# Cấu trúc sóng chức năng trong điện lý thuyết P4

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## Cấu trúc sóng chức năng trong điện lý thuyết P4

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We shall consider the scattering of a plane, linearly polarized monochromatic wave by a perfectly conducting prolate spheroid immersed in a homogeneous isotropic medium. Solution of the EM scattering by the oblate spheroid can be obtained by the transformations 5 --+ it and c --+ -ic. It is assumed that the surrounding medium is nonconducting and nonmagnetic. The geometry of the configuration is shown in Fig. 4.1, and the surface of the spheroid is given by

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## Nội dung Text: Cấu trúc sóng chức năng trong điện lý thuyết P4

1. Spheroidal Wave Functions in Electromagnetic Theory Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong Copyright  2002 John Wiley & Sons, Inc. ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic) E.M Scattering by a 4 Conducting Spheroid 4.1 GEOMETRY OF THE PROBLEM We shall consider the scattering of a plane, linearly polarized monochromatic wave by a perfectly conducting prolate spheroid immersed in a homogeneous isotropic medium. Solution of the EM scattering by the oblate spheroid can be obtained by the transformations 5 --+ it and c --+ -ic. It is assumed that the surrounding medium is nonconducting and nonmagnetic. The geometry of the configuration is shown in Fig. 4.1, and the surface of the spheroid is given by (4 . 11 4.2 INCIDENT AND SCATTERED FIELDS Without loss of generality, the direction of propagation of the linearly polar- ized monochromatic incident wave is assumed to be in the z, z-plane, making an angle 80 with the z-axis, as shown in Fig. 4.1. At an oblique incidence (00 # 0), the polarized incident wave is resolved into two components: the TE mode, for which the electric vector of the in- cident wave vibrates perpendicularly to the X, z-plane, and the TM mode, in which the electric vector lies in the x, z-plane. Thus the plane-wave expres- sions for both modes are given by -jk.r (4.2a) ETE = ETEO@e 7 89
2. 90 EM SCATTERING BY A CONDUCTING SPHEROID Incident Wave (#o=4 Scattered Wave Fig. 4.1 Geometry of EM scattering by a conducting prolate spheroid.
3. INCIDENT AND SCATTERED FIELDS 91 ETM = ETM~ (-ii? cos 80 + 2 sin 0e)e -jk*r 3 (4.2b) where ETEO and ETMO are the amplitudes of the TE and TM fields respec- tively, and -k l r = Ic(zsin& + ZCOS~~), (4 . 3) with k being the wave number of the monochromatic radiation. Flammer [l] has obtained the plane-wave expansion in terms of prolate spheroidal wave functions as where N mn(~) is the normalization constant given in Eq. (3.9) and em is the Neumann number, em = 1 for m = 0 and em = 2 for m > 0. For simplicity in what follows, the argument c will be suppressed in the description of the functions. From the equations above the excitation can be described in terms of vector wave functions [l] as follows: ETE = ETEo fJ 2 Amn(~O)M~~~(C; ?j, 0. In other words, by Eq. (4.5c), A mn is non-zero only for m = 0 and the expansions are defined only for m = 0 for an axial incidence: Ei = (4.6)
