EXPERIMENTAL INVESTIGATION OF SOME CONDUCTING CROSSED GRATINGS by L I - H E |CAI B . A . S c , Tsinghua A THESIS SUBMITTED University ( C h i n a ) , 1970 I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE in THE FACULTY OF GRADUATE STUDIES ELECTRICAL ENGINEERING We a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the required standard THE UNIVERSITY OF B R I T I S H COLUMBIA January © 1985 L i - h e C a i , 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of requirements f o r an advanced degree at the the University o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it f r e e l y a v a i l a b l e f o r reference and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of department or by h i s or her representatives. my It i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my permission. Department of ^3 The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date )E-6 £fa/C^l- (3/81) s*t&& written ABSTRACT The properties of a crossed grating of square pyramids and a crossed grating with hemispherical c a v i t i e s to eliminate specular r e f l e c t i o n from a conducting surface are studied experimentally. Measurements were made in the microwave range of 35 GHz. The best performance i s that 99.94% of the power of a TM-polarized incident wave can be scattered into a single spectral order by a pyramidal crossed grating, while for TE polarization the reduction in specular r e f l e c t i o n can be as high as 98%. A n t i - r e f l e c t i o n properties of a crossed grating with hemispherical c a v i t i e s near normal incidence are also observed. Comparison between the behavior of triangular and pyramidal gratings of the same p r o f i l e i s made. E f f e c t s of the p r o f i l e parameters are investigated. B a s i c a l l y the experimental results agree with the t h e o r e t i c a l predictions. This investigation provides a set of experimental data to a s s i s t further numerical study. Table of Contents ABSTRACT i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS v ACKNOWLEDGEMENTS ix 1. INTRODUCTION 1 2. SCATTERING BY CROSSED GRATINGS 9 2.1 Formulation of the Problem 9 2.2 The Rayleigh Expansion for the D i f f r a c t e d F i e l d .11 2.3 Determination of the Directions of Propagation of the D i f f r a c t e d Waves 15 2.3.1 Grating with Period Smaller than Half of Wavelength ...16 2.3.2 Grating with Period between a Half Wavelength and One Wavelength 2.3.2.1 Under Non-oblique (*.=0) 17 Incidence 2.3.2.2 Under Oblique Incidence 2.3.3 Grating with Period Greater than Wavelength 17 18 19 2.4 Determination of the Angular Region Where a D i f f r a c t e d Order Exists 21 2.5 A Product Formula Linking Crossed and C l a s s i c a l Gratings 23 3. EXPERIMENTAL ARRANGEMENT AND PROCEDURE 25 4. EXPERIMENTAL RESULTS OF PYRAMIDAL CROSSED GRATINGS ..31 4.1 Introduction 31 4.2 Plate B9 In Comparison with i t s Single-periodic Equivalent 35 4.3 Plate C4 A Deeply-grooved Plate in Comparison with i t s Singly-periodic Equivalent ..47 4.4 Plate A4 an E s s e n t i a l l y Perfect Blazed Crossed Grating Surface for TM p o l a r i z a t i o n 58 4.5 Plate A9 Influence of the Ratio X/d 70 4.6 Plate B6 Influence of Apex Angle a 78 4.7 Plate A6 a Blazed Crossed Grating Surface for TE Polarization 85 EXPERIMENTAL RESULTS OF CROSSED GRATINGS WITH HEMISPHERICAL CAVITIES 91 5.1 Introduction 91 5.2 Plate R4 93 5.3 Plate R6 101 6. ERROR ANALYSIS 106 7. CONCLUSIONS 111 5. APPENDIX: NUMERICAL RESULTS FOR PERFECTLY CONDUCTING STRIPS OF SIX SYMMETRICAL TRIANGULAR CORRUGATIONS ...115 REFERENCES 122 iv LIST OF ILLUSTRATIONS F i g . 2.1. The s p e c i f i c a t i o n of the incident f i e l d E and of the d i f f r a c t e d wave vectors It in the (Oxyz) system v: 1 9 F i g . 2.2. The d i r e c t i o n s of the d i f f r a c t e d wave vectors scattered by a crossed grating with X/d=0.66 in normal incidence ..20 F i g . 3.1. Photograph of experimental set-up 26 F i g . 3.2. Experimental arrangement 27 F i g . 4.1. Photograph of a pyramidal crossed plate (plate C4) grating 32 F i g . 4.2. A c l a s s i c a l triangular (echelette) grating and a pyramidal crossed grating with the same p r o f i l e , and dimensions for a l l pyramidal plates 33 F i g . 4.3a. Relative r e f l e c t e d power (or e ) vs. angle of incidence for the singly periodic equivalent of plate B9 at f = 35 GHz, ^ = 0 0 36 F i g . 4.3b. Relative r e f l e c t e d power (or e o) s . angle of incidence for plate B9 at f=35 GHz, ^=0 ......37 v 0 F i g . 4.4. Reflected power vs. angle of rotation for plate B9 at f = 35 GHz, 6^ = 43.5° 43 F i g . 4.5. Diagram of azimuthal angular regions in which spectral orders exist for a crossed grating at X/d=1.37, 6^ = 43. 5° 44 F i g . 4.6. Reflected power vs. frequency for plate B9 at 6^ = 43.5°, ^ = 0 46 F i g . 4.7. Relative r e f l e c t e d power (or e ) vs. angle of incidence for the singly periodic equivalent of plate C4 at f = 35 GHz, 1 ^ = 0 48 0 F i g . 4.8. Reflected power vs. frequency for the singly periodic equivalent of plate C4 at 0.=38°, i// 0 = i F i g . 4.9. Relative r e f l e c t e d power (or e ) vs. angle of incidence for the singly periodic equivalent of plate C4 at f = 33.75 GHz, \/N=0 49 0 50 F i g . . 4 . 1 0 . Relative r e f l e c t e d power (or e ) vs. angle of incidence for plate C4 at f=33.75 GHz, ^=0 ..52 0 0 F i g . 4.11. Reflected power vs. angle of incidence for v plate C4: a. at f=33.75 GHz b. at f=35 GHz 53 F i g . 4.12. Reflected power vs. frequency for plate C4 at 0^38°, *.=0 54 F i g . 4.13. Reflected power vs. angle of rotation for plate C4 at f = 35 GHz, 0-^=38° 56 F i g . 4.14. Diagram of azimuthal angular regions in which spectral orders exist for a crossed grating at X/d=1.24, 0^38° 57 F i g . 4.15. Relative r e f l e c t e d power (or e ) vs. angle of incidence for the singly periodic equivalent of plate A4 at f=35 GHz, ^=0 ...59 0 F i g . 4.16. Reflected power vs. angle of incidence for plate A4 a t : a. f = 35 GHz, i£.=0 b. f = 33 GHz, ^=0 F i g . 4.17. Reflected power vs. frequency for plate A4 at 0^46°, ^=0 F i g . 4.18. Reflected power vs. angle of incidence for plate A4 at f = 37.5 GHz, ^ = 0 F i g . 4.19. Reflected power vs. angle of rotation for plate A4 at f = 37.5 GHz, 0^46° 60 ...61 ...63 65 F i g . 4.20. Diagram of azimuthal angular regions in which spectral orders exist for a crossed grating at X/d=1.00, 0^46° ...66 F i g . 4.21. Reflected power vs. angle of rotation for plate A4 at f = 35 GHz, 0^32.5° ...67 F i g . 4.22. Diagram of azimuthal angular regions in which spectral orders e x i s t for a crossed grating at X/d=1.07, 0^32.5° ...68 F i g . 4.23. Reflected power vs. angle of incidence for plate A9: a. at f=37.5 GHz (X/d=1.00) b. at f = 35 GHz (X/d=1.07) ...71 F i g . 4.24. Reflected power vs. angle of incidence for a. plate A9 at f=33 GHz (X/d=1.14) b. plate B9 at f = 35 GHz (X/d-1.37) ...72 F i g . 4.25. Reflected power vs. frequency for plate A9 at 0^32.5°, i^-O F i g . 4.26. Reflected power vs. angle of rotation for plate A9 at 0 ^ 3 5 ° , f = 35 GHz vi ...74 75 F i g . 4.27. Diagram of azimuthal angular regions in which spectral orders exist for a crossed grating at X/d=1 .07, 6^=35° 76 F i g . 4.28. Relative r e f l e c t e d power (or e ) vs. angle of incidence for the singly periodic equivalent of plate B6 at f = 35 GHz, ^ = 0 ...79 0 F i g . 4.29. Relative r e f l e c t e d power (or e ) vs. angle of incidence for plate B6 at f = 35 GHz, </^ = 0 ....80 0 0 F i g . 4.30. Reflected power vs. frequency for plate B6 at 6^ = 43.5°, ^ = 0 82 F i g . 4.31. Reflected power vs. angle of rotation for plate B6 at f = 35 GHz, 6^ = 43.5° 83 F i g . 4.32. Reflected power vs. angle of incidence for plate A6: a. at f=37.5 GHz (X/d=1.00) b. at f = 35 GHz (X/d=1.07) 86 F i g . 4.33. Reflected power vs. angle of incidence a. for plate A6, at f=33 GHz (X/d=1.14) b. for plate B6, at f = 35 GHz (X/d=1.37) 87 F i g . 4.34. Reflected power vs. angle of rotation for plate A6 at f = 35 GHz, 6^ = 35° 89 F i g . 5.1. The p r o f i l e of a crossed grating with hemispherical c a v i t i e s , and dimensions for such two plates investigated 92 F i g . 5.2. Measured r e f l e c t e d power vs. 9- for plate R4 at f = 35 GHz, 1 ^ = 0 94 F i g . 5.3 Measured r e f l e c t e d power vs. 9. for plate R4 at f = 33 GHz, ^.=0 96 F i g . 5.4. Measured r e f l e c t e d power vs. at f = 35 GHz, ^ = 10° 98 for plate R4 F i g . 5.5. Measured r e f l e c t e d power vs. ^. for plate R4 at f = 35 GHz, 6^ = 30° 100 F i g . 5.6. Measured r e f l e c t e d power vs. 9- for plate R6 at f = 35 GHz , 1 / ^ = 0 102 F i g . 5.7. Measured r e f l e c t e d power vs. 0- for plate R6 at f = 33 GHz, ^.«0 103 F i g . 5.8. Measured r e f l e c t e d power vs. e. for plate R6 at = i0°, f = 35 GHz F i g . 6.1. Relative power of TM component of the reflected wave when a TE-polarized wave i s vi i ..105 incident on plate C4 109 F i g . A.I. Comparison between numerical results for perfectly conducting s t r i p s of six symmetrical triangular corrugations and experimental results for the corresponding triangular grating plates of 30-39 grooves with the same p r o f i l e 116 F i g . A.2. Calculated power patterns for s t r i p A4 at 0^32.4° and f=35 GHz 117 F i g . A.3. Calculated power patterns for s t r i p B6 at 0^43.5° and f = 35 GHz 118 F i g . A.4. Calculated power patterns for s t r i p B9 at 0^43.5° and f = 35 GHz 119 F i g . A.5. Calculated power patterns for s t r i p C4 at 0^38.1° and f=35 GHz 120 F i g . A.6. Calculated power patterns for s t r i p C4 at 0^40.0° and f = 33.75 GHz 121 viii ACKNOWLEDGEMENTS I wish to express sincere thanks to my supervisor Dr. E. V. J u l l for his patience, guidance and encouragement during the research work of t h i s thesis. Special thanks are due to Dr. R. D e l e u i l (Universite de Provence, France) for the h e l p f u l advice and suggestions that he gave me during his two v i s i t s to Vancouver. I would l i k e to thank Mr. Derek Daines for his spending many hours m i l l i n g the p l a t e s . Two plates with hemispherical c a v i t i e s were made by Laboratoire de R a d i o e l e c t r i c i t e , Universite de Provence, France. F i n a n c i a l support, a research a s s i s t a n t s h i p from the Natural Sciences and Engineering Research Council of Canada and a teaching a s s i s t a n t s h i p from the Department of E l e c t r i c a l Engineering, U.B.C., i s grateful acknowledged. Chapter 1 INTRODUCTION During the past f i f t e e n years t h e study of d i f f r a c t i o n g r a t i n g s h a s made g r e a t p r o g r e s s . R i g o r o u s theories [ 2 0 ] h a v e been d e v e l o p e d s i n g l y - p e r i o d i c , or c l a s s i c a l electromagnetic t o e x p l a i n the behavior of gratings with a period i n the range of t h e i n c i d e n t wavelength, a n d t h e y seem c a p a b l e o f d e a l i n g w i t h a l l s t r u c t u r e s of p r a c t i c a l importance. In. t h e l a s t d e c a d e i n t e r e s t h a s grown i n t h e p r o p e r t i e s of d o u b l y - p e r i o d i c g r a t i n g s , o r c r o s s e d g r a t i n g s , w h i c h have regular two-dimensional s t r u c t u r e (pyramids, bumps, h e m i s p h e r i c a l c a v i t i e s , periodically it etc.) distributed on a p l a n e . I n a s e r i e s o f r e c e n t h a s been shown t h a t s u c h s e l e c t i v e a b s o r p t i o n of s o l a r non-absorptive reflection publications, s t r u c t u r e s have g r e a t in useful applcations, p a r t i c u l a r l y filtering bisinusoidal i n connection radiation potential with [5,19,24], r e d u c t i o n [ 2 4 ] , and microwave [ 1 5 ] . Thus, i t i s v e r y important t o have accurate t h e o r e t i c a l models of c r o s s e d g r a t i n g behavior and t o c o n f i r m these models w i t h c a r e f u l e x p e r i m e n t a l measurements. Theoretical s t u d i e s of the d i f f r a c t i o n g r a t i n g s are. s t i l l wait u n t i l d e v e l o p i n g . T h e i r commencement h a d t o the study of c l a s s i c a l highly-developed by c r o s s e d g r a t i n g s reached s t a g e . However, t h e p r o g r e s s w h i c h i s due m a i n l y t o the fact that i n going a h a s been from slow, classical to crossed g r a t i n g s , the s i z e of the a s s o c i a t e d numerical p r o b l e m i s more t h a n squared, and so r i g o r o u s 1 techniques 2 valuable f o r the former impracticable f o r the l a t t e r present computers). r e q u i r e the million become, when g e n e r a l i z e d , (given the c a p a b i l i t i e s For example, such t e c h n i q u e s a elementsf20]. diffraction f o r m a l i s m s have been p r o p o s e d o f e l e c t r o m a g n e t i c r a d i a t i o n by g r a t i n g s . The earlier [3] f o r the case formalisms upon t h e a v a i l a b i l i t y c a n be f o r the crossed ( f o r e x a m p l e t h o s e o f Chen of a c o n d u c t i n g screen p e r f o r a t e d periodically with either circular or r e c t a n g u l a r h o l e s ) of a complete superposed to s p e c i f y the f i e l d t h e b o u n d a r y c o n d i t i o n s on o n l y known a n a l y t i c a l l y conducting geometries rely s e t of modal f u n c t i o n s r e g i o n . T h e s e m o d a l f u n c t i o n s must s a t i s f y and would i n v e r s i o n o f c o m p l e x m a t r i c e s h a v i n g more t h a n Several different which of i n the aperture t h e wave e q u a t i o n t h e a p e r t u r e w a l l s , and f o r a s m a l l number o f [ 2 0 ] . R e c e n t l y Toro and are perfectly Deleuil [21,22] implemented a r i g o r o u s modal t h e o r y f o r the study electromagnetic d i f f r a c t i o n from a p e r f e c t l y conducting c r o s s e d g r a t i n g w i t h h e m i s p h e r i c a l c a v i t i e s . The s t u d y was realized polarization, but f o r any theoretical i n c i d e n c e and d i r e c t i o n the f i r s t numerical of of r e s u l t s were o n l y f o r the s i m p l e r case of normal i n c i d e n c e . Wirgin's of formalism [25] i s also restricted a perfectly-conducting crossed grating. This s i m p l e m e t h o d i s b a s e d on an a p p r o x i m a t i o n diffracted field above the s u r f a c e can R a y l e i g h p l a n e wave e x p a n s i o n . that to the case extremely the be e x p r e s s e d as the H e n c e , i t c a n o n l y be a p p l i e d 3 to very shallowly-grooved (h/d, groove depth over period of grating, in the range of 0.1) conducting surfaces. A few numerical results were obtained by t h i s method for a perfectly conducting crossed grating of a sinusoidal p r o f i l e with h/d=0.14 under normal incidence. Maystre and Neviere [17], as well as Vincent [23] worked out d i f f e r e n t i a l formalisms v a l i d for crossed gratings whose surface conductivity i s not very high and whose depth i s not very large compared with the wavelength. They obtained a few numerical results for a pyramidal grating with f i n i t e conductivity and h/d=0.25. Derrick et a l . [5,19] elaborated a formalism which i s quite d i f f e r e n t from others. They used a coordinate transformation technique which flattens the crossed grating p r o f i l e , combined with an i t e r a t i v e method for resolution of the very large system of complex equations expressing the boundary and outgoing wave conditions for the d i f f r a c t i o n problem of crossed gratings. They presented in [19] some numerical results for the sinusoidally-grooved crossed gratings and a few examples for pyramidal crossed gratings with f i n i t e conductivity. In a private communication [16], they gave some calculated curves for pyramidal crossed gratings with i n f i n i t e conductivity. Among a l l e x i s t i n g formalisms, the last one i s the most v e r s a t i l e and powerful, since i t can work throughout the whole range of values of surface conductivity, including i n f i n i t e conductivity. Much work had been devoted to this 4 formalism Recently i n order t o extend i t s range of a p p l i c a b i l i t y . i t was r e p o r t e d [ 1 9 ] t h a t a p r a c t i c a l been r e a c h e d : t h e t h e o r y c o u l d n o t be e x t e n d e d t o d e e p e r c r o s s e d g r a t i n g s i n w h i c h h/d i s g r e a t e r t h a n further l i m i t had s i g n i f i c a n t progress one, and required a completely new method. In comparison with t h e o r e t i c a l experimental has investigations, r e s e a r c h on t h e d i f f r a c t i o n been v e r y l i m i t e d . Wilson and Hutley by c r o s s e d g r a t i n g s [24] 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 e o p t i c a l p r o p e r t i e s of a r t i f i c i a l eye' antireflection kind of i r r e g u l a r They f o u n d visible s u r f a c e s w h i c h c a n be r e g a r d e d moth e y e s c a n s e l e c t i v e l y r a d i a t i o n and r e j e c t • i n f r a r e d made them p r o m i s i n g structure selective solar absorbers. B l i e k and D e l e u i l experiment. g r a t i n g s i n t h e microwave a n d t h e y a l s o began a n i n v e s t i g a t i o n crossed gratings, a conducting a conducting pyramidal w i t h l e s s than 1 am), t h e y d i d [ 1 ] u n d e r t o o k an e x t e n s i v e s t u d y on c l a s s i c a l very powerful Because of o f measuring a c c u r a t e l y t h e shape of t h e not g i v e a comparison of t h e o r y and range, absorb r e - e m i s s i o n and t h i s ( n o t i n g the spacing of l e s s than experimental a s some sinusoidally-grooved crossed gratings. that metal the d i f f i c u l t y 'moth o f two k i n d s o f g r i d w i t h c i r c u l a r h o l e s and grating. Their experimental s e t u p was f o r d i f f r a c t i o n m e a s u r e m e n t s w h i c h were made 3% r e l a t i v e e r r o r . U n f o r t u n a t e l y t h e y were n o t a b l e t o c o m p l e t e t h e i n v e s t i g a t i o n due t o t h e l o s s o f the experimental facility. 