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

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c  QUANTITATIVE LINEAR OPTICAL SCATTERING SPECTROSCOPY TWO-DIMENSIONALLY TEXTURED PLANAR WAVEGUIDES by WILLIAM JODY MANDEVILLE B . S . , U n i t e d States A i r Force A c a d e m y , 1988 M . S . , A i r Force I n s t i t u t e of Technology, 1992 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF  PHILOSOPHY in  T H E F A C U L T Y OF G R A D U A T E STUDIES ( D e p a r t m e n t of P h y s i c s .  and Astronomy) W e accept this thesis as c o n f o r m i n g to the required s t a n d a r d  T H E UNIVERSITY OF BRITISH  COLUMBIA  J u n e 2001 © W i l l i a m J o d y M a n d e v i l l e , 2001  OF  In p r e s e n t i n g this thesis i n p a r t i a l fulfilment of the requirements for a n a d v a n c e d degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l m a k e it freely available for reference a n d study. I further agree t h a t p e r m i s s i o n for extensive c o p y i n g of this thesis for s c h o l a r l y purposes m a y be granted b y the head of m y d e p a r t m e n t or b y his or her representatives. It is u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of this thesis for financial g a i n s h a l l not be a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n .  D e p a r t m e n t of T h e U n i v e r s i t y 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  Ju^  Z-O&l  Abstract  Linear white light spectroscopy in conjunction with rigorous computer modeling reveals the fundamental nature of the electromagnetic excitations associated with the simple lattice and defect superlattice texturing of 2D planar waveguides. By achieving unprecedented 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. A n evaluation of the usefulness of a newly developed diffraction measurement technique for probing band structure is presented in conjunction with data and simulations 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  2.2 2.3  3  4  Background 2.1.1 U n t e x t u r e d P l a n a r Waveguides 2.1.2 2 D Simple Lattices 2.1.3 Defect Superlattices Green's Function M o d e l  9 10 11 21 24  Spectroscopy  28  2.3.1  Specular Measurement Technique  29  2.3.2  Diffraction Measurement Technique  30  Experimental Design  37  3.1  38  Apparatus 3.1.1  L i g h t Source  38  3.1.2  Optics  40  3.1.3  Positioning Mechanics  43  3.1.4  Spectrometer  45  3.2  A l i g n m e n t a n d O p e r a t i o n of A p p a r a t u s  46  3.3  Alternate Configurations  47  Sample Preparation  48  4.1  G a A s Sample  48  4.1.1  Planar Waveguide G r o w t h  48  4.1.2  Electron Beam Lithography  49  4.1.3  Etching  50  iii  CONTENTS 4.1.4 4.2  iv Oxidation  52  Polymer Sample  53  5 Results and Discussion 5.1  5.2  56  Waveguides w i t h Simple 2 D Gratings  58  5.1.1  Square L a t t i c e  58  5.1.2  Triangular Lattice  62  Waveguides E n g i n e e r e d for Specific A p p l i c a t i o n s  74  5.2.1  2 D T e x t u r e d A z o - P o l y m e r Waveguides  74  5.2.2  F l a t P h o t o n i c B a n d s A l o n g the E n t i r e T - X L i n e of a 2 D Square Lattice  5.3  6  89  W a v e g u i d e s w i t h Defect Superlattices 5.3.1  S u p e r l a t t i c e S c a t t e r i n g from a L o w D i s p e r s i o n B a n d  5.3.2  S u p e r l a t t i c e D i f f r a c t i o n from T r i a n g u l a r L a t t i c e S t r u c t u r e s  5.3.3  T r u e Defect M o d e s  Conclusions and Recommendations Bibliography  94 97 . . . .  105 125  136 139  List o f Tables 3.1  E l l i p s o i d a l m i r r o r specifications  42  4.1  P M M A d e v e l o p i n g recipe  51  4.2  E C R recipe  52  5.1  K e y m o d e l i n g parameters for t e x t u r e d p l a n a r waveguide w i t h square l a t t i c e 62  5.2  K e y m o d e l i n g parameters for t e x t u r e d p l a n a r waveguide w i t h t r i a n g u l a r lattice  70  5.3  K e y m o d e l i n g parameters for t e x t u r e d p l a n a r p o l y m e r waveguide  5.4  K e y m o d e l i n g parameters for "flat b a n d " s t r u c t u r e  5.5  K e y m o d e l i n g parameters for T - 5  5.6  D e s i g n p a r a m e t e r s for t e x t u r e d p l a n a r waveguide w i t h defect m o d e  . . . .  77 91 109  v  . . .  128  List of Figures 2.1  S c h e m a t i c d i s p e r s i o n d i a g r a m for u n t e x t u r e d , a s y m m e t r i c p l a n a r waveguide 12  2.2  S c h e m a t i c of I D zone-folding  13  2.3  S c h e m a t i c d i s p e r s i o n d i a g r a m for a 2 D square l a t t i c e  16  2.4  M o m e n t u m space d i a g r a m for a square l a t t i c e  17  2.5  M o m e n t u m space d i a g r a m for a t r i a n g u l a r l a t t i c e  20  2.6  S c h e m a t i c d i s p e r s i o n d i a g r a m for a 2 D t r i a n g u l a r l a t t i c e  21  2.7  B a n d s t r u c t u r e for t h i n G a A s p l a n a r waveguide w i t h I D defect s u p e r l a t t i c e w i t h air filling fraction of 2.5% . . .  2.8 2.9  23  B a n d s t r u c t u r e for t h i n G a A s p l a n a r waveguide w i t h I D defect s u p e r l a t t i c e w i t h a i r filling fraction of 2 5 %  24  S c h e m a t i c d i a g r a m of a t e x t u r e d p l a n a r waveguide  25  2.10 G r a p h i c a l representation of F o u r i e r coefficient generation technique  . . .  2.11 S c h e m a t i c d i s p e r s i o n d i a g r a m for I D t e x t u r e d waveguide  28 29  2.12 R e p r e s e n t a t i v e reflectivity s p e c t r a from a t e x t u r e d p l a n a r waveguide . . .  31  2.13 S c h e m a t i c d i s p e r s i o n for diffraction p r o b i n g m e t h o d  32  2.14 S i m u l a t e d s-polarized K  D  diffraction s p e c t r u m from waveguide w i t h a de-  fect s u p e r l a t t i c e  35  2.15 M o m e n t u m space d i a g r a m for a square defect l a t t i c e  35  2.16 S i m u l a t e d s-polarized s p e c t r a for various i n - p l a n e wavevectors  36  3.1  E x p e r i m e n t a l a p p a r a t u s schematic  39  3.2  E l l i p s o i d a l m i r r o r design schematic  42  3.3  D e t a i l of E M I a n d E M 2 m i r r o r s  43  3.4  B l o c k d i a g r a m of s a m p l e alignment a p p a r a t u s  44  3.5  Sample mount detail  3.6  A l t e r n a t e s a m p l e m o u n t for cleaved edge v i e w i n g  47  4.1  G a A s - b a s e d t e x t u r e d p l a n a r waveguide schematic  49  4.2  E x p o s u r e p a t t e r n for p o l y m e r sample  54  5.1  S p e c u l a r reflectivity d a t a for waveguide w i t h s i m p l e square l a t t i c e  . . . .  5.2  D i s p e r s i o n d i a g r a m for s i m p l e square l a t t i c e d a t a a n d s i m u l a t i o n s  . . . .  5.3  S p e c u l a r reflectivity d a t a ( r ^ M ) for t r i a n g u l a r lattice  64  5.4  S p e c u l a r reflectivity d a t a (r-K)  65  •  for t r i a n g u l a r l a t t i c e  vi  46  60 61  LIST OF FIGURES  vii  5.5  S p e c u l a r reflectivity d a t a for 10° angle of incidence i n the M d i r e c t i o n . .  66  5.6  D i s p e r s i o n d i a g r a m for t r i a n g u l a r lattice d a t a a n d s i m u l a t i o n s  67  5.7  M o m e n t u m space diagrams for each m o d e of the t r i a n g u l a r l a t t i c e waveguide at 10° angle of incidence  5.8  M o m e n t u m space diagrams for each m o d e of the t r i a n g u l a r l a t t i c e waveguide near zone center  5.9  68 -  69  S i m u l a t i o n s of t r i a n g u l a r lattice w i t h a n d w i t h o u t defect s u p e r l a t t i c e  . .  5.10 S i m u l a t e d s p e c t r a for t r i a n g u l a r lattice sample w i t h t h i n oxide  71 72  5.11 S i m u l a t e d s p e c t r a for t r i a n g u l a r lattice sample w i t h t h i c k oxide  73  5.12 A F M m i c r o g r a p h of the surface of the p o l y m e r waveguide  76  5.13 N o r m a l i z e d specular reflectivity d a t a for p o l y m e r waveguide, X d i r e c t i o n  78  5.14 N o r m a l i z e d specular reflectivity d a t a for p o l y m e r waveguide, M d i r e c t i o n  79  5.15 S p e c u l a r reflectivity s i m u l a t i o n a n d d a t a for p o l y m e r waveguide for 10-30° angles of incidence, M d i r e c t i o n  81  5.16 D i s p e r s i o n d i a g r a m for 2 D p o l y m e r g r a t i n g  82  5.17 S i m u l a t i o n of specular reflectivity of p o l y m e r waveguide at n o r m a l incidence  84  5.18 S i m u l a t i o n of specular reflectivity of p o l y m e r waveguide at 1°, X d i r e c t i o n  85  5.19 D a t a a n d s i m u l a t i o n of specular reflectivity of p o l y m e r waveguide at 3 0 ° , M direction  86  5.20 S - p o l a r i z e d specular reflectivity d a t a for p o l y m e r waveguide at 10°, X direction  88  5.21 A F M m i c r o g r a p h s of two p o l y m e r square lattice gratings  89  5.22 T h e o r e t i c a l dispersion d i a g r a m for square lattice sample s h o w i n g "flat b a n d s "  91  5.23 S p e c u l a r reflectivity s i m u l a t i o n for "flat b a n d " sample  93  5.24 S- a n d p - p o l a r i z e d specular d a t a for "flat b a n d " sample  95  5.25 S E M m i c r o g r a p h of "flat b a n d " sample  98  5.26 U n p o l a r i z e d diffraction d a t a for "fiat b a n d " sample  100  5.27 D i f f r a c t i o n s i m u l a t i o n s for "flat b a n d " sample  101  5.28 D i s p e r s i o n d i a g r a m of the flat p - p o l a r i z e d b a n d s  102  5.29 M o m e n t u m space diagrams of low d i s p e r s i o n b a n d s  104  5.30 S c h e m a t i c d i a g r a m i l l u s t r a t i n g the diffraction p r o b e technique  107  5.31 S E M m i c r o g r a p h of T - 5  109  5.32 U n p o l a r i z e d -Kf 5.33 U n p o l a r i z e d +Kf  diffraction d a t a for T - 5  110  diffraction d a t a for T - 5  Ill  5.34 S i m u l a t i o n for -K® diffracted order from defect s u p e r l a t t i c e of T - 5 . . . .  112  5.35 S i m u l a t i o n for  113  diffracted order from defect s u p e r l a t t i c e of T - 5  . . .  5.36 S - p o l a r i z e d specular reflectivity d a t a for T - 5  114  5.37 S- a n d p - p o l a r i z e d specular reflectivity s i m u l a t i o n s for T - 5 5.38 U n p o l a r i z e d -Kf 5.39 U n p o l a r i z e d +Kf  d a t a for T - 7 d a t a for T - 7  115 ". . . .  117 118  5.40 S i m u l a t i o n of -K% s p e c t r a for T - 7  119  5.41 S i m u l a t i o n of +K°  120  s p e c t r a for T - 7  LIST OF FIGURES  •  5.42 Schematic zone diagram showing upward dispersing bands at top of second order TE-like gap 5.43 S-polarized specular reflectivity data for T-7 5.44 P-polarized specular reflectivity data for T-7 5.45 Triangular superlattice diffraction data compared with simulations for various oxide thicknesses 5.46 Dispersion diagram showing large T E gap 5.47 Simulated specular reflectivity for defect mode in first order gap 5.48 Simulated diffraction for defect mode in first order gap 5.49 Momentum space diagram for defect mode 5.50 Real space plot of electric field intensity on surface of the defect lattice waveguide  viii  121 123 124 126 128 130 131 132 134  Acknowledgements  I w o u l d like to t h a n k m y advisor, D r . Jeff F . Y o u n g , for his u n f a i l i n g s u p p o r t a n d i n s p i r e d guidance.  H e p a t i e n t l y a n d unselfishly p r o v i d e d m o t i v a t i o n , c l a r i f i c a t i o n a n d d i r e c t i o n  w i t h o u t regard for his pressing schedule. I a m i n d e b t e d to the members of D r . Y o u n g ' s lab (past a n d present) for t h e i r suggestions a n d assistance t h r o u g h o u t the course of this project. In p a r t i c u l a r , I w o u l d like t o t h a n k : D r . P a u l P a d d o n for e l o q u e n t l y r e i n t r o d u c i n g m e to the 'basics' after m y seven year h i a t u s from g r a d u a t e school; V i g h e n P a c r a d o u n i a n d A l l a n C o w a n , for the m a n y e n l i g h t e n i n g discussions we shared; H o n g M a for his enthusiastic c o n t r i b u t i o n s to the p o l y m e r research; a n d A l e x B u s c h for his i n v a l u a b l e g u i d a n c e w i t h the " h a r d  parts"-  math and LaTex. I also w o u l d like to acknowledge m y wife, C h r i s , for her tremendous s u p p o r t t h r o u g h out this o r d e a l , a n d m y sons, D u n c a n , J a c k a n d K i t , for t h e i r patience. dept of g r a t i t u d e t h a t I w i l l never be able t o repay.  ix  I owe t h e m a  Chapter 1  Introduction  P h o t o n i c c r y s t a l s are of great interest due to t h e i r p o t e n t i a l to p r o v i d e advances i n science a n d t e c h n o l o g y c o m p a r a b l e to the advances p r o v i d e d b y s e m i c o n d u c t o r s over the last 50 years.  P h o t o n i c crystals do for photons w h a t s e m i c o n d u c t o r c r y s t a l s do  for electrons. N a t u r e created s e m i c o n d u c t o r crystals such t h a t the a t o m s are a r r a n g e d i n a perfectly p e r i o d i c l a t t i c e spaced o n the order of the w a v e l e n g t h of the electrons propagating through them.  T h i s p e r i o d i c "electrical" p o t e n t i a l creates the r i c h b a n d  s t r u c t u r e w h i c h has been s t u d i e d a n d e x p l o i t e d since the 1950's. N a t u r e has p r o v i d e d few c r y s t a l s w i t h a n analogous " p h o t o n i c " p o t e n t i a l l a t t i c e spaced o n the order of the w a v e l e n g t h of l i g h t . [18,36] However, w i t h recent t e c h n o l o g i c a l advances, such c r y s t a l s are b e i n g m a n u f a c t u r e d . These m a n - m a d e , low-loss p e r i o d i c d i e l e c t r i c m e d i a are k n o w n as " p h o t o n i c c r y s t a l s . " [12] T h e t e r m " p h o t o n i c c r y s t a l " is used to describe s t r u c t u r e s i n w h i c h a p e r i o d i c m o d u l a t i o n of the i n d e x of refraction is used to c o n t r o l the p r o p a g a t i o n of l i g h t t h r o u g h B r a g g diffraction.  In general the p e r i o d i c i t y c a n be i n one d i m e n s i o n  ( I D ) , 2 D or 3 D . T h e u n u s u a l o p t i c a l properties of I D t e x t u r e d d i e l e c t r i c s were recognized 1  CHAPTER  1. Introduction  2  a n d e x p l o i t e d l o n g before the t e r m "photonic c r y s t a l " was i n t r o d u c e d . T h e f a m i l i a r "quarter-wave dielectric stack" s t r u c t u r e is essentially a p h o t o n i c c r y s t a l . It uses a l t e r n a t i n g layers of h i g h a n d low refractive i n d e x materials to efficiently reflect n o r m a l l y - i n c i d e n t light w i t h wavelengths i n a range of a p p r o x i m a t e l y t w i c e the o p t i c a l period  1  of the stack. T h e light w i t h i n this "forbidden gap" is diffracted b a c k w a r d s , so  t h a t 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, a n d the sharpness of the gap edges increases as the n u m b e r of layers increases.  F o r t y p i c a l dielectric m a t e r i a l s , such as a  q u a r t e r wave stack, A n < 0.5, a n d the center frequency a n d w i d t h of the gap v a r y r a p i d l y as the i n c i d e n t angle of the r a d i a t i o n is varied away from the n o r m a l . I n p a r t i c u l a r , the gap s h r i n k s a n d t h e n vanishes at angles greater t h a n ~  3 0 ° . [9] T h e s e effects of I D  p h o t o n i c crystals have h a d great i m p a c t o n o p t i c a l a p p l i c a t i o n s ; 2 D a n d 3 D p h o t o n i c crystals have the p o t e n t i a l to have a n even greater i m p a c t . In 1987, Y a b l o n o v i t c h [37] a n d J o h n [13] i n d e p e n d e n t l y recognized t h a t a 3 D - t e x t u r e d s t r u c t u r e w i t h a sufficiently large index-contrast c o u l d e x h i b i t a c o m p l e t e b a n d g a p :  that  is, s u c h a m a t e r i a l c o u l d i n h i b i t the p r o p a g a t i o n of electromagnetic r a d i a t i o n w i t h i n s o m e c o n t i n u o u s range o f frequencies, regardless of the d i r e c t i o n of p r o p a g a t i o n or t h e p o l a r i z a t i o n of the field.  T h e significance of this g e n e r a l i z a t i o n c a n p e r h a p s best be  a p p r e c i a t e d b y c o n s i d e r i n g t h a t the t o t a l p h o t o n i c density of states w i t h i n such a m a t e r i a l is d r a m a t i c a l l y altered from t h a t available i n b u l k dielectrics. I n the gap the d e n s i t y of states c a n a p p r o a c h zero, a n d near the b a n d edges it is c o n s i d e r a b l y e n h a n c e d over a 1  T h e optical period is the vacuum wavelength divided by the average index of the periodic dielectric.  CHAPTER  1. Introduction  r e l a t i v e l y n a r r o w range of frequencies.  3  T h i s s h o u l d p r o f o u n d l y affect the  fundamental  properties of electronic oscillators i n these materials, a n d these m o d i f i e d properties m a y well f o r m the basis of new o p t i c a 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 p h o t o n i c c r y s t a l b y d r i l l i n g a d i e l e c t r i c b l o c k full of holes on the order of centimeters i n d i a m e t e r . T h e r e s u l t i n g facecentered c u b i c ( F C C ) c r y s t a l structure e x h i b i t s a full b a n d g a p at m i c r o w a v e frequencies. B y i n s e r t i n g a n a p p r o p r i a t e defect site w i t h i n these crystals it is possible t o c o m p l e t e l y localize r a d i a t i o n at frequencies t h a t fall w i t h i n the b a n d g a p .  I n finite-sized c r y s t a l s ,  these l o c a l i z e d defect states act as h i g h q u a l i t y ( Q ) , dispersionless filters.  Numerous  a p p l i c a t i o n s for these microwave p h o t o n i c crystals have been d e m o n s t r a t e d , i n c l u d i n g n a r r o w b a n d n o t c h filters, h i g h l y d i r e c t i o n a l , low-loss antennae, a n d h i g h - t r a n s m i s s i o n I D defect waveguides t h a t c a n i n c l u d e a b r u p t , right-angle bends.  [4,7,11,34]  T h e p o t e n t i a l i m p a c t of these crystals is even greater i n the o p t i c a l a n d near-infrared frequency d o m a i n where t h e y m a y p r o v i d e the p l a t f o r m technology needed to achieve h i g h l y i n t e g r a t e d o p t i c a l processing c i r c u i t s . [1] C u r r e n t o p t i c a l fiber c o m m u n i c a t i o n systems are based l a r g e l y o n discrete b u l k or fiber o p t i c components (splitters, switches, etc.)  t h a t are difficult t o mass-produce i n a cost-effective m a n n e r .  Progress has been  m a d e t o w a r d s i n t e g r a t i n g these functions o n " o p t i c a l chips" based o n p l a n a r waveguide technologies. If the functions c o u l d be integrated u s i n g p h o t o n i c c r y s t a l s , it w o u l d represent the u l t i m a t e i n m i n i a t u r i z a t i o n . T h i s is because light p r o p a g a t i n g i n p h o t o n i c c r y s t a l s w i t h c o m p l e t e bandgaps c a n , i n p r i n c i p l e , be channeled t h r o u g h a n d c o u p l e d  CHAPTER  1. Introduction  4  between lossless defect-waveguides t h a t c o n t a i n bends w i t h effective r a d i i of c u r v a t u r e o n the order of a single wavelength of light. T h e d i s p e r s i o n of these defect guides, a n d the d i s p e r s i o n of the b a c k g r o u n d p h o t o n i c c r y s t a l , c a n be t u n e d t o achieve u n i q u e c o n t r o l over the p r o p a g a t i o n properties of the light i n 3 D . F u r t h e r m o r e , i f i m p l e m e n t e d i n a n e l e c t r o n i c a l l y resonant m e d i u m (such as a I I I - V s e m i c o n d u c t o r , like I n P or G a A s ) , m i c r o c a v i t y lasers c a n be realized t h a t take advantage of the s i n g u l a r p h o t o n i c d e n s i t y of states associated w i t h isolated defect modes (effectively, m i c r o c a v i t i e s w i t h effective v o l u m e s less t h a n 0.1 c u b i c m i c r o n s ) . [35] P h o t o n i c c r y s t a l s w i t h bandgaps i n the near infrared require c o n t r o l over the dielect r i c t e x t u r e o n lengthscales of ~ 2 0 0 - 5 0 0 n m . W h i l e some groups have r e a l i z e d 3 D c r y s t a l structures w i t h gaps i n the near infrared [3,16,39], they are very difficult t o fabricate, especially w h e n engineering defects into t h e i r s t r u c t u r e . W h i l e 2 D p h o t o n i c c r y s t a l s c a n not possess full bandgaps, t h e y do e x h i b i t m a n y of the a t t r a c t i v e features of t h e i r 3 D c o u n t e r p a r t s , w h i l e b e i n g easier t o fabricate.  A l t h o u g h the t o t a l d e n s i t y of p h o t o n i c  states c a n n o t be reduced t o zero i n "pure" 2 D crystals (i.e. ones t h a t are t r a n s l a t i o n a l l y i n v a r i a n t n o r m a l to the plane of the t e x t u r e ) , 2 D p h o t o n i c c r y s t a l s c a n e x h i b i t c o m p l e t e b a n d g a p s i f the out-of-plane m o m e n t u m of the light is restricted. I n fact, pure 2 D c r y s t a l s are i n a sense "better" t h a n 3 D , i n t h a t the two o r t h o g o n a l p o l a r i z a t i o n states (transverse electric ( T E ) a n d transverse m a g n e t i c ( T M ) ) are c o m p l e t e l y i n d e p e n d e n t of each other, m a k i n g it easier t o achieve large bandgaps. I n p r a c t i c e , there is always some out-of-plane  CHAPTER  1. Introduction  5  v a r i a t i o n of the fields even i n b u l k 2 D crystals, w h i c h means t h a t the pure 2 D electrom a g n e t i c s i m u l a t i o n s o n l y a p p r o x i m a t e reality. T h e errors i n p r o p a g a t i o n s i m u l a t i o n s are s m a l l i n the l i m i t of " t a l l " crystals a n d c o r r e s p o n d i n g l y large b e a m cross-sections, b u t t h i s is not always a p r a c t i c a l geometry outside of the l a b o r a t o r y . W i t h respect t o l i g h t - m a t t e r i n t e r a c t i o n s , pure 2 D models r e a l l y o n l y w o r k i f the d i p o l e d i s t r i b u t i o n is also p u r e l y 2 D , w h i c h severely restricts the relevance of such s i m u l a t i o n s . If the d i p o l e d i s t r i b u t i o n is not t r a n s l a t i o n a l l y i n v a r i a n t , there w i l l always be some c o m p o n e n t s of its r a d i a t i o n d i r e c t e d out-of-plane, a n d these w i l l , i n general, c o u p l e t o a finite d e n s i t y o f p h o t o n i c states. T w o - d i m e n s i o n a l l y t e x t u r e d slab waveguides are even s i m p l e r to fabricate because the t e x t u r e m u s t o n l y be as t h i c k as the waveguide, w h i c h is t y p i c a l l y less t h a n a few microns.  T h i s g e o m e t r y is also d i r e c t l y c o m p a t i b l e w i t h e x i s t i n g o p t o e l e c t r o n i c tech-  nologies. W i t h t h i s " 2 D + 1" geometry of t e x t u r e d p l a n a r waveguides, the t r a n s l a t i o n a l i n v a r i a n c e p e r p e n d i c u l a r t o the t e x t u r e d p l a n e is sacrificed f r o m the outset. T h i s i m m e d i a t e l y i n t r o d u c e s a richness i n the electromagnetic b a n d s t r u c t u r e t h a t is absent i n the pure 2 D case, yet d i s t i n c t from the pure 3 D case. T h e r e are some d r a w b a c k s w i t h this 2 D + 1 geometry: there c a n be n o c o m p l e t e p h o t o n i c bandgaps, even i n a r e s t r i c t e d sense; a n d it is difficult (but not impossible) t o avoid i n t r i n s i c out-of-plane c o u p l i n g to r a d i a t i o n modes. A l t h o u g h consequences such as these m a y seem severe, the p o t e n t i a l benefits of a r e l a t i v e l y s i m p l e f a b r i c a t i o n technology a n d the c o m p a t i b i l i t y w i t h i n t e g r a t e d o p t o electronics are significant. T h i s m o t i v a t e s the search for a q u a n t i t a t i v e u n d e r s t a n d i n g 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 highquality, 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 simulations based on the Green's function formalism described in Chapter 2. First, the  CHAPTER  1.  7  Introduction  t h e o r e t i c a l a n d e x p e r i m e n t a l results from two basic structures are used t o derive a c o m prehensive u n d e r s t a n d i n g of electromagnetic e x c i t a t i o n s associated w i t h 2 D t e x t u r e d p l a n a r waveguides. S e c t i o n 5.1.1 discusses the d i s p e r s i o n a n d p o l a r i z a t i o n s of l e a k y m o d e s i n a t e x t u r e d p l a n a r waveguide w i t h a s i m p l e square l a t t i c e of air holes. [21] T h i s w o r k represents the most t h o r o u g h q u a n t i t a t i v e d e s c r i p t i o n of the l o w - l y i n g resonant p h o t o n i c b a n d s of h i g h index-contrast t e x t u r e d p l a n a r waveguides to date. T h i s q u a l i t a t i v e a n d q u a n t i t a t i v e analysis of square lattices was extended to h i g h i n d e x - c o n t r a s t  triangular  lattices, a n d e x p e r i m e n t a l l y confirmed, as described i n S e c t i o n 5.1.2. C h a p t e r 5 also describes two examples of 2 D t e x t u r e d p l a n a r waveguides engineered w i t h specific a p p l i c a t i o n s i n m i n d . T h e first example, i n S e c t i o n 5.2.1, addresses the use of resonant c o u p l i n g t o effect a p o l a r i z a t i o n insensitive n o t c h filter i n l o w i n d e x - c o n t r a s t , 2 D t e x t u r e d p o l y m e r waveguides.  T h i s section provides the first q u a n t i t a t i v e  study  of the s c a t t e r i n g properties of these structures w h i c h were fabricated u s i n g a directw r i t e h o l o g r a p h i c technique i n a special a z o - p o l y m e r by P a u l R o c h o n ' s g r o u p at R o y a l M i l i t a r y C o l l e g e of C a n a d a .  the  T h e second example, discussed i n S e c t i o n 5.2.2,  e x p e r i m e n t a l l y a n d t h e o r e t i c a l l y demonstrates h o w a h i g h i n d e x - c o n t r a s t 2 D t e x t u r e d p l a n a r waveguide c a n be engineered to possess a b a n d t h a t is effectively flat a l o n g the entire T - X d i r e c t i o n of a 2 D square B r i l l o u i n zone. F l a t b a n d s have been t h e o r e t i c a l l y s h o w n to significantly enhance c e r t a i n non-linear o p t i c a l conversion processes i n b u l k p h o t o n i c crystals. I n the final section of C h a p t e r 5, a novel diffraction measurement technique, d e v e l o p e d  CHAPTER  1. Introduction  8  b y the a u t h o r for this research, is e x p l o r e d v i a the c h a r a c t e r i z a t i o n of three t e x t u r e d p l a n a r waveguides w i t h defect superlattices.  