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Electromagnetic scatteringby an aggregate of spheres

Yu-lin Xu

We present a comprehensive solution to the classical problem of electromagnetic scat tering by a ggregat es

of an a rbitrary number of arbitrarily configured spheres tha t a re isotropic and h omogeneous but may be

of different s ize and composit ion. The profile of incident electromagnetic waves is ar bitrary. The

analysis is based on the framework of the Mie theory for a s ingle sphere and the exist ing addit iontheorems for spherical vector wa ve funct ions. The classic Mie theory is generalized. Applying the

extended Mie theory t o all t he spherical const ituents in a n a ggregate s imultaneously leads to a set of

coupled linear equa tions in the unknown intera ctive coefficients. We propose an a symptotic iterat ion

technique to solve for these coefficients. The tota l scatt ered field of the entire ensemble is constructed

with the interact ive scat tering coeff icients by the use of the translat ional addit ion theorem a second

time. Rigorous an a lytical expressions are derived for the cross sections in a general case and for all theelements of the amplitude-sca ttering ma trix in a special case of a plane-incident wa ve propaga ting a long

t he z axis . As an illustra t ion, we present some of our preliminary numerical results and compare them

with previously published laboratory scattering measurements.

Key words: Scat t ering, part icles , aggregat es .

1. Introduction

Light scat ter ing b y a smal l b ody or a col lect ion ofs m a l l ob je ct s w h o se s iz es a r e com p a r a b le t o t h ewavelength of the incident radiation is a problem ofgrea t interest t o a broa d ra nge of scientifi c disciplines.

To dat e, exactly solvable problems include th e scat ter-ing of a plane electromagnetic wave by a few highlysymmetric types of single particles and by arbitraryconfi gurat ions of para l lel, infi ni te cylinders . P er-ha ps the scat tering theory tha t is most widely used isthat for a homogeneous sphere of arbitrary size andref ract iv e index, know n as Mie theory , w hich w asdeveloped by Lorenz1 and Mie.2 The complete solu-tion for homogeneous, infinitely long, circular cylin-ders was first given by Lord Rayleigh 3 for the case ofperpendicular incidence and by Wait 4 for the case ofoblique incidence. The problem of homogeneousspheroids was solved by Asano and Yamamoto.5 Asolution for the problem of scattering by a homoge-

neous sphere coat ed w i th a homogeneous layer ofunif orm thickness w as firs t ob tained b y Aden and-

Kerker.6 Th i s i s a n e xa m p le o f a p a r t i cl e w i t h aspatially variable refractive index; their theory canalso be generalized to a ra dially stra tifi ed sphere.

The mutual interaction complicates l ight scatter-ing by ensembles of part icles. The problem t here-

f or e r eq u ir es t h e u s e o f a n a d d i t ion t h e or em t otra nsform the relevant basis functions from a coordi-na te system centered on the scat terer to other refer-ence systems centered outside the sca tt erer. Usinga n a ddition th eorem for cylindrical w a ves, Twers ky 7–9

wa s a ble to calculate t he sca tt ered field produced by ap la n e w a v e s t r i ki ng a n a r b it r a r y con fi g u r a t i on ofpara llel infi nite cylinders.

For multiple scat tering by spheres, i t is necessaryt o d ecom pos e t h e s ca t t e r ed fi e l d of a s ph er e i n t ospherical waves that impinge on the other spheres.Although Mie theory expresses the sca tt ered fi eld inthe form of spherical vector w ave functions by usingthe center of the sphere as the origin of the reference

system, the problem of interacting spheres requirestha t one seek representa tions of the scat tered fi eld bythe use of the same set of basis spherical vector wavef un ct i on s t h a t r ef er t o a n y ot h e r a r b it r a r y or i gi n .In 1954, Friedman and Russek10 reported a deriva-tion of such an expansion, or addition theorems forspherical scalar-wa ve function. With t he develop-ment of addition th eorems for spherical vector w avefunctions in the 1960’s by St ein11a n d C r u z e n ,12 itbecame, in principle, feasible to solve theoretically for

The a u t ho r i s w i t h t he Dep a r t m ent o f As t r onom y, P .O . B o x

112055, University of Florida, Gainsveille, Florida 32611-2055.

Received 3 October 1994; revised ma nuscript received 8 Febru ar y

1995.

0003-6935@95@214573-16$06.00@0.

r 1995 Optical Society of America.

20 July 1995 @ Vol. 34, No. 21 @ APPLIED OPTICS 4573

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the scat ter ing propert ies of arb i trary aggregates ofspheres. Since then, considerable progress has beenm a d e t ow a r d a s ol ut i on f or m u lt i sp he r e s y s t em s ,especial ly b y Liang and Lo, 13 B r u n i n g a n d L o ,14,15

F u ll er a n d K a t t a w a r ,16,17 and Mackow ski .18 Th er ea d e r i s r e f er r ed t o a n a r t i cl e b y F u ll er 19 f or ahistorical review. A general l itera ture survey is be-yond the scope of this paper; we cite only the mostclosely rela ted pa pers.

In 1971, B runing a nd Lo published th e fir st compre-

hensive solution for a tw o-sphere chain, 14,15 completew ith experimenta l v erifi cat ion. These aut hors a p-plied the standard electromagnetic boundary condi-tions to the surfaces of the t wo spheres w ith a dditiontheorems and were able to solve the resulting linearequa tions for the intera ctive sca tt ering coefficients bythe use of direct mat rix inversion. La ter, Fuller andKat ta w a r introduced the order-of -scat ter ing meth-od 16,17 a s a n a l t e r n a t i v e a n d m o r e e f f e c t i v e w a y o fsolving for the coefficients in a n at tempt to reduce a2L -sphere scattering problem to a sequence of interact-ing pairs . Their s tra tegy w as to extend Bruning andLo’s solution pair wise t o the m ore complicat ed case ofa n a r b it r a r y cl us t er of s ph er es . I n 1991, M a ck -owski18 rederived the addition theorems for vectorspherical harmonics and obtained a set of recurrencerelat ions for effective calculat ion of th e ad dition coef-fi ci en t s . Th i s s c a t t e r in g p r ob le m o f i n t e r a c t i n gspheres is a lso independently inv est igated b y Bor-ghese et al.,20–22 w ho handled the v ector scat ter ingproblem t hrough Debye potentia ls a nd by Wan g a ndChew,23 w ho developed a recur sive T-ma tr ix approa chf or the solut ion. Borghese et al.20,21 appear to hav ebeen the first to present explicit expressions for thecross sections of th e sphere clust ers. Other resear chconducted on the scattering cross sections includestha t of G era rdy an d Ausloos24 and of Mackowski.18

G era rdy a nd Ausloos provided a corr ect expression forextinction cross sections of th e aggrega tes th rough a nintegral of the P oynting fl ux of the scat tered field.I n addi t ion to t he scat ter ing a nd ext inct ion crosssections, Mackowski also derived an expression forthe absorption cross sections of the multisphere sys-tems by integrating t he Poynting fl ux at t he surface ofeach sphere in the cluster. A brief discussion of theMuller ma tr ix of a t w o-spher e syst em can be found ina pa per by Fuller et al.25

However, the solution of the problem of intera ctingspheres appears to be stil l incomplete; missing is aderivation of explicit expressions of the amplitude-scat ter ing ma tr ix and theref ore the Mul ler mat r ix

an d the polariza tion properties. The calculation ofthe angula r distribution of the scat tered field and t hesta te of polariza tion is importa nt or even essential inma ny a pplicat ions.

I n this paper the intent is to present a completegenera l solution t o the problem of intera ctive electr o-magnetic scattering by clusters of arbitrarily config-ured nonidentical spheres. The profile of incidentb eams is arb i trary. The only assumption ma de onthe incident electroma gnetic field is tha t the incident

w a v es can b e expressed b y elementa ry sphericalw a v es . The f ramew ork of the mult isphere scat ter-ing theory follows closely from t he Mie theory. InSection 3, based on a superposition principle in term sof vector spherical harmonic expansions, we extendth e classic Mie th eory to a genera l ca se of a n ar bitra ryprofi le of incident beam s. In Section 4 the extendedMie theory is appl ied to a l l the const i tuents of thecluster of spheres and results in a l inear system inw hich one is to solv e f or the interact iv e scat ter ing

coeff icients . I n S ect ion 5 a n asymptotic i terat ionmethod is suggested to solv e the resul tant l inearequations for the interactive scattering coefficients,which uses a numerical factor to improve the conver-gence property of th e linear system. Section 6 brieflydiscusses the construction of the t otal scat tered fi eldof the entire cluster and provides expressions for theintern a l fields of each sphere in the prima ry referencesystem. S ect ion 7 is dev oted t o the discussions ofcross sect ions and the elements of the ampli tude-scat tering mat rix. Some comparison of our prelimi-nary numerical resul ts w i th lab oratory microw av emea surement s is given in Section 8.

2. Statement of the Problem

Consider a cluster of L isotropic, homogeneous, andnonintersect ing spheres w i th know n radi i a j a ndknown complex refractive indices N j , j 5 1, . . . , L .Throughout this pa per, any single integer in a right-hand superscript, such as j in a j , indicates that thequant i ty is rela t ed to the j th sphere. These spheresare confi ned to a fi ni te volume. I n a three-dimen-sional coordinate system whose origin is at the centerof the j 0th sphere 1below we refer to this coordinatesystem a s the primary system 2, t he position vector ofthe center of any other j th sphere is denoted by d j 0, j ,wh ich extends from th e center of the j 0th t o th e center

of the j t h s p h er e. F or a n y p a i r of s p he re s in t h ecluster, the j t h a n d t h e l th , the rela t iv e posi t ionvector is defi ned by d j ,l 5 d j 0, l 2 d j 0, j , and d j , l $ a j 1 a l .

