Electromagnetism in the Spherical-Wave Basis:

Transcription

1 Eectromagnetism in the Spherica-Wave Basis: A (Somewhat Random) Compendium of Reference Formuas Homer Reid August 1, 216 Abstract This memo consoidates and coects for reference a somewhat random hodgepodge of formuas and resuts in the spherica-wave approach to eectromagnetism that I have found usefu over the years in deveoping and testing scuff-em and buff-em. Contents 1 Vector Spherica Wave Soutions to Maxwe s Equations 3 2 Expicit expression for sma 6 3 Transation matrices 9 4 Spherica-wave expansion of incident fieds Pane waves Point sources at the origin Point sources not at the origin Scattering from a homogeneous dieectric sphere Sources outside the sphere Anaytica resuts in the ow-frequency imit Sources inside the sphere Frequencies of spherica resonant cavities Scattering from a sphere with impedance boundary conditions 17 7 Dyadic Green s functions 18 8 VSWVIE: Voume-integra-equation approach to scattering with vector spherica waves as voume-current basis functions Review of the standard VIE formaism Vector spherica waves as voume-current basis functions Soution of VIE equation in VSW basis

2 Homer Reid: E&M in the Spherica-Wave Basis Overap integras Fieds produced by VSW basis functions Matrix eements of G Aternative expression for tota fieds inside the body Homogeneous dieectric sphere Stress-tensor approach to power, force, and torque computation Sampe stress-tensor power cacuation Sampe stress-tensor force cacuation x-directed force density Tota x-directed force T-matrix eements and surface currents 34

3 Homer Reid: E&M in the Spherica-Wave Basis 3 1 Vector Spherica Wave Soutions to Maxwe s Equations Many authors define pairs of three-vector-vaued functions {M m (x), N m (x) describing exact soutions of the source-free Maxwe s equations namey, the vector Hemhotz equation pus the divergence-free condition in spherica coordinates for a homogeneous medium with wavenumber k, i.e. k 2 { M m N m =, { Mm N m =. (1) In some cases, the set {M, N m is augmented to incude a third function {L m that satisfies the vector Hemhotz equation but is now cur-free (and not divergenceess): k 2 L m =, L m =. (2) The function L m is not a soution of Maxwe s equations and is never needed in a basis of expansion functions for fieds, but must be retained in a basis for expanding currents in inhomogeneous and/or anisotropic media. In a cases, the {M, N, L functions invove spherica Besse functions and spherica harmonics, but the precise definitions (incuding sign conventions and normaization factors) vary from author to author. In this section I set down the particuar conventions that I use. In the next section I give expicit cosed-form expressions for sma. Vector spherica harmonics X m (θ, ϕ) i { Y m (θ, ϕ)r ( + 1) Z m (θ, ϕ) ˆr X m (θ, ϕ) More expicity, the components of X and Z are i im X m (θ, φ) = ( + 1) sin θ Y θ m Y m θ ϕ Z m (θ, φ) = i Ym ( + 1) θ θ + im sin θ Y m ϕ These are orthonorma in the sense that X X = Z Z = 1, X Z =. where the inner product is F G F G dω = π 2π F (θ, ϕ) G(θ, ϕ) sin θ dϕ dθ

4 Homer Reid: E&M in the Spherica-Wave Basis 4 Their divergences are: X m = m cot θ csc θy m(θ, ϕ) r ( + 1) Z m = i cot θ r m cot θy m (θ, ϕ) + ξ m e iϕ Y,m+1 (θ, ϕ) ( + 1) ( m)!( + m + 1)! ξ m ( m 1)!( + m)! (3a) (3b) (3c) Radia functions R outgoing (kr) h (1) (kr) (kr) h (2) (kr) R reguar (kr) j (kr). R incoming I aso define the shorthand symbos R (kr) 1 kr R (kr) + R (kr) ( + 1) \R (kr) R (kr) kr where R (kr) = d dz R (z) z=kr. Scaar Hemhotz soutions 2 + k 2 ψ m (r) = = ψ m (r, θ, ϕ) = R (kr)y m (θ, ϕ) where R is one of the radia functions defined above. Vector spherica wave functions M m (k; r) i { ψ m r ( + 1) = R (kr)x m (Ω) N m (k; r) 1 ik M m = ir (kr)z m (Ω) + \R (kr)y m (Ω)ˆr L m (k; r) 1 k ( + 1) ψ m = i R (kr) 1 Z m (Ω) + R kr (kr)y m (Ω)ˆr ( + 1) (4) L (k; r) = R (kr) ˆr 4π Cur Identities M = ikn, N = +ikm.

5 Homer Reid: E&M in the Spherica-Wave Basis 5 Genera soution of source-free Maxwe equations The genera soution of Maxwe s equations in a source-free medium with reative materia properties ɛ r, µ r then reads { E(x) = A M (k; r) + B N (k; r) (5a) H(x) = 1 { Z Z r B M (k; r) A N (k; r) (5b) where k = ɛ ɛ r µ µ r ω is the photon wavenumber in the medium, Z = µ /ɛ 377 Ω is the impedance of vacuum, Z r = µ r /ɛ r is the reative wave impedance of the medium, and we must choose the M, N functions to be reguar, incoming, or outgoing depending on the physica conditions of the probem. Unified notation for M, N waves I wi use the symbo W to refer coectivey to M and N waves; here 1 = (mp ) is a compound index with P = {M, N identifying the poarization. With this notation, expansions such as (5) read E = C W, H = 1 Z Z r C σ W (6) { = m= { P {M,N and where the bar on a compound index fips the poarization: mm = mn, mn = mm, σ mm = +1, σ mn = 1. Spherica-wave expansion of dyadic Green s function Let G(k; r) and C(k; r) be the usua homogeneous dyadic Green s functions, with Cartesian components G ij (k; r) = (δ ij + 1 ) e ik r k 2 i j 4π r, C ij(k, r) = + 1 ik ε ij G(k, r) (7) Then have the spherica-wave expansion G(x, x ) = 1 k 2 δ(x x )ˆrˆr + ik W out (x > )W reg (x < ), (8a) C(x, x ) = ik { W out (x)w reg (x ), x > x, σ W out (x )W reg (x), x < x (8b) 1 With this notation I am committing the faux pas of using the same symbo for the two-fod compound index (m) in (5) and the three-fod compound index (mp ) in (6), but whaddya gonna do.

