APPLICATIONS OF INDUCED GRATINGS IN NONLINEAR MEDIA. Dissertation. Submitted to. The School of Engineering of the UNIVERSITY OF DAYTON

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1 APPLICATIONS OF INDUCED GRATINGS IN NONLINEAR MEDIA Dissertation Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Doctor of Philosophy in Electro-Optics By Haburugala Vithanage Ujitha A. Abeywickrema UNIVERSITY OF DAYTON Dayton, Ohio May 2015

2 APPLICATIONS OF INDUCED GRATINGS IN NONLINEAR MEDIA Name: Abeywickrema, Haburugala Vithanage Ujitha A. APPROVED BY: Partha P. Banerjee, Ph.D. Advisory Committee Chairman Program Director Electro-Optics Program Joseph W. Haus, Ph.D. Committee Member Professor Electro-Optics Program Andrew Sarangan, Ph.D. Committee Member Professor Electro-Optics Program Sergei F. Lyuksyutov, Ph.D. Committee Member Professor Physics The University of Akron Georges Nehmetallah, Ph.D. Committee Member Assistant Professor Catholic University of America, EECS John G. Weber, Ph.D. Associate Dean School of Engineering Eddy M. Rojas, Ph.D., M.A., P.E. Dean School of Engineering ii

3 Copyright by Haburugala Vithanage Ujitha A. Abeywickrema All rights reserved 2015 iii

4 ABSTRACT APPLICATIONS OF INDUCED GRATINGS IN NONLINEAR MEDIA Name: Abeywickrema, Haburugala Vithanage Ujitha A. University of Dayton Advisor: Dr. Partha P. Banerjee Materials exhibiting effective nonlinearity through refractive index modulation at relatively low optical powers can be exploited for various applications. Examples of such materials include liquids where the refractive index is modified through heating, and photorefractives where the refractive index modulation is caused by the induced space charge field due to optically generated charges and their redistribution. Optical probing techniques of these and related effects include digital holography, holographic interferometry, and diffraction. First, the effect of self-phase modulation of a focused laser beam in a thermal medium such as a liquid is studied using a low power probe beam. Beyond self-phase modulation, thermal blooming occurs, due to bubbles generated in the liquid. These bubbles are characterized using the same probe and digital holography. An application of these bubbles to nanoparticle agglomeration and transport for drug delivery systems is proposed. iv

5 Next, the use of recording materials such as photorefractive lithium niobate for implementing real-time phase shifting holographic interferometry is examined in detail. Holographic interferometry is a convenient tool for 3D characterization of deformations of an object. The hologram of an object is first written in the material using a reference beam, and then read out by the same reference beam and light from the deformed object. It is shown that the use of both Bragg and non-bragg orders during conventional twobeam coupling in a photorefractive material facilitates the simultaneous generation of phase shifts necessary for this type of holographic interferometry. In certain applications involving liquid crystals, the spatial modulation of the director axis can yield improved energy coupling in hybrid liquid crystal photorefractive devices. Nanoscale engineering of the director axis is possible using the surface corrugation in photorefractives induced by the space charge field through the piezoelectric effect. This surface corrugation is characterized by monitoring the diffraction of a probe beam from the surface of the photorefractive material, taking care to eliminate any Fabry-Perot effects and diffraction from the volume grating in the material. v

6 Dedicated to my mother, father and sister, for all the support that made this journey possible vi

7 ACKNOWLEDGEMENTS First of all I would like to specially thank my advisor Dr. Partha Banerjee, for his wonderful guidance, vast knowledge, advice, and continuous encouragement and support throughout this work. Without him this would have been impossible. I would also like to thank Dr. Sergei F. Lyuksyutov for his encouragement and giving me a wonderful start in my graduate life. I must thank Dr. Joseph Haus for giving his valuable time for discussions. I would also like to thank Dr. Andrew Sarangan, for advice and continuous attention throughout this work. I would also like to thank Dr. Georges Nehmetallah for spending his time with me to help me with MATLAB. My special thanks should go to Dr. Logan Williams for spending his time in the lab with me helping on my experiments and for his tips to make the experiments successful. I am grateful to Dr. Dean Evans from AFRL who loaned me the rotation stage for part of the experiments with the photorefractive material. I am also thankful to Dr. Shiral Fernando for providing me TEM facility to characterize silver nanoparticles. I would also like to thank my roommate Diego Garcia for his support throughout this work. Also I would specially thank the EO staff for giving us continuous updates and to my group members, fellow students and friends for their wonderful support and making a pleasant environment. Last but not least, I would specially thank to my mother and sister for being my strength and stay behind me all the time. vii

8 TABLE OF CONTENTS ABSTRACT iv DEDICATION vi ACKNOWLEDGEMENTS...vii LIST OF FIGURES.x LIST OF TABLES xv LIST OF ABBREVIATIONS..xvi LIST OF SYMBOLS..xviii CHAPTER 1 INTRODUCTION AND OBJECTIVES Introduction Research Objectives Holography Digital Holography Holographic Interferometry Phase-shifting DH Essential Theory for DH and HI Conclusion Publications Resulting from this Work..14 CHAPTER 2 HOLOGRAPHIC ASSESSMENT OF SELF-PHASE MODULATION AND BLOOMING IN A THERMAL MEDIUM Introduction Essential Theory of Thermal Lensing and its Measurement Experimental Setups Experimental Results Probe Beam Measurement of Thermally Induced Phase Change Probe Beam Measurement of TB and Bubble Size Nanoparticle Synthesis and Agglomeration around Thermal Bubbles.35 viii

9 2.5 Conclusion 38 CHAPTER 3 PHASE-SHIFTING HOLOGRAPHIC INTERFEROMETRY USING PHOTOREFRACTIVE MATERIALS Introduction Initial Observations: Non-Bragg Orders and HI using PR Materials The PR Effect Heuristic Derivation of Bragg and Non-Bragg Orders during HI using PR Material Explicit Derivation of Bragg and Non-Bragg Orders during HI using PR Material Numerical Solutions for Bragg and Non-Bragg Orders Writing the Initial Grating Reading the Grating Theoretical Model of Interacting Angular Spectra Conclusion 69 CHAPTER 4 HOLOGRAPHIC SURFACE GRATINGS IN PHOTOREFRACTIVE MATERIALS Introduction Previous Related Work Surface Gratings on PR Materials Diffraction Efficiency of a Surface Grating The Fabry-Perot Effect Diffraction Behavior of a Surface Grating Formed on a Lithium Niobate Crystal Writing and Reading a Corrugated Surface Grating Experimental Results Numerical Filtering of the Fabry-Perot Effect Heuristic Explanation of Angular Dependence.87 CHAPTER 5 CONCLUSION AND FUTURE WORK Conclusion Future Work..92 REFERENCES.93 APPENDIX..99 Selected MATLAB Codes (algorithms) used in this Work..99 ix

10 LIST OF FIGURES Figure 1.1: (a) Recording a double exposure hologram, (b) reconstructing the hologram.7 Figure 1.2: Interferogram recorded on a CCD camera while heating a liquid.9 Figure 2.1: Experimental setup for observing SPM due to thermal lensing using focused 514 nm Ar ion pump traveling through nonlinear sample. Far field patterns are recorded by CCD 1. A He-Ne probe beam at 633 nm passes through the nonlinear sample contrapropagating to the pump beam. Far field patterns due to the probe beam are recorded by CCD 2. The 633 nm beam can be split in order to form a Mach-Zehnder interferometer. Left photograph shows cuvette with solution and with focused 514 nm light passing from left to right; right photograph is a zoomed version of the circled region..21 Figure 2.2: Phase change profile inside thermal medium for pump power and with,, 22 Figure 2.3: (a) Redrawn schematic showing focused pump incident on cuvette containing thermal medium, traveling nominally horizontally through the medium, (b) schematic of redesigned setup where the focused pump nominally travels vertically through the thermal medium, (c) schematic of setup to capture the holograms of bubbles using a 633 nm collimated He-Ne laser 27 Figure 2.4: Experimental far field diffraction patterns of pump and probe beams for different pump powers, along with simulation results for probe beam. Figures 2.4 (a) and (b) are far field patterns due to SPM for pump powers 200 mw and 300 mw, respectively. The width of the outermost ring in (a) is 15mm; (b) is shown to the same scale as in (a). Figures 4 (c) and (d) are far field patterns for the probe due for pump powers 200 mw and 300 mw, respectively. Figures 4 (e) and (f) are simulated far field patterns corresponding to (c) and (d), respectively 29 x

11 Figure 2.5: (a) Hologram using reference and probe captured with the 514 nm pump beam, (b) Intensity pattern without the pump beam, (c) hologram after subtracting (b) from (a) numerically. This is now used for numerical reconstruction of the induced phase...30 Figure 2.6: (a) Wrapped phase distribution after hologram reconstruction using Fresnel technique. The red box indicates the location of the induced refractive index. (b) A blow-up of the wrapped phase distribution inside the box in (a). (c),(d) are the unwrapped 2D and 3D plots of the induced phase distribution respectively. Unwrapped phase is approximately rad which is in a good agreement with the phase obtained by counting the rings..31 Figure 2.7: (a) Typical SPM far field ring patterns in the vertical container (cuvette), (b) SPM far field ring patterns with the new horizontal container. Radially symmetric rings are obtained due to the absence of gravity and buoyancy, (c) transition from SPM to TB, (d) ring patterns after the bubble forms (TB).32 Figure 2.8: (a) Recorded in-line hologram (around center of picture) of bubble just after it forms. Reconstructions of bubbles, using Fresnel technique, (b) immediately after formation, (c) after 60 s, (d) after 120 s. Sizes are monitored as a function of pump exposure time, starting from initial size of 231 μm, 752 μm after 60 s and 829 μm after 120 s.34 Figure 2.9: Schematic of microbubble steering arrangement. Moving the lens along z steers the bubble along x in the liquid within the modified container. Once formed, bubble can be steered with weaker laser beams.35 Figure 2.10: (a) Colloidal suspensions of Ag nanoparticles of increasing size from left to right. The third from the left, which is dark yellow in color, is the best batch, (b) TEM of nanoparticles obtained from the dark yellow batch.36 Figure 3.1: Experimental setup for the recording the hologram on LN:Fe crystal with two beams. Bragg and Non-Bragg orders are also shown as -1, +1 and -3, +3 respectively [8]. 514 nm collimated laser beams are splitted using a beam splitter in order to create equal intensity beams. At the process of recording, the object is replaced by a flat mirror 43 Figure 3.2: 9 non-bragg orders seen during self-diffraction in PR LN:Fe [8].44 xi

12 Figure 3.3: (a) Schematic of recording the hologram of a CD-ROM, before and after heating. Bragg orders are shown as -1 and +1 while the non-bragg orders (PC and PE) are shown as +3 and -3. (b) Lab setup. Note that the object and the reference are named S and R respectively [8] 45 Figure 3.4: (a) Fringes in diffracted order after heating CD-ROM with the blue focused laser, (b) cropped fringe region [8]..46 Figure 3.5: (a) Deformation details of the CD-ROM due to the heating, (b) comparison of the details; red curve-from a profiler, blue curve-from unwrapped data [8] 47 Figure 3.6: Illustration of the PR effect. (a) Excitation of the electrons to the conduction band with the photons, (b) Illuminated PR material with a sinusoidal coherent light, (c) generation of charges inside the PR material due to the illumination, (d) separation of charges and redistribution, (e) creation of a spacecharge field, (f) final refractive index modulation.48 Figure 3.7: (a) Using the object beam and the reference beams to record the hologram, (b) reconstruction with the same reference beam, (c) reconstruction with the deformed object, (d) total Bragg and non-bragg orders at the exit plane of the PR crystal [8]..50 Figure 3.8: (a) Intensity and (b) phase variation as a function of z, through the PR medium, as found through numerically solving Eqs. (3.15) and for initial intensity. The initial phase of the object is [67]...60 Figure 3.9: (a) Intensity and (b) phase variation as a function of z, through the PR medium, as found through numerically solving Eqs. (9) and for initial intensity. The initial phase of the object is [67]..62 Figure 3.10: (a) Spatial evolution of the different orders during reading of the grating written by reference and object wave with the parameters and initial conditions as in Figure 2.. (b) The intensities of the ±3 orders drawn to a different vertical scale to show their evolution with interaction distance [67]..63 xii

