Harmonic Expansion of Electromagnetic Fields

Transcription

1 Chapter 6 Harmonic Expansion of Eectromagnetic Fieds 6. Introduction For a given current source J(r, t, the vector potentia can in principe be found by soving the inhomogeneous vector wave equation, ( c t A (r, t = µ J (r, t, provided the Lorenz gauge is chosen. In source free region, the eectromagnetic fieds E and H can then be deduced from the vector potentia through ε µ E t B = A, = B, (in source-free region J =. In this Chapter, we deveop spherica harmonic expansion of the eectromagnetic fieds. In experiments, the eectromagnetic fieds, not the potentias, are measured. Radiation eectromagnetic fieds can be decomposed into two basic vector components, Transverse Magnetic (TM and Transverse Eectric (TE, where transverse is with respect to the direction of wave propagation r. These fundamenta modes are norma to each other and provide convenient base vectors in anaysis of radiation fieds. 6. Wave Equations and Green s Function The set of Maxwe s equations, E = ρ ε, (6.

2 E = B t, (6. B =, (6.3 ( E B = µ J + ε, t (6.4 can be reduced to two inhomogeneous wave equations for the two potentias Φ and A as foows. Substitution of into Eq. (6. yieds E = Φ A t, (6.5 Φ + t A = ρ ε. (6.6 Aso, substitution of B = A and Eq. (6.5 into Eq. (6.4 yieds If the Lorenz gauge A c A t ( A + Φ c = µ t J. (6.7 A + c Φ t =, (6.8 is chosen for A (i.e., the ongitudina component of the vector potentia, then Eqs. (6.6 and (6.7 are competey decouped, and reduce, respectivey, to If the Couomb gauge ( c t ( c t A =, Φ = ρ ε, (6.9 A = µ J. (6. is chosen instead, that is, if the ongitudina component of the vector potentia is chosen to be zero, such decouping cannot be achieved, A c A t Since the ongitudina current J satisfies the charge conservation aw, Φ C = ρ ε, (6. c t Φ = µ J. (6. ρ t + J =, (6.3 it foows that c t Φ C = µ J, (6.4

3 and the wave equation for the vector potentia in Couomb gauge invoves ony the transverse component of the vector potentia, A t c A t t = µ J t. (6.5 As briefy pointed out in Chapter 3, the scaar potentia Φ in Eq. (6. is not subject to retardation due to finite propagation speed of eectromagnetic disturbance. generated from the gradient of the scaar potentia E = Φ C A t, The part of the eectric fied is aso nonretarded because of instantaneous propagation. Physica (retarded eectric fied is contained in the vector potentia. In the Lorenz gauge, both potentias are appropriatey retarded and we wi therefore empoy Lorenz gauge. (As we wi see in the foowing section, the instantaneous Couomb fied is in fact canceed by a term in the retarded transverse vector potentia. The scaar potentia Φ and three cartesian components of the vector potentia A i a satisfy the wave equation in the form ( c t A i = f i (r, t, (6.6 where f i (r, t is a source function, either the charge density or the current density. The Green s function for the scaar wave equation G(r, t can be found as a soution for the foowing singuar wave equation, ( c with the boundary condition that t G(r r, t t = δ(r r δ(t t, (6.7 G = at r = and t = ±. Once the Green s function is found, the soution to the origina wave equation can be written down in the form of convoution, A i (r, t = G(r r, t t f i (r, tdv dt. (6.8 We seek a soution for the Green s function by the method of Fourier transform. Let the Fourier transform of G(r, t be g(k, ω, g(k, ω = dv dτe i(k R ωτ G(R, τ, (6.9 G(R,τ = (π 4 d 3 k dωg(k, ωe i(k R ωτ, (6. 3

4 where R = r r, τ = t t. Then, Eq. (6.7 in the Fourier-Lapace space (k, ω becomes or ( k + ω c g(k, ω = Substitution of g(k, ω into Eq. (6. yieds Integration over ω can be done easiy, G(R, τ = (π 4 g(k, ω =, k (ω/c. (6. dk dω ei(k R ωτ k (ω/c. (Note that with a new variabe s = iω, the integra reduces to ic i i e sτ π s ds = + (ck ck e iωτ k (ω/c dω = π c sin(kcτ. (6. k i πi i cke sτ π s ds = sin(kcτ, (6.3 + (ck ck which is the standard inverse Lapace transform. Lapace transform. Then, Causaity is appropriatey handed in inverse G(R, τ = c (π 3 π π dk dθ dφk sin(ckτe ikr cos θ sin θ, (6.4 where θ is the poar ange in the k space measured from the direction of R = r r, and R = r r. The integrations can be carried out as foows: G(R, τ = = = c (π π kdk sin(ckτ e ikr cos θ sin θdθ c 4π sin(ckτ sin(krdk R c [δ(cτ R δ(cτ + R], (6.5 4πR where in the fina step, use is made of cos(akdk = πδ(a. The function c R δ(cτ R = ( R δ τ R, (6.6 c 4

5 describes an impuse propagating radiay outward at a speed c, and the function c R δ(cτ + t = ( R δ τ + R, (6.7 c describes an impuse propagating radiay inward. For radiation fieds due to a source spatiay imited, ony the outgoing soution is physicay meaningfu and we adopt it as the Green s function, G(R, τ = = ( 4πR δ τ R c 4π r r δ ( t t r r c, (m s. (6.8 It is straightforward to check that G(R, t satisfies the inhomogeneous wave equation in Eq. (6.7 if the foowing identity is recaed, r r = 4πδ ( r r. Having found the Green s function for the wave equation, the genera soution to the inhomogeneous wave equation, ( c can be written down as A i (r, t = 4π dv t If the source function f(r, t is separabe in the form, Eq. (6.3 reduces to where k = ω/c. The spatia part of this soution, A i = f i (r, t, (6.9 dt f i(r, t ( r r δ t t r r. (6.3 c f i (r, t = f i (re iωt, (6.3 A i (r, t = e ik r r 4π e iωt r r f i(r dv, (6.3 A i (r = e ik r r 4π r r f i(r dv, (6.33 satisfies the Hemhotz equation, ( + k A i = f i (r. (6.34 Corresponding Green s function is G(r r = eik r r 4π r r, (6.35 5

6 which satisfies the Hemhotz equation, ( + k G ( r r = δ ( r r. (6.36 The formuation deveoped here agrees with that in the preceding Chapter which has been derived through a quaitative argument based on the retarded nature of eectromagnetic disturbance. In static cases ω =, k =, we triviay recover the static potentias, In the Green s function G(r r, t t = Φ(r = 4πε A(r = µ 4π ρ(r r r dv, J(r r r dv. ( 4π r r δ t t r r, (6.37 c the retarded nature of eectromagnetic disturbance can be ceary seen. For a source ocated at r and observer at r, eectromagnetic phenomena observed at time t is due to source disturbance created at the time r r /c seconds earier than t. 6.3 Couomb and Lorenz Gauges The divergence of the vector potentia A can be assigned an arbitrary scaar function without affecting the eectromagnetic fieds E and B. The Maxwe s equations do not specify A. In fact, potentia transformation invoving a scaar function λ, does not affect the eectromagnetic fieds, A = A + λ, Φ = Φ λ t, (6.38 E = Φ A t = Φ A t = E, In Lorenz gauge characterized by B = A = A = B. A+ c Φ t the function λ must satisfy the homogeneous wave equation, ( c t =, (6.39 λ L =, (6.4 6