6 Homer Reid: E&M in the Spherica-Wave Basis 6 2 Expicit expression for sma The first few radia functions R reguar (x) = sin x x R outgoing (x) = i eix x R reguar 1 (x) = sin x x cos x x 2 R outgoing 1 (x) = ieix x 2 ( 1 ix ) The first few reguar functions sinusoida functions: R reguar (x) = cos x x R outgoing (x) = eix x R reguar 1 (x) = x cos x + (x2 1) sin x x 3 R outgoing 1 (x) = ieix x 3 ( 1 ix x 2 ) In what foows, the ζ n are dimensioness ζ 1 (x) = sin x x cos x ζ 2 (x) = (1 x 2 ) sin x x cos x

7 Homer Reid: E&M in the Spherica-Wave Basis 7 L reguar (r) = ζ 1(kr) 4π(kr) 2 k kr 6 π M reguar 3 1,±1 (r) = 16π k M reguar 3 1, (r) = i 2π 1 1 ζ1 (kr) (kr) 2 kr 4 3π e±iϕ k 3 i 8π N reguar 3 1 1,±1 (r) = 16π (kr) 3 k e ±iϕ 12π N reguar 3 1 1, (r) = 8π (kr) 3 k 1 6π e ±iϕ 1 ±i cos θ 1 ±i cos θ ( ) sin kr kr cos kr (kr) 2 ζ(kr) (kr) 2 e ±iϕ ± sin θ ± cos θ i cos θ sin θ sin θ sin θ ±2ζ 1(kr) sin θ ζ 2 (kr) cos θ iζ 2 (kr) 2ζ 1 (kr) cos θ ζ 2 (kr) sin θ = 1 6π ẑ

8 Homer Reid: E&M in the Spherica-Wave Basis 8 The first few outgoing functions poynomia factors: In what foows, the Q n are dimensioness Q 1 (x) = 1 x Q 2a (x) = 1 x + x 2 Q 2b (x) = 3 3x + x 2 ( ) M outgoing 3 e ikr 1,±1 (r) = 16π k 2 r 2 ( ) M outgoing 3 e ikr 1, (r) = 8π k 2 r 2 ( ) N outgoing 3 e ikr 1,±1 (r) = 16π k 3 r 3 Q 3 (x) = 6 6x + 3x 2 x 3 iq 1 (ikr) ±Q 1 (ikr) cos θ Q 1 (ikr) sin θ 2(ikr)Q 1(ikr) sin θ ±iq 2a (ikr) cos θ Q 2a (ikr) e ±iφ e ±iφ ( ) N outgoing 3 e ikr 1, (r) = 2iQ 1(ikr) cos θ +iq 8π k 3 r 3 2a (ikr) sin θ ( ) M outgoing 5 e ikr 2,±2 (r) = 16π k 3 r 3 ( ) M outgoing 5 e ikr 2,±1 (r) = 16π k 3 r 3 ( ) M outgoing 15 e ikr 2, (r) = 8π k 3 r 3 ( ) N outgoing 5 e ikr 2,±2 (r) = 16π k 4 r 4 ( ) N outgoing 5 e ikr 2,±1 (r) = 16π k 4 r 4 e ±2iφ e ±iφ ±iq 2b (ikr) sin θ Q 2b (ikr) cos θ sin θ iq 2b (ikr) cos θ ±Q 2b (ikr) cos 2θ Q 2b (ikr) cos θ sin θ e ±2iφ e ±iφ 3iQ 2b(ikr) sin 2 θ iq 3 (ikr) cos θ sin θ ±Q 3 (ikr) sin θ 3iQ 2b(ikr) sin 2θ ±iq 3 (ikr) cos 2θ Q 3 (ikr) cos θ ( ) N outgoing 15 e ikr 2, (r) = iq 2b(ikr)(3 cos 2 θ 1) iq 8π k 4 r 4 3 (ikr) cos θ sin θ.

9 Homer Reid: E&M in the Spherica-Wave Basis 9 3 Transation matrices Transation matrices arise when we want to evauate the fieds produced by sources not ocated at the origin. Scaar case Athough we don t need it for eectromagnetism probems, the scaar-wave anaog of (4) is or, more specificay, ψ m (x) = R (kr)y m (θ, φ) ψm out (x) = R out (kr)y m (θ, φ), ψ reg m (x) = Rreg (kr)y m (θ, φ) Now consider a point source at x S whose fieds we wish to evauate at an evauation ( destination ) point x D, using a basis of spherica waves centered at an origin x O. Then waves emitted by the source, which appear to be outgoing in a coordinate system centered at x S, can be described as superpositions of reguar waves in a coordinate system centered at x O : ( xd x S) = ψ out β ( A β k; xs x O) ψ reg ( β xd x O) (9) where, β are compound indices (i.e. = {, m ) and A β (k, L) = 4π γ a βγ = i ( β+ γ) a γβ ψγ out (L) Y (Ω)Y β (Ω)Y γ (Ω) dω = ( 1) m (2 + 1)(2 β + 1)(2 γ + 1) 4π ( β γ ) ( β γ m m β m γ ). Vector case ( M N ) out ) = β ( B C ( M N ) reg C B β β B β (k, L) = 4π γ i ( β+ γ) k C β (k, L) = ( + 1) β ( β + 1) λ ± = ( + 1) + β ( β + 1) γ ( γ + 1) a γβ ψγ out (L) 2 ( + 1) β ( β + 1) ( ) Lx il y A+,β + λ ( ) Lx + il y A,β + m L z A,β 2 2 λ+ ( β )( ± β + 1), ± = {, m ± 1