13 Figure 3.11: Intensities at the exit plane of the crystal ( ) as a function of phase, derived from results such as shown in Figure 4. The discrete points are obtained from results such as in Figure 4 for different values of the phase. The solid lines are obtained by curve-fitting to sinusoidal functions with appropriate phase shifts [67]..65 Figure 4.1: Variation of reflected intensity to initial intensity ratio ( ) with the incident angle for lithium niobate.. is chosen to be.78 Figure 4.2: Variation of reflected intensity to initial intensity ratio ( ) with squared of the incident angle, for lithium niobate.. is chosen to be...79 Figure 4.3: Holographic grating written on a lithium niobate crystal using a 514 nm Ar-Ion laser. (a) Grating is imaged with a lens on to a CCD using a 633 nm He-Ne laser, (b) portion of the image captured by the CCD and a zoomed version of the image (shown in a red box)..81 Figure 4.4: (a) Recording intensities of diffraction orders as a function of the incident angle. Grating area is illuminated with a He-Ne laser (633 nm) and the power of the each order is measured using a power meter. Rotational stage with a stepper motor is used and controlled by connected to a personal computer through a stepper motor controller. Figure (b) the top view of (a) 82 Figure 4.5: A photograph of the reflected diffraction orders recorded on a screen. Grating is read by a 633 nm laser beam (1 mm width) and the distance between the grating and the screen is approximately 45 cm. Intensities of each order are recorded using a power meter.83 Figure 4.6: Diffracted ( ) and undiffracted (0 th ) order intensity variations as a function of the incident angle when it is read by a 633 nm He-Ne laser. Grating is written in a LN:Fe crystal using a 514 nm Ar-ion laser. The grating has a Bragg angle of approximately 0.5 degrees and a period of approximately about.84 Figure 4.7: Fourier transforms of diffracted and undiffracted orders before and after removing high frequencies. (a) FFT of Figure 4.6(b) +1 order before filtering, (b) after filtering; (c) FFT of Figure 4.6(d) before filtering, (d) after filtering..85 xiii

14 Figure 4.8: Intensity patterns after removing the Fabry-Perot effect. (a) +1 diffraction order, (b) 0 th undiffracted order. Comparing the peaks and valleys of (a) and (b) it can be seen that there is approximately phase difference between the +1 and the 0 orders.86 Figure 4.9: Vector diagrams for the near-bragg diffraction. (a) Incident light is shown in the solid red line while the ±1 orders are marked with bold dashed lines. Angular spacing of the diffracted orders is, where is the Bragg angle. denotes the grating vector and determined by the recording wave (514 nm, showing in green). 633 nm reading wave is shown in red. (b) Same vector diagram as (a) when the incident angle is positive.88 xiv

15 LIST OF TABLES Table 2.1: Measured transmitted power of 405 nm laser for 5 trials each for (a) no bubble and no nanoparticles, (b) no bubble and with nanoparticles, (c) bubble and no nanoparticles, and (d) bubble with nanoparticles.38 Table 3.1: Unknown coefficients obtained from the curve fitting for Eqs. (3.22ad) for the case when for both writing and reading, and 65 xv

16 LIST OF ABBREVIATIONS 3D AFM AgNO 3 APT TM Ar-Ion BaTiO 3 BS CCD CD-ROM dc DH FFT GaAs He-Ne HI InP KNbO 3 LN/ LiNbO 3 Three-dimensional Atomic Force Microscopy Silver nitrate Stepper motor controller (trademark) Argon Ion Barium Titrate Beam Splitter Charge-Coupled Device Compact Disk-Read Only Memory Direct current Digital Holography Fast Fourier Transform Gallium Asanite Helium-Neon Holographic Interferometry Indium Phosphate Potassium Niobate Lithium Niobate xvi

17 NaBH 4 NaCl PC PE PR PSDH PTP PUMA PVA PVP RHIM RHIN RI SPM TB TE TEM TM Sodium borohydride Sodium chloride Phase conjugate Phase enhanced Photorefractive Phase Shifting Digital Holography Photothermoplastics Phase Unwrapping Max-Flow/Min-Cut Polyvinyl alcohol Polyvinyl pyrolidone Real time Holographic Imaging Real time Holographic interfermetry Refractive Index Self-phase Modulation Thermal Blooming Transverse Electric Transmission Electron Microscopy Transverse Magnetic xvii

18 LIST OF SYMBOLS Gaussian beam profile-pump beam Gaussian beam profile-probe beam Radius of curvature-pump beam Radius of curvature-probe beam Complex field, +1 Bragg order Complex field, -1 Bragg order Complex field, +3 non-bragg order Complex field, -3 non-bragg order i th component, +1 Bragg order /angular spectrum i th component, +3 Bragg order /angular spectrum Field, +1 Bragg order/wave Field, -1 Bragg order/wave Field, i th component, -1 Bragg order /angular spectrum Field, +1 Bragg order, reading/wave Field, -1 Bragg order, reading/wave xviii

19 Field, +1 Bragg order, writing/wave Field, -1 Bragg order, reading/wave Field, +3 non-bragg order/wave Field, -3 non-bragg order/wave Field, i th component, +3 non-bragg order /angular spectrum Field, -3 non Bragg order, writing/wave Field, Reference field/wave Field after heating nonlinear sample Field before heating nonlinear sample Field-probe Space-charge field in PR crystal Initial intensity Intensity, +1Bragg order, writing/wave Intensity, -1Bragg order, writing/wave Intensity, +3non-Bragg order, writing/wave Intensity, -1non-Bragg order, writing/wave Intensity, pump beam Intensity, probe beam Total intensity, Bragg and non-bragg orders Intensity, after heating nonlinear sample Intensity, before heating nonlinear sample xix

20 Dark intensity of PR material Incident intensity Reflected Intensity Effective length, pump Effective length, probe Acceptor concentrations Donor concentrations Pump power Probe power Power without bubble and nanoparticles Power without bubbles, with nanoparticles Power with bubble, without nanoparticles Power with bubbles and nanoparticles Object wave amplitude Reference wave amplitude Coefficient of photovoltaic Coefficient of diffusive Temperature coefficient Free space wave vector Wave vector, pump Wave vector, probe Spatial frequency along axis xx

21 Spatial frequency along axis Natural logarithm Linear part of the refractive index Beam waist (radius), pump Beam waist (radius), probe Absorption coefficient, pump Absorption coefficient, probe Half of the recording angle Incident angle Transmitted angle Phase, +1 Bragg order Phase, -1 Bragg order Phase, +3 non-bragg order Phase, -3 non-bragg order Bragg angle Phase, object wave Phase, reference wave Complex conjugate Pixels sizes in the recording plane, Pixels sizes in the recording plane, Phase change due to pump beam Phase change due to probe beam xxi

22 Path difference, reference and probe Change in the temperature Gradient operator Inverse tangent Hologram function O R Object wave Reference wave Change in the intensity Grating vector step size Change in the refractive index Pixels sizes in the reconstruction plane, Pixels sizes in the reconstruction plane, Change in the phase Grating period Fringe periods in x and y Electric field-complex Reconstruction field initial state Reconstruction field final state Exponential integral The coefficient of finesse Fast Fourier transform Intensity Bessel function of the 1 st kind xxii

23 Magnitude of the grating vector Thickness of the material (LN crystal) Length of array, Heat generated per unit volume Radial distance at Reconstruction distance Corrugation depth Spatial derivative w.r.t, Change in time Step size along Step size along axis, recording plane axis, recording plane Exponential operator Imaginary number, Thermal conductivity Discretized Discretized Discretized Discretized Refractive index Elasto-optic coefficient Electron charge, C Amplitude reflection coefficient xxiii

24 Radial coordinates Time coordinates, reconstruction plane coordinates, recording plane coordinates, reconstruction plane coordinates, recording plane coordinates Grating vector Absorption coefficient Added reference phase Amplitude ratio of the reference/probe Thermal generation rate Euler-Mascheroni constant Recombination constant Free-space permittivity Diffraction efficiency Angle between recording beams Diffusivity Wavelength The number of, approximately Partial derivative Phase xxiv

25 CHAPTER 1 INTRODUCTION AND OBJECTIVES 1.1 Introduction When a material reacts nonlinearly to an optical field, it is usually called a nonlinear optical material. Nonlinear and linear properties of a material can be explained by studying how the atomic structure of that particular material behaves with light. The refractive index (RI) of a material usually depends on its atomic structure. When a material is exposed to light, there is competition between the electric field from the incident light and the intra-atomic fields inside the material. When the electric field from the optical wave is strong enough to perturb the intra-atomic field, the electric field distribution in the material is changed and that causes a polarization change in the material, sometimes leading to a RI change. Sometimes, the induced polarization is a nonlinear function of the electric field. If this dependence is cubic, the RI depends on the incident beam intensity [1]. This is referred to as the nonlinear Kerr effect. Self-phase modulation (SPM) is one of the nonlinear optical effects that the RI change is produced by the optical Kerr effect [2]. The RI change which is created by the Kerr effect is proportional to the magnitude squared of the electric field or in other words, to the intensity of the illuminated beam [3]. When an intense laser beam is focused in to a Kerr medium, a phase is induced and that leads to self-focusing or self-defocusing. Some of 1

26 the applications of SPM are spectral broadening [4] and temporal and spectral pulse compression [5,6]. In some other materials, the RI change is also proportional to the intensity, but happens in a different way. In thermal media, the absorbed optical power heats the material causing a temperature change, and thereby a change in the RI. The basic theory of SPM in thermal media and its consequences are described in Chapter 2. Thermal blooming (TB), which occurs beyond SPM, is also investigated. The ensuing bubble is characterized using digital holography (DH) and is used for nanoparticle agglomeration. In some electro-optic (EO) materials with donor impurities, incident light can cause the donors to be ionized and create electrons in the conduction band. The free electrons redistribute through the material due to diffusion or other effects, and can trapped by ionized impurities or other intrinsic traps. The ensuing charge redistribution causes an induced space-charge field which in turn causes a RI change due to the linear electro-optic (or Pockels) effect [2]. This phenomenon of RI change is called the photorefractive (PR) effect and those materials which show this effect are called PR materials. The PR effect was discovered in 1966 and observed in electro-optic crystals [1]. This effect can be seen in PR crystals such as LiNbO 3 (LN), BaTiO 3, KNbO 3, GaAs, InP, etc. PR materials are well known for dynamic holographic recording [2], which is a part of this study. Since PR materials are capable of recording holograms [7], they can be used to record temporary gratings. Recording the Bragg and the non-bragg orders in PR materials can be used for phase shifting holographic interferometry (HI) to measure small 2

27 deformations of the order of a wavelength [8,9]. HI is briefly discussed later in this Chapter and in more detail in Chapter 3. It has been also shown that when a holographic phase grating is written inside a PR material, a surface corrugation occurs due to the converse piezoelectric effect [10]. A study of these surface gratings is described in detail in Chapter 4. Such surface corrugations can be employed in director axis modulation of liquid crystals (LCs) in hybrid PR-LC structures, leading to potentially high two-beam coupling coefficient during two-beam coupling [11]. 1.2 Research Objectives The objective of this work is to study various optical effects in materials due to their induced change in RI. Specifically, optically induced nonlinearities are studied in liquids and in PR materials. Potential applications of the latter in dynamic DH are also investigated. For instance, in Chapter 2, the objective is to investigate SPM and thermal blooming due to thermal nonlinearity by analyzing the far-field pattern, as well as by using DH and holographic interferometry (HI) using a probe beam. A novel way of clustering nanoparticles using bubbles resulting from TB is also proposed. In Chapter 3, it is shown that Bragg and non-bragg orders can be generated during two-beam coupling (TBC) in a PR material due to the induced RI in the material. These orders can be used for a unique technique of determining the object amplitude and phase profile, as is commonly done using phase-shifting HI, or determining object 3

28 deformation. A numerical analysis is also proposed for modeling these non-bragg orders. A consequence of the PR effect is deformation of the material due to the converse piezoelectric effect. This leads to surface deformation on the PR material, which can be considered as a surface grating with grating period identical to that of the phase grating written inside the PR material. The corrugation depth is in the order of nanometers and has been detected using various techniques such as AFM [12]. In Chapter 4, the existence of the surface grating is proven by experimentally monitoring the diffraction orders of a probe beam reflected from the surface of a PR material in which a grating has been previously recorded, and thereafter using a novel numerical filtering technique. Since DH and HI are some of the tools that will be used to pursue the experimental and theoretical work throughout much of this dissertation, a brief introduction to holography, DH, HI and phase shifting DH is given below. A brief introduction to PR materials is provided in Chapter Holography When a regular photograph is recorded on a photographic film or using a digital camera, the recorded image only has the intensity information in it. However, in 1948 Gabor invented a method of recording both intensity and phase of an object and it is known as holography which originates from two Greek words holos (entire) and graphein (write) [13]. In holography, the recorded interference pattern has the information of intensity as well as phase which can describe the depth of the image. These interference patterns are created from the scattered light from the object wave and 4

