1 May 1998

Ž .Optics Communications 150 1998 1–4

The equivalence of applying the Extended Boundary Conditionand the continuity conditions

for solving electromagnetic scattering problems

K. Schmidt 1, T. Rother, J. WauerGerman Aerospace Center, German Remote Sensing Data Center, Kalkhorstweg 53, D-17235 Neustrelitz, Germany

Received 24 October 1997; revised 26 January 1998; accepted 20 February 1998

Abstract

The equivalence of applying the Extended Boundary Condition and the continuity conditions of the tangential fieldcomponents at sharp boundaries for deriving methods to solve electromagnetic scattering problems is demonstrated inspherical coordinates. We show that both conditions provide the same transition matrix to determine the unknown expansioncoefficients of the scattered field in terms of vector spherical wave functions. In this way, a generalization of the Separationof Variables Method to non-spherical scattering problems is achieved which allows a unified mathematical description ofdifferent numerical techniques. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 42.25.Fx; 94.10.GbKeywords: T-matrix; Non-spherical scattering; Separation of variables method; Ewald-Oseen extinction theorem

1. Introduction

When an electromagnetic wave propagating in amedium with a given velocity impinges on an obstaclewith a sharp dielectric boundary, a new wave with adifferent velocity arises inside the obstacle and the primaryincident wave is extinguished. The extinction of the inci-dent wave inside the obstacle is expressed by the Ewald-Oseen extinction theorem originally derived from molecu-

w x w xlar optics 1,2 . Some years ago, Pattanayak and Wolf 3provided a derivation of a more general form of theextinction theorem based exclusively on Maxwell’s theory.An analogous extinction theorem for non-relativistic quan-

w xtum mechanics was presented in Ref. 4 . They showedthat the extinction theorem can be interpreted as a non-lo-cal boundary condition subject to which Maxwell’s equa-tions provide a unique solution for the internal field of the

1 E-mail: schmidt@nz.dlr.de

scattering process. If the internal solution has been ob-tained, the scattered field can simply be calculated in termsof a surface integral. This formulation is widely applied inactual problems dealing with scattering and propagation of

w xwave fields 5 . In another series of papers Waterman haspresented a new formulation of electromagnetic scatteringknown as the Extended Boundary Condition MethodŽ . ŽEBCM or the ‘‘transition matrix approach’’ T-matrix

. w xapproach 6–8 . This method is based on the vectorHuygens’s principle and is, at present, one of the mostpowerful and widely used technique for rigorously com-

Žputing of non-spherical scattering for a review see, e.g.,w x.Ref. 9 . In 1976, Agarwal derived the Extended Bound-

Ž .ary Condition EBC of Waterman by an identical rewrit-ing of the generalized Ewald-Oseen extinction theoremw x10 , thus establishing the equivalence between both condi-tions. The advantages of the T-matrix approach comparedto other integral equation formulations are threefold. First,this method sidesteps the discussion of the range of valid-ity of Rayleigh’s hypothesis and the singular kernel prob-

0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0030-4018 98 00113-8

( )K. Schmidt et al.rOptics Communications 150 1998 1–42

lem of the conventional surface integral equation methodsw xdue to the non-local character of the EBC 11 . Second, it

does not suffer from singularities if scattering on metallicw xobjects is treated at internal resonance frequencies 6,11 .

Third, the T-matrix decouples the orientation of the scat-terer in the incident field from its physical and geometricalconfiguration. This results in a simple procedure to per-

w xform orientation averaging 12,9 . Despite these advan-tages, the T-matrix approach still remains a difficult methodfor practitioners, and Waterman himself has tried to sim-

w xplify the derivation of the T-matrix 13 .Recently, Rother and Schmidt have shown that the

conventional Mie theory for spherical scattering, which isŽ .nothing but a Separation of Variables Method SVM , can

be generalized to non-spherical geometries within thew xframework of a special finite difference scheme 14 , and

that this finite difference scheme itself can be replaced byw xa continuous picture 15 . In this way, a generalization of

the SVM is achieved. It can be applied to solve differentŽtypes of boundary value problems of physics eigenvalue

