UBC Theses and Dissertations

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UBC Theses and Dissertations

Scattering by a conducting periodic surface with a rectangular groove profile Heath, James William 1977

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SCATTERING BY A CONDUCTING PERIODIC SURFACE WITH A RECTANGULAR GROOVE PROFILE by James W i l l i a m H e a t h B . S c , Queen ' s U n i v e r s i t y , 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t he Depa r tment o f E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t he r e q u i r e d s t a n d a r d THE UNIVERSITY OF BR IT ISH COLUMBIA J u n e , 1977 @ James W i l l i a m H e a t h , 1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of E lec tr ioa l Engineering The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 6 J u l y 1 9 7 7 ABSTRACT The use o f a p e r i o d i c r e c t a n g u l a r g r oove p r o f i l e t o e l i m i n a t e s p e c u l a r r e f l e c t i o n f r o m a c o n d u c t i n g s u r f a c e i s s t u d i e d . The c o n c e n t r a -t i o n o f a l l s c a t t e r e d power i n t h e p r i n c i p a l b a c k s c a t t e r mode, i . e . , i n a d i r e c t i o n o p p o s i t e t o t h e i n c i d e n t wave , i s p o s s i b l e unde r p l a n e wave i l l u m i n a t i o n i f t h e p e r i o d d = -z . ^  „ : where 0. i s t h e a n g l e o f r 2 s x n 9. 1 l i n c i d e n c e f r o m t h e s u r f a c e n o r m a l and A i s t h e w a v e l e n g t h ; and t h e w i d t h and d e p t h o f t h e g r oove s a r e p r o p e r l y c h o s e n . An a n a l y t i c a l and n u m e r i c a l s t u d y o f t h e s c a t t e r i n g i s c a r r i e d . o u t f o r a r b i t r a r y p o l a r i z a t i o n t o d e t e r m i n e t h e s e g roove d i m e n s i o n s and t h e e f f e c t o f s y s t e m a t i c e r r o r s i n them. E x p e r i m e n t s we re p e r f o r m e d a t 35 GHz u s i n g b r a s s p l a t e s o f f i n i t e s i z e unde r n o n - p l a n e wave i l l u m i n a t i o n . The e x p e r i m e n t a l r e s u l t s show t h a t t he p l a t e s behave e s s e n t i a l l y as p r e d i c t e d . i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i L I ST OF TABLES v L I S T OF ILLUSTRATIONS v i ACKNOWLEDGMENTS i x 1. INTRODUCTION . 1 1.1 S c a t t e r i n g by a P e r i o d i c S t r u c t u r e 1 1.2 R e v i e w o f t he L i t e r a t u r e 4 1.3 T h e s i s O b j e c t i v e s 8 1.4 T h e s i s O u t l i n e 1 ° 2. ANALYSIS . 1 2 2.1 F o r m u l a t i o n o f t h e P r o b l e m 12 2.2 S o l u t i o n f o r TM P o l a r i z a t i o n 1 2 2 . 2 . 1 Boundary C o n d i t i o n s 1 3 2 . 2 . 2 G e n e r a l S o l u t i o n f o r t h e S c a t t e r e d F i e l d . . . . i 4 2 . 2 . 3 The A m p l i t u d e C o e f f i c i e n t s 16 2 .3 S o l u t i o n f o r TE P o l a r i z a t i o n 19 2.4 S o l u t i o n U s i n g t h e Optimum P e r i o d 2 3 3. NUMERICAL RESULTS 2 4 3.1 I n t r o d u c t i o n 2 4 3.2 N u m e r i c a l S o l u t i o n o f t h e P r o b l e m . 2 5 3.3 S p e c i a l R e c t a n g u l a r Groove P r o f i l e s . . 2 ^ 3.4 B l a z i n g Depth as a F u n c t i o n o f A n g l e : t TE P o l a r i z a t i o n . 31 3.5 B l a z i n g Dep th as a F u n c t i o n o f A n g l e : TM P o l a r i z a t i o n . 33 3.6 S i m u l t a n e o u s B l a z i n g f o r TE and TM P o l a r i z e d Waves . . . 3 8 3.7 O p e r a t i n g P o i n t B e h a v i o u r 4 2 3 . 7 1 1 . V a r i a t i o n s i n A n g l e 4 3 3 .7 .2 V a r i a t i o n s i n Groove Dep th 4 ^ SO 3 . 7 . 3 V a r i a t i o n s i n A s p e c t R a t i o i i i Page 3.8 S i m u l t a n e o u s B l a z i n g O p e r a t i n g P o i n t B e h a v i o u r 52 3.9 Summary 55 4 . EXPERIMENTAL RESULTS . . 57 4.1 I n t r o d u c t i o n 57 4.2 E x p e r i m e n t a l A r r angement . . 59 4 .3 P l a t e s B l a z e d f o r TM P o l a r i z a t i o n 62 4 .4 P l a t e s f o r S i m u l t a n e o u s B l a z i n g 65 4 .5 O b l i q u e I n c i d e n c e o 81 4.6 E r r o r s 85 5 . CONCLUSIONS 87 APPENDIX A : B l a z i n g Depth D a t a . 9 0 APPENDIX B: Graphs o f F r e q u e n c y Responses o f S u r f a c e s w i t h S y s t e m a t i c E r r o r s . . . 98 REFERENCES . . . . . . . 1 0 6 i v L I ST OF TABLES T a b l e Page I TE B l a z i n g Depths v s . 0 ; : a/d = 0.500 - 0.900 90 I I TE B l a z i n g Depths v s . 6 : a/d = 0.95 - 1.00 . . . . . . 91 I I I TM P r i m a r y B l a z i n g Depth s v s . 9 ± : a/d = 0.667 - 0.900 . 92 IV TM P r i m a r y B l a z i n g Depth s v s . Q± : a/d = 0.950 - 0.9999. 93 V TM P r i m a r y B l a z i n g Depth s v s . 6 : a/d = 1.0 94 VI TM P r i m a r y B l a z i n g Depths v s . 6 : a/d = 0.00001 - 0.100 95 V I I TM P r i m a r y B l a z i n g Depth s v s . 6 : a/d = 0.250 - 0.500 . 95 V I I I TM Seconda r y B l a z i n g Depth s v s . 6 : a/d = 0.250 - 0.667 9 6 I X TM Seconda r y B l a z i n g Depth s v s . 6 : a/d = 0.75 - 0.99 . 9 7 X a/d and P r i m a r y B l a z i n g Depth s v s . 0. f o r S i m u l t a n e o u s B l a z i n g 9 8 X I a/d and Seconda r y B l a z i n g Depths v s . 0^ f o r S i m u l t a n e o u s B l a z i n g . 9 8 v L I ST OF ILLUSTRATIONS F i g u r e Page 1.1 P l a n e Wave I n c i d e n t on a P e r i o d i c S u r f a c e w i t h a R e c t a n g u l a r Groove P r o f i l e , N o r m a l t o t h e Groove Edges . 1 1.2 D i r e c t i o n o f P r o p a g a t i o n o f P r i n c i p a l B a c k s c a t t e r Mode (m = - 1 ) and S p e c u l a r l y R e f l e c t e d Mode (m = 0) 4 1.3 Th ree Types o f P e r i o d i c P r o f i l e s . . . . 4 2 .1 TM P o l a r i z e d Wave I n c i d e n t on a S u r f a c e w i t h a R e c t a n -g u l a r G roove P r o f i l e 13 2.2 S i n g l e C e l l w i t h a R e c t a n g u l a r Groove P r o f i l e 14 2.3 TE P o l a r i z e d Wave I n c i d e n t on a S u r f a c e w i t h a R e c t a n g u l a r Groove P r o f i l e 19 3.1 Two R e c t a n g u l a r Groove P r o f i l e s 27 3.2 TM R e l a t i v e R e f l e c t e d Power v s . h/X f o r l a r g e A s p e c t R a t i o s : a/d = 0.9999 and a/d = 1.0 28 3.3 TM R e l a t i v e R e f l e c t e d Power v s . h/d f o r S m a l l A s p e c t R a t i o s : 9 ± = 55° 29 3.4 G e o m e t r i c a l O p t i c s R e f l e c t i o n o f I n c i d e n t TEM Waves . . 31 3.5 TE B l a z i n g Dep th Cu r ve s v s . 6± 32 3.6 TM P r i m a r y B l a z i n g Depth Cu r ve s v s . 6 . : L a r g e A s p e c t R a t i o s 34 3.7 TM P r i m a r y B l a z i n g Dep th Cu r ve s v s . 0 . : S m a l l A s p e c t R a t i o s ^ 3 5 3.8 TM S e c o n d a r y B l a z i n g Dep th Cu r ve s v s . 0^ 37 3.9 S u p e r p o s i t i o n o f TM and TE B l a z i n g Dep th Cu r ve s f o r a/d = 0 .99 39 3.10 A s p e c t and Groove Dep th R a t i o s f o r a S u r f a c e B l a z e d f o r B o t h TE and TM I n c i d e n t Waves. 1 9 . 5 ° < 6. < 5 9 . 4 ° . . . 4 0 l 3.11 A s p e c t and Groove Dep th R a t i o s f o r a S u r f a c e B l a z e d f o r B o t h TE and TM I n c i d e n t Waves. 1 9 . 5 ° < 9. < 7 3 . 1 2 ° . . . 4 1 l 3.12 TM R e f l e c t e d Power v s . 9. : 9. = 3 5 ° 4 4 l i o p 3.13 TM R e f l e c t e d Power v s . 9. : 9„ = 6 0 ° 4 6 l i o p v i F i g u r e Page 3.14 TM R e l a t i v e R e f l e c t e d Power v s . h/d f o r L a r g e A s p e c t R a t i o s : 9 = 85° . . 48 3.15 TM R e l a t i v e R e f l e c t e d Power v s . h/d f o r S m a l l A s p e c t R a t i o s : 0 = 85° 49 3.16 TM R e l a t i v e R e f l e c t e d Power v s . a/d fv 51 3.17 a / A and h / A f o r a S u r f a c e B l a z e d f o r B o t h TE and TM I n c i d e n t Waves: 1 9 . 5 ° < 9. < 5 9 . 4 ° 53 l 3.18 a / A and h / A f o r a S u r f a c e B l a z e d f o r B o t h TE and TM I n c i d e n t Waves: 1 9 . 5 ° < 6 < 73 .12° . 54 4.1 P r o f i l e o f E x p e r i m e n t a l S u r f a c e s and D i m e n s i o n s f o r A l l P l a t e s 58 4.2 E x p e r i m e n t a l S e t Up 6 0 4.3 P i c t u r e o f E x p e r i m e n t a l S e t Up 61 4.4 Mea su red and C a l c u l a t e d TM R e f l e c t e d Power v s . 0 . f o r P l a t e # 1 1 . . . . 6 3 4.5 Mea su red and C a l c u l a t e d TM R e f l e c t e d Power v s . 9. f o r P l a t e #22 x . . . . 6 4 4.6 Mea su red and C a l c u l a t e d R e f l e c t e d Power v s . 9. f o r P l a t e # 3: f = 35.0 GHz * 6 6 4.7 Mea su red F r e q u e n c y Response o f P l a t e #3 . 68 4.8 Mea su red and C a l c u l a t e d R e f l e c t e d Power v s . 9. f o r P l a t e # 3: f = .34.9 GHz X. 6 9 4.9 Mea su red and C a l c u l a t e d R e f l e c t e d Power v s . 0 . f o r P l a t e #4: f = 35.0 GHz . . * 7 0 4.10 Mea su red F r e q u e n c y Response o f P l a t e # 4 7 ^ 4.11 Mea su red and C a l c u l a t e d R e f l e c t e d Power v s . 0 . f o r P l a t e #4: f = 34.8 GHz * 7 3 4.12 Mea su red TM R e f l e c t e d Power v s . 0 . f o r P l a t e # 4: f = 34.7 GHz X. 74 4.13 Mea su red and C a l c u l a t e d R e f l e c t e d Power v s . 0 . f o r P l a t e #5 76 4.14 Mea su red and C a l c u l a t e d R e f l e c t e d Power v s . 0 f o r P l a t e #6 ' 77 v i i F i g u r e Page 4 .15 Mea su red and C a l c u l a t e d R e f l e c t e d Power v s . 0 . f o r P l a t e # 7: f = 35.0 GHz * . . . . . 79 4.16 F r e q u e n c y Response o f P l a t e # 7 80 4.17 Mea su red and C a l c u l a t e d R e f l e c t e d Power v s . 0 . : f = 35.2 GHz . . * 82 4.18 Mea su red R e f l e c t e d Power v s . 3^ f o r P l a t e # 3 . . . . . 83 4 .19 Mea su red R e f l e c t e d Power v s . f o r P l a t e # 4 84 B . l F r e q u e n c y Response s o f a S i m u l t a n e o u s l y B l a z e d Su r face r , w i t h 0 . = 4 5 ° , f o r S y s t e m a t i c E r r o r s i n t h e Groove D i m e n s i o n s . . . . . . . . 100 B.2 F r e q u e n c y Responses o f a S i m u l t a n e o u s l y B l a z e d S u r f a c e , w i t h 0 . = 2 0 ° , f o r S y s t e m a t i c E r r o r s i n t h e Groove D i m e n s i o n s 103 v i i i ACKNOWLEDGEMENTS I w o u l d l i k e t o t hank t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r t h e r e s e a r c h a s s i s t a n t s h i p I r e c e i v e d f r o m 1975 t o 1977 and t h e E l e c t r i c a l E n g i n e e r i n g Depa r tment a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r t h e t e a c h i n g a s s i s t a n t s h i p s I r e c e i v e d f r o m 1975 t o 1977. I w o u l d e s p e c i a l l y l i k e t o t hank my s u p e r v i s o r , D r . E.V. J u l l , t o whom I am i n d e b t e d f o r h i s p a t i e n c e and g u i d a n c e . I w o u l d a l s o l i k e t o t hank D r . E.V. Bohn , D r . L. Young , and D r . G . F . S c h r a c k f o r r e a d i n g t h e m a n u s c r i p t and t h e i r v a l u a b l e s u g g e s t i o n s . I w o u l d l i k e t o t hank De rek D a i n e s f o r s p e n d i n g t h o s e many h o u r s m i l l i n g t h e p l a t e s , as w e l l as A l M a c K e n z i e f o r h i s a s s i s t a n c e i n p r e p a r i n g t h e i l l u s t r a t i o n s and f o r s h a r i n g h i s c h e e r f u l n e s s w i t h me. I w o u l d l i k e t o t hank a l l o f e t h e t e c h n i c a l s t a f f , whose names a r e t o o numerous t o m e n t i o n , who a l l h e l p e d me and made my s t a y a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a more p l e a s a n t . I w o u l d l i k e t o t hank t h e t y p i s t s , M a r y - E l l e n F l a n a g a n and S a n n i f e r L o u i e who n o t o n l y t y p e d t h e m a n u s c r i p t and c h e e r f u l l y made t he c o r r e c t i o n s b u t a l s o h e l p e d ou t i n many o t h e r way s , a l w o u l d l i k e t o t hank a l l t h e s e c r e t a r i e s i n t h e o f f i c e who a l w a y s answer my q u e s t i o n s c o n c e r n i n g u n i v e r s i t y and d e p a r t m e n t a l p r o c e d u r e s and r e g u l a t i o n s , and who h e l p e d make my s t a y a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a a p l e a s a n t one by e n s u r i n g t h a t a l l r e l e v a n t f o rms and n o t i c e s were drawn t o my a t t e n t i o n . F i n a l l y , I am v e r y g r a t e f u l t o my w i f e , Gwen, t o whom I w o u l d l i k e t o d e d i c a t e t h i s t h e s i s . She n o t o n l y encou raged and g u i d e d me, b u t she a l s o t y p e d t h e o r i g i n a l d r a f t , h e l p e d i n t h e c o r r e c t i o n o f t h e m a n u s c r i p t , and d i d t h e l a y o u t f o r t h e i l l u s t r a t i o n s . I w o u l d l i k e t o i x thank her for the sacrifices she made both for leaving her friends and relatives in coming to Vancouver and for spending those many hours help-ing me. Without her help I doubt i f this thesis would have been completed. x 1. INTRODUCTION 1.1 Scattering by a Per iodic Structure A plane electromagnetic wave inc ident on a surface with a two-dimensional per iodic p r o f i l e gives r i s e to a scattered electromagnetic f i e l d . Above the grooves, th i s scattered f i e l d i s comprised of a f i n i t e number of homogenous plane waves and an i n f i n i t e number of evanescent waves. The study of these propagating modes i s important i n the design of d i f -f rac t ion gratings [1] and has po ten t ia l app l ica t ion to laser mirrors [2,3] or to the reduction of in t e r fe r ing r e f l e c t i o n from the sides of bui ld ings Analyzing the boundary value problem i s easier i f the coordinate axes are oriented so that one axis i s normal to the surface and another i s p a r a l l e l to the grooves i n the surface. Figure 1.1 exh ib i t s th i s o r ien ta -t ion of the axes when the grooves of the surface are rectangular. The two-dimensional problem i s easiest when the plane wave i s invar ian t i n the grooves' d i r ec t i on (the a d i r ec t ion as shown i n F i g . 1.1). SCATTERED HOMOGENEOUS J PLANE WA VES INCIDENT PLANE WA VE T h F i g . 1.1 Plane Wave Incident on a Pe r iod ic Surface wi th a Rectangular Groove P r o f i l e , Normal to the Groove Edges. In the free space region (where z > 0) of F i g . 1.1, the scattered plane waves from different grooves must add, so the «i phase difference over a per iod, d, must d i f f e r by mult iples of 2TT; i . e . 2. kdsin 9 = kdsin 0 . + 2i\m m = 0, ± 1 , ± 2 , . . . , (1.1) m x where k = 2TT/A i s the wave number of the incident wave, and 0 . is the i angle of incidence of the plane wave on the surface. Therefore, sin 0 = sin 6 . + mX/d m = 0, ± 1 , ± 2 , . . . . (1.2) m l The values of m for which | s in © m | _< 1, correspond to the homogenous plane waves and these propagate away from the surface at angles 0^ to the a z axis. The other values of m correspond to the in f in i te number of evanes-cent modes which propagate back and forth in the grooves and decay away from the surface. The value of m = 0 represents the specularly reflected wave which is always propagating at angle 0^. It is only the propagating modes which can carry energy away from the surface. Thus, because energy must be conserved, when the scattering is lossless the propagating waves with amplitudes are related to the incident wave's amplitude by I A. I2 cos 0 . = 7 fA f cos 0 . (1.3) 1 i ' l L 1 m1 m m The values of m include only those modes which propagate. If the period d, is small, only the specularly reflected mode propagates, carrying a l l the incident energy. As the period becomes larg-er, the principal backscatter mode (m =-1) appears. It begins to propa-gate when X/d = (sin 0^ - 1) and then travels along the surface at 0_^ = - 9 0 ° . As the period increases further, higher order forward and backscat-ter modes begin to propagate when X/d = (1 - s in 0^)/m. These modes i n -i t i a l l y travel along the surface at = ± 9 0 ° , depending on whether they are forward ( m > 0) or backscatter (m < 0) modes. As d increases further, a l l angles of propagation tend to that of the specularly reflected mode, (6 = 0 N . )• In the l imit as d + » a l l modes propagate in the direction 3. 0 = 0 . . The s c a t t e r e d f i e l d f r o m t h i s s u r f a c e i s a p l a n e wave r e f l e c t e d m i f r o m a p e r f e c t l y c o n d u c t i n g f l a t s u r f a c e . I n t h e a p p l i c a t i o n s g i v e n , i t i s d e s i r e d t o c o n c e n t r a t e a l l t h e i n c i d e n t ene r gy i n one o f t h e s c a t t e r e d p r o p a g a t i n g modes. A p e r i o d i c s u r f a c e i s s a i d t o have been b l a z e d i f a l l t he i n c i d e n t ene r g y has been c o n c e n t r a t e d i n t h e p r i n c i p a l b a c k s c a t t e r e d mode. A n e c e s s a r y c o n d i t i o n f o r a b l a z e d s u r f a c e i s t h a t t h e r e be maximum b a c k s c a t t e r . A s u r f a c e g i v e s maximum b a c k s c a t t e r i f t h e p e r i o d d i s g o ve rned by t h e B r a g g c o n -d i t i o n k d s i n 8^ = mr n = 1, 2 , 3 , . . . . ( 1 . 4 ) I t i s e a s i e s t t o b l a z e a s u r f a c e when t h e number o f p r o p a g a t i n g modes a r e f e w e s t ; i . e . when t h e p e r i o d d i s s m a l l e s t . T h i s i s t r u e i n e q u a t i o n 1.4 when n = 1 o r d = X / ( 2 s ± n 0 ) ( 1 . 5 ) E q u a t i o n 1.2 now r e d u c e s t o s i n 0 = (2m + 1) s i n 0. m = 0 , ± 1 , ± 2 ( 1 . 6 ) m i ' ' ' The m = - 2 and m = 1 modes do n o t p r o p a g a t e as l o n g as s i n 0^ > 1/3 o r 0^ > 1 9 . 5 ° . F o r a s u r f a c e w i t h p e r i o d d , g i v e n by e q u a t i o n 1 .5 , and w i t h a p l a n e wave i n c i d e n t a t an a n g l e g r e a t e r t h a n 1 9 . 5 ° , o n l y t h e p r i n c i p a l b a c k s c a t t e r e d and s p e c u l a r l y r e f l e c t e d modes p r o p a g a t e . The p r i n c i p a l b a ck s ca t t e r edmmode i s i n t he d i r e c t i o n o f i n c i d e n c e , as shown i n F i g . 1 .2. When d s a t i s f i e s ( 1 . 5 ) , t h e c o n s e r v a t i v e ene r g y e q u a t i o n ( 1 . 3 ) r e d u c e s t o ' K l 2 - | A j 2 + \A_±\2 ( 1 . 7 ) and t h i s may be u sed as a check on t h e s o l u t i o n o f t h e s c a t t e r e d f i e l d s . The B r a g g c o n d i t i o n i s a s u f f i c i e n t c o n d i t i o n f o r a b l a z e d s u r f a c e . I n e v e r y known c a s e o f d i f f r a c t i o n by a p e r i o d i c s t r u c t u r e , t h e p e r i o d had 4. INCIDENT^< WA VE x T?i e 1 2 D i r ec t ion of Propagated of P r i n c i p a l Backscatter Mode (m - -1) . ' And Specularly Reflected Mode (m = 0) . to sa t i s fy the Bragg condi t ion for the surface to be perfec t ly blazed. 1.2 Review of the L i t e ra tu re This two dimensional problem of scat ter ing by a per fec t ly con-ducting per iod ic surface has been studied for p ro f i l e s other than the rectangular groove (or lamellar) p r o f i l e shown i n F igs . 1.1 and 1.2. Three other types of per iodic p ro f i l e s that have been studied are shown i n F i g . 1.3 and they are the echelet te , the s inusoida l groove and the comb. a) b) c) F i g . 