## Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities

Optics Express, Vol. 10, Issue 16, pp. 752-776 (2002)

http://dx.doi.org/10.1364/OE.10.000752

Acrobat PDF (2942 KB)

### Abstract

The quasi-bound modes localized on stable periodic ray orbits of dielectric micro-cavities are constructed in the short-wavelength limit using the parabolic equation method. These modes are shown to coexist with irregularly spaced “chaotic” modes for the generic case. The wavevector quantization rule for the quasi-bound modes is derived and given a simple physical interpretation in terms of Fresnel reflection; quasi-bound modes are explictly constructed and compared to numerical results. The effect of discrete symmetries of the resonator is analyzed and shown to give rise to quasi-degenerate multiplets; the average splitting of these multiplets is calculated by methods from quantum chaos theory.

© 2002 Optical Society of America

## 1 Introduction

23. M. V. Berry, “Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard,” Eur. J. Phys. **2**, 91–102 (1981). [CrossRef]

*ϕ*and the angle of incidence sin

*χ*at each bounce. The three types of motion described above are illustrated for the quadrupole billiard in Fig. 1 both in phase space and in real space.

24. B. Li and M. Robnik, “Geometry of high-lying eigenfunctions in a plane billiard system having mixed type classical dynamics,” J. Phys. A **28**, 2799–2818 (1995). [CrossRef]

*kl*≫ 1 where

*k*is the wavevector and

*l*is a typical linear dimension of the resonator, e.g. the average radius. The modes associated with quasi-periodic families can be treated semiclassically by eikonal methods of the type introduced, e.g. by Keller [25

25. J. B. Keller and S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. **9**, 24–75 (1960). [CrossRef]

*stable*case explicit quantization rules can be found which are equivalent to our results for the closed cavity derived below [27

27. W. H. Miller, “Semiclassical quantization of nonseparable systems: A new look at periodic orbit theory,” J. Chem. Phys. **63**, 996–999 (1975). [CrossRef]

17. N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” Phys. Rev. Lett. **88**, art. no.094 102 (2002). [CrossRef]

19. C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, “Kolmogorov-Arnold-Moser transition and laser action on scar modes in semiconductor diode lasers with deformed resonators,” Opt. Lett. **27**, 824–826 (2002). [CrossRef]

20. S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. **88**, art. no.033903 (2002). [CrossRef] [PubMed]

21. E. J. Heller, “Bound-state eigenfunctions of classically chaotic hamiltonian-systems - Scars of periodic orbits,” Phys. Rev. Lett. **53**, 1515–1518 (1984). [CrossRef]

29. O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, “Signature of dynamical localization in the resonance width distribution of wave-chaotic dielectric cavities,” Phys. Rev. E **62**, 2078–2084 (2000). [CrossRef]

30. J. U. Nöckel, “Angular momentum localization in oval billiards,” Phys. Scr. **T90**, 263–267 (2001). [CrossRef]

*all*PO modes, due to evanescent leakage across a curved interface, even if all the bounces satisfy the total internal reflection condition.

*quasi-degeneracy*in which the spectrum consists of nearly degenerate multiplets, whose multiplicity depends in detail on the particular PO mode. This point is illustrated by the inset to Fig. 3(b). We will show below how to calculate the multiplicity of these quasi-degeneracies for a given PO and introduce a theoretical approach to estimate the size of the associated splittings.

## 2 Gaussian optical approach to the closed cavity

*X*,

*Z*). In this case the TM and TE polarizations separate and we have a scalar wave equation with simple continuity conditions at the boundary for the electric field (for TM) or magnetic field (for TE). For the closed case (for which the field is zero on the boundary) we can set the index

*n*= 1 and work simply with the Helmholtz equation.

*E*(

*X*,

*Z*) of the Helmholtz equation in two dimensions:

*D*, with a boundary

*∂D*on which

*E*= 0, leading to discrete real eigenvalues

*k*. We are interested in a subset of these solutions

*for asymptotically large values of k*for which the eigen-functions are localized around stable periodic orbits (POs) of the specularly reflecting boundary

*∂D*.

