Cấu trúc sóng chức năng trong điện lý thuyết P5

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Cấu trúc sóng chức năng trong điện lý thuyết P5

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EM Scattering by a Coated Dielectric Spheroid 5.1 GEOMETRY OF THE PROBLEM In this chapter we consider the scattering of a linearly polarized plane monochromatic wave by a homogeneous lossy/lossless dielectric spheroid with a confocal lossy/lossless dielectric coating immersed in a homogeneous isotropic medium. It is assumed that the surrounding medium is nonconducting and nonmagnetic

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  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) 5 EM Scattering by a Coated Dielectric Spheroid 5.1 GEOMETRY OF THE PROBLEM In this chapter we consider the scattering of a linearly polarized plane mono- chromatic wave by a homogeneous lossy/lossless dielectric spheroid with a confocal lossy/lossless dielectric coating immersed in a homogeneous isotropic medium. It is assumed that the surrounding medium is nonconducting and nonmagnetic. Results are presented only for the prolate spheroids, as the re- sults for the oblate cases can be obtained by the transformations < --+ i
  2. 116 EM SCATTERING BY A COATED DIELECTRIC SPHEROID Incident Wave $2 Wave Fig. 5.1 Scattering geometry of a coated spheroid.
  3. INCIDENT, TRANSMITTED AND SCATTERED FIELDS 117 62 c2 = -co* (5.lc) lr CO 5.2 INCIDENT, TRANSMITTED AND SCATTERED FIELDS Unlike the case of the perfectly conducting spheroid, the existence of fields inside the dielectric spheroid makes the problem more complicated, and it is necessary to use the magnetic fields in the boundary conditions. The magnetic field of a propagating wave is related to the electric field via H - &VxE, - (5 .2) where k is the wave number and 2 is the characteristic impedance of the medium. The incident electric fields for TE and TM polarizations are the same as those of the conductor case, and the corresponding magnetic fields are 00 HTE = jETE0 MO coseo n=m x m=O (5.3a) HTM = (5.3b) n=m m=O The scattered fields are also derived in the same way as in the conductor case, but a slight change in the notation of the scattering coefficients is now made: (5.4a) (5*4b) I for the TE case; and aM M+t4) &TM = mn *,m+l,ntCO) + PmM+l,n+l M$Z+l,n+l 00 M- M-c4) +>: Qcln oln CcO), (5.5a) n=l aM mn M- N-c4) (5.5b) %n oln (‘0)
  4. 118 EM SCATTERUVG BY A COATED DIELECTRIC SPHEROID for the TM case. The transmitted fields inside the core (t < &) can be expressed as (5.6a) for the TE case, and E2t,TM = d!Zn”~~~l,n(CZ) + dF+l 9n+l”Z(ltl+l 9 9n+l 00 n=mm=O L +c M-M-(‘) n=l %t oln (‘2)7 (5.7a) * oM mn N~Zil,n(c2) + 7mM+l,n+l Nfifz+l,n+l 4 1 +% OK- N,-,{Z) (c2) (5.7b) z2 n=l for the TM case. Note that the spheroidal radial functions of the first cind are now used instead because they are finite near the origin, and the transmitted wave has to be finite at the center of the spheroid. For the region & < c < &, the transmitted waves are represented as El t,TE = YEmn M + 6~nM~~!,(cl) (5.8a) (5.8b) for the TE case; and El t,TM = + 6mM+1 9n+l MZ’Z+l 9 9n+l(‘l)) n=mm=O L + KYrz M~E~l,n(cl) + P,“,l 7n+l MZ(Z-i+l9 7n+l ( )I‘1 M-M-(~) In oln (‘1) + Aln M- l”oln --w (cl) 1 9 (5.9a)
