ANALYSIS AND APPLICATIONS OF NOVEL OPTICAL SINGLE- AND MULTI-LAYER STRUCTURES. Dissertation. Submitted to. The School of Engineering of the

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1 ANALYSIS AND APPLICATIONS OF NOVEL OPTICAL SINGLE- AND MULTI-LAYER STRUCTURES Dissertation Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Doctor of Philosophy in Electro-Optics By Han Li UNIVERSITY OF DAYTON Dayton, Ohio December 2015

2 ANALYSIS AND APPLICATIONS OF NOVEL OPTICAL SINGLE- AND MULTI-LAYER STRUCTURES Name: Li, Han APPROVED BY: Partha P. Banerjee, Ph.D. Advisory Committee Chair Program Director Electro-Optics Program Joseph W. Haus, Ph.D. Committee Member Professor Electro-Optics Program Andrew M. Sarangan, Ph.D. Committee Member Professor Electro-Optics Program Monish R. Chatterjee, Ph.D. Committee Member Professor Electrical & Computer Engineering John G. Weber, Ph.D. Associate Dean School of Engineering Eddy M. Rojas, Ph.D., M.A., P.E. Dean School of Engineering ii

3 Copyright by Han Li All rights reserved 2015 iii

4 ABSTRACT ANALYSIS AND APPLICATIONS OF NOVEL OPTICAL SINGLE- AND MULTI-LAYER STRUCTURES Name: Li, Han University of Dayton Advisor: Dr. Partha P. Banerjee A thin film is a layer of a material or an assembly of multilayers of different materials ranging from nanometers to several micrometers in thickness. Optical coatings, optical filters, semiconductor lasers, quantum well structures, and nonlinear frequency convertors are some of the main applications of thin films. This dissertation is concerned with some aspects of optical propagation through, and at the interface of, some specific thin film structures, such as a double negative metamaterial and nonlinear photonic bandgap structures. A versatile imaging system that captures the near field radiation from subwavelength objects for high resolution imaging beyond the diffraction limit can be accomplished with new types of thin film materials not found in nature. One such material, termed a metamaterial with a double negative index, has been recently developed in our laboratory from a binary mixture of nanoparticles comprising silver and iv

5 silicon carbide. A nanoscale structure comprising the sub-wavelength object, the single layer thin-film metamaterial superlens imaging setup to accurately characterize super resolution imaging for arbitrary polarized illumination of the object has now been fabricated as part of this dissertation and will be tested as part of future work. Secondly, electromagnetic wave reflection at a single interface is one of the most basic optical phenomena presented in nature. Snell s laws and the Fresnel equations determine the wave vectors and amplitudes for reflection and transmission. However, when a focused electromagnetic beam is incident at the interface between two layers, the reflected and transmitted beams suffer spatial shifts such as the Goos-Hänchen and Imbert-Fedorov shifts. In this dissertation, the Imbert-Fedorov shift is theoretically and experimentally investigated at the interface between air and a double negative index metamaterial. Finally, nonlinear effects such as optical second order and third order harmonic generation in thin films such as photonic bandgap structures are attracting growing attention due the enhancement of the participating electromagnetic waves inside the structure, leading to vastly increased conversion efficiency. In this dissertation, a method based on the transfer matrix method is developed to numerically simulate the second harmonic and third harmonic generation in such periodic nonlinear photonic bandgap structures. v

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

7 ACKNOWLEDGEMENTS I would like to specially thank Dr. Partha Banerjee, for all his help and advice in my studies, for directing this dissertation and careful modifications. His patience, time and vast knowledge are the biggest encouragement for me. I would also like to thank my committee members Dr. Joseph Haus who offered significant time and attention to my research, Dr. Andrew Sarangan who allowed me to use his clean room facilities, and Dr. Monish Chatterjee for his assistance and helpful comments. Additionally, I also would like to thank Dr. Qiwen Zhan for offering me COMSOL software. Also I would specially thank all EO faculty for their great classes, and my group members and friends for their wonderful support and assistance. Finally, I would like to thank my father Chengjiang Li, mother Guiying Han and my family for their love, support and encouragement. Last but not least, I would especially thank to my wife Li Li and my son Allen Li for being my strength and for staying behind me all the time. vii

8 TABLE OF CONTENTS ABSTRACT iv DEDICATION vi ACKNOWLEDGEMENTS...vii LIST OF FIGURES.x LIST OF ABBREVIATIONS..xv LIST OF SYMBOLS xvii CHAPTER 1 INTRODUCTION AND OBJECTIVES Introduction Objectives and Organization Publications Resulting from this Work...8 CHAPTER 2 METAMATERIAL: FABRICATION AND TESTING OF NEAR- FIELD IMAGING Introduction Essential Theory of Near-field Lensing and its Measurement Model and Simulation of Near-field Imaging using Finite Element Method Fabrication of Periodic Object: Experimental Details Fabrication Results and Characterization Summary 27 CHAPTER 3 PHOTONIC SPIN HALL EFFECT IN DOUBLE NEGATIVE METAMATERIALS Introduction Theory of Spin Hall Effect of Gaussian Beam 30 viii

9 3.3 Observation of Spin Hall Effect for Double Negative Metamaterial Summary 42 CHAPTER 4 APPLICATION OF TRANSFER MATRIX METHOD TO SECOND HARMONIC GENERATION IN NONLINEAR PHOTONIC BANDGAP STRUCTURES: OBLIQUE INCIDENCE Introduction Theory of Obliquely Incident Fundamental Fields in Periodically Nonlinear Materials Results and Discussion for SHG Summary 65 CHAPTER 5 APPLICATION OF TRANSFER MATRIX METHOD TO THIRD HARMONIC GENERATION IN NONLINEAR PHOTONIC BANDGAP STRUCTURES: OBLIQUE INCIDENCE Third Harmonic Generation in Multilayer Structures: Oblique Incidence Results and Discussion for THG Summary CHAPTER 6 CONCLUSION AND FUTURE WORK Conclusion Future Work Superlens for Super-resolution Applications Fabrication Steps in Clean Room Super-resolution Testing using AFM Super-resolution Testing using NSOM TMM Simulations of SHG in HM Role of Self Phase Modulation in THG..89 REFERENCES.90 ix

10 LIST OF FIGURES Figure 2.1: (a) Schematic diagram of the near-field imaging structure with a subwavelength object and thin layer of double negative metamaterial. (b) Schematic diagram of a similar structure with the sub-wavelength object but without double negative metamaterial used for comparison with (a).13 Figure 2.2: (a) COMSOL simulation of E 2 for the structure in Figure 1(a) with TE polarized plane wave and normal incidence. (b) Same as Figure 2(a), but with different scales for the data to highlight the optical fields within and beyond the metamaterial superlens. The grid structure representing the 1D object comprises alternating layers of Mo and photoresist, starting from the left. (c) Plot of E 2 in 1D in the region of image plane. As expected, E 2 is higher behind the regions where there is no Mo. (d) - (f): Similar to (a)-(c), except that there is no metamaterial superlens. In this case, the object is incorrectly resolved, with intensity maxima occurring at the geometrical shadow of the object. In (a), (b), (d) and (e), the incident illumination is from the top.14 Figure 2.3: (a) COMSOL simulation of E 2 for the structure in Figure 1(a) with TE polarized plane wave and normal incidence. (b) Same as Figure 2(a), but with different scales for the data to highlight the optical fields within and beyond the metamaterial superlens. The grid structure representing the 1D object comprises alternating layers of Mo and photoresist, starting from the left. (c) Plot of E 2 in 1D in the region of image plane. As expected, E 2 is higher behind the regions where there is no Mo. (d)-(f): Similar to (a)-(c), except that there is no metamaterial superlens. In this case, the object is incorrectly resolved, with intensity maxima occurring at the geometrical shadow of the object. In (a),(b),(d) and (e), the incident illumination is from the top...15 Figure 2.4: (a) E 2 in 1D immediately behind the object for 405 nm TE illumination at normal incidence; (b) E 2 in 1D in the photoresist beyond the superlens at distances 0, 1, 2, 3, 4, 5, 6 nm for TE illumination; (c) E 2 in 1D immediately behind the object for 405 nm TM illumination at normal incidence; (d) E 2 in 1D in the photoresist beyond the superlens at distances 16, 20, 24, 28, 32, 36, 40 nm for TM illumination. The best imaging distance for TE is around 4 nm, while the best imaging distance for TM is around 24 nm...17 Figure 2.5: Schematic of proposed Mo object fabrication process. (a) A 50 nm Mo film is deposited on a flat glass substrate using sputter machine. (b) A 100 nm BARC layer is deposited on top of Mo film using spin coater. (c) A 400 nm DUV photoresist is coated on top of BARC by the spin coater. (d) Using 266 nm laser as the light source, to do the interference photo lithography on the DUV photoresist. (e) After the development exposed photoresist. A grating with period of sub-wavelength is completed. (f) Using the photoresist grating as he etch mask. x

11 The sub- wavelength object of Mo is etched by reactive ion etching in a fluorinated plasma. (g) The left photoresist and BARC on Mo object are stripped by pouring acetone and followed by the treatment in the O2 plasma etching system. Then a sub-wavelength object is completed.20 Figure 2.6: For rotation angle of θ = 30, (a) SEM image for the developed photoresist without the underlying BARC layer (b) SEM image for the developed photoresist with the underlying BARC layer.22 Figure 2.7: For rotation angle of θ = 35, SEM image (a) corresponding to the structure of Figure 2.5(e); (b) corresponding to the structure of Figure 2.5(f); (c) of the final Mo object corresponding to the structure of Figure 2.5(g); (d) showing top view of the Mo object. (e) Semi-transparent nature of final sample with Mo object..24 Figure 2.8: For rotation angle of =40, SEM images for (a) 1D (b) 2D Mo objects. In this case the period is approximately 200 nm...26 Figure 2.9: Simulation results (varying from near zero to 0.4) and experimental observations (varying from 0.2 to 0.4) for transmission at 405 nm as a function of polarization angle, which varies from 0 (TE) to 90 (TM). 27 Figure 3.1: Sketch of spin Hall effect of Gaussian beam at an air-medium interface. A linearly polarized beam reflects on a medium and splits into left- and right-handed circularly polarized light. The medium can be normal material or metamaterial..30 Figure 3.2: Schematic of setup for measurement of IF shift. A 405 nm fiber laser is used as light source, lens 1 and lens 2 are working for collimation. P1 and P2 are Glan laser polarizer to eliminate x and y polarized light.34 Figure 3.3: Spin Hall effect of p polarized Gaussian beam showing beam shift of the RCP and LCP components of a Gaussian beam. The interface media are selected to be air glass.35 Figure 3.4: Schematic of setup for measurement of IF shift. 405 nm fiber laser is used as light source, lens 1 and lens 2 are working for collimation. P1 and P2 are Glan laser polarizer to eliminate x and y polarized light. The CCD is used to detect the magnified split beam. Gaussian beam splitting happens on the interface of air - Ag+SiC...37 Figure 3.5: Experimental setup for weak measurement method. 405 nm fiber laser is used as light source, lens 1 and lens 2 are working for collimation. P1 and P2 are Glan laser polarizer to eliminate x and y polarized light. CCD is used to detect the magnified splits beam. Gaussian beam splitting happened on the interface of air-ag + SiC 38 xi

12 Figure 3.6: Experimental beam profiles of s polarized light recorded by CCD. (a) Experimental beam profile at Δ = 2 ; (b) Experimental beam profile at Δ = 1 ; (c) Experimental beam profile at Δ = 0 ; (d) Experimental beam profile at Δ = 1 ; (e) Experimental beam profile at Δ = 2.39 Figure 3.7: Comparison between theory and experiment at air-double negative metamaterial interface for different Δ(polarization angle)..40 Figure 3.8: Comparison between theory and experiment at air-double negative metamaterial interface for different incident angles..41 Figure 3.9: Comparison between theory and experiment at air-double negative metamaterial interface for different incident angles, and for different metamaterial thicknesses.42 Figure 4.1: Layered structure composed of nonlinear materials with refractive indices n (1) i for FF and n (2) i for SH. The corresponding χ (2) i represents the nonlinear susceptibilities in each layer. N represents the number of segment. The FF with TM polarization is assumed to be propagating in the incident medium with angle of θ Figure 4.2: Sandwich structure composed of one layer of nonlinear medium in the center and two linear media on either side. The red and black lines represent SH and FF, respectively. The FF is incident with angle of θ 0 (1), and the SH is reflected back with an angle of θ 0 (2) into the incident linear medium.50 Figure 4.3: Calculated conversion efficiency of the TE polarized SH versus the incident angle θ 0 (0 0 ~30 0 ) of TM polarized FF ( λ = μm, E field 1 V/μm ) and thickness d 2 of (linear) AlAs, for (nonlinear) Ga 0.7 Al 0.3 As thickness d 1 = μm and for d 14 = 94pm/V. The number of segments is taken to be N = 50. The maximum conversion efficiency of happens at d 2 = μm, and for incident angle θ 0 = (a) External conversion efficiency. (b) Corresponding maximum conversion efficiency inside the structure..60 Figure 4.4: Calculated conversion efficiency of the TE polarized SH versus the incident angle θ 0 (0 0 ~30 0 ) of TM polarized FF (λ = μm, E field 1 V/μm) and thickness d 1 of (nonlinear) Ga 0.7 Al 0.3 As, for (linear) AlAs thickness d 2 = μm and for d 14 = 94 pm/v. The number of segments is taken to be N = 50. The maximum conversion efficiency is at d 1 = μm, and for an incident angle θ 0 = (a) External conversion efficiency. (b) Corresponding maximum conversion efficiency inside the structure 61 Figure 4.5: Calculated intensity distributions of (a) FF and (b) SH for d 1 = μm, d 2 = μm, N = 50, d 14 = 94 pm/v, and for incident angle of FF θ 0 = xii

