analyzer was rotated about 360 degrees. The noise, here, is the standard deviation across several measurements taken at the same analyzer angle. The liquid crystal noise was difficult to detect at room temperature, but was finally visible when temperatures approached the clearing point (transition temperature from nematic phase to isotropic phase). Additionally, when the quarter-wave plate was removed, the noise from the liquid crystal can be easily seen, even at lower temperatures (Figure 5.8). It is this system configuration (without the quarterwave plate) that is initially studied to understand the effect of the liquid crystal director fluctuations. 77

101 Figure 5.8. Transmission and noise in Senarmont setup measured with and without quarter-wave plate. [MITLL] Liquid crystal is a uniaxial material. In the thermal pixels, the liquid crystal is aligned in a planar (homogeneous) orientation where the liquid crystal long-axis (director) is in the plane of the cell. The thermal fluctuation of these liquid crystal molecules fall into two types: (1) inplane (twist) fluctuations and out-of-plane (splay) fluctuations. For each of these, the liquid crystal is deflected by some angle with respect to the easy-axis (the preferred direction of the director imposed by the alignment layer on the substrate). If the alignment layer provides strong anchoring, then the liquid crystal near the substrate is assumed to be unable to deflect while the liquid crystal near the center of the cell will experience a maximum deflection. This decays 78

102 sinusoidally as the distance from the substrate is decreased. This holds true for both splay and twist fluctuations. Figure 5.9. Illustration of twist fluctuations in cell viewed from the front (left) and splay fluctuations in cell viewed from the side (right). The key difference between the two director fluctuation types is the effect they have on incident light. Twist fluctuations result in a slowly changing principle axis while the splay fluctuations result in a slowly changing effective birefringence of the liquid crystal. Each of these scenarios is illustrated in Figure 5.9. Figure Poincare sphere plot of a fixed director configuration in a liquid crystal cell with no fluctuation (left), in-plane fluctuation (center), or out-of-plane fluctuation (right). [Left: σt=0deg, σs=0deg; Center: σt=10deg, σs=0deg; Right: σt=0deg, σs=10deg] 79

103 Next, it becomes important to understand how each of these fluctuation types affects the polarization of the light as it travels through the liquid crystal cell. For this, we turn to the Poincare sphere (Appendix B). Using this sphere, each of the fluctuation types can be simulated (as described in Appendix B) to determine the change in the polarization state from a case with no liquid crystal deflection to a case with a large director deflection in the center of the cell that is either in-plane (twist) or out-of-plane (splay). In each Poincare sphere plot (Figure 5.10), the output polarization states are shown as dots whereas the principle axes for each simulated liquid crystal layer are shown as a line from the center of the sphere to the surface. The director fluctuations used in these simulations are chosen to be larger than expected to accentuate the effect on the polarization state. In the case of in-plane fluctuations, the polarization state becomes closer to the equator (approaching linear polarization) and also falls to the side of the S2-S3 plane, which changes the tilt of the elliptical polarization. In the case of out-of-plane fluctuations, as the average birefringence of the cell is now lessened, the polarization state is further away from the equation (approaching circular polarization), but the tilt of the ellipticity is unchanged. 80

104 Standard Deviation Standard Deviation Analyzer Angle (deg) Analyzer Angle (deg) Figure Poincare sphere (left) and noise (right) of a moving director model with inplane fluctuations (top) or out-of-plane fluctuations (bottom). [Top: σt=5deg, σs=0deg, nloops=50; Bottom: σt=0deg, σs=5deg, nloops=50] Another way to illustrate this would be to loop over multiple photons (that is, the simulation is repeated multiple times and the results for each are averaged and the standard deviation at each analyzer angle is calculated). This allows simulation of the expected noise from each of these fluctuation types. In this case, the Poincare sphere shows only the collection of output polarization states, shown in Figure The right-hand plot for each, shows not the signal but the standard deviation of the signal at each analyzer angle when a larger number of loops through the simulation are used (in this case, nloops=50) and a different deviation of the liquid crystal director in the cell is randomly assigned (on a Gaussian distribution) for each 81

105 simulation loop. Again, the magnitude of the liquid crystal noise is exaggerated to make the effect of this noise obvious. The maxima and minima for each type of noise are 45 degrees from one another; when the in-plane (twist) noise is at its maximum, the out-of-plane (splay) noise is at its minimum and vice versa Isolating Fluctuation Modes Figure Original experimental configuration without quarter-wave plate. Figure Modified setup with liquid crystal and polarizer perpendicular to one another. If the simulated plots for the standard deviation/noise from Figure 5.11 are compared to the experimental standard deviation/noise obtained by MIT Lincoln Laboratory in Figure 5.8, the 82

106 pattern of noise obtained experimentally matches most closely with the in-plane (twist) fluctuations. However, it is important to understand whether there is any contribution to the noise due to out-of-plane (splay) fluctuations. The next step, then, is to attempt to isolate one or both of these fluctuation modes experimentally. This has been done by looking at two experimental setups: (1) the original configuration (Figure 5.12) with circularly polarized light (obtained by the correct selection of wavelength so that the light output from the liquid crystal cell was circularly polarized) and (2) a modified setup in which the input polarizer was rotated so that its transmission axis was perpendicular to the principle axis of the liquid crystal (Figure 5.13). In the first of these, the wavelength (λ=566nm) was selected to result in circularly polarized light output from the liquid crystal cell (d=6.48µm, ne=1.7304, no=1.5335). Each type of fluctuation causes a shift of the output polarization state along a different axis of the Poincare sphere (Figure 5.14). For in-plane (twist) fluctuations, the collection of output polarization states shown on the Poincare sphere oscillate across the S2 plane along the S1 plane. For out-of-plane (splay) fluctuations, the collection of output polarization states shown on the Poincare sphere oscillate across the S1 plane along the S2 plane. This, in turn, results in a maximum contribution to the noise from each fluctuation type at 45 degrees apart from one another. In the case when both types of fluctuations are simulated together, the amplitude of this noise curve is minimized (in particular, the noise minima are far above zero). However, despite the same magnitude being used for both splay and twist fluctuations in this final simulation, the shape of this noise with analyzer angle still follows closely with the shape of the noise when only in-plane (twist) fluctuations are used (as in the left-hand plot). 83

107 Standard Deviation Analyzer Angle (deg) Analyzer Angle (deg) Analyzer Angle (deg) Figure Poincare sphere illustration (top) and noise (bottom) of circularly polarized light with in-plane fluctuations (left), out-of-plane fluctuations (center), or both fluctuation types (right). [Left: σt=5deg, σs=0deg, nloops=50; Center: σt=0deg, σs=5deg, nloops=50; Right: σt=5deg, σs=5deg, nloops=50] The simulated noise for each of the scenarios shown in Figure 5.14 can be compared to the experimentally detected noise for the same optical setup (Figure 5.15), which was provided by MIT Lincoln Laboratory. The overall pattern seen in the noise is seen to follow both the simulated cases with just in-plane (twist) fluctuations and with both fluctuation types. However, because the minima are far from zero, these results suggest that, indeed, both fluctuation modes are present in the liquid crystal. 84

108 Figure Experimentally detected noise for circularly polarized light using setup shown in Figure 5.12 [MITLL]. A second experiment, with the polarizer rotated so that its transmission axis is perpendicular to the liquid crystal principle axis (Figure 5.13) serves as an effective follow-up. In this case, the incident light will experience only the ordinary index of the liquid crystal and the effect of the out-of-plane (splay) fluctuations will be minimized, leaving only noise generated from the in-plane (twist) fluctuations. If the minima in this case approach zero, then this is further confirmation that the out-of-plane (splay) fluctuations are present. Figure 5.16 shows these experimental results (from MIT Lincoln Laboratory) as well as the simulated results with both fluctuation types simulated in this configuration. As expected, the minima are now significantly decreased due to the ability to neglect the out-of-plane (splay) fluctuation modes in 85

109 this optical configuration. Note that the two slightly higher minima in the experimental data are due to the shot noise contribution in this case. Analyzer Angle (deg) Figure Experimentally detected [MITLL] (left) and simulated (right) noise for setup shown in Figure Simulation included both fluctuation types. [Simulated: σt=5deg, σs=5deg, nloops=50] Addition of Quarter-wave Plate As mentioned previously, initial measurements with a quarter-wave plate in the experimental setup (Figure 5.8) made detection of the liquid crystal noise challenging. Only when temperatures approached the clearing point (phase transition temperature) or when the quarter-wave plate was removed was the liquid crystal noise detectable. At low temperatures with the quarter-wave plate, the noise followed closely with the shot noise of the light. To understand this, we again return to the Poincare sphere and consider elliptically polarized light. Similar to the scenarios shown in Figures 5.10 and 5.11, the addition of the quarter-wave plate is plotted using a blue dot for the final polarization state (Figures 5.17 and 86

110 5.18). For non-fluctuating liquid crystal or for out-of-plane fluctuations, the quarter-wave plate serves to rotate the polarization state down onto the equator of the Poincare sphere, resulting in linearly polarized light. When in-plane fluctuations are considered, the resultant polarization state is not quite linear. Figure Poincare sphere plot of a fixed director configuration of a liquid crystal cell (green) then a quarter-wave plate (blue) with no fluctuation (left), in-plane fluctuation (center), or out-of-plane fluctuation (right). [Left: σt=0deg, σs=0deg; Center: σt=10deg, σs=0deg; Right: σt=0deg, σs=10deg] For the case of in-plane fluctuations (or when both are simulated), the noise is decreased when the quarter-wave plate is added. This is easiest to understand when the equation used to calculate the intensity through the analyzer is considered based on the comparison of the vectors for the analyzer transmission axis and the output polarization state 54, which is given by I = 1 (cos(2ω) cos(φ 2 A φ S ) + 1), (5.3) 87

111 Standard Deviation where ω is the angle from the equation and φa and φs are the equatorial angle away from the +S1 axis. The ω term is independent of the analyzer angle, so only the second cosine in this equation affects the shape of the noise. That is, the shift in equatorial angle has the greatest effect on the noise. Analyzer Angle (deg) Analyzer Angle (deg) Analyzer Angle (deg) Figure Poincare sphere (top) and noise (bottom) of a moving director with in-plane fluctuations (left), out-of-plane fluctuations (center) or both (right). [Left: σt=5deg, σs=0deg, nloops=500; Center: σt=0deg, σs=5deg, nloops=500; Right: σt=5deg, σs=5deg, nloops=500] In-plane (twist) fluctuations contribute most to the noise; these fluctuations cause a shift in the equatorial angle while the out-of-plane (splay) fluctuations cause a shift in the azimuthal 88

112 angle. This is reversed by the quarter-wave plate, suppressing the effect of the in-plane fluctuations and minimizing the noise overall Maximizing Signal to Noise Obviously, the two ways to maximize the signal-to-noise ratio of a system are maximizing the signal and minimizing the noise. Maximizing the signal is best achieved by designing an optical system such that the output polarization state is linear; this can be achieved by addition of a quarter-wave plate just prior to the analyzer in the optical setup. This results in linear polarization regardless of the input wavelength or the polarization state output from the liquid crystal cell. Fortunately, the addition of the quarter-wave plate in this configuration also serves to reduce the noise. The other way to minimize the noise is to minimize the effect of the director fluctuations (or average them out through spatial or temporal averaging). Mathematical models for director fluctuations and the correlations between these modes, spatially and temporally, have been previously explored by a number of researchers. Andrienko at al 55 describe the allowable fluctuation modes in the liquid crystal cell. The dominant wave number, qz, for either twist or splay fluctuations, is largest in the center of the cell and smallest near the edges where surface anchoring plays a role. When the anchoring condition is considered strong, the dominant mode is given by q z,strong = π, (5.4) d where d is the cell thickness. The relaxation time of this fluctuation is given by τ = γ 2, (5.5) Kq z,strong 89

113 where γ is the viscosity and K is the relevant elastic constant. It is clear, then, that minimizing the relaxation time (minimizing the spatial averaging required) involves maximizing the wavenumber of the dominant fluctuation mode. For example, cutting the thickness in half results in a factor of 4 reduction in the relaxation time of the fluctuations. r (um) d (um) Figure Spatial correlation between points as it depends on the distance, r, between those points and the thickness, d, of the cell. The distortion of a single liquid crystal director will result in interactions with neighboring directors, which causes a propagation of this distortion in the plane of the cell (observed as a spatial distortion). The strength of the interaction between neighboring directors affects how far from the central director the distortion propagates radially outward. The spatial correlation can be understood from Zel dovich et al 56, given by H(ρ 1 )H(ρ 2 ) = k BT 4d K ( d π 3 2 ρ 1 ρ 2 )1/2 exp ( π ρ 1 ρ 2 d 90 ), (5.6)

114 where ρ1 and ρ2 are xy positions within the cell. The fall-off of the correlation between two points is exponential as the distance between those points increases. Additionally, the thicker the cell, the slower the decay of the spatial correlations with increasing distance between points. This relationship can be seen in the three-dimensional plot in Figure Minimizing the spatial correlations, then, requires minimizing the cell thickness. In this case, cutting the thickness in half results in, very roughly, and order of magnitude decrease in the spatial correlations. Anchoring Energy (x10-6 J/cm 2 ) Figure Relationship between anchoring energy, cell thickness, and dominant fluctuation mode (wavenumber) for the case of weak anchoring. The situation for weak anchoring is somewhat more complicated, with the dominant mode being given by 55 q z,weak = W Kd, (5.7) where W is the surface anchoring energy. The relaxation time of this dominant mode is given by τ γd2 12K + γd 2W + ζ W, (5.8) 91

115 where W is the surface anchoring energy and ζ is the surface orientational viscosity 57. In this case, the effect of thickness is lessened, with the dominant mode being inversely proportional to the square root of the thickness; it is also directly proportional to the square root of the anchoring energy; this relationship is shown in Figure If the goal is, again, to maximize the wave number (in order to minimize the relaxation time), then the cell thickness should be minimized or the anchoring energy should be maximized. Because a reduction in cell thickness also reduces the signal, it becomes imperative to improve signal-to-noise by increasing the anchoring energy of the alignment layer. This has provided motivation to the investigations described in Chapters 6 & 7. 92

116 CHAPTER 6 Reactive-Mesogen Stabilized Azo Dye Photoalignment 6.1 Introduction While rubbed alignment layers have been an industry standard, there are a number of disadvantages to this technique 58 ; these disadvantages include the introduction of particles or charging due to the mechanical rubbing step and the difficulty of generating complex alignment patterns. As such, researchers have sought to create alignment layers that do not require this mechanical rubbing step. Most typically, this has been achieved by using a photoalignment method Photoalignment Techniques Photoalignment, broadly defined, is any alignment technique which generates surface anisotropy using irradiation with light. There are a number of different photoalignment techniques which can be categorized by the way in which the irradiation (polarized or unpolarized) generates this surface anisotropy 59. In photo-polymerization, the irradiation results in polymerization with cross-linking occurring selectively along one direction. In photodegradation, it can result in selective destruction of the surface molecules oriented along one direction (this generates anisotropy by selectively leaving behind molecules oriented in the opposite direction). In photo-isomerization, the irradiation with light generates a conformational 93

117 change in the surface molecules. In photo-reorientation, the irradiation selectively excites molecules along one direction. The last two of these are most commonly accomplished using azo dyes, which selectively absorb well in the UV and visible ranges. While photo-isomerization typically has a very short life-time due to the gradual relaxation of molecules from the cis to the trans state 60, photo-reorientation can be quite stable if not re-exposed to light. In addition to the lower required irradiation energies when compared to both photo-polymerization and photodegradation, photo-reorientation can produce alignment layers with very high order parameter and anchoring energy Azo Dye Photoalignment In photo-reorientation, a dichroic dye, most often one containing at least one azo group, is irradiated with polarized light of a wavelength which is well absorbed by the dye. Figure 6.1 shows the structure of one such dye, Brilliant Yellow (BY) which is the one we have utilized for our studies. It is structurally quite similar to the dye used in much of the literature on this photoalignment technique, SD1. Figure 6.2 shows an example absorption spectrum for a spundown layer of BY. Figure 6.1. Molecular structure of Brilliant Yellow azo dye

118 Figure 6.2. Absorption spectrum of BY on glass substrate spun on using 1%wt BY in DMF. The probability that a given dye molecule will absorb this incident irradiation is proportional to cos 2 θ where θ is the angle between the incident polarization axis and the long axis (the absorption axis) of the dye molecule 62. Over time, this will result in an increase in the population of dye molecules aligned perpendicular to the incident polarization, where the probability of absorption is low. After a sufficient exposure dose, the order parameter of this dye layer can exceed even that of the liquid crystal material it is being used to align. Anchoring energies of the liquid crystal director on these layers have been measured to be similar in magnitude to the anchoring energies achieved through traditional rubbed polyimide alignment 63. The ability to generate a pretilt angle with this alignment layer has also been shown 64. This type of photoalignment has received a great deal of attention in LC photonics applications due to its ability to be utilized in non-standard geometries 65. For instance, photoalignment has been utilized in the creation of a tunable microresonator 66, however, the alignment layer in this case is applied through a standard spin-coating method. Great success has 95

119 also been shown in the use of photoalignment for tunable photonic crystal fibers (PCFs) In this case, the application of the photoalignment layer via spin-coating is impossible; instead, the fiber is filled with the photoalignment solution through capillary action, then excess solution is removed through a pressure gradient. These methods offer a solution for the application of a photoalignment layer with either a completely open geometry or two open ends. However, neither is usable for a configuration with only one entry/exit port as in the desired liquid crystal thermal imager. Another problem with these layers is their questionable stability, particularly against subsequent light exposure 68,71. This has kept photoalignment from being adopted for large-scale applications. As such, researchers have sought ways to improve the stability of alignment generated with these layers, typically through the incorporation of a polymerizable material, with mixed results. For instance, one approach utilizes a photoalignment material which is also polymerizable However, this method requires complex chemical synthesis and strict control of heat and light during processing. Additionally, while a number of tests have been used to verify thermal and photostability of these alignment layers, the anchoring energy and order parameter of these polymerizable dyes is questionable. In one case, Kondo et al state that the order parameter they are able to achieve is quite low 74. As a modification of this approach, one can consider blending of azo dye with pre-polymer. However, this method is only tested for ferroelectric LCs, which require quite specific anchoring conditions Surface-Localized Polymer Layer As shown by previous researchers, the surface-localized polymer layer is generated by first mixing a reactive mesogen pre-polymer into the liquid crystal material and filling this into 96

120 the cell. Next, the reactive mesogen is allowed to separate to the substrate surfaces. This can be assisted by the application of a voltage under certain circumstances, but this method is beyond the scope of this dissertation 76. Once the polymer is sufficiently localized, a voltage can be applied to change the liquid crystal orientation (if no pretilt is required, then no voltage need be applied). Next, the cell is irradiated by UV light, which is absorbed by a photoinitiator mixed in at low concentrations to the RM/LC mixture. The reactive mesogen is polymerized along the orientation of the liquid crystals during this step. Finally, once voltage is turned off, the liquid crystal orientation in the region with high concentration of RM is frozen in place. This process is illustrated in Figure 6.3. The molecular structure of the reactive mesogen we have utilized, RM257, is shown in Figure 6.4. Figure 6.3. (a) Monomers are present in the liquid crystal sample which exist in some distribution across the cell normal, localized near the surfaces. (b) Applied voltage will reorient the liquid crystal. (c) UV illumination will crosslink the monomers, locking in the liquid crystal orientation during this stage. (d) After UV illumination, this liquid crystal orientation is locked in (where the polymer network exists), even after voltage is removed. 97

121 Figure 6.4. Molecular structure of reactive mesogen RM Model for Verifying Surface-Localization Dielectric constant vs. voltage curves have been previously utilized to develop a better understanding of the nature of a surface-localized polymer layer 77. The key factor considered in this investigation was the network density gradient of the polymer normal to the plane of the cell. The LC director configuration in the cell, given a particular applied voltage, was determined numerically by utilizing the free energy density of the system, given by 77 f d = k 11 2 ( n )2 + k 22 2 (n n )2 + k 33 2 (n n )2 W 2 (n n o) (D E ) 2 (6.1) where k11, k22, k33 are the splay, twist, and bend elastic constants of the LC, respectively, D is the electric displacement, E is the electric field, n is the LC director at a particular point, no is the preferred direction of the director (at points along the polymer network), and W is the effective anchoring strength of the LC director in contact with the polymer (W=0 in regions without polymer). The preferred director, no, is determined by the director orientation at the time of polymerization, where the orientation is imprinted onto the polymer network as described in the previous section. The effect of the polymer distribution through the cell was modeled by making the anchoring parameter, W, effectively proportional to a polymer distribution given by X(z) = X 0 (e z/ξ + e (d z)/ξ ) (6.2) where X0 is considered as the polymer concentration at the substrate surface, d is the cell thickness, and ξ is the length scale for the decay of the concentration away from the surface. 98

122 Figure 6.5 shows the effect of the values of X0 and ξ on the dieletric constant vs. voltage curves where the polymer orientation, also equal to no, is determined by the director distribution in the cell with 10V RMS applied. Note that these plots show dielectric constant versus voltage the dielectric constant was calculated directly based on the simulated director configuration with the dielectric properties of the polymer assumed to be equal to that of the liquid crystal. It can be seen that, if the polymer is quite evenly distributed through the cell, the main effect is to see a shift in the threshold voltage of the device (the voltage below which no change in dielectric constant has occurred). However, if the polymer is more concentrated on the surface, the main effect is to see a shift in the saturation voltage of the device (the voltage above which the change in dielectric constant is saturated). (a) (b) Figure 6.5. Simulated dielectric data using (a) surface concentration X0=0.08 and decay length ξ=0.2d and (b) surface concentration X0=0.8 and decay length ξ=0.02d 77. On the other hand, when a simplified model was used in which it was assumed that the effect of the polymer was restricted to an infinitesimally thin layer (effectively a monolayer) at the surface and acts as an alignment layer with a pretilt (no polymer network exists in the liquidcrystal-filled region of the cell and the polymer interaction term is dropped), the effect on the 99

123 dielectric constant vs. voltage curve was simulated as shown in Figure 6.6. Here, there is little effect on the curve where the polymer is cured at 0V. The main effect is to lower the zero volt value of the dielectric constant for the case where the cell is cured at high voltage. The zero volt value of the dielectric constant will be related to the induced pretilt that results from the given value of the applied voltage, with higher voltages and/or higher polymer concentration yielding values of the zero volt dielectric constant that are higher, approaching the saturation value with the effective pretilt of 90 degrees. Figure 6.6. Simulated dielectric data in which the polymer network is assumed to be infinitesimally thin (i.e. highly localized) Verifying Surface-Localization of the Polymer Layer Experimental Details Because we are interested in the optical properties of our samples, we utilized transmission vs. voltage (TV) curves on cells prepared similarly to those referenced in the simulations discussed in the previous section. Phase vs. voltage for each sample will show 100

124 similar behavior to the dielectric constant vs. voltage curves; a change in the zero volt phase retardation will indicate an increase in the pretilt of the sample while a change in the threshold voltage will indicate that the polymer network is not surface-localized and exists in the bulk. Figure 6.7. Diagram of cells prepared for testing surface localization of reactive mesogen. Polymide alignment layers on each substrate were rubbed in opposite directions (as indicated). Five-micron thick cells with at least a 1 cm x 1 cm active area were prepared with rubbed polyimide alignment layers on the substrate surfaces (each substrate was rubbed in the opposite direction). A diagram of this cell is shown in Figure 6.7. Before infiltrating the cells, RM257 was mixed with the photoinitiator Irgacure 651 in which the photoinitiator was at 10% concentration by weight. This was then added to the LC BL006 such that the RM257/photoinitiator was at 1.5% concentration by weight. The complete mixture was then heated to 125C and vortexed for 3 minutes to produce a uniform mixture. Cells were then infused with either the 1.5% wt RM257/BL006 mixture or pure BL006. The RM/LC cells were then stored overnight in dark conditions to allow for separation of the RM to the substrate surfaces. Next, these samples were polymerized using 20 minutes of exposure to 3.5mW/cm 2 101

125 UV light (λ=365nm) provided by a Mightex collimated UV LED light source. Each cell was either polymerized at 0V or 20V (1 khz AC). Once cells were polymerized, TV curves were obtained by placing the sample between two polarizers. TV curves were taken with both crossed and parallel polarizers; in both cases, the cell was oriented with the alignment at 45 degrees to the input polarizer. A broadband Oriel fiber optic illuminator was used as a light source and an interference filter (λ=633nm) was utilized to produce monochromatic light. To neglect transmission losses for phase retardation calculations, the TV curves were adjusted so that the maximum and minimum transmissions through the cell (i.e. detected voltage) are taken to be equivalent to a transmission of 1 or 0, respectively. To produce a plot more comparable to the dielectric constant vs. voltage curves, each set of TV curves was further converted into a phase retardation vs. voltage profile. This utilizes the fact that the transmitted intensity between crossed polarizers is given by 33 I = I o (sin δ 2 )2, (6.3) with δ being the phase retardation of the LC sample. This transmitted intensity between parallel polarizers is similarly given by I = I o (cos δ 2 )2. (6.4) The phase retardation of the sample at a particular voltage, then, is given by the transmission ratio in these two plots, as or δ = Nπ + 2tan 1 I I, N = 0, 2, 4,, (6.5) δ = (N + 1)π 2tan 1 I I, N = 1, 3, 5,, (6.6) 102