7 and in the Couomb gauge choice A =, λ must satisfy Lapace equation λ C =. (6.4 In cassica eectrodynamics, the scaar function λ can be chosen to be zero because a reasonabe boundary condition λ L,C at r for both equations aows ony λ L,C =. ( c t λ L =, λ C =, In macroscopic eectromagnetic anaysis, Lorenz gauge defined by A + c Φ t =, is more convenient because a potentias and eectromagnetic fieds are expicity retarded. contrast, the scaar potentia in the Couomb gauge is not retarded and consequent Couomb eectric fied is aso nonretarded. Such instantaneous fieds are ceary unphysica, and shoud be canceed somehow by a counterpart. In this section, it is shown that the eectromagnetic fieds formuated in Lorenz gauge, which are a retarded appropriatey, are indeed consistent with those in the Couomb gauge. The instantaneous Couomb eectric fied emerging in the Couomb gauge is canceed. In Couomb gauge, the scaar potentia satisfies time dependent Poisson equation, Its soution is nonretarded, and resutant Couomb eectric fied is aso nonretarded, ρ(r, t Φ(r, t =. (6.4 ε Φ(r, t = ρ(r, t 4πε r r dv, (6.43 E(r, t = Φ(r, t (r r ρ(r, t = 4πε r r 3 dv. (6.44 As we have seen, the vector potentia in Couomb gauge ( A = is transverse and obeys the wave equation, ( c t A t = µ J t = µ (J J, (6.45 where J t is the transverse component of the current density. Note that the ongitudina current J and the charge density ρ are reated through the charge conservation aw, ρ t + J =. (6.46 In 7

8 Soution for the transverse vector potentia is retarded, A t (r, t = µ 4π The eectric fied in the Couomb gauge is thus give by E C (r, t = Φ A t t = 4πε (r r ρ(r, t r r 3 dv µ 4π ( r r J t r, t r r dv. (6.47 c t ( r r J t r, t r r dv. (6.48 c The instantaneous Couomb fied (the first term in the RHS must be somehow canceed by a term stemming from the time derivative of the transverse vector potentia for the eectric fied to be consistent with that from Lorenz gauge which guarantees that a fieds are retarded. It is convenient to work with spatia Fourier transforms of the potentias and fieds. Since the inverse Lapace transform of the Fourier-Lapace Green s function is g(k, ω = k (ω/c, (6.49 c e iτω sin(ckτ π ω dω = c, (6.5 (ck ck the particuar soution for an inhomogeneous wave equation can be given in the form of convoution, ( d c dt + k A(k, t = f(k, t, A(k, t = c k Appying this to the transverse vector potentia, we find A t (k, t = µ c k Since in the Couomb gauge, the scaar potentia is given by sin(ckτf(k, t τdτ. (6.5 sin(ckτ [J(k, t τ J (k, t τ] dτ. (6.5 Φ(k, t = ρ(k, t ε k, (6.53 8

9 the eectric fied becomes Noting E C (k, t = ik ε k ρ(k, t t A t(k, t = ik ε k ρ(k, t µ c sin(ckτ [J(k, t τ J (k, t τ] dτ. (6.54 k t t = = ck sin(ckτj (k, t τdτ sin(ckτ τ J (k, t τdτ cos(ckτj (k, t τdτ, (6.55 and we find J (k, t = ik k ρ(k, t, (6.56 t E C (k, t = ik ε k ρ(k, t µ c k t + ik ik c ρ(k, t ε k ε k = ik c ε k sin(ckτj(k, t τdτ sin(ckτρ(k, t τdτ sin(ckτρ(k, t τdτ µ c k t sin(ckτj(k, t τdτ. (6.57 Note that the instantaneous fieds (the first and third terms in the RHS exacty cance each other and the eectric fied is appropriatey retarded and consistent with the resut in the Lorenz gauge, where E L (r, t = ρ(r, τ 4πε r r dv µ J(r, τ 4π t r r dv, (6.58 Fourier transform of Eq. (6.58 indeed recovers E L (k, t = ik ε c k τ = t r r. c c sin(ckτρ(k, t τdτ µ k t which is identica to the fied formuated in the Couomb gauge. The magnetic fied is generated by the transverse vector potentia, sin(ckτj(k, t τdτ, (6.59 B = A = A t, (6.6 (note that A = by definition and is thus expicity retarded irrespective of the choice of the 9

10 gauge. 6.4 Eementary Spherica Waves For a current source osciating at a frequency ω, the vector potentia can be cacuated from Eq. (6.33, A(r,t = µ e ik r r 4π e iωt r r J(r dv. Here, we wish to expand the spatia Green s function, in terms of spherica harmonics as done in static cases, Static : G(r r = =,m G(r r = eik r r 4π r r, (6.6 4π r r (r + r + Y m(θ, φym (θ, φ, r > r. (6.6 Evidenty, the objective of the harmonic expansion is to identify eectric and magnetic mutipoe components in the radiation eectromagnetic fieds. The Green s function satisfies the singuar Hemhotz equation, G(r r = eik r r 4π r r, ( + k G = δ(r r. (6.63 We wish to construct a series expansion of the Green s function in terms of soutions to the homogeneous Hemhotz equation, ( + k f(r =. (6.64 Let the function f(r be separated in the form f(r = R(rF (θ, φ. Substitution into Eq. (6.64 yieds [ sin θ ( d dr + r d dr + k ( sin θ + θ θ sin θ ( + r R(r =, (6.65 ] + ( + F (θ, φ =, (6.66 φ

11 where ( + is a separation constant. The anguar function F (θ, φ is unchanged from the static case, To find soutions for the radia function R(r, et us assume F (θ, φ = Y m (θ, φ. (6.67 R(r = u(r r. (6.68 Substitution into Eq. (6.65 yieds the foowing equation for u(r, whose soutions are Besse functions of haf-integer order, d dr + d r dr + k ( + r u(r =, (6.69 It is customary to introduce spherica Besse functions defined by j (kr = u(r = J (kr, N (kr. ( π J + (kr kr, n (kr = π N + (kr kr, (6.7 where the numerica factor π/ is to et both functions approach the form of spherica wave, j (kr, n (kr kr Noting that the asymptotic forms of the ordinary Besse functions are we observe that j (x and n (x asymptoticay approach for kr. (6.7 ( J n (x πx cos x n + π, ( ( N n (x πx sin x n + π, ( j (x ( x cos x + π, x (6.75 n (x ( x sin x + π, x. (6.76 In radiation anayses, it is convenient to introduce spherica Hanke functions defined by First kind: h ( (x = j (x + in (x, (6.77