1.3 Three Types of Per iod ic P r o f i l e s (a) Echelette P r o f i l e (b) Sinusoidal P r o f i l e (c) Comb P r o f i l e 5. S u r f a c e s w i t h t h e s e p r o f i l e s must a l s o have p e r i o d s w h i c h s a t i s f y t h e B r a g g c o n d i t i o n t o be b l a z e d . The r i g h t a n g l e e c h e l e t t e has a p e r i o d i c t r i a n g u -l a r g r oove p r o f i l e whose f a c e t s i n t e r s e c t a t r i g h t a n g l e s ( a = 9 0 ° ) . The s c a t t e r i n g b e h a v i o r o f t h i s p r o f i l e has been i n v e s t i g a t e d by S t r o k e [1] and I t o h and M i t t r a [ 5 ] . F o r an i n c i d e n t p l a n e wave w h i c h i s H - p o l a r i z e d , t he s c a t t e r e d f i e l d can be a n a l y z e d i n te rms o f c o n s t r u c t i v e i n t e r f e r e n c e f r o m the f a c e t s w h i c h a r e n o r m a l t o t h e d i r e c t i o n o f p r o p a g a t i o n o f t h e i n c i d e n t wave . Because o f t h i s b e h a v i o r , i t i s p o s s i b l e t o o b t a i n a s i m p l e e x p r e s s i o n f o r t h e d e p t h o f c o r r u g a t i o n needed t o b l a z e t h i s TM i n c i d e n t wave . U n f o r t u n a t e l y , i t i s n o t p o s s i b l e t o d e s i g n a s u r f a c e l i k e t h i s t h a t i s c o m p l e t e l y b l a z e d when t he i n c i d e n t wave i s E - p o l a r i z e d . S t r o k e [1] recommends t h a t a l a r g e r v a l u e o f a be u sed l i k e 110° o r 1 2 0 ° . The s i n u s i o d a l s u r f a c e has a p r o f i l e w h i c h i s s i n u s o i d a l l y mod-u l a t e d . Z a k i and N e u r e U t h e r [ 6 , 7 ] a n a l y z e d t h e b e h a v i o r o f t h i s g r a t i n g f o r b o t h TE and TM i n c i d e n t p l a n e wave s . They f o r m u l a t e d t h e s c a t t e r i n g i n te rms o f c o u p l e d i n t e g r a l e q u a t i o n s and f i e l d s f o r a s i n g l e p e r i o d . F o r a g i v e n a n g l e o f i n c i d e n c e t h e y we re a b l e t o d e s i g n two b l a z e d s u r -f a c e s w h i c h h a d d i f f e r e n t a m p l i t u d e s o f m o d u l a t i o n . One s u r f a c e was b l a z e d f o r a TE i n c i d e n t p l a n e wave, w h i l e t h e o t h e r was b l a z e d f o r a TM i n c i d e n t p l a n e wave. The comb p r o f i l e c o n s i s t s o f an i n f i n i t e a r r a y o f i n f i n i t e l y t h i n p l a t e s w h i c h a r e p e r i o d i c a l l y s p a c e d and s h o r t e d a t t h e same d e p t h . Tseng [8] and T s eng , H e s s e l and O l i n e r [9] a n a l y z e d t h e comb g r a t i n g f o r b o t h p o l a r i z a t i o n s . They u sed a s c a t t e r i n g m a t r i x a p p r o a c h t o c o n v e r t t h e p r o b l e m t o one o f a wave i n c i d e n t on an i n f i n i t e a r r a y o f s e m i - i n f i n i t e p a r a l l e l p l a t e s . Tseng e t a l . needed t o assume t h a t t he f i n d e p t h was s u f f i c i e n t f o r c o m p l e t e a t t e n u a t i o n o f t h e e v a n e s c e n t modes i n t h e g r o o v e s . 6. DeSanto [10 ,11 ] u sed t h e m o d i f i e d c a l c u l u s o f r e s i d u e s t h e o r y t o o b t a i n a c o m p l e t e s o l u t i o n f o r t h e s c a t t e r e d f i e l d f o r t he comb p r o f i l e . B o t h DeSanto and Tseng e t a l . f o und i t was p o s s i b l e t o b l a z e t h e s u r f a c e com-p l e t e l y f o r e i t h e r a TM o r a TE p o l a r i z e d i n c i d e n t wave . Ebbeson [12 ,13 ] i n d e p e n d e n t l y , a l s o o b t a i n e d t h e s o l u t i o n o f t h e s c a t t e r e d d f i e l d f o r a comb p r o f i l e when t h e i n c i d e n t wave i s TM p o l a r -i z e d . He u sed an i n t e g r a l t r a n s f o r m t e c h n i q u e [14] and a s c a t t e r i n g m a t r i x a p p r o a c h s i m i l a r t o t h a t employed by Tseng e t a l . The o b j e c t o f h i s s t u d y was t o u se t h e comb p r o f i l e on t he s i d e s o f a i r p o r t h a n g a r s t o r e -duce r e f l e c t i o n f r o m t h e w a l l s . T h i s r e f l e c t i o n cau sed i n t e r f e r e n c e w i t h t h e I n s t r u m e n t L a n d i n g Sy s tem ( I L S ) u sed a t t h e a i r p o r t . Some e x p e r i m e n -t a l r e s u l t s f o r g r a t i n g s w h i c h were d e s i g n e d u s i n g E b b e s o n ' s d a t a f o r t h e comb g r a t i n g were p r e s e n t e d by Ebbeson [13] and J u l l and Ebbeson [ 4 ] . A l s o , r e s u l t s f o r a r e c t a n g u l a r g r oove g r a t i n g s d e s i g n e d u s i n g d a t a f r o m H e s s e l , Schmoys and Tseng [2] were r e p o r t e d by J u l l [ 1 6 ] . Becau se t h e e x p e r i m e n t a l s u r f a c e s u sed by Ebbeson and J u l l [ 12 ,15 ] had t o i n c l u d e f i n s o f f i n i t e t h i c k n e s s , t h e p r o f i l e was a r e c t a n g u l a r g r oove r a t h e r t h a n a comb. T h i s d i f f i c u l t y and t h e r e a l i z a t i o n t h a t t h e d e p t h o f c o r r u g a t i o n f o r a b l a z e d r e c t a n g u l a r g r oove s u r f a c e c o u l d be much s h a l l o w e r , a l l o w i n g e a s i e r c o n s t r u c t i o n , p rompted t h i s i n v e s t i g a t i o n o f t h e r e c t a n g u l a r g r o o v e p r o f i l e . The f i r s t a n a l y s i s o f t h e r e c t a n g u l a r g r oove p r o f i l e was done by W i r g i n [17] and W i r g i n and D e l e u i l [3] f o r b o t h p o l a r i z a t i o n s . They a n a l y z e d t h e s c a t t e r e d f i e l d by m a t c h i n g t h e f r e e s p a c e modes and t h e wavegu i de modes ( t h e modes i n t he g r o o v e s ) a c r o s s t h e z = 0 I n t e r f a c e . They p r e s e n t e d f o u r s e t s o f c a l c u l a t e d and e x p e r i m e n t a l c u r v e s c o r r e s p o n -d i n g t o f o u r d i f f e r e n t s u r f a c e s , each d e s i g n e d f o r a p l a n e wave i n c i d e n t 7. a t 6^ = 3 0 ° . They d i d n o t o b t a i n any i n s t a n c e s where t h e g r a t i n g was b l a z e d c o m p l e t e l y . H e s s e l , Schmoys and Tseng [2] a l s o u n d e r t o o k an e x t e n s i v e num-e r i c a l s t u d y o f t he b l a z i n g c a p a b i l i t i e s o f t he r e c t a n g u l a r g r oove g r a t i n g f o r i n c i d e n t waves o f b o t h p o l a r i z a t i o n s . They , t o o , a n a l y z e d t h e s c a t -t e r e d f i e l d b y mode m a t c h i n g a c r o s s t h e z = 0 i n t e r f a c e . They p l o t t e d c u r v e s o f t h e r a t i o o f t h e g r oove d e p t h t o p e r i o d needed t o e l i m i n a t e t he r e f l e c t e d power , i . e . t h e b l a z i n g d e p t h , f o r a g i v e n g r o o v e w i d t h t o p e r i o d r a t i o , o r a s p e c t r a t i o . They p r e s e n t e d a number o f t h e s e c u r v e s f o r the c a s e o f TE p o l a r i z a t i o n and one b l a z i n g d e p t h wave f o r TM p o l a r -i z a t i o n c o r r e s p o n d i n g t o an a s p e c t r a t i o o f 0 . 6 6 7 . The TM c u r v e s were more c o m p l i c a t e d t h a n t he TE c u r v e s and so H e s s e l e t a l . d i d n o t e v a l u a t e b l a z i n g d e p t h c u r v e s f o r o t h e r a s p e c t r a t i o s b e c a u s e o f t h e c o m p u t a t i o n a l t i m e r e q u i r e d . H e s s e l e t a l . we re e v i d e n t l y t h e f i r s t t o r e a l i z e t h a t a r e c t a n g -u l a r g r oove p r o f i l e w i t h t h e r i g h t g r oove d i m e n s i o n s g i v e s a s u r f a c e w h i c h i s p e r f e c t l y b l a z e d f o r a r b i t r a r y p o l a r i z a t i o n . They gave g r oove d i m e n -s i o n s f o r t h r e e s u ch s u r f a c e s . R o u m i g u i e r e s , M a y s t r e and P e t i t [18] p r e -s e n t e d b l a z i n g d e p t h c u r v e s f o r TM p o l a r i z e d i n c i d e n t waves f o r a " range o f a s p e c t r a t i o s f r o m 0.5 t o 0 . 9 5 . They f ound more d u a l b l a z e d s u r f a c e s and p r e s e n t e d some e x p e r i m e n t a l measurements made a t n e a r u l t r a v i o l e t f r e q u e n c i e s on a g r a t i n g w h i c h u sed t h e d i m e n s i o n s f o r s i m u l t a n e o u s b l a z i n g o f b o t h p o l a r i z a t i o n s a t = 36° w h i c h were g i v e n by H e s s e l e t a l . A g r e e -ment w i t h e x p e r i m e n t was n o t good , p r o b a b l y due t o t h e d i f f i c u l t y i n mak i ng s u r f a c e s t o s u f f i c i e n t a c c u r a c y a t t h e s e f r e q u e n c i e s as w e l l as t h e e f f e c t o f f i n i t e c o n d u c t i v i t y . B o t h t h e s i n u s o i d a l p r o f i l e and t h e comb p r o f i l e have one o p e r a t i n g p o i n t e a c h , where t he s i m u l t a n e o u s b l a z i n g o f 8. b o t h p o l a r i z a t i o n o c c u r s . The l a t t e r p r o f i l e ' s d u a l b a l z i n g p o i n t was o b t a i n e d by J u l l [15] f r o m t h e r e s u l t s o f T s eng , H e s s e l and O l i n e r [9] and Ebbeson [ 1 2 ] . J u l l , H e a t h and Ebbeson [19] p r e s e n t e d c o m p l e t e c u r v e s , ( b l a z i n g d e p t h and a s p e c t r a t i o c u r v e s ) , f o r t h e g r a t i n g d i m e n s i o n s needed f o r s i m u l t a n e o u s b l a z i n g f o r an a n g l e o f i n c i d e n c e be tween 1 9 . 5 ° and 5 9 . 4 ° u s i n g t h e p o i n t s g i v e n by H e s s e l e t a l . and p o i n t s o b t a i n e d i n a d d i t i o n t o and i n d e p e n d e n t l y o f R o u m i g u i e r e s e t a l . A l s o p r e s e n t e d we re e x p e r i m e n t a l r e s u l t s f o r t h r e e d i f f e r e n t g r a t i n g s w h i c h s i m u l t a n e o u s l y b l a z e b o t h TE and TM i n c i d e n t wave s . T h i s was t h e f i r s t e x p e r i m e n t a l d e m o n s t r a t i o n o f d u a l b l a z i n g . 1.3 T h e s i s O b j e c t i v e s 1) To s t u d y t h e b e h a v i o r o f t h e r e c t a n g u l a r g r oove p r o f i l e as t he g r oove w i d t h v a r i e s f r o m b e i n g i n f i n i t e l y t h i n t o i t s maximum v a l u e when i t i s e q u a l t o t h e p e r i o d d . By t h e c o n t i n u i t y o f t h e p h y s i c s o f t h e bounda r y v a l u e p r o b l e m , i t i s e x p e c t e d t h a t when t h e g r oove w i d t h a p p r o a c h e s t h e p e r i o d o f t h e s u r f a c e , t h e s c a t t e r e d f i e l d s h o u l d c o n v e r g e t o t h e s c a t -t e r e d f i e l d o f t he comb g r a t i n g whose p r o f i l e i t a p p r o x i m a t e s . The c o n -v e r g e n c e o f t h e s c a t t e r e d f i e l d o f t h e r e c t a n g u l a r g r oove p r o f i l e t o t h e s e e x p e c t e d f i e l d s w i l l be shown. 2) To a c q u i r e t h e b l a z i n g d e p t h c u r v e s f o r a f u l l s e t o f a s p e c t r a t i o s f o r TE and TM p o l a r i z e d i n c i d e n t wave s . One TM b l a z i n g d e p t h c u r v e was g i v e n by H e s s e l e t a l . [2] and a n o t h e r f o r t h e comb g r a t i n g , was g i v e n by Ebbeson [12 ,13 ] and i t was d e c i d e d t o f i l l i n more b l a z i n g d e p t h c u r v e s be tween t h e s e two c u r v e s and t o o b t a i n c u r v e s f o r a s p e c t r a t i o s a p p r o a c h i n g z e r o . Unknown t o t h e a u t h o r , R o u m i g u i e r e s e t a l . [18] had a l r e a d y o b t a i n e d some o f t h e s e TM b l a z i n g d e p t h c u r v e s . 3) To o b t a i n d e s i g n c u r v e s f o r t h e d i m e n s i o n s o f s u r f a c e s w h i c h 9. a r e s i m u l a t e n o u s l y b l a z e d f o r b o t h TE and TM p o l a r i z a t i o n s . Mos t g r a t i n g s w h i c h a r e i n u se a r e p o l a r i z a t i o n d e p e n d e n t ; t h e y m i g h t be b l a z e d f o r an i n c i d e n t wave o f one p o l a r i z a t i o n b u t n o t t h e o t h e r . I t i s shown t h a t i t i s p o s s i b l e i n p r i n c i p l e t o o b t a i n a b l a z e d g r a t i n g w h i c h wo rk s f o r an a r b i t r a r i l y p o l a r i z e d wave f o r any a n g l e o f i n c i d e n c e f r o m = 1 9 . 5 ° t o 6^ = 9 0 ° , a l t h o u g h t h e g r a t i n g d i m e n s i o n s do become i n c o n v e n i e n t l y deep a t g r a z i n g i n c i d e n c e . 4) To s t u d y t h e b l a z i n g d e p t h c u r v e s t o d e v e l o p c r i t e r i a t o h e l p choose o p e r a t i n g p o i n t s w h i c h g i v e s u r f a c e s whose s c a t t e r e d f i e l d b e s t s u i t s t h e r e q u i r e d a p p l i c a t i o n . H e s s e l , Schmoys, and Tseng [2] m e n t i o n e d t h a t a minimum i n some o f t he TE b l a z i n g d e p t h c u r v e s d e f i n e s a s u r f a c e whose r e f l e c t e d power i s r e d u c e d o v e r a r e l a t i v e l y w i d e r ange o f a n g l e s o r w a v e l e n g t h s . I t i s hoped t o e n l a r g e on t h i s , t o p r o v i d e o b s e r v a t i o n s t h a t h e l p p r e d i c t a s u r f a c e ' s g e n e r a l b e h a v i o r as any one o f t he v a r i a b l e s o f t h e i n c i d e n t wave (A o r 0_^ ) o r o f t h e s u r f a c e ( d , a , h ) i s v a r i e d . T h i s w o u l d h e l p i n d e s i g n i n g a s u r f a c e f o r a s p e c i f i c a p p l i -c a t i o n f r o m t h e b l a z i n g d e p t h c u r v e s . 5) To b u i l d t h e s e p e r i o d i c s u r f a c e s and measure t h e r e f l e c t e d f i e l d f r o m them o v e r a r ange o f f r e q u e n c i e s and a n g l e s o f i n c i d e n c e . S u r f a c e s d e s i g n e d u s i n g a w i d e r ange o f opt imum a n g l e s o f i n c i d e n c e and s u r f a c e d i m e n s i o n s w i l l be u s e d . The v a l i d i t y and a p p l i c a b i l i t y o f t h e o b s e r v a t i o n s m e n t i o n e d e a r l i e r w i l l be d i s c u s s e d i n c o n j u n c t i o n w i t h t h e e x p e r i m e n t a l r e s u l t s w h i c h i l l u s t r a t e them. 6) To o b s e r v e t he e f f e c t o f t h e d i f f e r e n c e s o f t h e e x p e r i m e n -t a l s e t up and t h e i d e a l c a se u sed i n t h e o r y . The e f f e c t o f l i m i t a t i o n s i n t he e x p e r i m e n t a l s u r f a c e s u ch as i t s f i n i t e s i z e and t h e e f f e c t o f t h e t o l e r a n c e o f t he g r o o v e s ' d i m e n s i o n s and t he m a t e r i a l ' s f i n i t e c o n d u c t i v i t y w i l l be d i s c u s s e d . The i l l u m i n a t i o n w i l l n o t be i d e a l e i t h e r and such 10 . e f f e c t s as o b l i q u e i n c i d e n c e and n o n - p l a n e wave i n c i d e n c e w i l l be d i s -c u s s e d a l s o . The e f f e c t o f s u ch d i f f e r e n c e s be tween t h e e x p e r i m e n t a l c a se and t he i d e a l one i s n o t a c c o u n t e d f o r i n t h e a n a l y s i s b u t can be i n v e s t i g a t e d e x p e r i m e n t a l l y . 1.4 T h e s i s O u t l i n e A tho rough s t u d y o f t h e b l a z i n g c a p a b i l i t i e s o f t h e r e c t a n g u l a r g r oove p r o f i l e i s u n d e r t a k e n h e r e . I n c h a p t e r 2 t h e s t a n d a r d method o f a n a l y s i n g t he s c a t t e r e d f i e l d , mode m a t c h i n g a c r o s s t h e z = 0 i n t e r f a c e , i s o u t l i n e d . A l s o p r e s e n t e d a r e c e r t a i n r e f i n e m e n t s i n t h e m a t r i x e q u a -t i o n o b t a i n e d f r o m t h i s mode m a t c h i n g t h a t l e a d t o g r e a t e r c o m p u t a t i o n a l e f f i c i e n c y . B l a z i n g d e p t h c u r v e s f o r a c o m p l e t e r ange o f a s p e c t r a t i o s a r e p r e s e n t e d i n c h a p t e r 3 , a l o n g w i t h t a b l e s o f v a l u e s u s ed t o p l o t t h e s e c u r v e s . T h i s w i l l e n a b l e t h e p l o t t i n g o f d e s i g n c u r v e s . I t i s shown t h a t t h e n u m e r i c a l v a l u e s c o n v e r g e t o t h o s e f o r t h e comb g r a t i n g s t u d i e d i n [ 8 - 1 3 ] ; b e c a u s e o f t h i s i t i s p o s s i b l e t o o b t a i n t h e s c a t t e r e d f i e l d f o r a comb p r o f i l e w i t h v e r y s m a l l f i n d e p t h s , ( w h i c h was n o t p o s s i b l e i n [ 8 - 1 3 ] ) , by u s i n g a r e c t a n g u l a r g r oove p r o f i l e w i t h an a s p e c t r a t i o o f a l m o s t u n i t y . The b l a z i n g d e p t h and a s p e c t r a t i o c u r v e s f o r d u a l b l a z i n g a r e r e p r i n t e d f rom J u l l , H e a t h and Ebbeson [ 1 9 ] . These c u r v e s c o v e r o n l y the a n g l e s o f i n c i d e n c e f r o m 1 9 . 5 ° t o 5 9 . 4 ° . A method o f f i n d i n g t he d u a l b l a z i n g c u r v e s f o r g r oove d e p t h and a s p e c t r a t i o f o r a n g l e s o f i n -c i d e n c e r i g h t up t o g r a z i n g i n c i d e n c e i s d i s c u s s e d . I n c l u d e d a l s o i n c h a p t e r 3 i s an a n a l y s i s o f t h e b l a z i n g d e p t h c u r v e s w h i c h d i s c u s s e s ways t o choose an o p e r a t i n g p o i n t w h i c h c o r -r e spond s t o t h e g roove d i m e n s i o n s o f a s u r f a c e w i t h some d e s i r a b l e p r o p -e r t y . T h i s p r o p e r t y may be t h e i n s e n s i t i v i t y o f t he r e f l e c t e d f i e l d s t o 11. s l i g h t changes i n t he a n g l e o f i n c i d e n c e o r f r e q u e n c y o f t h e i n c i d e n t wave , ( b roadband b e h a v i o r ) . I t may a l s o be a change i n one o f t h e g r oove d i m e n -s i o n s s u c h as t h e g r oove w i d t h o r d e p t h t o w h i c h t h e r e f l e c t e d wave must be i n s e n s i t i v e . C o n v e r s e l y , t h e a n a l y s i s c an a l s o be u sed t o h e l p p r e d i c t t he b e h a v i o r o f t h e r e f l e c t e d f i e l d f o r a s u r f a c e whose d i m e n s i o n s have been cho sen f r o m the b l a z i n g d e p t h c u r v e s . Cu r ve s a r e p r e s e n t e d f o r s u r -f a c e s w h i c h w o u l d e x h i b i t t h e s e d e s i r e d p r o p e r t i e s . I n c h a p t e r 4 e x p e r i m e n t a l r e s u l t s a r e p r e s e n t e d f o r a s e r i e s o f g r a t i n g s whose d i m e n s i o n s c o v e r a w i d e r ange o f p o s s i b l e o p e r a t i n g p o i n t s on t h e b l a z i n g d e p t h c u r v e s o f c h a p t e r 3 . The s u r f a c e s have been cho sen t o e x h i b i t many o f t h e o p e r a t i n g p o i n t p r o p e r t i e s l i s t e d above and d i s -c u s s e d i n c h a p t e r 3 . The d a t a v e r i f y t h e n u m e r i c a l p r e d i c t i o n s and show t he i m p o r t a n c e o f s o u r c e s o f e r r o r s l i k e f i n i t e s i z e o f t h e s u r f a c e and n o n - p l a n e wave i l l u m i n a t i o n . I n p a r t i c u l a r t h e e x p e r i m e n t a l d a t a f o r s u r f a c e s w h i c h a r e d u a l l y b l a z e d v e r i f y t h e n u m e r i c a l p r e d i c t i o n t h a t i t i s p o s s i b l e t o d e s i g n a n o n - r e f l e c t i n g c o n d u c t i n g s u r f a c e f o r a r b i t r a r i l y p o l a r i z e d wave s . I n c o n c l u s i o n i t i s n o t e d how t h e o b j e c t i v e s a r e met and what advan tage s a g r a t i n g w i t h a r e c t a n g u l a r g r oove p r o f i l e ha s o v e r a g r a t i n g w i t h a n o t h e r o f t h e p r o f i l e s m e n t i o n e d e a r l i e r . 