4. 92 EM SCATTERING BY A CONDUCTING SPHEROID where m-1 AOn = k Son(l)* To describe the scattered fields, we can use radial functions of the fourth kind, because the radiation condition at infinity must be satisfied. Radial functions of the fourth kind are suitable because of their asymptotic behavior. This ensures that at large distances from the spheroid the scattered wave be- haves as a spherical diverging wave, emanating from the center of the spheroid. The components of the scattered field must also have the same +-dependence as the corresponding components of the incident field (&matching). To satisfy these requirements, we can write ESTE = arnn M+(4) + YrnnM~$f& (4.7a) e,m+l,n n=m m=O &TM = 2 2 (~~n”i$?t-l,n + Pm+l,n+l Mz!z+l,n+l) n=m m=O (4.7b) n=l where ESTE and ESTM are, respectively, the scattered fields corresponding to the TE and TM incident fields, while omn, ymn, &n, and Pm+l,n+i are the unknown expansion coefficients to be determined using boundary conditions. An extra term is needed at the end of Eq. (4.7b), due to the nature of the &dependence of the odd vector wave functions. The arguments of the M vectors have been dropped for simplicity, a procedure that will be repeated from now on. This representation of the scattered field was used by Sinha and Sebak [lo, 341. It will lead to the use of a single matrix for finding the expansion coefficients, and this matrix is dependent only on the scattering body. Hence there is no need to find a new matrix for different angles of incidence of the exciting wave. Moreover, the matrix used for the TE polarization can be used to derive the matrix for the TM case without extra effort. All these advantages will be seen later as the solution is derived. 4.3 TRANSFORMATION OF INCIDENT FIELDS TO SCATTERED FIELDS 4.3.1 Imposing the Boundary Conditions Assuming the prolate spheroid to be perfectly conducting, the incident field Ei and the scattered field E, must satisfy the boundary conditions at c = &-J:
5. TRANSFORMATION OF 1NClDENT FIELDS TO SCATTERED FIELDS 93 (4.8a) (4.8b) where the suffixes q and 4I denote, respectively, the q- and &components of the incident and scattered 1fields. Equations (4.8a) and (4.8b) must hold for all allowed values of 0 < 4 < 27r and -1 < q < 1. Note that because of _ _ the nature of the problem, it suffices to concern ourselves with the continuity of the electric fields at the boundary surface. The continuity of the magnetic fields is not utilized in this problem. 4.3.2 TE Polarization for Oblique Incidence The method used to obtain the transformation matrix for the scattered fields is outlined here. Detailed steps are presented only for the TE case for oblique incidence because the method is the same for the TM case. The case of axial incidence is considered later as a special case. On substituting the field expressions (4.5a) and (4.7a) into Eqs. (4.8a) and (4.8b), we obtain ETE0 - g 2 c c A,,(Bo)M$; Amn(~O)M~~~ k coz& n=m m=O + fJ 2 ((YmnM~~!f-l,n ((YmnM~~~l,n + nlrnnM~~~) = O* 0. (4 l 9) n=m m=O We consider only the v-component here, as the same steps can be applied to the &component. Expanding the equation above, we obtain 1 2 - d (1) d +... ... cose() PC5O 1) ‘I2 sin 4 A&‘oo -&R,, + AOlSOl ( zR8)+-* > + + + .. . . . . d + 2(C,2- 1)1’2 cos rn+ sin 4 AmmSmm- Rgi 4 d (1) + Am,m+lSm,m+l-Rm 9 m+l + l ’ ’ &5 MO -2 - 2-l sin rn+ cos$(A,mSmrnRgb c 0 +A S m,m+l (1) Rm,m+l m,m+l + l ’ l ) . . . + d (1) + 2(5,2- 1)li2 cos(m + 34 sin 4 Am+2,m+2Sm+2,m+2 - Rm+2,m+2 de
6. 94 EM SCATTERING BY A CONDUCTING SPHEROID - 2b + 2>50 (1) sin(m + 2)4 ~0s $(Am+2,m+2Sm+2,m+2Rm+2,m+2 t: - 1 (1) + Am+2,m+3Sm+2,m+3Rm+2,m+3 + ’ ’ ) l + -1 = + ... -(g - 1)1/2 SW [a& (~oo-$#) +c& (sO~$R$) + ee] l d - (r,” - 1p2 Smm-R$i - 4 d (4) m50 (4) + 4-8 9m+l~sm,m+l-Rm 9m+l - g - _ 1 sm9m+lRm,m+l + ’ ’ ’ 4 > I + . . . 2r7 (4) +(I -$)W Sin4(&91&1 (4) + 7;2S1242 + l l l) + . . . (4) + %n+l,m+2Sm+l,m+2Rm+l,m+2 + l l ’ ) + l ** ) where ynn = (k/ETEO)Tmn and ahn = (k/ETEO)amn* We then make use of the orthogonality of the trigonometric functions cosm4 and sinm4, by multiplying throughout Eq. (4.9) by sin(m + 1)4, m > 0, and integrating them with respect to 4 from 0 to T to get i ( 1 (m + 2)
