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Quantitative linear optical scattering spectroscopy of two-dimensionally textured planar waveguides Mandeville, William Jody 2001

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c Q U A N T I T A T I V E L I N E A R O P T I C A L S C A T T E R I N G S P E C T R O S C O P Y O F T W O - D I M E N S I O N A L L Y T E X T U R E D P L A N A R W A V E G U I D E S by W I L L I A M J O D Y M A N D E V I L L E B . S . , U n i t e d States A i r Force Academy, 1988 M . S . , A i r Force Inst i tute of Technology, 1992 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Physics . and Astronomy) W e accept this thesis as conforming to the required s tandard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A June 2001 © W i l l i a m Jody Mandev i l l e , 2001 In presenting this thesis i n par t i a l fulfilment of the requirements for an advanced degree at the Univers i ty of B r i t i s h C o l u m b i a , I agree that the L i b r a r y shal l make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholar ly purposes may be granted by the head of m y department or by his or her representatives. It is unders tood that copying or pub l i ca t ion of this thesis for f inancial gain shal l not be al lowed wi thou t m y wr i t t en permission. Depar tment of T h e Unive r s i ty of B r i t i s h C o l u m b i a Vancouver , C a n a d a D a t e e g J u ^ Z-O&l Abstract Linear white light spectroscopy in conjunction with rigorous computer modeling re-veals the fundamental nature of the electromagnetic excitations associated with the sim-ple lattice and defect superlattice texturing of 2D planar waveguides. By achieving un-precedented agreement between experimentally measured and rigorously simulated band structures of leaky modes associated with the second, and up to the seventh, zone-folded Brillouin zones of square and triangular lattice structures, a thorough characterization of the polarization and dispersive properties of these electromagnetic modes has been achieved. An evaluation of the usefulness of a newly developed diffraction measurement technique for probing band structure is presented in conjunction with data and simula-tions for waveguides with defect superlattices. Textured planar waveguides, as a powerful medium for engineering devices which control the propagation of light, are explored via the thorough characterization of a novel polymer waveguide, and a GaAs waveguide that was engineered to possess a flat band for use in non-linear optics applications; in addition an original design is discussed for an angle and polarization insensitive notch filter based on a localized defect mode. ii Contents Abstract ii List of Tables v List of Figures vi Acknowledgements ix 1 Introduction 1 2 Linear Light Scattering Spectroscopy of Textured Planar Waveguides 9 2.1 B a c k g r o u n d 9 2.1.1 Un tex tu red P l a n a r Waveguides 10 2.1.2 2 D Simple Lat t ices 11 2.1.3 Defect Superlatt ices 21 2.2 Green 's F u n c t i o n M o d e l 24 2.3 Spectroscopy 28 2.3.1 Specular Measurement Technique 29 2.3.2 Diffract ion Measurement Technique 30 3 Experimental Design 37 3.1 A p p a r a t u s 38 3.1.1 L i g h t Source 38 3.1.2 Opt ics 40 3.1.3 Pos i t ion ing Mechanics 43 3.1.4 Spectrometer 45 3.2 A l i g n m e n t and Opera t ion of Appa ra tus 46 3.3 Al t e rna te Configurat ions 47 4 Sample Preparation 48 4.1 G a A s Sample 48 4.1.1 P l a n a r Waveguide G r o w t h 48 4.1.2 E l ec t ron B e a m Li thography 49 4.1.3 E t c h i n g 50 i i i CONTENTS i v 4.1.4 O x i d a t i o n 52 4.2 P o l y m e r Sample 53 5 Results and Discussion 56 5.1 Waveguides w i t h S imple 2 D Gra t ings 58 5.1.1 Square La t t i c e 58 5.1.2 Tr iangula r La t t i ce 62 5.2 Waveguides Engineered for Specific App l i ca t ions 74 5.2.1 2 D Tex tu red A z o - P o l y m e r Waveguides 74 5.2.2 F l a t Pho ton ic Bands A l o n g the En t i r e T - X L i n e of a 2 D Square La t t i ce 89 5.3 Waveguides w i t h Defect Superlatt ices 94 5.3.1 Superla t t ice Scat ter ing from a L o w Dispers ion B a n d 97 5.3.2 Superla t t ice Diffract ion from Tr iangular La t t i ce Structures . . . . 105 5.3.3 True Defect Modes 125 6 Conclusions and Recommendations 136 Bibliography 139 List o f Tables 3.1 E l l i p s o i d a l mi r ro r specifications 42 4.1 P M M A developing recipe 51 4.2 E C R recipe 52 5.1 K e y mode l ing parameters for textured planar waveguide w i t h square la t t ice 62 5.2 K e y mode l ing parameters for textured planar waveguide w i t h t r iangular la t t ice 70 5.3 K e y mode l ing parameters for textured planar po lymer waveguide . . . . 77 5.4 K e y mode l ing parameters for "flat band" structure 91 5.5 K e y mode l ing parameters for T-5 109 5.6 Des ign parameters for textured planar waveguide w i t h defect mode . . . 128 v List of Figures 2.1 Schematic dispersion d iagram for untextured, asymmetr ic p lanar waveguide 12 2.2 Schematic of I D zone-folding 13 2.3 Schematic dispersion d iagram for a 2 D square lat t ice 16 2.4 M o m e n t u m space d iagram for a square lat t ice 17 2.5 M o m e n t u m space d iagram for a t r iangular lat t ice 20 2.6 Schematic dispersion d iagram for a 2 D t r iangular lat t ice 21 2.7 B a n d structure for t h in G a A s planar waveguide w i t h I D defect superlat t ice w i t h air f i l l ing fraction of 2.5% . . . 23 2.8 B a n d structure for t h in G a A s planar waveguide w i t h I D defect superlat t ice w i t h air f i l l ing fraction of 25% 24 2.9 Schematic d iagram of a textured planar waveguide 25 2.10 G r a p h i c a l representation of Fourier coefficient generation technique . . . 28 2.11 Schematic dispersion d iagram for I D textured waveguide 29 2.12 Representat ive reflectivity spectra from a textured planar waveguide . . . 31 2.13 Schematic dispersion for diffraction probing method 32 2.14 S imula ted s-polarized KD diffraction spect rum from waveguide w i t h a de-fect superlat t ice 35 2.15 M o m e n t u m space d iagram for a square defect lat t ice 35 2.16 S imula ted s-polarized spectra for various in-plane wavevectors 36 3.1 E x p e r i m e n t a l apparatus schematic 39 3.2 E l l i p s o i d a l mi r ro r design schematic 42 3.3 D e t a i l of E M I and E M 2 mirrors 43 3.4 B l o c k d iagram of sample al ignment apparatus 44 3.5 Sample mount detai l • 46 3.6 Al t e rna te sample mount for cleaved edge v iewing 47 4.1 G a A s - b a s e d textured planar waveguide schematic 49 4.2 Exposure pat tern for po lymer sample 54 5.1 Specular reflectivity da ta for waveguide w i t h s imple square la t t ice . . . . 60 5.2 Dispers ion d iagram for s imple square lat t ice da ta and s imulat ions . . . . 61 5.3 Specular reflectivity da ta (r^M) for t r iangular lat t ice 64 5.4 Specular reflectivity da ta (r-K) for t r iangular lat t ice 65 v i LIST OF FIGURES v i i 5.5 Specular reflectivity da ta for 10° angle of incidence i n the M di rec t ion . . 66 5.6 Dispers ion d iagram for t r iangular lat t ice da ta and s imulat ions 67 5.7 M o m e n t u m space diagrams for each mode of the t r iangular la t t ice wave-guide at 10° angle of incidence 68 5.8 M o m e n t u m space diagrams for each mode of the t r iangular la t t ice wave-guide near zone center - 69 5.9 Simula t ions of t r iangular lat t ice w i t h and wi thou t defect superlat t ice . . 71 5.10 S imula ted spectra for t r iangular lat t ice sample w i t h t h i n oxide 72 5.11 S imula ted spectra for t r iangular lat t ice sample w i t h th ick oxide 73 5.12 A F M micrograph of the surface of the po lymer waveguide 76 5.13 N o r m a l i z e d specular reflectivity da ta for po lymer waveguide, X di rec t ion 78 5.14 N o r m a l i z e d specular reflectivity da ta for po lymer waveguide, M di rec t ion 79 5.15 Specular reflectivity s imula t ion and da ta for po lymer waveguide for 10-30° angles of incidence, M direct ion 81 5.16 Dispers ion d iagram for 2 D polymer gra t ing 82 5.17 S imula t ion of specular reflectivity of po lymer waveguide at no rma l incidence 84 5.18 S imula t ion of specular reflectivity of po lymer waveguide at 1°, X d i rec t ion 85 5.19 D a t a and s imula t ion of specular reflectivity of po lymer waveguide at 30° , M di rec t ion 86 5.20 S-polarized specular reflectivity da ta for po lymer waveguide at 10°, X di rec t ion 88 5.21 A F M micrographs of two polymer square lat t ice gratings 89 5.22 Theore t ica l dispersion d iagram for square lat t ice sample showing "flat bands" 91 5.23 Specular reflectivity s imula t ion for "flat band" sample 93 5.24 S- and p-polar ized specular da ta for "flat band" sample 95 5.25 S E M micrograph of "flat band" sample 98 5.26 U n p o l a r i z e d diffraction da ta for "fiat band" sample 100 5.27 Diffract ion simulat ions for "flat band" sample 101 5.28 Dispers ion d iagram of the flat p-polar ized bands 102 5.29 M o m e n t u m space diagrams of low dispersion bands 104 5.30 Schematic d iagram i l lus t ra t ing the diffraction probe technique 107 5.31 S E M micrograph of T-5 109 5.32 Unpo la r i zed -Kf diffraction da ta for T-5 110 5.33 U n p o l a r i z e d +Kf diffraction da ta for T - 5 I l l 5.34 S imu la t i on for -K® diffracted order from defect superlat t ice of T - 5 . . . . 112 5.35 S imula t ion for diffracted order from defect superlat t ice of T - 5 . . . 113 5.36 S-polarized specular reflectivity da ta for T-5 114 5.37 S- and p-polar ized specular reflectivity s imulat ions for T-5 115 5.38 U n p o l a r i z e d -Kf da ta for T - 7 " . . . . 117 5.39 U n p o l a r i z e d +Kf da ta for T - 7 118 5.40 S imu la t i on of -K% spectra for T - 7 119 5.41 S imu la t i on of +K° spectra for T -7 120 LIST OF FIGURES • v i i i 5.42 Schematic zone diagram showing upward dispersing bands at top of second order TE-l ike gap 121 5.43 S-polarized specular reflectivity data for T-7 123 5.44 P-polarized specular reflectivity data for T-7 124 5.45 Triangular superlattice diffraction data compared with simulations for var-ious oxide thicknesses 126 5.46 Dispersion diagram showing large T E gap 128 5.47 Simulated specular reflectivity for defect mode in first order gap 130 5.48 Simulated diffraction for defect mode in first order gap 131 5.49 Momentum space diagram for defect mode 132 5.50 Real space plot of electric field intensity on surface of the defect lattice waveguide 134 Acknowledgements I w o u l d l ike to thank m y advisor, D r . Jeff F . Y o u n g , for his unfai l ing support and inspi red guidance. He pat ient ly and unselfishly provided mot iva t ion , c lar i f icat ion and d i rec t ion wi thou t regard for his pressing schedule. I a m indebted to the members of D r . Young ' s lab (past and present) for their sugges-tions and assistance throughout the course of this project. In par t icular , I wou ld l ike to thank: D r . P a u l P a d d o n for eloquently re in t roducing me to the 'basics ' after m y seven year hiatus from graduate school; V i g h e n Pacradoun i and A l l a n C o w a n , for the m a n y enl ightening discussions we shared; H o n g M a for his enthusiastic contr ibut ions to the po lymer research; and A l e x Busch for his invaluable guidance w i t h the "hard p a r t s " -m a t h and L a T e x . I also wou ld l ike to acknowledge m y wife, Ch r i s , for her tremendous support th rough-out this ordeal , and m y sons, D u n c a n , Jack and K i t , for their patience. I owe t hem a dept of grat i tude that I w i l l never be able to repay. i x Chapter 1 Introduction Pho t on i c crystals are of great interest due to their potent ia l to provide advances i n science and technology comparable to the advances provided by semiconductors over the last 50 years. Pho ton ic crystals do for photons what semiconductor crystals do for electrons. Na tu re created semiconductor crystals such that the atoms are arranged i n a perfectly per iodic la t t ice spaced on the order of the wavelength of the electrons propagat ing through them. T h i s per iodic "electrical" potent ia l creates the r i ch band s tructure wh ich has been studied and exploi ted since the 1950's. Na tu re has prov ided few crystals w i t h an analogous "photonic" potent ia l la t t ice spaced on the order of the wavelength of l ight . [18,36] However, w i t h recent technological advances, such crystals are being manufactured. These man-made, low-loss per iodic dielectr ic m e d i a are k n o w n as "photonic crystals ." [12] T h e t e rm "photonic c rys ta l " is used to describe structures i n wh ich a per iodic modu la t ion of the index of refraction is used to control the propagat ion of l ight th rough B r a g g diffraction. In general the per iod ic i ty can be i n one d imens ion ( I D ) , 2 D or 3 D . T h e unusual opt ica l properties of I D textured dielectrics were recognized 1 CHAPTER 1. Introduction 2 and exploi ted long before the te rm "photonic crys ta l" was in t roduced. T h e famil iar "quarter-wave dielectric stack" structure is essentially a photonic c rys ta l . It uses a l te rna t ing layers of h igh and low refractive index materials to efficiently reflect normal ly- inc ident l ight w i t h wavelengths i n a range of approximate ly twice the op t i ca l p e r i o d 1 of the stack. T h e l ight w i t h i n this "forbidden gap" is diffracted backwards, so that the reflection process is lossless. T h e w i d t h of the gap scales as the difference i n the refractive indices, A n , of the constituent layers, and the sharpness of the gap edges increases as the number of layers increases. For typ ica l dielectric mater ials , such as a quarter wave stack, A n < 0.5, and the center frequency and w i d t h of the gap vary r ap id ly as the incident angle of the rad ia t ion is varied away from the normal . In par t icu lar , the gap shrinks and then vanishes at angles greater than ~ 30° . [9] These effects of I D photonic crystals have had great impact on opt ica l applicat ions; 2 D and 3 D photonic crystals have the potent ia l to have an even greater impact . In 1987, Yab lonov i t ch [37] and John [13] independent ly recognized that a 3D- tex tured s tructure w i t h a sufficiently large index-contrast could exhibi t a complete bandgap: that is, such a mater ia l cou ld inh ib i t the propagat ion of electromagnetic rad ia t ion w i t h i n some continuous range of frequencies, regardless of the d i rec t ion of propagat ion or the po la r iza t ion of the field. T h e significance of this general izat ion can perhaps best be appreciated by considering that the to ta l photonic density of states w i t h i n such a mate r ia l is d r ama t i ca l ly altered from that available i n bu lk dielectrics. In the gap the densi ty of states can approach zero, and near the band edges i t is considerably enhanced over a 1 The optical period is the vacuum wavelength divided by the average index of the periodic dielectric. CHAPTER 1. Introduction 3 rela t ively narrow range of frequencies. T h i s should profoundly affect the fundamental properties of electronic oscillators i n these materials, and these modif ied properties may wel l form the basis of new opt ica l technologies. Y a b l o n o v i t c h et a l . [38] were the first to realize a full 3 D photonic c rys ta l by d r i l l i n g a dielectr ic block full of holes on the order of centimeters in diameter. T h e resul t ing face-centered cubic ( F C C ) crysta l structure exhibi ts a full bandgap at microwave frequencies. B y inser t ing an appropriate defect site w i t h i n these crystals it is possible to comple te ly local ize rad ia t ion at frequencies that fall w i t h i n the bandgap. In finite-sized crystals , these local ized defect states act as h igh qual i ty (Q) , dispersionless filters. Numerous appl icat ions for these microwave photonic crystals have been demonstrated, i nc lud ing narrow band notch filters, h igh ly direct ional , low-loss antennae, and high- t ransmiss ion I D defect waveguides that can include abrupt , right-angle bends. [4,7,11,34] T h e potent ia l impact of these crystals is even greater i n the opt ica l and near-infrared frequency doma in where they may provide the pla t form technology needed to achieve h igh ly integrated op t ica l processing circui ts . [1] Curren t op t ica l fiber communica t ion systems are based largely on discrete bu lk or fiber opt ic components (splitters, switches, etc.) that are difficult to mass-produce i n a cost-effective manner. Progress has been made towards in tegrat ing these functions on "opt ical chips" based on planar waveguide technologies. If the functions could be integrated using photonic crystals , i t w o u l d rep-resent the u l t imate i n min ia tu r i za t ion . T h i s is because l ight propagat ing i n photonic crystals w i t h complete bandgaps can, i n pr inciple , be channeled th rough and coupled CHAPTER 1. Introduction 4 between lossless defect-waveguides that contain bends w i t h effective r ad i i of curvature on the order of a single wavelength of l ight. T h e dispersion of these defect guides, and the dispersion of the background photonic crys ta l , can be tuned to achieve unique con-t ro l over the propagat ion properties of the light i n 3 D . Fur thermore , i f implemented i n an electronical ly resonant med ium (such as a I I I - V semiconductor , l ike I n P or G a A s ) , mic rocav i ty lasers can be realized that take advantage of the singular photonic densi ty of states associated w i t h isolated defect modes (effectively, microcavi t ies w i t h effective volumes less t han 0.1 cubic microns) . [35] Ph o ton i c crystals w i t h bandgaps i n the near infrared require control over the dielec-t r i c texture on lengthscales of ~200-500 n m . W h i l e some groups have realized 3 D crys ta l structures w i t h gaps i n the near infrared [3,16,39], they are very difficult to fabricate, especially when engineering defects into their structure. W h i l e 2 D photonic crystals can-not possess full bandgaps, they do exhibi t many of the at t ract ive features of their 3 D counterparts , while being easier to fabricate. A l t h o u g h the to ta l densi ty of photonic states cannot be reduced to zero i n "pure" 2D crystals (i.e. ones that are t rans la t iona l ly invariant n o r m a l to the plane of the texture), 2 D photonic crystals can exhibi t complete bandgaps i f the out-of-plane momentum of the l ight is restricted. In fact, pure 2 D crystals are i n a sense "better" than 3 D , i n that the two or thogonal po la r iza t ion states (transverse electric ( T E ) and transverse magnetic ( T M ) ) are completely independent of each other, m a k i n g i t easier to achieve large bandgaps. In practice, there is always some out-of-plane CHAPTER 1. Introduction 5 var ia t ion of the fields even i n bu lk 2D crystals, which means that the pure 2 D electro-magnet ic s imulat ions on ly approximate reality. T h e errors i n propagat ion s imulat ions are smal l i n the l i m i t of " t a l l " crystals and correspondingly large beam cross-sections, but this is not always a prac t ica l geometry outside of the laboratory. W i t h respect to l ight -mat ter interactions, pure 2 D models real ly on ly work i f the d ipole d i s t r i bu t ion is also pure ly 2 D , wh ich severely restricts the relevance of such s imulat ions . If the d ipole d i s t r ibu t ion is not t rans la t ional ly invariant , there w i l l always be some components of its r ad ia t ion directed out-of-plane, and these w i l l , i n general, couple to a finite dens i ty of photonic states. Two-d imens iona l ly textured slab waveguides are even simpler to fabricate because the texture must on ly be as th ick as the waveguide, which is t yp i ca l ly less than a few microns. T h i s geometry is also di rect ly compat ible w i t h exis t ing optoelectronic tech-nologies. W i t h this "2D + 1" geometry of textured planar waveguides, the t rans la t iona l invariance perpendicular to the textured plane is sacrificed from the outset. T h i s imme-dia te ly introduces a richness i n the electromagnetic band structure that is absent i n the pure 2 D case, yet d is t inct from the pure 3D case. There are some drawbacks w i t h this 2 D + 1 geometry: there can be no complete photonic bandgaps, even i n a restr icted sense; and it is difficult (but not impossible) to avoid in t r ins ic out-of-plane coupl ing to rad ia t ion modes. A l t h o u g h consequences such as these may seem severe, the potent ia l benefits of a re la t ively s imple fabr icat ion technology and the compa t ib i l i t y w i t h integrated opto-electronics are significant. T h i s motivates the search for a quant i ta t ive unders tanding of CHAPTER 1. Introduction 6 light propagation in structures characterized by this 2D + 1 symmetry. Many of the concepts originally motivated by the pure 3D and 2D simulations can be retained in the 2D +1 geometry, if care is taken to properly engineer the structure to minimize the influence of radiation modes and polarization mixing. In addition, the radiation coupling that is in some cases to be avoided, can in fact be used to advantage. Since the Brillouin zone is 2D, and momentum perpendicular to the layers is manifestly not a good quantum number for the electromagnetic excitations, it is possible to directly probe a significant fraction of the total band structure simply by illuminating it from the vacuum with collimated light. Furthermore, 2D textured waveguides offer a powerful medium for engineering the coupling of radiation into and out of the plane from the vacuum half space, which is desirable in certain applications. The fundamental objective of the work described in this thesis is to acquire high-quality, broadband linear optical scattering data from a variety of 2D textured planar waveguide structures in order to reveal the fundamental nature of the corresponding electromagnetic excitations. This was achieved by building a special light-scattering apparatus, and seeking the best possible quantitative agreement between the acquired data and accurate electromagnetic models developed to simulate scattering in this 2D + 1 dielectric geometry. In Chapter 5, the results of linear white light spectroscopy experiments conducted using the light-scattering apparatus are presented and rigorously compared with sim-ulations based on the Green's function formalism described in Chapter 2. First, the CHAPTER 1. Introduction 7 theoret ical and exper imental results from two basic structures are used to derive a com-prehensive unders tanding of electromagnetic excitat ions associated w i t h 2 D textured p la -nar waveguides. Sect ion 5.1.1 discusses the dispersion and polar izat ions of leaky modes i n a tex tured planar waveguide w i t h a s imple square lat t ice of air holes. [21] T h i s work represents the most thorough quant i ta t ive descr ipt ion of the low- ly ing resonant photonic bands of h igh index-contrast textured planar waveguides to date. T h i s qual i ta t ive and quant i ta t ive analysis of square lattices was extended to h igh index-contrast t r iangular latt ices, and exper imenta l ly confirmed, as described i n Sect ion 5.1.2. Chap t e r 5 also describes two examples of 2D textured planar waveguides engineered w i t h specific appl icat ions i n m i n d . T h e first example, i n Sect ion 5.2.1, addresses the use of resonant coup l ing to effect a po la r iza t ion insensitive notch filter i n low index-contrast , 2 D tex tured po lymer waveguides. T h i s section provides the first quant i ta t ive s tudy of the scat ter ing properties of these structures wh ich were fabricated us ing a direct-wr i te holographic technique i n a special azo-polymer by P a u l Rochon ' s group at the R o y a l M i l i t a r y Col lege of Canada . T h e second example, discussed i n Sect ion 5.2.2, exper imenta l ly and theoret ical ly demonstrates how a h igh index-contrast 2 D tex tured planar waveguide can be engineered to possess a band that is effectively flat a long the entire T - X direct ion of a 2 D square B r i l l o u i n zone. F l a t bands have been theoret ical ly shown to significantly enhance cer tain non-linear opt ica l conversion processes i n bu lk photonic crystals . In the final section of Chap te r 5, a novel diffraction measurement technique, developed CHAPTER 1. Introduction 8 by the author for this research, is explored v i a the character izat ion of three tex tured planar waveguides w i t h defect superlattices. T h e efficacy of this me thod for p rob ing the band structure of h igh and low dispersion modes is evaluated. It is shown how this background-free measurement technique can be used to further the quant i ta t ive analysis of textured planar waveguides. T h i s section also presents and analyzes the engineering of a local ized defect mode i n the first order gap to realize a design for an angle and po la r i za t ion insensitive notch filter. Chapter 2 Linear Light Scattering Spectroscopy of Textured Planar Waveguides This chapter introduces some of the key physics of textured planar waveguides, describes the spectroscopic techniques used to probe the band structure of these waveguides, and discusses a Green's function model used to simulate the influence of periodic 2D texturing in planar waveguides. 