5 H e r e i s g i v e n an e x p e r i m e n t a l investigation c r o s s e d g r a t i n g w i t h s q u a r e p y r a m i d s and of a a metal metal crossed grating with hemispherical c a v i t i e s i n the microwave r e g i o n . The k i n d s of r e a s o n s t h a t we chose these two crossed g r a t i n g s are as f o l l o w s : T h e s e two s t r u c t u r e s are expected absorbing surfaces Numerical analysis is s t i l l deeply-grooved impossible i t w o u l d be v e r y of a t r i a n g u l a r the s i n g l y - p e r i o d i c e q u i v a l e n t of the groove g r a t i n g , which i s corresponding crossed g r a t i n g , i s well-known [ 9 ] , T h i s provides the o p p o r t u n i t y behavior o f t r i a n g u l a r and f o r a comparison between pyramidal g r a t i n g s of the the profile. A modal t h e o r y [ 2 2 ] was r e c e n t l y developed grating with hemispherical c a v i t i e s . verification The Experimental useful. behavior same solar for a crossed grating. The pyramidal efficient [19,20]. pyramidal i n v e s t i g a t i o n of t o be reason of rigorously i s that t r a n s f o r m a t i o n method can g r a t i n g s w h i c h i s not crossed Experimental valuable. t h a t the deep p y r a m i d a l c a n n o t be a n a l y z e d coordinate i t w o u l d be for a groove g r a t i n g ,on one hand, only analyze v e r y deep (h/d l e s s than o t h e r hand, a r i g o r o u s modal e x p a n s i o n f o r the b e t w e e n t h e p y r a m i d s c a n n o t be w r i t t e n . The the crossed one); on the fields s i d e s of the p y r a m i d s do n o t c o n s t i t u t e a s e t o f o r t h o g o n a l surfaces, and a spherical coordinate boundaries of system does not f i t the 6 the pyramids as i t does t h e s u r f a c e s Experimental important study of a hemisphere. i n t h e m i c r o w a v e r e g i o n h a s an advantage i n that a l l g r a t i n g dimensions a r e l a r g e e n o u g h t o be m i l l e d a c c u r a t e l y a n d t h u s a v a l i d between t h e o r y and experiment comparison i s possible. W i t h m e a s u r e m e n t s made i n t h e r a n g e o f 35 GHz, Jull, Ebbeson, H e a t h , B e a u l i e u and H u i have s u c e s s f u l l y compared their theoretical r e s u l t s concerning periodic conducting grating three types g r a t i n g s : comb g r a t i n g [ 1 0 ] , rectangular [ 8 , 1 1 ] a n d t r i a n g u l a r g r a t i n g [ 9 ] . They a t t e m p t e d t o f i n d a p e r f e c t l y blazed g r a t i n g which completely specular of s i n g l y reflection from a c o n d u c t i n g b a c k s c a t t e r , and they converts surface to showed n u m e r i c a l l y a n d e x p e r i m e n t a l l y t h a t t h i s b l a z e e f f e c t c a n be a c h i e v e d by a r e c t a n g u l a r g r a t i n g or a t r i a n g u l a r g r a t i n g with adequately chosen d i m e n s i o n s . They a l s o p r o p o s e d a p p l i c a t i o n s t o m u l t i p a t h i n t e r f e r e n c e r e d u c t i o n and radar The experimental represents study target of crossed a natural extension b l a z e e f f e c t o f some c l a s s i c a l reflection interesting of specular reflection s u r f a c e s when t h e d i r e c t i o n o f i n c i d e n c e i s a r b i t r a r y . The o b j e c t i v e s o f t h i s e x p e r i m e n t a l to e x p l o i t here work on t h e g r a t i n g s . Crossed f o r the suppression from c o n d u c t i n g gratings given of the previous g r a t i n g s w i t h s q u a r e symmetry a r e a l s o possibilities design. the p o s s i b i l i t y of a pyramidal study a r e : reflection g r a t i n g t o s c a t t e r a l l i n c i d e n t energy i n t o a s i n g l e spectral order, 7 to investigate experimentally the predicted c a p a b i l i t y of a crossed grating with hemispherical cavities to 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 around normal i n c i d e n c e , to study the r e l a t i o n between t h e b e h a v i o r o f s i n g l y - p e r i o d i c and i t s c o r r e s p o n d i n g d o u b l y - p e r i o d i c gratings, to observe the e f f e c t s of p r o f i l e parameters, incident a n g l e s , p e r i o d of g r a t i n g and i n c i d e n t wavelength reflection to on t h e behavior of these g r a t i n g s , c o m p a r e t h e m e a s u r e d r e s u l t s w i t h a few n u m e r i c a l predictions, to p r o v i d e a s e t of e x p e r i m e n t a l data pyramidal f o r deeply-grooved g r a t i n g s a n d o b t a i n some b e h a v i o r c u r v e s o f crossed grating with hemispherical cavities non-normal i n c i d e n c e , which numerical In assist further investigations. chapter expansion will under 2, t h r o u g h a d e r i v a t i o n of t h e R a y l e i g h f o r the d i f f r a c t e d field, equation and the R a y l e i g h wavelength the crossed grating equation are obtained. These p l a y an i m p o r t a n t p a r t i n e x p l a i n i n g t h e shape o f t h e measured b e h a v i o r c u r v e s of g r a t i n g s . Methods t o determine the p r o p a g a t i n g d i r e c t i o n and t h e e x i s t i n g a n g u l a r a diffracted wave by u s e o f t h e s e region of two e q u a t i o n s a r e discussed. Chapter measuring 3 d e s c r i b e s t h e e x p e r i m e n t a l arrangement and procedure. 8 Experimental results of six pyramidal crossed gratings and two c r o s s e d g r a t i n g with hemispherical c a v i t i e s , discussions presented on them a r e respectively. theoretical the B a s i c a l l y the i n C h a p t e r s 4 and 5 experimental and n u m e r i c a l p r e d i c t i o n s . behavior same p r o f i l e of triangular measuring parameters data of verify Comparison and p y r a m i d a l i s made. The e f f e c t s and gratings varying the between of the grating and are o b s e r v e d by c o m p a r i n g m e a s u r e d the s o u r c e s of curves. Chapter effects of 6 lists them. investigation are Finally, errors c o n c l u s i o n s from presented in Chapter 7 . and d i s c u s s e s this the Chapter 2 SCATTERING BY CROSSED GRATINGS 2.1 FORMULATION OF THE PROBLEM Consider the d i f f r a c t i o n problem in which a l i n e a r l y polarized harmonic electromagnetic plane wave with wavelength X i s incident upon a doubly-periodic surface, having orthogonal p e r i o d i c i t y axes and separating free space from a perfectly conducting metallic medium. Set up a rectangular coordinate system Oxyz, and l e t the Oy axis be F i g . 2 . 1 . The s p e c i f i c a t i o n of the incident f i e l d and of the d i f f r a c t e d wave vectors in the (Oxyz) system 9 10 perpendicular to the plane of the grating and Ox axis to be aligned with one of the axes of p e r i o d i c i t y . The two p e r i o d i c i t i e s of the crossed grating are defined by d,, the period along Ox, and d , the period along Oz. 2 As shown in F i g . 2.1, the d i r e c t i o n of the incoming plane wave i s s p e c i f i e d by the polar angles \[/^ and 6^, and is represented by the wave vector K " . The p o l a r i z a t i o n i s 1 s p e c i f i e d by the angle 6 between the e l e c t r i c f i e l d E 1 vector and the intersection of the plane of incidence with the plane perpendicular to Tc . If 6 equals ir/2, the incident 1 e l e c t r i c f i e l d i s p a r a l l e l to the surface plane and we say that the incident wave i s transverse e l e c t r i c - p o l a r i z e d , or TE-polarized. If 5 equals zero, the incident magnetic f i e l d is p a r a l l e l to the surface and we say that the incident wave is transverse magnetic-polarized, or TM-polarized. TE and TM polarizations are two fundamental p o l a r i z a t i o n cases, since an a r b i t r a r i l y polarized incident plane wave can be resolved into TE and TM components which scatter e s s e n t i a l l y independently from highly conducting c l a s s i c a l grating surfaces, but not, in general, for crossed gratings. Let the grating surface have the equation y=f(x,z), and we have f(x+d,,z+d )=f(x,z) , 2 (2.1) since the grating i s periodic along Ox and Oz. In the region of y<f(x,z), the f i e l d s are n u l l because the medium i s assumed p e r f e c t l y conducting. Above the surface, because of i t s double-periodicity, the crossed 11 grating waves are gives in the rise by t h e section that plane (i.e. a discrete various orders, represented next to the the of derivation similar is (p,q), the diffracted field y>max[f(x,z)]) plane waves to (i.e. that diffracted directions wave v e c t o r s Epg« region superposition s p e c t r u m of We w i l l above the of which show in surface may be w r i t t e n a s a Rayleigh expansion). The in [ 2 1 ] . 2.2 THE RAYLEIGH EXPANSION FOR THE DIFFRACTED F I E L D The incident plane wave fields are given by: E ^ E J e x p t j (-Jt^-r+wt) ] , H^HJexpf j(-K .r+o)t)], i where k* =a x-/3 y 7oZ, 1 0 v e c t o r s a l o n g the with: r=xx+yy+zz, + 0 x, y and z a x e s a = k s i n 0 ^ c o s v ^ > P =kcos8^ o k=2tr/X 0 is the (x, of and z a r e the unit respectively) 7o = k s i n 0 ^ s i n i / ^ , f wave number y, incident where wave, then, S =Eoexp[-j(a x-0 y 7oz)]exp(jwt). 1 + o In what exp(jo)t) total E the and E where the be t h e field E^(x,y,z) vector we w i l l the diffracted respectively. diffracted which Helmholz 2 time dependence 2 2 electric field and the Then g = g + E ^ . t field E 1 c , we seek a vector satisfies: equation V =9 /9x +3 /3y 2 suppress the fields. determine function 2. all electric To 1. follows for Let o 2 + 3 /9z 2 boundary c o n d i t i o n s at for y>f(x,z): (V +k )E 2 2 the grating surface: 2 =0, 12 S ^xn=-E xn, and V x E ^ n s - V x E «n, where n i s the normal ( 1 1 unit vector of the surface 3. the outgoing wave condition. The solution of t h i s problem should e x i s t for physical reasons, and i t should be unique according to the uniqueness theorem. Let us show that the function F(x,y,z)=E (x,y ,z)exp[ j (a x+7 z) ] (2.2) d 0 0 i s periodic along the directions Ox and Oz with the periods d, and d respectively. 2 F i r s t , we show F(x,y,z+d )=F(x,y,z). 2 Obviously, the t o t a l f i e l d E i s periodic along Oz, fc that i s , E (x,y,z+d )=E (x,y,z) . t t 2 Considering E = E + E , we have t 1 d E (x,y,z+d )+E (x,y,z+d )=E (x,y,z)+E (x,y,z) . 1 d X 2 d 2 But since E (x,y,z+d )=Ejexp{-j[a x-^ y 7o(z+d )]} 1 + 2 0 0 2 =Ejexp{-j[a x-j3 y 7o ] l e x p ( - j d 7 ) + 0 0 2 0 =E (x,y,z)exp(-jd 7o), X 2 then E (x,y,z+d )=E (x,y,z)exp(-jd 7o) d (2.4) d 2 2 =E (x,y ,z)exp[ j (a x-/3 +7oz) ]exp{- j [a x-0 y+7 (z+d ) ] d 0 0 o o O 2 We can write E (x,y,z+d )exp{j[a x-0 y 7o(z+d )]}=E (x,y,z)exp[j(a x-0 y+7 z)] Q + 2 o o a 2 o o (2.5) Recalling t h e d e f i n i t i o n o f F(x,y,z), t h e a b o v e i s O 13 F(x,y,z+d )=F"(x,y,z), 2 and we have demonstrated that F(x,y,z) i s periodic along the Oz d i r e c t i o n with the period d . S i m i l a r l y we can 2 demonstrate that F*(x,y,z) i s also periodic along the Ox d i r e c t i o n with the period d,. Since the function ¥ i s doubly-periodic along Ox and Oz with periods d 1 and d , we can now expand i t in a double 2 Fourier series: F(x,y,z)= ^ where P ^ +CO 4 0 0 ^ F (y)exp[-j(pK +qK z)] , pq 1 2 (y) i s a function of y only, K = 2 7 r / d , K = 2rr/d , 1 1 2 2 p=0,±1,±2,••• , q=0,±1,±2,.--. From the d e f i n i t i o n F(x, y, z) =E^(x, y, z) exp[ j (a x+7 z) ], we 0 0 have E (x,y,z)= £ d ^ E (y)exp[-j(a x+7 z) ] , pq p q where a =a +P'2ir/d, , 7q=7o q* 27T/d . + p 0 2 We can introduce t h i s expansion for equation (V +k )E =0 and get 2 2 d d F (y)/dy +(k -a -7 )E (y)=0. 2 2 2 2 p q p 2 q p q We obtain F pq y ( ) = g pq e x p -^ ( p q y> where v/k -a 2 2 p 0pq = . .—^ -jVa D 2 7 q , i f k -a -7 ^0 2 2 p =— +7 a k » otherwise. 2 q in the Helmholtz 14 T h e r e f o r e , a t any p o i n t of t h e h a l f - s p a c e y > m a x [ f ( x , z ) ] , t h e f i e l d d i f f r a c t e d by a c r o s s e d g r a t i n g c a n be e x p r e s s e d Rayleigh as a double sum o f p l a n e w a v e s , t h e s o - c a l l e d expansion: YL + 0 0 +00 £ E (x,y,z)= d 4q e x p ( - 0 0 q=-oo ~ W j l <-> 2 6 ? ) where Er^=a^x+/3__y+7_z pq p pq- 'q ' H r a =a +P'27r/d , p 0 (2.7) 1 0pq Vk -V-V, = (2.8) 2 7 =7o q-27r/d . (2.9) + q 2 Equation (2.8) i s c a l l e d the crossed grating We c a n s e e t h a t t h e o r d e r s o f d i f f r a c t i o n equation. f o r m e d by a c r o s s e d g r a t i n g a r e s p e c i f i e d by a p a i r o f i n t e g e r s ( p , q ) r a t h e r than the single singly periodic i n t e g e r adequate i n the case of g r a t i n g s . I f /3__ i s r e a l , the order (p,q) is pq r e f e r e d t o as being a propagating energy to infinity away f r o m order the grating , since i t can c a r r y s u r f a c e . I f 0__ i s pq imaginary, the order c a r r i e s no e n e r g y ( p , q ) i s s a i d t o be e v a n e s c e n t away f r o m p h y s i c a l phenomena c a l l e d vicinity the grating surface. t h e Wood a n o m a l i e s arise and Important i n the o f t h e t r a n s i t i o n p o i n t /3_=0. I f t h e i n c i d e n c e pq parameters 9^ a n d ^ are fixed, j3pg v a n i s h e s a t t h e R a y l e i g h w a v e l eXn g(tph, q ) = ( - B + ' B + A c o s 0 ) / A (2.10) 2 r where v 2 i 15 A=p /d +q /d , 2 2 2 2 2 B=sint? (cosxf/^ - p / d ^ s i n ^ . «q/d ). i 2 We define the e f f i c i e n c y of a grating in a given order (p,q) to be the r a t i o between the power P(p,q) d i f f r a c t e d into this order and the incident power P ( i ) . It i s easy to show that e pq = | E M W | 2 ( | E ° (2 | 2 / 3 o K Because of the conservation ' 11) of energy and because of the assumption of perfectly-conducting surface, we have P=-oo (J=-oo which i s reduced to +00 +00 ZE pq e = K ( 2 * 1 2 ) This equality, which means that the sum of the e f f i c i e n c i e s is equal to unity, i s generally c a l l e d the energy balance c r i t e r i o n and can be used to check the numerical r e s u l t s . 