Dyadic formulation of morphology-dependent resonances. III. Degenerate perturbation theory

154 J. Opt. Soc. Am. B/Vol. 19, No. 1 /January 2002 Ng et al.

Dyadic formulation of morphology-dependentresonances.

III. Degenerate perturbation theory

Sheung-wah Ng and Pui-tang Leung

Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China

Kai-ming Lee

Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

Received March 16, 2001

Based on the completeness of morphology-dependent resonances (MDRs) in a dielectric sphere and the asso-ciated MDR expansion of the transverse dyadic Green’s function, a generic perturbation theory is formulated.The method is capable of handling cases with degeneracies in the MDR frequencies, which are ubiquitous insystems with a specific symmetry. One then applies the perturbation scheme to locate the MDRs of a dielectricsphere that contains several smaller spherical inclusions. To gauge the accuracy and efficiency of the pertur-bation scheme, we also use a transfer-matrix method to obtain an eigenvalue equation for MDRs in these sys-tems. The results obtained from these two methods are compared, and good agreement is found. © 2002Optical Society of America

OCIS codes: 290.0290, 290.4020, 260.2110, 260.5740, 000.3860, 000.4430.

1. INTRODUCTIONWhen droplets or dielectric spheres of sizes comparablewith optical wavelengths were illuminated with laserlight, various intriguing optical phenomena, includingelastic scattering, resonant fluorescence, stimulated Ra-man scattering, stimulated Brillouin scattering, lasing,and enhanced energy transfer were observed.1–8 The un-derlying physical mechanism of these phenomena, inwhich enhanced reaction rates at certain frequencieswere found, is the cavity resonant effect that is due to themorphology-dependent resonances (MDRs) that areformed in dielectric microspheres.2,3 These MDRs are infact metastable states and reveal themselves directly assharp spikes in the elastic cross section whose widths areinversely proportional to their respective lifetimes.2,3,9

In mathematical terms, MDRs correspond to complex fre-quency poles of the scattering matrix and are eigensolu-tions of the Maxwell equations that are subject to theboundary condition of the outgoing wave at infinity.10–13

MDRs can also be conceived of as light waves reflected in-ternally near the surface of a dielectric sphere at nearglancing angle, and providing feedback to various linearand nonlinear optical phenomena in microspheres.2,3

MDRs constitute a powerful tool to analyze optical phe-nomena in microspheres2,3; however, the spatial wavefunctions of MDRs grow exponentially at largedistances.11–15 This exponential growth readily pre-cludes the formulation of a regular MDR mode expansionand a generic perturbation scheme applicable to MDRs,despite the fact that it is desirable to study ways in whichvarious physical factors can alter MDRs by perturbativeexpansion.

Recently, the present authors and others, in a series of

0740-3224/2002/010154-11$15.00 ©

papers, considered the properties of MDRs in dielectricspheres and showed that MDRs form a complete set un-der certain conditions.11–15 Based on the completenessrelation, a practical perturbative expansion for MDRswas obtained.12 However, the perturbation scheme de-veloped so far applies only to systems in which either de-generacy in the MDR frequencies is absent or degeneratestates are not coupled by the perturbation. For example,the MDRs of a uniform dielectric sphere that contains asingle spherical inclusion can be solved with this nonde-generate perturbative scheme as long as the quantizationaxis passes through the center of the sphericalinclusion.12 However, it is obvious that this scheme failsto locate the MDRs if the sphere is doped with more thanone inclusion, which would break the cylindrical symme-try of the system. To examine generically how the MDRsin a uniform dielectric sphere (e.g., a microdroplet) will beaffected by various kinds of perturbation, such as thepresence of bubbles, latex spheres, and shapedistortions,16–20 a robust perturbation theory that dulytakes care of the problem of degeneracies is called for.

Our aim in the present paper is to extend the pertur-bation theory established recently12 to handle cases inwhich the coupling between degenerate states is nonneg-ligible. By adopting the MDR expansion and rearrang-ing the expansion series for the dyadic Green’s function ofthe perturbed system,11,12,21 we show that the first-ordershifts in the MDR frequencies can be obtained from theeigenvalues of a finite-dimensional matrix that measuresthe coupling among MDRs in a degenerate subspace in away akin to the conventional degenerate perturbationmethod for normal modes. Likewise, one can obtainhigher-order corrections by summing the contributions

2002 Optical Society of America

Ng et al. Vol. 19, No. 1 /January 2002 /J. Opt. Soc. Am. B 155

from other possible transitions. Besides resolving theproblem of degeneracy, this method properly takes thevector nature of electromagnetic waves into account andthus applies to optical modes with different polarizations(TE or TM). We consider it a robust scheme for remedy-ing the drawbacks of other methods proposed prev-iously.12,22–24

The structure of this paper is as follows: First, in Sec-tion 2 we review the completeness relation of MDRs andthe MDR expansion of the transverse dyadic Green’s func-tion. Second, in Section 3 we introduce a matrix repre-sentation for the transverse dyadic Green’s function and,in turn, formulate and resum a perturbation series for it.Third, in Section 4, by diagonalizing the Green’s matrix inthe degenerate subspace and examining shifts in its com-plex frequency poles on perturbation, we obtain the eigen-frequency shifts of MDRs. To illustrate the working prin-ciple and feasibility of our scheme, in Section 5 we applythe perturbation series to locate MDRs of a dielectricsphere in which are embedded several eccentric inclu-sions, which acts as a prototype model for microdropletsdoped with multiple impurities.16–18 The results ob-tained from perturbation expansion are then comparedwith the numerical results obtained from a transfer-matrix (T-matrix) method, which is detailed in AppendixA for reference.25 Finally, we conclude our paper in Sec-tion 6 by briefly discussing some potential applications ofthe perturbation method whose development we have de-scribed.