12. 2. ANALYSIS 2.1 F o r m u l a t i o n o f t h e P r o b l e m The p r o f i l e o f t h e r e c t a n g u l a r g r oove g r a t i n g w h i c h s h a l l be a n a l y z e d i s shown i n F i g . 1.1. I t i s assumed t h e g r a t i n g i s o f i n f i n i t e e x t e n t i n t h e a and a d i r e c t i o n s and t h e m a t e r i a l i s p e r f e c t l y c o n d u c -y x t i n g . The g r oove s a r e o f w i d t h a , o f d e p t h h , and o f p e r i o d d . The p r o b l e m can be s t a t e d i n t h e f o l l o w i n g t e r m s ; a p l a n e TEM wave i s i n c i d e n t on t h e g r a t i n g n o r m a l t o t h e g r o o v e edges and a t an a n g l e 0_^  f r o m t h e s u r f a c e n o r m a l as shown i n F i g . 1.1. A s c a t t e r e d f i e l d r e s u l t s . T h i s s c a t t e r e d f i e l d depends on t h e d i m e n s i o n s o f t h e g r o o v e s , ( a , h and d) and on t h e a n g l e o f i n c i d e n c e 0^, t h e w a v e l e n g t h A, and t h e • p o l a r i z a t i o n o f t h e i n c i d e n t TEM wave . The e l e c t r o m a g n e t i c bounda r y v a l u e p r o b l e m can be s o l v e d f o r t h i s s c a t t e r e d f i e l d . I t i s e a s i e r t o h a n d l e t h e d i f f e r e n t bounda r y c o n d i t i o n s on t h e s u r f a c e by t r e a t i n g t h e c a s e s o f TM and TE p o l a r i z e d i n c i d e n t waves s e p a r a t e l y . The TM c a s e w i l l be d e a l t w i t h f i r s t and i n more d e p t h t h a n t h e TE case w h i c h i s g i v e n l a t e r . 2.2 S o l u t i o n f o r TM P o l a r i z a t i o n The i n c i d e n t wave i s TM p o l a r i z e d , i . e . t he m a g n e t i c f i e l d H, as shown i n F i g . 2.1, i s p a r a l l e l t o t h e g r o o v e s . The i n c i d e n t m a g n e t i c f i e l d i s (2.1) iP-r \ u i / x - A - j k ( x s i n 0 - z co s B ) H ( x , y , z ) = H y ( x , z ) a y = A ± d e J i i a y where a ^ i s a u n i t v e c t o r and d 2 i s s e p a r a t e f r o m A^ f o r l a t e r a l g e b r a i c c o n v e n i e n c e . The t i m e dependence o f a l l waves i s u n d e r s t o o d t o be e ^ U t . The e l e c t r i c f i e l d i s E ( x , y , z ) = E ( x , z ) a + E ( x , z ) a , (2.2) 1 3 . F i g . 2 .1 TM P o l a r i z e d Wave I n c i d e n t on a S u r f a c e w i t h a R e c t a n g u l a r G r oove P r o f i l e where a" and & a r e u n i t v e c t o r s i n t h e x and z d i r e c t i o n s . From M a x w e l l ' x z e q u a t i o n s , t he e l e c t r i c f i e l d a m p l i t u d e s a r e r e l a t e d t o t h e m a g n e t i c f i e l d by JZ„ 3H„ ( 2 . 3 ) j Z 3H V x - 2 > • - r r . < x - z ) E z ( x , z ) - I Z 3H k 3 z ( x , z ) ( 2 . 4 ) where Z = 120iTfi i s t h e c h a r a c t e r i s t i c impedance , o The f i e l d may be d i v i d e d i n t o two p a r t s : t h e f i e l d above t h e g r oove s o r t h e f r e e space r e g i o n ; and t h e f i e l d i n t h e g r o o v e s , t h e w a v e -g u i d e r e g i o n . The t o t a l f i e l d must be c o n t i n u o u s and so t h e s e two p a r t s must s a t i s f y t h e b o u n d a r y c o n d i t i o n s imposed on them a t z = 0 . T h i s d i -v i s i o n o f t he s p a c e i n t o two r e g i o n s a l l o w s t h e f i e l d i n each r e g i o n t o be w r i t t e n i n a f o r m w h i c h s a t i s f i e s t h e b o u n d a r y c o n d i t i o n s t h e r e . 2 .2 .1 Bounda ry C o n d i t i o n s Due t o p e r i o d i c i t y o f t h e s t r u c t u r e , t h e f i e l d s r e s u l t i n g f r o m a d j a c e n t c e l l s w i l l d i f f e r o n l y by t h e pha se f a c t o r e J ^ s i n ^ d ^ x h e n , o n l y t h e f i e l d f r o m a s i n g l e c e l l , l i k e t h e one i n F i g . 2 . 2 , need be c o n s i d e r e d . The b o u n d a r y c o n d i t i o n s t h a t have t o be met a r e t h a t on t h e c o n d u c t i n g s u r f a c e 14. E = 0, x Ry(x,z) -> 0 as | z| -»- » , and that E and H be continuous across the z = 0 interface, x y (2.5) (2.6) F i g . 2.2 Single C e l l with a Rectangular Groove P r o f i l e . 2.2.2 General Solution for the Scattered Fie ld (V 2 + k 2 ) H y (x,z) = 0 The expression for the f i e ld both above and below the z = 0 interface must satisfy the Helmholtz equation (2.7) subject to the boundary conditions of section 2.2.1. The scattered f i e ld may be written as an in f in i t e sum of plane waves. In the free space region, the plane waves travel with angle 0 , real or imaginary, given by (1.2). A form of the scattered f i e ld which satisf ies the Helmholtz equation (2.7) i s H S (x,z) = T A e y L m -jkzcos 0m m=-°° m (2.8) where A^ is the unknown amplitude coefficient of the plane wave, -iksxn 0 x e = d e m m (2.8a) and cos 0 = (1 - s in 0 )"2 m m sin 0 < 1 m 2 k - j (s in 0 -1) ' J m s in 0 > 1 1 m1 (2.8b) 15 . T h i s r o o t f o r cos 0^ e n s u r e s t h a t t h e e v a n e s c e n t modes decay e x p o n e n t i a l l y as z -> 0 0 . The d t e r m i n s i m p l i f i e s t h e a l g e b r a l a t e r . The t o t a l m a g n e t i c f i e l d i s j u s t t he sum o f t h i s s c a t t e r e d f i e l d and t he i n c i d e n t f i e l d o f ( 2 . 1 ) H ( x , z ) - A . e e J k z C 0 S 9 i + f A e e " ^ 2 0 0 3 9 m ' ( 2 " 9 ) y x o u m m J n==°° The o t h e r t r a n s v e r s e f i e l d i s o b t a i n e d f r o m t h e t o t a l m a g n e t i c f i e l d by ( 2 . 3 ) ; 00 T-. / \ A t , n j k z c o s 0 J , v A m - j k z c o s 0~ ,„ E ( x , z ) = - A . Z cos 0 . e J xe + > A Z cos 0 e J m e . ( 2 .10 ) x ' x o x o ^ m o m m m=-°o I n t h e w a v e g u i d e r e g i o n t h e t r a n s v e r s e f i e l d s a r e s t a n d i n g waves o f t he f o r m °° cos (k (z+h)) H ( x , z ) = 7 C e 2 _ , ( 2 .11 ) y L n n n c o s k h J . n=0 n » Z s i n (k (z+h)) E ( x , z ) = I C k e 2 - r , ( 2 .12 ) x v ' ' L n n i k n n c o s k h n=0 J n w h e r e , C n i s t h e unknown w a v e g u i d e a m p l i t u d e c o e f f i c i e n t , h 2 2 k = (k - (rnr/a) ) In ir/al < k n 1 1 — h = - j ( ( m r / a ) 2 - k 2 ) |nir/a| > k , ( 2 . 12a ) e = / e / a c o s ( n i T / a ( x + a / 2 ) ) , (2 .12b) n and e = 1 n = 0 e = 2 n = 1, 2 , 3 , . . . ( 2 . 1 2 c ) The n e g a t i v e i m a g i n a r y s q u a r e r o o t was cho sen f o r so t h a t t h e n o n - p r o p -a g a t i n g modes decay i n t he g r o o v e s . A l s o t h e bounda r y c o n d i t i o n (2 .5 ) i s t a k e n i n t o a c c o u n t a t z = - h . The t e r m /e/a i s i n c l u d e d i n t h e e terms n to n o r m a l i z e them so t h a t f u t u r e a l g e b r a i s s i m p l i f i e d . 16. 2 . 2 . 3 The A m p l i t u d e C o e f f i c i e n t s S e c t i o n 2 .2 .2 g i v e s t h e g e n e r a l f o r m o f t he t a n g e n t i a l f i e l d s i n each o f t h e r e g i o n s above and b e l o w t h e z = 0 i n t e r f a c e . A t z = 0 t h e t a n g e n t i a l m a g n e t i c f i e l d s i n b o t h r e g i o n s must match i n t h e wavegu i de H y ( x , 0 + ) = H ( x , 0 " ) |x| < a/2 , A . e + I A e = Y C § |x| < a/2 . ( 2 .13 ) 1 0 ^ m m ^ n n 1 1 m=-<» n=0 The t a n g e n t i a l e l e c t r i c f i e l d i n t h e f r e e s p a c e r e g i o n must match b o t h t h e wavegu ide t r a n s v e r s e e l e c t r i c f i e l d and a l s o t h e b o u n d a r y c o n d i t i o n ( 2 . 5 ) a l o n g t h e z = 0 i n t e r f a c e o v e r a f u l l p e r i o d b u t o u t s i d e t h e wavegu i de E x ( x , 0 + ) = E ( x , 0~ ) |x| < a/2 = 0 a/2 < |x| < d/2 , C O C O ] ^ - A . e cos 6. + y A cos Qme = J C i r t a n ( k h ) e Ixl < a/2 x o x Jl m m m ^ n j k n n 1 ' m=—oo n=0 = 0 a/2 < |x| < d/2. ( 2 .14 ) S i n c e {e^} fo rms an o r t h o g o n a l b a s i s o v e r t h e r ange - d/2 < x < d/2 , t h e n an i n n e r p r o d u c t , d e f i n e d as d/2 <f,g> = / f - g * d x , ( 2 .15 ) - d/2 can be a p p l i e d t o b o t h s i d e s o f ( 2 .14 ) t o o b t a i n 00 y C k t a n ( k h ) <£ ,e > = A cos 0 - 2A . co s 0 . 6 ° . ( 2 .16 ) LA n n n n m m m x x m n=0 ,-H j k s i n 0 x , „ / „+«. a/2 ? |x| < d/2, ( 2 .15 ) r e d u c e s t o Here 6 ° i s t h e K r o n e c k e r d e l t a , e * = d ^ e J S l n m X , and s i n c e E ( x , 0 ) = 0 , m m x a/2 <f,g> = / f - g * d x . ( 2 .17 ) - a / 2 17. A has been redefined for convenience as A = A. + A c , where A - is o o 1 ref ref the amplitude coefficient of the specularly reflectedramode, and was the former A of (2.13) and (2.14). Since {e } forms an orthonormal basis o n only over the range -a/2 < x < a/2, the inner product is defined by (2.17) and the expression for the magnetic f ields at z = 0 reduces to I A <e.,fi > = C , (2-18) u m m n « m=-and e* = e n n When one evaluates the expression for the inner product of both bases, given by (2.17), one obtains (2.19a) <e ,e > = /e/ad sin(ksin 6 a/2) 2ksin 6 m m n m (ksin 6 ) 2 - (mr/a) 2 m n = 0,2,4,6, = j l / / a d cos (ksin 0 a/2) 2ksin 6 (2.19b) m m (ksin 6 ) 2 - (mr/a) 2 m n = 1,3,5, . . . while <e ,e > = <§ ,e > m n n m n = 0,2,4,6, . . . (2.20a) = - -<e ,e > n m n = 1,3,5, . . . 2 2 When (ksin 6 ) = (nir/a) the inner product reduces to m (2.20b) <£ ,e > = J^/ea/d ( - j ) n n m i f ksin 6 = mr/a, m = V e l T d (j) i f ksin 6 = -mr/a, m and <e , § > = Jg/ea/d ( j ) n m n i f ksin 0 = nTr/a, m i f ksin 0 = -mr/a. m 18. Now (2.18), when substituted into (2.16), gives the doubly i n -f in i te matrix'equation °° , °° k tan(k h) I A , ( 6 m ; - I -4r S _ < e , £ > <g , e > , ^ m m u n nkcos 6 m n n m m '=-00 n=0 J m = 2 A . 6 ° . (2.21) 1 m However, solving (2.16) and (2.18) for the waveguide coefficients f i r s t , results in the need to invert a much smaller matrix. The saving in com-putation time due to inverting this smaller matrix, more than outweighs the increase in time due to reapplication of (2.16) to obtain the free space amplitude coefficients. If (2.16) is solved for A and this is substituted in (2.18), m after rearranging the terms one obtains 00 k , 0 0 1 (C:.,tan(k ,h))' {-£- I <& • ,e > <e ,e > - 6 n , cot(k ,h)} n'=0 n n J k m=-°o n m m n n n = -2 A. <e , § > . (2.22) i o n Equation (2.22) is a matrix equation of the form ( [ c . ] [b ] - [d , ]) [x ,] = [f ] (2.23) n m mn n n n n <§ , ,e > k , i _ i - n m n where b , mn ikcos 0 m c = <e > nm m n d , = 6 n ' cot(k ,h) nn n n x , = C , cot(k ,h) n n n 1 9 . and f , = - 2 A. <e ,6 > n i o n T h i s e q u a t i o n has t he a d v a n t a g e o f i s o l a t i n g a l l o c c u r e n c e s o f g r o o v e d e p t h h i n t h e d i a g o n a l m a t r i x t ^ i ^ * A f t e r s o l v i n g f o r x ^ one c an s u b s t i t u t e t h i s d i r e c t l y i n t o ( 2 . 16 ) t o f i n d t h e f r e e s p a c e c o e f f i c i e n t s . An a t t e m p t t o e v a l u a t e t he c o e f f i c i e n t s o f t h e p r o d u c t m a t r i x by a c o n t o u r i n t e g r a t i o n t e c h n i q u e o u t l i n e d i n C o l l i n [14 p .582 ] d i d n o t s u c c e e d . The t e rm cos 9 i n b , p r o v i d e s a b r a n c h c u t i n t e g r a l , w h i c h c o u l d n o t be m mn e v a l u a t e d . 2.3 S o l u t i o n f o r TE P o l a r i z a t i o n I f t h e i n c i d e n t wave i s a TE p o l a r i z e d wave t h e n t h e e l e c t r i c f i e l d , E, i s p a r a l l e l t o t h e g r oove s as shown i n F i g . 2 . 3 . F i g . 2 .3 TE P o l a r i z e d Wave I n c i d e n t on a S u r f a c e w i t h a R e c t a n g u l a r G r oove P r o f i l e . The i n c i d e n t e l e c t r i c f i e l d i s g i v e n by — i , N „ f s~ „ -h - j k ( z c o s 6. - x s i n 0 . K E ( x , y , z ) = E ( x , z ) a = .B .d e i l a y y i y and t h e m a g n e t i c f i e l d i s ( 2 . 24 ) H ( x , y , z ) = H x ( x , z ) a x + H z ( x , z ) a z . M a x w e l l ' s e q u a t i o n s r e l a t e t h e m a g n e t i c f i e l d t o t h e e l e c t r i c f i e l d by H ( x , z ) 2v . 9E ( x , z ) k z 3z ' ( 2 . 25 ) 20. . 9E and H ( x , z ) = ^ ( x , z ) . (2.26) z k z ox o As f o r t h e s c a t t e r e d H f i e l d o f t h e TM p o l a r i z e d c a s e , t h e s c a t -t e r e d e l e c t r i c f i e l d i n t h e f r e e s pace r e g i o n i s 00 „ s , . V TJ a z - j k z c o s 0 / 0 „ - . s E ( x , z ) = ) B d e m e . (2.27) y ^ m m where e , s i n 0 and cos 0 a r e t h e same as b e f o r e . The t o t a l e l e c t r i c m m m f i e l d above t h e g r oove s i s t h e sum o f t h e i n c i d e n t and s c a t t e r e d f i e l d s 00 _ , s „ j k c o s 0 . z . r D - j k z c o s 0 / 0 O Q . E ( x , z ) = B . e J x e + ) B e J m e . (2.28) y ' x o ^ m m J m=-°° The t r a n s v e r s e m a g n e t i c f i e l d , where z > 0 , o b t a i n e d f r o m (2.25) i s B. COS 0 . . . „ oo B cos 0 M n „ , . x x i k c o s 0 . z v m m - i k c o s 0 H ( x , z ) = e x e - > . e m y ' z o L z m=-°° ° (2.29) The t o t a l f i e l d s i n t h e wavegu i de s a r e oo s i n ( k (z+h)) E ( x , z ) = £ D e " . . , (2.30) y -t n n s x n ( k h) ' J n = l n oo j D cos (k (z+h)) and H ( x , z ) = 7 , e k / . — T - T — (2.31) x ' L . k z n n s x n ( k h) n = l o n — where k_n i s g i v e n by (2.12a). S i n c e t h e t r a n s v e r s e e l e c t r i c f i e l d , E^, i s z e r o a t t he s u r f a c e o f a p e r f e c t c o n d u c t o r , we r e d e f i n e e ^ as e_ = /27I sin(nTr/a(x+a/2)) , (2.32) n t h i s e n s u r e s t h a t E = 0 a t x = ±a/2. y M a t c h i n g t h e t a n g e n t i a l e l e c t r i c and m a g n e t i c f i e l d s a t z = 0 , and a l s o t a k i n g i n t o a c c o u n t E^ = 0 o u t s i d e t h e g r oove g i v e s m m T n n J B e = . 7 D fi , (tlxI < a/2 , n= l 0 , a/2 < |x| < d/2 , ( 2 . 33 ) 2 1 . and 0 0 0 0 j D 2B . co s 6 .e - 7 B cos 6 e = • 7 - p ^ § k t a n ( k h) ( 2 .34 ) x x o ^ m m m ^ k n n n m=-°° n = l | x| < a/2 As w i t h (2 .16 ) and ( 2 . 1 8 ) , B has been r e d e f i n e d as B = B £ + B . , where v ' o o r e f x B j. was t h e f o r m e r B o f ( 2 . 2 8 ) . r e f o S i n c e {e } and {e } a r e o r t h o n o r m a l b a s e s , one may t a k e t h e i n -n • m n e r p r o d u c t o f b o t h s i d e s o f ( 2 .33 ) w i t h e and (2 .34 ) w i t h e . As b e f o r e m n t he i n n e r p r o d u c t f o r b o t h b a s e s r e d u c e s t o ( 2 . 1 7 ) . T h i s y i e l d s 00 B = I <6 ,e > D , ( 2 .35 ) m ^ n m n> n = l oo and y j k c o s 6 B• <e , § > - 2B . co s 6. <e , § > L m m m n x x o n m=-°o = D k t a n ( k h) . ( 2 .36 ) n n n The i n n e r p r o d u c t when e v a l u a t e d g i v e s < § ,e > = ^2/ad 2 n T r / a — C o s ( k s i n . . 6 a / 2 ) , n = 1 , 3 , 5 , ( k s m e m ) - ( — ) = j/2~7ad j s i n ( k s i n 0 a/2) ( k s i n 9 ) - ( m r / a ) Z m m n = 2 , 4 , 6 , and <e ,g > = <e ,e >,\ n = 1 5 3 , 5 , . . . m n n m = - < £ ,e >, n = 2 , 4 , 6 , . . . n m 2 2 When ( k s i n 0 ) = (mr/a) t h e i n n e r p r o d u c t r e d u c e s t o m n~ 1 <£ ,e > = /a/2d ( j ) , i f k s i n 0 = - m r / a , n m ' m = - / a/2d ( - j ) n + 1 , i f k s i n = mr/a , 22 . w h i l e <e ,e > = - / a / 2 d ( - j ) n + 1 , i f k s i n 0 = -n iT/a, m n m = /a/2d ( j ) n ^ , i f k s i n 0 M = mr/a. I t i s p o s s i b l e i n p r i n c i p l e t o s o l v e f o r t h e f r e e s pace a m p l i -t ude s d i r e c t l y b u t due t o t h e a v a i l a b i l i t y o f a q u i c k e r method, o f s o l v i n g t h e s e e q u a t i o n s , t h i s was n o t done . I n s t e a d , t h e wavegu i de a m p l i t u d e s were e v a l u a t e d by s u b s t i t u t i n g ( 2 .35 ) i n t o (2 .36 ) t o g i v e 7 D , ( j k I cos 0 <i\ , ,e > <e , § , L , n L m n m m n n =1 m=-°° > - 6 N ' k t a n ( k h ) ) n n n = 2B. <e ,£ > cos 0 . ( 2 .37 ) l o n l T h i s can be r e p r e s e n t e d i n m a t r i x f o r m as n i " J mn nn n n where t h e s u p e r s c r i p t e d i s t i n g u i s h e s t h e s e m a t r i c e s f r o m t h e i r m a g n e t i c f i e l d c o u n t e r p a r t s . The m a t r i c e s a r e b e , = <e , § > n m m n c = <£ , , e > i k c o s 0 mn n m m d e , = <5n k t a n ( k h) nn n n n x , = D , n n f = 2B. <e ,e > cos 0 . . n i o n l E q u a t i o n ( 2 .37 ) a l s o has t h e o n l y t e rms c o n t a i n i n g t h e g r oove d e p t h h i s o l a t e d i n t he d i a g o n a l m a t r i x [d t ] . T h i s means t h a t changes i n t he g roove d e p t h c an be a c c o u n t e d f o r w i t h o u t r e c o m p u t i n g t h e p r o d u c t m a t r i x . T h i s p o i n t was u s ed t o a c h i e v e e f f i c i e n c y o f c c o m p u t a t i o n as e x -p l a i n e d i n t h e n e x t c h a p t e r . 23 . 2.4 S o l u t i o n U s i n g t he Optimum P e r i o d S i n c e , as m e n t i o n e d i n c h a p t e r 1, one o f t h e o b j e c t s o f t h i s t h e s i s was t o s t u d y t he b l a z i n g c a p a b i l i t i e s o f t h e r e c t a n g u l a r g r oove p r o f i l e , t h e p e r i o d d s h o u l d be d i c t a t e d by t h e B r a g g c o n d i t i o n , ( 1 . 4 ) . When t he p e r i o d i s g i v e n by ( 1 .5 ) and s i n 6^ r e d u c e s t o ( 1 . 6 ) t h e n t h e terms i n t he p r o d u c t m a t r i c e s o f ( 2 .23 ) become r e l a t e d as c = c , when n = 0 , 2 , 4 , 6 , . . . nm - ( n + l ) m c = - c , , 1 N when n = 1 , 3 , 5 , . . . nm - ( n + l ) m and b , = b , , . when n ' = 0 , 2 , 4 , 6 , . . . mn m— (,n +X) b . = - b * ? i i \ when n ' = 1 , 3 , 5 , . . . mn m-(n +1) T h i s r e s u l t s i n t h e p r o d u c t m a t r i x r e d u c i n g t o 00 a , = 2 T c b i f b o t h n ' and n a r e even o r nn u ~ nm mn . ,. , , , , , m=0 i f b o t h n and n a r e odd = 0 , o t h e r w i s e . T h i s a l l o w s t h e d e c o m p o s i t i o n o f ( 2 . 23 ) i n t o two m a t r i x e q u a t i o n s , one w h i c h g o ve rn s t h e odd wavegu i de modes and one w h i c h g o ve rn s t h e even w a v e -g u i d e modes. The m a t r i c e s i n e ach o f t h e s e e q u a t i o n s i s j u s t one q u a r t e r t h e s i z e o f t h e m a t r i x i n ( 2 .23 ) and i t i s now o n l y n e c e s s a r y t o compute t he i n n e r p r o d u c t f o r p o s i t i v e v a l u e s o f m. S i m i l a r r e s u l t s f o r t h e TE c a s e h o l d when t he p e r i o d d i s g i v e n by ( 1 . 5 ) . These s i m p l i f i c a t i o n s w i l l a l l o w much f a s t e r n u m e r i c a l c o m p u t a t i o n s . 24. 3. NUMERICAL RESULTS 3.1 I n t r o d u c t i o n The o b j e c t i v e h e r e i s t o e s t a b l i s h t h e g r oove d i m e n s i o n s r e q u i r e d f o r minimum s p e c u l a r r e f l e c t i o n . S i n c e , i n most a p p l i c a t i o n s , t h e a n g l e o f i n c i d e n c e 6^ and w a v e l e n g t h A o f t h e i n c i d e n t TEM wave a r e known, t h e p e r i o d d i s g i v e n by ( 1 . 5 ) . I t i s t h e n n e c e s s a r y t o choose a g r o o v e w i d t h a and t h e n a d j u s t t he g roove d e p t h h t o a c h i e v e no r e f l e c t e d power . E q u a -t i o n s ( 2 . 