INVESTIGATION OF LIQUID CRYSTAL SPATIAL LIGHT MODULATORS TO SIMULATE SPECKLE FIELDS

Transcription

1 INVESTIGATION OF LIQUID CRYSTAL SPATIAL LIGHT MODULATORS TO SIMULATE SPECKLE FIELDS Thesis Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree Master of Science in Electro-Optics by Jared Michael Cordray UNIVERSITY OF DAYTON Dayton, Ohio May 2010

2 INVESTIGATION OF LIQUID CRYSTAL SPATIAL LIGHT MODULATORS TO SIMULATE SPECKLE FIELDS APPROVED BY: Edward Watson, Ph. D. Advisory Committee Chairman Research Advisor AFRL/RY, WPAFB, OH Qiwen Zhan, Ph. D Committee Member Associate Professor, Electro-Optics Joseph W. Haus, Ph. D. Committee Member Professor & Director, Electro-Optics Malcolm W. Daniels, Ph.D. Tony Saliba, Ph. D. Associate Dean Dean, School of Engineering School of Engineering ii

3 ABSTRACT INVESTIGATION OF LIQUID CRYSTAL SPATIAL LIGHT MODULATORS TO SIMULATE SPECKLE FIELDS Name: Cordray, Jared M. University of Dayton Advisor: Dr. Edward Watson We investigate liquid crystal spatial light modulators as a means to simulate the speckle fields produced by laser light scattering off of rough surfaces. Of primary interest was the ability of these devices to accurately simulated the statistical properties of speckle fields. Characterization of the liquid crystal spatial light modulators was performed and a look-up table was created that specified the required voltage for a desired phased on a pixel-by-pixel basis. A model was created to simulate the field leaving the device and the resulting irradiance distribution in the far field. The 2nd and 4th moments of the field at the observation plane were calculated to determine the mean irradiance and contrast of the speckle pattern. Two random phase distributions that create the speckle patterns were investigated. These distributions were uniform phase distribution and "wrapped" Gaussian phase distribution. It was found that the devices are unable to simulate spatially stationary irradiance and contrast. Experimental investigations showed good agreement with the theoretical data except where σ < 0.1π. iii

4 ACKNOWLEDGMENTS I would like to take a moment to thank all the people that helped me on this thesis, without them I would not have able to complete it. First, I would like to thank my advisor Dr. Edward Watson who brought this project to me and his guidance and insight over the last few years. I would also like to thank Dr. Joseph Haus who spent many hours discussion problems that I had and whom had valuable insight for all matters of the work that was done. I would also like to thank Dr. Qiwen Zhan for being on my committee and answering technical questions with devices I was using. I would like to extend a special thanks to Nick Miller who also took time out of his schedule to let me bounce ideas off him and listen to every complaint I had along the way. I d also like to thank Drs. Igor Anisimov, Partha Banerjee, and everyone at LOCI who answered any questions along the way. I would also like to take the time to thank my family and friends whom without I would not be in the position I am today. This effort was supported in part by the U.S. Air Force through contract number FA , and the University of Dayton Ladar and Optical Communications Institute (LOCI). The views expressed in this article are those of the author and do not reflect on the official policy of the Air Force, Department of Defense or the U.S. Government. iv

5 TABLE OF CONTENTS ABSTRACT... III ACKNOWLEDGMENTS... IV LIST OF FIGURES... VII CHAPTER INTRODUCTION Problem Statement Previous Work Thesis Overview... 3 CHAPTER LIQUID CRYSTAL SPATIAL LIGHT MODULATOR CHARACTERIZATION Initial LC SLM Characterization Phase versus voltage characterization Temporal Frequency Response Polarization Response Pixel by Pixel Characterization CHAPTER FULLY DEVELOPED SPECKLE Model of LC SLM device Results for uniformly random distributed phase on each pixel Case when η = Case when η Numerical Discrete Phase results Experimental results v

6 CHAPTER PARTIALLY DEVELOPED SPECKLE Wrapped Gaussian Distribution Mutual Irradiance and 4 th Order Moment Speckle Simulations of the Wrapped Gaussian Distribution Experimental Results CHAPTER DISCUSSION/CONCLUSION Uniform Distribution Discussion Wrapped Gaussian Distribution Discussion Future Work Conclusion APPENDIX A nd Moment of the Field Mutual Irradiance APPENDIX B th Moment of the Field APPENDIX C MATLAB THEORETICAL CODE REFERENCES vi

7 LIST OF FIGURES Figure 2.1: Diagram of common path interferometer and setup Figure 2.2: Irradiance versus Digital Voltage plot of Hana device # Figure 2.3: Irradiance versus Digital Voltage plot of Hana device # Figure 2.4: Irradiance versus Digital Voltage plot of BNS device Figure 2.5: Phase versus Digital Voltage plot of Hana device #1. The blue dots represent the data and the red line is a best fit 3 rd degree polynomial Figure 2.6: Phase versus Digital Voltage plot of Hana device #2. The blue dots represent the data and the red line is a best fit 3 rd degree polynomial Figure 2.7: Phase versus Digital Voltage plot of BNS LC SLM device. The blue dots represent the data and the black line is a best fit 3 rd degree polynomial Figure 2.8: Irradiance fluctuation of the Hana devices Figure 2.9: Irradiance fluctuations of the BNS device Figure 2.10: Irradiance versus voltage for individual polarizations of x-(horizontal in) and y- (vertical in) only polarizations Figure 2.11: Diagram of setup with collimated beam and 4-f imaging system Figure 2.12: Variation in voltage map of BNS LC SLM response over its entire 491 x 500 array Figure 2.13: Variation in phase response over a 5 x 5 pixel area of the BNS LC SLM Figure 3.1: Example of speckle field without correlation of phase between points in the object. a) larger scale of speckle and b) a zoomed in version of (a)

8 Figure 3.2: Example of the a) mean speckle field without correlation of phase between points in the object and b) contrast for the speckle field with 1000 realizations Figure 3.3: Example diagram of LC device Figure 3.4: Far field irradiance pattern with η = 1. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations. The single realization case can be seen that the max intensity can be greater than 4 times the irradiance than the mean value but the average of the numerical coincides with the theoretical mean irradiance Figure 3.5: Far field irradiance pattern in a db scale with η = 1. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations Figure 3.6: Far field irradiance pattern in a db scale with η = 1 zoomed in around x=0. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations Figure 3.7: Contrast versus position plots (a) from theoretical (blue line) solution and numerical (red dots) simulations. (b) a zoomed in portion of the original data, note that the contrast never reaches Figure 3.8: Theoretical mean irradiance speckle pattern in 2-D on a normalized linear scale Figure 3.9: Theoretical mean irradiance speckle pattern in 2-D on a db scale. Notice the 2-D sinc pattern Figure 3.10: Theoretical mean irradiance speckle pattern in 2-D on a db scale (3-D examining of irradiance values) Figure 3.11: Far field irradiance pattern with η = 0.9. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations Figure 3.12: Far field irradiance pattern in a db scale with η = 0.9. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations viii

