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UBC Theses and Dissertations

Study of the scattering of electromagnetic waves from certain types of random media Olsen, Roderic Lloyd 1970

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A STUDY OF THE SCATTERING OF ELECTROMAGNETIC WAVES FROM CERTAIN TYPES OF RANDOM MEDIA by RODERIC LLOYD OLSEN B.A.Sc, The Un i v e r s i t y of B r i t i s h Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Research Supervisor Members of the Committee Acting Head of Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA ' Ju l y , 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t hou t my w r i t t e n p e r m i s s i o n . Department o f E./,rXC7r?lctft<L £KIGI K^eg (Sj N^G The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada ABSTRACT This work i s a study of the s c a t t e r i n g of electromagnetic waves from random media of d i s c r e t e s c a t t e r e r s . The object i s p r i m a r i l y the i n v e s t i g a t i o n of e x i s t i n g general d i s c r e t e - s c a t t e r e r theories and the development of more accurate ones, the technique of Monte Carlo computer simulation being employed to provide "exact" experimental r e s u l t s f o r comparison with t h e o r e t i c a l data. A one-dimensional model of randomly-positioned planar s c a t t e r e r s i s used as a t o o l i n the i n v e s t i g a t i o n and as a means of providing i n s i g h t into the p h y s i c a l and s t a t i s t i c a l c h a r a c t e r i s t i c s of d i s c r e t e - s c a t t e r e r media. The l i m i t a t i o n s of the one-dimensional forms of Twersky's theories for the coherent f i e l d are i l l u s t r a t e d by a presentation of r e s u l t s f o r a wide range of s c a t t e r i n g parameters, and requirements necessary f o r the approximate v a l i d i t y of these theories are given. Accurate s e r i e s expressions f o r several average f i e l d functions of i n t e r e s t i n the problem of plane-wave s c a t t e r i n g from d i s t r i b u t i o n s of uniformly-random planar s c a t t e r e r s are presented and v e r i f i e d from simulation r e s u l t s . The asymptotic s c a t t e r i n g behavior f o r a low average density of s c a t t e r e r s i s emphasized; a modification to the one-dimensional form of Twersky's free-space theory f o r the coherent transmitted f i e l d to give exact asymptotic behavior i s shown to be a considerable improvement f o r higher average d e n s i t i e s a l s o . The r e l a t i o n of the one-dimensional model theory and r e s u l t s to more complex three-dimensional models i s discussed where p o s s i b l e . Simulation.methods for the generation of a non-uniform d i s t r i b u t i o n of p l a n a r - s c a t t e r e r configurations weighted towards " p e r i o d i c i t y " are presented. Based on the s c a t t e r i n g r e s u l t s obtained, c r i t e r i a f o r the assumption of a uniform d i s t r i b u t i o n are given. The p h y s i c a l conditions necessary f o r the approximate v a l i d i t y of the b i v a r i a t e Gaussian d i s t r i b u t i o n i n describing the t o t a l f i e l d s t a t i s t i c s of the one-dimensional model are discussed and i i q u a n t i t a t i v e r e s u l t s based on the t h i r d and fourth f i e l d moments given. Also presented i s a new p h y s i c a l model of a random medium of d i s c r e t e s p h e r i c a l s c a t t e r e r s f o r use i n c o n t r o l l e d laboratory experiments at millimeter-wave frequencies. The main feature of t h i s model i s that the sc a t t e r e r s t a t i s t i c s are d i r e c t l y c o n t r o l l e d by an a p p l i c a t i o n of the Monte Carlo method. The r e s u l t s of an experimental i n v e s t i g a t i o n into the s u i t a b i l i t y of the model are given. i i i TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v i i i LIST OF TABLES x LIST OF SYMBOLS x i ACKNOWLEDGEMENTS x v i 1. GENERAL INTRODUCTION 1 2. THEORETICAL CONSIDERATIONS 6 2.1 I n t r o d u c t i o n 6 2.2 The One-Dimensional Model 8 2.3 B a s i c Formalism f o r S c a t t e r i n g from a Fixed C o n f i g u r a t i o n of A r b i t r a r y S c a t t e r e r s 11 2.4 E x p l i c i t Theories f o r S c a t t e r i n g from a Fixed Array of Planar S c a t t e r e r s 13 2.4.1 Wave Transmission M a t r i x Representation 14 2.4.2 Orders-of-Back-Scattering Representation 15 2.5 S c a t t e r i n g from an Ensemble of S c a t t e r e r C o n f i g u r a t i o n s 19 2.5.1 Average F i e l d Functions of I n t e r e s t 19 2.5.2 Some E x i s t i n g Theories f o r Average F i e l d Functions 23 2.6 S e r i e s Approximations f o r S c a t t e r i n g from an Ensemble of Uniformly-Random P l a n a r - S c a t t e r e r C o n f i g u r a t i o n s 25 2.6.1 The Coherent F i e l d s 26 2.6.2 The Average T o t a l I n t e n s i t i e s 28 2.6.3 The Covariant F i e l d s 31 2.7 Asymptotic Theories f o r S c a t t e r i n g from Low Average Density S c a t t e r e r D i s t r i b u t i o n s 33 2.7.1 Asymptotic Theory f o r the Coherent Transmitted F i e l d ... 33 2.7.2 Asymptotic Theories f o r the Average T o t a l F i e l d I n t e n s i t i e s 36 2.8 T h e o r e t i c a l Models f o r the P r o b a b i l i t y Density of the T o t a l F i e l d 40 i v Page 2.9 Other T h e o r e t i c a l Considerations 45 2.10 Summary 47 3. APPLICATION OF MONTE CARLO SIMULATION TO THE STUDY OF SCATTERING FROM RANDOM MEDIA 49 3.1 Introduction 49 3.2 Technique of Simulation Applied to a Random Medium of Discrete Scatterers 50 3.3 Random Number Generation 51 3.4 Accuracy i n Monte Carlo Simulation 53 4. THEORETICAL AND SIMULATION RESULTS FOR A UNIFORM PROBABILITY DENSITY OF PLANAR-SCATTERER CONFIGURATIONS 55 4.1 Introduction -• 55 4.2 Transmitted F i e l d Moments 57 4.2.1 The Coherent F i e l d 57 4.2.2 The Average Incoherent Intensity 63 4.2.3 The Variances and Covariance 66 4.2.4 Moments of the Amplitude and Phase 66 4.3 Reflected F i e l d Moments 68 4.3.1 The Coherent F i e l d 68 4.3.2 The Average Incoherent Intensity 71 4.3.3 The Variances and Covariance 74 4.4 D i s t r i b u t i o n of the T o t a l F i e l d 76 4.5 Summary and General Discussion of Results 80 5. SIMULATION OF A NON-UNIFORM PROBABILITY DENSITY OF PLANAR-SCATTERER CONFIGURATIONS WEIGHTED TOWARDS PERIODICITY 82 5.1 Introduction 82 5.2 Methods of Generating the D i s t r i b u t i o n 84 5.2.1 Method A 85 5.2.2 Method B 85 5.2.3 Results 86 v Page 5 . 3 V a r i a t i o n of the D i s t r i b u t i o n between the Limits of Uniform-Randomness and P e r i o d i c i t y f o r Fixed Scattering Parameters .... 90 5 . 4 Comparison of Simulation and Mixed-Space Theory Results f or Planar Scatterers of F i n i t e Thickness 99 5 . 4 . 1 The Coherent Transmitted F i e l d 99 5 . 4 . 2 The Coherent Reflected F i e l d 104 5 . 5 Summary and General Discussion of Results 107 5 . 5 . 1 Summary . .. 107 5 . 5 . 2 General Discussion 107 6 . EXPERIMENTAL INVESTIGATION 109 6 . 1 Introduction 109 6 . 2 The Ph y s i c a l Model 110 \ 6 . 2 . 1 Generation of Uniform D i s t r i b u t i o n 112 6 . 2 . 2 The Support-Medium 114 6 . 2 . 3 Comparison with the Sylvania Model 115 6 . 3 The Scattering Range, Antenna C h a r a c t e r i s t i c s , and Scanning Device 118 6 . 4 Experimental Apparatus, Measurement and Data Processing Methods 121 6 . 4 . 1 Experimental Apparatus and Procedures 121 6 . 4 . 2 Data Processing Methods 125 6 . 5 Experimental Results 127 6 . 5 . 1 Experiments on the Support-Medium 127 6 . 5 . 2 Experiments on T y p i c a l Scatterer D i s t r i b u t i o n s 128 6 . 5 . 3 Discussion of O v e r a l l Results 138 7 . CONCLUSIONS 140 APPENDIX A SUMMARY OF TWERSKY'S THEORIES FOR RANDOM MEDIA OF DISCRETE SCATTERERS 143 A . l Twersky's Free-Space Theory f o r the Coherent F i e l d .... 143 A.2 Twersky's Mixed-Space Theory f o r the Coherent F i e l d ... 145 v i Page A. 3 Theories f o r Other Average F i e l d Functions 148 APPENDIX B SCATTERING FROM A SINGLE DIELECTRIC SLAB 150 B. l Conventional Scattering Amplitudes 150 B.2 Mixed-Space Scattering Amplitudes 153 APPENDIX C VALIDITY OF THE DISCRETE POSITION APPROXIMATION IN SIMULATION STUDIES 155 APPENDIX D DESIGN OF MICROWAVE ANECHOIC CHAMBER AND POSITIONING DEVICE 166 D.l Design and Testing of Anechoic Chamber 166 D.2 Design of P o s i t i o n i n g Device 168 REFERENCES 170 v i i LIST OF ILLUSTRATIONS Figure Page t 2.1 The One-Dimensional Model 9 2.2 Wave Transmission Matrix Representation 14 2.3 Dominant M u l t i p l e - S c a t t e r i n g Processes i n O-B-S Representation 16 2.4 Phasor Diagram of the T o t a l F i e l d Resolution 20 4.1.a Phase of Coherent Transmitted F i e l d as a Function of 58 4.1.b Coherent Transmitted F i e l d Intensity as a Function of 59 4.2 Coherent Transmitted F i e l d as a Function of N (Asymptotic Results) 61 4.3 Coherent Transmitted F i e l d as a Function of w i^ (Asymptotic Results) 62 4.4 Average Incoherent Intensity of Transmitted F i e l d as a Function of d-^  64 4.5 Asymptotic Results f o r the Average Incoherent Intensity of the Transmitted F i e l d 65 4.6 Variances and Covariance of Transmitted F i e l d Components as Functions of d^ 67 4.7 Standard Deviations of the Transmitted F i e l d Amplitude and Phase as Functions of N (Asymptotic Results) 69 4.8 Coherent Reflected F i e l d as a Function of d^ 70 4.9 Average Incoherent Intensity of Reflected F i e l d as a Function of d^ 72 4.10 Asymptotic Results for the Average Incoherent Intensity of the Reflected F i e l d 73 4.11 Variances and Covariance of Reflected F i e l d Components as Functions of d^ 75 4.12 Skewness and Kurtosis C o e f f i c i e n t s as Functions of the -Phase Reference 77 4.13 Extreme Values of b and y a s Functions of w^r 78 5.1 One-Scatterer Normalized P r o b a b i l i t y Density Curves for N = 4 87 v i i i Figure Page 5.2 One-Scatterer Normalized P r o b a b i l i t y Density Curves f o r N = 5 88 5.3 Phase and Intensity of the Transmitted F i e l d as Functions of d^ f o r a P e r i o d i c Array 92 5.A Dependence of Average F i e l d Functions on 3 0 f o r Various Values of d x 93 5.5 3 0~Variation Curves for Values of d^ i n the Neighbourhood of Resonance at P e r i o d i c i t y 97 5.6 Coherent Transmitted F i e l d Results f o r High 101 5.7 Coherent Transmitted F i e l d Results f o r Low 103 5.8 Comparison of Results f o r the Coherent Reflected F i e l d ....... 105 6.1 S i m p l i f i e d Diagram of Scattering Geometry I l l 6.2 T y p i c a l Computer Output f o r Sphere Coordinates 114 6.3 View of the P h y s i c a l Model 116 6.4 Plan View of the Geometry of Antennas and Medium 119 6.5 View of the Receiving Antenna and Mixer 120 6.6 View of the Experimental Apparatus 122 6.7 Block Diagram of the Experimental Apparatus 123 6.8 C o r r e l a t i o n C o e f f i c i e n t Curves for Amplitude and Phase 134 B . l Scattering from a Single D i e l e c t r i c Slab 150 B. 2 Scattering Amplitudes as a Function of w t^ f o r e r = 2.0 152 C. l Dependence of Average F i e l d Functions on 3^  f o r the Non-Uniform D i s t r i b u t i o n 157 C.2 Dependence of Average F i e l d Functions on for the Uniform D i s t r i b u t i o n 158 C. 3 Single-Scattering Geometry for Three-Dimensional Model 160 D. l S i m p l i f i e d Plan-View Diagram of the Anechoic Chamber 167 D.2 View of the P o s i t i o n i n g Device 169 i x LIST OF TABLES Table Page 6.1 Results f o r Estimated Average F i e l d Functions 131 6.2 Accuracy C a l c u l a t i o n s Based on Equivalent Uncorrelated Samples 135 6.3 Accuracy Estimates Based on Twice the Standard E r r o r s of the Means 137 x LIST OF SYMBOLS ( F ) = ensemble average of function F Re F = r e a l part of complex function F Im F = imaginary part of F |F| = magnitude of F Arg F = argument of F F* = complex conjugate of F a n m = c o e f f i c i e n t s i n asymptotic expression f o r (]T|2) A Q = amplitude of constant phasor A g, A^ = amplitudes of random phasors b , b = c o e f f i c i e n t s of skewness f o r the d i s t r i b u t i o n s of T x and T y b n m = c o e f f i c i e n t s i n asymptotic expression f o r <^|R|2) B n m = binomial c o e f f i c i e n t s C = amplitude of coherent f i e l d C 2 = i n t e n s i t y of coherent f i e l d C x, Cy = rectangular components of coherent f i e l d = C cosa, C sina C n m = covariant r e f l e c t e d f i e l d c o e f f i c i e n t s d = width of slab region containing s c a t t e r e r centers d^ = width of slab region i n free-space wavelengths = d/X d r = distance of r e c e i v i n g antenna from center of medium d t = distance of transmitting antenna from center of medium D = function i n free-space and mixed-space theories e = distance of c l o s e s t approach between s c a t t e r e r centers e x = e/X f = a r b i t r a r y function x i F = e l e c t r i c or magnetic f i e l d vector g(f,k) = s c a t t e r i n g amplitude f o r a three-dimensional scatterer g + , g_ = forward- and back-scattering amplitudes f o r an i s o l a t e d planar s c a t t e r e r (normal incidence) = T l " L R l g+s> g_ s = g+, g_ f o r s c a t t e r e r s g^ ., g]_ = mixed-space planar-scatterer amplitudes associated with mixed-space theory f o r the coherent f i e l d I = amplitude of incoherent f i e l d I 2 = i n t e n s i t y of incoherent f i e l d I x , Iy .= rectangular components of incoherent f i e l d = I cos(9, I sin<0 3 k = propagation constant i n free space = 2TT/X K = propagation constant i n synthetic medium associated with the coherent f i e l d K' = propagation constant i n medium of s c a t t e r e r k, K = propagation vectors associated with k, K k', K' = propagation vectors k, K r e f l e c t e d i n slab-region face n = number of uncorrelated samples n D = resonance index N = number of sc a t t e r e r s p (s^ ,. . . ,§jj) = j o i n t p r o b a b i l i t y density function f o r an ensemble {s"j,...,Sjg} of s c a t t e r e r configurations p(zj,...,z^) = j o i n t p r o b a b i l i t y density'function f o r the planar-scatterer p o s i t i o n s i n the one-dimensional model p(z|,...,z^j) = j o i n t p r o b a b i l i t y density function f o r the "ordered-p o s i t i o n s " of the sc a t t e r e r s i n the one-dimensional model p'(T ,T ) = j o i n t p r o b a b i l i t y density function f o r the f i e l d components X 7 T x a n d T y Q = function i n free-space and mixed-space theories f = f i e l d point p o s i t i o n vector x i i r g = p o s i t i o n vector for s c a t t e r e r s R = o v e r a l l r e f l e c t i o n c o e f f i c i e n t for an array of planar s c a t t e r e r s Rj = r e f l e c t i o n c o e f f i c i e n t for an i s o l a t e d planar scatterer = g-R g = R 1 f o r s c a t t e r e r s R I I I = contributions to R from f i r s t and t h i r d orders-of-back-scattering, etc. s = covariant f i e l d phase s, Sj, s 2> etc. = indeces s p e c i f y i n g s c a t t e r e r s s, s 1 } s 2 , etc. §2, s 2 , . . ., SJJ = random vector v a r i a b l e s describing a d i s c r e t e s c a t t e r e r configuration S = covariant f i e l d amplitude t = time t = index s p e c i f y i n g a s c a t t e r e r t T = o v e r a l l transmission c o e f f i c i e n t f o r an array of planar s c a t t e r e r s = transmission c o e f f i c i e n t f o r an i s o l a t e d planar s c a t t e r e r - 1 + 8+ T = T, for s c a t t e r e r s s i T-J--J-, T-jry = contributions to T from second and fourth orders-of-back-scattering, etc. u = i s o l a t e d s c a t t e r e r function for a s c a t t e r e r s s U g = f i e l d scattered by s c a t t e r e r s Xi = t o t a l f i e l d scattered by a configuration of s c a t t e r e r s V = volume occupied by three-dimensional s c a t t e r e r s w = width of d i e l e c t r i c slab s c a t t e r e r w^  = w/X w^ , = w/X' X = a r b i t r a r y f i e l d function x, y, z = c a r t e s i a n coordinates x i i i z g = random variable describing position of planar scatterer s z g = random variable describing "ordered-position" of scatterer s ex = phase of coherent f i e l d $ Q = f r a c t i o n a l volume m = maximum physically allowable value of $ = occupation r a t i o Y x, Yy = coe f f i c i e n t s of kurtosis for the d i s t r i b u t i o n s of T x and Ty A' = single slab scatterer function e r = d i e l e c t r i c constant of scatterer medium n = bulk index of re f r a c t i o n of synthetic medium associated with the coherent f i e l d n' = r e f r a c t i v e index of scatterer medium • ^7 Gs, 0^ = phases of random phasors 0 = angle of incidence for an obliquely incident plane wave X = free-space wavelength X' = wavelength i n scatterer medium y = correla t i o n c o e f f i c i e n t for T x and Ty v = phase reference of t o t a l f i e l d £^ = space between planar scatterer boundaries p = average density of scatterers = N/d (one-dimensional model) = N/V (three-dimensional model) px = pX = N/d^ a = t o t a l scattering cross-section 0 £ = standard deviation of function f O f 2 = variance of function f = <*x2> 0 > 2 ° 4 - < y xiv T = phase of normalized t o t a l f i e l d T = amplitude of normalized t o t a l f i e l d T x, T = rectangular components of normalized t o t a l f i e l d = T COST, T sinT <(> = incident f i e l d <j>' = f i e l d r e f l e c t e d i n slab-region face $ = phase of incoherent f i e l d $ s = multiple-scattered f i e l d e x c i t i n g s c a t t e r e r s if) = t o t a l f i e l d = <|» + u to = angular frequency V 2 = Laplacian operator xv ACKNOWLEDGEMENTS I wish to express my sincere gratitude to my research supervisor Dr. M. Kharadly f o r h i s i n i t i a l i n s p i r a t i o n and f o r his continuous encourage-ment throughout the course of t h i s work. Gr a t e f u l acknowledgement i s made to the B r i t i s h Columbia Telephone Company for a graduate scholarship i n the academic year 1965-1966 and to the National Research Council of Canada for postgraduate scholarships throughout the period 1966-1969 and for support under grants A-6068 and A-3344. Thanks are due to members of the workshop, s t a f f who constructed the apparatus necessary i n the p r o j e c t , often making h e l p f u l suggestions regarding d e t a i l s of construction; Mr. C. Chubb did tne machine work on the p o s i t i o n i n g device and Mr. C. S h e f f i e l d , the wiring. Thanks are due also to Miss S. Haslin who developed the sampling programs f o r the PDP-9 computer and i n t e r f a c e . I wish to thank Messrs. B. Wilbee and P. 0'Kelly f or proofreading the f i n a l d r a f t . Also, I wish to thank my fellow students f o r many h e l p f u l discussions regarding the work. F i n a l l y , I would l i k e to thank my loving wife, S a l l y , f o r her continuous support throughout my work and f o r typing the i n i t i a l and f i n a l manuscripts. x v i 1 1. GENERAL INTRODUCTION The s c a t t e r i n g of electromagnetic waves from random media has i n recent years become the subject of an increasing amount of research i n a v a r i e t y of d i s c i p l i n e s . The subject enters into such areas of i n v e s t i g a t i o n as the propagation of radio waves through the atmosphere, radar, radio and o p t i c a l astronomy, and studies of the microstructure of gases, l i q u i d s , and s o l i d s . The s p e c t r a l range of i n t e r e s t correspondingly extends from frequencies below the broadcast bands to those i n the X-ray region. Although the random nature of many media has long been recognized and approximate s c a t t e r i n g theories applicable to some media have existed for a number of years, the demands of modern technology have focussed new attention on the development of more accurate theories. Two general models have been used to t h e o r e t i c a l l y describe a random medium: (a) the perturbed continuum model, and (b) the d i s c r e t e s c a t t e r e r model.^ In the perturbed continuum model z the medium i s considered to be continuous i n character and randomness i s accounted f o r by a s t a t i s t i c a l d e s c r i p t i o n of i t s p e r m i t t i v i t y and permeability. Fluctuations i n these parameters are assumed to occur about mean values which correspond to the parameters of an i d e a l i z e d homogeneous medium. D i f f e r e n t forms of t h i s model have been used i n the study of ionospheric and tropospheric s c a t t e r i n g of radio waves and i n analyzing s c i n t i l l a t i o n of radio and o p t i c a l c e l e s t i a l sources. In the d i s c r e t e s c a t t e r e r model a homogeneous medium i s assumed to be embedded with d i s c r e t e s c a t t e r i n g regions of d i f f e r e n t p e r m i t t i v i t y and permeability, with randomness accounted for by a s t a t i s t i c a l d e s c r i p t i o n of these d i s c r e t e - s c a t t e r e r c h a r a c t e r i s t i c s . This model has also been used i n the study of radio-wave s c a t t e r i n g i n the atmosphere where i t applies to various forms of p r e c i p i t a t i o n and as a f i r s t approximation to small-scale 3 i r r e g u l a r i t i e s i n the r e f r a c t i v e index. It has also been used as a model of. the microstructure of gases, l i q u i d s , and s o l i d s i n s c a t t e r i n g studies at o p t i c a l and X-ray frequencies. The present thesis i s r e s t r i c t e d to an i n v e s t i g a t i o n of c e r t a i n types of d i s c r e t e s c a t t e r e r models. In general a random medium i s dependent on both " s p a t i a l " v a r i a b l e s and time. In a d i s c r e t e s c a t t e r e r model the random medium i s represented at time t by an ensemble of " s p a t i a l " configurations {s^ , s 2 , . . . of N d i s c r e t e s c a t t e r e r s within an otherwise homogeneous medium. The " s p a t i a l " v a r i a b l e s S j,s 2, • . . , S j q are random, each i n c l u d i n g the s i g n i f i c a n t properties such as p o s i t i o n , v e l o c i t y , s i z e , shape, o r i e n t a t i o n , and p e r m i t t i v i t y of one p a r t i c u l a r s c a t t e r e r . The ensemble i s described by a j o i n t p r o b a b i l i t y density function pCsj^,. . . ,sjjjt) which s p e c i f i e s the p r o b a b i l i t y "weight" associated with a f i n i t e range of configurations; p ( s 1 , . . . , s N ; t ) d s 1 ••• d s N i s the p r o b a b i l i t y of f i n d i n g the s c a t t e r e r s at time t i n a configuration i n the "volume element" - - t around s 1,...,s^. This function s a t i s f i e s /• 0 0 y - 0 0 ••• / p ( s 1 , . . . , s N ; t ) ds x ••• d s N =1 (1.1) — C O J — C O The usual s c a t t e r i n g problem associated with random media of d i s c r e t e s c a t t e r e r s may be stated as follows: An electromagnetic wave F 1 ( f ) e - ^ a J t i s incident on each configuration s^,...,!"^ of the ensemble and gives r i s e to a r e s u l t ant f i e l d F(s" 1, . . . j S ^ r . O e - ^ 1 " which i s s p e c i f i e d at a point f and time t by F ( s 1 , . . . ) s N ; r , t ) = F 1 (f) + F s ( s 1 , . . . , s N ; r , t ) (1.2) ^The common pr a c t i c e of employing a s i n g l e symbol to represent both a random v a r i a b l e and the values i t assumes i s followed throughout the t h e s i s . Also used i s the convention that p may represent any number of p r o b a b i l i t y density functions, the type being i n d i c a t e d by the random-variable symbols with-i n the parentheses (before the semicolon). where F s e J w t i s the scattered f i e l d . It i s desired to determine the s t a t i s t i c s of the f i e l d F at f and t over the ensemble of s c a t t e r e r configurations, or more s p e c i f i c a l l y , the j o i n t p r o b a b i l i t y density function p(X,Y;r,t) of the f i e l d components X and Y, where F = |F| = X + jY or F = XeJ Y. Usually the s c a t t e r i n g process i s stationary (see reference 3 for d e f i n i t i o n s ) so that the medium i s s p e c i f i e d by a s i n g l e p r o b a b i l i t y density function p (s^ ,. . . ,s^j) and the f i e l d by p(X,Y;f). Since p(X,Y;f) i s a function of various s t a t i s t i c a l moments of X and Y over the ensemble of configurations, a subsidiary problem i s to determine these moments. The m-th moment of a f i e l d component X i s defined /oo f oo ••• / P ( s i s N ; t ) X m ( s 1 s N ; r , t ) ds x ••• d s N (1.3) - c o J — C O and i s independent of t f o r a stationary process. Often the s c a t t e r i n g process i s ergodic^ as w e l l as stationary so that estimates of ( x m ( f ) ) can be deter-mined by long-time averages of X m. Such conditions occur i n p r a c t i c e i f the time constants of the s c a t t e r e r motion are much l a r g e r than" the period of the i n c i d e n t wave ( i . e . , so that the f i e l d i s r e l a t i v e l y independent of the s c a t t e r e r v e l o c i t i e s ) but much shorter than the time i n t e r v a l of measurement. 4' In t h i s t h e s i s , emphasis i s placed on the i n v e s t i g a t i o n of e x i s t i n g general d i s c r e t e - s c a t t e r e r theories f o r the f i e l d moments of i n t e r e s t and the development of more accurate ones. Such theories are of importance because they e x p l i c i t l y involve the various p h y s i c a l and s t a t i s t i c a l parameters of the s c a t t e r i n g medium (e.g., s c a t t e r i n g c h a r a c t e r i s t i c s of the i n d i v i d u a l s c a t t e r e r s ; parameters of the d i s t r i b u t i o n s of s c a t t e r e r p o s i t i o n s , s i z e s , p e r m i t t i v i t i e s , and other " s p a t i a l " v a r i a b l e s ) . Thus, for example, i n s c a t t e r i n g studies of the microstructure of matter they provide a framework for the i n v e r s i o n of f i e l d measurements to y i e l d the i n d i v i d u a l s c a t t e r e r functions and d i s t r i b u t i o n s of ph y s i c a l i n t e r e s t . Furthermore, when s p e c i a l i z e d to the. na t u r a l l y - o c c u r r i n g s c a t t e r e r d i s t r i b u t i o n s of the atmosphere and the corres-ponding communication problem, they provide a basis f o r the appropriate choice of s i g n a l parameters to minimize the e f f e c t s of the f l u c t u a t i n g medium on coherent propagation. As already stated, the f i e l d moments also enter into t h e o r e t i c a l models f o r the complete p r o b a b i l i t y density of the f i e l d which are necessary to more completely characterize the s c a t t e r i n g process. Some consideration i s also given i n the thesis to t h i s more general problem area. As a means of i n v e s t i g a t i n g e x i s t i n g general d i s c r e t e - s c a t t e r e r theories and providing i n s i g h t i n t o the p h y s i c a l and s t a t i s t i c a l c h a r a c t e r i s t i c s of d i s c r e t e - s c a t t e r e r media, much weight i s attached i n the thesis to the use of a one-dimensional model pf randomly-positioned planar s c a t t e r e r s . As a t o o l i n the i n v e s t i g a t i o n , the Monte Carlo method i s used extensively for a computer simulation of s c a t t e r i n g from t h i s model. The t h e o r e t i c a l aspects of the s c a t t e r i n g problem are considered i n Chapter 2; the two general approaches to the s o l u t i o n of the problem are introduced and some e x i s t i n g theories f o r the f i e l d moments and the complete f i e l d d i s t r i b u t i o n are presented. Also i n Chapter 2, the t h e o r e t i c a l basis f o r the consideration of the one-dimensional model i s out l i n e d i n d e t a i l and several approximate s c a t t e r i n g theories developed for t h i s model i n the present work are given. In Chapter 3 the use of the Monte Carlo simulation i n the study of random media of d i s c r e t e s c a t t e r e r s i s discussed and procedures f o r i t s a p p l i c a t i o n o u t l i n e d . Results of a comparative study of t h e o r e t i c a l and simulation data f o r s c a t t e r i n g from ensembles of configurations of uniformly-random planar s c a t t e r e r s are given i n Chapter 4. From these r e s u l t s the approximate theories presented i n Chapter 2 are evaluated. In Chapter 5 simulation methods developed for the i n v e s t i g a t i o n of non-uniform d i s t r i b u t i o n s of f i n i t e - w i d t h planar s c a t t e r e r s are presented and s c a t t e r i n g r e s u l t s based on these methods given. Discussed i n Chapter 6 i s a p h y s i c a l model of a random medium of s p h e r i c a l s c a t t e r e r s which has been developed i n t h i s work for use i n con-t r o l l e d laboratory experiments at millimeter-wave frequencies. The Monte Carlo method i s employed i n t h i s model to con t r o l the p o s i t i o n - s t a t i s t i c s of the s c a t t e r e r s . Results of an experimental evaluation of the model are given. 2. THEORETICAL CONSIDERATIONS 2.1 Introduction Previous t h e o r e t i c a l research i n the s c a t t e r i n g of waves by random media may be divided into three general, r e l a t e d problem areas: (i) The development of theories f o r c e r t a i n average f i e l d functions of i n t e r e s t subject to various mathematical models of the random medium. ( i i ) The development of t h e o r e t i c a l models for the p r o b a b i l i t y density function of the f i e l d associated with s c a t t e r i n g from a given random medium. ( i i i ) The development of theories f o r i n v e r t i n g s t a t i s t i c a l estimates of average f i e l d q u a n t i t i e s to determine the p h y s i c a l composition of the random medium. As stated i n Chapter 1, the f i r s t two problem areas are considered i n t h i s t hesis with most of the emphasis being placed on the f i r s t . Two techniques have been used to obtain e x p l i c i t expressions for the average f i e l d functions of i n t e r e s t . K e l l e r ^ has c a l l e d these "honest" and "dishonest" methods. In an "honest" method, as applied to random media of d i s c r e t e s c a t t e r e r s , an e x p l i c i t expression f o r the desired f i e l d quantity i s f i r s t determined for a f i x e d configuration of s c a t t e r e r s . This expression i s then d i r e c t l y integrated over the ensemble of configurations using the d e f i n i t i o n (1.3). In a "dishonest" method, randomness i s u t i l i z e d before an e x p l i c i t expression for the desired f i e l d quantity i s obtained. With c e r t a i n h e u r i s t i c approximations being made, the d e f i n i n g equation (1.3) i s transformed into compact i n t e g r a l or d i f f e r e n t i a l equations i n the- desired average f i e l d f unction. These simpler equations are then solved subject to the boundary conditions of the p a r t i c u l a r problem at hand. The "dishonest" technique has been the main t h e o r e t i c a l approach for many-scatterer problems i n which multiple s c a t t e r i n g i s considered. It has 7 advantages over the "honest" technique i n that i t s i m p l i f i e s the problem to be solved and leads to closed-form expressions f o r the desired average f i e l d functions. Furthermore, the r e s u l t i n g expressions are sometimes s u f f i c i e n t l y general to be a p p l i c a b l e to d i s t r i b u t i o n s of one-, two-, or three-dimensional s c a t t e r e r s . This technique i s usually handicapped, however, by the need for employing unproven h e u r i s t i c approximations. The "honest" technique, because i t requires an e x p l i c i t f i e l d expression f o r a f i x e d configuration of s c a t t e r e r s and involves a m u l t i p l e i n t e g r a t i o n of t h i s expression, i s mainly l i m i t e d to problems i n v o l v i n g two or three s c a t t e r e r s or many-scatterer problems i n which mu l t i p l e s c a t t e r i n g has e i t h e r been completely neglected or only p a r t i a l l y included. Its a p p l i -c ation i s also l i m i t e d to a s p e c i f i c s c a t t e r e r model with the r e s u l t i n g expressions often being le s s general than expressions obtained by a "dishonest" approach and i n s e r i e s form rather than i n closed form. For c e r t a i n s c a t t e r e r models, however, the "honest" technique y i e l d s more accurate (although more cumbersome) expressions for the various average f i e l d functions of i n t e r e s t than does the "dishonest" technique. Also, approximations may be made on a s t r i c t l y p h y s i c a l basis (usually with respect to the number of m u l t i p l e -s c a t t e r i n g e f f e c t s included) for the p a r t i c u l a r s c a t t e r e r model being con-sidered . As stated i n Chapter 1, throughout the thesis most of the emphasis i s placed on the problem of plane-wave s c a t t e r i n g from an ensemble of one-dimensionally random configurations of planar s c a t t e r e r s . This model i s introduced i n s e c t i o n 2.2 and reasons are given for i t s consideration. For convenience, i t i s frequently c a l l e d the "one-dimensional model". The basic formalism f o r s c a t t e r i n g from a f i x e d configuration of a r b i t r a r y s c a t t e r e r s i s presented i n section 2.3. In section 2.4 are presented two e x p l i c i t theories for s c a t t e r i n g from a f i x e d array of planar s c a t t e r e r s , one of which has been developed i n t h i s work to provide a basis for approximate theories for an ensemble of planar-scatterer arrays. D e f i n i t i o n s of the average f i e l d functions of i n t e r e s t i n s c a t t e r i n g from an a r b i t r a r y ensemble of s c a t t e r e r configurations are given i n s e c t i o n 2.5. The e x i s t i n g general d i s c r e t e - s c a t t e r e r theories for average f i e l d functions which are studied i n d e t a i l i n t h i s work are also introduced i n section 2.5 and a b r i e f mention of some of the previous research i n t h i s area i s made. Presented i n s e c t i o n 2.6 are approximate s e r i e s expressions developed i n the present work f o r some of the average f i e l d functions of i n t e r e s t i n the problem of s c a t t e r i n g from an ensemble of uniformly-random planar-scatterer configurations. In section 2.7, asymptotic theories for these average f i e l d functions i n the l i m i t as the average density of s c a t t e r e r s goes to zero are given. Based on the asymptotic form f o r the coherent transmitted f i e l d i n the one-dimensional model, a mo d i f i c a t i o n to the one-dimensional form of an e x i s t i n g general d i s c r e t e - s c a t t e r e r theory i s proposed. In s e c t i o n 2.8 the more general problem of obtaining a complete s t a t i s t i c a l representation for the random f i e l d i n terms of a j o i n t p r o b a b i l i t y density function of i t s components i s discussed. Detailed consideration i s given to the one-dimensional model. Generalization of the theories presented i n the chapter i s discussed i n s e c t i o n 2.9 and a summary given i n s e c t i o n 2.10. 2.2 The One-Dimensional Model The model considered i n most d e t a i l i n the thesis i s that of an ensemble of configurations of p a r a l l e l planar s c a t t e r e r s of i n f i n i t e extent randomly positioned within a slab region of space according to a s p e c i f i e d p r o b a b i l i t y density function p ( z 1 9 . . . , z N ) of the s c a t t e r e r p o s i t i o n s z 1 , . . . , z N . A p a r t i c u l a r configuration from the ensemble i s represented i n f i g u r e 2.1 with "ordered-positions" z{,...,z^ s a t i s f y i n g 0 < z[ < z'2 < ... < z^ < d. A plane wave in c i d e n t from the l e f t i s scattered by each co n f i g u r a t i o n causing r e s u l t a n t r e f l e c t e d and transmitted waves as i n d i c a t e d . Although they are shown i n the diagram as i n f i n i t e l y t h i n sheets, the one-dimensional scat t e r e r s may represent the centers of homogeneous or inhomogeneous d i e l e c t r i c slabs of f i n i t e thickness. I I z=0 z{ z 2 z\ z ^ z=d Figure 2.1 The One-Dimensional Model The main reasons for p l a c i n g emphasis on t h i s one-dimensional model are as follows: ( i ) C e rtain behavior exhibited by the random f i e l d associated with s c a t t e r i n g from a one-dimensional random medium of d i s c r e t e s c a t t e r e r s i s also present i n more complex random media of d i s c r e t e s c a t t e r e r s . Thus, a study of the simpler model can to a certain-extent contribute to a b e t t e r understanding of the o v e r a l l problem. ( i i ) The a p p l i c a t i o n of Monte Carlo simulation to the problem of s c a t t e r -ing from a given ensemble can y i e l d numerical r e s u l t s f o r the average f i e l d q u a n t i t i e s which are exact to w i t h i n a c e r t a i n s t a t i s t i c a l e r r o r . These "exact" r e s u l t s provide a means of evaluating various approximate theories f o r the average f i e l d q u a n t i t i e s . The Monte Carlo technique also allows the complete s t a t i s t i c a l d i s t r i b u t i o n of the f i e l d (e.g., cumulative d i s t r i b u t i o n ) 10 to be sampled as we l l as the various f i e l d moments. ( i i i ) Certain approximate theories f o r average f i e l d functions which have previously been developed using the "dishonest" technique apply generally to the one-dimensional model and to more complex two- and three-dimensional models of random volume d i s t r i b u t i o n s of d i s c r e t e s c a t t e r e r s . Thus, evaluation of these theories f o r the one-dimensional model serves to p a r t i a l l y evaluate the theories i n general, since i n p r i n c i p l e the approximations involved are independent of any p a r t i c u l a r model. (iv) The "honest" technique may be employed f o r the one-dimensional model to obtain approximate s e r i e s expressions f o r the various average f i e l d functions. The approximations involved are made on a ph y s i c a l basis i n that only lower order m u l t i p l e - s c a t t e r i n g processes are considered. These approxi-mate theories are i n general better than e x i s t i n g theories based on the "dishonest" technique and may be used to advantage i n improving e x i s t i n g theories. It should be evident that the choice of t h i s p a r t i c u l a r one-dimensional model has been based mainly on t h e o r e t i c a l considerations. Similar considerations have governed the choice of d i f f e r e n t one-dimensional models by other workers.