Circular dielectric cavity and its deformations

Circular dielectric cavity and its deformations

R. Dubertrand,^1,

*

E. Bogomolny,¹N. Djellali,²M. Lebental,^1,2and C. Schmit^1,†

1Université Paris Sud, CNRS UMR 8626, Laboratoire de Physique Théorique et Modèles Statistiques, 91405 Orsay, France

2Ecole Normale Supérieure de Cachan, CNRS UMR 8537, Laboratoire de Photonique Quantique et Moléculaire, 94235 Cachan, France 共Received 30 August 2007; published 9 January 2008兲

The construction of perturbation series for slightly deformed dielectric circular cavity is discussed in detail.

The obtained formulas are checked on the example of cut disks. A good agreement is found with direct numerical simulations and far-field experiments.

DOI:10.1103/PhysRevA.77.013804 PACS number共s兲: 42.55.Sa, 03.65.Sq

I. INTRODUCTION

Dielectric microcavities are now widely used as mi- croresonators and microlasers in different physical, chemical, and biological applications 共see, e.g., 关1,2兴 and references therein兲. The principal object of these studies is the optical emission from thin dielectric microcavities of different shapes关3兴. Schematically such cavity can be represented as a cylinder whose height is small in comparison with its trans- verse dimensions共see Fig. 1兲. If the refractive index of the cavity isn₁ and the cavity is surrounded by a material with the refractive indexn₂⬍n₁ 共we assume that the permeabili- ties in both media are the same兲the time-independent Max- well’s equations take the form共see, e.g.,关4兴兲

ⵜជ _·Bជ_{j= 0, ⵜ}ជ_·nj²_Eជ_{j= 0,}

ⵜជ_⫻Bជ_{j= −in}²_jkEជ_{j, ⵜ}ជ_⫻Eជ_j=ikBជ_{j, 共1兲} where the subscript j= 1 共j= 2兲 denotes points inside 共out- side兲 the cavity and k is the wave vector in the vacuum.

These equations have to be completed by the boundary con- ditions which follow from the continuity of normalBជ_{␯ and} n²Eជ_␯ and tangentialEជ_␶andBជ_{␶components}

n₁²Eជ_1␯=n2²_Eជ_{2␯, B}ជ_1␯=Bជ_{2␯, E}ជ_1␶=Eជ_{2␶, B}ជ_1␶=Bជ_2␶. In the true cylindrical geometry, thezdependence of electro- magnetic fields is pure exponential:⬃e^iqz. Then the above Maxwell equations can be reduced to the two-dimensional Helmholtz equations for the electric field,Ejz, and the mag- netic fieldB_jz along the axe of the cylinder

共⌬+˜n_j²k²兲Ejz共x,y兲= 0, 共⌬+˜n_j²k²兲Bjz共x,y兲= 0 共2兲 with the following boundary conditions

E_1z=E_2z, B_1z=B_2z, ⳵^E1z

⳵␶ ⁼

⳵^E2z

⳵␶ ^,

⳵^B1z

⳵␶ ⁼

⳵^B2z

⳵␶ ^, and

1 n˜₁²

⳵^B1z

⳵␯ ⁻ 1

˜n₂²

⳵^B2z

⳵␯ ⁼

q共n₂²−n₁²兲 kn˜₁²n˜₂²

⳵^Ez

⳵␶^,

n₁² n˜₁²

⳵^E1z

⳵␯ ⁻ n₂²

˜n₂²

⳵^E2z

⳵␯ ⁼

q共n₂²−n₁²兲 kn˜₁²˜n₂²

⳵^Bz

⳵␶^{. 共3兲}

Here ˜n_j²=n²_j−q²/k² plays the role of the effective two- dimensional共in thex-yplane兲refractive index.

When fields are independent on z 共i.e., q= 0兲 boundary conditions共3兲do not mixBzandEzand the two polarizations are decoupled. They are called transverse electric共TE兲field whenE_z= 0 and transverse magnetic共TM兲field whenB_z= 0.

Both cases are described by the scalar equations

共⌬+˜n²_jk²兲⌿j共x,y兲= 0, 共4兲 where ⌿共x,y兲 stands for electric 共TM兲 or magnetic 共TE兲 fields with the following conditions on the interface between both media:⌿1=⌿2 and

⳵⌿1

⳵␯ ⁼

⳵⌿2

⳵␯ for TM polarization, 共5兲 1

n₁²

⳵⌿1

⳵␯ ⁼ 1 n₂²

⳵⌿2

⳵␯ for TE polarization. 共6兲 These equations are, strictly speaking, valid only for an infi- nite cylinder but they are widely used for a thin dielectric cavities by introducing the effective refractive index corre- sponding to the propagation of confined modes in the bulk of the cavity 共see, e.g., 关5兴兲. In practice, it reduces to small changes in the refractive indices 共which nevertheless is of importance for careful comparison with experiment关6兴兲. For simplicity, we will consider below two-dimensional equa- tions共4兲as the exact ones.

Only in very limited cases, these equations can be solved analytically. The most known case is the circular cavity共the

*remy.dubertrand@lptms.u-psud.fr

†Deceased.

x y

z

FIG. 1. Schematic representation of a dielectric cavity.

disk兲where variables are separated in polar coordinates. For other cavity shapes tedious numerical simulations are neces- sary.