12 The asymptotic forms of these functions are: Second kind: h ( (x = j (x in (x. (6.78 h ( (x ( i h ( (x i + eix x + e ix which describe outgoing and incoming spherica waves, respectivey. x, x, (6.79, x, (6.8 Having found eementary spherica harmonic soutions for the homogeneous Hemhotz equation, we are now ready to construct the Green s function in terms of those eementary soutions. Since any inear combinations of the four radia functions j (kr, n (kr, h ( (kr and h ( (kr satisfy the Hemhotz equation, and the Green s function shoud be bounded everywhere, we may assume G(r, r = = m= A m j (kr h ( (kry m (θ, φym (θ, φ, for r > r, (6.8 and G(r, r = = m= A m j (krh ( (kr Y m (θ, φym (θ, φ, for r < r, (6.8 where A m is an expansion coeffi cient of the (, m harmonic and continuity on the surface r = r is imposed. Note that for a sma argument, ony j (x remains bounded, j (x x, x. (6.83 ( +!! The spherica Besse function of the second kind n (x diverges at sma x as (!! n (x x +, x. (6.84 The radiay converging (incoming soution h ( (kr is discarded because we are interested in radiation of eectromagnetic fied from ocaized sources. The coeffi cient A m can be determined through the famiiar procedure expoiting the discontinuity in the radia derivative. Substituting Eqs. (6.8 and (6.8 into the origina equation ( + k G = δ(r r = δ(r r rr sin θ δ(θ θ δ(φ φ, mutipying both sides by Y m (θ, φ, and integrating the resut over the entire soid ange by noting the orthogonaity of the harmonic functions, Y m (θ, φy m (θ, φdω = δ δ mm, (6.85

13 we obtain where A m ( d dr + r d dr + k g (r, r = ( + r g (r, r = δ(r r rr, (6.86 j (kr h ( (kr, r > r, j (krh ( (kr, r < r (6.87 The derivative of the radia function g (r, r is discontinuous at r = r and thus second order derivative yieds the singuarity compatibe with the RHS, d dr g (r, r = ik (kr δ(r r, (6.88 where use is made of the Wronskian of the spherica Besse functions, j (x[h ( (x] j (xh( (x = i [ j (xn (x j (xn (x ] = i x. (6.89 We thus find a rather simpe resut, A m = ik, (6.9 and the desired spherica harmonic expansion of the Green s function is e ik r r 4π r r = ik = m= ik = m= j (krh ( (kr Y m (θ, φym (θ, φ, r < r, j (kr h ( (kry m (θ, φym (θ, φ, r > r. In the imit of static fieds k ( that is, ω, we indeed recover 4π r r = = m= = m= (r + r + Y m(θ, φym (θ, φ, r > r + (r + Y m(θ, φym (θ, φ, r < r. r (6.9 (6.9 The expansion in Eq. (6.9 aows us to write the vector potentia in the form A(r = µ e ik r r 4π r r J(r dv = ikµ h ( (kry m (θ, φ = m= J(r j (kr Y m (θ, φ dv. (6.93 3

14 For cartesian components A i (r, (i = x, y, z, Eq. (6.93 gives A i (r = ikµ = m= h(kry m (θ, φ J i (r j (kr Y m (θ, φ dv. However, for non-cartesian components (e.g., components in the spherica coordinates, it is necessary to singe out the component of the current density J(r in the same direction as the vector potentia A at the observing position A(r = ikµ = m= h ( (kry m (θ, φe A e A J(r j (kr Y m (θ, φ dv, (6.94 where e A is the unit vector in the direction of A, e A = A/A. The probem with this representation is that the direction of the vector potentia e A is not known a priori. It is more convenient if we can find basic eectromagnetic fied vectors which are norma to each other everywhere. The Transverse Eectric (TE and Transverse Magnetic (TM eigenvectors wi serve exacty for this purpose and form a convenient dyadic for radiation eectromagnetic fieds. 6.5 TE (Transverse Eectric and TM (Transverse Magnetic Base Vectors In eectromagnetic wave probems, boundary conditions are usuay imposed on the eectric and magnetic fieds E and B, rather than the potentias Φ and A. (Note that the eectromagnetic fieds are gauge invariant, whie the potentias are not. Aso, in experiments, what is usuay measured is the eectric fied. For these reasons, it is desirabe to formuate spherica harmonic expansion of the eectric fied. In Chapter 3 on magnetostatics, we saw that A = aows us to write the vector potentia in the form A = F, where F is a vector fied. In the Lorenz gauge, the divergence of A does not vanish, A = c Φ t, which suggests A = f + F, where f is a scaar fied. However, the fied f does not contribute to the magnetic fied B = A since f and we may continue to assume A = F even for time varying fieds. (The scaar fied f does affect the formuation of the eectric fied. The vector fied F may further be 4

15 decomposed into radia and transverse components as foows, F = rφ + r ψ, (6.95 where φ and ψ are scaar functions. If the vector potentia satisfies the Hemhotz equation, so do φ and ψ, ( + k φ =, ( + k ψ =, (6.96 and thus can be expanded in spherica harmonics. (Note that Lapacian and operator r commute. The magnetic fied in terms of the scaar functions is given by B = A = (r φ r ψ = (r φ + k r ψ, (6.97 where use is made of (r ψ =. (6.98 The second term in the RHS is evidenty transverse to r, that is, the scaar fied ψ is a generating function for transverse magnetic (TM modes. The scaar fied φ generates transverse eectric (TE modes, since in the current free region, the eectric fied is given by iω c E = B = r φ (r ψ = k [r φ + (r ψ]. (6.99 The TE and TM modes are the desired base vectors because vector functions r φ and (r ψ are norma to each other. To prove this, et us assume Expanding φ(r = A m g (ry m (θ, φ, ψ(r = B m g (ry m (θ, φ. (6. [g (rr Y m ] = g (r [r Y m (θ, φ] + g (r (r Y m = ( + g (r [r Y m (θ, φ] g (r Y m e r g (r Y m r = ( + g (r Y m e r d r dr [rg (r] Y m, (6. we see that the two vectors r Y m and [g (r r Y m ] are indeed norma to each other, since (r Y m r = and (r Y m Y m =. (The product r (r Y m Y m = im d dθ Y m, 5

16 evidenty does not vanish. Interpret this. The operator r Y m (θ, φ = ily m (θ, φ, L = ir, (6. is the anguar momentum operator and its expicit form is Noting we can readiy show that Y m r Y m (θ, φ = e θ sin θ φ + e Y m φ θ. (6.3 e φ φ = sin θe r cos θe θ, (r (r Y m = (r Y m = ( sin θ Y m + Y m sin θ θ θ sin θ φ = ( + Y m, (6.4 or L Y m = ( + Y m. (6.5 Other usefu properties are: ( L x = i y z z y = e r r i r L, (6.6 r (r Y m (r Y m dω = ( + δ δ mm, (6.7 (r Y m =, (6.8 L LY m = LL Y m, LY m = L Y m, L LY m = ily m, (6.9 (, L y = i z x x (, L z = i x z y y = i x φ, (6. L ± L x ± il y = e ±iφ ( ± θ + i cot θ φ, (6. L + Y m (θ, φ = ( + m (m + Y,m+ (θ, φ, (6. L Y m (θ, φ = ( + m (m Y,m (θ, φ, (6.3 L z Y m (θ, φ = my m (θ, φ. (6.4 We are now ready to expand the raw form of the vector potentia J(r e ik r r r r dv = µ ik A(r = µ 4π = m= h ( (kry m (θ, φ j (kr Y m (θ, φ J(r dv, (6.5 6