2. COMPLETENESS RELATION OF MDRsTo make this paper self-contained, we briefly review herethe completeness relation of MDRs in spherically sym-metric dielectric systems.12,13 Hereafter we shall adoptGaussian units and the convention that c 5 1. Besides,standard notation for Maxwell’s equations is alsoassumed.26 Consider a nonmagnetic dielectric spherewith dielectric constant e(r), which depends only on ra-dius r. In the absence of any free charge and current, atime-harmonic magnetic field B(r, t) 5 b(r)exp(2ivt)satisfies

¹ 3 F 1

e~r !¹ 3 bG 2 v2b 5 0. (1)

Wave equation (1) for magnetic field b(r), together withthe transversality condition ¹ • b 5 0 and the outgoingwave boundary condition at infinity, defines the MDRs ofthe system.11,12

Following directly from the transversality condition,the magnetic fields of TE and TM MDRs, respectively, canbe expressed as

b1jlm 51

iv1jl¹ 3 F f1jl~r !

rX lmG , (2)

b2jlm 5f2jl~r !

rX lm , (3)

where Xlm is the vector spherical harmonics with angularmomentum quantum numbers $lm%, n is the polarizationindex and n 5 1 (2) for the TE (TM) mode, and fnjl(r) and

vnjl are, respectively, the radial eigenfunction and theeigenfrequency, with j 5 1, 2,... labeling the radial quan-tum number such that 0 < Re vn1l , Re vn 2l , Re vn 3l, ... . Substituting this expression into Eq. (1), we findthat

F2d

drr~r !

d

dr1 r~r !

l~l 1 1 !

r2 2 r~r !e~r !vnjl2 G fnjl 5 0,

(4)where r(r) 5 1 for the TE mode and r(r) 5 1/e(r) for theTM mode. MDRs of the sphere are then defined as eigen-modes of Eq. (4), satisfying the usual outgoing-waveboundary condition at infinity.11,12 Eigenfrequency vnjlmust be a complex number to satisfy the boundary condi-tions. It can be shown from stability argument thatIm vnjl , 0 as long as e . 0 everywhere, which is as-sumed in our paper.

Moreover, we define the associated conjugate vectorfield, bnjlm

† , by

b1jlm† [

i

v1jl¹ 3 F f1jl~r !

rXlm* G , (5)

b2jlm† [

f2jl~r !

rXlm* , (6)

where Xlm* is the complex conjugate of Xlm . Hence theinner product between the magnetic fields of two MDRscan be defined as

^^bnjlmubn8j8l8m8&&

[ limX → `

Er,X

d3r bnjlm†

• bn8j8l8m8

1i

vnjl 1 vn8j8l8

e~X !21/2

3 E d3rd ~r 2 X !bnjlm†

• bn8j8l8m8 . (7)

It can be shown that the inner product is well defined andthat ^^bnjlmubn8j8l8m8&& } dnn8d jj8d ll8dmm8 .11 In the ensu-ing discussion we normalize the MDRs to unity; i.e.,^^bnjlmubn8j8l8m8&& 5 dnn8d jj8d ll8dmm8 .

The physical significance of the magnetic (or electric)fields of MDRs is that the fields form a complete set fortransverse vector functions under the condition that e(r)is discontinuous at certain values of r, say, r 5 a.11,12 Inparticular, they can be used as a convenient basis withwhich to expand the transverse dyadic d function (ortransverse projection operator) It(r, r8) that selects thedivergence-free component of any vector field and satis-fies the projection condition

E It~r, r8! • V~r8!d3r8 5 Vt~r!, (8)

where V(r) is an arbitrary vector field and Vt(r) is itsdivergence-free component, i.e., ¹ • Vt(r) 5 0 and¹ 3 (V 2 Vt) 5 0. For vector fields V(r) that vanish forr > a, it suffices to define the transverse dyadic d func-tion It(r, r8) for r, r8 , a, and it was correspondinglyshown that11

It~r, r8! 5 (njlm

bnjlm~r!bnjlm† ~r8!

2. (9)


This expansion clearly demonstrates the completeness ofMDRs for transverse vector functions and forms the foun-dation of the present paper.

Furthermore, the retarded dyadic Green’s function ofMaxwell’s equations,27,28 Gt

b(r, r8;v), defined by

¹ 3 F1

e¹ 3 Gt

b~r, r8;v!G 2 v2Gtb~r, r8;v! 5 It~r, r8!,

(10)

can also be expanded in terms of MDRs and takes the fol-lowing form11:

Gtb~r, r8;v! 5 (

njlm2

bnjlm~r!bnjlm† ~r8!

2vnjl~v 2 vnjl!, r, r8 , a.

(11)

Note that we have used a subscript t and a superscript bto signify that the Green’s function Gt

b is transverse in na-ture and governs the distribution of magnetic field b.11

This expansion is of vital importance to the analysis ofoptical phenomena in dielectric microspheres inasmuchas magnetic field B(r, t) 5 b(r)exp(2ivt) generated by aharmonic current source J(r, t) 5 j(r)exp(2ivt) can beneatly expressed in terms of the dyadic Green’s function:

b~r! 5 E Gtb~r, r8;v! • ¹r8 3 F4pj~r8!

e~r8!Gd3r8, (12)

which follows directly from Maxwell’s equations and thedefinition of the dyadic Green’s function.27–29 Once themagnetic field is known, the electric field can be obtainedin a straightforward manner. Thus the full set of MDRsis a powerful tool for determining the electromagneticfield generated by an arbitrary current source. Inas-much as each term in the MDR expansion of the dyadicGreen’s function is weighted with a resonance frequencydenominator 2vnjl(v 2 vnjl), MDR frequencies canequally be defined as poles of the transverse dyadicGreen’s function. We make use of this point to develop aperturbation series for the eigenfrequencies of MDRs.