29 another coherent beam of light which is usually called as the reference wave. Once this information is recorded on a photographic plate, a three dimensional (3D) image can be recreated by using the same reference beam which is used to create the interference pattern or by using another plane wave. In Gabor s setup the recorded hologram is illuminated by a plane wave such a way that the object and the reference waves travel parallel to each other [13]. In this method, the real and virtual images and the undiffracted portion of the reconstruction wave are on top of each other. This is one drawback of the Gabor s setup and the method is well known as in-line holography. This was improved later by Leith and Upatnieks who suggested that the real and virtual images could be spatially separated by using an off-axis reference wave [14,15] Digital Holography Before DH, hologram recording has been done using photographic plates and films. This process is time-consuming when it comes to the development of the holograms and faces many more obstacles such as dealing with chemicals. Along with the development of the computers, holography also has turned its way to a digital world which leads to digital holography. Numerical reconstruction of holograms was first implemented by Goodman and Lawrence [16] and by Yaroslavskii et al. [13]. They digitized a photographic recorded hologram and reconstructed it numerically. Later this reconstruction algorithm was improved and it was used for the particle measurements by Onural and Scott [17,18,19]. The use of charged couple devices (CCD) for recording Fresnel holograms has become a very common, fast and effective way in DH and this was developed by Schnars 5

30 and Juptner [20,21]. This does not require an intermediate recording step and the hologram can be recorded directly on the CCD and reconstructed numerically Holographic Interferometry HI is mainly used to measure the micro scale deformations in objects, and was developed by Stetson and Powel [22,23] and others in the late sixties. When an object is deformed, the optical path changes and by recording the optical path difference, the amount of deformation can be measured using HI. This method is highly accurate such that it can resolve 1/100 of a wavelength of the optical path [13]. HI can be used not only to measure the deformations of solid opaque and transparent objects, but also to measure the other properties in liquids and solids such as the change in the refractive indices. An experimental setup for recording and reconstructing a double exposure hologram is shown in Figure 1.1 and an interferogram recorded on a CCD camera while heating a liquid is shown in Figure

31 (a) Figure 1.1: (a) recording a double exposure hologram, (b) reconstructing the hologram (b) Two wavefronts scattered by the same object are recorded on the same recording device in double exposure HI [24]. The initial state is recorded with the original object and the final state is recorded with the deformed object. Reconstructed optical fields of the initial and the final states, and can be written as [13,24], (1.1), (1.2) 7

32 where is the real amplitude and is the phase distribution. is the phase change after the object is deformed which is usually called as the interference phase difference [24]. Thus the interference intensity on the CCD due to and becomes, (1.3) which reduces to, (1.4) where and. Note that Eqs. (1.3) and (1.4) assume identical amplitudes for the fields. If the two amplitudes are different, it is easy to show that the reconstructed interference intensity can be written as. (1.5) Equations (1.4) and (1.5) above have the information of the phase difference due to deformation as well as the amplitude/intensity information of the interference pattern. The intensity pattern is a cosine function with bright and dark fringes where the bright fringes correspond to a phase difference of even integer multiples of while dark fringes correspond to he multiples of odd integers of, which means that between each set of dark and bright fringes there is a phase jump. The exact variation of the deformation 8

33 phase can be used to measure parameters quantitatively such as the nonlinearly induced change of the RI, the deformation of an object, etc. [25]. Figure 1.2: Interferogram recorded on a CCD camera while heating a liquid Phase-shifting DH Determining the exact phase of an object which has its 3d information using DH requires numerical propagation of the hologram function. This may be often complicated because of the following reasons. Sometimes, as in the case of an off-axis hologram, the angle between the object wave and the reference wave causes additional fringes because of what is called the carrier spatial frequency. In-line holograms, on the other hand, gives superposed real and virtual images (dc term), and filtering is often required to separate these two images. A way to find the Fresnel propagated complex object wave on the plane of the recording medium (and thereby to reconstruct the complex object) without the above complications is to record the intensity patterns by changing the phase of the reference wave by a predetermined quantity; thus it is named phase-shifting DH 9

34 [13, 26-28]. A novel single shot phase- shifting HI scheme using dynamic holography in a PR material is applied to investigate the 3d shape of objects [25] and more details can be found in Chapter 3. In phase-shifting DH, a minimum of three holograms (usually four) should be recorded with known phases. Phase shifts can be obtained by using a piezoelectric translator by attaching a mirror and placing it either in the reference or in the object. Since the piezoelectric translator is very sensitive, very small path differences (fractions of a wavelength) can be introduced. By recording four intensities, one can calculate the phase of the object on the recording plane as shown below. The recorded hologram can be expressed canonically as, (1.6) where and, with now denoting the intensity of the reference and the object. Then after adding a phase of to the reference at each time, the four intensities can be written as, (1.7), (1.8). (1.9) 10

35 It can be shown that can be calculated as. (1.10) Equation (1.10) is a generalized equation. More details can be seen in Chapter 3. It should be noted that there are other techniques that can be used to obtain the phase from an interference fringe pattern such as Fourier transform method and heterodyne technique [13] but the phase shifting HI is the most popular one amongst them. 1.4 Essential Theory for DH and HI All the theoretical simulations for numerical propagation of optical fields including reconstruction of digital holograms are done by using MATLAB software in this work. The Fraunhofer approximation of the Fresnel-Kirchhoff integral is used to calculate the far field intensities in the case of SPM and optical correlation. The Fresnel- Kirchhoff integral can be written as follows [29]. (1.11) Equation (1.11) can be compared to the Fraunhofer approximation (far-field) of the Fresnel diffraction integral which is expressed as 11. (1.12)

36 In other words Equations (1.11, 1.12) are just the Fourier transforms of the quantities in second brackets evaluated at the spatial frequencies, (1.13) where is the free space wavenumber and and z are the wavelength and the propagation distance, respectively. represents the coordinates of the reconstruction plane while represents coordinates of the recording plane viz., CCD. Simulations are done using the Fast Fourier Transform (FFT) algorithm [13]. All the numerical reconstructions in this work are performed using the discrete Fresnel transform as described below [13].. (1.14) In Eq. (1.14), is the distance between the object and the recording plane (CCD). This is usually also the reconstruction distance. represents the hologram function (digital hologram recorded on the CCD),, and, are the discretized (Eq. (1.12)) and planes, respectively. and are the pixel sizes of the CCD. After 12

37 propagation, the pixel sizes of the recording plane should be rescaled. The new pixel sizes at the reconstruction plane can be written as [13]. In Eq. (1.14), last exponential term represents the Fourier transform. Also, since holograms are usually captured and reconstructed using plane waves as the reference and reading beam, the reference wave can be considered as unity. Moreover, for most cases, the exponential phase terms in front of the equation can be neglected when calculating the intensity which is the absolute value squared of the field. Thus Eq. (1.14) can be simplified to, (1.15) and rescaled according to the new pixel sizes mentioned above. Equation (1.15) is used to reconstruction the holograms throughout this work. Other relevant techniques, methods and experimental setups are introduced in subsequent Chapters as required. 1.5 Conclusion In this work, DH and HI are used to investigate induced gratings in nonlinear media such as liquids with high absorption coefficients and PR materials. In Chapter 2, measurements of the induced RI changes in a thermal medium are reported, which uses SPM results and, additionally, a probe beam. Thermal blooming, which occurs beyond SPM, is also characterized, and the resulting bubbles are used for clustering nanoparticles. In Chapter 3, induced gratings and resulting RI changes resulting in Bragg and non-bragg orders in a PR material are studied and used as a novel way of 13

38 implementing phase shifting DH and applied to the investigation of deformations of objects and shown in Chapter 3. In Chapter 4, a novel way to detect the surface corrugation on a PR material in which a grating has been stored is described. A summary of future work is given in Chapter Publications Resulting from this Work A number of conference publications and one journal paper have resulted from the work thus far. These are listed below: 1. P. Banerjee, G. Nehmetallah, U Abeywickrema, S. Lyuksyutov, and N. Kukhtarev, Non-Bragg diffraction orders in holographic recording and its application to one-shot phase-shifting holographic interferometry, Proc. SPIE 8644, (2013). 2. P. Banerjee and U. Abeywickrema, A simple optical probing technique for nonlinearly induced refractive index, Proc. SPIE 8847, 1-6 (2013). 3. U. Abeywickrema and N. Banerjee, Characterization and application of bubbles during thermal blooming, Proc. SPIE 9194, 1-7 (2014). 4. U. Abeywickrema and P. Banerjee, Phase-shifting holography using Bragg and non-bragg orders in photorefractive lithium niobate, Proc. SPIE 9200, 1-8 (2014). 5. U. Abeywickrema, P. Banerjee, and N. Banerjee, Holographic Assessment of Self-Phase Modulation and Blooming in a Thermal Medium, Appl. Opt. 54, (2015). 14

39 CHAPTER 2 HOLOGRAPHIC ASSESSMENT OF SELF-PHASE MODULATION AND BLOOMING IN A THERMAL MEDIUM 2.1 Introduction SPM, a nonlinear effect that can be observed by focusing a high power laser beam onto a nonlinear sample, has been extensively studied for over four decades [1,30-32]. The RI of the material is modified due to the nonlinearly induced intensity-dependent phase change caused by a focused laser beam through the optical Kerr effect and/or due to temperature change in the material due to thermal effects [1,31]. This nonlinearly induced phase change, and hence the RI change, can be roughly estimated by observing the far field diffraction ring patterns due to SPM. The effect of SPM has been also studied with a pump-probe interaction in a highly thermal absorbing medium [32]. It has been shown that a nonlinear sample made of a mixture of ethanol and chlorophyll with a pump beam of 100 mw can generate deflection angles on the order of 0.1 degrees of a probe beam traveling obliquely through the induced RI profile [32]. SPM can be also used to determine thermo-optic parameters such as the temperature coefficient ( ) of the refractive index change, and the non-radiative quantum efficiency, viz., in the case of CdSe/ZnS core shell nanoparticles and ink solutions [33]. Measurement of absorption coefficients of liquids has been performed using thermally generated SPM and holographic interferometry to detect the change in the RI [34]. Also the effects of gravity and convection in SPM far field diffraction patterns have been investigated experimentally and modeled 15

40 theoretically using Fresnel-Kirchhoff diffraction theory [35]. Spectral broadening [36] and temporal and spectral pulse compression [37,38] are some of the other effects and applications of SPM. When a thermal medium is heated beyond SPM with a focused laser beam (viz., by increasing the power of the pump), the far-field rings characteristic of SPM suddenly change to a completely different diffraction pattern. It has been shown that these new diffraction patterns are generated due to bubble formation and trapping inside the thermal lens, which is created by the focused high power laser beam [39,40]. This phenomenon is usually called TB and can be considered as the next step beyond SPM. When the thermal medium is heated up to a critical temperature, the solution starts boiling, thus forming bubbles inside the thermal lens. Now the pump beam encounters the bubble which is trapped inside the thermal lens. The trajectory of the rays of the pump beam is changed inside the thermal lens because of the trapped bubble, thus changing its diffraction pattern in the far field. The behavior of the bubble inside the thermal lens has been explained using geometrical optics [40]. There are several forces that keep the bubble stable inside the thermal lens. The thermal gradient of the lens helps to generate the surface tension gradient. Since the surface tension gradient is not uniform, the bubble tends to move to the maximum temperature point. If the focused laser beam is blocked, the bubble is free to move because of the sudden absence of the temperature gradient. In some cases, where maximum transmission of the beam through the induced refractive index is desired, compensation for thermal blooming has been achieved using adaptive optics [41]. In this Chapter, the use of a probe beam at 633 nm to characterize the thermally generated RI change leading to SPM of the pump, and subsequent bubble formation which causes TB, is discussed. The pump is a focused beam at 514 nm which is incident on a liquid, 16