T h e efficacy of this m e t h o d for p r o b i n g  the b a n d s t r u c t u r e of h i g h a n d l o w d i s p e r s i o n modes is evaluated. It is s h o w n h o w t h i s background-free measurement technique c a n be used to further the q u a n t i t a t i v e a n a l y s i s of t e x t u r e d p l a n a r waveguides. T h i s section also presents a n d analyzes the e n g i n e e r i n g of a l o c a l i z e d defect m o d e i n the first order gap to realize a design for a n angle a n d p o l a r i z a t i o n insensitive n o t c h 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 semiconductor 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  F i r s t , u n t e x t u r e d waveguides are reviewed.  N e x t , the p o l a r i z a t i o n a n d d i s p e r s i o n  properties of 2 D t e x t u r e d waveguides are discussed for square a n d t r i a n g u l a r lattices. T h e n , the effect of a defect superlattice o n the b a n d s t r u c t u r e is described.  2.1.1  Untextured Planar Waveguides  A p l a n a r waveguide is a s t r u c t u r e t h a t uses t o t a l i n t e r n a l reflection between the core a n d c l a d d i n g m a t e r i a l t o confine light i n the plane of p r o p a g a t i o n . T h e p h y s i c a l characteristics of the waveguide determine w h i c h modes of light are allowed t o p r o p a g a t e . S o l v i n g M a x w e l l ' s equations for a n a s y m m e t r i c waveguide yields independent solutions for t r a n s 1  verse electric ( T E ) a n d transverse magnetic ( T M ) p o l a r i z e d modes t h a t are b o u n d t o the waveguide. A T E mode's electric field is oriented i n the plane of p r o p a g a t i o n , a n d is thus transverse t o the d i r e c t i o n of p r o p a g a t i o n , whereas a T M m o d e has its magnetic field oriented in-plane, transverse to the d i r e c t i o n of p r o p a g a t i o n .  P l o t t i n g the solutions to  the scaler wave e q u a t i o n as a f u n c t i o n of in-plane wavevector (K\\) o n a d i s p e r s i o n d i a g r a m produces a g r a p h like the one shown i n F i g u r e 2.1. N o t e t h a t there are an infinite n u m b e r of discrete modes, b u t o n l y the two lowest modes are s h o w n . T h e lowest energy m o d e is always T E . A l s o p l o t t e d o n the d i s p e r s i o n d i a g r a m i n F i g u r e 2.1 are three straight lines t h a t represent the d i s p e r s i o n of light i n air, i n b u l k core m a t e r i a l , a n d i n b u l k c l a d d i n g m a t e r i a l . T h e slope of each of these lines is p r o p o r t i o n a l to the r e c i p r o c a l of the i n d e x 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 t h e c o r r e s p o n d i n g m a t e r i a l . T h e s e lines are referred t o as light lines. T h e s e l i g h t lines d i v i d e t h e g r a p h i n t o four regions. B e l o w the core light line is a f o r b i d d e n region where n o electromagnetic modes c a n exist. A b o v e t h e air light line is a region where a c o n t i n u u m of r a d i a t i o n modes exist. T h e modes i n this region represent r a d i a t i o n w h i c h is m u l t i p l y reflected b u t otherwise passes through the waveguide. B e t w e e n t h e a i r l i g h t line a n d t h e s u b s t r a t e light line is a region where c l a d d i n g modes occur. T h e s e c l a d d i n g modes c a n b e t h o u g h t of as passing back a n d forth between t h e c l a d d i n g m a t e r i a l a n d t h e waveguide core, b u t n o t passing from t h e core i n t o the a i r d u e t o t o t a l i n t e r n a l reflection. T h e final region, w h i c h occurs between the substrate light line a n d the core light line, is where t h e b o u n d or "slab" modes occur. M o d e s i n this region experience t o t a l i n t e r n a l reflection at b o t h t h e core-air interface a n d t h e c o r e - c l a d d i n g interface. S c h e m a t i c d i a g r a m s of t h e electric field profile for modes i n each of these regions are s h o w n i n this figure. W h e n a p l a n a r waveguide is p e r i o d i c a l l y t e x t u r e d ,  the d i s p e r s i o n of g u i d e d l i g h t  is altered, analogous t o t h e w a y i n w h i c h t h e d i s p e r s i o n of free electrons is a l t e r e d b y a s e m i c o n d u c t o r lattice. T h e next section discusses how t e x t u r i n g changes t h e properties of a p l a n a r waveguide, a n d describes i n d e t a i l two o f t h e m a n y possible l a t t i c e configurations: square a n d triangle.  2.1.2  2D Simple Lattices  A 2 D s i m p l e (no defects) l a t t i c e has a t w o - d i m e n s i o n a l u n i t cell possessing a single lattice site. W h e n a 2 D s i m p l e l a t t i c e is etched i n t o the surface of a p l a n a r waveguide, t h e  CHAPTER  2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 12  co/c  air —— <— core cladding >  K  F i g u r e 2.1: S c h e m a t i c d i s p e r s i o n d i a g r a m for a n u n t e x t u r e d , a s y m m e t r i c p l a n a r waveguide.  T h e two lowest energy modes are shown as b o l d lines. T h e air, s u b s t r a t e a n d  c l a d d i n g light lines are s h o w n .  S c h e m a t i c d i a g r a m s inset i n each r e g i o n i n d i c a t e t h e  electric field profile for modes i n t h a t region.  p e r i o d i c holes f o r m a p e r i o d i c dielectric s t r u c t u r e w h i c h changes the d i s p e r s i o n characteristics of the waveguide. Specifically, w h e n light propagates i n a p l a n a r waveguide w i t h a 2 D p e r i o d i c l a t t i c e , the light B r a g g scatters off the p e r i o d i c d i e l e c t r i c g r a t i n g , c a u s i n g a r e n o r m a l i z a t i o n of the slab modes. P e r B l o c h ' s theorem, the d i s p e r s i o n of the resultant modes represented o n a d i s p e r s i o n d i a g r a m can be folded i n t o the first B r i l l o u i n zone, a l l o w i n g the use of the reduced zone scheme. A s d e p i c t e d i n the s c h e m a t i c d i a g r a m of zone-folding s h o w n i n F i g u r e 2.2, the b a n d is folded back at the edge of the B r i l l o u i n zone (K\\ = Kg/2,  where K  g  is the g r a t i n g vector.). W h e n the b a n d s are zone-folded, p o r t i o n s  of t h e m fall above the a i r l i g h t line, represented b y the g r a y area i n t h e s c h e m a t i c . M o d e s i n t h i s region are considered "leaky" since they have F o u r i e r c o m p o n e n t s w h i c h r a d i a t e  CHAPTER  2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides  13  out of the waveguide. T h e modes b e l o w the air light line are s t i l l b o u n d t o the waveguide a n d have no r a d i a t i n g c o m p o n e n t . A t the first B r i l l o u i n zone b o u n d a r i e s ( 2 D ) , the n o r m a l slope of the d i s p e r s i o n (i.e. group v e l o c i t y ) goes to zero, w h i c h is the result of the B r a g g s c a t t e r i n g c r e a t i n g s t a n d i n g waves a n d o p e n i n g gaps i n the a l l o w e d frequencies. W h e n the gaps are sufficiently large, it is possible for a s m a l l e r 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 b o u n d a r y . I n t h i s case t h e r e is a "complete pseudo-gap" for t h a t p a r t i c u l a r set of r e n o r m a l i z e d slab modes.  F i g u r e 2.2: Schematic of I D zone-folding.  T h e size of the gaps at zone edge, a n d whether or not a gap is c o m p l e t e , d e p e n d u p o n the g r a t i n g m a t e r i a l s a n d l a t t i c e configuration of a 2 D t e x t u r e d p l a n a r waveguide. T h e size of a gap at zone edge is d e t e r m i n e d b y the difference i n energy between the two s t a n d i n g wave solutions w h i c h o c c u r at this p o i n t .  O n e s t a n d i n g wave has nodes  l o c a t e d i n the d i e l e c t r i c , w h i l e the other has nodes l o c a t e d i n the holes w h i c h c o m p r i s e  CHAPTER  2. Linear Light Scattering Spectroscopy  of Textured Planar Waveguides 14  the g r a t i n g . T h e higher t h e index-contrast is between t h e g r a t i n g layer a n d t h e m a t e r i a l c o m p r i s i n g t h e g r a t i n g (air i n this case), t h e larger t h e g a p w i l l be. [12] T h e size o f t h e gap is also affected b y t h e filling f r a c t i o n o f t h e g r a t i n g . S m a l l e r first order gaps o c c u r 2  w h e n t h e air f i l l i n g fraction is near zero or one; t h e first order gaps become p r o p o r t i o n a l l y larger as t h e filling fraction approaches 50%. T h e more t h e g r a t i n g p e r t u r b s t h e modes, the larger t h e gaps w i l l be. W h e t h e r or n o t a g a p is a complete pseudo-gap is r e l a t e d t o the size of t h e gaps at t h e zone boundaries a n d t h e s y m m e t r y of the l a t t i c e c o n f i g u r a t i o n . T h e m o r e s y m m e t r i c t h e l a t t i c e , a n d t h e larger t h e gap, t h e m o r e l i k e l y i t is t h a t there w i l l b e a c o m p l e t e pseudo-gap for a l l directions. W h e n a p l a n e wave is i n c i d e n t o n t h e surface o f a t e x t u r e d p l a n a r waveguide w i t h a well-defined i n - p l a n e wavevector, Ki, t h e i n c i d e n t field c a n b e w r i t t e n as  Ei(Ki\z)  where w  = E e- ° e ^ iw  z  (2.1)  iR  0  = yjuj — Kf, Co = LO/C, a n d p = xx + yy. [5] T h e i n c i d e n t plane wave m a y 2  0  have a n y p o l a r i z a t i o n . If t h e g r a t i n g has a well-defined 2 D p e r i o d i c t e x t u r e  represented  by r e c i p r o c a l l a t t i c e vectors, { G } , the electric field i n t h e g r a t i n g c a n b e c o n v e n i e n t l y T O  represented as [5]  E(p, z) = E E(Ki + G ; ) m  z  e  ^ + ^ .  (2.2)  T h e s e p l a n e waves, w i t h well-defined frequency a n d i n - p l a n e wavevector, u,K\\, are useful t o d e t e r m i n e t h e d i s p e r s i o n o f t h e l e a k y modes a t t a c h e d t o t e x t u r e d waveguides, 2  F i l l i n g 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  b o t h e x p e r i m e n t a l l y a n d t h r o u g h m o d e l l i n g . F o r some general w,K\\, the  non-specular  (G 7^ 0 ) field c o m p o n e n t s i n E q u a t i o n 2.2 w i l l be s m a l l . However, w h e n u>,K\\ coincides w i t h one of the allowed l e a k y eigenstates of the 2 D t e x t u r e d waveguide, one or m o r e of the scattered field components i n E q u a t i o n 2.2 becomes large. F o r the l o w - l y i n g b a n d s s t u d i e d t h r o u g h o u t this thesis, the general d i s p e r s i o n a n d p o l a r i z a t i o n properties  can  often be d e s c r i b e d u s i n g a s m a l l sub-set of the F o u r i e r c o m p o n e n t s of E q u a t i o n 2.2. It is u s u a l l y sufficient t o consider the zeroth order (specular) c o m p o n e n t , a n d one or t w o sets of nearest neighbors. T h e zeroth order c o m p o n e n t is u s u a l l y s m a l l , b u t is essential because it is the one responsible for c o u p l i n g light i n t o a n d out of the g u i d e d modes of the waveguide. T h e p u r e k i n e m a t i c effects of 2 D t e x t u r e o n the dispersion a n d p o l a r i z a t i o n properties of the l e a k y modes are presented i n this section, for b o t h square a n d t r i a n g u l a r l a t t i c e types. O n l y s-polarized scattered fields are considered i n E q u a t i o n 2.2. F o r weak t e x t u r e , a n d under resonant e x c i t a t i o n c o n d i t i o n s , these b a s i c a l l y represent T E - p o l a r i z e d slab modes. M o r e generally, the scattered components i n a 2 D t e x t u r e d p l a n a r waveguide are b o t h s- a n d p - p o l a r i z e d , c o r r e s p o n d i n g respectively a n d a p p r o x i m a t e l y t o T E a n d T M p o l a r i z e d slab modes. However, for m a n y of the G a A s - b a s e d s t r u c t u r e s e x a m i n e d here, the T E - T M s e p a r a t i o n is large enough to effectively decouple the two. W h e n a 2 D s c a t t e r i n g p o t e n t i a l is weak, the m a i n effect the g r a t i n g has o n the ext e n d e d 2 D d i s p e r s i o n of the waveguide is to zone-fold the slab modes i n t o the first ( 2 D ) B r i l l o u i n zone, as i l l u s t r a t e d i n F i g u r e 2.3. T h e general characteristics a n d shape of the  CHAPTER  2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 16  d i s p e r s i o n i n the l i m i t of weak t e x t u r i n g c a n be u n d e r s t o o d b y c o n s i d e r i n g t h a t the m a i n effect of the g r a t i n g is t h e n m e r e l y to i n t r o d u c e new in-plane m o m e n t u m c o m p o n e n t s t o the i n c i d e n t field, thus affording it the o p p o r t u n i t y to excite w h a t are effectively pure T E slab modes w i t h wavevectors given b y Ki +  {G }. m  M  X  F i g u r e 2.3: S c h e m a t i c dispersion d i a g r a m for a t h i n p l a n a r waveguide t e x t u r e d w i t h a 2 D square l a t t i c e .  T h e 's' a n d ' p ' designations refer to p o l a r i z a t i o n s of the p h o t o n i c  eigenstates, as discussed i n the text.  F i g u r e 2.4 is a m o m e n t u m space d i a g r a m of the nine d o m i n a n t F o u r i e r c o m p o n e n t s of the dielectric scattering p o t e n t i a l for a s i m p l e square lattice. T h e r e c i p r o c a l - l a t t i c e vectors of a square l a t t i c e are given by G = %K x + jK y g  are p o s i t i v e integers i n c l u d i n g zero.  g  where K  g  = 1/A, i and j  F i g u r e 2.4(a) corresponds to the i n c i d e n t plane-  wavevector, Ki, a l i g n e d a l o n g the X s y m m e t r y d i r e c t i o n . F i g u r e 2.4(b) corresponds  to  the i n c i d e n t plane-wavevector, Ki, aligned a l o n g the M s y m m e t r y d i r e c t i o n . F o r b o t h of these figures, the c e n t r a l p o i n t represents the zeroth order c o m p o n e n t of the s c a t t e r i n g  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 1 p o t e n t i a l , a n d Kj is represented b y the s m a l l d o t t e d vectors. T h e i n - p l a n e wavevectors of the T E field c o m p o n e n t s a n d t h e i r c o r r e s p o n d i n g p o l a r i z a t i o n s are represented b y the s o l i d arrows a n d g r a y arrows r e s p e c t i v e l y .  (a)  3  (b)  F i g u r e 2.4: M o m e n t u m space d i a g r a m for a square l a t t i c e . F i g u r e (a) represents i n c i d e n t r a d i a t i o n a l o n g the X s y m m e t r y d i r e c t i o n . F i g u r e (b) represents i n c i d e n t r a d i a t i o n a l o n g the M s y m m e t r y d i r e c t i o n . T h e dots represent the F o u r i e r c o m p o n e n t s of the d i e l e c t r i c s c a t t e r i n g p o t e n t i a l . T h e s m a l l d o t t e d vectors represent the i n - p l a n e wavevector of the incident radiation.  T h e large vectors represent the i n - p l a n e field vectors w h i c h result  w h e n the s c a t t e r i n g p o t e n t i a l adds m o m e n t u m to the i n c i d e n t field. T h e p o l a r i z a t i o n of each field c o m p o n e n t is represented b y the s m a l l gray vectors.  If the (isotropic) d i s p e r s i o n of the u n d e r l y i n g T E slab modes is d e n o t e d b y u)(K\\), the zone-folded d i s p e r s i o n e x p e c t e d for a very weak 2 D t e x t u r e w i t h square s y m m e t r y c a n be u n d e r s t o o d as follows. F o r a given incident wavevector Ki, the F o u r i e r c o m p o n e n t s of the d i e l e c t r i c t e x t u r e scatter the incident field into a discrete set of s c a t t e r e d states w i t h i n plane wavevectors K  t  + G. m  W h e n the incident frequency c o r r e s p o n d s t o one of the  —* — * slab m o d e frequencies at these i n - p l a n e wavevectors, cj(\Ki + G \), m  3  the s c a t t e r i n g w i l l be  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 1 resonantly enhanced, c o r r e s p o n d i n g to the e x c i t a t i o n of one of the l o c a l i z e d e x c i t a t i o n s of the 2 D t e x t u r e d slab. F o r the four smallest r e c i p r o c a l lattice vectors w i t h m a g n i t u d e Kg, these resonantly excited slab modes disperse w i t h K  iy  i n F i g u r e 2.3. In p a r t i c u l a r the m o d e at u>(\KiX + K x\)  as i l l u s t r a t e d s c h e m a t i c a l l y  increases m o n o t o n i c a l l y as Ki  g  increases a l o n g the X d i r e c t i o n . T h i s dispersive m o d e corresponds to the highest b a n d a l o n g the X d i r e c t i o n i n F i g u r e 2.3. T h e m o d e e x c i t e d at uj(\KiX — K x\) g  disperses d o w n  i n energy as Ki increases i n the X d i r e c t i o n , w h i l e the modes at ui{\KiX ± K y\) g  disperse  u p w a r d , b u t at a m o d e r a t e rate c o m p a r e d to the other two modes i l l u s t r a t e d i n F i g u r e 2.3. T h e p o l a r i z a t i o n labels o n the b a n d s shown i n F i g u r e 2.3 can be u n d e r s t o o d  using  the f o l l o w i n g s y m m e t r y argument. T h e p o l a r i z a t i o n (s, p or e l l i p t i c a l ) of the r a d i a t i o n reflected from the surface for a given incident plane wave at Ki is d e t e r m i n e d b y the F o u r i e r c o m p o n e n t of the p o l a r i z a t i o n density i n the g r a t i n g at the same K . t  F r o m the  t h e o r y described below (Section 2.2), for a t h i n t e x t u r e d g r a t i n g , t h i s c o m p o n e n t of the p o l a r i z a t i o n density i n the g r a t i n g can be expressed as [5]  x(G )E(Ki  P(KA =  m  - G) m  (2.3)  m U n d e r resonant e x c i t a t i o n conditions, one or two F o u r i e r c o m p o n e n t s of the field corr e s p o n d i n g to the resonantly e x c i t e d T E slab modes w i l l d o m i n a t e the scattered field everywhere except at  Ki=0. F o r the u p w a r d d i s p e r s i n g m o d e i n F i g u r e 2.3, t h i s cor-  responds t o a T E slab m o d e at K  t  + K x, g  w h i c h is p o l a r i z e d i n the y d i r e c t i o n .  From  E q u a t i o n 2.3 above, the p o l a r i z a t i o n density at Ki is t h e n p o l a r i z e d i n the y d i r e c t i o n ,  CHAPTER  2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 19  thus generating a n s-polarized r a d i a t i o n field i n the u p p e r half space. T h e same is t r u e for the r a p i d l y d o w n w a r d dispersing b r a n c h i n F i g u r e 2.3. T h i n g s are s l i g h t l y m o r e c o m p l i c a t e d for the m o d e r a t e l y d i s p e r s i n g branches because from s y m m e t r y , the i n c i d e n t field m u s t excite either s y m m e t r i c or a n t i s y m m e t r i c c o m b i n a t i o n s of these slab m o d e s (degenerate i n the absence of texture) at Ki ± K y. 9  F r o m the p o l a r i z a t i o n vectors s h o w n  i n F i g u r e 2.4, a s y m m e t r i c s u p e r p o s i t i o n of these scattered slab modes w i l l a d d as s h o w n i n E q u a t i o n 2.3 to y i e l d a p o l a r i z a t i o n density at i ? , oriented a l o n g the y d i r e c t i o n , c o r r e s p o n d i n g again to s-polarized r a d i a t i o n i n the u p p e r h a l f space. However, the a n t i s y m m e t r i c c o m b i n a t i o n leads t o a p o l a r i z a t i o n density oriented p a r a l l e l t o K  i}  which can  o n l y lead to p - p o l a r i z e d r a d i a t i o n i n the u p p e r half space, hence the p o l a r i z a t i o n labels i n F i g u r e 2.3. It follows t h a t there are 2 s- a n d 2 p - p o l a r i z e d modes, one each d i s p e r s i n g up and down, when K  t  is oriented a l o n g the M d i r e c t i o n . If Ki does not lie i n a plane  w i t h m i r r o r s y m m e t r y , the modes are, i n general, e l l i p t i c a l l y p o l a r i z e d , a n d c a n n o t  be  l a b e l l e d as p u r e s- or p - p o l a r i z e d . A n analysis s i m i l a r to the one j u s t presented c a n be done to d e t e r m i n e the d i s p e r s i o n a n d p o l a r i z a t i o n characteristics of the modes attached t o a p l a n a r waveguide t e x t u r e d w i t h a t r i a n g u l a r lattice. M o m e n t u m space d i a g r a m s for a t r i a n g u l a r l a t t i c e are s h o w n i n F i g u r e 2.5. F i g u r e 2.5(a) shows a n incident wavevector a l i g n e d a l o n g the M s y m m e t r y d i r e c t i o n w h i l e F i g u r e 2.5(b) shows an incident wavevector aligned a l o n g the K s y m m e t r y d i r e c t i o n . Since the g r o u p of "nearest neighbors" consists of s i x F o u r i e r c o m p o n e n t s , there are six b a n d s m a k i n g u p the second order T E - l i k e gap. T h e d i s p e r s i o n a n d p o l a r i z a t i o n s  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 2 of these s i x b a n d s are s h o w n i n F i g u r e 2.6. In the M d i r e c t i o n there are four s - p o l a r i z e d bands:  two d i s p e r s i n g u p i n energy a n d two d i s p e r s i n g d o w n i n energy.  In a d d i t i o n  there are two p - p o l a r i z e d bands: one dispersing u p i n energy a n d one d i s p e r s i n g d o w n i n energy. I n the K d i r e c t i o n there are two s- and two p - p o l a r i z e d b a n d s w h i c h disperse u p i n energy, a n d one s- a n d one p - p o l a r i z e d bands w h i c h disperse d o w n i n energy.  M  (a)  (b)  F i g u r e 2.5: M o m e n t u m space d i a g r a m for a t r i a n g u l a r l a t t i c e . F i g u r e (a) represents i n c i dent r a d i a t i o n a l o n g the M s y m m e t r y d i r e c t i o n . F i g u r e (b) represents i n c i d e n t r a d i a t i o n a l o n g the K s y m m e t r y d i r e c t i o n . T h e dots represent the F o u r i e r c o m p o n e n t s of the d i electric s c a t t e r i n g p o t e n t i a l . T h e s m a l l d o t t e d vectors represent the i n - p l a n e wavevector of the i n c i d e n t r a d i a t i o n . T h e large vectors represent the in-plane field vectors w h i c h result w h e n the s c a t t e r i n g p o t e n t i a l adds m o m e n t u m t o the i n c i d e n t field. T h e p o l a r i z a t i o n of each field c o m p o n e n t are represented b y the s m a l l gray vectors.  O n e of the significant findings of this thesis w o r k was t h a t the d i s p e r s i o n a n d pol a r i z a t i o n properties of the l o w - l y i n g leaky bands i n strongly  t e x t u r e d waveguides c a n  largely be i n t e r p r e t e d b y c o m b i n i n g this k i n e m a t i c p i c t u r e w i t h n o n - p e r t u r b a t i v e ,  strong  c o u p l i n g of these basic bands near zone boundaries, a n d near anticrossings t h a t m a y o c c u r 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  s,p s,p s,p  s,p  s,p  s  F i g u r e 2.6: S c h e m a t i c d i s p e r s i o n d i a g r a m for a t h i n p l a n a r waveguide t e x t u r e d w i t h a 2 D triangular lattice.  2.1.3  Defect Superlattices  A s discussed previously, t e x t u r e d p l a n a r waveguides m a d e w i t h gratings of s i m p l e l a t t i c e s c a n affect the p r o p a g a t i o n of photons, w h i c h is a k i n t o the a t o m s i n the c r y s t a l l a t t i c e of a s e m i c o n d u c t o r affecting the p r o p a g a t i o n of electrons. C o n t i n u i n g the a n a l o g y to s e m i c o n d u c t o r s , a d d i n g defects to the l a t t i c e i n a p l a n a r waveguide c a n allow modes to o c c u r i n the p h o t o n i c b a n d g a p , w h i c h is s i m i l a r t o a d d i n g i m p u r i t i e s t o a s e m i c o n d u c t o r t h a t give rise to states w i t h i n the electronic b a n d g a p . T h i s section discusses the i n c o r p o r a t i o n of a defect s u p e r l a t t i c e into a t e x t u r e d p l a n a r waveguide. W h e n a defect s u p e r l a t t i c e is i n c o r p o r a t e d into a t e x t u r e d p l a n a r waveguide, the u n i t cell for the l a t t i c e increases to i n c l u d e the defect. F o r e x a m p l e , for a defect  superlattice  i n a I D g r a t i n g t h a t leaves out every fifth lattice p o i n t , the u n i t cell w o u l d be five t i m e s 0  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides larger t h a n the u n i t cell of the "base" lattice. T h e r e c i p r o c a l l a t t i c e vectors for the defect s u p e r l a t t i c e are 1/5 the l e n g t h of t h a t of the base l a t t i c e (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 t h a t it is for the non-defect l a t t i c e , r e s u l t i n g i n m o r e frequent zone-folds.  F i g u r e 2.7 illustrates these changes b y s u p e r i m p o s i n g a  d i s p e r s i o n d i a g r a m for a defect superlattice on a d i s p e r s i o n d i a g r a m for a non-defect lattice:  the s o l i d lines represent the b a n d s t r u c t u r e for a non-defect base l a t t i c e ; the  dashed lines represent the b a n d s t r u c t u r e for the i d e n t i c a l l a t t i c e , w i t h the e x c e p t i o n t h a t a defect s u p e r l a t t i c e has been i n c o r p o r a t e d . T h i s figure shows the decrease i n size of the B r i l l o u i n zone a n d the increase i n zone-folds. F i g u r e 2.8 shows the d i s p e r s i o n for a waveguide w i t h a defect superlattice s u p e r i m p o s e d on a d i s p e r s i o n d i a g r a m for a base l a t t i c e t h a t produces a m u c h larger pseudo-gap t h a n i n the previous e x a m p l e . I n this case, the i n c o r p o r a t i o n of a defect s u p e r l a t t i c e does more t h a n m e r e l y increase the zone-folding: it i n t r o d u c e s a m o d e i n t o this large pseudo-gap.  W h i l e these i l l u s t r a t i o n s are for I D  t e x t u r e d p l a n a r waveguides, i n c o r p o r a t i n g a defect s u p e r l a t t i c e i n a 2 D t e x t u r e d p l a n a r waveguide produces q u a l i t a t i v e l y s i m i l a r results, as discussed b e l o w i n S e c t i o n 5.3.3. W h e n a 2 D t e x t u r e d p l a n a r waveguide w i t h a defect s u p e r l a t t i c e produces a m o d e i n the first order pseudo-gap, o n l y light of t h a t frequency is a l l o w e d to p r o p a g a t e w h i l e a l l other frequencies of light i n the gap are f o r b i d d e n to propagate. T h i s effect c a n a l l o w for the l o c a l i z a t i o n a n d c h a n n e l i n g of light i n the waveguide [14,40], m a k i n g it useful for m a n y a p p l i c a t i o n s . Lasers have been designed a n d d e m o n s t r a t e d u s i n g l o c a l i z a t i o n of l i g h t i n o p t i c a l m i c r o c a v i t i e s l o c a t e d at defects i n p h o t o n i c c r y s t a l lattices. [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°  0.5  g  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 attached to textured planar waveguides with and without defect superlattices. A rigorous computer model and white light spectroscopy are used in conjunction to study the electromagnetic response of these structures to radiation incident from the top half space. The computer code and the spectroscopic techniques used are described in the following 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 waveguides, 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  b y i m p l e m e n t i n g a G r e e n ' s function technique to solve M a x w e l l ' s equations.  