126 where N is the peak number in the TV curve (counted up from the high-voltage end of the plot). When the cell is almost completely switched, N= Results Figure 6.8 shows the phase retardation vs. voltage for the prepared cells. In this case, the sample polymerized at 0V shows no significant differences from the cell filled with pure LC neither the threshold voltage nor the saturation voltage is noticeably different. This indicates that the RM layer is sufficiently thin so as to have no effect on the bulk LC. In the case of the sample polymerized at 20V, though, the zero volt retardation has dropped and the threshold voltage has also decreased, indicating that the pretilt of the cell has increased. These results are very similar to the case of an infinitesimally thin RM layer, shown in Figure 6.6, indicating that our RM layer is quite thin. Figure 6.8. Phase profiles for 5µm planar cells filled with pure LC or LC with 1.5%wt RM257 which was polymerized at either 0V or 20V. 103

127 6.3 Photostability Testing Experimental Details To test the ability of our polymer layer to stabilize the alignment generated using a photoalignment layer, additional 7µm cells were constructed in which one substrate was coated with a rubbed polyimide alignment layer and the other was coated with a spun-on BY photoalignment layer. The BY was applied to the glass by mixing the dye at 2% concentration by weight into DMF, then vortexing for 1 minute to create a uniform solution. The glass was cleaned via ultrasonic and UV/O3 cleaning just prior to the application of the dye solution, which was passed through a 1µm filter as it was applied. The entire substrate was coated and the sample was spun at 1500rpm for 30 seconds to create an even dye layer coating. The substrate was then baked at 120C for 40 minutes to evaporate any remaining solvent. Test samples, once assembled, were exposed to 50mW/cm 2 polarized blue light for 7 minutes with the polarization direction aligned with the rubbed polyimide alignment direction. This exposure results in an approximately 90-degree twist with the photoalignment direction perpendicular to the rubbed alignment. This twisted cell configuration provides for fast visual determination of the degradation of alignment. When the cell is initially fabricated, the anchoring energies of the rubbed-polyimide and of the photoalignment layer are both sufficiently strong, so the twisted LC director configuration is observed. If the photoalignment layer is rewritten to a new angle, then the twist angle through the cell will change. If the photoalignment layer is degraded, the rubbed polyimide alignment direction will dominate and the cell will lose its twisted director configuration completely. The director field in the cell will then be co-planar and aligned with the axis 104

128 determined by the polyimide. When viewed between crossed polarizers, twisted regions will appear bright while non-twisted planar regions will appear dark. When viewed between parallel polarizers, non-twisted planar regions will appear bright while twisted regions will appear dark. This is presented diagrammatically in Figure 6.9. Figure 6.9. Expected visual appearance of cells (when viewed between crossed and parallel polarizers) depending on the condition of the photoalignment layer following exposure to either polarized blue light at 20mW/cm 2 oriented at 45 degrees to the photoalignment axis (top) or unpolarized blue light at 120mW/cm 2 (bottom). The samples were filled with either pure BL006 or the same 1.5%wt RM257/BL006 mixture as utilized in verification of surface-localization of the polymer layer, with storage and polymerization at 0V occurring as previously described. For photostability testing, samples were 105

129 exposed to the same blue LED used to align them. In one case, samples were exposed to 20mW/cm 2 polarized blue light at 45 degrees to the original photoexposure direction. In another case, samples were exposed to unpolarized light of 120mW/cm Results Without Polymer Layer a 45 b c d e Figure Sample filled with pure BL006 (a) before exposure or after (b) 50 minutes, (c) 100 minutes, (d) 150 minutes, and (e) 200 minutes of exposure to ~20mW/cm 2 blue light oriented at 45 degrees to the photoalignment axis. Before images shown between crossed (left) and parallel (right) polarizers. After images shown between parallel (left) and crossed (center) polarizers as well as polarized oriented at 45 degrees (right). Samples containing pure BL006 showed a complete loss of their original photoalignment direction in the low intensity polarized exposure (Figure 6.10). The alignment has been rewritten 106

130 within the first 50 minutes of exposure. Between polarizers oriented at 45 degrees (approximately the newly written twist angle of the cell), the sample exhibits a somewhat dark twist state. However, as the sample is exposed for longer, even this alignment is lost, with the sample failing to twist light at all. The photoalignment layer has been completely degraded and the intended alignment of the sample has been lost. Samples with pure BL006 also showed a rapid degradation of their photoalignment layer in the high intensity unpolarized exposure condition. These samples exhibited planar alignment after less than 20 minutes of exposure to this condition (Figure 6.11). a b Figure Samples filled with pure BL006 shown (a) before and (b) after exposure to unpolarized blue light at ~120mW/cm 2 for 20 minutes. Shown between parallel (left) or crossed (right) polarizers With Polymer Layer On the other hand, samples which contained the RM-stabilization layer exhibited a high degree of stability. In both the low intensity polarized condition and the high intensity unpolarized condition, samples maintained their 90-degree twisted alignment for 3 weeks with no sign of degradation of their alignment (Figure 6.12). 107

131 a c b d Figure Samples filled with 1.5%wt RM257 in BL006 shown between parallel (left) or crossed (right) polarizers. (a) Before and (b) after exposure to polarized blue light of ~20mW/cm 2 for 21 days and (c) before and (d) after exposure to unpolarized blue light of ~120mW/cm 2 for 21 days. We have also considered the electro-optic response of these samples. In this case, a sample filled with pure BL006 which had not been exposed, as well as a sample filled with 1.5%wt RM257 in BL006 which had been exposed to ~120mW/cm 2 unpolarized blue light for 3 weeks, were utilized. Samples were placed between crossed polarizers with the entrance and exit LC director aligned with the entrance and exit polarizer, respectively. TV curves for the samples taken in this configuration are shown in Figure There is no significant difference in the electro-optic response of these two cells indicating that, not only does the RM layer have no effect on this response, but also that this TV response remains stable against extended light exposure. 108

132 Figure TV curves of 5µm samples with rubbed polyimide alignment on one substrate and BY alignment on the other substrate, aligned in 90-degree twisted configuration. Curve taken between crossed polarizers with the sample aligned with the polarizers as shown in the embedded diagram. Sample either unexposure (pure BL006) or exposed to 120mW/cm 2 unpolarized blue light for 3 weeks (1.5%wt RM257 in BL006). 6.4 Anchoring Energy While it is well known that the anchoring energy of azo dye photoalignment layers is comparable to that of a rubbed polyimide layer, the effect of the RM-stabilization layer on this anchoring energy is unknown. To that end, we calculated the anchoring energy of our cells with and without the RM-stabilization layer Details of the Method The method begins with thin 90-degree twist cells. Ideally, this configuration will result in a 90-degree twist through the cell. However, a small cell gap means that the distance over 109

133 which the liquid crystal director must twist to match these boundary conditions is quite small, creating a significant competition between the anchoring energy on the substrate surfaces and the twist energy through the cell. This competition, in turn, means that the actual twist angle through the cell may be somewhat less than the intended twist of 90 degrees. The larger the difference between the actual twist and the intended twist, the lower the anchoring energy. Measuring the anchoring energy in this way was originally discussed by Akahane et al 78. Here, an expression for the transmission through the cell is given by 2 T = ( 1 1+( Γ φ )2 sin (φ 1 + ( Γ φ )2 ) sin(φ ψ p ) + cos (φ 1 + ( Γ φ )2 ) cos(φ ψ p )) ( Γ φ )2 sin2 (φ 1 + ( Γ φ )2 ) cos 2 (φ + 2ψ o ψ p ), (6.7) where Γ=πΔnd/λ is the retardation through the cell (Δn=ne-no is the birefringence of the liquid crystal, d is the cell gap, and λ is the wavelength of light used to probe the cell), φ is the measured twist through the cell, ψp is the angle between the exit and entrance polarizers and ψo is the angle of the liquid crystal director at the entrance of the cell with respect to the entrance polarizer (i.e. the angle between the liquid crystal director and the polarization state of light as it entered the liquid crystal cell). Once ψp and ψo are determined experimentally (with T=0), they can be used to solve for the twist angle of the cell, φ. From there, the anchoring energy can be calculated by A φ = 2K 2φ dsin2 φ, (6.8) where Δφ=(φ-φo)/2 is the deviation between the actual twist angle and the targeted twist angle (the factor of two accounts for the fact that a deviation in the liquid crystal director occurs at both surfaces). 110

134 6.4.2 Experimental Details In order to perform the measurements, a number of cell were prepared with two identical substrates with BY photoalignment layers prepared using the preparation techniques described in previous sections. The only exception, in this case, is that the glass substrates with photoalignment layers are aligned with polarized blue light prior to being assembled into cells. Samples are then assembled using 1.4μm spacers such that the alignment on each substrate is perpendicular. The actual cell gap is confirmed utilizing a plot of reflection versus wavelength in an empty cell (air gap, n=1). Cells were then filled with either pure E7 or 1.5%wt RM257 in E7 (with photoinitiator mixed at a weight concentration of 10% with respect to the RM). The RM/LC solution was then heated to 80C and vortexed for 10 seconds immediately prior to filling. Pure LC was also filed into cells at 80C. From this point, RM cells were stored and irradiated and previously described. Figure Diagram of experimental setup used to measure twist angle of TN cells for the purpose of determining anchoring energy. To perform the actual measurement, we place a TN cell between two polarizers, as shown in Figure Here, the entrance polarizer is fixed while both the TN cell and analyzer are 111

135 mounted on high precision rotation stages. The illumination is provided by a broadband Oriel fiber optic illuminator with an interference filter (λ=543nm) to produce monochromatic light. First, a transmission minimum is located by rotating the TN cell between parallel polarizers (this minimizes the first term in the equation). With this minimum found, both the TN cell and the analyzer are rotated (with the ratio between the angle of rotation of the TN cell and the analyzer being 1:2) to a transmission minimum Results The anchoring energies calculated for four cells either with photoalignment or RMstabilized photoalignment are shown in Table 6.1. Not only are these anchoring energies quite high (our measured anchoring energies for a polyimide alignment layers ranged from *10-4 J/cm 2 ), but the presence of the RM did not appear to affect the anchoring energy. Table 6.1. Calculated anchoring energies (in J/cm 2 ) for twist cells filled with either pure E7 (photoalignment) or RM257 in E7 (RM-stabilized photoalignment). BY BY + RM257 Sample 1 1.2* *10-4 Sample 2 2.5* *10-4 Sample 3 1.8* *10-4 Sample 4 2.0* * Conclusions In this chapter, a technique has been introduced to generate stable alignment utilizing a photodefinable dye and a surface-localized polymer layer. The polymer layer, introduced in this manner, is able to naturally localize in a thin region near the substrate surfaces and that it can 112

136 significantly improve the robustness of the alignment layer against subsequent light exposure, regardless of any degradation of the underlying photoalignment layer. The incorporation of the stabilization layer does not interfere with the high anchoring energies attainable using this alignment technique. 113

137 CHAPTER 7 Optimization of the Surface-Localized Polymer Stabilization Layer for Infused Photoalignment 7.1 Introduction Sonication Vortex & Heat None 10µm Figure 7.1. Monomers distribution along the cell gap direction under different mixing conditions. Bright areas mean the dye is more concentrated. 0.66% RM84, 0.08% dye 79. Chapter 6 demonstrated the use of a surface-localized polymer layer to stabilize the alignment generated by an azo dye photoalignment layer. This provided proof-of-concept of a surface-localized polymer layer which is mixed in with the liquid crystal and allowed to separate to the substrate surfaces. However, while this process was demonstrated, it was not in any way optimized. Previous research has shown that the ability of the polymer layer to surface-localize may depend heavily on the degree of mixing of the pre-polymer with the liquid crystal prior to filling into the cell. Figure 7.1 shows the cross-section of a cell with pre-polymer and liquid crystal in which a dye which selectively associated with the pre-polymer was introduced and images 114

138 Intensity (a.u.) Intensity (a.u.) Intensity (a.u.) across the cell thickness. The intensity variation across a random vertical cross-section of each of these images is also plotted in Figure 7.2. These images show a high symmetric surfacelocalization when vortexing (and heat) are used and a lower, less symmetric distribution when sonication is used. This lack of symmetry is even more pronounced when no mixing is used. None Vortex & Heat Sonication Z (um) Z (um) Figure 7.2. Monomers distribution along the cell gap direction under different mixing conditions. Intensity was measured along a random vertical cross-section of each image shown in Figure Z (um) This chapter explores how preparation conditions can affect the ability of the polymer to localize near the surfaces and how this, then, impacts the resultant photostability of an underlying photoalignment layer. 7.2 Experimental Details Surface-Localization For demonstrating stable infused photoalignment using the surface-localized polymer layer, bulk cells were initially prepared as described in Appendix C. Note that the infused cell process resulted in variation in cell thickness from cell to cell 1.4μm spacers were used. Once the cells were made, they were filled with the pre-polymer/liquid crystal mixture. This mixture was prepared by starting with RM257 and adding to it 10% by weight of the photoinitiator Irgacure 651 (I651). This was then added at various weight concentrations to the liquid crystal 115

139 E7. The mixture was then vortexed for 10 seconds or 6 minutes. Cells were polymerized at either 0V or 10V. Cells were then polymerized by exposure to unpolarized UVLED (λ=365nm) at 3.5mW/cm 2 for 10 minutes. Once prepared, transmission vs. voltage measurements were taken of the cells between both crossed and parallel polarizers (with the alignment axis of the cell oriented at 45 degrees to the polarizers). These transmission vs. voltage curves were then used to generate phase retardation vs. voltage curves for each cell as described in Chapter 6. The only exception is that the interference filter used in the setup in this chapter has a transmitted wavelength of λ=543nm Photostability in Infused Bulk Cells To test the photostability of RM-stabilized photoalignment, the 1.4μm cells prepared as described in the previous section utilizing infused photoalignment layers (only those that were polymerized at 0V) were subsequently exposed to 20mW/cm 2 blue light polarized at 45 degrees to the original photoalignment direction. This blue light was the same blue light used to align the photoalignment layer. Alignment quality was observed at various stages during this exposure. 7.3 Results Phase Retardation vs. Voltage Curves Samples Polymerized at 0V To start, the samples polymerized at 0V are compared with simulated phase retardation vs. voltage curves for a sample of equal thickness filled with pure liquid crystal. Note that the simulation details will be discussed in Section 7.4. Figure 7.3 shows these curves for samples containing 0.5%wt of pre-polymer with cells polymerized at 0V. At this concentration, there is 116

140 Retardation (Multiples of Pi) Retardation (Multiples of Pi) Retardation (Multiples of Pi) Retardation (Multiples of Pi) very little difference in the shape of the phase retardation vs. voltage curve of the actual cell and the simulated cell without a polymer network (i.e. a pure cell). There is no effect on the threshold voltage, though the samples in which the pre-polymer/liquid crystal was mixed using 6 minutes of vortexing show a slight effect as the retardation begins to decay with increasing voltage. In this case, it is possible that the extended period of mixing results in more effective mixing (than when the mixture was vortexed for 10 seconds) and that it is, therefore, less separated from the LC at the time of polymerization. Simulated Experimental Simulated Experimental Voltage (V) Voltage (V) Simulated Experimental Simulated Experimental Voltage (V) Voltage (V) Figure 7.3. Simulated (blue) and experimental (orange) data for samples filled with 0.5%wt RM257 in E7 vortexed for either 10 seconds (top) or 6 minutes (bottom) and separated for 1hr (left) or 1 day (right) for cells polymerized at 0V. 117

141 Retardation (Multiples of Pi) Retardation (Multiples of Pi) Retardation (Multiples of Pi) Retardation (Multiples of Pi) Figure 7.4, then, shows curves for 1%wt pre-polymer. Note that some of the experimental curves are shown only from 0V to 4V due to a problem with the measurement system at the time this data was taken. With only 10 seconds of vortexing, the experimental curves follow very closely to the simulated (pure) curves. With 6 minutes of vortexing, though, some differences between these curves are observable. Interestingly, this difference is greater with 1 day of wait time between filling and polymerization. Simulated Experimental Simulated Experimental Voltage (V) Voltage (V) Simulated Experimental Simulated Experimental Voltage (V) Voltage (V) Figure 7.4. Simulated (blue) and experimental (orange) data for samples filled with 1%wt RM257 in E7 vortexed for either 10 seconds (top) or 6 minutes (bottom) and separated for 1hr (left) or 1 day (right) for cells polymerized at 0V. Figure 7.5 shows the comparison of these phase retardation vs. voltage curves for samples with 2%wt pre-polymer. Again, some of these experimental curves are shown only 118

142 Retardation (Multiples of Pi) Retardation (Multiples of Pi) Retardation (Multiples of Pi) Retardation (Multiples of Pi) from 0V to 4V due to a problem with the measurement system at the time this data was taken. Here, the differences between the experimental and simulated (pure) curves has become quite distinct. The threshold voltages of the experimental curves are all higher than for the simulated (pure) data; the threshold is highest in the case of the sample in which the pre-polymer/liquid crystal Voltage (V) mixture was vortexed for 6 minutes and the sample was polymerized 1 day after filling. These curves also show a more gradual decay in retardation with increasing voltage than their simulated (pure) counterparts. In the two samples with data all the way out to 10V, the retardation at this point is still quite a bit higher as well. Simulated Experimental Simulated Experimental Voltage (V) Voltage (V) Simulated Experimental Simulated Experimental Voltage (V) Voltage (V) Figure 7.5. Simulated (blue) and experimental (orange) data for samples filled with 2%wt RM257 in E7 vortexed for either 10 seconds (top) or 6 minutes (bottom) and separated for 1hr for cells polymerized at 0V. 119

143 Normalized Retardation Normalized Retardation Normalized Retardation Samples Polymerized at 10V Next, the shape of the phase retardation vs. voltage curves for cells polymerized at 10V was compared, as shown in Figure 7.6. In order to make comparison among the various curves straightforward and account for the thickness variations among the cells, the curves have all been normalized by dividing the retardation measured at each voltage by the net retardation possible in the cell when the liquid crystal is in a planar orientation (Γnet=2Δnd/λ where λ=543nm and Δn=0.22). The value is determined by using the thickness of the cell measured after filling with the liquid crystal in its isotropic state. So, if the liquid crystals are oriented in their planar configuration, the normalized retardation would be 1. For a normalized retardation less than this, the liquid crystals are tilted up. At zero volts, this represents a pretilt in the cell. The lower the normalized retardation, the higher the tilt of the liquid crystals. Vort. 10s Wait 1 hr Vort. 10s Wait 1 d Vort. 6 m Wait 1 hr Vort. 6 m Wait 1 d Vort. 10s Wait 1 hr Vort. 10s Wait 1 d Vort. 6 m Wait 1 hr Vort. 6 m Wait 1 d Voltage (V) Voltage (V) Vort. 10s Wait 1 hr Vort. 10s Wait 1 d Vort. 6 m Wait 1 hr Vort. 6 m Wait 1 d Voltage (V) Figure 7.6. Normalized retardation vs. voltage for all samples polymerized at 10V. 120

144 As shown in the figure, the polymer network has a larger effect with increasing prepolymer concentration. There is a slight difference when pre-polymer concentration is increased from 0.5%wt to 1%wt and a very large effect when pre-polymer concentration is increased 2%wt. For each concentration, the sample in which the pre-polymer/liquid crystal mixture was vortexed for 6 minutes and polymerized 1 day after filling showed the least effect, as seen by the highest normalized retardation at 0 volts (the y-intercept). Samples in which the prepolymer/liquid crystal mixture was vortexed for 10 seconds show the largest effect (particularly the curve in which the sample was polymerized 1 hour after filling) Considering Polymer Layer Thickness One way to consider the polymer layers in these cells is to try to estimate their thickness. In this chapter, the polymer layer thickness has been estimated in two different ways. The first of these was to determine what the thickness of the polymer layer would be if it were to fully separate out of the liquid crystal. The second of these was to determine the minimum thickness necessary for the polymer layer in order to generate the effective pretilt as shown in Figure Thickness of Fully Separated Polymer For this calculation, it is assumed that the polymer is fully separated out of the liquid crystal. The weight concentration of the RM can be written as where the weight of each material is given by wt% RM = m RM m Tot, (7.1) m = ρ V, (7.2) 121

145 If this is reduced to a one-dimensional problem, the volume can be replaced with thickness. Since the reactive mesogen will aggregate on both substrate surfaces, each RM layer will be half as thick as the total RM across the thickness of the cell. So, the weight concentration can be rewritten as wt% RM = 2 d RM ρ RM d LC ρ LC. (7.3) For this equation, the density of the RM is given as ρrm=1.219 g/cm 3 and the density of E7 is given by ρlc=1.03 g/cm 3. This can then be solved for drm for any given pre-polymer concentration and cell thickness Thickness of Polymer to Generate Effective Pretilt Figure 7.7. Effective pretilt angle of 1.5μm cells filled with 0.5%, 1%, or 2%wt prepolymer in liquid crystal vortexed for 10 seconds or 6 minutes and polymerized at 10V 1 hour or 1 day after filling. 122

146 For samples polymerized at 10V, the retardation at 0 volts is less than would be expected for a sample filled with pure LC. This is due to the fact that the LC near the substrate surfaces is being locked into a tilted configuration by the polymer network. This can be approximated by an effective pretilt, which can be defined as the pretilt necessary for a LC cell with equal thickness and no polymer network to produce the same retardation at zero volts as the cell polymerized at 10V. These results for the experimental samples are summarized in Figure 7.7. Figure 7.8. Director angle in the first 20nm near the substrate surface in a simulated cell (no polymer) with 10V applied. Red shows the area near the surface through which the polymer network must exist (to lock in the 10.2 degree pretilt in this cell). The values of the effective pretilt in each cell, then, can be used to determine a minimum thickness required for the polymer network. To calculate this thickness, the director configuration through the cell at the time of polymerization (with 10V applied) is considered (with infinite anchoring energy). The director angle near the surface of the sample is as shown in 123

147 Figure 7.8. As an example, the effective pretilt of 10.2 degrees is considered. The figure shows the region of the cell from the surface to the point where the director angle equals 10.2 degrees shaded in red. In order for the effective pretilt of the cell to be 10.2 degrees, the polymer network must extend at least this far into the cell. For this particular example, that would be 8nm. For a weaker polymer network, it may extend significantly further. However, the polymer network cannot be any thinner than 8nm Results Table 7.1. Comparing theoretical RM thickness with distance from the surface required to Prepolymer wt % lock-in the measured pretilt in the cell. Sample d Fully Separated Vortex Time Time to Polymerization RM Thickness (nm) RM Thickness Based on Pretilt (nm) "Density" Average density sec 1 day sec 1 hr min 1 day min 1 hr sec 1 day sec 1 hr min 1 day min 1 hr sec 1 day sec 1 hr min 1 day min 1 hr Table 7.1 shows the comparison between the distance from the substrate where the liquid crystal director is frozen in (from Section ) and the thickness of the polymer layer if it were to be completely separated out of the LC (from Section ). Also included is the ratio 124

148 of the separated value to the calculated value. This ratio is intended to be representative of a density of the polymer network with a larger value indicating a higher density of the polymer near the surface. Note that the actual thicknesses are somewhat variable, which leads to some variation in the calculated polymer layer thickness for full separation among cells with the same pre-polymer concentration. As expected, the fully separated polymer layer thickness increases with increasing prepolymer concentration, with the exception of one cell with 1% pre-polymer due to the fact that it is significantly thicker than the target thickness. The density of the pre-polymer network for each pre-polymer concentration is lower when a 10 second vortexing time is used and higher when a 6 minute vortexing time is used. The lowest density at each concentration is for 10 seconds of vortexing and 1 hour separation. However, the variations in each polymer concentration make it somewhat difficult to observe any clear trends using these approximations. 7.4 Mathematical Model Development of Simulation In Chapter 6, previously generated simulated dielectric constant vs. voltage plots 80 were used to compare to measured phase retardation vs. voltage. For the purposes of the study presented here, a new model was developed based on the same principles as previously described, which directly simulated phase retardation vs. voltage for polymer-stabilized LC cells under various conditions. Due to some modifications from the previous method, some details are repeated in this chapter. In this model, the polymer network concentration gradient is considered to vary through the thickness of the cell, with the highest concentration near the substrates and the lowest 125

149 concentration in the center of the cell. In the previous model described in Chapter 6, the polymer concentration gradient was modeled by a two-sided exponential decay (Equation 6.2), but this causes the area under the curve across the entire cell to change significantly with decay length. Instead, the equation was modified to resemble a Laplace probability distribution, given by (e z W(z) = W ξ+e (d z) ξ avg 2ξ ), (7.4) where z is the position across the thickness of the cell, d is the cell gap, ξ is the decay length of the gradient, and Wavg is the average coupling strength between the liquid crystal director and the polymer director through the cell. Note that this decay equation is just a rough estimate meant to approximate the decay of the polymer concentration through the cell. In actuality, the coupling between a polymer fiber and the liquid crystal at any point in the cell would be a fixed value, which could be considered the anchoring energy of the liquid crystal on the polymer fiber. However, in any plane in the cell, there is a concentration of points in which the polymer fiber exists. The concentration of these points decreases away from the substrate surfaces. So, the fixed anchoring energy between the polymer fiber and the liquid crystal is combined with the gradient in concentration of polymer fiber in any cross-section of the cell into a single term which encompasses both, the coupling strength, W, and the concentration gradient. Wavg, then, is representative of the concentration of the pre-polymer in the liquid crystal (and should be the same for all cells prepared with the same concentration of pre-polymer. Note that the choice of the Laplace probability distribution to represent the concentration gradient results in imperfect normalization over the thickness of the cell (the Laplace probability distribution is normalized over ± infinity). If this is integrated from 0 to the cell gap d, the result is Wavg*(1-e -d/ξ ). For the range of decay lengths considered in this investigation, this extra term changes the value by anywhere from to 10-22, which is 126

150 negligible. However, this term could become significant if this analysis were expanded to other systems. Figure 7.9 shows the exponential decay through a 1.5μm thick cell using different values for the decay length, ξ. For very short decay lengths, the coupling strength falls off rapidly in the first tens of nanometers near the surface with a very high strength close to the surface. However, for longer decay lengths, the polymer network extends significantly into the bulk of the cell with a much weaker strength near the surface. Figure 7.9. Calculated exponential decay of the coupling strength of a surface-localized polymer network in a 1.5μm cell based on Equation 1 for various decay lengths, ξ. 127