22 ) and (2 .37) a r e w e l l s u i t e d t o t h i s a p p r o a c h s i n c e t h e g r o o v e d e p t h h may be changed w i t h o u t t he need t o recompute t h e p r o d u c t m a t r i c e s [c M b ,] and [b® J [ c f l . nm mn nm mn S i n c e t he r e c a n g u l a r g r o o v e p r o f i l e has t h r e e g r o o v e d i m e n s i o n s , t h e r e e x i s t s a c h o i c e f o r 1 9 . 5 ° < 0^ < 9 0 ° , when d e c i d i n g on t h e d i m e n s i o n o f a b l a z e d s u r f a c e , w h i c h t he comb and s i n u s o i d a l p r o f i l e s do n o t h a v e . T h i s c h o i c e i s somewhat l i m i t e d when d e s i g n i n g a b l a z e d s u r f a c e f o r an i n -c i d e n t TE p o l a r i z e d wave , b u t l e s s r e s t r i c t i o n s e x i s t when d i s c u s s i n g t h e TM p o l a r i z e d c a s e . These l i m i t a t i o n s w i l l become a p p a r e n t l a t e r . F o r a g i v e n 0^ and A, one can choose a l m o s t any a s p e c t r a t i o a/d be tween z e r o and one and f i n d a b l a z i n g d e p t h . ( P h y s i c a l l y t h e comb p r o -f i l e i s t h e l i m i t i n g c a se o f t h e r e c t a n g u l a r g r oove p r o f i l e as t h e a s p e c t r a t i o a p p r o a c h e s u n i t y . ) The d e m o n s t r a t i o n o f how t h e b e h a v i o r o f t h e r e -f l e c t e d f i e l d c o n v e r g e s t o t h e r e s u l t s e x p e c t e d f o r a comb p r o f i l e , when the a s p e c t r a t i o i s a l m o s t u n i t y , i s h a n d l e d f i r s t , a l o n g w i t h t h e r e s u l t s f o r a s u r f a c e whose a s p e c t r a t i o i s a l m o s t z e r o . Cu r ve s o f b l a z i n g d e p t h h / d v s . 0 . f o r v a r i o u s a s p e c t r a t i o s a i w i l l be g i v e n , a l o n g w i t h a d i s c u s s i o n o f how t h e s e b l a z i n g d e p t h c u r v e s change as t he a s p e c t r a t i o v a r i e s be tween z e r o and one . A l s o g i v e n w i l l be c u r v e s t h a t g i v e b l a z i n g d e p t h and a s p e c t r a t i o v s . 0^ f o r s u r f a c e s w h i c h 25 . a r e s i m u l t a n e o u s l y b l a z e d f o r b o t h TE and TM p o l a r i z e d i n c i d e n t wave s . I t i s t h e e x t r a deg ree o f f r eedom i n t h e c h o i c e o f g r o o v e d i m e n s i o n s t h a t g i v e s the r e c t a n g u l a r g r o o v e p r o f i l e t h e p r o p e r t y t h a t s i m u l t a n e o u s b l a z -i n g i s p o s s i b l e f o r most a n g l e s o f i n c i d e n c e , w h i l e b o t h t h e comb and s i n -u s o i d a l p r o f i l e s have o n l y one s u ch a n g l e a t w h i c h s i m u l t a n e o u s b l a z i n g o c c u r s . T h i s e x t r a d i m e n s i o n o f f e r s a c h o i c e when d e s i g n i n g a b l a z e d g r a t i n g w h i c h i s i l l u m i n a t e d by a wave i n c i d e n t a t an a n g l e 8_ .^ To u t i l i z e t h i s f r eedom o f c h o i c e , o b s e r v a t i o n s o f t h e b e h a v i o r o f t h e power r e f l e c t -ed by a b l a z e d s u r f a c e , when e i t h e r one o f 0^ o f X, o r one o f t h e g r oove d i m e n s i o n s a , h o r d i s v a r i e d , w i l l f o l l o w . The o p e r a t i n g p o i n t f o r a g i v e n b l a z e d s u r f a c e i s t h e p o i n t on t h e b l a z i n g d e p t h ( l u / d v s . 8. ) c u r v e D 1 t h a t c o r r e s p o n d s t o t he g r oove d i m e n s i o n s o f t h e s u r f a c e . These o b s e r v a -t i o n s s h a l l p r o d u c e g u i d e l i n e s t o h e l p i n t h e c h o i c e o f an o p e r a t i n g p o i n t (a/d and h/d) w h i c h c o r r e s p o n d s t o a b l a z e d s u r f a c e t h a t b e s t s u i t s t h e r e q u i r e d a p p l i c a t i o n . 3.2 N u m e r i c a l S o l u t i o n o f t he P r o b l e m I n s s o l v i n g f o r t h e power i n t h e r e f l e c t e d and b a c k s c a t t e r e d modes, ( 2 .22 ) and ( 2 .37 ) were s o l v e d t o f i n d t he TM o r TE wavegu i de mode a m p l i -t u d e s , w h i c h were u s e d , i n t u r n , a l o n g w i t h ( 2 .16 ) o r ( 2 . 3 5 ) , t o compute A T and A o r B - and B . The m a t r i c e s [c ] and [b 1 Iwere l i m i t e d i n - X o - 1 o nm nm s i z e t o N by M w h i l e [b ] and [c ] were M by N m a t r i c e s . mn mn The r e q u i r e d p r o d u c t m a t r i c e s were fo rmed and t he r e s p e c t i v e d i a g o n a l m a t r i c e s t h e n s u b t r a c t e d . T h i s N x N s y s t e m c o u l d t h e n be s p l i t as shown i n C h a p t e r 2 i n t o two N/2 x N/2 s y s t e m s , and t h e n s o l v e d t o o b t a i n t h e odd and even wavegu i de mode a m p l i t u d e c o e f f i c i e n t s . F o l l o w i n g t h e example o f H e s s e l , Schmoys and Tseng [ 2 ] , 50 f r e e space modes and 10 wavegu i de modes were cho sen f o r M and N r e s p e c t i v e l y 26. and and B \ we re cho sen t o be u n i t y . N u m e r i c a l c o m p u t a t i o n s u s i n g 150 f r e e s pace modes and 12 wavegu i de modes were made t o check t h a t t h e e r r o r i n t h e s i m u l t a n e o u s t r u n c a t i o n o f b o t h t h e sum i n t h e m a t r i x c o e f f i c i e n t s and t h e r a n k o f t h e m a t r i x i t s e l f was i n s i g n i f i c a n t . I n a l l c a s e s t e s t e d , t he d i f f e r e n c e be tween computed v a l u e s u s i n g t h e s m a l l e r number o f modes and t h o s e u s i n g t h e l a r g e r number o f modes was i n s i g n i f i c a n t . The more i n d i r e c t r o u t e o f s o l v i n g f o r t h e wavegu i de mode a m p l i -t ude s f i r s t i s much f a s t e r t h a n t he method o f s o l v i n g f o r t h e f r e e space mode a m p l i t u d e s d i r e c t l y . I n t he l a t t e r c a s e , one must s o l v e an M x M m a t r i x w h i l e t he f o r m e r method r e q u i r e s t h e e v a l u a t i o n o f two N/2 x N/2 m a t r i c e s . S i n c e t he computer t i m e r e q u i r e d t o s o l v e a J x J m a t r i x i s 3 p r o p o r t i o n a l t o J , t h i s means, f o r M = 50 and N = 10 , t h a t t h e f o r m e r method a l l o w s t he d e s i r e d unknowns t o be f o u n d 125 t i m e s f a s t e r , a l t h o u g h an e x t r a m u l t i p l i c a t i o n has t o be done t o o b t a i n A q and A _ ^ o r B q and B _ ^ . T h i s i n d i r e c t method a l s o a l l o w s t h e i s o l a t i o n o f t h e te rms i n v o l v i n g t h e g roove d e p t h i n t he d i a g o n a l m a t r i x , w h i c h i s v e r y a d v a n t a g e o u s . The g roove d e p t h i s t h e n changed u n t i l t h e r e l a t i v e r e f l e c t e d power i s r e d u c e d b e l o w some l e v e l ; i n t h i s c a s e , 10 ^ i s t a k e n as t h e l e v e l . T h i s g r o o v e d e p t h i s c a l l e d t h e b l a z i n g d e p t h , h^ . D 3.3 S p e c i a l R e c t a n g u l a r Groove P r o f i l e s The re a r e c e r t a i n r e c t a n g u l a r g r o o v e p r o f i l e s whose s c a t t e r e d f i e l d i s a l r e a d y known o r e x p e c t e d . F i g u r e 3.1 shows t h e p r o f i l e s o f r e c t a n g u l a r g r oove s u r f a c e s whose a s p e c t r a t i o s a r e n e a r t h e maximum o f u n i t y and t h e minimum o f z e r o r e s p e c t i v e l y . I t i s a p p a r e n t f r o m F i g . 3 . 1a t h a t as t h e a s p e c t r a t i o a p p r o a c h e s u n i t y , t he p r o f i l e becomes t h a t o f t h e comb g r a t i n g w h i c h i s shown i n F i g . 1 . 3 c , and as t he a s p e c t r a t i o goes t o z e r o , t he p r o f i l e becomes one o f a f l a t p l a t e w i t h i n f i n i t e l y t h i n g r o o v e s . 27 . a) F i g . 3.1 Two R e c t a n g u l a r G roove P r o f i l e s (a) a/d = 0.9 (b) a/d = 0.1 b) An e x a m i n a t i o n o f ( 2 .22 ) and (2 .37 ) shows t h a t when a/d = 1 o r a/d = 0 , one o b t a i n s A m = 0 , m 4 0 , ( 3 . 1 ) A _ = A . , r e f x B = 0 , m 4- 0 m B C = - B . . r e f x ( 3 . 2 ) T h i s shows, t h a t i n t h i s c a s e , t h e i n c i d e n t wave i s s i m p l y r e f l e c t e d by a f l a t p l a t e . The minus s i g n i n ( 3 . 2 ) a c c o u n t s f o r t h e 180° pha se s h i f t i n the e l e c t r i c f i e l d upon r e f l e c t i o n . However , the b e h a v i o r o f t h e comb g r a t i n g has been documented [ 8 -13 ] and i t i s n o t t he b e h a v i o r o f a f l a t p l a t e . F i g . 3.2 shows r e l a -t i v e power v s . h/d f o r a r e c t a n g u l a r g r o o v e s u r f a c e w i t h a/d = 0 .9999 and f o r a s u r f a c e w i t h a comb p r o f i l e f o r t h e same 6^ and TM p o l a r i z a t i o n . The d a t a f o r t h e comb p r o f i l e was o b t a i n e d f r o m Ebbe son [12] and i s r e p -r e s e n t e d by t h e dashed l i n e . T h i s r e c t a n g u l a r g r o o v e p r o f i l e i s a v e r y c l o s e a p p r o x i m a t i o n of t h e comb p r o f i l e a l t h o u g h t h e f i n s a r e n o t i n f i n -i t e l y t h i n l i k e t h e c o m b ' s . The o n l y p l a c e whe re t h e two c u r v e s d i f f e r s i g n i f i c a n t l y i s f o r s m a l l g r o o v e d e p t h s . Ebbe son m e n t i o n s i n h i s t h e s i s t h a t h i s r e s u l t s a r e n o t a c c u r a t e f o r f i n d e p t h s so s m a l l t h a t t h e n o n -p r o p a g a t i n g w a v e g u i d e modes i n t e r a c t w i t h t h e f r e e s pace modes. The d i f -f e r e n c e be tween t he two c u r v e s f o r l a r g e r f i n d e p t h s i s a t t r i b u t e d t o t h e d i f f e r e n c e i n t h e wavegu i de p r o p a g a t i o n c o n s t a n t s o f t h e two d i f f e r e n t 30 . p r o f i l e s . S i n c e t he f i n s o f t h e r e c t a n g u l a r g r oove p r o f i l e a r e n o t i n -f i n i t e l y t h i n , t h e wavegu i de p r o p a g a t i o n c o n s t a n t i s n o t q u i t e e q u a l t o t h a t i n t he comb ' s g r o o v e s . The phase d i f f e r e n c e f o r t h e s e two s u r f a c e s becomes g r e a t e r as t h e f i n d e p t h i n c r e a s e s and c au se s t h i s d i s c r e p a n c y be tween the two c u r v e s . Thus , as e x p e c t e d , t h e r e s u l t s f o r t h e r e c t a n g -u l a r g r oove p r o f i l e c onve r ge t o t h o s e f o r t h e comb p r o f i l e as t h e a s p e c t r a t i o app roache s u n i t y . What happens as t h e a s p e c t r a t i o goes t o z e r o ? The b e h a v i o r w o u l d be e x p e c t e d t o a p p r o a c h t h a t o f a f l a t p l a t e . R e l a t i v e power i n t h e s p e c u l a r l y r e f l e c t e d mode v s . h/d i s shown i n F i g . 3.3 f o r s u r f a c e s w i t h i n c r e a s i n g l y s m a l l a s p e c t r a t i o s . T h i s wave i s i n c i d e n t a t 6^ = 55° and i s TM p o l a r i z e d . As a/d 0 t he r e c t a n g u l a r g r oove s u r f a c e a c t s i n c r e a s -i n g l y l i k e a f l a t p l a t e , r e f l e c t i n g a l m o s t a l l t h e i n c i d e n t power e x c e p t o v e r an e v e r d e c r e a s i n g r ange o f h/d u n t i l t h e c u r v e f o r a/d = 0 .00001 has a v i r t u a l p o i n t d i s c o n t i n u i t y . The re i s z e r o r e f l e c t i o n o n l y a t t h e g roove d e p t h h = A/4 and a l m o s t t o t a l r e f l e c t i o n f o r a l l o t h e r g r oove d e p t h s . T h i s v a l u e o f b l a z i n g d e p t h , A M , i s t h e same f o r a l l 6^ and c o r r e s p o n d s t o an i n f i n i t e i n p u t impedance a t t h e s l o t . S i n c e t he r ange o f h/d o v e r w h i c h t h e r e f l e c t e d power has been g r e a t l y r e d u c e d i s v e r y s m a l l , b l a z e d s u r f a c e s w i t h v e r y n a r r o w g r oove s w o u l d be u s e f u l i f t h e a p p l i c a t i o n c a l l e d f o r a g r a t i n g w i t h a n a r r o w b a n d w i d t h . A p o i n t o f d i s c o n t i n u i t y i n t h e r e l a t i v e r e f l e c t e d power c u r v e s does n o t o c c u r f o r s u r f a c e s w i t h an a s p e c t r a t i o o f a l m o s t u n i t y . F i g . 3.4 g i v e s a q u a l i t a t i v e e x p l a n a t i o n o f t h i s u s i n g g e o m e t r i c o p t i c s . From 3 .4a i t i s a p p a r e n t t h a t t h e comb g r a t i n g has most o f t h e i n c i d e n t wave r e f l e c t e d o f f o f i t s f i n s o r t h e b o t t o m o f i t s g r o o v e s and t h i s p r o v i d e s s u b s t a n t i a l b a c k s c a t t e r e d power . The ca se f o r s m a l l a s p e c t r a t i o s , as 31. F i g . 3.4 G e o m e t r i c a l O p t i c s R e f l e c t i o n o f I n c i d e n t TEM Waves (a) a/d = 1 (b) a/d = 0 shown i n F i g . 3 . 4b , i s j u s t t h e o p p o s i t e f r o m a g e o m e t r i c a l o p t i c s p o i n t o f v i e w . He re a l m o s t none o f t he i n c i d e n t power i s i n c i d e n t on t h e g r o o v e s so v e r y l i t t l e power i s b a c k s c a t t e r e d . Becau se o f t h i s , v a r y i n g t h e g r o o v e d e p t h o f s u r f a c e s w i t h t h i n g r o o v e s has v e r y l i t t l e e f f e c t on t h e amount o f power r e f l e c t e d e x c e p t f o r g r oove d e p t h s v e r y n e a r a q u a r t e r w a v e l e n g t h . I n b o t h . F i g s . 3.2 and 3 . 3 , t h e r e l a t i v e r e f l e c t e d power becomes u n i t y as t h e g r o o v e d e p t h goes t o z e r o . T h i s i s c o n s i s t e n t w i t h t h e p h y -s i c a l i n t e r p r e t a t i o n o f t h e r e c t a n g u l a r g r o o v e s u r f a c e w i t h z e r o g r o o v e d e p t h as a f l a t p l a t e . 3.4 B l a z i n g Depth as a F u n c t i o n o f A n g l e : TE P o l a r i z a t i o n An e x a m i n a t i o n o f ( 2 . 3 7 ) , w h i c h g o v e r n s t h e w a v e g u i d e mode amp-l i t u d e s f o r a TE p o l a r i z e d wave , shows t h a t t h e w a v e g u i d e mode p r o p a -g a t e s o n l y i f k i s r e a l , o r n a/d ' > n s i n Q± ( 3 . 3 ) When s i n >a/d a l l w a v e g u i d e modes a r e e v a n e s c e n t and i t i s t h e r e f o r e i m p o s s i b l e t o b a c k s c a t t e r any o f t h e i n c i d e n t power f o r a n g l e s o f i n c i d e n c e l a r g e r t h a n t h i s c r i t i c a l a n g l e , 9 c = a r c s i n ( a / d ) . T h i s means t h a t o n l y b l a z i n g d e p t h c u r v e s f o r a s p e c t r a t i o s g r e a t e r t h a n s i n 6^ c an be u sed t o d e s i g n a b l a z e d s u r f a c e f o r TE p o l a r i z a t i o n . The c u r v e s o f b l a z i n g d e p t h ( h / d ) v s . a n g l e o f i n c i d e n c e ( 6 . ) , is I shown i n F i g . 3 . 5 , e x h i b i t t h e i m p o s s i b i l i t y o f a c h i e v i n g b l a z i n g when Fig . 3.5 TE Blazing Depth Curves vs. 9 . Curves for Most Aspect Ratios Have Already Appeared in [2,18]. Values for a/d = 1.0 from [9]. 33 . 6^ > 6 c by i n c r e a s i n g w i t h o u t bound as 9^ a p p r o a c h e s i t s c r i t i c a l v a l u e . Some o f t h e c u r v e s have a minimum v a l u e o f h_/d and t h e n a r i s i n g edge a t a s m a l l a n g l e s o f i n c i d e n c e . T h i s can be a t t r i b u t e d t o t h e t r a n s i t i o n f r o m one p r o p a g a t i n g wavegu i de mode t o two , w h i c h o c c u r s a t 9_^  = a r c s i n ( a / 2 d ) . T h i s o n l y o c c u r s a t an a n g l e g r e a t e r t h a n 1 9 . 5 ° i f a/d > 2/3. The c u r v e s f o r a s p e c t r a t i o s o f 0 . 5 , 0 . 6 67 , 0 . 7 5 , 0 .8 a n d . 0 . 9 were f i r s t p r e s e n t e d by H e s s e l , Schmoys and Tseng [2] and l a t e r by R o u m i -g u i e r e s , M a y s t r e and P e t i t [ 1 8 ] . They we re r e c a l c u l a t e d as a check and a r e r e p e a t e d h e r e f o r c o m p l e t e n e s s . The c u r v e s w i t h a/d = 0.6 and 0.99 were a d d e d , v a l u e s i n t h e c u r v e f o r t h e comb g r a t i n g were o b t a i n e d f r o m T seng , H e s s e l and O l i n e r [9]. The re i s a p p a r e n t l y a c o n t i n u i t y be tween t h e c u r v e s as t h e a s p e c t r a t i o v a r i e s , and t he c u r v e s c o n v e r g e t o t h e c u r v e f o r t h e comb g r a t i n g as a/d a p p r o a c h e s u n i t y . S i n c e , as d i s c u s s e d i n s e c t i o n 3 . 3 , i t i s i m -p o s s i b l e t o o b t a i n t he comb g r a t i n g b l a z i n g d e p t h s by s u b s t i t u t i n g a/d = 1.0 i n t o ( 2 . 3 7 ) , t h e v a l u e s f o r t h e comb p r o f i l e were n o t r e c a l c u l a t e d and so t h e c u r v e i s n o t as c o m p l e t e as t h e o t h e r s . A l l a v a i l a b l e v a l u e s u sed t o p l o t t h e s e c u r v e s a r e a v a i l a b l e i n A p p e n d i x A , T a b l e s I and II . 3.5 B l a z i n g Depth as a F u n c t i o n o f A n g l e : TM P o l a r i z a t i o n The e x i s t e n c e o f t h e n = 0 (TEM) wavegu i de mode i n ( 2 . 22 ) g u a r -a n t e e s the e x i s t e n c e o f a t l e a s t one p r o p a g a t i n g mode i n t h e g r o o v e s , r e -g a r d l e s s o f t he a n g l e o f i n c i d e n c e . Thus , a b l a z i n g d e p t h e x i s t s f o r a l l a n g l e s o f i n c i d e n c e , 9.^  > 1 9 . 5 ° , f o r any a s p e c t r a t i o . F i g s . 3.6 and 3.7 show t h i s , and a l s o , t he r a p i d i n c r e a s e i n g r oove d e p t h f o r 9^ j u s t b e l o w t h e c r i t i c a l a n g l e , 9 £ = a r s i n ( a / d ) , c u t o f f f o r t h e TM^Q mode. T h i s i s due t o t he more complex i n t e r f e r e n c e p a t t e r n r e s u l t i n g f r o m t h e e x i s t e n c e o f two p r o p a g a t i n g modes i n t h e w a v e g u i d e . 3 4 . Fig. 3.6 TM Primary Blazing Depth Curves vs. 6 i : Large Aspect Ratios. Most Curves Have Already Appeared in [2,18]. Values for a/d - 1.0 from [12]. 3 5 . rt , 60 ^•(degrees) F i g . 3.7 TM Primary Blazing Depth Curves vs. 0.: Small Aspect Ratios. Curves for a/d = 0.5 and a/d = 0.667 Save Already appeared [2,18]. 36. The c u r v e s f o r a/d = 0 . 5 , 0 . 7 5 , 0 .8 and 0.9 we re o b t a i n e d p r i o r t o t he p u b l i c a t i o n o f t he same d a t e by R o u m i g u i e r e s , M a y s t r e and P e t i t [ 1 8 ] . The c u r v e f o r a/d = 0.667 was r e c a l c u l a t e d a f t e r f i r s t a p p e a r i n g i n H e s s e l , Schmoys and Tseng [2] as a c h e c k , w h i l e t h e c u r v e f o r t h e comb g r a t i n g i s r e p r o d u c e d fromcchta i n E b b e s o n ' s M .A . S c . t h e s i s [12] w h i c h appea r s i n t h i s f o r m i n [ 1 6 ] . The c o r r e l a t i o n be tween t h e c u r v e s as t h e a s p e c t r a t i o v a r i e s i s e v i d e n t i n F i g . 3.6 and 3 .7 . I n F i g . 3.7 t h e c o n v e r g e n c e o f t h e b l a z i n g d e p t h c u r v e s , f o r t he r e c t a n g u l a r g r oove p r o f i l e t o t h a t o f t h e comb p r o -f i l e , i s s e e n . The d i f f e r e n c e be tween t h e c u r v e s f o r a/d = 0.9999 and a/d = 1.0 i s s l i g h t l y e x a g g e r a t e d f o r c l a r i t y . F i g . 3.8 g i v e s t h e b l a z i n g d e p t h c u r v e f o r v e r y s m a l l a s p e c t r a t i o s . The c u r v e f o r a/d = 0 .00001 was n o t p l o t t e d s i n c e t h e r e was v i r t u a l l y no d i f f e r e n c e be tween i t and t h e c u r v e f o r a/d = 0 . 