Digital Holographic Microscopy

Transcription

1 Digital Holographic Microscopy A Thesis Presented by Yujuan(Janice) Cheng to The Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering in the field of Fields, Waves and Optics Northeastern University Boston, Massachusetts May 28

2 c Copyright by Yujuan(Janice) Cheng All Rights Reserved

3 NORTHEASTERN UNIVERSITY Graduate School of Engineering Thesis Title: Digital Holographic Microscopy Author: Yujuan(Janice) Cheng Department: Electrical and Computer Engineering Approved for Thesis Requirements of the Doctor of Philosophy Degree Thesis Advisor: Prof. Anthony J. Devaney Date Thesis Reader: Prof. Charles A. DiMarzio Date Thesis Reader: Prof. Edwin A. Marengo Date Thesis Reader: Prof. Dana H. Brooks Date Department Chair: Prof. Ali Abur Date Director of the Graduate School Date

4 NORTHEASTERN UNIVERSITY Graduate School of Engineering Thesis Title: Digital Holographic Microscopy. Author: Yujuan(Janice) Cheng. Department: Electrical and Computer Engineering. Approved for Thesis Requirements of the Doctor of Philosophy Degree Thesis Advisor: Prof. Anthony J. Devaney Date Thesis Reader: Prof. Charles A. DiMarzio Date Thesis Reader: Prof. Edwin A. Marengo Date Thesis Reader: Prof. Dana H. Brooks Date Department Chair: Prof. Ali Abur Date Director of the Graduate School Date Copy Deposited in Library: Reference Librarian Date

5 Abstract A traditional microscope is not capable of imaging phase object or three dimensional object. Computational Microscopy is proposed to address these two limitations. In Computational Microscopy, the sample is under coherent illumination and the intensities of the diffracted light are recorded by a detector system (CCD camera in most cases). The recorded information is then numerically processed by computer to retrieve the complex field scattered by the object from which the quantitative properties of the object are reconstructed. Three dimensional imaging via Computational Microscopy is performed by illuminating the semi-transparent object from multiple directions and solving the inverse scattering problem via techniques of Optical Diffraction Tomography(ODT). Digital holographic microscopy(dhm) discussed in this thesis is a synthetic implementation of the Computational Microscopy and employs phase-shifting digital holography for phase retrieval. Different experimental schemes were tested and three numerical back-propagation methods were implemented for back-propagating the complex wave from measurement plane to near field. The most emphasized part of the thesis is the research in the algorithms for ODT. An inverse scattering algorithm based on distorted wave approximation for the geometry of ODT is introduced. We implement the algorithm to two Optical Diffraction Tomography configurations: free space and cylindrical symmetric background. For free space case both near field and far field inverse scattering problems are studied. For cylindrical symmetric background inverse scattering based on Distorted Wave Born Approximation and Distorted Wave Rytov Approximation are discussed and compared via computer simulation. In addition, an inverse scattering algorithm based on ray tracing is derived based on the similar idea. Experimental results are presented in which we have successfully reconstructed transmission functions of 2D semi-transparent objects and refractive index profile of a step index optical fiber.

6 Contents Introduction. Previous work Phase and 3D imaging modalities Phase retrieval Digital recording and numerical reconstruction of holograms 5..4 Inverse scattering in optical diffraction tomography Research reported in this thesis Phase Retrieval and Wavefront Back-propagation 2. Phase shifting digital holography Constant phase shifting algorithms Phase shifting calibration and Least-Squares algorithm Wavefront back-propagation Rayleigh Sommerfeld back-propagation algorithm(rs) Angular spectrum expansion(ase) Fresnel transform(frt) Discussion Scattering Theory Formulation of Optical Diffraction Tomography General introduction Problem formulation Distorted Wave Born Approximation ii

7 3..3 Distorted Wave Rytov Approximation Inverse scattering within the Distorted Wave Approximation Discussion Inverse problem in free space Problem formulation Computer simulation Noise effect Inverse Scattering for Cylindrical Background Media Problem formulation Inverse scattering in cylindrical symmetric background Distorted Wave Born Approximation Distorted Wave Rytov Approximation Inverse scattering algorithm for ODT in cylindrical symmetric media Born Approximation Computer simulations Comparison of DWBA/DWRA and FBP method Comparison of DWBA and DWRA method Effect of noise Inverse Scattering Algorithm Based on Ray Tracing Ray tracing Inversion based on ray tracing Computer simulations Forward model evaluation Inversion algorithm evaluation Experimental Results 5 6. Digital holographic microscope setup Experimental system Data acquisition Thin object imaging results Simple 2D objects Biological samples iii

8 6.3 Three dimensional imaging Complex field retrieval from holograms Interferometer error removal Background removal Distorted wave reconstruction Summary and Future Work Discussion of main results Digital holographic microscope Inverse scattering algorithms Experimental results Limitations Mathematical limitations Experimental limitations Direction of future work A Regularization 3 A. Singular Value Decomposition and pseudo-inverse A.2 The truncated Singular Value Decomposition A.3 Tikhonov-type variational regularization methods B Solving Boundary Problems 34 B. Background green function B.2 Incident wave B.3 Point-wise object scattered field B.4 Moment method C Experiment Procedure 4 D List of Publications 42 Bibliography 43 iv

9 List of Tables 3. Π matrix construction of near field Π matrix of Far field The error of reconstruction of object 4 of three algorithms: DWBA, DWBA far field and FBP rms errors of inversion for constant perturbation (piece-wise object) by four reconstruction strategies: reconstruction by DWBA algorithm from DWBA field, DWBA algorithm from DWBA field, DWRA algorithm from exact field, FBP algorithm from exact field rms errors of inversion for sinusoidally varying perturbation(general object,δ n =.3 ) by four reconstruction strategies v

10 List of Figures. Illustration of traditional and computational microscopy for the imaging of thin objects. (a) traditional microscopy which is illuminated by incoherence light; (b) computational microscopy illuminated by coherent light. Object transmission function is represented by T (ρ) and the image function is described by T (ρ). 9.2 Digital holographic microscopy subsystems and their corresponding outputs. The data acquisition process is accomplished by phase-shifting digital holography Two configurations of digital holographic microscopy (up) Mach- Zehnder Interferometer (down) Michelson Interferometer An example of phase shifting interferometry: interferogram generated under four different reference phase values Phase shift value distribution across the imaging aperture in one measurement. The distribution is obtained from a phase shifting holography experiments without any object placed in the interferometry configuration Coordinates system for digital holography. The object function at input plane is f(x, y) = ψ(x, y, ) and wave function at holography plane is U(x, y) = ψ(x, y, z ) vi

11 2.5 Simulated real and imaginary parts of the propagated fields of a one dimensional slit at data plane z = λ. In the figures, the blue solid lines represent the field calculated by RS integral, which is shown in both left and right figures. The dashed green line shows the result of ASE method and the red dots illustrates the result by ASE method with zero padding. And the purple dot shows the results of Fresnel transform. The top two figures illustrate the real part and the bottoms are for the imaginary parts Simulated real and imaginary parts of the propagated fields of a one dimensional slit at data plane z = 2λ. In the figures, the blue solid lines represent the field calculated by RS integral, which is shown in both left and right figures. The dashed green line shows the result of ASE method and the red dots illustrates the result by ASE method with zero padding. And the purple dot shows the results of Fresnel transform. The top two figures illustrate the real part and the bottoms are for the imaginary parts Wave propagation error of different methods. Left: error plots of the forward model; right: error plots of back-propagation model Schematic illustration of scattering phenomina in optical diffraction tomography: a weak scatter embedded in a known compact supported V. The object coordinates x y and illumination coordinates ξ η An example of singular values for near field Π An example of singular values for far field Π Testing objects; (a) Step index fiber (b)graded-index fiber (c) One disc and one rectangular (d) Head Phantom Scattered fields of a piece-wise constant object (Intensity). The core refractive index is n =. and the cladding refractive index is n =.5. The red line represents the exact field and the blue * line represents the approximated field Scattered fields of a piece-wise constant object (Intensity). The core refractive index is n =.3 and the cladding refractive index is n = vii

12 3.7 Example of point spread function in two dimension. (a) reconstructed image of a point scatterer by DWBA inverse scattering algorithm in near field; The image is zoomed in to show the details of the reconstruction; (b) Fourier transform of (a); (c) reconstructed image of a point scatterer by filtered back-propagation algorithm; The image is zoomed in to show the details of the reconstruction; (d) Fourier transform of (c); The number of the detectors is 4 and the object is sampled by a 2 2 grids, which is reflected on the x and y axes of the figures Inverse scattering reconstruction of step-index optical fiber for the limited data case. The number of incident angle is 8 and the number of detector is 5 at each detection angle. (a) original object function; (b) reconstructed result by DWBA inverse scattering method in near field; (c) reconstruction result by FBP method; (d) diagonal values of object refractive index from (a),(b) and (c) Inverse scattering reconstruction of graded-index optical fiber for the limited data case; The highest core refractive index is n core =.3 and the cladding refractive index is n cladding =.5. The number of incident angle is 8 and the number of detectors is 5 at each detection angle Inverse scattering reconstruction of arbitrary shapes for limited view case(angle:8;detector number:5); The refractive index of circular disc is n =.5 and the refractive index of rectangular area is n = Inverse scattering reconstruction of standard head phantom for limited view case(angles:8;detector number:5); Inverse scattering reconstruction of step-index fiber from dense data() Inverse scattering reconstruction of graded-index fiber from scattered data collected from dense data() Inverse scattering reconstruction of arbitrary shapes from scattered data collected from dense data(). The refractive index of circular disc is n =.5 and the refractive index of rectangular area is n = viii

13 3.5 Inverse scattering reconstruction of standard head phantom from scattered data collected from dense data() Inverse scattering reconstruction of step-index fiber from scattered data collected from dense data(2) Inverse scattering reconstruction of graded-index fiber from scattered data collected from dense data(2) Inverse scattering reconstruction of arbitrary shapes from scattered data collected from dense data(2). The refractive index of circular disc is n =.5 and the refractive index of rectangular area is n = Inverse scattering reconstruction of standard head phantom from scattered data collected from dense data(2) Point Spread Function of limited data case Point Spread Function of dense data case () Point Spread Function of dense data case (2) Scattered fields generated by reconstructed object function, 8 views and 5 detectors; (a) step-index fiber; (b) graded-index fiber; (c) arbitrary shapes; (d) phantom; The blue line represents the original data, the read * represents the data obtained from the new algorithm and the green o represents the data obtained from FBP Point spread function of the inverse reconstruction algorithm formulated for far field case; The number of the detectors is 4. (a) The reconstruction of a point scatter by far field DWBA algorithm (b) 2D fourier transform of (a) (c) The reconstruction of a point scatter by filtered back-propagation algorithm (d) 2D fourier transform of (c) The reconstruction of the object -4 by using far field DWBA- ODT algorithm and FBP algorithm. The number of views is 8 and the number of detectors is 5 ;(a) original object function (b) the reconstruction from far field DWBA-ODT (c) the reconstruction by FBP ix

14 3.26 The reconstruction of the object -4 by using far field DWBA- ODT algorithm and FBP algorithm. The number of views is 8 and the number of detectors is 4 ;(a) original object function (b) the reconstruction from far field DWBA-ODT (c) the reconstruction by FBP The reconstruction of the object -4 by using far field DWBA- ODT algorithm and FBP algorithm. The number of views is 32 and the number of detectors is ;(a) original object function (b) the reconstruction from far field DWBA-ODT (c) the reconstruction by FBP The reconstruction of object 3 from noisy scattered field data. The upper row is the results from DWBA near field algorithm, and the lower row shows the results from FBP algorithm Reconstruction result of object 3 from 32 illumination angles and detectors which are separated λ away from each other. (a) original (b) Near field DWBA inverse scattering (c) filtered backpropagation algorithm The simulation geometry of inverse scattering in cylindrical symmetric background. (a) illustrates a radially symmetric scattering object embedded in free space and illuminated with a plane wave. (b) The total field generated in the scattering experiment is measured over a CCD array located at the distance l from the center of the object. (c) A radial cut through the index distribution of the object x

15 4.2 Comparison of scattered field data generated by different scattering models for an object comprised of two concentric cylinders. The cladding refractive index is n c =. and the core refractive index is n =.2 (a) Real part (b) Imaginary part (c) Magnitude. The blue solid line represents the exact field calculated by bessel function expansion. The green dot line represents the DWBA scattered field and the magenta crossing shows the DWRA scattered filed. The Born scattered field data were generated using the core index n c for the background medium and is represented by cyan dashed lines Simulated reconstruction results of the piece-wise constant object with cladding refractive index to be n c =. and the core refractive index to be n =.2. The original refractive index distribution is represented by the dark star line. The reconstruction from Distorted Wave Born Approximation field by the DWBA inverse scattering algorithm is represented by green dot line. The reconstruction from the actual exact field by DWBA algorithm is represented by the blue solid line and the corresponding reconstruction from the exact field by the DWRA algorithm is shown by magenta crossing line. The cyan dashed line is used to represent the reconstruction result from exact field by FBP algorithm Comparison of scattered field data generated by different scattering models for piece-wise constant object with weak background scattering but strong core scattering(case 2). The cladding refractive index is n c =. and the core refractive index is n =.3 (a) Real part (b) Imaginary part (c) Magnitude. The blue solid line represents the exact field calculated by bessel function expansion. The green dot line represents the DWBA scattered field and the magenta crossing shows the DWRA scattered filed. The Born scattered field data were generated using the core index n c for the background medium and is represented by cyan dashed lines xi

16 4.5 Simulated reconstruction results of the piece-wise constant object with cladding refractive index to be n c =. and the core refractive index to be n =.3. The original refractive index distribution is represented by the dark star line. The reconstruction from Distorted Wave Born Approximation field by the DWBA inverse scattering algorithm is represented by green dot line. The reconstruction from the actual exact field by DWBA algorithm is represented by the blue solid line and the corresponding reconstruction from the exact field by the DWRA algorithm is shown by magenta crossing line. The cyan dashed line is used to represent the reconstruction result from exact field by FBP algorithm Comparison of scattered field data generated by different scattering models for piece-wise constant object with strong background scattering. The cladding refractive index is n c =. and the core refractive index is n =. (a) Real part (b) Imaginary part (c) Magnitude. The blue solid line represents the exact field calculated by bessel function expansion. The green dot line represents the DWBA scattered field and the magenta crossing shows the DWRA scattered filed. The Born scattered field data were generated using the core index n c for the background medium and is represented by cyan dashed lines Reconstruction results for the piece-wise constant object from the DWBA scattered field data shown in Fig All the legends are consistent with previous figures Comparison of scattered field data generated by different scattering models for an object having a sinusoidally varying perturbation with respect to its cylindrical background (n c =.44; Maximum n o =.3). (a) Real part (b) Imaginary part (c) Magnitude. The Born scattered field data were generated using the core index n c for the background medium Reconstruction results for the sinusoidally varying perturbation object from the scattered field data shown in Fig The legend is consistent with previous reconstruction illustration xii

17 4. Reconstruction results for the object having a sinusoidally varying perturbation(n c =.44; Maximum n o =.3). The top figure shows the reconstruction generated using the DWBA algorithm and the bottom the FBP algorithm with k b as background wave number Comparison of DWBA and DWRA: same weak scatterer embedded in different background media; The upper figure shows the reconstruction results from DWBA method and the lower image shows the results from DWRA method Comparison of DWBA and DWRA: reconstruction of 6 different circular objects which indicates the effect of weak scatterer s size and refractive index difference on DWBA and DWRA; This figure shows the scattered fields of a weak scatterer with radius 4λ; Solid line: Exact field, Dashed line: DWBA field, dash-dot line: DWRA field; Continue: recontruction of weak scatterer with radius 4λ; Solid line: Original; Dashed line: DWBA reconstruction, dash-dot line: DWRA reconstruction; Comparison of DWBA and DWRA continue: scattered fields of a weak scatterer with radius λ; Solid line: Exact field, Dashed line: DWBA field, dash-dot line: DWRA field; Continue: recontruction of weak scatterer with radius λ; Solid line: Original; Dashed line: DWBA reconstruction, dash-dot line: DWRA reconstruction; Comparison of DWBA and DWRA continue: scattered fields of a weak scatterer with radius 2λ; Solid line: Exact field, Dashed line: DWBA field, dash-dot line: DWRA field; Continue: recontruction of weak scatterer with radius 2λ; Solid line: Original; Dashed line: DWBA reconstruction, dash-dot line: DWRA reconstruction; Comparison of DWBA and DWRA continue: scattered fields of a weak scatterer with radius 4λ; Solid line: Exact field, Dashed line: DWBA field, dash-dot line: DWRA field; xiii