  5. RELATIONSHIP BETWEEN INCIDENT AND SCATTERED FIELDS 119 + XmMnN~?j-l,n(C1) + P,M+l 9n+lNz(z+l 7 9n+l ( )Ic1 YlM,- Niln (l)(cl) + AK- N,i’,2’(cl) (5.9b) I for the TM cMX3* 7mn9 6m+l,n+l7 Am727 and Pm+r,n+r are the unknown trans- mission coefficients. Note that in the coating region both first and second kinds of spheroidal functions are used, because down here, the wave consists of two components, one propagating inward and one outward. This is due to reflections at both boundary surfaces. 5.3 RELATIONSHIP BETWEEN INCIDENT AND SCATTERED FIELDS 5.3.1 Boundary Conditions The unknown scattering and transmission coefficients can be determined by applying the boundary conditions (i.e., continuity of the tangential compo- nents of the electric and magnetic fields across each of the spheroidal surfaces at < = 51 and 5 = &). Thus, we have where the suffixes q and 4 denote, respectively, the r)- and +-components of the fields. These equations must hold for all allowed values of 0 - 4 - 27r < < and -1 - q - 1. < < 5.3.2 TE Polarization for Nonaxial Incidence The same method is applied as for the conductor case in Chapter 4. This means that making use of the orthogonality of the trigonometric functions and spheroidal angular functions, and integrating over q, we will get -1 E E E (COS~O) (Qm>Am = (R,)S,, m = 0,172, l l l 7 (5.11) and -1 E E E (coseo) (Q+)A+ = (R+)S+, (5.12)
  6. 120 EM SCATTERING BY A COATED DIELECTRIC SPHEROID where ’ -X$(Q)) -V$(co) ’ -x:,, -v:‘, (co) (co) -xC4) -vZ& (co) (co) m2 -- --- --- -- (R a >= - y%(co) -w$(co) - Y 21 (co) -3 (co) -yap, -wt4) (co) m2 . I e2w I v:&(4 (1) xi!(cl) I vml(cl) x:$1, I v:&(Q) . . . . . I . -- -- --_ --_- CR) = b e-&l) I w:~ccl) ezlk4 I w:;(Q) Y?~2h~ I w:k4 . . . . . I .
  7. RELAT/ONSH/P BETWEEN /NC/DENT AND SCATTERED FIELDS 121 CR ) = d ------ , Bg&l) I Bg$(Cl) ’ Bg&l) I Bg)&) CR e > = e!oh) I $2&I) &!&l) I &2&l) . . . . . . \ I - - Wf) Bgo(cl) I l$i?&l) ~~,h) I J&?&l) . .. . . I .
  8. 122 EM SCATTERING BY A COATED DIELECTRIC SPHEROID f -q&2, 1 -V$(c2) -x(‘)ml (c2) 1 -v(l) (c2) ml . . . . . I . (R9 >= f 43~9~2) 1 -By, - G&2) I -g$(cz) . . . . . I . CR >= h w(2)(c1 Cl) 0 WfyCl’< 7 1) 0 . . . . . I . I ---------- -- - (4) -B,, (co,&) I B$)$a a I (1) -BiP,)(co&) I B,, &l) ‘) I . . . . I .. I WE> += _------- ---- - ------ ----me 0 (1) (2) I w,, (Q&> WlO (Cl 3 -wg(c2 C2) 0 (1) -W(‘)(cz’ 11 I ‘w,, (cl7c2> Wl”,‘(Cl, C2) ’ C2) . . . . . . . I I ---e--B- ---- - ----m- ---w-- 0 I B$)@l ?t2> @(Cl 9 (2) 49~2 SO 3 t2) (1) (1) 0 I q1 hb> B(2)(~~ Sl 9 t2) -B,, cc27 . . . . . . I
  9. RElATIOfUSHlP BETWEEN lNClDENT AND SCATTERED FIELDS 123 (Qa) = km _ _ i - _ c1 = (Q y~~o(co) -Y(l)cm0 (co) Y i% (co) -y::, (co) . . . I ’ Bg,(co) x~~oko) B (1) w (1) Xfml (‘0) am1 . . . . . . (Qb) = k, ----- 7 (Q d > = ----- (1) B cmow x($) (co) (1) B cm1 9 hd $21 (co) . . . . . \ . Sirn E’ cY,,,+l . . y(l) a10 (co 7 Sl) . (1) ----- ya,,bh) E’ l . P m+l,m+l . E’ -- --- P m+l,m+2 . . . X~$yco,h~ ----- (QE) += x;; (co, Cl) 7 SEml - - . YEImm . E’ . Yon,m+l ----- . . 0 . ----- -- --- 0 bEf m+l,m+l liEI m+l,m+2 . . .
