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Scattering of electromagnetic waves by long radially inhomogeneous isotropic cylinders Parkinson, Robert George 1969

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SCATTERING OP ELECTROMAGNETIC WAVES BY LONG RADIALLY INHOMOGENEOUS ISOTROPIC CYLINDERS  ROBERT GEORGE PARKINSON B.A.Sc., The U n i v e r s i t y o f B r i t i s h Columbia, 1 9 6 ^ A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  i n t h e Department o f E l e c t r i c a l  Engineering  We a c c e p t t h i s t h e s i s as c o n f o r m i n g required  to the  standard  Research Supervisor Members o f t h e Committee  Head o f Department Members o f t h e Department of E l e c t r i c a l  Engineering  THE UNIVERSITY OF BRITISH COLUMBIA February, @  1969  Robert George Parkinson  1969  \ \  In p r e s e n t i n g an  this  thesis  advanced degree at  the  Library  I further for  shall  the  his  of  this  agree that  written  of  be  British  available for for extensive  g r a n t e d by  the  It i s understood  for financial  gain  shall  Eu  £.c  TAl&Al*  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  Head o f my  be  Z£r^  -2-0  ,  6  <?  I agree  r e f e r e n c e and  for  that  Study.  this  thesis  Department  copying or  or  publication  allowed without  g AJ G) *J & £ AI *J 6>  Columbia  requirements  c o p y i n g of  that  not  the  Columbia,  permission.  Department of  Date  University  permission  representatives. thesis  f u l f i l m e n t of  make i t f r e e l y  s c h o l a r l y p u r p o s e s may  by  in p a r t i a l  my  ABSTRACT The problem o f n o r m a l - i n c i d e n c e  s c a t t e r i n g by  i s o t r o p i c cylinders with arbitrary r a d i a l p e r m i t t i v i t y vari a t i o n and by p e r f e c t l y c o n d u c t i n g  c y l i n d e r s surrounded by  r a d i a l l y inhomogeneous i s o t r o p i c s h e l l s i s s t u d i e d . types of approximation  a r e c o n s i d e r e d , namely  Two  ( i ) an approx-  i m a t i o n of. t h e p e r m i t t i v i t y v a r i a t i o n w h i c h a l l o w s t h e use o f a power s e r i e s s o l u t i o n o f t h e wave e q u a t i o n and  ( i i ) approx-  i m a t i o n of the c y l i n d e r by a l a y e r e d s t r u c t u r e .  For the l a t t e r  t y p e , c o m p u t a t i o n s a r e c a r r i e d out f o r homogeneous s h e l l s and for  shells with linearly-varying permittivities.  are compared w i t h those o b t a i n e d by n u m e r i c a l  The r e s u l t s  i n t e g r a t i o n of  the R i c c a t i - t y p e d i f f e r e n t i a l e q u a t i o n f o r impedance o r admittance.  I n g e n e r a l , the homogeneous-shell a p p r o x i m a t i o n  appears  to be e a s i e s t t o a p p l y and r e q u i r e s a r e l a t i v e l y s h o r t computation  time. I t i s shown t h a t the s c a t t e r e d - f i e l d  coefficients  can be c a l c u l a t e d from measurements o f the s c a t t e r e d , f i e l d a t a s i n g l e r a d i u s by a p p l y i n g a F o u r i e r l e a s t - s q u a r e s f i t t o the data.  The s c a t t e r e d f i e l d f o r plane-wave i n c i d e n c e can t h e r e -  f o r e be c a l c u l a t e d from t h a t f o r c y l i n d r i c a l - w a v e i n c i d e n c e , w h i c h s u g g e s t s a more compact system f o r e x p e r i m e n t a l  investi-  gation. C y l i n d e r s w i t h a "smoothly" v a r y i n g p e r m i t t i v i t y were c o n s t r u c t e d u s i n g a c e r t a i n t y p e o f a r t i f i c i a l d i e l e c t r i c . Measurements were c a r r i e d o u t i n a p a r a l l e l - p l a t e r e g i o n f o r ii  b o t h p l a n e and c y l i n d r i c a l wave i n c i d e n c e ; t h e r e s u l t s  obtained  agree w i t h computed r e s u l t s and d i s a g r e e w i t h some p r e v i o u s l y published t h e o r e t i c a l r e s u l t s . As an a p p l i c a t i o n , an i n v e s t i g a t i o n i s made o f t h e range o f v a l i d i t y o f a p l a n a r model when i n t e r p r e t i n g phase a n g l e measurements on d i e l e c t r i c - e n c l o s e d c y l i n d r i c a l plasmas.  iii  and - u n e n c l o s e d  TABLE OP CONTENTS Page LIST OP ILLUSTRATIONS  vii-I  LIST OP TABLES  x l i  LIST OP SYMBOLS  xiii  ACKNOWLEDGEMENT  x  i  x  1.  INTRODUCTION  1  2.  FORMULATION OF THE PROBLEM  6  2.1  S c a t t e r i n g "by a C y l i n d e r C o n s i s t i n g o f M C o a x i a l Inhomogeneous R e g i o n s - P l a n e Wave I n c i d e n c e 2.1-1  2.1- 2  Boundary C o n d i t i o n s A p p l i e d by D i r e c t Matching of T a n g e n t i a l F i e l d Components  1^  Approximation f o r a Thin S h e l l 2.2- 1  19  The Case o f a T h i n S h e l l on a C o n d u c t i n g Core  3.  9  S o l u t i o n U s i n g Impedance Boundary Conditions  2.2  7  25  2.3  Far-Zone S c a t t e r e d F i e l d  27  2.4  Cylindrical-Wave Incidence  28  CALCULATION 3.1  OP THE SCATTERED FIELD  Methods o f C a l c u l a t i n g t h e S c a t t e r e d - F i e l d Coefficients .......... 3.1-1 Power S e r i e s S o l u t i o n o f t h e Wave Equation 3 . 1 - 2 A p p r o x i m a t i o n o f an Inhomogeneous R e g i o n by Homogeneous S h e l l s 3 * 1 - 3 A p p r o x i m a t i o n o f an Inhomogeneous R e g i o n by S h e l l s w i t h L i n e a r l y Varying P e r m i t t i v i t y  iv  32 32 33 '39 hZ  3.1-4  3.1-5 3.2 4.  45  R e s u l t s , and Comparison o f t h e Methods  " 46  Number o f Terms R e q u i r e d F i e l d E x p a n s i o n ..'  i n the Scattered-  ol  CALCULATION OF THE COMPLETE SCATTERED FIELD FROM FIELD MEASUREMENTS  63  4.1  63  Method o f C a l c u l a t i n g t h e C o e f f i c i e n t s .... 4.1-1 4.1-2  " 4.2 5.  N u m e r i c a l I n t e g r a t i o n o f t h e Impedance and A d m i t t a n c e D i f f e r e n t i a l Equations .  A p p l y i n g E q u a l Weight t o a l l Points A p p l y i n g a D i f f e r e n t Weight t o Each  65  Point  67  Determination  of the Required  69  Order  EXPERIMENTAL INVESTIGATION 5.1 C o n s t r u c t i o n p f t h e Inhomogeneous C y l i n d e r s 5.1- 1 E v a l u a t i o n o f t h e A r t i f i c i a l  76  Dielectric 5.2.  Plane-Wave I n c i d e n c e - The P a r a l l e l P l a t e Transmission  5.2- 2 5.3  90  Line  Cylindrical-Wave Incidence Section .  R e s u l t s o f Measurements  ...  99 104  5.3- 1  Incident F i e l d  104  5.3-2  Scattered F i e l d  108  5.3-3  Examples o f Measured D i f f r a c t e d Fields  124  Comparison w i t h P u b l i s h e d : R e s u l t s ..  129  5.3-4 5.4  87  Measurement o f t h e S c a t t e r e d F i e l d s 5.2- 1  73 74  130  Summary v  6.  APPLICATION TO MEASUREMENTS ON CYLINDRICAL PLASMAS 6.1  6.2  " 132  O u t l i n e o f t h e P r o b l e m and Method o f Computation  132  6.1-1  B a s i s o f Comparison  135  6.1- 2  Method o f C a l c u l a t i o n  136" 136"  R e s u l t s and D i s c u s s i o n 6-.-2-1  Comparison o f R e s u l t s f o r t h e P o u r 137  Profiles 6.2- 2  E f f e c t o f a G l a s s Boundary  6.2-3  E f f e c t o f Changing t h e P o i n t o f  . I38  139  Observation 6.3 7.  151  Summary  153  DISCUSSION AND CONCLUSIONS  APPENDIX A.  Fields i n a Cylindrically-Stratifled 156  Medium A.l  H  2  = 0 (Parallel Polarization)  157  A. 2  E  z  = 0 (Perpendicular Polarization)  159  APPENDIX B.  Power S e r i e s  Solution  i n t h e Case o f 160  a Linear Permittivity Variation B. l  Parallel Polarization  160  B. 2  Perpendicular P o l a r i z a t i o n  162  B.2-1  168  APPENDIX C.  Range o f Convergence  Impedance and A d m i t t a n c e R e l a t i o n s f o r Homogeneous R e g i o n s when f&r~S  C l  « n  ...  I69  Parallel Polarization  C. 2 P e r p e n d i c u l a r P o l a r i z a t i o n APPENDIX D. Some A s p e c t s o f t h e N u m e r i c a l t i o n Method D. l P a r a l l e l P o l a r i z a t i o vn i  I69  I73 Integra-  175 175  D. 2  Perpendicular P o l a r i z a t i o n  1 ??  D. 2-1  179  APPENDIX E.  D e a l i n g w i t h Zeros of P e r m i t t i v i t y .  D e t e r m i n a t i o n o f the Required Order f o r Computing the S c a t t e r e d F i e l d  E. l E.2  Maximum Magnitude of High Order S c a t t e r e d Field Coefficients ....  I83  T r u n c a t i o n - E r r o r i n C a l c u l a t i n g the S c a t tered F i e l d  188  E. 2-1  E. 3  I83  Procedure used t o Determine the Order Required t o Achieve a Given Accuracy o f the S c a t t e r e d F i e l d ....  T r u n c a t i o n - E r r o r i n C a l c u l a t i n g the B i s t a t i c Scattering Cross-Section E.3-1  APPENDIX F.  Procedure used to Determine the Order Required t o Achieve a Given Accuracy o f S c a t t e r i n g C r o s s - S e c t i o n  General Behaviour o f  ±9^ 196  198  and y„ W i t h i n  an Inhomogeneous C y l i n d e r  201  F. l  Parallel Polarization  201  F.2  Perpendicular P o l a r i z a t i o n  206  APPENDIX G.  E f f e c t i v e P e r m i t t i v i t y o f the A r t i f i c i a l Dielectric  REFERENCES  vii  210 ' 217  LIST OP ILLUSTRATIONS Figure  Page  2.1  Cross-Section  2.2  Coordinate  2.3  Cascaded Non-Uniform Transmission-Line A n a l o g o f the Inhomogeneous C y l i n d e r  lo  2.4  D e f i n i t i o n of Parameters f o r the ThinS h e l l Problem  19  2.5  I l l u s t r a t i o n o f t h e Problem o f C y l i n d r i c a l Wave I n c i d e n c e  29  3.1  D i v i s i o n o f a Core R e g i o n f o r M a t c h i n g S e r i e s S o l u t i o n s about D i f f e r e n t P o i n t s  35  3.2  o f C y l i n d e r .....  8  System  8  ...  D i v i s i o n o f a S h e l l which has a Zero o f P e r m i t t i v i t y f o r Matching S e r i e s S o l u t i o n s about D i f f e r e n t P o i n t s  36  I l l u s t r a t i o n o f the two Methods of C h o o s i n g the S h e l l P a r a m e t e r s s and t  44  Computed and P u b l i s h e d V a l u e s of k„oi(0) f o r a M e t a l l i c C y l i n d e r w i t h an Inhomogeneous D i e l e c t r i c S h e l l , £V(r) = c C / k r - .  47  Computed V a l u e s of k 0 ^ ( 0 ) f o r a M e t a l l i c C y l i n d e r w i t h an Inhomogeneous D i e l e c t r i c S h e l l , £,(r) = <xA<>r  49  ;5.1  Method o f C o n s t r u c t i n g R a d i a l l y Inhomogeneous C y l i n d e r s  74  5.2  Waveguide C o n f i g u r a t i o n used i n E v a l u a t i n g the A r t i f i c i a l D i e l e c t r i c  76  5.3  System used f o r R e f l e c t i o n C o e f f i c i e n t Measurements  79  5.4  Section Constructed to Hold A r t i f i c i a l D i e l e c t r i c Specimens f o r R e f l e c t i o n Coeff i c i e n t Measurements  80  3.3 3.4  0  3.5  o  vili  Figure  Page  5.5  Specimens used f o r t h e Waveguide E v a l u a t i o n of the A r t i f i c i a l D i e l e c t r i c ...  81  5.6  B a s i c System used i n S c a t t e r e d - F i e l d Measurements  88  5.7a  System f o r R e c o r d i n g Magnitude and Phase ..  89  5.7b  ' System f o r R e c o r d i n g Q u a d r a t u r e Components  89  5.8  Assembly o f t h e P a r a l l e l - P l a t e Transmis-'. s i o n - L i n e System ..  94  5.9  Some C o n s t r u c t i o n D e t a i l s o f t h e T r a n s m i s s i o n - L i n e System  95  5*10  System f o r R o t a t i n g C e n t e r p i e c e  5.11  Construction of P a r a l l e l - P l a t e  ••••  101 R e g i o n used  f o r Cylindrical-Wave Incidence  102  5.12a  C o n s t r u c t i o n o f E x c i t i n g Probe  103  5.12b 5.13  C o n s t r u c t i o n o f M e a s u r i n g Probe E r r o r s , i n t h e Measured I n c i d e n t F i e l d f o r Plane-Wave I n c i d e n c e  103  5.14  Measured I n c i d e n t F i e l d a t r.= 4 cm. f o r Cylindrical-Wave Incidence  106  5.15  E r r o r s i n t h e Measured I n c i d e n t F i e l d f o r Cylindrical-Wave Incidence  107  5.16  S c a t t e r e d F i e l d f o r Plane-Wave I n c i d e n c e and E r r o r s u s i n g D a t a from P l a n e and C y l i n d r i c a l Wave I n c i d e n c e , r = 4 cm. ..... I l l  105  0  5.17 5.18 5.19  A r t i f i c i a l D i e l e c t r i c S h e l l used i n t h e Experiments . E r r o r s i n the Scattered F i e l d f o r Cylindrical-Wave Incidence  116  Scattered F i e l d f o r Cylindrical-Wave I n c i d e n c e and E r r o r s u s i n g D a t a w i t h r = 4 cm. and r„ = 8 cm. M e t a l r , = 3.5 cm.  118  Scattered F i e l d f o r Cylindrical-Wave I n c i d e n c e and E r r o r s u s i n g D a t a w i t h r = 4 cm. and r = 8 cm. C o r e - t <= 2.54, 3 ^ = 1.5 cm.; S h e l l - A r t i f i c i a l  121  0  5.20  114  0  Y  0  ix  Figure  Page  5o21  Measured D i f f r a c t e d  6»la  Cylindrical Configuration  133  6 lb  Planar Configuration  133  6.2  E l e c t r o n D e n s i t y P r o f i l e and P o s s i b l e P e r m i t t i v i t y V a r i a t i o n s i n the y = o Plane  13^  6.3  E l e c t r o n D e n s i t y P r o f i l e s Considered  137  6.4  V a r i a t i o n o f <Mr) - •&(<») w i t h Radius:, P r o f i l e (b) w i t h N / N = 0.5-  ' W  6.5  V a r i a t i o n o f <p {x) - ^ ( r , ) w i t h R a d i u s , P r o f i l e (b) w i t h N^'A, « 2  149  B.l  Regions o f Convergence o f t h e S e r i e s S o l u r t i o n s o f t h e Wave E q u a t i o n f o r P e r p e n d i c u l a r P o l a r i z a t i o n and £y - s r + t  168  D.l  Types o f P e r m i t t i v i t y Zeros  180  D. 2  A p p r o x i m a t i n g £ (r) by a L i n e a r V a r i a t i o n Near £^(r) ~ 0 i n Order t o D e a l w i t h t h e Singularity  180  Behaviour of / a - J J £ n, < n* < n  186  0  0  c  E. l  0  t  r  t  Z  f o r t h r e e V a l u e s o f n,  3  E.2  la,,!  E. 3  Comparison Between t h e Bound on ~B and t h e Bound on K f o r f =  F. l  125  F i e l d s , r - 4 cm.  < =0  and /a>, /  z;=&  as F u n c t i o n s o f  187  N  192  B e h a v i o u r o f Impedance W i t h i n a D i e l e c t r i c C y l i n d e r f o r a Core R e g i o n i n w h i c h ( £ (J/k ) - n / 3 f * ) Becomes P o s i t i v e a t z  y  "5 =  F.2  F. 3  0  •  B e h a v i o u r o f Impedance W i t h an I n i t i a l Value i n a R e g i o n Where ( ) - nV5 ) is Positive  203  Behaviour o f Admittance W i t h i n a D i e l e c t r i c C y l i n d e r and a D i e l e c t r i c S u r r o u n d i n g a C o n d u c t i n g Core when € < 0  208  L a y e r e d Medium o f t h e Type Used f o r t h e Artificial Dielectric  210  x  r  G. l  203  x  Figure G.2a G.2b  Page Maximum Percentage E r r o r i n t u s i n g Equation'(G.6)  Calculated  re  215  Maximum Percentage E r r o r i n ty. C a l c u l a t e d u s i n g E q u a t i o n (G.9) • e  xi  215  LIST OP TABLES Table  Page  3.1  3.2  •  R e l a t i v e E r r o r i n t h e V a l u e o f k^o^O) C a l c u l a t e d Using the Shell-Approximation Methods  52  O v e r a l l A c c u r a c y o f t h e S h e l l Methods  57  .....  3.3  R e l a t i o n s h i p Between A c c u r a c y and S h e l l Thickness  58  3.4  Comparison o f t h e Methods o f Computation ..  59  5.1  Comparison of Measured and T h e o r e t i c a l V a l u e s o f the 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 t h e Samples o f A r t i f i c i a l D i e l e c t r i c  82  Range o f D i f f e r e n c e Between T h e o r e t i c a l and Measured R e s u l t s f o r t h e Specimens o f A r t i f i c i a l D i e l e c t r i c w i t h T i 0.2 i n  84  Comparison Between V a l u e s o f B a c k s c a t t e r i n g Cross-Section Calculated T h e o r e t i c a l l y and from E x p e r i m e n t a l D a t a  115  Comparison Amongst P u b l i s h e d , T h e o r e t i c a l and E x p e r i m e n t a l V a l u e s o f B a c k s c a t t e r i n g Crocs-Section  129  6.1  Comparison Between c? (°°) f o r a Plasma C y l i n d e r and q>n f o r t h e P l a n a r Model  140  6.2  R a d i u s i n Wavelengths a t w h i c h IT(r) = N  ..  142  6.3  Comparison Between ^ ( r , ) f o r a P l a s m a C y l i n d e r and (p f o r t h e P l a n a r Model  143  6.4  Comparison Between <P (°°) and for a G l a s s - E n c l o s e d P l a s m a C y l i n d e r , P r o f i l e (b)  145  6.5  Comparison Between ^ ( r , ) and <PT f o r a G l a s s - E n c l o s e d P l a s m a C y l i n d e r , P r o f i l e (b)  146  5.2  5.3  5*4  r  c  r  R  xii  LIST OF SYMBOLS a, &i, A  =  constants  a>,  =  a  =  A„/E  a ^ , &g. " *  or & „  f n  g  0  ( C h a p t e r 4)  (Appendix E)  = n o r d e r c o e f f i c i e n t s o f t h e r e a l and imaginary parts of the scattered field a t r =. r t h  0  AY,  =  » e » A>,  order s c a t t e r e d - f i e l d  coefficient  e  =  A „ f o r p a r a l l e l p o l a r i z a t i o n w i t h p l a n e and c y l i n d r i c a l wave i n c i d e n c e , r e s p e c t i v e l y  £  =  A-,, f o r p e r p e n d i c u l a r p o l a r i z a t i o n w i t h p l a n e and c y l i n d r i c a l wave i n c i d e n c e , r e s p e c t i v e l y  =  constants  =  c o e f f i c i e n t of U  s.  constants  C-n  =  U  d  =  half-thickness dielectric  D  =  distance to line-source  =  l o c a t i o n o f t h e mid-plane o f medium 1 "of t h e a r t i f i c i a l d i e l e c t r i c (Appendix G)  =  Neumann c o n s t a n t ,  A  Ay./, » A-„  b* b<: » ~b-n> B C^ c,  c„  e-n  n  or  w  and V„ i n t h e : m  t h  region  VT,  o f medium 1 o f t h e a r t i f i c i a l (Section  2.4)  «= 1 f o r n = 0;  e-„ = 2 f o r n i l E E  , c  E , 1  E , E°, E'" s  v  =  electric  =  magnitude o f t h e e l e c t r i c f i e l d o f t h e i n c i d e n t p l a n e wave i n c i d e n t , s c a t t e r e d , d i f f r a c t e d and m region e l e c t r i c f i e l d s  =  field  xiii  th  E , K  E^., Ej.  E^  = r a d i a l , a n g u l a r and a x i a l components o f electric field = a x i a l component o f the s c a t t e r e d f i e l d truncated at n = N  f f,  f ,  f  5  x  0  F v, .. '  •  g Si»Ss»  So  =  scattered-field truncation  =  function  electric  errors  = r e a l p a r t o f the i n c i d e n t , s c a t t e r e d d i f f r a c t e d f i e l d s a t r = r„  and  -  P - v a r i a t e w i t h n , and n  =  v a l u e o f F * , , , - n , , s u c h t h a t the p r o b a b i l i t y o f e x c e e d i n g t h i s v a l u e i s cc  -  function  -  i m a g i n a r y p a r t o f the i n c i d e n t , and d i f f r a c t e d f i e l d s a t r = r  2  degrees o f freedom  scattered  0  = wave-number i n medium 2 o f t h e  h  artificial  dielectric = magnetic  H H  = magnitude o f t h e magnetic f i e l d o f t h e i n c i d e n t p l a n e wave = i n c i d e n t and s c a t t e r e d magnetic f i e l d s  0  H , H 1  field  5  H , EQ., E  =  H^  = n^* o r d e r H a n k e l f u n c t i o n o f the second k i n d th = n o r d e r m o d i f i e d B e s s e l f u n c t i o n o f the f i r s t kind  Z  K  In  2)  r a d i a l , a n g u l a r and a x i a l components o f magnetic f i e l d 1  JT,  = n^* o r d e r B e s s e l f u n c t i o n o f the f i r s t k i n d  k  =  K.„  = n n^* oordi r d e r m o d i f i e d B e s s e l f u n c t i o n o f the . second k i n d  1  to J/L <f~ 1  xiv  K™ , K™~  = ri^  +  1  mode wave Impedance o f t h e ^ ' i n c l d e n t wave"  and " r e f l e c t e d wave" i n the m  region  L  =  M  = number o f c o a x i a l r e g i o n s o r magnitude o f the e l e c t r i c f i e l d ( C h a p t e r s .4 and 5) = magnitude o f the i n c i d e n t , s c a t t e r e d and diffracted electric fields  Mjr, M , M s  N  D  '  (number o f d a t a p o i n t s ) / 2  =  maximum o r d e r used i n c a l c u l a t i n g the s c a t tered f i e l d or e l e c t r o n number-density ..'. ( C h a p t e r 6)  No  =  a x i a l e l e c t r o n number-density  N  =  c r i t i c a l e l e c t r o n number-density  p  =  wave number i n medium 1 o f t h e dielectric  K* Q»» S ? , C  =  f u n c t i o n s o f U , , , V^,, U„', V,,' i n the m  r  = r a d i a l coordinate  c  r  c  r r  '  r a d i u s at which £ ( r ) = 0  w  =  o u t e r r a d i u s o f the m^  a  =  r a d i a l c o o r d i n a t e w i t h r e s p e c t t o the source ( S e c t i o n 2 . 4 )  =  r a d i u s about w h i c h s e r i e s s o l u t i o n s a r e d e v e l o p e d ( C h a p t e r 3 and Appendix B)  =  r a d i u s a t w h i c h measurements a r e ( C h a p t e r s 4 and 5 )  =  g e n e r a l s o l u t i o n o f the r a d i a l wave e q u a t i o n  s,  s  w  K  1  ( N(r) = N  region  t h  =  RT,  *•'  artificial  c  )  region line-  taken  = slope of l i n e a r p e r m i t t i v i t y v a r i a t i o n  s,  = s/k  s£  = S„ /(2L  S*  = sujn o f the squares o f the r e s i d u e s f o r n order least-squares f i t  J$N  -  2  (Appendix B) - n - 1 )  e r r o r i n the b i s t a t l c s c a t t e r i n g xv  the  cross-section  time ( t i m e dependence s u p p r e s s e d t h r o u g h o u t ) t h i c k n e s s o f an Inhomogeneous r e g i o n  :.  ( S e c t i o n 3.1-5)  h a l f - t h i c k n e s s o f medium 2 o f t h e a r t i f i c i a l d i e l e c t r i c (Appendix G) thickness of a r t i f i c i a l ( C h a p t e r 5)  dielectric  specimens  l o c a t i o n o f t h e mid-plane o f medium 2 o f t h e a r t i f i c i a l d i e l e c t r i c (Appendix G) independent s o l u t i o n s o f t h e r a d i a l wave equation th weight applied t o the r e s i d u e a t t h e i data point i n the least-squares f i t U„(/2) v'(/3) U„ and V„  - u'C/?)  V„(/l)  Wronskian o f  cartesian coordinates n o r m a l i z e d a d m i t t a n c e v a r i a b l e s (Appendix P) n^* mode n o r m a l i z e d a d m i t t a n c e v a r i a b l e 1  curves i n the y - 5 plane along which d y / d ^ = 0 (Appendix P) to modi^normalized admittance v a r i a b l e i n the m r e g i o n (Appendix D) o  n  r  o r d e r B e s s e l f u n c t i o n o f t h e second k i n d  a d m i t t a n c e l o o k i n g i n w a r d s a t r = r„, n o r m a l i z e d impedance v a r i a b l e (Appendix F ) n^* mode n o r m a l i z e d impedance v a r i a b l e 1  curves i n the z - J plane along which d y / d j = 0 (Appendix P) ^/^\o mode n o r m a l i z e d impedance v a r i a b l e i n the m r e g i o n (Appendix D) o  r  tn  impedance l o o k i n g inwards a t r = r ^  xvi  kr,  ( S e c t i o n 2.2)  constant k  elsewhere  T*  krv  ( S e c t i o n 2.2)  sr/t  ( S e c t i o n 3.1-1 and Appendix  constant  B)  elsewhere  v a l u e o f fi about w h i c h s e r i e s s o l u t i o n s are developed  fU/jLL  ( S e c t i o n 2.2)  fi + 1 o r fi- fio (Appendix B and S e c t i o n 3.1-1) JtZ //Mo 0  1  _TM  I* J  determinants appearing i n the expressions f o r the -scattered f i e l d c o e f f i c i e n t s specified f i e l d S e c t i o n E.2-1)  e r r o r (Appendix E and  Kronecker d e l t a , S „ =' 1, m = M  S-**, » 0 f o r m ^ M;:  w  m a t r i x a p p e a r i n g i n t h e a r r a y s o f S and A permittivity e/i  0  free-space  permittivity  p e r m i t t i v i t y o f t h e m^  1  coaxial  region  e f f e c t i v e p e r m i t t i v i t y of the a r t i f i c i a l dielectric  angular coordinate xvii  &c  = /ri/L  = XE  f r e e - s p a c e wavelength  = w a v e l e n g t h i n a medium o f p e r m i t t i v i t y I  /L  =  perDieability  fly.  = J^/y^o  y*<,  =  t  free-space permeability th = p e r m e a b i l i t y o f the m coaxial region ( C h a p t e r 2) = k r or k^r  ^  =  /t " 7  c  r e f l e c t i o n c o e f f i c i e n t ( C h a p t e r 5) v a r i a b l e i n power s e r i e s (Appendix E)  &  =  b i s t a t i c scattering cross-section  G ~  =  error-variance  j7  ,7L ,i ,z  <j o Mt  & f  cty  ._  v a r i a n c e o f t h e e r r o r s i n t h e measured v a l u e s o f M, 0- and (p  t  =  <p (Px* <?s > (p  = phase a n g l e *= phase a n g l e o f t h e i n c i d e n t , s c a t t e r e d and diffracted fields = phase a n g l e o f t h e s c a t t e r e d f i e l d i n t h e d i r e c t i o n & = 0 and the d i f f r a c t e d f i e l d i n t h e d i r e c t i o n 0- = 180° r e f e r r e d back t o r = r , by a d d i n g k„(r - r, ) r a d i a n s  D  <Pt(r)  (pRt <p  O  T  ct - 0 ( S e c t i o n 2.2)  =  argument o f t h e r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s o f the planar-model c o n f i g u r a t i o n ( C h a p t e r 6)  =  angular frequency  xviii  ACKNOWLEDGEMENT . G r a t e f u l acknowledgement i s g i v e n t o the N a t i o n a l . Research for  C o u n c i l of Canada f o r a B u r s a r y i n the year 1 9 6 4 - 1 9 6 5 ,  S t u d e n t s h i p s i n the p e r i o d I965-I968 and f o r support  under grant A-3344. I wish t o thank my for of  r e s e a r c h s u p e r v i s o r , Dr. M.  h i s continued i n t e r e s t and a s s i s t a n c e d u r i n g the  Kharadly,  course  the p r o j e c t . Thanks are a l s o due  _ C. Chubb: Mr. Wilbee performed  to Messrs. B. Wilbee the computations  c o n f i g u r a t i o n c o n s i d e r e d i n Chapter manuscript;  Mr.  Chubb  built  and  f o r the p l a n a r  6 and p r o o f - r e a d the  the experimental  apparatus,  o f t e n making s u g g e s t i o n s r e g a r d i n g the d e t a i l s of c o n s t r u c t i o n . Finally,  I am indebted t o my  t y p i n g the manuscript  and,  wife, E l l i e ,  more important,  xix  for  f o r just being  Ellie.  1  1.  INTRODUCTION  Much of the i n t e r e s t i n inhomogeneous d i e l e c t r i c c y l i n d e r s stems from the f a c t t h a t , under c e r t a i n c o n d i t i o n s , a plasma appears as a r e g i o n of r e l a t i v e p e r m i t t i v i t y l e s s (ep i ) -  than u n i t y  x  &  '  .  E a r l i e r work was  t e r i n g hy meteor t r a i l s .  generally applied to scat-  More r e c e n t l y , however, t h e r e has  i n c r e a s i n g i n t e r e s t i n l a b o r a t o r y plasma d i a g n o s t i c s and  been  plasma  s h e a t h problems a s s o c i a t e d w i t h s p a c e c r a f t r e - e n t r y t o the. atmosphere. The main d i f f i c u l t y encountered when d e a l i n g  with  r a d i a l l y inhomogeneous c y l i n d e r s i s t h a t c l o s e d form s o l u t i o n s of t h e wave e q u a t i o n e x i s t f o r o n l y c e r t a i n t y p e s o f p e r m i t t i v i t y variation 6(r)  .  = a r  b  Most n o t a b l e i s the power law v a r i a t i o n ,  where the s o l u t i o n s are e x p r e s s i b l e i n terms of  Bessel functions i n general  o r , when b = -2,  algebraic functions.  D e t a i l e d i n v e s t i g a t i o n s i n s p e c i a l cases have b e ^ n - c a r r i e d out (3) (4~ 5) by s e v e r a l a u t h o r s : Yeh and K a p r i e l i a n ^ ' and Burman ' have considered  a m e t a l l i c c y l i n d e r w i t h an inhomogeneous d i e l e c t r i c  s h e a t h o f p e r m i t t i v i t y a/r and a / r * , r e s p e c t i v e l y , w h i l e  Negi^^  has d e a l t w i t h the r e l a t e d problem of s c a t t e r i n g by a c y l i n d e r w i t h a c o n d u c t i v i t y which i s p r o p o r t i o n a l t o r . A power s e r i e s s o l u t i o n of the wave e q u a t i o n (7) been used t o some e x t e n t .  F o r example, F e i n s t e i n  has used t h i s  method f o r a l i n e a r v a r i a t i o n of p e r m i t t i v i t y as have Yeh Rusch^* 'for 9  has  and  the more cumbersome problem of a plasma c y l -  i n d e r w i t h an e l e c t r o n d e n s i t y of t h e form N ( r ) = N ( 1 - ( r / r , )* ) e  under the i n f l u e n c e of a s t a t i c magnetic f i e l d .  2  Numerous approximate methods which a r e a p p l i c a b l e f o r a r b i t r a r y v a r i a t i o n s have been d e v e l o p e d .  However, t h e s e  g e n e r a l l y p l a c e some r e s t r i c t i o n on t h e d i m e n s i o n s and/or o v e r a l l p r o p e r t i e s of the c y l i n d e r . w h i c h t r e a t s t h e inhomogenlety  Thus, t h e B o r n  approximation^ ^, 1 1  as a p e r t u r b a t i o n , may  be used i f  t h e p e r m i t t i v i t y range i n the c y l i n d e r i s o n l y s l i g h t l y  different  from the v a l u e i n t h e s u r r o u n d i n g medium. T h i s has been done by (12) A l b i n i and N a g e l b e r g f o r an o b l i q u e l y i n c i d e n t p l a n e wave. F o r l a r g e c y l i n d e r s a g e o m e t r i c a l o p t i c s approach may be (13 14 IS) used  '  *  .  The l o w - f r e q u e n c y and h i g h - f r e q u e n c y a s y m p t o t i c  t h e o r y o f s c a t t e r i n g by a p l a s m a - c l a d m e t a l l i c c y l i n d e r has been g i v e n by M a k s i m o v ^ ^ . 16  An a p p r o x i m a t i o n which i s n o t dependent upon the prope r t i e s o f the c y l i n d e r i s t h e " l a y e r method" i n which t h e c y l i n d e r i s r e p l a c e d by a n o t h e r composed o f a number o f homogeneous shells.^  1 5  ' ' 1 7  1  ^ .  These t h r e e r e f e r e n c e s p r e s e n t computed r e -  s u l t s , the f i r s t two f o r normal i n c i d e n c e , t h e t h i r d f o r o b l i q u e incidence.  (The s o l u t i o n i n the case o f a c y l i n d e r composed  o f an a r b i t r a r y number o f homogeneous c o a x i a l r e g i o n s had been (19 ZO) - "• given previously ' ' f o r normal i n c i d e n c e and has-been v  extended'to o b l i q u e ' i n c i d e n c e ;using". a': t r a n s m i s s i o n m a t r i x app r o a c h i n R e f e r e n c e 18.)  ^  . N u m e r i c a l approaches  have proved u s e f u l because t h e y  are capable of producing v e r y accurate r e s u l t s .  Computations  in  t h e case o f s c a t t e r i n g by a c y l i n d e r o f e l e c t r o n s w i t h a G a u s s i a n (21 2 2 r a d i a l d i s t r i b u t i o n have been p r e s e n t e d by s e v e r a l a u t h o r s 25,2 4,25")^  Brysk  a n u  '  B u c h a n a n ^ ) , who use a p o t e n t i a l t h e o r y  '  3  f o r m u l a t i o n which applies  to p a r a l l e l p o l a r i z a t i o n only, provide  a comparison between t h e i r e x a c t r e s u l t s and t h o s e o b t a i n e d u s i n g several  approximate methods.  Other v a r i a t i o n s have a l s o 'been  c o n s i d e r e d : E x t e n s i v e r e s u l t s f o r a v a r i a t i o n o f t h e form J ( 2 . 4 r / r , ) have been g i v e n by E a u g e r a s 0  p o l a r i z a t i o n and a s t r i k i n g d i f f e r e n c e  '  for parallel  between t h i s case and t h e (2 4)  case w i t h a " G a u s s i a n v a r i a t i o n has been observed  .  A method  o f s o l u t i o n f o r o b l i q u e i n c i d e n c e has been g i v e n by K o r b a n s k i y and S t e l ' m a s h  b u t no. r e s u l t s were computed.  This; t h e s i s d e a l s w i t h b o t h t h e o r e t i c a l and  experi-  m e n t a l a s p e c t s o f t h e problem o f s c a t t e r i n g by an i s o t r o p i c c y l i n d e r a t normal i n c i d e n c e .  On t h e t h e o r e t i c a l s i d e ,  several  approaches t o t h e s o l u t i o n o f the problem a r e c o n s i d e r e d and compared.  On t h e e x p e r i m e n t a l s i d e , measurements on inhomogen-  eous c y l i n d e r s  c o n s t r u c t e d u s i n g an a r t i f i c i a l d i e l e c t r i c a r e  c a r r i e d out i n o r d e r t o s u b s t a n t i a t e t h e t h e o r e t i c a l r e s u l t s . The m a t e r i a l  i n Chapter 2 c o n s i s t s  i z a t i o n s o f p r e v i o u s ; r e s u l t s and d e r i v a t i o n s  mainly of generalof basic  relations.  The problem o f s c a t t e r i n g by a c y l i n d e r composed o f an a r b i t r a r y number o f c o a x i a l , inhomogeneous, i s o t r o p i c r e g i o n s i s formul a t e d ; two approaches, d i f f e r i n g i n t h e manner i n w h i c h t h e boundary c o n d i t i o n s a r e a p p l i e d , d i r e c t matching of the t a n g e n t i a l  are used.  f i e l d components i s u s e d , l e a d s  to a s e t of simultaneous equations. pedance concept i s a p p l i e d ,  The f i r s t , i n which  The second, i n w h i c h an i m -  g i v e s an I t e r a t i v e f o r m u l a w h i c h i s  s u i t a b l e f o r s t u d y i n g the e f f e c t o f a t h i n s h e l l .  Expressions  f o r the f a r f i e l d and b i s t a t l c s c a t t e r i n g c r o s s - s e c t i o n  are then  4 d e r i v e d and l a s t l y , t h e r e l a t i o n s h i p between t h e s c a t t e r e d - f i e l d c o e f f i c i e n t s f o r plane-wave i n c i d e n c e and t h o s e f o r c y l i n d r i c a l wave i n c i d e n c e i s g i v e n . I n Chapter 3, f o u r methods o f o b t a i n i n g t h e s c a t t e r e d f i e l d f o r a c y l i n d e r w i t h an a r b i t r a r y r a d i a l p e r m i t t i v i t y v a r i a t i o n are considered.  