41 viz., a solution of red dye in a 91% isopropyl alcohol solution, contained in a quartz cuvette. The probe beam counter-propagates with respect to the pump beam in our experiments. In the regime of SPM, inferences about the induced RI (or phase) obtained from the far-field diffraction pattern for the probe beam is compared with the far-field pattern of the pump beam. The results are in agreement with results from digital holographic recording and reconstruction using a 633 nm probe and reference. In order to eliminate effects of gravity, buoyancy and convection, a modified container for the solution is used, along with a revised optical setup, which allows for the pump and probe beams to travel vertically through the sample. This also aids in precisely observing, monitoring, and manipulating the bubble created during TB. The probe beam is now used to record in-line digital holograms of the bubble in order to determine its size. It is shown that the bubble can be steered using a focused low power beam. A potential application of bubbles created through TB, viz., nanoparticle agglomeration around the bubble, is discussed. The essential theory of thermal lensing and induced RI or phase due to SPM of the focused pump beam is first summarized, along with the induced phase change of the contrapropagating unfocused probe beam. The basics of the interferometric scheme involving the probe and the reference are also discussed. In experimental setup section, the experimental setups are explained, which includes the original setup as well as a revised setup which eliminates the effect(s) of gravity, buoyancy and convection, essential for observation of radially symmetric far-field diffraction patterns for the pump and probe during SPM, as well as for creating stable bubbles during TB. Moreover, experimental results of induced phase assessment using the probe beam is discussed, along with comparison with the inferences from the far-field diffraction pattern of the pump due to SPM. The induced phase results compare favorably with the interferometric experiment using a 633 nm probe and reference similar to that described in Ref. 17

42 [34], as well as a variation of the experiment which enables an accurate digital holographic reconstruction of the induced phase profile. Stable bubble formation due to TB is then observed using a modified container that eliminates the effects of gravity and buoyancy, and characterized using an in-line digital holographic setup using the probe. It is shown that bubble size increases with the exposure time of the pump beam. The steering of bubbles is achieved using a focused low power laser beam. Finally, the interaction between the nanoparticles and the thermally generated bubble are investigated qualitatively and the preliminary results are shown. The Chapter is concluded with a discussion of a potential application of the thermally generated bubbles and other future work. 2.2 Essential Theory of Thermal Lensing and its Measurement In this Section, the analytical results for the RI change in a thermal medium are first summarized. The change in the temperature from ambient conditions can be found using the heat equation [30], (2.1) where is the diffusivity, is the thermal conductivity, denotes time, denotes the radial coordinate, and is the (transverse) Laplacian. The heat per unit volume generated by the (pump) laser can be expressed as [30], (2.2) 18

43 where is the absorption coefficient for the pump beam in the sample, is the pump intensity, is the power of the pump laser, and is the (waist) radius of the pump beam amplitude within the sample. Using Eq. (2.2), the solution to the steady state version of Eq. (2.1) ( ) is obtained analytically as [30,42], (2.3) where represents the exponential integral [43], and where denotes the radial distance where [30]. Incidentally, at, Eq. (2.3) can be approximated as, (2.4) where is the Euler-Mascheroni constant [44]. The temperature change and the RI change of the sample are related through the following well known relation [30]. (2.5) In Eq. (2.5), is the RI profile and is the ambient part of the RI. As stated above, it is assumed in this work that the pump (with propagation constant ) propagates along +z and the probe (with propagation constant ) propagates contradirectional to 19

44 the pump, as shown schematically in Figure 2.1. If the longitudinal physical thickness of the sample (along z) is, the induced phase of the (focused) pump can be expressed as [45],, (2.6) and where it is assumed for simplicity that the Rayleigh range of the focused Gaussian pump beam is larger than the sample thickness, so that the material can be considered a thin sample. Similarly, the phase imparted on the (unfocused) probe beam is,, (2.7) with denoting the absorption coefficient for the probe. A typical plot for the phase change using pump ( ) power,,,,, is shown in Figure 2.2. The value of has been chosen to correspond to the number of rings from SPM of the pump in the far field, which will be discussed later. Also, the chosen value for represents the average radius of the pump beam amplitude within the sample. This has been determined by first estimating the beam waist from calculating the focal spot size (approximately ) within the sample, and cross-checked with measurement of the beam width at a distance much larger than the Rayleigh range corresponding to the focal spot size. Incidentally, the Rayleigh range is of the order of the sample thickness. This average radius is found from the variation of the width of the pump beam within the sample. 20

45 The magnitudes of the peak values for the temperature, refractive index, and phase changes are,, and rad, respectively. Figure 2.1: Experimental setup for observing SPM due to thermal lensing using focused (f = 200 mm) 514 nm Ar ion pump (continuous line, green, color online) traveling through nonlinear sample from left to right. BS1-4 represent beam splitters with splitting ratio of 50%. Far field patterns are recorded by camera denoted as CCD 1. A He-Ne probe beam at 633 nm (dashed line, red, color online) passes through the nonlinear sample contrapropagating to the pump beam. Far field patterns due to the probe beam are recorded by camera denoted as CCD 2. The 633 nm beam can be split in order to form a Mach-Zehnder interferometer. Interference between the 633 nm reference (dashed line, red, color online) not passing through the sample, and probe can also be recorded by CCD 2. A filter (not shown in figure) is used before CCD 2 to block any stray light from the 514 nm pump beam. Left photograph of nonlinear sample shows cuvette with solution and with focused 514 nm light passing from left to right; right photograph is a zoomed version of the circled region. 21

46 Phase change (rad) r (m) x 10-4 Figure 2.2: Phase change profile inside thermal medium for pump power,,. and with After knowing the phase change due to each beam, the far field diffraction patterns can be determined by obtaining the scaled two dimensional Fourier transforms of the product of the relevant Gaussian beams and the complex exponentials containing the phases. For example, SPM ring patterns for the pump beam in the far field can be determined by taking the scaled two dimensional Fourier transform of, where represents the Gaussian beam profile of the pump beam and is given by Eq. (2.6). The far field diffraction pattern for the probe beam passing through the RI change created by the pump can similarly be determined by taking the scaled two dimensional Fourier transform of, where represents the Gaussian beam profile of the probe beam and is given by Eq. (2.7). In both cases of the pump and the probe, an exponential factor of the form can be introduced in the amplitude to account for losses during the passage of the beams through the sample. Furthermore, the form of the Gaussian for the pump and the probe can be taken as 22

47 , (2.8) where is the power of the laser beam, is the width and is the radius of the curvature, with the subscripts P, p denoting the pump and the probe, respectively. As stated earlier, the far field diffraction pattern of the probe beam, for instance, is a scaled Fourier transform of the exit probe field from the sample and can be explicitly written as, (2.9) where and are the coordinates at the detector plane, and is the distance between the exit plane of the sample and the detector. Preliminary results of the diffraction patterns for the pump and probe beams have been reported by our group earlier [42]. As will be shown below, the results for the induced phase change obtained from the probe beam diffraction patterns are in agreement with that inferred from the diffraction patterns of the pump beam. For further checking our results for the induced phase change, a Mach-Zehnder interferometric scheme has been set up, as shown in Figure 2.1. The probe passing through the sample is interfered with a reference and the ensuing interference pattern is recorded, similar to the work of Clark and Kim [34]. The interfering optical fields before (b) and after (a) the pump beam is introduced through the sample can be written as, (2.10a), (2.10b) 23

48 where is the amplitude ratio of the reference to the probe, and denotes the path difference between the reference and the probe. Using Equations (2.10a) and (2.10b), the intensity difference at the recording plane (CCD 2, see Figure 2.1) before and after heating by the pump can be expressed as. (2.11) It can be seen from Eq. (2.11) that the information about the phase can be obtained by recording the interference patterns before and after the pump is applied to the sample. Note, from Eq. (2.11), the difference in intensities should give a trigonometric function with in its argument, so that by counting the fringes in, it is again possible to estimate the peak phase change induced by the thermal lens. 2.3 Experimental Setups As mentioned earlier, a Ar ion laser is used as the pump and a He-Ne laser is used as the probe, and also as the source for the Mach-Zehnder interferometer, as shown in Figure 2.1. The plano-convex lens focuses the pump to a spot size of approximately at the center of the 1cm x1cm x5 cm quartz cuvette with four transparent windows containing the sample. Although initial experiments have been performed using tea solution in water, a solution of red dye in 91% isopropyl alcohol has been used for most of our experiments since it has a lower boiling point, and is thus more favorable for bubble formation. It also turns out that for the initial experimental arrangement (shown in Figure 2.1 and schematically again in 24

49 Figure 2.3(a)), the effects of buoyancy, gravity and convection give rise to radially asymmetric far field patterns for the pump and the probe. To mitigate this, a redesigned setup has been used, where the sample is confined in a modified container and the pump travels nominally vertically through the liquid sample, as shown in Figure 2.3(b). The container is made from four 3 x1 glass slides. The glass slides are approximately 1 mm thick. Two 3 x1 glass slides are placed side by side and separated by approximately ½. The third 3 x1 slide is now placed symmetrically on top of these two slides and glued with silicone caulk. The fourth and remaining slide is similarly placed and glued at the bottom of the two slides directly below the third slide. The liquid is introduced into the 1 mm thick empty space contained between the 4 slides using a dropper, and the two open ends are then sealed using the same silicone caulk. Care is taken to ensure that there are no trapped air bubbles during the filling process. This new design also contains considerably less volume of the liquid compared to the cuvette, thus the TB effect can be observed within a short time and with considerably less pump power. For holographically recording the bubble which causes TB, the pump beam is applied at least till the onset of TB; thereafter, a collimated probe beam (of beam width much larger than the bubble size) is introduced to record an in-line hologram of the bubble, as shown in Figure 2.3(c). 2.4 Experimental Results Probe Beam Measurement of Thermally Induced Phase Change Far-field ring patterns for the pump and probe beams are monitored at a distance of approximately from the sample. For convenience of recording, these far-field patterns are imaged using a 200 mm focal length lens onto CCD-1 and CCD-2, respectively. Figures 2.4(a), 25

50 (b) show the ring patterns due to SPM of the pump and Figures 2.4(c),(d) show the ring patterns for the probe for pump powers equal to 200 mw and 300 mw, respectively. For convenience of comparison with numerical results, a typical dimension of the far-field patterns for the pump and the probe beams are indicated on Figure 2.4(a) and (c), respectively. Note that the number of bright rings in Figure 2.4(a) corresponds to a peak phase change of approximately rad, which is in close agreement with the theoretical prediction in Figure 2.2. The power loss due to refractive index change from air to the thermal medium is negligible, typically around 5%. 26

51 (a) (b) (c) Figure 2.3: (a) Redrawn schematic showing focused pump incident on cuvette containing thermal medium, traveling nominally horizontally through the medium, (b) schematic of redesigned setup where the focused pump nominally travels vertically through the thermal medium, (c) schematic of setup to capture the holograms of bubbles using a 633 nm collimated He-Ne laser. 27

52 The numerical results for the far field diffraction patterns of the probe are shown in Figures 2.4(e) and (f), respectively. This has been computed using the theory in Section 2.2 and with the appropriate pump powers, along with,, initial width. The probe beam (radius ) is chosen to be much wider than the pump beam (radius ) to make sure that it covers the entire region of the phase change (, see Figure 2.2) due to the pump. It is also assumed that the distance traveled by the probe from the laser to the sample is approximately and the distance from the sample to the far-field is approximately (as mentioned earlier). The numerical results for the far-field pattern of the probe are in agreement the experimental observations. For instance, both experiment and simulation show a central dark region enclosed with a set of bright rings followed by a set of fainter rings. It is conjectured that this occurs from the fact that the probe beam, which is much wider than the pump, sees an induced phase (or RI) profile as shown in Figure 2.2. The pump beam, on the other hand, being narrower (focal spot size ) predominantly sees an induced RI profile which is more parabolic in shape. It has been checked that the values of the induced phase, obtained using the diffraction patterns of the pump due to SPM and the diffraction of the probe, are indeed in agreement with results from the interferometric technique [34] described in the previous Section. Additionally, as a variation of the interferometric experiment above, an alternate and simple method is used to determine the induced phase profile. First, the interference patterns due to the reference and probe are recorded as shown in Figure 2.5(a) for the case when a 200 mw 514 nm pump is incident on the thermal medium. Thereafter, the intensity pattern is recorded without the pump beam (see Figure 2.5(b)). The two intensity patterns are then subtracted to remove any 28

