(2) Complex Modulation Code for Low-Resolution Modulation Devices A Dissertation Presented by Rodrigo Ponce Díaz. Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Information Technology and Communications. Thesis Committee: Alfonso Serrano Heredia, ITESM Campus Monterrey Victor Arrizón Peña, Instituto Nacional de Astrofísica Óptica y Electrónica. Carlos M. Hinojosa E., ITESM Campus Monterrey Carlos Pfeiffer C., ITESM Campus Monterrey Bahram Javidi, The University of Connecticut. Optics Center Instituto Tecnológico y de Estudios Superiores De Monterrey Campus Monterrey December 2006.

(3) Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Monterrey Graduate Program in Information Technology and Electronics. The committee members hereby recommend the dissertation presented by Rodrigo Ponce Díaz to be accepted as a partial fulfillment of requirements to be admitted to the Degree of Doctor of Philosophy in Information Technology and Communications, major in Electronics and Telecommunications. Committee members:. Dr. Alfonso Serrano Heredia Advisor. Dr. Victor Arrizón Peña Advisor. Dr. Carlos M. Hinojosa E. Dr. Carlos Pfeiffer C.. Dr. Bahram Javidi. Dr. Graciano Dieck Assad Director of Graduate Program in Information Technology and Electronics.

(4) Declaration. I hereby declare that I composed this dissertation entirely myself and that it describes my own research.. Rodrigo Ponce Díaz Monterrey, N.L., México December 2006.

(5) Dedication. To my Dad, Mom and Sister. Thanks for all your unconditional confidence, support, patience, and encouragement. You were my main motivation for pushing through this work.. i.

(6) Acknowledgements. I would first like to express my gratitude to my advisor Dr. Alfonso Serrano Heredia, who constantly supported my effort in the Ph. D program. I am deeply indebted to my co-advisor, Dr. Victor Arrizón Peña for his advice, guidance and support for lab work. I thank his hospitality every summer for visiting the labs at INAOE, Puebla. This dissertation would not be possible without all his support. I am very grateful to Dr. Bahram Javidi by his hospitality and support for a year of research at the University of Connecticut. I would like to thanks Dr. Julio Gutierrez Vega and Tecnológico de Monterrey Research Chair in Optics, México for their financial support in my visiting scholar year. To Dr. David Garza and all the authorities of ITESM that supported me in the Doctoral Program, special thanks to Dr. Oliver Probst for the support as teaching assistance. I would like to thanks all the support of my complete families Ponce and Díaz, I have in my mind all your words and good wishes. Finally I would like to thanks to a wonderful girl who has patience, confidence, and encouraged words. Thank Vanessita to stand by me in this part of my life, you were a light in my walk, finally my child dream becomes true.. ii.

(7) Abstract. Digital information technology is constantly developed using electronic devices. The three dimensional (3D) image processing is also supported by electronic devices to record and display signals. Computer generated holograms (CGH) and integral imaging (II) use liquid-crystal spatial light modulator (SLM). This doctoral dissertation studies and develops the application of a commercial twisted nematic liquid crystal display (TNLCD) in computer generated holography and integral imaging. The goal is to encode and reconstruct complex wave fronts with computer generated holograms, and 3D images using Integral Imaging systems. Light modulation curves are presented: amplitude and phase-mostly modulation. Holographic codes are designed and implemented experimentally with optimum reconstruction efficiency, maximum signal bandwidth, and high signal to noise ratio (SNR). The study of TNLCD into II is presented as a review of the basics techniques of display. A digital magnification of 3D images is proposed and implemented. 3D digital magnified images have the same quality of optical magnified images, but the magnified system is less complex. Recognition system for partially occluded object is solved using a 3D II volumetric reconstruction. 3D Recognition solution presents better performance than the conventional 2D image systems. The importance in holography and 3D II is supported by the applications as: optical tweezers, as dynamic trapping light configurations, invariant beams, and 3D medical images.. iii.

(8) Contents Committee Declaration. 3. Declaration. 5. Dedication. i. Acknowledgement. ii. Abstract. iii. List of Figures. viii. List of Tables. xiii. 1. Introduction 1.1 Motivation . . . . . . . . . . . . . 1.2 Problem Statement . . . . . . . . . 1.3 Research Questions . . . . . . . . . 1.4 Solutions Overview . . . . . . . . . 1.4.1 Holographic Codes . . . . . . 1.4.2 Integral Imaging . . . . . . . 1.5 Main contributions . . . . . . . . . 1.6 Literature Review and Background . 1.6.1 Computer Generated Holograms 1.6.2 Integral Images . . . . . . . . 1.7 Thesis Organization . . . . . . . . . 1.8 References. . . . . . . . . . . . . .. iv. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. 1 1 2 2 2 2 3 3 4 4 4 5 8.

(9) 2. Liquid Crystals 2.1 Properties of a Liquid Crystal Cell. . . . . . . . . . . 2.1.1 Calamitic Liquid Crystal Molecules . . . . . . 2.1.2 Discotic Liquid Crystal Molecules 2.1.3 Molecular structure . . . . . . . . . . 2.1.4 Electro Optic Effect . . . . . . . . . . 2.1.4 Gray Control using a Polarization State 2.2 Jones Matrix Formalism . . . . . . . . . . . 2.2.1 Representation of Light Polarization . . 2.2.2 Linearly Polarized Light. . . . . . . . 2.2.3 Circularly Polarized Light . . . . . . . 2.2.4 Matrix of a Linear Polarizer . . . . . . 2.2.5 Matrix of a Wave Plate Retarder . . . . 2.3 References . . . . . . . . . . . . . . . . .. 3 Mathematical Model for a Liquid Crystal Cell 3.1 Mathematical Approximation . . . . . 3.2 Matrix of a Liquid Crystal Cell . . . . 3.3 Mathematical Analysis using Jones matrix 3.4 References . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. 27 . . . 27 . . . 28 . . . 32 . . . 35. 4 Experimental Characterization for Complex Modulation of a Holoeye-LC2000 4.1 Technique for Characterization of Modulation Curves . . . . 4.2 Amplitude Modulation . . . . . . . . . . . . . . . . . . . 4.3 Phase Modulation . . . . . . . . . . . . . . . . . . . . . 4.3.1 Talbot Effect . . . . . . . . . . . . . . . . . . . . 4.3.2 Fractional-Talbot Effect . . . . . . . . . . . . . . . 4.3.3 Diffraction Measurement . . . . . . . . . . . . . . . 4.4 Optical Set up Configurations. . . . . . . . . . . . . . . . 4.4.1 Couple of Linear Polarize . . . . . . . . . . . . . . 4.4.2 Phase-Mostly Configuration . . . . . . . . . . . . . 4.4.3 Simplified optimal set up for mostly phase modulation.. i) Numerical Simulation of Phase Modulation . . . . . ii) Experimental Verification . . . . . . . . . . . . . 4.5 References . . . . . . . . . . . . . . . . . . . . . . . . .. v. . . . . . . . . . . . . .. 11 11 12 14 14 16 17 18 18 20 21 22 22 25. . . . . . . . . . . . . .. 37 37 38 39 39 42 47 49 50 51 52 53 55 57.

(10) 5 Amplitude Hologram Display into a TNLCD 5.1 Conventional Holography . . . . . . . . . . . . . . . 5.2 Amplitude Computer-Generated Hologram. . . . . . . 5.3 Signal to Noise Ratio Measurement . . . . . . . . . . 5.4 Bias Function Optimization . . . . . . . . . . . . . . 5.4.1 Filtered Bias Function for Amplitude Hologram . 5.4.2 Constants Bias Function for Amplitude Hologram Coupled Phase Modulation . . . . . . . . . . . 5.5 Numerical Evaluation of CGH under LCD with Coupled 5.5.1 High Order Bessel Function. . . . . . . . . . . 5.5.2 Laguerre-Gauss Beam . . . . . . . . . . . . . 5.6 References . . . . . . . . . . . . . . . . . . . . . . 6. 7. 8. Holographic Code of a Single Pixel 6.1 Computer Generated Holograms. . . . 6.2 Mathematical Construction One Single 6.3 Experimental Results . . . . . . . . . 6.4 References . . . . . . . . . . . . . .. . . . . . . Pixel Code . . . . . . . . . . . . .. . . . .. . . . .. . . . . .. 67 69 69 72 76. . . . .. . . . .. 77 77 78 85 90. . . . . . .. 91 . 91 . 92 . 93 . 97 . 99 . 103. . . . . . . . . . . . . . . . with . . . Phase . . . . . . . . .. . . . .. Blazed Holographic Code 7.1 Codes for a Phase-Mostly Modulation . . . . . . . . . . 7.2 Holographic Code for a Perfect Phase SLM. . . . . . . . 7.3 Mathematical Construction of a Blazed Holographic Code 7.3.1 Separating noise from reconstructed signal . . . . 7.4 Numerical Simulation on Performance of Blazed Code . . 7.5 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . .. Three Dimensional Images using a LCD and Integral Imaging Technique . . . . . . . . . . . . . . . . . . 8.1 Integral Imaging Technique . . . . . . . . . . 8.2 Three-Dimensional Object Recording . . . . . . . . . . . . . 8.3 Three-Dimensional Optical Reconstruction. . . . . . . . . . . . . . . . . 8.4 Three-Dimensional Digital. Reconstruction . . . . 8.5 References. vi. . . . . .. 59 59 62 64 66 66. . . . . .. 105 105 106 109 111 113.