18 4.9 Continue: recontruction of weak scatterer with radius 4λ; Solid line: Original; Dashed line: DWBA reconstruction, dash-dot line: DWRA reconstruction; Comparison of DWBA and DWRA: same phase shift over the weak scatterer(.4π) Comparison of DWBA and DWRA: same phase shift over the weak scatterer(.8π) Comparison of the performance of DWRA and DWBA: the errors for reconstruction with DWBA(solid) and DWRA(broken) are shown. Each plot contains results from three cylinders of radius 2λ,4λ and λ (a) Reconstructions of the object having a sinusoidal varying perturbation from noisy DWBA data having signal to noise ratios ranging from S/N = 5dB to S/N = 35dB. (b) shows the RMS error plotted versus the signal to noise ratio (a) Reconstructions of the object having a sinusoidal varying perturbation from noisy DWRA data having signal to noise ratios ranging from S/N = 5dB to S/N = 35dB.(b) shows the RMS error plotted versus the signal to noise ratio Weak scattering object embedded in a supporting potential whose dimensional information and refractive index distribution is known. In this thesis we use cylindrical symmetric background media Ray tracing geometry within the a cylindrical symmetric media boundary Ray tracing result of a circular object with refractive index n > 5.4 Comparing the phase values generated by exact field calculation and ray raying approximation.the object is a simulated stepindex optical fiber. The Core size is 2λ,n =.22, The cladding size is 4λ,n =.22 (a) Object Profile (b) Ray tracing in circular background media; (c) Scattered fields: red represents the scattered fields generated by exact field calculation; blue o represents the approximated field generated by ray tracing algorithm... xiv

19 5.5 Comparing the scattered field phase distribution generated by exact field calculation and ray raying approximation: a weak scattering background. The object is a simulated step-index optical fiber. The Core size is 2λ,n =.22, The cladding size is 4λ,n = Comparing the scattered field phase distribution generated by exact field calculation and ray raying approximation: a strong scattering background. n =.42 and other parameters remain the same as the previous two experiments; Optical step index fiber simulated reconstruction by ray tracing algorithm when the detector plane is uniformly sampled. (a) Refractive index distribution of the disturbance; (b) Reconstruction result by the ray model based DWBA; (c) Diagonal values of the reconstructed refractive index Optical graded index fiber simulated reconstruction by ray tracing algorithm when the detector plane is uniformly sampled. The center refractive index is n =.22 and the size of area is 2λ. The rest is the same as the previous example Two shapes disturbance reconstruction results. The refractive index of both shapes is n = Simulated reconstruction result of five discs disturbance by ray tracing inverse scattering algorithm. The refractive index, from smallest to largest: Experimental setup of digital holographic microscopy A picture of experimental setup of digital holographic microscopy Flowchart of data acquisition system for digital holographic interferometer A picture of experimental setup of PZT control system An example of data processing program interface created by MATLAB R GUI Thin object imaging: the upper image illustrates the traditional imaging method while the lower image shows the numerical imaging process xv

20 6.7 Slit imaging via digital holographic microscopy. (a) interferogram; (b) amplitude of scattered field at image plane; (c) transmission function generated via wave expansion back-propagation method; line-per-inch Ronchi-ruling imaging via digital holographic microscopy. (a) Transmission function generated via wave expansion back-propagation method; (b) Correctional plot of transmission function along the y direction; (c) Three dimensional presentation of Ronchi-ruling s transmission function line-per-inch Ronchi-ruling imaging via digital holographic microscopy. (a) Transmission function generated via wave expansion back-propagation method; (b) Correctional plot of transmission function along the y direction; (c) Three dimensional presentation of Ronchi-ruling s transmission function Three dimensional representation of line-per-inch Ronchiruling via digital holographic microscopy The DHM image of micro lens array (a) Hologram (b) Scattered field intensity only (c) Scattered field phase (d) 3D illustration of scattered field Transmission function of a cotton stem cell; Left: DHM image; right: conventional microscopic image Transmission function of a tendon; Left: DHM image; right: conventional microscopic image DHM images of a tendon. (a) intensity distribution of the scattered field of tendon at the camera plane; (b) hologram (c) Intensity of the back-propagated scattered field; (d) Phase of the back-propagated field The fields propagating in the interferometer; Five holograms generated by phase shifting holography for SMF- 28 fibers Phase calibration map for a SMF-28 fiber holographic imaging. A small area on the map is chosen for calculation; xvi

21 6.8 The averaged intensity and phae distribution of the object beam complex field ψ raw over the camera aperture plane for repeated measurements D illustration of the averaged phase distribution of the object beam ψ raw over the camera aperture plane for repeated measurements The averaged intensity and phase distribution of the incident field distribution over the camera aperture plane for repeated measurements D display of the modified phase distribution of fiber field Normalized intensity of object leg complex field Normalized intensity distribution of the scattered field component of the total field generated by the fiber over the detector plane. The dotted line represents simulated data obtained using the Bessel function expansion and the solid line is the actual experimental data Reconstruction results obtained using the DWBA based FBP algorithm. The ideal refractive distribution of the SMF-28 fiber is shown by the solid line while the dot-dash line represent the reconstruction result from the simulated scattered field data and the short dash circle lines shows the reconstruction from the experimental data xvii

22 Chapter Introduction Optical microscopy has been used to observe the micro-world for more than a hundred years. A traditional microscope is composed by an incoherent illuminating component that illuminates a semi-transparent object, an objective lens that forms an enlarged image of the object and an eyepiece that allows for observation by the naked eye or forms an image on a plane. An example of such a system is illustrated in Figure (.a)[]. One limitation of a normal optical microscope is that only optically thin objects can be imaged. This is due to the fact that in the imaging process three-dimensional structure of an object is collapsed into a two-dimensional image with an attendant loss of structural information. A second limitation is that it is not capable of imaging of phase objects, i.e., objects which only affect the phase of the illuminating light rather than its intensity. Computational Microscopy is proposed to address the above two problems and yield a microscope that can image optical thick as well as phase objects. As illustrated by Figure (.b), the sample is under coherent illumination and the intensities of the diffracted light are recorded by a detector system (CCD camera in most cases) in Computational Microscopy. The recorded information is then numerically processed by computer to retrieve the complex field scattered by the object from which the quantitative properties of the object are reconstructed. The two dimensional optical transmission function of a thin object is directly calculated by a back-propagation algorithm. Three dimensional imaging via Computational Microscopy is performed by illuminating the semi-transparent

23 2 object from multiple directions, back-propagating the scattered fields from each direction to the near field and then solving the inverse scattering problem via techniques of Optical Diffraction Tomography(ODT)[2]. There is no need of expensive focusing lenses as required in a normal microscope since the actual image formation process is performed numerically. The images thus are free of aberrations introduced by imperfections of optical lens. The research and development of Computational Microscopy involves two aspects, phase retrieval of complex field and tomographic reconstruction algorithm study. Both of them are explored and discussed in this thesis.. Previous work.. Phase and 3D imaging modalities A widely used analog technique for imaging phase objects is the Zernike method[3, 4], which is referred to as optical phase contrast microscopy in the literature [5]. Zernike based microscopes are effective in producing high-contrast images of transparent specimens, such as living cells (usually in culture) and fibers, but do not provide quantitative information of the specimen. It is due to the fact that the relationship between the irradiance of light and phase of the image field is generally nonlinear. Another popular phase imaging technique is Differential Interference Contrast(DIC) method[6]. It has several strength such as high resolution, light efficiency and strong highlighting of phase features. In DIC, phase signal is mixed with amplitude signal and not entirely linear. It is a very useful biomedical imaging strategy and has been actively researched[7]. The Confocal microscopy [8] provides reliable three dimensional information of semi-transparent object by using a spatial pinhole to eliminate out-of-focus light or flare in specimens that are thicker than the focal plane. It is mostly used to obtain 3D reflective image and problems of obtaining transmission images remain. This optical imaging technique may find applications in life sciences and semiconductor inspection. Optical near-field microscopy[9] is also potentially

24 3 useful for 3D imaging by exploiting the properties of evanescent waves. However, its application is limited due to the requirements of manipulating a probe in the nanometer vicinity of the object. Optical waves diffract (or scatter) when passing through an object. The process of undoing the effects of diffraction so as to reconstruct an object from its diffracted wave is called Optical Diffraction Tomography(ODT). It consists of illuminating the sample from many directions and collecting diffracted field from each direction. The ODT relies entirely on a numerical procedure to retrieve the three-dimensional map of refractive index of sample. Different algorithms and numerical methods have been developed in literature[ 3]...2 Phase retrieval Computational imaging of two and three dimensional objects is based on the wave equation that governs the scattering process and thus requires the knowledge of both the intensity as well as the phase of the scattered fields. It is well known that direct measurement of the phase of an optical wave is theoretically impossible. There has been extensive discussion in the literature on this celebrated phase problem[4, 5], i.e., loss of phase information from a physical measurement, and several solutions have been given. One approach for solving the phase problem is to employ either a prior information about the wave field or multiple intensity measurements to recover the field from its Fourier Transform[6 9]. The advantage of this method is that only intensity measurements are required and the experimental setup is simple. However, these schemes are generally iterative and there is no guarantee of a unique or a convergent solution of the iteration process required by the retrieval algorithms. A second approach ideally suited for computational microscopy is to estimate phase information from off-axis or in-line holograms. In off-axis holography, the phase of the scattered field is extracted from holograms by standard holography reconstruction but a reference wave is needed for holograms recording [2 26].

25 4 The resolution and the image area for reconstruction is limited in this technique because of the off-axis setup. In the earlier 9 s, Devaney and Maleki[, 27 3] proposed a new procedure which employs in-line (Gabor) holograms and records two in-line holograms at different distances from a weak scatterer. The algorithm is non-iterative and hence does not have the convergence problems of iteration and the computational cost is much less than that of iterative algorithms. Aside from the advantages of simple experimental setup, the in-line holography suffers from twin-image artifacts[3] if the holograms are not recorded in the far field of the scatterer. A third approach of measuring phase is the phase-shifting holography technique [, 25, 3 35], which is free from the artifacts of in-line and off-axis holography and no iteration is needed for numerical calculation. There are several different methods of introducing the phase shift between the object wave and the reference wave. The most popular one is mechanical translating a reference mirror by the movement of a stepping motor or a piezoelectric transducer[34 37]. Another widely used technique is phase retardation, which introduces the phase shift between the two arms of the interferometer based on polarization optics, such as employed in references [3, 38] and [39]. In [3] a quarter wave plate is placed in the reference arm and one has to rotate it manually between two measurements of holograms. In reference [38] a phase modulator introduces phase variation based on the Electro-optic effect. This method is ultrafast and thus insensitive to environmental noise. However, the non-linear effect of the phase modulator is noticeable for a piece of the material that is large enough for the application of DHM. In [39] the authors proposed a quadrature detection technique to reconstruct the complex field by recording 4 holograms simultaneously. There is a π/2 phase shift between each hologram and it is not sensitive to environmental noise due to the synchronization of four CCD cameras. In addition, the setup collect the diffracted from image plane which has obvious advantage in numerical aperture[4].

26 5..3 Digital recording and numerical reconstruction of holograms Holography has advanced to the digital era 5 years after its invention by Dennis Gabor[4, 42]. A big step for digital recording and processing of holograms was made in [23] and [24]. In digital holography there is no need of wet processing to record the holograms and it is convenient to evaluate the properties of the specimen structures quantitatively. Since the mid 9 s, digital holography has found many applications such as in optical metrology[43 45], encryption of information[46, 47] and microscopy[48]. The application of digital holography in microscopy is the gaol of this thesis. Fundamental work in the field was introduced in [49]. There are basically two ways to implement digital holographic microscopy[5]. One uses direct holography reconstruction at the resolution of the pixel size of the CCD camera. The other employs phase-shifting interferometry techniques. The current active research in digital holographic microscopy are in the following areas Improvements of experimental techniques [35, 5 54]; Short coherence length source [55 6]; Noise and abberation compensation [6 67]; Application in 2D biological sample imaging [68 72]; Frequency and numerical analysis [73 82]; Microstructure measurements [83 86]; Measurement of refractive index distribution within semi-transparent area[87, 88]...4 Inverse scattering in optical diffraction tomography All ODT is usually formulated as a linearized inverse scattering problem where the wave propagation and interaction of an incident wave with a scattering

27 6 potential is governed by the Lippmann Schwinger Equation(LSE)[89]. The development of a stable and reliable solution to the inverse scattering problem has always been challenging and the subject of a large number of studies. Fundamental to conventional diffraction tomography is the Fourier Diffraction Theorem, which states that, within a weak scattering model, the Fourier transform of the measured forward scattered data is related to the Fourier transform of the object. This theorem is based on the first Born Approximation and is valid for weak scattering objects. It was first discussed by Wolf[9] and later employed to derive the Filtered back-propagation algorithm by Devaney [9, 92] by applying the angular spectrum expansion[93] of the Green function to the Helmholtz equation [94]. It was also pointed out that the Rytov approximation can be used to linearize the wave equation in a similar way[95]. The validity of diffraction tomography based on the first Born and Rytov approximations was compared in [96]. Fourier based methods are beautifully described by closed form algorithms and take advantages of the Fast Fourier Transform(FFT). However, their performance is limited by the underlying weak scattering approximation and a dense data set is required for high quality reconstruction[97, 98]. Iterative methods, such as the Born Iterative Method(BIM)[99], Distorted Born Iterative Method(DBIM)[] and other similar algorithms [ 3], have been developed to give inversions of two-dimensional inverse scattering problems when the Born and Rytov approximations break down. These methods, if convergent, give good inversions and have a wide application. However, they suffer from high CPU times and often may not converge to the correct solution because of a local minimum. A third approach to solve the inverse scattering problem is the Moment method[4, 5]. It also works beyond the limitation of the first Born approximation and has a wide applicability in different background media. The disadvantage of the moment method is the possible inversion of large matrices and the inversion may be sensitive to the positions of detectors and measurement accuracy. In a recent paper[6] the inverse scattering problem for scalar wave potential scattering at fixed temporal frequency was addressed within the context of

28 7 the distorted wave Born approximation (DWBA). In that paper a scattering object compactly supported in some volume V was assumed to be embedded in a known background medium and interrogated in a suite of scattering experiments employing point sources and receivers arbitrarily distributed outside the support of the object. It was shown that within the DWBA the scattered field data so acquired uniquely determines a projection of the object s scattering potential onto a certain subspace of the Hilbert space of scattering potentials compactly supported within the scattering volume V. Using this result a reconstruction algorithm that generates a least squares, pseudo-inverse of the scattering potential was derived, implemented and tested in a simple computer simulation study for the cases of a perfect homogeneous background medium as well as a homogeneous half-space with a perfectly reflecting plane boundary. This is directly relevant to ODT since it is a standard procedure to embed the scattering object in a test tube filled with a fluid whose index of refraction is closely matched to the nominal index of refraction of the object. Because of this index matching the DWBA can be employed to accurately model the scattering experiments in digital holographic microscopy and the DWBA algorithm developed in references[6] and [7] can be employed to reconstruct the object from the scattered field data. In [8] and [9] some preliminary research results are presented..2 Research reported in this thesis In this thesis, the specific implementation of Computational Microscopy is Digital holographic microscopy(dhm), which employs digital holography for phase retrieval and the distorted wave approximation inverse scattering algorithms for 3D reconstruction. Figure (.2) illustrates the framework and subsystems of the DHM built and studied in this thesis. The details of the novel research work for each subsystem is summarized as following: Digital Holography In addition to the traditional microscopy setup, DHM technique introduces a reference wave during the data acquisition process and uses a CCD camera to record the interference pattern (hologram[4, 42]). A DHM system based on Mach-Zender interferometer was built in

29 8 this research to provide experimental data. Different phase shifting methods are compared and noise filtering for experimental data is investigated. Phase retrieval The performance of the existing algorithms [32] [] are compared and reformulated into complex variables format. Based on the comparison results, a phase retrieval strategy which has the best performance for the current DHM system is proposed and tested on experimental data. The input of this subsystems is a number of N holograms and the output of the system is the complex wave field of the sample arm. Numerical back-propagation algorithm The numerical back-propagation transforms the complex wave field at one position in space to another position. For thin objects the output of the transformation at object plane will give the optical transmission function of the object. For thick objects the back-propagation requires certain approximations to be valid. There are basically three methods of doing the back-propagation, respectively Rayleigh Sommerfeld formulas, Angular spectrum expansion and Fresnel approximation [2] [93]. This thesis will compare and analyze the performance of each method in terms of resolution, computational complexity and noise propagation. The input and the output of the subsystem is the complex wave field over two planes along the light propagation path in the space. Optical diffraction tomography Optical diffraction tomography techniques are used to compute the three dimensional refractive index distribution of optical thick objects. It is the most emphasized subarea in this thesis. In the theoretical analysis an inverse scattering algorithm for the general geometry of optical diffraction tomography is introduced and reformulated for different background cases. The free space background case is thoroughly studied for both near and far field [8]. If the inversion algorithm is derived from a linearized approximation to the Lippmann Schwinger equation [89], there are two approaches, namely the Distorted Wave Born Approximation(DWBA) and the Distorted Wave Rytov Approximation(DWRA)[]. Both approaches are discussed for cylindrical symmetric backgrounds [2] [3] [4] [5]. Validity of ODT

30 9 inverse scattering algorithms within the DWBA and DWRA is compared. In addition, a novel algorithm formulated from ray model by ignoring the diffraction effect of the weak scattering objects is also developed in this thesis. Computer simulation results are given for all new algorithms and experimental results of constructing a step index fiber are reported. The structure of the thesis is arranged as follows. In Chapter (2) the research results for phase retrieval and wave back-propagation are presented. In Chapter (3), a generalized diffraction tomography inverse scattering algorithm approaches is introduced and formulated for reconstructing objects in free space. In Chapter (4), the theoretical analysis and simulation of inverse reconstruction for an object imbedded cylindrical background is discussed and the performance of DWBA and DWRA is compared. In Chapter (5) the inverse scattering based on ray tracing is introduced and simulation results are presented. Chapter (6) is devoted to the experimental setup and results of digital holographic microscopy. Finally, Chapter (7) contains a summary of the results and conclusions obtained in the thesis and suggestions for future work. (a) Figure.: Illustration of traditional and computational microscopy for the imaging of thin objects. (a) traditional microscopy which is illuminated by incoherence light; (b) computational microscopy illuminated by coherent light. Object transmission function is represented by T (ρ) and the image function is described by T (ρ). (b)

31 Figure.2: Digital holographic microscopy subsystems and their corresponding outputs. The data acquisition process is accomplished by phase-shifting digital holography.