53 extraneous interference patterns (e.g., ones that may arise due to the glass cover of the CCD array, the presence of the liquid in the cuvette); the resulting intensity pattern is shown in Figure 2.5(c). Figure 2.4: Experimental far field diffraction patterns of pump and probe beams for different pump powers, along with simulation results for probe beam. Figures 2.4(a) and (b) are far field patterns for the pump due to SPM for pump powers 200 mw and 300 mw, respectively. The width of the outermost ring in (a) is 15mm; (b) is shown to the same scale as in (a). Figures 2.4 (c) and (d) are far field patterns for the probe due for pump powers 200 mw and 300 mw, respectively. The width of the central dark region in (c) is 4 mm; (d) is shown to the same scale as in (c). Figures 2.4(e) and (f) are simulated far field patterns corresponding to (c) and (d), respectively. 29

54 Pixels Pixels Pixels Pixels Pixels Pixels (a) (b) (c) Figure 2.5: (a) Hologram using reference and probe captured after application of the 514 nm 200 mw pump beam, (b) Intensity pattern of the interferometer before heating the sample using the setup shown in Figure 2.1, (c) hologram after subtracting (b) from (a) numerically using MATLAB. This is now used for numerical reconstruction of the induced phase. Pixel size of the camera is 6.7 μm. The two-dimensional wrapped phase distribution, obtained by using Fresnel reconstruction of the processed hologram in Figure 2.5(c), is shown in Figure 2.6(a). The reconstruction distance is approximately 44 cm in this case. The red box in Figure 2.6(a) indicates the cropped region within which the fringes due to the change in refractive index are primarily located; these are shown in more detail in Figure 2.6(b). Unwrapping of the phase is achieved using the PUMA algorithm developed by Bioucas et al. [46]. The two-dimensional unwrapped phase profile is shown in Figure 2.6(c) and the corresponding 3D phase map is shown in Figure 2.6(d). Note that when unwrapping, PUMA finds the phase jumps in the wrapped phase and unwraps that phase accordingly. Thus, if the phase jumps are not clearly identified by PUMA, the unwrapping process does not occur accurately. Also, as evident from Fig. 2.6(d), PUMA has unwrapped some unwanted phase at the corners of the phase map, which do not arise from the heating and should be disregarded. It is worthwhile to note that the remaining linear fringes in Figure 2.5(c) likely arise from the asymmetry in the thermal lens due to gravity; this should not be present in our modified setup and will be pursued in the future. The computed phase difference according to Figure 2.6(c) and (d) is approximately rad, which is in good 30

55 y (m) y (m) agreement with the value of rad based on the number of rings in Figure 2.4(a), as well as the approximate number of rings in Figure 2.6(a). x x (m) x 10-3 (a) (b) x x (m) x x y (m) x (m) 2 x 10-3 (c) (d) Figure 2.6: (a) Wrapped phase distribution after hologram reconstruction using Fresnel technique. The red box (color online) indicates the location of the induced refractive index. (b) A blow-up of the wrapped phase distribution inside the box in (a). (c),(d) are the unwrapped 2D and 3D plots of the induced phase distribution respectively, using PUMA algorithm [46]. Unwrapped phase is approximately rad which is in a good agreement with the phase obtained by counting the rings ( rad) Probe Beam Measurement of TB and Bubble Size As stated earlier, SPM due to thermal lensing evolves into TB as the pump power is increased. The liquid is confined in the modified setup as shown in Figure 2.3(b) to observe the 31

56 effect of TB. Figures 2.7(a), (b) show the recorded far-field diffraction patterns for the pump undergoing SPM due to thermal lensing, corresponding to the setups in Figures 2.3(a), (b), respectively. As discussed earlier, the latter setup yields radially symmetric ring patterns, as expected. Figures 2.7(c), (d) show the onset of TB due to bubble formation and using the experimental setup drawn schematically in Figure 2.3(b). Care is taken to keep the container level and horizontal to keep the bubble stationary inside the container. It is observed that the bubble remains stable even after removing the pump beam, thus the probe beam can now be introduced in the same direction as the pump beam (see Figure 2.3(c)) for bubble characterization. The bubble size depends on the time the pump beam is on, as evidenced by the holographic reconstructions of the bubbles, described below. (a) (b) (c) (d) Figure 2.7: (a) Typical SPM far field ring patterns in the vertical container (cuvette), (b) SPM far field ring patterns with the new horizontal container. Radially symmetric rings are obtained due to the absence of gravity and buoyancy, (c) transition from SPM to TB, (d) ring patterns after the bubble forms (TB). The probe beam is now used to record inline holograms of the bubbles, and these are recorded on a CCD camera. A typical hologram of the bubble, just after its formation, is shown in Figure 2.8(a), along with its reconstruction in Figure 2.8(b). The reconstructed bubbles formed 60 s and 120 s after the formation of the initial bubble are shown in Figures 2.8 (c), (d), respectively. If the reconstruction distance (distance from 32

57 the bubble to CCD) is, the new re-scaled pixel size on the reconstructed image plane (which is the same as the object plane) can be expressed as [13], (2.12) where is the number of pixels in the CCD, and is the pixel size of the CCD. By multiplying the number of pixels in the reconstruction by, the dimension of the actual bubble at the reconstructed image plane can be calculated precisely. As seen from Figures 2.8 (b)-(d), the bubble sizes range from approximately 230 m from the time it is first formed to approximately 830 m after about 120 s since the formation of the initial bubble. In calculating bubble sizes,,, and have been used. The precise dependence on the bubble size on the exposure to the pump will be pursued in the future. In passing, it is worthwhile to point out that the bubble can be steered using a focused laser beam. As shown in Figure 2.9, after the bubble is created by focusing the pump beam, it can be moved inside the container by slowly steering the pump beam. Steering the beam is readily achieved by moving the lens transverse to the propagation of the beam (indicated by arrows before the lens in Figure 2.9), causing the bubble to move along the container (shown by arrows on top of the container). Note that even if the bubble escapes from the beam, it can be recollected easily by directing the beam to the bubble. The bubble can be steered even with the low laser powers, such as a 5 mw 633 nm or 532 nm laser pointer. There are two plausible conjectures for the reason behind the bubble steering: (a) the bubble may move toward the region of the thermal medium with a lower density created through localized laser heating, or (b) optical 33

58 Pixels Pixels Pixels Pixels tweezing by the focused laser beam [47]. The exact reason for bubble steering will be pursued in the future Pixels Pixels Pixels Pixels Figure 2.8: (a) Recorded in-line hologram (around center of picture) of bubble just after it forms. Reconstructions of bubbles, using Fresnel technique [13] (b) immediately after formation, (c) after 60 s, (d) after 120 s. Sizes are monitored as a function of pump exposure time, starting from initial size of 231 μm, 752 μm after 60 s and 829 μm after 120 s. (a) (b) (c) (d)

59 Figure 2.9: Schematic of microbubble steering arrangement. Moving the lens along z steers the bubble along x in the liquid within the modified container. Once formed, bubble can be steered with weaker laser beams Nanoparticle Synthesis and Agglomeration around Thermal Bubbles The behavior of nanoparticles when bubbles are thermally generated in a colloidal suspension of nanoparticles is investigated in this section. In drug delivery systems, the ideal choice of nanoparticles is gold, as it is currently the only FDA-approved nanoparticle. Gold nanoparticles have been shown to permeate cellular membranes with ease [48,49]. Since gold nanoparticles are expensive and more complicated to synthesize, slver nanoparticles is used for our experiment. Silver is a good candidate for couple of reasons. Silver nanoparticles have similar properties to the gold nanoparticles and they are convenient and cost effective to synthesis. When synthesizing, the intension is to keep the size of the nanoparticles between 5 and 10 nm to avoid clustering them together. A recipe for making Ag nanoparticles starting from the raw materials silver nitrate (AgNO 3 ), sodium borohydride (NaBH 4 ), sodium chloride (NaCl), polyvinyl pyrolidone (PVP), and polyvinyl alcohol (PVA), is followed as enunciated in Ref. [50]. Several attempts have been done to achieve a solution that is the appropriate color, which signifies that we have created nanoparticles of desired size. According to the recipe, in 35

60 order to obtain nanoparticles in the order of 5-10 nm, the final solution should be dark yellow in color as can be seen in see Figure 2.10(a). The samples are characterized with a transmission electron microscope (TEM) and the appropriate sample is used for our experiments. Figure 2.10(b), a TEM analysis, is shown for the dark yellow solution which yields nanoparticles of less than 10 nm and of relatively consistent size. The best batch of nanoparticles has been used to mix with the alcohol-dye solution and introduce into the new container. (a) (b) Figure 2.10: (a) Colloidal suspensions of Ag nanoparticles of increasing size from left to right. The third from the left, which is dark yellow in color, is the best batch, (b) TEM of nanoparticles obtained from the dark yellow batch. It is hypothesized that the behavior of our silver nanoparticles in the dye solution is dominated by Van der Waal s forces, which would cause the nanoparticles to attach to the vapor-liquid interface on the edge of each bubble [51]. When the bubble is surrounded by nanoparticles it is not possible to characterize the whole system using TEM. Thus, a preliminary and novel technique has been developed for verifying the agglomeration of nanoparticles around the edges of bubbles. It has been shown that silver nanoparticles of approximately 10 nm size have a peak absorption due to surface 36

61 plasmon resonance around 400 nm [52]. The absorption peak of silver nanoparticles can be utilized to identify whether they occupy a specific region, and even their relative concentrations. If the intensity after passing through the sample is weaker in the presence of nanoparticles than that of the absence of nanoparticles, it can be postulated that the illumination area is surrounded by nanoparticles. Accordingly, a weakly focused 405 nm laser beam is introduced into our samples and the transmitted powers, respectively, are measured for the following four cases: (a) solution without bubble and without nanoparticles, (b) solution without bubble and with nanoparticles, (c) solution with bubble and without nanoparticles, and (d) solution with bubble and with nanoparticles. Care is taken to capture all the light after it passes through the container. Also, for (c) and (d), care is taken to focus the light onto the bubble. For each of the four cases listed, recorded data from five sets of measurements are shown in Table 1. A possibility of nanoparticles agglomeration of silver nanoparticles around the liquid-vapor interface of bubbles can be observed by examining the ratios and. The first ratio illustrates the absorption effect of nanoparticles that are randomly interspersed throughout the thermal medium in the absence of a bubble. As seen from Table 1, the average value of = The second ratio, in contrast to the first, illustrates the absorption effect of nanoparticles that have possibly agglomerated around the bubble due to the hypothesized attractive effects of the liquid-vapor interface. Again, as seen from Table 1, the average value of = It is clear that there is increased absorption from the nanoparticles in the presence of the bubble. Since the 405 nm beam 37

62 corresponds to the surface plasmon resonance wavelength of silver nanoparticles, higher absorption should correlate with an increased concentration of silver nanoparticles around the bubble. Table 2.1: Measured transmitted power of 405 nm laser for 5 trials each for (a) no bubble and no nanoparticles, (b) no bubble and with nanoparticles, (c) bubble and no nanoparticles, and (d) bubble with nanoparticles. No bubble, no nanoparticles P a (mw) No bubble & with nanoparticles P b (mw) Bubble, no nanoparticles P c (mw) Bubble with nanoparticles P d (mw) Trial Trial Trial Trial Trial Average Conclusion Through a simple setup using a low power beam, SPM and TB in a thermal medium has been monitored and characterized. First, the induced phase change during SPM due to a focused 514 nm beam has been characterized using a single 633 nm probe beam. The results are in agreement with SPM results, as well as interferometric measurements using a 633 nm reference and probe. Additionally, using a variation of the interferometric setup and holographic reconstruction, we have accurately mapped the induced phase profile. Bubble formation due to TB has been characterized using a single 633 nm probe beam and in-line holography. It is further shown that the bubble size increases with pump exposure, and that the bubble, once formed, can be steered using a focused laser beam. Further details of work reported in this Chapter can be found in Abeywickrema et al. [53]. 38