The purpose of this paper is to develop perturbation series for quasistationary spectrum and corresponding wave func- tions for general cavities which are small deformations of the disk. The obtained formulas are valid when an expansion parameter is small enough. The simplicity, the generality, and the physical transparency of the results make such approach of importance for technological and experimental applica- tions.

The plan of the paper is the following. In Sec. II the calculation of quasistationary states for a circular cavity is reviewed for completeness. Special attention is given to cer- tain properties rarely mentioned in the literature. The con- struction of perturbation series for eigenvalues and eigen- functions of small perturbations of circular cavity boundary is discussed in Sec. III. The conditions of applicability of perturbation expansions are discussed in Sec. IV. The ob- tained general formulas are then applied to the case of cut disks in Sec. V. Some technical details are collected in the Appendixes.

II. DIELECTRIC DISK

Let us consider a two-dimensional circular cavity of ra- diusR made of a material with n⬎1 refractive index. The region outside the cavity is assumed to be the air with a refractive index of one. The two-dimensional equations共4兲 for this cavity are

共⌬+n²k²兲⌿= 0 when r艋R,

共⌬+k²兲⌿= 0 whenr⬎R. 共7兲

These equations describe the propagation of the electro- magnetic field inside a dielectric cavity. They can also be considered as a quantum problem for a particle moving in the following “potential”:

V共xជ兲=

再

^{−0,共n2− 1兲k2, rr⬎艋RR,}

冎

and throughout this paper we will often refer to this analogy using vocabulary related to the quantum problem.

There is no true bound states for dielectric cavities. The physical origin of the existence of long lived quasibound states is the total internal reflection of rays with the incidence angle bigger than the critical angle

␪c= arcsin 1

n. 共8兲

To investigate quasibound states one imposes outgoing boundary condition at infinity, namely, we require that far from the cavity there exist only outgoing waves

⌿共xជ兲⬀e^ik兩xជ兩 when兩xជ兩→⬁, wherexជlies in the cavity plane.

In cylindrical coordinates 共r,␪兲, the general form of the solutions is the following:

⌿共r,␪兲=

再

^{bammHJmm共1兲共nkr兲e共kr兲eimim␪␪,, rr⬎艋R,R}

冎

^共9兲

wherem= 0 , 1 , . . . is an integer共the azimuthal quantum num- ber兲related to the orbital momentum.Jm共x兲 关H_m^共1兲共x兲兴 stands for the Bessel function共the Hankel function of the first kind兲 of order m. Due to rotational symmetry, eigenvalues with m⫽0 are doubly degenerated.

By imposing the boundary conditions共5兲 or共6兲one gets the quantization condition

n

␯ J_m⬘ Jm

共nkR兲=H_m^共1兲⬘

H_m^共1兲共kR兲, 共10兲 where

␯⁼

再

¹ for TM polarization

n² for TE polarization.

冎

^共11兲

The quasistationary eigenvalues of this problem depend on azimuthal quantum numbermand on other quantum number p related with radial momentum: k=kmp. They are complex numbers

k=kr+iki, 共12兲 wherek_rdetermines the position of a resonance andk_i⬍0 is related with its lifetime.

In Fig.2 we plot solutions of Eq.共10兲obtained numeri- cally for a circular cavity with refractive indexn= 1.5. Points are organized in families corresponding to different values of radial quantum number p. The dotted line in these figures indicates the classical lifetime of modes with fixedmandk

→⬁. Physically these modes correspond to waves propagat- ing along the diameter whose lifetime is given by

Im共kR兲= 1

2nln

冉

^{nn− 1+ 1}

冊

^⯝− 0.536 48. 共13兲 In the semiclassical limit and for Im共kR兲ⰆRe共kR兲 simple approximate formulas can be obtained from the standard ap- proximation of the Bessel and Hankel functions 关7兴 when m⬍z,

Jm共z兲=

冑

_␲²共z²−¹m²兲^1/4cos

冉 ^冑

^{z2−m2−marccosm}z −␲ 4

冊

共14兲 and whenm⬎z,

H_m^共1兲共z兲= −i

冑

_␲²共m^e−2冑−^mz^2−z2兲^21/4

冋

^{mz +}

^{冑 冉}

^mz

^冊

^{2− 1}

册

^m.

共15兲 Denoting u= Re共kR兲 and v= Im共kR兲 and assuming that v Ⰶu,uⰇ1, andm/n⬍u⬍mone gets共see, e.g.,关8兴兲that the

real part of Eq. 共10兲 can be transformed to the following form

冑

n²u²−m²−marccos m nu−␲

4

= arctan␯

冑

n^m2u²²⁻−^um²²+共p− 1兲␲^, 共16兲 where the integer p= 1 , 2 , . . . is the radial quantum number and␯is defined in Eq.共11兲. The imaginary part of Eq.共10兲is then reduced to

v⬇− 2

␲u共n²− 1兲兩Hm共1兲共u兲兩²␨^, 共17兲 where␨= 1 for TM waves and␨^=n2u2/关m²共n²+ 1兲−n²u²兴for TE waves. Whenuandmare large,vis exponentially small as it follows from Eq.共15兲.