(11) 9. Magnification of Integral Imaging 9.1 Scaling 3D Problem . . . . . . . . . . . . . 9.2 Optical Magnification . . . . . . . . . . . . . . . . . . 9.2.1 Lateral and Longitudinal Magnification 9.3 Digital Magnification Method . . . . . . . . 9.3.1 Direction Information . . . . . . . . . 9.4 Experimental Results. . . . . . . . . . . . . 9.5 References . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 115 115 116 119 121 123 126 131. 10 Digital Recognition of Occluded Objects using a Three Dimensional Integral Imaging Volumetric Reconstruction 133 10.1 Image Recognition . . . . . . . . . . . . . . . . . . . . . 133 10.2 Occluded Objects . . . . . . . . . . . . . . . . . . . . . 134 10.3 Recognition Systems . . . . . . . . . . . . . . . . . . . . 135 10.3.1 Design of the Recognition System . . . . . . . . . . 138 10.3.2 Optimized Filter . . . . . . . . . . . . . . . . . . 138 10.4 Experimental Results. . . . . . . . . . . . . . . . . . . . 139 10.4.1 2D Recognition . . . Results . . . . . . . . . . . . . . . . . . 139 10.4.2 3D II Recognition Results using Volumetric Reconstruction . . . . . . . . . . . . . . . . . . 141 10.5 References. . . . . . . . . . . . . . . . . . . . . . . . . 146 11 Conclusions 11.1 Overview of 3D Image . . .Processing . . . . . . . . . . . 11.2 Contributions . . . . . . . . . . . . . . . . 11.2.1 Modulation Curves . . . . . . . . . . 11.2.2 Amplitude Holograms. . . . . . . . . 11.2.3 Holographic Codes . . . . . . . . . . . 11.2.4 Integral Image Magnification . . . . . 11.2.5 Partially Occluded Object Recognition 11.3 Conclusions . . . . . . . . . . . . . . . . . 11.4 Future Work . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 149 149 150 150 151 152 152 153 153 154. Bibliography. 155. Biography. 163. vii.

(12) List of Figures 2.1 2.2. 2.3 2.4 2.5. 3.1 3.2 3.3 3.4 3.5 4.1. 4.2 4.3. Calamitic crystal molecule rotates and aligns with the electric field. . . . . . . 12 The index ellipsoid. The coordinates (x, y, z) are the principal axes and n1,n2, n3 are the principal refractive indices. The refractive indices of the normal modes of a wave traveling in the direction k are na and nb. . . . . . . . . . . . . . . . . 13 The nematic and columnar discotic liquid crystal phases. The flat discs represent the molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Molecular organization of different types of liquid crystals: (a) nematic, (b) smectic, (c) cholesteric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Orientation of an electric dipole by an electric field. (a) The dipole is along the long axis of the molecule. (b) The dipole lies across the long axis of the molecule. The electric field (E) causes forces (F) at the ends of the dipoles and rotation of the molecules as shown by the curved arrows. . . . . . . . . . . . . 16 Orientation of liquid crystal molecules in a cell of a TNLCD. . . . . . . . . . . 28 The crystal molecules are tilted by the same angle τ when a voltage is applied to the cell, along the z-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Elliptic polarization states generated by a TNLCD for different values of β. . 31 Rotation θ of generated elliptic polarization versus β. . . . . . . . . . . . . . . 31 Schematic figure of a liquid crystal cell . . . . . . . . . . . . . . . . . . . . 32 Optical Fourier systems. A signal (2D image) is located at a focal distance in front of a Fourier lens. The Fourier spectrum of the signal is obtained in the back focal plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Schematic representation of Talbot self-imaging effect. A pure phase grating produces an intensity distribution at a fractional Fresnel distance. . . . . . . . 42 Transmittance Grating. Graphic representation of the transmittance of a binary complex grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. viii.

(13) 4.4. 4.5 4.6. 4.7. 4.8. 4.9 4.10 5.1. 5.2. 5.3 5.4. 5.5 5.6. 5.7 5.8. Two level grating displayed on a TNLCD in the x direction. The low and high levels are 0 and L, respectively. (a) Transmittance of the normalized amplitude modulation, for a pure phase modulation the grating has the same amplitude value. (b) Transmittance of the phase modulation. . . . . . . . . . . . . . . . . 45 Corresponding intensity distribution at one quarter of the Talbot distance. . . . 46 Conventional configuration for amplitude modulation. The set up is composed by a linear polarizer P1 oriented 90° respect to the input TNLCD director axis and a analyzer at the output plane that is a second linear polarizer P2 oriented 0° respect to the input TNLCD director axis. . . . . . . . . . . . . . . . . . . . . . . . . 50 Experimental (a) amplitude and (b) phase modulations versus gray level g, and (c) phase versus amplitude modulation, measured for the TNLC-SLM with polarizers oriented at angles φ1=90°, φ2=0°.. . . . . . . . . . . . . . . . . . . . . . . . 51 Setup to obtain mostly-phase modulation with a TNLCD. The polarizer P1 and a wave retarder generated an elliptic polarization state at the input of the TNLCD. The polarizer P2 and a wave retarder detect the output polarization state. . . . 52 Simplified optimum mostly-phase setup. The output polarization state detector is changed by a linear polarizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Phase and amplitude modulation of the experimental phase-mostly setup optimized with a linear polarizer output detector. . . . . . . . . . . . . . . . . .56 Optical systems for image processing the so-called 4-f system. The input plane is a distance f from the first lens, the Fourier plane is located a distance f from the back focal plane of the first lens and a distance f from the second lens, the output plane is located a distance f from the back focal plane of the second lens. . . . . 61 First order Bessel beam: (a) Modulus and (b) Phase. . . . . . . . . . . . . . . 69 SNR versus uo for amplitude CGHs of types 1(segmented line), 2(doted line) and 3 (bold line) without couple phase. First order Bessel beam (ρo=∆u/20). . . . 70 Fourier signal spectrum for an encoded first order Bessel beam: (a) type 1 CGH with uo=∆u/11 (b) type 2 CGH with uo=∆u/8, and (c) type 3 CGH with uo=∆u/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Reconstructed Bessel signal using the CGH of type 2 and frequency carrier uo=∆u/11 for the best SNR value. (a) Modulus and (b) phase. . . . . . . . . . 71 SNR versus uo for CGHs encoding a first order Bessel beam (ρo=∆u/20): The types are 1(segmented line), 2(doted line) and 3 (bold line). The coupled phase slopes are (a) γ =π/100, (b) γ =π/20, (c) γ =π/10, (d) γ =π/6, (e) γ =π/4, and (f) γ =π/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Laguerre-Gauss beam with indices p=l=1: (a) Modulus and (b) phase. . . . . 73 SNR versus uo for amplitude CGHs of types 1(segmented line), 2(doted line) and 3 (bold line) without couple phase. Laguerre-Gauss beam (wo=40 δx) . . . . . 73. ix.