0 0 1 . The b l a z i n g d e p t h c u r v e f o r t h e s e s m a l l a s p e c t s i n 6. r a t i o s i s c o n v e r g i n g t o ^ w h i c h c o r r e s p o n d s t o h/A = 1/4. A l l t h e c u r v e s a p p r o a c h h / d = 0 a t 6. = 9 0 ° , and t h e s t e e p n e s s o f t h e c u r v e s n e a r g r a z i n g f o r t he l a r g e a s p e c t r a t i o s i n d i c a t e s t h e t r a n s i t i o n r e g i o n f r o m one t o two p r o p a g a t i n g wavegu i de modes. S i n c e t he t e r m i n v o l v i n g h u sed i n ( 2 .22 ) and ( 2 .37 ) i s p e r i o d i c , t h e r e e x i s t s more t h a n one v a l u e o f h a t w h i c h a l l r e f l e c t e d power i s e l i m i n a t e d . The s m a l l e s t v a l u e s o f b l a z i n g d e p t h f o r an a s p e c t r a t i o a r e r e p r e s e n t e d i n F i g . 3.6 and 3.7 and t h e v a l u e s u sed i n t h e s e c u r v e s a r e i n A p p e n d i x A , T a b l e s I I I - V I I . The n e x t l a r g e r v a l u e s o f g r o o v e d e p t h w h i c h b l a z e t h e s u r f a c e f o r an i n c i d e n t TM p o l a r i z e d wave a r e i n A p p e n d i x A, T a b l e s V I I I and I X and p l o t t e d i n F i g . 3 .8 . These s e c o n d a r y c u r v e s o f b l a z i n g d e p t h v s . a n g l e o f i n c i d e n c e were m e n t i o n e d i n R o u m i g u i e r e s , M a y s t r e and P e t i t [ 1 7 ] . The c u r v e s show 0 6j (degrees) F i g . 3.8 TM Secondary Blazing Depth Curves vs 38 . a l l t h e u s u a l c h a r a c t e r i s t i c s o f t h e p r i m a r y b l a z i n g d e p t h c u r v e s o f F i g . 3.6 and 3 . 7 . They have t he same f a s t r i s i n g edge when < and t h e y a l l a ppea r t o a p p r o a c h l u / d = 1 a t g r a z i n g i n c i d e n c e . T h i s means, t h a t o a t n e a r g r a z i n g i n c i d e n c e , t h e d i f f e r e n c e i n b l a z i n g d e p t h f o r t h e p r i m -a r y and s e c o n d a r y c u r v e i s abou t A/2. The d e p t h s o f t he g r oove s r e p r e s e n t e d by t h e s e s e c o n d a r y c u r v e s a r e t oo l a r g e t o be o f u se i n many a p p l i c a t i o n s where t h e s h a l l o w e r d e p t h i s p r e f e r a b l e f o r these same a s p e c t r a t i o s . However , t h e s e s e c o n d a r y c u r v e s w o u l d be u s e f u l when d e s i g n i n g s u r f a c e s t h a t a r e b l a z e d f o r b o t h TM and TE p o l a r i z e d waves i n c i d e n t a t l a r g e a n g l e s . I t w o u l d be n e c e s -s a r y a t g r a z i n g and n e a r g r a z i n g i n c i d e n c e t o go t o e ven h i g h e r o r d e r b l a z i n g d e p t h c u r v e s t o o b t a i n t h e s e p o l a r i z a t i o n i n d e p e n d e n t b l a z e d s u r f a c e s . 3.6 S i m u l t a n e o u s B l a z i n g f o r TE and TM P o l a r i z e d Waves The i n t e r s e c t i o n p o i n t s g i v e n by s u p e r i m p o s i n g a TE c u r v e f r o m F i g . 3.5 on a TM c u r v e o f e i t h e r F i g s . 3 . 6 , 3 . 7 , o r 3.8 w i t h t h e same a s p e c t r a t i o g i v e s t he g r oove d i m e n s i o n s f o r a s u r f a c e w h i c h w o u l d be b l a z e d when i l l u m i n a t e d by a p l a n e wave o f a r b i t r a r y p o l a r i z a t i o n a t t h e g i v e n a n g l e . F i g . 3.9 shows how t h e s e d u a l b l a z i n g p o i n t s a r e o b t a i n e d and a l s o d e m o n s t r a t e s t h a t b o t h t h e g roove d e p t h and w i d t h f o r t h e s e p o i n t s a r e d e t e r m i n e d by t h e a n g l e o f i n c i d e n c e o f t h i s p l a n e wave. The p l o t o f t h e a s p e c t and d e p t h r a t i o s as a f u n c t i o n o f a n g l e f o r t h e s e d u a l b l a z i n g p o i n t s i s F i g . 3 . 10 . T h i s p l o t was f i r s t p r e s e n t -ed i n J u l l , Hea th and Ebbeson [ 1 8 ] . The v a l u e s were o b t a i n e d w i t h o u t t h e u se o f t he s e c o n d a r y c u r v e s o f F i g . 3.8 and a r e i n c l u d e d i n A p p e n d i x A , T a b l e X . The d u a l b l a z i n g p o i n t f o r t h e comb p r o f i l e o c c u r s a t 6 = 5 9 . 4 ° and i s t h e l a r g e s t a n g l e a t w h i c h d u a l b l a z i n g o c c u r s , u s i n g j u s t t h e 39. F i g . 3.9 Superposition of TM and TE Blazing Depth Curves for a/d = 0.99. Dual Blazing at 6, = 5 8 . 5 ° and 9. = 7 3 . 1 2 ° . 6; (degrees) F i g . 3.10 Aspect and Groove Depth Ratios for a Surface Blazed for Both TE and TM Incident Waves. 1 9 . 5 ° < 6 J < 5 9 . 4 ° F i g . 3.11 Aspect and Groove Depth Ratios for a Surface Blazed for Both TE and TM Incident Waves. 19.5° < 9, < 73.12° 42 . p r i m a r y c u r v e s o f F i g s . 3.6 and 3 . 7 . I t was o r i g i n a l l y t h o u g h t t h a t t h i s m i g h t be t h e l a r g e s t p o s s i b l e a n g l e a t w h i c h d u a l b l a z i n g c o u l d o c c u r u s i n g t h i s g r a t i n g p r o f i l e , b u t , as s een i n F i g . 3 . 9 , i t i s p o s s i b l e t o o b t a i n d u a l b l a z i n g p o i n t s up t o 6^ = 73° u s i n g t h e s e c o n d a r y c u r v e s o f F i g . 3 . 9 . F i g . 3.11 has t h e p l o t o f t h e s e a s p e c t and d e p t h r a t i o s as a f u n c t i o n o f t h e a n g l e s and t h e s e v a l u e s may be f o u n d i n A p p e n d i x A, T a b l e X I . The dep th s o f t h e g r oove s r e q u i r e d a r e much g r e a t e r t h a n t h o s e i n F i g . 3 . 10 . Hence t h e s e d u a l b l a z -i n g v a l u e s o f g r oove d i m e n s i o n s a r e u n l i k e l y t o be o f p r a c t i c a l u s e , e x -c e p t p e r h a p s , f o r t h o s e a n g l e s i n t h e r ange 5 9 . 4 ° < 6^ < 7 3 ° , w h i c h a r e beyond t he r ange o f t h e v a l u e s i n F i g . 3 . 10 . I f d u a l b l a z i n g p o i n t s f o r 0^ > 73° were n e e d e d , t h e n by u s i n g h i g h e r o r d e r TM b l a z i n g d e p t h c u r v e s , i t i s p o s s i b l e t o o b t a i n them, b u t t h e g r e a t d e p t h o f t h e g r o o v e s r e n d e r s t h i s t e c h n i q u e i m p r a c t i c a l . 3.7 O p e r a t i n g P o i n t B e h a v i o r The re a r e r e g i o n s o f t h e b l a z i n g d e p t h c u r v e s o f F i g . 3.5 t o 3.11 where t h e g r oove d i m e n s i o n s needed f o r b l a z i n g d i f f e r l i t t l e as a v a r i a b l e ( 8 ^ , X, d , a , o r h) i s changed s l i g h t l y . S i n c e t h e f r e e s p a c e a m p l i t u d e c o e f f i c i e n t s (A and B ) a r e c o n t i n u o u s f u n c t i o n s o f t h e s e v a r i a b l e s , a m m s u r f a c e whose o p e r a t i n g p o i n t i s i n a r e g i o n , where d i m e n s i o n s a r e r e l -a t i v e l y i n s e n s i t i v e t o change i n a v a r i a b l e , w o u l d have t h e amount o f b a c k s c a t t e r e d power change o n l y s l i g h t l y as t h a t v a r i a b l e was changed . Some o f t h e t y p e s o f b e h a v i o r needed f o r a g i v e n a p p l i c a t i o n m i g h t be m i n i m a l r e f l e c t i o n o v e r a r a t h e r l a r g e o r n a r r o w r ange o f 8^ o r X. A l s o , t h e more s e v e r e t h e d e s i g n t o l e r a n c e s on t h e d i m e n s i o n s o f t h e s u r f a c e , t h e h a r d e r i t i s t o make t he s u r f a c e , so i t w o u l d be h e l p f u l i f i t were . p o s s i b l e t o choose an o p e r a t i n g p o i n t w h i c h y i e l d e d t h e b e s t r e s u l t s w i t h 43 . t he l e a s t s e v e r e t o l e r a n c e s . A c h o i c e o f o p e r a t i n g p o i n t s i s p o s s i b l e , as p o i n t e d ou t i n s e c t i o n 3 . 1 , and i t s h a l l be shown how t o u se t h i s c h o i c e when d e s i g n i n g a s u r f a c e w h i c h has o p e r a t i n g p o i n t b e h a v i o r t h a t b e s t s u i t s t he r e q u i r e d a p p l i c a t i o n o r i s e a s i e s t t o c o n s t r u c t . 3 .7 .1 V a r i a t i o n s i n A n g l e The b l a z i n g d e p t h c u r v e s o f F i g s . 3.5 t o 3.8 a r e f u n c t i o n s o f 0^, a/d and h/d . S i n c e d i s g i v e n by t h e B r a g g c o n d i t i o n ( 1 . 5 ) , t h e n f o r a f i x e d p e r i o d , A i s a f u n c t i o n o f 6 . . The minimum v a l u e o f h / d on t he TE c u r v e s o f F i g . 3 . 5 , w h i c h was m e n t i o n e d i n s e c t i o n 3 . 5 , g i v e s a r e g i o n where t he g roove d i m e n s i o n s a r e r e l a t i v e l y i n s e n s i t i v e t o changes i n 9_^  o r A. As p o i n t e d o u t by H e s s e l , Schmoys, and Tseng [ 2 ] , t h e s e p o i n t s y i e l d s u r f a c e s w h i c h have b r oadband b e h a v i o r and a l s o w o u l d g r e a t l y r e -duce t h e r e f l e c t e d power o v e r a w i d e r ange o f 9_^. S i m i l a r l y , t h e l o c a l maxima and m in ima o f t h e TM c u r v e s i n F i g s . 3.6 t o 3 . 8 , w h i c h o c c u r i n t h e i r m u l t i v a l u e d r e g i o n , w o u l d a l s o be s u i t a b l e i f t h e a p p l i c a t i o n c a l l e d f o r r e d u c t i o n o f t h e r e f l e c t e d power o v e r a w i d e r ange o f 9^ o r A. F i g . 3.12 shows t h e change i n t h e TM r e f l e c t e d power as t h e a n g l e o f i n c i d e n c e i s v a r i e d f r o m 30° t o 4 0 ° . The o p e r a t i n g p o i n t was cho sen n e a r t he l o c a l maximum on t h e b l a z i n g d e p t h c u r v e u sed i n F i g . 5 o f H e s s e l e t a l . [ 2 ] , ( 30° < 6 < 4 0 ° , h/d = 0 . 8 , a/d = 0 . 667 , A/d = 2 s i n 9 ), b u t t h i s t i m e A/d i s h e l d c o n s t a n t . As i n H e s s e l e t a l . [ 2 ] , t h e r e f l e c t e d power was r e d u c e d o v e r a w i d e r ange o f a n g l e s . Over t h e a n g u l a r r ange o f 1 0 ° , more t h a n 99.8% o f t he i n c i d e n t power i s b a c k s c a t t e r e d . Hence , any p a r t o f one o f t h e b l a z i n g d e p t h c u r v e s o f F i g s . 3.5 t o 3.8 where t he t a n g e n t i s h o r i z o n t a l w o u l d be a s u i t a b l e o p e r a t i n g p o i n t t o g i v e m i n i m a l r e f l e c t e d power o v e r a w i d e r ange o f a n g l e s o r w a v e l e n g t h s . R o u m i g u i e r e s e t a l . [17] a l s o p o i n t e d o u t , t h a t i f t h e o p e r a t i n g p o i n t i s i n t h e m u l t i v a l u e d r e g i o n o f a d e s i g n c u r v e , i t i s p o s s i b l e t o o b t a i n 45. minimal reflection for more than one 8 . for the same values of a/d and 1 h/d. It should also be noted that the angular response is in general more sensitive to changes in 8^  or A when the operating point is on a design curve for an aspect ratio near the l imits of zero or one. As shown in F ig . 3.13, the blazed surface designed for 60° using a/d = 0.667, (the so l id curve), is a broader curve than the one for the comb prof i le designed for the same angle. The dashed curve values were obtained by Ebbeson and appear in [4,12,16]. This increasingly wider angular response comes to a l imit as the aspect ratio goes to zero. Near this aspect ratio any sl ight variation off of the optimum angle results in pract ical ly total reflection of the incident wave. Aspect ratios midway between these two l imits y ie ld surfaces which give better wide angle response. Also, surfaces designed to operate at near grazing incidence, have a large amount of backseatter over only a very narrow range of 8^. This is because the angular range over which the backscattered mode pro-pagates becomes very narrow. The cutoff angle for the m = -1 is 8 c Q = arcsin(A/d-l) (3.4) which, i f the period is given by (1.5), becomes almost the optimum angle for near grazing incidence. At 8^ = 9 0 ° , the angular range over which the backscattered mode propagates shrinks to zero. 3.Ui.2 Variations in Groove Depth Just as a point on the blazing depth vs. 8^ curves, which has a nearly horizontal tangent, is a good operating point when designing a surface that greatly reduces reflected power over a wide range of 8^ or A, i t i s expected that an operating point on a steeply r i s ing part of a h^/dvs . 0 . design curve would give a surface whose reflected power is 46. F i g . 3.13 TM Reflected Power vs. 6. 0. = 6 0 ° 1 iop from [12]. 47. r e l a t i v e l y i n s e n s i t i v e t o changes i n g r oove d e p t h . T h i s i s . d e m o n s t r a t e d i n F i g . 3.14 where c u r v e s o f r e l a t i v e r e f l e c t e d power v s . h/d a r e p l o t t e d a t 9 ± = 85° f o r a/d = 0.95 t o a/d = 0 . 9999 . I n F i g . 3.6 i t was s een t h a t t he c u r v e s become s t e e p e r a t 0^ = 85° as t h e a s p e c t r a t i o i s i n c r e a s e d f r o m 0.95 t o 0 . 9 9 5 . T h i s s h o u l d mean t h a t t h e s e n s i t i v i t y t o changes i n g r oove d e p t h d e c r e a s e s as t h e a s p e c t r a t i o i n c r e a s e s . T h i s i s a p p a r e n t i n F i g . 3 . 14 ; t h e r ange o f v a l u e s o f h/d o v e r w h i c h t h e r e f l e c t e d power i s g r e a t l y r e d u c e d i s s m a l l e s t f o r a/d = 0 . 9 5 . The c u r v e f o r a/d = 0.99 i n F i g . 3.6 i s much s t e e p e r a t 6 i = 85° a n d , i n F i g . 3 . 14 , t h i s i s shown as a g r e a t e r r ange o f h/d o v e r w h i c h t h e r e f l e c t e d power i s e f f e c t i v e l y e l i m i n a t e d . The b l a z i n g d e p t h c u r v e f o r a/d = 0 .995 i s e ven s t e e p e r and the r ange o f h/d i n F i g . 3.14 o v e r w h i c h t h e r e f l e c t e d power i s m i n i m i z e d i s s t i l l l a r g e r . I n F i g . 3.6 t h e c u r v e f o r a/d = 0.9999 i s a l m o s t p e r -p e n d i c u l a r t o t h e 0^ = 85° l i n e so a s m a l l r ange o f h/d w h i c h e l i m i n a t e s t he r e f l e c t e d power m i g h t be e x p e c t e d , b u t f r o m F i g . 3 . 14 , t h e r ange i s v e r y l a r g e . T h i s n e a r c o n s t a n t r e d u c t i o n o f r e f l e c t e d power i s a t t r i b u -t e d t o t h e n e a r v e r t i c a l d r op i n t h e b l a z i n g d e p t h c u r v e o f F i g . 3.6 a t 89° w h i c h r un s p a r a l l e l t o t h e 6^ = 85° l i n e . C o n t i n u i t y o f r e f l e c t e d power when t he a n g l e changes e n s u r e s t h i s l o n g , a l m o s t c o n s t a n t r e d u c -t i o n o f t h e r e f l e c t e d power d e m o n s t r a t e d h e r e . I n p r e d i c t i n g t h e b e h a v i o r o f s u r f a c e s w i t h v e r y s m a l l a s p e c t r a t i o s , i t i s u s e f u l t o r e c a l l t h e c u r v e s shown i n F i g . 3 . 3 . F i g . 3.15 d e m o n s t r a t e s t h e s h r i n k i n g o f t h e r ange o f h/d o v e r w h i c h there i s a r e -d u c t i o n i n s p e c u l a r r e f l e c t i o n f o r v e r y s m a l l a s p e c t r a t i o s . A t 0^ = 8 5 ° , t h e a/d = 0.1 g raph shows a w i d e r r ange o f h/d t h a t more l a r g e l y r e d u c e s t he r e f l e c t e d power t h a n a t 0^ = 5 5 ° , shown i n F i g . 3 . 3 . A t t h i s l a t t e r a n g l e , t he b l a z i n g depfti i s much more n e a r l y h o r i z o n t a l . However , as t h e RELATIVE POWER Q CD CD .CD CD CD CD CD CD ^ CD ^ [V> C o O - i O i \ j O o UD CD •67 50. a s p e c t r a t i o goes t o z e r o , t h e r ange o f h/d o v e r w h i c h t h e r e f l e c t e d power i s g r e a t l y r e d u c e d a l s o s h r i n k s t o z e r o . 3.7.3 V a r i a t i o n s i n A s p e c t R a t i o F o r TM p o l a r i z a t i o n , t h e b l a z i n g d e p t h needed f o r a g i v e n a n g l e o f i n c i d e n c e d e c r e a s e s as t h e a s p e c t r a t i o goes f r o m 1.0 t o 0.667, as shown I n F i g . 3.6. F o r a g i v e n a n g l e o f i n c i d e n c e i n F i g . 3i. 7, t h e v a l u e o f b l a z i n g d e p t h r e a c h e s a minimum as t he a s p e c t r a t i o d e c r e a s e s . The b l a z -i n g d e p t h t h e n s t a r t s i n c r e a s i n g a g a i n t o t h e l i m i t o f ( s i n 6^)/2 a s a/d -> 0. T h i s minimum v a l u e o f b l a z i n g d e p t h and t h e c o r r e s p o n d i n g a s p e c t r a t i o w o u l d y i e l d an o p e r a t i n g p o i n t f o r t h a t a n g l e o f i n c i d e n c e w h i c h i s r e l a t i v e l y i n s e n s i t i v e t o changes i n t h e g r oove w i d t h . S i m i l a r l y , f o r the TE c a s e , o p e r a t i n g p o i n t s c ho sen where two b l a z i n g d e p t h c u r v e s w i t h a l m o s t e q u a l a s p e c t r a t i o s i n t e r s e c t w o u l d g i v e s u r f a c e s t h a t a r e a l s o i n s e n s i t i v e t o changes i n g r oove w i d t h . I n g e n e r a l , t h e r e g i o n s o f F i g s . 3.5 - 3.7 where t h e c u r v e s f o r s e q u e n t i a l a s p e c t r a t i o s a r e c l o s e l y s p a c e d , s u ch as a t n e a r g r a z i n g i n -c i d e n c e i n F i g . 3.6, y i e l d o p e r a t i n g p o i n t s whose b e h a v i o r i s r e l a t i v e l y i n s e n s i t i v e t o changes i n t h e a s p e c t r a t i o . F i g . 3.16 shows t h e r e l a t i v e r e f l e c t e d power a t = 85° w i t h h/d = 0.06 f o r t h e r ange o f a s p e c t r a t i o s f r o m 0 t o 1. A t l e a s t 95% o f t h e i n c i d e n t power i s b a c k s c a t t e r e d o v e r a range o f a s p e c t r a t i o s f r o m 0.12 t o 0.87. The re a r e two a s p e c t r a t i o s w h i c h g i v e z e r o r e f l e c t e d power , a/d = 0.333 and a/d = 0.8. Between t h e s e two n u l l s t h e r e i s u s u a l l y 99% r e d u c t i o n i n t h e r e f l e c t e d power f o r any a s p e c t r a t i o . As t h e a s p e c t r a t i o goes t o z e r o , t h e b e h a v i o r o f t h e s u r -f a c e f o l l o w s t h a t w h i c h i s o u t l i n e d i n s e c t i o n 3.3; i . e . a l l i n c i d e n t power i s r e f l e c t e d . 51. d3M0d 3AI1V13U 52. 3.8 S i m u l t a n e o u s B l a z i n g P o i n t B e h a v i o r A b e t t e r u n d e r s t a n d i n g o f t h e b e h a v i o r t o be e x p e c t e d a t an o p -e r a t i n g p o i n t where s i m u l t a n e o u s b l a z i n g f o r b o t h p o l a r i z a t i o n s o c c u r s i s o b t a i n e d i f a/A. and h/A a r e p l o t t e d v s . 0_^. The v a l u e s i n F i g . 3.17 c o r -r e s p o n d s t o t h o s e i n F i g . 3 . 9 . The i n t e r e s t i n g p o i n t h e r e i s t h a t f o r 50° < 0 < 5 9 . 4 ° , h/A and a/A a r e n e a r l y c o n s t a n t w h i l e d/A = l / ( 2 s i n 0 ^ becomes more c o n s t a n t . From a d e s i g n p o i n t o f v i e w , t h e s e a n g l e s w o u l d be most s u i t a b l e t o y i e l d l a r g e r e d u c t i o n i n r e f l e c t e d power o v e r a w i d e r ange o f a n g l e s . On t he o t h e r h a n d , i f a g r a t i n g was d e s i g n e d t o b l a z e b o t h p o l a r i z a t i o n s a t 0_^  = 2 0 ° , t h e r ange o f a n g l e s o f f r e q u e n c i e s o v e r w h i c h t h e r e f l e c t e d power i s r e d u c e d t o t h e same l e v e l , w o u l d be e x p e c t e d t o be n a r r o w e r , s i n c e h/A a/A and d/A a r e c h a n g i n g more r a p i d l y as 0^ change s . F i g . 