  10. 124 EM SCATTERING BY A COATED DIELECTRIC SPHEROID AE’ \ E’ F” P 00 XE E’ m,m+l P 01 . . . ----- ----- PiX’+l.,m+l liEI 00 Pit’+l,m+2 liE’01 . . . SE -- ----- 7 SE - ----- m2 +- 7 oE’ mm PoEd E’ cTm,m+l POE; . . . ----- ----- E’ E’ 7,+l,m+l 700 E’ E’ 7,+l,m+2 701 . . . . k m = 2 for nz = 0 and km = 1 for nz > 0, and The column matrices Am and A+ are the same as those defined for the conductor case in Chapter 4. 0 is the zero matrix. Besides the row matrices used in Chapter 4, the other row matrices are defined as follows: i i B 0 (c> = amN 0 (amNO (’ > .. . >7 (5.13) where - 77-a 1 + (i) a@) (c) mNn IAmNn + ICmNn - cc,” _ q1/2 Rm,m+n(cO)IBmNn - m(m + ‘> R(i) m,m+n(CO)IDmNn; (s,” - 1y2 B@) (c) hN = (b@) (c) mN0 b@) (c) mN1 b@) (c) mN2 . . 0)9 (5.14) where bci) (c) mNn IEmNn -IGmNn + (m + ‘j2” Rf$+l,m+n+l({ 0 )I HmNn ; c; - 1 I 0i (5.15) B cmN ( c) = (P mN0 (c) cCi) mN1 (c) P) mN2 (c) l l 0) 9
  11. RELATIONSHIP BETWEEN INCIDENT AND SCATTERED FIELDS 125 where . 1 0 cmNn 2 (c) = R:;m+n (tO)IImNn (g - 1y2 . 0 + (r,” _ml),/2 RG,m+n (b)IKmNn IJmNn t=t0 IJmNn; t= = (pmN0 (c) pmN1 (c) pmN2 (c) 3 (5.18) where 1 &~,(c) = - RE$2,rn+n+2 (
  12. 126 EM SCATTERING BY A COATED DIELECTRIC SPHEROID . . 0 h@) hb) x( h2N ) = (h lt0 Nl N2 l l l )7 (5.19) where . h0 AZ = IAmNn +kmNn ; m=l m=l and finally, 0i (,(i) & SE; . . .) 7 B sN = NO (5.20) where & - - 2(I es N n- &~sNn) . Here CO= s m,m+N d% (5.21d) I -1 - drl ICmNn = (5.21e) IC2mNn = ~m+2,m+n+2~m,m+N dq, (5.21f) IDmNn = w1g) ID2mNn = (5.21h) IEmNn = (1 - 172)1’2Em+l,m+n+1Sm,m+N dq, (5.21i) IGmNn = Fm+l,m+n+lSm,m+N dq, (5.21j) IHmNn = sm+l,m+n+lSm,m+N d% (5.21k) IImNn = Gm,m+nSm,m+N dq, (5.211)
  13. RELATIONSHlP BETWEEN INCIDENT AND SCATTERED FIELDS 127 +1 (5.2lm) b2mNn = J-1 Gm+2,m+n+2Sm,m+N d% +1 1 (Sm,m+n>sm,m+N d% (5.21n) IJmNn = J-1 (1 - q2y2 +1 1 (S m+2,m+n+2>Sm,m+N d% (5.210) IJ2mNn = J-1 (1 - q2y2 +l d IKmNn = J-1 & ( (1 - rl q-2)1/2s mym+n s m,m+N d% (5*2lP) +l d IK2mNn = J ( -1 &j (1 _ rl 72)1/2 s m+2,m-i-n+2 Sm,m+N d% ww +’ d (5.21r) IOmNn = J - -1 drl (S m+l,m+n+l )Sm,m+N d% +1 rl (5.21s) IPmNn = J-1 l-q2 P m+l,m+n+l 1s m,m+N dr]~ I esNn (5.21t) = J-1 +‘(I - ~2)1’2&ns1,1+N dq~ +1 (5.21~) IGmNn = J-1 (1 - ~2)1’2&haS1,1+N d’& Besides the spheroidal angular functions (Sm,m+n), the other functions used in the integrals are B m,m+n (5.22a) C m,m+n (5.22b) d E m+l,m+n+l = r7-Sm+l,m+n+l drl (5.22~) Fm+l,m+n+l = 9 (5.22d) G m,m+n = f ((I- q2)1’2$Sm,m+n) 3 (5.22e) Eon = r)- ivSOn$ (fiRC) 7 (5.22f)
  14. 128 EM SCATTERING BY A COATED DIELECTRIC SPHEROID The functions Bm+2 m+n+2, Cm+2 m+n+2, and Gm+2 m+n+2 are obtained from their counterparts (with subscripts m and nz + A) by substituting m bym+Zandm+nbym+n+Z. Equation (5.11) can be written as SE m = (G$)IE, m=O,l,Z ,... where (GE) = (RE)-l(QE) represents the transformation matrix to trans- form the incident vector 1: = Am(cos 00)-l into the column vector of the unknown scattering and transmitting coefficients. We also note that (GE) is dependent only on the scattering body, which is the same as that of the conducting body. 5.3.3 TM Polarization for Nonaxial Incidence Using the same principles as before, we will get for the TM case, (Q,M)Am = (Rf)S,M, m = O,l,Z,. .., (5.23) (5.24) (Q,M) = The matrices Qa, Qb, QC, Qd, (RE), Am, and A+ are those defined for the TE case of the coated spheroid. The elements of Sf are the same as those for Sz except that the superscripts are now A-4instead of E. The other
  15. RELATIONSHIP BETWEEN INCIDENT AND SCATTERED FIELDS 129 matrices which are required are I I I - I I . I Pm - - ----A 1 X&Cl(1) , b> I -x(l)610 (c2,
  16. 130 EM SCATTERlNG BY A COATED DIELECTRIC SPHEROlD -- f (({’ - 1)‘/2$Rj:),,, IJllVn c=to R(i) 1,1+n IJllVn, )I t=t0 (0 = 51, and/or t=& ,m=O A00 A01 (Q&) - (Qb >
  17. NUMERICAL COMPUTATION AND MATHEMATICA SOURCE CODE 131 Unlike the conducting case, the truncation number required to obtain a given accuracy in the computed cross sections depends on a number of factors: for example, the electrical size, the properties of the coating and the core material, and the thickness of the coating. It is very difficult to come up with a rule that can take all these factors into account. As such, for the sizes and permittivities of the spheroids and coatings considered here, it was found accurate enough to consider only using m = 0, 1, 2 and 72 = m, m + 1, . . . .. nz + 5. The reason is that due to the dimensions of the matrices involved, it takes much longer to compute the system for a given m than in the conducting