63 As shown in earlier work, thermally generated microbubbles can be used to agglomerate nanoparticles and steer them for targeted drug delivery [48], such as for cell destruction after nanoparticle endocytosis [54,55]. In related work, Ramachandran et al. [56] have used microbubbles as scavengers of carbon nanotubes in biologically relevant fluids. A preliminary work on precisely detecting and monitoring agglomerated nanoparticles around microbubbles is shown and in currently in progress as well. Also, work on the dependence of bubble size on focused pump exposure and the exact reason for steering of the microbubbles will be pursued in the near future. In the next Chapter, another instance of induced gratings are considered, viz., induced RI in PR materials, due to interference of two or multiple plane waves, and their application to phase shifting DH is studied. 39

64 CHAPTER 3 PHASE-SHIFTING HOLOGRAPHIC INTERFEROMETRY USING PHOTOREFRACTIVE MATERIALS 3.1 Introduction The change of the phase/ri inside a thermal medium due to thermal nonlinearity and its applications are discussed in Chapter 2. The RI in some crystals and materials can change when the material is illuminated with a coherent source of light by creating a space charge field inside the material. This effect is called the PR effect and explained below in some detail. In this Chapter, the use of PR materials for implementing phase shifting HI is discussed. In traditional phase-shifting DH, holograms of the object are successively recorded using reference waves with different phase shifts, as described in Chapter 1. This enables the computation of the phase of the object, which also contains its depth or 3d information. If the amplitudes of the reference and the object beams are and respectively, and and are their corresponding phases at the recording plane, Eq. (1.6) can be re-written in terms of and to obtain the intensity at CCD plane as. (3.1) 40

65 Assuming sequentially takes on values such as, four intensity patterns can be recorded. Using these four equations, the expression for can be easily obtained as [24]. (3.2) Once the phase and the amplitude of the optical field at the recording plane are known, the original object can be obtained by back-propagating using the Fresnel transformation. The concept of phase shifting holography has been applied to HI by Chang et al. [57]. The interpretation of the phase as determined through Eq. (3.2) or its generalization (for when the phase changes of the reference are different than being introduced through the deformation of the object. ) can be interpreted as It has been shown that continuous phase shifting of the reference and subsequent integration over time can remove the need for discrete phase shifts and the setup is less sensitive to vibrations and does not require calibration of the phase-shifter. HI also can be used to track the change of an object in real time. HI can be used for opaque objects with rough surfaces or transparent phase objects [24]. HI has been used in many areas such as engineering and medical to measure small deformation (in wavelength range) of objects and surfaces and can be measured with an accuracy of ±1 0. Image processing and phase shifting can be combined together with HI to track the changes in two specimens which are barely equal [58]. Phase shifting HI has 41

66 been used to record three-dimensional image by detecting the complex amplitude of a phase shifted interferogram on a CCD and then by using the Fresnel transformation to recreate the object. This method gives a better image quality without using imaging lenses and has the ability to reconstruct the image at any arbitrary plane [59]. Phase shifting HI has been also used in medical industry to detect breast cancers by using a pulsed ruby laser and a device that can introduce small pressure and stress [60]. Real time holographic imaging has also been used to track the beat of the human heart. This can be improved to store three dimensional data of real time movement of the human heart instead of reviewing echocardiograms which is time consuming [61]. Instead of providing the phase shift of the reference beam sequentially or continuously and digitally recording the intensities for phase shifting DH and HI, it is shown in this Chapter that it may be possible to obtain a similar effect and perform simultaneous recordings by using the concept of Bragg and non-bragg orders during the writing of holograms in a PR medium. 3.2 Initial Observations: Non-Bragg Orders and HI using PR Materials PR materials and photothermoplastics (PTP) have been used as holographic storage media for HI. It is found that both Bragg and non-bragg orders can be present during recording in thin holographic materials, or when the interaction length in the recording medium is small compared to the Rayleigh range of the object beam [62]. This can be considered as an asymmetric dynamic self-diffraction in a volume grating which is somewhat similar to Raman-Nath diffraction in thin gratings [63]. It has been also shown 42

67 experimentally that the non-bragg orders can be observed even in bulk PR materials, and phase conjugation and phase doubling also occur in non-bragg orders [64]. An experiment to demonstrate the generation of Bragg and non-bragg orders in PR media has been conducted in our lab. In this experiment, an iron doped lithium niobate (LN:Fe) crystal is used as the PR material. A collimated 514 nm Ar-Ion laser is used to write the grating with a power of approximately 200 mw. The angle between the two beams is 2 degrees in this case, as shown in Figure 3.1. The Bragg and Non-Bragg orders are directly recorded for real time holographic imaging. As can be seen in Figure 3.2, many higher non-bragg orders are observable during self-diffraction in LN:Fe. Figure 3.1: Experimental setup for the recording the hologram on LN:Fe crystal with two beams. Bragg and Non-Bragg orders are also shown as -1, +1 and -3, +3 respectively [8]. 514 nm collimated laser beams are splitted using a beam splitter in order to create equal intensity beams. At the process of recording, the object is replaced by a flat mirror. 43

68 Figure 3.2: 9 non-bragg orders seen during self-diffraction in PR LN:Fe [8]. Effects analogous to sequentially imparting the phase changes to the reference in traditional phase shifting DH can be automatically obtained through the interactions between Bragg and non-bragg orders in PR materials (and photothermoplastics). Similarly, if the object has an initial phase of 0 and later changes its phase to, the information of this phase change can also be derived from the intensities of the various diffracted orders. As will be shown first using heuristic theory, the intensities of the various diffracted orders are now parametrized by different phase shifts (albeit, different than multiples of ). Indeed, as stated above, this is the basis of HI, which can be effectively used to register dynamic deformations of objects. HI using PR materials can also be performed by recording the hologram of the original object, reconstructing it with the reference and the deformed object, and monitoring the interference fringes of the Bragg and non-bragg orders [8]. Alternatively, the simultaneous generation of Bragg and non-bragg orders can be exploited to extract information about object phase similar to phase-shifting HI [8]. As an example of HI using PR materials, the hologram of an object, viz., a CD- ROM disk, is recorded using a 633 nm laser. The CD-ROM is then deformed by heating 44

69 with another focused (blue) laser beam. After the deformation, the hologram is read out using the reference beam and light from the deformed object, as shown in Figure 3.3. (a) (b) Figure 3.3: (a) Schematic of recording the hologram of a CD-ROM, before and after heating. Bragg orders are shown as -1 and +1 while the non-bragg orders (PC and PE) are shown as +3 and -3. (b) Lab setup. Note that the object and the reference are named S and R respectively [8]. 45

70 (a) (b) Figure 3.4: (a) Fringes in diffracted order after heating CD-ROM with the blue focused laser, (b) cropped fringe region [8]. After recording the interference pattern, the 3D deformation information can be obtained by using phase unwrapping techniques. Figure 3.5 shows the unwrapped 3D deformation of the CD-ROM, along with results obtained from a stylus Veeco profiler measurement. As evident from Figure 3.5, the two results are in excellent agreement with each other. 46

71 (a) (b) Figure 3.5: (a) Deformation details of the CD-ROM due to the heating, (b) comparison of the details; red curve-from a profiler, blue curve-from unwrapped data [8]. 3.3 The PR Effect It is appropriate at this stage to briefly describe how a grating can form in a PR material due to the PR effect. This can be illustrated step by step using a set of figures as shown in Figure 3.6. It can be assumed that the impurities in a PR material are identical and have an energy level somewhere between the conduction and the valance band as illustrated in Figure 3.6(a). When a PR material is illuminated with a coherent light source, with an alternating pattern (dark and bright regions) as can be seen in Figure 3.6(b), the electrons are photogenerated in bright regions of the material, assuming donor type impurities (Figure 3.6(c)). Generated electrons are then excited to the conduction band and diffuse in the crystal toward the darker regions, as shown in Figure 3.6(d). Because of the redistribution of the charges, an electrostatic field, called a space-charge field, is created (Figure 3.6(e)), thereby inducing a RI modulation in the material as shown in Figure 3.1(f) through the EO effect. Thus a phase grating is recorded inside the 47

72 PR material. In some materials, the photovoltaic effect also plays a role in space charge generation and hence RI modulation. Furthermore, since many PR materials are also piezoelectric, the space charge field creates stresses/strains in the materials, as well as a corrugation on the surface [65]. Figure 3.6: Illustration of the PR effect. (a) Excitation of the electrons to the conduction band with the photons, (b) Illuminated PR material with a sinusoidal coherent light, (c) generation of charges inside the PR material due to the illumination, (d) separation of charges and redistribution, (e) creation of a space-charge field, (f) final refractive index modulation. The generation of the space charge field and induced RI profile in PR materials can be modeled using the Kukhtarev equations [65]. An approximate expression for the space charge field can be written as, (3.3) 48

73 where and are the coefficients resulting from photovoltaic and diffusive effects, respectively in the PR medium [66]. 3.4 Heuristic Derivation of Bragg and Non-Bragg Orders during HI using PR Material It is assumed in this case that the hologram is written with two plane waves (reference and object wave), and read out by the reference and the object wave that has a certain phase (corresponding to depth). The writing (recording) process is first considered. It is assumed that the reference and the object fields are in phase and can be written as where, is the propagation constant, and is the angle between the object and reference waves (see Figure 3.7(a)), so that. Assuming diffractive coupling due to photovoltaic effect only, Eq. (3.3) becomes. (3.4) 49

74 (a) (b) (c) (d) Figure 3.7: (a) Using the object beam and the reference beams to record the hologram, (b) reconstruction with the same reference beam, (c) reconstruction with the deformed object, (d) total Bragg and non-bragg orders at the exit plane of the PR crystal [8]. For simplicity, it is now assumed that the reading waves are same as writing waves, except that has additional phase. The thin grating approximation is used and it is assumed that the grating is illuminated by each wave separately. Thus, the effect of illumination by the reference wave illumination with the deformed object wave alone is considered followed by alone. It is noted that the phase hologram can be written as a summation of a Bessel series as follows:, (3.5) where the Jacobi-Anger formula has been used, and L is the thickness of the PR material. Hence for the first case while reading with the reference beam, the Bragg (R [+1] and O [-1]) and first non-bragg (phase conjugate PC [+3] and phase enhanced PE [-3]) terms are: 50

75 (PC), (3.6a) (R), (3.6b) (O), (3.6c) (PE), (3.6d) where (see Figure 3.7(b)) and where represents the phase mismatch between the Bragg and the non-bragg orders (viz., 1 and 3; -1 and -3). The derivation of appears in Section 3.4. Similarly, for the second case while reading with the deformed object beam, the different diffracted orders are (see Figure 3.7(c)): (PC), (3.7a) (R), (3.7b) (O), (3.7c) (PE). (3.7d) The total intensities of the different Bragg and non-bragg orders at the exit plane of the PR material can be defined as: (PC), (3.8a) (R), (3.8b) (O), (3.8c) (PE). (3.8d) 51

76 where the T subscript denotes total intensities (see Figure 3.7(d)). After some algebra Eqs. (3.8a-d) become: (PC), (3.9a) (R), (3.9b) (O), (3.9c) (PE), (3.9d) where, is used. After some algebra we can find from Eqs. (3.9a-d) to be:. (3.10) Equation (3.10) is reminiscent of Eq. (3.2) which describes the phase of an object in terms of recorded intensities with different phases (viz., ) imparted to the reference beam. The phase precisely, the additional phase has the connotation of the phase of the object, or, more incurred by the object under a deformation. Therefore Eq. (3.10) shows that it is feasible to obtain the phase information by monitoring the intensities of the various simultaneously generated diffracted orders which may have phases imparted to them by the material, in this case through the phase mismatch between the Bragg and the non-bragg orders during interaction in the material. 52

77 It can be seen from Eq. (3.10) that the object phase can be obtained by recording the intensities of Bragg and non-bragg orders. In other words the induced RI in the PR or recording material helps to determine the phase (or phase deformation) in a single shot through the simultaneous generation of Bragg and non-bragg orders. One further advantage of using the PC image is that it helps to examine various longitudinal planes in the object. Phase conjugate fringes give the current information of the deformation which is more useful than the Fresnel diffraction pattern of the object observed from the Bragg image and this is more significant when the feature size is small. Displacement details of the object superposed on the real image of the object by investigating the fringes of the PC (+3). The intensity of the phase conjugate (+3) can be written as (3.11) and the fringe periods in and becomes. (3.12) 53