.and scattering problems, as well in a straight-forward waysince it is based exclusively on the differential equationformulation of the problem. The essential conditions in theSVM are the continuity conditions of the tangential elec-tromagnetic field components at the scatterer surface. Wewant to demonstrate now that, using these conditions,Waterman’s T-matrix can be derived directly without tak-ing the detour of the EBCM. This simplifies the derivation

Žof the T-matrix drastically and shows that the EBC or the.extinction theorem and the continuity conditions of the

tangential electromagnetic field components are equivalentin their meaning for non-spherical scattering methods. In

w xwhat follows, Waterman’s notation 8 is used.

2. Mathematical developments

Consider a homogeneous and isotropic, dielectric scat-terer with an arbitrary permittivity e

X and permeability mX

Ž .embedded in a homogeneous and isotropic medium e ,m .The scatterer can be of arbitrary shape with a single-valuedboundary surface with respect to a given coordinate sys-tem. On the scatterer, an arbitrary, incident field E inc witha eyiv t time dependence impinges. Let

c X r sc rŽ . Ž .n as m n

ay1s g k == kr Y u ,f h kr 1' Ž . Ž . Ž . Ž .mn s m n n

be the vector wave function obeying the vector waveequation

====c X r yk 2 c X r s0 . 2Ž . Ž . Ž .n n

Ž .In Eq. 1 , the spherical harmonics Y and the normal-s m n

ization constant g are given bymn

cos mf P m cosu , sse evenŽ . Ž . Ž .nY u ,f sŽ .s m n m½ sin mf P cosu , sso oddŽ . Ž . Ž .n

2nq1 nym !Ž .Ž .g se ,mn m 4n nq1 nqm !Ž .Ž .

e s1 , e s2 for m/0 ,0 m

where P m designates the associated Legendre functions.n

The index nX represents a collection of the indices as1,2;sse,o; ms0, . . . ,n and ns1,2, . . . . Note that, cas1

and c are proportional to the vector wave functions Mas2w xand N, respectively, given in Ref. 16 . h denotes then

spherical Hankel functions of the first kind taking intoaccount the radiation condition at infinity. Thus, the vector

Ž .wave functions 1 are appropriate to describe the scatteredfield E s in the outer medium with the wave number

Ž .1r2 intksv em . In order to describe the field E inside theX Ž X X.1r2scatterer with the wave number k sv e m , the regu-

lar vector wave functions Rgc X are required, i.e., we haven

to use the spherical Bessel functions j instead of then

Hankel functions. It should be mentioned that the functionsY h and Y j are the eigenfunctions of the scalars m n n s m n n

Helmholtz equation in spherical coordinates obtained bymeans of the SVM. Consequently, the following expan-sions of the scattered, internal and incident field E inc interms of vector wave functions represent expansions in

Ž .terms of eigenfunctions of the vector wave equation 2 ,

E inc k r s a Rgc k r , 3Ž . Ž . Ž .Ý n nn

E s k r s f c k r , 4Ž . Ž . Ž .Ý n nn

E int kXr s a Rgc kXr . 5Ž . Ž . Ž .Ý n nn

Ž . Ž . Ž .From Eqs. 3 , 4 , 5 , and Faraday’s induction law thecorresponding magnetic fields are readily obtained:

1incH k r s a ==Rgc k r , 6Ž . Ž . Ž .Ý n nivm n

1sH k r s f ==c k r , 7Ž . Ž . Ž .Ý n nivm n

1X XintH k r s a ==Rgc k r . 8Ž . Ž . Ž .ÝX n nivm n

Next, insert these expressions into the continuity condi-tions of the tangential field components at the scatterersurface given by

n= E inc qE s sn=E int , 9Ž . Ž .ˆ ˆ

n= H inc qH s sn=H int , 10Ž . Ž .ˆ ˆ

( )K. Schmidt et al.rOptics Communications 150 1998 1–4 3

with r lying on the scatterer surface and n being theˆoutward directed unit normal vector at the surface. Theresult is