2.3 DETERMINATION OF THE DIRECTIONS OF PROPAGATION OF THE DIFFRACTED WAVES A plane wave incident on a crossed grating excites a discrete, rather than continuous, spectrum of d i f f r a c t e d plane waves. The wave vector of the d i f f r a c t e d wave of the (p,q) order can be specified by *pq=V W V+ + It i s easy to show that 16 a p = k s i n 0 pq cos\i/p q '. B 0 sin\I> p q = k c o s 0p q '. a n d qy = k s i n pq° pq F Thus, from 7 ( 2 . 7 ) a n d ( 2 . 9 ) , we 1 w obtain ksin0pqCOSi//pq=a P'2?T/d, =ksin0^cosi//^+p* 2 i r / d , , + o and k s i n 0 _ s i n ^ _ = 7 o + q ' 2 7 r / d = k s i n 0 . sini//. +q« 2 7 r / d , rv rT 2 2 w h i c h c a n be r e d u c e d t o sin0 gCOSvVpq=sin0 cos^ +P'X/d , p i i (2.13) 1 sin0„sin\// =sin0. sin\(/. + q « X / d . r Dividing (2.13) (2.14) 2 into ( 2 . 1 4 ) , we g e t t a n t / v = ( s i n 0 . s i n i K +q«X/d )/(sin0.cos<£. +p«X/d, ) . (2.15) 2 fc-K-i X X X X T h e s e t h r e e e q u a t i o n s c a n be u s e d t o d e t e r m i n e t h e d i r e c t i o n s o f t h e d i f f r a c t e d waves i n t h e v a r i o u s orders. I n t h e f o l l o w i n g d i s c u s s i o n , we assume d,=d =d f o r 2 simplicity. 2.3.1 GRATING WITH PERIOD SMALLER THAN HALF OF WAVELENGTH S i n c e now X/d>2, t h i s r e q u i r e s p=0 a n d q=0 i n ( 2 . 1 3 ) a n d ( 2 . 1 4 ) . Then we h a v e t h e s e t sin0„„cos^„=sin0; c o s ^ pq "pq l *i sin0 sini// =sin6 sin^ , p g p q w h i c h l e a d s t o ^^=^1 pq solution i 1 i a n d 0_=0- . T h i s i s t h e o n l y pq 1 real f o r d<X/2. T h e r e f o r e , f o r a c r o s s e d g r a t i n g w i t h t h e p e r i o d d<X/2, o n l y the s p e c u l a r l y t h e ( 0 , 0 ) o r d e r wave (i.e., r e f l e c t e d wave) p r o p a g a t e s , c a r r y i n g a l l the i n c i d e n t energy, and t h e s u r f a c e behaves as a p l a n e c o n d u c t o r no m a t t e r what k i n d o f p r o f i l e the g r a t i n g 17 has. For l a r g e p e r i o d s s p e c u l a r r e f l e c t i o n scatter i s reduced by i n t o other s p e c t r a l o r d e r s . 2.3.2 GRATING WITH PERIOD BETWEEN A HALF WAVELENGTH AND ONE WAVELENGTH 2.3.2.1 Under Non-oblique Incidence ( ^ = 0) We o f t e n study the case i n non-oblique i . e . , the case when ^=0, incidence, the i n c i d e n t beam i s p e r p e n d i c u l a r t o one of the axes of the g r a t i n g p e r i o d i c i t y . Then we get sinf3 =sinfl. +p«X/d cost// (2.16) sin»5 sini// =q.X/d pq S i n c e now l<X/d^2, from q=0, which If and (2.17) pq requires #,-,0=0, 0^ = 0, which e pq = u (2.17), we can conclude that o r ^pq ®' = i t i s easy t o d e r i v e from (2.16) that p=0 i s the normally i n c i d e n t c a s e . Otherwise, i//pg=0. From (2.16), we have sin which 0_ =sin0.+p«X/d p, u 1 (2.18) n i s of the same form as the c l a s s i c a l grating linear formula. Thus we can see t h a t the t h r e e - d i m e n s i o n a l g r a t i n g problem where X/2<d^X and i / ^ = 0 i s reduced t o the two-dimensional g r a t i n g problem, d i s c u s s i o n f 1 1 ] of which can be used f o r t h i s special crossed grating case. In some a p p l i c a t i o n s , reflection i t i s d e s i r e d t o reduce from a s u r f a c e . T h i s can be done by i n c r e a s i n g b a c k s c a t t e r . Maximum constructive some 18 interference i n the d i r e c t i o n the d i r e c t i o n of incidence, occurs sin0 =mX/(2d), (2.19) i s e a s i e r t o b l a z e a s u r f a c e when t h e number o f propagating small,but in f o r kdsin0^=mff o r m=1,2,**». i It o f b a c k s c a t t e r , o r back i n orders greater a r e f e w ; i . e . , when t h e p e r i o d d i s than half of wavelength. This i s true ( 2 . 1 9 ) when m=1 o r d=X/(2sin0 ) (2.20) i which i s c a l l e d equation sin and with ( 2 . 1 9 ) , we g e t 0 ^ =(p+l/2)X/d, p o f o rperiods (2.21), reflection Q n a n d 0_ t9_ Q =-6K. T h e r e f o r e 1 ( t h e order (-1,0)) e x i s t . (0,0)) By a d j u s t i n g o t h e r We now c o n s i d e r 0<d^<ir/2 (2.21) n real , and from only (2.20) specular (the order parameters of the i n b a c k s c a t t e r and a decrease i n specular 2.3.2.2 U n d e r O b l i q u e 1 and b a c k s c a t t e r g r a t i n g , we c a n g e t a i n c r e a s e corresponding t h e above p=0,±1,±2, ••• i n t h e r a n g e X/2<d^X t h e o n l y t o (2.21) a r e 6 solutions and the Bragg c o n d i t i o n . Combining reflection. Incidence oblique a n d 0<^<TT/2 w i t h o u t 1^X/d^2, we c a n c o n c l u d e i n c i d e n c e when t/>^*0. Assume l o s i n g g e n e r a l i t y . Since f r o m E q . 2.13 a n d E q . 2.14 t h a t o n l y p=0,-1 a n d q=0,-1 a r e p o s s i b l e s o l u t i o n s , s o t h e r e are four p o s s i b l e orders (0,-1), For of d i f f r a c t e d waves: (0,0), (-1,0) a n d ( - 1 , - 1 ) . example, i n t h e c a s e o f X/d=1.24, 0^=38° a n d ^.=30°, w h i c h i s one o f t h e e x p e r i m e n t a l cases, after 19 some c a l c u l a t i o n using Eq. 2.13 and Eq. 2.14, we find that in t h i s case of oblique incidence, besides the order (0,0), only the order (-1,0) occurs, the d i r e c t i o n of which can be determined by finding 6_y Q =50.5° and 1 2.3.3 Q=156.5°. GRATING WITH PERIOD GREATER THAN WAVELENGTH We consider the cases of non-oblique incidence (^ = 0). Again we have Eq. 2.16 and Eq. 2.17, from which we can see that q=±1 and possible higher orders are excited since now X/d<1. For instance, we can calculate the d i r e c t i o n s of a l l the d i f f r a c t e d waves scattered by a crossed grating with hemispherical c a v i t i e s which we have investigated. The case i s in normal incidence, and X=8.57 mm, d=13 mm. Since 0^=0, we get sin0 cosi// =p-X/d, (2.22) sinepgSini/Zp^q.X/d. (2.23) pg pq and Now that X/d=0.66, p and q can only take values of 0,1 or -1, and we obtain nine sets of equations from the above two equations. Solving them, the d i r e c t i o n s of nine orders of the d i f f r a c t e d waves c a n be found as follows: order (0,0): specular r e f l e c t i o n , 6Q Q =0; order (0,1): ^ ^=90°, Q 6 Q } =41.3°; 20 order (0,-1): order (1,0): e tf _ =270°, 0f 1 tf Q= °' 0 1f order (-1,0): n *i,0 lr1 } order (1,-1): order (-1,1): * _ = 4 1 =l80°, 0 _ order (1,1): <// =45°, 6 - °? 3 =41.3°; 1fO =69°; t 6 1rl ^ =41.3°; Q } , =69°; - l 3 5 ° , 0..,^ =69°; order (-1,-1): ^ _ _ = 2 2 5 ° , 0_ ^_ 1f 1 1 1 =69°. From the above r e s u l t s , i t turns out that except for the order (0,0), which i s back i n the d i r e c t i o n of incidence, the d i f f r a c t e d wave vectors l i e on two cones F i g . 2.2. The d i r e c t i o n s of the d i f f r a c t e d wave vectors scattered by a crossed grating with X/d=0.66 in normal incidence 21 whose axes are superposed on the incident wave vector and whose half-angles are 41.3° and 69° respectively (see F i g . 2.2). 2.4 DETERMINATION OF THE ANGULAR REGION WHERE A DIFFRACTED ORDER EXISTS In general, we can use the Rayleigh wavelength equation (2.10) as a guide to find the region of the azimuthal angle i//^ where the d i f f r a c t e d order (p,q) e x i s t s : for fixed incident angles 9^ and \}/^, i f the actual wavelength X i s not greater than the Rayleigh wavelength X (p,q) given by f (2.10), the d i f f r a c t e d order (p,q) can exist at t h i s p o s i t i o n . Let us consider the range 0<\j/^<90° and a crossed grating with X/2^d^X. For the order (-1,0), we have from (2.10) X (0,-1 )/d=sin0 cos^ + /sin e cos ^ +cos 0 2 r i i v 2 i 2 i i which i s a decreasing function of 6^ in the range (2.24) 0<\^^<90°, and ( s i n 0 + 1) | i ^X (-1,O)/d^cos0 | r i ^ = 90° (2.25) If we now assume 0^<sin~ (X/d-1) which i s the necessary 1 condition that the order (-1,0) exists at <^=0, and increase ^ from zero while holding 6^ at that constant value, we can see that the d i f f r a c t e d order (-1,0) w i l l remain existing u n t i l \}/^ reaches a certain value which makes the right side of (2.24) equal to the actual value of X/d. This i s the t r a n s i t i o n point where the (-1,0) Wood anomaly occurs and the order (-1,0) ceases to propagate. 22 For the spectral order (0,-1), we obtain from (2.10) \(0,-1 )/d=sinr3 sin^ Vsin f3 sin i// +cos f9 2 2 i i 2 i i which i s an increasing function of ^ i (2.26) in the range 0^\^<90°, and cos0.| 1 <X(0,-1)/d^(1+sin0.)| i^-O ^ = 90° r (2.27) 1 At ^ = 0, the order (0,-1) does not exist since we have assumed 1^X/d£2. As ^ becomes large enough while 0^ i s fixed, the d i f f r a c t e d order (0,-1) appears. It begins to propagate at the point where the actual value of X/d i s equal to the right side of (2.26), and the (0,-1) Wood anomaly happens. For the d i f f r a c t e d order (-1,-1), the normalized Rayleigh wavelength can be expressed as X (-1 ,-1 )/d={/2sin0 cos(«/' -45 ) V s i n 0 ( s i n 2 ^ - 1 )+2}/2. (2.28) o r i 2 i i i When ^ = 0 or 90°, we get X (-1 ,-1 ) / d = ( s i n 0 V 2 - s i n 0 ) / 2 (2.29) 2 r i i which i s always less than one when 0^0^90°. Thus, in the case of 1£X/d^2, the order (-1,-1) does not exist around ^ = 0 or 90°, t h i s agrees with the previous discussion. The maximum value of X (-1,-1)/d can be e a s i l y found from (2.28) as max[X (-1 ,-1 )/d] = ( 1+sin0 )//2", r i at ^ = 4 5 ° . If 0^ i s large enough, the order (-1,-1) can exist around \^=45°, and two (-1,-1) Wood anomalies, located symmetrically about i / ^ = 45 , defines the region where the 0 order (-1,-1) e x i s t s . (2.30) 23 In t h i s way, we can find out the range of ^ where a d i f f r a c t e d order (p,q) can propagate. We w i l l apply t h i s method when discussing experimental results from the cases of oblique incidence. For the case of non-oblique incidence (^=0 fixed), the range of 6^ where a d i f f r a c t e d order (p,q) can exist w i l l be s i m i l a r l y determined. 2.5 A PRODUCT FORMULA LINKING CROSSED AND CLASSICAL GRATINGS It i s interesting to investigate any simple r e l a t i o n between the behavior of singly periodic grating and doubly periodic grating having the same p r o f i l e and use i t as a guide to the design of crossed grating surfaces since the theories for the d i f f r a c t i o n by c l a s s i c a l grating are now capable of dealing with a l l structures of p r a c t i c a l importance. Here we introduce an empirical equivalence formula l i n k i n g crossed and c l a s s i c a l gratings which was suggested by Derrick et a l . [5]. Let us suppose that the crossed grating i s of the form y=f(x,z)=u(x)+v(z). The equivalence formula indicates that the e f f i c i e n c y e m n of the crossed grating in the order (m,n) is given by e mn=<^n/ ' R < ' ' 2 31 where r j and T?* are the e f f i c i e n c i e s of the c l a s s i c a l m gratings with p r o f i l e equations y=u(x) and y=v(z) in orders m and n respectively, under the same conditions of incidence and polarization as for the crossed grating , and R i s the Fresnel reflectance of a plane surface with the same 24 conditions of incidence. This formula in the case of the order (0,0) can be derived roughly by use of Taylor series for s u f f i c i e n t l y shallow crossed gratings under normally incident radiation [ 2 0 ] . Indeed, comparisons [5,19] between the rigorous t h e o r e t i c a l results and those obtained using the equivalence formula showed that t h i s product formula gave to a good accuracy the e f f i c i e n c y e o of some sinusoidally-modulated 0 crossed gratings under normally incident radiation at o p t i c a l frequencies. It i s worth while pointing out that the agreement i s s t i l l very good for a crossed grating with h/d=1 [19, F i g . 1] although the product formula i s supposed to be v a l i d for very shallow crossed gratings. We are mostly concerned about investigating the r e f l e c t i o n free property of a crossed grating and i f t h i s product formula were always true, a crossed grating would have the r e f l e c t i o n free property when i t s singly-periodic equivalent grating can backscatter a l l the incident energy ( i . e . , e = 0 i f T7o or rj§ equals to zero). This would be a oo simple way to find a r e f l e c t i o n - f r e e crossed grating since i t i s possible now to design a blazed c l a s s i c a l grating by numerical methods. We w i l l investigate t h i s p o s s i b i l i t y experimentally. Chapter 3 EXPERIMENTAL ARRANGEMENT AND PROCEDURE For our experimental investigation, s i x metal crossed gratings of square pyramids and two metal crossed gratings with hemispherical c a v i t i e s were made in a l l . The d e t a i l s of dimensions of these plates w i l l be given in the following two chapters. Each of these surfaces consisted of a matrix of more than 15x15 i d e n t i c a l elements with the period d, the value of which i s i n the range of the wavelength of radiation used. The experimental arrangement was similar to that used in [8], A photograph of the experimental set up i s shown in F i g . 3.1 and a diagram of the experimental arrangement i s shown in F i g . 3.2. Reflection measurements were made in a microwave anechoic area. Transmitting and receiving antennas were two i d e n t i c a l pyramidal horns with 24.7 dB gain and E-plane 3 dB beamwidth of 9°, the orientation of which can be changed to have TE or TM polarization of the incident wave and to analyze the polarization of the d i f f r a c t e d wave. Microwave absorbers suspended between the horns prevented d i r e c t transmission, while absorbers were positioned around the plate to eliminate r e f l e c t i o n from any surface except the top of the grating plate investigated. Absorbers were also placed at the end of the grating to reduce transmission under the plate. The incident angle 8^ could be varied from approximately 5° to 90°, but i t was d i f f i c u l t to eliminate coupling of two antennas near the 5° l i m i t and direct 25 F i g . 3.1. Photograph of experimental set-up, receiving horn in foreground 26 N CRYSTAL CURRENT METER CURHFNT METER CRYSTAL DC TECTORp ISOLATOR CRY'TrTi DE TEECTOR ST^> \ / RECISIOH VARIABLE ATTENUATOR N PLATE 3 (b) F i g . 3.2. Experimental arrangement a. Range b. C i r c u i t KLYSTRON 28 transmission between the antennas near the 90° limit. Therefore, our measurements were usually taken in the range of incident angles from 10° to 80° or less, with an reading accuracy of about 1°. The distance from the grating surface to either of the antennas was 1.38 m. This non-plane wave illumination, as reported by J u l l and Ebbeson [10] and Heath [8], has very l i t t l e effect on the grating performances, and the r e s u l t s , as far as r e f l e c t i o n reduction i s concerned, could only be better under plane wave i l l u m i n a t i o n . A diagram of the experimental c i r c u i t i s shown schematically in F i g . 