The above equations cannot be applied for the most con- fined levels 共similar to the “whispering gallery” modes in closed billiards兲for whichnuis close tom. In Appendix A it is shown that the real part of such quasi-stationary eigenval- ues withO共m⁻¹兲precision is given by the following expres- sion

x_m,p=m n +␩p

n

冉

^m2

冊

^1/3−_␯

冑

n¹²− 1+3␩2p

20n

冉

^m2

冊

^1/3

+ n²␩p

2␯共n²− 1兲^3/2

冉

3²␯^{2− 1}

冊冉

m²

冊

^{2/3, 共18兲}

where␩pis the modulus of thepth zero of the Airy function 共A2兲.

A more careful study of Eq.共10兲reveals that there exist other branches of eigenvalues with large imaginary part not visible in Fig.2. Some of them are indicated in Fig.3. These states can be called external whispering gallery modes as their wave functions are practically zero inside the circle. So they are of minor importance for our purposes. They can also be identified with above-barrier resonances. In Appendix A it is shown that in the semiclassical limit these states are re- lated with the complex zeros of the Hankel functions and they are well described asymptotically关withO共m⁻¹兲error兴as follows:

0 20 40

Re(kR)

−2

−1.5

−1

−0.5 0

Im(kR)

0 20 40

Re(kR)

−2

−1.5

−1

−0.5 0

Im(kR)

(a)

(b)

FIG. 2. Quasistationary eigenvalues for a circular cavity with n= 1.5.共a兲TM polarization;共b兲 TE polarization. Filled circles are deduced from direct numerical resolution of Eq.共10兲, while open squares indicate semiclassical approximation for these eigenvalues based on Eqs.共16兲and共17兲when兩kR兩⬍m⬍n兩kR兩.

0 20 40 60 80 100

Re(kR)

−10

−8

−6

−4

−2

Im(kR)

FIG. 3. 共Color online兲Additional branches of quasistationary eigenvalues for a circular cavity withn= 1.5. Circles indicate the TM modes and squares show the position of the TE modes. Solid and dashed lines represent the asymptotic result 共19兲 for respec- tively the TM and the TE modes.

x_m,p=m+

冉

^m2

冊

^{1/3␩pe−2i␲/3−}

冑

n^i2␯− 1+ 3e^−4i␲/3␩p2

20

冉

^m2

冊

^1/3

+ i␯␩pe^−2i␲/3

2共n²− 1兲^3/2

冉

^{1 −23␯2}

冊冉

^m2

冊

^{2/3, 共19兲}

with the same␩pas in Eq. 共18兲.

Similar equations have been obtained in关9兴.

III. PERTURBATION TREATMENT OF DEFORMED CIRCULAR CAVITIES

In the previous section we have considered the case of a dielectric circular cavity. It is one of the rare cases of inte- grable dielectric cavities in two dimensions. The purpose of this section is to develop a perturbation treatment for a gen- eral cavity shape which is a small deformation of the circle 共see Fig.4兲. We consider a cavity whose boundary is defined as

r=R+␭f共␪兲 共20兲 in the polar coordinates共r,␪兲. Here ␭ is a formal small pa- rameter aiming at arranging perturbation series.

Our main assumption is that the deformation function

␭f共␪兲is small,

兩␭f共␪兲兩ⰆR. 共21兲

Of course, for the quantum mechanical perturbation theory this condition is not enough. It is quite natural共and will be demonstrated below兲that the criterion of applicability of the quantum perturbation theory is, roughly,

␦^{ak2Ⰶ1, 共22兲}

where␦^a is the area where perturbation “potential” ␦^{n2 is} nonzero共represented by dashed regions in Fig.4兲.

To construct the perturbation series for the quasistationary states, we use two complementary methods. In Sec. III A we adapt the method proposed in 关10,11兴 for diffraction prob- lems. The main idea of this method is to impose the required boundary conditions共5兲or共6兲not along the true boundary of the cavity but on the circler=R. Under the assumption共22兲 this task can be achieved by perturbation series in␭. In Sec.

III B we use a more standard method based on the direct perturbation solution of the required equations using the Green function of the circular dielectric cavity. Both methods lead to the same series but they stress different points and may be useful in different situations.

For clarity we consider only the TM polarization where the field and its normal derivative are continuous on the di- electric interface. For the TE polarization the calculations are more tedious but follow the same steps. To simplify the dis- cussion we assume that the deformation function f共␪兲 is symmetric: f共−␪兲=f共␪兲 共as in Fig. 4兲. In this case the qua- sistationary eigenfunctions are either symmetric or antisym- metric with respect to this inversion. Then in polar coordi- nates, they can be expanded either in cos共p␪兲 or sin共p␪兲 series. The general case of nonsymmetric cavities is analo- gous to the case of degenerate perturbation series and can be treated correspondingly.