151 The free energy of the liquid crystal now includes a term which takes into account this coupling between the liquid crystal director and the polymer network as given by 77, f d = k 11 2 ( n )2 + k 22 2 (n n )2 + k 33 2 (n n )2 W 2 (n n o) (D E ) 2 (7.5) where k11, k22, k33 are the splay, twist, and bend elastic constants of the LC, respectively, D is the electric displacement, E is the electric field, n is the LC director at a particular point, no is the preferred direction of the director (at points along the polymer network), and W is the effective anchoring strength of the LC director in contact with the polymer (W=0 in regions without polymer). The preferred director, no, is determined by the director orientation at the time of polymerization. While this model is an approximation, it provides a quick way to develop an understanding of how these phase retardation vs. voltage plots relate to the degree of surfacelocalization of the pre-polymer. In these simulations, once the polymer network is determined (by determining a director concentration at the polymerization voltage), the liquid crystal free energy as described in Equation 7.5 can be used to determine a director configuration through the cell at various voltages and phase vs. voltage curves can be simulated directly Simulated Results Figure 7.10 shows simulated phase retardation vs. voltage plots with a constant Wavg but varying decay lengths for two different coupling strengths. These plots are compared side-byside for samples cured at 0V and samples cured at 10V. Note that the decay lengths shown in the plots with Wavg=1000 extend higher than those shown with Wavg= The effect of an increased decay length is to decrease the retardation at zero volts (i.e. increase the effective pretilt of the cell) in samples polymerized at 10V. When polymerized at 0V, the increasing 128

152 Normalized Retardation Normalized Retardation Normalized Retardation decay length initially has only a small effect on the shape of the phase vs. retardation curve. However, at longer decay lengths, as the polymer network begins to extend into the bulk, the shape of the curve is significantly changed the initially rapid decrease in retardation through the cell becomes more gradual. Particularly in the case of a stronger coupling strength, the threshold voltage increases and the retardation achieved with 10V applied is much higher. Of course, it may be desirable to generate a polymer network that has little effect on the curve when polymerized at 0V to maintain optimal electro-optic properties. Voltage (V) Voltage (V) Figure Simulated phase curves of samples with either a weaker (top) or stronger (bottom) average coupling strength, Wavg, and varying values of the decay length, ξ, for cells polymerized at either 0V (left) or 10V (right). 129

153 In analyzing results from these simulations, an interesting phenomenon was observed when the polymer network became highly surface-localized. This effect is demonstrated in Figure 7.11 by showing the zero volt retardation of several simulated phase vs. voltage curves for Wavg=6000 and different decay lengths, ξ, for samples polymerized at 10V. Note that, while the retardation at zero volts initially decreases (pretilt increases) as the decay length is decreased (the polymer network becomes more surface-localized), as previously described, the decay length eventually becomes short enough that the retardation at zero volts begins to increase again (pretilt decreases). Figure Zero volt retardation vs. decay length for simulated RM-stabilized cell polymerized at 10V with Wavg=6000. This effect can be understood by considering the director configuration through the cell at the time of polymerization. When the decay length is long, and the polymer network extends well into the bulk, as previously discussed, the strength of the polymer network near the surface is weak. So while polymer network is frozen into this director configuration, it is too weak to fully hold the liquid crystal into this position. When the voltage is turned off, the liquid crystal 130

154 relaxes partially back to its planar orientation. As the decay length decreases and the polymer becomes more surface-localized, the strength of the polymer network near the surface increases and the liquid crystal is less capable of relaxing to its planar state, generating a higher pretilt. However, as the decay length is further decreased and the polymer network becomes highly surface-localized, not enough polymer exists far enough out into the cell while the polymer network holds the liquid crystal strongly in place when the voltage is turned off, it can only do this so far into the thickness of the cell. After this point is reached, the more surface-localized the polymer becomes, the higher the retardation at zero volts (lower the effective pretilt) the cell will have. 7.5 Comparing Simulation and Experiment Now that a mathematical model has been developed to consider the polymer network in the experimental LC cells, the simulation parameters with respect to the polymer network gradient, Wavg and ξ, through the cell can be adjusted in an attempt to match the simulated and experimental curves for a given data set. First, though, we return to the approximated polymer layer thicknesses as described in Section As previously stated, the polymer network gradient through the cell is approximated as an exponential decay, as given by Equation 7.4. This equation can be integrated to give the proportion of the polymer which exists between the substrate surface and a point a particular distance into the cell. With this distance defined as z=a, the polymer concentration gradient can be integrated from 0 to a to give a (e W avg 0 z ξ+e (d z) ξ) 2ξ dz = 1 W 2 avge (a+d) a ξ (e ξ 1) (e a ξ + e d ξ ). (7.6) The percent of the total RM in the cell which exists between the substrate surface and the point, a, is given by 131

155 RM fraction RM fraction RM % (a) = e (a+d) a ξ (e ξ 1) (e a ξ + e d ξ). (7.7) 2 (1 e d ξ) Figure 7.12 shows this equation plotted in across a 1.5μm thick cell for two different values of the decay length, ξ. Note that, because the polymer network is localized near both the top and bottom substrates, the proportion of the polymer existing between the substrate and the center of the cell is 50%; the proportion existing through the entire cell is 100% Integrated from left surface to this point (um) Integrated from left surface to this point (um) Figure Proportion of pre-polymer between the substrate and various points through the cell with ξ=0.025 (left) or ξ=0.25 (right). Equation 7.7, then, can be used to estimate an appropriate starting value for the decay length, ξ, in simulations to match the experimental data. If the value for the minimum polymer layer thickness necessary to generate the effective pretilt of cells polymerized at 0V is used as the point, a, at which, for starters, 49.5% of the polymer network must exist between this point and the surface, then the necessary decay length can be calculated. That is, RM%(a) is set to 0.495, and a is set to the values provided in Table 7.1, then an estimated decay length can be given for each experimental condition; these values are summarized in Table 7.2. These were 132

156 then used as a starting point to develop a match between simulated and experimental data. Also shown in Table 7.2 are the average coupling strength, Wavg, and decay length, ξ, which allowed for a rough match between simulation and experiment. The actual values of the decay length that are used to match the data are quite close to the predicted values calculated based on the effective pretilt of these cells. In most cases, the simulation decay lengths are slightly longer than the estimated decay lengths. Table 7.2. Estimated decay lengths for the various experimental cells based on the effective pretilt calculated for cells polymerized at 10V and average coupling strength, Wavg, and decay length, ξ, used to match simulated and experimental data for the various Pre-Polymer Concentration (wt%) Vortex. Time experimental conditions. Time from Fill to Polymerization ξpretilt Wavg ξ s 1d s 1h m 1d m 1h s 1d s 1h m 1d m 1h s 1d s 1h m 1d m 1h

157 Retardation (multiples of Pi) Retardation (multiples of Pi) Retardation (multiples of Pi) Retardation (multiples of Pi) Simulated Experimental Simulated Experimental Retardation (multiples of Pi) Retardation (multiples of Pi) Voltage (V) Voltage (V) Simulated Simulated Experimental Experimental Voltage (V) Voltage (V) Simulated Simulated Experimental Experimental Voltage (V) Voltage (V) Figure Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) for a selected sample at each pre-polymer concentration. Figure 7.13 shows an example for both 0V and 10V polymerized cells at each prepolymer concentration. Note that a full set of these curves is included in Appendix E. While the matching is not exact here, it is quite close. In particular, the magnitude of the retardation at zero volts in the cases where the cells are polymerized at 10V match closely even though these curves 134

158 Retardation (Multiples of Pi) Retardation (Multiples of Pi) have a difference in shape as they decay with increasing voltage. At 2%wt pre-polymer, there appears to be a somewhat more significant difference between the simulated and experimental curves at 10V polymerization as the experimental curve is somewhat flat whereas the simulated curve decays to a lower retardation at higher voltages. While this was not fully investigated, it is suspected that the high level of scattering in these cells made it difficult to get accurate data (scattering may have raised the measured retardation of these cells by increasing the measured transmission, which should be near zero). For the purposes of this particular study, getting a close match between the retardation at zero volts of the simulated and experimental curves was chosen to represent a good match (this is also the value that was used to calculate the effective pretilt). It is possible, also, that using a different model for the polymer network gradient would provide a closer match in these scenarios. Simulated Experimental Simulated Experimental Voltage (V) Voltage (V) Figure Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 6 minutes and separated for 1 hr prior to polymerization. Experimental data simulated with Wavg=16000 and ξ=

159 Retardation (Multiples of Pi) Retardation (Multiples of Pi) As with the effective pretilts estimated from these cells, the decay lengths used to match the simulated data to the experimental data were similar among cells with 1%wt and 0.5%wt prepolymer, though, for the most part, the 0.5%wt pre-polymer samples had lower decay lengths. However, once the pre-polymer concentration is increased to 2%wt, the decay lengths more than double. This, along with the observation of scattering in these cells, indicates that 2%wt prepolymer cannot fully separate in these cells and extends significantly into the bulk. Also, in the case of this high pre-polymer concentration, getting a close match between the simulated and experimental data is somewhat more challenging. Figure 7.14 shows the matching that was obtained in the case of 2%wt pre-polymer vortexed for 6 minutes and separated for 1 hour. In this case, even though the simulated retardation at zero volts for the cell polymerized at 10V is not quite as low as the experimental value, the case of the sample polymerized at 0V shows a greater effect on the phase retardation vs. voltage curve in the simulated case than in the experimental case. Simulated Experimental Simulated Experimental Voltage (V) Voltage (V) Figure Comparison between simulated and experimental phase vs. retardation curves for samples with 2%wt RM vortexed for 10 seconds and separated for 1 day polymerized at 0V (left) or 10V (right) with Wavg=40000 and ξ=

160 Figure 7.15, then, shows a variation in the simulated data for the sample with 2%wt prepolymer vortexed for 10 seconds and separated for 1 day prior to polymerization using an average coupling strength that is quite a bit higher than the value used in Table 7.2. The match obtained for this sample was shown in Figure At higher average coupling strength, a good match can still be obtained in the case of 0V polymerization, though the decay length used to generate this match is much lower than the estimated decay length given in Table 7.2. For 10V polymerization, the retardation at zero volts is slightly higher for the simulated data than for the experimental data. Increasing the decay length, in this case, did not result in a decrease in the retardation at zero volts for the simulated data. As described in Figure 7.11, this is due to the fact that, with this decay length, the simulated polymer network is too surface localized to be able to generate the pretilt achieved in the experimental cell. As such, this value for the average coupling strength is too high. The estimated decay lengths described in Table 7.2, therefore, become quite useful in restricting the decay lengths which are usable in the simulation. 7.6 Photostability To test the photostability of RM-stabilized photoalignment, the cells prepared as described in Section (only those that were polymerized at 0V) were subsequently exposed to 20mW/cm 2 blue light polarized at 45 degrees to the original photoalignment direction. This blue light was the same blue light used to align the photoalignment layer. For comparison, a set of samples filled with pure E7 were identically prepared and exposed. 137

161 0.5%wt RM Vortexed 10 s 0.5%wt RM Vortexed 6m Before Photoexposure Before Photoexposure Sep. 1 hour Sep. 1 day Sep. 1 hour Sep. 1 day After 20 min. Photoexposure After 20 min. Photoexposure No RM Before Photoexposure After 20 min. Photoexposure Figure Infused photoaligned cells filled with pure E7 (no RM) or 0.5%wt pre-polymer in E7 vortexed for 10 seconds or 6 minutes and polymerized at 0V 1 hour or 1 day after filling. Shown on the light table between crossed polarizers with the photoalignment axis oriented along the original photoalignment axis. Each cell is shown before and after 20 min exposure to 20mW/cm 2 blue (447nm) light polarized at 45 degrees to the original photoalignment axis. In the case of the pure samples as well as all samples filled with 0.5%wt RM (regardless of mixing and wait times), all samples were rewritten within 20 minutes of exposure (Figure 7.16). However, unlike previously tested spun photoalignment layers, which were initially rewritten and completely destroyed, these photoalignment layers stayed rewritten even after days of exposure. Note that the twist of the samples filled with pure E7 is now approximately 45 degrees (written by the 45-degree polarized photoexposure of the stability testing) while the samples 138

162 filled with 0.5%wt RM257 in E7 have been rewritten to approximately at 20 degree twist. While the generation of the twist is likely due to the ability of the liquid crystal to modify the polarization state of the blue light before it strikes the back photoalignment layer, the decrease of this twist angle in the 0.5%wt RM257 samples (compared to the samples with pure E7), provides evidence of the presence of polymerized RM. a b c d Figure Infused photoaligned samples filled with 1%wt RM257 in E7 (a) vortexed for 10 seconds and polymerized 1 hour after filling, (b) vortexed for 10 seconds and polymerized 1 day after filling, (c) vortexed for 6 minutes and polymerized 1 hour after filling, or (d) vortexed for 6 minutes and polymerized 1 day after filling. Shown on the microscope at 5X between crossed polarizers with the original photoalignment axis along one of the polarizers (i.e. the dark state). (Top) Before photostability testing; (bottom) after 5 weeks of exposure to 20mW/cm 2 blue light polarized at 45 degrees to the initial photoalignment axis. Figure 7.17 shows the 1%wt samples both before and after 5 weeks of exposure to this same irradiation condition. These samples have maintained their original alignment, indicating that the alignment direction has been fully stabilized in all of these cells. Again, this is 139

163 regardless of the preparation and degree of surface-localization of the polymer layer. The 2%wt samples also exhibited stability after 5 weeks but are not shown here because of the high degree of scattering due to the polymer network extending into the bulk. 7.7 Conclusion This chapter has shown the development of the use of a surface-localized polymer layer to stabilize infused photoalignment. As developed, both the photoalignment layer and the surface-localized polymer layer can be infused into the cell, allowing this method to be used even in photonic applications where standard cell assembly is not possible. Experimentally collected phase retardation vs. voltage curves were used to investigate the effects of the various tested process parameters including pre-polymer concentration, degree of mixing, and amount of wait time between filling this mixture into the cell and polymerization. These curves were directly compared to simulated curves without a polymer network which showed that, with increasing pre-polymer concentration, the phase retardation vs. voltage curves deviated more and more from those of pure liquid crystal. The large difference between the curves for samples with 2%wt pre-polymer indicate that the polymer network has begun to extend into the bulk in these cases. Next, these phase retardation vs. voltage curves were used to try to quantify some of the differences between the cells. First, rough estimates of how thick the polymer layer might be in these cells were developed. Next, a polymer network gradient through the cell was used to generate simulated data in order to match to the experimental data. The rough estimates of polymer layer thickness were used to inform the simulations and provided a good starting point for the simulation parameters. The decay lengths obtained from the simulation were quite close 140

164 to those estimated based on the effective pretilts achieved in experimental cells polymerized at 10V. The simulated matches obtained vary from experimental curves most with 2%wt prepolymer. This may be due to scattering in these cells but may also indicate that a better polymer network gradient model may be found with additional testing/simulation. Photostability testing revealed that only cells prepared with 0.5%wt pre-polymer were not sufficiently stable against subsequent light exposure. All cells prepared using 1%wt or 2%wt pre-polymer were stable, regardless of the preparation condition. It is possible that this test suggests that concentration is the controlling factor that affects whether or not a cell will be stable against subsequent light exposure. However, it could also be that resultant layer thickness is more important. Estimated layer thickness when the pre-polymer is assumed to separate fully from the liquid crystal show that 0.5%wt pre-polymer would result in less than a 5nm thick polymer layer in all cases (shown in Table 7.1). At these values, the polymer may not be able to form a uniform layer. Additional testing is required to determine whether concentration is polymer layer thickness/density is the controlling factor in terms of resultant photostability. Even from this limited data set, some conclusions regarding selection of the proper conditions may be evident. As previously stated, 2%wt pre-polymer was too high a concentration as it resulted in significant scattering. In the phase retardation vs. voltage curves for these samples, the effect of a bulk polymer network is apparent, especially when compared to phase retardation vs. voltage curves for pure liquid crystal. Additionally, the effective pretilt generated in these cells required a very high decay length compared to cells prepared with a lower concentration of pre-polymer. Therefore, it may be possible that the ideal pre-polymer concentration is the maximum concentration at which no significant evidence of a polymer network in the bulk exists, as evidenced by comparing phase retardation vs. voltage curves to 141

165 those of pure liquid crystal and determining a required decay length based on the effective pretilt of cells polymerized at high voltage. For the cells prepared in this chapter, that concentration would be 1%wt pre-polymer. These cells are stable while exhibiting phase retardation vs. voltage curves for cells polymerized at 0V that are close to those of pure cells and requiring a decay length ξ<0.01 in order to generate the effective pretilt for cells polymerized at 10V. It may be possible to refine these guidelines with additional study. 142

166 CHAPTER 8 Conclusion This dissertation has described work on two key applications of liquid crystals to infrared imaging. First, work was presented on the development of a wide angle polarization intereference Fourier transform spectrometer particularly for use in the near-infrared. In this case, existing designs had a significant limitation in the achievable spectral resolution due to the narrow acceptance angle of such devices. This dissertation provided work to describe the incorporation of a design modification based on the principles of a wide-view Lyot filter. Mathematical modeling was used to show that this modification provided significant gains in the achievable angle of incident for this device, even at high spectral resolutions. However, another consideration in these devices is of the variable elements; these have been neglected in previous modeling of these devices. Polarization rotators are required between the fixed stages, and these elements can have a significant impact on the angular limit and performance of these devices. Additionally, some researchers have proposed the use of variable retarders to replace some of the smaller fixed birefringent stages in order to reduce mechanical tolerancing and allow for faster switching. The effect of these elements on incident angle limits have also not been previously considered. Therefore, the consideration of both of these types of elements has been explored. Examples of TN cells for polarization rotators and ECB cells for 143

167 variable retarders are used to illustrate the analysis of these elements and the trade-offs between speed and performance for each. One analysis that was demonstrated but not fully explored was the field-of-view limitation that may be introduced by the walk-off of the beam. That is, due to the different path that the ordinary and extraordinary modes take through the system, there can be a significant spatial difference in them when they exit the system. This can cause difficulty in measuring the path difference between the two components as it is possible that they would strike different detector elements. Walk-off, then, requires an additional trade-off between the spectral and spatial resolutions of the device. Full modeling and analysis of this effect must be explored to fully understand the field-of-view effects and limitations of the PIFTS design. In the second application, liquid crystals are used to develop a new uncooled thermal imaging device capable of replacing the existing microbolometers. One important aspect in the development of this imager is the generation of a high quality, uniform alignment layer inside the liquid crystal microcavities. As liquid crystal director fluctuations provide the major noise source in this device, higher anchoring energy is necessary to effectively suppress the director fluctuations and limit the noise. This has motivated the work presented in Chapter 6-7 in which an alignment layer was developed that can be infused into the fully assembled microcavity geometry. A surfacelocalized polymer layer was used to stabilize this infused photoalignment against subsequent light exposure. This surface-localized polymer layer is first demonstrated in cells with twisted alignment in which one substrate had a rubbed polyimide alignment layer and the other a spun photoalignment layer. Once this showed proof that this polymer could effectively stabilize alignment, it was further demonstrated in cells with two infused photoaligned substrates in planar 144

168 configuration. In this case, the liquid crystal/pre-polymer mixture was prepared in a number of different conditions and the degree of surface-localization of the polymer layer investigated. There were subtle differences in these curves under different mixing conditions and times between filling and polymerization. 1%wt pre-polymer provided an optimal concentration to limit the extension of the polymer network into the bulk while maximizing the effect of the polymer network on the effective pretilt of the cell in cells polymerized at 10V. Additionally, the resultant photostability of these cells was most greatly affected by the pre-polymer concentration used, with no effect of the preparation conditions of the sample. 145

169 Appendix A Fourier-Transform Simulations A.1 Introduction This Appendix provides some of the details (as well as the codes) necessary to simulate the various Fourier-transform system designs and generate some of the output, as discussed in Chapter 2-4. In general, the system size, step size, and input wavelength(s) are definable at the beginning of all FT simulations. All input signals are declared in terms of wavelength and converted to wavenumber (in cm -1 ). Note that the codes included in this Appendix are not an exhaustive collection of all simulation codes utilized in this dissertation but rather covers all key factors considered. For instance, a different simulation was utilized for each Hybrid Analog design, but only one of these codes is provided here because the key aspects of the code are identical. The first section in this Appendix A.2 Michelson Interferometer In the case of the Michelson interferometer, the equation necessary to determine the interferogram was described in Chapter 6 and is given by I(x) = A(ν) cos(2πν x), (A.1) where ν is the wavenumber and Δx is the current path difference between the two optical paths (one with a fixed mirror and one with a movable mirror). A is the intensity of the signal and is assumed, for all inputs, to be unity. The on-axis path difference can be calculated for each step by multiplying the step size (0.2μm) by the current simulation loop. To consider off-axis propagation, this path difference is simply divided by the cosine of the angle of the off-axis propagation with respect to the normal. This gives the Δx utilized in Equation A.1. Once the full interferogram is generated by looping over all input wavelengths and steps through the optical system, the Fast Fourier Transform algorithm built-into MatLab is used to transform the interferogram into an output spectrum. Because this information is, essentially, unitless, scaling factors are applied so that this output spectrum can be accurately plotted against wavenumber. A.3 Digital Crystal Polarization-Interference FTS The basic principle of this model is the same as that of the Michelson interferometer. That is, an interference pattern is generated and then Fourier-transformed and scaled to provide output power vs. wavenumber. The simulation is broken into a number of steps. Note that the 146

170 code for a dual crystal system is described and included; the code for a single crystal system is identical with the exception that only a single crystal is used for each stage. A.3.1 Determining Crystal Sequence In this case, the first step is to determine the crystal sequence (that is, which polarization rotators are rotating and which are not). This is done by starting at the maximum retardation and working backward from there step by step. At the maximum retardation, all of the crystal stages are on so none of the polarization rotators are in their rotating state. Next, the smallest crystal rotator is switched from off to on. In fact, any time the smallest crystal rotator is off, it will be switched on for the next (backward) step. If it was on, then it must be switched off and the polarization rotator for the next crystal stage must be switched. This continues for each larger crystal stage (if the polarization rotator was off, it switches on; if it was on, it switches off and the next larger one switches on). A.3.2 Generating the Interferogram Once the crystal (polarization rotator) sequence for a given step through the system has been determined, the input signals can be propagated through the system. As described in Chapter 3, this utilizes Extended Jones matrix method. As this method was described in detail in the main text, it will not be repeated here. However, greater detail of the general flow of this method and its incorporation into the FTS models is provided in the flowcharts in the following section. A.4 Simulating LC Cells Simulating the LC cells was done by using the Frank free energy, as described in Chapter 3. The flowcharts in the following section describes how this is utilized to calculate the director configuration at a given applied voltage. In the case of the ECB cells utilized in the hybrid models, which will be described next, the ECB cells must be tuned to a number of specific net retardations. Therefore, a simulation was created which looped the cell over many voltages and calculated the net retardation through the cell at each. If the net retardation was one of those necessary in the main simulation, then this director configuration was stored. A.5 Hybrid Polarization Interference FTS These models essentially pull together the PIFTS simulation and the simulation of ECB cells. In this case, the smaller stages are replaced with the ECB cells. The sequence at each step through the scan, originally described in A.3.1 must be adjusted to take these ECB cells and their calculated director configurations (described in A.5) into consideration. The essential logic of this is presented in the flowchart in the following section. 147

171 A.6 Simulation Flowcharts This section provides a diagrammatic representation of the various models using to simulate the FTS systems described in this dissertation. 148

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177 A.7 Simulation Code A.7.1 Michelson Interferometer 1 % % 2 % Michelson FTS % 3 % By: Valerie Finnemeyer % 4 % Updated: 02/16/2013 % 5 % % 6 % Calculates the detector interference pattern and output spectrum of the % 7 % Michelson interferometer FTS. Variable system size, wave signals, and % 8 % polar angle. % 9 % % 10 close all 11 clear all lamd = 0.8; % design wavelength (um) 14 DHStage=3; % System size based on DeHoog analysis 15 Deg = 0:89; % Polar angle array (deg) 16 Pmax=2^DHStage*lamd; % maximum retardation (um) 17 Pstep = lam/4; % retardation step size (um) 18 Npath = ceil(pmax/pstep)+1; % # of path steps 19 wm=num2str(lam*1e3); 20 lam = 0.8; % input signals (um) 21 N=1/(lam.*10^-4); % convert to wavenumber (cm^-1) 22 Numwaves = length(n); % # of signals % % 25 % Main Program % 26 % % 27 Combos = length(deg); 28 Sall=zeros(Npath+2,Combos); 29 PowerFile=['Michelson_P_',wM,'nm_DH',num2str(DHStage),'.xls']; 30 IntFile=['Michelson_I_',wM,'nm_DH',num2str(DHStage),'.xls']; 31 f=1; 32 nyquist = 5; for a=1:length(deg) % loop over all polar angles 35 q=deg(a) 36 Thi=(q )*pi/180; 37 cost=cos(thi); 38 S = zeros(npath,1); % Reset Signal 39 for i=1:npath % loop over all path steps 40 x=0.2*(i-1)*10^-4-pmax; % Ideal path difference (cm) 41 del=x/cost; % Actual path difference (cm) 42 for j=1:numwaves % loop over all wavelengths 43 S(i)=S(i)+A(j)*cos(2*pi*N(j)*del); % detected signal 44 end 45 end Sall(2,f)=q; 48 Sall(3:Npath+2,f)=S(1:Npath); 49 Y=fft(S); 50 n=length(y); 51 power = abs(y(1:floor(n/2))).^2; 52 if a==1 53 m=length(power); 54 Pall=zeros(m+2,Combos); 55 freq = (1:n/2)/(n/2)*nyquist; 56 end 57 for g=1:m 58 Pall(g+2,f)=power(g); 59 end 60 Pall(2,f)=q; 154