a Pn=Rgc k r q f Pn=c k rw xŽ . Ž .ˆ ˆÝ n n n nn

s a Pn=Rgc kXr , 11Ž . Ž .ˆÝ n nn

1a Pn= ==Rgc k rŽ .Ž .ˆÝ n n

mr n

qf Pn= ==c k rŽ .Ž .ˆn n

1Xs a Pn= ==Rgc k r . 12Ž . Ž .Ž .ˆÝX n n

mr n

Here, m and mX , respectively, denote the relative perme-r r

Ž . Ž .ability. Eqs. 11 and 12 can be used to derive anequation system for calculating the unknown expansioncoefficients f and a of the scattered and the internaln n

field. This system is obtained by applying a Method ofMoment scheme to both equations. In order to derive aT-matrix that relates only the coefficients of the scatteredto those of the incident field, the Method of Momentscheme has to be taken in such a way that the right hand

Ž . Ž .sides of Eqs. 11 and 12 become equal, thus eliminatingthe coefficients a of the internal field. This can ben

Ž . Žachieved by multiplying Eq. 11 from the left with ==Ž X .. Ž . Ž X .X XRgc k r and Eq. 12 with Rgc k r . Afterwards,n n

both equations are integrated over the scatterer surface S.In using the interchange property of the parallelopipedalproduct the result reads as follows,

XXa dS nP ==Rgc k r =Rgc k rŽ . Ž .Ž .ˆ Ž .EÝ n n n

Sn

XXqf dS nP ==Rgc k r =c k rŽ . Ž .Ž .ˆ Ž .En n n

S

s a dS nP ==Rgc X kXr =Rgc kXr ,Ž . Ž .Ž .ˆ Ž .EÝ n n nSn

13Ž .

1X

Xa dS nP Rgc k r = ==Rgc k rŽ . Ž .Ž .ˆ Ž .EÝ n n nm Sr n

XXqf dS nP Rgc k r = ==c k rŽ . Ž .Ž .ˆ Ž .En n n

S

1X

Xs a dS nP Rgc k rŽ .ˆ ŽEÝX n nm Sr n

= ==Rgc kXr . 14Ž . Ž .Ž . .n

By means of the divergence theorem, some vector identi-Ž .ties, and the vector wave equation 2 it can be shown that

Ž . Ž .the right hand sides of Eqs. 13 and 14 are, apart fromŽ X .the factor y1rm , equal. As a result we obtain ther

following single equation.

a PRgQ X sy f PQ X , 15Ž .Ý Ýn n n n n nn n

with

1X

X XQ s dS nP P == Rgc k r =c k rŽ . Ž .Ž .ˆE Xn n n nmS r

1X

Xq PRgc k r = ==c k r , 16Ž . Ž . Ž .Ž .n nmr

1X

X XRg Q s dS nP P ==Rgc k r =Rgc k rŽ . Ž .Ž .ˆE Xn n n nmS r

1X

Xq PRgc k r = ==Rgc k r .Ž . Ž .Ž .n nmr

17Ž .In order to get a finite system for calculating a finitenumber of unknown expansion coefficients f of the scat-n

Ž .tered field we truncate the sum over n in Eq. 15 at acertain n and let nX run up to n . Note that, n canmax max max

be determined by increasing the number of expansionŽ . Ž .terms in 3 – 8 until the fields or other scattering quanti-

ties converge within a chosen accuracy. In writing theresulting n equations in matrix form, i.e.max

fsTPa , 18Ž .a T-matrix is introduced that is defined by

T:syQy1 PRg Q . 19Ž .Ž .The matrix elements of Q and RgQ given by Eqs. 16 and

X XŽ .17 are very similar to the elements Qn n and Rg Qn n

w xderived by Waterman 8 within the framework of theEBCM,

k 1X

X XQ s dS nP P == Rgc k r =c k rŽ . Ž .Ž .ˆEn n X n np mS r

XXyRgc k r = ==c k r , 20Ž . Ž . Ž .Ž .n n

k 1X

X XRg Q s dS nP P == Rgc k rŽ .Ž .ˆEn n X np mS r

=Rgc k r yRgc X kXrŽ . Ž .n n

= ==Rgc k r . 21Ž . Ž .Ž .n

Ž . Ž .A comparison with our expressions 16 and 17 showsŽ .that the main difference consists in the factor y1rm .r

