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Quantitative phase imaging microscopy with multi-wavelength optical phase unwrapping

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Title:
Quantitative phase imaging microscopy with multi-wavelength optical phase unwrapping
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English
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Warnasooriya, Nilanthi
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University of South Florida
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Tampa, Fla
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Subjects / Keywords:
Interferometry
Phase contrast microscopy
Interference microscopy
Phase shifting
Dissertations, Academic -- Physics -- Doctoral -- USF   ( lcsh )
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non-fiction   ( marcgt )

Notes

Summary:
The results presented here are divided into three main categories based on the source of illumination; light emitting diodes, laser diodes and a ring dye laser. Results for both two-wavelength optical unwrapping and three-wavelength optical unwrapping techniques are demonstrated. The interferographic images using broadband sources such as light emitting diodes are significantly less affected by coherent noise compared to images obtained using lasers. Our results show that the three wavelength optical phase unwrapping can also be effectively applied to unwrap phase images obtained using coherent light sources such as lasers and laser diodes, without amplifying phase noise in the final phase image. We have successfully shown that our multi-wavelength phase-shifting technique extends the range free of 2π ambiguities in the phase map without using conventional computation intensive phase unwrapping methods.This phase imaging technique can be used to measure physical thickness or height of both biological and other microscopic samples, with nanometer axial resolution. An added advantage of the multi-wavelength optical phase unwrapping technique is that the beat wavelength can be tailored to match height variations of specific samples.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
Additional Physical Form:
ABSTRACT: This dissertation presents a quantitative phase imaging microscopy technique that combines phase-shifting interferometry with multi-wavelength optical phase unwrapping. The technique consists of a Michelson-type interferometer illuminated with any of three types of light sources; light emitting diodes, laser diodes and a ring dye laser. Interference images are obtained by using a 4-frame phase shifting method, and are combined to calculate the phase of the object surface. The 2π ambiguities are removed by repeating the experiment combining two and three different wavelengths, which yields phase images of effective wavelength much longer than the original. The resulting image is a profile of the object surface with a height resolution of several nanometers and range of several microns. To our knowledge, this is the first time that a three wavelength optical phase unwrapping method with no amplified phase noise has been presented for full-frame phase images.
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by Nilanthi Warnasooriya.
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Title from PDF of title page.
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Document formatted into pages; contains 130 pages.
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Includes vita.

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Quantitative Phase Imaging Microscopy w ith Multi-Wavelength Optical Phase Unwrapping by Nilanthi Warnasooriya A dissertation submitted in partial fulfillment of the requirements for the degree of Department of Physics College of Arts and Sciences University of South Florida Major Professor: Myung K. Kim, Ph.D. Srikanth Hariharan, Ph.D. Chun-Min Lo, Ph.D. Sarath Witanachchi, Ph.D. Date of Approval: August 21, 2008 Keywords: interferometry, phase contrast microscopy, interference microscopy, phase shifting Copyright 2008 Nilanthi Warnasooriya

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Dedication To my parents.

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Acknowledgments First I would like to take th is opportunity to express my heartfelt gratitude to my research advisor Dr. Kim for his immeasur able mentoring, patience and encouragement throughout this project. I have been pr ivileged to be a student in his lab. I would also like to thank all my comm ittee members Dr. Srikanth Hariharan, Dr. Chun-Min Lo, and Dr. Sarath Witanachchi fo r serving on my dissertation committee. Thanks also go to both my previous and current colleagues at Digital Holography and Microscopy Laboratory; Dr. Lingfeng Yu, Dr. Christopher Mann, Dr. Leo Krzewina, William Ash, Alex Khmaladze and Mariana Potc oava for their helpful suggestions and assistance in numerous ways. Most importantly I would like to tha nk my parents and sisters for their unconditional love and support. It is the thought of them that motivates me to go another step further during difficult times. My deepest thanks also go to all my dear friends, who are always there for me when I need them. I would also like to thank the faculty of the Department of Physics at USF for offering me an assistantship to pursue my st udies. Without this assistance I would have never come so far. Finally, I would like to thank the staff of the Physics department for their help in many ways.

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Table of Contents List of Tables.......................................................................................................................v List of Figures....................................................................................................................vi ABSTRACT....................................................................................................................... ix CHAPTER 1. INTRODUCTION.1 1.1. Optical Microscopy...... 1 1.1.1. Modern Optical Microscopy..................................................... 2 1.2. Phase Contrast Microscopy... 1.2.1. Zernike Phase Contrast Microscopy................................................. 4 1.2.2. Normaski Differential Interference Contrast Microscopy................ 5 1.2.3. Hoffman Modulation Contrast Microscopy..................................... 6 1.3. Interference Microscopy.......................................................................... 7 1.3.1. Michelson Interferometer..7 1.3.2. Mach-Zehnder Interferometer.............................................. 8 1.4. Phase-Shifting Interference Microscopy................................................. 9 1.5. Phase Unwrapping ................................................. 9 1.5.1. Multi-wavelength Phase Unwrapping 1.6. Light Sources.............................................................................10 1.7. Research Contributions............................................................................. 11 1.8. Thesis Organization..... i

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CHAPTER 2. PHASE SH IFTING INTERFEROGRAPHY.......... 2.1. Interference................................................................................ 14 2.2. Coherence.................................................................................. 16 2.2.1. Temporal Coherence..... 2.2.2. Spatial Coherence..... 2.2. Principle of Phase Shifting Interferography...17 2.3. Phase Shifting Methods............................................................................. 18 2.3.1. Moving the Reference Mirror..................................................... 18 2.3.2. Using a Tilted Glass Plate... 18 2.3.3. Using a Phase Plate..... 18 2.4. Phase Shifting Algorithms......19 2.4.1. Three-frame Phase Shifting Algorithm....................................... 20 2.4.2. Four-frame Phase Shifting Algorithm.... 20 2.5. Moving the Reference Mirror.....24 2.6. Phase Shifting Interference Microscopy Using Single Wavelength .27 CHAPTER 3. EXPE RIMENTAL SETUP..........29 3.1. Interferometer........29 3.2. Camera and Image Acquisition..33 3.3. Computer and Programs........ 33 3.4. Light Sources......... 34 3.4.1. Light Emitting Diodes..... 34 3.4.2. Laser Diodes................................................................................... 36 3.4.3. Ring Dye Laser....... 38 ii

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CHAPTER 4. MULTI-WAVELENGTH PHASE SHIFTING INTERFERENCE MICROSCOPY.......39 4.1. Numerical Algorithms Phase Unwrapping. 4.2. Multi-Wavelength Phase Unwrapping 4.3. Principle of Two-Waveleng th Optical Phase Unwrapping ....44 4.4. Two-Wavelength Optical Ph ase Unwrapping Simulations 4.4.1. Noise Reduction of Coarse Map..... 48 4.5. Principle of Three-Wavele ngth Optical Phase Unwrapping...50 4.6. Three-Wavelength Optical Unwrapping Simulations.52 CHAPTER 5. MULTI-WAVELENGTH OPTICAL PHASE UNWRAPPING USING LIGHT EMITTING DIODES....57 5.1. Two-Wavelength Optical Phas e Unwrapping with Light Emitting Diodes.......57 5.1.1. Resolution Target... 58 5.1.2. Onion Cells..................... 62 5.1.3. Micro-Electro-Mechanical System (MEMS)................................. 66 5.1.4. Transmission Electro Micr oscope (TEM) Grids............................ 69 5.1.5. Cheek Cells..72 5.2. Three-Wavelength Optical Phas e Unwrapping with Light Emitting Diodes...74 5.2.1. Resolution Target.... 74 5.2.2. Onion Cells...................... ....... 77 5.2.3. Micro-Electro-Mechanical System (MEMS) Biosensor..... 80 iii

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5.2.4. Transmission Electro Micr oscope (TEM) Grids............................. 83 5.2.5. Cheek Cells...86 5.3. Discussion and Conclusions......... 88 CHAPTER 6. MULTI-WAVELENGTH OPTICAL PHASE UNWRAPPING USING LASER DIODES......89 6.1. Two-Wavelength Optical Phas e Unwrapping with Laser Diodes ..89 6.2. Three-Wavelength Optical Phase Unwrapping with Laser Diodes.... 91 6.2.1. Micro-electro-mechanical Biosensor.. 91 6.2.2. Long Playing (LP) Record Grooves 6.3. Discussion and Conclusions........... 97 CHAPTER 7. THREE-WAVELENGTH OPTICAL PHASE UNWRAPPING USING RING DYE LASER......98 7.1. Cheek Cells......98 7.2. Long Playing (LP) Record Grooves..........100 7.3. Discussion and Conclusions............. 103 CHAPTER 8. CONCLU SIONS AND FUTURE WORK....104 References .. 108 Bibliography .. 116 Appendices..... 118 Appendix A: Comput er Programs ...... 119 Appendix B: List of Accomplishments....... 129 About the Author ...... End Page iv

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List of Tables Table 3.1. Characteristics of LEDs....................................................................................36 Table 3.2. Characteristics of laser diodes..........................................................................36 Table 5.1. Specification table of USAF 1951 resolution target.................58 v

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List of Figures Figure 1.1. Zernike Phase Contrast Microscope..................................................................5 Figure 1.2. Normaski differential in terference contrast microscope...................................5 Figure 1.3. Hoffman modulati on contrast microscope........................................................6 Figure 1.4. Michelson interferometer..................................................................................8 Figure 1.5. Phase map with 2 discontinuities.................................................................10 Figure 2.1. Several methods of phase shifting...................................................................19 Figure 2.2. Experimental setup..........................................................................................21 Figure 2.3. Five image frames of quadrature phase shifting..............................................23 Figure 2.4. An interferogram and intensity distribution....................................................24 Figure 2.5. Ramp waveform..............................................................................................25 Figure 2.6. Acquiring five im ages by skipping frames......................................................26 Figure 2.7. An interferogram.............................................................................................27 Figure 2.8. Wrapped phase imag e and its surface profile..................................................28 Figure 3.1. Experimental setup..........................................................................................30 Figure 3.2. Polarization of a light beam.............................................................................31 Figure 3.3. Effect of quarter wave pl ates and polarizer-analyzer pair...............................32 Figure 3.4. A light emitting diode......................................................................................35 Figure 3.5. Spectra of light emitting diodes.......................................................................35 Figure 3.6 A laser diode.....................................................................................................37 vi

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Figure 3.7. Spectra of laser diodes.....................................................................................37 Figure 3.8. The ring dye laser............................................................................................38 Figure 4.1. Phase vs. Distance...........................................................................................44 Figure 4.2. Simulation of two-wavele ngth optical phase unwrapping..............................47 Figure 4.3. Noise reduction of two-wa velength optical phase unwrapping......................49 Figure 4.4. Phase noise in co arse map and fine map.........................................................50 Figure 4.5. Simulation of three-wave length optical phase unwrapping............................53 Figure 4.6. Continuation of three-waveleng th optical phase unwrapping simulations......55 Figure 4.7. Noise reduction in three-wa velength optical phase unwrapping.....................56 Figure 5.1. Resolution target..............................................................................................59 Figure 5.2. Two-wavelength optical phase unwrapping of resolution target....................60 Figure 5.3. Surface profiles of resolution target................................................................61 Figure 5.4. Interferogram of onion cells............................................................................62 Figure 5.5. Two-wavelength optical phase unwrapping of onion cells.............................64 Figure 5.6. Surface profiles of onion cells.........................................................................66 Figure 5.7. Micro electrode array biosensor......................................................................67 Figure 5.8. Two-wavelength optical phase unwrapping of biosensor...............................67 Figure 5.9. Surface profiles of biosensor...........................................................................68 Figure 5.10. TEM grids......................................................................................................69 Figure 5.11. Two-wavelength optical phase unwrapping of TEM grids...........................70 Figure 5.12. Surface profiles of TEM grids.......................................................................71 Figure 5.13. Direct imag e of a cheek cell..........................................................................72 Figure 5.14. Two-wavelength optical phase unwrapping of cheek cell............................73 vii

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Figure 5.15. Three wavelength optical pha se unwrapping of re solution target.................74 Figure 5.16. Surface profiles of resolution target..............................................................76 Figure 5.17. Three-wavelength optical phase unwrapping of onion cells.........................77 Figure 5.18. Surface profiles of onion cells.......................................................................79 Figure 5.19. Three-wavelength optical phase unwrapping for biosensor..........................80 Figure 5.20. Surface profiles of biosensor.........................................................................82 Figure 5.21. Three-wavelength optical phase unwrapping of TEM grids.........................83 Figure 5.22. Surface profiles of TEM grids.......................................................................85 Figure 5.23. Three-wavelength optical phase unwrapping of cheek cells.........................87 Figure 6.1. Two-wavelength OPU of ch eek cells using laser diodes................................90 Figure 6.2. Three-wavelength OPU of MEMS sensor using laser diodes.........................93 Figure 6.3. Surface profiles of MEMS sensor...................................................................94 Figure 6.4. Three-wavelength OPU of LP record grooves using laser diodes..................96 Figure 7.1. Three-wavelength OPU of ch eek cells using ring dye laser............................99 Figure 7.2. Three-wavelength OPU of LP record grooves using ring dye laser..............101 Figure 7.3. Surface profiles of LP record grooves...........................................................102 Figure 8.1. Modified setup w ith fiber optics cables.........................................................106 viii

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Quantitative Phase Imaging Microscopy with Multi-wavelength Optical Phase Unwrapping Nilanthi Warnasooriya ABSTRACT This dissertation presents a quantitativ e phase imaging microscopy technique that combines phase-shifting interferometry with multi-wavelength optical phase unwrapping. The technique consists of a Michelson-type in terferometer illuminated with any of three types of light sources; light emitting diodes, la ser diodes and a ring dye laser. Interference images are obtained by using a 4-frame pha se shifting method, and are combined to calculate the phase of the object surface. The 2 ambiguities are removed by repeating the experiment combining two and three different wavelengths, which yields phase images of effective wavelength much longer th an the original. The resulting image is a profile of the object surface w ith a height resolution of several nanometers and range of several microns. To our knowledge, this is th e first time that a th ree wavelength optical phase unwrapping method with no amplified pha se noise has been presented for fullframe phase images. The results presented here are divided into three main categories based on the source of illumination; light emitting diodes, laser diodes and a ring dye laser. Results for ix

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both two-wavelength optical unwrapping a nd three-wavelength optical unwrapping techniques are demonstrated. The interferographic images using br oadband sources such as light emitting diodes are significantly less affected by coherent noise compared to images obtained using lasers. Our results show that the th ree wavelength optical phase unwrapping can also be effectively applied to unwrap phase images obtained using coherent light sources such as lasers and laser diodes, without amp lifying phase noise in the final phase image. We have successfully shown that our multi-wavelength phase-shifting technique extends the range free of 2 ambiguities in the phase map without using conventional computation intensive phase unwrapping met hods. This phase imaging technique can be used to measure physical thickness or height of both biological and other microscopic samples, with nanometer axial resolution. An added advantage of the multi-wavelength optical phase unwrapping technique is that the beat wavelength can be tailored to match height variations of specific samples. x