Suppose tha t t he incident beams il luminating ea chsphere in th e cluster can be represented by elemen-ta ry spherical w a ves about the center of each sphere.The incident electromagnetic waves are monochro-ma tic but arbitra ry in profile. The scatt ering proper-ties of such a cluster of spheres are to be examinedanalytically.

3. Generalization of the Mie Theory

To investiga te t he scat tering propert ies of th e clust erof i n t er a c t in g s ph er es , w e n ee d t o d et e r mi n e t h einteract iv e scat ter ing coeff icients f or each sphereindiv idual ly , s imilar to w hat Mie theory does f or as ingle sphere. H ow ever, in this case the sphericalconstituents in the cluster can no longer be consid-er ed a s i sol a t ed , a n d t h e a s s um pt ion of a pl a n eincident wa ve is no longer valid. We need to ha ndlea more general case than t he classic Mie theory. Theproblem of generalizing the Mie theory was consid-ered by G ouesbet et al.26,27 as well as by B ar ton et al.28

These aut hors studied t he scatt ering properties of a

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s ph er e l oca t e d i n a G a u s si a n l a s er b ea m , u s in g aB romwich scalar formulat ion. Our study of the mul-t isphere case leads to the same resul t , though in atota lly dif ferent manner. The basic derivation is asfollows:

A. Expansion of Electromagnetic Fields

To solve scat ter ing problems, it is d esira ble to expresselectromagnetic fields in terms of infinite-series expan-sions at a ll points in space. In our case the fields ar e

t o b e e xp a n d ed i n t e r ms of e le me nt a r y s ph er i ca lwaves about a fixed center.

1. Spheri cal Vector Wave Fu nctions

Time-harmonic electric and magnetic fields E a nd Hin a sourceless, isotropic, and homogeneous mediumar e divergence free an d must sat isfy the vector wa veequations

= 3 = 3 E 2 k 2E 5 0, = 3 = 3 H 2 k 2H 5 0, 112

w here k 2 5 v2Eµ, k i s the w av e numb er, v i s t h e

circular f requency of the w a v e, E is the dielectricconstant , a nd µ is the permeab il ity of the medium.I n spherical polar coordinates 1r , u , f2, t h e l i n e a r

independent vector field solutions of the vector waveequations are the vector spherical functions M a nd Ntha t, in t he specifi c component form, ma y be writ tena s

Mm n 1J 2 5 3iui pm n 1cos u2 2 iftm n 1cos u 24z n

1J 21k r 2ex p1i m f2,

Nm n 1J 2 5 ir n 1n 1 1 2P n

m 1cos u2z n

1J 21k r 2

k r exp1i m f2

1 3iutm n 1cos u2 1 ifi pm n 1cos u24

31

k r

d

d r 3r z n

1J 21k r 24exp1i m f2, 122

w here i

r , i

u, i

f a r e u n it v ect or s i n t h e s ph er i ca lpolar -coordina te sy stem; z n 1J 2 is appropriately selected

from any of the four spherical Bessel functions: thefi rst kind j n , the second kind y n , or th e third kind 1alsocal led spherical H a nkel f unct ions of t he fi rs t andsecond kin d 2 h n

112 a nd h n 122, denoted by J 5 1, 2, 3, or 4,

respectively; P n m 1cos u2 i s t h e a s s oci a t e d L eg en d r e

function of the first kind and of degree n a nd order m ,n , and m a re integers with 1 # n , ` a nd 2n # m # n ,an d wit h defi nitions for the functions

pm n 1cos u2 5m

sin u P n

m 1cos u 2,

tm n 1cos u2 5

d

d u P n m

1cos u2. 132

Recursion formulas for pm n a nd tm n a r e g iv en i nAppendix A. Here both th e vector spherica l ha rmon-ics 1Mm n , Nm n 2 and the a ngular functions 1tm n , pm n 2 a redenoted by the order m fi r s t a n d t h e d e gr ee n l as t .This notat ion agrees w i th that used b y Bohren andH uf f man,29 but th e one tha t is commonly used in th ear bitrar y-incident-beam literature 28,30 has the orderof m a nd n reversed.

2. E x pansi on of the S cattered, In tern al , and

In ci dent Fi el ds

In terms of the spherical vector wave functions, thescattered field 1Es , Hs 2 and the internal field 1E I , H I 2 ofa n i n di vi du a l s ph er e, s a y, t h e j t h s p h e r e i n t h eclust er, can be expa nded a s

Es 1 j 2 5 on 51

`

om 52n

n

i E m n 3a m n j Nm n

132 1 b m n j Mm n

132 4,

Hs 1 j 2 5k

vµ on 51

`

om 52n

n

E m n 3b m n j Nm n

132 1 a m n j Mm n

132 4,

E I 1 j 2 5 2 on 51

`

om 52n

n

i E m n 3d m n j Nm n

112 1 c m n j Mm n

112 4,

HI 1 j 2 5 2k j

vµ j on 51

`

om 52n

n

E m n 3c m n j Nm n

112 1 d m n j Mm n

112 4, 142

w here

E m n 5 0 E 0 0 i n 12n 1 1 21n 2 m 2!

1n 1 m 2!

. 152

The introduction of E m n is desired for keeping theformulat ion of the mult isphere-scat tering t heory con-sistent w i th tha t of the Mie theory. I t ensures thata ll the expressions in the mult isphere theory turn outto be identical to those in the Mie theory when one isdealing with a cluster containing only one sphere andilluminat ed by a single plane wa ve. When m 5 1,

E 1n 5 E n 5 0 E 0 0 i n 2n 1 1

n 1n 1 1). 162

This is what appears in the Mie theory 1see B ohrenand H uf f man,29 chapter 42. The superscript 132 or 112appended to the vector spherical harmonics indicatesthat the generating function is specified by the Han-k el f un ct i on of t h e fi r s t k in d , h n

112, or t h e B es selfunction of the fi rst kind, j n , respectively. Also, k j isthe wa ve number inside the j th sphere, and µ j is thepermeab il ity of t he same sphere. Throughout thispaper a s ingle integer in parentheses ha s t he samemeaning as in a r ight-hand superscript . I n Eqs . 142,we a ssociat e the int erna l coefficient d with th e scat ter-ing coefficient a a nd t he intern a l coefficient c w i t h t h escat terin g coefficient b . Ag a in , t h i s a g rees w it hB ohren and Huffman. The readers should, however,be a wa re of a different nota tion; for example, both vande Hulst 31 and Kerker 32 associate c w i t h a a nd d w i t hb .

Similarly, the incident fi eld that strikes the surfaceof th e j th sphere is assum ed to have the form

E i 1 j 2 5 2on 51

`

om 52n

n

i E m n 3p m n j Nm n

112 1 q m n j Mm n

112 4,

Hi 1 j 2 5 2k

vµ on 51

`

om 52n

n

E m n 3q m n j Nm n

112 1 p m n j Mm n

112 4, 172


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wh ere the incident w a ves from all the possible sourcesare included. H ere the ha rmonic t ime dependenceexp12i w t 2 of the fi elds is a ssumed a nd suppressed.

B. Extension of the Mie Theory to a General Application

S ti l l using t he j th sphere as an example, we imposethe s tandard b oundary condi t ions a t the surf ace ofthe sphere:

E i 1 j 2 1 Es 1 j 2 2 E I 1 j 24 3 i r j

5 3Hi 1 j 2 1 Hs 1 j 2 2 HI 1 j 24 3 ir j 5 0. 182

In a component form, these boundary conditions at

r j

5 a j

a reE i u1 j 2 1 E s u1 j 2 5 E I u1 j 2, E i f1 j 2 1 E s f1 j 2 5 E I f1 j 2,

H i u1 j 2 1 H s u1 j 2 5 H I u1 j 2, H i f1 j 2 1 H s f1 j 2 5 H I f1 j 2.

192

From the orthogonality of exp 1i m f2, together with t heexpansions 142 a nd 172 and the expressions f or thespherical vector harmonics 122, the ab ov e b oundaryconditions give rise to four l inear equat ions conta in-ing t he int era ctive coefficients:

j n 1m j x j 2c m n j 1 h n

1121x j 2b m n j 5 q m n

j j n 1x j 2,

µ3m j x j j n 1m j x j 248c m n j 1 µ j 3x j h n 1121x j 248b m n

j

5 q m n j µ j 3x j j n 1x j 248,

µm j j n 1m j x j 2d m n j 1 µ j h n

1121x j 2a m n j 5 p m n

j µ j j n 1x j 2,

3m j x j j n 1m j x j 248d m n j 1 m j 3x j h n

1121x j 248a m n j

5 p m n j m j 3x j j n 1x j 248.