2 .1 Background Light traveling through a textured planar waveguide experiences dispersion, much like electrons traveling through a semiconductor crystal. Just as electrons in a semiconduc-tor experience a periodic electrical potential from each atom in the crystal lattice, the photons in a textured planar waveguide experience a periodic photonic potential. This section presents some of the key characteristics associated with the dispersion of photons traveling through a textured planar waveguide. 9 CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 10 Fi r s t , untextured waveguides are reviewed. Next , the po la r iza t ion and dispersion properties of 2 D textured waveguides are discussed for square and t r iangular lat t ices. T h e n , the effect of a defect superlat t ice on the band structure is described. 2.1.1 Untextured Planar Waveguides A planar waveguide is a structure that uses to ta l internal reflection between the core and c ladd ing mater ia l to confine light i n the plane of propagat ion. T h e phys ica l characteris-t ics of the waveguide determine which modes of light are al lowed to propagate. S o l v i n g M a x w e l l ' s equations for an a symmet r i c 1 waveguide yields independent solutions for trans-verse electric ( T E ) and transverse magnetic ( T M ) polar ized modes that are b o u n d to the waveguide. A T E mode's electric field is oriented i n the plane of propagat ion, and is thus transverse to the d i rec t ion of propagat ion, whereas a T M mode has its magnetic field oriented in-plane, transverse to the direct ion of propagat ion. P l o t t i n g the solutions to the scaler wave equat ion as a function of in-plane wavevector (K\\) on a dispersion d ia -gram produces a graph like the one shown i n F igure 2.1. Note that there are an infinite number of discrete modes, but only the two lowest modes are shown. T h e lowest energy mode is always T E . A l s o p lo t ted on the dispersion d iagram i n F igure 2.1 are three straight lines tha t rep-resent the dispersion of l ight i n air, i n bu lk core mater ia l , and in bu lk c ladd ing mater ia l . T h e slope of each of these lines is propor t iona l to the reciprocal of the index of refraction 1 A n asymmetric waveguide is one in which the cladding material on top of the core has a different index of refraction than the cladding on the bottom. CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 11 i n the corresponding mater ia l . These lines are referred to as light lines. These l ight lines d iv ide the graph into four regions. Be low the core l ight line is a forbidden region where no electromagnetic modes can exist. A b o v e the air l ight line is a region where a con t inuum of rad ia t ion modes exist. T h e modes i n this region represent rad ia t ion which is m u l t i p l y reflected but otherwise passes through the waveguide. Between the air l ight l ine and the substrate l ight line is a region where c ladding modes occur. These c l add ing modes can be thought of as passing back and forth between the c ladding mater ia l and the waveguide core, but not passing from the core into the air due to to ta l in ternal reflection. T h e final region, wh ich occurs between the substrate light line and the core l ight l ine, is where the b o u n d or "slab" modes occur. Modes i n this region experience to ta l in ternal reflection at b o t h the core-air interface and the core-cladding interface. Schematic diagrams of the electric field profile for modes i n each of these regions are shown i n this figure. W h e n a planar waveguide is per iodica l ly textured, the dispersion of guided l ight is altered, analogous to the way i n which the dispersion of free electrons is al tered by a semiconductor lat t ice. T h e next section discusses how tex tur ing changes the properties of a p lanar waveguide, and describes i n deta i l two of the many possible la t t ice configurations: square and triangle. 2.1.2 2D Simple Lattices A 2 D simple (no defects) lat t ice has a two-dimensional uni t cel l possessing a single lat-tice site. W h e n a 2 D simple lat t ice is etched into the surface of a p lanar waveguide, the CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 12 co/c > —— <— core cladding air K F igure 2.1: Schematic dispersion diagram for an untextured, a symmet r i c p lanar wave-guide. T h e two lowest energy modes are shown as bo ld lines. T h e air , substrate and c l add ing l ight lines are shown. Schematic diagrams inset i n each region indicate the electric field profile for modes i n that region. per iodic holes form a per iodic dielectric structure wh ich changes the dispers ion charac-teristics of the waveguide. Specifically, when light propagates in a p lanar waveguide w i t h a 2 D per iodic la t t ice , the light B r a g g scatters off the per iodic dielectr ic gra t ing, causing a renormal iza t ion of the slab modes. Per Bloch ' s theorem, the dispers ion of the resultant modes represented on a dispersion d iagram can be folded into the first B r i l l o u i n zone, a l lowing the use of the reduced zone scheme. A s depicted i n the schematic d i ag ram of zone-folding shown i n F igure 2.2, the band is folded back at the edge of the B r i l l o u i n zone (K\\ = Kg/2, where Kg is the grat ing vector.). W h e n the bands are zone-folded, por t ions of t hem fal l above the air l ight l ine, represented by the gray area i n the schematic. M o d e s i n this region are considered "leaky" since they have Fourier components wh ich radiate CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 13 out of the waveguide. T h e modes below the air l ight l ine are s t i l l b o u n d to the waveg-uide and have no rad ia t ing component. A t the first B r i l l o u i n zone boundaries (2D) , the normal slope of the dispersion (i.e. group velocity) goes to zero, wh ich is the result of the B r a g g scat ter ing creat ing s tanding waves and opening gaps in the al lowed frequencies. W h e n the gaps are sufficiently large, it is possible for a smaller gap to exist w i t h i n some range of frequencies everywhere on the first B r i l l o u i n zone boundary. In th is case there is a "complete pseudo-gap" for that par t icular set of renormal ized slab modes. F igure 2.2: Schematic of I D zone-folding. T h e size of the gaps at zone edge, and whether or not a gap is complete, depend upon the gra t ing materials and lat t ice configuration of a 2 D tex tured p lanar waveguide. T h e size of a gap at zone edge is determined by the difference i n energy between the two s tand ing wave solutions which occur at this point . One s tand ing wave has nodes located i n the dielectr ic , while the other has nodes located i n the holes wh ich comprise CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 14 the grat ing. T h e higher the index-contrast is between the gra t ing layer and the mate r i a l compr i s ing the gra t ing (air i n this case), the larger the gap w i l l be. [12] T h e size of the gap is also affected by the fi l l ing f rac t ion 2 of the grat ing. Smaller first order gaps occur when the air f i l l ing fraction is near zero or one; the first order gaps become p ropor t iona l ly larger as the filling fraction approaches 50%. T h e more the gra t ing per turbs the modes, the larger the gaps w i l l be. Whe the r or not a gap is a complete pseudo-gap is related to the size of the gaps at the zone boundaries and the symmet ry of the la t t ice configurat ion. T h e more symmet r ic the lat t ice, and the larger the gap, the more l ike ly i t is tha t there w i l l be a complete pseudo-gap for a l l directions. W h e n a plane wave is incident on the surface of a textured planar waveguide w i t h a well-defined in-plane wavevector, Ki, the incident field can be wr i t t en as Ei(Ki\z) = E0e-iw°zeiR^ (2.1) where w0 = yjuj2 — Kf, Co = LO/C, and p = xx + yy. [5] T h e incident plane wave m a y have any pola r iza t ion . If the gra t ing has a well-defined 2 D periodic texture represented by reciprocal la t t ice vectors, { G T O } , the electric field i n the gra t ing can be convenient ly represented as [5] E(p, z) = E E(Ki + Gm; z ) e ^ + ^ . (2.2) These plane waves, w i t h well-defined frequency and in-plane wavevector, u,K\\, are useful to determine the dispersion of the leaky modes at tached to textured waveguides, 2 Fi l l ing fraction is defined as the area of the grating material in a unit cell divided by the area of the unit cell. CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 15 bo th exper imenta l ly and through model l ing . For some general w,K\\, the non-specular (G 7^  0) field components i n E q u a t i o n 2.2 w i l l be smal l . However, when u>,K\\ coincides w i t h one of the al lowed leaky eigenstates of the 2 D textured waveguide, one or more of the scattered field components i n E q u a t i o n 2.2 becomes large. For the low- ly ing bands s tudied throughout this thesis, the general dispersion and po la r iza t ion properties can often be described using a smal l sub-set of the Fourier components of E q u a t i o n 2.2. It is usual ly sufficient to consider the zeroth order (specular) component , and one or two sets of nearest neighbors. T h e zeroth order component is usual ly smal l , but is essential because it is the one responsible for coupl ing l ight into and out of the guided modes of the waveguide. T h e pure k inemat ic effects of 2 D texture on the dispersion and po la r i za t ion propert ies of the leaky modes are presented i n this section, for bo th square and t r iangular la t t ice types. O n l y s-polarized scattered fields are considered i n E q u a t i o n 2.2. Fo r weak texture, and under resonant exci ta t ion condit ions, these basical ly represent T E - p o l a r i z e d slab modes. M o r e generally, the scattered components in a 2 D textured planar waveguide are b o t h s- and p-polar ized, corresponding respectively and approximate ly to T E and T M polar ized slab modes. However, for many of the GaAs-based structures examined here, the T E - T M separation is large enough to effectively decouple the two. W h e n a 2 D scat ter ing potent ia l is weak, the m a i n effect the gra t ing has on the ex-tended 2 D dispersion of the waveguide is to zone-fold the slab modes in to the first (2D) B r i l l o u i n zone, as i l lus t ra ted i n F igure 2.3. T h e general characteristics and shape of the CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 16 dispersion i n the l i m i t of weak tex tur ing can be unders tood by considering that the m a i n effect of the gra t ing is then merely to introduce new in-plane m o m e n t u m components to the incident field, thus affording i t the oppor tun i ty to excite what are effectively pure T E slab modes w i t h wavevectors given by Ki + {Gm}. M X Figure 2.3: Schematic dispersion d iagram for a t h i n planar waveguide tex tured w i t h a 2D square lat t ice. T h e 's' and 'p ' designations refer to polar izat ions of the photonic eigenstates, as discussed i n the text. F igure 2.4 is a momen tum space d iagram of the nine dominant Fourier components of the dielectr ic scat ter ing potent ia l for a s imple square lat t ice. T h e reciprocal- la t t ice vectors of a square lat t ice are given by G = %Kgx + jKgy where Kg = 1 / A , i and j are posi t ive integers inc lud ing zero. F igure 2.4(a) corresponds to the incident plane-wavevector, Ki, a l igned along the X symmet ry direct ion. F igure 2.4(b) corresponds to the incident plane-wavevector, Ki, aligned along the M symmet ry di rect ion. For b o t h of these figures, the central point represents the zeroth order component of the scat ter ing CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 17 potent ia l , and Kj is represented by the smal l dot ted vectors. T h e in-plane wavevectors of the T E field components and their corresponding polar izat ions are represented by the sol id arrows and gray arrows respectively. 3 (a) (b) F igure 2.4: M o m e n t u m space d iagram for a square lat t ice. F igure (a) represents incident rad ia t ion along the X symmet ry direct ion. Figure (b) represents incident r ad ia t ion a long the M symmet ry d i rec t ion . T h e dots represent the Fourier components of the dielectr ic scat ter ing potent ia l . T h e smal l dot ted vectors represent the in-plane wavevector of the incident rad ia t ion . T h e large vectors represent the in-plane field vectors wh ich result when the scat ter ing potent ia l adds momentum to the incident field. T h e po la r i za t ion of each field component is represented by the smal l gray vectors. If the (isotropic) dispersion of the under ly ing T E slab modes is denoted by u)(K\\), the zone-folded dispersion expected for a very weak 2 D texture w i t h square s y m m e t r y can be unders tood as follows. For a given incident wavevector Ki, the Four ie r components of the dielectr ic texture scatter the incident field into a discrete set of scat tered states w i t h i n plane wavevectors Kt + Gm. W h e n the incident frequency corresponds to one of the —* —* slab mode frequencies at these in-plane wavevectors, cj(\Ki + Gm\), the scat ter ing w i l l be 3 A similar diagram can be made for the T M field components and their corresponding polarization vectors. CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 18 resonantly enhanced, corresponding to the exci ta t ion of one of the local ized exci ta t ions of the 2 D tex tured slab. For the four smallest reciprocal lat t ice vectors w i t h magni tude Kg, these resonantly excited slab modes disperse w i t h Kiy as i l lus t ra ted schemat ica l ly i n F igu re 2.3. In par t icular the mode at u>(\KiX + Kgx\) increases monoton ica l ly as Ki increases a long the X direct ion. T h i s dispersive mode corresponds to the highest band along the X di rect ion i n F igure 2.3. T h e mode exci ted at uj(\KiX — Kgx\) disperses down i n energy as Ki increases i n the X direct ion, whi le the modes at ui{\KiX ± Kgy\) disperse upward , but at a moderate rate compared to the other two modes i l lus t ra ted i n F igu re 2.3. T h e po la r iza t ion labels on the bands shown i n F igure 2.3 can be unders tood us ing the fol lowing symmet ry argument. T h e polar iza t ion (s, p or e l l ip t ical) of the rad ia t ion reflected from the surface for a given incident plane wave at Ki is determined by the Fourier component of the polar iza t ion density i n the gra t ing at the same Kt. F r o m the theory described below (Section 2.2), for a t h in textured grat ing, this component of the po la r i za t ion densi ty i n the gra t ing can be expressed as [5] P(KA = x(Gm)E(Ki - Gm) (2.3) m Under resonant exci ta t ion condit ions, one or two Fourier components of the field cor-responding to the resonantly exci ted T E slab modes w i l l dominate the scattered field everywhere except at Ki=0. For the upward dispersing mode i n F igure 2.3, this cor-responds to a T E slab mode at Kt + Kgx, wh ich is polar ized i n the y d i rec t ion. F r o m E q u a t i o n 2.3 above, the po la r iza t ion density at Ki is then polar ized i n the y d i rec t ion , CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 19 thus generating an s-polarized rad ia t ion field i n the upper half space. T h e same is true for the r ap id ly downward dispersing branch i n F igure 2.3. Th ings are s l ight ly more com-pl ica ted for the moderate ly dispersing branches because from symmetry , the incident field must excite either symmet r ic or an t i symmetr ic combinat ions of these slab modes (degenerate i n the absence of texture) at Ki ± K9y. F r o m the po la r iza t ion vectors shown i n F igure 2.4, a symmet r ic superposi t ion of these scattered slab modes w i l l add as shown i n E q u a t i o n 2.3 to y i e ld a polar iza t ion density at i ? , oriented along the y d i rec t ion , corresponding again to s-polarized rad ia t ion i n the upper hal f space. However, the ant i -symmet r i c combina t ion leads to a po lar iza t ion density oriented paral le l to Ki} w h i c h can on ly lead to p-polar ized rad ia t ion i n the upper half space, hence the po la r i za t ion labels i n F igure 2.3. It follows that there are 2 s- and 2 p-polar ized modes, one each dispers ing up and down, when Kt is oriented along the M direct ion. If Ki does not lie i n a plane w i t h mi r ro r symmetry , the modes are, i n general, e l l ip t ica l ly polar ized, and cannot be label led as pure s- or p-polar ized. A n analysis s imi la r to the one just presented can be done to determine the dispers ion and po la r iza t ion characteristics of the modes at tached to a planar waveguide tex tured w i t h a t r iangular lat t ice. M o m e n t u m space diagrams for a t r iangular la t t ice are shown i n F igu re 2.5. F igure 2.5(a) shows an incident wavevector al igned along the M symmet ry d i rec t ion whi le F igure 2.5(b) shows an incident wavevector al igned along the K symmet ry di rect ion. Since the group of "nearest neighbors" consists of s ix Fourier components , there are six bands m a k i n g up the second order T E - l i k e gap. T h e dispersion and polar iza t ions CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 20 of these s ix bands are shown in F igure 2.6. In the M di rec t ion there are four s-polarized bands: two dispers ing up i n energy and two dispersing down i n energy. In add i t i on there are two p-polar ized bands: one dispersing up i n energy and one d ispers ing d o w n i n energy. In the K d i rec t ion there are two s- and two p-polar ized bands w h i c h disperse up i n energy, and one s- and one p-polar ized bands which disperse down i n energy. M (a) (b) Figure 2.5: M o m e n t u m space d iagram for a t r iangular lat t ice. F igu re (a) represents i n c i -dent rad ia t ion along the M symmet ry direct ion. F igure (b) represents incident r ad ia t ion along the K symmet ry direct ion. T h e dots represent the Fourier components of the d i -electric scat ter ing potent ia l . T h e smal l dotted vectors represent the in-plane wavevector of the incident rad ia t ion . T h e large vectors represent the in-plane field vectors w h i c h re-sult when the scat ter ing potent ia l adds momentum to the incident field. T h e po la r i za t ion of each field component are represented by the smal l gray vectors. One of the significant findings of this thesis work was that the dispers ion and po-la r iza t ion properties of the low- ly ing leaky bands i n strongly t ex tured waveguides can largely be interpreted by combin ing this k inemat ic picture w i t h non-per turbat ive , s t rong coupl ing of these basic bands near zone boundaries, and near anticrossings that m a y occur anywhere i n the first B r i l l o u i n zone. CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 21 s,p s,p s,p s s,p s,p Figure 2.6: Schematic dispersion d iagram for a t h i n planar waveguide tex tured w i t h a 2 D t r iangular lat t ice. 2.1.3 Defect Superlattices A s discussed previously, textured planar waveguides made w i t h gratings of s imple lat t ices can affect the propagat ion of photons, which is ak in to the atoms i n the c rys ta l la t t ice of a semiconductor affecting the propagat ion of electrons. C o n t i n u i n g the analogy to semi-conductors , add ing defects to the lat t ice i n a planar waveguide can al low modes to occur i n the photonic bandgap, which is s imi lar to adding impuri t ies to a semiconductor that give rise to states w i t h i n the electronic bandgap. T h i s section discusses the incorpora t ion of a defect superlat t ice into a textured planar waveguide. W h e n a defect superlat t ice is incorporated into a textured planar waveguide, the uni t cel l for the la t t ice increases to include the defect. For example, for a defect superlat t ice i n a I D gra t ing that leaves out every fifth lat t ice point , the uni t cell wou ld be five t imes 0 CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 22 larger than the uni t cell of the "base" latt ice. T h e reciprocal la t t ice vectors for the defect superlat t ice are 1/5 the length of that of the base lat t ice (Kg — \Kg). A d d i n g the defect decreases the B r i l l o u i n zone to 1/5 the size that it is for the non-defect la t t ice, resul t ing i n more frequent zone-folds. F igure 2.7 i l lustrates these changes by super impos ing a dispersion d iagram for a defect superlatt ice on a dispersion d iagram for a non-defect lat t ice: the sol id lines represent the band structure for a non-defect base lat t ice; the dashed lines represent the band structure for the ident ical lat t ice, w i t h the except ion that a defect superlat t ice has been incorporated. T h i s figure shows the decrease i n size of the B r i l l o u i n zone and the increase i n zone-folds. F igure 2.8 shows the dispersion for a waveguide w i t h a defect superlatt ice superimposed on a dispersion d iagram for a base lat t ice that produces a much larger pseudo-gap than i n the previous example. In this case, the incorpora t ion of a defect superlat t ice does more than merely increase the zone-folding: i t introduces a mode into this large pseudo-gap. W h i l e these i l lus t ra t ions are for I D tex tured planar waveguides, incorpora t ing a defect superlat t ice i n a 2 D tex tured planar waveguide produces qual i ta t ive ly s imi lar results, as discussed below i n Sect ion 5.3.3. W h e n a 2 D textured planar waveguide w i t h a defect superlat t ice produces a mode i n the first order pseudo-gap, only light of that frequency is al lowed to propagate whi le a l l other frequencies of l ight i n the gap are forbidden to propagate. T h i s effect can al low for the loca l iza t ion and channel ing of l ight i n the waveguide [14,40], m a k i n g i t useful for many appl icat ions. Lasers have been designed and demonstrated using loca l iza t ion of l ight i n op t ica l microcavi t ies located at defects i n photonic crys ta l latt ices. [22] A d d i t i o n a l l y , CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 23 °'1 *VK°g 0.5 Figure 2.7: Band structure for a thin GaAs planar waveguide with ID defect superlattice of air "holes." The air filling fraction of the texturing is 2.5%. The solid lines show the dispersion for this structure without the superlattice, while the dashed ones represent the modified dispersion of this structure when every fifth "hole" is omitted. analysis has been done on channeling light via ID arrays of defect sites in a photonic crystal lattice. [2] This