3.18 i s t h e p l o t o f h/A and a/A v s . 0^ c o r r e s p o n d i n g t o F i g . 3 . 10 . The v a l u e s o f a/A change even l e s s r a p i d l y w h i l e t h e v a l u e s o f h/A a r e f a i r l y c o n s t a n t t h o u g h ' l a r g e . I n A p p e n d i x B a r e l o c a t e d F i g s . B . l and B .2 . These a r e t h e p l o t s o f r e f l e c t e d power v s . f r e q u e n c y f o r a d u a l b l a z e d s u r f a c e a t 0^ = 45° and 0_^  = 2 0 ° , r e s p e c t i v e l y . Each o f t h e p a i r s o f c u r v e s i s f o r x a s u r f a c e whose g r o o v e d e p t h and w i d t h v a r i e s w i t h i n 0 .03 mm (0 .001 i n . ) o f t h e d i m e n s i o n s needed f o r s i m u l t a n e o u s b l a z i n g . These c u r v e s a r e u sed t o i n v e s t i g a t e t h e e f f e c t o f s y s t e m a t i c e r r o r s i n e i t h e r a o r h . The TE and TM c u r v e s move r e l a t i v e t o each o t h e r as t h e g r oove d e p t h i s i n c r e a s e d . I n F i g . B . l t h e TM c u r v e moves t o t h e r i g h t r e l a t i v e t o t h e TE c u r v e w h i l e i n F i g . B . 2 , t h e TM c u r v e s h i f t s t o t h e l e f t o f t he TE c u r v e . The v a l u e o f t h e g r oove d e p t h d e t e r m i n e s t h e s e p a r a t i o n be tween t he TM and TE f r e q u e n c i e s a t w h i c h maximum r e d u c t i o n i n r e f l e c t e d power o c c u r s . I n F i g . B . l , t h e c l o s e r t h e f r e q u e n c y a t w h i c h maximum r e d u c t i o n o c c u r s i s t o t h e opt imum f r e q u e n c y o f 35 GHz, t h e g r e a t e r t h e r e d u c t i o n . F i g . 3.17 a/A and h/A for a Surface Blazed for Both TE and TM Incident Waves. 1 9 . 5 ° < 0. < 5 9 . 4 ° x 54. F ig . 3.18 a/A and h/A for a Surface Blazed for Both TE AND TM Incident Waves. 1 9 . 5 ° < G. < 7 3 . 1 2 ° 5 5 . No s i m i l a r o b s e r v a t i o n can be made i n F i g . B.2 s i n c e t h e maximum r e d u c t i o n , i s a lway s g r e a t e r t h a n 50 db . The b a s i c d i f f e r e n c e be tween w o r k i n g a t 8^ = 20° and 8^ = 45° i s t h a t f o r 8^  = 20° t h e c u r v e s a r e n a r r o w e r and d e e p e r . The n a r r o w n e s s o f t h e c u r v e s i n F i g . B.2 c o n f i r m s t h e e a r l i e r p r e d i c t i o n abou t d u a l b l a z e d s u r f a c e s d e s i g n e d f o r 6^ = 2 0 ° . The n a r r o w n e s s o f t h e c u r v e s means t h a t t h e measured r e d u c t i o n w i l l p r o b a b l y be much l e s s t h a n what i s p r e d i c t e d s i n c e a s m a l l e r r o r i n one o f t h e p a r a m e t e r s ( 8 ^ , a o r h ) w o u l d g r e a t l y d e c r e a s e t h e amount t h e r e f l e c t e d power r e d u c e d . T h i s i s e s p e c i a l l y t r u e f o r t h e TM c a s e b e c a u s e t h e c u r v e s a r e n a r r o w e r t h a n t h e c o r r e s p o n d i n g TE c u r v e s . I f t h e p e r i o d d were changed , t h e n , s i n c e a/d and h/d needed f o r a d u a l b l a z e d s u r f a c e r e m a i n t he same, o n l y t h e f r e q u e n c y a t w h i c h s i m u l t a n e o u s b l a z i n g o c c u r s w o u l d change . The change i n t h e f r e q u e n c y i s g o ve rned by t h e B r a g g c o n d i t i o n . 3.9 Summary The n u m e r i c a l r e s u l t s p r e s e n t e d i n t h i s c h a p t e r d e m o n s t r a t e t he f o l l o w i n g : 1) As a s u r f a c e has i t s a s p e c t r a t i o a p p r o a c h u n i t y , t h e n u m e r i -c a l r e s u l t s c onve r ge t o t ho se f o r a f i e l d s c a t t e r e d by a s i m i l a r s u r f a c e w i t h a comb p r o f i l e unde r t h e same i l l u m i n a t i o n . 2) As a s u r f a c e has i t s a s p e c t r a t i o d e c r e a s e d t o z e r o , t h e r ange o f h/d o v e r w h i c h t h e r e f l e c t e d power i s e f f e c t i v e l y e l i m i n a t e d s h r i n k s u n t i l i t i s a p o i n t d i s c o n t i n u i t y . A t h/A = w h i c h i s t h e b l a z i n g d e p t h f o r a s u r f a c e w i t h i n f i n i t e l y t h i n g r o o v e s , t h e r e i s no r e f l e c t e d power b u t a t a l l o t h e r g r o o v e d e p t h s t h e r e i s a l m o s t c o m p l e t e r e f l e c t i o n . 56 . 3) The n u m e r i c a l r e s u l t s , d emon s t r a t e ' t h a t a r e c t a n g u l a r g roove p r o f i l e w i t h z e r o g r o o v e ' ' d e p t h r e f l e c t s t h e i n c i d e n t wave . 4) F o r a TE p o l a r i z e d i n c i d e n t wave t h e r e e x i s t s a b l a z i n g d e p t h w h i c h i n c r e a s e s w i t h o u t bound as 8^ a p p r o a c h e s t h e c r i t i c a l a n g l e (8 = a r s i n ( a / d ) ) where c u t o f f o f t h e T E ^ Q wavegu i de mode t a k e s p l a c e . 5) F o r a TM p o l a r i z e d i n c i d e n t wave , s i n c e a TEM wavegu i de mode a lways c an p r o p a g a t e , t h e r e e x i s t s a b l a z i n g d e p t h f o r e v e r y a s p e c t r a t i o and 8. > 1 9 . 5 P . The b l a z i n g d e p t h c an be m u l t i v a l u e d when 8. < 8 1 e> f i cc b e c a u s e o f t h e p r e s e n c e o f t h e T M ^ Q wavegu i de mode. 6) F o r an a r b i t r a r i l y p o l a r i z e d wave i n c i d e n t a t an a n g l e g r e a t -e r t h a n 1 9 . 5 ° , a d u a l b l a z e d s u r f a c e may be c o n s t r u c t e d , a l t h o u g h t h e g r oove s become i m p r a c t i c a l l y deep a t n e a r g r a z i n g i n c i d e n c e . 7) O p e r a t i n g p o i n t s c ho sen on b l a z i n g d e p t h c u r v e s , f o r a s p e c t r a t i o s n e a r z e r o o r one , y i e l d s u r f a c e s t h a t r e d u c e t h e r e f l e c t e d power o v e r a n a r r o w r ange o f 0^ o r A . B e t t e r b r o a d b a n d b e h a v i o r i s a l s o o b s e r -ved f o r o p e r a t i n g p o i n t s cho sen where t h e t a n g e n t t o a b l a z i n g d e p t h c u r v e i s n e a r l y h o r i z o n t a l . 8) O p e r a t i n g p o i n t s c ho sen where t h e b l a z i n g d e p t h c u r v e r i s e s o r f a l l s s t e e p l y w i t h 8^ y i e l d s u r f a c e s whose b e h a v i o r i s a f f e c t e d l i t t l e by s m a l l changes i n t he g r oove d e p t h . 9) O p e r a t i n g p o i n t s c h o s e n i n r e g i o n s where t h e b l a z i n g d e p t h c u r v e s a r e c l o s e l y s p a c e d o r i n t e r s e c t c o u l d y i e l d s u r f a c e s whose b e h a v i o r i s r e l a t i v e l y i n s e n s i t i v e t o changes i n t h e a s p e c t r a t i o . 10) U s i n g t h e c u r v e s o f a / A and h / A v s . 8_^  f o r s i m u l t a n e o u s l y b l a z i n g i t i s p o s s i b l e t o p r e d i c t w h i c h a n g l e s o f i n c i d e n c e g i v e a d u a l b l a z e d s u r f a c e w h i c h ha s t he r e f l e c t e d power r e d u c e d o v e r a w i d e r ange o f 8^ o r A , and i s c o r r e s p o n d i n g l y l e a s t s u b j e c t t o e r r o r s i n t he p a r a m -e t e r s . 4. EXPERIMENTAL RESULTS 4.1 I n t r o d u c t i o n I n t h i s c h a p t e r t h e measurements o f t h e power r e f l e c t e d by a p e r i o d i c s u r f a c e a r e p r e s e n t e d . A l l t h e s u r f a c e s have r e c t a n g u l a r g r o o v e p r o f i l e s w h i c h were d e s i g n e d u s i n g t he d a t a o f t h e l a s t c h a p t e r . The e x -p e r i m e n t a l v a l u e s a r e p l o t t e d as c i r c l e s and a d j a c e n t c i r c l e s a r e j o i n e d by s t r a i g h t l i n e s t o g i v e a p i e c e w i s e l i n e a r c u r v e . The v a l u e s p r e d i c t e d by t h e n u m e r i c a l methods o f c h a p t e r 2 a r e p l o t t e d as a smooth c u r v e . The re were s e v e n p l a t e s whose r e f l e c t e d power was measured o v e r a r ange o f o r f r e q u e n c y f o r b o t h p o l a r i z a t i o n s t h e s e b e i n g t h e o n l y p a r a m e t e r s t h a t can be changed f o r a p a r t i c u l a r p l a t e . F i g . 4 .1 shows t h e t y p i c a l p r o f i l e o f a p l a t e and g i v e s t h e g roove d i m e n s i o n s o f each o f t h e p l a t e s . These d i m e n s i o n s were u sed i n p r e d i c t i n g t h e r e f l e c t e d power and t h e s u r f a c e d i m e n s i o n s a r e m i l l e d w i t h i n ± 0 . 0 3 mm ( ± 0 . 0 0 1 i n . ) o f t h e s e g i v e n d i m e n s i o n s . The p l a t e s ' d i m e n s i o n s were cho sen t o t e s t t he b e h a v i o r o f s u r f a c e s whose • o p e r a t i n g p o i n t s c o v e r a w i d e r ange o f a s p e c t r a t i o s and ang l e s o f i n c i d e n c e . The s u r f a c e s were d e s i g n e d f o r an i n c i d e n t wave a t a f r e q u e n c y o f 35 GHz. F i g . 4.1 l i s t s t he p l a t e d i m e n s i o n s , t h e number o f g r oove s and 6^ 0p> t h e a n g l e t he p l a t e i s b l a z e d f o r when t h e i l l u m i n a t i o n i s TM p o l a r i z e d . P l a t e s 3-7 a l s o b a c k s c a t t e r most o f t h e power o f an i n c i d e n t TE p o l a r i z e d wave . P l a t e s 2, 5 and 6 were d e s i g n e d f rom, ,data i n [2] and [12 ,13 ] and t h e e x -p e r i m e n t a l d a t a have been p r e s e n t e d a l r e a d y i n [ 4 , 1 6 , 1 9 ] . The p r e d i c t e d r e f l e c t e d power c u r v e s f o r t h e s e p l a t e s were c a l c u l a t e d and a r e p r e s e n t e d h e r e w i t h t he e x p e r i m e n t a l r e s u l t s f o r t he f i r s t t i m e . The c a l c u l a t e d v a l u e s we re u n a v a i l a b l e when t h e e x p e r i m e n t a l r e s u l t s were f i r s t p r e s e n t e d . 58. Bl -d ~ h - a T h PZ./A7"E DIMENSIONS d-4.95 mm a = 3.30 mm h = 1.30 m m d** 475mm a--=120 mm h = l78mm d= 6.07mm q =4.98 mm n = 5.13mm d = 12.54mm a - 4.90mm h = 5.58 mm GROOVES 3 4 5 d a h 8.57mm 4.95mm 6.14mm d = 5.23 m m c = 4.97mm h = 4.80 mm d=558mm a = 5.00 mm h = 485mm 54 56 44 21 31 51 48 (degrees) 60.0 64.25 45.0 20.0 30.0 55,0 50.0 F i g . 4.1 Prof i le of Experimental Surfaces and Dimensions for A l l Plates . . Experimental Values for Plates / / l , 5 and 6 have been Reported Ear l i er in [4,16,19]. 59 . Due t o t h e o r i g i n a l o r i e n t a t i o n o f t h e s t u d y t o t h e ILS a p p l i c a -t i o n , t h e p l a t e s were b r a s s w i t h a p p r o x i m a t e o v e r a l l d i m e n s i o n s 26.65 cm x 11.24 cm x 1.269 cm. The l e n g t h o f t he p l a t e was r ounded o f f t o t h e n e a r e s t i n t e g r a l number o f p e r i o d s and an e x t r a r i d g e was a d d e d . T h i s f i x i n g o f t h e p l a t e l e n g t h meant t h a t t h e number o f g r oove s v a r i e d f r o m 21 f o r p l a t e #5, d e s i g n e d f o r 8 = 2 0 ° , t o 56 f o r p l a t e #2, d e s i g n e d t o o p e r a t e a t 8_^  = 6 4 . 2 5 ° . The e x p e r i m e n t a l a r r angement was e s s e n t i a l l y t h e same as t h a t u sed i n [ 1 2 , 1 3 ] , as m o d i f i e d i n [ 1 6 ] , and t h e r e f o r e , t h e a r r angement w i l l be d e s c r i b e d o n l y b r i e f l y h e r e . 4 .2 E x p e r i m e n t a l A r r angement A d i a g r a m o f t h e e x p e r i m e n t a l s e t - u p i s shown i n F i g . 4 .2 w h i l e F i g . 4 .3 i s a p i c t u r e o f t h e s e t up . F i g . 4 .2 and 4 .3 show how t h e r e -c e i v i n g a n t e n n a ( i n t h e f o r e g r o u n d i n F i g . 4 .3 ) i s p o s i t i o n e d t o measure the power r e f l e c t e d by t he p l a t e . The a b s o r b e r was hung be tween t h e two an tenna s t o e l i m i n a t e d i r e c t t r a n s m i s s i o n . The two a n t e n n a s were i d e n -t i c a l p y r a m i d a l h o r n s w i t h a 24.7 db g a i n and an E - p l a n e beamwidth o f 9 ° . The a b s o r b e r s were p o s i t i o n e d a round t h e p l a t e t o r e d u c e r e f l e c t i o n f r o m any s u r f a c e e x c e p t t h e t op o f t h e p l a t e . A b s o r b e r a n d , i n some i n -s t a n c e s , a m e t a l p l a t e were p l a c e d a t one end o f t he g r a t i n g t o e l i m i n a t e t r a n s m i s s i o n under t h e p l a t e . The an tenna s we re 1.38 m f r o m the c o r r u -g a t e d s u r f a c e . The d i s t a n c e needed f o r p l a n e wave i l l u m i n a t i o n was t a k e n 2 as ('2D ) / A where D = C cos 8. . ( 4 .1 ) l S i n c e C i s a p p r o x i m a t e l y 26.65 cm, t h e r ange needed f o r p l a n e wave i l l u m -i n a t i o n a t 35 GHz v a r i e s f r o m 14.64 m a t 8. = 20° t o 3.13 m a t 8. = 6 4 . 2 5 ° . l l The r ange needed f o r p l a n e wave i l l u m i n a t i o n o f t h e p l a t e v a r i e s g r e a t l y and i s u n r e a l i s t i c when w o r k i n g i n d o o r s f o r most a n g l e s o f i n c i d e n c e . 6 0 . \ CRYSTAL CURRENT METER CRYSTAL DE TECTOR CRYSTAL DETEC w CRYSTAL CURRENT MEIER Rx TORfif ISOL A TOR,—N ^ G 3 @ /PRECISION VARIABLE ATTENUATOR KL YSTRON PLATE (b) F i g . 4.2 Experimental Set-up (a) Range (b) Circu i t Fig. 4 .3 Picture of Experimental Set Up. Receiving Horn is in the Foreground. 62 . As shown by J u l l [16] and J u l l and Ebbeson [4] t h e r e s u l t s , no m a t t e r how good a t t h i s r a n g e , c o u l d o n l y be b e t t e r unde r p l a n e wave i l l u m i n a t i o n . The a n g l e o f i n c i d e n c e c o u l d be r e a d t o an a c c u r a c y o f = 1° and c o u l d be v a r i e d f r o m a p p r o x i m a t e l y 5° t o 9 0 ° , a l t h o u g h i t was d i f f i c u l t t o work n e a r e i t h e r o f t h e s e two l i m i t s . The e x p e r i m e n t a l c i r c u i t d i a g r a m i s shown i n F i g . 4 .2 b). W i t h a f l a t c o n d u c t i n g p l a t e o f t h e same a r e a as t h e c o r r u g a t e d s u r f a c e i n p l a c e , t he c r y s t a l c u r r e n t r e a d i n g o f t h e r e c e i v i n g a n t e n n a was n o t e d . When t h i s r e f e r e n c e p l a t e was removed, t h e c r y s t a l c u r r e n t r e a d i n g f e l l b u t was r e t u r n e d t o i t s f o r m e r l e v e l b y . a d j u s t i n g t h e p r e c i s i o n v a r i a b l e a t t e n u -a t i o n . The power r e d u c t i o n due t o t h e p l a t e i s j u s t t he d i f f e r e n c e b e -tween t h e f o r m e r and l a t t e r p r e c i s i o n v a r i a b l e a t t e n u a t i o n r e a d i n g s . The k l y s t r o n o u t p u t l e v e l was a l s o m o n i t o r e d so t h a t e r r o r s i n t h e r e a d i n g s due t o f l u c t u a t i o n s i n t h e t r a n s m i t t i n g l e v e l s c o u l d be d e t e c t e d . The p l a t e s were mounted on t h r e e a d j u s t a b l e s c r ews so t h a t t h e y w i l l be l e v e l , t hu s i n c r e a s i n g t he a c c u r a c y o f t he 6^ measu rement s . The p l a t f o r m on w h i c h t h e s c r ews were mounted a l s o r o t a t e d so t h a t t h e e f f e c t o f o b l i q u e i n c i d e n c e c o u l d be i n v e s t i g a t e d . 4 .3 P l a t e s B l a z e d f o r T M . P o l a r i z a t i o n The measurements shown i n F i g . 4 .4 were made on p l a t e #1, whose o p e r a t i n g p o i n t f a l l s on t h e a/d = 0.667 TM b l a z i n g d e p t h c u r v e . The g r oove d e p t h f o r t h i s p l a t e was r e a d o f f o f t he same c u r v e g i v e n by H e s -s e l , Schmoys, and Tseng [2] and t h e e x p e r i m e n t a l d a t a was f i r s t r e p o r t e d by J u l l [ 1 6 ] . The c a l c u l a t e d c u r v e and e x p e r i m e n t a l c u r v e c o r r e s p o n d v e r y c l o s e l y . The maximum r e d u c t i o n i n t he e x p e r i m e n t a l c u r v e o c c u r s a t 0. = 59° r a t h e r t h a n a t 0. = 60° where t h e p r e d i c t e d maximum r e d u c t i o n i s . T h i s d i s c r e p a n c y i s w i t h i n t h e r e a d i n g e r r o r o f 0^.. The re i s a 23 dB 63 . F i g . 4 . 4 Measured and Calculated TM Reflected Power vs. 8^ for Plate #1. —o-— from [4] F i g . 4.5 Measured and Calculated TM Reflected Power vs. Q± for Plate #2. 65. r e d u c t i o n i n r e f l e c t e d power o v e r 54° < 0^ < 6 6 ° . Th i s r e d u c t i o n , o v e r a w i d e r ange o f a n g l e s , i s what w o u l d be e x p e c t e d s i n c e t h e a/d = 0 .667 c u r v e i s n o t s t e e p a t ®^ 0 ^ = 6 0 ° . T h i s a c c o u n t s f o r t he c l o s e c o r r e s p o n d e n c e be tween t h e p r e d i c t e d and . e x p e r i m e n t a l v a l u e s . The a n g u l a r r e s p o n s e o f p l a t e #2 i s shown i n F i g . 4 . 5 . The g r oove d i m e n s i o n s were c h o s e n t o compare r e s u l t s f o r a/d = 0 . 2 5 , a s a check on t h e a c c u r a c y o f t he n u m e r i c a l r e s u l t s f o r s m a l l a s p e c t r a t i o s . The two c u r v e s compare q u i t e f a v o r a b l y , t h e m i n ima o e c u r i n g a t t h e same a n g l e , 0^ = 6 4 . 5 ° . The maximum measured r e d u c t i o n i s 31 .5 dB w h i c h i s pe rhap s t h e l i m i t o f what c an be measured w i t h t h e power o u t p u t o f t h e k l y s t r o n and t he n o i s e l e v e l o f t h e a m p l i f i e r . The p l a t e ' s r e s p o n s e i s r e l a t i v e l y i n s e n s i t i v e t o changes i n 0_^  w i t h a r e d u c t i o n i n r e f l e c t e d power o f 23 dB o v e r t h e r ange 6 1 . 5 ° < 0^ < 6 7 ° . 4 .4 P l a t e s f o r S i m u l t a n e o u s B l a z i n g As d i s c u s s e d i n s e c t i o n 3 . 8 , t h e b e h a v i o r o f a s u r f a c e d e s i g n e d t o s i m u l t a n e o u s l y b l a z e an i n c i d e n t T E ^ r TM wave v a r i e s g r e a t l y w i t h the a n g l e o f i n c i d e n c e cho sen f o r minimum s p e c u l a r r e f l e c t i o n . S u r f a c e s were mach ined t h a t o p e r a t e a t 0_^  = 20° and 0^ = 4 5 ° , t h e two c a s e s d i s -c u s s e d i n s e c t i o n 3.8 and i l l u s t r a t e d i n F i g s . B . l and B .2 . Measurements o f t he a n g u l a r and f r e q u e n c y r e s p o n s e o f t h e s e p l a t e s w i l l t e s t t h e p r e -d i c t i o n s o f s e c t i o n 3 . 8 . The measured a n g u l a r r e s p o n s e o f t h e power r e f l e c t e d by p l a t e #3 i s shown i n F i g . 4.6 f o r b o t h p o l a r i z a t i o n s a t 35 GHz. The re i s a i r e d u c t i o n i n t h e r e f l e c t e d power a t 0^ = 45° o f 31 dB o r 99 .9% f o r t h e TM c a s e , and 21 dB o r 99 .2% f o r t h e TE c a s e . The maximum measured r e -d u c t i o n does n o t o c c u r a t 0 . f o r e i t h e r c a s e b u t a t 0. = 4 3 . 5 ° f o r t h e i o p 1 TE c a s e and a t 0_. = 4 6 ° f o r t h e TM c a s e . T h i s d i s c r e p a n c y i s p r o b a b l y 01 o\ F i g . 4.6 Measured and Calculated Reflected Power vs. 9. for Plate #3. f = 35.0 GHz 1 (a) TM Polarizat ion (b) TE Polarizat ion 67. due t o e r r o r s irv.:the g roove d i m e n s i o n s . A b e t t e r i d e a o f t h e e f f e c t o f e r r o r s c an be s e e n b y l o o k i n g a t t h e f r e q u e n c y r e s p o n s e c u r v e s o f p l a t e #3 shown i n F i g . 4 . 7 . The c u r v e s i n F i g . 4.7 a r e most s i m i l a r t o t h o s e i n F i g . B . l f ) , whe re h i s t o o l a r g e . I n b o t h F i g . 