7. TRANSFORMATION OF INCIDENT FIELDS TO SCATTERED FIELDS 95 ( d MO (4) + a; m+1 Sm,m+rR4),+1 7 - - _ 502 1 ‘m,m+lRm,m+l. + l l ’ 4 > I (,‘t lb7 (4) + (1 _ +y/2 (~~+l,m+lSm+l~m+lRm+l,m+l 0 + %n+l,m+2Sm+l,m+2Rm+l,m+2 + ” 9 form>l,and - d (1) (t2 - 1p2 AOOSOO d (1) 0 -go0 + AOlSOl ( l zRol + l ** > - ;(
8. 96 EM SCATTERING BY A CONDUCTING SPHEROID We can now combine all the equations derived so far into the following matrices: -1 E E E (coseo) (Q,)Am = (R,)S,, n-a = O,lJ, l l l , (4.10) and E (coseo) (&+>A+ = CR+ E -1 E >S+’ (4.11) where (QE)= ((QZ,,) ’ (Qg,m+s)) with - X (1) (1) X cm0 m0 - X (1) (1) X cm1 - XT8 m2 X (1) cm2 . . . - E --Me 7 (4 m,m+2 > = - y(l) (1) Y cm0 am0 - y(l) (1) Y cm1 am1 - y(l) (1) am2 Y cm2 A mm A m,m+l A m,m+2 . . . A m= ----- A m+2,m+2 A m+2,m+3 (4) Y am1 A m+2,m+4 Y (4) . . am2 . . . . . l l :> All WI 4 A12 WI A+ = 0 7 9 A13 WI . . . . . . i
9. TRANSFORMATION OF INCIDENT FIELDS TO SCATTERED FIELDS 97 / darn ’ / am,m+l I cym,m+2 SE - rn- 9 SE+- - L+l,m+l %2+l,m+2 %2+l,m+3 while km = 2 for m = 0 and km = 1 for nz > 0. Xz$, Xc:& (where i = 1,4), and the like are row matrices which are defined as follows: i x(mN) - - 0i 0i X mN1 i) x( mN2 (4.12) F mN0 ” ’> 9 where i x( mNn) i) x(cmN = (X0i i) x(cmN1 0i X cmN2 (4.13) cmN0 “* >3 i) x(cmNn = i) Y(amN - (Y' i ) amN0 y(i) amN1 y(i) amN2 l ” >7 (4.14) 0i YamNn 0i Y cmN i) (Y’cmN0 0i YcmN1 0i YcmN2 >9 (4.15) = “’ 10. 98 EM SCATTERlNG BY A CONDUCTING SPHEROID 0i YcmNn = -(G - 1)$R$)+2 ?m+n+2(E)l I8mNn 11. TRANSFORMATION OF INCIDENT FIELDS TO SCATTERED FIELDS 99 I5mNn = +l(l - ~2)1’2S,+l,m+n+lSm,m+N d% (4.19e) J -1 +1 d I6rnNn = ?j(l - ~2)1’2;i;;(sm+l,mfn+l)Sm,m+N d% (4.19f) J -1 +1 I7mNn = J = J+1 %-n+2,rn+n+2Sm,m+N d% (4*19g) -1 18mNn 7&n+2,m+n+2Sm,m+N &‘, (4.19h) -1 (4.19i) IlO,Nn = J+1 -1 +b - ~2)1’2sOn%,N+1 dq, (4.19j) Ill,Nn = J -1 q(1 - q2)li2 l,N+l drl. (4.19k) After all these intermediates are obtained, the scattering column vector SE can be obtained from the incident column vector IE via the transformation SE = (GE)IE, (4.20) TE = A (cosOo)-’ SE A+ SIi 0 A0 SE 1 A - - Al 7 7 SE 2 A2 . . . . ) 0 . . 0 -1 E ( RE+ ) (Q+ > -1 0 E 0 . . . w E- - 0 ( RE > 0 (Q > 0 ’ 0 0 (Rfj-1 (Qf) ::. . . . . . . . . . . . . . . . 4.3.3 TM Polarization for Oblique Incidence By matching the boundary conditions (4.8a) and (4.8b) in Eqs. (4.5b) and (4.i’b), and using the orthogonality of the trigonometric functions and the spheroidal angular functions as in section 4.3.2, we can get (Q,M)Am = (RE)S,M, m = 0,1,2,. . ., (4.21) 12. 100 EM SCATTERING BY A CONDUCTING SPHEROlD and (Q$%+ = (R$?sy, (4.22) where (Q:) = ((Qgm) i - (Qg,,,,,) 7 x(l) x(1> 610 (Qtf) - bll Pm - i I X (1) b12 . . I ’ / \ P mm / P mm+1 1 P n-v-M-2 SM- m- - 7 SM - + PA+l,m+l Plm+l,m+3 with I P mn = (k/ETMO)Pmn, P&n = (k/ETMO)Pmn, PFi = (k/ETMO)Pcn, . . x(bZmN ) = (X( ) i&NO x(i) brnN1 ’ x( t&N2) l *’ > (4.23) and further with . 0 X bfnNn = 6% - 1)1’2$ Rg;m+n(
13. TRANSFORMATION OF INCIDENT FIELDS TO SCATTERED FIELDS 101 and -1 (RM > -k (aM -1> + 0 M 0 . . . (GM) = ; ( RE > 0 (& > 0 0 (R;)-l (Qf”) ::: ’ . . . . . . . . . i . . . . . . i To determine (GM), it is observed that (Qg) is readily obtained from (Qg), and the inverse matrices (RE)-’ are the same as those for (GE) ex- cept for (Ry)? Thus the matrices obtained for (GE) will readily yield (GM), and vice versa, which provides great convenience in the computation of scattered fields for both polarizations. Furthermore, the transformation matrix (GE) or (GM) is independent