32 Chapter 2 Phase Retrieval and Wavefront Back-propagation This chapter describes how to retrieve the complex wave field generated by a scattering object under plane wave illumination in digital holography and how to reconstruct the complex scattered wave at any arbitrary plane given the distribution over one plane. It covers the theoretical part and the description of the real experimental setup of digital holography will be presented in Chapter (6). 2. Phase shifting digital holography There are two possible digital holographic configurations implemented using an interferometer: via a Michelson and Mach Zehnder interferometer, as illustrated in Figure (2.). In both configurations a set of controlled phase shifts are introduced between the reference wavefront and sample wavefront so that a set of interference patterns are then generated. An example of phase shifting holograms is illustrated by Figure (2.2), where the phase shift value between two consecutive holograms is π/2. In mathematical terms, the wave fronts of the reference and the object beam are expressed as ψ r (x, y, n) = A r (x, y) exp ı[φ r (x, y) + δ n ] ψ o (x, y) = A o (x, y) exp ıφ o (x, y) (2.a) (2.b)

33 2 with A r and A o to be the amplitudes of the reference and object wave. δ n stands for the phase shifting value of the nth hologram and is assumed to be the same for all points on the aperture plane. The resulting interferogram intensity patterns at the measurement plane are given by I n (x, y) = ψ o (x, y) + ψ r (x, y, n) 2 (2.2) = A 2 o + A2 r + A oa r exp ı[φ(x,y)+δn] +A o A r exp ı[φ(x,y)+δn] (2.3) I n (x, y) are digital holograms which can be recorded by a CCD camera and φ(x, y) = φ o (x, y) φ r (x, y). Define three unknowns x,x and x 2 : x = A 2 o + A 2 r x = A o A r exp ı[φ(x,y)] x 2 = A o A r exp ı[φ(x,y)] (2.4a) (2.4b) (2.4c) These unknowns can be solved if we have more than three measurements (n > 3). A r and A o are related to x,x via A r + A o = x + 2 x A o A r = ± x 2 x (2.5a) (2.5b) The phase of the complex object wave is solved from cos(φ(x, y)) = x + x 2 2 x (2.6) The quality of the data is evaluated by the modulation matrix γ(x, y), which is defined by γ(x, y) = 2 x x (2.7) A data modulation near one is good and a low modulation is not desirable. Data points with low modulation below some threshold will have insufficient signal to noise ratio and should be dropped for further processing. There are several frequently used algorithms for deriving the object complex

34 3 field from a series measurements I, I,, I n introduced in []. In this thesis these basic algorithms are adapted to digital holographic microscopy applications and a strategy which is proven effective in providing high precision phase data for further numerical reconstructions is proposed. 2.. Constant phase shifting algorithms Two phase-shifting method The intensities of the reference and object waves can be directly measured by CCD camera without the other beam. The unknown x is removed and only two holograms with phase shift values and δ are needed for phase retrieval. A more detailed discussion of this algorithm can be found in [3]. If the object field intensity is represented by I o and reference field represented by I r, the scattered complex field and its complex conjugate are the solutions to the linear equations, [ ] I I o I r = [ ] [ I r I 2 I o I r exp (ıδ) exp ( ıδ) where δ can be any value except for π/2. ψ o ψ o ] (2.8) Three phase-shifting method There are three unknowns in Eq.(2.2), hence at least three measurements are needed before one can determine the complex field of the object wave. For the measurement case when n = 3, I I I 2 = exp ı[δn] exp ı[ δn] exp ı[2δn] exp ı[ 2δn] x x x 2 (2.9) The object intensity and phase value is derived from Eq.(2.5a), Eq.(2.5b) and Eq.(2.6).

35 Phase shifting calibration and Least-Squares algorithm In this section we assume that the phase shift value is unknown and not uniform across the illumination aperture. This is usually the case when there is a change in the slope of the PZT displacement versus voltage curve or the beam is converging/ diverging. Therefore our data processing can be divided into two steps: phase shifting calibration and least-square estimation of the complex field. For phase shift value calibration the Hariharan Algorithm is used in this thesis, which will be implemented via phase shifting holography. This method uses five holograms and initially assumes a linear phase shift of α between different holograms: δ n = 2α, α,, α, 2α; n =, 2, 3, 4, 5 (2.) The intensities of five interferograms are written as I (x, y) = I (x, y) + I (x, y) cos [φ(x, y) 2α] I 2 (x, y) = I (x, y) + I (x, y) cos [φ(x, y) α] I 3 (x, y) = I (x, y) + I (x, y) cos [φ(x, y)] I 4 (x, y) = I (x, y) + I (x, y) cos [φ(x, y) + α] I 5 (x, y) = I (x, y) + I (x, y) cos [φ(x, y) + 2α] (2.a) (2.b) (2.c) (2.d) (2.e) with I = A 2 o + A 2 r and I = A o A r. The solution for α at each point across the hologram plane from the above five equations is [ ( )] α(x, y) = cos I5 I (2.2) 2 I 4 I 2 Figure (2.3) illustrates the statistics of phase shift values for the measurement points in one measurement when there is no object placed in the interferometer. The hardware system is then adjusted until the histogram of phase shift value is narrowed to a desired value. It is used as a standard of evaluating the phase

36 5 shifting system as well. For simplicity and computational efficiency, the statistical mean of the phase shift values across the aperture plane is used for future calculation: δ n = E[α(x, y)] (2.3) The next step is to calculate the complex object field from the 5 holograms. The linear square algorithm is chosen to make the full use of the five holograms.it is found that the measured intensity at a given location on the hologram is a sinusoidal function of the phase shifts δ n computed in step with three unknowns. In particular we can rewrite Eq.(2.2) as I n = a + a cos(δ n ) + a 2 sin(δ n ) (2.4) where a = x, a = R(x ) and a = I(x 2 ) in Eq.(2.5). Solving a,a and a 2 is equivalent to finding a least-square fit for a given series of values I n, which is given by the following matrix equations a a a 2 = A (δ n )B(δ n ) (2.5) where A and B is A(δ n ) = N cos(δn ) sin(δn ) cos(δn ) cos 2 (δ n ) cos(δn )sin(δ n ) cos(δn ) cos(δn ) sin(δ n ) (2.6) sin 2 (δ n ) B(δ n ) = In In cos(δ n ) In sin(δ n ) (2.7) and all the summations are from to N. N is the number of hologram and equal to five here. Once the a, a and a 2 are determined at each measurement location, the unknowns in Eq.(2.4) x and x is specified by x = a, x = a ıa 2, x 2 = a + ıa 2 (2.8)

37 6 Hence the object and the reference complex fields ψ o and ψ r can be calculated from Eq. (2.5). However, to identify which of the two solutions to Eq. (2.5) is the object field ψ r, one can use either of the following two methods Compare the standard deviation of the 2D magnitude distribution of two fields and assign the higher one to the object field; Block the reference arm of the interferometer and record the object scattered field only; Since we assume the phase value of the reference field are all zeros at the detector plane, we simply assign the phase value difference between the two field to be the object phase field. 2.2 Wavefront back-propagation As illustrated in Figure (2.4), the complex wave function ψ(x, y, z) is determined at the plane z = z from phase shifting holography. This quantity is described as a two-dimensional matrix U(x, y). The numerical back-propagation is then used for calculating the complex wave function at the plane z =, represented by f(x, y ). The distance between the two planes is z. In this thesis we discussed three methods used for free space back-propagation of the complex wave field at the CCD camera plane to the plane of interest Rayleigh Sommerfeld back-propagation algorithm(rs) A freely propagating wave field in free space satisfies the three dimensional homogeneous Helmholtz equation [94], [ ] 2 x y z + 2 k2 ψ(r) = (2.9) where r = (x, y, z) is the position vector. The propagating wave ψ(r) satisfying the Eq.(2.9) and the Sommerfeld radiation condition [6] from an arbitrary data plane z = z can be computed using the well know Rayleigh Sommerfeld

38 7 formulas, which is given by ψ(r) = ds ψ(r ) z =z z z G +(r r ) z =z (2.2) where the minus sign is used for propagation into the r.h.s. with z > z and the plus sign for propagation into the half space where z < z. In these equations G + is the three-dimensional outgoing wave free space Green function G + (r r ) = exp ık r r 4π r r The normal derivative of the G + is found to be z G +(r r ) = [ exp ık r r ık z z 4π r r r r + z z ] r r 2 (2.2) (2.22) which when substituted into Eq.(2.2) yields the propagation formula ψ(r) = [ ds ψ(r exp ık r r ) z 2π =z ık z z z r r r r + z z ] r r 2 z =z (2.23) The propagated field is essentially spatially bandlimited to the wavenumber k so that the sampling interval is δx = λ/2 with λ/2 being the Nyquist sampling interval Angular spectrum expansion(ase) The complex wave function at the position z = z can be written as the superposition of plane waves propagating in different directions. Each diffracted plane wave component carries sample information at a specific spatial frequency. The propagation of the plane wave can be simply described as U(x, y, z ) = U(x, y, z 2 ) exp ık(z z 2 ) (2.24) Thus for the propagation of complex wave field in paraxial field, one is able to first decompose the wave into plane waves, propagate the plane waves to the designated position and finally combine them together. Described in the

39 8 language of mathematics, the amplitude of each plane wave is given by ˆψ(K) = drψ(r) z=z exp ık r (2.25) the output complex wave field at the plane z has the form ψ(r) = dk (2π) ˆψ(K) exp ık r + γ(z z 2 ) (2.26) with γ = k 2 K 2. In addition to the band limitation introduced by 2π/λ, the resolution of this algorithm is limited by the sampling frequency and the number of pixels on the detector plane Fresnel transform(frt) The propagation of waves in free space can be viewed as a transform in a linear system. The relation between the input function ψ(x, y, ) = f(x, y) and the output function ψ(x, y, z ) = U(x, y) can be described as U(x, y) = f(x, y )h(x x, y y )dx dy (2.27) with h(x, y) the transfer function of free space, which in the Fresnel approximation is h(x, y) = If we apply Eq.(2.28) to Eq.(2.27), U(x, y) = ı [ ] exp ( ıkz ) exp ık (x2 + y 2 ) λz 2z (2.28) ı [ exp ( ıkz ) f(x, y ) exp ı k ] ((x x ) 2 + (y y ) 2 ) dx dy λz 2z (2.29) It is easily reformulated to U(x, y) = U (x, y) ı k ] (x 2 + y 2 ) 2z [ exp ı k ] (xx + yy ) dx dy (2.3) z [ f(x, y ) exp

40 9 with U (x, y) = ı [ exp ( ıkz ) exp ı k ] (x 2 + y 2 ) ; (2.3) λz 2z Equation (2.3) is called the Fresnel Approximation. For digitizing Eq.(2.3) we make the substitution which transforms Eq.(2.3) to µ = x λz ; ν = [ U(µ, ν) = U (µ, ν) f(x, y ) exp ı k ] (x 2 + y 2 ) exp [ı2π(x µ + y ν)]dx dy 2z (2.32) with U (µ, ν) = y λz ı λz exp [ ıπλz (µ 2 + ν 2 )] by omitting the factor exp ( ıkz ) which is simply a constant phase shift. Equation (2.32) is in the form of an inverse two-dimensional Fourier Transform up to a spherical phase factor, and can be written in the form [ U(µ, ν) = U (µ, ν)f {f(x, y) exp ı k ] (x 2 + y 2 ) } (2.33) 2z which is called Fresnel Transform. A special property of the Fresnel transform is that the sampling interval on the observation plane increases linearly with the propagating distance. If the sampling interval at the original plane is δx, δy, due to the FFT we have that K x = k z z x, K y = k z z y so that δx = z z δk x = λ z z k Nδx δy = z z δk y = λ z z k Nδy (2.34a) (2.34b)

41 Discussion To compare the performance of all three algorithms, we first consider a freely propagating field in two dimensional space. At the data plane z = the field is a slit function which is sampled at an interval δx = λ/4. The width of the slit is 2λ. Figure (2.5) and (2.6)shows the simulated fields at the data planes z = λ and z = 2λ respectively. Both the real and imaginary parts are shown for the simulation results from the three methods. It is found that angular spectrum expansion works for short distance but will have aliasing effect in larger distance. The aliasing can be avoided by zero padding the original field, which increase the fourier transform window.for larger propagation distances where the Fresnel approximation is valid, the Fresnel transform approximates the RS Dirichlet propagation better, but not as good as angular spectrum expansion with zero padding. The exact RS formula is valid for all propagation distances in forward model but there is no solution for back-propagation in RS formula. Another method of evaluating the performance of algorithms is to compute their error rates. For the forward model, the error is defined as E = [w(n) RS(n)] 2 [RS(n)2 ] (2.35) where n is the index of each sampling point, w(n) is either the ASE or the FRT forward propagated fields. For the same slit function, we generated an forward propagation error map illustrated by the left figure of Figure (2.7). We may draw the same conclusion as we did in the previous section. The error of the back-propagation is defined as the normalized square difference between the back-propagated field and the original slit function. We calculated the diffracted fields of the same slit by RS method and back-propagated the fields by ASE, FRT and ASE with zero padding. The error plot of back-propagation is shown by the right figure of Figure (2.7). The ASE and ASE zero padding has the similiar performance is because that there is no aliasing effect for converging fields.