thesis seeks to quantitatively characterize the electromagnetic excitations at-tached to textured planar waveguides with and without defect superlattices. A rigorous computer model and white light spectroscopy are used in conjunction to study the elec-tromagnetic response of these structures to radiation incident from the top half space. The computer code and the spectroscopic techniques used are described in the follow-ing sections. CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 24 0.1 K,|/K°g 0.5 Figure 2.8: Band structure for a thin GaAs planar waveguide with a I D defect superlattice of air "holes". The air filling fraction of the texturing is 25%. The solid lines show the dispersion for this structure without the superlattice, while the dashed ones represent the modified dispersion of this structure when every fifth "hole" is omitted. K° is the grating vector of the "base" lattice. Notice that in this case the superlattice does more than simply zone-fold the base lattice modes: it introduces defect modes in both the first- and second-order pseudo-gaps. 2.2 Green's Function Model In order to explore the photonic band structure associated with textured planar wave-guides, a computer code was used to model the waveguide's response to electromagnetic radiation. The results of this computer code have been validated by comparison with other models [21] as well as by comparison with experimental results [20]. This code mathematically models the electric fields associated with textured planar waveguides CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 25 by implement ing a Green 's function technique to solve M a x w e l l ' s equations. For the mode l ing of defect superlattices reported i n this thesis, modif icat ions were made to the code wh ich are discussed below. T h e computer code is designed to accurately mode l the scat ter ing of l ight incident on a tex tured planar waveguide, such as the one depicted schemat ica l ly i n F igu re 2.9. T h e waveguide depicted has four layers: substrate, lower c ladding , core and upper c ladd ing . T h e code has the capab i l i ty to mode l any number of layers, i nc lud ing semi-infinite upper and lower layers. T h e code is also capable of model ing any angle of incident l ight (6 and (j>) on the surface of the waveguide. upper cladding (air) core Figure 2.9: Schematic d iagram of a textured planar waveguide. T h e code implements a solut ion of Maxwel l ' s equations i n the waveguide geometry by t a k i n g the per iodic po lar iza t ion density in the textured region as the d r i v i n g t e rm for a Green 's funct ion. [5] T h e self-consistent solut ion for a single Four ie r component of the CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 26 field i n the gra t ing is given by E(u,Kn,z) = Ehom(L>,Kn,z) + Jdz'g (Kn,u,z,z') £ * i m (-u,u>)E(u),Km,z') (2.4) 771 when a plane wave w i t h frequency u = ui/c and in-plane wavevector Ki, is incident from the upper hal f space, and where Km = Ki — Gm is the in-plane wavevector of the Four ier components of the field, and Gm are the reciprocal lat t ice vectors. [5] T h e per iodic sus-cep t ib i l i ty is represented as a Fourier series w i t h coefficients Xnm that couple the various Fourier components of the scattered field. T h e infinite set of vector integral equations imp l i ed i n E q u a t i o n 2.4 is reduced to 3 N scaler algebraic equations by t runca t ing the Four ier series descr ibing the in-plane components of the field at N . T h i s also requires that the textured region is sufficiently t h i n so that the fields can be taken as constant over its extent, thus e l imina t ing the integral over dz' in E q u a t i o n 2.4. Gra t ings that are too thick for this assumpt ion can be modeled by d i v i d i n g them into thinner regions that each satisfy the constant field approximat ion . T h i s technique is described i n de ta i l for a single layer s t ructure i n [5]. In the or ig inal code, the reciprocal lat t ice vectors as wel l as the Fourier coefficients were hand-coded for each lat t ice investigated. T h i s was sufficient when mode l ing a s imple la t t ice (i.e. one that has a single lat t ice point i n the uni t cel l) , wh ich typ ica l ly requires fewer t han ten Fourier coefficients to proper ly mode l the low- ly ing bands. T h e calculat ions for defect superlattices can require more than 200 Fourier coefficients, CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 27 m a k i n g hand-coding imprac t i ca l and inefficient. T o facili tate s imulat ions of defect su-perlatt ices, subroutines were wr i t t en which calculate the reciprocal la t t ice vectors and Fourier coefficients. T h e Fourier coefficients wh ich are used by the code to represent the per iodic po la r iza t ion i n the gra t ing layer are normal ized to range between 0 and 1, where 0 represents no hole and 1 represents a hole. For a s imple lat t ice described by !(7f) = YJCnmei3^ (2.5) nm the Fourier coefficients, Cnm, are given by Cnm^^—f dxy/b2 - x2 cos(Gnmx) = -^—^—Ji{bGnm) (2.6) where Gnm = Go y/n2 + m2 — 2nm cos(7r — a) (2.7) and where b is the radius of the holes, n and m are the base lat t ice uni t vectors and a is the angle between them. Gnm is the reciprocal lat t ice vector. T o generate a Fourier series for a defect superlat t ice, first the above equations are used to generate a Fourier series wh ich describes the "base" simple (non-defect) lat t ice, (a), shown i n a real-space plot i n F igure 2.10. Next , a Fourier series is generated wh ich represents the loca t ion and size of those holes to be omi t t ed to form the defect, (b). F i n a l l y , the Fourier series for (b) is subtracted from the Fourier series for (a) p roduc ing the Fourier series for the defect superlatt ice (c), that for this example omits every fifth hole i n the base lat t ice pat tern. Opt iona l ly , an addi t iona l Fourier series can be generated CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 28 in order to add holes of a selected radius into the "empty" superlattice spaces, resulting in a defect comprised of holes which are smaller or larger than those of the base lattice, rather than a defect comprised of omitted holes. • • • * ( to ai # % • • • • • < » • • • • * • • • • < i » • • * • « • • • * I ^ i • A • i • • • • to \ m i 1 • V 1 1 IP • • • • * < i • m m m • • • • • < t W% 'fft • • • • m n • • • 1 • • • • » • « ( a ) (b) • l l l l l l l l :H 0 n $1 • • « » ( 0 Figure 2.10: Graphical representation of Fourier coefficient generation technique, (a) represents the base lattice, (b) represents the defect location, (c) is the resulting defect superlattice. (c)=(a)-(b). 2.3 Spectroscopy In conjunction with the Green's function code, white light spectroscopy is used to charac-terize the textured planar waveguides. White light spectroscopy consists of illuminating the waveguide with a broadband source at a well-defined angle of incidence, and using a spectrometer to analyze the scattering spectrum to infer the band structure. For the re-search presented here, two types of spectroscopic technique are used: a well-documented specular measurement technique, and a novel diffraction measurement technique devel-oped by the author. CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 29 2.3.1 Specular Measurement Technique T o characterize the textured planar waveguides for this thesis, the band s t ructure was mapped and i n some cases the lifetimes were measured. A specular measurement tech-nique was used to accomplish this for the waveguides w i t h 2 D simple latt ices. T h i s technique involves sh in ing white l ight on the surface of a textured planar waveguide at a well-defined angle of incidence and then ana lyz ing the spectra of the specular ly reflected l ight . T h i s technique provides a straight-forward means for m a p p i n g the leaky photonic modes of tex tured planar waveguides. F igure 2.11: Schematic dispersion d iagram for a I D textured waveguide. B o l d lines indicate locat ions probed w i t h the specular measurement technique. W h e n whi te l ight is incident on the surface of a textured planar waveguide at a well-defined angle of incidence, each energy, oo, of the incident l ight corresponds to a specific in-plane wavevector, K\\, according to the relat ion K\\ = to sin.0. Genera l i z ing the k inemat ic arguments given above, coupl ing into a guided mode eigenstate w i l l occur t CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 30 whenever K\\±nK9 stimulates a mode. Here n is any integer and Kg is the grating vector. Figure 2.11 shows a zone-folded schematic diagram of the dispersion for a waveguide with a ID textured layer clad by air on top and bottom. The dotted line represents the air light line. Any modes which are above the air light line (when folded into the first Brillouin zone) are accessible with this specular measurement technique. The solid lines represent the parameter space probed with this technique for a specific angle of incidence. At each point where this line intersects a band, a feature is detected in the specular reflectivity spectrum. A simulated example of the spectra that can be obtained with this technique is shown in Figure 2.12. The narrow features are Fano-resonances which are a result of coupling into guided modes of the waveguide. A mathematical fitting technique can be used to extract the modes' positions and lifetimes from this type of spectra. [21] By recording and analyzing the spectra for multiple angles of incidence, the band structure of the waveguide's photonic modes can be mapped and the lifetimes can be measured. 2.3.2 Diffraction Measurement Technique The specular measurement technique works well for characterizing the band structure of many textured planar waveguides, but the mathematical fitting technique used to extract the mode energy and linewidth from the specular background is not well-suited for use when the modes are closely spaced. If the modes could be observed free of the specular background, the characteristics could be analyzed without complicated mathematical fitting. The author developed a background-free probe technique based on the light CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 31 8.0 8.4 8.8 9.2 9.6 10.0 10.8xl03 Energy (cm *) Figure 2.12: Representative reflectivity spectra from a textured planar waveguide w i t h a square la t t ice of holes. So l id lines represent s-polarizat ion; dashed lines represent p-po la r iza t ion . T h e narrow features are Fano-resonances which are a result of coup l ing into guided modes of the waveguide. diffracted, rather than reflected, from samples conta ining a superlat t ice of defects. T h i s diffraction measurement technique is described below. T o implement this new diffraction measurement technique, l ight at a par t icu la r angle of incidence st imulates a guided mode by adding an integer mul t ip le of defect la t t ice gra t ing vectors to the paral le l component of the incident l ight . T h i s is represented by the equat ion K = K\\ ± nK®. T h e light is scattered out of the waveguide by add ing or sub t rac t ing an integer mul t ip le of defect lat t ice gra t ing vectors from the guided mode. W h i l e the specular technique uses the same number of gra t ing vectors to s t imulate the mode as to scatter out of the mode, the diffraction technique uses an unequal number . T h i s diffraction measurement technique is shown schematical ly i n F igure 2.13. T h e parameter space probed by the incident beam is represented by the bo ld line. T h i s CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 32 schematic corresponds to a I D lat t ice w i t h a defect located at every fifth site. In this case the defect la t t ice gra t ing vector is 1/5 the size of the base la t t ice gra t ing vector. C o u p l i n g to the guided mode results by adding or subt rac t ing 5 defect lattice g ra t ing vectors. T h e l ight now scatters out of the waveguide not on ly i n the specular d i rec t ion , but also i n any of the directions associated w i t h the lines paral le l to the bo ld l ine labeled '0 ' on F igure 2.13. T h e locat ion of the -K® diffracted order is ind ica ted on the d i ag ram F igure 2.13: Schematic dispersion for the diffraction p rob ing method . T h e d i rec t ion of the specular reflection is labeled 0, while the d i rec t ion of the -K^ diffracted order is labeled -1 . Theoret ica l ly , the diffraction technique should al low observation of modes above the l ight l ine free from specular background. Significantly, it cou ld also a l low for the v i ewing of modes from the base lat t ice wh ich are below the light line. Conven t iona l methods for by a circle. -1 0 CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 33 prob ing modes below the l ight line involve elaborate techniques for get t ing l ight into and out of the waveguide. T h e diffraction measurement technique offers the potent ia l to map these modes by s imp ly i l l umina t i ng them from the upper half space, a l lowing analysis of the scattered l ight which reveals the band structure of the bound modes. T o investigate whether the diffraction measurement technique has these capabi l i t ies , the Green 's funct ion code was used to simulate light diffracting from a tex tured planar waveguide w i t h a defect superlatt ice. T h e s imula t ion was done for a free-standing G a A s waveguide, 80 n m thick, w i t h a square lat t ice of circular holes completely penet ra t ing the core, spaced 400 n m apart w i t h a radius of 110 n m . T h e defect superlat t ice was created by o m i t t i n g every fifth hole i n the x and y directions. F igure 2.14 shows a s imula t ion of the first order diffraction which results when broadband light is incident no rma l to the surface of this waveguide. It is readily apparent that the modes i n this spectra are free of the specular background. It is not obvious, however, where these modes originate. If these modes originate below the l ight line of the base latt ice, then i t should be possible to reproduce their energies by look ing at the dispersion of the defect-free base lat t ice, as discussed below. A m o m e n t u m space d iagram for the waveguide conta ining the superlat t ice (Figure 2.15) i l lustrates wh ich specific in-plane wavevectors occur w i t h i n the reduced zone of the base lat t ice. In this d iagram, the large gray dots represent the rec iprocal lat t ice vectors of the base (non-defect) lat t ice. T h e square identifies the first B r i l l o u i n zone of this base lat t ice, and the tr iangle delineates the reduced zone. T h e smal l gray dots represent the CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 34 rec iprocal la t t ice vectors that are in t roduced by the defect superlat t ice. T h e da rk dots w i t h the smal l squares around them represent reciprocal lat t ice vectors from the defect la t t ice occur r ing w i t h i n the reduced zone of the base lat t ice. W h e n l ight is incident no rma l to the surface of the waveguide, the in-plane wavevector of the incident r ad ia t ion is zero, so the in-plane wavevectors of the field scattered by the defect superlat t ice are just given by the reciprocal lat t ice vectors of the defect superlatt ice. F igu re 2.15 shows that there are five dis t inct in-plane wavevectors (dark dots) where the defect superlat t ice samples the first B r i l l o u i n zone of the base latt ice. F igu re 2.16 shows the calculated specular reflectivity spectra from the defect-free base lat t ice at these five wavevectors. Poles i n these spectra represent true bound modes that lie below the l ight l ine i n the base lat t ice band structure. Labels i n Figures 2.14 and 2.16 relate the peaks observed in the background-free diffraction spectra from the defect superlat t ice sample w i t h the poles observed i n the reflectivity spectra at the relevant i n -plane wavevectors of the base lat t ice. T h i s confirms that , theoretically, the use of defect superlatt ices does offer the ab i l i ty to probe the dispersion of true b o u n d modes that exist below the l ight l ine i n the defect-free B r i l l o u i n zone. T h i s diffraction measurement technique can be used whenever the energy of the mode to be detected radiates a diffracted signal above the light l ine. A n y lat t ice that has per iodic spacing longer than the wavelength of the probe l ight w i l l have at least one diffracted order. In waveguides where a diffracted signal does not exist above the l ight l ine, a defect superlat t ice can be incorporated to create one. Tex tu red planar waveguides CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 35 0.20 0.15 go.io 0.05 -\ 0.00 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 Energy (cm ') Figure 2.14: Simulated s-polarized K® diffraction spectra from a waveguide with a defect superlattice with a pitch 5x the pitch of the base lattice. Base lattice pitch is 400 nm. Incident light is normal to the surface of the waveguide. Note: The feature labeled ' A ' does not diffract above the surface of the waveguide. •X o o o O 0 O O o Figure 2.15: Momentum space diagram for a square defect lattice where every fifth lattice site contains a defect. Large dots indicate lattice sites related to the "base" lattice. Dots with small squares indicate defect lattice sites that lie within an irreducible portion of the base lattice reduced zone. with defect superlattices are used to evaluate this diffraction method in Chapter 5. CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 36 Figure 2.16: S imula ted s-polarized spectra for the defect-free lat t ice described i n the text . T h e notat ions along the right side indicate the in-plane wavevectors for each spectra. Poles indicate bound modes that exist below the air l ight l ine. Chapter 3 Experimental Design T h i s chapter describes an apparatus designed and bui l t to implement the l inear l ight scat ter ing techniques described i n Chap te r 2. T h e apparatus makes i t possible to char-acterize leaky photonic modes of textured planar waveguides. Its function is to b r ing whi te l ight incident on a textured planar waveguide at a well-defined angle and to col-lect, cond i t ion and focus the l ight scattered from the waveguide into a spectrometer for analysis. T h e basic design consists of a fiber opt ic whi te l ight source, three el l ipsoidal mi r -rors, and a concentric ro ta t ion stage. Reflective optics were chosen throughout for their ab i l i t y to avoid the chromat ic aberrations inherent i n refractive optics. T h e apparatus was designed specifically to work w i t h a B o m e m D A 8 Fourier Transform Interferometer ( F T I R ) . T h e key challenge i n the design was to ensure achromatic performance over the range of 6,000 - 13,000 c m - 1 ( ~ 0 . 8 n m - 1.7/im) whi le loca l iz ing the scat ter ing area to ~ 1 0 0 u m x 100 u m , the typ ica l size of the textured planar waveguides s tudied here. For the overall design of the apparatus, ease of use was a p r imary objective. Samples can be mounted and aligned qu ick ly and characterized easily. 37 CHAPTER 3. Experimental Design 38 3.1 Apparatus T h e pr incip le components of the experimental apparatus are a whi te l ight source, focusing optics, pos i t ion ing mechanics, and a spectrometer. A schematic of the op t ica l layout is shown i n F igure 3.1. Unpo la r i zed whi te l ight is delivered v i a a fiber opt ic cable fixed to a large ro ta t ing r ing which is concentric w i t h a smal l ro ta t ion stage that holds a textured planar waveguide sample. T h e whi te l ight emi t ted from the output facet of the fiber is imaged onto the sample by an el l ipsoidal m i r ro r ( E M I ) . T h e l ight s t r i k i n g the sample is reflected, t ransmit ted , and /o r diffracted. One of these emissions is collected by a second el l ipsoidal mi r ror ( E M 2 ) and subsequently refocused to an image plane where an adjustable field stop is used i n conjunct ion w i t h a removable C C D camera to e l iminate extraneous l ight . T h e l ight then passes through a l inear polar izer where the s-or p -po la r iza t ion is allowed to continue to a t h i rd el l ipsoidal mi r ro r ( E M 3 ) . T h i s f inal e l l ipsoidal mi r ro r focuses the l ight into the B o m e m F T I R for spectral analysis. A detai led descr ipt ion of the exper imental subsystems follows. 3.1.1 Light Source A fiber-coupled l ight source was chosen for this experiment to facil i tate easy pos i t ion ing of the incident l ight . T h e textured region of the planar waveguide is t yp i ca l ly on ly 90 pm x 90 pm, therefore the diameter of the l ight on the sample was chosen to be 200 pm to overfill the pa t te rn slightly. T o achieve the 200 pm diameter spot, the output facet of a 100 pm diameter core opt ica l fiber (opt imized for N I R transmission) is imaged by CHAPTER 3. Experimental Design 39 Bomem FTIR EM3 2 cm Figure 3.1: Schematic of the experimental apparatus used for op t i ca l character iza t ion of textured planar waveguides. E M I , w h i c h was designed to provide a 2x magnif icat ion. A 100 wat t tungsten quar tz halogen bu lb housed i n a radiometr ic fiber optic source (Or ie l 77501) was chosen for the l ight source. T h e electronics in this l ight stabilize the output to be that of a 3200° K e l v i n b lackbody. T h e broadband light emit ted from the output facet of the 100 pm d iameter fiber was measured to be approximate ly 1 m W a t t between 400 n m and 1.1 pm. CHAPTER 3. Experimental Design 40 3.1.2 Optics T h e funct ion of the optics i n this apparatus is to efficiently focus whi te l ight onto a sample, collect l ight emit ted from the sample, condi t ion the l ight and focus it into the F T I R . T h e design of the optics was constrained by several factors: the parameters of the exis t ing F T I R and the opt ica l fiber chosen to deliver the l ight , the necessity to create an enlarged image for use w i t h a field stop, and the decision to use whi te l ight . Specifically, the l ight entering the F T I R had to have a full angle divergence of ~ 1 4 ° ( F / 4 ) to proper ly fi l l its internal optics. Secondly, the optics had to accommodate the 100 pm diameter core opt ica l fiber. T h i r d l y , the optics had to create a lOx image, wh ich is of adequate size for use w i t h the field stop. F ina l ly , the choice to use whi te l ight necessitated the use of reflective optics throughout the apparatus i n order to avoid the chromat ic aberrations inherent i n refractive optics. T h e function of the optics cou ld be achieved and the constraints cou ld be accommodated by us ing either e l l ipsoidal mir rors or parabolo ida l mirrors . E l l i p s o i d a l mirrors were chosen over paraboloidal mirrors for their ab i l i ty to provide point- to-point focusing using a single mirror . A single e l l ipsoidal mi r ro r focuses l ight coming from one focus to the other focus, whereas a parabolo ida l mi r ro r collects l ight from its focus and coll imates i t . A second paraboloidal mi r ro r must be used to take the co l l imated l ight and focus it to form an image. Since some l ight is lost u p o n reflection, a design w i t h fewer mirrors is preferable when m a x i m u m opt ica l throughput is required. CHAPTER 3. Experimental Design 41 Off-the-shelf e l l ipsoidal mirrors are not generally available, so mirrors were cus tom de-signed and fabricated for this experiment. T h e following describes how these mir rors were designed. T h e fol lowing discussion of the mi r ro r design refers to the ellipse p ic tured i n F igu re 3.2. T h e three e l l ipsoidal mirrors were designed to function interdependently (i.e. a change i n a parameter of one mi r ro r necessitated dependent changes i n the others). T h e mir rors were designed as e l l ipsoidal mirrors where the central cross-section is a s imple ellipse. T o define the ellipse, the i n i t i a l considerat ion was the desired magnif icat ion produced by the mir ror . Ma themat i ca l ly , the magnif icat ion of a s imple lens or mi r ro r is determined by the equat ion Mt = Si/SD [9] where Mt is the transverse magnif icat ion, and Si and S0 are the image and object distances respectively. For these mirrors the magnif ica t ion is given by M ( = r 2 / r i . A specific value of either ?