9 Figure 3.13: Far field irradiance pattern in a db scale with η = 0.9 zoomed in around x=0. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations Figure 3.14: Contrast versus position plots (a) from theoretical (blue line) solution and numerical (red dots) simulations. (b) a zoomed in portion of the original data, again notice again the contrast never reaches Figure 3.15: Theoretical speckle field irradiance in 2-D on a db scale with η = Figure 3.16: Theoretical speckle field irradiance in 2-D on a db scale with η = 0.9. (3-D examining of irradiance values) Figure 3.17: Linear numerical results (red) with 5 equally spaced discrete phase values between 0 and 2π were applied to LC SLM against theoretical (blue). a) is a single realization and b) is the mean of 1000 realizations Figure 3.18: Logarithmic numerical results (red) with 5 equally spaced discrete phase values between 0 and 2π were applied to LC SLM against theoretical (blue). a) is a single realization and b) is the mean of 1000 realizations Figure 3.19: Experimental setup for imaging the far field create from the LC SLM Figure 3.20: a) Theoretical speckle field irradiance in 2-D on a db scale with η = 0.9. b) Experimental far field average irradiance pattern of 100 realizations. Notice the irradiance range between diffraction orders (0 th and 1 st ) is about 20 db for both the theoretical and experimental results Figure 3.21: Cross-section of experimental data in Figure a) y = m and b) y = m Figure 3.22: (a) Theoretical speckle field irradiance in 2-D on a db scale with η = 0.9. (b) is the experimental results showing an overall continuous smooth transition between points Figure 3.23: Cross-section of Figure 3.22b with y=0.001 m Figure 3.24: (a) Theoretical speckle field irradiance in 2-D on a db scale with η = 0.9. (b) is the experimental results where the range of irradiance value of the diffraction and surround sinc pattern is the same as the theoretical ix

10 Figure 3.25: Experimental speckle contrast results. a) 2-D imaged results from the CCD camera and b) cross-section of (a) at approximately y = 0.001m Figure 4.1. PDF for different values of σ in Wrapped Gaussian Figure 4.2: Plots for σ = 5π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position Figure 4.3: Plots for σ = π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled Figure 4.4: Plots for σ = 0.005π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled Figure 4.5: Plots for σ = 0.01π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled Figure 4.6: Plots for σ = 0.05π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled Figure 4.7: Plots for σ = 0.1π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled Figure 4.8: Plots for σ = 0.5π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled Figure 4.9: Experimental results of far field with σ = 0.1π where the range of irradiance values between the diffraction orders is about 17 db Figure 4.10: Experimental result of far field with σ = 0.05π where the 1 st diffraction order still contributes away from Y=0 axis x

11 Figure 4.11: Experimental result of far field with σ = 0.01π where the 1 st diffraction order no longer contributes away from Y=0 axis xi

12 CHAPTER 1 INTRODUCTION Optical radars based on transmitted and received laser pulses (laser radar or ladar) can measure many features of an object of interest. Ladar has been used in place of radar to gather more information about the object of interest since the wavelength is much smaller. Ladar can generate high resolution topological images and can also generate three dimensional images of distant objects. In laser radar systems, speckle is a main artifact seen in the measurements [1]. Speckle is the rapid variation in irradiance that is observed when a rough surface is illuminated with highly coherent laser light. This rapid variation appears much like that of noise in a system. The roughness of these surfaces is microscopic but could be many wavelengths in height variation for laser illumination. Therefore, when coherent light is incident onto the surface of a material, the random variations in the roughness causes different optical path differences associated with them. At the receiver the light from a rough surface will interfere but the optical path differences cause a highly random irradiance pattern. The effects of so-called fully developed speckle can be quite severe, producing signal-to-noise ratios of 1. 1

13 1.1 Problem Statement It is desirable to simulate the effects of speckle as part of a ladar test bed. We are interested in developing a technique to simulate such effects in particular the variable and dynamic effects of speckle on moving sensors. A simulator that can be controlled electronically would be very beneficial and eliminate the need to have all materials that are desired to be tested on hand and manual insert each into test bed for each specific test. Electronic control will also allow us to simulate the dynamic effects of motion of a material. In this work we wish to develop a speckle field simulation tool that will produce the statistical properties of surfaces of interest. Specifically, we investigate liquid crystal spatial light modulator (LC SLM) devices as a means to impart desired statistical properties to laser illumination. LC SLMs will first need to be studied and characterize to determine and quantify their ability to electronically control phase modulator across each pixel for producing speckle fields. This will be done first by characterizing the device for phase response to input voltage, temporal response, and discreteness of pixels and available phase values. Once characterization has been completed, models of the device s far field irradiance pattern will be evaluated and compared to experimental results from the LC SLM device. 1.2 Previous Work Previous work has investigated the use of phase modulators for laser beam steering [2][3]. Such operation does not use the phase control in a statistical manner. The use of such devices has also been postulated in the statistical 2

14 control of coherence and polarization properties of illumination beams[1], but realistic limitation of the devices were not considered or demonstrated. We now want to assess their utility as random phase modulators to simulate speckle fields in a ladar system. Three properties of the LC SLM are of interest with the first being their ability to impart the correct statistics to the illumination wavefront (we will eventually look at statistics up to the second order). The second property is the temporal effects that the device may impart to the illumination wavefront. The final property is the effect of the discreteness of the device, both in terms of the spatial pixels as well as the finite number of phase values that can be imparted. While not considering application to electronically controlled SLMs, a recent paper has looked at binary gratings with random heights variations [9]. The results in this paper are essentially a subset of the work done in this thesis. In our work, the random phase imparted by the active part of the LC SLM pixel can be considered the same as the phase change in a random binary grating. The work in Ref. [9] studied the far field pattern and the effects on the diffraction orders, where as our focus was on the speckle field created from this randomness effect. 1.3 Thesis Overview This paper is organized in the following manner. In Chapter 2, LIQUID CRYSTAL SPATIAL LIGHT MODULATOR CHARACTERIZATION, we discuss the LC SLM devices under test and their characterization. Characterization is done on phase versus input voltage, temporal response and polarization effects. In Chapter 3, FULLY DEVELOPED SPECKLE, we describe some aspects of 3

15 speckle phenomena with a uniform random distribution of phase. A model for LC SLM devices is derived as used to calculate the mutual irradiance and its contrast when uniformly distributed phase is imparted to the device. Numerical simulations are compared to theoretical calculations and experimental results. Lastly, experimental data was gathered with the devices to compare to the theoretical solution. In Chapter 4, PARTIALLY DEVELOPED SPECKLE, we discuss partially developed speckle resulting from a Gaussian distribution of phase. In Chapter 5, DISCUSSION AND CONCLUSION, we come to the conclusions of the work and discuss future work that could be done. 4

16 CHAPTER 2 LIQUID CRYSTAL SPATIAL LIGHT MODULATOR CHARACTERIZATION In examining the use of liquid crystal (LC) devices for a speckle simulator, more needs to be understood about them. Liquid crystals are a state of matter containing properties of both a liquid and a solid. The molecules still have mobility, yet have a fixed long-range orientational order like that of a crystalline solid. There are different types of schematics for liquid crystal structures but the one used in our LC SLMs are nematic LC, in which the molecules do not align along a particular axis but are oriented in the approximately the same direction. LC SLM devices are driven by electrical signals, in which the orientation of the liquid crystals aligns in the direction of the electric field and is dependent on voltage. Our LC SLM device creates a phase-only modulator for incident light with polarization aligned with the extraordinary axis. The constant index of refraction, n o, is the ordinary axis (fast-axis, i.e. x-axis) and the changing index, n e, is the extraordinary axis (slow-axis, i.e. y-axis) where the LC molecules rotate within the y-z plane [8]. Therefore, in theory each electrode (pixel) applying voltage across the LC creates a phase modulation which can be used to simulate 5