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CHAPTER 1 INTRODUCTION 1.1. Optical Microscopy The technique of using a single lens or a se ries of lenses to magnify features of an object is known as microscopy. The main goal of a microscope is to resolve and magnify minute details in a specimen so that they ar e visible to the human eye. It is not known when the first microscope was invented; a single lens has been used as a magnifying device for hundreds of years. The first co mpound microscope was built in 1590 by two Dutch eyeglass makers; Hans Jansen and Zacharias Jansen. Two well known users of single lens and compound microscopes ar e Anton Leeuwenhoek and Robert Hooke. Anton Leeuwenhoek (1632-1723), later known as the father of microscopy, used a single convex lens to study samples placed on an adjustable needle. He was the first to observe and describe microorganisms. R obert Hooke (16351703) used a simple compound microscope consisting of two lenses one lens as the objective and the other as the eyepiece. He wrote the book Micrographia describing his observations of microscopic structures using the term cell for the first time and confirming Leeuwenhoeks observations of living organisms in water. 1

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During the 18th and 19th centuries, compound microscopes rapidly gained popularity and were improved with the new knowledge in optics. In the 1730s Chester More Hall combined a convex lens made of crown glass wi th a concave lens made of Flint glass and invented the achromatic lens to solve chroma tic aberrations. However, he did not publish his invention. In 1759, a telescope maker name d John Dolland used Halls idea to make an achromatic lens and obtained a patent. Another major milestone of microscopy is the solution to spherical aberration. In 1830, Jose ph Jackson Lister mathematically showed how to minimize spherical aberration of an optical system. Microscopes with Listercorrected lenses were produced a few years later. In 1873, Ernst Abbe demonstrated a formula that gives the resolving power of a microscope. According to Abbes formula, the minimum resolved distance is related to the wavelength of the light and the numerical aperture. The numerical aperture is a combination of the refractive index of the medium and the half angle of the cone of light that enters the lens. In other words, Abbes formula states that in order to get the maximum resolution, the cone of light that enters the lens must be maximum. In 1893, August Koehler developed an illumination method that improves the resolution of the microscope by using an evenly illuminated field of view. Today the method is known as Koehler illumination and is used by all modern microscope users. 1.1.1 Modern Optical Microscopy Modern optical microscopes have surpassed 19th century microscopes in numerous ways. They are equipped with advanced lenses that minimize aberrations, high numerical apertures for better resolution, digital image acquisition and processing 2

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methods and ergonomical designs. Different techniques such as scanning microscopy, confocal microscopy, fluorescence microscopy, phase contrast microscopy and digital holographic microscopy have been used in bi omedical, engineering and materials science fields to investigate biological cells, micromet er scale electronic and mechanical devices and nanometer scale polymer composites. E ach technique has advantages and unique features that are suitable for specific applic ations. For example, when studying a living organism, a phase contrast microscope or digital holographic microscope can be used without a sample preparation, while a sca nning electron microsc ope or a fluorescence microscope needs a vacuum or staining. 1.2 Phase Contrast Microscopy In microscopy, specimens can be divided in to two main categories; amplitude objects and phase objects. Amplitude objects ab sorb light to create sufficient intensity contrast to be observed with bright fiel d microscopy. If the specimen doesnt absorb enough light, it is not visible to the human ey e. Phase objects do not absorb light to produce visible intensity contrast. Since the hum an eye is capable of seeing only color or intensity differences, phase changes have to be converted to intensity changes. Phase contrast microscopy converts phase changes into amplitude changes. The technique was first introduced by Fritz Zern ike in 1942. There are several phase contrast microscopy techniques; Zernike phase contrast (ZPC), differential interference contrast (DIC), Hoffman modulation contrast (HMC) etc. 3

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1.2.1 Zernike Phase Contrast Microscopy Zernike phase contrast microscopy, intr oduced by Fritz Zernike [1, 2], converts changes in phase into corresponding changes in amplitude by using a phase plate. It uses common path interferometry, where a partially coherent light beam passes through the specimen. The light that is diffracted due to phase variations of the specimen, and the light that passes without diffrac tion are then focused to form a phase contrast image of the specimen. A schematic diagram of Zernike phase contrast microscope is shown in Figure (1.1). Figure 1.1 : Zernike phase contrast microscope. 4

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1.2.2 Normaski Differential Int erference Contrast Microscopy In 1950s George Normaski modified the conventional Wollaston prism used in differential interference contrast microscope so that it can be placed outside the objective [3]. The Normaski DIC microscope converts re fractive index changes of the sample into amplitude changes by splitting a polarized b eam into two perpendicularly polarized beams less than a micrometer apart. The phase of the two beams changes after passing through the sample, due to variations of the refraction index. The two beams are recombined using a Wollaston (Normaski) prism. The recombined beam is sent through an analyzer where it passes only the circularly and elliptically polar ized light beams to form an image of the specimen. A schematic diagram of Normaski differential interference contrast microsc ope is shown in Figure (1.2). Figure 1.2 : Normaski differential interf erence contrast microscope. 5

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1.2.3 Hoffman Modulation Contrast Microscopy In 1975, Robert Hoffman invented a new phase contrast technique, named Hoffman modulation microscopy [4]. HMC micr oscopy converts optic al phase gradients into amplitude using a spatial filter called modulator, which has three light-passing zones, a slit plate and a circular polarizer. Di fferent specimen thickne sses deflect the light to different modulator zones thus controlling the contrast of the image. A schematic diagram of Hoffman modulation contrast microscope is shown in Figure (1.3). Figure 1.3: Hoffman modulation contrast microscope. 6

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1.3 Interference Microscopy In interference microscopy, an image is formed by the interference of two light beams; one beam reflected from (or transmitted through) the specimen, and one beam reflected from a plane mirror known as reference mirror. In 1887, Albert Michelson and Edward Mo rley invented a device that uses the interference of light to measure the difference in velocity of two perpendicular light beams [5]. Today, this device is known as the Michelson interferometer and is extensively used in a range of in terference microscopy techniques. Most quantitative microscope systems ar e based on the Michel son interferometer or the Mach-Zehnder interferom eter or their variations. 1.3.1 Michelson Interferometer In the basic Michelson interferometer, a light beam is divided into two beams of equal intensity and these two beams illuminate two mirrors. A compensator is used so that light beams on both arms travel the same optical path lengths. The two reflected beams are then combined again to form interference fringes. In the Michelson type microscope, the object to be analyzed is pla ced in one arm, replacing one of the mirrors. The phase information of the object ca n be extracted from interferograms. 7

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Figure 1.4 : A schematic diagram of th e Michelson interferometer. A variation of the Michelson interferomet er, where two microscope objectives are used to obtain higher magnification is known as the Linnik interferometer. The objectives are placed in front of the object and of the re ference mirror. This configuration is widely used in modern microscopy systems to acquire desired magnifications. The Michelson type configurations are best suitable for studying reflective and semi-transparent samples. For fully transp arent samples a Mach-Zehnder interferometer can be used. 1.3.3 Mach-Zehnder Interferometer The Mach-Zehnder interferometer, invent ed by Ernst Mach and Ludwig Zehnder, uses two beam splitters (or two half silver mi rrors) and two regular mirrors to divide the light beam into two paths and then recombin es them to produce interference. The sample 8

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can be placed in one of the arms so that the light beam can traverse once through the sample. 1.4 Phase-shifting Interference Microscopy The basis of the phase-shif ting interference microscope (PSIM) is a phase-shifting interferometer (PSI), where the phase difference between two beams of the interferometer is changed in equal steps. At each step, th e intensity of the in terference pattern is recorded and then the interferograms are combined to obtain the phase map of the object. Phase shifting can be done by using several differe nt techniques as described in Chapter 2. Depending on the researchers needs, PSI can be built in a Michelson, Linnik or MachZehnder configuration. 1.5 Phase Unwrapping Each fringe in an interferogram repres ents an area of data ranging from 0 to2 Therefore, the final phase map obtained from a series of interferograms also contains 2 ambiguities. Such phase maps are called wrapped phase maps, and are needed to be unwrapped by removing 2 ambiguities. Figure (1.5) shows a phase map with 2 discontinuities where the black represents pha se value of zero and the white represents phase value of2 9

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Figure 1.5 : phase map with 2 discontinuities. Black represen ts phase value of zero and white represents phase value of2 Once these 2 ambiguities are removed, a continuous surface profile of the test object can be obtained. Such a surface profile provides height information of surface features. Generally, phase unwrapping is done by using numerical algorithms. Most of these numerical algorithms are computationa lly intensive and can fail when there are irregularities in the test object. 1.5.1 Multi-wavelength Phase Unwrapping Two or more wavelengths can be used to extend the range free of 2 ambiguities in the phase map, thus avoiding the need fo r phase unwrapping. In two-wavelength phase unwrapping, two wavelengths are used to pr oduce a longer waveleng th called the beat wavelength. For two wavelengths1 and2 the beat wavelength is defined by12 21 21 12 By choosing closer wavelengths, th e beat wavelength can be extended to suit the surface heights of the object. One problem of the two-wavelength phase unwrapping method is noise amplification. Th e phase noise in each single wavelength 10

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phase map is amplified by a factor equal to the magnification of the wavelengths. This problems has been successfully addressed in this thesis and the multi-wavelength phase unwrapping has been extended to three wa velengths without noi se amplification. 1.6 Light Sources The source of light is a very important pa rt of any microscopy technique. An ideal light source for a microscope should provide a constant lumi nosity without voltage drop variances. It should be small enough to fit in a microscope, cost effective and should have a long life span. In this experiment, light emitting diodes (LED), laser diodes (LD) and a ring dye laser are used as light sources. Specifications of each light source are described in detail in Chapter 3. 1.7 Research Contributions The microscope is an indispensable tool in many areas of modern science. The need for microscopy techniques that provi de high magnification and resolution has become vital with the progress of nano t echnology and bio-medical fields. Studying properties and characteristics of new compositi ons, cells and tissues has been easier with the development of phase imaging microscope techniques. As the use for phase imaging techniques increases, so does the need for simple, user friendly phase unwrapping methods. For the first time, to our knowledge, we ha ve presented full-frame phase images by using a three wavelength optical phase unwrapping method with no amplified phase noise. We present unwrapped phase images using both two-wavelength and three11

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wavelength phase unwrapping methods. The e xperimental setup is a widely known Michelson type phase shifting interferometer but the use of incoherent light sources (light emitting diodes) and low coherent light sources (laser diodes) provides significantly less speckle noise than laser light sources. The added advantages are reduced apparatus dimensions, low cost a nd ease of operation. Such qualities are important factors of a user friendly microscope system. We have successfully shown that our multi-wavelength phase-shifting technique extends the range free of 2 ambiguities in the phase map without using conventional computational intensive phase unwrapping methods. This phase imaging technique can be used to measure physical thickness or height of samples, with nanometer axial resolution. 1.8 Thesis Organization This thesis consists of 8 chapters an d 2 appendices. Chapter 1 provides a general introduction of phase imaging and phase unwrapping techniques included in the study. Chapter 2 consists of basic concepts and principles of phase shifting interferography describing interference of two light beams, the principle of phase shifting interference microscopy, phase shifting methods, 3 and 4 frame phase shifting algorithms and examples of single wavelength phase images. In Chapter 3 the experimental setup is described beginning with the interferometer. It also de scribes the camera and image acquisition process, computer programs used in the experiment and properties of the light sources. Chapter 4 presents multi-wavelengt h phase shifting interference microscopy beginning with numerical algorithms for phase unwrapping and then describing multi12

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wavelength phase unwrapping, the principle of two-wavelength optical phase unwrapping, two-wavelength optical phase unwrapping simulations, the principle of three-wavelength optical phase unwrapping and three-wavele ngth optical phase unwrapping simulations. Experimental results of multi-wavelength optical phase unwrapping using light emitting diodes are presented in Chapter 5. Chapter 6 presents experimental results of multiwavelength optical phase unwrapping using laser diodes. Chapte r 7 presents experimental results of multi-wavelength optical phase unw rapping using a ring dye laser. Conclusions and future work are presented in Chapter 8. 13

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CHAPTER 2 PHASE SHIFTING INTERFEROGRAPHY This chapter presents several fundamental principles of optics. The first section describes principles of interference of light waves which is the base of this experiment. Then Section (2.2) presents the principle of phase shifting inte rference microscopya technique that utilizes both interference phenomenon and phase shifting for imaging. Section (2.3) describes different phase shifting methods. 2.1 Interference Interference is the superposition of two or more electromagnetic waves. In optics, this means addition of two or more light wave s. Since light is a type of electromagnetic wave, the electric field at a point can be given as E),,( zyx (,,,),,,,,,ixyztExyztAxyzte (2.1) Here A is the amplitude and is the phase of the light wave For linearly polarized light, in which the electric vector of the light vibrates only in one plane, the electric field can be written as 14

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,,,,,,,itxyzExyztAxyze where is the average frequency. (2.2) The intensity of the field can be obtained by time averaging the modulus value of square of the field E 2,,,, I xyzExyz (2.3) When two waves E1 and E2 interferes, the resultant intensity ,,, I xyzt is equal to 22 12121,,,2 I xyztEEEEEE or 121212,,, I xyztIIEEEE (2.4) For two linearly polarized waves using Eq. (2 .2), the resultant in tensity is given by 1212121 2,,,2cos ,,),, I xyztIIAA txyzxyz (2.5) If two linear polarizat ions are parallel, 1212AAII Then 1212121 2,,,2cos ,,,, I xyztIIII txyzxyz (2.6) If both waves have the same angular frequency, Then 12121 2,,,2cos,,,, I xyztIIIIxyzxyz (2.7) 15

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1212,,,2cos IxyztIIII (2.8) Here is the phase difference. Phase difference or phase shift betw een two light beams can therefore be determined by interfering the two beams. If one of the beams passes through an object or reflected from an object, that beam carries the information of the object. By combining this beam with a reference beam, one can de termine the phase shift of the two beams due to the object. 2.2 Coherence The ability of light beams to interfere is called coherence. Coherence is the correlation of phase at two points separated in ti me or space in the two interfering beams. 2.2.1 Temporal Coherence Temporal coherence is a measurement of how well a wave interferes with itself at different times. Consider two points in the two interfering waves. The maximum distance between the two points while maintaining a constant phase difference is called temporal coherence length The time required for waves to trav el the coherence length is called coherence timeclc The temporal coherence length depends on spectral band width and wavelength of the light source. In the two beam interferometry, interference fringes can 16

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only be seen when the optical path difference between reference and object arms are within the coherence length cl 2.2.2 Spatial Coherence Spatial coherence is a measurement of how well two points on the same wavefront can interfere, and is defined by the mutual correlation of different points of the same wavefront. In other words, it descri bes the correlation betw een wave fronts at different points in space. The spatial cohe rence depends on both the geometry of the interferometer and the properties of the lig ht source. A misaligned interferometer can cause wavefront mismatch, thus allowing low spatial coherence and therefore poor interference visibility. An incandescent bulb, in which each point of the filament acts as an individual light source, is an example of an incoherent light source. 2.3 Principle of Phase Shifting Interference Microscopy Phase shifting interference microscopy is a widely used technique for quantitative measurements of microscopic objects. It can be used for both transparent and reflective samples without special sample preparation [6 -12]. Different phase shift methods can be used in an interferometer setup to produce a phase shift between th e object beam and the reference beam. Typically, phase difference between two beams is changed in steps and intensity values of the interference beam at each step are recorded. The phase difference between object beam and reference beam can be calculated by combining such intensity values. 17