3.2(a). With the corrugated surface exposed, the c r y s t a l current reading of the receivng antenna was noted. When the grating was covered by a f l a t and (1 mm thin thickness) conducting plate of the same area as the grating surface, the c r y s t a l current reading increased but was restored by adjusting a precision variable attenuator. The power reduction in surface r e f l e c t i o n due to the grating is the difference between the former and l a t t e r precision variable attenuation readings, accurate to about ±0.2 dB. The tuning range of the millimeter reflex klystron i s 33 to 37.5 GHz mm), (or the wavelength X i s from 9.09 mm to GHz 8.00 and the klystron output was monitored continuously to detect output l e v e l changes during the measurements. To investigate the e f f e c t s of oblique illumination, the grating surfaces were mounted on a platform which could be rotated around the v e r t i c a l axis by an azimuthal angle After each rotation, the grating surface was . re-leveled by 29 adjusting three screws which were mounted on the platform, thus increasing the accuracy of the 6^ measurements. Although the emphasis of our investigation was upon measuring the r e l a t i v e r e f l e c t e d power by a grating plate under non-oblique incidence, we also measured the other two kinds of responses of each grating: azimuthal angular response and frequency response in r e f l e c t e d power. Normally, the measurement procedure was as follows: 1. With the incident beam perpendicular to one of the axes of the grating p e r i o d i c i t y , the r e l a t i v e reflected power was measured by varying the incident angle from 10° to 80° with increments of 2.5°. More measurement points were taken near a reduction peak to increase the accuracy of the position and the value of the maximum reduction of r e f l e c t e d power. 2. With a fixed incident angle which was usually the angle where the maximum r e f l e c t i o n reduction occured or was the Bragg angle (0£=sin~ (X/(2a)), 1 an azimuthal angular response was recorded by rotating the plate horizontally from i//^=-45° to ^ = 45°. Since the crossed gratings investigated were square symmetric, the f u l l i//^ response between 0° to 360° could be derived from the measurements taken between 0° and 45°. Nevertheless, we also measured the response from -45° to 0° in order to check the symmetry of the response curve . 3. With the same incident angle 9^ and ^-=0° (non-oblique incidence), the frequency response was obtained by 30 varying the incident wave frequency from 33.0 GHz to 37.5 GHz (the tuning range of the klystron). Whenever a more satisfactory result was found at a new frequency, measurements in steps 1 and 2 were repeated at that frequency. In a l l the measurements, the performance for TM polarization and for TE polarization was recorded i n d i v i d u a l l y by changing the orientation of the antenna horns. The experimental values for TM p o l a r i z a t i o n were plotted as small c i r c l e s which were joined by straight l i n e s , while those for TE polarization as small crosses connected by broken l i n e s . Normally we drew curves of variation of the reflected power l e v e l in terms of dB by using the measured data d i r e c t l y . Sometimes for convenience we also plotted curves of the e f f i c i e n c y in the order (0,0); i . e . , the r e l a t i v e r e f l e c t e d power. Chapter 4 EXPERIMENTAL RESULTS OF PYRAMIDAL CROSSED GRATINGS 4.1 INTRODUCTION In t h i s chapter measurements of the power r e f l e c t e d by a pyramidal crossed grating are presented. F i g . 4.1 shows a photograph of one of the pyramidal crossed grating plates studied experimentally, which i s composed of conducting pyramids with a square base. The pyramids were constructed by r u l i n g consecutively, in orthogonal d i r e c t i o n s , symmetrical two triangular groove grating having the same groove spacing d and the same apex angle a. F i g . 4.2 shows a c l a s s i c a l triangular groove grating (echelette grating) and a pyramidal crossed grating with the same p r o f i l e , and lists the plate dimensions (period d and pyramid height h), apex angle a, the depth-to-period r a t i o h/d and the numbers of pyramids for a l l pyramidal grating plates investigated experimentally. These were m i l l e d within ±0.03 mm (±0.001 in.) of the given dimensions. The grating periods were designed for an incident wave at a frequency of 35 (X=8.57 mm) GHz to s a t i s f y the r e l a t i o n of \/2<d£X, and hence only specular r e f l e c t i o n (the order (0,0)) and the d i f f r a c t e d order (-1,0) existed under non-oblique according to the discussion in Section incidence 2.3. Since in general i t i s only possible to obtain by using current theories numerical results for a crossed grating whose depth i s less than i t s period (also depending on 31 32 F i g . 4 . 1 . Photograph of a pyramidal crossed grating plate (plate C 4 ) a. covered with a f l a t conducting plate b. crossed grating exposed REFLECTION GRATINGS CROSSED PYRAMIDAL ECHELETTE PLATE APEX ANGLE (degrees) DIMENSIONS d(mm) h(mm) h/d PYRAMIDS A4 45 8.00 9.66 1.21 30x30 A6 60 8.00 6.93 0.87 30x30 A9 90 8.00 4.00 0.50 30x30 B6 60 6.23 5.40 0.87 39x39 B9 90 6.23 3.12 0.50 39x39 C4 44 6.95 8.60 1 .24 37x37 F i g . 4.2. A c l a s s i c a l triangular (echelette) grating and pyramidal crossed grating with the same p r o f i l and dimensions for a l l pyramidal plates 34 wavelength) [19], i t i s interesting to get experimentally d i f f r a c t i o n patterns for crossed gratings which are deeply-grooved. Therefore, each grating depth which we chose was equal to or greater than half of the period, or, in other words, the apex angle a of each grating was equal to or less than 90°. The surfaces of the f i r s t set (set A) consisted of a matrix of 30x30 i d e n t i c a l pyramids with base dimension d=8.00 mm and apex angles a=90°, 60° and 45°. These were machined on 240x240 mm about 20 mm. brass plates whose thickness was A second set (set B) consisted of 39x39 pyramids with d=6.23 mm. Plate B9 with apex angle of 60° i s made of brass while Plate B6 with apex angle of 60° i s made of aluminium, but no problem in the measurements should arise from using these two d i f f e r e n t materials because both can be treated as e s s e n t i a l l y p e r f e c t l y conducting in the microwave region. Two plates of set B were chosen for comparison with e a r l i e r work [16] for which some numerical results are a v a i l a b l e . There was only one brass grating plate with d=6.95 mm and a=44° in set C, which was designed to observe how specially well the r e f l e c t i o n free properties of a singly-periodic surface carry over to i t s doubly-periodic equivalent. Unlike previous experiments on c l a s s i c a l gratings where numerical data were used to design the gratings, we had l i t t l e suitable theoretical results on pyramidal crossed gratings with i n f i n i t e conductivity available for predicting 35 our investigation. Therefore, i n order to search for the best operating point where the maximum r e f l e c t i o n reduction occured, we t r i e d to change between l i m i t s the parameters which could be changed, i . e . , the parameters of the grating (period d, and apex angle a (or height h)) and the parameters of the measurement (incident angle 6^ , azimuthal incident angle i / ^ , and frequency of the incident wave). 4.2 PLATE B9 IN COMPARISON WITH ITS SINGLE-PERIODIC EQUIVALENT In our departmental workshop, a symmetrical triangular groove grating was f i r s t m i l l e d across the brass plate in making a square pyramidal crossed grating of the same p r o f i l e . One such comparison i s shown in F i g . 4.3 for 39 symmetrical triangle grooves with apex angle of 90° ( F i g . 4.3a) and the corresponding plate B9 ( F i g . 4.3b) of 39x39 square pyramids at the frequency of 35 GHz (X/d=1.37). The experimental results have already been presented in [12], For TM p o l a r i z a t i o n , the triangular grooves are e s s e n t i a l l y perfectly blazed to the n=-1 spectral order at the Bragg angle 0^=sin~ (X/(2d))=43.5° where a reduction in 1 the r e f l e c t e d power i s about 47 dB (e =0.00002) or 99.998% o of the incident energy i s r e f l e c t e d back in the d i r e c t i o n of incidence. Over a wide angular range of 3O°^0^7O°, the zero order e f f i c i e n c y i s below 0.01 and specular r e f l e c t i o n i s almost eliminated. TE polarized r e f l e c t i o n i s approximately halved (e =0.45 at ^=43.5°) by the grooves for Bragg angle o F i g . 4 . 3 a . Relative r e f l e c t e d power (or e ) vs. angle of incidence for the singly periodic equivalent of plate B9 at f*35 GHz, 4^=0 0 37 0 30 60 90° F i g . 4.3b. Relative r e f l e c t e d power (or e o) s . angle of incidence for plate B9 at f«35 GHz, ^=0, v 0 Predicted for i n f i n i t e surfaces: — T E case, • TM case 38 incidence. We also obtained from a computer program by Facq [17] two numerical results of e =0.04 for TM polarization o and e =0.64 for TE p o l a r i z a t i o n at 0^=43.5° and f=35 GHz for o a p e r f e c t l y conducting s t r i p of six symmetrical triangular grooves with the same period and apex angle. The calculated power patterns were also obtained (see F i g . A.2, Appendix). After taking into account the e f f e c t of the larger size (39 grooves) of the experimental grating, the correspondence between the calculated and measured values i s good. With the corresponding pyramidal grating F i g . 4.3b shows that TM polarized specular r e f l e c t i o n i s s t i l l s u b s t a n t i a l l y reduced over a wide range of incident angles where at least 80% of incident energy i s d i f f r a c t e d into the order (-1,0). At t h i s point, i t seems that the TM polarized r e f l e c t i o n free property of the corresponding singly-periodic surface c a r r i e s over in a certain extent to crossed grating plate B9. TE polarized r e f l e c t i o n i s only s l i g h t l y affected by the surface. F i g . 4.3b also shows that the experimental results for both TM and TE cases agree with predicted results [16] indicated by bold s o l i d (TM) and broken (TE) curves. A s l i g h t l y upward s h i f t of the experimental TM curve with respect to the calculated TM curve i s probably due to the site-.reflections. P. Blick and R. Deleuil had reported their experimental results [10, F i g . 12] obtained from the same kind of pyramidal crossed grating with the same apex angle of 90° under the same incident conditions. These are the 39 only pair of experimental curves available for comparison with our work. Their results almost coincided with the numerical curves presented here. Our measuring arrangement i s much simpler and incapable of the same precision (their r e l a t i v e accuracy of the grating e f f i c i e n c y measurement was better than 3% as reported), but s t i l l has provided adequate experimental accuracy to v e r i f y theoretical predictions for millimeter-wave r e f l e c t i o n gratings, as proved in the previous work on c l a s s i c a l gratings [9-12], For c l a s s i c a l gratings maximum r e f l e c t i o n reduction always occurs at the Bragg angle, which can also be seen in F i g . 4.3a. However, from F i g . 4.3b, i t seems that b a s i c a l l y there i s no Bragg angle effect for crossed grating plate B9. It i s also noted that the experimental points are in good agreement with the theoretical predictions concerning the so-called Wood anomaly, which occurs at any wavelength where a d i f f r a c t e d order ceases to propagate, that i s , at the Rayleigh wavelength introduced in Section 2.2. Using (2.10), the Rayleigh wavelength X f of the order (p=-l,q=0) under non-oblique incidence (i^ = 0) can be found as X (-1,O)=(1+sin0 )d, r i from which we deduce that when the incident angle 6^ i s less than e ^ s i n - (X/d-1 ), 1 (4.1) the d i f f r a c t e d order(-1,0) ceases to propagate and only the specular r e f l e c t i o n e x i s t s . 40 In the case of c l a s s i c a l gratings, by applying the grating equation sine =sin0 +nX/d, n where & n i (n=0,±1,±2,...) i s the d i f f r a c t i o n angle from the surface normal, and considering that when 6> =-ir/2,the n=-1 spectral order n w i l l cease to propagate, we can get e a s i l y the value of incident angle where the n=-1 Wood anomaly occurs which i s also fl^sin" (X/d-1), 1 the same as (4.1). Therefore,the (-1,0) Wood anomaly for plate B9 and the n=-1 Wood anomaly for i t s singly-periodic equivalent at 35 GHz should both occur at S ^ s i n " ( 1 .375-1 )=22°. 1 Back to the experimental results presented in F i g . 4.3a or 4.3b, we can see that there i s indeed an abrupt drop at about d^=22° in the r e l a t i v e r e f l e c t e d power curve for TM p o l a r i z a t i o n , which i s due to a rapid exchange of power between specularly reflected and d i f f r a c t e d modes — in a change of only 2° in 6^, 80% of the incident energy switches from the order (0,0) (or the n=0 order for c l a s s i c a l grating ) to the order (-1,0) (or the n=-1 order for c l a s s i c a l grating) which can only exist when 0^>22°. Thus, the agreement between theory and experiment i s very good. In the case of TE p o l a r i z a t i o n , the Wood anomaly i s s t i l l evident for the c l a s s i c a l grating,but i s not clear for the crossed grating. 41 Let us concentrate our study on the crossed grating. For angles of incidence less than the l i m i t value 22°, only the order (0,0) propagates and the e f f i c i e n c y e 0 0 in the two polarizations TM and TE should be i d e n t i c a l and equal to unity. The fact that c F i g . 4.3b 0 0 i s only near unity when 0^<22° in i s due to experimental error. The effect of polarization becomes apparent rapidly as soon as the incident angle i s greater than 20°. e o 0 for TM p o l a r i z a t i o n f a l l s rapidly to about 0.1 and stays below 0.2,while e 0 0 for TE polarization remains close to one. Thus, t h i s structure behaves l i k e a polarizer,because i t can d i f f r a c t e d most of the TM component of an a r b i t r a r i l y polarized incident wave into a single order (-1,0) and r e f l e c t specularly most of the TE component. F i g . 4.3b also shows that when the incident angle i s greater than 65°, the r e l a t i v e reflected power i s below 0.1 and even reach as low as 0.007 at 0^=82°. Measurements for 0£>8O° are d i f f i c u l t due to d i r e c t transmission between transmitting and receiving horns, but i t has been established that e 0 0 at 6^=82° i s at most 0.007 or less (since the effect of d i r e c t transmission i s to make the measured value of e o 0 larger than i t s true value). Also, t h i s near-zero value of e o agrees with the tendency 0 of both the calculated and the experimental curves for TM p o l a r i z a t i o n in F i g . 4.3b when the incident angle i s greater than 50°. It seems that e o would approach zero and almost 0 t o t a l cancellation of specular r e f l e c t i o n would occur over a 42 wide range of 9^ for near-grazing incidence, which i s a desirable property for some applications (for example, multipath interference suppression,since most multipath interference occurs for near-grazing incidence). Thus t h i s kind of crossed grating appears to be a promising structure for those applications. The experimental plot of the r e l a t i v e r e f l e c t e d power as a function of the angle of rotation (^) for plate B 9 at an angle of incidence 8^ =43.5° i s exhibited i n F i g . 4.4. The measurements were taken at 3 5 GHz. The curves for TE and TM cases are e s s e n t i a l l y symmetrical expected about \ ^ = 0 , which i s from the square symmetry of t h i s crossed grating and w i l l be found to be true b a s i c a l l y for a l l plates investigated. Hence we w i l l consider mainly the region 0<,\l/^45° when we discuss the cases of oblique incidence for a l l p l a t e s . As shown in F i g . 4 . 4 , in a broad range -42°£i/^£42°,the reduction in TM polarized specular r e f l e c t i o n i s at least 7 . 