EFFECT OF A THIN OPTICAL KERR MEDIUM ON A LAGUERRE-GAUSSIAN BEAM AND THE APPLICATIONS WEIYA ZHANG

Transcription

1 EFFECT OF A THIN OPTICAL KERR MEDIUM ON A LAGUERRE-GAUSSIAN BEAM AND THE APPLICATIONS By WEIYA ZHANG A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Physics DECEMBER 2006 c Copyright by WEIYA ZHANG, 2006 All Rights Reserved

2 c Copyright by WEIYA ZHANG, 2006 All Rights Reserved

3 To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of WEIYA ZHANG find it satisfactory and recommend that it be accepted. Chair ii

4 ACKNOWLEDGMENTS We acknowledge the financial support of NSF (ECS ), the Summer Doctoral Fellows Program provided by Washington State University, and Wright Patterson Air Force Base. I would like to thank my adviser, Mark Kuzyk, who has always been encouraging and patient, for teaching me how to be a scientist by exploring crazy ideas. I also thank the faculty and the staff of the Physics Department for their help during my education. I am grateful to my classmates and friends at WSU, for the many joys that we have shared. Finally, I would like to thank my parents, sisters, and my wife, Wen. My endeavour in the world of physics would have been meaningless without their love. iii

5 EFFECT OF A THIN OPTICAL KERR MEDIUM ON A LAGUERRE-GAUSSIAN BEAM AND THE APPLICATIONS Abstract by Weiya Zhang, Ph.D. Washington State University December 2006 Chair: Mark G. Kuzyk Using a generalized Gaussian beam decomposition method we determine the propagation of a Laguerre-Gaussian (LG) beam after it has passed through a thin nonlinear optical Kerr medium. The orbital angular momentum per photon of the beam is found to be conserved while the component beams change. We apply our theory to using LG 1 0 beams to measure the nonlinear refractive index coefficient of the medium with high sensitivity, such as the Z-scan and I-scan techniques, and to a new optical limiting geometry. We test the validity of the theory and demonstrate the applications experimentally using a dye-doped polymer, disperse red 1 (DR1) doped poly(methyl methacrylate) (PMMA) (DR1/PMMA). In order to do that, we investigate the mechanisms of the nonlinear refractive index change in DR1/PMMA (trans-cis-trans photoisomerization and photoreorientation) by a three-state model and a holographic volume index gratings recording experiment, and determine the conditions under which DR1/PMMA acts as an optical Kerr medium. iv

6 Contents Acknowledgments iii Abstract iv List of Figures ix List of Tables xvi 1 Introduction 1 2 Theory Introduction A review of the Laguerre-Gaussian beams Optical Kerr effect and trans-cis-trans photoisomerization and photoreorientation Optical Kerr effect Mechanisms of trans-cis-trans photoisomerization and photoreorientation Effect of a thin optical Kerr medium on an LG beam Introduction Field of the beam immediately after the sample Propagation of the beam after the sample v

7 2.4.4 Examples assuming small nonlinear phase distortion Application: Z scan Review of the traditional Z scan using a LG 0 0 beam Z scan using a LG 1 0 beam Effect of the aperture size: the off-axis normalized transmittance Application: optical limiting Introduction Effect of the position of the nonlinear thin film Large nonlinear phase distortion Application: Measuring the nonlinear refractive index Motivation I scan Φ max scan to measure samples with large n Experiment Introduction Generating the higher order Laguerre Gaussian beams The principles Making the hologram Examining the phase singularity Fabricating the DR1/PMMA Samples Solvent-polymer-dye method polymerization-with-dye method Recording of high efficiency holographic volume index gratings in DR1/PMMA Background Experimental setup Experiments with the LG 1 0 beam vi

8 4 Results and discussion Properties of DR1/PMMA Absorption spectrum of DR1/PMMA Recording of high efficiency holographic volume index gratings in DR1/PMMA Conditions for DR1/PMMA as optical Kerr media Z-scan measurement using a LG 1 0 beam I-scan measurement Optical limiting Conclusion 140 Appendices 144 A Generalized Gaussian Beam Decomposition 144 A.1 General Derivation A.2 Examples A.2.1 LG 0 0 beam A.2.2 LG 1 0 beam B Simplifying the normalized Z-scan transmittance T 158 B.1 LG 0 0 beam B.2 LG 1 0 beam C Evaluating the normalized optical limiting transmittance T 164 C.1 LG 0 0 beam C.2 LG 1 0 beam D Intensity and and power of an LG beam 173 D.1 Intensity of an LG beam vii

9 D.2 Power of an LG beam viii

10 List of Figures 1.1 The intensity profiles (upper) and the wavefront (lower) of a fundamental gaussian beam ((a) and (c)) and a LG 1 0 beam ((b) and (d)) Self-lensing of a fundamental gaussian beam when it traverses a thin optical Kerr medium. O.T., optical thickness. Upper: a fundamental gaussian beam, whose radial intensity distribution is shown to the left, traverses a thin optical Kerr medium. Lower: (a) when n 2 < 0, the medium resembles a concave lens; (b) when n 2 > 0, the medium resembles a concave lens Ray diagram of the effect of a positive lens on the propagation of a fundamental gaussian beam. (a) If the lens is placed before the minimum beam waist, the far-field pattern of the beam is more spread out; (b) If the lens is placed after the beam waist, the far-field pattern of the beam is more confined The optical thickness of a thin optical Kerr medium illuminated by a LG 1 0 beam. O.T., optical thickness. Upper: a LG 1 0 beam, whose radial intensity distribution is shown to the left, traverses a thin optical Kerr medium. Lower: the optical thickness of the medium when (a) n 2 < 0 and (b) n 2 > Intensity profiles of some Laguerre Gaussian beams of different orders. l is the angular mode number and p is the radial moder number ix

11 2.2 Comparison of the phase profiles of a LG 0 0 beam and a LG 1 0 beam near their beam waists. (A) The phase profile on the transverse plane of the LG 0 0 beam (the transverse plane is also the plane of equal phase in the LG 0 0 beam.); (B) The phase profile on the transverse plane of the LG 1 0 beam; (C) The plane of equal phase of the LG 1 0 beam Isomers of the DR1 molecule Schematic energy diagram of the photoisomerization process. Path 1: trans isomers with absorption cross section σ t jump to the excited state by absorbing photons; Path 2: molecules in the trans excited state relax to the cis ground state with a quantum yield (or probability) of Φ tc ; Path 3: at room temperature, cis isomers relax to the trans isomer thermally with a rate of γ; Path 4: cis isomers with absorption cross section σ c jump to the excited state by absorbing photons; Path 5: molecules in the cis excited state relax towards the trans ground state with a quantum yield (or probability) of Φ ct The dynamics of n in DR1/PMMA at short time scales as calculated from Eq.(2.39) with the following parameters: γ = 1 s 1, ξ tc I = 0.01 s 1, η tp = 2, and η c = Configuration of the LG beam propagation problem Schematic diagram of the Z scan experiment. L: lens, S: sample, A: aperture, and D: detector A typical Z-scan trace for positive (solid line) and negative (dotted line) Φ A typical LG 1 0 Z-scan trace for positive (solid line) and negative (circles) Φ 0. The T = 1 level is indicated by the dashed line Comparison of a typical LG 1 0 Z-scan trace with a typical LG 0 0 Z-scan trace. The values of Φ 0 are chosen such that the major peaks (valleys) of the two traces almost overlap. Also shown is the T=1 line x

12 2.11 The Z-scan normalized transmittance for a LG 1 0 beam as a function of transverse coordinate R. ( Φ 0 = 0.1) The Z-scan normalized transmittance for a LG 1 0 beam for R = 0, R = 0.05 and R = 0.1. ( Φ 0 = 0.1) Illustration of the transmittance of the optical limiter Schematic diagram of optical limiting using the LG beam. L1: focusing lens, S: nonlinear thin film, L2: Fourier transform lens, A: small aperture, D: optical component to be protected, f2: focal length of L Typical curves of normalized optical limiting transmittance T vs. position Z. Φ max = 0.1. The circled line is for the LG 0 0 beam, the solid line is for the LG 1 0 beam, and the dashed line shows T = The normalized optical limiting transmittance T versus the maximum nonlinear phase distortion Φ max in the sample when the incident beam is a LG 0 0 beam. The position Z of the sample for each of the curve is indicated by the number along the curve. A sample of negative n 2 is assumed The normalized optical limiting transmittance T versus the maximum nonlinear phase distortion Φ max in the sample when the incident beam is a LG 1 0 beam. The position Z of the sample for each of the curves is indicated by the number along that curve The maximum nonlinear phase distortion Φ max as a function of the normalized transmittance T. The incident beam is a LG 0 0 beam and the position of the sample is Z=-3 and Z=3 for the upper and lower curve, respectively. The dots are the calculated results and the lines are the linear fits xi

13 2.19 The maximum nonlinear phase distortion Φ max as a function of the normalized transmittance T. The incident beam is a LG 1 0 beam and the position of the sample is Z=-1.73 and Z=1.73 for the upper and lower curve, respectively. The dots are the calculated results and the lines are the linear fits The maximum nonlinear phase distortion Φ max as a function of the normalized transmittance T. The incident beam is a LG 1 0 beam and the position of the sample is Z=-8.55 and Z=8.55 for the upper and lower curve, respectively. The dots are the calculated results and the lines are the linear fits Solid lines: selected curves of the normalized transmittance T versus the maximum nonlinear phase distortion Φ max in the sample when the incident beam is a LG 1 0 beam. The position Z of the sample for each of the curves is indicated by the number along the curve. Dotted line: the coordinates of the valleys of the T Φ max curves The sample position Z versus the T coordinate of the valley of the corresponding T Φ max curve. The arrows represent the useful range of the Z T curve for determining the position from the transmittance The sample position Z versus the T coordinate of the valley of the corresponding T Φ max curve. Circles: calculated results. Line: best fit using an inverse Gauss function (see text for details) Normalized transmittance, T, versus the maximum nonlinear phase distortion, Φ max, in the sample when the incident beam is a LG 1 0 beam for selected curves whose Z are between 0.61 and The position, Z, of the sample for each of the curve is indicated by the number along the curve.. 84 xii

14 2.25 Normalized transmittance, T, versus the maximum nonlinear phase distortion, Φ max, in the sample when the incident beam is a LG 0 0 beam for selected curves whose Z values are larger than 0. The position, Z, of the sample for each of the curve is indicated by the number along the curve Schematic diagram of a hologram that converts a LG 0 0 beam into a LG 1 0 beam Typical holographic pattern that converts a LG 0 0 beam to a LG 1 0 beam The multiple orders of beams generated by the binary amplitude hologram Schematic diagram of the interference experiment to exam the phase dislocation of a LG 1 0 beam. M: mirror; BS: beam splitter; DP: dove prism Typical self-interference pattern of a LG 1 0 beam with a dove prism placed in one arm. The three-prong fork in the center is evidence that the angular mode number l of the incident beam is 1 (or -1) The alumina-filled column used to remove the inhibitor from the MMA Diagram of the squeezer that is used to press thick polymer films Diagram of the diffraction of a light beam in an index grating Illustration of forming the index grating by two-beam coupling Setup of the holographic volume index grating recording in DR1/PMMA and the in-situ diffraction efficiency measurement system Schematic diagram of the setup for the experiments using a LG 1 0 beam ( Z-scan measurement, I-scan measurement and optical limiting). WP: half wave plate, P1,P2: polarizers, CGH: computer generated hologram, AP1,AP2: apertures, M1,M2: mirrors, L1-L4: lenses, PH: pin hole, BS: beam splitter, D1,D2: detectors Absorption spectrum of DR1/PMMA. The arrow shows the wavelength which is used in our experiments. OD, optical density xiii

15 4.2 Diffraction efficiency as a function of time n 1 as a function of time in the grating recording experiment. Upper: the data and the best-fit with a single exponential onset function, Lower: the data and the best-fit with a biexponential onset function Saturation values n 1 as a function of the amplitude of intensity modulation at the front surface of the sample Experimental (squares) and theoretical (solid curve) results of the Z-scan of a DR1/PMMA sample using a LG 1 0 beam. Also shown is the theory for a LG 0 0 Z-scan trace (dashed curve) Φ 0 vs. power of the incident beam. The circles are the experimental data. The line is a linear fit of the data. Also shown is a data point (the square) obtained for a beam power higher than the range within which the sample responds like an optical Kerr medium Normalized transmittance T as a function of the maximum beam intensity at the front surface of a DR1/PMMA sample placed at Z = 1.6. The circles are the experimental data. The line is a linear fit of the data The maximum nonlinear phase distortion Φ max as a function of the normalized transmittance T. The incident beam is a LG 1 0 beam and the position of the sample is Z=-1.6. The dots are the calculated results and the line is the linear fit Optical limiting using a LG 1 0 beam and a DR1/PMMA sample placed at Z = 0.6. The circles are the experimental data showing the normalized transmittance (T) as a function of the maximum beam intensity (I max, bottom axis) at the front surface of the sample. The curve shows the theory for T vs. the magnitude of the maximum nonlinear phase distortion ( Φ max, top axis) on the sample assuming the sample is an optical Kerr medium xiv

16 4.10 Power transfer curve of the optical limiting system with sample position Z = 0.6. The dots are the experimental data, and the line shows the response of the system with no optical limiting Comparison of the optical limiting effect with the sample in front of the beam focus (negtive Z) and behind the beam focus (positive Z). Left: experimental results of the normalized transmittance (T) as a function of the maximum beam intensity (I max ) in DR1/PMMA, where the squares are data at Z= 0.6, and the circles are data at Z= 7. Right: calculated results of the normalized transmittance (T) as a function of the magnitude of the maximum nonlinear phase distortion in the sample ( Φ max ), assuming an optical Kerr medium. The solid line is for Z= 0.6, and the broken line is for Z= xv

17 List of Tables 4.1 Time constants determined from a biexponential onset function fit to grating data Radius of beam waist, ω 0, obtained by Z-scan curve fit xvi

18 Chapter 1 Introduction In this dissertation we study the interaction between a nonlinear optical Kerr medium in the form of a thin film and a special, somewhat mysterious, laser beam called a twisted light beam, which has a spiral wave front, a dark center in the transverse plane, and carries orbital angular momentum (OAM). It is well known that laser cavities can produce laser beams of different modes, each of which is a solution of the wave equation under the constrain of the cavity resonator. 1 3 In cylindrical coordinates (r, φ, z), a complete set of solutions, known as the Laguerre- Gaussian (LG) beams, 1 can be obtained. Each member of the set is characterized by two mode numbers, the angular mode number, l (l = 0, ±1, ±2,...), and the transverse radial mode number, p (p = 0, 1, 2,...), written as LG l p. Among the LG beams, the LG 0 0 beam is probably familiar to most readers. It is more frequently referred to as the fundamental gaussian beam, or just a gaussian beam when there is no ambiguity, due to the gaussian distribution ( exp( r 2 )) of its intensity in the beam s transverse plane (r, φ plane). Its phase profile has no φ dependence, and resembles that of a plane wave near the beam waist. The high order LG beams are like the fundamental gaussian beam in many aspects: their beam radii all reach a minimum at the beam focus and diverge when getting away from the focal point. Meanwhile, there are 1