78 3.5 Explicit Derivation of Bragg and Non-Bragg Orders during HI using PR Material In our preliminary work, the differential equations for the Bragg and non-bragg orders have been derived under the plane wave assumption and assuming that the higher orders (non-bragg) are not interacting with the Bragg orders. In this Section, these interactions are not neglected, yielding a more exact theory. The starting point is the Helmholtz equation,, (3.13) where has been defined in Eq. (3.3). Once again, purely diffractive coupling, i.e., is assumed. With the assumption for simplicity, the total optical field in the PR medium can be expressed as, (3.14) where ( ) are the first and higher orders amplitudes respectively, and where K is the spatial frequency of the Bragg grating. Now, using Eq. (3.13) with Eq. (3.3) (with ) along with Eq. (3.14), the differential equations for the Bragg and non-bragg orders can be expressed as 54

79 , (3.15a), (3.15b), (3.15c), (3.15d) with, (3.16) where is the angle between the object wave and the reference wave. In deriving Eqs. (3.15), it has been assumed that the change in refractive index is primarily brought about by the interference between the ±1 orders. As a check, it is to be noted that for conventional two-wave coupling, i.e., between ±1 orders, the first two of the coupled set of equations in (3.15) can be written as (3.17a) (3.17b) where the substitution 55

80 (3.18) has been used to simplify. This set of complex equations can be separated in to magnitude and phase by setting (3.19) and writing out the imaginary and real parts separately. For instance, the amplitude and phase for the +1 and -1 orders can be written as, (3.20a), (3.20b), (3.20c). (3.20d) Analytical solutions can be obtained for the magnitude and the phase of the Bragg orders, and can be expressed as, (3.21a), (3.21b), (3.21c) 56

81 . (3.21d) It is to be pointed out that for an arbitrary incident optical field from the object, needs to be decomposed in terms of its angular plane wave spectral components. In this case, all interacting (Bragg and non-bragg) orders also need to be decomposed into their respective spectra. Work on this is in progress and is outlined later in this Chapter. Equations (3.15a-d) can be numerically solved using Runge-Kutta methods (ode45 in MATLAB ) to obtain the exact solutions for the complex fields, and hence, intensities, of the Bragg and non-bragg diffraction orders. After the grating is stored in the PR medium, it is assumed that the grating is re-illuminated with the reference and the object, albeit with a different phase. The objective is to check whether the intensities of the interacting orders during readout of the original grating can be expressed in terms of relations similar to Eq. (3.1), as is the case in traditional phase-shifting DH, i.e., in the form, (3.22a), (3.22b), (3.22c), (3.22d) 57

82 where is the phase of the object wave (-1) w.r.t. that of the reference wave (1) (assumed to be zero phase for simplicity). 3.6 Numerical Solutions for Bragg and Non-Bragg Orders Writing the Initial Grating First, Eqs. (3.15a-d) are solved using MATLAB to obtain the intensities and phases of the Bragg and non-bragg orders as a function of crystal thickness, z. Solutions are shown in Figure 3.8 for the case where the initial intensities of the incident waves are taken to be. The initial phase of the reference (+1) is taken to be zero, and the initial phase of the object (-1) is taken to be zero as well. The angle between the two waves is taken to be 2 degrees, corresponding to a grating period of 14 microns. The wavelength is 514 nm. The thickness of the interaction region is taken to be L=1 mm and the value of the nonlinearity parameter is taken to be. Note that the intensities and the nonlinearity parameters can be readily scaled as long as their product is a constant. For instance, an incident intensity of along with a nonlinearity parameter of should yield similar results. It can be readily verified from Eqs. (3.15a-d) that. (3.23) 58

83 This is also numerically checked from Figure 3.8(a), verifying conservation of energy. In PR physics, for diffractive two-beam coupling ( ), there should be no energy exchange between the beams. Indeed, this is true for this case as evident from Figure 3.8(a). Higher or non-bragg orders are generated at the expense of energy from the +1 and -1 Bragg orders, and the intensities of the +3 and -3 (non-bragg) orders are equal. Also, while the phases, of the Bragg orders are identical and decrease almost linearly throughout the interaction region from their initial value of zero, the phases, of the non-bragg orders acquire the values of after generation and remain identical through the interaction region. 59

84 Phase Intensity 2.5 I 1 I -1 I 3 I -3 I T z (m) x 10-3 (a) z (m) x 10-3 (b) Figure 3.8: (a) Intensity and (b) phase variation as a function of z, through the PR medium, as found through numerically solving Eqs. (3.15) and for initial intensity. The initial phase of the object is [67]. The intensity and phase variations for are shown in Figure 3.9, once again with the initial phases set equal to zero. Once again, in this case, there is no energy exchange between the Bragg orders, with the non-bragg orders generated from the energy of the Bragg orders. In this case, the phases, of 60

85 the Bragg orders, while not identical, decrease almost linearly throughout the interaction region from their initial value of zero, while the phases, of the non-bragg orders acquire the values of after generation but change differently throughout the PR material Reading the Grating An explanation for the reading process of the grating can be explained using the set of differential equations (Equations (3.15a-d)). In the second stage, the grating formed during the interaction of the reference and the object is read by the same reference wave and the object wave which now has a phase. Contributions to, for instance, the object wave come from diffraction from the gratings originally written by the object (-1) and the reference (+1) [ ] and the object (-1) and the -3 non-bragg order [ ], and read out by the reference wave, as noted from the first term in square brackets and the last term on the RHS of Eq. (3.15b). The remaining contribution to the object wave from gratings originally written in the PR medium comes from, which is read out by the new object wave (with phase ) and represents the equivalent of the undiffracted order in the heuristic treatment of Ref. [8]. Accordingly, during readout, the spatial evolution of the -1 order can be modeled by the equation (modified from Eq. (3.15b)): 61

86 Phase Intensity I 1 I -1 I 3 I -3 I T z (m) x 10-3 (a) z (m) x 10-3 (b) Figure 3.9: (a) Intensity and (b) phase variation as a function of z, through the PR medium, as found through numerically solving Eqs. (9) and for initial intensity. The initial phase of the object is [67]., (3.24) 62

87 Intensity Intensity where the subscripts w and r have been inserted to distinguish between the writing and reading waves. Similar modifications are done to the remaining three equations from the set of Eqs. (3.15a-d). Figures 3.10 (a,b) show the spatial distribution of the ±1 and ±3 orders for the case when the grating written by the reference and object waves as shown in Figure 3.9 is now re-illuminated with the same reference and object waves, but with the object wave now, for instance, having a phase shift of. The spatial evolution of the Bragg and non-bragg orders are different from that shown in Figure I 1 I -1 I 3 I I 3 I z (m) x 10-3 (a) z (m) x 10-3 (b) Figure 3.10: (a) Spatial evolution of the different orders during reading of the grating written by reference and object wave with the parameters and initial conditions as in Figure 2. The reading is done with the reference and object waves with the same respective intensities, and the phase of the object wave is. (b) The intensities of the ±3 orders drawn to a different vertical scale to show their evolution with interaction distance [67]. The intensities at the exit plane of the crystal for each order viz., and are now plotted as a function of. Results are shown in Figure The discrete points are values of the intensities obtained from changing the phase and reading off their values from figures such as Figure 3.10 at the exit plane. Next, the intensity variations postulated in Eqs. (3.22a-d) 63

88 are matched with the variations shown in Figure 3.11 using curve fitting. This has been done using MATLAB with a least squares algorithm, starting from initial guesses of the parameters. The fitted curves are indicated with continuous lines and superposed on the calculated variations of intensities vs.. The unknown parameters obtained for this case (viz., when the initial object and the reference intensities are the same and equal to 1) are shown below in Table 3.1. As evident from Figure 3.10, the values of and are determined once the physical parameters of the PR material and the initial intensities of the reference and the object waves are known. It is to be noted that for the example above, the phase shifts of three of the orders happen to be close to multiples of, similar to the case of traditional phase shifting DH as seen from the result written in Eq. (3.2) which is a specific case when the reference phase is changed by 0,, and. The amplitudes are also all different, and not according to the simple relations for traditional phase shifting DH as expressed in Eq. (3.1). However, it is important to note that in spite of this, it is theoretically possible to deduce the phase of the object, as was for instance done in Eqs. (3.9a-d). Knowing the initial intensities of the writing and reading waves and the material parameters, it is possible to determine all amplitudes and phases. Work on the dependence of the intensities as a function of phase for different intensity ratios of the object to the reference is currently in progress. Phase calculation of an arbitrary object will be also done experimentally in near future to verify the simulation results. 64

89 Intensity( ) Table 3.1: Unknown coefficients obtained from the curve fitting for Eqs. (3.22a-d) for the case when for both writing and reading, and I 1 I -1 I 3 I Phase Figure 3.11: Intensities at the exit plane of the crystal ( ) as a function of phase, derived from results such as shown in Figure 4. The discrete points are obtained from results such as in Figure 4 for different values of the phase. The solid lines are obtained by curve-fitting to sinusoidal functions with appropriate phase shifts [67]. 3.7 Theoretical Model of Interacting Angular Spectra All of the calculations in the previous Sections are based on the plane wave assumption of the Bragg and non-bragg orders. Further it has been assumed that the change in refractive index is primarily brought about by the interference between the ±1 orders. If all interacting orders are included in the intensity, the revised set of equations become 65

90 , (3.25a), (3.25b), (3.25c), (3.25d) Then by considering each order as a collection of plane waves, the complex field for each order can be written as, (3.26a), (3.26b), (3.26c), (3.26d) where is the magnitude of the fundamental spatial frequency component in the angular plane wave spectrum of the profile. Thus the total field inside the PR material can be expressed as 66

91 . (3.27) The total intensity is calculated based on the following equation:. (3.28) Then using the Helmholtz equation and following the same steps similar to the previous Sections, after some algebra, a new set of differential equations can be derived for each order. For example, the differential equation for the +1 order can be expressed as. (3.29) Similar equations can be written for the -1 and ±3 orders. Note that in Eq. (3.29), each term contains a triple sum which can be simplified using the Cauchy product for triple sum, viz.,. Then the differential equations for the th angular component for each order become 67

92 , (3.30a), (3.30b), (3.30c). (3.30d) The above set of equations should be numerically solved using MATLAB and is being done currently. 68

93 3.8 Conclusion It has been shown that information from the Bragg and non-bragg orders in a PR material can be used for phase shifting HI, where the gratings initially written by the object and a reference can be read out by light from the displaced or distorted object and the same reference. While this has been shown using two incident plane waves, extension to the case of an angular plane wave spectrum representing the object is currently under way. 69

94 CHAPTER 4 HOLOGRAPHIC SURFACE GRATINGS IN PHOTOREFRACTIVE MATERIALS 4.1 Introduction In Chapter 3, the use of PR materials for phase shifting holography has been discussed in detail. As mentioned in the previous Chapter, a phase grating can be induced in the PR material due to the induced space charge field via the electro-optics effect by interfering two beams on to a PR material. The use of the Bragg and non-bragg orders for implementing real-time phase shifting HI is examined and the advantages of using the Bragg and non-bragg orders over traditional phase shifting HI are explained in Chapter 3. It has been also shown that a surface corrugation can occur on the PR material when a holographic grating is written [10,12]. The induced space charge field along with the converse piezoelectric effect can cause these surface deformations [10]. This surface deformation can be considered as a surface grating with a grating period identical to the phase grating written inside the PR material. Typically the maximum corrugation depth of the surface grating is in the order of nanometers [10,12]. Existence of the surface grating has been proved using a diffraction of a reflected probe beam and high resolution phase-shifted interferometric profilometry [10]. 70

95 The purpose of this Chapter is to carefully study the surface grating associated with a holographic volume grating written in a PR material. A volume hologram is written using two beams as described in Chapter 3 and it is read using a low power reading beam. Reflection from the surface is analyzed as a function of the incidence angle. As will be shown, by considering the variation of the intensities of diffracted (+1 and -1) and undiffracted orders, and by removing the Fabry-Perot effect, the existence of the surface grating can be proved without using a profilometer. A novel signal processing technique is used to remove the contributions from the Fabry-Perot effect. LN:Fe is used as the PR material as in Chapter Previous Related Work Diffraction properties of corrugated gratings have been studied theoretically and experimentally for many years [10,12,68,69]. A coupled wave theory for thick hologram gratings has been derived by Kogelnik [70]. Algebraic formulae for angular and wavelength dependence on diffraction efficiencies of thick hologram gratings has been derived for both reflection and transmission holograms. Diffraction characteristics for dielectric corrugated gratings have been investigated using rigorous coupled wave theory for various parameters such as arbitrary grating profiles (viz, square-wave, sinusoidal, triangular, sawtooth), incident angles, wavelenthghs and groove depths [68]. It has been also shown that gratings should be made with symmetric profiles to obtain large diffraction efficiencies [68]. The behavior of diffraction from such gratings has been also investigated by solving Maxwell s equations and compared with experimental results. A differential method has been used to solve the equations numerically and 71