1102

I n E q s . 1102 the prime indicat es a dif ferentiation withrespect to the argument in parentheses. x j a nd m j

a r e t h e s i z e p a r a m e t e r a n d t h e r e l a t i v e r e f r a c t i v eindex of th e j th sphere, respectively, an d a re given by

x j 5 k a j 52pN 0a j

l, m j 5

k j

k 5

N j

N 0, 1112

w here l is the wa velength of th e incident wa ves in thesurrounding medium a nd N 0 is the refra ctive index of

the surrounding medium. The f our s imulta neouslinear equat ions 1102 ca n be solved for t he four intera -tive coefficients: a m n

j , b m n j , c m n

j , d m n j . The intera c-

tive scat tering coefficients a re given by

S imilar ly , the coef f ic ients of the internal field aregiven by

Equations 1122 a nd 1132 can be writt en as

a m n j 5 a n j p m n j , b m n j 5 b n j q m n j , 1142c m n

j 5 c n j q m n

j , d m n j 5 d n

j p m n j , 1152

w here a n j , b n

j , c n j , d n

j , are exactly the Mie coefficientsf or the isolated j th sphere 3s e e B o r h e n a n d H u f f -m a n ,29 equations 14.522 a nd 14.5324.

This result is th e same a s tha t obtained by G oues-bet et al. and b y Ba rton et al. I t clearly reveals thatthe radiative scattering response of a homogeneoussphere to an a rbitra ry beam can be directly relat ed tothe Mie scattering properties of that sphere and theprofi le of the electromagn etic waves t ha t ar e incidentupon i t . No matt er how complicat ed the incidentfi eld, th e scat tering coefficients 1a n d t h e r e f o r e t h e

scattering properties 2 of a sphere, w heth er isolat ed ornot isolated, can be easily determined with E qs. 1142 ifthe expansion coefficients of the incident field can beexplicitly found. The sca tt ering coefficient s in a gen-e ra l ca s e a r e ju s t t h e l in ea r m od ifi c a t i on s of M iecoefficients by the expansion coefficients of the inci-dent field. Thus, to investigat e the scatt ering behav-ior of a sphere in a general case, the only new ta skinvolved is to determine the expansion coefficientstha t describe the incident fi eld.

a m n j 5

µ1m j 22 j n 1m j x j 23x j j n 1x j 248 2 µ j j n 1x j 23m j x j j n 1m j x j 248

µ1m j 22 j n 1m j x j 23x j h n 1121x j 248 2 µ j h n

1121x j 23m j x j j n 1m j x j 248p m n

j ,

b m n j 5

µ j j n 1m j x j 23x j j n 1x j 248 2 µ j n 1x j 23m j x j j n 1m j x j 248

µ j j n 1m j x j 23x j h n 1121x j 248 2 µh n

1121x j 23m j x j j n 1m j x j 248q m n

j , 1122

c m n j 5

µ j j n 1x j 23x j h n 1121x j 248 2 µ j h n

1121x j 23x j j n 1x j 248

µ j j n 1m j x j 23x j h n 1121x j 248 2 µh n

1121x j 23m j x j j n 1m j x j 248q m n

j ,

d m n j 5

µ j m j j n 1x j 23x j h n 1121x j 248 2 µ j m j h n

1121x j 23x j j n 1x j 248

µ1m j 22 j n 1m j x j 23x j h n 1121x j 248 2 µ j h n

1121x j 23m j x j j n 1m j x j 248p m n

j . 1132


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4. Description of the Total Incident Field for a Spherical

Constituent

From t he discussions in S ection 3, one can see th a t forthe multisphere problem at ha nd, we need to seek a nexplicit description of the total incident field for eachsphere in the cluster. To determine the interactivecoefficients 1a m n

j , b m n j , c m n

j , d m n j 2 of th e j th sphere, we

n e ed t o d et e r m in e t h e e xp a n s ion coe ffi ci en t s1 p m n

j , q m n j 2 o f the incident field of the same sphere.

The electromagnetic field that is incident upon the

surfa ce of th e j th sphere consists of two part s: 112 t heoriginal incident waves and 122 the scattered fields ofa l l the other spheres in the c luster , w hich can b ew ri t ten as

E i 1 j 2 5 E01 j 2 1 ol fi j

Es 1l , j 2,

Hi 1 j 2 5 H01 j 2 1ol fi j

Hs 1l , j 2. 1162

Throughout this paper, two integers separated by acomma in parentheses or in a right-hand superscriptl , j i m pl y a t r a n sl a t ion f rom t h e l t h t o t h e j t hcoordina te system. To obta in ana lytical expressionsfor the total incident field striking the surface of eachsphere, the ini t ia l incident w av es , and a l l the scat-tered fi elds of t he other spheres must b e expandedabout the center of the sphere. The last of these canbe accomplished by t he use of the t ra nslat iona l ad di-tion th eorem for vector spherical ha rmonics.

A. Expansion of the Initial Incident Waves

The origina l incident w a ves are a ssumed to be expan d-able about t he center of each sphere and, for the j t hsphere, i.e., in th e j th coordinate system, the incidentfi eld has the form

E01 j 2 5 2on 51

`

om 52n

n

i E m n 3 p m n j , j Nm n

112 1 q m n j , j Mm n

112 4,

H01 j 2 5 2k

vµon 51

`

om 52n

n

E m n 3q m n j , j Nm n

112 1 p m n j , j Mm n

112 4. 1172

For use below, w e intr oduce

p m n 0 5 p m n

j 0, j 0, q m n 0 5 q m n

j 0, j 0, 1182

w here superscript 0 indicates that the quanti ty isrelated to the initial incident waves and the primarysyst em. All th ese expansion coefficients of th e initia l

incident w a v es tha t dr ive the w hole scat ter ing pro-cess a re supposed t o be known for the problem underconsideration. An exam ple is as follows.

S u p p o s e a s s h o w n i n F i g . 1 , t h a t t h e c l u s t e r i sil luminated by a plane wave characterized by a wavevector k ,

k 5 k 1ix s in a cos b 1 iy s in a s in b 1 iz cos a2, 1192

which indica tes tha t t here is an incident an gle a w i t hrespect t o the z axis and a n angle b between the x axis

and the projection of k on t he x –y pl a n e. I n E q . 1192ix , iy , a n d iz a r e t h e u n i t v e c t o r s o f t h e C a r t e s i a ncoordinate system. U sual ly, tw o di ff erent polar iza-tion modes of the incident plan e wa ve are considered:t h e t r a n s v e r s e m a g n e t i c m o d e 1TM 2, w hich corre-sponds to the case in w hich the electr ic vector vibra tesin t he incident plane, and the tra nsv erse electr icmode 1TE 2, which corresponds to the case in w hich thema gnetic vector vibrat es in the incident pla ne. Theincident plane is defi ned by the z a xis and t he incidentwave vector k . F or t h e TM m o d e, t h e e xp a n s ioncoefficients are

p m n j , j 5 exp 1i k · d j 0, j 2p m n

0 , q m n j , j 5 exp 1i k · d j 0, j 2q m n

0 ,

1202

w here

p m n 0 5

1

n 1n 1 1 2 3tm n 1cos a2cos b 2 i pm n 1cos a 2sin b4,

q m n 0 5

1

n 1n 1 1 2 3tm n 1cos a 2cos b 2 i pm n 1cos a2sin b4.

1212

B. Addition Theorems for Spherical Vector Wave Functions

To tra nsf orm the w a v es scat tered b y a n indiv idualsphere into the incident wa ves of oth er spheres in thecluster, we need to describe th e sam e sca tt ered fi eld inalternative forms, each form referring to a dif ferentcoordinate system but with exactly the same commonbasis vector functions. The connections betw een th ea l t e rn a t i v e r e pr es en t a t i on s of t h e s a m e fi e l d a r eprovided by th e add ition theorems, i.e., th e expan sionof the basis set of one representa tion in terms of thebasis set of an other. Such addition theorems for thespherical vector wave functions have been obtained

Fig. 1. G eometry of the multisphere scatt ering problem.


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by St ein 11 a n d C r u z a n .12 The tra nslat iona l addit ionth eorem is represented by 12

Mm n 5 on50

`

oµ52n

n

1A 0µnm n M8µn 1 B 0µn

m n N8µn2,

Nm n 5 on50

`

oµ52n

n

1B 0µnm n M8µn 1 A 0µn

m n N8µn2, 1222

w here Mm n

a n d Nm n

are the b asis v ector sphericalwave functions about an origin O and M8µn a nd N8µn a reabout a nother diplaced origin O 8; M8µn a nd N8µn are ofthe same form as Mm n a nd Nm n ; A 0µn

m n a nd B 0µnm n are the

so-called translation or addition coefficients, given inAppendix B. These tra nslat ion coefficients dependon the rela t iv e separat ion and direct ion of the dis-placed origin O8 wit h respect t o O.

C. Total Incident Field of a Constituent Sphere

From the addi t ion theorem, the translat ion of thevector spherical wa ve functions from the l th to the j t hcoordinate system takes the form

Mm n 132 1l 2 5on50

`

oµ52n

n

3A 0µnm n 1l , j 2Mµn

112 1 j 2 1 B 0µnm n 1l , j 2Nµn

112 1 j 24,

Nm n 132 1l 2 5 o

n50

`

oµ52n

n

3B 0µnm n 1l , j 2Mµn

112 1 j 2 1 A 0µnm n 1l , j 2Nµn

112 1 j 24.