3 dB, while the reduction i n TE polarized specular r e f l e c t i o n i s less 1 dB. Thus the behavior of a polarizer i s conserved i/^ except by ^^=45°. over the f u l l range of for a narrow range of about six degrees centered This exception i s caused by the (-1,0) and (0,-1) Wood anomalies. By applying the method of Rayleigh wavelength equation discussed in Section 2 . 4 , we can determine which d i f f r a c t e d order can exist and where i t can exist in the f u l l range of . F i g . 4 . 5 shows schematically the results of c a l c u l a t i o n which i s corresponding to F i g . F i g . 4.4. Reflected power vs. angle of rotation for plate B9 at f = 35 GHz, 0^43.5° (0,0) (1.0) (0.1) r U I -18b 5 1 - F i g . 4.5. (o.-n (-1.0) 9 t U 1I f t f / iI AL \ 9rf . U Diagram of a z i m u t h a l a n g u l a r r e g i o n s i n which spectral orders exist for a crossed grating a t X/d-1.37, 0.-43.5° region boundaries: *f{«43 . 9 , 0 Vi = 46.1° (1.0) I IBCf 45 4.4 and can be applied to a crossed grating having any type of square-symmetrical p r o f i l e with the incident conditions of X/d=1.37 and 0^=43.5°. Since the pyramidal surface i s square-symmetrical, the regions for the orders (-1,0),(0,-1),(0,1) and (1,0) have the same width and are symmetrical about 0°, 90°, -90° and 180° respectively, as indicated in F i g . 4.5. For 0^=43.5° fixed, at any value of there i s at most one more d i f f r a c t e d order which can exist besides the order (0,0). According to the c a l c u l a t i o n , the order (-1,0) ceases to propagate at ^=43.9° where the (-1,0) Wood anomaly occurs, and the order (0,-1) begins to propagate at ^=46.1° where the (0,-1) Wood anomaly e x i s t s . Thus there i s only specular r e f l e c t i o n in a very narrow azimuthal range of about two degrees around \/^ = 45 . This 0 explains the abrupt r i s e of TM polarized power from 40° to 45° in F i g . 4.4: the energy in the order (-1,0) switches t o t a l l y to the order (0,0). Ideally the r e l a t i v e r e f l e c t e d power at i^=45° should be one in this case. The experimental discrepancy i s attributed to positioning the plate. The measured frequency response curves of plate B9 at 0^=43.5° under non-oblique incidence are shown in F i g . 4.6. The broad band nature of t h i s p o l a r i z e r - l i k e plate i s r e a d i l y seen. Within the tuning range of the klystron, the reduction in TM polarized specular r e f l e c t i o n remains about 8dB,or 84% of the TM-polarized incident power i s scattered, while the TE-polarized incident wave i s almost reflected. completely FREQUENCY(GHZ) F i g . 4.6. Reflected power vs. frequency for plate B9 at 6^ = 43.5°, tf.=0 47 4.3 PLATE C4 A DEEPLY-GROOVED PLATE IN COMPARISON WITH ITS SINGLY-PERIODIC EQUIVALENT It i s interesting to investigate experimentally deeper groove surfaces which cannot be handled t h e o r e t i c a l l y at present while seem most l i k e l y to be e f f e c t i v e in some applications [20]..It had been reported that e s s e n t i a l l y perfect blazing to the n=-1 spectral order for a r b i t r a r y p o l a r i z a t i o n should occur for symmetrical triangular groove r e f l e c t i o n gratings when the apex angle i s 44° and the angle of incidence i s 0^=38.1° [4,9]. Such a surface was first milled in making i t s doubly-periodic equivalent plate C4. The experimental values in F i g . 4.7, which were obtained from t h i s c l a s s i c a l grating at f=35 GHz, show that although e s s e n t i a l l y perfect blazing occurs for TM p o l a r i z a t i o n at the Bragg angle 6^=38° where e = 0.00003 or the reduction i s o 45 dB, the result for TE p o l a r i z a t i o n (e =0.039 or the o reduction i s 14 dB) i s less s a t i s f i e d . F i g . 4.8 shows the measured frequency response of t h i s grating at 0^=38° under non-oblique incidence. The intersection of the TM and TE curves proves the same e f f e c t for a r b i t r a r y p o l a r i z a t i o n , and the corresponding frequency i s 33.75 GHz where the reduction i s about 23 dB, or the d i f f r a c t i o n e f f i c i e n c y to the n=-1 order for both p o l a r i z a t i o n s i s about 99.5%. The angular response of t h i s triangular grating at the frequency 33.75 GHz in F i g . 4.9 new shows that at the new Bragg angle 0^=40°, near-perfect dual blazing i s achieved, a good improvement over that obtained at f=35 GHz. Numerical 48 F i g . 4.7. Relative r e f l e c t e d power (or e ) vs. angle of incidence for the singly periodic equivalent of plate C4 at f=35 GHz, ^.=0 0 Oj 1 _J 3a0 I 1 i 1 i L i L_ 340 350 FREQUENCY(GHZ) \ 1 i L_ 3ao F i g . 4.8. Reflected power vs. frequency for the singly periodic equivalent of plate C4 at 0^=38°, *.-0 U3 4.9. Relative r e f l e c t e d power (or e ) vs. angle of incidence for the singly periodic equivalent of plate C4 at f = 33.75 GHz, ^=0 0 51 results from Facq's program agree with these experimental results (see Appendix). F i g . 4.10 shows the measured r e l a t i v e r e f l e c t e d power as a function of the incident angle for the corresponding pyramidal grating plate C4. Both TM and TE polarized r e f l e c t i o n s near the Bragg angle 0^=40° are only halved by the crossed grating. Apparently, the r e f l e c t i o n free properties of the singly-periodic surface are lost for i t s doubly-periodic equivalent . Therefore, we can conclude from t h i s example of plate C4 that the product formula (2.31) l i n k i n g shallow crossed and c l a s s i c a l gratings cannot in general be applied to deeper gratings under non-normal incidence. In F i g . 4.10, the (-1,0) Wood anomaly occurred around 0^=sin~ (X/d-1)=17° i s evident only as a change in the 1 gradient of the TE and TM curves, while for the c l a s s i c a l grating, F i g . 4.9 shows the corresponding n=-1 Wood anomaly around the same incident angle as being an abrupt energy transfer from the specular r e f l e c t i o n to the n=-1 order. The measured angular response at f=35 GHz i s represented in F i g . 4.11, along with the response at f=33.75 GHz which i s the same as that in F i g . 4.10 except that now i t i s plotted in terms of dB for easy comparison. At both frequencies, the behavior for TM polarization i s similar to that for TE case, and no maximum reduction i s observed around the Bragg angle. The experimental frequency at 0.=38° shown in F i g . 4.12 confirms further the response 52 F i g . 4.10. Relative r e f l e c t e d power (or e o) vs. angle of incidence for plate C4 at f=33.75 GHz, ^.=0 0 F i g . 4 . 1 1 . Reflected power vs. angle of incidence for plate C 4 : b. at f=35 a. at GHz f=33.75 GHz _J 33J0 I I 340 I I 1 35J0 1 36.0 1 - J — 370 FREQUENCY (GHZ) F i g . 4.12. Reflected power vs. frequency for plate C4 at 0^38°, ^ = 0 55 i n s e n s i t i v i t y of t h i s crossed grating around 35 GHz. to frequency changes Increasing or decreasing by 2 GHz from 35 GHz does not a f f e c t the d i f f r a c t i o n e f f i c i e n c y of plate C4 very much and the reduction of r e f l e c t e d power i s always below 5 dB. F i g . 4.13 function shows the measured r e f l e c t e d power as a of rotation angle \^ for plate C4 at f=35 GHz and 0^=38°. Again, the curves are b a s i c a l l y symmetrical about v//^ = 0°. Although the d i f f r a c t i o n e f f i c i e n c y of the grating insensitive to p o l a r i z a t i o n under non-oblique incidence, is the e f f e c t of p o l a r i z a t i o n become apparent in a remarkably rapid manner as the plate i s rotated apart from the ^=0° p o s i t i o n . For TM p o l a r i z a t i o n , the reduction gets larger and reaches i t s maximum 13 dB at about \^=35°, while the reduction in TE polarized 2 dB. The calculated specular r e f l e c t i o n remains under region diagram for a l l spectral orders d i f f r a c t e d by plate C4 with X/d=1.24 and F i g . 4.14. and We can see that only two (-1,0) exist between i//^=-37° and F i g . 4.13, 95% of the 0^38° i s shown in spectral orders i/^ = 37°, and hence from incident energy i s d i f f r a c t e d into order (-1,0) at 0^=35°, the d i r e c t i o n of which can determined by in F i g . 4.14 (0,0) (2.13) and (2.14). The the be results of c a l c u l a t i o n predict that the regions of orders (-1,0) and (0,-1) are overlapped around i/^ = 45° and anomaly should occur at $^=31° where the the (0,-1) Wood (0,-1) order starts to propagate. This Wood anomaly causes a r e - d i s t r i b u t i o n of incident power among three orders (0,0), (-1,0) and (0,-1) F i g . 4.13. Reflected power vs. angle of rotation for plate C4 at f=35 GHz, 0^38° (0.0) (0,-1) (0,1) 180° -90° 90P F i g . 4.14. Diagram of azimuthal angular regions in which spectral orders exist for a crossed grating at X/d=1 .24, 0^38° region boundaries: ^=53°, Vi. = 37° I8<f 58 when ^£>37°. This explains the r i s e of the experimental TM curve from 35° to 40°. 4.4 PLATE A4 AN ESSENTIALLY PERFECT BLAZED CROSSED GRATING SURFACE FOR TM POLARIZATION Plate A4 i s also a deeply-grooved surface with the period d=8 mm and the apex angle a=45°. F i g . 4.15 shows experimental values of the reduction in specular r e f l e c t i o n by the singly-periodic equivalent of plate A4, with 30 triangular grooves at f=35 GHz. The maximum reduction 16.1 dB for TM polarization occurs almost exactly at the Bragg angle fl^sin" (X/(2d))=32.4°, 1 where the d i f f r a c t i o n e f f i c i e n c y (e.,) of t h i s c l a s s i c a l grating i s 97.5%, whereas for TE p o l a r i z a t i o n there i s no maximum reduction observed at the Bragg angle. The behavior of the corresponding crossed grating plate A4 at the same frequency of 35 GHz i s exhibited in F i g . 4.16a. The Bragg angle e f f e c t for TM case now becomes not evident and the reduction in the specular r e f l e c t i o n for both TM and TE polarizations i s always below 4 dB. The experimental results of plate A4 measured at f=33 GHz i s shown in F i g . 4.16b, which shows a decrease by about 6% in frequency does not change the behavior of t h i s surface very much. Measured frequency v a r i a t i o n of TE and TM polarized r e f l e c t i o n from plate C4 at 0^=46° under non-oblique incidence i s shown in F i g . 4.17. For TE p o l a r i z a t i o n , the g 'or LU £ Q LU »— O UJ _J li_ UJ or F i g . 4.15. Relative reflected power (or e ) vs. angle of incidence for the singly periodic equivalent of plate A4 at f = 35 GHz, ^ = 0 0 cn 60 CO Q -5 CC UJ £ o -10 111 t— o \®r| UJ _J u_ UJ AAAM -15h or M H -201 J 30 , P l L 6: 60 r vs. incidence for p.*lected power vs. angle angxe of « * c a g . 4.16. »«^; j; ? . .35 GHz, *-0 B s a b. f = 33 GHz, £ ^=0 1 33.0 • I 34.0 I I 35.0 FREQUENCY (GHZ) 1 360 F i g . 4.17. Reflected power vs. frequency for plate A4 at 6^=46°, ^ = 0 I 37.0 I I 62 reduction i s below 3 dB and the curve i s quite f l a t within the f u l l frequency range. The reduction for TM p o l a r i z a t i o n from f=32.75 GHz to f=36.5 GHz i s a l i t t l e less than that for TE p o l a r i z a t i o n . However, when the frequency becomes larger than 36.5 GHz, TM curve descends steeply and the reduction reaches as large as 32 dB at f=37.52 GHz which i s the highest attainable frequency of the klystron. Here, for TM p o l a r i z a t i o n , a increase by about 3% in frequency completely changes the properties of this crossed grating from r e f l e c t i n g specularly 100% of the incident power at f=36.25 GHz to d i f f r a c t i n g e s s e n t i a l l y t o t a l l (99.94%) of the incident power into the (-1,0) order at f=37.52 GHz. This i s the best result that we got from a l l plates investigated, as far as t o t a l c a n c e l l a t i o n of specular r e f l e c t i o n i s concerned. Therefore, we demonstrated experimentally near perfect blazing of the pyramidal crossed grating for TM p o l a r i z a t i o n . Fig. 4.18 shows the measured r e l a t i v e reflected power as a function of 8^ under non-oblique incidence at f=37.5 GHz. Very i n t e r e s t i n g behavior of plate A4 i s observed when the incident wavelength i s equal to the period. There are two remarkable reduction peaks for TM p o l a r i z a t i o n . The larger one occurs at 0^=46° where the reduction i s 27 dB, or 99.8% of the incident power i s d i f f r a c t e d to the order (-1,0), the d i r e c t i o n of which can be found as 0_ 1 and \{/_] Q=180° by using angular Q =16.3° (2.13) and (2.14). Over a narrow range of about 3°, the reduction i s greater than 20 64 dB or 99%, and the specular r e f l e c t i o n i s e s s e n t i a l l y eliminated. The smaller peak i s at 0^=20° where the reduction for TM p o l a r i z a t i o n i s 16.7 dB. I t i s surprising that at the Bragg angle 0 =sin" (X/(2d))=30°, a l o c a l 1 i minimum reduction i s observed, instead of a maximum reduction of specular reduction. This behavior i s quite d i f f e r e n t from that of c l a s s i c a l gratings. The reduction in TE polarized specular r e f l e c t i o n i s much less than that for TM polarization and i s below 4 dB, except for a peak at 0^80° where the reduction i s 13.1 dB or 95.1%. Holding the frequency at 37.5 GHz and the incident angle at 46° where maximum reduction for TM polarization occurs under non-oblique incidence, we recorded reflected power as a function of angle of rotation by rotating plate A4 h o r i z o n t a l l y (see F i g . 4.19). Both TM and TE curves somewhat complicated look in t h i s case where the r a t i o of wavelength over period i s one. For TM p o l a r i z a t i o n , the plate could be rotated up to about three degrees in both directions and s t i l l give about 20 dB or 99% reduction. Within t h i s narrow azimuthal angular range, the surface i s blazed to the spectral order (-1,0). F i g . 4.20 shows the corresponding diagram of azimuthal angular regions where spectral orders e x i s t , which i s the results of c a l c u l a t i o n by applying Rayleigh wavelength equation. Theory predicts that the (-1,-1) Wood anomaly occurs at \^=3.2° where the d i f f r a c t e d order (-1,-1) begins to propagate. This prediction corresponds to the sudden r i s e of the 65 4 19 Reflected power vs. angle of rotation for plate A4 at f = 37.5 GHz, 6^=46° (0,0) (0,1) (0,-1) (-1,0) (1.0) (-1.1) (1.1) ' 1 (1.0) (1.-1) (-1,-1) -air ' rfSr> * so* 1 • 1 . ...A . I F i g . 4.20. Diagram of azimuthal angular regions in which spectral orders exist for a crossed grating at X/dk1.00, 0."46° region boundaries: Vv-68.9, i/i=21.1°, ^ = 3.2° ' wcf 1.._ (0,0) Pig. 4.22. Diagram of azimuthal angular regions in which spectral orders exist for a crossed grating at X/d=1 .07, 0^32.5° region boundaries: ^=67.8°, Vi=22.2°, ^ = 31.5° 69 experimental TM curve at about ^ = 3 ° in F i g . 4.19, where the appearance of the spectral order (-1,-1) causes a r e - d i s t r i b u t i o n of incident power among the three existing orders (0,0),(-1,0) and (-1,-1). S i m i l a r l y , we a t t r i b u t e the small peak of the experimental TM curve and the dip of the experimental TE curve, both at i/^ = 20°, to the (0,-1) Wood anomaly at ^=21.1° indicated in F i g . 4.20, where a fourth d i f f r a c t e d order appears. In t h i s case, the agreement between theory and experiment i s very good for both TM and TE p o l a r i z a t i o n s . There exist four spectral orders around </^ = 45°, instead of only two orders at ^.=0°. The azimuthal angular response of plate A4 at f=35 GHz and 6^=32.5° i s presented in F i g . 4.21. The general shape of the TE curve i s somewhat similar to that at f=37.5 GHz 0^=46° shown in F i g . 4.19, and while for TM p o l a r i z a t i o n , the reduction in specular r e f l e c t i o n becomes much l e s s . The downward peak of TE curve at i//^ = l8° and the upward peak of TM curve at \p^ = 20° are caused by the (0,-1) Wood anomaly which should occur at ^=22.2° according to the results of c a l c u l a t i o n shown in F i g . 