13 Figure 4.6: Calculated transmission spectra around the FF and SH wavelengths for 100 layers with d1= µm, d2= µm, incident FF angle θ = (a) FW and (b) SH 64 Figure 5.1: Layered structure composed of nonlinear materials with refractive indices n (1) i for FF and n (3) i for TH. The corresponding χ (3) i represents the nonlinear susceptibilities in each layer. N represents the number of segments. The FF with TM polarization is assumed to be propagating from the incident medium with angle of θ Figure 5.2: Calculated conversion efficiency of the TE polarized TH versus thickness d 1 of TiO2 and thickness of d 2 SiO 2 for TM polarized FF (λ = 1.55 μm, E field 1 V/μm). The number of segments is taken to be N = 20. The maximum conversion efficiency of happens at d 1 = μm and d 2 = μm, and for incident angle θ 0 = 0 0. (a) External conversion efficiency on forward direction. (b) Corresponding maximum conversion efficiency inside the structure..75 Figure 5.3: Calculated E 2 distributions of (a) FF and (b) TH for d 1 = μm, d 2 = μm, N = 20, and for incident angle of FF θ 0 = Figure 5.4: Calculated transmission spectra around the FF and SH wavelengths for N=20 for d 1 = μm, d 2 = μm, N = 20, and for incident angle of FF θ 0 = Figure 5.5: Calculated conversion efficiency of the TE polarized TH versus the incident angle θ 0 (0 0 ~15 0 ) of TM polarized FF ( λ = 1.55 μm, E field 1 V/μm) and thickness d 2, for thickness d 1 = μm. The number of segments is taken to be N = 20. The maximum conversion efficiency is atd 2 = μm, and for an incident angle θ 0 = 0 0. (a) Corresponding maximum conversion efficiency inside the structure. (b) External conversion efficiency.80 Figure 5.6: Calculated conversion efficiency of the TE polarized TH versus the incident angle θ 0 (0 0 ~15 0 ) of TM polarized FF (λ = 1.55 μm, E field 1 V/μm) and thickness d 2, for thickness d 1 = μm. The number of segments is taken to be N = 20. The maximum conversion efficiency is at d 1 = μm, and for an incident angle θ 0 = 0 0. (a) Corresponding maximum conversion efficiency inside the structure. (b) External conversion efficiency..81 Figure 6.1: Schematic of proposed imaging process. (a) A 50 nm thick layer of PR-955 is spin coated on the object for working as a spatial layer and serving as object distance. (b) A 50 nm metamaterial of Ag + SiC is co-sputtered on the previous structure. (c) A 70 nm image plane of Photoresist is spin coated on top of the metamaterial.86 Figure 6.2: The fabricated sample is illuminated by 405 nm light as the exposure procedure, then the exposed photoresist is developed...87 xiii

14 Figure 6.3: The exposed and developed sample is tested by AFM...87 Figure 6.4: The performance of superlens is tested by NSOM.88 Figure 6.5: Isofrequency surfaces of extraordinary waves in hyperbolic metamaterials. (a) Type I, (b) Type II [28] 89 xiv

15 LIST OF ABBREVIATIONS AFM Ag AR CCD EM FEM FF GH HMs IF Mo NSOM PBEs PBG QPM SEM SH SHG Atomic force microscopy Silver Anti-reflective Charge-Coupled Device Electromagnetic Fundamental frequency Goos-Hänchen Hyperbolic metamaterials Finite element method Imbert-Fedorov Molybdenum Near-field scanning optical microscope Photonic band edges Photonic bandgap Quasi-phase matching Scanning electron microscope Second harmonic Second harmonic generation xv

16 SiC TE TEM TH THG TM TMM WLM Silicon carbide Transverse Electric Transmission electron microscope Third harmonic Third harmonic generation Transverse magnetic Transfer matrix method White light microscopy xvi

17 LIST OF SYMBOLS χ D i (1) p y r± Nonlinearity coefficient Dynamical matrix for fundamental field Spin split of p polarized incident Gaussian beam s y r± Spin split of s polarized incident Gaussian beam ΔY r s E E i E r p E r s E r E r k i k r Λ (1) P i r p r s Magnified shift for s polarization field Electric field Incident electric field Reflected electric field Reflected electric field for p polarization Reflected electric field for s polarization Reflected field at far field Wave vector of reflected plane wave Wave vector of incident plane wave Period of the interference pattern Propagation matrix for fundamental field Reflection coefficient for p polarization Reflection coefficient for s polarization xvii

18 θ i (1) θ i (2) u i Propagation angle for fundamental field Propagation angle for second harmonic field Angular spectrum of the incident Gaussian beam u r Angular spectrum of the reflected Gaussian beam xviii

19 CHAPTER 1 INTRODUCTION AND OBJECTIVES 1.1 Introduction Thin films play a very important role in the field of optics. Remarkable reflective and transmissive properties can be realized by using single or multiple layers of thin films, typically of thickness comparable to the optical wavelength (approximately 500 nm). These properties occur due to interference of light and the difference of refractive index between the thin film layer(s), air, and the substrate. Effects of thin-film interference can be observable in everyday life, such as in soap bubbles and oil slicks. The conventional glass lens is pervasive throughout our society and in the sciences. It is one of the fundamental tools of optics. Anti-reflective coating (also called AR coating or anti-glare coating) is a microscopically thin multilayer coating that eliminates reflections from the front and back surface of eyeglass lenses, thereby reducing the glare caused by light reflected from the lenses, providing better vision for nighttime driving. Thin-film layers are also common in nature. For instance, the same principle behind soap bubbles applies to butterfly wings. Lepidoptera, or scaled wings, which is also the scientific Latin name for butterflies, comprise multilayers which form photonic bandgap (PBG) structures [1] responsible for their vivid colors when light is reflected from the structure. The conventional lenses mentioned above can only image waves that propagate. When an object is illuminated by a light wave, the scattered optical field comprises both 1

20 propagating (far-field) and non-propagating (near field) components. When light is captured by a glass lens, useful resolution details (sub-wavelength scale) are not observed, because they are hidden in the near field. They remain localized, staying much closer to the light emitting object, unable to travel, and unable to be captured by the conventional lens. Today, lots of novel techniques have been invented such as scanning electron microscope (SEM) [2], transmission electron microscope (TEM) [3], near-field scanning optical microscope (NSOM) and white light microscopy (WLM) [4,5]. They are used to capture the information of sub-wavelength objects. However, all of systems are high-cost, expensive to maintain, and each technique has its disadvantages in different applications. Cost-effective noninvasive optical microscopy techniques with high transverse spatial resolution is always desirable for imaging nano-scale objects such as viruses, biological proteins, quantum structures and nanocomposite materials. A versatile microscope system that captures the near field radiation, for high resolution, can be accomplished with new types of material not found in nature, termed metamaterials, which allow for details of images that surpass the limitations imposed by the wavelength of light. Metamaterials are artificial materials fabricated to have properties that are not existent in nature. They are assemblies of multiple individual elements fashioned from conventional materials such as dielectric materials and conductive materials, and the materials are usually constructed into repeating patterns, often with sub-wavelength structures. Metamaterials derive their properties not from the compositional properties of the base materials, but from their designed structures. Their precise shape, geometry, size, orientation and arrangement can affect electromagnetic (EM) waves (or other waves such as sound waves) in both near field and far field in a manner not observed in natural 2

21 materials. The lens made of this metamaterial is called a superlens, which has led to many applications, such as the display of biological cell interactions, in a real-time, natural environment, subwavelength imaging, etc. Recently, our group has designed and fabricated a new thin-film metamaterial which has a negative refractive index at a wavelength of 405 nm, and with low absorption ( n = 1 + i0.6 ). Compared with the most metamaterials, the co-sputtered Ag+SiC nanoparticle based metamaterial has a rather simple, quick, and cost-effective fabrication process, and also works at visible wavelengths [6]. This metamaterial can act as a superlens for imaging sub-wavelength objects through super-resolution and for arbitrarily polarized optical illumination. A compact nanoscale structure comprising the subwavelength object, the single layer thin-film metamaterial superlens made from cosputtered Ag + SiC nanoparticles and a novel imaging setup to accurately characterize super-resolution imaging for arbitrary polarized illumination of the object is being fabricated and will be tested as part of the proposed work. This is discussed in detail in Chapter 2. When an EM wave is reflected at the interface of two media, the reflected propagation direction can be determined by the Snell s law for a plane wave. However, for beam propagation, such as fundamental and high order Gaussian beam propagation, transverse and longitudinal shifts happen when it reflects at the interface. The longitudinal shift during total internal reflection is called the Goos-Hänchen (GH) shift [7]. The associated transverse shift is called the Imbert-Fedorov (IF) shift [8]. The IF shift can be resolved through the use of quantum weak measurement techniques [9]. With the development of the weak measurement method, it is also predicted that angular 3

22 shifts happen in the incident plane, these are called the GH and IF angular shifts. In the case of double negative metamaterials, i.e., those that have a negative permittivity and permeability simultaneously and consequently a negative refractive index such as the type mentioned above, the GH shift is reversed [10-11]. However, the IF shift does not reverse due to the unreversed angular momentum in double negative refractive index material [12-14]. As discussed in Chapter 3, the IF shifts at the interface of air and a single layer double negative index thin-film material such as Ag + SiC will be experimentally studied and compared with theoretical predictions, as part of the proposed work. One of the major research areas in nonlinear optics are second harmonic (SH) and third harmonic (TH) generations. SH generation (SHG) is due to degenerate three wave mixing [15-17] using the second order or quadratic nonlinearity of a material. The SH conversion efficiency can be improved using phase matching, which can be achieved using interacting frequencies with different polarizations in a birefringent crystal [18], or by quasi-phase matching (QPM) by alternating the orientation of crystal segments [19]. Furthermore, by exploiting the properties of periodic alternating layers of different materials or PBG structures, enhancement can be achieved by choosing the fundamental frequency (FF) near the photonic band edges (PBEs) [20]. The combination of QPM and PBE effect in a nonlinear structure with periodically poled crystals can significantly increase SHG conversion efficiency [21-24]. Periodic multilayer structures are most effectively analyzed using the transfer matrix method (TMM) [25, 26]. TMM can be also modified to study SHG in PBGs, and is especially straightforward in the case of nondepleted FF. When the SH is generated inside a periodically nonlinear material, its 4

23 spatial evolution inside the PBG structure is coupled to the spatial distribution of the forward and backward propagating FF [21-24]. However, until now, TMM has been utilized for calculating the spatial variation of the FF and SH for normal incidence of the FF [24, 25]. Local field enhancement in PBG structure by using FF with oblique incidence angles may be larger than for normal incidence due to its ability to tune the phase matching and local field enhancements for the FF. TMM is employed for calculating the EM fields in linear PBGs (e.g., with positive and negative refractive indices) with oblique incident angles [26]. In Chapter 4 and 5, TMM is used to account for reflections and interferences between all forward and backward propagating (nondepleted) FF, SH and TH waves. Another class of metamaterials, also comprised of metal-dielectric structures, are called hyperbolic metamaterials (HMs) and have attracted significant interest recently [27-32]. The HM structure reveals open curved hyperbolic dispersion which can be used in sub-wavelength imaging [33] or optical cloaking [34]. In HMs, EM field enhancement may occur significantly [35]. The field of SHG and third harmonic generation (THG) [36] using HMs remain relatively unexploited. In chapter 6, a TMM analysis of the nonlinear processes of SHG and THG using HMs are proposed for future work, along with other proposed future work based on the progress reported in Chapters 2, 3, 4 and Objectives and Organization In Chapter 2, the design of a compact nanoscale structure comprising a subwavelength object and the Ag + SiC nanoparticle based metamaterial superlens for nearfield imaging is described as well as the imaging system without metamaterial. Based on 5

24 these designs, the models are simulated by using COMSOL, a finite element method (FEM) [37]. By using the parameters of negative permittivity and permeability of Ag + SiC, in Section 2.3, the image is clearly resolved in the structure with metamaterial and contrastively, the image is incorrectly resolved for the imaging system without metamaterial. The corresponding optical fields at different distances beyond the metamaterial superlens have been plotted, which helps to identify position of the best imaging in near field. In Section 2.4, the fabrication technique details of 1D and 2D subwavelength objects from Mo deposited on a glass substrate using interference lithography is discussed. In Section 2.5 many SEM image are represented to show the quality of the sub-wavelength Mo object. In Section 2.6 a summary of the metamaterial superlens is presented. In Chapter 3, Spin Hall effect for Gaussian beam at double negative metamaterial is theoretically studied. In Section 3.1, a review of related works and significance of the spin Hall effect is demonstrated. In Section 3.2, the theory and equation for Spin Hall effect is studied and derived. In Chapter 4, RTMM is used to account for reflections and interferences between all forward and backward propagating FF and SH waves at oblique incidence for the phase matching condition, and assuming non-depleted FF. FF and SH wave spatial distributions inside and outside of the PBG crystal are calculated. In Section 4.2 the previous work related to TMM for metamaterial is represented. In Second 4.3 the theory of obliquely incident fundamental fields in periodically nonlinear materials is analyzed. In Section 4.4 the FF and SH fields distributions inside and outside of periodically nonlinear structure are calculated and plotted. At oblique incidence, the conversion 6

25 efficiency is meshed in 3D map, which helps to lock the incident angle and thickness of the one layer where has the maximal conversion efficiency. In Section 4.5 a summary for the use of TMM to analysis SH periodically nonlinear structure with oblique incident angle are shown. In Chapter 5, a systematic method using TMM to analyze THG in nonlinear PBGs is discussed, again under the assumption of pump non-depletion. The exact distributions of the FF and the harmonics are computed, along with the prediction of optimum layer thicknesses and angle of incidence for maximum conversion efficiency. In Second 5.1 the theory of obliquely incident FF and TH in periodically nonlinear materials are analyzed. In Section 5.2 the FF and TH fields distributions inside and outside of periodically nonlinear structure are numerically simulated and plotted. At oblique incidence, the conversion efficiency is meshed in 3D map, which helps to lock the incident angle and thickness of the one layer where has the maximal conversion efficiency. In Section 5.3, a summary for the use of TMM to analysis TH periodically nonlinear structure with oblique incident angle are shown. In Chapter 6, the conclusion and future work corresponding research reported in Chapter 2 to Chapter 5 are presented. 7