^ Indeed, many theories developed f o r models of continuous random media ( i . e . , perturbed continuum models) have involved randomness i n one-dimension only. E x p l i c i t numerical r e s u l t s i n the thesis are given only f o r the s p e c i a l case of normal plane-wave incidence on an ensemble of configurations of N l o s s l e s s i d e n t i c a l planar s c a t t e r e r s . L i t t l e i s l o s t by t h i s s p e c i a l -i z a t i o n and i t allows d i f f e r e n t s t a t i s t i c a l d i s t r i b u t i o n s f o r the s c a t t e r e r p o s i t i o n s to be studied more conveniently. Methods of extending the theory and simulation to the more general cases of oblique incidence, lossy s c a t t e r e r s with random s c a t t e r i n g amplitudes, and random N are in d i c a t e d . 11 2 . 3 Basic Formalism for Scattering.from a Fixed Configuration  of A r b i t r a r y Scatterers For convenience i n t h i s chapter and throughout the thesis a s c a l a r formalism for the f i e l d i s used. This i s a common p r a c t i c e , even for some 8 three-dimensional electromagnetic problems, and i s completely v a l i d f o r the one-dimensional model. The time f a c t o r e^ U 3 t i s also removed from a l l equations. For a f i e l d <}>(?) incident on a configuration of N d i s c r e t e s c a t t e r e r s whose p o s i t i o n s and s c a t t e r i n g c h a r a c t e r i s t i c s are s p e c i f i e d by S j , . . . , ! ^ , the t o t a l f i e l d at a point f outside the s c a t t e r e r s ' surfaces i s represented by i K s j , . . . , s N ; f ) = <f>(f) + i ( ( s 1 , . . . , s N ; f ) ( 2 . 1 ) where H i s the t o t a l scattered f i e l d . The r e s t of the s c a t t e r i n g problem i s s p e c i f i e d by the conditions at the boundaries of the s c a t t e r e r s , the conditions at i n f i n i t y , and the s c a l a r Helmholtz equation (V 2 + k 2 ) i | / = 0 , k = 2TT/X ( 2 . 2 ) Although X i s assumed to be the free-space wavelength, i t may equally w e l l be the wavelength of any other medium i n which the s c a t t e r e r s are embedded. The t o t a l scattered f i e l d may be represented by N U (s1,...,sN;f) = £ U s ( s 1 , . . . , s N ; f ) ( 2 . 3 ) s=i where U g i s the c o n t r i b u t i o n from s c a t t e r e r s. Several d i f f e r e n t represent-ations for U s and i t s vector equivalent e x i s t and are u s e f u l f o r three-9 10 dimensional problems. ' The many-scatterer problem can also be formulated i n terms of i s o l a t e d s c a t t e r e r functions. The t o t a l f i e l d i s written N i|)(s 1,. .. , s N ; r ) = <j>(f) + Y  us^~ fs) $ s »•••»% 5*s) ( 2- 4> s=l where u g ( r ) i s the scattered f i e l d due to plane-wave e x c i t a t i o n of sc a t t e r e r s i f i t were situated alone at the o r i g i n . For a three-dimensional s c a t t e r e r , u s i s of the form of an outgoing s p h e r i c a l wave u s ( f ) ~ g(f,k) (2.5) as r + The three-dimensional " s c a t t e r i n g amplitude" g(r,k) i s a function only of the d i r e c t i o n of the incident plane wave and the d i r e c t i o n of obser-vation as represented by the unit vectors k and r. The quantity $ g i s the multiple-scattered " e x c i t i n g f i e l d " f o r sca t t e r e r s and i s represented by N $ s ( s 1 , . . . , s N ; f s ) = <K?S) + Y  ut^s~^t / $ t ^ l ' • ' • » % ; f t ^ ^ 2* 6) t=i#s E s s e n t i a l l y , equations (2.4) and (2.6) are operational forms written on the basis of the superposition p r i n c i p l e . Since the response of a s i n g l e s c a t t e r e r to a plane wave i s known, the response to the multiple-scattered e x c i t i n g f i e l d given by (2.6) can i n p r i n c i p l e be determined by a plane-wave i n t e g r a l expansion of $ g. The compact forms of (2.4) and (2.6) can be i t e r a t e d i n terms of the i s o l a t e d s c a t t e r e r functions u g and the following expanded form f o r if; obtained as an i n f i n i t e s e r i e s of "o r d e r s - o f - s c a t t e r i n g " : *(§!,...,§!,;?) = K r ) + E  n s ^ s ) *<fs> + E I U s < r - V u t < W *(r t) s s t^s + E E E u s < r - f s ) u t < W W f m > + •••• (2 .7 ) s t^s m#t A s i m i l a r form can be obtained for the f i e l d i n t e n s i t y l^ l2. ' Because these expansions for ty and |^| 2 are quite general and indepen-dent of any p a r t i c u l a r s c a t t e r i n g model, they have proved u s e f u l i n t h e o r e t i c a l i n v e s t i g a t i o n s of the processes of m u l t i p l e s c a t t e r i n g as they e f f e c t a c o n f i g u r a t i o n of s c a t t e r e r s or an ensemble of c o n f i g u r a t i o n s . ^ Because of the necessity f o r plane-wave i n t e g r a l expansions of the e x c i t a t i o n terms f o r other than one-dimensional s c a t t e r e r s , however, i t i s exceedingly d i f f i c u l t f o r e x p l i c i t s e r i e s solutions to be obtained. Furthermore, for p a r t i c u l a r s c a t t e r i n g models they are not always quickly convergent nor n e c e s s a r i l y even convergent. A more quickly convergent i n f i n i t e s e r i e s expression f o r the one-dimensional model w i l l be given i n the following s e c t i o n . 2.4 E x p l i c i t Theories for Scattering from a Fixed Array of Planar Scatterers Consider now the s c a t t e r i n g of a normally i n c i d e n t plane wave <f>(z) = e -J k z f rom an array of planar s c a t t e r e r s located at z j , . . . , z ^ as shown i n f i g u r e 2.1. For l a t e r development of theory for ensembles of arrays i t i s assumed that the s c a t t e r e r s are located between the planes z = 0 and z = d as i n d i c a t e d . The i s o l a t e d s c a t t e r e r functions for e x c i t a t i o n by <}>(z) are u s ( z - z s ) = g + s <Kz-zs) (z > z'g) (2.8) u s (z-z s ) = g_ s cf>'(z-zs) ( z < Z g ) where <j>'(z) - e?^Z. The q u a n t i t i e s g + s and g_ s, termed the forward- and back-s c a t t e r i n g amplitudes, are r e l a t e d to the s i n g l e s c a t t e r e r transmission and r e f l e c t i o n c o e f f i c i e n t s T s and R g by T s = 1 + g + s and R s = g_ s. These qu a n t i t i e s are used interchangeably throughout the t h e s i s . Although the theories to follow are v a l i d f or configurations of non-i d e n t i c a l s c a t t e r e r s , l a t e r numerical r e s u l t s for ensembles of s c a t t e r e r configurations are based on configurations of i d e n t i c a l s c a t t e r e r s . Numerical values f o r the s c a t t e r i n g amplitudes g + = Tj - 1 and g_ = Rj of these i d e n t i -c a l s c a t t e r e r s are taken to be those f o r actual d i e l e c t r i c slabs of f i n i t e thickness. E x p l i c i t expressions f o r g + and g_ are given i n Appendix B for a d i e l e c t r i c slab. In a l a t e r development of theory f o r s c a t t e r i n g from an ensemble of uniformly-random planar s c a t t e r e r s , i t i s assumed that the scatt e r e r s are i n f i n i t e l y t h i n but have f i n i t e s c a t t e r i n g amplitudes. For the present development, however, the question of f i n i t e thickness i s immaterial since g + and g_ are re f e r r e d to the scat t e r e r ' s center. 2.4.1 Wave Transmission Matrix Representation An exact representation f o r the t o t a l f i e l d can be obtained using 12 wave transmission matrices. This theory provides the basis f o r the "exact" simulation r e s u l t s i n the t h e s i s . Consider the array of planar s c a t t e r e r s shown i n fi g u r e 2.2. I I I i l l I I I I I z=0 z[ z'2 Z3 ZJJ z=d Figure 2.2 Wave Transmission Matrix Representation The matrix expression r e l a t i n g the complex wave amplitudes c^,b^ and c Q , b Q at the planes z = 0 and z = d i s 15 where N n TC s=i A l l A 1 2 A A 21 A 2 2 0 e - 2 j k ( d - Z p (2.9) A l l A 1 2 A 2 1 A 2 2 N -n s=l 0 e - 2 J k ( z s - z s - l ) (2.10) (for s = 1, z g = 0). Thus, the t o t a l f i e l d i n the forward- and back-s c a t t e r i n g regions due to the plane wave inc i d e n t from the l e f t i s i K z J , • • • ,z^;z) = T ( z J , . . . , z ^ ) (j>(z) ( z > d) where and i K z { , . . . ,z^;z) = <j>(z) + R(z{,...,z^) <f>'(z) (z < 0) T(z^,... »ZJJ) e^ N b,-R(z|,.. . ,z^j) -b Q=0 s=l "21 b Q=0 A n n v A n (2.11) (2.12) (2.12) (2.14) are the o v e r a l l transmission and r e f l e c t i o n c o e f f i c i e n t s f o r the array as re f e r r e d to the plane z = 0. 2.4.2 Orders-of-Back-Scattering Representation Another e x p l i c i t representation f o r the t o t a l f i e l d has been developed i n t h i s work. Termed the orders-of-back-scattering (0-B-S) representation, i t i s u s e f u l as a basis f o r the development of approximate theories f o r s c a t t e r i n g from an ensemble of planar-scatterer configurations. Consider the s c a t t e r i n g diagrams f o r three s c a t t e r e r s shown i n f i g u r e 2.3. Figure 2.3a shows the two dominant m u l t i p l e - s c a t t e r i n g processes c o n t r i b u t i n g to the transmitted f i e l d . The f i r s t process, termed the zeroth-order-of-back-scattering (Z-O-B-S), involves no r e f l e c t i o n s , while the second, termed the second-order-of-back-scattering (S-O-B-S), involves two r e f l e c t i o n s . Figure 2.3b shows the two dominant m u l t i p l e - s c a t t e r i n g processes c o n t r i b u t i n g to the r e f l e c t e d f i e l d . These processes, c a l l e d the f i r s t - o r d e r - o f - b a c k - s c a t t e r i n g (F-O-B-S) and the third-order-of-back-scattering (T-O-B-S) , involve one and three r e f l e c t i o n s r e s p e c t i v e l y . The symbol -»-|-> , i n z s the diagrams, represents a transformation of the f i e l d along a ray path by T g and the symbol T\ , a transformation by Rg. : 1 a) oo I PQ I O I EN] 00 I PQ I O I 00 00 I PQ I o I fx* 00 I PQ I O I H (b) Figure 2.3 Dominant M u l t i p l e - S c a t t e r i n g Processes i n O-B-S Representation The m u l t i p l e - s c a t t e r i n g processes shown i n f i g u r e 2.3 give the f i r s t two terms i n an i n f i n i t e s e r i e s of orders-of-back-scattering for the trans-mitted and r e f l e c t e d f i e l d s . Induction y i e l d s the general term corresponding to the nth order-of-back-scattering. For normal incidence on an array of non-i d e n t i c a l s c a t t e r e r s , the r e s u l t i n g expressions are of the form 17 T(zJ,...,z^) - T Q + T l I + T I V + .... + T n t h + R ( z | , . . . , z p = R x + R I I ] t + R v + . . . . + R n t h + (2.15) where N TO - n TV p=i N n T P Lp=l N s - 1 E E S=2 t=l s - t - 1 n q=0 Lt+q R s R t e - 2 J k ( z s - Z t > , (2.16) Lnth N n TF N s l - 1 N E E E L- p=l J s ^ s 2 = l s 3=s 2+l V - l " 1 E s n = 1 S ^ - s ^ l n T s 22+PI Pi=0 r S3~ s«f -1 p =b S ^ 2 J r s n - l - s n ~ l TT Ts__,+p_ Pn/2=° n 1 n e - 2 J k ( z s r z s 2 + z s 3 - "zs,> 'n /2 (n even) R „ R „ R„ • • • • R s l s 2 s3 and R-r - E N r s - i S=l n t, q=0 R e 2 j k z s , T 0 ^ 1, etc. R N s - 1 N i n = E E E s=2 t=l u=t+l r s - i i r u - t - i n T i q=0 n *t2+r r=o R s R t R u e - 2 J k ( z s - Z t + z u > (2.17) N s l - 1 N ^ t h " E E E s x=2 s 2=l s 3=s 2+l N E n T s n = s n - l + 1 L P i = 0 Pi s 3 - s 2 - l FT T s 2 + p 2 P2=0 • • • • " s n _ s n _ i - l n L P ( n + l ) / 2 = 0 s n - l + P ( n + l ) / 2 R_ R_ R_ . . . R„ e b l b2 3 n -2jk(z! - z ' + z ' - . . . + z ' ) s l s2 s3 n (n odd) These expressions are v a l i d not only f o r an array of d i e l e c t r i c s l a b s separated by spaces of the embedded medium, but a l s o f o r a stack of d i e l e c t r i c s l a b s w i t h no spaces between t h e i r boundaries. S i m p l i f i c a t i o n f o r arrays of i d e n t i c a l s c a t t e r e r s i s , of course, s t r a i g h t f o r w a r d . The O-B-S r e p r e s e n t a t i o n of equations (2.15), (2.16), and (2.17) was obtained by the d i r e c t p h y s i c a l approach o f s u c c e s s i v e l y i n t r o d u c i n g l e s s dominant m u l t i p l e - s c a t t e r i n g processes c o n t r i b u t i n g to the t o t a l f i e l d . I t may a l s o be obtained (but l e s s e a s i l y ) by regrouping terms of the or d e r s - o f -s c a t t e r i n g r e p r e s e n t a t i o n of equation (2.7) on the same p h y s i c a l b a s i s . Thus, the Z-O-B-S contains terms from up to the Nth o r d e r - o f - s c a t t e r i n g , the S-O-B-S contains terms from up to the (3N - 2 ) t h o r d e r - o f - s c a t t e r i n g , and the f o i ^ r t h -o r d e r - o f - b a c k - s c a t t e r i n g (FO-O-B-S) contains terms from up to the (5N - 4)th o r d e r - o f - s c a t t e r i n g , e t c . S i m i l a r l y , terms from up to the (2N - 1 ) , (4N - 3 ) , and (6N - 5 ) t h o r d e r - o f - s c a t t e r i n g c o n t r i b u t e to the three dominant O-B-S f o r the r e f l e c t e d f i e l d . E s s e n t i a l l y the same p h y s i c a l approach i n terms of ray paths was 13 used by Marcus to o b t a i n a general s e r i e s expression f o r T based a l s o on the r e f l e c t i o n and t r a n s m i s s i o n c h a r a c t e r i s t i c s of the i n d i v i d u a l " d i s c o n t i n u i t i e s " but c o n t a i n i n g a d i f f e r e n t grouping of terms. He considered the c o m b i n a t o r i a l aspects of the problem more thoroughly than has been done i n the present work, showing that the s e r i e s f o r T could be reduced to the closed-form expression r e s u l t i n g from a matrix approach. In the present work, however, the e x p l i c i t o r d e r s - o f - b a c k - s c a t t e r i n g forms given f o r both T and R more r e a d i l y a l l o w the necessary m u l t i p l e - s c a t t e r i n g approximations to be made i n t h e i r a p p l i c a t i o n to an ensemble of s c a t t e r e r c o n f i g u r a t i o n s . From the numerical r e s u l t s of Chapter 4, convergence of the O-B-S s e r i e s on an average b a s i s appears to be quick f o r i d e n t i c a l planar s c a t t e r e r s w i t h f a i r l y l a r g e b a c k - s c a t t e r i n g c r o s s - s e c t i o n s . As shown by Kay and Silverman,^ however, the convergence of the ensemble average of a s e r i e s expression for a fixed c o n f iguration can be f a s t e r because of the e f f e c t s of incoherent s c a t t e r i n g i n the random case. Both the wave transmission matrix theory of the previous section and the O-B-S representation can be generalized for plane-wave incidence at an a r b i t r a r y angle 0 with the normal by the replacement of T g and R g with t h e i r corresponding values for oblique incidence (see Appendix B) and by the s u b s t i t u t i o n of k cosG for k. 2.5 Scattering from an Ensemble of Scatterer Configurations 2.5.1 Average F i e l d Functions of Interest The f i e l d s t a t i s t i c s i n the problem of s c a t t e r i n g from an ensemble of s c a t t e r e r configurations are generally r e l a t e d to a normalized f i e l d quantity designated Te^ T. For a random medium having the slab-region geometry of f i g u r e 2.1 ( i . e . , bounded by the planes z = 0 and z = d), t h i s quantity i s defined i K s , , . . . , s N ; r ) T e J T & - - (z > d) T(s^ , . .. >SJJ) ( 2 . 1 8 ) i n the forward-scattering region and i K s . , . . . ,Sx,;r) - <f>(r) K(s s M ; r ) T e j x 4 1 l2 _ 1 iN 4>?(r) *'(r) = R ( s 1 , . . . , s N ) (z < 0) ( 2 . 1 9 ) i n the back-scattering region. I t i s commonly c a l l e d the t o t a l f i e l d . Throughout the thesis i t w i l l be c l e a r from the context whether use of the term, t o t a l f i e l d , r e f e r s to the a c t u a l t o t a l f i e l d or the normalized t o t a l f i e l d as defined i n equations (2.18) and (2.19). The t o t a l f i e l d i s customarily separated into two components^ Te J T = Ce*a + IeJ* such that < T e J T ) = CeJ a, <IeJ < 5>= 0 (2.20) (2.21) The component Ce^a i s c a l l e d the "coherent f i e l d " and the component Ie^^, the "incoherent" or "variant f i e l d " . The t o t a l , coherent, and incoherent f i e l d s are divided into t h e i r rectangular components as, for example, such that T e J T = T COST + j T sinx = T x + J T y C x = <Tx>> S = <Ty> <*x> - 0. <Iy> = 0 tana = C y/C x (2.22) (2.23) The i n t e r r e l a t i o n of the functions of (2.20) and (2.23) may be represented on a phasor diagram as shown i n f i g u r e 2.4. * ( r ) , +*(r) Figure 2.4 Phasor Diagram of the T o t a l F i e l d Resolution The i n t e n s i t i e s of the qua n t i t i e s are rela t e d by < T 2> = C 2 + <I2> (2.24) where ( T 2 ) i s the "average t o t a l i n t e n s i t y " , C 2 the "coherent i n t e n s i t y " , and ( l 2 ) the "average incoherent i n t e n s i t y " . The variances and covariance of the t o t a l f i e l d components T x and Ty are given by °T* - <y> = <y> - s 2 ( 2- 2 5 ) ^ a T x a T y = ( V y ) = <T xT y> - C xC y where u i s the c o r r e l a t i o n c o e f f i c i e n t between T X and T Y . These second c e n t r a l moments may also be expressed i n terms of the average incoherent i n t e n s i t y ( l 2 ) and a complex function defined^ 2 S 2 e l 2 s ft < I 2 e i 2 « ) = <(TeJ T - C e J a ) 2 > = < T 2 e i 2 T > - C 2 e J 2 a Thus, < Ix 2> = i< l 2> + S 2 cos 2s <V> = i<I 2> - S 2 cos 2s ( l x I y > = S 2 s i n 2s and <I 2> = <I X 2> + <I y 2> S 2 cos 2s = I (<I X 2> - < I y 2 » Twersky^ has termed Se^ s the "covariant f i e l d " , S 2 the "covariant i n t e n s i t y " , and s the "covariant phase". (2.26) (2.27) (2.28) Further d e s c r i p t i o n of the t o t a l f i e l d s t a t i s t i c s i s given by higher c e n t r a l moments of T x and T y. In th i s work the t h i r d and fourth c e n t r a l momen ts ( l x 3 ) > {\ 3} a n d ( I x' t)> (^y1*) a r e used as a measure of the extent to which the j o i n t d i s t r i b u t i o n of T X and T Y corresponds to a b i v a r i a t e Gaussian d i s t r i b u t i o n (see section 2.8). S p e c i f i c a l l y , the " c o e f f i c i e n t s of skewness" ^ x 3 ) <Iy3> b x = N X , , b y = V y ' t (2.29) ( I ^ 2 ) 3 / 2 7 < I y 2 ) 3 / 2 and the " c o e f f i c i e n t s of k u r t o s i s " Y = : 3, Y = 3 (2.30) X <Ix 2> 2 7 <Iy 2> 2 are o b t a i n e d . ^ Expressions f o r a l l the higher c e n t r a l moments have been given by Twersky^'^ i n terms of complex functions ( l n e ^ n 2 m ^ ) [where m ranges from 0 to n/2 for the even moments and to (n-l)/2 f o r the odd]. Twersky^'^ has also given generalized expressions f o r the c e n t r a l moments for a change v i n the phase of the reference f i e l d [ i . e . , <f>(r) and <J>'(r) i n equations (2.18) and (2.19) replaced by <j)(r)eJV and <J>'(f)e^V]. Moments of the amplitude T and phase x of the t o t a l f i e l d are also of i n t e r e s t . In t h i s work the average f i e l d amplitude ( T ) and the average phase (x ) are obtained, as are the variances a T 2 = < T 2 > - <T>2 (2.31) 0 2 = ( T 2 > _ < T > 2 Twersky^ has obtained the following expansions of these moments to second-order terms i n the previously defined moments (T) = C + ~ X ' 2C I ( I2 ) - S 2 cos 2(s-a) + .... (2.32) S 2 (x) = a s i n 2(s-a) + .... (2.33) C 2 a T 2 = i <I 2) + S 2 cos 2(s-a) + (2.34) 2 = _ i ° T C 2 ( T2S - S 2 cos 2(s-a) (2.35) 2.5.2 Some E x i s t i n g Theories for Average F i e l d Functions In t h i s section some of the e x i s t i n g contributions to the develop-ment of approximate theories f o r average f i e l d fucntions are b r i e f l y mentioned and two coherent f i e l d theories f o r which numerical r e s u l t s are given i n the present work are introduced. A more d e t a i l e d account of previous research i s 15 16 given i n the survey papers by Burke and by Twersky. L. F o l d y ^ appears to have been the f i r s t researcher to apply p. "full-wave" treatment to the s c a t t e r i n g of waves by random d i s t r i b u t i o n s of d i s c r e t e s c a t t e r e r s . In h i s 1945 paper, Foldy introduced the concept of obtaining averages of f i e l d q u a n t i t i e s of i n t e r e s t over an ensemble of s c a t t e r e r configurations, e s t a b l i s h i n g the basis for nearly a l l subsequent formulations. Through the use of h e u r i s t i c approximations for the f i e l d $ s e x c i t i n g a s c a t t e r e r , he obtained i n t e g r a l expressions for the coherent f i e l d , the average t o t a l i n t e n s i t y , and the average energy f l u x f o r s c a t t e r i n g from uniformly-random d i s t r i b u t i o n s of i s o t r o p i c point s c a t t e r e r s . Foldy's t r e a t -ment of the problem was l a t e r generalized by M. L a x ^ ' ^ to include s c a t t e r i n g from d i s t r i b u t i o n s of a n i s o t r o p i c point s c a t t e r e r s . Lax introduced further h e u r i s t i c approximations for the e x c i t i n g f i e l d i n order to obtain the necessary i n t e g r a l equations. 20 21 More recent t h e o r e t i c a l work has been done by Waterman et a l . , ' 72 8 11 23—28 Mathur and Yeh, and Twersky. ' ' Several of Twersky's theories are of p a r t i c u l a r i n t e r e s t i n t h i s thesis because they are s u f f i c i e n t l y general to be a p p l i c a b l e to random d i s t r i b u t i o n s of one-, two-, or three-dimensional s c a t t e r e r s . Much of Twersky's work, furthermore, has been i n i t i a t e d as a r e s u l t of experimental research with a p h y s i c a l model of a random medium of d i s c r e t e s c a t t e r e r s . This type of research i s the subject of Chapter 6. Considered i n some d e t a i l i n the present work are Twersky's " f r e e -pi O A space" and "mixed-space" (or "two-space") theories for the coherent f i e l d . These theories, which are based on a "dishonest" approach, apply generally to slab-region volume.distributions of i d e n t i c a l one-, two-, or three-dimensional s c a t t e r e r s . The one-dimensional forms of the theories are summarized i n Appendix A and the assumptions and h e u r i s t i c approximations on which they are based are o u t l i n e d . Twersky's free-space theory i s most v a l i d for ensembles of s c a t t e r e r configurations which conform c l o s e l y to a uniform d i s t r i b u t i o n described by p ( r x ? N) = p ( r x ) p ( r 2 ) .p(r N) = (p/N) N (2.36) where p i s the average density of s c a t t e r e r s i n one, two, or three dimensions. A m o d i f i c a t i o n to the free-space theory f o r the coherent transmitted f i e l d i s proposed i n section 2.7 based on the exact asymptotic behavior of Ce J as p -»• 0. Numerical r e s u l t s f o r the one-dimensional form of the theory f o r the coherent transmitted and r e f l e c t e d f i e l d s and the m o d i f i c a t i o n to the theory fo r the coherent transmitted f i e l d are compared with "exact" Monte Carlo simulation r e s u l t s f o r a d i s t r i b u t i o n of uniformly-random planar s c a t t e r e r s i n Chapter 4. o / As shown by Twersky, the mixed-space theory can approximately describe c e r t a i n dense d i s t r i b u t i o n s of f i n i t e - s i z e s c a t t e r e r s i f the average density p i s c o r r e c t l y interpreted (see Appendix A). To i l l u s t r a t e the requirements f o r the approximate v a l i d i t y of the mixed-space theory with p interpreted i n the correct manner, numerical r e s u l t s f or i t s one-dimensional form are compared with "exact" simulation r e s u l t s f or a non-uniform d i s t r i -bution of f i n i t e - w i d t h planar s c a t t e r e r s i n Chapter 5. For completeness and 25 f o r l a t e r comparison w i t h the r e s u l t s of Chapter 5, r e s u l t s f o r the mixed-space theory w i t h the a c t u a l average d e n s i t y p = N/d are a l s o given f o r u n i f o r m l y - d i s t r i b u t e d planar s c a t t e r e r s i n Chapter 4. Other general d i s c r e t e - s c a t t e r e r t h e o r i e s f o r the average incoherent i n t e n s i t y and other average f i e l d f u n c t i o n s have been developed by Twersky but have not been n u m e r i c a l l y evaluated f o r the one-dimensional model i n the present work. These are b r i e f l y discussed i n Appendix A. T h e o r e t i c a l c o n t r i -butions to the study of s c a t t e r i n g from non-uniform d i s t r i b u t i o n s of s c a t t e r e r s w i t h c o r r e l a t i o n between the s c a t t e r e r p o s i t i o n s are discussed i n Chapter 5. 2.6 S e r i e s Approximations f o r S c a t t e r i n g from an Ensemble of Uniformly-Random  P l a n a r - S c a t t e r e r C o n f i g u r a t i o n s Considered i n t h i s s e c t i o n are c e r t a i n approximate s e r i e s expressions developed i n the present work f o r some of the average f i e l d f u n c t i o n s of i n t e r e s t i n the problem of s c a t t e r i n g from a d i s t r i b u t i o n of uniformly-random pl a n a r s c a t t e r e r s . These s e r i e s expressions were obtained by the "honest" technique of d i r e c t l y i n t e g r a t i n g approximate r e p r e s e n t a t i o n s f o r the f i e l d f u n c t i o n s over the ensemble of s c a t t e r e r c o n f i g u r a t i o n s . Although more cumbersome, such s e r i e s expressions are i n general b e t t e r than closed-form t h e o r i e s developed by means of "dishonest" methods. The primary value of the present t h e o r i e s , however, i s considered to be t h e o r e t i c a l . They provide a p o s s i b l e means of i n v e s t i g a t i n g and e l i m i n a t i n g the im p e r f e c t i o n s of e x i s t i n g closed-form t h e o r i e s based on h e u r i s t i c approximations, an approach p r e v i o u s l y i l l u s t r a t e d by Twersky.^ A l s o , they l e a d n a t u r a l l y to the development of exact asymptotic forms f o r the average f i e l d f u n c t i o n s i n the l i m i t of p •> 0 (see s e c t i o n 2.7). D i r e c t i n t e g r a t i o n over the ensemble of p l a n a r - s c a t t e r e r configurations i s most easily, performed with the use of the j o i n t p r o b a b i l i t y density function f o r the "ordered-positions" z j , . . . , zj-j, i . e . , 2 ^ p (zj,...,z^) = ^  = N!j|)N (0 < z{ < z' < ... < z^ < d) (2.37) The N! m u l t i p l i e r i n t h i s function expresses the fact that there are Nl permutations of the s t a t i s t i c a l l y independent s c a t t e r e r p o s i t i o n s z^,...,z^ i n the underlying uniform d i s t r i b u t i o n described by p(z 1,...,z^) = (p/N)^. The technique of obtaining approximate s e r i e s expressions, v a l i d f o r a r b i t r a r y N, i s i l l u s t r a t e d f i r s t for the coherent r e f l e c t e d and transmitted f i e l d s . 2.6.1 The Coherent F i e l d s In order f o r an e x p l i c i t s e r i e s expression f or (ty) to be obtained, the r e f l e c t e d f i e l d was approximated by the contribution from the F-O-B-S and the transmitted f i e l d by the contributions from the Z-O-B-S and the S-O-B-S. Thus, for configurations of i d e n t i c a l s c a t t e r e r s , N R ( z i , . . . , z £ ) = £ T2(*-U e - 2 J k z s (2.38) s=l and T ( z } , . . . , z f i ) = T X N 1 + R l 2 £ S £ T i 2 ( s - t - l ) e - 2 j k ( z ' - z ' ) s=2 t=l (2.39) The i n t e g r a l representations f o r the ensemble averages of these approximate expressions are then < R > = ! i i i f T i 2 ( . - i ) r d r * . . . . d z , . . . d z , " ( 2 . 4 0 ) d N s = l J 0 J 0 J 0 L d S=2 t=i J0 J0 J0 2 e-2jk(z s-z£) dz{ ••• dz^ (2.41) The procedure used to obtain general s e r i e s expressions for (R), ( T ) , and other average f i e l d functions, v a l i d f o r a r b i t r a r y N, was to evaluate the i n t e g r a l s involved for N = 2, N = 3 , N = 4 , etc., u n t i l by induction the general forms could be recognized. Thus, the coherent f i e l d expressions determined are ( 2 . 4 2 ) / T S - T N / I J - P 2 e 1 N l I J l " n m 2 v i i - 2 [, 2(N-n) -2jkd ( T ) - T 1 ^ l + R 1 J - - — ^ — J ( 1 - T l ) [ ( n - l ) T l ' e + (N-n)T/ - (N-1) } Since N! (N-n)I I2kd 4TT) ( d ) I1 N ) ! 1 N ) 1 -n-1 N (2.43) (2.44) i t i s evident that these expressions are f i n i t e power se r i e s i n the average density of s c a t t e r e r s per wavelength = NA./d. In the l i m i t as 0, they reduce to and <T> = T / (2.45) (2.46) The r e s u l t of equation (2.46), that the coherent transmitted f i e l d i n the l i m i t of 0 i s composed only of the term due to in-phase forward s c a t t e r i n g ( i . e . , Z-O-B-S), i s exact. The higher O-B-S terms neglected i n equation (2.41) a l l involve exponentials and consequently when integrated over the ensemble would give only terms to f i r s t and higher order i n p^. The asymptotic expression of (2.46) i s discussed further i n section 2.7. Further examination of the O-B-S representations f o r R and T reveals that the terms of equations (2.42) and (2.43) f or f i r s t and higher orders of p^ are the i n i t i a l terms of i n f i n i t e s e r i e s i n powers of R^  corresponding to higher O-B-S. Thus, the expressions obtained for (R) and ( T ) can be expected to be good approximations f o r scatt e r e r s with small | R J | . This, of course, i s also implied by the i n i t i a l approximations f o r R and T. Numerical r e s u l t s given i n Chapter 4 confirm t h i s reasoning. The seri e s expressions of equations (2.42) and (2.43) and those given i n the following sections can be generalized f o r a wave obliquely incident at an a r b i t r a r y angle 0 with the normal to the slab-region boundaries by a replacement of a l l occurrences of k with kcos©. The oblique-incidence expres-sions f o r Rj and T 1 are given i n Appendix B. 2.6.2 The Average T o t a l I n t e n s i t i e s The F-O-B-S approximation was also used to obtain a s e r i e s expression fo r the average t o t a l i n t e n s i t y of the r e f l e c t e d f i e l d . Thus, f or c o n f i g -urations of i d e n t i c a l s c a t t e r e r s , l * | 2 " M 2 E E ?l2(S~l) T t 2 ( t _ 1 ) e " 2 J k < z s - z t > s=l t=l (2.47) = | R j 2 4N-, 1 - l T i i L 1 - | T j 4 + l R J 2 E E T ^ 8 - 1 ) T* 2^' 1) e - 2 J k ( z s " z t > s=l t=l#s and 2\ i„ |2 <|R|Z>= K 1 - T 4 N l N: IR 12 N N ',% r d rzi + i <r-> 1 - | T j 4  „, ,. n, f d f : E E 2 ( ^ » T f ( t - i , r r d N s = l t=TVs J 0 J 0 •N zl 2 e - 2 j k ( z s - z p d z » . . . . d z . ( 2 > 4 8 ) 0 Evaluation of the i n t e g r a l of (2.48) gives 29 < | R | 2 > = K 1 - T, 1 - T, ' 1 1 n ^ (N-n)l I2kdl n+3 ^odd_nH-l)_ 2 ^ n+2 (even,n)(-l) 2 Re f (-Dm+1 Bnm (_> nm m=l 2m i _ | T |4(N-m) I x l I  i - I T J 4 T 2(N-n+m) -"•1 1 _ |x | 4(n-m) i - I T J 4 3-2jkd (2.49) where the B n m are the well-known binomial c o e f f i c i e n t s defined by nm n - l i m-l (n-1)! (n-m)!(m-l): (2.50) n+3 n+2 The factors (-1) 2 and (-1) 2 i n equation (2.49) provide the signs f o r the terms of the summation and are used a l t e r n a t e l y for odd n and even n. S i m i l a r l y , f or odd n the imaginary part of the second summation i s taken, and for even n the r e a l part i s taken. The f i r s t term of equation (2.49) gives the approximate form for (|R| 2) and also <I 2) = <|R| 2> - |< R >|2 i n the l i m i t of px ->» 0. The exact i n f i n i t e s e r i e s form f o r a l l 0-B-S contributing to the r e f l e c t e d f i e l d i s given i n s e c t i o n 2.7. Results from the numerical evaluation of equation (2.49) are compared with "exact" simulation r e s u l t s i n Chapter 4. The t o t a l i n t e n s i t y of the transmitted f i e l d f o r a S-O-B-S approxi-mation applied to configurations of i d e n t i c a l planar s c a t t e r e r s can s i m i l a r l y be w r i t t e n |T| 2 = | T 1 | 2 N ' [ l + 2Re(R 1 2C 1) + a ^ R j 4 + C J R J 4 ] (2.51) where a l l - L L i T j 4 ^ - " = Z ( N - D l T j 4 ^ " 1 ) s=2 t=l ' " N-1 i=l (2.52) h " E E1 T ^ 3 " ^ e - 2 J k ( z s - z t > (2.53) S=2 t=l ?2 = E L E ' E1' T ^ 8 " ' " 1 * T * 2 ^ " 1 ) e - 2 J k ( z s - z t - z u + z ; ) (2.54) s=2 t=l u=2 v=l [The primes on the t h i r d and fourth summation of (2.54) i n d i c a t e that u and v cannot simultaneously equal s and t respectively.] Thus, the average t o t a l i n t e n s i t y can be represented by < |T| 2) = | T l | 2 N [ l + 2Re(R 1 2< ? 1>) + a n | R l | 4 + ( s ^ R j 4 ] (2.55) The summation average (C^) can e a s i l y be obtained from the expression for ( T ) , i . e . , <^ 1> " nE 7 ^ ) T ( i f " " V)n-2 [ ( n - l ) T l 2 ( N - ) e - 2 J k d + ( N - n ) T l 2 - (N-1)] (2 .56 ) The summation average (C2)» represented by ,T, N s-1 N u-1 „. ' . „. rd r zl, d s=2 t=l u=2 v=l J 0 J0 Z2 _S e - 2 j k ( z s - z ^ - z u + z ; ) d z , .... d z , ( 2 > 5 7 ) 0 has not been determined i n the present work because of the d i f f i c u l t y involved i n recognizing the general expression for a r b i t r a r y N by the induction procedure. Instead, r e s u l t s f or (|T|2) have been obtained by an a p p l i c a t i o n of Monte Carlo simulation to the S-O-B-S approximation f o r T. The t h i r d term of equation (2.55) gives the approximate form for ( l 2 > = (|T|2) - |<T)| 2 i n the l i m i t of -* 0. The exact i n f i n i t e s e r i e s form f o r a l l O-B-S c o n t r i b u t i n g to the transmitted f i e l d i s given i n section 2.7 The f i r s t two terms of (2.55) contribute to the coherent i n t e n s i t y only. The• fourth term contributes both to the coherent and incoherent i n t e n s i t i e s . With-out i t s evaluation, only the zeroth-order term i n the power se r i e s for ( i 2 ) [ i . e . , t h i r d term of equation (2.55)] i s a v a i l a b l e . 2.6.3 The Covariant F i e l d s It i s evident from equations (2.27) that an e x p l i c i t expression f o r the complex function 2 S 2 e J 2 s = ( T 2 e J 2 x ) - c 2 e J 2 a provides a means of evaluating the second c e n t r a l moments of the t o t a l f i e l d . Thus, since approxi-mate se r i e s have already been given f o r the coherent f i e l d s , only the function remains. In the present work, a se r i e s expression has been obtained for the r e f l e c t e d f i e l d function < T 2 e J 2 T ) = (R2> based on the F-O-B-S approximation. For configurations of i d e n t i c a l planar s c a t t e r e r s , the basic equations are R2 = R,2 £ e " 4 J k z s + R X 2 f; £ T l 2 ( s + t - 2 > e - 2 J k ( z s + z t > s=l S=l t=l * S (2.58) = R ^ C g + ^ \ and (R2) = R^CCa ) + R ! 2 < ^ > < t j)4E !,«<-» /7 ' A . . . . / , 5 . -*^i d,J....^ (2.59) d s=i JO -JO -JO <?„> -4 E i ^Hs+t-2) ff 4 - rXJ '-2w» •••• d s = 1 t=<y>s JO J 0 JO The s e r i e s expression f o r ( C 3 ) can be obtained by inspection of the expression determined f o r (R). Straightforward i n t e g r a t i o n and induction must be used to 32 evaluate ( C u) f o r a r b i t r a r y N. Thus, « 3 > - E ark ( i f » - '.V'1 [^(N-n) ^  -*] «•«•) N „, / , \ n ; i - T 2' N~ 2 n=l (r > = T 2 V — 3_ I L . f x 2(N-n) e - 2 j k d _ A li L ( N _ n ) : 2kd 2 L 1 ! 6 M Cnm [T, 2^-^- 1) e " 2 J k d - T12<n-,n-1>l (2.61) m=l J The constant c o e f f i c i e n t s C n m are defined i n terms of the binomial c o e f f i c i e n t s Bnm ^ C n l = 1 (n = 2,...,N) (2.62) Cnm = Cn(m-1) + Bnm (n = 3,...,N; m = 2,...,n-1) They may also be generated from the recurrence r e l a t i o n s C n l = 1 (n = 2,...,N) Cnm = C ( n - l ) ( m - l ) + C ( n - l ) m (n = 3,...,N; m = 2,...,n-2) (2.63) Cn(n-1) = 2 C ( n - l ) ( n - 2 ) + 1 (n = 3,...,N) Approximate s e r i e s expressions have not been determined f o r ( T 2 e J 2 x ) = (T 2) i n the present work. Instead, r e s u l t s have been obtained f o r the second c e n t r a l moments of the transmitted f i e l d i n the S-0-B-S approxi-mation by employment of the simulation technique. The future development of approximate s e r i e s f o r both <T 2 e J 2 x ) and ( T 2 ) , however, would allow both the approximate evaluation of the second c e n t r a l moments for T and T and a l s o , x y from equations (2.32) to (2.35), the approximate evaluation of the f i r s t two moments of the amplitude T and phase T . 