F o r the  m o d e l i n g of defect superlattices r e p o r t e d i n this thesis, m o d i f i c a t i o n s were m a d e t o the code w h i c h are discussed below. T h e c o m p u t e r code is designed to accurately m o d e l the s c a t t e r i n g of l i g h t i n c i d e n t o n a t e x t u r e d p l a n a r waveguide, such as the one d e p i c t e d s c h e m a t i c a l l y i n F i g u r e 2.9. T h e waveguide d e p i c t e d has four layers: substrate, lower c l a d d i n g , core a n d u p p e r c l a d d i n g . T h e code has the c a p a b i l i t y to m o d e l any n u m b e r of layers, i n c l u d i n g semi-infinite u p p e r a n d lower layers. T h e code is also capable of m o d e l i n g any angle of i n c i d e n t l i g h t (6 a n d (j>) o n the surface of the waveguide.  upper cladding (air) core  F i g u r e 2.9: S c h e m a t i c d i a g r a m of a t e x t u r e d p l a n a r waveguide.  T h e code i m p l e m e n t s a s o l u t i o n of M a x w e l l ' s equations i n the waveguide g e o m e t r y b y t a k i n g the p e r i o d i c p o l a r i z a t i o n density i n the t e x t u r e d region as the d r i v i n g t e r m for a G r e e n ' s f u n c t i o n . [5] T h e self-consistent s o l u t i o n for a single F o u r i e r c o m p o n e n t of the  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 2 field i n t h e g r a t i n g is given b y  E(u,K ,z) n  = E (L>,K ,z) + Jdz'g hom  n  £  (K ,u,z,z') n  * i m (-u,u>)E(u),K ,z') m  (2.4)  771  w h e n a plane wave w i t h frequency u = ui/c a n d in-plane wavevector Ki, is i n c i d e n t from the u p p e r h a l f space, a n d where K  m  c o m p o n e n t s of t h e field, a n d G  m  = Ki — G  m  is t h e in-plane wavevector of t h e F o u r i e r  are t h e r e c i p r o c a l l a t t i c e vectors. [5] T h e p e r i o d i c sus-  c e p t i b i l i t y is represented as a F o u r i e r series w i t h coefficients X m t h a t couple t h e various n  F o u r i e r c o m p o n e n t s of the scattered  field.  T h e infinite set of vector i n t e g r a l equations  i m p l i e d i n E q u a t i o n 2.4 is reduced t o 3 N scaler algebraic equations b y t r u n c a t i n g t h e F o u r i e r series d e s c r i b i n g t h e in-plane c o m p o n e n t s of t h e field a t N . T h i s also requires t h a t t h e t e x t u r e d region is sufficiently t h i n so t h a t the fields c a n b e t a k e n as c o n s t a n t over its extent, thus e l i m i n a t i n g t h e integral over dz' i n E q u a t i o n 2.4. G r a t i n g s t h a t are t o o t h i c k for this a s s u m p t i o n c a n be m o d e l e d b y d i v i d i n g t h e m i n t o t h i n n e r regions t h a t each satisfy t h e constant field a p p r o x i m a t i o n . T h i s technique is d e s c r i b e d i n d e t a i l for a single layer s t r u c t u r e i n [5]. I n t h e o r i g i n a l code, t h e r e c i p r o c a l l a t t i c e vectors as well as t h e F o u r i e r coefficients were h a n d - c o d e d for each l a t t i c e investigated. T h i s was sufficient w h e n m o d e l i n g a s i m p l e l a t t i c e (i.e. one t h a t has a single l a t t i c e p o i n t i n t h e u n i t cell), w h i c h t y p i c a l l y requires fewer t h a n t e n F o u r i e r coefficients t o p r o p e r l y m o d e l t h e l o w - l y i n g bands. T h e c a l c u l a t i o n s for defect superlattices c a n require more t h a n 200 F o u r i e r coefficients,  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides m a k i n g h a n d - c o d i n g i m p r a c t i c a l a n d inefficient.  T o facilitate s i m u l a t i o n s of defect su-  perlattices, s u b r o u t i n e s were w r i t t e n w h i c h c a l c u l a t e the r e c i p r o c a l l a t t i c e vectors  and  F o u r i e r coefficients. T h e F o u r i e r coefficients w h i c h are used b y the code to represent the p e r i o d i c p o l a r i z a t i o n i n the g r a t i n g layer are n o r m a l i z e d to range between 0 a n d 1, w h e r e 0 represents no hole a n d 1 represents a hole. F o r a s i m p l e l a t t i c e d e s c r i b e d b y  !(7 ) = Y C e ^ f  (2.5)  i3  J  nm  nm the F o u r i e r coefficients, C ,  are given b y  nm  Cnm^^—f  dxy/b - x cos(G x) = 2  2  nm  -^—^—Ji{bG ) nm  (2.6)  where G  = Go y/n  2  nm  + m  2  — 2nm cos(7r — a)  (2.7)  a n d w h e r e b is the r a d i u s of the holes, n a n d m are the base l a t t i c e u n i t vectors a n d a is the angle between t h e m . G  nm  is the r e c i p r o c a l l a t t i c e vector.  T o generate a F o u r i e r series for a defect s u p e r l a t t i c e ,  first  the above e q u a t i o n s are  used to generate a F o u r i e r series w h i c h describes the "base" s i m p l e (non-defect) l a t t i c e , (a), s h o w n i n a real-space plot i n F i g u r e 2.10. N e x t , a F o u r i e r series is generated w h i c h represents the l o c a t i o n a n d size of those holes to be o m i t t e d t o f o r m the defect,  (b).  F i n a l l y , the F o u r i e r series for (b) is s u b t r a c t e d from the F o u r i e r series for (a) p r o d u c i n g the F o u r i e r series for the defect superlattice (c), t h a t for this e x a m p l e o m i t s every  fifth  hole i n the base l a t t i c e p a t t e r n . O p t i o n a l l y , a n a d d i t i o n a l F o u r i e r series c a n be generated  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 2 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 • • V 1 1 • m i IP • • • * <i • m m m• • ••< •• • • •m• • n• » • •« • • • • • • • •  •  *  •  *  to  \  •  i 1  •  •  W%  t  •  A  i  •  l  l  l  l  l  l :  l  l  H 0 n $1  •  •  «  »  'fft  1  (a)  (b)  (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 characterize 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 research presented here, two types of spectroscopic technique are used: a well-documented specular measurement technique, and a novel diffraction measurement technique developed by the author.  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar  2.3.1  Specular Measurement Technique  T o characterize the t e x t u r e d p l a n a r waveguides for this thesis, the b a n d s t r u c t u r e was m a p p e d a n d i n some cases the lifetimes were measured.  A specular m e a s u r e m e n t tech-  n i q u e was used to a c c o m p l i s h this for the waveguides w i t h 2 D s i m p l e lattices.  This  technique involves s h i n i n g w h i t e light o n the surface of a t e x t u r e d p l a n a r waveguide at a well-defined angle of incidence a n d t h e n a n a l y z i n g the s p e c t r a of the s p e c u l a r l y reflected light. T h i s technique provides a straight-forward means for m a p p i n g the l e a k y p h o t o n i c modes of t e x t u r e d p l a n a r waveguides.  t  F i g u r e 2.11: S c h e m a t i c d i s p e r s i o n d i a g r a m for a I D t e x t u r e d waveguide.  B o l d lines  i n d i c a t e l o c a t i o n s p r o b e d w i t h the specular measurement technique.  W h e n w h i t e light is i n c i d e n t on the surface of a t e x t u r e d p l a n a r waveguide at a well-defined angle of incidence, each energy, oo, of the i n c i d e n t light corresponds t o a specific i n - p l a n e wavevector, K\\, a c c o r d i n g to the r e l a t i o n K\\ = to sin.0. G e n e r a l i z i n g the k i n e m a t i c arguments given above, c o u p l i n g i n t o a g u i d e d m o d e eigenstate w i l l o c c u r  Waveguides 29  CHAPTER  2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides 30  whenever K\\±nK  9  stimulates a mode. Here n is any integer and K is the grating vector. g  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 3  8.0  8.4  8.8  9.2  9.6  10.0  10.8xl0  3  Energy (cm *) F i g u r e 2.12: R e p r e s e n t a t i v e reflectivity s p e c t r a from a t e x t u r e d p l a n a r waveguide w i t h a square l a t t i c e of holes. S o l i d lines represent s - p o l a r i z a t i o n ; d a s h e d lines represent pp o l a r i z a t i o n . T h e n a r r o w features are Fano-resonances w h i c h are a result of c o u p l i n g i n t o g u i d e d modes of the waveguide.  diffracted,  r a t h e r t h a n reflected, from samples c o n t a i n i n g a s u p e r l a t t i c e of defects.  This  diffraction measurement technique is described below. T o i m p l e m e n t this new diffraction measurement technique, light at a p a r t i c u l a r angle of i n c i d e n c e s t i m u l a t e s a g u i d e d m o d e b y a d d i n g a n integer m u l t i p l e of defect l a t t i c e g r a t i n g vectors to the p a r a l l e l c o m p o n e n t of the incident light. T h i s is represented b y the e q u a t i o n K = K\\ ± nK®.  T h e light is scattered out of the waveguide b y a d d i n g or  s u b t r a c t i n g a n integer m u l t i p l e of defect l a t t i c e g r a t i n g vectors from the g u i d e d m o d e . W h i l e the s p e c u l a r technique uses the same n u m b e r of g r a t i n g vectors t o s t i m u l a t e the m o d e as to scatter out of the m o d e , the diffraction technique uses a n u n e q u a l n u m b e r . T h i s diffraction measurement technique is s h o w n s c h e m a t i c a l l y i n F i g u r e 2.13. T h e p a r a m e t e r space p r o b e d b y the incident b e a m is represented b y the b o l d line.  This  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides s c h e m a t i c corresponds t o a I D l a t t i c e w i t h a defect l o c a t e d at every fifth site. I n t h i s case the defect l a t t i c e g r a t i n g vector is 1/5 the size of the base l a t t i c e g r a t i n g vector. C o u p l i n g to the g u i d e d m o d e results b y a d d i n g or s u b t r a c t i n g 5  defect lattice  grating  vectors. T h e light n o w scatters out of the waveguide not o n l y i n the s p e c u l a r d i r e c t i o n , b u t also i n a n y of the directions associated w i t h the lines p a r a l l e l to the b o l d line l a b e l e d ' 0 ' o n F i g u r e 2.13. T h e l o c a t i o n of the -K® diffracted order is i n d i c a t e d o n the d i a g r a m b y a circle.  -1  0  F i g u r e 2.13: S c h e m a t i c dispersion for the diffraction p r o b i n g m e t h o d . T h e d i r e c t i o n of the specular reflection is l a b e l e d 0, w h i l e the d i r e c t i o n of the -K^  diffracted order is  labeled -1.  T h e o r e t i c a l l y , the diffraction technique s h o u l d allow o b s e r v a t i o n of modes above the light line free f r o m specular b a c k g r o u n d . Significantly, it c o u l d also a l l o w for the v i e w i n g of modes f r o m the base l a t t i c e w h i c h are below the light line. C o n v e n t i o n a l m e t h o d s for  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides p r o b i n g modes b e l o w the light line involve elaborate techniques for g e t t i n g light i n t o a n d out of the waveguide. T h e diffraction measurement technique offers the p o t e n t i a l to m a p these modes b y s i m p l y i l l u m i n a t i n g t h e m from the u p p e r h a l f space, a l l o w i n g a n a l y s i s of the scattered light w h i c h reveals the b a n d s t r u c t u r e of the b o u n d modes. T o investigate w h e t h e r the diffraction measurement technique has these c a p a b i l i t i e s , the G r e e n ' s f u n c t i o n code was used t o s i m u l a t e light diffracting from a t e x t u r e d p l a n a r waveguide w i t h a defect superlattice. T h e s i m u l a t i o n was done for a free-standing G a A s waveguide, 80 n m t h i c k , w i t h a square l a t t i c e of c i r c u l a r holes c o m p l e t e l y p e n e t r a t i n g the core, spaced 400 n m a p a r t w i t h a r a d i u s of 110 n m . T h e defect s u p e r l a t t i c e was created b y o m i t t i n g every fifth hole i n the x a n d y directions. F i g u r e 2.14 shows a s i m u l a t i o n of the first order diffraction w h i c h results w h e n b r o a d b a n d light is i n c i d e n t n o r m a l t o the surface of this waveguide. It is r e a d i l y apparent t h a t the modes i n this s p e c t r a are free of the s p e c u l a r b a c k g r o u n d . It is not obvious, however, where these modes o r i g i n a t e . If these modes o r i g i n a t e below the light line of the base lattice, t h e n i t s h o u l d be possible to r e p r o d u c e t h e i r energies b y l o o k i n g at the d i s p e r s i o n of the defect-free base l a t t i c e , as discussed b e l o w . A m o m e n t u m space d i a g r a m for the waveguide c o n t a i n i n g the s u p e r l a t t i c e ( F i g u r e 2.15) i l l u s t r a t e s w h i c h specific in-plane wavevectors o c c u r w i t h i n the r e d u c e d zone of the base l a t t i c e . I n this d i a g r a m , the large gray dots represent the r e c i p r o c a l l a t t i c e vectors of the base (non-defect) l a t t i c e . T h e square identifies the first B r i l l o u i n zone of this base l a t t i c e , a n d the t r i a n g l e delineates the reduced zone. T h e s m a l l gray dots represent the  CHAPTER 2. Linear Light Scattering Spectroscopy of Textured Planar Waveguides r e c i p r o c a l l a t t i c e vectors t h a t are i n t r o d u c e d b y the defect s u p e r l a t t i c e . T h e d a r k d o t s w i t h the s m a l l squares a r o u n d t h e m represent r e c i p r o c a l l a t t i c e vectors from the defect l a t t i c e o c c u r r i n g w i t h i n the reduced zone of the base l a t t i c e .  W h e n light is i n c i d e n t  n o r m a l t o the surface of the waveguide, the i n - p l a n e wavevector of the i n c i d e n t r a d i a t i o n is zero, so the in-plane wavevectors of the field scattered b y the defect s u p e r l a t t i c e are j u s t given b y the r e c i p r o c a l l a t t i c e vectors of the defect s u p e r l a t t i c e . F i g u r e 2.15 shows t h a t there are five d i s t i n c t i n - p l a n e wavevectors (dark dots) where the defect s u p e r l a t t i c e samples the first B r i l l o u i n zone of the base lattice. F i g u r e 2.16 shows the c a l c u l a t e d specular reflectivity s p e c t r a from the defect-free base l a t t i c e at these five wavevectors. Poles i n these s p e c t r a represent t r u e b o u n d modes t h a t lie b e l o w the l i g h t l i n e i n the base l a t t i c e b a n d structure.  L a b e l s i n F i g u r e s 2.14 a n d  2.16 relate the peaks observed i n the background-free diffraction s p e c t r a from the defect s u p e r l a t t i c e s a m p l e w i t h the poles observed i n the reflectivity s p e c t r a at the relevant i n p l a n e wavevectors of the base l a t t i c e . T h i s confirms t h a t , theoretically, the use of defect superlattices does offer the a b i l i t y to probe the d i s p e r s i o n of t r u e b o u n d modes t h a t exist b e l o w the light line i n the defect-free B r i l l o u i n zone. T h i s diffraction measurement technique c a n be used whenever the energy of the m o d e t o be detected radiates a diffracted signal above the light line.  A n y l a t t i c e t h a t has  p e r i o d i c s p a c i n g longer t h a n the wavelength of the probe light w i l l have at least one diffracted order. I n waveguides where a diffracted signal does not exist above the light line, a defect s u p e r l a t t i c e c a n be i n c o r p o r a t e d t o create one. T e x t u r e d p l a n a r 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  F i g u r e 2.16: S i m u l a t e d s-polarized s p e c t r a for the defect-free l a t t i c e d e s c r i b e d i n the t e x t . T h e n o t a t i o n s a l o n g the right side i n d i c a t e the in-plane wavevectors for each s p e c t r a . Poles i n d i c a t e b o u n d modes t h a t exist below the air light line.  Chapter 3 Experimental Design  T h i s chapter describes a n a p p a r a t u s designed a n d b u i l t to i m p l e m e n t the l i n e a r light s c a t t e r i n g techniques described i n C h a p t e r 2. T h e a p p a r a t u s makes it possible t o characterize l e a k y p h o t o n i c modes of t e x t u r e d p l a n a r waveguides.  Its f u n c t i o n is to b r i n g  w h i t e light i n c i d e n t o n a t e x t u r e d p l a n a r waveguide at a well-defined angle a n d to collect, c o n d i t i o n a n d focus the light scattered from the waveguide i n t o a s p e c t r o m e t e r for analysis. T h e basic design consists of a fiber o p t i c w h i t e light source, three e l l i p s o i d a l m i r rors, a n d a concentric r o t a t i o n stage. Reflective optics were chosen t h r o u g h o u t for t h e i r a b i l i t y t o a v o i d the c h r o m a t i c aberrations inherent i n refractive o p t i c s . T h e a p p a r a t u s was designed specifically t o w o r k w i t h a B o m e m D A 8 F o u r i e r T r a n s f o r m Interferometer ( F T I R ) . T h e key challenge i n the design was to ensure a c h r o m a t i c performance over the range of 6,000 - 13,000 c m  - 1  ( ~ 0 . 8 n m - 1.7/im) w h i l e l o c a l i z i n g the s c a t t e r i n g area to  ~ 1 0 0 u m x 100 u m , the t y p i c a l size of the t e x t u r e d p l a n a r waveguides s t u d i e d here. F o r the overall design of the apparatus, ease of use was a p r i m a r y objective. S a m p l e s c a n be m o u n t e d a n d a l i g n e d q u i c k l y a n d c h a r a c t e r i z e d easily. 37  CHAPTER 3. Experimental Design  3.1  38  Apparatus  T h e p r i n c i p l e c o m p o n e n t s of the e x p e r i m e n t a l a p p a r a t u s are a w h i t e light source, f o c u s i n g optics, p o s i t i o n i n g mechanics, a n d a spectrometer.  A schematic of the o p t i c a l l a y o u t is  s h o w n i n F i g u r e 3.1. U n p o l a r i z e d w h i t e light is delivered v i a a fiber o p t i c cable fixed to a large r o t a t i n g r i n g w h i c h is concentric w i t h a s m a l l r o t a t i o n stage t h a t holds a t e x t u r e d p l a n a r waveguide sample. T h e w h i t e light e m i t t e d from the o u t p u t facet of the fiber is i m a g e d o n t o the s a m p l e b y a n e l l i p s o i d a l m i r r o r ( E M I ) . s a m p l e is reflected, t r a n s m i t t e d , a n d / o r diffracted.  T h e l i g h t s t r i k i n g the  O n e of these emissions is c o l l e c t e d  b y a second e l l i p s o i d a l m i r r o r ( E M 2 ) a n d subsequently refocused to a n i m a g e p l a n e where a n adjustable field stop is used i n c o n j u n c t i o n w i t h a removable C C D c a m e r a t o e l i m i n a t e extraneous l i g h t . T h e light t h e n passes t h r o u g h a l i n e a r p o l a r i z e r where the sor p - p o l a r i z a t i o n is allowed to continue to a t h i r d e l l i p s o i d a l m i r r o r ( E M 3 ) .  T h i s final  e l l i p s o i d a l m i r r o r focuses the light i n t o the B o m e m F T I R for s p e c t r a l analysis. A d e t a i l e d d e s c r i p t i o n of the e x p e r i m e n t a l subsystems follows.  3.1.1  Light Source  A fiber-coupled light source was chosen for this e x p e r i m e n t to facilitate easy p o s i t i o n i n g of the i n c i d e n t l i g h t .  T h e t e x t u r e d region of the p l a n a r waveguide is t y p i c a l l y o n l y  90 pm x 90 pm, therefore the d i a m e t e r of the light on the sample was chosen to be 200 pm to overfill the p a t t e r n slightly. T o achieve the 200 pm d i a m e t e r spot, the o u t p u t  facet  of a 100 pm d i a m e t e r core o p t i c a l fiber ( o p t i m i z e d for N I R t r a n s m i s s i o n ) is i m a g e d b y  CHAPTER 3. Experimental Design  39  Bomem FTIR  EM3  2  cm  F i g u r e 3.1: S c h e m a t i c of the e x p e r i m e n t a l a p p a r a t u s used for o p t i c a l c h a r a c t e r i z a t i o n of t e x t u r e d p l a n a r waveguides.  E M I , w h i c h was designed to p r o v i d e a 2x m a g n i f i c a t i o n . A 100 w a t t t u n g s t e n  quartz  halogen b u l b housed i n a r a d i o m e t r i c fiber o p t i c source ( O r i e l 77501) was chosen for the light source. T h e electronics i n this light stabilize the o u t p u t to be t h a t of a 3 2 0 0 ° K e l v i n b l a c k b o d y . T h e b r o a d b a n d light e m i t t e d from the o u t p u t facet of the 100 pm fiber was m e a s u r e d t o be a p p r o x i m a t e l y 1 m W a t t between 400 n m a n d 1.1 pm.  diameter  CHAPTER 3. Experimental Design  3.1.2  40  Optics  T h e f u n c t i o n of the optics i n this a p p a r a t u s is to efficiently focus w h i t e l i g h t o n t o a sample, collect light e m i t t e d from the sample, c o n d i t i o n the light a n d focus it i n t o the F T I R . T h e design of the optics was c o n s t r a i n e d b y several factors:  the p a r a m e t e r s of  the e x i s t i n g F T I R a n d the o p t i c a l fiber chosen to deliver the l i g h t , the necessity  to  create a n enlarged image for use w i t h a field stop, a n d the decision t o use w h i t e l i g h t . Specifically, the light entering the F T I R h a d to have a full angle divergence of ~ 1 4 ° ( F / 4 ) t o p r o p e r l y fill its i n t e r n a l optics. Secondly, the optics h a d t o a c c o m m o d a t e  the  100 pm d i a m e t e r core o p t i c a l fiber. T h i r d l y , the optics h a d t o create a l O x image, w h i c h is of adequate size for use w i t h the field stop.  F i n a l l y , the choice to use w h i t e light  necessitated the use of reflective optics t h r o u g h o u t the a p p a r a t u s i n order t o a v o i d the c h r o m a t i c aberrations inherent i n refractive optics. T h e f u n c t i o n of the optics c o u l d be achieved a n d the constraints c o u l d be a c c o m m o d a t e d b y u s i n g either e l l i p s o i d a l m i r r o r s or p a r a b o l o i d a l m i r r o r s . E l l i p s o i d a l m i r r o r s were chosen over p a r a b o l o i d a l m i r r o r s for their a b i l i t y to p r o v i d e p o i n t - t o - p o i n t focusing u s i n g a single m i r r o r .  A single e l l i p s o i d a l m i r r o r focuses l i g h t  c o m i n g from one focus to the other focus, whereas a p a r a b o l o i d a l m i r r o r collects l i g h t from its focus a n d collimates i t . A second p a r a b o l o i d a l m i r r o r must be used t o t a k e the c o l l i m a t e d l i g h t a n d focus it t o form a n image. Since some l i g h t is lost u p o n reflection, a design w i t h fewer m i r r o r s is preferable w h e n m a x i m u m o p t i c a l t h r o u g h p u t is r e q u i r e d .  CHAPTER  3. Experimental Design  41  Off-the-shelf e l l i p s o i d a l m i r r o r s are not generally available, so m i r r o r s were c u s t o m designed a n d fabricated for this experiment.  T h e following describes how these m i r r o r s  were designed. T h e f o l l o w i n g discussion of the m i r r o r design refers to the ellipse p i c t u r e d i n F i g u r e 3.2. T h e three e l l i p s o i d a l m i r r o r s were designed t o function i n t e r d e p e n d e n t l y (i.e. a change i n a p a r a m e t e r of one m i r r o r necessitated dependent changes i n the others). T h e m i r r o r s were designed as e l l i p s o i d a l m i r r o r s where the central cross-section is a s i m p l e ellipse. T o define the ellipse, the i n i t i a l c o n s i d e r a t i o n was the desired m a g n i f i c a t i o n p r o d u c e d b y the m i r r o r . M a t h e m a t i c a l l y , the m a g n i f i c a t i o n of a s i m p l e lens or m i r r o r is d e t e r m i n e d b y the e q u a t i o n M  = Si/S  t  S  0  D  [9] where M  t  is the transverse m a g n i f i c a t i o n , a n d Si a n d  are the image a n d object distances respectively. F o r these m i r r o r s the m a g n i f i c a t i o n  is g i v e n b y M  (  = r /ri. 2  A specific value of either ?*i or r  2  was selected based o n the  desired p h y s i c a l d i m e n s i o n s of the a p p a r a t u s . A d d i n g r\ a n d r  2  the ellipse (26 = r\ + r ). 2  gives the m a j o r axis of  N e x t the eccentricity of the ellipse must be d e t e r m i n e d .  do this, the angle between r% a n d r  2  To  must be selected; to facilitate a l i g n m e n t , 90° was  selected for a l l three m i r r o r s . F i n a l l y , i n order t o specify the p h y s i c a l size of the m i r r o r s , the angles Q\ a n d 9  2  must be selected.  F o r this experiment a s m a l l cone angle of l i g h t  i l l u m i n a t i n g the sample was desired i n order to a p p r o x i m a t e c o l l i m a t e d l i g h t . See T a b l e 3.1 for the specifications of each m i r r o r . N o t e the b o l d values i n d i c a t e the i n i t i a l design c o n s t r a i n t s from w h i c h a l l other values were c a l c u l a t e d . In a d d i t i o n to the specifications delineated i n T a b l e 3.1, there was a n a d d i t i o n a l design  CHAPTER 3. Experimental Design  42  2b  F i g u r e 3.2: S c h e m a t i c d i a g r a m of e l l i p s o i d a l m i r r o r design. L i g h t e m i t t i n g from a p o i n t source at one focus w i l l be i m a g e d at the other focus. r e q u i r e m e n t for m i r r o r E M I . A p o r t i o n of the m i r r o r b l a n k was r e m o v e d t o a l l o w m i r r o r E M 2 t o be p l a c e d i n close p r o x i m i t y to m i r r o r E M I , i n order to enable v i e w i n g of the specular reflection as close to n o r m a l incidence as possible (see F i g u r e 3.3). M i r r o r s E M I , E M 2 , a n d E M 3 were fabricated out of a l u m i n u m b y L u m o n i c s C o r p o r a t i o n . T h e f o l l o w i n g section discusses the c o m p o n e n t s for p o s i t i o n i n g the m i r r o r s a n d sample.  T a b l e 3.1: E l l i p s o i d a l m i r r o r specifications. N o t e : B o l d values i n d i c a t e design constraints. EMI  EM 2  EM3  r i (cm)  5  15  150  r  10  150  2  M  2  10  0.013  0i  3.8°  2.0°  0.19°  2.0°  0.19°  14.39°  2  (cm) t  (F/4)  CHAPTER  3. Experimental  Design  43  EM2  Light F i g u r e 3.3: D e t a i l of E M I a n d E M 2 m i r r o r s , s h o w i n g shape of E M I w h i c h allows v i e w i n g of specular reflection near n o r m a l incidence.  