10 Homer Reid: E&M in the Spherica-Wave Basis 1 4 Spherica-wave expansion of incident fieds 4.1 Pane waves For a scattering probem in which the incident fied is a z-directed pane wave, i.e. E inc = E e ikz, H inc = 1 ẑ E e ikz Z the spherica-wave expansion coefficients in (14) take the foowing forms for various possibe poarizations: E = ˆx + iŷ (right circuar poarization) : P m = δ m,+1 P E = ˆx iŷ (eft circuar poarization) : P m = δ m, 1 P ) E = ˆx (inear poarization) : P m = 1 2 (δ m,+1 + δ m, 1 P where in a cases I have P = i 4π(2 + 1), Q,±1 = ip. 4.2 Point sources at the origin Let E(x; p) be the eectric fied at evauation point x due to an eectric dipoe p at the origin. The spherica-wave expansion of this fied invoves ony N- functions with = 1, i.e. E(x; p) = where the ξ coefficients are 1 m= 1 ξ 1m (p)n outgoing 1m (x) p = p xˆx ξ 1,1 = ξ 1, 1 = i p x 2 3π ɛ, ξ 1, = p = p y ŷ ξ 1,1 = +ξ 1, 1 = 1 p y 2 3π ɛ k 3 k 3 ξ 1, = p = p z ẑ ξ 1,1 = ξ 1, 1 =, ξ 1, = ik3 6π p z ɛ Here ɛ = ɛ ɛ r is the absoute permittivity of the medium. Simiary, the magnetic fieds of a magnetic dipoe m at the origin are H(x; m) = ξ (m)n outgoing (x) where the ξ coefficients are the same as the ξ coefficients above with the repacement p ɛ m µ.

11 Homer Reid: E&M in the Spherica-Wave Basis Point sources not at the origin The fieds of point sources not at the origin may be obtained by appying the transation matrices of Section to the fieds of Section 4.2. If the point source ies at x S (here S stands for source ), then its fieds at destination point x D read { E(x D ; x S, p) = β H(x D ; x S, p) = 1 { Z β E(x D ; x S, m) = Z β H(x D ; x S, m) = { β ξ C β M reguar β (x D ) + ξ B β N reguar β (x D ) ξ C β N reguar β (x D ) + ξ B β M reguar β (x D ) { ξ B β M reguar β (x D ) + ξ C β N reguar β (x D ) ξ C β M reguar β (x D ) + ξ B β N reguar β (x D ) (1a) (1b) (1c) (1d) Here {B, C β are eements of the transation matrices B(k, x S ), C(k, x S ). Note that, for a given source point x S, I ony have to assembe the transation matrices B and C once (at a given frequency), after which I can get the fieds at any number of destination points x D from equation (1).

12 Homer Reid: E&M in the Spherica-Wave Basis 12 5 Scattering from a homogeneous dieectric sphere I consider scattering from a singe homogeneous sphere with reative permittivity and permeabiity ɛ r, µ r in vacuum irradiated by spherica waves emanating from within our outside the sphere. Irrespective of the origin of the incident fieds, the scattered fieds inside and outside the sphere take the form (n = ɛ r µ r, Z r = µ r /ɛ r ) Inside the sphere: E scat (x) = { A M reg (nk ; r) + B N reg (nk ; r) H scat (x) = 1 Z Z r { B M reg (nk ; r) A N reg (nk ; r) (11a) (11b) Outside the sphere: E scat (x) = { C M out (k ; r) + D N out (k ; r) H scat (x) = 1 { D M out (k ; r) C N out (k ; r) Z (12a) (12b) The {A, B, C, D coefficients are proportiona to the spherica-wave expansion coefficients of the incident fieds, with the proportionaity constants determined by enforcing continuity of the tangentia components of the tota fieds {E, H tot = {E, H inc + {E, H scat at r = r, ˆr E tot r r = + ˆr H tot r r = + ˆr E tot r r ˆr H tot r r (13a) (13b) 5.1 Sources outside the sphere If the sources of the incident fied ie outside the sphere (the usua Mie scattering probem), then I can expand the incident fieds in the form E inc (x) = { P M reguar (k ; x) + Q N reguar (k ; x) (14a) H inc (x) = 1 { Q M reguar (k ; x) P N reguar (k ; x). (14b) Z

13 Homer Reid: E&M in the Spherica-Wave Basis 13 Matching tangentia fieds at the sphere surface then determines the scatteredfied expansion coefficients in terms of the incident-fied expansion coefficients (a = k r ): R reg (a)r out (a) R reg (a)r out (a) A = R reg (na)r out (a) 1 Z R reg P (15a) (na)r out (a) r R reg (a)r out (a) R reg (a)r out (a) B = R reg (na)r out (a) 1 Z r R reg (na)r out (a) R reg (a)r reg (na) Z r R reg (a)r reg (na) C = Z r R reg (na)r out (a) R reg (na)r out (a) {{ T M R reg (a)r reg (na) Z r R reg (a)r reg (na) D = Z r R reg (na)r out (a) R reg (na)r out (a) {{ T N Q P Q (15b) (15c) (15d) In (15c,d) I have identified the quantities C /P and D /Q as eements of the T-matrix for the M- and N- poarizations Anaytica resuts in the ow-frequency imit The coefficients (15) may be expressed in cosed form, e.g. A 1 2a 3 e ia n 3 Z = P 1 ((1 ia) (a 2 n 2 1) ( 1 + a(a + i))nz) sin(an) + an(( 1 + a(a + i))nz ia + 1) cos(an) B 1 2a 3 e ia n 3 Z = Q 1 ((1 ia)z (a 2 n 2 1) ( 1 + a(a + i))n) sin(an) + an(( 1 + a(a + i))n iaz + Z) cos(an) where a = (k r ) is the dimensioness Mie size parameter. The ow-frequency imiting forms (assuming µ = 1:) are A 1 P 1 = 2 ɛ + A 2 P 2 = 2 ɛ + (ɛ 1) 3 ɛ (ɛ 1) 5ɛ a 2 + O(a 3 ) a 2 + O(a 3 ) B 1 Q 1 = B 2 Q 2 = 6 ( 3 ɛ 2 ɛ ɛ 1 ) 5(ɛ + 2) 2 a 2 + O(a 3 ) 1 ( 5a 2 2ɛ 2 + 5ɛ 7 ) + ɛ(2ɛ + 3) 7 a 2 + O(a 3 ) ɛ(2ɛ + 3) 2 2 The T-matrix mutipies a vector of reguar-wave incident-fied coefficients to yied a vector of outgoing-wave scattered-fied coefficients. If, instead of the reguar-wave incident fied (14), I irradiated the sphere with a superposition of incoming waves as the incident fied, then the resuting modified versions of equations (15c,d) woud instead define eements of the S-matrix (scattering matrix).