(14) 5.9. Fourier signal spectrum of an encoded Laguerre-Gauss beam: (a) CGH of type 1 with uo=∆u//8, (b) CGH of type 2 with uo=∆u/15, and (c) CGH of type 3 with uo=∆u/6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.10. Reconstructed Laguerre-Gauss signal using the CGH of type 2 and frequency carrier uo=∆u/15 that corresponds to the best SNR value: (a) Modulus and (b) phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.11 SNR versus uo for encoded Laguerre-Gauss beam (wo=40 δx) onto CGHs of types 1(segmented line), 2(dot mark line) and 3 (bold line) with coupled phase (a) γ =π/100 (b) γ =π/60 (c) γ =π/44 (All CGH has almost the same performance) (d) γ =π/40 (e) γ =π/30 (f) γ =π/10 (The best performance is for CGH type 3). . . . 75 6.1 Phase and amplitude modulation of the experimental mostly-phase setup. . . . 78 6.2. 1 2 1 Two possible definitions for vectors M nm and M nm : (a) M nm appears at right. R 1 position (and is redefined as M nm ), and (b) M nm appears at the left position (and L R L ). (c) Positions of vectors M nm and M nm . . . . . . . . . . 83 is redefined as M nm. 6.3 6.4 6.5 6.6 6.7 6.8 7.1 7.2 7.3 7.4 8.1 8.2 8.3 8.4 8.5. SLM modulation values (g) for the CGH encoding the finite first order Bessel beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Intensity distribution generated with the CGH (a) at 5 cm and (b) at 90 cm from the second transforming lens of the 4-f spatial filtering setup. . . . . . . . . . 87 SLM modulation values (g) for the CGH encoding the finite first order Bessel beam with a circular beam support to generate two beams. . . . . . . . . . . . 87 Intensity distribution generated with the CGH (a) at 5 cm and (b) at 90 cm from the second transforming lens of the 4-f spatial filtering setup. . . . . . . . . . . 88 SLM modulation values (g) for the CGH encoding the finite modified first order Bessel beam, in order to reconstruct two reinforced rings. . . . . . . . . . . . 89 Intensity distribution generated with the CGH (a) at 5 cm and (b) at 90 cm from the second transforming lens of the 4-f spatial filtering setup. Mostly-phase modulation curve. . . . . . . . . . . . . . . . . . . . . . . . . 100 First order Bessel beam: (a) Modulus and (b) Phase. . . . . . . . . . . . . . 101 Encoded CGH using a blaze holographic code. . . . . . . . . . . . . . . . . 101 Numerical reconstructed complex signal of a first order Bessel beam with the blazed CGH code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Lenslet array: 1mm diameter, 3mm focal length. . . . . . . . . . . . . . . . 107 Conventional 2D image using a single lens. . . . . . . . . . . . . . . . . . . . 107 A group of 4x4 elemental images extracted from an elemental image array. . 108 3D image recording system is composed by a lenslet array and a CCD sensor.108 3D image II reconstruction using a lenslet array and a LCD as a panel display.109. x.

(15) 8.6 8.7. 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7 9.8 9.9. The display procedure using 3D projection II with a micro-convex mirror array or a lenslet array with low reflectivity. 2D image projectors are not shown. . . . 110 3D computational volumetric reconstruction scheme with a virtual pinhole array. The optical reconstruction is simulated using a virtual pinhole array in front o the elemental image array. 2D images of the 3D object are reconstructed along the z direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 The principle of the Optical Magnification (OM) method. (a) The pickup procedure of the OM method using a Movement Array Lenslet Technique (MALT) . (b) The display procedure of the OM method using a stationary lenslet array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 The optical magnification produces over-sampled elemental image array using MALT and rearrange the elemental images in a single array that produce 3D magnified reconstructed II image. . . . . . . . . . . . . . . . . . . . . . . 118 Longitudinal and transversal magnification by controlling spatial ray sampling rate. The object consists of two points A and B. The lateral distance between the points is a and the longitudinal is b. (a) Pickup with sampling rate 1/p. (b) Display with sampling rate 1/p'. . . . . . . . . . . . . . . . . . . . . . . . . 119 The digital magnification algorithm uses a single elemental image to calculate the over sampled elemental images. Each square is an elemental image and fractional numbers represent elemental images calculated digitally. . . . . . . . . . . . 123 Three objects are recorded in two elemental images and analyzed by ray sampling. Object 2 in white area is similar information for upper and lower elemental-image. Object 1 and 3 are different information for each other elemental-image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 For the implementation method of the DM method, from the upper and lower elemental images we identify the similar information and average this part, also different information in order to replicate. The final digital elemental image is cut at both edges as D/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Experimental objects: button with a footprint on it and button with a peace mark on it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Elemental image arrays that are magnified by factor two. (a) The Optical Magnification method. (b) Digital Magnification Method. . . . . . . . . . . 127 The Digital 3D Volumetric Optical reconstruction shows the reconstruction planes for button with a footprint on it and button with a peace mark on it, in order to calculate the depth of the scene (a) No magnification ∆z=8.5 mm (b) The optical magnification method moves the object plane two times farther from the virtual pin-hole than original reconstruction ∆z= 17 mm. (c) the digital magnification method has the same characteristics than optical magnification method ∆z=17 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 xi.

(16) 9.10. Optical reconstructions of 3D II images. (a) No magnification (b) The optical magnification method. (c) the digital magnification method. . . . . . . . . . 130 10.1 An occlusion is a barrier in front of a 3D object. This barrier partially blocks the object image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10.2 Integral Imaging pickup system for a scene with a car and an occlusion. . . . 135 10.3 3D computational volumetric reconstruction scheme that represents a computer synthesized pinhole array. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 10.4 2D images for objects with occlusion and without occlusion (a) Original Blue Car, (b) Occluded Blue Car, (c) Original Green Car, (d) Occluded Green Car , (e) Original Red Car, (f) Occluded Red Car . . . . . . . . . . . . . . . . . . . . 137 10.5 3D II Volumetric reconstruction for objects with occlusion and without occlusion (a) Original Blue Car, (b) Occluded Blue Car, (c) Original Green Car, (d) Occluded Green Car , (e) Original Red Car, (f) Occluded Red Car . . . . . . 137 10.6 Two dimensional Images of the experimental cars, (a) original scene, from left to right red, green, and blue, (b) occluded scene, from left to right red, green, and blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 10.7 Correlation peaks of the optimum filter for the 2D scene with the occlusion. . 140 10.8 Elemental image arrays of the experimental cars, (a) original scene red car (b) original scene green car (c) original scene blue car, (d) occluded scene red car, (e) occluded scene green car and (f) occluded scene blue car. . . . . . . . . . . 141 10.9 Three dimensional images reconstruction planes of the scene at distances (a) 10.7 mm (b) 44.94 mm (c) 51.36 mm (d) 72.76 mm. . . . . . . . . . . . . . . . . 142 10.10 Correlation peaks of the optimum filter for the reconstruction plane 44.94 mm at the 3D computational volumetric reconstruction. . . . . . . . . . . . . . . . . 143 10.11 Plot of the maximum correlation peak at each reconstruction plane, considering the 3D computational volumetric reconstruction. . . . . . . . . . . . . . . . . 144. xii.

(17) List of Tables Table 10.1 Correlation output peak intensities and PSR values for conventional 2D image correlation and 3D computational volumetric reconstructed image correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145. xiii.

(18) Chapter 1 Introduction. 1.1 Motivation Modern digital technology is constantly developing, in order to improve electronic systems like signal processing, telecommunication, and image processing. Among these systems, image processing has taken an important priority into the technology research and applications, so that better image systems for acquisition and display procedures can be implemented. Nowadays a cellular phone or personal data assistant has the possibility of picking up and displaying digital two-dimensional (2D) images in seconds and with good quality. Personal computers can storage 2D digital color images and make basic processing actions as brightness and contrast control. The industry uses 2D image recognition systems, measurement systems and quality systems by image analysis. This development is also supporting the study of three dimensional (3D) images as holography, integral imaging and stereoscopic images. Holography and Integral Imaging are 3D image techniques that were proposed before the development of electronic systems.. 1.

(19) 2. CHAPTER 1. INTRODUCTION. Conventional holography uses an interferometric setup to record the 3D information into transparent photographic film in the form of an amplitude grating. On the other hand, the integral imaging system records the 3D information with a group of 2D images pick up with different perspectives.. 1.2 Problem Statement The main problem of these two 3D image systems is the display method: an amplitude hologram does not have the best reconstruction efficiency and it can not be changed in real time. The integral imaging should display each 2D image into the same lens used in the pick up stage. The display element should have good spatial resolution and dimension of the picked up image. Also, the display should only modulate the amplitude of the image and avoid the phase modulation.. 1.3 Research Questions This thesis studies and develops the application of a commercial twisted nematic liquid crystal display (TNLCD) in computer generated holography and integral imaging. The goal is to encode and reconstruct complex wave fronts with computer generated holograms, and 3D images using Integral Imaging systems.. 1.4 Solution Overview The development of Coupled Charged Devices (CCD) and Liquid Crystal Displays (LCD) gave new research areas for both 3D systems.. 1.4.1 Holographic Codes The holography uses the LCD’s as a dynamic amplitude modulator device, where interference gratings are displayed. The computer generated holography can generate complex interference patterns to be display into a LCD. The quality of reconstruction also depends of the algorithm to encode the information. Holographic codes have been developed in order to improve the quality of reconstruction,.