These methods i n v o l v e  s e r i e s s o l u t i o n o f t h e wave e q u a t i o n c y l i n d e r by homogeneous s h e l l s  ( i ) u s i n g a power  ( i i ) approximating the  ( i i i ) approximating the c y l i n d e r  by s h e l l s w i t h l i n e a r l y - v a r y i n g p e r m i t t i v i t i e s and  ( i v ) numer-  i c a l i n t e g r a t i o n of the R i c c a t i - t y p e d i f f e r e n t i a l equation for  impedance o r a d m i t t a n c e .  -  A comparison o f t h e s e methods  i s made on t h e b a s i s o f d i f f i c u l t y o f a p p l i c a t i o n , l i m i t a t i o n s , a c c u r a c y and c o m p u t a t i o n time r e q u i r e d .  The t r u n c a t i o n - e r r o r  i n c a l c u l a t i n g t h e s c a t t e r e d f i e l d i s i n v e s t i g a t e d and t e s t s f o r determining  t h e number o f terms r e q u i r e d t o a c h i e v e a s p e c i f i e d  accuracy are given. A method o f o b t a i n i n g t h e s c a t t e r e d - f i e l d c o e f f i c i e n t s from measurements o f t h e s c a t t e r e d f i e l d i s developed i n Chapter  4.  Accordingly,  t h e complete s c a t t e r e d f i e l d may be c a l c u -  l a t e d from d a t a t a k e n a t a r e l a t i v e l y s m a l l number o f p o i n t s . A f u r t h e r r e s u l t , which i s a p p l i e d i n t h e e x p e r i m e n t a l  investi-  g a t i o n , i s t h a t t h e s c a t t e r e d f i e l d f o r plane-wave i n c i d e n c e be c a l c u l a t e d from measurements w i t h c y l i n d r i c a l - w a v e The c o e f f i c i e n t s a r e computed by a p p l y i n g a F o u r i e r f i t t o t h e measured d a t a and hence t h e f i e l d  incidence.  least-squares;  c a l c u l a t e d from t h e  c o e f f i c i e n t s i s "smoother" t h a n t h a t measured d i r e c t l y ; t h e c h o i c e o f t h e number o f c o e f f i c i e n t s w h i c h w i l l g i v e t h e b e s t r e p r e s e n t a t i o n of  the f i e l d i s d i s c u s s e d .  may  5 The  experimental  work, d i s c u s s e d i n Chapter 5» i n -  v o l v e d the c o n s t r u c t i o n o f inhomogeneous c y l i n d e r s u s i n g a  type  of a r t i f i c i a l d i e l e c t r i c and the measurement o f d i f f r a c t e d  field  i n a p a r a l l e l - p l a t e r e g i o n under c o n d i t i o n s o f p l a n e and i n d r i c a l wave i n c i d e n c e .  cyl- .  A p r e l i m i n a r y e v a l u a t i o n o f the  accur-  acy w i t h which the a r t i f i c i a l d i e l e c t r i c a p p r o x i m a t e s the a c t u a l inhomogeneous d i e l e c t r i c was  a c c o m p l i s h e d by t a k i n g r e f l e c t i o n '  c o e f f i c i e n t measurements i n r e c t a n g u l a r waveguide. the e r r o r s i n the v a l u e s of the d i r e c t l y - m e a s u r e d  P l o t s of scattered  f i e l d and the smoothed v a l u e s computed u s i n g the t h e o r y g i v e n i n the p r e v i o u s c h a p t e r are shown f o r s e v e r a l c y l i n d e r s .  The  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 and the plane-wave s c a t t e r e d f i e l d computed from measurements w i t h c y l i n d r i c a l - w a v e i n c i d e n c e  are  compared w i t h t h o s e computed from d i r e c t measurements w i t h plane-wave i n c i d e n c e . I n C h a p t e r 6,  the range of v a l i d i t y of a p l a n a r model  i n t e r p r e t a t i o n of phase a n g l e measurements on an inhomogeneous c y l i n d r i c a l plasma column i s i n v e s t i g a t e d .  The  e f f e c t s of  .(i) t h e r a d i u s o f the column, ( i i ) t h e a x i a l e l e c t r o n d e n s i t y , ( i i i ) the form o f the e l e c t r o n d e n s i t y v a r i a t i o n , ( i v ) an ing  g l a s s tube and  (v) the 'point o f o b s e r v a t i o n a r e  Conclusions c u s s i o n are p r e s e n t e d  drawn from t h e work and i n C h a p t e r 7«  enclos  considered.  a general  dis-  6  . 2.  FORMULATION OF  The  THE  PROBLEM  problem of o b t a i n i n g the s c a t t e r e d f i e l d  of  a c y l i n d e r c o n s i s t i n g of an a r b i t r a r y number o f c o a x i a l horn{.1.5  ogeneous r e g i o n s has been c o n s i d e r e d 13,20) ^ general  j  n  -y^g  '  y  '  are inhomogeneous. ::Two' approaches  t h e s e d i f f e r i n the manner i n w h i c h boundary  c o n d i t i o n s are a p p l i e d *  I n the f i r s t approach, d i r e c t  m a t c h i n g of t a n g e n t i a l f i e l d s i s used w h i l e i n the  second,  an Impedance c o n c e p t i s : employed. / i W h i l e the:.former I s the more u s u a l a p p r o a c h ^ f' * » 7  ^ \, 5  the l a t t e r  r e a d i l y a p p l i e d t o s t u d y i n g the case o f a t h i n  :?-y.-:::  inherently  y i e l d s a c o n v e n i e n t i t e r a t i v e m a t c h i n g p r o c e d u r e and  is  shell,which  r e s u l t s i n a R i c c a t i - t y p e d i f f e r e n t i a l e q u a t i o n f o r impedance. This equation proves u s e f u l f o r numerical i n t e g r a t i o n purposes and  f o r the a n a l y s i s of the t r u n c a t i o n - e r r o r w h i c h i s  discussed  i n C h a p t e r 3.  then considered  18  c h a p t e r the problem i s f o r m u l a t e d i n t h e  case where t h e s e r e g i o n s  are c o n s i d e r e d ;  by s e v e r a l a u t h o r s  ~L 7  The  and f i n a l l y  far-zone scattered f i e l d i s i t i s shown t h a t a s i m p l e r e -  l a t i o n s h i p e x i s t s between the c o e f f i c i e n t s of the f i e l d f o r an i n c i d e n t u n i f o r m c y l i n d r i c a l wave and f o r an i n c i d e n t p l a n e wave.  scattered those  7  2.1  Scattering by a Cylinder Consisting of M Coaxial Inhomogeneous Regions - Plane-Wave Incidence A plane wave i s perpendicularly incident on an  i n f i n i t e l y long c i r c u l a r c y l i n d e r consisting of M coaxial regions as I l l u s t r a t e d l n Figure 2.1  e  The m  region i s  characterized by a constant p e r m e a b i l i t y ^ and a permitt i v i t y £ ( r ) which may vary smoothly with r a d i u s . m  formulation /t  m  In the  i s made d i f f e r e n t from yu j; t h i s allows the 0  d i r e c t d e r i v a t i o n of r e l a t i o n s pertinent to the case of perpendicular p o l a r i z a t i o n from those of p a r a l l e l p o l a r i z a t i o n . The core region may be perfectly.conducting rather than dielectrico The coordinate system used i s shown i n Figure 2.2. The axis of the cylinder coincides with the z-axls and the incident wave t r a v e l s i n the negative x d i r e c t i o n . and perpendicular p o l a r i z a t i o n s w i l l be considered  Parallel separately.  The s o l u t i o n f o r a r b i t r a r y l i n e a r p o l a r i z a t i o n may be expressed as a superposition of these two cases.  A time de-  pendence of the form e ^ ^ i s assumed and i s suppressed w  throughout. The incident f i e l d , scattered f i e l d and the f i e l d i n the m® region are denoted by superscripts ( I ) , (S) and 1  (m) respectively. y  The cases of p a r a l l e l and perpendicular  p o l a r i z a t i o n are denoted by the subscripts (e) and (h), respectively.  These subscripts w i l l be omitted unless  i s a p o s s i b i l i t y of confusion.  there  8  Region  ;  Outside Radius = r Inside Radius = r  m m  +  1  Permittivity Permeability  Figure 2.1  Cross-Section of Cylinder.  Figure 2.2  Coordinate System.  = cT(r  9  2.l-l  Boundary C o n d i t i o n s A p p l i e d by D i r e c t M a t c h i n g o f Tangential F i e l d  Components  Parallel Polarization I n t h i s case t h e f i e l d Eo-o  components  are E  z f  H  r P  E x p a n d i n g the i n c i d e n t wave i n c y l i n d r i c a l f u n c t i o n s  the e l e c t r i c f i e l d  ,  3 k o X  oo  J  e,  (3)  (2.1a)  J „ ( k r ) cos(ne-) 0  The symbols a p p e a r i n g i n e q u a t i o n quent r e l a t i o n s  7)  i s g i v e n "by  E*(r,e-) = E o e • . s E,  (Z  (2,1a) and a l l s u b s e -  a r e defined" i n the l i s t o f symbols:.  The s c a t t e r e d f i e l d i s e x p r e s s e d  i n the form /  E (r,eO = X 2  The c o e f f i c i e n t s  e  (2.1b)  » » H r ( k r ) cos(ne-) A  0  o f the s c a t t e r e d f i e ' d ,  A„, are  arbi-  t r a r y c o n s t a n t s t o be d e t e r m i n e d from the boundary If  the s e p a r a t i o n  of v a r i a b l e s technique  a p p l i e d ; to the wave e q u a t i o n i n the m t h a t t h e dependence on r s a t i s f i e s A  where n y  2  + _ i  1_  = separation  +  c «  f  rr/V  \  is  r e g i o n , i t i s . found  t h  the  conditions.  equation (2.2)  £ l  constant  ~ k r M  E q u a t i o n (2.2) i s d e r i v e d i n Appendix A .  The f i e l d  in  10  the  m*  h  region can he written as  Er(r,e-)  « J  e, T B ^ T U k ^ r ) + 0™ (1- .$„„ ) V„ ( k r ) w  1 < m < M where 5  W M  » 0 f o r m 4- M,  S  mM  cos(ne-)  1  (2,1c)  = 1 for m = M  U» and V„ are two independent solutions of equation (2,2). When considering the core region U,, must be analytic at the  origin.  The constants B™ and  are determined by the  boundary conditions. The magnetic f i e l d i s obtained from V X E  = -Jco^'Ho  Thus , f o r the ©- component, we have -  H*(r,©-) = - - ^ - E  6  Y. ^  UT  Jn(k.r) cos(nO-)  (2.3a)  »> =0 H;(r,e).= - - ^ - l e ,  A  H;(r,G-) = - ^ l e ,  [  .  H „ (k„r) cos(ne-) (  w  2)  (2.3b)  U^k.r) > c : ( l - U v X r ) ] 1 < m < M  cos(nG-)  (2.3c)  where the ( ) indicates a derivative with respect to the x  argument. The c o e f f i c i e n t s A„ are now found by matching the  tangential f i e l d components at r = r ^ , m = 1, 2,  M. "til  We obtain the following system of 2M equations f o r the n mode:  11  - H ^ ( k r , ) A„  H^(k r 0  ;  ) A  w  + V ( k , r , ) o;  + l U k , r , ) B;  0  +-^- U » ' ( k r (  (  ) B'„  n  +-~r-  v'Oc.r, ) o'r, = (3)" E. J (k„r, ) n  m « 1 -  (2.4)  U,(Jc,.,pJ -  V„  (k .,  ) 0?"  m  + U„ ( k ^ ) B™  + (!-&,„> ^ ( k ^ ) 0* = 0 jfA-i  ^ntkyn.jr^) By,  ~r  V^k^.rJ  c r ' +77~  u ' t k ^ r j B;  Mr Mr  2 < m < M Cramer's r u l e  gives (2.5a)  E,  where  2m-2  A = ^2 2m-l  AM  J„ ( k r , )  U„(k r, )  a  J'(k.r, )  (  V^Ck.r, )  - ^ r - u ' ( k , r , ) -^r- V*'(k,r, )  - V. (k,., r j "JJ^-I  (k^_,  r^)  t U k ^ ) U (k>„r^) w  2 $ m < M-1  M k ^ r J j^ytV 1  ' O C A )  12  A  *  M  - U, (k „ r ) ' M  _ ~ jut*'  1  - V„ (k _, r )  M  (k^_, r )  M  U (k r )  w  n  M  *" Jix^i-l ^n-^-nn-i ~X-M ^ ~JjJ*  M  M  ^ »  5 «= determinant of the array formed by r e p l a c i n g •. 7 :,. in  j  J ^ k . r , ) by  I l f (k r, ) and. j ' ( k r , ) by; H^'tk.r, ) p  0  i n the f i r s t column of the A array  In order to indicate p a r a l l e l p o l a r i z a t i o n , equation (2.5a) may be written as  e  The other c o e f f i c i e n t s B^, B*, may be determined  . . . , B*,C^  i n a s i m i l a r manner.  t  C„,  Cy,  Since we are only  concerned with the scattered f i e l d , however, the expressions f o r these c o e f f i c i e n t s are not given. I f the core i s a perfect conductor, the f i e l d s are i d e n t i c a l l y zero f o r r < r . M  A zero tangential e l e c t r i c  f i e l d at r - r ^ . requires • ^ ( k A»,  e  M  , r J  B r  - V„(k _,r ) <-' M  M  =0  i n "this case i s given by equation (2.5b) with the  arrays f o r £\ and Se. modified by removing t h e i r l a s t e  row  and column. The scattered' f i e l d c o e f f i c i e n t s are of the form  (22>)  This was noted previously by Adey  ' i n the case of a  c y l i n d e r made up of two coaxial homogeneous regions.  ^ M ^ V V J  13  Perpendicular P o l a r i z a t i o n For perpendicular p o l a r i z a t i o n , the f i e l d ponents are H , E , B <, z  &  r  The a n a l y s i s procedure  com-  i s : the same  as i n the case o f p a r a l l e l p o l a r i z a t i o n and i n f a c t ,  equations-  (2.1) t o (2.5a), with the e x c e p t i o n o f (2.2), a p p l y i f E i s r e p l a c e d by H,: H by -E, 8™ and £  by JJ.™ and jx-o, and ft and  c  ,  S  /<.„ by £™(r) and  00  E q u a t i o n (2.2) i s now r e p l a c e d by  (2.6) which i s d e r i v e d i n Appendix A.  (2.6)  We note t h a t equation  has s i n g u l a r i t i e s a t zeros o f £^(r) i n a d d i t i o n t o t h e s i n g u l a r i t y a t the o r i g i n .  As a r e s u l t , t h i s case p r e s e n t s  s o l u t i o n d i f f i c u l t i e s n o t encountered  i n the p r e v i o u s  case.  E q u a t i o n (2.5b) becomes  Kh = - U f i r - H  (2.7)  0  Oh  When the core i s a p e r f e c t conductor, of  zero t a n g e n t i a l e l e c t r i c f i e l d  at r=r  - <(k .,r ) Bj-' - V M  M  the boundary c o n d i t i o n  ^  requires  M  r  J cr'  As a r e s u l t , / \ becomes M  A  M  = [ -  vU*«-, r  M  ) - <(k _,rj] M  « 0  14  2.1-2  Solution Usln^ Impedance Boundary Conditions /  An alternative method of c a l c u l a t i n g the scatt e r e d - f i e l d c o e f f i c i e n t s i s to apply an impedance boundary c o n d i t i o n at the surface of the cylinder and use nonuniform transmission-line theory as given by Schelkunoff to. calculate t h i s impedance.  Wait  5  (ZB)  has used t h i s  method, which he notes i s desirable i f an i t e r a t i v e procedure i s to be used.  Again, we w i l l deal with the two p o l a r i -  zations separately. Parallel Polarization The tangential e l e c t r i c and magnetic f i e l d s i n the region outside the cylinder are found by adding equation (2.1a)  to ( 2 . 1 b )  and equation ( 2 . 3 a )  to ( 2 . 3 b ) .  Thus we  have, from ( 2 . 1 a ) and ( 2 . 1 b )  (2.8) and from ( 2 . 3 a )  and  (2.3b)  j. . He  (2.9)  Por the n E  2  and  mode, we define an impedance r e l a t i n g  at r=r, by (2 10) C  15 Substituting from equations (2.8) and (2.9) i n t o (2.10) gives  equation  -  E  (.if M k r , ) + A, Hff.(lc.r. 1  (2.11)  c  0  Solving f o r Ar, we obtain An  ~  "  k^r  (3)" E,  (2.12)  f  k,r, ) + 3 z£Au~H ? T  The impedance Z^, i s determined by using an analogy between the concentric cylinders and cascaded sections of non-uniform transmission-line.  The transmission-line prob-  lem i s i l l u s t r a t e d i n Figure 2.3. (23)  Following Schelkunoff  , we define an "incident  wave" (+) and a " r e f l e c t e d wave" (-).in the m^  1  EST*  region by  V.dc^r)  Thus, the wave impedances are v-*#v  >  E  ""  ^  M^r)  (2.13a)  (2.13b) I f E ^ and H^, denote the t o t a l tangential e l e c t r i c and magnetic f i e l d s respectively, the impedance  til  looking into the m  region, i n the d i r e c t i o n of decreasing  16  n  Figure 2.3  Cascaded Non-Uniform T r a n s m i s s i o n - L i n e A n a l o g o f t h e Inhomogeneous  Cylinder  We c a n w r i t e t h e Impedance l o o k i n g i n t o t h e "til  m  "fcll  r e g i o n i n terms o f t h a t l o o k i n g i n t o t h e  (m+1)  r e g i o n as  (2.14)  where  17  K;"(A)  C  +  V„(A.)  TC^W,  When t h e e x p r e s s i o n s f o r K™  +  (2.13) a r e s u b s t i t u t e d  into  and  v'(*U) ,r„) =  g i v e n by equations  (2.14)  equation  and the r e -  s u l t s i m p l i f i e d , we o b t a i n  Z? - 3*1 rJ  where  -rn + \  p^=/M«»0  v'(A.)  1 -S -m < M  (2.15)  +"3 T.-'T* -"M**,) u ' ( A . )  s;>u'(«^) v ' ( A J u'(/?J C = » ( A ) <(«*.) - v,, (/£J u'(* ) u  w  I n t h e core t h e r e l s no " r e f l e c t e d wave" and therefore z; =  x r ( ^ ) = 3-1.  (2.16)  - 7 7 —  Now, by t h e s y s t e m a t i c a p p l i c a t i o n  o f equation  (2.15)  18  we obtain  K  ,  »  For a p e r f e c t l y conducting  • ••»  core, Z^, = 0. Perpendicular P o l a r i z a t i o n Tne solution i n the case of perpendicular p o l a r i z a t i o n may be easily derived from that i n the case of p a r a l l e l p o l a r i z a t i o n by the same interchange of variables used i n Section 2.1-1.  In addition, the transmission-line  analog i s replaced by i t s dual and, as a r e s u l t , the impedances become admittances.  Thus, the scattered f i e l d !  c o e f f i c i e n t s are given by  A„  where  ~- (3) Y^ =  J->,(kr, ) + 3 Y^/r 0  H,  H  Z7?  0  H ™ (k,r, ) +  3  J„(lc r, )  Y ; / ^ O  0  (2ol7)  (k r, ) 0  (r,,e-)  E ,(r, ,0-) er  The i t e r a t i o n formula ( equation (2.15) ) becomes  Yr =• 3 2 U r ) w  r«1 + i S™ ri *! + 3 y n , ( r  1 5 j  7  w+l  ) T •n  < M  (2.18)  where the expressions f o r P™, Q™, S™, and T™ are as given i n equation ( 2 . 1 5 ) , U-w and Vy, being the solutions of equation (2.6) i n t h i s case. For a d i e l e c t r i c core> Yy,  —  3  ( ^ftl )  /  IM*„)  and f o r a p e r f e c t l y conducting core, Y " = co •  (2.19)  19  2  «  2  Approximation f o r a T h i n , S h e l l The  case of a t h i n * Inhomogeneous s h e l l can  be  r e a d i l y a n a l y s e d by u s i n g t h e i t e r a t i v e f o r m u l a s o f S e c t i o n 2,1-2.  By c o n s i d e r i n g a s h e l l o f v a n i s h i n g t h i c k -  n e s s , a R l c c a t i - t y p e d i f f e r e n t i a l e q u a t i o n f o r impedance or admittance i s d e r i v e d .  This equation i s convenient to  use i f a n u m e r i c a l s o l u t i o n i s d e s i r e d and i n a d d i t i o n p r o v e s u s e f u l i n d e t e r m i n i n g upper l i m i t s on t h e magnitudes of high-order s c a t t e r e d - f i e l d c o e f f i c i e n t s . problem  i s i l l u s t r a t e d i n F i g u r e 2.4  The  and t h e s u b s c r i p t (m)  i s dropped i n o r d e r t o s i m p l i f y t h e n o t a t i o n .  F i g u r e 2.4  D e f i n i t i o n of Parameters  compared w i t h t h e wavelength p e r m i t t i v i t y i n the s h e l l  f o r t h e T h i n - S h e l l Problem  c o r r e s p o n d i n g t o t h e maximum  20  Parallel Polarization Equation  (2.15)  &ay be rewritten as  Z* P^ + zj,  where  =  5\  Z* S„  3  (2 20a)  —  e  + J -T. T„  P„ = TM«.) v'(/2) - V^ar) "Q„ = u^o?)  Qr,  \  U^V)-  V„(*) - V^{0) U  u'(rf) v'(/) -'v'(°0 TY= M/?). v'(tf) - VL,(/?) Sn =  (<*)  Tj'(/£)  u'(a)  For a t h i n s h e l l , T-oc0 iff.••small and the following  ex-  pansions give good approximations::  /  /"  where "C^ may be either  //  or V^.  (2.21)  i t i s : therefore possible  to eliminate OL as an argument of these functions i n (2 20a).  equation  g  Substituting from ( 2 . 2 1 ) i n t o the. expressions for P„ , Q„, S,, and we  and neglecting terms with a power  T%  obtain  (2.20b)  where  Sn ^  ( nV/  Tn «  (1  = TM/?). V „ V )  - M / W  - V^) - u'^)  >^ V^)  ) f  w  .  21  amd Tr,  I n a r r i v i n g a t the e x p r e s s i o n s f o r i n e q u a t i o n (2,20b),  second  d e r i v a t i v e terms are e l i m i n -  ated by making use o f the f a c t t h a t U^, and Vn s a t i s f y equation ( 2 . 2 )  c  With these approximations,  equation  (2„20a)  becomes z; ^ 3 ^  Z* + 3 \ ^ ~  (2.22a)  Z* ( n V ^  A  - SrifA)  - )r  + 3 >l ( I - W  or (2.22b) 1 - [ ( where  t {J/k.) - n V /  2  r  )  z * + V / ] t  z' = Zj./J'T. n  <  Z*/3 ^  =  Note t h a t equations p e r m i t t i v i t y a t r=r^ o n l y .  (2.22) i n v o l v e the r e l a t i v e  The change i n impedance i s  t h e r e f o r e the same as i f we had considered: a t h i n homogeneous s h e l l of p e r m i t t i v i t y Ke now  £(r*).  consider a very t h i n s h e l l .  I f t i s small  enough, then f o r f i n i t e v a l u e s o f z* ( i f z* i s l a r g e we may d e a l w i t h y*=l/z*) we have  and t h e r e f o r e l -  [(  - W  ^  ) <  1 + [ (  +  ] t  EyWA) - n V / ?  z  ) z* +  l / * ] r  22  Again n e g l e c t i n g terms i n t , e q u a t i o n ( 2 . 2 2 b ) may now be 2  s i m p l i f i e d to z4 -  < *[  </0 + (  l +  £ (/?A) - * i ^ ) z j * 2  z  r  J  r  (2.23)  I f we c o n s i d e r the impedance t o be a f u n c t i o n o f the v a r i a b l e j=kr, e q u a t i o n ( 2 . 2 3 ) may be r e w r i t t e n as  #  Z„(5+A0 - Z „ ( f )  L 1 + z (t)/t  A "Jf = tr  where  /  :  D i v i d i n g both s i d e s of t h i s e q u a t i o n by A f and t a k i n g the l i m i t as A ^ approaches zero we o b t a i n the d i f f e r e n t i a l equation dz,, = 1 + Z./5-+ (  £ ( f r  A)  »:nV5 /) 2  z*  (2.24a)  T h i s e q u a t i o n w i l l u s u a l l y have to be i n t e g r a t e d humer~ ically. y^l/z^.  Thus, when z* i s l a r g e i t i s convenient t o work w i t h The admittance  — - d 5'  [ Ji  +• 7*/t  s a t i s f i e s the e q u a t i o n  + M?A)  " n7^  z  ]  (2.24b)  ( 2 . 2 4 a ) and (2.24b) may be d e r i v e d  Equations  by a d i r e c t s u b s t i t u t i o n i n t o e q u a t i o n (2.2) as has been (31) done by B i s b l n g medium.  v  ' i n the case of a s p h e r i c a l l y  stratified  Thus, f o r example, .*  w  - R y  JJ~  (2.25)  23 Perpendicular P o l a r i z a t i o n The approximate r e s u l t f o r Y-^ i n the case of perpendicular p o l a r i z a t i o n i s not simply the dual of that f o r derived f o r p a r a l l e l p o l a r i z a t i o n .  This i s due to the  f a c t that TJ-,, and v.* s a t i s f y d i f f e r e n t d i f f e r e n t i a l equations, I t i s found that the approximate expressions f o r Pn,  Qn,  and ST, are of the same -form but that Tn i s replaced, by  1 „ ( x//3 - ^  ) t  ¥ 1  M / ? A )  As a r e s u l t ,  x/,  » '  *  Y  ( M#)  +  3  «=  *  (2 26a)  i T i )  + 3  ) Y*t  [1  tf(r,)  i f-jC £(/?A) ) -j r  .  £ (y#k>  J  r  or *  *  +  (2,26b)  £,(/?A) 1 - f ( 1 - n*/( U*A> + ( 1/^ -  r  ) ) y-J )]  .  ?  where . y„' =' - 3 T. ? i y* =  -  H  The d e r i v a t i o n of the d i f f e r e n t i a l equations f o r admittance and impedance follows the pattern used i n the case of p a r a l l e l p o l a r i z a t i o n .  For  24  ( 1 -  t «  ) y* + ( l / *  £,(/?A) P  1  elm) (•2.27)  we have  M*A)  < i  y, + 1  •n ~  £ (/?A>/?  2.  "  "7)  r  + (i/*-^ Eliminating  (  W  }  £,(/?A)  )y-r  +  £ 04*>t r  £,.(a;A) by using the expansion £,(^A) » £ (/?A) + J ^ ^ A ) r  f  gives::  - y* ~  WA)  + 71//S +  • ) y*  ( 1  (2.28)  and  therefore  dy»  )  7 i  *  (2.29a)  L e t t i n g z ^ l / y ^ we obtain  dT  ( 1 -  £,(?A) T  ) + 2T„/f + £J[S/lc)2* (2.29b)  Equations ( 2 . 2 9 ) may also be derived by substituting  25  v  =  (2.30a)  -1-1—  dR„/d? •  • .  cVdRn/a? 2  Into equation 2.2-1  =  •  —  (2.6).  MJA)  (2.30b)  RM  The Case of a Thin S h e l l on a Conducting Core We now  consider the e f f e c t which a t h i n dielec«  t r i e sheath with a r e l a t i v e permeability of unity has on the s c a t t e r i n g properties of a conducting cylinder. be the radius of the conducting  Letting r*  cylinder, the r e s u l t  may  be derived by s e t t i n g fUyu and Z*=0 l n Section 2 . 2 . 0  Parallel  Polarization With Z*=0, equation ( 2 . 2 2 a )  l  becomes  " l « X//3  I f the s h e l l i s t h i n compared to the radius of the conand hence Z ^ j - ^ r . S u b s t i -  ducting cylinder, we have  t u t i n g t h i s value f o r zj, i n equation ( 2 . 1 2 ) , A  '•  (3)  E  (k*r  (  .  P  - v  ) :  J^Ck^r,  we have  > -  —r  (kr, ) - T H g ' U c r , ) 0  Since f i s small we may  ^(k^) H * (k,r £  z  write  v Jntk^r, ) - tr j'(k„r, ) ) ^ H^(k.r  ;  ) ~ f H^k.r, )  •:?.;•:: (2.31a)  26  and thereby s i m p l i f y equation (2.31a) to  (2.31b)  I  (k„r, )  The r i g h t hand side of equation (2.31b) i s the th scattered f i e l d c o e f f i c i e n t of the n  mode  f o r a conducting  cylinder of radius r*.. ¥e conclude therefore that, i n the case of p a r a l l e l p o l a r i z a t i o n , the scattered f i e l d of a conducting cylinder i s not altered appreciably by a t h i n d i e l e c t r i c shell.  Evidence of t h i s i s the f a c t that a plot of back-  scattering cross section as a function of sheath radius s t a r t s with a zero, slope at the surface of the conducting cylinder as can be seen i n Figure 3« '» i,  13)  Yen and K a p r i e l i a n  v  ' have derived equation  and from computed r e s u l t s i n the s p e c i a l case  (2.31a)  £. (T)=o/r t  have observed that the e f f e c t of a t h i n s h e l l i s small.  Their  r e s u l t s do not agree with present c a l c u l a t i o n s i n general however, as shown i n Figure 3,4, and the "small e f f e c t " observation i s v i o l a t e d i n t h e i r r e s u l t s shown i n Figure (3.4b). Perpendicular P o l a r i z a t i o n L e t t i n g Y* approach i n f i n i t y i n equation  (2.26a)  gives:  3 V(r, ) * ~  (  E i/?Ao) r  - n  2  / ^ ) t  Substituting t h i s value f o r Yt, i n equation (2.17), we have  27  J-n(k.r, )  A„^  - (j)  E  - —  .frto*  >  }  -  0  —• (k.r, )  H^Uc.r, ) (2.32)  No s i g n i f i c a n t s i m p l i f i c a t i o n i s evident i n t h i s case and i t appears that the t h i n sheath has a noticable e f f e c t .  In  Figure 3*5 i t i s seen that the curves have a non-zero slope at the surface of the conducting cylinder i n contrast to the case of p a r a l l e l p o l a r i z a t i o n (Figure 3 . 4 ) . 2*3  Far-Zone Scattered F i e l d I t is; often the case that the scattered f i e l d at  large distances from the cylinder i s required.  The expression  for  the scattered' f i e l d may then be s i m p l i f i e d by using the (32) large argument asymptotic form of the Hankel function .  )(;-v -i) r  Substituting f o r H^ (kor) i n equation  ( 2 . 1 b ) we have, f o r  J  parallel polarization, ~  lim  -  J (k r 0  E^(r,G-) = \T e„ A„ / - § — e  —  •k r  ~e :  e  - 3  •  - { ) cos(ne-)  /r  ( k.r - f ) * I  e , ( J ) " A,, cos(nS-) (2.33)  A quantity which l s often given i s the b i s t a t l c scattering cross section per u n i t length cf(&).  28  This i s defined as follows* p  s  CT-(e-) - - r g <•  where p  5  E  lim 2/rr  E  r—*-co  S  (2.3*0  l  = t o t a l power reradiated per u n i t length of an i d e a l omnidirectional scatterer that maintains the same f i e l d at a r a d i a l distance r f o r a l l values of G- as that maintained by the actual scattering cylinder i n the d i r e c t i o n e-  S  L  s= r e a l magnitude of the Poyntlng vector of the incident plane wave.  Substituting the expressions f o r E (2.1a)  and  cT(e-) -  (2.33)  into  1  4'  E*  ko  —  (2.34)  1  and E  5  given i n equations  we have x  oo  I  «=o  e„ ( J f A„ cos(ne-)  (2.35)  For perpendicular p o l a r i z a t i o n , the above expressions hold with E replaced by 2.4  Cylindrical-Wave  H.  Incidence  In many situations where experiments are carr i e d out under presumably plane-wave conditions, the ture of the incident wavefront cannot be neglected i t s e f f e c t has to be accounted f o r .  and  This e f f e c t has been  investigated by F a r a n ( ) and by Z i t r o n and D a v i s ^ ^ 34  curva-  35  in  the related problem of scattering of a c o u s t i c a l waves by  29  h a r d and  s o f t c y l i n d e r s . Paran  has  shown t h a t a  r e l a t i o n s h i p e x i s t s between t h e s c a t t e r e d - f i e l d for  simple  coefficients  plane-wave i n c i d e n c e and t h o s e f o r c y l i n d r i c a l - w a v e  incidence,  A s i m i l a r r e l a t i o n h o l d s i n the  electromagnetic  c a s e , as w i l l be shown below, a n d ' p r o v i d e s t h e b a s i s f o r t h e experimental  method u s i n g c y l i n d r i c a l - w a v e I n c i d e n c e  dis-  cussed i n S e c t i o n 5»2-2.  F i g u r e 2.5  I l l u s t r a t i o n o f the P r o b l e m o f  Cylindrical-Wave  Incidence Consider  an e l e c t r i c l i n e - s o u r c e p a r a l l e l t o  the z - a x i s l o c a t e d o u t s i d e the c y l i n d e r at a d i s t a n c e D a l o n g t h e x - a x i s as I l l u s t r a t e d i n F i g u r e 2 . 5 .  I f the  i n c i d e n t e l e c t r i c f i e l d has z e r o phase and an a m p l i t u d e a t r=0,  then  A  30  El(i\>) =  A H  (  z) o  (k r„ ) 0  Hf(k D) 0  where r = d i s t a n c e from the l i n e e  Since T - / D  - 2 D r cos©- + r , we may use t h e expansion  z  2 -  0  k  J D  l  0  source.  - 2 D r cose- + r  to o b t a i n  ) =X  2  (k^D) ^ ( k ^ r ) cos(ne-)  ,  Ef(r,e-) = —  fie?, H ? ( k D ) J„(k,r) cos(ne-) }  e  (k*D)  (2.36) f o r c y l i n d r i c a l -  (2.1a) f o r plane-wave i n c i d e n c e  wave i n c i d e n c e and equation  i s that A Htf^k^Dj/H^tk^D) replaces E 0  (2.36)  ^*  The d i f f e r e n c e between equation  that E  (3Z)  (J) .  I f we n o t e  1 7  c  (j))* appears as a f a c t o r i n equation  (2,5b) f o r the  s c a t t e r e d - f i e l d c o e f f i c i e n t s , i t i s evident that  «  A  A E  (k,D)  —  e  ( J ) " H f (k.D)  Ane  (2.37)  where A „ = o r d e r n s c a t t e r e d - f i e l d c o e f f i c i e n t f o r a c y l i n e  d r i c a l i n c i d e n t wave produced by an e l e c t r i c  line-  source. F o r a magnetic l i n e - s o u r c e producing magnetic f i e l d B H « (KXo ) £  r  Ht(r  0  ) =  H^(k.D)  an i n c i d e n t  31  we have  where A-„ = o r d e r n s c a t t e r e d - f i e l d H  c y l i n d r i c a l incident netic  line-source.  coefficient for a  wave produced by a mag-  32  3«  CALCULATION OP THE  The  f i r s t s t e p i n c a l c u l a t i n g the  i s the d e t e r m i n a t i o n of the  SCATTERED FIELD scattered  c o e f f i c i e n t s A^.  field  I t i s then  n e c e s s a r y to d e c i d e upon the number of terms, to be used\ i n the ified  e x p r e s s i o n f o r the f i e l d i n order to a c h i e v e a s p e c accuracy.  investigated The  I n t h i s chapter, these two  i n the  points  are  case of g e n e r a l p e r m i t t i v i t y variations„  r e l a t i v e permeability  of each r e g i o n  i s assumed to  be  unity, 3.1  Methods of C a l c u l a t i n g the S c a t t e r e d - F i e l d I f c l o s e d form s o l u t i o n s  existed  i n each r e g i o n  of the wave equation  of the inhomogeneous c y l i n d e r ,  s c a t t e r e d - f i e l d c o e f f i c i e n t s could, i n a straightforward  of course, be  the  determined  manner u s i n g the r e l a t i o n s i n S e c t i o n  However, t h i s i s seldom the  case.  I n t h i s s e c t i o n , we  c o n s i d e r f o u r methods, b a s i c a l l y f o r use these closed  Coefficients  form s o l u t i o n s are not  t h i r d involve  the  available.  I n the  In the f o u r t h ,  linearly-varying permittivity,  a n u m e r i c a l i n t e g r a t i o n of the  The  The  shells respectively.  impedance  and/or admittance d i f f e r e n t i a l equations i s used e s s e n t i a l l y exact r e s u l t s .  first,  c a l c u l a t i o n of the c o e f f i c i e n t s : -  of an approximating c y l i n d e r formed of c o n c e n t r i c w i t h constant and  will  i n r e g i o n s where  a power s e r i e s s o l u t i o n of the wave equation i s used. second and  giving  methods are d i s c u s s e d w i t h  r e g a r d to d i f f i c u l t y of a p p l i c a t i o n , l i m i t a t i o n s , accuracy, and  computer time  required.  2,1  e  33  3»1-1  Power Series Solution of the Wave Equation Solutions of the r a d i a l wave equation i n the *  .54.  neighbourhood of a regular p o i n t may be obtained as a 1  power s e r i e s *  3 6  \ and i n f a c t t h i s method has been applied  by several authors*  ?  J  6  3  9  >  1  0  K  In order to carry out  the solution, one must use an expression of £ (r) i n the form r  of a r a t i o of polynomials.  When £ (r) i s not of t h i s form, r  a Taylor series expansion may be developed and., by using a s u f f i c i e n t number of terms i n the expansion, as accurate as desired can be obtained. cedure i s simple.  solutions  B a s i c a l l y the pro-  However, i n practice, the following two  drawbacks exist: (a) As the degree of the polynomials describing £ (r) increases, the number of terms i n the recurrence r e r  l a t i o n f o r c a l c u l a t i n g the c o e f f i c i e n t s i n the series increases (very r a p i d l y i n the case of perpendicular p o l a r i z a t i o n ) . (b) An upper l i m i t i s placed on r and also on the magnitude of the c o e f f i c i e n t s i n the polynomials such that terms of excessively large magnitude r e l a t i v e to the sum of the series are avoided (these reduce the number of s i g n i f i c a n t figures i n the sum).  S i n g u l a r i t i e s of 6 (r) must be poles of order 1 or 2. Note however that the p e r m i t t i v i t y of the cylinder i s assumed f i n i t e and hence s i n g u l a r i t i e s of £„(r) cannot occur within the region of i n t e r e s t . The behaviour of £>.(r) outside t h i s region i s of no consequence. r  3^  The d e t a i l s of the series solution f o r a l i n e a r permittivity variation  £ (r) r  = sr + t are given i n Appendix B.  Several of the r e l a t i o n s i n the appendix were derived p r e v i ously by Feinstein^ ^ \  We w i l l now  which the solutions are developed and i n t h i s special case and  examine the manner i n the problems encountered  then make further comments on  the  method i n general„ Parallel Polarization When £ (r) = sr + t, the series solution about r  r = 0, i s convergent f o r a l l values or r, s and  t.  