1232

Because Nm n a nd Mm n in ei ther the l t h o r t h e j t hcoordinate system hav e exact ly the same f orm, thecoordina te syst em indicator for these tw o ba sis vectorf unctions is suppressed b elow . Also, in pract icala pplicat ions of our problem, w e do not need t o includethe modes with index n 5 0, based on the forms of thevector field solutions discussed in Subsection 3.A.1.Consequently, the scattered field of the l th sphere ha san expan sion in the j th coordinate system as follows:

Es 1l , j 2 5 on 51

`

om 52n

n

i E m n Aa m n l 5o

n51

`

oµ52n

n

3A0µnm n 1l , j 2Nµn

112

1 B 0µnm n 1l , j 2Mµn

112 461 b m n

l 5on51

`

oµ52n

n

3B 0µnm n 1l , j 2Nµn

112

1 A 0µnm n 1l , j 2Mµn

112 46B ,

Hs 1l , j 2 5k

vµon 51

`

om 52n

n

E m n Ab m n l 5on51

`

oµ52n

n

3A 0µnm n 1l , j 2Nµn

112

1 B 0µnm n 1l , j 2Mµn

112 461 a m n

l 5on51

`

oµ52n

n

3B 0µnm n 1l , j 2Nµn

112

1 A 0µnm n 1l , j 2Mµn

112 46B . 1242

After interchanging 1m , n 2 w i th 1µ, n2 and rearra ngingth e expansion coefficients, we h a ve

Es 1l , j 2 5 2 on 51

`

om 52n

n

i E m n 3 p m n l , j Nm n

112 1 q m n l , j Mm n

112 4,

Hs 1l , j 2 5 2k

vµon 51

`

om 52n

n

E m n 3q m n l , j Nm n

112 1 p m n l , j Mm n

112 4, 1252

w here

p m n l , j 5 2 o

n51

`

oµ52n

n

3a µnl A m n

µn 1l , j 2 1 b µnl B m n

µn 1l , j 24 1l fi j 2,

q m n l , j 5 2 o

n51

`

oµ52n

n

3a µnl B m n

µn 1l , j 2 1 b µnl A m n

µn 1l , j 24 1l fi j 2.

1262

S pecial a t tent ion should b e paid to the translat ioncoefficients in Eqs. 1262: A 0m n

µn a nd B 0m n µn have been

replaced by A m n µn a nd B m n

µn , respectively. The relat ions

between 1A 0m n µn , B 0m n µn 2 a nd 1A m n µn , B m n µn 2 ar e as follows:

A m n µn 5

E µn

E m n

A 0m n µn 5 i n2n

12n 1 121n 1 m 2! 1n 2 µ2!

12n 1 1 21n 2 m 2! 1n 1 µ2! A 0m n

µn ,

B m n µn 5

E µn

E m n

B 0m n µn 5 i n2n

12n 1 121n 1 m 2! 1n 2 µ2!

12n 1 121n 2 m 2! 1n 1 µ2! B 0m n

µn .

1272

According to Eq s. 172, 1162, 1172, 1252, a n d 1262, t h eexpansion coefficients of th e tota l incident fi eld of th e j th sphere a re given by

p m n j 5o

l 51

L

p m n l , j , q m n

j 5ol 51

L

q m n l , j 1282

or in deta il,

p m n j 5 p m n

j , j 2ol fi j

11, L 2

on51

`

oµ52n

n

3a µnl A m n


µn 1l , j 24,

q m n j 5 q m n

j , j 2ol fi j

11, L 2

on51

`

oµ52n

n

3a µnl B m n


µn 1l , j 24,

1292

w here the fi rs t term on the r ight-hand s ide ref ers tothe initial incident wa ves and the second t erm refersto the fields scattered by other spheres.

5. Solution of the Interactive Coefficients

A. Linear System of the Interactive Coefficients

By insert ing Eqs . 1292 into Eqs. 1142, we arrive at a setof l in ea r e q ua t i on s t h a t con t a i n s t h e i n t er a c t iv e


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scat tering coefficients:

a m n j 5 a n

j 5 p m n j , j 2o

l fi j

11,L 2

on51

`

oµ52n

n

3a µnl A m n

µn 1l , j 2

1 b µnl B m n

µn 1l , j 246 ,

b m n j 5 b n

j 5q m n j , j 2o

l fi j

11,L 2

on51

`

oµ52n

n

3a µnl B m n

µn 1l , j 2

1 b µnl A m n

µn 1l , j 246 . 1302

Lo and his colleagues 1Liang and Lo, Bruning and Lo 2w ere to our know ledge the firs t to deriv e a s imilarresult for the interactive coefficients of two-spheres y st e m s. F u ll er a n d K a t t a w a r l a t e r ex t en d ed t h eapplicat ion of the same f orm of l inear equat ions toclusters of spheres. I t is no surprise tha t the intera c-tive coefficients of a sphere in t he cluster a re deter-mined by three ma jor factors: 112 th e physical proper-t i es of t h e s ph er e i t se lf , r ep re se nt e d b y t h e M iecoefficients; 122 the interacting properties of th e clus-ter determined by th e configura tion of the cluster andt h e p hy s ica l p r op er t i es of a l l t h e s ph er es i n t h eclust er, implied in p m n

j a nd q m n j ; a nd 132 the profi le of

the incident w av es , implied in p m n j , j a nd q m n

j , j . Tw othings a re worth noting here:

1. We need only to solve the linear system for thescat tering coefficients a m n

j a nd b m n j , because there

exist simple relations between the interactive scatter-ing coefficients a nd th e int erna l coefficients:

d n j a m n

j 2 a n j d m n

j 5 0, c n j b m n

j 2 b n j c m n

j 5 0. 1312

2. The highest order N required for convergencefor the cluster is approximately the largest value in

Mie calculat ions for all the constituents. For a singlehomogeneous sphere of size pa ra meter x , Bohren a ndH uf f man suggest a v alue s l ight ly larger than x besufficient. In their code B HMIE t hey use a criterion1x 1 4 x 1@3 1 2 2. E x a m in in g E q s . 1142 s how s t h a t t h esame criterion should hold for a cluster of spheres ift h e l a r g e s t s i z e p a r a m e t e r i s t a k e n a m o n g a l l t h espheres.

I n t he l inear system 1302, all the Mie coefficients, a n j ,

b n j , c n

j , and d n j , are calculated by the Mie theory for all

the individual spheres, p m n j , j a nd q m n

j , j ar e the expansioncoeff ici en t s of t h e i n ci de nt w a v e s, a n d t h e re a r e2L N 1N 1 2 2 unknow n interact iv e scat ter ing coeff i-

cients in 2L N 1N 1 2 2 l inear equations, provided thatthe system is convergent af ter n 5 N . Th i s l in ea rs ys tem ca n be w r it t en in t h e f or m of a m a t rixequat ion in w hich the tra nslat ion coefficients consti-t u t e t h e coe ffi ci en t m a t r i x. I n p r in ci pl e, t h e u n -known interactive coefficients can be solved by theuse of direct mat r ix inv ersion. But the direct solu-t ion is of ten not f easib le in pract ice w henev er thenumb er of spheres or the s ize parameters of somespheres in the cluster is large. Ful ler and Ka tta w a r

introduced the order-of-scattering technique to solvethe l inear system. H ow ever, a ctual computat ionsindicat e tha t convergence of t he order-of-scat teringtechnique is not guaranteed once the number or thesizes of the spheres exceed a certa in l imit .18 Analt ernat ive wa y to solve for t he interactive scat teringcoeff ici en t s i s t o u s e a n i t er a t i on m e t h od t h a t i sdiscussed below.

B. Asymptotic Iteration Method for Solving the Coefficients

To solve for th e inter a ctive scat ter ing coefficient s, a m n j

a nd b m n j , an asymptotic i terat ion method that uses

E q s . 1142 a nd 1292 repeat edly can be used. The itera -tion may take the following form:

1. S t a r t in g w i t h 0a m n j 5 0b m n

j 5 0. F r om E q s . 1292it follows that

1p m n j 5 p m n

j , j , 1q m n j 5 q m n

j , j , 1322

an d from Eqs. 1142

1a µnl

5 a nl 1p µn

l

, 1b µnl

5 b nl 1q µn

l

. 1332

I n Eqs . 1322 a nd 1332 the prescript int eger indica tes th esequentia l number of the iterat ion; the nota tion 1µ, n , l 2i s e q ui va l en t t o 1m , n , j 2. O nl y M ie s ca t t e ri ng i sinvolved a t t his sta ge.

2. I n a s im il a r m a n n er, w e n ex t u s e t h e n ew values of th e scat tering coefficients in E qs. 1292, a n dth e procedure is continued by th e process

i p m n j 5 p m n

j , j 2ol fi j

11, L 2

on51

`

oµ52n

n

3i 21a µnl A m n

µn 1l , j 2

1 i 21b µnl B m n

µn 1l , j 24,

i q m n j 5 q m n

j , j 2ol fi j

11,L 2

on51

`

oµ52n

n

3i 21a µnl B m n

µn 1l , j 2

1 i 21b µnl A m n

µn 1l , j 24, 1342

i a µnl 5 11 2 f 2i 21a µn

l 1 f a nl i p µn

l ,

i b µnl 5 11 2 f 2i 21b µn

l 1 f b nl i q µn

l , 1352

until all the coefficients show no significant improve-ment .

The nu merical fa ctor f 10 , f # 1 2 is intr oduced in Eqs.