(20) CHAPTER 1. INTRODUCTION. 3. reducing the noise, improving the reconstruction efficiency. The CGH study is focused into the development of new holograms codes.. 1.4.2 Integral Imaging The display limitation of integral imaging was solved using the LCD as amplitude modulator. The group of integral images is displayed into a single panel and project through a microlens array. The LCD made possible the experimental setup for 3D reconstruction. The improvement of Integral Imaging display is growing, but the research is still focused into the basics of the technique because the recently introduction of electronics devices. The any study of Integral Imaging should start into the bases of image processing.. 1.5 Main Contribution The contributions of this thesis research are into both 3D image systems. The main contribution is the characterization of a commercial TNLCD. Curves of amplitude and phase modulation were obtained [1.1]. From the understanding of the modulation curves, an improvement of the conventional amplitude computer generated hologram was obtained, because the coupled phase modulation was considered [1.2]. As far as we know, a novel signal to noise ratio (SNR) measurement was proposed, in order to evaluate the quality of the complex wavefront reconstruction [1.2]. Two holographic codes were studied and implemented taking advantages of the phasemostly modulation of the TNLCD. The experimental results have relevance, because the quality is good enough considering a low resolution device and phase-mostly modulation, it means that it is not a perfect phase only modulation [1.3]. The contribution to Integral Imaging was related with the digital processing. Scaling problem was solved using a digital interpolation showing the same quality as optical solution, but with less complexity [1.4]. This solution was the first digital magnification solution..

(21) 4. CHAPTER 1. INTRODUCTION. The last contribution was the solution of occluded object recognition [1.5]. This problem was solved using a 3D volumetric digital reconstruction and then the partial occlusion can be isolated. This approach shows advantages compared with a conventional 2D recognition system.. 1.6 Literature Review The theory review includes the computer generated holograms and the principles of 3D recording and reconstruction using integral image technique.. 1.6.1 Computer Generated Holograms A binary hologram was first proposed by Brown and Lohmann in 1966 [1.6]. They described a detour phase method for making binary computer-generated hologram for complex spatial filtering. A sampled wavefront represented by a complex valued function can be recorded into a hologram, where each complex point value is encoded into an aperture compose by a transparent cell and a black rectangle inside. Each aperture is determined by three parameters: its height, its width, and its center with respect the center of the cell. From this first coding technique, other studies have been developed in order to improve the efficiency of the complex wavefront reconstruction. Nowadays, these computergenerated holograms have been display in real time into spatial light modulators. Recently Arrizón et al analyzed the attributes of CGHs implemented with amplitude LCSLMs and discussed their optimization [1.7]. This new research studies have been supported by applications in optical information processing [1.8]-[1.9], invariant beams and optical tweezers [1.7], among others applications [1.11]-[1.12].. 1.6.2 Integral Images The so-called Integral Imaging (II), first proposed by Lippmann [1.13] in 1908, has been studied in the last couple of decades using the new technology of sensor and display devices. The renewed II has been proposed as a real solution to display 3D images which includes 3D TV [1.14]-[1.16]. Compared with other 3D systems, II presents advantages such as continuous view points, full parallax, no special eyewear devices for users are needed, and there is no convergence-accommodation conflict..

(22) CHAPTER 1. INTRODUCTION. 5. Many studies have been performed in order to solve the resolution limitation of II [1.17][1.20], increase depth of focus [1.21]-[1.22], and many other analysis and challenges in II [1.23]-[1.26], because of the interest of improving 3D image quality.. 1.7 Thesis Organization The research work was divided into three parts: the first part studies the characterization of a commercial TNLCD, the second part is the implementation of computer generated holograms that reconstruct a complex wavefront, and the final part is the research of integral imaging applications using a real time display. In order to report the research work, this thesis is presented by chapters. In chapter 2, the theory of liquid crystal cell behavior is presented. The understanding of the LC needs the study of different concepts as: molecular structure crystal electro-optic effect, light polarization state, and gray control at each pixel by electronic devices. In chapter 3, the TNLCD is mathematical modeled in order to understand the behavior of the liquid crystal cell when a voltage is applied. Jones matrix approximation is used to obtain a simulation of the TNLCD. The mathematical model helps to obtain configuration for different types of light modulation: amplitude, phase only or phasemostly modulation. The experimental modulation curves are presented in chapter 4. The modulation of the light can be only amplitude, only phase, mostly-phase, and real modulation. The amplitude modulation setup is basically composed by a couple of linear polarizers, but it contains a coupled phase modulation. A mostly-phase modulation is composed by a elliptic polarization state in the input plane of a TNLCD, which is obtained adding a wave plate retarder. Amplitude and phase modulation curves are obtained for the best configurations. In chapter 5, the amplitude modulation is used to implement an amplitude hologram. The coupled phase to the amplitude affects the reconstruction of a given mathematical description of the wavefront or an object represented by an array of points. The best computer generated hologram is obtained considering the effect of the coupled phase, and it also has a relation with the complex function encoded..

(23) 6. CHAPTER 1. INTRODUCTION. The phase mostly modulation is implemented using a holographic code of a single pixel. Chapter 6 describes the computer-generated hologram for encoding arbitrary complex modulation, based on a commercial twisted-nematic liquid-crystal display. This hologram is implemented with the constrained complex modulation provided by the display in the mostly-phase configuration. The hologram structure and transmittance are determined in order to obtain on-axis signal reconstruction, maximum hologram bandwidth, optimum efficiency, and high signal to noise ratio. In chapter 7, a new holographic code is proposed in order to use a curve modulation with incomplete phase range. The most efficient hologram codes use only phase modulation displays to encode complex wavefronts. The phase range should be complete 2πrange and constant amplitude. The proposed algorithm, called blazed code, uses a phase mostly modulation with small amplitude variation and a phase range small of 2π. The so-called Integral Imaging (II) is presented in chapter 8. The renewed II has been proposed as a real solution to display 3D images which includes 3D TV. Basics concepts of recording 3D objects and reconstruction 3D images are described. Compared with other 3D systems, II presents advantages such as continuous view points, full parallax, no special eyewear devices for users are needed, and there is no convergenceaccommodation conflict. An analysis of II is presented in chapter 9, in order to solve a scaling problem digitally. The scaling problem in 3D II was studied and solved by Song et. al using an optical technique[1.27]. In this chapter, a new solution is proposed by using computational processing. Optically the elemental image array is increased by moving the lenslet array. The proposal presented picks up a conventional elemental image array, and digitally increase the number of elemental images. The goal of this digital technique is to obtain a magnified elemental image with the same quality of an optical magnification method. Final chapter 10 presents another application of II research, as a solution of occludes object recognition. The occluded objects have been studied recently using II systems with computational reconstruction as a 3D volumetric image. The proposal recognition system uses a 3D volumetric reconstruction image of the partially occluded object. Digitally the object is insolated from the occlusion and matched filters can be applied to recognize the original object, improving the performance of pattern recognition systems..

(24) CHAPTER 1. INTRODUCTION. 7. Conclusions of the research work are presented as a final part of this thesis. The work reported in this thesis has purpose to understand the behavior of a TNLCD as a light modulator (amplitude and phase). Applications of those characteristics are also obtained into the computer generated holography and Integral Imaging..

(25) 8. CHAPTER 1. INTRODUCTION. 1.8 References [1.1] [1.2]. [1.3]. [1.4]. [1.5]. [1.6] [1.7]. [1.8] [1.9] [1.10] [1.11] [1.12] [1.13] [1.14] [1.15] [1.16] [1.17] [1.18]. R. Ponce, V. Arrizón, A. Serrano-Heredia, “Simplified Optimum Phase-Only Configuration for a TNLCD” Proc. SPIE Vol. 5556, p. 206-213, (2004) R. Ponce-Díaz, Victor Arrizón, Julio C. Gutierrez-Vega, Alfonso Serrano-Heredia “Experimental synthesis of general complex fields using an amplitude modulator” Proc. SPIE Vol. 6311 Optical Information Systems IV Editor(s): Bahram Javidi, Demetri Psaltis, H. John Caulfield ISBN: 0-8194-6390-6 (2006). (In press) V. Arrizón, L. A. González, R. Ponce, and A. Serrano-Heredia "Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquid-crystal displays" Appl. Opt. Vol. 44 N 9 1625-1634(2005) Rodrigo Ponce-Díaz, M. Martínez-Corral, R. Martínez-Cuenca, B. Javidi, Y. W. Song “Digital Magnification of Three-Dimensional Integral Images” Journal of Display Technology IEEE-OSA.Vol. 2, No.3, 284-291 (2006) B. Javidi, R. Ponce, Seung-Hyun Hong “Three-Dimensional Recognition of Occluded Objects Using Integral Imaging Volumetric Reconstruction.” Opt. Lett. Vol. 31,No. 8, 1106-1109 (2006) . B. R. Brown and A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967 (1966). V. Arrizón, G. Mendez and D. Sanchez de la Llave “Accurate Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators” Opt. Express 13, 7913-7927 (2005) L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inform. Theory IT-6, 386-400 (1960). J. W. Goodman, “Holography,” in Introduction to Fourier Optics,McGraw-Hill, pp. 295-392 (1996). A. Vasara, J. Turunen, and A. Friberg, “Realization of general nondiffracting beams with computer generated holograms,” J. Opt. Soc. Am. A 6, 1748 (1989). M. A. Bandres, J. C. Gutierrez-Vega, and S. Chavez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44 (2004). G. Rodriguez-Morales and S. Chavez-Cerda, “Exact nonparaxial beams of the scalar Helmholtz equation,” Opt. Lett. 29, 430 (2004). Lippmann, “La photographie integrale,” C. R. Acad. Sci. 146, 446-451 (1908). B. Javidi and F. Okano, eds., Three Dimensional Television, Video, and Display Technology (Springer-Verlag, Berlin, 2002). S. A. Benton, ed., Selected Papers on Three-Dimensional Displays (SPIE Optical Engineering Press, Bellingham. WA, 2001). T. Okoshi, “Three-dimensional display,” Proc. IEEE 68, 548-564 (1980). C. B. Burckhardt, “Optimum parameters and resolution limitation of integral photography,” J. Opt. Soc. Am. 58, 71-76 (1968). J.-S. Jang and B. Javidi, “Improved viewing resolution of three-dimensional integral imaging by use of nonstationary micro-optics,” Opt. Lett. 27, 324-326 (2002).