19 Figure 1.1: The intensity profiles (upper) and the wavefront (lower) of a fundamental gaussian beam ((a) and (c)) and a LG 1 0 beam ((b) and (d)). many differences between the different modes, among which the most predominant ones are the intensity and phase profiles. For example (see Fig. 1.1), the LG 1 0 beam s phase profile has a factor of exp(iφ), making the phase at the center (r = 0) undetermined and forming a screw dislocation, 4 and the wavefront is screw-shaped, unlike the flat wavefront of a plane wave. Corresponding to the phase singularity, the intensity profile is characterized by a null in the center, in contrast to the bright center of the fundamental LG 0 0 beam. Sometimes, the non-fundamental LG beams are referred to as twisted light beams due to their twisted wavefront. In section 2.2, we have a more detailed review of the properties of the LG beams. Historically the fundamental Gaussian beam LG 0 0 has been the most commonly studied LG mode in both theory and experiment, probably because among all the modes it is the only one that resembles a plane wave near the beam waist, and it is widely available though commercial lasers. However, recently higher order LG beams, especially those 2

20 with higher angular mode number l, are attracting more attention, because of their intriguing properties associated with the non-plane-wave-like phase and intensity profiles. For example: The unique intensity profiles of the high order LG beams allow them to be used as optical levitators 5 and optical tweezers that can trap small particles having not only an elevated refractive index but also a lower refractive index than the surrounding medium. 6 The spiral interference pattern formed by LG beams can be used to control the rotation of the trapped small particles. 7 9 The well-defined null throughout the beam axis helps to align the beam. 10 The screw-like phase singularity makes LG beams suitable for the study of optical vortices, 11, 12 which are named from their similarity to the vortices in fluids. Some standard experiments have been revisited by researchers with high LG beams, such as scattering 13 and double slit interference. 14 Among the many studies using LG beams, one must mention the work by Allen et al, who revealed that the LG beam possesses well defined orbital angular momentum (OAM) of l h per photon. 15 Their work triggered a series of investigations involving the OAM carried by the LG beams. In the mechanical aspect, experiments have been done to transfer OAM from LG beams to microscopic particles, 16 to rotate the microscopic particles using LG beams like optical spanners, 17 and to observe the rotational Doppler effect. 18 In the area of nonlinear optics, several wave mixing processes have been investigated using high-order LG beams, including second-harmonic generation, four wave mixing, 22 and parametric down conversion In particular, the parametric down conversion experiments used the OAM state of the photons in high order LG beams to realize 3

21 multi-dimensional entanglement of quantum state, 24, 27 which provides a practical route to multi-dimensional quantum computation and communication. Within this perspective, some techniques to measure 28, 29 and to store 30 the OAM information carried by the photons have been proposed and tested. There are also studies of OAM spectra of beams 31 and imaging with OAM. 32 One of our research efforts in the Nonlinear Optics Laboratory (NOL) at Washington State University (WSU) is to search for new phenomena that result from the interaction between intense light and a material, and to apply such phenomena to build novel alloptical devices. As such, when the rising importance of high-order LG beams caught our attention, we were immediately motivated to study their interaction with a nonlinear material. In conventional (linear) optics, the properties of a medium, such as the refractive index and the absorption coefficient, are assumed to be constants for beams of given frequency. The presence of one beam can not alter the properties of the media nor can it mediate the interaction between beams. In a linear medium, light beams obey the principle of superposition, i.e., the response to multiple input beams is a sum of the responses to each one of the beams. In general linear optics works well if the electric field strength of the incident beam is weak compared to the internal fields. However, when the electric field strength of the incident beam gets sufficiently strong, superposition fails. For example, the refractive index or the absorption coefficient of a medium may become intensity dependent, so a strong beam may be able to manipulate the behavior of a weak beam, and under proper configurations, two input beams may be combined together and become one beam whose frequency is the sum or difference of the two input beam frequencies. Such phenomena are beyond the scope of linear optics, but are the subject of nonlinear optics. In this work we focus on the nonlinear optical phenomena that result from the interaction between beams and media whose refractive indices are intensity-dependent. In general the way that the refractive index of a nonlinear optical medium depends on the 4

22 intensity can be of any form. But the simplest form of the dependence is called the optical Kerr type, which requires that the refractive index depend linearly on the intensity, or n = n 0 + n 2 I, where n 0 is the conventional refractive index and n 2 is called the nonlinear refractive index coefficient. Materials that have an optical Kerr type refractive index are called optical Kerr media. Because of its simple mathematical form, the optical Kerr medium is often used as a simple case in theoretical derivations. And many materials act as optical Kerr media under proper conditions. Figure 1.2: Self-lensing of a fundamental gaussian beam when it traverses a thin optical Kerr medium. O.T., optical thickness. Upper: a fundamental gaussian beam, whose radial intensity distribution is shown to the left, traverses a thin optical Kerr medium. Lower: (a) when n 2 < 0, the medium resembles a concave lens; (b) when n 2 > 0, the medium resembles a concave lens. A well studied phenomenon is the self-lensing effect of a fundamental gaussian beam when it traverses a thin optical Kerr medium. As illustrated in Fig. 1.2, when an intense LG 0 0 beam propagate through an optical Kerr medium, the refractive index, and therefore 5

23 the optical thickness (or the optical path length) of the medium is changed according to the beam s transverse intensity distribution, which is a gaussian function (shown in the top left in the figure). If n 2 > 0, the beam induces a higher optical thickness in the center than in the periphery like a convex lens, causing the beam to converge, or self-focuse ; If n 2 < 0, the sample will have a lower optical thickness in the center than in the periphery like a concave lens, causing the beam to diverge, or self-defocuse. Figure 1.3: Ray diagram of the effect of a positive lens on the propagation of a fundamental gaussian beam. (a) If the lens is placed before the minimum beam waist, the far-field pattern of the beam is more spread out; (b) If the lens is placed after the beam waist, the far-field pattern of the beam is more confined. The self-lensing effect has several important applications. For example, using ray optics, it s easy to show that depending on the position of the nonlinear sample, i.e., the introduced lens, with respect to the location of the beam waist, the far-field pattern of the beam can appear to be either dilated or constricted. The case of a positive lens is illustrated as an example in Fig Based on this phenomenon, a high-sensitivity n 2 measurement technique called Z-scan was developed. 33, 34 If the induced lens causes the beam to focus more tightly, then an increase in beam intensity will result in less light in the beam center in the far field. Thus optical limiting can be achieved by, for example, placing an aperture around the beam axis in the far field and only observing the light 6

24 35, 36 through the aperture. Figure 1.4: The optical thickness of a thin optical Kerr medium illuminated by a LG 1 0 beam. O.T., optical thickness. Upper: a LG 1 0 beam, whose radial intensity distribution is shown to the left, traverses a thin optical Kerr medium. Lower: the optical thickness of the medium when (a) n 2 < 0 and (b) n 2 > 0. Following the example of the LG 0 0 beam, one would naturally speculate about the consequence of a high-order LG beam transversing an optical Kerr medium. However, the intensity profile of a high-order LG beam is more complex than the fundamental gaussian beam. For example, Fig. 1.4 shows the transverse intensity profile of a LG 1 0 beam, as well as the optical thickness of an optical Kerr medium under the illumination of the beam. Clearly, the sample neither acts as a concave nor a convex lens. As such, a simple ray diagram can not be used to determine the far field profile after a nonlinear sample as was the case of the fundamental gaussian beam. And whether or not high-order LG beams can be used in the Z-scan measurement or optical limiting is unclear. Furthermore, since high-order LG beams may carry OAM, it is natural to wonder if the OAM of the beam will 7

25 be changed by the nonlinear interaction. These questions, to the best of our knowledge, have never been previously discussed and are worth exploring. A large portion of this dissertation is thus devoted to developing a theory that can answer the above questions. In doing so, we build on the existing study of the fundamental gaussian beam. First, some of the theoretical approaches that have been implemented to study the case of the fundamental gaussian beam can be generalized to study the case of LG beams of arbitrary orders. To be specific, our theory is developed with a method which is a generalization of the gaussian beam decomposition method used by Weaire, et. al.. 37 Secondly, as one of the modes of all LG beams, the fundamental gaussian beam should conform to the generalized theory. Therefore, the case of the fundamental gaussian beam can always be used as an initial test of the validity of the generalized theory. Third, a comparative study of the case of the fundamental gaussian beam and high-order LG beams would be helpful in determining the merits and shortcomings of each, especially when considering their applications. We also carry out experiments to test the validity of our theory as well as to demonstrate the proposed applications. We use disperse red 1 (DR1) doped poly(methyl methacrylate) (PMMA) (DR1/PMMA) samples as the optical Kerr medium in our experiments. As a dye doped polymer, DR1/PMMA has the advantages of low cost, ease of fabricating thin films, and ease of mechanical processing, compared to inorganic materials. Previously in the NLO lab, we found that DR1/PMMA can show big nonlinear intensity-dependent refractive index change with off-resonant beams (i.e., where the material is transparent), and we have successfully demonstrated several nonlinear optical processes that require an intensitydependent refractive index In this work, however, we apply theoretical modeling and experimental measurement to determine the conditions under which the DR1/PMMA sample can be treated as an optical Kerr medium. We then design experiments that at least approximately obey these conditions for LG beams. The dissertation is organized as follows: Chapter 2 presents the theory and the princi- 8

26 ples of the applications. We start with a brief review of the Laguerre-Gaussian beam (Sec. 2.2) and the optical Kerr medium (Sec. 2.3), including a study of the mechanisms of the optical nonlinearity in DR1/PMMA samples, namely, the trans-cis-trans photoisomerization and photoreorientation mechanisms. Then in section 2.4, we present our theory on the effect of a thin optical Kerr medium on a Laguerre-Gaussian beam. Subsequently, we propose several applications based on our theory, including new methods to measure the nonlinear refractive index coefficient (the Z-scan technique in section 2.5 and the I-scan and Φ max -scan techniques in section 2.7) and optical limiting (section 2.6). Chapter 3 describes the details of the experiments, including how to generate high order LG beams (Sec. 3.2), sample fabrication (Sec. 3.3), a holographic volume index gratings recording experiment for the purpose of studying the properties of the DR1/PMMA samples (Sec. 3.4), and most importantly, the experiments that implement LG beams (Sec. 3.5). In chapter 4 we show the experimental results and discuss their implications. We first summarize the properties of our DR1/PMMA samples, particularly the conditions under which they can be treated as an optical Kerr medium (Sec. 4.1). We then test the validity of our theory and show that the Z-scan (Sec. 4.2) and the I-scan (Sec. 4.3) techniques using the LG 1 0 beams can measure correctly the nonlinear refractive index coefficients. Finally in Sec. 4.4 we demonstrate optical limiting in DR1/PMMA using a LG 1 0 beam and discuss limitations of our technique as well as show the advantages of using LG 1 0 beams over LG 0 0 beams. We concludes the dissertation with Chapter 5. 9

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32 Chapter 2 Theory 2.1 Introduction The purpose of this dissertation is to study the propagation of light beams with orbital angular momentum (OAM) in a nonlinear optical material. As such, the two key elements in our theory are the Laguerre-Gaussian (LG) beam 1 and the optical Kerr medium. 2 Before we elaborate on our theory, we spend the first two sections (2.2 and 2.3) of this chapter reviewing these two concepts. Because we use disperse red 1 (DR1) doped poly(methyl methacrylate) (PMMA) (DR1/PMMA) samples as the optical Kerr media in our experiments, we also study in section 2.3 the mechanisms of the optical nonlinearity in DR1/PMMA samples, namely, trans-cis-trans photoisomerization and photoreorientation. 3 5 In specific, we develop a three-state model to formulate the mechanisms and point out the conditions under which the DR1/PMMA samples can be treated as the optical Kerr media. In section 2.4, we present our theory on the effect of a thin optical Kerr medium on a Laguerre-Gaussian beam. After that, we propose several applications according to our theory, including the methods to measure the nonlinear refractive index coefficient (the Z scan technique in section 2.5 and the I scan and Φ max scan technique in section 2.7) and 15

33 optical limiting (section 2.6). 2.2 A review of the Laguerre-Gaussian beams Laguerre-Gaussian (LG) modes were discovered soon after the invention of the laser. 1 Under the slowly varying envelope approximation, the paraxial wave solution of the wave equation, U( r,t) = A( r) exp ( ikz) exp (i2πνt), can be obtained by solving the scalar paraxial Helmholtz equation 6 2 TA i2k A z = 0, (2.1) where A( r) is a slowly varying complex function of position which characterizes the amplitude of a wave component, T = 2 / x / y 2, k is the wave vector, and ν is the frequency. For simplicity, we will not write exp (i2πνt) explicitly in the following. In cylindrical coordinates (r,φ,z), a complete set of solutions, known as the LG beams, can be obtained. Each member of the set is called a mode specified by two mode numbers, the angular mode number and the transverse radial mode number. An LG beam with angular mode number l (l = 0, ±1, ±2,...) and transverse radial mode number p (p = 0, 1, 2,...) can be written as: ( ) ( ) l ( ) LG l ω0 2r 2r p (r,φ,z) = L l 2 p ω (z) ω (z) ω 2 (z) ( ) ( ) r 2 kr 2 z exp exp i ω 2 (z) 2 (z 2 + z ( )) r) 2 exp (i (2p + l + 1) tan 1 zzr exp ( ilφ) exp ( ikz), (2.2) where ω 0 is the beam waist radius, z r = kω 2 0/2 is the Rayleigh length, ω(z) = ω 0 (1+z 2 /z 2 0)

34 is the beam radius at z, and L l p is the associated Laguerre polynomial defined as 7 L l p(x) = p ( 1) m (p + l)! (p m)! (l + m)!m! xm, l > 1. (2.3) m=0 When l = 0 and p = 0, the LG beam becomes the familiar fundamental Gaussian beam ( ω0 ) ( ) r LG (r,φ,z) = exp ω (z) ω 2 (z) ( ) ( )) kr 2 z exp i exp (i tan 1 exp ( ikz). 2 (z 2 + zr) zzr 2 (2.4) High order ( l > 0 or p > 0) LG beams share a few important properties with the fundamental Gaussian beam, such as: 1. The intensity distributions in the transverse planes in one beam are similar regardless of the beam propagation, with the beam radius, ω(z), as the scale factor at the position z. 2. The beam divergence, or how the beam radius, ω(z), changes as a function of z, is totally determined by its minimum value, ω 0, which is reached at a place called beam waist. 3. A thin convex or a concave lens can focus or defocus an LG beam without affecting its mode numbers. In contrast, there are important differences between the high order LG beams and the fundamental Gaussian beam, among which are: 1. The intensity distributions in the transverse planes are different from one mode to another. The typical patterns consist of concentric bright rings and/or dark rings with a bright center (if l = 0) or a dark center (if l 0). And p gives the number of dark rings, which is the reason why it is called the radial mode number. Figure 2.1 shows the intensity distributions of some LG beams. 17

35 Figure 2.1: Intensity profiles of some Laguerre Gaussian beams of different orders. l is the angular mode number and p is the radial moder number. 2. While the fundamental gaussian beam has plane-wave-like wavefronts near the beam waist, the high l order LG beams have screw shaped wavefronts, like a twisted (fundamental gaussian) beam. Figure 2.2 shows the phase profiles of a LG 0 0 beam and a LG 1 0 beam near their beam waists. 3. The Guoy phase, which is the extra on-axis phase retardation of the beam in comparison with a plane wave as the wave propagates, is described in a more general form (2p + l + 1) tan 1 (z/z r ). 18

36 Figure 2.2: Comparison of the phase profiles of a LG 0 0 beam and a LG 1 0 beam near their beam waists. (A) The phase profile on the transverse plane of the LG 0 0 beam (the transverse plane is also the plane of equal phase in the LG 0 0 beam.); (B) The phase profile on the transverse plane of the LG 1 0 beam; (C) The plane of equal phase of the LG 1 0 beam. 19

