"Applied Science, Faculty of"@en . "Electrical and Computer Engineering, Department of"@en . "DSpace"@en . "UBCV"@en . "Parkinson, Robert George"@en . "2011-06-23T20:18:42Z"@en . "1969"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "The problem of normal-incidence scattering by isotropic cylinders with arbitrary radial permittivity variation\r\nand by perfectly conducting cylinders surrounded by radially inhomogeneous isotropic shells is studied. Two types of approximation are considered, namely (i) an approximation\r\nof the permittivity variation which allows the use of a power series solution of the wave equation and (ii) approximation\r\nof the cylinder by a layered structure. For the latter type, computations are carried out for homogeneous shells and for shells with linearly-varying permittivities. The results are compared with those obtained by numerical integration of the Riccati-type differential equation for impedance or admittance.\r\nIn general, the homogeneous-shell approximation appears to be easiest to apply and requires a relatively short computation\r\ntime.\r\nIt is shown that the scattered-field coefficients can be calculated from measurements of the scattered, field at a single radius by applying a Fourier least-squares fit to the data. The scattered field for plane-wave incidence can therefore\r\nbe calculated from that for cylindrical-wave incidence, which suggests a more compact system for experimental investigation.\r\nCylinders with a \"smoothly\" varying permittivity were constructed using a certain type of artificial dielectric. Measurements were carried out in a parallel-plate region for both plane and cylindrical wave incidence; the results obtained agree with computed results and disagree with some previously published theoretical results.\r\nAs an application, an investigation is made of the range of validity of a planar model when interpreting phase angle measurements on dielectric-enclosed and -unenclosed cylindrical plasmas."@en . "https://circle.library.ubc.ca/rest/handle/2429/35706?expand=metadata"@en . "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 of 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 the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the re q u i r e d standard Research Supervisor Members ofthe Committee Head of Department Members of the 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 t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Eu \u00C2\u00A3.c TAl&Al* g AJ G) *J & \u00C2\u00A3 AI *J 6> The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date Z\u00C2\u00A3r^ -2-0 , 6 r 49 ;5.1 Method of C o n s t r u c t i n g R a d i a l l y Inhomogen-eous 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 S e c t i o n 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 Coef-f i c i e n t Measurements 80 v i l i F i g u r e Page 5.5 Specimens used f o r the Waveguide Evalua-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 Recording Magnitude and Phase .. 89 5.7b ' System f o r Recording Quadrature Components 89 5.8 Assembly of the 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 of the Transmission-Line System \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 95 5*10 System f o r R o t a t i n g Centerpiece 101 5.11 C o n s t r u c t i o n of P a r a l l e l - P l a t e Region used f o r C y lindrical-Wave Incidence 102 5.12a C o n s t r u c t i o n of E x c i t i n g Probe 103 5.12b C o n s t r u c t i o n of Measuring Probe 103 5.13 Errors, i n the Measured I n c i d e n t F i e l d f o r Plane-Wave Incidence 105 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 the Measured I n c i d e n t F i e l d f o r Cylindrical-Wave Incidence 107 5.16 Scattered F i e l d f o r Plane-Wave Incidence and E r r o r s u s i n g Data from Plane and C y l i n d r i c a l Wave Incidence, r 0 = 4 cm. ..... I l l 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 the Experiments 114 5.18 . E r r o r s i n the Scattered F i e l d f o r Cylindrical-Wave Incidence 116 5.19 Scattered F i e l d f o r Cyl i n d r i c a l - W a v e Incidence and E r r o r s u s i n g Data w i t h r 0 = 4 cm. and r\u00E2\u0080\u009E = 8 cm. Metal r , = 3.5 cm. 118 5.20 Scattered F i e l d f o r Cylindrical-Wave Incidence and E r r o r s u s i n g Data w i t h r0 = 4 cm. and r 0 = 8 cm. Core- tY <= 2.54, 3 ^ = 1.5 cm.; S h e l l - A r t i f i c i a l 121 i x F i g u r e Page 5o21 Measured D i f f r a c t e d F i e l d s , r 0 - 4 cm. 125 6\u00C2\u00BBla C y l i n d r i c a l C o n f i g u r a t i o n 133 6 0 l b P l a n a r C o n f i g u r a t i o n 133 6.2 E l e c t r o n Density 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 of , / as Functions of 187 < = 0 z;=& E. 3 Comparison Between the Bound on ~BN and the Bound on K f o r f = 192 F. l Behaviour of 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 Region i n which ( \u00C2\u00A3y(J/k0) - n z / 3 f * ) Becomes P o s i t i v e a t \"5 = \u00E2\u0080\u00A2 203 F . 2 Behaviour of Impedance With an I n i t i a l x Value i n a Region Where ( ) - nV5 ) i s P o s i t i v e 203 F. 3 Behaviour of 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 Surrounding a Conducting Core when \u00E2\u0082\u00ACr < 0 208 G. l Layered Medium of the Type Used f o r the A r t i f i c i a l D i e l e c t r i c 210 x Figure Page G.2a Maximum Percentage Error i n tre Calculated using Equation'(G.6) 215 G.2b Maximum Percentage Error i n ty.e Calculated using Equation (G.9) \u00E2\u0080\u00A2 215 x i LIST OP TABLES Table Page 3.1 R e l a t i v e E r r o r i n the Value of k^o^O) Ca l c u l a t e d Using the Shell-Approximation Methods 52 3.2 \u00E2\u0080\u00A2 O v e r a l l Accuracy of the S h e l l Methods ..... 57 3 .3 R e l a t i o n s h i p Between Accuracy and S h e l l Thickness 58 3.4 Comparison of the Methods of Computation .. 59 5.1 Comparison of Measured and T h e o r e t i c a l Values of 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 the Samples of A r t i f i c i a l D i e l e c t r i c 82 5.2 Range of D i f f e r e n c e Between T h e o r e t i c a l and Measured Re s u l t s f o r the Specimens of 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 5 . 3 Comparison Between Values of B a c k s c a t t e r i n g Cross-Section C a l c u l a t e d T h e o r e t i c a l l y and from Experimental Data 115 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 Crocs-Section 129 6 .1 Comparison Between c?r(\u00C2\u00B0\u00C2\u00B0) f o r a Plasma C y l -i n d e r and q>n f o r the P l a n a r Model 140 6 .2 Radius i n Wavelengths at which IT(r) = N c . . 142 6.3 Comparison Between ^ ( r , ) f o r a Plasma C y l -i n d e r and (pr f o r the P l a n a r Model 143 6.4 Comparison Between , = a f n or &g\u00E2\u0080\u009E (Chapter 4) = A \u00E2\u0080\u009E / E 0 (Appendix E) a ^ , &g. = n t h order c o e f f i c i e n t s of the r e a l and \" * imaginary parts of the s c a t t e r e d f i e l d at r =. r 0 AY, = 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 A \u00C2\u00BBe \u00C2\u00BB A>,e = A \u00E2\u0080\u009E 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 plane 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 Ay./, \u00C2\u00BB A-\u00E2\u0080\u009E\u00C2\u00A3 = A-,, f o r perpendicular p o l a r i z a t i o n w i t h plane 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 b* b<: \u00C2\u00BB ~b-n> B = constants C^ = c o e f f i c i e n t of U w and V\u00E2\u0080\u009E i n the:m t h r e g i o n c , c \u00E2\u0080\u009E s. constants C-n = U n or VT, d = h a l f - t h i c k n e s s of medium 1 of the a r t i f i c i a l d i e l e c t r i c D = d i s t a n c e to l i n e - s o u r c e ( S e c t i o n 2.4) = l o c a t i o n of the mid-plane of medium 1 \"of the a r t i f i c i a l d i e l e c t r i c (Appendix G) e-n = Neumann constant, \u00C2\u00AB= 1 f o r n = 0; e-\u00E2\u0080\u009E = 2 f o r n i l E , = e l e c t r i c f i e l d E c = magnitude of the e l e c t r i c f i e l d of the i n c i d e n t plane wave E 1 , E s , E\u00C2\u00B0, E'v\" = 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 t h r e g i o n e l e c t r i c f i e l d s x i i i E K , E^., Ej. = r a d i a l , angular and a x i a l components of e l e c t r i c f i e l d E^ = a x i a l component o f the s c a t t e r e d e l e c t r i c f i e l d t runca ted a t n = N = 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 s f = f u n c t i o n fx, f 5 , f 0 = r e a l p a r t of the i n c i d e n t , s c a t t e r e d and d i f f r a c t e d f i e l d s a t r = r\u00E2\u0080\u009E F v, .. - P - v a r i a t e w i t h n , and n 2 degrees o f freedom = va lue o f F * , , , - n , , such tha t the p r o b a b i l i t y ' \u00E2\u0080\u00A2 o f exceeding t h i s va lue i s cc g - f u n c t i o n S i \u00C2\u00BB S s \u00C2\u00BB So - imag ina ry pa r t o f the i n c i d e n t , s c a t t e r e d and d i f f r a c t e d f i e l d s a t r = r 0 h = wave-number i n medium 2 o f the a r t i f i c i a l d i e l e c t r i c H = magnetic f i e l d H 0 = magnitude of the magnetic f i e l d o f the i n c i d e n t plane wave H 1, H 5 = 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 H K, EQ., EZ = r a d i a l , angular and a x i a l components of magnetic f i e l d H^2) = n^*1 o rde r Hankel f u n c t i o n of the second k i n d t h I n = n o rder modi f i ed B e s s e l f u n c t i o n of the f i r s t k i n d JT, = n^*1 o rder 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 = to J/L =0 H;(r ,e).= - - ^ - l e , A w H(\u00E2\u0080\u009E2) (k\u00E2\u0080\u009Er) cos(ne-) (2.3b) H;(r,G-) = - ^ l e , [ U^k.r) . > c : ( l - U v X r ) ] cos(nG-) 1 < m < M (2.3c) where the ( x ) indicates a derivative with respect to the argument. The coefficients A\u00E2\u0080\u009E are now found by matching the tangential f i e l d components at r = r^, m = 1, 2, M. \" t i l We obtain the following system of 2M equations for the n mode: 11 - H ^ ( k 0 r , ) A\u00E2\u0080\u009E + l U k , r , ) B; + V n ( k , r , ) o ; H ^ ( k 0 r ; ) A w +-^- U \u00C2\u00BB ' ( k (r ( ) B'\u00E2\u0080\u009E +-~r- v'Oc.r, ) o'r, - U,(Jc,.,pJ = (3)\" E. Jn(k\u00E2\u0080\u009Er, ) m \u00C2\u00AB 1 (2.4) - V \u00E2\u0080\u009E (km., ) 0?\" + U\u00E2\u0080\u009E ( k ^ ) B\u00E2\u0084\u00A2 + (!-&,\u00E2\u0080\u009E> ^ ( k ^ ) 0* = 0 j f A - i ^ntkyn.jr^) By, ~r V ^ k ^ . r J c r ' +77~ u ' t k ^ r j B; Mr Mr 2 < m < M Cramer's r u l e g ives where A = 2m-l ^2 2m-2 AM J\u00E2\u0080\u009E ( k a r , ) U \u00E2\u0080\u009E ( k ( r , ) J ' ( k . r , ) - ^ r - u ' ( k , r , ) E, V^Ck.r, ) -^r- V*'(k,r, ) (2 .5a) - V. (k,., r j t U k ^ ) M k ^ r J \"JJ^-I (k^_, r^) U w (k>\u00E2\u0080\u009Er^) j^yt 2 $ m < M-1 1 V ' O C A ) 12 AM * - U, (k M\u00E2\u0080\u009E r M ) ' - V\u00E2\u0080\u009E (kM_, r w ) Un ( k M r M ) _ ~ jut*'1 (k^_, r M ) *\" Jix^i-l ^n-^-nn-i ~X-M ^ ~JjJ* ^ \u00C2\u00BB ^ M ^ V V J 5 \u00C2\u00AB= determinant of the array formed by replacing .\u00E2\u0080\u00A27 :,. j J^k.r, ) by I l f (k pr, ) and. j'(k 0r, ) by; H^'tk.r, ) in i n the f i r s t column of the A array In order to indicate parallel polarization, equation (2.5a) may be written as e The other coefficients B^, B*, . . . , B * , C ^ t C\u00E2\u0080\u009E , Cy, may be determined i n a similar manner. Since we are only concerned with the scattered f i e l d , however, the expressions for these coefficients are not given. If the core i s a perfect conductor, the f i e l d s are identically zero for r) This was noted previously by Adey ' i n the case of a cylinder made up of two coaxial homogeneous regions. 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 com-ponents are H z, E r , B&<, The analysis procedure is: the same as 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 and i n f a c t , equations-(2.1) to (2.5a), with the exception of (2.2), apply i f E i s replaced by H,: H by -E, 8\u00E2\u0084\u00A2 and \u00C2\u00A3 c by JJ.\u00E2\u0084\u00A2 and jx-o, and ft and , /<.\u00E2\u0080\u009E by \u00C2\u00A3\u00E2\u0084\u00A2(r) and S00 Equation (2.2) i s now replaced by (2.6) which i s derived i n Appendix A. We note that equation (2.6) has s i n g u l a r i t i e s at zeros of \u00C2\u00A3^(r) i n addition to the sing-u l a r i t y at the o r i g i n . As a r e s u l t , t h i s case presents so l u t i o n d i f f i c u l t i e s not encountered i n the previous case. Equation (2.5b) becomes Kh = - U f i r - H 0 (2.7) Oh When the core i s a perfect conductor, the boundary condition of zero tangential e l e c t r i c f i e l d at r=r M requires - < (k M.,r M) Bj-' - V ^ r J c r ' \u00C2\u00AB 0 As a r e s u l t , /\ M becomes A M = [ - vU*\u00C2\u00AB-, r M ) - < ( k M _ , r j ] 14 2 . 1 - 2 Solution Usln^ Impedance Boundary Conditions / An alternative method of calculating the scat-tered-field coefficients i s to apply an impedance boundary condition at the surface of the cylinder and use non-method, which he notes i s desirable i f an iterative procedure i s to be used. Again, we w i l l deal with the two polari-zations separately. Parallel Polarization The tangential electric and magnetic fields i n the region outside the cylinder are found by adding equation ( 2 . 1 a ) to ( 2 . 1 b ) and equation ( 2 . 3 a ) to ( 2 . 3 b ) . Thus we have, from ( 2 . 1 a ) and ( 2 . 1 b ) uniform transmission-line theory as given by Schelkunoff to. calculate this impedance. Wait 5 has used this (ZB) ( 2 . 8 ) and from ( 2 . 3 a ) and ( 2 . 3 b ) j . . He ( 2 . 9 ) Por the n mode, we define an impedance relating E 2 and at r=r, by ( 2 C 1 0 ) 15 Substituting from equations (2.8) and (2.9) into equation (2.10) gives -E 0 (.if M k c r , ) + A, Hff.(lc.r. 1 (2.11) Solving for Ar, we obtain A n ~ \" (3)\" E, k^r f k,r, ) + 3 z\u00C2\u00A3Au~HT? (2.12) The impedance Z^, i s determined by using an analogy be-tween the concentric cylinders and cascaded sections of non-uniform transmission-line. The transmission-line prob-lem i s illustrated i n Figure 2.3. (23) Following Schelkunoff , we define an \"incident wave\" (+) and a \"reflected wave\" (-).in the m^1 region by EST* V.dc^r) Thus, the wave impedances are v-*#v > E \" \" ^ M ^ r ) (2.13a) (2.13b) If E ^ and H^, denote the total tangential electric and magnetic fi e l d s respectively, the impedance til looking into the m region, i n the direction of decreasing 16 n F i g u r e 2 . 3 Cascaded Non-Uniform Transmission-Line Analog of the Inhomogeneous C y l i n d e r We can w r i t e the Impedance l o o k i n g i n t o the \"til \"fcll m r e g i o n i n terms of t h a t l o o k i n g i n t o the (m+1) r e g i o n as where (2.14) 17 K;\"(A) + C V\u00E2\u0080\u009E(A.) v'(*U) TC^W, ,r\u00E2\u0080\u009E) = When the expressions f o r K\u00E2\u0084\u00A2+ and given by equations (2.13) are substituted i n t o equation (2.14) and the r e -s u l t s i m p l i f i e d , we obtain Z? - 3*1 rJ-rn + \ 1 -S -m < M +\"3 T.-'T* where p^=/M\u00C2\u00AB\u00C2\u00BB0 v ' ( A . ) -\"M**,) u ' ( A . ) s;>u'(\u00C2\u00AB^ ) v ' ( A J - u'(/?J C = u \u00C2\u00BB ( A ) <(\u00C2\u00AB*.) - v,, (/\u00C2\u00A3J u'(*w) (2.15) In the core there l s no \"r e f l e c t e d wave\" and therefore z ; = x r ( ^ ) = 3-1. - 7 7 \u00E2\u0080\u0094 (2.16) Now, by the systematic ap p l i c a t i o n of equation (2.15) 18 we obtain , K \u00C2\u00BB \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00C2\u00BB For a perfectly conducting core, Z^ , = 0. Perpendicular Polarization Tne solution i n the case of perpendicular pol-arization may be easily derived from that i n the case of parallel polarization 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 result, the im-pedances become admittances. Thus, the scattered field! coefficients are given by A\u00E2\u0080\u009E ~- (3) H, J->,(k0r, ) + 3 Y^/r0 J\u00E2\u0080\u009E(lc0r, ) H\u00E2\u0084\u00A2 (k,r, ) + 3 Y ; / ^ O (k 0r, ) where Y ^ = H Z 7 ?(r , ,e-) E e r,(r, ,0-) (2ol7) The iteration formula ( equation (2.15) ) becomes Yr =\u00E2\u0080\u00A2 3 2 U r w ) r\u00C2\u00AB1 + i ri7*! S\u00E2\u0084\u00A2 + 3 yn,(r w + l ) T \u00E2\u0080\u00A2n 1 5 j < M (2 .18) where the expressions for P\u00E2\u0084\u00A2, Q\u00E2\u0084\u00A2, S\u00E2\u0084\u00A2, and T\u00E2\u0084\u00A2 are as given i n equation (2.15), U-w and Vy, being the solutions of equation (2.6) i n this case. For a dielectric core> Yy, \u00E2\u0080\u0094 3 ( ^ ftl ) / I M * \u00E2\u0080\u009E ) and for a perfectly conducting core, Y \" = co \u00E2\u0080\u00A2 (2.19) 19 2 \u00C2\u00AB 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 analysed by u s i n g the i t e r a t i v e formulas of 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 of v a n i s h i n g t h i c k -ness, a R l c c a t i - t y p e d i f f e r e n t i a l equation f o r impedance or admittance i s d e r i v e d . This equation i s convenient t o use i f a numerical 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 proves u s e f u l i n determining upper l i m i t s on the magni-tudes 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 . The problem i s i l l u s t r a t e d i n F i g u r e 2.4 and the s u b s c r i p t (m) i s dropped i n order to s i m p l i f y the 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 f o r the T h i n - S h e l l Problem compared w i t h the wavelength corresponding t o the maximum p e r m i t t i v i t y i n the s h e l l 20 Parallel Polarization Equation ( 2 . 1 5 ) &ay be rewritten as Z* P^ + 3 \ Qr, zj, = 5 \ \u00E2\u0080\u0094 ( 2 e 2 0 a ) Z* S\u00E2\u0080\u009E + J -T. T\u00E2\u0080\u009E where P\u00E2\u0080\u009E = TM\u00C2\u00AB.) v'(/2) - V^ar) U^V)-\"Q\u00E2\u0080\u009E = u^o?) V\u00E2\u0080\u009E(*) - V^{0) U (<*) Sn = u'(rf) v'(/) -'v '(\u00C2\u00B00 T j ' ( / \u00C2\u00A3 ) TY= M/?). v'(tf) - VL,(/?) u'(a) For a thin shell, T-oc0 iff.\u00E2\u0080\u00A2\u00E2\u0080\u00A2small and the following ex-pansions give good approximations:: / / \" // ( 2 . 2 1 ) where \"C^ may be either or V^. i t is: therefore possible to eliminate OL as an argument of these functions i n equation ( 2 g 2 0 a ) . Substituting from ( 2 . 2 1 ) into the. expressions for P\u00E2\u0080\u009E , Q\u00E2\u0080\u009E, S,, and and neglecting terms with a power T% we obtain Sn ^ ( n V / - M / W ) f w . Tn \u00C2\u00AB (1 - V ^ ) >^ where = TM/?). V \u00E2\u0080\u009E V ) - u ' ^ ) V ^ ) ( 2 . 2 0 b ) 21 In a r r i v i n g at the expressions f o r amd Tr, i n equation (2,20b), second derivative terms are elimin-ated by making use of the f a c t that U^ , and Vn s a t i s f y equation (2.2) c With these approximations, equation (2\u00E2\u0080\u009E20a) becomes Z* + 3 \ ^ z ; ^ 3 ^ ~ (2.22a) Z* ( n V ^ A - SrifA) - )r + 3 >l ( I - W or (2.22b) 1 - [ ( tr{J/k.) - n V / 2 ) z * + V / ] t where z'n = Zj./J'T. < = Z*/3 ^ Note that equations (2.22) involve the r e l a t i v e p e r m i t t i v i t y at r=r^ only. The change i n impedance i s therefore 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 \u00C2\u00A3(r*). Ke now 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 values of z* ( i f z* i s large we may deal with y*=l/z*) we have and therefore l - [ ( - W ) < + ] t ^ 1 + [ ( EyWA) - nV/? z ) z* + l / * ] r 22 Again neglecting terms i n t 2 , equation ( 2 . 2 2 b ) may now be s i m p l i f i e d to z4 - < * [ l + J or * + (2,26b) * \u00C2\u00A3,(/?A) 1 - f ( 1 - n*/( U*A> ) ) y-J . + ( 1/^ - r ) ] ? where . y\u00E2\u0080\u009E' =' - 3 T. ? i y* = - H The derivation of the d i f f e r e n t i a l equations for admittance and impedance follows the pattern used i n the case of parallel polarization. For 24 ( 1 -\u00C2\u00A3,(/?A) P ) y* + ( l / * elm) t \u00C2\u00AB 1 (\u00E2\u0080\u00A22.27) we have \u00E2\u0080\u00A2n ~ M * A ) y,1 + < i \u00C2\u00A3r(/?A>/? 2. \" \" 7 ) + ( i / * - ^ ( W } ) y - r + \u00C2\u00A3 r 0 4 * > t \u00C2\u00A3,(/?A) Eliminating \u00C2\u00A3,.(a;A) by using the expansion \u00C2\u00A3,(^A) \u00C2\u00BB \u00C2\u00A3 r(/?A) + J ^ ^ A ) f gives:: - y * ~ W A ) + 71//S + ( 1 \u00E2\u0080\u00A2 ) y* (2.28) and therefore dy\u00C2\u00BB ) 7 i * ( 2 . 2 9 a ) Letting z ^ l / y ^ we obtain dT ( 1 -\u00C2\u00A3,(?A) T ) + 2T\u00E2\u0080\u009E / f + \u00C2\u00A3J[S/lc)2* ( 2 . 2 9 b ) Equations ( 2 . 2 9 ) may also be derived by substituting 25 v = -1-1\u00E2\u0080\u0094 ( 2 . 3 0 a ) dR\u00E2\u0080\u009E/d? \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 . cVdRn/a? 2 = \u00E2\u0080\u0094 \u00E2\u0080\u00A2 ( 2 . 3 0 b ) MJA) R M Into equation (2.6). 2 . 2 - 1 The Case of a Thin Shell on a Conducting Core We now consider the effect which a thin dielec\u00C2\u00AB t r i e sheath with a relative permeability of unity has on the scattering properties of a conducting cylinder. Letting r* be the radius of the conducting cylinder, the result may be derived by setting fUyu0 and Z*=0 l n Section 2 . 