of 80 and depends only on CO(the scat- terer). Hence it needs to be determined once, and the scattering vector SE or SM is obtained for any incident vector by the transformation SEW - (GEyM) IEyMe (4.25) 4.3.4 Fields at Axial Incidence In the case where the propagating wave is incident along the axis (at 80 = 0), the solution becomes much simpler by virtue of Eq. (4.6). If the electric field is polarized along the positive y-axis and the magnetic field along the positive z-axis, the unknown scattering coefficients would just be aen and ~1~. Using the same methods as before, we would get the following: (QoEo)AO (R,E)S& = (4.26) (1) - X 00 (1) - X 01 -X&j A00 A01 ---mm 7 A0 - E (62 00 A02 ; -Y&i . . , . 1 -Y&j -Y$J 14. 102 EM SCATTERING BY A CONDUCTING SPHEROID (RE> = 0 9 SE - 0- Ya (4) I W (4) 00 ya?‘) W (4) 01 I 01 . while XEi, Yaz$, Wz&, and VEi denote row matrices for the TE case. 4.3.5 TE Fields with Incidence Angle go0 In order to obtain the limiting form of Eq. (4.5a) at 00 = n/2, we proceed as follows. Taking the gradient of, and then the vector product by Z?on both sides of Eq. (4.4) and substituting for cos00 = 0 on the left-hand side of the resulting equation, we get cc 00 00 A,,M$; = 0, cos e. = o . (4.27) n=m m=O Since Smn(0) = 0 for odd values of (n - m), from Eqs. (4.27) and (4.5c), we can get (n-m)tWXl x &nM$i = 0, cos 80 = 0. (4.28) By substituting the expression of Smn(cos00) from Eq. (2.14) into (4.5a), we get (n-m)even (n-m>odd ETE = -ETEO x AmnM$L + 2 cos 80 x Dmn 9 @*W k cos e. m,n m,n where Cmjnwl D mn = - (1 - ~08~ eo)m12 1 - cos2eo)M$g . N mn k=O I Then from Eqs. (4.28) and (4.29), in the limit as co& -+ 0, we will get ETEO 00 ETE = 2x m=O BrnnM~~~, (4.30) k n=m
15. FAR-NE1 D EXPRESSIONS 103 where B mn = 0, (n - m)even, B mn = (n-m-l)lz(n + m + l)! = p/-l C-1) mn 2n (n--;--l>! (n+;+l),’ (n-m)odd* The scattering column vector is now given by SE = (GE)IE, O. = n/2, where IE is obtained from A by replacing Am, by Bmn. 4.4 FAR-FIELD EXPRESSIONS Equations (4.7a) and (4.7b) give rise to the scattered fields (corresponding to the two principal polarizations of the incident field) in both the near-field (Fresnel) and far-field (Fraunhofer) scattering zones. In this book, we are interested only in the wave behavior in the far zone. For very large distances from the scatterer (i.e., in the limit as cc --) oo), the polar angle 8 and the spherical radial coordinate T are related, respectively, to the spheroidal angle coordinate 7 and radial coordinate c by the formulas r) = co&, (4.31) Fc = r, cc = kF< = kr; (4.32) and the radial functions of the fourth kind become 1 R( 4 ) mn _ - +n++jc( 7 (4.33a) cc d 4 1 q-y-jc
16. 104 EM SCATTERING BY A CONDUCTING SPHEROID e- jkr ESTE = -ErEo g 2 +&(q) sin@ + 1)8)? kr I n=m m=O XSm+l,n+l rl) ‘OsCrn+ ‘)+ ( I 00 - >&in&(1 - ~2>‘/2son(~) 3 (4.34) n=O 11 and e- jkr &TM = -&MO g 2 ~n~rnn(~)cos(m+ l)(b kr 00 - x jn+k,b)}ij+ F4 [{ n=m m=O n=l { 2 j”[~V%&7) n=m m=O - jpk+l 7n+l(l - ~2)1/2Sm+l,n+1(17) 1 sin(m + l)4 >I 4 9 (4.35) where e and 3 are unit vectors in the direction of increasing q and 4, respec- t ively. For the case of axial incidence, the scattered wave would be Es = x&,n+l(rl) I 11 CO@ 3 l (4.36) The bistatic radar cross section is defined as 47r times the ratio of the scattered power delivered per unit solid angle in the direction of the receiver in the far zone to the power per unit area incident at the scatterer and is obviously independent on r. This is given mathematically as IES312 a(@, 4) = lim 4m2- i2 ’ (4.37) r+oo IE I where ? represents the polarization of the receiver at the observation point (5 6 49.