37 4. When l 1, the LG beam possesses well defined orbital angular momentum of l h per photon. 8 The phase term exp ( ilφ) of the LG beams for l > 0 accounts for many of of the above differences. For example, it is the cause of the twisted wavefronts; it makes the on-axis (r = 0) phase undefined, which makes it a singularity or a screw dislocation, 9 forcing the intensity to be zero at the center; and it is closely related to the orbital angular momentum possessed by the LG beams. Finally we introduce a notation that we will use in the rest of this dissertation. For a given LG beam mode, z r (or ω 0 ) alone is sufficient to characterize the relative amplitude and phase of the electric field of the beam. When multiple beams are involved, it is often necessary to specify the waist locations of each beam. For convenience we use C LG l p(r,φ,z z w ;z r ) to describe an LG beam unambiguously, where z w is the waist location on the z axis and C is a complex constant that gives the amplitude and the initial phase. 2.3 Optical Kerr effect and trans-cis-trans photoisomerization and photoreorientation In this section, we first review the optical Kerr effect and the intensity dependent refractive index. In the second part we use a three-state ( (1) molecules in trans form parallel and (2) perpendicular to the polarization of the incident beam; and (3) in the cis form) model to describe the trans-cis-trans photoisomerization and molecule reorientation effect, in which emphasis is placed on the conditions under which the response of the material can be treated as the optical Kerr effect. 20

38 2.3.1 Optical Kerr effect Historically different systems of units have been used in nonlinear optics, such as the gaussian system and several versions of MKS systems. 2, 6, 10 Although the expressions and even some definitions are different from one to another, the physics content is the same. In this brief review we mainly follow the convention in reference 6, which is one of the versions using the MKS system. It is well known that in linear optics, the polarization P(t) of a material system depends linearly on the applied optical field E(t) i.e., P (t) = ǫ 0 χ (1) E (t), (2.5) where P (t) = 1 2 ( P (ω)e iωt + c.c. ), (2.6) E (t) = 1 2 ( E (ω)e iωt + c.c. ), (2.7) and χ (1) is the linear susceptibility. For simplicity, we have treated P(t) and E(t) as scalars and assumed that the material is lossless and dispersionless. In general, however, the relation between the two is not necessarily linear. If expressed as a power series in the electric field strength, the polarization can be written as P (t) = ǫ 0 χ (1) E (t) + 2χ (2) E 2 (t) + 4χ (3) E 3 (t) +, (2.8) where, χ (n) describes the nth-order nonlinear effect and is called the nth-order nonlinear optical susceptibility. The high-order terms in Eq. (2.8) are usually much smaller than the first order susceptibility, so the linear relationship is a good approximation when the electric field strength is sufficiently weak. When the electric field strength gets stronger, however, it is necessary to include some of the higher order susceptibilities. It can be 21

39 shown that if the material possesses inversion symmetry, the even order nonlinear optical susceptibilities vanish. Then to the lowest order nonlinear term P (t) = ǫ 0 χ (1) E (t) + 4χ (3) E 3 (t). (2.9) Substituting Eq. (2.7) into the above equation, we have P (t) = 1 2 ǫ 0χ (1) E (ω)e iωt χ(3) E 3 (ω)e i3ωt χ(3) E (ω) 2 E (ω)e iωt + c.c.. (2.10) One immediately sees that the induced polarization has a new frequency component 3ω, which is the third-harmonic frequency of the applied field. Third harmonic generation requires a material that responses in an optical cycle, which is not the case for the photoisomerization/reorientation mechanisms that are the focus of this work. Here we focus on the ω frequency component of the polarization, which can be expressed as P (ω) = ( ǫ 0 χ (1) + 3χ (3) E (ω) 2) E (ω). (2.11) Defining the effective susceptibility as χ eff = χ (1) + 3χ(3) E (ω) 2 ǫ 0, (2.12) then P (ω) = ǫ 0 χ eff E (ω). (2.13) The above equation together with the Maxwell equations can be solved using the usual procedures, 11 which then give us the refractive index of the material n = 1 + χ eff. (2.14) 22

40 In the linear case, we have n 0 = 1 + χ (1), (2.15) with the help of Eq. (2.14), this can be rewritten as n = n χ(3) E (ω) 2 ǫ 0 n 2 0 n 0 + 3χ(3) E (ω) 2 2ǫ 0 n 0. (2.16) 6, 11 Recall the intensity of light is given by I = 1 2η E (ω) 2, (2.17) where η = µ ǫ = η 0 n 0 (2.18) is the impedance of the material and η 0 = (µ 0 /ǫ 0 ) 1/2. After replacing E (ω) 2 with the intensity of the light, Eq. (2.16) becomes n = n 0 + 3η 0χ (3) I. (2.19) ǫ 0 n 2 0 Defining the nonlinear refractive index coefficient as n 2 3η 0 χ (3), (2.20) ǫ 0 n 2 0 then n = n 0 + n 2 I. (2.21) 23

41 The effective refractive index of the material is a linear function of the intensity of the incident light. This effect is known as the optical Kerr effect. The optical Kerr effect can be observed in many materials. For materials that posses inversion symmetry including centro-symmetry, it is often the lowest-order nonlinear effect that appears when the intensity of the incident beam is increased beyond the linear regime. Due to the intensity-dependent feature, the refractive index of the material can be modified by the incident optical beam. As a consequence, the propagation of the beam itself is affected by the modified refractive index. A lot of interesting phenomena result from this interaction. As shown later, the major part of this dissertation is devoted to the interaction between a thin optical Kerr medium and an LG beam Mechanisms of trans-cis-trans photoisomerization and photoreorientation One of the nonlinear materials frequently used in the nonlinear optics lab at Washington State University is disperse red 1 (DR1) dye doped poly(methyl methacrylate) (PMMA) polymer (DR1/PMMA) due to its big nonlinear effect and easiness of synthesis and processing. The big nonlinear effect of DR1/PMMA is due to the photo-induced trans-cis-trans isomerization of DR1 molecules followed by reorientation in the direction perpendicular to the polarization of the laser beam. In this section, we develop a simple theory to explain the mechanisms of photoisomerization and photoreorientation. The theory is not intended to be a precise description of all aspects of the real physical system. For example, the geometry of a real sample is three-dimensional, but our theory is a highly idealized model that approximates the dynamics of the real system. The purpose is to help understand the experimental observations qualitatively without complex mathematics or numerical calculations. Emphasis is placed on the conditions under which the response of the material can be treated as an optical Kerr effect. The model is an improvement of the one in reference 5 24

42 where the cis population is totally ignored and only the effects of photoreorientation (but not photoisomerization ) are considered. DR1 molecules and photoisomerization Figure 2.3: Isomers of the DR1 molecule. A DR1 molecule can exist in two geometric forms, or isomers, as shown in Fig In the trans form, the two substituent groups are oriented on the opposite sides of the nitrogen double bonds, while in the cis form, the two substituent groups are oriented on the same side of the nitrogen double bonds. Due to the difference in their shapes, the response of the two isomers to electric fields and light are different. The trans isomer, having a shape like a cigar, is anisotropic in 25

43 response to the external field because it forms a larger dipole if the polarization of the applied field is parallel to the axis of the cigar than if it is perpendicular. A cis isomer, more like a ball, responds to the external field more isotropically. The energy levels of the two isomers are slightly different. The trans isomer has lower energy levels than the cis isomer, as indicated in Fig Therefore, most DR1 molecules are in the trans form at the room temperature. However, if an optical field at the proper wavelength is applied, a trans isomer can jump to the excited state by absorbing a photon, where it either relaxes back to the trans ground state or to the cis ground state. A cis isomer, once formed, decays to the trans isomer through thermal relaxation. Or a cis isomer can be excited to a higher energy level by absorbing a photon, then can relax to either the trans ground state or the cis ground state. The process of trans to cis, then back to trans, is called photoisomerization. Fig. 2.4 shows a schematic energy diagram of the photoisomerization process. Figure 2.4: Schematic energy diagram of the photoisomerization process. Path 1: trans isomers with absorption cross section σ t jump to the excited state by absorbing photons; Path 2: molecules in the trans excited state relax to the cis ground state with a quantum yield (or probability) of Φ tc ; Path 3: at room temperature, cis isomers relax to the trans isomer thermally with a rate of γ; Path 4: cis isomers with absorption cross section σ c jump to the excited state by absorbing photons; Path 5: molecules in the cis excited state relax towards the trans ground state with a quantum yield (or probability) of Φ ct. 26

44 DR1/PMMA and photoreorientation A polymer such as PMMA consists of many entangled long molecular chains. The entanglement is statistically random, leaving many small empty space, or voids between chains. The distribution of the sizes of these voids depend on the polymerization conditions such as the amount and type of initiator (which chemically cause the polymer to form), the temperature, and the pressure. An example in daily life is the sponge with a lot of small pores. Just as the sponge can hold water, we can dope PMMA with molecules such as DR1 through a special process (see the experimental part of this dissertation). The DR1 molecules are then trapped inside those voids and may have limited freedom of mobility depending on the size and shape of each individual void. A fresh (having not been exposed to light) DR1/PMMA sample is usually homogeneous with the orientation of the trans isomers evenly distributed. The refractive index of the sample is therefore isotropic. If we pump a DR1/PMMA sample with a linearly polarized light beam, two things happen. First, some of the trans isomers are converted to cis isomers due to the photoisomerization. Second, the trans isomers whose long axis are parallel to the polarization of the beam are more likely to be excited and converted to cis isomers than those are not. The cis isomers, being smaller than trans isomers, move and rotate much more easily in the PMMA voids than the trans isomers. As a consequence, when a cis isomer relaxes back to the trans states, its orientation is not necessarily the same as before. In the long run, more and more trans isomers that are oriented along the polarization direction of the incident beam are depleted and converted to trans isomers oriented in other directions. This is called photoreorientation. Both the photoisomerization and the reorientation result in changes of the properties of the material, including the mechanical 5 and the optical properties. 4, 12, 13 In this dissertation, we focus on the change of the refractive index of the material as seen by the incident beam. 27

45 The idealized three-state model To catch the key dynamics of the processes without overly complicated mathematics, we use three-state model to approximate the photoisomerizing system. The model has the following approximations: 1. A DR1 molecule in DR1/PMMA can only be in one of the following three states: (a) a trans isomer parallel to the polarization of the incident light (assuming linearly polarized beam); (b) a trans isomer perpendicular to both the polarization and the wave vector of the incident beam; (c) a cis isomer which is isotropic. 2. A trans isomer interacts with light only if it is oriented parallel to the polarization of the incident light. 3. When relaxing back to the trans form from the cis form, a molecule has equal possibilities to be oriented in either of the two orientations. 4. An entropic process independent of the light intensity always tries to equalize the populations of the trans isomers in both orientations. By making the above assumptions, we mainly ignore the following facts about the real material system: 1. The trans isomer can orient in all directions in the three-dimensional space, interacting with the light differently depending on the orientation. 2. The cis isomer is not perfectly isotropic. 28

46 3. The dynamical behavior of the molecules, such as the entropic decay of the orientation of the trans isomers, are affected by their environment, i.e. the PMMA voids surrounding them, and may vary form site to site. Now let s define the following quantities: 1. N tp : the fraction of molecules in the trans form that is oriented parallel to the polarization of the incident light beam. 2. N c : the fraction of molecules in the cis form. The fraction of molecules in the trans form oriented perpendicular to both the polarization and the wave vector of the incident beam is thus N ts = 1 N tp N c. 3. I: the intensity of the light beam. 4. ξ tc : the probability rate per unit intensity of light in the material that a trans isomer will be converted into a cis isomer. 5. ξ ct : the probability rate per unit intensity that a cis isomer will be converted into a trans isomer. 6. γ: the thermal relaxation rate of the cis isomer. 1/γ thus gives the lifetime of the cis isomer in darkness. 7. β: the entropic decay rate of the anisotropy due to the trans isomer orientation. With these definitions, we are ready to formulate the processes. But before that, we point out that from N tp (t) and N c (t), we can determine the change of the refractive index n(t) along the light s polarization. Taking the differential of Eq. (2.14), it can be shown that for small χ eff, n is given by n χ eff 2n 0. (2.22) 29

47 But χ eff is connected to the change of the isomer populations N tp and N c by χ eff = χ tp N tp + χ c N c, (2.23) where χ tp and χ c are the contributions to the total effective susceptibility from the trans isomers parallel to the polarization of the incident beam and the cis isomers, respectively. Therefore, we have n χ tp N tp + χ c N c 2n 0 (2.24) = η tp N tp + η c N c, where we have introduced the new constant coefficients η tp and η c for simplicity. Assuming first-order kinetics, the dynamics of the photoisomerization and photoreorientation processes are governed by dn tp dt dn c dt = ξ tc IN tp ξ ctin c γn c + β(1 2N tp N c ), (2.25) = ξ tc IN tp ξ ct IN c γn c, (2.26) where (1 2N tp N c ) is the population fraction difference between the parallel and perpendicular trans isomers. We assume that we start with a fresh sample, so the the initial conditions are: N tp (t = 0) = 1, N 2 c(t = 0) = 0. (2.27) Equations (2.25) and (2.26) can be solved rigorously, yielding general solutions characterized by two exponentially decay functions with different time constants. Here we are interested in the special case in which further approximations can be made according to the properties of the DR1/PMMA samples under our experimental conditions. 30

48 Dynamics over short time scales First we focus on a time scale that is short enough that only a small fraction of isomers is converted (a few seconds). The entropic decay of the anisotropy of the trans isomer orientation in DR1/PMMA is a slow process (hours) compared to the photoisomerization process (seconds), which means β << γ. And at short time scales, the population fraction difference between the parallel and perpendicular trans isomers is small, so (1 2N tp N c ) is a small quantity. Under these conditions, it is reasonable to drop the last term in Eq. (2.25), which yields the solution: N tp (t) = ξ tc I (ξ ct I + γ) (ξ tc I) 2 + (ξ ct I + γ) 2 e λ1t ξ tc I (ξ ct I + γ) (ξ tc I) 2 + (ξ ct I + γ) 2 e λ 2t, (2.28) and N c (t) = ξ tc I ( e λ 2 t e ) λ 1t, (2.29) 2 (ξ tc I) 2 + (ξ ct I + γ) 2 where λ 1 = 1 2 ( ) ξ tc I + (ξ ct I + γ) + (ξ tc I) 2 + (ξ ct I + γ) 2, (2.30) and λ 2 = 1 2 ( ) ξ tc I + (ξ ct I + γ) (ξ tc I) 2 + (ξ ct I + γ) 2. (2.31) 31

49 If the intensity I is low such that ξ tc I << γ and ξ ct I << γ, then the above expressions can be simplified, yielding: N tp (t) ξ ( tci 1 4γ e λ 1t + 2 ξ ) tci e λ2t, (2.32) 4γ and N c (t) ξ tci 2γ ( e λ 2 t e λ 1t ), (2.33) where λ 1 (γ + ξ ct I) ξ tci, (2.34) and λ ξ tci. (2.35) We see that the population dynamics of the isomers are characterized by the two exponential decay functions with time constants 1/λ 1 and 1/λ 2. 1/λ 1, which is dominated by the contribution from γ, is the time needed to build up the population equilibrium between the trans isomers and the cis isomers through photoisomerization. 1/λ 2 gives the time scale for the trans isomers parallel to the polarization of the light to be totally depleted to the perpendicular direction. Because we have ignored the entropic process that reverses such a depletion, 1/λ 2 will need to be modified if we include β into the equations, as will be discussed shortly. It s obvious that 1/λ 1 << 1/λ 2 since we have assumed ξ tc I << γ and ξ ct I << γ. If we are interested only in the short time period within which λ 1 t is no greater than the order of 1, then λ 2 t << 1. Also we assume λ 1 = γ for simplicity. With these further 32