2 . Parallel Polarization With Z*=0, equation ( 2 . 2 2 a ) becomes l \" l \u00C2\u00AB X//3 I f the shell i s thin compared to the radius of the con-ducting cylinder, we have and hence Z ^ j - ^ r . Substi-tuting this value for zj, i n equation ( 2 . 1 2 ) , we have A - '\u00E2\u0080\u00A2 (k*r( ) - v J^ Ck^ r, > - \u00E2\u0080\u00A2:?.;\u00E2\u0080\u00A2:: (3 ) E P . : \u00E2\u0080\u0094 r ( 2 . 3 1 a ) (k0r, ) - T H g ' U c r , ) Since f i s small we may write ^(k^) v Jntk^ r, ) - tr j'(k\u00E2\u0080\u009Er, ) H \u00C2\u00A3 * (k,rz ) ^ H^(k.r ; ) ~ f H^k.r, ) 26 and thereby simplify equation (2.31a) to The right hand side of equation (2.31b) i s the th scattered f i e l d coefficient of the n mode for a conducting cylinder of radius r*.. \u00C2\u00A5e conclude therefore that, i n the case of parallel polarization, the scattered f i e l d of a con-ducting cylinder i s not altered appreciably by a thin dielectric s h e l l . Evidence of this i s the fact that a plot of back-scattering cross section as a function of sheath radius starts with a zero, slope at the surface of the conducting cylinder as can be seen i n Figure 3\u00C2\u00ABi,'\u00C2\u00BB 13) Yen and Kaprielian v ' have derived equation (2.31a) and from computed results i n the special case \u00C2\u00A3.t(T)=o/r have observed that the effect of a thin shell i s small. Their results do not agree with present calculations i n general however, as shown i n Figure 3,4, and the \"small effect\" observation i s violated i n their results shown i n Figure (3.4b). Perpendicular Polarization Letting Y* approach i n f i n i t y i n equation (2.26a) gives: 3 V(r, ) * ~ ( Eri/?Ao) - n 2 / ^ ) t Substituting this value for Yt, i n equation (2.17), we have I (k\u00E2\u0080\u009Er, ) (2.31b) 27 J-n(k.r, ) - \u00E2\u0080\u0094 . f r t o * > } A \u00E2\u0080\u009E ^ - (j) E0 - \u00E2\u0080\u0094\u00E2\u0080\u00A2 (k.r, ) H^Uc.r, ) ( 2 . 3 2 ) No significant simplification i s evident i n this case and i t appears that the thin sheath has a noticable effect. 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 parallel polarization (Figure 3 . 4 ) . 2*3 Far-Zone Scattered Field It 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 simplified by using the (32) large argument asymptotic form of the Hankel function . ) ( ; - v r - i ) Substituting for H^ J(kor) i n equation ( 2 . 1 b ) we have, for parallel polarization, ~ - J ( k 0 r - - { ) lim E^(r,G-) = \T e\u00E2\u0080\u009E A\u00E2\u0080\u009E / - \u00C2\u00A7 \u00E2\u0080\u0094 e \u00E2\u0080\u00A2 cos(ne-) /r - 3 ( k.r - f ) \u00E2\u0080\u00A2k r : e ( 2 . 3 3 ) \u00E2\u0080\u0094 ~e * I e, ( J ) \" A,, cos(nS-) A quantity which l s often given i s the b i -statlc scattering cross section per unit length cf(&). 28 This i s defined as follows* p s E S CT-(e-) - - r lim 2/rr g <\u00E2\u0080\u00A2 r\u00E2\u0080\u0094*-co E l (2.3*0 where p 5 = total power reradiated per unit length of an ideal omnidirectional scatterer that maintains the same f i e l d at a radial distance r for a l l values of G- as that maintained by the actual scattering cylinder i n the direction e-SL s= real magnitude of the Poyntlng vector of the incident plane wave. Substituting the expressions for E 1 and E 5 given i n equations ( 2 . 1 a ) and ( 2 . 3 3 ) into ( 2 . 3 4 ) we have 1 4' oo x cT(e-) - \u00E2\u0080\u0094 I e\u00E2\u0080\u009E ( J f A\u00E2\u0080\u009E cos(ne-) ( 2 . 3 5 ) E * ko \u00C2\u00AB=o For perpendicular polarization, the above ex-pressions hold with E replaced by H. 2 . 4 Cylindrical-Wave Incidence In many situations where experiments are car-ried out under presumably plane-wave conditions, the curva-ture of the incident wavefront cannot be neglected and i t s effect has to be accounted for. This effect has been investigated by Faran( 3 4) and by Zitron and Davis^ 3 5^ i n the related problem of scattering of acoustical waves by 29 hard and s o f t c y l i n d e r s . Paran has shown t h a t a simple r e l a t i o n s h i p e x i s t s between 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 f o r plane-wave i n c i d e n c e and those 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 , A s i m i l a r r e l a t i o n holds i n the electromagnetic case, as w i l l be shown below, and'provides the b a s i s f o r the experimental method u s i n g c y l i n d r i c a l - w a v e Incidence d i s -cussed i n S e c t i o n 5\u00C2\u00BB2-2. F i g u r e 2 .5 I l l u s t r a t i o n of the Problem of C y l i n d r i c a l - W a v e 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 to the z - a x i s l o c a t e d outside the c y l i n d e r at a d i s t a n c e D along the 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 zero phase and an amplitude A a t r=0, then 30 El(i\>) = A H ( o z )(k 0r\u00E2\u0080\u009E ) H f ( k 0 D ) where r e = distance from the l i n e source. Since T0 - / D z - 2 D r cos\u00C2\u00A9- + r 2 - , we may use the expansion (3Z) k0 J Dl - 2 D r cose- + r 2 ) =X (k^D) ^ ( k ^ r ) cos(ne-) to obtain , Ef(r,e-) = \u00E2\u0080\u0094 fie?, H? }(k eD) J\u00E2\u0080\u009E(k,r) cos(ne-) (2.36) (k*D) ^ * The difference between equation (2.36) f o r c y l i n d r i c a l -wave incidence and equation (2.1a) f o r plane-wave incidence i s that A H t f^k^Dj / H^tk^D) replaces E c ( J ) 1 7 . I f we note that E 0 (j))* appears as a fact o r i n equation (2,5b) f o r the sc 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 (k,D) A \u00C2\u00AB \u00E2\u0080\u0094 Ane (2.37) E e ( J ) \" H f (k.D) where A \u00E2\u0080\u009E e = order 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 -d r i c a l incident wave produced by an e l e c t r i c l i n e -source. For a magnetic line-source producing an incident magnetic f i e l d r B H \u00C2\u00A3\u00C2\u00AB (KXo ) Ht(r 0 ) = H^(k.D) 31 we have where A-\u00E2\u0080\u009EH = order 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 d r i c a l i n c i d e n t wave produced by a mag-n e t i c l i n e - s o u r c e . 32 3\u00C2\u00AB CALCULATION OP THE SCATTERED FIELD The f i r s t step i n c a l c u l a t i n g the scattered f i e l d i s the determination of the c o e f f i c i e n t s A^. I t i s then necessary to decide upon the number of terms, to be used\ i n the expression f o r the f i e l d i n order to achieve a spec-i f i e d accuracy. In t h i s chapter, these two points are investigated i n the case of general p e r m i t t i v i t y variations\u00E2\u0080\u009E The r e l a t i v e permeability of each region i s assumed to be unity, 3.1 Methods of Calculating the Scattered-Field C o e f f i c i e n t s I f closed form solutions of the wave equation existed i n each region of the inhomogeneous cylinder, 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 could, of course, be determined i n a straightforward manner using the r e l a t i o n s i n Section 2,1e However, t h i s i s seldom the case. In t h i s section, we w i l l consider four methods, b a s i c a l l y f o r use i n regions where these closed form solutions are not a v a i l a b l e . In the f i r s t , a power series solution of the wave equation i s used. The second and t h i r d involve the c a l c u l a t i o n of the coefficients:-of an approximating cylinder formed of concentric s h e l l s with constant and 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 y , r e s p e c t i v e l y . In the fourth, a numerical integration of the impedance and/or admittance d i f f e r e n t i a l equations i s used giving e s s e n t i a l l y exact r e s u l t s . The methods are discussed with regard to d i f f i c u l t y of application, l i m i t a t i o n s , accuracy, and computer time required. 33 3\u00C2\u00BB1-1 Power Series Solution of the Wave Equation Solutions of the radial wave equation in the .54. * neighbourhood of a regular point 1 may be obtained as a power s e r i e s * 3 6 \ and i n fact this 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 \u00C2\u00A3r(r) i n the form of a ratio of polynomials. When \u00C2\u00A3r(r) i s not of this form, a Taylor series expansion may be developed and., by using a sufficient number of terms i n the expansion, solutions as accurate as desired can be obtained. Basically the pro-cedure i s simple. However, i n practice, the following two drawbacks exist: (a) As the degree of the polynomials describing \u00C2\u00A3 r(r) increases, the number of terms i n the recurrence re-lation for calculating the coefficients i n the series increas-es (very rapidly i n the case of perpendicular polarization). (b) An upper l i m i t i s placed on r and also on the magnitude of the coefficients i n the polynomials such that terms of excessively large magnitude relative to the sum of the series are avoided (these reduce the number of significant figures i n the sum). Singularities of 6 r(r) must be poles of order 1 or 2. Note however that the permittivity of the cylinder i s assumed f i n i t e and hence singularities of \u00C2\u00A3\u00E2\u0080\u009E(r) cannot occur within the region of interest. The behaviour of \u00C2\u00A3>.(r) outside this region i s of no consequence. 3^ The details of the series solution for a linear permittivity variation \u00C2\u00A3r(r) = sr + t are given i n Appendix B. Several of the relations i n the appendix were derived previ-ously by Feinstein^ ^ \ We w i l l now examine the manner in which the solutions are developed and the problems encountered i n this special case and then make further comments on the method i n general\u00E2\u0080\u009E Parallel Polarization When \u00C2\u00A3r(r) = sr + t, the series solution about r = 0, i s convergent for a l l values or r, s and t. This w i l l always be the case i f \u00C2\u00A3^ .(r) has no singularities for 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 detail when deal-ing with perpendicular polarization. An inspection of the recurrence relation i n the series solution rev.eals that, i f - f r ( r ) i s a polynomial, there-i s a linear relationship between the number of terms i n the recurrence relation and the order of the polynomial. This makes the series method quite attractive for regions of small radius, i n which case the series may be accurately summed, particularly i n the core region where only one solu-tion i s required. Perpendicular Polarization In this case, the series solution when \u00C2\u00A3t(r).- sr + t i s relatively complicated because: ' \"\" . (a) There are more terms i n the recurrence relation. 35 (b) I t Is not always possible to f i n d a point about which a solu t i o n w i l l converge throughout the required range. Referring to Figure B . l i t i s seen that there are two s i t u a t i o n s i n which a series s o l u t i o n about a single point i s i n a d e q u a t e . These are (1) r = 0 and r = - t/s within the region ( i l ) s and t of opposite sign, r ~ - t/s\u00E2\u0080\u00A2 and r = - 2t/s within the region. In such s i t u a t i o n s , the region concerned may ba divided in t o two sub-regions and then the solutions about d i f f e r e n t points matched using equation (2.18). The procedure w i l l be i l l u s t r a t e d by two: examples. Example 1 Core region with s > 0, t > 0, rM > t/s as shown i n Figure 3.1. \u00E2\u0080\u0094r- \u00E2\u0080\u00A2 1 r 1 1 s>-- V s r M 1 t / s r c r M r Figure 3.1 D i v i s i o n of a Core Region f o r Matching Series Solutions about D i f f e r e n t Points 36 The procedure i s as follows: ( i ) Define a new core region by r = r M ( , 0 < r M / < t / s . (11) Determine the admittance Y\"' looking i n at r = rMI from equation (2.19) where U-\u00E2\u0080\u009E i s given by the ex-pansion about r = 0 ( equation (B.14) ). ( i i i ) Calculate Y\" using equation (2.18) with :.\" = Y\"' \u00E2\u0080\u00A2 Ur, and are series solutions about r = r<> > t/s given by equations (B.19) with /80 = s r 0 / t . Example 2 S h e l l with t < 0 , s > 0, r w + )< - t / s , r ^ > - 2t/s> as i l l u s t r a t e d i n Figure 3.2. . c % ( r ) . A Figure 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 f o r Matching Series Solutions about D i f f e r e n t Points 37 Assuming Y\u00E2\u0080\u009E has been calculated, we proceed as follows: (1) Divide, the region into two at r = r w , - t/s < r w , < - 2t/s:. ( i i ) Determine the admittance Y\u00E2\u0084\u00A2 1 looking inwards: at r = r\u00E2\u0080\u009E, by, applying equation (2.18) with Uy, and V^ the solutions about r.= - t/s given by equation (B .17). ( i i i ) Calculate Y^ using equation (2.18) with Y\u00E2\u0084\u00A2 +' = Y\u00E2\u0084\u00A2' \u00E2\u0080\u00A2 U,, and V^ are series solutions about r = r 0 , r 0 > |(r > n+t/s) - t/s - K r^t/s), given by equations (B.19) with /30 = s r c / t . , We note that i n the expansion about r = ~ t / s , y is:: p o s i t i v e f o r r < \u00C2\u00AB t/s and negative f o r r > - t / s . The solu t i o n i n the l a t t e r case i s complex due to the ln(20 term and therefore Y\u00E2\u0084\u00A2 becomes complex rather than, purely imaginary. In order to determine the correct sign of the imaginary part of the logarithm, we consider a physical s i t -uation i n which a small l o s s i s present and determine the sign of the imaginary part of y at r = - t / s . A suitable example f o r the above s i t u a t i o n l s that of a plasma;where c o l l i s i o n losses are considered. The e l e c t r i c p o l a r i z a t i o n , P, i s given by* 1 ^ X p = \u00E2\u0080\u009E So E 1 - 3Z -:.:X,y. V N e 2 where X.= : ' \u00C2\u00A30 me CJZ 38 N =\u00E2\u0080\u00A2 electron number density e = electron charge mc= electron mass Z = v/co V = electron c o l l i s i o n frequency. The 3 Z term represents the loss and we assume Z \u00C2\u00AB 1. Since \u00C2\u00A3 =\u00C2\u00A3o+ \u00C2\u00A5/\u00E2\u0082\u00AC<,, a linear variation of electron density with radius corresponds to a linear variation of permittivity. We may write Ey (r) = sr + t ar + b = 1 ^ - (ar + b - 1) (1 + 3Z) - 3Z l - 3Z Here:, the constants s and t are assumed complex. Writing them as s = s 1 3 s \" and t = t* + 3 t I f , we find the follow-ing relations: . s* = -a s* 1 = s* Z t ' = 1 - b t\u00C2\u00BB 1 = Z ( t 1 - 1 ) Substituting these relations into the expression for / we obtain \u00E2\u0080\u00A2 2f = sr/t + 1 & ( s'r/t' + 1 ) + 3Z The imaginary part of y i s positive indicating that an argument of (-t-rr) must be used for representing y< 0 i n the series. '39 If \u00C2\u00A3,-(r) i s of a more complex form, the method of matching solutions about various points may s t i l l be applied. The recurrence relations i n the series rapidly become unmanageable, however. For example, i f \u00C2\u00A3r(r) i s a polynomial, the number of terms i n the recurrence i n -creases as the square of the polynomial degree as can be seen by examining equation ( B o l l ) , The case of a quadratic variation of permittivity has been considered to some extent* 8 * 3 * 1 0 ) 3\u00C2\u00BBl-2 Approximation of an Inhomogeneous Region by Homogeneous Shells. An inhomogeneous region may be approximated by a large number of thin homogeneous s h e l l s * 1 5 f l 7 ' ^ ^ \u00E2\u0080\u00A2 Real solutions of equations ( 2 , 2 ) and ( 2 . 6 ) , when \u00C2\u00A3 r(r) i s constant, are Bessel functions J^{/\u00C2\u00A3^k\u00E2\u0080\u009Er) and Y ^ ( 7 ^ k 0 r ) for \u00C2\u00A3\u00C2\u00BB- > 0 and the modified Bessel functions I ^ v O ^ ^ r ) and K\u00E2\u0080\u009E(/T\u00C2\u00A3k f fr) for Sr < 0. : The calculation of integer-order Bessel func-tions i s rapid using recurrence relations and polynomial 13 Z) approximations for the zero and f i r s t orders, . Large arguments (\u00E2\u0080\u00A2//erl k 0 r ) present no: problem (asymptotic forms may be used i f necessary) as i s the case with the method of series solution. Small arguments and large orders to-gether may present some computational d i f f i c u l t i e s because 3-n and approach zero and YT, and K,, approach i n f i n i t y . These may be d e a l t w i t h as discussed i n Appendix C where l t i s shorn t h a t the l i m i t i n g case of a zero p e r m i t t i v i t y I s p a r t i c u l a r l y simple. Lunow and T u t t e r v suggest t h a t the computational problems may be avoided by u s i n g J^isfJJj k 0 r ) 8 1 1 ( 1 1 \u00E2\u0080\u00A2\u00C2\u00BB(//\u00C2\u00A3,\u00E2\u0080\u00A2/\" k o r ) a s s o l u t i o n s i n a r e g i o n near a permit-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 \u00C2\u00A3 r. T h i s method seems to be unfounded mathematically and gives d i f f e r e n t r e s u l t s from those d e r i v e d i n Appendix 0. Choosing the S h e l l Parameters A method of determining the optimum t h i c k -ness and p e r m i t t i v i t y f o r the s h e l l s i s not obvious, i f at a l l p o s s i b l e . Here, the choice of these parameters i s : done i n two steps, f i r s t choosing the th i c k n e s s and then d e c i d i n g on the p e r m i t t i v i t y . Two approaches t o choosing the t h i c k n e s s are suggested ( i ) Choose the r a d i i such t h a t the change of B g n ( \u00C2\u00A3 K(r) ) \u00E2\u0080\u00A2 l\u00C2\u00A3r(r)l* , cc being a constant, i s equal f o r each s h e l l . 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 the power i n order to avoid d i f f i c u l t i e s when \u00C2\u00A3r(r) changes s i g n w i t h -i n the inhomogeneous r e g i o n . The s p e c i a l cases and cc = 1 correspond t o equal increments of r e f r a c t i v e index and p e r m i t t i v i t y , r e s p e c t i v e l y . This method tends to concentrate the approximating s h e l l s i n r egions of 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 places no r e s t r i c t i o n on th i c k n e s s where the 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 of SY{T) i s not monotomic throughout the entire region, the method can be applied to sub-regions i n which t h i s i s the case. For cC-O, the regions are of equal thickness. When <*- = i , the i n t e g r a l i s s i m i l a r to the phase i n t e g r a l of geomet-r i c a l optics and we may therefore think of t h i s case as equalizing an equivalent e l e c t r i c a l thickness f o r each s h e l l . Although t h i s choice of oc Is 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 not simple i n general because of the necessity of evaluating the i n t e g r a l and subsequently solving f o r r ^ . The case of oc~l corresponds to equalizing the areas under the Sy.(x) vs. r curve. .1 to each approximating s h e l l . There are several p o s s i b i l -i t i e s which w i l l give apparently reasonable r e s u l t s . For example we can use \u00E2\u0080\u00A2/\u00C2\u00A3,,(*\")/ dr Is equal f o r a l l s h e l l s . The second step i s to assign a p e r m i t t i v i t y ( J Er(r^) + / \u00C2\u00A3 K ( r m + f ) 2 (3 .1a) \u00C2\u00A3 K ( r w ) + Er ( r m + I ) (3.1b) 2 The c r i t e r i a used to determine \u00C2\u00A37 i n equations (3 .1a) and 42. (3\u00C2\u00BBlb) are s i m i l a r to those used i n the two sp e c i a l cases of (1) f o r determining the s h e l l thickness. Choices which are i n the same respect s i m i l a r to the special cases i n ( i i ) are \u00C2\u00A3 \u00C2\u00AB i t ) ( 3 . 1 c ) Z sgn( \u00C2\u00A3\u00E2\u0080\u009E(r) ) y/\u00C2\u00A3r(r)7 dr ) (3..W) ( 3 . 1 e ) Treating the problem from a curve f i t t i n g point of view, the least-squares f i t method may be applied i n either the S'y\r) vs. r or the 7 \u00C2\u00A3K(r j vs. r plane. I f \u00C2\u00A3'(r) does not have large v a r i a t i o n s within the s h e l l , a l l of the methods suggested above w i l l give, approximately the same value f o r \u00C2\u00A3\u00E2\u0084\u00A2 when the s h e l l s are t h i n . Since i t i s reasonable to assume that the r e s u l t s w i l l be i n s e n s i t i v e to small changes i n the p e r m i t t i v i t y of the s h e l l s , the ease of app l i c a t i o n should be the deciding f a c t o r . 3*1-3 Approximation of an Inhomogeneous Region by Shells with Linearly-Varying P e r m i t t i v i t y (37) Wilbee has considered the one dimensional problem of approximating a planar inhomogeneous region by slabs with constant and 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 . 43 Accurate results of the reflection and transmission co-efficients at the boundary of the region were obtained 'with significantly fewer slabs of the latter type. It was decided to examine the merits of a similar approximation i n the case of a cylinder. Since neither equation ( 2 . 