1352 to improve the convergence property of the linea rs y st e m . I n ou r a c t u a l ca l cu l a t i on s , s om e m u l t i -sphere systems do not converge if f 5 1 , but they doconverge wh en th e value of f is redu ced to, sa y, 0.7.

6. Total Scattered Field and the Internal Fields

Once all t he int era ctive coefficient s, a m n j , b m n

j , c m n j , a n d

d m n j , a re f ound, the t ota l scat tered fi eld of the entire

cl us t er i n t h e p ri m a r y coor d in a t e s y st e m ca n b e


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ob tained w hen the translat ion addi t ion theorem isa pplied aga in, wh ich is expressed in the forms

Es 5 on 51

`

om 52n

n

i E m n 3a m n Nm n 132 1 b m n Mm n

132 4,

Hs 5k

vµon 51

`

om 52n

n

E m n 3b m n Nm n 132 1 a m n Mm n

132 4, 1362

w here

a m n 5 ol 51

L

on51

`

oµ52n

n

3a µnl A m n


µn 1l , j 024,

b m n 5 ol 51

L

on51

`

oµ52n

n

3a µnl B m n


µn 1l , j 024. 1372

I n Eqs . 1372 use ha s been ma de of th e properties

A m n µn 1 j 0, j 02 5 dµm dnn , 1382

B m n µn 1 j 0, j 02 5 0, 1392

w here dµm or dnn is the Kronecker del ta . Also, w eassume t ha t t he series expansion 1372 of the scatt eredfi eld is uniformly convergent. Therefore we can ter-minate the series a t n 5 N , a nd the resul t ing errorw i l l b e a r b i t r a r i l y s m a l l i f N is sufficiently large.This representa tion of the tota l scatt ering coefficientsdefined by Eqs. 1372 has been discussed by Borghese et

al .21 an d by Mackowski.18

In a similar ma nner, the expressions for the inter-nal fields of each sphere in the primary coordinatesyst em can be obta ined by the use of

E I 1 j , j 02 5 2 on 51

`

om 52n

n

i E m n 3d m n j , j 0Nm n

112 1 c m n j , j 0Mm n

112 4,

HI 1 j , j 02 5 2k j

vµ j on 51

`

om 52n

n

E m n 3c m n j , j 0Nm n

112 1 d m n j , j 0Mm n

112 4,

1402

w here

c m n j , j 0 5 o

n51

`

oµ52n

n

3c µn j A m n

µn 1 j , j 02 1 d µn j B m n

µn 1 j , j 024,

d m n j , j 0 5 o

n51

`

oµ52n

n

3d µn j B m n

µn 1 j , j 02 1 c µn j A m n

µn 1 j , j 024. 1412

The translation coefficients in either Eqs. 1372 or 1412a r e e ss en t i a l ly t h e s a m e a s t h os e u s ed i n E q s . 1302except tha t the f ormer are gov erned b y the H ankelfunction of the first kind and the latter are governedby the B essel function of the fi rst kind .

7. Scattering Properties of the Sphere Cluster

With a ll the int era ctive sca tt ering coefficients solved,we now proceed to derive expressions for the scatter-ing properties of the cluster in terms of these coeffi-

cients . When electromagnetic w av es i l luminate acollection of particles, the electromagnetic energy ofthe incident radiat ion is usual ly not only scat teredbut a lso absorbed by the pa rticles. The presence ofthe part icles resul ts in ext inct ion of the incidentw a v es . S imilar to the approach used b y Bohren andH uf f man29 for deriving th e expressions for cross sec-tions of a sin gle sphere, we obta in t he expressions forcross sections of a cluster of spheres by calculating thenet ra te a t w hich electromagn etic energy crosses th e

surface of an imaginary sphere enclosing the wholecluster. Here we consider only the case in which thesurrounding medium is nonabsorbing.

An electromagnetic wave of arbitrary polarizationcan be represented by a column vector, the Stokesv ector , the f our elements of w hich are the S tokesparameters . I n general , the s ta te of polar izat ion ofa n i n ci de nt w a v e i s ch a n g ed on i n t er a c t ion w i t hscat terers, w ith the exception of perfectly symmetr icpar ticles. Thus it is possible to represent the scat ter-ing propert ies of scat terers b y a 4 3 4 scat ter ingma trix, known as the Mu ller m at rix, tha t describesthe rela t ion b etw een the incident and the scat teredS tokes v ectors . All the informat ion a b out a ngular

scattering by a collection of particles is contained ini t s s ca t t e ri ng m a t r i x. Th e f or m of t h e s c a t t e ri n gmat r ix refl ects general propert ies of the scat terers.This scattering matrix is related to the 2 3 2 a mpli-tude-scat ter ing ma tr ix tha t describes the rela t ionb et w e en t h e a m p li t u de s o f t h e i n ci de nt a n d t h escat tered fields . A deriv at ion of a nalyt ical expres-sions for the four elements of the am plitude-sca tt er-ing mat rix is given below. Also, a derivation of theexpressions for the cross sections from the far-fieldapproximat ion a nd t he optical theorem is given, andit confirms that the results are consistent with thoseobta ined with th e exact integrat ion.

A. Extinction and Scattering Cross Sections

Once we have obta ined the interna l and t he scat teredelectromagnetic fields, we are able to determine theP oynting vector a t any point in space. For an y pointoutside the cluster, the complex Poynting vector 1i.e.,the time-avera ged P oynting vector 2 S can b e w ri t tena s29

S5 Si 1 Ss 1 Sext , 1422

w here

Si 5 1 ⁄2 R e1E i 3 H*i 2, Ss 5 1 ⁄2 R e1Es 3 H*s 2,

Sext 5 1 ⁄2 R e1E i 3 H*s 1 Es 3 H*i 2. 1432

The superscript * s tands , as usual , f or the complexconjugate. Si , the complex Poynting vector associ-ated with the incident wave, is independent of posi-t ion i f t he surrounding medium is nonab sorb ing.Ss i s the complex P oynting v ector of the scat teredfield. Sext ma y be interpreted as t he term that a risesf rom the interact ion b etw een the incident and thescat tered w a v es . Let W a be th e rat e at w hich electr o-


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ma gnetic energy is absorbed by t he scatt erers, W s b et h e r a t e a t w h ich en er g y is s ca t t e red a cr os s t h esurface A , a n d W ext be the sum of the energy absorp-tion and the energy scattering ra tes. Then we canwrite29

W a 5 W ext 2 W s , 1442

w here

W ext 5 2

eA

Sext · ir d A , W s 5 2

eA

Ss · ir d A , 1452

a nd A i s t h e s u r f a c e o f a n i m a g i n a r y s p h e r e t h a tencloses the scat terers . Equa t ions 1452 can a lso b ewrit ten in the form 29

W ext 51

2 R e e

0

2p

e0

p

1E i fH *s u 2 E i uH *s f

2 E s uH *i f 1 E s fH *i u2r 2 sin ud udf,

W s 51

2 R e e

0

2p

e0

p

1E s uH *s f 2 E s fH *s u2r 2 si n ududf,

1462

w here r , t h e r a d iu s of t h e im a g in a r y s ph er e, i sar bitrar y. According to Eq s. 1172 a nd 1362 and expres-sions 122, t h e com p on en t s of t h e i nci d en t a n d t h escat tered fi elds are, respectively,

E i u 5on 51

`

om 52n

n

E m n 12i p m n 0 c8n tm n 1 q m n

0 cn pm n 2ex p1i m f2

k r ,

E i f 5on 51

`

om 52n

n

E m n 1i q m n 0 cn tm n 1 p m n

0 c8n pm n 2exp1i m f2

k r ,

H i u 5k

vµ0 on 51

`

om 52n

n

E m n

3 1i p m n

0

cn tm n 2 q m n

0

c8n tm n 2

exp1i m f2

k r ,

H i f 5k

vµ0 on 51

`

om 52n

n

E m n

3 1i q m n 0 c8n pm n 1 p m n

0 cn tm n 2exp 1i m f2

k r , 1472

E s u 5on 51

`

om 52n

n

E m n 12i a m n j8n tm n 2 b m n jn pm n 2exp1i m f2

k r ,

E s f 5on 51

`

om 52n

n

E m n 12i b m n jn tm n 2 a m n j8n pm n 2exp1i m f2

k r ,

H s u 5k

vµ0 on 51

`

om 52n

n

E m n

3 1i a m n jn pm n 1 b m n j8n tm n 2exp 1i m f2

k r ,

H s f 5k

vµ0 on 51

`

om 52n

n

E m n

3 1i b m n j8n pm n 2 a m n jn tm n 2exp 1i m f2

k r , 1482

where the a rgument cos u of the functions pm n a nd tm n

ha s been suppressed for succinctness, a nd t he Ricca ti–Bessel functions are defined by

cn 1r2 5 r j n 1r2, jn 1r2 5 rh n 1121r2. 1492

By sub st i tut ing the series expansions 1482 i n t o t h eintegral 1462 for W s a n d i n t eg r a t i n g t h e r es u lt i n gproduct series term by term , we obta in

W s 52p 0E 0 0 2

k vµ0 on 51

`

om 52n

n

n 1n 1 1 212n 1 1 2 1n 2 m 2!1n 1 m 2!

3 R e12i j*n j8n a m n a *m n 1 i jn j8*n b m n b *m n 2, 1502

where we ha ve used the relat ions

E m n E *m n 5 0 E 0 0 212n 1 1 2231n 2 m 2!