A. Boundary shift

The condition of continuity of the wave function at the dielectric interface states

⌿1关R+␭f共␪兲,␪兴=⌿2关R+␭f共␪兲,␪兴, 共23兲 where subscripts 1 and 2 refer respectively to wave function inside and outside the cavity. Expanding formally ⌿1,2 into powers of␭ one gets

关⌿1−⌿2兴共R,␪兲= −␭f共␪兲

冋

^⳵⳵^⌿r1−

⳵⌿2

⳵^r

册

^{共R,␪兲−12␭2f2共␪兲}

⫻

冋

^⳵⳵^2⌿r21−

⳵²⌿2

⳵^r2

册

^{共R,␪兲+ ¯. 共24兲}

For the TM polarization the conditions 共5兲 imply that the derivatives of the wave functions inside and outside the cav- ity along any direction are the same. Choosing the radial direction, one gets the second boundary condition

⳵⌿1

⳵^{r 关R+␭f共}␪兲,␪兴=⳵⌿2

⳵^{r 关R+␭f共}␪兲,␪兴, 共25兲 which can be expanded over␭ as follows:

冋

^⳵⳵^⌿r1−

⳵⌿2

⳵^r

册

^{共R,␪兲= −␭f共␪兲}

冋

^⳵⳵^2⌿r21−

⳵²⌿2

⳵^r2

册

^共R,␪兲

−1

2␭²f²共␪兲

冋

^⳵⳵^3⌿r31−

⳵³⌿2

⳵^r3

册

^共R,␪兲

+ ¯. 共26兲

We find it convenient to look for the solutions of Eqs.共24兲 and共26兲in the following form:

⌿1共r,␪兲=Jm共nkr兲

Jm共nx兲 cos共m␪兲+

兺

p⫽m

ap

Jp共nkr兲

Jp共nx兲 cos共p␪兲, 共27兲 FIG. 4. Example of a deformed circular cavity. Shaded areas

represent the regions where the refractive index differs from the one of the circular cavity.

⌿2共r,␪兲=共1 +bm兲H_m^共1兲共kr兲

H_m^共1兲共x兲 cos共m␪兲 +

兺

p⫽m

共ap+bp兲H_p^共1兲共kr兲

H_p^共1兲共x兲 cos共p␪兲. 共28兲 Here, and for all which follows,x stands forkR. These ex- pressions correspond to symmetric eigenfunctions. For anti- symmetric functions all cos共¯兲 have to be substituted by sin共¯兲.

From Eqs.共24兲and共26兲one concludes that the unknown coefficientsap, andbp have the following expansions:

ap=␭␣p+␭²␤p+ ¯, bp=␭²␥p+ ¯. 共29兲 Correspondingly, the quasistationary eigenvalue,kR⬅x, can be represented as the following series

x=x₀+␭x₁+␭²x₂+ ¯. 共30兲 Herex₀is the complex solution of Eq.共10兲which we rewrite in the form

Sm共x0兲= 0 共31兲 introducing for a further use the notation for allmandx,

Sm共x兲=nJ_m⬘ Jm

共nx兲−H_m^共1兲⬘

H_m^共1兲共x兲. 共32兲 The explicit construction of these perturbation series is pre- sented in Appendix B. The results are the following. The perturbed eigenvalue共30兲is

x=x₀

冋

^{1 −␭Amm+␭2}

^冉

^{12共3Amm2 −Bmm兲+x0共Amm2 −Bmm兲}

⫻H_m^共1兲⬘

H_m^共1兲共x0兲−共n²− 1兲x0_k

兺

⫽m

Amk

1

S_k共x₀兲Akm

冊 册

^{+O共␭3兲.}

共33兲 The coefficients of the quasistationary eigenfunction 共27兲 and共28兲are

ap=␭x0共n²− 1兲 1

Sp共x0兲

冠

^Apm+␭

再

^ApmAmm

^冉

^{Sx0p⳵⳵Sxp− 1}

^冊

+1

2Bpm

冋

^{1 +x0}

^冉

^{HHm共m共1兲1兲⬘+HHp共p共1兲1兲⬘}

^冊 册

^{+x0共n2− 1兲}

⫻_k

兺

⫽m

Apk

1

S_k共x₀兲Akm

冎 ^冡

^{+O共␭3兲 共34兲}

and

bp=␭²1

2x₀²共n²− 1兲Bpm+O共␭³兲. 共35兲 In these formulasAmnandBmn are the Fourier harmonics of the perturbation function f共␪兲and its square, given by Eqs.

共B5兲 and 共B10兲, respectively. The above expressions are quite similar to usual perturbation series andSp共x0兲plays the role of the energy denominator. In Eq.共34兲 and in the fol-

lowing we may omit the argument of the function when it is equal tox₀.

It is instructive to calculate the imaginary part of the per- turbed level from the knowledge of the first order terms only.

Assuming that Im共x0兲ⰆRe共x0兲and using the Wronskian re- lation共see, e.g.,关7兴, 7.11.34兲

H_m^共1兲⬘共x兲Hm共2兲共x兲−H_m^共2兲⬘共x兲Hm共1兲共x兲= 4i

␲^{x 共36兲} one gets from Eq.共33兲

Im共x兲= Im共x₀兲共1 +␳兲−␭²2x₀共n²− 1兲

␲ _p

兺

_⫽m兩Sp^A_H^mp2

p共1兲兩², 共37兲 where 关assuming that m⬎x₀ so H_m^共1兲⬘/H_m^共1兲 is real, cf. Eq.