3.1.3  Positioning  Mechanics  T h e p r i m a r y f u n c t i o n of the p o s i t i o n i n g mechanics is to enable consistent a n d precise p l a c e m e n t of the s a m p l e a n d light source relative t o each other a n d t o the c o l l e c t i o n optics. F o r this e x p e r i m e n t , the F T I R a n d associated c o l l e c t i o n o p t i c s r e m a i n s t a t i o n a r y w h i l e the s a m p l e a n d source light rotate. A concentric, 9 — 29 design is used t o a c c o m p l i s h t h i s . T h i s c o n c e n t r i c s y s t e m consists of a n inner r o t a t i o n stage w h i c h h o l d s the s a m p l e , a n d a n outer r o t a t i n g r i n g w h i c h holds E M I as well as the e n d of the o p t i c a l fiber. T h e s y s t e m is a l i g n e d so t h a t the light is focused precisely at the s h a r e d center of r o t a t i o n . P o s i t i o n i n g a t e x t u r e d p l a n a r waveguide sample precisely o n the center of r o t a t i o n of the t w o r o t a t i o n stages enables the sample to be r o t a t e d to different v i e w i n g angles  CHAPTER 3. Experimental Design  44  w i t h o u t r e q u i r i n g the v i e w i n g optics ( E M 2 a n d E M 3 ) to be r e a l i g n e d . T o achieve the precise p o s i t i o n i n g of the sample, a five stage s a m p l e m o u n t was designed (see F i g u r e 3.4). T h e b o t t o m t r a n s l a t i o n stage allows the center of r o t a t i o n of t h e s a m p l e m o u n t t o be p o s i t i o n e d c o n c e n t r i c w i t h the outer r o t a t i n g r i n g . T h e h o r i z o n t a l r o t a t i o n stage is used i n c o n j u n c t i o n w i t h the outer r o t a t i o n r i n g to c o n t r o l the angle of i n c i d e n t l i g h t , 9. T h e X Y Z t r a n s l a t i o n stage is used to p o s i t i o n the s a m p l e o n the center of r o t a t i o n of the s a m p l e stage a n d t o p o s i t i o n the height of the s a m p l e r e l a t i v e t o the o p t i c s . T h e t i l t stage is used t o ensure the sample is m o u n t e d vertically.  F i n a l l y , the v e r t i c a l r o t a t i o n  stage is used t o adjust the a z i m u t h a l angle of the s a m p l e (<j>).  T h i s five stage s y s t e m  allows for samples t o be m o u n t e d r a p i d l y as well as accurately.  Tilt  X Y Z Translation Horizontal Rotation Ring X Y Translation  F i g u r e 3.4: B l o c k d i a g r a m s h o w i n g sample m o u n t i n g a n d a l i g n m e n t a p p a r a t u s .  CHAPTER 3. Experimental Design  45  T h e m o u n t , s h o w n i n F i g u r e 3.5, was designed to be used i n c o n j u n c t i o n w i t h the v e r t i c a l r o t a t i o n r i n g d e p i c t e d i n F i g u r e 3.4.  T h i s m o u n t was designed to h o l d  the  s a m p l e i n the center o f the v e r t i c a l r o t a t i o n r i n g a l l o w i n g for the m a x i m u m p o s s i b l e angle of i n c i d e n c e for b o t h reflection a n d t r a n s m i s s i o n measurements w i t h o u t n e c e s s i t a t i n g adjustment  t o the p o s i t i o n of the sample.  B a s e d o n the p h y s i c a l size of the v e r t i c a l  r o t a t i o n r i n g a n d the 2° divergence of i n c i d e n t l i g h t , the m a x i m u m angle o f i n c i d e n c e for reflection a n d t r a n s m i s s i o n measurements is ~ 6 5 ° . T h e m o u n t was designed for ease of use i n affixing various samples. the s a m p l e is affixed.  T h e m o u n t has a flat, p o l i s h e d surface onto w h i c h  T h e s a m p l e is h e l d i n place w i t h a n adhesive m e d i u m , s u c h as  v a c u u m grease for the lighter weight G a A s samples, a n d two-sided t a p e for the heavier p o l y m e r samples. W h e n p l a c i n g a sample onto the m o u n t , it is not necessary to p o s i t i o n i t precisely, because, as p r e v i o u s l y discussed, the t r a n s l a t i o n stages are used t o p e r f o r m a l l necessary adjustments to b r i n g the sample into proper a l i g n m e n t . F u r t h e r , the s a m p l e c a n be m o u n t e d such t h a t the area of interest is h e l d above the m o u n t , a l l o w i n g for t r a n s m i s s i o n as w e l l as reflection.  3.1.4  Spectrometer  T h e l i g h t s c a t t e r e d from the t e x t u r e d p l a n a r waveguide s a m p l e is a n a l y z e d w i t h a B o m e m F T I R . T h i s spectrometer consists of a M i c h e l s o n interferometer w i t h a q u a r t z b e a m s p l i t t e r a n d a n I n G a A s detector. T h e light is focused i n t o the entrance a p e r t u r e of the  CHAPTER 3. Experimental Design  Front view  46  Side view  F i g u r e 3.5: D e t a i l of sample mount used i n v e r t i c a l r o t a t i o n r i n g .  F T I R b y m i r r o r E M 3 , w i t h the F / 4 needed t o p r o p e r l y fill the i n t e r n a l o p t i c s . T h e I n G a A s detector a n d q u a r t z b e a m s p l i t t e r allow spectral measurements f r o m a p p r o x i m a t e l y 6,000 c m " t o 13,000 c m " . 1  3.2  1  Alignment and Operation of Apparatus  T h e f o l l o w i n g is the procedure for a l i g n i n g the a p p a r a t u s for use, l i s t e d i n t h e p r e s c r i b e d order. P l a c e the s a m p l e t o be characterized on the s a m p l e m o u n t . A l i g n the centers of r o t a t i o n of the i n n e r r o t a t i o n stage a n d the outer r o t a t i n g r i n g t o c o i n c i d e . F o c u s E M 2 on this shared center of r o t a t i o n , u s i n g the image o n the C C D c a m e r a as a guide. A d j u s t E M I t o focus the l i g h t o n t o this center of r o t a t i o n , u s i n g the C C D c a m e r a t o c o n f i r m p r o p e r placement. U s e t h e X Y Z t r a n s l a t i o n stage t o p o s i t i o n t h e s a m p l e at the s h a r e d  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  Alternate Configurations  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 e x p e r i m e n t a l a p p a r a t u s described i n C h a p t e r 3 is used to characterize p l a n a r waveguides.  textured  T w o types of t e x t u r e d p l a n a r waveguides are i n v e s t i g a t e d i n t h i s  thesis: G a A s o n oxide, a n d p o l y m e r o n glass. T h i s chapter discusses the f a b r i c a t i o n of these t e x t u r e d p l a n a r waveguides.  4.1  GaAs Sample  T h e r e are four phases i n the f a b r i c a t i o n of a G a A s - b a s e d t e x t u r e d p l a n a r waveguide s u c h as the one d e p i c t e d i n F i g u r e 4.1: g r o w t h of the p l a n a r waveguide, e l e c t r o n b e a m l i t h o g r a p h y , etching, a n d o x i d a t i o n . A d e s c r i p t i o n of each of these phases follows.  4.1.1  Planar Waveguide Growth  T h e first step i n f a b r i c a t i n g a G a A s - b a s e d t e x t u r e d p l a n a r waveguide is t o m a k e a p l a n a r waveguide. T h i s is done b y u s i n g m o l e c u l a r b e a m e p i t a x y ( 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 waveguides, 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 % P o l y m e t h y l m e t h a c r y l a t e ( P M M A ) 9 5 0 K dissolved i n a chlorobenzene s o l u t i o n . T h e samples were s p u n at 8,000 r p m for 40 seconds, y i e l d i n g a thickness of a p p r o x i m a t e l y 200 n m .  F i n a l l y the samples are baked on a hot plate at 1 7 5 ° 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 , a n d the samples are r e a d y for lithography. E l e c t r o n b e a m l i t h o g r a p h y is a process b y w h i c h an electron b e a m transfers a pres c r i b e d p a t t e r n onto a resist. A H i t a c h i 4100 c o m p u t e r - c o n t r o l l e d t h e r m a l - e m i s s i o n scann i n g electron m i c r o s c o p e ( S E M ) was used t o p e r f o r m the l i t h o g r a p h y o n the G a A s s a m ples. L i t h o g r a p h y software ( N P G S ) [19] directs the S E M to m a k e a p a t t e r n i n the P M M A by e x p o s i n g the resist at pre-determined coordinates w i t h a specified dose. T h e electron b e a m breaks the l o n g p o l y m e r chains of the P M M A . T h e b r o k e n chains are t h e n r e m o v e d from the s a m p l e b y developing, w h i c h involves i m m e r s i n g the sample i n chemicals t h a t selectively e l i m i n a t e the shorter p o l y m e r chains w i t h o u t affecting the longer chains. T h e longer the electron b e a m dwells i n a p a r t i c u l a r l o c a t i o n d u r i n g l i t h o g r a p h y , the  more  chains are b r o k e n , r e s u l t i n g i n a larger hole once developed. T h e d e v e l o p i n g recipe for this e x p e r i m e n t is s h o w n i n T a b l e 4.1. T h e P M M A resist has now been t r a n s f o r m e d i n t o a n e t c h i n g mask.  4.1.3  Etching  T o transfer the p a t t e r n from the m a s k to the sample, an e t c h i n g process is used.  The  e t c h i n g for t h i s experiment is based on a d r y p l a s m a technique a n d is done w i t h a  CHAPTER 4. Sample Preparation  51  T a b l e 4.1: P M M A d e v e l o p i n g recipe Chemical  Time  MIBK  90 sec  Propanol  30 sec  D l Water  30 sec  Oxyethylmethenol  15 sec  Methanol  30 sec  P l a s m a Q u e s t e l e c t r o n - c y c l o t r o n resonance ( E C R ) etcher. I n the etcher, w i t h i n a s t r o n g , s t a t i c m a g n e t i c field, a m i c r o w a v e source drives the electrons i n a l o w pressure gas at t h e i r c y c l o t r o n resonance, c r e a t i n g a p l a s m a . T h e r a d i o frequency ( R F ) bias drives ions from the p l a s m a c l o u d a l o n g the m a g n e t i c field lines to strike the s a m p l e n o r m a l to the surface, where t h e y k i n e m a t i c a l l y a n d c h e m i c a l l y etch the exposed G a A s . T h e specific etch recipe used is s h o w n i n T a b l e 4.2. T h i s recipe, developed based o n research presented by S a h [27], etches c y l i n d r i c a l holes (or other desired p a t t e r n ) w i t h v e r t i c a l sidewalls i n the G a A s at a p p r o x i m a t e l y 100 n m per m i n u t e . D u r i n g etching, the m a s k e d p o r t i o n s of the G a A s are p r o t e c t e d b y the P M M A , w h i l e the chemicals etch the r e m a i n d e r of the G a A s . T h e P M M A is also etched b y the chemicals, b u t the e t c h i n g t i m e is short e n o u g h (i.e.  less t h a n 150 seconds) t h a t the chemicals do not fully ablate the m a s k . A f t e r the  e t c h i n g process is complete, the sample is rinsed w i t h acetone a n d m e t h a n o l to remove the r e m a i n i n g P M M A . T h e sample is now ready for o x i d a t i o n .  CHAPTER 4. Sample Preparation  52  T a b l e 4.2: E C R recipe Cl  BC1  4.1.4  2.0 seem  2  3  2.0 seem  Ar  20.0 seem  Microwave  100 W a t t s  R F Bias  100 V o l t s  R F Power  25 W a t t s  Chuck Temp  5°C  Process Pressure  10 m T o r r  Backside He  5 Torr  Time  145 sec  Oxidation  T h e final step i n f a b r i c a t i n g the G a A s t e x t u r e d p l a n a r waveguides used i n t h i s thesis is to o x i d i z e the a l u m i n u m c l a d d i n g layer of the waveguide u s i n g a n o x i d a t i o n furnace. F o r a d e t a i l e d d e s c r i p t i o n of the design a n d c o n s t r u c t i o n of the o x i d a t i o n furnace used here, refer to Reference [32].  T o oxidize the a l u m i n u m layer, the s a m p l e is sealed i n  the o x i d a t i o n furnace w h i c h is p u r g e d w i t h d r y n i t r o g e n for 1 hour.  S u b s e q u e n t l y the  t e m p e r a t u r e is increased from r o o m t e m p e r a t u r e to 4 2 5 ° C over a p e r i o d of 30 m i n u t e s . T h e t e m p e r a t u r e is m a i n t a i n e d at 4 2 5 ° C for 40 minutes, d u r i n g w h i c h t i m e 100 seem of n i t r o g e n is b u b b l e d t h r o u g h 9 5 ° C water a n d r o u t e d t h r o u g h the furnace.  This warm,  m o i s t e n v i r o n m e n t oxidizes the a l u m i n u m layer t h r o u g h the etched holes. A f t e r the 40 m i n u t e o x i d a t i o n p e r i o d , the 100 seem of n i t r o g e n is re-routed d i r e c t l y i n t o the furnace,  CHAPTER  53  4. Sample Preparation  b y p a s s i n g the b u b b l e r , slowly p u r g i n g the m o i s t u r e out of the furnace w h i l e the t e m p e r a t u r e is reduced at a rate of 30° per hour u n t i l reaching r o o m t e m p e r a t u r e , w h i c h completes the o x i d a t i o n process. T h e purpose of o x i d i z i n g the a l u m i n u m layer is to change its i n d e x of refraction so t h a t it is significantly lower t h a n t h a t of the core layer. P r i o r t o o x i d a t i o n , the i n d e x of refraction of the a l u m i n u m layer is n^/=3.6; o x i d a t i o n reduces this to n ie=^-Q- [33] oxi(  T h e resultant G a A s t e x t u r e d p l a n a r waveguide has a h i g h i n d e x core (ncore ~ 3 . 5 ) w i t h low i n d e x c l a d d i n g ( n j = 1 . 0 , noxide=1.6). T h e f a b r i c a t i o n process is n o w complete, a n d O  r  the sample is ready to be m o u n t e d i n the e x p e r i m e n t a l a p p a r a t u s for c h a r a c t e r i z a t i o n .  4.2  Polymer Sample  T h e 2 D t e x t u r e d p o l y m e r waveguides used i n this i n v e s t i g a t i o n were f a b r i c a t e d b y P a u l R o c h o n ' s group at the R o y a l M i l i t a r y College of C a n a d a . T h e p o l y m e r m a t e r i a l is an a z o a r o m a t i c p o l y m e r film, specifically p o l y [ ( 4 - n i t r o p h e n y l ) [4-[[20(methacryloyloxy)ethyl]ethylomino]phenyl]diazene] ( p D R I M ) . [10,15,24] T o f a b r i cate this waveguide, a piece of B K 7 glass was spin-coated w i t h this p o l y m e r t o a thickness of ~ 4 0 0 n m . T h e process of t e x t u r i n g this p o l y m e r waveguide is unique, due to the p o l y m e r ' s u n u s u a l response t o intense l i g h t . T h i s p h o t o f a b r i c a t i o n process is a d i r e c t - w r i t e technique w h i c h transfers a h o l o g r a p h i c p a t t e r n d i r e c t l y to the p o l y m e r .  A 514 n m argon laser  b e a m is e x p a n d e d a n d c o l l i m a t e d , a n d used to form a linear interference p a t t e r n i n the  CHAPTER 4. Sample Preparation p o l y m e r layer of the waveguide.  54  T h i s p a r t i c u l a r p o l y m e r has a w e l l k n o w n trans-cis-  t r a n s p h o t o i s o m e r i z a t i o n t h a t causes a change i n o r i e n t a t i o n of the p o l y m e r molecules. T h e p o l y m e r moves as macromolecules w h e n " a c t i v a t e d " b y light at a m o d e s t i n t e n s i t y ( ~ m W / c m ) , well b e l o w the glass t r a n s i t i o n t e m p e r a t u r e of the p o l y m e r . T h i s has a 2  m i g r a t i o n effect since the molecules t h a t h a p p e n to move i n t o the d a r k areas stop m o v ing.  T h e p o l y m e r becomes t h i c k e r i n the areas of d e s t r u c t i v e interference a n d t h i n n e r  i n the areas of c o n s t r u c t i v e interference.  T h e resultant p a t t e r n is a l i n e a r v a r i a t i o n i n  the m o r p h o l o g y of the p o l y m e r w h i c h d i r e c t l y corresponds to the interference  pattern  used. T o m a k e a 2 D square l a t t i c e g r a t i n g , a linear p a t t e r n is m a d e , t h e n t h e process is repeated w i t h the s a m p l e r o t a t e d 9 0 ° . T h e exposure p a t t e r n for t h i s p o l y m e r g r a t i n g is s h o w n i n F i g u r e 4.2.  F i g u r e 4.2:  E x p o s u r e p a t t e r n for the p o l y m e r sample.  R e g i o n s (1) a n d (3) are  ID  gratings, w h i l e the o v e r l a p p i n g region (2) is a 2 D square l a t t i c e .  T h i s f a b r i c a t i o n t e c h n i q u e c a n be used to make gratings i n the p o l y m e r w i t h a d e p t h of m o d u l a t i o n of several h u n d r e d nanometers.  G r a t i n g s w r i t t e n i n t h i s w a y are easily  erased b y h e a t i n g the s a m p l e t o the glass t r a n s i t i o n t e m p e r a t u r e , b u t are t h o u g h t t o be  CHAPTER 4. Sample Preparation stable at r o o m t e m p e r a t u r e . [25]  55  Chapter 5 Results and Discussion  T h i s chapter presents the results of linear w h i t e light spectroscopy e x p e r i m e n t s o n several 2 D t e x t u r e d p l a n a r waveguides.  T h e e x p e r i m e n t a l d a t a is r i g o r o u s l y c o m p a r e d w i t h  s i m u l a t i o n s based o n the G r e e n ' s f u n c t i o n f o r m a l i s m d e s c r i b e d i n C h a p t e r 2. Together, the t h e o r e t i c a l a n d e x p e r i m e n t a l results from some basic structures are used to derive a c o m p r e h e n s i v e u n d e r s t a n d i n g of electromagnetic e x c i t a t i o n s associated w i t h b o t h l o w a n d h i g h i n d e x - c o n t r a s t 2 D t e x t u r e d p l a n a r waveguides. T h i s forms the f o u n d a t i o n for u s i n g the same techniques t o design a n d characterize structures w i t h p a r t i c u l a r p r o p e r t i e s t h a t m a y find a p p l i c a t i o n s as l i n e a r a n d non-linear o p t i c a l c o m p o n e n t s . S e c t i o n 5.1 presents the d i s p e r s i o n a n d p o l a r i z a t i o n properties of modes near  the  second order gap of two s i m p l y - t e x t u r e d p l a n a r G a A s waveguides as d e t e r m i n e d from specular r e f l e c t i v i t y spectra. O u t s t a n d i n g q u a n t i t a t i v e agreement between the m o d e l a n d e x p e r i m e n t a l results is d e m o n s t r a t e d for b o t h square a n d t r i a n g u l a r l a t t i c e structures i n t h i n ( ~ 8 0 n m ) G a A s slabs. T h e results from these h i g h i n d e x - c o n t r a s t structures reveal s u b s t a n t i a l r e n o r m a l i z a t i o n of the electromagnetic modes a t t a c h e d to the p o r o u s slab 56  CHAPTER 5. Results and Discussion  57  over a range of frequencies t h a t s p a n a p p r o x i m a t e l y 10% of the "center frequency."  The  d i s p e r s i o n , lifetimes, a n d c o m p o s i t i o n of these l e a k y B l o c h states are i n t e r p r e t e d u s i n g a s i m p l e p i c t u r e : the 2 D t e x t u r e couples w h a t are effectively the slab modes c h a r a c t e r i s t i c of a n u n t e x t u r e d s t r u c t u r e w i t h the core layer replaced b y a u n i f o r m layer of refractive i n d e x g i v e n a p p r o x i m a t e l y b y the average i n d e x of the t e x t u r e d core. S e c t i o n 5.2 reports d i s p e r s i o n properties of two q u i t e different structures t h a t were considered because of t h e i r p o t e n t i a l relevance i n p r a c t i c a l a p p l i c a t i o n s . S e c t i o n 5.2.1 discusses a l o w i n d e x - c o n t r a s t 2 D t e x t u r e d waveguide f o r m e d i n a n o v e l p o l y m e r m a t e r i a l t h a t s p o n t a n e o u s l y develops a deep ( ~ 2 0 0 n m ) t e x t u r e w i t h s u b - m i c r o n p e r i o d s u p o n b e i n g exposed to h o l o g r a p h i c laser  fields.  T h e s e p o l y m e r structures were d e v e l o p e d i n  K i n g s t o n O n t a r i o , a n d the results r e p o r t e d here represent the first c o m p r e h e n s i v e s t u d y of t h e i r s c a t t e r i n g properties over a b r o a d range of frequencies.  S e c t i o n 5.2.2 r e p o r t s  the successful r e a l i z a t i o n of a h i g h index-contrast t e x t u r e d p l a n a r waveguide specifically designed t o possess a n e x t r e m e l y flat b a n d a l o n g the entire T - X a x i s o f the B r i l l o u i n zone.  square  S u c h flat bands are e x p e c t e d t o offer significant advantages i n some  non-linear optical applications. S e c t i o n 5.3 e x p e r i m e n t a l l y a n d t h e o r e t i c a l l y evaluates the use of defect s u p e r l a t t i c e s as a n effective a l t e r n a t i v e for p r o b i n g the electromagnetic B l o c h states associated w i t h porous slab waveguides. T h e e v a l u a t i o n process identifies instances where a new diffract i o n measurement  technique offers considerable advantages due to its background-free  nature, b u t also identifies some l i m i t a t i o n s of the a p p r o a c h , at least w h e n i m p l e m e n t e d  CHAPTER 5. Results and Discussion  58  w i t h a l o w power w h i t e light source. F i n a l l y , S e c t i o n 5.3.3 describes a design for a n angle insensitive n o t c h filter based o n a h i g h index-contrast s t r u c t u r e t h a t i n c o r p o r a t e s a defect s u p e r l a t t i c e a n d e x h i b i t s a complete first-order pseudo-gap.  5.1  Waveguides with Simple 2D Gratings  T h e p u r p o s e of this section is to explore the s y m m e t r y a n d p o l a r i z a t i o n p r o p e r t i e s of l e a k y modes b o u n d to t e x t u r e d p l a n a r waveguides, a n d to q u a n t i t a t i v e l y v a l i d a t e the c o m p u t e r m o d e l i n g code. Specifically, the specular r e f l e c t i v i t y technique is used t o p r o b e the modes near the second order gap of waveguides w i t h square a n d t r i a n g u l a r lattices t h a t were designed a n d fabricated for this purpose. B o t h waveguides c o n t a i n h i g h i n d e x contrast gratings.  5.1.1  Square Lattice  T o investigate the characteristics of a p l a n a r waveguide w i t h a s i m p l e , h i g h i n d e x - c o n t r a s t g r a t i n g , a square l a t t i c e of r o u n d holes was etched i n t o a G a A s waveguide, like the one d e p i c t e d i n F i g u r e 4.1.  T h i s waveguide does not have a c o m p l e t e p h o t o n i c pseudo-  gap, b u t was selected for pedagogical purposes because it i l l u s t r a t e s k e y features of the d i s p e r s i o n of p h o t o n i c modes i n this t y p e of waveguide. T h e t e x t u r e d p l a n a r waveguide sample discussed i n this section was f a b r i c a t e d b y another student i n the group, V . P a c r a d o u n i , u s i n g A S U 7 2 1 . T h e s a m p l e is c o m p r i s e d of: 40 n m of G a A s o n t o p of 40 n m of Alo.3Gao.7As, c l a d b y air above a n d 1.8 yum of fully  CHAPTER  59  5. Results and Discussion  o x i d i z e d Alo.9sGao.02As below, on a G a A s substrate. A square l a t t i c e of c i r c u l a r holes 550 n m a p a r t w i t h a radius of 165 n m was etched c o m p l e t e l y t h r o u g h the waveguide core, e x t e n d i n g i n t o the u n d e r l y i n g c l a d d i n g . T h e p a t t e r n e d region is 90 u m x 90 yum.  The  oxide c l a d d i n g layer is sufficiently t h i c k t h a t the evanescent F o u r i e r c o m p o n e n t s of the B l o c h states i n the v i c i n i t y of the second order gap are l i m i t e d to the oxide layer. T h u s , the finite lifetimes of the modes are d e t e r m i n e d entirely b y a single F o u r i e r c o m p o n e n t of the p o l a r i z a t i o n w h i c h radiates into the upper a n d lower h a l f spaces. T h i s s a m p l e was characterized b y V . P a c r a d o u n i u s i n g the a p p a r a t u s designed a n d b u i l t b y the author. A d e t a i l e d i n v e s t i g a t i o n of this sample is presented i n Reference [21]. W h e n the p h o t o n i c modes of this sample are p r o b e d i n the T - X d i r e c t i o n u s i n g the specular m e a s u r e m e n t technique, the s p e c t r a s h o w n i n F i g u r e 5.1 are o b t a i n e d . [20] T h e dashed s p e c t r u m at the b o t t o m of F i g u r e 5.1 was s i m u l a t e d for p - p o l a r i z e d r a d i a t i o n i n c i d e n t at 5 ° . T h e second line from the b o t t o m is the c o r r e s p o n d i n g p - p o l a r i z e d d a t a t a k e n f r o m the G a A s waveguide. also s h o w n .  S-polarized d a t a for various angles of incidence are  T h e l o w frequency oscillations are F a b r y - P e r o t fringes, w h i c h are due t o  the interference of l i g h t reflecting off the t o p a n d b o t t o m surfaces of the oxide layer. T h e sharp F a n o - l i k e resonances i n d i c a t e c o u p l i n g i n t o one of the leaky eigenstates of the t e x t u r e d waveguide. These resonances are n a r r o w for modes w i t h l o n g lifetimes a n d w i d e for modes w i t h short lifetimes. T o extract the energy a n d lifetime of the modes from these spectra, a m a t h e m a t i c a l f i t t i n g technique was used.  T h i s technique fits an A i r y f u n c t i o n to the  Fabry-Perot  CHAPTER 5. Results and Discussion  60  F i g u r e 5.1: S p e c u l a r reflectivity d a t a for a t e x t u r e d p l a n a r waveguide w i t h a s i m p l e square l a t t i c e of holes etched t h r o u g h the core of A S U 7 2 1 . A n g l e of i n c i d e n c e is l a b l e d on each s p e c t r u m . T h e b o t t o m two s p e c t r a are a c o m p a r i s o n of s i m u l a t i o n ( d o t t e d line) a n d d a t a (solid line) for p - p o l a r i z e d reflectivity. T h e upper nine s p e c t r a are s - p o l a r i z e d specular r e f l e c t i v i t y d a t a . [20]  oscillations a n d a F a n o - f u n c t i o n to each mode. F o r a d e t a i l e d d e s c r i p t i o n of this f i t t i n g technique refer to Reference [21]. T h e energy of the modes a n d the angles of i n c i d e n c e are used t o convert these s p e c t r a i n t o a d i s p e r s i o n d i a g r a m . F i g u r e 5.2 shows the d i s p e r s i o n d i a g r a m for this G a A s waveguide sample. T h i s d i a g r a m depicts the s- a n d p - p o l a r i z e d bands for b o t h the m o d e l a n d the e x p e r i m e n t a l d a t a .  CHAPTER  61  5. Results and Discussion •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 .. , ^gp.  i  -  B o  9 E?  A-  8:  _  M  -*- p polarization •  Model Experiment -0.3  -0.2  -0.1  0.0  0.2  0.1  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 lowdispersion bands in the T - X direction near 10,000 c m . These bands have a very low - 1  group velocity and are therefore good candidates as hosts for certain non-linear optical conversion processes. [28-30] A s 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, w h i c h are also i n remarkable agreement w i t h the s i m u l a t i o n s a n d p r e d i c t i o n s based o n s y m m e t r y arguments.  T h e q u a n t i t a t i v e agreement between the d a t a a n d the  c o m p u t e r m o d e l for the dispersion, as well as for the lifetimes of the modes, t o t h i s a u t h o r ' s k n o w l e d g e is the best r e p o r t e d for 2 D t e x t u r e d p l a n a r waveguide structures. T a b l e 5.1: K e y m o d e l i n g parameters for t e x t u r e d p l a n a r waveguide w i t h square l a t t i c e Parameter  Value  Pitch  550 n m  H o l e radius  161 n m  T h i c k n e s s of core  83 n m  Core composition  6 0 % 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 d e m o n s t r a t i n g a c o m p l e t e gap i n a pure 2 D p h o t o n i c c r y s t a l of "infinite" rods arranged i n a square l a t t i c e [12], a t t e m p t s have p r o v e n unsuccessful at c r e a t i n g a complete pseudo-gap i n a p l a n a r waveguide w i t h a square l a t t i c e of holes such as the one described above. A more s y m m e t r i c l a t t i c e c o n f i g u r a t i o n makes it easier t o achieve a c o m p l e t e pseudo-gap i n a p l a n a r waveguide.  