Finally, electric field e (and its conjugate e†) of a MDRcan also be defined by means of Maxwell’s equations,11,26

namely,

enjlm 5i

e~r !vnjl¹ 3 bnjlm ,

enjlm† 5

2i

e~r !vnjl¹ 3 bnjlm

† . (13)

3. DEGENERATE PERTURBATION THEORYIn this section we generalize the nondegenerate perturba-tion theory for MDRs that was developed previously byour group of researchers11,12 to systems in which transi-tions among degenerate states, e.g., dielectric spheresdoped with multiple inclusions, have to be considered.

First we consider a spherically symmetric dielectricsystem characterized by a dielectric constant e0(r) thatdepends only on radius r and satisfies the condition of dis-continuity introduced above at r 5 a.11,12 We further as-sume that this system is simple enough that its MDRs

can be obtained without much difficulty and use it as astandard for comparison, namely, as an unperturbed sys-tem. As was discussed above, the transverse dyadicGreen’s function Dt

b for this system satisfies

¹ 3 F 1

e0~r !¹ 3 Dt

b~r, r8!G 2 v2Dtb~r, r8! 5 It~r, r8!

(14)

and can be expanded in terms of corresponding MDRs forr, r8 , a:

Dtb~r, r8! 5 2(

am

bam~0 ! ~r!bam

~0 !†~r8!

2va~0 !@v 2 va

~0 !#, (15)

where bam(0) (r), bam

(0)†(r8), and va(0) are, respectively, the

normalized magnetic field, the normalized conjugate mag-netic field, and the eigenfrequency of the (am)th MDR ofthe unperturbed system.11,12 Here a is a combined pa-rameter that stands for $njl% and m is the magnetic quan-tum number. As the unperturbed system is sphericallysymmetric, eigenfrequency va

(0) is independent of the val-ues of m.

Furthermore, it is also beneficial to introduce matrixnotation to simplify relevant equations. First we intro-duce a matrix representation for the Green’s function, D,defined as

Dam,bn 5 21

2

1

va~0 !@v 2 va

~0 !#dabdmn . (16)

Here am (or bn) is a combined matrix index. Similarly,we also define a row matrix B (0) and a column matrixB (0)†, respectively, by

Bam~0 ! 5 bam

~0 ! ~r!, (17)

Bam~0 !† 5 bam

~0 !†~r8!. (18)

It is then obvious that the dyadic Green’s function can beexpressed as the product of these three matrices, namely,

Dtb~r, r8! 5 B ~0 !DB ~0 !†. (19)

We further apply this matrix notation in the following dis-cussion.

We now proceed to consider the MDRs of a more com-plicated system, the perturbed system, described by an-other dielectric constant e(r), which probably depends onthe angular variables. In particular, the degeneracyamong the 2l 1 1 states with the same set of quantumnumbers $njl% can be broken. However, the transversedyadic magnetic Green’s function Gt

b(r, r8;v) for this sys-tem still obeys

¹ 3 F 1

e~r!¹ 3 Gt

b~r, r8;v!G 2 v2Gtb~r, r8;v! 5 It~r, r8!,

(20)

and its complex frequency poles again yield the MDR fre-quencies. So, by searching for the poles of Gt

b(r, r8;v),we can locate the MDRs of the perturbed system.

Here we show how to construct the Green’s functionGt

b(r, r8) of the perturbed system from that of the unper-turbed system in an iterative way. First we write


Gtb~r, r8! [ Dt

b~r, r8! 1 dGtb~r, r8! (21)

and substitute it into Eq. (20) to solve for dGtb(r, r8). It

is then readily seen that

dGtb~r, r8! 5 E

s,ad3s Dt

b~r, r8! • ¹

3 @dr~s!¹ 3 Gtb~s, r8!#, (22)

with dr(r) 5 e0(r)21 2 e(r)21 measuring the differencebetween the perturbed and the unperturbed systems.We also assume that dr(r) 5 0 for r . a throughout thispaper. In principle, Eq. (22) can be solved by iteration,yielding

Gtb 5 Dt

b 1 Dtb• DDt

b 1 Dtb• DDt

b• DDt

b 1 ¯ ,(23)

where the shorthand notation

~Dtb• DDt

b!~r, r8! 5 Es,a

d3s Dtb~r, s! • ¹

3 @dr~s!¹ 3 Dtb~s, r8!# (24)

is adopted. By virtue of the completeness of the MDRs inthe unperturbed system, the dyadic Green’s function Gt

b

can be expanded by these MDRs:

Gtb 5 (

am,bnbam

~0 ! ~r!Gam,bnbbn~0 !†~r8!, (25)

and, following directly from Eq. (23), the Green’s matrix Gin Eq. (25) is given explicitly by

G 5 D 1 DDD 1 DDDDD 1 .... (26)

Here D is the matrix that represents the perturbation,namely,

Dam,bn 5 Es,a

d3sbam~0 !†~s! • ¹ 3 @dr~s!¹ 3 bbn

~0 !~s!#.

(27)

Whereas D is a diagonal matrix, D and G are likely to ac-quire nonvanishing off-diagonal matrix elements. Inparticular, in an asymmetric perturbed system, e.g., a di-electric sphere doped with multiple inclusions, transitionsbetween degenerate modes with the same quantum num-bers njl and hence the same eigenfrequency but with dif-ferent magnetic quantum numbers m are allowed, and de-generacy would be lifted by the perturbation.