50 approximations, the above results become N tp (t) 1 2 ξ tci 4 t ξ tci ( ) 1 e γt, (2.36) 4γ and N c (t) ξ tci 2γ ( 1 e γt ). (2.37) Using Eq. (2.24) with Eqs. (2.36) and (2.37), the change of the refractive index of the sample as seen by the incident beam is n(t) (2η c η tp ) ξ tci 4γ ( ) 1 e γt ξ tc I η tp t. (2.38) 4 This shows that at any time instant t, n depends linearly on the intensity I. Hence we draw one important conclusion: on short time scales (t 1/γ) the material can be treated as an optical Kerr medium if the intensity of the incident light beam is not too strong. We rewrite Eq.(2.38) to better illustrate the two contributors to the change of the refractive index as n(t) (η tp η c ) ξ tci 2γ ( ) ( 1 e γt ξtc I η tp 4 t ξ tci ( )) 1 e γt, (2.39) 4γ where the first term is the change of the refractive index due to photoisomerization and the second is due to the photoreorientation. A plot of n as well as the two components as a function of time is shown in Fig. 2.5, where the values of the parameters are assumed to be: γ = 1 s 1, ξ tc I = 0.01 s 1, η tp = 2, and η c = 1. The figure shows that at the beginning, both mechanisms contribute to n significantly, but in the long run (after t > 1/γ), photoreorientation wins out. 33

51 0.000 n (arbitrary unit) total n n due to photoisomerization n due to photoreorientation Figure 2.5: The dynamics of n in DR1/PMMA at short time scales as calculated from Eq.(2.39) with the following parameters: γ = 1 s 1, ξ tc I = 0.01 s 1, η tp = 2, and η c = 1. t(s) Dynamics at long time scales For long time scales (t >> 1/γ), the entropic decay of the anisotropy of the trans isomer orientation in DR1/PMMA plays an important role, so the β term in Eq.(2.25) must be retained. Using Eq: (2.26), Eq.(2.25) can be written as dn tp dt = 1 2 ξ tcin tp 1 dn c + β(1 2N tp N c ). (2.40) 2 dt We limit our discussion to low intensity such that ξ tc I << γ and ξ ct I << γ. According to Eq. (2.33), the population of the cis isomers is no more than ξ tc I/(2γ) at any time. 34

52 Specifically, if t >> 1/γ, then N c (t) ξ tci 2γ e λ 2t << 1 (2.41) and dn c (t) dt ξ tci 2 N c(t). (2.42) So N c is a small quantity compared with (1 2N tp ) when t >> 1/γ, as the latter is the population difference between the parallel and the perpendicular trans isomers, which increases with time and approaches 1. Also N c is much smaller than N tp until the majority of the population of the parallel trans isomers are depleted. Thus in the following discussion, we ignore the cis population, yielding from Eq. (2.40), dn tp dt = 1 2 ξ tcin tp + β(1 2N tp ). (2.43) The solution of the above equation is easily obtained, giving N tp (t) = 1 2 ( ( ( ξ tc I 1 exp 2β + ξ ) )) tci t. (2.44) 2 (4β + ξ tc I) 2 Using Eq. (2.24) and ignoring the population of the cis isomers, the change of the refractive index of the sample as seen by the incident beam is ξ tc I n(t) η tp 2 (4β + ξ tc I) ( ( ( 1 exp 2β + ξ ) )) tci t. (2.45) 2 We see that in general, the way n changes with I is not of the optical Kerr type, but saturates in high intensity. Moreover, the saturation time constant also depends on the intensity. However, there are two situations under which the material can be approximated 35

53 as an optical Kerr medium. First, if the intensity is very weak such that ξ tc I << 4β, then n(t) η tp ξ tc I 8β (1 exp ( 2βt)). (2.46) Secondly, if (2β + ξ tc I/2)t << 1 or t << 1/ (2β + ξ tc I/2), then ξ tc I n(t) η tp t. (2.47) 4 We note that usually ξ tc I << 4β is a more strict approximation than ξ tc I << γ for DR1/PMMA since γ >> β. Therefore the amplitude of n using Eq. (2.46) which is limited by η tp ξ tc I/8β is fairly small. Using Eq. (2.47) the behavior overlaps with the short time scale result given by Eq. (2.38), where the requirement of λ 2 t << 1 or t << 1/ξ tc I/2 is now replaced by t << 1/ (2β + ξ tc I/2). Summary We conclude this section by summarizing the conditions under which a DR1/PMMA sample can be treated as a optical Kerr medium. 1. If the intensity of the incident beam is very weak such that ξ tc I << β, the sample can be approximated by an optical Kerr medium at any time. However, the change of refractive index is small because the population of both the cis isomers and the reoriented trans isomers are rather small. 2. If the intensity of the incident beam is not very strong such that ξ tc I << γ (but possibly ξ tc I > β.), the sample can be approximated by an optical Kerr medium for the time range of t << 1/ (2β + ξ tc I/2). The introduced change of the refractive index in this case can grow substantially larger than the previous case. 36

54 2.4 Effect of a thin optical Kerr medium on an LG beam Introduction When an intense light beam propagates through a nonlinear material with an intensitydependent refractive index, the beam will modify the refractive index of the material. As a consequence, the propagation of the optical beam itself will be affected by the modified refractive index. Such phenomenon have been studied extensively for the case that the incident beam is the fundamental Gaussian beam. 2, 14, 15 When the sign of the change of the refractive index is positive(i.e. n 2 > 0), the nonlinear sample acts as a convex lens, causing the beam to converge, or self-focus ; When the sign of the change of the refractive index is negative(i.e. n 2 < 0), the nonlinear sample acts as a concave lens, causing the beam to diverge, or self-defocus. Furthermore, how the Gauss beam will change its shape also depends on the position of the nonlinear sample with respect to the location of the beam waist. For example, for a sample having positive intensity-dependent refractive index, the far-field pattern of the beam will appear to be dilated if the sample is placed before the beam waist, while it will appear to be contracted if the sample is placed after the beam waist. Based on this phenomenon, a high-sensitivity n 2 measurement technique 15, 16 called Z-scan was proposed by Sheik-bahae, etc. An interesting question pertains to what would happen if the incident beam is a high order LG beam. In general, the transverse intensity profile of a high order LG beam is more complex than the fundamental Gaussian beam. For example, the transverse intensity profile of a LG 1 0 beam is like a donut, so the resulting nonlinear refractive index change would neither act as a concave nor a convex lens. As a consequence, how would the beam reshape itself after the nonlinear sample? A schematic diagram of this problem is shown in Fig An LG beam E(r,φ,z) = 37

55 Figure 2.6: Configuration of the LG beam propagation problem. E 0 LG l 0 p0 (r,φ,z;z r ) is focused by a convex lens. The waist of the focused beam is located at z = 0. A nonlinear sample of thickness d is placed at position z = z s along the optical axis of the beam. Our purpose is to analyze the propagation of the beam after it passes through the nonlinear sample. We assume the nonlinearity of the sample is of the optical Kerr type, i.e. n = n 0 + n(i) with n(i) = n 2 I. Historically several approaches have been developed to investigate the propagation of laser beams (mostly fundamental Gaussian beams) inside and through a nonlinear material. Some of them can be modified to the case of the higher order LG beams. We choose to follow the derivation procedures used in references 15 and 16 because it is simple yet practical. We make essential modifications such that the method can apply to the LG beams Field of the beam immediately after the sample We assume that the sample is very thin such that the intensity pattern of the beam does not change within the sample. Two conditions are required to guarantee this assumption. (i): The sample thickness is much shorter than the beam s diffraction length, or d << z r, so linear diffraction within the sample can be neglected. A phase shift of Φ l = n 0 kd will be introduced due to the linear refractive index of the sample. But since Φ l is constant in the beam s transverse plane, it won t affect the beam s propagation other than 38

56 a trivial phase shift, so this effect will be ignored in the following discussion. (ii): d << zr Φ max, where Φ max is the maximum value of the nonlinear phase distortion Φ across the beam s transverse plane due to the nonlinear refractive index change of the sample. This is sometimes referred to as the external self-action condition, which assures that nonlinear refraction can be neglected within the sample. The nonlinear phase distortion Φ depends on the intensity distribution of the beam in the transverse plane and will be taken into account when considering the beam propagation after the sample. Under the above thin sample assumption, Φ is governed by d Φ dz = n(i)k. (2.48) The intensity, I, varies in the sample due to absorption according to di dz = α(i)i, (2.49) where α(i) is the absorption coefficient of the sample material. In general, α(i) includes linear and nonlinear parts and can be written as: α(i) = α + α(i), (2.50) where α is the linear absorption coefficient and is a constant. α(i) is the nonlinear coefficient and depends on the intensity. We assume α(i) = βi, (2.51) where β is the first-order nonlinear absorption coefficient. Using Eq. (2.49) - Eq. (2.51), we can solve for the intensity of the beam as a function 39

57 of distance of propagation through the sample, I = I incidente αz, (2.52) 1 + q where I incident is the intensity at the input surface of the sample, z is the propagation depth in the sample, and q = βi incident z eff, (2.53) where z eff = 1 e αz. (2.54) α Substituting Eq. (2.52) into Eq. (2.48), we can solve the nonlinear phase shift due to the sample: Φ = kn 2 β ln (1 + q). (2.55) The natural logarithm in the above equation can be expanded about q. When q << 1, which is satisfied when either β is very small or the sample is very thin, the total nonlinear phase shift due to the sample is approximately: 1 e αd Φ = kn 2 I incident. (2.56) α The intensity of the beam at the exit surface of the sample is obtained by replacing z with the sample thickness d in Eq.(2.52): I = I incident e αd, (2.57) where q is dropped under the condition q << 1. Eq.(2.56) and Eq.(2.57) together deter- 40

58 mine the complex electric field immediately after the sample, E = E incident e αd 2 e i Φ. (2.58) When the incident beam is an LG beam, E(r,φ,z) = E 0 LG l 0 p0 (r,φ,z;z r ), and the sample is at z = z s on the z axis, Eq.(2.58) becomes: E (r,φ,z s ) = E (r,φ,z s )e αd 2 e i Φ(r,φ,z s). (2.59) Expressing I incident in Eq.(2.56) with the electric field (refer to Appendix D.1), we find the nonlinear phase distortion Φ (r,φ,z s ) obeys: Φ (r,φ,z) = Φ z 2 /z 2 r ( 2r 2 ω 2 (z) ) l0 ( ( )) L l 0 2r 2 2 ( ) 2r 2 p 0 exp, (2.60) ω 2 (z) ω 2 (z) where Φ 0 = π λ cǫ 0n 0 n 2 E 0 21 e αd α (2.61) is a constant proportional to the maximum nonlinear phase change Φ max (z s ) in the sample. This coefficient depends on the radial and angular mode numbers of the LG beam as well as the position of the sample z s. When the incident beam is a LG 0 0 beam, we simply have Φ 0 = Φ max (z s = 0) Propagation of the beam after the sample Eq.(2.59) gives the complex electric field of the beam immediately after it traverses the sample. In principle the propagation of the beam thereafter can be analyzed using the standard methods that evaluate field propagation in free space, for example, the Fresnel diffraction integral and the angular spectrum method. However, an analytic result is 41

59 hard to obtain using these methods. The Gaussian decomposition method used by Weaire 19 could give analytic solution under certain approximations and provides a more clear physical interpretation. Weaire used this method to analyze the propagation of a fundamental gaussian beam that traverses a nonlinear sample. Here we generalize this method to deal with the LG beams. The exponential in Eq. (2.59) can be expanded in a Taylor series as e i Φ(r,φ,zs) = ( i Φ (r,φ,z s )) m. (2.62) m! m=0 The complex electric field of the incident beam after it passes through the sample can be written as a summation of the electric fields of a series of LG beams of different modes as E (r,φ,z) = p m m=0 p=0 l= C p,l,m LG l p (r,φ,z z wm ;z rm ), (2.63) where z wm and z rm are the waist location and the Rayleigh length, respectively, of the corresponding beam mode and C p,l,m is the amplitude and phase of the component beam. These parameters are determined by letting z = z s in Eq. (2.63) and comparing it with Eq. (2.59) with the exponential replaced by Eq. (2.62). The details of the decomposition calculation are included in Appendix A. Here we write the result: where E (r,φ,z) = p m m=0 p=0 C p,m LG l 0 p (r,φ,z z wm ;z rm ), (2.64) 4m (m + 1)Z z wm = z r Z 2 + (2m + 1) 2, (2.65) 42

60 z rm = z r (2m + 1) (Z 2 + 1) Z 2 + (2m + 1) 2, (2.66) and C p,m = D p,m F p,m where Z is defined as Z = z s /z r, and F p,m = E 0 e αd/2 ( i Φ 0) m (2m + 1) 2 + Z 2 m! (2m + 1) (1 + Z 2 ) 2m+1 exp exp ( i (2p 0 + l 0 + 1) tan 1 (Z) ) exp ( ) 4m (m + 1)Z ikz r Z 2 + (2m + 1) 2 ( i (2p + l 0 + 1) tan 1 ( Z 2m + 1 )), (2.67) and p m and D p,m are determined through p m p=0 D p,m L l 0 p (x) = ( x m l 0 L l ( 0 x ) ) 2m+1 p 0 2m+1 (2m + 1) 2m+1 2 l 0, (2.68) where x is an arbitrary real variable. Our theoretical results are embodied in Eq. (2.64). It is worth noting that all the component LG beams have the same angular mode number l 0 as that of the incident beam, which reflects the conservation of the photon s orbital angular momentum. Therefore the effect of the Kerr material on the incident LG beam is to generate new LG beams of different radial modes. These results are important in applications that leverage mode sensitivity Examples assuming small nonlinear phase distortion The Taylor expansion in Eq. (2.62) always converges and the speed of convergence depends on the value of Φ max (z s ). So does the expansion of Eq. (2.64). In practice, m only needs to be summed up to a certain finite value in order to achieve a given precision. If the nonlinear phase distortion is very small (e.g., Φ max (z s ) << 1) such that only a few terms in the summation are needed to make a good approximation, we can write out the result analytically. 43

61 To illustrate, assume that the incident beam is a LG 0 0 beam and the maximum nonlinear phase distortion in the sample at position Z is Φ max (Z) = Φ 0 << 1. (2.69) 1 + Z2 It is sufficient to keep the first two terms (m = 0 and m = 1) in Eq. (2.62) and neglect the higher order terms, yielding ( see Appendix A: Example 1 for details of the derivation.) E (r,φ,z) C 0,0 LG 0 0 (r,φ,z z w0 ;z r0 ) + C 0,1 LG 0 0 (r,φ,z z w1 ;z r1 ), (2.70) where C 0,0 = E 0 e αd/2 ; C 0,1 = E 0 e αd/2( i Φ ( ) 0) 9 + Z 2 3 (1 + Z 2 ) 3 exp 8Z ikz r Z exp ( i tan 1 (Z) ) ( ( )) Z exp i tan 1 ; 3 and z w0 = 0; 8Z z w1 = z r Z ; z r0 = z r ; z r1 = z r 3 (Z 2 + 1) Z Next we show an example of the higher order LG beam. Assume the incident beam is 44

62 a LG 1 0 beam and the maximum nonlinear phase distortion in the sample at position Z is Φ max (Z) = Φ 0 e (1 + Z 2 ) << 1. (2.71) Again we keep the first two terms in the Eq. (2.62) and neglect the higher order terms, yielding ( see Appendix A: Example 2 for details of the derivation.) E (r,φ,z) F 0,0 LG 1 0 (r,φ,z z w0 ;z r0 ) F 0,1LG 1 0 (r,φ,z z w1 ;z r1 ) F 1,1LG 1 1 (r,φ,z z w1 ;z r1 ), (2.72) where F 0,0 = E 0 e αd/2 ; F 0,1 = E 0 e αd/2( i Φ ( ) 0) 9 + Z 2 3 (1 + Z 2 ) 3 exp 8Z ikz r Z exp ( i2 tan 1 (Z) ) ( ( )) Z exp i2 tan 1 ; 3 F 1,1 = E 0 e αd/2( i Φ ( ) 0) 9 + Z 2 3 (1 + Z 2 ) 3 exp 8Z ikz r Z exp ( i2 tan 1 (Z) ) exp ( i4 tan 1 ( Z 3 )), 45