17 different surfaces. The number of pixels on the LC SLM device determines the number of individual phase modulators and the phase variability in the field profiles that can be applied. LC SLM devices also have problems with thermal excitation causing the liquid crystals to have a random orientation/movement affecting the response of the LCs. This can be neglected if the device is heated to a constant temperature which will be unaffected by the surrounding environment during each use creating consistent and repeatable results. The two device types we investigate in this work have been used as phase modulators for laser beam steering [2][3]. The use of such devices has also been postulated in the control of coherence and polarization properties of illumination beams [4]. In this thesis we want to assess their utility as random phase modulators to simulate rough objects in a ladar system. Three properties are of interest. The first is the ability of the modulators to impart the correct statistics to the illumination wavefront (we will eventually look at statistics up to second order). The second property is the temporal effects that the device may impart to the illumination wavefront. The final property is the effect of the discreteness of the device, both in terms of the spatial pixels as well as the finite number of phase values that can be imparted. The devices that were investigated were two Hana Microdisplay LCD, that were modified to perform as laser beam steering devices, and a Boulder Nonlinear Systems XY Phase Series Model P512 spatial light modulator. The Hana devices consist of an array of 1024 x 768 of pixels with an active area of 20 mm x 15mm giving a pixel (electrode) pitch of approximately 19.5 µm x 19.5 µm. The 6

18 Hana devices are controlled by inputting a gray scale image from a computer video output to the control box of the device. The control box converts the 8-bit grayscale digital image to an analog voltage across the corresponding electrode on the device. The BNS LC SLM is an array of 512 x 512 of pixels (electrodes) with an active area of 7.68 mm x 7.68 mm giving a pixel (electrode) pitch of approximately 15 µm x 15 µm. Again, 8-bit grayscale images are uploaded to the BNS control electronics to define the voltages to be applied to the pixels. Grayscale values ranging from were used to capture the complete dynamic range of the device. The other bit in the 8-bit control is used to DC balance the device. The BNS LC SLM has a heating element embedded to keep the LC at a constant temperature and ensure a consistent phase response is being applied from day to day in our experiments. 2.1 Initial LC SLM Characterization Phase versus voltage characterization One important characterization of the liquid crystal devices is to understand the optical path difference of the slow axis, and hence, the resulting phase produced as a result of the applied voltage. We desire to simulate random phase profiles with specific distributions on the LC SLM. For example, we may want a uniformly distributed phase between 0 and 2π to be implemented across the device. However, unless there is a linear relationship between the applied voltage and the resulting phase, a uniformly distributed voltage will not result in a uniformly distributed phase[6]. The goal of this experiment was to quantify the basic phase 7

19 versus voltage of the device over an area the approximate size of the laser beam diameter. To perform this characterization we use a common path interferometer as illustrated in Figure 2.1. This type of interferometer has been used by Harris [6] and only measures the difference between the two polarizations which neglects all other variations seen by both polarizations. Since the LC device has a fast- (ordinary) and slow-axis (extraordinary), an interferometer system is setup up to have input and output polarizations that are crossed at a relative ± 45 o to the slow-axis (y-axis of the system). The x-component is unaffected by the SLM, whereas the y-component experiences phase modulation as a function of the voltage applied to the device. Jones calculus was used to show that after the second polarizer the phase-modulated y component is essentially mixed with the unmodulated x component to produce an interference pattern. The difference is phase between the y-polarization (slow axis) and the x-polarization(fast axis) will determine the irradiance of light that is measured at the detector of the system. The phase change that occurs to the y-polarization can then be calculated using the formula [6] I( φ ( V ) ( V ) 1 2cos( φ ) ) = A + B (2.1) 2 8

20 Figure 2.1: Diagram of common path interferometer and setup. where A is the DC irradiance, B the varied irradiance, the phase, V is the digital voltage, and is the measured irradiance at the output of the system. Derivation of Equation (2.1) assumed that the energy of both the x- and y-polarization are equal. Subtracting out the minimum output irradiance and normalizing, the values A and B are removed from the formula and algebraically solving for, the following formula is found φ( V ) = cos 1 (1 2 * I( φ( V ))) (2.2) The measured irradiance is a function of the digital volts (voltage index) that has been applied to the LCD, therefore, is a function of the digital volts. A HeNe laser at 543 nm was used to illuminate a small section of each device under test. An EOT silicon PIN fast detector coupled to an SRS Model SR570 current amplifier recorded the output irradiance. Stepping through all 256 input voltages for the HANA LC SLM (128 for the BNS), the irradiance was found at 9

21 each voltage. The normalized irradiance was found as a function of digital volts by subtracting the minimum irradiance and normalizing. Typical examples for each device are shown in Figure Figure 2.4. Note that the BNS device used only 128 digital volt step to achieve its full dynamic range. As seen below, all three devices have different responses to the input digital volts. Both Hana devices had noncontiguous sections in their irradiance functions which needed to be taken into consideration when determining the phase change that is occurring at a specific digital voltage. The normalized irradiance was used in Equation (2.2) to find the phase response of each of the devices. The phase values where then unwrapped into a one-toone function by adding each 0 to π section to the previous. The phase for the Hana devices and the BNS device are shown in Figure Figure 2.7, respectively. It becomes apparent what the discontinuities to do the phase profile in looking at Figure 2.5 and Figure 2.6 where no portions of the Hana LC SLM devices have a continuous region over a range of 2π. Figure 2.2: Irradiance versus Digital Voltage plot of Hana device #1. 10

22 Figure 2.3: Irradiance versus Digital Voltage plot of Hana device #2. Figure 2.4: Irradiance versus Digital Voltage plot of BNS device. Figure 2.5: Phase versus Digital Voltage plot of Hana device #1. The blue dots represent the data and the red line is a best fit 3 rd degree polynomial. 11

23 Figure 2.6: Phase versus Digital Voltage plot of Hana device #2. The blue dots represent the data and the red line is a best fit 3 rd degree polynomial. Figure 2.7: Phase versus Digital Voltage plot of BNS LC SLM device. The blue dots represent the data and the black line is a best fit 3 rd degree polynomial Temporal Frequency Response It was observed while investigating the phase versus voltage characteristics of the Hana devices that there were temporal phase fluctuations. Temporal fluctuations of the voltage lead directly to fluctuations of the phase that is being applied to the device. The voltage that is used to apply a desired phase to a pixel is a pulsed signal and the electric field with decrease between pulses 12

24 causing phase to change (phase drooping). We examined this in more detail with a Lecroy WaveJet 354 oscilloscope coupled to a detector with sample rates up to 2 GS/s. The experimental setup shown in Figure 2.1 was used. The digital voltage was held at a constant value. The irradiance was recorded from the output of the system as a function of time to be able to find the temporal response of the devices. As shown in Figure 2.8, it was found that the Hana devices produce a 60 Hz irradiance fluctuation along with harmonics of that frequency. We believe this is due to the way in which the voltages applied to these devices are refreshed (keep in mind these devices are manufactured to be display devices, so they are refreshed at a common video rate). The BNS device was found to produce irradiance fluctuations as shown in Figure 2.9, with the lowest frequency being at 500 Hz. Figure 2.8: Irradiance fluctuation of the Hana devices. 13