178 61 f=f+1; 62 end 63 xlswrite(powerfile,pall) 64 xlswrite(intfile,sall) A.7.2 Digital Crystal PIFTS 1 % % 2 % Digital Bi-Crystal FTS with Real (TN) Polarization Rotators % 3 % By: Valerie Finnemeyer % 4 % Updated: 02/16/2013 % 5 % % 6 % Calculates the detector interference pattern and output spectrum of the % 7 % digital bi-crystal FTS with TN polarization rotators. Variable system % 8 % size, wave signals, and angles. % 9 % % 10 function varargout = FTIR_RealBC_TN(varargin) clear all 13 close all % % 16 % Define Variables % 17 % % DHStage=10; % Stages based on DeHoog model 20 DegMax=10; % Maximum polar angle (deg) 21 DegStep=DegMax/100; % Polar angle step size (deg) lamd=0.8; % design wavelength (um) 24 U = 3; % Interference min of TN cells 25 lam =0.8; % input waves (um) 26 lamtn=1.65; % design wavelength of TNs (um) %-----Define Parameters of Polarizers-----% 29 nopreal = 1.5; % no Polarizer-real component 30 nopimag = 1e-20; % no Polarizer-imaginary component 31 nepreal = 1.5; % ne Polarizer-real component 32 nepimag = ; % ne Polarizer-imaginary component 33 dp = 200; % thickness of polarizer (um) %------Define Parameters of TN Cells------% 36 notn = 1.5; % no of TN cell 37 netn = 1.75; % ne of TN cell 38 delntn=netn-notn; % birefringence of TN cell 39 Von = 0; % Voltage for twisting (V) 40 Voff = 20; % Voltage for non-twisting (V) 41 layers = 50; % # of simulated layers uvals=[ ]; 44 u2=uvals(u); 45 dtn=sqrt(u2)*lamtn/(2*delntn); 46 LayerD = dtn/layers; % TN layer thickness (um) rmax=2^dhstage*lamd; % maximum retardation (um) 49 rstep = lamd/4; % retardation step size (um) 50 StepNum = ceil(rmax/rstep); % # of steps (excluding 0) 51 Npath = StepNum+1; % # of retardation steps 52 Nwaves = length(lam); % number of waves in input signal 53 Cryst=DHStage+4; % Actual number of stages 54 J = zeros(2,2); % Jones matrix 55 S = zeros(npath*2-1,1); % Signal array 56 P = zeros(cryst,1); % Ideal retardation of stages (um)

179 58 nop = nopreal+nopimag*1i; % no Polarizer 59 nep = nepreal+nepimag*1i; % ne Polarizer 60 npave = (2*noPreal+nePreal)/3; % average index of polarizer 61 ThP=( )*pi/180; % Polarizer-polar angle of axis (deg) 62 PhiP=45*pi/180; % Polarizer-azimuthal angle axis (deg) %-Define Parameters of Birefringent Crystals-% 65 nec1=1.4797; % ne (Calcite) 66 noc1=1.6423; % no (Calcite) 67 delnc1 = abs(nec1-noc1); % Birefringence (Calcite) 68 PhiC1 = ( )*pi/180; % Polar angle (deg) 69 ThC1 = ( )*pi/180; % Azimuthal angle (deg) nec2=1.5428; % ne (Quartz) 72 noc2=1.5341; % no (Quartz) 73 delnc2 = abs(nec2-noc2); % Birefringence (Quartz) 74 PhiC2 = ( )*pi/180; % Polar angle (deg) 75 ThC2 = ( )*pi/180; % Azimuthal angle (deg) %---Define path difference of crystals----% 78 for j=3:cryst 79 P(j)=rstep*2^(j-3); 80 end 81 P(1)=rstep/2; 82 P(2)=rstep/2; % Configuration of TN cell in on and off positions 85 [OnB,OnPhi]=TN_config(dTN,Von,layers,LayerD); % Twisting 86 [OffB,OffPhi]=TN_config(dTN,Voff,layers,LayerD); % Non-twisting % % 89 % Begin Main Program % 90 % % 91 PDeg=0; % Azimuthal angle array (deg) 92 Deg=0:DegStep:DegMax; % Polar angle array (deg) 93 Combos=length(PDeg)*length(Deg); 94 Sall=zeros(Npath+2,Combos); 95 wm=num2str(lam*1e3); 96 PowerFile=['RealBC_Power_',wM,'nm_DH',num2str(DHStage),'.xls']; 97 IntFile=['RealBC_Int_',wM,'nm_DH',num2str(DHStage),'.xls']; f=1; 100 for b=1:length(pdeg) % loop over azimuthal angles 101 t=pdeg(b) 102 PhiI = (t )*pi/180; 103 for a=1:length(deg) % loop over polar angles 104 q=deg(a) 105 Thi = (q )*pi/180; 106 Lold = zeros(cryst,1); 107 L = zeros(cryst,1); 108 Thk=asin(sin(Thi)/nPave); 109 costhi=cos(thi); 110 costhk=cos(thk); 111 A=1; 112 B=1; 113 C=1; 114 D=1; 115 Jent=[A 0; 0 B]; 116 Jexit=[C 0; 0 D]; 117 PathCalc=zeros(Npath,2); 118 S=zeros(Npath*2-1,1); for r = 1:Npath % loop over retardation steps 156

180 121 % determine crystal sequence for current step (except first step) 122 if r>1 123 Lold=L; 124 if r==maxstep 125 for j=1:cryst 126 L(j)=1; % all crystals are on 127 end 128 elseif Lold(2)==0 129 L(2)=1; % if Cr2 was off, turn it on 130 else 131 CR=0; 132 j=3; 133 while CR==0 134 if Lold(j)==0 135 CR=j; 136 end 137 j=j+1; 138 end 139 L(CR)=1; 140 for j=2:cr L(j)=0; 142 end 143 end 144 end for w = 1:Nwaves % loop over all wavelengths 147 % for the positive half of the measurement 148 wave=lam(w); 149 k0=2*pi/wave; 150 kx=k0*sin(thk); % Exit polarizer 153 Jtemp=ExJones(dP,k0,kx,neP,noP,PhiP,PhiI,ThP); 154 J=Jexit*Jtemp; 155 for c=cryst:-1:1 % for each birefringent crystal 156 d=p(c)/delnc1/2; 157 Jtemp=ExJones(d,k0,kx,neC1,noC1,PhiC1,PhiI,ThC1); 158 J=J*Jtemp; 159 d=p(c)/delnc2/2; 160 Jtemp=ExJones(d,k0,kx,neC2,noC2,PhiC2,PhiI,ThC2); 161 J=J*Jtemp; 162 end 163 if c==1 164 before=1; 165 else before=l(c-1); % state of previous crystal 166 end 167 after=l(c); % state of current crystal 168 if before==after 169 %No twist 170 Jtemp=Jones_TN(OffB,OffPhi,layers,LayerD,noTN,neTN,k0,kx,PhiI); 171 J=J*Jtemp; 172 else 173 % Twist 174 Jtemp=Jones_TN(OnB,OnPhi,layers,LayerD,noTN,neTN,k0,kx,PhiI); 175 J=J*Jtemp; 176 end 177 end % Entrance polarizer 180 Jtemp=ExJones(dP,k0,kx,neP,noP,PhiP,PhiI,ThP); 181 J=J*Jtemp*Jent; E=J*[1; 0]+J*[0; 1]; 157

181 184 T=(E(1)*conj(E(1))+cosThk^2*E(2)*conj(E(2)))/(1+cosThk^2); 185 S(r)=S(r)+abs(T); 186 rspot=(npath-1)*2-(r-1); % record measurement negative path diff 187 S(rspot)=S(rspot)+abs(T); % symmetric scan end % waves 190 end % r Sall(1,f)=t; 194 Sall(2,f)=q; 195 Sall(3:Npath+2,f)=S(1:Npath); 196 Y=fft(S); 197 n=length(y); 198 power = abs(y(1:floor(n/2))).^2; 199 if a==1&&b==1 200 m=length(power); 201 Pall=zeros(m+2,Combos); 202 nyquist = 2.5; 203 freq = (1:n/2)/(n/2)*nyquist; % scaling of output signal 204 end 205 for g=1:m 206 Pall(g+2,f)=power(g); 207 end 208 Pall(1,f)=t; 209 Pall(2,f)=q; 210 f=f+1; end % q 213 end % t xlswrite(powerfile,pall) 216 xlswrite(intfile,sall) A.7.3 TN Cells 1 function TNJones=Jones_TN(Th,Phi,layers,LayerD,noT,neT,k0,kx,PhiI) 2 3 a=layers; 4 ThLC=Th(a); ThLC=pi/2-ThLC; 5 PhiLC=Phi(a); 6 Jones=ExJones(LayerD,k0,kx,neT,noT,PhiLC,PhiI,ThLC); 7 8 for a=layers-1:-1:1 9 ThLC=Th(a); 10 PhiLC=Phi(a); 11 ThLC=pi/2-ThLC; 12 tempj=exjones(layerd,k0,kx,net,not,philc,phii,thlc); 13 Jones=Jones*tempJ; 14 end TNJones=Jones; function [ThConfig,PhiConfig] = TN_config(d,V,layers,LayerD) % TN cell parameters 22 e0 = 8.85; % epsilon-naught 23 K1 = 10; % splay elastic constant 24 K2 = 7; % twist elastic constant 25 K3 = 18; % bend elastic constant 26 epl = 15; % epsilon-parallel 158

182 27 epd = 10; % epsilon-perpendicular 28 dele = epl-epd; % dielectric anisotropy 29 Pretilt = 1*pi/180; % Pretilt (from degrees) % simulation parameters 32 dt_g_i = ; % initial time step/viscosity 33 jmax = 10000; % maximum time steps for relaxation loop 34 stability =.1; % maximum allowed angle change (radians) 35 relaxed = ; % max allowed angle for cell to be relaxed (rad) % other variables 38 z = zeros(layers,1); % position of lattice sites 39 beta_tn = zeros(layers,1);% polar angle of director (from substrate surface) 40 Phi_TN = zeros(layers,1); % azimuthal (twist) angle of director Newbeta = zeros(layers,1); 43 NewPhi = zeros(layers,1); % Establish initial director configuration 46 for a = 1:layers % loop over all layers in cell 47 Pos = LayerD/2+(a-1)*LayerD; 48 z(a) = Pos; 49 A=60*pi/180; 50 B=Pos-LayerD/2; 51 C=(layers-1)*LayerD; 52 D=sin(B*pi/C); 53 E=-A*D; 54 F=E+pi/2; 55 G=F-Pretilt; 56 beta_tn(a) = G; 57 Phi_TN(a) = (pi/2)*(z(a)/d); 58 end % % 61 % Main Program Loop % 62 % % 63 j = 0; 64 dt_g = dt_g_i; while j<jmax 67 InvSum = 0; 68 for k=1:length(z) 69 InvSum = InvSum+(1/(epl*(cos(beta_TN(k)))^2+epd*(sin(beta_TN(k)))^2)); 70 end 71 Dz = V*e0*layers/(d*InvSum); MaxDbeta = 0; % reset maximum polar angle change 74 MaxDPhi = 0; % reset maximum azimuthal angle change % impose boundary conditions 77 Newbeta(1) = beta_tn(1); 78 Newbeta(length(z))=beta_TN(length(z)); 79 NewPhi(1)=0; 80 NewPhi(length(z))=90*pi/180; for a=2:(length(z)-1) 83 % set first and second derivative for beta and phi at all sites 84 B=beta_TN(a); 85 Bp1=beta_TN(a+1); 86 Bm1=beta_TN(a-1); 87 db=(1/2)*(bp1-bm1)/(layerd); 88 d2b=(bp1+bm1-2*b)/(layerd^2); 89 P=Phi_TN(a); 159

183 90 Pp1=Phi_TN(a+1); 91 Pm1=Phi_TN(a-1); 92 dphi=(1/2)*(pp1-pm1)/(layerd); 93 d2phi=(pp1+pm1-2*p)/(layerd^2); % Euler-Lagrange equations 96 SinB=sin(B); 97 CosB=cos(B); 98 ELB=-(K1*SinB^2+K3*CosB^2)*d2B-(K1-K3)*SinB*CosB*dB^2+(2*K2- K3)*SinB^3*CosB*dPhi^2+K3*SinB*CosB^3*dPhi^2+Dz^2*dele*SinB*CosB/(e0*(epl*CosB^2+e pd*sinb^2)^2); 99 ELPhi=-(d2Phi*(K2*SinB^4+K3*SinB^2*CosB^2)+2*dPhi*dB*((2*K2- K3)*SinB^3*CosB+2*K3*SinB*CosB^3)); % Update angles 102 delb=dt_g*elb; 103 delphi=dt_g*elphi/(sinb^2); 104 Newbeta(a)=beta_TN(a)-delB; 105 OldAngle=Phi_TN(a); 106 NewAngle=OldAngle-delPhi; 107 NewPhi(a)=NewAngle; % Update stability condition 110 if abs(delb)>maxdbeta 111 MaxDbeta=abs(delB); 112 end if abs(delphi)>maxdphi 115 MaxDPhi=abs(delPhi); 116 end 117 end beta_tn=newbeta; 120 Phi_TN=NewPhi; % Increase viscosity if unstable; determine max angle change 123 if MaxDbeta>MaxDPhi % whichever is bigger, set to max change 124 MaxD=MaxDbeta; 125 else 126 MaxD=MaxDPhi; 127 end 128 if MaxD>stability % If unstable, increase viscosity by factor of dt_g=dt_g/2; 130 end 131 dt_g = dt_g*1.001; % incremenetally decrease viscosity each step % exit loop if system is fully relaxed 134 if j> if MaxD<relaxed 136 j=9999; 137 end 138 end 139 j=j+1; 140 end ThConfig=beta_TN; 143 PhiConfig=Phi_TN; 160

184 A.7.4 ECB Cells 1 % % 2 % Director Configuration % 3 % % 4 % This program calculates the director configuration of % 5 % a cell given defined surface anchoring on both % 6 % substrates and a given applied voltage. % 7 % % 8 9 close all 10 clear all 11 degtorad = pi/180; 12 radtodeg = 180/pi; 13 jmax = 10000; % maximum time steps for relaxation 14 stability = 0.1; % maximum allowed angle change between steps (rad) 15 relaxed = ; % maximum allowed angle for cell to be relaxed (rad) % % 18 % Initial Conditions % 19 % % % User-Defined Parameters % 22 e0 = 8.85; % epsilon-naught % Set Initial Cell Type (set desired type to 1, other to 0) 25 ECB = 1; % for electrically controlled birefringence cell condition 26 TN = 0; % for twisted-nematic cell condition % Universal cell parameters 29 K1 = 11; % splay elastic constant (E7) 30 K2 = 10.2; % twist elastic constant (E7) 31 K3 = 16.9; % bend elastic constant (E7) 32 epl = 19.6; % epsilon parallel (E7) 33 epd = 5.1; % epsilon perpendicular (E7) 34 d = 5; % cell thickness (in microns) 35 layers = 1000; % number of layers in simulation cell 36 dt_g_i = ; % initial time step/viscosity V = 50; % Applied voltage (V) 39 Pt = 0; % pretilt % Constants % 42 q = 0; % natural pitch of material 43 dele = epl-epd; % dielectric anisotropy 44 LayerD = d/layers; % layer thickness 45 z = zeros(layers); % position of lattice sites 46 Th = zeros(layers); % angle of director at each lattice site 47 Phi = zeros(layers);% twist angle of director at each lattice site 48 NewTh = zeros(layers); 49 NewPhi = zeros(layers); % Theta angle and positions for all cell conditions 53 for i = 1:layers 54 z(i) = LayerD/2+(i-1)*LayerD; 55 Th(i) = -(60*degtorad)*sin((z(i)-LayerD/2)*pi/((layers-1)*LayerD))+90*degtorad-Pt; 56 if TN==1 57 Phi(i) = (pi/2)*(z(i)/d); 58 end 59 end % % 161

185 63 % Main Loop % 64 % % 65 j = 0; 66 dt_g = dt_g_i; 67 while j < jmax 68 InvSum = 0; 69 for k = 1:length(z) 70 InvSum = InvSum + (1/(epl*(cos(Th(k)))^2+epd*(sin(Th(k)))^2)); 71 end 72 Dz = V*e0*layers/(d*InvSum); 73 MaxDTh = 0; % reset maximum change in theta 74 MaxDPhi = 0; % reset maximum change in phi 75 NewTh(1) = Th(1); 76 NewTh(length(z))=Th(length(z)); 77 % impose twisted boundary condition for TN cell condition 78 if TN==1 79 NewPhi(1) = 0; 80 NewPhi(length(z))=90*degtorad; 81 end 82 for i = 2:(length(z)-1) 83 % set first and second derivatives for Theta and Phi at all 84 % sites 85 dth=(1/2)*(th(i+1)-th(i-1))/(layerd); 86 d2th=(th(i+1)+th(i-1)-2*th(i))/((layerd)^2); 87 dphi=(1/2)*(phi(i+1)-phi(i-1))/(layerd); 88 d2phi=(phi(i+1)+phi(i-1)-2*phi(i))/((layerd)^2); 89 % Euler-Lagrange equations 90 sinth=sin(th(i)); 91 costh=cos(th(i)); 92 ELTh=-(K1*sinTh^2+K3*cosTh^2)*d2Th-(K1-K3)*sinTh*cosTh*dTh^2+(2*K2- K3)*sinTh^3*cosTh*dPhi^2+K3*sinTh*cosTh^3*dPhi^2+Dz^2*dele*sinTh*cosTh/(e0*(epl*c osth^2+epd*sinth^2)^2); 93 ELPhi=-(d2Phi*(K2*sinTh^4+K3*sinTh^2*cosTh^2)+2*dPhi*dTh*((2*K2- K3)*sinTh^3*cosTh+2*K3*sinTh*cosTh^3)); 94 % Update angles 95 delth = dt_g*elth; 96 delphi = dt_g*elphi/(sinth^2); 97 NewTh(i)=Th(i)-delTh; 98 NewPhi(i)=Phi(i)-delPhi; 99 % Update stability conditions 100 if abs(delth)>maxdth 101 MaxDTh=abs(delTh); 102 end 103 if abs(delphi)>maxdphi 104 MaxDPhi=abs(delPhi); 105 end 106 end 107 Th = NewTh; 108 Phi = NewPhi; 109 % Decrease time step/viscosity if system is unstable 110 % Determine maximum change in angle (ignoring phi boundary) 111 if MaxDTh>MaxDPhi 112 MaxD = MaxDTh; 113 else 114 MaxD = MaxDPhi; 115 end 116 if MaxD>stability 117 dt_g=dt_g/2; 118 end 119 dt_g = dt_g*1.001; % increase time step/viscosity 120 ThDeg = Th*radtodeg; 121 PhiDeg = Phi*radtodeg; 122 % Prepare plot 162

186 123 %if mod(j,50)==0 124 % figure(1) 125 % drawnow 126 % plot(z,thdeg) 127 % hold on 128 % plot(z,phideg,'r') 129 % hold off 130 %end % exit loop is system is fully relaxed 133 if j> if MaxD < relaxed 135 j=9999; 136 end 137 end 138 j=j+1; 139 end % j loop 140 figure(2) 141 hold on 142 plot(z,thdeg) 143 plot(z,phideg,'r') 144 hold off 145 title('final Director Orientation vs. Vmax'); 146 xlabel('z axis of cell (microns)'); 147 ylabel('angle (degrees)'); 148 figure(3) 149 hold on 150 plot(z,thdeg) 151 plot(z,phideg,'r') 152 hold off 153 title('final Director Orientation vs. Vmax'); 154 xlabel('z axis of cell (microns)'); 155 xlim([0 0.1]); 156 ylabel('angle (degrees)'); A.7.5 Hybrid PIFTS 1 % % 2 % Hybrid Analog Bi-Crystal FTS with Ideal Polarization Rotators % 3 % By: Valerie Finnemeyer % 4 % Updated: 02/16/2013 % 5 % % 6 % Calculates the detector interference pattern and output spectrum of the % 7 % hybrid analog bi-crystal FTS with ideal polarization rotators, with % 8 % eight crystal stages replaced with four hybrid analog ECB cells. % 9 % Variable system size, wave signals, and angles. % 10 % % 11 function varargout = HA5_Flip_DH(varargin) clear all 14 close all % % 17 % Define Variables % 18 % % 19 DHStage=10; % Stages based on DeHoog model 20 DegMax=10; % Maximum polar angle (deg) 21 DegStep=DegMax/100; % Polar angle step size (deg) 22 lamd=0.8; % design wavelength (um) 23 lam =0.8; % input waves (um) 163

187 %-----Define Parameters of Polarizers-----% 27 nopreal = 1.5; % no Polarizer-real component 28 nopimag = 1e-20; % no Polarizer-imaginary component 29 nepreal = 1.5; % ne Polarizer-real component 30 nepimag = ; % ne Polarizer-imaginary component 31 dp = 200; % thickness of polarizer (um) % Define ECB Parameters % 34 noecb = 1.5; % no of ECB 35 neecb = 1.75; % ne of ECB 36 delnecb=neecb-noecb; % birefringence of ECB 37 layersecb = 50; % # of layers of ECB 38 ECBConfig=xlsread('ECB_Th_Configs_HA5.xls'); % Stored director configuration 39 decb=14; % Thickness of ECB (um) 40 RetComp = 4*0.3; % Residual retardation compensation (um) %-Define Parameters of Birefringent Crystals-% 43 nec1=1.4797; % ne (Calcite) 44 noc1=1.6423; % no (Calcite) 45 delnc1 = abs(nec1-noc1); % Birefringence (Calcite) 46 PhiC1 = ( )*pi/180; % Polar angle (deg) 47 ThC1 = ( )*pi/180; % Azimuthal angle (deg) nec2=1.5428; % ne (Quartz) 50 noc2=1.5341; % no (Quartz) 51 delnc2 = abs(nec2-noc2); % Birefringence (Quartz) 52 PhiC2 = ( )*pi/180; % Polar angle (deg) 53 ThC2 = ( )*pi/180; % Azimuthal angle (deg) rmax=2^dhstage*lamd; % maximum retardation (um) 56 rstep = lamd/4; % retardation step size (um) 57 StepNum = ceil(rmax/rstep); % # of steps (excluding 0) 58 LayerECB=dECB/layersECB; % Layer thickness of ECB (um) 59 Npath = StepNum+1; % # of retardation steps 60 Nwaves = length(lam); % number of waves in input signal 61 Cryst=DHStage+4; % Actual number of stages 62 J = zeros(2,2); % Jones matrix 63 S = zeros(npath*2-1,1); % Signal array 64 P = zeros(cryst,1); % Ideal retardation of stages (um) 65 nop = nopreal+nopimag*1i; % no Polarizer 66 nep = nepreal+nepimag*1i; % ne Polarizer 67 npave = (2*noPreal+nePreal)/3; % average index of polarizer 68 ThP=( )*pi/180; % Polarizer-polar angle of axis (deg) 69 PhiP=45*pi/180; % Polarizer-azimuthal angle axis (deg) P = [ ]; % % 74 % Begin Main Program % 75 % % PDeg=0; % Azimuthal angles (deg) 78 Deg=0; % Polar angles (deg) Combos=length(PDeg)*length(Deg); 81 Sall=zeros(Npath+2,Combos); 82 wm=round(lam(1)*1e3); 83 PowerFile=['HAF5_P_DH',num2str(DHStage),'.xls']; 84 IntFile=['HAF5_I_DH',num2str(DHStage),'.xls']; 85 f=1; 86 for b=1:length(pdeg) % Loop over all azimuthal angles 164

188 87 t=pdeg(b) 88 PhiI = (t )*pi/180; 89 for ab=1:length(deg) % Loop over all polar angles 90 q=deg(ab) 91 Thi = (q )*pi/180; 92 % Calculate entrance and exit matrices for system 93 Thk=asin(sin(Thi)/nPave); 94 costhi=cos(thi); 95 costhk=cos(thk); 96 A=1; 97 B=1; 98 C=1; 99 D=1; 100 Jent=[A 0; 0 B]; 101 Jexit=[C 0; 0 D]; 102 PathCalc=zeros(Npath,2); 103 S=zeros(Npath*2,1); 104 ECBMax=33; % # different ECB configurations 105 ECB=[1 1]; % row of director configurations for LC cells 106 Stage=1; % Stores which set of ECB cells is being controlled 107 Switch=0; 108 Lold = zeros(cryst,1); % Crystal sequence for previous path step 109 L = zeros(cryst,1); % Crystal sequence for current path step 110 L(Cryst)=1; % last crystal is always on 111 for r = 1:Npath % Loop over all path steps 112 if Switch==1 % If crystal sequence needs to be recalculated 113 Lold=L; 114 if r==maxstep 115 for j=1:cryst 116 L(j)=1; % all crystals are on 117 end 118 else 119 CR=0; 120 j=1; 121 while CR==0 122 if Lold(j)==0 123 CR=j; 124 end 125 j=j+1; 126 if j==(cryst+1) % all crystals on 127 CR=100; 128 end 129 end 130 if CR< L(CR)=1; 132 for j=1:cr L(j)=0; 134 end 135 end 136 end 137 Switch=0; 138 end 139 for w = 1:Nwaves % Loop over all wavelengths 140 % for the positive half of the measurement 141 wave=lam(w); 142 k0=2*pi/wave; 143 kx=k0*sin(thk); 144 % Exit polarizer 145 Jtemp=ExJones(dP,k0,kx,neP,noP,PhiP,PhiI,ThP); 146 J=Jexit*Jtemp; 147 for c=cryst:-1:2 % Loop over all crystal stages 148 d=p(c)/delnc1/2; 149 Jtemp=ExJones(d,k0,kx,neC1,noC1,PhiC1,PhiI,ThC1); 165