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2.4 Phase Shifting Methods Phase shifting can be used in any two b eam interferometer and the phase can be shifted in many different ways [13]. Severa l such methods, including the method used in our research, are described below. 2.4.1 Moving the Reference Mirror In this method, the reference mirror is moved by using a piezoelectric transducer (PZT). When the PZT moves the position of the reference mirror, it produces a phase change. This is shown in Figure (2.1a). 2.4.2 Using a Tilted Glass Plate In this method, a phase shift is introduced by placing a glass plate in the path of the collimated light beam as shown in Figure (2.1b). When the glass plate makes an angle 1 with the optical axis then the phase shift is given by 21cos(cos()) t n k Where, t is the thickness of the glass plate, is refractive index of the glass,n 2 k,1 and 2 are angles between the normal to the plat e and the light rays outside and inside the plate respectively. The optical path di fference can be increased by increasing the angle1 2.4.3 Using a Phase Plate In this method, phase shift is introdu ced by a rotating phase plate. When a circularly polarized light beam passes the half wave phase plate as shown in Figure (2.1c), 18

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the direction of polarization is reversed. When the plate is rotated by an angle then the phase change is given by 2 Figure 2.1: Several methods of phase shifting. (a) using a moving mirror; (b) using a tilted glass plate; (c) us ing a half wave plate. 2.5 Phase Shifting Algorithms Once a phase shift is introduced to the experimental setup using one of above mentioned methods, a phase shifting algorithm is needed to determine the relative phase between the reference mirror and the object. There are several phase shifting algorithms, and three-frame phase shifting algorithm re quires the minimum number of frames. The general expression for the intensity of interference fringes can be given as 19

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(BmiIII ) (2.1) where IB is the background intensity, Im is fringe modulation intensity, is the relative phase between the object and the reference mirror and i is the phase shift. 2.5.1 Three-frame Phase Shifting Algorithm At minimum, three intensity values are needed at phase interval of 900. If intensity values of the interference beam are recorded at 00, 900 and 1800 phase intervals, Equation (2.1) can be written as 0 /2cos cos(/2)sin cos()cosBm Bm Bm Bm BmIII IIIII IIIII (2.2) From Equations (2.2), the phase can be calculated as II III Tan0 2/0 12 (2.3) 2.5.2 Four-frame phase shifting algorithm In this experiment, four step phase shif ting algorithm is applied to the interference microscope. Though the minimum number of intensity values needed for phase calculation is three, even a small error in measurements can cause a large phase error. Taking four intensity measurements can reduc e this effect. In order to acquire phase images, the Michelson interferometer is used as the experimental setup as shown in Figure (2.2). 20

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Figure 2.2: Experimental setup. A standard Michelson-type interferometer consists of the reference arm and the object arm. The Intensity I(x,y) of the light captured by CCD can be written as; ),(cos),(),(2),(),(),(),( yx yxIyxIyxIyxIyxIyxIi R O R B O (2.4) Here IO(x, y) is the part of the beam reflected by the object that is coherent with respect to IR(x, y) the intensity of the beam reflected by the reference mirror. IB(x, y) is part of reflection from the object that is incoherent with respect to the reference i.e. outside the coherence length. (,) x y is the relative phase be tween the object and the reference mirror and i is the phase shift introduced by moving the reference mirror by quarter wavelength intervals. The reference mirror is mounted on a piezo transducer and is dithered by applying a ramp wavefo rm. Four images are acquired at 0,/2,i 21

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and3/ 2 Intensity distributions corresponding to the four images can be given as follows; sin2 cos2 sin2 cos22/3 2/ 0 RO RBO RO RBO RO RBO RO RBOIIIIII IIIIII IIIIII IIIIII (2.5) The phase map of the object is given by; II II Tan0 2/2/3 1 (2.6) And the amplitude of the image of the object is given by; R OI IIII I 162 2/32/ 2 0 (2.7) Once a phase profile of a specimen is obtained, it can be used to determine the height profile of the specimen. The optical path difference (OPD) between the object wave and reference wave is given by 2 OPD (2.8) 22

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Here is the wavelength of the light beam and is the relative phase between the object and reference mirror. In the given Michelson ty pe interferometer, the height profile of the object is half the OPD because the light travels towards th e object, reflects and travels back. Therefore, the height profile is related to the phase h as follows. 1 22 h (2.9) The reference mirror is dithered by a distance equal to the wavelength of the light source, by supplying a ramp wave from the si gnal generator. Four images are acquired when the distance = /4,/2,3/4 /4 and The accuracy of these images can be easily tested by acquiring an additional image so that the first image is equal to the fifth image as shown in Figure (2.3). The difference of phase values of the first image and the fifth image must be zero and this is also check ed to verify the accuracy of the procedure. The intensity distributions of th e first and the fifth frames are plotted in the same graph to check the accuracy of phase sh ifting. If the two distributions coincide as shown in Figure (2.4), they have the same phase value. Figure (2.3): Five image frames acquired when the shift of reference mirror is 0, /4,/2,3/4 /4 and 23

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Figure 2.4: An interferogram (left) and intensity di stribution of the first image frame (I=0) and the fifth image frame (I = 2 ) (right). If two distributions coincide the first and fifth frames have same the same phase value. 2.6 Moving the Reference Mirror Reference mirror has to be dithered in sp ecific amount so that five images can be captured within one ramp cycle. The applied ramp waveform is shown in Figure (2.5). The CCD camera captures 30 frames per second. Here a green light with 530 nm wavelength is used for an example. A LabV IEW program is used to capture 5 intensity images, each at quarter wavelength intervals, an d then use first four images to calculate the final phase image. The program is shown in Appendix A. 24

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Figure 2.5: The ramp waveform used to dither the reference mirror. A Stanford Research System M odel DS345 30 MHz Synthesized Function Generator is used as the vo ltage supply. The function outpu t is connected to the piezo electric transducer, which is glued to a plane mirror. This mirror is used as the reference mirror. The sync output of the generator is connected to the computer so that the LabVIEW program and voltage signal are s ynchronized. Once the LabVIEW program is started, it provides a trigger signal. The program is written so that the first image is captured five frames after the trigger signal and next four images are captured at 11th, 16th, 21st and 26th frames by skipping 4 frames at a time. This is shown in Figure (2.6). Figure 2.6: Acquiring five images by skipping frames. 25

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Since each image is captured at the 5th frame and they should be at quarter wavelength intervals, 5 frames = /4 = 0.53 m /4 1 frame = 0.53 m /20 = 0.0265 m Time for 1 frame = 1/30 seconds Length per second = 0.0265 m/frame x 30 frames/second =0.795 m/second 1.8 V needs to travel one wavelength. (This is by observation of fringes) 1.8V/0.53 m x 0.795 m/second = 2.7V/second If voltage need is V, V/1 second = 2.7 V/second V = 2.7 Volts Therefore, for 0.53 m wavelength, 2.7 Volts should be given to the reference mirror, in order to dither it by quarter wavelength intervals. Using the above method, reference mirro r can be dithered by quarter wavelength intervals for different wavelengths. Figure (2.7 ) shows a sample of interferogram. In the interferograms, optical path length between two fringes is and this is equal to a physical distance of /2 because of the round trip of light in reflection mode. 26

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Figure 2.7: An interferogram. The fringes appear when the object beam and the reference beam interfere. The optical pa th length between two fringes is equal to one wavelength. 2.7 Phase Shifting Interference Microscopy Using Single Wavelength The experimental setup is shown in Figure (2.2). A standard 1951 USAF resolution target is used as the object. Elements 2-6 of group 7 are imaged. The image size is 314 by 264 pixels and 100 m by 120 m. After obtaining four interferograms at quarter wavelength intervals, they are combin ed according to Equation (2.5). Figure (2.8) shows the final phase image using green LED ( = 550.18 nm) and the cross section of the surface profile. 27

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Figure 2.8: Wrapped phase image of a resolution ta rget (left) and its surface profile along the dotted line (right). The light source is a light emitting diode of ( = 550.18 nm). 28

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CHAPTER 3 EXPERIMENTAL SETUP This chapter describes the experimental setup and computer software used to acquire data in detail. The first section explai ns how each part of the interferometer setup works. Section (3.2) is a desc ription of the camera and imag e acquisition. Section (3.3) describes the computer software used to acquire and analyze phase images. These programs are shown in Appendix (A). Section (3.4) presents characteristics of light sources used in the experiment. 3.1 Interferometer The experimental setup is shown in the Figure (3.1). A set of light emitting diodes (driven by i-XitaniumTM LED electronic driver) illu minates a Michelson type interferometer. The light is collimated by lens L1 and then linearly polarized by the polarizer P. 29

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Figure 3.1: Experimental setup. MO, microscope objective; L1, collimating lens; L2, focusing lens; P, polarizer; PBS, polarized beam splitter; QW1,QW2, quarter-wave plates; A, analyzer; OBJ, object; REF, reference mirror; PZT, piezo-electric transducer; CCD, charged-coupled device. 30

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Figure 3.2: Unpolarized light comi ng from the LED is lin early polarized along the transmission axis of polarizer P. The polarized beam splitter PBS splits the incoming beam into an S-polarized (polarization plane is perpendicular to polarization axis) ray and a P-polarized (polarization plane is parallel to polarization axis) ray. This is shown in Figure (3.3). Spolarized beam is reflected at the PBS to illuminate the sample object OBJ and Ppolarized beam is transmitted through PBS to illuminate the reference mirror REF. When S-polarized light passes thr ough quarter wave plate QW1, phase changes by 90 and it becomes circularly polarized. After reflecti ng at the Mirror and going through another 90 phase shift at QW1, the light becomes P-polar ized. This change from S-polarization to Ppolarization avoids light trave ling back to LED and directs al l reflected light to the CCD. Similarly, P-polarized light illuminating the REF changes to S-polarized light and travels to the CCD. At the analyzer A, the two S-polarized and P-polarized light beams are changed into a common polarization state so that interference can occur on the CCD plane. This is shown in Figure (3.3). 31

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Figure 3.3: Effect of quarter wave plates and polarizer-analyzer combination on incoming light. The polarizer-analyzer pair also allows continuous variation of the relative intensity between the two arms. In order to acquire images with high resolution, two 20X microscope objectives are placed in front of the object and the reference mirror. The reference mirror is mounted on a piezo-electr ic transducer (PZT). A function generator supplies a ramp signal to the PZT to dither the reference mirror by a distance of one wavelength. Images are recorded at quarter wavelength intervals. 32

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3.2 Camera and Image Acquisition Images acquired by the CCD are sent to an image acquisition board (National Instruments IMAQ PCITM-1407) installed in the computer. The CCD (Charged Coupled Device) used in the experiment is Sony XC -ST50 black & white camera module. It has a 6.4 mm 4.8 mm sensing area, 768 494 pixels with 8.4 m 9.8 m pixel size. 3.3 Computer and Programs Intel Pentium 4 CPU 2.80 GHz com puter with Microsoft Windows XP Professional Version 2002 is used in the expe riment. Image analysis and calculations are done by using LabVIEW programs. The first LabVIEW program used in the experiment records interference images at quarter wavelength intervals and calculat es the final phase image of the object being imaged. For each color LED, a final phase image is produced and saved in the computer. Another LabVIEW program is used to combine single wavelength phase images to produce an unwrapped phase image. This program consists of optical phase unwrapping based on both two wavelength a nd three wavelength phase unwrapping methods. These programs are shown in Appendix A. 33

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3.4 Light Sources Three types of light sources are used in the experiment; light emitting diodes, laser diodes and ring dye laser. Light emitting diodes have very small coherence length and are categorized as an incoherent light source. Laser diodes have a larger coherence length compared to that of light emitting di odes, but still smaller than the coherence length of ring dye laser. 3.4.1 Light Emitting Diodes Light emitting diodes (LED) have been used as in interferometric light sources in order to reduce the speckle noise inherent to lasers [14-17]. Inte rference of coherent waves produces speckle noise, which limits the phase maps information. Since a LEDs coherence length is in the micron range, speckle noise is greatly reduced. LEDs also cost much less than lasers, are easy to use and replace and can reduce overall apparatus dimensions. The LEDs used in this experiment are LuxeonTM Emitter diodes from Lumileds Lighting LLC. All of the LEDs us ed herein have a Lambertian (high dome) radiation pattern. And their spectra are show n in Figure (3.4). These spectra were taken with Ocean Optics SD-1000 fiber optics spectrometer. 34

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Figure 3.4: A light emitting diode us ed in the experiment. Figure 3.5: Spectra of light emitting diodes. The peak wavelengths, luminous flux, cal culated and measured coherence lengths for the red, amber and green LEDs used in this experiment are shown in Table (3.1). The calculated coherence length of a light source is given by //2ln22Cl, where is the mean wavelength and is the full width half ma ximum (FWHM) of Gaussian 35

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spectrum [18]. The coherence length was dire ctly measured here by counting the number of fringes in the interference of the tilted mirror object. Table 3.1: Characteristics of LEDs. Luminous flux values are at 350 mA, Junction Temperature TJ =250C [19] Color Luminous Flux lm) [14] Peak Wavelength (nm) Spectral Width (nm) Calculated Coherence Length ( m) Measured Coherence Length ( m) Red 44 653.83 0.07 27.24 0.15 6.91 0.04 9.15 2.45 RedOrange 55 643.42 0.07 23.21 0.14 7.85 0.05 10.29 2.57 Amber 36 603.48 0.03 17.53 0.05 9.14 0.03 10.86 2.56 Green 25 550.18 0.09 38.93 0.19 3.42 0.02 3.85 1.46 3.4.2 Laser Diodes Laser diodes (LD) have been widely used as a light source in interferometry due to their frequency tunability, smaller size and cost compared to those of lasers [20-24]. They have shorter coherence lengths, typica lly few centimeters, compared to coherence length of lasers. The laser diodes used in th is experiment have following properties as shown in Table (3.2). Measured wavelengt hs are obtained from laser diode spectra. Spectra are shown in Figure (3.5). 36

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Figure 3.6: A laser diode used in the experiment. Figure 3.7: Spectra of laser diodes. 37

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Table 3.2: Characteristics of laser diodes. Name Power (mW) Manufacturers typical wavelength (nm) Measured wavelength (nm) Imatronic LDM145 1 633 636.89 0.01 DL3148-025 Red Laser Diode 5 635 653.22 0.01 DL3147-060 Red Laser Diode 7 650 659.77 0.01 HL6724MG AlGaInP Laser Diode 5 670 677.81 0.01 3.4.3 Ring Dye Laser The ring dye laser used in the experiment is Coherent CR699 Ring Dye Laser and uses Rhodamine 6G (R6G) dye. It is pumped by Spectra-Physics Millenia V diode pumped solid state (DPSS) lase r with an output of 5 W, 532 nm light. With R6G dye, the ring laser has a scan range of 560 nm 625 nm. The beam diameter is 0.75 mm [25]. Figure 3.8: The ring dye laser used in the experiment. 38

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CHAPTER 4 MULTIWAVELENGTH PHASE SHIFTI NG INTERFERENCE MICROSCOPY Multi wavelength phase shifting interfer ence microscopy is a technique that combines phase shifting interference micr oscopy and optical phase unwrapping. Phase shifting methods were described in the pr evious chapter. In this chapter phase unwrapping methods are de scribed in detail. When the optical depth of an object is greater than the wavelength, the phase image contains 2 discontinuities. Therefore, phase data has to be unwrapped from (, + ) or (0, 2 ) interval to (, + ) interval before one can obtain unambiguous optical thickness profile. Phase unwrapping is not difficult when there is no noise present in the phase map and the absolute value of the phase difference between adjacent phase samples is less than 4.1 Numerical Algorithms for Phase Unwrapping In basic phase unwrapping method, phase image is divided to horizontal lines and these lines are unwrapped sepa rately by scanning pixels and adding an offset to each pixel. At each discontinuity a 2 offset is added or subtract ed. After all horizontal lines are unwrapped, they are connect ed vertically and the unwrapping process is done along vertical lines. There are many phase unwrapping methods to remove 2 discontinuities 39