4.22. The experimental discrepancy i s due to locating the plate. The e f f e c t of the (-1,-1) Wood anomaly which should exist at 1 ^ = 31.5° i s not sor evident in the observed curves. 70 4.5 PLATE A9 INFLUENCE OF THE RATIO X/d Plate A9 has the same apex angle of 90° as that of plate B9, but with a different period. Therefore, we group together in F i g . 4.23 and F i g . 4.24 four cases of the experimental results obtained from these two plates in order to investigate the effect of the r a t i o X/d on the behavior of crossed gratings. The reflected power curves as a function of incident angle for plate A9, measured at frequencies 37.5 GHz (X/d=1.00), 35 GHz (X/d=1.07) and 33 GHz (X/d=1.14), are presented in F i g . 4.23a, F i g . 4.23b and F i g . 4.24a respectively. The curves at X/d=1.37 in F i g . 4.24b are for plate B9 and are the same as those in F i g . 4.3b, except that the reduction i s expressed in terms of dB here for easy comparison. From these four pairs of curves we can see that the value of the r a t i o X/d a f f e c t s the d i f f r a c t i o n e f f i c i e n c y greatly. The behavior of the grating with X/d=1.00 (see F i g . 4.23a) looks quite d i f f e r e n t from those with other values of X/d. At X/d=1.00, the reduction in specular r e f l e c t i o n for both polarizations fluctuates very much as the angle of incidence varies, and the reduction in TE polarized specular r e f l e c t i o n i s the largest among these four cases and reaches i t s maximum 13.8 dB at 0^=8.5°, where 96% of incident energy i s scattered to the spectral order (-1,0), noticing that only two spectral orders (0,0) and (-1,0) exist in the f u l l range of 0^ since order (-1,0) i s cut off at 0^=sin" (X/d-1)=0°. The maximum reduction for TM 1 p o l a r i z a t i o n occurs at 0.=32.5° which i s close to the Bragg F i g . 4.23. Reflected power vs. angle of incidence for plate A9: a. at f=37.5 GHz (X/d=1.00) b. at f=35 GHz (X/d=1.07) 4.24. R e f l e c t e d power v s . a n g l e o f i n c i d e n c e f o r a . p l a t e A9 a t f=33 GHz (X/d=1.14) b. p l a t e B9 a t f=35 GHz (X/d=1.37) 73 angle 0^=sin~ (X/(2d))=30°. On the contrary, there appears a 1 minimum reduction at 0^=30° for TE p o l a r i z a t i o n . The curves in F i g . 4.23b, F i g . 4.24a and F i g . 4.24b looks somewhat similar, and the behavior of a polarizer i s conserved over a wide range of 0^, where the reduction for TE p o l a r i z a t i o n i s small, and that for TM p o l a r i z a t i o n i s roughly 10 dB or 90%. From the point of view of a good p o l a r i z e r , the best operating range i s found in F i g . 4.24a to be lO°^0 <27.5°, where the surface can scatter 90% of the i TM components of an a r b i t r a r i l y polarized incident wave, while TE polarization i s t o t a l l y r e f l e c t e d . In F i g . 4.24b, the very small reduction for both TM and TE polarizations below 20° i s due to the fact that t h e o r e t i c a l l y the (-1,0) spectral order w i l l cease to propagate as long as 0^ i s smaller than sin" (X/d-1)=22° where the (-1,0) Wood anomaly 1 occurs, and the surface should act as i t were a f l a t metal plane, as discussed in Section 4.2. For X/d=1.07 and the (-1,0) Wood anomaly should occur at 0^ = 4° and 1.14, 7.8° respectively, but these two positions are out of our range of convenient measurements. The Bragg angles are 32.3°, 34.6° and 43.4° for the values of X/d respectively, 1.07, 1.14 and 1.38 their e f f e c t s are not evident as found in Figures 4.23b, 4.24a and Measured frequency b. response of plate A9 at 0^=32.5° under non-oblique incidence i s shown in F i g . 4.25. the range 33 GHz<f<37.5 GHz, p o l a r i z e r - l i k e behaviour Within or 1.136<X/d<1, the i s s t i l l preserved. This i s CD Q or LU O CL Q UJ u_ UJ CC 330 340 350 FREQUENCY (GHZ) 360 F i g . 4.25. Reflected power vs. frequency for plate A9 at 0.=32.5°, *.=0 -0 F i g . 4.26. Reflected power vs. angle of rotation for plate A9 at f=35 GHz,0^=35° (0,0) (0,1) (1,0) AO * -1K? II 1 i (0.-1) j (-1,0) t (-1-1). .(-1,1). . 11 . I 1 11 ^90° , li I 1 J L .('.-1)j 1 1 v, 90° , LL Cf { \ (1.0) U ' . * F i g . 4.27. Diagram of azimuthal angular regions in which spectral orders exist for a crossed grating at X/d=1 .07, 0^35° region boundaries: V; 67.2 , T/£ = 22.8°, \^ = 24° = 0 U 1 I8(f 77 consistent with the frequency response of plate B9 in F i g . 4.6, where X/d varies from 1.46 to 1.28. It seems that the apex angle of 90° determines the p o l a r i z e r - l i k e behavior of plates A9 and B9. Therefore, the p r o f i l e of a pyramidal crossed grating has great effect on the general behavior of the grating, while at some values of X/d, a small change in X/d may change the behavior of the grating tremendously (for example, from X/d=1.00 to 1.07 in F i g . 4.23). The experimental results i n F i g . 4.26 display the r e f l e c t e d power as a function of angle of rotation for plate A9 at f=35 GHz and 0^=35°. The curves are b a s i c a l l y symmetrical about ^ = 0 ° as before. It i s interesting that when ^ becomes larger than 10°, the reduction for TE p o l a r i z a t i o n increases rapidly and reaches 12.8 dB or 95% at i//^ = 20°. Then, there appears an abrupt rise in TE curve from 20° to 25°. We attribute t h i s r i s e to the (0,-1) Wood anomaly which should occur at \^=22.8° according to our c a l c u l a t i o n (see F i g . 4.27). The appearance of the spectral order, (0,-1) at t h i s position causes a r e - d i s t r i b u t i o n of incident energy between three orders (0,0), (-1,0) and (0,-1). In F i g . 4.26, the TE curve changes suddenly again at i/^ = 25°. This change i s due to the appearance of the order (-1,-1) at 1^=24°, as shown in F i g . 4.27, where the (-1,-1) Wood anomaly should e x i s t . Again, we have very good agreement between theoretical predictions and experimental r e s u l t s . For TM p o l a r i z a t i o n , the effect of these two Wood anomalies i s not so obvious. 78 4.6 PLATE B6 INFLUENCE OF APEX ANGLE a Together with plate B9, Plate B6 was designed to check with some e a r l i e r numerical r e s u l t s . F i g . 4.28 shows the measured r e f l e c t e d power as a function of angle of incidence for the singly-periodic equivalent of plate B6 at f=35 GHz (X/d=1.37) When 0^ i s small, there i s no reduction in specular r e f l e c t i o n for both p o l a r i z a t i o n s , and the surface behaves as a plane conductor. This i s because only the n=0 order exists when 0^<sin" (X/d-1)=22°. For angles of 1 incidence greater than t h i s l i m i t value, the reduction for TE p o l a r i z a t i o n remains small, but for TM p o l a r i z a t i o n , there appears a sudden drop between 20° and 25°, which i s caused by the n=-1 Wood anomaly occured at 0^=22°. Also, a minimum r e l a t i v e r e f l e c t e d power of 0.28 i s observed at 0^=42.5° which i s close to the Bragg angle ©^sin- (X/(2d))=43.5°. 1 F i g . 4.29 presents the experimental corresponding results for the crossed grating plate B6 under non-oblique incidence. We can see from the figure that the behavior of t h i s doubly-periodic grating i s quite d i f f e r e n t from that of i t s singly-periodic equivalent. For the incident angle greater than twenty degrees, the reduction in TM polarized specular r e f l e c t i o n becomes much larger than that for the c l a s s i c a l grating, whereas for TE p o l a r i z a t i o n the reduction gets smaller. It i s worth notice that the influence of the Wood anomaly occured at 0^=22° i s .remarkable for TM p o l a r i z a t i o n , instead of for TE p o l a r i z a t i o n i n the case of 79 4 28. Relative r e f l e c t e d power (or e ) vs. angle o incidence for the singly periodic equivalent of plate B6 at f = 35 GHz, 0 ia 4.29. Relative reflected power (or e ) vs. angle incidence for plate B6 at f=35 GHz, ^=0 0 0 81 i t s c l a s s i c a l g r a t i n g . No maximum r e d u c t i o n i s observed around the Bragg angle (43.5°) Instead, t h e r e i s a l o c a l minimum r e d u c t i o n at 37.5°, around o v e r l a p r o u g h l y . The still which TE and TM curves r e d u c t i o n f o r both p o l a r i z a t i o n s i s s m a l l when the i n c i d e n t angle i s l e s s than 20°, s i n c e only the s p e c u l a r r e f l e c t i o n e x i s t s i n t h i s range where the r e l a t i v e r e f l e c t e d power should be u n i t y t h e o r e t i c a l l y . discrepancy i s probably due i n F i g . 4.29, The to the experimental e r r o r s . A l s o f o r comparison with the e x p e r i m e n t a l results, we d i s p l a y the c a l c u l a t e d curves [16] o b t a i n e d by u s i n g the c o o r d i n a t e - t r a n s f o r m a t i o n method mentioned i n Chapter Although 1. the behavior of the two curves i n some r e g i o n s (e.g., i n the range 22°£0 ^62° f o r TE p o l a r i z a t i o n ) i s i s i m i l a r t o each other, the agreement i s much l e s s than found that i n the case of p l a t e B9. N o t i c i n g t h a t p l a t e B6 with apex angle of 60° i s more deeply-grooved, the great d i s c r e p a n c y between numerical p r e d i c t i o n and r e s u l t s i s presumably due grooves experimental to the f a c t t h a t f o r deeper the numerical r e s u l t s converge very s l o w l y , or not at a l l [ 5 ] . The measured frequency response f o r p l a t e B6 at 0^=43.5° under non-oblique incidence i s i l l u s t r a t e d 4.30, reduction in specular r e f l e c t i o n is where no s i g n i f i c a n t in F i g . found. F i g . 4.31 shows r e f l e c t e d power curves as a f u n c t i o n of angle of r o t a t i o n Although f o r p l a t e B6 at f = 35 GHz the energy and 6^=43.5°. p r o p e r t i e s of the g r a t i n g are F i g . 4.30. R e f l e c t e d power v s . f r e q u e n c y f o r p l a t e B6 at 0 ^ 4 3 . 5 ° , ^ = 0 83 F i g . 4.31. Reflected power vs. angle of rotation for plate B6 at f*35 GHz, 0^43.5° 84 p o l a r i z a t i o n independent around i/^ = 0 ° , the e f f e c t s of p o l a r i z a t i o n become apparent as i/^ gets greater than 1 5 ° . The reduction in TE polarized specular r e f l e c t i o n becomes smaller and reaches zero at I / N = 4 5 ° , while the reduction i n TM polarized specular r e f l e c t i o n gets larger , reach a maximum of 8 . 5 dB at 4 0 ° , and then suddenly decreases to zero at ^ = 4 5 ° . Since the incident conditions here are the same as those in F i g . (X/d=1.37 and 6^=43.5°) B9, we can use the schematic diagram in F i g . 4 . 5 to explain 4.4 for plate the above behavior of plate B 6 . As discussed before for plate B9, the d i f f r a c t e d order (-1,0) i s cut off at ^ = 4 3 . 9 ° according to the c a l c u l a t i o n , and only specular r e f l e c t i o n (the order (0,0)) narrow range of for both polarizations exists in a very centered by \J^=45°, where the crossed grating should behave l i k e a p e r f e c t l y conducting f l a t surface even with such a noticeable roughness. Hence, we a t t r i b u t e the sharp r i s e of TM polarized r e f l e c t e d power from 4 0 ° to 4 5 ° in F i g . 4 . 3 1 to a sudden t o t a l transfer from the order (-1,0) to the order energy (0,0), and the experimental result that no reduction for both polarizations is observed at i/^ = 4 5 ° in F i g . 4 . 3 1 i s exactly what the theory predicts. The azimuthal angular responses of plate B 6 in F i g . 4 . 3 1 and plate B 9 in F i g . 4 . 4 give us good examples to look into the influences of the apex angle of pyramidal grating and the r a t i o X/d. I t seems that the apex angle has an important effect on the general behavior of the grating, 85 whereas the value of X/d defines some c r i t i c a l points, for example, the positions of Wood anomalies, where remarkable changes in the shape of the curves often occur. Therefore, both of apex angle a and r a t i o X/d determine the properties of the grating at a p a r t i c u l a r incident condition. 4.7 PLATE A6 A BLAZED CROSSED GRATING SURFACE FOR TE POLARIZATION Since plate A6 and plate B6 have the same p r o f i l e with an apex angle of 60°, we c o l l e c t their experimental curves together in Figures 4.32 and 4.33 to see how the behavior of the grating changes as the value of X/d v a r i e s . Figures 4.32a, 4.32b and 4.33a are from the measurements of plate A6 at frequencies 37.5 GHz GHz (X/d=1.00), 35 GHz (X/d=1.07) and 33 (X/d=1.14), respectively. F i g . 4.33b i s for plate B6 and i s the same as F i g . 4.29, except that the reduction i s expressed in terms of dB here. Again, the experimental results at X/d=1.00 in F i g . 4.32a looks most i n t e r e s t i n g . In F i g . 4.32a, although the reduction for both p o l a r i z a t i o n i s weak when the incident angle i s small, i t becomes larger after 9^ i s greater than 40°. The reduction in TE polarized specular r e f l e c t i o n reaches a maximum of 17.2 dB at 0^=77.5°, where 98% of the incident energy i s d i f f r a c t e d into a single order (-1,0), the d i r e c t i o n of which can be determined as 8 (-1,0)=1.4° and ^ = 1 8 0 ° by Equations (2.13) i and (2.14). In the point of view of elimination of specular r e f l e c t i o n , t h i s i s the best result for TE p o l a r i z a t i o n 86 - - zviutt 4 32 i-fv%- b. at f=35 GHz (X/d=1.07) n n » o , 87 F i g . 4.33. Reflected power vs. angle of incidence a. for plate A 6 , at f«33 GHz (X/d=1.14) b. for plate B 6 , at f«35 GHz (X/d=1.37) 88 which we got from a l l the crossed grating plates investigated. Also in F i g . 4.32a, the intersection of TM curve with TE curve at 0^=64° provides an equal reduction of 6.6 dB for both p o l a r i z a t i o n s . At this p a r t i c u l a r incident angle, plate A6 scatters about 80% of the energy of an a r b i t r a r i l y polarized incident wave into a single order (-1,0). This i s the best performance achieved with regard to equal reduction for both p o l a r i z a t i o n s . The behavior curves of t h i s grating at X/d=1.07 and X/d=1.14 are a l i k e as seen in F i g . 4.32b and F i g . 4.33a. In both cases, the reduction in TM polarized specular r e f l e c t i o n i s always less than that for TE p o l a r i z a t i o n . The behavior of the grating with a apex angle of 60° at X/d=1.37 has been discussed in last section. The drop of reflected power for TM polarization in F i g . 4.33b i s due to the (-1,0) Wood anomaly. The Bragg angles are 30°, 32.3°, 34.6° and 43.4° for X/d=1.00, 1.07, 1.14 and 1.37 respectively. As observed in F i g . 4.32 and F i g . 4.33, no maximum reduction occurs at the Bragg angle in these four cases. The reflected power as a function of angle of rotation for plate A6 at f=35 GHz and 0^35° i s i l l u s t r a t e d in F i g . 4.34. In order to analyze the shape of the curves, we can apply the calculated results presented in F i g . 4.27 to t h i s case, because of the same incident conditions here (X/d=1.07 and 0^=35°) as those in F i g . 4.26 for plate A9. The most remarkable feature in F i g . 4.34 i s a very steep f a l l i n g down of reflected power for TM polarization from 20° to 22.5°. 89 90 According to the c a l c u l a t i o n , t h i s i s due to the appearance of the d i f f r a c t e d order (0,-1) at ^=22.8°. where the (0,-1) Wood anomaly occurs. The (-1,-1) Wood anomaly at \/>^ = 24°, which i s close to the (0,-1) Wood anomaly, causes the TM polarized reflected power to increase somewhat from 22.5° to 30°. For TE p o l a r i z a t i o n , the influence of Wood anomalies i s not obvious. It i s interesting to take a comparison between the azimuthal angular behavior of plate A6 and plate A9 having the same period but d i f f e r e n t apex angle. In the case of plate A6 (see F i g . 