189 150 J=J*Jtemp; 151 d=p(c)/delnc2/2; 152 Jtemp=ExJones(d,k0,kx,neC2,noC2,PhiC2,PhiI,ThC2); 153 J=J*Jtemp; 154 before=l(c-1); % state of previous crystal 155 after=l(c); % state of current crystal 156 if before==after 157 else 158 % send through TN cell 159 Jtemp=[0-1; 1 0]; 160 J=J*Jtemp; 161 end 162 end 163 % send the ECB stage 164 % First set of ECB cells 165 Volt=(ECBMax+1)-ECB(1); 166 for g=layersecb:-1:1 167 ECBTh=ECBConfig(Volt,g); 168 ECBTh=pi/2+ECBTh; 169 Jtemp=ExJones(LayerECB,k0,kx,neECB,noECB, ,PhiI,ECBTh); 170 J=J*Jtemp; 171 end 172 for g=layersecb:-1:1 173 ECBTh=-ECBConfig(Volt,g); 174 ECBTh=pi/2+ECBTh; 175 Jtemp=ExJones(LayerECB,k0,kx,neECB,noECB, ,PhiI,ECBTh); 176 J=J*Jtemp; 177 end 178 % Second set of ECB cells 179 Volt=(ECBMax+1)-ECB(2); 180 for g=layersecb:-1:1 181 ECBTh=ECBConfig(Volt,g); 182 ECBTh=pi/2+ECBTh; 183 Jtemp=ExJones(LayerECB,k0,kx,neECB,noECB, ,PhiI,ECBTh); 184 J=J*Jtemp; 185 if L(1)==1 186 end 187 for g=layersecb:-1:1 188 ECBTh=-ECBConfig(Volt,g); 189 ECBTh=pi/2+ECBTh; 190 Jtemp=ExJones(LayerECB,k0,kx,neECB,noECB, ,PhiI,ECBTh); 191 J=J*Jtemp; 192 end 193 % Send through residual birefringence compensator 194 d=retcomp/delnc1/2; 195 Jtemp=ExJones(d,k0,kx,neC1,noC1,(pi/2+PhiC1),PhiI,ThC1); 196 J=J*Jtemp; 197 d=retcomp/delnc2/2; 198 Jtemp=ExJones(d,k0,kx,neC2,noC2,(pi/2+PhiC2),PhiI,ThC2); 199 J=J*Jtemp; 200 % Send through associated TN cell 201 if L(1)==1 202 Jtemp=[0-1; 1 0]; 203 J=J*Jtemp; 204 else 205 end 206 % Entrance polarizer 207 Jtemp=ExJones(dP,k0,kx,neP,noP,PhiP,PhiI,ThP); 208 J=J*Jtemp*Jent; 209 E=J*Ein; 210 T=(E(1)*conj(E(1))+E(2)*conj(E(2)))/2; 211 S(r)=S(r)+T; 212 rspot=(npath)*2-(r-1); 166

190 213 S(rspot)=S(rspot)+T; 214 % Adjust ECB Cells 215 if w==nwaves&&r>1 216 if L(1)==0 % contribution is negative 217 Control=ECB(Stage); 218 if Control==ECBMax 219 if Stage==1 220 Stage=2; 221 ECB(Stage)=ECB(Stage)+1; 222 else 223 Stage=1; 224 ECB(Stage)=ECB(Stage)-1; 225 Switch=1; 226 end 227 else 228 ECB(Stage)=ECB(Stage)+1; 229 end else % contribution is positive 232 Control=ECB(Stage); 233 if Control==1 234 if Stage==1 235 Stage=2; 236 ECB(Stage)=ECB(Stage)-1; 237 else 238 Stage=1; 239 ECB(Stage)=ECB(Stage)+1; 240 Switch=1; 241 end 242 else 243 ECB(Stage)=ECB(Stage)-1; 244 end 245 end 246 end 247 if w==nwaves&&r==1 248 ECB(Stage)=ECB(Stage)+1; end 251 if w==nwaves&&ecb(1)==1&&ecb(2)==1 252 Switch=1; 253 Stage=1; 254 end 255 end % loop over waves (w) 256 end % loop over phase steps (r) 257 Sall(1,f)=t; 258 Sall(2,f)=q; 259 Sall(3:Npath+2,f)=S(1:Npath); 260 Y=fft(S); 261 n=length(y); 262 power = abs(y(1:floor(n/2))).^2; 263 if ab==1&&b==1 264 m=length(power); 265 Pall=zeros(m+2,Combos); 266 nyquist = 2.5; 267 freq = (1:n/2)/(n/2)*nyquist; 268 end 269 for g=1:m 270 Pall(g+2,f)=power(g); 271 end 272 Pall(1,f)=t; 273 Pall(2,f)=q; 274 f=f+1; 275 end % q 167

191 276 end % t xlswrite(powerfile,pall) 279 xlswrite(intfile,sall) A.7.6 Extended Jones Method 1 function JMatrix = ExJones(PhiA,Theta,d,ne,no,Phi,k,k0) 2 % % 3 % Extended Jones Matrix method % 4 % for each layer of material % 5 % % 6 7 % Calculate in advance to decrease runtime 8 P=Phi+PhiA; 9 T=pi/2-Theta; 10 CosT=cos(T); 11 SinT=sin(T); 12 CosP=cos(P); 13 SinP=sin(P); 14 dn2=ne^2-no^2; % dielectric tensor 17 e11=no^2+dn2*cost^2*cosp^2; 18 e12=dn2*cost^2*sinp*cosp; 19 e21=e12; 20 e13=dn2*sint*cost*cosp; 21 e31=e13; 22 e22=no^2+dn2*cost^2*sinp^2; 23 e23=dn2*sint*cost*sinp; 24 e32=e23; 25 e33=no^2+dn2*sint^2; % Matrix calculations 28 L1=sqrt(no^2-k^2); 29 L2=-e13*k/e33+sqrt((no*ne)^2/e33-(e11*e33-e13^2)*k^2/e33^2); 30 r1=(k*sint-l1*cost*cosp)/(l1*cost*sinp); 31 r2=(-k*l2*sint+l1^2*cost*cosp)/(no^2*cost*sinp); 32 A=exp(1i*k0*L2*d); 33 B=exp(1i*k*L1*d); 34 JMatrix=(1/(1-1/(r1*r2)))*[-B/(r1*r2)+A (B-A)/r1; (A-B)/r2 -A/(r1*r2)+B]; A.7.7 OPD Calculation for Fixed Stages 1 % % 2 % New OPD Calculation Program % 3 % By: Valerie Finnemeyer % 4 % Last Updated: 03/05/2013 % 5 % % 6 7 lam=0.8; 8 9 % Calcite 10 no1=1.658; 11 ne1=1.486; 12 deln1=abs(ne1-no1); % Quartz 15 no2=1.544; 16 ne2=1.553; 17 deln2=abs(ne2-no2);

192 19 for DH=10:-1:3 20 FOV=90; 21 q=0; 22 while FOV==90 23 Cryst=DH+4; 24 Path=2^DH*lam; 25 Thi=(q+1e-20)*pi/180; 26 d1=path/deln1/2; 27 d2=path/deln2/2; % P-polarized wave 30 % First interface (Air-Calcite) 31 Tha1=asin(sin(Thi)/no1); 32 da1=d1/cos(tha1); 33 WOa1=d1*tan(Tha1); 34 % Second interface (Calcite-Quartz) 35 Tha2=asin(no1*sin(Tha1)/ne2); 36 da2=d2/cos(tha2); 37 WOa2=d2*tan(Tha2); 38 % Totals 39 WOa=WOa1+WOa2; 40 Patha=no1*da1+ne2*da2; % S-polarized wave 43 % First interface (Air-Calcite) 44 for a=1:10 45 neff=sqrt(no1^2*ne1^2/(ne1^2*(sin(thi))^2+no1^2*(cos(thi))^2)); 46 Thb1=asin(sin(Thi)/neff); 47 end 48 db1=d1/cos(thb1); 49 WOb1=d1*tan(Thb1); 50 % Second interface (Calcite-Quartz) 51 Thb2=asin(neff*sin(Thb1)/no2); 52 db2=d2/cos(thb2); 53 WOb2=d2*tan(Thb2); 54 % Totals 55 WOb=WOb1+WOb2; 56 Pathb=neff*db1+no2*db2; % Adding it all up 59 WO=WOb-WOa; 60 if WO>0 61 Extra=WO*sin(Thi); 62 PathTot=Patha+Extra-Pathb; 63 PathDiff=abs(PathTot-Path); 64 else 65 WO=-WO; 66 Extra=WO*sin(Thi); 67 PathTot=Pathb+Extra-Patha; 68 PathDiff=abs(PathTot-Path); 69 end if PathDiff>(lam/4) 72 FOV=q; 73 Field(DH-2)=FOV; 74 q 75 DH 76 end 77 q=q+0.1; 78 end 79 end 80 'the end' 169

193 A.7.8 OPD Calculations for Hybrid Models 1 % % 2 % Define Variables % 3 % % 4 5 HA=12; 6 StageMin=HA; 7 if StageMin<7 8 StageMin=7; 9 end P = [ ]; 12 dlc = [ ]; 13 decb=dlc(ha); 14 lam=0.8; % ECB parameters 17 noecb = 1.5; 18 neecb = 1.75; 19 layersecb = 50; 20 dlayer=decb/layersecb; neq=1.5428; 23 noq=1.5341; 24 nec=1.4797; 25 noc=1.6423; dnq=abs(neq-noq); 28 dnc=abs(nec-noc); % Define ECB Cell Orientation 31 for g=1:layersecb 32 ECBConfig(g)=45*pi/180*sin(pi*(g-1)/layersECB); 33 end for Stages=14:-1:StageMin 37 Stages; 38 if Stages>HA 39 CPath=0; 40 for a=(ha+1):stages % Loop over all crystal stages 41 CPath=CPath+P(a); % Sum total crystal stages 42 end 43 dc=cpath/dnc/2; % Thickness of Calcite (half Crystal retardation) 44 dq=cpath/dnq/2; % Thickness of Quartz (half Crystal retardation) 45 end Thi=0; 48 FOV=90; 49 while FOV==90; 50 Th=Thi*pi/180; % Positive birefringent materials out-of-plane 53 Patha=0; 54 Pathb=0; 55 WOa=0; 56 WOb=0; 57 if Stages>HA 58 % s-pol 59 % Quartz (ordinary) 60 ThaQ=asin(sin(Th)/noQ); 61 daq=dq/cos(thaq); 62 WOaQ=dQ*tan(ThaQ); 170

194 63 % Calcite (extraordinary--in plane) 64 ThaC=ThaQ; 65 for a=1:10 66 neff=sqrt(nec^2*noc^2/(nec^2*(sin(thac))^2+noc^2*(cos(thac))^2)); 67 ThaC=asin(sin(Th)/neff); 68 end 69 dac=dc/cos(thac); 70 WOaC=dC*tan(ThaC); 71 % Totals 72 Patha=daQ*noQ+daC*neff; 73 WOa=WOaQ+WOaC; % p-pol 76 % Quartz (extraordinary--out of plane) 77 ThbQ=asin(sin(Th)/neQ); 78 dbq=dq/cos(thbq); 79 WObQ=dQ*tan(ThbQ); 80 % Calcite (ordinary) 81 ThbC=asin(sin(Th)/noC); 82 dbc=dc/cos(thbc); 83 WObC=dC*tan(ThbC); 84 % Totals 85 Pathb=dbQ*neQ+dbC*noC; 86 WOb=WObQ+WObC; 87 end % Stages > HA % Path through LC cells 90 dlccell=decb; 91 % Cell 1 92 % Ordinary 93 ThoLC=asin(sin(Th)/noECB); 94 dolc=dlccell/cos(tholc); 95 WOoLC=dLCcell*tan(ThoLC); 96 PathoLC=doLC*noECB; 97 % Extraordinary (out-of-plane) 98 TheLC=asin(sin(Th)/neECB); 99 delc=dlccell/cos(thelc); 100 WOeLC=dLCcell*tan(TheLC); 101 PatheLC=deLC*neECB; % Cell % Ordinary 105 WOoLC=WOoLC+dLCcell*tan(ThoLC); 106 PathoLC=PathoLC+doLC*noECB; 107 % Extraordinary (out-of-plane) 108 TheLC=asin(sin(Th)/neECB); 109 delc=dlccell/cos(thelc); 110 WOeLC=WOeLC+dLCcell*tan(TheLC); 111 PatheLC=PatheLC+deLC*neECB; % Cell % Ordinary 115 WOoLC=WOoLC+dLCcell*tan(ThoLC); 116 PathoLC=PathoLC+doLC*noECB; 117 % Extraordinary (out-of-plane) 118 for g=1:layersecb 119 ECBTh=ECBConfig(g); 120 kth=ecbth; 121 neff=sqrt(neecb^2*noecb^2/(noecb^2*(cos(kth))^2+neecb^2*(sin(kth))^2)); 122 TheLC=asin(sin(Th)/neff); 123 delc=dlayer/cos(thelc); 124 WOeLC=WOeLC+dlayer*tan(TheLC); 125 PatheLC=PatheLC+deLC*neff; 171

195 126 end % Cell % Ordinary 131 WOoLC=WOoLC+dLCcell*tan(ThoLC); 132 PathoLC=PathoLC+doLC*noECB; 133 % Extraordinary (out-of-plane) 134 for g=1:layersecb 135 ECBTh=ECBConfig(g); 136 kth=ecbth; 137 neff=sqrt(neecb^2*noecb^2/(noecb^2*(cos(kth))^2+neecb^2*(sin(kth))^2)); 138 TheLC=asin(sin(Th)/neff); 139 delc=dlayer/cos(thelc); 140 WOeLC=WOeLC+dlayer*tan(TheLC); 141 PatheLC=PatheLC+deLC*neff; 142 end % Add to Crystals 145 PathA=Patha+PathoLC; 146 PathB=Pathb+PatheLC; 147 WOA=WOa+WOoLC; 148 WOB=WOb+WOeLC; % Bringing it all together 151 WO=abs(WOA-WOB); 152 Extra=abs(WO*sin(Th)); 153 if WOA>WOB 154 OPD=abs(PathB+Extra-PathA); 155 else 156 OPD=abs(PathA+Extra-PathB); 157 end OPD; 160 if Thi==0 161 PathOoP=OPD; 162 else 163 PathDiffOoP=abs(OPD-PathOoP); 164 end % Positive birefringence in-plane 167 Patha=0; 168 Pathb=0; 169 WOa=0; 170 WOb=0; 171 if Stages>HA 172 % s-pol 173 % Quartz (ordinary) 174 ThaQ=asin(sin(Th)/noQ); 175 daq=dq/cos(thaq); 176 WOaQ=dQ*tan(ThaQ); 177 % Calcite (extraordinary--out of plane) 178 ThaC=asin(sin(Th)/neC); 179 dac=dc/cos(thac); 180 WOaC=dC*tan(ThaC); 181 % Totals 182 Patha=daQ*noQ+daC*neC; 183 WOa=WOaQ+WOaC; % p-pol 186 % Quartz (extraordinary--in plane) 187 ThbQ=ThaQ; % initial guess 188 for a=1:10 172

196 189 neff=sqrt(neq^2*noq^2/(neq^2*(sin(thbq))^2+noq^2*(cos(thbq))^2)); 190 ThbQ=asin(sin(Th)/neff); 191 end 192 neff; 193 dbq=dq/cos(thbq); 194 WObQ=dQ*tan(ThbQ); 195 % Calcite (ordinary) 196 ThbC=asin(sin(Th)/noC); 197 dbc=dc/cos(thbc); 198 WObC=dC*tan(ThbC); 199 % Totals 200 Pathb=dbQ*neff+dbC*noC; 201 WOb=WObQ+WObC; 202 end % Crystal stages % Path through LC cells 205 dlccell=decb; 206 % Cell 1 (planar--consider as 1 layer) 207 % Ordinary 208 ThoLC=asin(sin(Th)/noECB); 209 dolc=dlccell/cos(tholc); 210 WOoLC=dLCcell*tan(ThoLC); 211 PathoLC=doLC*noECB; 212 % Extraordinary (in-plane) 213 TheLC=ThoLC; 214 for a=1: neff=sqrt(neecb^2*noecb^2/(neecb^2*(sin(thelc))^2+noecb^2*(cos(tholc))^2)); 216 TheLC=asin(sin(Th)/neff); 217 end 218 delc=dlccell/cos(thelc); 219 WOeLC=dLCcell*tan(TheLC); 220 PatheLC=deLC*neff; % Cell 2 (planar--consider as 1 layer) 223 % Ordinary 224 WOoLC=WOoLC+dLCcell*tan(ThoLC); 225 PathoLC=PathoLC+doLC*noECB; 226 % Extraordinary (in-plane) 227 TheLC=ThoLC; 228 for a=1: neff=sqrt(neecb^2*noecb^2/(neecb^2*(sin(thelc))^2+noecb^2*(cos(tholc))^2)); 230 TheLC=asin(sin(Th)/neff); 231 end 232 delc=dlccell/cos(thelc); 233 WOeLC=WOeLC+dLCcell*tan(TheLC); 234 PatheLC=PatheLC+deLC*neff; % Cell 3 (planar for now) 237 % Ordinary (consider as 1 layer) 238 WOoLC=WOoLC+dLCcell*tan(ThoLC); 239 PathoLC=PathoLC+doLC*noECB; 240 % Extraordinary (in-plane--layers) 241 for g=1:layersecb 242 ECBTh=ECBConfig(g); 243 %ECBTh=0; 244 TheLC=ThoLC; 245 for a=1: kth=ecbth+thelc; 247 neff=sqrt(neecb^2*noecb^2/(noecb^2*(cos(kth))^2+neecb^2*(sin(kth))^2)); 248 TheLC=asin(sin(Th)/neff); 249 end 250 delc=dlayer/cos(thelc); 251 WOeLC=WOeLC+dlayer*tan(TheLC); 173

197 252 PatheLC=PatheLC+deLC*neff; 253 end % Cell 4 (planar for now) 256 % Ordinary (consider as 1 layer) 257 WOoLC=WOoLC+dLCcell*tan(ThoLC); 258 PathoLC=PathoLC+doLC*noECB; 259 % Extraordinary (in-plane--layers) 260 for g=1:layersecb 261 ECBTh=ECBConfig(g); 262 %ECBTh=0; 263 TheLC=ThoLC; 264 for a=1: kth=ecbth-thelc; 266 neff=sqrt(neecb^2*noecb^2/(noecb^2*(cos(kth))^2+neecb^2*(sin(kth))^2)); 267 TheLC=asin(sin(Th)/neff); 268 end 269 delc=dlayer/cos(thelc); 270 WOeLC=WOeLC+dlayer*tan(TheLC); 271 PatheLC=PatheLC+deLC*neff; 272 end % Add to Crystals 275 PathA=Patha+PathoLC; 276 PathB=Pathb+PatheLC; 277 WOA=WOa+WOoLC; 278 WOB=WOb+WOeLC; % Bringing it all together 281 WO=abs(WOA-WOB); 282 Extra=abs(WO*sin(Th)); 283 if WOA>WOB 284 OPD=abs(PathB+Extra-PathA); 285 else 286 OPD=abs(PathA+Extra-PathB); 287 end 288 OPD; 289 if Thi==0 290 PathIP=OPD; 291 else 292 PathDiffIP=abs(OPD-PathIP); 293 PathDiff=max([PathDiffIP PathDiffOoP]); if PathDiff> FOV=Th; 297 Thi 298 end 299 end % Only do if not normal incidence 300 Thi=Thi+0.005; 301 end % FOV Calc 302 end % Loop over system sizes 174

198 Appendix B Modeling Director Fluctuations on the Poincare Sphere B.1 Introduction to the Poincare Sphere The Poincare sphere is a useful tool for visualizing the change in polarization state as it travels through an optical system. This method of visualization has been chosen because it allows the development of an intuitive sense of how the liquid crystal director fluctuations will affect the output polarization. Each point on the surface of the Poincare sphere represents a different polarization state of light; all possible polarization states exist on this sphere. The equator represents all possible linear polarization states while the poles represent the two opposite-handed circular polarization states. The remainder of the sphere accounts for all elliptical polarization states. This is illustrated in Figure B.1. Figure B.1. Poincare Sphere The Stokes parameters S1, S2, and S3 produce the x, y, and z coordinates for each of these polarization states on the surface of the sphere. For example, S = (1, 0, 0) represents horizontal polarization while S = (-1, 0, 0) represents vertical polarization. Alternatively, S = (0, 0, 1) represents right-circular polarization while S = (0, 0, -1) represents left-circular polarization. Finally, S = (0, 1, 0) represents linearly polarized light at 45-degrees. 175

199 Next, elliptical light is defined by the angle of the S vector away from the +S1 axis. The azimuthal angle, 2ω, represents the ellipticity of the light, given by tanω = b a, (B.1) where b and a are the long and short axes of the ellipse, respectively. The equatorial angle, 2α, represents the tilt angle of the ellipse (simply α). This is all represented diagrammatically in Figure B.2. Figure B.2. Azimuthal and equatorial angles on the Poincare sphere. Finally, the effect optical elements have on the S vector of the light can be considered. In the same way that the polarization state of the light is defined by S, the principle axis of an optical element can be represented as a vector in the Poincare sphere (emanating from the center of the sphere to the surface). Keeping the angle between this retarder axis vector and S constant, the S vector is rotated about the retarder axis by an angle equal to the phase retardation of the retarder. The new S vector is the polarization state output from the retarder. B.2 Simulations Using the Poincare Sphere To best visualize the polarization states through the optical system of interest on the Poincare sphere, two separate simulations were utilized; one which illustrated the polarization state evolution as it travelled through the optical system for a single simulation photon, and one which showed only the collection of output polarization states for several simulated photons. So that all figures included are clear, the development of each of these simulations will be explained separately. For the purpose of this discussion, the polarization state as it travels through the optical system shown in Figure B.3 will be analyzed. The input polarization is at 45-degrees and the principle axis of the liquid crystal cell is 45 degrees from this axis (90 degrees in space). Finally, a quarter-wave plate is placed after the liquid crystal cell with its principle axis parallel to the input polarizer. The analyzer is rotated around with the angle shown 176

200 (45 degrees in space) being considered an analyzer angle of zero degrees (this angle is relative to the input polarizer). Figure B.3. Simplified schematic of optical system. B.2.1 Fixed Director Model An example of the output from the Fixed Director Poincare sphere model is shown in Figure B.4. Each optical element in the system is represented by a different color; the input polarizer is represented by black, the liquid crystal cell is represented by green, and the quarterwave plate is represented by blue. The principle axis of each element is represented as a line from the center of the sphere to the surface and the polarization state output from each element is shown as a dot on the surface of the sphere. Figure B.4. Example Poincare sphere for a fixed director simulation. So, for example, the black dot shown on the +S2 axis is the polarization state directly after the input polarizer, which is also the direction of its principle axis (which is difficult to see inside the sphere). The path of the polarization state as it travels through the liquid crystal cell is shown fading slowly from black to green. Because the liquid crystal cell is represented by a series of layers, each dot on this arc around the circle is the polarization state after a single layer. Finally, the blue dot shows the polarization state output from the quarter-wave plate (with its principle axis along the +S2 axis as well). The liquid crystal cell, then, is split up into a number of layers (numlayers=31) with each layer having its own defined principle axis. The liquid crystal central layer is given a maximum 177

201 deflection angle (splay and/or twist) while the liquid crystal layers at each substrate are assumed to have no deflection. The deflection decays sinusoidally from the central layer to the outer layers. Figure B.5 shows an example of this (using only in-plane fluctuations) which results in a spread of polarization states from the S1 axis. Figure B.5. Example Poincare sphere fixed director simulation with in-plane director fluctuations. [σt=10deg, σs=0deg]. Using both the output polarization state from the liquid crystal cell (green) and from the quarter-wave plate (blue), the output intensity for a given analyzer angle (along the equator) is calculated using the dot product between the vectors for the transmission axis of the analyzer and the polarization state of the light, as given by I = 1 2 (S A + 1). (B.2) To match experimental conditions, the plots of transmission versus analyzer angle show the analyzer angle as zero when the analyzer is aligned at 45 degrees (parallel to the input polarizer in the configuration shown in Figure B.1). An example of this simulated signal is shown in Figure B

202 Figure B.6. Example of transmitted intensity versus analyzer angle for single photon Poincare sphere model without quarter-wave plate (green) and with quarter-wave plate (blue). B.2.2 Moving Director Model In this case, the simulation is looped over the optical system multiple times. A new liquid crystal configuration is simulated each time. All expected output polarization states are plotted on the sphere (Figure B.7) while the polarization state path and the principle axes of the optical elements are no longer shown. Figure B.7. Example Poincare sphere moving director simulation [σt=5deg, σs=5deg, nloops=50]. For each fluctuation type (splay or twist), the angle of deflection of the central layer director is defined separately as given by θ, φ = nrand σ T,S, (B.3) where nrand generates a random number on a Gaussian distribution with a standard deviation of 1 and σt,s is the standard deviation of the twist and splay deflection angles, respectively. For the in-plane (twist) fluctuations, this translates directly to a change in the principle axis of each 179