### 2.1 The Parabolic equation approximation

*∂D*are the set of ray orbits which close upon themselves upon reflecting specularly N times. The shape of the boundary defines a non-linear map from the incident angle and polar angle at the

*m*

^{th}bounce (

*ϕ*

_{m}, sin

*χ*

_{m}) to that at the

*m*+ 1 bounce. Typical trajectories of this map are shown in the surface of section plot of Fig. 1. The period-N orbits are the fixed points of the

*N*

^{th}iteration of this map. For a given period-N orbit (such as the period four “diamond” orbit shown in Fig. 5, let the length of

*m*th segment (“arm”) be

*l*

_{m}, the accumulated distance from origin be

*L*

_{M}=

*l*

_{m}, and

*L*=

*L*

_{N}be the length of the entire PO. We are looking for modal solutions which are localized around the PO and decay in the transverse direction, hence we express Eq. (1) in Cartesian coordinates (

*x*

_{m},

*z*

_{m}) attached to the PO, where

*z*

_{m}-axis is aligned with

*m*th arm and

*x*

_{m}is the transverse coordinate. We also use

*z*to denote the cumulative length along the PO, which varies in the interval (-∞, +∞).

*z*

_{m}=

*z*along the PO, and we do so (see Fig. 5).

_{m}is the Laplacian expressed in local coordinate system. This reduction is possible as long as the solutions are well-localized, and the bounce points of the PO are well-separated (with respect to

*k*

^{-1}), even if the PO were to self-intersect. These assumptions will be justified by the ensuing construction.

*ikz*] which varies on the scale of the wavelength

*λ*, the

*z*-dependence of

*u*is slow, i.e.

*u*

_{z}~

*u*/

*l*,

*u*

_{zz}~

*u*/

*l*

^{2}, where

*l*is a typical linear dimension associated with the boundary, e.g. a chord length of the orbit or the curvature at a bounce point. In the semiclassical limit

*l*≫ λ. The transverse (

*x*) variation of

*u*on the other hand is assumed to occur on a scale

*l*; hence the transverse variation of

*u*(

*x*,

*z*) is much more rapid than its longitudinal variation. This motivates us to introduce the scaling

*x̃*= √

*kx*to treat this boundary layer, which leads to

*u*

_{zz}term due to the condition

*kl*≫ 1, and obtain a partial differential equation of parabolic type:

*i∂*

_{z}. Next, we make the ansatz

*z*-derivative. Next, making the substitution

13. A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and q spoiling in lasing droplets,” Phys. Rev. Lett. **75**, 2682–2685 (1995). [CrossRef] [PubMed]

*Q*(

*z*) =

*αz*+

*β*; we will be able to interpret

*Q*(

*z*) below as describing a ray nearby the periodic orbit with relative angle and intercept determined by

*α*,

*β*.

### 2.2 Boundary Conditions

*l*we can express this arc on the boundary

*∂D*as an arc of a circle of radius

*ρ*(the curvature at the reflection point). We express the boundary condition in a (scaled) common local coordinate system for the incident and reflected fields (

*ξ̃*

_{1},

*ξ̃*

_{2}) pointing along the tangent and the normal at the bounce point (see Fig. 5). Because the boundary condition must be satisfied on the entire arc it follows the phases of each term must be equal,

*ξ̃*

_{1}=

*O*(1), and we will carry out the solution of these equations to

*O*(1/√

*k*). Note that, it is sufficient to take

*Q*(

*z*)|

*∂D*≈

*Q*(

*l*

_{m}), at this level. In each segment we have three constants which determine our solution:

*c*

_{m},

*α*

_{m},

*β*

_{m}(where

*Q*

_{m}(

*z*) =

*α*

_{m}

*z*+

*β*); however due to its form our solution is uniquely determined by the two ratios

*β*

_{m}/

*α*

_{m},

*c*

_{m}/√

*α*

_{m}. Therefore we have the freedom to fix one matching relation for the

*Q*

_{m}by convention, which then determines the other two uniquely. To conform with standard definitions we fix