26 1.3 Publications Resulting from this Work The following publications have resulted from prior work: Journal papers: (1) H. Li, J. W. Haus and P. P. Banerjee, Analysis of harmonic generation in photonic bandgap structures using the transfer matrix method, invited paper, Asian J. Phys., submitted (2015). (2) H. Li, J. W. Haus and P. P. Banerjee, Application of transfer matrix method to second harmonic generation in nonlinear photonic bandgap structures: oblique incidence, J. Opt. Soc. Amer. B 32, (2015). (3) R. Aylo, G. Nehmetallah, H. Li, and P. P. Banerjee, Multilayer periodic and random metamaterial structures: analysis and applications, invited paper, IEEE Access 2, (2014). (4) G. Nehmetallah, R. Aylo, P. Powers, A. Sarangan, J. Gao, H. Li, A. Achari, and P.P. Banerjee, Co-sputtered SiC + Ag nanomixtures as visible wavelength negative index metamaterials, Opt. Express 20, (2012). Conference papers: (1) H. Li, P. P. Banerjee, and J. W. Haus, "Third harmonic generation in multilayer structures: oblique incidence," paper JTu4A.41, Frontiers in Optics OSA Technical Digest (2015). (2) H. Li, J. W. Haus and P. P. Banerjee, Second harmonic generation at oblique angles in photonic bandgap structures, Proc. SPIE 9347, 93470Z (2015). 8

27 (3) H. Li, A. Sarangan, G. Rui, H. Yu, and P. P. Banerjee, Nano-scale patterns of molybdenum on glass substrate for use in super-resolution imaging with metamaterials, Proc. SPIE D (2014). (4) P. P. Banerjee, R. Aylo, G. Nehmetallah, H. Li, A. Sarangan, and P. Powers, Binary nanoparticle dispersed metamaterial implementation and characterization, Proc, SPIE 8268, (2012). (5) P. P. Banerjee, H. Li, R. Aylo, and G. Nehmetallah, Transfer matrix approach to propagation of angular plane wave spectra through metamaterial multilayer structures, invited paper, Proc. SPIE 8093, 80930P (2011). 9

28 CHAPTER 2 METAMATERIAL: FABRICATION AND TESTING OF NEAR-FIELD IMAGING 2.1 Introduction In EM and optics, the diffraction limit has been considered to be a barrier in imaging. SEM, TEM, near-field NSOM and WLM are widely used for nano-scale imaging, along with atomic force microscopy (AFM) to determine the nature of the surface at the nanoscale [38, 39]. For this reason, near-field superlensing using double negative index metamaterial has been proposed [40], which can amplify evanescent waves and form sub-wavelength images without the need for scanning. An experiment demonstrating super-resolution in the near field has been shown by Fang et al. [41]. However, super-resolution is different from superlensing, since the only criterion in super-resolution is to resolve sub-wavelength features, without imaging them; furthermore Fang et al. s approach only super-resolves for a specific polarization of light since plasmonic materials are used to fabricate the metamaterial. Recently, a new metamaterial which has double negative refractive indices at a wavelength of 405 nm, and with a low absorption, has been designed and fabricated by our group [6]. Compared to other fabrication techniques, the cosputtered Ag + SiC nanoparticle based metamaterial has a rather simple, quick, and costeffective fabrication process, and also works at visible wavelengths. It has been shown that at 405 nm, the permittivity (ε) and permeability (µ) are negative, and the refractive index is negative. 10

29 In this Chapter, the design of a compact nanoscale structure comprising a sub-wavelength object and the Ag + SiC nanoparticle based metamaterial superlens for near-field imaging is described. The object is a molybdenum (Mo) structure comprising periodic gratings with period size of the order of 200 nm. In our design, it is assumed that there is a dielectric layer of thickness about 50 nm on the top of object to act as the object distance for the double negative index metamaterial made of Ag + SiC nanoparticles of approximate thickness 70 nm, which is to be cosputtered on top of the dielectric layer. Near-field super resolution and imaging is expected for both TE and TM polarized light. Following Fang et al. [41], photoresist will be spin-coated on top of the metamaterial. Upon exposure of the structure to 405 nm light, near-field imaging should leave the signature of the object in the developed photoresist, which can be then analyzed using AFM. Subsequently, a more precise estimation of imaging should also be performed using NSOM. In the first part of the this Chapter, simulations based on this design for both TM and TE polarization conditions using finite element method (FEM) software such as COMSOL is reported. It is shown that the presence of the metamaterial superlens is essential in proper superresolution and imaging of a 1D object fabricated from Mo with periodicity of 200 nm for both TE and TM polarized 405 nm illuminations. In the second part of this Chapter, the fabrication technique of 1D and 2D sub-wavelength objects from Mo deposited on a glass substrate using interference lithography is discussed. A 266 nm laser is used as the deep UV (DUV) source in interference lithography due to its excellent coherence properties. This enables feature sizes as small as 133 nm to be realized. 11

30 2.2 Essential Theory of Near-field Lensing and its Measurement A well-known design used for characterizing super resolution utilizing Ag ((Re(ε) < 0)) thin film as the (plasmonic) metamaterial [41] has been adapted for our near-field imaging setup. In such a structure, shown schematically in Figure 2.1(a), the Ag is replaced by Ag + SiC, which is a double negative refractive index metamaterial ((Re(ε) < 0), (Re(μ) < 0)). In the visible range, most plasmonic metallic nanoparticles like Ag exhibit negative permittivity ( Ag < 0). The negative effective permeability (µsic < 0) of SiC nanoparticles is due to the enhancement of the azimuthal component of the displacement current in the SiC spheres due to their large permittivity ( SiC >> 1). This current is responsible for enhanced magnetic activity near TE resonance, finally resulting in a magnetic flux density which is opposite to the direction of the magnetic field of the optical wave. The negative effective permittivity and permeability creates an effective negative refractive index (Re(n eff ) < 0). Specifically, the proposed design contains a sub-wavelength periodic object of period 200 nm, where the width of the Mo is 100 nm and the thickness is 50 nm, and the space between the Mo fingers (100 nm) is filled with photoresist. The same photoresist is also assumed to exist over the Mo photoresist structure to be used as a spacer layer of thickness 50 nm. This can be regarded as the object distance for our near-field imaging system. It is assumed that a 70 nm thick Ag + SiC layer is sputtered on the photoresist to act as the negative index superlens. Finally, it is assumed that there is a photoresist layer (of up to 100 nm thickness) beyond the superlens to serve as the image distance. For comparison, the same structure but with the superlens replaced by photoresist is shown in Figure 1(b). 12

31 (a) Figure 2.1. (a) Schematic diagram of the near-field imaging structure with a subwavelength object and thin layer of double negative metamaterial. (b) Schematic diagram of a similar structure with the sub-wavelength object but without double negative metamaterial used for comparison with (a). (b) 2.3 Model and Simulation of Near-field Imaging using Finite Element Method COMSOL is used to model the proposed structure and simulate the electromagnetic fields inside the structure, using the values for the permittivity and permeability of the Ag + SiC superlens layer derived by Nehmetallah et al. [6]. It has first been checked that for both TE and TM illumination, only the structure in Figure 1(a) using the superlens can reproduce the general shape of the optical field expected immediately behind the object. For instance, as shown in Figures 2.2(a)-(c) for TE polarization, the locations of the maxima of the optical field are faithfully reproduced for the structure of Figure 2.1(a), while a contrast reversal is observed for the structure of Figure 2.1(b) as evident from Figures 2.2(d)-(f). Parameters used for simulation at 405 nm are: ɛ Mo = i [42], ɛ photorsesist =2.7225, ɛ Ag+SiC = i, μ Ag+SiC = i ). 13

32 (a) (b) (c) (d) (e) (f) Figure 2.2. (a) COMSOL simulation of E 2 for the structure in Figure 1(a) with TE polarized plane wave and normal incidence. (b) Same as Figure 2(a), but with different scales for the data to highlight the optical fields within and beyond the metamaterial superlens. The grid structure representing the 1D object comprises alternating layers of Mo and photoresist, starting from the left. (c) Plot of E 2 in 1D in the region of image plane. As expected, E 2 is higher behind the regions where there is no Mo. (d)-(f): Similar to (a)-(c), except that there is no metamaterial superlens. In this case, the object is incorrectly resolved, with intensity maxima occurring at the geometrical shadow of the object. In (a),(b),(d) and (e), the incident illumination is from the top. Furthermore, as expected, for a normally incident TM polarized plane wave, the metamaterial works as the superlens which can resolve the object correctly, but the structure without metamaterial cannot resolve the object in the image plane. This is shown in Figure 2.3. The parameters used for this simulation is identical to those used in Figure

33 (a) (b) (c) (d) (e) (f) Figure 2.3 (a) COMSOL simulation of E 2 for the structure in Figure 1(a) with TE polarized plane wave and normal incidence. (b) Same as Figure 2(a), but with different scales for the data to highlight the optical fields within and beyond the metamaterial superlens. The grid structure representing the 1D object comprises alternating layers of Mo and photoresist, starting from the left. (c) Plot of E 2 in 1D in the region of image plane. As expected, E 2 is higher behind the regions where there is no Mo. (d)-(f): Similar to (a)-(c), except that there is no metamaterial superlens. In this case, the object is incorrectly resolved, with intensity maxima occurring at the geometrical shadow of the object. In (a),(b),(d) and (e), the incident illumination is from the top. It is interesting to compare the optical fields immediately behind the object and at different distances behind the metamaterial superlens for both TE and TM polarizations. Figure 2.4 (a) is a plot of the optical intensity at a plane immediately behind the object for 15

34 TE incidence, plotted over one transverse period. In Figure 2.4 (b), the corresponding optical intensities at different distances beyond the metamaterial superlens have been plotted. As is visually evident from Figure 2.4 (b), the best imaging occurs at a distance of 4 nm beyond the superlens. Similarly, Figure 2.4 (c) is a plot of the optical intensity at a plane immediately behind the object for TM incidence, again plotted over one transverse period. In Figure 2.4 (d), the corresponding optical fields at different distances beyond the metamaterial superlens have been plotted. As is visually evident from Figure 2.4 (d), the best imaging now occurs at a distance of approximately 24 nm beyond the superlens. A more critical study of the pattern, involving comparison of the Fourier coefficients of the optical field and intensity patterns at the object and image planes will be pursued as part of the proposed work. 2.4 Fabrication of Periodic Object: Experimental Details Initially, a 50 nm thick Mo is coated on the substrate of glass by sputter deposition. Before coating with the photoresist, an anti-reflection layer is spin-coated on top of Mo for avoiding the multiple reflections from the Mo surface. Then a nm thick negative tone photoresist is spin-coated on top of anti-reflection layer to ensure the desired dry etch resistance. For achieving a periodic pattern at the negative tone photoresist, interference lithography is used. It provides a method to achieve 1D, 2D and 3D periodic nanostructures without a mask or high numerical aperture optics [43, 44]. Here, a 266 nm laser is used as the source for DUV interference lithography. The pitch of the interference pattern can be close to half of the wavelength (i.e., 133 nm). The development, post-bake and hard-bake procedures make the negative tone photoresist cross-linked for the plasma etch process [45, 46]. Fabrication of the periodic patterns on 16

35 Mo is performed by reactive ion etching in fluorinated plasma. After the plasma etch, the resist left on top of the Mo periodic structure is stripped with acetone followed by oxygen plasma. Finally, a clean Mo object grating is achieved and SEM images are taken to visualize the object. Thereafter, the objects are characterized using TE and TM polarized illumination. (a) (b) (c) (d) Figure 2.4. (a) E 2 in 1D immediately behind the object for 405 nm TE illumination at normal incidence; (b) E 2 in 1D in the photoresist beyond the superlens at distances 0, 1, 2, 3, 4, 5, 6 nm for TE illumination; (c) E 2 in 1D immediately behind the object for 405 nm TM illumination at normal incidence; (d) E 2 in 1D in the photoresist beyond the superlens at distances 16, 20, 24, 28, 32, 36, 40 nm for TM illumination. The best imaging distance for TE is around 4 nm, while the best imaging distance for TM is around 24 nm. 17

36 Schematics of proposed Mo object fabrication process, which includes sputtering, spin coating, interference lithography and etching, is shown in Figure 2.5. First, as shown in Figure 2.5(a), a 50 nm Mo thin film is sputtered on the surface of glass substrate at a pressure of 4 mtorr, and with a deposition rate of about 0.72 Å/s using a radio frequency (RF) gun. Then the sample is cleaned by acetone, methanol and IPA. To avoid multiple reflections from the Mo surface during DUV interference lithography exposure, an anti-reflection coating (BARC) is spin-coated on top of the Mo layer (Figure 2.5(b)). The BARC (BSI.N0889A) is spin-coated at a speed of 1500 rpm for 60 sec. followed by 1 min. post-bake at 190 C. Thereafter, a negative tone DUV photoresist (UVN , diluted 3:2 with photoresist thinner) is spun on at 2000 rpm for 60 sec, followed by a 90 C post-bake on a hot plate for 60 sec, to yield a 350 nm thick photoresist thin layer, as shown in Figure 2.5(c). A custom DUV interference lithography system comprising a 266 nm laser and a Lloyd mirror interferometer [47-49] is used for making the periodic photoresist nano-patterns (1D or 2D) with a large exposure area and for a low cost (Figure 2.5(d)). In the process of the exposure, the Lloyd mirror interferometer can be rotated for varying pattern periodicities ranging from half of wavelength to infinity based on the relation Λ = λ 2nsinθ (2.1) where is the period of the interference pattern, is the wavelength of the source light, n is the refractive index of the air, and is the rotation angle of the Lloyd mirror. After the 18

37 exposure, it is necessary to do a 95 C post-bake for 75 sec. It is then developed in MEGAPOSIT MF 319 at 21 C for 30 sec, and then submerged in de-ionized (DI) water. After exposure and development, the cross-linked photoresist looks as shown schematically Figure 2.5(e), which shows that the periodic patterns are created on the top of the BARC. This pattern can be used as the mask in the dry etching process due to its high resistance to the underlying Mo layer. Several hours of 100 C hard-bake is necessary before the dry-etch process. In the process of reactive ion etching (RIE), the most commonly used fluorinated gases are CF4 and SF6. Oxygen is generally added to SF6 or CF4 to enhance the creation of fluorine radicals. The etch rate is also influenced by the RF power and pressure. Here SF6 and O2 are mixed with the RF power of 100 W and under a pressure of 40 mtorr. An etching rate of 5.5 Å/sec is achieved. The resulting pattern is shown schematically in Figure 2.5(f). Finally, the remaining photoresist and BARC on the Mo object are stripped by pouring acetone, followed by treatment in an O2 plasma etching system. A clean sub-wavelength object is realized, as shown in Figure 2.5(g). 19