96 calculations have been performed for both TE and TM cases [69]. It is concluded that higher diffraction efficiencies (viz., 90%) can be achieved when the wavelength to grating period ratio is in between 0.75 and 1.85 [69]. A new approach to investigate resonance domain surface relief gratings has been also developed by obtaining a closed form analytical solution and also verified with experimental results [71]. This model explains the unique diffraction efficiency peak, and it has been shown that an efficiency of more than 85% can be achieved by optimizing important grating parameters such as the grove spacing and the grove depth. Diffraction properties of index and surface-relief gratings formed on a methacrylate photopolymers has been studied experimentally by Kojima et al. [12]. When a dual grating is formed on a methacrylate photopolymer, it has been shown that surface-relief grating governs the diffraction [12]. Related to PR space charge fields and energy coupling is two-beam energy exchange in a hybrid photorefractive inorganic-cholesteric liquid crystal (LC) cell, where a cholesteric LC cell is placed between two inorganic PR windows [11]. When a weak and strong light beam are incident on the LC cell, the interfering light beams induce a periodic space-charge field in the PR windows which penetrates into the LC, inducing a diffraction grating written on the LC director. It has been shown that enhanced gain can be obtained due to the director reorientation effect. By comparing theoretical results for exponential gain coefficients with experimental results for hybrid cells filled with various cholesteric mixtures, it has been deduced that that enhanced gain can be obtained due to the director reorientation effect resulting from the PR space charge field. However, in this analysis, the effect of surface corrugation is not included. It may be possible that the 72

97 induced surface corrugation can pre-tilt the LCs at the surface of the PR windows, thereby enhancing the two-beam coupling in the LC hybrid device. 4.3 Surface Gratings on PR Materials Since PR LN is also piezoelectric and elasto-optic, it seems reasonable to surmise that during two-beam coupling and generation of induced refractive index profile, there should also be strains in the material and on its surface from the contributions of the piezoelectric effect (due to the space charge field) and the elasto-optic effect (due to the intensity profile of the light in the medium). Indeed, surface corrugation in PR LN has been observed by Sarkisov et al. [10], and in other PR materials [72]. Cook et al. [73] have investigated the effect of the piezoelectric and elasto-optic effects on two-beam coupling in PR KNbO 3 in a self-pumped reflection geometry, mainly concentrating on the change in the effective electro-optic coefficient and the effective dielectric constant. Photorefractive gain dependence due to piezoelectricity and photoelasticity in BaTiO 3 has been similarly investigated by Mathey [74]. A simple derivation of the amount of surface displacement in unclamped PR LN can be determined starting from the Kukhtarev equations and incorporating the revised constitutive equations and the elastodynamic equation [72]. The surface displacement due to a sinusoidal intensity profile can be expressed as, (4.1) 73

98 where, is the free-space permittivity, is the refractive index, is the characteristic impedance of LN, is the elasto-optic coefficient, is the electron charge, is the thickness of the material, are the acceptor and donor concentrations, is the elasto-optic constant, is the absorption coefficient, is the recombination constant, is the thermal generation rate, is the elastic constant, and is the photovoltaic constant. It is assumed that which is less than the dark intensity. With typical parameters for PR LN substituted [72,66], the approximate estimate for the surface displacement is of the order of a few nm, which is in agreement with the experimental observations of Sarkisov et al. [10]. An experiment similar to that outlined in Chapter 3 has been set up to see the effect of surface displacement. It is surmised that once the grating is formed, a different light source (e.g., 633 nm from a He-Ne laser) when incident on the surface of PR LN at an arbitrary angle should diffract in the far field into three orders, similar to what is expected from a surface relief grating. Experimental details and results are discussed below. 4.4 Diffraction Efficiency of a Surface Grating As mentioned in the Introduction, when a phase grating is formed inside a PR material due to the space charge field generated by interfering two beams, a periodic corrugation occurs on the surface of the PR material. This surface corrugation can act as a surface grating with a grating period similar to the phase grating inside the PR material. Since the maximum corrugation depth is usually in the order of nanometers [10], this can be considered as a very thin grating which has a sinusoidal profile. When such a grating 74

99 is illuminated with a reading beam, diffraction efficiency is an important parameter to be considered. The diffraction efficiency can be simply defined by. (4.2) Since this can be considered as a thin grating, the first order diffraction efficiency can be expressed using the Raman-Nath diffraction efficiency formula for a thin sinusoidal grating as [12], (4.3) where is the diffraction efficiency of the order and is the Bessel function of the first kind of order, is the internal angle of incidence that can be calculated using the Snell s law. is the corrugation depth and is the difference between the refractive indices of air and the material. Equation (4.3) is based on the assumption that the grating has a sinusoidal profile. For a PR material, when the diffusion term is neglected, the space-charge field can be written as [66], (4.4) 75

100 where the intensity profile is given by. If, however,, the induced field may be non-sinusoidal, implying that application of Eq. (4.3) may not always be appropriate. An alternative expression for the diffraction efficiency, which is easier to use, but for a (non-sinusoidal) rectangular surface relief grating, for near-normal incidence is [75] ; ; 0 otherwise. (4.5) 4.5 The Fabry-Perot Effect Since the surface grating is formed on a 1.8 mm thick transparent LN:Fe crystal, diffracted and un-diffracted order intensities should be affected by the multiple reflections inside the crystal. Thus the Fabry-Perot effect should be taken in to account when monitoring the diffracted intensities. When a coherent light source travels through a thick transparent, partially reflective material, the total reflected light and the transmitted light consist of the multiple reflections which occur inside the material. The relation between the incident angle and the reflected intensity can be obtained from the well known relation [76], (4.6) with 76

101 , (4.7) and. (4.8) In Eqs. (4.6), (4.7) and (4.8), is the amplitude reflection coefficient and can be calculated using the well known Fresnel equations. is the initial intensity, is the wave number, is the refractive index of the material and is the thickness of the material. is the angle of refraction which can be deduced from the Snell s law. Substituting Eq. (4.7) and Eq. (4.8) in to Eq. (4.6), the reflected intensity can be expressed in terms of the incident angle as, (4.9) where is the refractive index of air. The variation of reflected intensity to initial intensity ratio ( ) with the incident angle is shown in Figure 4.1 for the case of LN. 77

102 I r /I i incident angle i (deg) Figure 4.1: Variation of reflected intensity to initial intensity ratio ( ) with the incident angle for lithium niobate.. is chosen to be. This Fabry-Perot effect is embedded in the diffracted and un-diffracted orders and should be removed in order to see the reflection just from the surface. A simple numerical method to remove the Fabry-Perot effect is introduced here. Note that the reflected intensity varies with the incident angle according to an approximately chirp function, as can also can be seen from Eq. (4.8) for small incident angles. Therefore, the reflected intensity has a sinusoidal variation with as can be seen in Figure 4.2. The Fourier transformation of a sinusoidal function represents two delta functions which indicate the main frequency of that function. By taking the Fourier transform of this sinusoidal function and by filtering the frequencies which corresponds to this sinusoidal variation, the Fabry-Perot effect can be removed numerically. After numerical filtering, the signal due to the surface reflection can be retrieved by taking the inverse Fourier 78

103 transform of the filtered signal. This technique is used later in this Chapter to interpret experimental results I r /I i i 2 Figure 4.2: Variation of reflected intensity to initial intensity ratio ( ) with squared of the incident angle, for lithium niobate.. is chosen to be. 4.6 Diffraction Behavior of a Surface Grating Formed on a Lithium Niobate Crystal Writing and Reading a Corrugated Surface Grating Since the surface grating is associated with the phase grating, the same procedure as mentioned in Chapter 3 can be followed to write the surface grating. Since the corrugation is in the order of nanometers and it may decay with time, a new grating should be written just before the diffraction efficiency measurements. Before writing the grating, the LN:Fe crystal is placed on a pyrex Petri dish and cooked in an oven to a maximum temperature of C, for about one hour. It is then carefully taken for the experiment after it reaches the room temperature. This is done to make sure to erase any 79

104 old gratings written from previous experiments. Experiments are done in a dark room and diffraction intensities are recorded just after the grating is written to minimize the effect of decaying as mentioned earlier. The 0 th and ±1 order diffraction intensities are measured as a function of the incident angle. The grating is written following the same procedure mentioned in Chapter 3 (see Figure 3.2) using light from an Ar laser of wavelength 514 nm. After the grating is written, it has been approximately imaged on to a CCD using a 633 nm He-Ne probe beam with a magnification of 2. The pixel size of the CCD is. The image hologram of the grating is shown in Figure 4.3. Intensity variations from a periodic phase grating can be observed due to the Talbot effect. If the half angle between the two writing beams is denoted by, the grating spacing,, can be calculated using the relation [77], (4.10) where, is the wavelength of the writing beam. For a half angle of 0.5 degrees and for the writing beams, the calculated grating period is. This can also be confirmed with Figure 4.4 by counting the distance between two successive dark or bright regions. In this particular case, the pixels between the two dark regions is approximately 9 which is a length of about. Since the magnification factor is 2 in this case the grating period is which agrees with the calculated value. 80

105 (a) (b) Figure 4.3: Holographic grating written on a lithium niobate crystal using a 514 nm Ar- Ion laser. (a) Grating is imaged with a lens on to a CCD using a 633 nm He-Ne laser, (b) portion of the image captured by the CCD and a zoomed version of the image (shown in a red box). The PR sample is now mounted on a 360 degree continuous rotation stage with stepper motor actuator (ThorLabs-NR360S), as shown in Figure 4.4. The motor is controlled by APT TM stepper motor controller which has a resolution of 1 arcsecond. In this part of the experiment the grating is rotated with a step size of 0.1 degrees and intensities of the diffracted orders (+1 and -1) and the 0 th order, are measured with a power meter. An iris is used in front of the detector in order to illuminate the detector with one order at a time. The incident angle is measured starting from 2 0 which is the minimum angle between the power meter and the +1 order diffraction. Intensities are recorded from 2 0 to 5 0. The distance between the crystal and the detector is kept constant as well as the other lab conditions. 81

106 (a) (b) Figure 4.4: (a) Recording intensities of diffraction orders as a function of the incident angle. Grating area is illuminated with a He-Ne laser (633 nm) and the power of the each order is measured using a power meter. Rotational stage with a stepper motor is used and controlled by connected to a personal computer (PC) through the APT TM stepper motor controller. Rotational stage is rotated increments of 0.1 degrees. Figure (b) shows the top view of (a) (power meter, PC and the APT TM stepper motor controller are not shown) Experimental Results As discussed above, the grating written in PR LN:Fe is read by a 633 nm He-Ne laser beam which has a width of approximately 1mm. Diffraction patterns recorded on an observation screen located approximately 45 cm from the sample is shown in Figure 4.5. As mentioned above, intensities of each order are then measured using a power meter as a function of the incident angle. 82

107 Figure 4.5: A photograph of the reflected diffraction orders recorded on a screen. Grating is read by a 633 nm laser beam (1 mm width) and the distance between the grating and the screen is approximately 45 cm. Intensities of each order are recorded using a power meter. The recorded intensities of the diffracted orders and the un-diffracted 0 th order are plotted as a function of the incident angle, and shown in Figure 4.6. Note that the horizontal axis is kept as the absolute intensity. It can be seen that the 0 th order intensity is approximately 160 times larger than the diffracted order intensities Numerical Filtering of the Fabry-Perot Effect It is observed from Figures 4.6(a) and 4.6(c) that the 0 th order un-diffracted order has more oscillations than diffracted orders. This shows that the 0 th order is affected by the Fabry-Perot effect more than orders. It can be removed by using the filtering technique mentioned in Section 4.5. By using the Fast Fourier transform (FFT) algorithm in MATLAB, Fourier transforms of Figures 4.6(b) and 4.6(d) are obtained, as shown in Figures 4.7(a) and (c), for the +1 and 0 orders, respectively. Note that here only the +1 diffracted order is considered since +1 and -1 shows the similar behavior with respect to the incident angle (see Figure 4.6(a)). 83