14 Homer Reid: E&M in the Spherica-Wave Basis 14 Interior fieds For a sphere irradiated by a ineary-poarized pane wave, the fieds inside the body to second order in a = kr read E x = 3 ɛ + 4 (ɛ 1)(35ɛ + 46) E 2 + ɛ + ikz ɛ 5(ɛ + 2) 2 k 2 x 2 (3ɛ + 4) E y = E E z E = H x Z E = H y Z E = 1 + H z Z E = (ɛ 1)( 2ɛ ɛ + 42) + 5(ɛ + 2) 2 (3ɛ + 4) 2(ɛ 2 1) k 2 xy 5(2 + ɛ)(4 + 3ɛ) ɛ 1 ikx + 2ɛ + 3 (ɛ 1) 2 k 2 xy 5(2ɛ + 3) 2ɛ + 1 ikz 2 + ɛ (ɛ 1)(ɛ 6) 15(3 + 2ɛ) ɛ 1 ɛ + 2 Fied derivatives: iky + k 2 y 2 (ɛ 1)(7ɛ + 12) 5(2 + ɛ)(4 + 3ɛ) k 2 x 2 + (ɛ 1)(ɛ + 4) 5(3 + 2ɛ) ɛ 1 k 2 xz 15 k 2 yz 14ɛ 3 + 3ɛ ɛ (ɛ + 2) 2 (3ɛ + 4) k 2 y 2 z E = ( ikc 1 2k 2 C 2 z )ˆx + k 2 C 3 x ẑ z H = ( ikc 4 2k 2 C 5 z ) ŷ + k 2 C 6 y ẑ 2ɛ ɛ + 27 k 2 z 2 3(3 + 2ɛ) k 2 z 2 C 1 = ɛ ɛ, C 2 = 14ɛ3 + 3ɛ ɛ (ɛ + 2) 2 (3ɛ + 4) (ɛ 1)(7ɛ + 12) C 3 = 5(2 + ɛ)(4 + 3ɛ) C 4 = 2ɛ ɛ C 5 = 2ɛ2 + 46ɛ (3 + 2ɛ) (ɛ 1)(ɛ + 4) C 6 = 5(3 + 2ɛ)

15 Homer Reid: E&M in the Spherica-Wave Basis Sources inside the sphere If the sources of the incident fied ie inside the sphere, then I can expand the incident fied in the form E inc (x) = { P M out (nk ; x) + Q N out (nk ; x) (16) The tota fieds inside and outside then read E in (x) = { P M out (x) + Q N out (x) + { A M reg (x) + B N reg (x) H in (x) = 1 { Z Z r P N out (x) Q M out (x) 1 { Z Z r C N reg (x) D M reg (x) E out (x) = { C M out (x) + D N out (x) H out (x) = 1 { C N out (x) D M out (x) Z (17) (18) (19) (2) Now equate tangentia components of E in,out and H in,out at the sphere surface (r = r ), take inner products with M and N, and use the orthogonaity reations to find R out (nkr )P + R reg (nkr )A = R out (kr )C R out (nkr )P + R reg (nkr )A = Z r R out (kr )C R out (nkr )Q + R reg (nkr )B = R out (kr )D R out (nkr )Q + R reg (nkr )B = Z r R out (kr )D which we sove to obtain the coefficients of the scattered fied outside the sphere in terms of the incident-fied coefficients: R out (na)r reg (na) R out (na)r reg (na) C = R out (a)r reg (na) Z r R out P (a)r reg (21a) (na) R out (na)r reg (na) R out (na)r reg (na) D = (a)r reg (na) Z r R out (a)r reg Q. (21b) (na) R out 5.3 Frequencies of spherica resonant cavities Frequencies at which the denominator of one of the scattering coefficients (15) vanishes correspond to cavity resonances, in which the fieds inside the sphere

16 Homer Reid: E&M in the Spherica-Wave Basis 16 can be nonzero even for vanishing incident-fied coefficients {P, Q. For a sphere of given frequency-dependent refractive index n(ω) = ɛ(ω), the vaues of a = ωr c at which resonances occur may be abeed by a wave type (M or N) and a pair of integers {, p, where a M,p and an,p (p = 1, 2, ) are the pth smaest-magnitude roots of the equations (assuming here the non-magnetic case µ = 1) R reg nr reg ( na M n ) R out ( na N n ) R out ( a M n ) nr reg ( a N n ) R reg ( na M n ) R out ( na N n ) R out ( a M n ) = ( a N n ) =. This can be soved numericay using the mathematica code shown beow, with resuts tabuated in Tabe 1 for the particuar case of a ossess dieectric sphere with frequency-independent reative permittivity ɛ 4. ROutL_, a_:=sphericahankeh1l,a; RBarOutL_, a_ := ROutL,a/a + (DROutL,aa,aa /. {aa->a); RRegL_, a_:=sphericabessejl,a RBarRegL_, a_ := RRegL,a/a + (DRRegL,aa,aa /. {aa->a); amequationl_,n_,a_:=\ (RRegL,n*a*RBarOutL,a - n*rbarregl,n*a*routl,a); anequationl_,n_,a_:=\ (RBarRegL,n*a*ROutL,a - n*rregl,n*a*rbaroutl,a); aequationmn_,l_,n_,a_:=\ IfMN==, amequationl,n,a, anequationl,n,a; (* find an M- or N-type root near a *) anearmn_,l_,n_,a_:=\ FindRoot aequationmn,l,n,a==, {a,a, AccuracyGoa->1, PrecisionGoa->1, WorkingPrecision->2; Figure 1: mathematica code for computing resonant frequencies of a spherica dieectric cavity.