42 2 Figure 2.: Two configurations of digital holographic microscopy (up) Mach- Zehnder Interferometer (down) Michelson Interferometer Figure 2.2: An example of phase shifting interferometry: interferogram generated under four different reference phase values

43 22 4 x mean=.2673 std= rad Figure 2.3: Phase shift value distribution across the imaging aperture in one measurement. The distribution is obtained from a phase shifting holography experiments without any object placed in the interferometry configuration Figure 2.4: Coordinates system for digital holography. The object function at input plane is f(x, y) = ψ(x, y, ) and wave function at holography plane is U(x, y) = ψ(x, y, z )

44 λ λ λ λ Figure 2.5: Simulated real and imaginary parts of the propagated fields of a one dimensional slit at data plane z = λ. In the figures, the blue solid lines represent the field calculated by RS integral, which is shown in both left and right figures. The dashed green line shows the result of ASE method and the red dots illustrates the result by ASE method with zero padding. And the purple dot shows the results of Fresnel transform. The top two figures illustrate the real part and the bottoms are for the imaginary parts

45 24.5 RS ASE ASE zero padding.5 RS FRT λ λ λ λ Figure 2.6: Simulated real and imaginary parts of the propagated fields of a one dimensional slit at data plane z = 2λ. In the figures, the blue solid lines represent the field calculated by RS integral, which is shown in both left and right figures. The dashed green line shows the result of ASE method and the red dots illustrates the result by ASE method with zero padding. And the purple dot shows the results of Fresnel transform. The top two figures illustrate the real part and the bottoms are for the imaginary parts

46 Figure 2.7: Wave propagation error of different methods. Left: error plots of the forward model; right: error plots of back-propagation model. 25

47 Chapter 3 Scattering Theory Formulation of Optical Diffraction Tomography This chapter presents the wave theory underlying optical diffraction tomography and new inverse scattering algorithms for reconstructing three dimensional refractive index distributions of semi-transparent objects. The Lippmann Schwinger integral equation [89] underlying the propagation and scattering of a probing wave in a penetrable medium is presented and specialized to Optical Diffraction Tomography. The role of multiple scattering is discussed and the distorted wave Born [89, 7] and Rytov approximations [] that account for multiple scattering within the ODT measurement configuration are presented. The reconstruction algorithms for weak scattering object embedded in free space are introduced in this chapter as well. 3. General introduction 3.. Problem formulation We consider a scattering medium that consists of an unknown scatterer embedded within a known background as illustrated in Figure (3.). The medium is illuminated by a sequence of plane waves having different angles α and the scattered field is measured by a CCD camera placed l = l away from the center 26

48 Figure 3.: Schematic illustration of scattering phenomina in optical diffraction tomography: a weak scatter embedded in a known compact supported V. The object coordinates x y and illumination coordinates ξ η 27

49 28 of the scatterer for each illuminating plane wave. We will work entirely in the frequency domain and limit our attention to optical wave fields. For a single frequency plane wave a scalar wave model can be employed and one finds that the field generated at any fixed frequency in any given experiment must satisfy the Lippmann Schwinger equation [89] ψ(r, α) = ψ (in) (r, α) + d 3 r G b (r, r )O(r )ψ(r, α). (3.) V The first term on the r.h.s. of the above equation represents the incident wavefield that is parameterized by α and that propagates in the known background medium and the second term represents the scattered wave that is generated by the interaction of the incident wave with the embedded scatterer. r, r represents the coordinate vectors of the detector and the scatterer respectively. Here, G b (r, r ) is the background Green function and O(r) is the scattering potential ( object profile ) of the embedded scatterer, assumed to be compactly supported in V, and is the quantity to be determined from the scattered field data. When the background is free space, O(r) is related to the scatterer s refractive index by O(r) = k 2 [ n 2 (r)] (3.2) where k represents the wavenumber of free space. We have suppressed the frequency variable ω in the arguments of all the field variables since we will work exclusively in the frequency domain throughout the thesis. By the assumption that there is no z variation of the object, the problem is simplified to be twodimensional. The Lippmann Schwinger integral equation Eq (3.) defines a non-linear mapping between the object profile O(r) and the field and in order to generate useful and computationally efficient algorithms for the inverse scattering problem it is customary to linearize this mapping. There are generally two methods for linearization, which are known as the Distorted Wave Born Approximation (DWBA) and Distorted Wave Rytov Approximation (DWRA).

50 Distorted Wave Born Approximation The Distorted Wave Born Approximation linearization is obtained by simply replacing the total field ψ(r, α) in the integrand of Eq (3.) by the known incident field ψ (in) (r, α) propagating in the background thus generating a linear mapping between O and ψ which can be inverted using linear schemes. The DWBA thus takes the form ψ (s) (r, α) = d 2 r G b (r, r )O(r )ψ (in) (r, α). (3.3) V The DWBA Eq (3.3) will be accurate only if the scattered field as represented by the second term on the r.h.s. of Eq (3.) is small compared to the incident field. For the normal Born approximation (homogeneous background) to be valid, it is known that the change in phase between the incident field and the wave propagating through the object be less than π, which can be expressed mathematically as [98] rn δ < λ (3.4) 4 where r is the effective radius of the supporting volume and n δ is the average refractive index difference between the object and the background. Although there are no known necessary and sufficient conditions for the DWBA case, it is certainly necessary that the object profile O be point-wise small relative to the square of the background wavenumber k 2. This is then translated into the obvious requirement that the known background be point-wise as close as possible to the actual scattering object being interrogated Distorted Wave Rytov Approximation The Distorted Wave Rytov Approximation is explored to get around the breakdown of the DWBA for large extended objects in which the scattered field continually grows in amplitude as the incident wave propagates through the sample. DWRA is a linearization of the mapping between the object profile and the complex phase of the total field. If the incident and total field are represented in

51 3 their complex phase forms, ψ (in) (r, α) = e ıkw (in) (r,α) ıkw (r,α) ψ(r, α) = e we find that the phase W (r, α) also admits a perturbation expansion similar in form to the Born expansion W (r, α) W R (r, α) = W (in) (r, α) + V d2 r G b (r, r )O(r )ψ (in) (r, α) ıkψ (in) (r, α) (3.5) where the subscript R on the phase indicates that this is the Rytov approximation to the actual (exact) phase. The numerator of the second term in the above equation is recognized as being the Born scattered field as represented by the second term in Eq (3.3) so that both the Born and Rytov scattering models have the same overall functional form. However, they have a totally different range of validity. The Rytov model is superior to the Born model for large extended samples and the necessary condition for the validity of the DWRA is[8] δn < [ φ s λ] 2 (3.6) 2π which means that the change in the scattered phase φ s over one wavelength is more important than the total change of the phase Inverse scattering within the Distorted Wave Approximation Within the DWBA and DWRA the scattered field data in Eq (3.3) and Eq (3.5) uniquely determines a projection of the object s scattering potential O(r) onto a certain subspace of the Hilbert space of scattering potentials compactly supported within the scattering volume V. For each incident angle α q and a point r on the detector, Eq (3.3) and Eq (3.5) is simplified to D n = d 2 r G b (r, r )O(r )ψ (in) (r, α) = P O (3.7) V

52 3 where D n is ψ (s) (r, α) in DWBA and ıkψ (in) (r, α)(w R (r, α) W (in) (r, α)) in DWRA. n is the index of the detector and r is dependent on n. Define a the projection operation P P = d 2 r πn (r ) (3.8) V with πn(r, r ) = G b (r, r )ψ (in) (r ) (3.9) and n =, 2,..., N α N d, where N α is the number of incident wave directions and N d is the number of detectors. The set of Eq (3.9) can be inverted using either the Singular Value Decomposition (SVD) [7] or the generalized linear inverse method [6]. SVD inversion Using the singular value decomposition(svd) a reconstruction algorithm that generates a least squares, pseudo-inverse of the scattering potential can be derived. More specifically, since the data D n only determines the projection of O into N = N α N d sets πn, it follows that a minimum L2 norm solution of the inverse scattering problem within the DWBA and DWRA is then given by the sum N Ô(r) = C n π n (r) (3.) n= where the coefficients C n are solutions to the coupled set of equations N D n = C n < π n, π n > Hγ, n =,..., N (3.) n = which simplifies to the form ΠC = D (3.2) Here Π is a N α N d matrix having components < π n, π n > Hν.The matrix Π is easily shown to be Hermitian and non-negative definite and thus has the diagonal representation Π = UΛU (3.3)

53 32 The pseudo-inverse solution of C n is found to be given by Ĉ = UΛ + U D (3.4) Point Spread Function The point spread function(psf) is a useful tool to evaluate the performance of reconstruction algorithms. From above derivation, we find that if D in Eq (3.4) is replaced by Eq (3.7) and the result is applied to Eq (3.), we obtain Ô(r) = d 3 r O(r )π (r )U Λ + U T π(r) = d 3 r H(r, r )O(r ) (3.5) where we have defined the column vector π = [π, π 2,, π N ] T. Hence the PSF of the DWBA inverse algorithm is found to be [6] H(r, r ) = π (r )U Λ + U T π(r) (3.6) The PSF of the algorithm is the reconstruction image of a point scatterer located at the point r and a function of the background Green function and scattering geometry as well as the location of the point scatterer. It is also a function of the number of incident waves and detector elements. The closer the reconstruction image to the point scatter, the better the performance of the algorithm Discussion The inverse scattering algorithm introduced in this chapter will be referred as DWA-ODT algorithm hereafter in this thesis. We mention that the inversion returned by the DWBA/DWRA algorithm will be guaranteed to have the correct object support due to the fact that all the spanning functions π n (r) vanish identically outside the known object volume V. In many applications of Optical Diffraction Tomography the object support is known quite accurately and should be employed as a constraint to improve the quality of the reconstruction algorithm. This is an advantage of this algorithm that is not shared by the Filtered Back-propagation(FBP) [9] algorithm.

54 Inverse problem in free space 3.2. Problem formulation When the background is taken to be uniform; e.g., free space, the above DWBA reduces to the normal Born approximation [89]. The normal Born approximation has the advantage that the free space Green function is known in a simple closed form which results in a beautiful and simple inverse scattering theory known as diffraction tomography [98]. In this special case the Lippmann Schwinger equation takes the form Ψ (s) (r, α) = d 2 r G (r, r )O(r )A e ıkα r (3.7) V where O = k 2 k2 (r) and the G is the free space Green function which in two space dimensions is given by G (r, r ) = ı 4 H (k r r ) (3.8) We shall employ two rectangular Cartesian coordinate systems relative to which we can represent O(r). The first system is defined by an arbitrary reference axis x, such as indicated in Figure (3.). The center of (x, y) coordinates is located in the object. The second system (ξ, η) is obtained from the (x, y) by a counterclockwise rotation of the angle α with α being the angle subtended by the unit propagation vector s, which means s = ˆη (3.9) and the x axis. The η axis is aligned with s. The coordinates of the pixels on the CCD in the defined system are: r = ξ ˆξ + lˆη, d ξ d (3.2)

55 34 with 2d the width of CCD detector. The transformation between the two coordinates system is: ˆx = ˆξ cos θ ˆη sin θ ŷ = ˆξ sin θ + ˆη cos θ (3.2a) (3.2b) and ˆξ = ˆx cos θ + ŷ sin θ ˆη = ˆx sin θ + ŷ cos θ (3.2c) (3.2d) In practice that the sample is rotated, which is mathematically equivalent to keeping the sample fixed and rotating the incident plane wave direction and camera axis. We will use the second interpretation here. Near field In the near field the Green function of the background takes the form of zero order Hankel function. By applying Eq (3.8) and plane wave incidence to Eq (3.9), the expansion functions π n take the form { G π n (r, w) = (r, r)e ıks r if r v (3.22) else wise and the projection is defined as P = d 3 r πn(r ) (3.23) Accordingly, the matrix Π is constructed as v Π =< π n, π n > = d 2 r[g (r, r)e ıks r ][G (r, r)e ıks r ] (3.24) In computer simulation, Π is a matrix constructed according to table (3.). The object profile is readily solvable from Eq (3.) and Eq (3.4).

56 35 n θ {}}{ θ 2 {}}{ θ N {}}{ n θ { ξ ξ ξ ξ ξ.. ξ θ 2 { ξ ξ ξ ξ ξ.. ξ θ N { ξ ξ ξ ξ ξ.. ξ Table 3.: Π matrix construction of near field Far field Calculation of Π from Eq (3.24) is computationally intensive in general. In this section, we propose a computationally efficient method appropriate for the far field data (l λ), which is typical for most optical diffraction tomography application. The Green function in free space admits the angular spectrum representation G (r, r ) = ı dk 4π γ eı[k(ξ ξ )+γ(η η )] (3.25) where r = (ξ, η ) and r = (ξ, η ), γ = { k2 K 2, K k; ı K 2 k 2, K > k. (3.26) When K k, the plane wave are of ordinary type. However,for K > k, γ becomes imaginary and the waves become evanescent waves. Evanescent waves usually are of no significance beyond a few wavelengths from the scatterer. Therefore Eq (3.25), limited to only homogeneous plane waves, is a good approximation to Green function in ODT. Substituting Eq (3.25) into Eq (3.7),

57 36 the scattered field is then given by Ψ (s) B (r, s ) = d 2 r G (r, r )O(r )e ıks r V [ = d 2 r ı k ] dk 4π γ ei[k(ξ ξ )+γ(η η )] O(r )e ikη k (3.27) we can transform Eq (3.27) to the form Ψ (s) B (r, s ) = ı k dk 4π k γ ei(kξ +γη ) = ı 4π = 2π k k k k We have used the fact that dk γ eı(kξ +γη ) Õ(k(s s )) d 2 r O(r )e ı h Kξ +(γ K)η i V }{{} Õ(k(s s )) D(s s )e ıkξ dk (3.28) Kξ + γη = ks r (3.29) and defined The following Fourier transform pair exists: D(s s ) = 2iγ eiγη Õ(k(s s )) (3.3) DFT(Ψ (s) B (r, s )) = D(s s ) (3.3) IDFT(D(s s )) = Ψ (s) B (r, s ) (3.32) It leads to the famous Fourier Diffraction Theorem [98], which states that when an object is illuminated with a plane wave as shown in Figure (3.), the Fourier transform of the forward scattered field Ψ (s) B (r, s ) measured on line l ˆη gives the values of the 2-D transform Õ along a semicircular arc in the frequency domain k (s s ).

58 37 Now Õ(k(s s )) is found to be the linear projection of O(r) from Hilbert space to Fourier transform space. A similar definition of projection as in Eq (3.23) is defined as and π n is π n (r) = P = { v d 2 r e ık(s s ) r e ık(s s ) r if r v else wise (3.33) (3.34) Following the same procedure as described in the previous section, the matrix Π is defined according to table (3.2). n θ {}}{ θ 2 {}}{ θ N {}}{ n θ { K K K K K.. K θ 2 { K K K K K.. K θ N { K K K K K Table 3.2: Π matrix of Far field.. K It is noticed that Π has a closed form and the computation efficiency is much better than that of general case. To facilitate computation, the inner product k (s s ) is expanded as following, k(s s ) = Kξ + (γ K)η = K(x cos θ + y sin θ) + (γ K)( x sin θ + y cos θ) = [K cos θ (γ K) sin θ]x + [K sin θ (γ K) cos θ]y (3.35) let define [K cos θ (γ K) sin θ]x = ss x (3.36a)

59 38 [K sin θ (γ K) cos θ]y = ss y (3.36b) The Π integral reduces to Π = dx dy e ı(ssx ss x ) e ı(ssy ss y ) (3.37) One can get the closed form of this integral using the following procedure. for ss x ss x,ss y ss y, then Π = ı ss x ss x e ı(ssx ss x ) xmax x min ı ss y ss y e ı(ssy ss y ) ymax y min 2. for ss x = ss x and ss y ss y, ı Π = (x max x min ) ss y ss y e ı(ssy ss y) ymax y min 3. for ss x ss x and ss y = ss y, ı Π = (y max y min ) e ı(ssx ss x) xmax ss x ss x min x 4. for ss x = ss x and ss y = ss y, Π = (x max x min ) (y max y min ) Thus the computational efficiency of calculating Π matrix is a lot better than that of general form. However the expansion substitution introduces a small difference between the r.h.s and l.h.s of Eq (3.27). This theoretical error is inevitable and might make the inverse problem unstable. To solve this problem usually regularization methods were applied[9][2] instead of simple SVD. Regularization In this section we focus on how to find the solution for Eq (3.2), i.e., calculating the inverse of matrix Π. The most popular tool is Singular Value Decomposition. However, it was found in simulations that the direct pseudo-inverse of the

60 39 Π is very sensitive to noise. A small disturbance to scattered field data leads to meaningless solution. The reason is that most of the singular values of Π matrix are usually very small. Figure (3.2) and Figure (3.3) illustrate two examples of singular values of Π matrix. The former is for near field data and the latter figure is generated from far field data. The two figures indicate an ill-posed inverse problem and hence regularization is necessary. In this work several regularization methods are applicable, namely SVD and Pseudo-inverse, Truncated Singular Value Decomposition and Tikhonov-type Variational Regularization Methods, which are presented in the appendix (A). From later simulation results we will find that regularization process significantly improves the performance of the algorithm. Point Spread Function of free space background case According to the well known Fourier Diffraction Theorem, when an object is illuminated with a plane wave and the background is homogeneous, the Fourier transform of the forward scattered fields gives values of the 2D spatial fourier transform of the object profile on an arc called the Ewald Circle[8]. Since the profile of a point scatterer is the impulse response function, one is expected to observe Ewald Circles on the fourier transform of the reconstruction image. The number of the Ewald Circles is equal to number of views. The more we are able to cover the frequency domain, the more accurate the reconstruction would be. Traditional Filtered Backpropogation method The Distorted wave based algorithm will be compared with the traditional filtered back-propagation(fbp) algorithm of optical diffraction tomography. The FBP algorithm is first introduced by A.J.Devaney [9] in 982 and has been widely used since. From the derivation presented in the previous section, Õ(k(s s )) = Õ(K, γ k) = 2ıγe iγη Ψ(s) B (K ξ, s ) (3.38)

61 Figure 3.2: An example of singular values for near field Π Plot of Singular Values of Far Field Pi Figure 3.3: An example of singular values for far field Π

62 4 The low-pass filtered reconstruction of the object profile from its Fourier transform takes the form where the vector K is defined as O LP (r) = d 2 KÕ(K)e ik r (3.39) (2π) 2 K 2k K = k(s s ) (3.4a) The vectors s and s are given by s = (cos χ, sin χ) s = (cos φ, sin φ ) (3.4b) (3.4c) and vector (K, γ) k cos χ = K; k sin χ = γ; (3.4d) Substituting Eq (3.4a) Eq (3.4d) into Eq (3.39), we obtain O LP (r) = = π dφ 2π π k π 2(2π) 2 i 2π π k k dφ k k h dk Kξ K Õ(K, γ +(γ K)η i k)ei (3.4) γ Ψ (s) dk K B (K ξ, s ) }{{} F iltering e i h Kξ +(γ K)η i } {{ } BackP ropagation (3.42) This algorithm will be compared with the distorted wave based algorithms in the computer simulation section Computer simulation The purpose of computer simulations is to test and evaluate the distorted wave based algorithm and to compare it with traditional FBP algorithm. We will first look at near field data and then present the simulation results for far field data. The effect of noise and robustness of the two algorithms will be studied as well.

