Electromagnetic scattering by 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 con fi gured spheres tha t a re isotropic and h omogeneous but may be of different size and composi tion. The pro fi le of inc ident electromagnetic waves is ar bitrary. The analysis is based on the framework of the Mie theory for a single sphere and the existing addition theorems for spherical vector wa ve functions. The classic Mie theory is generalized. Applying the extended Mie theory to all the spherical constituents in a n a ggregate simultaneously leads to a set of coupled linear equa tions in the unknown interactive coeffic ients. We propose an a symptotic iterat ion technique to solve for these coefficients. The total scatt ered fi eld of the entire ensemble is c onstructed with the interactive scattering coefficients by the use of the translational addition theorem a second time. Rigorous an a lytical expressio ns are derived for the c ross sections in a general case and for all the elements of the amplitude-sc a ttering ma trix in a special case of a plane-incident wa ve propaga ting a long t he zaxis. As an illustra tion, we present some of our preliminary numerical results and compare them with previously published laboratory scattering measurements. Key words: Scatt ering, particles, aggregat es. 1. Int rod uct ion Light scattering by a small body or a colle ct i on of small obj ec ts whose sizes are comparable to the wavelength of the incident radiation is a problem of grea t interest t o a broad range of scienti fi c disciplines. To date, exactly solvable problems include th e scatter- ing of a plane electromagnetic wave by a few highly symmetric types of single particles and by arbitrary con fi gurations of parallel, in fi nite cyl inders. Per- ha ps the scat tering theory tha t is most widely used is that for a homogeneous sphere of arbitrary size and refractive index, known as Mie theory, which was developed by Lorenz 1 and Mie. 2 The complete solu- tion for homogeneous, in fi nitely long, circular cylin- ders was fi rst given by Lord Rayleigh 3 for the case of perpendicular incidence and by Wait 4 for the case of oblique incidence. The problem of homogeneous spheroids was solved by Asano and Yamamoto. 5 A solution for the problem of scattering by a homoge- neous sphere coated with a homogeneous layer of uniform thickness was fi rst obtained by Aden and- Kerker. 6 T his is a n example of a particle with a spatially variable refractive index; their theory can also be generalized to a ra dially stra ti fi ed sphere. The mutual interaction complicates light scatter- ing by ensembles of particles. T he problem t here- fore requi res the use of a n addition theorem to tra 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 e r e r. Using a n a ddition theorem for cylindrical w aves, Twers ky 7–9 wa s able to calculate the sca ttered fi eld produced by a plane wave striking an arbi trary con fi guration of para llel in fi nite cylinders. For multiple scat tering by spheres, it is necessary to decompose the scattered fi eld of a sphere into spherical waves that impinge on the other spheres. Although Mie theory expresses the sca tt ered fi eld in the form of spherical vector wave functions by using the center of the sphere as the origin of the reference system, the problem of interacting spheres requires tha t one seek representa tions of the scat tered fi eld by the use of the same set of basis spherical vector wave functions that refer to any other arbitrary origin. In 1954, Friedman and Russek 10 reported a deriva- tion of such an expansion, or addition theorems for spherical scalar-wa ve function. With t he develop- ment of addition th eorems for spheric al vector w ave functions in the 1960’s by St ein 11 and Cruzen, 12 it became, in principle, feasible to solve theoretically for The author is with the Department of Astronomy, P.O. Box 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 @ APP LI ED O PT I CS 45 73
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
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
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
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:
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
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.
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 2 1 b µnl B 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 2 1 b µnl A 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
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
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 02 1 b µnl B m n
µn 1l , j 024,
b m n 5 ol 51
L
on51
`
oµ52n
n
3a µnl B m n
µn 1l , j 02 1 b µnl A 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-
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
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
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.
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,
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.
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.
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-
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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