17 Homer Reid: E&M in the Spherica-Wave Basis 17 a M 1, i a M 1, i a N 1, i a N 1, i a M 2, i a M 2, i a N 2, i a N 2, i Tabe 1: Lowest resonant frequencies a = ωr c for a spherica dieectric cavity of (ossess, frequency-independent) reative permittivity ɛ = 4. 6 Scattering from a sphere with impedance boundary conditions For a sphere characterized by a surface-impedance boundary condition with reative surface impedance 3 η, the continuity condition (13) is repaced by a reationship between the tangentia E and H fieds at the sphere surface: ) E = ηz (ˆr H at r = R. Equations (15c,d) for the T-matrix eements are repaced by R reg (a) R reg (a) C = iηr out P, D = (a) R out (a) iηr out (a) + R out Q, (22) (a) {{{{ T M T N In particuar, taking η yieds the T-matrix eements for a perfecty eectricay conducting (PEC) sphere. 3 Note that η is dimensioness; the absoute surface impedance is ηz where Z 377 Ω is the impedance of vacuum.

18 Homer Reid: E&M in the Spherica-Wave Basis 18 7 Dyadic Green s functions The scattering part of the eectric dyadic Green s function G EE (x D, x S ) is a 3 3 matrix whose i, j component Gij EE(xD, x S ) is the (appropriatey normaized) 4 i component of the scattered eectric fied at x D due to a j-directed point eectric dipoe source at x S. (The superscripts on x stand for destination and source ). If I take the eectic-dipoe fieds Section 4.3 equations (1a,b) to be the incident fieds in the externay-sourced scattering probem of Section 5.1 so that, for exampe, the coefficient of M reg in the incident-fied expansion (14) is P = β ξ βc β, then I need ony mutipy by T-matrix eements equation (15) to get the outgoing-wave coefficients in the scattered-fied expansion (12). Thus the E- and H-fieds at x D due to an eectric dipoe source p at x S are { E scat (x D ; x S, p) = ξ (p) C β (x S )T M β M out β (x D ) + B β (x S )T N βn out β (x D ) H scat (x D ; x S, p) = 1 Z β β β { ξ (p) + C β (x S )T M γβm out β (x D ) + B β (x S )T N γβn out β (x D ) The E- and H-fieds at x D due to a magnetic dipoe source m at x S are E scat (x D ; x S, m) = Z { ξ (m) + C β (x S )T M β M out β (x D ) + B β (x S )T N βn out β (x D ) β { H scat (x D ; x S, m) = ξ (m) B β (x S )T N βm out β (x D ) + C β (x S )T M β N out β (x D ). In writing out these equations, I have used the fact that the T-matrix of a homogeneous sphere is diagona. However, simiar equations coud be written down for the DGFs of any arbitrary-shaped object; in this case the T-matrix woud not be diagona and the doube sums woud become tripe sums, but such a representation might nonetheess be usefu in some cases. 4 The normaization just invoves dividing by dimensionfu prefactors to ensure that the components of G have units of inverse ength and are independent of the point-source magnitude.

19 Homer Reid: E&M in the Spherica-Wave Basis 19 8 VSWVIE: Voume-integra-equation approach to scattering with vector spherica waves as voume-current basis functions 8.1 Review of the standard VIE formaism Consider a materia body with spatiay-varying reative permittivity tensor ɛ(x) iuminated by incident radiation with eectric fied E inc (x) at frequency ω = kc in vacuum. In the usua voume-integra-equation formuation of the scattering probem, the scattered fied E scat (x) is understood to arise from an induced voume current distribution J(x), which is itsef proportiona to the oca tota (incident+scattered) fied at each point: E scat = ikz G J, J = ik (ɛ 1) (E inc + E scat) Z Combining these yieds 1 + VG J(x) = i VE inc (23) kz where the diagona operator V(x, x ) k 2( ɛ 1 ) δ(x, x ) is the potentia. Another way to write this is V + G J(x) = i E inc (24) kz ( ) 1δ(x where (x, x ) 1 k ɛ 1 x ). 2 Upon approximating the induced current as an expansion in a finite set of vector-vaued basis functions, N BF V N BF J(x) j B (x) (25) =1 the integra equation (24) becomes an N BF N BF inear system: V + G j = i kz e where j is the vector of expansion coefficients in (25) and V e i E inc B, kz ab = 1 1 k 2 B (ɛ 1) Bβ G ab = 1 k 2 B G Bβ.

20 Homer Reid: E&M in the Spherica-Wave Basis Vector spherica waves as voume-current basis functions For a compact materia body confined within a sphere of radius R, I use the reguar vector spherica waves to define a basis of voume-current basis functions:, x > R B P m (x) = M reg N reg m (x), m (x), L reg m x R, P = M x R, P = N (x), x R, P = L. I use these basis functions to expand currents and fieds in (23): (26) J(x) = j B, E inc (x) = e B. (27) Note that the e coefficients have dimensions of eectric fied strength. 8.3 Soution of VIE equation in VSW basis Inserting (27) into (23) yieds a reation between the vectors of voume-current expansion coefficients and incident-fied expansion coefficients: j = i 1Ve S + VG (28) kz where the eements of the SV, G matrices are R S β = B (x)b β (x) dx, V β = k 2 x <R G β = R R B (x)g(x, x )B β (x ) dx dx B (x) ɛ(x) 1 B β (x) dx, and e is the vector of incident-fied projections onto the {B basis. If e is expressed as an expansion in the usua (non-normaized) reguar VSWs, as in equation (16), then the entries of e are just the P, Q coefficients in that equation: 8.4 Overap integras e M m = P m, e N m = Q m. The overap matrix is diagona in the and m indices and independent of m: S P m;p m S P P δ, δ m,m It is aso partiay diagona in the P index, in the sense that the M functions are orthogona to the N and L functions, but the N and L functions have nonvanishing overap.