63 42 In all simulations the geometric setup in Figure (3.) was used. The performance of the algorithms will be measured by the normalized root-mean-squared (rms) error between the reconstructed value of the object s index of refraction profile n r (r) and its exact profile n(r) as defined by[2] E = [ j n ] r(r) n(r) 2 /2 (3.43) j n(r) 2 We used four test objects to evaluate the performance of algorithms λ.3.2 λ (a) 2 (b) 2 2 λ.5. λ (c) 2 2 (d) Figure 3.4: Testing objects; (a) Step index fiber (b)graded-index fiber (c) One disc and one rectangular (d) Head Phantom. Step-index fiber (object ) a point-wise constant object whose refractive index distribution function is given by n core r r core n(r) = n cladding r core < r r r > r, (3.44) The step-index fiber is used as a test object in real experiments as well as simulations. An example of its refractive distribution is illustrated in

64 43 Figure (3.4a). 2. Graded-index fiber (object 2) an optical fiber whose core has a refractive index that decreases with increasing radial distance from the fiber axis (the central axis running down the length of the fiber). Its index of refreaction is defined by n core (n core n cladding )/r core r r r core n(r) = n cladding r core < r r (3.45) r > r, and illustrated by Figure (3.4b). 3. Geometric shapes (object 3) which consists of circular and rectangular shapes embedded in a homogeneous background. The parameters are the refractive indices of each components, n r and n c and their relative position. The object is illustrated in Figure (3.4c). 4. Head phantom (object 4) is widely used in testing the numerical accuracy of tomographic reconstruction algorithms. It is a gray-scale distribution image that consists of one large ellipse (representing the brain) containing several smaller ellipses (representing features in the brain). The Head Phantom is illustrated in Figure (3.4d). Near field data The exact scattered field of a piece-wise constant cylindrical object can be computed using a Bessel function expansion. One is able to use this class of object to evaluate the accuracy of the first order Born approximation. In the simulation we set the wavelength λ = and the CCD camera was placed λ away from the center of object with the detector pixels separated λ from each other. The size of the piece-wise constant cylindrical object is 6λ 6λ which was sampled at the rate of δ = λ/8. Figure (3.5) shows the scattered fields when n core =. and n cladding =.5. Figure (3.6) shows the case when n core =.3 and n cladding =.5. From Eq (3.4) we learned the maximum refractive index difference does not exceed δn =.6 for Born approximation of this object

65 44 to be valid, which explains the facts that more accurate scattered field data is obtained from the latter case. As discussed in a previous section, the point spread function is the reconstruction of a point scatterer located at origin. The spatial Fourier transform of the PSF is the 2-D Fourier Transform of the reconstruction. Each Ewarld circle on the transform corresponds to the data collected from one incident angle of an incident plane wave. The parameters in previous experiments are employed with the exception that the number of pixels on CCD is 4. The simulated results were illustrated by Figure (3.7). It is observed that sharper image and more complete Ewarld circle were obtained by linear mapping method. For simplicity and illustration purpose only one incident angle is used in this experiment. We tested the algorithms for three different cases. In all these experiments the distance between the center of object and the detector plane is λ, the camera pixel size is λ and the object supporting area is 6λ 6λ with the sampling rate of δ = λ/8.. Limited data: figure(3.8) to Figure (3.) shows the reconstruction results from limited scattered data collection. The scattered fields are collected from 8 directions and measured at 5 CCD pixels. One is able to tell the basic structures of objects from the reconstructions by DWBA-ODT method while it is not the case with FBP method; 2. Dense data() :Figure (3.2) to Figure (3.5) shows the reconstruction results from dense scattered data. The scattered fields are again collected from 8 directions but measured at 4 CCD pixels. This time we have sufficient large numerical aperture to cover most propagation directions of diffracted wave. One finds out the performance of DWBA-ODT algorithm is still better than FBP in this case. 3. Dense data(2) : Figure (3.6) to Figure (3.9) shows the reconstruction results from another sets of dense scattered data, where the scattered fields are collected from 32 views and CCD pixels. The computation load of this set is the same as that of dense data() case(8 4 = 32 ),

66 45 the reconstructing performance a lot better for both cases. The PSFs of above cases, which are illustrated by Figure (3.2), Figure (3.2) and Figure (3.22) respectively, predicts the difference of performance between different parameter sets and algorithms. 32 views is sufficient to cover a full area of disk with radius decided by numerical aperture of the imaging system. In this case the performance of the DWBA-ODT and FBP is close to each other. It is safe to draw the conclusion that for DWBA-ODT works much better under the limited measured data case, while FBP is preferable in dense data case due to its computational efficiency. To further prove the feasibility of new algorithm, the scattered field generated from reconstructed profile are calculated and presented by Figure (3.23). The measurement parameters are 8 views and 5 detectors. The scattered field via DWBA-ODT approximates the original values very well, which is not the case for FBP method, especially for object 3. Far field We repeated the above experiments for the inverse scattering algorithm formulated for the far field measurement configuration. The scattered field is calculated via Eq (3.3). Figure (3.24) shows the point spread function of the reconstruction algorithm when the number of the receiver is 4. Comparing the PSF of the far field DWBA and the FBP algorithm, it is found that the a better formed Ewald circle is obtained from far field DWBA inverse reconstruction algorithm, which indicates a better performance.. Limited data: Figure (3.25) shows the reconstruction results for the four objects from limited measurement data. The number of the detector is 5 and the number of the incident angle is 8. The object 3 are well constructed by both methods and according to error Table (3.2.2), the performance of far field DWBA is better than that of FBP. The reconstruction of object 4 failed for both case because the low-pass filtering effect of the algorithms such that high frequency components of the object can not be reconstructed.

67 46 DWBA Far DWBA FBP Limited Data Dense Data() Dense Data(2) Table 3.3: The error of reconstruction of object 4 of three algorithms: DWBA, DWBA far field and FBP 2. Dense data(): Figure (3.26) shows the reconstruction results under dense measurement data: the number of views is 8 and the number of the detector is Dense data(2):figure (3.27) shows the reconstruction results under dense measurement data: the number of views is 32 and the number of the detector is. The simulated results again shows advantages of the far field DWBA algorithm in limited data case. The Tikhonov-type regularization methods were used to generate the results illustrated in Figure (3.25) to Figure (3.27) Noise effect In order to evaluate the robustness of algorithms to noise, simulated Gaussion noised is added to scattered field and the dirtiness of data is described by SNR, signal to noise ratio. We defined the SNR via the equation SNR = 2 log max ψ(s) (y n ) σ n (3.46) where σ n is the standard deviation of the additive noise. The polluted signal is written as ψ (s) n (y n ) = ψ (s) (y n ) + G r + ıg i (3.47) where the G r and G i are random variables with mean and standard deviation σ n and generated independently. Figure (3.28) illustrates the results of reconstruction from scattered fields with SNR from 5dB to 3dB.

68 47 3 Born Approximation Exact Field Intensity CCD pixel Figure 3.5: Scattered fields of a piece-wise constant object (Intensity). The core refractive index is n =. and the cladding refractive index is n =.5. The red line represents the exact field and the blue * line represents the approximated field.8.7 Born Approximation Exact Field.6.5 Intensity CCD pixel Figure 3.6: Scattered fields of a piece-wise constant object (Intensity). The core refractive index is n =.3 and the cladding refractive index is n =.5.

69 48 x (a) (b).5..5 x (c) (d) 2 Figure 3.7: Example of point spread function in two dimension. (a) reconstructed image of a point scatterer by DWBA inverse scattering algorithm in near field; The image is zoomed in to show the details of the reconstruction; (b) Fourier transform of (a); (c) reconstructed image of a point scatterer by filtered back-propagation algorithm; The image is zoomed in to show the details of the reconstruction; (d) Fourier transform of (c); The number of the detectors is 4 and the object is sampled by a 2 2 grids, which is reflected on the x and y axes of the figures (a) (b) DWBA FBP Original (c) Figure 3.8: Inverse scattering reconstruction of step-index optical fiber for the limited data case. The number of incident angle is 8 and the number of detector is 5 at each detection angle. (a) original object function; (b) reconstructed result by DWBA inverse scattering method in near field; (c) reconstruction result by FBP method; (d) diagonal values of object refractive index from (a),(b) and (c)

70 (a) (b) DWBA FBP Original (c) Figure 3.9: Inverse scattering reconstruction of graded-index optical fiber for the limited data case; The highest core refractive index is n core =.3 and the cladding refractive index is n cladding =.5. The number of incident angle is 8 and the number of detectors is 5 at each detection angle (a) (b) (c) Figure 3.: Inverse scattering reconstruction of arbitrary shapes for limited view case(angle:8;detector number:5); The refractive index of circular disc is n =.5 and the refractive index of rectangular area is n =.7

71 (a) (b) (c) Figure 3.: Inverse scattering reconstruction of standard head phantom for limited view case(angles:8;detector number:5);

72 (a) (b) DWBA FBP Original (c) Figure 3.2: Inverse scattering reconstruction of step-index fiber from dense data() (a) (b) DWBA FBP Original (c) Figure 3.3: Inverse scattering reconstruction of graded-index fiber from scattered data collected from dense data().

73 (a) (b) (c) Figure 3.4: Inverse scattering reconstruction of arbitrary shapes from scattered data collected from dense data(). The refractive index of circular disc is n =.5 and the refractive index of rectangular area is n = (a) (b) (c) Figure 3.5: Inverse scattering reconstruction of standard head phantom from scattered data collected from dense data().

74 (a) (b) DWBA Far Field FBP Original (c) (d) Figure 3.6: Inverse scattering reconstruction of step-index fiber from scattered data collected from dense data(2) (a) (b) DWBA Far Field FBP Original (c) (d) Figure 3.7: Inverse scattering reconstruction of graded-index fiber from scattered data collected from dense data(2).

75 (a) (b) (c) Figure 3.8: Inverse scattering reconstruction of arbitrary shapes from scattered data collected from dense data(2). The refractive index of circular disc is n =.5 and the refractive index of rectangular area is n = (a) (b) (c) Figure 3.9: Inverse scattering reconstruction of standard head phantom from scattered data collected from dense data(2).

76 55 real part of reconstructed point scatter 2 x 3 2 2D fft of (a) (a) (b) reconstruction from FBP (c) x D fft of (c) (d) Figure 3.2: Point Spread Function of limited data case. real part of reconstructed point scatter D fft of (a) (a) (b) reconstruction from FBP (c) x D fft of (c) (d) Figure 3.2: Point Spread Function of dense data case ()

77 56 real part of reconstructed point scatter 2D fft of (a) (a) (b) reconstruction from FBP x D fft of (c) (c) (d) Figure 3.22: Point Spread Function of dense data case (2) (a) HSM method FBP Original (b) (c) (d) Figure 3.23: Scattered fields generated by reconstructed object function, 8 views and 5 detectors; (a) step-index fiber; (b) graded-index fiber; (c) arbitrary shapes; (d) phantom; The blue line represents the original data, the read * represents the data obtained from the new algorithm and the green o represents the data obtained from FBP

78 57 x (a) (b) x (c) (d) Figure 3.24: Point spread function of the inverse reconstruction algorithm formulated for far field case; The number of the detectors is 4. (a) The reconstruction of a point scatter by far field DWBA algorithm (b) 2D fourier transform of (a) (c) The reconstruction of a point scatter by filtered back-propagation algorithm (d) 2D fourier transform of (c)

79 object (a) 2 5 object (b) 2 5 object (c) object 2(a) object 2(b) object 2(c) object 3(a) object 3(b) object 3(c) object 4(a) object 4(b) object 4(c) Figure 3.25: The reconstruction of the object -4 by using far field DWBA- ODT algorithm and FBP algorithm. The number of views is 8 and the number of detectors is 5 ;(a) original object function (b) the reconstruction from far field DWBA-ODT (c) the reconstruction by FBP

80 object (a) object (b) object (c) object 2(a) object 2(b) object 2(c) object 3(a) object 3(b) object 3(c) object 4(a) object 4(b) object 4(c) Figure 3.26: The reconstruction of the object -4 by using far field DWBA- ODT algorithm and FBP algorithm. The number of views is 8 and the number of detectors is 4 ;(a) original object function (b) the reconstruction from far field DWBA-ODT (c) the reconstruction by FBP

81 object (a) 2 5 object (b) 2 5 object (c) object 2(a) 2 5 object 2(b) 2 5 object 2(c) object 3(a) object 3(b) object 3(c) object 4(a) object 4(b) object 4(c).2.5. Figure 3.27: The reconstruction of the object -4 by using far field DWBA-ODT algorithm and FBP algorithm. The number of views is 32 and the number of detectors is ;(a) original object function (b) the reconstruction from far field DWBA-ODT (c) the reconstruction by FBP

82 6 4 SNR=5 4 SNR=2 4 SNR=25 4 SNR= DWBA Algorithm FBP Figure 3.28: The reconstruction of object 3 from noisy scattered field data. The upper row is the results from DWBA near field algorithm, and the lower row shows the results from FBP algorithm (a) (b) (c) Figure 3.29: Reconstruction result of object 3 from 32 illumination angles and detectors which are separated λ away from each other. (a) original (b) Near field DWBA inverse scattering (c) filtered back-propagation algorithm

83 Chapter 4 Inverse Scattering for Cylindrical Background Media In this chapter, inverse scattering algorithms derived for object embedded in cylindrical symmetric background media are proposed. We develop the basic inverse scattering theory from both DWBA and DWRA, which account for the non-uniform background media and are more accurate compared to the uniform background models employed in the previous chapter. The performance of DWBA and DWRA for cylindrical background media are compared by a sets of computer simulations and the new DWBA algorithms is compared with the cylindrical background FBP algorithm. The motivation for studying such a background model is that in optical diffraction tomography, it is a standard procedure to embed the scattering object in a cylindrical test tube filled with a fluid whose index of refraction is closely matched to the nominal index of refraction of the object. To simplify the problem, we will limit our attention to circularly symmetric scattering objects such as non-uniform circularly symmetric optical fibers centered in a circular test tube so that the analysis and computer simulations can be reduced to two dimensional space. 62

84 63 4. Problem formulation We consider the case of two-dimensional cylindrically symmetric media where the material properties and incident and scattered fields vary only over a plane which is labelled by polar coordinates (ρ, φ). The object being illuminated consists of a known circularly symmetric background part having radius a b and (possibly complex) wavenumber k b (ρ) and an unknown inner circularly symmetric perturbation δk(ρ) that we wish to determine from a suite of scattered field data. The object is embedded in free space and illuminated by an incident plane wave propagating perpendicular to the axis of rotation of the object. The experimental geometry associated with optical diffraction tomography is illustrated in Figure (4.). This measurement geometry is appropriate for a number of applications that include ultrasound tomography as well as optical diffraction tomography. Because of the circular symmetry of the object and the assumption of plane wave illumination it is only necessary to perform a single scattering experiment using, for example, a plane wave propagating along the x axis of the Cartesian coordinate system. We can thus dispense with displaying the parameter α that characterizes the incident wave throughout the remainder of this section. For this scattering scenario the Lippmann Schwinger Eq (3.) simplifies to become ψ(ρ, φ) = ψ (in) (ρ, φ) + ab 2π ρ dρ dφ G b (ρ, ρ ; φ, φ )O(ρ )ψ(ρ, φ ) (4.) where ψ (in) (ρ, φ) is the incident plane wave propagating in the cylindrical background medium, which results from the scattering of the plane wave field ψ (in) (ρ, φ) = exp(ik ρ cos φ) with the background and thus satisfies the homogeneous Helmholtz equation [ 2 + k 2 b(ρ)]ψ (in) (ρ, φ) =, (4.2) as well as the usual Sommerfeld radiation condition [2] ψ (in) (ρ, φ) e ik ρ cos φ + f b (φ) eik ρ ρ,

85 64 y Plane Waves air n c D Recorder y n n(r) x n l (a) (b) r (c) Figure 4.: The simulation geometry of inverse scattering in cylindrical symmetric background. (a) illustrates a radially symmetric scattering object embedded in free space and illuminated with a plane wave. (b) The total field generated in the scattering experiment is measured over a CCD array located at the distance l from the center of the object. (c) A radial cut through the index distribution of the object.