4 .7 and B . l f ) , t h e TE c u r v e i s d e e p e r t h a n t h e TM c u r v e , b u t t he o b s e r v e d s e p a r a t i o n be tween t h e c u r v e s f o r e a ch p o l a r i z a t i o n i s g r e a t -e r i n F i g . 4 . 7 . T h i s i s most p r o b a b l y due t o t h e e r r o r s i n t r o d u c e d i n m i l l i n g p l a t e #3. The f r e q u e n c y w h i c h g i v e s t h e same r e d u c t i o n f o r b o t h p o l a r i z a -t i o n s i s 34.9 GHz. The a n g u l a r r e s p o n s e o f p l a t e #3 a t 34.9 GHz i s shown i n F i g . 4 . 8 . The measured c u r v e s c o r r e s p o n d b e t t e r t o t h e p r e d i c t e d c u r v e s t h a n was t h e c a s e i n F i g . 4.7 b u t ' t h e r e d u c t i o n a t 0^ = 45° i n c r e a s e d o n l y t o 26 dB o r 99.8% f o r TE p o l a r i z a t i o n and d e c r e a s e d t o 28 dB o r 99.8% f o r TM p o l a r i z a t i o n , a s m a l l imp rovemen t . None o f t h e measu red a n g u l a r r e s p o n s e c u r v e s were as deep as the p r e d i c t e d c u r v e s , a l t h o u g h t h e a c t u a l d i f f e r e n c e be tween t he two c u r v e s was a l w a y s l e s s t h a n 1% o f t h e i n c i d e n t power . F o r l a r g e r e d u c t i o n s , h oweve r , t h e a c c u r a c y o f t h e measurements a r e l i m i t e d b y t h e s i g n a l t o n o i s e r a t i o . The s h i f t i n t h e a n g l e s o f maximum r e d u c t i o n f o r b o t h p o l a r i z a t i o n s i n F i g . 4 .8 i s a t t r i b u t e d t o e x p e r i m e n t a l e r r o r i n t h e measurement o f 0_^ . The a n g u l a r r e s p o n s e o f p l a t e #4, a p l a t e d e s i g n e d f o r o p e r a t i o n a t 0^  = 2 0 ° , i s shown i n F i g . 4 .9 f o r an i n c i d e n t wave o f e i t h e r p o l a r i z -a t i o n a t 35 GHz. The measu red and c a l c u l a t e d c u r v e s d i f f e r more t h a n d i d t h e a n g u l a r r e s p o n s e c u r v e s f o r p l a t e #3. T h i s a g r e e s w i t h t h e p r e d i c t i o n i n s e c t i o n 3. 8 t h a t t h i s a n g l e o f i n c i d e n c e , whe re a l l t h e g r o o v e d i m e n s i o n s change most r a p i d l y w i t h 0^, w o u l d be h a r d e r t o o p e r a t e at , due t o t h e 68. REFL EC TED PO WER fdb) ft s o o H H N N r r r r era c o (D CO c H CO s O Co o o CD Cu fsJ CD l-ti M CO o r r ro o CD CO M l o l-i P> r r CD CvJ REFLECTED POWER(db) '69 70. 10 20 J0~ 8, (degrees) ST) ( a) 40 SO 20 30 Gj (degrees) (b) F ig . 4.9 Measured and Calculated Reflected Power vs. 0 for Plate #4 f - 35.0 GHz (a) TM Polarization (b) TE Polarization 71 . c r i t i c a l n a t u r e o f t h e g r oove d i m e n s i o n s . The c a l c u l a t e d TM c u r v e shows t h e n a r r o w a n g u l a r r e s p o n s e o f an i d e a l s u r f a c e w i t h t h e g r oove d i m e n s i o n s o f p l a t e #4. The measured r e d u c t i o n a t 0 = 20° o f 23 dB o r 99 .5% f o r TE p o l a r i z a t i o n and o n l y 10 .5 dB o r 91.8% f o r TM p o l a r i z a t i o n shows t h a t t h e b e h a v i o r i s s t i l l good b u t t h a t t h e e x p e r i m e n t a l e r r o r s have a g r e a t e r e f f e c t a t t h i s o p e r a t i n g a n g l e , e s p e c i a l l y f o r TM p o l a r i z a t i o n . The measured f r e q u e n c y r e s p o n s e c u r v e s f o r p l a t e #4 a t 0^ = 20° a r e shown i n F i g . 4 . 1 0 . The TM f r e q u e n c y c u r v e i n d i c a t e s t h e n a r r o w band n a t u r e o f t h e p l a t e f o r t h i s p o l a r i z a t i o n . F i g . 4 .10 i s most s i m i l a r t o F i g . B.2 c) where h i s t o o l a r g e . I n b o t h F i g s . 4 .10 and B.2 c) t h e TM c u r v e i s n a r r o w e r t h a n t h e c o r r e s p o n d i n g TE c u r v e , however t h e f r e q u e n c y a t w h i c h TE and TM r e d u c t i o n a r e e q u a l i s 35 GHz i n F i g . B.2 c ) and 6 n l y 34.8 GHz i n F i g . 4 . 1 0 . T h i s c o r r e s p o n d s t o a s u r f a c e w h i c h has a l a r g e r s y s t e m a t i c e r r o r i n b o t h a and h t h a n what i s c o n s i d e r e d i n F i g . B .2 . The d i s c r e p a n c i e s be tween t h e two p a i r s o f c u r v e s i s a t t r i b u t e d t o e r r o r s i n t h e g r oove d i m e n s i o n s . The a n g u l a r b e h a v i o r o f p l a t e #4 a t 34 .8 GHz i s e x h i b i t e d i n F i g . 4 . 1 1 . As was t h e ca se f o r p l a t e #3, t h e measured a n g u l a r r e s p o n s e c u r v e s a t t h i s new f r e q u e n c y c o r r e s p o n d more c l o s e l y t o t h e c a l c u l a t e d c u r v e s t h a n was t he c a s e a t 35 GHz. A l l t h e measured a n g u l a r r e s p o n s e c u r v e s show t h e complex b e h a v i o r p r e d i c t e d by t h e c a l c u l a t e d c u r v e s . T h i s comp lex n a t u r e o f t he c u r v e s i s b e c a u s e t h e o p e r a t i n g p o i n t f o r p l a t e #4 f a l l s i n t he m u l t i v a l u e d r e g i o n o f t he b l a z i n g d e p t h c u r v e s where i t i s p o s s i b l e w i t h t h e same g r oove d i m e n s i o n s t o o b t a i n m i n i m a l r e f l e c t i o n a t more t h a n one a n g l e . A t 9^ = 2 0 ° , t h e r e d u c t i o n i s 21 dB o r 99 .2% f o r TM p o l a r i z a t i o n and 19 dB o r 98 .75% f o r TE p o l a r i z a t i o n b u t t he n a r r o w n e s s o f t h e TM c u r v e d i c t a t e s t h e r ange o f a n g l e s o v e r w h i c h t h i s p l a t e w o u l d be e f f e c t i v e . 73. iO JO 6) (degrees) (a) 10 20 30 Q; (degrees) (b) 40 F i g . 4.11 Measured and Calculated Reflected Power vs . 9 i for P la te #4 f = 34.8 GHz (a) TM P o l a r i z a t i o n (b) TE P o l a r i z a t i o n 7 5 . The a n g u l a r r e s p o n s e o f p l a t e # 3 a t 3 4 . 7 GHz, t h e f r e q u e n c y where t h e r e i s minimum r e f l e c t i o n f o r a TM p o l a r i z e d wave , i s e ven n a r r o w e r t h a n a t 3 4 . 8 GHz. T h i s i s shown i n F i g . 4 . 1 2 . A change i n 6^ o f 1 ° f r o m t h e opt imum a n g l e o f 2 0 ° changes t h e amount o f r e d u c t i o n s i g n i f i c a n t l y . T h i s w o u l d be an i d e a l s u r f a c e i f t he a p p l i c a t i o n needed a v e r y s e n s i t i v e i a n g l e o r f r e q u e n c y r e s p o n s e . P l a t e # 5 was d e s i g n e d t o g i v e a b l a z e d s u r f a c e f o r a wave o f a r b i t r a r y p o l a r i z a t i o n i n c i d e n t at90_^ = 3 0 ° . A t t h i s a n g l e o f i n c i d e n c e i n F i g . 3 . 1 7 t h e g r oove d e p t h and w i d t h c u r v e s change s l o w l y w i t h changes i n 9_^ . T h i s o p e r a t i n g p o i n t s h o u l d y i e l d a s u r f a c e t h a t g i v e s b e t t e r r e d u c t i o n o v e r a w i d e r r ange o f 9. t h a n was o b s e r v e d a t 9. = 4 5 ° o r 6. = 2 0 ° . The a n g u l a r r e s p o n s e o f p l a t e #5 a t 3 5 GHz i s shown i n F i g . 4 . 1 3 . The e x p e r i m e n t a l r e s u l t s have a l r e a d y been p r e s e n t e d i n [ 1 9 ] . The c a l -c u l a t e d and measured r e s p o n s e c u r v e s a r e w i d e r t h a n o b s e r v e d a t 3 5 GHz f o r p l a t e s # 3 and # 4 , c o n f i r m i n g t h e p r e d i c t i o n o f l a r g e r e d u c t i o n o v e r a w i d e r ange o f 0^. P l a t e # 5 was d e s i g n e d b e f o r e t h e r e e x i s t e d p r e c i s e n u m e r i c a l r e s u l t s f o r a l l d u a l b l a z i n g p o i n t s ; hence t h e a c t u a l g r oove d e p t h s h o u l d have been 6 . 1 1 mm i n s t e a d o f t he 6 . 1 5 mm u sed h e r e . T h i s e r r o r a c c o u n t s f o r t h e p r e d i c t e d r e d u c t i o n a t 3 0 ° b e i n g o n l y 3 8 dB. The TM c u r v e s shown i n F i g . 4 . 1 3 a) a r e v e r y w i d e b e c a u s e t h e o p e r a t i n g p o i n t , w h i c h has a/d = 0 . 5 6 6 , i s i n t h e m u l t i v a l u e d r e g i o n o f t h e TM b l a z i n g d e p t h c u r v e a t 6^ = 3 0 ° . Hence , f o r TM p o l a r i z a t i o n , p l a t e # 5 s h o u l d have an a n g u l a r r e s p o n s e w h i c h c l o s e l y r e s e m b l e s t h a t shown i n F i g . 3 . 1 2 . P l a t e #6 was d e s i g n e d by Ebbeson [ 1 2 ] t o compare h i s n u m e r i c a l r e s u l t s f o r a comb w i t h t h o s e o f a f i n p r o f i l e f o r TM p o l a r i z a t i o n . Due t o t he need t o u se f i n s o f f i n i t e t h i c k n e s s t h e s e d i m e n s i o n s were cho sen u s i n g a method o u t l i n e d i n [ 1 2 ] and t h e measured TM r e s u l t s were f i r s t 01 &! (degrees) (a) 0\ 6j (degrees) (b) F ig . 4.13 Measured and Calculated Reflected Power vs. 6 for Plate o from [19] f = 35.0 GHz (a) TM Polarization (b) TE Polarization U' ft) TJ h i o o !— 1 h- 1 CO f» CO Co O a O O rt CO Cu hj O ro H < H i O i-i h j I—1 Co r t CD REFLECTED N o O POWER (db) hd 09 n> CO CO c •1 ro cu CO a Cu o (-> o c M Co rt CD Cu itf CD REEL EC TED P0 WER (db) C D C D LL 78. p r e s e n t e d t h e r e . The TE r e s u l t s were f i r s t p r e s e n t e d i n [ 1 6 ] . The p l a t e s ' d i m e n s i o n s a r e n o t e x a c t l y opt imum f o r d u a l b l a z i n g a t 0^  = 5 5 ° , t h e y s h o u l d have b e e n ; a =. 5.04 mm and h = 4 . 8 3 mm, n o n e t h e l e s s , t h e r e d u c t i o n as shown i n F i g . 4 .14 i s s t i l l 23 dB o r 99 . 5% f o r t h e TM c a s e o r 31 dB o r 99 . 1% f o r t h e TE c a s e . The p r e d i c t e d r e d u c t i o n i s n o t as good as i t w o u l d be i f t h e opt imum d i m e n s i o n s were u s e d , b u t t he c o r r e s p o n d e n c e b e -tween t he c a l c u l a t e d and measured v a l u e s i s s t i l l v e r y good. T h i s i n d i -c a t e s 0^  = 55° f a l l s i n t he r e g i o n o f F i g . 3.17 where d u a l b l a z e d p l a t e s a r e e a s i e s t t o make. P l a t e #7 was d e s i g n e d t o o p e r a t e a t 0^  = 50° and i t s d i m e n s i o n s a r e as c l o s e as p o s s i b l e t o t h o s e g i v e n f o r a s i m u l t a n e o u s l y b l a z e d s u r f a c e a t t h i s a n g l e . T h i s a n g l e i s i n t h e r e g i o n where i t was f e l t t h a t b r o a d -band b e h a v i o r and b e s t p e r f o r m a n c e o f a s i m u l t a n e o u s l y b l a z e d s u r f a c e c o u l d be a c h i e v e d . F i g . 4 .15 shows t he measured and c a l c u l a t e d a n g u l a r r e s p o n s e o f p l a t e #7 a t 35 GHz. F i g . 4 .15 e x h i b i t s t h e l a r g e s t measured r e d u c t i o n o b t a i n e d f o r b o t h p o l a r i z a t i o n s s i m u l t a n e o u s l y . A t = 50° t h e r e i s a 28 dB o r 99.8% r e d u c t i o n i n t h e TM power and 30 dB o r 99 .9% r e d u c t i o n i n t he TE power . The r e s p o n s e o f t he p l a t e i s w i d e r t h a n a t 0_^  = 20° o r a t 0^  = 4 5 ° , t h e r e b e i n g a 23 dB r e d u c t i o n f o r b o t h p o l a r i z a t i o n s o v e r t h e r ange 4 9 ° « 0 i < 5 3 ° , a l t h o u g h t h i s i s n o t as w i d e as a t Q± = 3 0 ° . B o t h measured c u r v e s a r e s h i f t e d t o t h e r i g h t o f t h e c a l c u l a t e d c u r v e s by l e s s t h a n a d e g r e e , b u t t h e c a l c u l a t e d and measured c u r v e s do c o r r e s p o n d t o t he e x t e n t t h a t b o t h TM c u r v e s a r e n a r r o w e r and d e e p e r t h a n t h e i r r e s p e c t i v e TE c u r v e s . The c a l c u l a t e d and measured f r e q u e n c y r e s p o n s e c u r v e s o f p l a t e //7 a r e shown i n F i g . 4 . 1 6 . The measured c u r v e s a r e s h i f t e d upwards i n f r e q u e n c y by 0.1 GHz w i t h r e s p e c t t o t he c a l c u l a t e d c u r v e s and a r e n o t as * J H -09 • • U l • — N M l c r CO nT —y N ' II CO en O J c H W • CD o Cu o o o CO M w Co CO N 0 -H H- O N N 0) £0 CO r-» r t r t O H - H-O O M fo r t (0 P-. 5 « ro i—* CD O r t CO Cu h j O ro H <! CO • a> H i o H f" r t ro REFLECTED Co CD POWER C D C D I U l C D C D REFLECTED CD POWER (db) C D C D '6L REFLECTED POWER (db) 09 I—1 cr a) n o g ro pi It 3 M 23 O n o) "< c t) ro ro rtia.cn ro xs pu o o c 0 n >-< w c < ro H ro < 0) o ro H I to M REFLECTED POWER (db) i *08 8 1 . smooth . The c a l c u l a t e d f r e q u e n c y c u r v e s do n o t g i v e b e s t r e s u l t s a t 35 GHz as e x p e c t e d b u t a r e b e t t e r n e a r 35.1 GHz, i n d i c a t i n g t h e g r e a t e r s e n s i t i -v i t y o f t h i s p l a t e t o i t s g r o o v e s ' d i m e n s i o n s t h a n e i t h e r p l a t e #3 o r #4. The b e s t measured r e s u l t s were o b t a i n e d a t 35.2 GHz. F i g . 4 .17 shows t h e a n g u l a r r e s p o n s e o f p l a t e #7 a t 35.2 GHz. The measured r e s u l t s a r e t h e b e s t so f a r , t h e r e b e i n g 31 dB and 35 dB r e d u c t i o n f o r TM and TE p o l a r i z a t i o n , r e s p e c t i v e l y , a t 0^ = 5 0 ° . B o t h TM c u r v e s a r e s t i l l deepe r and n a r r o w e r t h a n t he TE c u r v e s . I n f i g . 4 .17 b ) f o r t he f i r s t t i m e a measured maximum r e d u c t i o n g r e a t e r t h a n any p r e -d i c t e d r e d u c t i o n i s o b s e r v e d , b u t t h e d i f f e r e n c e i s o n l y 0.04% o f t h e i n -c i d e n t power . B o t h e x p e r i m e n t a l c u r v e s a r e s t i l l s h i f t e d t o t he r i g h t w i t h r e s p e c t t o the c a l c u l a t e d c u r v e s and a maximum r e d u c t i o n o f a l m o s t 40 dB was measured f o r each p o l a r i z a t i o n . 4.5 O b l i q u e I n c i d e n c e The e f f e c t o f o b l i q u e i n c i d e n c e on t h e r e f l e c t e d power has n o t been p r e d i c t e d h e r e , b u t was i n v e s t i g a t e d e x p e r i m e n t a l l y . T h i s was done by h o l d i n g t he a n g l e o f i n c i d e n c e a t 9^op a n d r o t a t i n g t h e p l a t e t h r o u g h an a n g l e 3^. The b e h a v i o r a t o t h e r a n g l e s , 8_ ,^ w o u l d be o f t h e same g e n e r a l f o r m as t h e b e h a v i o r a t 9. l o p F i g s . 4 .18 and 4.19 show t h e measured r e d u c t i o n as a f u n c t i o n o f 3^ f o r p l a t e s #3 and #4. The measurements i n F i g . 4 .18 were t a k e n a t 34.9 GHz w h i l e t h o s e i n 4.19 a) and 4.19 b ) were t a k e n a t 34.7 and 35 GHz r e s p e c t i v e l y . A l l t h e c u r v e s t e n d t o be s y m m e t r i c a l abou t 3^ = - 3 ° . Symmetry abou t 3^ = 0° i s e x p e c t e d and t h i s d i s c r e p a n c y i s e v i d e n t l y due t o p o s i t i o n i n g o f t he p l a t e . The i n s e n s i t i v i t y t o 3^ i s r e a d i l y s e e n s i n c e e i t h e r o f t h e s e p l a t e s c o u l d be r o t a t e d up t o 10° and s t i l l g i v e 20 dB r e d u c t i o n . 09 —1 M l o o M [-3 N (U 0! CD CO CO c H CD 9 K w b N a. o fo o c H-» Co rt ro CD M l M CD o rt ro o-|-d o sd ro H < CO flEFZ. ECTED POWER i REFLECTED POWER ' 3 8 83. F i g . 4.18 Measured Reflected Power vs. ^ for Plate #3 (a) TM Polarization (b) TE Polarization 84. F i g . 4.19 Measured Reflected Power vs. 8. for Plate #4 (a) TM Polarization (b) TE Polarization 85 . 4.6 E r r o r s The re a r e two m a j o r s o u r c e s o f t h e d i s c r e p a n c i e s be tween t h e measured power r e f l e c t e d by a r e a l i z a b l e s u r f a c e and t h e r e f l e c t e d power p r e d i c t e d f o r an i d e a l s u r f a c e ; e x p e r i m e n t a l e r r o r s and t h e n o n - i d e a l n a t -u r e o f t he s u r f a c e . The e x p e r i m e n t a l e r r o r s a r i s e f r o m e r r o r s i n r e a d i n g s , t he n o n - p l a n e wave n a t u r e o f t h e i l l u m i n a t i n g beam, o b l i q u e i n c i d e n c e , t h e n o i s e o f t h e a m p l i f i e r , and s i t e r e f l e c t i o n s . The r e a l i z a b l e s u r f a c e d i f -f e r s f r o m t h e i d e a l one i n t h a t i t has f i n i t e d i m e n s i o n s , f i n i t e c o n d u c t i -v i t y as w e l l as e r r o r s i n t he g r o o v e s ' d i m e n s i o n s due t o m i l l i n g . S i n c e t he a b s o r b e r s work b e s t a t n e a r n o r m a l i n c i d e n c e , t h e s i t e r e f l e c t i o n s t e n d t o be l a r g e r f o r l a r g e r 0_^ . These r e f l e c t i o n s a c c o u n t f o r t h e o s c i l -l a t o r y b e h a v i o r o f some o f themmeasured c u r v e s and a r e a l a r g e r s o u r c e o f e x p e r i m e n t a l e r r o r . The e r r o r s i n r e a d i n g t h e a n g l e s a t w h i c h t h e measurements were t a k e n c o u l d a c c o u n t f o r a l m o s t a l l t h e s h i f t s i n t h e measured c u r v e s w i t h r e s p e c t t o t h e p r e d i c t e d c u r v e s . The n o n - p l a n e wave n a t u r e o f t h e i l l u m -i n a t i o n r e d u c e s t h e amount o f r e d u c t i o n v e r y l i t t l e as shown by J u l l [16] and J u l l and Ebbeson [ 4 ] , b u t when t h e measurements f a l l b e l o w abou t 30 dB i t i s t he r a t i o o f t h e r e c e i v e d s i g n a l t o t h e n o i s e o f t h e a m p l i f i e r w h i c h l i m i t s t h e a c c u r a c y o f t he measu rement s . As s een i n s e c t i o n 4 . 5 , o b l i q u e i n c i d e n c e has a m i n i m a l e f f e c t on t h e p e r f o r m a n c e o f t h e s u r f a c e . The u se o f a mode l f r e q u e n c y o f 35 GHz means t h a t a l l g r oove d i m e n s i o n s a r e l a r g e enough t o be m i l l e d t o a s u f f i c i e n t a c c u r a c y f o r most p l a t e s . S u r f a c e r oughnes s and f i n i t e c o n d u c t i v i t y e f f e c t s a r e n e g -l i g i b l e a t t h i s f r e q u e n c y . These e f f e c t s a r e t h e l a r g e s t s o u r c e o f e r r o r due t o g r a t i n g i m p e r f e c t i o n s ( R o u m i g u i e r e s , M a y s t r e and P e t i t [ 1 8 ] ) , when u s i n g t h e s e g r a t i n g s a t o p t i c a l f r e q u e n c i e s , b u t t h e y po se no p r o b l e m a t t h i s f r e q u e n c y . 86. The effect of systematic errors in the groove dimensions can be investigated, (Figs. B . l and B.2), but thererrors in a plate's groove dim-ensions tend to be both random and systematic. To reduce discrepancies due to errors in the groove dimensions, the actual dimensions given in F ig . 4.1 were used to calculate the curves. These errors account for the shifting of the measured frequency response curves.from what is predicted. However, these errors do not seem to pose too great a problem although they seem more important for surfaces that are very narrowband in nature. The effect of the f in i te size of the plates also seems to be not too great. The best correspondence occurs between angular response curves which are wide. These results seem least affected by the errors in the plates of experimental set up. 87. 