*Q*

_{m+1}=

*Q*

_{m}, then the amplitude and phase equality conditions give the relations:

*Q*. These conditions allow us to propagate any solution through a reflection point via the reflection matrix ℜ

_{m}and Eq. (16). Note that with these conventions the reflection matrix is precisely the standard “ABCD” matrix of ray optics for reflection at arbitrary incidence angle (in the tangent plane) from a curved mirror [33]. It follows that

*Q*

_{m}and

*Q*

_{m+1}can be interpreted as the transverse coordinates of the incident and reflected rays with respect to the PO, and that then Eq. (15) describes specular ray reflection at the boundary, if the non-linear dynamics is linearized around the reflection point.

### 2.3 Ray dynamics in phase space

*Q*and the conjugate momentum

*P*=

*Q*′, with

*z*playing the role of the time. Let’s introduce the column-vector

*W*(

*Q*′,

*Q*) ≠ 0 reduces to det ∏ ≠ 0. Then, for example, the Euler equation Eq. (13) in each arm can be expressed as

*z*-independent column vector

*h*. Any ray of the

*m*th arm in the solution space of Eq. (13) can then be expressed by a z-independent column vector

*h*

_{m}

^{-1}(

*z*) = ∏(-

*z*). Then if

*Q*

_{m}(

*z*) =

*α*

_{m}

*z*+

*β*

_{m}, we have

*z*+ 1 <

*L*

_{m}where

*L*

_{m-1}<

*z*<

*L*

_{m}) is induced by the ‘propagator’ ∏

*z*→

*z*′ propagation for

*L*

_{m-1}<

*z*<

*L*

_{m}<

*z*′ <

*L*

_{m+1}is

*𝚳*(

*z*), which propagates rays a full round-trip, i.e. by the length

*L*of the corresponding PO and is given by

*L*

_{m-1}<

*z*<

*L*

_{m}. Note that

*l*

_{m+N}=

*l*

_{m}and ℜ

_{m+N}= ℜ

_{m}. Although the specific form of

*𝚳*(

*z*) depends on the choice of origin,

*z*, it is easily shown that all other choices would give a similar matrix and hence the eigenvalues of the monodromy matrix are independent of this choice. We will suppress the argument

*z*below.

### 2.4 Single-valuedness and quantization

*Q*,

*P*an arbitrary distance around the periodic orbit we can generate a solution of the parabolic equation which satisfies the boundary conditions by an arbitrary initial choice

*Q*(0),

*P*(0). However an arbitrary solution of this type will not reproduce itself after propagation by L (one loop around the PO). Recalling that our solution is translated back into the two-dimensional space (

*X*,

*Z*) by Eq. (2), and that the function

*E*(

*X*,

*Z*) must be single-valued, we must require periodicity for our solutions

*E*(

*x*,

*z*) =

*E*

_{m}(

*x*

_{m},

*z*

_{m}) and

*u*(

*x*,

*z*) =

*u*

_{m}(

*x*

_{m},

*z*

_{m}) whenever

*L*

_{m-1}<

*z*<

*L*

_{m}. Since the phase factor in

*E*advances by exp[

*ikL*] with each loop around the periodic orbit, single-valuedness implies that

*k*and will lead to our quantization rule for the PO modes.

*u*in Eq. (9) we see that the phase Ω(

*z*) =

*Q*′/

*Q*will be unchanged if we choose (

*Q*,

*P*) to be an eigenvector (

*q*

_{1},

*p*

_{1})), (

*q*

_{2},

*p*

_{2}) of the monodromy matrix as in this case the

*Q*′(

*z*+

*L*) = λ

_{1,2}

*Q*′(

*z*),

*Q*(

*z*+

*L*) = λ

_{1,2}

*Q*(

*z*) and the ratio Ω(

*z*) = Ω(

*z*+

*L*). The monodromy matrix is unimodular and symplectic and its eigenvalues come in inverse pairs, which are either purely imaginary (stable case) or purely real (unstable and marginally stable cases). If the PO is unstable and the eigenvalues are real, then the eigenvectors and hence Ω(

*z*) are real. But a purely real Ω(

*z*) means that the gaussian factor in

*u*(

*x*,

*z*) is purely imaginary and the solution does not decay in the direction transverse to the PO, contradicting the initial assumption of the parabolic approximation to the Helmholtz equation.