203 layer. For the out-of-plane (splay) fluctuations, this results in a change in the effective birefringence of each layer; the effective extraordinary index for each layer is given by 2 n eff = n e 2 n o 2 n e 2 sin 2 φ+n o 2 cos 2 φ (B.4) To obtain a signal, similar to that shown in Figure B.6, the average intensity across several program loops is taken. For each analyzer angle, then, the standard deviation of the intensities is taken as the noise in this signal and plotted, as shown in Figure B.8. Figure B.8. Example of noise vs. analyzer angle for the moving director Poincare sphere model without quarter-wave plate (green) and with quarter-wave plate (blue). [σt=5deg, σs=0deg, nloops=500]. B.3 Simulation Code B.3.1 Fixed Director Model 1 close all 2 clear all 3 4 Temp=29.4; 5 lam=.500; 6 dlc=2; % cell thickness (microns) 7 QW=1; % quarter-wave? (1=yes, 0=no) 8 ThMax=0; % maximum possible deflection angle 9 PhMax=0; 10 numlayer=31; % # LC layers (must be odd) 11 Rax=1; % Draw retarder axes (1=yes, 0=no) 12 Rcirc=1; % Draw retarder circles (1=yes, 0=no) 13 VA=210; 14 PA=20; 15 PAlign=45; 16 QWAlign=45; 17 LCAlign=90; DTR= /180; % degrees-to-radians 20 IPS=[1; 0; 0]; % input polarization state 21 RotP=[cos(2*PAlign*DTR); sin(2*palign*dtr); 0]; 22 OPS=RotP; % current polarization state 23 DFang=ThMax; % mid-deflection equals max deflection 24 Waves=zeros(1,numLayer); A = ; 180

204 27 B = 5.79*10^-4; 28 Dno = ; 29 Beta = ; 30 Tc = 306.6; 31 TK=Temp ; % Temperature in Kelvin 32 ne=a-b*tk+2*dno/3*(1-tk/tc)^beta; 33 no=a-b*tk-dno/3*(1-tk/tc)^beta; 34 R=zeros(numLayer+1,2); % Create retarder stack % 37 % Define retarder angles for each LC layer 38 for x=1:numlayer 39 R(x,1)=LCAlign+DFang*sin(pi*(x-1)/(numLayer-1)); 40 Phang=PhMax*DTR*sin(pi*(x-1)/(numLayer-1)); 41 neff=sqrt((ne^2*no^2)/(ne^2*sin(phang)^2+no^2*cos(phang)^2)); 42 Waves(x)=(neff-no)*dLC/(lam*numLayer); 43 end 44 R(1:numLayer,2)=360*Waves(:); % Retardation of each LC layer (degrees) 45 NR=numLayer+1; % total number of retarder layers % Add quarter-wave plate to stack 48 if QW==1 49 R(NR,1)=QWAlign; % angle of QWP in lab 50 R(NR,2)=-90; % retardation angle of QWP 51 end % Create sphere plot % 54 figure('position',[ ]); 55 [X,Y,Z] = sphere(20); 56 X = X; 57 Y = Y; 58 Z = Z; 59 Hs = mesh(x,y,z,'facecolor','w','edgecolor',[ ]); % set grid facecolor to white 60 caxis([ ]); % set grid to appear like all one color 61 alpha(0.70); % set opacity of sphere to 70% 62 axis equal; % make the three axes equal so the ellipsoid looks like a sphere 63 set(gcf,'renderer','opengl'); 64 hold on; 65 % Draw x- and y- and z-axes 66 Hx = plot3([ ], [0 0], [0 0],'k-'); 67 set(hx,'linewidth',2,'linestyle','-','color','k'); 68 ht_x = text(1.75,0,0,'+s_1','fontweight','bold','fontsize',12,'fontname','arial'); 69 Hy = plot3([0 0], [ ], [0 0],'k-'); 70 set(hy,'linewidth',2,'linestyle','-','color','k'); 71 ht_y = text(0.1,1.6,0,'+s_2','fontweight','bold','fontsize',12,'fontname','arial'); 72 Hz = plot3([0 0], [0 0], [ ],'k-'); 73 set(hz,'linewidth',2,'linestyle','-','color','k'); 74 ht_z = text(-0.05,0,1.35,'+s_3','fontweight','bold','fontsize',12,'fontname','arial'); 75 ht_lcp = text(-0.05,0.0,1.1,'rcp','fontweight','bold','fontsize',12,'fontname','arial','color','k'); 76 % Draw a bold circle about the equator (2*epsilon = 0) 77 x_e = (-1:.01:1); 78 for i = 1:length(x_e) 79 z_e(i) = 0; 80 y_e_p(i) = +sqrt(1 - x_e(i)^2); 81 y_e_n(i) = -sqrt(1 - x_e(i)^2); 82 end 83 He = plot3(x_e,y_e_p,z_e,'k-',x_e,y_e_n,z_e,'k-'); 84 Set(He,'linewidth',2,'color','k'); 85 % Draw a bold circle about the prime meridian (2*theta = 0, 180) 86 y_pm = (-1:.01:1); 87 for i = 1:length(x_e) 88 x_pm(i) = 0; 181

205 89 z_pm_p(i) = +sqrt(1 - y_pm(i)^2); 90 z_pm_n(i) = -sqrt(1 - y_pm(i)^2); 91 end 92 Hpm = plot3(x_pm,y_pm,z_pm_p,'k-',x_pm,y_pm,z_pm_n,'k-'); 93 set(hpm,'linewidth',2,'color','k'); 94 view(va,pa); % change the view angle H=plot3(OPS(1,1),OPS(2,1),OPS(3,1),'o'); 97 set(h,'markersize',10,'markeredgecolor',[0 0 0],'markerfacecolor',[0 0 0],'color',[0 0 0],'linewidth',0.1); % propagate light through retarder layers 100 for a=1:nr 101 % plot retarder axis 102 if Rax==1 103 if a<nr % excludes QWP 104 Hr=plot3([cos(2*R(a,1)*DTR) 0], [sin(2*r(a,1)*dtr) 0], [0 0]); 105 shade=a/numlayer; 106 set(hr,'linewidth',2,'linestyle','-','color',[0 shade 0]); 107 elseif a==nr&&qw==1 108 Hr=plot3([cos(2*R(a,1)*DTR) 0], [sin(2*r(a,1)*dtr) 0], [0 0]); 109 set(hr,'linewidth',2,'linestyle','-','color','b'); 110 end 111 end 112 % propagate through retarder layer 113 if a<nr QW==1 114 RP=[cos(2*R(a,1)*DTR) sin(2*r(a,1)*dtr) 0]; % S for retarder layer 115 ROPS=[cos(2*R(a,1)*DTR) sin(2*r(a,1)*dtr) 0; -sin(2*r(a,1)*dtr) cos(2*r(a,1)*dtr) 0; 0 0 1]*OPS; % pol state rotated to retarder frame 116 RNPS=[1 0 0; 0 cos(r(a,2)*dtr) -sin(r(a,2)*dtr); 0 sin(r(a,2)*dtr) cos(r(a,2)*dtr)]*rops; % pol state rotated by retarder (in ret. frame) 117 NPS=[cos(2*R(a,1)*DTR) -sin(2*r(a,1)*dtr) 0; sin(2*r(a,1)*dtr) cos(2*r(a,1)*dtr) 0; 0 0 1]*RNPS; % pol state rotated back to lab frame 118 end 119 % % draw retarder circle 120 % plot polarization states 121 if a<nr 122 H=plot3(NPS(1,1),NPS(2,1),NPS(3,1),'o'); 123 shade=a/numlayer; 124 set(h,'markersize',10,'markeredgecolor',[0 shade 0],'markerfacecolor',[0 shade 0],'color',[0 shade 0],'linewidth',0.1); 125 elseif QW==1 126 H=plot3(NPS(1,1),NPS(2,1),NPS(3,1),'o'); 127 set(h,'markersize',10,'markeredgecolor',[0 0 1],'markerfacecolor',[0 0 1],'color',[0 0 1],'linewidth',0.1); 128 end % draw retarder circle -- from one dot to the next 131 if Rcirc==1 132 for b=1: if a<nr QW==1 134 p=r(a,2)/50*b; 135 RCP=[1 0 0; 0 cos(p*dtr) -sin(p*dtr); 0 sin(p*dtr) cos(p*dtr)]*rops; % points on circle in retarder frame between ROPS and RNPS 136 CP=[cos(2*R(a,1)*DTR) -sin(2*r(a,1)*dtr) 0; sin(2*r(a,1)*dtr) cos(2*r(a,1)*dtr) 0; 0 0 1]*RCP; % rotates points to lab frame 137 end 138 if a<nr % excludes QWP 139 Hc=plot3(CP(1),CP(2),CP(3),'g.'); 140 shade=a/numlayer; 141 set(hc,'marker','.','markersize',4.5,'color',[0 shade 0]); 142 elseif a==nr&&qw==1 143 Hc=plot3(CP(1),CP(2),CP(3),'g.'); 144 set(hc,'marker','.','markersize',4.5,'color','b'); 145 end 182

206 146 end 147 end % Rcirc OPS=NPS; % Calculate intensity after LC 152 if a==numlayer 153 for c=1: ca=c+44; 155 A=[cos(2*cA*DTR+pi) sin(2*ca*dtr+pi) 0]; % analyzer angle on sphere 156 ILC(1,c)=0.5*(dot(A,NPS)+1); 157 end 158 end % Calculate intensity after QWP 161 if a==nr 162 for c=1: A=[cos(2*c*DTR+pi) sin(2*c*dtr+pi) 0]; % analyzer angle on sphere 164 IQWP(1,c)=0.5*(dot(A,NPS)+1); 165 end 166 end 167 end B.3.2 Moving Director Model 1 clear all 2 close all 3 4 % % 5 % Set Experimental Conditions % 6 % % 7 8 % Simulation Parameters % 9 10 lam=0.566; 11 dlc=6.48; % cell thickness (microns) numlayer=31; % # LC layers (must be odd) 14 NLoops=100; % Simulation loops 15 DegStep=1; % Analyzer step size (deg.) 16 PAlign = 45; % polarizer alignment 17 LCAlign = 90; % LC Cell alignment 18 QWAlign = 45; IntMax= ; 21 ThMax=2.75; % Standard deviation of director fluctuation (deg.) 22 PhiMax=2.75; % Standard deviation of splay angle (normal dist.) % Liquid Crystal Cell % fluc=1; % Model out-of-plane fluctuations? (0=no, 1=yes) % % 29 % Other Variables % 30 % % 31 Ndeg=360/DegStep; % # angle steps 32 DTR= /180; % degrees-to-radians 33 neloop=zeros(1,numlayer); % extraordinary index at each layer 34 Waves=zeros(1,numLayer); % Waves of retardation per layer RotP=[cos(2*PAlign*DTR); sin(2*palign*dtr); 0]; 37 % % 183

207 38 % Begin Main Program % 39 % % 40 % Loop over temperature 41 ILC=zeros(NLoops,Ndeg); % Intensity w/o QWP 42 ILCave=zeros(1,Ndeg); % Ave. intensity w/o QWP 43 IQWP=zeros(NLoops,Ndeg); % Intensity w/ QWP 44 ne=1.7304; 45 no=1.5335; % Loop over director fluctuations 48 for L=1:NLoops 49 for D=1:Ndeg 50 lamloop=lam; 51 OPS=RotP; % current polarization state 52 PhiAng=randn*PhiMax; if fluc==1 % Simulate out-of-plane fluctuations 55 for x=1:numlayer 56 DAng=PhiAng*DTR*sin(pi*(x-1)/(numLayer-1)); 57 neloop(x)=sqrt(ne^2*no^2/(ne^2*(sin(dang))^2+no^2*(cos(dang))^2)); 58 end 59 else 60 neloop(1:numlayer)=ne; 61 end c=d*degstep; % Angle for simulation loop 64 DFang=randn*ThMax; % random angle LC deviation (normal) 65 for x=1:numlayer 66 R(x,1)=LCAlign+DFang*sin(pi*(x-1)/(numLayer-1)); 67 end % Create retarder stack % 70 % Define retarder angles for each LC layer 71 for x=1:numlayer 72 Waves(x)=(neLoop(x)-no)*dLC/(lamloop*numLayer); % # Waves of retardation 73 end 74 R(1:numLayer,2)=360*Waves(:); % Retardation of each LC layer (degrees) 75 NR=numLayer+1; % total number of retarder layers % Add quarter-wave plate to stack 78 R(NR,1)=QWAlign; % angle of QWP in lab 79 R(NR,2)=-90; % retardation angle of QWP % propagate light through retarder layers 82 for a=1:nr 83 % propagate through retarder layer 84 RP=[cos(2*R(a,1)*DTR) sin(2*r(a,1)*dtr) 0]; % S for retarder layer 85 ROPS=[cos(2*R(a,1)*DTR) sin(2*r(a,1)*dtr) 0; -sin(2*r(a,1)*dtr) cos(2*r(a,1)*dtr) 0; 0 0 1]*OPS; % pol state rotated to retarder frame 86 RNPS=[1 0 0; 0 cos(r(a,2)*dtr) -sin(r(a,2)*dtr); 0 sin(r(a,2)*dtr) cos(r(a,2)*dtr)]*rops; % pol state rotated by retarder (in ret. frame) 87 NPS=[cos(2*R(a,1)*DTR) -sin(2*r(a,1)*dtr) 0; sin(2*r(a,1)*dtr) cos(2*r(a,1)*dtr) 0; 0 0 1]*RNPS; % pol state rotated back to lab frame 88 OPS=NPS; % Calculate intensity after LC 91 if a==numlayer 92 ca=c+44; 93 Ana=[cos(2*cA*DTR+pi) sin(2*ca*dtr+pi) 0]; % analyzer angle on sphere 94 ILC(L,D)=0.5*(dot(OPS,Ana)+1)*IntMax; 95 end

208 97 % Calculate intensity after QWP 98 if a==nr 99 ca=c+44; 100 Ana=[cos(2*cA*DTR+pi) sin(2*ca*dtr+pi) 0]; % analyzer angle on sphere 101 IQWP(L,D)=0.5*(dot(OPS,Ana)+1)*IntMax; 102 end end % Retarders 105 end % Angles 106 end % Loops 107 ILCave=mean(ILC); 108 IQWPave=mean(IQWP); 109 ILCStd=std(ILC,1); 110 IQWPStd=std(IQWP,1); 111 LCmin=min(ILCStd); 112 LCmax=max(ILCStd); 113 IMax=max(ILCave); Angle=1:1:360; 116 figure(1) 117 plot(angle,ilc(1,:),'color',[0 1 0],'linewidth',1.5) 118 hold on 119 plot(angle,iqwp(1,:),'color',[0 0 1],'linewidth',1.5) 120 hold off 121 xlim([0 360]) figure(2) 124 plot(angle,ilcstd(1,:),'color',[0 1 0],'linewidth',1.5) 125 hold on 126 plot(angle,iqwpstd(1,:),'color',[0 0 1],'linewidth',1.5) 127 hold off 128 xlim([0 360]) IMax 131 LCmin 132 LCmax 185

209 Appendix C Development of Infused Photoalignment Layers for Photonic Applications C.1 Introduction Photoalignment layers which operate on the principle of photoreorientation, as discussed in Chapter 6, have received a great deal of attention in LC photonics applications due to the ability of these layers to be utilized in non-standard geometries 65. For instance, photoalignment has been utilized in the creation of a tunable microresonator 66, however, the alignment layer in this case is applied through a standard spin-coating method. Great success has also been shown in the use of photoalignment for tunable photonic crystal fibers (PCFs) In this case, the application of the photoalignment layer via spin-coating is impossible; instead, the fiber is filled with the photoalignment solution through capillary action, then excess solution is removed through a pressure gradient. These methods offer a solution for the application of a photoalignment layer with either a completely open geometry or two open ends. However, neither is usable for a configuration with only one entry/exit port as in the desired liquid crystal thermal imager. Additionally, one of the key benefits of these azo dye photoalignment layers over other photoalignment techniques is their high ordering and anchoring energies. The values of these parameters have been measured in spun photoalignment layers, but never in infused layers, such as those used for tunable PCFs and thermal imager cavities. This Appendix describes the development of the infused photoalignment layers utilized in this dissertation for application to thermal imaging. The issues which came up during the development of these layers are discussed. Finally, preliminary results of the anchoring energy of these layers is presented. C.2. Developing Infused Photoalignment in Bulk Cells Microcavity samples, which will also be discussed in this Appendix, are the best analog to the desired liquid crystal thermal imager cavities, but the supply of these, as well as our ability to measure relevant parameters, is limited. Therefore, the bulk of the investigations in this Appendix are conducted using bulk liquid crystal cells (these cells have at least a 1cm x 1cm active area). C.2.1 Methods The general process for fabricating infused photoalignment cells is outlined in Table C.1. The various steps in this table have been color-coded to indicate the room/lab in which these steps are typically conducted (relevant for humidity monitoring). Note that, for certain experiments, some of these steps were varied; steps are identical to those shown in this table unless otherwise noted. Samples were prepared using patterned ITO-coated glass that was well 186

210 cleaned using ultrasonic bath and UV/O3 cleaning immediately prior to assembling using 1.4µm spacers. Samples were then placed under vacuum at 100C for 1-2 hours (finalized process utilized a 2 hour pre-bake step) to bake off any water that adsorbed onto the substrate surface between UV/O3 cleaning and this step. After pre-bake, the sample chamber was vented using dry nitrogen, samples were moved onto an insulating plate (still inside the vented chamber) and immediately filled with a freshly prepared solution of 3% by weight BY in DMF. Once the BY solution fully filled the samples, the nitrogen was turned off and the samples were moved out of the vacuum chamber and covered for 1 hour. This step was used to allow the dye time to adsorb onto the surface and is herein referred to as the soak step. Table C.1. Outline of processing steps for fabrication of infused photoaligned bulk cells. Step Temperature Time Zenith Ultrasonic 65 C 15 min DI Water Rinse RT ~3-5 min IPA Rinse RT ~1-3 min Bake 90 C ~10-15 min UV/O3 cleaning RT (causes 20 min heating) Cell Assembly RT ~10 min Glue Cure (UV), vacuum pressed RT 10 min Measure cell gap, vacuum pressed RT 5 min Prebake, under vacuum 100 C 1 hr N2 vent chamber (maintain vent through dye infusion) Move samples to insulating plate Infuse dye solution into samples Allow dye to soak inside samples RT 1 hr Bake to evaporate solvent, under vacuum 100 C 1 hr Photoalign RT 5 min/sample Glue seal one open edge, vacuum pressed RT 10 min Samples on hotplate plate, degas liquid crystal 80 C 10 min under vacuum (off hot plate) Fill LC into samples, under vacuum 80 C 10 min Move samples to hot aluminum plate to cool 80 C RT 30 min After soaking, the samples are placed back into the vacuum chamber at 100C, allowing the dye solution to rapidly evaporate out of the sample. Samples were left baking in this state for 1 hour. Next, samples were removed from the vacuum chamber and cooled, then exposed to polarized blue light at 40mW/cm 2 for 5 minutes to align the dye; this method is similar to the one described in Chapter 6. Once the dye was aligned, samples were filled with pure E7 at 80C and allowed to cool gradually. 187

211 C.2.2 Results C Effects of Moisture Figure C.1. Infused photoaligned samples prepared with 2-day-old (left) or fresh (right) 3%wt BY in DMF. Shown on the light table between crossed polarizers oriented with the photoalignment layer along one of the polarizers. During development of the infused photoalignment layer, a number of important parameters were discovered. Most important, strict control of moisture is absolutely critical for high-quality (i.e. low scattering/high uniformity) alignment. For instance, Figure C.1 shows two different infused photoaligned cells filled with pure E7 on a light table between crossed polarizers in their dark state. These cells were prepared identically and at the same time. The only exception is that the dye solution used in one case was prepared several days prior; this resulted in a non-uniform scattering texture. The sample prepared using fresh dye solution (prepared minutes prior to infusing into the cell) appears dark in this image (with the exception of some dye clumps caused by nucleation of bubbles during the evaporation of the dye solution). This same phenomenon was observed when the bottle of DMF had been used for an extended period of time. The DMF purchased was anhydrous, but continued use likely led to moisture contamination. Once the scattering texture was observed (typically, this occurred after approximately 1 month of use during humid summer months), it was necessary to utilize a new, unopened bottle of DMF to again generate high-quality photoalignment layers. Figure C.2. Infused photoaligned cells with identical preparation except preparation occurred when fabrication room was at 37% relative humidity (left) or 45% relative humidity (right). Shown on the light table between crossed polarizers oriented with the photoalignment axis along one of the polarizers. This same phenomenon was observed when excess humidity was present in the laboratory environment. Figure C.2 shows two cells prepared identically with the exception that 188

212 the relative humidity of the fabrication room was 48% or 39%. The cell prepared at higher humidity exhibited the non-uniform scattering texture while the cell prepared at lower humidity did not. Figure C.3. (Top) Images of an infused cell at 10 seconds into the solvent evaporation stage. (Bottom) The same cell shown after complete processing and fill on the light table between crossed polarizers oriented with the photoalignment axis along one of the polarizers. When observing the evaporation of the solvent out of these cells, this cloudy texture was most likely to appear in areas from which the solvent evaporated first. This effect is shown in Figure C.3, which shows the image of a sample 10 seconds into the solvent evaporation phase of the cell preparation process as well as at the end of the process; the appearance of the nonuniform scattering texture coincides well with the areas from which the solvent has already evaporated. (a) (b) Glass on both ITO on one ITO on both (c) ITO on one Figure C.4. (a) DI water selectively wetting plain glass region and dewetting from central region where ITO-coating remains. (b) Infused photoaligned prepared using glass which exhibited this selective wetting phenomenon. Shown filled with pure E7 on light table between crossed polarizers with photoalignment axis along one of the polarizers (dark state). (c) The same cell shown on the microscope at the edge of the ITO-coated region. Also shown in the dark state between crossed polarizers. Another observation of this effect came when some of the prepared substrates exhibited differential wetting of the plain glass and ITO-regions, as shown in Figure C.4. During the water rinsing step conducted after ultrasonic cleaning, the DI water can be seen clinging to the plain 189

213 glass region while fully dewetting the central ITO-coated region (Figure C.4a). Once the cells are completed and observed on the light table (Figure C.4b) or under the microscope (Figure C.4c), the ITO-coated region is well-aligned while the plain-glass region indicates a partially- Schlieren texture indicative of unaligned liquid crystal; this suggests that no dye adhered in these regions which were well-wetted by DI water in the initial cleaning phase. C Other Relevant Parameters During experimentation to develop an optimal process for infused photoalignment, several observations were made that provided some evidence as to the importance of other experimental parameters. Most specifically, this included dye concentration and sample bake temperature. Figure C.5 shows several infused photoaligned cells prepared using different weight concentrations of BY in DMF ranging from 3% to 7%. While 3%wt of BY generated alignment which appears black between crossed polarizers, the two higher concentrations generated very poor alignment. Figure C.5. Infused photoalignment cells prepared with ITO-coated glass Figure C.6 shows a sample prepared using a lower vacuum bake temperature of 70C to evaporate out the solvent. Despite keeping all other parameters identical, cells prepared with this reduced temperature evaporation exhibit very poor alignment quality. Cells prepared with elevated bake temperatures (up to 150C was utilized), did not show any improvement or degradation. Figure C.6. Infused photoalignment cells with a 1 hour bake in a vacuum bake at a reduced temperature of 70C. Shown on the light table between crossed polarizers oriented with the photoalignment axis along one of the polarizers. 190

214 BC2.3 Textures in Photoalignment Layers Figure C.7 show the microscopic view of the scattering texture exhibited in cells discussed in the previous section. The scattering observed macroscopically appears as microdomains under the microscope. Each of these micro-domains appears as a slightly different color, which is particularly evident at 50X, and are several microns across. While dark-state textures can be observed in other alignment layers, such as polyimide, due to the slight spatial variation in the dark-state orientation, the individual micro-domain textures observed in poorly aligning infused photoaligned cells do not exhibit dark states at different orientations; in fact, many of these domains do not go dark at any orientation. Figure C.7. Microscopic texture in poorly aligned regions of infused photoaligned liquid crystal cells. Shown on the microscope between crossed polarizers (with photoalignment axis oriented along one of the polarizers) at 5X under normal camera exposure conditions (left) or 50X with 10-times increase in camera exposure time (right). Similar textures are observed when this BY dye is allowed to dry on a substrate, as shown in Figure C.8. In this case, 3%wt BY in DMF was applied to an ITO-coated glass substrate and this droplet was allowed to dry under ambient conditions. Various textures were observed with larger smaller domains observed near the center of the droplet and larger domains apparent near the edges. 191

215 Figure C.8. Various textures of a dried dye droplet on glass. Shown on the microscope at 50X between crossed polarizers. C.3 Infused Photoalignment in Microcavities For the liquid crystal thermal imager project, the liquid crystal cell is an approximately 20μm diameter microcavity, as shown in Figure C.9. While the thickness of this microcavity is similar to that of the bulk infused samples utilized in the previous section, the actual volume of these microcavities is much smaller. This difference means that there can be significant differences in the way the infused photoalignment process works in this cell geometry. Therefore, the infused photoalignment process was also verified in the microcavity geometry. 2µm (a) (b) 2µm ~16µm Figure C.9. Example microcavity geometry; (a) side-view and (b) top-view. C.3.1 Methods The substrates utilized for microcavity samples are fabricated at MIT Lincoln Laboratory. These samples start with a silicon or fused silica wafer on which a 1μm thick layer of molybdenum is deposited. Next a nm layer of silicon nitride is deposited as the top layer of the microcavity. A pattern of 2μm diameter holes are then etched in the silicon nitride layer to 192