*i or r2 was selected based on the desired phys ica l dimensions of the apparatus. A d d i n g r\ and r2 gives the major axis of the ellipse (26 = r\ + r2). Nex t the eccentricity of the ellipse must be determined. T o do this, the angle between r% and r 2 must be selected; to facili tate al ignment , 90° was selected for a l l three mirrors . F ina l ly , i n order to specify the phys ica l size of the mir rors , the angles Q\ and 92 must be selected. For this experiment a smal l cone angle of l ight i l l u m i n a t i n g the sample was desired i n order to approximate co l l imated l ight . See Tab le 3.1 for the specifications of each mir ror . Note the bo ld values indicate the i n i t i a l design constraints from which a l l other values were calculated. In add i t ion to the specifications delineated i n Table 3.1, there was an add i t iona l design CHAPTER 3. Experimental Design 42 2b Figure 3.2: Schematic d iagram of el l ipsoidal mir ror design. L i g h t emi t t i ng from a point source at one focus w i l l be imaged at the other focus. requirement for mi r ro r E M I . A por t ion of the mi r ro r b lank was removed to a l low mi r ro r E M 2 to be placed i n close p rox imi ty to mi r ro r E M I , i n order to enable v iewing of the specular reflection as close to normal incidence as possible (see F igure 3.3). M i r r o r s E M I , E M 2 , and E M 3 were fabricated out of a l u m i n u m by Lumonics Corpo ra t ion . T h e fol lowing section discusses the components for pos i t ioning the mirrors and sample. Table 3.1: E l l i p s o i d a l mi r ro r specifications. Note: B o l d values indicate design constraints. E M I E M 2 E M 3 r i (cm) 5 15 150 r 2 (cm) 10 150 2 Mt 2 10 0.013 0i 3.8° 2.0° 0.19° 2.0° 0.19° 14.39° (F /4) CHAPTER 3. Experimental Design 43 E M 2 Light Figure 3.3: D e t a i l of E M I and E M 2 mirrors , showing shape of E M I w h i c h allows v i ewing of specular reflection near normal incidence. 3.1.3 P o s i t i o n i n g M e c h a n i c s T h e p r i m a r y funct ion of the pos i t ioning mechanics is to enable consistent and precise placement of the sample and light source relative to each other and to the col lec t ion optics. For this experiment , the F T I R and associated col lec t ion optics remain s ta t ionary whi le the sample and source l ight rotate. A concentric, 9 — 29 design is used to accompl i sh this . T h i s concentric sys tem consists of an inner ro ta t ion stage w h i c h holds the sample, and an outer ro ta t ing r ing which holds E M I as wel l as the end of the op t i ca l fiber. T h e system is al igned so that the l ight is focused precisely at the shared center of ro ta t ion . Pos i t i on ing a textured planar waveguide sample precisely on the center of ro ta t ion of the two ro ta t ion stages enables the sample to be rotated to different v i ewing angles CHAPTER 3. Experimental Design 44 wi thou t requi r ing the v iewing optics ( E M 2 and E M 3 ) to be real igned. T o achieve the precise pos i t ion ing of the sample, a five stage sample mount was designed (see F igu re 3.4). T h e bo t tom t rans la t ion stage allows the center of ro ta t ion of the sample mount to be posi t ioned concentr ic w i t h the outer rota t ing r ing. T h e hor izonta l ro ta t ion stage is used i n conjunct ion w i t h the outer ro ta t ion r ing to cont ro l the angle of incident l ight , 9. T h e X Y Z t rans la t ion stage is used to posi t ion the sample on the center of ro ta t ion of the sample stage and to posi t ion the height of the sample relat ive to the opt ics . T h e t i l t stage is used to ensure the sample is mounted vert ically. F i n a l l y , the ver t ica l ro ta t ion stage is used to adjust the az imutha l angle of the sample (<j>). T h i s five stage system allows for samples to be mounted rap id ly as well as accurately. Tilt X Y Z Translation Horizontal Rotation Ring XY Translation Figure 3.4: B l o c k d iagram showing sample moun t ing and a l ignment apparatus. CHAPTER 3. Experimental Design 45 T h e mount , shown i n F igure 3.5, was designed to be used i n conjunct ion w i t h the ver t ica l ro ta t ion r ing depicted i n F igure 3.4. T h i s mount was designed to ho ld the sample i n the center of the ver t ical ro ta t ion r i n g a l lowing for the m a x i m u m possible angle of incidence for bo th reflection and transmission measurements wi thou t necessi tat ing adjustment to the pos i t ion of the sample. Based on the phys ica l size of the ver t ica l ro ta t ion r i n g and the 2° divergence of incident l ight , the m a x i m u m angle of incidence for reflection and t ransmiss ion measurements is ~ 6 5 ° . T h e mount was designed for ease of use i n affixing various samples. T h e mount has a flat, pol ished surface onto wh ich the sample is affixed. T h e sample is held i n place w i t h an adhesive m e d i u m , such as vacuum grease for the l ighter weight G a A s samples, and two-sided tape for the heavier po lymer samples. W h e n p lac ing a sample onto the mount , i t is not necessary to pos i t ion i t precisely, because, as previously discussed, the t rans la t ion stages are used to perform a l l necessary adjustments to b r ing the sample into proper al ignment. Fur ther , the sample can be mounted such that the area of interest is held above the mount , a l lowing for t ransmiss ion as wel l as reflection. 3 . 1 . 4 S p e c t r o m e t e r T h e l ight scattered from the textured planar waveguide sample is ana lyzed w i t h a B o m e m F T I R . T h i s spectrometer consists of a Miche l son interferometer w i t h a quar tz beam spl i t ter and an I n G a A s detector. T h e l ight is focused into the entrance aperture of the CHAPTER 3. Experimental Design 46 Front view Side view Figure 3.5: D e t a i l of sample mount used i n ver t ica l ro ta t ion r ing . F T I R by mi r ro r E M 3 , w i t h the F / 4 needed to proper ly fill the in te rna l optics. T h e In-G a A s detector and quar tz beam spli t ter al low spectral measurements from approx imate ly 6,000 c m " 1 to 13,000 c m " 1 . 3.2 Alignment and Operation of Apparatus T h e fol lowing is the procedure for a l igning the apparatus for use, l i s ted i n the prescr ibed order. P l ace the sample to be characterized on the sample mount . A l i g n the centers of ro ta t ion of the inner ro ta t ion stage and the outer ro ta t ing r ing to coincide. Focus E M 2 on this shared center of rota t ion, using the image on the C C D camera as a guide. Ad jus t E M I to focus the l ight onto this center of rotat ion, using the C C D camera to conf i rm proper placement. Use the X Y Z t ransla t ion stage to pos i t ion the sample at the shared CHAPTER 3. Experimental Design 47 center of rotation and at the proper height. Remove the C C D camera from the optics path. Back-illuminate the detector aperture inside the Bomem F T I R and use to co-align with the light reflected from the sample. Remove back illumination source. To collect data at a desired angle of incidence, use the vernier scales on the outer rotating ring, the inner rotation stage and the vertical rotation ring to adjust the angle of incident light and the angle of collected light accordingly. 3.3 A l t e r n a t e Con f i gu ra t i ons This experimental apparatus is versatile, allowing for the substitution of light sources as well as collection optics. Mounts were designed to accommodate an Erbium doped fiber amplifier ( E D F A ) as an alternate light source. Also, an additional sample mount was designed to be used in place of the vertical rotation ring to allow the viewing of light emitted from the cleaved edge of a sample which would otherwise be blocked by the vertical rotation ring. This mount is shown in Figure 3.6. Front view Side view Figure 3.6: Sample mount specifically designed to enable the viewing of light coupling out of the cleaved edge of a waveguide. Chapter 4 Sample Preparation T h e exper imenta l apparatus described i n Chap te r 3 is used to characterize t ex tured planar waveguides. T w o types of textured planar waveguides are invest igated i n this thesis: G a A s on oxide, and po lymer on glass. T h i s chapter discusses the fabr icat ion of these tex tured planar waveguides. 4.1 GaAs Sample There are four phases i n the fabricat ion of a GaAs-based textured planar waveguide such as the one depicted i n F igure 4.1: growth of the planar waveguide, electron beam l i thography, etching, and oxida t ion . A descr ipt ion of each of these phases follows. 4.1.1 Planar Waveguide Growth T h e first step i n fabricat ing a GaAs-based textured planar waveguide is to make a p lanar waveguide. T h i s is done by us ing molecular beam epi taxy ( M B E ) to grow Alo.9sGao.02As 48 CHAPTER 4. Sample Preparation 49 GaAs core Figure 4.1: GaAs-based textured planar waveguide schematic. on a GaAs substrate. The waveguide core is then grown on top of the Alo.98Gao.02As layer. Dr. Shane R. Johnson from Arizona State University grew two separate planar waveg-uides, ASU721 and ASU506, which were used for the experiments reported here. ASU721 was grown on a 500 um GaAs substrate; the Alo.9gGao.02As layer is ~1.6/mi thick and the core is 80 nm thick, consisting of two layers: a 40 nm GaAs cap on top of 40 nm of Alo.3Gao.7As. ASU506 was grown on a 100 /xm GaAs substrate; the Alo.9sGao.02As layer is 1.0 /xm thick and the single layer core of GaAs is 155 nm thick. Once complete, the planar waveguide wafer is cleaved into smaller samples for further processing. The typical sample size used for these experiments was 5 mm x 5 mm. 4.1.2 Electron Beam Lithography To prepare the waveguide samples for lithography, they are cleaned and an electron beam resist is applied to the top surface of each. For this experiment, ASU721 and ASU506 were cleaned with acetone and methanol. The resist applied was approximately 3 drops of CHAPTER 4. Sample Preparation 50 4% Po lymethy lmethac ry la te ( P M M A ) 9 5 0 K dissolved i n a chlorobenzene solut ion. T h e samples were spun at 8,000 r p m for 40 seconds, y ie ld ing a thickness of approx imate ly 200 n m . F i n a l l y the samples are baked on a hot plate at 175°C for a m i n i m u m of 2 hours. T h i s process bakes off the solvents i n the P M M A , and the samples are ready for l i thography. E l e c t r o n beam l i thography is a process by which an electron beam transfers a pre-scr ibed pat tern onto a resist. A H i t a c h i 4100 computer-control led thermal-emission scan-n ing electron microscope ( S E M ) was used to perform the l i thography on the G a A s sam-ples. L i t hog raphy software ( N P G S ) [19] directs the S E M to make a pa t te rn i n the P M M A by exposing the resist at pre-determined coordinates w i t h a specified dose. T h e electron beam breaks the l ong po lymer chains of the P M M A . T h e broken chains are then removed from the sample by developing, wh ich involves immers ing the sample i n chemicals that selectively e l iminate the shorter polymer chains wi thou t affecting the longer chains. T h e longer the electron beam dwells i n a par t icular loca t ion dur ing l i thography, the more chains are broken, resul t ing i n a larger hole once developed. T h e developing recipe for this experiment is shown i n Table 4.1. T h e P M M A resist has now been t ransformed into an etching mask. 4 . 1 . 3 E tch ing T o transfer the pat tern from the mask to the sample, an e tching process is used. T h e etching for this experiment is based on a dry p lasma technique and is done w i t h a CHAPTER 4. Sample Preparation 51 Table 4.1: P M M A developing recipe C h e m i c a l T i m e M I B K 90 sec P ropano l 30 sec D l Water 30 sec Oxye thy lmethenol 15 sec M e t h a n o l 30 sec P l a s m a Quest electron-cyclotron resonance ( E C R ) etcher. In the etcher, w i t h i n a s trong, stat ic magnet ic field, a microwave source drives the electrons i n a low pressure gas at their cyc lo t ron resonance, creat ing a plasma. T h e radio frequency ( R F ) bias drives ions from the p lasma c loud along the magnetic field lines to strike the sample n o r m a l to the surface, where they k inemat ica l ly and chemical ly etch the exposed G a A s . T h e specific etch recipe used is shown i n Table 4.2. T h i s recipe, developed based on research presented by Sah [27], etches cy l ind r i ca l holes (or other desired pattern) w i t h ver t ica l sidewalls i n the G a A s at approximate ly 100 n m per minute . D u r i n g etching, the masked port ions of the G a A s are protected by the P M M A , whi le the chemicals etch the remainder of the G a A s . T h e P M M A is also etched by the chemicals, but the etching t ime is short enough (i.e. less t h a n 150 seconds) that the chemicals do not fully ablate the mask. Af te r the etching process is complete, the sample is rinsed w i t h acetone and methanol to remove the remain ing P M M A . T h e sample is now ready for oxidat ion . CHAPTER 4. Sample Preparation 52 Table 4.2: E C R recipe C l 2 2.0 seem B C 1 3 2.0 seem A r 20.0 seem Microwave 100 Wat t s R F B ia s 100 Vol t s R F Power 25 Wat t s Chuck Temp 5°C Process Pressure 10 m T o r r Backside He 5 Tor r T i m e 145 sec 4 . 1 . 4 O x i d a t i o n T h e final step i n fabr icat ing the G a A s textured planar waveguides used i n this thesis is to oxidize the a l u m i n u m c ladding layer of the waveguide using an ox ida t ion furnace. For a detai led descr ipt ion of the design and const ruct ion of the ox ida t ion furnace used here, refer to Reference [32]. T o oxidize the a l u m i n u m layer, the sample is sealed i n the ox ida t ion furnace which is purged w i t h dry ni t rogen for 1 hour. Subsequently the temperature is increased from room temperature to 4 2 5 ° C over a per iod of 30 minutes. T h e temperature is mainta ined at 4 2 5 ° C for 40 minutes, du r ing which t ime 100 seem of ni t rogen is bubbled through 9 5 ° C water and routed through the furnace. T h i s w a r m , moist environment oxidizes the a luminum layer through the etched holes. Af ter the 40 minute ox ida t ion per iod , the 100 seem of ni trogen is re-routed d i rec t ly into the furnace, CHAPTER 4. Sample Preparation 53 bypassing the bubbler , s lowly purging the moisture out of the furnace whi le the tem-perature is reduced at a rate of 30° per hour un t i l reaching room temperature, wh ich completes the ox ida t ion process. T h e purpose of ox id iz ing the a l u m i n u m layer is to change its index of refraction so tha t i t is s ignif icantly lower t han that of the core layer. P r i o r to ox ida t ion , the index of refraction of the a l u m i n u m layer is n^/=3.6; ox ida t ion reduces this to noxi(ie=^-Q- [33] T h e resultant G a A s textured planar waveguide has a h igh index core (ncore ~3 .5) w i t h low index c ladd ing (n O j r =1 .0 , noxide=1.6). T h e fabricat ion process is now complete, and the sample is ready to be mounted i n the exper imental apparatus for character izat ion. 4.2 Polymer Sample T h e 2 D tex tured po lymer waveguides used i n this invest igat ion were fabricated by P a u l Rochon ' s group at the R o y a l M i l i t a r y College of Canada . T h e po lymer mater ia l is an azoaromatic po lymer f i lm, specifically poly[(4-ni t rophenyl) [4-[[20(methacryloyloxy)ethyl]ethylomino]phenyl]diazene] ( p D R I M ) . [10,15,24] T o fabri-cate this waveguide, a piece of B K 7 glass was spin-coated w i t h this po lymer to a thickness of ~ 4 0 0 n m . T h e process of t ex tu r ing this po lymer waveguide is unique, due to the po lymer ' s un-usual response to intense l ight . T h i s photofabricat ion process is a di rect-wri te technique which transfers a holographic pat tern di rect ly to the polymer . A 514 n m argon laser beam is expanded and col l imated , and used to form a linear interference pa t te rn i n the CHAPTER 4. Sample Preparation 54 po lymer layer of the waveguide. T h i s par t icular po lymer has a wel l k n o w n trans-cis-trans photo isomer iza t ion that causes a change in or ienta t ion of the p o l y m e r molecules. T h e po lymer moves as macromolecules when "activated" by light at a modest in tensi ty ( ~ m W / c m 2 ) , wel l below the glass t rans i t ion temperature of the po lymer . T h i s has a migra t ion effect since the molecules that happen to move into the da rk areas stop mov-ing. T h e po lymer becomes thicker i n the areas of destructive interference and th inner i n the areas of construct ive interference. The resultant pat tern is a l inear var ia t ion i n the morphology of the po lymer which direct ly corresponds to the interference pa t te rn used. T o make a 2 D square lat t ice grat ing, a linear pa t te rn is made, then the process is repeated w i t h the sample rotated 90° . T h e exposure pa t te rn for this po lymer g ra t ing is shown i n F igure 4.2. F igure 4.2: Exposure pat tern for the polymer sample. Regions (1) and (3) are I D gratings, whi le the over lapping region (2) is a 2 D square lat t ice. T h i s fabr icat ion technique can be used to make gratings i n the po lymer w i t h a dep th of m o d u l a t i o n of several hundred nanometers. Gra t ings wr i t t en i n this way are easily erased by heat ing the sample to the glass t ransi t ion temperature, but are thought to be CHAPTER 4. Sample Preparation 55 stable at r o o m temperature. [25] Chapter 5 Results and Discussion T h i s chapter presents the results of l inear whi te light spectroscopy experiments on several 2 D tex tured planar waveguides. T h e exper imental da ta is r igorously compared w i t h s imulat ions based on the Green's function formal ism described i n Chap t e r 2. Together, the theoret ical and exper imental results from some basic structures are used to derive a comprehensive unders tanding of electromagnetic exci tat ions associated w i t h b o t h low and h igh index-contrast 2 D textured planar waveguides. T h i s forms the foundat ion for us ing the same techniques to design and characterize structures w i t h par t icu la r propert ies that may find appl icat ions as l inear and non-linear opt ica l components. Sect ion 5.1 presents the dispersion and polar iza t ion properties of modes near the second order gap of two s imply- textured planar G a A s waveguides as determined from specular reflect ivi ty spectra. Ou t s t and ing quant i ta t ive agreement between the mode l and exper imenta l results is demonstrated for bo th square and t r iangular la t t ice structures i n t h i n ( ~ 8 0 n m ) G a A s slabs. T h e results from these high index-contrast structures reveal substant ia l renormal iza t ion of the electromagnetic modes at tached to the porous slab 56 CHAPTER 5. Results and Discussion 57 over a range of frequencies that span approximate ly 10% of the "center frequency." T h e dispersion, lifetimes, and composi t ion of these leaky B l o c h states are interpreted us ing a s imple picture: the 2 D texture couples what are effectively the slab modes character is t ic of an untextured structure w i t h the core layer replaced by a uni form layer of refractive index given approximate ly by the average index of the textured core. Sect ion 5.2 reports dispersion properties of two quite different structures that were considered because of their potent ia l relevance i n prac t ica l appl icat ions. Sect ion 5.2.1 discusses a low index-contrast 2 D textured waveguide formed i n a novel p o l y m e r ma te r i a l that spontaneously develops a deep ( ~ 2 0 0 n m ) texture w i t h sub-micron periods u p o n be ing exposed to holographic laser fields. These po lymer structures were developed i n K i n g s t o n Onta r io , and the results reported here represent the first comprehensive s tudy of their scat ter ing properties over a broad range of frequencies. Sect ion 5.2.2 reports the successful rea l iza t ion of a h igh index-contrast textured planar waveguide specifical ly designed to possess an extremely flat band a long the entire T - X axis of the square B r i l l o u i n zone. Such flat bands are expected to offer significant advantages i n some non-l inear op t ica l appl icat ions. Sect ion 5.3 exper imenta l ly and theoret ical ly evaluates the use of defect superlat t ices as an effective al ternative for p rob ing the electromagnetic B l o c h states associated w i t h porous slab waveguides. T h e evaluat ion process identifies instances where a new diffrac-t ion measurement technique offers considerable advantages due to its background-free nature, but also identifies some l imi ta t ions of the approach, at least when implemented CHAPTER 5. Results and Discussion 58 w i t h a low power whi te l ight source. F ina l ly , Sect ion 5.3.3 describes a design for an angle insensit ive notch filter based on a h igh index-contrast structure that incorporates a defect superlat t ice and exhibi ts a complete first-order pseudo-gap. 5.1 Waveguides with Simple 2D Gratings T h e purpose of this section is to explore the symmet ry and po la r iza t ion propert ies of leaky modes bound to textured planar waveguides, and to quant i ta t ive ly val idate the computer mode l ing code. Specifically, the specular reflectivity technique is used to probe the modes near the second order gap of waveguides w i t h square and t r iangular lat t ices that were designed and fabricated for this purpose. B o t h waveguides conta in h igh index-contrast gratings. 5.1.1 Square Lattice T o investigate the characteristics of a planar waveguide w i t h a s imple, h igh index-contrast gra t ing, a square lat t ice of round holes was etched into a G a A s waveguide, l ike the one depicted i n F igure 4.1. T h i s waveguide does not have a complete photonic pseudo-gap, but was selected for pedagogical purposes because it i l lustrates key features of the dispersion of photonic modes i n this type of waveguide. T h e textured planar waveguide sample discussed i n this section was fabricated by another student i n the group, V . Pacradouni , using A S U 7 2 1 . T h e sample is comprised of: 40 n m of G a A s on top of 40 n m of Alo.3Gao.7As, c lad by air above and 1.8 yum of ful ly CHAPTER 5. Results and Discussion 59 oxid ized Alo.9sGao.02As below, on a G a A s substrate. A square la t t ice of c i rcu lar holes 550 n m apart w i t h a radius of 165 n m was etched completely through the waveguide core, extending in to the under ly ing c ladding. T h e pat terned region is 90 u m x 90 yum. T h e oxide c l add ing layer is sufficiently th ick that the evanescent Fourier components of the B l o c h states i n the v i c in i ty of the second order gap are l imi t ed to the oxide layer. T h u s , the finite lifetimes of the modes are determined entirely by a single Four ier component of the po la r iza t ion which radiates into the upper and lower half spaces. T h i s sample was characterized by V . Pac radoun i us ing the apparatus designed and bui l t by the author. A detai led invest igat ion of this sample is presented i n Reference [21]. W h e n the photonic modes of this sample are probed i n the T - X d i rec t ion us ing the specular measurement technique, the spectra shown i n F igure 5.1 are obta ined. [20] T h e dashed spec t rum at the bo t t om of F igure 5.1 was s imulated for p-polar ized rad ia t ion incident at 5° . T h e second l ine from the bo t tom is the corresponding p-polar ized d a t a taken from the G a A s waveguide. S-polarized da t a for various angles of incidence are also shown. T h e low frequency oscil lat ions are Fabry-Pero t fringes, wh ich are due to the interference of l ight reflecting off the top and b o t t o m surfaces of the oxide layer. T h e sharp Fano-l ike resonances indicate coupl ing into one of the leaky eigenstates of the textured waveguide. These resonances are narrow for modes w i t h long lifetimes and wide for modes w i t h short lifetimes. T o extract the energy and lifetime of the modes from these spectra, a ma thema t i ca l f i t t ing technique was used. T h i s technique fits an A i r y funct ion to the Fabry-Pero t CHAPTER 5. Results and Discussion 60 F igure 5.1: Specular reflectivity da ta for a textured planar waveguide w i t h a s imple square la t t ice of holes etched through the core of A S U 7 2 1 . A n g l e of incidence is labled on each spect rum. T h e bo t tom two spectra are a comparison of s imula t ion (dot ted line) and da ta (solid line) for p-polar ized reflectivity. T h e upper nine spectra are s-polarized specular reflectivity data . [20] oscil lat ions and a Fano-funct ion to each mode. For a detai led descr ipt ion of this f i t t ing technique refer to Reference [21]. T h e energy of the modes and the angles of incidence are used to convert these spectra into a dispersion d iagram. F igure 5.2 shows the dispersion d iagram for this G a A s waveguide sample. T h i s d ia -g ram depicts the s- and p-polar ized bands for bo th the model and the exper imenta l da ta . CHAPTER 5. Results and Discussion 61 B o E? 9 8 •n n 11 m 11111111 rp i n | r m 11 r i rp 111 , , i , • [ 1 1 j I M 111 r 111111 Tf 11 i i j n r t | t111 [% 111\ 11 .. , g^p. [ i -A - -*- p polarization • : Model Experiment _ M -0 . 