86 65 as ρ where k is the free space wavenumber and f b (φ) is the scattering amplitude of the background medium. We will also need the Green function of the background medium (the background Green function ) which is defined in the usual way as the outgoing wave solution to the 2D inhomogeneous Helmholtz equation [ 2 + k 2 b (ρ)]g b(ρ, ρ ; φ, φ ) = δ(ρ ρ, φ φ ) ρ The object profile appearing in Eq (4.5)is given by O(ρ) = k 2 b(ρ) (4.3) k 2 (ρ) {}}{ [k b (ρ) + δk(ρ)] 2 = 2k b (ρ)δk(ρ) δk 2 (ρ), (4.4) with k(ρ) = k b (ρ) + δk(ρ) being the complex wave number of the composite medium (background plus scatterer). We consider the inverse scattering problem of estimating the object function O(ρ) from measurements of the field ψ(ρ, φ) performed over the set of discrete points y = y n = ρ n sin φ n, n =, 2, N along the line x = l > a b that lies outside the support of the object as illustrated in Figure ( 4.). 4.2 Inverse scattering in cylindrical symmetric background 4.2. Distorted Wave Born Approximation The mapping from the object function to the total field is non-linear as defined by the Lippman Schrodinger equation. We can express this equation in the form ab 2π ψ(ρ, φ) = ψ (in) (ρ, φ)+ ρ dρ dφ G b (ρ, ρ ; φ, φ )O(ρ )(ψ (in) (ρ, φ)+ψ (s) (ρ, φ)) (4.5) It is a standard practice to linearize the above mapping by neglecting the high order scattering generated by the first order scattered field. This approximation is referred to as the Born Approximation, or more properly, the Distorted Wave Born Approximation if the background is not free space. We thus obtain the

87 66 approximate forward model ψ(ρ, φ) ψ (in) (ρ, φ) + ab 2π ρ dρ dφ G b (ρ, ρ ; φ, φ )O(ρ )ψ (in) (ρ, φ ). (4.6) Since the incident field ψ (in) (ρ, φ) is assumed to be known we will take the field data to be the scattered field component of the total field which, within the DWBA so that Eq (4.6) is given by ψ (s) (y n ) ab 2π ρ dρ dφ G b (ρ n, ρ ; φ n, φ )O(ρ )ψ (in) (ρ, φ ). (4.7) where ψ (s) is the scattered field component of ψ on the measurement line x = l. We emphasize that the distorted wave Born approximation includes all multiple scattering processes that are present in the empty ODT measurement configuration. What is ignored are the multiple scattering that occurs between the sample itself and between the configuration and the sample and which are modelled by the higher order terms in Eq (4.5). The validation range of the DWBA is that the total phase shift through the core area between the incident field and the wave propagation is less than π, expressed mathematically as a o δn < λ 4 (4.8) where the δn is the largest refractive index difference between the known background and the embedded object Distorted Wave Rytov Approximation The Distorted Wave Rytov Approximation is employed to avoid the break-down of the DWBA for large extended object. DWRA is a linearization of the mapping between the object profile and the complex phase of the total field. If the total field is represented in its complex phase form, we find that the phase φ(ρ, φ) also admits a perturbation expansion similar in form to the Born expansion ψ(ρ, φ) = e φ(ρ,φ) = e [φin (ρ,φ)+φ s (ρ,φ)] (4.9)

88 67 where φ in (ρ, φ) is the phase of incident wave and φ s (ρ, φ) is the phase of the perturbation component φ s (ρ, φ) introduced by the scatterers. Within Distorted Wave Rytov Approximation, φ s (ρ, φ) is connected to the object s index perturbation by φ (s) (y n )ψ (in) (y n ) ab 2π ρ dρ dφ G b (ρ n, ρ ; φ n, φ )O(ρ )ψ (in) (ρ, φ ) (4.) The right hand side of the above equation is the same as Eq (4.7). Both the DWBA and DWRA scattering models have the same overal functional form. However, they have a totally different range of validity. The Rytov model is superior to the Born model for large extended samples and the necessary condition for the validity of the DWRA is δn < [ φ s λ] 2 (4.) 2π which means that the change in the scattered phase φ s over one wavelength is more important than the total change of the phase. Unfortunately, the DWRA has two deficiencies which limits its practical use. The first limitation is that it saturates under free space propagation and can only be used for field points located in the near field of the scatterer. To get around this problem, one needs to first backpropagate the scattered field data to the near field of the object and then use the DWRA. The backpropagation is performed using the free space Green function. The second weakness of the DWRA is that, in practice, the field amplitudes and not their phases are to be directly measured, which brings up the famous phase unwrapping problem Inverse scattering algorithm for ODT in cylindrical symmetric media In this section, we introduce an inverse reconstruction algorithm given either the DWBA or DWRA scattered field. We can write Eq (4.7) and Eq (4.) in a

89 68 unified and simplified form T ab ρ dρ O(ρ )π n (ρ ), (4.2a) with T to be ψ (s) (y n ) in DWBA and φ (s) (y n )ψ (in) (y n ) in DWRA as well as π n (ρ ) = 2π dφ G b(ρ n, ρ ; φ n, φ )ψ (in) (ρ, φ ), ρ a b, (4.2b) where y n = ρ n sin φ n and where we are free to define π n (ρ ) = for ρ > a b. As discussed in the previous chapter, Eq (4.2a) is in the form of a standard inner product in the Hilbert space H V of square integrable functions of ρ supported in V = {ρ < a b }; i.e., T (y n ) < π n, O > HV (4.3a) where < f, f 2 > HV = ab ρdρ f (ρ)f 2(ρ). (4.3b) The inverse scattering problem within the Distorted Wave Approximations thus reduces to estimating the unknown object profile O(ρ) V from knowledge of the set of inner products < π n, O > HV with the functions π n (ρ) H V, n =, 2,, N. As discussed earlier pseudo-inverse solution Ô to this problem is given by the projection of the object profile onto the subspace of H V spanned by the set of functions π n (ρ), n =, 2,, N. The least squares pseudo-inverse can be expressed in the form Ô(ρ) = N C n π n (ρ), n= (4.4a) where C n are the least squares, pseudo-inverse solution to the set of equations N T (y n ) = C n < π n, π n > HV. (4.4b) n = Eqs.(4.4b) are a coupled set of N equations for N unknowns that are readily solved using standard matrix schemes. The inverse scattering procedure outlined

90 69 above is conceptually simple but requires computation of the spanning functions π n (ρ), n =, 2,, N as well as the inner product matrix Π =< π n, π n > V Born Approximation The Born Approximation is actually a special case of DWBA when the background is air. For cylindrically symmetric scatterers, the Born approximation takes the form ψ(ρ, φ) e ik ρ cos φ + ab 2π ρ dρ dφ G (ρ, ρ ; φ, φ )O (ρ )e ik ρ cos φ. (4.5) where O = k 2 k2 (ρ) and G is the free space Green function which in two space dimensions is given by G (ρ, ρ ; φ, φ ) = ı 4 H (k r r ) (4.6) where r = (x, y) and r = (x, y ) are position vectors in the (x, y) plane. The filtered backpropagation(fbp) algorithm generates a pseudo-inverse of the object profile under the normal Born approximation. For the special scattering and measurement geometry employed here this algorithm reduces to the form [22] Ô (ρ) = ık π k KdK ψ (s) (K)e iγl J ( 2k (k γ)ρ) (4.7a) where γ = k 2 K 2 and ψ (s) (K) = dy ψ (s) (y)e ıky (4.7b) is the spatial Fourier transform of the scattered field over the measurement line. In practice the integrals in Eq (4.7a) and (4.7b) are discretized and Eq (4.7b) is implemented using an FFT algorithm. Thus, the inversion within the Born approximation using the FBP algorithm is very efficient.

91 7 The disadvantage of the normal Born/Rytov approximation is that a constant medium background model is very restrictive so that the approximation is not accurate for many important inverse scattering applications. In the distorted wave approaches, it is assumed that the known solution is already perturbed relative to some ideal, simple model. As discussed earlier, the object profile O(r) = k[ 2 n 2 (r)] within the first order approximation will be point-wise larger than the corresponding object profile O(r) = k 2[n2 background (r) n2 (r)] within the distorted wave approach so that the accuracy of the Born/Rytov approximation can be expected to be less than that of the distorted wave approximations. It then follows that inverse scattering algorithms based on the normal Born/Rytov approximation will be inherently less dependable and useful than their distorted wave counterparts. 4.3 Computer simulations The computer simulations in this section give and compare the performance of inverse reconstruction algorithms based on DWBA, DWRA and traditional Born approximation. algorithms is given by A quantitative evaluation of the performance of these E = [ j n ] r(ρ j ) n(ρ j ) 2 /2 j n(ρ (4.8) j) 2 where ρ j are the sample values generated by the reconstruction algorithms. The computation of the spanning functions π n (ρ), n =, 2,, N requires knowledge of both the background Green function, G b, and the incident field ψ (in) propagating in the background medium. In order to simplify the simulations we will assume that the background medium consists of a constant index core defined by k b ρ a b, k b (ρ) = (4.9) k ρ > a b.

92 7 Fortunately, for the class of backgrounds defined in Eq (4.9) both the background Green function as well as the incident wave propagating in the background medium can be analytically computed using series of Bessel functions. We have relegated these calculations to the Appendicies B. and B.2. It is expected that the performance of the DWBA should not depend on the values of k b. However it may not be the case for DWRA. We will show this later by testing the algorithms on weak scattering background and strong scattering background, respectively. We will employ Distorted Wave Born Approximated scattering data, DWRA data and (in principle) exact scattering data in the simulations. The DWBA data are easily generated using Eq (4.7) for arbitrary symmetric object profiles O(ρ). On the other-hand the computation of exact scattering data and DWRA data is more difficult and different computational schemes are required depending on the specific nature of the scatterer. In the computer simulations we will employ both a Bessel function expansion, which is applicable to piecewise constant scatterers as well as a moment method, which is applicable to more general object profiles. The detailed procedure of calculating the field will be presented in the Appendix B. The DWRA field is generated from the phase of the exact field via Eq (4.9). T DW RA = φ (s) ψ (in) = [ ln (ψ) ln (ψ (in) ) ] ψ (in) (4.2) In the end, we note that in implementing the FBP algorithm we selected the background wave number to be that of the core k b rather than the free space wave number k. The use of the core wave number as background for the FBP inversion is common in ODT applications since it yields a scattering model that is much superior to the model that results from using free space as background. Indeed, we found in the simulation study that the use of the free space background in the FBP algorithm yielded results that were far worse than those obtained using k b as background. We have thus opted to show only the better FBP results obtained using k b rather than k for the sake of fairness to this algorithm.

93 Comparison of DWBA/DWRA and FBP method Two example objects with different refractive index distributions are used for algorithm testing. Dimensional and measurement parameters are mostly consistent in all of our simulations. The unknown component of the object was chosen to have a radius of a = λ and was assumed to be embedded in a background cylinder (core) having a radius a b = 2 λ. The object is enclosed within a 2λ 2λ square grid divided into 4 4 cells and illuminated by a plane wave with wavelength λ =. Scattered field measurements were assumed to be performed right outside the cladding l = 2 λ with a sample-to-sample spacing of.2λ. Piece-wise constant object The first example is a uniform perturbation on the background defined by the index perturbation δk ρ a δk(ρ) = (4.2) ρ > a, relative to the constant core background component defined in Eq (4.9). The exact field computations were performed using the eigenfunction expansion described in Appendix B.3. We tested the algorithms on three backgroundperturbation combinations,. n b =., δn =. 2. n b =., δn =.2 3. n b =., δn =. Figure (4.2) shows the scattered fields of case generated by four different methods: the Bessel function expansion, the DWBA, the DWRA and the Born Approximation implemented using the wave number k b as background. The scattered fields generated from case No.2 and No.3 are illustrated by Figure (4.4) and Figure (4.6) respectively. We applied the inverse scattering algorithm introduced in section (4.2.3) and the FBP algorithms to the fields data and generated the results shown in Figure (4.3), (4.5) and (4.7). As mentioned above

94 73 Table 4.: rms errors of inversion for constant perturbation (piecewise object) by four reconstruction strategies: reconstruction by DWBA algorithm from DWBA field, DWBA algorithm from DWBA field, DWRA algorithm from exact field, FBP algorithm from exact field case DWBA/DWBA field DWRA/exact DWBA/exact FBP/exact we implemented the FBP algorithm using an infinite homogeneous background whose wave number was taken to be the core wave number k b. We also show the FBP reconstruction only over the support of the object since this is the region of interest in the study.the RMS errors for the above described experiments are listed in Table( 4.3.). As mentioned above the use of the cladding wave number yields a scattering model that is much superior to the Born model that results from using free space as background. The scattered fields simulations, Figure (4.2),(4.4) and Figure (4.6), show that both DWBA and DWRA fields provides models that are very close to the exact one which indicates that the distorted wave inversion should work quite well. The Born model, on the other-hand, has poor agreement with the (exact) data, especially for the strong scattering background case. This is due primarily to the fact that this model neglects multiple scattering that occurs within the cylindrical core. It is observed that an ideal reconstruction is obtained with DWBA field for the three testing cases due to the way the forward model is constructed. For weak scattering background (case,2), as shown in Figure (4.3) and Figure(4.7), the FBP algorithm returned similar result as DWBA. For stronger scattering background(case 3), the DWBA reconstruction from the exact field returns reasonable result while the FBP algorithms failed in this case. This shows the advantages of distorted wave algorithm for accurately modelling the scattering experiment when the refractive index of cladding is very different from the free space.

95 74 General object We also examined the DWBA, DWRA, FBP algorithms performance on a more general object having a sinusoidal perturbation along its radial axis given by δn sin(4πρ/a ) ρ a δn(ρ) = (4.22) ρ > a. with a = 6λ and δ n =.3. In this case the Bessel function expansion can no longer be used and we employed the moment method described in the Appendix B.4 to compute the exact scattered field. Figure (4.8) shows the simulated scattered field data generated using the moment method, DWBA, DWRA and Born approximation again using the k b as background. In this case k b =.44. As in the earlier example, the DWBA/DWRA data is again very close to the exactly computed data indicating that the DWBA and DWRA inversion algorithm should work quite well. The reconstruction obtained using the DWBA/DWRA algorithm from the exact field as well as the reconstruction result from FBP method are presented in Figure(4.9). As expected the DWBA algorithm from DWBA field returns an exact reconstruction. The reconstruction by DWBA and DWRA from the exact field is somewhat similar. Due to the small refractive index different, the FBP algorithm also gives a good reconstruction result. If we increase the background refractive index to be n c =.44, as shown in Figure (4.), the reconstruction from FBP fails again. The inversion from DWBA field by DWBA method yielded essentially exact reconstructions while the reconstructions by DWRA/DWBA algorithms returned an acceptable estimated object profile. The RMS reconstruction errors are listed in Table ( 4.3.) Comparison of DWBA and DWRA method The performance of DWBA algorithm from exact scattered field is almost the same for the object with the same perturbation but different backgrounds, which shows that only the refractive perturbation from the background matters in DWBA reconstruction algorithm. It is not the case for the DWRA algorithm due to the fact that the phase values of a stronger scatterer might go beyond 2π

96 75 Table 4.2: rms errors of inversion for sinusoidally varying perturbation(general object,δ n =.3 ) by four reconstruction strategies. object DWBA/DWBA field DWRA/exact DWBA/exact FBP/exact n c = n c = which brings up the phase unwrapping problem. An example of this is shown in Figure (4.). Three weak scatters having radius of λ is embedded in three different background media respectively. The refractive index of the background media are n =.,.5,. and the refractive index difference of the scatterer and background remain the same n =.5. Figure (4.2) to Figure (4.9) show the simulated results for 6 reconstructions using the distorted wave Born and Rytov approximations. All the objects are embedded in a cylindrical symmetric background with radius 5λ, refractive index n =.. The object is sampled at interval of.5λ and the distance between the center of object and the detector plane is 5λ. All the singularities in the DWRA fields are removed before reconstruction computation. For small 4λ objects, good reconstruction results are obtained for all refractive index changes with DWBA. But all DWRA reconstructions failed. The reason is that there are too many phase singularities. For all the other objects, DWBA is good as long as the phase shift of the incident field within the scatterer is less than π as predicted by phase shift <= 4 na/λ for DWBA. DWRA is more sensitive to refractive index but produces better results when object get larger. According to Figure (4.2) and (4.2), the performance of the Rytov approximation is similar to DWBA with phase shift.4λ and superior to the Born approximation for the reconstruction of objects with phase shift.8π. To further compare the performance of DWBA and DWRA, a quantitative study of the error in the reconstruction was also performed. As a measure of error we use Eq (4.8). In this study the background size is 2λ and the refractive index

97 76 change varies in between..2. Three object sizes are tested, which are in radius 2λ,4λ and λ respectively. The Figure (4.22) shows the error plots. In each case the error for the Born approximation is shown as a solid line while the Rytov reconstruction is shown as a broken line. For weak scattering background nc =. and nc =.5, the error of DWBA and DWRA is the same for small object a = 2λ and DWRA is superior to DWBA for larger object a = 4λ, λ. For strong scattering background, DWRA may fail while DWBA maintain the same error properties as it has for weak scattering background. For nc =., DWRA failed for all 4λ objects. For nc =.5, the error of DWRA for λ object increases very quickly. The failures of DWRA are caused by phase singularities Effect of noise The effect of noise was studied by adding a Gaussian random signal to the scattered fields data shown in Figure(4.8) and applying a regularized version of the distorted wave inversion algorithm to the noisy data. A Tikhonov-type[9] variational regularization method was employed to keep the oscillation caused by noise under control. Instead of directly using the Singular Value Decomposition compute the coefficients C n in Eq (4.4a) these coefficients were selected to satisfy Ĉ = argmin C D ΠC λ LC 2 2 (4.23) which has a close form solution Ĉ = (Π T Π + λl T L) Π T D (4.24) The regulation parameter λ is chosen using the so-called L-curve method[2]. The robust of the algorithm was tested on the DWBA field and DWRA field respectively. Figure (4.23) shows the reconstruction results from noisy DWBA approximated data with different S/N values. We can see that the algorithm can easily tolerate S/N = 5dB noise and hence is quite robust. Figure (4.24) shows the reconstruction result from the noisy DWRA data when the S/N = 35dB,S/N = 3dB and S/N = 25dB. The performance of reconstruction shows

98 77 that for data with S/N = 25dB, one is able to get reasonable inversion by using DWRA.