38 (a) (b) (b) (d) (e) (f) (g) Figure 2.5. Schematic of proposed Mo object fabrication process. (a) A 50 nm Mo film is deposited on a flat glass substrate using sputter machine. (b) A 100 nm BARC layer is deposited on top of Mo film using spin coater. (c) A 400 nm DUV photoresist is coated on top of BARC by the spin coater. (d) Using 266 nm laser as the light source, to do the interference photo lithography on the DUV photoresist. (e) After the development on the exposed photoresist. A grating with period of sub-wavelength is completed. (f) Using the photoresist grating as the etch mask. The sub- wavelength object of Mo is etched by reactive ion etching in a fluorinated plasma. (g) The left photoresist and BARC on Mo object are stripped by pouring acetone and followed by the treatment in the O2 plasma etching system. Then a sub-wavelength object is completed. 20

39 2.5 Fabrication Results and Characterization Most of the intermediate fabrication results corresponding to Figure 2.5 and the final Mo object on the substrate of glass are analyzed by SEM. Samples analyzed by SEM are usually sputtered with a thin layer of gold before SEM to avoid surface charging. For instance, Figure 2.6 shows SEM images of a 1D photoresist nano-structure of period 270 nm created by DUV interference lithography with =30 and illumination dose at 10.5 mj. As is clear from Figure 2.6(a), without using anti-reflection layer (BARC), the exposed and developed photoresist has almost lost its sinusoidal shape due to significant reflection from Mo layer. In contrast, as shown in Figure 2.6(b), upon using a 100 nm BARC anti-reflection layer, only 1% of the light reflects back and the SEM image shows a good shape with a period at 270 nm and a high aspect ratio. (a) 21

40 (b) Figure 2.6. For rotation angle of θ = 30, (a) SEM image for the developed photoresist without the underlying BARC layer (b) SEM image for the developed photoresist with the underlying BARC layer. (a) 22

41 (b) (c) 23

42 (d) Figure 2.7. For rotation angle of θ = 35, SEM image (a) corresponding to the structure of Figure 2.5(e); (b) corresponding to the structure of Figure 2.5(f); (c) of the final Mo object corresponding to the structure of Figure 2.5(g); (d) showing top view of the Mo object. (e) Semi-transparent nature of final sample with Mo object. (e) 24

43 Figures 2.7 (a)-(c) show the SEM images corresponding to the steps in Figures 2.5 (e)-(g). For = 35 and 10 mj dose of illumination, Figure 2.7 (a) is the SEM image for the developed photoresist with the underlying BARC layer. Figure 2.7 (b) shows the structure after dry etch. The high aspect ratio observed for both Figure 2.7 (a) and Figure 2.7 (b) may indicate that the RIE etching rate for Mo and the photoresist mask are almost the same for certain conditions. Since some of photoresist and BARC are still present on top of the Mo object due to incomplete RIE etching, the sample is stripped by using acetone followed by an oxygen plasma etching, which finally yields a clear Mo object as shown in Figure 2.7 (c). Figure 2.7 (d) shows the SEM image for the top view of this object. As evident from Figure 2.7 (e), the sample now appears visually semitransparent. In order to reduce the period of the object down to 200 nm, the rotation angle is set to 40. The corresponding 1D periodic pattern of the Mo object as seen from the SEM image is shown in Figure 2.8 (a). Recall that the simulations in the first part of the paper have been performed with objects with similar dimensions in order to study nearfield imaging for TE and TM polarizations. 2D periodic patterns using Mo are also feasible by the same fabrication procedure. For the 2D pattern shown in Figure 2.8(b), 2 exposures with different oritations are required; however, each of the exposures only need half of the amount of the dose as the 1D case. 25

44 (a) (b) Figure 2.8. For rotation angle of = 40, SEM images for (a) 1D (b) 2D Mo objects. In this case the period is approximately 200 nm. In order to further characterize the performance of the Mo object, transmission at normal incidence for different polarizations of incident light, using a 405 nm laser, is 26

45 studied. For the 1D periodic object, the transmission increases as the polarization of the optical field changes from a polarization angle of 0 (TE) to 90 (TM). The reason for this is that the 1D periodic metal structure is naturally a micro-polarizer which reflects TE polarization but transmits TM polarization. COMSOL simulations and experimental measurements are compared in Figure 2.9. The mismatch may be due to the roughness of the fabricated structure; however, the trend in transmission is similar. 0.4 Transmission Simulation Result Experiment Result T(1=100%) Polarization Angle (deg) at normal incident condition Figure 2.9. Simulation results (varying from near zero to 0.4) and experimental observations (varying from 0.2 to 0.4) for transmission at 405 nm as a function of polarization angle, which varies from 0 (TE) to 90 (TM). 2.6 Summary The design of a near-field optical system for imaging sub-wavelength objects using double negative metamaterial superlens using SiC + Ag operating at 405 nm has been presented. Simulation results show the extent of TE and TM imaging of the optical field immediately behind the object, which is a 1D periodic Mo structure. With 27

46 experimental verification as the goal, large area Mo 1D and 2D structures with periodicities as small as 200 nm have been fabricated on a glass substrate using 266 nm DUV interference lithography and reactive-ion etching. As is clear from the fabrication process, different periodicities can be realized by adjusting the interference angles. Compared to electron beam and focused ion beam lithography, this technique allows for making nano-scale patterns on Mo at significantly lower cost. Our fabricated objects are examined using SEM, and optically characterized using TE and TM illumination. 28

47 CHAPTER 3 PHOTONIC SPIN HALL EFFECT IN DOUBLE NEGATIVE METAMATERIALS 3.1 Introduction When EM wave reflection occurs at an air-medium interface, the reflected propagation direction can be determined by the Snell s low for a plane wave. However, for beam propagation, such as fundamental and high order Gaussian beam propagation, transverse and longitudinal shifts happen when it reflects on the interface. The longitudinal shift during total internal reflection is called the GH shift. The transverse shift is the IF shift. The IF shift is resolved through the use of quantum weak measurement technique. With the development of the weak measurement method, it is predicted that angular shifts happen in the incident plane; these are named GH and IF angular shifts. The GH shift at an air-negative refractive index (or double negative) metamaterial interface is reversed compared to air- positive refractive index material interface. However, IF shift for these two conditions does not reverse due to the unreversed angular momentum in double negative refractive index material. Experimental measurement of the magnified shifts using double negative metamaterials are done. The weak measurement technique is used to find the IF shift of air-ag+sic structure experimentally. 29

48 3.2 Theory of Spin Hall Effect of Gaussian Beam In this Section, the theory of the spin Hall effect of Gaussian beam in both normal materials and double negative materials is first summarized. As a light beam reflects from an air-medium interface, two different kinds of shifts occur. One is the GH shift, which happens in the incident plane. Another is IF shift, which is perpendicular to the incident pane. The eigenmodes of the IF shift are left-handed circularly polarized (LCP) light and right-handed circularly polarized (RCP) wave. Now linearly polarized light can be visualized as being split into a LCP and a RCP spatially with the same amplitude yet different handedness to different directions after reflection at an interface. This effect is called spin Hall effect of light (SHEL). As sketched in Figure 3.1, linearly polarized light with spin-up and spin-down photons impinge onto the air-medium surface. To meet the angular momentum conservation in z direction, spin-up photons shift to the -y direction, while spin-down photons shift to +y direction. The split direction along y is perpendicular to the direction of the gradient of the refractive index z. Figure 3.1. Sketch of spin Hall effect of Gaussian beam at an air-medium interface. A linearly polarized beam reflects on a medium and splits into left- and right-handed circularly polarized light. The medium can be normal material or metamaterial. 30

49 To study the spin Hall effect of a Gaussian beam, the angular spectrum method is applied to analyze the reflected and transmitted fields. First, the Gaussian beam is decomposed into a spectrum of angular plane waves. Secondly, the reflection and transmission of each angular plane wave component is analyzed. Each wave component has distinctive propagation direction, and has thus different reflection and transmission coefficients. The Fresnel coefficients are expended to the first order by using Taylor series. Third, the reflected and transmitted fields are recomposed in spatial domain. The angular spectra of the incident field E i, reflected field E r can be expressed as: E i = [αx i + βy i 1 k (αk x i + βk yi )z i] u i, (3.1) E r = {[α (r p r p θ i k xr k ) + β(r p + r s )cotθ i k yr k ] x r + [β (r s r s θ i k xr k ) α(r p + r s )cotθ i k yr k ] y r 1 k (αr pk xr + βr s k yr )z r} u r, (3.2) The corresponding expressions in spatial coordinates can be derived from inverse Fourier transform, which can be seen in Ref. [12]. In the equations above, [α, β] T is the Jones vector, [x i,r, y i,r, z i,r ] T are unit vectors in incident and reflected coordinate systems, respectively. u r = u i(γ r k xr,t, k yr ), u iis the angular spectrum of the incident Gaussian beam and given as u i = w 0 exp [ w (k xi +kyi ) ], γ w(z) 4 r = 1, θ i is the incidence angle. r p,s are the reflection coefficients of p and s polarized light, w 0 is the beam waist size. Thus u r = w 0 exp [ w (k xi +kyi ) ] exp [ jz ((k w(z) 4 2k x 2 r +k 2 yr )] and k i = (k xi x i + k yi y i + k zi z i), k r = (k xr x r + k yr y r + k zr z r), are the wave vectors of the incident and reflected plane waves. They satisfy the following boundary conditions: k xr = k xi, k yr = k yi, k zr = k zi. 31

50 As it is stated in the previous paragraphs, SHEL implies that linearly polarized light is split after reflection at an air-medium interface. When the incoming light is p polarized (α=1, β = 0), the reflected optical field is E p r = [(r p r p k x r ) θ i k x r k y r (r p + r s )cotθ i k y r r pk x r z r] u r. (3.3) k The reflected optical field can be written as a combination of RCP and LCP component E p r = E p r+ + E p r, where the LCP and RCP components can be written as E p r± = 1 [r 2 p r p k x r ± i(r k y r θ i k p + r s )cotθ i k r pk x r ] k [x r ± iy r ( k xr ± i k yr ) z r] u r. k k (3.4) Upon using method to calculate centroid of the beam in special frequency domain in [12, 50], the spin split of p polarized incident Gaussian beam can be expressed as y p r± = [ cotθ i k (1 + r s r p cos(φ p φ s ))] {1 + θ [1 + (1 + 4 r s r p + 2 r s cos(φ r p φ s )) cotθ 2 i + ( ln r 2 p ) + ( φ 2 1 p ) ]}, p θ i θ i (3.5) where θ 0 is the divergence angle of the incident beam, r p = r p exp(iφ p ), r s = r s exp(iφ s ). When the incidence angle is far away from the Brewster angle, the above equation can be simplified as 32

51 y p r± = [ cotθ i k (1 + r s r p cos(φ p φ s ))]. (3.6) When the incoming light is s polarized (α=0, β = 1), the reflected optical field is E r s = [(r p + r s )cotθ i k y r k x r + (r s r s k x r ) θ i k y r r sk y r z r] u r. (3.7) k Similarly, by using the method described in Equations (3.4) (3.6) for p polarization, the simplified spin split of s polarized incident Gaussian beam can be expressed as: s y r± = [ cotθ i k (1 + r p r s cos(φ s φ p ))]. (3.8) Equations (3.6) and (3.8) are the main results of this theoretical part. The differences between the SHEL of Gaussian beam in normal materials and metamaterials can be analyzed from the reflection coefficients r p and r s. The influence of dispersion on SHEL can also be derived from the reflection coefficients r p and r s. In both negative refractive index metamaterials and normal materials, the reflection coefficients are: r p = εr μr cosθ i 1 1 εrμr sin2 θ i, (3.9) ε r μr cosθ i+ 1 1 εrμr sin2 θ i 33

52 r s = cosθ i ε r μr 1 μ r 2 sin2 θ i cosθ i + ε r μr 1 μ r 2 sin2 θ i, (3.10) where ε r and μ r are the relative permittivity and permeability, respectively. Figure 3.2 shows the schematic of the setup for the weak measurement. Equations (3.6) and (3.8) describe the beam splitting on the interface of air- Ag+SiC. For simplifying the calculation and experiment, the radius of the beam is set to be beam waist at the interface. Figure 3.2. Schematic of setup for measurement of IF shift. A 405 nm fiber laser is used as light source, lens 1 and lens 2 are used for collimation. P1 and P2 are Glan laser polarizer to eliminate x and y polarized light. Based on the condition of alignment of Figure 3.2, Figure 3.3 shows the numerical simulation corresponding to Equation (3.6). Air - glass interface is considered instead of metamaterial, the refractive index of glass is assumed to be 1.53 at the wavelength of 405 nm. Due to the effect of the approximation, the shifts around the Brewster angle are not valid, thus they can be ignored. The dimension of shifts at the media interface are from 0 to 400 nm and the RCP beam is highly overlapping with LCP 34

53 beam, which make it hard to detect experimentally. These shifts will be magnified because of the property of diffraction at far-field. Then, weak measurement method will be used later in detail to calculate the magnified shifts at far-field. Figure 3.3. Spin Hall effect of p polarized Gaussian beam showing beam shift of the RCP and LCP components of a Gaussian beam. The interface media are selected to be air glass. 3.3 Observation of Spin Hall Effect for Double Negative Metamaterial As presented in Chapter 2, the method for fabricating the double negative metamaterial ( Ag + SiC ) has already been developed. In Section 3.2, the spin Hall effect for double negative metamaterial has been introduced, and the expression of the spin split of s and p polarized incident Gaussian beam at the surface of double negative metamaterial is shown in Equations (3.6) and (3.8). The amount of split is simulated in 35

54 Figure 3.3 in p polarizations for normal material. However, the detection of this split can be only achieved using the weak measurement technique. The proposed weak measurement experiment for measuring this spin Hall effect at air-double negative metamaterial interface is shown schematically in Figure 3.4. Compared with Figure 3.2, a diode laser is used to couple the 405 nm source to single mode fiber as light source. Lenses 1 and 2 are used for focusing and re-collimation, respectively. Polarizers P1 is used to generate linear polarized light and analyzer P2 is used to filter the necessary power, respectively. A CCD coupled to a beam analyzer is used to detect the magnified split beam. The reflected optical field after P2 is derived by using Jones matrix in Ref. [62]. E r = r p sin 2 Δ [1 lnr pk xr Θ i k (1 + r s r p ) cot Θ i cot Δ k y r k ] u r x r +r p sin Δ cos Δ [1 lnr pk xr Θ i k r p sin 2 Δ( k xr k (1 + r s r p ) cot Θ i cot Δ k y r k ] u r y r +cot Δ k yr k )u r z r, (3.10) where Δ is defined as the difference angle between the two polarizers from the orthogonal condition, and the other parameters are defined in Section 3.2 already. After the second lens, the magnified shift for s polarized light is given by ΔY r s = f 2 kzr2 (r p 2 +r p r s ) cot Θ i sin Δ cos Δ sin 2 Δr p kzr2 cos2 Δ cot 2 Θ i (r p +r s ) 2. (3.11) 36