As remarked above, dyadic Green’s function Gtb and

therefore matrix G are singular at the MDR frequenciesof the perturbed system. In what follows, we shall locatethe MDRs by searching for the poles of matrix G. Con-sider a degenerate eigenfrequency vg

(0) of the unperturbedsystem, whose eigenstates, labeled by quantum numbers$gm% 5 $njlm%, form a (2l 1 1)-dimensional subspace.Henceforth we call it the g subspace. When we switch onthe perturbation, we expect that the perturbed frequen-cies vgp , where p 5 1, 2,..., 2l 1 1 is a new quantumnumber that labels the perturbed states in the subspace,will be close to the unperturbed frequency vg

(0) for suffi-ciently small perturbations. To locate the associated

poles of Gtb , one has to examine its behavior near

v ' vg(0) . However, we immediately see from Eq. (16)

that the dominant contributions to Eq. (26) are connectedwith the modes in the g subspace, and, more importantly,that these contributions increase indefinitely with the or-der of iteration in Eq. (26). Therefore, instead of truncat-ing the perturbation series at any finite order, we have toresum the infinite series to ensure that all the dominantcontributions are being counted correctly. That is, all thetransitions that link states in the degenerate g subspacehave to be considered.

To achieve this purpose, we write matrix D as the sumof a regular part H and a singular part K, i.e., D 5 H1 K. Here K is the principal part of D in Eq. (16) atv 5 vg

(0) , given explicitly by

Kam,bn 5 21

2

1

vg~0 !@v 2 vg

~0 !#dmn (28)

for a 5 b 5 g, and Kab 5 0 otherwise. Note that K infact describes free propagation of MDRs within the g sub-space, which is (2l 1 1)-fold degenerate. The regularpart H reads as

Ham,bn 5 21

2

1

va~0 !@v 2 va

~0 !#dmn (29)

for a 5 b Þ g and Hab 5 0 otherwise. Obviously, H cor-responds to free propagation of MDRs that do not belongto the g subspace. Besides, the self-energy matrix, W, isdefined by

W 5 D 1 DHD 1 DHDHD 1 .... (30)

After carefully counting and keeping track of the in-verse powers of v 2 vg

(0) , we can rewrite G in terms of D,K, H, and W as

G 5 D@I 1 WK 1 ~WK !2 1 ...#~I 1 WH !. (31)

The physical meaning of this expression is clear. Eachterm (WK) i (i 5 0, 1, 2,...) inside the brackets representsan infinite partial sum in Eq. (26), which is of the ith or-der in @v 2 vg

(0)#21. In other words, we have rearrangedthe series for G according to the expansion parameter@v 2 vg

(0)#21. Thus, summing the infinite series in Eq.(31) yields a more-compact form for G:

G 5 D~I 2 WK !21~I 1 WH !. (32)

Matrix G is a function of v and, following directly from itsdefinition, has poles at the MDR frequencies of the per-turbed system.

In general, G may contain nonzero off-diagonal ele-ments; however, we can, at least in principle, transform itinto a diagonal form with a similarity transformation.That is, there exists a matrix P such that matrix G, de-fined by

G 5 P21GP, (33)

is diagonal. As a consequence, the dyadic Green’s func-tion of the perturbed system assumes the following com-pact form:

Gtb~r, r8! 5 BGB†, (34)


where B 5 B (0)P and B† 5 P21B (0)† are, respectively, therow and the column vectors formed by the MDR eigen-functions of the perturbed system. Thus, by transform-ing matrix G with matrix P and looking for the poles ofthe diagonal elements of matrix G, one can then locatethe MDRs of the perturbed system.30

4. PERTURBATIVE SCHEMEInstead of carrying the diagonalization scheme in the fullinfinite-dimensional Hilbert space, we first diagonalizematrix G within the g subspace and in turn obtain a per-turbative expansion for the MDR frequencies. First, werewrite Eq. (32) as follows:

G 5 ~D21 2 WKD21!21~I 1 WH !. (35)

It is worthwhile to note that the matrix elements of D21

are explicitly given by

@D21#am,bn 5 22va~0 !@v 2 va

~0 !#dabdmn , (36)

and consequently the product KD21 [ G is the projectionoperator of the g subspace; i.e., Gam,bn 5 dabdmn fora 5 b 5 g and Gam,bn 5 0 otherwise.

Arranging the matrix entries such that the first2l 1 1 entries represent states in the g subspace andadopting block-matrix notation, we have

G 5 F Ig 0

0 0G , (37)

where Ig is the identity matrix of the g subspace, and

W [ FWg W2

W3 W4G . (38)

Consequently, the inverse of the first factor in Eq. (35),D21 2 WKD21, takes the following form:

D21 2 WKD21 5 D21 2 FWg 0

W3 0G , (39)

which is amenable to a perturbative expansion that wedetail below.

Let Pg be a matrix in the g subspace that transformsWg into a diagonal (or lower-triangular) matrix.30,31

Thus the matrix D21 2 WKD21 can accordingly be trans-formed into a lower-triangular form in the full space:

FPg21 0

0 1G ~D21 2 WKD21! FPg 0

0 1G5 D21 2 FPg

21WgPg 0

W3Pg 0G . (40)

This matrix will become singular if any one of the diago-nal elements in the upper-left block, namely, (D21)gg

2 Pg21Wg Pg , vanishes. Hence the MDR frequencies of

the perturbed system, vg , are given by the solutions ofthe following equation:

v 5 vg~0 ! 2 @lp/2vg

~0 !#, (41)

where lp 5 (Pg21Wg Pg)pp ( p 5 1, 2 ,..., 2l 1 1) are the

diagonal elements and hence the eigenvalues of Wg .The task of evaluating the MDR frequencies then reduces

to straightforward diagonalization of self-energy matrixWg , which is of dimension (2l 1 1) 3 (2l 1 1). How-ever, inasmuch as matrix Wg and therefore its eigenval-ues lp are usually frequency dependent, Eq. (41) is actu-ally an implicit equation for the MDR frequencies. Inwhat follows, we establish a perturbative expansion forits solution.