17. FAR-FIELD EXPRESSIONS 105 To obtain the bistatic radar cross section in this project, Eqs. (4.34) to (4.36) can be rewritten as Es = (4.38) and the normalized bistatic cross section is then v =IF@, I&+@, +)I2 +>i2, + (4.39) where F&9, qb) = 2 2 -‘$-a&&,&cod) sin(m + l)& n=m m=O 2 gj7fakn Ep(eAN = cos BS,,(COS e) - jTA+l,n+l sine n=m m=O X Sm+l,n+l (COS COS(m + 1>4 0) I cxl - x jny& sin BSo,(cos e) for TE polarization with oblique incidence; Fe(8,#) 2 2 = ~&nSmn(COSB) L, COS(m -I- l)(b n=m m=O 00 --/ - P In ln &(cos e), x 3 2 n=l - - 2 gj-[Ky2 COS OS,,(COS e) - jpk+l,n+l sin e n=m m=O Sm+l,n+l(COS~) sin(m + l)+ 1 for TM polarization at oblique incidence; and Fe@ 4) = 2 -$&Son(cos 0) sin 4, n=O for TM polarization at axial incidence.
18. 106 EM SCATTERING BY A CONDUCTING SPHEROID When 0 = 00 and + = 0, we obtain the normalized backscattering cross section for different values of the incident angle 80. 4.5 NUMERICAL COMPUTATION AND MATHEMATICA SOURCE CODES As mentioned previously, all the matrices and seriesare infinite in extent; that is, the integer index m varies from 0 to 00, and for each m the integer index n varies from m to 00. To obtain numerical results by a computer we must truncate the matrices and series according to the requirement of the desired accuracy of results. As discussedby Sinha and MacPhie [lo, 1301,to obtain two to three signif- icant figures of accuracy, it is sufficient to take the truncation number nt for the index n to be Integer(]lca] + 4). For this chapter, the truncation number mt for m (where m = 0, 1,2,. . . , rnt - 1) and the truncation number nt for n belonging to each nz (where n = nz, m + 1, . . . ,772 nt - 1) were each taken + to be Integer(]ka] + 4). The truncation number Nt for N [where (N + 1) is the number of rows of each submatrix] is also set to Integer(]lca] + 4), where N = 0, 1,2, . . . , NT - 1. This will ensure that all sub-matrices formed will have the same number of rows and columns. To test the convergence, computations were also made with truncation number mt = nt = NT =Integer( ]ka] + 6) for the case of ka = 1 and ka = 2 and the axial ratios a/b of 10, 2, and 1.01 at angles of incidence 80 ranging from 0’ to 90’. The results for the two schemes matched at least to four significant figures and in most casesto five significant figures. This ensures that the scheme of truncation above is proper within the controlled accuracy. Two Mathematics subroutine packages were written for the conducting spheroid. The first is Conductor,back.nb, which contains the main user’s module, Conductorback[ka-, ratio-], for the computation and plot of the normalized backscattering crosssection. The second package, Conductor-bi- static.nb, contains two user’s modules, Conductorbistatzero[ka,, ratio-], for the computation and plot of the normalized bistatic cross section at axial incidence, and Conductorbistat [ka-, ratio-, thetao-, phi-], for the bista- tic cross section at oblique incidence. Even though the latter module is also capable of computation for the case of axial incidence, it is not used for this purpose because a more efficient method is available. To compute the cross sections, a user needs to load the relevant packages and call any of the three modules. The module Conductorback[ka-, ra- tio,] takes in two arguments: ka (where /c is the wave number and a is the semimajor axis length) and ratio. ka represents the electrical size of the spheroid and ratio denotes the axial ratio. The output is a logarithmic plot
19. NUMERICAL COMPUTATION AND MATHEMATICA SOURCE CODES 107 of the backscattering cross section against the incident angle, for both the TE and TM cases. The module Conductorbistat[ka-, ratio,, thetaO_, phi-], on the other hand, computes the bistatic cross section for a user-specified angle of incidence (thetao-) at a user-specified azimuthal angle of observa- tion (phi-), and outputs a plot of the cross section against 8 (the angle of observation) from 0’ to 180’ for both the TE and TM cases. For Conduc- torbistatzero[ka-, ratio-], the output is a plot of the bistatic cross section of the E-plane (4 = 90°) and the H-plane (4 = 0’)) also against 0 (angle of observation) from 0’ to 180”. It is not necessary to explain every single step of the program flow. Hence, only the salient points of this and subsequent routine packages will be de- scribed. Moreover, the two packages mentioned above are very similar in their structures, except for the last part, when they are used to compute the various scattering cross sections. So they will be described as a single entity, and can be summarized as follows: 1. Based on the user’s input, the parameters, 50 and c, are calculated. 