2 ) nor ( 2 . 6 ) has a closed form solution when \u00C2\u00A3 r = sr -f- t, the power series solution developed i n Section 3 . 1 - 1 w i l l be used. As a re-sult, the allowed permittivity variation and cylinder radius: w i l l be limited by the series summation errors discussed i n that section. The advantage of this method over the direct power series solution method i s that the application to arbitrary permittivity variations i s immediate. The statements made i n Section 3 . 1 - 2 concern-ing the choice of the thickness of the approximating shells: naturally apply i n this case. We w i l l therefore only dis-cuss the choice of the values of s and t for each sh e l l . Two simple methods are illustrated i n Figure 3 . 3 . Method 1 ) For the m region we have +1 t w - \u00C2\u00A3 r(r ,J + s ^ r w ( 3 . 2 b ) Method i l ) For the approximation shown i n Figure 3 . 3 1 1 , sm i s given by equation , 3 .2a and t w = ti - -| ( s\u00E2\u0080\u009Er0 + t'm - \u00C2\u00A3 K(r e) ) (1) ( i i ) F i g u r e 3.3 I l l u s t r a t i o n of the two Methods of Choosing the S h e l l Parameters s and t . where =.- H r ( r w ) + s m r r n + , r 0 ~-h ( r^, + r v n J I t i s c l e a r t h a t t h i s method gives the more accurate approx-i m a t i o n of the p r o f i l e . Other c r i t e r i a may he used f o r choosing t\u00E2\u0080\u009E, w h i l e l e a v i n g s w f i x e d . For example, as suggested I n S e c t i o n 3\u00E2\u0080\u00A21-2, the \" e l e c t r i c a l t h i c k n e s s \" of the approximate and a c t u a l medium may be e q u a l i z e d . Another reasonable choice f o r the slope i s s^ = \u00C2\u00A3'(r 0). From a somewhat d i f f e r e n t p o i n t of view, a l e a s t - s q u a r e s f i t may be performed thereby 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 of these 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 than method l i ) above, which has the a d d i t i o n a l advantage of s i m p l i c i t y . 45 3.1-4 Numerical I n t e g r a t i o n of the Impedance and Admittance D i f f e r e n t i a l Equations -The v a r i a t i o n of impedance or admittance w i t h i n an inhomogeneous reg i o n may be a c c u r a t e l y c a l c u l a t e d by numer-i c a l i n t e g r a t i o n . Since e i t h e r z\u00E2\u0080\u009E or 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 the equation i n v o l v i n g the smal l e r of the two, equation (2.24a) or (2.24b) 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 and (2 . 29a ) or (2 . 29b) i n the case ( 3 1 ) 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 (\u00C2\u00A3l Z3) sphere. Other authors ' have obtained 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 equation. f 3 &) I n t h i s work, a Runge-ICutta^ ; method of order f o u r was used i n the numerical i n t e g r a t i o n . The determination of an i n i t i a l value f o r i n t e g r a t i o n i n , the m^ *1 r e g i o n and d i f f i c u l t i e s which a r i s e i n performing'the i n t e g r a t i o n . a r e discussed i n Appendix D. The method of d e a l i n g w i t h a zero of p e r m i t t i v i t y i n the case of perp 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 approximation i n the;neighbourhood of the r e s u l t i n g s i n g u l a r i t y i s of p a r t i c u l a r , importance. (Z3) Gal and Gibson avoid t h i s s i n g u l a r i t y by c o n s i d e r i n g s m a l l l o s s e s when c a r r y i n g out computations i n v o l v i n g zeros/ of p e r m i t t i v i t y . 46 3\u00C2\u00BBl-5 R e s u l t s and Comparison of the Methods Since the f o u r methods of computation described i n S e ctions 3\u00C2\u00AB1-1 to 3,l~b- are e s s e n t i a l l y independent, agreement between r e s u l t s c a l c u l a t e d by each method provides a check on each of them. Fu 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 \u00C2\u00A3 r(r) \u00C2\u00AB ar f c and by comparison vrith p u b lished r e s u l t s . The pub l i s h e d r e s u l t s considered were, f o r p a r a l l e l p o l a r i z a t i o n 1) Adey*^ 5^ - f i e l d of Ahomogeneous d i e l e c t r i c c y l i n d e r , \u00C2\u00A3 r * 2.56 i i ) Tang*^ 0^ - 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 of a metal core w i t h a homogeneous d i l e c t r i c s h e l l , \u00C2\u00A3 r = 2.541 ( 3 ) i i i ) Yeh and K a p r i e l i a n v ' - 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 of a metal core w i t h an inhomogeneous d i e l e c t r i c s h e l l , \u00C2\u00A3*-(r) = a/r and f o r perp e n d i c u l a r p o l a r i z a t i o n i v ) Gal and G i b s o n ^ 3 ) - 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 of an inhomogeneous d i e l e c t r i c c y l i n d e r , \u00C2\u00A3V(r) = 1 - a ( r / r , f With the exception of i i i ) , agreement between computed and published r e s u l t s was obtained. A r e p r o d u c t i o n of some of the curves given i n Reference 3 together w i t h corresponding computed r e s u l t s are shown i n F i g u r e 3\u00C2\u00AB^\u00C2\u00AB I t I s r e a d i l y seen t h a t there l s no resemblance between the two r e s u l t s i n general, although they do c o i n c i d e f o r k 0 r , < 3 i n F i g u r e 3.4a. The corresponding r e s u l t s i n the case of pe r p e n d i c u l a r p o l a r -i z a t i o n are shown i n F i g u r e 3.5. 47 \5 0 Figure 3.4 Computed and Published Values of k o c ^ ( 0 ) 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 , \u00C2\u00A3r(r) = cc/k0r. _. , Computed ~~77~.T~. Published (3) 48 49 30 o 20 10 0 / ^ cC=0.5 / / / r \ \ \ \ 5 \u00E2\u0080\u00A2 / / / / v. s v. / / 0. 2 4 6 8 10 k ?r, (a) o o 50 40 30 20 10 0 A 1 v 1 \ I 1 \ V \ \ \ kcc=3 /\ / I O V ^ 1\u00E2\u0080\u00941 , v / v / 0 2 4 6 (b) 8 10 Figure 3.5 Computed Values of k / J O ) for a Metallic Cylinder with an Inhomogeneous D i e l e c t r i c S h e l l , Sr(r) = c 6 / k 0 r . a) k cr^ = 0.5 b) k 0 r * = 3.0 50 Accuracy of the Shell-Approximation Methods I n order to compare the r e l a t i v e accuracy achieved f o r a given number of s h e l l s i n the sh e l l - a p p r o x i m a t i o n methods, computations were made w i t h v a r i o u s numbers of approximating . 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 of \u00C2\u00A3 r(r) > 0 and the r e s u l t s compared w i t h those c a l c u l a t e d u s i n g numerical i n t e g r a t i o n . She l l s : of equal t h i c k n e s s were used; i n the homogeneous s h e l l approximation, \u00C2\u00A3\u00E2\u0084\u00A2 was c a l c u l a t e d u s i n g equation ( 3 .1c ) and i n the l i n e a r l y - v a r y i n g s h e l l approximation, the two methods of choosing s^ and t m given by equations ( 3 \u00C2\u00BB 2 ) and (3\u00C2\u00AB3) were considered. These are designated 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 the b a s i s of comparison. The c y l i n d e r s considered were 1) \u00C2\u00A3 r(r) = a + b f ( r / r , ) where a = 0 . 2 5 , 0 . 5 , 2 . 0 , 4 . 0 -b = 1 - a r, = 0 . 2 5 , 0 . 5 , 0 . 7 5 . 1 .0 Ao f ( r / r , ) = ( r / r , )* , Jr/r~ \u00C2\u00A5e note t h a t i n the case of perpendicular 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 , equation (B.22) must o f t e n be used f o r the l i n e a r l y v a r y i n g s h e l l s i f E'r{x)>0. Since the s e r i e s summation i s r a t h e r time consuming even i n the simpler case of p a r a l l e l p o l a r i z a t i o n , numerical i n t e g r a t i o n i n the l i n e a r s h e l l s was used r a t h e r than the s e r i e s s o l u t i o n f o r perp e n d i c u l a r p o l a r i z a t i o n 51 i i ) S h e l l : ^ ( r ) = a + b f ( r ) Core: ^ ( r ) = c where a and b are such t h a t \u00C2\u00A3^(r, )' = 1 , \u00C2\u00A3,(rz) = c r, = 0 . 5 , 0 . 7 5 , 1 - 0 , 1 . 2 5 *o r ^ ~ 0 . 2 5 )i0 f ( r ) = / F , c = 0 . 5 f ( r ) = 1/r, c = 2 f ( r ) = 1/r*, c = 4 \u00E2\u0080\u00A2 The r e l a t i v e e r r o r s i n the approximate r e s u l t s together w i t h the c o r r e c t values i n a few r e p r e s e n t a t i v e cases: are shown i n Tables 3\u00C2\u00ABl a to e. An e v a l u a t i o n of the o v e r a l l accuracy i s given i n Tables 3 . 2 a and b. Table 3 . 2 a a p p l i e s i n the cases considered except when \u00C2\u00A3 K oc J~Y\ r i 0 , 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 are obtained; these are shown i n Table 3\u00C2\u00BB2b . The q u a l i t y of 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 < 1% Good - g,G; E r r o r s 1-5% F a i r - f , F ; E r r o r s 5 - 1 0 $ Poor - p,P; E r r o r s 1 0 - 2 5 $ Very Poor - v,V; E r r o r s > 25$ * This type of 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 the others considered i n th a t \u00C2\u00A3^(r) becomes i n f i n i t e at r=0 Table 3.1 R e l a t i v e E r r o r i n the Value of ko0\0) C a l c u l a t e d Using the S h e l l Approximation Methods V \ N r , / X 0 0.25 0.50 0.75 1.00 S h e l l s S w Par Perp Par Perp Par Perp Par Perp 1 HOMO 0 . 194 9 0 . 0 72 0 - 0 . 7 6 8 2 - 0 . 2 7 98 3 3 . 1 7 0 7 10 2 1 . 2641 17 7 . 3 3 7 5 4 4 . 9 6 6 1 1 LIM i . 4 6 1 2 - 0 . 5660 - 0 . 6 311 - 0 . 6 1 6 1 -C .2 05 1 7 7 . 4 402 6 . 0 0 2 9 2 . 2 8 1 9 1 L IN i i - 0 . 2 07 5 - 0 . 2 790 - 0 . 4 9 08 - 0 . 4 7 1 3 0 . 2 4 5 8 185 . 3488 1 5 . 2 2 42 7 . 3 7 77 2 HCMG - 0 . C 6 7 5 -0 . 0805 - 0 . 5 8 50 - 0 . 4 9 9 9 4 .4 435 810 .9352. 5 7 . 4 9 5 9 2 1 . 4 ^ 6 4 2 LIM i - 0 . 106 8 - 0 . 1 4 9 6 - 0 . 1 5 13 - 0 . 2 0 3 1 . - o . 2 8 1 5 5 . 4 3 1 9 - 0 . 9 9 7 0 - 0 . 9 5 5 0 2 LIM i i - 0 . 0 3 1 7 - 0 . 0 4 5 7 - 0 . 0 4 9 2 - 0 . 0 6 53 - o . 004 4 1 . 0 5 3 2 - 0 . 9 3 6 5 - 0 . 8 8 8 0 3 HOMO - 0 . 0 37 7 - 0 . 0435 - 0 . 2 720 - 0 . 2 1 7 3 0 . 6 8 3 9 110 . 9 0 5 3 3 . 6 0 3 2 - C . 4 0 1 6 3 LIN i - 0 . 04 6 9 - 0 . 0 6 7 1 - 0 . 0 6 9 4 - 0 . 0 9 6 8 - 0 . 1 7 3 2 0 . 16 78 - 0 . 3 8 5 0 - 0 . 2 8 7 5 3 LIN i i - 0 . 0 1 2 7 -0 . 01 8 3 - 0 . 0 1 9 4 - 0 . 0 2 6 7 - 0 . 0 3 7 7 - 0 . 1 0 8 0 - 0 . 2 6 4 5 - 0 . 1 6 84 4 HOMO -0 . 0227 - 0 . 0259 . - 0 . 1 5 4 6 - 0 . 1 2 15 \"\u00E2\u0080\u00A2' 0 . 2 0 0 3 \u00E2\u0080\u00A2 30 . 8 0 0 7 0 . 5 6 5 6 - 0 . 5 1 5 0 4 L I N i - 0 . 0 2 6 3 - 0 . 0 3 79 - 0 . 0 3 9 5 - 0 . 0 5 59 - 0 . 1 1 0 1 - 0 . 2458 - 0 . 1.630 - 0 . 1 2 8 5 4 L IN i i - 0 . C 0 6 9 - 0 . 0099 - 0 . 0 1 0 5 - 0 . 0 1 4 7 - 0 . 0 2 6 L \u00E2\u0080\u0094 o . 1 1 6 3 - 0 . 0 3 3 7 - 0 . 0 6 2 8 -> 5 HOMO - 0 . 0 14 9 - 0 . 0170 - 0 . 0 9 9 4 - 0 . 0 7 75'. 0 . 0 8 3 1 L2 . 6 2 6 2 0 . 1 3 5 4 - 0 . 3 8 34 5 L I ,\' i \ - 0 . 0 1 6 8 - o . 0 2 43 - 0 . 0 2 54 - 0 . 0 3 6 2 - 0 . 0 7 3 3 . - 0 . 2 4 5 8 . - 0 . 0 9 5 I - 0 . 0 7 46 5 L IN i i v - 0 . G04 3 - 0 . 0063 - 0 . 0 0 6 6 - 0 . 0 0 9 4 - 0 . 0 1 7 8 - 0 . ( .84 7 - 0 . 0 4 1 6 - 0 . 0 3 0 2 10 HOMO \u00E2\u0080\u0094 1.1 . 0 0 3 9 - 0 . 0 04 4 - 0 . 0 2 50 - 0 . 0 1 9 3 0 . 0 0 7 8 \u00E2\u0080\u00A2 1. . 2 4 7 5 - 0 . 0 1 56 . - 0 . 1 G S 8 10 LIM i - 0 . C04 2 - 0 . 0 0 6 1 - 0 . 0 0 6 4 - 0 . 0 0 9 2 - 0 . 0 1 9 2 - 0 . 0 8 8 0 - 0 . 0 2 02 - 0 . 0 1 6 5 10 L IN i i ' - 0 . 0 0 1 1 -0 . 0 01 5 - 0 . 0016 - 0 . 0 0 2 3 - 0 . 0 0 4 8 - 0 . 0 2 3 3 - 0 . 0 0 6 1 - 0 . 0 0 43 15 HOMO - 0 . 0 0 1 7 - 0 . 0 0 1 9 - 0 . 0 1 1 1 - 0 . 0 0 86 0 . 0 0 2 5 0 . 4 1 0 6' - 0 . 0 1 0 1 - 0 . 0 4 93 \u00E2\u0080\u00A2 20 HOMO - 0 . 0 0 1 0 ' - 0 . 0 0 1 1 - 0 . 0 0 6 3 - 0 . 0 0 4 8 0 . 0 0 1 2 0 - 0 . 0 0 6 3 - 0 . 0 2 7 9 . . k o O l 0 ) . 3 3 8 9 0 . 3 0 1 9 0 . 4 3 08 0 . 4 7 8 1 0 . 0 3 5 5 G . 0006 0 . 0 1 3 7 0 . 0 4 1 2 (a) \u00C2\u00A3,.(r) = 0.25 + 0.75(r/r, f 0.25 0.50 0.75 1.00 S h e l l s V P a r P e r p P a r P e r p P a r P e r p P a r P e r p 1 HOMO . i .0731 1 4.665 2. 0 . 5240 6.6741 16.4110 0.2 56 0 5.2 2 80 2.43 32 . 1 LIN' i -0 2.0155 -0.7423 -0.6443 2 . 2 2 2 0 -0.9003 -0.4062 -0.6536 1 U N i i -o . 3000 - 0 . 6734 -0.3 8 73 - 0 . 52.43 . -0 . 1390' -0.4017 0.2831 - 0 . 5 0 5 8 . 2 HOiYG \u00E2\u0080\u00A20 .221 a 1 . 346 0 - 0.9366 0.7173 7.1933 0.5651 0.4319 -0.06 64 2 L IN i -0 . 2 1 59 -0.51 74 . - 0 . 1 5 3 7 -0.0295 1 .3262 -0.3 544 0.6693 0.3 9 06 2 L IN i i -0 . 0651 -0.2415 -0.04 4 2 -0.0671 0.6552 -0.1465 0.768 2 0.0189 3 HOMO 0 .0761 0.4335 -0.2883 -0.3208 7.3721 -0.6565 16.3281 -0.9^83 3 L IN i -0 .1049 - 0 . 3 3 4 8 - 0 . 0 6 4 1 0.0032 0 .1982 -0.1510 - 0.22 0 L 0.2 924 3 L IN i i -0 . 0284 -0.10 34 -0.0177 -0.0133 0.0574 -0.0489 - 0 . 1 134 C . 0 8 2 6 4 HOMO 0 .0 330 0.1 9 7 3 - 0 . 1 4 3 8 -0.2476 1.8064 -0.2 60 0 0.8775 0.0 0 43 4 L I N i -c . 0 607 -0.3079 -0.0342 0.00 40 \u00E2\u0080\u00A2 0 .0695 -0.0809 - 0 . 2 0 8 9 0.19 42 4 L IN i i -0 .015 9 '-0 .-0 6 0 ? \u00E2\u0080\u00A2. - 0 . 0038 -0.0040 0 . 02 5 3 -0.0 244 -0.0834 0.0 6 65 5 1-iOMG 0 . 0 2 30 0.10 9 7 - 0. 0 8 5 2 -0.1795 1.0719 -0.1714 1. 1000 -0.2038 5 L IN i -0 . 0 3 '5 3 -0,1>,R0 -0.0214 0.0030 0.0293 -0.0496 -0.1'-03 0.1178 5 U N i i -0 . 0 10 1 -0 .03)3 2 -0.0054 -0 . 0014 0 .0094 -0.0137 -0 .0 504 0.0 374 10 HOMO 0 . 005 3 0.0205 -0.0190 -0.0529 0 . 2.2 2 5 -0.0417 0.19 S7 -0.0341 10 L I N i -0 .010 0 -0.0 3 64 - 0.0052 0.0009 0 .003 4 -0.0119 -0.0356 0.02 67 10 L I N i i -0 .002 5 -0.0093 - 0 . 0 0 1 3 0.0001 0.0009 -0.0030 -0.0095 0.00 69 : 15 HOMO 0 .0023 0.0 084 - 0.008 3 -0.02 43' 0.09 51 -0.0184 0.0845 -0.0143 20 'HCMG 0 .0013 0.0 04 6 -0.0046 -0.0139 0.0528 -0.0 1 0 3 0.04 6 8 -0.0079 koOKo) 1 .5519 0 . 0 77 0 6.62 29 1\u00E2\u0080\u00A2874S 0.8 54 6 .9.1889 2.4433 6.5957 ( b ) \u00C2\u00A3,(r) = 4 - 3 ( r / r , f \ r ( / X0 0.25 0.50 0.75 1 .00 ShellsX. Par Perp.: Par Perp Par Perp Par Perp 1 HOMO \u00E2\u0080\u0094 0 . A 5 2 3 - 0 . 5 648 1 .7 2 46 1 .3728' 5 .9455 3.CA79 3. OS.5 A 1.7.5 55 1 LIM i 1 .3086 2 . 0 02 6 2.4383 2.4566 -0 . 0 1 2 9 0 . 0 5 7 6 0.965 7 1.19 64 ! 1 LIN i i 0 . 685 A 0. 94 4 2 2.21 8 5 2.2072 -0.7 791 . -0.8375 0.7805 0.98 59 2 HCMG _ r. . 0 6 6A -0 . 0 6 1 3 - 0 . 152 5 -0.0575 0 .442 6 -0.33A2 4 . 075 2 5.4418 \" 2 L IN i 0 . 3- 8 6 5 o. 5 8 6 2 . 1 . 5 9 ^ 3 2.5547 3 .0711 .3. 5 07 9 2.0 70 2 2.2188 2. LIM i i 0 . 107 8 0. 1 7 80 0.6060 1.0261 2.. 0 5 1 5 2.3732 1 .9731 2.2 0 90 3 HOMG -0 . C 2 0 A - 0 . 0 0 8 5 -0.0122 0.0790 0.1916 - 0 . 01.96 -0.7590 -0.7644 3 L IN i 0 . 1 64 3 0. 25(1 0.7212 1 . 2089 1 . 7 3 A A 2 . 2 0 10 1.7226 2.3724 3 LIM i i . o . OA0 3 0 . 0715 0.2049 0.3757 0 .6546 0.8709 0.8839 1.2 4 33 \u00E2\u0080\u00A2 4 HOMO -0 . 0086 0. 0 0 ]. 5 0. 0.103 0.0810 0.1A 79 0.065A -0.3 596 -0.33 71 A L IN i 0 . 08 8 9 0. 142 7 0.3905 0.66 81 0.9827 1.2922 1 .1047 1.5940 A L IN i i 0 . 0 2.1 1 0. 03 30 0.1001 0.19 06 0.29A7 ' 0.A179 0.4 066 0.6131 5 HOMO -0 .0 04 A 0. 0 0 3 6 0.0142 0.0 6 78 0.1125 0.0758 -0.1948 -0.1610 5 L I N i 0 . 055 1 0 . 0 9 u 0 0.2412 .0 .4186 . ...0.6116 0.8206 0.7219 1.06 22 ' 5 L I N i i 0 . 0 1. 2 8 0. C 2 3 2 0.05 90 0.1146 0.16A3 0 . 2 A1 5 0.2232 0.3508 10 HOMG -0 . C C 0 7 0 . 0021 0.00 77 0 .02 66 0 .0379 0.0 38 2 -0.0292 ' -0.00 61 10 L I N i 0 .0124 0. 0 2 1 0 0.0 5 39 0.09 76 0 . 13A5 0.1892 0. 1672 0.2 5 66 10 L I N i i 0 . 0026 0. 0047 0.0124 0.02 46 0 .0303 0.0A69 0.0399 0.06 74 ' 15 'HOMO -0 .0004 0. 0 0 0 8 0.0040 0.0134' 0.0170 0.0191 -0.010 2 . 0.00 27 20 HOMO -0 . 0004. 0. 0 0 0 1 0.00 2 3 0.0078 0 .0097 0.0105 -0.0053 0.00 25 ; 0 .0765 0. 1157 .0. 0456 0.0531 0.02 3 6 0.0AA7 0.049 0 0.0615 (c) EAr) = 0.25 + 0.75 / r / r , 0.25 0.50 0.75 1 .00 Shells^ v Par Perp Par Perp Par Perp Par Perp 1 HOMO -0 . fc 2 5 6 - 0 . 3 3 6 2 17. 90 2 5 167 .3034 - 4 .4252 9.0 06 8 14.5874 -0.64 92 1 L I N i 0. 190 9 -0.3107 ' 6. 6783 12 6 . 5 2 37 '5 .1342 14.4629 1.9011 9.9 8 39 i 1 L I N i i 0.6 5\u00C2\u00B00 0.4 5 92 5 . 4 7 07 2 .4 503 1 .802 2 2 2.0200 \u00E2\u0080\u00A2 1 . 2344 1-7.1938 2 HOMO -0.3013 -0.4034 0. 0278 -0 . 06 8 8 -o .3610 2.3435 5.7049 -0.6992 2 LI M i 0 .\u00E2\u0080\u00A23 74 7 0.2135 o. 17 35 10 .2195 -0 .6730 4.7959 -0.319 1 0 . 1041 2 LIM i i 0 . 1625 0.0 9 09 -o. 5 8 90 8 . 30 01 -0 .1347 3.6419 1.0710 0.0916 3 HOMO - 0 . 1224 -0.1810 - 0 . 2757 1 .8482 -0 .6550 0.5154 1 .9032 3.8 3 99 3 LIM i 0.1893 0. 1087 - 0 . 4756 8 . 5319 -0 . 0 6 4 9 0.8673 - 0.1740 -0.67 79 3 L I N i i 0.0580 0 . 0306 -0 . 39 89 3 . 269 4 0 .2595 -0.4554 -0.0682 -0.5021 4 HOMO -0.0 64 5 - 0 . 1 0 1 r - 0 . 2198 1 . 2004 \" -o .376 5 -0.2044 -0.4326 -0.9571 \u00E2\u0080\u00A24 1. IN i 0. 106-3 0 . 0 6 c a - 0 . 414 0 4 . 8373 o .1701 - 0 . 1410 -0.14 51 -0.16 89 4 LI N i i 0. 028 5 0.0147 - 0 . 2 1 5 8 I .3184 0 .208 3 ' -0.44 3 0 -0.1461 0.2 8 48 5 HOMO -0.0394 -0.0642 - 0 . 16 5 6 \u00E2\u0080\u00A2 0 .7918 -0 . 222 3 -0.2009 -0.4351 -0.7518 5 L IN i 0.0671 0.0379 -o. 3134 .2 . 8371 0 .19 3 8 -0.3007 -0.1728 0. 1376 5 L I N i i 0 . 0 1 6 6 0.00 8 5 -o. 1261 0 .67 57 0 .1260 -0.2949 -0.0773 0.32 11 10 HOMO -0.0 09 1 -0.0L58 - 0 . 0 5 4 3 0 .2067 -0 .0454 -0.0754 - 0 . 1 2 5 3 -0.1749 10 L I N i 0 . 0 1 5 0 0.0 08 3 - 0 . 0895 0 . 5201 0 .0689 - 0 . 1506 -0.0501 0.1640 10 LIM i i 0 .0080 0.0016 0232 0 .1115 o .0197 -0.0469 -0.0128 0.06 33 15 HOMO -0.0 04 2 --C.0 071 02 60 0 . 09 35 -0 .0191 '-0.035 8 -0.0572 - C O 7 42 20.HOMO -0 . 0026 -0 .0041 - 0 . 0149 0 .05 26 -0 .0109 -0.0196 -0.0 32 4 -0.04 11 koCf(0) 0.7 08 3 0.3 3 70 0. 2166 0 . 00 52 0 .4453 0.0 590 0. 500 1 0 . 2 0 8 0 (d) \u00C2\u00A3\u00E2\u0080\u009E(r) = 4 - 3 J7JT~, Shells >v Par 0.25 Perp Par 0.50 Perp Par 0.75 Perp Par 1.00 Perp 1 HOMO 1 L I N i - 0 . 9 46 0 -0 . 3 64 6 -0 .934 2 -0 .7018 27.9120 7.5041 1 1.4761 -0 .3628 -0 .8140 3.6263 - 0 . 3 299 - 0 . 6 2.3 0 2 9.5149 17.7065 10.0186 1.1398 ; 1 L I M i i -0 .1073 -0 .2016 10.6194 0.2 3 52 10.3016 2.2 52 2 29.9043 2.4 0 93 2 HOMG 2 L I N i 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 30.0 3 68 -0 .3491 - 0 . 4 6 96 3 HUMG 3 L I N i 3 L I M i i - 0 . 1929 -0 .0196 - 0 . 0 0 3 4 \u00E2\u0080\u00A2 - 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 . 3 0 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 L I N i - 0 . 1105 - 0 . 0 09 0 -0 .1445 -0 .0184 -0 .2966 0.0316 -0.6891 0.03 76 ' -0 .9403 -0 .4263 -0 .9406 -0 .2814 -0 .0352 -0 .6218 -0 .6525 -0 .7277 . 4 LIN i i -0 . 0 02 0 - 0 . 0 0 4 0 0.0173 0.02 91 -0 .096 6 - 0 . 0443 -0 . 3407 -0 .2446 5 HOMG 5 L I N i 5 L I M i i - 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 . 5 5 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 . C G 0 3 - 0 . 0 2 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 \u00E2\u0080\u00A2 -0 . 0080 -0 .0106 -0 .0347 -0 .0613 -0 .1070 -0 .1028 -0 .2397 - 0 . 2 147 i 20 HOMO - 0 . 0 04 5 - 0 . 0 06 0 -0 .0198 -0 .0347 -0 .0607 -0 .0584 - 0 . 1390 - C . 1 2 3 7 0.8537 0.2.51R 0.034 0 0.02 79 0 .0755 0.0676 0.0234 0.03 04 (e) Core - \u00C2\u00A3 r = 2, r 2 = 0.25 A\u00E2\u0080\u009E? Shell - \u00C2\u00A3 r(r) = a/r + b O N . . 57 Table 3 .2 O v e r a l l Accuracy of the S h e l l Methods a) Cases Considered Except 6r(r) = a J T + b, r > 0 b) \u00C2\u00A3,.(r) = a / r + b , r > 0 S h e l l s 1 2 3 A 5 10 20 S h e l l s t//)c Homo 0.25 0.75 Lin 1 0.25 0 . 7 5 v-V v-V P-V f-P g-F e-G e-E e-E v - v v-V v-V v-V v-V f-P f-F g-G v-V p-V f-P g-F g-G e-G v-V v-V v-V p-V f-P g-G (a) Homo 0 . 2 5 0 . 7 5 Lin i 0 . 2 5 0 . 7 5 Lin i i 0 . 2 5 0.75 v - v f-P g-F e-G e-G e-E v-V v-V f-V g-P g-F e-G Lin i i 0 . 2 5 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 lower case l e t t e r s apply f o r \u00C2\u00A3 r < l and the upper case f o r \u00C2\u00A3 r > l . I n some cases, the e r r o r s observed were l a r g e r than i n d i c a t e d i n Table 3 . 2 . These anomalous cases are u n d e r l i n e d i n Table 3*1 and i t i s seen t h a t they g e n e r a l l y occur near what appears to be minima of 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 s . Although no d i f f e r e n t i a t i o n i s made between the 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 the accuracy achieved f o r a given number of 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 accuracy i s i n general r e l a t e d to the max-imum e l e c t r i c a l t h i c k n e s s of the s h e l l s as shown i n Table 3 \u00C2\u00BB 3 ' Table 3 . 3 R e l a t i o n s h i p Between Accuracy and S h e l l Thickness E r r o r s Maximum S h e l l Thickness Homogeneous Li n e a r 0 . 2 5 A > 25 % 10 - 25\" % 0 . 1 A 10 - 25 % 1-5% 0 . 0 5 A 1-5% < 1 % 0 . 0 2 5 A < 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 of computation i s presented i n Table 3 . 4 . Table 3.4 Comparison of the Methods of Computation Power Series Homogeneous Shells Linear Shells Numerical Integration Application to Ar-bitrary Permittiv-i t y Variations Impractical since complexity of ser-ies is so depen-dent on variation Straightforward Straightforward Inherent Limitations Size and permit-tivity variation limited due to series summation errors c>0 f o r ( i ) perp. pol. \u00C2\u00A3^0 when .