17 in terms of the TE and TM base vectors. For TE modes, the vector potentia shoud be in the form A TE = r φ. (6.6 Therefore, it is necessary to identify TE component of the source current J(r, which can be effected by Jm TE (r = j (kr(r Ym J(r. (6.7 The TE component of the vector potentia can thus be assumed in the form A TE (r = µ ik = m= a m h ( (krr Y m (θ, φ j (kr [r Y m (θ, φ ] J(r dv, (6.8 where a m is expansion coeffi cient. Note that in the summation over, the monopoe term = vanishes because Y =. It is convenient to choose a m so that a m (r Y m (r Ym dω =, (6.9 or Corresponding TE base vector is a m = ( +. (6. u TE m (r = ( + r Y m (θ, φ, (6. which is normaized as u TE [ute m (r m (r] dω =. (6. The TE component of the vector potentia is therefore given by A TE (r = µ ik = µ ik = m= = m= ( + h( (krr Y m (θ, φ ( + h( (krr Y m (θ, φ j (kr [r Y m (θ, φ ] J(r dv j (kr ( r J(r Ym (θ, φ (6.3 dv, The current density J(r may be separated into the conduction current J c and magnetization current J m = M (r, where M (r is the magnetic dipoe moment density. J(r = J c (r + M (r, (6.4 For the TM modes, it is convenient to et the TM base vector have the same dimensions as TE 7

18 base vector. For this purpose, we rewrite the decomposition of the vector potentia in the form A = r φ + (r ψ. (6.5 k Foowing the same procedure deveoped for the TE modes, we find A TM (r = iµ k = m= ( + [h( (krr Y m (θ, φ] Corresponding TM base vector in the radiation zone is [j (kr r Y m (θ, φ ] J(r dv. (6.6 u TM m (r = ( + e r [r Y m (θ, φ], (6.7 which is aso normaized as u TM [utm m (r m (r] dω =. (6.8 Note that in Eq. (6.6, [j (kr r Y m (θ, φ ] J(r (6.9 is the TM component of the current density. For a sma source kr, the spherica Besse function j (kr can be approximated by j (kr (kr ( +!!. (6.3 Then, the moment for the TE modes becomes (the primes are omitted for brevity [j (krr Y m (θ, φ] J c (rdv k = r (r J c Ym ( +!! dv k = (r Ym ( +!! (r J cdv. (6.3 This contains the current circuation r J c which produces magnetic mutipoe moments. magnetic mutipoe moment is in genera defined by m m = + The (r Y m (r J cdv. (6.3 In terms of the magnetic moment m m, the integra may be rewritten as ( + k [j (krr Y m (θ, φ] J c (rdv = ( +!! m m. (6.33 8

19 The contribution from the magnetization current M can be cacuated in a simiar manner, m m = r Y m MdV, and the (, m component of the TE vector potentia is given by A TE k + m (r = µ i ( +!! (m m + m m h( (krr Y m (θ, φ. (6.34 The radiation power associated with the vector potentia can readiy be cacuated as foows: P TE m = r Z H TE dω, (6.35 where the magnetic fied H TE m (r in the radiation region kr is approximatey given by H TE m (r ik A TE m µ m = k+ ( i (m m + m m ( +!! r ei(kr ωt n [r Y m (θ, φ]. (6.36 Then Noting H TE m = k (+ [( +!!] m m + m m r Y m (θ, φ. (6.37 r Y m (θ, φ dω = ( +, (6.38 we find the radiation power in the form Pm TE = µ + ε = = 4π 4πε c k (+ [( +!!] m m + m m + k (+ [( +!!] m m + m m (6.39 4πc + 4πε [( +!!] mg m + m G m, (6.4 k (+ where the atter expression faciitates comparison with the formuation in the Gaussian unit system in which the magnetic mutipoe moment is defined by m G m = c + (r Ym (r JdV, (6.4 Eectric mutipoes radiate TM modes. This may be seen from the TM component of the current 9

20 density, [j (krr Ym (θ, φ] J(rdV = j (kr[r Ym (θ, φ] JdV = [j (krrym ] JdV = j (krym r [ ( J]dV = j (krym r [ ( J J ] dv = Ym d dr [rj (kr] JdV k j (krym r JdV, (6.4 where it is noted that the function j (krym (θ, φ satisfies the Hemhotz equation, ( + k j (krym (θ, φ =. Again, if the current is separated into the condution current and magnetization currents, J = J c + M, Eq. (6.4 can be rewritten as = Ym Ym d dr [rj (kr] JdV k d dr [rj (kr] J c dv k j (kry m r JdV j (kry m r J cdv + k j (krym (r M dv. (6.43 For a sma source kr, the first term in the RHS containing eectric mutipoe moment is dominant. Recaing and we may approximate Eq. (6.43 by Ym j (kr d dr [rj (kr] J c dv k ( + k+ ic ( +!! ρ t + J c =, (kr ( +!! for kr, j (kry m r J cdv + k j (krym (r M dv ( qm + q m, (6.44 where the second term is of order ka (a being the source size and thus ignorabe, q m is the

21 eectric mutipoe moment defined earier, q m = r Ym (θ, φρ(rdv, (6.45 and q m is an effective eectric quadrupoe moment created by the magnetic dipoe moment density, q m = ik ( + c The radiation vector potentia of TM mode is thus given by A TM k m (r = µ c ( +!! q m and corresponding radiation power is r Ym (θ, φ (r M dv. (6.46 [ ] h ( (krr Y m (θ, φ, (6.47 Pm TM = µ c 3 k (+ + [( +!!] q m + q m k (+ + = 4πc q 4πε [( +!!] m + q m. ( Radiation of Anguar Momentum In chapter 5, radiation of anguar momentum from a circuating charge was discussed. In genera, if a charge undergoes circuar motion, it radiates (oses anguar momentum as we as energy, for the two quantities are intimatey reated. In this section, a genera formuation for radiation of anguar momentum is given. Since the momentum fux density associated with eectromagnetic fieds is given by c E H, (6.49 The anguar momentum fux density associated with eectromagnetic fieds is c r (E H. (6.5 In the radiation zone, this must be proportiona to /r if radiation of anguar momentum accompanies radiation of energy. Therefore, terms proportiona to /r 3 in the Poynting fux, which have been ignored in the cacuation of radiation power, shoud be retained. As a concrete exampe, et us consider TM modes due to eectric mutipoes. The vector potentia of TM mode is A TM k m (r = µ c ( +!! q m [ ] h ( (krr Y m (θ, φ. (6.5