5.1.2  Triangular Lattice  T o explore the b a n d s t r u c t u r e of a p l a n a r waveguide w i t h a m o r e s y m m e t r i c l a t t i c e , a h i g h i n d e x - c o n t r a s t t r i a n g u l a r g r a t i n g was designed a n d fabricated. T h i s waveguide was specifically designed i n order to characterize the p o l a r i z a t i o n a n d d i s p e r s i o n p r o p e r t i e s of  CHAPTER 5. Results and Discussion  63  the bands near the second order gap u s i n g the specular r e f l e c t i v i t y technique. T h i s sect i o n d e m o n s t r a t e s the characteristics specific to a t r i a n g u l a r l a t t i c e , a n d shows the b a n d s t r u c t u r e t o be richer t h a n t h a t of the p r e v i o u s l y discussed square l a t t i c e .  Although  t h i s p a r t i c u l a r waveguide was not engineered to have a c o m p l e t e second order pseudogap, a t r i a n g u l a r l a t t i c e does have a sufficiently h i g h degree of s y m m e t r y t o p r o d u c e a c o m p l e t e pseudo-gap, as w i l l be discussed i n S e c t i o n 5.3.3. U n l i k e the square l a t t i c e s t r u c t u r e discussed above, the Alo.9sGao.02As layer b e n e a t h the core slab layer i n t h i s t r i a n g u l a r l a t t i c e s t r u c t u r e was o n l y p a r t i a l l y o x i d i z e d , t o a d e p t h of ~ 2 0 0 n m . N e v e r theless, the s i m u l a t i o n s again agree r e m a r k a b l y w e l l w i t h the e x p e r i m e n t a l l y d e t e r m i n e d d i s p e r s i o n , as discussed i n the present section, a n d w i t h the lifetimes, as discussed b e l o w i n S e c t i o n 5.3.2. T o fabricate t h i s t e x t u r e d p l a n a r waveguide, a 90 pm x 90 pm  t r i a n g u l a r l a t t i c e of  c i r c u l a r holes 600 n m a p a r t w i t h a r a d i u s of 80 n m was etched c o m p l e t e l y t h r o u g h the core of a n A S U 7 2 1 waveguide, o m i t t i n g every seventh h o l e .  1  T h e Alo.9sGao.02As layer  was p a r t i a l l y o x i d i z e d , p r o d u c i n g a t h i n oxide layer, o n the order of 200 n m t h i c k , o n t o p of the r e m a i n i n g ~ 1 6 0 0 n m of u n - o x i d i z e d Alo.9sGao.02As. P r o b i n g the p h o t o n i c modes of this waveguide w i t h the specular measurement  tech-  nique produces the s p e c t r a s h o w n i n F i g u r e 5.3 for the M d i r e c t i o n , a n d the s p e c t r a i n F i g u r e 5.4 for the K d i r e c t i o n . T h e F a n o - l i k e resonances are evident, as w e l l as F a b r y P e r o t o s c i l l a t i o n s . C l o s e r e x a m i n a t i o n of the s p e c t r a for a 10° angle of i n c i d e n c e i n the M d i r e c t i o n (shown i n F i g u r e 5.5) reveals a beat i n the F a b r y - P e r o t o s c i l l a t i o n s , w h i c h x  The 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 o p t i c a l cavities ( A l - o x i d e a n d Alo.98Gao.02As) created b y the p a r t i a l o x i d a t i o n . T h e r e are four s- a n d two p - p o l a r i z e d resonances i n the s p e c t r a , w h i c h are labeled o n this d i a g r a m .  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 i g u r e 5.3:  S p e c u l a r reflectivity d a t a for the T - M d i r e c t i o n from the G a A s  p l a n a r waveguide w i t h a t r i a n g u l a r lattice discussed i n the text.  textured  S o l i d lines represent  s - p o l a r i z a t i o n ; dashed lines represent p - p o l a r i z a t i o n .  T h e m o d e energies were e x t r a c t e d from the s p e c t r a to p r o d u c e the d a t a points i n the dispersion d i a g r a m shown i n F i g u r e 5.6. In the M d i r e c t i o n near zone center, there are four s-polarized bands: two dispersing up i n energy, a n d two d i s p e r s i n g d o w n . T h e r e are also t w o p - p o l a r i z e d bands: one dispersing up, a n d one d o w n . In the K d i r e c t i o n near  CHAPTER  ^  65  5. Results and Discussion  "o  ,.'••50  —  D  \-  <  .-  0  30  .\> o o q=l o  3  20'  Pi  10  3  \ 1 7.0  7.5  8.0  8.5  9.0  9.5  1  10.0  10.5  11.0  12.0x10  Energy (cm )  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  p2  r  At? *'  y  /  1  1  j  if  si  / s2  v  y  f s,3  1  /  s4 6.0  6.5  7.0  7.5  8.0  8.5  9.0  9.5  10.0 10.5 11.0  12.0xl0  3  Energy (cm ') F i g u r e 5.5: S p e c u l a r reflectivity d a t a for a 10° angle of incidence i n the M d i r e c t i o n . S o l i d lines represent s - p o l a r i z a t i o n ; dashed lines represent p - p o l a r i z a t i o n . N o t i c e t h e two F a b r y - P e r o t frequencies w h i c h are due t o i n c o m p l e t e o x i d a t i o n of the Alo.9sGao.02As layer. T h e higher frequency is a p p r o x i m a t e l y 1 , 0 0 0 c m w h i c h corresponds to ~ l , 3 0 0 n m of u n o x i d i z e d Alo.9sGao.02As. S i x modes are a n n o t a t e d . - 1  six resonant modes c l e a r l y seen i n the d a t a for this angle of incidence are l a b e l e d i n F i g u r e 5.5. T h e four s-polarized modes are l a b e l e d s l - s 4 , w h i l e the t w o p - p o l a r i z e d m o d e s are labeled p l a n d p2. T h e strength of the F o u r i e r components of the field (see E q u a t i o n 2.2) v a r y from one m o d e to the next. F i g u r e 5.7 depicts the relative strengths of each of these F o u r i e r components for each of the s i x modes. the same two d o m i n a n t components.  F i g u r e s (a) a n d (d) b o t h show  These components are "away" from the d i r e c t i o n  of the i n c i d e n t wavevector a n d therefore produce d o w n w a r d p r o p a g a t i n g modes.  These  c o m p o n e n t s a d d s y m m e t r i c a l l y c r e a t i n g an s-polarized mode, a n d a n t i - s y m m e t r i c a l l y c r e a t i n g a p - p o l a r i z e d mode. F i g u r e s (b) a n d (e) show two d o m i n a n t c o m p o n e n t s each, however i n this case the d o m i n a n t components are "toward" the d i r e c t i o n of p r o p a g a t i o n . T h e s e two s i t u a t i o n s give rise to the u p w a r d d i s p e r s i n g s- a n d p - p o l a r i z e d modes. In the  CHAPTER 5. Results and Discussion  0.5  0.4  0.3  0.2  — K F i g u r e 5.6:  0.1  67  0.0  0.1  0.2  0.3  r  0.4  0.5 M  D i s p e r s i o n d i a g r a m for the p l a n a r waveguide w i t h the t r i a n g u l a r  discussed i n the text. p-polarized data.  T h e crosses represent s-polarized d a t a .  lattice  T h e circles represent  T h e s o l i d lines represent s-polarized s i m u l a t i o n s .  T h e d a s h e d lines  represent p - p o l a r i z e d s i m u l a t i o n s .  final  two figures, (c) a n d (g), there is a single d o m i n a n t c o m p o n e n t w h i c h gives rise  to one s-polarized m o d e d i s p e r s i n g q u i c k l y d o w n w a r d a n d another d i s p e r s i n g q u i c k l y upward.  It is i m p o r t a n t to note these results are for modes o c c u r r i n g a p p r o x i m a t e l y  20% of the w a y across the B r i l l o u i n zone, w h i c h is far from zone center where there is s t r o n g c o u p l i n g between several of the field components. T h i s is not e x p l a i n e d b y s i m p l e k i n e m a t i c arguments. T o i l l u s t r a t e the s t r o n g c o u p l i n g near zone center, m o m e n t u m space d i a g r a m s were c o n s t r u c t e d for the modes close t o zone center (K\\ — O.Ol-Kg) but s t i l l s l i g h t l y d e t u n e d i n the M d i r e c t i o n . T h e s e d i a g r a m s are s h o w n i n F i g u r e 5.8. T h e s e d i a g r a m s reveal s t r o n g  CHAPTER 5. Results and Discussion  68  M  F i g u r e 5.7:  (a) p i  (b) p2  (c) s i  (d) s2  (e) s3  (f) s4  M o m e n t u m space d i a g r a m s for each B l o c h m o d e of the t r i a n g u l a r l a t t i c e  waveguide w h e n p r o b e d w i t h light at a 10° angle of incidence. F o u r i e r c o m p o n e n t is i n d i c a t e d b y the size of the dot.  T h e s t r e n g t h of each  T h e labels for each d i a g r a m  c o r r e s p o n d to the labels o n the modes s h o w n i n F i g u r e 5.5  c o u p l i n g between m u l t i p l e field c o m p o n e n t s , w h i c h is very n o n - k i n e m a t i c .  A w a y from  zone center, the highest energy b a n d s d i s p l a y anti-crossings w i t h modes d e s c e n d i n g f r o m a h i g h e r order gap.  T h e size of the gaps is s i m i l a r to the zone-center g a p , suggesting  s t r o n g b a n d - m i x i n g , w h i c h cannot be e x p l a i n e d w i t h s i m p l e k i n e m a t i c s . A l t h o u g h this waveguide has a defect superlattice, the d i s p e r s i o n properties a n d p o l a r i z a t i o n s of these b a n d s are fully consistent w i t h t h a t of a waveguide w i t h a s i m p l e (non-defect) t r i a n g u l a r lattice. T h e s i m u l a t e d b a n d s t r u c t u r e o b t a i n e d u s i n g the parameters i n T a b l e 5.2, is represented as lines o n the d i s p e r s i o n d i a g r a m i n F i g u r e 5.6. A s w i t h the square l a t t i c e s a m p l e ,  CHAPTER  69  5. Results and Discussion  • •  • • ©  (a) p i  (b) p2  (c) si  (d) s2  (e) s3  (f) s4  F i g u r e 5.8: M o m e n t u m space d i a g r a m s for each B l o c h m o d e of the t r i a n g u l a r l a t t i c e at K||=0.01K  9  i n the M d i r e c t i o n .  T h e M d i r e c t i o n is t o w a r d the top of the page.  s t r e n g t h of each F o u r i e r c o m p o n e n t is i n d i c a t e d b y the size of the dot.  The  T h e labels for  each d i a g r a m c o r r e s p o n d to the labels on the modes s h o w n i n F i g u r e 5.5  there is r e m a r k a b l e q u a n t i t a t i v e agreement between the m o d e l a n d the d a t a w i t h r e g a r d to d i s p e r s i o n a n d p o l a r i z a t i o n s . T h e o n l y p a r a m e t e r of the waveguide w h i c h was a l t e r e d from its n o m i n a l v a l u e for this s i m u l a t i o n was the thickness of the core; i t was m o d e l e d 2  as a single layer, 73 n m t h i c k , c o m p r i s e d of Alo.15Gao.85As, w h i c h was used to s i m u l a t e the c o m b i n a t i o n of the 40 n m t h i c k layer of Alo.30Gao.70As a n d the 40 n m t h i c k layer of G a A s i n the sample. C o m p u t e r m o d e l i n g was performed to determine the effect the defect  superlattice  has o n the d i s p e r s i o n characteristics of this s t r u c t u r e . F i g u r e 5.9 shows a c o m p a r i s o n o f 2  The deviation from the nominal value is within the margin of error.  CHAPTER 5. Results and Discussion  70  T a b l e 5.2: K e y m o d e l i n g parameters for t e x t u r e d p l a n a r waveguide w i t h t r i a n g u l a r l a t t i c e Parameter  Value  Pitch  600 n m  Hole radius  80 n m  T h i c k n e s s of core  73 n m  Core composition  Alo.15Gao.85As  O x i d e thickness  200 n m  Hole depth  73 n m  the s p e c t r a for the waveguide at 10° w i t h a n d w i t h o u t the defect s u p e r l a t t i c e . T h e r e is essentially no difference between the results of the m o d e l i n g w i t h defects a n d w i t h o u t t h e m . T h u s , i n t h i s c o n t e x t , the defect s u p e r l a t t i c e i n c o r p o r a t e d i n this s a m p l e represents o n l y a weak p e r t u r b a t i o n . T h e u t i l i t y of t h i s defect s u p e r l a t t i c e is e x p l a i n e d i n S e c t i o n 5.3.2. C o m p u t e r m o d e l i n g was also performed to s t u d y the effect of t h e thickness of t h e oxide layer for t h i s s t r u c t u r e .  F i g u r e 5.10 shows the s i m u l a t e d s p e c t r a for t h i s s a m p l e  w i t h 200 n m of o x i d e , i.e. a t h i n oxide c l a d d i n g . F i g u r e 5.11 shows the s i m u l a t e d s p e c t r a for t h i s s a m p l e w i t h a t h i c k oxide (1600 n m of fully o x i d i z e d Alo.9sGao.02 A s ) c l a d d i n g . T h e s p e c t r a for t h e t h i c k o x i d e sample shows well-defined, n a r r o w F a n o resonances, w h i c h go to u n i t y reflectivity. [6] T h e s p e c t r a for the t h i n oxide shows m u c h broader, less w e l l defined resonances, w h i c h do not go to u n i t y reflectivity.  T h e b r o a d n a t u r e (i.e.  short  lifetime) of these modes is due to the fact t h a t the evanescent c o m p o n e n t s of the B l o c h states of these modes penetrate the t h i n oxide a n d r a d i a t e i n t o t h e s u b s t r a t e t h r o u g h  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  10.4x10  3  Energy ( c m ' ) F i g u r e 5.9: C o m p a r i s o n of s i m u l a t i o n s of the t r i a n g u l a r l a t t i c e waveguide w i t h a n d w i t h out a defect s u p e r l a t t i c e at a 10° angle of incidence i n the M d i r e c t i o n of s y m m e t r y . T h e s o l i d lines represent s-polarized a n d the dashed lines represent p - p o l a r i z e d reflectivity s p e c t r a . T h e defect s u p e r l a t t i c e s p e c t r a appear b e l o w the non-defect s p e c t r a .  CHAPTER 5. Results and Discussion  72  the oxide. T h i s issue is q u a n t i t a t i v e l y addressed i n S e c t i o n 5.3.2.  It is i n t e r e s t i n g t o  note t h a t this q u a l i t a t i v e change i n the n a t u r e of the modes has h a r d l y a n y effect o n the d i s p e r s i o n or the energy of the modes, b u t a d r a m a t i c effect o n the lifetimes.  7.0  7.5  8.0  8.5  9.0  9.5  10.0  10.5  11.0  12.0x10  3  Energy (cm" ) 1  F i g u r e 5.10: S i m u l a t e d s p e c t r a for the M d i r e c t i o n of a t r i a n g u l a r l a t t i c e s a m p l e w i t h 200 n m of oxide, i.e. t h i n oxide layer. T h e s p e c t r a are for angles of i n c i d e n c e of 1 0 ° , 2 0 ° , 3 0 ° , 4 0 ° , a n d 50° from the b o t t o m up. S o l i d lines represent s - p o l a r i z a t i o n ; dashed lines represent p - p o l a r i z a t i o n .  T h i s section has d e m o n s t r a t e d the a b i l i t y to a c c u r a t e l y fabricate a n d characterize text u r e d p l a n a r waveguides, a n d to a c c u r a t e l y predict w i t h a rigorous c o m p u t e r m o d e l the c o m p l e x b a n d s t r u c t u r e o f these waveguides. O u t s t a n d i n g q u a n t i t a t i v e agreement has been s h o w n between the c o m p u t e r m o d e l i n g results a n d the d a t a for b o t h of the s i m p l e  CHAPTER 5. Results and Discussion  7.0  7.5  8.0  8.5  9.0  9.5  73  10.0  10.5  11.0  12.0x10  3  Wavenumber (cm ) F i g u r e 5.11: S i m u l a t e d s p e c t r a for the M d i r e c t i o n of a t r i a n g u l a r l a t t i c e sample w i t h 1600 n m of oxide, i.e. t h i c k oxide layer. T h e s p e c t r a are for angles of incidence of 1 0 ° , 2 0 ° , 3 0 ° , 4 0 ° , a n d 50° from the b o t t o m up. S o l i d lines represent s - p o l a r i z a t i o n ; dashed lines represent p - p o l a r i z a t i o n .  l a t t i c e configurations e x a m i n e d . T h e configuration a n d m a t e r i a l s of t e x t u r e d waveguides have been s h o w n t o s t r o n g l y influence the p r o p a g a t i o n of l i g h t i n these structures.  The  a b i l i t y to p r e d i c t t h r o u g h c o m p u t e r m o d e l i n g the effects of these factors o n the elect r o m a g n e t i c e x c i t a t i o n s of these waveguides, c o m b i n e d w i t h a t h o r o u g h  understanding  of the u n d e r l y i n g n a t u r e of the modes, p r o v i d e the basis for engineering desired b a n d s t r u c t u r e s u s i n g the proven f a b r i c a t i o n a n d c h a r a c t e r i z a t i o n techniques.  CHAPTER 5. Results and Discussion  5.2  74  Waveguides Engineered for Specific Applications  P l a n a r t e x t u r e d waveguides offer a powerful m e d i u m for engineering devices w h i c h c o n t r o l the p r o p a g a t i o n characteristics of light, b o t h i n the waveguide a n d i n the s u r r o u n d i n g h a l f spaces.  T h i s section describes two examples of t e x t u r e d p l a n a r waveguides w h i c h  have been engineered for specific a p p l i c a t i o n s . T h e first is a l o w i n d e x - c o n t r a s t t e x t u r e d waveguide w h i c h uses resonant c o u p l i n g t o effect a p o l a r i z a t i o n insensitive n o t c h filter for r a d i a t i o n i n c i d e n t from the u p p e r h a l f space. T h i s is just one of m a n y e x a m p l e s of s i m p l e passive o p t i c a l devices t h a t m i g h t be easily fabricated from the i n t r i g u i n g a z o - p o l y m e r s d e s c r i b e d i n C h a p t e r 4. T h e results i n S e c t i o n 5.2.1 serve as m u c h to characterize the properties of these t e x t u r e d p o l y m e r s as they do t o d e m o n s t r a t e a p a r t i c u l a r o p t i c a l funct i o n a l i t y . T h e second e x a m p l e is a h i g h index-contrast waveguide engineered to possess a n e x t r e m e l y b r o a d , flat b a n d , w h i c h has been p r e d i c t e d to be useful for significantly e n h a n c i n g the second h a r m o n i c o p t i c a l conversion process for modes p r o p a g a t i n g inside p h o t o n i c c r y s t a l s . [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 m a t e r i a l w h i c h c a n be q u i c k l y a n d s i m p l y m a d e i n t o large area, l o w i n d e x - c o n t r a s t , p e r i o d i c a l l y t e x t u r e d waveguides u s i n g a d i r e c t - w r i t e h o l o g r a p h i c technique, as d e s c r i b e d i n C h a p t e r 4. Because waveguides f a b r i c a t e d w i t h this p o l y m e r do not have the stringent p u r i t y requirements t h a t t h e i r s e m i c o n d u c t o r c o u n t e r p a r t s do, f a b r i c a t i o n costs are m u c h less for devices m a n u f a c t u r e d w i t h this m a t e r i a l . T h i s section  CHAPTER 5. Results and Discussion  75  reports the first i n - d e p t h q u a n t i t a t i v e s t u d y of the b r o a d b a n d s c a t t e r i n g properties of this t y p e of a z o - p o l y m e r waveguide. T h e reflective characteristics p e c u l i a r to this t y p e of waveguide make it p a r t i c u l a r l y well-suited for use as a n o p t i c a l filter. P o l y m e r s are c u r r e n t l y used i n numerous o p t i c a l a p p l i c a t i o n s , such as coatings, filters a n d  fibers.  As  a representative e x a m p l e of a p o l y m e r waveguide a p p l i c a t i o n , this section presents the s i m u l a t i o n of a p D R I M t e x t u r e d p l a n a r waveguide as a p o l a r i z a t i o n insensitive n o t c h filter. P r i o r t o t e x t u r i z a t i o n , the p D R I M p o l y m e r waveguide consisted of a n o m i n a l l y 400 n m t h i c k film on B K 7 glass. T h e i n d e x of refraction of the p o l y m e r was m e a s u r e d t o be 1.66 i n the v i s i b l e w i t h a n A b b e refractometer, a n d is e s t i m a t e d t o be 1.65 i n the near-infrared. [26] T h e i n d e x of refraction of the glass is 1.507 at lpm. t e x t u r e d w i t h a square l a t t i c e of p e r i o d 6 5 9 ± 2 n m  3  [8] T h e core was  a n d a n o m i n a l average m o d u l a t i o n  a m p l i t u d e of 250 n m . T h e u n i t cell of this l a t t i c e consists of a p a r a b o l o i d a l - l i k e "rise" i n the core m e d i u m , r a t h e r t h a n a c y l i n d r i c a l hole w i t h v e r t i c a l sidewalls. T h e result is a g r a t i n g of s i g n i f i c a n t l y different m o r p h o l o g y t h a n the p r e v i o u s l y discussed G a A s t e x t u r e d p l a n a r waveguides, as the a t o m i c force m i c r o g r a p h of the surface of the t e x t u r e d p o l y m e r shows i n F i g u r e 5.12. Since the g r a t i n g of the p o l y m e r waveguide is s t r u c t u r a l l y different from the l i t h o g r a p h i c a l l y defined gratings discussed t h r o u g h o u t this thesis, the parameters of the p o l y mer g r a t i n g layer are defined differently for the Green's function code. P r e v i o u s l y , where the code m o d e l e d a layer of G a A s w i t h "holes" of air, for this waveguide the code m o d e l s 3  T h e period of the lattice was measured using HeNe diffraction.  CHAPTER 5. Results and Discussion  70  77  CHAPTER 5. Results and Discussion  a layer of air w i t h "holes" of p o l y m e r . E a c h u n i t cell of the p o l y m e r g r a t i n g was m o d e l e d as a t h i c k v e r t i c a l c y l i n d e r , r a t h e r t h a n the a c t u a l shape, w h i c h was s o m e w h a t uniform and asymmetric.  non-  T h e p a r a b o l o i d a l shape of the m o u n d s was not entered i n t o  the code because m o d e l i n g t h e m as v e r t i c a l c y l i n d e r s w i t h the a p p r o p r i a t e f i l l i n g f r a c t i o n is sufficient to a c c u r a t e l y reproduce the measured dispersion of a waveguide w i t h a l o w index-contrast g r a t i n g such as this.  T h e s i m u l a t e d s p e c t r a for the p o l y m e r waveguide  were generated b y the c o m p u t e r code using the values s h o w n i n T a b l e 5.3.  T a b l e 5.3: K e y m o d e l i n g parameters for t e x t u r e d p l a n a r p o l y m e r waveguide Parameter  Nominal  Modeled  P i l l a r radius  NA  300 n m  T h i c k n e s s of t e x t u r e d layer  250 n m  250 n m  T h i c k n e s s of u n t e x t u r e d layer  250 n m  400 n m  Period  660 n m  660 n m  n  1.507  1.507  1.65  1.65  g  n  p  T h e s p e c t r a i n F i g u r e 5.13 were o b t a i n e d b y p r o b i n g the p h o t o n i c modes of the p o l y m e r waveguide at a range of incident angles a l o n g the T - X axis u s i n g the specular m e a s u r e m e n t technique.  N o t i c e t h a t s u p e r i m p o s e d o n the r e l a t i v e l y l o w b a c k g r o u n d  there are sets of h i g h reflectivity peaks of s- a n d p - p o l a r i z e d pairs.  T h e r e appears t o  be one s- a n d one p - p o l a r i z e d p a i r of modes d i s p e r s i n g s t r o n g l y u p i n energy, a n d one p a i r d i s p e r s i n g s t r o n g l y d o w n i n energy.  T h e r e m a i n i n g s- a n d p- b a n d s consist of a n  unresolved g r o u p of modes t h a t disperse m o d e r a t e l y u p i n energy.  F i g u r e 5.14 shows  CHAPTER  78  5. Results and Discussion  the s p e c t r a i n the M d i r e c t i o n , w h i c h show two apparent s- a n d p - p o l a r i z e d b a n d pairs, one d i s p e r s i n g u p i n energy a n d the other d o w n . T h e s p e c t r a generated b y the  computer  m o d e l reveals t h a t each of these apparent pairs a c t u a l l y consists of two s-polarized m o d e s a n d two p-, t o t a l l i n g eight modes i n the M d i r e c t i o n . F u r t h e r m o r e , the m o d e l i n g for the X d i r e c t i o n shows t h a t the m i d d l e group of unresolved modes is a c t u a l l y c o m p r i s e d of two s- a n d two p - p o l a r i z e d modes. T h i s m o d e l confirms the r e m a i n i n g modes i n the X d i r e c t i o n t o be one s- a n d one p - p o l a r i z e d p a i r of modes d i s p e r s i n g s t r o n g l y u p i n energy, a n d one p a i r d i s p e r s i n g s t r o n g l y d o w n i n energy.  Energy ( c m ' ) F i g u r e 5.13: N o r m a l i z e d specular reflectivity d a t a for the t e x t u r e d p o l y m e r waveguide for the X d i r e c t i o n . S o l i d lines represent s-polarization; dashed lines represent p - p o l a r i z a t i o n .  CHAPTER  6.0  5. Results and Discussion  7.0  8.0  79  9.0  10.0  11.0  12.0  13.0xl0  3  Energy (cm ') Figure 5.14: Normalized specular reflectivity data for the textured polymer waveguide 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 corresponding 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  s p e c t r o m e t e r , b u t is a t r u e representation of the m o d e shape. 4  A p u b l i s h e d character-  i z a t i o n s t u d y of a s i m i l a r p o l y m e r waveguide at 632.8 n m reveals a s i m i l a r d i s c r e p a n c y between the t h e o r e t i c a l a n d the a c t u a l m o d e w i d t h , w h i c h was p a r t i a l l y a t t r i b u t e d  to  a b s o r p t i o n . [25] C a l c u l a t i o n s for the p o l y m e r waveguide characterized here i n d i c a t e t h a t a b s o r p t i o n o n l y accounts for a p p r o x i m a t e l y 10% of the observed b r o a d e n i n g .  I n the  p r e v i o u s p u b l i c a t i o n , it is suggested t h a t birefringence c o n t r i b u t e s t o the b r o a d e n i n g , b u t s i m u l a t i o n s w i t h birefringence c o n d u c t e d here do not show sufficient b r o a d e n i n g . F u r t h e r , it is suggested i n the p u b l i c a t i o n t h a t the b r o a d e n i n g m a y be due t o o p t i c a l l y i n d u c e d changes t o the p o l y m e r caused by t h e i r 100 / / W probe b e a m . Since t h i s p o l y m e r has the inherent characteristic of c h a n g i n g w h e n exposed t o intense light (see C h a p t e r 4 ) , it is reasonable t h a t the p r o b i n g light c o u l d affect changes i n the waveguide t h a t w o u l d p r o d u c e noticeable changes i n the s p e c t r a l response. T h e reason t h a t the s p e c t r a consist of eight b a n d s r a t h e r t h a n four (as p r e v i o u s l y s h o w n for a square lattice) is because the lowest order T E a n d T M slab modes for t h i s waveguide are n e a r l y degenerate:  four b a n d s therefore come f r o m the m i x i n g of the  T E - l i k e slab modes, a n d four b a n d s come from the m i x i n g of the T M - l i k e slab modes. T h e b a n d s t r u c t u r e is s h o w n i n F i g u r e 5.16.  T h i s d i a g r a m depicts the d i s p e r s i o n for  the e x p e r i m e n t a l d a t a as w e l l as for the c o m p u t e r s i m u l a t i o n of the p o l y m e r waveguide o b t a i n e d u s i n g the parameters listed i n T a b l e 5.3. T h e c o m p u t e r s i m u l a t i o n of the d i s p e r s i o n of the modes agrees w e l l w i t h the e x p e r i m e n t a l d a t a a n d shows a n apparent convergence of the bands near 9,800 c m 4  The spectral data reported in this thesis was acquired at 7 c m  - 1  resolution.  - 1  , minimal  CHAPTER 5. Results and Discussion  81  Energy (cm ) F i g u r e 5.15: S i m u l a t i o n (narrow features) a n d d a t a ( b r o a d features) for the t e x t u r e d p o l y m e r waveguide for 1 0 ° , 20° a n d 30° angles of incidence a l o n g the M d i r e c t i o n . N o t e : the d a t a are not p l o t t e d o n a n absolute scale. S - p o l a r i z e d s p e c t r a are represented b y t h e s o l i d lines, a n d p - b y the dashed lines.  CHAPTER 5. Results and Discussion  82  F i g u r e 5.16: D i s p e r s i o n d i a g r a m for t e x t u r e d p l a n a r waveguide w i t h 2 D p o l y m e r grati n g discussed i n the t e x t .  