1n 1 m 2! 42

, 1512

e0

2p

exp1i m f23ex p1i m 8f24*df

5 2pdm m 8

, 1522

e0

p

1pm n pm n 1 tm n tm n2sin udu

5 dnn

2n 1n 1 1 2

2n 1 1

1n 1 m 2!

1n 2 m 2!. 1532

I n E q . 1502, j*n j8n a nd jn j8*n are f unct ions of r , t h earb i trary ra dius of an imaginary sphere. When r issufficient ly lar ge, the spherical Ha nkel function of thefirst kind is asymptotically given by

h n 112

1k r 2 ,

12i 2n ex p1i k r 2

i k r , k r :n 2, 1542

an d t he Riccat i–Bessel function jn a nd i ts derivat ivej8n with respect to k r a re given by

jn , 12i 2n 11ex p1i k r 2, j8n , 12i 2n exp1i k r 2, 1552

whence

i jn j8*n 5 2 i j*n j8n 5 1. 1562

The scat tering cross section is t hus

C sca 5

W s

I i

54p

k 2 o

n 51

`

om 52n

n

n 1n 1 1 212n 1 1 2

31n 2 m 2!

1n 1 m 2! 1 0a m n 0

2 1 0 b m n 022, 1572

w here I i is the incident irradiance, which is k 0 E 0 02@2vµ0. Similarly, by substituting the expansions 1472


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an d (48) into th e integra l 1462 f or W ext and using Eqs .1512, 1522, a n d 1532, w e ob t a i n t h e e xt i n ct i on cr os ssection a s

C ext 5W ext

I i

54p

k 2 o

n 51

`

om 52n

n

n 1n 1 1 212n 1 1 2

31n 2 m 2!1n 1 m 2!

R e1 p m n 0* a m n 1 q m n

0* b m n 2. 1582

Here we ha ve used the relat ions

Re1 p m n 0 a *m n 2 5 R e1 p m n

0 *a m n 2,

R e1q m n 0 b *m n 2 5 R e1q m n

0 *b m n 2. 1592

Also, when r is sufficiently la rge,

cn 1r2 , cos 3r 2n 1n 1 1 2p

2 4 ,

c8n 1r2 , 2 s in 3r 2n 1n 1 1 2p

2 4 ,2i 1c8n j*n 1 c*n j8n 2 5 i 1cn j8*n 1 c8*n j8n 2 5 1. 1602

The absorption cross section is simply obtained by

C a bs 5 C ext 2 C sca . 1612

B. Expressions for the Far Field and the

Amplitude-Scattering Matrix

The scatt ered fi eld in th e far zone of the ensemble isof t en of p r a ct i ca l i nt e r es t . Wh en t h e a s y m pt ot i c

expressions 1552 of th e Riccat i–Bessel functions for t hescat tered electric field ar e substituted int o the series1482, the resulting t ra nsverse components ar e

E s u , E 0

exp1i k r 2

2i k r on 51

`

om 52n

n

12n 1 1 21n 2 m 2!

1n 1 m 2!

3 1a m n tm n 1 b m n pm n 2ex p1i m f2,

E s f , E 0

ex p1i k r 2

2i k r on 51

`

om 52n

n

12n 1 1 21n 2 m 2!

1n 1 m 2!

3 1a m n pm n 1 b m n tm n 2i exp1i m f2. 1622

In practical applications, these series are supposed tob e truncat ed a t some n 5 N , and the resulta nt errorincurred because of the truncation is assumed to bei n si gn i fi ca n t . H ow e ve r, w e s t i ll w r i t e t h e s u m ma -tion ov er a range of 11, ` 2 a bout n . B e ca u s e of t helinearity of the boundary conditions, the amplitude ofthe scat tered fi eld is a l inear f unct ion of t he a mpli-t u d e o f t h e i nci de nt fi e l d. Th e r el a t i on b et w e enincident a nd scat tered fi elds is conveniently wr itten-

in a ma trix form,

(63)1 E s

E 's 2 5 ex p3i k 1r 2 z 24

2i k r 3S 2 S 3

S 4 S 141 E i

E 'i 2 ,

where th e elements S j 1 j 5 1, 2, 3, 4 2 of th e a mplitude-scat ter ing mat r ix S depend, in genera l, on both u a ndf. The components 1E s , E 's 2 a nd 1E i , E 'i 2 o f t h es ca t t e r ed a n d t h e i nci de nt fi e l ds a r e p a r a l le l a n dperpendicular to the scattering plane, respectively.The scattering plane is uniquely defined by the direc-tion of the incident wave vector k and the scat ter ingdirection ir . F or a g en er a l c a s e, t h e se pa r a l l el a n dperpendicular components do not have simple rela-tions with the transverse-field components 1E u, E f2.H ow ever, there is a special case in w hich explici te xp r es s ion s f or t h e e le me n t s of t h e a m p l it u d e-scat tering ma trix can be found.

If a single plane-incident wave is considered andthe incident wave vector k defines the z axis of thereference system, the two components of the incidentelectric field for a n a rbitra ry l inear polar izat ion a ngleb a re

E i 5 E 01cos f cos b 1 s in f s in b2,

E 'i 5 E 01sin f cos b 2 cos f s in b2, 1642

w here b 5 0 cor r es pon d s t o a ca s e of t h e p la n ex -p ol a r i ze d i n ci d en t w a v e a n d b 5 90° t o t h ey -polarized incident wa ve. Thus we ha ve

1 E s

E 's 2 5 1

E s u

2E s f2

5exp3i k 1r 2 z 24

2i k r 3S 2 S 3

S 4 S 143E 0 cos 1f 2 b2

E 0 s in 1f 2 b24 . 1652

This case is i llustra ted in Fig. 2. Using expressions1622 and the equat ions

p2m ,n 5 1212m 111n 2 m 2!

1n 1 m 2! pm n ,

t2m ,n 5 1212m 1n 2 m 2!

1n 1 m 2! tm n , 1662

leads to

S 2 cos 1f 2 b21 S 3 s in 1f 2 b2

5 on 51

`

om 50

n

1Cm n cos m f 1 Fm n i sin m f2,

S 4 cos 1f 2 b21 S 1 s in 1f 2 b2

5 i on 51

`

om 50

n

1Um n cos m f 1 Jm n i si n m f2,

1672


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w here

Cm n 52n 1 1

1 1 d0m 31n 2 m 2!

1n 1 m 2! 1a m n tm n 1 b m n pm n 2

1 1212m 1a 2m n tm n 2 b 2m n pm n 24 ,

Fm n 52n 1 1

1 1 d0m

31n 2 m 2!

1n 1 m 2! 1a m n tm n 1 b m n pm n 2

2 1212m 1a 2m n tm n 2 b 2m n pm n 24 ,

Um n 52n 1 1

1 1 d0m 31n 2 m 2!

1n 1 m 2! 1a m n pm n 1 b m n tm n 2

2 1212m 1a 2m n pm n 2 b 2m n tm n 24 ,

Jm n 52n 1 1

1 1 d0m 31n 2 m 2!

1n 1 m 2! 1a m n pm n 1 b m n tm n 2

1 1212m 1a 2m n pm n 2 b 2m n tm n 2

4 . 1682

I t f o l low s that the f our elements of the ampli tude-

scat tering ma trix can be derived to be

S 21u, f2 5 on 51

`

om 50

n

5Cm n cos 31m 2 1 2f 1 b4

1 i Fm n sin 31m 2 1 2f 1 b46,

S 31u, f2 5 2on 51

`

om 50

n

5Cm n sin 31m 2 1 2f 1 b4

2 i Fm n cos 31m 2 1 2f 1 b46,

S 41u, f2 5 2on 51

`

om 50

n

5i Um n cos 31m 2 1 2f 1 b4

2 Jm n sin 31m 2 1 2f 1 b46,

S 11u, f2 5 on 51

`

om 50

n

5i Um n si n 31m 2 1 2f 1 b4

1 Jm n cos 31m 2 1 2f 1 b46. 1692

With the four elements of the amplitude-scattering

ma trix known, the sca tt ering ma trix, i .e., t he Mu llermat r ix, w hich rela t es the incident and the scat teredSt okes para meters, can be found ea sily 3for example,s e e B o h r e n a n d H u f f m a n , 29 equations 13.1224. Onemust a lw a ys b ear in mind that E qs . 1692 are va lid onlyw hen the z axis is para llel to the direction of propaga -tion of the plan e-incident wa ve. For this part icularcase, we derive the cross sections from the opticaltheorem as a check of Eqs . 1572 a nd 1582, w h i c h w eobta ined for a genera l case in S ubsection 7.A. Weuse V for the vector scattering amplitude, which isrelated to the scala r am plitude-scat tering ma trix S a sfollows:

V 5 3S 2 cos 1f 2 b2 1 S 3 s in 1f 2 b24iu

2 3S 4 cos 1f 2 b2 1 S 1 s in 1f 2 b24if. 1702

Accordingly, th e cross sections can be obtained by t heuse of th e followin g equa tions 29:

C sca 5 e0

2p

e0

p 0V 0 2

k 2 sin ududf, 1712

C ext 54p

k 2 R e31V ·iV2u504, 1722

w here

iV 5 ix cos b 1 iy s in b 5 s in u cos 1f 2 b2ir

1 cos u cos 1f 2 b2iu 2 s in 1f 2 b2if. 1732

B y i ns er t i ng E q s . 1672 a nd 1682 i n t o E q s . 1712, w eaga in obta in the scat tering cross section, which is ofe xa c t ly t h e s a m e f or m a s i n E q . 1572. Wit h u 5 0,

Fig. 2. When the direct ion of propagat ion of a plane-incident

wa ve is para llel to the z axis, th e components of the scat tered fi eld

1E s , E 's 2 ar e r a t h e r s im p ly r e la t e d t o t h e c om p on e n t s of t h e

incident wa ve 1E i , E 'i 2. I n t h i s ca s e E s 5 E us , E 's 5 2E fs a nd

the incident electric vector is in the x –y plane.