21 Homer Reid: E&M in the Spherica-Wave Basis 21 The overap integras are S P P (R) R R P P (r) dr where R P P (r) r 2 Expicit forms of the integrand function are B P m(r, Ω)B P m(r, Ω) dω. R MM r 2 j (kr) 2 R NN ( + 1) j k 2 (kr) 2 r + 2 k j (kr)j (kr) j k 2 (kr) 2 which may be simpified? to read ( ) ( ) + 1 R NN r 2 j (kr) + r 2 j (kr) R LL 1 k 2 j (kr) 2 + rj (kr)2 ( + 1) R NL 2r k j (kr)j (kr) 1 k 2 j2 (u) Expicit forms for the overap integras are S MM (R) = j2 (kr) j 1(kR)j +1 (kr) (29a) { 2 1 S NN (R) = ( + 1) j 1 (kr) 2 ( + 1)j (kr)j 2 (kr) 2(2 + 1) j (kr)j +2 (kr) + j +1 (kr) 2 (29b) I aso need another type of overap integra: Ŝ P P (R) R P P (r) dr where RP P R B P m(r, Ω) B P m(r, Ω) dω. The R integrands are simiar to the R integrands given above, except that one of the two j factors in each term is repaced by h (1) : R MM j (kr)h (1) (kr) R NN r 2 j (kr)h (1) ( + 1) (kr) + k 2 j (kr)h (1) (kr) R LN r k j (kr)h (1) (kr) r k j (kr)h (1) (kr)

22 Homer Reid: E&M in the Spherica-Wave Basis Fieds produced by VSW basis functions I wi use the symbo E (x) to denote 1/(iωɛ ) times the E-fied at x due to a current distribution produced by basis function B popuated with unit strength. The quantity E has dimensions of ength 2. Evauation using dyadic Green s function E (x) G(x, x )B (x ) dx x <R The integra here may be easiy evauated using the standard eigenexpansion of G?? : G(x, x ) = ˆrˆr k 2 δ(x x ) + ik The resut is: Evauation point outside sphere ( x > R): { M (r > )M (r < ) + N (r > )N (r < ) E Mm (r) = iks MM (R) M(r) E Nm (r) = iks NN (R) N(r) E Lm (r) = iks NL (R) N(r) Evauation point inside sphere ( x < R): E Mm (r) = iks MM (r) M(r) + ikŝmm(r, R)M(r) (3a) E Nm (r) = 1 k 2 ˆr N m ˆr + iks NN (r) N(r) + ikŝnn(r, R)N(r) (3b) E Lm (r) = 1 k 2 ˆr L m ˆr + iks NL (r) N(r) + ikŝnl(r, R)N(r) (3c) Evauation using scaar Green s function I can aso evauate the fieds using the scaar Green s function, whose eigenexpansion reads G (r, r ) = ik R reg (kr < )R out (kr > )Y (θ, φ)y (θ, φ )

23 Homer Reid: E&M in the Spherica-Wave Basis 23 In genera, the E-fied produced by a current distribution J reads E(x) = G (x x )J(x ) dv + 1 V k {{ 2 x G (x x ) J(x ) dv V {{ E 1 E 2 1 k 2 x G (x x )J r (x )da V {{ E 3 The third integra here captures the effect of the surface charge ayer due to the discontinuous dropoff of the currents at the sphere surface. In evauating E 1 integras I need to use the 3x3 matrices that convert spherica vector components to cartesian vector components and vice versa: V x V y V z V r V θ V ϕ = = sin θ cos ϕ cos θ cos ϕ sin ϕ sin θ sin ϕ cos θ sin ϕ cos ϕ cos θ sin θ {{ Λ S2C (θ,ϕ) sin θ cos ϕ sin θ sin ϕ cos θ cos θ cos ϕ cos θ sin ϕ sin θ sin ϕ cos ϕ {{ Λ C2S (θ,ϕ) Reevant integras incude: Y (θ, φ)λ C2S (θ, φ) {Y β (θ, φ)ˆr dω = I then find: E 1 Mm(x) 8.6 Matrix eements of G E 2 Mm(x) = because M E 3 Mm(x) = because M r = G P P B P m G BP m B P m(x) E P (x) dx x <R V r V θ V ϕ V x V y V z

24 Homer Reid: E&M in the Spherica-Wave Basis 24 Using equation (3), one finds G MM = 2ik = 2ik R R r R R G NN = 1 k 2 r x <R R ( + 1) = k 4 G LN = + 1 k 3 R + ik R MM (r) R MM (r ) dr dr j (kr) 2 j (kr )h (1) (kr ) r 2 r 2 dr dr (31a) Nmr (x) R 2 dx + 2ik R R 1 G LL = ( + 1)k 2 j (kr) 2 dr + 2ik R R rj (kr)j (kr) dr + ik r R R r R r R R NN (r)r LN (r ) dr dr rj (kr) 2 dr + 2ik R r R NN (r) R NN (r ) dr dr R NN (r) R NN (r ) dr dr R NN (r) R LN (r ) dr dr R r (31b) (31c) R LN (r) R LN (r ) dr dr (31d) The convenient thing about the VSW basis is that the G matrix is diagona (and m-independent) in this basis, with eements that may be computed in cosed form: G P m;p m = δ P P δ δ mm δ P N 3k 2 + ig P R G P = 2k S P (R) 2k S M (R) = 2k S N (R) S P (r)r P (r) r 2 dr R R r 2 j (kr) 2 dr, P = M r 2 j (kr) 2 + \j (kr) 2 dr, P = N. In the = 1 sector one finds k 2 G M1m,M1m = i cos(2a) + 2a 2 + a sin(2a) 4a = i 45 a5 i 315 a7 + O ( a 9) k 2 G N1m,N1m = i 4a 3 2 ( a 4 a 2 1 ) a ( a 2 4 ) sin(2a) 2 ( a 2 1 ) cos(2a) = i 9 a3 2i 45 a5 + O(a 7 )