Let us first note that Waterman has used the Gaussiansystem in his derivation so that the internal magnetic fieldŽ . Ž X . Ž X.8 has the prefactor 1rikm instead of 1rivm . It canr

be furthermore seen from his surface currents n=H thatˆ qthe prefactor of his incident and scattered magnetic fieldsŽ . Ž . Ž . Ž .6 and 7 is y1rik instead of 1rivm . Taking these

( )K. Schmidt et al.rOptics Communications 150 1998 1–44

modifications in our derivation into account we end upŽ . Ž . Ž . Ž .with the expressions 20 and 21 instead of 16 and 17 .

If we vary the procedure described above slightly, weare able to arrive at other numerical methods. For instance,choosing the incident field at the scatterer surface directlyŽ Ž . Ž ..instead of using the expansions 3 and 6 and multiply-

Ž . Ž .ing Eqs. 11 and 12 with spherical harmonics only, thenthe characteristic equation system of the method described

w xin Refs. 14,15,17,18 is obtained. It has the advantage ofallowing the treatment of scattering of focused beams onnon-spherical particles in a simple way, as discussed by

w x Ž .Barton and Alexander 17 . Otherwise, multiplying 11Ž .and 12 with appropriate delta distributions, we end up

Žwith the conventional Point Matching procedure see, e.g.,w x.Ref. 19 . Other variations are possible. Additionally, the

limiting case of spherical scatterers reduces the T-matrix toa diagonal matrix due to the orthogonality of the sphericalharmonics. From this matrix, the analytical results of the

w xconventional Mie theory can be exactly reproduced 12,9 .

3. Conclusion

The results obtained above demonstrate that the waydescribed in this paper offers a unified mathematical de-scription of different numerical techniques dealing withnon-spherical scattering in solving Maxwell’s equationsrigorously. Some of these methods have been consideredto be not related with each other, so far. Additionally it canbe seen that, although the differential equation and theintegral equation formulation may differ significantly inconcept and provide different points of view to a certainphysical problem, the numerical procedures derived fromthese different formulations are equivalent. This knownmathematical fact is most obviously expressed by theGreen’s function technique since, if the Green’s functionof a certain problem is known, we can switch between theintegral and differential equation formulation without lossof generality. This aspect is sometimes overlooked indiscussions concerning advantages and disadvantages ofdifferent numerical techniques. In most cases, difficultiesin the rigorous numerical methods can be attributed todifficulties in technical performance and not to a lack inmathematical rigour.

One of those controversial discussions is concernedwith the range of applicability of the scattered field expan-

Ž .sion given in Eq. 4 . The validity of this expansion isknown as the Rayleigh hypothesis since first assumed byRayleigh in dealing with scattering from sinusoidal sur-

Ž w x.faces for an overview see Ref. 20 . Especially the inte-

gral equation methods are thought to be ‘‘exact’’, i.e., theyare not influenced by this hypothesis. On the other hand,for methods which apply this expansion directly to the

Žcontinuity conditions and that is exactly what we have.done in this paper , there are derived upper limits up to

w xwhich this hypothesis is valid 20 . Surprisingly, theseupper limits are well exceeded in numerical considerationsw x21 . All difficulties could be explained by technical prob-lems like the problem of ill-conditioned matrices, forinstance. From the result of this paper, this is, of course,not a surprising fact. Furthermore, for the discrete version

Ž .of our method we could already show that Eq. 4 iscorrect as long as the boundary surface of the scatterer is a

w xstar-shaped one 14 . A more detailed consideration of thisproblem will be published elsewhere.

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