This  w i l l always be the case i f £^.(r) has no s i n g u l a r i t i e s f o r r 4= 0.  The solution when the wave equation has. a singular-  i t y at more than one point i s considered i n d e t a i l when dealing with perpendicular p o l a r i z a t i o n . An inspection  of the recurrence r e l a t i o n i n the  series solution rev.eals that, i f - f ( r ) i s a polynomial, t h e r e r  i s a l i n e a r r e l a t i o n s h i p between the number of terms i n the recurrence r e l a t i o n and the order of the polynomial.  This  makes the series method quite a t t r a c t i v e f o r regions of small radius, i n which case the series may  be  accurately  summed, p a r t i c u l a r l y i n the core region where only one tion i s  solu-  required.  Perpendicular P o l a r i z a t i o n In t h i s case, the series solution when £ (r).t  i s r e l a t i v e l y complicated because:  ' ""  sr + t .  (a) There are more terms i n the recurrence r e l a t i o n .  35  (b) I t I s not always p o s s i b l e to f i n d  a point  about which a s o l u t i o n w i l l converge throughout the r e q u i r e d range. R e f e r r i n g t o F i g u r e B . l i t i s seen t h a t t h e r e a r e two s i t u a t i o n s i n which a s e r i e s s o l u t i o n about a s i n g l e point i s inadequate. (1)  These a r e  r = 0 and r = - t / s w i t h i n the r e g i o n  ( i l ) s and t o f o p p o s i t e r  sign, r  ~ - t/s• and  = - 2 t / s w i t h i n the r e g i o n .  I n such s i t u a t i o n s , the r e g i o n concerned may  ba d i v i d e d  i n t o two sub-regions and then the s o l u t i o n s about d i f f e r e n t p o i n t s matched u s i n g e q u a t i o n ( 2 . 1 8 ) .  The procedure w i l l  be i l l u s t r a t e d by two: examples. Example 1 i n Figure  Core r e g i o n w i t h s > 0, t > 0, r  > t / s as shown  3.1.  —r-Vs  F i g u r e 3.1  M  •  1  r  M 1  r  t/s  1  r  c  1  r  M  s>-  r  D i v i s i o n o f a Core Region f o r Matching S e r i e s S o l u t i o n s about D i f f e r e n t P o i n t s  36  The procedure  i s as f o l l o w s :  (i) 0 < r  M  /  D e f i n e a new  core r e g i o n by r = r  MI  (  ,  < t/s. (11) Determine the admittance  r = r  M  from  Y"'  looking i n at  e q u a t i o n (2.19) where U-„ i s g i v e n by the  ex-  p a n s i o n about r = 0 ( e q u a t i o n (B.14) ). ( i i i ) C a l c u l a t e Y" u s i n g e q u a t i o n (2.18) w i t h :." = Y"' •  Ur, and  g i v e n by equations Example 2  are s e r i e s s o l u t i o n s about r = r<> > t / s (B.19) w i t h /8  0  S h e l l with t < 0 ,  as i l l u s t r a t e d  i n F i g u r e 3.2.  =  sr /t.  s > 0, r  0  w+ )  < - t / s , r ^ > - 2t/s>  .  c%(r). A  F i g u r e 3.2  D i v i s i o n of a S h e l l which has a Zero of P e r m i t t i v i t y for  Matching  Points  S e r i e s S o l u t i o n s about D i f f e r e n t  37  Assuming Y „  has been c a l c u l a t e d , we proceed (1)  D i v i d e , the r e g i o n i n t o two  as f o l l o w s : at r = r  w  ,  - t / s < r , < - 2t/s:. w  (ii) a t r = r„,  Determine the admittance  Y™  l o o k i n g inwards:  1  by, a p p l y i n g equation (2.18) with Uy, and V^ the  s o l u t i o n s about r.= - t / s g i v e n by equation  (B.17).  ( i i i ) C a l c u l a t e Y^ u s i n g equation (2.18) w i t h Y ™ ' = Y™' • +  r  0  U,, and V^ are s e r i e s s o l u t i o n s about r = r , 0  > |(r +t/s) - t/s -  w i t h /3  >n  K r ^ t / s ) , g i v e n by equations (B.19)  = sr /t.  0  ,  c  We note t h a t i n the expansion  about r = ~ t / s , y  is:: p o s i t i v e f o r r < « t / s and n e g a t i v e f o r r > - t / s . s o l u t i o n i n the l a t t e r term and  case i s complex due  to the  The  ln(20  t h e r e f o r e Y™ becomes complex r a t h e r than, p u r e l y  imaginary.  I n o r d e r to determine the c o r r e c t s i g n of the  imaginary p a r t of the l o g a r i t h m , we  consider a p h y s i c a l s i t -  u a t i o n i n which a s m a l l l o s s i s p r e s e n t and determine the s i g n of the imaginary p a r t of y a t r = - t / s . A s u i t a b l e example f o r the above s i t u a t i o n l s t h a t of a plasma;where c o l l i s i o n l o s s e s are c o n s i d e r e d . electric  p o l a r i z a t i o n , P,  i s g i v e n by* X  p = „ So  E 1 -  -:.:X,y.  V  where  X.= '  N  e  2  :  £  0  m  e  CJ  Z  3Z  1  ^  The  38  N =• electron number density e = electron charge m= c  electron mass  Z = v/co V = electron c o l l i s i o n frequency. The 3  term represents the loss and we assume Z «  Z  Since  £ =£o+  a l i n e a r v a r i a t i o n of electron  ¥/€<,,  1. density  with radius corresponds to a l i n e a r v a r i a t i o n of p e r m i t t i v i t y . We may write E  y  (r) = sr + t ar + b  = 1  l  - 3Z  ^ - (ar + b - 1) (1 + 3Z) - 3Z  Here:, the constants s and t are assumed complex. them as s = s  Writing  3 s " and t = t* + 3 t , we f i n d the follow-  1  I f  ing r e l a t i o n s : . s* = -a t' = 1 - b  s* = s* Z 1  t»  1  = Z ( t  1  - 1 )  Substituting these r e l a t i o n s into the expression f o r / we obtain • 2f = s r / t + 1 & ( s ' r / t ' + 1 ) + 3Z The imaginary part of y i s p o s i t i v e i n d i c a t i n g that an argument of series.  (-t-rr)  must be used f o r representing  y< 0 i n the  '39  If  £,-(r) i s of a more complex form, the method  of matching solutions about various points may applied.  s t i l l be  The recurrence r e l a t i o n s i n the series r a p i d l y  become unmanageable, however.  For example, i f £ (r) i s r  a polynomial, the number of terms i n the recurrence  in-  creases as the square of the polynomial degree as can be seen by examining equation ( B o l l ) ,  The case of a quadratic  v a r i a t i o n of p e r m i t t i v i t y has been considered to some extent* 3»l-2  8  *  3  *  1 0 )  Approximation  of an Inhomogeneous Region by  Homogeneous Shells. An inhomogeneous region may be approximated by a large number of t h i n homogeneous s h e l l s * Real solutions of equations is  for  (2,2)  and  (2.6),  constant, are Bessel functions J^{/£^k„r)  1 5 f l 7  '^^•  when £ (r) r  and  Y^(7^k r) 0  £»- > 0 and the modified Bessel functions I ^ v O ^ ^ r ) and  K„(/T£k r) for f f  S  r  <  0.  : The c a l c u l a t i o n of integer-order Bessel functions i s rapid using recurrence r e l a t i o n s and  polynomial  13 Z) approximations  f o r the zero and f i r s t orders,  .  Large  arguments (•//e l k r ) present no: problem (asymptotic forms r  0  may be used i f necessary) as i s the case with the method of series s o l u t i o n .  Small arguments and large orders t o -  gether may present some computational d i f f i c u l t i e s because 3-n and  approach zero and  Y T , and K,, approach i n f i n i t y .  These may be d e a l t w i t h as d i s c u s s e d i n Appendix C where l t i s shorn t h a t the l i m i t i n g Is p a r t i c u l a r l y simple.  case o f a z e r o p e r m i t t i v i t y  Lunow and T u t t e r  v  suggest t h a t  t h e c o m p u t a t i o n a l problems may be a v o i d e d by u s i n g J^isfJJj k r ) 0  8 1 1 ( 1  1  •»(//£,•/" o ) k  r  a s  s o l u t i o n s i n a r e g i o n near a p e r m i t -  t i v i t y zero i r r e s p e c t i v e of the s i g n of £ . r  T h i s method  seems t o be unfounded m a t h e m a t i c a l l y and g i v e s d i f f e r e n t r e s u l t s from t h o s e d e r i v e d i n Appendix 0. Choosing t h e S h e l l  Parameters  A method o f d e t e r m i n i n g t h e optimum t h i c k n e s s and p e r m i t t i v i t y f o r t h e s h e l l s i s n o t o b v i o u s , i f at a l l p o s s i b l e .  Here, t h e c h o i c e o f t h e s e parameters i s :  done i n two s t e p s , f i r s t c h o o s i n g t h e t h i c k n e s s and t h e n d e c i d i n g on t h e p e r m i t t i v i t y . Two approaches t o c h o o s i n g t h e t h i c k n e s s a r e suggested (i) Bgn(  Choose t h e r a d i i such t h a t t h e change o f  £ ( r ) ) • l£r(r)l* K  each s h e l l .  , cc b e i n g  a constant, i s equal f o r  The s i g n i s a p p l i e d s e p a r a t e l y from t h e power  i n o r d e r t o a v o i d d i f f i c u l t i e s when £ (r) changes s i g n w i t h r  i n t h e inhomogeneous r e g i o n . The s p e c i a l c a s e s  and cc = 1 c o r r e s p o n d t o e q u a l  i n c r e m e n t s o f r e f r a c t i v e i n d e x and p e r m i t t i v i t y ,  respectively.  T h i s method tends t o c o n c e n t r a t e t h e a p p r o x i m a t i n g  shells  i n r e g i o n s o f r a p i d p e r m i t t i v i t y v a r i a t i o n and p l a c e s no restriction  on t h i c k n e s s where t h e p e r m i t t i v i t y v a r i e s s l o w l y .  kl  I f the v a r i a t i o n o f S {T) i s n o t monotomic throughout the Y  e n t i r e r e g i o n , the method can be a p p l i e d t o sub-regions i n which  t h i s i s the c a s e .  •/£,,(*")/ For  cC-O,  d r I s equal f o r a l l s h e l l s . the r e g i o n s a r e o f equal t h i c k n e s s .  When <*- = i ,  the i n t e g r a l i s s i m i l a r t o the phase i n t e g r a l o f geometr i c a l o p t i c s and we may t h e r e f o r e t h i n k o f t h i s case as e q u a l i z i n g an e q u i v a l e n t e l e c t r i c a l t h i c k n e s s f o r each  shell.  Although t h i s c h o i c e o f oc I s i n t u i t i v e l y a t t r a c t i v e , the procedure i s n o t simple i n g e n e r a l because  o f the n e c e s s i t y  of e v a l u a t i n g t h e i n t e g r a l and subsequently s o l v i n g f o r r ^ . The case o f the  oc~l  corresponds t o e q u a l i z i n g the areas under .1  Sy.(x) v s . r curve.  The second s t e p i s t o a s s i g n a p e r m i t t i v i t y to each approximating s h e l l .  There are s e v e r a l  possibil-  i t i e s which w i l l g i v e a p p a r e n t l y r e a s o n a b l e r e s u l t s .  For  example we can use  (  J E (r^)  K  2  £ (r ) + E K  +/£ (r  r  w  2 The c r i t e r i a used t o determine  r  (r  m + I  m + f  )  (3.1a)  ) (3.1b)  £7 i n equations ( 3 . 1 a ) and  42.  (3»lb) a r e s i m i l a r t o those used i n t h e two s p e c i a l of  (1) f o r determining the s h e l l t h i c k n e s s .  are  cases  Choices which  i n the same r e s p e c t s i m i l a r t o t h e s p e c i a l  cases i n  ( i i ) are  £  )  « it  (3.1c)  sgn(  £„(r)  ) y/£ (r)7 d r r  Z (3..W)  )  (3.1e)  Treating  t h e problem from a curve f i t t i n g p o i n t o f view,  the l e a s t - s q u a r e s f i t method may be a p p l i e d  i n e i t h e r the  S' \r) v s . r o r t h e 7 £ (r j v s . r p l a n e . y  K  If the  £'(r) does n o t have l a r g e v a r i a t i o n s  s h e l l , a l l o f the methods suggested above w i l l  within give,  approximately t h e same v a l u e f o r £™ when t h e s h e l l s a r e t h i n . S i n c e i t i s r e a s o n a b l e t o assume t h a t  the r e s u l t s w i l l be  i n s e n s i t i v e t o s m a l l changes i n the p e r m i t t i v i t y the 3*1-3  ease o f a p p l i c a t i o n  should be t h e d e c i d i n g  of the s h e l l s ,  factor.  Approximation o f an Inhomogeneous Region by S h e l l s with Linearly-Varying  Permittivity  (37) Wilbee  has c o n s i d e r e d the one d i m e n s i o n a l  problem o f approximating a p l a n a r inhomogeneous r e g i o n by s l a b s w i t h c o n s t a n t and l i n e a r l y - v a r y i n g  permittivities.  43  Accurate r e s u l t s of the r e f l e c t i o n and transmission coe f f i c i e n t s at the boundary of the region were obtained 'with s i g n i f i c a n t l y fewer slabs of the l a t t e r type.  I t was  decided  to examine the merits of a s i m i l a r approximation i n the case of a c y l i n d e r . Since neither equation ( 2 . 2 ) closed form s o l u t i o n when £  r  nor ( 2 . 6 )  has a  = sr -f- t , the power series  solution developed i n Section 3 . 1 - 1  w i l l be used.  As a r e -  s u l t , the allowed p e r m i t t i v i t y v a r i a t i o n and cylinder radius: w i l l be l i m i t e d by the series summation errors discussed i n that section.  The advantage of t h i s method over the d i r e c t  power series s o l u t i o n method i s that the a p p l i c a t i o n to a r b i t r a r y p e r m i t t i v i t y v a r i a t i o n s i s immediate. The statements made i n Section 3 . 1 - 2  concern-  ing the choice of the thickness of the approximating n a t u r a l l y apply i n t h i s case.  shells:  We w i l l therefore only d i s -  cuss the choice of the values of s and t f o r each s h e l l . Two  simple methods are i l l u s t r a t e d i n Figure  Method 1 )  3.3.  region we have <f (r) = s „ , r + t  For the m  where  r  (3.2a) •*••»•> +1  t Method i l )  w  - £ (r,J + s ^ r r  (3.2b)  w  For the approximation shown i n Figure 3 . 3 1 1 ,  i s given by e q u a t i o n , 3 . 2 a t  w  = ti  and  - -| ( s„r  0  + t'  m  - £ (r ) ) K  e  s  m  (1) F i g u r e 3.3  (ii)  I l l u s t r a t i o n o f t h e two Methods o f C h o o s i n g t h e S h e l l P a r a m e t e r s s and t .  where  =.- H ( r ) + r  r  ~-h  0  w  s r  ( r^, + r  m  v n  r n +  ,  J  I t i s c l e a r t h a t t h i s method g i v e s t h e more a c c u r a t e  approx-  i m a t i o n o f the p r o f i l e . O t h e r c r i t e r i a may he used f o r c h o o s i n g t„, w h i l e leaving s  w  fixed.  F o r example, as s u g g e s t e d I n S e c t i o n 3•1-2,  t h e " e l e c t r i c a l t h i c k n e s s " o f t h e a p p r o x i m a t e and a c t u a l medium may be e q u a l i z e d . s l o p e i s s ^ = £'(r ). 0  Another reasonable choice f o r the  From a somewhat d i f f e r e n t p o i n t o f  v i e w , a l e a s t - s q u a r e s f i t may be performed t h e r e b y f i x i n g both  and t  r n  . I t i s d o u b t f u l whether any o f t h e s e a l t e r -  n a t i v e s w i l l , produce s i g n i f i c a n t l y b e t t e r r e s u l t s t h a n method l i ) above, w h i c h has t h e a d d i t i o n a l advantage o f simplicity.  45  3.1-4  Numerical I n t e g r a t i o n  of t h e Impedance and A d m i t t a n c e  D i f f e r e n t i a l Equations  -  The v a r i a t i o n o f impedance o r a d m i t t a n c e w i t h i n an inhomogeneous r e g i o n may be a c c u r a t e l y i c a l integration.  c a l c u l a t e d by numer-  S i n c e e i t h e r z„ o r jy, may become i n f i n i t e ,  the i n t e g r a t i o n i s c a r r i e d out on t h e e q u a t i o n i n v o l v i n g t h e smaller  o f the two, e q u a t i o n (2.24a) o r (2.24b) i n t h e case  o f p a r a l l e l p o l a r i z a t i o n and ( 2 . 2 9 a ) o r ( 2 . 2 9 b ) i n t h e case (31)  of p e r p e n d i c u l a r p o l a r i z a t i o n . B i s b i n g has used t h i s approach i n d e a l i n g w i t h s c a t t e r i n g by a r a d i a l l y inhomogeneous (£l Z3) sphere.  Other a u t h o r s  '  have o b t a i n e d r e s u l t s f o r r a d i -  a l l y inhomogeneous c y l i n d e r s by n u m e r i c a l l y  i n t e g r a t i n g the  wave e q u a t i o n .  f3 I n t h i s work, a Runge-ICutta^  &) ;  method o f o r d e r  f o u r was used i n t h e n u m e r i c a l i n t e g r a t i o n .  The d e t e r m i n a t i o n  of an i n i t i a l v a l u e f o r i n t e g r a t i o n i n , t h e m^*  1  d i f f i c u l t i e s which a r i s e i n performing'the d i s c u s s e d i n Appendix D.  region  and  integration.are  The method of d e a l i n g w i t h a z e r o  o f p e r m i t t i v i t y i n the case o f p e r p e n d i c u l a r p o l a r i z a t i o n by u s i n g a s e r i e s s o l u t i o n a p p r o x i m a t i o n i n the;neighbourhood of t h e r e s u l t i n g s i n g u l a r i t y i s o f p a r t i c u l a r , i m p o r t a n c e . (Z3) G a l and Gibson  avoid  t h i s s i n g u l a r i t y by  considering  s m a l l l o s s e s when c a r r y i n g o u t c o m p u t a t i o n s i n v o l v i n g z e r o s / of p e r m i t t i v i t y .  46  3»l-5  R e s u l t s and Comparison o f t h e Methods S i n c e t h e f o u r methods o f c o m p u t a t i o n d e s c r i b e d  i n S e c t i o n s 3«1-1 t o 3,l~b- a r e e s s e n t i a l l y i n d e p e n d e n t , agreement between r e s u l t s c a l c u l a t e d a check on each o f them.  by each method  provides  F u r t h e r checks were made by com-  p a r i n g the r e s u l t s w i t h exact r e s u l t s f o r a v a r i a t i o n of the form £ (r) « a r  and by comparison v r i t h p u b l i s h e d r e s u l t s .  fc  r  p u b l i s h e d r e s u l t s c o n s i d e r e d were, f o r p a r a l l e l 1) cylinder, £  r  dielectric  5  A  ii) a metal  polarization  Adey*^ ^ - f i e l d of homogeneous *  The  2.56 Tang*^ ^ - 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 o f 0  core w i t h a homogeneous d i l e c t r i c s h e l l ,  £  r  =  2.541  ( 3 )  i i i ) Yeh and K a p r i e l i a n s e c t i o n of a metal  v  ' - backscattering cross-  c o r e w i t h an inhomogeneous d i e l e c t r i c  shell,  £*-(r) = a / r and f o r p e r p e n d i c u l a r iv)  polarization  G a l and G i b s o n ^ ) - b a c k s c a t t e r i n g c r o s s 3  s e c t i o n o f an inhomogeneous d i e l e c t r i c £V(r)  cylinder,  = 1 - a(r/r, f  With the exception of i i i ) ,  agreement between computed and  p u b l i s h e d r e s u l t s was o b t a i n e d .  A r e p r o d u c t i o n o f some o f  the c u r v e s g i v e n i n R e f e r e n c e 3  together with  computed r e s u l t s a r e shown i n F i g u r e 3«^«  corresponding  I t Is readily  seen t h a t t h e r e l s no resemblance between t h e two r e s u l t s i n g e n e r a l , a l t h o u g h they do c o i n c i d e f o r k r , < 3 i n F i g u r e 0  The c o r r e s p o n d i n g  3.4a.  r e s u l t s i n t h e case o f p e r p e n d i c u l a r p o l a r -  i z a t i o n a r e shown i n F i g u r e  3.5.  47  \5 0  Figure  3.4  Computed Metallic Shell, £ _.  and P u b l i s h e d V a l u e s of k c ^ ( 0 ) f o r a C y l i n d e r w i t h an Inhomogeneous D i e l e c t r i c (r) = cc/k r. , Computed ~~77~.T~. P u b l i s h e d(3) o  r  0  48  49  30  / /  20  o  / /  \ \ \  5  •  \  r  10  ^ cC=0.5  /  v.  /  v.  /  0 0.  /  /  s  /  2  4  6  8  10  k r, ?  (a) 50  A  1 v  1  40  \ \  I  1  V  \  30  \  o  \  cc=3  k  o 20  I  /\  10  O  V  ^  /  1— 1  ,  v / v/  0 0  2  4  6  8  10  (b) F i g u r e 3.5  Computed Values of k / J O ) f o r a M e t a l l i c C y l i n d e r with an Inhomogeneous D i e l e c t r i c S h e l l , S (r) = c 6 / k r . a) k r ^ = 0.5 b) k r * = 3.0 r  c  0  0  50  Accuracy of the Shell-Approximation  Methods  I n o r d e r t o compare t h e r e l a t i v e a c c u r a c y  achieved  f o r a g i v e n number o f s h e l l s i n t h e s h e l l - a p p r o x i m a t i o n methods, c o m p u t a t i o n s were made w i t h v a r i o u s numbers o f a p p r o x i m a t i n g . s h e l l s f o r s e v e r a l v a r i a t i o n s o f £ ( r ) > 0 and t h e r e s u l t s r  compared w i t h t h o s e c a l c u l a t e d u s i n g n u m e r i c a l i n t e g r a t i o n . S h e l l s : o f e q u a l t h i c k n e s s were used; i n t h e homogeneous approximation,  £™ was c a l c u l a t e d u s i n g e q u a t i o n  i n the l i n e a r l y - v a r y i n g s h e l l approximation, o f c h o o s i n g s ^ and t were c o n s i d e r e d .  m  g i v e n by e q u a t i o n s  These a r e d e s i g n a t e d  shell  ( 3 . 1 c ) and  t h e two methods  ( 3 » ) and (3«3) 2  l i n e a r ( i ) and  l i n e a r ( i i ) , r e s p e c t i v e l y . The 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 was used as t h e b a s i s o f c o m p a r i s o n . The c y l i n d e r s c o n s i d e r e d 1) where  were  £ (r) = a + b f ( r / r , ) r  a = 0.25, 0.5, 2.0, 4.0  -  b = 1 - a r, =  0.25,  0.5,  0.75.  f ( r / r , ) = ( r / r , )* ,  1.0  Ao  Jr/r~  ¥e n o t e t h a t i n t h e case o f p e r p e n d i c u l a r p o l a r i z a t i o n , the most cumbersome s e r i e s s o l u t i o n , e q u a t i o n (B.22) must o f t e n be used f o r t h e l i n e a r l y v a r y i n g s h e l l s i f E' {x)>0. S i n c e t h e s e r i e s summation i s r a t h e r time consuming even i n t h e s i m p l e r case o f p a r a l l e l p o l a r i z a t i o n , n u m e r i c a l i n t e g r a t i o n i n t h e l i n e a r s h e l l s was used r a t h e r t h a n t h e series solution f o r perpendicular p o l a r i z a t i o n r  51  i i ) Shell: Core:  ^ ( r ) = a + b f(r) ^(r) = c  where a and b a r e such t h a t £^(r, )' = 1 , r, = 0 . 5 , r^ ~ 0.25  0.75, )i  1-0,  1.25  £,(r ) = c z  *o  0  f ( r ) = / F , c = 0.5 f (r) = 1/r, c = 2 f ( r ) = 1/r*,  c = 4  •  The r e l a t i v e e r r o r s i n t h e a p p r o x i m a t e r e s u l t s t o g e t h e r w i t h t h e c o r r e c t v a l u e s i n a few r e p r e s e n t a t i v e cases: a r e shown i n T a b l e s 3 « l  a  t o e.  An e v a l u a t i o n o f t h e  o v e r a l l a c c u r a c y i s g i v e n i n T a b l e s 3 . 2 a and b. T a b l e 3 . 2 a a p p l i e s i n the cases considered  e x c e p t when  £ oc J~Y\ K  where n o t i c a b l y d i f f e r e n t r e s u l t s a r e o b t a i n e d ; shown i n Table 3 » 2 b .  r i 0,  these are  The q u a l i t y o f agreement has been  c l a s s i f i e d as f o l l o w s : E x c e l l e n t - e,E; E r r o r s Good -  g,G; E r r o r s 1-5%  Fair -  f,F; Errors 5-10$  Poor -  p,P; E r r o r s 1 0 - 2 5 $  V e r y P o o r - v,V; E r r o r s  *  < 1%  > 25$  T h i s t y p e o f v a r i a t i o n i s s i g n i f i c a n t l y d i f f e r e n t from t h e o t h e r s c o n s i d e r e d i n t h a t £^(r) becomes i n f i n i t e a t r=0  Table  3.1  \ r , / X ShellsSw  Relative Error i n the Value A p p r o x i m a t i o n Methods 0.25  V  N  •  o f k 0\0)  0  Par  o  Calculated  Using  0.50 Perp  Par  the S h e l l  0.75 Perp  1.00  Par  Perp  Perp  17 7 . 3 3 7 5 6.0029 1 5 . 2 2 42  44.9661 2.2819 7 . 3 7 77  1 HOMO 1 LIM i 1 LIN i i  0 . 194 9 . 461 2 - 0 . 2 07 5  0 . 0 72 0 - 0 . 5660 - 0 . 2 790  -0.7682 - 0 . 6 311 - 0 . 4 9 08  - 0 . 2 7 98 -0.6161 -0.4713  2 HCMG 2 LIM i 2 LIM i i  -0 . C675 - 0 . 106 8 - 0 . 03 1 7  -0 . 0 8 0 5 - 0 .14 9 6 - 0 .045 7  - 0 . 5 8 50 - 0 . 1 5 13 -0.0492  -0.4999 -0.2031 - 0 . 0 6 53  4 .4 435 .-o .2815 - o . 004 4  810 . 9 3 5 2 . 5 .4319 1 .0532  57.4959 -0.9970 - 0 .9365  2 1 .4^64 -0.9550 -0.8880  3 3 3  HOMO LIN i LIN i i  - 0 . 0 37 7 - 0 . 04 6 9 - 0 .0127  - 0 . 0435 - 0 .0671 -0 . 01 8 3  - 0 . 2 720 -0.06 94 -0.0194  -0.2173 -0.0968 -0.0267  0 .6839 - 0 .1732 - 0 .0377  110 . 9 0 5 3 0 . 16 78 - 0 .1080  3.6032 -0.3850 -0.2645  -C.4016 -0.2875 - 0 . 1 6 84  4 HOMO 4 LINi 4 LIN i i  -0 . 0 2 2 7 - 0 .0263 - 0 .C069  - 0 . 0259 - 0 . 0 3 79 - 0 . 0099  -0.1546 -0.039 5 -0.0105  - 0 . 1 2 15 - 0 . 0 5 59 -0.0147  0 .2003 - 0 .1101 - 0 .026 L  • 30. 8 0 0 7 - 0 . 2458 — o .116 3  0.5656 - 0 . 1.630 -0.0337  -0.5150 -0.1285 -0.0628  5 HOMO 5 L I ,\' i \ 5 LIN i i v  - 0 . 0 14 9 - 0 .0168 - 0 . G04 3  - 0 . 0170 - o . 0 2 43 - 0 . 0063  -0.09 94 - 0 . 0 2 54 -0.0066  - 0 . 0 7 75'. -0.0362 -0.0094  0 .0831 - 0 .0733. - 0 .0178  L2 . 6 2 6 2 -0 . 245 8 - 0 .(.84 7  0.1354 .-0.095 I -0.0416  - 0 . 3 8 34 - 0 . 0 7 46 -0.0302  - 0 . 0 1 56 - 0 . 0 2 02 -0.0061  .-0.1GS8 -0.0165 - 0 . 0 0 43  .  3 3 . 1 7 0 7 10 2 1 . 2641 7 7 . 4 402 -C . 2 05 1 185 . 3488 0 .2458  Par  "•'  10 10 10  HOMO LIM i LIN i i '  . 0039 - 0 . C04 2 -0 . 001 1  - 0 . 0 04 4 - 0 .0061 -0 . 0 01 5  - 0 . 0 2 50 -0.0064 - 0 . 0016  -0.0193 -0.0092 -0.00 23  0 .0078 - 0 .0192 - 0 .0048  15  HOMO  - 0 .0017  - 0 .0019  -0.0111  - 0 . 0 0 86  0 .0025  0 . 4 1 0 6'  -0.0101  - 0 . 0 4 93  20 HOMO  - 0 .0010  - 0 .0011  -0.00 63  -0.0048  0 .0012  0  -0.0063  -0.0279  0 . 4 3 08  0.4781  0 .0355  G. 0006  0.0137  0.0412  k Ol0) o  — 1.1  . 33 8 9  '  0 .3019 (a)  £,.(r) = 0.25 + 0 . 7 5 ( r / r ,  f  • 1. . 2 4 7 5 - 0 .0880 - 0 .0233  > -  . .  0.50  0.25 ShellsV  Perp  Par  Perp  1 4 . 6 6 5 2. 2.0155 - 0 . 6734  0 . 5240 -0.7423 - 0 . 3 8 73  6.6741 -0.6443 - 0 . 52.43 .  1.00  0.75 Par  Perp  Par  Perp  16.4110 2 . 222 0 -0 . 1390'  0.2 56 0 -0.9003 -0.4017  5.2 2 80 -0.4062 0.2831  0.7173 -0.0295 -0.0671  7.1933 1 .3262 0.6552  0.5651 - 0 . 3 544 -0.1465  0.4319 0.6693 0.768 2  -0.2883 -0.064 1 -0.0177  -0.3208 0.0032 -0.0133  7.3721 0 .1982 0.0574  -0.6565 -0.1510 -0.0489  16.3281 -0.22 0 L - 0 . 1 134  -0.9^83 0.2 9 2 4 C . 0 826  -0 .015 9  0.1 9 7 3 -0.3079 '-0 .-0 6 0 ?  -0.1438 -0.0342 •. - 0 . 0 0 3 8  -0.2476 0.00 40 -0.0040  - 0 . 2 60 0 -0.0809 - 0 . 0 244  0.8775 -0.2089 -0.0834  0.0 0 43 0.19 42 0.0 6 65  0 . 0 2 30 -0 . 0 3 '5 3 -0 . 0 10 1  0.10 9 7 -0,1>,R0 - 0 .03)3 2  - 0. 0 8 5 2 -0.0214 -0.0054  -0.1795 0.0030 -0 . 0 0 1 4  1.0719 0.0293 0 .0094  -0.1714 -0.0496 -0.0137  1. 1 0 0 0 -0.1'-03 -0 .0 504  -0.2038 0.1178 0.0 3 7 4  0. 005 3 -0 .010 0 -0 .002 5  0.0205 -0.0 3 64 -0.0093  -0.0190 -0.0052 -0.0013  -0.0529 0.0009 0.0001  0 . 2.2 2 5 0 .003 4 0.0009  -0.0417 -0.0119 -0.0030  0.19 S7 -0.0356 -0.0095  -0.0341 0.02 67 0.00 69  .0731  1 HOMO . 1 LIN' i 1 UN i i  -o .  2 HOiYG 2 L IN i 2 L IN i i  •0 .221 a - 0 . 2 1 59 -0 . 0651  1 . 346 0 - 0 . 5 1 74 . -0.2415  -0.9366 -0.1537 -0.04 4 2  3 HOMO 3 L IN i 3 L IN i i  0 .0761 -0 . 1 0 4 9 -0 . 0 2 8 4  0.4335 -0.3348 -0.10 34  4 HOMO 4 LIN i 4 L IN i i  -c . 0 607  0 .0 330  5 1-iOMG 5 L IN i 5 UN i i 10 HOMO 10 L I N i 10 L I N i i :  Par i  -0  3000  1.8064 • 0 .0695 0 . 02 5 3  2.43 32 . -0.6536 -0.5058. - 0 . 0 6 64 0.3 9 06 0.0189  15 HOMO  0.0023  0.0 084  -0.008 3  - 0 . 0 2 43'  0.09 51  -0.0184  0.0845  -0.0143  20 'HCMG  0.0013  0.0 04 6  -0.0046  -0.0139  0.0528  -0.0103  0.04 6 8  -0.0079  koOKo)  1 .5519  0 . 0 77 0  6.62 29  1•874S  0.8 54 6  .9.1889  2.4433  (b)  £,(r) = 4 - 3 ( r / r ,  f  6.5957  \ r / X0 ShellsX. (  Par  0.25  Par  Perp.:  - 0 .5 6 4 8 2 .0 02 6 0. 94 4 2  1 HOMO 1 LIM i 1 LIN i i  —0.A52 3 1 .3086 0 . 685 A  2 HCMG 2 L IN i 2. L I M i i  _ r.  3 HOMG 3 L IN i 3 LIM i i .  -0 . C 2 0 A 0 . 1 64 3 o . OA0 3  -0. 0 0 8 5 0. 25(1 0 .0 7 1 5  -0.0122 0.7212 0.2049  -0 . 0 0 8 6 0 . 08 8 9 0 . 0 2.1 1  0. 0 0 ]. 5 0. 142 7 0. 03 30  5 HOMO 5 LIN i 5 LIN i i  -0 .0 04 A 0 . 055 1 0 . 0 1. 2 8  10 HOMG 10 L I N i 10 L I N i i  • 4 HOMO A L IN i A L IN i i  ' 15 'HOMO 20 HOMO  1 .7 2 46 2.4383 2.21 8 5  0.50  Perp  1 .3728' 2.4566 2.2072  Par  0.75  5 .9455 -0.0129 -0.7 791  Perp  3.CA79 0. 0576 . -0.8375  Par  1 .00  Perp  3. OS.5 A 0.965 7 0.7805  1.7.5 55 1.19 64 0.98 59  . 075 2 2.0 70 2 1 .9731  5.4418 " 2.2188 2.2 0 90  !  0 .442 6 3 .0711 2.. 0 5 1 5  -0.33A2 .3. 5 07 9 2.3732  0.0790 1 . 2089 0.3757  0.1916 1 . 73AA 0 .6546  - 0 . 01.96 2 . 2 0 10 0.8709  -0.7590 1.7226 0.8839  -0.7644 2.3724 1.2 4 33  0. 0.103 0.3905 0.1001  0.0810 0.66 81 0.19 06  0.1A 79 0.9827 0.29A7  0.065A 1.2922 ' 0.A179  - 0 . 3 596 1 .1047 0.4 0 6 6  - 0 . 3 3 71 1.5940 0.6131  0. 0 0 3 6 0 .0 9 u 0 0. C 2 3 2  0.0142 0.2412 0.05 90  0.0 6 78 .0 . 4 1 8 6 0.1146  0.1125 . ...0.6116 0.16A3  0.0758 0.8206 0 . 2 A1 5  -0.1948 0.7219 0.2232  -0.1610 1.06 22 ' 0.3508  -0 . C C 0 7 0.0124 0. 0026  0 .0 0 2 1 0. 0 2 1 0 0. 0 0 4 7  0.00 77 0.0 5 39 0.0124  0 .02 66 0.09 76 0.02 46  0 .0379 0 . 13A5 0 .0303  0.0 38 2 0.1892 0.0A69  -0.0292 ' 0. 1 6 7 2 0.0399  - 0 . 0 0 61 0.2 5 66 0.06 74  -0 .0004  0. 0 0 0 8  0.0040  0.0134'  0.0170  0.0191  -0.010 2  -0 . 0 0 0 4 .  0. 0 0 0 1  0.00 2 3  0.0078  0 .0097  0.0105  -0.0053  0.00 25  0 .0765  0. 1 1 5 7  .0. 0 4 5 6  0.0531  0.02 3 6  0.0AA7  0.049 0  0.0615  . 0 6 6A -0 . 0 6 1 3 o. 5 8 6 2 0 . 3- 8 6 5 0. 1 7 80 . 107 8 0  - 0 . 152 5 .1.59^3 0.6060  (c)  -0.0575 2.5547 1.0261  EAr) = 0.25 + 0.75 / r / r ,  4  .  0.00 27 ;  Shells^v  Par  0.25  Perp  Par  0.50  Perp  Par  0.75  1 .00  Perp  Par  9.0 06 8 14.4629 2 2.0200 •  14.5874 1.9011 1 . 2344  - 0 . 6 4 92 9.9 8 39 1-7.1938  2.3435 4.7959 3.6419  5.7049 -0.319 1 1.0710  -0.6992 0 . 1041 0.0916  0.5154 0.8673 -0.4554  1 .9032 - 0.1740 -0.0682  3.8 3 99 - 0 . 6 7 79 -0.5021  -0.4326 - 0 . 1 4 51 -0.1461  -0.9571 - 0 . 1 6 89 0.2 8 48  Perp  1 HOMO 1 LIN i 1 LIN i i  -0 .fc2 5 6 0. 190 9 0.6 5°0  - 0 . 3362 -0.3107 0.4 5 92  17. 90 2 5 ' 6.6 7 8 3 5 .4 7 07  167 . 3 0 3 4 12 6 . 5 2 37 2 .4 5 0 3  4 .4252 '5 . 1 3 4 2 1 .802 2  2 HOMO 2 LI M i 2 LIM i i  -0.3013 0 .•3 74 7 0 . 1625  -0.4034 0.2135 0.0 9 09  0. 0 2 7 8 o. 17 35 -o. 5 8 90  -0 . 06 8 8 10 .2195 8 . 30 01  -o . 3 6 1 0  3 HOMO 3 LIM i 3 LIN i i  - 0 . 1224 0.1893 0.0580  -0.1810 0. 1 0 8 7 0 . 0306  - 0 . 2757 - 0 . 4756 -0 . 39 89  1 .8482 8 . 5319 3 . 269 4  4 HOMO •4 1. IN i 4 LIN i i  - 0 . 0 64 5 0. 106-3 0. 0 2 8 5  -0.101r  - 0 . 2198 - 0 . 414 0 -0. 2 1 5 8  1 . 2004 4. 8373 I .3184  .376 5 " -o o. 1 7 0 1 0 .208 3  -0.2044 - 0 . 1410 ' -0.44 3 0  5 HOMO 5 L IN i 5 LIN i i  -0.0394 0.0671 0. 01 66  -0.0642 0.0379 0.00 8 5  -o. 3 1 3 4 -o. 1 2 6 1  • 0 .7918 .2 . 8 3 7 1 0 .67 57  -0 . 222 3 0 .19 3 8 0.1260  -0.2009 -0.3007 -0.2949  -0.4351 -0.1728 -0.0773  -0.7518 0. 1 3 7 6 0.32 11  10 HOMO 10 L I N i 10 L I M i i  - 0 . 0 09 1 0. 01 50 0 .0080  -0.0L58 0.0 08 3 0.0016  -0. 0 5 4 3 - 0 . 0895 0232  0.2067 0 . 5201 0 .1115  -0 .0454 0 .0689 o.0197  -0.0754 - 0 . 1506 -0.0469  -0.1253 -0.0501 -0.0128  -0.1749 0.1640 0.06 33  15 HOMO  - 0 . 0 04 2  --C.0 071  02 60  0 . 09 35  -0 . 0 1 9 1  '-0.035 8  -0.0572  - C O 7 42  20.HOMO  -0 . 0 0 2 6  -0 . 0 0 4 1  -0. 0149  0 .05 26  -0 . 0 1 0 9  -0.0196  - 0 . 0 32 4  - 0 . 0 4 11  0.7 08 3  0.3 3 70  0. 2 1 6 6  0 . 00 52  0 .4453  0.0 590  0. 500 1  k Cf(0) o  0. 06 ca 0.0147  - 0 . 16 5 6  (d)  -0 . 6 7 3 0 -0 . 1 3 4 7 -0 . 6 5 5 0 -0 . 0 6 4 9 0 .2595  £„(r) = 4 - 3 J7JT~,  0. 2080  i  S h e l l s >v  Par  0.25  0.50  Par  Perp  Perp  Par  1.00  0.75 Perp  Par  Perp  1 HOMO 1 LINi 1 LIMii  - 0 . 9 46 0 -0 . 3 64 6 -0.1073  -0.934 2 -0.7018 -0.2016  27.9120 7.5041 10.6194  1 1.4761 -0 .3628 0.2 3 52  -0.8140 3.6263 10.3016  - 0 . 3 299 - 0 . 6 2.3 0 2.2 52 2  2 9.5149 17.7065 29.9043  10.0186 1.1398 2 . 4 0 93  2 HOMG 2 LINi 2 LIM i i  -0.4083 -0.0579 -0.0030  -0.5173 -0.102 7 - 0 . 0 177  0.9170 0.9551 0.2290  -0.5043 -0.6196 - 0 . 17.14  ' 0.3146 -0.1321 -0.394 1  0.3136 -0.6865 -0.9230  74.4158 4.2804 3.0973  3 0 . 0 3 68 -0.3491 - 0 . 4 6 96  3 HUMG 3 LINi 3 LIMii  -0.1929 • -0.0196 -0.0 03 4  - 0 . 2 502 -0.036 1 ' -0.0072  -0.2438 0.0866 0 . 0 416  -0.9540 -0.0474 0.0362  -0.8014 -0.7353 -0.30 5 6  .-0.8732 -0.6641 -0.2192  1.3816 0.9575 - 0 . 1 542  0. 7 5 75 - 0 . 8 7 83 - 0 . 3 9 50  4 HQMC 4 LINi 4 LIN i i  - 0 . 1105 - 0 . 0 09 0 - 0 . 0 02 0  -0.1445 -0.0184 - 0 . 004 0  -0.2966 0.0316 0.0173  -0.6891 0 . 0 3 76 ' 0.02 91  - 0 .9403 -0.4263 - 0 .096 6  -0.9406 -0.2814 - 0 . 0443  -0.0352 -0.6218 - 0 . 3407  -0.6525 -0.7277 . -0.2446  5 HOMG 5 LINi 5 LIMii  - 0 . 0 713 -0.0059 - 0 . 0013  - 0 . 0 936 -0.0113 - 0 . 0 02 6  - 0 . 2 376 0 . 01 9 9 0.0093  - 0 . 4 8 50 0.0429 0.0144  - 0 . 7507 -0.2416 - 0 .0459  -0.7344 -0.1306 -0.0142  -0.0055 -0.55 8 1 -0.1375  - 0 . 9 9 62 - 0 . 3 5 33 - 0 . 0 4 87  10 HOMD 10 L I N i 10 LIN i i  - 0 . 0 180 - 0 . 0014 -0.CG03  -0.02 3 7 -0.0026 -0.0006  -0.0752 0.00 5 7 0.0017  - 0 . 1354 0.0165 0.0047  - 0 . 2 3 30 - 0 .0433 - 0 .0093  -0.2244 -0.01.39 -0.0018  - 0 . 4 89 3 -0.096 8 -0.0164  - 0 . 4 492 - 0 . 0 103 0.0046  15 HOMO •  - 0 . 0080  -0.0106  -0.0347  -0.0613  -0.1070  -0.1028  -0.2397  - 0 . 2 147  20 HOMO  - 0 . 0 04 5  - 0 . 0 06 0  -0.0198  -0.0347  - 0 .0607  -0.0584  - 0 . 1390  - C . 12 37  0.8537  0.2.51R  0.034 0  0.02 79  0 .0755  0.0676  0.0234  0 . 0 3 04  (e)  Core - £ = 2, r r  2  = 0.25 A„? S h e l l - £ ( r ) = a/r + b r  ON. .  ;  i  57  Table 3 . 2 O v e r a l l A c c u r a c y of the S h e l l Methods a) Cases C o n s i d e r e d Except 6 (r) = a J T b) £,.(r) = a / r + b , r > 0 r  Lin i i  Lin 1  Homo Shells  + b, r > 0  0.25  0.75  0.25  0.75  0.25  0.75  1  v-V  v-v  v-V  v-V  v-v  v-V  2  v-V  v-V  p-V  v-V  f-P  v-V  3  P-V  v-V  f-P  v-V  g-F  f-V  A  f-P  v-V  g-F  p-V  e-G  g-P  5  g-F  v-V  g-G  f-P  e-G  g-F  e-G  f-P  e-G  g-G  e-E  e-G  e-E  f-F  e-E  g-G  10  20  (a)  Shells  Lin i i  Lin i  Homo t//)c  0.25  0.75  0.25  0.75  0.25  0.75  1  v-V  v-V  v-V  v-V  v-V  v-V  2  f-V  v-V  v-V  v-V  v-V  v-V  3  g-P  p-V  p-P  v-V  f-F  v-V  g-P  p-V  p-P  v-V  g-G  ' v-V  5  e-F  p-V  f-F  p-P  .g-G  p-P  10  e-G  g-F  g-G  p-P  e-E  g-G  15  e-E  g-G  20  e-E  e-G (b)  58  The l o w e r case l e t t e r s a p p l y f o r for  £ >l. r  £ < l and t h e upper case r  I n some c a s e s , t h e e r r o r s observed were l a r g e r  t h a n i n d i c a t e d i n Table 3 . 