D a t a values are s h o w n as circles a n d crosses for s- a n d p -  p o l a r i z a t i o n s respectively.  S i m u l a t i o n s are shown as solid lines a n d dashed lines for s-  a n d p - p o l a r i z a t i o n s respectively. T h e light lines are represented b y d o t - d a s h e d lines: the u p p e r is the glass light line, a n d the lower is the p o l y m e r light line.  c u r v a t u r e of the lower a n d u p p e r bands, a n d a slight u p w a r d curve of the c e n t r a l b a n d s i n the X d i r e c t i o n . I n the M d i r e c t i o n the u p w a r d a n d d o w n w a r d d i s p e r s i n g b a n d s show m i n i m a l curvature. A d i s p e r s i o n d i a g r a m of this p o l y m e r waveguide  without  t e x t u r i n g w o u l d show a  s l i g h t l y c u r v i n g b a n d following the glass light line at l o w energies a n d t r a n s i t i o n i n g g r a d u a l l y t o w a r d the p o l y m e r light line at higher energies.  A s evident i n F i g u r e 5.16,  the  CHAPTER 5. Results and Discussion d i s p e r s i o n of this  textured  83  waveguide has the same basic characteristics as i f it were u n -  t e x t u r e d , w i t h the e x c e p t i o n of zone-folding. T h u s the t e x t u r i n g is not g r e a t l y p e r t u r b i n g the b a n d s t r u c t u r e .  W h e n a g r a t i n g serves as a weak p e r t u r b a t i o n e v i d e n c e d b y s m a l l  gaps at zone b o u n d a r i e s , such as is the case w i t h this p o l y m e r waveguide, the d i s p e r s i o n c a n be u n d e r s t o o d to follow s i m p l e k i n e m a t i c s . T h i s means t h a t the g r a t i n g serves o n l y to i m p a r t m o m e n t u m i n integer m u l t i p l e s to allow zone-folding, a n d does not s t r o n g l y affect the d i s p e r s i o n characteristics of the waveguide. T h i s occurs i n this case essentially because the modes are quite w e a k l y confined to the p o l y m e r due t o the s m a l l d i e l e c t r i c contrast between the p o l y m e r a n d the glass. F o r this p a r t i c u l a r p o l y m e r waveguide, the d i s p e r s i o n follows s i m p l e k i n e m a t i c s . T h i s c h a r a c t e r i z a t i o n of the t e x t u r e d p o l y m e r waveguide has d e m o n s t r a t e d t h a t t h i s t y p e of s t r u c t u r e c a n e x h i b i t r e l a t i v e l y h i g h resonant reflectivity a n d l o w non-resonant reflectivity. T h e s e t r a i t s m a k e it well-suited for use as a n o p t i c a l filter, a n d i n fact s i m i l a r gratings of other materials are c u r r e n t l y b e i n g used as p o l a r i z a t i o n insensitive n o t c h  filters  at n o r m a l incidence. [23] These devices s t r o n g l y reflect a specific frequency range, w h i l e t r a n s m i t t i n g the r e m a i n d e r of the i n c i d e n t l i g h t . T h e y take advantage of the fact t h a t there is n o d i s t i n c t i o n between s- a n d p - p o l a r i z a t i o n at n o r m a l incidence for a s y m m e t r i c t w o - d i m e n s i o n a l l a t t i c e . F i g u r e 5.17 shows a s i m u l a t i o n of the reflectivity s p e c t r a for t h i s p o l y m e r waveguide at n o r m a l incidence. N o t e the h i g h , n a r r o w b a n d r e f l e c t i v i t y a n d the low, non-resonant reflectivity, w h i c h w o u l d m a k e this waveguide a n excellent p o l a r i z a t i o n insensitive n o t c h filter. However, F i g u r e 5.18 shows s p e c t r a for the same waveguide at a  CHAPTER  5. Results and Discussion  84  1° angle of i n c i d e n c e . A change of j u s t one degree causes considerable d i s p e r s i o n , m a k i n g a l l eight modes c l e a r l y d i s t i n c t , a n d r e n d e r i n g it ineffective as a p o l a r i z a t i o n i n s e n s i t i v e n o t c h filter. A l t h o u g h progress has been m a d e i n decreasing the angle s e n s i t i v i t y o f t h i s t y p e of filter [23], these designs are c o n s t r a i n e d to f u n c t i o n at n e a r - n o r m a l i n c i d e n c e due to the inherent d i s p e r s i o n characteristics of l o w i n d e x - c o n t r a s t waveguides w i t h t w o dimensional texturing. 1.0  0.8 - \  , °- 1 6  0.4 H  0.2 H  :  9.4  9.5  1  1  1  9.6  9.7  9.8  i  9.9  1 — f = i  10.0  10.2xl0  3  Energy (cm ')  F i g u r e 5.17: S i m u l a t e d specular reflectivity of the t e x t u r e d p o l y m e r waveguide at n o r m a l incidence.  T h e p o l y m e r g r a t i n g presented here has p o l a r i z a t i o n insensitive r e f l e c t i v i t y at angles  CHAPTER 5. Results and Discussion 1.0  85  1  0.8  0.6  oi  -  0.4  J  0.0 9.4  9.5  9.6  {  •  — 1/ "  9.7  9.8  l 9.9  10.0  10.2xl0  3  Energy (cm"') F i g u r e 5.18: S i m u l a t e d specular reflectivity of the t e x t u r e d p o l y m e r waveguide at 1° i n the X d i r e c t i o n . S o l i d lines represent s-polarization; dashed lines represent p - p o l a r i z a t i o n .  far from n o r m a l incidence due t o the significant overlap of the s- a n d p - p o l a r i z e d modes. F i g u r e 5.19 shows an enlargement of the s p e c t r a from this sample at a 30° angle of incidence, i n c o m p a r i s o n w i t h the s i m u l a t e d results at the same angle. N o t e a g a i n t h a t the m o d e l shows the modes t o be d i s t i n c t a n d separate, w h i c h w o u l d i n d i c a t e t h a t t h i s device w o u l d not be p a r t i c u l a r l y useful.  However, i n the a c t u a l d a t a the modes are  c l e a r l y o v e r l a p p i n g . T h i s o v e r l a p p i n g of s- a n d p - p o l a r i z e d modes, i n c o n j u n c t i o n w i t h the s t r o n g resonant reflection a n d low non-resonant reflection, m a k e the p D R I M p o l y m e r  CHAPTER 5. Results and Discussion  86  t e x t u r e d p l a n a r waveguide p a r t i c u l a r l y well suited as a p o l a r i z a t i o n insensitive device w h i c h filters (i.e. reflects) a specific frequency range of incident light. I n a d d i t i o n , the selected frequencies c a n be t u n e d b y c h a n g i n g the angle of incidence, w h i l e m a i n t a i n i n g p o l a r i z a t i o n insensitivity.  7\p  -  I  J ,  0  1  7800  8000  1  1  8200 Energy (cm"')  i  8400  8600  F i g u r e 5.19: S p e c u l a r reflectivity of the p o l y m e r waveguide at a 30° angle of incidence i n the M d i r e c t i o n . T h e top s p e c t r a are d a t a , a n d the b o t t o m are s i m u l a t e d . N o t e t h a t the scale is specifically for the s i m u l a t i o n . T h e s-polarized s p e c t r a are represented b y s o l i d lines; p- p o l a r i z e d s p e c t r a are represented by dashed lines. T h e peak reflectivities i n the s i m u l a t i o n s go to unity.  A d d i t i o n a l l y , d a t a was collected for the two o r t h o g o n a l X s y m m e t r y d i r e c t i o n s . T h e  CHAPTER 5. Results and Discussion  87  s-polarized specular reflectivity d a t a for a 30° angle of incidence is s h o w n i n F i g u r e s 5.20 (a) a n d (b).  I n each s p e c t r u m the three m o d e groups are evident.  H o w e v e r , there is  a significant difference i n o s c i l l a t o r strength between the t w o o r t h o g o n a l d i r e c t i o n s . I n F i g u r e (a) the c e n t r a l modes are m u c h stronger t h a n the side modes, w h i l e i n F i g u r e (b) the s i t u a t i o n is reversed. N o t e t h a t the energies are the same for b o t h o r i e n t a t i o n s . B a s e d on the k i n e m a t i c s discussion i n C h a p t e r 2, the h i g h a n d l o w energy modes w h i c h disperse q u i c k l y u p a n d d o w n i n energy are p r i m a r i l y due t o c o u p l i n g w i t h the F o u r i e r c o m p o n e n t s p a r a l l e l t o the d i r e c t i o n of i n c i d e n t r a d i a t i o n , w h i l e the modes w h i c h disperse s l o w l y u p i n energy are p r i m a r i l y due to c o u p l i n g w i t h the F o u r i e r c o m p o n e n t s p e r p e n d i c u l a r to the d i r e c t i o n of i n c i d e n t r a d i a t i o n . B y r o t a t i n g the waveguide 9 0 ° , the axes are s w i t c h e d . T h i s is a clear sign t h a t there is a difference i n the s t r e n g t h of the B r a g g s c a t t e r i n g i n the two o r t h o g o n a l X directions. T h i s is c o r r o b o r a t e d b y A F M m i c r o g r a p h s of this s a m p l e w h i c h show t h a t i n one d i r e c t i o n the t e x t u r i n g p e r i o d i c i t y is v e r y regular, whereas i n the other d i r e c t i o n it is less regular. F u r t h e r , p D R I M appears t o change over t i m e . F i g u r e 5.21 shows t w o A F M m i c r o graphs of the same p D R I M g r a t i n g t a k e n before a n d after a three m o n t h i n t e r v a l . A c o m p a r i s o n of the two images shows a change i n the g r a t i n g m o r p h o l o g y .  In F i g u r e  5.21(a), the a m p l i t u d e of m o d u l a t i o n is r e l a t i v e l y u n i f o r m i n b o t h d i r e c t i o n s , w h i l e i n F i g u r e 5.21(b), the a m p l i t u d e is m u c h larger i n one d i r e c t i o n t h a n it is i n the other. T h e cause for this change is not k n o w n , b u t c o u l d be due to p r o l o n g e d exposure to a m b i e n t  CHAPTER 5. Results and Discussion  0.20  88  0.20  0.00 8.0  9.0  10.0 3  11.0  8.0  9.0  Energy (10 cm ) (a)  10.0 3  -1  11.0 -1  Energy (10 cm ) (b)  F i g u r e 5.20: S - p o l a r i z e d specular reflectivity d a t a for the p o l y m e r waveguide at a 10° angle of incidence a l o n g the X s y m m e t r y d i r e c t i o n .  T h e s p e c t r u m i n F i g u r e (a) was  t a k e n o r t h o g o n a l l y (i.e. a z m u t h a l angle of incidence was r o t a t e d 9 0 ° ) to the s p e c t r u m i n F i g u r e (b).  light. C u r r e n t l y , the m e c h a n i s m b y w h i c h this p o l y m e r moves is not c o m p l e t e l y unders t o o d . U n t i l further research on p D R I M determines the effects of l o n g - t e r m exposure t o a m b i e n t l i g h t , it m a y not be possible to c o m p l e t e l y prevent s t r u c t u r e s m a d e w i t h t h i s m a t e r i a l from d e g r a d i n g , w h i c h m a y serve to l i m i t its usefulness i n a p p l i c a t i o n s . O v e r a l l , the p D R I M p o l y m e r is a n i n t r i g u i n g new m a t e r i a l w h i c h holds p r o m i s e for  CHAPTER  5. Results and Discussion  (a) Fresh polymer grating  89  (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 provides 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 for  engineering  90  a fiat b a n d (or bands) t h a t e x t e n d across the first B r i l l o u i n zone. It has  been s h o w n t h a t a b a n d e x h i b i t i n g l o w group v e l o c i t y  (v = 5u>/5k) g  offers a n increase  i n the efficiency of c e r t a i n non-linear processes inside b u l k p h o t o n i c c r y s t a l s . [28-30] Therefore a waveguide was designed w i t h a square l a t t i c e i n such a w a y as to o p t i m i z e the flatness of one of the bands.  T h i s exercise served to test the a b i l i t y to design a n d  realize a t e x t u r e d p l a n a r waveguide w i t h a specific, i n t e n d e d b a n d s t r u c t u r e . In d e s i g n i n g this s t r u c t u r e , a 150 n m t h i c k waveguide core was selected, w h i c h is t h i c k e r . t h a n t h a t used i n the p r e v i o u s l y described G a A s samples, because a t h i c k e r waveguide provides a richer b a n d s t r u c t u r e t h a n a t h i n n e r one, i n c r e a s i n g the p o t e n t i a l for a c h i e v i n g flat bands. T h i s richer b a n d s t r u c t u r e occurs because i n a t h i c k e r waveguide the T E a n d T M slab modes are closer together i n energy. W h e n a t h i c k e r waveguide is t e x t u r e d w i t h a h i g h index-contrast g r a t i n g , the modes from the T E - l i k e gap a n d the T M - l i k e gap are closer together, p r o v i d i n g anti-crossings closer to zone center.  With  more bands anti-crossing, there is more i n t e r a c t i o n between the bands a n d thus m o r e p o t e n t i a l for flat bands t o occur. M o d e l i n g this s t r u c t u r e r e q u i r e d the use of the m o d i f i c a t i o n t o the code w h i c h a l l o w s for the m o d e l i n g of t h i c k gratings.  W i t h this m o d i f i c a t i o n , the 150 n m t h i c k core was  m o d e l e d as t w o layers. T h e specific values used for the m o d e l i n g are given i n T a b l e 5.4. T h e c o m p u t e r - m o d e l e d b a n d s t r u c t u r e of the waveguide is s h o w n i n F i g u r e 5.22. I n this t h e o r e t i c a l d i s p e r s i o n d i a g r a m , the lowest four bands near zone center c a n be loosely associated w i t h the T E slab m o d e gap. A s expected for a square l a t t i c e , there are  CHAPTER  5. Results and Discussion  91  T a b l e 5.4: K e y m o d e l i n g parameters for "flat b a n d " s t r u c t u r e  t  Parameter  Value  Pitch  390 n m  H o l e radius  90 n m  T h i c k n e s s of t e x t u r e d layer  2x70 n m  T h i c k n e s s of oxide layer  1000 n m  _--  ^  »-• » - «, „ * *  a o  <-> 10-  S=! W . 9  #  *  •  •  •  •  •  M  — p polarization - - s polarization  *,  r  0.2  •  0.1  0.1  K„/K  0.2  0.3 x -  g  F i g u r e 5.22: T h e o r e t i c a l d i s p e r s i o n d i a g r a m for the 150 n m t h i c k t e x t u r e d p l a n a r waveguide w i t h the square l a t t i c e discussed i n the text. N o t e the p - p o l a r i z e d , l o w - d i s p e r s i o n b a n d s near 10,000 c m - . [17] 1  four b a n d s i n the X d i r e c t i o n : two dispersive s-polarized bands, as w e l l as one s- a n d one p - p o l a r i z e d b a n d t h a t are less dispersive. T h e next higher set of b a n d s c a n be loosely associated w i t h the T M slab m o d e gap. A t the T M - l i k e gap the t w o dispersive m o d e s  CHAPTER 5. Results and Discussion  92  are e x p e c t e d to be p - p o l a r i z e d , a n d the two less dispersive modes to be one s- a n d one p p o l a r i z e d . [20] O n l y the three lower-energy bands are p l o t t e d because the highest energy p - p o l a r i z e d b a n d occurs above the G a A s a b s o r p t i o n edge (11,400 c m  - 1  ) a n d thus does  not show u p i n the s i m u l a t i o n . N o t e the e x t r e m e l y flat "band," t h a t a c t u a l l y consists of two p - p o l a r i z e d bands, w h i c h occurs i n the X d i r e c t i o n a n d originates at the t o p 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 b a n d , t h e n curves d o w n w a r d i n energy. B e y o n d the anti-crossing the second b a n d becomes n e a r l y d i s p e r s i o n free at a s l i g h t l y higher energy (10,445 c m  - 1  ) t h a n the  first b a n d . T h i s b a n d continues across the r e m a i n d e r of the B r i l l o u i n zone, where at zone edge it has a n energy of 10,452 c m  - 1  . T h u s the overall d i s p e r s i o n is o n l y 62 c m  - 1  across  the entire B r i l l o u i n zone. T h e flatness of this b a n d is a consequence of the p - p o l a r i z e d b a n d from the T E - l i k e gap anti-crossing w i t h the d o w n w a r d d i s p e r s i n g p - p o l a r i z e d b a n d from the T M - l i k e gap. T h e s i m u l a t e d specular reflectivity s p e c t r a for this design is s h o w n i n F i g u r e 5.23. A t e x t u r e d p l a n a r waveguide for this flat b a n d was fabricated u s i n g A S U 5 0 6 , w h i c h consists of 150 n m of G a A s o n top of 1.0 pm of Alo.9sGao.02As o n a G a A s substrate. A square l a t t i c e of c i r c u l a r holes 390 n m apart w i t h a radius of 90 n m was etched c o m p l e t e l y t h r o u g h the waveguide core. E v e r y fifth hole was o m i t t e d , c r e a t i n g a defect s u p e r l a t t i c e . T h e Alo.98Gao.02As layer was fully o x i d i z e d . T h e specular reflectivity d a t a at various angles of incidence a l o n g the T - X d i r e c t i o n  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 i g u r e 5.23:  S i m u l a t i o n of specular reflectivity for the 150 n m t h i c k t e x t u r e d  waveguide w i t h the square l a t t i c e discussed i n the text.  planar  T h e spectra correspond  angles of incidence of 10°, 2 0 ° , 3 0 ° , 4 0 ° a n d 50° from the b o t t o m up.  to  T h e s o l i d lines  represent s-polarized s i m u l a t i o n s . T h e dashed lines represent p - p o l a r i z e d s i m u l a t i o n s .  are s h o w n i n F i g u r e 5.24. A l l the modes from the T E - l i k e gap a n d the lowest p - p o l a r i z e d m o d e from the T M gap are evident. T h e r e is a n s-polarized m o d e t h a t disperses d o w n i n energy from ~ 9 , 5 0 0 c m near ~ 1 0 , 0 0 0 c m . - 1  _ 1  to ~ 8 , 3 0 0 c m . T h e r e is a r e l a t i v e l y flat s - p o l a r i z e d m o d e - 1  T h e r e is also a faint s-polarized m o d e b e g i n n i n g at  ~10,600cm  a n d d i s p e r s i n g u p i n energy, w h i c h is lost i n the G a A s a b s o r p t i o n above ~ l l , 4 0 0 c m T h e r e is one p - p o l a r i z e d m o d e s t a r t i n g at ~ l l , 1 0 0 c m  - 1  _ 1  - 1  .  and dispersing down, w h i c h anti-  crosses w i t h the p - p o l a r i z e d m o d e at ~ 1 0 , 4 0 0 c m , as p r e d i c t e d i n the s i m u l a t i o n . D u e - 1  CHAPTER  94  5. Results and Discussion  to the b r o a d n a t u r e of the Fano-resonances associated w i t h the fiat b a n d , i t is difficult t o fit these s p e c u l a r d a t a to the a c c u r a c y needed i n order t o d e t e r m i n e the precise d i s p e r s i o n of this b a n d .  Therefore a defect s u p e r l a t t i c e was i n c l u d e d i n the design so t h a t m o r e  precise measurements c o u l d be t a k e n w i t h the new diffraction measurement  technique  (as w i l l be discussed i n S e c t i o n 5.3.1). Q u a l i t a t i v e l y , the d i s p e r s i o n of the b a n d s f r o m this t e x t u r e d p l a n a r waveguide w i t h a defect s u p e r l a t t i c e agrees well w i t h the t h e o r e t i c a l p r e d i c t i o n s for the d i s p e r s i o n of a waveguide w i t h a s i m p l e l a t t i c e . T o s u m m a r i z e , the waveguide discussed herein e x h i b i t s a n e a r l y d i s p e r s i o n free (<1%) m o d e across the entire B r i l l o u i n zone i n the X d i r e c t i o n . T h i s flat b a n d was observed e x p e r i m e n t a l l y u p t o a 50° angle of incidence, w h i c h is more t h a n 3 0 % of the w a y across the B r i l l o u i n zone.  However, due to the close p r o x i m i t y a n d b r e a d t h of the modes,  the exact m o d e p o s i t i o n s a n d lifetimes c a n not be r e l i a b l y e x t r a c t e d u s i n g the n u m e r i c a l f i t t i n g t e c h n i q u e m e n t i o n e d i n S e c t i o n 5.1.1. T o measure the precise p o s i t i o n a n d lifetimes of these t y p e s of modes, the background-free measurement technique d e s c r i b e d i n C h a p t e r 2 was used. T h e s e results are discussed i n the following section.  5.3  Waveguides with Defect Superlattices  T h e 2 D t e x t u r e d p l a n a r waveguides i n the previous sections were c h a r a c t e r i z e d u s i n g s p e c u l a r reflectivity to p r o b e the l o w - l y i n g resonant bands.  W h i l e this  measurement  technique has p r o v e n useful i n terms of q u a n t i t a t i v e c o m p a r i s o n s between m o d e l a n d e x p e r i m e n t , i t has some l i m i t a t i o n s . T h e F a n o - l i k e features i n the reflectivity s p e c t r a  CHAPTER 5. Results and Discussion  95  < o o cm o  11.0  11.5x10  Energy (cm ) F i g u r e 5.24: S - p o l a r i z e d a n d p - p o l a r i z e d specular reflectivity d a t a , represented b y s o l i d a n d dashed lines respectively, for the 150 n m t h i c k t e x t u r e d p l a n a r waveguide w i t h a square l a t t i c e referred to i n the t e x t . T h e spectra c o r r e s p o n d t o angles of incidence of 10°, 2 0 ° , 3 0 ° , 40° a n d 50° from the b o t t o m up. N e a r 10,400 c m  - 1  these s p e c t r a e x h i b i t  flat p - p o l a r i z e d bands, as i n d i c a t e d b y the dot-dashed lines.  used to e x t r a c t the m o d e profiles are sometimes difficult t o d i s t i n g u i s h from the n o n resonant b a c k g r o u n d reflectivity.  I n a d d i t i o n , this technique probes modes above  the  light line only, a n d does not offer access t o the b a n d s t r u c t u r e b e l o w the light line. B o t h of these l i m i t a t i o n s c a n p o s s i b l y be addressed b y i n c o r p o r a t i n g i n the t e x t u r e p a t t e r n a "defect s u p e r l a t t i c e " lattice."  t h a t w e a k l y p e r t u r b s the b a n d s t r u c t u r e of the u n d e r l y i n g "base  CHAPTER  96  5. Results and Discussion  A defect s u p e r l a t t i c e couples a s m a l l a m o u n t of light out of the waveguide i n a n o n specular d i r e c t i o n v i a the F o u r i e r coefficients of the defect s u p e r l a t t i c e , w h i c h i n t h e o r y allows m a p p i n g of the b a n d structure of base l a t t i c e modes below the light line, w h i l e at the same t i m e e n a b l i n g a background-free m e t h o d for v i e w i n g the modes above a n d below the light line.  It is desirable to characterize modes below the light line of the  base l a t t i c e because these modes d o not' have r a d i a t i v e c o m p o n e n t s , a n d thus have the p o t e n t i a l t o efficiently t r a n s m i t i n f o r m a t i o n in-plane for device a p p l i c a t i o n s . In some cases, a defect s u p e r l a t t i c e introduces a "defect b a n d " t h a t appears i n s i d e the pseudo-gap c h a r a c t e r i s t i c of the base l a t t i c e (see S e c t i o n 2.1.3). T h i s b a n d c a n o c c u r w h e t h e r or not the defect s u p e r l a t t i c e has a significant affect o n the u n d e r l y i n g base lattice structure. T h e a u t h o r is unaware of any other p u b l i s h e d w o r k t h a t considers this a p p l i c a t i o n of defect superlattices as a means of e x t e n d i n g the u t i l i t y of w h i t e light s c a t t e r i n g as a powerful p r o b e of the  entire b a n d s t r u c t u r e of p l a i n p h o t o n i c c r y s t a l s . T h i s section  describes w o r k a i m e d at e v a l u a t i n g the p r a c t i c a l u t i l i t y of this novel concept.  This  m e t h o d of p r o b i n g the b a n d s t r u c t u r e of t e x t u r e d p l a n a r waveguides is e x p l o r e d b o t h e x p e r i m e n t a l l y a n d t h e o r e t i c a l l y v i a c h a r a c t e r i z a t i o n of three waveguides: the p r e v i o u s l y discussed "flat b a n d " square l a t t i c e as well as the p r e v i o u s l y discussed t r i a n g u l a r l a t t i c e , a n d a n a d d i t i o n a l waveguide w i t h a t r i a n g u l a r l a t t i c e .  T h e m e t h o d proves e s p e c i a l l y  useful i n c h a r a c t e r i z i n g l o w - d i s p e r s i o n bands above the light line. T h e usefulness of the technique i n p r o b i n g other bands, such as i n the t r i a n g u l a r l a t t i c e structures, is d e p e n d a n t  CHAPTER 5. Results and Discussion  97  u p o n the degree to w h i c h the defect superlattice p e r t u r b s the modes of the u n d e r l y i n g base l a t t i c e . F i n a l l y , t h i s section describes a t e x t u r e d p l a n a r waveguide design i n w h i c h a defect b a n d exists w i t h i n a c o m p l e t e pseudo-gap. T h e p o t e n t i a l use of such a s t r u c t u r e as the basis for a b r o a d b a n d angle a n d p o l a r i z a t i o n insensitive n o t c h 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 a n a l y z e the G a A s "flat b a n d " t e x t u r e d p l a n a r waveguide s a m p l e d e s c r i b e d i n S e c t i o n 5.2.2. R e c a l l t h a t this waveguide has a square base lattice of holes w i t h a defect s u p e r l a t t i c e t h a t o m i t s every fifth hole of the base lattice i n the x a n d y d i r e c t i o n . T h i s g r a t i n g p a t t e r n is c l e a r l y evident i n t h e s c a n n i n g electron m i c r o g r a p h of this s a m p l e s h o w n i n F i g u r e 5.25. N o t e t h a t the o r i e n t a t i o n of the lattice i n this p i c t u r e is i n d i c a t i v e of the p a t t e r n o n the a c t u a l sample: the l a t t i c e is on a 90 / m i x 90 / m i p o r t i o n of the waveguide, a n d the square p a t t e r n is r o t a t e d 2 2 . 5 ° w i t h i n the 9 0 / / m x 9 0 / / m square.  T h i s was done t o prevent t h e s q u a r e  a p e r t u r e diffraction p a t t e r n from interfering w i t h the c o l l e c t i o n optics d u r i n g d i f f r a c t i o n data collection. T o o b t a i n the diffraction d a t a for the region o f the b a n d s t r u c t u r e c o n t a i n i n g t h e flat bands, the e x p e r i m e n t a l a p p a r a t u s was oriented t o collect the diffraction s p e c t r a f r o m the -K  D  diffracted order at a n energy of 10,000 c m  flat b a n d s ) . T h i s required,adjustment  - 1  (the a p p r o x i m a t e l o c a t i o n of the  to the s a m p l e m o u n t , as well as the i n c i d e n t l i g h t ,  for each angle of incidence. T o calculate the angles for the o r i e n t a t i o n of the a p p a r a t u s  98  CHAPTER 5. Results and Discussion  •.• *  .'••*.•• .•••*«  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 G a A s core is 150 nm. the following equations are used  Ksme -K  = Ksmd  D  in  out  (5.1)  where  K  D  9  = — \D  (5.2) v  '  and  K = \ and where A  D  is the spacing between defects.  (5.3)  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 -K  D  diffracted order while Figure 5.27 shows the simulated -K  D  diffracted order. The spectra presented here are raw data, as opposed to the normalized  5  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 Specular reflectivity data is normalized by dividing the collected spectra by the spectra from bare  5  GaAs.  CHAPTER 5. Results and Discussion  9.6  9.8  10.0  10.2  100  10.4  10.6  ll.OxlO  3  Energy (cm ')  F i g u r e 5.26: D i f f r a c t i o n s p e c t r a for the -K  D  diffracted order from the defect s u p e r l a t t i c e  of the t e x t u r e d p l a n a r waveguide w i t h a square l a t t i c e for i n c i d e n t angles of 1 0 ° , 2 0 ° , 3 0 ° , 4 0 ° a n d 5 0 ° , from the b o t t o m up.  c a l c u l a t e d w i t h the m a t h e m a t i c a l fitting technique, as d e s c r i b e d i n S e c t i o n 5.1.1. T h i s m e t h o d is not w e l l - s u i t e d for c a l c u l a t i n g the l i n e w i d t h for closely spaced m o d e s , such as the ones associated w i t h the anti-crossing seen here. W h e n u s i n g the new diffraction measurement technique, the l i n e w i d t h is i m m e d i a t e l y available i n the r a w d a t a .  