55 Equation (3.11) is the main result of this theoretical part for weak measurement method, which could be in order of 1000 µm at far-field. In Figure 3.4, a collimated 405 nm Gaussian beam is focused by Lens 1 and the beam waist is hitting the interface of airmetamaterial. Polarizer 2 is working as an analyzer and is orthogonal with Polarizer 1 at original status. Figure 3.4. Schematic of setup for measurement of IF shift. 405 nm fiber laser is used as light source, lens 1 and lens 2 are working for collimation. P1 and P2 are Glan laser polarizer to eliminate x and y polarized light. The CCD is used to detect the magnified split beam. Gaussian beam splitting happens on the interface of air - Ag+SiC. Corresponding to Figure 3.4, the experiment is done in laboratory and is shown in Figure 3.5. A 600 nm thick of double negative metamaterial ( Ag + SiC ) is co-sputtered on the substrate of glass in clean room. Its refractive index is verified in Ref. [51]. With this thickness, because of the large absorption, the sample can be treated as a bulk material without worrying about any multi-reflections from the substrate. The weak measurement experiment is realized along the 37

56 purposed setup in Figure 3.4. At the incident angle of 45, the Gaussian beam the displacement of the centroids are recorded by a CCD. Figure 3.5. Experimental setup for weak measurement method. 405 nm fiber laser is used as light source, lens 1 and lens 2 are working for collimation. P1 and P2 are Glan laser polarizer to eliminate x and y polarized light. CCD is used to detect the magnified splits beam. Gaussian beam splitting happened on the interface of air- Ag+SiC. Figures 3.6 (a) (e) show the experimental beam profiles of s polarized light at CCD according to the setup in Figure 3.4. The centroids of the beam profiles are moved along with different values of Δ and these beam displacements only occur along the y direction which is perpendicular to the direction of the gradient of the refractive index z. The values of these centroids of the beam profiles can be recorded by a CCD. 38

57 (a) (b) (c) (d) (e) Figure 3.6. Experimental beam profiles of s polarized light recorded by CCD. (a) Experimental beam profile at Δ = 2 ; (b) Experimental beam profile at Δ = 1 ; (c) Experimental beam profile at Δ = 0 ; (d) Experimental beam profile at Δ = 1 ; (e) Experimental beam profile at Δ = 2. 39

58 The results of Figure 3.7 show the comparison between theory and experiment at air-double negative metamaterial interface for different Δ(polarization angle). A great agreement between experiment and theory is achieved. First, the refractive index value of Ag+SiC is verified. Secondly, Compare to what was concluded in Ref. [52], SHEL in double negative metamaterial is unreversed, although the sign of refractive index gradient is reversed. The physics underlying this counterintuitive effect is that the spin angular momentum of photons is unreversed Magnifited beam shift vs. Polarization angle Delta ys Simulation Delta ys Experiment Magnified beam shift (microns) Polarization angle Figure 3.7. Comparison between theory and experiment at air-double negative metamaterial interface for different Δ (polarization angle). In Equation (3.11), the magnified beam shift is also a function of beam incident angle. To see this, at Δ = 0.4, the beam displacements for different incident angles are recorded and plotted in Figure 3.8. It should be noted that the experiment results are in 40

59 good agreement with the theoretical simulations when the film thickness is 600 nm. The results provide further evidence for what is conclude in Figure Delta ys Simulation data1 Experiment data2 Experiment data3 Experiment Magnifited beam shift vs. incidence angle Magnified beam shift(micron) Incidence angle (degree) Figure 3.8. Comparison between theory and experiment at air-double negative metamaterial interface for different incident angles. After analysis of the experimental results, it is concluded that may a need to be reexamine our results to see if the thickness of the metamaterial should be incorporated in the calculations, along with the fact that there is a substrate on which the metamaterial is deposited. A 100 nm thick (thin film) metamaterial is therefore co-sputtered on the substrate of glass and the same experiment described before is performed again. From Figure 3.9, it is clear there is a significant difference for thickness of 600 nm metamaterial (black) and thickness of 100 nm metamaterial (red) at the incident angle of 41

60 45. SHEL for multilayer structures is still under exploration. The reduction of SHEL for 100 nm may come from different effective reflection coefficients of p and s polarizations for thin films, with multilayer reflections being a contributing factor. Figure 3.9. Comparison between theory and experiment at air-double negative metamaterial interface for different incident angles, and for different metamaterial thicknesses. 3.4 Summary In this Chapter, SHEL of a Gaussian beam at normal material-air interface and double negative is analyzed theoretically. The general method for realizing SHEL is used in laboratory and a great agreement between experiment and theory is achieved. First, the refractive index value of Ag + SiC is verified. Secondly, it is also verified that SHEL occurs for a double negative metamaterial, although the sign of refractive index is 42

61 reversed. The spin angular momentum is unreversed for a double negative refractive index materials, and correspondingly, the SHEL beam displacement on that interface is unreversed. SHEL may lead to potential applications for characterizing bulk and thin film materials. 43

62 CHAPTER 4 APPLICATION OF TRANSFER MATRIX METHOD TO SECOND HARMONIC GENERATION IN NONLINEAR PHOTONIC BANDGAP STRUCTURES: OBLIQUE INCIDENCE 4.1 Introduction In the last five decades, much of the research in nonlinear optics (NLO) has been in the understanding and applications of three- wave mixing, both non degenerate and degenerate cases [15-18]. This has led to practical NLO devices for generating or detecting coherent radiation from terahertz frequencies to the extreme-ultraviolet regimes of the electromagnetic spectrum [19, 20]. Low conversion efficiencies have been an impediment to practical applications; hence field enhancement techniques that improve nonlinear conversion efficiency is important. The phase matching in wave mixing (including second harmonic (SH) studies) is a common strategy for improving the conversion efficiency. Common methods to achieve SHG phase matching include the use of crystal birefringence [21] to mix waves with different polarizations and quasi phase matching (QPM) by alternating the molecular orientation of crystal segments by 180 o [22, 23]. Alternatively, a large enhancement of the SH conversion efficiency can be obtained by using a one-dimensional photonic band gap (PBG) structure as a Bragg mirror, where the enhancement comes from the simultaneous availability of a high density of states, high field localization and the improvement of effective coherence length near the photonic band edges 44

63 (PBEs) [24]. The combination of the QPM and PBE effects in a nonlinear structure with periodically poled crystals can significantly increase SH conversion efficiency, often 3-4 orders of magnitude compared to using QPM only [25 28]. The transfer matrix method (TMM) is a fast and accurate approach that can numerically analyze both phases and amplitudes of all electromagnetic (EM) waves inside and outside of multilayer structures [53-57]. TMM has been widely used to analyze the linear wave propagation of EM waves in PBG structures [19]. It has been modified to study nonlinear optical phenomena in PBGs and in particular, SHG evolution inside the structure. When the SH wave is generated inside the periodically nonlinear photonic crystal, its spatial evolution inside the PBG is coupled to the spatial distribution of the forward and backward fundamental frequencies (FFs) [24 26, 53]. However, until now, TMM has been utilized for calculating the spatial variation of the FF and SH in a PBG structure for normal incidence of the FF only. Local field enhancement in a PBG structure by introducing the FF at oblique incidence angles may be larger than with normal incidence. Oblique incidence provides a simple technique to tune the phase matching and local field enhancements for the sample. TMM has been previously used for calculating EM field propagation in linear PBGs (e.g., with positive and negative refractive indices) with oblique incident angles [56]. In this Chapter, TMM is used to account for reflections and interferences between all forward and backward propagating FF and recursive transfer matrix method (RTMM) is used to account SH waves at oblique incidence for the phase matching condition. FF and SH wave spatial distributions inside and outside of the PBG crystal are calculated. It is shown that widely different conversion efficiencies can be obtained for various incident angles of the FF and various 45

64 thicknesses of the linear and nonlinear material. The assumption of a nondepleted FF within the nonlinear PBG structure structure can be maintained by restricting the incident FF intensity. 4.2 Theory of Obliquely Incident Fundamental Fields in Periodically Nonlinear Materials As is well known, for an incident plane wave with a particular polarization state, TMM can be used for calculating its electric (E) and magnetic (H) fields inside and outside of multilayer structures, which also determines the transmittance and reflectance for these structures. Figure 4.1 shows a nonlinear PBG structure with N (assumed to be even) layers. The i th layer is (1,2) assumed to have a thicknesses d i, refractive index n i (with superscripts 1, 2 denoting FF and SH, respectively), and nonlinearity coefficients χ (2) i. In this paper we will treat the case where alternate layers have the same thickness and material properties, i.e., for two alternating materials d i(odd) = d 1, d i(even) = d 2 ; (4.1a) n i(odd) (1,2) = n 1 (1,2), n i(even) (1,2) = n 2 (1,2). (4.1b) The condition χ i(odd) (2) = χ (2), χ i(even) (2) = χ (2), (4.1c) denotes periodic poling that is often used in QPM nonlinear optics for enhancing SHG [58]; however, in our example (see below), the alternating nonlinear (Ga 0.7 Al 0.3 As) and 46

65 linear (GaAs) layers will be assumed. The refractive indices n i (1) and n i (2) correspond FF and SH, all layers i (odd) have thicknesses d 1 and i (even) have thicknesses d 2. The combination of layer i (odd) and layer i + 1 (even) will be called a segment. The incident EM wave is assumed to be TM polarized and with the arbitrary incident angle θ from the incident medium labeled with a subscript 0. The substrate is indicated by the subscript s. n 0 and n s represent the refractive indices of the incident medium and substrate medium, respectively for both FF and SH. Figure 4.1. Layered structure composed of nonlinear materials with refractive indices n (1) i for FF and n (2) i for SH. The corresponding χ (2) i represents the nonlinear susceptibilities in each layer. N represents the number of segment. The FF with TM polarization is assumed to be propagating in the incident medium with angle of θ 0. In general, the component of the incident FF E field along the boundary is: E (1) (x, z, ω (1) ) = E (1) (z)e j(ω(1) t k 0x x). (4.2) In Equation (4.2), the transverse component of the wave vector is k x0 = ω n (1) 0 sin θ (1) c 0 where c is the velocity of the EM wave in vacuum. For the multilayer structure as shown in Figure 4.1, the interfaces of the structure are in the x - y plane. At 47

66 the boundaries between each region the transverse wave vectors must match. Furthermore, in each layer, the x-component of the E field consists of right-traveling and left-traveling waves defined as: E (1) (1) (1) (1) i (z) = A i cos θi e jk iz z (1) (1) (1) + B i cos θi e jk iz z ; z i 1 < z < z i. (4.3) A i (1) and B i (1) represent the forward and backward propagating fundamental amplitude coefficient in the i th layer, and k (1) ω iz = n (1) i c cos θ (1) (1) i = [(k i ) 2 (1) (k ix ) 2 ] 1 2. By using Faraday s law E = jkh, the H field with propagating amplitude coefficients A i (1) and B i (1) in the i th layer is expressed as: H (1) (1) (1) (1) i (z) = n i Ai e jk iz z (1) (1) (1) n i Bi e jk iz z ; z i 1 < z < z i. (4.4) Since the number of layers is N, from Equations. (4.3) and (4.4), the relation between the incident amplitude coefficients and the substrate amplitude coefficients can be written in terms of the matrix equation: ( A (1) 0 (1) (1)) = M (1) ( A s (1)); M (1) = [D (1) B 0 B 0 ] 1 ( N (1) (1) i=1 Di P i [D (1) i ] 1 ) D (1) s, (4.5) s 48

67 where D i (1) is the dynamical matrix in the i th layer. In the TM case this is defined as D (1) i = D (1) itm = ( cos θ (1) i n i (1) (1) cos θ i ). P (1) i is the propagation matrix defined as n i (1) (1) P (1) i = ( ejk iz z 0 0 e jk (1) ); z i 1 < z < z i. iz z The approach for non-depleted FF propagation in the PBG structure using TMM has been presented above. However, in a nonlinear PBG, the SH generation process couples with the FF. First, to ensure that the phase along the boundary is maintained for all fields the transverse wave vectors must satisfy the usual phase matching conditions. In Figure 4.2, a sandwich structure composed of one layer nonlinear medium in the center and two linear media on either hand is used as an illustration. The FF wave is incident from the linear medium on the left with an incident angle of θ (1) 0. The coupling between the FF and SH requires that k (2) 1 sin θ (2) 1 = 2k (1) 1 sin(θ (1) 1 ) in the nonlinear medium, where θ (1) 1 can be calculated using Snell s law sin θ (1) 1 n 1 = sin θ (1) 0 n 0 at the first boundary for the FF wave. Then the generated SH wave will be bounced back from the nonlinear medium with the reflected angle of θ (2) 0 in the incident linear medium, which can be calculated using Snell s law at the first boundary for SH wave again: sin( θ (2) 1 )n (2) 1 = sin θ (2) 0 n 0. Using these conditions, the propagation angle for the SH in the incident linear medium is given by θ 0 (2) = sin 1 ( 2k 1 (1) n1 (2) sin(θ 1 (1) ) k 1 (2) n0 ), (4.6a) 49

68 moreover, using Snell s law, the propagation angles of the SH wave inside of the PBG structure is given by sin θ (2) i = sin θ (2) 0 n0 n (2). (4.6b) i Under phase-matching condition, the propagation direction of the SH wave obeys Equation (4.6b). The RTMM matrices for SH are generated by using a procedure similar to that used to derive Equations (4.3) - (4.5). In this PBG structure, using boundary condition, the plane wave coefficients of the SH for one layer can be written by a simple matrix and then the whole PBG transfer matrices can be derived based upon the simple one layer matrix by using the recursion algorithm. Figure 4.2. Sandwich structure composed of one layer of nonlinear medium in the center and two linear media on either side. The red and black lines represent SH and FF, respectively. The FF is incident with angle of θ 0 (1), and the SH is reflected back with an angle of θ 0 (2) into the incident linear medium. 50