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and most can be categorized into two t ypes; path-dependent methods and pathindependent methods [26]. Path-dependent me thods detect positions of edges and phase discontinuities in images and use this information to calculate phase offset values. In path-independent methods, areas that can cau se errors in unwrapping are identified and eliminated before the unwrapping process starts. In 1994 Ghiglia and Romero used a leas t squares integration method with phase unwrapping [27]. In this method, which is know n as least squares in tegration of phase gradient method, the phase gradient is obt ained as wrapped phas e differences along two perpendicular directions and the gradient fi eld is least squares integrated to obtain continuous phase. However, this method is no t effective for phase maps with high noise [28]. P. G. Charette and I. W. Hunter proposed a robust phase unwrapping method for phase images with high noise content [26]. The basic concept behind this method is to identify contiguous areas that are not on or close to a fringe boundary by locally fitting planes to the phase data. Then these areas ar e phase shifted with re spect to one another by multiples of 2 to unwrap the phase. Software algorithms that ex ist for detecting and removing 2 discontinuities are mostly computational-intensive and prone to errors when the phase profile is noisy or when the object has irregular ities. Multi-wavelength phase unwrapping is an easy method that can be used to eliminate 2 ambiguities in phase maps without such problems. 40

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4.2 Multi -Wavelength Phase Unwrapping The basis of multi-wavelength phase unwrapping method is the idea of beat wavelength. For two wavelengths 1 and2 the beat wavelength 12 is defines as 12 12 12 (4.1) For years it has been known that a l onger wavelength light source produces fewer fringes over a given object than will a short wavelength light source, thus reducing the number of 2 ambiguities in the phase image. However, the drawback is the need of infrared light sources instead of visible li ght sources. J. C. Wyant has shown that two wavelengths of visible region can be used in the cont ext of holography to produce a longer beat wavelength [29]. Using various pairs of wave lengths from an Argon and HeNe lasers an aspheric optic element was tested First, a hologram of the test target was obtained by using a wavelength1 Then the hologram was processed and placed at the original position of the interferometer and illuminated by a second wavelength2 The resultant interferogram was identical to the interferogram that would have been illuminated by a light source of12 ; the beat wavelength of wavelengths 1 and2 The term two-wavelength interferometry was first used by C. Polhemus in a paper where he introduced a two-wavelength tec hnique for interferometric testing [30]. In the method of static interferometry, a fri nge pattern obtained w ith a light source of wavelength 1 is recorded and replaced into the sy stem as a moir reference mask. Then the light source is replaced by one with wavelength2 The resultant moir fringe pattern is identical to the fringe pattern that w ould have been obtained with a light source of12 41

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Polhemus modified the method to apply in r eal time systems. In the method of dynamic interferometry, light sources of 1 and2 are operated simultaneously in the interferometer setup, giving a resultant fringe pattern of 12 In 1984, two-wavelength phase shifting interferometry was introduced as an optical phase unwrapping method [31]. In this method, phase shifting interferometry and two-wavelength interferometry were combined to extend the phase measurement range of single-wavelength phase shifting interferometry. Two methods were introduced to solve 2 ambiguities by using two-wavelength phase shifting interferometer. In the first method, two sets of wrapped phase data were obtained with wavelengths 1 and2 The data is then used to calcul ate the phase difference between pixels for beat wavelength 12 Then all phase difference values are in tegrated to calculate the relative surface height of the test object. In the second method, two phase maps of different wavelengths are used to produce a phase map of beat wa velength. The beat wavelength phase map is then used as a reference to correct 2 ambiguities in the single wavelength phase m Both methods were used to measure 1-D surf ace heights. Two-wavelength phase shif interferometry has also been used to obtain three dimensional contour maps of aspheric surfaces with an accuracy of ap. ting 12 /100 [32]. However, a disadvantage of this optical phase unwrapping is that the phase noise in each wa velength is magnified by a factor equal to the magnification of the wavelengths. This pr oblem has been addressed by J. Gass, A. Dakoff and M. K. Kim [33] In the context of digital holography, two phase maps of wavelengths 1 and2 are used to produce a phase map called coarse map with beat wavelength Then one of the single wavelength phase maps is used to reduce the 12 42

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amplified phase noise of coarse map. The resu ltant fine map has noise similar to single wavelength phase map, with a larger axial range free of 2 ambiguities. The two-wavelength phase unwrapping method has been extended to multiple wavelengths; enabling measurements of steep surfaces without software phase unwrapping. A hierarchical phase unwrappi ng algorithm that chooses a minimum number of wavelengths to increase the accuracy of optical unwrapping has been introduced by C. Wagner, W. Osten and S. Seebacher [34]. The basic principle of this method is to start with a larger beat wavelength. Then a systema tic reduction of beat wavelengths is used to improve the accuracy of the measurement while the information of the preceding measurements is used to eliminate 2 ambiguities. A similar version of hierarchical phase unwrapping has been presented by U. Sc hnars and W. Jueptner [35], however it has not been used experimentally. Threewavelength phase unwra pping algorithms have been introduced in both interferometry and digital holography enabling measurements of steep surfaces without software phase unw rapping [9-10, 36-39]. However, while the principle of multi-wave phase unwrapping has been known in interferometry, known applications have been confined to optical profilers with raster-scanned pointwise interferometry. Other than recent digital ho lography experiments [33, 38, 39], multi-wave phase unwrapping has not yet been applied to full-frame phase images in interferometry. 43

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4.3 Principle of Two-Wavelength Optical Phase Unwrapping When an object is imaged by using a wavelength smaller than the objects height, the phase image contains 2 discontinuities, as shown in Figure (4.1). Figure 4.1: Phase Vs Distance. 2 ambiguities occur when the distance is a multiple of the wavelength. From Figure (4.1) it is clear that ther e are many distance values for a given phase value. In order to obtain unambiguous optical thickness profile, we need to have only one z distance for a given phase. These 2 discontinuities are eliminated by using multiwavelength optical phase unwrapping method. For the mth wavelength m the surface profile Zm of an object is relate d to the phase difference m as follows; 2mm mZ (4.2) It is apparent that unambiguous range of Z can be increased by using a longer Consider two single wavelength phase maps 1 and 2 with wavelengths 1 = 530 nm and 2 = 470 nm respectively .The beat wavelength 12 for 1 and2 is given 44

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by 121212 = 4.151 m. The 12 can be increased by choosing closer values of 1 and2 The phase map for 12 is obtained by subtracting one single wavelength phase map from the other and then adding 2 whenever the resultant value is less than zero. This phase map is called coarse map12 The surface profile for coarse map 12 is given by 1212122 Z However, the phase noise in eac h single wavelength phase map is magnified by the same factor as the magnification of wavelengths. In the twowavelength optical phase unwrappi ng method introduced by J. Gass et.al [33], the phase noise is reduced by using the following steps. First, the surface profile Z12 is divided into integer multiples of a single wavelength, say1 Then, the result is added to th e single wavelength surface profile Z1.This significantly reduces the phase noise in the coarse map. Ho wever, at the boundaries of wavelength intervals1 the noise of the single wavelengt h phase map appears as spikes. These spikes can be removed by comparing the result with the coarse map surface profile Z12. If the difference is more than half of1 addition or subtraction of one 1 depending on the sign of the difference removes the spik es. The final result fine map has a noise level equal to that of single wavelength surface profile. If a single wavelength phase map m contains a phase noise of2m the two-wavelength phase unwrapping method works properly for 124mm Using a lager beat wavelengt h reduces the maximum noise limit. 45

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4.4 Two-Wavelength Optical Phase Unwrapping Simulations Following simulations were done for green (530 nm) and blue (470nm) wavelengths with 1 = 2 = 0.02. Each step is shown in Figure (4.2). The object is a tilted mirror with a height 5 m as shown in Figure (4.2a). Surface profiles of green and blue wavelengths are shown in Figures (4.2b) and (4.2c). Their phase maps are shown in Figures (4.2d) and (4.2e). Figure (4.2f) shows 12 ; subtraction of two phase maps. Adding 2 whenever the result is less than zero produces the coarse map 12 as shown in Figure (4.2g). The surface profile of coarse map is shown in Figure (4.2h). Coarse map has a longe r axial range without 2 ambiguities and the range is equal to the beat wavelength 4.15 m. 46

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Figure 4.2: Simulation of two-wavelength optical phase unwrapping. (a) the slanted plane of height h = 5 m; (b) surface profile Z1(x) for 1=0.53 m; (c) surface profile Z2(x) for 2=0.47 m; (d) phase map 1(x) for 1=530 nm; (e) phase map 2(x) for 2=470nm; (f) subtraction of two phase maps 12 '(x) = 12; (g) phase map 12(x); (h) coarse map Z12 (x) with beat wavelength 12 = 4.15 m. 47

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4.4.1 Noise Reduction of Coarse Map Once coarse map Z12 (x) is obtained, noise has to be reduced. Figure (4.3) shows simulation results of noise reduction. Figure (4.3a) is surface profile of coarse map. Figure (4.3b) shows the result after coarse map was divided by integer multiples of 1. In Figure (4.3c), the surface profile of 1 is added to the result of Figure (4.3b). Now, Figure (4.3c) is compared with the surface profile of coarse map Z12 by subtracting Figure (4.3c) from Z12 and the result is shown in Figure (4.3d). If the difference is more than half of 1, one 1 is added or subtracted depending on the sign of the difference as shown in Figure (4.3e). The final result fine map is obtained by adding the result to Figure (4.3c). 48

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Figure 4.3: Noise reduction for two-wave length phase unwrapping. (a) coarse map Z12(x); (b) coarse map Z12(x) is divided into integer multiples of 1; (c) Z1(x) is pasted on result (b); (d) comparison of (c) with coarse map; (e) adding or subtracting 1 to Z(c); (f) fine map. 49

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Figure (4.4) shows phase noise in coarse map and fine map. For 1 = 530 nm and 2 = 470 nm, the maximum noise limit is m ~ 3.2 %. Using a lager beat wavelength reduces the maximum noise limit. Figure 4.4: Phase noise in coarse map and fine map. (a) noise in the coarse map;(b) noise in the fine map. 4.5 Principle of Three-Wavelength Optical Phase Unwrapping The advantage of three wavelength pha se unwrapping method is that the beat wavelength can be increased without reduc ing the maximum noise limit. Suppose the three chosen wavelengths are 1 = 625 nm, 2 = 590 nm, and 3 = 530 nm. The first two wavelengths give beat wavelength 12 = 10.53 m. Instead of using the surface profile Z12 of = 10.53 m, which has a high noise, an id entical surface profile can be produced by using surface profiles Z13 and Z23 with beat wavelengths = 3.49 m and = 5.21 m. The resultant coarse map of coarse maps 12132313 23 with surface profile Z13-23 also has the same beat wavelength 13231323 23 13 = 10.53 m. 50

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The noise reduction is done as follows. In the first step, the quantity of integer multiples of present in the range131323Z is calculated. The result () Z a is given by 1323 13 13()intZ Za (4.3) In the next step, the result is added to the surface profile 13 Z 13()() Z bZaZ (4.4) The resultant map () Z b is then compared with1323Z If the difference () Z c is more than half of one is added or subtracted depending on the sign difference. 1313 (4.5) 13 13 13 13 13 13()()/2 ()() /2()/2 ()()/2ZcifZc ZdZcifZc ZcifZc The resultant surface profile () Z d is called intermediate fine map and has significantly reduced noise. Any remaining noise is due to the noise in the phase map13 The remaining noise in() Z dis reduced by using a single wavelength, say1 First, the intermediate fine map () Z dis divided into integer multiples of1 1 1() ()Zd ZeInt (4.6) Then the result is added to th e single wavelength surface profile1 Z 1()() Z fZeZ (4.7) 51

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The resultant map () Z f is then compared with1 Z If the difference is more than half of1 one1 is added or subtracted depending on the sign difference. The noise in the final map is equal to the noise in th e single wavelength surface profile1 Z The maximum noise level m in the single wavelength phase map for the three wavelength phase imaging to work is given by the smaller value of 13/412 ~ 8.3% or 1/413 ~ 4.5%. Therefore, the three wavelength phase unwrapping method increases the beat wavelength without magnifying the noise in the final phase map. 4.6 Three-Wavelength Optical Phase Unwrapping Simulations Simulations were done for 1 = 625 nm, 2 = 590 nm and 3 = 530 nm wavelengths with 123 0.04 as shown in Figure (4.5). The object is a tilted mirror with a height 12 m as shown in Figure (4.5a). Surface profiles of 1 = 625 nm, 2 = 590 nm and 3 = 530 nm wavelengths are shown in Figures (4.5b), (4.5c) and (4.5d). The surface profil e of coarse map of 1 = 625 nm, 2 = 590 nm with beat wavelength 12 = 10.53 m is shown in Figure (4.5e). The fine map Z12 produced by following the procedure described in section (4 .4.1) is shown in Figure (4.5f). The phase noise in the fine map Z12 is too large. 52

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Figure 4.5: Simulation of three-wavelength optical phase unwrapping. (a) the slanted plane of height 12 m; (b) surface profile Z1 for 1 = 625 nm; (c) surface profile Z2 for of 2 = 590 nm; (d) surface profile Z3 for 2 = 530 nm; (e) coarse map Z'12 with beat wavelength 12 = 10.53 m; (f) fine map Z12. 53

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Figure (4.6) shows the next steps of si mulations of the three wavelength optical unwrapping method. Surface profiles Z13 and Z23 with beat wavelengths = 3.49 m and = 5.21 m are shown in Figure (4.6a) and Figure (4.6b). The surface profile of coarse map of coarse maps with beat wavelength 13231323 = 10.53 m is shown in Figure (4.6d). Figures (4.6e) and (4.6f) show interme diate fine map and the final fine map after applying the noise reduction. 54

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Figure 4.6: Continuation of three-wavelength op tical phase unwrapping simulations. (a) coarse map Z'13 with beat wavelength 12 =3.49 m; (b) coarse map Z'23 with beat wavelength 32 = 5.21 m; (c) coarse map of coarse maps with the beat wavelength 10.53 m ; (d) intermediate fine map Z"13-23 where Z'13 is pasted on Z'13-23 ( Z'12); (e) Final fine map Z13-23 where Z1 is pasted on Z"13-23 55

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Figure 4.7: Noise reduction in three-wavelength op tical phase unwrapping. (a) noise in coarse map of coarse maps Z13-23; (b) noise in intermediate fine map; (c) noise in final fine map. Figure (4.7) shows how the three wave length optical phase unwrapping method effectively reduces phase noise in the final phase map. All three graphs are normalized for better comparison. 56

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CHAPTER 5 MULTI-WAVELENGTH OPTICAL PHASE UNWRAPPING USING LIGHT EMITTING DIODES This chapter presents results of multi -wavelength optical phase unwrapping (OPU) using light emitting diodes. First section presents results of two-wavelength optical phase unwrapping and the second section presents re sults of three-wavelength optical phase unwrapping. Subsections of each section presen t different samples describing image size, wavelengths used for unwrapping proce ss and noise in the phase image. All samples shown in this chapter are im aged with 20X microscope objectives in reference and object arms of the experimental setup. 5.1 Two-Wavelength Optical Phase Unwrappi ng with Light Emitting Diodes In this section two light emitting diodes of different wavelengths are used to obtain a larger beat wavelength. A phase image is obtained using each wavelength and then two phase images are combined to produce a coarse map and a fine map as described in the Chapter 4. The samples used in the experiment are; a resolution target, onion cells, a micro electro-mechanical system (MEMS), a transmission electron microscope (TEM) grid and cheek cells. 57