(26) CHAPTER 1. INTRODUCTION [1.19] [1.20] [1.21]. [1.22]. [1.23]. [1.24]. [1.25] [1.26]. 9. F. Jin, J.-S. Jang and B. Javidi, “Effects of device resolution on three-dimensional integral imaging,” Opt. Lett. 29, 1345-1347 (2004). J.-S. Jang, F. Jin and B. Javidi, “Three-dimensional integral imaging with large depth of focus by use of real and virtual image fields,” Opt. Lett. 28, 1421-1423 (2003). M. Martínez-Corral, B. Javidi, R. Martínez-Cuenca, and G. Saavedra, “Integral imaging with improvement depth of field by use of amplitude modulated microlens array,” Appl. Opt. 43, 5806-5813 (2004) S. W. Min, B. Javidi, and B. Lee, "Enhanced 3D Integral Imaging System by use of double display devices," Journal of Applied Optics-Information Processing 42, 41864195 (2003). R. Martínez-Cuenca, G. Saavedra, M. Martínez-Corral and B. Javidi, "Enhanced depth of field integral imaging with sensor resolution constraints," Opt. Express 12, 52375242 (2004). P. Ambs, L. Bigue, R. Binet, J. Colineau, J.-C. Lehureau and J.-P. Huignard, "Image reconstruction using electro-optic holography," in Proc. of the 16th Annual Meeting of the IEEE Lasers and Electro-Optics Society, LEOS 2003, vol. 1 (IEEE, Piscataway, NJ, 2003) pp. 172-173. N. Davies, M. McCormick and M. Brewin, "Design and analysis of an image transfer system using microlens array," Opt. Eng. 33, 3624-3633 (1994). M. Martínez-Corral, B. Javidi, R. Martínez-Cuenca and G. Saavedra, "Multifacet structure of observed reconstructed integral images," J. Opt. Soc. Am. A 22, 597-603 (2005).. [1.27]. Y.-W. Song, F. Jin and B. Javidi, “3D object scaling in integral imaging display by varying the spatial ray sampling rate,” Opt. Express, 13, 3242-3251 (2005)..

(27) 10. CHAPTER 1. INTRODUCTION.

(28) Chapter 2 Liquid Crystals. 2.1. Properties of a Liquid Crystal Cell.. Liquid Crystal Displays (LCD) are composed by an array of liquid crystal (LC) cells each one represents a picture element (pixel). The LC cell is a rectangular transparent box filled with liquid crystal and a couple of transparent electric terminals. The liquid crystal state is a state of matter with elongated molecules typically cigar shape [2.1]. The molecules have an orientation order like crystals, but lack positional order like liquids. The liquid crystal molecules are anisotropic material whose optical properties depend on the direction of propagation as well as polarization of the light waves [2.2]. Also the orientation order can be modified by external forces and induced electric fields, and then the effect of LC over a light wave is nontrivial. In this chapter, the theory of liquid crystal cell behavior is presented. The understanding of LC needs the study of different concepts as: molecular structure, crystal electro-optic effect, light polarization state, and gray control at each pixel by electronic devices.. 11.

(29) 12. 2.1.1. CHAPTER 2. LIQUID CRYSTALS. Calamitic Liquid Crystal Molecules. Most of liquid crystals are former by small uniaxial crystal that we call molecules. The most common molecules shape is cylindrical or cigar shape, as is shown in Fig. 2.1. These molecules are call calamitic [2.1]. The crystal molecules have characteristics of uniaxial crystal, as those related with the transmission of light through it. The birefringence or double refraction is observed as the natural crystal anisotropy, and artificial birefringence produced by an electric field [2.4][2.5].. Figure 2.1 Calamitic crystal molecule rotates and aligns with the electric field.. The birefringence of the crystal produces two waves of different velocities propagating through a crystal with a given wave normal; moreover, these waves are plane polarized. The refractive index n can be defined as the ratio of c v where c is the light speed in vacuum and v is speed of light of the wave in the crystal. The refractive index of anisotropic materials depends on the polarization. Waves with different polarizations therefore travel at different velocities and undergo different phase shift, so that the polarization ellipse is modified as the wave advances. This property is used in the design of many optical devices..

(30) CHAPTER 2. LIQUID CRYSTALS. 13. For the two waves in the crystal the refractive index as a function of the direction of their common wave normal, are obtained by drawing an ellipsoid known as the indicatrix [2.6]. If x1, x2, x3 are the principal axes of the dielectric constant; the indicatrix is defined by the equation x12 x22 x32 + + =1 n12 n22 n32. (2.1). where n1 = k1 , n2 = k2 , n3 = k3 ,and k1, k2, k3 are the principal dielectric constants, as shown in Fig. 2.2.. Figure 2.2 The index ellipsoid. The coordinates (x, y, z) are the principal axes and n1,n2, n3 are the principal refractive indices. The refractive indices of the normal modes of a wave traveling in the direction k are na and nb.. For crystal molecules cigar shape, the x-axis, where they are oriented, is called the director axis, drawing with a vector n. The extraordinary axis is perpendicular to the principal..

(31) 14. 2.1.2. CHAPTER 2. LIQUID CRYSTALS. Discotic Liquid Crystal Molecules. Other forms of crystal molecules were discovered in 1977 by Indian researchers [2.3]. These new disclike molecules also form liquid crystal phases in which the axis perpendicular to the plane of the molecules tends to orient along a specific direction. These phases and the molecules that form them are called discotic liquid crystals, as is shown in Fig. 2.3.. Figure 2.3 The nematic and columnar discotic liquid crystal phases. The flat discs represent the molecules.. Most of the commercial display devices use calamatic crystals, cigar shape, and then from now on the discussion about liquid crystal will be focused on that type of molecule.. 2.1.3. Molecular structure. Many substances can exist in more than one state of mater if we consider external conditions as pressure and temperature. The liquid crystal can be described as substance in a middle state between solid and liquid. The crystal molecules neither occupy a specific average position nor remain oriented in a particular way. The random orientation and movement of the crystal molecules depends on the temperature, and then different phases can be obtained..

(32) CHAPTER 2. LIQUID CRYSTALS. 15. Figure 2.4 Molecular organization of different types of liquid crystals: (a) nematic, (b) smectic, (c) cholesteric.. The Calamatic crystal below Tm fusion point is solid, crystalline and anisotropic, above this point with temperatures Tm>Tc is an isotropic crystal. In the mesophase between Tm and Tc, it has the liquid appearance, but it keeps an order at each phase[2.6]. Along this temperature range different molecular structure, related with phases, can be observed. Nematic (from Greek yarns of fiber knitting). In this type of structure, the liquid crystals the molecules tend to be parallel but their positions are random. The Nematic phase has one-dimensional order it means the material is enrolled. All these phases are anisotropic, as in solid state. The nematic phase is used for the most popular Display Devices Twisted Nematic (TN) active matrix cells, and for the Super Twist Nematic (STN) passive matrix cells [2.1]-[2.4]. Smetic (from Greek smEktikos soap). The first phase just above the temperature Tc, is call Smetic C, the order is bi-dimensional, it means by layers, but molecules are arrange randomly at each layer so that they have positional order in only one dimension. In the next phase, at a higher temperature, is call Smetic A the direction of director axis is against the direction perpendicular to the plane of the molecules layer [2.1]-[2.4]. Cholesteric phase is a distorted form of the nematic phase in which the orientation undergoes helical rotation about an axis. If you add a chiral component as esters cholesterols, the nematic phase of a liquid crystal changes to a cholesteric phase, and show a helicoidal structure, at each layer the direction of director axis changes [2.1][2.4]..