3 -0 . 2 - 0 . 1 0 . 0 0 . 1 0 .2 0 .3 0 . 4 x K||/Kg Figure 5.2: Dispersion diagram of the data and simulations for the 2D textured planar waveguide with the square lattice described in the text. [20] The key computer modeling parameters used are shown in Table 5.1. There are two low-dispersion bands in the T - X direction near 10,000 c m - 1 . These bands have a very low group velocity and are therefore good candidates as hosts for certain non-linear optical conversion processes. [28-30] As a consequence of the symmetry of a square lattice (as discussed in Chapter 2), one p- and three s-polarized bands are evident near the second order gap in the T - X direction, while there is one s-polarized band and one p-polarized band dispersing up, and one s-polarized band and one p-polarized band dispersing down in the T - M direction. This grating produces a large second order gap at zone center, ~10% of the center frequency. Reference [20] elaborates on the dispersion of the mode CHAPTER 5. Results and Discussion 62 lifetimes, wh ich are also i n remarkable agreement w i t h the s imulat ions and predict ions based on symmet ry arguments. T h e quant i ta t ive agreement between the da ta and the computer mode l for the dispersion, as wel l as for the lifetimes of the modes, to this author 's knowledge is the best reported for 2 D textured planar waveguide structures. Table 5.1: K e y mode l ing parameters for textured planar waveguide w i t h square la t t ice Parameter Va lue P i t c h 550 n m Hole radius 161 n m Thickness of core 83 n m Core composi t ion 60% Alo.3Gao.7As 40% G a A s O x i d e thickness 1900 n m Hole depth 433 n m W h i l e others have been successful i n demonst ra t ing a complete gap i n a pure 2 D photonic c rys ta l of "infinite" rods arranged i n a square la t t ice [12], a t tempts have proven unsuccessful at creat ing a complete pseudo-gap i n a planar waveguide w i t h a square la t t ice of holes such as the one described above. A more symmet r ic la t t ice configurat ion makes i t easier to achieve a complete pseudo-gap i n a planar waveguide. 5.1.2 Triangular Lattice T o explore the band structure of a planar waveguide w i t h a more symmet r i c la t t ice , a h igh index-contrast t r iangular gra t ing was designed and fabricated. T h i s waveguide was specifically designed i n order to characterize the polar iza t ion and dispersion propert ies of CHAPTER 5. Results and Discussion 63 the bands near the second order gap using the specular reflectivity technique. T h i s sec-t ion demonstrates the characteristics specific to a t r iangular lat t ice, and shows the band structure to be richer than that of the previously discussed square lat t ice. A l t h o u g h this par t icu lar waveguide was not engineered to have a complete second order pseudo-gap, a t r iangular la t t ice does have a sufficiently h igh degree of symmet ry to produce a complete pseudo-gap, as w i l l be discussed i n Sect ion 5.3.3. Un l ike the square la t t ice s tructure discussed above, the Alo.9sGao.02As layer beneath the core slab layer i n this t r iangular la t t ice structure was only par t ia l ly oxid ized , to a depth of ~ 2 0 0 n m . Never-theless, the s imulat ions again agree remarkably wel l w i t h the exper imenta l ly determined dispersion, as discussed i n the present section, and w i t h the lifetimes, as discussed below i n Sect ion 5.3.2. T o fabricate this textured planar waveguide, a 90 pm x 90 pm t r iangular la t t ice of c i rcular holes 600 n m apart w i t h a radius of 80 n m was etched completely th rough the core of an A S U 7 2 1 waveguide, o m i t t i n g every seventh hole . 1 T h e Alo.9sGao.02As layer was pa r t i a l ly oxid ized , p roduc ing a t h i n oxide layer, on the order of 200 n m thick, on top of the remain ing ~ 1 6 0 0 n m of un-oxidized Alo.9sGao.02As. P r o b i n g the photonic modes of this waveguide w i t h the specular measurement tech-nique produces the spectra shown i n F igure 5.3 for the M direc t ion, and the spectra i n F igure 5.4 for the K direct ion. T h e Fano-l ike resonances are evident, as wel l as Fabry -Perot oscil lat ions. Closer examina t ion of the spectra for a 10° angle of incidence i n the M d i rec t ion (shown i n F igure 5.5) reveals a beat i n the Fabry-Pero t osci l lat ions, wh ich xThe reason for the omission of every seventh hole will be explained in Section 5.3 CHAPTER 5. Results and Discussion .64 is a result of the two I D opt ica l cavities (Al -ox ide and Alo.98Gao.02As) created by the pa r t i a l ox ida t ion . There are four s- and two p-polar ized resonances i n the spectra, wh ich are labeled on this d iagram. 50° 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 12 .0xl0 3 Energy (cm ') F igure 5.3: Specular reflectivity da ta for the T - M direct ion from the G a A s tex tured planar waveguide w i t h a t r iangular lat t ice discussed i n the text. So l id lines represent s-polar izat ion; dashed lines represent p-polar iza t ion. T h e mode energies were extracted from the spectra to produce the da t a points i n the dispersion d iagram shown i n F igure 5.6. In the M direct ion near zone center, there are four s-polarized bands: two dispersing up i n energy, and two dispersing down. There are also two p-polar ized bands: one dispersing up, and one down. In the K d i rec t ion near CHAPTER 5. Results and Discussion 65 < .\> o o q=l o Pi ^ " o ,.'••50 — D \- .-30 0 20' 3 10 3 \ 1 1 7.0 7.5 8.0 8.5 9.0 9.5 Energy (cm ) 10.0 10.5 11.0 12.0x10 Figure 5.4: Specular reflectivity data for the T -K direction from the GaAs textured planar waveguide with a triangular lattice discussed in the text. Solid lines represent s-polarization; dashed lines represent p-polarization. zone center there are also six bands, however three are s- and three are p-polarized. Two s- and p-polarized pairs disperse up in energy, one pair more strongly than the other. A third pair disperses down in energy. For both the M and K directions, the bands away from zone center and away from the anticrossings are good illustrations of the kinematic dispersion properties discussed in Section 2.1. To further investigate the characteristics of these modes, the computer model was used to simulate incident radiation at a 10° angle of incidence in the M direction. The CHAPTER 5. Results and Discussion 66 r At? *' si / s2 s y / 1 v y f 1 ,3 p2 / s4 if 1 j 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 12.0xl03 Energy (cm ') F igure 5.5: Specular reflectivity da ta for a 10° angle of incidence i n the M di rec t ion . So l id lines represent s-polarizat ion; dashed lines represent p-polar iza t ion . N o t i c e the two Fabry-Pero t frequencies wh ich are due to incomplete ox ida t ion of the Alo.9sGao.02As layer. T h e higher frequency is approximate ly 1 , 0 0 0 c m - 1 which corresponds to ~ l , 3 0 0 n m of unoxid ized Alo.9sGao.02As. S ix modes are annotated. s ix resonant modes clearly seen i n the da ta for this angle of incidence are labeled i n F igu re 5.5. T h e four s-polarized modes are labeled s l - s 4 , whi le the two p-polar ized modes are labeled p l and p2. T h e strength of the Fourier components of the field (see E q u a t i o n 2.2) vary from one mode to the next. F igure 5.7 depicts the relative strengths of each of these Fourier components for each of the s ix modes. Figures (a) and (d) b o t h show the same two dominant components. These components are "away" from the d i rec t ion of the incident wavevector and therefore produce downward propagat ing modes. These components add symmet r i ca l ly creat ing an s-polarized mode, and an t i - symmet r ica l ly creat ing a p-polar ized mode. Figures (b) and (e) show two dominant components each, however i n this case the dominant components are "toward" the d i rec t ion of propagat ion . These two si tuat ions give rise to the upward dispersing s- and p-polar ized modes. In the CHAPTER 5. Results and Discussion 67 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 — K r M Figure 5.6: Dispers ion d iagram for the planar waveguide w i t h the t r iangular la t t ice discussed i n the text. T h e crosses represent s-polarized data. T h e circles represent p-polar ized data . T h e sol id lines represent s-polarized s imulat ions. T h e dashed lines represent p-polar ized simulat ions. final two figures, (c) and (g), there is a single dominant component wh ich gives rise to one s-polarized mode dispersing qu ick ly downward and another dispers ing qu i ck ly upward . It is impor tan t to note these results are for modes occur r ing approx imate ly 20% of the way across the B r i l l o u i n zone, wh ich is far from zone center where there is s t rong coup l ing between several of the field components. T h i s is not expla ined by s imple k inemat ic arguments. T o i l lust ra te the s t rong coupl ing near zone center, m o m e n t u m space diagrams were constructed for the modes close to zone center (K\\ — O.Ol-Kg) but s t i l l s l ight ly detuned i n the M direct ion. These diagrams are shown i n F igure 5.8. These diagrams reveal s t rong CHAPTER 5. Results and Discussion 68 M (a) p i (b) p2 (c) s i (d) s2 (e) s3 (f) s4 F igure 5.7: M o m e n t u m space diagrams for each B l o c h mode of the t r iangular la t t ice waveguide when probed w i t h light at a 10° angle of incidence. T h e strength of each Four ier component is ind ica ted by the size of the dot. T h e labels for each d i ag ram correspond to the labels on the modes shown i n F igure 5.5 coupl ing between mul t ip le field components, wh ich is very non-kinemat ic . A w a y from zone center, the highest energy bands display anti-crossings w i t h modes descending from a higher order gap. T h e size of the gaps is s imi lar to the zone-center gap, suggesting s t rong band-mix ing , wh ich cannot be explained w i t h s imple kinemat ics . A l t h o u g h this waveguide has a defect superlatt ice, the dispersion properties and polar izat ions of these bands are fully consistent w i t h that of a waveguide w i t h a s imple (non-defect) t r iangular la t t ice . T h e s imula ted band structure obtained using the parameters i n Table 5.2, is repre-sented as lines on the dispersion d iagram i n F igure 5.6. A s w i t h the square la t t ice sample, CHAPTER 5. Results and Discussion 69 • • • • © (a) pi (b) p2 (c) si (d) s2 (e) s3 (f) s4 Figure 5.8: M o m e n t u m space diagrams for each B l o c h mode of the t r iangular la t t ice at K | | = 0 . 0 1 K 9 i n the M direct ion. T h e M direct ion is toward the top of the page. T h e s trength of each Fourier component is indicated by the size of the dot. T h e labels for each d i ag ram correspond to the labels on the modes shown i n F igure 5.5 there is remarkable quant i ta t ive agreement between the model and the da ta w i t h regard to dispersion and polar izat ions. T h e only parameter of the waveguide wh ich was al tered from its n o m i n a l va lue 2 for this s imula t ion was the thickness of the core; i t was modeled as a single layer, 73 n m thick, comprised of Alo.15Gao.85As, wh i ch was used to s imulate the combina t ion of the 40 n m thick layer of Alo .30Gao.70As and the 40 n m th ick layer of G a A s i n the sample. C o m p u t e r mode l ing was performed to determine the effect the defect superlat t ice has on the dispersion characteristics of this s tructure. F igure 5.9 shows a compar i son o f 2The deviation from the nominal value is within the margin of error. CHAPTER 5. Results and Discussion 70 Table 5.2: K e y mode l ing parameters for textured planar waveguide w i t h t r iangular la t t ice Parameter Va lue P i t c h 600 n m Hole radius 80 n m Thickness of core 73 n m Core composi t ion A l o . 1 5 G a o . 8 5 A s Oxide thickness 200 n m Hole depth 73 n m the spectra for the waveguide at 10° w i t h and wi thout the defect superlat t ice. There is essentially no difference between the results of the model ing w i t h defects and wi thou t them. Thus , i n this context, the defect superlatt ice incorporated i n this sample represents on ly a weak per turba t ion . T h e u t i l i t y of this defect superlat t ice is expla ined i n Sect ion 5.3.2. C o m p u t e r mode l ing was also performed to s tudy the effect of the thickness of the oxide layer for this structure. F igure 5.10 shows the s imulated spectra for this sample w i t h 200 n m of oxide, i.e. a t h i n oxide c ladding. F igure 5.11 shows the s imula ted spectra for this sample w i t h a th ick oxide (1600 n m of fully oxidized Alo .9sGao.02 A s ) c ladding . T h e spectra for the th ick oxide sample shows well-defined, narrow Fano resonances, wh ich go to un i ty reflectivity. [6] T h e spectra for the t h in oxide shows much broader, less wel l -defined resonances, wh ich do not go to uni ty reflectivity. T h e broad nature (i.e. short lifetime) of these modes is due to the fact that the evanescent components of the B l o c h states of these modes penetrate the t h in oxide and radiate into the substrate th rough CHAPTER 5. Results and Discussion 71 0.5 H 8.6 8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.4x103 Energy (cm' ) F igure 5.9: C o m p a r i s o n of s imulat ions of the t r iangular lat t ice waveguide w i t h and w i t h -out a defect superlat t ice at a 10° angle of incidence i n the M direct ion of symmetry . T h e sol id lines represent s-polarized and the dashed lines represent p-polar ized reflect ivi ty spectra. T h e defect superlat t ice spectra appear below the non-defect spectra. CHAPTER 5. Results and Discussion 72 the oxide. T h i s issue is quant i ta t ive ly addressed i n Section 5.3.2. It is interest ing to note that this qual i ta t ive change i n the nature of the modes has ha rd ly any effect on the dispersion or the energy of the modes, but a d ramat ic effect on the lifetimes. 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 12.0x103 Energy (cm"1) F igure 5.10: S imula ted spectra for the M direct ion of a t r iangular la t t ice sample w i t h 200 n m of oxide, i.e. t h i n oxide layer. T h e spectra are for angles of incidence of 10°, 20° , 30° , 4 0 ° , and 50° from the b o t t o m up. Sol id lines represent s-polar izat ion; dashed lines represent p-polar iza t ion . T h i s section has demonstrated the ab i l i ty to accurately fabricate and characterize tex-tured p lanar waveguides, and to accurately predict w i t h a rigorous computer mode l the complex band s t ructure of these waveguides. Ou t s t and ing quant i ta t ive agreement has been shown between the computer model ing results and the da ta for b o t h of the s imple CHAPTER 5. Results and Discussion 73 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 12.0x103 Wavenumber (cm ) F igure 5.11: S imula ted spectra for the M direct ion of a t r iangular la t t ice sample w i t h 1600 n m of oxide, i.e. th ick oxide layer. T h e spectra are for angles of incidence of 10°, 20° , 30° , 40° , and 50° from the b o t t o m up. So l id lines represent s-polar izat ion; dashed lines represent p-polar iza t ion . lat t ice configurations examined. T h e configuration and materials of textured waveguides have been shown to s trongly influence the propagat ion of l ight in these structures. T h e ab i l i ty to predict th rough computer model ing the effects of these factors on the elec-t romagnet ic exci tat ions of these waveguides, combined w i t h a thorough unders tanding of the under ly ing nature of the modes, provide the basis for engineering desired b a n d structures us ing the proven fabricat ion and character izat ion techniques. CHAPTER 5. Results and Discussion 74 5.2 Waveguides Engineered for Specific Applications P l a n a r textured waveguides offer a powerful med ium for engineering devices wh ich con-t ro l the propagat ion characteristics of l ight, bo th i n the waveguide and i n the su r round ing ha l f spaces. T h i s section describes two examples of textured planar waveguides wh ich have been engineered for specific applicat ions. T h e first is a low index-contrast t ex tured waveguide which uses resonant coup l ing to effect a po la r iza t ion insensit ive notch filter for rad ia t ion incident from the upper hal f space. T h i s is just one of many examples of s imple passive op t ica l devices that might be easily fabricated from the in t r igu ing azo-polymers described i n Chap te r 4. T h e results i n Sect ion 5.2.1 serve as much to characterize the properties of these textured polymers as they do to demonstrate a par t icu la r op t i ca l func-t ional i ty. T h e second example is a h igh index-contrast waveguide engineered to possess an extremely broad, flat band, which has been predicted to be useful for s ignif icantly enhancing the second harmonic opt ica l conversion process for modes propagat ing inside photonic crystals . [28,29,31] 5.2.1 2D Textured Azo-Polymer Waveguides P o l y m e r p D R I M is a novel mater ia l wh ich can be qu ick ly and s imply made into large area, low index-contrast , per iodica l ly textured waveguides us ing a direct-wri te holographic technique, as described i n Chap te r 4. Because waveguides fabricated w i t h this po lymer do not have the stringent pur i ty requirements that their semiconductor counterparts do, fabr icat ion costs are much less for devices manufactured w i t h this mater ia l . T h i s section CHAPTER 5. Results and Discussion 75 reports the first in-depth quant i ta t ive s tudy of the broadband scat ter ing properties of this type of azo-polymer waveguide. T h e reflective characteristics pecul iar to this type of waveguide make i t pa r t i cu la r ly well-suited for use as an opt ica l filter. Po lymers are cur rent ly used i n numerous opt ica l applicat ions, such as coatings, filters and fibers. A s a representative example of a po lymer waveguide appl ica t ion , this section presents the s imula t ion of a p D R I M textured planar waveguide as a po la r iza t ion insensit ive notch filter. P r i o r to tex tur iza t ion , the p D R I M polymer waveguide consisted of a n o m i n a l l y 400 n m th ick film on B K 7 glass. T h e index of refraction of the po lymer was measured to be 1.66 i n the visible w i t h an A b b e refractometer, and is est imated to be 1.65 i n the near-infrared. [26] T h e index of refraction of the glass is 1.507 at lpm. [8] T h e core was tex tured w i t h a square lat t ice of per iod 6 5 9 ± 2 n m 3 and a nomina l average m o d u l a t i o n ampl i tude of 250 n m . T h e uni t cell of this lat t ice consists of a paraboloidal - l ike "rise" i n the core med ium, rather t han a cy l ind r i ca l hole w i t h ver t ical sidewalls. T h e result is a gra t ing of signif icantly different morphology than the previously discussed G a A s tex tured planar waveguides, as the a tomic force micrograph of the surface of the tex tured po lymer shows i n F igu re 5.12. Since the gra t ing of the po lymer waveguide is s t ruc tura l ly different from the l i tho-graphica l ly defined gratings discussed throughout this thesis, the parameters of the po ly -mer gra t ing layer are defined differently for the Green's function code. Previous ly , where the code modeled a layer of G a A s w i t h "holes" of air, for this waveguide the code models 3 The period of the lattice was measured using HeNe diffraction. CHAPTER 5. Results and Discussion 70 CHAPTER 5. Results and Discussion 77 a layer of air w i t h "holes" of polymer . E a c h uni t cell of the po lymer gra t ing was modeled as a th ick ver t ica l cyl inder , rather than the actual shape, wh ich was somewhat non-uni form and asymmetr ic . T h e paraboloidal shape of the mounds was not entered into the code because mode l ing them as ver t ical cylinders w i t h the appropr ia te f i l l ing fract ion is sufficient to accurately reproduce the measured dispersion of a waveguide w i t h a low index-contrast g ra t ing such as this. T h e s imulated spectra for the po lymer waveguide were generated by the computer code using the values shown i n Table 5.3. Table 5.3: K e y model ing parameters for textured planar po lymer waveguide Parameter N o m i n a l M o d e l e d P i l l a r radius N A 300 n m Thickness of textured layer 250 n m 250 n m Thickness of untextured layer 250 n m 400 n m Pe r iod 660 n m 660 n m ng 1.507 1.507 np 1.65 1.65 T h e spectra i n F igure 5.13 were obta ined by probing the photonic modes of the po lymer waveguide at a range of incident angles along the T - X axis us ing the specular measurement technique. Not ice that superimposed on the relat ively low background there are sets of h igh reflectivity peaks of s- and p-polar ized pairs. There appears to be one s- and one p-polar ized pair of modes dispersing s t rongly up i n energy, and one pai r dispersing s t rongly down i n energy. T h e remain ing s- and p- bands consist of an unresolved group of modes that disperse moderately up i n energy. F igure 5.14 shows CHAPTER 5. Results and Discussion 78 the spectra i n the M direct ion, wh ich show two apparent s- and p-polar ized band pairs, one dispersing up i n energy and the other down. T h e spectra generated by the computer mode l reveals tha t each of these apparent pairs ac tua l ly consists of two s-polarized modes and two p-, to ta l l ing eight modes i n the M direct ion. Fur thermore, the mode l ing for the X d i rec t ion shows that the midd le group of unresolved modes is ac tua l ly comprised of two s- and two p-polar ized modes. T h i s mode l confirms the remain ing modes i n the X di rec t ion to be one s- and one p-polar ized pai r of modes dispersing s t rongly up i n energy, and one pai r dispersing s t rongly down i n energy. Energy (cm' ) F igure 5.13: Norm a l i zed specular reflectivity da ta for the textured po lymer waveguide for the X di rect ion. So l id lines represent s-polarizat ion; dashed lines represent p -po la r iza t ion . CHAPTER 5. Results and Discussion 79 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0xl0 3 Energy (cm ') Figure 5.14: Normalized specular reflectivity data for the textured polymer waveg-uide for the M direction. Solid lines