2 . u n d e r l i n e d i n T a b l e 3*1  These anomalous cases a r e  and i t i s seen t h a t t h e y g e n e r a l l y  o c c u r n e a r what appears t o be minima o f t h e b a c k s c a t t e r i n g cross-sections.  A l t h o u g h no d i f f e r e n t i a t i o n i s made between  t h e r e s u l t s f o r the two p o l a r i z a t i o n s , i t appears t h a t t h e a c c u r a c y a c h i e v e d f o r a g i v e n number o f s h e l l s i s s l i g h t l y b e t t e r f o r p a r a l l e l than f o r p e r p e n d i c u l a r . The a c c u r a c y i s i n g e n e r a l r e l a t e d t o the maximum e l e c t r i c a l t h i c k n e s s o f the s h e l l s as shown i n T a b l e  Table 3.3  3»3'  R e l a t i o n s h i p Between A c c u r a c y and S h e l l T h i c k n e s s  Maximum S h e l l Thickness  0.05  Homogeneous  A  0.25 0.1  Errors  A  10  A  > 25  %  -  %  25  1-5%  0.025 A  Linear 10  -  25" %  1-5% < 1 %  < 1 %  O v e r a l l Comparison An o v e r a l l e v a l u a t i o n of the f o u r methods o f c o m p u t a t i o n i s p r e s e n t e d i n Table  3.4.  Table 3.4  Comparison of the Methods of Computation  Power Series  Application to Arbitrary Permittivi t y Variations  Limitations  Impractical since complexity of series i s so dependent on variation Size and permitt i v i t y variation limited due to series summation errors  Computation Time  Homogeneous Shells  Straightforward  c>0 f o r perp. pol.  Straightforward  £ ^ 0 when .£,.-0^ for perp. pol.  ( i )  10 shells, r,=0.5 >io Both pols. »0.6 sees.  Inherent  for^ perp. pol. EJO  Also power series summation l i m i tation 20 shells, r, = 0.5 )\o  r,=0.5 ^  Par. pol. &7  sec8.  u i ;  Perp. p o l . ^ * * ^  * on an I.B.M. 7044 computer  Numerical Integration  Linear Shells  0  Both pols. **4 to 8 sees, for cylinders considered  60  (i)  A power s e r i e s s o l u t i o n about the  of r a t which £>.(r) = 0 may  be used i n c o n j u n c t i o n w i t h each  o f t h e s e methods, t h e r e b y e l i m i n a t i n g (ii) out u s i n g 16  The  summation of the  significant figures.  mittivity variations  s e r i e s was  For  carried  cylinders  with pergive  corresponding time i s decreased  by  half. ( i l l ) The  i c a l integration are  this limitation.  such t h a t 8 s i g n i f i c a n t f i g u r e s  a c c u r a t e summation, the a l m o s t one  value  s o l u t i o n was  instead  a c c o m p l i s h e d u s i n g numer-  o f the power s e r i e s .  The  series  c o n s i d e r a b l y more cumbersome i n t h i s case t h a n f o r p a r a l l e l  p o l a r i z a t i o n and  hence the  (iv)  The  time would be s i g n i f i c a n t l y l o n g e r .  time r e q u i r e d ' f o r the n u m e r i c a l I n t e g r a -  t i o n i s v e r y dependent upon the time r e q u i r e d t o the f u n c t i o n  describing  the i n t e g r a t i o n  the p e r m i t t i v i t y v a r i a t i o n and  step s i z e .  The  upon  l a t t e r seems t o depend more  on the range of p e r m i t t i v i t y w i t h i n actual  evaluate  the  c y l i n d e r . t h a n on  the  variation.  Summary" The  homogeneous s h e l l method appears t o be  a t t r a c t i v e f o r c a l c u l a t i n g the s c a t t e r e d  field  w i t h an a r b i t r a r y v a r i a t i o n of p e r m i t t i v i t y . l i n e a r s h e l l method g e n e r a l l y fewer s h e l l s , the roundoff errors pedance and  g i v e s the  of a  quite cylinder  Although  the  same a c c u r a c y f o r  c o m p u t a t i o n time i s c o n s i d e r a b l y l o n g e r ;  i n a p p l y i n g the i t e r a t i o n e q u a t i o n s f o r  a d m i t t a n c e produced a r e l a t i v e e r r o r of  less  im-  61  than 10  100  i n the 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 f o r up to  s h e l l s and hence the r e d u c t i o n i n s h e l l numbers seems unimportant.  The n u m e r i c a l  I n t e g r a t i o n method may  d e s i r a b l e i f v e r y accurate cases c o n s i d e r e d ,  r e s u l t s are required.'' Prom the  i t appears t h a t the homogeneous s h e l l method  w i l l be f a s t e r f o r a c c u r a c i e s of up It  should be p o i n t e d  v a r y i n g s h e l l s may  sometimes be  to  0.1$.  out t h a t the use  of l i n e a r l y -  be more advantageous than homogeneous s h e l l s  when c o n s i d e r i n g the i n v e r s e problem ( d e t e r m i n a t i o n  of  the  p e r m i t t i v i t y v a r i a t i o n from s c a t t e r e d f i e l d measurements) because of the s m a l l e r number of s h e l l s r e q u i r e d to achieve given 3»2  a  accuracy. Number of Terms Required i n the S c a t t e r e d - F i e l d Expansion When c a l c u l a t i n g the s c a t t e r e d f i e l d ,  the number  of terms, N, used i n the expansion must be l a r g e enough to make the e f f e c t of the h i g h e r  order terms i n s i g n i f i c a n t .  i s d e s i r a b l e , of course,  to use  the case of a conducting Ul)  c y l i n d e r , a v a l u e of N ^ 2 k r ,  been suggested for  as few  It  terms as p o s s i b l e .  has'  0  ; i t appears t h a t no v a l u e has been  In  given  a dielectric cylinder. A procedure f o r choosing  the v a l u e  i s g i v e n l n Appendix E, S e c t i o n s E . 2 - 1 scattered f i e l d  (Section E . 2 - 1 ) ,  of N r e q u i r e d  and E . 3 - 1 .  For  the  N i s determined such t h a t  e r r o r a t a s p e c i f i e d r a d i u s i s below a given v a l u e .  For  the  the  s c a t t e r i n g c r o s s - s e c t i o n , i t i s not p o s s i b l e t o s a t i s f y e i t h e r  62  an a b s o l u t e e r r o r c o n d i t i o n o r a r e l a t i v e e r r o r c o n d i t i o n f o r a l l values of c r o s s - s e c t i o n . F o r c y l i n d e r s w i t h a low p e r m i t t i v i t y and" conducti n g c y l i n d e r s , "unexpectedly  large" high-order  coefficients  do n o t o c c u r and t h e a c c u r a c y o f t h e r e s u l t s may be e s t i m a t e d (1  from t h e magnitude o f t h e h i g h e s t o r d e r term  .  This i s not  so when h i g h p e r m i t t i v i t i e s a r e c o n s i d e r e d ( s a y £ ^ > 1 0 ) , and i t i s n e c e s s a r y t o f o l l o w t h e procedure dix*. E«  o u t l i n e d i n Appen-  I t s h o u l d be p o i n t e d o u t t h a t i t i s always, d e s i r a b l e  t o know t h e r e q u i r e d o r d e r b e f o r e h a n d ,  p a r t i c u l a r l y when u s i n g  the homogeneous-shell method o f S e c t i o n 3•1-2.  The r e a s o n  f o r t h i s i s ; t h a t s e r i o u s e r r o r s o c c u r i f J«(z) i s c a l c u l a t e d u s i n g an upwards r e c u r r e n c e when n > z and hence i t i s advantageous t o c a l c u l a t e a l l t h e J-„ f u n c t i o n s a t t h e same t i m e .  63  .  4  CALCULATION OP THE COMPLETE SCATTERED FIELD FROM FIELD MEASUREMENTS  I n t h i s c h a p t e r , we g i v e a method hy w h i c h t h e scattered-field field  c o e f f i c i e n t s and hence t h e complete  scattered  can be d e t e r m i n e d from measurements o f t h e s c a t t e r e d ( o r  diffracted) f i e l d  a t one r a d i u s .  known, t h e s c a t t e r e d calculated  field for  Since the c o e f f i c i e n t s are o.-.t plane-wave i n c i d e n c e c a n be  from measurements t a k e n w i t h c y l i n d r i c a l - w a v e  dence and v i c e v e r s a .  inci-  T h i s c o u l d r e s u l t i n a s i m p l e r experimen-  t a l method f o r t h e i n v e s t i g a t i o n  o f plane-wave  scattering.  As i n C h a p t e r 3, two s t e p s a r e i n v o l v e d i n t h e p r o cess, 1)  D e t e r m i n a t i o n o f t h e c o e f f i c i e n t s and  i i ) Determination of the order f o r which the best representation of the f i e l d i s obtained. The  derivation  applies  i s c a r r i e d o u t f o r p a r a l l e l p o l a r i z a t i o n and  d i r e c t l y t o perpendicular p o l a r i z a t i o n i f E is. replaced  by H. 4.1  Method o f C a l c u l a t i n g The  scattered  the Coefficients  f i e l d of a cylinder f o r either  or c y l i n d r i c a l wave i n c i d e n c e i s o f t h e form CO  E (r,6-) = I s  e , A, H ^ ( k r ) cos(ne-) 0  plane  64  T r u n c a t i n g the s e r i e s i n t h i s e q u a t i o n a t n = N we have V (r,&) S  »  E^(r,e-) = Zl e,,  A  A t a f i x e d r a d i u s r = r , we may  2  + *£_  a.  0  cos(nG-) ) (4.2) a  +  **  -j ( —*.  21  +  2  where  (4.1)  write  0  E„(0-) = ( — -  cos(ne-)  H^(k r)  fl  A„ =  —  a^r  »='  cos(nG-) )  511  0 < n £ N  (4.2)  The r e a l and i m a g i n a r y p a r t s o f e q u a t i o n F o u r i e r c o s i n e s e r i e s i n t h e v a r i a b l e 8-.  are t r u n c a t e d  Thus,- i f we  determine  (from measurements) t h e r e a l and i m a g i n a r y p a r t s of t h e s c a t tered f i e l d  at s e v e r a l angles, estimates of a  and a ^  may  be  4-3)  (4-2  o b t a i n e d by a p p l y i n g a l e a s t - s q u a r e s f i t  f n  '  t o the  I f we have 2L d a t a p o i n t s , t h e a p p r o x i m a t i n g  data. func-  t i o n s a r e chosen from t h e s e t 1,  and t h e r e f o r e we  cos(e-),  cos(Le-)  cos(2©-),  are l i m i t e d to N 6 L .  F o r s i m p l i c i t y , we  assume t h a t t h e d a t a i s t a k e n a t t h e e q u a l l y spaced v a l u e s &;  m  = —  i = 0.  1.  2.  2L-1  will  65 D e f i n i t i o n of q u a n t i t i e s Measured v a l u e s w i l l be d e s i g n a t e d  by t h e s i g n  (~)  Thus, f o r t h e measured d i f f r a c t e d f i e l d we w r i t e E°(Gi) = M where  M  Di  e '^ J  d :  ;  =t-  oL  + J  t  (^.3)  o i  = magnitude o f t h e d i f f r a c t e d f i e l d a t ©- = ©-<• = phase o f t h e d i f f r a c t e d f i e l d a t & = 0- t  = g Either M  o i  oi  cos(#>;)  = M : sin(<p„<; ) P  and tf  Di  or f  and g . may be measured d i r e c t l y .  o i  p  S i m i l a r d e f i n i t i o n s h o l d f o r t h e i n c i d e n t and s c a t t e r e d where I and S a r e used i n p l a c e o f D.  fields  7  Note t h a t i f t h e i n c i d e n t  f i e l d i s a c l o s e enough a p p r o x i m a t i o n t o t h e t h e o r e t i c a l i n c i d e n t field,  t h i s may be used i n s t e a d o f t h e measured v a l u e s .  Also,  i n g e n e r a l , t h e s c a t t e r e d f i e l d i s n o t measured d i r e c t l y b u t i s calculated  using  *—*  ~  .—  ^  4.1-1  A p p l y i n g E q u a l Weight t o a l l P o i n t s Because o f t h e o r t h o g o n a l  cos(ne-)  property  o f the functions:  (and sin(ne-) a l s o ) , t h e c o e f f i c i e n t s i n a F o u r i e r l e a s t -  s q u a r e s f i t may be e a s i l y d e t e r m i n e d when a l l t h e d a t a p o i n t s are weighted e q u a l l y * » ^ . 38  4  A l t h o u g h t h e expected v a l u e o f t h e  d e t e r m i n e d c o e f f i c i e n t s i s always c o r r e c t , a c h o i c e o f e q u a l weight gives the best estimates  i n t h e minimum-variance sense  66 o n l y when t h e v a r i a n c e &i  o f t h e measurement a t t h e l ^  i s t h e same f o r a l l i , s a y cr/ c\ =  assumption i f l  , g.,Z  o C  c  and g  zL  Q  x  point,  This i s a reasonable  1  t  1  e  xi  measured d i r e c t l y and  angular p o s i t i o n errors are negllgable.. We w r i t e t h e e s t i m a t i n g f u n c t i o n o f t h e s c a t t e r e d field  a t r = r as 0  EJ(G-) = ( —  a , (1 ~TK  +Z  fT  ) cos(nG-) )  L  (4.5) (1 -  + 3 C — '+2 a2 • »»i " J  The  —  coefficients a  Frt  L  i  =  0  _ a,„ = - Z L  L =  of the  °  )  (38 4-4-)  —  12L-t  estimate  J cos(ne-)  and a ^ a r e g i v e n by  a. = ~Z  The  ^  '  _ i n rr fsi •••?6'st^d£L-)I L  g  (4.6a)  n=0,l,2,...,N  i n rr cos( )  5 i  (4.6b)  L  scattered f i e l d c o e f f i c i e n t i s  f» + 3 a,  a Ay  >  - - ,(2) 2(1 . +•. 6r, ) H ^ ( k r ) L  By s u b s t i t u t i n g A^ f o r  e  0  i n e q u a t i o n (4.1) we may o b t a i n t h e  s c a t t e r e d f i e l d f o r a l l r > r , and G-. A l s o , t h e r e l a t i o n (2.37) may be a p p l i e d t o d e t e r m i n e e s t i m a t e s  of  o r A^e, depending  on whether p l a n e o r c y l i n d r i c a l wave i n c i d e n c e was used i n the; measurements.  67  We n o t e t h a t , because o f t h e symmetry o f t h e f i e l d , i t i s o n l y n e c e s s a r y t o t a k e measurements i n t h e range 0 f ©- £ rr i n order t o determine the c o e f f i c i e n t s .  However, by r e d u c i n g t h e  number o f p o i n t s over which t h e l e a s t - s q u a r e s f i t i s a p p l i e d , we d e c r e a s e t h e a c c u r a c y o f t h e r e s u l t s . 4.1-2  ' .  A p p l y i n g a D i f f e r e n t Weight t o Each P o i n t ' Yfrien t h e v a r i a n c e o f t h e measurements i s n o t t h e  same a t a l l p o i n t s ( a s w i l l c e r t a i n l y be t h e case i n t h e p r e s ent a p p l i c a t i o n when e r r o r s o c c u r i n t h e v a l u e s o f &i o r i f t h e f i e l d i s measured i n terms o f magnitude and p h a s e ) , t h e bes.t e s t i m a t e o f t h e c o e f f i c i e n t s i s o b t a i n e d when a w e i g h t w; o c l / c r / i s applied to the residue at the i ^ *  1  point^^'^ ^.  The c o e f -  5  f i c i e n t s &f. and a ^ i n e q u a t i o n ( 4 . 5 ) a r e determined b y s o l v i n g n  the N simultaneous  equations 1  0 A = Y  Zw^cosCG-)  where '  C  0  C  Y. W i c o s ( e j )  < N £ L  . . . ]T  '  L  Z w i C o s ( G ^ ... 2  Y  IL w^cos(N©- ) £  a,/2 a  Zw< f  (  A =  Y_  L  (4.7)  U + 4,,) 11  W  '  _Jii  cosOJ  Y =•  I^y.  cos(NG-)  cos(NG^)  r \  c  o  s  (%)  eos(Hefr)  cos (Ne;-) 2  68  0 < i < 2L-1  W;  = ¥ .  Or  W^  a*  - a  or  a „  0 s n ± N  j  = fi  or  g^  0 i  c  f  f T 1  s  r  i f 2L-1  An e s t i m a t e o f t h e v a r i a n c e o f f r and g S(  s i  a t each  p o i n t may he d e r i v e d as f o l l o w s : i)  Errors i n 87  We assume t h a t t h e e r r o r s i n 6 7 a r e u n c o r r e l a t e d v a r i a n c e c£ and t h a t f L = £ £  or t  e  measured d i r e c t l y .  and g. = g_, o r g  xi  z i  with  are  We have  var(f£) =  2>£  be-  e- = e-  z  *g var(g ) - I £  S i n c e t h e t r u e v a l u e s o f G-^,  hf/d&  and  use t h e measured v a l u e s as e s t i m a t e s . c u l a t e d from e q u a t i o n  a r e unknown, we  }g/b&  I f f ^ and Zsi a r e c a l 5  ( 4 . 4 ) , we have 2  -I  var(f ; ) s  ( 4 . 8 a )  e- = e-/ var(g )  bs  0  bi  .2.  -1  x  <9-  5 i  be-  bo-  e-=e-  (4.8b)  69  i l ) Measurements o f M and <P Assume t h a t t h e e r r o r s i n t h e measured v a l u e s o f magnitude and phase a r e u n c o r r e l a t e d w i t h v a r i a n c e s <3C* and o^,  respectively.  We have  var(f;)  V  2>q  dVL  ^ cos ((Pi)<?»  + Iii s i n ((Pc) &<?  (4.9a)  2  and s i m i l a r l y (4.9b)  v a r ( g - ) « s i n ( & ) cf* + M- c o s ( ) < / * 2  2  where a g a i n t h e a p p r o x i m a t i o n r e s u l t s because measured v a l u e s : M/jand  a r e used I n s t e a d o f t h e a c t u a l v a l u e s o f magnitude and  phase (which a r e unknown).  I f f - and g ^ a r e c a l c u l a t e d by st  ( 4 . 4 ) , we have  v a r ( f . ) = var(f - ) + var(f - )  (4.10a)  vartg^) =var(g  (4.10b)  s  Dt  I6  o i  ) + varfg^ )  I f . e r r o r s i n t h e v a l u e s o f 0^ o c c u r i n t h i s  case,  e s t i m a t e s o f t h e r e s u l t i n g v a r i a n c e s o f t h e magnitude and phase must f i r s t be c a l c u l a t e d as shown i n ( i ) . iance a t the i " * b  1  The a p p r o p r i a t e v a r -  p o i n t i s t h e n used f o r cf^ and  i n equa-.  t l o n s ( 4 . 9 a ) and ( 4 . 9 b ) . 4.2  D e t e r m i n a t i o n o f t h e R e q u i r e d Order I n Chapter  3 , a v a l u e o f N was determined  the e r r o r caused by t r u n c a t i n g t h e s e r i e s i n e q u a t i o n  such t h a t (2.1b)  70  at n - N was below some s p e c i f i e d v a l u e .  By i n c r e a s i n g N, t h e  e r r o r c o u l d be made as s m a l l as desired.; e r r o r s i n t h e c o e f f i c i e n t s were assumed n e g l i g i b l e .  When d e a l i n g w i t h  experimental  d a t a , however, t h e e r r o r s i n t h e c o e f f i c i e n t s , and hence i n t h e s c a t t e r e d f i e l d , w h i c h a r e a s s o c i a t e d w i t h measurement e r r o r s cannot be n e g l e c t e d . C o n s i d e r , f o r example, t h e case o f e q u a l •sse r r o r - v a r i a n c e a t a l l p o i n t s . The v a r i a n c e o f t h e f i t t e d s c a t t e r e d - f i e l d value given b y * ^ ^ var(  E*(ei) ) = (N  + 1)  tY /ZL z  i n c r e a s e s w i t h N and as a r e s u l t , I f t h e c o n t r i b u t i o n t o t h e cf /2L,  adding  term w i l l a c t u a l l y g i v e l a r g e r e r r o r s i n t h e f i t t e d  value.  f i e l d due t o t h e (N -f l )  In equation  t  h  term i s l e s s than  z  this;  o r d e r t o d e t e r m i n e t h e optimum v a l u e o f N- i n  (4.5), we may assume t h a t t h e e r r o r s i n j f ^ and  a r e n o r m a l l y d i s t r i b u t e d and u s e t h e v a r i a n c e - r a t i o  zi s  t e s t of  (43 45)  the n u l l hypothesis i n normal r e g r e s s i o n theory  '  . I n  the p r e s e n t a p p l i c a t i o n we a r e t e s t i n g t h e h y p o t h e s i s H, : a ^ = 0  N-:< n ^ L ,  -oo<a  Y 7  < o 0  n < N  against the a l t e r n a t i v e H  *  2  : a •»,• jfc 0  N < n < L,  - co < a „ < «*>  n .£ N  The t h e o r y i n t h i s s e c t i o n a p p l i e s t o t h e minimum-variance e s t i m a t o r s and hence t h e w e i g h t i n g used i s g i v e n by w; oc \/o~c  71  where n = L l s t h e maximum number o f c o e f f i c i e n t s w h i c h may be determined  from 2L d a t a p o i n t s . *  I f t h e h y p o t h e s i s H, i s t r u e ,  t h e n we have •  j>  Nm-N, 2L-N „-1 y  ,nz  where F  ( S t - S ; , ) / ( K , , - H ) S /  ^/(2L -  1)  = F - v a r i a t e w i t h n , and n , degrees o f f r e e d o m ^ »' 3  n» For equal e r r o r - v a r i a n c e  S*  ZL-,  N  = z I y ? " L l t = £>  a*  (4.12a)  O  and f o r u n e q u a l e r r o r - v a r i a n c e *  2L-i  S* = Z  2 L _ I & 7 1  W; y j - Z  Z  w< JL c o s ( n ^ )  (4.12b)  The e x p r e s s i o n s f o r S*^ a r e o b t a i n e d by r e p l a c i n g N by U e q u a t i o n s (4.12a) and (4.12b).  w  in  Now l e t P  "„ ."denote" t h e »n such t h a t t h e p r o b a b i l i t y o f e x c e e d i n g t h i s : £  v a l u e o f P„ v a l u e i s oi (F  may be o b t a i n e d from t a b l e s ni,n  ' or readily  2  computed a p p r o x i m a t e l y  (4-6)  ). I f the value of P c a l c u l a t e d u s i n g  e q u a t i o n (J/-.11) i s l e s s : t h a n P^.JJ _JJ 2L-N - 1 '  w  e  ma  «  5r  a  s  s  u  m  e  w i t h a c o n f i d e n c e ( 1 - cc) t h a t t h e c o e f f i c i e n t s f o r n > N a r e zero.  A d i s c u s s i o n o f t h e power o f t h i s t e s t i s g i v e n i n Ref-  e r e n c e 4-,3-.  I n p r a c t i c e , a maximum v a l u e < L i s u s u a l l y assumed. F o r example, I f t h e r a d i u s o f t h e c y l i n d e r i s known, t h e e r r o r l i m i t s d i s c u s s e d i n Appendix E may be used t o e s t i m a t e a v a l u f o r N , t h e c o e f f i c i e n t s f o r n > J\ b e i n g n e g l i g i b l e T  TO  W  72  A l t h o u g h t h e method o f d e t e r m i n i n g N i s t h e same f o r b o t h e q u a l and u n e q u a l e r r o r - v a r i a n c e a t a l l p o i n t s , t h e a p p l i c a t i o n i n t h e former  case i s c o n s i d e r a b l y e a s i e r because  i ) t h e v a l u e s o f t h e c a l c u l a t e d c o e f f i c i e n t s a r e independent o f N and hence i t i s n o t n e c e s s a r y t o r e c a l c u l a t e t h e complete s e t o f c o e f f i c i e n t s f o r each v a l u e o f N t r i e d and z  2  pressions f o r S  N  and S  Wrv)  are r e l a t i v e l y simple.  We n o t e t h a t i f t h e h y p o t h e s i s a c o r r e c t , t h e v a l u e o f s* = S*/(2L - n - 1 ) o f n f o r n > N.  i i ) t h e ex-  w  = 0 for n > N i s  s h o u l d be i n d e p e n d e n t  As a r e s u l t , an e s t i m a t e o f N ma.y be o b t a i n e d  by o b s e r v i n g t h e b e h a v i o u r o f s* as n i s i n c r e a s e d  ; the  v a l u e o f s* s h o u l d d e c r e a s e r a p i d l y and then become c o n s t a n t f o r n > N.  T h i s c r i t e r i o n , a l t h o u g h n o t an e x a c t one, i s u s e f u l  because o f i t s s i m p l i c i t y o f a p p l i c a t i o n and has proved i n t h e experiments  carried out.  adequate  73  5.  EXPERIMENTAL INVESTIGATION  Previous experimental i n v e s t i g a t i o n s  of r a d i a l l y  inhomogeneous c y l i n d e r s w i t h a smoothly v a r y i n g p e r m i t t i v i t y (ee have g e n e r a l l y been c o n f i n e d t o t h e s t u d y o f p l a s m a s \  &  47)  *  . I n  t h e s e c a s e s , t h e p e r m i t t i v i t y p r o f i l e I s seldom known a p r i o r i t o an adequate degree o f a c c u r a c y t o a l l o w d i r e c t with theory.  comparison  I n t h i s c h a p t e r , we c o n s i d e r s c a t t e r i n g  by  c y l i n d e r s made o f a t y p e o f a r t i f i c i a l d i e l e c t r i c whose r a d i a l p e r m i t t i v i t y v a r i a t i o n can be c o n t r o l l e d  and i s a c c u r a t e l y  known. The measurements a r e c a r r i e d out i n the r e g i o n between two p a r a l l e l c o n d u c t i n g p l a t e s s e p a r a t e d by l e s s h a l f a free-space wavelength, c y l i n d r i c a l wave i n c i d e n c e .  under c o n d i t i o n s o f p l a n e  than and  T h i s t y p e of system r e s t r i c t s  t h e measurements t o cases where t h e e l e c t r i c f i e l d i s p a r a l l e l t o the a x i s o f t h e c y l i n d e r , p e r p e n d i c u l a r t o the :  plates.  Measurements a r e t a k e n a t a c o n s t a n t r a d i u s n e a r t h e  cylinder  and from t h e s e t h e s c a t t e r e d - f i e l d c o e f f i c i e n t s a r e  calculated  u s i n g the method o f C h a p t e r 4.  I n t h e case o f c y l i n d r i c a l -  wave i n c i d e n c e , the c o e f f i c i e n t s f o r t h e plane-wave case are c a l c u l a t e d u s i n g the r e l a t i o n s g i v e n i n S e c t i o n  2.4.  74 5*1  C o n s t r u c t i o n of the Inhomogeneous C y l i n d e r s I n o r d e r t o a p p r o x i m a t e a smooth r a d i a l v a r i a t i o n  of p e r m i t t i v i t y , we may  construct a cylinder  eous s h e l l s of a p p r o p r i a t e p e r m i t t i v i t i e s .  from t h i n homogen« For  cylinders  o f l a r g e d i a m e t e r compared t o w a v e l e n g t h , a l a r g e number o f s h e l l s and  hence a wide range of m a t e r i a l s would be  needed  i n o r d e r t o a c h i e v e a r e a s o n a b l y good a p p r o x i m a t i o n .  This  may  variety  prove impractical".  A n o t h e r method r e q u i r i n g  l n m a t e r i a l s i s t o c o n s t r u c t the tapered layers  c y l i n d e r from  less  alternating  of tvro d i e l e c t r i c s as shown l n F i g u r e 5»la. cylinder axis  cylinder axis •  (b) F i g u r e 5*1  Method of C o n s t r u c t i n g R a d i a l l y Inhomogeneous C y l i n d e r s a) U s i n g two  Dielectrics  when the V a r i a t i o n  b) U s i n g t h r e e D i e l e c t r i c s  of P e r m i t t i v i t y i s  large.  75  I f t h e maximum t h i c k n e s s of each l a y e r i s much s m a l l e r t h a n t h e w a v e l e n g t h i n the m a t e r i a l , the r e s u l t i s a t y p e of a r t i f i c i a l d i e l e c t r i c s i m i l a r to that considered  by H o r i t a  and C o h n ^ ^ i n m a t c h i n g d i e l e c t r i c l e n s e s , and by C o l l i n ^ 8  5  f o r c o n s t r u c t i n g an a n i s o t r o p i c medium. mittivity,  The  effective  per-  £ (r), a t any r a d i u s depends on the r e l a t i v e e  thick-  ness:, of t h e two d i e l e c t r i c s a t t h a t r a d i u s , t h e r e l a t i o n d e r i v e d i n Appendix G.  By u s i n g t h e a p p r o p r i a t e  of the two d i e l e c t r i c s , one  should  o b t a i n any  being  profile  effective  per-  m i t t i v i t y v a r i a t i o n as a f u n c t i o n of r a d i u s . Thus, l t would appear t h a t we c o u l d c o n s t r u c t c y l i n d e r w i t h a d e s i r e d p e r m i t t i v i t y v a r i a t i o n i f we available  a  had  two m a t e r i a l s , one w i t h a p e r m i t t i v i t y above, t h e  o t h e r w i t h a p e r m i t t i v i t y below the range encountered i n t h e cylinder.  T h i s c o u l d be r a t h e r o p t i m i s t i c , however, f o r  two  reasons: i)  I f the two p e r m i t t i v i t i e s used a r e  vastly  d i f f e r e n t , the f i e l d d i s t r i b u t i o n w i l l be v e r y n o n - u n i f o r m . ii)  E r r o r s due  t o m a c h i n i n g have a l a r g e  when t h e p e r m i t t i v i t i e s a r e v e r y  effect  different.  A compromise must be made between t h e number of v a r i o u s m a t e r i a l s and the a c c u r a c y of a p p r o x i m a t i o n to.be achieved.  For a c y l i n d e r w i t h a l a r g e p e r m i t t i v i t y  t h r e e o r more m a t e r i a l s may  variation,  be used as shown l n F i g u r e 5«lh.  76  5.1-1  E v a l u a t i o n of the A r t i f i c i a l D i e l e c t r i c Because t h e measurements on c y l i n d e r s  a r e t o be  t a k e n i n a p a r a l l e l - p l a t e r e g i o n , o n l y t h e TE mode, c o r responding  t o t h e case' o f p a r a l l e l p o l a r i z a t i o n i s a p p l i c a b l e .  I n t h i s case, an analogous waveguide mode (LSM o r l o n g i (50) tudinal-sectlon  magnetic  ) exists.  The s o l u t i o n when  l a y e r s of constant thickness are placed i n a rectangular 5/)  [SO  waveguide has been d i s c u s s e d f u l l y  7  and i t i s f o u n d  t h a t t h e e f f e c t i v e p e r m i t t i v i t y i s t h e same as when t h e l a y e r s : are i n f r e e s p a c e .  Thus, t h e c h a r a c t e r i s t i c s o f t h e a r t i -  f i c i a l d i e l e c t r i c may be c o n v e n i e n t l y a s c e r t a i n e d by waveg u i d e measurements u s i n g t h e c o n f i g u r a t i o n shown i n F i g u r e 5<>-» 2  \  1  i  Artificial Dielectric The r e f l e c t i o n c o e f f i c i e n t , / ' , was used as a  measure o f t h e c h a r a c t e r i s t i c s o f t h e specimens because, f o r t h e t y p e s c o n s i d e r e d , i t p r o v e s t o be s e n s i t i v e t o changes  77 i n the p e r m i t t i v i t y v a r i a t i o n and  because i t can be measured  S i n c e the magnitudes of P  accurately.  encountered are  of  the (5  o r d e r of 0 . 1 ,  the W e i s s f l o c h - F e e n b e r g ( n o d a l - s h i f t )  i s s u i t a b l e f o r the measurements.  The  2)  method  t h e o r e t i c a l r e s u l t s were  computed u s i n g the assumed p e r m i t t i v i t y v a r i a t i o n , i n a manner s i m i l a r t o t h a t used i n the homogeneous-shell approxi m a t i o n of S e c t i o n 3 . 1 - 2 ; vided into'very mission-line  each inhomogeneous r e g i o n was  t h i n homogeneous s l a b s .  t h e o r y may  be a p p l i e d  ( Elementary  ditrans-  to homogeneous d i e l e c t r i c  (51) s l a b s p e r p e n d i c u l a r to the  a x i s of the waveguide  ).  A b l o c k diagram of the system used f o r the  re-  f l e c t i o n c o e f f i c i e n t measurements i s shown i n F i g u r e 5 » 3 « The  s p e c i a l l y c o n s t r u c t e d waveguide s e c t i o n  samples a r e p l a c e d i s shown i n F i g u r e 5«^» making t h i s s p e c i a l s e c t i o n between the sample and  the  r e s i d u a l r e f l e c t i o n s i n the due  t o one  i n which The  i s t h a t t h e r e i s : no  the  advantage i n discontinuity  short-circuiting piston.  The  system are thus, r e d u c e d t o t h o s e  waveguide J u n c t i o n and  the s l o t t e d s e c t i o n .  Fur-  thermore, because t h e r e are no d i s c o n t i n u i t i e s b e h i n d specimen, an approximate c o r r e c t i o n  can be made f o r the  i n the measured v a l u e s of r e f l e c t i o n c o e f f i c i e n t s due residual reflections. s m a l l we  can  error  to  the  Assuming the r e s i d u a l r e f l e c t i o n s  are  write PV|  where  the  ^  /^M /  ~~ Pp.  = measured v a l u e of the complex r e f l e c t i o n c o e f f i c i e n t /°  R  = complex 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  residual  reflections /°r* «= /^II c o r r e c t e d f o r e r r o r due  to r e s i d u a l  reflections  78  Specimens o f t h e t y p e s shown i n F i g u r e 5»5& were made u s i n g p o l y s t y r e n e , polystyrene,  t  Y  = 1.5.  E  r  = 2.56  and, f o r t y p e I I I , expanded  The v a l u e s o f T used f o r each t y p e  a r e i n d i c a t e d i n Table 5.1 •  I n a l l c a s e s , the v a l u e s  were such t h a t an i n t e g r a l number o f specimens f i l l e d  chosen the  0.400 d i m e n s i o n o f the waveguide; the r e s u l t i n g c o n f i g u r a t i o n s a r e i l l u s t r a t e d i n F i g u r e 5«5D.  These were p l a c e d i n the  mounting s e c t i o n as i n d i c a t e d i n F i g u r e  5«^« 8.5»  Measurements were t a k e n a t t h r e e f r e q u e n c i e s : 10.0  and 11.5  GHz.  The r e s u l t s a r e shown i n T a b l e s 5»la t o 5«lc  together w i t h t h e o r e t i c a l values.  The  e n t r i e s i n column /?  are  t h e t h e o r e t i c a l v a l u e s f o r i d e a l specimens. Those i n column / r c a r e v a l u e s c a l c u l a t e d t a k i n g i n t o a c c o u n t an e s t i m a t e d c h i n i n g e r r o r f o r type I and u s i n g  t  r  - 1.55  as t h e  ma-  relative  p e r m i t t i v i t y o f the expanded p o l y s t y r e n e f o r type I I I ( t h i s i s ; d i s c u s s e d f u l l y i n i ) and 11) o f the f o l l o w i n g s e c t i o n on accuracy  of r e s u l t s ) .  The  agreement between the c a l c u l a t e d  and measured v a l u e s l s poor f o r T = 0.4  i n . e x c e p t a t 10.0  GHz,  i n w h i c h case we must assume t h e c l o s e agreement t o be c o i n cidental. i n Table  The r a n g e o f e r r o r s f o r T $• 0.2  i n . are  listed  5.2. I t appears t h a t the a r t i f i c i a l d i e l e c t r i c proposed  provides a reasonable  approximation  t o the d e s i r e d inhomogene-  ous d i e l e c t r i c when the t h i c k n e s s o f the samples i s 0.2in. o r less.  The  observed e r r o r s a r e comparable w i t h t h o s e  expected  MODULATOR 1 KHz _ n _ T L  POWER  CRYSTAL  SUPPLY  DETECTOR  FREQUENCY  KLYSTRON  BUFFER ATTENUATOR  F i g u r e 5.3  METER  ISOLATOR  20 db  V  0 k  SLOTTED;  SPECIMEN  SECTION  MOUNT  System used f o r R e f l e c t i o n C o e f f i c i e n t Measurements  SPECIMEN-  S.C. PLUNGER  OTHER HALF OF SECTION IS LOCATED BY 4,-%-DIA PINS AND HELD  BY S, NoWxl^-24  i  ; ®  MACH SCR  HOLES FOR MICROMETER-HEAD MOUNT (MATCH UG-39/U FLANGE)  ©  ©  i  <  - ^°=_  C3 p  I O C ©  T  #  ©  ^— t= =» " =* =  HOLES MATCH UG-33/U FLANGE  Figure  5.4  Section Constructed t oHold A r t i f i c i a l R e f l e c t i o n C o e f f i c i e n t Measurements  Dielectric  Specimens f o r  81 10 STEPS-  TYPE ->j.S /  EP-EXPANDED  cm ^1  P  \  —>| 12 cn\*—2.5 cm ->|/. 2 c/nr*-  TYPE I  P-POLYSTYRENE  U  E =2.56  TYPE  r  POLYSTYRENE  m  E =1.5 r  (a)  (c)  ure  5.5  Specimens used f o r t h e Waveguide E v a l u a t i o n o f t h e Artificial Dielectric a) Types o f Specimens b) M e t h o d o f S t a c k i n g S p e c i m e n s c) Machining E r r o r i n Type I S p e c i m e n s  82.  Table  5.1  •Comparison o f - M e a s u r e d - a n d T h e o r e t i c a l V a l u e s o f t h e 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 t h e Samples o f A r t i f i c i a l D i e l e c t r i c  Type I ....  n in  T  Mag  0.400  0.191  -140  0.206  97  0.208  0.200  0.109  108  -0.115  111  0.125  108  0.133  0.099  110  0.098  114  0.114  110  0.100  0.116  116  0.093  115  0.113  110  0.200  0.106  105  0.116  112  0.133  0.095  113  0.100  115  0.200  0.055  107  0.026  126  0.055  120  Arg  (a)  Mag  f = 8.5 GHz  Arg  Mag  Arg 95  A  Type I  II  III  Ac  T  Mag  Arg  Mag  Arg  Mag  Arg  0.400  0.122  117  0.110  126  0.121  124  0.200  0.120  147  0.096  148  0.112  145  0.133  0.124  149  0.102  152  0.123  148  0.100  0.127  157  0.104  154  0.132  148  0.200  0.118  140  0.085  149  0.133  0.114  146  0.091  153  0.200  0.149  156  0.130  166  0.123  159  Mag  Arg  0.048  -19  (b)  f = 10.0 GHz  fr  Type I  II  III  T  Mag  Arg  Mag  0.400  0.314  -32  0.060  0.200  0.058  20  0.073  14  O.063  11  0,133  0.048  8  0.078  20  0.064  15  0.100  0.054  27  0.081  22  0.062  16  0.200  0.049  8  O.O63  15  0.133  0.059  10  0.068  21  0.200  O.076  16  0.089  36  0.081  29  (c)  f = 11.5 GHz  Arg .  -17  84  due to c o n s t r u c t i o n a l d i f f i c u l t i e s  o r , f o r type I I I , an  e r r o r i n the p e r m i t t i v i t y of the medium used.  Table 5 - 2  Range o f D i f f e r e n c e Between T h e o r e t i c a l and Measured R e s u l t s f o r the Specimens o f . . A r t i f i c i a l D i e l e c t r i c with T < 0.2 i n .  Type  -0.024  I  to  II  0.030  -0.008 to 0 . 0 1 6  -0.033 to 0 . 0 1 0  III  to  Accuracy 1)  Arg(/V) -Arg(/>J  \Prl-\Pnl  -0.030 0.013  to  -0.026 0.005  to  -6.0 12.0  to  -11.0 11.0  Arg( PTc ) -Arg(/k)  to  -10.0 7.0  to  3.0 13.0  10.0  to 2 0 . 0  o f the R e s u l t s  Machining E r r o r s F o r the i n d i v i d u a l specimens, the e r r o r s i n o v e r a l l  dimensions  and the t h i c k n e s s o f the s t e p s f o r samples o f  type I I were a l i t t l e l e s s than ±0.0005 i n .  The e r r o r s i n  the l e n g t h o f the tapered s e c t i o n s were about ±0.005 i n . g e n e r a l , these e r r o r s were random and hence t h e i r were d i f f i c u l t  t o determine.  effects^  In  85  F o r specimens of type I , one e r r o r was  observed.  except f o r T = 0.4  s y s t e m a t i c machining  T h i s occurred due to the f a c t t h a t , i n . , a sharp edge could not he machined  without f r a c t u r i n g the samples.  The f i n i s h i n g . w a s t h e r e f o r e  done w i t h sandpaper, the r e s u l t b e i n g t h a t the ends of the t a p e r were somewhat rounded.  I n order to e v a l u a t e the  effect  of t h i s type of e r r o r , c a l c u l a t i o n s were made f o r the t a p e r shown i n F i g u r e  5»5c, the  column of Table 5*1•  The  r e s u l t s b e i n g g i v e n i n the e f f e c t was  and when taken i n t o account,  definitely  Pro  significant  the r e s u l t s were g e n e r a l l y c l o s e r  to the measured v a l u e s than those c a l c u l a t e d u s i n g the  ideal  taper. il)  E r r o r i n the P e r m i t t i v i t y of the Expanded P o l y s t y r e n e The  specimens was  expanded p o l y s t y r e n e used i n c o n s t r u c t i n g the c u t from "a 24 x 18 x 1 i n . sheet w i t h a nom-  inal relative permittivity by measuring £  K  £^=1.6.  T h i s v a l u e was  checked  f o r ' s e v e r a l s m a l l p i e c e s of the m a t e r i a l u s i n g (53)  the method of Roberts e r r o r b e i n g ±0.01. was 1.5  obtained.  and Von H i p p e l  , the  A range of v a l u e s between 1.48  of the m a t e r i a l .  The  taken to be the r e l a t i v e  r  I.63  •= 1.55)  type I I I specimens.  permittivity  e f f e c t of a d e v i a t i o n from t h i s  i s i n d i c a t e d by comparing the e n t r i e s i n the P (t  and  The m a j o r i t y of the samples gave a v a l u e near  and t h i s v a l u e was  Table 5.1  estimated  rc  value  column of  w i t h those i n the /° column, f o r the r  86 iii)  Measurement E r r o r s There are f o u r e f f e c t s c a u s i n g e r r o r s i n the  measured v a l u e s o f the r e f l e c t i o n c o e f f i c i e n t s .  These  are  (a) System i r r e g u l a r i t i e s such as s l i g h t changes i n the -width of the waveguide. (b) E r r o r s i n p o s i t i o n i n g the s h o r t (±0.0001 i n . ) , measuring slotted-section  circuit  the motion of the probe i n the  ( ±0.001 cm.)  and d e t e r m i n i n g the p o s i t i o n  of the probe r e l a t i v e t o the sample (±0.005  cm.).  (c) R e s i d u a l r e f l e c t i o n s . (d) Frequency d r i f t  (which causes an a d d i t i o n a l  nodal-shift). The measurements were c o r r e c t e d to (c) and  (d).  f o r the e r r o r s  I t i s expected t h a t the e r r o r i n the exper-  i m e n t a l l y determined v a l u e s of the r e f l e c t i o n c o e f f i c i e n t i s l e s s than ±0.0025 f o r the magnitude and ± 3 " f o r the argument.  due  8?.  5*2  Measurement of the S c a t t e r e d F i e l d s The  b a s i c m e a s u r i n g system, shown i n F i g u r e  i s quite conventional.  The  frequency synchronizer  eliminates  the n e c e s s i t y o f e q u a l i z i n g the e l e c t r i c a l l e n g t h s nal  and  reference  may  be u s e d .  1)  Measurement of Magnitude and The  Model 1751»  paths.  Two  5*6,  of t h e  methods of r e c o r d i n g the  sig-  field  Phase  phase-amplitude r e c e i v e r , a S c i e n t i f i c A t l a n t a  has meters w h i c h i n d i c a t e the a m p l i t u d e of t h e mea-  s u r e d s i g n a l , M, a reference  and i t s phase a n g l e , <p, r e l a t i v e t o t h a t  signal.  I n a d d i t i o n , a d.c. v o l t a g e  t o t h e phase a n g l e and a 1 KHz  of  proportional  s i n e wave o u t p u t whose a m p l i t u d e  I s p r o p o r t i o n a l t o the s i g n a l a m p l i t u d e are a v a i l a b l e . magnitude and phase of the f i e l d may  The  t h e r e f o r e be r e c o r d e d .  The  5.7a.  system used i s shown i n F i g u r e  i i ) D i r e c t Measurement of Quadrature Components The  phase of the 1 KHz  a m p l i t u d e - o u t p u t of  the  r e c e i v e r i s p r o p o r t i o n a l t o the phase of t h e microwave s i g n a l . By a p p l y i n g t h i s o u t p u t t o two i n F i g u r e 5.7b,  a d i r e c t measurement of the q u a d r a t u r e com-  ponents of the f i e l d of Chapter 4)  l o c k - i n a m p l i f i e r s as shown  may  (necessary  be o b t a i n e d .  f o r the c u r v e - f i t t i n g method The  o u t p u t of each l o c k - i n  a m p l i f i e r i s p r o p o r t i o n a l t o the p r o d u c t of t h e  amplitude  FREQUENCY SYNCHRONIZER  RECEIVER  POWER  CRYSTAL  SUPPLY  DETECTOR 3 db PHASE MIXER  KLYSTRON  MIXER  SHIFTER  BUFFER  FREQUENCY  ATTENUATOR  METER  ATTENUATOR  V 20 db  ISOLATOR  Figure  5.6  Basic  J  '7—  System used  10 db  J  E-H  TRANSMISSION  TUNER  LINE  i n S c a t t e r e d - F i e l d Measurements  89 AMP  1 KHz TUTTED  AC - DC  OUT  AMPLIFIER  CONVERTER  M CHART RECORDER  PHASE  <P  OUT  LOCK-IN  M cos(0)  AMPLIFIER CHART  AMP OUT  RECORDER LOCK-IN  M sln(^)  AMPLIFIER  l KHz_n_n_  Ref. Phase i s +90"  REFERENCE  Figure 5»?  &) System f o r Recording  Magnitude and Phase  b) System f o r Recording  Quadrature Components  o f t h e measured s i g n a l and t h e c o s i n e o f t h e phase between i t and a r e f e r e n c e  angle  ( a 1 KHz square wave s u i t a b l e  f o r the r e f e r e n c e i s p r o v i d e d by t h e r e c e i v e r ) .  By adjust-  i n g t h e r e f e r e n c e phase o f one a m p l i f i e r so t h a t i t l e a d s t h a t of t h e o t h e r by 9 0 ° , we o b t a i n M cos(<^) and M These can be read from meters o r r e c o r d e d .  sin(00.  90  5.2-1  Plane-Wave I n c i d e n c e - The  P a r a l l e l - P l a t e Transmission  Line The  p a r a l l e l - p l a t e t r a n s m i s s i o n - l i n e has  been  used by s e v e r a l authors i n making d i f f r a c t i o n and backfee 3 9 4-0 5-f ) s c a t t e r i n g c r o s s - s e c t i o n measurements  &  *  '  '  The  advantage-of u s i n g t h i s c o n f i g u r a t i o n , as compared w i t h (3 3 )  image-plane or f r e e - s p a c e  configurations  c y l i n d e r r e q u i r e d i s l e s s than present  A<,/2  ', i s t h a t  long.  T h i s , i n the  s i t u a t i o n , i s of prime importance because of  d i f f i c u l t y encountered  and  the  the  the p r e c i s i o n r e q u i r e d i n making  the inhomogeneous c y l i n d e r s .  However, the use  of t h i s  f i g u r a t i o n l i m i t s the i n v e s t i g a t i o n to the case of  con-  parallel  polarization. When p l a n n i n g i t was  the p r o b i n g  thought t h a t e x t e n s i v e  considerable  facilities  d i s t a n c e away from the c y l i n d e r would be  With the development o f the theory  t h i s was  unnecessary.  I t may  line,  f i e l d measurement up to a  quired.  be p o i n t e d  re-  i n Chapter  applicable  example i n the case of a square c y l i n d e r )  extensive  measurements are indeed n e c e s s a r y l n order to determine scattered f i e l d  and  the e x t r a p r o b i n g  the v e r s a t i l i t y of the  line.  4,  out t h a t f o r s i t u -  a t i o n s i n which the theory i n Chapter 4 i s not (for  of the  facilities  add  to  the  91  O v e r a l l Dimensions o f the T r a n s m i s s i o n - L i n e I t was decided at x«=band.  t o c a r r y out a l l measurements  T h i s i s convenient  s i n c e the wavelength i s l o n g  enough t o make the machining accuracy n e c e s s a r y  i n construct"  Ing t h e c y l i n d e r s and t r a n s m i s s i o n - l i n e p r a c t i c a l l y a b l e , and s h o r t enough t o r e s u l t i n reasonable dimensions o f the l i n e .  achiev-  overall  These dimensions were decided  upon  p a r t l y from b a s i c c o n s i d e r a t i o n s and p a r t l y from a compar55)  (39  i s o n w i t h l i n e s p r e v i o u s l y used  '  •  The b a s i c f a c t o r s d e t e r m i n i n g  t h e width o f t h e  l i n e a r e the r a d i u s o f t h e l a r g e s t c y l i n d e r t o be c o n s i d e r e d and  t h e g r e a t e s t d i s t a n c e from the c e n t e r a t which measureThese dimensions were s e t a t 2 A  ments a r e t o be taken. (about  6 cm.) and 10 /) (about 0  incident f i e l d at  0  30 cm.), r e s p e c t i v e l y . The  i s r e q u i r e d t o be e s s e n t i a l l y constant  l e a s t the diameter o f the c y l i n d e r .  over  Guided by the f i e l d  (55) patterns p r e v i o u s l y given  , a l i n e width o f 3 f t . was  used. A compromise i s n e c e s s a r y  between t h e d i r e c t i v i t y  lof o n gt henough o make tehneg ht o e r t ul ri en e .cover e horn t and the l h r no fa p the Thel el si sn e than mustone be (3  F r e s n e l zone  9)  a t the measuring r e g;ion. ion.  of width ¥, t h i s d i s t a n c e i s g i v e n by  (56)  F o r an a p e r t u r e  V  92  Kharadly  found t h a t a l i n e l e n g t h c o r r e s p o n d i n g  d i s t a n c e o f about 4 R  f  was r e q u i r e d t o o b t a i n . a (39)  f i e l d a l o n g t h e l i n e w h i l e Adey i n g t o about 4/3 R .  24 f t . corresponded t o about 2 R  f  Probing  •  •  correspond-  The chosen maximum l i n e l e n g t h o f  f  of 2 f t .  constant  • used a l e n g t h  v  to a  f o r t h e chosen a p e r t u r e  •  Considerations When d e a l i n g w i t h c i r c u l a r c y l i n d e r s , I t i s  d e s i r a b l e t o make f i e l d measurements a t f i x e d r a d i i and a t f i x e d angles.  The use o f a number o f s l o t s , i n w h i c h a  s l i d i n g probe may be mounted, c u t a t v a r i o u s a n g l e s sented  severe c o n s t r u c t i o n d i f f i c u l t i e s .  pre-  A s i m p l e r method  was t o u s e i n d i v i d u a l probe h o l e s d r i l l e d a t s e l e c t e d r a d i i and a n g l e s .  P l u g s were i n s e r t e d i n a l l t h e h o l e s e x c e p t t h e  one where a measurement was t a k e n . I t was thought r e a s o n a b l e  t o r e q u i r e measurements  a t 0.1 ?\ ( 0 . 3 cm.) i n t e r v a l s i n t h e r a d i a l d i r e c t i o n . . I t 0  seemed i m p r a c t i c a l , however, t o space t h e h o l e s l e s s t h a n 1 cm. a p a r t .  T h i s problem was overcome by p r o v i d i n g a  f a c i l i t y f o r p o s i t i o n i n g the c y l i n d e r s l i g h t l y o f f the z e r o - r a d i u s p o i n t and a p p l y i n g a s u i t a b l e c o r r e c t i o n f o r t h e change i n phase o f t h e i n c i d e n t wave.  By p o s i t i o n i n g t h e  c y l i n d e r a t z e r o , and t h e n a t 0 . 2 5 , 0 . 5 and 0 . 7 5 » c m  from  z e r o , r e p e a t i n g t h e measurements a t 1 cm. i n t e r v a l s each t i m e , t h e f i e l d c o u l d be o b t a i n e d a t 0 . 2 5 cm. i n t e r v a l s .  93  I t was  r a t h e r d i f f i c u l t t o e s t i m a t e how  small  an  a n g u l a r i n c r e m e n t i s n e c e s s a r y s i n c e t h i s i s v e r y dependent upon the p r o p e r t i e s and s i z e of the c y l i n d e r i n q u e s t i o n . F o r a homogeneous c y l i n d e r , we  e x p e c t the most r a p i d v a r i a t i o n  w i t h angle to occur f o r a l a r g e c y l i n d e r of h i g h p e r m i t t i v ity.  With  t = 2^5  and k r , = 1 0 ,  r  0  2°  i n t e r v a l s of about  s h o u l d be used i n o r d e r t o o b t a i n a r e a s o n a b l e  representation.  Thus, even i n t h i s case where q u i t e moderate parameters a r e u s e d , the probe h o l e s would be too c l o s e . decided  t o use 1 0 °  I t was  therefore  i n t e r v a l s and hence o n l y o b t a i n "samples"  of t h e a n g u l a r f i e l d p a t t e r n .  However, by means of a r o t a t -  a b l e d i s c i n the c e n t e r , measurements may a n g l e f o r r a d i i between 2 and 7 cm.  be t a k e n a t  a t 1 cm.  intervals?.  C o n s t r u c t i o n o f the P a r a l l e l - P l a t e T r a n s m i s s i o n - L i n e The  t r a n s m i s s i o n - l i n e system may  be  d i v i d e d i n t o t h r e e main s e c t i o n s : f i e h o r n - l e n s l a u n c h i n g s e c t i o n , the t r a n s m i s s i o n - l i n e and section. 5.8  and  any  System  conveniently system o r  t h e measuring  The d e t a i l s of c o n s t r u c t i o n a r e shown i n F i g u r e s 5-9.  The Horn-Lens System The  t r a n s m i s s i o n - l i n e was  f e d by an H-plane  s e c t o r i a l h o r n w i t h an a p e r t u r e d i m e n s i o n of % x 2k i n . A d i e l e c t r i c l e n s was  used t o o b t a i n a c o n s t a n t  across the aperture.  The  t a k e n as one t o f i v e .  phase f r o n t  s l o p e of the s i d e s o f the h o r n  was  No.8*j--32 UNC SOCKET HD  '^"'•f'f  AL ANGLE No.tO*j--32UNF  HACH SCR  CRS -* Pisces  Na.t2*-j- RO HO WO SCR S^j-CHIPBOARD  IS GA AL SH -2 PIECES "j-AL  -2 PIECES  B  33 1  -25 CA AL SH SECTION A-A,  M  PROBE HOLE MOS DRILL $OEEP .128S DRILL THROUGH  FIGURE 5.8  (b)  -LG SO THAT BOT IS FLUSH WHEN PLUG IS INSERTED No. 9-1-32 UNC FH HACH SCR (At) BOT AL SH IS THREADED J28*i SCR IS CUT AND FILED FLUSH  -AL BUSH. .HO ID, J50 OD j£ LONG •A 0/A*ALG INSERT IN  8 CA AL SH  •f  CRS FORMS A FRAMEWORK TO MAKE PLATE RIGID S FLAT  FIN WITH No. iOO SANDPAPER (c)  Figure  5.9  PIN LOCATES CENTERPIECE  77 DRILL 'W-IA PIN THROUGH THIS HOLE LOCATES THE •J2S5 DRILL CYLINDERrfem.FROM THE CENTER C'BORE .2S0OIA*-faDEEP d-0, .25, .50, .75 cm.  M  Some C o n s t r u c t i o n D e t a i l s o f t h e T r a n s m i s s i o n L i n e S y s t e m a) H o r n b) J o i n i n g S e c t i o n s o f t h e L i n e c) M e a s u r i n g S e c t i o n Showing D i raensions of S l o t d) P r o b e H o l e e) M e t h o d o f L o c a t i n g C y l i n d e r s  96  The h o r n was made of two p i e c e s o f 16 guage a l uminum sheet s e p a r a t e d by | x | i n . aluminum bars: as shown i n Figure 5.9a.  A I x | i n . c o l d r o l l e d s t e e l b a r , screwed  a l o n g the top and bottom o f each edge, e f f e c t i v e l y clamped the s h e e t s t o t h e aluminum b a r s w h i c h formed the s i d e s o f the horn.  T h i s ensured  a good, e l e c t r i c a l c o n t a c t a l o n g the com-  p l e t e l e n g t h o f each j o i n t . plywood was  A 6 i n . wide s t r i p o f % i n .  g l u e d t o the t o p and bottom a t t h e a p e r t u r e o f  t h e h o r n and l i x l i x f i n . aluminum a n g l e , used as a f l a n g e t o connect the h o r n t o t h e l i n e , was  screwed t o each o f t h e s e  strips.  The i x 0 . 9 0 0 i n . d i m e n s i o n  of t h e t h r o a t o f the  h o r n was  t a p e r e d t o the 0 . 4 0 0 x 0.900 i n . i n s i d e  o f RG - 52/U waveguide by an adaptor w h i c h was  dimension  tapped  to  a c c e p t screws i n s e r t e d t h r o u g h a UG - 39/U f l a n g e . The l e n s was made-of p o l y e t h y l e n e ( E  y  =  2,26),  and d e s i g n e d from g e o m e t r i c a l o p t i c s c o n s i d e r a t i o n s ^ * ^ ) . 5 6  Because o f a problem w i t h mode g e n e r a t i o n , i t was t o match the l e n s .  T h i s was  expanded p o l y s t y r e n e ( E  Y  accomplished  = 1.5)J  5  necessary  u s i n g s t r i p s of  a ' s t r i p of constant thickness:  a l o n g each f a c e produced f a v o u r a b l e r e s u l t s .  I d e a l l y , of  c o u r s e , p r o p e r m a t c h i n g r e q u i r e s a s t r i p of v a r y i n g p e r m i t t i v i t y and t h i c k n e s s a l o n g t h e curved f a c e . The  Transmission-Line The  t r a n s m i s s i o n - l i n e c o n s i s t e d o f up t o t h r e e  8 f t . s e c t i o n s , the m a t e r i a l used b e i n g 3 x 8 f t . p i e c e s o f i n . c h i p b o a r d w i t h a s h e e t o f 28 gauge aluminum l a m i n a t e d  97  to  one  side.  Each s e c t i o n o f the l i n e c o n s i s t e d of  p a r a l l e l s h e e t s , aluminum f a c e i n w a r d s , pered wooden wedges a l o n g the s i d e s . connected t o each o t h e r and by means of I f  x If  separated  The  two  by t a -  s e c t i o n s were  t o the h o r n and m e a s u r i n g s e c t i o n  x £ i n . aluminum a n g l e screwed on a t b o t h s h e e t as. shown i n F i g u r e 5»9"b.  ends o f each c h i p b o a r d The M e a s u r i n g S e c t i o n The  b a s i c c o n s t r u c t i o n was  3 x 3 f t . sheets  p l a t e s o f t h e m e a s u r i n g s e c t i o n , two 10  guage aluminum s e p a r a t e d  c o l d r o l l e d s t e e l bar l a i d  the same f o r b o t h of  by a network of 3 / 8 x 5/8 on edge.  Each p l a t e was  in.  held to-  g e t h e r by screws w h i c h passed t h r o u g h t h e o u t s i d e s h e e t t h e b a r s and were t h r e a d e d trated i n Figure 5»9c.  and  i n t o the i n s i d e s h e e t as i l l u s -  By c a r e f u l l y c u t t i n g o f f the ends  -  o f t h e s c r e w s , f i l i n g them f l u s h and f i n i s h i n g t h e assembled s t r u c t u r e w i t h f i n e sandpaper, a smooth s u r f a c e was T h i s c o n s t r u c t i o n was  favoured  obtained.  over t h a t of u s i n g a s i n g l e  t h i c k aluminum p l a t e because t h i c k p l a t e s were found t o have a r a t h e r wavy s u r f a c e w i t h c o n s i d e r a b l e t h i c k n e s s v a r i a t i o n (extremely uniform probe-penetration  t h i c k n e s s was  r e q u i r e d t o ensure  a t a l l p o i n t s ) and f a c i l i t i e s f o r m a c h i n i n g  a 3 x 3 f t . p i e c e were n o t r e a d i l y . A 15  equal  cm.  available.  d i a m e t e r h o l e , t h r o u g h w h i c h the  d e r s were i n s e r t e d , and  a l s o h o l e s and  a s l o t f o r probing  the f i e l d were l o c a t e d i n the top p l a t e assembly. h o l e s were 0 . 1 2 8 i n . i n d i a m e t e r and  cylin-  the s l o t was  The 3/16  probe in.  98  wide "by 28 cm. l o n g .  T h e i r p o s i t i o n i n g i s shown i n F i g u r e  and c o n s t r u c t i o n d e t a i l s a r e shown i n F i g u r e s 5»9c and The  s l o t was  5«3  d.  i n c l u d e d t o a l l o w the t a k i n g o f s t a n d i n g wave  r a t i o measurements ( f o r example as a method o f d e t e r m i n i n g backscattering cross-section^^')• f i e l d measurements, i t was  When t a k i n g d i f f r a c t e d  plugged w i t h a s t r i p o f aluminum  w h i c h had h o l e s f o r the p r o b e . The 15  cm. d i a m e t e r h o l e i n the c e n t e r was  fitted  w i t h an aluminum d i s k w h i c h had probe h o l e s l o c a t e d 1  cm.  a p a r t a l o n g a d i a g o n a l and a 1 i n . d i a m e t e r h o l e i n i t s c e n t e r . Various 1 i n . diameter i n s e r t s f o r l o c a t i n g the c y l i n d e r at 0, O . 2 5 ,  0.5  o r O.75  The l o c a t i o n was a 1/32  cm. f r o m the c e n t e r were made ( F i g u r e 5 » 9 c ) .  accomplished  by a p i n w h i c h passed  through  i n . d i a m e t e r h o l e i n the i n s e r t i n t o a s i m i l a r h o l e  i n the center of the c y l i n d e r . The p l a t e s were s e p a r a t e d by s m a l l i i n . t h i c k wooden b l o c k s .  S t r i p s c u t f r o m B.F.  G o o d r i c h VHP-4 and  VHP-6 microwave a b s o r b e r were i n s e r t e d a l o n g t h e s i d e s and  end,  respectively.  by  The whole s e c t i o n was  connected  t o the l i n e  I f x l g x i i n . aluminum a n g l e f l a n g e s . The probe was made by a d d i n g an e x t e n s i o n t o a Narda Model 229B probe. l e n g t h was  The d i a m e t e r was  0.02  i n . and  a d j u s t a b l e t o a maximum o f about i i n . ( a 0.04 i n .  p e n e t r a t i o n g e n e r a l l y proved m a l l y used w i t h the probe was  satisfactory).  The  c r y s t a l nor-  removed and r e p l a c e d by  a d a p t e r w h i c h p r o v i d e d a microwave o u t p u t t o an H-type connector.  the  an female  99  5«2-2  Cylindrical-Wave Incidence The  Section  long l i n e necessary  incidence i s not necessary  t o o b t a i n "plane-wave"  when u s i n g a c y l i n d r i c a l wave  and hence a compact measuring u n i t i s p o s s i b l e i n t h i s The  system was  designed  case.  s p e c i f i c a l l y f o r t a k i n g measure-  ments t o w h i c h t h e t h e o r y o f C h a p t e r 4 may  be a p p l i e d , namely  a t a c o n s t a n t r a d i u s n e a r the c y l i n d e r . I n o r d e r t o p l o t t h e measurements d i r e c t l y as  a  f u n c t i o n of a n g l e on a c h a r t r e c o r d e r , a synchronous motor d r i v e was  used t o r o t a t e a c e n t e r p i e c e on w h i c h the probe  was  mounted.  Two  gear r a t i o s a r e p r o v i d e d  150/minute. 2.5,  5*0,  5*10.  The d r i v e mechanism i s shown i n F i g u r e  g i v i n g speeds o f 5 0 % i i a u t e , a n d  These speeds a r e s u i t a b l e f o r p l o t t i n g on  15.0  and  the  3°»0 cm./minute c h a r t speeds o f t h e r e -  corder used. The  c o n s t r u c t i o n d e t a i l s o f the  r e g i o n " a r e shown i n F i g u r e 5 « H .  I t was  parallel-plate  unnecessary to  use  the sandwich c o n s t r u c t i o n d e s c r i b e d p r e v i o u s l y ; s i n c e the probe was  o n l y i n s e r t e d t h r o u g h the c e n t e r p i e c e , t h e  v a r i a t i o n i n t h e p l a t e s c o u l d be t o l e r a t e d . two  thickness  The  p l a t e s were  30 x 44 i n . s h e e t s o f § i n . aluminum and were  by t e n f - NC  separated  b o l t s l o c a t e d as shown i n the f i g u r e .  t o r t s ^ c o u l d be used t o f o r c e the p l a t e s t o g e t h e r and s l i g h t warps were c o r r e c t e d so t h a t a c o n s t a n t  Other thereby  separation  was  o b t a i n e d , p a r t i c u l a r l y o v e r the a r e a I n w h i c h the c y l i n d e r s were l o c a t e d .  P i e c e s c u t from B.F.  Goodrich  VHP-6 microwave  1 0 0  absorber  were i n s e r t e d a l o n g the p e r i m e t e r . o f  order to prevent f i e l d was  r e f l e c t i o n s at the edges.  The i n c i d e n t  produced by an e x c i t a t i o n probe, shown i n F i g u r e  T h i s probe was  designed  to be matched when  the d a t a given by King^ "^;. 5  was  the s e c t i o n i n  ?\ = ff cm.  the e f f e c t of the conducting  three element c o l i n e a r d i p o l e a r r a y . probe 25 or 50 cm.  plates  p a r t of a  Holes f o r l o c a t i n g . t h e  from the c e n t e r of the c y l i n d e r were p r o v i d -  ed; however, o n l y the 50 cm. iments c a r r i e d  using  0  approximated by assuming t h a t the probe was  5«12a.  l o c a t i o n was  used i n the  exper-  out.  The  c e n t e r p i e c e was  w i t h i n a f i x e d brass r i n g .  made of brass and  T h i s allowed  would be p o s s i b l e with an all-aluminum c e n t e r p i e c e was  rotated  a b e t t e r f i t than  construction.  The  s l i g h t l y t h i c k e r than the r i n g to g i v e  clear-  ance to the s p e c i a l l y c o n s t r u c t e d probe, d e s c r i b e d below, when measurements were taken near the r i n g ' s edge. (0.100  Probe-holes  i n . i n diameter) were l o c a t e d so t h a t measurements  could be taken a t the c e n t e r and tween 1 . 5  cm.  and  i n v e s t i g a t i o n was  8.0  cm.  a t 0 . 5 cm.  intervals  from the c e n t e r .  The  be-  c y l i n d e r under  l o c a t e d by a p i n p l a c e d i n the c e n t e r probe-  hole. The  c o n s t r u c t i o n of the measuring probe i s shown  i n Figure 5»12b.  The waveguide adapter  i n g to the c r i t e r i o n i n Reference 5 9 . impedance was  maintained  b e i n g achieved The u n i t was f o u r screws.  u s i n g two  was  designed  A 50 ohm  accord-  characteristic  throughout, the change i n diameters quarter-wave ( a t A  0  - ft cm.)  sections.  attached f i r m l y to the c e n t e r p i e c e by means of The  p e n e t r a t i o n of the probe was  changing the l e n g t h of the 0 . 0 1 8  i n . diameter  changed wire.  by  101  F i g u r e 5.10  System f o r R o t a t i n g  Centerpiece  GOODRICH VHP-S  MICROWAVE  ABSORBER  T  0 6$-  D/A  PROBE HOLE AT CENTER ANO AT 1cm INTERVALS FROM 1.5 TO 7.5cm AND 2.0 TO 8.0cm FROM CENTER.  $j-D/A B4DIA  No 8-32NC * 4-DEEP HOLES AROUND EACH PROBE )UN SHOWN IN FIG SJ3b HOLE. LOCATION  SECTION B-B PROBE HOLE J09S5 DRILL THROUGH .1040 DRILL $DEEP ALCHAN  , FIGS.I2 , =H FIXED h= -I PLATE BOLTED 1  -TWO PIECES 2J-*fj-*-k ANGLE IRON  1  L  1  FRAME ~ m "  r  1  J  SECTION A-A  -12'  •Ki-  JL  JL  JL S  H O  ro  Figure  5.11  Construction  of  Parallel-Plate  Region  used  for  Cylindrical-Y/ave  Incidence  Figure  5.12  Construction  of  a) E x c i t i n g  Probe  b) M e a s u r i n g  Probe  O  104  5.3  R e s u l t s o f Measurements A l l measurements were t a k e n a t a f r e q u e n c y o f  9 . 5 4 2 6 GHz (k<> « 1 . 9 9 9 9 4  cmT),. t h e measured 1  magnitude and phase o f t h e f i e l d .  q u a n t i t i e s being the  F o r plane-wave I n c i d e n c e ,  v a l u e s were o b t a i n e d a t 1 0 ° i n t e r v a l s .  For cylindrical-wave  i n c i d e n c e , t h e f i e l d s were r e c o r d e d w i t h a c h a r t r e c o r d e r and the v a l u e s r e q u i r e d f o r computations  were t a k e n from t h e graph;  b o t h 5 ° and 1 0 ° i n t e r v a l s were c o n s i d e r e d . The c o e f f i c i e n t s were c a l c u l a t e d u s i n g e q u a l w e i g h t i n g ("equation  ( 4 . 6 ) ) and t h e r e q u i r e d o r d e r was determined by  the s i m p l i f i e d method suggested culated using equation ( 4 . 1 2 a ) .  i n S e c t i o n 4 . 2 , s^ b e i n g  cal-  The v a l u e o f s* decreased  slow-  l y a t f i r s t , t h e n r a p i d l y , and f i n a l l y became q u i t e c o n s t a n t ; the v a l u e used f o r N was such t h a t a d d i n g a n o t h e r term caused a d e c r e a s e o f l e s s t h a n 2%  i n the value of s  N  f o r both the r e a l  and i m a g i n a r y p a r t s . 5.3-1  Incident F i e l d The e r r o r s i n t h e measured v a l u e s o f t h e i n c i d e n t  f i e l d a t r = 4 cm. and r = 1 1 cm. f o r plane-wave i n c i d e n c e a r e shown i n F i g u r e s 5 . 1 3 a and b. field  A p l o t o f t h e measured i n c i d e n t  a t r = 4 cm. f o r c y l i n d r i c a l - w a v e i n c i d e n c e i s shown i n  F i g u r e 5 . 1 ^ and t h e e r r o r s i n t h e r e a d i n g s a t lo"  *  intervals  Note t h a t t h e magnitude curve l a g s t h e phase curve by 3 * i n the experimental p l o t s  0.02  0  -0.02  F i g u r e 5.13  E r r o r s i n the Measured I n c i d e n t F i e l d f o r P l a n e Wave I n c i d e n c e  Figure  5.14  Measured  Incident  Field  at  r =  4  cm.  for Cylindrical—Wave  Incidence  10? 0.04 CD  0.02  -P  •H  Cl  bO  ce  -0.02  10 CD  bo o 0)  CO  cd XI PL,  0.05 (D  -P •rl Cl  bo cd  a  -0.05  CO CD  bo 0) CO  cd XI PL,  -10 &• , degrees b) F i g u r e 5.15  x = 8 -cm.  E r r o r s i n the Measured Incident F i e l d f o r Wave Incidence  Cylindrical-  108  a t r = 4 cm. In  and r ~ 8 cm.  a r e shown i n F i g u r e s 5 « 1 5 a and  b o t h c a s e s , some asymmetry i n t h e f i e l d i s e v i d e n t ,  u l a r l y a t the l a r g e r r a d i i .  The r e a d i n g s a t t h e 4 cm.  i n d i c a t e t h a t the e r r o r i n the f i e l d  over the r e g i o n  by t h e c y l i n d e r i s about ± 0 . 0 2 i n magnitude and ± 2 ° 5.3-2.  b.  particradius  occupied i n phase.  Scattered F i e l d (1)  Comparison o f the two  Systems  Most o f the r e s u l t s were o b t a i n e d u s i n g wave i n c i d e n c e s i n c e t h i s system was t h a n t h e plane-wave system.  cylindrical-  c o n s i d e r a b l y more  convenient  A c o m p a r i s o n between the r e s u l t s  o b t a i n e d from measurements u s i n g p l a n e and c i d e n c e i s g i v e n i n F i g u r e 5.Tb  c y l i n d r i c a l wave i n -  for  a) A m e t a l l i c c y l i n d e r , r , = 1.5 b) A d i e l e c t r i c c y l i n d e r , c) A m e t a l l i c c o r e , r  z  £  r  = 1.5  cm.  = 2 . 5 4 * , r , = 2.9 cm.  cm.  w i t h the a r t i f i c i a l  d i e l e c t r i c s h e l l shown i n F i g u r e 5 » 1 7 « I t i s seen from t h e e r r o r c u r v e s i n F i g u r e s to  c t h a t the a c c u r a c y  5.16&  a c h i e v e d u s i n g t h e c y l i n d r i c a l - w a v e system  i s comparable w i t h o r b e t t e r than t h a t a c h i e v e d u s i n g the wave s y s t e m , ( t h e r e s u l t s f o r the d i e l e c t r i c c y l i n d e r a r e a t i v e l y poor f o r the plane-wave s y s t e m ) . i s . much s m a l l e r t h a n the e x p e r i m e n t a l  The  planerel-  truncation error  e r r o r i n e i t h e r case.  *• The m a t e r i a l used was c r o s s - l i n k e d p o l y s t y r e n e w h i c h had e r i o r m a c h i n i n g q u a l i t i e s t o the p o l y s t y r e n e used i n the waveguide e v a l u a t i o n o f the a r t i f i c i a l d i e l e c t r i c  sup-,  109  A comparison between t h e t h e o r e t i c a l v a l u e s backscattering  of the  c r o s s - s e c t i o n and t h o s e d e t e r m i n e d from e x p e r i -  m e n t a l d a t a i s g i v e n i n T a b l e 5«3» a c c u r a c y of t h e r e s u l t s o b t a i n e d  A g a i n i t appears t h a t t h e  by t h e two e x p e r i m e n t a l  systems  i s comparable. (II) The field, T  C  Measurement E r r o r s e r r o r s i n t h e measured v a l u e s  - 4 cm., w i t h c y l i n d r i c a l - w a v e i n c i d e n c e f o r t h e t h r e e  c y l i n d e r s considered The  of the scattered  above a r e shown i n F i g u r e s 5.18a t o c.  s c a t t e r e d f i e l d i s n o t shown s i n c e i t I s n o t v e r y d i f f e r e n t  from t h a t f o r plane-wave i n c i d e n c e , F i g u r e s 5«l6a t o c. s c a t t e r e d f i e l d and f i e l d  The  e r r o r s a t r = 4 cm. f o r  d) A m e t a l l i c c y l i n d e r , r , = 3«5 cm. e) A d i e l e c t r i c c o r e  £ - 2.54, r = 1 . 5 cm. w i t h t h e z  r  a r t i f i c i a l d i e l e c t r i c s h e l l shown i n F i g u r e  5*17  are shown i n F i g u r e s 5«l9h and 5»20b, r e s p e c t i v e l y . The  errors i n the directly-measured  magnitudes; a r e  about t h e same f o r a l l t h e c y l i n d e r s , g e n e r a l l y l e s s t h a n ±0.05. The  phase e r r o r s on t h e o t h e r hand a r e s t r o n g l y dependent upon t h e  t y p e o f c y l i n d e r , r a n g i n g from + 3 ° f o r t h e 3*5 cm. r a d i u s m e t a l l i c c y l i n d e r t o over ±20* f o r t h e d i e l e c t r i c - c o r e d a r t i f i c i a l s h e l l . The  l a r g e phase e r r o r s g e n e r a l l y appear where t h e phase I s chang-  i n g r a p i d l y , c o r r e s p o n d i n g t o minima i n t h e magnitude. of e r r o r s i n l i s  The range  and g ^ w i l l t h e r e f o r e tend t o be t h e same f o r  a l l cylinders. ( i i i ) E f f e c t o f Changing T  A  A comparison between t h e r e s u l t s o b t a i n e d  from  ) 110  measurements w i t h r  0  inders  = 4 cm. and T  (d) and (e) i n F i g u r e s  = 8 cm. i s g i v e n f o r C y l -  0  5*19 and 5 . 2 0 ,  respectively. For  both c y l i n d e r s , t h e value of N determined w i t h r one l e s s t h a n t h a t w i t h r  der  (d) and r  f t  u s i n g the data w i t h r  were about t h e same f o r b o t h v a l u e s  r  c  0  curves  1  I t i s r e a d i l y seen t h a t b e t t e r  = 8 cm. f o r C y l i n d e r ( e ) .  