\u00C2\u00A3,.-0^ for perp. pol. Also power series summation limi-tation EJO f o r ^ perp. pol. Computation Time 10 shells, r,=0.5 >io Both pols. \u00C2\u00BB0.6 sees. 20 shells, r, = 0.5 )\o Par. pol. &7 s e c 8 . u i ; Perp. pol.^**^ r,=0.5 ^ 0 Both pols. **4 to 8 sees, for cylinders con-sidered * on an I.B.M. 7044 computer 60 ( i ) A power s e r i e s s o l u t i o n about the value of r at which \u00C2\u00A3>.(r) = 0 may be used i n c o n j u n c t i o n w i t h each of these methods, thereby e l i m i n a t i n g t h i s l i m i t a t i o n . ( i i ) The summation of the s e r i e s was c a r r i e d out u s i n g 16 s i g n i f i c a n t f i g u r e s . For c y l i n d e r s w i t h per-m i t t i v i t y v a r i a t i o n s such t h a t 8 s i g n i f i c a n t f i g u r e s g i v e accurate summation, the corresponding time i s decreased by almost one h a l f . ( i l l ) The s o l u t i o n was accomplished u s i n g numer-i c a l i n t e g r a t i o n i n s t e a d of the power s e r i e s . The s e r i e s are c o n s i d e r a b l y more cumbersome i n t h i s case than 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 time would be s i g n i f i c a n t l y l o n g e r . ( i v ) The time required' f o r the numerical I n t e g r a -t i o n i s very dependent upon the time r e q u i r e d to evaluate the f u n c t i o n d e s c r i b i n g the p e r m i t t i v i t y v a r i a t i o n and upon the i n t e g r a t i o n step s i z e . The l a t t e r seems to depend more on the range of p e r m i t t i v i t y w i t h i n the c y l i n d e r . t h a n on the a c t u a l v a r i a t i o n . Summary\" The homogeneous s h e l l method appears to be q u i t e 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 f i e l d of a c y l i n d e r 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 . Although the l i n e a r s h e l l method g e n e r a l l y gives the same accuracy f o r fewer s h e l l s , the computation time i s c o n s i d e r a b l y longer; roundoff e r r o r s i n a p p l y i n g the i t e r a t i o n equations f o r im-pedance and admittance produced a r e l a t i v e e r r o r of l e s s 61 than 10 i n the backscattering cross-section f o r up to 100 s h e l l s and hence the reduction i n s h e l l numbers seems unim-portant. The numerical Integration method may sometimes be desirable i f very accurate r e s u l t s are required.'' Prom the cases considered, i t appears that the homogeneous s h e l l method w i l l be f a s t e r f o r accuracies of up to 0 . 1 $ . I t should be pointed out that the use of l i n e a r l y -varying sh e l l s may be more advantageous than homogeneous s h e l l s when considering the inverse problem (determination of the p e r m i t t i v i t y v a r i a t i o n from scattered f i e l d measurements) be-cause of the smaller number of s h e l l s required to achieve a given accuracy. 3 \u00C2\u00BB 2 Number of Terms Required i n the Scattered-Field Expansion When c a l c u l a t i n g the scattered f i e l d , the number of terms, N, used i n the expansion must be large enough to make the e f f e c t of the higher order terms i n s i g n i f i c a n t . I t i s desirable, of course, to use as few terms as possible. In the case of a conducting cylinder, a value of N ^ 2 k 0r, has' Ul) been suggested ; i t appears that no value has been given f o r a d i e l e c t r i c c y l i n d e r . A procedure f o r choosing the value of N required i s given l n Appendix E, Sections E . 2 - 1 and E . 3 - 1 . For the scattered f i e l d (Section E . 2 - 1 ) , N i s determined such that the error at a s p e c i f i e d radius i s below a given value. For the sc a t t e r i n g cross-section, i t i s not possible to s a t i s f y either 62 an absolute e r r o r c o n d i t i o n or 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 . For 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\" conduct-i n g c y l i n d e r s , \"unexpectedly l a r g e \" high-order c o e f f i c i e n t s do not occur and the accuracy of the r e s u l t s may be estimated (1 from the magnitude of the highest order term . This i s not so when high p e r m i t t i v i t i e s are considered (say \u00C2\u00A3^>10), and i t i s necessary to f o l l o w the procedure o u t l i n e d i n Appen-dix*. E\u00C2\u00AB I t should be pointed out t h a t i t i s always, d e s i r a b l e to know the r e q u i r e d order beforehand, p a r t i c u l a r l y when u s i n g the homogeneous-shell method of S e c t i o n 3\u00E2\u0080\u00A21-2. The reason 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 occur i f J\u00C2\u00AB(z) i s c a l c u l a t e d u s i n g an upwards recurrence when n > z and hence i t i s advan-tageous to c a l c u l a t e a l l the J-\u00E2\u0080\u009E f u n c t i o n s at the same time. 63 . 4 CALCULATION OP THE COMPLETE SCATTERED FIELD FROM FIELD MEASUREMENTS In t h i s chapter, we give a method hy which 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 hence the complete s c a t t e r e d f i e l d can be determined from measurements of the s c a t t e r e d (or d i f f r a c t e d ) f i e l d at one r a d i u s . Since the c o e f f i c i e n t s are known, the s c a t t e r e d f i e l d f o r o.-.t plane-wave inci d e n c e can be c a l c u l a t e d from measurements taken w i t h c y l i n d r i c a l - w a v e i n c i -dence and v i c e v e r s a . This could r e s u l t i n a simpler experimen-t a l method f o r the i n v e s t i g a t i o n of plane-wave s c a t t e r i n g . As i n Chapter 3, two steps are i n v o l v e d i n the pro-cess, 1) Determination of the 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 r e p r e s e n t a t i o n of the f i e l d i s obtained. The d e r i v a t i o n i s c a r r i e d out f o r p a r a l l e l p o l a r i z a t i o n and a p p l i e s d i r e c t l y to perpendicular p o l a r i z a t i o n i f E i s . r eplaced by H. 4.1 Method of C a l c u l a t i n g the C o e f f i c i e n t s 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 f o r e i t h e r plane or c y l i n d r i c a l wave i n c i d e n c e i s of the form CO E s(r,6-) = I e, A, H ^ ( k 0 r ) cos(ne-) 6 4 Truncating the s e r i e s i n t h i s equation at n = N we have VS(r,&) \u00C2\u00BB E^(r,e-) = Zl e,, A f l H ^ ( k 0 r ) cos(ne-) (4.1) At a f i x e d r a d i u s r = r 0 , we may w r i t e E\u00E2\u0080\u009E(0-) = ( \u00E2\u0080\u0094 - + *\u00C2\u00A3_ a. cos(nG-) ) 2 (4.2) a** + -j ( \u00E2\u0080\u0094*. + 21 a^r cos(nG-) ) 2 \u00C2\u00BB=' 5 1 1 where A\u00E2\u0080\u009E = \u00E2\u0080\u0094 0 < n \u00C2\u00A3 N The r e a l and imaginary p a r t s of equation (4.2) are truncated F o u r i e r cosine s e r i e s i n the v a r i a b l e 8-. Thus,- i f we determine (from measurements) the r e a l and imaginary p a r t s of the s c a t -tered f i e l d at s e v e r a l angles, estimates of a f n and a ^ may be (4-2 4-3) obtained 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 ' to the data. I f we have 2L data p o i n t s , the approximating func-t i o n s are chosen from the set 1, cos(e-), cos(2\u00C2\u00A9-), cos(Le-) and t h e r e f o r e we are l i m i t e d to N 6 L . For s i m p l i c i t y , we w i l l assume th a t the data i s taken at the e q u a l l y spaced values m &; = \u00E2\u0080\u0094 i = 0. 1. 2. 2L-1 65 D e f i n i t i o n of q u a n t i t i e s Measured values w i l l be designated by the s i g n (~) Thus, f o r the measured d i f f r a c t e d f i e l d we w r i t e E\u00C2\u00B0(Gi) = M d : e J'^ ; =t-oL + J t o i (^.3) where M D i = magnitude of the d i f f r a c t e d f i e l d a t \u00C2\u00A9- = \u00C2\u00A9-<\u00E2\u0080\u00A2 = phase of the d i f f r a c t e d f i e l d a t & = 0-t-= cos(#>;) g o i = M P: sin( - - ,(2) 2(1 . +\u00E2\u0080\u00A2. 6r,L) H ^ ( k e r 0 ) By s u b s t i t u t i n g A^ f o r i n equation (4.1) we may o b t a i n the 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 , the r e l a t i o n (2.37) may be a p p l i e d to determine estimates of or A^e, depending on whether plane or c y l i n d r i c a l wave incid e n c e was used i n the; measurements. 67 We note t h a t , because of the symmetry of the f i e l d , i t i s only necessary to take measurements i n the range 0 f \u00C2\u00A9- \u00C2\u00A3 rr i n order to determine the c o e f f i c i e n t s . However, by reducing the number of p o i n t s over which the 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 decrease the accuracy of the 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 to Each P o i n t ' Yfrien the v a r i a n c e of the measurements i s not the same a t a l l p o i n t s (as w i l l c e r t a i n l y be the case i n the pres-ent a p p l i c a t i o n when e r r o r s occur i n the values of &i or i f the f i e l d i s measured i n terms of magnitude and phase), the bes.t estimate of the c o e f f i c i e n t s i s obtained when a weight w; o c l / c r / i s a p p l i e d to the re s i d u e at the i ^ * 1 p o i n t ^ ^ ' ^ 5 ^ . The coef-f i c i e n t s &f.n and a ^ i n equation ( 4 . 5 ) are determined by s o l v i n g the N simultaneous equations 0 A = Y 1 < N \u00C2\u00A3 L ( 4 . 7 ) where 0 Zw^cosCG-) . . . ]T C ' C ' L U + 4,,) cos(NG-) Y. Wicos ( e j ) ZwiCos 2(G^ ... Y_ 1 1 W ' r \ c o s ( % ) eos(Hefr) IL w^cos(N\u00C2\u00A9-\u00C2\u00A3) Y _ J i i cos2(Ne;-) A = a,/2 a ( Y =\u00E2\u0080\u00A2 Zw< fL cosOJ I^y. cos(NG^) 68 W; = \u00C2\u00A5 f. Or W^ a * - a f T 1 o r a r\u00E2\u0080\u009E jc = fsi o r g ^ 0 < i < 2L-1 0 s n \u00C2\u00B1 N 0 i i f 2L-1 An estimate of the var i a n c e of f S (r and g s i at each p o i n t may he deriv e d as f o l l o w s : i ) E r r o r s i n 8 7 We assume t h a t the e r r o r s i n 6 7 are u n c o r r e l a t e d w i t h v a r i a n c e c\u00C2\u00A3 and t h a t f L = \u00C2\u00A3 e\u00C2\u00A3 or txi and g. = g_, or g z i are measured d i r e c t l y . We have var(f\u00C2\u00A3) = 2>\u00C2\u00A3 be-e- = e-z *g v a r ( g \u00C2\u00A3 ) - I Since the true values of G-^ , hf/d& and }g/b& are unknown, we use the measured values as estimates. I f f 5 ^ and Zsi are c a l -c u l a t e d from equation ( 4 . 4 ) , we have v a r ( f s ; ) e- = e-/ 2 -I ( 4 . 8 a ) v a r ( g 5 i ) bs0 be-b i x bo-.2. -1 <9-e-=e- ( 4 . 8 b ) 69 i l ) Measurements of M and

q ^ cos ((Pi) n .\u00C2\u00A3 N * The theory i n t h i s s e c t i o n a p p l i e s to the minimum-variance estimators and hence the weighting used i s given by w; oc \/o~c 71 where n = L l s the maximum number of c o e f f i c i e n t s which may be determined from 2L data po i n t s . * I f the hypothesis H, i s t r u e , then we have j> \u00E2\u0080\u00A2 ( S t - S ; , ) / ( K , , - H ) Nm-N, 2L-Ny\u00E2\u0080\u009E-1 S / ^ / ( 2 L - 1) where F = F - v a r i a t e w i t h n, and n, degrees of freedom^ 3\u00C2\u00BB' n\u00C2\u00BB ,nz For equal e r r o r - v a r i a n c e ZL-, N z t = \u00C2\u00A3> O S* = I y ? \" L l a* ( 4 . 1 2 a ) and f o r unequal e r r o r - v a r i a n c e 2 L-i * & 7 1 2 L _ I S* = Z W; y j - Z Z w< JL cos(n^) (4.12b) The expressions f o r S*^ are obtained by r e p l a c i n g N by U w i n equations (4.12a) and (4.12b). Now l e t P \"\u00E2\u0080\u009E .\"denote\" the \u00C2\u00BBn \u00C2\u00A3 value of P\u00E2\u0080\u009E - such t h a t the p r o b a b i l i t y of exceeding t h i s : value i s oi (F may be obtained from t a b l e s ' or r e a d i l y n i , n 2 (4-6) computed approximately ). I f the value of P c a l c u l a t e d u s i n g equation (J/-.11) i s le s s : than P^.JJ _JJ 2L-N -1' w e m a\u00C2\u00AB 5 r a s s u m e w i t h a confidence (1 - cc) t h a t the c o e f f i c i e n t s f o r n > N are zero. A d i s c u s s i o n of the power of t h i s t e s t i s given i n Ref-erence 4-,3-. I n p r a c t i c e , a maximum value < L i s u s u a l l y assumed. For example, I f the r a d i u s of the c y l i n d e r i s known, the e r r o r l i m i t s discussed i n Appendix E may be used to estimate a v a l u f o r NTO, the c o e f f i c i e n t s f o r n > J\ T W being n e g l i g i b l e 72 Although the method of determining N i s the same f o r both equal and unequal e r r o r - v a r i a n c e at a l l p o i n t s , the a p p l i c a t i o n i n the former case i s c o n s i d e r a b l y e a s i e r because i ) the values of the c a l c u l a t e d c o e f f i c i e n t s are independent of N and hence i t i s not necessary to r e c a l c u l a t e the complete set of c o e f f i c i e n t s f o r each value of N t r i e d and i i ) the ex-2 z p r e s s i o n s f o r SN and S W r v ) are r e l a t i v e l y simple. We note t h a t i f the hypothesis a w = 0 f o r n > N i s c o r r e c t , the value of s* = S*/(2L - n - 1 ) should be independent of n f o r n > N. As a r e s u l t , an estimate of N ma.y be obtained by observing the behaviour of s* as n i s incre a s e d ; the value of s* should decrease r a p i d l y and then become constant f o r n > N. This c r i t e r i o n , although not an exact one, i s u s e f u l because of i t s s i m p l i c i t y of a p p l i c a t i o n and has proved adequate i n the experiments c a r r i e d out. 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 47) have g e n e r a l l y been confined to the study of plasmas\ &* . I n these cases, the 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 of accuracy to a l l o w d i r e c t comparison w i t h theory. I n t h i s chapter, we consider s c a t t e r i n g by c y l i n d e r s made of a type of 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 are 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 conducting p l a t e s separated by l e s s than h a l f a free-space wavelength, under c o n d i t i o n s of plane and c y l i n d r i c a l wave i n c i d e n c e . This type of system r e s t r i c t s the measurements to cases where the e l e c t r i c f i e l d i s par-a l l e l t o the a x i s : of the c y l i n d e r , p e r p e n d i c u l a r to the p l a t e s . Measurements are taken at a constant r a d i u s near the c y l i n d e r and from these 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 are c a l c u l a t e d u s i n g the method of Chapter 4. I n the case of 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 the 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 given 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 order to approximate 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 c o n s t r u c t a c y l i n d e r from t h i n homogen\u00C2\u00AB 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 . For c y l i n d e r s of l a r g e diameter compared to wavelength, a l a r g e number of 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 order to achieve a reasonably good approximation. This may prove i m p r a c t i c a l \" . Another method r e q u i r i n g l e s s v a r i e t y l n m a t e r i a l s i s to c o n s t r u c t the c y l i n d e r from a l t e r n a t i n g tapered l a y e r s of tvro d i e l e c t r i c s as shown l n F i g u r e 5\u00C2\u00BBla. c y l i n d e r a x i s \u00E2\u0080\u00A2 c y l i n d e r a x i s (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) Using two D i e l e c t r i c s b) Using three D i e l e c t r i c s when the V a r i a t i o n of P e r m i t t i v i t y i s l a r g e . 75 I f the maximum thic k n e s s of each l a y e r i s much sma l l e r than the wavelength i n the m a t e r i a l , the r e s u l t i s a type 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 t h a t considered by H o r i t a and C o h n ^ 8 ^ i n matching d i e l e c t r i c l e n s e s , and by C o l l i n ^ 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. The e f f e c t i v e per-m i t t i v i t y , \u00C2\u00A3e(r), a t any r a d i u s depends on the r e l a t i v e t h i c k -ness:, of the two d i e l e c t r i c s at t h a t r a d i u s , the r e l a t i o n being d e r i v e d i n Appendix G. By u s i n g the appropriate p r o f i l e of the two d i e l e c t r i c s , one should o b t a i n any e f f e c t i v e 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 could c o n s t r u c t a 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 had a v a i l a b l e 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, the other w i t h a p e r m i t t i v i t y below the range encountered i n the c y l i n d e r . This could 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 are v a s t l y 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 non-uniform. i i ) E r r o r s due to machining have a l a r g e e f f e c t when the p e r m i t t i v i t i e s are very d i f f e r e n t . A compromise must be made between the number of v a r i o u s m a t e r i a l s and the accuracy of approximation 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 v a r i a t i o n , three or more m a t e r i a l s may be used as shown l n Figure 5\u00C2\u00ABlh. 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 the measurements on c y l i n d e r s are to be taken 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 the TE mode, c o r -responding to the case' of 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 or l o n g i -(50) t u d i n a l - s e c t l o n magnetic ) e x i s t s . The s o l u t i o n when l a y e r s of constant t h i c k n e s s are placed i n a r e c t a n g u l a r [SO 5/) waveguide has been discussed f u l l y 7 and i t i s found t h a t the e f f e c t i v e p e r m i t t i v i t y i s the same as when the l a y e r s : are i n f r e e space. Thus, the c h a r a c t e r i s t i c s of the a r t i -f i c i a l d i e l e c t r i c may be co n v e n i e n t l y a s c e r t a i n e d by wave-guide measurements u s i n g the c o n f i g u r a t i o n shown i n F i g u r e 5<>2-\u00C2\u00BB \ 1 i A r t i f i c i a l D i e l e c t r i c 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 of the c h a r a c t e r i s t i c s of the specimens because, f o r the types considered, i t proves to be s e n s i t i v e to 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 a c c u r a t e l y . Since the magnitudes of P encountered are of the ( 5 2 ) order of 0 . 1 , the Weissfloch-Feenberg ( n o d a l - s h i f t ) method i s s u i t a b l e f o r the measurements. The 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 to t h a t used i n the homogeneous-shell approx-i m a t i o n of S e c t i o n 3 . 1 - 2 ; each inhomogeneous r e g i o n was d i -v i d e d i n t o ' v e r y t h i n homogeneous s l a b s . ( Elementary t r a n s -m i s s i o n - l i n e theory may be a p p l i e d 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 r e -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\u00C2\u00BB3\u00C2\u00AB The s p e c i a l l y constructed waveguide s e c t i o n i n which the samples are placed i s shown i n F i g u r e 5\u00C2\u00AB^\u00C2\u00BB The advantage i n making t h i s s p e c i a l s e c t i o n i s t h a t there i s : no d i s c o n t i n u i t y between the sample and the s h o r t - c i r c u i t i n g p i s t o n . The r e s i d u a l r e f l e c t i o n s i n the system are thus, reduced to those due to one 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 there are no d i s c o n t i n u i t i e s behind the specimen, an approximate c o r r e c t i o n can be made f o r the e r r o r i n the measured values of r e f l e c t i o n c o e f f i c i e n t s due t o the r e s i d u a l r e f l e c t i o n s . Assuming the r e s i d u a l r e f l e c t i o n s are s m a l l we can w r i t e PV| ^ /^M / ~~ Pp. where = measured value of the complex r e f l e c t i o n c o e f f i c i e n t /\u00C2\u00B0 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 r e s i d u a l r e f l e c t i o n s /\u00C2\u00B0r* \u00C2\u00AB= /^ I I 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 r e f l e c t i o n s 78 Specimens of the types shown i n F i g u r e 5\u00C2\u00BB5& were made u s i n g p o l y s t y r e n e , Er = 2.56 and, f o r type I I I , expanded po l y s t y r e n e , tY = 1 . 