represent s-polarization; dashed lines represent p-polarization. Figure 5.15 shows the downward dispersing branch of modes in the M direction. In this figure the experimentally derived spectra are superimposed on the correspond-ing simulated spectra. The model, which assumes a perfectly periodic, infinite grating, shows four distinct modes that rise to unity. However, in the data these modes are broad and overlapping. This broadening of the modes is not due to the resolution limit of the CHAPTER 5. Results and Discussion 80 spectrometer 4 , but is a true representation of the mode shape. A publ i shed character-iza t ion s tudy of a s imi lar po lymer waveguide at 632.8 n m reveals a s imi la r discrepancy between the theoret ical and the actual mode w i d t h , wh ich was pa r t i a l ly a t t r ibu ted to absorpt ion. [25] Ca lcu la t ions for the polymer waveguide characterized here indicate that absorpt ion on ly accounts for approximate ly 10% of the observed broadening. In the previous publ ica t ion , it is suggested that birefringence contributes to the broadening, but s imulat ions w i t h birefringence conducted here do not show sufficient broadening. Fur ther , it is suggested i n the publ ica t ion that the broadening may be due to op t i ca l ly induced changes to the po lymer caused by their 100 / / W probe beam. Since this po lymer has the inherent characterist ic of changing when exposed to intense l ight (see Chap t e r 4) , i t is reasonable that the probing light could affect changes i n the waveguide that wou ld produce noticeable changes i n the spectral response. T h e reason that the spectra consist of eight bands rather than four (as previously shown for a square lat t ice) is because the lowest order T E and T M slab modes for this waveguide are nearly degenerate: four bands therefore come from the m i x i n g of the T E - l i k e slab modes, and four bands come from the m i x i n g of the T M - l i k e slab modes. T h e band s t ructure is shown i n F igure 5.16. T h i s d iagram depicts the dispers ion for the exper imenta l da ta as wel l as for the computer s imula t ion of the po lymer waveguide obta ined us ing the parameters l isted i n Table 5.3. T h e computer s imula t ion of the dispersion of the modes agrees wel l w i t h the experi -menta l da ta and shows an apparent convergence of the bands near 9,800 c m - 1 , m i n i m a l 4 The spectral data reported in this thesis was acquired at 7 c m - 1 resolution. CHAPTER 5. Results and Discussion 81 Energy (cm ) Figure 5.15: S imula t ion (narrow features) and da ta (broad features) for the tex tured po lymer waveguide for 10°, 20° and 30° angles of incidence along the M direct ion. Note : the da ta are not p lo t ted on an absolute scale. S-polarized spectra are represented by the sol id lines, and p- by the dashed lines. CHAPTER 5. Results and Discussion 82 F igure 5.16: Dispers ion d iagram for textured planar waveguide w i t h 2 D po lymer grat-ing discussed i n the text. D a t a values are shown as circles and crosses for s- and p-polar iza t ions respectively. Simulat ions are shown as solid lines and dashed lines for s-and p-polar izat ions respectively. T h e light lines are represented by dot-dashed lines: the upper is the glass l ight l ine, and the lower is the po lymer l ight l ine. curvature of the lower and upper bands, and a slight upward curve of the central bands i n the X di rect ion. In the M direct ion the upward and downward dispers ing bands show m i n i m a l curvature. A dispersion d iagram of this polymer waveguide without t ex tu r ing wou ld show a s l ight ly cu rv ing band fol lowing the glass light l ine at low energies and t rans i t ion ing grad-ua l ly toward the po lymer light l ine at higher energies. A s evident i n F igu re 5.16, the CHAPTER 5. Results and Discussion 83 dispersion of this textured waveguide has the same basic characteristics as i f i t were un-textured, w i t h the exception of zone-folding. Thus the t ex tur ing is not great ly pe r tu rb ing the band structure. W h e n a gra t ing serves as a weak per turba t ion evidenced by smal l gaps at zone boundaries, such as is the case w i t h this po lymer waveguide, the dispers ion can be unders tood to follow simple kinematics . T h i s means that the gra t ing serves on ly to impar t m o m e n t u m i n integer mult iples to al low zone-folding, and does not s t rongly affect the dispersion characteristics of the waveguide. T h i s occurs i n this case essentially because the modes are quite weakly confined to the po lymer due to the smal l dielectr ic contrast between the po lymer and the glass. For this par t icular po lymer waveguide, the dispersion follows simple kinematics . T h i s character izat ion of the textured po lymer waveguide has demonstra ted that this type of s t ructure can exhibi t re lat ively high resonant reflectivity and low non-resonant reflectivity. These trai ts make i t well-suited for use as an opt ica l filter, and i n fact s imi la r gratings of other materials are current ly being used as po la r iza t ion insensitive notch filters at n o r m a l incidence. [23] These devices s trongly reflect a specific frequency range, whi le t r ansmi t t i ng the remainder of the incident l ight . T h e y take advantage of the fact tha t there is no d i s t inc t ion between s- and p-polar iza t ion at no rma l incidence for a symmet r i c two-dimensional lat t ice. F igure 5.17 shows a s imula t ion of the reflectivity spectra for th is po lymer waveguide at no rma l incidence. Note the h igh, narrow band reflect ivi ty and the low, non-resonant reflectivity, wh ich wou ld make this waveguide an excellent po la r i za t ion insensit ive notch filter. However, F igure 5.18 shows spectra for the same waveguide at a CHAPTER 5. Results and Discussion 84 1° angle of incidence. A change of just one degree causes considerable dispersion, m a k i n g a l l eight modes clearly dis t inct , and rendering it ineffective as a po la r iza t ion insensit ive notch filter. A l t h o u g h progress has been made in decreasing the angle sensi t iv i ty of this type of filter [23], these designs are constrained to function at near-normal incidence due to the inherent dispersion characteristics of low index-contrast waveguides w i t h two-dimensional tex tur ing . 1.0 0.8 - \ , °-61 0.4 H 0.2 H : 1 1 1 i 1 — f = i 9.4 9.5 9.6 9.7 9.8 9.9 10.0 10.2xl03 Energy (cm ') Figure 5.17: S imula ted specular reflectivity of the textured po lymer waveguide at n o r m a l incidence. T h e po lymer gra t ing presented here has po la r iza t ion insensitive reflect ivi ty at angles CHAPTER 5. Results and Discussion 85 1.0 0.8 0.6 oi 0.4 0.0 -J { • 1 l — 1/ " 9.4 9.5 9.6 9.7 9.8 9.9 10.0 10.2xl03 Energy (cm"') F igure 5.18: S imula ted specular reflectivity of the textured po lymer waveguide at 1° i n the X di rect ion. So l id lines represent s-polarizat ion; dashed lines represent p -po la r iza t ion . far from norma l incidence due to the significant overlap of the s- and p-polar ized modes. F igure 5.19 shows an enlargement of the spectra from this sample at a 30° angle of incidence, i n compar ison w i t h the s imulated results at the same angle. No te again that the mode l shows the modes to be dis t inct and separate, wh ich wou ld indicate that this device wou ld not be par t i cu la r ly useful. However, i n the ac tual da ta the modes are c lear ly over lapping. T h i s over lapping of s- and p-polar ized modes, i n conjunct ion w i t h the strong resonant reflection and low non-resonant reflection, make the p D R I M po lymer CHAPTER 5. Results and Discussion 86 textured planar waveguide par t i cu la r ly wel l suited as a po la r iza t ion insensit ive device wh ich filters (i.e. reflects) a specific frequency range of incident l ight . In add i t ion , the selected frequencies can be tuned by changing the angle of incidence, whi le ma in t a in ing po la r i za t ion insensit ivi ty. 0 -J , 7 \ p I 1 1 1 i 7800 8000 8200 8400 8600 Energy (cm"') F igure 5.19: Specular reflectivity of the polymer waveguide at a 30° angle of incidence i n the M direct ion. T h e top spectra are data, and the bo t tom are s imulated. No te that the scale is specifical ly for the s imula t ion . T h e s-polarized spectra are represented by sol id lines; p- polar ized spectra are represented by dashed lines. T h e peak reflectivities i n the s imulat ions go to unity. A d d i t i o n a l l y , da ta was collected for the two or thogonal X symmet ry direct ions. T h e CHAPTER 5. Results and Discussion 87 s-polarized specular reflectivity da ta for a 30° angle of incidence is shown i n Figures 5.20 (a) and (b). In each spect rum the three mode groups are evident. However, there is a significant difference i n oscil lator strength between the two or thogonal direct ions. In F igure (a) the central modes are much stronger than the side modes, whi le in F igure (b) the s i tua t ion is reversed. Note that the energies are the same for bo th orientat ions. Based on the k inemat ics discussion i n Chap te r 2, the h igh and low energy modes wh ich disperse qu i ck ly up and down i n energy are p r imar i l y due to coup l ing w i t h the Four ier components para l le l to the d i rec t ion of incident radia t ion, whi le the modes which disperse s lowly up i n energy are p r i m a r i l y due to coupl ing w i t h the Fourier components perpendicular to the d i rec t ion of incident radia t ion. B y ro ta t ing the waveguide 90° , the axes are switched. T h i s is a clear sign that there is a difference i n the strength of the B r a g g scat ter ing i n the two or thogonal X directions. T h i s is corroborated by A F M micrographs of this sample which show that i n one direct ion the tex tur ing per iodic i ty is very regular, whereas i n the other d i rec t ion i t is less regular. Fur ther , p D R I M appears to change over t ime. F igure 5.21 shows two A F M micro-graphs of the same p D R I M grat ing taken before and after a three m o n t h interval . A compar ison of the two images shows a change i n the gra t ing morphology. In F i g u r e 5.21(a), the ampl i tude of modu la t ion is relat ively uni form i n bo th directions, whi le i n F igu re 5.21(b), the ampl i tude is much larger i n one di rec t ion than it is i n the other. T h e cause for this change is not known, but could be due to prolonged exposure to ambient CHAPTER 5. Results and Discussion 88 0.20 0.00 0.20 8.0 9.0 10.0 11.0 3 -1 Energy (10 cm ) 8.0 9.0 10.0 11.0 3 -1 Energy (10 cm ) (a) (b) Figure 5.20: S-polarized specular reflectivity da t a for the po lymer waveguide at a 10° angle of incidence along the X symmet ry direct ion. T h e spec t rum i n F igure (a) was taken or thogonal ly (i.e. azmutha l angle of incidence was rotated 90°) to the spec t rum i n F igure (b). l ight . Cur ren t ly , the mechanism by which this po lymer moves is not complete ly under-s tood. U n t i l further research on p D R I M determines the effects of long-term exposure to ambient l ight , i t may not be possible to completely prevent structures made w i t h this mater ia l from degrading, wh ich may serve to l i m i t its usefulness i n appl icat ions . Overa l l , the p D R I M polymer is an in t r igu ing new mater ia l wh ich holds promise for CHAPTER 5. Results and Discussion 89 (a) Fresh polymer grating (b) Three month old polymer grating Figure 5.21: Atomic force micrographs of two polymer square lattice gratings. The sample shown in Figure (a) was recently manufactured, while the one shown in Figure (b) was exposed to ambient light for approximately 3 months. Each contour line is 50 nm. optical applications. The low index-contrast polymer waveguide characterized here pro-vides an innovative medium for engineering low-cost devices requiring large area gratings. As an example, this polymer has been shown to exhibit properties that could be further engineered to realize a polarization insensitive notch filter. It should be noted, however, that this sample was not optimized for this purpose. 5.2.2 Flat Photonic Bands Along the Entire T-X Line of a 2D Square Lattice From analyzing the previously discussed triangular lattice and square lattice waveguides with high index-contrast gratings, the square lattice seems to offer the greater potential CHAPTER 5. Results and Discussion 90 for engineering a fiat band (or bands) that extend across the first B r i l l o u i n zone. It has been shown that a band exh ib i t ing low group veloci ty (vg = 5u>/5k) offers an increase i n the efficiency of cer tain non-linear processes inside bulk photonic crystals . [28-30] Therefore a waveguide was designed w i t h a square lat t ice i n such a way as to op t imize the flatness of one of the bands. T h i s exercise served to test the ab i l i t y to design and realize a textured planar waveguide w i t h a specific, intended band structure. In designing this structure, a 150 n m thick waveguide core was selected, wh ich is t h i c k e r . t h a n that used in the previously described G a A s samples, because a thicker waveguide provides a richer band structure than a thinner one, increasing the potent ia l for achieving flat bands. T h i s richer band structure occurs because i n a thicker waveguide the T E and T M slab modes are closer together i n energy. W h e n a thicker waveguide is tex tured w i t h a h igh index-contrast grat ing, the modes from the T E - l i k e gap and the T M - l i k e gap are closer together, p rov id ing anti-crossings closer to zone center. W i t h more bands anti-crossing, there is more interact ion between the bands and thus more potent ia l for flat bands to occur. M o d e l i n g this s t ructure required the use of the modif ica t ion to the code which al lows for the mode l ing of th ick gratings. W i t h this modif ica t ion, the 150 n m th ick core was modeled as two layers. T h e specific values used for the mode l ing are given i n Table 5.4. T h e computer-modeled band structure of the waveguide is shown i n F igure 5.22. In this theoret ical dispersion d iagram, the lowest four bands near zone center can be loosely associated w i t h the T E slab mode gap. A s expected for a square lat t ice, there are CHAPTER 5. Results and Discussion 91 Table 5.4: K e y model ing parameters for "flat band" s tructure Parameter Value P i t c h 390 n m Hole radius 90 n m Thickness of textured layer 2x70 n m Thickness of oxide layer 1000 n m a o <-> 10-S=! W 9 . t _ - - ^ » - • » - «, „ * * # * • • • M r • • • — p polarization * , - - s polarization 0.2 0.1 0.1 0.2 0.3 x -K„/K g Figure 5.22: Theore t ica l dispersion d iagram for the 150 n m th ick textured planar wave-guide w i t h the square la t t ice discussed i n the text. Note the p-polar ized , low-dispersion bands near 10,000 c m - 1 . [17] four bands i n the X direct ion: two dispersive s-polarized bands, as wel l as one s- and one p-polar ized band that are less dispersive. T h e next higher set of bands can be loosely associated w i t h the T M slab mode gap. A t the T M - l i k e gap the two dispersive modes CHAPTER 5. Results and Discussion 92 are expected to be p-polar ized, and the two less dispersive modes to be one s- and one p-polar ized . [20] O n l y the three lower-energy bands are p lo t ted because the highest energy p-polar ized band occurs above the G a A s absorpt ion edge (11,400 c m - 1 ) and thus does not show up i n the s imula t ion . Note the extremely flat "band," that ac tua l ly consists of two p-polar ized bands, which occurs i n the X di rec t ion and originates at the top of the T E - l i k e gap. [17] M o d e l i n g shows the first of these two bands to originate at 10,390 c m - 1 at zone center. It remains flat across 25% of the B r i l l o u i n zone, where it anti-crosses w i t h the second band, then curves downward i n energy. B e y o n d the anti-crossing the second band becomes nearly dispersion free at a s l ight ly higher energy (10,445 c m - 1 ) than the first band . T h i s band continues across the remainder of the B r i l l o u i n zone, where at zone edge i t has an energy of 10,452 c m - 1 . Thus the overall dispersion is on ly 62 c m - 1 across the entire B r i l l o u i n zone. T h e flatness of this band is a consequence of the p-polar ized band from the T E - l i k e gap anti-crossing w i t h the downward dispersing p-polar ized band from the T M - l i k e gap. T h e s imulated specular reflectivity spectra for this design is shown i n F igure 5.23. A textured planar waveguide for this flat band was fabricated us ing A S U 5 0 6 , wh ich consists of 150 n m of G a A s on top of 1.0 pm of Alo.9sGao.02As on a G a A s substrate. A square lat t ice of c i rcular holes 390 n m apart w i t h a radius of 90 n m was etched complete ly through the waveguide core. E v e r y fifth hole was omit ted , creat ing a defect superlat t ice. T h e Alo.98Gao.02As layer was fully oxidized. T h e specular reflectivity da ta at various angles of incidence along the T - X d i rec t ion CHAPTER 5. Results and Discussion 93 0 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5xl0 3 Energy (cm ') F igure 5.23: S imula t ion of specular reflectivity for the 150 n m th ick textured planar waveguide w i t h the square la t t ice discussed i n the text. T h e spectra correspond to angles of incidence of 10°, 20° , 30° , 40° and 50° from the b o t t o m up. T h e sol id lines represent s-polarized simulat ions. T h e dashed lines represent p-polar ized s imulat ions . are shown i n F igure 5.24. A l l the modes from the T E - l i k e gap and the lowest p-polar ized mode from the T M gap are evident. There is an s-polarized mode that disperses down i n energy from ~ 9 , 5 0 0 c m _ 1 to ~ 8 , 3 0 0 c m - 1 . There is a relat ively flat s-polarized mode near ~ 1 0 , 0 0 0 c m - 1 . There is also a faint s-polarized mode beginning at ~ 1 0 , 6 0 0 c m _ 1 and dispers ing up i n energy, wh ich is lost i n the G a A s absorpt ion above ~ l l , 4 0 0 c m - 1 . There is one p-polar ized mode s tar t ing at ~ l l , 1 0 0 c m - 1 and dispersing down, w h i c h ant i -crosses w i t h the p-polar ized mode at ~ 1 0 , 4 0 0 c m - 1 , as predicted i n the s imula t ion . D u e CHAPTER 5. Results and Discussion 94 to the b road nature of the Fano-resonances associated w i t h the fiat band, i t is difficult to fit these specular da ta to the accuracy needed i n order to determine the precise dispersion of this band. Therefore a defect superlatt ice was inc luded i n the design so that more precise measurements could be taken w i t h the new diffraction measurement technique (as w i l l be discussed in Sect ion 5.3.1). Qual i ta t ive ly , the dispersion of the bands from this textured p lanar waveguide w i t h a defect superlat t ice agrees wel l w i t h the theoret ical predict ions for the dispersion of a waveguide w i t h a s imple lat t ice. T o summarize , the waveguide discussed herein exhibi ts a nearly dispersion free (<1%) mode across the entire B r i l l o u i n zone i n the X di rect ion. T h i s flat band was observed exper imenta l ly up to a 50° angle of incidence, which is more than 30% of the way across the B r i l l o u i n zone. However, due to the close p rox imi ty and breadth of the modes, the exact mode posit ions and lifetimes can not be re l iably extracted us ing the numer ica l f i t t ing technique mentioned i n Sect ion 5.1.1. T o measure the precise pos i t ion and lifetimes of these types of modes, the background-free measurement technique described i n Chap t e r 2 was used. These results are discussed i n the fol lowing section. 5 .3 Waveguides with Defect Superlattices T h e 2 D tex tured planar waveguides i n the previous sections were characterized us ing specular reflectivity to probe the low- ly ing resonant bands. W h i l e this measurement technique has proven useful i n terms of quant i ta t ive comparisons between mode l and experiment , i t has some l imi ta t ions . T h e Fano-l ike features i n the reflect ivi ty spectra CHAPTER 5. Results and Discussion 95 < o o cm o 11.0 11.5x10 Energy (cm ) F igure 5.24: S-polarized and p-polar ized specular reflectivity data , represented by sol id and dashed lines respectively, for the 150 n m th ick textured planar waveguide w i t h a square la t t ice referred to i n the text. T h e spectra correspond to angles of incidence of 10°, 20° , 30° , 40° and 50° from the bo t tom up. Near 10,400 c m - 1 these spectra exhib i t flat p-polar ized bands, as indicated by the dot-dashed lines. used to extract the mode profiles are sometimes difficult to d is t inguish from the non-resonant background reflectivity. In addi t ion , this technique probes modes above the l ight line only, and does not offer access to the band structure below the l ight l ine. B o t h of these l imi ta t ions can possibly be addressed by incorpora t ing i n the texture pa t te rn a "defect superlat t ice" that weakly perturbs the band structure of the under ly ing "base lat t ice." CHAPTER 5. Results and Discussion 96 A defect superlat t ice couples a smal l amount of l ight out of the waveguide i n a non-specular d i rec t ion v i a the Fourier coefficients of the defect superlat t ice, wh ich i n theory allows m a p p i n g of the band structure of base la t t ice modes below the l ight l ine, whi le at the same t ime enabl ing a background-free method for v iewing the modes above and below the l ight l ine. It is desirable to characterize modes below the l ight l ine of the base la t t ice because these modes do not' have radiat ive components, and thus have the potent ia l to efficiently t ransmit informat ion in-plane for device appl icat ions. In some cases, a defect superlat t ice introduces a "defect band" that appears inside the pseudo-gap characterist ic of the base lat t ice (see Sect ion 2.1.3). T h i s band can occur whether or not the defect superlat t ice has a significant affect on the under ly ing base la t t ice s tructure. T h e author is unaware of any other publ ished work that considers this app l ica t ion of defect superlatt ices as a means of extending the u t i l i t y of whi te l ight scat ter ing as a powerful probe of the entire band structure of p l a in photonic crystals . T h i s section describes work a imed at evaluat ing the prac t ica l u t i l i t y of this novel concept. T h i s me thod of p rob ing the band structure of textured planar waveguides is explored b o t h exper imenta l ly and theoret ical ly v i a character izat ion of three waveguides: the previously discussed "flat band" square la t t ice as well as the previously discussed t r iangular la t t ice , and an add i t iona l waveguide w i t h a t r iangular lat t ice. T h e me thod proves especial ly useful i n character iz ing low-dispersion bands above the l ight l ine. T h e usefulness of the technique i n p rob ing other bands, such as i n the t r iangular la t t ice structures, is dependant CHAPTER 5. Results and Discussion 97 upon the degree to which the defect superlatt ice perturbs the modes of the under ly ing base lat t ice. F i n a l l y , this section describes a textured planar waveguide design i n w h i c h a defect band exists w i t h i n a complete pseudo-gap. T h e potent ia l use of such a s t ructure as the basis for a broad band angle and polar iza t ion insensitive notch filter is discussed. 5 . 3 . 