4.34), the effect of Wood anomalies i s much more apparent for TM p o l a r i z a t i o n than that for TE p o l a r i z a t i o n , whereas in the case of plate A9 (see F i g . 4.26) the reverse i s true. In both cases, the positions where Wood anomalies occur are very close to what the theory predicts. From the comparison, we can see again, as we have discussed for plates B6 and B9 in the end of last section, that the p r o f i l e of the grating (that i s , the apex angle for pyramidal grating) a f f e c t s the general behavior of grating very much, while the grating period (more accurately, the r a t i o X/d) determines the positions of Wood anomalies, where great change in grating behavior always happens. Chapter 5 EXPERIMENTAL RESULTS OF CROSSED GRATINGS WITH HEMISPHERICAL CAVITIES 5.1 INTRODUCTION This chapter presents the r e s u l t s of the experiments c a r r i e d out on crossed gratings with hemispherical are regularly spaced along two orthogonal conducting plane surface. F i g . 5.1 c a v i t i e s which d i r e c t i o n s on a i l l u s t r a t e s the p r o f i l e of t h i s crossed grating, and gives the dimensions of such two plates investigated. Each surface consisted of i d e n t i c a l hemispherical 208x208 mm 16x16 c a v i t i e s which were hollowed on a aluminium plate with thickness of 20 mm. The crossed gratings have the same period, but d i f f e r e n t cavity radius. It i s worth notice that the period 13 mm i s greater than the wavelength within our experimental range (8 mm 9.09 mm). According d>X, For example, at f=35 GHz, the r a t i o X/d to the discussion in Section 2.3, is to 0.66. in the case of generally more than three d i f f r a c t e d orders are excited. For instance, i f either of these two gratings with hemispherical crossed c a v i t i e s i s illuminated normally (6^=0°) by a incident plane wave with X=8.57 mm (f = 35 GHz), there w i l l exist as many as nine d i f f r a c t e d orders (see the c a l c u l a t i o n in the end of Section 2.3). Therefore, blazing to a single spectral order i s not possible for these two crossed gratings in our measurements, although elimination of specular r e f l e c t i o n i s possible. This i s quite d i f f e r e n t 91 PLATE DIMENSIONS HEMISPHERICAL CAVITIES d (nun) R (mm) R4 13 4 16x16 R6 13 6 1 6x1 6 F i g . 5.1. The p r o f i l e of a crossed grating with hemispherical c a v i t i e s , and dimensions for such two plates investigated 93 from our measurements on the pyramidal gratings. In the l a t t e r case, the periods were chosen to be in the range of X/2<d£X and hence blazing to a single order (-1,0) i s possible. These two gratings were made by Laboratoire de R a d i o e l e c t r i c i t e , Universite de Provence, and were s p e c i a l l y designed numerical for comparison with^newTy^\ork [22] for which some results are available for the simple case of normal incidence. The numerical work for the non-normal incidence case i s being performed [56] and our experimental study on t h i s case provides the opportunity for a check on the r e l i a b i l i t y of the theory. Since these two plates were available in our laboratory for a short period of f i v e days, only limited measurements had been taken. 5.2 PLATE R4 The measure reflected power as a function of 8^ for plate R4 under non-oblique incidence at f=35 GHz (X/d=0.66) i s presented in F i g . 5.2. It i s interesting to see that when the incident angle i s less than 30°, the reduction in specular r e f l e c t i o n for both p o l a r i z a t i o n s i s greater than 5 dB. It was reported [22] that the e f f i c i e n c y for both polarizations in the order (0,0) for a i n f i n i t e crossed grating of this p r o f i l e under normal incidence (8^=0°) has a value of 0.08 (or -10.8 dB in terms of reduction), which i s indicated as Point P in F i g . 5.2. This i s the only available numerical result for t h i s grating. We could not take 94 F i g . 5.2. Measured r e f l e c t e d power vs. 0. for plate R4 at f = 35 GHz, ^=0; 1 and calculated angular regions of ^ e x i s t i n g d i f f r a c t e d orders (on top) for a l l 95 measurements near normal incidence as mentioned in Chapter 3. However, from F i g . 5.2, we can s t i l l observe that t h i s calculated result agrees quite well with the tendency of both TM and TE curves when the incident angle becomes smaller and approaches 0°. This more than 10 dB reduction in specular r e f l e c t i o n for normal incidence means that 90% of the incident energy of an a r b i t r a r i l y polarized plane wave is scattered into the other eight spectral orders and only 10% of that i s specularly r e f l e c t e d . In F i g . 5.2, on top of the experimental data, we also display schematically the results of our c a l c u l a t i o n by using Rayleigh wavelength equation (2.10), which indicate the regions where various spectral orders can e x i s t . One can see that there are at least seven spectral orders at any position of 0^. Four Wood anomalies exist within our range of measurement (1O £0 ^75°). The (1,0) Wood anomaly at o i 0^=20° (calculated value) probably cause the small dip in the TM curve at 0^=22.5°. The e f f e c t s of others are not evident. One w i l l see later that the e f f e c t s of Wood anomaly on the behavior of these two crossed grating with hemispherical c a v i t i e s are less apparent, which probably due to the multi-diffracted-order nature when the grating period i s greater than the incident wavelength. F i g . 5.3 shows the experimental results for the same plate at a d i f f e r e n t frequency 33 GHz (X/d=0.70). A frequency deviation of about 6% from 35 GHz does not change the behavior of the grating very much. The reduction for 96 (0.0) . (-1.il), (-1.0) (0.11) (-2,0) (1 0) , (1,±1) l 1 t 17.5* , 23-5* 30* I • (-2,±1) I I I 90° 43 45.5 TM -5 CQ Q cr UJ -10 P i Q LU I— O LU _J U_ LU or -15 5" = -20-25 070 R = 4mm 3d* 6i 60° 90 Pig. 5.3 Measured r e f l e c t e d power vs. 0. for plate R4 at f=33 GHz, ^=0; and calculated angular regions of ^ for a l l e x i s t i n g d i f f r a c t e d orders (on top) ' 97 both polarizations i s s t i l l greater than 5 dB when 0^ i s smaller than 30°, and i t also seems that the reduction in specular r e f l e c t i o n for both polarizations at 0^ = 0° would be more than 10 dB according to the tendency of both TM and TE curves, as seen in F i g . 5.3. This i s consistent with the results shown in F i g . 5.2. Small changes in gradient of TE and TM curves are observed in F i g . 5.3, some of which we attribute to the corresponding Wood anomalies, the calculated positions of which are shown on top of the experimental curves. For example, a small peak in experimental TM curve at 0^=17.5° i s probably due to the (1,0) Wood anomaly at the same angle as calculated, and a small change in gradient of TE curve at 0^=45° corresponds to the disappearance of the spectral orders (0,±1) at -0^45.5°. F i g . 5.4 presents the reflected power as a function of angle of rotation (^) for plate R4 at f=35 GHz and 0^10°. The experimental curves are roughly symmetrical about ^.=0°. A remarkable feature of the behavior in t h i s case i s that within the f u l l range of , the reduction for both polarizations i s greater than 5 dB. Again, the diagram showing the calculated angular region of \(/^ for a l l existing d i f f r a c t e d orders i s displaced d i r e c t l y on top of the experimental curves for easy comparison. Only the results in the range of 0°<\l/^<^5° are shown because of the symmetry of the behavior curves. According to the c a l c u l a t i o n , seven spectral order exist in the range of 0°^\//.^45 o and a 8th 98 (0.0H0.±1),(*1.0).(-1 ±1) J (1,-1) 27.5* 45 m Q ffi i Q UJ f— O UJ UJ or + 45 F i g . 5.4. Measured r e f l e c t e d power vs. at f = 35 GHz, 6^10) for plate R4 and calculated angular regions of ^- for a l l e x i s t i n g d i f f r a c t e d orders (on top) 99 order (1,-1) begins to propagate at ^=27.5° which corresponds to a smooth change of the experimental TE curve near this position. The azimuthal angular responses of plate R4 at the same frequency of 35 GHz, but at a different fixed 8^ of 30° are shown in F i g . 5.5. While the curves here look more complicated than those in F i g . 5.4, they have a common feature that the reduction in specular r e f l e c t i o n for both polarizations i s about 5 dB or more. Relating t h i s with the fact shown in F i g . 5.2 that the r e l a t i v e r e f l e c t e d power i s predicted 8% (or -10.8 dB) for normal incidence (0^=0°) and keeps lower than 32% (or -5 dB) when 8^ i s smaller than 30° under non-oblique incidence (^ = 0°). We can conclude reasonably that t h i s crossed grating w i l l r e f l e c t specularly at most 32% (-5 dB) of the incident energy of an a r b i t r a r i l y polarized plane wave whenever the incident angle 8^ from the normal of the surface plane i s less than 30° (^». i s a r b i t r a r y ) . This a n t i - r e f l e c t i o n property of plate R4 around normal incidence i s of great interest to the proposed application on solar energy absorption where non-tracking c o l l e c t o r s are used. On top of the experimental curves in F i g . 5.5, the c a l c u l a t i o n results show that four Wood anomalies occur in the range of 0°<\J/^<45°, which are responsible for the complication of the curves. The (-1,1) Wood anomaly at 1 ^ = 37.5° accounts for the reduction peak at \//^ = 37° for TM p o l a r i z a t i o n , while the very close (-2,0) and (-2,-1) Wood F i g . 5.5. Measured r e f l e c t e d power vs. \b. for D l a t e at f = 35 GHz, ^ = 30°; 1 and calculated angular regions of d i f f r a c t e d orders (on top) R4 for a l l existina existing 101 anomalies at \^=41° and ^ = 41.25°, respectively, correspond to the reduction peak at ^ = 4 4 ° for TE p o l a r i z a t i o n . The discrepancy i s due 5.3 to positioning the plate. PLATE R6 Plate R6 has the same period as that of plate R4, larger cavity radius of 6 mm. In Figures 5.6 but a and 5.7, present the variation of r e f l e c t e d power with the we incident angle 8^ for plate R6 under non-oblique incidence at f=35 GHz (X/d=0.66) and f=33 GHz (X/d=0.70) respectively. The behavior curves in both cases are remarkably d i f f e r e n t from those for plate R4 in Figures 5.2 and 5.3. Thus, we find experimentally that the cavity depth R a f f e c t s the shape of the curves greatly. The only available numerical result for plate R6 i s indicated as point P in F i g . 5.6, where the r e l a t i v e r e f l e c t e d power for both polarizations in the case of normal incidence (0^=0°) i s 0.8 (or -0.97 dB by means of reduction). This value corresponds quite well with the tendency of experimental TE and TM curves shown in F i g . The R4 and R6 largest reduction i s 20.1 in F i g . 5.7, in specular 5.6. r e f l e c t i o n for plates dB as found at ^ = 7 6 ° for TM p o l a r i z a t i o n where 99% of the incident TM-polarized energy i s scattered into other seven spectral orders. This shows that e s s e n t i a l elimination of specular r e f l e c t i o n for TM p o l a r i z a t i o n can be achieved by using t h i s kind of crossed grating, although the available angular range i s very narrow 102 (go), (-i.fi), (-1.0) (-2,0) (O.il) (-2,i1) (10) (Ul) 1 0 P CO Q or ., 1— i I (-3,0) I I I 60" 90" r -5 - LU o 0L Q LU I— U LU _ J U_ LU or -10 -15 - -20 90 F i g . 5.6. Measured r e f l e c t e d power vs. 0- for plate R6 at f=35 GHz, 1^=0; and calculated angular regions of ^ for a l l e x i s t i n g d i f f r a c t e d orders (on top) ; 103 (0.0) . (-1.il), (-1.0) (0,±1) (-2,0) (1,0) , d.tD -I—1—, (-2,i1) 1 1 L_ : i 1 1 ' ' 90 F i g . 5.7. Measured r e f l e c t e d power vs. ^ at f = 33 GHz, ^=0; for plate R6 and calculated angular regions of ^ for a l l e x i s t i n g d i f f r a c t e d orders (on top) ; 104 here. By using the calculated results shown schematically on top of Figures 5.6 and 5.7, we can investigate the e f f e c t s of Wood anomalies. The (0,±1) and (-2,±1) Wood anomalies can account for the changes of the TM curve at the corresponging positions in Figures 5.6 and 5.7. The influence of the other Wood anomalies are not evident here. The two reduction peaks for TM p o l a r i z a t i o n at 0^=65° and 0^=75° in F i g . 5.6 are not caused by Wood anomalies, since two similar peaks can be found at 0^67.5° and 0^76° in F i g . 5.7 and there i s no Wood anomaly at a l l in that range of 0^. The Bragg angle effect i s not observed either in F i g . 5.6 or in F i g . 5.7 (the Bragg angles are 19.3° for X/d=0.66 and 20.5° for X/d=0.70). F i g . 5.8 shows the measured reflected power as a function of angle of rotation (^) for plate R6 at f=35 GHz and 0^=10°. It i s observed that the reduction in TE polarized specular r e f l e c t i o n i s always somewhat greater than that for TM p o l a r i z a t i o n in t h i s case. This i s very d i f f e r e n t from the behavior of plate R4 under the same incident condition, shown in F i g . 5.4. Again, we can see that the cavity depth R has an important influence on the behavior of t h i s kind of crossed gratings. 105 (o,o),(g*i),(*i,o).(-ui) (1,-1) and calculated angular regions of d i f f r a c t e d orders (on top) for a l l existing Chapter 6 ERROR ANALYSIS There are mainly two kinds of errors: experimental errors and errors from the non-ideal nature of the surface. Experimental errors occur in positioning the plate, readings, non-plane wave illumination, and s i t e r e f l e c t i o n s . The surface i s non-ideal from the fact that i t has f i n i t e conductivity, f i n i t e size, and m i l l i n g errors in groove dimensions. The error in positioning the plate accounts for the s h i f t of symmetry of the measured azimuthal angular response curves with respect to the predicted symmetry about 1/^ = 0. The non-plane wave illumination has l i t t l e influence on the performance of grating as shown by J u l l and Ebbeson [ 1 0 ] . The s i t e r e f l e c t i o n s are believed to cause the o s c i l l a t o r y behavior of some of the experimental curves and they are a large source of experimental error. By using microwave radiation in the range of 35 GHz, a l l groove dimensions which had been designed in the range of the incident wavelength were large enough to be milled to a s u f f i c i e n t accuracy, and also we can work with near-perfectly conducting models ( i t was reported [ 2 0 ] that the model of i n f i n i t e conductivity for metal gratings i s very well adapted even when the wavelength exceeds 4 am). Hence, the e f f e c t s of f i n i t e conductivity, surface roughness and oxide layers which are the largest source of error in the o p t i c a l region [ 1 ] , are n e g l i g i b l e in t h i s range of 106 107 frequency. It i s also believed that the f i n i t e size of grating only a f f e c t s the performance of grating s l i g h t l y . The measurements on some p r o f i l e s of singly periodic gratings in the microwave region conducted by D e l e u i l [1,20] showed that, as soon as the number of corrugations i s greater than about twelve, gratings behave as i f they were i n f i n i t e l y wide. Further numerical analysis by J u l l and Hui [14] shows that for high e f f i c i e n c y d i f f r a c t i o n gratings, as the grating size decreases, the d i f f r a c t e d beam broadens but the d i f f r a c t i o n e f f i c i e n c y can remain high u n t i l the number of corrugations i s very small. For instance, Facq's numerical results [7] for TM-polarized scatter from f i n i t e near-optimum rectangular-groove surfaces at 0^=30° show a reduction in d i f f r a c t i o n e f f i c i e n c y from 98.8% i n f i n i t e surface to 97.4% for an for 10 corrugations and 96% for 5 . Therefore very few corrugations are needed for e f f e c t i v e reduction of specular r e f l e c t i o n . The above conclusion i s for singly periodic gratings, but i t seems quite reasonable to believe that the situation for crossed gratings i s similar to that for c l a s s i c a l gratings, and for the nine crossed grating plates investigated experimentally, there must be enough corrugations for g r a t i n g - l i k e behavior. It i s worth while pointing out that there i s another kind of experimental error. In our measurements i t had been actually assumed that when the incident wave was TE-polarized (say), the specularly reflected wave would be 108 s t i l l t o t a l l y TE-polarized. Therefore the orientation of the receiving horn was always the same as that of the transmitting horn when the reflected power was measured. This assumption i s true for c l a s s i c a l gratings in the case of non-oblique incidence (^=0°). For crossed gratings, there i s s t i l l no t h e o r e t i c a l analysis on t h i s problem. In order to investigate the above assumption, we took some measurements on plate C4: by holding both horns v e r t i c a l l y - o r i e n t e d , we measured the power of TE component of the reflected wave when the incident wave i s TE-polarized; then by changing the receiving horn from v e r t i c a l to horizontal, we measured the power of TM component of the r e f l e c t e d wave. The r e s u l t s , presented in F i g . 