33 2.7 Asymptotic Theories f o r Scattering from Low Average Density  Scatterer D i s t r i b u t i o n s A study of s c a t t e r i n g from random media of di s c r e t e s c a t t e r e r s i n the asymptotic region of p -»• 0 i s important f o r two reasons: (i) The constituent s c a t t e r e r s are only sparsely d i s t r i b u t e d for many na t u r a l l y - o c c u r r i n g media of i n t e r e s t . ( i i ) The further development of theories for more densely-packed s c a t t e r e r d i s t r i b u t i o n s would be f a c i l i t a t e d through the understanding of the s c a t t e r i n g behavior of such d i s t r i b u t i o n s i n the region of low p. In t h i s s e c t i o n , asymptotic theories developed i n the present work f o r the one-dimensional ensemble of planar-scatterer configurations are given and, based on these asymptotic theories, modifications to e x i s t i n g theories applicable to higher density d i s t r i b u t i o n s are proposed. Consideration i s also given to s i m i l a r i t i e s between the asymptotic coherent transmitted f i e l d theory f o r the one-dimensional d i s t r i b u t i o n and an approximate theory developed by Twersky to describe c e r t a i n d i s t r i b u t i o n s of three-dimensional s c a t t e r e r s . 2.7.1 Asymptotic Theory f o r the Coherent Transmitted F i e l d As shown i n the previous s e c t i o n , f o r = NA/d 0, the coherent transmitted f i e l d f o r the one-dimensional ensemble of uni f o r m l y - d i s t r i b u t e d i d e n t i c a l planar sca t t e r e r s reduces to the simple form <T> <f> = (1 + g+) N <1> = <J> (2.64) the c o n t r i b u t i o n to the t o t a l f i e l d from the Z-O-B-S. The contributions from the higher 0-B-S are therefore e n t i r e l y incoherent. This i s a p h y s i c a l l y reasonable r e s u l t since the actu a l phases of the higher order terms vary over many lengths of the basic phase cycle (0 to 2TT) , making the equivalent phases on the basic phase cycle e f f e c t i v e l y uniformly d i s t r i b u t e d (see section 2.8). Twersky^ has obtained a s i m i l a r r e s u l t f o r a uniform slab-region d i s t r i b u t i o n of three-dimensional "forward-type" scatterers such that the s c a t t e r i n g amplitude i s " t i g h t l y peaked" around the forward-scattering d i r e c t i o n . In the i n t e g r a l version of the o r d e r s - o f - s c a t t e r i n g s e r i e s (2.7) f o r the coherent f i e l d , he ignored those terms corresponding to a "back-and-f o r t h " i n t e r a c t i o n between pai r s of s c a t t e r e r s ; equivalently, he retained only those successive s c a t t e r i n g terms i n which a l l scatterers are d i f f e r e n t . Using the method of stationary phase f o r performing the integrations i n the coordinates p a r a l l e l to the slab region, he obtained an e x p l i c i t s e r i e s form f o r (ty> which reduced to the closed-form expression <T> ty = I + 2irpdg(z,z) k 2N N l + 2iTg(z,z) k 2A N • (2.65) where g(z,z) i s the forward-scattering amplitude for three-dimensional s c a t t e r e r s , p = N/V = N/Ad i s the average volume density, V i s the volume of the f i n i t e slab region, and A i t s area. Twersky also showed t h i s r e s u l t to be v a l i d f o r a s p h e r i c a l source wave ty. Twersky has compared the expression i n (2.65) to that obtained f o r planar s c a t t e r e r s whose forward-scattering amplitudes are g + = 2ug(z,z)/k2A and whose back-scattering amplitudes are zero. In view of the present r e s u l t s for planar s c a t t e r e r s , the comparison does not require the r e s t r i c t i o n to planar s c a t t e r e r s with zero back-scattering amplitudes i n the l i m i t of p ->• 0. This would seem to suggest also that the r e s t r i c t i o n to "forward-type" scat t e r e r s i n the three-dimensional medium i s also unnecessary for p -> 0. Indeed, the p h y s i c a l reasoning used to explain equation (2.64) appears to be equally v a l i d i n the three-dimensional case. The m u l t i p l e - s c a t t e r i n g terms neglected by Twersky i n the d e r i v a t i o n of (2.65) are a l l higher-order extensions of those included, containing i n addition one or more orders of s c a t t e r i n g between one or more pai r s of s c a t t e r e r s . For an average path length between s c a t t e r e r s of many wavelengths, these a d d i t i o n a l terms should be completely incoherent. As shown by Twersky, as N -»• 0 0 the r e s u l t of ( 2 . 6 5 ) becomes i d e n t i c a l to the corresponding r e s u l t obtained by using the "forward-type" s c a t t e r e r r e s t r i c t i o n i n the approximate i n t e g r a l equation f o r (ty), i . e . , -»• e A d as N -> 0 0 ( 2 . 6 6 ) where A = 2iTpg(z, z ) / k 2 . The form ( T ) = e A d i s a s p e c i a l case of Twersky's f r e e -space theory equation with e i t h e r the "forward-type" s c a t t e r e r r e s t r i c t i o n or the r e s t r i c t i o n p - 0 . The r e s u l t of ( 2 . 6 6 ) therefore i l l u m i n a t e s one of the main l i m i t a t i o n s of the approximate i n t e g r a l equation and the free-space theory based on i t . The one-dimensional equivalent of the form <T> = e A d i s <T> = e N§+, the asymptotic form f o r p^ •> 0 given i n equation (A.14) of Appendix A. However, f o r the one-dimensional planar-scatterer d i s t r i b u t i o n , ( T ) = e^+ becomes equivalent to the exact r e s u l t of equation ( 2 . 6 4 ) only i n the t r i v i a l case of ( T ) ->- 0. Since equation ( 2 . 6 4 ) i s the exact expression f o r f i n i t e N and p^ -> 0, i t seems p l a u s i b l e to modify the general free-space theory equation <T> = D ( l - Q 2) e - J ( N _ 1 ) K D ( 2 . 6 7 ) [given also as equation (A.10) i n Appendix A; see section A . l for d e f i n i t i o n of symbols] to give the correct r e s u l t i n the l i m i t p^ -> 0. The modification required i s contained i n the equation <T) = ( 1 + g + ) N + [ D ( 1 - Q 2) e - J ^ - 1 ) ^ - e N § + ] ( 2 . 6 8 ) As shown i n Chapter 4, t h i s modification f o r f i n i t e N gives numerical r e s u l t s very close to the actual r e s u l t s (as obtained by Monte Carlo simulation) over 1 + Ad N 36 a wide range of p^. The r e s u l t s , i n f a c t , are better than those obtained from the S-O-B-S approximation f o r ( T ) as given i n equation (2.43). Further t h e o r e t i c a l i n v e s t i g a t i o n should reveal the reason f o r the improvement f or high as w e l l as low p^. Because of the improvement acquired by an asymptotic modification to the free-space theory f o r the one-dimensional medium with f i n i t e N, i t i s tempting to suggest a s i m i l a r m odification to the free-space theory f o r a three-dimensional medium based on equation (2.65). However, despite the marked s i m i l a r i t i e s , the coherent f i e l d properties of a three-dimensional medium are not i d e n t i c a l to those of a one-dimensional medium. In the one-dimensional medium, f o r example, (T) 0 as N ->• °°; i n the three-dimensional medium (T) remains f i n i t e f o r N -»• 0 0 with p = N/V constant. In reference 11 Twersky indi c a t e s that corrections f o r f i n i t e N are of i n t e r e s t but states i n reference 27 that no p r a c t i c a l error r e s u l t s f o r "forward-type" scatterers and low N i n using e i t h e r equation (2.65) or the free-space theory form e ^ . The reason f o r t h i s r e s u l t i s r e a d i l y apparent, since c a l c u l a t i o n s f o r the "forward-type" scat t e r e r s considered by Twersky give |Ad/N| << 1. S i m i l a r l y , f o r planar s c a t t e r e r s with |g +| << 1, e N§+ = (1 + g+) N even f or f i n i t e N, as i s v e r i f i e d by the numerical r e s u l t s of Chapters 4 and 5. Further i n v e s t i g a t i o n i s required f o r three-dimensional d i s t r i b u t i o n s , however, to determine which p r a c t i c a l combinations of d i s t r i b u t i o n parameters and i n d i v i d u a l s c a t t e r e r cross-sections make f i n i t e - N corrections necessary and whether the suggested modification to the free-space theory f o r such d i s t r i b u t i o n s i s v a l i d . 2.7.2 Asymptotic Theories f o r the Average T o t a l F i e l d I n t e n s i t i e s The problem of determining the asymptotic expressions f o r the average t o t a l i n t e n s i t i e s (and correspondingly the average incoherent i n t e n -s i t i e s ) i s a d i f f i c u l t one because an i n f i n i t e number of m u l t i p l e - s c a t t e r i n g processes contribute to the incoherent f i e l d . Emphasis i s therefore again placed on the one-dimensional model. The t o t a l i n t e n s i t y of the transmitted f i e l d for a f i x e d array of planar s c a t t e r e r s can be written |T| 2 = T Q(T§ + T*! + T* v + ...) + T i r ( T * + + T f v + ...) + .... (2.69) where the components TQ, T-J-J-, etc., are given i n equation (2.16). From the re s u l t s of section 2.6 i t i s evident that the ensemble averages of the terms i n (2.69) are of the i n f i n i t e s e r i e s form ,2 c Q + C l k d ) + c 2 | k d ) + < 2 ' 7 0 > where the c o e f f i c i e n t s c 0 , c^, c 2 , etc., are functions of the planar-scatterer parameters. Only the terms of (2.69) i n v o l v i n g no exponentials contribute to the c Q c o e f f i c i e n t s , and thus i t i s these terms which contribute to the asymptotic form f o r (|T|2). The asymptotic expression may therefore be represented as <|T| 2) = T 0T* + As [I'XI-'-II] + A s [ 2 R e ( T I v T l I ) + T I V T * V ] + As[2Re(T V IT* I) + 2Re(T V IT* v) + T ^ T ^ ] + (2.71) where "As" implies that the asymptotic form of the expression i n brackets i s to be taken. For an ensemble of i d e n t i c a l s c a t t e r e r s , the e x p l i c i t form of equation (2.71) i s <|T| 2>= | T l | 2 N { l + a u | R l | 4 + [ a 2 1 R e ( R l T f ) 2 | R, | 4 + a 2 2 | R j | 8 ] + . . . j (2.72) oo n " | T 1 ! 2 N [ l + E E a n m * e < R l T l > 2 ( n ~ m ) l R l l 4 m ] n = l m = l where the c o e f f i c i e n t s a n m are r e a l functions of T 1 and N. The f i r s t three coefficients are a l l " E L i T j 4 ^ " ^ = ^ (N-DlTj^1-1-) (N - 2,3,...) (2.73) s = 2 t=l i=l ' (N = 2) a 2 1 = <! N_! s_! N (2.74) 2 E E E i T i l 4 ^ ^ (N = 3,4,...) s= 2 t=l u=s+l N s-1 N u-1 a 2 2 = E E E E i T j 4 ^ - ^ " 2 ) . (N=2,3,...) (2.75) s = 2 t=l u=t+l v=l As seen from equation (2.72), the asymptotic form cannot be reduced to a sum of contributions from each O-B-S; there are also "cross terms" (i.e., m * n) contributed by sets of two O-B-S. From equations (2.64) and (2.72), the asymptotic expression for the average incoherent intensity in the S-O-B-S approximation i s <l2> ~- | T l | 2 N a i J R l | 4 ( 2 > 7 6 ) As shown in Chapter 4, this approximation gives results in close agreement with "exact" results for a wide range of the parameters |RjJ and N. The infinite-series asymptotic form for the average total intensity of the reflected f i e l d is obtained in a similar manner to that for the trans-mitted f i e l d . The explicit expression for an ensemble of identical planar scatterers i s <IR|2>= E E b n m Re(R1T*)2(n-m) | R l|2(2m-l) ( 2 > 7 7 ) n=l m=l where b n l = 0 for n > 2 (i.e., a l l cross terms between the F-O-B-S and higher O-B-S are zero) and the other coefficients b are real functions of | X | and N. The f i r s t two c o e f f i c i e n t s are N 1 _ I T |4N . b " " s?i | T ' I ' , < S ~ 1 ) b22 =[ [ E i T j * ^ - ^ ) (2.79) s=2 t=l u=t+l Since |< R >|2 •+ 0 as -»• 0, the asymptotic series for ( I 2) i s also given by equation (2.77). As shown i n Chapter 4, the F-O-B-S term of (2.77), <I2> - b n | R 1 | 2 (2.80) gives results i n close agreement with "exact" results for a wide range of |RjJ and N . Better agreement i s obtained with the addition of the T-O-B-S term, i.e. <I2> - b n | R 1 | 2 + b 2 2 | R j 6 (2.81) The F-O-B-S approximation for (|R|2) of equation (2.49) can be improved for a wide range of with the use of more accurate asymptotic terms, such as that of (2.81). The res u l t of this asymptotic modification to the F-O-B-S theory i s shown i n Chapter 4. An examination of equations (2.76) and (2.80) reveals that to f i r s t order, ( l 2 > <* | R J 4 = | g_ | 4 for the transmitted f i e l d and ( i 2 ) « | R J 2 = |g_| 2 for the reflected f i e l d . This result i s i n marked contrast to that given by equations (A.22) and (A.23) of Appendix A for Twersky's approximation based on the conservation of energy p r i n c i p l e . As shown by these equations for the one-dimensional ensemble, the incoherent power i s divided approximately equally between the transmitted and reflected f i e l d s , with ( I 2) for the transmitted f i e l d also approximately proportional to |g_| 2. Since the O-B-S approximations for the average i n t e n s i t i e s do not s a t i s f y the energy p r i n c i p l e , but give very accurate r e s u l t s as shown i n Chapter 4, i t i s evident that l i t t l e importance should be attached to complete adherence to t h i s p r i n c i p l e except under s p e c i a l conditions. For the "forward-type" scat t e r e r s considered by Twersky^ t h i s p r i n c i p l e must be s a t i s f i e d f o r the accuracy of the theories f o r |(T)|2 and (|T|2) to be equivalent. 2.8 T h e o r e t i c a l Models for the P r o b a b i l i t y Density of the T o t a l F i e l d The s t a t i s t i c s of the t o t a l f i e l d associated with s c a t t e r i n g from a random medium cannot completely be defined u n t i l the j o i n t p r o b a b i l i t y density function, p(T x,T ) or p(T,x), of the f i e l d components has been obtained. Because the problem of completely s p e c i f y i n g p(T x,Ty) i s a p a r t i c u l a r l y d i f f i c u l t one, e s p e c i a l l y f o r the one-dimensional model where multiple s c a t t e r i n g i s important, more emphasis has been placed i n the thesis on the problem of determining the f i e l d moments. A p a r t i a l i n v e s t i g a t i o n has been c a r r i e d out f o r an ensemble of uniformly-random planar s c a t t e r e r s , however, to determine the extent to which T x and T y conform to a b i v a r i a t e Gaussian d i s t r i b u t i o n f o r c e r t a i n ranges of parameters. The r e s u l t s are given i n Chapter 4. The commonly used approach i n the development of t h e o r e t i c a l models fo r the f i e l d s t a t i s t i c s has been to i n v e s t i g a t e the properties of random phasor sums of the form N T e J T = A e3 eo + V A e J 6 s (2.82) O i_j s S=l on the basis of known s t a t i s t i c a l properties of the A g and 6 S. In t h i s expression A QeJ®o i s usually a constant phasor and the phasors A geJ^s are random and i n general s t a t i s t i c a l l y dependent. Many workers have investigated t h i s problem area, in c l u d i n g Beckmann,-^ 30-32 Bremmer,-^ Hoyt,^ 4 Nakagami,-^ Norton et a l . , 3 6 and R i c e . 3 7 The general method used i n the s o l u t i o n of the problem has been to begin with a study of the s t a t i s t i c a l properties of the components N T x = T COST = A Q cos9 0 + £ A g cos0 c s=l N Ty = T sinT = A Q s i n 0 Q + £ A s s i n G s s=l (2.83) If the terms of the sums i n these expressions are s t a t i s t i c a l l y independent, i f N i s large, and i f the t o t a l variances of T x and Ty are much larger than the variances of the i n d i v i d u a l terms ( i . e . , the conditions of the Central Limit 3 Theorem), then T and T are j o i n t l y Gaussian with p r o b a b i l i t y density function P(T x,T y) 27r/<I 2><I 2>(l-u 2) exp < 2 ( l - u 2 ) ( T - C U 2 lx ~x _ 2 y ( y c x ) ( T y - c y ) | ( T y - C y ) 2 /<I 2)<I 2> <Iy> (2.84) For T x and Ty conforming to t h i s d i s t r i b u t i o n , a v a r i e t y of amplitude d i s t r i b u t i o n s p(T) and phase d i s t r i b u t i o n s P(T) are po s s i b l e . These are obtained by transforming to polar coordinates from P ( T , T ) = T P ( T X , T ) (2.85) and using the r e l a t i o n s p(T) p ( T , T ) d T (0 < T < 2TT) p(x) = / p(T,T) dT (0 < T) 0 (2.86) The most general expression f o r p(T) under the given conditions has been obtained by Nakagami. Less general r e l a t i o n s for s p e c i f i c d i s t r i b u t i o n s p(A s,8 g) of the components of the random phasor sum have been obtained or studied by the other researchers mentioned. Of s p e c i f i c i n t e r e s t i n th i s work i s the d i s t r i b u t i o n p(T,x) = -4? e t 2 / 2 ° 2 (0 < T, 0 < T < 2TT) (2.87) <l£> E <A|> = 5 2 , y = 0 s=l which occurs when (a) the constant phasor of equation (2.82) i s zero, (b) the phases 6 g are uniformly d i s t r i b u t e d over the basic phase c y c l e , i . e . , p ( 6 „ ) = \ - (0 ^  6 S < 2TT) (2.88) b 2TT B and (c) the A g and 8 S are mutually uncorrelated. Under these conditions the phase x i s uniformly d i s t r i b u t e d over the basic phase cycle and the amplitude T follows the well-known Rayleigh d i s t r i b u t i o n p ( T ) = I _ e - T 2 / 2 5 2 (2.89) a*1 Also of i n t e r e s t i s the Nakagami-Rice d i s t r i b u t i o n f o r the a m p l i t u d e 3 5 ' 3 7 p ( T ) = I _ e - ( T 2 + A o ) / 2 a 2 I |TAr a 2 °1 a 2 (2.90) where I i s the modified Bessel function of order zero. This d i s t r i b u t i o n a r i s e s when the constant phasor i n equation (2.82) i s not equal to zero. The 33 corresponding d i s t r i b u t i o n f o r the phase i s i A2/o^2 r r2 i A n COS(T-0 o) p ( T ) = i_ e ' V 2 0 |1 + G/WeG (1 + e r f G ) l , G = _ — : (2.91) 2TT L J /2a 4 3 The s t a t i s t i c a l behavior of the t o t a l f i e l d f o r the one-dimensional model can to a c e r t a i n extent be predicted by observation of the f i r s t and second O-B-S approximations f o r the r e f l e c t e d and transmitted f i e l d s . For i d e n t i c a l s c a t t e r e r s these are re s p e c t i v e l y N T e J T = Rx £ T ^ 3 " 1 ) e - 2 i k z s (2.92) s=l and T e J T = T jN + T iN R^2 £ S £ T i 2 ( s - t - l ) e - 2 j k ( z s - z [ ) ( 2 > 9 3 ) S=2 "t=l The r e s u l t i n g amplitude and phase components f or the r e f l e c t e d f i e l d corresponding to the A g and 6 g of equation (2.82) are therefore A Q = 0 A s = I R J I T J 2 ^ " 1 ) (S = 1,...,N) • (2,94) 6 S = ArgR x + 2(s- l ) ArgT 1 - 2kz g (s = 1,...,N) The components f or the transmitted f i e l d can be obtained by rew r i t i n g (2.93) as N(N-l)/2 T e J T = + T ^ R ^ £ T ^ ^ i " 1 ) e " 2 J k z i (2.95) i=l where z^ = z g-z^. and q^ = s- t . Thus, A eJ 9o = T N . o l A± = |R 1| 2|T X j N + 2 < ^ i - 1 > [ i = l,...,N(N-l)/2] (2.96) Q± = 2 ArgR 1 + [N + 2 ( q ± - l ) ] ArgT 1 - 2kz[ [ i = 1,...,N(N-1)/l] Since the ordered-positions z',...,zA i n equation (2.92) are s t a t i s t i c a l l y dependent, the problem of p r e d i c t i n g the s t a t i s t i c a l behavior of the r e f l e c t e d f i e l d i s seen to be a d i f f i c u l t one i n general. For s p e c i f i c ensembles of planar s c a t t e r e r s i n which the 6 g assume values over many lengths of the basic phase cy c l e , however, i t i s f e l t that this dependence can to a c e r t a i n extent be ignored. Under t h i s condition, which occurs f o r low average dens i t i e s of sc a t t e r e r s ( i . e . , << 1), i t i s u s e f u l to write the i n d i v i d u a l phase d i s t r i b u t i o n s over the basic phase cycle i n the form p(8 s) = ^ + c s ( 6 s ) (0 < 6 S < 2TT; s = 1,...,N) (2.97) Since |e g| << 1/2TT f o r << 1, the equivalent 8 g are e f f e c t i v e l y uniformly d i s t r i b u t e d over the basic phase cycle. They should also be les s c o r r e l a t e d than the act u a l 6 g. As | T J -»• 1, T x and T y w i l l be uncorrelated and normally d i s t r i b u t e d f o r large N, and T w i l l correspondingly be Rayleigh d i s t r i b u t e d . As jX1| -* 0, however, the s = 1 term i n (2.92) contributed by the f i r s t s c a t t e r e r w i l l predominate over the others, eventually breaking the "variance condition" of the Central Limit Theorem and making the r e s u l t i n g r e f l e c t e d f i e l d d i s t r i b u t i o n more complex. The problem of p r e d i c t i n g the transmitted f i e l d d i s t r i b u t i o n i s even more d i f f i c u l t . The terms i n equation (2.95) corresponding to = 1 have equal amplitudes A^ = I T J ^ R - J 2 and w i l l predominate over the other terms f o r | T J | -> 0. Under t h i s condition i t can be expected that the d i s t r i b u t i o n s of T x and Ty w i l l be approximately Gaussian f o r large N. However, the equivalent phases 8^ over the ba s i c phase cycle are highly c o r r e l a t e d , i r r e s p e c t i v e of the number of cycles over which the actual phases vary, since the 8^ f o r the q.j_ = 1 terms add to give the 6^ for the les s dominant terms. As | T J J ->• 1 these other terms w i l l therefore become more important, making the d i s t r i b u t i o n more complex. 45 2.9 Other T h e o r e t i c a l Considerations Generalization of the theories presented i n t h i s chapter to the cases of oblique incidence and lossy s c a t t e r e r s i s straightforward and the changes have been indicated where ap p l i c a b l e . Two other extensions not yet discussed involve the cases of random N and random g_j_s and g_ s. The case of a random number of s c a t t e r e r s N within the incident beam i s an important consideration i n such problems as s c a t t e r i n g from inhomo-geneities i n the atmosphere and meteor t r a i l s . A more immediate example o f t h i s s i t u a t i o n i s i l l u s t r a t e d with the p h y s i c a l model of a random d i s c r e t e -s c a t t e r e r medium discussed i n Chapter 6. Here the number of spheres illuminated by a narrow-beam transmitting antenna v a r i e s randomly as the medium i s "scanned". In the development of most approximate theories f o r the average f i e l d functions based on the "dishonest" approach, the random-N consideration i s not important. The fixed-N requirement inherent i n the d e f i n i t i o n (1.3) i s bypassed i n the transformation of the problem to one i n v o l v i n g the s o l u t i o n of i n t e g r a l equations. In the r e s u l t i n g equations, only the average density p appears e x p l i c i t l y . In random media problems i n v o l v i n g a f i n i t e number of s c a t t e r e r s , however, the random-N consideration cannot be ignored. Furthermore, i n theories based on the "honest" approach, N appears e x p l i c i t l y i n the equations. The extension of the theories developed i n the present work for a f i x e d number of planar s c a t t e r e r s (and, indeed, f o r any d i s t r i b u t i o n of a f i x e d number of sca t t e r e r s ) to the case of random N i s made possible by the theorem of t o t a l O 1 p r o b a b i l i t y . For p (z 1,...,z N|N) the j o i n t c o n d i t i o n a l p r o b a b i l i t y density function of an ensemble of fixed-N configurations [ i . e . , the function p(z.,...,z N) previously used i n the chapter] and p(z.,...,z N,N) the corresponding j o i n t p r o b a b i l i t y density function for the ensemble of random-N' configurations i n which the fixed-N ensemble i s contained, t h i s theorem gives the r e s u l t oo p( Z l,...,z N,N) = £ P(N) p,(z 1,...,z N|N) (2.98) N=0 where P(N) i s the p r o b a b i l i t y d i s t r i b u t i o n of N. The average of a f i e l d function F over the ensemble of random-N configurations i s therefore given by ( F > = £ POO • •••• F ( Z l , . . . , z N ) p( Z l,...,z N|N) d z : N=0 JO JO d z N (2.99) = £ P(N) <F> F N=0 where ( F ) N are the averages for f i x e d N. Thus, the theories f o r fixed N are the basis of more general theories f o r random N, the a d d i t i o n a l function necessary being the p r o b a b i l i t y d i s t r i b u t i o n f o r N. The d i s t r i b u t i o n function f o r N of most immediate i n t e r e s t i s the Poisson d i s t r i b u t i o n given by P ( N ) = e-<N> (2.100) N: This d i s t r i b u t i o n , f o r example, describes the p r o b a b i l i t y of f i n d i n g a given number of sc a t t e r e r s N within the s c a t t e r i n g volume when those of the e n t i r e medium are uniformly d i s t r i b u t e d throughout a much larger volume. It i s a p p l i c a b l e , therefore, to the p h y s i c a l model discussed i n Chapter 6. The e f f e c t of random N on the d i s t r i b u t i o n of the f i e l d components T x and T y i s also of i n t e r e s t . Beckmann ' has developed a c r i t e r i o n f o r T x and Ty to be considered normally d i s t r i b u t e d f o r random N given that they are normally d i s t r i b u t e d f o r f i x e d N s u f f i c i e n t l y large. This c r i t e r i o n , which can be expressed as 47 (N) N <N>2 - 1 0 (2.101) e f f e c t i v e l y states that the d i s t r i b u t i o n P(N) must assume s i g n i f i c a n t values only i n an i n t e r v a l about (N) small compared to the s i z e of (N). Since °N 2 = ( ^ ^ o r a P ° l s s o n d i s t r i b u t i o n , t h i s c r i t e r i o n i s s a t i s f i e d f o r (N) >> 1. Generalization of the one-dimensional model theories for the case of random i n d i v i d u a l s c a t t e r e r amplitudes i s d i f f i c u l t and a study of the problem i s beyond the scope of t h i s work. Even i n the simplest case of i d e n t i c a l l y d i s t r i b u t e d and s t a t i s t i c a l l y independent amplitudes, i t i s evident that Rj and T^ i n the given equations cannot simply be replaced by (R^) and (T^) , etc. In the F-O-B-S approximation f o r (R), for example, while R^  can be replaced by ( R 1 ) , ( T x 2 ) (not <T 1> 2) must be substituted f o r T^ 2. Generalization of the S-O-B-S approximation f o r (T) further involves the i n d i v i d u a l s c a t t e r e r averages (R J T J ) , ( T ^ , and ( T ^ ) . The Z-O-B-S term T j N of th i s approximation, however, can be replaced by ( T j ) N , making |(T 1) | 2 N the dominant term of the coherent i n t e n s i t y . Since the dominant term of (|T| 2) would then be ( I T ^ 2 ) 1 ^ (not i n general equal to Klj)!21*), i t i s evident that the incoherent trans-mitted f i e l d would contain a Z-O-B-S component. Although the t h e o r e t i c a l complexity makes i n s i g h t into the problem d i f f i c u l t to a t t a i n , "experimental" studies i n v o l v i n g Monte Carlo simulation provide an a l t e r n a t i v e future approach. 2.10 Summary summarized as follows: (i ) An e x p l i c i t s e r i e s representation i n orders-of-back-scattering has been given f o r the t o t a l f i e l d i n plane-wave s c a t t e r i n g from a f i x e d array of non - i d e n t i c a l planar s c a t t e r e r s . The new t h e o r e t i c a l developments considered i n t h i s chapter may be ( i i ) Approximate s e r i e s expressions based on the O-B-S representation have been obtained for several average f i e l d functions of i n t e r e s t i n the problem of s c a t t e r i n g from an ensemble of configurations of uniformly-random i d e n t i c a l planar s c a t t e r e r s . These expressions have been shown to be useful i n p r e d i c t i n g the exact or approximate asymptotic behavior of the average f i e l d functions i n the l i m i t p -> 0 and i t i s believed that they may also prove u s e f u l i n further t h e o r e t i c a l research dir e c t e d towards the improvement of general d i s c r e t e - s c a t t e r e r theories. ( i i i ) The exact asymptotic forms f o r p -> 0 i n the planar-scatterer model have been obtained for the coherent transmitted f i e l d and the average t o t a l and incoherent i n t e n s i t i e s of both the transmitted and r e f l e c t e d f i e l d s . In p a r t i c u l a r , the exact asymptotic form (1 + g + ) ^ for the coherent transmitted f i e l d has been used to modify the one-dimensional form of Twersky's f r e e -space theory and the p o s s i b i l i t y of a s i m i l a r f i n i t e - N c o r r e c t i o n to the three-dimensional form of Twersky's theory has been suggested. (iv) Based on the O-B-S approximation f o r the transmitted and r e f l e c t e d f i e l d s and the e x i s t i n g theory of random phasor sums, p h y s i c a l conditions necessary for the approximate v a l i d i t y of the b i v a r i a t e Gaussian d i s t r i b u t i o n i n d e s c r i b i n g the t o t a l f i e l d s t a t i s t i c s of the one-dimensional model have been discussed. Conditions necessary f o r the occurrence of a Rayleigh-distributed incoherent f i e l d amplitude with u n i f o r m l y - d i s t r i b u t e d phase have also been considered. 3. APPLICATION OF MONTE CARLO SIMULATION TO THE STUDY OF SCATTERING FROM RANDOM MEDIA 3.1 Introduction The method of Monte Carlo simulation i s an important t o o l i n the present i n v e s t i g a t i o n . Although Monte Carlo methods have been used extensively i n the f i e l d s of nuclear physics and operations research, they have only been 38 3 9 A 0 used s p o r a d i c a l l y i n other f i e l d s . Hochstim and Martens » have recently applied a Monte Carlo method to the study of radar s c a t t e r i n g from a one-dimensionally random slab region with d i s c r e t e p e r m i t t i v i t y v a r i a t i o n s . Their work has l a r g e l y been d i r e c t e d , however, towards the i n v e s t i g a t i o n of s c a t t e r -ing theories for continuous random media ( i . e . , the perturbed continuum model). The present work, while also i n v o l v i n g random media with d i s c r e t e p e r m i t t i v i t y v a r i a t i o n s , i s mainly dir e c t e d towards the i n v e s t i g a t i o n of theories based on a di s c r e t e - s c a t t e r e r formalism. The Monte Carlo method i s e s s e n t i a l l y an "experiment" with random numbers. Problems handled by the method are ei t h e r p r o b a b a l i s t i c , as i n th i s 3 8 th e s i s , or det e r m i n i s t i c i n nature. For eith e r type of problem the a p p l i -cation of the method i s greatly f a c i l i t a t e d by the use of a d i g i t a l computer with a means of generating a large quantity of suitably-random numbers. Two d i s t i n c t types of Monte Carlo simulation are considered i n t h i s work. For both types of simulation, the random numbers generated represent the random po s i t i o n s of d i s c r e t e s c a t t e r e r s i n a d i s c r e t e - s c a t t e r e r model. The di f f e r e n c e i n the two types of simulation i s i n the amount of computer involve-ment. In the f i r s t type, as i l l u s t r a t e d i n Chapters 4 and 5 with the planar-s c a t t e r e r model, a computer i s used for a l l phases of the simulation with an exact or approximate theory being required for the f i e l d i n s c a t t e r i n g from a fi x e d c onfiguration of s c a t t e r e r s . In the second type, as discussed i n Chapter 6, a computer i s used only to generate s c a t t e r e r configurations from the desired d i s t r i b u t i o n and to s t a t i s t i c a l l y analyze the s c a t t e r i n g r e s u l t s ; the t o t a l f i e l d s due to s c a t t e r i n g from the generated configurations are obtained by experimental measurements on a p h y s i c a l model of the random medium. Several a p p l i c a t i o n s of Monte Carlo simulation to the study of s c a t t e r i n g from random media are i l l u s t r a t e d i n this t h e s i s . As i n the work of Hochstim and Martens, "exact" simulation r e s u l t s f o r average f i e l d functions ( i . e . , those based on an exact theory for s c a t t e r i n g from a f i x e d configuration of s c a t t e r e r s ) are used to determine t h e ^ v a l i d i t y of various approximate theories f or these f i e l d functions over a wide range of s c a t t e r i n g parameters. In the present work, however, "exact" simulation r e s u l t s are also used to determine the extent to which c e r t a i n t h e o r e t i c a l models for the p r o b a b i l i t y density function of the f i e l d components describe actual behavior. In a. further a p p l i c a t i o n , "approximate" simulation r e s u l t s based on approximate theories f o r s c a t t e r i n g from a f i x e d configuration of sca t t e r e r s are employed to v a l i d a t e the corresponding theories f or the ensemble. A po s s i b l e future a p p l i c a t i o n of Monte Carlo simulation l i e s i n i t s use to check inv e r s i o n techniques for determining the p h y s i c a l and s t a t i s t i c a l composition of a random medium from sampled estimates of the f i e l d moments. 2 R Twersky has developed techniques based on approximate s c a t t e r i n g theory which require measurement of f i r s t and second f i e l d moments. 3.2 Technique of Simulation Applied to a Random Medium of Discrete Scatterers The object i n Monte Carlo simulation, as applied to s c a t t e r i n g from a random medium of d i s c r e t e s c a t t e r e r s , i s to approximate the exact i n t e g r a l expression /• 00 /- 00 ••• / p ( s x , . . . , s N ) X ( s 1 , . . . , s N ; f ) dsj • • • d s N (3.1) —00 J _QO 51 by l n <X(r)) = - £ X . ^ i N ; r ) (3.2) i=l where are the i n d i v i d u a l samples of the f i e l d function X for n d i f f e r e n t s c a t t e r e r configurations from the ensemble. By the "law of large numbers", 3 the approximation (3.2) w i l l converge to the t h e o r e t i c a l value represented by (3.1) i n the l i m i t as n -* °°. The accuracy of the estimation a f t e r a f i n i t e number of samples i s discussed i n section 3.4. Each random sample of a f i e l d function requires the generation of a sequence of random numbers from the desired s t a t i s t i c a l d i s t r i b u t i o n p(s-, Sjp . I f , for example, each s c a t t e r e r i s described by an ^-component random vector s^ = ( s ^ l >s^2,. . . , s ^ ) whose elements are the p o s i t i o n coordinates and other s c a t t e r e r parameters (e.g., s i z e , p e r m i t t i v i t y , e t c . ) , a sequence of N£ random numbers must be generated f o r the configuration of N s c a t t e r e r s associated with each f i e l d sample. For the one-dimensional and three-dimensional models of i d e n t i c a l s c a t t e r e r s considered i n the present work, the necessary sequences are of length N and 3N r e s p e c t i v e l y , since only p o s i t i o n coordinates are random. 3.3 Random Number Generation The random numbers used i n the Monte Carlo simulation of t h i s work are not t r u l y random, but "pseudo-random", since they are generated by an a r i t h -metic method but manage to pass c e r t a i n s t a t i s t i c a l tests for randomness. The generators employed produce u n i f o r m l y - d i s t r i b u t e d numbers on the u n i t i n t e r v a l (0, 1), as i s most common. » Such generators often form the basis f o r the generation of numbers from more complex d i s t r i b u t i o n s , since i t i s sometimes p o s s i b l e by a transformation to derive other d i s t r i b u t i o n s from a uniform d i s t r i b u t i o n . For the simulation r e s u l t s of Chapter 4, the uniformly-d i s t r i b u t e d numbers could of course be used d i r e c t l y . The simulation was performed on an IBM 7044 computer during the e a r l i e r part of the work and on an IBM 360/67 during the l a t e r part. The random number program RAND as supplied for each computer by the Univ e r s i t y of B r i t i s h Columbia Computing Centre was used. The generator for the IBM 7044 was of the m u l t i p l i c a t i v e congruential type ' with m u l t i p l i e r constant 2 7 + 1 and add i t i v e constant l l l l j , The algorithm for the IBM 360/67 generator i s given i n reference 43. Various d e t a i l e d tests have been used i n determining the "randomness" of sequences generated by a v a r i e t y of random number generators, * although such tests f or the sequences themselves do not guarantee that a random number generator w i l l be s u i t a b l e f o r a p a r t i c u l a r a p p l i c a t i o n . Results obtained during the course of t h i s work, however, have v e r i f i e d the s u i t a b i l i t y of the generators used. Much evidence i s provided by the fa c t that, f or a low average density of s c a t t e r e r s , "exact" simulation r e s u l t s f o r several average f i e l d functions agree with the corresponding numerical data for the asymptotic theories (see Chapter 4). The agreement between "approximate" simulation r e s u l t s and t h e o r e t i c a l r e s u l t s based on the same approximation over a wide range of s c a t t e r i n g parameters provides further evidence. Another method used to check the s u i t a b i l i t y of the generator RAND for the IBM 7044 computer was to compare the simulation r e s u l t s with r e s u l t s obtained using a second generator. No deviations larger than the s t a t i s t i c a l accuracy of estimation were noticed between the corresponding r e s u l t s . The second generator used was a modification of the generator RAND as follows: The generator RAND was used i n i t i a l l y to f i l l a large array (ten times l a r g e r than the length of a sequence of numbers describing a si n g l e configuration of scat t e r e r s ) with pseudo-random numbers; whenever a number was needed, two more were generated by RAND, one determining the element of the array to be used and the other r e p l a c i n g i t . This technique has been a p p l i e d by G e b h a r d t 4 4 to a very poor b a s i c generator and shown to y i e l d good r e s u l t s . The b a s i s of the method i s that i t e f f e c t i v e l y e l i m i n a t e s any c o r r e l a t i o n which may e x i s t between c l o s e l y adjacent numbers i n a sequence by s h u f f l i n g the sequence. A s i m i l a r 45 technique was o r i g i n a l l y proposed by MacLaren and M a r s a g l i a whereby two sep-arate generators are used, one f o r f i l l i n g the a r r a y and the other f o r s p e c i f y -i n g the sequence order. 