22 Corresponding magnetic and eectric fieds are and H TM m (r = A TM m µ k + = c ( +!! q mh ( (krr Y m (θ, φ, (6.5 E TM m (r = i H TM m ωε where use is made of the identity = ic ωε k + ( +!! q m [h ( (krr Y m (θ, φ], (6.53 [ ] h ( (krr Y m (θ, φ =. The expressions for the eectromagnetic fieds are exact. The radiation fux of anguar momentum associated with an (, m mode is c r (E m H m TM = ic k (+ ωε [( +!!] q m r = ic ωε = ic ωε [ [ (h ( ] (krr Y m (θ, φ] [h ( (krr Ym (θ, φ] k (+ ( [( +!!] q m r [h ( (krr Y m (θ, φ] h ( (krr Ym (θ, φ k (+ [( +!!] q m ( + h ( (kr Y m (θ, φr Ym (θ, φ. (6.54 In radiation zone kr, this reduces to ic + ωε [( +!!] q m Y m (θ, φr Ym (θ, φ r, (6.55 k (+ and the integration over the soid ange yieds the rate of anguar momentum radiation, dl m dt k (+ = + 4πc 4πε [( +!!] q m m ω e z, (6.56 where use is made of Y m (θ, φr Ym (θ, φdω = im Y m (θ, φym (θ, φdωe z = ime z, (6.57

23 Comparing with the radiation power given in Eq. (6.48, we see that they are reated through a simpe ratio, L z P = m ω. (6.58 Evidenty, this is not consistent with the famiiar quantum mechanica resut, L z P = ( + ( + =. (6.59 ω ω The discrepancy is not surprising because the quantum mechanica formua is for a singe photon whie the resut obtained in cassica eectrodynamics pertains to (infinitey many photons. Quantum mechanica formua for N photons is L z N P = m + N( + m, (6.6 Nω and the cassica resut can be recovered in the imit of N. 6.7 Some Exampes Exampe Spherica Dipoe Antenna Figure 6-: Spherica antenna. Two ideay conducting hemispheres of radius a insuated from each other at the equator with a sma gap are connected to an osciating votage source V e iωt. In this case, the surface current fows in the θ direction and creates an azimutha magnetic fieds B φ. The resutant radiation fieds 3

24 are thus TM (Transverse Magnetic and we may assume the magnetic fied in the form B = r ψ, (6.6 where ψ is a scaar function. Since the system is axiay symmetric without dependence on the azimutha ange φ, the scaar function ψ can be expanded in spherica harmonics as ψ(r, θ = a h ( (krp (cos θ, r > a, (6.6 which yieds the magnetic fied B = r ψ = a h ( (krp (cos θe φ, r > a. (6.63 Note that d dθ P d (cos θ = sin θ d(cos θ P (cos θ = P (cos θ. (6.64 The eectric fied associated with the magnetic fied can be found from the Maxwe s equation, ε µ E t = B, (6.65 which yieds E(r = ic ω [ r sin θ θ (sin θb φe r r ] r (rb φe θ. (6.66 Substituting B φ (r in Eq.(6.63 yieds the transverse component of the eectric fied, E θ (r, θ = ic ω [ ] a r h( (kr kh ( (kr P (cos θ, (6.67 where the foowing recurrence formua for the spherica Besse functions has been used, d dx h( (x = x h( (x h ( + (x = h ( (x + x h( (x. (6.68 Since the sphere is ideay conducting, the θ component of the eectric fied shoud vanish except at the gap. Therefore, at the sphere surface r = a, the eectric fied may be approximated by E θ (r = a, θ = V (θ a δ π, (6.69 4

25 which satisfies the obvious requirement that Then, for arbitrary ange θ, the foowing reation must hod V (θ a δ π = ic ω π a E θ (a, θdθ = V. (6.7 [ ] a r h( (ka kh ( (ka P (cos θ. (6.7 Mutipying both sides by P (cos θ sin θ and integrating over θ, we readiy find the expansion coeffi cient a, where a = ω + ic ( + P ( = and use has been made of the integra P ( h ( (ka kah ( (kav, (6.7 (!! (!!, odd, even (6.73 P (xp (xdx = + ( + δ. (6.74 The disappearance of even harmonics is expected because of the up-down anti-symmetry of the probem. (Note that P (x is an odd function of x if is odd. The surface current density J s on the sphere surface can be found from the boundary condition for the magnetic fied, J s = n H φ (r = a, θ The current at the gap θ = π/ where the votage V appears is = H φ (a, θe θ = a h ( µ (kap (cos θe θ. (6.75 I = πaj s (θ = π/ = πa a h ( µ (kap ( = i πaω µ c + ( + h ( (ka h ( (ka kah ( (ka [ P ( ] V. (6.76 5

26 This defines the radiation admittance of the spherica dipoe antenna, Y = I V ε + = iπ µ ( + kah ( (ka h ( (ka kah ( (ka [ P ( ]. (6.77 A spherica antenna has imited practica appications. However, the procedure deveoped in this exampe is appicabe to more practica probems such as haf waveength dipoe antennas. A thin rod antenna can be anayzed rigorousy using the proate spheroida coordinates. Exampe Circuar Loop Antenna Figure 6-: Circuar oop antenna. ka is arbitrary. Let us consider a circuar osciating current I e iωt carried by a thin conductor ring of radius a. The current density may be written as iωt δ(r a ( J = I e δ θ π e φ. (6.78 a As shown in Chapter 5, if the ring radius is much smaer than the waveength ka = ω a, the c probem reduces to radiation by a magnetic dipoe with a dipoe moment m z (t = πa I e iωt. For an arbitrary vaue of ωa/c, higher order mutipoe fieds must be retained. Since the oop current radiates TE modes, the TE vector potentia in Eq. (6.8 can be directy appied with a resut A TE (r = µ ik ( + h( (krr Y (θ, φm, (6.79 6

27 where M = j (kr [r Y (θ ] J(r dv, (6.8 is the TE component of the current density. Note that because of the axia symmetry, ony m = components are nonvanishing. Since we find r Y (θ = dy dθ e φ The vector potentia is therefore given by = + 4π P (cos θ e φ, + M = πai 4π j (kap (. (6.8 A TE (r = iµ I ka + ( + j (kah ( (krp (P (cos θe φ, (6.8 and the magnetic fied in the radiation zone kr is H(r µ ik A TE e ikr + + = aki ( i r ( + j (kap (P (cos θe θ. (6.83 The radiation power associated with the -th harmonic mode is P = r Z H dω ( + = Z (aki [ j (kap ( + (] π π [P (cos θ] sin θdθ = πz (aki [ j (kap (] +, =, 3, 5, (6.84 ( + In the ong waveength imit ka, the dipoe term ( = is dominant, and we recover P = πz (aki (ka 6 ω 4 m = 4πε 3c 5, (6.85 where m = πa I is the magnetic dipoe moment of the ring current. 7

28 6.8 Spherica Harmonic Expansion of a Pane Wave A pane wave of panar poarization is characterized by constant ampitudes of eectromagnetic fieds and unidirectiona propagation assumed here in the z-direction, E e ikz e x, H e ikz e y. (6.86 The purpose of this section is to decompose the pane wave into spherica harmonics. Once achieved, such a representation wi be very usefu in anayzing scattering of eectromagnetic wave by an object paced in the pane wave. Scattered waves are evidenty no onger pane waves but they consist of many (often infinitey many spherica harmonic waves. A key observation to be made is that there is one-to-one correspondence between spherica harmonic component in the incident pane wave and the one in the scattered wave. This is because a scattered harmonic mode characterized by mode numbers (, m can ony be produced by a mode in the incident wave having exacty the same anguar dependence. For exampe, TM (, m harmonic component in the scattered wave is generated by the same TM harmonic component contained in the incident wave. Figure 6-3: Scattered eectromagnetic waves consist of spherica waves. The incident pane wave can be decomposed into spherica waves to faciitate anaysis. The first step is to expand the propagator function e ikz = e ikr cos θ in terms of spherica harmonics. This can be effected by considering the imiting case of the scaar Green s function, G(r, r = eik r r 4π r r = ik = m= j (krh ( (kr Ym (θ, φy m(θ, φ, r > r. (6.87 In the imit of r, that is, when the source is far away, h ( (kr approaches h ( (kr + eikr ( i kr. (6.88 8