25 Figure 2.9: Irradiance fluctuations of the BNS device Polarization Response To further investigate the variations observed in the BNS device we looked at depolarization resulting from reflection from the device. The configuration shown in Figure 2.1 was used, but the first polarizer was rotated so that the input to the device was polarized parallel with the x-axis(fast axis) of the device. The output polarizer was crossed with respect to the input polarizer. In this configuration there should be no variation in the measured irradiance as the digital voltage to the device is varied. However, it was observed that there is indeed a variation in the measured irradiance. Next, the input polarizer was rotated so that the illumination was polarized parallel to the y-axis(slow axis) of the device, and the output polarizer was crossed with respect to the input polarizer. Again, no variation in irradiance was expected as a function of digital voltage applied. However, variations were again observed. The data from these two experiments are shown in Figure The polarization effects are most likely due to the liquid crystals not perfectly twisting in the slow- and optical-axis plane (yz-plane) 14

26 and could need to be examined further to determine the exact affects that could arise from this problem. Figure 2.10: Irradiance versus voltage for individual polarizations of x-(horizontal in) and y-(vertical in) only polarizations. 2.2 Pixel by Pixel Characterization Figure 2.11: Diagram of setup with collimated beam and 4-f imaging system. To characterize the LC SLM devices for the application of interest requires an understanding of the phase versus voltage for every pixel in modulator. To 15

27 accomplish this characterization we set up an imaging version of the common path interferometer as illustrated in Figure A 4-f imaging system was used to image the BNS LC SLM onto the Pointgrey Grasshopper CCD camera. To improve the sampling of the wavefront we adjusted the magnification of the imaging system by adjusting the ratio of the focal lengths in the 4-f system. The first lens had a focal length of f = 200 mm and the second a focal length of f = 175 mm, to get a magnification that mapped one BNS LC SLM pixel onto an area slightly larger than a 2x2 array of pixels on the CCD which satisfies the Nyquist criterion. The BNS LC SLM was placed on a rotational stage along the z-axis (a.k.a. optical path) such that the coordinates of the LC device were aligned with those of the CCD camera. The laser illumination was spatially filtered with a 25 μm pinhole and then collimated by a doublet lens with focal length of f c = 500 mm. The collimated beam was large enough to illuminate the entire surface of the BNS LC SLM. A uniform digital voltage was applied to the BNS device and the irradiance at the CCD camera was recorded. The camera averaged 50 data images for each digital voltage. The CCD was aligned by capturing the top left corner of the BNS LC SLM to ensure proper mapping of the pixels. The images then had to be resized (with bilinear interpolation) and crop out the area of the image that did not pertain to the BNS LC SLM. Therefore each pixel in the final image represents a pixel on the BNS LC SLM though not all pixels could be captured on the devices leaving us with a 491 x 500 array of measured pixels. The phase was then found as a function of digital voltage for each pixel. It was observed that there were variations in the phase versus voltage curves from pixel 16

28 to pixel. Figure 2.12 shows the LUT values for a constant phase of π/2 for the LC SLM device where the colorbar represents the 8-bit voltage to be applied to the pixel. It is seen that the voltage mask values vary over a range of 20 digital voltages and therefore a default characterization of the LC SLM to describe the phase response of the device in its entirety cannot be done. The known RMS WFE is approximately 0.01λ and other constant phase masks are on the same order of magnitude showing that the discrete phase values do not have much impact on the phase profile. Figure 2.12: Variation in voltage map of BNS LC SLM response over its entire 491 x 500 array. Looking at a 5 by 5 area of pixels, a scatter plot was created to look at changes over a small area of the BNS LC SLM. Figure 2.13 shows an example of the variation observed. As seen in the figure, there is larger variation around the voltage indices corresponding to the maximum irradiances. A simplified phase versus voltage curve for small regions could not be created for the LC SLM 17

29 device. Therefore, a look-up table (LUT) for each pixel was saved to be able to apply desired phase. Figure 2.13: Variation in phase response over a 5 x 5 pixel area of the BNS LC SLM. 18

30 CHAPTER 3 FULLY DEVELOPED SPECKLE When continuous-wave lasers became commercially available in the early 1960 s, scientist using these lasers noticed at the time a strange phenomenon which has now been defined as speckle[1]. In the use of coherent light, when the light is reflected off a surface that is rough on a scale that is comparable to the wavelength, surface a granular pattern is noticed in the observation plane. The granular pattern is made up of random bright and dark spots created from constructive and destructive interference of light reflected from multiple regions of the surface. Note that interference is dependent on polarization and throughout the remaining part of this thesis the coordinates will be chosen such that the polarization is in one dimension, i.e. just y-polarized light. When light is incident on a rough surface with height fluctuations that vary randomly over multiple wavelengths, the phase term in the signal at the object plane becomes a function of position. Observing the light after propagation into the far field, a Fourier transform, the light becomes a speckle field as the random phase terms from the object plane sum together at an observation point. Picking a point in the observation field, there will be a summation of different wavefronts. The 19

31 summation of these wavefronts, will define the electric field at that point. The following derivation has already been done by Goodman [5] but is being shown here to illustrate the concept of the work being done. At a particular point in an observation plane, the electric field can be written as (3. 1) where is the frequency of the light, is time, is the position vector ( ), and is represented as (3. 2) where is the amplitude of the field and is the phase. The field at a point will be the summation of a large number of wavefronts, which would be in the form of (3. 3) where N is the number of wavefronts incident on the point, n is the number of the wavefront, is the amplitude of the individual wavefront, and is the phase of the individual wavefront. As mentioned earlier, will be a random variable. Goodman assumes that a n is Gaussian distributed and φ n is uniformly distributed over 2π. Also that the two variables are independent of each other and other elements and well as N. The analysis of the amplitude and, thus, the intensity has been done by Goodman in [5] and is shown below. The wavefronts can be broken into real, R, and imaginary, I, parts which are sums of independent random (contributions) 20

32 variables. With a large enough sample, N., the central limit theorem [5] states that both, R and I, will be Gaussian distributed random variables. They can be represented by (3. 4) (3. 5) Looking at the average of the random variable, using the assumptions that a n and θ n are independent random variables, we can see that the average of R is given by (3. 6) If θ n is a uniformly randomly distributed variable from -π to π, then the assemble average of cos θ n is 0 due to averaging over a circle. Therefore, the mean of the random variable, R, is also zero. The second moment of R can been written (3. 7) For the case when m not equal to n, then the two cosine terms become separable due to being independent of other elements and lead to a value of 0. For the case when m = n, then the term becomes a cosine square which leads to. So 21

33 (3. 8) where a = <a n >, assuming a n and a m for all m n are independent. A similar analysis for random variable I can easily be shown to produce the same results. (3. 9) Calculating the correlation between R and I, we have (3. 10) It is easily shown that when m = n and m n, the average of the θ terms goes as (3. 11) Now examining the joint density function of the two parts, real and imaginary, implies they are uncorrelated, therefore, the Gaussian PDF becomes (3. 12) where. A transform of variables can be made using the definition of a = (r 2 + i 2 ) 1/2 and θ = tan -1 (i/r). The Jacobian for the transform is 22

34 (3. 13) A key assumption of fully developed speckle is that the distribution of phase is uniform. Then (3. 14) Being that θ is only defined from -π to π, integration of Equation (3. 12) can be done over θ to get the PDF in terms of just the variable a, which looks like (3. 15) Again, doing a transform of variables and setting I = f(a) = A 2 and p(u) = p(f - 1 (v))dv/du, the PDF is (3. 16) The instantaneous intensity obeys a negative exponential probability density function. This distribution has the important property that its standard deviation σ I = <I> = 2σ 2, so the final form of the PDF is (3. 17) This derivation leads to spatially stationary results. Speckle can be characterized by a term to show its severity call speckle contrast. Speckle contrast, C, is defined as C = 1/SNR = 1/μ/σ = σ/μ [1]. C = 1 would also mean that the SNR is also one, and is the case for fully developed speckle. The smaller the speckle contrast, the less fluctuation of irradiances values at a point, therefore called partially developed speckle. In this case, the signal can still be 23