63 and z w0 = 0; 8Z z w1 = z r Z ; z r0 = z r ; z r1 = z r 3 (Z 2 + 1) Z This outgoing electric field includes the generated LG 1 0 and LG 1 1 beam. 2.5 Application: Z scan Our theory applies to several important applications. In this section, we discuss the Z-scan measurement Review of the traditional Z scan using a LG 0 0 beam Figure 2.7: Schematic diagram of the Z scan experiment. L: lens, S: sample, A: aperture, and D: detector. The Z-scan measurement was first reported by Sheik etc. as a highly sensitive technique to measure the optical nonlinearities using a single LG 0 0 beam. 16 Figure 2.7 shows the 46

64 schematic diagram of the Z-scan experiment. The placement of the incident beam and the nonlinear sample is the same as in Figure 2.6. The incident beam is focused by a convex lens. The waist of the beam is located at z = 0. An optically nonlinear sample of thickness d is placed at position z = z s along the optical axis of the beam. In addition, a small aperture is placed on axis of the beam in the far field. The power passed through the aperture is recorded by a detector as a function of sample position z. To better explain the Z-scan procedures, it s useful to define the on-axis normalized 15, 16 Z-scan transmittance T (Z, Φ 0 ) = E (r 0,φ,z ) 2 E (r 0,φ,z ) 2, (2.73) Φ0 =0 which characterizes the on axis light power transmitted though the small aperture in the far field. Applying Eq. (2.70) we find (see Appendix B for details on the derivation.) T (Z, Φ 0 ) = 1 + 4Z (1 + Z 2 ) (9 + Z 2 ) Φ 0 (2.74) when Φ 0 << 1. In a Z-scan, one measures the trace of the normalized transmittance T as a function of the sample position z, which we call a Z-scan trace. Figure 2.8 shows a typical Z-scan trace for positive (solid line) and negative (dotted line) Φ 0. A typical Z-scan trace has a peak (maximum) and a valley (minimum). The positions of the peak and valley can be obtained by solving the equation dt (Z, Φ 0 ) dz = 0, (2.75) 47

65 peak 0 = =-0.1 Normalized Transmittance T valley T p-v Figure 2.8: A typical Z-scan trace for positive (solid line) and negative (dotted line) Φ 0. Z yielding Z peak(valley) = ± ±0.859, (2.76) where the sign is + ( ) for the peak (valley) and (+) for the valley (peak) when Φ 0 > 0 ( Φ 0 < 0). Substituting result (2.76) into Eq. (2.74), we can calculate the difference between the peak and the valley T p v = Φ 0. (2.77) The above relationship provides a handy way to determine Φ 0 from the Z-scan trace, 48

66 from which one can calculate the nonlinear refractive index n 2, e.g., through Eq. (2.61). If the experimental apparatus is able to resolve the normalized transmittance change of T p v = 1%, then it can measure Φ 0 as small as 0.025, corresponding to wavefront distortion of λ/250. Thus the Z-scan technique has very high sensitivity Z scan using a LG 1 0 beam Following the example of the LG 0 0 Z-scan, we calculate the normalized transmittance T for a LG 1 0 incident beam. Using Eq. (2.72) and Eq. (2.73) we find (see Appendix B for details on the derivation.) T (Z, Φ 0 ) = 1 + 8Z ( Z2 Z 4 ) (1 + Z 2 ) (9 + Z 2 ) 3 Φ (9 + Z 2 ) 3 Φ2 0. (2.78) The last term can be dropped if Φ 0 << 1, yielding T (Z, Φ 0 ) = 1 + 8Z ( Z2 Z 4 ) (1 + Z 2 ) (9 + Z 2 ) 3 Φ 0. (2.79) This equation shows a similar relationship between T and Φ 0 as Eq. (2.74), suggesting that it is possible to do a Z-scan experiment using the LG 1 0 beam to measure the nonlinearity of a thin sample. Figure 2.9 shows the theoretical typical LG 1 0 Z-scan traces for positive (solid line) and negative (dotted line) Φ 0. The shape of the Z-scan curve using the LG 1 0 beam differs from the traditional one using the LG 0 0 beam in that the former has an extra peak and valley (indicated in the figure by the arrows). The extra valley brings down the tail of the major peak to below T = 1 and the extra peak brings up the tail of the major valley up above T = 1, while the tails in the LG 0 0 Z-scan trace never cross the T = 1 line. The differences are clearly seen in Figure 2.10 which shows a LG 1 0 Z-scan trace and a LG 0 0 Z-scan trace simultaneously. 49

67 Normalized Transmittance T = =-0.1 T= Figure 2.9: A typical LG 1 0 Z-scan trace for positive (solid line) and negative (circles) Φ 0. The T = 1 level is indicated by the dashed line. Z (2.79). The coordinates of the peaks and valleys can be calculated using Eq. (2.75) and Eq. Z major peak(valley) = ±0.902 sign( Φ 0 ), (2.80) T major peak(valley) = 1 ± Φ 0, (2.81) and Z minor peak(valley) = 5.24 sign( Φ 0 ), (2.82) T minor peak(valley) = 1 ± Φ 0, (2.83) 50

68 Normalized Transmittance T LG 0 1 Z-scan LG 0 0 Z-scan T= Figure 2.10: Comparison of a typical LG 1 0 Z-scan trace with a typical LG 0 0 Z-scan trace. The values of Φ 0 are chosen such that the major peaks (valleys) of the two traces almost overlap. Also shown is the T=1 line. Z where sign( Φ 0 ) = 1 if Φ 0 > 0 and sign( Φ 0 ) = 1 if Φ 0 < 0. The amplitudes of the minor peak and valley are much smaller than the major ones in a Z-scan trace and are less important in determining Φ 0 and n 2. However, they have significance in other applications such as optical limiting, which will be discussed later. The major peak and valley are important in the Z-scan measurement. We can use the difference between them ( T p v ) to determine Φ 0, which then lets us calculate n 2. From Eq. (2.81) it s easy to get T p v = Φ 0. (2.84) 51

69 We point out that there is a subtle difference between the Φ 0 in Eq. (2.77) and in Eq. (2.84). The Φ 0 in Eq. (2.77) equals Φ max (Z = 0) (refer to Eq. (2.69)), the maximum nonlinear phase distortion introduced by the LG 0 0 beam in the transverse plane of the sample. The Φ 0 in Eq. (2.84) is equal to e Φ max (Z = 0) (refer to Eq. (2.71)), more than the maximum nonlinear phase distortion introduced by the LG 1 0 beam in the sample. So the Φ 0 in the LG 0 0 case is a real phase distortion that indeed happens in the sample, while the Φ 0 in the LG 1 0 case is an imaginary phase distortion, bigger than any real phase distortion in the sample. In order to make a fair comparison of the sensitivity of the two Z-scan methods, we express T p v in terms of the real maximum nonlinear phase distortion. For the LG 0 0 Z-scan, T p v = Φ max (Z = 0), (2.85) and for the LG 1 0 Z-scan, T p v = Φ max (Z = 0). (2.86) Therefore, the sensitivity of the LG 1 0 Z-scan is slightly higher than the LG 0 0 Z-scan providing the same maximum nonlinear phase distortion is produced in the sample. A more important difference between the two Z-scan measurements is that in the LG 1 0 Z-scan experiment the detector is placed at the beam center where the intensity is the weakest due to the screw phase dislocation while in the LG 0 0 Z-scan experiment the center intensity is the strongest. As a result the former shows a much bigger deviation from the normal value if any phase or intensity distortion that destroys the symmetry of the beam profile is present. This suggests that the LG 1 0 Z-scan experiment is more sensitive to changes of the n 2 of the sample. 52

70 2.5.3 Effect of the aperture size: the off-axis normalized transmittance One may question how it is possible to measure the on-axis transmittance of a LG 1 0 beam since the on-axis intensity is always zero. In fact, what s really being measured in the Z scan is the power transmitted through a small aperture centered on the optical axis. Thus not only the on-axis light but also some off-axis (but near the axis) light are collected and measured. A follow-up question is then: does the off-axis light behave the same as the on-axis light in terms of the normalized transmittance? This question can be answered by examining the off-axis normalized transmittance. We use the LG 1 0 beam as an example. Using a procedure similar to what is described in Appendix B, we can calculate the off-axis normalized transmittance for a LG 1 0 beam, yielding [ ( T (R,Z, Φ 0 ) = 1 i2 3 Φ 0 exp (1 + ) ] Z2 ) 8Z + i R Z Z 2 (1 + iz) 2 (1 (3 + iz) Z2 (1 + Z 2 ) (3 + iz) 2 3 (1 + ) 2 Z2 ) (3 + iz) 2 R2, (2.87) where, R r/ω(z ) is the radial distance from the beam center normalized by the beam radius at the far field. Fig shows a 3D plot of the Z-scan normalized transmittance for a LG 1 0 beam as a function of Z as well as R. The important information from the figure is that T changes very slowly as R increases. Therefore we expect that because the off-axis intensity profile is relatively smooth, the size of the aperture has little effect on the accuracy of the Z-scan result. Fig further illustrates our conclusion, where the Z-scan normalized transmittance for a LG 1 0 beam at R = 0.05 and R = 0.1 are plotted in comparison with at R = 0. The differences between the three curves are hardly distinguishable in the graph. The numerical values show that the difference of T between R = 0 and R = 0.05 is less than 2%, and 53

71 1.01 T Z R Figure 2.11: The Z-scan normalized transmittance for a LG 1 0 beam as a function of transverse coordinate R. ( Φ 0 = 0.1) about 6% for R = 0.1. In the far field, the beam radius is very large, so a small aperture usually corresponds to a very small R. In practice, we can treat all the light that passes through the aperture as on-axis light. 2.6 Application: optical limiting Introduction With the development of laser technology, the output power and intensity of lasers has been substantially increased. For example, lasers operating on the principle of chirped pulse amplification (CPA) can produce ultrashort laser pulse up to the petawatt(

72 T R=0 R=0.05 R= Figure 2.12: The Z-scan normalized transmittance for a LG 1 0 beam for R = 0, R = 0.05 and R = 0.1. ( Φ 0 = 0.1) Z watt) level. 20 Higher laser power can induce large effects and certainly enables more novel applications. However, high powers and intensities inevitably lead to increasing the possibilities of damaging optical components. It is often desirable to apply a protector to those expensive components such that under normal power or intensity, the protector passes all the light and the protected components work normally; if the power or intensity of incident light is higher than the safe level, the protector blocks the dangerous light. The optical limiter is one such protector. Figure 2.13 illustrates the transmittance of an optical limiter. Without a limiter, the output power equals the input power as shown by the dashed line. An ideal limiter (the solid line) has no influence on the system under low incident power, but prohibits a further increase of the output power when the incident power is more than a threshold value. In 55

73 practice, an ideal limiter can not be achieved. A practical limiter s response approaches that of the ideal limiter, e.g., the triangles in the figure. 60 Output Power (a.u.) no limiter ideal limiter practical limiter P threshold Incident Power (a.u.) Figure 2.13: Illustration of the transmittance of the optical limiter. In this section, we discuss a new optical limiting geometry using the LG beam and a thin nonlinear optical film. In fact, the valleys of the curves, and more generally, the portions of the curves that fall below T = 1 in figure 2.9 imply the optical limiting phenomenon. Recall that T p v is proportional to the nonlinear phase distortion Φ. For those portions below T = 1, it means that the bigger Φ becomes, the lower the value of T will be. But Φ is proportional to the intensity I. So the higher the intensity, I, the lower the transmittance, T. The key requirement of an optical limiter is thus met, suggesting that it s possible to make an optical limiter using a setup similar to the Z- scan setup in figure 2.7. The detector in the Z-scan setup is replaced with the optical 56

74 component to be protected. Alternatively (and sometimes more practically) a second lens can be placed after the thin film as shown in figure 2.14, acting as a Fourier transform lens to bring the far field closer. Figure 2.14: Schematic diagram of optical limiting using the LG beam. L1: focusing lens, S: nonlinear thin film, L2: Fourier transform lens, A: small aperture, D: optical component to be protected, f2: focal length of L2. There are, however, questions to be answered before we put this new optical limiting geometry into practice Effect of the position of the nonlinear thin film One question pertains to the best place to put the thin film. In the Z-scan measurement, the sample is scanned over a wide range of Z along the optical axis, so there is no such issue. In the optical limiting application, however, the position of the sample must be fixed. So sample placement is important. We therefore need to compare the limiting efficiency of the thin film at different positions, Z, to obtain the optimal placement. In order to make such comparisons meaningful, we should use the same illuminating conditions at different values of Z. Specifically, the maximum intensity and therefore the maximum nonlinear phase distortion in the sample should be kept the same. This ensures that the sample is under the same challenge since the damage threshold of the material 57

75 is often determined by the maximum intensity it can tolerate. This should not be confused with the Z-scan measurement, in which the maximum intensity in the sample has to change when the sample is placed at different Z s because the power of the beam is fixed but the beam s radius varies with position. In the optical limiting case, for the purpose of making a fair comparison, we intentionally force the maximum intensity in the sample to be the same even though the sample is at different Z s. So the power of the incident beam must be changed for each Z. One may argue that when comparing the limiting efficiencies, the power of the incident beam should be kept the same. This is not appropriate because by changing the intensity at the sample, e.g., using different focal lenses, we can get quite different limiting efficiencies even if the power of the incident beam is fixed. So we rewrite the normalized Z-scan transmittance T (Z, Φ 0 ), using the maximum nonlinear phase distortion at position Z, Φ max (Z), as a parameter instead of using Φ 0 or the maximum nonlinear phase distortion at position Z = 0, Φ max (Z = 0). And we will compare the values of T at different positions using the same value of Φ max (Z). Since Φ max (Z) will be treated as a constant, for convenience we write Φ max (Z) as Φ max. We call this new form of transmittance T (Z, Φ max ) the normalized optical limiting transmittance to distinguish it from the normalized Z-scan transmittance. Below we show some examples. LG 0 0 beam Substituting Φ 0 = ( 1 + Z 2) Φ max (2.88) into Eq. (2.74), we get T (Z, Φ max ) = Z 9 + Z 2 Φ max. (2.89)

76 LG 1 0 beam Substituting Φ 0 = e ( 1 + Z 2) Φ max (2.90) into Eq. (2.79), we get T (Z, Φ max ) = 1 + 8e Z ( Z2 Z 4 ) (9 + Z 2 ) 3 Φ max. (2.91) 1.2 Normalized Transmittance T max = -0.1 LG 0 0 LG 0 1 T= Figure 2.15: Typical curves of normalized optical limiting transmittance T vs. position Z. Φ max = 0.1. The circled line is for the LG 0 0 beam, the solid line is for the LG 1 0 beam, and the dashed line shows T = 1. Typical curves of normalized optical limiting transmittance T vs. position Z are shown Z 59