69 Under the conditions placed on the transverse wave vectors the spatial evolution of the FF (E (1) ) and the SH (E (2) ) can be analyzed by solving the coupled set of equations: ( d2 + (k (1) 2 (1) dz 2 iz ) ) Ei = (1) (k0z ) 2 χ (2) i E (1) i E (2) i, (4.7) ( d2 + (k (2) 2 (2) dz 2 iz ) ) Ei = (2) (k0z ) 2 χ (2) i (E (1) i ) 2, (4.8) where χ (2) i stands for the second order nonlinear susceptibility in the i th layer, k (1) iz = n i (1) k 0 (1) cos(θ i (1) ), k (2) iz = n (2) i k (2) 0 cos(θ (2) i ), k (1) 0 = ω(1), k (2) c 0 = 2ω(1), k (1) c 0z = k (1) 0 cos θ (1) 0, k (1) 0z = k (1) 0 cos θ (1) 0, k (2) 0z = k (2) 0 cos θ (2) 0, θ (1) (2) i and θ i stand for the propagation angles with respect to the z axis in the i th layer for FF and SH, respectively. In general, the solution of Equations (4.7) and (4.8) are formidable. However, using TMM one can take into account the boundary conditions for multilayers and FF and SH plane wave propagation with arbitrary incident angles. It is concentrated on the case (1) (2) where the SH is weak, i.e., E iz Eiz (non-depleted pump approximation) so that there is no back-coupling of power from the SH to the fundamental. In this case, Equation (4.7) can be rewritten as: ( d2 + (k (1) dz 2 iz ) 2 ) E (1) i = 0, (4.9) 51

70 and its solution is similar to that outlined in Equations (4.3)-(4.6) above. For the TM polarization case, the solution of Equation (4.9) can be written as: (1) E i (z) = (1) Ai e jk (1) iz (z) +B (1) i e jk (1) iz (z) ; z i 1 < z < z i, (4.10) substituting Equation (4.10) in Equation (4.8), E field of the SH with TE polarization can be derived: E (2) (2) (2) (2) i (z) = A i e ( jk iz z) (2) + B i e (jk iz z) (2) (2) 2 (2) (1) (1) 2 (k 0 cos θ0 ) χi e 2jk iz z (A i ) (2) 2 (1) 2 (k iz ) 4(kiz ) (2) (2) 2 (2) (1) (1) 2 (k 0 cos θ0 ) χi e 2jk iz z (B i ) (2) 2 (1) 2 (k iz ) 4(kiz ) 2(k (2) (2) 2 (2) (1) (1) 0 cos θ0 ) χi Ai Bi (2) 2 ; (k iz ) z i 1 < z < z i, (4.11) 52

71 where A i (2) and B i (2) represent the forward and backward propagating amplitude coefficient in the i th layer of the linear part of SH wave and which will be finally calculated by a recursion algorithm for multilayer PBG structure. The tangential magnetic field component for the SH can be represented by propagating amplitude coefficients A (2) (2) i cos θ i, (2) (2) Bi cos θ i, (1) (2) Ai cos θ i and B i (1) cos θ i (2) using Equation (4.11) and Faraday s law: H (2) (2) (2) (2) (2) i (z) = n i cos θi Ai e jk iz z (2) (2) (2) (2) n i cos θi Bi e jk iz z + (1) 2k iz (2) (2) k (A i 0 cos θ0 (2) (2) 2 (2) (1) (1) 2 (k 0 cos θ0 ) χi e 2jk iz z ) (2) 2 (1) 2 (k iz ) 4(kiz ) 2k (1) iz (1) 2 (k 0 (2) (2) (B k 0 cos θ0 i ) (2) (2) 2 (1) (2) 2jk z cos θ0 ) χi e iz (2) 2 (1) 2 ; (k iz ) 4(kiz ) z i 1 < z < z i. (4.12) Equations (4.11) and (4.12) can be summarized in matrix form as follows: ( E (2) i (z) H (2) i (z) ) = (2) (2) e jk iz z 1 1 ( (2) (2) (2) (2)) ( A i n i cos θi n i cos θi (2) B i e jk iz (2) z ) 53

72 1 (1) 2k iz + ( (2) (2) k 0 cos θ0 1 (1) 2k iz ) (2) (2) k 0 cos θ0 ( (A i (1) ) 2 (k0 (2) cos θ0 (2) ) 2 χi (2) (1) z (2) 2 (1) 2 e 2jk iz (k iz ) 4(kiz ) (1) 2 2 (2) (2) (2) (B i ) (k0 cos θ0 ) χ i (2) 2 (1) 2 e 2jk iz (k iz ) 4(kiz ) ( 1 0 ) 2(k (2) (2) 2 (2) (1) (1) 0 cos θ0 ) χi Ai Bi (2) 2 ; (k iz ) (1) z ) z i 1 < z < z i. (4.13) (2) In Equation (4.13), the first term with the phase of k iz z represents the part of (1) SH due to propagation. The second term with the phase of 2k iz z is the part of SH which is nonlinearly generated from the FF. The last term is contributed to by the interaction between the forward and backward traveling components of the FF; the SH generated from this term is phase-locked to the FF and effectively has the refractive index of the FF [59, 60]. The last term is contributed to by the interaction between the forward and backward traveling components of the FF; interpreted as a drive dipole in the volume it can radiate SH waves perpendicular to the propagation axis. The y component of E i (2) and x component of H i (2) are always continuous at each boundary. As in Equation (4.5), for a multilayer nonlinear PBG structure, the matrix T i (2) can be derived using the recursion algorithm. In Equation (4.13), both electric field and magnetic field are defined 54

73 (1,2) (1,2) in terms of coefficients of A i and Bi and the terms in the 1 2 matrix represent the forward and backward propagation components of the SHG. So, it is convenient to redefine the forward and backward components in Equation (4.13) as E (2)+ i (z) and E (2) i (z). For simplifying the RTMM calculation, other teams are defined as below: 1 1 D 0 =( (2) (2)), D (2) 1 1 n 0 cos θ 0 n 0 cos θ i = ( (2) (2) (2) (2)), 0 n i cos θi n i cos θi F i (1) = 2 ( (k (2) (2) 2 (2) 0 cos θ0 ) χi (2) 2 (1) 2 (k (2) (2) 2 (2) 0 cos θ0 ) χi (2) 2 (1) 2 (k iz ) 4(kiz ) (k iz ) 4(kiz ) (1) k iz (2) (2) k 0 cos θ0 (k 0 (2) cos θ0 (2) ) 2 χi (2) (2) 2 (k iz ) 4(kiz (1) ) 2 2 (1) k iz (2) (2) k 0 cos θ0 (k 0 (2) cos θ0 (2) ) 2 χi (2) (2) 2 (1) 2 (k iz ) 4(kiz ) ), (1) P i = ( e 2jk (1) iz z 1 0 (1) 0 e 2jk (1) ), P i = ( e jk (2) iz z 1 0 iz z 0 e jk (2) ), I i = 2(k (2) (2) 2 (2) (1) (1) 0 cos θ0 ) χi Ai Bi iz z (2) 2. (k iz ) For one layer of nonlinear medium inside of linear of incident and substrate media, it is found at the left boundary: D 0 ( E (2)+ 0 (z) (2) ) = D (2) E 0 (z) i ( A (2) 1 (1) [A (2)) + F B i ( 1 ) + (1) 1 [B 1 ] 2 (1 0 )I 1, (4.14) (1) ] 2 and on the right boundary: (2) D 0 ( E (2)+ s (z) (2) ) = D (2) E s (z) i P (2) i ( A 1 (1) (2)) + F (1) B i Pi ( [A 1 ) + (1) 1 [B 1 ] 2 (1 0 )I 1. (4.15) 55 (1) ] 2

74 To extend from this signal layer nonlinear condition to multilayer nonlinear condition, by having the recursion algorithm method, the SH wave in the incident and substrate media can be related as: ( E (2)+ s (z) (2)+ (z) + (2) )=T (2) E s (z) i ( E 0 (2) )+( R i E 0 (z) R i ), (4.16) where: T i (2) =D 0 1 D i (1) F i (1) [Di (1) ] 1 D 0 T i 1 (2), (4.17) R + i ( R i 1 + )=D R 0 1 D (1) (1) i F (1) i [Di ] 1 ( )+D i R 0 1 [(D (2) (2) i F (1) (1) i -Di F (1) i [Di ] 1 D (2) i ) ( A i i 1 (1) B i (1)) +(I-D i (1) F i (1) [Di (1) ] 1 ) ( 1 0 )I i]. (4.18) From Figure 4.2, it is found the forward propagation SH wave and backward propagation SH wave are zero, then Equation (3.16) can be written as: 0 ( (2) E s (z) )=T (2) i ( E 0 (2)+ (z) 0 )+( R i + R i ). (4.19) 56

75 Both E s (2) (z) and E 0 (2)+ (z) can be solved in Equation (4.19), then either of them can be a seed to calculate all the unknown coefficients in multilayer nonlinear structure by using TMM. 4.3 Results and Discussion for SHG In this Section, periodic Ga 0.7 Al 0.3 As (nonlinear medium) / AlAs (linear medium) structures with high and low refractive indices are assumed. The incident fundamental E (1) field is TM polarized (polarized in x z plane) and the H (1) field is polarized along the y direction. The corresponding second order susceptibilities for Ga 0.7 Al 0.3 As media are χ (2) = 2d 14 sin θ cos θ and zero for AlAs, where θ denotes the propagation angle with respect to the z axis inside the structure. It is known that the maximal conversion efficiency of SHG should be obtained when the phase matching condition is satisfied in a homogeneous nonlinear structure. However, in the BPG structure, the FF and SH will encounter significant multiple reflections caused by the variation of the refractive indices. Therefore, the conventional phase-matching mechanism for calculating conversion efficiency is not valid since the phase cannot be readily calculated. In this case, we numerically impose the phase matching condition by adjusting incident angle of FF and the thickness of the linear and nonlinear structures. For normal incidence of the FF wave, the method for numerically calculating conversion efficiency of the SHG is amply discussed in the literature [26, 53]. For oblique incidence of the FF wave, the effective phase matching condition and photonic band edge effects are used to predict the conversion efficiency [61]. In related work, a recursion algorithm for third harmonic generation under the condition of pump depletion is considered to analyze the conversion 57

76 efficiency at oblique incidence for a multilayer nonlinear structure [62]. In our nonlinear PBG structure, additional insight into the SH generation inside the structure and propagation outside the structure can be obtained using our TMM analysis and by varying the incident angle of the FF and the layer thicknesses. In this Chapter, the optimum incident angle and the optimum thicknesses of the linear and nonlinear layers in the PBG structure are determined for maximum conversion efficiency. Although all computations are done with an incident FF amplitude of 1V/μm (corresponding to approximately 300kW/cm 2 ), it is clear that since the SH conversion efficiency generally scales as the square of the FF amplitude, higher FF amplitudes (as is used in practice) will cause SH amplitudes to be large enough that the nondepleted FF approximation may not be valid. The validity of the nondepleted FF assumption can therefore be predicted from the calculations [63]. Figure 4.3(a) shows the 3 D map of the computed SH conversion efficiency immediately outside the nonlinear PBG structure as a function of incident angle θ 0 (0 0 ~30 0 ) of the FF (λ = μm) and the thickness d 2 (0.092 ~ μm) of each linear (AlAs) layer, and the thickness of each nonlinear (Ga 0.7 Al 0.3 As) layer is kept fixed at d 1 = μm. The refractive indices in each segment for the FF are n (1) 1 = 3.231, n (1) 2 = 2.902, while that for the SH are n (2) 1 = 3.466, n (2) 2 = Also, it has been assumed that n 0 = 1, n s = 1 for the incident and substrate media, respectively. The nonlinearity coefficient is taken as d 14 = 94 pm/v. The number of segments N is taken as 50 as an illustration. It is well known that the SH conversion efficiency in a nonlinear PBG structure is dependent not only on the power of the incident FF and the second order susceptibility, but also on the enhancement of the FF and SH inside the 58

77 structure. From Figure 4.3(a), a conversion efficiency of approximately outside the PBG structure is found for a layer thickness d 2 = μm, and for an incident angle θ 0 = It is clear that the incident angle has a significant impact on the conversion efficiency at incident angles from 0 0 to around 5 0 since χ (2) is a function of sin θcos θ. Also, by varying the thickness of the linear layer simultaneously, it is clear that the properties of photonic band gap also contribute significantly to the SH conversion efficiency because both the FF and SH waves get strongly enhanced in this PBG structure which acts as a resonator. The corresponding 3-D map of the peak conversion efficiency inside of the structure is shown in Figure 4.3(b), which is about two orders of magnitude higher than the output conversion efficiency. The consistency of Figures 4.3(a) and (b) indicates that the high conversion efficiency is a result of the enhancement of the SH wave inside the nonlinear PBG structure. If, instead, the thickness d 2 of each linear ( AlAs ) layer is kept fixed at d 2 = μm and d 1, the thickness of each nonlinear (Ga 0.7 Al 0.3 As) layer is varied from to μm, Figures 4.4 (a), (b) are obtained. It is clear that the results obtained from Figure 4.4 are consistent with those obtained from Figure 4.3, as expected. Indeed, the values of the conversion efficiency, and the values of the optimum incident angle and layer thicknesses are the same, as evident from Figures 4.3(a) and 4.4(a). 59

78 (a) (b) Figure 4.3. Calculated conversion efficiency of the TE polarized SH versus the incident angle θ 0 (0 0 ~30 0 ) of TM polarized FF ( λ = μm, E field 1 V/μm ) and thickness d 2 of (linear) AlAs, for (nonlinear) Ga 0.7 Al 0.3 As thickness d 1 = μm and for d 14 = 94pm/V. The number of segments is taken to be N=50. The maximum conversion efficiency at d 2 = μm, and for incident angle θ 0 = (a) External conversion efficiency. (b) Corresponding maximum conversion efficiency inside the structure. 60