35 detected in the irradiance pattern. For example, if there is a specular component on the incident light, the waves incident on the point will have a more stable irradiance. Goodman shows that if a constant phasor is imbedded in a random field then the irradiance pattern will have a modified Rician density function[1] and the results will no longer be spatially stationary. Figure 3.1: Example of speckle field without correlation of phase between points in the object. a) larger scale of speckle and b) a zoomed in version of (a). Figure 3.2: Example of the a) mean speckle field without correlation of phase between points in the object and b) contrast for the speckle field with 1000 realizations. The default speckle pattern in Figure 3.1 shows one realization of a speckle field with no correlation between points in the object. Figure 3.2 shows that the mean speckle field and contrast is spatially stationary such that points taken anywhere 24

36 along the spatial coordinate has the same statistics. The contrast can also be seen to tend to Model of LC SLM device Figure 3.3: Example diagram of LC device. First, equations were derived to get the field just after the BNS LC device which differs from Goodman s derivation which was for fully developed speckle and had spatially stationary results. In deriving a model for the LC SLM it was noticed that it could be related to that of a binary grating with random heights [9]. Unlike that of a binary grating modeled in the reference, the LC device is a 2 dimensional array that has a periodic structure. Also, the grating has a binary phase variation on which the random phase is impressed, while our device is a mirror (flat surface/full aperture) on which the random phase is impressed. The 1-D periodic structure is illustrated in Figure 3.3. A 1-D solution is evaluated for simplicity and can easily be converted to a 2-D equation from the 1-D solution, as will be shown later. In defining the field directly after the LC device, it would be a summation of rectangle functions and at a 100% fill factor would look like 25

37 (3.18) where x is the position in the SLM plane, d is the width of the pixel (pixel pitch), and n is an integer that represents each pixel. Notice that the phase is constant on all pixels. In practical LC devices, the pixel cannot be completely filled due to that if the electrodes were touching they would both hold the same voltage and be unable to both produce unique phase modulation. Therefore, a LC pixel will have a fill factor which will be call η. With this information, Equation (3.18) can be modified to include η. Also, since different voltages/phases will be applied to each pixel a random phase term must also be added. The equation that now represents the active area of the BNS LC SLM is as follows (3.19) where is random phase difference at the n th pixel from the off state. Now, there are areas of the BNS LC SLM that are not represented by the equation, which are the inactive areas of the device (i.e. non-working area) which is actually 1 fill factor, η, so now modifying the original Equation (3.18), we can represent the inactive area as (3.20) 26

38 The equation models the full pixel pitch subtracting out the active area. Notice, that there is no phase term since the phase defined in the active pixel equation, (3.19), is the difference between that pixel and the default constant phase incident on the device. Now the summation of Equations (3.19) and (3.20) represents the field directly following BNS LC device in its entirety. Hence (3.21) where x n = (2n-1)d/2. The field is then propagated to the far field (i.e. Fourier Transform). Applying the Fraunhofer kernel and separating the terms gives (3.22) where ξ x = x 2 /λz, λ is the wavelength, z is the distance from object (SLM) plane to observation plane, and x 2 is the spatial coordinate at the observation plane. Using Fourier transformation tables, three useful transforms are: (3.23) (3.24) (3.25) The transform can easily be done to give 27

39 (3.26) where the sinc function has been defined as sinc(x) = sin(πx)/πx. Multiplying this result with its complex conjugate will give the instantaneous irradiance pattern for the specific phases that were applied to the pixels. To get the average irradiance pattern, the second moment of the field, <A n ( )A * m( )>, must be calculated. Since, the irradiance is wanted we set ξ = ξ and the equation becomes (3.27) The algebra is straight forward but extensive and is shown in APPENDIX A. The final form is 28

40 (3.28) In looking at the possibility of obtaining fully developed speckle results, a random uniformly distribution of phases will need to be produced at the object plane. As showed earlier, fully developed speckle should have a contrast of 1. Adding a random uniformly distributed phase to each pixel of the LCD will simulate to some degree a uniform beam incident onto a rough surface as Goodman had assumed previously. The terms and then both go to zero due to the uniform distribution over 2π and averaging over a circle. In looking at the term, it is seen that when n = m, the ensemble average is one (with N equal terms) and when n m it is zero. We define a short-hand notation (3.29) where N is the total number of pixels and taking note that if b is omitted then b = 1. This equation denotes the far field diffraction pattern of the total device. Equation (3.28) then becomes 29

41 (3.30) To extend to a 2-D device we define the field directly after the LC device as a direct extension of the 1-D case: (3.31) where n and j are indexes representing individual pixels. Noting that the equation is separable in x and y, the procedure to define the far field irradiance is the same as in the 1-D case. The final result of the 2-D mutual irradiance (mean irradiance) in the far field is (3.32) where ξ x = x 2 /λz and ξ y = y 2 /λz and x 2 and y 2 are the coordinates of the observation plane. Again for the case that the phase at each pixel is uniformly random distributed over 2π region, the equation simplifies to 30

42 (3.33) Finding the contrast of the resulting speckle pattern involves even more elaborate algebra than the average irradiance since the fourth moment of the field must be calculated, which is the second moment of the irradiance. The fourth moment of the field is (3.34) The equation has 15 cases that need to be examined to get the final form of the fourth moment. In looking at a uniformly random distributed phase it can be simplified greatly. The math is shown in APPENDIX B with the final form of the 4 th moment for a uniform distribution as follows 31

43 (3.35) For contrast, the standard deviation is needed, this is found by (3.36) Using Equations (3.30) and (3.35), the standard deviation, σ is found to be (3.37) The final form for the contrast of the LC SLM device is (3.38) where N is number of pixels along one dimension. Goodman has shown that a constant and random phasor sum with a distribution that follows circular Gaussian statistics can be defined in a much simpler matter[1]. Note that our distribution is a uniformly randomly distributed phase with unit amplitude. The PDF of his solution can be simplified to (3.39) 32

44 (for I 0) where is the average intensity of the random phasor, I 0 is the intensity of the constant phasor, is the radio of I 0 to (I 0 / ) (also defined at beam ratio in holography). Looking back at the mean irradiance in Equation (3.30), it is seen that there is a portion that has a constant assemble average factor of 1 which is the constant (known) irradiance and a portion defined by the random variables which is the average irradiance of the random phasor. Therefore, the beam ratio can be written as (3.40) Goodman showed contrast in this case to be [1] (3.41) Substituting Equation (3.40) into Equation (3.41) gives (3.42) Simplifying Equation (3.42) gives (3.43) The only difference between the contrast defined by Equation (3.38) and in Equation (3.43) is a factor of (-N). It can be seen that as N, then the 33