29 Noting aso k r r kr k r, r r, we find e ik r = 4πi = m= ( i + j (krym (θ, φy m(θ, φ, (6.89 where the anges (θ, φ are those of the wavevector k = (k, θ, φ. Since we have assumed a pane wave propagating in the z-direction, it foows that θ = and φ becomes irreevant. Noting P m ( = δ m, and taking the compex conjugate of Eq. (6.89 yieds the foowing identity, e ikz = e ikr cos θ = i ( + j (krp (cos θ. (6.9 = The eectric fied assumed to be in the x-direction can be converted into components in the spherica coordinates as E e x = E (r sin θ cos φ = E (sin θ cos φe r + cos θ cos φe θ sin φe φ. (6.9 This fied can be represented as a sum of TE and TM modes. To effect such a representation, we assume the foowing expansion, E e ikr cos θ e x = a m j (krr Y m (θ, φ + k m or substituting the expansion in Eq. (6.9 for e ikr cos θ in the LHS, b m [j (krr Y m (θ, φ], m = E i ( + j (krp (cos θ(sin θ cos φe r + cos θ cos φe θ sin φe φ = = m= a m j (krr Y m (θ, φ + b m [j (krr Y m (θ, φ], (6.9 k m where a m and b m are expansion coeffi cients for TE and TM modes, respectivey. The TE coeffi cient a m can be determined by mutipying both sides by r Ym (θ, φ and integrating the resut over the soid ange. Expoiting the foowing orthogonaity reationships, [r Y m (θ, φ] [r Y m (θ, φ]dω = ( + δ δ mm, [r Y m (θ, φ] [j (krr Y m (θ, φ =, 9

30 we find a m = E i ( + ( + P (cos θ(cos θ cos φe θ sin φe φ (r Ym dω. (6.93 The presence of cos φ and sin φ functions in the integra makes ony m = ± components nonvanishing. For m = +, we find where a, = E i ( + ( + P (cos θ(cos θ cos φe θ sin φe φ (r Y, dω, (6.94 Y, (θ, φ = Y, (θ, φ + = 4π( + P e iφ. (6.95 Since P (cos θ(cos θ cos φe θ sin φe φ (r Y, dω + π ( cos θ = iπ P (cos θ 4π( + sin θ P dp (cos θ + (cos θ dθ sin θdθ, (6.96 and π = = dp π P π (cos θ P (cos θ sin θdθ dθ (cos θ d dθ [P (cos θ sin θ]dθ P (cos θp (cos θ cos θdθ + π [P (cos θ] sin θdθ, (6.97 the integra reduces to = = Then, finay, π π ( cos θ P (cos θ sin θ P [P (cos θ] sin θdθ dp (cos θ + (cos θ dθ sin θdθ ( +. ( a, = i + π( + ( + E. (6.99 3

31 The coeffi cient a, can be found in a simiar manner, a, = a, = i + π( + ( + E. (6. The appearance of m = ± components in the spherica harmonic expansion of a pane wave is understandabe, for a pane wave can be decomposed into two circuary poarized waves of opposite heicity. The coeffi cient b m of the TM mode can be found in a simiar manner. It is more convenient to work with the magnetic fied, B = iω E = a,± [j (krr Y,± ] + k iω iω = where use is made of the identity, b,± j (krr Y,±, (6. [j (krr Y,± ] = [j (krr Y,± ] [j (krr Y,± ] The magnetic fied associated with the pane wave is Then, B e ikr cos θ e y = E c E c = iω = = + k j (krr Y,±. (6. i ( + j (krp (cos θ (sin θ sin φe r + cos θ sin φe θ + cos φe φ. i ( + j (krp (cos θ (sin θ sin φe r + cos θ sin φe θ + cos φe φ = = a,± [j (krr Y,± ] + k iω b,± j (krr Y,±. (6.3 Mutipying both sides by r Y,± and integrating the resut over the soid ange, we find b,± = E i + ( + ( + = i + π( + ( + E ( P (cos θ cos θ sin θ,± sin φ Y,± φ,± + cos φ Y θ dω = ±a,±. (6.4 (Cacuation steps are eft for exercise. Then the desired spherica harmonic expansion of a pane 3

32 wave is E e ikz e x = E,± i [ 4π( + j (krr Y,± (θ, φ ± ] i ( + k [j (krr Y,± (θ, φ]. The accompanying magnetic fied is expanded as B e ikz e y = E c,± (6.5 [ ] i π( + ( + k [j (krr Y,± (θ, φ] ± j (krr Y,± (θ, φ. (6.6 These expansions of the pane wave greaty faciitate anaysis of scattering of pane eectromagnetic waves by an object. An important observation to be made is that the eectromagnetic waves re-radiated (scattered by an object aso consist of TE and TM modes in the form TE mode: h ( (krr Y,± (θ, φ, (6.7 TM mode: k [h( (krr Y,± (θ, φ], (6.8 since the boundary conditions for eectric and magnetic fieds at the object require that scattered wave shoud have exacty the same anguar dependence as the incident pane wave. In exampes to foow, we anayze scattering of a pane wave by ideay conducting sphere and by dieectric sphere. 6.9 Scattering by an Ideay Conducting Sphere Let us consider a highy conducting sphere of radius a paced in a pane eectromagnetic wave as shown in Fig.6-4. In two extreme cases, ka (ow frequency imit and ka (high frequency imit, the probem can readiy be soved without detaied anaysis based on the spherica harmonic expansion. If ka, the dipoe approximation can be used. As we have seen, the eectric dipoe moment of a conducting sphere paced in a static eectric fied E is p = 4πε a 3 E, (6.9 and the magnetic dipoe moment of a superconducting sphere paced in a static magnetic fied H is m = πa 3 H. (6. These expressions remain vaid in osciating fieds as ong as the osciation frequency is suffi cienty sma so that ka = ωa/c. Each dipoe radiates ineary independent modes. The radiation powers of the modes are thus additive. The power radiated by the eectric dipoe is P E = 4πε ω 4 p 3c 3 = 4πε ω 4 a 6 3c 3 E, (6. 3