77 in figure 2.15, where the maximum nonlinear phase distortion Φ max = 0.1 is selected as an example, corresponding to material with negative nonlinear refractive index n 2. When Φ max > 0, it s easy to verify that the curves are the mirror images of their negative counterparts with maximum nonlinear phase distortion being Φ max. For the curve associated with the LG 0 0 beam, Z > 0 and T < 1. Thus, the Z > 0 regime can be used for optical limiting if the nonlinear film has a negative n 2. The most sensitive place is Z = 3, where T reaches the valley with a value of (1 + 2 Φ max /3), (or in the figure) as Φ max = 0.1. The extremum is solved by analyzing Eq. (2.89). The curve of the LG 1 0 beam has two regions whose normalized transmittance T falls below 1. one is when Z is less than approximately 3.49 with a valley at ( 8.55, Φ max ). The other is between Z = 0 and approximately Z = 3.49 with a valley at (1.73, Φ max ). These values are obtained by analyzing Eq. (2.91). In the figure, the extrema are ( 8.55, 0.846) and (1.73, 0.895), respectively, as Φ max = 0.1. Both valleys are lower than the one for the LG 0 0 beam. It s obvious that not only the extrema but also the shapes of the curves in figure 2.15 are different from the Z-scan traces in which Φ 0 is held constant. We emphasize that only by keeping Φ max constant does the curve reflect the true potential of the thin film for optical limiting at different positions, as we discussed earlier. The normalized optical limiting transmittance, rather than the normalized Z-scan transmittance, should be used in evaluating where is the best place to put the thin film for optical limiting Large nonlinear phase distortion In the Z-scan measurement, the nonlinear phase distortion is assumed to be very small, e.g., Φ << 1. The incident intensity of the beam is thus limited by this assumption. Eqs. (2.74), (2.89), (2.79) and (2.91) are all derived under this assumption. However, in the optical limiting application, the incident intensity usually varies over a much larger range and 60

78 the resulting nonlinear phase distortion is not necessarily small. Whether optical limiting takes place when the incident intensity is high remains unknown. To answer this question, a calculation of the normalized optical limiting transmittance, T, for arbitrary nonlinear phase distortion is necessary. In Appendix C, we derive the normalized optical limiting transmittance for arbitrary nonlinear phase distortion for the LG 0 0 and LG 1 0 beams. We find that in general, a numerical calculation is required. We have developed Mathematica codes to complete the numerical calculations which are included in Appendix C. Here we present the calculated results. LG 0 0 beam Figure 2.16 shows the series of curves of the normalized optical limiting transmittance, T, versus the maximum nonlinear phase distortion, Φ max, in the sample, where each curve corresponds to a different sample position, Z, indicated by the number along the curve. The nonlinear refractive index of the sample, n 2, is assumed to be negative. As shown in the figure, the curves can be divided into several groups according to their behavior near Φ max = 0: Group I ( < Z 3): Initially T increases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T increases as the value of Z increases. Group II ( 3 Z 0): Initially T increases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T decreases as the value of Z increases. Group III (0 Z 1.11) and Group IV (1.11 Z 3): Initially T decreases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T decreases as the value of Z increases. Among them the curve of Z = 1.11 is special because it s value of T reaches 0 when Φ max

79 I Normalized Transmittance T III Maximum nonlinear phase distortion II Figure 2.16 (I-III) 62

80 Normalized Transmittance T IV V Maximum nonlinear phase distortion Figure 2.16: The normalized optical limiting transmittance T versus the maximum nonlinear phase distortion Φ max in the sample when the incident beam is a LG 0 0 beam. The position Z of the sample for each of the curve is indicated by the number along the curve. A sample of negative n 2 is assumed. 63

81 Group V (3 Z < ): Initially T decreases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T increases as the value of Z increases. It s easy to seen that the curves in Group III-IV are candidates for optical limiting since their transmittance T s decrease as the nonlinear phase distortion Φ max increases. The slope of T vs. Φ max curve depends on the position Z. For example, the curve of Z = 3 has the most steep slope. In practice, we should choose the curve based upon the requirement of the system. For example, curves with greater slope provide smaller limiting threshold, while curves with smaller slope have wider transmittance range for the weak intensity. It s important to point out that after T reaches the minimum value, it will turn back to increase as Φ max further increases. Thus, optical limiting operates up to the minimum value of T. The minimum value of T and the corresponding value of Φ max also depend on the position Z. For example, the curve of Z = 1.11 is the only one whose value of T reaches 0 when Φ max In practice, there is usually an upper limit of the incident intensity in the system, like the maximum intensity that a laser can emit. Furthermore, the nonlinear sample has its own damage threshold which limits the maximum intensity it can tolerate. Thus the maximum nonlinear phase distortion in a practical system is always limited. Therefore, the turning back of the transmittance T can be avoided by selecting the proper curves whose turning points are out of the range of operation. Finally, we note that some of the curves in Group II can be used to perform a special type of optical limiting. Curves like the one with Z = 0.75, which initially increase as Φ max increases, soon turn back to decrease. These curves can be used to make a system that enhances low intensity transmittance but limits high intensity transmittance. 64

82 LG 1 0 beam When the incident beam is a LG 1 0 beam, the situation is more complex. Figure 2.17 shows the series of curves of the normalized optical limiting transmittance T versus the maximum nonlinear phase distortion Φ max in the sample, where each curve corresponds to a different sample position Z indicated by the number along the curve. The nonlinear refractive index of the sample n 2 is assumed to be negative. As shown in the figure, the curves can be divided into several groups according to their behavior near Φ max = 0: Group I ( < Z 8.55): Initially T decreases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T decreases as the value of Z increases. Group II ( 8.55 Z 7.42) and Group III ( 7.42 Z < 3.49): Initially T decreases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T increases as the value of Z increases. The curve of Z = 7.42 is special because it s value of T reaches 0 when Φ max Group IV ( 3.49 Z 1.73): Initially T increases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T increases as the value of Z increases. Group V ( 1.73 Z < 0): Initially T increases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T decreases as the value of Z increases. Group VI (0 Z 0.61) and Group VII (0.61 Z 1.73) : Initially T decreases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T decreases as the value of Z increases. The curve of Z = 0.61 is special because it s value of T reaches 0 when Φ max

83 Normalized Transmittance T III I II Maximum nonlinear phase distortion -4 Figure 2.17 (I-III) 66

84 Normalized Transmittance T V IV Maximum nonlinear phase distortion Figure 2.17 (IV-V) 67

85 Normalized Transmittance T VI 0.61 VII VIII Maximum nonlinear phase distortion Figure 2.17 (VI-VIII) 68

86 Normalized Transmittance T IX Maximum nonlinear phase distortion X Figure 2.17: The normalized optical limiting transmittance T versus the maximum nonlinear phase distortion Φ max in the sample when the incident beam is a LG 1 0 beam. The position Z of the sample for each of the curves is indicated by the number along that curve. 69

87 Group VIII (1.73 Z < 3.49): Initially T decreases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T increases as the value of Z increases. Group IX (3.49 Z 8.55): Initially T increases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T increases as the value of Z increases. Group X (8.55 Z < ): Initially T increases from 1 as Φ max increases from 0. And for fixed value of Φ max near Φ max = 0, the value of T decreases as the value of Z increases. The curves in Group I-III and Group VI-VII can be implemented for optical limiting since their transmittances decrease as the nonlinear phase distortion Φ max increases. Compared to using a LG 0 0 beam, using a LG 1 0 beam to achieve optical limiting has more choices. We can put the sample in front of the focus of the beam, utilizing the curves in Group I-III, or put the sample behind the focus of the beam, making use of the curves in Group VI-VII. This is more flexible than the LG 0 0 beam case in which the sample has to be placed behind the focus of the beam. This added flexibility is important when designing certain optical systems. More importantly, some of the curves in the LG 1 0 beam case have steeper slopes than the LG 0 0 beam case. For example, the curve of Z = 7.42 in the LG 1 0 beam case decreases to 0 when Φ max 1.42, while it s counterpart in the LG 0 0 beam case, the curve of Z = 1.11 does not drop to 0 until Φ max reaches approximately Thus using a LG 1 0 beam can achieve a smaller limiting threshold than using a LG 0 0 beam provided the rest of the conditions are held the same. As in the LG 0 0 beam case, the limiting curves of the LG 1 0 beam also turn back to increase as Φ max further increases after T reaches the minimum value. Thus the same kind of considerations should be taken when we design the optical limiting systems. 70

88 Some of the curves in Group V can be used to perform a special kind of optical limiting we proposed in the case of the LG 0 0 beam. Curves like the one with Z = 0.5, although they initially increase as Φ max increases, soon turn back to decrease. These curves can be used to make a system that enhances low intensity transmittance but limit the high intensity transmittance. 2.7 Application: Measuring the nonlinear refractive index Motivation The Z-scan technique is very useful for it s high sensitivity and relatively simple setup using a single beam. However, the requirement of moving the sample along the z axis has drawbacks. One problem is the error due to re-alignment. When the sample is moved to a new Z position, the sample must be re-aligned to be perpendicular to the incident beam and the aperture may need to be re-aligned to the center of the far field. These procedures not only cause errors but are also generally time-consuming. The problem becomes more serious when the sample is not uniform. Then not only the sample needs to be perpendicular to the optical axis, but also the exact same spot in the sample must overlap with the beam. The latter is very difficult without a precise mechanical adjusting system. Also, in order to move the sample along the Z axis, there must be enough clear space to operate, which is not possible when making a small device. Another requirement of Z scan is that Φ 0 must be less than 1, which could be a problem under certain circumstances. For example, if we want to measure how the n 2 of a sample with a slow nonlinear mechanism increases with time, a larger incident intensity is used to let T (and Φ 0 ) increase at a larger rate, such that the resolution of n 2 as a function of time is good. The traditional Z scan technique can not handle this kind of measurement once 71

89 Φ 0 increases to larger than 1. In this section, we explore alternative techniques to measure the nonlinear refractive index that do not require moving the sample along the Z axis, yet retain as much as possible the merits of the Z-scan technique, such as high sensitivity and single beam simplicity. One of the new techniques, namely the Φ max scan, can handle the situations when Φ > I scan The inspiration comes from Figure 2.16 and Figure 2.17, which are originally plotted to examine the optical limiting abilities of the corresponding setups by showing the series of curves of the normalized optical limiting transmittance T versus the maximum nonlinear phase distortion Φ max in the sample at different position Z. One of the features of the curves is that at a given position Z, T is totally determined by the value of Φ max. The reverse is not always true because knowing the value of T generally does not suffice to give the value of Φ max. However, between Φ max = 0 and the first turning point of each curve, Φ max and T are in one-to-one correspondence. In fact, if Φ max is small, T depends on Φ max almost linearly, and vice versa. Thus, if we know the sample position Z and measure normalized T, we should be able to determine the nonlinear phase distortion in the sample, which combined with other information such as intensity I gives the nonlinear refractive index. Figures (2.18) to (2.20) show the maximum nonlinear phase distortion Φ max as a function of the normalized transmittance T. The dots are calculated using our methods developed in the previous section. The lines are the best linear fits. The incident beam is a LG 0 0 beam in Figure (2.18) and the sample positions are Z = ±3. The incident beam in Figure (2.19) and Figure (2.20) is a LG 1 0 beam and the positions of the sample are Z = ±1.73 and Z = ±8.55, respectively. As can be seen in the figures, the lines fit the 72

90 Maximum nonlinear phase distortion z=-3 calculated result linear fit: max = -1.48*(T-1) z=3 calculated result linear fit: max = 1.53*(T-1) Normalized Transmittance T Figure 2.18: The maximum nonlinear phase distortion Φ max as a function of the normalized transmittance T. The incident beam is a LG 0 0 beam and the position of the sample is Z=-3 and Z=3 for the upper and lower curve, respectively. The dots are the calculated results and the lines are the linear fits. 73

91 Maximum nonlinear phase distortion calculated result linear fit: max = -0.93*(T-1) Z= 1.73 Z= calculated result linear fit: max = 0.99*(T-1) Normalized Transmittance T Figure 2.19: The maximum nonlinear phase distortion Φ max as a function of the normalized transmittance T. The incident beam is a LG 1 0 beam and the position of the sample is Z=-1.73 and Z=1.73 for the upper and lower curve, respectively. The dots are the calculated results and the lines are the linear fits. 74

92 Maximum nonlinear phase distortion Z= calculated result linear fit: max = -0.63*(T-1) Normalized Transmittance T calculated result linear fit: max = 0.67*(T-1) Z= 8.55 Figure 2.20: The maximum nonlinear phase distortion Φ max as a function of the normalized transmittance T. The incident beam is a LG 1 0 beam and the position of the sample is Z=-8.55 and Z=8.55 for the upper and lower curve, respectively. The dots are the calculated results and the lines are the linear fits. 75

93 data very well. Examination of the numerical values shows that the maximum deviation of the data from the linear fit lines is less than 4%. Although the same kind of behavior occurs in many other positions of the sample, the positions chosen in the figures give the most sensitive change of T for a given amount of change of Φ max among their classes. In other words, they provide the most high sensitivity when measuring the nonlinear refractive index. Among the plotted curves, using a LG 1 0 beam and letting Z = 8.55, the highest sensitivity results. The linear fit gives Φ max = 0.63(T 1). If the detector is able to measure a change of T = 1%, then we are able to determine a change of Φ max = , equivalant to a precision of about λ/1000, better than that of the traditional Z-scan. In theory, one measurement of T other than T = 1 would allow us to determine the corresponding Φ max and then the nonlinear refractive index. In practice, we can change Φ max over a range, e.g., by changing the intensity of the incident beam, (hence the name I scan is given.) to get multiple measurements of T in order to minimize the errors. As long as T is within the plotted region, the linear relationship holds well. Compared with the Z-scan method, the I-scan method discussed here uses the same kind of setup and provides competitive sensitivity, but does not require moving the sample along the z axis. The price paid for this advantage is that the position of the sample must be set exactly in order to make use of the corresponding Φ max T curve. Thus before actually measuring the nonlinear refractive index of the sample, the position of the waist and the Rayleigh length of the incident beam must be measured which then allows one to determine the position Z of the sample. Alternatively a calibration curve relating Φ max to T at a certain position can be measured using a sample with known nonlinear refractive index. The curve can then be used to measure the nonlinear refractive index of other samples that are placed at the exact same position, obviating the need of knowing sample position Z explicitly. In the next section, we propose a method that determines the position Z of the sample at 76

94 the same time when T is measured. The procedure of pre-calibrating the system mentioned above is then unnecessary Φ max scan to measure samples with large n 2 We start with Figure 2.21, a re-plot of Figure 2.17 which shows the normalized transmittance T versus the maximum nonlinear phase distortion Φ max in the sample when the incident beam is a LG 1 0 beam, with selected curves of different sample positions. The position Z of the sample for each of the curves is indicated by the number along the curve. The curves plotted in Figure 2.21 all show one and only one extreme point (valley) within the plotted range. These valleys are crossed by the dotted curve which is the set of minima of the T Φ max curves with Z ranging from to One important feature immediately seen from Figure 2.21 is that the T values of the valleys are in one-to-one correspondence with the position Z of the sample. In other words, if we can measure the T value of the valley of the T Φ max curve, we should be able to deduce the position Z of the sample. Experimentally, measuring the T value of the valley is possible. Although Φ max is not directly measurable, the increase or decrease of its value is controllable. For example, if the nonlinear mechanism of the sample is a fast process, we can increase Φ max by increasing the intensity of the incident beam; If the nonlinear mechanism of the sample is a slow process, we can fix the intensity of the incident beam and prolong the exposure time, simply waiting for Φ max to increase with time, assuming that the nonlinearity of the sample builds up monotonously with the exposure time (which is often true). Initially we shall see the decrease of T with Φ max increasing. At some point, Φ max reaches and passes the valley. Subsequently T turns back and increases with Φ max with further increase of Φ max. The value of T at the turning point is determined from the height of the minimum. In practice, we can measure the T I curve (normalized transmittance vs. 77

95 Normalized Transmittance T Maximum nonlinear phase distortion Figure 2.21: Solid lines: selected curves of the normalized transmittance T versus the maximum nonlinear phase distortion Φ max in the sample when the incident beam is a LG 1 0 beam. The position Z of the sample for each of the curves is indicated by the number along the curve. Dotted line: the coordinates of the valleys of the T Φ max curves. 78