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5.1.1 Resolution Target The sample is a USAF 1951 resolution target in reflection mode. It consists of vacuum deposited chromium bars on a 1.5 mm thickness of soda lime glass substrate. The dimensions are 2"". Bars are orga nized in groups and elements. Each group consists of 6 elements and an each element c onsists of 3 horizontal and 3 vertical equally spaced bars. Table (5.1) shows the specifica tion table of USAF 1951 resolution target. Table 5.1: Specification table of US AF 1951 resolution target Line pairs / mm in USAF Resolution Target 1951 Group Number Element 0 1 2 3 4 5 6 7 1 1.00 2.00 4.00 8.00 16.00 32.0 64.0 128.0 2 1.12 2.24 4.49 8.98 17.95 36.0 71.8 144.0 3 1.26 2.52 5.04 10.10 20.16 40.3 80.6 161.0 4 1.41 2.83 5.66 11.30 22.62 45.3 90.5 181.0 5 1.59 3.17 6.35 12.70 25.39 50.8 102.0 203.0 6 1.78 3.56 7.13 14.30 28.50 57.0 114.0 228.0 Red and green light emitting diodes with wavelengths 1 = 653.83 nm and 2 = 550.18 nm are used to obtain a beat wavelength 12 of 3.47 m. Image area is 0.1 mm 0.12 mm. 58

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Figure 5.1: Resolution target. The direct image is of the elements of group 7. The experimental results for two-wave length optical phase unwrapping are shown in Figure (5.2). Red ( = 653.83 nm) and green ( 2 =550.18 nm) LEDs are used as the two wavelengths. The beat wavelength 12 = 3.47 m. Figure (5.2a) and (5.2b) show single wavelength phase maps 1 and 2 with 1 = 653.83 nm and 2 =550.18 nm respectively. The coarse map 12 with 12 = 3.47 m is shown in Figure (5.2c) and the final phase map with reduced noise is shown in Figure (5.2d). Figure (5.2e) shows the 3D rendering of final phase map. Figure (5.3) shows cross s ections of phase maps along the lines shown in Figure (5.2) and the phase noise in the chosen regions Figure (5.3a) is a cross section of single wavelength phase map with 1 = 653.83 nm and Figure (5.3b) is a cross section of coarse map with 12 = 3.47 m. A cross section of the fine ma p with reduced noise is shown in Figure (5.3c). For maps (a), (b) and (c), the vertical axis is 4 m. The root mean square (rms) noise of the single wa velength phase map is 4.65 nm. This is shown in Figure (5.3d). Figure (5.3e) shows the noise in coarse map. The root mean square (rms) noise is 42.89 nm. Final fine phase map with reduced noise is shown in Figure (5.3f). The rms phase noise in the final phase map is 7.16 nm. 59

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Figure 5.2: Results of two-wavelength optical unwr apping for resolution target. (a) phase map with 1 = 653.83 nm; (b) phase map with 2 = 550.18 nm; (c) coarse map with beat wavelength = 3.512 m; (d) fine map with reduced noise; (e) 3-D rendering of (d). 60

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Figure 5.3: Surface profiles of resolution target. (a ) single wavelength surface profile; (b) surface profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (d) no ise of single wavelength phase ma p in the region between the two markers in plot (a). Rms noise is 4.65 nm; (e) noise of coarse map in the area shown in plot (b). Rms noise is 42.89 nm; (f) noise of final unwrapped phase map in the area shown in (c). Rms noise is 7.16 nm. 61

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5.1.2 Onion Cells A fresh layer of onion skin is placed on a plane mirror and used as the object. Red and amber light emitting diodes with wavelengths 1 = 653.83 nm and 2 = 603.48 nm are used to obtain an beat wavelength of 7.84 m. Onion cells are rectangular in shape and the size ranges from 250 m to 400 m. Figure (5.4) show s an interferogram of a sample of onion cells. Figure 5.4: Interferogram of onion cells. Figure (5.5) shows the experimental re sults of two-wavelength optical phase unwrapping of onion cells. Images are of a 184 m 184 m area. Figure (5.5a) and (5.5b) show single wa velength phase maps 1 and 2 with 1 = 653.83 nm and 2 =603.48 nm respectively. The coarse map 12 with 12 = 7.84 m is shown in Figure (5.5c) and the final phase map with reduced noise is s hown in Figure (5.5d). Figure (5.5e) shows the 3-D rendering of final phase map. 62

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Figure (5.6) shows cross s ections of phase maps along the lines shown in Figure (5.5) and the phase noise in the chosen regions Figure (5.6a) is a cross section of single wavelength phase map with 1 = 653.83 nm and Figure (5.6b) is a cross section of coarse map with 12 = 7.84 m. A cross section of the fine ma p with reduced noise is shown in Figure (5.6c). For maps (a), (b) and (c), the vertical axis is 8 m. The root mean square (rms) noise of the single wa velength phase map is 9.04 nm. This is shown in Figure (5.6d). Figure (5.6e) shows the noise in coarse map. The root mean square (rms) noise is 139.04 nm. Final fine phase map with reduced noise is shown in Figure (5.6f). The rms phase noise in the final phase map is 9.04 nm. 63

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Figure 5.5: Results of two-wavelength optical un wrapping for onion cells. Images are of a 184 m 184 m area; (a) phase map with 1 = 653.83 nm; (b) phase map with 2 = 603.48 nm; (c) coarse map with beat wavelength 12 = 7.84 m; (d) fine map with reduced noise; (e) 3-D rendering of (d). 64

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Figure 5.6: Surface profiles of onion cells. (a) si ngle wavelength surface profile with1 = 653.83 nm; (b) surface profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (d) noise of si ngle wavelength phase map in the region between the two markers in plot (a). Rms noise is 9.04 nm; (e) noise of coarse map in the area shown in plot (b). Rms noise is 139.04 nm; (f) noise of final unwrapped phase map in the area shown in (c). Rms noise is 9.04 nm. 65

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5.1.3 Micro-Electro-Mechanic al System (MEMS) The object here in Figure (5.7) is a microelectrode array biosenso r. It consists of 16 gold electrodes on a Pyrex glass substrate. The center is a 125m diameter circle with an approximate thickness of 2m [40]. The center of the device was imaged and the experimental results for two-wavelength opt ical phase unwrapping are shown in Figure (5.8). Red (= 653.83 nm) and green (2 =550.18 nm) LEDs are used as the two wavelengths. The beat wavelength 12 = 3.47 m. Images are of a 184 m 184 m area. Figure (5.8a) shows a single wavelength phase map 1 with 1 = 653.83 nm. The coarse map 12 with 12 = 3.47 m is shown in Figure (5.8b) and the final phase map with reduced noise is shown in Figure (5.8c). Figur e (5.9) shows cross sections of phase maps along the lines shown in Figure (5.8) and the phase noise in the chosen regions. Figure (5.9a) is a cross section of single wavelength phase map with 1 = 653.83 nm and Figure (5.9b) is a cross section of coarse map with 12 = 3.47 m. A cross section of the fine map with reduced noise is shown in Figure (5.9 c). For maps (a), (b) and (c), the vertical axis is 4 m. The root mean square (rms) noise of the coarse map is 43.27 nm. This is shown in Figure (5.9d). Figure (5.9e) shows th e reduced noise in fine phase map. Since the center of the MEMS device has a curvature, a paraboloid is fitted to the data. The red dotted line is the best-fit parabolic curve. Af ter subtracting the curv ature from the data, the Figure (5.9f) shows the corrected phase noise of 10.29 nm. 66

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Figure 5.7: micro-electrode array bio sensor. Figure 5.8: Results of two-wavelength optical ph ase unwrapping for biosensor. (a) single wavelength phase map; (b) twowavelength coarse map; (c) two-wavelength fine map with reduced noise. 67

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Figure 5.9: Surface profiles of biosensor. (a) single wavele ngth surface profile; (b) surface profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (d) noise of the coarse map in the region between the two markers in plot (b). Rms noise is 43.27 nm; (e) noise of fi nal unwrapped phase map in the area shown in (c). Red dotted line is the best fit parabolic curvature and black solid line is data; (f) corrected phase noise of the unw rapped phase map, after subt racting the curvature of the object. Rms noise is 10.29 nm. 68

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5.1.4 Transmission Electron Microscope (TEM) Grids The object here is a transmission electron microscope (TEM) grid with a hexagonal 150 mesh pattern. The grid is ma de of copper and the diameter is 3.05 mm diameter with a solid bord er. Typical hole width is 130 m and bar width is 35 m. Figure 5.10: TEM grids. Image is taken by Natio nal DC2-156-S Digital Microscope. The length of the green line is 125.7 m. Figure (5.11) shows the results of tw o-wavelength optical phase unwrapping using red and green light emitting diodes with wavelengths 1 = 653.83 nm and 2 = 550.18 nm are used to obtain a beat wavelength 12 of 3.47 m. Images are of a 184 m 184 m area. Figure (5.11a) shows a single wavelength phase map 1 with 1 = 653.83 nm. The coarse map 12 with 12 = 3.47 m is shown in Figure (5.11b) and the final phase map with reduced noise is shown in Figure (5.11c ). Figure (5.12) shows cross sections of phase maps along the lines shown in Figure (5 .11) and the phase noise in the chosen regions. Figure (5.12a) is a cross sec tion of single wavelength phase map with 1 = 69

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653.83 nm and Figure (5.12b) is a cr oss section of coarse map with 12 = 3.47 m. A cross section of the fine map w ith reduced noise is shown in Figure (5.12c). For maps (a), (b) and (c), the vertical axis is 4 m. The root mean square (r ms) noise of the coarse map is 99.01 nm. This is shown in Figure (5.12e). Figure (5.12f) shows the reduced noise in fine phase map. The rms noise is 10.88 nm. Figure 5.11: Results of two-wavelength optical phase unwrapping for TEM grids. (a) single wavelength phase map; (b) twowavele ngth coarse map; (c) two-wavelength fine map with reduced noise. 70

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Figure 5.12: Surface profiles of TEM grids. (a) single wavelength surface profile with1 = 653.83 nm; (b) surface profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (d) noi se of single wavelength phase map in the region between the two markers in plot (a). Rms noise is 10.88 nm; (e) noise of coarse map in the area shown in plot (b). Rms noise is 99.01 nm; (f) noise of final unwrapped phase map in the area shown in (c). Rms noise is 10.88 nm. 71

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5.1.5 Cheek Cells In this section experimental results of two-wavelength opt ical phase unwrapping using a sample of basal mucosa (commonl y known as cheek cells) are presented. Figure 5.13: Direct image of a cheek cell. Red and green light emitting diodes with wavelengths 1 = 653.83 nm and 2 = 550.18 nm are used to obtain an beat wavelength of 3.5 m. Image size is 66.9 m 77.8 m with 233 271 pixels. Experimental results of two-wavelength optical unwrapping are shown in Figure (5.14). Figure (5.14a) show s a single wavelength phase map 1 with 1 = 653.83 nm. The coarse map 12 with 12 = 3.47 m is shown in Figure (5.14b) and the final phase map with reduced noise is shown in Figure (5.14c). Figure (5.14d) shows the 3-D rende ring of final phase map. 72

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Figure 5.14: Results of two-wavelength optical un wrapping for cheek cell. (a) phase map with 1 = 653.83 nm; (b) phase map with 2 = 550.18 nm; (c) coarse map with beat wavelength = 3.4712 m; (d) fine map with reduced noise; (e) 3-D rendering of (d). 73

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5.2 Three-Wavelength Optical Phase Unwrappi ng with Light Emitting Diodes In three-wavelength optical phase unwrapping, three light emitting diodes of different wavelengths are used. Here the beat wavelength is increased by choosing closer wavelengths, and the thir d wavelength is used to redu ce the noise in phase maps. The samples used in the experiment are; a resolution target, a micro electro-mechanical system (MEMS), a transmission electron mi croscope (TEM) grid and cheek cells. 5.2.1 Resolution Target The object here is the same resolution target used in section (5.1.1). Red (1= 653.83 nm), amber (2= 603.48 nm) and green (3 = 550.18 nm) LEDs are used as the three wavelengths. The beat wavelength 13-23 = 7.84 m. Figure (5.15) shows the experimental results of three-wavelength optical phase unwrapping. Image area is 0.1 mm 0.12 mm. Figure (5.15a) show s a single wavelength phase map 1 with 1 = 653.83 nm. The coarse map 12 with 12 = 7.84 m is shown in Figure (5.15b) and the final phase map with reduced noise is shown in Figure (5.15c). Figure 5.15: Results of three-wavelength OPU for resolution target. (a) single wavelength phase map; (b) threewavelength coarse map; (c) threewavelength fine map with reduced noise. 74

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Cross section of each phase map is taken along the lines shown in Figure (5.15). These cross sections and phase noise of coarse and fine maps are shown in Figure (5.16). Figures (5.16a) (5 .16c) show surface profiles of single wavelength phase map, coarse map and fine map respectively. According to Figure (5.16c), the thickness of the verti cal bars of group 7 element 4 is ~ 70 nm. Vertical axis for each map is 7 m. Figure (5.16d) shows 144.6 nm rms noise of the coarse map. The reduced phase noise in the fine map is 3.98 nm as shown in Figure (5.16e). 75

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Figure 5.16: Surface profiles resolution target. (a) single wavelength surface profile; (b) surfac e profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (d) noise of coarse map in the area shown in (b). Rms noise is 144.6 nm; (e) noise of final unwrapped phase map in the area shown in (c).Rms noise is 3.98 nm. 76

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5.2.2 Onion Cells The object here is the same onion cell sample used in section (5.1.2). Red (1= 653.83 nm), amber (2= 603.48 nm) and green (3 = 550.18 nm) LEDs are used as the three wavelengths. The beat wavelength 13-23 = 7.84 m. Image area is 184 m 184 m. Figure (5.17a) is the singl e wavelength phase map with 1 = 653.83 nm. The three wavelength coarse map is shown in Fi gure (5.17b) with a beat wavelength 13-23 = 7.84 m. The final fine map with reduced noise is shown in Figure (5.17c). Figure 5.17: Results of three-wavelength OPU for onion cells. (a) single wavelength phase map; (b) three-wavelength coarse ma p; (c) threewavelength fine map with reduced noise. Figure (5.18) shows cross sections of pha se maps along the lines shown in Figure (5.17) and the phase noise in the chosen re gions. Figure (5.18a) is a cross section of single wavelength phase map with 1 = 653.83 nm and Figure (5.18b) is a cross section of coarse map with 12 = 7.84 m. A cross section of the fine map with reduced noise is shown in Figure (5.18c). For maps (a), (b) and (c), the vertical axis is 8 m. The root mean square (rms) noise in the selected re gion of the coarse map is 349.67 nm. This is 77

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shown in Figure (5.18d). Figure (5.18e) shows the reduced noise in fine phase map. The rms noise is 31.43 nm. 78

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Figure 5.18: Surface profiles of onion cells. (a ) single wavelength surface profile with1 = 653.83 nm; (b) surface profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (d) noi se of coarse map in the area shown in plot (b). Rms noise is 349.67 nm; (e) noise of final unwrapped phase map in the area shown in (c). Rms noise is 31.43 nm. 79