3.4 Accuracy i n Monte Carlo S i m u l a t i o n The accuracy of the estimated mean ( x ) = ^ X^/n can be r e a d i l y x = l obtained s i n c e i t s sampling d i s t r i b u t i o n i s Gaussian f o r l a r g e n w i t h the 38 standard e r r o r r e l a t e d to by °X O/YN = — (3.3) Thus, f o r example, at the 95% l e v e l of confidence, the e r r o r of the Monte Carlo estimate w i t h respect to the true mean given by equation (3.1) i s l e s s than twice the standard e r r o r 2 a ( x ) * The f a c t o r i n the denominator of (3.3) i m p l i e s t h a t i n order to double the s t a t i s t i c a l accuracy of the estimated mean, four times as many samples must be obtained, e t c . Because of the e f f e c t of t h i s f a c t o r on the computation time i n a complete computer s i m u l a t i o n , the number of samples n must be l i m i t e d . Most of the s i m u l a t i o n r e s u l t s obtained i n the present work were based on 1,000 samples, although some r e s u l t s were based on 4,000 samples to improve the estimates of c e r t a i n f u n c t i o n s of very s m a l l magnitude. Because of the l a r g e number of samples taken, b i a s e d estimates of the second, t h i r d , and f o u r t h c e n t r a l moments of the f i e l d components were used w i t h only a very s m a l l e r r o r i n t r o d u c e d . Biased estimates of the v a r i a n c e s , f o r example,differ from unbiased estimates by a m u l t i p l i c a t i o n f a c t o r of ( n - l ) / n . E r r o r s due to the use of biased estimate formulas are s i m i l a r l y s m a l l f o r the t h i r d and f o u r t h c e n t r a l moments and the corresponding c o e f f i c i e n t s of skewness 46 and k u r t o s i s when n i s l a r g e . In the computer s i m u l a t i o n w i t h the one-dimensional model (Chapters 4 and 5 ) , accuracy estimates were obtained f o r a l l the f i r s t and second f i e l d moments based on twice the estimated standard e r r o r s of the means. For e s t i -mated f i e l d q u a n t i t i e s such as C and a whose sampling v a r i a n c e s could not be obtained d i r e c t l y , "worst-case" accuracy estimates based on approximate r e l a t i o n s w i t h the d i r e c t l y obtained v a r i a n c e s of the b a s i c components were determined. For example, from a Taylor s e r i e s expansion to f i r s t - o r d e r terms^ 4 C x 2 c r T x + S S ' + 2 C x C y y a T x J T y [ l C x K x + l C y l ° T y ] ' C ~ 2 a 2 a : i_ < : ( 3 > 4 ) nC 2 nC 2 and C 2 a 2 + C 2 a 2 + 2C xC„ya T a f|C v|cr + | c x|a T I 2 o Y Jx y x 'y L x xyJ . o /• ~ _ _ < KJ-J) a nC 4 nCk w i t h the upper bounds of these expressions g i v i n g "worst-case" estimates ( i . e . , C xC yy = |c xC y|) of the sampling v a r i a n c e s . Because of the l a r g e number of data p o i n t s obtained i n the complete computer s i m u l a t i o n (and the inaccuracy of c e r t a i n of the "worst-case" e s t i m a t e s ) , accuracy estimate r e s u l t s are not given i n the t h e s i s . A good i n d i c a t i o n of the accuracy i s given by the " s t a t i s t i c a l s c a t t e r " of the data p o i n t s ( r e s u l t i n g from the generation of a d i f f e r e n t psuedo-random number sequence f o r each set of data) on many of the graphs i n Chapters 4 and 5. In the s i m u l a t i o n w i t h the three-dimensional p h y s i c a l model discussed i n Chapter 6, c e r t a i n of the sampling v a r i a n c e s f o r complete se t s of data were estimated from the v a r i a n c e s of the means f o r s e c t i o n s of the data. 4. THEORETICAL AND SIMULATION RESULTS FOR A UNIFORM PROBABILITY DENSITY OF PLANAR-SCATTERER CONFIGURATIONS 4.1 Introduction The uniform p r o b a b i l i t y density of s c a t t e r e r p o s i t i o n s has been the basis of most approximate s c a t t e r i n g theories so f a r developed, being the easiest to apply and adequately describing s i t u a t i o n s where a sparse d i s t r i -bution of f i n i t e - s i z e s c a t t e r e r s e x i s t s ( i . e . , where the volume occupied by the s c a t t e r i n g material i s much less than the volume of the containing region). Studies of d i s t r i b u t i o n s of uniformly-random sc a t t e r e r s have furthermore provided a s t a r t i n g point f o r studies of more complex denser d i s t r i b u t i o n s of s c a t t e r e r s . In t h i s chapter, simulation and approximate t h e o r e t i c a l r e s u l t s are given f o r the s c a t t e r i n g of a normally incident plane wave from an ensemble of configurations of uniformly-random i d e n t i c a l planar s c a t t e r e r s . The requirements f o r the approximate v a l i d i t y of the uniform p r o b a b i l i t y density f o r d i s t r i b u t i o n s of f i n i t e - w i d t h planar sca t t e r e r s are discussed i n d e t a i l i n Chapter 5. The s c a t t e r i n g parameters for the one-dimensional model introduced i n Chapter 2 are N, d^, g + , and g_. Since g + and g_ for the i n f i n i t e l y - t h i n s c a t t e r e r s are taken to be those f o r l o s s l e s s d i e l e c t r i c slabs of f i n i t e t h i c k -ness, the s c a t t e r i n g parameters are equivalently N, d^, w^ , and e r . In order f o r an evaluation of the approximate theories to be most e a s i l y made and the p h y s i c a l behavior of the random medium i l l u s t r a t e d , r e s u l t s are given f o r the v a r i a t i o n of one of these parameters at a time. The complexity of the p h y s i c a l behavior of the medium i s l a r g e l y dependent on the average number of s c a t t e r e r s per wavelength p^ = N/d^, and increases as p^ increases. The e f f e c t of p^ v a r i a t i o n over a wide range of values on the various f i r s t and second f i e l d moments of i n t e r e s t i s best i l l u s t r a t e d by the v a r i a t i o n of d^ rather than N, since e f f e c t s r e s u l t i n g from N v a r i a t i o n overshadow those r e s u l t i n g from d^ v a r i a t i o n . The e f f e c t s of v a r i a t i o n of the parameters N, g + , and g_ on the p h y s i c a l behavior of the medium and on the accuracy of the approximate theories i s adequately displayed by r e s u l t s f o r a few of the average f i e l d functions i n the l i m i t of p^ -* 0. The f i x e d parameters f o r the d^-variable r e s u l t s have the values N = 10, e r = 2.0, and w^  = 0.1/V2 (or w^ t = 0.1, where A' i s the wavelength i n the d i e l e c t r i c material of the equivalent slab s c a t t e r e r s ) . The r e s u l t i n g values f o r the s c a t t e r i n g amplitudes, g + = 0.2107 /-101.7° and g_ = 0.2035 /-102.2°, i n d i c a t e that the i n d i v i d u a l s c a t t e r e r s considered are almost one-dimensional monopoles. In f a c t , the " t h i n - s l a b " approximation f o r the s c a t t e r i n g amplitudes gives g + ~ g_ ~ 0.2276 /-102.5° (see Appendix B). These values have been chosen because the e f f e c t s of higher orders of multiple s c a t t e r i n g are s u f f i c i e n t l y large to display f i n e differences i n the various simulation and t h e o r e t i c a l r e s u l t s . They are used also f o r the N-variable r e s u l t s . For the g+,g_ -v a r i a b l e r e s u l t s given, N = 10. The simulation r e s u l t s f o r d^ v a r i a t i o n are based on 4,000 sample configurations and those f o r v a r i a t i o n of the other parameters on 1,000 sample The execution time on the IBM 360/67 computer f o r generation and s t a t i s t i c a l a n a lysis of 1,000 sample configurations i s about 0.24 minutes. Throughout the chapter, simulation r e s u l t s f o r the various O-B-S approximations considered i n Chapter 2 are given where necessary to v a l i d a t e the theories based on the same approximations. Results f o r the various average transmitted f i e l d functions of i n t e r est are given i n section 4.2 and r e s u l t s f o r the r e f l e c t e d f i e l d functions i n section 4.3. Emphasis i s placed on the complete s t a t i s t i c a l d i s t r i b u t i o n of t o t a l f i e l d i n section'4.4 and qu a n t i t a t i v e r e s u l t s based on the t h i r d and fourth f i e l d moments are presented. A general discussion and summary of the r e s u l t s i s given i n section 4.5. 4.2 Transmitted F i e l d Moments 4.2.1 The Coherent F i e l d - CeJ" Results f o r the phase and i n t e n s i t y of the coherent transmitted f i e l d are given i n figures 4.1a and 4.1b f o r a v a r i a t i o n of d^. From the "exact" simulation r e s u l t s , C e J a -»• (1 + g+)^ for •> 0 (or d^ ->- °°) as was shown a n a l y t i c a l l y i n Chapter 2. In the other l i m i t of + 0 0 (or d^ -> 0) , C e l a tends asymptotically to the value of the transmitted f i e l d f o r a p e r i o d i c array of s c a t t e r e r s . The behavior of the transmitted f i e l d f o r a p e r i o d i c array i s both o s c i l l a t o r y and p e r i o d i c i n d^, with a resonance condition occurring when the spacing between the s c a t t e r e r s i s approximately a multiple of A/2 (see Chapter 5). As seen from figures 4.1a and 4.1b, approximately the same o s c i l l a t o r y behavior remains i n Cel for high p x although because of incoherent s c a t t e r i n g i t becomes i n c r e a s i n g l y damped out as p ^ increases. As discussed i n Chapter 2, the asymptotic l i m i t of Ce^ a as p ^ -> 0 for Twersky's free-space and mixed-space theories i s e^+. For the present f i x e d parameters, t h i s form gives a r e s u l t appreciably d i f f e r e n t from the exact r e s u l t of (1 + g + ) ^ , p a r t i c u l a r l y i n the coherent i n t e n s i t y C 2. Because of t h i s discrepancy i n the asymptotic l i m i t , the free-space theory does not accurately describe the a c t u a l r e s u l t s f or high p ^ e i t h e r . On the other hand, the modified free-space theory presented i n section 2.7.1, containing the correct asymptotic l i m i t , shows very good agreement with exact r e s u l t s over the e n t i r e range of d^. As seen from figures 4.1a and 4.1b, Twersky's mixed-space theory does not accurately describe the a c t u a l c h a r a c t e r i s t i c s of C e l a for uniformly-random planar s c a t t e r e r s . The l i m i t i n g behavior of C 2 -> 1 as p 0 0, however, i s i n approximate agreement with the p h y s i c a l behavior of c e r t a i n dense d i s t r i b u t i o n s / / / / / L / / o Exact S i m u l a t i o n & & S-O-B-S Si m u l a t i o n and Theory J. Arg (1 + g+) J l Arg e NS+ Twersky's Free-Space Theory — — — Twersky's Mixed-Space Theory — Modified Free-Space Theory P e r i o d i c Array Theory N = 10, wX' = 0.1, c r = 2.0 Figure 4.1.a Phase of Coherent Transmitted F i e l d as a.Function of d 0.01 0.1 n u n o i a\ 1 1 f 1.0 10 100 Figure A.l.b Coherent Transmitted Field Intensity as a Function of d x 60 of f i n i t e - s i z e s c a t t e r e r s . The a p p l i c a t i o n of the mixed-space theory to such, d i s t r i b u t i o n s i s considered i n Chapter 5. The S-O-B-S approximation for Ce^a developed i n the present work i s also shown i n figures 4.1a and 4.1b. The portion of the theory curves for small d^ could not be computed because the higher-order terms of the s e r i e s expression obtained "blow up", r e s u l t i n g i n i n s u f f i c i e n t accuracy of comput-at i o n . The approximate simulation r e s u l t s , however, give the remaining portion of the curves and v e r i f y the correctness of the approximate theory. As seen from the curves, the .S-O-B-S theory gives good agreement with exact r e s u l t s f o r a wide range of d^. Over most of the range of d^, the agreement i s approxi-mately the same as that for the modified free-space theory. For very large p x , however, the S-O-B-S theory gives r e s u l t s which d i f f e r markedly from the exact r e s u l t s f o r C 2. In f a c t , the trend of behavior i n C 2 as p x -»• 0 0 i s s i m i l a r to that f o r the mixed-space theory. I t i s i n t e r e s t i n g that the free-space and modified free-space theories give reasonably good agreement with exact r e s u l t s f o r p x ->• <» while the S-O-B-S theory does not. The reason for t h i s i s not immediately apparent and further t h e o r e t i c a l i n v e s t i g a t i o n of these theories i s thus required. Figure 4.2 shows the e f f e c t on C e J a of a v a r i a t i o n i n N and i l l u s -t rates the e f f e c t of t h i s parameter on the accuracy of the asymptotic form e^g-K " T ^ g loss of coherent transmitted f i e l d energy for increasing N corresponds to an increase i n the energy of the incoherent transmitted and r e f l e c t e d f i e l d s , as i s i l l u s t r a t e d by r e s u l t s of following sections. As discussed i n Chapter 2, e^§+ -> (1 + g+)^ as N -»- ». For the one-dimensional model, however, this i s an u n s a t isfactory r e s u l t , making necessary the c o r r e c t i o n f o r f i n i t e N contained i n the modified free-space theory. The e f f e c t of a v a r i a t i o n of the t o t a l s c a t t e r i n g c ross-section a= -2 Reg, on the accuracy of the form e N8+ i s shown i n f i g u r e 4.3. The 61 63 cross-section i s changed by v a r i a t i o n of w^i, rather than e r, which i s f i x e d at a value of 2.0. The aim i s only to determine the e f f e c t of increased a and t h i s can be done by e i t h e r a v a r i a t i o n of e r or w^i. The curves of figures 4.3a and 4.3b are shown only up to w^i =0.25, or a h a l f period i n | l + g +|. Figure 4.3c gives r e s u l t s f o r C 2 up to w^i = 4. Curves f o r the rectangular components of g + i n the range w^i = 0 to 0.5 are given i n Appendix B. Results for higher w^i with e r = 2.0 show that a (and |&4_|) varies p e r i o d i c a l l y i n w approximately every 3.4A', reaching i t s f i r s t maximum at about w^i = 1.7. The free-space theory form e 2 ^ R e § + for C 2 corres-pondingly has the same p e r i o d i c behavior as seen from fig u r e 4.3c, but with i t s maxima occurring at the minima of c. It i s thus required that o be small i n order that e^+ - (1 + g _ P ^ f o r f i n i t e N. For d i e l e c t r i c slab s c a t t e r e r s t h i s occurs f o r very t h i n slabs ( i . e . , w^i << 0.25) and at p e r i o d i c values i n w for thick slabs (e.g., w - 3.4A', 6.8A' f o r e r = 2.0). This r e s u l t i s discussed further i n Chapter 5. 4.2.2 The Average Incoherent Intensity - ( i 2 ) Simulation r e s u l t s f o r ( i 2 ) as a function of d^ are given i n f i g u r e 4.4. The approximate r e s u l t s from the S-O-B-S are i n good agreement with the exact r e s u l t s although the o s c i l l a t o r y e f f e c t s f o r high p^ are not contained i n the approximate r e s u l t s . Maximum ( i 2 ) occurs f o r p^ ->• 0 since only the Z-0-B-S component | l + g_|_ | 2 N i s c o n t r i b u t i n g to the coherent f i e l d , a l l higher O-B-S contributions being diverted to the incoherent f i e l d . Figure 4.5 shows the e f f e c t of N v a r i a t i o n and w^i v a r i a t i o n on ( i 2 ) i n the l i m i t of p^ -»• 0 and displays the accuracy of the asymptotic S-O-B-S theory presented i n section 2.7.2. The simulation r e s u l t s were obtained for d^ = 10 5. As seen by these curves, the l e v e l of ( i 2 ) i s quite small, being approximately p r o p o r t i o n a l to j g 0.04 O . O l 0 L i. & n A < O A < i A A > O o A A A ° o ° A < o > a < o o ° o o < o 0 < 0 ( o 1 O 0 A A O ) A A i < 0 > < < ° o ° / A 1 A A A A A 1 ( , A— n o 1 o i o & \ A \ A L 0.01 0.1 1.0 10 100 Figure 4.4 Average Incoherent I n t e n s i t y of Transmitted F i e l d as a Function of N = 10, w^, = 0.1, e r = 2.0; o Exact S i m u l a t i o n , A S-O-B-S Simul a t i o n 0.12 0.10 0.08 0.06 0.04 0.02 A o > K/ o rO / ° / ° > 0.12 10 N 15 20 25 (a) ( i 2 ) versus N for w^  ? = 0.1, e r = 2.0 0.20 0.25 (b) ( i 2 ) versus wx, f o r N = 10, e r = 2.0 Figure 4.5 Asymptotic Results f o r the Average Incoherent Intensity of the Transmitted F i e l d o Exact Simulation with d x = 10 5, A S-O-B-S Simulation with d x = 10 5, Asymptotic S-O-B-S Theory as Cn 4.2.3 The Variances and Covariance - ( I x 2 ) , ( i y 2 ) , ( l x I y ) Simulation results for the rectangular components ( ( l x 2 ) - ( l y 2 ) ) / 2 and ( l x l y ) of the complex function 2S 2eJ 2s defined in section 2.5.1 are shown in figure 4.6. In agreement with the elementary theory of that section, ( I x 2 ) = ( i y 2 ) when | ( l x I y ) | i s maximum and the difference between ( l x 2 ) and ( i y 2 ) is greatest when ( l x I y ) = 0. The exact results and those for the S-O-B-S approximation differ most for high although both show that ( l x 2 ) ( i y 2 ) and ( l x l y ) 0 as p A -> 0. These functions are important because they give the parameters of a bivariate Gaussian distribution for the total f i e l d components T x and Ty (see section 2.8) when the scattering behavior of the medium is such that this distribution i s valid. The asymptotic behavior for p^ -> 0 as shown in figure 4.6 is particularly important because, under the condition of jointly Gaussian T v and T„, the total f i e l d amplitude is described by the well-known x y Nakagami-Rice distribution (or equivalently the incoherent f i e l d amplitude is Rayleigh distributed). The applicability of these distributions to the scattering behavior of the one-dimensional model is discussed i n section 4.4. 4.2.4 Moments of the Amplitude and Phase - (T), O-J>, ( T ) , O T Simulation results were also obtained for the f i r s t and second moments of the transmitted f i e l d amplitude and phase for a variation of d^. These results showed the amplitude and phase moments to also be oscillatory in behavior for high p^, with a T and a T reaching their maximum values as p^ ->- 0. The agreement between the "exact" simulation results and those based on the S-O-B-S approximation was comparable with that already shown for the f i r s t and second moments of T x and Ty. The accuracy of the approximate relations for (T), ( i ) , a T 2 , and o T 2 based on equations (2.32) to (2.35) to second-order terms was also investigated by a comparison of the direct simulation results for these functions with those 67 0.010 0.005 -0.005 ( o 0 A •*M ° o O i A © ° A a J A k o ) > A A 0 ^ £ ( r ° j AAA ° * J ' S O fl A ^ •o-u J o 2 o o o k O A ° o o A J A A A ° • ^ O a o —**—y A * J — 0 0.25 ' 0.50 0.75 1.00 1.25 1.50 1.75 2.00 (a) (<I X 2> - <Iv2>)/2 versus d A 0.010 0.005 -0.005 -0.010 o o • o < o A A » A ' A « k ° ° > A O A O . A * A o A 1 o ° A A o A, A/ A ' 2 o A /«v A A A ^ A o * o A o o A -A < A i ^ O > A O l A u A J ^ £ 8. i 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 (b) ( l x I ) versus d^ Figure 4.6 Variances and Covariance of Transmitted F i e l d Components as Functions of d^. N = 10, w^ , = 0.1, e r = 2.0; o Exact Simulation, A S-O-B-S Simulation i n d i r e c t l y obtained from the values of the f i r s t and second moments of T x and T y using these r e l a t i o n s . The accuracy of the r e l a t i o n s for (T> and (T) was good over the e n t i r e range of p^, s t a t i s t i c a l l y s i g n i f i c a n t , but n e g l i g i b l e differences occurring only for high p x where the covariant i n t e n s i t y S 2 i s greatest. Differences between the d i r e c t l y and i n d i r e c t l y obtained r e s u l t s for and o T were larger but s t i l l quite small. Since differences were greatest f o r p x -> 0, the accuracy of the approximate r e l a t i o n s for o T and a T i s best indicated by the asymptotic r e s u l t s f o r a v a r i a t i o n of N and w^i . Shown i n f i g u r e 4.7 are the r e s u l t s for a v a r i a t i o n of N, the i n d i r e c t r e s u l t s based on the approximate r e l a t i o n s l a b e l l e d as the "second moment approximation". Shown also are the asymptotic theory curves based on both the S-O-B-S approxi-mation and the second moment approximation. As seen from fig u r e 4.7b, these two approximations p a r t i a l l y cancel one another f o r o T. Results f o r a v a r i a t i o n of w^ , were s i m i l a r , the greatest deviation between the d i r e c t and i n d i r e c t r e s u l t s occurring for the l a r g e s t values of |g_|. 4.3 Reflected F i e l d Moments 4.3.1 The Coherent F i e l d - C e J a Results for the phase and i n t e n s i t y of the coherent r e f l e c t e d f i e l d are given i n f i g u r e 4.8 for the f i r s t three wavelengths i n d. The coherent i n t e n s i t y i s p l o t t e d i n decibels below the l e v e l of the incident f i e l d i n t e n s i t y . As seen from f i g u r e 4.8, C e l a i s a damped o s c i l l a t o r y function, i t s amplitude decreasing to zero f o r •> 0 as predicted by theory (simulation r e s u l t s consequently become in c r e a s i n g l y inaccurate as p^ decreases). Minima i n C 2 occur when the phase c a n c e l l a t i o n of the m u l t i p l e - s c a t t e r i n g contributions from the i n d i v i d u a l s c a t t e r e r s i s greatest; maxima occur when i t i s l e a s t . The r a p i d l y decreasing amplitude as + 0 ( i . e . , C « 1/d) i s due to the f a c t that, as the phases of the m u l t i p l e - s c a t t e r i n g contributions from the i n d i v i d u a l 0.25 0.20 0.15 0.10 0.05 ^ A / ° / A 10 15 N (a) a,p versus N 20 25 (b) o*T versus N Figure 4.7 Standard Deviations of the Transmitted F i e l d Amplitude and Phase as Functions of N (Asymptotic Results), w^ , = 0.1, e r = 2.0; o Exact Simulation with d^ = 10 5, A Exact Simulation (d^ = 10 5) with Second Moment Approximation, S-O-B-S Theory with Second Moment Approximation 250° 70 200c -a 150° 100c • t °°/ A/ h'l —• o $\. y A / $1 , An ? l\ \ V 1 rA> t > 1 b J ?l i'V/ UP/ 1 I I / v> A \ r/ W 1/ 1 V ; / ° / / i \l o * o o 0.5 1.0 1.5 2.0 2.5 3.0 -10 6C O -20 -30 -40 (a) -a versus \\ \ \ ) <"> AF ' I / H o ¥ / tO 1 .1/ 1 II \v \ / <-w TO / \l' II II 1 u II ¥ V 1 o \ / / ?«/ w 0.5 1.0 1.5 2.0 2.5 3.0 (b) 10 log C 2 versus d^ Figure 4.8 Coherent Reflected F i e l d as a Function of d^. N = 10, , = 0.1, e r = 2.0; o Exact Simulation, A- A F-O-B-S Simulation and Theory, Twersky's Free-Space Theory, Twersky's Mixed-Space Theory 71 sca t t e r e r s become more uniformly d i s t r i b u t e d , coherent energy i s diverted to the incoherent f i e l d (see section 2.8). Although not shown i n figure 4.8, the behavior of Ce^a tends to that of the r e f l e c t e d f i e l d f o r a p e r i o d i c array of scatte r e r s as p^ -> °°. It i s evident from fi g u r e 4.8 that Twersky's free-space theory gives very good agreement with "exact" r e s u l t s i n the l o c a t i o n of the maxima and minima of C 2 and a. Like the theory f o r the coherent transmitted f i e l d , i t also accurately describes the behavior f o r p^ -»• °°. The F-O-B-S theory (note agreement between F-O-B-S theory and simulation r e s u l t s ) gives better agree-ment i n the magnitudes of the maxima and minima but less accurately describes t h e i r l o c a t i o n s f o r high p^. The mixed-space theory f o r the coherent r e f l e c t e d f i e l d , as f o r the coherent transmitted f i e l d , does not describe the physi c a l behavior of the uniformly-random ensemble of configurations as p^ -*• 0 0. Its s i g n i f i c a n c e i s that i t can approximately describe the behavior of c e r t a i n d i s t r i b u t i o n s of f i n i t e - s i z e s c a t t e r e r s i f the average density p i s c o r r e c t l y i n t e r p r e t e d . This i s shown i n Chapter 5, section 5.4. 4.3.2 The Average Incoherent Intensity - ( i 2 ) Results f o r ( i 2 ) as a function of d^ are given i n fi g u r e 4.9. The F-O-B-S theory gives r e l a t i v e l y good agreement with exact r e s u l t s over a wide range of although f o r large p^ i t does not describe the o s c i l l a t o r y behavior evident i n the exact r e s u l t s . (The theory curve i s shown to a point where the computational accuracy breaks down.) The modified F-O-B-S theory based on in c l u d i n g the e f f e c t of the T-O-B-S i n the asymptotic term of the F-O-B-S se r i e s gives b e t t e r agreement with exact r e s u l t s except f o r p-^  -> 0 0 (not shown). This again i l l u s t r a t e s the e f f e c t that an "asymptotic c o r r e c t i o n " can have on the accuracy of c e r t a i n theories applicable to low and mid-range values of p^. Figure 4.10 gives r e s u l t s f o r N v a r i a t i o n and w^i v a r i a t i o n i n the l i m i t of p, -> 0, d i s p l a y i n g the r e l a t i v e accuracies of the asymptotic F-O-B-S 0.4 0.3 <I2> 0.1 o <> 0 < o O a o o o ) c L—«— _ _ o—-°-— A ( o o <; 0< /O L f s o < 1 o- A. 4 ( 0 O I o A > 0.01 0.1 1.0 10 100 Figure 4.9 Average Incoherent Intensity of Reflected Field as a Function of d^ N = 10, wx, =0.1, e r = 2.0; o Exact Simulation, A A F-O-B-S Simulation and Theory, Modified F-O-B-S Theory N3 0.6 0.4 <I 2> 0.3 0.2 0.1 o o \ < O > V' V 0.6 10 15 N 20 25 (a) ( i 2 ) versus N for w^ , = 0.1, e r = 2.0 0.5 0.4 <I 2> 0.3 0.2 0.1 < o o , ° ° / / / / ^ A - &~ < 'A v/ '/ // f V 0.05 0.10 0.15 0.20 0.25 (b) ( i 2 ) versus wx , f o r N = 10, e r = 2.0 Figure 4.10 Asymptotic Results for the Average Incoherent Intensity of the Reflected F i e l d o Exact Simulation with d^ = 10 5 , A F-O-B-S Simulation with d^ = 10 5, Asymptotic F-O-B-S Theory, Asymptotic T-O-B-S Theory CO and T-O-B-S theories. Maximum deviation between approximate t h e o r e t i c a l and "exact" simulation r e s u l t s occurs f o r the la r g e s t N and largest |g_| ( i . e . , l a r g e s t w ^ i ) employed. As i s evident by a comparison of these r e s u l t s with those of figure 4.5 on page 65, the average incoherent i n t e n s i t y of the r e f l e c t e d f i e l d i s much greater than that of the transmitted f i e l d , being approximately proportional to |g_| 2 rather than to I g . l 4 . For the d^-v a r i a t i o n parameters i n the l i m i t of p^ -> 0, the "signal-to-noise r a t i o " C 2 / ( l 2 ) of the transmitted f i e l d i s approximately 17.9 whereas f or the e n t i r e medium ( i . e . , ( i 2 ) for both transmitted and r e f l e c t e d f i e l d s included as noise) i t i s approximately 1.92. 4.3.3 The Variances and Covariance - ( l x 2 ) > ( l y 2 ) j ( l x l y ) Results f o r the complex function 2 S 2 e J 2 s of the variances and covariance of T x and T y are shown i n fi g u r e 4.11. The behavior of the rectangular components of t h i s function f or varying d^ i s seen to be s i m i l a r to that f o r the corresponding transmitted f i e l d functions. The asymptotic behavior as d^ ->• 0 0 i s not reached as quickly, however, and the period of o s c i l l a t i o n i s smaller. The computational accuracy of the F-O-B-S theory curves based on equations (2.59) to (2.61) f o r ( T 2 e J 2 x ) = <R2) and the already v a l i d a t e d equation (2.42) f o r C e J a = (R) appears to break down i n the v i c i n i t y of d^ = 0.75. The poorer computational accuracy f o r lower values of d^ occurs f o r the ( R 2 ) equations because of the la r g e r denominator constants i n the terms of the s e r i e s . However, even f o r d^ > 0.75, the theory curves appear to deviate s i g n i f i c a n t l y from the F-O-B-S simulation r e s u l t s , although they c o r r e c t l y describe the o s c i l l a t o r y behavior. Thus, although the serie s expression f o r (R 2) has been c a r e f u l l y checked, i t s v a l i d i t y has not yet been established. 75 0.010 0.005 -0.005 -0.010 o A 0 o o o i o \ A A O A / o \ . A A O O A A p 0 A o o ° A O 0 o 0 < p. A \ f o > ^ o \ „ » / o 0 * J A o yL NJ? A 0 < » o< o o 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 (a) « I X 2 > - < I 2 » / 2 versus d x 0.015 0.010 o o o o o o o o o A ° I o 1° & A I ° A , < A A A ' k / A k A / \ 0 k A o o/ / A \ o y \ o X o o \ A J & I \ ° / o \ ^ A V o < o V f o o 0.005 -0.005 0 (b) ( l x I y > versus d x Figure 4.11 Variances and Covariance of Reflected F i e l d Components as Functions of d x . N = 10, wx i = 0.1, e r = 2.0; o Exact Simulation, A F-O-B-S Simulation, -F-O-B-S Theory 4.4 D i s t r i b u t i o n of the T o t a l F i e l d The object of t h i s section i s to determine the extent to which the transmitted and r e f l e c t e d f i e l d components T„ and T„ deviate from Gaussian y behavior i n the region of low p^ f o r a range of the parameters w A i and N. Quantitative r e s u l t s are given by means of the c o e f f i c i e n t s of skewness ( b v , b„7 x y and k u r t o s i s (Y x> Yy)> obtained from the t h i r d and fourth f i e l d moments re s p e c t i v e l y (see section 2.5.1). A l l r e s u l t s are based on 1,000 f i e l d samples T y p i c a l p l o t s of b x and Y x for a v a r i a t i o n of the phase reference v are given i n figures 4.12a and 4.12b f o r the transmitted f i e l d . The plo t f o r by i s i d e n t i c a l to that f o r b x but displaced by 90°. S i m i l a r l y the p l o t f o r Yy i s displaced 90° from that f o r Y x * Shown on the same graph as b x i n figu r e 4.12a i s the in-phase component C x of the coherent f i e l d . A comparison of t h i s curve with that f o r b v shows that the d i s t r i b u t i o n of T i s symmetrical f o r X X C x - 0 and most highly skewed for maximum C x. The shortest t a i l of the d i s t r i b u t i o n i s that c l o s e s t to the p h y s i c a l truncation l i m i t s of +1 or -1 for T . Plots of the equivalent q u a n t i t i e s f o r the r e f l e c t e d f i e l d are given i n f igures 4.12c and 4.12d. The p e r i o d i c behavior of the t h i r d and fourth f i e l d moments with v as shown by these curves has been t h e o r e t i c a l l y predicted by Twersky^ and s i m i l a r l y checked by means of experiments on a p h y s i c a l model of a d i s t r i b u t i o n of s p h e r i c a l s c a t t e r e r s . 5 Plots of the maximum value of |b| reached over a f u l l period of v v a r i a t i o n as a function of w^i are given i n f i g u r e 4.13a for the transmitted f i e l d . The s c a t t e r i n g parameters are N = 10, £ r = 2.0, and d^ = 100. The l i m i t within which 95% of the sampled values of b x and by would be expected to be found f o r Gaussian behavior of the f i e l d components i s shown on t h i s graph.1^ Plots of the maximum and minimum values of Y obtained with the corresponding 95% l i m i t s f o r Gaussian behavior are given i n figu r e 4.13b. •1.5 1.0 0.5 x 0 -0.5 •1.0 -1.5 i y N / \ / \ / \ V / X 1/ f / \ \ \ iif \ \ \\ • /I ' • • /i • • \ \ \ ^ / 1 y i " • • 9 \ \ / \ / \ / - V (a) b , C x versus -v - Transmitted F i e l d 400° 0.075 0.050 0.025 o -0.025 -0.050 -0.075 A / / / # / / / ~ \ '/ / / \\ / / y \\ \ v -y"°^ y i 1 \ \ \ V \ \ 1 1 ? \ 0° 100° 200° 300° -v 400c (c) b x versus -v - Reflected F i e l d 'x / / "N \ \ / \ / / / \ \ \ \ / / / \ \ \ \ V 100c 200' -v 3001 400' (b) Y x versus -v - Transmitted F i e l d -0.25 -0.50 -0.75 — • " 100' 200 -v 300 (d) Y x versus -v - Reflected F i e l d 400' Figure 4.12 Skewness and Kurtosis C o e f f i c i e n t s as Functions of the Phase Reference. N = 10, d A = 100, w^ , = 0.08, e r = 2.0; Exact Simulation, Approximate Simulation 'max 'max m m 2.0 1.0 0 O A O A A 0 A A A A 0 ( > O O ° 0 0.2 0 0.05 0.10 0.15 wx, 0.20 (a) | b | m a x versus wx, - Transmitted F i e l d 6 -2 A 0 A o A A A A A 6 -A o O A i A O A A A A A « O * « O u ° O O o o 0 (b) Y. 0.05 0.10 0.15 0.20 wx, , Y versus W j i - Transmitted F i e l d max mm A 'max 0.1 A A « i A i A O O ( k O o ) A O O A 0.5 'max ' m m -0.5 -1.0 -1.5 0 (d) Y 0.05 0.10 0.15 wx, 0.20 (c) | b | m a x versus wx? - Reflected F i e l d A I A O A A " A A A A A A O o O 1 > A A A A O ° ° < < > O r o ° o O o o o 0.05 0.10 0.15 0.20 max 'mm A versus w,, - Reflected F i e l d Figure 4.13 Extreme Values of b and y as Functions of w x i . N = 10, e r = 2.0, d x = 100; o Exact Simulation, A Approximate Simulation, 95% Confidence Limits f o r Gaussian Behavior oo As seen by these figures and as discussed i n section 2.8, the d i s t r i b u t i o n of the f i e l d components i s more nearly Gaussian f o r large values of |g_|. The re s u l t s show, however, that the d i s t r i b u t i o n of the actual f i e l d i s les s highly skewed and less sharply peaked than that of the f i e l d as approximated by the S-O-B-S. The corresponding r e s u l t s f o r the r e f l e c t e d f i e l d are given i n figures 4.13c and 4.13d. As seen by these r e s u l t s and as discussed i n section 2.8, the d i s t r i b u t i o n s of T x and Ty are more nearly Gaussian f o r low values of |g_|. The d i s t r i b u t i o n s of the act u a l f i e l d components, however, are somewhat f l a t t e r than those of the approximate f i e l d components f o r the F-O-B-S, as shown by the plo t s f o r y. Furthermore, deviation of the act u a l f i e l d from Gaussian behavior occurs more quickly f o r an increase of |g_| than does that of the approximate f i e l d . From the s t a t i s t i c a l s c a t t e r of the data points i n figu r e 4.13, i t appears that the variances of the b and y s t a t i s t i c s are greater f o r the approximate f i e l d than f o r the act u a l f i e l d . Curves are not given f o r the e f f e c t of N v a r i a t i o n on the skewness and k u r t o s i s c o e f f i c i e n t s . As expected, the r e s u l t s obtained showed a trans-mitted f i e l d approaching Gaussian behavior f o r large N. The rate of approach, however, was quite slow. With s c a t t e r i n g parameters of w^i = 0.06, e r = 2.0, and d^ = 100, maxima i n the | b | m a x and Y m a x , Y m i n p l o t s occurred at approxi-mately N = 12. For these same parameters, however, the d i s t r i b u t i o n s of the r e f l e c t e d f i e l d components appeared to change l i t t l e beyond about N = 6, being somewhat f l a t t e r than Gaussian f o r lower N. This r e s u l t i s of course p h y s i c a l l y reasonable since contributions to the r e f l e c t e d f i e l d from scat t e r e r s further removed from the o r i g i n become i n c r e a s i n g l y smaller. As seen from the r e s u l t s of sections 4.2.3 and 4.3.3, T x and Ty become uncorrelated f o r low p^. Thus, i t can be assumed that as the d i s t r i -butions of T and T approach the Gaussian d i s t r i b u t i o n , the d i s t r i b u t i o n s of the incoherent transmitted and r e f l e c t e d f i e l d amplitudes w i l l approach a • Rayleigh d i s t r i b u t i o n . Likewise, the d i s t r i b u t i o n of the t o t a l transmitted f i e l d amplitude w i l l approach the Nakagami-Rice d i s t r i b u t i o n . Further i n v e s t i -gation i s required, however, to determine the extent to which deviations of T and T from Gaussian behavior cause a s i m i l a r deviation of the incoherent x y f i e l d amplitude I from Rayleigh behavior. Plots of the cumulative d i s t r i b u t i o n s of T x and T y i n standard normal deviates^- 4 were also obtained as an a d d i t i o n a l check for deviation from Gaussian behavior. Such p l o t s , however, are le s s s e n s i t i v e to deviations than are the b and y c o e f f i c i e n t s . S t r a i g h t - l i n e p l o t s f o r the exact r e f l e c t e d f i e l d cumulative d i s t r i b u t i o n were obtained up to w^i = 0.08, for example; the values f o r y of fi g u r e 4.13d i n d i c a t e a somewhat flat t e n e d d i s t r i b u t i o n at t h i s point. 4.5 Summary and General Discussion of Results The main developments of t h i s chapter may be summarized as follows: (i) Results have been given demonstrating the accuracy of the various O-B-S theories developed i n the present work. This opens the way for future t h e o r e t i c a l work di r e c t e d toward a comparison of these theories with the one-dimensional forms of more general d i s c r e t e - s c a t t e r e r theories. ( i i ) The importance of the low average density asymptotic behavior of the medium on the evaluation and improvement of e x i s t i n g theories applicable to more dense s c a t t e r e r d i s t r i b u t i o n s has been q u a n t i t a t i v e l y i l l u s t r a t e d . ( i i i ) Results have been given f o r the one-dimensional forms of Twersky's theories f o r the coherent f i e l d f o r a wide v a r i a t i o n of parameters, showing the l i m i t a t i o n s of these theories i n a manner not as e a s i l y allowed by experiments on p h y s i c a l models of d i s c r e t e - s c a t t e r e r d i s t r i b u t i o n s . Since the uniform-randomness of the s c a t t e r e r s has not been an approximation as i t must be f o r p h y s i c a l d i s t r i b u t i o n s of f i n i t e - s i z e s c a t t e r e r s , the e f f e c t of the h e u r i s t i c approximations contained i n these theories has been separated from the e f f e c t of a u n i f o r m - d i s t r i b u t i o n approximation. Results have also been given which demonstrate the improvement contained i n the modified free-space theory f o r the coherent transmitted f i e l d presented i n Chapter 2. (iv) A quantitative analysis of the t o t a l f i e l d d i s t r i b u t i o n based on the t h i r d and fourth f i e l d moments has been made and some e f f e c t s of multiple s c a t t e r i n g on t h i s d i s t r i b u t i o n i l l u s t r a t e d . P h y sical conditions necessary f o r the approximate v a l i d i t y of the b i v a r i a t e Gaussian d i s t r i b u t i o n as discussed i n Chapter 2 have been v e r i f i e d . (v) Monte Carlo simulation has been demonstrated to be a useful t o o l i n the study of random media of d i s c r e t e s c a t t e r e r s and associated s c a t t e r i n g theory. The "exact" simulation r e s u l t s presented i n t h i s chapter provide a basis f o r future evaluation of the one-dimensional forms of other general theories developed by means of the "dishonest" technique discussed i n Chapter 2. Further r e s u l t s f o r oblique angles of incidence, lossy s c a t t e r e r s , and Poisson-distributed random N would complement those already obtained. The exponential d i s t r i b u t i o n f o r the spaces between the sc a t t e r e r centers, p ( q ) = pe-P^i, p = <N>/d ( i = 1,2,...,N + 1) (4.1) required to give Poisson-distributed N i s e a s i l y generated f r o m 4 7 «1 = " J l o8e< zi> <4'2> where the z. are uniformly-random numbers from the unit i n t e r v a l . 