96 intensity) or the T t curve (normalized transmittance vs. time) instead of the T Φ max curve. Although the shape of these curves are not the exactly the same, they all have one valley, and the T coordinates of the valley are the same Position of sample Z Normalized Transmittance T Figure 2.22: The sample position Z versus the T coordinate of the valley of the corresponding T Φ max curve. The arrows represent the useful range of the Z T curve for determining the position from the transmittance. Once the T value of the valley is obtained, the sample position Z can be found, e.g., using Figure 2.22, where Z is plotted as a function of T value of the valley of the corresponding T Φ max curve. Note that for given resolution of T (which is determined by the detecting system), the resolution of Z is not uniform over the plotted range. At both ends, the Z T curve is very steep, making the resolution of Z poor. However, the middle part of the curve, roughly from Z = 6 to Z = 4 as indicated by the arrows in Figure 2.22, has smaller and nearly uniform slope, providing better resolution of Z. Therefore the middle part of the curve is more useful in practice. We re-plot the middle part as shown in 79

97 Figure 2.23, in which we also show a best fit line with an analytic function. The function -4.0 Position of sample Z calculated fitted Normalized Transmittance T Figure 2.23: The sample position Z versus the T coordinate of the valley of the corresponding T Φ max curve. Circles: calculated results. Line: best fit using an inverse Gauss function (see text for details). we choose is an inverse Gauss function defined as y = y 0 + ω 0.5 ln ( ) A. (2.92) x It is named such because it is the inverse function of the Gauss function ( ( ) ) 2 y y0 x = A exp 2. (2.93) ω The fitting result from Figure 2.23 is Z = ln (0.98/T). (2.94) 80

98 Examination of the numerical values shows that the maximum deviation of the fit from the calculated data is less than 0.3%. Therfore Eq. (2.94) gives a good approximation of the value of Z given the value of T of the valley. Now that the sample position Z is found, we can apply the method introduced in the previous section, i.e., using the Φ max - T curve at this Z to determine the values of Φ max from the measured T. From Φ max we determine the nonlinear refractive index coefficient n 2. Following the name of Z scan, we name this method Φ max scan because during the process of searching the valley, Φ max is changed gradually from 0 to some value beyond the valley point. As mentioned earlier, Φ max can be changed by changing the intensity I or the exposure time t, depending on the nonlinear mechanism of the sample. If it is the intensity I, then we obtain a n 2 I curve from the measured T I curve; if it is the time t, then we obtain a n 2 t curve from the measured T t curve. Thus, one inherent advantage of the Φ max scan is revealed: it can measure the speed of the development of n 2 (through the n 2 t curve) automatically for a slow nonlinear mechanism. When we determine Φ max from T, there is a major difference between the I-scan method introduced in the previous section and the Φ max scan. In the previous section, the change of T is well inside the linear region of the Φ max - T curve and Φ max and T are in one-to-one correspondence, so we can use the best fit formula to calculate Φ max from T easily. In the Φ max scan, however, T has to extend out of the linear region of the Φ max - T curve in order to reach the valley point. Moreover, once passing the valley, T is not in one-to-one correspondence with Φ max because for each value of T there could be two Φ max values (see Figure 2.21). The issue of the one-to-one correspondence is not a problem because during the measurement Φ max is always allowed to increase monotonously. So if the same value of T occurs twice, the first must correspond to the smaller Φ max before the valley and the second corresponds to the bigger Φ max after the valley. The correspondence can be identified unambiguously from the graph or by a computer program. The nonlinear attribute of the Φ max - T curves makes it difficult to 81

99 find an analytic function to fit the curve, especially when Z is also a parameter. We handle this problem by calculating T as a function of Φ max numerically and interpolating the value of Φ max for the experimental values of T. All these can be achieved automatically by a computer program. The procedures of the Φ max scan are summarized as follows. 1. Place the sample to be measured somewhere between Z = 6 and Z = 4 along the optical axis, using a setup similar to the Z-scan measurement with a LG 1 0 beam. 2. Record the T I (or the T t) curve untile the valley is passed. 3. Calculate the sample position Z by substituting the T coordinate of the valley point into Eq. (2.94). 4. Calculate T as a function of Φ max with the Z from the above step. Make a two column data table, and fill one column with Φ max increasing with certain step and the other column with the corresponding values of T. 5. Transfer the T I (or the T t) curve into the Φ max I (or the Φ max t) curve by interpolating the above data table. 6. Calculate the n 2 from the Φ max and obtain the n 2 value (or the n 2 t curve). The advantages of the Φ max -scan method are obvious in comparison with the Z-scan or the I-scan method. There is no need to move the sample along the Z axis. It s not necessary to pre-determine the sample position accurately or use another sample with known nonlinear refractive index to calibrate the system. In the case of a slow nonlinear mechanism, the Φ max -scan method can measure how the n 2 increases with time automatically. There are, however, trade-offs. To be able to reach the valley point requires a relatively large Φ max : when Z = 4, the Φ max coordinate of the valley is about -0.5, and when Z = 6, the Φ max coordinate of the valley is about Although Φ max can 82

100 be increased in several ways, such as using a thicker sample, larger intensity, or longer exposure time for a slow nonlinear process, they all eventually have an upper limit, due to the thin sample assumption, the damage threshold of the sample, or the time period in which the nonlinearity can be treated as the optical Kerr effect. So materials that have a larger n 2 are more appropriately to be measured by the Φ max -scan method. Another issue is that the resolution of n 2 is not uniform for given resolution of T, which can be seen from Figure 2.21, where around the valley in the T Φ max curve, Φ max shows the biggest uncertainty for given uncertainty of T, giving the lowest resolution of Φ max (and therefore n 2 ) compared to other part of the curve. This should be kept in mind when interpreting the final result. Finally we note that the region between Z = 4 and Z = 6 is not the only one that a Φ max scan can be carried out. In principle any group of T Φ max curves that show a clear turning point of T where Φ max increases from 0 can be used to do the Φ max - scan measurement, provided that the T coordinate of the turning point is in one-to-one correspondence to Z. For example, the curves whose Z values are between 0.61 and 3.49 using a LG 1 0 beam have the valleys as shown Figure 2.24, and the curves whose Z are larger than 0 using a LG 0 0 beam have the peaks as shown Figure In practice the selection of these groups should be governed by their overall performances, including the resolution and the minimum Φ max required, which depend on the shape of the curves and the location of the turning points, respectively. Among them, the one used as the example in this section (i.e., the curves whose Z are between -4 and -6 using a LG 1 0 beam) has the least minimum Φ max requirement, yet gives good overall resolution. 83

101 Normalized Transmittance T Maximum nonlinear phase distortion Figure 2.24: Normalized transmittance, T, versus the maximum nonlinear phase distortion, Φ max, in the sample when the incident beam is a LG 1 0 beam for selected curves whose Z are between 0.61 and The position, Z, of the sample for each of the curve is indicated by the number along the curve. 84

102 5 Normalized Transmittance T Maximum nonlinear phase distortion Figure 2.25: Normalized transmittance, T, versus the maximum nonlinear phase distortion, Φ max, in the sample when the incident beam is a LG 0 0 beam for selected curves whose Z values are larger than 0. The position, Z, of the sample for each of the curve is indicated by the number along the curve. 85

103 Bibliography [1] H. Kogelnik and T. Li, Laser beams and resonators, Appl. Opt. 5, 1550 (1966). [2] R. W. Boyd, Nonlinear optics (Academic Press, Boston, 1992). [3] M. Dumont, Z. Sekkat, R. Loucifsaibi, K. Nakatani, and J. A. Delaire, Photoisomerization, photoinduced orientation and orientational relaxation of azo dyes in polymeric films, Nonlinear Opt. 5, 395 (1993). [4] W. Zhang, S. Bian, S. Kim, and M. Kuzyk, High efficiency holographic volume index gratings in dr1-dopped pmma, Opt. Lett. 27, 1105 (2002). [5] S. Bian, D. Robinson, and M. Kuzyk, Optical activated cantilever using photomechanical effects in dye-doped polymer fibers, J. Opt. Soc. Am. B 23, 697 (2006). [6] B. E. A. Saleh and M. C. Teich, Fundamentals of photonics, Wiley series in pure and applied optics (Wiley, New York, 1991). [7] G. B. Arfken and H. J. Weber, Mathematical methods for physicists, 5th ed. (Harcourt Academic Press, San Diego, Calif., 2001), george B. Arfken, Hans J. Weber. [8] L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Orbital angular momentum of light and the trasformation of laguerre-gaussian laser modes, Phys. Rev. A 45, 8185 (1992). 86

104 [9] J. Nye and M. V. Berry, Dislocations in wave trains, Proc. R. Soc. Lond. A. 336, 165 (1974). [10] A. Yariv, Quantum electronics, 3rd ed. (Wiley, New York, 1989). [11] J. D. Jackson, Classical electrodynamics, 3rd ed. (Wiley, [New York], 1999). [12] Z. Sekkat, D. Morichere, M. Dumont, R. Loucifsaibi, and J. A. Delaire, Photoisomerization of azobenzene derivatives in polymeric thin-films, J. Appl. Phys. 71, 1543 (1992). [13] R. Loucifsaibi, K. Nakatani, J. A. Delaire, M. Dumont, and Z. Sekkat, Photoisomerization and 2nd harmonic-generation in disperse red one-doped and one-functionalized poly(methyl methacrylate) films, Chem. Mater. 5, 229 (1993). [14] J. A. Hermann, Simple model for a passive optical power limiter, J. Mod. Opt. 32, 7 (1985). [15] M. Sheik-bahae, S. A. A., and V. S. E. W., High-sensitivity, single-beam n 2 measurement, Opt. Lett. 14, 955 (1989). [16] M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagen, and E. W. Van Stryland, Sensitive measurement of optical nonlinearities using a single beam, IEEE J. Quantum Electron. 26, 760 (1990). [17] E. W. Van Stryland and M. Sheik-Bahae, Z-scan, in Characterization techniques and tabulations for organic nonlinear optical materials, C. W. Dirk and M. G. Kuzyk, eds., (Marcel Dekker, New York, 1998), p [18] A. E. Kaplan, external self-focusing of light by a nonlinear layer, Radiophys. Quant. Electron. 12, 692 (1969). 87

105 [19] D. Weaire, B. Wherrett, D. Miller, and S. Smith, Effect of low-power nonlinear refraction on laser-beam propagation in insb, Opt. Lett. 4, 331 (1979). [20] M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown, S. Herman, B. Golick, M. Kartz, J. Miller, H. T. Powell, M. Vergino, and V. Yanovsky, Petawatt laser pulses, Opt. Lett. 24, 160 (1999). 88

106 Chapter 3 Experiment 3.1 Introduction This chapter is devoted to introduce the experiments that we carry out in order to test the validity of our theory, as well as to demonstrate the proposed applications. First in Section 3.2, we describe how we generate high-order LG beams. This is important because the-high order LG beams are usually not available from commercial lasers. We also include in this section an experimental method that can identify the orbital angular momentum carried by an LG beam. Sample fabrication is another important factor to the success of our experiments. So in Section 3.3 we explain how we synthesize the DR1/PMMA material, and how we make the samples that are suitable for our experiments. As pointed out in the theory part, DR1/PMMA acts like an optical Kerr medium only under certain conditions. The time that the sample is illuminated can not be long, and the beam intensity can not be too high. In order to get an estimation of this time scale and the beam intensity, we measure the dynamics of the nonlinear refractive index change in DR1/PMMA through a holographic volume index gratings recording experiment. The details of the experiment are explained in Section

107 Finally in Section 3.5, we describe the experiments that use the LG 1 0 beams, including the Z-scan measurement, the I-scan measurement, and optical limiting. As the setups for the three experiments are similar, we explain them together and make notes wherever there are differences. 3.2 Generating the higher order Laguerre Gaussian beams Several methods have been reported to produce higher-order Laguerre Gaussian beams, including directly from specially designed laser cavities, 1 4 by converting Hermite-Gaussian modes using cylindrical-lens mode converters, 5 7 by using spiral phase plates, 8, 9 and by computer-generated holograms In our experiment, we use the computer-generated hologram to generate the LG 1 0 beam, because it is simple, demands only limited equipments, yet gives reasonable efficiency and good beam quality. In this section, we first review the principles of designing the computer-generated hologram, and then describe how we make the hologram and generate the LG 1 0 beam The principles A hologram is a recording of the interference pattern generated by two light beams, namely, the signal beam and the reference beam. The signal beam can be reconstructed by illuminating the hologram with the reference beam. By using a computer, we can simulate the interference pattern and produce the hologram without using real beams. Consider the following hologram which converts a LG 0 0 beam into a LG 1 0 beam as shown in Fig The hologram is placed in the xy plane at z = 0. The LG 0 0 beam propagates along the z axis. The wave vector of the LG 1 0 beam lies in the xz plane and makes an angle, α, with the z axis such that it can be better separated from the incident beam in 90

108 Figure 3.1: Schematic diagram of a hologram that converts a LG 0 0 beam into a LG 1 0 beam. practice. Although it is not necessary, for simplicity we assume the waists of both beams are at z = 0. The complex amplitude of the two beams at z = 0 can be written as ) E LG 0 0 = exp ( r2, (3.1) ω0 2 and E LG 1 0 = ) 2r exp ( r2 exp ( iφ) exp ( ik ω 0 ω0 2 x x) exp ( iψ 0 ), (3.2) respectively, where k x = k sin (α) is the x component of the wave vector k of the LG 1 0 beam, ψ 0 is a possible relative phase shift between the LG 0 0 and the LG 1 0 beam, r, φ and z are the cylindrical coordinates, and we have ignored the time-dependent factor exp (i2πνt). The conversion can be achieved if the hologram gives a complex transmittance function T(r,φ) such that E LG 0 0 T(r,φ) = E LG 1 0. The expression of T(r,φ) is easily obtained by using Eqs. (3.1) and (3.2), yielding T(r,φ) = 2r ω 0 exp ( i (φ + k x r cos φ + ψ 0 )). (3.3) As can be seen, a pure conversion requires the hologram to be able to modify both the 91

109 amplitude and the phase of the incident beam, which is demanding but doable. As a compromise, an amplitude hologram, which has a relaxed requirement, can be used, by sacrificing the conversion efficiency and the mode purity of the converted beam. For example, an amplitude transmittance function T(r,φ) = (1 + cos (φ + k xr cos φ + ψ 0 )) 2 (3.4) can also be written as T(r,φ) = exp ( i (φ + k xr cos φ + ψ 0 )) exp (i (φ + k xr cos φ + ψ 0 )), (3.5) where the second term has the phase modulation required by Eq. (3.3) and generates the target beam. The first term generates a beam propagating along the direction of the incident beam. The third term generates a beam whose wave vector is the mirror image of that of the target beam with respect to the yz plane, and whose angular mode number is 1, opposite to that of the target beam in sign. Note that for an amplitude hologram, the transmittance function is real and generally within the range of 0 to 1. Therefore, some useless terms in Eq. (3.5) do not generate the desired mode. As a consequence, not all power of the transmitted light is converted to the target beam and a significant amount is wasted on other beams. For the same reason, the amplitude variation 2r/ω 0 in Eq. (3.3) should be modified with care such that it is between 0 to 1. However, it has been shown that for all practical purposes, this variation 10, 13 can be neglected for convenience at the price of slight mode impurity. The fabrication is further simplified if the amplitude hologram is binary, i.e. the values 92