共15兲兴

␳^{= −}␭Amm+␭²

冋

^{12共3Amm2 −Bmm兲}

+x₀共A_mm² −B_mm兲H_m^共1兲⬘

H_m^共1兲共x₀兲

册

^{. 共38兲}

Expression共37兲without the␳^correction关which is multiplied by Im共x0兲and so negligible for well-confined levels兴can be independently calculated from the following general consid- erations. From Eq.共7兲it follows that inside the cavity

n²共k²−k^*2兲

冕

V

兩⌿共xជ兲兩²dxជ=J, 共39兲

where

J=

冕

B

冉

^⌿*⳵␯^⳵ជ^⌿−⌿

⳵

⳵␯ជ^⌿*

冊

^{d␴ជ. 共40兲}

In the first integral the integration is performed over the vol- ume of the cavity,V, while the second integral is taken over the boundary of the cavity,B.␯ is the coordinate normal to the cavity boundary andJrepresents the current through the boundary. For the TM polarization, it equals the current at infinity.

From Eqs. 共28兲and共34兲 it follows that the field outside the cavity in the first order of the perturbative expansion is

⌿2共r,␪兲=H_m^共1兲共kr兲

H_m^共1兲共x兲 cos共m␪兲+␭共n²− 1兲x₀

⫻_p

兺

_⫽m^ApmH_S^共p_p¹_H^兲共kr兲

p

共1兲 cos共p␪兲.

The first order term proportional to cos共m␪兲 is taken into account by calculating H_m^共1兲共x兲 instead of H_m^共1兲共x0兲. The cur- rent can be directly calculated from Eq. 共36兲 or from the asymptotic of the Hankel functions. Then the current can be written, neglecting first order variation for the cos共m␪兲term:

J= 4i

冋

^{兩H1m共1兲兩2+␭2共n2− 1兲2x02p}

^兺

^{⫽m兩SpAHpm2m共1兲兩2}

册

^{. 共41兲}

To calculate the integral over the cavity volume in the lead- ing order, the nonperturbed function can be used inside the cavity. Then the integration over the circler=R leads to

冕

V

兩⌿共xជ兲兩²dxជ⬇ ␲ J_m²共nx0兲

冕

0

R

J_m²共nkr兲r dr. 共42兲 The last integral is共see关7兴, 7.14.1兲

冕

0 R

J_m²共nkr兲r dr=R²

2

冋

^{Jm⬘2共nkR兲+Jm2共nkR兲}

冉

^{1 −}共nkR^m2兲²

冊 册

^.

From the eigenvalue equation共10兲and the asymptotic共15兲, it follows that

nJ_m⬘共nkR兲

Jm共nkR兲⬇−

冑

m²/x²− 1.

Therefore,

冕

0 R

J_m²共nkr兲r dr⬇J_m²共nx₀兲共n²− 1兲R²

2n² . 共43兲 Combining these equations leads to

Imx= − 2

␲

冋

^{共n2− 1兲1x0兩Hm共1兲兩2+␭2共n2− 1兲x0p}

^兺

^{⫽m兩SpAHpm2m共1兲兩2}

册

^.

共44兲 According to Eq.共17兲the first term of this expression is the imaginary part of the unperturbed quasistationary eigenvalue 关assuming that Im共x0兲ⰆRe共x0兲兴and one gets Eq.共37兲using only the first order corrections. The missing terms are pro- portional to the imaginary part of the unperturbed level and can safely be neglected for the well confined levels.

These calculations clearly demonstrate that the deforma- tion of the cavity leads to the scattering of the initial well confined wave function with Re共kR兲⬍m⬍nRe共kR兲 into all possible states with different p momenta. Among these states, some are very little confined or not confined at all.

These states withp⬍Re共kR兲give the dominant contribution to the lifetime of perturbation eigenstates. Such scattering picture becomes more clear in the Green function approach discussed in Sec. III B.

The important quantity for applications is the far-field emission. It is calculated using the coefficientsa_p共34兲andb_p 共35兲in Eq.共28兲and substituting its asymptotic forH_p^共1兲共kr兲:

H_p^共1兲共kr兲→

r→⬁

冑

_␲²kre^{i共kr−␲p/2−␲/4兲}. 共45兲

Then one gets

⌿2共r,␪兲→

r→⬁

冑

_␲²kre^{i共kr−␲/4兲}F共␪兲, 共46兲 where

F共␪兲=共1 +bm兲e^−i␲m/2

H_m^共1兲共x兲cos共m␪兲 +_p

兺

⫽m

共ap+bp兲e^−i␲p/2

H_p^共1兲共x兲cos共p␪兲. 共47兲 The boundary shift method discussed in this section is a simple and straightforward approach to perturbation series expansions for dielectric cavities. As it is based on Eqs.

共23兲–共26兲, it first shrinks to zero the regions where the re- fractive index differs from its value for the circular cavity.

Consequently, the calculation of the field distribution in these regions remains unclear. Besides the direct continuation of perturbation series 共34兲 inside these regions diverges. To clarify this point, we discuss in the next section a different method without such a drawback.

B. Green function method

Fields in two-dimensional dielectric cavities obey the Helmholtz equations共2兲which can be written as one equa- tion in the whole space for TM polarization

关⌬+k²n²共xជ兲兴⌿共xជ兲= 0 共48兲 with position dependent “potential”n²共xជ兲. For perturbed cav- ity共20兲

n²共xជ兲=n₀²共xជ兲+␦ⁿ²共xជ兲 共49兲 wheren₀²共xជ兲is the “potential” for the pure circular cavity

n₀²共xជ兲=

再

^{n12 whenwhen兩x兩xជជ兩兩⬍⬎RR,}

冎

^共50兲

and the perturbation␦ⁿ²共xជ兲is equal to

共n²− 1兲 when f共␪兲⬎0 andR⬍兩xជ兩⬍R+␭f共␪兲,

−共n²− 1兲 when f共␪兲⬍0 andR+␭f共␪兲⬍兩xជ兩⬍R, 0 in all other cases.