99 (a) (b) Exact DWBA DWRA Born (c) Figure 4.2: Comparison of scattered field data generated by different scattering models for an object comprised of two concentric cylinders. The cladding refractive index is n c =. and the core refractive index is n =.2 (a) Real part (b) Imaginary part (c) Magnitude. The blue solid line represents the exact field calculated by bessel function expansion. The green dot line represents the DWBA scattered field and the magenta crossing shows the DWRA scattered filed. The Born scattered field data were generated using the core index n c for the background medium and is represented by cyan dashed lines.

100 Original DWBA field by DWBA Exact field by DWBA Exact field by DWRA Exact field by BA.2 Refractive Index Radius,r(λ) Figure 4.3: Simulated reconstruction results of the piece-wise constant object with cladding refractive index to be n c =. and the core refractive index to be n =.2. The original refractive index distribution is represented by the dark star line. The reconstruction from Distorted Wave Born Approximation field by the DWBA inverse scattering algorithm is represented by green dot line. The reconstruction from the actual exact field by DWBA algorithm is represented by the blue solid line and the corresponding reconstruction from the exact field by the DWRA algorithm is shown by magenta crossing line. The cyan dashed line is used to represent the reconstruction result from exact field by FBP algorithm.

101 (a) (b) Exact DWBA DWRA Born (c) Figure 4.4: Comparison of scattered field data generated by different scattering models for piece-wise constant object with weak background scattering but strong core scattering(case 2). The cladding refractive index is n c =. and the core refractive index is n =.3 (a) Real part (b) Imaginary part (c) Magnitude. The blue solid line represents the exact field calculated by bessel function expansion. The green dot line represents the DWBA scattered field and the magenta crossing shows the DWRA scattered filed. The Born scattered field data were generated using the core index n c for the background medium and is represented by cyan dashed lines.

102 Original DWBA field by DWBA Exact field by DWBA Exact field by DWRA Exact field by BA.35.3 Refractive Index Radius,r(λ) Figure 4.5: Simulated reconstruction results of the piece-wise constant object with cladding refractive index to be n c =. and the core refractive index to be n =.3. The original refractive index distribution is represented by the dark star line. The reconstruction from Distorted Wave Born Approximation field by the DWBA inverse scattering algorithm is represented by green dot line. The reconstruction from the actual exact field by DWBA algorithm is represented by the blue solid line and the corresponding reconstruction from the exact field by the DWRA algorithm is shown by magenta crossing line. The cyan dashed line is used to represent the reconstruction result from exact field by FBP algorithm.

103 (a) (b) 2 5 Exact DWBA DWRA Born (c) Figure 4.6: Comparison of scattered field data generated by different scattering models for piece-wise constant object with strong background scattering. The cladding refractive index is n c =. and the core refractive index is n =. (a) Real part (b) Imaginary part (c) Magnitude. The blue solid line represents the exact field calculated by bessel function expansion. The green dot line represents the DWBA scattered field and the magenta crossing shows the DWRA scattered filed. The Born scattered field data were generated using the core index n c for the background medium and is represented by cyan dashed lines Original DWBA field by DWBA Exact field by DWBA Exact field by DWRA Exact field by BA.2.5 Refractive Index Radius,r(λ) Figure 4.7: Reconstruction results for the piece-wise constant object from the DWBA scattered field data shown in Fig All the legends are consistent with previous figures.

104 (a) (b) Exact DWBA DWRA Born (c) Figure 4.8: Comparison of scattered field data generated by different scattering models for an object having a sinusoidally varying perturbation with respect to its cylindrical background (n c =.44; Maximum n o =.3). (a) Real part (b) Imaginary part (c) Magnitude. The Born scattered field data were generated using the core index n c for the background medium..7.6 Original DWBA field by DWBA Exact field by DWBA Exact field by DWRA Exact field by BA.5 Refractive Index Radius,r(λ) Figure 4.9: Reconstruction results for the sinusoidally varying perturbation object from the scattered field data shown in Fig The legend is consistent with previous reconstruction illustration

105 Refractive Index..8.6 Original DWBA field by DWBA Exact field by DWBA Exact field by DWRA Exact field by BA Radius,r(λ) Figure 4.: Reconstruction results for the object having a sinusoidally varying perturbation(n c =.44; Maximum n o =.3). The top figure shows the reconstruction generated using the DWBA algorithm and the bottom the FBP algorithm with k b as background wave number. 6 x r(λ) 8 x Original nc=. nc=.5 nc= r(λ) Figure 4.: Comparison of DWBA and DWRA: same weak scatterer embedded in different background media; The upper figure shows the reconstruction results from DWBA method and the lower image shows the results from DWRA method

106 85.5 dn=..5 dn= dn=..5 dn= Figure 4.2: Comparison of DWBA and DWRA: reconstruction of 6 different circular objects which indicates the effect of weak scatterer s size and refractive index difference on DWBA and DWRA; This figure shows the scattered fields of a weak scatterer with radius 4λ; Solid line: Exact field, Dashed line: DWBA field, dash-dot line: DWRA field;.5 dn=..5 dn= r(λ) dn= r(λ) r(λ) dn= r(λ) Figure 4.3: Continue: recontruction of weak scatterer with radius 4λ; Solid line: Original; Dashed line: DWBA reconstruction, dash-dot line: DWRA reconstruction;

107 86.2 dn=..8 dn= dn=. 3 dn= Figure 4.4: Comparison of DWBA and DWRA continue: scattered fields of a weak scatterer with radius λ; Solid line: Exact field, Dashed line: DWBA field, dash-dot line: DWRA field;.25 dn=.. dn= r(λ).5..5 dn= r(λ) 5 5 r(λ) dn= r(λ) Figure 4.5: Continue: recontruction of weak scatterer with radius λ; Solid line: Original; Dashed line: DWBA reconstruction, dash-dot line: DWRA reconstruction;

108 87.4 dn=..5 dn= dn=. 6 dn= Figure 4.6: Comparison of DWBA and DWRA continue: scattered fields of a weak scatterer with radius 2λ; Solid line: Exact field, Dashed line: DWBA field, dash-dot line: DWRA field;.25 dn=.. dn= r(λ).5..5 dn= r(λ) 5 5 r(λ) dn= r(λ) Figure 4.7: Continue: recontruction of weak scatterer with radius 2λ; Solid line: Original; Dashed line: DWBA reconstruction, dash-dot line: DWRA reconstruction;

109 88.8 dn=. 3 dn= dn=. 2 dn= Figure 4.8: Comparison of DWBA and DWRA continue: scattered fields of a weak scatterer with radius 4λ; Solid line: Exact field, Dashed line: DWBA field, dash-dot line: DWRA field;.25 dn=..8 dn= r(λ) dn= r(λ) r(λ) dn= r(λ) Figure 4.9: Continue: recontruction of weak scatterer with radius 4λ; Solid line: Original; Dashed line: DWBA reconstruction, dash-dot line: DWRA reconstruction;

110 r(λ) r(λ) Figure 4.2: Comparison of DWBA and DWRA: same phase shift over the weak scatterer(.4π) r(λ) r(λ) Figure 4.2: Comparison of DWBA and DWRA: same phase shift over the weak scatterer(.8π)

111 9.2 nc=..2 nc= nc= λ DWBA 4λ DWBA λ DWBA 2λ DWRA 4λ DWRA λ DWRA nc= Figure 4.22: Comparison of the performance of DWRA and DWBA: the errors for reconstruction with DWBA(solid) and DWRA(broken) are shown. Each plot contains results from three cylinders of radius 2λ,4λ and λ.

112 9 Reconstructed n Original S/N=5 S/N=5 S/N= (a) radius r (λ)..8 rms error (b) S/N Figure 4.23: (a) Reconstructions of the object having a sinusoidal varying perturbation from noisy DWBA data having signal to noise ratios ranging from S/N = 5dB to S/N = 35dB. (b) shows the RMS error plotted versus the signal to noise ratio. Recomstruted n Original S/N=5 S/N=5 S/N= (a) radius r(λ) 4 x 3 2 rms error (b) S/N Figure 4.24: (a) Reconstructions of the object having a sinusoidal varying perturbation from noisy DWRA data having signal to noise ratios ranging from S/N = 5dB to S/N = 35dB.(b) shows the RMS error plotted versus the signal to noise ratio.

113 Chapter 5 Inverse Scattering Algorithm Based on Ray Tracing In this chapter a novel inverse scattering algorithm based on ray tracing and vector space mapping is introduced and simulation results are presented. It is applicable to the reconstruction of weak scatterers embedded in a supporting potential which is known in terms of geometric shape and refractive index distribution. In the new algorithm it is assumed that the incident light is refracted at the boundary of the background media and the weak scatterer does not change the direction of the light path, as indicated in Figure (5.). The forward scattering of the probing plane wave within the supporting potential is modeled based on ray tracing, i.e., the propagation of the light within the sample follow the refracted light path within the air/background media boundary. The phase change of the incident plane wave from incident plane to the measurement plane is the integral of the refractive index along the ray path illustrated in Figure (5.). δφ = n(r)dl (5.) Each pixel on the CCD detector corresponds to one optical path l which can be determined based on the geometrical information and refractive index distribution of the media. One can thus get a phase change distribution along the detector plane via calculating a set of line integrals. This distribution can be 92

114 93 Figure 5.: Weak scattering object embedded in a supporting potential whose dimensional information and refractive index distribution is known. In this thesis we use cylindrical symmetric background media used to approximate the real phase change from incident plane to the detector plane. By doing this we will obtain a forward scattering model that can then be used as a basis for an inversion algorithm of reconstruction. It is found in Eq (5.) that there is a linear mapping relationship defined between refractive index distribution of weak scatterer and the phase change. It is possible to utilize the idea of vector space mapping for inverse reconstruction. The algorithm is applicable to any background media whose geometric information is available. For simplicity, in this thesis we use cylindrical symmetric background media as an example to show the feasibility of the algorithm. We also assume that the weak scatterer is homogeneous in z direction. In this way we reduce the three dimensional reconstruction problem to two dimensions.

115 94 5. Ray tracing In this section the ray tracing functions within the cylindrical symmetric background media are calculated. In the two dimensional space where the inverse problem is defined, we set up the coordinate system in a way that the origin of the coordinate is the center of the background media, the y axis is parallel to the detector plane and the x axis is parallel to the incident plane wave. The detailed geometry is illustrated in Figure (5.2). The radius of the background media is r. Assume we have an incident light crossing the boundary of the background media at point (x, y ). It forms an angle α with the norm of the boundary and the corresponding refraction angle is α. The normal of the boundary at (x, y ) makes an angle with x axis of γ. Given the radius of boundary circle r, refractive index inside the circle n and incident ray height y, the following relationships exist: x = r 2 y 2 (5.2a) γ = π arctan (y /x ) α = π γ ; α = arcsin [sin (α )/n] (5.2b) (5.2c) (5.2d) If we use y = m x + b to represent the ray trace function within the background media boundary, the slope m and y-intercept b is solvable from above equations and given as m = tan (θ ) = tan (γ + α ); b = y m x ; (5.3a) (5.3b) The refracted light inside the boundary is illustrated by green color in Figure (5.2). It crosses with boundary circle again at another point (x 2, y 2 ), the values of which could be solved by applying line equation y = m x + b into circle

116 95 Figure 5.2: Ray tracing geometry within the a cylindrical symmetric media boundary equation x 2 + y 2 = r 2. x 2 = [ ] 2m 2( + m 2 y + 2m 2 ) x + 2 2m y x y 2 m2 x2 + r2 + m 2 r2 (5.4a) y 2 = m x 2 + b At the point (x 2, y 2 ), the light is refracted again. (5.4b) The angle formed by the normal of the circle at (x 2, y 2 ) and the x axis is represented by γ 2 and it is known that γ 2 = arctan (y 2 /x 2 ) (5.5a) α 2 = π α (γ γ 2 ); α 2 = arcsin [n sin α ] (5.5b) α 2 and α 2 is the incident and refractive angles at (x 2, y 2 ). The parameters of the ray path between the background media boundary and the detector plane, i.e., the slope and y-intercept can be expressed as m 2 = tan (θ 2 ) = tan (γ 2 + π α 2); b 2 = y 2 m 2 x 2 ; (5.6a) (5.6b)

117 96 The coordinates of the crossing between the ray path and the detector plane is (x 3, y 3 ). In practice, we usually start from detector coordinates and backpropagate the light to incident plane by inversion the process described above. An example of ray tracing is presented in Figure (5.3). 5.2 Inversion based on ray tracing The weak scatter under study is not cylindrical symmetric. It is necessary to illuminate the scatterer and collect the scattered light from different direction in 36 degrees. Let assume we keep the object fixed and rotate the illuminationdetector system to Q different directions evenly distributed in 36 degrees. At each direction the scattered field is sampled at P points. Thus each combination of incident plane wave direction and sampling point coordinates defines an element in a multistatic data matrix φ p,q, which is written as φ p,q = n(l p,q )dl p,q = n(r)π (p, q)dr (5.7) where π (p, q) is defined as π (r) = {, r on the ray path (p,q), elsewhere. (5.8) The ray path (p, q) is defined by one combination of incident angle β q, q =,, Q and sampling points (x 3, y 3 ) p, p =,, P. The refractive index distribution of a general object is n(r) = n b (r) + δn(r) (5.9) apply to Eq (5.7) we learn that φ p,q = φ b p + δn(r q )π (p, q)dr (5.) where φ b p = n b (r)π(p, )dr is the phase disturbance introduced by the background media and is the same for all illumination angles in cylindrical symmetric

118 97 background. The distribution of φ b p over the detector plane is either directly measurable by removing the object or simulated given the information of background media. Thus second term of Eq (5.) is the phase disturbance generated by the weak scatterer and for this reason the algorithm is called inversion within distorted wave ray tracing approximation (DWRTA). φ s p,q = δn(r q )π (p, q)dr (5.) A minimum L 2 DWBA is then given by the sum norm solution of the inverse scattering problem within the p q ˆn(r) = C n π(p, q) (5.2) where the coefficient C n are solutions to the coupled set of equations D n = p q n = n= C n < π(p, q), π(p, q ) > Hυ (5.3) where the D n, n =,, P Q is the elements of φ s p,q. For simplicity,the Eq (5.3) can be written in the form of ΠC = D (5.4) where Π =< π(p, q), π(p, q ) > (5.5) should be easily computed.