*Hence for unstable POs our construction is inconsistent and we cannot find a solution of this form localized near the PO*.

*q*(

*z*),

*p*(

*z*)) such that

*Im*Ω > 0. Note that the overall magnitude of the Wronskian is determined by our choice of normalization of the eigenvectors and is here chosen to be unity.

*c*

_{m+N}=

*e*

^{-iπN}

*c*

_{m}we obtain

*m*is an integer,

*N*is the number of bounces in the PO,

*φ*is the Floquet phase obtained from the eigenvalues of the monodromy matrix and

*N*

_{μ}is an integer known as the Maslov index in the non-linear dynamics literature[34

34. V. P. Maslov and M. V. Fedoriuk, *Semiclassical Approximations in Quantum Mechanics* (Reidel, Boston, USA, 1981). [CrossRef]

*𝚳*. However our solution, Eq. (9), involves the

*square root*of

*q*(

*z*) and will be sensitive to the number of times the phase of

*q*(

*z*) wraps around the origin as

*z*goes from zero to

*L*. If this winding number (or Maslov index) is called

*N*

_{μ}, then the actual phase advance along the PO is

*φ*+ 2

*πN*

_{μ}; if

*N*

_{μ}is odd this leads to an observable

*π*phase shift in the solution not included by simply diagonalizing

*𝚳*to find the Floquet phase.

*N*

_{μ}may be directly calculated by propagating

*q*(

*z*):

*m*gives rise to a free spectral range Δ

*k*

_{long}= 2

*π*/

*L*for gaussian modes of a stable PO of length

*L*. The Floquet phase

*φ*/2

*L*is the zero-point energy associated with the transverse quantization of the mode and we will shortly derive excited transverse modes with spacing

*φ*/

*L*. In Fig. 6, we plot for comparison the analytic solutions for the bow-tie resonance just derived in comparison to a numerical solution of the same problem; both the intensity patterns and the quantized value of

*k*agree extremely well.

### 2.5 Transverse excited modes

*L*(

*u*) = 0, i.e. eigenfunctions of the differential operator with eigenvalue zero. It is natural following the analogy to quantum oscillators to seek additional solutions by defining lowering and raising operators.

^{†},

*L*) form an algebra. Namely, that [Λ

^{†},

*L*] = [Λ,

*L*] = 0 and furthermore that [Λ, Λ

^{†}] =

*i*(

*p*

^{*}

*q*-

*q*

^{*}

*p*) = 1. Defining the “ground-state” solution we have found as

*u*

^{(0)}, the commutation condition implies that (Λ

^{†}

*u*

^{(0)})(

*x*,

*z*) is also a solution of

*L*(

*u*) = 0. Further it can be checked that while (Λ

*u*

^{(0)}) = 0, (Λ

^{†})

*u*

^{(0)}is a non-trivial solution[32

32. F. Laeri and J. U. Nöckel, Nanoporous compound materials for optical applications - Microlasers and microresonators, in *Handbook of Advanced Electronic and Photonic Materials*, H. S. Nalwa, ed. (Academic Press, San Diego, 2001). [CrossRef]

^{†}

*u*

^{(0)}(

*x*,

*z*) upon performing one loop around the orbit (Λ

^{†}

*u*

^{(0)}(

*x*,

*z*+

*L*). Noting that

*q*

^{*}(

*z*+

*L*) =

*e*

^{-iφ}

*q*

^{*}(

*z*),

*p*

^{*}(

*z*+

*L*) =

*e*

^{-iφ}

*p*

^{*}(

*z*), we find that Λ

^{†}(

*z*+

*L*) =

*e*

^{-iφ}Λ

^{†}(

*z*). Thus this solution will acquire an additional phase -

*φ*with respect to

*u*

^{(0)}. This means that the

*l*

^{th}family of solutions

*l*transverse quanta Δ

*k*

_{trans}=

*φ*/

*L*to the energy of the ground state gaussian solution. Thus (in two dimensions) the general gaussian modes have two mode indices (

*l*,

*m*) corresponding to the number of transverse and longitudinal quanta respectively and two different uniform spacings in the spectra (see Fig. 3(b)).