5 . The values of T used f o r each type are i n d i c a t e d i n Table 5.1 \u00E2\u0080\u00A2 I n a l l cases, the values chosen were such t h a t an i n t e g r a l number of specimens f i l l e d the 0.400 dimension of 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 are i l l u s t r a t e d i n F i g u r e 5\u00C2\u00AB5D. These were placed 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\u00C2\u00AB^\u00C2\u00AB Measurements were taken at three f r e q u e n c i e s : 8.5\u00C2\u00BB 10.0 and 11 .5 GHz. The r e s u l t s are shown i n Tables 5\u00C2\u00BBla to 5\u00C2\u00ABlc together w i t h t h e o r e t i c a l v a l u e s . The e n t r i e s i n column /? are the t h e o r e t i c a l values f o r i d e a l specimens. Those i n column / r c are values c a l c u l a t e d t a k i n g i n t o account an estimated ma-c h i n i n g e r r o r f o r type I and u s i n g tr - 1.55 as the r e l a t i v e p e r m i t t i v i t y of the expanded polys t y r e n e f o r type I I I ( t h i s i s ; discussed f u l l y i n i ) and 11) of 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 values l s poor f o r T = 0.4 i n . except at 10.0 GHz, i n which case we must assume the c l o s e agreement to be c o i n -c i d e n t a l . The range of e r r o r s f o r T $\u00E2\u0080\u00A2 0.2 i n . are l i s t e d i n Table 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 to 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 of the samples i s 0.2in. or l e s s . The observed e r r o r s are comparable w i t h those expected MODULATOR 1 KHz _n_TL POWER SUPPLY KLYSTRON BUFFER ATTENUATOR ISOLATOR CRYSTAL DETECTOR FREQUENCY METER 20 db V 0 k SLOTTED; SECTION SPECIMEN MOUNT Figure 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 SPECIMEN-OTHER HALF OF SECTION IS LOCATED BY 4,-%-DIA PINS AND HELD BY S, NoWxl^-24 MACH SCR C3 I O C \u00C2\u00A9 p T HOLES MATCH UG-33/U FLANGE i ; \u00C2\u00AE \u00C2\u00A9 S.C. PLUNGER HOLES FOR MICROMETER-HEAD MOUNT (MATCH UG-39/U FLANGE) \u00C2\u00A9 <- ^ \u00C2\u00B0 = _ i # \u00C2\u00A9 ^\u00E2\u0080\u0094 t= =\u00C2\u00BB \" =* = F i g u r e 5.4 S e c t i o n C o n s t r u c t e d t o H o l d A r t i f i c i a l D i e l e c t r i c S p e c i m e n s f o r R e f l e c t i o n C o e f f i c i e n t M e a s u r e m e n t s 81 10 STEPS-TYPE I P-POLYSTYRENE Er=2.56 EP-EXPANDED POLYSTYRENE Er=1.5 (a) TYPE U ->j.S cm1^-/ P \ \u00E2\u0080\u0094>| 12 cn\*\u00E2\u0080\u00942.5 cm ->|/. 2 c/nr*-TYPE m (c) u r e 5.5 S p e c i m e n s u s e d f o r t h e Waveguide 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 D i e l e c t r i c a) Types o f S p e c i m e n s 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 ) M a c h i n i n g E r r o r i n Type I S p e c i m e n s 82. T a b l e 5 . 1 \u00E2\u0080\u00A2Comparison of-Measured-and 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 the Samples o f A r t i f i c i a l D i e l e c t r i c Type T Mag Arg Mag Arg Mag Arg I 0.400 0 .191 -140 0 .206 97 0.208 95 .... 0 . 2 0 0 0 .109 108 - 0 . 1 1 5 111 0 . 1 2 5 108 0 . 1 3 3 0 .099 110 0 . 098 114 0.114 110 0 . 1 0 0 0 .116 116 0 . 0 9 3 115 0 . 1 1 3 110 n 0 .200 0 .106 105 0 . 1 1 6 112 0 . 1 3 3 0 . 0 9 5 113 0 . 1 0 0 115 in 0 . 2 0 0 0 . 0 5 5 107 0 . 0 2 6 126 0 . 0 5 5 120 ( a ) f = 8 . 5 G H z A Ac Type T Mag Arg Mag Arg Mag Arg I 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 II 0.200 0.118 140 0.085 149 0.133 0.114 146 0.091 153 III 0.200 0.149 156 0.130 166 0.123 159 (b) f = 10.0 GHz fr Type T Mag Arg Mag Arg . Mag Arg I 0.400 0.314 -32 0.060 -17 0.048 -19 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 II 0.200 0.049 8 O.O63 15 0.133 0.059 10 0.068 21 III 0.200 O.076 16 0.089 36 0.081 29 (c) f = 11.5 GHz 84 due to constructional d i f f i c u l t i e s or, f o r type I I I , an error i n the p e r m i t t i v i t y of the medium used. Table 5-2 Range of Difference Between Theoretical and Measured Results f o r the Specimens of . . 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 \Prl-\Pnl Arg ( / V ) -Arg(/>J Arg( PTc ) -Arg(/k) I -0.024 -0.008 - 6 . 0 - 1 0 . 0 to 0 . 0 3 0 to 0 .016 to 1 2 . 0 to 7 . 0 II - 0 . 0 3 3 - 1 1 . 0 to 0 .010 to 1 1 . 0 III - 0 . 0 3 0 - 0 . 0 2 6 1 0 . 0 3 . 0 to 0.013 to 0 . 0 0 5 to 2 0 . 0 to 1 3 . 0 Accuracy of the Results 1) Machining Errors For the i n d i v i d u a l specimens, the errors i n o v e r a l l dimensions and the thickness of the steps f o r samples of type II were a l i t t l e l e s s than \u00C2\u00B10.0005 i n . The errors i n the length of the tapered sections were about \u00C2\u00B10.005 i n . In general, these errors were random and hence t h e i r effects^ were d i f f i c u l t to determine. 85 For specimens of type I, one systematic machining error was observed. This occurred due to the f a c t that, except f o r T = 0.4 i n . , a sharp edge could not he machined without f r a c t u r i n g the samples. The finishing.was therefore done with sandpaper, the r e s u l t being that the ends of the taper were somewhat rounded. In order to evaluate the e f f e c t of t h i s type of error, calculations were made f o r the taper shown i n Figure 5\u00C2\u00BB5c, the r e s u l t s being given i n the Pro column of Table 5*1\u00E2\u0080\u00A2 The e f f e c t was d e f i n i t e l y s i g n i f i c a n t and when taken into account, the r e s u l t s were generally closer to the measured values than those calculated using the i d e a l taper. i l ) E r ror i n the P e r m i t t i v i t y of the Expanded Polystyrene The expanded polystyrene used i n constructing the specimens was cut from \"a 24 x 18 x 1 i n . sheet with a nom-i n a l r e l a t i v e p e r m i t t i v i t y \u00C2\u00A3^=1.6. This value was checked by measuring \u00C2\u00A3 K for' several small pieces of the material using ( 5 3 ) the method of Roberts and Von Hippel , the estimated error being \u00C2\u00B10.01. A range of values between 1.48 and I.63 was obtained. The majority of the samples gave a value near 1.5 and t h i s value was taken to be the r e l a t i v e p e r m i t t i v i t y of the material. The e f f e c t of a deviation from t h i s value i s indicated by comparing the entries i n the Prc column of Table 5.1 ( tr \u00E2\u0080\u00A2= 1.55) with those i n the /\u00C2\u00B0r column, f o r the type III specimens. 86 i i i ) Measurement Errors There are four effects causing errors i n the measured values of 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) Errors i n po s i t i o n i n g the short c i r c u i t (\u00C2\u00B10.0001 i n . ) , measuring the motion of the probe i n the slotted-section ( \u00C2\u00B10.001 cm.) and determining the p o s i t i o n of the probe r e l a t i v e to the sample (\u00C2\u00B10.005 cm.). (c) Residual r e f l e c t i o n s . (d) Frequency d r i f t (which causes an addi t i o n a l n o d a l - s h i f t ) . The measurements were corrected f o r the errors due to (c) and (d). I t i s expected that the error i n the exper-imentally determined values 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 \u00C2\u00B10.0025 f o r the magnitude and \u00C2\u00B1 3 \" f o r the argument. 8?. 5*2 Measurement of the Scattered F i e l d s The b a s i c measuring system, shown i n F i g u r e 5*6, i s q u i t e c o n v e n t i o n a l . The frequency synchronizer e l i m i n a t e s the n e c e s s i t y of e q u a l i z i n g the e l e c t r i c a l lengths of the s i g -n a l and reference paths. Two methods of r e c o r d i n g the f i e l d may be used. 1) Measurement of Magnitude and Phase The 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 Model 1751\u00C2\u00BB has meters which i n d i c a t e the amplitude of the mea-sured s i g n a l , M, and i t s phase angle, \u00C2\u00AB 1 .99994 cmT1),. the measured q u a n t i t i e s b e i n g the magnitude and phase of the f i e l d . For plane-wave Incidence, values were obtained at 1 0 \u00C2\u00B0 i n t e r v a l s . For c y l i n d r i c a l - w a v e i n c i d e n c e , the f i e l d s were recorded w i t h a ch a r t recorder and the values r e q u i r e d f o r computations were taken from the graph; both 5 \u00C2\u00B0 and 1 0 \u00C2\u00B0 i n t e r v a l s were considered. 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 equal weighting (\"equation ( 4 . 6 ) ) and the r e q u i r e d order was determined by the s i m p l i f i e d method suggested i n S e c t i o n 4 . 2 , s^ being c a l -c u l a t e d u s i n g equation ( 4 . 1 2 a ) . The value of s* decreased slow-l y a t f i r s t , then r a p i d l y , and f i n a l l y became q u i t e constant; the value used f o r N was such that adding another term caused a decrease of l e s s than 2% i n the value of sN f o r both the r e a l and imaginary p a r t s . 5 . 3 - 1 I n c i d e n t F i e l d The e r r o r s i n the measured values of the 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 are shown i n Fig u r e s 5 . 1 3 a and b. A p l o t of the measured i n c i d e n t f i e l d 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 Fi g u r e 5 . 1 ^ and the e r r o r s i n the readings at lo\" i n t e r v a l s * Note t h a t the magnitude curve l a g s the phase curve by 3 * i n the experimental p l o t s 0.02 0 -0.02 Figure 5.13 E r r o r s i n the Measured Incident F i e l d f o r P l a n e -Wave Incidence F i g u r e 5.14 M e a s u r e d I n c i d e n t F i e l d a t r = 4 cm. f o r C y l i n d r i c a l \u00E2\u0080\u0094 W a v e I n c i d e n c e 10? CD - P \u00E2\u0080\u00A2H Cl bO ce 0.04 0.02 -0.02 0.05 (D -P \u00E2\u0080\u00A2rl Cl bo cd a -0.05 10 CD bo o 0) CO cd X I PL, CO CD bo 0) CO cd X I PL, -10 &\u00E2\u0080\u00A2 , degrees b) x = 8 -cm. Figure 5.15 Errors in the Measured Incident F i e l d for C y l i n d r i c a l -Wave Incidence 108 at r = 4 cm. and r ~ 8 cm. are shown i n Figures 5 \u00C2\u00AB 1 5 a and b. In both cases, some asymmetry i n the f i e l d i s evident, p a r t i c -u l a r l y a t the l a r g e r r a d i i . The readings at the 4 cm. r a d i u s 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 occupied by the c y l i n d e r i s about \u00C2\u00B1 0 . 0 2 i n magnitude and \u00C2\u00B1 2 \u00C2\u00B0 i n phase. 5.3 -2 . Scattered F i e l d (1 ) Comparison of the two Systems Most of the r e s u l t s were obtained u s i n g c y l i n d r i c a l -wave i n c i d e n c e s i n c e t h i s system was c o n s i d e r a b l y more convenient than the plane-wave system. A comparison between the r e s u l t s obtained from measurements u s i n g plane and c y l i n d r i c a l wave i n -cidence i s given i n F i g u r e 5.Tb f o r a) A m e t a l l i c c y l i n d e r , r , = 1 .5 cm. b) A d i e l e c t r i c c y l i n d e r , \u00C2\u00A3r = 2 . 5 4 * , r , = 2 . 9 cm. c) A m e t a l l i c core, r z = 1 .5 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\u00C2\u00BB17\u00C2\u00AB I t i s seen from the e r r o r curves i n F i g u r e s 5.16& to c t h a t the accuracy achieved u s i n g the c y l i n d r i c a l - w a v e system i s comparable w i t h or b e t t e r than t h a t achieved u s i n g the plane-wave system,(the 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 are r e l -a t i v e l y poor f o r the plane-wave system). The t r u n c a t i o n e r r o r i s . much sma l l e r than the experimental e r r o r i n e i t h e r case. *\u00E2\u0080\u00A2 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 which had sup-, e r i o r machining q u a l i t i e s to the polystyrene used i n the 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 1 0 9 A comparison between the t h e o r e t i c a l values of 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 those determined from e x p e r i -mental data i s given i n Table 5\u00C2\u00AB3\u00C2\u00BB Again i t appears t h a t the accuracy of the r e s u l t s obtained by the two experimental systems i s comparable. ( I I ) Measurement E r r o r s The e r r o r s i n the measured values of the s c a t t e r e d f i e l d , TC - 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 the three c y l i n d e r s considered above are shown i n Fig u r e s 5.18a to c. The s c a t t e r e d f i e l d i s not shown s i n c e i t I s not very 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\u00C2\u00ABl6a to c. The s c a t t e r e d f i e l d and f i e l d e r r o r s at 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\u00C2\u00AB5 cm. e) A d i e l e c t r i c core \u00C2\u00A3 r - 2.54, r z = 1 . 5 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*17 are shown i n F i g u r e s 5\u00C2\u00ABl9h and 5\u00C2\u00BB20b, r e s p e c t i v e l y . The e r r o r s i n the directly-measured magnitudes; are about the same f o r a l l the c y l i n d e r s , g e n e r a l l y l e s s than \u00C2\u00B10.05. The phase e r r o r s on the other hand are s t r o n g l y dependent upon the type of c y l i n d e r , ranging from +3\u00C2\u00B0 f o r the 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 to over \u00C2\u00B120* f o r the 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 the phase I s chang-i n g r a p i d l y , corresponding to minima i n the magnitude. The range of e r r o r s i n l s i and g ^ w i l l t h e r e f o r e tend t o be the same f o r a l l c y l i n d e r s . ( i i i ) E f f e c t of Changing TA A comparison between the r e s u l t s obtained from ) 110 measurements w i t h r0 = 4 cm. and T0 = 8 cm. i s given f o r C y l -i n d e r s (d) and (e) i n Figures 5*19 and 5 . 2 0 , r e s p e c t i v e l y . For both c y l i n d e r s , the value of N determined w i t h r c = 8 cm. was one l e s s than t h a t w i t h r0 = 4 cm.; the. t r u n c a t i o n e r r o r curves 1 are f o r the sma l l e r value of N. I t i s r e a d i l y seen t h a t b e t t e r r e s u l t s are obtained u s i n g the data w i t h r 0 - 4 cm. f o r C y l i n -der (d) and r f t = 8 cm. f o r C y l i n d e r ( e ) . R e s u l t s f o r C y l i n d e r (b) were about the same f o r both values of rc wh i l e f o r C y l i n d e r (c) the r e s u l t s w i t h r 0 = 4 cm. were again more accurate than w i t h r c \u00C2\u00AB= 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 to the r a p i d decrease i n the magnitude of the high-order Hankel f u n c t i o n s near the c y l i n d e r , the num-ber of s i g n i f i c a n t terms tends to decrease 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 note the r e l a t i v e l y small t r u n c a t i o n e r r o r at r = 8 cm. compared w i t h t h a t at r = 3*5 cm. f o r the m e t a l l i c c y l i n d e r . 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 to use a sm a l l value of r 0 , t h i s may not always be the case. I f the e r r o r s i n f 5 i and g 3; were the same at both r a d i i considered, the e r r o r s i n a ^ and Hg-n would al s o be the same but, s i n c e the magnitude of the Hankel 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-\u00E2\u0080\u009E would be l a r g e r w i t h r 0 = 8 cm. Thus, the e r r o r s i n the c a l -c u l a t e d f i e l d would decrease w i t h decreasing r 0 . The good r e -s u l t s f o r the r 0 = 8 cm. data shown i n F i g u r e s 5 \u00C2\u00AB 2 0 a to c l n d i -cate that- the e r r o r s i n lsi and g s \u00C2\u00A3 depend on rad i u s and i t MAG ERROR oq P H fD \u00E2\u0080\u00A2 P C/3 pb p e+ O e+ <<\u00E2\u0080\u00A2 ro I - H H - re B p . H \u00C2\u00BBrj o ce P K-i p o < H P CD pi p o \u00C2\u00BB ' fD H * ra ct-Pu 1 P CD t\u00E2\u0080\u0094' 0 p %\u00E2\u0080\u00A2 o < fD fD >i II o. n . H -II P- \u00E2\u0080\u00A2 fD o o O fD B B \u00E2\u0080\u00A2 ....... p 0 P-o M P t-i *d 4 <+ O 1 H- ti 1 O 01 0 a o w \u00E2\u0080\u00A2 1 a trioq P OQ fc\u00C2\u00BB fD p - p O B \u00E2\u0080\u00A2-d t\u00E2\u0080\u00941 P 0 fD SCAT F L D MflG .0 .5 a ' ARG ERROR 10.0 .0 CT)o m a rn.-o I I I ro cr a 10.0 _J a m a c p g \" a ro cr a L.O SCAT F L D ARG ( X l O 1 ) -20.0 .0 20.0 J o RNGLE (OEG) 200.0 RNGLE (DEG) b) lr = 2.54, r- C M I .0 50.0 100.0 150.0 200.0 RNGLE (DEG) r, = 2.9 cm. H MflG ERROR a SCRT FLD MflG .0 L.O 2.0 SCAT FLD ARG ( X l O 1 ) -20.0 .0 20.0 ro 114 Caption A p p l y i n g to the F i e l d E r r o r s I n F i g u r e s 5 . l 6 , 5.18, 5 .19 and 5 . 20 Y O E r r o r s i n measured values, & =0 to \u00C2\u00B1180*; plane-wave Incidence x A E r r o r s i n measured values, 0- = 0 to \u00C2\u00B1180\u00C2\u00B0; c y l i n d r i c a l -wave Incidence - \u00E2\u0080\u0094 \u00E2\u0080\u0094 E r r o r due to t r u n c a t i n g the 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 the e x p e r i -mental data \u00E2\u0080\u0094 \u00E2\u0080\u0094 E r r o r i n the s c a t t e r e d f i e l d c a l c u l a t e d from data u s i n g plane-wave i n c i d e n c e , r 0 = 4 cm. 1 \u00E2\u0080\u00A2 E r r o r i n the s c a t t e r e d f i e l d c a l c u l a t e d from data u s i n g c y l i n d r i c a l - w a v e Incidence, r e = 4 cm. As above w i t h r a = 8 cm. o C O 1.5 cm. Cross-Linked P o l y s t y r e n e , ( \u00C2\u00A3 r = 2 . 5 4 ) 0 0 o 3.4 cm. F i g u r e 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 the Experiments Table 5.3 Comparison Between Values of Backscattering Cross-Section Calculated Theoretically and from Experimental Data Cylinder Theoretical Experimental Plane Cylindrical Metal, r, =1.5 cm. 4.930 -.4.97 -.5.34 4.51 - 5.21 Metal, r, =3.5 cm. 11.117 9.93 11.01 - 11.88 ly = 2.54, r, = 2.0 an. -6.491 6.61 \u00C2\u00A3 r = 2.54, r,= 2.9 cm. 22.012 21.43 21.12 - 22.81 Core- Sy = 2.54, r,. \u00C2\u00BB 1.5 Shell- Artific i a l (Figure cm. 5.17) 0.158 0.06 - 0.07 Core-Metal, rz*=1.5 cm. Shell- A r t i f i c i a l (Figure 5.17) 7.822 8.00 - 8.06 8.06 - 8.08 CEin - 5 - 0 1 I .0 50.0 100.0 RNGLE (DEG) 150.0 200.0 .0 a) Metal, r, = 1.5 cm. 5D.Q 1D0.0 RNGLE (DEG) 150.0 200.0 Figure 5.18 Errors i n the Scattered 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 ERROR MflG ERROR -.1 .0 .1 -.1 .0 .1 CD O MflG ERROR SCAT FLD MflG -.1 .0 .1 .0 .5 1.0 MflG ERROR SCAT F L D MflG -.1 .0 .1 .0 .5 1.0 MflG ERROR SCAT F L D MflG -.1 .0 .1 .0 1.0 2.0 MAG ERROR -.1 .0 .1 SCAT FLD MAG .0 1.0 2.0 ro J a \u00E2\u0080\u00A2 a 124 appears that the r e l a t i v e l y s m a ll phase e r r o r s i n the range 0 - 8- - 1 0 0 \u00C2\u00B0 may he a c o n t r i b u t i n g f a c t o r (note t h a t the phase v a r i a t i o n i s smoother at r = 8 cm. than at r = 4 cm.). 5 . 3 - 3 Examples of 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 measured w i t h the c y l i n d r i c a l - w a v e system, r 0 = 4 cm., f o r the f o l l o w i n g c y l i n d e r s are given i n F i g u r e s 5\u00C2\u00AB18a to d: a) Metal, r , = 3 . 5 cm. b) Core- Metal, r z \u00C2\u00AB 1 . 5 cm.;; ... S h e l l - A r t i f i c i a l ( F igure 5 . 1 7 ) c) <% = 2 . 5 4 , r , = 2.9 cm. d) Core- Et = 2 . 5 4 , r x = 1 . 5 cm.; S h e l l - A r t i f i c i a l ( F igure 5 -17) Some b a s i c c h a r a c t e r i s t i c s of the d i f f r a c t e d f i e l d s of these c y l i n d e r s should be pointed: out. For C y l i n d e r ( a ) , we note the broad shadow r e g i o n where the f i e l d magnitude i s very small (the geometrical o p t i c s shadow extends from about 1 5 8 \u00C2\u00B0 to 202\u00C2\u00B0). For C y l i n d e r ( b ) , the magnitude v a r i a t i o n i s qu i t e I r r e g u l a r . For C y l i n d e r s (c) and ( d ) , there i s a r a p i d o s c i l l a t i o n of magnitude w i t h 6- and both have a maximum at & = 180\u00C2\u00B0. There i s a st r o n g c o n t r a s t between (c) and ( d ) , how-ever, i n tha t whereas the magnitude of the o s c i l l a t i o n i s l a r g e at a l l angles f o r ( c ) , the magnitude' i s small near 0- = 0 and inc r e a s e s s t e a d i l y w i t h \u00C2\u00A9- f o r ( d ) . MAGNITUDE 5s J 5 MAGNITUDE \u00E2\u0080\u00A2 2ZT 129 5.3-4 Comparison w i t h P u b l i s h e d R e s u l t s I t seemed d e s i r a b l e to o b t a i n experimental r e s u l t s f o r d i r e c t comparison w i t h the p r e v i o u s l y published r e s u l t s 1 ' which were i n disagreement w i t h computed v a l u e s . C y l i n d e r s which approximate the f o l l o w i n g cases were constructed: a) Mr) = 10Ac.r; k 0 r ^ = 3, = 3.7, 6 .8, 8.0 b) \u00C2\u00A3 K(r) = 5 A 0 r ; k ^ r * \u00C2\u00AB 3, k<,r, = 4.2 The comparison i s given i n Table 5 .4. Table 5\u00C2\u00AB4 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 C y l i n d e r Published^ 3 ^ T h e o r e t i c a l * ; Experimental** \u00C2\u00A3r(r) = l 0 A o r k,r, = 3.7 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 \u00C2\u00A3r(r) = 5/k0r k e r , = 4.2 1.0 4.826 . 4.94 * C a l c u l a t i o n discussed i n Chapter 3 ** Using 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 130 5.4 Summary A method of 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 of a r t i f i c i a l d i e l e c t r i c has been developed. 