the r e s u l t s w i t h r  = 8 cm. was  = 4 cm.; the. t r u n c a t i o n e r r o r  0  are f o r t h e s m a l l e r v a l u e o f N. r e s u l t s are obtained  c  0  - 4 cm. f o r C y l i n -  R e s u l t s f o r C y l i n d e r (b)  of r c while f o r C y l i n d e r (c)  = 4 cm. were a g a i n more a c c u r a t e  than w i t h  «= 8 c i , The  lower value of N determined f o r the l a r g e r radius-  i s expected because, due t o t h e r a p i d d e c r e a s e i n t h e magnitude of t h e h i g h - o r d e r  Hankel f u n c t i o n s n e a r t h e c y l i n d e r , t h e num-  ber o f s i g n i f i c a n t terms t e n d s t o d e c r e a s e w i t h i n c r e a s i n g r a d i u s . I n p a r t i c u l a r , we n o t e t h e r e l a t i v e l y s m a l l t r u n c a t i o n e r r o r a t r = 8 cm. compared w i t h t h a t a t r = 3*5 cm. f o r t h e m e t a l l i c cylinder. For the c y l i n d e r s considered,  i t appears t h a t , a l -  though i t i s g e n e r a l l y b e t t e r t o u s e a s m a l l v a l u e o f r , t h i s 0  may n o t always be t h e c a s e .  I f the errors i n f  the same a t b o t h r a d i i c o n s i d e r e d ,  5  i  and g ; were 3  t h e e r r o r s i n a ^ and Hg-n would  a l s o be t h e same b u t , s i n c e t h e magnitude o f t h e H a n k e l f u n c t i o n s decreases w i t h i n c r e a s i n g r a d i u s , the corresponding errors: i n A-„ would be l a r g e r w i t h r  0  = 8 cm.  Thus, t h e e r r o r s i n t h e c a l -  c u l a t e d f i e l d would d e c r e a s e w i t h d e c r e a s i n g s u l t s f o r the r  0  r . 0  The good r e -  = 8 cm. d a t a shown i n F i g u r e s 5 « 2 0 a t o c l n d i -  c a t e that- t h e e r r o r s i n l  si  and g  s £  depend on r a d i u s and i t  SCAT FLD  MAG ERROR  .0  MflG  L.O  .5  a  oq  P  H  fD  •  m P  C/3  pb  p  a  e+ O e+  cpg"  <<• ro  H  I-  re p.  H-  B  H  »rj  o  ce  P K-i p  o  < H  P  CD pi o  p »  H*  ra  Pu  1  0  p < fD  CD o fD  o.  II  n  H-  ' fD ct-  o O fD B ....... p  ro cr a  P t—' %• >i  .  P-  fD  a  A R G ERROR  II  10.0  a'  • o B  •  0 P-  o  P  M t-i O ti 01  o  aw  0  CT)o  4  *d <+ HO  1 m 1 •  a  a rn.1  trioq  P  OQ fc» fD p -  p  O B  o  •-d  t— P 1  0  fD  ro  III  cr a  .0  10.0  _J  SCAT FLD -20.0  ARG .0  (XlO  1  ) 20.0 J  o r-  CM  I  .0  R N G L E (OEG)  50.0  100.0  RNGLE (DEG)  150.0  200.0  200.0  RNGLE (DEG) b)  l  r  = 2.54, r, = 2.9 cm.  H  MflG ERROR  S C R T F L D MflG .0 L.O  2.0  SCAT FLD ARG ( X l O ) -20.0 .0 20.0 1  ro  a  114  Caption Applying 5.19  Y  and  5.20  O  E r r o r s i n measured v a l u e s , wave  x  A  5.l6,  5.18,  & = 0 t o ±180*;  plane-  t o the F i e l d Errors I n Figures  Incidence  E r r o r s i n measured v a l u e s , 0- = 0 t o ±180°; c y l i n d r i c a l wave  Incidence  -— — E r r o r  due t o t r u n c a t i n g t h e t h e o r e t i c a l l y c a l c u l a t e d  f i e l d a t the Value of n = N determined f o r t h e e x p e r i mental data —  —  E r r o r i n the scattered f i e l d u s i n g plane-wave i n c i d e n c e , r  c a l c u l a t e d from d a t a 0  = 4 cm. 1  • Error i n the scattered f i e l d  c a l c u l a t e d from d a t a  using cylindrical-wave Incidence, As above w i t h r  a  r  e  = 4 cm.  = 8 cm.  Cross-Linked  Polystyrene ,  o  (£r=2.54)  CO  00  o 1.5 cm.  3.4 cm.  Figure 5.17  A r t i f i c i a l D i e l e c t r i c S h e l l used i n t h e E x p e r i m e n t s  Table 5.3  Comparison Between Values of Backscattering Cross-Section Calculated Theoretically and from Experimental Data  Cylinder  Theoretical  Metal, r, =1.5 cm.  4.930  Metal, r, =3.5 cm.  11.117  ly = 2.54, r, = 2.0 an. £  r  = 2.54, r,= 2.9 cm.  Experimental Cylindrical  -.4.97 -.5.34 9.93  22.012 0.158  Core-Metal, r *=1.5 cm. Shell- A r t i f i c i a l (Figure 5.17)  7.822  4.51 - 5.21 11.01 - 11.88 6.61  -6.491  Core- Sy = 2.54, r,. » 1.5 cm. Shell- A r t i f i c i a l (Figure 5.17) z  Plane  21.43  21.12 - 22.81 0.06 - 0.07  8.00 - 8.06  8.06 - 8.08  CEin -5-0  1  I  .0  50.0  100.0  150.0  200.0  .0  5D.Q  1D0.0  150.0  200.0  RNGLE (DEG)  RNGLE (DEG) a)  Metal,  r, = 1.5 cm.  F i g u r e 5.18 E r r o r s i n the S c a t t e r e d F i e l d f o r C y l i n d r i c a l Wave-Incidence, r  0  = 4 ca.  H  H ON  MAG  -.1  CD  ERROR  .0  MflG  .1  -.1  O  ERROR  .0  .1  MflG ERROR -.1  .0  SCAT FLD MflG .1  .0  .5  1.0  MflG ERROR -.1  .0  SCAT .1  .0  F L D MflG .5  1.0  MflG ERROR -.1  .0  SCAT .1  .0  F L D MflG 1.0  2.0  MAG ERROR  -.1  .0  SCAT FLD MAG  .1  .0  ro J a •  a  1.0  2.0  124  appears t h a t t h e r e l a t i v e l y s m a l l phase e r r o r s i n t h e range 0 - 8- - 1 0 0 ° may he a c o n t r i b u t i n g f a c t o r ( n o t e t h a t t h e phase v a r i a t i o n i s smoother a t r = 8 cm. t h a n a t r = 4 cm.). 5.3-3  Examples o f Measured D i f f r a c t e d F i e l d s Experimental p l o t s of the d i f f r a c t e d f i e l d  w i t h t h e c y l i n d r i c a l - w a v e system, r  0  measured  = 4 cm., f o r t h e f o l l o w i n g  c y l i n d e r s a r e g i v e n i n F i g u r e s 5«18a t o d: a) M e t a l , r , = 3 . 5 cm. b) C o r e - M e t a l , r  z  « 1 . 5 cm.;;  ...  S h e l l - A r t i f i c i a l (Figure 5.17) c) <% = 2 . 5 4 , d) C o r e - E  t  r , = 2.9 cm.  = 2.54, r  x  = 1 . 5 cm.;  S h e l l - A r t i f i c i a l (Figure 5-17) Some b a s i c c h a r a c t e r i s t i c s o f t h e d i f f r a c t e d o f t h e s e c y l i n d e r s s h o u l d be p o i n t e d : o u t .  fields  For Cylinder (a),  we n o t e t h e b r o a d shadow r e g i o n where t h e f i e l d magnitude i s v e r y s m a l l ( t h e g e o m e t r i c a l o p t i c s shadow e x t e n d s from about 1 5 8 ° t o 202°).  F o r C y l i n d e r ( b ) , t h e magnitude v a r i a t i o n i s  quite Irregular.  F o r C y l i n d e r s ( c ) and ( d ) , t h e r e i s a r a p i d  o s c i l l a t i o n o f magnitude w i t h 6- and b o t h have a maximum a t & = 180°.  There i s a s t r o n g c o n t r a s t between ( c ) and ( d ) , how-  e v e r , i n t h a t whereas t h e magnitude o f t h e o s c i l l a t i o n i s l a r g e a t a l l a n g l e s f o r ( c ) , t h e magnitude' i s s m a l l n e a r 0- = 0 and i n c r e a s e s s t e a d i l y w i t h ©- f o r ( d ) .  5  s  MAGNITUDE J  5  MAGNITUDE •  2ZT  129  5.3-4  Comparison w i t h P u b l i s h e d  Results  I t seemed d e s i r a b l e t o o b t a i n e x p e r i m e n t a l  results  f o r d i r e c t comparison w i t h t h e p r e v i o u s l y p u b l i s h e d r e s u l t s w h i c h were i n d i s a g r e e m e n t w i t h computed v a l u e s .  1  '  C y l i n d e r s which  a p p r o x i m a t e t h e f o l l o w i n g cases were c o n s t r u c t e d : a) Mr)  = 10Ac.r; k r ^ = 3,  b) £ (r)  = 5A r;  K  = 3.7,  0  6 . 8 , 8.0  k ^ r * « 3, k<,r, = 4.2  0  The comparison i s g i v e n i n T a b l e 5 . 4 .  T a b l e 5«4  Comparison Amongst P u b l i s h e d , T h e o r e t i c a l , and Experimental Values of B a c k s c a t t e r i n g Cross-Section  Cylinder  Published^  3  ^  Theoretical*  ;  Experimental**  £r(r) = l 0 A r o  3.7  k,r, =  £ (r) r  =  8.0  4.837  4.44 -  5.02  6.8  5.5  0.695  0.74 -  0.89  8.0  46.3  8.425  9.14  9.50  1.0  4.826  5/k r 0  k r , = 4.2 e  -  * C a l c u l a t i o n d i s c u s s e d i n Chapter 3 ** U s i n g c y l i n d r i c a l - w a v e i n c i d e n c e system  . 4.94  130  5.4  Summary A method o f c o n s t r u c t i n g c y l i n d e r s w i t h a smoothly,  v a r y i n g p e r m i t t i v i t y u s i n g a type o f a r t i f i c i a l d i e l e c t r i c has been d e v e l o p e d .  R e f l e c t i o n c o e f f i c i e n t measurements on s p e c i -  mens p l a c e d i n r e c t a n g u l a r waveguide and d i f f r a c t e d - f i e l d measurements on v a r i o u s c y l i n d e r s i n d i c a t e t h a t computations u s i n g t h e e f f e c t i v e p e r m i t t i v i t y o f t h e a r t i f i c i a l medium a r e v a l i d .  The  measured r e s u l t s on c y l i n d e r s c o n f i r m computed r e s u l t s ( a l s o d i s a g r e e w i t h some p u b l i s h e d  results).  P a r a l l e l - p l a t e systems f o r measuring t h e d i f f r a c t e d f i e l d under c o n d i t i o n s o f p l a n e and c y l i n d r i c a l wave i n c i d e n c e were c o n s t r u c t e d .  F o r b o t h systems, t h e method o f c a l c u l a t i n g  the s c a t t e r e d - f i e l d c o e f f i c i e n t s g i v e n i n Chapter 4 was a p p l i e d to  the experimental  data.  The c o e f f i c i e n t s f o r plane-wave  s c a t t e r i n g were c a l c u l a t e d from t h o s e o b t a i n e d w i t h wave i n c i d e n c e and i t was found t h a t t h e a c c u r a c y tered f i e l d  cylindrical-  of the s c a t -  c a l c u l a t e d u s i n g these c o e f f i c i e n t s was comparable  w i t h t h a t o b t a i n e d from measurements u s i n g plane-wave i n c i d e n c e . I t s h o u l d be p o i n t e d out t h a t t h e a c c u r a c y  of the r e -  s u l t s i s a f f e c t e d by t h e r e l a t i v e l y crude form o f t h e t h e o r y g i v e n i n Chapter 4 which was a p p l i e d t o t h e e x p e r i m e n t a l and i s l i m i t e d by two d e p a r t u r e s have n o t been accounted f o r . (i)  data,  from t h e i d e a l i z e d model which  These a r e :  The d e v i a t i o n o f t h e i n c i d e n t f i e l d from t h e  h y p o t h e t i c a l v a l u e over t h e r e g i o n o c c u p i e d w i l l cause e r r o r s i n t h e s c a t t e r e d f i e l d .  by t h e c y l i n d e r  131  (11)  S i n c e t h e a c t u a l i n c i d e n t f i e l d i s composed  of d i r e c t r a d i a t i o n from t h e s o u r c e and r a d i a t i o n r e f l e c t e d from the edges o f t h e p a r a l l e l - p l a t e r e g i o n , t h e p r e s e n c e o f "the c y l i n d e r changes t h e i n c i d e n t  field.  132  6.  APPLICATION TO MEASUREMENTS ON CYLINDRICAL PLASMAS The  determination  of the electron, density  t i o n i n l a b o r a t o r y plasmas by microwave p r o b i n g current I n t e r e s t ^ considered  e s  *  6 0  distribu-  i s a topic of  ^ and t h e phase a n g l e o f t h e s i g n a l i s .  t o be t h e most s i g n i f i c a n t q u a n t i t y .  When d e t e r m i n -  i n g t h e r a d i a l e l e c t r o n d e n s i t y p r o f i l e i n a c y l i n d r i c a l plasmacolumn, t h e measurements a r e g e n e r a l l y i n t e r p r e t e d u s i n g a p l a n a r model.  The v a l i d i t y o f t h i s model has been i n v e s t i g a t e d i n a  s p e c i a l case b y comparing r e s u l t s f o r a c y l i n d r i c a l plasma cons i s t i n g o f two homogeneous r e g i o n s  and a s u r r o u n d i n g (61)  w i t h those i n t h e c o r r e s p o n d i n g p l a n a r case  .  g l a s s tube  In this  chapter,  a comprehensive comparison between t h e phase a n g l e s o f t h e s c a t t e r e d s i g n a l s computed f o r an Inhomogeneous plasma c y l i n d e r and  t h o s e f o r t h e c o r r e s p o n d i n g p l a n a r model i s p r e s e n t e d . The  e f f e c t s of the f o l l o w i n g f a c t o r s are considered:  .i)  R a d i u s o f t h e plasms, column r e l a t i v e t o 7\  li)  Maximum e l e c t r o n d e n s i t y (assumed t o o c c u r on  0  the a x i s o f t h e c y l i n d e r ) -  i i i ) Form o f t h e e l e c t r o n d e n s i t y v a r i a t i o n  6.1  iv)  S u r r o u n d i n g g l a s s tube  v)  Location of the point of observation  O u t l i n e o f t h e Problem and Method o f Computation The  Figure 6 . 1 .  two c o n f i g u r a t i o n s c o n s i d e r e d  are i l l u s t r a t e d i n  I n both of these, the i n c i d e n t f i e l d i s a plane  wave w i t h e l e c t r i c v e c t o r E  z  t r a v e l l i n g i n the negative  x direction.  133  (a)  I n c i d e n t Wave T r a n s m i t t e d Wave T (  e  -^o(x+r, )  R e f l e c t e d Wave > 3k (x-r, )  R  e  0  -2rr •2r  r  T, = f T , | e ^  (b)  F i g u r e 6.1  (a) C y l i n d r i c a l Configuration Configuration  (b) P l a n a r  T  134  For convenience, the reference  phase i s t a k e n t o be t h a t o f t h e  i n c i d e n t f i e l d a t x = r, ( n o t a t x - 0 as was done p r e v i o u s l y ) . The c y l i n d e r c o n s i s t s o f a g l a s s t u b e , w i t h and  Inner r a d i i of r  (  and x'  iy  outer  r e s p e c t i v e l y , c o n t a i n i n g a plasma  r e g i o n whose- e l e c t r o n d e n s i t y v a r i e s as N ( r ) = N„( 1 - f ( r / r ) 2  F i g u r e 6.2  )  E l e c t r o n D e n s i t y P r o f i l e and P o s s i b l e V a r i a t i o n s i n the y - 0 Plane  (6.1)  Permittivity  135  In the c o r r e s p o n d i n g p l a n a r  c o n f i g u r a t i o n we have  N(x) = N ( 1 - f ( x / r ) ) c  x  = N„( 1 - f ( - x / r * ) )  x > 0  (6.2a)  x 5 0  (6.2b)  We assume t h a t the c o l l i s i o n l o s s e s i n the plasma can be neg l e c t e d so t h a t I t may be represented m i t t i v i t y , 6y(r)  - 1 - N(r)/ft . c  as a r e g i o n of r e a l per-  A hypothetical electron density  p r o f i l e and the c o r r e s p o n d i n g p e r m i t t i v i t y v a r i a t i o n i n the y « 0 plane are shown i n F i g u r e 6.1-1  6.2.  B a s i s of Comparison A n a t u r a l choice  of q u a n t i t i e s f o r the p l a n a r  f i g u r a t i o n i s the argument o f the r e f l e c t i o n , ^ , argument of the t r a n s m i s s i o n  con-  and the  c o e f f i c i e n t , (?T I these a r e compared  with (P {oo) and ^ ( r , ), r e s p e c t i v e l y , f o r the c y l i n d r i c a l c o n f i g r  t  u r a t i o n , where <?Y(r) i s the phase o f the s c a t t e r e d f i e l d  i n the  ©• «= 0 d i r e c t i o n r e f e r r e d back t o r = r, by adding k ( r - r, ) 0  radians  and (p± ( r ) i s the phase of the d i f f r a c t e d f i e l d i n the  &• = rr d i r e c t i o n r e f e r r e d back t o r = r, by adding k ( r - r , ) 0  radians..  The v a l u e of (Pr(r)  a t r = «? I s used s i n c e (pr{r)  near the c y l i n d e r but becomes constant f o r l a r g e r , w h i l e i s most i n d i c a t i v e of the c h a r a c t e r i s t i c s o f a wave through the c y l i n d e r .  varies  <J> (.T ) t  transmitted  t  136  6.1-2  Method o f C a l c u l a t i o n The v a l u e s o f CP and (P were s p e c i a l l y computed by n  T  TCilbee u s i n g a wave t r a n s m i s s i o n  m a t r i x approach and homogeneous(3 7)  l a y e r approximation of the electron density p r o f i l e where e x a c t s o l u t i o n s were a v a i l a b l e ( £ (r) = a r r  merical  field  ), t h e n u -  c o e f f i c i e n t s i n the c y l i n d r i c a l  The phases were t h e n c a l c u l a t e d  Except  3 » 1 - ^ was used t o c a l c u -  i n t e g r a t i o n method o f S e c t i o n  l a t e the scattered  1 ,  .  case.  using  Cp (r)• = k ( r - 2 r , ) + A r g E 0  where ^ ( r ) denotes e i t h e r (py(r)  or  2  and E  (?±{T)  z  i s g i v e n by  i)  E q u a t i o n ( 2 . 1 b ) when <?(r) = (p (v),  ii)  E q u a t i o n ( 2 . 3 3 ) when ^'(r) = ?V(r), r = »  r  T  <oo  i i i ) E q u a t i o n ( 2 . 8 ) when 0>(r) = <?*(r), r < a> ' 6.2  Results  and D i s c u s s i o n  The c o m p u t a t i o n s were performed f o r t h e f o l l o w i n g functions  f ( r / r ) i n equation 2  a) f ( r / r ) z  (6.1):  = 3(r/r,e f  -2(r/r )  3  2  b) f ( r / r , ) - ( r / r * ) * c) f ( r / r * ) -  { T / x z f  d) f ( r / r j = 0 The forms o f t h e c o r r e s p o n d i n g e l e c t r o n d e n s i t y p r o f i l e s a r e shown i n F i g u r e 6 . 3 .  A l l have a z e r o s l o p e a t r = 0 .  I n ad-  d i t i o n (a) has a z e r o s l o p e a t r = r . V a r i a t i o n s o f t h e forms z  (b) and ( c ) a p p r o x i m a t e d e n s i t y p r o f i l e s measured i n l a b o r a t o r y plasmas^  6  2  and  (d) i s c o n s i d e r e d f o r c o m p a r i s o n p u r p o s e s .  137.  I M A  F i g u r e 6.3  6.2-1 i)  Electron Density P r o f i l e s  Considered  Comparison o f R e s u l t s f o r t h e F o u r P r o f i l e s .  ,  Comparison o f (P [po) and Y  The v a l u e s o f <Py{°°)  and (p f o r t h e f o u r p r o f i l e s n  c o n s i d e r e d a r e g i v e n i n T a b l e s 6.1a  t o 6 . I d . A stepped  line,  below which we have "agreement" between Q>t.{co) and (p , has been n  drawn i n each o f t h e s e t a b l e s .  Agreement i s t a k e n t o mean t h a t  the d i f f e r e n c e i s l e s s t h a n 6°.  We n o t e t h a t f o r a g i v e n v a l u e  o f r , t h e s i g n o f (P (») - ($) changes from n e g a t i v e t o p o s i t i v e A  r  n  138  as 2^e/Wo i n c r e a s e s from 0 . 2 5 t o 1 g i v i n g v e r y good agreement f o r intermediate values of N /N . c  The range o f N / N c  ensured  0  0  and r  depends on the t y p e o f p r o f i l e and i t appears t h a t t h e  c u t o f f r a d i u s , r , a t which N(r) = N c  The v a l u e s o f r r  c  = r  over •which agreement i s  x  A  c  c  i s an i m p o r t a n t  factor.  f o r the f i r s t t h r e e p r o f i l e s ( f o r P r o f i l e ( d ) ,  ) a r e g i v e n i n T a b l e s 6 . 2 a t o 6 . 2 c and stepped  lines  c o r r e s p o n d i n g t o those i n T a b l e s 6 . 1 a t o 6 . 1 c a r e drawn. I t i s seen t h a t agreement i s g e n e r a l l y o b t a i n e d f o r r except near N / N c  il)  0  c  > 0.5  = 1 .  Comparison o f <Pt(r, ) and Q  r  The v a l u e s o f <? (r, ) and (p a r e g i v e n i n T a b l e s 6 . 3 a t o t  6.3d.  r  Agreement between <?*(r, ) and Q> t o w i t h i n 6 ° i s c o n s i s 7  t e n t l y o b t a i n e d i n t h e . r e g i o n below the stepped l i n e s drawn i n these t a b l e s .  I t i s e v i d e n t t h a t t h e range- o f parameters over  w h i c h we have agreement i s s i g n i f i c a n t l y l a r g e r f o r P r o f i l e ( b ) than f o r the other three p r o f i l e s . I t appears t h a t the t r a n s m i s s i o n r e s u l t s a r e  sen-  s i t i v e t o t h e form o f e l e c t r o n d e n s i t y v a r i a t i o n ; the more t h e c o n c e n t r a t i o n n e a r t h e a x i s o f t h e c y l i n d e r and t h e h i g h e r t h e g r a d i e n t w i t h i n t h e c y l i n d e r , the p o o r e r the agreement between Q>t(r, ) and 0 . T  6.2-2  E f f e c t o f a G l a s s Boundary A l a b o r a t o r y plasma i s g e n e r a l l y e n c l o s e d b y a g l a s s  tube and we now c o n s i d e r t h e e f f e c t w h i c h t h i s h a s on the range  139  o f v a l i d i t y of t h e p l a n a r model. (i) r, - r  considered,  - ?i /2,  t h i c k n e s s e s of g l a s s , a r e  which i s t r a n s p a r e n t f o r the  £  z  Two  p l a n a r c o n f i g u r a t i o n and t h u s has no e f f e c t on Cp and (p n  (il)  r, - r  --- ^ / 4 ,  2  configurations.  T  and  w h i c h s h o u l d have a l a r g e e f f e c t f o r b o t h  The c o m p u t a t i o n s were c a r r i e d out f o r a l l t h e  p r o f i l e s u s i n g a r e l a t i v e p e r m i t t i v i t y of 4 . 8 4 Pyrex g l a s s ) f o r the g l a s s . shown i n T a b l e s 6 . 4  and  The r e s u l t s f o r P r o f i l e (b) a r e  6.5»  For the r e f l e c t e d f i e l d , r e g i o n i s e s s e n t i a l l y unchanged duced s h a r p l y f o r r , - r  (near t h a t o f  z  the previous  for r, - r  = ^ /4. e  agreement-  = ?i /2 b u t i s r e -  z  £  For the t r a n s m i t t e d  field,  t h e r e a r e no r e g i o n s of agreement between <Pt(r, ) and Q>r, a l though d i f f e r e n c e s w i t h i n about 30 ° a r e g e n e r a l l y observed over the p r e v i o u s agreement r e g i o n f o r r , - r r,  - r  2  = z^/4,  2  = ?) /2; f o r £  t h e r e i s l i t t l e resemblance between <Pt(r, ) and  (Pr.  6.2-3  E f f e c t o f Changing the P o i n t o f  Observation  The e f f e c t w h i c h t h e p o i n t o f o b s e r v a t i o n has on the r e s u l t s i s c o n v e n i e n t l y shown by p l o t t i n g ( CPr{v) - Q^ipo) ) and ( (pt ( r ) - <p (r, ) ) as a f u n c t i o n o f ( r - r ). t  (  2  Computations  were c a r r i e d out f o r a l l t h e p r e v i o u s p r o f i l e s and some p l o t s for  P r o f i l e (b) w i t h (1)  ( i i i ) r, - r  2  =  r, - r  2  = 0, ( i i ) r , - rz  are given i n F i g u r e s 6.4  = ?i /2 and  and 6 . 5  £  as an  illustration. For the r e f l e c t e d f i e l d s , the r e s u l t s g e n e r a l l y  7  Table 6.1  Comparison Between  0.25 0.5  <Pr{°°) -12.6 -5.2 <?«.  1.0  P («) r  <PR  2.0  <p H r  <PR  5.0  <PM)  0.50 -60.3 -56.5  (p (°o)  0.75  r  for a Plasma Cylinder and  0.85  -93.9 -104.5 -103.2 -121.3  0.90  0.95  1.00  ty  R  for the Planar Model  1.25  2.00  -113.7 -117.9 -134.8 -163.1 -138.2 -146.1 -177.1 . 141.7  -109.3 -130.1  4.00 170.9 113.9  -98.6 -95.3  174.2 175.9'  95.0 88.6  64.6 48.9  50.2 27.3  36.4 6.5  23.3 -13.9  87.4 88.9  -96.4 -95.0  97.5 95.5  28.6 21.2  -6.3 -20.2  -40.8 -65.6  -73.9 -112.9  161.8 90.3  163.2 -68.2  -137.5 -175.3  -94.1 -93.4  170.7 171.5  109.7 109.3  -68.5 -70.7  -163.8 -168.8  94.8 81.2  -10.3 -52.5  -23.2 -92.9  -177.3 -126.1  -168.7 146.8  1.25  2.00  4.00  1  -30.1 -114.2 -172.5 -87.5 -168.8 137.4  Profile (a) 0.25 0.5 %(<*>)  45.2 51.6  1.0 (?M) <P*.  10.5 13.8  2.0 %{<»)'.  -47.3 -45.6  <PR ...  5.0 <&.(»).  134.7 135.2  0.75  0.85  0.90  0.95  1.00  -35.4 -39.9  -48.3 -60.2  -54.3 -71.3  -60.1 -82.3  -91.2 -93.5  -57.0 -129.6 -164.8 -55.1 -134.2 -177.8  176.1 154.8  156.3 123.1  140.9 89.2  -1.0 -1.1  -124.8 -135.0  -172.0 164.5  0.50 0.9 5.4  -178.7 -178.2  -81.5 -85.9  132.4 . 53.1 -158.7 53.4 -161.0 133.2  -88.7 -120.4 -141.4 158.8 30.3 -23.7  -86.5- -137.3 -26.6 -133.9  -85.6 -118.8 140.1 -96.1 89.5 -141.1 -178.5 -143.2  76.1 ; -64.9 139.1 -87.3 -87.3 -72.6 89.7 -130.4 -131.7 73.1  Profile (b)  -44.5 -58.2  -87.7 -133.1  0.25 0.5  <Pr<po) 68.7 75.0 45.8 49.0:  1.0  0.50  0.75  0.85  0.90  0.95  1.00  1.25  2.00  4.00  31.4 36.6  -0.3 -0.5  -12.2 -17.6  -18.1 -27.0  -23.8 -37.2  -29.3 -48.4  -54.4 -107.9  -79.3 -7.0  -12.9 -51.6  -3.2 -0.5  -51.3 -51.5  -73.6 -77.6  -86.1 -94.2  -99.8 -115.6  -114.7 -147.1  -97.1 -142.0  -59.7 -104.0  -137.6 -13.4  2.0  <?(») <Pn  9.4 11.0  -64.6 -63.3  -158.2 -159.4  144.1 H0.3  105.7 100.0  57.6 47.3  -2.0 -35.0  -115.2 -21.3  -73.9 -119.0  -71.1 -116.9  5.0  <p,W  -78.3 -77.7  87.1 88.5  -176.2 -176.8  33.0 32.7  -57.5 -59.0  -167.8 -173.7  6.2 -32.0  -116.6 -12.1  -110.9 -163.1  -33.8 -67.3  r  P r o f i l e (c)  0.5  0.25  0.50  0.75  0.85  0.90  0.95  1.00  1.25  2.00  4.00  114.2 120.0  84.6 90.0  57.9 60.0  47.1 45.9  41.7 37.7  36.2 28.4  13.6 17.7  3.6 -65.2  42.5 14.6  9.7 -41.5  117.1 120.0  87.1 90.0  58.1 60.0  45.0 45.6  37.8 36.9  29.9 26.0  6.7 9.0  8.0 -43.6  -8.2 -57.6  52.9 6.4  (P H  118.5 120.0  88.6 90.0  58.8 60.0  44.8 45.6  36.7 36.9  27.1 . 25.8  3.7 4.5  26.7 -10.5  15.8 -26.9  -24.4 -76.9  GVH  119.4 120.0  89.4 90.0  59.5 60.0  45.1 45.6  36.5 36.9  25.9 25.8  1.2 1.8  -0.6 -76.6  -41.1 63.1  72.8 32.0  <p (») r  <PR  1.0 (p (oo) r  (PR  2.0 5.0  r  (PR  P r o f i l e (d)  H  142  T a b l e 6.2  R a d i u s i n Wavelengths a t w h i c h l\ (r) = 2f T  c  0.25  0.50  0.75  0.85  0.90  0.95  0.5  0o337  O.250  O.I63  0.122  0.098  0.068  1.0  0.674  0.500  O.326  0.244  0.196  0.135  2.0  1.3^7  1.000  0.653  0.489  0.392  0.271  5.0  3.368  2.500  1.632  1.222  0.979  0.677  P r o f i l e (a)  0.5  0.433  0.35^  0.250  0.19^  0.158  0.112  1.0  0.866  0.707  0.500  0.387  O.316  0.224  2.0  I.732  1.414  1.000  0.775  O.632  0.447  5.0  4.330  3.536  2.500  1.936  1.581  1.118  P r o f i l e (b)  0.5  0.465  0.420  0.35^  0.311  0.281  O.236  1.0  0.931  0.841  0.707  0.622  0.562  0.473  2.0  I.861  1.682  1.414  1.245  1.125  0.9^6  5.0  4.653  4.204  3.536  3.112  2.812  2.364  P r o f i l e (c)  my  Table 6 . 3  Comparison Between <P (r, ) f o r a P l a s m a C y l i n d e r and cp f o r t h e P l a n a r Model t  T  1 . 0 0  0.5  <P (r,) t  A  ,  1 . 0  2.0  <P*(r,) <Pr ,  5.0  36,4  36.2  129.9  116.7  -136.8  - 1 4 7 . 7  -100.2  <P (r,) t  1.25  1.05  2.00  4 . 0 0  3 3 . 7  29.8  23.3  1 1 . 9  9 3 . 1  7 3 . 6  51.7  23.9  1 3 4 . 3  90.7  172.1 -177.5  -122.6  1.50  1 ^ 3 . 0  39.2  101.2  47.4  -I66.O  91.^  107.7 159.5  8 1 . 6  -8.7  -84.6  113.7  - 0 . 4  -76.2  -158,4  171.7  72.6  171.9  -15.7  1 4 1 , 6  - 1 2 4 . 7  9 3 . 7  177.1  - 1 1 . 6  1 4 4 . 0  -123.3  1.50 ,  2,00  4 . 0 0  - 1 4 1 . 2  9 * K 8  P r o f i l e (a)  1 . 0 0  0.5  <&(r,)  - 4 5 . 9  176.8 36.4  1 . 0  - 0 . 7  2.0  5.0  <P*(r,) <Pt(r,)  1 2 8 . 9  -67.8  8 1 . 0  1 6 3 . 2  I69.6  63.3  46.2  21.7  9 9 . 4  68.9  31.8  1 4 1 . 3  65.I  - 1 2 1 . 3 , -I65.6  -32.0  -0.5  - 0 . 3  76.5  9 0 . 6  I65.8  -85.1 -70.9  63.I  1.25  1.05  -113.7 •  126.4  -167.5 2 4 . 0  1 2 8 . 8  23.6  137.7 139.1  -122.1  P r o f i l e (b)  - 1 2 4 . 7  I36.O - 8 8 c 9 - 8 8 . 6  63.4 1 2 4 . 0  126.8  - 4 1 . 3  - 4 4 . 0  - 4 1 . 8  - 4 3 . 2  144  1.00  .1.05  -127.3  -345.3  n/>io 0.5  -138.3 103.6 122.5  1.0  2.0  <?t(r,)  5.0  1.25  1.50  143.0  103.2 121.2  69.3 83.O  —111.8  179.6  -150.2  162.1  67.6  -43.6  74.6  -170.1 -125.0  111.9  -169.5 -122.1  -102.6  109.5  -90.4  4.00 30.4 38.4  85.6  166.1  76.6  115.5  -26.4  112.2  -29.0  152.2 153.2  102.3  108.4  -52.1  -122.4  -IO3.3 83.O  . -111.3  2.00  98.8  77.9  106.9  23.0 22.7  Profile (c)  70  0.5 1.0  2.0  5.o  1.05  1.25  1.50  -W.7  -107.8 -85.2  -137.2 -155.2  -179.1 148.6  119.6  93.3 -81.0  -159.4-  65.5 46.4  -28.9 -59.^  -132.9  -131.2  63.3  -112.1  67.9  85.3 79.5  53.3  -49.0  -148.8  71.0 82.7  -85-5  <Pt(r.) Or  /  1.00  -72.3  <P*(r.)  -  -59.3 -88.1 •  -13^.6  -79.3  -166.6  P r o f i l e (d)  -108.7  2.00  104.6 -147.6 63.I  -34.2 -26.9  4.00 49.1  48.5 109.0  96.4  -17^.3 -I66.9  ,129.7  133.0  Table 6.4- Comparison Between <P(«v and (pn for a Glass-Enclosed Plasma Cylinder, Profile (b) r  0.25 0.5 <&.(«>)  58.0 51.6  1.0  10.8 13.8  (? (oo) r  0.75  0.85  0.90  0.95  1.00  -42.3 -39.9  -62.8 -60.2  -72.7 -71.3  -82.4 -82.3  -65.2 -93.5  -57.7 -130.6 -164.9 -55.1 -134.2 -177.8  176.9 154.8  158.6 123.1  136.7 89.2 137.2 89.5  0.50 8.2 5.4  2.0 #.(«>)  -47.5 -178.5 -4.5.6 -178.2  -0.8 -1.1  -82.9 -85.9  5.0 %(°°)  134.8 135.2  132.6 133.2  53.5 53.4  -158.9 -161.0 (i)  0.75  0.85  0.90  0.95  -152.3 171.3  -153.6 166.3  -154.1 163.2  -154.6 159.9  -137.5 -168.6  -147.6 -179.3  1.0  <Z>(«>)  -171.3 -177.2  178.2 167.8  168.0 170.0  18.8 7.5  2.0 5.0  <?(«>) r  2.00  4.00  -129.4 -178.6 158.8 -141.4  164.5 -58.2  74.4 -23.7  -116.6 -133.9  -171.0 -26.6  -109.7 -141.1  -153.6 -178.5  -86.5 -143.2  137.1 -123.6 -108.1 89.7 -130.4 -131.7  50.9 -133.1  r, -  0.50  ?rH r  76.9 , -66.0 -72.6 73.1  0.25 0.5  (PH  -126.2 -172.2. -135.0 164.5  1.25  162.9 -166.2 10.3 127.9 179.0 179.6  •175.9 158.2  -129.9 -139.3 -170.3 -126.8 -129.4 -168.1  170.7 77.7 (ii)  1.00  1.25  2.00  • 4.00  -125.0 171.2  -152.3 148.9  -176.1 -163.7 -137.3 -178.8 -144.9 -95.1 -139.2 -158.7 ."178.0 —156.8  -164.8 168.9  177.5 126.8  -155.4 -154.3 •156.1 156.3  -171.8 -160.7 -67.3 -158.1  -169.1 179.0  -170.0 -178.4 -174.8 -178.0 -162.4 -162.8 -157.8 179.6  r, - r = ^/4 2  -178.9 -148.3 -179.4 -163.4 143.7 -155.9  -143.2 -157.0  146 Comparison Between (P {T, ) and q> f o r a G l a s s E n c l o s e d Plasma C y l i n d e r , P r o f i l e ( b )  T a b l e 6.5  t  <P*(r,)  0.5 1.0  1.00  1.05  -85.0 176.8  165.8 . 128.9  -51.6*  -0.7  <P*Cr)  2.0  (  -35.2 -0.5  -40.8  5.0  -G.3  (PT  0  0.5 1.0  <?*(r.) <Pr >  2.0  Mr,)(PT  5.0  Mr,\ <PT  :  1.50  1.25 56.0  53.7  54.7 99.4  2.00  49.1 68.9  4.00  36.I  31.8  64.?  -136.4 -II3.7  -178.8 -I67.5  136.0  63.4  -81.7  109.0 128.8  15o3 23.6  -81.2 -88.6  137.5 126.8  133.6  132.8 139.1  -122.1  -98.2  -63.3  -41.8  -£2.2 -43.2  4.00  -72.8 -32.0.  -70.9 I69.6  (1)  n/*  T  r,  -  r  ?i /2  =  2  129.7  a  I.25  1.50  2.00  114.3  117.8 -85.7  107.7 -98.9  98.0 -121.0  -68.9 45.1  -80.5  -90.4 36.O  -69.I  -37.8 -101.I  50.7 65.3  83.3 -15.5  84.3  -140.3  53.7 -130.9  6.6 90.6  83.2-73.5  -80.5  133.5  -96.4  88.7 76.1  -18.0 114.1  -118.9  65.1  -159.6 -130.2  1.00  1.05  -27.2 -65.8  112.0 -77.7  -I6.8 65.7  (ii)  -129.8 -21.9  r, - r  2  - ^/4  -66.8  113.1  150  250  _100 I 0  —  -»  0.5  1  — .  L  1  1.0 1.5 (r - r , ) / to iii)  r, - r = z  ?\ /4 S  2.0  j  2.5  151 i n d i c a t e t h a t where agreement between (p {a>) and r  i s obtained,  l i t t l e v a r i a t i o n o f <?,-(r) o c c u r s w i t h r a d i u s and v i c e v e r s a w i t h one  exception*.  I t may t h e r e f o r e be p o s s i b l e t o a s c e r t a i n  whether t h e p l a n a r model i s a p p l i c a b l e b y m e a s u r i n g t h e v a r i a t i o n o f (?V(r) w i t h r a d i u s .  There a l s o appears t o be a c o r -  r e l a t i o n between t h e v a r i a t i o n o f C? (r) w i t h r a d i u s n e a r t h e t  c y l i n d e r and t h e q u a l i t y o f agreement between (Pt(r) improved agreement i s g e n e r a l l y observed w i t h  and L? ;: T  decreasing  /(?'(*»,)/ . 6.3  Summary I t h a s been shown t h a t a p l a n a r model i n t e r p r e t a t i o n  of r e f l e c t e d - f i e l d phase measurements on a c y l i n d r i c a l plasma column " i s g e n e r a l l y v a l i d i f r  c  i s g r e a t e r t h a n about 0.5/}*,  independent o f t h e a c t u a l e l e c t r o n d e n s i t y v a r i a t i o n .  For the  t r a n s m i t t e d f i e l d , t h e range o f v a l i d i t y depends on t h e v a r i a t i o n ; i n p a r t i c u l a r , a h i g h e l e c t r o n c o n c e n t r a t i o n around t h e a x i s o r a l a r g e g r a d i e n t r e d u c e s t h e range o f r a d i u s and Ne/Uo o v e r w h i c h t h e p l a n a r model i s v a l i d . Although a half-wavelength  g l a s s boundary has l i t t l e  e f f e c t on t h e r e f l e c t e d f i e l d , i t s e f f e c t on t h e t r a n s m i t t e d f i e l d i s quite noticeable.  The e f f e c t o f a  quarter-wavelength  g l a s s boundary i s l a r g e i n b o t h cases and i n f a c t l t appears  * When N c / N o > 1 and t h e r e i s no g l a s s boundary, t h e v a r i a t i o n of (pr{T) w i t h r a d i u s i s s m a l l b u t (pt(<») and <pR do n o t agree  152  t h a t the p l a n a r model i s no l o n g e r v a l i d f o r measurements. field  I t should  be p o i n t e d  transmission  out t h a t i f the  scattered-  c o e f f i c i e n t s were d e t e r m i n e d f o r the c y l i n d r i c a l c o n f i g -  u r a t i o n , i t would be p o s s i b l e t o a c c o u n t f o r the e f f e c t of  the  g l a s s by the f o l l o w i n g p r o c e d u r e : i)  Compute z\  ii)  Determine  s o l v i n g equation  from the c o e f f i c i e n t s ( t h e i n p u t impedance a t r = T ) by Z  (2.15) f o r z V  i n terms o f Z™ w i t h m =  1.  i i i ) C a l c u l a t e the s c a t t e r e d - f i e l d c o e f f i c i e n t s and then the s c a t t e r e d f i e l d The  o f the plasma c y l i n d e r .  l o c a t i o n o f the p o i n t o f o b s e r v a t i o n  i s unimpor-  t a n t f o r the r e f l e c t e d f i e l d when t h e p l a n a r model i s : a p p l i c a b l e F o r the t r a n s m i t t e d f i e l d , t h e measurements must be t a k e n as n e a r as p o s s i b l e t o the c y l i n d e r .  153  7. The k n o w l e d g e on  the  subject  the  scattered  mittivity  main c o n t r i b u t i o n s  the  On  DISCUSSION AND  may  be  theoretical  field  of  CONCLUSIONS ' of  t h i s work t o  summarized  side,  a cylinder  four  and  methods were e s s e n t i a l l y i n d e p e n d e n t and computed r e s u l t s  provided  a c h e c k on  follows.  methods o f  w i t h an  v a r i a t i o n have been a p p l i e d  as  evaluated.  each of  them.  ( i ) an  of  the  of  solution  of  the  cylinder  by  a layered  tations  wave e q u a t i o n and  were c a r r i e d  with l i n e a r l y - v a r y i n g obtained was  by  easy to  computation time, ical  integration The  derivation  cylinder  and  permittivity impedances a t felt  that  of  of  (rather  tered  field  The  type,  required  n e c e s s a r y t o use  of  with angle,  the  short  the  numer-  were  required.  bounds r e q u i r e d  of  the the  example),  normalized was  behaviour of  guided  by  the  the  mode-  established.  