6.1, show that when the incident wave i s TE-polarized, the power of the TM component of the reflected wave i s at most 5% (-13 dB) of that of the TE component, either in the case of non-oblique incidence (see F i g . 6.1a) or in the case of oblique incidence ( see F i g . 6.1b). One similar measurement was also taken on Plate R4 in the case of 0^=30°, ^ = - 1 5 ° , and f=35 GHz. The measured r e l a t i v e power of TM component of the r e f l e c t e d wave i s 2.2% (-16.5 dB) of that of the TE component when the incident wave i s TE-polarized. From the above sample measurements, i t seems reasonable that in general, when the incident wave i s TE-polarized, the measured power of TE component of the reflected wave can approximate the t o t a l r e f l e c t e d power. It i s believed that a V; =0 6i=38° Fig. 6.1. Relative power PTM of TM component of the reflected wave when a TE-polarized wave i s incident on plate C4 (that of TE component i s 0 dB) a. non-oblique case (^=0) b. oblique case ( 9; =38° f ixed) o 110 similar approximation i s v a l i d for a TM incident wave. In other words, the crossed-polarized reflected power can be neglected when compared with the co-polarized reflected power. Hence, our measurements on reflected power from crossed gratings seem within acceptable accuracy as far as the above problem i s concerned. Chapter 7 CONCLUSIONS To our knowledge, this work i s the f i r s t systematic experimental investigation of the scattering of an electromagnetic wave by a conducting pyramidal crossed grating, concentrated on the case of a higher ratio of groove depth to period of grating, for which numerical results are s t i l l very d i f f i c u l t to obtain. Some measurements were also taken on the conducting crossed grating with hemispherical c a v i t i e s . Summarized below are the conclusions drawn: 1. It i s demonstrated experimentally for the f i r s t time that elimination of specular r e f l e c t i o n from a conducting pyramidal surface can be achieved. The best performance i s that 99.94% of the power of a TM-polarized incident wave can be scattered into a single spectral order (-1,0) by plate A4. For TE p o l a r i z a t i o n , the best r e s u l t i s a reduction of 98%, achieved by plate A6. The same plate can also d i f f r a c t at the other incident angle 80% of the energyof an a r b i t r a r i l y polarized incident wave, and t h i s i s the best result from the point of view of equal reduction for both polarizations. The above results are less satisfactory i f compared with those obtained from the two-dimensional with rectangular grooves gratings [11]. However, the p o s s i b i l i t i e s of obtaining better performances of 1 11 1 12 pyramidal crossed gratings s t i l l exist since our investigation i s just the beginning. 2. It i s worth noting that very interesting phenomena occur when the incident wavelength i s equal to the period of the pyramidal crossed grating, and in t h i s case of X/d=1 our three best results just mentioned were obtained. We suggest that further experimental and t h e o r e t i c a l investigation for crossed gratings with d i f f e r e n t p r o f i l e (for example, crossed gratings with hemispherical c a v i t i e s ) be conducted under the condition of X/d=1. 3. For situation in which the angle of incidence i s near grazing (0^80°), reduction in TM-polarized specular r e f l e c t i o n can be as large as 99.3%, which i s obtained by plate B9. This i s a desirable property for some applications. 4. The p o l a r i z e r - l i k e behavior of some pyramidal crossed gratings i s observed. The best performance i s that 90% of the TM components of an a r b i t r a r i l y polarized incident wave can be scattered by a pyramidal surface (plate A9), while TE p o l a r i z a t i o n i s completely reflected. 5. It i s shown experimentally that the product formula l i n k i n g shallow crossed and c l a s s i c a l gratings i s not v a l i d for deeply-grooved crossed gratings, and i n general the r e f l e c t i o n free properties of a singly-periodic surface cannot carry over to i t s 113 doubly-periodic equivalent. The Wood anomaly often causes remarkable changes in the shape of behavior curves of grating, the position of which i s determined by the values of incident angles (both 0£ and yp^), period of grating and incident wavelength (or the r a t i o X/d). The experimental points are in good agreement with the t h e o r e t i c a l predictions concerning the Wood anomalies. For c l a s s i c a l gratings with period in the range X/2<d^X, the largest reduction in specular r e f l e c t i o n in the non-oblique case (\^=0°) always occurs at the Bragg angle 0^=sin~ (X/(2a)). 1 However, t h i s Bragg angle e f f e c t is not evident for crossed gratings. Because of the square-symmetry of the pyramidal grating, i t was o r i g i n a l l y thought that t h i s kind of crossed grating was an interesting p r o f i l e for the reduction of specular r e f l e c t i o n when the d i r e c t i o n of incidence i s a r b i t r a r y . However, our measured azimuthal angular responses have shown that the reduction in specular r e f l e c t i o n also depends on the value of \[/^ greatly. It seems that an e f f e c t i v e wide, range of i//^ where the specular r e f l e c t i o n i s eliminated i s at least d i f f i c u l t to find, i f not impossible. The major parameter of the grating p r o f i l e (the apex angle a in the case of pyramidal crossed gratings, the cavity depth R in the case of crossed gratings with hemispherical c a v i t i e s ) has an important effect on the 114 general behavior of grating, and the value of the r a t i o X/d defines the c r i t i c a l positions of Wood anomalies. Thus, the grating p r o f i l e and the r a t i o X/d both determine the properties of grating at a p a r t i c u l a r incident condition. 10. Our experimental results prove i n d i r e c t l y the t h e o r e t i c a l prediction that a crossed grating with hemispherical c a v i t i e s (plate R4) can eliminate 90% of specular r e f l e c t i o n for an a r b i t r a r i l y polarized incident wave in the case of normal incidence, and i t i s also observed that t h i s plate can s t i l l keep cancelling at least 68% of specular r e f l e c t i o n when the incident wave i s deviated from the normal of surface by an angle as large as 30°. This i s very interesting for the study of solar energy captation with a roughened surface. 11. It i s demonstrated experimentally that a crossed grating with hemispherical c a v i t i e s (plate R6) can eliminate 99% of TM polarized specular r e f l e c t i o n , though the e f f e c t i v e angular range i s very narrow. In general, our experimental results agree with the t h e o r e t i c a l predictions, and provide useful information for an understanding of the behavior of these two kinds of crossed gratings. This may a s s i s t and give new impetus to further numerical and experimental investigations. APPENDIX: NUMERICAL RESULTS FOR PERFECTLY CONDUCTING STRIPS OF SIX SYMMETRICAL TRIANGULAR CORRUGATIONS We obtained from a computer program by Facq [17] numerical results for p e r f e c t l y conducting s t r i p s of six symmetrical triangular corrugations, with the same p r o f i l e s as those of the c l a s s i c a l grating plates investigated experimentally. A comparison between numerical and experimental results i s presented in F i g . A.1. After taking into account the e f f e c t of the larger size of a experimental grating of 30-39 grooves, which increases the d i f f r a c t i o n e f f i c i e n c y , the correspondence between the calculated and measured values i s good. There are only two exceptions where the measured value is s l i g h t l y less than the calculated value. Also obtained were the calculated power patterns which are exhibited in Figures A.2-A.6. Since a grating i s f i n i t e there i s always a forward-scatter lobe in 9O°<0<18O°. In Figures A.5 and A.6 there i s also a backscatter lobe but e s s e n t i a l l y no specular r e f l e c t i o n , indicating near t o t a l d i f f r a c t i o n to the n=-l spectral order. 115 116 ECHEIETTE STRIP d a (mm) DIFFRACTION EFFICIENCY £ - 1 (GHz) NUMERICAL EXPERIMENTAL TM TE TM TE A4 8.00 45° 35 32.4° .8488 .7281 .975 .620 B6 6.23 60° 35 43.5° . 1962 .6566 .073 .718 B9 6.23 90° 35 43.5° .9605 .3603 .99998 .553 C4 6.9 44° 35 38. 1° .9526 .9704 .99997 .9611 C4 6.9 44° 33.75 40.0° .9701 .9489 .9967 .9946 F i g . A.1. Comparison between numerical results for perfectly conducting s t r i p s of six symmetrical triangular corrugations and experimental r e s u l t s for the corresponding triangular grating plates of 30-39 grooves with the same p r o f i l e 12 um 4-> •o cn i • REFERENCES [I] P.Bliek and R.Deleuil, "Experimental investigation in microwave range of d i f f r a c t i o n by c l a s s i c a l and crossed gratings", Proc. SPIE, 240, 263-271, 1980. [2] L. Cai, E. V. J u l l and R. D e l e u i l , "Scattering by pyramidal r e f l e c t i o n gratings", Intern. IEEE AP-S Symposium, Boston, June, 1984. [3] C-C. Chen, "Transmission of microwaves through perforated f l a t plates of f i n i t e thickness", IEEE Trans. MTT-21,1-6, 1973. [4] L. S. Cheo, J . Shmoys and A. Hessel, "On simultaneous blazing of triangular groove d i f f r a c t i o n gratings", J . Opt. Soc. Am., 67, 1686-1688, 1977. [5] G. H. Derrick, R. C. McPhedran, D. Maystre and M. Neviere, "Crossed gratings: a theory and i t s applications", Applied Physics, J_8, 39-52, 1979. [6] G. R. Ebbeson, "The use of fin-corrugated periodic surfaces for the reduction of Interference from large r e f l e c t i n g surfaces", M. A. Sc. Thesis, University of B r i t i s h Columbia, 1974. [7] P. Facq, "Application des matrices de Toeplitz a l a theorie de l a d i f f r a c t i o n par les structures cylindriques periodiques limitees", D. Sc. Thesis, University of Limoges, 1977. [8] J . W. Heath, "Scattering by a conducting periodic surface with a rectangular groove p r o f i l e " , M. A. Sc. Thesis, University of B r i t i s h Columbia, 1977. [9] E. V. J u l l , N. C. Beaulieu and D. C. W. Hui, "Perfectly blazed triangular groove r e f l e c t i o n gratings", J . Opt. Soc. Am., J_, 180-182, Feb. 1984. [10] E. V. J u l l and G. Ebbeson, "The reduction of interference from large r e f l e c t i n g surfaces", IEEE Trans. AP-25, 656-670, 1977. [II] E. V. J u l l and J . W. Heath, "Conducting surface corrugations for multipath interference suppression", PROC. I EE, 21 ., 1321-1326, Dec. 1978. 5 [12] E. V. J u l l and J . W. Heath, "Reflection grating p o l a r i z e r s " , IEEE Trans. AP-28, 586-588, July 1980. [13] E. V. J u l l , J . W. Heath and G. R. Ebbeson, "Gratings that d i f f r a c t a l l incident energy", J . Opt. Soc. Am., 122 123 67, 557-560, A p r i l 1977. [14] E. V. J u l l , D. C. W. Hui and P. Facq, "Scattering by dual blazed corrugated conducting s t r i p s and large r e f l e c t i o n gratings", to be published in J . Opt. Soc. Am., 1985. [15] J . Y. L. Ma and L. C. Robinson, "Night moth eye window for the millimetre and sub-millimetre wave region", Optica Acta, 30, 1685-1695, 1983. [16] D. Maystre, R. McPhedran and M. Neviere, Private communication. [17] D. Maystre and M. Neviere, "Electromagnetic theory of crossed gratings", J . Optics (Paris), 9, 301-306, 1978. [18] D. Maystre, M. Neviere, and P. Vincent, "On the properties of f i n i t e l y conducting crossed gratings (some applications)", Proceedings of ICO-11 Conference, Madrid, Spain, 1978. [19] R. C. McPhedran, G. H. Derrick, M. Neviere and D. Maystre, "Metallic crossed gratings", J . Optics (Paris), J_3, 209-218, 1982. [20] R. P e t i t , ed., Electromagnetic Theory of Gratings, Springer-Verlag: B e r l i n , 1980. [21] E. Toro, "Etude de l a d i f f r a c t i o n des ondes electromagetiques par un reseau metallique croise forme de cavites hemispheriques", D. Sc. thesis, Universite de provence, France, 1983. [22] E. Toro and R. D e l e u i l , "Application of a rigorous theory to electromagnetic d i f f r a c t i o n from a biperiodic rough surface", submitted for publication, 1984. [23] P. Vincent, "A f i n i t e - d i f f e r e n c e method for d i e l e c t r i c and conducting crossed gratings", Opt. Commun., 26, 293-296, 1978. [24] S. J . Wilson and M. C. Hutley, "The o p t i c a l properties of 'moth eye' a n t i r e f l e c t i o n surfaces". Optica Acta, 29, 993-1009, 1982. [25] A. Wirgin, " D i f f r a c t i o n of light by a perfectly conducting b i a x i a l periodic surface", Opt. Commun., 45, 221-225, A p r i l 1983. [26] A. Wirgin and R. D e l e u i l , "Theoretical and experimental investigaton of a new type of blazed grating", J . Opt. Soc. Am., 59, 1348-1357, Oct. 1969.
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Experimental investigation of some conducting crossed gratings Cai, Li-He 1985
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Title | Experimental investigation of some conducting crossed gratings |
Creator |
Cai, Li-He |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | The properties of a crossed grating of square pyramids and a crossed grating with hemispherical cavities to eliminate specular reflection from a conducting surface are studied experimentally. Measurements were made in the microwave range of 35 GHz. The best performance is that 99.94% of the power of a TM-polarized incident wave can be scattered into a single spectral order by a pyramidal crossed grating, while for TE polarization the reduction in specular reflection can be as high as 98%. Anti-reflection properties of a crossed grating with hemispherical cavities near normal incidence are also observed. Comparison between the behavior of triangular and pyramidal gratings of the same profile is made. Effects of the profile parameters are investigated. Basically the experimental results agree with the theoretical predictions. This investigation provides a set of experimental data to assist further numerical study. |
Subject |
Diffraction gratings |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0096228 |
URI | http://hdl.handle.net/2429/25056 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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