To solve Eq. (41), we first replace matrix D by mD,where m is a formal expansion parameter that measuresthe strength of the perturbation. Perturbed MDR fre-quency vgp can then be expanded as a power series in m:

vgp 5 vg~0 ! 1 (

s51

`

msvgp~s ! . (42)

Furthermore, we introduce a similarity transformationPd that transforms D into a diagonal (or lower-triangular)matrix in g subspace while it leaves other states outsidethe g subspace intact, and we define D 5 Pd

21DPd . It isworthwhile to note that D and Pd as well are frequencyindependent. Therefore, the eigenvalues of Wg can beexpanded as

lp 5 mDpp 1 m2~DHD !pp 1 O~m3! 1 ... . (43)

Substituting these two expansions into Eq. (41), we ob-tain the coefficients vgp

(s) order by order. The explicitforms of the two leading orders are

vgp~1 ! 5 2

vg~0 !

2Vgp,gp ,

vgp~2 ! 5

vg~0 !

4 (bÞg

(q

Vgp,bq

vb~0 !

@vg~0 ! 2 vb

~0 !#Vbq,gp ,

(44)

where the reduced matrix element Vgp,bq is defined byVgp,bq 5 Dgp,bq /@vg

(0)vb(0)#. In working out the second-

order expansion, we have already assumed that the first-order perturbation completely lifts the degeneracy; i.e.,that the differences between any two diagonal elements ofD are large compared with those of other, off-diagonal el-ements. However, this is not a stringent condition andcan be relaxed when necessary.

5. APPLICATIONIn this section we apply the degenerate perturbationtheory developed above to locate the MDRs of a dielectricsphere that contains several smaller dielectric sphericalinclusions. As shown in Fig. 1, the host sphere, centeredat the origin of the coordinate system, has a radius a anda uniform dielectric constant eI [ nI

2. There are N inclu-sions inside the host sphere (N 5 3 in Fig. 1). The ithinclusion (i 5 1, 2, ..., N) has a radius bi and is centeredat a point with spherical coordinates (di , u i , f i). With-out loss of generality, we assume that all inclusions havethe dielectric constant eII [ nII

2 that is different from thatof the host. Besides, we adopt a convention such that theinnermost inclusion lies on the z axis and d1 < d2 < ...< dN . This system is simple yet realistic. As a matterof fact, several attempts have been made to study MDRsof a droplet that contains many tiny latex inclusions, and


interesting experimental observations have been found inthese doped-droplet systems.16–18

To locate the MDRs of this system, we choose the hostsphere as the unperturbed system. This is a sphericallysymmetric system, and its MDRs form a complete set.The eigenfrequencies for MDRs with an angular momen-tum l satisfy11

%

rjl~nIvr !

d@rjl~nIvr !#

drU

r5a

51

rhl~1 !~vr !

d@rhl~1 !~vr !#

drU

r5a

,

(45)

where % 5 1(1/nI2) for the TE (TM) modes and jl(x) and

hl(1)(x) are, respectively, spherical Bessel and Hankel

functions of the first kind. This eigenvalue equation isindependent of magnetic quantum number m, and thusthe frequencies are in general 2l 1 1-fold degenerate.The normalized wave function of the magnetic field forr , a is

b1jlm 51

N1jlv1j¹ 3 @ jl~nIv1jlr !Xlm#,

b2jlm 51

N2jlj l~nIv2jlr !Xlm , (46)

where the normalization constants are given by

N1jl2 5 ~nI

2 2 1 !a3

2jl

2~nIv1jla !,

Fig. 1. Schematic diagrams showing a host dielectric spheredoped with three latex inclusions. Two different configurationsare shown here.

N2jl2 5 S 1 2

1

nI2D F S jl~nIv2jla !

jl~nIv2jla !1

1

nIv2jlaD 2

1l2 1 l

~v2jla !2G a3

2jl

2~nIv2jla !. (47)

When the inclusions are embedded in the host sphere,they introduce a constant perturbation dr 5 1/eI 2 1/eIIwithin the volumes that they occupy, which are denoted vi(i 5 1, 2, ..., N). To examine how these perturbationsaffect the MDRs in a degenerate multiplet, we apply thefirst-order perturbation theory developed here to this sys-tem. It is readily shown that matrix element Dam,bn is,to first order in de 5 eII 2 eI , equal to12

Dam,bn 5 vavb~eII 2 eI!(i51

N

Iam,bn~i ! , (48)

where

Iam,bn~i ! [ E

vi

d3r eam† ~r! • ebn~r! (49)

is the contribution of the ith inclusion to the matrix ele-ment.

As a concrete example, we consider here a TE multiplet(i.e., the g subspace defined above), accordingly, the MDRwave functions are11

egm† ~r! 5

1

Ng

jl~kgr !Xlm* ~V!, (50)

egn~r! 51

Ng

jl~kgr !Xln~V!, (51)

where kg [ nIvg . To evaluate Igm,gn(i) , we first rotate our

coordinate system such that the z axis of the new systempasses through the center of the ith inclusion and refer toit as the ith rotation. Using the rotational transforma-tion formula,32,33 we find that

Igm,gn~i ! 5

1

Ng2 (

m8(m9

Dm8ml* ~i !Dm9n

l~i !