(33) 16. 2.1.4. CHAPTER 2. LIQUID CRYSTALS. Electro Optic Effect. A uniaxial crystal can be assumed with constants refractive indexes in normal conditions. If an electric field is induced along the crystal a change in permittivity values is observed. Small changes of permittivity at optical frequencies are equivalent to small changes in refractive index, and these can be measured with great precision by using the effects of birefringence and by optical interferometry [2.3],[2.4]. This change in refractive index of a crystal produced by an electric field is known as the electro-optical effect. The molecular structure is also affected. Any neutral charged molecule in presence of an electric field behaves as an induced electric dipole. Orientation of an electric dipole by an electric field can be one of two types, as show in Fig 2.5. Assuming that the dipole is along the long axis of the molecule the electric field (E) causes forces (F) at the end of the dipole and the molecules rotate clockwise. A second possibility is that the electric dipole lies across the long axis and the rotation of the molecule is counterclockwise.. Figure 2.5 Orientation of an electric dipole by an electric field. (a) The dipole is along the long axis of the molecule. (b) The dipole lies across the long axis of the molecule. The electric field (E) causes forces (F) at the ends of the dipoles and rotation of the molecules as shown by the curved arrows.. It is not difficult to understand how crystal molecules behave in presence of an electric field. If the molecule is one that tends to orient its long axis along the electric field, then an electric field causes the molecules of the liquid crystal to tend to lie along the field..

(34) CHAPTER 2. LIQUID CRYSTALS. 17. The orientation order is not greater than in the absence of an electric field; the difference is that the electric field causes the director of the liquid crystal molecule to orient along the field. If the molecule tends to orient perpendicular to the electric field, then the presence of the electric field causes the director of the liquid crystal molecule to lie perpendicular to the field.. 2.1.4. Gray Control using a Polarization State. The modulation of each LC cell considers the control of each polarization state. A linear polarization along the director axis produces an elliptic polarization at the output plane, rotated as much as the twist of the molecules. A couple of linear polarizer at the input and output plane can be used for the optical configuration, where both of them should be aligned with the director axis and crossed each other. The induced electric field produces a rotation of the molecule structure and changes of the refractive indexes. For each value of induced electric field, or voltage between electrodes, it has its related elliptic polarization state, ellipticity and orientation. After the output linear polarizer, a variation of intensity can be observed; also a difference of phase can be obtained. The applied voltage controls the amplitude of the output light, then the number of gray values is as much as the number of voltages steps. The LCD has 256 gray values using two linear polarizers configuration. This configuration is used as only amplitude modulation; in spite of the small phase modulation attached. Any change in the orientation of the linear polarizers produces different curves of modulation in amplitude and range. In general, light modulation can be defined as: amplitude only [2.7], phase only [2.8], real [2.9], mostly-phase[2.10]-[2.11]. Each modulation has it specific application. The introduction of wave plate retarders in the setup configuration improved the performances of modulations, for example the phase only modulation. A linear polarizer and a wave plate retarder can be used as a generator of an elliptic polarization state or as a detector of elliptic polarization state. An optimized setup uses only one elliptic polarization state and improves a phase only modulation, useful for display of complex fields..

(35) 18. CHAPTER 2. LIQUID CRYSTALS. 2.2 Jones Matrix Formalism Mathematical analysis of an optical setup should consider the light with a polarization state and the effect of optical elements over the original polarization. It means that the changes of polarization through linear polarizers, wave plate retarders and a twisted nematic liquid crystal cell. The theoretical analysis helps to simulate the modulation curves for different orientations of optical elements. In this work, the light and optical elements are studied with Jones Matrix formalism. The Jones calculus, invented in 1940 by R.C. Jones [4.11]-[4.13], is a powerful 2x2 matrix method in which the state of polarization is represented by a two components vector while each optical element is represented by a 2x2 matrix. The overall matrix for the whole system is obtained by multiplying all matrices, and the polarization state of the transmitted light is computed by multiplying the vector representing the input beam by the overall matrix.. 2.2.1. Representation of Light Polarization. The polarization of light is determined by the time course of the direction of the electric – field vector E(r,t) [2.6]. For monochromatic light, the three components of E(r,t) vary sinusoidally with time with amplitude and phase, so that each point r the end point of the vector, moves in a plane and trace and ellipse. The wave is said to be elliptically polarized. When the ellipse degenerates into a straight line or become a circle, the wave is said o be linearly polarized or circular polarized. Consider a monochromatic plane wave of frequency v traveling in the z direction with velocity c. The electric field lies in the x-y plane and is generally described by, ⎧ ⎡ ⎛ z ⎞⎤ ⎫ E ( z , t ) = Re⎨ A exp ⎢i 2πv⎜ t − ⎟⎥ ⎬ , ⎝ c ⎠⎦ ⎭ ⎣ ⎩. (2.2). where the complex envelope. A = Ax xˆ + Ay yˆ ,. (2.3).

(36) CHAPTER 2. LIQUID CRYSTALS. 19. is a vector with complex components Ax y Ay. To describe the polarization of the wave, we trace the endpoint of the vector E(z,t) at each position z as a function of time. Expressing Ax, and Ay in terms of their magnitude and phases, Ax = ax exp(iϕ x ) and. Ay = a y exp(iϕ y ) , and substituting into equations (2.2) and (2.3), we obtain E ( z, t ) = Ex xˆ + E y yˆ ,. (2.4). ⎡ ⎛ z⎞ ⎤ E x = ax cos ⎢2πv⎜ t − ⎟ + ϕ x ⎥ ⎣ ⎝ t⎠ ⎦. (2.5). ⎡ ⎛ E y = a y cos ⎢2πv⎜ t − ⎣ ⎝. (2.6). ⎤ z⎞ ⎟ + ϕy ⎥ t⎠ ⎦. are the x and y components of the electric field vector E(z,t). The components Ex and Ey are periodic function with period t – z / c oscillating at frequency v. Equations 4a and b are the parametric equation of the ellipse, 2. 2. EE ⎛ Ex ⎞ ⎛ E y ⎞ ⎜⎜ ⎟⎟ + ⎜ ⎟ − 2 x y cos ϕ = sin 2 ϕ ⎜ ⎟ ax a y ⎝ ax ⎠ ⎝ a y ⎠. (2.7). where ϕ= ϕy - ϕ x is the phase difference. The state of polarization of the wave is determined by the shape of the ellipse (ellipticity and direction of major axis), which depends of two parameters, the ratio of the magnitude ax/ay and the phase difference. The size of the ellipse, determine the intensity of the wave. (a I=. 2 x. + a y2 ) 2η. (2.8). where η is the impedance of the medium. The monochromatic wave is characterized by the complex envelopes Ax=ax exp(iϕ) and Ay=ay exp(iϕ) of the components of the electric field..

(37) 20. CHAPTER 2. LIQUID CRYSTALS. It is convenient to write these complex quantities in the form of a column matrix ⎡ Ax ⎤ J =⎢ ⎥ ⎣ Ay ⎦. (2.9). know as the Jones vector. The total intensity of the wave is given by. (A I=. 2. + Ay. x. 2. ). (2.10). 2η. and used the ratio ay / ax = Ay / Ax and the phase difference. ϕ= ϕy - ϕ x = arg(Αy)-. arg(Αy) to determined the orientation and shape of the polarization ellipse.. The ellipticity of a polarization ellipse is defined as. e=±. b a. (2.11). where a and b are the lengths of the principal axes. Let φ (0≤φ<π) be the angle between the direction of the major axis x’ and the x axis. Then the lengths of the principal axes are given by. 2.2.2. a 2 = Ax2 cos 2 φ + Ay2 sin 2 φ + 2 Ax Ay cos δ cos φ sin φ. (2.12). b 2 = Ax2 sin 2 φ + Ay2 cos 2 φ − 2 Ax Ay cos δ cos φ sin φ. (2.13). Linearly Polarized Light. If one of the components vanishes (ax=0 for example), the light is linearly polarized in the direction of the other components (y direction). It is also, linearly polarized if the phase difference is zero or π [2.6]. The wave is also said to have planar polarization..