33 and that due to the magnetic dipoe is P M = ω 4 m 4πε 3c 5 = 4 4πε ω 4 a 6 3c 3 E, (6. where H = E /Z, Z = µ /ε is substituted in the magnetic dipoe moment. The tota reradiated (scattered power is and corresponding scattering cross-section is given by ω 4 a 6 ( P = 4πε 3c 3 E +, (6.3 4 σ = P = P S i cε E = ( 8π 3 + π 3 a (ak 4. (6.4 The dependence σ k 4, which is common to a dipoe radiation by objects much smaer than the waveength of incident wave, is known as Rayeigh s aw. The reader may wonder why the magnetic dipoe radiates a power comparabe with that by eectric dipoe in this case. As we have seen, the radiation fieds due to a magnetic dipoe is of higher order by a factor (ka compared with those due to eectric dipoe. The reason is as foows. The ow order vector potentia A(r µ 4πr eikr J(r ( ik r dv, (6.5 must be appied separatey for eectric and magnetic dipoes because the surface currents on the conducting sphere induced by the incident eectric and magnetic fieds are entirey different. The current due to the eectric fied fows between the poes whie the current induced by the magnetic fied is azimutha. The surface current in the poar direction is of order J sθ aωε E = akh, (6.6 whie the azimutha current is of order J sφ H. (6.7 The poar current is smaer than the azimutha current by a factor ak. (This is not surprising because current fows in the azimutha direction without any hindrance whie in the poar direction, current fow resuts in charge accumuation at the poes. Therefore, the radiation fieds from the eectric and magnetic dipoes are comparabe. In the opposite imit of ka (high frequency or short waveength imit, geometric and physica optics approximation can be appied. The probem reduces to refection by the iuminated surface and diffraction by the other haf surface in the shadow. Each surface contributes equay to the scattering cross-section, σ = πa, (6.8 33

34 athough the anguar distribution of scattered Poynting fuxes are entirey different. We wi revisit this probem in chapter 7. Figure 6-4: Geometry assumed in anaysis of scattering by a conducting sphere. For arbitrary vaue of ka, the spherica harmonic expansion of the pane wave can be expoited as foows. The scattered wave can be decomposed into TE and TM modes having the same anguar dependence as those contained in the incident pane wave. expansion for the eectric and magnetic fieds of the scattered wave, E sc (r = E,± B sc (r = B,± where M,± and N,± are the TE and TM base vectors, We therefore assume the foowing i + π( + ( + (A M,± ± B N,±, r > a (6.9 i π( + ( + (A N,± ± B M,±, r > a M,± (r = h ( (krr Y,±, (6. N,± (r = k [h( (krr Y,± ], (6. and A and B are expansion coeffi cients to be determined. If the sphere is ideay conducting, the boundary conditions at the sphere surface are that the tangentia component of the eectric fied and norma component of the magnetic fied both vanish. Since the tota fied at the surface is the sum of incident and scattered waves, the expicit forms of the boundary conditions are: (E inc +E sc t =, at r = a, (6. and (B inc +B sc n =, at r = a. (6.3 34

35 In fact, the coeffi cients A and B can be determined from (E inc +E sc t = aone, where B = A = j (ka h ( (ka, (6.4 d dr [rj (kr] r=a [kaj (ka] d = (kr] [kah ( (ka], r=a (6.5 dr [rh( Note that for g (kr = j (kr or h ( (kr, [xj (x] = d dx [xj (x], [xh ( (x] = d dx [xh( (x]. (6.6 [g (krr Y,± ] = ( + g (kr Y,± e r d r dr [rg (kr] Y,±, (6.7 and the coeffi cient B has been determined from the tangentia component of the eectric fied of the TM mode. The boundary condition for the magnetic fied is automaticay satisfied. (Verify this statement x 6 8 Figure 6-5: Normaized scattering cross-section of an ideay conducting sphere. function of ka. σ/(πa as a The radiation power of the -th harmonic mode can be readiy found as P = cε E π( + ( A ( + r = cε E M dω + B N dω π( + k ( A + B, (6.8 35

36 where the factor accounts for an equa contribution from the (, m = ± modes. scattering cross-section is given by Then the σ(ka = = π k P cε E = π k ( + ( A + B = ( + j (ka [kaj (ka] = h ( + (ka [kah (, (m. (6.9 (ka] σ(ka normaized by πa (this is the scattering cross-section in the geometrica optics imit ka is potted in Fig.?? in the range < x(= ka <. In the ong waveength imit, ka, the spherica Besse functions approach j (x x ( +!!, [xj (x] + ( +!! x for x, (6.3 h ( (!! (x i x +, [xh ( (x] (!! i x + for x. (6.3 The owest order terms of = remain finite in this imit, which yieds a scattering cross-section A = i 3 (ka3, B = i 3 (ka3, (6.3 σ π 3 k4 a 6, ka. (6.33 In short waveength imit ka, the figure indicates that σ indeed approaches πa. In radar engineering, the Poynting fux scattered back toward a radar is of main interest. In the geometry assumed in Fig. 6-4, the direction of back scattering corresponds to θ = π. Since Y, (θ, φ + Y, (θ, φ = i + 4π( + P (cos θ sin φ, (6.34 dp dθ + sin θ P =, and dp dθ the scattered fied at θ = π becomes E sc (r = E,± e ikr = ie kr = ( ( + i + π( + ( + (A M,± ± B N,± at θ = π, ( ( (A B e x. (6.36 At θ = π, the scattered fied is pane poarized in the same direction as the incident eectric fied. 36

37 The Poynting fux at θ = π is S r (θ = π = cε E 4r k The radar scattering cross section is defined by where 4π is the tota soid ange. ( ( + (A B σ radar = 4π cε E r S r (θ = π = π k ( + ( (A B Exampe 3 Scattering by a Dieectric Sphere., (6.37 Scattering by a sphere of dieectric and magnetic properties can be anayzed in a simiar manner. We assume a sphere of radius a having reative permittivity ε r and reative permeabiity µ r paced in a pane wave. In genera, ε r and µ r are both compex to account for dissipation (absorption of eectromagnetic energy. The incident wave is described by the fieds and E e ikz e x = E,±,± [ i + π( + j (krr Y,± (θ, φ ± ] ( + k [j (krr Y,± (θ, φ], H e ikz e y = E [ ] i π( + Z ( + k [j (krr Y,± (θ, φ] ± j (krr Y,± (θ, φ. where Z = µ /ε. The scattered wave continues to be assumed in the form E e (r = E,± (6.38 (6.39 i π( + ( + (A M,± ± B N,±, r > a (6.4 H e (r = E i π( + Z ( + (A N,± ± B M,±, r > a (6.4,± where Z = µ /ε is the impedance in the free space. The fieds inside the sphere shoud be bounded at r = and thus may be assumed as E i (r = E,± ( i π( + C j (k rr Y,± ± D ( + k [j (k rr Y,± ], (6.4 H i (r = E ( i π( + C Z s ( + k [j (k rr Y,± ] ± D j (k rr Y,±, (6.43,± 37