45 equation (3.38) converges to Equation (3.43). N is defined as the number of individual wavefront contributors that are included in the sum of the field at the observation plane. In the case of the LC SLM, N is a discrete number of wavefront contributors, the BNS LC SLM has N = 512x512 = , and the pixels are larger than a wavelength producing a correlated area across each pixel. In real natural objects, the wavefront contributions come from points smaller than the wavelength and the number of points increases rapidly with slight increases in the illuminated area of the surface. Therefore, with a illumination spot size that would cover the entire LC SLM would have magnitudes of more point contributing to the observation plane. As was stated earlier in the chapter, the default speckle pattern derived by Goodman (without correlations of phase between points of the object) is spatially stationary, but individual pixels in the LC SLM device will be correlated across their whole area causing the irradiance pattern to not be spatially stationary. 3.2 Results for uniformly random distributed phase on each pixel In addition to the theoretical model, a numerical solution was found by creating an object in MATLAB which was an array of one s in length resulted from modeling each pixel as 20 elements long, hence 20x512 = The randomly distributed phase was then applied to the pixels with the fill factor determining how many of the 20 elements implemented the randomly distributed phase. The far field irradiance distribution was calculated using the FFT (Fast Fourier Transform) function in conjunction with the FFTSHIFT function to shift and center the results in MATLAB. In using the FFT, a scaling factor for position 34

46 was done to get the correct coordinates for the results. The position (x-axis) scale was found by using the fact that the FFT uses two points to do its sampling. Therefore, the highest frequency, f max, is 1/(2*dx), where dx is the distance between two points in the created object. To convert the coordinates back to spatial, multiply f max by λz to get x max and dx = x max /(N/2), where N is the number of points that were used in the FFT. By assigning f max to the end of the position array and decrease the preceding elements by a value of dx, a spatial position array the same size of your results for the FFT is created. An average over 1000 realizations was performed in the numerical solution to get a reliable result for the average intensity. The mutual irradiance and numerical solution were evaluated for two cases, one when η = 1 and another when η = Case when η = 1 Using the model described by Equation (3.30) with η = 1, the mean speckle irradiance pattern in the far field simplifies to a single sinc function representing the irradiance pattern associated with an aperture the size of a pixel width of the LC device do to the phase being correlated over this area. The results are shown as the solid blue line in the figures below. A numerical solution was also calculated. The y-axis of the numerical solution was scaled by fitting the average to a sinc function, knowing that the data represents a sinc function, and then normalizing the data to the average amplitude. The results are shown in Figure Figure 3.6, with a) representing one realization in the numerical simulation and b) the average over 1000 realizations. 35

47 Figure 3.4: Far field irradiance pattern with η = 1. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations. The single realization case can be seen that the max intensity can be greater than 4 times the irradiance than the mean value but the average of the numerical coincides with the theoretical mean irradiance. Figure 3.5: Far field irradiance pattern in a db scale with η = 1. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations.. 36

48 Figure 3.6: Far field irradiance pattern in a db scale with η = 1 zoomed in around x=0. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations. It is seen that there is good agreement between the numerical and analytical results. Notice that the mean irradiance value is not the same over all spatial coordinates like that done in Goodman s derivation that are shown in Figure 3.1 and Figure 3.2. The mean contrast for spatial coordinates using Equation (3.38) is shown by the blue line and the numerical simulations represented by red stars are shown in Figure 3.7. A zoomed in version of these results is shown in Figure 3.7b, notice that the contrast is theoretically less than one, this is due to the factor of N discussed previously between Equations (3.38) and (3.43). Ten different 1000 realizations simulations were done and the mean contrast was calculated to be where the theoretical value was

49 Figure 3.7: Contrast versus position plots (a) from theoretical (blue line) solution and numerical (red dots) simulations. (b) a zoomed in portion of the original data, note that the contrast never reaches 1. In examining the 2-D solution, a computing problem arose. Due to limited memory and computing power, numerical results were unable to be done for the 2-D case. Also, the resulting theoretical spatial distances had to be shortened due to the same reason. Comparing the 2-D to the 1-D theory, it is easily seen that the 2-D results are two 1-D cases, x and y coordinates, multiplied together. The results for the 2-D mean irradiance speckle pattern are shown in Figure Figure Figure 3.8: Theoretical mean irradiance speckle pattern in 2-D on a normalized linear scale. 38

50 Figure 3.9: Theoretical mean irradiance speckle pattern in 2-D on a db scale. Notice the 2-D sinc pattern. Figure 3.10: Theoretical mean irradiance speckle pattern in 2-D on a db scale (3-D examining of irradiance values) Case when η 1 A realistic LC SLM will not have unity fill factor. Using the model described by Equation (3.30), the case of η 1, specifically η = 0.9, is examined. (Note that the SLM device used in the experiments has η = ). One thousand realizations were done numerically with uniformly distribution random phase across the pixels. The average was normalized by the peak which was at the 0 th diffraction order, so that the scale was the same as the theoretical average. The 39

51 same scaling factor was used to go back and normalize the single realization. Again, the mean irradiance of the far field is seen as a solid blue line in the Figure Figure 3.13, below with a) showing one realization of the numerical simulation and b) showing the average of 1000 realizations. Figure 3.11: Far field irradiance pattern with η = 0.9. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations. Figure 3.12: Far field irradiance pattern in a db scale with η = 0.9. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations. 40

52 Figure 3.13: Far field irradiance pattern in a db scale with η = 0.9 zoomed in around x=0. The theoretical average is in blue and the numerical data red. (a) a single realization of a numerical and (b) is the average over all 1000 realizations. Diffraction orders are now observed in the irradiance pattern like that of a grating due to the periodic structure of active area of the pixels having different phases and the inactive area having a constant phase. The location of the diffraction orders can be calculated using the period of pixel pitch, d, and using the formula (3.44) where m is the order of diffraction (integer), λ is the wavelength, z distance from LC SLM to observation plan, and x diff is the distance from the 0 th diffraction order in the observation plane. The average speckle pattern beneath the diffraction orders is again a sinc function based on the size of the active pixel width. The zeros of the squared sinc function can be found by the equation (3.45) where n is an integer, η is the fill factor, and x sinc is the distance from the center of observation where zero irradiance occurs. Since, x diff and x sinc are measureable; a check can be done to confirm the fill factor and pixel pitch of the SLM. If looking at the initial case of η = 1, the diffraction orders and the zeros of 41

53 the sinc function occur at the same point, leading to the fact that the diffraction orders would not exist. The mean contrast for spatial coordinates can be found using Equation (3.38) with η = 0.9 and is represented by a blue line and the numerical contrast is represented by red stars in Figure A zoomed in version of these results is shown in Figure 3.14b, notice that the contrast, theoretically, again never reaches 1. In addition to contrast for η 1, there are spatial coordinates where the contrast drops close to zero. The positions where the numerical results match the theoretical occur at the diffraction orders because it is a constant (specular) phasor which the positions can be verified with Equation (3.44). The other positions where the contrast is very low in the analytical solution is due to the zeros in the sinc pattern associate with the pixel size and can be verified with Equation (3.45). Though the results are easily seen in the theoretical data, they in fact occur at very narrow regions about the sinc s zero s and with the finite sampling in the numerical simulations (and also in experiment) they are not observed. 42

54 Figure 3.14: Contrast versus position plots (a) from theoretical (blue line) solution and numerical (red dots) simulations. (b) a zoomed in portion of the original data, again notice again the contrast never reaches 1. The mean irradiance speckle pattern for the 2-D case in far field is shown in Figure 3.15 and Figure Note the linearly scale data has not been shown due to the diffraction orders being orders of magnitude higher. A point to notice is the region between diffraction orders where the contrast would all be approximately 1 could be a preferable region to evaluate. Figure 3.15: Theoretical speckle field irradiance in 2-D on a db scale with η =