110 of T(r,φ) are either 1 or 0. For this purpose, Eq. (3.4) can be modified as follows T(r,φ) = 0, 0 (1+cos(φ+kxr cos φ+ψ 0)) 2 < 1 2 ; 1, 1 2 (1+cos(φ+kxr cos φ+ψ 0)) 2 1. (3.6) This actually describes a square wave with ν = φ + k x r cosφ + ψ 0 as the variable, 0, T(ν) = 1, ( ) ( 2n π < ν < 2n + 3 2) π; ( ) ( ) (3.7) 2n 1 2 π ν 2n π, where n is an integer. Thus, using the Fourier series representation of a square wave, we have T(r,φ) = n=1 ( sinc n π ) cos (n (φ + k x r cos φ + ψ 0 )), (3.8) 2 among which the term associated with n = 1 is the desired one. The conversion efficiency of the binary amplitude hologram can be estimated using the coefficient of the term. The ratio of the power of the generated LG 1 0 beam to that of the incident beam is approximately ( 1 ( π ) ) 2 2 sinc 10%, (3.9) 2 where the factor 1/2 arises from the power of the LG 1 0 mode being shared with the LG 1 0 beam that is generated by the same term. The angle α in Fig. 3.1 determines how well the different generated beams separate from each other in the far field. The larger it is, the less the target beam overlaps the other beams. The minimum value of α must be larger than half the divergence angle of the gaussian beam, thus we require α >> λ/πω 0. 93

111 3.2.2 Making the hologram The following Mathematica codes are used to calculate and plot the pattern of the binary amplitude hologram. l = 1; (*angular mode number *) Lamda = 647*10^-9; (*wavelength*) alpha = 0.1/180*Pi; (*inclined angle *) kx = 2*Pi/Lamda*Sin[alpha]; (* x component of the wave vector *) T[x_, y_] = Sign[Cos[kx*x + l*arctan[x, y] + Pi/2]] + 1; (* transmittance function of the binary amplitude hologram *) DensityPlot[T[x, y], {x, , }, {y, , }, PlotPoints -> 3000, Mesh -> False, Frame -> False] (* plot the pattern*) Figure 3.2: Typical holographic pattern that converts a LG 0 0 beam to a LG 1 0 beam. Figure 3.2 shows the typical pattern of a binary hologram. The pattern appears like a grating but with a fork in the center, which is a result of the phase singularity. The pattern is printed on an A4 paper with a laser printer and then reduced to about a 7 mm 94

112 Figure 3.3: The multiple orders of beams generated by the binary amplitude hologram. by 7mm area onto a transparency film using a high resolution photocopy machine. We then cut the transparency film to the appropriate size and attach it to an optical mount, which is used to make fine adjustments of the hologram s position. We use the fundamental gaussian beam at 647 nm from a Coherent Innova 70C argon CW laser by using the smaller output aperture of the laser. We then let the beam strike the center of the hologram at normal incidence. In the far field we observe a pattern of the different orders of the diffracted beams, which is shown in Figure 3.3. In the center of the multiple beams is the fundamental gaussian beam (LG 0 0), as can be judged by its bright center. Nearby are the LG 1 0 and LG 1 0 beams whose centers are dark. Also shown are the LG ±2 0 and the LG ±3 0 beams. The LG ±2 0 beams are weak, consistent with Eq. 3.8 in which the coefficient is zero when n is a even number. We use a screen with a window of appropriate size as a spatial filter to let through the LG 1 0 beam and block the other beams. 95

113 3.2.3 Examining the phase singularity The intensity profile of the generated LG 1 0 beam is easily verified by looking at the pattern on a screen or more precisely, using a beam profiler. The screw-like phase profile, however, is not directly observable. Specifically we need to verify that the angular mode number l of the beam is indeed equal to 1, or, equivalently, that the beam carries orbital angular momentum of h per photon. This can be done by the following interference 10, 14 experiment. Figure 3.4: Schematic diagram of the interference experiment to exam the phase dislocation of a LG 1 0 beam. M: mirror; BS: beam splitter; DP: dove prism. The setup is basically a modified Mach-Zehnder interferometer as shown in Figure 3.4. The beam is separated into two paths by a beam splitter. One of the beams travels through a dove prism, making the profile of the transmitted beam by reflection the mirror image of the incident beam. As a result, the handedness of the phase screw of the beam is changed, i.e., the LG 1 0 beam becomes a LG 1 0 beam, and vice versa. The transmitted 96

114 beam is recombined with the beam from the other arm by the second beam splitter. The two beams, now having opposite signs in their angular mode number l, interfere and give us an pattern that can be used to characterize the value of l. Figure 3.5: Typical self-interference pattern of a LG 1 0 beam with a dove prism placed in one arm. The three-prong fork in the center is evidence that the angular mode number l of the incident beam is 1 (or -1). Figure 3.5 shows the typical interference pattern from the beam that we generated using the binary amplitude hologram. The three-prong fork in the center indicates that the angular mode number l of the incident beam is 1 (or -1) Fabricating the DR1/PMMA Samples Two methods are used in our lab to make the DR1/PMMA samples. The solvent-polymerdye method is used to make thin (several micrometers) films that are suitable to measure the absorption spectrum, while the polymerization-with-dye method is used to make thicker bulk samples (a few millimeters) and are used in our nonlinear optics experiments Solvent-polymer-dye method The idea of this method is to make a homogeneous mixture of the polymer and the dye by dissolving them in certain solvents which are subsequently removed through evaporation. The ingredients are weighted by an electronic balance according to the following formula (all the percentage are in weight): 97

115 1. The solution is made of 85% solvents and 15% solids. 2. The solvents include 67% propylene Glycol Methyl Ether Acetate (PGMEA) and 33% γ-buterolactone. 3. The solids consist of the PMMA and the DR1 with the desired proportions, e.g. 1% DR1 and 99% PMMA. The ingredients are then put into a bottle with a magnetic stirrer in it in the following order, stirring between each ingredient: 1. PGMEA, 2. DR1, 3. γ-buterolactone, 4. PMMA. We use a Corning micro slide as the substrate of the sample, clean it with methanol and place it on a spin coater. After all the ingredients are thoroughly mixed, the solution is filtered through a 0.2 µm filter using a syringe, deposited directly on the center of the substrate, forming a small puddle. We then use the spin coater to cast the solution over the surface of the substrate to form a thin film. The thickness of the film decreases with faster spin speed or longer spin time. The substrate is then baked in a convection oven at a temperature of 95 C for one hour to evaporate the solvents. Finally, the sample is cooled slowly to room temperature polymerization-with-dye method The solvent-polymer-dye method works well to make thin films because the surface area is large relative to the volume, allowing the solvents to evaporate. However, this method can 98

116 not be used to make bulk samples because the solvents would be trapped in the sample and would form bubbles. Instead, we use the polymerization-with-dye method. The basic idea is to mix the dye with the liquid monomer and then polymerize it to form the solid polymer with embedded dye, eliminating the need for solvent. Figure 3.6: The alumina-filled column used to remove the inhibitor from the MMA. The monomer used to make PMMA is methyl methacrylate (MMA), and is commercially available. It arrives in bottles with the inhibitor, which is a chemical added to prevent the monomer from polymerization during transportation and storage. The inhibitor needs to be removed to allow for polymerization. The inhibitor is removed by passing the MMA through a column which is filled with alumina powders (see Fig. 3.6) and collecting the resulting liquid one drop at a time. As a result, the inhibitor is trapped in the column and 99

117 pure MMA goes through. Because MMA can be polymerized by room light, we wrap the container of the MMA with aluminum foil after the inhibitor is removed. Next we add the desired amount of DR1 powder into the MMA liquid and use a magnetic stirrer to mix them until the powder is thoroughly dissolved in the liquid. When the concentration of the DR1 is high, greater than 1%, we find that using an ultrasonic bath is needed to help the dye to dissolve into the solution. Subsequently the plasticizer (dibutyl phthalate, 0.5%-1% by weight), the chain transfer agent (butanethiol, 2.2 µl/ml solution), and the initiater (ter-butyl peroxide, 2.2 µl/ml solution), are added into the solution sequentially. The plasticizer adds flexibility to the polymer so that the polymer is easier to be shaped mechanically. It works by separating the polymer chains, which increases the size of the voids in the polymer as a consequence. The initiater starts the polymerization by making the monomer molecules chemically active and links them together one by one like a chain, until a CTA molecule is met, which terminates the growth of the chain. Thus, the amount of CTA limits the average length of the polymer molecules. We note that all the operations must be conducted in the chemical fume hood to avoid the strong unpleasant smells, but more importantly, to protect against the associated respiration hazard and toxicity. After the solution is thoroughly mixed, we filter it through a 0.2 µm filter by using a syringe and inject it into test tubes. This removes possible undissolved solid particles, ensuring that a homogeneous sample results. The test tubes are then sealed tightly using their caps and placed vertically in a metal test tube rack. The rack together with the test tubes are put into a convection oven which is set to 90 C. After half an hour of heating, we open the caps of the test tubes to allow the generated gases to escape and to reduce the increased pressure. This step is helpful in decreasing bubbling in the polymer. It also prevents the test tubes from exploding due to the high gas pressure inside. The caps are then re-tightened and the samples are left in the oven for further polymerization. After about 48 to 72 hours, when the sample should be totally polymerized, we remove 100

118 the test tubes from the oven and immediately put them in a box into a freezer. The sudden temperature drop separates the polymer from the wall of the test tube so that it is easy to remove the polymer undamaged by braking the test tube. The resulting sample, called a preform, is a rod-shaped solid about 1.2 cm in diameter and 6-8 cm in length. Figure 3.7: Diagram of the squeezer that is used to press thick polymer films. To make film samples a squeezing process is used. The preform is cut into small chunks about 2cm to 4 cm in length in the machine shop. A chunk is then sandwiched between two pieces of glass plates with the sides of the cylinder touching the glass. The glass plates together with the sample are fixed in a squeezer (as shown in Fig. 3.7) and are put into the convection oven which is set at about 120 C. After about half an hour, when the sample becomes soft, we remove the squeezer from the oven and start to tighten the set screws (There are a total of four of them, one at each corner of the square-shaped metal plate.) to press down the sample. This must be done quickly as the sample is cooling down and becoming stiff. Care must be taken not to apply too high a torque to the screws, preventing the glass plates from breaking. Once the screws can not be tightened further, we put the squeezer back into the oven for another half hour. This process is repeated until the sample is close to the desired thickness. At this point, we insert the spacers of 101

119 the desired thickness in between the glass plates to control further squeezing. Once both glass plates come in contact with the spacers, no more squeezing is necessary. The oven temperature is then reduced to about 90 C and the samples are allowed to relax at this temperature for several hours. Finally we turn off the oven and let the sample cool down slowly (in the oven) to room temperature, which often takes one night. The sample is then removed from the squeezer and the glass plates. Samples with good surface quality are obtained from this process. The thickness of the sample is decided by the spacer, usually a few millimeters. 3.4 Recording of high efficiency holographic volume index gratings in DR1/PMMA Using the DR1/PMMA film made by the squeezing process, we have successfully recorded high efficiency (up to 80% ) holographic volume index gratings using off-resonant writing beams (such as 633 nm and 647 nm). 15 In this section we introduce this experiment. Although achieving such a high efficiency is significant, the main reason that we include this experiment is that it allows us to determine the conditions under which the nonlinear process in DR1/PMMA can be treated as the optical Kerr effect, as will be discussed below Background A grating is an optical component whose dielectric constant is periodic in space. When only on the real part of the refractive index is periodic, the grating is an index grating. A grating can diffract a light beam to some other direction at certain incident angles. The theory on gratings can be found in many references. 16, 17 Here we briefly summarize some important points. 102

120 Assume that an index grating has a refractive index of the form n(z) = n 0 + n 1 cos(kx), (3.10) where n 0 is the normal refractive index of the medium, n 1 is the amplitude of the periodic modulation of the refractive index, and K is called the grating wave number. For simplicity, the index variation is assumed to be sinusoidal. The grating spacing or period Λ is thus equal to 2π/K. Figure 3.8 shows one such grating, where the thickness of the grating is assumed to be d. When 2πλd/n 0 Λ 2 >> 1, the grating is said to be a thick or volume grating. 17 Our discussion focuses on the volume grating. Figure 3.8: Diagram of the diffraction of a light beam in an index grating. The Bragg angle is defined by ( ) λ θ B = sin 1. (3.11) 2n 0 Λ For a volume grating (refer to Fig. 3.8, where the wave vector is in the xz plane.), the 103

121 grating can diffract part or all of the energy of the incident beam to another direction and form a new beam if and only if a beam is incident to the grating with the incident angle θ 1 = θ B. 16, 17 The angle between the diffracted beam and the normal of the incident plane, θ 2, also equals θ B, but the direction of the x component of the wave vector of the diffracted beam is opposite to that of the incident beam. The diffraction efficiency, which is the ratio of the intensity of the diffracted beam to 16, 17 that of the incident beam, can be proved to be ( ) η = sin 2 πn1 d. (3.12) λ cos θ B Figure 3.9: Illustration of forming the index grating by two-beam coupling. If the refractive index of the material is intensity dependent, such as it is in an optical Kerr medium, an index grating can be formed by letting two coherent beams interference inside the material. 18 Figure 3.9 illustrates such a configuration, where the two coherent beams intersect inside the material and form an interference pattern whose intensity is 104

122 sinusoidal in the x direction. Consequentially the refractive index of the material in the intersection region is modulated by the intensity pattern and becomes periodic, forming an index grating. It s easy to show that the directions of the two beams automatically satisfies the Bragg angle θ B of the grating. In other words, if one of the beams is blocked, the grating diffracts the other beam to the direction of the blocked beam. If we let one of the beams (signal beam) to carry information, and the other (reference beam) is a plane wave, then the information will be embedded in the formed grating and can be recovered by letting the reference beam strike the grating at the Bragg angle. Therefore the grating is like a hologram, and can be called a holographic grating. A good holographic volume grating demands high diffraction efficiency, which in turn requires relatively big refractive index modulation and grating thickness according to Eq. (3.12). For materials with an intensity dependent refractive index, given the same intensity of the light beam, bigger refractive index modulation means bigger nonlinear response. The azo-dye doped polymer materials, including DR1/PMMA, are well known for having big nonlinear refractive index change due to the trans-cis-trans photoisomerization with subsequent molecular reorientation Using these materials to record holographic gratings have been studied extensively The previous research had used the wavelengths at the strong resonant absorption band of the azo-dye in order to maximize the photo-isomerization efficiency. However, because of azo-dye s very high optical absorption at these wavelengths, the thickness of the gratings are limited to several microns, which makes a volume grating difficult. Our group proposed the use of off-resonant writing beams (such as 633 nm and 647 nm) to generate volume gratings in DR1/PMMA. 15 When absorption is weak, the light can travel much deeper into the sample before being attenuated, so thick films (on the order of mm) can be used to record the volume gratings. On the other hand, we find that light at these wavelengths can still introduce a large change of refractive index. We have observed diffraction efficiencies in excess of 80% in a 2 mm thick DR1/PMMA bulk 105

123 material by using 647 nm writing beams Experimental setup We build an in-situ diffraction efficiency measurement system to monitor the diffraction efficiency of a grating while it is forming. As the grating results directly from the nonlinear refractive index, the system also allows us to observe the dynamics of the underlying nonlinear mechanisms, which in DR1/PMMA is mainly photoreorientation of the DR1 molecules. As such, the observed dynamics let us determine the time scales within which we can treat the DR1/PMMA as a Kerr medium. Figure 3.10: Setup of the holographic volume index grating recording in DR1/PMMA and the in-situ diffraction efficiency measurement system. The setup is illustrated in Figure A laser beam of wavelength of 647 nm from an Innova 90K Krypton/Argon mixed gas laser is expanded and collimated by a beam expander so that the beam closely resembles a plane wave. A polarizer is used to ensure 106