magnitude of  a  the  between changes i n  cylinder  study of  variation for  were  impedance b e h a v i o u r w i t h i n  changes i n the  surface  compushells  a relatively  variation  and  the  h o m o g e n e o u s - s h e l l method  relationship  than the  series  "Exact" r e s u l t s  a qualitative  the  a power  latter  truncation-error  the  a quantitative  ances  the  permittivities.  a p p l y and  types  approximation  method when v e r y a c c u r a t e r e s u l t s  investigation  Two  u s i n g homogeneous s h e l l s and  a l t h o u g h i t was  detailed  is  For  numerical integration.  generally  amongst  ( i i ) approximation of  structure. out  use  per-  The  hence agreement  a p p r o x i m a t i o n were c o n s i d e r e d , n a m e l y the  obtaining  arbitrary radial  of  p e r m i t t i v i t y which allowed  existing  It  these the  imped-  scat-  qualitative  15k  r e s u l t s d e r i v e d , may prove u s e f u l i n c h a r a c t e r i z i n g v a r i o u s typ.es o f p e r m i t t i v i t y v a r i a t i o n .  J  On t h e e x p e r i m e n t a l s i d e , a newr approach ha,s been developed  which a l l o w s t h e complete p l a h e - w a v e . s c a t t e r e d  to be determined incidence.  field  from measurements u s i n g c y l i n d r i c a l - w a v e  The e x p e r i m e n t a l system proved  t o be compact and  e f f i c i e n t , and t h e a c c u r a c y o f t h e r e s u l t s o b t a i n e d i n t h i s manner was comparable w i t h t h a t o b t a i n e d from measurements w i t h plane-wave i n c i d e n c e .  I t s h o u l d be p o i n t e d o u t t h a t t h e method  " i s a p p l i c a b l e t o r a d i a l l y inhomogeneous ( o r homogeneous c i r c u l a r ) cylinders only.  (An e x t e n s i o n o f t h e t h e o r y t o i n c l u d e t h e  a n i s o t r o p i c case c o r r e s p o n d i n g t o a plasma c y l i n d e r w i t h a cons t a n t . a x i a l magnetic f i e l d i s p o s s i b l e by i n c l u d i n g s i n e terms when a p p l y i n g the P o u r i e r l e a s t - s q u a r e s f i t used t o c a l c u l a t e t h e scattered f i e l d  coefficients.)  j  Inhomogeneous c y l i n d e r s used i n t h e experiments constructed using a simple a r t i f i c i a l d i e l e c t r i c .  were  An a c c u r a t e  a p p r o x i m a t i o n t o the assumed p e r m i t t i v i t y v a r i a t i o n i s a c h i e v e d "as"shown'by"the' good"agreement o f t h e e x p e r i m e n t a l r e s u l t s w i t h predicted values. for  ( C y l i n d e r s o f t h i s type s h o u l d prove u s e f u l  e x p e r i m e n t a l I n v e s t i g a t i o n s i n cases where a t h e o r e t i c a l  1  i  -analysis i s not possible.) Some of. t h e t h e o r e t i c a l a s p e c t s o f t h e work have been a p p l i e d t o s t u d y t h e v a l i d i t y o f the o f t e n - u s e d p l a n a r model for  i n t e r p r e t i n g phase measurements on c y l i n d r i c a l plasma  columns.  F o r r e f l e c t e d f i e l d s i t has been found  that the planar  155  model i s v a l i d i f the c u t o f f - r a d i u s i n the column i n g r e a t e r than h a l f a free-space  w a v e l e n g t h , however, no s p e c i f i c  i o n c o u l d he found f o r t r a n s m i t t e d f i e l d s . s t r o n g l y a f f e c t e d by a d i e l e c t r i c boundary. u l a r l y t r u e f o r the t r a n s m i t t e d  The r e s u l t s may be This i s  partic-  f i e l d s and i n one case no  resemblance c o u l d be seen between the r e s u l t s f o r the c a l and p l a n a r c o n f i g u r a t i o n s .  criter-  cylindri-  A p o s s i b l e method o f d e t e r m i n i n g  whether the p l a n a r model i s v a l i d i n a p a r t i c u l a r s i t u a t i o n i s suggested.  156  APPENDIX  A  F i e l d s i n a Cylindrically„Stratified Medium We c o n s i d e r the s o l u t i o n o f Maxwell's  for  vxH  - j CJ £(r) E  V X E  =  W  equations (A.la)  / t H  (A.lb)  f i e l d s which a r e independent o f the z - c o o r d i n a t e  medium i n which the p e r m e a b i l i t y i s constant  in a  and the p e r m i t -  t i v i t y v a r i e s smoothly w i t h r a d i u s . Taking  the c u r l o f equations  ( A . l a ) and ( A . l b )  gives v x v x H  = j w v x 6 ( r )  E  *= j 6 j £ ( r ) v x E  +36jv£(r) x E  - k* £ ( r ) H + 3 co E\T) r  V.XVXE  = -3 ( j y t v x H  = k  2  r X E  € (r) y  (A.2a)  E  (A.2b)  where r = u n i t v e c t o r i n the r - d i r e c t i o n . The v e c t o r  identity  v x v x V r = v ( v - V ) - v*V i s now a p p l i e d t o the l e f t hand s i d e o f equations and  (A.2b).  z-components  We o b t a i n the f o l l o w i n g equations o f E and H:  (A,2a)  f o r the  + k*  V  2  E, + k  2  E  -t  r  H  = 0  2  (A.3a)  = 0  z  S i n c e t h e s e e q u a t i o n s a r e uncoupled,  (A.3b) t h e f i e l d may be ex-  p r e s s e d as t h e sum o f two p a r t i a l f i e l d s , one w i t h H the other w i t h E A.l  H  = 0  z  2  2  =• 0,  = 0.  ( Parallel Polarization )  We a p p l y t h e s e p a r a t i o n o f v a r i a b l e s to equation (A.3h).  technique  Letting  E,(r,e-) = R ( r ) -G-(e) we o b t a i n //  r  z  /  R  R  R  R  //  - + r - +r  k  z  z  C + y  -0-  =0  and t h e r e f o r e (A.4a)  -r = - n' -e-v, f/  R„ + —  2  +•( k  n £ - - —.•) R„ = 0 r  (A.4b)  where -n = s e p a r a t i o n c o n s t a n t . Letting  k r , e q u a t i o n (A.4b) may be w r i t t e n i n t h e form d d?  Id 2  5 dT  n*  (A.5)  158  The  s o l u t i o n , of e q u a t i o n  (A.4a) may  be  expressed  as -Q- N  S i n c e the f i e l d  a ^ cos(nG-) •+• b^  sln(nG-)  must be unchanged by a 2 T change i n  n must take on o n l y I n t e g r a l v a l u e s . t i o n of equation  A c l o s e d form  (A.4b) e x i s t s f o r o n l y a few  solu-  types of p e r -  m i t t i v i t y v a r i a t i o n s , most n o t a b l y when cTj-(r) - a r . b  I n t h i s case, the s o l u t i o n s are^ ^ ^ /2k  £  I  \  /a"  b+2r  b' + 2  \  2.  ]  v  /  2 n  =—  (A.6)  • b + 2  where ;£ denotes a B e s s e l f u n c t i o n . v  Other cases i n which s o l u t i o n s e x i s t may converting equation  be determined  (A.4b) t o the standard  IK(u) + f ( u ) U(u)  =  by  form  0  (A.7) (63)  and  c o n s u l t i n g the t a b l e g i v e n by R i c h a r d s  s o l u t i o n s of e q u a t i o n (A.7) The  which  lists  f o r various functions f ( u ) .  g e n e r a l s o l u t i o n of e q u a t i o n  (A.3b) i s of the  form  CO E (r,9-) z  =  »  X.  --co  ( a * cos(nO) + b , sin(nO) T  ( c.„ U where U „  (Ac4b).  e  and V  M e  are two  w e  ( r ) + d„ V  independent  n e  ) (r)  )  s o l u t i o n s of e q u a t i o n  159 A.2  E x= 0  ( Perpendicular Polarization )  By a p p l y i n g t h e same p r o c e d u r e t o e q u a t i o n (A.3a) we f i n d t h e g e n e r a l s o l u t i o n CO  H (r,6) =  Y-  2  ( * &  cos(n©-) +  sln(n©-) )  « ( c>, U ^ t r ) +. d„ V„ (r) ) h  where V  and V„j, a r e i n d e p e n d e n t s o l u t i o n s o f  vh  , .1 +  (  _  .  r  ) c  R  M  +  ^TSy -  (  r  r  o r , i n terms o f t h e v a r i a b l e  l  d  n  )• R „ = 0  (A.8)  *  f  § 3 £,(?A)— d) + £„( JA) -n' ^(?A) d? %  T-X  R,,($)  =  0  (A.9)  As i n t h e case H  2  = 0, c l o s e d form s o l u t i o n s o f  equation ( A . 8 ) r a r e l y e x i s t .  ^ V \  b + 2  '  v  a r  P o r F: (r) •= a r r  y  b  these are  * = ( 4 n * +' b* )/( b + 2 ) (A.10)  ;  l6o  APPENDIX B Power S e r i e s S o l u t i o n I n t h e Case o f a L i n e a r P e r m i t t i v i t y Variation  , C l o s e d form s o l u t i o n s  o f t h e r a d i a l wave e q u a t i o n  do n o t e x i s t f o r a l i n e a r v a r i a t i o n o f p e r m i t t i v i t y £ (r) = s r + to r  I n t h i s appendix s o l u t i o n s  i n t h e form o f a power s e r i e s S o m e been d e r i v e d by P e l n s t e i n ^ Bel  are obtained  o f t h e r e s u l t s have  ^ ,  Parallel Polarization Substituting  £,,(?A) = s  5/k. + t i n t o e q u a t i o n (A.5)  we o b t a i n  (B.l)  where s, = s/k Assume t h e power s e r i e s  solution CO  The i n d i c l a l  equation  (36)  (B.2) i s found t o be c ( c - l ) + c » n* = 0  which y i e l d s n  and t h e r e c u r r e n c e r e l a t i o n i s  (B.3)  l6l  Choosing the p o s i t i v e s i g n i n e q u a t i o n  (B„3) g i v e s the.  s o l u t i o n which i s a n a l y t i c a t the o r i g i n .  The r e c u r r e n c e  r e l a t i o n becomes  a  1  ——-—— ( t a, o -H s,a, o )  =  i ( 2n+i )  1  l m m d  1  (B.5)  3  and thus the s o l u t i o n i s g i v e n by  J  2-(2n+2)  —  ?  3-(2n+3)  (B.6)  t* 2-4-(2n+2)  If  (2n+4)  + ••  T h i s i s e s s e n t i a l l y the r e s u l t o b t a i n e d by P e l n s t e i n . ( B . 3 ) i s chosen,  I f the negative sign i n equation  we f i n d t h a t a zero appears i n t h e denominator o f e q u a t i o n ( B . 4 ) when I - 2 n , making a „ I n f i n i t e .  A second s o l u t i o n  2  may  be obtained i n t h e f o r m ^ ^ 3 6  V, = A U , l n ( Y )  CO X*T-  +  • ^  (  B  *  7  )  l=o The c o n s t a n t A i s chosen such t h a t b n becomes i n d e t e r m i n 2  ate ( o f the form 0/0). choosing b  2 7 J  The v a l u e o f b „ i s now a r b i t r a r y ; ; 2  = 0 adds a term b , 2T  TJ.,, t o the  \>  2yi  = 0 solution.  We f i n d  1 A  . .  -  s  2 n  B.  0  (  t  b  2  n  -  2  + s,b  2 n  „  3  )  (B.8)  162  and t h e r e c u r r e n c e  relations  i 1 ( 2n-i )  ( * ^1-2  +  s  'i-3 ^  i < 2n  b  (Bo9) 1  = - — ( "t b, o + s, i ( i~2n ) 1  4  « + 2 A '(l-n)«a ,  1  1  i > 2n  When n = 0, A i s undetermined by e q u a t i o n  1  ^  o„ ) n  (B.8) and may  be chosen a r b i t r a r i l y . v.-  .The s e r i e s i n e q u a t i o n s  ( B . 6 ) and ( B . 7 ) converge  f o r a l l f i n i t e v a l u e s o f ^5. B.2  Perpendicular Polarization S u b s t i t u t i n g £ ( J/k) r  = s,  + t i n t o e q u a t i o n (A.8)  + s, f  +1  we o b t a i n d* d?  1 + ( _ 2  3  s, s, J +"t  a  ) —  d?  »  n*  — (BclO)  I  Comparing t h i s e q u a t i o n w i t h e q u a t i o n  ( B . l ) i t i s seen t h a t  an a d d i t i o n a l r e g u l a r s i n g u l a r i t y o c c u r s a t ~§ -  t / s , and  t h e r e f o r e a power s e r i e s s o l u t i o n v a l i d f o r a l l v a l u e s o f (.7 )  5" i s n o t p o s s i b l e .  Felnstein  , who has i n v e s t i g a t e d  t h i s problem i n some d e t a i l , g i v e s t h e s o l u t i o n about t h e s i n g u l a r i t y a t f = «• t / s , .  His analysis l s not general,  however, s i n c e h i s p o s s i b l e : v a l u e s o f s, and t a r e c o n s t r a i n e d such t h a t  163  t  B , f +  1  <  J  <  ?,  s, f + t = 1 ;  These c o n s t r a i n t s f o l l o w e d  from t h e f a c t t h a t he was  i n g w i t h a model of a meteor t r a i l h a v i n g a z e r o density  deal-  electron  a t i t s o u t s i d e boundary. The s o l u t i o n o f e q u a t i o n ( B . 1 0 ) i n t h e s p e c i a l  cases s, = 0 and t = 0 i s g i v e n by e q u a t i o n (A.6) w i t h b = 0 and b = 1 r e s p e c t i v e l y .  W i t h t h e r e s t r i c t i o n s s, =£0, t ^ 0 ,  a more c o n v e n i e n t form o f e q u a t i o n ( B . 1 0 ) i s o b t a i n e d by t h e change o f v a r i a b l e d —, 2  +  I  fi-s,~%/t. d  The r e s u l t i n g e q u a t i o n i s  • + a(fi  + 1)  «»  n  2  R^ = 0  (B.lla)  or d*  t.fi  d  R„ = 0  d/2  (B.llb)  where cc = t / s f 3  The s i n g u l a r i t i e s a r e a t  = 0 and fi = - 1 .  We now  consider  power s e r i e s s o l u t i o n s about t h e s e two s i n g u l a r i t i e s and about some o t h e r p o i n t , s a y  fi-fic.  where  fi  0  i s outside  t h e range -1 t o 0 , E x p a n s i o n about fi - 0 Assume a power s e r i e s s o l u t i o n o f t h e form R ; = / f  0  l  (B.12)  164  The l n d l c i a l  e q u a t i o n g i v e s t h e v a l u e s o f c, c =  tn  The r e c u r r e n c e r e l a t i o n l s [ ( c + l - l ) ' C c + i - 2 ) - n .] & 2  (c+i+n)•(c+i~n)  ) +cc( a  + i  -  2 a  2  j_„3  +  a  ±mml  i - 4 ) (B.1'3)  When c - n, e q u a t i o n (B.13)  becomes  ( l + 2 n i - 3 i - 3n+2 ) & z  a 1• =  i*(2n+i) /  +Qi  (-'- j__2  +  a  2  a  i-3  +  a  ^  i - 4^ (B.14)  and t h e s o l u t i o n may be w r i t t e n as n 1 + _ / 3 2n+l  1 ( 2-(2n+l)  n • • ©  2n+2 (B.15)  As i n S e c t i o n B . l , t h e second s o l u t i o n cannot be o b t a i n e d by s i m p l y l e t t i n g c = ~ n . A s o l u t i o n o f t h e form ( B . 7 ) i n t h e v a r i a b l e /3 must be u s e d .  The r e c u r r e n c e r e l a t i o n i n  t h i s case i s found t o be ( i  b; = l-(2n-i)  )  - 2 n i - 31 + 3n + 2 ) b i-l  -hcc{  +  2  b  i-3  +  b  i-4 ^  i  < 2n  165  ( i i-(i-2n)£  2  - 2ni - 31 + 3n + 2 )  +^ (  +  2 t ,  + Af2(i-n) a  i-3  +  b  b ^  i-4.)  + ( 2(i-n) - 3 ) a ^ ^ J  1 - 2 n  1 > 2n (B.l6)  W  h  6  r  e  A  2naT t  =  (  3  n  °  i s chosen so t h a t b  2  )  b  2n-l " *  (  2n-2 + 2 n ~  b  2 b  i s arbitrary.  2 n  ^k  + 3  )  ]  When, n = 0, A i s  arbitrary. Expansion  about /3 = -1 With the f u r t h e r change o f v a r i a b l e ,  t - /3 + 1,  e q u a t i o n ( B . l l b ) becomes  y (-2T-  Proceeding  l )  2  — dtf  1) — + ct d 2T  + 2  .l)  2 ,  - n  2  y (B.l?)  e x a c t l y as b e f o r e w i t h  t = o  we f i n d c = 0 o r 2.  Again, i n order t o o b t a i n two s o l u t i o n s ,  the forms = t  z  a  r  £  (B.18a) (B,18b)  must be used.  The r e c u r r e n c e r e l a t i o n s are  (i+1)»(21-1) a i--(i+2-) ;  - <*( a  i  =  3  -  2  i - ] L  A  - ( i - ( i - l ) - n* ) a  ^  4 „ ^  +  1  A I  ^2  -  ^  i  -  2  )  (B.19)  166 b; =  (1-1)-(21-5) b ^ 1-(1-2)  "  0 6  ( i„3 ~ b  2 b  ~ ( ( 1 - 2 ) . (1-3) - n  i~4  - A ( 2(1-1) a  1 = > 2  +  2  ) b  l i - a  .  )  - (41-7)  * (21-5)  a ^  (B.20) where A =  l s chosen so that b?. I s a r b i t r a r y *  Expansion about Letting  H> =/3 ~ fi i n equation ( B o l l b ) we 0  ' '+*(  d  d  dT  d#  (  +1)  2  obtain  - n* (y + + D  which may be w r i t t e n as'  d  (  A  f +  2  2T + A,  y  d  + A.) —  +< * +  ~  dy + B* ^*'+  B  2f + B  X + 2  5  3  a  dy B, 2C" + B0  - C, 2T - 0< R-« = (B.21)  where A  2  A, A,  c,  = 3A+1  - 4,(3 A  B* + 2)  = 2 a(2  +1)  B  3  = A ( A + 1)  B  2  = n*  B,  * 2 a A ( 2 A ' + 3 A + 1)  B„  =  l  - a*( A  + 1)  Upon s u b s t i t u t i n g  = * (6/?* + 6/?.  + 1)  2 A + 1)  167  we f i n d c = 0 or 1,  When c = 0, "both B . and a, a r e a r b i C  When c = 1,  t r a r y and hence we have the complete s o l u t i o n .  the s e r i e s i s t h e same as would be obtained f o r c - 0 and the c h o i c e o f constants a. = 0,  a, a r b i t r a r y , .  0  two Independent s o l u t i o n s ,  We may w r i t e  each w i t h one a r b i t r a r y  by l e t t i n g a, = 0 when c = 0.  constant  The r e s u l t i s (B.22a)  v -x r  where the f o l l o w i n g  z b^ y  (B.22b)  recurrence r e l a t i o n s  (1 A, - 2A, + jS ) a  (i-l) a; =  k  i.(l-l)  0  apply:  0  + [ (1-2). ( i A  i - ] L  - 3 A + 1) + B  2  Z  .+ [ ( 1 - 3 X 1 - 4 ) + B, - 0, ] a _ ±  + B  *  a  i-4  +  B  3 i«5 a  +  B  *  a  -C.J  e  3  i„6 1 > (B.23)  i - ( i A , - A, +.0o ) b i - ,  b; = -  A  0  1.(1+1)  + [ ( i - l ) ( i - A - 2A + 1) z  + B  2  + [ ( i - 2 ) - ( i - 3 ) + B, - 0, J b _ ±  +B  2  b  ±  ^ + B  3  b _ ±  5  -+B, b  i  -  - C J  0  3  6  i (B.24)  >  168  B.2-1  Range o f Convergence. The range o f convergence o f t h e t h r e e e x p a n s i o n s  i s e a s i l y d e t e r m i n e d by o b s e r v i n g t h e d i s t a n c e f r o m t h e p o i n t about w h i c h t h e e x p a n s i o n i s made t o t h e n e a r e s t s i n g ularity.  The r e l a t i o n s h i p between g i v e n v a l u e s o f s and t  and t h e range o f r o v e r w h i c h each e x p a n s i o n i s v a l i d may be s e e n by i n s p e c t i n g F i g u r e B . l , remembering  t h a t we a r e  o n l y concerned w i t h r ^- 0.  s>0, t<  s>0, t>0  Figure B . l .  Regions: o f Convergence o f t h e S e r i e s S o l u t i o n s of t h e Wave E q u a t i o n f o r P e r p e n d i c u l a r P o l a r i - . z a t i o n and £ = sr. -*- t . a ) s and t o f O p p o s i t e S i g n b) s and t o f t h e Same S i g n . r  1 6 9  APPENDIX C Impedance and A d m i t t a n c e R e l a t i o n s , f o r Homogeneous R e g i o n s when  fe7  T «  n  I t was  out i n S e c t i o n 3.1-2  pointed  that,  be-  cause o f the b e h a v i o u r of B e s s e l f u n c t i o n s f o r s m a l l a r g u ments, c o m p u t a t i o n a l  d i f f i c u l t i e s a r i s e when e i t h e r  r , or both, are s m a l l .  By  E  or  r  c o n s i d e r i n g the s m a l l argument  b e h a v i o u r of B e s s e l f u n c t i o n s , the l i m i t i n g forms of equa=» t i o n s ( 2 . 1 5 ) and  ( 2 . 1 8 )  can be d e r i v e d .  A  method of d e a l i n g  w i t h s m a l l arguments i n g e n e r a l i s e a s i l y deduced bystudying these l i m i t i n g forms. 0.1  We  assume t h a t  jJT"-jj.  oa  Parallel Polarization P o r a homogeneous r e g i o n o f p e r m i t t i v i t y , Q™,  S™ and  where a** - JT?  Equation  T„ i n e q u a t i o n  £™  >  0,  (2.15) become  "5",  (2.16) becomes J>, ( a ) M  J-  (a )y^ M  ( C 2 )  170  For a  m  and b  w  approaching  ment l i m i t i n g forms o f J„  z e r o , we may (3 2) and  use t h e s m a l l argu-  1  l i m ^ ( z . ) = -~ ( i-z ) z 0 n /  n  l i m Y ( z ) = 2/n-ln(z) z-»0 , (n-1)/  n =  0  n  1  n  TT  W i t h the e x c e p t i o n o f J ( z ) , i n w h i c h case the 0  >  (C.3)  relation  J(f(z) = - J; ( z ) must "be u s e d , t h e l i m i t i n g forms f o r t h e derivatives  may  be o b t a i n e d d i r e c t l y from t h e s e expressions:,  Substituting i n t o equations we  (Cd)  t h e v a l u e s from e q u a t i o n s  and  (C.2)  and s i m p l i f y i n g  (C.3)  the r e s u l t ,  obtain  = 0 n  n i l  In  n = 0 V>1  n i l  2  c  r  5rn  1  n  5™  +l  f  n  = 0 vn-i  n  2 1  n  n > 1 71  (0.4)  K ~ - 3\ • '  «  J >>  ~z— C  f  n =  5M  n > 1  — 0  0  (C.5)  n  E q u a t i o n s ( 0 . 4 ) and ( C . 5 ) a r e e x a c t when £ = 0. r  In  computation, i t i s u n l i k e l y t h a t extremely  small values of icant.  and i~  r  0 ( s a y £ 10  3  ) w i l l be s i g n i f -  As a r e s u l t , unmanageable v a l u e s o f t h e B e s s e l  f u n c t i o n s w i l l o n l y o c c u r when h i g h o r d e r s a r e r e q u i r e d . may d e a l w i t h t h e s e cases by removing  certain factors  We  from  the B e s s e l f u n c t i o n s when p e r f o r m i n g t h e c o m p u t a t i o n s , as e x p l a i n e d bslow*  P o r a c o r e r e g i o n , i t i s e v i d e n t from  e q u a t i o n ( C . 2 ) t h a t an a r b i t r a r y f a c t o r may be  chosen,,the  r e s u l t i n g v a l u e o f z " a p p r o a c h i n g t h a t g i v e n by e q u a t i o n ( C . 5 ) P o r a s h e l l , we d i s t i n g u i s h between two c a s e s . i)  £^  small but  ^ 1«  T h i s means t h a t  t h e s h e l l t h i c k n e s s I s much l e s s t h a n i t s i n n e r r a d i u s .  If  we remove a f a c t o r P x ^ ( b ^ ) from t h e B e s s e l f u n c t i o n s o f t h e f i r s t k i n d and a f a c t o r 1/E? from t h o s e o f t h e second k i n d , t h e r e s u l t i n g v a l u e s w i l l be manageable; t h e m o d i f i e d f u n c t i o n s may be used d i r e c t l y i n e q u a t i o n s ( C . l ) .  Prom  e q u a t i o n s ( C . 4 ) we see t h a t t h e p r o d u c t s w i l l be o f t h e o r -  der of (Xyi^.f . 71  172  ii)X,/^„  +  ,»  1 causing  f  l a r g e t o be h a n d l e d by t h e computer. i n J,, ( a , J and Y (b^,) w i l l dominate.  I n t h i s c a s e , terms N e g l e c t i n g the o t h e r  vi  terms, i t i s seen t h a t a r b i t r a r y f a c t o r s may  imated  be removed  N e g l e c t i n g t h e X.+, /J^ terms i n  from t h e s e f u n c t i o n s . e q u a t i o n s (C.4)  t o be too  we f i n d t h a t e q u a t i o n ( C . l ) l s now  approx-  by  Z7'  •„  J ^  —  :  -  :  <-r  2  ir-  3*1.  • -  «  to  O •>« +|  1  f  \  J -  :  T ^ n ^  0 (0*6)  ZT, '.  n  —  n  c-1  When £™ < 0, J,, and (C.2)  i n e q u a t i o n s (0.1)  and  a r e r e p l a c e d by t h e m o d i f i e d B e s s e l f u n c t i o n s I„  K>, a n d a b s o l u t e v a l u e o f E A  r  i s used.  and  The s m a l l argument  l i m i t i n g forms o f t h e s e f u n c t i o n s a r e ^ ^ 3  l i m l„(z:) = 1/h/  (IzT  n > 0  lim ^ ( z ) = - ln(z) = l(n-l)/  n = 0  (Iz)"  71  n i l (0.7)  l i m I„(z)  - |z  2->0  The o t h e r d e r i v a t i v e s may  be o b t a i n e d d i r e c t l y from  l i m i t i n g forms o f the f u n c t i o n s .  the  The approximate  v a l u e s o f P„ , Q„, S„ and T„ a r e  t h e n e g a t i v e s o f t h o s e g i v e n hy e q u a t i o n (C.4) and t h e v a l u e o f z " i s g i v e n by e q u a t i o n (G.5)«  approximate __  r  must be m a i n t a i n e d i n b o t h c a s e s .  The  Because o i  the s i m i l a r i t y i n the behavior of the f u n c t i o n s i n v o l v e d , t h e d i s c u s s i o n on d e a l i n g w i t h s m a l l arguments and l a r g e o r d e r s remains a p p l i c a b l e Ce2  c  Perpendicular Polarization When  £™ > 0,  e q u a t i o n (2.18) may be w r i t t e n as  p.; +  HoC  Q  •m (0.8)  where P^ , Q™, S™ and T™ a r e as d e f i n e d i n e q u a t i o n ( C l ) . 1  The s m a l l argument r e s u l t i s g i v e n i m m e d i a t e l y by a s u b s t i t u t i o n o f v a l u e s from e q u a t i o n ( C . 4 ) . computational procedures of p a r a l l e l  The s m a l l argument  a r e thus t h e same as i n t h e case  polarization. The case  £™ = 0 r e q u i r e s f u r t h e r i n v e s t i g a t i o n .  F o r n £ 1 we have t h e i n t e r e s t i n g r e s u l t t h a t Y ™ = 0 r e g a r d l e s s o f Y^ '» +  I f n = 0, however, a c a n c e l l a t i o n o f t h e £  f a c t o r o c c u r s and we have  r  174  Under no c o n d i t i o n does  "become complex r a t h e r t h a n pure  i m a g i n a r y as vras observed w i t h a l i n e a r l y v a r y i n g r e g i o n w i t h a zero of p e r m i t t i v i t y .  I t appears t h e r e f o r e t h a t a.  homogeneous s h e l l a p p r o x i m a t i o n i n an inhomogeneous r e g i o n i n which a zero of p e r m i t t i v i t y occurs w i l l n o t give a correct result. I n p l a c e o f e q u a t i o n (C.2) we have  (c.9)  The l i m i t i n g r e s u l t f o r a ^ a p p r o a c h i n g z^ero i s found t o be  K = - 3X=3^ C  2/X J«A  n = 0 n  >  1  This: h o l d s f o r 6™ e i t h e r p o s i t i v e o r n e g a t i v e .  (C.10)  175  APPENDIX D Some Aspects of the Numerical I n t e g r a t i o n  Method;  I n t h i s appendix, the d e t e r m i n a t i o n of values f o r a numerical i n t e g r a t i o n  initial  of the impedance or  admittance d i f f e r e n t i a l equations i s d i s c u s s e d and of d e a l i n g ln  the  w i t h the  s i n g u l a r i t y a t a z:ero of  i s c a r r i e d out  i n terms of impedance i n  case of p a r a l l e l p o l a r i z a t i o n and perpendicular p o l a r i z a t i o n .  admittance l n the  I n e i t h e r case, the  z„ or y^ i s used when p e r f o r m i n g the D.l  v a l u e of z^ a t r = r  where z^" ' = +  the case of  smaller  of  integration.  When m = M,  w  i n the  must be known.  ( m - l ) ^ region,  The  i n t e g r a t i o n v a r i a b l e i n the  g i v e s the r e q u i r e d  the  relation  (m-l)  region  ./  i n i t i a l v a l u e i f Z™;has been determlned.  t h i s becomes z  0  a c o n d u c t i n g core  IVl -/ for  The  Parallel Polarization In order to i n t e g r a t e  for  permittivity  case of p e r p e n d i c u l a r p o l a r i z a t i o n i s g i v e n .  discussion  a method  a d i e l e c t r i c core.  (D.2a)  and  tCCS)  (D.2b)  176  When V  ne  I s not a v a i l a b l e , i n t e g r a t i o n i n the core r e g i o n  i s r e q u i r e d and an i n i t i a l v a l u e o f z" f o r a s m a l l v a l u e of "J.must be f o u n d i of  c (x)  I f i t l s assumed t h a t t h e b e h a v i o u r  a t t h e o r i g i n c a n be d e s c r i b e d by-  r  C(r)= a r ,  Urn  b  b 2- 0  r->0  ( w h i c h s h o u l d i n c l u d e t h e cases o f i n t e r e s t ) t h e n an approx* imate i n i t i a l v a l u e i s g i v e n by  (D.3)  J = K*L  fe  )  J.v'( a  5  kr 0  <K c  1  ^2n  where v =  b + 2  fe+z  2k, f (I)  =  b + 2  (  T  A  )  2  The approximation corresponds t o r e p l a c i n g  the a c t u a l diel«=  e c t r i c i n the r e g i o n r £ r ^ by one w i t h a v a r i a t i o n a r . b  The s p e c i a l case b = 0 a p p l i e s whenever £™(0) 4= 0. The behaviour of z£ a t the o r i g i n i s determined by u s i n g the f i r s t term i n the s e r i e s expansions f o r J Jy i n equation (D.3).  Urn  v  and  The r e s u l t i s  zZ = ~ = TA  We n o t e t h a t f o r n > 1, z "  b + 2 n = 0 n > 1  i s independent  (D.4)  of p e r m i t t i v i t y .  177 If very small i n i t i a l e q u a t i o n (D.3)  v a l u e s o f ~% c o u l d be u s e d ,  would g i v e a c c u r a t e r e s u l t s .  (Arbitrarily  s m a l l - v a l u e s o f ^- cannot be used because o f t h e ( 1 / ? ) and (-  n  z  / J ) r  terms i n t h e e q u a t i o n s  the c o m p u t a t i o n ,  ( 2 . 2 4 ) ) .  I n c a r r y i n g out  i t i s found t h a t a r e l a t i v e l y l a r g e I n i t i a l  v a l u e o f "5 o r a r e l a t i v e l y s m a l l i n t e g r a t i o n s t e p s i z e must be used when n i s l a r g e i n o r d e r t o a v o i d an i n s t a b i l i t y i n the numerical i n t e g r a t i o n . initial  The use o f r e l a t i v e l y l a r g e  v a l u e s i s j u s t i f i e d because, as n i n c r e a s e s , t h e  v a l u e o f z^ f o r s m a l l "5 becomes l e s s s e n s i t i v e t o p e r m i t t i v i t y . In  most cases o f i n t e r e s t , t h e power s e r i e s  method o f S e c t i o n 3,1-1 may be a p p l i e d t o o b t a i n a c c u r a t e initial in  values of z . n  Due t o t h e s i m p l i c i t y o f t h e s e r i e s  t h e case o f p a r a l l e l p o l a r i z a t i o n , t h i s method g e n e r a l l y  r e q u i r e s l i t t l e more e f f o r t t h a n t h e a p p l i c a t i o n o f e q u a t i o n (D.3) D.2  0  Perpendicular Polarization The i n i t i a l  (m-l)  t ] l  v a l u e f o r an i n t e g r a t i o n l n t h e  region i s = Y^/jy.  0.5)  T h i s reduces t o = oo  5 = k r, 0  for  a c o n d u c t i n g c o r e , and  (D.6a)  .178 =  —;  EA**,)  (D.6b)  —  f o r a d i e l e c t r i c core. When c"^(J"/k: ) = £"(r)'» a r ^ i n the neighbourhood o f t h e 0  o r i g i n , an approximate i n i t i a l v a l u e f o r i n t e g r a t i o n i n the core i s  h  y" » a ( T A J  where  v  2  - (kn + 2 k  f(?)  =  D  J ( f(?) ) '/a (lAo)* v  (D.7)  T - - ,  /  b* ) / (b + 2 / y a / r \-r~  — - (~ / b + 2 \k /  i s the same as i n equation  (D«3)  0  U s i n g the f i r s t  lim ?-*0  term i n the s e r i e s f o r J  v  and J y we have  t 7 (b•+ 2 ) y* = - a ( J A J — r • 7 a ( T A J - V 2 (b + 2 )  n - 0  11  (D,8a)  = a (TAo)  T  1 2  ,  n  /y  (n* + -b /k) z  /z  i  l  (D.8b)  + b/2  Due t o the dependence o f y™ on £ ( r ) r  i n equation  ( D . 8 b ) , ' r e l a t i v e l y . l a r g e s t a r t i n g v a l u e s o f J may n o t a l ways be s u i t a b l e when n » t h i s dependence appears  1.  Note, however, t h a t when n » b/2,  o n l y as the f a c t o r  a(TAo)^»  p l i f i c a t i o n which occurs by u s i n g o n l y the f i r s t  The sim-  term i n  the B e s s e l f u n c t i o n s e r i e s i s a c c u r a t e f o r i n c r e a s i n g v a l u e s of T as n i s i n c r e a s e d .  T h i s suggests t h a t f o r n »  1 and  179  » b / 2 , l a r g e r I n i t i a l v a l u e s o f Y may "be used i n equation  n  (D.7) the  i f the m u l t i p l y i n g true p e r m i t t i v i t y As  factor  a( J/LO^  f^(TAo).  i s r e p l a c e d by  i n t h e case o f p a r a l l e l p o l a r i z a t i o n , the  power s e r i e s method o f S e c t i o n 3»1-1 obtain i n i t i a l values.  may a l s o be used t o  S i n c e the s e r i e s s o l u t i o n i s more  cumbersome i n t h i s case, i t i s g e n e r a l l y I m p r a c t i c a l t o use  a p o l y n o m i a l o f degree g r e a t e r then u n i t y  t o approximate  ErM.  3).2-1  D e a l i n g w i t h Zeros o f P e r m i t t i v i t y When £y(r)  = 0,  equations ( 2 . 2 9 a )  and ( 2 . 2 9 b )  both have a s i n g u l a r i t y f o r n ^ 1 due t o the l / £ ( r ) f a c t o r K  i n t h e c o e f f i c i e n t of y * . As a r e s u l t , i n c o n t r a s t t o the case o f p a r a l l e l p o l a r i z a t i o n ,  a numerical  integration  cannot be used i n the neighbourhood o f a p e r m i t t i v i t y  zero.  I f t h e s i n g u l a r i t y i s i s o l a t e d i n a t h i n sub-region, then, by u s i n g power s e r i e s s o l u t i o n s transfer culated.  i n e q u a t i o n ( 2 . 1 8 ) , the  o f admittance a c r o s s the s i n g u l a r i t y may be c a l When an approximation t o £ {r) r  i s used f o r t h i s  purpose, i t must have the same behaviour as £ ( r ) i n t h e r  neighbourhood o f the z e r o .  F o r example, the minimum degree  of a p o l y n o m i a l which may be used t o approximate a v a r i a t i o n of t h e type ( i ) , (11) is  1, 2 and 3 ,  or ( i i i ) i l l u s t r a t e d i n Figure D . l respectively.  I f we approximate £>.(r) by a l i n e a r v a r i a t i o n t {x) r  .= 0 as i n t h e case i l l u s t r a t e d i n F i g u r e D . 2 , the  near  F i g u r e D.2  Approximating £ (r) t  Mr)  by a L i n e a r V a r i a t i o n Near  = 0 i n Order t o L e a l w i t h t h e S i n g u l a r i t y  181  s e r i e s developed  i n Appendix B may "be used.  The p r o c e d u r e  i s as f o l l o w s : I) region r  I s o l a t e t h e s i n g u l a r i t y by d e f i n i n g a s u b -  £r 5 r  3  z  and approximate  £>.(r) by a l i n e a r v a r i -  ation s r + t . I I ) S t a r t i n g a t a s m a l l v a l u e o f ~S, c a l c u l a t e y« p ~ y-n  5  ' "  , =  ^  b y a n u m e r i c a l i n t e g r a t i o n o f e q u a t i o n (2.29a) .3  ^  I i i ) L e t Y^ =3 tf y ^ p and use e q u a t i o n (2.18) 0  w i t h m = 2 and U,, and V„ t h e s e r i e s s o l u t i o n s about r = r  c  g i v e n b y e q u a t i o n s (B.18a) and (B.18b) t o c a l c u l a t e Y* . i v ) Use t h e i n i t i a l v a l u e y • to integrate numerically rather  to r = r,.  '„ Y* / J y  o  S i n c e Y* i s complex  t h a n p u r e i m a g i n a r y , y,, i s complex.  I n order t o  c a r r y o u t t h e n u m e r i c a l i n t e g r a t i o n , we s u b s t i t u t e y-n - 7r,t-  +  3 7-ni i n t o e q u a t i o n (2.29a) and s e p a r a t e t h e r e a l  and i m a g i n a r y p a r t s .  _  •= S (J/k. ) r  0  We o b t a i n t h e f o l l o w i n g c o u p l e d n  + y„ / j + ( l -  z  r  z  d  t (£/k. ) r  0  ) (y„ - y £ ) r  (D.9)  n* d 5  equations:  5  I n a s i m i l a r manner, e q u a t i o n (2.29b) y i e l d s d .z, d?  n  2  ( i - 7 — — — ) + W f + f (?A») (<r~<^ 2  r  182  d z no d  5  1/5 + 2 z „ £ ( 5 A o )  = z  where z„ = z^  r  r  +  3 z„ t  r  (D.10)  183  APPENDIX E D e t e r m i n a t i o n o f t h e R e q u i r e d Order f o r Computing t h e Scattered -Field When computing t h e s c a t t e r e d f i e l d , i t i s n e c e s s a r y t o d e c i d e upon t h e number of terms t o be used i n t h e e x p a n s i o n such t h a t the e r r o r due t o n e g l e c t i n g h i g h e r o r d e r terms i s below a s p e c i f i e d v a l u e .  I n t h i s appendix, we  de-  t e r m i n e l i m i t s f o r t h i s t r u n c a t i o n e r r o r and as a d i r e c t consequence  s t a t e t e s t s f o r determining the r e q u i r e d order.  The  e r r o r l i m i t s depend upon t h e f a c t t h a t upper bounds f o r t h e magnitude  o f h i g h - o r d e r c o e f f i c i e n t s can be f i x e d p r o v i d e d  t h a t z' o r y^ s a t i s f y c e r t a i n c o n d i t i o n s . n  I t i s found  that,  a l t h o u g h the e r r o r bounds a p p l y f o r any p e r m i t t i v i t y v a r i a t i o n i n t h e case o f p a r a l l e l p o l a r i z a t i o n , t h e y a r e l i m i t e d t o £ (r) K  > 0 i n t h e case o f p e r p e n d i c u l a r p o l a r i z a t i o n .  development  The  w h i c h i s c a r r i e d out i n the f o r m e r case i s a p p l i c -  a b l e t o the l a t t e r s u b j e c t t o t h e above r e s t r i c t i o n , i f E replaces E Eel  c  and a d m i t t a n c e s r e p l a c e  c  impedances.  