25 Homer Reid: E&M in the Spherica-Wave Basis Aternative expression for tota fieds inside the body E tot (x) = ikz V 1 J(x) = ikz V 1 j B (x). 8.8 Homogeneous dieectric sphere For a homogeneous dieectric sphere irradiated by a incident fied of the form (16), the soution of (28) is ( ) i j = kz k 2 (ɛ 1) 1 k 2 (ɛ 1)G 1 N {P Q Tota fieds inside: E tot (x) = 1 1 k 2 (ɛ 1)G {P, Q {M, N reg (x) Scattered fieds outside: E scat (x) = ik 3 (ɛ 1) 1 k 2 S {P, Q {M, N out (x) (ɛ 1)G Mode-matching soution From the discussion of Section 5.1 with P regions are = 1, the tota fieds in the two E tot = VSWVIE soution { A M reg (nk ; r), r < R M reg (k ; r) + C M out (k ; r), r > R Now I sove the same probem using the VSWVIE formaism of equation (28). The RHS vector e in (28) contains a singe nonzero entry: e = S For a homogeneous sphere, the V matrix is proportiona to the identity matrix, V = k 2 (ɛ 1) 1, and since G is aso aways diagona in the VSW basis it foows that the entirety of equation (28) is diagona, so we have ony a singe nonzero voume-current coefficient, S j = ik (ɛ 1) Z 1 k 2 (ɛ 1)G

26 Homer Reid: E&M in the Spherica-Wave Basis 26 The scattered fied is E scat = ikz G J = ikz j G B k 2 (ɛ 1) = 1 k 2 C reg M reg (k ; r) + D out M out (k ; r) (ɛ 1)G 1 R C = 3k 2 + ik R out (kr )R reg (kr )r 2 dr, r < R r, r > R min(r,r) 2r D = ik R reg (kr ) 2 dr

27 Homer Reid: E&M in the Spherica-Wave Basis 27 9 Stress-tensor approach to power, force, and torque computation Power The power radiated away from (or, the negative of the power absorbed by) the sphere is obtained by integrating the outward-pointing normay-directed Poynting vector over any bounding surface containing the sphere. For convenience we wi take the bounding surface to be a sphere of radius r b > r (denote this sphere by S b ). Then the power is P = 1 2 S Re ˆr E (r) H(r) da b = r2 b 2 Re Ĥ (r b, Ω) ˆr E(r b, Ω) dω = r2 b E (H ˆr ) + H (ˆr E ) dω (32) 4 The integrand here may be expressed as a 6-dimensiona vector-matrix-vector product: ( ) ( ) ( ) = r2 b E P E dω (33) 4 H P H where, in our shorthand, ( E H with ) ( P P ) ( E H ) P = E r E θ E ϕ H r H θ H ϕ If we now insert the expansions (19) and (2) into (33), we obtain the tota radiated power as a biinear form in the {C, D m coefficients: Force The ith Cartesian component of the time-average force experienced by the sphere is obtained by integrating the time-average Maxwe stress tensor over a sphere with radius r b > r (ca this sphere S b :) F x = 1 2 Re r2 b T ij (r b, Ω)ˆn j (Ω)dΩ. (34) E r E θ E ϕ H r H θ H ϕ

28 Homer Reid: E&M in the Spherica-Wave Basis 28 For definiteness I wi consider the x-component of the force, i = x. The reevant quantity invoving the stress tensor is T xj n j = ɛ E 1 x 2 n x n y n z E y 1 2 n x E x E y + µ (E H) E z 1 2 n x E z where a fieds are to be evauated just outside the sphere surface. The timeaverage x-directed force per unit area is f x = 1 2 Re T xjn j = 1 T xj n j + (T xj n j ) 4 = ɛ E x E y n x n y n z n y n x 4 E z n z n x which we may write in the shorthand form E x E y E z + µ (E H) f x = ɛ 4 EC N x E C + µ 4 HC N x H C (35) where {E, H C are three-vectors of cartesian fied components (the superscript C stands for cartesian ) and the 3 3 matrix N x is N x = n x n y n z sin θ cos φ sin θ sin φ cos θ n y n x = sin θ sin φ sin θ cos φ n z n x cos θ sin θ cos φ (36) where the atter form is appropriate for points on the surface of a spherica bounding surface. On the other hand, the Cartesian and spherica components of the fied are reated by E x E y E z or, in shorthand, = sin θ cos φ cos θ cos φ sin φ sin θ sin φ cos θ sin φ cos φ cos θ sin θ E r E θ E φ (37) E C = Λ E S (38) where Λ is the 3 3 matrix in equation (38). Inserting (37) into (35) yieds f x = ɛ 4 ES F x E S + µ 4 HS F x H S (39) where {E, H S are 3-dimensiona vectors of spherica fied components and F x is a product of three matrices: F x = Λ N x Λ.

29 Homer Reid: E&M in the Spherica-Wave Basis 29 Working out the matrix mutipications, one finds sin θ cos φ cos θ cos φ sin φ F x = cos θ cos φ sin θ cos φ (4) sin φ sin θ cos φ and, proceeding simiary for the y- and z-directed force, sin θ sin φ cos θ sin φ cos φ F y = cos θ sin φ sin θ sin φ (41) cos φ sin θ sin φ cos θ sin θ F z = sin θ cos θ. (42) cos θ If I now insert expressions (19) and (2) into equation (39), I obtain the x- directed force per unit area as a biinear form in the C, D coefficients: { f x (x) = ɛ ( ) C 4 C β + DD β M (x)f x M β (x) + N (x)f x N β (x) β + ( C D β D C β ) M (x)f x N β (x) N (x)f x M β (x) (43) The tota x-directed force on the sphere is the surface integra of (43) over the fu sphere S b : F x = f x (x) dx S b { = ɛ ( ) C 4 C β + DD F Mβ F Nβ β M x + N x β + (C D β D C ) F Nβ F Mβ β M x N x (44) where the inner products invove integras over the radius-r b spherica bounding surface, i.e. F Mβ M x = rb 2 M (r b, Ω)F x M β (r b, Ω) dω. (45) 9.1 Sampe stress-tensor power cacuation As a specific exampe of a radiated-power computation, et s consider a pointike dipoe source at the center of a ossy sphere and ask for the tota power radiated away from the sphere.