case. Moreover, the convergence has been verified by computing until m = 4 for most cases. In all instances, the results matched at least to four significant figures and in most cases to five significant figures. This ensures that the scheme of truncation above is proper. Three Mathematics packages were written for the coated dielectric spher- oid. The first is Coatedback.nb, which contains the user module Coat- edback [a2-, ratio2-, thickness-, epsilonrl,, epsilonr2J for computing and plotting the normalized backscattering cross section. The second pack- age, Coatedbistatic.nb, contains the module Coatedbistatic[a2-, ratio2,, thickness-, epsilonr l-, epsilonr2,, thetaO-, phi-] for the computing and plotting the normalized bistatic cross section at nonaxial incidence. The last package, Coatedbizero.nb, with the module Coatedbistatzero[a2-, ra- tiofl,, thickness,, epsilonrl-, epsilonr2-1, is developed for the bistatic cross section at axial incidence. Note that the formulas for the various cross sections are the same as those given for the conductor case. To compute the cross sections, the user needs to load the relevant packages and call any of the three modules. The arguments a2 and ratio2 denote the semimajor axis length (in terms of X0) and the axial ratio of the core, respectively. The argument thickness denotes the thickness of the dielec- tric coating (also in terms of &), while epsilonrl and epsilonr2 are the relative permittivities of the two regions (see Fig. 5.1). In addition, Coat- edbistatic.m computes the bistatic cross section for a user-specified angle of incidence (thetao-) at a user-specified azimuthal angle of observation (phi-). For all three cases, the output is formatted in the same way as the conducting case. The packages mentioned above are very similar in their structures, except for the last part, when they compute the different scattering cross sections. So they shall be described as one entity. The program flow in the packages can be summarized as follows: 1. Based on the user’s input, the parameters 50,
  18. 132 EM SCATTERING BY A COATED DlELECTRlC SPHEROID 3. The spheroidal radial functions of the first, second and fourth kinds and their first and second order derivatives are evaluated next. All of these are done with the following supporting modules: l PSpheroidRl l PSpheroidR2 l PSpheroidR2forsmallc l PSpheroidR2forlargec 4. The various functions and integrals required for formulation of the ma- trices are then defined. For the integrals, they are computed in a manner similar to that for the conducting case. 5. The matrices for the TE and TM cases are then computed using the “ Table” command. 6. Once the scattering coefficients are obtained, they are substituted into the relevant expressions for the scattering cross sections, and a plot is generated with the corresponding title and label. 5.5 RESULTS AND DISCUSSION One advantage of the source code for coated dielectric spheroids is that it allows us to model the scattering by homogeneous dielectric spheroids. This is done by letting the relative permittivity of the coated layer be unity. Fig- ure 5.2 shows the variation of the bistatic cross sections (at an axial incidence of 60 = 0) of two dielectric spheroids of the same axial ratios and permittivities but different semimajor axis lengths. The solid circles represent the results from Asano and Yamamoto [24]. The close agreement once again verifies the accuracy of the Mathematics source code used. In Fig. 5.2, for the smaller spheroid (top), the scattering intensity is al- most a constant in the plane perpendicular to the plane of polarization of the incident wave (H-plane), whereas the scattering in the plane parallel to the plane of polarization (E-plane) is approximately proportional to cos20. This behavior is similar to that of the Rayleigh limit for the scattering by very small spheres, whose radii are much smaller than for the incident wave- length. In such cases(Rayleigh scattering), the intensity is a constant in the plane perpendicular to the plane of polarization and varies as cos20 in the parallel plane [132]. With the increase in size (bottom), the magnitude of the scattered intensity increases, and the forward scattering amplitude becomes much greater than the backscattering amplitude. Again, this is similar to the Mie effect observed in scattering by spheres [133]. In addition, the patterns in power intensity distributions become more and more complicated, showing oscillating fluctuations with 0.