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5.2.3 Micro-Electro Mechanical System (MEMS) Biosensor The experimental results for threewavelength optical phase unwrapping are shown in Figure (5.19). Red (1= 653.83 nm), amber (2= 603.48 nm) and green (3 = 550.18 nm) LEDs are used as the three wavelengths. The beat wavelength 13-23 = 7.84 m. The object is the center of the same microelectrode array bio-sensor used in section (5.1.3). Images are of a 184 m 184 m area. Figure (5.19a) is the single wavelength phase map with 1 = 653.83 nm. The three wavelength coarse map is shown in Figure (5.19b) with an beat wavelength 13-23 = 7.84 m. The final fine map with reduced noise is shown in Figure (5.19c). Figure 5.19: Results of three-wavelength OPU for bi osensor. (a) single wavelength phase map; (b) three-wavelength coarse map; (c ) threewavelength fine map with reduced noise. Cross section of each phase map is taken along the lines shown in Figure (5.19). These cross sections and phase noise of coarse and fine maps are s hown in Figure (5.20). Figures (5.20a)-(5.20c) show surface profiles of single wavelength phase map, coarse map and fine map respectively. Ve rtical axis for each map is 11 m. Figure (5.20d) shows 105.79 nm rms noise of the coarse map. Because of the curvature of the object 80

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surface, a paraboloid is fitted with the final fine map data. This is shown in Figure (5.20e). The black line is data and the red dotted line shows the bestfit parabolic curve. Corrected phase noise in the fine map is 4.78 nm, which is shown in Figure (5.20f). 81

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Figure 5.20: Surface profiles of biosensor. (a) single wavelength su rface profile; (b) surface profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (d) nois e of coarse map in the area show n in (b). rms noise is 105.79 nm; (e) noise of final unwrapped phase map in th e area shown in (c). Red dotted line is the best fit parabolic curv ature and black solid line is data ; (f) Final noise of the unwrapped phase map, after subtracti ng the curvature of the object Rms noise is 4.78 nm. 82

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5.2.4 Transmission Electron Microscope (TEM) Grids In this section, experimental result s of three wavelength optical unwrapping are presented with the same TEM grid used in section (5.1.4). Three wavelengths are 1= 653.83 nm, 2= 603.48 nm and 3 = 550.18 nm. The beat wavelength is 7.84 m. Images are of a 184 m 184 m area. Figure (5.21a) is the single wavelength phase map with 1 = 653.83 nm. The three wavelength coarse map is shown in Figure (5.21b) with an beat wavelength 13-23 = 7.84 m. The final fine map w ith reduced noise is shown in Figure (5.21c). Figure 5.21: Results of three-wavelength OPU fo r TEM grids. (a) single wavelength phase map; (b) three-wavelength coarse ma p; (c) threewavelength fine map with reduced noise. Cross section of each phase map is taken along the lines shown in Figure (5.21). These cross sections and phase noise of coarse and fine maps are s hown in Figure (5.22). Figures (5.22a)-(5.22c) show surface profiles of single wavelength phase map, coarse map and fine map respectively. Ve rtical axis for each map is 7 m. Figure (5.22d) shows 32.26 nm rms noise in the single wavelength phase map. Figure (5.22e) shows 303.77 nm 83

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rms noise of the unwrapped coarse map. Fi nal phase map with reduced phase noise is shown in Figure (5.22f). The reduced rms noise in the final phase map is 32.11 nm 84

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Figure 5.22: Surface profiles of TEM grids. (a) single wavelength surface profile with1 = 653.83 nm; (b) surface profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (d) noi se of single wavelength phase map in the region between the two markers in plot (a). Rms noise is 32.26 nm; (e) noise of coarse map in the area shown in plot (b). Rms noise is 303.77 nm; (f) noise of final unwrapped phase map in the area shown in (c). Rms noise is 32.11 nm. 85

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5.2.5 Cheek Cells In this section experimental results of three-wavelength optical phase unwrapping using a sample of basal mucosa (commonly known as cheek cells) are presented with 1= 653.83 nm, 2= 603.48 nm and 3 = 550.18 nm. The beat wavelength is 7.84 m. Image size is 66.9 m 77.8 m with 233 271 pixels. Figure (5.23a) is the single wavelength phase map with 1 = 653.83 nm. The three wave length coarse map is shown in Figure (5.23b) with an beat wavelength 13-23 = 7.84 m. The final fine map with reduced noise is shown in Figure (5.23c). Figure (5.23d) shows the 3-D rendering of the final fine map. 86

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Figure 5.23: Results of three-wavelength OPU fo r cheek cells. (a) phase map with 1 = 653.83 nm; (b) coarse map with beat wavelength 1323 = 7.84 m; (c) fine map with reduced noise; (d) 3-D rendering of (c). 87

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5.3 Discussion and Conclusions Both two wavelength and three wavele ngth optical phase unwrapping techniques can be successfully used to unwrap phase images obtained using light emitting diodes. Since the coherence length of light emitting diodes is in micro meter range, images are free of speckle noise. How ever, the height profiles are limited by the coherence length and the features that extend beyond coherence length appear as noise Therefore, light emitting diodes are most suitable for imaging objects with features smaller than the coherence length of the diode. Fo r objects with larger features, laser diodes or lasers can be used as described in next chapters. 88

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CHAPTER 6 MULTI-WAVELENGTH OPTICAL PHASE UNWRAPPING WITH LASER DIODES This chapter presents results of multi-wavelength optical phase unwrapping using laser diodes. Diode were chosen so that the combination of diodes provides a much larger beat wavelength that that of LEDs. First section presents results of two-wavelength optical phase unwrapping and the second sect ion presents results of three-wavelength optical phase unwrapping. Subsections of each section present different samples describing image size and waveleng ths used for unwrapping process. 6.1 Two-Wavelength Optical Phase Unwrapping with Laser Diodes In this section experimental results of two-wavelength opt ical phase unwrapping (OPU) using a sample of basal mucosa (c ommonly known as cheek cells) are presented with 1= 679.68 nm and 2= 660.86 nm. The beat wavelength 13-23 is 23.87 m. Image size is 92.42 m and 448 pixels per side. Figure (6.1 a) is the single wavelength phase map with 1 = 653.83 nm. The three wavelength coar se map is shown in Figure (6.1b) with a beat wavelength 13-23 = 23.87 m. The final fine map with reduced noise is shown in Figure (6.1c). Figure (6.1d) show s the 3-D rendering of final fine map. 89

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Figure 6.1: Results of two-wavelength OPU for cheek cells. (a) phase map with 1 = 679.68 nm; (b) coarse map with beat wavelength 12 = 23.87 m; (c) fine map with reduced noise; (d) 3-D rendering of (c). 90

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6.2 Three-wavelength Optical Phase Un wrapping with Laser Diodes In three-wavelength optical phase unwrapping, three light emitting diodes of different wavelengths are used. Here the beat wavelength is increased by choosing closer wavelengths, and the thir d wavelength is used to redu ce the noise in phase maps. The samples used in the experiment are; a micro electro-mechanical system (MEMS), and a sample of LP grooves. 6.2.1 Micro Electro Mechanical (MEM) Biosensor The object here is a micro-electrode a rray biosensor. It consists of 16 gold electrodes on a Pyrex glass s ubstrate. The center is a 125m diameter circle with an approximate thickness of 2m [40]. The unwrapped phase map shows the grainy surface of electrodes. The three wavelengths are 1 = 677.81 nm, 2 = 639.37 nm and 3 = 636.89 nm with a beat wavelength of 1323 = 11.27m Figure (6.2) shows the results of threewavelength optical phase unwrappi ng. Figure (6.2a) is the single wavelength phase map with 1 = 677.81 nm. The three wavelength coarse map is shown in Figure (6.2b) with a beat wavelength 13-23 = 11.27 m. The final fine map w ith reduced noise is shown in Figure (6.2c). Figure (6.2d) shows th e 3-D rendering of final fine map. Cross section of each phase map is taken along the lines shown in Figure (6.2). These cross sections and phase noise of coarse and fine maps are s hown in Figure (6.3). Figures (6.3a)-(6.3c) show surface profiles of single wavelength phase map, coarse map and fine map respectively. Vertical axis for each map is 14 m. Figure (6.3d) shows 326.9 nm rms noise of the unwrapped coarse map. Final phase map with reduced phase 91

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noise is shown in Figure (6.3e). The reduced rms noise in the final phase map is 257.08 nm. 92

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Figure 6.2: Three-wavelength OPU of MEMS sensor. Image size is 86.6 m and 300 pixels per side. (a) a single wavelength phase map with = 677.81 nm; (b) Coarse map of coarse map produced by 1 = 677.81 nm, 2 = 639.37 nm and 3 = 636.89 nm with = 11.27 1323m ; (c) Final fine map with reduced noise, (d) 3-D rendering of (c). 93

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Figure 6.3: Surface profiles of MEMS sensor. (a ) single wavelength surface profile; (b) surface profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (d) noise of coar se map in the area shown in (b). rms noise is 326.9 nm; (e) noise of final unwrapped phase map in the area shown in (c). Rm s noise is 257.08 nm. 94

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6.2.2 Long Playing (LP) Record Grooves The object is a piece of 33 1/3 r.p.m. long playing (LP) record. For 33 1/3 r.p.m. records the typical width at the top of the groove ranges from 25.4 m to 76.2 m and the groove spacing is 84 m -127 m [41]. The sample is coated with a layer of 200 nm Aluminum for better reflectivity. The three wavelengths are 1 = 677.81 nm, 2 = 659.77 nm and 3 = 636.89 nm with a beat wavelength of 1323 = 24.77 m. Figure (6.4) shows the results of three-wavelength optical phase unwrapping. Image size is 147 m 110 m, 640 480 pixels. Figures (6.4a)-(6.4c) show single wavelength phase map with 1 = 677.81 nm, coarse map of coarse map with beat wavelength 13 23 = 24.77 m and the final fine map with reduced noise. Figure (6. 4d) is the cross sec tion taken along the line shown in Figure (6.4b). The rms noi se in the region shown is 1.9 m. Figure (6.4e) shows the cross section along the line shown in Figure (6.4c). The rms noise in the region is 0.6 m. 95

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Figure 6.4: Three-wavelength OPU of LP grooves. Image size is 147 m 110 m, 0 480 pixels. (a) a single wavelength phase map with 64 =677.81 nm; (b) Coarse ma f coarse map with 1323=7 m p o 24.7 ; (cinal fine map with reduced noise; (d) surface profile of coarse map; (c) surface profile of final unwrapped phase map with reduced noise; (e) noise of coarse map in th e area shown in (b). rms noise is1.9 ) F m; (f) surface profile of final unwrapped phase map in the area shown in (c); (g) final noise of the unwrapped phase map. Rms noise is 0.6 m. 96

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6.3 Discussion and Conclusions Laser diodes can be effectively used as a light source for multi-wavelength optical phase unwrapping. Small size and low price comp ared to those of lasers are advantages of laser diodes. The coherence length of laser diodes is larger than that of light emitting diodes and therefore laser diodes can be used to image objects that cannot be successfully used with light emitting diodes. 97

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CHAPETR 7 THREE-WAVELENGTH OPTICAL PHASE UNWRAPPING USING RING DYE LASER This chapter presents experimental results of three-wavelength optical phase unwrapping obtained with a ring dye laser. With ring dye laser there are more wavelength choices and the uncertainty of wavelength measurements is 0.1 nm. Cheek cells and aluminum coated LP grooves are used as the samples. 7.1 Cheek Cells Using wavelengths 1 = 579 nm, 2 = 577 nm and 3 = 574 nm, a cheek cell is imaged. The beat wavelength is 167m Image size is 102m and 448 pixels per side. 98

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Figure 7.1: Three-wavelength OPU of cheek cells. Image size is 102m and 448 pixels per side.(a) direct image of cheek cell; (b) a single wavelength phase map ( = 579 nm); (c) coarse map of coarse map produced by 1 = 579 nm, 2 = 577 nm and 3 = 574 nm with = 167.04 1323m ; (d) Final fine map with reduced noise; (e) 3-D rendering of (d). 99

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7.2 LP Record Grooves The object is a piece of 331/3 r.p.m. LP record, coated with 200 nm Aluminum layer. For 331/3 r.p.m. records the typical width at the top of the groove ranges from 25.4m to 76.2m and the groove spacing is 84m -127m [41]. The three wavelengths used for optical unwrapping process is 1 = 577 nm, 2 = 575 nm and 3 = 570 nm with a beat wavelength of 166m Figure (7.2a) is the single wavelength phase map with 1 = 577 nm. The three wavelength coarse map is shown in Figure (7.2b) with beat wavelength 13-23 = 166 m. The final fine map w ith reduced noise is shown in Figure (7.2c). Figure (7.2d) is the 3-D rendering of final fine map. In the final unwrapped phase map, the width of the top of the groove is measured along the line shown in Figure (7.2d). The measured width is 44 m. Cross sections and phase noise of coarse and fine maps are shown in Figure (7.3). Figure (7.3a) is the unwrapped coarse map and Figure (7.3b) is the final fine map with reduced noise. Figure (7.3c) is the surface pr ofile of coarse map along the line shown in Figure (7.3a). The rms noise in coarse map in the area shown in (a) is 2.12 m and this is shown in Figure (7.3d). Figure (7.3e) s hows the surface profile of fine map along the line shown in (b). The groove depth h = 18 m. Figure (7.3f) shows the noise of the fine map in the selected area. Rms noise is 1.36 m. 100

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Figure 7.2: Three-wavelength OPU of LP reco rd grooves. Image size is 102 m and 448 pixels per side. (a) a singl e wavelength phase map with = 577 nm; (b) Coarse map of coarse map produced by 1 = 577 nm, 2 = 575 nm and 3 = 570 nm with 13 23 = 165.89 m; (c) Final fine map with reduced noise ; (d) 3-D rendering of (c). The groove width is 44 m. 101

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Figure 7.3: Surface profiles of LP re cord grooves. (a) Coarse map of coarse map with = 165.89 1323 m; (b) final fine map with reduced noise; (c) surface pr ofile of coarse map along the line shown in (a); (d) noise in coarse map in the area shown in (a). Rms noise is 2.12 m; (e) surface profile of fine map al ong the line shown in (b). The groove depth h = 18 m; (f) noise of the fine map in the area shown in (b). Rms noise is 1.36 m. 102

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7.3 Discussion and Conclusions Effectiveness of multi-wavelength optical phase unwrapping using a ring dye laser as the light source is shown. The tech nique is a valuable tool for imaging both biological samples and other microscopic samples. 103

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CHAPETR 8 CONCLUSIONS AND FUTURE WORK In this research we have successfully demonstrated the effectiveness of the multiwavelength optical unwrapping method. To our know ledge this is the fi rst time that three wavelengths have been used for phase unw rapping without increasing phase noise. Conventional software unwrapping methods fail when there is high phase noise and also cannot be used for objects with irregula rities. The multi-wavelength optical phase unwrapping method is free of such problems. Software unwrapping algorithms can take more than ten minutes to unwrap phase images This is a disadvantage when one needs to study live samples in real time or near real time. The multi-wavelength optical unwrapping method is significantly faster than software algorith ms and can be effectively used to study live samples in real time. A nother advantage is that the optical phase unwrapping method is free of complex algorith ms and needs less user intervention. The advantage of three wavelengt h optical phase unwrapping over two wavelength optical phase unwrapping is that, the use of three wavelengths increases the beat wavelength without increasing phase noise in the final unwrapped phase image. The ability to extend the beat wavelength is impor tant because this allows studying samples with height variations of hundred micrometers or more. 104