5. SIMULATION OF A NON-UNIFORM PROBABILITY DENSITY OF PLANAR-SCATTERER CONFIGURATIONS WEIGHTED TOWARDS PERIODICITY 5.1 Introduction The uniform p r o b a b i l i t y density function and s c a t t e r i n g theory based on i t are most v a l i d f or " g a s - l i k e " s c a t t e r e r d i s t r i b u t i o n s of low average density. For d i s t r i b u t i o n s of higher average den s i t i e s the uniform p r o b a b i l i t y density function i s inappropriate and more accurate functions must be con-sidered. I d e a l l y , a . p r o b a b i l i t y density function i s required which i s applicable over the e n t i r e range of s c a t t e r e r concentrations from a "rare gas", through a " l i q u i d " , to the other l i m i t of a " c r y s t a l - l i k e s o l i d " . One t h e o r e t i c a l approach to the problem has been to l i m i t consid-e r a t i o n to the two-scatterer p r o b a b i l i t y density function p ( f s , f t ) = p ( r s ) p ( r t | r s ) (5.1) S p e c i f i c a l l y , a two-scatterer c o n d i t i o n a l p r o b a b i l i t y density function of the form p ( r t | f g ) = h ( | r s - f j ) (5.2) ( i . e . , a function of the.separation of two scatterers) has been pursued such that h reduces to the case of a "rare gas" and a " c r y s t a l " i n the appropriate l i m i t s . This approach i s suited to the usual s c a t t e r i n g theories based on a "dishonest" method i n which only a two-scatterer p r o b a b i l i t y density function i s required. For the case of a "one-dimensional l i q u i d " , a s u i t a b l e h e x i s t s and has been applied to s c a t t e r i n g by one-dimensionally random d i s t r i b u t i o n s of c y l i n d e r s . 4 8 This function, however, seems l i m i t e d to the case of an i n f i n i t e number of s c a t t e r e r s i n the development of approximate s c a t t e r i n g theory. For three-dimensional s c a t t e r e r d i s t r i b u t i o n s , there e x i s t no e x p l i c i t forms of h that cover the f u l l range from "rare gas" to any one of the appropriate " c r y s t a l s " . Another t h e o r e t i c a l approach to the problem has been to consider a "two-phase" system whose population of s c a t t e r e r s i s divided between a "gas-l i k e " phase and a " c r y s t a l - l i k e " phase. 2 5 The scatterers i n both phases contribute to the coherent f i e l d , but only those i n the gas phase contribute to the incoherent f i e l d . A t h i r d more h e u r i s t i c approach has been to assume uniform-randomness for more dense sc a t t e r e r d i s t r i b u t i o n s and use an average density p based on the volume a v a i l a b l e to the s c a t t e r e r s ( i . e . , volume of the containing region less the volume occupied by the f i n i t e - s i z e s c a t t e r e r s ) . This approach, discussed i n Appendix A (section A.2), has been used by Twersky fo r h i s mixed-space theory for the coherent f i e l d . In the present work, the t h e o r e t i c a l problem r e q u i r i n g a s u i t a b l e p r o b a b i l i t y density function for more dense s c a t t e r e r d i s t r i b u t i o n s has been ' bypassed. In order that the s c a t t e r i n g c h a r a c t e r i s t i c s of dense d i s t r i b u t i o n s of f i n i t e - s i z e planar sca t t e r e r s might be i n v e s t i g a t e d , a computer simulation technique has been used to generate an appropriate one-dimensional d i s t r i b u t i o n ranging between the l i m i t s of uniform-randomness and " p e r i o d i c i t y " ( i . e . , p e r i o d i c a l l y - p o s i t i o n e d s c a t t e r e r s ) . Such an approach i s not l i m i t e d to the one-dimensional model; s i m i l a r techniques could be used i n Monte Carlo simulation studies with more complex mathematical models or with p h y s i c a l models of the type discussed i n Chapter 6. Two s i m i l a r methods have been developed i n t h i s work f o r generating a s u i t a b l e one-dimensional d i s t r i b u t i o n of s c a t t e r e r s ranging between the l i m i t s of uniform-randomness and p e r i o d i c i t y . These methods are e s s e n t i a l l y " r e j e c t i o n " t e c h n i q u e s ; 4 7 however, random numbers are generated to conform to a p h y s i c a l requirement rather than a t h e o r e t i c a l d i s t r i b u t i o n . The requirement i s that no s c a t t e r e r s be allowed to approach one another more 84 c l o s e l y than a distance e between t h e i r centers. This parameter may be the p h y s i c a l width of the s c a t t e r e r or some hypothetical "distance of c l o s e s t approach". The methods of generating the d i s t r i b u t i o n are discussed i n section 5.2 and r e s u l t s given which i l l u s t r a t e the type of d i s t r i b u t i o n generated. In section 5.3 r e s u l t s are given f o r a v a r i a t i o n of the d i s t r i b u t i o n parameter e between the l i m i t s of uniform-randomness and p e r i o d i c i t y with the s c a t t e r i n g parameters remaining f i x e d . C r i t e r i a f o r the v a l i d i t y of assuming the planar s c a t t e r e r s to be uniformly d i s t r i b u t e d are presented, based on the average density per wavelength and the f r a c t i o n a l "volume" 3 0 = Ne/(d + e) occupied. In section 5.4 a comparison i s made between numerical r e s u l t s obtained from Twersky's mixed-space theory and simulation r e s u l t s f o r d i s t r i b u t i o n s of f i n i t e - w i d t h planar s c a t t e r e r s . A general discussion and summary of r e s u l t s i s given i n section 5.5. 5.2 Methods of Generating the D i s t r i b u t i o n The uniform random number generator provided the basis f o r the two methods used to generate a non-uniform d i s t r i b u t i o n with the desired l i m i t s of uniform-randomness and p e r i o d i c i t y . The i n i t i a l steps of the procedure were: (i) N uniformly-random numbers were generated on the unit i n t e r v a l corresponding to the normalized positions of the s c a t t e r e r s ( i . e . , z^/d, z 2/d,...,z^/d). These numbers were placed i n an "array" of computer memory i n the sequence of generation; i . e . , f o r an array A(I) with I = 1,2,...,N, the f i r s t number occupied p o s i t i o n 1, the second, p o s i t i o n 2, etc. ( i i ) The N random numbers were then sorted within memory so that they occupied the array A(I) i n order of s i z e ; i . e . , i n the sequence of the ordered-p o s i t i o n s z | , , . . . , z ^ . Once these i n i t i a l steps had been completed the p h y s i c a l requirement that no 85 adjacent p a i r of numbers be closer than e/d, the normalized distance of close s t approach, was applied. The two methods developed d i f f e r i n the manner i n which numbers which did not s a t i s f y t h i s requirement were rejected and new ones generated. 5.2.1 Method A In t h i s method a new random number was generated a f t e r each r e j e c t i o n . The steps of the procedure were as follows: (i) The array A(I) was scanned beginning at p o s i t i o n 1 u n t i l an adjacent p a i r of numbers were found to be closer than e/d. Depending on the value of the previous random number generated, e i t h e r the larger or the smaller of the p a i r was rejected. I f the previous number generated was less than 0.5, the smaller number of the p a i r was rejected; i f i t was 0.5 or greater, then the lar g e r was rejected. Thus, both numbers were rejected with equal p r o b a b i l i t y . ( i i ) For each random number rejected a new one was generated to take i t s place i n the array. This new number was then merged with the remaining N - 1 by an interchange procedure to place i t i n the correct order of s i z e . ( i i i ) Steps ( i ) and ( i i ) were then repeated with a change i n the scanning d i r e c t i o n of step ( i ) , scanning beginning at element N of the array rather than at element 1. This process of r e j e c t i n g and generating one number at a time, with scanning of the array beginning a l t e r n a t e l y with elements 1 and N, was repeated u n t i l a s u i t a b l e configuration of N numbers was obtained, with no adjacent p a i r c l o s e r than e/d. 5.2.2 Method B In t h i s method the e n t i r e array was scanned with one of each p a i r of numbers breaking the required condition being simultaneously rejected and replaced by a new one. The steps of t h i s second procedure were as follows: ( i ) The array A(I) was scanned beginning with element 1 and a l l pa i r s of numbers not s a t i s f y i n g the distance-of-closest-approach c r i t e r i o n were recorded. ( i i ) New random numbers were generated corresponding to the number of unacceptable p a i r s . These were stored i n another array B(J), with J = 1,...,L (L being the number of unacceptable p a i r s ) . ( i i i ) Depending upon the value of the new random number i n the storage l o c a t i o n J of B(J), e i t h e r the smaller or the larger of the J-th p a i r of unacceptable numbers i n A(I) was rejected. The equa l - p r o b a b i l i t y r e j e c t i o n r u l e of method A was again used. In the event that two unacceptable number-pa i r s were adjacent to one another ( i . e . , a sequence of three numbers) and the one rejected from each p a i r was common ( i . e . , the center number of the sequence), one less number from the array B(J) was required. (iv) The required numbers from the array B(J) were then sorted i n order of s i z e and merged with those not rejected from A ( I ) . (v) The preceding steps were repeated u n t i l a s u i t a b l e configuration of N numbers was obtained. 5.2.3 Results The one-scatterer normalized p r o b a b i l i t y density function p(z g/d) = p(z g/d)d/N was determined f o r both methods A and B f o r several values of N and distances of cl o s e s t approach e/d. Plots of th i s function f o r three values of e/d are given i n figures 5.1 and 5.2 f o r N = 4 and N = 5 res p e c t i v e l y The p e r i o d i c p o s i t i o n of each s c a t t e r e r and the "excluded region" surrounding i t are shown at the top of each graph f o r comparison with the p r o b a b i l i t y density curves determined. The graphs were obtained by drawing smooth curves through the experimental points r e s u l t i n g from 20,000 sample configurations and f i f t y histogram i n t e r v a l s . As seen from figures 5.1 and 5.2, as e/d i s increased the p r o b a b i l i t y density curves become more highly peaked. For N = 5 the peaks of the curves fo r both methods correspond c l o s e l y to the p e r i o d i c p o s i t i o n s , but for N = 4, only f o r method B. For N = 4 and low values of e/d the curves f o r method A 0 z s/d z s / d (b) e/d = 1/6 (B 0 = 0.572) (c) e/d = 2/9 (B 0 = 0.728) Figure 5.1 One-Scatterer Normalized P r o b a b i l i t y Density Curves f o r N = 4 Method A, Method B oo 89 contain an extra peak at the center of the d i s t r i b u t i o n . This center peak, however, gradually disappears as e/d i s increased, leaving the remaining peaks to correspond c l o s e l y to the p e r i o d i c p o s i t i o n s . Both methods give a d i s t r i -bution symmetrical about the center of the slab region containing the s c a t t e r e r s . This was achieved i n method A by the procedure of a l t e r n a t e l y scanning the array i n opposite d i r e c t i o n s to detect u n s a t i s f a c t o r y p a i r s of numbers. (An i n i t i a l test of the method with scanning i n only one d i r e c t i o n showed the d i s t r i b u t i o n to become i n c r e a s i n g l y non-symmetric for increasing e/d.) Similar r e s u l t s were obtained f o r higher values of N. Other differences i n the one-scatterer p r o b a b i l i t y density curves fo r the two methods are apparent. The peaks corresponding to the outermost sc a t t e r e r s are higher f o r method B than method A, and the heights of the peaks fo r method B gradually diminish towards the center of the d i s t r i b u t i o n . Such differences are to be expected, however, because of the d i f f e r e n t manner i n which the d i s t r i b u t i o n s are generated. In method A the random number r e j e c t i o n process proceeds gradually from the outer ends of the array towards the center, whereas i n method B i t i s applied "uniformly" over the e n t i r e array. Although some differences i n the one-scatterer p r o b a b i l i t y d e n s i t i e s generated by the two methods are evident f o r mid-range values of e, these dif f e r e n c e s of course disappear f o r e -> 0 and i n p r i n c i p l e must disappear f o r e -> d/(N - 1). Towards the p e r i o d i c l i m i t , the peaks of the one-scatterer d i s t r i b u t i o n s generated by both methods must separate e n t i r e l y f o r d - Ne < 0 or e q u i v a l e n t l y 3 Q = Ne/(d + e) > N/(N + 1), providing c e r t a i n regions i n which no s c a t t e r e r can be located. In the p e r i o d i c l i m i t , the one-scatterer p r o b a b i l i t y density functions must be of the form 1 N i - 1 \ p ( z j = - V 6 z q d ( s = l , . . . , N ) (5.3) s N ' N - 1 ' where 6 i s the Dirac d e l t a function. Although both methods can i n p r i n c i p l e generate a d i s t r i b u t i o n approaching the p e r i o d i c l i m i t , i n p r a c t i c e they are s u i t a b l e only i f the f r a c t i o n a l "volume" 3 Q i s below a c e r t a i n value. For N = 10, the computer time involved increased sharply f o r 3 Q > 0.65 because of the great number of r e j e c t i o n s required. There appeared, furthermore, to be a l i m i t beyond which no s u i t a b l e configuration could be obtained with the a v a i l a b l e generator. This i s considered possible because the generator has a f i n i t e population of numbers. 4^-' 4 2 For N .= 10 t h i s l i m i t appeared to be about 3 Q = 0.8 for the IBM 7044 computer generator RAND. The l i m i t appeared furthermore to decrease with increasing N. Method A usually required the r e j e c t i o n of more random numbers than method B before a s u i t a b l e configuration could be obtained, although method A, being a somewhat simpler procedure, was always f a s t e r . Such d i f f e r e n c e s , however, were not large and i n the factors discussed i n t h i s s e ction the two methods are considered to be approximately equivalent. Test runs of the two methods for the type of r e s u l t s of the following sections furthermore showed them to d i f f e r very l i t t l e f o r the average f i e l d functions investigated. The r e s u l t s given are therefore based on method A. The disadvantage that e x p l i c i t r e s u l t s cannot be obtained for 0.8 < 3 0 < 1.0 i s not serious because the behavior i n t h i s i n t e r v a l can be i n f e r r e d from the r e s u l t s f o r mid-range values of 3 Q and the r e s u l t f or the p e r i o d i c l i m i t i t s e l f ( i . e . , 3 Q = 1). Methods of e l i m i n a t i n g t h i s disadvantage and generating a s u i t a b l e d i s t r i b u t i o n on a more t h e o r e t i c a l basis are discussed i n section 5.5. 5.3 V a r i a t i o n of the D i s t r i b u t i o n Between the Limits of Uniform-Randomness  and P e r i o d i c i t y f o r Fixed Scattering Parameters In t h i s s e c t i o n "exact" simulation r e s u l t s are given which show the e f f e c t of a v a r i a t i o n of the d i s t r i b u t i o n between the l i m i t s of uniform-91 randomness and p e r i o d i c i t y on the coherent f i e l d and average incoherent i n t e n s i t y . Based on these r e s u l t s , c r i t e r i a are developed for the assumption of u n i f o r m l y - d i s t r i b u t e d " f i n i t e - w i d t h " planar sca t t e r e r s ( i . e . , e # 0) to be v a l i d . For convenient comparison with the r e s u l t s of Chapter 4, the f i x e d s c a t t e r i n g parameters N = 10, e r = 2.0, and w^i =0.1 are again used. A l l simulation r e s u l t s are f o r 1,000 sample configurations. Shown i n fi g u r e 5.3 are the phase and i n t e n s i t y d ^ - v a r i a t i o n curves of the transmitted f i e l d f o r a p e r i o d i c array of the s c a t t e r e r s . These curves are of importance i n .the following discussion of r e s u l t s f o r the " p e r i o d i c i t y -weighted" random d i s t r i b u t i o n whose s c a t t e r i n g behavior becomes " p e r i o d i c " as e -»- d/(N - 1) or equivalently $ Q = Ne/(d + e) -> 1. The well-known resonance behavior which occurs at p e r i o d i c i n t e r v a l s i n d A i s s p e c i f i e d on the x-axis of the graphs by the index n^ which takes the values n R = 0,1,2,.... For the present parameters the resonance i n t e r v a l i n d A i s 4.5. This gives a separation between sc a t t e r e r s at resonance of approximately n-g\/2, a s l i g h t deviation from t h i s value occurring because of the d i f f e r e n t wavelength A' i n the s c a t t e r e r material. The f i r s t set of r e s u l t s i l l u s t r a t e the e f f e c t of e v a r i a t i o n on the average f i e l d functions for a number of widely-spaced slab-region widths d^. Curves f o r the coherent phase and i n t e n s i t y of the transmitted f i e l d are given i n f igures 5.4a and 5.4b; curves f o r the average incoherent i n t e n s i t y of the r e f l e c t e d f i e l d are given i n f i g u r e 5.4c. The f i v e values of d^ chosen to display the r e s u l t s are a l l approximately mid-way between adjacent resonance values and give i d e n t i c a l p e r i o d i c l i m i t s f o r Ce3 a. Smooth curves have been drawn through a l l "experimental" points except those f o r d^ = 97 where the act u a l curves are too o s c i l l a t o r y i n nature to display accurately from the present r e s u l t s . The points are included f o r d A = 2.5 and d A = 7. Curves f o r ( l 2 ) of the transmitted f i e l d are not shown because they are s i m i l a r i n form -a 130° 125C 120° 1151 93 . \ ^ o \ ^ 1 • v . o 7^*0- o/cN o \ T • — y * j~x in i r \ Y 0.2 0.4 „ 0.6 e 0 0.8 1.0 (a) -a versus 3 0 - Transmitted F i e l d 1.0 0.9 0.8 0.7 0.6 0.5 ^ — 1 - ^ » ' X • 7 / • , { / / ^ — ""^ > & ° \ •• \ \-* / / / / J 0.2 0.4 0.6 So 0.8 (b) C 2 versus 6 Q - Transmitted F i e l d 0.4 <I 2> 0.2 0.1 0.2 0.4 „ 0.6 0.8 (c) ( l 2 ) versus 6 Q - Reflected F i e l d 1.0 < > "> - - * ^ • / V \ \ \ -j n (-) - ^ ^ - l \J \J t\ N — y \ \ ^ \ ~^^"X 1.0 Figure 5.4 Dependence of Average F i e l d Functions on B Q for Various Values of d x . N = 10, wx, = 0.1, e r = 2.0; • • d x = 2.5, A A d x = 7, d x = 11.5, d x = 16, o d x = 97, — Asymptotic Values for p x -> 0, x P e r i o d i c i t y Values to those f o r the r e f l e c t e d f i e l d . The section of each curve for 3 Q 1 has been in t e r p o l a t e d from the a v a i l a b l e r e s u l t s , the beginning of the section being ind i c a t e d by a "dotted" break i n the curve. The asymptotic behavior of the in t e r p o l a t e d sections i n the p e r i o d i c l i m i t has been assumed on the basis of the occurrence of the same behavior f o r d A -> 0. As seen from fi g u r e 5.4, the given curves are very o s c i l l a t o r y i n character. The number of primary o s c i l l a t i o n s over the i n t e r v a l 0 < 3 0 ^ 1 increases for increasing d A (or decreasing p^) but the o s c i l l a t i o n s decrease i n magnitude f o r increasing d^. For the p e r i o d i c spacing between scat t e r e r s i n the i n t e r v a l dA nR + 1 (5.4) — < < 2 N - 1 2 the number of apparent r e l a t i v e maxima or minima i n the curves (not in c l u d i n g the p e r i o d i c value) i s given by the "resonance index" n^. The r e s u l t s f or d^ = 2.5 i n d i c a t e the possible presence of secondary o s c i l l a t i o n s i n the curves, although more accurate r e s u l t s are necessary f o r t h i s to be established. Such behavior i n the periodicity-weighted d i s t r i b u t i o n would seem possible because of the secondary o s c i l l a t i o n s i n the curves of f i g u r e 5.3 f o r a p e r i o d i c array. The r e s u l t s of f i g u r e 5.4 show that as n R increases the d i s t r i b u t i o n does not begin to e x h i b i t " p e r i o d i c " behavior u n t i l i n c r e a s i n g l y higher values of 3 Q. For d^ < (N - l ) / 2 the curves proceed d i r e c t l y to the p e r i o d i c l i m i t s as 3 Q increases; f o r d^ >> (N - l ) / 2 (e.g., d^ = 97) the d i s t r i b u t i o n behaves as though the sca t t e r e r s were uniformly d i s t r i b u t e d up to a very high value of 3 0. The reasons for the observed behavior can be explained i n terms of the theory of random phasor sums discussed i n section 2.8. It i s u s e f u l to write the transmitted f i e l d as N T e j x = A o e J 8 o + £ A s e J e s = A 0 e J e o + T'eJ T' (5.5) s=i where the constant phasor A QeJ^o = ( l + g + ) ^ i s the Z-O-B-S contr i b u t i o n to T e J T and the random phasor A geJ^s i s the m u l t i p l e - s c a t t e r i n g contribution from s c a t t e r e r s composed of a l l other even O-B-S. (The values of QQ and A 2 for the present parameters and that f o r ( i 2 ) as p A -> 0 are shown on the scales of the graphs i n fi g u r e 5.4.) Then, . C e J a = <TeJ T) = A 0eJ°o + C'eJ 0 1' (5.6) where N C ' e J a = < T ' e J T ) = T <A seJ 6s> . (5.7) i _ j x fa ' s=l Thus, maxima i n the curves f o r C 2 occur when C'e^01 i s i n phase with A 0eJ^° ( i . e . , a = a' = 8 Q ) ; minima occur when C'e^01 i s 180° out of phase ( i . e . , a = a' - TT = 8 ). The energy l o s t from the coherent transmitted f i e l d when C 2 i s minimum reappears i n the incoherent f i e l d s . The theory of section 2.8 showed that the random components of the 8 g represent approximately double the " e l e c t r i c a l lengths" between adjacent s c a t t e r e r s . P h y s i c a l reasoning then requires that f o r e = 0, °~x' " ° 8 S ~ 4 7 r d A / ( N - 1) C 5* 8)' S i m i l a r l y , f o r e # 0, Thus, f o r a p e r i o d i c spacing between scat t e r e r s i n the i n t e r v a l given by (5.4), 2ir n R ( l - B Q) < ajt < 2 i r ( n R + l ) ( l - g Q) (5.10) I f a T i >> 2TT, the basic phase cycle equivalent of x' i s e f f e c t i v e l y uniformly o T t - 4ir d i s t r i b u t e d , the random component T'eJ 1 of the t o t a l f i e l d then completely incoherent ( i . e . , C'eJ a = 0), and therefore Ce3 a = (1 + g^N. The f act that t h i s s i t u a t i o n occurs for large n R ( l - 8 ) agrees with the r e s u l t s of fi g u r e 5. The o s c i l l a t o r y behavior of the curves develops because, as B Q varies between zero and one, the weighting of the equivalent d i s t r i b u t i o n of T' towards d i f f e r e n t sections of the basic phase cycle varies i n a p e r i o d i c manner causing the phasor C'e^01 to p e r i o d i c a l l y rotate. For l a r g e r n R , aT, varies over a l a r g e r number of basic phase cycles as 8 0 v a r i e s between zero and one and the curves therefore contain more o s c i l l a t i o n s as shown. Furthermore, as n R ( l - 8 0 ) i s decreased (either by a decrease i n n R or an increase i n 8 0 ) , the deviation of T' from a uniform d i s t r i b u t i o n on the basic phase cycle must increase, causing the magnitude of the o s c i l l a t i o n s to become l a r g e r . This i s also apparent from the curves of figure 5.4. Similar reasoning can be used to explain an o s c i l l a t o r y behavior i n the coherent r e f l e c t e d f i e l d f o r a v a r i a t i o n of 8 0 i n the i n t e r v a l 0 < B 0 < 1. Curves are not given because of the inaccuracy of simulation r e s u l t s f o r the small magnitudes involved and because the equivalent behavior for a v a r i a t i o n of d^ was i l l u s t r a t e d i n Chapter 4. For p e r i o d i c spacing of the sca t t e r e r s i n the neighbourhood of a resonance condition, the s c a t t e r i n g behavior of the d i s t r i b u t i o n f o r B Q -* 1 i s highly v a r i a b l e f o r s l i g h t changes i n d^. Figure 5.5 i l l u s t r a t e s the e f f e c t i n a s e r i e s of curves f o r C 2 of the transmitted f i e l d and ( i 2 ) of the r e f l e c t e d f i e l d i n the neighbourhood of the second resonance ( i . e . , n R = 1). Actual resonance occurs at about d^ = 4.2. These curves also i l l u s t r a t e the beginning of the large-amplitude o s c i l l a t o r y behavior a f t e r the second resonance. The observed d i v e r s i o n of a large f r a c t i o n of energy to the incoherent r e f l e c t e d f i e l d j u s t above resonance (e.g., at d^ = 4.6) i s of i n t e r e s t although a reason f o r t h i s behavior i s not immediately evident. A s i m i l a r 1.0 O l : 1 1 1 1 1 0 0.2 0.4 0.6 0.8 1.0 (a) C 2 versus 8 - Transmitted F i e l d 0.6 <I 2> \ / / ft \ ' y \ Wt ~~ • m . ^ 0 0.2 0.4 0.6 0.8 1.0 (b) ( l 2 ) versus 3 0 - Reflected F i e l d Figure 5.5 B 0 ~ V a r i a t i o n Curves for Values of d x i n the Neighbourhood of Resonance at P e r i o d i c i t y . N = 10, w x i = 0.1, e r = 2.0; d x = 3, d x = 3.8, d x = 4.2, d x = 4.6, d x = 5.4 e f f e c t occurs f o r the incoherent transmitted f i e l d . The r e s u l t s of t h i s section c l e a r l y i n d i c a t e when i t i s v a l i d to assume that a d i s t r i b u t i o n of planar sca t t e r e r s of f i n i t e width i s uniform. For high p A the average spacing between the scatterers must be much greater than the minimum spacing. Since the average spacing i s approximately equal to the spacing d^/(N - 1) at p e r i o d i c i t y , this condition can be wri t t e n d A dx 1 N - 1 N p^ or » ex (5.11) 8 n « 1 (5.12) T y p i c a l l y , as indicated by the r e s u l t s of figu r e 5.4, for 0.4 < p x < 4, i t i s required that 8 0 < 0.1 f o r the assumption of uniform-randomness to be reasonably v a l i d f o r the given s c a t t e r e r s . For p A > 100, as seen by the r e s u l t s of Chapter 4, i t makes l i t t l e d i f f e r e n c e whether the sc a t t e r e r p o s i t i o n s are uniformly-random or p e r i o d i c and thus the value of 3 0 i s not important. For d i s t r i b u t i o n s of low p A the requirement of (5.12) f o r uniform-randomness i s too st r i n g e n t , as i s indic a t e d by the r e s u l t s of figu r e 5.4 for d A = 97. A less stringent condition i s that the standard phase deviation O T T of the random component of the transmitted f i e l d be much greater than 2TT radians, or equivalently from equation (5.9), (1 - 3 0 ) / p x » 1 (5.13) Thus, f o r even a large f r a c t i o n of the containing slab region f i l l e d by the excluded regions of the s c a t t e r e r s , uniform-randomness may be assumed as long as p^ i s small. For d i s t r i b u t i o n s of three-dimensional sca t t e r e r s a c r i t e r i o n s i m i l a r to that of (5.12) has often been used ( i . e . , 3 Q << 3 m, where 3 m i s the maximum 3 Q p h y s i c a l l y achievable). In view of the v a l i d i t y of the less stringent c r i t e r i o n (5.13) for low-p^ d i s t r i b u t i o n s of planar s c a t t e r e r s , the question of the possible v a l i d i t y of a s i m i l a r c r i t e r i o n f or d i s t r i b u t i o n s of three-dimensional s c a t t e r e r s a r i s e s ( i . e . , where i s the average density per cubic wavelength). The v a l i d i t y of such a c r i t e r i o n would seem possible f o r the mu l t i p l e - s c a t t e r i n g contributions to the random component of the f i e l d i n a three-dimensional d i s t r i b u t i o n since even for large f3Q a low average density of s c a t t e r e r s per cubic wavelength would give a large OQS for each con t r i b u t i o n and a r e s u l t i n g uniform d i s t r i b u t i o n f o r the equivalent 0 S on the basic phase cy c l e . As shown i n Appendix C, however, the O Q s for the dominant s i n g l e -s c a t t e r i n g contributions to the forward-scattered f i e l d i n a three-dimensional d i s t r i b u t i o n can be quite small, even f o r a low p per cubic wavelength. Thus, the more stringent c r i t e r i o n appears to be necessary. 5.4 Comparison of Simulation and Mixed-Space Theory Results f o r Planar  Scatterers of F i n i t e Thickness In t h i s s ection a comparison i s made between numerical r e s u l t s f o r the coherent f i e l d obtained from Twersky's mixed-space theory and "exact" r e s u l t s based on the simulation of a non-uniform d i s t r i b u t i o n of planar s c a t t e r e r s of f i n i t e width. For the simulation, the width e of the s c a t t e r e r excluded region i s set equal to the p h y s i c a l width w of the d i e l e c t r i c slab s c a t t e r e r s employed. As discussed i n Appendix A (section A.2), the modified form p = N / ( d + w - Nw) of the average density i s used i n the mixed-space theory and the slab-region width d containing the sca t t e r e r centers i s replaced by d + w, that containing the sc a t t e r e r boundaries. Results are also given fo r the mixed-space coherent r e f l e c t e d f i e l d theory with only the modified p. 5.4.1 The Coherent Transmitted F i e l d Results f o r the coherent transmitted f i e l d are based on a "compression 100 process" s i m i l a r to that used f o r a physical-model d i s t r i b u t i o n of a slab region of s p h e r i c a l s c a t t e r e r s . 4 ^ The width of the planar s c a t t e r e r s and the width of the slab region containing them are f i x e d so that as an increasing number are placed within the region the d i s t r i b u t i o n i s gradually compressed, the s c a t t e r e r s eventually f i l l i n g the region. The parameters have been chosen so that the slab region i s f i l l e d f o r N = 25. The f i r s t set of curves of fi g u r e 5.6 give r e s u l t s f o r a high p x occurring i n the l i m i t of 3 Q = Nw/(d + w) -> 1. The s i n g l e s c a t t e r e r parameters fo r these curves are .again e r = 2.0 and w^i = 0.1. The w^i value and the N = 25 l i m i t f o r 3 Q = 1 were chosen to give the l i m i t i n g r e s u l t C 2 = 1 ( i . e . , since dA + W A ' = 2 ,-")> although t h i s was not necessary for the mixed-space theory to give exact r e s u l t s i n the 3 0 = 1 l i m i t . Shown also for comparison i n fi g u r e 5.6 are simulation r e s u l t s f or u n i f o r m l y - d i s t r i b u t e d i n f i n i t e l y - t h i n s c a t t e r e r s having the same s c a t t e r i n g amplitudes. Both sets of simulation r e s u l t s are based on 1,000 samples. As seen from f i g u r e 5.6, the general trend of the coherent f i e l d behavior f o r increasing N i s approximately predicted by the mixed-space theory. The coherent i n t e n s i t y at f i r s t decreases (as f o r a uniform d i s t r i b u t i o n ) , reaches a minimum, and then increases as the d i s t r i b u t i o n becomes more "ordered" and energy i s diverted from the incoherent f i e l d s . Approximately the same behavior has been measured on the model d i s t r i b u t i o n of s p h e r i c a l s c a t t e r e r s , although the experimental r e s u l t s obtained agree more c l o s e l y with the mixed-space theory r e s u l t s f o r mid-range values of 3 Q. 4^ The discrepancy i n the present r e s u l t s f o r the one-dimensional model i s due to the fa c t that the t o t a l s c a t t e r i n g c ross-section a = -2 Reg + i s s u f f i c i e n t l y large that the asymptotic form ( T ) = e^§+ f o r the mixed-space theory i s not accurate. This was shown i n f i g u r e 4.3 on page 62. As pointed out i n Chapter 4, only c e r t a i n parameters give s u f f i c i e n t l y small a that adequate agreement i n the asymptotic 101 (b) C 2 versus N Figure 5.6 Coherent Transmitted F i e l d Results for High p A . wA, = 0.1, e r = 2.0, d A = 1.697; o Non-Uniform D i s t r i b u t i o n Simulation, — Uniform D i s t r i b u t i o n Simulation, Twersky's Mixed-Space Theory, x P e r i o d i c i t y Values 102 forms can be obtained f o r f i n i t e N (e.g., f o r w^  i << 0.25 or at p e r i o d i c values i n w^i f o r thick s l a b s ) . The s i t u a t i o n may not be as c r i t i c a l for three-dimensional s c a t t e r e r s . In f i g u r e 5.6 the apparently correct l i m i t i n g behavior predicted by the mixed-space theory f o r 8 Q -»•• 1 i s achieved because the scatt e r e r s completely f i l l the containing region and are s u f f i c i e n t l y " t h i n " that g_/g_J_ •+ 1 and n n' (see Appendix A, section A. 2, and Appendix B, section B.2). If the excluded-region width e i s chosen to be la r g e r than w ( i . e . , spaces e x i s t i n g between the sc a t t e r e r boundaries f o r B Q = 1), the mixed-space theory does not give exact r e s u l t s i n the " p e r i o d i c " l i m i t . This s i t u a t i o n i s somewhat equivalent to that occurring f o r d i s t r i b u t i o n s of spheres discussed by Beard et a l . , 4 ^ where the slab region cannot be completely f i l l e d with s c a t t e r i n g material. Exact r e s u l t s i n the " p e r i o d i c " l i m i t are also not achieved i f the l i m i t s g!_/g\. ->- 1 and n -> n 1 f o r B Q -* 1 do not hold. This i s i l l u s t r a t e d by the r e s u l t s of f i g u r e 5.7 f o r low p^. For these r e s u l t s the s c a t t e r e r width of W^T = 6.9 has been chosen to give s u f f i c i e n t l y small a that e^§+ - (1 + g_p^, as shown i n fi g u r e 5.7. The r e s u l t i n g s c a t t e r i n g amplitudes are now g+ = 0.1044 7-104.5° and g_ = 0.2035 /84.Q70, but the value of | l + g+| 2 N remains the same as that f o r w^i = 0.1. Again the " p e r i o d i c " l i m i t f o r C 2 i s unity f o r the chosen parameters. The simulation r e s u l t s of f i g u r e 5.7 follow c l o s e l y the asymptotic curves as obtained from (1 + g + ) N , even f o r r e l a t i v e l y high values of 8 . This, of course, i s i n agreement with the r e s u l t s of section 5.3. Because the present methods of generating the d i s t r i b u t i o n do not allow higher values of 8 Q to be reached, the exact point at which the d i s t r i b u t i o n begins to e x h i b i t marked " p e r i o d i c " behavior i s not known, although i t i s estimated to be i n the neighbourhood of N = 23. As seen from f i g u r e 5.7, the mixed-space theory does 103 150° -a 100° 50° 1 X 0 5 10 15 20 25 N (a) -a versus N 1.0 0.4 0.2 \ ^ - s. 0 5 10 15 20 25 N (b) C 2 versus N Figure 5.7 Coherent Transmitted F i e l d Results f or Low p x . w^i = 6.9 e r = 2.0, d^ = 117.1; o Non-Uniform D i s t r i b u t i o n Simulation, Twersky's Mixed-Space Theory, CeJ a = (1 + g + ) N , CeJ a = e N§+, x P e r i o d i c i t y Values 104 not give the correct " p e r i o d i c " l i m i t although i t gives good agreement with simulation r e s u l t s f or most of the range of 3 0 . Several other sets of numeri-c a l r e s u l t s obtained for the mixed-space theory appear to i n d i c a t e that approximately the correct l i m i t i n g behavior occurs only for " t h i n " slabs. In summary, i t has been shown that the requirement for the "one-dimensional" mixed-space theory to give adequate agreement with t y p i c a l simulation r e s u l t s f or f i n i t e N and low or mid-range values of 3 0 i s that | | << 1 (or a << 1 f o r the sca t t e r e r s used). For exact r e s u l t s i n the l i m i t of " p e r i o d i c i t y " , the s c a t t e r i n g material must completely f i l l the slab region a v a i l a b l e and the requirements g2./g+ 1 a n d n n' as 3Q 1 must be f u l f i l l e d . If these " p e r i o d i c i t y " conditions are not s a t i s f i e d , however, the mixed-space theory can s t i l l adequately describe the d i s t r i b u t i o n behavior f o r low and mid-range values of 3Q i f the forward amplitude condition i s s a t i s f i e d . The required conditions f o r the approximate v a l i d i t y of the mixed-space theory f o r f i n i t e - N d i s t r i b u t i o n s of three-dimensional scatterers are believed to be i d e n t i c a l to those given f o r d i s t r i b u t i o n s of one-dimensional s c a t t e r e r s , except that the forward amplitude condition must be replaced by 12Trpdg(z ,z)/k 2N | << 1 (see section 2.7.1). This forward amplitude condition has previously been implied by Twersky, 2 7 and Beard et a l . 4 ^ have shown that the " p e r i o d i c i t y " conditions need not be s a t i s f i e d for the theory to be approximately v a l i d f o r low and mid-range values of 3 0 . 