共51兲 Hence the integral of ␦ⁿ²共xជ兲 with an arbitrary function F共xជ兲⬅F共r,␪兲 can be calculated as follows:

冕

^{␦n2共xជ兲F共xជ兲dxជ=共n2− 1兲}

冕

^d␪

冕

R R+␭f共␪兲

F共r,␪兲r dr. 共52兲 Equation 共48兲 with “potential” 共49兲can be rewritten in the form

关⌬xជ+k²n₀²共xជ兲兴⌿共xជ兲= −k²␦ⁿ²共xជ兲⌿共xជ兲. 共53兲 Then its formal solution is given by the following integral equation

⌿共xជ兲= −k²

冕

^{G共xជ,yជ兲␦n2共yជ兲⌿共yជ兲dyជ, 共54兲}

whereG共xជ,yជ兲 is the Green function of the equation for the dielectric circular cavity which describes the field produced

at pointxជby the delta-function source situated at pointyជ. The explicit expressions of this function are presented in Appen- dix C.

It is convenient to divide the xជ plane into three circular regionsr⬍R₁,R₁⬍r⬍R₂, andr⬎R₂where

R₁= min_␪关R,R+␭f共␪兲兴,

R₂= max_␪关R,R+␭f共␪兲兴. 共55兲 The boundaries of these regions are indicated by dashed circles in Fig. 4. Notice that the deformation “potential”

␦ⁿ²共xជ兲is nonzero only in the second regionR₁⬍r⬍R₂. Due to singular character of the Green function共cf. Appendix C兲, wave functions inside each region are represented by differ- ent expressions.

Let共r,␪兲be the polar coordinates of pointxជ. For simplic- ity we assume for a moment that f共␪兲艋0. Using Eq.共C10兲, Eq.共54兲in the regionr⬍R₁can be rewritten in the form

⌿共xជ兲=

兺

_p ^J_J^p_p^共_共^nkr_nx兲^{兲cos共p␪兲Lˆp关⌿兴, 共56兲}

whereLˆp关⌿兴 is the following integral operator:

Lˆp关⌿兴=x²共n²− 1兲

R²

冕

^{d␾cos共p␾兲}

⫻

冕

R R+␭f共␾兲

␳^d␳

冋

^{2␲xSJpp共共x兲Jnk␳p兲共nx兲}

+ i

4关H共1兲p 共kn␳兲Jp共nx兲−H_p^共1兲共nx兲Jp共kn␳兲兴

册

⫻⌿共␳^,␾兲. 共57兲 Assuming that we are looking for corrections to a quasista- tionary state of the nonperturbed circular cavity with the mo- mentum equal to m, one concludes that the quantized eigenenergies are fixed by the condition that the perturbation terms do not change zeroth order function共see, e.g., 关14兴兲, i.e.,

Lˆm关⌿兴= 1 共58兲

which can be transformed into S_m共x兲=x²共n²− 1兲

R²

冕

^{d␾cos共p␾兲}

⫻

冕

R R+␭f共␾兲

⌿共␳^,␾兲␳^d␳

冋

^{2␲JmxJ共nkm共nx兲␳兲 +iSm4共x兲}

⫻关Hm共1兲共kn␳兲Jm共nx兲−H_m^共1兲共nx兲Jm共kn␳兲兴

册

^{. 共59兲}

To perform the perturbation iteration of Eqs.共56兲and共59兲, integrals like the following must be calculated:

Vpm⬅ 1

Jm共nx兲Lˆp关Jm共kn␳兲cos共m␾兲兴. 共60兲 For small␭the integral over␳can be computed by expand- ing the integrand into a series of␦^r=␳^−R,

冕

R R+␭f共␾兲

F共␳兲d␳⬇␭f共␾兲F共R兲+1

2␭²f²共␾兲F⬘^共R兲+ ¯ . 共61兲 Notice that this method is valid only outside the second re- gionR₁⬍r⬍R₂ which shrinks to zero when ␭→0 关cf. Eq.

共55兲兴. In such a manner, it leads to

Vpm=x²共n²− 1兲共␭Vpm共1兲+␭²V_pm^共2兲兲, 共62兲 where

V_pm^共1兲= 1

xS_p共x兲A_pm 共63兲 and

V_pm^共2兲= B_pm

2xS_p共x兲

冋

^{1 +x}

^冉

^{HHp共p共11兲兲⬘共x兲共x兲+ HHm共m共11兲⬘兲共x兲共x兲}

^冊

^{− 2xSm共x兲}

册

^.

共64兲 HereApm andBpmare defined in Eqs.共B5兲and共B10兲.