119 Computer simulations 5.3. Forward model evaluation The performance of the forward model based on DWRTA is tested on a pointwise constant object embedded in a known cylindrical symmetric background which simulates a step-index optical fiber. The core of the fiber is a weak scatterer and the cladding is regarded as cylindrical symmetric background. The reason of using this object is that the exact scattered field can be computed by bessel expansion algorithm presented in appendix (B.3) so that we have a benchmark to evaluate the performance of forward model. The forward model based on DWRTA is tested on three cases of cladding with radius 4λ, namely weak(n =.2), medium(n =.22), and strong(n =.42). The core refractive is.22 and the radius of the core area is 2λ. The results of simulated phase distribution at the plane right outside the background area are illustrated in Figure (5.4) Figure (5.6). It is observed that the weaker the background scattering, the better the approximation of DWRTA. In general, it is safe to draw the conclusion that the ray tracing algorithms works pretty well in approximating the phase of the scattered field from weak scattering object embedded in a circular symmetric background. The algorithm can be easily extended to other shape of background as long as the contour of the area is known Inversion algorithm evaluation The inversion algorithm based on ray tracing model is evaluated by reconstruction of four different weak scatterers embedded in a medium scattering background (n =.22). We assume that the weak scatterers are illuminated from 32 different angles and the scattered fields are measured at a plane right outside the background area. The distance between the detector plane and object center is then the radius of circular background area, which is 4λ in this case. The effective pixel number on the detector plane is 83 in all the experiments and the pixels are separated at distance of.5λ from each other. The background area is divided into grids with the object sampling rate to be.4λ at both

120 99 x and y axis. The reconstruction results are illustrated in Figure (5.7) Figure (5.). In those figures the original refractive index distribution of the disturbance is shown side by side with the reconstruction from the new inversion algorithm. Step-index optical fiber The refractive index and the size of the fiber core is n =.22 and 2λ respectively; The reconstruction result is shown in Figure (5.7); Graded index optical fiber The disturbance area is still circular symmetric with radius 2λ, but not homogeneous. The refractive index decreases along the radius direction and the center refractive index is n =.22. The result is shown in Figure (5.8). Two randomly placed shapes There are two randomly placed shapes inside the circular symmetric area. One is rectangular and the other is circular. Both shapes are homogeneous and the refractive index of both them is n =.22. The result is shown in Figure (5.9). Five discs The fives discs are evenly placed inside the circular background area. The refractive index of the discs are different from one to the other and range between The result is shown in (5.); All the experiments gives decent reconstruction results which indicates that the new inversion algorithm can be used to reconstruct weak scatterer as long as the scatterer does not refract the light within the resolution of detectors.

121 Figure 5.3: Ray tracing result of a circular object with refractive index n >

122 Object Profile 5 (a) Ray tracying inside fiber cladding (b) Scaled Scattered Field Comparison:Phase λ (c) Figure 5.4: Comparing the phase values generated by exact field calculation and ray raying approximation.the object is a simulated step-index optical fiber. The Core size is 2λ,n =.22, The cladding size is 4λ,n =.22 (a) Object Profile (b) Ray tracing in circular background media; (c) Scattered fields: red represents the scattered fields generated by exact field calculation; blue o represents the approximated field generated by ray tracing algorithm.

123 2 Object Profile Ray tracying inside fiber cladding (a) (b).8 Scaled Scattered Field Comparison:Phase Exact field Ray Tracing Approximation λ (c) Figure 5.5: Comparing the scattered field phase distribution generated by exact field calculation and ray raying approximation: a weak scattering background. The object is a simulated step-index optical fiber. The Core size is 2λ,n =.22, The cladding size is 4λ,n =.2 Object Profile.4 4 Ray tracying inside fiber cladding (a) (b).8 Scaled Scattered Field Comparison:Phase Exact field Ray Tracing Approximation λ (c) Figure 5.6: Comparing the scattered field phase distribution generated by exact field calculation and ray raying approximation: a strong scattering background. n =.42 and other parameters remain the same as the previous two experiments;

124 3 Object function x 3 Reconstructed object function x (a) 5 x 3 diagonal values of object function (b) original reconstruction (c) Figure 5.7: Optical step index fiber simulated reconstruction by ray tracing algorithm when the detector plane is uniformly sampled. (a) Refractive index distribution of the disturbance; (b) Reconstruction result by the ray model based DWBA; (c) Diagonal values of the reconstructed refractive index Object function (a) x Reconstructed object function diagonal values of object function (b) original reconstruction x (c) Figure 5.8: Optical graded index fiber simulated reconstruction by ray tracing algorithm when the detector plane is uniformly sampled. The center refractive index is n =.22 and the size of area is 2λ. The rest is the same as the previous example

125 Object function (a) x Reconstructed object function diagonal values of object function (b) original reconstruction (c) Figure 5.9: Two shapes disturbance reconstruction results. The refractive index of both shapes is n =.22 Object function x Reconstructed object function x (a) 8 x 3 diagonal values of object function (b) original reconstruction (c) Figure 5.: Simulated reconstruction result of five discs disturbance by ray tracing inverse scattering algorithm. The refractive index, from smallest to largest:.22.27

126 Chapter 6 Experimental Results In this chapter the experimental setup of a digital holographic microscopy is described and the results of imaging are presented. For thin object we showed the numerical reconstruction of their 2D transmission functions. For threedimensional object we discussed the processing of the raw experimental data and showed the inverse reconstruction results. 6. Digital holographic microscope setup 6.. Experimental system The Mach-Zehnder interferometer system shown in Figure (6.) is set up for digital holographic imaging and phase retrieving. An Uniphase R He-Ne laser is used as illumination source and its wavelength is λ = 633nm. The laser light is expanded by a beam expander which comprises of a 8.mm focal length objective, a 5µm pin-hole spatial filter and a 2mm focal length doublet. The expanded light is divided by a plate beam splitter into two beams. The plate beam splitter is coated with an anti-ghost film. One light beam is reflected by a plane mirror and then propagates through the imaging target and is referred as the object beam. The other beam is simply reflected by two mirrors and is referred as the reference beam hereafter. The two beams join at another beam splitter and the final interferogram is captured by a Pulnix 3 CCD camera. One of the two mirrors in reference arm is mounted on a Piezo-electronic translation(pzt) stage such that the optical path difference between two beams can 5

127 6 be modulated. A picture of the system is shown by figure (6.2). PZT is a mechanical phase modulator that changes its size upon applying electrical energy, which make it possible to automate the phase shifting process via computer. The flowchart of controlling PZT is illustrated by figure (6.3) and the actual system is illustrated by figure (6.4). One uses MATLAB R data acquisition toolbox to interface with a National Instrument R analog input-output unit(ni-daq) and modulates the output voltage. The output of NI-daq is connected with Thorlabs R piezo-controller, which amplifies the 5 volts output of NIdaq to range. More detailed information of each components is described as following:. Piezo electric actuator: it transforms electrical energy into precisely controlled mechanical displacements. It is mounted carefully to a very light, high precision mirror by a room temperature epoxy. Any bending force has to be avoided and the mechanical load on the piezo-electric faces need to be centered. The maximum displacement of µm is achieved at Volts and applying a drive voltage in excess of the Volts will lead to probable failure of the actuator. 2. NI-USB68: It is a 2-Bit, ks/s multifunction data acquisition device which interacts with computer via USB port. The unit is used for analog output in the range of [ 5V ] and is compatible with MATLAB data acquisition functions. 3. Piezo-controller: The unit offers master scan controls that allow internal & external control. Computer control is available via an RS-232 interface. It has a selectable output voltage ranges: 75V, V and 5V. Please refer to figure (6.3) for the function description of the control knobs Data acquisition The DHM system was set up using standard interferometer building procedure. The detailed data acquisition procedure is included in Appendix (C). The described procedure generated a great amount of data to process and figure (6.5) Device information is available at

128 7 shows an example of data processing program interface created by MATLAB R GUI. Different phase shifting algorithms are compared with this program and the back-propagation of the fields at any position specified can be visualized. Both simulation and experimental data processing can be done in this program. 6.2 Thin object imaging results Thin object refers to the class of object that the complex wave field right out of the object is the multiplication of incident field and object transmission function. As illustrated by figure (6.6), the imaging of thin object in digital holographic microscopy is to use CCD camera and back-propagation numerical calculation to replace the imaging lens and imaging plane in traditional imaging. If the incident field is plane wave and represented by ψ in = A, the transmission function of the object T (ρ) B[ T (ρ)] (6.) where T (ρ) is the complex wave field at the detector plane and B represents the back-propagation transform Simple 2D objects This section presents the imaging results of simple objects. Those objects are characterized by known transmission function and hence could be used to evaluate the feasibility of the system and the performance of phase retrieval algorithms. Slit The slit used in this experiment is 5µm width and 3mm long and mounted on a inch support. It is placed 55mm away from the detector plane of the CCD camera. The reconstruction result is shown in figure (6.7). Ronchi-ruling Ronchi-ruling is generally used for evaluating the quality of an imaging system. If placed l = 6mm away from the CCD plane, the system has a numerical

129 8 aperture of.. Hence the diffraction limit of the system is about 6.7µm. Three resolutions of ronchi-ruling are tested in our experiments, lpi, 5lpi and lpi respectively. The reconstruction results are illustrated by figure (6.8), figure (6.9) and figure (6.). The results show that the system does a good job for all of them. Microlens Micronlens array is a pure phase object which only changes the phase of the light that propagates though. Figure (6.) shows the DHM images of a microlens array. It is seen that no structure can be identified from the intensity image but very clear lens shape can be observed from phase image Biological samples A great area of application for DHM is the imaging of biological samples. In this section, both the imaging results from DHM and conventional microscopy of several biological samples are presented. Figure (6.2) illustrates the transmission function of a cotton stem cell. The sample is dyed and the phase map does not provide additional information, which is not shown for this object. The right side of figure (6.2) is the conventional microscopic image captured by the same CCD camera. One can see that the DHM provides as much the same level of detail as the analog microscope. Same result is shown for tendon in figure (6.3) and figure (6.4) shows the holographic images of the samples as well as the back-propagated intensity and phase. 6.3 Three dimensional imaging We employed the same setup to obtain experimental data to test our inversion algorithms and 3D imaging capability of the digital holographic microscopy. The object chosen for imaging is SMF-28 optical fiber which has known refractive index distribution and simple structure. The fiber has a step index distribution

130 9 and is circularly symmetric. The core of the fiber has a diameter 8.2µm and refractive index.453. The cladding of the fiber is 25µm in diameter and.4496 in refractive index. The distance between the center of the fiber core and the camera aperture is l = 32mm. The optical signal processing process for 3D thick object imaging is more complicated than that of optical transmission function imaging case due to the high signal to noise ratio required for inversion algorithms. We break the process into several steps Complex field retrieval from holograms As illustrated in configuration (6.), a SMF-28 fiber is placed in one of the interferometer arms, which will be referred as object arm hereafter. By moving the piezo-electronic transducer placed at the other arm, named as reference arm, 5 holograms are formed and recorded at the camera plane, which are two dimensional matrixes labelled as I, I 2,..., I 5. The first step is to retrieve the complex field from the object arm, referred as raw object field ψ raw = A raw exp ıφraw in the future discussion. Figure (6.6) shows the five holograms I, I 2,..., I 5 generated by varying the phase shift in the reference arm of the interferometer as, α,..., 4α. The phase shift value α is unknown and is calculated by the algorithm introduced in chapter (2). The phase calibration map for the SMF-28 fiber is shown in figure (6.7). A small area on the phase calibration map is chosen for averaged phase shift computation. The reason for not choosing the whole image is to avoid singular values in Eq. (2.2). The computed phase shift value is then used to calculate the complex field φ raw. We repeat the process for groups of measurement to remove the influence of air turbulence and vibration by averaging the calculated data. The intensity and the phase has to be averaged separately due to the complex nature of data points. It is necessary to do phase unwrapping for each calculated phase map

131 before averaging due to inconsistent phase jump. The averaged intensity and the phase distribution of the object beam complex field ψ raw is illustrated by figure (6.8). A three dimensional view of phase map is also shown by figure (6.9) Interferometer error removal The computed complex wave field ψ raw consists errors from interferometer, as illustrated by figure (6.5). The measured phase φ raw of the raw object field ψ raw consists of three parts. φ raw = φ background + φ fiber + k l (6.2) where φ fiber is the phase variation introduced by the fiber, φ background is the phase difference of the two arms of the interferometer. k is the wave number and k l is the additional phase increase at the reference arm if we set the center of the optical fiber as zeros phase plane. φ background can be identified by obtaining another sets of holograms without any object placed in the interferometer object arm. The background intensity A background and the phase φ background distributions of the interferometer setup is illustrated by figure (6.2). Hence φ correct = φ raw φ background k l (6.3) Due to unbalanced illumination and DC components of CCD camera, the intensity of ψ raw need to be normalized and is given by A norm = (A raw a m )/a s (6.4) where a m and a s is the mean and standard deviation of the four corners of the two dimensional matrix A raw. Therefore the complex field at the object total scattered field is ψ correct = A norm exp ıφ fiber (6.5) The ψ correct is the total field comprising incident plane wave, scattered field generated by the whole fiber and diffracted wave generated by the fiber slit effect(background field).

132 6.3.3 Background removal In this section the method of removing the incident illumination field is introduced.the incident background field ψ in at the optical fiber center position can be obtained by simply blocking the reference light after removing the fiber. The phase of the incident field is zero. The measured amplitude distribution A in is normalized by its mean and standard deviation due to the shifting of the DC values of the camera. The computed total field ψ correct and incident field ψ in is not at the same plane in the space. Here we propagated the incident field from the center of the fiber to the camera plane and subtract the propagated field from the normalized field. The resulted field ψ t is the total field which consists of the scattered field generated by the whole fiber: core as well as the cladding. ψ t = ψ correct ψ in bp (6.6) which is illustrated by figure (6.23) as well as the simulated fields Distorted wave reconstruction In this section we consider using distorted wave inverse algorithm for cylindrical symmetric object to reconstruct the index refraction distribution of the fiber core. Assuming that the parameters of the circular symmetric background is known, one can estimate the background field generated by the fiber cladding ψ sc and remove it from the total field. The resulted field ψ B = ψ t ψ sc is the ψ B in equation (4.7) and is used to reconstruct the refractive index distribution of the core by DWBA generalized FBP algorithm. Due to the presence of noise, the Tikhonov regularization method is used in reconstruction algorithm and the results are shown in figure (6.24).

133 2 Figure 6.: Experimental setup of digital holographic microscopy Figure 6.2: A picture of experimental setup of digital holographic microscopy

134 3 Figure 6.3: Flowchart of data acquisition system for digital holographic interferometer Figure 6.4: A picture of experimental setup of PZT control system

135 4 Figure 6.5: An example of data processing program interface created by MATLAB R GUI Figure 6.6: Thin object imaging: the upper image illustrates the traditional imaging method while the lower image shows the numerical imaging process.

136 (a) (b) (c) Figure 6.7: Slit imaging via digital holographic microscopy. (a) interferogram; (b) amplitude of scattered field at image plane; (c) transmission function generated via wave expansion back-propagation method; (a) (b) Figure 6.8: line-per-inch Ronchi-ruling imaging via digital holographic microscopy. (a) Transmission function generated via wave expansion backpropagation method; (b) Correctional plot of transmission function along the y direction; (c) Three dimensional presentation of Ronchi-ruling s transmission function (c)

137 6 Figure 6.9: 5 line-per-inch Ronchi-ruling imaging via digital holographic microscopy. (a) Transmission function generated via wave expansion backpropagation method; (b) Correctional plot of transmission function along the y direction; (c) Three dimensional presentation of Ronchi-ruling s transmission function lpi Ronchi Ruling, 3D representation pixels(6.7µ m) pixels(6.7µ m) Figure 6.: Three dimensional representation of line-per-inch Ronchiruling via digital holographic microscopy.

138 7 Figure 6.: The DHM image of micro lens array (a) Hologram (b) Scattered field intensity only (c) Scattered field phase (d) 3D illustration of scattered field Figure 6.2: Transmission function of a cotton stem cell; Left: DHM image; right: conventional microscopic image.

139 8 Figure 6.3: Transmission function of a tendon; Left: DHM image; right: conventional microscopic image (a) (b) (c) (d) Figure 6.4: DHM images of a tendon. (a) intensity distribution of the scattered field of tendon at the camera plane; (b) hologram (c) Intensity of the backpropagated scattered field; (d) Phase of the back-propagated field

140 Figure 6.5: The fields propagating in the interferometer; 9

141 2 Figure 6.6: Five holograms generated by phase shifting holography for SMF-28 fibers Please select a valid area for phase calibration Figure 6.7: Phase calibration map for a SMF-28 fiber holographic imaging. A small area on the map is chosen for calculation;

142 2 Figure 6.8: The averaged intensity and phae distribution of the object beam complex field ψ raw over the camera aperture plane for repeated measurements Figure 6.9: 3D illustration of the averaged phase distribution of the object beam ψ raw over the camera aperture plane for repeated measurements

143 22 Figure 6.2: The averaged intensity and phase distribution of the incident field distribution over the camera aperture plane for repeated measurements Figure 6.2: 3D display of the modified phase distribution of fiber field

144 Normalized Intensity Detector Position Counter Figure 6.22: Normalized intensity of object leg complex field

145 Figure 6.23: Normalized intensity distribution of the scattered field component of the total field generated by the fiber over the detector plane. The dotted line represents simulated data obtained using the Bessel function expansion and the solid line is the actual experimental data.