(38) CHAPTER 2. LIQUID CRYSTALS. 21. The Jones vector representation is:. ⎡cos(ψ )⎤ J =⎢ ⎥ ⎣ sin (ψ )⎦. (2.14). where ψ is the azimuth angle of the oscillation direction with respect to the x axis.. 2.2.3. Circularly Polarized Light. If ϕ =± π/2 and ax = a y = a0 , gives. ⎡ ⎛ E x = a0 cos ⎢2πv⎜ t − ⎣ ⎝. ⎤ z⎞ ⎟ + ϕx ⎥ t⎠ ⎦. (2.15). ⎡ ⎛ z⎞ ⎤ E y = m a0 sin ⎢2πv⎜ t − ⎟ + ϕ y ⎥ t⎠ ⎣ ⎝ ⎦. (2.16). from which E y2 + E x2 = a02 , which is the equation of a circle. In the case ϕ =– π/2 the electric field at a fixed position z rotates in a counterclockwise direction by and observer facing the approaching when viewed from the direction toward which the wave is approaching. The light is then said to be right circularly polarized. The case ϕ = + π/2 corresponds to counterclockwise rotation and left circularly polarized light. The jones vectors for a circularly polarized light are: 1 ⎡1⎤ Lˆ = ⎢ ⎥, 2 ⎣i ⎦ (a). 1 ⎡1⎤ Rˆ = ⎢ ⎥ 2 ⎣− i ⎦. (2.17). (b). The ellipticiy is taken as positive when the rotation is of the electric field is right handed and negative otherwise..

(39) 22. 2.2.4. CHAPTER 2. LIQUID CRYSTALS. Matrix of a Linear Polarizer. A polarizer is a device that transmits the component of the electric field in the direction of its transmission axis and blocks the orthogonal component [2.6]. This preferential treatment of the two components of the electric field is achieved by selective absorption, selective reflection from an isotropic medium, or selective reflection/refraction at the boundary of an anisotropic medium. A polarizer can be defined as a system represented by the Jones matrix,. ⎡1 0⎤ T =⎢ ⎥ ⎣0 0 ⎦. (2.18). Which transforms a wave of components (A1x,A1y) into a wave of components (A1x,0), thus polarizing the wave along the x direction. The system is a linear polarizer with transmission axis pointing in the x direction. Using a coordinate transformation, we can write a Jones Matrix, for any angle orientation of a linear polarizer, giving by, ⎡ cos 2 θ T =⎢ ⎣sin θ cosθ. sin θ cosθ ⎤ ⎥ sin 2 θ ⎦. (2.19). that shows a linear polarizer with a transmission axis making an angle θ with the x axis.. 2.2.5. Matrix of a Wave Plate Retarder. Wave plate retarders are often made of anisotropic materials [2.12]. The main characteristics are the retardation Γ and its fast and slow axis. The normal modes are linearly polarized waves, these waves are polarized in the directions of the axes and the velocities are different. The directions of polarization for these eigenwaves are mutually orthogonal are called “slow” and “fast” axes of the crystal for that direction of propagation..

(40) CHAPTER 2. LIQUID CRYSTALS. 23. Retardation plates are usually cut in such a way that the c axis lies in the plane of the plate surfaces. Thus the propagation direction of normally incident light is perpendicular to the c axis. The polarization state of a light beam can be converted to any other polarization state by means of a suitable retardation plate. Let ns and nf be the refractive indices for the slow and fast components respectively. The Jones Matrix formulation for a wave retarder is:. ⎤ ⎡ ⎛ − iΓ ⎞ 0 ⎥ ⎢exp⎜ 2 ⎟ ⎠ ⎥ W0 = exp(− iφ )⎢ ⎝ ⎛ iΓ ⎞ ⎥ ⎢ 0 exp⎜ ⎟ ⎢⎣ ⎝ 2 ⎠⎥⎦ The phase retardation is given by the difference between the Γ = (ns − n f ). (2.20). ωl. where l is c the thickness of the plate and ω is the frequency of the light beam. This is a measure of the relative change in phase, not the absolute change. The birefringence of a typical crystal retardation plate is small, that is, ns − n f ⟨⟨ ns , n f . Consequently, the absolute. change in phase caused by the plate may be hundreds of times greater than the phase retardation. Let φ be the mean absolute phase change,. φ=. 1 2. (n. s. + nf ). ωl c. (2.21). The propagation of light in the wave plate needs the decomposition of the incident beam, and the coordinate transformation of the emerging beam, in order to obtain the polarization state at the output plane. Then, we can write the transformation due to the retardation plate as: W = R(− ψ )W0 R(ψ ). (2.22).

(41) 24. CHAPTER 2. LIQUID CRYSTALS. Where R(ψ) is the rotation matrix. ⎡ cosψ R(ψ ) = ⎢ ⎣− sinψ. sinψ ⎤ cosψ ⎥⎦. (2.23). The phase factor e-iφ can be neglected if interference effects are not important, or not observable. A retardation plate is characterized by the phase retardation Γ and its azimuth angle ψ..

(42) CHAPTER 2. LIQUID CRYSTALS. 25. 2.3 References [2.1] [2.2] [2.3] [2.4] [2.5] [2.6] [2.7] [2.8] [2.9] [2.10] [2.11] [2.12] [2.13]. B. Bahadurl “Liquid Crystals -Applications and Uses. Types and Classification of Liquid Crystals” (D Demus) Litton Systems, Toronto Canada (1998). E. Lueder, “Liquid Crystal Displays”, chapter 2, John Wiley & Sons, New York, 2001. P. J. Collings, “Liquid Crystals” Princeton University Press. 2nd Ed. pp.1-45 Oxford UK (2002) J. F. NYE “Physical Properties of Crystals. Their Representation by Tensors and Matrices” Oxford Science Publications, pp. 2-49, Oxford UK.(1985) D. W. Berreman, “Dynamics of liquid-crystal twist cells”, Appl. Phys. Lett. 25, 12-15 (1974). B.E.A. Saleh, M.C.Teich “Fundamental of Photonics” John Wiley & Sons, pp 193-237, 696-736 USA(1991) C. Iemmi, S. Ledesma, J. Campos and M. Villareal, “Gray level computer generated hologram filters for multiple object correlation,” Appl. Opt. 39, 1233 (2000). J. A. Davis, D. M. Cottrell, J. Campos, M.. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004-5013 (1999). I. Juvells, A. Carnicer, , S. Vallmitjana, and J. Campos, “Implementation of real filters in a joint transform correlator using positive-only display,” J. Optics 25, 33 (1994). V. Arrizón, “Optimum on-axis computer-generated hologram encoded into lowresolution phase-modulation devices,” Opt. Lett. 28, 2521-2523 (2003). K. Lu, B. E. .A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng., 29, 240-246 (1990). Yariv Ammon, Yeh Pochi, “Optics of Liquid Crystal Displays Propagation and control of laser radiation” Ed. John Wiley & Sons 121-154 USA (2003) E. Hecht “Optics”, Addison Wesley; 4th Edition, pp. 680 (2001).

(43) 26. CHAPTER 2. LIQUID CRYSTALS.

(44) Chapter 3 Mathematical Model for a Liquid Crystal Cell. 3.1 Mathematical Approximation The characterization of a LCD starts with a mathematical model in order to obtain operational configurations by numerical simulation. The model proposed by Yariv [3.1] considers a twisted anisotropic media composed by layers of crystal molecules. This model has been used to simulate the behavior of a TNLCD and obtain the orientation values for specific modulation. Each group of orientation values is called configuration for a specific light modulation [3.2]. The main drawback of this mathematical model is that the numerical simulation has a lower accuracy when those results are implemented experimentally. Modulation curves obtained experimentally do not correspond to numerical curves. We expend most of the lab time finding the correct orientation. Marquez et. al. has been proposed a model based on the Jones matrix theory, but assuming two free parameter that change with the applied voltage inside of the liquid crystal panel [3.3]. The effective birefringence located in the central part of the panel and the birefringence of the edges layers. The difference between these two parts of the liquid. 27.