216 serve as the entry ports into the liquid crystal microcavities. At this stage, the microcavity samples are transferred to LCI for testing. The basic preparation process of these microcavity samples from start to finish is outlined in Table C.2. Samples are first submerged into hydrogen peroxide which selectively etches the molybdenum and generates a hole centered on the holes in the silicon nitride. The timing of this process varies and can be reduced by increasing the temperature during this etch step to 90C. Once the microcavities are of the desired diameter, the sample is removed from the hydrogen peroxide and immediately submerged into DI water, which clears the hydrogen peroxide out of the microcavities and halts the etch process. Next, the sample is submerged in IPA, then removed and dried (either allowed to dry on the table top or with application of dry nitrogen). The use of IPA in this step is to reduce the capillary force on the silicon nitride during drying. Table C.2. Outline of processing steps for fabrication of infused photoaligned microcavities. Step Temperature Time Hydrogen peroxide etch 90C ~30 min Submerge in DI water RT ~15 sec Submerge in IPA RT ~15 sec Dry RT ~10-15 min Submerge in BY in DMF RT 15 min Bake to evaporate solvent 150C 15 min Photoalign (40mW/cm 2 ) RT 5 min/sample Samples on hotplate plate, degas liquid crystal 80 C 1 hour under vacuum (off hot plate) Fill LC into samples, under vacuum 80 C ~ 5 min Move samples to hot aluminum plate to cool 80 C RT 30 min At this point, a solution of BY in DMF is prepared in the same way as described for spun samples, with the exception that the BY is mixed in at 0.5%wt (this concentration can be as high as 1%wt). Once the dye solution has been filtered, the sample is fully submerged and allowed to soak, covered, for 15 minutes. Then, the sample is removed from the dye solution and a fresh razor blade is used to quickly remove excess dye solution from the surface of the sample. The sample is immediately baked at 150C for 15 minutes to evaporate out the solvent. This temperature has been chosen because it is just below the boiling point of the DMF (153C). Temperatures higher than the boiling point of DMF caused damage to the silicon nitride top layer. Once the dye layer has been deposited into the microcavities in this way, the photoalignment layer can be aligned by 5 minutes exposure to polarized blue light at ~40mW/cm 2. Next, the desired liquid crystal (or liquid crystal with pre-polymer) can be infused into the microcavities. This is done by placing the samples on a hot plate under vacuum the liquid crystal (solution) is also placed inside the vacuum chamber and allowed to degas for 1 hour before being deposited onto the microcavity sample so as to cover the entire surface of the sample. Finally, the sample is allowed to cool and excess liquid crystal is removed from the substrate surface using a fresh razor blade. 193

217 C.3.2 Results (a) (b) (c) (d) 20µm Figure C.10. Bright (a, c) and dark (b, d) state of photoaligned LC microcavities, between crossed polarizes, which operate in either reflective (a, b) or transmissive (c, d) mode. Photoalignment layer prepared using 0.5%wt BY in DMF. Microcavity diameter is ~12μm (top) or ~20μm (bottom). Bright and dark images for each taken with equal exposure/light settings. Figure C.10 shows several examples of liquid crystal alignment generated in the microcavity samples using this infused photoalignment technique. Each of the samples shown are on different substrate materials (silicon or fused silica) and with different microcavity diameters (~12μm or ~20μm). However, both samples exhibit a uniform bright state and dark state in the majority of the microcavities. This process is shown to be quite repeatable. Additionally, there has been no observed effect of excess moisture from the lab environment affecting the quality of the alignment in these microcavities. These samples do not require a prebake step and have been prepared in lab humidities up to 50% relative humidity without the need to work in vacuum at any point during the process. 194

218 C.4 Anchoring Energy C.4.1 Bulk Cells C Methods Figure C.11. Infused photoalignment cell which has been disassembled then reassembled into a twisted configuration. Shown on the light table between crossed (left) and parallel (right) polarizers. To determine anchoring energy of the infused photoalignment layers, the cells needed to be assembled into a twisted configuration. Because the photoalignment layers are typically infused and aligned after the cell has been assembled, the fabrication process for these cells needed to be modified. In this case, the substrates were clipped together instead of glued together during the dye infusion and bake-out process. Once the dye layer was deposited, cells were disassembled and the glass substrates exposed to polarized blue light, then glued together in the 90-degree twist configuration. A cell prepared in this fashion and filled with pure E7 is shown in Figure C.11. C Results This 90-degree twist cell is then utilized as described by Akahane et al 78 to calculate the actual twist angle of the cell, which is expected to deviate from the designed twist angle (90 degrees). The magnitude of the difference between the designed twist angle and the actual twist angle of the cell provides the anchoring energy. This work is completed by Colin McGinty and method is described in somewhat more depth in Chapter 6. The anchoring energy for this twist cell was calculated to be 4.8*10-5 J/m 2. Note that this is only a preliminary measurement and has not yet been fully validated or reproduced. C.4.2 Microcavities C Defect Lines and Anchoring Energy Often times, when liquid crystal cells are fabricated, small lines or threads can be observed in the nematic phase. In fact, nema is Greek for thread, as this characteristic feature unique to the nematic phase is what gave it its name 80. The formation of these defects is primarily caused during quenching from the isotropic to the nematic phase. The energy barrier between the two phases results in a first-order phase transition which is primarily a process of nucleation and growth. Nematic bubbles will occur randomly through the sample and will collide with one another, growing into larger and larger domains until the sample has fully transitioned. As the bubbles collide with one another, the molecules in the intermediate shrinking isotropic region may not be able to properly reorient as they transition from isotropic to nematic. This results in a 180 degree rotation of the director, known as a Néel wall 81 ; the director configuration through such a wall is shown in Figure C

219 Figure C.12. Liquid crystal director configuration through a defect line. This director rotation is, of course, not energetically favorable. The wider a defect line (wall), the less energy is associated with the director distortion. However, the substrate surface imposes a preferred direction on the liquid crystals. Deviation from this preferred direction increases the free energy of the system. As such, the smaller the wall, the less energy is associated with the anchoring distortion. This is in direct competition with the distortion energy of the director rotation through the wall. As such, the stronger the anchoring energy, the more it will be able to compete with the director distortion energy and the thinner the wall will be. A number of researchers have attempted to quantify this relationship. Kleman 82 and Ryschenkow 83 proposed that the anchoring energy is approximately described as W s ~ hk d 2, (C.1) where h is the thickness of the cell, d is the width of the defect, and K is the elastic constant of the liquid crystals. Kleman supposed that this K should be the sum of the splay and bend constants (K1 and K3) while Ryschenkow supposed that this K should be the twist elastic constant K2. In either case, both researchers point to the fact that this is only a proportional relationship and cannot provide absolute values of the anchoring energy. This allows us to use defect line width to qualitatively determine anchoring energy for various alignment layers in microcavities. C Defect Line Width for Various Alignment Layers C Measurement Method Microscope images of defect lines were taken for a number of samples at 10X. Images were also taken of a metric ruler with 1mm spacing and the software package ToupView was utilized to set a scale for the number of image pixels equal to 1mm (1800 pixels). The number of pixels across each defect line was determined using this same software and the relationship used to determine the approximate width of the defect lines. C Results Table C.3 shows the defect line width for various alignment layers, including the mechanical grooves and infused photoalignment generated in microcavities as well as an evaporated SiOx alignment layer (generating planar alignment). Based on these line widths, the infused photoalignment layer has anchoring energy similar to that of an evaporated SiOx layer and approximately three times higher than that of the alignment currently being utilized in the liquid crystal thermal pixels. 196

220 Table C.3. Defect line width measured in liquid crystal microcavities with various alignment layers. Alignment Type Line Width (μm) Photoalignment 0.61 Evaporated SiOx 0.56 Scratched Alignment 1.78 C.5 Conclusion This Appendix has described the development of infused photoalignment layers for application in a liquid crystal thermal imager. These layers were first developed in bulk cells and then demonstrated in microcavity samples of similar dimension to the actual liquid crystal thermal pixels. A number of experimental parameters were identified which significantly affected the resultant quality of these alignment layers, particularly those related to lab humidity during cell preparation and the wetting behavior of the substrates. Other things such as dye concentration and vacuum bake temperature during solvent evaporation were observed to also affect dye layer quality high dye concentration or low bake temperature significantly reduced alignment quality. Infused photoalignment was demonstrated to be quite repeatable in microcavity samples and was much less dependent on lab environment during preparation. 197

221 Appendix D Photostability and Polymer Separation in Other Experimental Conditions D.1 Introduction During the course of the studies presented in this dissertation, the ability of a surfacelocalized polymer to stabilize the alignment provided by a photoalignment layer was studied in a variety of conditions. This Appendix provides a summary of some work which studied polymer surface localization and photostability which have not been presented in the main text. D.2 Surface-Localization Chapter 7 presented an investigation of the surface-localization of polymer in thin cells with an infused photoalignment layer and the liquid crystal E7. Phase retardation vs. voltage curves were also taken for thicker samples (approximately 5μm) with both BL006 and E7 and rubbed polyimide alignment layers. This data is presented in the following Figures. Note that the y-axis of these curves is normalized retardation. Phase retardation vs. voltage curves for samples polymerized at 0V are shown alongside samples filled with pure liquid crystal for comparison. Unless otherwise noted, these samples were prepared using 1.5%wt pre-polymer and were polymerized for 10 minutes using 3.5mW/cm 2 UV light. Figure D.1. 5µm planar samples filled with 1.5%wt RM257 in either E7 or BL006 vortexed for 6 minutes then separated for 30 minutes, 2 hours, or 1 day prior to polymerization at 50V (1kHz) using 3.5mW/cm 2 UV light for 10 minutes. Regardless of the liquid crystal used, or the amount of vortexing used prior to filling, none of these solutions exhibited a dependence on the amount of time between filling and polymerization. Figure D.1 shows 1.5%wt RM257 in both E7 and BL006 (vortexed for 6 minutes) when polymerized after 30 minutes, 2 hours, or 1 day. This lack of dependence was true across all trials (i.e. for other mixing conditions). 198

222 Figure D.2. 5µm cells filled with 1.5%wt RM257 in BL006 vortexed for 10 seconds or 6 minutes, then separated for 30 minutes prior to polymerization at 50V (1kHz) using UV (365nm) light for 10 minutes at either 3.5mW/cm 2 or 14mW/cm 2. In the case of BL006, the pretilt depends on the amount of vortexing. Figure D.2 shows the 1.5%wt RM257 in BL006 which has been vortexed for either 10 seconds or 6 minutes. The solution vortexed for 6 minutes shows a lower pretilt than the solution vortexed for 10 seconds. The better the mixing, the poorer the separation. However, this trend is not observed in the 1.5%wt RM257 in E7 solution. Figure D.3 shows the same set of plots for mixtures using E7. These results, for both BL006 and E7, are independent of the UV intensity used. Figure D.3. 5µm cells filled with 1.5%wt RM257 in E7 vortexed for 10 seconds or 6 minutes, then separated for 30 minutes prior to polymerization at 50V (1kHz) using UV (365nm) light for 10 minutes at either 3.5mW/cm 2 or 14mW/cm 2. While the intensity does not affect the dependence on vortexing time, it does affect the actual pretilt achieved. This can be seen in both Figures D.2 and D.3 where polymerization at 3.5mW/cm 2 results in a higher pretilt than polymerization at 14mW/cm 2. The lower intensity results in a slower polymerization process, which allows more of the monomers to diffuse toward the surface during the polymerization. This appears independent of the liquid crystal material. Finally, concentration plays a significant role in pretilt formation. Figure D.4 shows the phase profiles for 5µm or 1.4µm cells filled with either 1%wt or 2%wt RM257 in BL006, vortexed for 10 seconds and separated for 2 hours prior to polymerization. In both cases, the 1%wt solution results in a significantly lower pretilt. This is likely due to insufficient thickness of RM near the cell surface at the time of polymerization. At 2%wt, the pretilt is much higher, 199

223 particularly in the case of the 5µm cell. For the 1.4µm cell, the pretilt improvement is not as significant. Again, this could be due to lack of RM material. However, higher concentrations in these thin cells could result in polymerization in the bulk rather than isolated to near the substrate surfaces. Figure D.4. 5µm or 1.4µm planar cells filled with either 1%wt or 2%wt RM257 in BL006, vortexed for 10 seconds then separated for 2 hours prior to polymerization at 50V (for 5µm) or 10V (for 1.5µm) using 3.5mW/cm 2 UV (365nm) light. These results have helped highlight the important process parameters when preparing RM-stabilized liquid crystal mixtures. While the separation time appears insignificant in this set of studies, the vortexing time could be significant, with shorter vortexing times producing higher pretilts (better separation). Polymerization intensity is also significant, with lower intensities producing better separation. Finally, concentration is important, with 1%-1.5% producing sufficiently high pretilts without concern for bulk polymerization and/or light scattering. D.3 Photostability Date Table D.1 Summary of RM-stabilized Photostability Results Thickness LC RM% Prep. Results (μm) V3m, 1d wait 05/22/14 5μm BL %, 1.2%, 1.5% 10/09/14 7μm BL % V3m, 1d wait 10/23/14 7μm BL % V3m, 1d wait 11/06/14 7μm BL % V3m, 1d and wait 1.5% % unstable, 1.2%, 1.5% stable 2 days (3-15mW/cm 2 unpol.) Stable 27 days (3-15mW/cm 2 unpol.) Stable 28 days (3-15mW/cm 2 unpol.) Stable 160min (120mW/cm 2 unpol) 12/18/14 1.4μm BL % V3m, 1d Stable 14 days (20mW/cm 2 pol or 120mW/cm 2 unpol) 02/05/15 5μm E7 1.5% V10s, V1m, V3m, V6m, wait 30m or 1d Stable 5 weeks (20mW/cm 2 pol or 120mW/cm 2 unpol)

224 Table D.1 presents a summary of experiments conducted to test photostability in cells stabilized using a surface-localized polymer layer. All of these samples were prepared as described in Chapter 6. That is, one of the substrates in each sample utilized a spun photoalignment layer and the other utilized a rubbed polyimide layer. With the exception of the samples in this table prepared using 0.9%wt pre-polymer, all of these samples were stable against subsequent light exposure. This was regardless of the thickness of the sample, the liquid crystal used, the preparation condition tested, or the photoexposure intensity or polarization state. As presented in Chapter 7, cells prepared using two infused photoaligned substrates were stable when pre-polymer concentration was at least 1%wt. In a final experiment, 5μm thick samples were prepared using spun photoalignment on both substrates (prepared with 1%wt BY in DMF). These samples were filled with 0.1%wt, 0.5%wt, 1%wt, and 2%wt RM257 in E7, sonicated for 1 minute and separated for 1-2 hours prior to polymerization using 3.5mW/cm 2 UV light for 10 minutes (the variation in separation time here is due to the time it takes to polymerize each cell). These samples were then placed on life testing. Figure D.5 shows the 0.1%wt and 0.5%wt pre-polymer samples before and after photoexposure (45-degree polarized) for 20 minutes. Both of these samples were degraded in this time they are now exhibiting a twisted alignment. Figure D.5. 5μm thick spun photoalignment samples filled with 0.1%wt (left) or 0.5%wt (right) RM257 in E7 shown before (top) or after (bottom) 20 minutes of photoexposure at 20mW/cm 2 polarized at 45-degrees to the photoalignment axis. Figure D.6 shows the 1%wt pre-polymer sample before life testing and at various stages of photoexposure. In this case, the sample is showing partial stability, degrading more slowly than a sample filled with pure LC (this occurs in 20 minutes). The sample prepared using 2%wt pre-polymer showed no sign of degradation after 1 day of photoexposure, but was not tested beyond this point. 201

225 a b c d Figure D.6. 5μm thick spun photoalignment sample filled with 1%wt RM257 in E7 shown before (a) or after 20 minutes (b), 3 hours (c), or 1 day (d) of photoexposure at 20mW/cm 2 polarized at 45-degrees to the photoalignment axis. This recent result appears somewhat contradictory to the photostability exhibited from a variety of hybrid samples (different alignment on both sides) and double-sided infused photoaligned samples, which has been described in this Appendix and in the main body of this dissertation. Additional work is necessary to develop a clear understanding of the reason for this observed difference. D.4 Conclusion This Appendix provides a summary of some work done during our efforts to understand the effect of the surface-localization of a polymer layer and the resultant photostability of an underlying photoalignment layer. Generally speaking, the mixing conditions appeared to matter much more when BL006 was used as the liquid crystal than when E7 was used. However, the one factor that still showed a significant affect was the intensity of the UV light used during polymerization. The polymer layer showed a much better degree of surface-localization when 3.5mW/cm 2 UV intensity was used than when 14mW/cm 2 was used. In studies of photostability in surface-localized polymer-stabilized photoalignment, there is significant evidence of the repeatability of photostability using at little as 1.2%wt pre-polymer in hybrid aligned samples (with rubbed polyimide on one substrate and spun photoalignment on the other). Infused photoalignment was also shown to be stable with as little as 1%wt prepolymer. However, a set of samples prepared with spun photoalignment showed a lack of stability that conflicts with these previous results. This phenomenon requires extensive investigation which is currently underway. 202

226 Appendix E Details of Surface Localization Modeling E.1 Introduction This Appendix describes the surface-localized polymer code utilized in Chapter 7 to consider the degree of surface-localization of the polymer in the various systems studied. E.2 Surface-Localized Polymer Simulation The bulk of this simulation is based on the minimization of the Frank free energy, as described in Chapter 7. Note that, in this case, this free energy is given an extra term as f d = k 11 2 ( n )2 + k 22 2 (n n )2 + k 33 2 (n n )2 W 2 (n n o) (D E ) 2 (E.1) where k11, k22, k33 are the splay, twist, and bend elastic constants of the LC, respectively, D is the electric displacement, E is the electric field, n is the LC director at a particular point, no is the preferred direction of the director (at points along the polymer network), and W is the effective anchoring strength of the LC director in contact with the polymer (W=0 in regions without polymer). The simulation is built to allow direct comparison between the simulated and experimental data. Because there are some thickness variations among the experimental cells, the thickness of the experimental cell which is serving as a comparison is set as the thickness of the simulated cell. The simulation then starts with the original form of the free energy where the fourth term in this equation is not included (or W=0). This allows for the simulation of the director configuration through the cell at the time of polymerization for a given applied voltage (typically either 0V or 10V). This is then taken to be the director configuration of the polymer network (which defines no) in Equation E.1. Next, the liquid crystal director configuration is determined over a wide range of applied voltages (looping over each voltage). In this case, the fourth term in Equation E.1 is included with no defined by the polymer network configuration and W being defined by the polymer network gradient through the cell. Once the director configuration for a given applied voltage is determined, the phase retardation through the cell is calculated. That is, the liquid crystal cell has been divided into a number of layers, each layer with its own director. The angle of the director in each layer is used to determine the effective extraordinary index, and this provides the effective birefringence of each layer. The phase retardation of each layer is summed to provide the total retardation through the cell for a given applied voltage. This allows the generation of a plot of phase retardation vs. voltage. 203

227 E.3 Matching Between Simulation and Experiment Figures E.1 through Figure E.12 show a comparison of the simulated and experimental data for all of the tested conditions. In each case, the left-hand plots shows the phase retardation vs. voltage curves for samples polymerized at 0V and the right-hand plot shows the curves for samples polymerized at 10V. Figure E.1. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 0.5%wt RM vortexed for 10 seconds and separated for 1 hr prior to polymerization. Experimental data simulated with Wavg=4000 and ξ= Figure E.2. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 0.5%wt RM vortexed for 10 seconds and separated for 1 day prior to polymerization. Experimental data simulated with Wavg=4000 and ξ=

228 Figure E.3. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 0.5%wt RM vortexed for 6 minutes and separated for 1 hr prior to polymerization. Experimental data simulated with Wavg=4000 and ξ= Figure E.4. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 0.5%wt RM vortexed for 6 minutes and separated for 1 day prior to polymerization. Experimental data simulated with Wavg=4000 and ξ= Figure E.5. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 1%wt RM vortexed for 10 seconds and separated for 1 hr prior to polymerization. Experimental data simulated with Wavg=8000 and ξ=

229 Figure E.6. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 1%wt RM vortexed for 10 seconds and separated for 1 day prior to polymerization. Experimental data simulated with Wavg=8000 and ξ= Figure E.7. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 1%wt RM vortexed for 6 minutes and separated for 1 hr prior to polymerization. Experimental data simulated with Wavg=8000 and ξ= Figure E.8. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 1%wt RM vortexed for 6 minutes and separated for 1 day prior to polymerization. Experimental data simulated with Wavg=8000 and ξ=

230 Figure E.9. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 10 seconds and separated for 1 hr prior to polymerization. Experimental data simulated with Wavg=16000 and ξ= Figure E.10. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 10 seconds and separated for 1 day prior to polymerization. Experimental data simulated with Wavg=16000 and ξ= Figure E.11. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 6 minutes and separated for 1 hr prior to polymerization. Experimental data simulated with Wavg=16000 and ξ=

231 Figure E.12. Comparison between simulated and experimental phase retardation vs. voltage curves for samples polymerized at 0V (left) or 10V (right) with 2%wt RM vortexed for 6 minutes and separated for 1 day prior to polymerization. Experimental data simulated with Wavg=16000 and ξ= E.4 Flow Chart The following flow-chart outlines the basic function of the polymer simulation described in this Appendix and utilized in Chapter 7. The director relaxation follows the same flow as the simulation described in Appendix A with the exception that the free energy includes an additional term, as given in Equation E

232 209

233 E.5 Simulation Code 1 function varargout = PolymerRelax2(varargin) 2 3 close all 4 clear all 5 degtorad = pi/180; 6 7 % % 8 % Polymer Relax % 9 % % 10 % By: Valerie Finnemeyer % 11 % Last Updated: 03/31/2015 % 12 % % % % 15 % Initial Conditions % 16 % % % Applied voltages VMin = 0; % Minimum voltage for phase profile 22 VMax = 10; % Maximum voltage for phase profile 23 VStep = 0.02; % Voltage step size for phase profile % Coupling parameters 26 Xsi = 0.018; % Concentration Decay (in microns) 27 W = 55000; 28 % Coupling energy (J/m^3) % File info 31 PhaseFile=(['NewMatch_2V10s1d_W',num2str(W),'_Xsi',num2str(Xsi),'.xls']); 32 ExperimentalFile=('2V10s1d.xlsx'); % User-defined cell parameters 35 Pt = 0; % pretilt 36 lam = 0.543; % wavelength of light (in microns) % Read data from experimental data file 40 ExpData=xlsread(ExperimentalFile); % Liquid crystal constants 43 K1 = 11; % splay elastic constant (E7) 44 K3 = 16.9; % bend elastic constant (E7) 45 epl = 19.6; % epsilon parallel (E7) 210

234 46 epd = 5.1; % epsilon perpendicular (E7) 47 ne = 1.75; % extraordinary index of refraction (E7) 48 no = ; % ordinary index of refraction (E7) % Stability Conditions 51 layers = 1000; % number of layers in simulation cell 52 jmax = ; 53 relaxed = ; if layers== dt_g_i= ; 57 stability=0.001; 58 elseif (layers>100&&layers<1000) 59 dt_g_i= ; 60 stability=0.001; 61 elseif layers== dt_g_i= ; 63 stability=0.0006; 64 end % Initialize empty variables and calculate terms 67 Vi = 0; Volt=VMax:-VStep:VMin; 70 ThP = zeros(layers,1); 71 ThLCHigh = zeros(layers,1); 72 ThLCLow = zeros(layers,1); 73 Coupling=zeros(layers/2,1); 74 PhaseHigh=zeros(length(Volt),1); 75 PhaseLow=zeros(length(Volt),1); 76 z = zeros(layers,1); % position of lattice sites 77 ThLCLast = zeros(layers,1); % angle of director at each lattice site % % 81 % Main Loop % 82 % % % HIGH VOLTAGE % 85 VPoly = 10; % Applied voltage during polymerization d=expdata(1,3)*lam/(2*(ne-no)); % Define coupling between polymer network and LC for each LC layer 90 % This is the term I am adjusting to fit experimental data 91 LayerD = d/layers; % layer thickness 211

235 92 for i=1:layers/2 % Loop over half layers 93 dist = LayerD*(i-1); 94 % Coupling(i)=W*(exp(-dist/Xsi)+exp(-(d-dist))/Xsi); % Exponential 95 Coupling(i)=W*d/(2*Xsi)*cosh((dist-d/2)/Xsi)/sinh(d/(2*Xsi)); % Hyperbolic 96 Coupling(layers+1-i)=Coupling(i); 97 end % End loop over half layers % Theta angle and positions for all cell conditions 100 for i = 1:layers 101 z(i) = LayerD/2+(i-1)*LayerD; 102 ThLCLast(i) = -(30*degtorad)*sin((z(i)-LayerD/2)*pi/((layers-1)*LayerD))+90*degtorad- Pt; 103 end % Determine director configuration during polymerization 106 NoCoupling = zeros(layers,1); % No coupling to polymer network 107 ThPoly = DirectorRelax(NoCoupling,ThLCLast,z,VPoly,ThP,d,K1,K3,epl,epd,layers,LayerD,dt_g_i, jmax,stability,relaxed,pt); % Loop over voltage 110 for Voltage=VMax:-VStep:VMin 111 Vi=Vi+1; 112 Voltage % Find director configuration with polymer network 115 ThLCHigh = DirectorRelax(Coupling,ThLCLast,z,Voltage,ThPoly,d,K1,K3,epl,epd,layers,LayerD,dt_g_ i,jmax,stability,relaxed,pt); 116 ThLCLast=ThLCHigh; % Determine phase through LC cell 119 PhaseSum=0; 120 for i=1:layers % Loop over all layers 121 Thi=ThLCHigh(i); 122 neff=sqrt(ne^2*no^2/(ne^2*(cos(thi))^2+no^2*(sin(thi))^2)); 123 deln=neff-no; 124 PhaseSum=PhaseSum+deln*LayerD; 125 end 126 PhaseHigh(Vi)=2*PhaseSum/lam; 127 end % End loop over voltage % Initialize empty variables and calculate terms 130 Vi = 0; % LOW VOLTAGE % 212