Thus,  there is l i t t l e u n c e r t a i n t y i n this measurement, a n d artifacts are not i n t r o d u c e d b y the fitting  p r o c e d u r e . Specifically for the flat b a n d , the s i m u l a t i o n shows the l i n e w i d t h for  CHAPTER 5. Results and Discussion  101  h -*'  i  1 i  !  /I-  u  ! r |,  -  i ;  1  9.6  UL  1  \\ 1  1  9.8  7'.  — i — — i 1— "— i — ...... 10.0 10.2 10.4 10.6  ,  i  11.0xl0  J  Energy (cm ') F i g u r e 5.27:  S i m u l a t e d p - p o l a r i z e d diffraction s p e c t r a (dashed lines) a n d s - p o l a r i z e d  diffraction s p e c t r a (solid lines) for the -K  D  diffracted order f r o m the defect  superlattice  of the "flat b a n d " sample for incident angles of 10°, 2 0 ° , 3 0 ° , 4 0 ° a n d 5 0 ° , f r o m the bottom up.  the p - p o l a r i z e d b a n d to be 9 7 c m  - 1  the p - p o l a r i z e d b a n d is 1 5 0 c m  at 10°. T h e m o d e is broader i n the d a t a t h a n i n the  - 1  at 1 0 ° . I n the e x p e r i m e n t a l d a t a , the l i n e w i d t h for  s i m u l a t i o n , w h i c h is l i k e l y due to the modes b e i n g effected b y p h y s i c a l i m p e r f e c t i o n s w h i c h are not t a k e n i n t o account b y the s i m u l a t i o n . S l i g h t irregularities i n t h e t e x t u r i n g of t h i s waveguide are suspected of c a u s i n g some degree of b r o a d e n i n g of the modes, as  CHAPTER 5. Results and Discussion  102  was suspected to be the case for the p o l y m e r waveguide discussed i n S e c t i o n 5.2.1.  The  diffraction measurement technique removes one significant source of uncertainty, a l l o w i n g a m o r e precise, quantifiable e v a l u a t i o n a n d c o m p a r i s o n of the lifetimes of these modes. The ~ 5 0 c m  differential i n the measured a n d s i m u l a t e d b a n d w i d t h s is consistent w i t h  _ 1  the l i n e w i d t h comparisons done b y V . P a c r a d o u n i on the square lattice sample of S e c t i o n 5.1.1, u s i n g o n l y specular d a t a .  11.2x10  H  10.8 i  J l  10.4-  S-H  10.0 9.69.2 0  r F i g u r e 5.28: data.  o.i  0.2  0.3  K„/K  0.4  0.5 X  D i s p e r s i o n d i a g r a m of the flat p - p o l a r i z e d b a n d s s h o w i n g s i m u l a t i o n a n d  D o t t e d line represents the s i m u l a t i o n for the base lattice.  L a r g e dots represent  c e n t r a l l o c a t i o n of the m o d e d e r i v e d e x p e r i m e n t a l l y v i a the -K® diffracted order. E r r o r bars o n d a t a p o i n t s represent the w i d t h of gaussian fits t o m o d e profile from diffraction data.  Q u a l i t a t i v e l y , the p o s i t i o n of the modes i n the s i m u l a t i o n has g o o d agreement w i t h  CHAPTER 5. Results and Discussion  103  the modes i n the d a t a . T h i s c o m p a r i s o n is m a d e more easily quantifiable b y the diffract i o n measurement technique. W i t h specular reflectivity d a t a , the m a t h e m a t i c a l  fitting  technique c a n be used to a p p r o x i m a t e the p o s i t i o n of the modes. H o w e v e r , w h e n modes are p o s i t i o n e d close together, this m e t h o d is p r o b l e m a t i c . T h e diffraction t e c h n i q u e facilitates reliable, precise p o s i t i o n e x t r a c t i o n w i t h o u t c o m p l i c a t e d m a t h e m a t i c a l  fitting,  thus a l l o w i n g easy c a l c u l a t i o n of dispersion. F o r the p - p o l a r i z e d b a n d d e p i c t e d i n F i g ure 5.26, the d a t a show a 1.0% d i s p e r s i o n over more t h a n 3 0 % of the B r i l l o u i n zone, w h i l e the s i m u l a t i o n predicts 0.6%. T h e r e is o n l y 0.4% difference between the t h e o r e t i c a l a n d m e a s u r e d d i s p e r s i o n ; the level of a c c u r a c y to w h i c h this is measured is not possible w i t h the s p e c u l a r measurement technique. F i g u r e 5.28 shows the s i m u l a t i o n of the flat p - p o l a r i z e d b a n d s a n d the m o d e l o c a t i o n s from the diffraction d a t a . T o further investigate these l o w d i s p e r s i o n bands, the c o m p u t e r code was used t o e x t r a c t the s t r e n g t h of the p - p o l a r i z e d F o u r i e r field c o m p o n e n t s for v a r i o u s values of i n p l a n e wavevector. F i g u r e 5.29 shows m o m e n t u m space d i a g r a m s of these results. F i g u r e s (a)-(c) are for the u p p e r b a n d , t h a t originates from the T M - l i k e gap a n d anticrosses, b e c o m m i n g flat, whereas F i g u r e s (d)-(f) are for the lower b a n d , t h a t originates from the T E - l i k e gap a n d t h e n disperses d o w n i n energy after the a n t i c r o s s i n g . P r i o r to the anticrossing, the u p p e r b a n d is p r i m a r i l y c o m p r i s e d of T M field c o m p o n e n t s . the a n t i c r o s s i n g it is c o m p o s e d m a i n l y of T E field c o m p o n e n t s .  After  T h i s is evident for  this p - p o l a r i z e d b a n d because the single d o m i n a n t F o u r i e r coefficient lies o n the line of s y m m e t r y before the anticrossing. A f t e r the a n t i c r o s s i n g the s t r o n g F o u r i e r coefficients  CHAPTER 5. Results and Discussion  104  o c c u r i n s y m m e t r i c pairs, w h i c h a d d a s y m m e t r i c a l l y to p r o d u c e a p - p o l a r i z e d m o d e . T h e lower b a n d starts near zone center p r i m a r i l y c o m p o s e d of T E field c o m p o n e n t s a n d t r a n s i t i o n s t o b e i n g c o m p o s e d p r i m a r i l y of a T M field c o m p o n e n t after the a n t i c r o s s i n g .  t X  (a) K = 0.15K , upper band n  g  (b) = 0.25K , upper band  (c) K = 0A0K , upper band  (e) K\\ = lower band  (f) K\\ = lower band  g  u  g  X  (d) #|| = lower band  0.l5K , g  0.25K , g  0A0K , g  F i g u r e 5.29: M o m e n t u m space d i a g r a m s of low d i s p e r s i o n bands before a n d after  the  anticrossing. T h e s t r e n g t h of each F o u r i e r c o m p o n e n t is i n d i c a t e d b y the size of the d o t . F i g u r e s (a)-(c) c o r r e s p o n d to the b a n d t h a t originates at the b o t t o m of the T M - l i k e gap. F i g u r e s ( d ) - ( f ) c o r r e s p o n d t o the b a n d t h a t originates near the t o p of the T E - l i k e gap.  W h e n c o m p a r i n g the s p e c t r a o b t a i n e d u s i n g the diffraction technique w i t h t h a t obt a i n e d w i t h the specular measurement technique, there are t w o n o t a b l e discrepancies. F i r s t , the d o w n w a r d dispersive p - p o l a r i z e d m o d e evident i n the s p e c u l a r r e f l e c t i v i t y d a t a is c o n s p i c u o u s l y m i s s i n g from the diffraction d a t a .  T h i s m o d e is also absent from the  s i m u l a t e d diffraction spectra. A l t h o u g h the b a n d itself is not v i s i b l e , at 4 0 ° there is clear  CHAPTER 5. Results and Discussion evidence of i t anti-crossing w i t h the flat b a n d .  105  Secondly, the lower energy s - p o l a r i z e d  feature t h a t appears faintly i n the specular reflectivity d a t a is clearly s h o w n i n the diffract i o n d a t a to be a d o u b l e t . S i m u l a t i o n s of this m o d e w i t h w i d e r defect s p a c i n g show t h i s d o u b l e t t o be a single mode, as i t has to be i n the absence of the s u p e r l a t t i c e . T h e diffraction measurement technique has been s h o w n here t o enable b a c k g r o u n d free p r o b i n g of these l o w d i s p e r s i o n b u l k modes.  In addition, using this technique to  further characterize flat bands has i m p r o v e d the level of quantifiable results u s i n g t h i s t y p e of w h i t e light p r o b e technique.  5.3.2  Superlattice Diffraction from Triangular Lattice Structures  F o r the p l a n a r waveguide discussed i n the p r e v i o u s section, b o t h the e x p e r i m e n t a l a n d the t h e o r e t i c a l diffraction results were c o m p l e t e l y d o m i n a t e d b y modes associated w i t h flat bands l y i n g above the air light line. T o investigate the use of the diffraction m e a s u r e m e n t technique t o p r o b e more dispersive modes, a p l a n a r waveguide was selected w h i c h is t e x t u r e d w i t h the p r e v i o u s l y d e s c r i b e d t r i a n g u l a r l a t t i c e c o n f i g u r a t i o n , k n o w n to e x h i b i t r e l a t i v e l y d i s p e r s i v e bands.  T h i s section reports e x p e r i m e n t a l a n d t h e o r e t i c a l studies  of defect diffraction from t w o samples of this waveguide w h i c h differ o n l y i n the l a t t i c e constant of the defect superlattice. F o r the technique to p r o v i d e a n effective means of p r o b i n g the u n d e r l y i n g base l a t t i c e b a n d s t r u c t u r e , the defect s u p e r l a t t i c e m u s t act o n l y as a weak p e r t u r b a t i o n .  " W e a k " here i m p l i e s t h a t the a c t u a l b a n d s t r u c t u r e rendered  CHAPTER 5. Results and Discussion  106  i n the first B r i l l o u i n zone of the s u p e r l a t t i c e c a n be i n t e r p r e t e d as t r i v i a l , k i n e m a t i c zone-folding of the b a n d s characteristic of the base l a t t i c e s t r u c t u r e (as i l l u s t r a t e d i n F i g u r e 2.7). F i g u r e 5.30 illustrates the same ideas i n a s l i g h t l y different way. H e r e "weak" sup e r l a t t i c e is d e s c r i b e d as c a u s i n g a s m a l l r e n o r m a l i z a t i o n of the (already r e n o r m a l i z e d ) b a n d s c h a r a c t e r i s t i c of the base l a t t i c e . If the gaps at the defect zone b o u n d a r i e s are s m a l l , t h e n the diffraction from the various defect l a t t i c e vectors s h o u l d follow the base l a t t i c e d i s p e r s i o n . T h e d a r k line i n F i g u r e 5.30 l a b e l l e d "0" represents the  parameter  space s t i m u l a t e d d i r e c t l y b y w h i t e light at a well-defined angle of incidence. T h i s is the p a r a m e t e r space p r o b e d b y the specular measurement technique, as d e s c r i b e d i n S e c t i o n 2.3.1.  T h e lines p a r a l l e l to the z e r o t h order represent a d d i t i o n a l p o i n t s i n p a r a m e t e r  space p r o b e d b y a d d i n g or s u b t r a c t i n g integer m u l t i p l e s of the defect l a t t i c e g r a t i n g vector. T h e s p e c t r a o b t a i n e d from a n y of these diffracted orders ( i n c l u d i n g specular) s h o u l d show signs of c o u p l i n g i n t o this m o d e v i a various m u l t i p l e s of K  D  the intersections. F o r this e x a m p l e , the +K  D  represented b y a l l of  diffracted s p e c t r a m i g h t be e x p e c t e d t o  d o m i n a n t l y show this m o d e background-free at a higher energy t h a n seen i n the s p e c u l a r reflectivity spectra; this m o d e s h o u l d c o n t i n u o u s l y disperse u p i n energy as the angle of incidence of the probe b e a m is increased. I n the -K  D  diffracted order, w h e n the p r o b e  b e a m is near n o r m a l incidence the m o d e s h o u l d appear higher i n energy t h a n i t does i n the specular; as the angle of i n c i d e n t light is increased, the m o d e s h o u l d disperse d o w n i n energy u n t i l zone center, where i t s h o u l d " t u r n a r o u n d " a n d disperse u p i n energy.  CHAPTER 5. Results and Discussion  107  +M F i g u r e 5.30: S c h e m a t i c d i a g r a m i l l u s t r a t i n g the diffraction p r o b e technique. T h e b o l d , straight l i n e at 0 represents the parameter space d i r e c t l y s t i m u l a t e d b y the i n c i d e n t l i g h t . T h e d a s h e d line represents the u p p e r most p - p o l a r i z e d b a n d , a n d the s o l i d represents the u p p e r m o s t s - p o l a r i z e d b a n d of the t r i a n g u l a r l a t t i c e s t r u c t u r e (see F i g u r e 5.6).  A t some p o i n t as the defect l a t t i c e s p a c i n g becomes c o m p a r a b l e t o the base l a t t i c e spacing, this s i m p l e i n t e r p r e t a t i o n of r e n o r m a l i z i n g the a l r e a d y r e n o r m a l i z e d base-lattice modes c a n n o t be used effectively.  Instead, the b a n d s t r u c t u r e m u s t be i n t e r p r e t e d i n  terms of the r e n o r m a l i z a t i o n of the slab modes b y a c o m p l e x u n i t cell w h i c h i n c l u d e s the defect s u p e r l a t t i c e . F o r e x a m p l e , consider a t r i a n g u l a r l a t t i c e w i t h a defect s u p e r l a t t i c e w h i c h o m i t s every fifth hole.  T h e u n i t cell for this l a t t i c e contains nineteen sites, as  o p p o s e d t o the non-defect version of this lattice, w h i c h contains o n l y one.  So, w h e r e  p r e v i o u s l y seven F o u r i e r coefficients were sufficient to describe the b a n d s t r u c t u r e near  CHAPTER 5. Results and Discussion  108  the second order gap, now ninety-one F o u r i e r coefficients must be used to describe the r e n o r m a l i z e d b a n d s t r u c t u r e t o the same accuracy. T o evaluate the diffraction technique's efficacy i n r e v e a l i n g the b a n d s t r u c t u r e of the u n d e r l y i n g base l a t t i c e , t w o samples are investigated: the G a A s waveguide d e s c r i b e d i n S e c t i o n 5.1.2, w h i c h has a defect s u p e r l a t t i c e w i t h every seventh hole o m i t t e d , referred to as T - 7 ; a n d a s i m i l a r G a A s waveguide w i t h every fifth hole m i s s i n g , referred t o as T - 5 . A s c a n n i n g electron m i c r o g r a p h of T - 5 is s h o w n i n F i g u r e 5.31. T h e e x p e r i m e n t a l a p p a r a t u s was o r i e n t e d t o collect diffraction d a t a from ~ 1 0 , 0 0 0 c m  _ 1  because t h i s is t h e  range of energies where the second order gap from the base l a t t i c e g r a t i n g is l o c a t e d , as d e t e r m i n e d v i a specular reflectivity measurements. s a m p l e were c a l c u l a t e d u s i n g E q u a t i o n 5.1 w i t h  T h e angles r e q u i r e d t o orient the  = ^  0  .  T h e samples were o r i e n t e d  so t h a t the light was incident a l o n g the M d i r e c t i o n . T h e diffraction d a t a was c o l l e c t e d w i t h o u t a p o l a r i z e r due to the l o w s i g n a l strength. T h e -K  D  diffraction d a t a for T - 5 are s h o w n i n F i g u r e 5.32. T h e m o d e s w h i c h are  evident i n the b o t t o m four s p e c t r a ( 2 ° , 4 ° , 6° a n d 8°) are d i s p e r s i n g d o w n i n energy. N o d a t a c o u l d be c o l l e c t e d between 8° a n d 14° since the s i g n a l for these angles is p h y s i c a l l y b l o c k e d b y the a p p a r a t u s .  T h e t o p four s p e c t r a , w h i c h c o r r e s p o n d t o angles o f i n c i d e n t  light of 1 4 ° , 1 6 ° , 18° a n d 2 0 ° , show modes w h i c h are generally d i s p e r s i n g u p i n energy. F i g u r e 5.33 shows the +K  D  diffraction d a t a from sample T - 5 . T h i s d a t a was c o l l e c t e d  for angles o f i n c i d e n t light o f 2°, 4 ° , 6 ° , 8 ° , a n d 1 0 ° . T h e r e are n o well-defined modes, b u t there is a n u p w a r d t r e n d i n the d i s p e r s i o n of the weak structures i n these s p e c t r a .  The  CHAPTER 5. Results and Discussion  109  F i g u r e 5.31: S c a n n i n g electron m i c r o g r a p h of T - 5 sample: a p l a n a r waveguide  textured  w i t h a t r i a n g u l a r base l a t t i c e , w i t h a defect superlattice 5x the p e r i o d of the base l a t t i c e . T h i s waveguide was fabricated u s i n g A S U 7 2 1 , a n d the p i t c h of the l a t t i c e is 600 n m .  diffraction signal i n the +K  D  d i r e c t i o n is ~ 5 x weaker t h a n t h a t i n the -K  D  direction.  T a b l e 5.5: K e y m o d e l i n g parameters for T - 5 Parameter  Value  Pitch  560 n m  H o l e radius  70 n m  T h i c k n e s s of t e x t u r e d layer  73 n m  T h i c k n e s s of oxide layer  300 n m  S i m u l a t i o n s were performed for T - 5 using the p a r a m e t e r values i n T a b l e 5.5. s i m u l a t i o n results are s h o w n i n F i g u r e 5.34 for the -K  D  diffraction directions. F o r the -K  D  a n d F i g u r e 5.35 for the  The +K  D  diffracted order, a group of s- a n d p - p o l a r i z e d m o d e s  disperses d o w n i n energy for incident angles from 2° t o 1 2 ° , t h e n " t u r n s a r o u n d "  and  CHAPTER 5. Results and Discussion  8.8  9.0  9.2  9.4  9.6 9.8 Energy (cm )  110  10.0  10.4xl0  3  F i g u r e 5.32: U n p o l a r i z e d -K® diffraction d a t a for T - 5 . S p e c t r a from the b o t t o m u p are for i n c i d e n t angles of 2 ° , 4 ° , 6 ° , 8 ° , 14°, 16°, 18°, a n d 2 0 ° .  disperses u p i n energy t h r o u g h 2 0 ° . T h i s " t u r n a r o u n d " is l o c a t e d at the p o i n t where these b a n d s pass t h r o u g h the first defect B r i l l o u i n zone b o u n d a r y , w h i c h is consistent w i t h the d a t a i n F i g u r e 5.32. F o r the +K®  diffracted order, there are several m o d e s of  b o t h s- a n d p - p o l a r i z a t i o n w h i c h disperse up i n energy for a l l angles of i n c i d e n c e f r o m 2° to 2 0 ° . T h i s is q u a l i t a t i v e l y consistent w i t h the b e h a v i o r of the weak peaks i n F i g u r e 5.33. T h e c a l c u l a t e d i n t e n s i t y of the s i m u l a t e d modes for the +K® weaker t h a n those for the -Kg  d i r e c t i o n are ~ 5 x  simulations.  T h e s - p o l a r i z e d specular reflectivity d a t a t a k e n i n the M d i r e c t i o n for T - 5 are s h o w n  CHAPTER 5. Results and Discussion  8.8  9.0  F i g u r e 5.33: U n p o l a r i z e d  9.2  9.4  111  9.6 9.8 Energy (cm )  10.0  10.4xl0  3  diffraction d a t a for T - 5 . S p e c t r a f r o m the b o t t o m u p are  for i n c i d e n t angles of 2 ° , 4 ° , 6 ° , 8 ° , a n d 10°.  i n F i g u r e 5.36. C o r r e s p o n d i n g s i m u l a t i o n s for s- a n d p - p o l a r i z a t i o n s for various angles of i n c i d e n c e are s h o w n i n F i g u r e 5.37. A t a gross level, b o t h the d a t a a n d the s i m u l a t i o n s are v e r y s i m i l a r to those c o r r e s p o n d i n g to a c o m p a r a b l e structure w i t h no defect s u p e r l a t t i c e . However, evidence of the p e r t u r b a t i o n is e x h i b i t e d c l e a r l y i n the v i c i n i t y of the b r o a d , h i g h energy modes at 2.5° a n d 5° angles of incidence.  These additional undulations  are m o r e p r o n o u n c e d i n the s i m u l a t i o n s t h a n i n the d a t a , especially at large angles of incidence. I n s u m m a r y , the T - 5 diffraction d a t a i n the  +K a n d -K® d i r e c t i o n s e x h i b i t s m a n y of D  g  CHAPTER 5. Results and Discussion  5x10  112  H  10.8x10 Energy (cm ) F i g u r e 5.34:  S i m u l a t i o n for -K  D  diffracted order from the defect s u p e r l a t t i c e of T - 5 .  P - p o l a r i z e d is represented b y the dashed lines, a n d s-polarized b y the s o l i d .  CHAPTER 5. Results and Discussion  113  rR7L560r70t20~  J  'R7L560r70tl8  "R7L560r70tl6~  .Li  "R7L560r70tl4  _  <L>  i  "R7L560r70tl2  -  'R7L560r70tlO"  "R7L560r70t8"  "R7L560r70t6"  "R7L560r70t4~  1x10  L  -4 "R7L560r70t2"  Li  0 8.8  9.2  9.6  10.0  10.8x10  Energy (cm *) F i g u r e 5.35: S i m u l a t i o n for +K  D  diffracted order from the defect s u p e r l a t t i c e of T - 5 .  P - p o l a r i z e d is represented b y the dashed lines, a n d s-polarized b y the s o l i d .  CHAPTER 5. Results and Discussion  F i g u r e 5.36:  114  S - p o l a r i z e d specular reflectivity d a t a for T - 5 , t a k e n at various angles of  incidence i n the M d i r e c t i o n .  F i g u r e 5.37: S - p o l a r i z e d (solid lines) a n d p - p o l a r i z e d (dashed lines) specular r e f l e c t i v i t y s i m u l a t i o n s for T - 5 , at various angles of incidence i n the M d i r e c t i o n .  CHAPTER 5. Results and Discussion  116  the q u a l i t a t i v e features p r e d i c t e d by the s i m u l a t i o n s . G o i n g away from n o r m a l i n c i d e n c e i n the M d i r e c t i o n , the -K® d a t a contains several modes t h a t disperse d o w n i n energy out to ~ 1 0 ° angle of incidence, b e y o n d w h i c h the g r o u p starts t o disperse b a c k u p i n energy. I n contrast, the modes i n the +K  D  d i r e c t i o n m o n o t o n i c a l l y increase i n energy  away from n o r m a l incidence out to at least ~ 2 0 ° . T h e l i n e w i d t h s of the modes i n the -K® d i r e c t i o n are w i t h i n a factor of t w o of the s i m u l a t e d m o d e s ' l i n e w i d t h s , a n d the s c a t t e r i n g s t r e n g t h i n the -Kg  d i r e c t i o n is s u b s t a n t i a l l y stronger t h a n i n the +K ° g  direction. T h e  p r i n c i p a l difference between the e x p e r i m e n t a n d s i m u l a t i o n s lies i n the r e l a t i v e s t r e n g t h s of the various modes t h a t appear i n a n y given order of diffraction.  T h i s can largely  be a t t r i b u t e d t o the fact, as borne out b y s i m u l a t i o n s , t h a t the d i s p e r s i o n efficiency i n a n y g i v e n o r d e r seems t o be e x c e e d i n g l y sensitive t o v a r i a t i o n s i n i n c i d e n t angle a n d the s t r u c t u r e ' s precise p h y s i c a l characteristics. M o r e discussion r e g a r d i n g the n a t u r e of the modes observed i n these s p e c t r a follows the presentation of a s i m i l a r s t u d y of s a m p l e T-7. U s i n g the diffraction measurement technique to p r o b e s a m p l e T - 7 p r o d u c e s the -Kg d a t a s h o w n i n F i g u r e 5.38. T h e s e s p e c t r a were o b t a i n e d at angles of i n c i d e n t light of 2 ° , 4 ° , 6° a n d 1 5 ° . N o d a t a c o u l d be collected from 8° to 14° because the s i g n a l for these angles is p h y s i c a l l y b l o c k e d b y the a p p a r a t u s . T h e b o t t o m three s p e c t r a show a single, well-defined m o d e d i s p e r s i n g d o w n i n energy. A t 15° there is a single m o d e l o c a t e d at a p p r o x i m a t e l y the same energy as the m o d e at 2 ° . F i g u r e 5.39 shows the +K®  diffraction  d a t a for t h i s s a m p l e at 2° a n d 4 ° . O n e m o d e is apparent; i t disperses u p from 2° t o 4 ° .  CHAPTER 5. Results and Discussion A g a i n , the diffraction s i g n a l for the +K  D  9.4  9.6  9.8  117  is ~ 5 x weaker t h a n t h a t for the  10.0  10.4xl0  -K '. D  3  Energy (cm *) F i g u r e 5.38: U n p o l a r i z e d -K  D  d a t a for T - 7 , taken at 2, 4, 6, a n d 15° i n the M d i r e c t i o n .  S i m u l a t i o n s were performed for T - 7 w i t h the same values used p r e v i o u s l y to m o d e l this waveguide, as described i n S e c t i o n 5.1.2.  T h e s i m u l a t i o n s for the -K  D  diffracted  order, s h o w n i n F i g u r e 5.40, e x h i b i t one s- a n d one p - p o l a r i z e d group of modes d i s p e r s i n g d o w n i n energy from 2 ° , flattening out between 6° a n d 10°, a n d t h e n d i s p e r s i n g u p i n energy t h r o u g h 18°. T h e " t u r n a r o u n d " i n the d i r e c t i o n of d i s p e r s i o n occurs between 6° a n d 10°, w h i c h corresponds w i t h the c o n d i t i o n where the modes encounter the first B r i l l o u i n zone b o u n d a r y of the defect l a t t i c e (just as s h o w n for T - 5 ) , w h i c h occurs for this s t r u c t u r e at 7.9° for 10,000 c m  - 1  . T h e two dashed lines i n d i c a t e the l o c a t i o n of the  base l a t t i c e modes, as i n d i c a t e d b y the specular reflectivity s i m u l a t i o n s for the non-defect l a t t i c e (shown p r e v i o u s l y i n F i g u r e 5.10). T h e s i m u l a t i o n s e x h i b i t q u a l i t a t i v e agreement  CHAPTER 5. Results and Discussion  118  <  9.4  9.6  9.8  10.0  10.4xl0  3  Energy (cm" ) 1  F i g u r e 5.39: U n p o l a r i z e d  +K  D g  d a t a for T - 7 , for 2 a n d 4° i n the M d i r e c t i o n .  w i t h the d a t a w i t h regard t o the intensity of the modes, a n d t o the general d i s p e r s i o n ; however there is o n l y one m o d e apparent i n the d a t a .  T h e s i m i l a r i t y between the 2°  a n d 15° s p e c t r a observed i n the d a t a is also s h o w n i n the s i m u l a t i o n s .  F o r the  +K  D g  s i m u l a t i o n s , s h o w n i n F i g u r e 5.41, one s- a n d one p - p o l a r i z e d m o d e are each d i s p e r s i n g c o n t i n u o u s l y u p i n energy for this range. T h e dashed line indicates the l o c a t i o n of the u p w a r d d i s p e r s i n g s-polarized m o d e , as i n d i c a t e d by specular r e f l e c t i v i t y s i m u l a t i o n s . T h e continuous u p w a r d dispersion of the two modes i n the  +K  D g  s i m u l a t i o n s corresponds  to the u p w a r d dispersion of the single m o d e shown i n the d a t a .  T h e w i d t h s of the  modes also c o m p a r e well. O v e r a l l , the diffraction d a t a a n d s i m u l a t i o n s for s a m p l e T - 7 e x h i b i t m a n y of the same features seen i n the T - 5 sample, b u t the n u m b e r of m o d e s 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  10.4xl0  3  Energy (cm ) l  F i g u r e 5.40:  S i m u l a t i o n of -K  D  s p e c t r a for T - 7 .  S o l i d lines represent  s-polarization;  d a s h e d lines represent p - p o l a r i z a t i o n . S i m u l a t i o n is for the M d i r e c t i o n a n d for angles of 2° to 18° from the b o t t o m u p .  T h e i n t e r p r e t a t i o n t h a t follows makes reference to the s i m u l a t i o n s s h o w n i n F i g u r e s 5.40 a n d 5.41, a n d the schematic d i a g r a m of the defect-zone-folded  band structure i n  F i g u r e 5.42. T h e zone-folding s h o w n i n F i g u r e 5.42 involves o n l y the base l a t t i c e s- a n d p - p o l a r i z e d b a n d s (represented b y solid lines i n the figure) t h a t disperse u p w a r d a c o m m o n o r i g i n at 9,750 c m  - 1  , as i n F i g u r e 5.6.  Furthermore,  from  it o n l y includes those  parts of these b a n d s t h a t are folded b y r e c i p r o c a l lattice vectors oriented a l o n g the M direction.  T h a t is, it is effectively a I D r e d u c t i o n t h a t ignores b a n d s folded b y defect  l a t t i c e vectors t h a t do not lie o n 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 light b e a m c o r r e s p o n d i n g to the t h i c k gray l i n e i n  CHAPTER 5. Results and Discussion  9.6  9.8  120  10.0  10.2  10.4x10  3  E n e r g y ( c m *) F i g u r e 5.41: S i m u l a t i o n of +K®  s p e c t r a for T - 7 .  