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 placed 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 measure-ments 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 the 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 medium are v a l i d . The measured r e s u l t s on c y l i n d e r s confirm computed r e s u l t s ( a l s o disagree w i t h some published r e s u l t s ) . P a r a l l e l - p l a t e systems f o r measuring the 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 of plane and c y l i n d r i c a l wave i n c i d e n c e were constructed. For both systems, the method 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 c o e f f i c i e n t s given 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 those obtained w i t h c y l i n d r i c a l -wave in c i d e n c e and i t was found t h a t the accuracy of the s c a t -tered f i e l d 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 obtained from measurements u s i n g plane-wave i n c i d e n c e . I t should be pointed out t h a t the accuracy of the r e -s u l t s i s a f f e c t e d by the r e l a t i v e l y crude form of the theory given i n Chapter 4 which was a p p l i e d to the experimental data, and i s l i m i t e d by two departures from the i d e a l i z e d model which have not been accounted f o r . These are: ( i ) The d e v i a t i o n of the i n c i d e n t f i e l d from the h y p o t h e t i c a l value over the r e g i o n occupied by the c y l i n d e r w i l l cause e r r o r s i n the s c a t t e r e d f i e l d . 131 (11) Since the 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 the source and r a d i a t i o n r e f l e c t e d from the edges of the p a r a l l e l - p l a t e r e g i o n , the presence of \"the c y l i n d e r changes the i n c i d e n t f i e l d . 132 6 . APPLICATION TO MEASUREMENTS ON CYLINDRICAL PLASMAS The determination of the electron, d e n s i t y d i s t r i b u -t i o n i n l a b o r a t o r y plasmas by microwave probing i s a t o p i c of current I n t e r e s t ^ e s * 6 0 ^ and the phase angle of the s i g n a l i s . considered to be the most s i g n i f i c a n t q u a n t i t y . When determin-i n g the 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 plasma-column, the measurements are 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 planar model. The v a l i d i t y of 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 by comparing r e s u l t s f o r a c y l i n d r i c a l plasma con-s i s t i n g of two homogeneous regions and a surrounding g l a s s tube (61) w i t h those i n the corresponding planar case . I n t h i s chapter, a comprehensive comparison between the phase angles of the s c a t -tered s i g n a l s computed f o r an Inhomogeneous plasma c y l i n d e r and those f o r the corresponding planar model i s presented. 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 ) Radius of the plasms, column r e l a t i v e to 7\ 0 l i ) Maximum e l e c t r o n d e n s i t y (assumed to occur on the a x i s - o f the c y l i n d e r ) i i i ) Form of 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 ) Surrounding g l a s s tube v) L o c a t i o n of the p o i n t of obs e r v a t i o n 6 . 1 O u t l i n e of the Problem and Method of Computation The two c o n f i g u r a t i o n s considered are i l l u s t r a t e d i n Fi g u r e 6 . 1 . 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 d i r e c t i o n . 133 (a) Transmitted Wave T ( e - ^ o ( x + r , ) -2rr \u00E2\u0080\u00A22r r I n c i d e n t Wave R e f l e c t e d Wave > R e 3 k 0(x-r, ) T, = f T , | e ^ T (b) F i g u r e 6.1 (a) C y l i n d r i c a l C o n f i g u r a t i o n (b) P l a n a r C o n f i g u r a t i o n 134 For convenience, the re f e r e n c e phase i s taken to be t h a t of the i n c i d e n t f i e l d a t x = r, (not at 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 of a glas s tube, w i t h outer and Inner r a d i i of r ( and x'iy 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\u00E2\u0080\u009E( 1 - f ( r / r 2 ) ) (6.1) 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 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 - 0 Plane 135 In the corresponding planar configuration we have N(x) = N c( 1 - f ( x / r x ) ) x > 0 (6.2a) = N\u00E2\u0080\u009E( 1 - f ( - x/r*) ) x 5 0 (6.2b) We assume that the c o l l i s i o n losses i n the plasma can be ne-glected so that I t may be represented as a region of r e a l per-m i t t i v i t y , 6y(r) - 1 - N ( r ) / f t c . A hypothetical electron density p r o f i l e and the corresponding 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 \u00C2\u00AB 0 plane are shown i n Figure 6.2. 6.1-1 Basis of Comparison A natural choice of quantities f o r the planar con-f i g u r a t i o n i s the argument of the r e f l e c t i o n , ^ , and the argument of the transmission c o e f f i c i e n t , (?T I these are compared with (Pr{oo) and ^ t ( r , ), respectively, f o r the c y l i n d r i c a l config-uration, where t(.Tt ) 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 of a wave transmitted through the cylin d e r . 136 6 . 1 - 2 Method of C a l c u l a t i o n The values of CPn and (PT were s p e c i a l l y computed by TCilbee u s i n g a wave tr 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 e l e c t r o n d e n s i t y p r o f i l e . Except where exact s o l u t i o n s were a v a i l a b l e ( \u00C2\u00A3 r(r) = a r 1 , ), the nu-m e r i c a l i n t e g r a t i o n method of S e c t i o n 3 \u00C2\u00BB 1 - ^ was used to 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 i n the c y l i n d r i c a l case. The phases were then c a l c u l a t e d u s i n g Cp (r)\u00E2\u0080\u00A2 = k 0 ( r - 2 r , ) + Arg E 2 where ^ ( r ) denotes e i t h e r (py(r) or (?\u00C2\u00B1{T) and E z i s given by i ) Equation ( 2 . 1 b ) when (r) = ' 6.2 R e s u l t s and D i s c u s s i o n The computations were performed f o r the f o l l o w i n g f u n c t i o n s f ( r / r 2 ) i n equation ( 6 . 1 ) : a) f ( r / r z ) = 3(r/r,e f - 2 ( r / r 2 ) 3 b) f ( r / r , ) - ( r / r * ) * c) f ( r / r * ) - { T / x z f d) f ( r / r j = 0 The forms of the corresponding e l e c t r o n d e n s i t y p r o f i l e s are shown i n Figu r e 6 . 3 . A l l have a zero slope a t r = 0 . I n ad-d i t i o n (a) has a zero slope a t r = r z . V a r i a t i o n s of the forms (b) and (c) approximate 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 p l a s m a s ^ 6 2 a n d (d) i s considered f o r comparison purposes. 137. I M A F i g u r e 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 6.2-1 Comparison of R e s u l t s f o r the Four P r o f i l e s . , i ) Comparison of (PY[po) and The values of t.{co) and (pn, has been drawn i n each of these t a b l e s . Agreement i s taken to mean th a t the d i f f e r e n c e i s l e s s than 6\u00C2\u00B0. We note t h a t f o r a given value of r A , the s i g n of (Pr (\u00C2\u00BB) - ($)n changes from negative to p o s i t i v e 138 as 2^ e/Wo i n c r e a s e s from 0.25 to 1 g i v i n g very good agreement f o r intermediate values of N c/N 0. The range of N c/N 0 and r x over \u00E2\u0080\u00A2which agreement i s ensured depends on the type of p r o f i l e and i t appears t h a t the c u t o f f r a d i u s , r c , at which N(r) = N c i s an important f a c t o r . The values of r c f o r the f i r s t three p r o f i l e s ( f o r P r o f i l e ( d ) , r c = r A ) are given i n Tables 6.2a to 6.2c and stepped l i n e s corresponding to those i n Tables 6 .1a to 6.1c are drawn. I t i s seen that agreement i s g e n e r a l l y obtained f o r r c > 0.5 except near N c/N 0 = 1 . i l ) Comparison of 7 to w i t h i n 6 \u00C2\u00B0 i s c o n s i s -t e n t l y obtained i n the.re 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 evident t h a t the range- of parameters over which 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 are sen-s i t i v e t o the form of 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 the conce n t r a t i o n near the a x i s of the c y l i n d e r and the higher the gradient w i t h i n the c y l i n d e r , the poorer the agreement between Q>t(r, ) and 0 T . 6.2-2 E f f e c t of a Glass Boundary A l a b o r a t o r y plasma i s g e n e r a l l y enclosed by a glass tube and we now consider the e f f e c t which t h i s has on the range 139 of v a l i d i t y of the p l a n a r model. Two thi c k n e s s e s of glass, are considered, ( i ) r , - r z - ?i\u00C2\u00A3/2, which i s transparent f o r the planar c o n f i g u r a t i o n and thus has no e f f e c t on Cpn and (pT and ( i l ) r , - r 2 --- ^ / 4 , which should have a l a r g e e f f e c t f o r both c o n f i g u r a t i o n s . The computations were c a r r i e d out f o r a l l the 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 (near t h a t of Pyrex g l a s s ) f o r the g l a s s . The r e s u l t s f o r P r o f i l e (b) are shown i n Tables 6 . 4 and 6 . 5 \u00C2\u00BB For the r e f l e c t e d f i e l d , the previous agreement-r e g i o n i s e s s e n t i a l l y unchanged f o r r , - r z = ?i\u00C2\u00A3/2 but i s r e -duced s h a r p l y f o r r , - r z = ^ e / 4 . For the t r a n s m i t t e d f i e l d , there are no regi o n s of agreement between r, a l -though d i f f e r e n c e s w i t h i n about 30 \u00C2\u00B0 are g e n e r a l l y observed over the previous agreement r e g i o n f o r r , - r 2 = ?)\u00C2\u00A3/2; f o r r , - r 2 = z ^ / 4 , there i s l i t t l e resemblance between ) 45.2 51.6 0.9 5.4 -35.4 -39.9 -48.3 -60.2 -54.3 -71.3 -60.1 -82.3 -91.2 -93.5 -88.7 -141.4 -120.4 158.8 -44.5 -58.2 1.0 (?M) io 1 . 0 0 .1.05 1.25 1.50 2 . 0 0 4.00 0.5 -127 .3 -138.3 -345 . 3 - 1 5 0 . 2 143.0 1 6 2 . 1 1 0 3 . 2 121.2 69 . 3 83.O 30 . 4 38 .4 1 . 0 103 .6 122 .5 67.6 74.6 -43.6 - 5 2 . 1 \u00E2\u0080\u0094111.8 -122 .4 179 .6 1 6 6 . 1 85.6 76 .6 2 . 0 ) 58.0 51.6 8.2 5.4 -42.3 -39.9 -62.8 -60.2 -72.7 -71.3 -82.4 -82.3 -65.2 -93.5 -129.4 -141.4 -178.6 158.8 164.5 -58.2 1.0 (?r(oo) 10.8 13.8 -57.7 -55.1 -130.6 -134.2 -164.9 -177.8 176.9 154.8 158.6 123.1 136.7 89.2 74.4 -23.7 -116.6 -133.9 -171.0 -26.6 2.0 #.(\u00C2\u00AB>) -47.5 -4.5.6 -178.5 -178.2 -0.8 -1.1 -82.9 -85.9 -126.2 -135.0 -172.2. 164.5 137.2 89.5 -109.7 -141.1 -153.6 -178.5 -86.5 -143.2 5.0 %(\u00C2\u00B0\u00C2\u00B0) 134.8 135.2 132.6 133.2 53.5 53.4 -158.9 -161.0 76.9 73.1 , -66.0 -72.6 137.1 89.7 -123.6 -130.4 -108.1 -131.7 50.9 -133.1 (i) r, -0.25 0.50 0.75 0.85 0.90 0.95 1.00 1.25 2.00 \u00E2\u0080\u00A2 4.00 0.5 ?rH -137.5 -168.6 -147.6 -179.3 -152.3 171.3 -153.6 166.3 -154.1 163.2 -154.6 159.9 -155.4 \u00E2\u0080\u00A2156.1 -154.3 156.3 -125.0 171.2 -152.3 148.9 1.0 r(\u00C2\u00AB>) (PH -171.3 -177.2 178.2 167.8 162.9 127.9 -166.2 10.3 -176.1 -95.1 -163.7 -139.2 -137.3 -158.7 -178.8 .\"178.0 -144.9 \u00E2\u0080\u0094156.8 -164.8 168.9 2.0 ) 168.0 170.0 18.8 7.5 179.0 179.6 \u00E2\u0080\u00A2175.9 158.2 177.5 126.8 -171.8 -67.3 -160.7 -158.1 -169.1 179.0 -178.9 -179.4 -148.3 -163.4 5.0 -129.9 -126.8 -139.3 -129.4 -170.3 -168.1 170.7 77.7 -170.0 -162.4 -178.4 -162.8 -174.8 -157.8 -178.0 179.6 143.7 -155.9 -143.2 -157.0 (ii) r, - r 2 = ^/4 146 Table 6.5 Comparison Between (Pt{T, ) and q>T f o r a G l a s s -Enclosed Plasma C y l i n d e r , P r o f i l e (b) 1 . 0 0 1 . 0 5 1 .25 1 .50 2 . 0 0 4 . 0 0 0 . 5 - I 6 . 8 6 5 . 7 - 6 8 . 9 4 5 . 1 -80.5 - 2 1 . 9 -90 .4 36.O - 6 9 . I - 6 6 . 8 - 3 7 . 8 -101. I 2 . 0 Mr,)-(PT 50.7 6 5 . 3 8 3 . 3 - 1 5 . 5 84.3 -140.3 53.7 - 1 3 0 . 9 6 . 6 9 0 . 6 83.2-- 7 3 . 5 5 . 0 Mr,\ ) and i s obtained, l i t t l e v a r i a t i o n of 1 and there i s no gl a s s boundary, the v a r i a t i o n of (pr{T) w i t h r a d i u s i s small but (pt(<\u00C2\u00BB) and , U ^ t r ) +. d\u00E2\u0080\u009E V\u00E2\u0080\u009E h(r) ) where Vvh and V\u00E2\u0080\u009Ej, are independent s o l u t i o n s of , .1 n + ( _ . ) R M + ( ^TSy - )\u00E2\u0080\u00A2 R\u00E2\u0080\u009E = 0 r c r r * ( A . 8 ) or, i n terms of the v a r i a b l e f d l \u00C2\u00A7 3 \u00C2\u00A3,(?A) % d n' \u00E2\u0080\u0094 ) + \u00C2\u00A3\u00E2\u0080\u009E( JA) - T-X ^(?A) d? R , , ( $ ) = 0 (A.9) As i n the case H 2 = 0, closed form s o l u t i o n s of equation ( A . 8 ) r a r e l y e x i s t . Por F:r(r) \u00E2\u0080\u00A2= a r b these are ^ V \ b + 2 ' v a r y * = ( 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 the Case of a 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 , do not e x i s t 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 \u00C2\u00A3 r(r) = s r + to I n t h i s appendix s o l u t i o n s are obtained i n the form of a power s e r i e s S o m e of the r e s u l t s have been d e r i v e d by P e l n s t e i n ^ ^ , B e l P a r a l l e l P o l a r i z a t i o n S u b s t i t u t i n g \u00C2\u00A3,,(?A) = s 5/k. + t i n t o equation (A.5) we o b t a i n Closed form s o l u t i o n s of the r a d i a l wave equation ( B . l ) where s, = s/k Assume the power s e r i e s s o l u t i o n CO (B.2) The i n d i c l a l equation (36) i s found t o be c ( c - l ) + c \u00C2\u00BB n* = 0 which y i e l d s n (B.3) and the recurrence r e l a t i o n i s l 6 l Choosing the p o s i t i v e sign i n equation (B\u00E2\u0080\u009E3) gives the. soluti o n which i s ana l y t i c at the o r i g i n . The recurrence r e l a t i o n becomes 1 a = \u00E2\u0080\u0094\u00E2\u0080\u0094-\u00E2\u0080\u0094\u00E2\u0080\u0094 ( t a, o -H s,a, o ) (B .5) 1 i ( 2n+i ) l m m d 1 3 and thus the so l u t i o n i s given by J \u00E2\u0080\u0094 ? 2-(2n+2) 3-(2n+3) If + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 t * (B.6) 2 - 4 - ( 2 n + 2 ) (2n+4) This i s e s s e n t i a l l y the r e s u l t obtained by Pelnstein. I f the negative sign i n equation (B .3) i s chosen, we f i n d that a zero appears i n the denominator of equation (B .4) when I - 2 n , making a 2\u00E2\u0080\u009E I n f i n i t e . A second solut i o n may be obtained i n the form^ 3 6^ CO \u00E2\u0080\u00A2 V, = A U , ln ( Y ) + X*T- ^ ( B * 7 ) l = o The constant A i s chosen such that b 2 n becomes indetermin-ate (of the form 0/0). The value of b 2\u00E2\u0080\u009E i s now arbitrary;; choosing b 2 7 J = 0 adds a term b 2 T, TJ.,, to the \>2yi = 0 s o l u t i o n . We f i n d 1 A . s . - ( t b 2 n - 2 + s , b 2 n \u00E2\u0080\u009E 3 ) (B.8) 2 n B.0 162 and the recurrence r e l a t i o n s i ( * ^1-2 + s' b i - 3 ^ i < 2n 1 ( 2n-i ) (Bo9) 1 = - \u00E2\u0080\u0094 ( \"t b, o + s, \u00C2\u00AB +2 A , '( l-n)\u00C2\u00ABa 1 o\u00E2\u0080\u009E ) i ( i~2n ) 1 4 1 1 ^ n i > 2n When n = 0, A i s undetermined by equation (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 equations (B . 6 ) and (B . 7 ) converge f o r a l l f i n i t e values of ^5. B.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 S u b s t i t u t i n g \u00C2\u00A3r( J/k) = s, + t i n t o equation (A.8) we o b t a i n d* 1 s, a n* + ( _ ) \u00E2\u0080\u0094 + s, f +1 \u00C2\u00BB \u00E2\u0080\u0094 d ? 2 3 s, J +\"t d ? I (BclO) Comparing t h i s equation w i t h equation ( 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 occurs a t ~\u00C2\u00A7 - t / s , and th 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 values of (.7 ) 5\" i s not p o s s i b l e . F e l n s t e i n , 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 , gives the s o l u t i o n about the s i n g u l a r i t y at f = \u00C2\u00AB\u00E2\u0080\u00A2 t / s , . His a n a l y s i s l s not genera l , however, si n c e h i s possible: values of s, and t are constrained such t h a t 163 B , f + t < 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 the f a c t t h a t he was d e a l -i n g w i t h a model of a meteor t r a i l having a zero e l e c t r o n d e n s i t y at i t s outside boundary. The s o l u t i o n of equation (B . 1 0 ) i n the s p e c i a l cases s, = 0 and t = 0 i s given by equation (A.6) w i t h b = 0 and b = 1 r e s p e c t i v e l y . With the r e s t r i c t i o n s s, =\u00C2\u00A30, t ^ 0 , a more convenient form of equation (B . 1 0 ) i s obtained by the change of v a r i a b l e fi-s,~%/t. The r e s u l t i n g equation i s d 2 I d n 2 \u00E2\u0080\u0094, + \u00E2\u0080\u00A2 + a(fi + 1) \u00C2\u00AB\u00C2\u00BB R^ = 0 ( B . l l a ) or d* d t.fi d/2 R\u00E2\u0080\u009E = 0 ( B . l l b ) where cc = t 3 / s f The s i n g u l a r i t i e s are 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 these two s i n g u l a r i t i e s and about some other p o i n t , say fi-fic. where fi0 i s outside the range -1 t o 0 , Expansion about fi - 0 Assume a power s e r i e s s o l u t i o n of the form R ; = / f 0 l (B . 1 2 ) 164 The l n d l c i a l equation gives the values of c, c = t n The recurrence r e l a t i o n l s [ (c+l- l ) 'Cc+i -2) - n2.] &\u00C2\u00B1mml (c+i+n)\u00E2\u0080\u00A2(c+i~n) ) +cc( a i - 2 + 2 a j _ \u00E2\u0080\u009E 3 + a i - 4 ) (B.1'3) When c - n, equation (B .13) becomes a \u00E2\u0080\u00A2 = 1 ( lz + 2 n i - 3 i - 3n+2 ) & ^ i * ( 2 n + i ) / + Q i ( - ' - a j__2 + 2 a i - 3 + a i - 4 ^ (B.14) and the s o l u t i o n may be w r i t t e n as ( n n 1 1 + _ / 3 2n+l 2-(2n+l) 2n+2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00C2\u00A9 (B.15) As i n Se c t i o n B . l , the second s o l u t i o n cannot be obtained by simply l e t t i n g c = ~n. A s o l u t i o n of the form (B.7) i n the v a r i a b l e /3 must be used. The recurrence r e l a t i o n i n t h i s case i s found to be b; = ( i - 2 n i - 31 + 3n + 2 ) b i - l l - ( 2 n - i ) ) -hcc{ + 2 b i - 3 + b i - 4 ^ i < 2n 165 ( i 2 - 2ni - 31 + 3n + 2 ) b ^ i - ( i - 2 n ) \u00C2\u00A3 + ^ ( + 2 t , i - 3 + b i - 4 . ) + Af2(i-n) a 1 - 2 n + ( 2(i-n) - 3 ) a ^ ^ J 1 > 2n (B . l 6 ) W h 6 r e A = 2naT t ( 3 n \u00C2\u00B0 2 ) b2n-l \" * ( b2n-2 + 2 b2n~ 3 + ^k ) ] i s chosen so that b 2 n i s a r b i t r a r y . When, n = 0, A i s a r b i t r a r y . Expansion about /3 = -1 With the further change of var i a b l e , t - /3 + 1, equation (B.llb) becomes y (-2T- l ) 2 \u00E2\u0080\u0094 + 1) \u00E2\u0080\u0094 + ct . l ) 2 , - n 2 y dt f 2 d 2T (B.l?) Proceeding exactly as before with t = o we f i n d c = 0 or 2. Again, i n order to obtain two solutions, the forms = t z a \u00C2\u00A3 r (B.18a) (B,18b) must be used. The recurrence r e l a t i o n s are (i+1)\u00C2\u00BB(21-1) a i - ] L - ( i - ( i - l ) - n* ) a i - 2 2 A 4 \u00E2\u0080\u009E ^ + A I - ^ ) i--(i+2-) ; - <*( a i = 3 - ^ 1 ^2 (B.19) 166 b; = 1-(1-2) (1-1)-(21-5) b ^ ~ ( (1-2). (1-3) - n 2 ) b l i - a . \" 0 6 ( b i \u00E2\u0080\u009E 3 ~ 2 b i ~ 4 + ) - A ( 2(1-1) a 1 = > 2 - (41-7) * (21-5) a ^ (B.20) where A = l s chosen so that b?. Is a r b i t r a r y * Expansion about L e t t i n g H> =/3 ~ fi0 i n equation (Bollb) we obtain d d dT d # ' '+*( ( +1) 2 - n* (y + + D which may be written as' d d ( f + A 2 2T + A, y + A.) \u00E2\u0080\u0094 +< * + ~ d y d y + B* *^'+ B 3 2f 5 + B a X 2 + B, 2C\" + B0 - C, 2T - 0< where Upon sub s t i t u t i n g R-\u00C2\u00AB = (B.21) A 2 = 3 A + 1 B* A, - 4 ,(3 A + 2) B 3 = 2 a(2 +1) A, = A l( A + 1) B 2 = * (6/?* + 6/?. + 1) c, = n* B, * 2 a A ( 2 A ' + 3A + 1) - a*( A + 1) B\u00E2\u0080\u009E = 2 A + 1) 167 we f i n d c = 0 or 1, When c = 0, \"both B . C and a, are a r b i -t r a r y and hence we have the complete s o l u t i o n . When c = 1, the series i s the same as would be obtained f o r c - 0 and the choice of constants a.0 = 0, a, arbitrary,. We may write two Independent solutions, each with one a r b i t r a r y constant by l e t t i n g a, = 0 when c = 0. The r e s u l t i s vr - x z b^ y (B.22a) (B.22b) where the following recurrence r e l a t i o n s apply: a; = k0 i . ( l - l ) b; = -A 0 1.(1+1) ( i - l ) (1 A, - 2A, + jS0 ) a i - ] L + [ (1-2). ( i A 2 - 3 A Z + 1) + B e - C.J .+ [ (1-3X1-4) + B, - 0, ] a \u00C2\u00B1 _ 3 + B * a i - 4 + B 3 a i \u00C2\u00AB 5 + B * a i \u00E2\u0080\u009E 6 1 > (B.23) i - ( i A , - A, +.0o ) b i - , + [ ( i - l ) ( i - A z - 2A 2 + 1) + B 0 - C J +[(i-2)-(i - 3 ) + B, - 0, J b \u00C2\u00B1 _ 3 + B 2 b \u00C2\u00B1 ^ + B 3 b \u00C2\u00B1 _ 5 -+B, b i - 6 i > (B.24) 168 B.2-1 Range of Convergence. The range of convergence of the three expansions i s e a s i l y determined by observing the d i s t a n c e from the p o i n t about which the expansion i s made to the nearest s i n g -u l a r i t y . The r e l a t i o n s h i p between given values of s and t and the range of r over which each expansion i s v a l i d may be seen 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 are onl y concerned w i t h r -^ 0. s>0, t< s>0, t>0 Fi g u r e B . l Regions: of Convergence of the S e r i e s S o l u t i o n s of the Wave Equation 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 \u00C2\u00A3 r = sr. -*- t . a) s and t of Opposite . Sig n b) s and t of the Same Sig n . 