236 133 VPoly = 0; % Applied voltage during polymerization d=expdata(1,2)*lam/(2*(ne-no)); % Define coupling between polymer network and LC for each LC layer 138 % This is the term I am adjusting to fit experimental data 139 LayerD = d/layers; % layer thickness 140 for i=1:layers/2 % Loop over half layers 141 dist = LayerD*(i-1); 142 % Coupling(i)=W*(exp(-dist/Xsi)+exp(-(d-dist))/Xsi); % Exponential 143 Coupling(i)=W*d/(2*Xsi)*cosh((dist-d/2)/Xsi)/sinh(d/(2*Xsi)); % Hyperbolic 144 Coupling(layers+1-i)=Coupling(i); 145 end % End loop over half layers % Theta angle and positions for all cell conditions 148 for i = 1:layers 149 z(i) = LayerD/2+(i-1)*LayerD; 150 ThLCLast(i) = -(30*degtorad)*sin((z(i)-LayerD/2)*pi/((layers-1)*LayerD))+90*degtorad- Pt; 151 end % Determine director configuration during polymerization 154 NoCoupling = zeros(layers,1); % No coupling to polymer network 155 ThPoly = DirectorRelax(NoCoupling,ThLCLast,z,VPoly,ThP,d,K1,K3,epl,epd,layers,LayerD,dt_g_i, jmax,stability,relaxed,pt); % Loop over voltage 158 for Voltage=VMax:-VStep:VMin 159 Vi=Vi+1; 160 Voltage % Find director configuration with polymer network 163 ThLCLow = DirectorRelax(Coupling,ThLCLast,z,Voltage,ThPoly,d,K1,K3,epl,epd,layers,LayerD,dt_g_ i,jmax,stability,relaxed,pt); 164 ThLCLast=ThLCLow; % Determine phase through LC cell 167 PhaseSum=0; 168 for i=1:layers % Loop over all layers 169 Thi=ThLCLow(i); 170 neff=sqrt(ne^2*no^2/(ne^2*(cos(thi))^2+no^2*(sin(thi))^2)); 171 deln=neff-no; 172 PhaseSum=PhaseSum+deln*LayerD; 173 end 213

237 174 PhaseLow(Vi)=2*PhaseSum/lam; 175 end % End loop over voltage ExpHigh=zeros(length(ExpData),1); 178 ExpLow=zeros(length(ExpData),1); 179 ExpV(:,1)=ExpData(:,1); 180 ExpHigh(:,1)=ExpData(:,3); 181 ExpLow(:,1)=ExpData(:,2); % Plot resultant phase profile 184 figure(1) 185 plot(volt,phasehigh,'r') 186 hold on 187 plot(volt,phaselow,'b') 188 xlabel('voltage(v)') 189 ylabel('retardation') 190 title(['w=',num2str(w),'; Xsi=',num2str(Xsi)]) 191 plot(expv,exphigh,'k',expv,explow,'g') 192 legend 193 hold off % Write Phase Profile to File 196 for i=1:length(phaselow) 197 PhaseSave(i,1)=Volt(i); 198 PhaseSave(i,2)=PhaseLow(i); 199 PhaseSave(i,3)=PhaseHigh(i); 200 end 201 xlswrite(phasefile,phasesave); % % 204 % % function Theta = DirectorRelax(W,Th,z,V,Th0,d,K1,K3,epl,epd,layers,LayerD,dt_g_i,jmax,stability,relaxed, Pt) 207 % % 208 % Director Relax % 209 % Last Updated: 03/31/2015 % 210 % % 211 % This program calculates the director configuration of % 212 % a cell given defined surface anchoring on both % 213 % substrates and a given applied voltage with or without % 214 % the presence of a polymer network with a given % 215 % configuration. % 216 % % 217 % Inputs: % 214

238 218 % - W = Coupling between polymer and LC % 219 % - V = Applied voltage % 220 % - Th0 = Configuration of polymer network % 221 % % 222 % Outputs: % 223 % - Theta = Relaxed director configuration % 224 % % degtorad = pi/180; 227 % % 228 % Initial Conditions % 229 % % % Constants % 232 e0 = 8.85; % epsilon-naught 233 dele = epl-epd; % dielectric anisotropy NewTh = zeros(layers,1); % % 238 % Main Loop % 239 % % 240 j = 0; 241 dt_g = dt_g_i; 242 while j < jmax 243 InvSum = 0; 244 for k = 1:length(z) 245 InvSum = InvSum + (1/(epl*(cos(Th(k)))^2+epd*(sin(Th(k)))^2)); 246 end 247 Dz = V*e0*layers/(d*InvSum); 248 MaxDTh = 0; % reset maximum change in theta 249 NewTh(1) = Th(1); 250 NewTh(length(z))=Th(length(z)); 251 for i = 2:(length(z)-1) 252 % set first and second derivatives for Theta and Phi at all 253 % sites 254 dth=(1/2)*(th(i+1)-th(i-1))/(layerd); 255 d2th=(th(i+1)+th(i-1)-2*th(i))/((layerd)^2); 256 % Euler-Lagrange equations 257 ThI=Th(i); % director angle for current layer 258 Th0I=Th0(i); % polymer angle for current layer 259 sinth=sin(thi); 260 costh=cos(thi); 261 sindiff=sin(thi-th0i); 262 cosdiff=cos(thi-th0i); 263 Wi=W(i); 215

239 264 ELTh=-(K1*sinTh^2+K3*cosTh^2)*d2Th-(K1- K3)*sinTh*cosTh*dTh^2+Dz^2*dele*sinTh*cosTh/(e0*(epl*cosTh^2+epd*sinTh^2)^2)+ Wi*sinDiff*cosDiff; 265 % Update angles 266 delth = dt_g*elth; 267 NewTh(i)=Th(i)-delTh; 268 % Update stability conditions 269 if abs(delth)>maxdth 270 MaxDTh=abs(delTh); 271 end 272 end 273 Th = NewTh; if MaxDTh>stability 276 dt_g=dt_g/2; 277 end 278 dt_g = dt_g*1.001; % increase time step/viscosity 279 % exit loop is system is fully relaxed 280 jstab=j; 281 if j> if MaxDTh < relaxed j=jmax-1; end 287 end % if mod(j,5000)==0 290 % Th(layers/2) 291 % end j=j+1; 294 end % j loop 295 jstab 296 Theta = Th; 297 Th(layers/2); 216

240 Appendix F Experimental Procedures and Methods F.1 Bulk Cell Preparation Processes F.1.1 Processes for Cells without Infused Photoalignment Layer F Prepare Glass for Alignment Scribe glass (ITO-coated) into 7 x7 plates Place glass in Zenith Ultrasonic tank for at least 10 minutes. Ultrasonic tank should include a Caviclean Ultrasonic detergent by Mettler Electronics and be kept at 90C. Rinse glass thoroughly with DI water then IPA Place glass in oven at 90C to evaporate any remaining IPA from surface Remove from oven and allow glass to cool to room temperature o Note that ITO can be patterned at this point but has not been unless otherwise noted. Place glass in UV/O3 clean, set to clean for 5 minutes. Apply desired alignment layer (see following procedures). F Photoaligned Substrates (Spin-coating) Prepare 2%wt BY in DMF (vortexed for 30 seconds-1 minute and filtered through 1µm filter) Shortly after UV/O3 clean, place glass plate on Brewer spinner and coat with dye solution, applied through a 1µm filter Once entire glass surface is covered, spin glass at 1500rpm for 30 seconds (Brewer Run Program 5) Move glass to hot plate at 120C and bake for 40 minutes-1 hour to evaporate excess solvent Allow glass to cool slowly to room temperature Glass plate is now ready for additional processing. F Rubbed Polyimide Substrates Shortly after UV/O3 clean, place glass plate on Brewer spinner and coat with PI2555, mixed at 9:1, applied through a 1µm filter Once entire glass surface is covered, spin glass at 1500rpm for 30 seconds (Brewer Run Program 5) Move glass to hot plate at 95C and bake for at least 90 seconds to evaporate excess solvent Bake in oven at 275C for 1 hour o I utilize the VWR PI Bake Oven, program 1, which ramps up to 275C over 45 minutes 217

241 Allow glass to cool slowly to room temperature Rub glass with metal block covered in velvet cloth o Velvet cloth is Yoshikawa YA-19R o Rub 10 times in the same direction (mark glass to indicate rub direction) Glass plate is now ready for additional processing F Assembling Cells Place one glass plate with desired alignment layer in spacer sprayer chamber o If assembling hybrid cells, apply spacers to the photoaligned glass Place one glass plate with desired alignment layer on top of 106 mask o If assembling hybrid cells, use the rubbed polyimide substrate Using a fine-point Sharpie, mark locations of squares and crosses from bottom edge of mask onto the glass o This step allows use of array processing without having to pattern the ITO layer and can be skipped if the ITO layer has been patterned. Place same glass plate onto Asymtek XY glue dispenser Apply glue seal o Utilize program for 106 mask with vacuum glue seal (found in Doug s directory: DRB) Place same glass plate on 12 ceramic vacuum chuck and turn on vacuum to the chuck Remove other glass plate from spacer sprayer chamber and place on top of glued glass, careful to align them as much as possible o If utilizing two rubbed polyimide substrates, they should be assembled with the rub directions anti-parallel to one another o If utilizing any photoaligned substrate, cover the central region of each cell with a small square of electrical tape (~ 1 cm x 1 cm) to protect the photoalignment from UV exposure. Cover the entire vacuum chuck and sample with plastic wrap o This usually takes two strips of plastic wrap Carefully press down on the glass, making sure not to disrupt the alignment. Continue applying pressure across various areas of the glass until it appears to be well pressed onto the bottom glass o This should be indicated by a gentle widening of the glue lines and of uniformity of thickness, shown by limited color variation Place Electrolite UV on top of glass Expose glass to Electrolite UV for at least 8 minutes Use VPI Scriber program 106 to scribe the 7 x7 array into samples o If using two photoalignment layers or thin cells (<2µm), it is recommended to increase the pressure settings on the VPI scriber to 28psi and 30psi. o Discard any samples with significant non-uniformity or significantly disrupted glue seal (disrupted glue seal with appear as rainbow color) F Filling Cells Place sample(s) on the heat stage at a temperature somewhat higher than LC transition temperature. 218

242 o 50C for 5CB o 80C for E7 o 125C for BL006 Place liquid crystal in fill vial (if RM/LC, place fill vial in fill chamber as far away from heat stage as possible). Place fill chamber under vacuum Allow liquid crystal to degas for at least 10 minutes. Fill liquid crystal into cells. o If using RM/LC mixture, only allow filling for 5-10 minutes to reduce heat-induced polymerization of the RM Vent chamber and remove samples from vacuum, placing on an insulating surface to slow cooling. o Polyimide and spun photoalignment layers have relatively high anchoring energies and can be cooled fairly rapidly, so they do not need to cool on the aluminum plate for 30 minutes, which was the case for infused photoalignment layers F.1.2 Infused Photoaligned Cells F Cell Preparation (Including Dye Infusion) Using VPI scriber, cut 14 x16 ITO-coated glass down to 4 7 x7 substrates. Place glass substrates in Zenith Ultrasonic tank for at least 10 minutes (Utilizes Caviclean Ultrasonic Detergent by Mettler Electronics). Utilize DI water sprayer to rinse substrates for several minutes. Rinse substrates with IPA to remove any remaining droplets of water. Place in oven at 90C for at least 10 minutes to evaporate IPA. Using Brewer spinner, apply photoresist (S1818 mixed 3:1). Spin at 1500rpm for 30 seconds, then bake at 90C for 90 seconds. Place glass in UV exposure with mask 106 on top and place under vacuum. Set to 10.0 and expose. Once photoresist has been patterned, place in developer for 1 minute with constant agitiation, then rinse in DI water. 219

243 Place glass plate in acid etchant for approximately 7 minutes to etch ITO, then rinse in DI water o Etch time should be calibrated prior to each time glass plates are processed. Place glass plate in solution to remove remaining photoresist for 1-2 minutes. Agitate gently near the end of this time. Place glass susbtrate back in Zenith Ultrasonic tank for 5 minutes Remove from Zenith Ultrasonic and place each glass sheet (one at a time) on a wiper, ITOside up. Using a high-quality wiper wet with Ultrasonic bath solution, rub gently across glass plate in all directions. o Cleanwipe: 100% knitted polyester, sealed edge, Berkshire Ultra-seal Item# US o Diagram below shows wiping motion along the vertical direction (repeat horizontally and along both diagonals). Once all plates are complete, return to Zenith Ultrasonic tank for minutes. Utilize DI water sprayer to rinse substrates for several minutes. Rinse substrates with IPA to remove any remaining droplets of water. Place in oven at 90C for at least 10 minutes to evaporate IPA. Using VPI scriber program 106, cut glass into individual substrates (align fiducials with writing upside-down but forward-facing). Place in UV/O3 cleaner and clean for 20 minutes (UVOCS Model T16x16/OES, lamp ~2 from glass). All steps above take place in the cleanroom. All steps below take place in the back fab room with the exception of photoexposure, which takes place in our main lab. Blow off all substrates with dry N2. To one substrate of each cell, apply glue seal by hand (NOA68). On the other, apply desired spacers using nebulizer and N2 pressure in spacer sprayer (standard: 1.4µm spacers). Observe substrate with spacers in bright light. If any large particles remain, blow off with N2 pressure and reapply spacers. It may also be necessary to blow off glued substrates at this point. Place glued substrates on vacuum chuck and place other substrate on top, aligning with a significant ledge at the end of the glue lines (for filling) and apply gentle force. 220

244 Cover entire chuck with plastic wrap, turn on vacuum, and press down firmly to increase cell uniformity. Place in cure chamber with both lights on for 10 minutes (2 blacklights; UV intensity ~0.88mW/cm 2 at samples) Place cells in vacuum fill chamber at 100C. Pump to vacuum to bake-out samples for 1 hour. In a glass vial, prepare fresh dye solution of BY in DMF (standard = 3%wt). Vortex for 30 seconds 1 minute to mix. Vent vacuum fill chamber using dry N2, continue flowing N2 until cells are all filled with dye solution. o Using gauge at tank, set line pressure to 75psi. For knob controlling flow to fill chamber, open all the way to maximize vent pressure. Keeping samples inside fill chamber (under flow of N2), remove from hot plate and place on insulator to lower cell temperature. Using a glass pipette without bulk, obtain a small quantity of dye solution and apply to ledge of each cell. Allow it to completely fill the cell (adding a little more may be necessary). Once all cells are filled, turn off N2, remove cells from fill chamber and cover to reduce solvent evaporation. Allow dye solution to sit (soak) inside samples for 1 hour. Place samples back in vacuum fill chamber at 100C and place under vacuum. Allow solvent to evaporate in this condition for 1 hour. Remove from vacuum fill chamber and photoexpose for 5 minutes at 40mW/cm 2. Using a KimWipe, wipe off all dye residue on both ledges of glass. Continue wiping with KimWipe until the wipe comes away clean. F RM-Stabilized Infused Photoalignment On the ledge originally used for filling the dye into the sample, apply a bead of NOA68 across, making sure to connect to the two original glue lines. Masking the rest of the cell with aluminum foil, place in cure chamber with both lights on for 10 minutes (2 blacklights; UV intensity ~0.88mW/cm 2 at samples). While cells are curing, prepare RM/LC mixture (see appropriate procedure). Let sit for 10 minutes. Then vortex as desired and place in vial in fill chamber. Pump to vacuum for 10 minutes. Once cells are cured, take vacuum chuck and vacuum to main lab. With cell pulled to vacuum, use Ocean Optics spectrometer to measure reflection spectra (to be used to calculate cell thickness). Place cells on hotplate at 80C in fill chamber along with aluminum plate and place everything under vacuum again. Fill cells with LC, limiting fill time to approximately 5 minutes. Open fill chamber, place cells on aluminum plate, and remove plate fromfill chamber. Allow cells 30 minutes to cool. Wipe excess LC from fill ledge of samples. Place sample in bladder press and apply approximately 12 psi of pressure using N2 line. Let sit for ~5-10 minutes Using an acetone-soaked KimWipe, clean any remaining LC residue from ledge. 221

245 Use a toothpick to apply 5-minute epoxy across the entire ledge. Let sit until glue is cured. Remove cells from bladder and solder leads onto them. Cover leads with hot glue. o Once glue is cooled, store cells in dark until it is time to polymerize. Mount cell securely in UV exposure station. This station utilizes a Mightex UV LED (365nm) o Mightex collimated UVLED Part Number: LCS UV LED Lens Plastic Bag/Diffuser Sample Mount Slowly ramp voltage from 0V to 10V (1kHz, applied using a function generator). o From 0V to 2V at 0.1V per 3 seconds. o From 2V-5V at 0.1V per 2 seconds. o From 5V-10V at 0.1V per 1 second. Polymerize with exposure to 3.5mW/cm 2 UV LED (365nm) for 10 minutes. o If curing at 0V, turn off voltage prior to UV exposure. o If curing at 10V, leave voltage on through UV exposure. o Intensity variation at sample is shown below (measured values were determined using a pin-hole ~1 diameter) F.2 Microcavity Processes F.2.1 Photoalignment in Microcavities Etch: o If sample is coated with photoresist, use acetone and gentle rubbing to remove. o Place sample in hydrogen peroxide to etch microcavities. With fresh/high quality hydrogen peroxide, this takes ~25-30 minutes at room temperature or 5-7 minutes at 90C. With old/degraded hydrogen peroxide, I found this process took ~25 minutes at 90C. o Remove sample from hydrogen peroxide and place in DI water bath. 222

246 o Remove sample from DI water bath and place in IPA bath. o Upon removal from IPA bath, rinse briefly with IPA. o Tap sample against wipe to remove excess IPA on surface, then set down and allow to dry slowly at room temperature o Place one sample on microscope to verify cavities are of the proper size. If they are not large enough, place back into hydrogen peroxide and continue etching. Dye Infusion o In a vial, prepare solution of 0.6%wt BY in DMF. The amount should be sufficient to fully submerge samples in dye solution. Current results suggest that this concentration can be higher o Vortex solution for 30 seconds 1 minute to dissolve dye. o Filter solution (using 1µm filter) into a small pyrex dish. o Place sample(s) in dye solution bath, cover, and allow to soak for 15 minutes (at room temperature). o Remove the sample from the dye solution and hold from the top at a slight angle away from the vertical with the bottom edge resting against a wipe. o Using a fresh razor blade (cleaned with acetone), draw the blade across the sample at a slight angle from the sample. Only a gentle pressure should be used. The razor should be drawn from the top down so that excess dye solution can be pulled into the wipe. Repeat 2-3 times as needed to remove all dye. It may be necessary to flip sample so that wiping the area previously being held is possible. This entire cleaning process should be done as quickly as possible to limit solvent evaporation during this step. o Immediately place sample(s) in the oven (or on heat stage) at 150C to evaporate solvent from cavities. Leave the sample(s) to anneal at this temperature for 15 minutes. Photoalignment o Remove sample from oven and affix to glass slide using double-sided tape. o Affix glass slide to polarizer in exposure setup. 223

247 o Expose sample to high intensity (~40mW/cm 2 ) polarized blue light for ~ 5 minutes. Higher exposure intensities require shorter exposure time, but exact required time is unknown. o Turn off exposure and remove sample of exposure station. F.2.2 LC Infusion into Microcavities Place sample(s) on the heat stage at a temperature somewhat higher than LC transition temperature. Place aluminum plate on heat stage (used for cooling step later) o 50C for 5CB o 80C for E7 o 125C for BL006 Place RM/LC mixture in fill vial as far away from hotplate as possible. Place under vacuum for 1 hour to degas the liquid crystal. o House vacuum at Lincoln does not appear sufficiently strong to degas liquid crystal Apply liquid crystal to the top sample surface. o LC does not have to be isotropic; it will heat to isotropic once it touches the sample surface. o LC should be sufficient to cover the entire desired fill area. It may be useful to gently move fill vial around to spread the LC out (this will reduce the volume of LC required to coat the entire surface). Immediately at 5 minute fill, vent chamber, place samples on aluminum plate, and remove aluminum plate from heat stage to allow it to cool gradually. o Short fill time is necessary to reduce potential polymerization of RM in cavities. This is not necessary when pure liquid crystal is used. o Cooling should take ~ 30 minutes. Once cooled, affix sample to glass slide using double-sided tape. Using a fresh razor blade (cleaned with acetone) to clean LC residue from top surface of sample. o Fresh blade should be used on up to 2 LC samples. Clean blade with acetone and turn it over between samples. o Applying gentle pressure with the razor blade at a slight angle to the sample, wipe the blade across the surface to remove the bulk of the LC (collect at the edge of the sample using a KimWipe). o Now repeat this process two times in each of the four directions across the sample, wiping the blade with a KimWipe after each time (acetone cleaning if needed) Any remaining LC on the surface should be cleaned by running the sample under a gentle flow of DI water o Adjust DI water faucet to lowest possible flow that results in a continuous stream of water 224

248 o Holding the sample securely with tweezers, gently oscillate sample between vertical and horizontal under the flow of DI water o Continue this process until LC residue appears to be completely removed o Allow sample to dry (or blow dry with gentle dry air) o Repeat rinsing step as necessary o Note: this process is still not finalized as my method still results in removal of some of the LC from inside the cavities in a number of samples. o Note: For RM/LC mixtures, I have not yet determined whether this step should occur before or after polymerization. F.3 Miscellaneous Processes F.3.1 Preparing RM/LC Mixture Numbers provided as examples to prepare small quantity of 1.5%wt RM solution To a small brown vial, add ~0.0002g Irgacure 651 Add RM257 so that the total mass is now ~0.002g (I651 should be 10%wt of current mix). o For other I651 masses, multiply I651 mass by 10 to get target total mass at this stage. Add liquid crystal of choice so that the total mass is now ~0.1333g (RM257/I651 mix should now be 1.5%wt of current mix). o For other RM257/I651 masses or desired weight concentrations, divide RM257/I651 total mass by desired weight concentration to get total target mass at this stage. Vortex vial on Fisher Vortex Genie 2 at setting ~6.5 o Vortex time varies among trials, but standard vortexing time is currently 3 minutes for BL006 and 10 seconds for E7 225

249 Immediately fill RM/LC mix into cells in the isotropic state. o Note: RM/LC mix may first need to be placed into fill jar and degassed under vacuum before filling into cells (see individual filling processes for details). F.4 Measurement and Testing F.4.1 Photostability Testing Samples are placed in individual channels of the photostability test bed, which are separated with velvet cloth. Each channel has a polarizer (at 45 degrees) just prior to the sample Exposure intensity is ~20mW/cm 2 Parts (from Luxeon) included in this setup are: o Heat Sink, Part Number: LPD25-20B o 9-degree Collimating Lenses, Part Number: C11916_SATU-S o Flexblock LED Drivers (allowing current-controlled driving of 4 Tri-Star LEDs by a single voltage source), Part Number: A011-D-V-700 o Luxeon Royal Blue Tri-Star LEDs (λmax=447nm), Part Number: SP-03-V4 226

250 F.4.2 Measuring Transmission vs. Voltage Curves Voltage Multiplier Light Source Color Filter Detecto Lens Sample Polarizer Polarizer 1. In EOM setup, verify that color filter is in place in front of light source. a. Current color filter being used is 543nm 2. Turn on light source. 3. Mount cell in EOM setup. 4. Using detector viewport, verify that detected region (dark spot) is in the center of the cell. Detected region (dark 5. Using knob on top of detector and looking through detector viewport, adjust so that the cell appears in focus. 227

251 6. Looking through detector viewport, rotate the cell until the transmission through it is minimized with the polarizers crossed. The cell is now aligned with one of the polarizers this is the dark state configuration. 7. Rotate the cell 45 degrees from the dark state configuration this is the bright state configuration. 8. Attach electrodes to the cell. 9. If using Voltage Multiplier (10X), turn it on and verify that electrical connection travels through the Voltage Multiplier. 10. On the computer, open EOM software from desktop. 11. Select Static Waveform. The following screen should appear. 12. On this screen, select Configure Waveform Driver. This will cause an additional screen to appear, as shown below: 228

252 13. From the Drive Mode drop-down menu, select Ramp. 14. Enter in Starting Voltage and Ending Voltage. Once both are entered, the actual values may automatically adjust slightly. a. If scanning from 20V to 0V using voltage multiplier, enter Starting Voltage = and Ending Voltage = b. If scanning from 10V to 0V without voltage multiplier, enter Starting Voltage = and Ending Voltage = Enter Increment Voltage as desired (should be negative). I typically use 0.05V or smaller a. If using voltage multiplier, remember that this will be one-tenth of the actual voltage step. 16. Select Load Waveform. From the file menu, select asquare. This waveform is shown below. a. Note that this square wave has 5 repetitions. The Waveform Repetition Rate on the Waveform Driver menu is actually five times smaller than the true repetition rate. It will automatically be set to 200 reps/s (actual frequency of 1kHz). 17. Select Quit to close this menu and return to the Static Response Measurement menu. 18. Select Configure Detector Driver. This will cause an additional screen to appear, as shown below: 229

253 19. Select Load Detector. From the file menu, select Chan0_Rse_Unipolar.det. Press Detector Off to take a measurement (which will appear in box under Detector Voltage ) to verify that everything is working properly. 20. Select Quit to close this menu and return to the Static Response Measurement menu. 21. Select Acquire to start data acquisition. a. Acquire button will turn red while data acquisition is taking place and read Abort b. Acquire button will return to original appearance once data acquisition is complete. c. If you select abort to stop data acquisition, subsequent data files will save improperly. To avoid this, you must exit out of all EOM windows and reinitialize the program (start from step 10) 22. Once data acquisition is complete, select Save File and follow commands in dialog to save the collected data. F.4.3 Converting TV Curves to Phase Retardation vs. Voltage Curve To produce a phase vs. voltage profile, two transmission versus voltage curves are taken, one with parallel polarizers and one with crossed polarizers. In both cases, the cell is oriented 230