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E q s . 1672 become

S 210, f2cos 1f 2 b2 1 S 310, f2si n 1f 2 b2

5 on 51

` 2n 1 1

2

3 3a 1n 1 b 1n 2 n 1n 1 1 21a 21n 2 b 21n 24cos f

1 on 51

` 2n 1 1

2

33a 1n 1 b 1n 1 n 1n 1 1 21a 21n 2 b 21n 24i sin f,

S 410, f2cos 1f 2 b2 1 S 110, f2si n 1f 2 b2

5 2 on 51

` 2n 1 1

2

3 3a 1n 1 b 1n 1 n 1n 1 1 21a 21n 2 b 21n 24i cos f

1 on 51

` 2n 1 1

2

3 3a 1n 1 b 1n 2 n 1n 1 121a 21n 2 b 21n 24sin f, 1742

wh ere we ha ve used the followin g properties:

pm n 112 5 51

2m 5 21

n 1n 1 1 2

2m 5 1

0 ot h er w ise,

pm n 112 5

52

1

2m 5 21

n 1n 1 1 2

2m 5 1

0 ot h erw ise.

1752

From Eqs. 1702, 1732, and 1742 it follows t ha t

Re3S 1024 5 R e31V · iv 2u504

5 on 51

`

12n 1 1 2Re3 p 1n 0*a 1n 1 q 1n

0*b 1n

1 n 21n 1 1221 p 21n 0* a 21n 1 q 21n

0* b 21n 24, 1762

w h er e w e h a v e a l so u sed t h e r el a t ion s f rom t h eproperties of complex numbers

Re31a 21n 1 b 21n 2exp12i f24 5 R e31a *21n 1 b *21n 2exp1i f24,

R e1 p 21n 0 a *21n 2 5 R e1 p 21n

0* a 21n 2,

R e1q 21n 0 b *21n 2 5 R e1q 21n

0* b 21n 2, 1772

a n d E q s . 1212, wh ich give rise to

p 1n 0 5 q 1n

0 5exp12i b2

2,

p 21n 0 5 2

exp12i b2

2n 1n 1 1 2, q 21n

0 5ex p12i b2

2n 1n 1 1 2. 1782

As a result , from Eq s. 1722, or equiva lently from

C ext 54p

k 2 R e3S 1024, 1792

w e arr iv e a t

C ext 54p

k 2 o

n 51

`

12n 1 1 2Re3 p 1n 0*a 1n 1 q 1n

0*b 1n

1 n 21n 1 1 221 p 21n 0* a 21n 1 q 21n

0* b 21n 24, 1802

This agrees with what we obtained in Eq. 1582, whichturns out to be exactly the sa me result because p m n

0 5q m n

0 5 0 for all norma l modes wit h m fi 61 in th is ca se.

C. Summary of the Expressions for Scattering Properties of

the Clusters

The results obta ined so far in t his section ar e summa -rized a s follows:

1a 2 For the general case, the cross sect ions aregiv en b y Eqs . 1572, 1582, a n d 1612; no general explicite xp r es s ion s f or t h e e le m en t s of t h e a m p l it u d e-scat tering mat rix are given.

1b 2 When th e direction of propaga tion of a plane-incident wa ve is par allel to the z axis, the formula forthe scat ter ing cross sect ion remains the same a s in

E q . 1572; the expression for the extinction cross sectionis simplified as shown in Eq. 1802, because the expan-sion coefficients for all the modes other than m 5 61v a n is h i n t h i s c a s e, a n d t h e e xp r es si on s f or t h eelement s of the amplitude-scat tering ma tr ix a re givenb y E q s . 1692 w i t h 1682 f or an arb i trary polar izat ionangle of b. Wh en b 5 0, the incident plane wave is x

polarized, and when b 5 90°, i t is y polar ized. Forthese tw o part icular s ta tes of l inear polar izat ion ofthe plane-incident wave, the explicit expressions forthe extinction cross section and the four elements ofthe amplitude5scatt ering ma trix ca n be writ ten as

C ext

x

5

2p

k 2 on 51

`

12n 1 1 2

3 R e3a 1n 1 b 1n 2 n 1n 1 1 21a 21n 2 b 21n 24,

C exty 5

2p

k 2 o

n 51

`

12n 1 1 2

3 R e5i 3a 1n 1 b 1n 1 n 1n 1 121a 21n 2 b 21n 246,

1812


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S 2x 1u, f2 5 o

n 51

`

om 50

n

3 3Cm n cos 1m 2 1 2f 1 i Fm n si n 1m 2 1 2f4,

S 3x 1u, f2 5 2 o

n 51

`

om 50

n

3 3Cm n si n 1m 2 1 2f 2 i Fm n cos 1m 2 1 2f4,

S 4x 1u, f2 5 2 o

n 51

`

om 50

n

3 3i Um n cos 1m 2 1 2f 2 Jm n si n 1m 2 1 2f4,

S 1x 1u, f2 5 o

n 51

`

om 50

n

3 3i Um n sin 1m 2 1 2f 1 Jm n cos 1m 2 1 2f4,

1822

S 2y

1u, f2 5 2 on 51

`

om 50

n

3 3Cm n si n 1m 2 1 2f 2 i Fm n cos 1m 2 1 2f4,

S 3y 1u, f2 5 2 o

n 51

`

om 50

n

3 5Cm n cos 31m 2 12f4 1 i Fm n s in 1m 2 12f6,

S 4y 1u, f2 5 o

n 51

`

om 50

n

3 3i Um n sin 1m 2 1 2f 1 Jm n cos 1m 2 1 2f4,

S 1y 1u, f2 5 o

n 51

`

om 50

n

3 3i Um n cos 1m 2 1 2f 2 Jm n si n 1m 2 1 2f4,

1832

w i t h Cm n , Fm n , Um n , a n d Jm n given by Eqs. 1682, a n dthe superscript x or y indicates the state of polariza-tion of the incident pla ne wa ve.

8. Some Preliminary Numerical Results

The f ormulat ion described in this paper has b eenimplemented in a computer code. We do not intendto discuss the numerical resul ts in detai l . S ome ofour numerical calculat ions a re presented for i l lustra -tion. The theoretical predictions are compar ed withthe laboratory scattering measurements obtained byWang and Gusta f son 33 through a microw a v e ana logtechnique. The examples shown here ar e the angu-lar distributions 1phase functions i 11 a nd i 222 a t a fi x e d

orienta tion for six sets of sphere chains, ea ch consist-i n g of t w o or t h r e e i d en t i ca l s ph er es i n v a r i ou sintersphere separat ions . For a l l the cases , the axisof symmetry of each sphere chain is perpendicular toeither the scattering plane 1x –z plan e2 or the in cidentwa ve vector 1along the z axis 2. I n ot h e r w o rd s , i t isa lw a ys paral lel to the y axis. The polariza tion com-ponents of scattering intensities i 11 a nd i 22 corr espondto the scattered-field components, perpendicular orpara llel to the sca tt ering plane, respectively. In our

calculations,

i 11 5 0 S 1y 1u, 02 0 2,

i 22 5 0 S 2x 1u, 02 0 2. 1842

These dimensionless quantities are independent ofthe measur ement or computa tional units used. Thephysical and geometric parameters of t he chains ofspheres are l is ted in Tab le 1. Figure 3 show s thecomparison of our theoretical calculations against thecorresponding experimenta l data f or these spherechains.

9. Remarks

1a 2 The Mie theory is a special case of the mult i-sphere theory described a b ov e. For the Mie case,i .e. , t he case of a l inear ly polarized plane-incidentw av e propagat ing a long the z axis , a l l t he f ormula-tions turn out to be in complete agreement with theMie f ormulat ion w hen L 5 1. Wh en t h e p la n e-incident wa ves is x polarized, Eq s. 1142, 1182, 1292, a n d1782 show tha t

a 1n 5a n

2, b 1n 5

b n

2,

a 21n 5 2a n

2n 1n 1 1 2, b 21n 5

b n

2n 1n 1 1 2, 1852

Table 1. Sphere-System Parameters

I D

No. of

Spheres

in theC ha i n

n

Single-

Sphere

SizePa r a m et er

x

Complex

RefractiveIndex

m 5 m 8

2 i m 9

Dimensionless

SeparationPa r a m et er

k s a

1 2 3.083 1.61 2 i 0.004 6.1662 2 3.083 1.61 2 i 0.004 8.030

3 2 4.346 1.63 2 i 0.010 9.9404 2 4.346 1.63 2 i 0.010 10.760

5 3 3.083 1.61 2 i 0.004 6.1666 3 3.083 1.61 2 i 0.004 7.520

a s is the center-to-center separation distance between the twoneighboring spheres.