1 Superlattice Scattering from a Low Dispersion Band T h i s section explores the use of the diffraction measurement technique to further analyze the G a A s "flat band" textured planar waveguide sample described i n Sect ion 5.2.2. R e c a l l that this waveguide has a square base lat t ice of holes w i t h a defect superlat t ice that omits every fifth hole of the base lat t ice i n the x and y direct ion. T h i s gra t ing pa t te rn is c lear ly evident i n the scanning electron micrograph of this sample shown i n F i g u r e 5.25. N o t e that the or ienta t ion of the lat t ice i n this picture is indicat ive of the pa t te rn on the ac tua l sample: the la t t ice is on a 90 / m i x 90 / m i por t ion of the waveguide, and the square pa t te rn is ro ta ted 22.5° w i t h i n the 9 0 / / m x 9 0 / / m square. T h i s was done to prevent the square aperture diffraction pa t te rn from interfering w i t h the col lect ion optics dur ing diffract ion da ta col lect ion. T o ob ta in the diffraction da t a for the region of the band s t ructure con ta in ing the flat bands, the exper imenta l apparatus was oriented to collect the diffraction spectra f rom the -KD diffracted order at an energy of 10,000 c m - 1 (the approximate loca t ion of the flat bands). T h i s required,adjustment to the sample mount , as wel l as the incident l ight , for each angle of incidence. T o calculate the angles for the or ienta t ion of the apparatus CHAPTER 5. Results and Discussion 98 • . • * .'••*.•• .•••*« Figure 5.25: S E M micrograph of the textured planar waveguide with a square lattice fabricated using ASU506. The air filling fraction is 18%. The pitch of the lattice is 390 nm. The thickness of the GaAs core is 150 nm. the following equations are used Ksmein-KD = Ksmdout (5.1) where and KD = — (5.2) 9 \D  v ' K = \ (5.3) and where AD is the spacing between defects. CHAPTER 5. Results and Discussion 99 Diffraction spectra were collected for the same angles of incident light as were used to obtain the specular reflectivity spectra shown in Figure 5.24. Figure 5.26 shows the experimental data for the -KD diffracted order while Figure 5.27 shows the simulated -KD diffracted order. The spectra presented here are raw data, as opposed to the normalized5 data presented for the specular reflectivity spectra. The non-dispersive (flat) s- and p-polarized bands from the TE-like gap, as well as the downward dispersive p-polarized band from the TM-like gap, occur within the selected energy range. The simulation for this energy range, shown in Figure 5.27, produces two modes: a higher energy p-polarized mode and a lower energy s-polarized doublet. For the experimental data, a polarizer was used to confirm that the higher energy mode is p-polarized and the lower energy doublet is predominantly s-polarized. In both the data and the simulation, the p-polarized mode near 10,400 c m - 1 is clearly evident. Slightly lower in energy there is a less intense s-polarized doublet. Both the s- and p-polarized modes show little dispersion up to a 30° angle of incidence, where the s- modes become undetectable. At approximately 40° the p-polarized mode becomes a broad doublet, then at 50° returns to a (now asymmetric) single mode, centered at a slightly higher energy. Overall there is striking qualitative agreement between the data and the simulation. It is now possible to quantitatively compare the linewidth of the modes from the simulation with the actual linewidths from the raw data. Previously, when using data taken with the specular measurement technique, the linewidth would have had to be 5Specular reflectivity data is normalized by dividing the collected spectra by the spectra from bare GaAs. CHAPTER 5. Results and Discussion 100 9.6 9.8 10.0 10.2 10.4 10.6 l l .Ox lO 3 Energy (cm ') Figure 5.26: Diffract ion spectra for the -KD diffracted order from the defect superlat t ice of the tex tured planar waveguide w i t h a square lat t ice for incident angles of 10°, 20° , 30° , 40° and 50° , from the b o t t o m up. ca lcula ted w i t h the mathemat ica l fitting technique, as described i n Sect ion 5.1.1. T h i s me thod is not well-sui ted for ca lcula t ing the l inewid th for closely spaced modes, such as the ones associated w i t h the anti-crossing seen here. W h e n using the new diffraction measurement technique, the l inewid th is immedia te ly available i n the raw data . T h u s , there is l i t t le uncer ta inty i n this measurement, and artifacts are not in t roduced by the fitting procedure. Specif ical ly for the flat band, the s imula t ion shows the l i newid th for CHAPTER 5. Results and Discussion 101 h -*' i 1 i ! / I -u ! -U L r | , i ; \\ 1 7'. 1 " 1 1 1 — i — — i — — i — ...... , i 9.6 9.8 10.0 10.2 10.4 10.6 11.0xl0 J Energy (cm ') F igure 5.27: S imula ted p-polar ized diffraction spectra (dashed lines) and s-polarized diffraction spectra (solid lines) for the -KD diffracted order from the defect superlat t ice of the "flat band" sample for incident angles of 10°, 20° , 30° , 40° and 50° , from the b o t t o m up. the p-polar ized band to be 9 7 c m - 1 at 10°. In the exper imenta l da ta , the l i newid th for the p-polar ized band is 1 5 0 c m - 1 at 10°. T h e mode is broader i n the da ta t h a n i n the s imula t ion , wh ich is l ike ly due to the modes being effected by phys ica l imperfect ions wh ich are not taken in to account by the s imula t ion . Slight irregulari t ies i n the t ex tu r ing of this waveguide are suspected of causing some degree of broadening of the modes, as CHAPTER 5. Results and Discussion 102 was suspected to be the case for the polymer waveguide discussed i n Sect ion 5.2.1. T h e diffraction measurement technique removes one significant source of uncertainty, a l lowing a more precise, quantifiable evaluat ion and comparison of the lifetimes of these modes. T h e ~ 5 0 c m _ 1 differential i n the measured and s imulated bandwid ths is consistent w i t h the l i newid th comparisons done by V . Pac radoun i on the square lat t ice sample of Sect ion 5.1.1, us ing on ly specular data . 11.2x10 H 10.8 -i J l 10.4-10.0 -9.6-S-H 9.2 0 r o . i 0.2 0.3 K„/K 0.4 0.5 X Figure 5.28: Dispers ion d iagram of the flat p-polar ized bands showing s imula t ion and data . D o t t e d line represents the s imula t ion for the base lat t ice. Large dots represent central loca t ion of the mode derived exper imenta l ly v i a the -K® diffracted order. E r r o r bars on da ta points represent the w i d t h of gaussian fits to mode profile from diffraction data . Qual i ta t ive ly , the pos i t ion of the modes i n the s imula t ion has good agreement w i t h CHAPTER 5. Results and Discussion 103 the modes i n the data . T h i s comparison is made more easily quantifiable by the diffrac-t ion measurement technique. W i t h specular reflectivity data, the ma themat i ca l fitting technique can be used to approximate the posi t ion of the modes. However, when modes are posi t ioned close together, this method is problemat ic . T h e diffraction technique fa-cil i tates reliable, precise posi t ion extract ion wi thou t compl ica ted mathemat ica l fitting, thus a l lowing easy ca lcula t ion of dispersion. For the p-polar ized band depicted i n F i g -ure 5.26, the da t a show a 1.0% dispersion over more than 30% of the B r i l l o u i n zone, whi le the s imula t ion predicts 0.6%. There is only 0.4% difference between the theoret ical and measured dispersion; the level of accuracy to wh ich this is measured is not possible w i t h the specular measurement technique. F igure 5.28 shows the s imula t ion of the flat p-polar ized bands and the mode locations from the diffraction data . T o further investigate these low dispersion bands, the computer code was used to extract the strength of the p-polar ized Fourier field components for various values of i n -plane wavevector. F igu re 5.29 shows momentum space diagrams of these results. F igures (a)-(c) are for the upper band, that originates from the T M - l i k e gap and anticrosses, becomming flat, whereas Figures (d)-(f) are for the lower band, tha t originates from the T E - l i k e gap and then disperses down i n energy after the anticrossing. P r i o r to the anticrossing, the upper band is p r imar i l y comprised of T M field components . Af t e r the ant icrossing it is composed ma in ly of T E field components. T h i s is evident for this p-polar ized band because the single dominant Fourier coefficient lies on the l ine of symmet ry before the anticrossing. Af ter the anticrossing the s t rong Four ier coefficients CHAPTER 5. Results and Discussion 104 occur i n symmet r ic pairs, which add asymmetr ica l ly to produce a p-polar ized mode. T h e lower band starts near zone center p r i m a r i l y composed of T E field components and t ransi t ions to being composed p r imar i ly of a T M field component after the anticrossing. t X (a) Kn = 0.15Kg, (b) = 0.25Kg, (c) Ku = 0A0Kg, up-upper band upper band per band X (d) #|| = 0.l5Kg, (e) K\\ = 0.25Kg, (f) K\\ = 0A0Kg, lower band lower band lower band F igure 5.29: M o m e n t u m space diagrams of low dispersion bands before and after the anticrossing. T h e strength of each Fourier component is indica ted by the size of the dot . Figures (a)-(c) correspond to the band that originates at the b o t t o m of the T M - l i k e gap. Figures (d)-(f) correspond to the band that originates near the top of the T E - l i k e gap. W h e n compar ing the spectra obtained using the diffraction technique w i t h that ob-ta ined w i t h the specular measurement technique, there are two notable discrepancies. F i r s t , the downward dispersive p-polar ized mode evident i n the specular ref lect ivi ty d a t a is conspicuously miss ing from the diffraction data . T h i s mode is also absent from the s imulated diffraction spectra. A l t h o u g h the band itself is not vis ible , at 40° there is clear CHAPTER 5. Results and Discussion 105 evidence of i t anti-crossing w i t h the flat band. Secondly, the lower energy s-polarized feature that appears faint ly i n the specular reflectivity da ta is clearly shown i n the diffrac-t i on da t a to be a doublet . Simulat ions of this mode w i t h wider defect spacing show this doublet to be a single mode, as i t has to be i n the absence of the superlat t ice. T h e diffraction measurement technique has been shown here to enable background-free p rob ing of these low dispersion bulk modes. In addi t ion , us ing this technique to further characterize flat bands has improved the level of quantifiable results us ing this type of whi te l ight probe technique. 5.3.2 Superlattice Diffraction from Triangular Lattice Struc-tures For the p lanar waveguide discussed i n the previous section, bo th the exper imenta l and the theoret ical diffraction results were completely domina ted by modes associated w i t h flat bands l y i n g above the air l ight l ine. To investigate the use of the diffraction measurement technique to probe more dispersive modes, a planar waveguide was selected wh ich is tex tured w i t h the previously described t r iangular lat t ice configurat ion, k n o w n to exhibi t re la t ively dispersive bands. T h i s section reports exper imenta l and theoret ical studies of defect diffraction from two samples of this waveguide which differ on ly i n the la t t ice constant of the defect superlatt ice. For the technique to provide an effective means of p rob ing the under ly ing base lat t ice band structure, the defect superlat t ice must act only as a weak per turba t ion . "Weak" here implies that the ac tual band structure rendered CHAPTER 5. Results and Discussion 106 i n the first B r i l l o u i n zone of the superlat t ice can be interpreted as t r i v i a l , k inemat ic zone-folding of the bands characterist ic of the base lat t ice structure (as i l lus t ra ted i n F igure 2.7). F igure 5.30 i l lustrates the same ideas i n a s l ight ly different way. Here "weak" su-perlat t ice is described as causing a smal l renormal iza t ion of the (already renormalized) bands characterist ic of the base lat t ice. If the gaps at the defect zone boundaries are smal l , then the diffraction from the various defect lat t ice vectors should follow the base lat t ice dispersion. T h e dark line i n F igure 5.30 label led "0" represents the parameter space s t imula ted d i rec t ly by whi te light at a well-defined angle of incidence. T h i s is the parameter space probed by the specular measurement technique, as described i n Sect ion 2.3.1. T h e lines paral le l to the zeroth order represent addi t iona l points i n parameter space probed by adding or subt rac t ing integer mul t ip les of the defect la t t ice gra t ing vec-tor. T h e spectra obtained from any of these diffracted orders ( inc luding specular) should show signs of coupl ing into this mode v i a various mult iples of KD represented by a l l of the intersections. For this example, the +KD diffracted spectra might be expected to dominan t ly show this mode background-free at a higher energy than seen i n the specular reflect ivi ty spectra; this mode should continuously disperse up in energy as the angle of incidence of the probe beam is increased. In the -KD diffracted order, when the probe beam is near no rma l incidence the mode should appear higher i n energy than i t does i n the specular; as the angle of incident light is increased, the mode should disperse down i n energy u n t i l zone center, where i t should " turn around" and disperse up i n energy. CHAPTER 5. Results and Discussion 107 + M Figure 5.30: Schematic d iagram i l lus t ra t ing the diffraction probe technique. T h e bo ld , straight l ine at 0 represents the parameter space d i rec t ly s t imula ted by the incident l ight . T h e dashed l ine represents the upper most p-polar ized band, and the sol id represents the upper most s-polarized band of the t r iangular la t t ice structure (see F igu re 5.6). A t some point as the defect lat t ice spacing becomes comparable to the base la t t ice spacing, this s imple interpretat ion of renormal iz ing the already renormal ized base-lattice modes cannot be used effectively. Instead, the band structure must be interpreted i n terms of the renormal iza t ion of the slab modes by a complex uni t cel l wh ich includes the defect superlat t ice. For example, consider a t r iangular lat t ice w i t h a defect superlat t ice which omits every fifth hole. T h e uni t cell for this lat t ice contains nineteen sites, as opposed to the non-defect version of this lat t ice, which contains on ly one. So, where previously seven Fourier coefficients were sufficient to describe the band s t ructure near CHAPTER 5. Results and Discussion 108 the second order gap, now ninety-one Fourier coefficients must be used to describe the renormal ized band s t ructure to the same accuracy. T o evaluate the diffraction technique's efficacy i n revealing the band structure of the under ly ing base lat t ice, two samples are investigated: the G a A s waveguide described i n Sect ion 5.1.2, wh ich has a defect superlat t ice w i t h every seventh hole omi t t ed , referred to as T-7 ; and a s imi la r G a A s waveguide w i t h every fifth hole miss ing, referred to as T - 5 . A scanning electron micrograph of T - 5 is shown i n F igure 5.31. T h e exper imenta l apparatus was oriented to collect diffraction da ta from ~ 1 0 , 0 0 0 c m _ 1 because this is the range of energies where the second order gap from the base lat t ice gra t ing is located, as determined v i a specular reflectivity measurements. T h e angles required to orient the sample were ca lcula ted us ing E q u a t i o n 5.1 w i t h = ^ 0 . T h e samples were oriented so that the l ight was incident along the M direct ion. T h e diffraction da ta was collected wi thou t a polar izer due to the low signal strength. T h e -KD diffraction da t a for T - 5 are shown i n F igu re 5.32. T h e modes w h i c h are evident i n the b o t t o m four spectra (2° , 4° , 6° and 8°) are dispersing down i n energy. N o da t a cou ld be collected between 8° and 14° since the signal for these angles is phys ica l ly b locked by the apparatus. T h e top four spectra, wh ich correspond to angles of incident l ight of 14°, 16°, 18° and 20° , show modes which are generally dispersing up i n energy. F igure 5.33 shows the +KD diffraction da ta from sample T - 5 . T h i s da ta was collected for angles of incident l ight of 2° , 4° , 6° , 8° , and 10°. There are no well-defined modes, but there is an upward t rend i n the dispersion of the weak structures i n these spectra. T h e CHAPTER 5. Results and Discussion 109 F igure 5.31: Scanning electron micrograph of T-5 sample: a planar waveguide t ex tu red w i t h a t r iangular base lat t ice, w i t h a defect superlatt ice 5x the pe r iod of the base la t t ice . T h i s waveguide was fabricated using A S U 7 2 1 , and the p i t ch of the la t t ice is 600 n m . diffraction signal i n the +KD d i rect ion is ~ 5 x weaker than that i n the -KD d i rec t ion . Table 5.5: K e y model ing parameters for T - 5 Parameter Va lue P i t c h 560 n m Hole radius 70 n m Thickness of textured layer 73 n m Thickness of oxide layer 300 n m Simula t ions were performed for T-5 using the parameter values i n Tab le 5.5. T h e s imula t ion results are shown i n F igure 5.34 for the -KD and F igu re 5.35 for the +KD diffract ion directions. For the -KD diffracted order, a group of s- and p-polar ized modes disperses down i n energy for incident angles from 2° to 12°, then "turns a round" and CHAPTER 5. Results and Discussion 110 8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.4xl0 3 Energy (cm ) F igure 5.32: Unpo la r i zed -K® diffraction da ta for T - 5 . Spect ra from the b o t t o m up are for incident angles of 2° , 4° , 6° , 8° , 14°, 16°, 18°, and 20° . disperses up i n energy through 20° . T h i s " turn around" is located at the point where these bands pass through the first defect B r i l l o u i n zone boundary, wh ich is consistent w i t h the da ta i n F igure 5.32. For the +K® diffracted order, there are several modes of b o t h s- and p-polar iza t ion which disperse up i n energy for a l l angles of incidence from 2° to 20° . T h i s is qual i ta t ive ly consistent w i t h the behavior of the weak peaks i n F igu re 5.33. T h e calcula ted intensi ty of the s imulated modes for the +K® d i rec t ion are ~ 5 x weaker t han those for the -Kg s imulat ions. T h e s-polarized specular reflectivity da ta taken i n the M di rec t ion for T - 5 are shown CHAPTER 5. Results and Discussion 111 8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.4xl0 3 Energy (cm ) F igu re 5.33: U n p o l a r i z e d diffraction da ta for T - 5 . Spect ra from the b o t t o m up are for incident angles of 2° , 4 ° , 6° , 8° , and 10°. i n F igure 5.36. Cor responding simulat ions for s- and p-polar izat ions for various angles of incidence are shown i n F igure 5.37. A t a gross level, b o t h the da ta and the s imulat ions are very s imi lar to those corresponding to a comparable structure w i t h no defect superlat t ice. However, evidence of the per turba t ion is exhibi ted clearly i n the v i c i n i t y of the broad , h igh energy modes at 2.5° and 5° angles of incidence. These add i t iona l undula t ions are more pronounced i n the s imulat ions than i n the data , especially at large angles of incidence. In summary, the T - 5 diffraction da ta i n the +KgD and -K® direct ions exhibi ts many of CHAPTER 5. Results and Discussion 112 5x10 H 10.8x10 Energy (cm ) Figure 5.34: S imula t ion for -KD diffracted order from the defect superlat t ice of T-5 . P -po la r i zed is represented by the dashed lines, and s-polarized by the sol id . CHAPTER 5. Results and Discussion 113 rR7L560r70t20~ <L> 1x10 -4 0 'R7L560r70tl8 "R7L560r70tl6~ "R7L560r70tl4_ "R7L560r70tl2-'R7L560r70tlO" "R7L560r70t8" "R7L560r70t6" "R7L560r70t4~ "R7L560r70t2" J .Li i L L i 8.8 9.2 9.6 10.0 Energy (cm *) 10.8x10 Figure 5.35: S imula t ion for +KD diffracted order from the defect superlat t ice of T - 5 . P -po la r i zed is represented by the dashed lines, and s-polarized by the sol id . CHAPTER 5. Results and Discussion 114 F igure 5.36: S-polarized specular reflectivity da ta for T - 5 , taken at various angles of incidence i n the M direct ion. Figure 5.37: S-polarized (solid lines) and p-polar ized (dashed lines) specular ref lect ivi ty s imulat ions for T - 5 , at various angles of incidence i n the M direct ion. CHAPTER 5. Results and Discussion 116 the qual i ta t ive features predicted by the s imulat ions. G o i n g away from n o r m a l incidence i n the M direct ion, the -K® da ta contains several modes that disperse down i n energy out to ~ 1 0 ° angle of incidence, beyond which the group starts to disperse back up i n energy. In contrast , the modes in the +KD d i rect ion monotonica l ly increase i n energy away from norma l incidence out to at least ~ 2 0 ° . T h e l inewidths of the modes i n the -K® direc t ion are w i t h i n a factor of two of the s imulated modes' l inewidths , and the scat ter ing s trength i n the -Kg d i rect ion is substant ia l ly stronger than i n the +Kg° d i rec t ion . T h e p r inc ipa l difference between the experiment and simulat ions lies i n the relat ive strengths of the various modes that appear i n any given order of diffraction. T h i s can largely be a t t r ibu ted to the fact, as borne out by simulat ions, that the dispersion efficiency i n any given order seems to be exceedingly sensitive to variat ions i n incident angle and the structure 's precise physical characteristics. M o r e discussion regarding the nature of the modes observed i n these spectra follows the presentation of a s imi lar s tudy of sample T - 7 . U s i n g the diffraction measurement technique to probe sample T - 7 produces the -Kg da ta shown i n F igure 5.38. These spectra were obtained at angles of incident l ight of 2° , 4° , 6° and 15°. N o da ta could be collected from 8° to 14° because the s ignal for these angles is phys ica l ly blocked by the apparatus. T h e b o t t o m three spectra show a single, well-defined mode dispersing down i n energy. A t 15° there is a single mode located at approximate ly the same energy as the mode at 2° . F igure 5.39 shows the +K® diffraction da t a for this sample at 2° and 4° . One mode is apparent; i t disperses up from 2° to 4° . CHAPTER 5. Results and Discussion 117 A g a i n , the diffraction signal for the +KD is ~ 5 x weaker than that for the -KD'. 