The second order terms can also be expressed through Vmp:

⌿共xជ兲=Jm共knr兲

Jm共nx兲 cos共m␪兲+_p

兺

_⫽m^J_J^p_p^共knr兲_共nx兲 ^{cos共p␪兲}

⫻

冋

^Vpm+k

^兺

^⫽mVpkVkm

册

^{. 共65兲}

The quantization condition 共58兲 in the second order states that

Vmm+

兺

k⫽m

VmkVkm= 1, 共66兲 which can be expressed as

Sm共x兲=x共n²− 1兲

冉

^{␭Amm+12␭2}

冋

^{1 + 2xHHm共1兲m共1兲⬘共x兲共x兲}

册

^Bmm

^冊

+␭²x²共n²− 1兲²_k

兺

_⫽m^A_S^mk_k共x兲^{Akm. 共67兲}

Writing as in the previous sectionx=x₀+␭x1+␭²x₂wherex₀ is a zero ofS_m共x兲and using Eq.共B1兲, one obtains the same series as Eq.共33兲. Other expansions up to the second order also coincide with the ones presented in Sec. III A.

To calculate the higher terms of the perturbation expan- sion, the wave function must be known in the regions where the perturbation “potential”␦ⁿ²共xជ兲is nonzero. But exactly in these regions the Green function differs from the one used in Eq. 共57兲. In other words, a method must be found for the continuation of the wave functions defined in the first region r⬍R₁共or in the third oner⬎R₂兲into the second regionR₁

⬍r⬍R₂.

The straightforward way of such a continuation is to use explicit formulas for the Green function in the second region and to perform the necessary calculations. As the radial de- rivative of the Green function is discontinuous, delta- function contributions will appear in certain domains when calculating the integrals as in Eq.共61兲. One can check that this singular contribution appears in the bulk only in the third order in␭in agreement with Eqs.共33兲and共34兲.

The expansion of wave functions into series of the Bessel functions 共56兲, in general, diverges when r⬎R₁ and the Green function method gives the correct continuation inside this region. Another equivalent method consists in a local expansion of wave function into power series in small devia- tion from the boundary of convergence. As the value of the function and its radial derivative are assumed to be known along this boundary共r=R₁in our case兲the knowledge of the wave equation inside and outside the cavity determines uniquely the wave function in the both regions.

IV. APPLICABILITY OF PERTURBATION SERIES In the previous section the formal construction of pertur- bation series has been performed for quasibound states in slightly deformed dielectric cavities. The purpose of this sec- tion is to discuss in detail the conditions of validity of such an expansion.

From Eq. 共B5兲 it follows that the coefficients Apm obey the inequality

兩Apm兩艋2␰^, 共68兲

where␰^{stands for}

␰⁼

冕 冏

^f共R␪兲

冏

^d␪⬇ ␲^{␦Ra2. 共69兲} Here␦^ais the surface where the perturbation “potential”␦ⁿ² is nonzero and␲^R2is the full area of the unperturbed circle.

The last equality is valid when Eq.共21兲is fulfilled which we always assume.

Consequently, the perturbation formulas can be applied providing

␰共n²− 1兲u

冓 ^冏

^Sp1共u兲

^冏 冔

^{Ⰶ1, 共70兲}

where具F_p典indicates the typical value ofF_pandustands for Re共kR兲at a first approximation.

The usual arguments to estimate this quantity for largex₀ are the following. In the strict semiclassical approximation, states with a corresponding incident angle larger than the critical angle have a very small imaginary part and are prac- tically true bound states. For closed circular cavities, the mean number of states 共counting doublets only once兲 is given by the Weyl law

N共Ej⬍n²k²兲=An²

8␲^{k2+O共k兲, 共71兲} whereA=␲^R2is the full billiard area andnis the refractive index. The latter appears because by definition inside the

cavity the energy isE=n²k². For a dielectric circular cavity with radiusR, the condition that the incidence angle is larger than the critical angle leads to the following effective area 关12兴

A_eff共n兲=␲^R2sn, 共72兲 where

sn= 2 R²

冕

R/n

R

冉

^{1 −}␲^{2 arcsin} R

nr

冊

^{r dr 共73兲}

= 4

␲^n2x

冕

x

nx

冑

x²n²−m²dm

= 1 − 2

␲

冉

^{arcsin1n +1n}

^冑

^{1 −n12}

冊

^{. 共74兲}

Here the first integral共73兲corresponds to the straightforward calculation of phase-space volume such that the incident angle is larger than the critical one and the second integral 共74兲 is obtained from Eq. 共16兲 taking into account that x

⬍m⬍nx. Forn= 1.5,sn⬇0.22.

Consequently, the typical distance between two eigen- states is

␦^x⬃ 4

n²snx. 共75兲 The eigenmomenta of nonconfining eigenstates have imagi- nary parts of the order of unity 关cf. Eq. 共13兲兴 and will be ignored.