5. CONCLUSIONS The a i m o f t h i s s t u d y was t o i n v e s t i g a t e t h e c a p a b i l i t y o f a p e r i o d i c s u r f a c e w i t h a r e c t a n g u l a r g r oove p r o f i l e t o s c a t t e r a l l i n c i -den t e n e r g y i n t o t h e m = -1 s p e c t r a l o r d e r . To t h i s end a n u m e r i c a l i n v e s t i g a t i o n o f t h e power r e f l e c t e d by a r e c t a n g u l a r g r oove p r o f i l e was p r e s e n t e d , a l o n g w i t h e x p e r i m e n t a l r e s u l t s w h i c h were t o be a check o f t h e n u m e r i c a l p r e d i c t i o n s . An e x a m i n a t i o n o f t h e s e r e s u l t s a l l o w s t h e f o l l o w i n g c o n c l u s i o n s t o be d rawn. (1) I t i s p o s s i b l e t o d e s i g n s u r f a c e s w h i c h b a c k s c a t t e r a l m o s t a l l t h e power o f an a r b i t r a r i l y p o l a r i z e d i n c i d e n t wave . The s u r f a c e s d e s i g n e d h e r e a p a r t f r o m p l a t e #2, b a c k s c a t t e r a t l e a s t 99% o f t h e i n c i -den t power a n d i i n t ' t h e b e s t c a se 9 9 . 9% . Even b e t t e r r e s u l t s we re r e p o r t e d f o r s u r f a c e s d e s i g n e d t o b l a z e o n l y an i n c i d e n t TM wave . The e f f i c i e n c y a c h i e v e d w o u l d make t h e s e s u r f a c e s s u i t a b l e f o r u se as l a s e r m i r r o r s o r as a means o f e l i m i n a t i n g r e f l e c t i o n f r o m c o n d u c t i n g s u r f a c e s . I t i s advan tageou s t o u se a r e c t a n g u l a r g r oove p r o f i l e i n s t e a d o f a comb p r o f i l e when t r y i n g t o r e d u c e r e f l e c t i o n s . The g r oove d e p t h s a r e s h a l l o w e r , a l l o w i n g e a s i e r c o n s t r u c t i o n , and t h e a n g u l a r r e s p o n s e c u r v e s a r e w i d e r , w h i c h r e s u l t s i n t h e e x p e r i m e n t a l r e s u l t s more c l o s e l y r e s e m b l i n g t he p r e d i c t e d r e s u l t s . U s i n g a r e c t a n g u l a r g r oove p r o f i l e as a l a s e r m i r r o r has. t h e d i s a d v a n t a g e t h a t t h e g roove s u r f a c e s a r e t o o r ough a t t he o p e r a t i n g f r e q u e n c i e s f o r t h e e f f e c t o f f i n i t e c o n d u c t i v i t y t o be n e g l i g i b l e , [ 1 8 ] . The r e c t a n g u l a r g r oove p r o f i l e does o f f e r , howeve r , t h e p o t e n t i a l advan tage t h a t i t can be d e s i g n e d t o b l a z e an a r b i t r a r i l y p o l a r i z e d i n c i d e n t wave . (2) The b l a z i n g d e p t h c u r v e s (hg/d v s . 6^) f o r a f u l l r a n ge o f a s p e c t r a t i o s f o r b o t h p o l a r i z a t i o n s were p r e s e n t e d f o r u se as d e s i g n 88. c u r v e s t o make t h e s e s u r f a c e s . The e x t r a p a r a m e t e r i n h e r e n t i n t h e r e c t a n g u l a r g r oove p r o f i l e means t h a t t h e r e i s a c h o i c e o f g r oove d i m e n -s i o n s when d e s i g n i n g a b l a z e d s u r f a c e . E x p e r i m e n t a l measurements we re made on s u r f a c e s whose d i m e n s i o n s c o v e r e d a w i d e r ange o f p a r a m e t e r s as a check on t h e v a l i d i t y o f t h e n u m e r i c a l r e s u l t s . These r e s u l t s a g r e e d , w i t h i n e x p e r i m e n t a l e r r o r , w i t h t h e n u m e r i c a l r e s u l t s . (3) T h i s e x t r a p a r a m e t e r means t h a t . i t i s p o s s i b l e t o d e s i g n a s u r f a c e w h i c h i s b l a z e d f o r a p o l a r i z a t i o n , i f 0^ > 1 9 . 5 ° . D e s i g n c u r v e s g i v i n g t h e g r oove d i m e n s i o n s n e c e s s a r y f o r d u a l b l a z i n g , were p r e s e n t e d f o r a n g l e s o f i n c i d e n c e up t o 7 3 ° . (4) The b l a z i n g d e p t h c u r v e s c o n v e r g e t o t h e c u r v e f o r t h e comb p r o f i l e as a/d 1, and t o t h e c u r v e ( s i n 0 ^ ) / 2 as a/d •-»- 0 . Thus i t i s p o s s i b l e t o a p p r o x i m a t e t h e r e s u l t s f o r t h e comb p r o f i l e w i t h a r e c t a n g u l a r g roove p r o f i l e whose a s p e c t r a t i o i s a l m o s t u n i t y . As t h e a s p e c t r a t i o d e c r e a s e s t o z e r o , t h e t o l e r a n c e on t h e g roove d e p t h needed t o b l a z e t h e s u r f a c e becomes more a c u t e u n t i l f i n a l l y a s u r f a c e w i t h any g r oove d e p t h , e x c e p t t he b l a z i n g d e p t h , r e f l e c t s a l l o f t h e i n c i d e n t power . (5) The d e s i g n c u r v e s were s t u d i e d t o f i n d r e g i o n s where t h e o p e r a t i n g p o i n t s have a c e r t a i n t y p e o f f r e q u e n c y o r a n g u l a r r e s p o n s e . I t was c o n c l u d e d t h a t o p e r a t i n g p o i n t s on a b l a z i n g d e p t h c u r v e , where t h e t a n g e n t i s h o r i z o n t a l , g i v e g r e a t l y r e d u c e d r e f l e c t e d power o v e r a w i d e r ange o f 0^ o r A . An o p e r a t i n g p o i n t on a b l a z i n g d e p t h c u r v e f o r an a s p e c t r a t i o n e a r e i t h e r z e r o o r one does n o t have as w i d e a r ange o f 0^ o r A , o v e r w h i c h t he r e f l e c t e d power i s r e d u c e d , as an o p e r a t i n g p o i n t on a b l a z i n g d e p t h c u r v e f o r an a s p e c t r a t i o midway be tween t h e s e two e x t r e m e s . D u a l b l a z e d s u r f a c e s have b e s t b r oadband p e r f o r m a n c e f o r 89 . 5 0° < 8 < 5 9 . 4 ° and n a r r o w band p e r f o r m a n c e i f t h e y a r e d e s i g n e d f o r 0. = 2 0 ° . x (6) I t was a l s o p o s s i b l e t o f i n d r e g i o n s where t h e r e f l e c t e d power i s r e l a t i v e l y i n s e n s i t i v e t o changes i n t h e g roove d e p t h o r w i d t h . I n s e n s i t i v i t y t o g roove d e p t h o c c u r s whe re a d e s i g n c u r v e hg/d v s . 0^ r i s e s q u i c k l y , w h i l e i n s e n s i t i v i t y t o g r oove w i d t h o c c u r s whe re t h e d e s i g n c u r v e s a r e c l o s e l y s p a c e d o r i n t e r s e c t . A s t u d y o f t h e e f f e c t o f s y s t e m a t i c and random e r r o r s i n t h e g r oove d i m e n s i o n s was p r e s e n t e d . I t was seen b o t h e x p e r i m e n t a l l y and n u m e r i c a l l y t h a t t h e s e e f f e c t s a r e o n l y s i g n i f i c a n t where t h e a n g u l a r o r f r e q u e n c y r e s p o n s e c u r v e s a r e n a r r o w . (7) As o b s e r v e d by Ebbeson [ 1 2 ] , i t was f o u n d t h a t t h e e f f e c t o f n o n - p l a n e wave i n c i d e n c e , o b l i q u e i n c i d e n c e , and t h e f i n i t e s i z e o f t he p l a t e s do n o t l i m i t t h e a p p l i c a t i o n o f t h e t e c h n i q u e . As a l r e a d y s shown, [ 4 , 1 6 ] , t h e r e s u l t s , though good h e r e , c o u l d o n l y be b e t t e r unde r p l a n e wave i l l u m i n a t i o n . I t was seen t h a t r o t a t i o n o f t h e s u r f a c e s up t o ± 10° cause s i n s i g n i f i c a n t changes i n t h e amount o f power r e f l e c t e d . The f i n i t e s i z e seems t o have l i t t l e e f f e c t on t h e amount o f power r e -f l e c t e d f o r s u r f a c e s w i t h more t h a n abou t 20 c o r r u g a t i o n s . 90 . APPENDIX A: B l a z i n g Depth D a t a The f o l l o w i n g i s a c o l l e c t i o n o f t h e b l a z i n g d e p t h s needed f o r each a s p e c t r a t i o and a n g l e o f i n c i d e n c e . Mos t o f t h i s d a t a a p p e a r s i n g r a p h i c a l f o r m i n C h a p t e r 3. e^s. a/d d e g r e e s x ^ 0.500 0.600 0.667 01750 0.800 0 .900 19.50 0 .6050 20.00 0.2644 0.2400 0.2456 0.2775 0.3412 0 .6025 20.65 0 .6025 21.46 0 .2962 21.88 0 .3900 22.05 0 .3775 24.22 0 .3501 25.00 0.5150 0.3575 0.3269 0.3175 0.3214 0.3525 28.00 1.0125 28.50 1.2275 30.00 0 .5800 0 .4613 0.4062 0.3937 0.3962 34.00 1.1025 35.00 1.4700 0.7225 0.5406 0.4975 0.4650 37.50 1.0050 38.50 1.2050 40.00 1.7900 0.7775 0.6525 0.5550 42.00 0.9425 43.00 1.0575 45.00 1.4300 0.6775 46.00 1.7700 1.0225 50.00 1.7700 0.8550 52.50 0 .9850 55 .00 1.1600 57.50 1.4300 60 .00 1.9200 T T a b l e I: TE B l a z i n g Depths v s . 6 ±: a/d = 0.500 - 0 .900 6>v a/d 0.95 0 .99 1.00 ( d e e r e e s ) \ . 19.50= 0.7200 0.7250 20,00 0.6487 21.46. 0 .6450 22.05 0.6375 22.50 0.6825 22.70 0 .4650 23.00 0.6725 24.00 0 .4500 24 .22 0.3875 25 .00 0.3825 0 .4225 3 0 . 0 0 0 .4100 0.4300 35 .00 0.4675 0.4787 35 .50 0.5000 40 .00 0.5412 0.5425 42.86 0.6000 45 .00 0.6350 0.6212 48 .75 0.7000 50 .00 0.757.5 0.7175 53.75 0.8000 55 .00 0.9300 0.8425 57 .50 1.0450 57 .18 0.9000 60 .00 1.0075 60 .41 1.0000 65 .00 1.2450 70.00 1.6200 75.00 2 .3500 T a b l e I I : TE B l a z i n g Depths v s . 6 ^ a/d = 0.95 - 1. (degrees) 0.667 0.750 0.800 0.900 25.0 1.0500 25.7 1.0830 25.7 .0.7440 28.6 1.1490 28.6 1.0200 28.6 0.7310 30.0 0.7470 1.1927 1.2148 1.6123 31.0 0.8161 1.3447 32.5 1.3247 33.0 0.8000 0.8251 1.2381 33.0 0.9056 35.0 0.80.63 0.8421 0.8836 1.3170 36.7 0.8090 36.7 0.5570 36.7 0.3590 39.0 0.8624 0.8930 40.0 0.3333 1.0475 42.0 0.4762 42.5 0.3261 0.9865 45.0 0*3190 0.4037 0.7206 0.9456 48.0 0.4769 0.9056 50.0 0.3090 55.0 0.3244 0.3734 0.7575 60.0 0.2620 0.5719 65.0 0.2408 0.2669 0.4203 75.0 0.1333 0.1475 0.1634 0.2303 85.0 0.0430 0.0526 0.0603 0.0866 Table III: TM Brimary Blazing Depths vs. 6 : a/d = 0.667 - 0.900 93 . e J ^ V a / d (degrees) 0.9500 0.9900 0.9950 0.9999 32.5 1.6300 1.7180 1.7259 35 . 0 1.3772 1.6594 1.6928 1.7153 40 .0 1.3217 1.3936 1.4010 1.4083 45 .0 1.0303 1.3575 1.3787 1.3967 48 .0 0.9617 1.0312 1.0475 1.0687 55 .0 0 .9004 0.9521 0.9580 0.9637 65 .0 0 .7788 0.9425 0.9528 0.9622 75 .0 0 .3500 0.9128 0.9491 0 .9741 85 .0 0 .1228 0.2600 0.3925 0.9794 87.5 0.9675 89 .0 0 .7400 T a b l e IV : TM P r i m a r y B l a z i n g Depth s v s . 9 : a/d = 0 . 9 5 0 0 - 0.9999 1 ( d e g r e e s ) x . 1.0 31.76 1.7430 33.75 1.7440 34.85 1.7260 35.08 1.7240 35.31 1.7110 35.55 1.6700 35.79 1.6140 36.03 1.5060 36 .84 1.4508 37 .04 1.4400 38.68 1.4200 41 .81 1.4170 43 .60 1.4140 45 .58 1.4000 46 .44 1.3770 46 .88 1.3580 47 .33 1.2940 47.79 1.1050 50.28 1.0049 56.44 0.9692 59.99 0 .9673 65 .38 0 .9713 68 .88 0.9754 78.64 0.9869 80.40 0.9884 81 .24 0 .9893 85.42 0.9920 T a b l e V: TM P r i m a r y B l a z i n g Dep th s v s . 6 : a/d = 1.0 ( f r o m [ 12 , 13 ] ) 0.00001 0.001 0.010 0.050 0.100 25.0 0.2113 0.2101 011999 0.1734 0.1581 35.0 0.2868 0.2857 0.2763 0.2530 0.2400 45.0 0.3535 0.3525 0.3434 0.3206 0.3081 55.0 0.4096 0.4085 0.3995 0.3772 0.3644 65.0 0.4531 0.4521 0.4431 0.4206 0.4069 75.0 0.4829 0.4819 0.4730 0.4500 0.4312 85.0 0.4981 0.4971 0.4881 0.4550 0.3800 87.5 0.4995 0.4250 0,2250 T a b l e V I : TM P r i m a r y B l a z i n g Dep th s v s . 9 ± : a/d = 0.00001 - 0.100 ( d e g r e e s ) ^s. 0.250 0.333 0.500 24.0 0.1443 0.1550 0.5857 26Q0 0.1643 0.1739 0.2192 26.0 0.6285 26.0 0.3423 27.0 0.2209 28.0 0.1823 0.1909 0.2246 30.0 0.1989 0.2067 0.2341 32.0 0.2147 0.2216 0.2441 35.0 0.2371 0.2426 0.2587 40.0 0.2713 0.2741 0.2798 45.0 0.3018 0.3011 0.2949 55.0 0.3496 - 0.3375 0.2975 65.0 0.3714 0.3342 0.2462 75.0 0.3781 0.2387 0.1458 85.0 0.0902 0.0602 0.0447 T a b l e V I I : TM P r i m a r y B l a z i n g Depth s v s . B±i a/d = 0.250 - 0.500 96. a/d e i \ ( deg ree s ) 1 \ 0 .250 0.333 0 .500 0 .667 24 .00 0.5509 0.5612 26.00 0.6026 0 .6118 28.00 0.6516 0.6599 1.1525 28.50 1.1690 28.75 1.1756 29 .00 1.1800 29 .00 0 .8831 29 .00 0 .7375 29.25 1.1600 29.25 0 .7287 29 .50 0 .7287 30 .00 0.6989 0.7063 0 .7341 32.00 0.7446 0.7512 0.7697 34 .00 0 .7890 0.7947 0 .8083 37 .50 1.5200 38.50 1.5362 40.00 0 .9141 0.9167 0.9179 1.5425 42 .00 0 .9930 42 .50 0 .9950 45 .00 1,0089 1.0080 0.9972 1.0075 50 .00 1.0941 1.0887 55 .00 1.1687 1.1562 1.1100 1.0700 60 .00 1.2312 1.2087 1.1362 65 .00 1.2775 1.2403 1.1375 1.0737 70.00 1.3025 1.2425 1.1150 75.00 1.2950 1 .2025 1.0787 1.0437 80 .00 1.2200 1.1188 1.0412 85.00 1.0762 1.0347 1.0159 1.0216 T a b l e V I I I : TM Seconda r y B l a z i n g Depth s v s . 6 : a/d = 0 .250 - 0 .667 I ( d e g r e e s ) 0 .75 0 .80 0 .90 0.99 40.0 2.1625 2.3025 42.5 1.7150 2.2850 45 .0 1.7150 47.5 1.1650 1.8250 50 .0 1.1013 1.7725 2.4675 52 .5 1.0900 1.2225 2.3175 55 .0 1.0875 1.1512 2.0850 57 .5 1.9650 60 .0 1.8550* 2 .6581 62 .5 •1.5375 65 .0 1.0712 1.0825 1.2850 2.2814 70.0 2.0286 75 .0 1.0450 1.0512 1.0900 2 .0020 80 .0 1.9162 85 .0 1.0300 1.0373 1.0597 2.0286 T a b l e IX : TM Seconda r y B l a z i n g Depth s v s . 0 : a/d = 0.75 - 0.99 98. 6. l ( d e g r e e s ) h/d a/d h / A a / A 19 .50 0.398 0.386 0.596 0.579 20.00 0.446 0.391 0.652 0.572 26.15 0 .631 0 .500 0.716 0.566 30.00 0.715 0.567 0.715 0 .567 35 .94 0.807 0.667 0 .687 0.568 40 .00 0.836 0.736 0 .650 0 .572 45 .00 0 .844 0 .821 0.597 0 .581 50 .00 0.867 0.896 0.566 0.585 59 .40 0 .968 1.000 0 .562 . 0 .581 1 T a b l e X: a/d and P r i m a r y B l a z i n g Depth s v s . 9. f o r S i m u l t a n e o u s B l a z i n g 0. 1 ( d e g r e e s ) h/d . a/d h / A al A 19 .50 1>. 065 0.386 1.595 0.578 28 .37 1.164 0 .500 1.225 0.526 39 .54 1.548 0.667 1.216 0 .524 45.85 1.702 0.750 1.186 0 .523 50 .00 1.772 0 .800 1.157 0.522 59 .81 1.866 0 .900 1.079 0.521 73,12 2.008 0 .990 1.049 9 ,573 T a b l e X I : a/d and Seconda r y B l a x i n g Depth s v s . 6 f o r S i m u l t a n e o u s B l a z i n g 99 . APPENDIX B: Graphs o f F r e q u e n c y Responses o f S u r f a c e s w i t h S y s t e m a t i c E r r o r s . The f o l l o w i n g a r e g raphs w h i c h show t h e f r e q u e n c y r e s p o n s e o f two s u r f a c e s as t h e g roove d i m e n s i o n s a r e s y s t e m a t i c a l l y changed . E a ch g raph c o r r e s p o n d s t o a s u r f a c e w h i c h has a d i f f e r e n t s y s t e m a t i c e r r o r i n t h e g roove d i m e n s i o n s . to II II • • O Ul 2* P il II Ul U) Ul cu II II Ul .£> Ul Ul 3 B 09 & . C D »rj H- H ii ro II ,a c -P- ro Ui 3 o O - ^ H i O fD f-{ W X) cn o ^ 3 cn cn rr CD ro cn 3 as a Hi W CO S C I—" rr Co ro o rr C cn ro r-1 o hi td O M O CO <; N ro ro an H H i cn co H- O o ro •3 » CO H-rt I C3 6x o Q o<3 •nm REFLECTED POWER (db) Co NJ C3 C5 I CD C3 REFL EC TED P0 V/ER (db) O C3 5 rrj oS 4?s tL til J . C3 O O C3 C3_ O REFLECTED POVfER (db) 3* to II II Ul f> Ul CO CT> ^ O *1 CM O. o 4 H- H II II 3* (D II II Ul -C-Ul CO CO --J 11 a & c to M o • o y Ul o O 3 H, >d 3 o ro H 01 Co o VI a cn w rt p a o O Hi M H-3 H H o H rr cn to p - ro 3 O c rt cn 3* t-» ro vj H H 1 o to o N <! co ro cv o cn H- c 3 H ro HI to o • ro 3 cn H-O 3 s: cn H-rt 3* to II II Ul -P-<T> CO I 1 REFLECTED POWER (db) 09 3* Co II II Ul £~ Ul vo as ro 3 3 § 3 rj* co II II Ul Ui V D 00 N > g g 3 3 co II ll Ul j> Os V O r-1 M ! i i cn o I CD REFLECTED POWER (db) £ N O I • IX CD REFLECTED POWER (db) CO No CD CD I CD ' SOI 106. REFERENCES [1] G.W. S t r o k e , " D i f f r a c t i o n G r a t i n g s " i n Handbuch d e r P h y s i k , v o l . 29 , p p . 570 -583 ( S p r i n g e r , B e r l i n , 1 967 ) . [ 2 ] A . H e s s e l , J . Schmoys and D.Y. T s eng , " B r a g g - A n g l e B l a z i n g o f D i f f -r a c t i o n G r a t i n g s " , J . Op t . Soc . Am., v o l . 6 5 , p p . 380 -384 , A p r i l 1975. [3] A. W i r g i n and R. D e l e u i l , " T h e o r e t i c a l and E x p e r i m e n t a l I n v e s t i g a t i o n o f a New Type o f B l a z e d G r a t i n g " , J . O p t . Soc . Am., v o l . 5 9 , p p . 1348 -1357 , O c t . 1969. [4] E .V. J u l l and G.R. E b b e s o n , " The R e d u c t i o n o f I n t e r f e r e n c e f r o m L a r g e R e f l e c t i n g S u r f a c e s " , IEEE T r a n s . An tenna s and P r o p a g a t i o n , A P - 2 5 , J u l y 1977 ( i n p r e s s ) . [ 5 ] T. I t o h and R. M i t t r a , " A n A n a l y t i c a l S t u d y o f t h e E c h e l e t t e G r a t i n g w i t h A p p l i c a t i o n t o Open R e s o n a t o r s " , I EEE T r a n s . MTT-17, p p . 3 1 9 -328, June 1969. [6] K.A. Z a k i and A .R . N e u r e u t h e r , " S c a t t e r i n g f r o m a P e r f e c t l y Conduc -t i n g S u r f a c e w i t h a S i n u s o i d a l H e i g h t P r o f i l e : TE P o l a r i z a t i o n " , I EEE T r a n s , on An tenna s and P r o p a g a t i o n , v o l . A P - 1 9 , p p . 208 - 214 , Ma rch 1971 . [7] K.A. Z a k i and A .R . N e u r e u t h e r , " S c a t t e r i n g f r o m a P e r f e c t l y Conduc -t i n g S u r f a c e w i t h a S i n u s o i d a l H e i g h t P r o f i l e : TM P o l a r i z a t i o n " , I EEE T r a n s , on An tenna s and P r o p a g a t i o n , v o l . A P - 1 9 , p p . 7 47 - 751 , November 1971. [8] D.Y. T seng , " G u i d i n g and S c a t t e r i n g o f E l e c t r o m a g n e t i c F i e l d s b y C o r r u g a t e d S t u r c t u r e s " , P h .D . T h e s i s ( P o l y t e c h n i c I n s t i t u t e o f B r o o k l y n , 1 967 ) . [9] D.Y. Tseng, A . H e s s e l , and A . A . O l i v e r , " S c a t t e r i n g by a M u l t i m o d e C o r r u g a t e d S t r u c t u r e w i t h A p p l i c a t i o n t o P Type Wood A n o m a l i e s " , A l t a . F r e q . , v o l . 38 , p p . 8 2 - 8 8 , 1969. [10] J . A . D e S a n t o , " S c a t t e r i n g f r o m a P e r i o d i c C o r r u g a t e d S t r u c t u r e : T h i n Comb w i t h S o f t B o u n d a r i e s " , J . M a t h . P h y s . , v o l . 12 , pp.1.1913-1 9 2 3 , S e p t . 1971 . [11] J . A . D e S a n t o , . . , . " S c a t t e r i n g f r o m a P e r i o d i c C o r r u g a t e d S t r u c t u r e I I : T h i n Comb w i t h H a r d B o u n d a r i e s " , J . M a t h . P h y s . , v o l . 1 3 , p p . 3 36 -341 , M a r c h 1972 . [12] G.R. E b b e s o n , "The Use o f F i n - C o r r u g a t e d P e r i o d i c S u r f a c e s f o r t he R e d u c t i o n o f I n t e r f e r e n c e f r o m L a r g e R e f l e c t i n g S u r f a c e s " , M . A . S c .  T h e s i s , ( U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1974 ) . [13] G.R. E b b e s o n , "TM P o l a r i z e d E l e c t r o m a g n e t i c S c a t t e r i n g f r o m F i n -C o r r u g a t e d P e r i o d i c S u r f a c e s " , J . Op t . S o c . Am. , v o l . 6 6 , p p . 1 3 6 3 -1367, Dec . 1976. 107. [14] R .E . C o l l i n , F i e l d Theo r y o f G u i d e d Waves, M c G r a w - H i l l : New Y o r k , 1960. [15] E .V. J u l l , " R e d u c t i o n o f I n t e r f e r e n c e f r o m L a r g e R e f l e c t i n g S u r f a c e s I", MOT R e s e a r c h S t udy C o n t r a c t R e p o r t , D e p t . o f E l e c t . E n g . , UBC, June 1974. [16] E . V . . J u l l , " R e d u c t i o n o f I n t e r f e r e n c e f r o m L a r g e R e f l e c t i n g S u r f a c e s I I " , MOT R e s e a r c h S tudy C o n t r a c t R e p o r t , D e p t . o f E l e c t . E n g . , UBC, Aug . 1975. [17] A . W i r g i n , "On t h e Theo ry o f S c a t t e r i n g f r o m Rough L a m e l l a r S u r f a c e s " , A l t a . F r e q . , v o l . 38 , p p . 3 27 - 331 , 1969. [18] J . L . R o u m i g u i e r e s , D. M a y s t r e and R. P e t i t , "On t h e E f f i c i e n c i e s o f R e c t a n g u l a r - G r o o v e G r a t i n g s " , J . O p t . S o c . Am. , v o l . 6 6 , p p . 772 -775 , A u g . 1976. [19] E .V. J u l l , J .W. H e a t h , and G.R. E b b e s o n , " G r a t i n g s t h a t D i f f r a c t A l l I n c i d e n t Energy '/ , J . O p t . Soc . Am., v o l . 67 , p p . 5 5 7 - 5 6 0 , A p r i l 1977. 

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