55 Figure 3.16: Theoretical speckle field irradiance in 2-D on a db scale with η = 0.9. (3-D examining of irradiance values) Numerical Discrete Phase results Using a LC SLM, only a discrete set of phase values can be applied based on a discrete set of voltages. The effects the discrete phase values have on the speckle far field irradiance pattern were examined. Only the numerical solution was used to examine the discrete phase values as the theoretical solution has not taken in to account the PDF of a discrete phase distribution. An extreme discrete case was examined to find the effect the discrete phase would induce, which used 5 equally spaced phase values between 0 and 2π. 44

56 Figure 3.17: Linear numerical results (red) with 5 equally spaced discrete phase values between 0 and 2π were applied to LC SLM against theoretical (blue). a) is a single realization and b) is the mean of 1000 realizations. Figure 3.18: Logarithmic numerical results (red) with 5 equally spaced discrete phase values between 0 and 2π were applied to LC SLM against theoretical (blue). a) is a single realization and b) is the mean of 1000 realizations. It is easily seen that the discrete random phase values in Figure 3.17 and Figure 3.18 do not play a large role in the outcome of the far field irradiance pattern when comparing to Figure 3.11 and Figure The mean irradiance matches that of the theoretical results for a continuous uniform distribution of phase from 0 to 2π. 45

57 3.3 Experimental results Figure 3.19: Experimental setup for imaging the far field create from the LC SLM. A default setup for the data collection system is shown in Figure The polarizer is aligned with the slow axis of the LC SLM. The CCD camera is mounted on a 3 axis stage, x,y, and z, to ensure proper focus and position. The second lens in the system images the far field pattern created by the first lens and gives a magnification of 2x. The speckle size will be will be diffraction limited by the imaging system and measured fields will be under sampled. One hundred unique uniformly distributed phase masks were applied to the LC SLM. Each mask represents a single realization, and one image/data set was taken for each mask. These 100 images were then averaged on a point by point basis to give a mean irradiance pattern. With the setup shown in Figure 3.19, only one diffraction order is captured within a camera field. Therefore, 9 separate image sets were taken to provide a wider FOV. The CCD camera pitch pixel was

58 μm x 6.45 μm and the images captured were 1008x1024, therefore, the image covered an area of x mm 2.The first set included the 0 th diffraction order. The camera was then moved along the x- and y-axis at increments of 5.08 mm which allowed some overlap between sets. The goal was to capture the 1 st diffraction orders along x and y direction. The overlay between images was then used to stitch all the images into one larger image. As an example, consider just the 2 images moved along the x-direction with the first starting at the coordinates of 0 mm, 0 mm and second at 5.08 mm, 0 mm. Therefore, the images overlap by mm, which is approximately 236 pixels. The second image is cropped to exclude the first 236 pixels and then appended to the first image. This process was done for all images and the final result is shown in Figure Two cross sections were also taken (Figure 3.21) of the experimental results and compared to the theoretical solution noting the fact that the theoretical result have been shifted to try and compensate for the saturation of irradiance at the 0 th order of diffraction on the CCD camera. Figure 3.20: a) Theoretical speckle field irradiance in 2-D on a db scale with η = 0.9. b) Experimental far field average irradiance pattern of 100 realizations. Notice the irradiance range between diffraction orders (0 th and 1 st ) is about 20 db for both the theoretical and experimental results. 47

59 Figure 3.21: Cross-section of experimental data in Figure a) y = m and b) y = m. Note that the irradiance pattern away from the diffraction order follows the same irradiance pattern as that of the theoretical solution and are of the same magnitude, that being approximately 20 db difference between the area around the 0 th and 1 st diffraction orders. Two issues arose with the analysis of this data. First, there is a read out error in the CCD camera that cause values on the left side of the CCD array to read out higher values than the right as seen by the discontinuity in Figure 3.20b. To test for this, the second lens was removed to change the system magnification. The 0 th and most of the 1 st diffraction order could then be captured within one image. The image was 1384x1036 which captured both 1 st diffraction orders in the longer direction and only one in the other and is shown in Figure Again, the data is saturated for the diffraction orders and partially for the underlying sinc pattern. Also, the image now includes many aberrations as is easily noticed in the shape of the diffraction orders. The main point to notice is that the sinc pattern is a smooth transition over all directions. 48

60 Figure 3.22: (a) Theoretical speckle field irradiance in 2-D on a db scale with η = 0.9. (b) is the experimental results showing an overall continuous smooth transition between points. Figure 3.23: Cross-section of Figure 3.22b with y=0.001 m Second, the irradiance on and around the diffraction orders saturated the camera at the exposure time used. We found the irradiance difference between the 0 th diffraction order and the start of the sinc irradiance pattern due to the pixel size exceeded the dynamic range of the CCD camera and so both could not be captured within one exposure time. The dynamic range of the CCD camera is approximately 25 db. Therefore, experiments were repeated multiple times at different shutter speeds to capture a larger range of irradiance values. (It has been assumed that the CCD camera has a linear mapping across the 8-bit signal 49

61 that is captured.) A simple procedure was use to compile the data from different shutter speeds into one overall set of data. The longest exposure time, T 0, was used as a base. The lower irradiance spots are visible at this shutter speed and the higher irradiances are saturated. The exposure time was then decreased to T n to capture values for the previously saturated values, and this process was continued until the image captured contained no saturated values. To combine the information from all data sets, each set first had to be multiplied by the scalar factor of T 0 /T n where T n is the exposure time for the set. Therefore, starting at the longest exposure time, all saturated values were replaced by the values from the data set with the next longest exposure time. The final image gave a range of approximately 100 db. Figure 3.24b shows the final result of the 0 th diffraction order and its relative irradiance to the surrounding irradiance. The peak is approximately between 35 and 45 db over the surrounding area, compared to the theoretical results, Figure 3.24a, of roughly 41 db. It is also seen that there is an artifact noticed in the results which is due to retroreflection within the beam splitter being used causing two specular phasor seen as the two bright circles above the diffraction order in Figure 3.24a. 50

62 Figure 3.24: (a) Theoretical speckle field irradiance in 2-D on a db scale with η = 0.9. (b) is the experimental results where the range of irradiance value of the diffraction and surround sinc pattern is the same as the theoretical. Next, speckle contrast of the experimental results were calculated and shown in Figure The surrounding area is shown to have a contrast around a value of 1 which is predicted by the theoretical results. It can be seen that the diffraction order and retroreflections from the beam splitter (specular phasor) have a lower contrast than the rest of the results showing specular (constant) phasors at those points. Figure 3.25: Experimental speckle contrast results. a) 2-D imaged results from the CCD camera and b) cross-section of (a) at approximately y = 0.001m. 51

63 CHAPTER 4 PARTIALLY DEVELOPED SPECKLE Though fully developed speckle is the most widely studied and analyzed, partially developed speckle can occur and needs to be studied further. Partially developed speckle is defined when speckle contrast is less than 1, in other words SNR is greater than 1. In this chapter we derive and study a phase distribution that will be called a Wrapped Gaussian distribution. It was created and used in the distribution of phase values for the LC SLM simulations and experiments. The distribution is a function of the variable σ and has as its limiting cases either a uniform distribution or a constant. 4.1 Wrapped Gaussian Distribution A reasonable distribution to expect in nature would be a Gaussian (normal) distribution. It might occur, for example, for a surface or aperture in which one is attempting to control the phase. In attempting to simulate this distribution we see that the LC SLM has a limit of only being able to apply a range of approximately 2π values. Being that phase is periodic on 2π, values can be wrapped back within the same 2π interval to get a distribution that is within the limits of the LC device. Setting the interval of phase to be -π < π, a simple phase term can 52