AsSm共x0兲= 0, we estimate that forp⫽m

冓

^S1p

冔

^⬃

^冏

^S⬘1␦x

^冏

^{. 共76兲}

Using Eq.共B1兲one finds that this value is of the order of

冓

^S1p

冔

^{⬃Cx, 共77兲}

where constantC⬃0.25n²sn/共n²− 1兲. With Eq.共70兲it leads to the conclusion that for typicalSpthe criterion of applica- bility of perturbation series is

sn

␦^a

8␲^{k2n2Ⰶ1, 共78兲}

which up to a numerical factor agrees with Eq.共22兲. But this statement is valid only in the mean. If there exist quasi-degeneracies of the nonperturbed spectrum关i.e., there exist p for which 1/Sp共x兲 is considerably larger than the estimate 共77兲兴 then the standard perturbation treatment re- quires modifications. As circular cavities are integrable, the real parts of the strongly confined modes are statistically dis- tributed as the Poisson sequences 关15兴 and they do have a large number of quasidegeneracies even for smallk. For in- stance, these are double quasicoincidences for the dielectric circular cavity withn= 1.5,

x_14,2= 16.7170 − 0.03895i, x_11,4= 16.6976 − 0.4695i,

x_15,3= 17.5042 − 0.37540i, x_12,4= 17.5232 − 0.4612i, x_17,4= 21.5715 − 0.41621i, x_14,5= 21.5106 − 0.4712i, x_25,1= 19.4799 − 0.00254i, x_21,2= 19.4830 − 0.1211i.

We notice also a triple quasicoincidence x_46,1= 34.3110 − 2.2206⫻10⁻⁶i,

x_41,2= 34.3167 − 0.001982i,

x_37,3= 34.317 − 0.06408i. 共79兲 The existence of these quasidegeneracies means that the per- turbation series require modifications close to these values of kR for any small deformations of a circular cavity with n

= 1.5.

Double quasidegeneracy is the simplest case because there is only one eigenvalue of the dielectric circle, with quantum number, say p, whose eigenvalue is close to the eigenvalue x₀ corresponding to the quantum number m. In such a situation, instead of the zeroth order equation共31兲, the system of the following two equations must be considered

Sm关x0共1 +␦x兲兴a1=M₁₁a₁+M₁₂a₂,

S_p关x₀共1 +␦^x兲兴a2=M₂₁a₁+M₂₂a₂, 共80兲 where in the leading orderMij=x₀共n²− 1兲Aij.

Expanding theSpfunctions and using the dominant order 共B1兲forS_p⬘, the system共80兲can be transformed into

共s₁−␦^x兲a₁=A₁₂a₂,

共s2−␦x兲a2=A₂₁a₁ 共81兲

where

s₁= Sm共x0兲

x₀共n²− 1兲−A₁₁, s₂= Sp共x0兲

x₀共n²− 1兲−A₂₂. 共82兲 Our usual choice is Sm共x0兲= 0 but for symmetry we do not impose it. The compatibility of the system共81兲 leads to the equation

冏

^␦xA−21^s1 ␦^xA−12s2

冏

^{= 0. 共83兲}

Its solution which tends tos₁whenA₁₂A₂₁→0 is

␦^x=1

2共s1+s₂兲+1

2共s1−s₂兲

冑

^{1 +}共s^4A1¹²−^As₂²¹兲²

=s₁−A₁₂A₂₁ s₂−s₁

2

1 +

冑

1 + 4A₁₂A₂₁/共s₁−s₂兲². 共84兲 WhenA₁₂A₂₁/共s1−s₂兲² is small,␦^{x in Eq.}共84兲tends to the usual contribution of the second order共33兲. Therefore, in this approximation, expression共33兲may be used for allp except the one which is quasidegenerate withx₀. For this later one, the whole expression 共84兲 共without s₁兲 has to be used. A useful approximation proposed in关16兴consists in taking the

modification共84兲for all the nondegenerate levels which re- duces the numerical calculations.

As the circular billiard is integrable, the probability of having three and more quasidegeneracies is not negligible 关cf. Eq.共79兲兴. The necessary modifications can be performed for any number of levels but the resulting formulas become cumbersome. In Appendix D we present the formulas for three quasidegenerate levels.

V. CUT DISK

As a specific example, we consider a deformation of the circular cavity which is useful for experimental and techno- logical points of view 关17兴. Namely, a circle is cut over a straight line共see Fig.5兲. Such a deformation is characterized by the parameter⑀Ⰶ1 which determines the distance from the cut to the circular boundary. This shape corresponds to the following choice of the deformation function f共␪兲

f共␪兲=R

冉

^{1 −cos1 −⑀}␪

冊

^⬇R

冉

^{⑀− ␪22+⑀␪22−524␪4}

冊

^{, 共85兲}

when 兩␪兩⬍␪mand f共␪兲= 0 for other values of ␪^{. Here} ␪mis the small angle

␪m= arccos共1 −⑀兲 ⯝

冑

2⑀⁺

冑

2

12⑀^3/2. 共86兲 Using the formulas discussed in the preceding sections, we compute all the necessary quantities and compare them with the results of the direct numerical simulations based on a boundary element representation similar to the one discussed in关13兴.

The spectrum of quasi-bound states for a cut disk with⑀

= 0.05 is plotted in Fig. 6. To get a good agreement in the region Re共kR兲⯝15– 20, it is necessary to take into account double quasidegeneracies and, in the region close to Re共kR兲= 35, triple degeneracy共79兲has been considered. The agreement is quite good even in the region of largekRwhere many families intersect.

Two wave functions of this cut disk are plotted in Fig.7.

The first one is obtained by direct numerical simulations and the second one corresponds to the perturbation expansion.

Even tiny details are well reproduced by perturbation com- putations. In the direct numerical simulations, the wave func- tions are reconstructed from the knowledge of the boundary currents. This procedure requires the integration of the Han-

2θ

m

εR θ

FIG. 5. Cut disk cavity.⑀Ris the distance between the cut and the circular boundary.