38 where Z s = µr ε r Z, k = ε r µ r k, (6.44 to account for the impedance and index of refraction of the sphere. Continuity of the tangentia components of the eectric fied E and magnetic fied H yieds the foowing conditions: where Soving for the coeffi cients A and B, we find j (ka + A h ( (ka = C j (k a, (6.45 [kaj (ka] + B [kah ( (ka] = εr µ r D [k aj (k a], (6.46 j (ka + B h ( εr (ka = D j (k a, (6.47 µ r [kaj (ka] + A [kah ( (ka] = µ r C [k aj (k a]. (6.48 [xj (x] = d dx [xj (x], [xh ( (x] = d dx [xh( (x]. (6.49 A = µ rj (k a[kaj (ka] j (ka[k aj (k a] h ( (ka[k aj (k a] µ r j (k a[kah ( (ka], (6.5 B = ε rj (k a[kaj (ka] j (ka[k aj (k a] h ( (ka[k aj (k a] ε r j (k a[kah ( (ka]. (6.5 The case of ideay conducting sphere can be recovered in the imit ε r and µ r. The scattering cross-section can readiy be found, σ = π k ( ( + A + B. (6.5 = In the ong waveength imit, ka, k a, the eading order terms are the dipoes, =, A = i µ r 3 µ r + (ka3, B = i 3 ε r ε r + (ka3. (6.53 For a dieectric sphere with µ r =, the scattering cross-section in the ow frequency regime is thus given by σ 8π 3 ε r ε r + k 4 a 6, ka. (6.54 In the imit of ε r, we recover the eectric dipoe portion of the cross-section of a conducting sphere, σ = 8π 3 k4 a 6. (6.55 Note that the case of conducting sphere can be recovered, mathematicay, in the imit µ r = for 38

39 A and ε r = for B. The case of conducting sphere shoud be anayzed by incorporating the impedance iωµ Z =, iωε + σ c where σ c is the conductivity. Idea conductor is characterized by Z = which can be reaized by etting σ c, or µ, or ε. 6. Scattering by a Cyinder In this section, we consider scattering of eectromagnetic waves by a cyindrica object having a ength suffi cienty onger than the waveength, k. If the incident wave propagates perpendicuar to the cyinder axis, the probem becomes two dimensiona. For genera incident ange, the incident wave can be decomposed into norma and axia components. The axia component suffers itte scattering and it is suffi cient to consider ony the case of norma incidence. Figure 6-6: Two possibe poarizations of the incident fied reative to the cyinder axis, E i parae to the axis (top and perpendicuar (bottom. Scattering by a conducting cyinder can be anayzed in a manner simiar to the case of conducting sphere. If the incident waveength is much shorter than the radius of the cyinder, ka, the geometrica optics approximation appies. Regardess of the poarization of the incident wave reative to the cyinder axis, the scattering cross-section per unit ength of the cyinder is given by σ = a + a = 4a, ka, (6.56 where a is the contribution form the iuminated surface and another a is the contribution from the shadow surface due to forward diffraction. Anaysis for this case, which is quite parae to the 39

40 case of a conducting sphere, is eft for an exercise.. In ong waveength imit ka, the poarization direction of the incident wave becomes important. This is understandabe because the current induced on the cyinder surface sensitivey depends on the orientation of the incident eectric fied. As intuitivey expected, the surface current fows much more easiy aong the axis than in azimutha direction. Therefore, the scattering crosssection when E i e z shoud be much arger than the case E i e z. The axia components of the eectric and magnetic fieds satisfy the foowing -dimensiona scaar Hemhotz equation, ( ρ + ρ ρ + ( ρ φ + Ez k =. (6.57 Genera soutions can be found by assuming a form R(ρe imφ, where the radia function satisfies the Besse equation ( d dρ + d ρ dρ m ρ + k H z R(ρ = J m (kρ, N m (kρ. R(ρ =, (6.58 For wave anaysis, it is convenient to define the Hanke functions of the first and second kinds by The asymptotic form of the first kind is H ( m (kρ = J m (kρ + in m (kρ, (6.59 H ( m (kρ = J m (kρ in m (kρ. (6.6 ( H m ( (kρ πkρ exp ikρ i m + π, (6.6 4 which has an ampitude dependence / ρ appropriate for cyindrica waves. H ( m (kρ describes an outgoing wave and H ( m (kρ an incoming wave. We first anayze the case of incident eectric fied aong the cyinder axis E i e z. The incident wave is assumed to be propagating in the negative x-direction, The scattered eectric fied is aso in the z-direction and we assume E i (r = E e ikρ cos φ e z. (6.6 E sc z = m a m H ( m (kρe imφ. (6.63 If the cyinder is ideay conducting, the boundary condition is that the tangentia component of 4

41 the eectric fied vanish at the cyinder surface, E e ika cos φ + m a m H ( m (kae imφ =. (6.64 Mutipying by e im φ and integrating over φ, we find the expansion coeffi cient a m, a m = and the scattered eectric fied is E sc z E πh ( m (ka The Poynting fux in the radiation zone kρ is and the tota scattered power is given by The scattering cross-section is The function π e i(ka cos φ+mφ dφ = ( i m J m(ka H ( m (ka E, (6.65 = E ( i m J m(ka H m ( (ka H( m (kρe imφ. (6.66 m S ρ = E J m (ka Z πkρ H m ( (ka m π P = ρ S ρ dφ = E 4 J m (ka Z k H m ( (ka m m σ = 4 J m (ka k H m ( (ka f(x = J m (x x H m ( (x m., (6.67. (6.68, (6.69 is potted beow. In the short waveength regime it approaches unity and the scattering cross section is σ/ = 4a as expected from the geometrica optics approximation. In the ong waveength regime ka, m = mode is dominant and the cross-section diverges in a manner σ π k [ n ( ka ], + γ E 4

42 where γ E =.577 is the Euer s constant. Note that in the imit x, J (x and H ( (x i [ ( x ] n + γ π E, x. Physicay, the symmetric mode (m = corresponds to radiation by a ong cyindrica antenna having a ength much arger than the waveength. Fig.6-7 shows σ/ normaized by 4a as a function of x = ka. x m= J m (x J m (x + jy m (x ka Figure 6-7: Scattering cross-section per unit ength of a ong conducting cyinder of radius a. σ/4a as a fucntion of x = ka. The eectric fied is poarized aong the cyinder. If the incident eectric fied is perpendicuar to the cyinder axis, it is more convenient to use the magnetic fied which is axia. The incident magnet fied is and we assume the scattered magnetic fied in the form B sc z (r = m B e ikρ cos φ e z, (6.7 b m H ( m (kρe imφ. (6.7 Corresponding eectric fied can be found from the Maxwe s equation, E sc c t = B sc. (6.7 4

43 The φ component of the scattered eectric fied is thus E sc φ (ρ, φ = ic m b m k ρ [H( m (kρ]e imφ, (6.73 whie the φ component of the incident eectric fied is E i φ (ρ, φ = cb cos φe ikρ cos φ. (6.74 The boundary condition of vanishing tangentia component of the eectric fied at the cyinder surface yieds B cos φe ika cos φ i m Mutipying by e im φ and integrating over φ, we obtain b m k b m = B ( i m [J m(ka] a [H( m (ka]e imφ =. (6.75 [H ( m (ka], (6.76 where the prime means differentiation with respect to the argument ka and the foowing transformation is used, π A resutant scattering cross-section is cos φe i(ka cos φ+mφ dφ = i( i m π[j m (ka]. σ = 4 [J m (ka] k [H m ( (ka] m. (6.77 Fig.6-8 shows σ/ normaized by 4a. In contrast to the preceding case E i = E e z, the cross-section in the ong waveength regime ka is bounded and given since if ka, the dominant harmonics are σ 3π 4 k3 a 4, (6.78 b = iπb 4 (ka, b ± = ± πb 4 (ka. (6.79 In short waveength regime ka, the cross-section approaches 4a as in the case of axia poarization. 43