S o l i d lines represent s - p o l a r i z a t i o n ;  dashed lines represent p - p o l a r i z a t i o n . S i m u l a t i o n is for the M d i r e c t i o n a n d for angles of 2° t o 18° f r o m the b o t t o m u p .  F i g u r e 5.42, m a n y b a n d s are e x c i t e d at l o c a t i o n s i n d i c a t e d b y the o p e n circles. A l l of the zone-folded b a n d s i n this figure c a n be e x c i t e d w i t h a b e a m i n c i d e n t at angles between 0 a n d ~ 7 . 9 ° . T h e modes t h a t show u p c l e a r l y i n the -Kg  s i m u l a t i o n s for s a m p l e T - 7 over  this a n g u l a r range c o r r e s p o n d t o the two d o w n w a r d dispersive b a n d s s h o w n as d a s h e d lines i n the  figure.  T h e modes t h a t show up c l e a r l y i n the +K®  s i m u l a t i o n s over t h i s  a n g u l a r range c o r r e s p o n d to the t w o of the u p w a r d dispersive bands.  Specifically, the  lower energy m o d e corresponds t o the u p w a r d d i s p e r s i n g s-polarized b a n d f r o m the base l a t t i c e (solid line i n F i g u r e 5.42 l a b e l e d ' s ' ) w h i l e the higher energy m o d e  corresponds  to the dash-dot line w h i c h is the second zone-fold of the u p p e r d i s p e r s i n g p - p o l a r i z e d  121  CHAPTER 5. Results and Discussion co (cm ) 1  T  K /2  K  D  g  D g  F i g u r e 5.42: S c h e m a t i c zone d i a g r a m s h o w i n g u p w a r d d i s p e r s i n g b a n d s at t o p of second order T E - l i k e gap i n the M d i r e c t i o n . T h e dashed lines represent the d o w n w a r d d i s p e r s i n g b a n d s a n d the s o l i d lines represent the u p w a r d d i s p e r s i n g bands. b a n d f r o m the base l a t t i c e .  O t h e r bands, most n o t a b l y the lowest u p w a r d d i s p e r s i n g  p - p o l a r i z e d b a n d , are a l m o s t c o m p l e t e l y absent i n the -K  D  diffraction s i g n a l . L o o k i n g  back at the T - 5 s i m u l a t i o n s , some of the peaks c o r r e s p o n d to these same bands.  Note,  however, t h a t the lowest u p w a r d d i s p e r s i n g p - p o l a r i z e d b r a n c h is v i s i b l e , as are h i g h e r l y i n g bands.  T h e p r i n c i p a l difference between T - 7 a n d T - 5 samples is t h a t there are  numerous other b a n d s i n T - 5 t h a t c o n t r i b u t e to the diffraction, a l t h o u g h t h e y are weaker t h a n those i n T - 7 . M o s t of these a d d i t i o n a l b a n d s i n T - 5 are effectively zone-folded b y r e c i p r o c a l l a t t i c e vectors oriented away from the T - X axis. T h u s it appears t h a t for d i l u t e superlattices the signals diffracted v i a different orders of the defect l a t t i c e p r o v i d e a direct, background-free probe of the I D zone-folded base  CHAPTER 5. Results and Discussion  122  b a n d s , as a n t i c i p a t e d . M o r e dense superlattices cause a more significant p e r t u r b a t i o n t o the base b a n d s t r u c t u r e , r e s u l t i n g i n more c o m p l i c a t e d spectra.  Although in principle  the d a t a c o u l d be "unfolded" i n 2 D , this w o u l d represent a significant challenge.  The  biggest difficulty appears t o be the huge v a r i a b i l i t y i n the diffraction efficiency,  both  w i t h i n a b a n d (as a f u n c t i o n of i n - p l a n e wavevector) a n d between different bands. T h e s e efficiencies are also v e r y sensitive to the g e o m e t r i c a l parameters of the structures. W i t h this u n d e r s t a n d i n g , the diffraction d a t a from T - 7 c a n also be b e t t e r i n t e r p r e t e d . F i g u r e 5.43 shows the s-polarized, a n d F i g u r e 5.44 the p - p o l a r i z e d s p e c u l a r r e f l e c t i v i t y d a t a c o l l e c t e d i n the M d i r e c t i o n for T - 7 . T h e h i g h energy m o d e i n the s - p o l a r i z e d d a t a a n d the h i g h energy m o d e i n the p - p o l a r i z e d d a t a are degenerate where t h e y intersect zone center at 9,750 c m T h e m o d e seen i n the  - 1  . T h e s e modes disperse u p i n energy a w a y f r o m zone center.  +K ° diffraction d a t a ( F i g u r e 5.39) also has a value of 9,750 c m  - 1  g  at zone center, a n d disperses up i n energy away from zone center. T h e m o d e i n the  +Kg  d a t a is a background-free representation of the s-polarized base l a t t i c e m o d e seen i n the s p e c u l a r reflectivity spectra.  R e f e r r i n g to the -K®  m o d e disperses d o w n from 9 , 9 0 0 c m  - 1  to 9 , 7 5 0 c m  diffraction d a t a ( F i g u r e 5.38), the - 1  as i t approaches 8 ° , w h i c h is the  intersection of the m o d e w i t h zone edge. T h i s m o d e i n the -K  D  d a t a is a b a c k g r o u n d -  free representation of the p - p o l a r i z e d base l a t t i c e m o d e seen i n the specular reflectivity d a t a . T h e r e l a t i v e diffraction efficiencies are different from the s i m u l a t i o n s , b u t the b a s i c i n t e r p r e t a t i o n is the same. T h u s , the s u p e r l a t t i c e diffraction measurement technique does p r o v i d e a background-free probe of the b a n d s t r u c t u r e of t e x t u r e d p l a n a r waveguides  CHAPTER 5. Results and Discussion  123  w h e n t h e defect s u p e r l a t t i c e is d i l u t e .  50 deg  < G  k30 deg  p25  deg  H20 deg*"  6  111111111111111111111111111111111111111 j 111111111111111111111111111111111111111 [ 9 10 11 12 13 14x10 7  Energy (cm ) F i g u r e 5.43: S - p o l a r i z e d specular reflectivity d a t a for T - 7 t a k e n i n t h e M d i r e c t i o n .  T h e diffraction measurement technique c a n e x t e n d t h e level of q u a n t i t a t i v e agreement between e x p e r i m e n t a l results a n d t h e o r e t i c a l p r e d i c t i o n s b y p r o v i d i n g 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  F i g u r e 5.44: P - p o l a r i z e d specular reflectivity d a t a for T - 7 t a k e n i n t h e M d i r e c t i o n .  m o d e profiles.  Specifically i n t h e case of T - 7 , this aids i n t h e d e t e r m i n a t i o n  of t h e  thickness of t h e oxide layer b e n e a t h the waveguide core. T h e +K® diffraction d a t a t a k e n w i t h light incident at 2° is shown i n the b o t t o m g r a p h of F i g u r e 5.45 w i t h s i m u l a t i o n s of  CHAPTER 5. Results and Discussion  125  this s t r u c t u r e done for various thicknesses of the oxide layer. T h e b o t t o m s i m u l a t i o n is for a f u l l y - o x i d i z e d layer, the next is for 900 n m of oxide, the t h i r d is for 300 n m , a n d the top is for 200 n m . B a s e d o n the l i n e w i d t h , i t c a n be c o n c l u d e d t h a t the oxide layer for T - 7 is a p p r o x i m a t e l y 200 n m t h i c k , w h i c h agrees w i t h the F a b r y - P e r o t s p a c i n g evident i n F i g u r e 5.5. T o c o n c l u d e this section, the diffraction measurement technique c a n p r o v i d e a background-free p r o b e of dispersive modes, as well as flat bands.  T h e scattering strength  of the flat bands tends to be stronger t h a n t h a t from dispersive bands. T h i s technique' w o r k s best for dispersive modes w h e n the defect s u p e r l a t t i c e represents a weak p e r t u r b a t i o n t o the base l a t t i c e modes.  W h e n the p e r t u r b a t i o n is s t r o n g , the t e c h n i q u e is  not as useful for p r o b i n g the base l a t t i c e modes because the defect s u p e r l a t t i c e causes a c o m p l i c a t e d r e n o r m a l i z a t i o n of the slab modes, r e n d e r i n g the d a t a difficult t o decipher. W h i l e t h i s technique does not replace the specular reflectivity technique for s t u d y i n g the d i s p e r s i o n of l e a k y p h o t o n i c eigenstates, it does further the q u a l i t a t i v e a n d q u a n t i t a t i v e u n d e r s t a n d i n g of t e x t u r e d p l a n a r waveguide b a n d s t r u c t u r e b y p r o v i d i n g background-free measurements of the m o d e s ' lineshapes, a n d hence t h e i r lifetimes.  5.3.3  True Defect Modes  T h e previous sections have discussed the i n c o r p o r a t i o n of defect s u p e r l a t t i c e s i n t o text u r e d p l a n a r waveguides as a means of revealing the l e a k y - m o d e b a n d s t r u c t u r e characteristic of the u n d e r l y i n g base l a t t i c e . T h i s section discusses the design of a t e x t u r e d  CHAPTER 5. Results and Discussion  126  F i g u r e 5.45: T r i a n g u l a r superlattice diffraction d a t a c o m p a r e d w i t h s i m u l a t i o n s for various o x i d e thicknesses.  S i m u l a t i o n s a n d d a t a are for -K®  diffraction w i t h i n c i d e n t l i g h t  at 2° i n the M d i r e c t i o n . T h e oxide thickness for the s i m u l a t i o n s are, from the b o t t o m u p , 1600 n m , 900 n m , 300 n m , a n d 200 n m .  CHAPTER 5. Results and Discussion  127  p l a n a r waveguide i n w h i c h the base l a t t i c e possesses a complete pseudo-gap. T h e defect s u p e r l a t t i c e c o n f i g u r a t i o n is s i m i l a r t o t h a t of the p r e v i o u s l y discussed T - 7 s a m p l e . T h i s s u p e r l a t t i c e gives rise t o a defect b a n d w i t h i n the c o m p l e t e pseudo-gap t h a t e x h i b i t s v i r t u a l l y no d i s p e r s i o n . A s discussed above, such a s t r u c t u r e makes possible a w h o l e range of passive as well as active o p t i c a l devices, such as c h a n n e l waveguides [12,40], n o t c h filters a n d lasers [22]. S i m u l a t i o n s of the specular reflectivity a n d s u p e r l a t t i c e d i f f r a c t i o n from this s a m p l e show clear evidence of b o t h the defect b a n d a n d other base l a t t i c e b a n d s t h a t lie b e l o w the l i g h t line. T h e s e results t h e o r e t i c a l l y v a l i d a t e the o r i g i n a l concept of u s i n g defect superlattices not o n l y as a background-free p r o b e of  leaky  localized modes  (as discussed i n d e t a i l i n Sections 5.3.1 - 5.3.2), b u t also as a means of p r o b i n g m o d e s b e l o w the light line, a n d l o c a l i z e d defect bands. In order to realize a c o m p l e t e pseudo-gap i n a p e r i o d i c a l l y t e x t u r e d p l a n a r waveguide, the l a t t i c e must possess a h i g h degree of s y m m e t r y , a n d have a n air f i l l i n g f r a c t i o n far from zero or one. T o this end, a t e x t u r e d p l a n a r waveguide was designed w i t h a t r i a n g u l a r l a t t i c e a n d a n air filling fraction of 34%. T h e other design parameters for t h i s s t r u c t u r e are d e t a i l e d i n T a b l e 5.6.  T h i s design includes a defect s u p e r l a t t i c e t h a t o m i t s every  seventh hole. T h e s i m u l a t e d d i s p e r s i o n d i a g r a m for the T E b a n d i n this s t r u c t u r e is s h o w n i n F i g u r e 5.46. N o t e t h a t this d i s p e r s i o n is for the base l a t t i c e only, i n order to c l e a r l y d e m o n s t r a t e the m a g n i t u d e of the c o m p l e t e first order T E pseudo-gap. T h i s large gap extends f r o m 7565 c m " to 9120 c m 1  - 1  , w h i c h is 18.6% of the center frequency. T h e first order T M gap  CHAPTER 5. Results and Discussion  128  T a b l e 5.6: D e s i g n parameters for t e x t u r e d p l a n a r waveguide w i t h defect m o d e Parameter  Value  Pitch  375 n m  H o l e radius  115 n m  T h i c k n e s s of core  150 n m  Core composition  GaAs  O x i d e thickness  1000 n m  Hole depth  150 n m  is m u c h s m a l l e r a n d higher i n energy, a n d does not overlap this region.  T h e i n c l u s i o n of the defect superlattice i n this waveguide creates a m o d e w i t h i n t h e large first order pseudo-gap.  S i m u l a t i o n s of the specular r e f l e c t i v i t y for this s t r u c t u r e  are s h o w n i n F i g u r e 5.47 for a range of i n c i d e n t angles a l o n g the M d i r e c t i o n .  The  CHAPTER 5. Results and Discussion p r o n o u n c e d feature at 8 , 3 0 0 c m  - 1  is evidence of the existence of this m o d e .  129  Although  this is s t r i c t l y a T E gap, the defect m o d e is evident i n b o t h the s- a n d p - p o l a r i z a t i o n s . N o t e t h a t there is v i r t u a l l y no dispersion of this defect b a n d . F i g u r e 5.48 shows the s i m u l a t e d -K  diffracted signal for this structure. O n c e a g a i n ,  D  the defect m o d e is a p p a r e n t at 8,300 c m  - 1  i n b o t h the s- a n d p - p o l a r i z a t i o n s for a l l angles  of i n c i d e n c e a l o n g the M d i r e c t i o n . These background-free s p e c t r a h i g h l i g h t several other s m a l l e r features, w h i c h were not as apparent i n the specular r e f l e c t i v i t y s i m u l a t i o n s . T h e s e features o c c u r p r i m a r i l y near the edges of the pseudo-gap, a n d are associated w i t h the b a n d edges of the base l a t t i c e w h i c h lie below the light line as discussed i n S e c t i o n 2.3.2.  T h e s e features are s i m i l a r i n n a t u r e t o the defect-zone-folded signatures of the  base l a t t i c e m o d e s discussed extensively above, b u t here the c o r r e s p o n d i n g base l a t t i c e b a n d s lie b e l o w the air light line. It is h i g h l y desirable to be able to d i r e c t l y p r o b e the d i s p e r s i o n of the t r u l y b o u n d modes t h a t lie b e n e a t h the light line, a n d these s i m u l a t i o n s d e m o n s t r a t e t h a t defect superlattices offer this ability, i n p r i n c i p l e . F i g u r e 5.49 is a m o m e n t u m space d i a g r a m i l l u s t r a t i n g the relative strengths of the F o u r i e r coefficients t h a t comprise this defect m o d e w h e n e x c i t e d at a 10° angle of i n c i dence.  T o m a k e this d i a g r a m , the F o u r i e r coefficients for each r e c i p r o c a l l a t t i c e vector  were e x t r a c t e d at the central energy of the defect m o d e for light i n c i d e n t at 10° angle of incidence a l i g n e d i n the M d i r e c t i o n . E a c h dot represents a r e c i p r o c a l l a t t i c e vector (by p o s i t i o n ) a n d the relative m a g n i t u d e (by dot diameter) of the c o r r e s p o n d i n g F o u r i e r field c o m p o n e n t . T h i s n u m b e r of r e c i p r o c a l l a t t i c e vectors was r e q u i r e d i n order to m o d e l  CHAPTER,5. Results and Discussion  6500  7000  7500  8000  130  8500  9000  9500  10000  Energy (cm ) l  F i g u r e 5.47: S i m u l a t e d specular reflectivity for a t e x t u r e d p l a n a r waveguide w i t h a t r i a n g u l a r lattice, e x h i b i t i n g a dispersion-free defect b a n d i n the first order pseudo-gap, s h o w n for angles of incidence of 10°, 2 0 ° , 3 0 ° , 40° a n d 50° a l o n g the M d i r e c t i o n . T h e solid lines represent s-polarized a n d the dashed lines represent p - p o l a r i z e d reflectivity spectra.  CHAPTER 5. Results and Discussion  6500  7000  7500  8000  131  8500  9000  9500  10000  Energy (cm ) l  F i g u r e 5.48: S i m u l a t e d -K  D  diffraction for a t e x t u r e d p l a n a r waveguide w i t h a t r i a n g u l a r  l a t t i c e , e x h i b i t i n g a dispersion-free defect m o d e i n the first order pseudo-gap, s h o w n for angles of incidence of 10°, 20,° 30,° 40° a n d 50° a l o n g the M d i r e c t i o n . T h e s o l i d lines represent s-polarized a n d the d a s h e d lines represent p - p o l a r i z e d diffraction s p e c t r a .  CHAPTER 5. Results and Discussion  132  a s t r u c t u r e w i t h a defect s u p e r l a t t i c e such as this. T h e s t r o n g F o u r i e r coefficients (i.e. the larger dots) t h a t c o n t r i b u t e to this defect m o d e o c c u r p r i m a r i l y at a r a d i u s halfway between zone center a n d the smallest  base  l a t t i c e vectors, w h i c h c o r r e s p o n d to the  seventh r i n g of r e c i p r o c a l l a t t i c e vectors i n this d i a g r a m . T h i s makes sense because the energies of the base l a t t i c e 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. R e c a l l t h a t the c o r r e s p o n d i n g d i a g r a m s for dispersive modes s h o w n i n S e c t i o n 5.1.2 were a l l d o m i n a t e d b y just one or two c o m p o n e n t s . T h e r e l a t i v e l y large n u m b e r of significant components associated w i t h the defect b a n d s also makes sense, because a n i n f i n i t e l y l o c a l i z e d state w o u l d c o r r e s p o n d to a u n i f o r m dot size i n t h i s figure.  (a) s-polarization  (b) p-polarization  F i g u r e 5.49: M o m e n t u m space d i a g r a m for the defect m o d e i n the waveguide w i t h the t r i a n g u l a r l a t t i c e discussed i n the text.  M d i r e c t i o n is u p i n the d i a g r a m . T h e defect  s u p e r l a t t i c e for this waveguide o m i t s every seventh hole. T h e size of the dots shows the relative strengths of the F o u r i e r coefficients of the fields e x c i t e d at a 10° angle of i n c i d e n c e a l o n g the M d i r e c t i o n .  CHAPTER 5. Results and Discussion  133  U s i n g these F o u r i e r coefficients a n d r e c i p r o c a l l a t t i c e vectors to p l o t the i n t e n s i t y of the s - p o l a r i z e d fields of the defect m o d e yields the real space d i a g r a m s h o w n i n F i g u r e 5.50. T h i s figure shows significant l o c a l i z a t i o n of the fields o n the l o c a t i o n s of the defect sites w h e n i l l u m i n a t e d w i t h w h i t e light at a 10° angle of i n c i d e n c e a l o n g the M direct i o n . T h i s i l l u s t r a t e s the fact t h a t light w i t h i n the pseudo-gap is not p e r m i t t e d t o t r a v e l c l a s s i c a l l y t h r o u g h the waveguide, b u t c a n t u n n e l from defect site t o defect site. It has been s h o w n t h a t w h e n these defect sites are arranged i n p a r t i c u l a r p a t t e r n s (e.g. s t r a i g h t lines, straight lines w i t h bends, etc.), light at the frequency of the defect m o d e c a n be m a d e to "follow the defects." [2]. A t e x t u r e d p l a n a r waveguide was fabricated to the specifications d e l i n e a t e d above. A s the p a r a m e t e r s for this s t r u c t u r e are s i m i l a r to the other G a A s t e x t u r e d waveguides fabricated for the research presented i n this thesis, the r e a l i z a t i o n of t h i s waveguide design s h o u l d be no less successful t h a n the other structures. H o w e v e r , c h a r a c t e r i z a t i o n of this waveguide has revealed o n l y a b r o a d , u n s t r u c t u r e d e m i s s i o n c o v e r i n g the entire first order pseudo-gap. T h e reason for this is not k n o w n . A l t h o u g h this design has not been e x p e r i m e n t a l l y realized, a p a t e n t a p p l i c a t i o n is i n the process of b e i n g filed for its use as a p o l a r i z a t i o n a n d angle insensitive n o t c h filter.  R e c a l l t h a t the n o t c h filter referred t o i n the context of the p o l y m e r waveguide  i n v o l v e d p o l a r i z a t i o n - i n s e n s i t i v e response, w h i c h is desirable; b u t the center frequency of the n o t c h v a r i e d c o n t i n u o u s l y w i t h the i n c i d e n t angle, w h i c h m a y or m a y not be desirable,  CHAPTER 5. Results and Discussion  134  F i g u r e 5.50: R e a l space p l o t of the m a g n i t u e d squared of the t o t a l v e c t o r field at the surface of the defect l a t t i c e waveguide discussed i n the t e x t , for a 10° angle of i n c i d e n c e a l o n g the M d i r e c t i o n . T h e r e is s t r o n g l o c a l i z a t i o n of the field at the defect sites.  d e p e n d i n g u p o n t h e a p p l i c a t i o n . Here, the use of a h i g h i n d e x - c o n t r a s t t e x t u r e d waveguide c a p a b l e of s u p p o r t i n g a c o m p l e t e pseudo-gap, together w i t h the defect s u p e r l a t t i c e , offers p o l a r i z a t i o n insensitive filtering at a fixed frequency, i n d e p e n d e n t of i n c i d e n t angle. T o s u m m a r i z e t h i s section, the theoretical m o d e l used i n p r e v i o u s sections t o q u a n t i t a t i v e l y e x p l a i n the s c a t t e r i n g properties of several different 2 D t e x t u r e d waveguides was used to design a different s t r u c t u r e w i t h a c o m p l e t e T E pseudo-gap.  The model  p r e d i c t s t h a t clear signatures of near-dispersionless gap modes a n d base l a t t i c e m o d e s  CHAPTER 5. Results and Discussion  135  b e l o w the light line s h o u l d be observed i n w h i t e light s c a t t e r i n g s p e c t r a w h e n a defect sup e r l a t t i c e is i n c o r p o r a t e d i n the s t r u c t u r e . A l t h o u g h the first a n d o n l y a t t e m p t t o verify these p r e d i c t i o n s was not successful, this is almost c e r t a i n l y a consequence of f a b r i c a t i o n difficulties, not a n i n d i c a t i o n t h a t the design is flawed.  Chapter 6 Conclusions and Recommendations  T h e objective of the w o r k presented i n this thesis, t o reveal a n d q u a n t i f y the b r o a d b a n d l i n e a r o p t i c a l s c a t t e r i n g properties of 2 D t e x t u r e d p l a n a r waveguide structures, has been achieved. T h e factors c o n t r i b u t i n g t o this success were a versatile, accurate a n d easy-to-use b r o a d b a n d spectroscopy a p p a r a t u s , a rigorous yet efficient c o m p u t e r m o d e l , a n d the a b i l i t y to fabricate the relevant samples. T h e a u t h o r processed the m a j o r i t y of the samples, was solely responsible for the a p p a r a t u s , a n d e x t e n d e d the m o d e l i n g code b y d e v e l o p i n g a n d i m p l e m e n t i n g the routines necessary to m o d e l defect  superlattices.  O v e r a l l , o u t s t a n d i n g agreement was achieved between the e x p e r i m e n t a l c h a r a c t e r i z a t i o n of the samples u s i n g the a u t h o r ' s a p p a r a t u s a n d the results o b t a i n e d w i t h the c o m p u t e r m o d e l i n g code.  Together the t h e o r e t i c a l a n d e x p e r i m e n t a l results p r o v i d e a c o m p r e -  hensive e x a m i n a t i o n of electromagnetic e x c i t a t i o n s associated w i t h 2 D t e x t u r e d p l a n a r waveguides. B y a c h i e v i n g unprecedented agreement of measured a n d c a l c u l a t e d b a n d  structures  of l e a k y modes associated w i t h the second, a n d u p to the seventh, zone-folded B r i l l o u i n 136  CHAPTER 6. Conclusions and Recommendations  137  zones of square a n d t r i a n g u l a r l a t t i c e structures, a t h o r o u g h c h a r a c t e r i z a t i o n of the p o l a r i z a t i o n a n d dispersive properties of these l o c a l i z e d electromagnetic modes has been achieved. A l l of the results c a n be interpreted u s i n g s y m m e t r y arguments a n d a p i c t u r e i n w h i c h the t r u e B l o c h states are the result of t e x t u r e - i n d u c i n g m i x i n g of T E - l i k e a n d T M - l i k e slab modes characteristic of the u n d e r l y i n g "average" slab waveguide. T h e p u r e k i n e m a t i c effects of 2 D t e x t u r i n g are revealed i n the s c a t t e r i n g s p e c t r a from a n o v e l azop o l y m e r - b a s e d s t r u c t u r e . M o r e s u b s t a n t i a l r e n o r m a l i z a t i o n effects are c l e a r l y evident i n the G a A s - b a s e d structures: second order gaps at zone center ~ 1 0 % ; c o m p l e t e first order pseudo-gaps ~ 2 0 % of the center frequencies; significant anti-crossings away f r o m zone center; a n d n e a r l y dispersion-free modes across the entire B r i l l o u i n zone, b o t h o u t s i d e a n d inside a c o m p l e t e pseudo-gap. T h e w o r k presented i n this thesis has c o n t r i b u t e d significantly t o the q u a n t i t a t i v e a n a l y s i s of the d i s p e r s i o n characteristics of 2 D t e x t u r e d p l a n a r waveguides due, i n large p a r t , t o the use of the specular reflectivity a n d diffraction measurement techniques i m p l e m e n t e d i n c o m b i n a t i o n w i t h the s p e c i a l l y designed e x p e r i m e n t a l a p p a r a t u s . T h e specular measurement technique has p r o v e d to be a n i n v a l u a b l e overall c h a r a c t e r i z a t i o n t o o l for p r o b i n g the l e a k y p h o t o n i c modes attached to these p l a n a r waveguides.  T h e diffrac-  t i o n measurement technique, developed b y the author, has been s h o w n to enhance t h i s c h a r a c t e r i z a t i o n of l e a k y modes b y p r o v i d i n g background-free s p e c t r a , e s p e c i a l l y f r o m modes w i t h l o w d i s p e r s i o n . It has also been s h o w n to p r o v i d e background-free s p e c t r a from more h i g h l y dispersive l e a k y modes of a t e x t u r e d p l a n a r waveguide w i t h a defect  CHAPTER 6. Conclusions and Recommendations  138  s u p e r l a t t i c e , w h e n the defects are spaced w i d e l y enough so as not to s t r o n g l y p e r t u r b the r e n o r m a l i z e d base l a t t i c e modes. I n a d d i t i o n , this diffraction measurement  technique  has been e x p l o r e d for its p o t e n t i a l to u n i q u e l y supplement specular r e f l e c t i v i t y charact e r i z a t i o n b y e n a b l i n g the p r o b i n g of a l l base l a t t i c e modes, i n c l u d i n g those b e l o w the light line. A l t h o u g h c h a r a c t e r i z a t i o n of these b o u n d modes has not been e x p e r i m e n t a l l y d e m o n s t r a t e d , the m o d e l s i m u l a t i o n s give confidence t h a t the m e t h o d is s o u n d . R e c o m m e n d a t i o n s for future w o r k i n c l u d e c o n t i n u e d efforts to use the d i f f r a c t i o n measurement technique to characterize modes below the light line. T h i s c o u l d be m a d e easier b y r e p l a c i n g the l o w power w h i t e light source w i t h a higher power arc l a m p . O n c e a g o o d s a m p l e is fabricated w i t h a l o c a l i z e d defect m o d e , a near-field o p t i c a l s c a n n i n g m i c r o s c o p e ( N S O M ) c o u l d be used to m a p the p h y s i c a l extent of the l o c a l i z e d m o d e .  Bibliography  [1] J o s e p h F . A h a d i a n a n d J r . C l i f t o n G . F o n s t a d .  Epitaxy-on-electronics technology  for m o n o l i t h i c optoelectronic i n t e g r a t i o n . Optical  Engineering,  [2] M e h r n e t B a y i n d i r , B . T e m e l k u r a n , a n d E . O z b a y . hopping:  37:3161, 1998.  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