Maximum Magnitude o f High-Order S c a t t e r e d F i e l d . Coefficients We f i r s t c o n s i d e r the dependence o f /A„/ on t h e  n o r m a l i z e d i n p u t impedance z' - Z^,/;)-^. n  E q u a t i o n (2.12)  may be r e w r i t t e n as  A, = -  (3f  E  MI,) e  —7  - <  <<T,> 7  1  •  , (E.l)  184 The zl,  / k^/E j  extrema of may  when c o n s i d e r e d as a f u n c t i o n of  0  be found by s e t t i n g the d e r i v a t i v e w i t h r e s p e c t to  z^ equal to z e r o .  We have F*  /aj = 2  where  a ^ = A ,/E 7  -  YAf,  - -^r  (E.2)  7  0  ) - <  Y'(J,)  D i f f e r e n t i a t i n g w i t h r e s p e c t to z'^ g i v e s  )>_ ^<  Z K  _ *V  Gy 2(  '  (?, )  ; U  ) - Y„  (F*  +  (T, )  ),)  G*)  (E.3) The numerator of equation (E.3)  may  be s i m p l i f i e d u s i n g the  Wronskian r e l a t i o n ^ )  •51  '  giving  (F  S e t t i n g equation (E.4)  2  +  G^)  equal to zero and s o l v i n g f o r z'^  obtain z' = — , — — <(?,)  or  z„ = — Y^(5, )  we  185  We n o t e t h a t = 0  , _ 3-nCS:,) = i  /aJ'  (E.5)  < i oo  z,  and t h e r e f o r e z ; = J „ ( 5 , ) / j ' C ? , ) and <  = Y^C?,) A ' ( 3 , )  y i e l d t h e minimum and maximum v a l u e o f /a^,/, r e s p e c t i v e l y . I n o r d e r t o o b t a i n a good a p p r o x i m a t i o n t o t h e f i e l d , we e x p e c t t h a t terms up t o a " l a r g e " v a l u e o f n w i l l be r e q u i r e d .  W i t h t h i s i n mind; we t e n t a t i v e l y impose t h e  r e s t r i c t i o n n >f . and observe t h e b e h a v i o u r o f t  as a  f u n c t i o n o f z' . The r e s u l t , which i s e a s i l y ' d e d u c e d u s i n g t h e n  p r o p e r t i e s o f B e s s e l f u n c t i o n s and e q u a t i o n s (E.2) and ( E . 5 ) i s s k e t c h e d i n F i g u r e ( E . l ) which shows t h e e f f e c t o f i n c r e a s i n g n. We now determine  t h e maximum v a l u e o f /a*,/ f o r  p o s i t i v e v a l u e s o f z^,. The. r e a s o n f o r t h i s i s t h a t , as d i s cussed i n Appendix F, i f n l s l a r g e enough, z^ ^ 0 f o r a d i e l e c t r i c c y l i n d e r or a d i e l e c t r i c c y l i n d e r with a conducting core.  From F i g u r e E . l ,  i t i s e v i d e n t t h a t t h e maximum must  be reached e i t h e r a t z ' = 0 o r as z' -> <x>.  The v a l u e s o f  186  F i g u r e E . l B e h a v i o u r o f l&J 6. n , < n  z  f o r three Values  < n . 3  o f n,  The argument o f a l l B e s s e l  f u n c t i o n s I s 7, 'Ti  l*J'  UJ' Z  ing  / a„/  "  _  >  C  O  UJ  (E.6a)  UJ  (E.6b)  ~ /H?U)/  a r e p l o t t e d as f u n c t i o n s o f i n F i g u r e E.2.  T1  Each /a„/ J  , f o r various values of n a  .  c u r v e i s above t h e correspond-  . c u r v e e x c e p t when n i s c l o s e t o ~S, (where  the value given f o r / a J i s ' r e l a t i v e l y l a r g e ) .  ¥e may w r i t e  (E.?a)  n > ^, +1 6  (E.7b)  187  188  A good e s t i m a t e o f the v a l u e g i v e n by e q u a t i o n (E.7a) (3 ?-)  may  be o b t a i n e d by u s i n g t h e f o l l o w i n g a p p r o x i m a t i o n s  (the  e r r o r i s about 1% f o r n = 2 and d e c r e a s e s w i t h i n c r e a s i n g n ) : Jjn)  *  n  — / 3  0.7748  Y»(n)  n'  /3  We f i n d 0.4473 '  /J„(n)/  /H^(n)/  m  (0.4473* + 0.774&V*  c» o«5  I t i s e v i d e n t t h a t t h e r e s t r i c t i o n n i f, i s j u s t i f i e d ' s i n c e a c o n s i d e r a b l y l a r g e r v a l u e o f n must be used t o ensure a s m a l l value o f /a^/. E.2  Truncation-Error i n Calculating the Scattered F i e l d The l i m i t s on Iky,} w h i c h were developed  i nSection E . l  a r e now used t o f i x a bound on the t r u n c a t i o n e r r o r i n t h e computed s c a t t e r e d f i e l d .  A t e s t procedure f o r d e t e r m i n i n g  the o r d e r which ensures t h a t t h i s e r r o r i s w i t h i n a s p e c i f i e d value i s then g i v e n . The  error  i n t h e magnitude o f t h e s c a t t e r e d  f i e l d , caused by t r u n c a t i n g t h e s e r i e s i n e q u a t i o n  (2.1b)  a t n = N, n o r m a l i z e d w i t h r e s p e c t t o t h e magnitude o f t h e i n c i d e n t f i e l d , i s g i v e n by  189 E„=  /Z  e , A , / E H ^ C ? ) cos(n©-)/ 0  - / Z  e^A^/Eo  (I ) cos(nG-) /  (E.8a)  OO  ±2. I Z  ( 7 ) cos(nG-) /  a.  n  (E.8b)  CO <Z'ZT-  / a , / . / H ? ( ? ) / - /cos(ne-)/  (E.8c)  /a /../H« (I)/  (E.8d")  CO  *  2 X  }  W  S u b s t i t u t i n g t h e maximum v a l u e o f /a„/ from e q u a t i o n  (E.?b)>  we have  (E.9) (E.9) i s inconvenient  Equation  i s I n t h e form o f an i n f i n i t e s e r i e s .  t o u s e because i t  A s i m p l i f i c a t i o n can  / j ' ( J,) / / / H™'( X ) /  be mafte i f we r e p l a c e  by J  ( £ ) / / / H ? (-£  T h i s change i s g e n e r a l l y s m a l l ( c e r t a i n l y i f max |a-„| i s — 4-  s m a l l , s a y max |a-n| < 10 We now d e f i n e  ) as c a n be seen f r o m F i g u r e E.2.  cf^ as Oo  - Z  b„  (E.10)  /j>,(?,)/  By making use o f t h e f o l l o w i n g B e s s e l f u n c t i o n p r o p e r t i e s : a)  / < J ( z ) / and d/dv(/<J ( z ) /) a r e d e c r e a s i n g v  V  functions  of V f o r v > z b)  / H J ' ( Z ) /  of  V  for  and d/dy~(/H * (z) /) a r e d e c r e a s i n g f u n c t i o n s , (  V  > z  }  190  /H"(Z)/  c)  and d/dz ( / H ^ ( Z ) / ) a r e d e c r e a s i n g f u n c t i o n s  o f z, we o b t a i n t h e c o n d i t i o n s by,, > b  f o r n, < n  ri2  >  f  by,*  o  r n  (E.lla)  2  < n .£ n  j  z  < n  3  (Eollb)  +  b^  I n o r d e r t o e v a l u a t e a l i m i t . o n *S ,  consider the  N  power s e r i e s S„ = b „  + I  (E.12a)  Z  1 B  *  f o r / ^ 1  _  (E.12b)  1 - /° where  / =b >  w + 2  /b  The f i r s t two terms o f e i t h e r c „ o r S i > 1,  inequality  a r e s u l t , "c? < S^. w  (E.30b) i m p l i e s  Equation  w  are b  \> /°  ,, b  N +  >^>  N+I  N+l+  N + z  .  For  i and, as  (E.12b) i s a p p l i c a b l e s i n c e ,  from i n e q u a l i t y ( E . l l a ) , ft- 1.  We may t h e r e f o r e w r i t e  (E.13)  = 1 - b*+* / b , v+  At t h e surface of the c y l i n d e r ,  ^  and ( E . 1 3 )  2 ^ , ( 1 , )  s  i - «W(X)/J„ , +  a,)  gives  (E.14)  191  As X I n c r e a s e s , b  /b„  decreases,  +I  equal t o ( J „  + 2  for  The e f f e c t o f t h e c o r r e s p o n d i n g  y~fooo  ( X )/J  w + 2  /v + )  becoming  ( X ) ) - ( / H^, ( X ) / / / H ^ ( X ) /  )  i n c r e a s e i n "the  denominator o f i n e q u a l i t y (E.13) i s s m a l l compared t o t h e d e c r e a s e due t o on c i s  / H ^ d f )/  i n t h e numerator.  The bound  t h e r e f o r e n o t r a i s e d s i g n i f i c a n t l y i f we r e p l a c e  w  (-E.13) by  £  < —  —  (E.15)  ~  i - j a)/j , (X) w+z  v+  The q u a n t i t y "S^ has been i n t r o d u c e d as an approxi m a t i o n t o t h e bound on t h e e r r o r £ . w  The a c t u a l bound on "£ J" = X )  a t t h e s u r f a c e o f t h e c y l i n d e r ( g i v e n by (E.9) w i t h i s p l o t t e d i n F i g u r e E . 3 t o g e t h e r w i t h t h e bound on t'  cal-  N  c u l a t e d u s i n g (E.14).  I t i s seen t h a t t h e bound on ~E i s N  somewhat l a r g e r t h a n t h a t on !>'„ f o r r e l a t i v e l y l a r g e e r r o r s . However, f o r 10 ~  < max ~c? < 1 0 ~ , w h i c h would seem t o be a  6  reasonable  2  w  range i n t h e p r e s e n t c o m p u t a t i o n s ,  a r e e s s e n t i a l l y t h e same.  I f s m a l l e r v a l u e s t h a n those  a r e c o n s i d e r e d , t h e bound on c two ( f o r N-><» , t  l N  t h e two. v a l u e s  N  shown  becomes t h e l a r g e r o f t h e  I s t h e bound on  The l i m i t on /a,,/ used i n t h e above d e r i v a t i o n a p p l i e s when z' > 0 . n  Since the l i m i t i s required f o r a l l  n > N, I t i s : seen from t h e d i s c u s s i o n i n Appendix F t h a t t h e v a r i a b l e z„  +(  must be p o s i t i v e w i t h i n t h e c y l i n d e r .  goes t o +oo , t h r o u g h  the z  n  If  z, N+  < 0 r e g i o n i n t o t h e z^ > 0  r e g i o n , t h e n z^ can be n e g a t i v e f o r a h i g h e r o r d e r , w i t h t h e  N  193 r e s u l t t h a t /a^/ may r e a c h I t s maximum v a l u e o f u n i t y When - oo < z'„ ,  <Y  +  N+I  (X  ( 1 , ) and z „  + /  has o n l y one  change o f s i g n ( + t o -) w i t h i n t h e c y l i n d e r , t h e e r r o r bounds r e q u i r e o n l y a s i m p l e m o d i f i c a t i o n , however. Under t h e s e c o n d i t i o n s , as n i N+1 increases:, w i l l d e c r e a s e t o a l a r g e n e g a t i v e v a l u e , Jump t o a l a r g e p o s i t i v e v a l u e and t h e n d e c r e a s e towards z e r o .  Referring to  F i g u r e E . l , we n o t e t h a t f o r - co < z' < Y^ ( ) / Y ' ( T , ) we n  have As zl, d e c r e a s e s f r o m Y^( J,) /Y'(  i) proaches  / j'(  '(If ) / (  more r a p i d l y as n i n c r e a s e s .  The v a l u e z ^ = Y^C?,) A - ' ( T ; )  ii)  )» /a*,/ ap-  a t which / a , , / = 1  increases with n. As a r e s u l t , when z^ i s i n t h e range - co < z ^ < Y^C  )/Y' CS, ), X  / a ^ / d e c r e a s e s more r a p i d l y w i t h i n c r e a s i n g n t h a n does  /J'C?,) / / i K f a , ) i . (  The m o d i f i c a t i o n o f t h e e r r o r bound i s d e r i v e d as f o l l o w s .  F o r v a l u e s o f n such t h a t - oo -c z'^ < Y^CT,) /Y'( Ifi)»  we r e p l a c e b ^ by /a„ / */H^ (f)I i n t h e i n e q u a l i t i e s  (E.ll).  The r a t e o f d e c r e a s e o f t h e changed terms i s f a s t e r t h a n t h a t of t h e o r i g i n a l ones and we may t h e r e f o r e u s e t h e s e r i e s  *  Note t h a t /da„/dz/ i n c r e a s e s r a p i d l y w i t h n > ~S, f o r v a l u e s o f z; n e a r Y^ (3r, ) ) (see F i g u r e E..1). As a r e s u l t , t h e e r r o r i n A>, >> / J ^ ( T , ) / // due t o an e r r o r i n z^ when z^ has t h e s e v a l u e s Trill be r e l atively large.  194  ex> i n place of t h a t given i n equation  H  w  (E.12a),' t h e r e b y o b t a i n i n g  2 /a ,/./H??C?)/ w+  <  ; 1 — b 2  (E.l6)  /b +/  w+  w  C o r r e s p o n d i n g t o i n e q u a l i t y (E.14) o r ( E . 1 5 ) we have  2 /a /./H«(I)/ w+J  t<  -  N  -  rr-  :  for  1 - J „ ( X ) A v , (I,) +2  •  (E.17)  •  +  Note t h a t (E.16) and  ( E . 1 7 ) r e q u i r e the knowledge of  /a ,/ w +  and a r e t h e r e f o r e used t o determine whether a l a r g e enough o r d e r has been c o n s i d e r e d  r a t h e r than to estimate  the r e -  quired order, E.2-1  P r o c e d u r e used t o Determine the Order R e q u i r e d t o A c h i e v e a G i v e n A c c u r a c y of the S c a t t e r e d  Field  I t i s r e q u i r e d t o d e t e r m i n e N such t h a t where A i s a s p e c i f i e d e r r o r . N such t h a t  ^„<A  is  satisfied.  a)  Estimate  We  f i r s t obtain a value  of t ±&  and t h e n check t h a t the c o n d i t i o n  N  R e f e r r i n g t o i n e q u a l i t y ( E . 1 5 ) , choose N  N:  such t h a t J  w + I  -  l - J b)  IxZ,  <?,) w + Z  Check b e h a v i o u r of z  N+)  ( * > / / / C (7,)/  (X)/J„ , (T.) " +  :  A  < _  - 2  " (E.18)  Determine the b e h a v i o u r o f  w i t h i n the c y l i n d e r o r , i f p o s s i b l e , by u s i n g  z  w + l  ( i i ) or ( i i i )  195  i n Section P . l . i)  There a r e t h r e e p o s s i b i l i t i e s  z  remains p o s i t i v e (from ( i ) i n S e c t i o n P . l ,  N+l  we n o t e t h a t t h i s c o n d i t i o n i s a u t o m a t i c a l l y s a t i s f i e d i f z  N  r e m a i n s p o s i t i v e and hence t h e c a l c u l a t i o n o f z , may n o t Kf+  be n e c e s s a r y ) .  Equation  i i ) z^  (E.18) a p p l i e s I n t h i s  changes from ( + ) t o (-)•.  +l  must be i n c r e a s e d by one and s t e p (b) r e p e a t e d  z  »+,  /a  <  M + l  /  /  w +  )/r ' ( X ) N +1  (E.l9b)  )  Note t h a t  (E.l9a)  and ( E . 1 9 b ) i s s a t i s -  e v a l u a t e d u s i n g z' i n s t e a d o f z'r,+,  the i n e q u a l i t y . be  +  unless  (E..19a)  - ( 1 -  <  i s s a t i s f i e d i f z' < Y„ , Cf, N  The v a l u e of N  -•-  The l a t t e r i n e q u a l i t y r e s u l t s from ( E . l ? ) .  fied i f /a ,/  case.  N  The c a l c u l a t i o n o f z.„ , +  satisfies,  may t h e r e f o r e n o t  necessary. iii)  z  N+l  changes s i g n more than once.  N must  be i n c r e a s e d b y one and s t e p (b) r e p e a t e d . c)  Check  Use i n e q u a l i t y ( E . 9 )  ~ZAJ±A:  the a c t u a l e r r o r c o n d i t i o n , ~B ^A,  i s satisfied.  N  shown i n F i g u r e E . 3 ,  t o d e t e r m i n e whether  t h e bound on  p r o v i d e s a good e s t i m a t e  o f t h e bound on 1z , t h i s s t e p i s u n n e c e s s a r y N  S i n c e , as  i n most  appli-  cations. We n o t e t h a t r e l a t i o n ( l i l ) i n S e c t i o n F . l i m p l i e s t h a t the p o s i t i v e c o n d i t i o n on z\ i s a l w a y s s a t i s f i e d f o r a N  196  plasma c y l i n d e r when N 2. X eliminated E.3  •  As a r e s u l t , s t e p (b) may  i n t h i s v e r y Important  Truncation-Error  be  case»  i n C a l c u l a t i n g the B i s t a t i c  Scattering  Cross-Section I f j$  N  denotes the magnitude  of the e r r o r i n the  c a l c u l a t e d v a l u e o f <?(&) which i s caused by t r u n c a t i n g  the  s e r i e s i n e q u a t i o n ( 2 . 3 5 ) a t n = N, then we have  ^ = 7 ~ E  1 4  j8« where E  -- ]T  =  T  E  e„(3)  c  2  —  i^i j  c  (E.20)  —  k  (  + /Ej  / B /  ) -//E/ - / E j /  0  A-  E N = fl e „  / /E /  k  e  cos(nG-)  A „ cos(ne-)  •n = o I f we  assume t h a t the e r r o r i s s m a l l ,  / E / + / E J ** 2 / E j  and we have ^  *  IT —" /E /*//E/ E k w  0  ^  1  8  ~  —  B _  /E J  - /E - E ^ /  0  16  k  - /EJ/  0  0  / E J .  —  E  0  .J^  «,  -o=+i w  , A, , (-j) — cos(ne-) /  E  (E.21)  197 16 |E„/  - — k  0  — E  ' 21  Eo F o r z^ , > +  Since  0  /  I a-nl 'lcos(riQr) j  n-N+i  0  »=/>/+/  > J,  H  J>f(T» ) / / H ^  16" |E„J  , «  k  '  0  E  (^f/) /  0  J,'(f,) /<>  ,  (?,)!'  and i t s d e r i v a t i v e w i t h r e s p e c t to  n are d e c r e a s i n g f u n c t i o n s of n, a l i m i t on the summation i n equation (E.21) may be obtained by comparison w i t h a power s e r i e s i n the same manner used i n S e c t i o n E„2. 16 < —  A  k<?  where ^  - J.'(T,  JEJ ~ — E  1  e  c„.  M  (E.23)  — —  ON+Z  / N+I C  )// H^'cf,)/  (The v a l u e s o f j ' ( X ) / / H^'t"?/) / various n > X  The r e s u l t i s  as a f u n c t i o n of ?, f o r  are, p l o t t e d i n F i g u r e E.2)  Since / E / i s w  r e l a t e d to the c a l c u l a t e d b i s t a t i c s c a t t e r i n g c r o s s - s e c t i o n ^(G-) by  equation  (E.23) may be r e w r i t t e n as c <&N  <  8//K>'Jcr {e-) N  • 1  ~  CrJ+z/Crj + i  U s i n g the same argument as i n S e c t i o n E.2, we may a s i n g l e change of s i g n i n z  N+l  (E.24)  +  and o b t a i n  allow  198  'MJ&T  £ S/fk,  A  N  (E.25) 1  p r o v i d e d z'  N+i  E«3-l  < Y„  +l  (X ) / r '  +(  ~  £ N+Z /&hJ+l  (X)  P r o c e d u r e used t o Determine the Order R e q u i r e d t o Achieve a Given Accuracy of S c a t t e r i n g C r o s s - S e c t i o n The a b s o l u t e  cross-section,^,  e r r o r i n the b i s t a t i c  i s p r o p o r t i o n a l to f o ^ .  we cannot s p e c i f y an o r d e r w h i c h w i l l absolute  scattering As a  result,  ensure e i t h e r a g i v e n  error or a given r e l a t i v e e r r o r , ^ / c ^ , for a l l w  v a l u e s o f <f.  The e r r o r c r i t e r i o n chosen w i l l g e n e r a l l y  de-  pend upon the expected range of cf * If  t h e minimum s i g n i f i c a n t v a l u e o f <?N i s 10"  Cl  ,  we may determine N such t h a t the r e l a t i v e e r r o r i s - l e s s : t h a n 10~  Cz  l  z  (hence cT i s c o r r e c t t o ( i z - 1 ) s i g n i f i c a n t f i g u r e s when N  i s an i n t e g e r ) .  as l / / c £  as (f  N  We n o t e t h a t t h e r e l a t i v e e r r o r  increases:.  Thus, i f xhe r e l a t i v e e r r o r  d i t i o n i s s a t i s f i e d f o r a c e r t a i n v a l u e o f oU » i t i s fled for a l l larger values. c r i t e r i o n g i v e n above i s  </ = 1 0 ~ ^  r e l a t i v e e r r o r i s not the l i m i t i n g The t e s t p r o c e d u r e i s as D e t e r m i n e N such t h a t  con-  satis-  I t i s seen t h a t t h a t v a l u e f o r 6 /  ~ ^. 6  F o r cr* < 10  the number o f s i g n i f i c a n t f i g u r e s i s d e c r e a s e d  a)  decreases  quantity. follows;  and hence  the  ~  L z5  the  199  8  'N-i-l  •{Li-Cj.)/z  c/ =10  1  -  CN+Z/CN +  i  10  1  w  This  gives  10  8  1 — CtJ+Z. /^N + i  Note t h a t t h e r e l a t i v e e r r o r i n E  N  (E.26)  i s s m a l l even a t t h e  s m a l l e s t v a l u e o f o^, ( t h i s f a c t i s r e q u i r e d i n t h e a p p r o x i m a t i o n /E/-+ / E J « 2 / E J ) u n l e s s 1^ < 2.  Prom e q u a t i o n  (E.21)  we have /£/ " / E J  At  /Ej  8  /E/ - / E J  10  0W  </„ = 10  1  J E J  b)  / E j /E*  Determine t h e b e h a v i o u r o f z 1)  quired  repeated  w  +  /  xl -"  = 0.25 x 10 / i 5  0  w i t h i n the c y l i n d e r ,  remains p o s i t i v e .  N i s then the r e -  order. ii)  inder.  z  N+l  7.  Z  N+I  changes from ( + ) t o (-) w i t h i n t h e c y l -  The v a l u e o f N must be i n c r e a s e d by one and s t e p b) unless:  (X)  (E.2?a)  200  ± - —  '  1  •  0  8  111)  z  (1 -  N+)  -(L,-l )M  ) • max t  N+l  }  z  \10  changes s i g n more t h a n once.  /  N must  be I n c r e a s e d by one and s t e p b) r e p e a t e d . I t may appear t h a t t h e r u l e g i v e n would be i n e f f i c i e n t f o r l a r g e c^.  N o t e , however, t h a t s i n c e  decrea-  ses v e r y r a p i d l y w i t h i n c r e a s i n g n when n i s l a r g e r t h a n 3P/, the number o f i n s i g n i f i c a n t terms c a l c u l a t e d w i l l be s m a l l .  The u s e o f fo^  helps avoid unnecessary  i n r e l a t i o n (E.27b)  actually  when  cf  c a l c u l a t i o n when cfn i s . l a r g e .  N  >10~^  201  APPENDIX F General Behaviour of  and y  W i t h i n an,Inhomogeneous  n  Cylinder The  b e h a v i o u r o f z' o r y^ w i t h i n c r e a s i n g n ^ f, n  l s i m p o r t a n t when d e r i v i n g bounds f o r t r u n c a t i o n e r r o r s I n t h i s a p p e n d i x , we c o n s i d e r t h e b e h a v i o u r o f  c  and y,, 1  w i t h i n a c y l i n d e r and hence d e t e r m i n e t h e dependence o f z^ and y^ on n > 0, 7 , » and ^ ( T / k J . placed  on £ ^ ( ? A J  No r e s t r i c t i o n s a r e  i n t h e case o f p a r a l l e l p o l a r i z a t i o n ;  however, i t i s r e q u i r e d t h a t £ (?/k: J ^ 0 ( e x c e p t f o r a r  homogeneous r e g i o n o f zero p e r m i t t i v i t y ) i n t h e case o f perpendicular p o l a r i z a t i o n . P.l  Parallel Polarization We w i l l c o n s i d e r i n d e t a i l t h e case o f a d i e l -  e c t r i c c y l i n d e r ; the extension i s q u i t e s t r a i g h t forward. we r e w r i t e e q u a t i o n  t o i n c l u d e a conducting core  D r o p p i n g t h e s u b s c r i p t n on z„,  (2.24a) as  dz —. = f ( n , ? , z) d? where f ( n , T ,  (Pel)  z ) = 1 + z/S + ( E ( 7 A J r  - n*/T  2  ) z  z  I t i s seen t h a t d z / d ? i s i n d e p e n d e n t o f z- and 7 f o r z -> 0, and  that the sign of ( £ ( J A J  haviour  r  - n /*?) 2  o f dz/d;? f o r l a r g e v a l u e s o f z.  governs t h e b e For f i n i t e  values  202  of p e r m i t t i v i t y lim ^->0  (^(FAJ v  .  - n /? ) < 2  z  '•  0,  n  > 0  We t h e r e f o r e c o n s i d e r f i r s t a case i n which n and ^ ( J / f c J a r e such t h a t  >S,(IAJ <M7A*)  nVf  •where X^. i s determined Setting  c o<"f<7*  (F.2b)  5*<7£5.  (P.2c)  "by e q u a t i o n (F.2a).  d z / d T = 0 we have  U.dAo) - n 7 | ) z + z/f+ 1 =• 0 2  (F.3)  2  The r o o t s o f e q u a t i o n ( F . 3 ) a r e  1/7 V  z ±  V ? ^ 4 (nVf*-  ( n /5 2  =  2  z  By p l o t t i n g t h e v a l u e s o f z z  1  £ (?A.)) r  ( p  .  •  M7A.)) as a f u n c t i o n o f 7 i n t h e  v s . f p l a n e , we o b t a i n c u r v e s a l o n g w h i c h  dz/dT  =  0. The  t y p e o f c u r v e s expected a r e s k e t c h e d l n F i g u r e F . l and t h e s i g n o f dz/d"5 i n t h e v a r i o u s r e g i o n s bounded b y t h e s e i s indicated.  The v a l u e  ~$ =  curves  a t which the r o o t s o f  e q u a t i o n (F.4) c o i n c i d e , i s o b t a i n e d by s o l v i n g t h e e q u a t i o n  For  "J 5"fe» >  z  +  z ~ a r e complex.  F i g u r e F . l B e h a v i o u r o f Impedance W i t h i n a D i e l e c t r i c C y l i n d e r f o r a Core R e g i o n i n w h i c h ( e ( ? A ' ' ) - ' / ? * ) Becomes Positive at. dz/d."S = 0 A l o n g t h e Curves z and z . r  +  F i g u r e P.2  B  B  -  B e h a v i o u r o f Impedance W i t h an I n i t i a l V a l u e i n a R e g i o n Where ( M ^ A J - n V ^ ) Positive. 2  i  s  204  (3»7)> i t i s seen t h a t as 7 approach-  From e q u a t i o n  es z e r o , z. i s p o s i t i v e and approaches z e r o .  The b e h a v i o u r  of z_ w i t h i n t h e c y l i n d e r i s t h e r e f o r e o f t h e form shown: i n F i g u r e F.1, c u r v e s I t o I V . Any number o f c r o s s i n g s o f t h e z = 0 a x i s a r e p o s s i b l e i n t h e range f none c a n o c c u r f o r f  b  < 7 £ "?, ; however,  . We n o t e f u r t h e r t h a t o < /? - ? o  0 < z < c o  (F.5)  Effect of Increasing n From e q u a t i o n  ( F . l ) i t i s seen t h a t f o r  and z  fixed, dz  dz  dl  ~  d~?  n = n, The  e q u a l i t y a p p l i e s when z = 0.  n  z  > n,  n = n. Thusv, t h e b e h a v i o u r o f z  as n I s i n c r e a s e d f o l l o w s t h e p a t t e r n o f p r o g r e s s i n g from c u r v e I V t o c u r v e I i n F i g u r e F . l . F o r n l a r g e enough, a v a r i a t i o n s i m i l a r t o c u r v e I w i l l always be a t t a i n e d .  The  v a r i a t i o n f o r s m a l l e r n > 0 may be o f any t y p e shown d e p e n d i n g on t h e c y l i n d e r parameters o r may have more "J-axis c r o s s i n g s :than curve;:IVv< E f f e c t of Decreasing  £r(J/k. )  Decreasing dz;/d? as i n c r e a s i n g n .  0  cV(TAo) has a  s i m i l a r e f f e c t on  The. p r o g r e s s i o n from c u r v e I V t o  curve I I n Figure F . l i s therefore r e p r e s e n t a t i v e of the corresponding  p o s s i b l e b e h a v i o u r and a g a i n more Jr  axis  205  c r o s s i n g s may  occur.  We now c o n s i d e r an i n i t i a l v a l u e o f z, say a t  '"Si » i n a r e g i o n where n / ? " < 2  2  S^'TAo)-  i s : d e f i n e d as t h e v a l u e o f "S f o r w h i c h n / ^ 2  2  again  =  Xo.  ^ A J J  we i n v e s t i g a t e t h e "behaviour o f z i n t h e range  ~§i £ "f £ "5,  where  -5,  > £ (TA.)" r  P r o c e e d i n g as b e f o r e we p l o t z i n Figure F.2.  For z  ^  ,.  as a f u n c t i o n o f f *  ±  ,<  z  b  v a r i a t i o n s o f z a r e shown ( m u l t i p l e n o t shown, and f o r z  >  as shown  = - 2"?* , f o u r p o s s i b l e - axis crossings are  Zk two a r e shown.  As "before,  a d e c r e a s e i n t h e numlbers on t h e c u r v e s c o r r e s p o n d s  t o an  i n c r e a s e I n n o r a decrease i n £ . r  By s t u d y i n g t h e above two s i t u a t i o n s ,  illustrated  i n F i g u r e s F . l and F . 2 , we may I n f e r t h e e f f e c t w h i c h i n creasing n or decreasing Let c  A  £> has on z' = z  = (number o f t i m e s w h i c h z. (number o f t i m e s z J ^  where z  J  i n generals:  5=5/ goes t o + oo  r e t u r n s from  )  + oo )  I m p l i e s z e v a l u a t e d f o r n = a o r £ {J/k ) r  How. i f or w i l l write  c  A ;  z'/  = c >  A Z  , z'/  z  '/  "  z  'l  0  0 - 7 -  8 (TAo) rA  206  U s i n g t h i s n o t a t i o n , we have 1)  I f n , < n , then z'/  ii)  If  >  2  £,,(IAJ  z'l  4  Equation(p 5) gives the f u r t h e r 0  i i i ) I fn /f* > z  0 < 5 £ ? / ,  <S {l/k ) rz  0  then  W  relation  6 (lA ) r  z'/  e  0  ^ T - "S, , t h e n  0 < z'<#>  The b e h a v i o u r o f z i n a medium s u r r o u n d i n g a c o n d u c t i n g c o r e i s r e a d i l y seen by i m a g i n i n g c u r v e s  similar  t o t h o s e s h o r n i n F i g u r e s F . l and F . 2 s t a r t i n g a t a p o i n t on t h e f » a x i s .  Clearly,  i ) , i i ) and i i i ) a r e a p p l i c a b l e ,  t h e r a n g e o f "5 b e i n g l i m i t e d t o  where  f r  F.2  z  z  ~ kY 0  z  - radius of conducting core.  Perpendicular Polarization R e w r i t i n g e q u a t i o n ( 2 . 2 9 a ) i n a form s i m i l a r t o  ( F . l ) , we have  dy  ~  d^  (F.6)  = g ( n , ~§ , y ) n*  where g ( n , J, y ) =  M ? A * ) + y/S + (  1 - —  . ) y*  M?AJ? Setting  g = 0 and s o l v i n g f o r y, we o b t a i n t h e f o l l o w i n g  e q u a t i o n s f o r c u r v e s a l o n g which d y / d f = 0 :  207  1/7  ^  +  t j l / f  y - M?A«) -  -  1  ^  ( nV?  :—7~ 2  ( n*/T  - M?A.)  )  —:  - MIA*)  )  (F.7) The r i g h t hand s i d e o f e q u a t i o n (P.7) f r o m t h a t o f e q u a t i o n (P.4) o n l y by t h e f a c t o r  differs  £>(7Ae>)-  S i n c e y "becomes complex i f a z e r o o f p e r m i t t i v i t y o c c u r s , we l i m i t t h e d i s c u s s i o n t o cases i n w h i c h t i s r  e i t h e r p o s i t i v e or negative. £f>  0  Por  MIAJ >  0,  0  <*?  < f,»  a p l o t o f y and +  y~ v s . f i s s i m i l a r t o t h e c u r v e s f o r z"*" and z ~ shown I n F i g u r e s F . l and F . 2 , equation  ( 3 . 1 1 b ) ,  The v a l u e o f y as f - > 0 , g i v e n "by  i s p o s i t i v e and approaches z e r o , and hence  the o t h e r c u r v e s i n F i g u r e s F . l and F . 2 . a l s o r e p r e s e n t p o s s i b l e v a r i a t i o n s o f y.  A d e c r e a s e i n t h e number o f t h e  c u r v e a g a i n i n d i c a t e s t h e e f f e c t o f I n c r e a s i n g n o r dec r e a s i n g Ef-o decreases: from whether  I f a conducting core i s present, y i n i t i a l l y + oo o r i n c r e a s e s from -co  depending on  i s l e s s than o r g r e a t e r than " ? r e s p e c t i v e l y . a  I t i s r e a d i l y seen t h a t r e l a t i o n s 1), i i ) , a p p l y i n t h i s case w i t h y'= y /  and i i i )  replacing z l  / : ? = -?/ £>  <  0  Plotting y  +  and y~ i n t h e y v s . f p l a n e , we ob-  t a i n c u r v e s o f t h e t y p e shown i n F i g u r e F . 3 .  T y p i c a l behav-  i o u r o f y f o r a d i e l e c t r i c c y l i n d e r ( c u r v e s I and I I ) and  208  F i g u r e F«3  Behaviour  o f Admittance W i t h i n a D i e l e c t r i c  C y l i n d e r ( I and I I ) and a D i e l e c t r i c a Conducting  Surrounding  Core ( i and i i ) when. S  dy/d'§ .= 0 a l o n g y  r  +  < 0,  and y " .  f o r a d i e l e c t r i c c y l i n d e r w i t h a c o n d u c t i n g core (curves i and i i ) a r e a l s o shown ( s i n c e t h e c o n d i t i o n • £ {f/k\o)'^ n / y i s 2  z  r  s a t i s f i e d f o r a l l values of 7 , slhilities).  Figure F.3  shows a l l pos~  A decrease i n the number o f the curve  cates the e f f e c t which an i n c r e a s e i n n o r £»- has behaviour o f y .  o  n  indithe  As a r e s u l t , we g e t the f o l l o w i n g r e l a t i o n s  comparable w i t h i ) ,  i i ) and i i i ) i n S e c t i o n F . l :  209  i)  I f n, < n , then y ' /  >  .ii)  I f £„(TAJ  ) f o r 0 S\ 1 < X ,',then  z  0 > z ' > «  r  >7'l  7'I iii)  < £?.( A  co  y' /  APPENDIX G E f f e c t i v e P e r m i t t i v i t y of the A r t i f i c i a l D i e l e c t r i c \Je c o n s i d e r a wave p r o p a g a t i n g i n t h e medium shown i n F i g u r e G . l .  i n the z - d l r e c t i o n  The f o r m o f t h e r e q u i r e d ;  s o l u t i o n i s e a s i l y o b t a i n e d by comparing t h e problem w i t h that of propagation  o f a s u r f a c e wave a l o n g a d i e l e c t r i c  (4-3)  slab  .  ¥e a r e guided i)  by t h e f o l l o w i n g c o n s i d e r a t i o n s :  The l a y e r s a r e t h i n enough t h a t o n l y t h e f u n -  damental mode p r o p a g a t e s . .  i i ) The t r a n s v e r s e f i e l d  components s h o u l d be  much g r e a t e r t h a n t h e l o n g i t u d i n a l f i e l d component and s h o u l d be " c o n s t a n t " i n each r e g i o n i n t h e t r a n s v e r s e  plane.  e,  D  T ± T  2t 2d  F i g u r e G . l L a y e r e d Medium o f t h e Type Used f o r t h e A r t l ficlal  Dielectric  211  TM Mode The TM .mode has t h e f i e l d components H , E„, E . y  I f we assume  t h e n t h e r e q u i r e d forms o f H  z  y  i n the  two r e g i o n s a r e Region  (1)  H  * A cosh(. p(x-D) ) e " ^ J  y  x  Jx-D f < d Region  (2)  H  v  = B Cos( h(x-T) ) e ~  (G.la)  j / ? 2  (G.lh) /x-T/ < t where D and T denote t h e p o s i t i o n o f t h e c e n t r a l p l a n e o f a l a y e r o f d i e l e c t r i c 1 and 2, r e s p e c t i v e l y d = h a l f - t h i c k n e s s o f Region  (1)  t = h a l f - t h i c k n e s s of Region  (2)  The wave numbers p and h and t h e p r o p a g a t i o n constant  a r e c o n s t r a i n e d by t h e wave e q u a t i o n as f o l l o w s : Region  (1)  p  Region  (2)  - h  The r e q u i r e m e n t  2  - /?  + kl €  -  + kl  z  =  ri  E  Yz  0  (G.2a).  = 0  (G.2b)  t h a t t h e p r o p a g a t i o n c o n s t a n t be t h e same  i n both regions y i e l d s p  2  + h* = ( E  rz  - E  From t h e r e l a t i o n V X H  ri  ) kl  (G.3)  = 3u£E we o b t a i n , f o r E  z  212  R e g i o n (1)  E  z  -  R e g i o n (2)  E  2  =-  A p s i n h ( p(x-D) ) e  3»£  J  B h s i n ( h(x-T) ) e "  J / 3 z  and f o r E -A  1  R e g i o n (1)  E  = 7— A 0 cosh( p(x-D) ) e~  x  jp:  Wei  ia*  1  R e g i o n (2)  E  - —  x  B ft cos( h(x-T) ) e"  JP  Note t h a t as d and t approach z e r o , the maximum value of E  2  approaches z e r o due t o t h e s i n h ( p(x-D) ) and  s i n ( h(x-T) ) f a c t o r s w h i l e H ues i n each r e g i o n . f i e l d components  y  and E  x  approach -constant  val-  Thus f o r t h i n l a y e r s , t h e t r a n s v e r s e  a r e indeed  dominant.  A p p l y i n g t h e boundary c o n d i t i o n s o f c o n t i n u i t y o f magnetic and e l e c t r i c f i e l d s a t a t y p i c a l boundary, s a y !  x = D + d gives A cosh(pd) = B cos( h ( - t ) ) •= B c o s ( h t ) f,Ap  (G.4b)  s i n h ( p d ) = £, B h s i n ( h t )  On d i v i d i n g ' e q u a t i o n ( G . 4 b ) by e q u a t i o n  (G.4a)  ( G . 4 a ) , we o b t a i n a  second r e l a t i o n s h i p between p and h namely £ -p t a n h ( p d ) = S, h t a n ( h t ) z  (G.5a)  or p tanh(pd) = £  K )  h tan(ht)  (G.5b)  213  E q u a t i o n s ( G . 3 ) and ( G . 5 b ) determine of  p and h.  the eigenvalues  The c o r r e s p o n d i n g p r o p a g a t i o n c o n s t a n t may  be c a l c u l a t e d u s i n g e i t h e r ( G . 2 a ) o r ( G . 2 b ) .  then  Thus, f o r  example, ( G . 2 a ) g i v e s p  2  + £  n  k*  (G.6)  I f we d e f i n e t h e e f f e c t i v e r e l a t i v e p e r m i t t i v i t y by  = k  0  t  re  ,  we have  A l t h o u g h t h e s o l u t i o n o f e q u a t i o n s ( G . 3 ) and must, i n g e n e r a l , be performed an approximate  (G.5b)  g r a p h i c a l l y or numerically,  s o l u t i o n i s e a s i l y obtained u s i n g the s e r i e s  e x p a n s i o n f o r tanh(pd) and t a n ( h t ) i f pd and h t a r e b o t h much l e s s t h a n u n i t y . we a r e concerned  These c o n d i t i o n s w i l l be s a t i s f i e d s i n c e  with thin layers.  The r e q u i r e d e x p a n s i o n s  are  tanh(pd) = pd -  (pd)  3  *•  tan(ht)  = ht +  3  - ...  15  3 (ht)  (pd)""  3  (ht) " 5  15  U s i n g t h e f i r s t terms o n l y , e q u a t i o n ( G . 5 b ) becomes  214  Substituting f o r h p  from e q u a t i o n (G.6) g i v e s = k* E  z  ( £^  r)  - i  )  ri  t/( E  ri  t + E, J ) r  (G.8) A more a c c u r a t e r e s u l t i s o b t a i n e d u s i n g t h e f i r s t two terms o f t h e e x p a n s i o n s .  I n t h i s case, e q u a t i o n (G.5t>)  becomes  V  P  P d  z  hV  2  )  2  E, (  3  £„( h*t +  -  3  )  and t h e r e f o r e p  2  7b* - 4ac )/2a  = ( b -  (G.9)  a = t / 3 + £ d /3  where  3  3  R  b « t + £„ d + 2 £ c = E  D  £R  -  ( t +  &r£ / S  k* t / 3 3  B  k* t / 3 ) 3  n  The maximum percentage; e r r o r i n the:, e f f e c t i v e p e r m i t t i v i t y c a l c u l a t e d u s i n g p* g i v e n by e q u a t i o n s (G.6) and (G.9) i s p l o t t e d as a f u n c t i o n o f E. . /E . v  iL  r  i  f o r various values '  o f (d + t j / ^ e , i n F i g u r e s G.2a and b. TE Mode T h i s mode i s c h a r a c t e r i z e d by t h e f i e l d E , H y  x  and H . 2  components  The s o l u t i o n i s f o r m a l l y t h e same as f o r  the TM mode ease.  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