  19. RESULTS AND DISCUSSION 133 -20 -40 f \ \ f \ f -60 0 20 40 60 80 100 120 140 160 180 8 (“1 (a) a2 = - l x0 4 0 -10 -20 -30 -40 -50 I I I ) I I 1 I 1 I 1 i I 0 20 40 60 80 100 120 140 160 180 $9 co) (b) a2 = -X0 d Fig. 5.2 Bistatic cross sections of dielectric spheroids with a& = 2 and a relative permittivity of 1.769, but different sizes at an axial incidence of 80 = O”.
  20. 134 EM SCATTERING BY A COATED DIELECTRIC SPHEROID Figures 5.3 and 5.4 show the variation of the bistatic cross sections with different angles of incidence for a dielectric spheroid. For these cases, the plane of observation is taken to be the X, z-plane. The cross sections for the TE mode do not change dramatically 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 increasesand very strong backscattering is observed. The maximum in the TM scattering also increases, as the surface area available for scattering reaches a maximum at t90= 90’. This is similar to what was observed for a conducting spheroid in Chapter 4. Figure 5.5 illustrates the difference between a lossy dielectric and lossless dielectric on the backscattering cross sections. Notice that the shape of the cross sections depends mainly on the dimensions of the spheroid. Since both spheroids are of the sameaxial ratio and semimajor axis, they have roughly the same shape. A smaller amplitude is observed from the lossy one as expected. Figure 5.6 depicts the Mie scattering effect and the effect of dimensions on the bistatic cross sections of dielectric spheres. The results were compared with those (in bullets) by Van de Hulst [134]. In both cases, the forward- scattering cross section amplitude is larger than the backscattering amplitude. For a bigger sphere, the scattered intensity increases, and the pattern exhibits more oscillations [1331. Figure 5.7 shows the bistatic cross sections of a single dielectric spheroid with a confocal coating for two values of the axial ratio . The results were compared with those (in bullets) by Cooray and Ciric [33]. Here we observe that the variation of both the E- and H-plane patterns is almost the same for the two cases. There are signs of Rayleigh scattering, but the Mie scattering effect is also coming into play. The difference between the two casesis that the magnitudes of scattering cross sections are smaller for the “thinner” spheroid (axial ratio of 5) than for the “fatter” spheroid. This is because the area available for scattering is lesswhen the axial ratio is 5 than that when it is 2. Figure 5.8 illustrates two different backscattering cross sections of two coated spheroids of the same type of material but of totally different dimen- sions. The one at the bottom has a much thicker coating relative to the inner core, and it is also fatter (axial ratio of 2) than the one at the top (axial ratio of 5). The one with the thinner coating exhibits a TM backscatter- ing that is always higher than the TE backscattering. For the one with the thicker coating, the TM backscattering starts off initially larger than the TE backscattering, but the TE backscattering increases steadily after 00 = 45” until it is largest at broadside incidence (00 = 900). Here, the bistatic cross sections for dielectric-coated spheroids of oblique angles of incidence were also computed and discussed. Figures 5.9 and 5.10 show the results for a coated dielectric spheroid, which is basically the product of adding a confocal coating of thickness 0.02& to the dielectric spheroid of Figs. 5.3 and 5.4. As such, the crosssections for both casesare almost identical
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