2. Using the value of c, the expansion coefficients dmn and the eigenvalues A,, are computed using the supporting modules Getdmn and Eigen- Common, respectively. The number of these values computed depends on the truncation schemefor m and n. 3. The spheroidal radial functions of the first, second, and fourth kinds and their derivatives are evaluated next, through the supporting mod- ules PSpheroidRl, PSpheroidR2, PSpheroidR2forsmallq and PSpheroidR2forlargec. 4. Various functions in Eqs. (4.12) to (4.18) and integrals in Eqs. (4.19a) to (4.19k) required for forming the matrices are then defined. For the in- tegrals, the product (to be integrated) involving two spheroidal angular functions is first expanded out in terms of 7. The resulting expression is an infinite polynomial in q and/or a function of 77. The integration is then performed only on those terms whose coefficients have a magni- tude larger than 1 x 10-lo . This criterion is chosen only after extensive testing to ensure that the accuracy of the integrals is sufficient. In the process of evaluation, the spheroidal angular functions are called and computed via the PSpheroidS module. 5. The matrices for the TE and TM cases are then computed using the command “ Table” . 6. Once the scattering coefficients are obtained, they are substituted into the relevant expressions for the scattering cross sections, and a plot with the corresponding titles and legends is then generated.
20. 108 EM SCATTERING BY A CONDUCTING SPHEROID 46 . RESULTS AND DISCUSSION This section consists of results obtained for perfectly conducting spheroids with different dimensions relative to the incident wavelength. Figure 4.2 shows the variation of the bistatic cross sections for axial inci- dence (00 = 0) with two such spheroids of the same semi-major axis length (1/2~)& (ka = 1) but different axial ratios. The solid circles are the results from Sinha [34,130] and Sebak [131]. A s can be seen, the results obtained here are in very good agreement with the existing work. This verifies the ac- curacy of the Mathematics source code developed. For the two spheroids, the backscatter is larger than the forward scatter. The scattering cross section for the“fatter” spheroid (a/b = 2) is larger than that of the “thinner” spheroid, presumably due to a larger surface area for scattering. Another difference is that the minimum in the E-plane occurs at different angles. Figures 4.3 and 4.4 show the variation in bistatic cross sections at differ- ent angles of incidence for the same spheroid. For these casesthe plane of observation is taken to be the x, z-plane. The cross sections remain about the same as the incident angle varies, but the minimum in the TM scattering continues to move toward the other end of the spheroid as the incident angle increases. The maximum in the TM scattering also increases, as the surface area available for scattering reaches a maximum at 80 = 90’. Figure 4.5 illustrates the effect of axial ratio on the backscattering cross sections. For both spheroids [with a semimajor axis length of (1/27r)Xo], the backscatter for TE polarization does not show much variation as the incident angle changes, whereas that of TM polarization shows a gradual increase up to broadside incidence (0, = 90’). Once again, we see that the “thinner” spheroid offers a smaller area for scattering than the “fatter” one. From Fig. 4.6 we also see that as the spheroid becomes bigger with the increase in ka, the backscattering cross section also becomes more oscillatory. All the results also agree closely with those of Sinha [lo]. If we put the axial ratio as 1.Ol , the source code for the spheroid could be used to approximately model a sphere. Figure 4.7 shows the backscattering cross sections for two spheresof different sizes. They show an almost constant cross section, independent of the incident angle and polarization, which is a necessary result due to the symmetries of the sphere. This further verifies the accuracy of the source code developed as well as its suitability for modeling spheres.