3 Evi

d3r • jl~kgri!Xlm8* ~V i! • jl~kgri!Xlm9~V i!,

(52)

where Dm8ml (i) is the Wigner D function for the ith

rotation.32

Second, we further translate the origin of the rotatedsystem to the center of the ith inclusion. Applying thetranslation formulas for the Bessel functions and vectorspherical harmonics, namely,12,32,33

jl~kgr !Xlm* ~V! 5 (l8

Tl8mlm~i ! jl8~kgr8!Xl8m

* ~V8! 2 Vl8mlm~i ! ¹

3 @ jl8~kgr8!Xl8m* ~V8!#, (53)

jl~kgr !Xlm~V! 5 (l8

Tl8mlm~i ! jl8~kgr8!Xl8m~V8! 1 Vl8mlm

~i ! ¹

3 @ jl8~kgr8!Xl8m~V8!#, (54)


where

Tl8m8lm~i !

5 dmm8

~21 !mil82l

2 F ~2l 1 1 !~2l8 1 1 !

l~l 1 1 !l8~l8 1 1 !G1/2

3 (l9

il9@l8~l8 1 1 ! 1 l~l 1 1 !

2 l9~l9 2 1 !#~2l9 1 1 !S l8 l l9

0 0 0 D3 S l8 l l9

m 2m 0 D jl9~kgdi!, (55)

Vl8m8lm~i !

5 dmm8~21 !mil82l11dimF ~2l 1 1 !~2l8 1 1 !

l~l 1 1 !l8~l8 1 1 !G1/2

3 (l9

il9~2l9 1 1 !S l8 l l9

0 0 0 D3 S l8 l l9

m 2m 0 D jl9~kgdi!, (56)

we obtain

Igm,gn~i ! 5

1

Ng2 (

m8

Dm8ml* ~i !Dm8n

l~i !

3 H(l8

@Tl8m8lm8~i !

#2K1~kgbi!

2 @Vl8m8lm8~i !

#2K2~kgbi!J , (57)

where standard notation for 3jm symbols is adopted,32

and

K1~kgbi! 5 Eri8,bi

d3ri8jl8~kgri8!Xl8m* ~V i8!

• jl8~kgri8!Xl8m~V i8!

5bi

3

2@ jl8

2~kgbi! 2 jl811~kgbi!jl821~kgbi!#,

(58)

K2~kgbi! 5 Eri8,bi

d3ri8¹ 3 @ jl8~kgri8!Xl8m* ~V i8!#

• ¹ 3 @ jl8~kgri8!Xl8m~V i8!#

5 biH F ~kgbi!2

21 1G jl8

2~kgbi!

1 kgbi jl8~kgbi!jl88 ~kgbi!

2~kgbi!

2

2jl811~kgbi!jl821~kgbi!J . (59)

It then completes the calculation of matrix D, serving asthe input of the diagonalization scheme that yields thefirst-order shifts in the MDR frequencies.

In what follows, we present relevant numerical resultsfor a host dielectric sphere doped with three latex inclu-

sions, each with radius b. The refractive index of thehost sphere is nI 5 1.5, and b/a 5 0.15. As shown inFig. 1, the centers of the first two inclusions are located,respectively, at $0.17a, 0, 0% and $0.49a, p/2, 0%, whereasthat of the third inclusion is at $0.81a, p/2, 2p/2% [Fig.1(a)] or $0.81a, 2p, 0% [Fig. 1(b)]. This configuration isadopted because eigenfrequencies of the MDRs can alsobe obtained numerically from a transfer-matrix theory,which we detail in Appendix A.24,25

Figures 2 and 3, respectively, show the shifts in theMDR frequencies, dv 5 v 2 v (0), versus perturbationparameter dn 5 nII 2 nI for a TE MDR withv (0) 5 (2.68186, 20.42285), l 5 2, and j 5 2. We havedeliberately chosen a MDR with a large leakage rate todemonstrate the validity of the present scheme even inrather extreme cases, compared with other schemes forwhich a small leakage limit is assumed.22,23 The resultsshown in Figs. 2(a) and 3(a) are quite similar to those inFigs. 2(b) and 3(b). In fact, their difference is only mar-ginally observable. The unperturbed state, which is five-fold degenerate, splits into five MDRs with different fre-quencies on the introduction of the perturbation.However, dv is small for two of these five MDRs, hencethey are hardly distinguishable. In Figs. 2 and 3 we alsocompare the results obtained from the perturbation

Fig. 2. (a), (b) Real and imaginary parts, respectively, of theshift in the MDR frequencies dv (in units of a21) versus pertur-bation parameter dn for a TE MDR withv (0) 5 (2.68186, 20.42285), l 5 2, and j 5 2. The three inclu-sions are centered at $0.17a, 0, 0%, $0.49a, p/2, 0%, and$0.81a, p/2, 2p/2% [i.e., Fig. 1(a)]. Dotted curves and cross, re-sults obtained from the first-order perturbation theory and theT-matrix method, respectively.


method and the T-matrix method and note that the first-order perturbation formula in fact suffices to yield accu-rate results for not-too-large perturbations.

6. DISCUSSION AND CONCLUSIONTo augment the perturbation theory for MDRs developedrecently,11,12 which is based on the completeness of MDRsand properly takes the vector nature of electromagneticwaves into consideration from the outset, a useful pertur-bation scheme for degenerate MDRs has been establishedin the present paper. The scheme consists of a two-stageprocess. First, the self-energy matrix was transformedwith a new basis in which matrix D is diagonalized withinthe degenerate subspace, thus yielding an eigenvalueequation for the MDRs. Second, the eigenvalue equationwas solved by use of an ordinary perturbation method,and a Rayleigh–Schrodinger-like perturbation series forthe degenerate MDRs was established. We have appliedour scheme to study the MDRs in a dielectric sphere thatcontained several inclusions and found that even thefirst-order scheme sufficed to produce satisfactory results.