5.4.2 The Coherent Reflected F i e l d Results f o r the coherent r e f l e c t e d f i e l d can best be shown by a "compression process" i n which the width of the slab region containing a f i x e d number of planar sca t t e r e r s i s gradually decreased. Curves are given i n fig u r e 5.8 for v a r i a b l e d^ with s c a t t e r i n g parameters N = 10, e r = 2.0, and w^i = 0.1. For these parameters, " p e r i o d i c i t y " occurs when d^ i + w i^ = 1 or, to four figures of accuracy, d^ = 0.6364. The simulation r e s u l t s are - 1 0 ~ i - 3 0 - 4 0 •/ / \ V: 1 \ 4 9 qf / V, .- vo. \ t 1 # • i k V /VVi i A f o l l :\ 1 : 1 1 ! i ll • \: V? \ I ' ' ': A o k i \:> A/ ^ \y / •: \ \ / ^  f> /A \w H II II H i i i A :: o V\j \ / / 0.6364 1.0 1.5 , 2.0 2.5 3.0 (b) 10 log C 2 versus dx Figure 5.8 Comparison of Results for the Coherent Reflected F i e l d . N = 10, wA i = 0.1, e r = 2.0; A Non-Uniform D i s t r i b u t i o n Simulation, o Uniform D i s t r i b u t i o n Simulation, Twersky's Mixed-Space Theory with Modified p and d, Mixed-Space Theory with Modified p, Periodic Array Theory 106 based on 4,000 sample configurations. Curves are given for Twersky's mixed-space theory with and without d being replaced by d + w. Since a replacement of d with d + w i n the mixed-space equation for ( R ) (termed "modified d" i n fi g u r e 5.8) changes the phase reference with respect to that used f o r the simulation, an a d d i t i o n a l m u l t i p l i e r e ~ l k w has been inserted i n the equation to allow comparison of r e s u l t s f o r the coherent phase a. As seen by a comparison of r e s u l t s f or the modified-d form of the mixed-space theory with r e s u l t s f or the theory f o r a p e r i o d i c array, the mixed-space theory appears to have the correct l i m i t i n g behavior as B Q -> 1 ( i . e . , d^ -> 0.6364). (Results f o r the mixed-space theory were obtained down to d^ = 0.644, beyond which the i t e r a t i o n method employed to solve the mixed-space theory equations did not converge.) The mixed-space theory with unmodified d does not, of course, have the correct l i m i t i n g behavior f o r 8 Q 1 but reduces to the form displayed i n Chapter 4 (figure 4.8, page 70) for B q -> 0 since p = N/(d + w - Nw) -> N/d as d^ -> °°. Although both forms of the theory displayed i n f i g u r e 5.8 give Ce^a -»- 0 as d^ ->- °°, a displacement i n the curves for both a and C 2 must remain even for large d^. As expected, the simulation r e s u l t s given i n f i g u r e 5.8 for the periodicity^weighted d i s t r i b u t i o n d i f f e r i n c r e a s i n g l y from those f o r the uniform d i s t r i b u t i o n f o r decreasing d^, approaching the r e s u l t s f o r a p e r i o d i c array as B Q ->• 1. Each form of the mixed-space theory gives r e s u l t s approaching the simulation r e s u l t s i n one of the two l i m i t s . I t i s i n t e r e s t i n g that the form of the theory with only modified p gives almost exact agreement with the simulation r e s u l t s f o r the uniform d i s t r i b u t i o n ; a reason for t h i s i s not immediately evident. The present work i s believed to be the f i r s t comparison of Twersky's mixed-space theory f o r (R) with "experimental" r e s u l t s . Further experimental 107 and t h e o r e t i c a l research with t h i s theory and that f o r the coherent trans-mitted f i e l d applied to other s c a t t e r e r d i s t r i b u t i o n s ( p a r t i c u l a r l y three-dimensional d i s t r i b u t i o n s ) i s required. 5.5 Summary and General Discussion of Results 5.5.1 Summary The main contributions of t h i s chapter may be summarized as follows: ( i ) Two simulation methods of generating non-uniform one-dimensional s c a t t e r e r d i s t r i b u t i o n s weighted towards p e r i o d i c i t y have been developed. ( i i ) A study has been made of the v a r i a t i o n of a planar s c a t t e r e r d i s t r i b u t i o n between the l i m i t s of uniform-randomness and p e r i o d i c i t y . C r i t e r i a based on the average density of scatt e r e r s per wavelength and the f r a c t i o n a l "volume" occupied have been given f o r the v a l i d i t y of assuming planar sca t t e r e r s of f i n i t e width to be uniformly-random. ( i i i ) The one-dimensional form of Twersky's mixed-space theory f o r the coherent f i e l d has been investigated and the requirements f o r i t s approximate v a l i d i t y c l e a r l y outlined. The requirements necessary f o r the v a l i d i t y of the three-dimensional form of the theory have been considered. 5.5.2 General Discussion The r e j e c t i o n methods developed f o r generating a s u i t a b l e non-uniform d i s t r i b u t i o n of N random va r i a b l e s s u f f e r the same p r a c t i c a l disadvantage of most r e j e c t i o n methods f o r si n g l e random variables i n that they are wasteful of computation time. In ad d i t i o n , they do not generate the s c a t t e r e r p o s i t i o n v a r i a b l e s from known t h e o r e t i c a l d i s t r i b u t i o n s and cannot give r e s u l t s f o r a f r a c t i o n a l "volume" approaching unity. These l a s t two disadvantages (at least) could be overcome by allowing a random number of scatterers N within the containing slab region and generating the spaces between t h e i r boundaries rather than t h e i r p o s i t i o n s . A s u i t a b l e p r o b a b i l i t y density function f o r the 108 spaces £^ between the s c a t t e r e r boundaries i s the exponential function PC?.) = pe-P^i, p = < N ^ ( i = 1,2,...,N + 1) (5.14) x d - (N)e where p i s e s s e n t i a l l y the modified average density used i n th i s chapter f o r fixed N, e i s the minimum spacing between s c a t t e r e r s , and d i s the width of the slab region containing the boundaries of the random number of scatterers ( i . e . , d i s equivalent to the d + e used i n t h i s chapter). The exponential function (5.14) i s the "adjacent-scatterer" form of the more general two-sc a t t e r e r conditional, p r o b a b i l i t y density function used by Twersky. 4^ For e •+ 0 i t reduces to the form of equation (4.1) f o r i n f i n i t e l y - t h i n s catterers and N becomes Poisson d i s t r i b u t e d . For e ->• d/(N), the scat t e r e r s become p e r i o d i c a l l y positioned and N i s no longer random. The spaces £^ between the sc a t t e r e r boundaries can be generated using the transformation of equation (4.2). The generation of a random-N d i s t r i b u t i o n i n t h i s manner should give r e s u l t s d i f f e r i n g l i t t l e from those presented i n t h i s chapter. The t h e o r e t i c a l problem of determining a s u i t a b l e two-scatterer p r o b a b i l i t y density function of fixed-N s c a t t e r e r configurations f o r the, basis of an approximate s c a t t e r i n g theory, however, s t i l l remains. T h e o r e t i c a l work dire c t e d to obtaining such a function and applying i t to the O-B-S approximations f o r the f i e l d developed i n the present work would be of i n t e r e s t . 109 6. EXPI21IMENTAL INVESTIGATION 6.1 Introduction In t h i s chapter a p h y s i c a l model of a random medium of d i s c r e t e s c a t t e r e r s f o r possible use i n d e t a i l e d s c a t t e r i n g studies i s proposed and the r e s u l t s of i n i t i a l experiments to determine the s u i t a b i l i t y of t h i s model are given. The model consists of three-dimensional scat t e r e r s randomly positioned within a slab region according to s t a t i s t i c s generated by computer from a desired d i s t r i b u t i o n . - A narrow microwave beam scans this slab region and the r e s u l t i n g f l u c t u a t i n g f i e l d i s measured at a point outside the region and s t a t i s t i c a l l y analyzed. I n i t i a l measurements of the forward-scattered f i e l d from a "uniformly-random" configuration of one-half inch diameter polyethylene spheres have been made at 0.86 centimeter wavelength. A seri e s of experiments on a s i m i l a r p h y s i c a l model have previously been c a r r i e d out by a group of researchers at Sylvania E l e c t r o n i c Defence. Laboratories. 5> 4^~54 As i l l u s t r a t e d by the Sylvania group, experiments on a c o n t r o l l e d - d i s t r i b u t i o n model, unlike experiments on uncontrolled n a t u r a l l y -occurring s c a t t e r e r d i s t r i b u t i o n s , provide the p h y s i c a l and s t a t i s t i c a l c h a r a c t e r i s t i c s of the d i s t r i b u t i o n i n add i t i o n to the s c a t t e r i n g data of i n t e r e s t . These known p h y s i c a l and s t a t i s t i c a l c h a r a c t e r i s t i c s can be used i n various approximate s c a t t e r i n g theories and the t h e o r e t i c a l p r e d i c t i o n s compared with s c a t t e r i n g measurements on the same d i s t r i b u t i o n to determine t h e i r v a l i d i t y . Thus, the advantages of experiments on a p h y s i c a l model con-structed and c o n t r o l l e d to conform to a mathematical model are s i m i l a r to those gained by Monte Carlo computer simulation with a mathematical model. Such experiments have the a d d i t i o n a l advantages, however, that more complex models can be studied, r e a l - l i f e antenna beams used, and new measurement techniques investigated. 110 The main f e a t u r e of the p h y s i c a l model presented i n t h i s chapter i s ' that the s t a t i s t i c s of the s c a t t e r e r p o s i t i o n s are d i r e c t l y c o n t r o l l e d as i n a complete Monte Carlo computer s i m u l a t i o n . Thus, the s c a t t e r e r s t a t i s t i c s are known f rom the begi n n i n g , u n l i k e those of the S y l v a n i a model where they are c o n t r o l l e d i n d i r e c t l y by a p h y s i c a l process^ and monitored by means of movie f i l m s or a method r e q u i r i n g i n i t i a l knowledge of the s c a t t e r i n g medium.5>^9,50 The primary advantage of such d i r e c t c o n t r o l , however, i s that i t presents the . p o s s i b i l i t y of experimental s t u d i e s on other than "uniform" d i s t r i b u t i o n s . The c o n s t r u c t i o n d e t a i l s of the present p h y s i c a l model are o u t l i n e d i n s e c t i o n 6.2 and a more thorough a n a l y s i s of i t s advantages and disadvantages i s g iven. The s c a t t e r i n g range, antenna c h a r a c t e r i s t i c s , and scanning device are described i n s e c t i o n 6.3. The experimental apparatus, measurement techniques, and data processing methods are discussed i n s e c t i o n 6.4. The r e s u l t s of the i n i t i a l experiments to i n v e s t i g a t e c e r t a i n aspects of the model s u i t a b i l i t y are presented and discussed i n s e c t i o n 6.5. 6.2 The P h y s i c a l Model The present p h y s i c a l model c o n s i s t s of s p h e r i c a l s c a t t e r e r s (e.g., polyethylene b a l l s ) p o s i t i o n e d a t "uniformly-random" d i s c r e t e l o c a t i o n s w i t h i n a slab r e g i o n of a r e l a t i v e l y transparent support-medium constructed of l a y e r s of po l y s t y r e n e foam. A narrow millimeter-wave beam i l l u m i n a t e s a sm a l l volume r e g i o n of the f i x e d c o n f i g u r a t i o n of s c a t t e r e r s and the r e s u l t a n t tThe S y l v a n i a model c o n s i s t s of a s l a b - r e g i o n Styrofoam container whose bottom and top are g r i d s which a l l o w f o r the passage of t u r b u l e n t a i r streams. For low average d e n s i t i e s of l i g h t - w e i g h t s c a t t e r e r s ( u s u a l l y Styrofoam spheres), a r e l a t i v e l y uniform d i s t r i b u t i o n i s developed by a system of blowers, t u r b u l e n c e - c r e a t i n g wedges, and c o l l i s i o n processes. The e r g o d i c i t y of the process allows a d i r e c t comparison between the time averages of f i e l d q u a n t i t i e s measured and approximate t h e o r e t i c a l estimates of the corresponding ensemble averages. s c a t t e r i n g process produces a random f i e l d i n the space both i n s i d e and out-side the slab region containing the s c a t t e r e r s . A procedure of moving the slab region i n front of a transmitting antenna i n a d i r e c t i o n perpendicular to that of the incident beam therefore causes a random f l u c t u a t i n g f i e l d at the l o c a t i o n of a r e c e i v i n g antenna. The desired components of the random f i e l d produced by t h i s "scanning" process are measured and sampled at d i s c r e t e distance i n t e r v a l s . A processing of these sampled signals by standard d i g i t a l averaging methods r e s u l t s i n estimates of the ensemble averages of the desired f i e l d q u a n t i t i e s . A s i m p l i f i e d diagram of the s c a t t e r i n g geometry used f o r the experiments described i n t h i s chapter i s given i n fig u r e 6.1 (see section 6.3 for distances of antennas from the s c a t t e r i n g region). Although measurements re c e i v i n g antenna scanning d i r e c t i o n s 10.5" transmitting antenna slab region occupied by spheres Figure 6.1 S i m p l i f i e d Diagram of Scattering Geometry were made of the forward-diffracted f i e l d on the axis of the transmitting antenna, s i m i l a r measurements may be obtained f o r the forward f i e l d i n various d i r e c t i o n s o f f the axis and f o r the back-scattered f i e l d . The transverse dimensions of the d i s t r i b u t i o n (56" x 30") were decided on the basis of the width of the anechoic chamber a v a i l a b l e and on the v e r t i c a l movement of the 112 device f o r scanning the region (see section 6.3). The 10.5 inch dimension p a r a l l e l to the beam axis i s approximately the same as that of the Sylvania model.+ In order f o r the spheres to be placed i n p o s i t i o n within the slab region, the support-medium was divided into twenty-one v e r t i c a l layers of one-h a l f inch width. This therefore made necessary a d i s c r e t e d i s t r i b u t i o n i n at l e a s t the d i r e c t i o n p a r a l l e l to that of the incident beam, each layer con-t a i n i n g a plane of randomly-positioned spheres. The layer width was chosen as a matter of convenience f o r the i n i t i a l experiments described i n se c t i o n 6.5 and to maximize the number of layers possible f o r the sphere diameter of one-h a l f inch. As a furth e r convenience i n plac i n g the spheres i n p o s i t i o n , the transverse coordinates of each layer were also truncated at one-half inch i n t e r v a l s . The v a l i d i t y of such a " d i s c r e t e p o s i t i o n approximation" i s discussed i n Appendix C with r e s u l t s from s i m i l a r approximations applied to the one-dimensional model of planar s c a t t e r e r s being used to provide i n s i g h t i n t o the problem. 6.2.1 Generation of Uniform D i s t r i b u t i o n A "uniformly-random" array of d i s c r e t e sphere posi t i o n s throughout the slab region was generated as follows: Each layer of the support-medium was divided i n t o s i x equal-area sections of 28" x 10", making a t o t a l of 126 sections i n a l l . A sequence of uniformly-random numbers from the unit i n t e r v a l , equal i n number to the desired number of spheres within the slab region, was generated. Each number was then m u l t i p l i e d by 126 and truncated to designate one of the sections f o r a sphere to be placed. +The aim i n the o r i g i n a l Sylvania experiments was that measurements be comparable with the plane-wave theory e x i s t i n g at the time.50 One requirement was that the inverse distance v a r i a t i o n of the i l l u m i n a t i n g f i e l d be small over the depth of the d i s t r i b u t i o n f o r a given distance from the transmitting antenna to the d i s t r i b u t i o n center. 113 Corresponding to the number of spheres to be placed i n each section, p a i r s of d i s c r e t e uniformly-random "x and y" coordinates were generated. Because of the p h y s i c a l requirement that no more than one sphere occupy any one p o s i t i o n , each successive p o s i t i o n was chosen uniformly from those remaining. Thus, the o v e r a l l d i s t r i b u t i o n was not t r u l y uniform, although i t was e f f e c t -i v e l y uniform f o r a low average number of a v a i l a b l e locations occupied.''" Immediately a f t e r the sphere coordinates were generated for each s e c t i o n , they were obtained i n a p i c t o r i a l form as shown i n figu r e 6.2 for two of the sections ("stars" correspond to sphere p o s i t i o n s ) . This picture of sphere p o s i t i o n s f o r each s e c t i o n , and a rectangular metal g r i d with holes at one-half inch i n t e r v a l s placed over the desired section of one of the support-medium l a y e r s , provided a means of quickly marking the p o s i t i o n s . The uniformly-random numbers were obtained by means of the modified generator discussed i n section 3.3, the basic generator RAND being used to generate a d i f f e r e n t i n i t i a l array of 1,000 numbers for each section and then to randomly choose the desired number of coordinates from these. The generator was designed so that i t could be " i n i t i a l i z e d " by one number and so that the i n i t i a l array of random numbers f o r each section would be the same regardless of the number of coordinates required. Thus, i f a set of experiments were being c a r r i e d out i n which the number of spheres within the slab region progressively increased, each new experiment would only require that a d d i t i o n a l spheres be added to those f o r the previous experiment. A second +The s i z e of the equal-area sections i s a r b i t r a r y and was chosen to keep the amount of computer memory required by the program within a reasonable l i m i t , since t h i s i s approximately proportional to the number of d i s c r e t e p o s i t i o n s i n a s i n g l e s e c t i o n . In t h i s method of generation the section designation i s e s s e n t i a l l y the "z" coordinate; f o r sections equal i n area to the support-medium layers i t would correspond to the "z" coordinate i n the d i s t r i b u t i o n . M O P - N • « O 0 0 N « • 1 2 6 N T • 6 0 N 2 • 2 0 I R N •«•*« or S P H E B E P O S I T I O N S F O R s t e m * u s H U K B E B of S P H E R E S . j * I S 1 0 I S 2 0 2 > 1 0 I S 4 0 6 S S O S I 6 0 1 c c o o o o o o o c o c o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 o c .) a 6 { 6 6 6 6 i 6 a a a C a a t 1 c * c II Si j j i iii I i I si 1i ll IU ll S i iI i ll ll ll ll I ll H ll ll UiiiUiiiUi Win I* C C O C O O C » O C C C C C C C C O O O O O « O O C O O O O « O O O O O O O O O O O O O O O O O O O C O O O O O O O O C 1 * M S5SgS88g;£5g558SSSSS88g5SgSSSSS;gSSgggg HUM 111111 S-frfK- S 8 S1» 1 0 I S A R R 4 T C F S P H E R E P O S I T I O N S F O B S E C T I O N 1 1 6 \ M I N C E * O F S P H E R E S • I S 1 0 I S 2 0 2 S 1 0 3 5 s sgggsssgggggssggssggggsssggggggggggggsgggggggs-irawf-g-s-fw-6 C C C C 8 gSgg88g8ggg888ggggggggg;855g5ggggggS5gSg:gggggg S WW.W-H-*-10 o o o o o o o o o o « c o o c c c o o o c e o o o c c e e o o o o o o o o o c c o o o o o o o o o o o o o o o o o O ' 0 0 i o 11 12 11 I l l l i l l l l l S l l i l l l l l S I l l l I I l I I l l i t t t S f H i " i i r i ^ ~ ~ 18 O O O O O O O O O O O O O O O O O O O O O C O O O O C O C P O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 19 C C C C O O 0 O » 0 C C O O 0 O 0 O O O O O 0 O O 0 O O O O O O » O O O O O 0 O 0 O O O O 0 C C O 0 O O 0 0 O O 0 O O O 1 1 C O O O O O O O . O O O O O O O O O O O O O O O O O O O O O O O C O O O O O O O O O O O O O O C C O O O O O O O O O O O l l o 12 Figure 6.2 T y p i c a l Computer Output f o r Sphere Coordinates program was developed to d i s t i n g u i s h between " o l d " and "new" spheres by means of d i f f e r e n t characters to minimize the labour required to i n s e r t a d d i t i o n a l spheres for-each experiment. 6.2.2 The Support-Medium The scatterer-supporting material used f or the i n i t i a l experiments described i n section 6.5.2 was a l o c a l l y obtained "beaded" v a r i e t y of poly-styrene foam sheet of one-half inch width and density of 1.0 l b / f t 3 . The experimental i n v e s t i g a t i o n which led to the choice of t h i s material i s discussed i n section 6.5.1. For the one-half inch diameter spheres used i n the i n i t i a l experiments, holes of s l i g h t l y smaller diameter were f i r s t d r i l l e d i n the support-medium layers at the computer-generated p o s i t i o n s . This allowed the spheres to stay 115 e a s i l y i n place with a minimum compression of the surrounding material. A box of the same material with two-inch thick sides was used to hold the sphere-loaded layers i n place. One of the wide faces was l e f t unfastened so that the layers could be e a s i l y inserted or removed. I t was fastened into place with small spikes i n s e r t e d from the sides of the box. The sides of the box were glued together at only a few locations to minimize unwanted d i f f r a c t i o n . Figure 6.3 shows the medium i n place with the side removed and one of the layers with spheres inserted open to view. 6.2.3 Comparison with the Sylvania Model The main advantages of the present model over the Sylvania model are the following: ( i ) Unlike the Sylvania model, the present model lends i t s e l f f a i r l y w e l l to the experimental study of other complex, but s t i l l t h e o r e t i c a l l y p r e d i c t a b l e , s c a t t e r e r d i s t r i b u t i o n s besides the uniform d i s t r i b u t i o n . Generalization of the technique used f o r generating a uniform d i s t r i b u t i o n to one which would allow the average density of s c a t t e r e r s to vary non-uniformly i n the d i r e c t i o n p a r a l l e l to that of the incident beam, according to e i t h e r a mathematically or e m p i r i c a l l y s p e c i f i e d p r o b a b i l i t y density, i s straightforward. The i n c l u d i n g of s c a t t e r e r s i z e , shape, o r i e n t a t i o n , and d i e l e c t r i c properties as random v a r i a b l e s i n the d i s t r i b u t i o n function i s also p o s s i b l e , although i t would increase the labour involved i n p l a c i n g the s c a t t e r e r s i n p o s i t i o n . A more d i f f i c u l t type of d i s t r i b u t i o n for possible i n v e s t i g a t i o n i s one i n which the s c a t t e r e r p o s i t i o n s are c o r r e l a t e d i n some manner (e.g., a d i s t r i b u t i o n i n which the s c a t t e r e r p o s i t i o n s are weighted towards c e r t a i n " p e r i o d i c p o s i t i o n s " of higher p r o b a b i l i t y as i n the one-dimensional d i s t r i b u t i o n investigated i n Chapter 5). The type of c o r r e l a t i o n which could be used, however, and the manner i n which the desired d i s t r i b u t i o n could be generated are problems which would require much future research. Figure 6.3 View of the Physical Model ( i i ) Heavier s c a t t e r e r s , usually having higher p e r m i t t i v i t y , may be used i n the present model since they are embedded i n a r i g i d supporting material and need not s a t i s f y a "buoyancy" requirement as i n the Sylvania model. ( i i i ) Because the s t a t i s t i c a l d i s t r i b u t i o n of the scatterers i n the present model i s d i r e c t l y controlled by a known pseudo-random process, i t need not be monitored as i n the Sylvania model. (iv) An experiment on the present model can be repeated using the same scatterer configurations (as i n a complete Monte Carlo simulation), making i t r e l a t i v e l y easy to determine measurement errors introduced by equipment alone. The main drawbacks of the present model i n comparison with the Sylvania model, with possible methods of minimizing them, are as follows: (i ) The accuracy of the estimated ensemble averages obtainable with the present model i s governed l a r g e l y by i t s transverse dimensions since t h i s determines the number of uncorrelated samples. In the Sylvania model, only the unimportant time f a c t o r governs the accuracy. Because of p r a c t i c a l con-s i d e r a t i o n s such as the s i z e of room and scanning device a v a i l a b l e , the s i z e and ease of handling of the p l a s t i c foam sheets a v a i l a b l e , and the labour involved i n s e t t i n g up the medium, the transverse dimensions of the model must be l i m i t e d . One of the main objects of the experiments described i n section 6.5.2 was to determine, for t y p i c a l s c a t t e r e r d i s t r i b u t i o n s , the s t a t i s t i c a l accuracy obtainable from a model of a given s i z e . Without completely changing the sphere p o s i t i o n s , i t i s possible to improve the accuracy of the r e s u l t s f o r a uniform d i s t r i b u t i o n . The method proposed involves randomly s h u f f l i n g , s l i d i n g , or changing the o r i e n t a t i o n ( a l l four edges reversed) of the sphere-loaded layers with respect to one another to obtain new configurations, or a combination of these operations. With the exception of the s l i d i n g , which would require a l a r g e r container, t h i s method was used to improve the accuracy of the r e s u l t s i n section 6.5.2. ( i i ) The necessity f o r a d i s c r e t e p o s i t i o n approximation appears to l i m i t the present model to r e l a t i v e l y low average density d i s t r i b u t i o n s of s c a t t e r e r s . The d e t a i l s of t h i s disadvantage are discussed i n Appendix C. ( i i i ) The necessity f o r a support-medium places heavy importance on the "uniformity-of-propagation" c h a r a c t e r i s t i c s of the material used. The r e s u l t s and recommendations from an experimental i n v e s t i g a t i o n of t h i s problem are given i n s e c t i o n 6.5.1. This support-medium fa c t o r also means that s c a t t e r e r s having a low r e l a t i v e d i e l e c t r i c constant cannot be used with the same guarantee of accuracy as i n the Sylvania model. (iv) The long term s t a b i l i t y of the s i g n a l source during experiments on the present model i s a more important f a c t o r than f o r the Sylvania model, since the time required to record the data i s much longer. Fortunately, t h i s 118 problem can be minimized w i t h e x i s t i n g equipment. The e f f e c t s of phase i n s t a b i l i t y are q u a n t i t a t i v e l y analyzed i n s e c t i o n 6.5.2. (v) The time and labour i n v o l v e d i n s e t t i n g up the d i s t r i b u t i o n i n the present model i s much grea t e r than i n the S y l v a n i a model, because of the n e c e s s i t y of p h y s i c a l l y p o s i t i o n i n g each s c a t t e r e r to a mathematically s p e c i f i e d p o s i t i o n . I t i s f e l t , however, that t h i s i s a minor disadvantage as the time i n v o l v e d i s s t i l l only a f r a c t i o n of that r e q u i r e d to analyze the experimental r e s u l t s . 6.3 The S c a t t e r i n g Range, Antenna C h a r a c t e r i s t i c s , and Scanning Device A general purpose microwave anechoic chamber was designed and c o n s t r u c t e d f o r the experiments, f o r p r e l i m i n a r y antenna p a t t e r n measurements, and f o r f u t u r e s c a t t e r i n g and propagation s t u d i e s . D e t a i l s of the design and t e s t i n g of t h i s chamber are given i n Appendix D. The millimeter-wave region of the spectrum was considered the most s u i t a b l e f o r experiments w i t h the present model. This s p e c t r a l r e g i o n provides a f a i r l y l a r g e range of sphere-diameter-to-wavelength r a t i o s ( f o r e a s i l y handled spheres from one-eighth i n c h to one-half i n c h i n diameter) and allo w s narrow beams to be e a s i l y acquired f o r m a i n t a i n i n g the transverse dimensions of the model w i t h i n reasonable l i m i t s . Because of a v a i l a b l e equip-ment c o n s i d e r a t i o n s , the band 26.5-40.0 GHz was chosen as being most s u i t a b l e f o r the i n i t i a l experiments. The r e s u l t s of s e c t i o n 6.5 were obtained at a frequency of approximately 35 GHz. The geometry of antennas and medium i s given i n f i g u r e 6.4. In f i g u r e 6.3 on page 116, which shows the s c a t t e r i n g medium at one s i d e of i t s h o r i z o n t a l t r a v e l , the t r a n s m i t t i n g antenna p r o t r u d i n g from the f r o n t w a l l of the chamber i s hidden behind the r i g h t hand edge of the medium. Shown i n Figure 6.4 Plan View of the Geometry of Antennas and Medium figur e 6.5 i s a view of the r e c e i v i n g antenna and mixer mounted on a pedestal i n f ront of the back w a l l . A v a i l a b l e horn-lens combinations designed for 35 GHz were considered to have s u i t a b l y narrow beamwidths f o r use as both transmitting and r e c e i v i n g antennas i n the i n i t i a l experiments. The measured antenna parameters at t h i s frequency were aperture dimensions: 2.97" x 2.38" half-power beamwidth: H-plane - 7.4°, E-plane - 7.3° beamwidth to f i r s t n u l l : H-plane - 18.3°, E-plane - 16.0° l e v e l of f i r s t side-lobe: H-plane - 22db down, E-plane - 16db down The diameter of the approximately c i r c u l a r area of the medium center i l l u m i n -ated by the half-power beam of the transmitting antenna i s 3.6 inches. The f a c t that the diameter Dp = 2 A d t = 6.1 inches of the f i r s t Fresnel zone of the transmitter on the medium center i s greater than the diameter of the "3db c i r c l e " previously assumed to be the necessary l i m i t f o r the approximate v a l i d i t y of plane-wave theory^O i s unimportant f o r the experimental ,120 Figure 6.5 View of the Receiving Antenna and Mixer investigation described i n section 6.5.2. The consideration of a spherical incident wave i n more recent theoretical work » would also seem to eliminate the necessity of s a t i s f y i n g any plane-wave requirement i n possible future studies with the present model. The distance of d t = 27.5 inches from the transmitting antenna to the center of the medium was set on the basis of the commonly-used f a r - f i e l d c r i t e r -ion d t > D 2 A,55 where D i s the largest dimension of the horn aperture. The receiving antenna distance of d r = 110 inches corresponds to that obtained by satisfying the f a r - f i e l d c r i t e r i o n d r 2. Dp2/A = 4d t, with the f i r s t Fresnel zone regarded as a radiating aperture. Or i g i n a l l y this c r i t e r i o n was considered to be necessarily s a t i s f i e d for the observation of the f a r - f i e l d form of the average incoherent intensity;^0 more recent measurements, however, have shown 121 (.I2) to be r e l a t i v e l y constant f o r a wide v a r i a t i o n i n d t and d r.51 Approxi-mate theory accounting f o r f i n i t e d t and d r now allows the v a r i a t i o n of these parameters i n studies comparing t h e o r e t i c a l and experimental results.5>27 A general purpose p o s i t i o n i n g device was designed and constructed to serve as a scanning platform f o r the model and as an antenna p o s i t i o n e r f o r tests on the anechoic chamber and for pattern measurements. Design d e t a i l s f o r t h i s device, which can be remotely c o n t r o l l e d from outside the anechoic chamber, are given i n Appendix D. The dimensions of the area of the slab region scanned were 40.8" x 12.0". The 40.8 inch dimension was set by the width of the anechoic chamber and the extra width of the slab region considered necessary to minimize d i f f r a c t i o n at the edges. The twelve inch dimension was set by the v e r t i c a l movement of the scanning device. The transverse dimensions of the slab region (56" x 30") were designed so that the width of the edges not illuminated by the f i r s t Fresnel zone at the l i m i t s of the scanning area were approximately equal to the diameter of t h i s zone. The i n t e n s i t y of i l l u m i n a t i o n at the edges i s below -20db, minimizing edge d i f f r a c t i o n (see section 6.5.1). 6.4 Experimental Apparatus, Measurement and Data Processing Methods 6.4.1 Experimental Apparatus and Procedures A l l the s i g n a l generation and measurement apparatus i s located out-side the anechoic chamber as shown i n fi g u r e 6.6. A block diagram of the millimeter-wave equipment with superheterodyne receiver i s given i n f i g u r e 6.7a and a block diagram of the s i g n a l processing and recording equipment i n f i g u r e 6.7b. The equipment and measurement techniques are s i m i l a r to those used already by the Sylvania group.5>49,51,53 xhe main change i s the introduction of commercial r e c e i v i n g and s i g n a l processing equipment, i n c l u d i n g the 122 Figure 6.6 View of the Experimental Apparatus S c i e n t i f i c - A t l a n t a wide-range phase/amplitude receiver (model 1751) and the two PAR l o c k - i n amplifiers (model 120). The receiver i s a two-channel device containing two intermediate-frequency (IF) stages and employing automatic phase control c i r c u i t s . For operation i n the millimeter-wave s p e c t r a l region, external harmonic mixers are used to derive the f i r s t IF of 45 MHz. The phase and amplitude measurements are both made at 1 KHz, the second IF i n the double-conversion chain. Recorder outputs are provided for both the amplitude channels (1 KHz) and the phase channel (0 to -3.6 vdc for ranges ±180°, ±45°, or ±18°). As shown i n figure 6.7a, the RF si g n a l from the k l y s t r o n i s s p l i t into two main paths, one path (direct path) passing through the sc a t t e r i n g medium and the other (reference path) passing through a v a r i a b l e attenuator synchro-n i z e r k l y s t r o n power supply lOdb to o s c i l l o s c o p e 1 wave- c r y s t a l meter dectector 3db k l y s t r o n i s o l a t o r d i r e c t i o n a l o s c i l l a t o r coupler 6db v a r i a b l e . v a r i a b l e harmonic attenuator phase-shifter mixer transmitting antenna medium rec e i v i n g antenna (a) Millimeter-Wave Apparatus receiver ch.A-lKHz reference ch.B-lKHz amplifier s i g n a l phase-dc phase s e n s i t i v e d e tector l o c k - i n amplifiers 1 ono p . s . d. amplifier ac-dc converter from scanning device T COST c o n t r o l ch. T s i n T FM tape recorder chart recorder (b) Signal Processing Apparatus Figure 6.7 Block Diagram of the Experimental Apparatus U l 124 and phase-shifter. The reference path provides a phase reference s i g n a l f or use i n measuring the phase s h i f t T of the wave passing through the s c a t t e r e r d i s t r i b u t i o n and for d e r i v i n g the phase quadrature components T x = T COST and Ty = T sinT (see s e c t i o n 2.5.1). The phase of the reference channel i s adjusted as follows: For a wave passing through free space only, the v a r i a b l e phase-shifter i n the reference path and a 0.l°-increment d i g i t a l c o n t r o l on the r e c e i v e r are set so that the phase-shift i n d i c a t e d on the r e c e i v e r phase-meter i s zero. At the same time the phase controls on the l o c k - i n a m p l i f i e r s are adjusted so that the l e v e l from the in-phase detector i s maximum and that from the in-quadrature detector i s zero. A p o s i t i v e phase-shift i s then introduced i n the reference path to o f f s e t that introduced from the unloaded support-medium by readjusting the v a r i a b l e phase-shifter to the desired value (see section 6.5.2) using the receiver phase i n d i c a t o r s . Thus, x^hen the medium loaded with spheres i s introduced, the changes i n the f i e l d components T x, Ty, T, and T are due to the spheres alone. As shown i n f i g u r e 6.7b, the s i g n a l s representing the four f i e l d components were recorded on an FM tape recorder. The machine used f o r the present experiments was a seven channel Ampex SP-300. The amplitude and phase were also recorded on chart paper f o r an instantaneous v i s u a l check of the r e s u l t s during the experiments and for l a t e r comparison with the s i g n a l s recorded on magnetic tape. The phase-lock feedback loop containing the k l y s t r o n synchronizer shown i n f i g u r e 6.7a was intended to be used f o r s t a b i l i z i n g the output frequency of the k l y s t r o n and hence the phase v a r i a t i o n between the s i g n a l s i n the reference and d i r e c t paths. The synchronizer, an FEL model 136-AK, proved however to be u n s a t i s f a c t o r y f o r a number of klystrons at d i f f e r e n t frequencies. I t was only capable of providing a short-term phase-lock (approximately f i v e 125 minutes), not long enough f o r the desired measurements. Although i t i s f e l t that a s u i t a b l e synchronizer should be used for future more-precise measurements, the frequency s t a b i l i t y obtainable without the synchronizer was considered s u f f i c i e n t f o r the i n i t i a l experiments. The problem was minimized through the use of a water cooling-system, c o n s i s t i n g of a c o i l of copper tubing immersed i n a jacket of the low melting point metal Cerrobend (Wood's metal) surrounding the k l y s t r o n . I n i t i a l s t a b i l i t y measurements showed a maximum amplitude v a r i a t i o n of 2% and a maximum phase v a r i a t i o n of le s s than 3° over an hour period with the EMI R9521 k l y s t r o n . At the time of the experiments described i n section 6.5.2, however, a phase stab-i l i t y of only 6° over a one-half hour period could be achieved. The q u a n t i t a t i v e e f f e c t on the r e s u l t s i s estimated i n that se c t i o n . 