79 (a) (b) Figure 4.4. Calculated conversion efficiency of the TE polarized SH versus the incident angle θ 0 (0 0 ~30 0 ) of TM polarized FF (λ = μm, E field 1 V/μm) and thickness d 1 of (nonlinear) Ga 0.7 Al 0.3 As, for (linear) AlAs thickness d 2 = μm and for d 14 = 94 pm/v. The number of segments is taken to be N = 50. The maximum conversion efficiency is at d 1 = μm, and for an incident angle θ 0 = (a) External conversion efficiency. (b) Corresponding maximum conversion efficiency inside the structure. 61

80 By using the optimal thicknesses of the layers inferred from Figures 4.3 and 4.4, the intensity distribution of the FF and SH inside and outside of nonlinear PBG structure are simulated in Figures 4.5 (a) and (b), respectively. As is clear from Figure 4.5 (b), the SH is enhanced to approximately ( V2 2) inside the PBG structure. The FF and SH intensities μm oscillate inside the PBG structure due to interference from multiple reflections. From Figure 4.5 (b), it is observed that there are no oscillations in the incident medium and the substrate for the SH because there is no incident wave of SH in the incident medium and no backpropagating wave of SH in the substrate medium. Also, the transmitted SH intensity is larger than the reflected SH intensity. (1) E 2 tm z ( m) (a) 62

81 10 x (2) E te z ( m) (b) Figure 4.5. Calculated intensity distributions of (a) FF and (b) SH for d 1 = μm, d 2 = μm, N = 50, d 14 = 94 pm/v, and for incident angle of FF θ 0 = In a linear PBG structure, maximum transmission (and, hence, minimal reflection) of a particular wavelength occurs when it is at the maximum T 1 of the transmission band. The FF in this case is also expected to be maximally enhanced in the PBG structure, which should help to increase the conversion efficiency of SH. However, from Figure 4.5(a), it is clear that there is appreciable reflection of the FF which creates standing waves of the FF in the incident medium. Thus, in our case, maximal enhancement of the FF is not needed for maximizing the SH. Indeed, as shown in Figure 4.6(a), the transmission spectrum for the FF (1.548 μm) at the incident angle of θ 0 = shows that the FF is not located at the edge of the bandgap and at the transmission maximum, which is in agreement with our conclusions above. 63

82 0.8 X: Y: T Wavelength ( m) (a) 0.8 X: Y: T Wavelength ( m) (b) Figure 4.6. Calculated transmission spectra around the FF and SH wavelengths for 100 layers with d 1 = µm, d 2 = µm, incident FF angle θ 0 = (a) FF and (b) SH. 64

83 Figure 4.6(b) shows the transmission spectrum for the SH. In Equation (4.11), the first term on the right hand side should play a dominant role in the enhancement of the SH in the PBG structure. The SH is, indeed, located at the peak of the band edge [26, 62], signifying maximum transmission, and, correspondingly, maximal enhancement of the SH in the PBG structure, in agreement with the above results. The density of modes of the linear part of the SH is large and correspondingly, the group velocity is low, which enables maximal enhancement of the SH. In passing, it is to be reiterated that the SH conversion efficiency depends on the FF intensity. For instance, upon increasing the value of incident FF to be 100 V/µm at this incident angle without changing the thickness of the structure can make the conversion efficiency inside of the structure to get above 10%, which implies that the SH is relatively large enough to affect the FF. In this case, the non-depleted assumption for the FF is no longer valid. 4.4 Summary In conclusion, under phase-matching conditions, SH generated by an obliquely incident nondepleted FF in a nonlinear PBG structure comprising alternating layers of nonlinear Ga 0.7 Al 0.3 As and linear AlAs is analyzed by using TMM. Multiple reflections from interfaces and interference effects are taken into account while deriving the TMM for both the FF and the SH. Type I interaction where the FF is TM polarized and SH is TE polarized is assumed. The optimum layer thicknesses and the incident angle for maximal conversion efficiency are numerically determined for 50 alternating layers. The spatial profiles of the FF and SH both inside and outside the nonlinear PBG structure are also simulated, and it is shown that the internal SH conversion efficiency can be as large as 100 times the external conversion efficiency. The SH wavelength, however, is located at the maximum of the band edge, indicating SH field 65

84 enhancement under optimal layer thicknesses and incident angle. Since the SH conversion efficiency depends on the FF intensity, it is shown that the nondepleted pump approximation, used in this analysis is valid as long as the FF amplitude is lower than approximately 100V/μm. Oblique incidence allows for maximization of TE polarized SH conversion efficiency starting from TM polarized FF due to the combined effects of phase matching and effective nonlinearity coefficient. The coherence length, given by l c = π, k = k k 2 2k 1, is of the order of 1-2 microns. However, it is observed that the maximum conversion efficiency occurs when layer thickness is approximately quarter wavelength for the FF and half wavelength for the SH. Work on optimization of SH conversion efficiency for different number of layers and thicknesses is currently in progress, and for the general case of depleted pump intensities are currently in progress. 66

85 CHAPTER 5 APPLICATION OF TRANSFER MATRIX METHOD TO THIRD HARMONIC GENERATION IN NONLINEAR PHOTONIC BANDGAP STRUCTURES: OBLIQUE INCIDENCE 5.1 Third Harmonic Generation in Multilayer Structures: Oblique Incidence In this Chapter, RTMM is derived to account for reflections and interferences between all forward and backward propagating FF and TH waves at oblique incidence for the phase matching condition. The spatial distributions of the FF and TH inside and outside of the multilayer structure can be calculated using RTMM [64]. As in the previous Chapter, Figure 5.1 shows a nonlinear PBG structure with N (assumed to be even) layers. The i th (1,3) layer is assumed to have a thicknesses d i, refractive index n i (with superscripts 1, 3 denoting FF and TH, respectively), and nonlinearity coefficients χ (3) i. In this Chapter, the alternate layers have the same thickness and material properties, i.e., the (1) (3) refractive indices n i and n i correspond FF and TH, all layers i (odd) have thicknesses d 1 and i (even) have thicknesses d 2. The combination of layer i (odd) and layer i +1 (even) will be called a segment. The incident EM wave is assumed to be TM polarized and with the arbitrary incident angle θ O from the incident medium labeled with a subscript 0. The substrate is indicated by the subscript s. Also, n 0 and n s represent the refractive indices of the incident medium and substrate medium, respectively, for both FF and TH. 67

86 Figure 5.1. Layered structure composed of nonlinear materials with refractive indices n (1) i for FF and n (3) i for TH. The corresponding χ (3) i represents the nonlinear susceptibilities in each layer. N represents the number of segments. The FF with TM polarization is assumed to be propagating from the incident medium with angle of θ 0. Assuming a nondepleted FF, and under the conditions placed on the transverse wave vectors the spatial evolution of the FF (E (1) ) and the TH (E (3) ) can be analyzed by solving the coupled set of equations ( d2 + (k (1) 2 (1) dz 2 iz ) ) Ei = 0 (5.1) ( d2 + (k (3) 2 (3) dz 2 iz ) ) Ei = (3) (k0z ) 2 χ (3) i ω (3) (E (1) i ) 3, (5.2) where ω (3) is the frequency of TH (E (3) ). The solutions corresponding to Equations (5.1) and (5.2) for calculating FF (E (1) ) and the TH (E (3) ) are given as E (1) i (z) = A (1) i cos θ (1) i e jk (1) iz z + B (1) i cos θ (1) i e jk (1) iz z ; (5.3) 68

87 E (3) (3) (3) (3) i (z) = A i e ( jk iz z) (3) + B i e (jk iz z) ψ (3)+ i e 3jk (1) iz z ψ (3) i e 3jk (1) iz z ψ (1)+ i e jk (1) iz z ψ (1) i e jk (1) iz z, (5.4) where z i 1 < z < z i, χ i (3) stands for the third order nonlinear susceptibility in the i th layer, k (3) iz = n (3) i k (3) 0 cos(θ (3) i ), k (3) 0 = 3ω(1), k (3) 0z = k (3) 0 cos θ (3) 0, θ (3) i stand for the propagation angles with respect to the z axis in the i th layer for FF and TH, respectively. In Equation (5.4), c ψ (3)+ i = [(A (1) 3 (3) (3) (3) i ) χi (k0 cos θ0 ) 4 (1) 3 (3) (3) (3) (A i ) χi (k0 cos θ0 ) 2 (1) 2 (k iz ) ] (1) 4 (1) 2 (3) 2 (3) (3) 9(k iz ) 10(kiz ) (kiz ) +(k0 cos θ0 ) 4, (5.5a) ψ (3) i = [(B (1) 3 (3) (3) (3) i ) χi (k0 cos θ0 ) 4 (1) 3 (3) (3) (3) (B i ) χi (k0 cos θ0 ) 2 (1) 2 (k iz ) ] (1) 4 (1) 2 (3) 2 (3) (3) 9(k iz ) 10(kiz ) (kiz ) +(k0 cos θ0 ) 4, (5.5b) ψ (1)+ i = [3(A (1) 2 (1) (3) (3) (3) i ) Biz χi (k0 cos θ0 ) 4 (1) 2 (1) (3) (3) (3) 27(A i ) Bi χi (k0 cos θ0 ) 2 (1) 2 (k iz ) ] (1) 9(k iz ) 4 (1) 2 (3) 2 (3) (3) 10(k iz ) (kiz ) +(k0 cos θ0 ) 4, (5.5c) 69

88 ψ (1) i = [3(B (1) 2 (1) (3) (3) (3) i ) Ai χi (k0 cos θ0 ) 4 (1) 2 (1) (3) (3) (3) 27(B i ) Ai χi (k0 cos θ0 ) 2 (1) 2 (k iz ) ] (1) 9(k iz ) 4 (1) 2 (3) 2 (3) (3) 10(k iz ) (kiz ) +(k0 cos θ0 ) 4, (5.5d) where the signs + and - denote the forward and backward propagating waves, respectively. The FF wave is incident from the linear medium on the left with an incident angle of θ 0 (1). It is assumed that THG is in TE polarized and its propagation angle in the incident medium is θ 0 (3). The coupling between the FF and TH requires that k (3) 1 sin θ (3) 1 = 3k (1) 1 sin(θ (1) 1 ) in the nonlinear medium, where θ (1) 1 can be calculated using Snell s law sin θ (1) (1) 1 n 1 = sin θ (1) 0 n 0 at the first boundary for the FF wave. Then the generated TH wave will be bounced back from the nonlinear medium with the reflected angle of θ (3) 0 in the incident linear medium, which can be calculated using Snell s law at the first boundary for TH wave again: sin( θ (3) 1 )n (3) 1 = sin θ (3) 0 n 0. Using these conditions, the propagation angle for the TH in the incident linear medium is given by θ (3) 0 = sin 1 ( 3k (1) 1 n1 (3) (1) sin(θ 1 ) (3) ), (5.6 a) k 1 n0 moreover, using Snell s law, the propagation angles of the TH wave inside of the PBG structure is given by sin θ i (3) = sin θ 0 (3) n0 n i (3). (5.6 b) 70

89 Under phase-matching condition, the propagation direction of the TH wave obeys Equation (5.5b). The RTMM matrices for TH are generated by using a procedure similar to that used to derive Equations (4.11) - (4.19). In this PBG structure, using boundary condition, the plane wave coefficients of the TH for one layer can be written by a simple matrix and then the whole PBG transfer matrices can be derived based upon the simple one layer matrix by using the recursion algorithm. Using boundary conditions and considerable algebra for one layer nonlinear structure, it can be shown, for instance, that the forward and backward traveling components of the electric field in the substrate can be expressed as ( E (3)+ s (z) (3) ) = E s (z) (D (3) 0 ) 1 (3)+ (3) E [R 1 D 0 ( 0 (z) (1)+ (1) (1) (1) ψ (3) ) + (F E 0 (z) 1 K1 R1 F 1 ) ( 1 ψ 1 (1) ) + (G (1) 1 L (1) 1 R 1 G (1) 1 ) ( ψ (3)+ 1 (3) )], (5.7) ψ 1 where R i = D i (3) H i (3) (D i (3) ) 1 and 71

90 D (3) i = ( (3) (3) (3) (3)), G (1) n i cos θi n i cos θi i = ( 3k iz (1) (3) k 0z 1 1 (1) ), F (1) i = ( 3k iz (3) k 0z (1) k iz (3) k 0z 1 (1) ), k iz (3) k 0z (3) (1) H (3) i = ( e jk iz di 0 0 e jk (3) ), K (1) iz di i = ( e jk iz di 0 0 e jk (1) ), L (1) iz di i = (1) ( e 3jk iz di 0 0 e 3jk (1) ), k (1) (1) (1) iz di iz = k i cos θi, (3) (3) (3) kiz = k i cos θi, i =1,2,3. (5.8) For this multilayer structure, the RTMM can be derived by using mathematical induction from Equations (5.8) as ( E (3)+ s (z) i=1 (3)+ (z) (3) ) =(D (3) E s (z) 0 ) 1 [( i=n R i ) D (3) 0 ( E 0 (3) ) + ( i E 0 (z) M )], (5.9) i + where M )=R i ( M + i 1 ) + (F (1) i i K (1) i R i F (1) i ) ( ψ (1)+ i (1) ) + (G (1) ψ i L (1) i R i G (1) i ) ( ψ (3)+ i (3) ), (5.10) i ψ i ( M i + M i 1 where ( M 0 + M 0 ) = ( 0 0 ). 72

91 Using the boundary conditions E s (3) (0) =0 and E 0 (3)+ (0) =0, all of the coefficients for A i (3) and Bi (3) can be computed by using this RTMM numerically. This technique will be used to calculate the TH fields inside and outside of nonlinear multilayer structure. Optimum conditions for TH generation can be accordingly predicted. 5.2 Results and Discussion for THG In this Section, periodic TiO 2 (nonlinear medium) / SiO 2 (nonlinear medium) structures with high and low refractive indices are assumed. The incident fundamental E (1) field is TM polarized. The corresponding third order constant susceptibilities for TiO 2 media are χ (3) TiO2 = um 2 /V 2 and χ (3) SiO2 = um 2 /V 2 for SiO 2. The number of segments N is taken as 20 as an illustration. Figure 5.2(a) shows the 3D map of the computed TH conversion efficiency immediately outside the nonlinear PBG structure as a function of the thickness d 1 (0.104~0.11 μm) of each TiO 2 and the thickness d 2 ( ~0.081 μm) of each SiO 2 layer. As in Chapter 4 for the case of SHG, an incident FF amplitude of 1V/μm (corresponding to approximately 300kW/cm 2 ) is assumed; THG conversion efficiency scales as the fourth power of the FF amplitude. The incident angle is assumed to be 0 degrees (i.e., normal incidence) at first. The refractive indices in each segment for the FF are n (1) 1 = 2.45, n (1) 2 = at wavelength of 1.55 μm, while that for the TH are n (3) 1 = 3, n (3) 2 = at wavelength of μm. Also, it has been assumed that n 0 = 1, n s = 1 for the incident and substrate media, respectively. It is well known that the TH conversion efficiency in a nonlinear PBG structure is dependent not only on the power of the incident FF and the 73