9.4 9.6 9.8 10.0 10.4xl03 Energy (cm *) Figure 5.38: U n p o l a r i z e d -KD da ta for T -7 , taken at 2, 4, 6, and 15° i n the M di rec t ion . S imula t ions were performed for T - 7 w i t h the same values used previously to mode l this waveguide, as described i n Section 5.1.2. T h e s imulat ions for the -KD diffracted order, shown i n F igure 5.40, exhibi t one s- and one p-polar ized group of modes dispers ing down i n energy from 2° , f lat tening out between 6° and 10°, and then dispers ing up i n energy through 18°. T h e "turn around" i n the direct ion of dispersion occurs between 6° and 10°, wh ich corresponds w i t h the condi t ion where the modes encounter the first B r i l l o u i n zone boundary of the defect lat t ice (just as shown for T -5 ) , wh ich occurs for this s t ructure at 7.9° for 10,000 c m - 1 . T h e two dashed lines indicate the loca t ion of the base la t t ice modes, as indicated by the specular reflectivity s imulat ions for the non-defect la t t ice (shown previously i n F igure 5.10). T h e s imulat ions exhibi t qual i ta t ive agreement CHAPTER 5. Results and Discussion 118 < 9.4 9.6 9.8 10.0 10.4xl03 Energy (cm"1) Figu re 5.39: Unpo la r i zed +KgD da ta for T -7 , for 2 and 4° i n the M direct ion. w i t h the da ta w i t h regard to the intensity of the modes, and to the general dispersion; however there is only one mode apparent i n the data. T h e s imi la r i ty between the 2° and 15° spectra observed i n the da t a is also shown i n the s imulat ions . For the +KgD s imulat ions , shown i n F igure 5.41, one s- and one p-polar ized mode are each dispers ing cont inuously up i n energy for this range. T h e dashed line indicates the loca t ion of the upward dispersing s-polarized mode, as indicated by specular reflect ivi ty s imulat ions . T h e continuous upward dispersion of the two modes i n the +KgD s imulat ions corresponds to the upward dispersion of the single mode shown in the data. T h e widths of the modes also compare wel l . Overa l l , the diffraction da t a and s imulat ions for sample T - 7 exhibi t many of the same features seen i n the T-5 sample, but the number of modes w i t h significant diffraction efficiency is less i n T -7 , m a k i n g it easier to interpret the results. CHAPTER 5. Results and Discussion 119 c 9.6 9.8 10.0 10.2 10.4xl03 Energy (cm l) Figure 5.40: S imu la t i on of -KD spectra for T -7 . So l id lines represent s-polar izat ion; dashed lines represent p-polar iza t ion . S imula t ion is for the M direct ion and for angles of 2° to 18° from the bo t tom up. T h e in terpreta t ion that follows makes reference to the s imulat ions shown i n Figures 5.40 and 5.41, and the schematic d iagram of the defect-zone-folded band s tructure i n F igure 5.42. T h e zone-folding shown i n F igure 5.42 involves only the base lat t ice s- and p-polar ized bands (represented by sol id lines i n the figure) that disperse upward from a c o m m o n or ig in at 9,750 c m - 1 , as i n F igure 5.6. Fur thermore, it on ly includes those parts of these bands that are folded by reciprocal lat t ice vectors oriented along the M direct ion. T h a t is, i t is effectively a I D reduct ion that ignores bands folded by defect la t t ice vectors that do not lie on the T - X axis. B y i l l u m i n a t i n g the sample w i t h a l ight beam corresponding to the th ick gray l ine i n CHAPTER 5. Results and Discussion 120 9.6 9.8 10.0 10.2 10.4x103 E n e r g y ( c m *) Figure 5.41: S imula t ion of +K® spectra for T -7 . Sol id lines represent s-polar izat ion; dashed lines represent p-polar iza t ion . S imula t ion is for the M direct ion and for angles of 2° to 18° from the b o t t o m up. F igure 5.42, many bands are excited at locations indica ted by the open circles. A l l of the zone-folded bands i n this figure can be excited w i t h a beam incident at angles between 0 and ~ 7 . 9 ° . T h e modes that show up clearly i n the -Kg s imulat ions for sample T - 7 over this angular range correspond to the two downward dispersive bands shown as dashed lines i n the figure. T h e modes that show up clearly i n the +K® s imulat ions over this angular range correspond to the two of the upward dispersive bands. Specifically, the lower energy mode corresponds to the upward dispersing s-polarized band from the base lat t ice (solid line i n F igure 5.42 l a b e l e d ' s ' ) whi le the higher energy mode corresponds to the dash-dot l ine wh ich is the second zone-fold of the upper dispersing p-polar ized CHAPTER 5. Results and Discussion 121 co (cm1) T Kg D/2 K g D Figure 5.42: Schematic zone d iagram showing upward dispersing bands at top of second order T E - l i k e gap i n the M direct ion. T h e dashed lines represent the downward dispers ing bands and the sol id lines represent the upward dispersing bands. band from the base lat t ice. Other bands, most no tab ly the lowest upward dispers ing p-polar ized band , are almost completely absent i n the -KD diffraction s ignal . L o o k i n g back at the T - 5 s imulat ions, some of the peaks correspond to these same bands. Note , however, that the lowest upward dispersing p-polar ized branch is vis ible , as are higher l y i n g bands. T h e p r inc ipa l difference between T - 7 and T - 5 samples is that there are numerous other bands i n T-5 that contr ibute to the diffraction, a l though they are weaker than those i n T - 7 . M o s t of these addi t iona l bands i n T - 5 are effectively zone-folded by reciprocal la t t ice vectors oriented away from the T - X axis. Thus i t appears that for d i lu te superlattices the signals diffracted v i a different orders of the defect la t t ice provide a direct, background-free probe of the I D zone-folded base CHAPTER 5. Results and Discussion 122 bands, as ant ic ipated. M o r e dense superlattices cause a more significant pe r tu rba t ion to the base band structure, resul t ing i n more compl ica ted spectra. A l t h o u g h i n pr inc ip le the da ta could be "unfolded" i n 2 D , this would represent a significant challenge. T h e biggest difficulty appears to be the huge var iab i l i ty i n the diffraction efficiency, b o t h w i t h i n a band (as a funct ion of in-plane wavevector) and between different bands. These efficiencies are also very sensitive to the geometrical parameters of the structures. W i t h this unders tanding, the diffraction da ta from T - 7 can also be better interpreted. F i g u r e 5.43 shows the s-polarized, and F igu re 5.44 the p-polar ized specular ref lect ivi ty da ta collected i n the M direct ion for T -7 . T h e high energy mode i n the s-polarized d a t a and the h igh energy mode i n the p-polar ized da ta are degenerate where they intersect zone center at 9,750 c m - 1 . These modes disperse up i n energy away from zone center. T h e mode seen i n the +Kg° diffraction da ta (Figure 5.39) also has a value of 9,750 c m - 1 at zone center, and disperses up i n energy away from zone center. T h e mode i n the +Kg da ta is a background-free representation of the s-polarized base la t t ice mode seen i n the specular reflectivity spectra. Referr ing to the -K® diffraction da t a (Figure 5.38), the mode disperses down from 9 , 9 0 0 c m - 1 to 9 , 7 5 0 c m - 1 as i t approaches 8° , wh ich is the intersection of the mode w i t h zone edge. T h i s mode i n the -KD da t a is a background-free representation of the p-polar ized base lat t ice mode seen i n the specular reflect ivi ty data . T h e relat ive diffraction efficiencies are different from the s imulat ions, but the basic in terpreta t ion is the same. Thus , the superlatt ice diffraction measurement technique does provide a background-free probe of the band structure of textured planar waveguides CHAPTER 5. Results and Discussion 123 when the defect superlat t ice is di lute . < G 50 deg k30 deg p 2 5 deg H20 deg*" 111111111111111111111111111111111111111 j 111111111111111111111111111111111111111 [ 6 7 9 10 11 12 13 14x10 Energy (cm ) Figure 5.43: S-polarized specular reflectivity da ta for T - 7 taken i n the M direc t ion . T h e diffraction measurement technique can extend the level of quant i ta t ive agreement between exper imenta l results and theoretical predict ions by p rov id ing background-free CHAPTER 5. Results and Discussion 124 < +^ 'in S3 I I I I 111111111111 I I I11II111 II1111 111111 I | I 111111 111111 111111 | I I I I 111111I 1111 I I I 11 10 11 12 13 14xl0 3 Energy (cm l) Figure 5.44: P -po la r i zed specular reflectivity da ta for T -7 taken i n the M direc t ion . mode profiles. Specif ical ly i n the case of T -7 , this aids i n the de te rmina t ion of the thickness of the oxide layer beneath the waveguide core. T h e +K® diffraction da t a taken w i t h l ight incident at 2° is shown i n the bo t tom graph of F igure 5.45 w i t h s imulat ions of CHAPTER 5. Results and Discussion 125 this s t ructure done for various thicknesses of the oxide layer. T h e b o t t o m s imula t ion is for a ful ly-oxidized layer, the next is for 900 n m of oxide, the t h i r d is for 300 n m , and the top is for 200 n m . Based on the l inewid th , i t can be concluded that the oxide layer for T - 7 is approximate ly 200 n m thick, which agrees w i t h the Fabry-Pero t spacing evident i n F igu re 5.5. T o conclude this section, the diffraction measurement technique can provide a back-ground-free probe of dispersive modes, as wel l as flat bands. T h e scat ter ing s trength of the flat bands tends to be stronger than that from dispersive bands. T h i s technique' works best for dispersive modes when the defect superlatt ice represents a weak pertur-ba t ion to the base la t t ice modes. W h e n the per turba t ion is strong, the technique is not as useful for p rob ing the base lat t ice modes because the defect superlat t ice causes a compl ica ted renormal iza t ion of the slab modes, rendering the da t a difficult to decipher. W h i l e this technique does not replace the specular reflectivity technique for s tudy ing the dispersion of leaky photonic eigenstates, it does further the qual i ta t ive and quant i ta t ive unders tanding of textured planar waveguide band structure by p rov id ing background-free measurements of the modes ' lineshapes, and hence their lifetimes. 5 . 3 . 3 True Defect Modes T h e previous sections have discussed the incorpora t ion of defect superlatt ices in to tex-tured p lanar waveguides as a means of revealing the leaky-mode band structure char-acterist ic of the under ly ing base lat t ice. T h i s section discusses the design of a t ex tured CHAPTER 5. Results and Discussion 126 F igure 5.45: Tr iangula r superlat t ice diffraction da ta compared w i t h s imulat ions for var-ious oxide thicknesses. Simulat ions and da ta are for -K® diffraction w i t h incident l ight at 2° i n the M direct ion. T h e oxide thickness for the s imulat ions are, from the b o t t o m up, 1600 n m , 900 n m , 300 n m , and 200 n m . CHAPTER 5. Results and Discussion 127 planar waveguide i n which the base lat t ice possesses a complete pseudo-gap. T h e defect superlat t ice configurat ion is s imi lar to that of the previously discussed T - 7 sample. T h i s superlat t ice gives rise to a defect band w i t h i n the complete pseudo-gap that exhibi ts v i r -t ua l ly no dispersion. A s discussed above, such a structure makes possible a whole range of passive as wel l as active opt ica l devices, such as channel waveguides [12,40], no tch filters and lasers [22]. Simulat ions of the specular reflectivity and superlat t ice diffract ion from this sample show clear evidence of bo th the defect band and other base la t t ice bands that lie below the l ight line. These results theoret ical ly val idate the or ig inal concept of us ing defect superlatt ices not only as a background-free probe of leaky loca l ized modes (as discussed i n de ta i l i n Sections 5.3.1 - 5.3.2), but also as a means of p rob ing modes below the l ight l ine, and local ized defect bands. In order to realize a complete pseudo-gap i n a per iodica l ly textured planar waveguide, the la t t ice must possess a h igh degree of symmetry, and have an air f i l l ing fract ion far from zero or one. T o this end, a textured planar waveguide was designed w i t h a t r iangular la t t ice and an air f i l l ing fraction of 34%. T h e other design parameters for this s t ructure are detai led i n Table 5.6. T h i s design includes a defect superlat t ice that omits every seventh hole. T h e s imula ted dispersion d iagram for the T E band i n this s tructure is shown i n F igure 5.46. Note that this dispersion is for the base lat t ice only, i n order to clear ly demonstrate the magni tude of the complete first order T E pseudo-gap. T h i s large gap extends from 7565 c m " 1 to 9120 c m - 1 , wh ich is 18.6% of the center frequency. T h e first order T M gap CHAPTER 5. Results and Discussion 128 Table 5.6: Des ign parameters for textured planar waveguide w i t h defect mode Parameter Value P i t c h 375 n m Hole radius 115 n m Thickness of core 150 n m Core composi t ion G a A s Ox ide thickness 1000 n m Hole depth 150 n m is much smaller and higher i n energy, and does not overlap this region. T h e inc lus ion of the defect superlatt ice i n this waveguide creates a mode w i t h i n the large first order pseudo-gap. Simulat ions of the specular reflectivity for this s t ructure are shown i n F igure 5.47 for a range of incident angles along the M di rec t ion . T h e CHAPTER 5. Results and Discussion 129 pronounced feature at 8 , 3 0 0 c m - 1 is evidence of the existence of this mode. A l t h o u g h this is s t r i c t ly a T E gap, the defect mode is evident i n bo th the s- and p-polar iza t ions . Note that there is v i r t ua l l y no dispersion of this defect band. F igure 5.48 shows the s imulated -KD diffracted signal for this structure. Once again, the defect mode is apparent at 8,300 c m - 1 i n bo th the s- and p-polar izat ions for a l l angles of incidence along the M direct ion. These background-free spectra highl ight several other smaller features, wh ich were not as apparent i n the specular reflect ivi ty s imulat ions . These features occur p r imar i l y near the edges of the pseudo-gap, and are associated w i t h the band edges of the base lat t ice which lie below the l ight line as discussed i n Sect ion 2.3.2. These features are s imi lar i n nature to the defect-zone-folded signatures of the base la t t ice modes discussed extensively above, but here the corresponding base la t t ice bands lie below the air l ight l ine. It is h igh ly desirable to be able to d i rec t ly probe the dispersion of the t ru ly bound modes that lie beneath the l ight l ine, and these s imulat ions demonstrate that defect superlattices offer this abil i ty, i n pr inciple . F igure 5.49 is a momen tum space d iagram i l lus t ra t ing the relat ive strengths of the Fourier coefficients that comprise this defect mode when exci ted at a 10° angle of i nc i -dence. T o make this d iagram, the Fourier coefficients for each reciprocal la t t ice vector were extracted at the central energy of the defect mode for l ight incident at 10° angle of incidence al igned i n the M direct ion. E a c h dot represents a rec iprocal lat t ice vector (by posi t ion) and the relative magni tude (by dot diameter) of the corresponding Four ie r field component . T h i s number of reciprocal lat t ice vectors was required i n order to mode l CHAPTER,5. Results and Discussion 130 6500 7000 7500 8000 8500 9000 9500 10000 Energy (cm l) Figure 5.47: S imula ted specular reflectivity for a textured planar waveguide w i t h a t r i an-gular lat t ice, exh ib i t ing a dispersion-free defect band i n the first order pseudo-gap, shown for angles of incidence of 10°, 20° , 30° , 40° and 50° a long the M direct ion. T h e sol id lines represent s-polarized and the dashed lines represent p-polar ized reflectivity spectra. CHAPTER 5. Results and Discussion 131 6500 7000 7500 8000 8500 9000 9500 10000 Energy (cm l) Figure 5.48: S imula ted -KD diffraction for a textured planar waveguide w i t h a t r iangular lat t ice, exh ib i t ing a dispersion-free defect mode i n the first order pseudo-gap, shown for angles of incidence of 10°, 20,° 30,° 40° and 50° along the M direct ion. T h e sol id lines represent s-polarized and the dashed lines represent p-polar ized diffraction spectra. CHAPTER 5. Results and Discussion 132 a s t ructure w i t h a defect superlat t ice such as this. T h e strong Fourier coefficients (i.e. the larger dots) that contr ibute to this defect mode occur p r i m a r i l y at a radius half-way between zone center and the smallest base lat t ice vectors, wh ich correspond to the seventh r ing of reciprocal la t t ice vectors i n this d iagram. T h i s makes sense because the energies of the base la t t ice T E slab modes at this average wavevector fall w i t h i n a large T E - l i k e pseudo-gap. Reca l l that the corresponding diagrams for dispersive modes shown i n Sect ion 5.1.2 were a l l domina ted by just one or two components. T h e re la t ively large number of significant components associated w i t h the defect bands also makes sense, because an inf ini te ly local ized state would correspond to a uni form dot size i n this figure. (a) s-polarization (b) p-polarization F igure 5.49: M o m e n t u m space d iagram for the defect mode i n the waveguide w i t h the t r iangular la t t ice discussed i n the text. M direct ion is up i n the d iagram. T h e defect superlat t ice for this waveguide omits every seventh hole. T h e size of the dots shows the relative strengths of the Fourier coefficients of the fields excited at a 10° angle of incidence along the M direct ion. CHAPTER 5. Results and Discussion 133 U s i n g these Fourier coefficients and reciprocal lat t ice vectors to plot the intensi ty of the s-polarized fields of the defect mode yields the real space d iagram shown i n F igu re 5.50. T h i s figure shows significant loca l iza t ion of the fields on the locat ions of the defect sites when i l l umina ted w i t h whi te l ight at a 10° angle of incidence along the M direc-t ion . T h i s i l lustrates the fact that l ight w i t h i n the pseudo-gap is not pe rmi t t ed to t ravel classical ly th rough the waveguide, but can tunnel from defect site to defect site. It has been shown that when these defect sites are arranged i n par t icu lar patterns (e.g. straight lines, straight lines w i t h bends, etc.), l ight at the frequency of the defect mode can be made to "follow the defects." [2]. A textured planar waveguide was fabricated to the specifications del ineated above. A s the parameters for this structure are s imi lar to the other G a A s tex tured waveguides fabricated for the research presented i n this thesis, the real izat ion of this waveguide design should be no less successful than the other structures. However, character iza t ion of this waveguide has revealed only a broad, unst ructured emission covering the entire first order pseudo-gap. T h e reason for this is not known. A l t h o u g h this design has not been exper imental ly realized, a patent app l i ca t ion is i n the process of being filed for its use as a po la r iza t ion and angle insensit ive notch filter. R e c a l l that the notch filter referred to i n the context of the po lymer waveguide involved polar izat ion-insensi t ive response, wh ich is desirable; but the center frequency of the notch var ied cont inuously w i t h the incident angle, wh ich may or may not be desirable, CHAPTER 5. Results and Discussion 134 F igure 5.50: R e a l space plot of the magni tued squared of the to t a l vector field at the surface of the defect la t t ice waveguide discussed in the text , for a 10° angle of incidence along the M direct ion. There is s trong local izat ion of the field at the defect sites. depending u p o n the appl ica t ion . Here, the use of a h igh index-contrast t ex tured waveg-uide capable of suppor t ing a complete pseudo-gap, together w i t h the defect superlat t ice, offers po la r i za t ion insensit ive fi l tering at a fixed frequency, independent of incident angle. T o summar ize this section, the theoretical mode l used i n previous sections to quan-t i t a t ive ly exp la in the scat ter ing properties of several different 2 D tex tu red waveguides was used to design a different structure w i t h a complete T E pseudo-gap. T h e mode l predicts that clear signatures of near-dispersionless gap modes and base la t t ice modes CHAPTER 5. Results and Discussion 135 below the l ight l ine should be observed i n whi te l ight scat ter ing spectra when a defect su-per lat t ice is incorporated i n the structure. A l t h o u g h the first and on ly a t tempt to verify these predict ions was not successful, this is almost cer ta inly a consequence of fabr ica t ion difficulties, not an ind ica t ion that the design is flawed. Chapter 6 Conclusions and Recommendations T h e objective of the work presented i n this thesis, to reveal and quantify the broad-band l inear op t ica l scat ter ing properties of 2D textured planar waveguide structures, has been achieved. T h e factors cont r ibu t ing to this success were a versatile, accurate and easy-to-use b roadband spectroscopy apparatus, a rigorous yet efficient computer mode l , and the ab i l i ty to fabricate the relevant samples. T h e author processed the major i ty of the samples, was solely responsible for the apparatus, and extended the mode l ing code by developing and implement ing the routines necessary to mode l defect superlatt ices. Ove ra l l , ou t s tand ing agreement was achieved between the exper imenta l charac ter iza t ion of the samples using the author 's apparatus and the results obtained w i t h the computer mode l ing code. Together the theoretical and exper imental results provide a compre-hensive examina t ion of electromagnetic excitat ions associated w i t h 2 D textured planar waveguides. B y achieving unprecedented agreement of measured and calcula ted band structures of leaky modes associated w i t h the second, and up to the seventh, zone-folded B r i l l o u i n 136 CHAPTER 6. Conclusions and Recommendations 137 zones of square and t r iangular lat t ice structures, a thorough character izat ion of the po-la r i za t ion and dispersive properties of these local ized electromagnetic modes has been achieved. A l l of the results can be interpreted using symmet ry arguments and a p ic ture i n wh ich the true B l o c h states are the result of texture- inducing m i x i n g of T E - l i k e and T M - l i k e slab modes characterist ic of the under ly ing "average" slab waveguide. T h e pure k inemat ic effects of 2 D tex tu r ing are revealed i n the scat tering spectra from a novel azo-polymer-based structure. M o r e substant ial renormal iza t ion effects are clearly evident i n the GaAs-based structures: second order gaps at zone center ~ 1 0 % ; complete first order pseudo-gaps ~ 2 0 % of the center frequencies; significant anti-crossings away from zone center; and nearly dispersion-free modes across the entire B r i l l o u i n zone, b o t h outside and inside a complete pseudo-gap. T h e work presented i n this thesis has cont r ibuted significantly to the quant i ta t ive analysis of the dispersion characteristics of 2 D textured planar waveguides due, i n large part , to the use of the specular reflectivity and diffraction measurement techniques imple -mented i n combina t ion w i t h the special ly designed exper imental apparatus. T h e specular measurement technique has proved to be an invaluable overal l character izat ion too l for p rob ing the leaky photonic modes attached to these planar waveguides. T h e diffrac-t i on measurement technique, developed by the author, has been shown to enhance this character izat ion of leaky modes by p rov id ing background-free spectra, especial ly from modes w i t h low dispersion. It has also been shown to provide background-free spectra from more h igh ly dispersive leaky modes of a textured planar waveguide w i t h a defect CHAPTER 6. Conclusions and Recommendations 138 superlat t ice, when the defects are spaced widely enough so as not to s t rongly per turb the renormal ized base lat t ice modes. In addi t ion , this diffraction measurement technique has been explored for its potent ia l to uniquely supplement specular reflect ivi ty charac-te r iza t ion by enabl ing the probing of a l l base lat t ice modes, i nc lud ing those below the l ight l ine. A l t h o u g h character izat ion of these bound modes has not been exper imenta l ly demonstrated, the mode l s imulat ions give confidence that the method is sound. Recommendat ions for future work include continued efforts to use the diffract ion measurement technique to characterize modes below the light line. T h i s cou ld be made easier by replacing the low power white light source w i t h a higher power arc l amp . Once a good sample is fabricated w i t h a local ized defect mode, a near-field op t ica l scanning microscope ( N S O M ) could be used to map the physical extent of the local ized mode. Bibliography [1] Joseph F . A h a d i a n and J r . C l i f t on G . Fonstad. Epi taxy-on-elect ronics technology for monol i th ic optoelectronic integrat ion. Optical Engineering, 37:3161, 1998. [2] Mehrne t B a y i n d i r , B . Temelkuran , and E . Ozbay. P ropaga t ion of photons by hopping: A waveguiding mechanism through local ized coupled cavities i n three-dimensional photonic crystals. Physical Review B, 61(18):855-858, 2000. 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