6.4.2 Data Processing Methods The recorded four-channel analog data was sampled and processed on a DEC PDP-9 computer having a remote i n t e r f a c e which contains multiplexing and a n a l o g / d i g i t a l (A/D) conversion equipment. During an experiment, a two-level c o n t r o l voltage was recorded on a f i f t h channel (see f i g u r e 6.7, page 123) to designate those sections of the four analog signals (corresponding to scanning of the medium) to be sampled. The "sampling l e v e l " was switched on and o f f with the power to the h o r i z o n t a l drive motor i n the scanning device. The sampling program was designed so that sampling of the four s i g n a l channels and the c o n t r o l channel would be a con-tinuous process once begun, with the presence of the sampling l e v e l on the c o n t r o l channel s p e c i f y i n g which s i g n a l samples to be retained. The m u l t i -plexer i n the i n t e r f a c e provided very quick switching between channels so that e f f e c t i v e l y a l l f i v e channels were sampled instantaneously by the A/D converter. The desired time i n t e r v a l between each set of samples was s p e c i f i e d i n the sampling program. Samples were i n i t i a l l y stored i n memory during the sampling 126 process and then copied on to magnetic tape (DEC tape) a f t e r each block of analog data had been sampled. The d i g i t a l data corresponding to each scan of the medium was contained i n separate f i l e s on DEC tape. In the present procedure the sampling process stops a f t e r each block of data has been obtained and the operator must r e s t a r t the process for the next block of data and s p e c i f y the desired f i l e . Once the d i g i t a l data had been obtained on DEC tape i t was normalized according to predetermined reference l e v e l s and processed by means of other computer programs to determine the means, variances, a u t o c o r r e l a t i o n functions, and other average f i e l d functions of the f i e l d components recorded. The reference l e v e l s corresponded to the values of the f i e l d components obtained with only the unloaded support-medium i n place. These l e v e l s were recorded on the FM recorder before the beginning of a set of data scans and were sampled i n the same manner as the a c t u a l data to determine s u i t a b l e values for use i n s c a l i n g (see section 6.5.2). The a v a i l a b l e A/D converter has a twelve-bit r e s o l u t i o n c a p a b i l i t y (4096 l e v e l s of quantization) and, with accuracy of only one b i t l e s s , i s s u i t a b l e f o r sampling the data obtained from the present type of experiment. Quantization noise was l e s s than that generated by the FM tape recorder, as observed by a comparison of sampled values with the corresponding analog records reproduced on chart paper from the tape recorder. The amount of noise generated by the tape recorder was not considered s u f f i c i e n t l y high to require the use of low-pass f i l t e r i n g before the i n t e r f a c e . Because of the c a p a b i l i t y of sampling a l l four s i g n a l channels simultaneously, two of these channels were redundant. Thus, a comparison of the d i r e c t l y sampled data against the corresponding computed data f o r both sets of channels provided one method of checking f o r measurement or data processing errors (see section 6.5.2). 6.5 Experimental Results 6.5.1 Experiments on the Support-Medium The main problem associated with the support-medium was to obtain a s u i t a b l e material and a s u i t a b l e o r i e n t a t i o n of the constituent layers to present a minimum of d i s c o n t i n u i t y to the millimeter-wave beam as i t passed through. I n i t i a l measurements of the transmitted wave amplitude and phase f or hor i z o n t a l l y - s t a c k e d layers of Dow Styrofoam FR (1.9 l b / f t 3 ) showed wide v a r i a t i o n s of these q u a n t i t i e s i n the scanning d i r e c t i o n perpendicular to the la y e r s . Further measurements on blocks of the same material having a s i n g l e j o i n t p a r a l l e l to the d i r e c t i o n of propagation showed a marked d i f f r a c t i o n pattern as the beam crossed the j o i n t f o r both p a r a l l e l and perpendicular p o l a r i z a t i o n . This e f f e c t , apparently caused by a large r e f l e c t i o n at the j o i n t due to almost grazing incidence, precludes the use of any j o i n t s p a r a l l e l to the beam except close to the edges of the medium where the beam i n t e n s i t y i s low. The best arrangement of the support-medium layers was therefore found to be a stacked array perpendicular to the d i r e c t i o n of the incident beam. V e r t i c a l layers of Styrofoam r e s u l t e d i n very uniform transmission character-i s t i c s , although the necessity f o r j o i n t s p a r a l l e l to the beam i n the narrow widths of board obtainable (twenty-four inches maximum) made t h i s material unsuitable. The "beaded" v a r i e t y of polystyrene foam (1.0 l b / f t 3 ) supplied a reasonably s a t i s f a c t o r y s o l u t i o n to the problem. Sheets of t h i s material can be obtained i n a v a r i e t y of lengths, widths, and thicknesses s u i t a b l e to the requirements. Because of the bead-bead i n t e r f a c e s i n t h i s material, however, i t i s not as uniform i n transmission properties as Styrofoam. A set of seven h o r i z o n t a l scans (separated by two inches) of the unloaded support-medium, 128 c o n s i s t i n g of the two inch thick sides of the container and the twenty-one one-half inch thick l a y e r s , showed a maximum v a r i a t i o n i n the reference l e v e l amplitude of 4% and i n the phase of 5°. This v a r i a t i o n was considered to be low enough f o r the experiments described i n the next section where the sphere d i s t r i b u t i o n s presented a f a i r l y large s c a t t e r i n g cross-section to the beam. Experiments were also performed to measure the d i e l e c t r i c constant of the "beaded" polystyrene foam and to determine how close the transmitting antenna could approach the edge of the support-medium before the onset of a noticeable d i f f r a c t i o n pattern. A d i e l e c t r i c constant of 1.018 at 35.1 GHz was obtained by measuring the phase s h i f t of the transmitted wave through a two-inch thick slab of the m a t e r i a l . In the other experiment, a noticeable d i f f r a c t i o n pattern was obtained with the beam center l e s s than about nine inches from the edge of the medium, consistent with the i n i t i a l design of the model. For future more precise experiments, the use of a support-medium material having smaller and more well-packed beads than the present material (average bead diameter i n the one-eighth to three-sixteenth inch range) should r e s u l t i n more uniform transmission c h a r a c t e r i s t i c s . The problem might also be a l l e v i a t e d somewhat with the sanding of the material surfaces (surfaces produced by a hot-wire cutter are not as smooth) to allow the layers to be pressed more f i r m l y together. The use of a non-beaded material such as polyeurethane foam or p o l y - v i n y l - c h l o r i d e (PVC) foam could also provide a more s a t i s f a c t o r y s o l u t i o n . Both these materials, and p a r t i c u l a r l y PVC foam, are considerably more expensive than polystyrene foam, however, and the d i e l e c t r i c properties at millimeter-wavelengths do not appear to be tabulated. 6.5.2 Experiments on T y p i c a l Scatterer D i s t r i b u t i o n s Experiments were performed on two d i f f e r e n t average density d i s t r i b u t i o n s of one-half inch diameter polyethylene spheres at a frequency of 129 35.1 GHz. These two average d e n s i t i e s of 183 and 366 spheres per cubic foot (scf) are t y p i c a l values from a range of d i s t r i b u t i o n d e n s i t i e s which could be studied i n a more d e t a i l e d experiment such as one c a r r i e d out by the Sylvania group.^9 They correspond to t o t a l s of 2,000 and 4,000 spheres i n a slab region four inches longer than that used, the two-inch edge being replaced by the sides of the foam container. The one-half inch sphere diameter gives a t y p i c a l ka-factor (4.66 at 35.1 GHz) from a range of values of i n t e r e s t and i s con-venient f o r use i n the present model. As i n d i c a t e d i n section 6.2.3, data was obtained for more than one configuration of spheres to increase the s t a t i s t i c a l accuracy of the average f i e l d functions estimated i n the experiment. This was done by a combination of s h u f f l i n g the support-medium layers according to a random computer-generated permutation^ 6 and at the same time randomly adjusting the o r i e n t a t i o n of the layers to give an equal p r o b a b i l i t y f o r two possible o r i e n t a t i o n s : r i g h t side up or a l l four edges reversed. Horizontal scans of 40.8 inch length were taken at two-inch v e r t i c a l i n t e r v a l s with v e r t i c a l E - f i e l d p o l a r i z a t i o n on both sides of three d i f f e r e n t l a y e r - c o n f i g u r a t i o n s , giving a t o t a l of f o r t y -two scans (seven scans per medium side) i n a l l . The analog s i g n a l recordings of the four f i e l d components of i n t e r e s t were sampled at the rate of 304 samples per scan (sampling i n t e r v a l of 0.134 inch) to produce accuracy close to that obtainable from an analog averaging method. The phase of the reference path s i g n a l was s h i f t e d 147° with respect to the direct-path s i g n a l through free space to o f f s e t that introduced by the unloaded support-medium. Because t h i s value was obtained by v i s u a l l y averaging the phase s h i f t of the transmitted wave through the unloaded support-medium, i t cannot be considered to provide an accurate absolute reference with respect to t h i s medium. Such a reference, however, was not considered necessary f o r the present experiments. 130 Results f o r estimated average f i e l d functions. The average f i e l d functions estimated i n the experiment are given i n Table 6.1. Section (a) of t h i s table l i s t s those functions l e a s t susceptible to phase-drift errors (due to frequency d r i f t of klystron) and errors i n the phase reference s e t t i n g between the two sets of channels; section (b) l i s t s the functions most susceptible to these e r r o r s . In processing mode 1, the components T x and T y were computed from the sampled values of T and x, while i n mode 2, T and x were computed from T v and T,,. x y As seen from Table 6.1a, a greater deviation between the mode 1 and mode 2 r e s u l t s occurs f o r the 183 scf density (maximum of 7.7% f o r C 2 / ( l 2 ) ) than f o r the 366 scf density (maximum of 1.5% for C 2 ) . The improved agreement between the corresponding r e s u l t s f o r the 366 scf density resulted from a more c a r e f u l s c a l i n g procedure ( i . e . , the f i r s t set of scale factors were obtained by sampling each of two sets of reference l e v e l s 500 times to average out the e f f e c t s of tape recorder noise; the second set were obtained by sampling each of three sets of reference l e v e l s - recorded a f t e r every 14 scans - 1,500 times). The discrepancy between the mode 1 and mode 2 r e s u l t s i n Table 6.1b i s due mainly to an i n i t i a l error i n phase s e t t i n g between the x channel and the T x, Ty channels (except f o r o T where the small discrepancy i s due to the inaccuracy i n s c a l i n g ) . The functions ( l x 2 ) , ( l y 2 ) , ( l x I y ) , and uare affected by t h i s discrepancy because of t h e i r p e r i o d i c behavior with the v a r i a t i o n of the phase r e f e r e n c e . 5 ' 5 ^ A thorough c a l i b r a t i o n of the l o c k - i n a m p l i f i e r s with the receiver phase detector and a more c a r e f u l i n i t i a l adjustment should minimize t h i s discrepancy i n future work. As seen by the values f o r the average t o t a l i n t e n s i t y ( T 2 ) i n Table 6.1a, the f r a c t i o n of t o t a l average power transmitted i n the a x i a l d i r e c t i o n with the medium i n place i s considerably le s s than that without the medium. This indicates a high degree of s c a t t e r i n g by the present spheres i n 131 TABLE 6.1 RESULTS FOR ESTIMATED AVERAGE FIELD FUNCTIONS (a) Functions Least Susceptible to Phase Errors Average Density Mode* C <T> <T2> C 2 ( I 2 ) C 2/<I 2> 183 f scf 1 0.657 0.666 0.455 0.431 0.0237 18.2 0.106 2 0.645 0.656 0.441 0.417 0.0247 16.9 0.107 366 + scf 1 0.447 0.464 0.227 0.200 0.0265 7.55 0.107 2 0.444 0.460 . 0.224 0.197 0.0265 7.45 0.109 (b) Functions Most Susceptible to Phase Errors Average Density Mode* a (T) <I y2> y ax 183 T scf 1 -0.26° -0.67° 0.0115 0.0122 0.0029 0.24 9.9° 2 1.35° 0.95° 0.0117 0.0130 0.0027 0.22 10.3° 366 + scf 1 2.28° 1.42° 0.0119 0.0146 0.0027 0.20 16.0° 2 5.00° 5.94° 0.0132 0.0133 -0.0032 -0.24 15.9° Tresults f o r 183 scf density based on 35 data scans; r e s u l t s f o r 366 scf density based on 39 data scans *mode 1: T x and T y computed from T and x; mode 2: T and T computed from T x and T 132 the off-forward and backward d i r e c t i o n s , i n contrast to the r e s u l t s f o r s i m i l a r average d e n s i t i e s of large Styrofoam spheres obtained by the Sylvania group.^ This d i f f e r e n c e i s of course to be expected and h i g h l i g h t s the necessity f o r future experiments with a wide range of constituent s c a t t e r e r parameters. Results of accuracy estimates. As indicated previously, the primary purpose of the present experiments was to determine the s u i t a b i l i t y of the ph y s i c a l model with respect to the s t a t i s t i c a l accuracy obtainable. This was accomplished by means of two d i f f e r e n t methods used to compute the sampling variances of the four estimated mean values: ( T x ) , (Ty), ( T ) and ( x ) . The f i r s t method was based on computation of the s p a t i a l a u t o c o r r e l -a t i o n and autocovariance functions f o r T x, Ty, T and x. The autocovariance fo r a function f i s defined K f(X) = <f(0)f(X)> - ( f ) 2 (6.1) = R f(X) - ( f > 2 where Rf(X) i s the au t o c o r r e l a t i o n function and X i s the separation distance. The a u t o c o r r e l a t i o n functions were f i r s t computed f or each data scan using the formula 5^ ns-m R.(mAX) = T f(iAX) f(iAX + mAX) (6.2) t n -m .<—> s 1=1 where AX i s the distance between adjacent samples, m i s the separation distance i n terms of samples, and n g i s the number of samples per scan. The o v e r a l l estimates f o r the R^ were then obtained by averaging the r e s u l t s over a l l scans, and the o v e r a l l estimates f o r the were obtained from the d e f i n i t i o n (6.1). The c o r r e l a t i o n c o e f f i c i e n t s , or normalized autocovariance functions, Kf(mAX) K'(mAX) = (6.3) f K f(0) 133 for the amplitude T and phase T are given i n f i g u r e 6.8. The curves f o r T^ and Ty are not given because they are almost i d e n t i c a l to those f o r T and T r e s p e c t i v e l y . The autocovariance curves obtained are only accurate f o r separation distances m up to about 5% or 10% of the number of samples per data scan ( i . e . , n s = 304).->7 The sections of the curves i n fi g u r e 6.8 computed f o r separation distances beyond those given showed very erroneous behavior due to the inherent unequal weighting of c e r t a i n sections of samples with the use of formula (6.2), and the los s of accuracy from fewer samples. I t can be assumed, however, that the true c o r r e l a t i o n beyond a separation distance of about t h i r t y samples i s very close to zero because of the f a c t that the half-power beam of the transmitting antenna i l l u m i n a t e s a c i r c u l a r area of only 3.6 inches diameter at the center of the medium. The combining of a l l data scans as a s i n g l e data segment to obtain greater accuracy of the autocovariance function t a i l s could be e a s i l y accomplished on a computer with a l a r g e r memory capacity than the present PDP-9. Estimates of the sampling d i s t r i b u t i o n variances for ( T x ) , (Ty), (T) CO' and (x), based on the c o r r e l a t i o n data, were obtained from the formula^ 2 1 °<f> = n7 c n c - l °> 2 + 2 E I1 - H Kf ( 1 A X ) i = l (6.4) where n c i s the t o t a l number of correlated samples. The equivalent number of uncorrelated samples n f o r each f i e l d component was then obtained from V 0 < i > - - r < 6 - 5 ) Table 6.2 (method 1) gives the computed r e s u l t s using t h i s approach i n the form of the equivalent number of uncorrelated samples per scan and the corresponding distance between these samples for each of the four f i e l d 1.0 0.8 0.6 Separation Distance mAX, inches 0 1 2 3 4 5 6 0.4 0.2 -0.2 0 V \ \ \\ \v 1 1 1 1 \ \\ \\ V \\ \\ \\ \\ V \\ \\ \\ • V \ \ \ s. N X \ V / / y s V 10 20 30 40 Separation Distance m, samples (a) Amplitude 50 1.0 0.8 0.6 0.4 0.2 -0.2 Separation Distance mAX, inches 1 2 3 4 5 6 \ 1 1 1 1 1 A"* N 10 20 30 40 Separation Distance m, samples (b) Phase 50 Figure 6.8 Co r r e l a t i o n C o e f f i c i e n t Curves for Amplitude and Phase 183 s c f , 366 scf CO TABLE 6.2 ACCURACY CALCULATIONS BASED ON EQUIVALENT UNCORRELATED SAMPLES (a) Equivalent Number of Uncorrelated Samples per Scan of T x, T y, T, and T. Average Density Method1" T x T y T T 183 1 14 16 15 17 scf 2 20 5 22 5 366 1 18 19 18 19 scf 2 25 10 24 9 (b) Distance i n Inches between Equivalent Uncorrelated Samples o f T , T , T, and T. Average Density Method1" T x T y T T 183 1 2.8 2.6 2.7 2.3 scf 2 2.0 9.2 1.8 7.6 366 1 2.3 2.1 2.3 2.2 scf 2 1.6 4.2 1.7 4.6 "^Method 1 - using covariance c a l c u l a t i o n s and equation (6.4) Method 2 - using variance c a l c u l a t i o n s of means over i n d i v i d u a l scans 136 components. The r e s u l t s are based on approximating the summation i n (6.4) by the sum of p o s i t i v e terms only, up to and i n c l u d i n g i = 30. The accuracy obtainable from an analog average of the data scans can be estimated by dropping, the o^ 2 term i n the c a l c u l a t i o n of a ( f ) i n equation (6.4). The approximately one more uncorrelated sample per scan which would be added from such a scheme ind i c a t e s that a f a s t e r sampling rate f o r the d i g i t a l technique would be unwarrented. The second method used i n determining the accuracy was to estimate the variances of the mean values over the i n d i v i d u a l data scans, d i v i d i n g by the number of scans to obtain estimates of the variances over a l l scans. The r e s u l t s obtained by t h i s approach, given also i n Table 6.2 (method 2 ) , are i n good agreement with the corresponding r e s u l t s f or method 1, except f o r the components T^ and T. The reason f o r the discrepancy i s that the f i r s t method, with the dropping of terms from the t a i l s of K T and K i n the computations y based on (6.4), i s much less s e n s i t i v e to phase d r i f t than the second method. The second set of r e s u l t s can therefore be considered to be more accurate under the actual conditions of considerable phase d r i f t (see section 6.4.1). This r e s u l t strongly points to the need f o r improved phase s t a b i l i t y i n future experiments. The values for some of the f i e l d functions and t h e i r accuracy estimates obtained by the second method are given i n Table 6.3. These fi g u r e s are based i n the normal manner on twice the standard errors of the means (see se c t i o n 3.4). Two sets of accuracy values are given, one assuming a l l data scans contribute to the accuracy and the other assuming h a l f the data scans contribute to the accuracy. The reason f o r t h i s i s that the f i r s t set of estimates are based on the assumption that a l l data scans are mutually uncorrelated. Since h a l f the scans were obtained by scanning the back sides of the three l a y e r - c o n f i g u r a t i o n s , however, the mean f i e l d component values f o r 137 TABLE 6.3 ACCURACY ESTIMATES BASED ON TWICE THE STANDARD ERRORS OF THE MEANS (a) Results f o r 183 scf Average Density (35 data scans) S (T) ( T ) ( I x 2 ) <v> ° T 2 Function Estimate • 0.645 0.0153 0.666 -0.67° 0.0117 0.0130 0.0112 97.2 Accuracy Estimate 1 0.008 0.018 0.008 1.4° 0.0010 0.0008 0.0011 6.4 Accuracy^ Estimate 2 0.012 0.026 0.011 2.0° 0.0014 0.0012 0.0016 9.0 (b) Results f o r 366 scf Average Density (39 data scans) - c x c y (T> ( T ) <Ix2> < I y 2 ) Of2 Or2 Function Estimate 0.442 0.0387 0.464 1.42° 0.0132 0.0133 0.0114 257 'Accuracy Estimate 1 0.007 0.012 0.007 1.7° 0.0009 0.0008 0.0009 16 Accuracy ^ Estimate 2 0.010 0.017 0.010 2.4° 0.0013 0.0012 0.0012 23 ^accuracy estimates with maximum c o r r e l a t i o n between f r o n t - s i d e scans and back-side scans assumed 138 the corresponding scans were probably highly c o r r e l a t e d . The second set of estimates can therefore be assumed to be "worst-case" estimates under the con-d i t i o n of maximum c o r r e l a t i o n between f r o n t - s i d e data scans and back-side scans. The accuracy of estimation could be increased s t i l l f urther by taking data scans of more la y e r - c o n f i g u r a t i o n s . It should be emphasized, however, that c a l c u l a t i o n s of o v e r a l l accuracy are only accurate i f the data scans are taken i n a manner so as to be r e l a t i v e l y uncorrelated. One anomaly i n the present technique for obtaining new sphere-configurations i s that the density of spheres i n the center region of the medium remains the same a f t e r the support-medium layers have been s h u f f l e d and oriented randomly. The e f f e c t of the higher weight placed on t h i s p a r t i c u l a r density has been v i s u a l l y observed by obtaining averages of the field-component signals over a l l data scans. Although t h i s anomaly probably e f f e c t s the accuracy estimates of t h i s section very l i t t l e because of the s i z e of the medium, i t could be eliminated by further introducing a random s l i d i n g between layers as suggested i n section 6.2.3. 6.5.3 Discussion of O v e r a l l Results The tests on the support-medium i n d i c a t e that the problem of obtaining uniform transmission c h a r a c t e r i s t i c s of the incident beam i s solvable. Two other possible problems associated with the support-medium which could not be investigated because of the d i f f i c u l t y involved are: (a) the e f f e c t of the j o i n t s between the layers on the scattered f i e l d s of the constituent s c a t t e r e r s , and (b) the e f f e c t on the scattered f i e l d s of the unavoidable inhomogeneity of the region surrounding the i n s e r t e d s c a t t e r e r s . In general, because of the presence of the support-medium, the v a l i d i t y of any comparison between ex p e r i -mental and t h e o r e t i c a l r e s u l t s i s a problem which requires f u r t h e r research. The experimental r e s u l t s on t y p i c a l s c a t t e r e r d i s t r i b u t i o n s i n d i c a t e that the proposed p h y s i c a l model i s reasonably s u i t a b l e from the point of view 1 of s t a t i s t i c a l accuracy obtainable. Under conditions of n e g l i g i b l e phase reference i n s t a b i l i t y and measurement e r r o r s , the accuracy should only be dependent on the s i z e of the medium and the degree of c o r r e l a t i o n between samples. The r e s u l t s show that considerable overlap between the c i r c u l a r regions of half-power beamwidth i l l u m i n a t i o n i s p o s s i b l e before the corre-sponding f i e l d samples become hig h l y c o r r e l a t e d . The s t a t i s t i c a l accuracy appears to be s u f f i c i e n t that trends i n the r e s u l t s f o r the f i r s t two f i e l d moments (for a v a r i a t i o n i n the average density of s c a t t e r e r s , f or example) could be established from a set of data scans from one side of a model having transverse dimensions comparable with those of the present one. More accurate r e s u l t s can be obtained for uniform d i s t r i b u t i o n s by changing the l a y e r - c o n f i g u r a t i o n of the support-medium. For other d i s t r i b u t i o n s which do not allow t h i s procedure (e.g., d i s t r i b u t i o n s i n which the s c a t t e r e r p o s i t i o n s i n other than a s i n g l e layer are c o r r e l a t e d ) , a slab region of l a r g e r transverse dimensions would be necessary. The maximum dimensions p r a c t i c a b l e are probably about 8' x 4'. From the two sets of r e s u l t s obtained, the accuracy of estimation seems l i t t l e affected by the average density of the d i s t r i b u t i o n . The parameter most a f f e c t i n g such accuracy i s probably the transmitting antenna beamwidth, although further experiments with d i f f e r e n t beamwidths are required to determine the exact behavior. The present experimental r e s u l t s give only an i n d i c a t i o n of the accuracy obtainable f o r p a r t i c u l a r d i s t r i b u t i o n and incident beam parameters. Any future experiments performed using the present model must -always include some means of estimating s t a t i s t i c a l accuracy. 140 7. CONCLUSIONS The main developments of th i s thesis which are considered to be contributions to the subject of s c a t t e r i n g from random media of di s c r e t e s c a t t e r e r s may be summarized as follows: I. The One-Dimensional Model Extensive use has been made of the one-dimensional model of randomly-positioned planar sca t t e r e r s as a t o o l i n the i n v e s t i g a t i o n of general d i s c r e t e -s c a t t e r e r theories and as a basis f o r providing further knowledge of the ph y s i c a l and s t a t i s t i c a l c h a r a c t e r i s t i c s of d i s c r e t e - s c a t t e r e r media: A. T h e o r e t i c a l Developments (i ) An e x p l i c i t s e r i e s representation i n orders-of-back-scattering has been developed f o r the t o t a l f i e l d i n plane-wave s c a t t e r i n g from a fi x e d array of n o n - i d e n t i c a l planar s c a t t e r e r s . ( i i ) Approximate s e r i e s expressions based on the O-B-S representation have been obtained f o r several average f i e l d functions of i n t e r e s t i n the problem of sc a t t e r i n g from an ensemble of configurations of uniformly-random i d e n t i c a l planar s c a t t e r e r s . These expressions have been shown to be useful i n p r e d i c t i n g the exact or approximate asymptotic behavior of the average f i e l d functions i n the l i m i t as p -> 0 and i t i s believed that they may also prove u s e f u l i n further t h e o r e t i c a l research directed towards the improvement of general d i s c r e t e -s c a t t e r e r theories. ( i i i ) The exact asymptotic forms f o r p -* 0 i n the planar-scatterer model have been obtained f o r the coherent transmitted f i e l d and the average t o t a l and incoherent i n t e n s i t i e s of both the transmitted and r e f l e c t e d f i e l d s . The importance of the asymptotic forms i n the improvement of theories applicable to higher p has been i l l u s t r a t e d . In p a r t i c u l a r , the exact asymptotic form (1 + g +)N for the coherent transmitted f i e l d ( T ) has been used to modify the 141 one-dimensional form of Twersky's free-space theory; the p o s s i b i l i t y of a s i m i l a r f i n i t e - N c o r r e c t i o n to the three-dimensional form of Twersky's theory has been suggested. (iv) Based on the O-B-S approximations f o r the transmitted and r e f l e c t e d f i e l d s and the e x i s t i n g theory of random phasor sums, p h y s i c a l conditions neces-sary f o r the approximate v a l i d i t y of the b i v a r i a t e Gaussian d i s t r i b u t i o n i n describing the t o t a l f i e l d s t a t i s t i c s of the one-dimensional model have been given. Conditions necessary for the occurrence of a Rayleigh-distributed inco-herent f i e l d amplitude with u n i f o r m l y - d i s t r i b u t e d phase have also been outlined. B. Monte Carlo Simulation ( i ) "Exact" simulation r e s u l t s f o r use i n the evaluation of approximate theories f o r the one-dimensional model have been obtained. ( i i ) Monte Carlo simulation applied to the approximate O-B-S represent-ations f o r the f i e l d has been used to v a l i d a t e the approximate s e r i e s expres-sions f o r the average f i e l d functions obtained; the expression derived f o r ( R 2) remains to be v e r i f i e d . A comparison of "exact" simulation r e s u l t s and t h e o r e t i c a l data has shown c e r t a i n of these theories to be i n the main better than the one-dimensional forms of e x i s t i n g general d i s c r e t e - s c a t t e r e r theories. ( i i i ) The l i m i t a t i o n s of the one-dimensional form of Twersky's free-space theory for the coherent f i e l d and conditions necessary for i t s approximate v a l i d i t y have been i l l u s t r a t e d by a presentation of r e s u l t s f o r a wide range of. s c a t t e r i n g parameters. The improvement contained i n the "asymptotic c o r r e c t i o n " to the free-space theory f o r ( T ) has been v e r i f i e d . The comparison of theoret-i c a l data for the free-space theory for (R) with "experimental" r e s u l t s i s believed to be the f i r s t . (iv) A q uantitative analysis of the t o t a l f i e l d d i s t r i b u t i o n based on the t h i r d and fourth f i e l d moments has been made and c e r t a i n e f f e c t s of multiple s c a t t e r i n g i l l u s t r a t e d . The stated p h y s i c a l conditions necessary for the 142 approximate v a l i d i t y of the b i v a r i a t e Gaussian d i s t r i b u t i o n have been v e r i f i e d . (v) Simulation methods f o r the generation of a non-uniform d i s t r i b u t i o n of planar-scatterer configurations weighted towards p e r i o d i c i t y have been developed. Based on the s c a t t e r i n g r e s u l t s obtained, c r i t e r i a for the assump-ti o n of uniform-randomness have been determined. The l i m i t a t i o n s of the one-dimensional form of Twersky's mixed-space theory for the coherent f i e l d have been i l l u s t r a t e d and conditions necessary f o r i t s approximate v a l i d i t y given. I I . The Three-Dimensional P h y s i c a l Model (i ) A new three-dimensional p h y s i c a l model of s p h e r i c a l s c a t t e r e r s i n which the Monte Carlo method i s employed to co n t r o l the s c a t t e r e r s t a t i s t i c s has been developed f o r use i n laboratory experiments at millimeter-wave frequencies. I t i s believed that t h i s model may be of value i n t h e o r e t i c a l -experimental i n v e s t i g a t i o n s of the type performed previously on the Sylvania p h y s i c a l model and i n t h i s work on the one-dimensional model. ( i i ) An i n i t i a l experimental i n v e s t i g a t i o n i n t o the s u i t a b i l i t y of the proposed model has been c a r r i e d out. Experiments have been performed on the support-medium and measurements of the forward-diffracted f i e l d i n s c a t t e r i n g from t y p i c a l d i s t r i b u t i o n s have been obtained and analyzed to determine the importance of disadvantages associated with the model. From the r e s u l t s obtained, the disadvantages investigated appear to be minimal. ( i i i ) An i n i t i a l i n v e s t i g a t i o n i n t o the v a l i d i t y of the d i s c r e t e p o s i t i o n approximation used i n the three-dimensional model has been c a r r i e d out employing the r e s u l t s f o r s i m i l a r approximations applied to the one-dimensional model. A basis f o r a p a r t i a l comparative-evaluation of the approximation i n the two models has been found and factors important i n the future development of more fi r m c r i t e r i a o u t l i n e d . Approximate theory necessary i n the comparison has been developed. 143 APPENDIX A SUMMARY OF TWERSKY'S THEORIES FOR SCATTERING FROM RANDOM MEDIA OF DISCRETE SCATTERERS Q A . l Twersky's Free-Space Theory for the Coherent F i e l d This theory i s app l i c a b l e to the problem of a plane wave <J>(f) = e ^  ^ obliquely incident at an a r b i t r a r y angle 0 on a slab-region d i s t r i b u t i o n ( i . e . , bounded by the planes z = 0 and z = d) of one-, two-, or three-dimensional sca t t e r e r s random i n one, two, or three dimensions r e s p e c t i v e l y . I t contains the following assumptions: (i) The sca t t e r e r s are i d e n t i c a l and s i m i l a r l y aligned, ( i i ) The one-scatterer p r o b a b i l i t y density function, defined p ( r x ) = j" J" p ( r 1 , . . . , r N ) d r g d r 3 .... d r N i s of the form p ( r : ) = p(r x)/N (A.l) (A. 2) and i s the same for a l l s c a t t e r e r s [p(fj) i s the average density of sca t t e r e r s at f ^  . . ( i i i ) The two-scatterer p r o b a b i l i t y density function i s the product of one-sca t t e r e r forms, i . e . , 1 ( r 1 , r 2 ) =y ^ ( r j ,. . . , r N ) d r 3 dr^ d r N = ' P ( r 2 ) " N N (A.3) or e q u i v a l e n t l y , the two-scatterer c o n d i t i o n a l p r o b a b i l i t y density function defined :1> *\P<?1 p ( r , | r 1 ) = / p(r, , r 2 ) d r 2 (A.4) i s equal to p ( r 1 ) . The free-space theory equations are also based on the approximate r e l a t i o n N £ p(f | f s ) = (N-1) s=l*t .P(r s) N 8 P ( f s > (A.5) v a l i d f or large N, and the h e u r i s t i c approximation (A . 6 ) where (ty)s i s the average t o t a l f i e l d with the p o s i t i o n vector r g held f i x e d , etc. These two approximations can be combined to give the s i n g l e approximate equation 0 > s = 010 + (U> s (A. 7) where (u) g i s the average scattered f i e l d from a s c a t t e r e r with i t s p o s i t i o n vector f held f i x e d . Recognizing that equation (A.7) i s the form of the s o l u t i o n for a s i n g l e object excited by a set of plane waves and s c a t t e r i n g into free space, and using the following two a d d i t i o n a l assumptions, Twersky solved the r e s u l t -ing i n t e g r a l equations to obtain e x p l i c i t expressions for the coherent f i e l d : (iv) The coherent f i e l d i n t e r n a l to the medium i s of the form <i|/) = A(z)e j k ' r + B(z)e J k ' ' r (0 < z < d) (A.8) where A and B are unknown functions of p o s i t i o n within the medium (k' has the magnitude k and the d i r e c t i o n of the i n c i d e n t f i e l d r e f l e c t e d i n the s l a b -region face) . (v) The average density p of s c a t t e r e r s within the slab region i s constant. Altogether, because of the assumptions and approximations required for mathe-matical manageability, the expressions obtained by Twersky are most v a l i d f or u n i f o r m l y - d i s t r i b u t e d a n i s o t r o p i c point s c a t t e r e r s ( l i n e s c a t t e r e r s i n two dimensions, plane sca t t e r e r s i n one dimension), i . e . , p(r l f...,E N) = (p/N) N (A.9) For the problem of a plane wave normally incident on a slab-region d i s t r i b u t i o n of i d e n t i c a l planar s c a t t e r e r s , the free-space theory gives the following equations f o r the coherent transmitted and r e f l e c t e d f i e l d s : <T > = D(l - Q2) e " J ( T 1 - 1 ) k d . (z > d) (A.10) where <R> = -QD(1 - e 2 j n k d ) (z < 0) (A. 11) P g - n 1 N /» 1 0 s , D = P = - (A. 12) Pg+ - Jk(n + D 1 - Q2 e _ 2Jnkd d n 2 = [ l - p(g+ + g _ ) / j k ] [ 1 - p(g + - g _ ) / j k ] (A.13) 8 As discussed by Twersky, these approximate expressions are i d e n t i c a l to those p e r t a i n i n g to the f i e l d produced when a normally i n c i d e n t plane wave scatters from a homogeneous d i e l e c t r i c slab of width d and r e f r a c t i v e index n. The asymptotic forms of these equations i n the l i m i t as p -»- 0 are ( T ) - e N g + (A. 14) <R)~ pg_(l - e 2 N g + ~ 2 j k d ) / 2 j k (A.15) Q ~ jpg_/k(n + 1) (A.16) n ~ 1 + jPg +/k (A. 17) Other d e t a i l s of the free-space theory are discussed i n reference 8. 