1 6 9 APPENDIX C Impedance and Admittance Relations, f o r Homogeneous Regions cause of the behaviour of B e s s e l f u n c t i o n s f o r small argu-ments, computational d i f f i c u l t i e s a r i s e when e i t h e r Er or r , or both, are s m a l l . By c o n s i d e r i n g the s m a l l argument behaviour 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=\u00C2\u00BB 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 general i s e a s i l y deduced by-studying these l i m i t i n g forms. We assume t h a t jJT\"-jj.oa 0.1 P a r a l l e l P o l a r i z a t i o n when fe7 T \u00C2\u00AB n I t was pointed out i n S e c t i o n 3.1-2 t h a t , be-Por a homogeneous r e g i o n of p e r m i t t i v i t y \u00C2\u00A3\u00E2\u0084\u00A2 > 0, , Q\u00E2\u0084\u00A2, S\u00E2\u0084\u00A2 and T\u00E2\u0080\u009E i n equation (2.15) become where a** - JT? \"5\", Equation (2.16) becomes J>, ( a M ) J- ( a M ) y ^ ( C 2 ) 170 For a m and b w approaching zero, we may use the s m a l l argu-(3 2) ment l i m i t i n g forms of J\u00E2\u0080\u009E and 1 l i m ^ ( z . ) = -~ ( i-z ) z 0 n / n l i m Y n ( z ) = 2/n-ln(z) z-\u00C2\u00BB0 , (n-1)/ TT n = 0 n > 1 (C.3) With the exception of J0 ( z ) , i n which case the r e l a t i o n J(f(z) = - J; (z) must \"be used, the l i m i t i n g forms f o r the d e r i v a t i v e s may be obtained d i r e c t l y from these expressions:, S u b s t i t u t i n g the values from equations (C.3) i n t o equations ( C d ) and (C.2) and s i m p l i f y i n g the r e s u l t , we o b t a i n = 0 n n i l I n V>1 n = 0 n i l 2 c r n 5rn +l 5\u00E2\u0084\u00A2 1 f n = 0 vn-i n 2 1 n 71 n > 1 (0 . 4 ) K ~ - 3 \ ~z\u00E2\u0080\u0094 n = 0 C f 5 M \u00E2\u0080\u00A2 ' \u00C2\u00AB J >> \u00E2\u0080\u0094 n > 1 0 n (C . 5 ) Equations (0 . 4 ) and (C . 5 ) are exact when \u00C2\u00A3 r = 0. I n computation, i t i s u n l i k e l y t h a t extremely small values of and i~r 0 (say \u00C2\u00A3 10 3 ) w i l l be s i g n i f -i c a n t . As a r e s u l t , unmanageable values of the B e s s e l f u n c t i o n s w i l l o n l y occur when h i g h orders are r e q u i r e d . We may d e a l w i t h these cases by removing c e r t a i n f a c t o r s from the B e s s e l f u n c t i o n s when performing the computations, as explained bslow* Por a core r e g i o n , i t i s evident from equation (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 value of z\" approaching t h a t given by equation (C .5 ) Por a s h e l l , we d i s t i n g u i s h between two cases. i ) \u00C2\u00A3^ s m a l l but ^ 1\u00C2\u00AB This means t h a t the s h e l l t h i c k n e s s I s much l e s s than i t s i n n e r r a d i u s . I f we remove a f a c t o r P x ^ ( b ^ ) from the B e s s e l f u n c t i o n s of the f i r s t k i n d and a f a c t o r 1/E? from those of the second k i n d , the r e s u l t i n g values w i l l be manageable; the modified f u n c t i o n s may be used d i r e c t l y i n equations ( C . l ) . Prom equations (C . 4 ) we see t h a t the products w i l l be of the o r -der of (Xyi^.f71. 172 i i ) X , / ^ \u00E2\u0080\u009E + , \u00C2\u00BB 1 causing f to be too l a r g e to be handled by the computer. I n t h i s case, terms i n J,, (a,J and Yvi(b^,) w i l l dominate. N e g l e c t i n g the other 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 be removed from these f u n c t i o n s . N e g l e c t i n g the X.+, /J^ terms i n equations (C.4) we f i n d t h a t equation ( C . l ) l s now approx-imated by Z7' \u00E2\u0080\u00A2\u00E2\u0080\u009E J ^ \u00E2\u0080\u0094 : : - : 2 <-r 3 * 1 . T ^ -ir- \u00E2\u0080\u00A2 - \u00C2\u00AB t o 1 f O \u00E2\u0080\u00A2>\u00C2\u00AB +| \ J -ZT, \u00E2\u0080\u0094 n c-1 '. n n ^ 0 (0*6) When \u00C2\u00A3\u00E2\u0084\u00A2 < 0, J,, and i n equations (0.1) and (C.2) are r e p l a c e d by the modified B e s s e l f u n c t i o n s I\u00E2\u0080\u009E and K>, and Aabsolute value of Er i s used. The s m a l l argument l i m i t i n g forms of these f u n c t i o n s a r e ^ 3 ^ l i m l\u00E2\u0080\u009E(z:) = 1/h/ (IzT n > 0 l i m ^ ( z ) = - l n ( z ) n = 0 = l ( n - l ) / (Iz)\"71 n i l (0.7) l i m I\u00E2\u0080\u009E(z) - |z 2 ->0 The other d e r i v a t i v e s may be obtained d i r e c t l y from the l i m i t i n g forms of the f u n c t i o n s . The approximate values of P\u00E2\u0080\u009E , Q\u00E2\u0080\u009E, S\u00E2\u0080\u009E and T\u00E2\u0080\u009E are the negatives of those given hy equation (C.4) and the approximate value of z \" i s given by equation (G.5)\u00C2\u00AB The __ r must be maintained i n both cases. 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 , the d i s c u s s i o n on d e a l i n g w i t h small arguments and l a r g e orders remains a p p l i c a b l e c Ce2 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 When \u00C2\u00A3\u00E2\u0084\u00A2 > 0, equation (2.18) may be w r i t t e n as where P^1, Q\u00E2\u0084\u00A2, S\u00E2\u0084\u00A2 and T\u00E2\u0084\u00A2 are as defined i n equation ( C l ) . The s m a l l argument r e s u l t i s given immediately by a s u b s t i -t u t i o n of values from equation (C . 4 ) . The sm a l l argument computational procedures are thus the same as 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 case \u00C2\u00A3\u00E2\u0084\u00A2 = 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 . For n \u00C2\u00A3 1 we have the i n t e r e s t i n g r e s u l t t h a t Y\u00E2\u0084\u00A2 = 0 r e g a r d -l e s s of Y^ +'\u00C2\u00BB I f n = 0, however, a c a n c e l l a t i o n of the \u00C2\u00A3 r f a c t o r occurs and we have p.; + H o C Q \u00E2\u0080\u00A2m (0.8) 174 Under no c o n d i t i o n does \"become complex r a t h e r than pure imaginary 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 approximation 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 not give a c o r r e c t r e s u l t . The l i m i t i n g r e s u l t f o r a^ approaching z^ero i s found to be I n place of equation (C.2) we have (c.9) K = - 3 X - 2 / X n = 0 (C.10) = 3 ^ C J \u00C2\u00AB A n > 1 This: holds f o r 6\u00E2\u0084\u00A2 e i t h e r p o s i t i v e or n e g a t i v e . 175 APPENDIX D Some Aspects of the Numerical Integration Method; In t h i s appendix, the determination of i n i t i a l values f o r a numerical int e g r a t i o n of the impedance or admittance d i f f e r e n t i a l equations i s discussed and a method of dealing with the s i n g u l a r i t y at a z:ero of p e r m i t t i v i t y l n the case of perpendicular p o l a r i z a t i o n i s given. The discussion i s c a r r i e d out i n terms of impedance 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 and admittance l n the case of perpendicular p o l a r i z a t i o n . In either case, the smaller of z\u00E2\u0080\u009E or y^ i s used when performing the i n t e g r a t i o n . D.l P a r a l l e l P o l a r i z a t i o n In order to integrate i n the ( m - l ) ^ region, the value of z^ at r = r w must be known. The r e l a t i o n where z^\"+' = i n t e g r a t i o n variable i n the (m-l) region ./ gives the required i n i t i a l value i f Z\u00E2\u0084\u00A2;has been determlned. When m = M, t h i s becomes z 0 (D.2a) fo r a conducting core and IVl -/ tCCS) (D.2b) fo r a d i e l e c t r i c core. 176 When Vne 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 value of z\" f o r a s m a l l value of \"J.must be foundi I f i t l s assumed th a t the behaviour of cr(x) at the o r i g i n can be described by-Urn C ( r ) = a r b , b 2- 0 r->0 (which should i n c l u d e the cases of i n t e r e s t ) then an approx* imate i n i t i a l v alue i s given by where v = ^2n b + 2 J = K*L fe J.v'( ) a 5 (D.3) k0rc 1 (D .4) We note t h a t f o r n > 1, z\" i s independent of p e r m i t t i v i t y . 177 I f v e ry small i n i t i a l values of ~% could be used, equation (D.3) would give accurate r e s u l t s . ( A r b i t r a r i l y s mall-values of -^ cannot be used because of the ( 1 / ? ) and (- n z / J r ) terms i n the equations ( 2 . 2 4 ) ) . I n c a r r y i n g out the computation, 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 lue of \"5 or 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 step s i z e must be used when n i s l a r g e i n order to avoid 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 . The use of r e l a t i v e l y l a r g e i n i t i a l values i s j u s t i f i e d because, as n i n c r e a s e s , the value of z^ f o r s m a l l \"5 becomes l e s s s e n s i t i v e to p e r m i t t i v i t y . I n most cases of i n t e r e s t , the power s e r i e s method of 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 accurate i n i t i a l values of z n . Due to the s i m p l i c i t y of the s e r i e s 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 , 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 than the a p p l i c a t i o n of equation (D.3) 0 D . 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 The i n i t i a l v alue f o r an i n t e g r a t i o n l n the ( m - l ) t ] l r e g i o n i s = Y ^ / j y . 0.5) This reduces to = oo (D.6a) 5 = k 0 r , f o r a conducting core, and .178 = EA**,) \u00E2\u0080\u0094; \u00E2\u0080\u0094 ( D . 6 b ) f o r a d i e l e c t r i c core. When c\"^(J\"/k:0) = \u00C2\u00A3\"(r)'\u00C2\u00BB a r ^ i n the neighbourhood of the o r i g i n , an approximate i n i t i a l value f o r int e g r a t i o n i n the core i s h Jv( f(?) ) '/a (lAo)* y\" \u00C2\u00BB a ( T A J / T - - , (D . 7 ) where v 2 - (kn + b* ) / (b + 2 / 2 k D y a / r \-r~ f ( ? ) = \u00E2\u0080\u0094 - ( ~ / i s the same as i n equation (D\u00C2\u00AB3) b + 2 \k 0/ Using the f i r s t term i n the series f o r J v and J y we have t 7 (b\u00E2\u0080\u00A2+ 2 ) l i m y * = - a ( J A J \u00E2\u0080\u0094 r n - 0 ? - * 0 1 1 \u00E2\u0080\u00A2 7 a ( T A J - V 2 (b + 2) (D,8a) = a ( T A o ) 1 2 T ,/y n i l (D.8b) (n* + -bz/k)/z + b/2 Due to the dependence of y\u00E2\u0084\u00A2 on \u00C2\u00A3 r ( r ) i n equation (D.8b),'relatively.large s t a r t i n g values of J may not a l -ways be suitable when n \u00C2\u00BB 1. Note, however, that when n \u00C2\u00BB b/2, t h i s dependence appears only as the f a c t o r a(TAo)^\u00C2\u00BB The sim-p l i f i c a t i o n which occurs by using only the f i r s t term i n the Bessel function series i s accurate f o r increasing values of T as n i s increased. This suggests that f o r n \u00C2\u00BB 1 and 179 n \u00C2\u00BB b / 2 , larger I n i t i a l values of Y may \"be used i n equation (D . 7 ) i f the multiplying f a c t o r a( J/LO^ i s replaced by the true p e r m i t t i v i t y f^(TAo). As 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 power series method of Section 3\u00C2\u00BB1-1 may also be used to obtain i n i t i a l values. Since the series s o l u t i o n i s more cumbersome i n t h i s case, i t i s generally Impractical to use a polynomial of degree greater then unity to approximate ErM. 3).2-1 Dealing with Zeros of P e r m i t t i v i t y When \u00C2\u00A3y(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 to the l/\u00C2\u00A3 K(r) f a c t o r i n the c o e f f i c i e n t of y*. As a r e s u l t , 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 , a numerical i n t e g r a t i o n cannot be used i n the neighbourhood of a p e r m i t t i v i t y zero. I f the s i n g u l a r i t y i s i s o l a t e d i n a th i n sub-region, then, by using power series solutions i n equation ( 2 . 1 8 ) , the transfer of admittance across the s i n g u l a r i t y may be c a l -culated. When an approximation to \u00C2\u00A3r{r) i s used f o r t h i s purpose, i t must have the same behaviour as \u00C2\u00A3 r(r) i n the neighbourhood of the zero. For example, the minimum degree of a polynomial which may be used to approximate a v a r i a t i o n of the type ( i ) , (11) or ( i i i ) i l l u s t r a t e d i n Figure D.l is 1, 2 and 3 , r e s p e c t i v e l y . I f we approximate \u00C2\u00A3>.(r) by a l i n e a r v a r i a t i o n near tr{x) .= 0 as i n the case i l l u s t r a t e d i n Figure D . 2 , the F i g u r e D.2 Approximating M r ) by a L i n e a r V a r i a t i o n Near \u00C2\u00A3t(r) = 0 i n Order to L e a l w i t h the 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 procedure i s as f o l l o w s : I ) I s o l a t e the s i n g u l a r i t y by d e f i n i n g a sub-re g i o n r 3 \u00C2\u00A3 r 5 r z and approximate \u00C2\u00A3>.(r) by a l i n e a r v a r i -a t i o n s r + t . I I ) S t a r t i n g a t a sma l l value of ~S, c a l c u l a t e y\u00C2\u00AB p ~ y-n ' \" ^ .3 , by a numerical i n t e g r a t i o n of equation (2.29a) 5 = ^ I i i ) L e t Y^ =3 tf0 y^p and use equation (2.18) w i t h m = 2 and U,, and V\u00E2\u0080\u009E the s e r i e s s o l u t i o n s about r = r c given by equations (B.18a) and (B.18b) t o c a l c u l a t e Y* . i v ) Use the i n i t i a l value y \u00E2\u0080\u00A2 '\u00E2\u0080\u009E Y* / J y o to i n t e g r a t e n u m e r i c a l l y t o r = r , . Since Y* i s complex r a t h e r than pure imaginary, y,, i s complex. I n order to ca r r y out the numerical 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 equation (2.29a) and separate the r e a l and imaginary p a r t s . We o b t a i n the f o l l o w i n g coupled equations: n z _ \u00E2\u0080\u00A2= Sr(J/k.0) + y\u00E2\u0080\u009E r / j + ( l - z ) (y\u00E2\u0080\u009E r - y \u00C2\u00A3 ) d n* d 5 tr(\u00C2\u00A3/k.0) 5 I n a s i m i l a r manner, equation (2.29b) y i e l d s d .z, (D.9) d ? n 2 ( i - 7 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 2 ) + W f + f r(?A\u00C2\u00BB) ( 0 i n the 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 . The development which i s c a r r i e d out i n the former case i s a p p l i c -able t o the l a t t e r s u b ject t o the above r e s t r i c t i o n , i f Ec r e p l a c e s E c and admittances r e p l a c e impedances. E e l Maximum Magnitude of High-Order Scattered F i e l d . C o e f f i c i e n t s We f i r s t c onsider the dependence of /A\u00E2\u0080\u009E/ on the normalized i n p u t impedance z'n - Z^,/;)-^. Equation (2.12) may be r e w r i t t e n as M I , ) - < < 1 \u00E2\u0080\u00A2 , \u00E2\u0080\u00947 7 ( E . l ) A, = - (3f E e 184 The extrema of / k^/E0j when considered as a function of zl, may be found by se t t i n g the derivative with respect to z^ equal to zero. We have F* / a j 2 = -r-^-7 (E.2) where a^ = A7,/E0 - YAf, ) - < Y'(J,) D i f f e r e n t i a t i n g with respect to z'^ gives )>_ Z _ *V Gy 2( (?, ) ) - Y\u00E2\u0080\u009E (T, ) ),) ^< K ' ; U (F* + G * ) (E.3) The numerator of equation (E.3) may be s i m p l i f i e d using the Wronskian r e l a t i o n ^ ) \u00E2\u0080\u00A251 ' giving ( F 2 + G ^ ) Setting equation (E.4) equal to zero and solving f o r z'^ we obtain z' = \u00E2\u0080\u0094 , \u00E2\u0080\u0094 \u00E2\u0080\u0094 - or z\u00E2\u0080\u009E = \u00E2\u0080\u0094 < ( ? , ) Y^(5, ) 185 We note t h a t , _ 3-nCS:,) = 0 /aJ' = i (E . 5 ) z, oo < i and t h e r e f o r e z ; = J \u00E2\u0080\u009E ( 5 , ) / j ' C ? , ) and < = Y^ C ? , ) A ' ( 3 , ) y i e l d the minimum and maximum value of /a^,/, r e s p e c t i v e l y . I n order to o b t a i n a good approximation to the f i e l d , we expect t h a t terms up to a \" l a r g e \" value of n w i l l be r e q u i r e d . With t h i s i n mind; we t e n t a t i v e l y impose the r e s t r i c t i o n n >ft. and observe the behaviour of as a f u n c t i o n of z'n. The r e s u l t , which i s easily'deduced u s i n g the p r o p e r t i e s of B e s s e l f u n c t i o n s and equations (E.2) and (E . 5 ) i s sketched i n Figu r e ( E . l ) which shows the e f f e c t of i n -c r e a s i n g n. We now determine the maximum value of /a*,/ f o r p o s i t i v e values of z^ ,. The. reason 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 w i t h a conducting core. From F i g u r e E . l , i t i s evident t h a t the maximum must be reached e i t h e r at z' = 0 or as z' -> . The values of 186 F i g u r e E . l Behaviour of l&J f o r three Values of n, 6. n, < nz < n 3 . The argument of a l l B e s s e l f u n c t i o n s I s 7, l*J' ' T i UJ' Z T 1 \" _ > C O UJ UJ ~ /H?U)/ (E .6a) (E .6b) are p l o t t e d as f u n c t i o n s of , f o r v a r i o u s v a l u e s of n a . i n F i g u r e E.2. Each /a\u00E2\u0080\u009E/ J curve i s above the correspond-i n g / a \u00E2\u0080\u009E / . curve except when n i s c l o s e to ~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 ) . \u00C2\u00A5e may w r i t e 6 n > ^ , +1 (E.?a) (E .7b) 187 188 A good estimate of the value given by equation (E .7a) (3 ?-) may be obtained by u s i n g the f o l l o w i n g approximations (the e r r o r i s about 1% f o r n = 2 and decreases w i t h i n c r e a s i n g n): J j n ) * \u00E2\u0080\u0094 n / 3 Y\u00C2\u00BB(n) 0.7748 n'/3 We f i n d /J\u00E2\u0080\u009E(n)/ 0.4473 m ' c\u00C2\u00BB o\u00C2\u00AB5 / H ^ ( n ) / (0.4473* + 0.774&V* I t i s evident t h a t the 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 value of n must be used to ensure a sma l l value of /a^/. E.2 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 Scattered F i e l d The l i m i t s on Iky,} which were developed i n S e c t i o n E . l are now used to f i x a bound on the t r u n c a t i o n e r r o r i n the computed s c a t t e r e d f i e l d . A t e s t procedure f o r determining the order 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 e r r o r i n the magnitude of the 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 the s e r i e s i n equation (2 . 1 b ) a t n = N, normalized w i t h respect to the magnitude of the i n c i d e n t f i e l d , i s given by 189 (E.8a) E\u00E2\u0080\u009E= / Z e,A,/E 0 H^C?) cos(n\u00C2\u00A9-)/ - / Z e^A^/Eo (I ) cos(nG-) / OO \u00C2\u00B12. I Z a.n ( 7 ) cos(nG-) / (E.8b) CO we have (E . 9 ) Equation (E . 9 ) i s inconvenient to use because i t i s I n the form of an i n f i n i t e s e r i e s . A s i m p l i f i c a t i o n can be mafte i f we re p l a c e / j ' ( J,) / / / H\u00E2\u0084\u00A2'( X ) / by J ( \u00C2\u00A3 ) ///H? (-\u00C2\u00A3 This 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-\u00E2\u0080\u009E| i s \u00E2\u0080\u0094 4-s m a l l , say max |a-n| < 10 ) as can be seen from Fig u r e E.2. We now de f i n e cf^ as Oo - Z b\u00E2\u0080\u009E (E.10) /j>,(?,)/ By making use of the 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) / z b) / H J ' ( Z ) / and d/dy~(/H (* }(z) /) are decreasing functions, of V f o r V > z 190 c) / H \" ( Z ) / and d/dz ( / H ^ ( Z ) / ) are decreasing f u n c t i o n s of z, we o b t a i n the c o n d i t i o n s by,, > b r i 2 f o r n, < n 2 ( E . l l a ) > f o r n j < n z. \u00C2\u00A3 n3 < n + (Eollb) by,* b ^ I n order to evaluate a l i m i t . o n *SN, consider the power s e r i e s S\u00E2\u0080\u009E = b \u00E2\u0080\u009E + I Z (E .12a) 1 B * _ f o r / ^ 1 (E . 12b) 1 - / \u00C2\u00B0 where / > = b w + 2 / b The f i r s t two terms of e i t h e r c\u00E2\u0080\u009E or S w are b N + , , b N + z . For i > 1, i n e q u a l i t y (E . 3 0 b ) i m p l i e s \>N+I/\u00C2\u00B0 >^>N+l + i and, as a r e s u l t , \"c?w < S^. Equation (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 the surface of the c y l i n d e r , ^ and (E . 1 3 ) gives 2 ^ , ( 1 , ) s i - \u00C2\u00AB W ( X ) / J \u00E2\u0080\u009E + , a,) (E.14) 191 As X Increases, b w + 2 /b\u00E2\u0080\u009E + I decreases, becoming equal to ( J \u00E2\u0080\u009E + 2 (X )/J / v + ) ( X ) ) - ( / H^, ( X ) / / / H ^ ( X ) / ) f o r y~fooo The e f f e c t of the corresponding i n c r e a s e i n \"the denominator of i n e q u a l i t y (E.13) i s small compared to the decrease due to / H ^ d f )/ i n the numerator. The bound on c w i s t h e r e f o r e not 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 (-E.13) by \u00C2\u00A3 < \u00E2\u0080\u0094 \u00E2\u0080\u0094 ~ (E . 1 5 ) i - j w + za)/j v +, (X) The q u a n t i t y \"S^ has been introduced as an approx-i m a t i o n to the bound on the e r r o r \u00C2\u00A3 w . The a c t u a l bound on \"\u00C2\u00A3 N at the surface of the c y l i n d e r (given by (E.9) w i t h J\" = X ) i s p l o t t e d i n F i g u r e E . 3 together w i t h the bound on t'N c a l -c u l a t e d u s i n g (E.14). I t i s seen t h a t the bound on ~EN i s somewhat l a r g e r than t h a t on !