It is customary to use MDRs to analyze various opticalphenomena in microdroplets or other dielectric micro-spheres, where degeneracies in MDR frequencies are fre-quently encountered.1–8 The perturbation theory re-ported here is a robust method for studying ways in whichasymmetric perturbations remove degeneracy. For in-stance, MDRs in a perfect sphere with orbital angular mo-

Fig. 3. Similar to Fig. 2, except that the third inclusion is nowcentered at $0.81a, 2p, 0% instead of at $0.81a, p/2, 2p/2% [i.e.,Fig. 1(b)].

mentum l are degenerate with a degeneracy of 2l 1 1;however, these MDRs acquire different eigenfrequenciesif inclusions (or other asymmetric perturbations) are in-troduced into the system. This effect was clearly demon-strated in the numerical example discussed above. Al-though the spreading of degenerate MDRs onperturbation can also be obtained from an exact T-matrixmethod (which is introduced in Appendix A), the pertur-bation method provides a more convenient and a physi-cally more transparent scheme for examining ways inwhich the properties and locations of the inclusions mayaffect the distribution of the MDRs. It is also worthwhileto remark that the perturbation scheme works nicely ir-respectively of the configuration of the inclusions,whereas the T-matrix method has to be modified to caterto specific configurations.

We have already applied our theory to study the MDRsin droplets that contain thousands (or even as many asmillions) of inclusions and discovered that our methodsucceeded in interpreting essential features of experimen-tal observations.18

APPENDIX A: TRANSFER-MATRIXMETHODTo verify the validity of the perturbation method and alsoto gauge the accuracy of the results obtained, here we de-velop a transfer-matrix method for locating exact posi-tions of MDRs for the system considered above.24,25 De-spite the fact that the perturbation scheme is completelygeneral, for simplicity we consider only a layered-inclusion system in which the inclusions are placed insidea host sphere in such a way that (i) no other inclusion canoccupy the space inside the spherical shell defined bydi 2 bi , r , di 1 bi except the ith inclusion and (ii)the center of the host sphere is free of inclusions. Detailsof this method can be found elsewhere;34 the idea behindthis scheme is straightforward and is briefly outlined asfollows.

Consider a dielectric sphere with several inclusions, asshown schematically in Fig. 4. We use B1i , B2i , and B3ito denote, respectively, the spherical surfaces r 5 di2 bi and r 5 di 1 bi and the surface of the ith inclusion.

Fig. 4. Schematic diagram showing a layered-inclusion system.The large circle with center O is the host sphere, and the smallercircles are the inclusions. The ith inclusion, centered at a dis-tance di from O and inscribed between the spherical shells B1iand B2i , has a radius bi and a spherical surface B3i .


Besides, the region bounded by B1i and B2i but excludingthe inclusion, and the region inside the inclusion, are de-noted, respectively, by regions Ii and IIi . Without loss ofgenerality, we assume that the center of the innermost in-clusion is located on the z axis. In what follows, we workout the electric field of an MDR inside the host sphere.

First, we expand the electric field in terms of multipolefields in three different regions26:

(i) r , d1 2 b1

eIu1 5 (lm

alm~1 !

jl~nIky !

jl~nIka !Xlm~Vy! 1 blm

~1 !¹

3 F jl~nIky !

jl~nIka !Xlm~Vy!G

1 (lm

clm~1 !

hl~nIky !

hl~nIka !Xlm~Vy!

1 dlm~1 !¹ 3 F hl~nIky !

hl~nIka !Xlm~Vy!G ; (A1)

(ii) d2 2 b2 . r . d1 1 b1

eIu2 5 (lm

alm~2 !

jl~nIky !


~2 !¹

3 F jl~nIky !

jl~nIka !Xlm~Vy!G 1 (

lmclm

~2 !hl~nIky !

hl~nIka !Xlm~Vy!

1 dlm~2 !¹ 3 F hl~nIky !

hl~nIka !Xlm~Vy!G ; (A2)

(iii) the region inside inclusion 1 (i.e., region II1)

eII 5 (l8m8

Al8m8II

jl8~nIIky8!

jl8~nIIkb1!Xl8m8~Vy8! 1

Bl8m8II

nII2 ¹8

3 F jl8~nIIky8!

jl8~nIIkb1!Xl8m8~Vy8!G , (A3)

where $ y8, Vy8% denote a new spherical coordinate systemwhose origin is located at the center of this inclusion.

Second, we use the dyadic Green’s theorem for vectorfields to obtain a formal integral expression for the elec-tric field in region I1, yielding12,27,28,34

eI~ y ! 5 (lm

alm~2 !

jl~nIky !


~2 !¹

3 F jl~nIky !

jl~nIka !Xlm~Vy!G

1 (lm

clm~2 !

hl~nIky !

hl~nIka !Xlm~Vy! 1 dlm

~2 !¹

3 F hl~nIky !

hl~nIka !Xlm~Vy!G

1 (lm

Al8m8II Gl8

hl8~nIIky !

jl8~nIIka !Xl8m8~Vy8!

1 Bl8m8II Hl8¹ 3 Fhl~nIIky !

jl~nIIka !Xl8m8~Vy8!G ,

(A4)

where

Gl8 5 inIka@nIkb1 jl8~nIIkb1!jl88 ~nIkb1!