*H*

_{l}are the Hermite polynomials[32

32. F. Laeri and J. U. Nöckel, Nanoporous compound materials for optical applications - Microlasers and microresonators, in *Handbook of Advanced Electronic and Photonic Materials*, H. S. Nalwa, ed. (Academic Press, San Diego, 2001). [CrossRef]

## 3 Opening the cavity - The dielectric resonator

*∂D*, defined by the discontinuity in

*n*. Here Ψ =

*E*(Ψ =

*B*) for

*TM*(

*TE*) modes. We will only consider the case of a uniform dielectric in vacuum for which the index of refraction

*n*(

**r**∈

*D*) =

*n*and

*n*(

**r**∉

*D*) = 1. Thus we have the Helmholtz equation with wave vector

*nk*inside the dielectric and

*k*outside. The solutions to the wave equation for this case cannot exist only within the dielectric, as the continuity conditions at the dielectric interface do not allow such solutions. On physical grounds we expect solutions at every value of the external wavevector

*k*, corresponding to elastic scattering from the dielectric, but that there will be narrow intervals

*δk*for which these solutions will have relatively high intensity within the dielectric, corresponding to the scattering resonances. A standard technique for describing these resonances as they enter into laser theory is to impose the boundary condition that there exist only outgoing waves external to the cavity. This boundary condition combined with the continuity conditions cannot be satisfied for real wavevectors

*k*and instead leads to discrete solutions at complex values of

*k*, with the imaginary part of

*k*giving the width of the resonance. These discrete solutions are called quasi-bound modes or quasi-normal modes[36

36. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, T. S. S., and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. **70**, 1545–1554 (1998). [CrossRef]

*k*complex (assuming real index

*n*). It is worth noting that there are well-known corrections to specular reflection and refraction at a dielectric interface, for example the Goos-Hanchen shift[37

37. J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math **24**, 396–413 (1973). [CrossRef]

*kl*.

*E*

_{m}(

*x*

_{m},

*z*

_{m}) attached to each segment of the PO and define local coordinates (

*x*

_{m},

*z*

_{m}) attached to the PO. Now in addition we need an outside solution at each reflection point

*E*

_{mt}with its own coordinate system (

*x*

_{mt},

*z*

_{mt}) rotated by an angle given by Snell’s law applied to the direction of the incident ray (see Fig. 5). The ansatz of Eq. (3) thus applies, where now the sum will run over 2N components, which will include the transmitted fields. Introducing the slowly varying envelope approximation Eq. (2) and the scalings

*x̃*

_{mt}= √

*kx*,

*k*. The boundary conditions close to the

*m*

^{th}bounce point will take the form

*∂*

_{n}is the normal derivative at the boundary. The alternative indices

*i*,

*r*,

*t*stand for

*m*,

*m*+ 1 and

*mt*, respectively. Since the parabolic equation Eq. (8) is satisfied in appropriately scaled coordinates within each segment, we write all solutions in the general form

*E*

_{M}=

*A*

_{M}exp(

*iΦ*

_{M}) where

*M*stands for

*m*or

*mt*and

*n*

_{m}=

*n*,

*n*

_{mt}= 1. As for the closed case, we need to determine

*Q*

_{M},

*Q*

_{M}′ and

*c*

_{M}, so that the boundary conditions are satisfied, and then impose single-valuedness to quantize

*k*.

*ξ̃*

_{1},

*ξ̃*

_{2}) along the tangent and normal to the boundary at the reflection point, as before. We can again expand the boundary as the arc of a circle of radius

*ρ*, the curvature at the bounce point. Since the equations are of the same form for each reflection point it is convenient at this point to suppress the index

*m*and use the indices

*i*,

*r*,

*t*to denote the quantities associated with the incident, reflected and transmitted wave at the

*m*

^{th}bounce point.