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Multi-wavelength optical phase unwrapping method can be used successfully with any type of light source; incoherent or co herent. This provides the user a greater freedom of choosing a light source suitable for sample features and research goals. Incoherent light sources such as light emitting diodes reduce the speckle noise inherent to lasers. However, light emitting diodes are availa ble only in several different wavelengths. Therefore, wavelength combina tions that produce large beat wavelengths are limited. Because of small coherence le ngths of light emitting diodes, imaging phase profiles of samples with features larger than the coherence range is not possible. In this case, laser diodes which have larger coherenc e lengths than that of light emitting diodes can be used. Both light emitting diodes and laser diodes are small in size and provide reduced apparatus dimensions. This is an ideal feature for a compact, portable microscope system. If larger beat wavelengths are needed, a ring dye laser can be used to extend the beat wavelength to more than hundred micrometers. The three wavelength optical unwrapping method successfully eliminates 2 ambiguities while reducing phase noise in the final unwrapped phase profile to the order of several nanometers, regardless of the light source. The technique can be optimized by minimizing possible errors in the phase shifting procedure and by modifying the expe rimental setup with a horizontal sample mount and a color CCD. Here we suggest several modifications to the setup. In the phase shifting interferometry, th e accuracy of phase shifting depends on the phase shifting device. In this experiment, a function generator is used to send a ramp waveform to the PZT mounted reference mirror. The amplitude of the waveform is calculated according to the wavelength of the light source, so that the reference mirror is 105

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moved by quarter-wavelength inte rvals. The practical phase sh ift can be different from the theoretical value because of several fact ors; accuracy of the function generator and PZT, air turbulences and vibra tions of the setup. Many methods have been introduced to compensate the phase shift e rror [42-47]. Kinnstaetters me thod is based on a Lissajous figure formed by using the interference pattern. The Lissajous figure is used to detect the accuracy of the calibration of the phase sh ifter and the phase steps, the non-linear characteristics of detectors and mechanical vibrations [42]. Ma ny error compensating methods use averaging techniques where more than four interferograms are used to calculate the final phase profile [44, 46]. In our experiment five frames, each 900 apart are used and the intensity profiles of the first and the fifth frames are checked to de termine the accuracy of phase shifting. If the PZT is calibrated well enough, the first and the fifth fr ames should coincide. The breadboard on which the experimental setup built is also placed on inner tubes to minimize vibration effects. Furthermore, the entire setup is covered while measurements are taken to reduce the e ffect of air currents. The setup can be further improved by usi ng a horizontal sample holder instead of the current vertical once. This facilitates us ing samples in the setup easier. A compact version of the setup is also needed to make it portable. At present the desired wavelength is chosen by moving the light sources. Only one light source can be turned on at a time since stray light can adversely affect the interference. Moving light sources can cause collimating and focusing errors if not done care fully. Three fiber optic cables as shown in Figure (8.1) can be used to cha nge the wavelengths efficiently. This also reduces the time 106

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that is needed to acquire single wavelength phase profiles since changing wavelengths can be done by simply turning off two unwanted light sources. Figure 8.1: Modified setup with fiber optic cables. The Michelson-type setup is most suitable for reflective samples or transparent samples mounted on a reflective surface. The technique can also be easily used in a Mach-Zehnder type setup, which is more suitable for transparent samples. The images presented in this study have not been subjected to image enhancing techniques, except for pseudo color rendering of 3-dimensional phase images, since our goal was to demonstrate the effectiveness of multi-wavelength optical phase unwrapping technique. If user wishes image enhanci ng techniques can be applied to improve the quality of phase images once they are unwrapped. 107

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REFERENCES [1] F. Zernike, Phase contrast, a ne w method for the micros copic observation of transparent objects, Physica 9, 686-698 (1942). [2] F. Zernike, Phase contrast, a new method for the microscopic observation of transparent objects Part II, Physica 9, 974-986 (1942). [3] R. D. Allen, G.B. David, and G. No marski, The Zeiss-Nomarski differential interference equipment fo r transmitted-light microscopy, Z. Wiss. Mikrosk. 69,193-221 (1969). [4] Robert Hoffman and Leo Gross, Modulat ion contrast microscope, App. Opt. 14, 1169-1176 (1975). [5] Albert A. Michelson and Edward W. Morl ey, On the relative mo tion of the earth and the luminiferous ether, Am. Jour. Sci. Third Series, XXXXIV, No. 203, 333-345, (1887). 108

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[6] Sanjit K. Debnath, Mahendra P. Kothiyal, Joanna Schmit, Prameswaran Hariharan, Spectrally resolved white-light phase-shi fting interference micr oscopy for thicknessprofile measurements of transparent thin film layers on pattern substrates, Opt. Express, 14, 4662-4667 (2006). [7] Jun Chen, Junji Endo, Yoshiaki Niino, Hi royuki Fujita, Phase-shifting interference microscopy using a Fresnel's biprism, Proceedings of SPIE -Volume 4416 Optical Engineering for Sensing and Nanotechnology (ICOSN 2001), Koichi Iwata, Editor, May 2001, pp. 158-161. [8] Maitreyee Roy, Collin J. R. Sheppard, Guy Cox, Parameswaran Hariharan, Whitelight interference microscopy: a way to obtain high lateral resolution over an extended range of heights, Opt. Express, 14, 6788-6793 (2006). [9] N. Warnasooriya and M. K. Kim, Mul ti-wavelength Phase Imaging Interference Microscopy, Proceedings of SPIE Volume 6090 Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, Joe-Angel Conchello, Carol J. Cogswell. Tory W ilson, Editors, January 2006, pp. 60900U-1 60900U-8. [10] N. Warnasooriya, M. K. Kim, LED-based multi-wavelength phase imaging interference microscopy, Opt. Express, 15, 9239-9247 (2007). 109

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[11] Xinhong Li, Toyohiko Yamauchi, Hidenao Iwai, Yutaka Yamashita, Haijun Zhang and Teruo Hiruma, Full-field quantitative pha se imaging by white-light interferpmetry with active phase stabilizati on and its application to biological samples, Opt. Lett. 31, 1830-1832 (2006). [12] Andrew Lewis, Measur ement of length, surface form and thermal expansion coefficient of length bars up to 1.5 m using multiple-wavelength phase-stepping interferometry, Meas. Sci. Technol. 5, 694-703 (1994). [13] D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical testing (Marcel Dekker, Inc., New York, 1998). [14] M. Tziraki, R. Jones, P. M. W. French, M. R. Melloch and D. D. Nolte, Photorefractive holography for imaging throug h turbid media using low coherent light, Appl. Phy. B, 70, 151-154 (2000). [15] S. Dilhaire, S. Grauby, S. Jorez, L. D. P. Lopez, J. Rampnoux and W. Claeys, Surface displacement imaging by interferometry with a light emitting diode, Appl. Opt. 41, 4996-5001 (2002). [16] L. Repetto, E. Piano and C. Pontiggi a, Lensless digital holographic microscope with light-emitting diode illumination, Opt. Lett. 29, 1132-1134 (2004). 110

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[17] L. Deck and F. Demarest, Two-color light-emitting-diode source for high-precision phase-shifting interferometry, Opt. Lett. 18, 1899-1901 (1993). [18] A. Fercher, W. Drexler, C. K. Hitzenberger and T. Lasser, Optical coherence tomography-principles and applications, Rep. Prog. Phys. 66, 239-303 (2003). [19] LuxeonTM Emitter and Star sample information AB11, 2 (Feb 2002). [20] Jiyuan Liu and Ichirou Yamaguchi, Surfa ce profilometry with laser-diode optical feedback interferometer outside optical benches, App. Opt. 39, 104-107 (2000). [21] Yukihiro Ishii and Ribun Onodera, Phase-extraction al gorithm in laser-diode phaseshifting interferometry, Opt. Lett., 20, 1883-1885 (1995). [22] Ribun Onodera and Yukihiro Ishii, Phase -extraction analysis of laser-diode phaseshifting interferometry that is insensitive to changes in laser power, J. Opt. Soc. Am. A 13, 139-146 (1996). [23] Peter de Groot and Stanley Kishner, Synthetic wavelength stabilization for twocolor laser-diode interferometry, App. Opt. 30, 4026-4033 (1991). 111

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[24] Andrei N. Lukashkin, Mikhail E. Ba shtanov, Ian J. Russell,A sef-mixing laserdiode interferometer for measuring basila membrane vibrations without opening the cochlea, J. Neurosci. Methods 148, 122-129 (2005). [25] Coherent Ring Dye Laser User Ma nual, 0159-216-00-699, Coherent Inc. [26] P. G. Charette, I. W. Hunter, R obust phase-unwrapping method for phase images with high noise content, App. Opt., 35, 3506-3513 (1996). [27] Dennis C. Ghiglia, Louise A. Rome ro, Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods, J. Opt. Soc. Am. A, 11,107-117 (1994). [28] M. Servin, J. L. Marroquin, D. Malacara and F.J. Cuevas, Phase unwrapping with a regularized phase-tracki ng system, Appl. Opt. 37, 1917-1923 (1998). [29] J. C. Wyant, Testing aspherics usi ng two-wavelength holography, Appl. Opt. 10, 2113-2118 (1971). [30] C. Polhemus, Two-wavelen gth interferometry, Appl. Opt. 12, 2071-2074 (1973). [31] Yeou-Yen Cheng, James C. Wyant, Two-wavelength phase shif ting interferometry, App. Opt., 23, 4539-4543 (1984). 112

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[32] Katherine Creath, Yeou-Yen Cheng, James C. Wyant, Countoutring aspheric surfaces using two-wavelength phase-shifting interferometry, Optica Acta, 32, 14551464 (1985). [33] J. Gass, A. Dakoff, M. K. Kim, Phase imaging without 2 ambiguity by multiwavelength digital holography, Opt. Lett. 28, 1141-1143 (2003). [34] Christope Wagner, Wolfgang Oste n, Soenke Seebacher, Direct shape measurements by digital wavefront reconstruction and multiwavelength countoutring, Opt. Eng. 39, 79-85 (2000). [35] U. Schnars and W. Jueptner, Digital Holography Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, Berlin Heidelberg, 2005). [36] Yeou-Yen Cheng, James C. Wyant, Multiple-wavelength phase-shifting interferometry, Appl. Opt. 24, 804-807 (1985). [37] Karl Meiners-Hagen, Volker Burgar th, Ahmed Abou-Zeid, Profilometry with a multi-wavelength diode laser interferometer, Meas. Sci. Technol., 15, 741-746 (2004). [38] Frederic Montfort, Tristan Colomb, Florian Charriere, Jonas Kuhn, Pierre Marquet, Etienne Cuche, Sylvain Herminjard, Christ ian Depeursinge, Sub micrometer optical 113

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tomography by multi-wavelength digital holographic microscopy, Appl. Opt. 45, 82098217 (2006). [39] D. Parshall and M. K. Kim, Digita l holographic microscopy with dual wavelength phase unwrapping, App. Opt., 45, 451-459 (2006). [40] A. R. A. Rahman, C. M. Lo, S. Bh ansali, A MEMS Micr o-Electrode Array Biosensor for Impedance Spectroscopy of Hu man Umbilical Vein Endothelial Cells," Sensors & Actuators B 118, 115-120 (2006). [41] Vitaliy Fadeyev and Carl Haber, Recons truction of mechanically recorded sound by image processing, LBNL Report 51983 (2003). [42] K. Kinnstaetter, Adolf W. Lohmann, Johannes Schwider and Norbert Streibl, Accuracy of phase shifting interferometry, App. Opt. 27, 5082-5089 (1988). [43] Yeou-Yen Cheng and James C. Wyant, P hase shifter calibrati on in phase-shifting interferometry, App. Opt. 24, 3049-3052 (1985). [44] Joanna Schmit and Kather ine Creath, Extended averag ing technique for derivation of error-compensating algorithms in phase-shifting interferometry, App. Opt. 34, 36103619 (1995). 114

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[45] Peter L. M. Heydemann, Determina tion and correction of quadrature fringe measurement errors in interferom eters, App. Opt. 20, 3382-3384 (1981). [46] J. Schwider, R. Burow, K. E. Elssn er, J. Grzanna, R. Spolaczyk and K. Merkel, Digital wave-front measuring interferometry: some systematic error sources, App. Opt. 22, 3421-3432 (1983). [47] Shuqun Zhang, A non-iterative method for phase-shift estimation and wave-front reconstruction in phase-shifting di gital holography, Opt. Commun. 268, 231-234 (2006). 115

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BIBLIOGRAPHY Thomas E. Jones, History of the Light Microscope, http://web.archive.org/web/2002080 6183837/http://web.archive.org/web/2002080618383 7/www.utmem.edu/~thjones/tj.htm (Last Visited 10/02/2008). Michael Bass, Editor in Chief, Eric W. Van Stryland, David R. Williams, William L. Wolfe, Associate Editors, Handbook of Optics Fundamentals, Techniques, & Design, Second Edition, Volume I, McGraw-Hill, Inc., New York, NY (1995). Michael Bass, Editor in Chief, Eric W. Van Stryland, David R. Williams, William L. Wolfe, Associate Editors, Handbook of Optics Device, Measurements, & Properties, Second Edition, Volume II, McGraw-Hill, Inc., New York, NY (1995). Elizabeth M. Slayter, Optical Methods in Biology, Wiley-Interscience, ISBN 0471796700 (1970). 116

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Joseph W. Goodman, Introduction to Fourier Optics, Second Edition, McGraw-Hill Companies, Inc., New York, NY (1988). Bahram Javidi, Pietro Ferraro, Seung-Hyun H ong, Sergio De Nicola, Andrea Finizio, Domenico Alfieri, Giovanni Pierattini, Three-dimensional image fusion by use of multiwavelength digital holography, Opt. Lett. 30, 144-146 (2005). 117

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APEENDICES 118

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119 APPENDIX A COMPUTER PROGRAMS A.1 LabView Program for Quadrature Phase Shifting When the PZT is dithered by supplying a ramp waveform from the function generator, this LabView program also receives a signa l from the function generator and starts capturing 5 frames as describes in the section (2.5). Then it combines the frames to calculate the final phase image of the object.