30 Homer Reid: E&M in the Spherica-Wave Basis 3 Assuming the dipoe is z-directed, i.e. p = p z ẑ, the ony nonvanishing spherica mutipoe coefficient of the incident fied equation (16) is Q 1, = icz (nk ) 3 ɛ p z 6π where k = ω/c is the free-space wavenumber. The tota fieds outside the sphere are E tot (x) = D 1, N out 1, (x), H tot (x) = 1 Z D 1, M out 1, (x) where D 1, is given by (21): D 1, = 1 cz e ia n 3 a 6 p z 6π n 2 (a 3 + a + i) a i sin(na) + nain 2 ( 1 + a(a + i)) + a + i cos(na) (46) where the dimensioness size parameter is a = k R. The radiated power is P rad = 1 2 Re (E H) ˆr da = D 1, 2 ( ) r 2 Re N 2Z 1,(r, Ω) M 1, (r, Ω) ˆr dω {{ =1/(k r) 2 = D 1, 2 2k 2 Z (47) Sanity check As a sanity check, et s first try putting ɛ = 1. Then equation (46) reads D 1, (ɛ = 1) = icz k 3 p z 6π and equation (47) reads P rad = c2 Z k 4 12π p2 z in agreement with Jackson equation (9.24). Nontrivia exampes As ess trivia exampes, consider putting (a) ɛ = 3 and (b) ɛ = 3 + 6i.

31 Homer Reid: E&M in the Spherica-Wave Basis 31 1 Eps=3 Eps=3+6i PRad / PRad(Eps=1) Omega (SCUFF units) Figure 2: Power radiated by a dipoe at the center of a dieectric sphere, normaized by the power radiated by a dipoe in free space. 9.2 Sampe stress-tensor force cacuation The simpest incident-fied configuration that gives rise to a nonvanishing tota force on the sphere is a superposition of (1, ) and (2, 1) spherica waves, corresponding to coherent dipoe and quadrupoe sources at the origin. Thus, in the incident-fied expansion (16) we take P (1,) = P (2,1) = 1, P = for a other, Q = for a. (48) The coefficients in expansions (19, 2) for the fieds outside the sphere are then simiary given by C (1,) = C (2,1) = nonzero, C = for a other, D = for a. (49) The actua vaues of C (1,) and C (2,1), which are ess important for our immediate goas, are determined by equation (21) for a specific frequency, dieectric constant, and sphere radius. For exampe, for the particuar case {ω, r, ɛ = { rad/sec, 1 µm, 1 we find C (1,) = i, C (2,1) =.1 +.1i.

32 Homer Reid: E&M in the Spherica-Wave Basis x-directed force density At a point x = (r b, Ω) on the surface of the bounding sphere of radius r b, the x-directed force per unit area is, from (39), f x (x) = ɛ {C 1C 1 M 4 1(x)F x (Ω)M 1 (x) + N 1(x)F x (Ω)N 1 (x) +C1C 21 M 1(x)F x (Ω)M 21 (x) + N 1(x)F x (Ω)N 21 (x) +C21C 1 M 21(x)F x (Ω)M 1 (x) + N 21(x)F x (Ω)N 1 (x) +C21C 21 M 21(x)F x (Ω)M 21 (x) + N 21(x)F x (Ω)N 21 (x) 9.4 Tota x-directed force The tota force is obtained from equation (3): F x = ɛ 4 {C 1C F M21 F N21 21 M 1 x + N 1 x + CC (5) where CC stands for compex conjugate. The inner product here is defined by equation (45). With some effort, we compute F M21 F N M 1 x + N 1 x = i 1 k 2 and thus the tota force (5) reads F x = ɛ 3 ( ) 2k 2 1 Im C1C 21 (51) To make sense of the units here, suppose that fied-strength coefficients ike P, Q, C, D in (16) and (19) are measured in typica scuff-em units of V/µm, whie k is measured in units of inverse µm. Then the units of (51) are Use ɛ = 1 Z c Use Z = V/A: units of (51) = ɛ V 2 µm 2 µm 2 where c is the vacuum speed of ight: = 1 Z c V2 = V A µm s 1

33 Homer Reid: E&M in the Spherica-Wave Basis 33 Now use 1 V A=1 watt, 1 watt 1 s = 1 joue, 1 joue / 1 µm = 1 6 Newtons: = Newtons..1 XForce.dat u 1:(abs($2)).1 X component of tota force on sphere.1 1e-5 1e-6 1e-7 1e-8 1e-9 1e Omega Figure 3: x-component of tota force on sphere irradiated from within by an incident fied of the form (16) with coefficients (49).

34 Homer Reid: E&M in the Spherica-Wave Basis 34 1 T-matrix eements and surface currents The T-matrix for a body reates the coefficients of the outgoing spherica-wave expansion of the scattered fied to the coefficients of the reguar spherica-wave expansion of the incident fied; thus, if we write E inc = c inc W reguar, E scat = c scat W outgoing where the W notation was defined by equation (6) then the coefficient vectors are reated by c scat = Tc inc. Individua T-matrix eements have the significance T β = { coefficient of outgoing -type wave due to irradiation by unit-ampitude incoming β-type wave. Now consider a scattering geometry irradiated by a unit-ampitude reguar wave, and et K and N be the eectric and magnetic surface currents induced by this excitation. The scattered fied is E scat = ikzg K + ikc N with k and Z the wavevector and (absoute) wave impedance of the exterior medium. Insert the expansions (8): E scat (x) = k { 2 Z W reg K W out (x) + σ W reg N W out (x) or, rearranging sighty, E scat (x) = k 2 Z W reg K σ W reg N out W (x). (52) {{ T β This yieds a prescription for computing T-matrix eements directy from surface currents.