*χ*=

*χ*

_{m},

*ρ*=

*ρ*

_{m}and all quantities are evaluated at

*z*=

*L*

_{m}. Recalling that

*n*sin

*χ*

_{i}= sin

*χ*

_{t}we get up to

*O*(1/√

*k*)

*μ*= cos

*χ*

_{i}/cos

*χ*

_{t}and the relation

*Q*

_{i}=

*μQ*

_{t}is a convention similar to

*Q*

_{r}=

*Q*

_{i}. Again, the matrix in Eq. (37) is just the ABCD matrix for transmission of rays through a curved dielectric interface at arbitrary angle of incidence in the tangential plane[33].

*Q*

_{r}=

*Q*

_{i},

*Q*

_{t}=

*Q*

_{i}/

*μ*)

*c*

_{i}and

*c*

_{r}as we had in the closed case. This is provided by the normal derivative boundary condition Eq. (36). Keeping only the leading terms this condition becomes:

*∂*

_{n}Φ

_{i}=

*nk*cos

*χ*,

*∂*

_{n}Φ

_{r}= -

*nk*cos

*χ*,

*∂*

_{n}Φ

_{t}=

*k*cos

*χ*

_{t}, leading to

*k*.

*k*is complex. We have the condition

*Q*(

*z*),

*P*(

*z*) to be the appropriate eigenvector of the monodromy matrix (note that

*𝚳*is unchanged from the closed case as it only pertains to the propagation of the phase) and with this choice the quantization condition becomes

*k*predicted by Eq. (46) for three different indices of refraction. Note that the best agreement is for the case far from total internal reflection, and the worst agreement is the case near TIR.

## 4 Symmetry Analysis and Quasi-Degeneracy

*d*

_{m}is the dimensionality and

*χ*

_{m}(

*g*) is the character of the m

^{th}irreducible representation and the solution so obtained, denoted by

*E*

_{m}(x) is the resulting symmetry-projected solution. For a given irreducible representation there are as many symmetrized solutions as the dimension of that irreducible representation of

*G*. We will focus here on the case of the closed resonator, but the general principles apply to the open case as well.

### 4.1 Symmetrized modes for the quadrupole

*G*=

*C*

_{2}⊗

*C*

_{2}= {1,

*σ*

_{X},

*σ*

_{Z},

*σ*

_{X}

*σ*

_{Z}}, the group of reflections about the

*X*and

*Z*axes. This group has four one-dimensional representations only, and thus cannot have

*any*exactly degenerate solutions (barring accidental degeneracy). The existence of four irreducible representations means that given the one solution

*E*(

*X*,

*Z*) we have constructed to the Helmholtz equation, we can generate four linearly independent solutions by projection according to Eq. (49) above. We will label the representations by

*m*= (

*r*,

*s*), where

*r*,

*s*= ± denotes the action of inversion of

*X*and

*Z*respectively. The symmetrized solutions are then

*E*(

*X*,

*Z*) cannot be an exact solution, as it does not transform as

*any*irreducible representation of the symmetry group. A further important point is that while we can always construct a number of symmetrized solutions equal to the sum of the dimensions of the irreducible representations, there is no guarantee that such a projection will yield a non-trivial solution. In fact in the case of the quadrupole we will show below that for each quantized value of

*k*only two of the projected solutions are non-trivial, leading to quasi-degenerate doublets in the spectrum. We will present below a simple rule which allows one to calculate the quasi-degeneracy given the periodic orbit and the symmetry group of the resonator.