(45) 28. CHAPTER 3. MATHEMATICAL MODEL. crystal cell is that molecules in the edges of the panel can not be tilted, in spite of applying voltage. These considerations correct the mathematical model of liquid crystal panel, but we still have uncertainty between numerical simulation and lab implementation of element orientations. The objective is to find a numerical model with a higher accuracy to simulate the behavior of a TNLCD cell. This will help us to find different configurations, for a complex, real, or phase-mostly modulation.. 3.2 Matrix of a Liquid Crystal Cell The transmission of light through a twisted nematic liquid crystal is a typical jones matrix analysis of anisotropic material. It is modeled as a structure with the number of plates, N, tending to infinity and the plate thickness tending to zero as Γ/N. Each plate is assumed to be a wave plate with phase retardation Γ and an azimuth angle. The overall Jones Matrix can be obtained by multiplying together the entire matrix that is associated with these plates. In a cell of a TNLCD, with no applied voltage, the liquid crystal molecules are parallel to the glass plates containing the liquid crystal [3.4]-[3.5] as is shown in Fig. 3.1. The molecules are anchored to rubbing groves at the inner faces of the glass plates. In Fig. 3.1, these groves appear oriented with the x-axis; at the input plane, and with the y-axis, at the output plane. Such orientation of the groves forces the molecules to form a helix with the z-direction as the axis. If an electric field is applied between the glass plates, the molecules tend to obtain the orientation of the field. The applied voltage to each cell of a TNLCD is controlled by an electronic interface, with the purpose of modifying the optical properties of the cell.. Figure 3.1 Orientation of liquid crystal molecules in a cell of a TNLCD..

(46) CHAPTER 3. MATHEMATICAL MODEL. 29. A simplified physical model of the TNLCD cell assumes that the twist is a linear function of z. The Jones matrix of the cell, obtained under this assumption and the additional consideration that no electric field is applied to the cell [3.1], can be expressed as sin γ ⎡ cos γ − iβ ⎢ γ M ( β ) = e −iβ R( θ T )⎢ ⎢ θ T sin γ ⎢⎣ γ. sin γ. ⎤ ⎥ γ ⎥ , sin γ ⎥ cos γ + iβ γ ⎥⎦ − θT. (3.1). where β is the birefringence of the liquid crystal, given by,. β=. πd (n e − n o ) , λ. (3.2). d is the thickness of the liquid crystal layer in the cell, no and ne are the ordinary and extraordinary refraction indices for light propagated along the z-axis, and γ is defined by γ = θT2 + β 2 .. (3.3). In addition, θT is the total twist angle of the molecules in the cell, and R(θT) denotes the rotation matrix,. ⎡cos θT sin θT ⎤ R(θT ) = ⎢ ⎥. ⎣− sin θT cos θT ⎦. (3.4). To complement the above simplified physical model of a TNLCD cell, it is assumed that when a voltage is applied to the cell, along the z-axis, all the molecules are tilted by the same angle τ, toward the applied electric field..

(47) 30. CHAPTER 3. MATHEMATICAL MODEL. Figure 3.2 The crystal molecules are tilted by the same angle τ when a voltage is applied to. the cell, along the z-axis. For this simple model the tilted cell can also be described by the Jones matrix in Eq. (3.23) by the substitution of the extraordinary refractive index ne by the new index ne(τ), given by 1 sin 2 τ cos 2 τ = + n e2 (τ) n o2 n e2. (3.5). For each value of the tilt angleτ, the above parameters β and γ adopt different values, providing different modulation properties to the LC cell. An interesting example is the modification of the polarization state of a monochromatic plane wave with linear polarization, illuminating the LC cell. If the polarization of this beam is aligned with the optical axis at the input plane of the cell, the output elliptic polarization states, computed with Eqs. (3.23-3.26) for several increasing values of β, are shown in Fig. 3.2..

(48) CHAPTER 3. MATHEMATICAL MODEL. 31. Figure 3.3 Elliptic polarization states generated by a TNLCD for different values of β.. As noted in this figure, the large axis of the generated elliptic polarization increases its orientation angle as a function of β. A plot giving the orientation angles of the generated elliptic states, versus β, is shown in Fig. 3.4.. Figure 3.4 Rotation θ of generated elliptic polarization versus β.. The reference system, at which the above Jones matrices are referred, has the rectangular axes shown in Fig. 3.1. In this frame, the orientation of the polarizing components is measured following the counter-clockwise rotation. The Jones matrix model of the TNLCD that is summarized in Eqs. (3.1) to (3.5) is only an approximation, based on assumptions about the twist and tilt of the liquid crystal.

(49) 32. CHAPTER 3. MATHEMATICAL MODEL. molecules. Although these assumptions are not exactly true6-7, we have found that the above simple model is quite useful to obtain rough predictions of the behavior of the TNLCD in different situations. The above simple model of the TNLCD is employed to perform the computational optimization of our simplified array version of the optimum phase-only setup for a TNLCD.. 3.3 Mathematical Analysis using Jones matrix Another mathematical model proposed by Marquez et. al, considers that the liquid crystal cell can be divided in three different parts [3.3]. Figure 3.5 shows a scheme of the assumptions made by the model. The total thickness of the cell is d=d1+d2+ d3, where d1 and d3 are the thickness of the layers next to the surfaces and d2 is the thickness of the central part.. Figure 3.5 Schematic figure of a liquid crystal cell.. This model assumes that the crystal layers, next to the surfaces of the crystal cell, do not move and they behave like wave plate retarders then the twist angle remains constant and ∆n does not change with the voltage. The central part of the liquid crystal cell is modeled as twisted nematic crystal layers, as study by Yariv [3.1]. In this part the birefringence changes with the applied voltage as show in Eq. (3.2). The mathematical analysis of this part is exactly as Yariv approach. Finally, the proposed model considers that the individual thickness will be affected by the applied voltage. The wave plate retarders at the edges increase their thickness d1 and d3.

(50) CHAPTER 3. MATHEMATICAL MODEL. 33. when the voltage increases. The total thickness of the liquid crystal cell will remain constant, as a consequence the central part will decrease the thickness d2 when the voltage increases. From equation 3.2, the birefringence of the central part will change by the thickness change. The birefringence of wave plate parts will be given by. δ (V ) =. π d1 (ne − no ) , λ. (3.6). then the wave plates at the edges also change their birefringence values. The Jones matrix that describes the proposed liquid crystal cell considers three important parameters: the twist angle from the input to the output surfaces, the birefringence of the central part and the birefringence of the edges layers. The Jones matrix formalism will be used to obtain the mathematical expression for each element, in order to obtain a resultant matrix that is given by, M LCD (α , β , δ ) = WR(δ )R(θT )M (β )R(θT )WR(δ ) .. (3.7). WR(δ) denotes the matrix of a wave plate retarder, M(β) denotes the twisted nematic crystal cell matrix and R(θΤ) the rotational matrix, with a total twist θT.. The complete matrix should be:. (. M LCD. ⎡exp − i Γ 2 ⎢ 0 ⎢ =⎣. ). β sin X sin X ⎤ ⎡ φ − − cos X i ⎤ ⎥ ⎢ 2 X X ⎥ R(θT )⎢ β sin X ⎥ sin X exp − i Γ ⎥ ⎥ ⎢ φ + cos X i 2⎦ 2 X ⎦ X ⎣ ⎡exp − i Γ ⎤ 0 2 ⎥ × R(θT )⎢ 0 exp − i Γ ⎥ ⎢⎣ 2⎦. (. 0. ). (. ). (. (3.8). ). Reducing the expression to, ⎡ X − iY M LCD = exp[− i(β + 2δ )]R(− α ) × ⎢ ⎣ −Z. Z ⎤ X + iY ⎥⎦. (3.9).

(51) 34. CHAPTER 3. MATHEMATICAL MODEL. where terms X and Y are given by, X = cos γ cos 2δ −. β sin γ sin 2δ γ. (3.10). Y = cos γ sin 2δ −. β sin γ cos 2δ . γ. (3.11). Term Z is given by, Z=. α sin γ γ. (3.12). with the definition of γ = α 2 + β 2 . Experimental values of the liquid crystal cell should be obtained in order to have an appropriated simulation. The experimental values are the orientation angle of the director axis at the entrance face, the twist angle α, and the value of the birefringence when the liquid crystal cell is in the off state βOFF. In the off state the birefringence of the wave plates retarders δ(V) is nearly zero and the birefringence of the twisted nematic crystal layers may be considered as the maximum value the display can reach..

(52) CHAPTER 3. MATHEMATICAL MODEL. 35. 3.4 References [3.1] Yariv Ammon, Yeh Pochi “Optics of Liquid Crystal Displays Propagation and control of laser radiation” Ed. John Wiley & Sons 121-154 USA (2003) [3.2] K. Lu, B. E. .A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng., 29, 240-246 (1990). [3.3] Andres Marquez, Claudio lemmi, Ignacio Moreno, Jeffrey Davis, Juan Campos, Maria J. Yzuel “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model” Op. Eng. (40) 2558-2564 (2001) [3.4] E. Lueder, “Liquid Crystal Displays”, chapter 2, John Wiley & Sons, New York, (2001) [3.5] B.E.A. Saleh, M.C.Teich “Fundamental of Photonics” John Wiley & Sons, pp 193-237, 696-736 USA(1991).

(53) 36. CHAPTER 3. MATHEMATICAL MODEL.