64 be written as, where n is an integer, to describe all phase values. The following steps show how the phase term can be wrapped to a 2π interval. Since (-1) 2n is always equal to 1 being that 2n is even, the phase term equals where -π < π. Hence the phase from all values can be wrapped within a 2π interval. Therefore, a Gaussian phase distribution, with mean μ equal to zero and standard deviation σ can be wrapped into the interval -π to π. This wrapped probability density function, therefore, demonstrates the same effects as the original Gaussian and is found to be equivalent. Figure 4.1 shows curves for the Wrapped Gaussian distribution at different values of standard deviation, σ. It can be seen that with larger σ, the more the distribution tends toward a uniform phase distribution. Note the wrapped Gaussian PDF is almost uniform when σ = 2π. Figure 4.1. PDF for different values of σ in Wrapped Gaussian. 53

65 4.2 Mutual Irradiance and 4 th Order Moment The mutual irradiance (mean irradiance speckle pattern) and the 4 th order moment become more complicated for a distribution other than a uniform distribution of phase. Recall Equation (3.30), for the mutual irradiance in 1-D, (3.30) where ξ x = x 2 /λz, η = fill factor, d = pixel pitch, x n = (2n-1)d/2 and sinc(x) = sin(π x)/(π x). The ensemble averages need to be re-calculated for a Gaussian distribution. The calculation of the mutual irradiance for the Gaussian distribution is shown in Appendix A and the result is (4.1) where M(ω) is the characteristic function. The characteristic function for a Gaussian distributed variable in an exponential is given by [1] where σ is the standard deviation of the distribution. 54 (4.2)

66 To solve for the contrast the 4 th order moment of the field is needed. Goodman solved a problem much like the one we needed in Ref [1] though this solution is generalized to include the periodicity of the SLM. The 4 th order moment is calculated in Appendix B with the results being (4.3) where and. The contrast is defined as (4.4) The closed form of the contrast equation is not shown but can be solved for by substituting Equation (4.1) and Equation (4.3) into Equation (4.4) 55

67 4.3 Speckle Simulations of the Wrapped Gaussian Distribution The mutual (mean) irradiance of the far field defined by Equation (4.1) was evaluated for various values of sigma. Also, numerical simulations were done, preformed in the same fashion as in Chapter 3 except now with the wrapped Gaussian distribution of random phase values. When evaluating the different σ, two limiting cases were considered. The limiting case when σ > 2π, the results in Figure 4.2 (note σ = 5π for results) tend to the same as for a uniform distribution shown in Figure 3.12 and Figure The limiting case when σ < π, the results tend towards specular reflection (all pixels are held at the same constant phase) shown in Figure 4.3. The theoretical curve in Figure 4.3 is the diffraction pattern one would expect from a full aperture the size and shape of the LC SLM device. Figure 4.2: Plots for σ = 5π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position 56

68 Figure 4.3: Plots for σ = π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled. Notice in the case when σ < π, the numerical solution does not closely agree with the theoretical result. This is due to an under sampling in the numerical solution that is unable to resolve the tight sinc pattern observed from the overall size of the LC SLM and could not be fixed to due to limited computing power. The numerical results are associated with the irradiance pattern due to the finite size of the pixel (correlated area). The energy from the pattern associated with the size of the pixel is such that it will not contribute to the tight sinc irradiance pattern from the size of the full aperture of the LC SLM device with σ < π. With the loss of this information in the discrete Fourier transform in Matlab, the numerical simulation for contrast also loses values and it is easily seen in Figure 4.3b. One thing to note about the contrast for the case when σ < π is that there are points above 1. The points are due to two factors. One is that the mean irradiance values of the diffraction orders make a negligible contribution to the overall mean irradiance pattern, yet the diffraction orders will have a high variation giving a value of σ larger than the mean irradiance at that point. The other is from what would be harmonics of the 57

69 diffraction orders and can be seen in Equation (4.3) for the 2 nd moment of the irradiance. The Ω 2 (ξ) have twice as many peaks as that of Ω 1 (ξ). The effect from these points occurs, as in Ch. 3., at very isolated region and with finite sampling in the numerical and experiment results, they are not observed. The results for the limiting cases are expected results; for the limiting case when σ > 2π, Figure 4.1 shows the Wrapped Gaussian starts to converges to a uniform distribution and the results in Figure 4.2a is in agreement with the results for a uniform distribution of phase in Figure For the case when σ < π, the values of the distribution are approximately the same (constant), and Figure 4.3 represents the far field pattern of an aperture the size of the SLM. Looking at the mean irradiance pattern in Figure 4.3a, the theoretical solution shows a tight sinc pattern, whereas the numerical solution shows a wider sinc (underlying sinc) pattern. The tight sinc pattern is determined by the overall size of the SLM whereas the underlying sinc pattern and the diffraction orders are determined by the finite size (correlated area) of the individual pixels and then periodicity. Now to look at the effect the sinc irradiance pattern associated with the correlated area over the finite size of a pixel holds with different values of σ. Various values of σ, σ = 0.005π, 0.01π, 0.05π, 0.1π, and 0.5π were examined and are shown in Figure Figure 4.8, respectively. It can be seen that as σ increases the intensity of the sinc associated with the finite size of a pixel increases and also the diffraction orders become observable when it rises above the sinc pattern associated with the full aperture. When the magnitude of the sinc pattern associated with the finite pixel size rises above that associated 58

70 with the full aperture, the diffraction orders become observable and the theoretical and numerical begin to start being in agreement. Notice the positions where the contrast stays at approximately zero is where the sinc pattern associated with the finite pixel size is zero. Figure 4.4: Plots for σ = 0.005π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled. Figure 4.5: Plots for σ = 0.01π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled. 59

71 Figure 4.6: Plots for σ = 0.05π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled. Figure 4.7: Plots for σ = 0.1π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled. Figure 4.8: Plots for σ = 0.5π where the theoretical result is in blue and the numerical solution is in red. a) Irradiance pattern in far field. b) Contrast versus position where the numerical solution is under sampled. 60

72 4.4 Experimental Results The experiment was setup up again like that in Section (Experimental results) with a minor change of removing the second lens and moving the CCD camera a focal length after the first imaging lens. This gives a smaller magnification enabling us to capture one 1 st order of diffraction along with the 0 th order. Masks were made for 3 different Wrapped Gaussian distributions with σ = 0.1π, 0.05π, and 0.01π and tested with the results shown in Figure Figure 4.11, respectively. As σ decrease the absolute magnitude of the theoretical solution also decreases, so the theoretical data has been shifted on the y-axis in the figures below to make it easier for comparison. It can be seen in Figure 4.9 at σ = 0.1π that the data is in approximate agreement with the theoretical data. In Figure 4.10 when σ = 0.05π, the irradiance pattern drops below the noise of the CCD camera along with Figure When the value of σ was decreased below σ = 0.1π, the results of the irradiance pattern were not as expected which is easily seen looking at Figure 4.11, the results are very pixilated and discontinuous. This is due to the discrete set of phase values obtainable from the LC SLM device are going to limit the ability to simulate small variations of phase values of the phase distribution making the limiting case for the BNS LC SLM device to be a wrapped Gaussian with σ = 0.1π. 61

73 Figure 4.9: Experimental results of far field with σ = 0.1π where the range of irradiance values between the diffraction orders is about 17 db. Figure 4.10: Experimental result of far field with σ = 0.05π where the 1 st diffraction order still contributes away from Y=0 axis. 62

74 Figure 4.11: Experimental result of far field with σ = 0.01π where the 1 st diffraction order no longer contributes away from Y=0 axis. 63