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Quadrature Phase ImageAAA.vi Y :\AAA KimLab Folders\folder 2004 N ilanthi W\programs Nilanthi\ QuadraturePhase\Quadratu re Phase ImageAAA.vi Q uadrature Phase Ima g eAAA.vi 0Array Quadrature Phase Image Amplitude Image 4 -4 -3 -2 -1 0 1 2 3 Time480 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 Plot 0 Phase Graph abs show B&W coor 0.00vmin 0.00 vmax saveplot params 3.14 -3.14 -1.57 0.00 1.57 PhaseIma g e 260 0 20 40 60 80 100 120 140 160 180 200 220 240 650 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 Pot 0I(2pi) I(0) and I(2p) 3.14 -3.14 1.57 480 0 50 100 150 200 250 300 350 400 450 640 0 100 200 300 400 500 Phase Image 255 10 132 I(0)

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Quadrature Phase ImageAAA.vi Y :\AAA KimLab Folders\folder 2004 N ilanthi W\programs Nilanthi\ QuadraturePhase\Quadratu re Phase ImageAAA.vi p lot p arams PhaseIma g e ZScale.MarkerVals [] T o save p hase ima g es Dis p Save-a.vi Phase Ima g e 2 [ 0..2 ] Interface IMAQ Init.vi IMA Q Ima g e T yp e Tri gg er start of ac q uisition 1 External tri gg er 0 0 IMA Q Create Format Into Strin g CCD %d IMAQ Ima g eToArra y IMA Q Se q uence.vi Ski p frames before ac q uire each buffer A rra y 5 5 5 5 5 5 0 Ac q uire ma g es 0 [ 0..2 ]

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Quadrature Phase ImageAAA.vi Y :\AAA KimLab Folders\folder 2004 N ilanthi W\programs Nilanthi\ QuadraturePhase\Quadratu re Phase ImageAAA.vi Index Arra y 0 Index Arra y Index Arra y Index Arra y Index Arra y 1 2 3 4 IMAQ Ima g eToArra y IMA Q Ima g eToArra y IMA Q Ima g eToArra y IMA Q Ima g eToArra y IMAQ Ima g eToArra y I ( 0 ) -I ( P ) I ( 3Pi/2 ) -I ( Pi/2 ) Phase Ima g e Phase Gra p h Index Arra y I(0) I (p /2 ) I(pi) I ( 3 p i/2 ) I ( 2 p ) I ( 0 ) and I ( 2 p i ) I ( 0 ) I (p i ) I ( 3 p i/2 ) I(2pi) Phase Ima g e Cursor.PosY Phase Ima g e I ( 0 ) 1 [ 0..2 ]

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A.2 LabView Program for Two-Wavelength Optical Phase Unwrapping After obtaining two wrapped phase images, each with a different wavelength, they are combined in this program to produce the final unwrapped phase image with a larger beat wavelength. The method is desc ribed in the section (4.3). 123

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2WavelengthExp-STANDARD.vi K:\AAA KimLab Folders\folder 2004 Nilanthi W\programs Nilanthi\ 2WavelengthExp\2WavelengthExp-STANDARD.vi 530 WL 1(nm) 470 WL 2 (nm) z from phi 2 z from phi 2 9859.02 0.00 2464.76 4929.51 7394.27 450 0 50 100 150 200 250 300 350 400 Time448 0 50 100 150 200 25 300 350 400 9859.02 0.00 2464.76 4929.51 7394.27 450 0 50 100 150 200 250 300 350 400 Time448 0 50 100 150 200 250 300 350 400 absshow b&w color 0.00vmin 0.00ax 0.00a y 0.00vmax plot params 10000 0 2000 4000 6000 8000 Time48 0 0 50 100 150 200 250 300 350 400 450 CROSS SECTION 480 0 50 100 150 200 250 300 350 400 450 SAVE 6000 6000 -4000 -2000 0 2000 4000 Noise in CROSS SECTION 100 Sawtooth Start 200 Sawtooth End 0.00 rms value Z1 & L1 For Z12 Fine Map SAVE 6000 6000 -4000 -2000 0 2000 4000 Noise in Cross Section (parabola-data) 0.00 rms value SAVE SAVE SAVE 6000 6000 -4000 -2000 0 2000 4000 Noise & parabola noise parabola 0 Sawtooth Start 4 0 Sawtooth End 4

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2WavelengthExp-STANDARD.vi K:\AAA KimLab Folders\folder 2004 Nilanthi W\programs Nilanthi\ 2WavelengthExp\2WavelengthExp-STANDARD.vi ReadBMP.vi 0 [0..1] WL 1(nm) WL 2 (nm wlwn.vi wlwn.vi shit2D.vi x shft y shit 1 0 Def z2Dfine.vi z2Dcoarse.vi wnumber1 wnumber 2 Z1 Z 2 disp2Dcont.vi dsp2Dcont.vi Intensty Graph1 Intensit 2 Unbunde Unbunde Unbundle ZScale.MarkerVals[] YScae.Ofst&Mult XScae.Ofst&Mult Intensity Graph1 ZScae.MarkerVals[] YScale.Ofst&Mult XScale.Ofst&Mult Intensity Graph 2 pot params Cross Secton ndex (row) new fie set ext.vi mp Fatten Pixmap.vi Intensty Graph1 Write BMP File.vi ZScae.Maxmum ZScaleMinimum ColorTbl Intensty Graph1 255 Fle Daog True Save Noise in Cross Secton Sawtooth Start Sawtooth End Array Subset Signals Locations best ft resdual mean squared err o error n (no error) error out intercept slope Curve Ftting Index Array rms value 0 De WFGraph (strict) Image Data Image Data BG Color BG Color Image Depth Image Depth Get Image Get Image Wrie BMP Fie.vi new fie set ext.vi mp WaveformGraph Nose n Cross Section (parabola-data) g True SAVE 2 [0..3] Signals Locatons best fit residual mean squared err error in (no error) error out a2 a1 a0 Curve Fitting2 Array Subset Noise n Cross Section (paraboa-data) rms value Error usng parabola Noise & parabola Signals Locatons best ft residual mean squared err o error n (no error) error out a2 a1 a0 Curve Fittng3 Array Subset Sawooth Start 4 Sawtooth End 4 a2x^2a1xa0 Image 1 Image 2 data params data params 2 255 255

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A.3 LabView Program for Three-Wavele ngth Optical Phase Unwrapping After obtaining three wrapped phase images, each with a different wavelength, they are combined in this program to produce the final unwrapped phase image with a larger beat wavelength. The method is desc ribed in the section (4.5). 126

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3WLExp-Corrected.vi K:\AAA KimLab Folders\folder 2004 Nilanthi W\programs Nilanthi\ 3WavelengthExp\3WLExp-Corrected.vi 0 WL 1 (nm) 0 WL 2 (nm) 0 WL 3 (nm) z from phi 2 0 -30000 -25000 -20000 -15000 -10000 -5000 480 0 50 100 150 200 250 300 350 400 450 28194 0 7048 14097 21145 300 0 50 100 150 200 250 300 0 50 100 150 200 250 480 0 50 100 150 200 250 300 350 400 450 15070 0 3767 7535 11302 300 0 50 100 150 200 250 300 0 50 100 150 200 250 16000 0 2000 4000 6000 8000 10000 12000 14000 240 0 20 40 60 80 100 120 140 160 180 200 220 ph(23) 0 L13 0 L23 0 L13-23 0 L12 SAVE 2 SAVE 1 6000 -4 000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000 0 Sawtooth Start 0 Sawtooth End 0.00 rms value 25000 -15000 -10000 -5000 0 5000 10000 15000 20000 0 Sawtooth Start 2 0 Sawtooth End 2 000 rms value 2 0max value 2 0 mn value 2 Z1L1 For Fnal Fne Map Z'13 L13 For Intermedate Fne Map Z1 & L1 For Z12 Fne Map Z1 & L1 For Z13 Fne Map Z2 & L2 For Z23 Fne Map 20000 -2 0000 -15000 -10000 -5000 0 5000 10000 15000 0.00 rms value 3 30 -4 0 -30 -20 -10 0 10 20 noise paraboa 0 Sawtooth Start 4 0 Sawtooth End 4 Nose in cross section (st line-data) Noise n cross section-(paraboa-data) Noise n cross section(st lne-data) SAVE SAVE Noise&Parabola SAVE nose SAVE cross secton SAVE nose (paraboa-data) 0K13 0K23

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3WLExp-Corrected.vi K:\AAA KimLab Folders\folder 2004 Nilanthi W\programs Nilanthi\ 3WavelengthExp\3WLExp-Corrected.vi ReadBMP.vi 2 [ 0.2 ] data params data params 2 data p arams 3 Image 1 Image 2 Ima g e 3 256 256 256 z2Dcoarse.vi z2Dcoarse.vi z2Dcoarse.vi WL 1 ( nm ) WL 2 (nm) WL 3 (nm phase 1 p hase 2 phase 3 Z12 Z'13 Z'23 wlwn.vi wlwn.vi wlwn.vi Z1 Z2 Z3 z2Dfne.vi z2Dfine.vi z2Dfne.vi dis p 2Dcont.vi Unbundle Index Array ZScale.MarkerVals [] YScae.Ost&Mult XScae.Ost&Mult 2 Beatwavelen g thvi wlwn.vi wlwn.vi Beatwavelength.vi Coarse map of coarse mapsZ13-23 z2Dfine.vi 2 dis p 2Dcont.vi Index Array ZScale.MarkerVals[] YScae.Ofst&Mult XScae.Ofst&Mult Unbundle z2Dfne.vi Internmedate Fine MapZ13-23 L13 L23 Beatwavelength.vi L13-23 Beatwavelen g thvi L12 Fnal Fine Map new fle set ext.vi mp Flatten Pixmap.vi ZScale.Maxmum ZScale.Minimum ColoTbl Intensit y Gra p h 2 255 Intensity Graph 2 Fie Dialo g True SAVE Phase Ma p s SAVE 2 new file set ext.vi mp Flaten Pixmap.vi 255 Intensit y Gra p h ZScale.Maxmum ZScale.Minimum ColorTbl Intensity Graph Fle Dialo g True SAVE SAVE 1 phase 13 p hase 23 Noise in Cross Section Array Subset S g nas Locations best fit resdual mean squared err o error in ( no eror ) error out interce p t sope Curve Ftting Index Array Sawtooth Sart Sawtooth End rms value Nose n sin g leWL Array Subset Signals Locatons best ft residual mean squared err error in (no error) error out nterce p t slo p e Curve Fitting2 Index Arra y Sawtooth Start 2 Sawtooth End 2 rms value 2 plot params max value 2 mn value 2 1 1 2 Z1 0 1 0 0 Sgnas Locations best fit resdual mean s q uared err error in ( no error ) error out a2 a1 a0 Curve Fttin g 3 Array Subset Noise n Cross Secton 3 rms value 3 Error usng parabola Noise & p arabola Sgnas Locations best fit resdual mean s q uared err error in (no error) error out a2 a1 a0 Curve Fttin g 4 Array Subset a2x^2a1xa0 WFGraph (strict) Image Data Image Data BG Color BG Color Image Depth Image Depth Get Image Get Image Wrte BMP File.vi new fie set ext.vi mp WaveformGra p h Cross Secton SngleWL File Daog True 4 [0.4] Save Gra p hs K13 K23 z2Dcoarse.vi

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APPENDIX B LIST OF ACCOMPLISHMENTS Peer Review Journals 1. N. Warnasooriya and M. K. Kim, LED-based multi-wavelength phase imaging interference microscopy", Optics Express 15, 9239-9247 (2007). 2. N. Warnasooriya and M. K. Kim, Quantitative phase imaging using three-wavelength optical phase unwrapping, Journal of Modern Optics (in review). Conference Presentations 1. N. Warnasooriya and M. K. Kim, Optical Phase Unwrapping with Laser-diode Phase Shifting Interferometry, OSA Digital Holography and Three Dimensional Imaging Topical Meeting, St. Petersburg, Florida, March 19, 2008 Oral presentation. 2. N. Warnasooriya and M. K. Kim, Phase-Shifting Interference Microscopy with Multi-Wavelength Optical Phase Unwrapping, OSA Digital Holography and Three-Dimensional Imaging, Vancouver, Canada, June 18, 2007 Oral presentation. 3. N. Warnasooriya and M. K. Kim, Quantitative Phase Microscopy by Multi-Wavelength Phase-Shifting Interference Microscopy, The Conference on Lasers and Electro-Optics (CLEO), Baltimore, MD, May 8, 2007 Oral presentation. 4. N. Warnasooriya and M. K. Kim, LED-based Phase-Shifting Interference Microscopy with Multi-wavelength Optical Phase Unwrapping, Graduate Research Symposium, University of South Florida, Tampa, FL, March 20, 2007 Oral presentation. 5. N. Warnasooriya and M. K. Kim, Phase Shifting Interference Microscopy with Multi-wavelength Optical Phase Unwrapping Microscopy, 3 rd Annual Integrative Graduate Education and Research Traineeship (IGERT) Symposium, University of South Florida, Tampa, FL, April 11, 2007 Poster presentation. 129

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6. N. Warnasooriya and M. K. Kim, Multi-wavelength Phase Imaging Interference Microscopy, Graduate Research Symposium, University of South Florida, Tampa, FL, April 20, 2006 Poster presentation. 7. N. Warnasooriya and M. K. Kim, LED-based Phase Imaging Interference Microscopy with Multi-wavelength Optical Phase Unwrapping, 2 nd Annual Integrative Graduate Education and Research Traineeship (IGERT) Symposium, University of South Florida, Tampa, FL, April 5, 2006 Poster presentation. 8. N. Warnasooriya and M. K. Kim, LED-based Phase Imaging Interference Microscopy with Multi-wavelength Optical Phase Unwrapping, OSA Biomedical Optics Conference, Ft. Lauderdale, FL, March 19-22, 2006 Poster presentation. 9. N. Warnasooriya and M. K. Kim, Multi-wavelength Phase Imaging Interference Microscopy, Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, part of the Biomedical Optics (BiOS) symposium, The International Society for Optical Engineering, San Jose, CA, January 21-26, 2006 Poster presentation. 10. N. Warnasooriya and Janet Seger, Search for X (1750) in Ultra Peripheral Collisions at STAR, The Nebraska Academy of Sciences on April 25, 2003 Oral presentation. 11. N. Warnasooriya and Janet Seger, Search for X (1750) in Ultra Peripheral Collisions at STAR, Poster presentation for St. Alberts Day, Creighton University on November 11, 2002. 12. N. Warnasooriya and Janet Seger, Ultra Peripheral Collision Program at STAR, The Nebraska Academy of Sciences on April 26, 2002 Oral presentation. 130

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About the Author Nilanthi Warnasooriya grew up in Colom bo, Sri Lanka. She received a Bachelor of Science degree in Physics from the Univer sity of Colombo, Sri Lanka in 2000 and a Master of Science degree in Physics with High Energy Nuclear Physics from Creighton University, Nebraska, USA in 2003. In Fa ll 2003 she entered the PhD program in Applied Physics at the Univers ity of South Florida, USA. Nilanthi joined Digital Holography & Microscopy Laboratory at the Department of Physics, USF for her PhD research under Professor M. K. Kim and has presented her work at many conferences including The Inte rnational Society for Optical Engineering (SPIE), The Conference on Lasers and Electro -Optics (CLEO) and topical meetings of Optical Society of America (OSA). She has su bmitted her work to peer reviewed journal Optics Express. She completed an industria l practicum at Varioptic SA, Lyon, France. Nilanthi currently resides in Tampa a nd enjoys traveling, reading and gardening as pastimes.


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Quantitative phase imaging microscopy with multi-wavelength optical phase unwrapping
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ABSTRACT: This dissertation presents a quantitative phase imaging microscopy technique that combines phase-shifting interferometry with multi-wavelength optical phase unwrapping. The technique consists of a Michelson-type interferometer illuminated with any of three types of light sources; light emitting diodes, laser diodes and a ring dye laser. Interference images are obtained by using a 4-frame phase shifting method, and are combined to calculate the phase of the object surface. The 2 ambiguities are removed by repeating the experiment combining two and three different wavelengths, which yields phase images of effective wavelength much longer than the original. The resulting image is a profile of the object surface with a height resolution of several nanometers and range of several microns. To our knowledge, this is the first time that a three wavelength optical phase unwrapping method with no amplified phase noise has been presented for full-frame phase images.
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The results presented here are divided into three main categories based on the source of illumination; light emitting diodes, laser diodes and a ring dye laser. Results for both two-wavelength optical unwrapping and three-wavelength optical unwrapping techniques are demonstrated. The interferographic images using broadband sources such as light emitting diodes are significantly less affected by coherent noise compared to images obtained using lasers. Our results show that the three wavelength optical phase unwrapping can also be effectively applied to unwrap phase images obtained using coherent light sources such as lasers and laser diodes, without amplifying phase noise in the final phase image. We have successfully shown that our multi-wavelength phase-shifting technique extends the range free of 2 ambiguities in the phase map without using conventional computation intensive phase unwrapping methods.This phase imaging technique can be used to measure physical thickness or height of both biological and other microscopic samples, with nanometer axial resolution. An added advantage of the multi-wavelength optical phase unwrapping technique is that the beat wavelength can be tailored to match height variations of specific samples.
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