92 third-order susceptibility, but also on the enhancement of the FF and TH inside the structure. From Figure 5.2(a), at forward direction, a conversion efficiency of approximately outside the PBG structure is found for a layer thicknessd 1 = μm, d 2 = μm, and for an incident angle θ 0 = 0 0. By varying the thickness of two material simultaneously, it is clear that the properties of photonic band gap also contribute significantly to the TH conversion efficiency because both the FF and TH waves get strongly enhanced in this PBG structure which acts as a resonator. The corresponding 3D map of the peak conversion efficiency inside of the structure is shown in Figure 5.2(b), which is about 1 order of magnitude higher than the output conversion efficiency. The consistency of Figures 5.2(a) and (b) indicates that the high conversion efficiency is a result of the enhancement of the TH wave inside the nonlinear PBG structure. (a) 74

93 (b) Figure 5.2. Calculated conversion efficiency of the TE polarized TH versus thickness d 1 of TiO 2 and thickness of d 2 SiO 2 for TM polarized FF (λ = 1.55 μm, E field 1 V/μm). The number of segments is taken to be N = 20. The maximum conversion efficiency of happens at d 1 = μm and d 2 = μm, and for incident angle θ 0 = 0 0. (a) External conversion efficiency on forward direction. (b) Corresponding maximum conversion efficiency inside the structure. By using the selected thicknesses of the layers gathered from Figures 5.2, the intensity distribution of the FF and TH inside and outside of nonlinear PBG structure are simulated in Figures 5.3 (a) and (b), respectively. As is clear from Figure 5.3 (a) and (b), the E 2 of TH is enhanced to ( V2 μm 2) inside the PBG structure and the forward output is ( V2 μm 2). The FF and TH intensities oscillate inside the PBG structure due to interference from multiple reflections. From Figure 5.3 (b), it is observed that there are no oscillations in the incident medium and the substrate for the TH because there is no standing wave of TH in the incident medium and substrate medium. From Figure 5.3(a), the standing wave is found in the incident medium which shows that the transmission of FF is not 1 and this phenomenon is similar to SHG 75

94 in Chapter 4. The peak TH conversion efficiency does not occur at the position where maximum enhancement of FF happens, although the enhancement of the FF does play a role in the high TH conversion efficiency. (a) (b) Figure 5.3. Calculated E 2 distributions of (a) FF and (b) TH for d 1 = μm, d 2 = μm, N = 20, and for incident angle of FF θ 0 =

95 In a linear PBG structure, maximum field enhancement usually happens at the edge of the band and the transmission is 1 [61]. The FF in this case is expected to be maximally enhanced in the PBG structure because it is a driver part for THG, which help to increase the conversion efficiency of TH. Simultaneously, larger enhancement of TH is also desired in PBG as well. Ideally, both of the FF and TH located on the band edges generate maximum conversion efficiency. However, from Figure 5.3(a), it is clear that there is small reflection of the FF which creates standing waves of the FF in the incident medium which means FF is not enhanced maximally. Thus, under dispersion, maximal enhancement of the FF is not necessary for maximizing the TH conversion efficiency, as found by our numerical simulation method. Indeed, as shown in Figure 5.4(a), the transmission spectrum for the FF (1.55 μm) at the incident angle of θ 0 = 0 0 shows that the FF is not located at the edge of the bandgap and at the transmission maximum, which is in agreement with our conclusions above. From Figure 5.4(b), it is found that 100% of T is achieved and is also close to the edge of the band, which means the TH field is enhanced very much in this PBG structure. Examination of the (linear) transmission spectra alone for the FF and TH suggests that maximum TH conversion efficiency is possible even when the FF and TH are not exactly at their respective band edges. 77

96 (a) (b) Figure 5.4. Calculated transmission spectra around the FF and SH wavelengths for N = 20 for d 1 = μm, d 2 = μm, N=20, and for incident angle of FF θ 0 =

97 Figure 5.5(a) shows the 3D map of the computed TH conversion efficiency immediately outside the nonlinear PBG structure as a function of incident angle θ 0 (0 0 ~15 0 ) of the FF (λ = 1.55 μm) and the thickness d 2 ( 0.8~0.81 μm) of each nonlinear (SiO 2 ) layer, and the thickness of each nonlinear (TiO 2 ) layer is kept fixed at d 1 = μm. The number of segments N is taken as 20 as well. It is clear that the incident angle has a significant impact on the conversion efficiency at incident angles from 0 0 to around 15 0 since the enhancement of the field is died at certain incident angles. In this case, the maximal conversion efficiency happens at normal incidence; also, by varying the thickness of the linear layer simultaneously, it is clear that the properties of photonic band gap also contribute significantly to the TH conversion efficiency because both the FF and TH waves get strongly enhanced in this PBG structure at normal incidence. Unlike the case of the SH, the maximum at normal incidence is likely due to the fact that the third order nonlinearity is not a function of the incident angle. If, instead, the thickness d 2 of each SiO 2 layer is kept fixed at d 2 = μm and d 1, the thickness of each TiO 2 layer is varied from to 0.11 μm, Figures 5.6(a), (b) are obtained. It is clear that the results obtained from Figure 5.6 are consistent with those obtained from Figure 5.5, as expected. Indeed, the values of the conversion efficiency, and the values of the optimum incident angle and layer thicknesses are the same, as evident from Figures 5.5(a) and 5.6(a). 79

98 (a) (b) Figure 5.5. Calculated conversion efficiency of the TE polarized TH versus the incident angle θ 0 (0 0 ~15 0 ) of TM polarized FF ( λ = 1.55 μm, E field 1 V/μm) and thickness d 2, for thickness d 1 = μm. The number of segments is taken to be N = 20. The maximum conversion efficiency is atd 2 = μm, and for an incident angle θ 0 = 0 0. (a) Corresponding maximum conversion efficiency inside the structure. (b) External conversion efficiency. 80

99 (a) (b) Figure 5.6. Calculated conversion efficiency of the TE polarized TH versus the incident angle θ 0 (0 0 ~15 0 ) of TM polarized FF (λ = 1.55 μm, E field 1 V/μm) and thickness d 2, for thickness d 1 = μm. The number of segments is taken to be N = 20. The maximum conversion efficiency is at d 1 = μm, and for an incident angle θ 0 = 0 0. (a) Corresponding maximum conversion efficiency inside the structure. (b) External conversion efficiency. 81

100 5.3 Summary In conclusion, a systematic method using RTMM to analyze THG in nonlinear PBGs has been discussed, under the assumption of pump non-depletion. The exact distributions of the FF and the harmonics can be computed, along with the prediction of optimum layer thicknesses and angle of incidence for maximum conversion efficiency. The limits of the non-depleted pump approximation can be readily found. It is also shown that for maximum conversion efficiency, the location of the TH should be at near its transmission peak of the PBG structure. The reason for FF to be at zero incidence angle for maximum conversion efficiency may be due to the independence of the nonlinearity parameter on the incident angle. 82

101 CHAPTER 6 CONCLUSION AND FUTURE WORK 6.1 Conclusion In this dissertation, different optical thin film structures and their behavior have been studied, along with possible applications. Specifically, in Chapter 2, the design of a near-field optical system for imaging sub-wavelength objects using double negative metamaterial superlens using SiC + Ag operating at 405 nm has been presented. Simulation results show the extent of TE and TM imaging of the optical field immediately behind the object, which is a 1D periodic Mo structure. With experimental verification as the goal, large area Mo 1D and 2D structures with periodicities as small as 200 nm have been fabricated on a glass substrate using 266 nm DUV interference lithography and reactive-ion etching. As is clear from the fabrication process, different periodicities can be realized by adjusting the interference angles. Compared to electron beam and focused ion beam lithography, this technique allows for making nano-scale patterns on Mo at significantly lower cost. Our fabricated objects are examined using SEM, and optically characterized using TE and TM illumination. In Chapter 3, SHEL of a Gaussian beam at normal material-air interface and double negative is analyzed theoretically. The general method for realizing SHEL is used in laboratory and a great agreement between experiment and theory is achieved. First, 83

102 the refractive index value of Ag + SiC is verified. Secondly, it is also verified that SHEL in double negative metamaterial, although the sign of refractive index gradient is reversed. The spin angular momentum is unreversed in double negative refractive indices, then the SHEL beam displacement on that interface is unrevised. SHEL may lead a potential application for characterizing bulk and thin film materials. In Chapters 4 and 5, under phase-matching conditions, SH and TH generated by an obliquely incident nondepleted FF in a nonlinear PBG structure comprising alternating layers of structures is analyzed by using RTMM. In the simulation, multiple reflections from interfaces and interference effects are taken into account while deriving the RTMM for both the FF and the high order nonlinear generations. The optimum layer thicknesses and the incident angle for maximal conversion efficiency are numerically determined. The spatial profiles of the FF, SH and TH both inside and outside the nonlinear PBG structure is also simulated. Work on TH conversion efficiency for metal dielectric periodic structure is currently in progress. 6.2 Future Work are presented. Selected future work corresponding to the research reported in Chapters 2 and Superlens for Super-resolution Applications As mentioned in Chapter 2, a structure for testing the superlens has been designed. Also, previous work on Mo subwavelength object fabricated on a glass substrate has been discussed. The object has been characterized by SEM and its transmission for TE and TM polarized light has been studied. For observing super- 84

103 resolution imaging, there are certain steps that need to be finished in the clean room. AFM and NSOM will be used to test the performance of the super-resolution Fabrication Steps in Clean Room As presented in Chapter 2, the technique for fabricating the sub-wavelength Mo object has been established. There are still three main steps to finish the superlens structure designed in Figure 2.1. Figures 6.1(a) (c) represent these three steps in sequence. Figure 6.1(a) shows that a thin layer of PR-955 (spacer) will be spin coated on top of the Mo object, at certain spin speed, and the concentration of PR-955 can be adjusted by using a photoresist thinner. Then the exposure and development for PR-955 will be performed by using a UV lamp and photoresist developer until it gets crosslinked. If the thickness of this thin layer cross-linked PR-955 is much larger than 50 nm, O2 plasma will be used to etch down the photoresist to its desired thickness. The function of this spacer is to provide the desired object distance for the Mo object. Next, as shown in Figure 6.1(b), a 70 nm thick Ag + SiC metamaterial (superlens) will be cosputtered on top of spacer. This novel technique has been shown to create a highly uniform metamaterial. Finally, on top of this metamaterial, as shown in Figure 6.1(c), another layer of positive photoresist will be spin coated to serve as the image distance during illumination of the object by a 405 nm laser, which is the optimum wavelength for the metamaterial. 85

104 (a) (b) (c) Figure 6.1. Schematic of proposed imaging process. (a) A 50 nm thick layer of PR-955 is spin coated on the object for working as a spatial layer and serving as object distance. (b) A 50 nm metamaterial of Ag + SiC is co-sputtered on the previous structure. (c) A 70 nm image plane of Photoresist is spin coated on top of the metamaterial Super-resolution Testing using AFM As proposed in Section 6.1.1, a subwavelength structure will be fabricated in the clean room, and the fabricated samples with and without the metamaterial superlens will be illuminated by a 405 nm laser. A pattern is expected to be generated in the developed photoresist corresponding to the Mo object (see Figure 6.2). AFM will be used to test the surface of the photoresist in the two cases in order to test if near field imaging is indeed achieved using the superlens. A schematic of the AFM setup for the expected near field imaging case is shown in Figure

105 Figure 6.2. The fabricated sample is illuminated by 405 nm light as the exposure procedure, then the exposed photoresist is developed. Figure 6.3. The exposed and developed sample is tested by AFM Super-resolution Testing using NSOM Because of the nonlinear response of photoresist to incident illumination [65], the pattern on the developed photoresist may not be exactly proportional to the optical intensity, which may in turn affect the result expected with regard to near-field imaging. Therefore, another method to test the action of the superlens is proposed. Instead of 87

106 Figure 6.1(c), the exact profile of the optical field will be tested using NSOM, as shown schematically in Figure 6.4. Figure 6.4. The performance of superlens is tested by NSOM TMM Simulations of SHG in HM As mentioned in Chapter 4 and 5, TMM along with an iterative algorithm are used to calculate SHG and THG field distribution inside and outside of periodically nonlinear multilayer structure for oblique incidence of the FF, and the nonlinear conversion efficiency inside and outside of the structure has been calculated as well. Recently, HMs have attracted a lot of attention because of their special properties which do not exist in nature SHG in such materials should be analyzable using TMM as well. Furthermore, TMM methods developed for SHG can be also extended to THG in periodic 3 rd order nonlinear multilayer structures. HMs are comprised of multilayered metal-dielectric structures, have attracted significant interest recently. Figure 6.5 [28] shows an open-curved hyperbolic dispersion, which has one or two components of the permittivity tensor having the opposite sign to the others [67]. Due to continuity of the normal component of the electric displacement, 88

107 significant electric field enhancement occurs inside of the structure. Current research includes spontaneous emission enhancement and with applications in heat transport [31]. A significant advantage of PBG structures involving multilayered HMs is that HMs are easier to fabricate than double negative metamaterial. HMs may lead to the design of novel nonlinear electromagnetic devices, with exciting applications, such as all-optical switches and tunable subwavelength second and higher harmonic imaging microscopy systems [68]. Application of TMM to PBG structures comprising alternating layers of linear and nonlinear HMs for SHG is proposed as part of future research. Figure 6.5. Isofrequency surfaces of extraordinary waves in hyperbolic metamaterials. (a) Type I, (b) Type II [28] Role of Self Phase Modulation in THG In Chapter 5, THG has been investigated using TMM using the same nonlinearity parameters for THG and self-phase modulation (SPM) which also occurs during third order nonlinear interactions. In an actual system, the nonlinearity coefficients responsible for THG and SPM may be different. This will be simulated in future research, and should enable the understanding of the role of SPM in THG as well. 89