2 A A.2 Twersky's Mixed-Space Theory f o r the Coherent F i e l d This theory i s also a p p l i c a b l e to the problem of a plane wave inc i d e n t on a slab-region d i s t r i b u t i o n of scatt e r e r s but i s based on "two-space' or "mixed-space" i s o l a t e d s c a t t e r e r amplitudes rather than the conventional amplitudes; In a mixed-space i s o l a t e d s c a t t e r e r problem the incident wave tra v e l s i n one medium (propagation constant K) and scatters from a s i n g l e s c a t t e r e r into another medium (e.g., free space with propagation constant k ). The mixed-space theory includes a l l the assumptions and approxi-mations of the free-space theory except assumption ( i v ) , the form of the coherent f i e l d i n t e r n a l to the medium. In the mixed-space theory (ty ) i s assumed to be of the form (ty > = Ae~iK'r + Be _ ; i K' * r (0 < z < d) (A. 18) where A and B are now constants, independent of p o s i t i o n z, and K i s the "bulk propagation constant" of the medium. For a coherent f i e l d of t h i s form, equation (A.7) represents a mixed-space i s o l a t e d s c a t t e r e r problem. For the one-dimensional ensemble of planar s c a t t e r e r s , the mixed-space formalism r e s u l t s i n the free-space theory equations (A.10) and (A.11) for ( T ) and (R) . However, for the mixed-space theory (n - D g l Q = — — (A.19) (n + l ) g | ' and the bulk r e f r a c t i v e index n = K/k s a t i s f i e s the f u n c t i o n a l equation F(n) = n 2 + — (g! - gl)n + — (g! + g'J - 1 = 0 (A.20) jk j k In these equations g^ _ and g^ are the mixed-space forward- and back-scattering amplitudes f o r a wave normally incident i n "K-space" on an i s o l a t e d planar s c a t t e r e r which sca t t e r s into "k-space". E x p l i c i t expressions for g^ _ and g^ for a d i e l e c t r i c slab of f i n i t e width are given i n Appendix B. The asymptotic free-space theory equations (A.14) to (A.17) apply also to the mixed-space theory since g^ _ -> g + , g^ _ -> g_, and n •> 1 as p -> 0. The s o l u t i o n of the mixed-space equations i n v o l v i n g n and the 147 mixed-space s c a t t e r i n g amplitudes i s straightforward for c e r t a i n types of s c a t t e r e r s . Twersky has obtained e x p l i c i t approximate solutions f or the separate cases of small s p h e r i c a l s c a t t e r e r s and large tenuous s c a t t e r e r s . In Appendix B an approximate s o l u t i o n i s given f o r " t h i n " d i e l e c t r i c slab s c a t t e r e r s and i t i s shown for these sca t t e r e r s (as shown by Twersky f o r small spheres) that g^ _ = g'_ 0 and n -> n' as p •> » (where n' i s the r e f r a c t i v e index of the slab m a t e r i a l ) . More generally, the exact s o l u t i o n of the mixed-space theory equations i s d i f f i c u l t and a numerical method must be used. A numerical s o l u t i o n of equation (A. 20) and those of g^J_ and %]_ for d i e l e c t r i c slab s c a t t e r e r s of f i n i t e width was performed i n the present work and i s discussed i n Appendix B. s e c t i o n B.2. The d i f f e r e n c e between the mixed-space and free-space theories a r i s e s 2 A f o r high p. As shown by Twersky, the mixed-space theory can approximately describe c e r t a i n dense d i s t r i b u t i o n s of f i n i t e - s i z e s c a t t e r e r s i f p i s i n t e r -preted as p = N/(V - NV ), where V i s the volume of the containing region and V g the volume occupied by a s i n g l e s c a t t e r e r . For slab-region d i s t r i b u t i o n s of small spheres he has shown that the bulk parameter equations of the mixed-space theory reduce to e x i s t i n g forms and that the l i m i t i n g behavior of such d i s t r i b u t i o n s and d i s t r i b u t i o n s of large tenuous s c a t t e r e r s as p -> 0 0 i s approximately c o r r e c t . Experimental measurements on a model d i s t r i b u t i o n of large tenuous s c a t t e r e r s have furthermore confirmed the approximate v a l i d i t y 49 of the mixed-space theory for that p a r t i c u l a r case. For a d i s t r i b u t i o n of " t h i n " d i e l e c t r i c slabs such that the approxi-mate expressions i n Appendix B for g_J_, g^ _, and n are v a l i d , the mixed-space theory gives almost exact r e s u l t s i n the l i m i t as p + 0 0 i f p = N/(d4w-Nw) and i f d i n the equations f o r (T), (R), and D i s replaced by d+w, the width of the slab region occupied by the sca t t e r e r s i n c l u d i n g t h e i r boundaries. This i s e a s i l y seen from equations (A.10), (A.11), and (A.19), since g'/g!_ - 1 f o r 148 " t h i n " slabs and -Q i s the Fresnel r e f l e c t i o n c o e f f i c i e n t f o r the boundary between the medium of the incident f i e l d and the slab region f i l l e d with s c a t t e r e r s of r e f r a c t i v e index n = n'. It i s confirmed from the numerical r e s u l t s i n Chapter 5, section 5.4. A.3 Theories for Other Average F i e l d Functions General d i s c r e t e - s c a t t e r e r theories for other average f i e l d functions have also been of i n t e r e s t i n the present work but have not been numerically evaluated f o r the one-dimensional model. One theory for ( l 2 ) developed by 8 11 Twersky ' i s based on the conservation of energy p r i n c i p l e . The general r e l a t i o n obtained by Twersky for a slab region of u n i f o r m l y - d i s t r i b u t e d i d e n t i c a l s c a t t e r e r s ( I 2 > = P J I u g s | 2 d r s (A.21) (u g i s the i s o l a t e d s c a t t e r e r function defined i n section 2.3) can be r e a d i l y evaluated for the one-dimensional model. The ( i 2 ) expressions obtained for the transmitted and r e f l e c t e d f i e l d s involve the free-space theory functions Q, D, and ri. In the l i m i t of p -> 0, these expressions reduce to < I 2 > ~ < I 2 ) ~ Thus, from the forward amplitude theorem f o r l o s s l e s s planar scatterers (see equation B.4 of Appendix B) and from the asymptotic forms for the coherent i n t e n s i t i e s as obtained from equations (A.14) and (A.15), i t i s r e a d i l y observed that energy i s conserved. This f a c t does not r e s u l t i n an accurate theory f o r the one-dimensional model, however, as shown by the t h e o r e t i c a l r e s u l t s of s e c t i o n 2.7. Another theory for (|^|2) developed by Twersky^ which also Jit!! (1 - e2NReg + ) > 2 Reg + J i J ! (1 - e 2 N R e S + ) (z < 0) (A. 23) 2 Reg + s a t i s f i e s the energy p r i n c i p l e i s the (^)-consistent approximation. This theory was obtained by the in t r o d u c t i o n of s i m i l a r approximations into the se r i e s representation for (|^|2) as were shown to e x i s t i n the expanded form of the compact representation f o r (i>). Numerical r e s u l t s for t h i s theory as applied to a d i s t r i b u t i o n of large tenuous scat t e r e r s have been shown to com-pare w e l l with experimental r e s u l t s from a p h y s i c a l model of the d i s t r i b u t i o n . Similar r e s u l t s f o r the a p p l i c a t i o n of t h i s theory to the one-dimensional model could also be of value. APPENDIX B SCATTERING FROM A SINGLE DIELECTRIC SLAB B.1 Conventional Scattering Amplitudes For a plane wave normally incident i n free space on a l o s s l e s s d i e l e c t r i c slab of width w = 2a and r e f r a c t i v e index n 1 = /e~ as shown i n fi g u r e B . l , the forward- and back-scattering amplitudes g + and g_ are given r e s p e c t i v e l y by 1 + g+ = T1 = i tU-i. (B.l) and where A' g_ = R, = (n'2 - l ) ( e - 2 ^ ' ' k a - e 2 J n ' k a ) e 2 J k a A' A' = (n 1 + l ) 2 e 2 J ^ k a - (n 1 - l ) 2 e" 23n' ka (B.2) (B.3) The s i n g l e slab transmission and r e f l e c t i o n c o e f f i c i e n t s and Rj are re f e r r e d to the slab center as are the s c a t t e r i n g amplitudes. The forward Q amplitude theorem 0 f o r the slab s c a t t e r e r , r e l a t i n g the t o t a l s c a t t e r i n g c ross-section a = | g + | 2 + |g_| 2 to the forward-scattering amplitude g + i s |g+|2 + |g_| 2 = -2 Reg + (B.4) Expansion of the exponentials i n the expressions f o r g + and g_ y i e l d s (l+g+)e-J kz Figure B . l Scattering from a Single D i e l e c t r i c Slab 151 the approximate expression f o r " t h i n " slabs ( i . e . , n'ka << 1) 8+ a g_ = ~ ( n ' 2 - D k a [ ( n ' 2 - D k a + j ] & g m (B.5) Thus, f o r a "thin" s l ab, the back-scattering cross-section 21g_|2 and the forward-scattering cross-section 2|g +| 2 are approximately equal. For a slab of d i e l e c t r i c constant e r = 2.0 ( i . e . , the value used f o r the numerical r e s u l t s i n the t h e s i s ) , the rectangular components of g + , g_ and g m are plo t t e d as functions of the slab width w^i ( i . e . , width i n wavelengths A' within the slab material) i n f i g u r e B.2. For a plane wave obliquely incident on the slab at an angle 0 with the i n t e r f a c e normal, the s c a t t e r i n g amplitudes are more generally given by 4Z' e2jkacos0 (1 + Z ' ) 2 e 2 i n ' k a c o s 0 r - (1 - Z ' ) 2 e-2jn'kacos0 r 1 + g+ = „ „,_„ s ~ - — (B.6) ci _ 7'2w -2jn'kacos0 r _ 2jn'kacos0 r s 2jkacos0 g_ = U Z K £ }—, (B.7) (1 + Z ' ) 2 e 2 j n kacos0 r _ ( i _ z ' ) 2 e " 2 J n k a c o s 0 r where cos 0 r = /n' 2 - sin 2© (B.8) For a wave of perpendicular p o l a r i z a t i o n ( i . e . , e l e c t r i c f i e l d vector perpendicular to the plane containing the i n t e r f a c e normal and the propagation v e c t o r ) , COS0 Z' = / , 9 o (B.9) /n' 2 - s i n 2 0 For a wave of p a r a l l e l p o l a r i z a t i o n , 153 A l l the given equations can be generalized to include losses i n the slab by replacement of e r with the complex quantity e r ( l - j tan5), where tan6 i s the loss tangent of the slab material. B.2 Mixed-Space Scattering Amplitudes For a plane wave <J>(z) = e J n k z normally incident on the slab within a medium of r e f r a c t i v e index n, and with the scattered waves again t r a v e l l i n g 23 i n free space, the more general s c a t t e r i n g amplitudes are given by g| = ^ [ 2 n ' ( l + n ) e J n k a + (n« - l ) (n ' - n ) e-J(n+2n')ka A (B.ll) - (n' + D(n' + n ) e-j ( n-2n')ka] g- = ^ 7 - [ zn 'd - tfe-t"** - (n' + D(n' - n ) e J ( n + 2 n ' > k a (B.12) + (n' - D(n* + n ) e J ( n - 2 T 1 ' ) k a ] For a th i n slab, these mixed-space s c a t t e r i n g amplitudes are given by the approximate expression g| - gl - - (n' 2 - n 2)ka [(n' 2 - l) k a + j ] = gm (B.13) corresponding to the r e s u l t of equation (B.5) for the conventional amplitudes. For n -> 1, these equations a l l reduce to the conventional forms. In Twersky's mixed-space theory f o r the coherent f i e l d , n i s i d e n t i f i e d with the bulk r e f r a c t i v e index of the random medium, given by equation (A.20) of Appendix A. Substit u t i o n of the " t h i n " slab equation (B.13) into equation (A.20) y i e l d s 2 1 + 2ap [ 1 - j ( n ' 2 - Dka] n' 2 rr = z — (B.14) 1 + 2ap [ 1 - j ( n ' 2 Dka] Thus, as seen by t h i s expression and that of equation (B.13), n H' and g+ ~ 8- ~ S m 0 a s P 0 0• F o r P interpreted as p = N/(d + w - Nw) i n the manner discussed i n Appendix A (where w = 2a i s the scatterer width), equation (B.14) can be rewritten i n terms of the f r a c t i o n a l "volume" 8 0 = Nw/(d + w) as 6 0 ( n ' 2 - 1) [ 1 - j ( n ' 2 - Dka] n - 1 + -1 " j B 0 ( n - Dka (B.15) - 1 + 3 0 ( n ' 2 - .1) [ 1 - j ( l - e 0 ) ( n ' 2 - Dka] Exact s o l u t i o n of the mixed-space theory equations ( B . l l ) , (B.12), and (A.20) i s possible only by means of a numerical technique. For comparison of mixed-space theory r e s u l t s with "exact" simulation r e s u l t s i n Chapters 4 59 and 5, the Newton-Raphson method was employed. This well-known i t e r a t i v e technique makes use of the equation F<ni-i> 1,1 - "t-1 - F^ T ) ( B - 1 6 ) where i s the value f o r n a f t e r i i t e r a t i o n s and F'(n) i s the d e r i v a t i v e of F(n) with respect to n- The i n i t i a l value n 0 used i n the i t e r a t i o n was the free-space theory value given by equation (A.13). For the s p e c i f i c s c a t t e r i n g parameters chosen f o r study, t h i s i n i t i a l value was s u f f i c i e n t l y close to the actual mixed-space value f o r quick convergence of equation (B.16). 155 APPENDIX C VALIDITY OF THE DISCRETE POSITION APPROXIMATION IN SIMULATION STUDIES The v a l i d i t y of a d i s c r e t e p r o b a b i l i t y density approximation to a continuous p r o b a b i l i t y density of s c a t t e r e r p o s i t i o n s i s of i n t e r e s t i n t h i s work mainly as i t pertains to the construction of the three-dimensional p h y s i c a l model discussed i n Chapter 6. However, the a p p l i c a t i o n of such an approximation i n computer simulation studies of mathematical models may also allow the use of s i m p l i f i e d e f f i c i e n t techniques for processing the random numbers involved (e.g., i n the s o r t i n g or r e j e c t i o n procedures used) and i t s v a l i d i t y i s therefore of more general i n t e r e s t . In order that i n s i g h t i n t o the problem might be obtained, the d i s c r e t e p o s i t i o n approximation (DPA) has been applied to the one-dimensional model considered i n previous chapters. The r e s u l t s of the study are given i n t h i s section and r e l a t e d where possible to the three-dimensional model. Results are given for two types of d i s c r e t e p r o b a b i l i t y d e n s i t i e s of s c a t t e r e r p o s i t i o n s . In the f i r s t type ( l a b e l l e d " d i s c r e t e non-uniform"), the one-dimensional equivalent of that used for the three-dimensional model, the s c a t t e r e r p o s i t i o n s are chosen uniformly at random from those a v a i l a b l e under the condition that at most one s c a t t e r e r occupy any one p o s i t i o n ( i . e . , Fermi-Dirac " s t a t i s t i c s " i n s t a t i s t i c a l mechanics). In the second type ( l a b e l l e d " d i s c r e t e uniform"), the p o s i t i o n s are chosen uniformly at random from a l l those a v a i l a b l e with no r e s t r i c t i o n on the number of the N s c a t t e r e r s per p o s i t i o n ( i . e . , Bose-Einstein " s t a t i s t i c s " i n s t a t i s t i c a l mechanics). The average f i e l d functions shown f o r both types of d i s c r e t e d i s t r i b u t i o n are p l o t t e d against the "occupancy r a t i o " 6^ = N/n^, where n^ i s the number of equally-spaced posi t i o n s a v a i l a b l e . To most c l e a r l y display the e f f e c t of the DPA, n^ i s v a r i e d rather than N, which i s f i x e d at N = 10. The planar s c a t t e r e r amplitudes employed, g + = 0.2107 7-101.7° and g_ = 0.2035 /-1Q2.20, 156 are the same as those used e a r l i e r . The r e s u l t s are based on the exact wave • matrix theory f o r a f i x e d configuration with 1,000 sample configurations. Results f o r the d i s c r e t e non-uniform d i s t r i b u t i o n are given i n fig u r e C l . Shown are curves of the phase and i n t e n s i t y of the coherent trans-mitted f i e l d and the average incoherent i n t e n s i t y of the r e f l e c t e d f i e l d f o r a s e r i e s of d^ values. The e n t i r e curves for d^ = 2, 7, and 12 are shown for completeness although i t i s the f i r s t monotonically-varying portions which are of present i n t e r e s t . Smooth curves have been drawn through a l l "experimental" points (included f or .d^  = 7) except those f o r small values of n^ where s t r a i g h t -l i n e segments are used to i n d i c a t e the d i s c r e t e nature of the r e s u l t s . Curves for ( I 2 ) of the transmitted f i e l d are not given because they are s i m i l a r i n form to those f o r the r e f l e c t e d f i e l d . The r e s u l t s of fi g u r e C l show the combined e f f e c t of the DPA and the s i n g l e - s c a t t e r e r - p e r - p o s i t i o n requirement. They may be likened to those of section 5.3 for the continuous non-uniform d i s t r i b u t i o n and e s s e n t i a l l y the same arguments may be applied to explain the o s c i l l a t o r y behavior displayed i n the curves. The main differ e n c e i n the present r e s u l t s i s that resonance phenomena occur f o r c e r t a i n mid-range values of 3^ where the d i s c r e t e - p o s i t i o n i n t e r v a l d/n^ i s approximately equal to an i n t e g r a l multiple of A/2. For l a t e r comparison with the d i s c r e t e non-uniform r e s u l t s f o r p^ = 5 ( i . e . , d^ = 2) , r e s u l t s f o r a continuous non-uniform d i s t r i b u t i o n are also given i n fi g u r e C l . To provide a v a l i d comparison, these r e s u l t s have been obtained using method A of Chapter 5 with the a d d i t i o n a l requirement that the distance of clo s e s t approach of zj and z^ with the slab region boundaries be e/2, where e = d/n^. The f r a c t i o n a l "volume" 3 Q = Ne/d f o r the continuous d i s t r i b u t i o n i s then equivalent to the occupation r a t i o 3^ = N/n^ for the d i s c r e t e d i s t r i b u t i o n . The e f f e c t of the DPA alone i s displayed by the d i s c r e t e uniform d i s t r i b u t i o n r e s u l t s of f i g u r e C.2. Although resonance phenomena occur also 157 -a 150c 130' 110' 90' • 1 j / \ / i I / ' 1 ' I I J i J • • 1 /" ! / i / i i i i i« \ A f / 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 (a) -a versus 8^ - Transmitted F i e l d 1.0 — — — y ^ C o - ^ < r 4 o - x r _ 0 . \ • y \ / \ / V M i i i i f / \ \ / * \ / v i i i i I \! / / 0.2 0.4 0.6 0.8 1.0 0.5 0.4 0.3, <I2> 0.2 0.1 (b) C 2 versus 8 d - Transmitted F i e l d ; 4 / /! — O—_Q__  1 ! O-0-K>-^  \ * \ ' » ^ ^ ^ ^ ^ r - ^ ^ \y \ \ i / i ~~~~~ \ 0.2 0.4 0.6 0.8 1.0 (c) ( i 2 ) versus 8 d - Reflected F i e l d Figure C l Dependence of Average F i e l d Functions on 8 d for the Non-Uniform D i s t r i b u t i o n . N = 10, wx? = 0.1, e r = 2.0; — d-y = 2 ( d i s c r e t e ) , -o dx = 7, d A = 12, 2 (continuous), 17, dx = 97 Figure C.2 Dependence of Average F i e l d Functions on Bj f o r the Uniform D i s t r i b u t i o n . N = 10, w^ , = 0.1, e r = 2.0; d x = 2, d A = 7, d x = 12, d A = 17 for t h i s d i s t r i b u t i o n , only the f i r s t monotonically-varying portions of the curves are given. A comparison of figures C l and C.2 shows the s i n g l e -s c a t t e r e r - p e r - p o s i t i o n requirement of the non-uniform d i s t r i b u t i o n to be the predominant cause of deviation from the continuous d i s t r i b u t i o n l i m i t ( i . e . , B d = 0) f o r low values of 3 d and high p-^ , but the DPA to be i n c r e a s i n g l y more dominant for decreasing p^. Since the p h y s i c a l differences between the one-dimensional planar-s c a t t e r e r model and the three-dimensional s p h e r i c a l - s c a t t e r e r model are considerable, the e f f e c t of the DPA i n the l a t t e r can be expected to be some-what d i f f e r e n t . Some comparison can be made, however, once the main s c a t t e r i n g processes i n each model are i d e n t i f i e d and written i n the form of random phasor sums. The transmitted f i e l d , as discussed i n section 2.8, can be written as N I T e J T = A o e J 9 o + £ A s e l 9 s ( C l ) s=i T fl T fl where A Q e J 0 i s a constant phasor, the A g e J s are random phasors representing the s i g n i f i c a n t s c a t t e r i n g contributions to the random component of the f i e l d , and i s the number of these contributions. The s i g n i f i c a n t s c a t t e r i n g processes present i n the one-dimensional model have already been i d e n t i f i e d and the approximate transmitted f i e l d w r i tten i n the form of ( C l ) i n section 2.8. For r e l a t i v e l y low p i n the three-dimensional model, i t i s w e l l known that "i fl s i n g l e s c a t t e r i n g i s the only s i g n i f i c a n t process. In t h i s case, A Q e J o = 1 and, f o r a f i e l d point on the beam axis (see f i g u r e C.3), G(a * ) | f ( Y s , * s ) | ( d t + d ) A = — - — — (C.2) s G ( 0 , 0 ) t s v s fi« = "k(t_ + v - d t - d_) + Argf(Y„,<U (C.3) where G ( a s > ^ s ^ = f i e l d pattern f a c t o r of transmitting antenna i n the d i r e c t i o n s p e c i f i e d by a g , <(> G(0,0) = f i e l d pattern f a c t o r along the beam axis fCYgj'f'g) = ~ g ( v s 5 t s ) / j k = s c a t t e r i n g amplitude (unnormalized) of s p h e r i c a l s c a t t e r e r i n the d i r e c t i o n s p e c i f i e d by y s > <j>s cf>3 = polar coordinate i n x,y-plane t s = / ( d t + z s ) 2 + r s 2 v s = / ( d r - z s ) 2 + r s 2 160 Figure C.3 Single-Scattering Geometry f o r the Three-Dimensional Model The number of s i g n i f i c a n t s c a t t e r i n g contributions N-j- i s equal to the number of s c a t t e r e r s i n the volume of i l l u m i n a t i o n i n which the A g are s i g n i f i c a n t l y large. Although the A g are functions of the s c a t t e r e r amplitudes as w e l l as the r a d i a t i o n pattern of the transmitting antenna, the l i m i t s of the s i g n i f i c a n t volume may usually be assumed to l i e between the l i m i t s of that i l l u m i n a t e d by the half-power beam and that by the main lobe. I t seems reasonable that the v a l i d i t y of the DPA i s r e l a t e d to the following interconnected f a c t o r s : (i) the number of basic phase cycles over which the 8 g vary, ( i i ) the number of d i s c r e t e values of the 9 g within a basic phase cycle 161 of v a r i a t i o n , ( i i i ) the number of di s c r e t e values of the A g over t h e i r range of v a r i a t i o n , and (iv) the average number of s i g n i f i c a n t s c a t t e r i n g contributions Nj to the random component of the f i e l d . Since the A g are constant i n the one-dimensional model and random i n the three-dimensional model, i t i s i n the r e l a t i v e "discreteness" of the phases 0 S that comparisons between the two models can best be made. In the one-dimensional model the range of 6S v a r i a t i o n i s governed by d^ and N (or P A ) . In p a r t i c u l a r , i t has been shown i n Chapter 5 that CT0S ~ ^Trd^/ (N - 1) f o r uniformly-distributed s c a t t e r e r s . As seen from the re s u l t s of fig u r e C.2, the required number of di s c r e t e values of 0G per basic cycle f o r a good approximation to a continuous d i s t r i b u t i o n decreases f o r increasing O"Qs • With p^ = 5 and O Q s = 0.89TT, f o r example, a good approximation fo r the average f i e l d functions investigated i s maintained f o r 3^  as high as 0.6, or the number of d i s c r e t e values of 6G per 2ir radians as low as four. For p x = 0.103 ( i . e . , d x = 97) and O Q s = 43TT, a minimum of s l i g h t l y greater than one d i s c r e t e value of 0G per basic phase cycle r e s u l t s i n a good approxi-mation and, except f o r a small i n t e r v a l of 3^  values i n the neighbourhood of resonance ( f o r these parameters the f i r s t resonance occurs f o r 3^ - 0.05), a di s c r e t e i n t e r v a l i n 0G even larger than 2ir r e s u l t s i n a reasonably good approximation (see figure C l for these r e s u l t s ) . In the three-dimensional model the range of the 0G v a r i a t i o n s i s governed mainly by the average density of scatt e r e r s p, the distances of the source and f i e l d points from the medium, and the average locati o n s of the sc a t t e r e r s within the s i g n i f i c a n t volume. This may be shown as follows: The s p h e r i c a l s c a t t e r e r s are i d e n t i f i e d with average c y l i n d r i c a l - c o o r d i n a t e l o c a t i o n s (<r g) , (<J>g) , ( z g ) ) i n a manner analogous to that i n which the planar 162 scatterers were considered i n terms of t h e i r ordered-positions and e s s e n t i a l l y i d e n t i f i e d with t h e i r average posi t i o n s ( z s ) . For slowly-varying Argf (Ys>4>s) > v a r i a t i o n i n 8 g i s caused mainly by deviations i n r g and z g from t h e i r means; to f i r s t - o r d e r terms 14 3 6 g « r s > , < z s » 3 r c a- 2 4-36 « 0 , < z R » ' 8z c (C.4) P h y s i c a l reasoning or evaluation of both terms i n (C.4) shows further that v a r i a t i o n i n 6 g i s caused mainly by v a r i a t i o n i n r g and hence that 3 9 f i ( < 0 , < 0 ) 3r c Thus, from equation (C.3), 2 7 r < r s > ° r g r + / ( d r + <z„>) 2 + < r „ ) 2 /(d_ - <z„)) 2 + r„ 2 . ( C 5 ) (C.6) For the uniform-randomness of the s c a t t e r e r s i n the present model, the number of s c a t t e r e r s i n a volume much smaller than that of the slab region i s approximately Poisson d i s t r i b u t e d ; or more s p e c i f i c a l l y , the d i s t r i b u t i o n of the increase i n an a r b i t r a r i l y - s h a p e d volume V s about an average l o c a t i o n before a s c a t t e r e r i s reached i s approximately exponential with p(V g) = pe D ^ s , In s p h e r i c a l polar coordinates (rg,6',<j>s) about the average l o c a t i o n s , V g = 47rr g 3/3 and , 2 = ( r „ ) 2 + r : 2 s i n 2 0 ' - 2<r_>r' s i n e ! cos<j>' ( C 7 ) Thus, 2 _ <rs>' /•00 r TT r 2TT o -'o e-47T Pr g 3/3 [ r ^ 2 s i n 2 g [ r g 2 s i n 2 e g - 2 < r g ) r g sine' cos<i»s] r ; 2 s i n 0 g d*; dO s d r g /• o 0 r,k e-47T Pr s 3/3 d r . c 2p 2 / 3 (C.8) or -1/3 * 2 ^ - 1 / 3 a = cp 1 1 3, c = -'•s 3 4lT\ _ 1 / 3 , — J /r(2/3) (C.9) where r i s the well-known Gamma function. I t i s of i n t e r e s t to compare c e r t a i n numerical r e s u l t s based on equations (C.6), (C.9.), and the experimental parameters s p e c i f i e d i n Chapter 6 with those given for the one-dimensional model. The number m of equivalent d i s c r e t e values of 0 g per basic phase c y c l e , where m^  i s the number of d i s c r e t e p o s i t i o n s per inch, i s given by 2irm.a 1 r s m = (CIO) s s For p = 183 scf and an average s c a t t e r e r l o c a t i o n on the x,y-plane ( i . e . , (z ) = 0) and the edge of the volume illuminated by the main lobe, a a = l . l i r b a s and m = 3.6; for p = 366 s c f , O g = 0.9TT and m = 3.6 (independent of p). For s p = 183 s c f and an average l o c a t i o n on the edge of the volume illuminated by the half-power beam, OQ = 0.47TT and m = 8.5; for p = 366 s c f , O g = 0.38TT. s s For an average l o c a t i o n at ( z s ) = 0 and ( r g ) = a with p = 183 s c f , s O g = 0.27TT and m = 15; with p = 366 s c f , O g = 0.18-rr and m = 18. s s Although a g ^ i n the three-dimensional model i s a function of the average p o s i t i o n of each s c a t t e r e r , i t i s evident from the preceding r e s u l t s that the average O g g would be of the same order of magnitude as the value O g =0.89 obtained f o r p^ = 5 i n the one-dimensional model. If i t i s assumed that the r e l a t i o n s h i p between O g g and m f o r the DPA to be v a l i d i s the same as i n the one-dimensional model, the d i f f e r e n c e i n U g ^ for each s c a t t e r e r does not matter since smaller a g are accompanied by larger m. I f i t i s further 164 assumed that a c t u a l values of a A and m i n the one-dimensional model can be °s used to predict the v a l i d i t y of the DPA i n the three-dimensional model, the values of m f o r the three-dimensional O Q ^ obtained appear to be approximately within the necessary l i m i t s . It i s i n fac t possible that the numerical r e l a t i o n s h i p between m and O g g for the v a l i d i t y of the DPA might need to be even les s stringent than i n the one-dimensional model, since the unequal truncation i n t e r v a l s i n 0 S would seem to preclude the p o s s i b i l i t y of strong resonance e f f e c t s as i n the one-dimensional model. The v a l i d i t y of the d i s c r e t e non-uniform p r o b a b i l i t y density which i s made necessary i n a DPA accounting f o r f i n i t e s c a t t e r e r s i z e i s also of i n t e r e s t . As seen by the one-dimensional model r e s u l t s of figures C l and C.2 fo r p^ = 5, the e f f e c t of the DPA i s noticeable f o r lower values of 3^ i n the di s c r e t e non-uniform d i s t r i b u t i o n than i n the dis c r e t e uniform d i s t r i b u t i o n . The explanation f o r t h i s would seem to be that OQS remains r e l a t i v e l y constant for increasing i n the uniform case whereas i n the non-uniform case i t does not, being given approximately by OQ^ - 4 7 r ( l - 3 ( j ) d X / ( N - l ) = 4TT(1-B ( J ) / P x i n analogy to equation (5.8) for the continuous non-uniform d i s t r i b u t i o n . Thus, fo r a given 3^, a f l i s smaller f o r the non-uniform d i s t r i b u t i o n than f o r the uniform d i s t r i b u t i o n while m i s unchanged. For the three-dimensional model i t would seem p l a u s i b l e to account f o r the non-uniformity r e s u l t i n g from f i n i t e s c a t t e r e r s i z e i n the same manner. While the replacement f o r ar = c p - 1 ^ 3 i n equation (C.6) would be a r = c [ ( l - 3 0 / 3 m ) / p ] 1 ^ 3 (where 3 m i s the maximum 3 0 p h y s i c a l l y possible) i n the continuous case, i t would be a = c [ ( 1 - 8 J ) / p ] 1 ^ 3 i n the d i s c r e t e case. I f s t h i s one-third power (or any f r a c t i o n a l power) co r r e c t i o n i n 1-3^ i s a c t u a l l y v a l i d , higher values of 3^ i n the three-dimensional model than i n the one-dimensional model might be tolerated before breakdown of the DPA. Any estimation of the maximum t o l e r a b l e 3^ i s complicated by the f a c t that the 165 v a l i d i t y of the s i n g l e - s c a t t e r i n g approximation i s also dependent on Bj, however, and i s thus impossible at present. The low values of B d = 0.013 and B d = 0.026 for the experimental d i s t r i b u t i o n parameters of Chapter 6 would i n any case seem e a s i l y t o l e r a b l e since they change the numerical values already given for Og very l i t t l e . Much further work incl u d i n g a l l four factors mentioned i s necessary to e s t a b l i s h d e f i n i t e c r i t e r i a f o r the v a l i d i t y of the DPA i n the three-dimensional model. U n t i l such c r i t e r i a are developed, however, one precaution can be taken to minimize the DPA error f or higher density d i s t r i b u t i o n s than those used for the experimental r e s u l t s of Chapter 6. This i s to use a d i s c r e t e - p o s i t i o n i n t e r v a l i n the coordinate d i r e c t i o n s transverse to the beam smaller than the diameter of the s c a t t e r e r s , the one-scatterer-per-position requirement being modified accordingly. If no DPA were employed i n these coordinate d i r e c t i o n s at a l l , a two-dimensional " r e j e c t i o n " technique s i m i l a r to those used i n Chapter 5 for the one-dimensional model could be employed. Although the necessity of support-medium layers i n the present model l i m i t s the d i s c r e t e - p o s i t i o n i n t e r v a l p a r a l l e l to the beam to be no smaller than the s c a t t e r e r diameter, i t i s evident from the preceding discussion that the r e s u l t i n g e r r o r would be le s s important than that f o r the transverse coordinate d i r e c t i o n s as long as s i n g l e s c a t t e r i n g remains the only s i g n i f i c a n t process. It i s the presence of s i g n i f i c a n t m u l t i p l e - s c a t t e r i n g e f f e c t s and the necessity for a DPA i n the coordinate d i r e c t i o n p a r a l l e l to the beam that must ultimately l i m i t the range of average d e n s i t i e s possible f or study with the present model. 166 APPENDIX D DESIGN OF MICROWAVE ANECHOIC CHAMBER AND POSITIONING DEVICE D.1 Design and Testing of Anechoic Chamber A small microwave anechoic chamber has been designed and constructed fo r general use i n s c a t t e r i n g experiments of the type described i n Chapter 6 and i n antenna pattern measurements. The design i s standard, a wedge-shaped back w a l l and l o n g i t u d i n a l l y - b a f f l e d side walls being employed to achieve a c e n t r a l "quiet" volume region, and therefore only l i m i t e d d e t a i l s are given. A s i m p l i f i e d plan-view diagram of the chamber i s shown i n f i g u r e D . l . Inside dimensions (not i n c l u d i n g absorbing material) are approximately 15*4" x 9'4" x 10' (height). The chamber s h e l l i s constructed of two-by-four framing and plywood sheeting i n four foot sections bolted together which allows for easy extension i n length. Power receptacles are located at s u i t a b l e places both i n s i d e and outside, s w i v e l - f i x t u r e l i g h t s are located i n the four upper corners, and v e n t i l a t i o n i s provided by means of an exhaling fan and vents. A l l i n s i d e f i x t u r e s are located at l e a s t c r i t i c a l p o s i t i o n s to minimize r e f l e c t i o n s . The l e a s t c r i t i c a l surface areas are covered with Emerson & Cuming Eccosorb FR330 absorbing m a t e r i a l . B. F. Goodrich VHP-4 absorbing material covers the c r i t i c a l back-wall area and a s i x foot wide area of the side w a l l s . The minimum r e f l e c t i v i t y l e v e l s f or these absorbers at X-band are -20db and -45db r e s p e c t i v e l y . The FR330 material l i m i t s the operating frequency range to frequencies above 2.3 GHz. The l o n g i t u d i n a l l y - b a f f l e d side walls and wedge-shaped back wall produce a thirty-one inch diameter "quiet zone" ( i . e . , volume i n which s p e c u l a r l y - r e f l e c t e d contributions to the t o t a l f i e l d involve no fewer than two surface r e f l e c t i o n s ) running the e n t i r e length of the chamber to within nineteen inches of the back-wall apex. Det a i l e d "one-way transmission" tests were performed on the chamber 167 ige of absorbing material on b a f f l e zzzzzzzzz: 16' Figure D.l S i m p l i f i e d Plan-View Diagram of the Anechoic Chamber at 9.32 GHz and 35.0 GHz using the B. F. Goodrich "free-space VSWR" technique.60,61 with the remotely-controlled p o s i t i o n e r described i n the following section used as a mount f o r the rec e i v i n g antenna ( v e r t i c a l l y p o l a r i z e d ) , traverses along two-foot h o r i z o n t a l and v e r t i c a l r a d i i from the room axis were made at a distance of twenty-five inches from the back-wall apex, with the aspect angle 6 of the antenna (see figure D.l) set at 10° increments between 40° and 320°. The 9.32 GHz measurements were made with 16db Narda type 640 transmitter and receiver horns and the 35.0 GHz measurements with the horns used f o r the experiments described i n Chapter 6. Calculations based on the standing-wave patterns obtained showed an average r e f l e c t i v i t y l e v e l ^ within a two-foot diameter s e c t i o n of the quiet zone of -53db at 9.32 GHz (averaged over a l l aspect angles and one foot h o r i z o n t a l and v e r t i c a l r a d i i ) and better than -66db at 35.0 GHz. This l e v e l dropped to only -51db at 9.32 GHz along one-foot r a d i i sections centered 1.5 feet from the chamber a x i s . Plots of the r e f l e c t i v i t y l e v e l versus aspect angle showed the largest source of r e f l e c t e d power to be the back w a l l , as expected. For an aspect angle 168 of 6 = 0° (as used f o r the Chapter 6 experiments), no r e f l e c t i o n s from the side walls were evident and for other forward aspect angles the r e f l e c t i v i t y l e v e l was considerably lower than the average. D.2 Design of Remotely-Controlled P o s i t i o n i n g Device A remotely-controlled p o s i t i o n i n g device has been designed and constructed f o r general use as a scanning platform i n the type of experiments described i n Chapter 6 and as an antenna p o s i t i o n e r f o r pattern measurements and tests on the anechoic chamber. This device i s capable of a four-foot h o r i z o n t a l movement, a one-foot v e r t i c a l movement, and a 390° azimuthal movement. The photograph shown i n fi g u r e D.2 i l l u s t r a t e s the main features of the mechanical design: The vertical-movement system i s mounted on the azimuth ro t a t o r and both these systems are f i x e d to a platform which i s propelled on wheels along two h o r i z o n t a l tracks. A screw coupled to a dc gear motor provides the h o r i z o n t a l drive and a screw coupled to a gear t r a i n and ac servo motor, the v e r t i c a l d r i v e . An ac servo motor and gear t r a i n also drives the azimuth r o t a t o r . The tubular center shaft allows e i t h e r a s c a t t e r i n g medium or an antenna to be e a s i l y mounted i n place. Accurate open-loop c o n t r o l of the three coordinate posi t i o n s i s achieved i n the device by means of the speed-controllable motors and transmit-receive synchro systems. The co n t r o l box, located outside the anechoic chamber, i s shown i n the center of the photograph on page 122. P o s i t i o n readout of the h o r i z o n t a l and v e r t i c a l coordinates i s derived by means of counters driven by s i n g l e synchro r e c e i v e r s . Dual synchro transmitters and receivers i n the azimuth system, with gear r a t i o s of 1:1 and 36:1, provide both a course and f i n e i n d i c a t i o n of azimuth on two graduated c i r c u l a r d i a l s . P o s i t i o n readout r e s o l u t i o n i s 0.00625 inches i n the h o r i z o n t a l d i r e c t i o n , 0.005 inches i n the v e r t i c a l d i r e c t i o n , and 0.1° i n azimuth; accuracy i s only Figure D.2 View of the P o s i t i o n i n g Device s l i g h t l y l e s s . Potentiometers coupled to a l l three coordinate systems i n the positioner provide an e l e c t r i c a l output of p o s i t i o n f or c o n t r o l l i n g an X-Y p l o t t e r used i n the measurement of free-space standing-wave patterns and antenna patterns. A relay-battery c i r c u i t connected to the h o r i z o n t a l motor input provides a two-level control voltage for use i n processing the data recorded during scanning of the physical model described i n Chapter 6. Limit switches incorporated i n a l l three motor-drive c i r c u i t s remove power to the motors when the physical l i m i t s i n the positioner are reached. 170 REFERENCES 1. S i l v e r , S. 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