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which, together with E qs. 1572, 1812, and 1822, lea ds to

C scax 5

2p

k 2 o

n 51

`

12n 1 1 21 0a n 02 1 0 b n 0

22,

C ext

x

5

2p

k 2 on 51

`

12n 1 1 2Re1a n 1 b n 2,

S 2x 1u2 5 S 2

x 1u, f2

5on 51

` 2n 1 1

n 1n 1 1 2 1a n tn 1 b n pn 2,

S 3x 1u2 5 S 4

x 1u2 5 S 3x 1u, f2 5 S 4

x 1u, f2 5 0,

S 1x 1u2 5 S 1

x 1u, f2

5on 51

` 2n 1 1

n 1n 1 1 2 1a n pn 1 b n tn 2, 1862

w here w e ha v e defi ned pn 5 p1n , tn 5 t1n . Wh en t h eplane-incident wa ve is y polar ized, we ha ve

a 1n 5 2i a n

2, b 1n 5 2

i b n

2,

a 21n 5 2i a n

2n 1n 1 1 2, b 21n 5

i b n

2n 1n 1 1 2, 1872

and f rom Eqs. 1572, 1812, a n d 1832 similar expressionscan be obtained for th e cross sections and the a mpli-tude-scat ter ing mat r ix. These results are exact lyth e same a s in the Mie th eory.

1b 2 As shown in Section 7, Eqs. 1692 are valid onlywhen the direction of propagation of a linearly polar-ized plane-incident w a v e defines the z a x i s o f t h ereference coordina te system. In fact, for th e case of aplane-incident w a v e t hat is usual ly considered inpractice, this restriction does not lose any generality.When th e wa ve vector is initially not pa ra llel to the z

a x i s , w e c a n a l w a y s r o t a t e t h e p r i m a r y c o o r d i n a t es ys tem s o t h a t t h e z a x is of t h e n e w s y st em i sconsistent wit h the incident wa ve vector. This is a

trivial m at ter of relative geometric orientat ion1c2 Although this paper is basically devoted to a

general scat ter ing t heory f or mult isphere systemsan d does not specifically discuss the n umerical tech-niques needed in practical applications, i t is worthmentioning here tha t the calculat ion of t he tra nsla-tion coefficients is critica l to th e problem. As point edout above, the addition theorems play a key role indeveloping the scat tering t heory for a multispheresystem. I n pract ical applicat ions , calculat ing thevector a ddition coefficient s, i.e., the tr a nsla tion coeffi-cients, is not wit hout dif ficulty. The evaluat ion ofthese coefficients requires the determination of theso-called Gaunt coefficients by the use of a definite

integral of a product of three a ssociated Legendref unctions , as show n in Appendix B . These Gauntcoefficients ar e closely related to Clebsch–G ordancoefficients, which are often encountered in quantummechanics,34,35 especially in the calculation of transi-t ion amplitudes. Mackow ski18 has obtained a set ofrecursion relations for the tra nslat ion coefficientstha t bypass th e calculat ion of the G aun t coefficients.We ha v e dev eloped an a lgori thm to calculate theG a unt coefficients directly, wh ich enables us to evalu-at e the vector ad dition coefficients expeditiously.Also, a new microw av e lab oratory has b een set uprecently in our department by Gustafson, and the new fa cility is expected to provide more scat terin g measur e-

m en t s o f b et t e r q u a l i t y s oon . O ur n u m er i ca l a n dexperimental results will be discussed in details inpapers that a re now in prepara tion.

1d 2 In S ection 6, we discussed the total scatt eredfi eld from the sphere cluster as a w hole. The single-field representation described there is based on reex-pansions of all the individual scattered fields from thespherical constituents about a common origin. Ouractua l calculations show t ha t t his may be not the bestwa y to construct the tota l scattered field. We encoun-t e r ed s om e n u m er i ca l p r ob le m s i n s om e ca s e s.Further research needs to b e done b ef ore usef ulconclusions can be dra wn .

Appendix A: Recurrence Formulas for pmn , tmn

Definitions

pm n 5m P n

m 1x 2

11 2 x 221@2,

tm n 5 11 2 x 221@2d P n

m 1x 2

d x , 21 # x # 1. 1A12

Fig. 3. Angular dis tribut ions of six sphere chains. The para m-

eters of these sphere chains are listed in Table 1. 112– 162, identifi ca-

tion numbers of the sphere cha ins involved. The dotted curve in

each pa nel is the t heoretical prediction for i 11, an d the solid curve is

for i 22. The open circles in each panel are the laboratory scat ter-

ing measurements for i 11, and t he filled circles ar e for i 22.


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Recurrence Rela tions

pm n 11 52n 1 1

n 2 m 1 1 x pm n 2

n 1 m

n 2 m 1 1 pm n 21,

pm 11n 521m 1 1 2x

11 2 x 221@2 pm n

21m 1 1 21n 1 m 21n 2 m 1 1 2

m 2 1

pm 21n ,

m fi 1,

pnn 5 11 2 x 21@2n 12n 2 12

n 2 1 p1n 2121n 212, n fi 1,

1A22

tm n 5n 2 m 1 1

m pm n 11 2

n 1 1

m x pm n , m fi 0,

t0n 11 52n 1 1

n x t0n 2

n 1 1

n x t0n 21, 1A32

p2m n 5 1212m 11 1n 2 m 2!1n 1 m 2!

pm n ,

pm n 12x 2 5 1212n 1m pm n 1x 2, 1A42

t2m n 5 1212m 1n 2 m 2!

1n 1 m 2! tm n ,

tm n 12x 2 5 1212n 1m 11tm n 1x 2. 1A52

Special Values

p00 5 0, p015 0, p10 5 0, p11 5 1,

t00 5 0, t0,1 5 211 2 x 221@2, t10 5 0, t11 5 x ,

1A62

tm n 1612 5 5162n 111122 m 5 21

162n 113n 1n 1 1 2

2 4 m 5 1

0 ot herw ise,

tm n 1612 5

52162n 1122 m 5 21

162n 3n 1n 1 1 2

2 4 m 5 1

0 ot herw ise.

1A72

Appendix B: Translation Coefficients A0mn

mn , B 0mn

mn

A 0µnm n 1l , j 2 a nd B 0µn

m n 1l , j 2 are the tra nslat ion coeff i-cients needed for the transformation from the l t h t ot he j th coordina te system. 19

A 0µnm n 1l , j 2 5 1212µi n2n

2n 1 1

2n1n 1 1 2 op 50n 2n0

n 1n

12i 2 p

3 3n 1n 1 1 2 1 n1n 1 1 2 2 p 1 p 1 1 24

3 a 1m , n , 2 µ, n, p 2h p 1121k d l , j 2

3 P p m 2µ1cos ul , j 2exp3i 1m 2 µ2fl , j 4,

B 0µn

m n 1l , j 2 5 1212µi n2n 2n 1 1

2n1n 1 12

3 op 50n 2n0

n 1n

12i 2 p b 1m , n , 2µ, n, p , p 2 12

3 h p 112 1k d l , j 2P p

m 2µ1cos ul , j 2exp3i 1m 2 µ2fl , j 4,

1B 12

w here

b 1m , n , 2µ, n, p , p 2 1 2

52p 1 1

2p 2 131n 2 µ21n 1 µ1 12a 1m , n , 2µ2 1, n, p 2 12

2 1 p 2 m 1 µ21 p 2 m 1 µ 2 1 2

3 a 1m , n , 2µ 1 1, n, p 2 1 2

1 2µ1 p 2 m 1 µ2a 1m , n , 2µ, n, p 2 1 24, 1B 22

a n d t h e a 1m , n , µ, n , p 2 terms are defin ed by

P n m 1cos u 2P n

µ1cos u2

5 op 50n 2n0

n 1n

a 1m , n , µ, n, p 2P p m 1µ1cos u2, 1B 32

which ca n be also writ ten a s

a 1m , n , µ, n, p 2 52p 1 1

2

1 p 2 m 2 µ2!

1 p 1 m 1 µ2!

3 e21

1

P n m 1x 2P n

µ1x 2P p m 1µ1x 2d x . 1B 42

I n Eqs . 1B 12, 1d l , j , u l , j , f l , j 2 are the spherical coordinatesof the center of the l th sphere in the j th coordinatesystem. For the tra nslat ional coeff icients used inE q s . 1372 a nd 1412, h

p

112 is replaced by j p .

For the scattering theory presented in this paper,th e tra nsla tion coefficient s defin ed above need a slightmodifi cation, which is shown by Eq s. 1272.

The aut hor tha nks B o Å. S. G usta fson for inspiringthis pa per a nd for ma ny useful discussions a nd S . F.D e r m o t t , t h e C h a i r m a n o f t h e D e p a r t m e n t o f A s -tronomy of the University of Florida, for continuingsupport an d encouragement. Special tha nks also goto S. F. Dermott and Bo Å. S. Gustafson for improv-


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ing th e En glish. The a uth or a lso expresses his a ppre-ci a t i on t o g r a d u a t e s t u d e nt s S . J a y a r a m a n a n d S .Kortenka mp for t heir ca reful review of the ma nuscript.The author is deeply grateful to an anonymous ref-eree for his valua ble comment s an d careful rea ding ofa p re vi ou s v er s ion of t h i s p a p er. Th i s w or k w a ssupported by NASA t hrough gra nts NAG W-2482,NAG W-1923, an d NAG W-2775.

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