2 nIIkb1 jl8~nIkb1!jl88 ~nIIkb1!#, (A4)

Hl8 5 inIkb1F jl8~nIkb1!jl8~nIIkb1!S 1

nI2 2

1

nII2 D

1kb1

nIjl8~nIIkb1!jl8

8 ~nIkb1!

2kb1

nIIjl8~nIkb1!jl8

8 ~nIIkb1!G . (A4)

The next step is to find a matrix that relates the coef-ficients at the inner boundary B11 , @alm

(1) , blm(1) , clm

(1) , dlm(1)],

to those at the outer boundary B21 , @alm(2) , blm

(2) , clm(2) , dlm

(2)],by requiring that all boundary conditions be satisfied onB11, B21, and B31. To do this we have to invoke thetranslational formulas for the vector sphericalharmonics.28,34 After carrying out lengthy algebraic ma-nipulations, we get six sets of equations with ten un-knowns: alm

(1) , blm(1) , clm

(1) , dlm(1) , alm

(2) , blm(2) , clm

(2) , dlm(2) , Alm

II ,and Blm

II . By eliminating Al8m8II and Bl8m8

II , we get foursets of equations that relate the coefficients on innerboundary B11 to those on outer boundary B21, which canbe expressed in matrix form:

QF al~2 !

bl~2 !

cl~2 !

dl~2 !

G 5 PF al~1 !

bl~1 !

cl~1 !

dl~1 !

G . (A5)

Note that we have deliberately suppressed the m depen-dence in these equations, inasmuch as modes with differ-ent values of m are decoupled in this special case. Thusthe coefficients on boundary B21 are related to those onthe boundary B11 by


F al~2 !

bl~2 !

cl~2 !

dl~2 !

G 5 Q 2 1PF al~1 !

bl~1 !

cl~1 !

dl~1 !

G 5 M~v!F al~1 !

bl~1 !

cl~1 !

dl~1 !

G . (A6)

Explicit expressions for matrices P and Q and transfermatrix M(v) are highly complicated and are not reportedhere.34

After obtaining the transformation matrix that relatesthe field on inner boundary B11 to that on outer boundaryB21, we now rotate the coordinate system in such a waythat the new z axis passes through the center of the sec-ond innermost inclusions (i.e. i 5 2). We call it the sec-ond coordinate. The reason for doing this is that, afterrotating the coordinate, we can use a similar transforma-tion to relate the fields on boundaries B12 and B22. Ap-plying this strategem repeatedly, we obtain

F alm~n !

blm~n !

clm~n !

dlm~n !

G 5 MnDn,n21¯M3D3,2M2D2,1M1F alm~1 !

blm~1 !

clm~1 !

dlm~1 !

G ,

(A7)

where alm(n) , blm

(n) , clm(n) , and dlm

(n) are expansion coefficientsin the region dN 1 bN , r , a, Mi is the transfer ma-trix for the ith inclusion, and Di,i21 is the rotation matrixthat transforms the coefficients from the (i 2 1)th coor-dinate system to those of the ith coordinate system.

Finally, we expand the electric field outside the hostsphere as

eout 5 (lm

alm~o !

jl~nokr !

jl~noka !Xlm~V! 1 blm

~o !¹

3 F jl~nokr !

jl~noka !Xlm~V!G 1 (

lmclm

~o !hl~nokr !

hl~noka !Xlm~V!

1 dlm~o !¹ 3 F hl~nokr !

hl~noka !Xlm~V!G . (A8)

Matching appropriate boundary conditions at r 5 a, weobtain

F alm~o !

blm~o !

clm~o !

dlm~o !

G 5 S~v!F alm~n !

blm~n !

clm~n !

dlm~n !

G , (A9)

where S(v) is another matrix that depends on the prop-erties of the host sphere. Therefore we find a transfor-mation that relates the fields in the innermost region andthose outside the host:

F alm~o !

blm~o !

clm~o !

dlm~o !

G 5 SMnDn,n 2 1¯M3D3,2M2D2,1M1F alm~1 !

blm~1 !

clm~1 !

dlm~1 !

G5 T~v!F alm

~1 !

blm~1 !

clm~1 !

dlm~1 !

G . (A10)

Inasmuch as the eigenfunction of MDR satisfies theregular boundary condition at the origin and the outgoingboundary condition outside the sphere, we require that

clm~1 ! 5 0,

dlm~1 ! 5 0,

alm~o ! 5 0,

blm~o ! 5 0, (A11)

for a MDR eigenfunction. We rewrite overall transfermatrix T in Eq. (A10) in block form:

T~v! 5 FT11~v! T12~v!

T21~v! T22~v!G , (A12)

where Tij are matrices. It is then readily shown that thecondition for MDR frequency v is that the matrix T11 besingular. Exact numerical results can then be obtainedby solution of the eigenvalue equation uT11u 5 0.

ACKNOWLEDGMENTSOur study is supported in part by a grant from the HongKong Research Grants Council (grant CUHK4282/00P).K. M. Lee is thankful to be the recipient of a Chinese Uni-versity postdoctoral fellowship.

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29. Our definition for the transverse dyadic Green’s function isslightly different from the conventional one, which is re-lated to ours through direct differentiation. In addition toyielding the magnetic field of a localized current source, theGreen’s function defined by us is symmetric for the trans-position of the field and source points and can be expandedin terms of the tensor products of relevant MDR fields.

30. In general, both G and D are non-Hermitian and sometimescannot be diagonalized. However, it is always possible totransform these matrices into a triangular form, and ourderivation in this paper still holds.

31. See, e.g., S. S. M. Wong Computational Methods in Physicsand Engineering (Prentice-Hall, Englewood Cliffs, N. J.,1992).

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Dyadic formulation of morphology-dependent resonances. III. Degenerate perturbation theory