>'\u00E2\u0080\u009E 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 ~6 < max ~c?w < 1 0 ~ 2 , which would seem to be a reasonable range i n the present computations, the two. values are e s s e n t i a l l y the same. I f sm a l l e r values than those shown are considered, the bound on cN becomes the l a r g e r of the two ( f o r N-><\u00C2\u00BB , tlN I s the bound on The l i m i t on /a,,/ used i n the above d e r i v a t i o n a p p l i e s when z'n > 0 . Since the l i m i t i s r e q u i r e d f o r a l l n > N, I t i s : seen from the d i s c u s s i o n i n Appendix F that the v a r i a b l e z\u00E2\u0080\u009E + ( must be p o s i t i v e w i t h i n the c y l i n d e r . I f zN+, goes to +oo , through the zn < 0 r e g i o n i n t o the z^ > 0 r e g i o n , then z^ can be negative f o r a higher order, w i t h the 193 r e s u l t that /a^/ may reach I t s maximum value of u n i t y When - oo < z'\u00E2\u0080\u009E+, < YN+I (X ( 1 , ) and z \u00E2\u0080\u009E + / has only one change of s i g n ( + to -) w i t h i n the c y l i n d e r , the e r r o r bounds r e q u i r e only a simple m o d i f i c a t i o n , however. Under these c o n d i t i o n s , as n i N+1 increases:, w i l l decrease t o a l a r g e negative v a l u e , Jump to a l a r g e p o s i t i v e value and then decrease towards zero. R e f e r r i n g t o Fi g u r e E . l , we note t h a t f o r - co < z'n < Y^ ( ) / Y ' ( T , ) we have i) As zl, decreases from Y^( J,) /Y'( )\u00C2\u00BB /a*,/ ap-proaches / j ' ( '(If () / more r a p i d l y as n i n c r e a s e s . i i ) The value z^ = Y^C?,) A - ' ( T ; ) at which /a,, / = 1 i n c r e a s e s w i t h n. As a r e s u l t , when z^ i s i n the range - co < z^ < Y^C )/Y'XCS, ), /a^/ decreases more r a p i d l y w i t h i n c r e a s i n g n than does / J ' C ? , ) / / i K ( f a , ) i . The m o d i f i c a t i o n of the e r r o r bound i s derived as f o l l o w s . For values of n such t h a t - oo -c z'^ < Y^CT,) /Y'( Ifi)\u00C2\u00BB we r e p l a c e b^ by /a\u00E2\u0080\u009E / */H^ (f)I i n the i n e q u a l i t i e s ( E . l l ) . The r a t e of decrease of the changed terms i s f a s t e r than t h a t of the o r i g i n a l ones and we may th e r e f o r e use the s e r i e s * Note t h a t /da\u00E2\u0080\u009E/dz/ in c r e a s e s r a p i d l y w i t h n > ~S, f o r values of z; near Y^ (3r, ) ) (see F i g u r e E..1). As a r e s u l t , the e r r o r i n A>, >> / J ^ ( T , ) / // due to an e r r o r i n z^ when z^ has these values Trill be r e l -a t i v e l y l a r g e . 194 ex> i n place of t h a t given i n equation (E .12a),' thereby o b t a i n i n g 2 /aw+,/./H??C?)/ H w < ; (E.l6) 1 \u00E2\u0080\u0094 b w + 2 /bw+/ Corresponding to i n e q u a l i t y (E.14) or (E . 15) we have 2 /aw+J/./H\u00C2\u00AB(I)/ tN< - - r r - : f o r \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 (E . 1 7 ) 1 - J\u00E2\u0080\u009E + 2(X)Av +, (I,) Note t h a t (E.16) and (E . 1 7 ) r e q u i r e the knowledge of / a w + , / and are t h e r e f o r e used to determine whether a l a r g e enough order has been considered r a t h e r than to estimate the r e -quired order, E . 2 - 1 Procedure used to Determine the Order Required to Achieve a Given Accuracy of the Scattered F i e l d I t i s r e q u i r e d to determine 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 . We f i r s t o b t a i n a value of N such t h a t ^\u00E2\u0080\u009E / / / C (7,)/ A \" - < _ (E.18) l - J w + Z (X)/J\u00E2\u0080\u009E+, (T.) \" - 2 b) Check behaviour of zN+) : Determine the behaviour of z w + l w i t h i n the c y l i n d e r or, i f p o s s i b l e , by u s i n g ( i i ) or ( i i i ) 195 i n S e c t i o n P . l . There are three p o s s i b i l i t i e s i ) zN+l remains p o s i t i v e (from ( i ) i n S e c t i o n P . l , we note that 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 zN remains p o s i t i v e and hence the c a l c u l a t i o n of zKf+, may not be necessary). Equation (E.18) a p p l i e s I n t h i s case. i i ) z^+l changes from ( + ) to (-)\u00E2\u0080\u00A2. The value of N must be increased by one and step (b) repeated unless z\u00C2\u00BB+, < / - \u00E2\u0080\u00A2 - (E..19a) / a M + l / < - ( 1 - ) ( E . l 9 b ) 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 ? ) . Note that ( E . l 9 a ) i s s a t i s f i e d i f z'N < Y\u00E2\u0080\u009E+, Cf, )/rN'+1 (X ) and (E . 1 9 b ) i s s a t i s -f i e d i f / a w + , / evaluated u s i n g z'N i n s t e a d of z'r,+, s a t i s f i e s , the i n e q u a l i t y . The c a l c u l a t i o n of z.\u00E2\u0080\u009E+, may th e r e f o r e not be necessary. i i i ) zN+l changes s i g n more than once. N must be increased by one and step (b) repeated. c) Check ~ZAJ\u00C2\u00B1A: Use i n e q u a l i t y (E . 9 ) to determine whether the a c t u a l e r r o r c o n d i t i o n , ~BN^A, i s s a t i s f i e d . Since, as shown i n Figure E . 3 , the bound on provides a good estimate of the bound on 1zN, t h i s step i s unnecessary i n most a p p l i -c a t i o n s . We note that 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\N i s always s a t i s f i e d f o r a 196 plasma cylinder when N 2. X \u00E2\u0080\u00A2 As a r e s u l t , step (b) may be eliminated i n t h i s very Important case\u00C2\u00BB E.3 Truncation-Error i n Calculating the B i s t a t i c Scattering Cross-Section I f j$N denotes the magnitude of the error i n the calculated value of /+/ For z^+, > 0 / H > J, 16\" |E\u00E2\u0080\u009EJ , \u00C2\u00AB J,'(f,) , k 0 E 0 ' /<> ( ? , ) ! ' Since J>f(T\u00C2\u00BB ) / / H ^ (^f/) / and i t s deri v a t i v e with respect to n are decreasing functions of n, a l i m i t on the summation i n equation (E.21) may be obtained by comparison with a power series i n the same manner used i n Section E\u00E2\u0080\u009E2. The r e s u l t i s 16 J E J c\u00E2\u0080\u009E. M A < \u00E2\u0080\u0094 ~ \u00E2\u0080\u0094 \u00E2\u0080\u0094 (E . 2 3 ) k X are, plotted i n Figure E.2) Since /E w/ i s rela t e d to the calculated b i s t a t i c s cattering cross-section ^(G-) by equation (E.23) may be rewritten as c <&N < 8//K>'JcrN{e-) \u00E2\u0080\u00A2 + (E.24) 1 ~ CrJ+z/Crj + i Using the same argument as i n Section E.2, we may allow a sing l e change of sign i n zN+l and obtain 198 AN \u00C2\u00A3 S/fk, 'MJ&T (E .25) 1 ~ \u00C2\u00A3 N+Z /&hJ+l prov ided z'N+i < Y\u00E2\u0080\u009E+l (X )/r ' + ( ( X ) E\u00C2\u00AB3 - l Procedure used to Determine the Order Required to 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 The abso lu te e r r o r i n the 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 , ^ , i s p r o p o r t i o n a l to f o ^ . As a r e s u l t , we cannot s p e c i f y an order which w i l l ensure e i t h e r a g i v e n abso lu te e r r o r o r a g iven r e l a t i v e e r r o r , ^ w / c ^ , f o r a l l va lue s o f 10~^ helps avoid unnecessary c a l c u l a t i o n when cfn i s . l a r g e . 201 APPENDIX F General Behaviour of and y n W i t h i n an,Inhomogeneous C y l i n d e r The behaviour of z'n or y^ wit h i n c r e a s i n g n ^ f, l s important 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 c I n t h i s appendix, we consider the behaviour of and 1 y,, w i t h i n a c y l i n d e r and hence determine the dependence of z^ and y^ on n > 0, 7 , \u00C2\u00BB and ^ ( T / k J . No r e s t r i c t i o n s are placed on \u00C2\u00A3 ^ ( ? A J 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 ; however, i t i s r e q u i r e d t h a t \u00C2\u00A3r(?/k: J ^ 0 (except f o r a homogeneous r e g i o n of zero p e r m i t t i v i t y ) i n the case of perpen d i c u l a r p o l a r i z a t i o n . P . l P a r a l l e l P o l a r i z a t i o n We w i l l consider i n d e t a i l the case of a d i e l -e c t r i c c y l i n d e r ; the extension to i n c l u d e a conducting core i s q u i t e s t r a i g h t forward. Dropping the s u b s c r i p t n on z\u00E2\u0080\u009E, we r e w r i t e equation (2.24a) as dz \u00E2\u0080\u0094. = f ( n , ? , z) (Pel) d ? where f ( n , T , z) = 1 + z/S + ( E r ( 7 A J - n * / T 2 ) z z I t i s seen that dz/d? i s independent of z- and 7 f o r z -> 0, and t h a t the s i g n of ( \u00C2\u00A3 r ( J A J - n 2/*?) governs the be-haviour of dz/d;? f o r l a r g e values of z. For f i n i t e values 202 of p e r m i t t i v i t y l i m (^(FAJ - n 2 / ? z ) < 0, n > 0 ^->0 v . ' \u00E2\u0080\u00A2 We t h e r e f o r e consider f i r s t a case i n which n and ^ ( J / f c J are such t h a t n V f >S,(IAJ c o<\"f<7* (F.2b) 5\"fe\u00C2\u00BB z + z ~ are complex. F i g u r e F . l Behaviour of 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 Region i n which ( e r ( ? A B ' ' ) - B ' / ? * ) Becomes P o s i t i v e a t . dz/d.\"S = 0 Along the Curves z + and z - . Fi g u r e P.2 Behaviour of Impedance With an I n i t i a l Value i n a Region Where ( M ^ A J - n V ^ 2 ) i s P o s i t i v e . 204 From equation (3\u00C2\u00BB7)> i t i s seen t h a t as 7 approach-es zero, z. i s p o s i t i v e and approaches zero. The behaviour of z_ w i t h i n the c y l i n d e r i s t h e r e f o r e of the form shown: i n Fi g u r e F.1, curves I to IV. Any number of cr o s s i n g s of the z = 0 a x i s are p o s s i b l e i n the range f b < 7 \u00C2\u00A3 \"?, ; however, none can occur f o r f . We note f u r t h e r t h a t 0 < z < c o o < / ? - ? o (F.5) E f f e c t of I n c r e a s i n g n From equation ( F . l ) i t i s seen t h a t f o r and z f i x e d , dz d l dz ~ d~? n = n, n z > n, n = n. The e q u a l i t y a p p l i e s when z = 0. Thusv, the behaviour of z as n I s increased f o l l o w s the p a t t e r n of pr o g r e s s i n g from curve IV t o curve I i n Fi g u r e F . l . For 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 curve 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 of any type shown de-pending on the c y l i n d e r parameters or 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 \u00C2\u00A3r(J/k.0) Decreasing cV(TAo) has a s i m i l a r e f f e c t on dz;/d? as i n c r e a s i n g n. The. pro g r e s s i o n from curve IV to curve I I n Fi g u r e F . l i s th e r e f o r e 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 behaviour and again more Jr a x i s 205 c r o s s i n g s may occur. We now consider an i n i t i a l value of z, say at '\"Si \u00C2\u00BB i n a r e g i o n where n 2 / ? 2 \" < S^'TAo)- again Xo. i s : d e f i n e d as the value of \"S f o r which n 2 / ^ 2 = ^ A J J we i n v e s t i g a t e the \"behaviour of z i n the range ~\u00C2\u00A7i \u00C2\u00A3 \"f \u00C2\u00A3 \"5, where > \u00C2\u00A3r(TA.)\" - 5 , > Zk two are shown. As \"before, bers on the curves corresponds to an Proceeding as before we p l o t z\u00C2\u00B1 as a f u n c t i o n o f f * as shown i n F i g u r e F . 2 . For z ^ ,. , < z b = - 2\"?* , f o u r p o s s i b l e v a r i a t i o n s of z are shown ( m u l t i p l e - a x i s c r o s s i n g s are not shown, and f o r z a decrease i n the numl i n c r e a s e I n n or a decrease i n \u00C2\u00A3 r. By studying the above two s i t u a t i o n s , i l l u s t r a t e d i n F i g u r e s F . l and F . 2 , we may I n f e r the e f f e c t which i n -c r e a s i n g n or decreasing \u00C2\u00A3> has on z' = z L e t c A = (number of times which z. goes to + oo ) (number of times z J ^ r e t u r n s from + oo ) 0 - 7 -where z J Implies z evaluated f o r n = a or \u00C2\u00A3r{J/k0) - 8rA(TAo) How. i f or c A ; = c A Z , z i n generals: 5=5/ '/ \" z ' l w i l l w r i t e z'/ > z'/ 206 Using t h i s n o t a t i o n , we have 1) I f n, < n 2 , then z'/ > z'/ i i ) I f \u00C2\u00A3 , , ( IAJ 6 r ( l A e ) 0 ^ T - \"S, , then 0 < z'<#> The behaviour of z i n a medium surrounding a conducting core i s r e a d i l y seen by imagining curves s i m i l a r to those shorn i n Fig u r e s F . l and F . 2 s t a r t i n g at a p o i n t on the f\u00C2\u00BB a x i s . C l e a r l y , i ) , i i ) and i i i ) are a p p l i c a b l e , the range of \"5 being l i m i t e d to where f z ~ k0Yz rz - r a d i u s of conducting core. F.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 R e w r i t i n g equation ( 2 . 2 9 a ) i n a form s i m i l a r to ( F . l ) , we have dy ~ = g(n, ~\u00C2\u00A7 , y) ( F . 6 ) d^ n* where g(n, J, y) = M?A*) + y/S + ( 1 - \u00E2\u0080\u0094 . ) y* M ? A J ? S e t t i n g g = 0 and s o l v i n g f o r y, we o b t a i n the f o l l o w i n g equations f o r curves along which dy/df =0: 207 + ^ 1 / 7 t j l / f ^ ( n V ? - M ? A . ) ) y1 - M?A\u00C2\u00AB) - - : \u00E2\u0080\u0094 7 ~ \u00E2\u0080\u0094: 2 ( n*/T - M I A * ) ) (F.7) The r i g h t hand s i d e of equation (P.7) d i f f e r s from t h a t of equation (P.4) only by the f a c t o r \u00C2\u00A3>(7Ae>)-Since y \"becomes complex i f a zero of p e r m i t t i v i t y occurs, we l i m i t the d i s c u s s i o n to cases i n which tr i s e i t h e r p o s i t i v e or n e g a t i v e . \u00C2\u00A3 f > 0 Por MIAJ > 0 , 0 < * ? < f,\u00C2\u00BB a p l o t of y + and y~ v s . f i s s i m i l a r to the curves f o r z\"*\" and z~ shown I n Fi g u r e s F . l and F . 2 , The value of y as f - > 0 , given \"by equation ( 3 . 1 1 b ) , i s p o s i t i v e and approaches zero, and hence the other curves i n Fig u r e s F . l and F . 2 . a l s o represent pos-s i b l e v a r i a t i o n s of y. A decrease i n the number of the curve again i n d i c a t e s the e f f e c t of I n c r e a s i n g n or de-c r e a s i n g Ef-o I f a conducting core i s present, y i n i t i a l l y decreases: from + oo or i n c r e a s e s from -co depending on whether i s l e s s than or greater than \" ? a r e s p e c t i v e l y . 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 ) , and i i i ) apply i n t h i s case w i t h y'= y / r e p l a c i n g z l / :? = -?/ \u00C2\u00A3 > < 0 P l o t t i n g y + and y~ i n the y v s . f plane, we ob-t a i n curves of the type shown i n F i g u r e F . 3 . T y p i c a l behav-i o u r of y f o r a d i e l e c t r i c c y l i n d e r (curves I and I I ) and 208 Figure F\u00C2\u00AB3 Behaviour of Admittance Within a D i e l e c t r i c Cylinder (I and II) and a D i e l e c t r i c Surrounding a Conducting Core ( i and i i ) when. Sr < 0, dy/d'\u00C2\u00A7 .= 0 along y + and y\". f o r a d i e l e c t r i c cylinder with a conducting core (curves i and i i ) are also shown (since the condition\u00E2\u0080\u00A2 \u00C2\u00A3r{f/k\o)'^ n 2 / y z i s s a t i s f i e d f o r a l l values of 7 , Figure F . 3 shows a l l pos~ s l h i l i t i e s ) . A decrease i n the number of the curve i n d i -cates the e f f e c t which an increase i n n or \u00C2\u00A3\u00C2\u00BB- has o n the behaviour of y. As a r e s u l t , we get the following r e l a t i o n s comparable with i ) , i i ) and i i i ) i n Section F . l : 2 0 9 i ) I f n, < n z , then y ' / > y' / . i i ) I f \u00C2\u00A3 \u00E2\u0080\u009E ( T A J < \u00C2\u00A3r?.( A ) for 0 S\ 1 < X ,',then 7'I >7'l i i i ) 0 > z ' > \u00C2\u00AB co 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 consider a wave propagating i n the z - d l r e c t i o n i n the medium shown i n F i g u r e G . l . The form of the required; s o l u t i o n i s e a s i l y obtained by comparing the problem w i t h t h a t of propagation of a su r f a c e wave along a d i e l e c t r i c (4-3) s l a b . \u00C2\u00A5e are guided by the f o l l o w i n g c o n s i d e r a t i o n s : i ) The l a y e r s are t h i n enough t h a t o n l y the f u n -damental mode propagates. . i i ) The t r a n s v e r s e f i e l d components should be much gre a t e r than the l o n g i t u d i n a l f i e l d component and should be \"constant\" i n each r e g i o n i n the tr a n s v e r s e plane. e, D T \u00C2\u00B1 T 2t 2d F i g u r e G.l Layered Medium of the Type Used f o r the A r t l f i c l a l D i e l e c t r i c 211 TM Mode The TM .mode has the f i e l d components H y, E\u00E2\u0080\u009E, E z . I f we assume then the r e q u i r e d forms of H y i n the two regions are Region (1) H y * A cosh(. p(x-D) ) e\" J^ x (G.la) Jx-D f < d Region (2) H v = B Cos( h(x-T) ) e ~ j / ? 2 (G.lh) /x-T/ < t where D and T denote the p o s i t i o n of the c e n t r a l plane of a l a y e r of 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 of 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 the propagation constant are constrained by the wave equation as f o l l o w s : Region (1) p 2 - /? + kl \u00E2\u0082\u00ACri = 0 (G.2a). Region (2) - h z - + kl EYz = 0 (G.2b) The requirement t h a t the propagation constant be the same i n both regions y i e l d s p 2 + h* = ( Erz - Eri ) kl (G.3) From the r e l a t i o n V X H = 3u\u00C2\u00A3E we o b t a i n , f o r E z 212 Region (1) E z - A p sinh( p(x-D) ) e J 3\u00C2\u00BB\u00C2\u00A3 and f o r E Region (2) E 2 = - B h s i n ( h(x-T) ) e \" J / 3 z 1 -A Region (1) E x = 7\u00E2\u0080\u0094 A 0 cosh( p(x-D) ) e~jp: W e i 1 ia* Region (2) E x - \u00E2\u0080\u0094 B ft cos( h(x-T) ) e\"JP Note that as d and t approach zero, the maximum value of E 2 approaches zero due to the sinh( p(x-D) ) and si n ( h(x-T) ) f a c t o r s w h i l e H y and E x approach -constant v a l -ues i n each r e g i o n . Thus f o r t h i n l a y e r s , the transverse f i e l d components are indeed dominant. App l y i n g the boundary c o n d i t i o n s of c o n t i n u i t y of 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, say x = D + d gives A cosh(pd) = B cos( h ( - t ) ) \u00E2\u0080\u00A2= B cos(ht) (G .4a) f,Ap sinh(pd) = \u00C2\u00A3, B h s i n ( h t ) (G .4b) On d i v i d i n g ' e q u a t i o n (G .4b) by equation (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 \u00C2\u00A3z-p tanh(pd) = S, h tan ( h t ) (G.5a) or p tanh(pd) = \u00C2\u00A3 K ) h tan(h t ) (G.5b) 213 Equations (G . 3 ) and (G . 5b ) determine the eigenvalues of p and h. The corresponding propagation constant may then be c a l c u l a t e d u s i n g e i t h e r (G . 2a ) or (G . 2 b ) . Thus, f o r example, (G . 2a ) gives p 2 + \u00C2\u00A3 n k* (G.6) I f we d e f i n e the 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 tre , we have Although the s o l u t i o n of equations (G . 3 ) and (G . 5b ) must, i n general, be performed g r a p h i c a l l y or n u m e r i c a l l y , an approximate 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 expansion f o r tanh(pd) and tan ( h t ) i f pd and h t are both much l e s s than u n i t y . 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 we are concerned w i t h t h i n l a y e r s . The r e q u i r e d expansions are (pd) 3 (pd)\"\" tanh(pd) = pd - *\u00E2\u0080\u00A2 - ... 3 15 t a n ( h t ) = h t + ( h t ) 3 ( h t ) 5 \" comes 3 15 Using the f i r s t terms only, equation (G . 5b ) be-214 S u b s t i t u t i n g f o r h from equation (G .6) gives p z = k* Er) ( \u00C2\u00A3^ - i ri ) t / ( Eri t + Er, J ) (G .8) A more accurate r e s u l t i s obtained u s i n g the f i r s t two terms of the expansions. I n t h i s case, equation (G.5t>) becomes P V 2 h V E,z( P 2d ) - \u00C2\u00A3\u00E2\u0080\u009E( h*t + ) 3 3 and t h e r e f o r e p 2 = ( b - 7b* - 4ac )/2a (G .9) where a = t 3 / 3 + \u00C2\u00A3 R d 3 / 3 b \u00C2\u00AB t + \u00C2\u00A3\u00E2\u0080\u009E d + 2 \u00C2\u00A3 B k* t 3 / 3 c = ED ( t + k* t 3 / 3 ) \u00C2\u00A3 R - &r\u00C2\u00A3 / S n The maximum percentage; e r r o r i n the:, e f f e c t i v e per-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* given by equations (G .6) and (G .9) i s p l o t t e d as a f u n c t i o n of E.v.iL/Er.i f o r v a r i o u s values ' of (d + t j / ^ e , i n Figures G.2a and b. TE Mode This mode i s c h a r a c t e r i z e d by the f i e l d components E y , H x and H 2. The s o l u t i o n i s f o r m a l l y the same as f o r the TM mode ease. We s t a r t w i t h E y of the form 215 216 Region (1) Ey ~ A cosh( p(x-D) ) e~J/*z -/x-D / ^ d Region (2) E y = B cos( h(x-T) ) e~jAz /x-T / 1 t The equations determining the eigenvalues of p and h are p + h - c P k\u00E2\u0080\u009E p tanh(pd) = h tan(ht) and those, corresponding to (G.8) and (G.9), giving the approximations f o r p* are p z = t \u00C2\u00A3 p/(t + d) (G.10) and p* = ( b - J b* - 4ac )/2a ( G . l l ) where \" a = t 3 / 3 + d 3 / 3 . b = t + d + 2 \u00C2\u00A3 c k 0 t 3 / 3 c = \u00C2\u00A3 p( t + \u00C2\u00A3 p k 0 ) t 3 / 3 217 REFERENCES 1. Budden, K.G., \"Radio Waves i n the Ionosphere\", Cambridge-. U n i v e r s i t y P r e s s , 1 9 6 1 . 2 . Burman, R . , \"Some E l e c t r o m a g n e t i c Wave F u n c t i o n s f o r Pro p a g a t i o n i n C y l i n d r i c a l l y S t r a t i f i e d Media\", IEEE Trans, on Ant, and Prop., V o l . AP -13 , No. . 6 , p. T^oTT9^57~ 3. 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Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Scattering of electromagnetic waves by long radially inhomogeneous isotropic cylinders"@en . "Text"@en . "http://hdl.handle.net/2429/35706"@en .