_{1}and ℓ

_{2}be the lengths of the vertical and diagonal legs of the bow-tie, so that

*L*= 2(ℓ

_{1}+ ℓ

_{2}) is the total length. Then,

*g*= 1*g*=*σ*_{X}: (*z*→*L*/2 +*z*,*x*→ -*x*)*g*=*σ*_{Z}:*z*→ ℓ-*z*,*x*→*x*)*g*=*σ*_{X}*σ*_{Z}: (*z*→*L*/2 + ℓ_{1}-*z*,*x*→ -*x*)

*ζ*= 1/2 (π -

*φ*/2 +

*kL*). Here we use the fact that

*𝚳*

_{L}=

*𝚳*

_{L/2}

*𝚳*

_{L/2}for the bow-tie orbit, where

*M*

_{L}is the monodromy matrix for the whole length

*L*. It follows that

*q*(

*z*+

*L*/2) =

*e*

^{iφ/2}

*q*(

*z*). Note also the appearance of the factors

*k*-dependent and must be evaluated for the quantized values of

*k*. Referring to the quantization condition Eq. (27) we find that the phase

*ζ*=

*mπ*where

*m*is the longitudinal mode index of the state. Hence

*σ*

_{Z}form the doublets (see inset Fig. 3(b)), and these two parity types alternate in the spectrum every free spectral range. Note that while we have illustrated the analysis for the ground state (

*l*= 0) one finds exactly the same result for the

*l*

^{th}transverse mode, with doublets paired according to the index

*m*, independent of

*l*.

*E*(

*X*,

*Z*) localized on the PO. Second, one generates the symmetrized solutions from knowledge of the irreducible representations of the symmetry group. Third, one evaluates these solutions for the quantized values of

*k*; the non-zero solutions give one the quasi-degeneracy and the symmetry groupings (e.g. (++) with (+-) in the above case). The same principles apply to mirror resonators with the same symmetry group. Note that in the case of a high symmetry resonator (or mirror arrangement) e.g a square or a hexagon, for which there exist two dimensional irreducible representations, exact degeneracy is possible and can be found by these methods.

### 4.2 Simple Rule for Quasi-Degeneracy

38. J. M. Robbins, “Discrete symmetries in periodic-orbit theory,” Phys. Rev. A. **40**, 2128–2136 (1989). [CrossRef] [PubMed]

*one period in the reduced resonator*. The symmetry-reduced resonator has boundaries which correspond to lines of reflection symmetry in the original problem. Anti-symmetric solutions with respect to each of these lines of symmetry correspond to Dirichlet boundary conditions; symmetric solutions must have zero derivative corresponding to Neumann boundary conditions. The boundary conditions at the true boundary of the resonator don’t affect the symmetry pairing. For each symmetry choice one can evaluate the phase accumulated in the reduced resonator at each bounce, assigning a phase shift π to each bounce off a “Dirichlet” internal boundary, and zero phase shift for each bounce off a “Neumann” internal boundary. If two symmetry types lead to the same final phase shift (modulo 2

*π*) then those two symmetry types will be paired and quasi-degenerate, otherwise not. A subtle issue is the question of how to count bounces at the corner between two boundaries. The answer is that the semiclassical method really sums over orbits nearby the PO which will then hit both boundaries and experience the sum of the two phase shifts.

*X*axis and two boundary bounces. The boundary bounces don’t matter as they will give the same phase shift for all symmetry types. The

*X*axis bounce will give phase shift 0 for the + symmetry of

*σ*

_{Z}and

*π*for the - symmetry. The corner bounce sums the two shifts and gives: (+, +) → 0, (+, -) →

*π*, (-, +) →

*π*, ( -, -) → 2

*π*. Adding these two shifts modulo 2

*π*gives [(+, +), (+, -)] → 0, [(-, -), (-, +)] →

*π*corresponding to the symmetry pairing we found above. In Table 2, these two rules are applied to a number of relevant orbits in the quadrupole. It should be emphasized however that the group-theoretic projection method combined with the quantization rule which we illustrated in this section will work for any symmetry group and the rules that we have stated are just useful shortcuts.

### 4.3 Evaluation of Mode Splittings

39. M. J. Davis and E. J. Heller, “Multidimensional wave functions from classical trajectories,” J. Chem. Phys. **75**, 246 (1981). [CrossRef]

40. O. Bohigas, S. Tomsovic, and D. Ullmo, “Manifestations of classical phase space structures in quantum mechanics,” Phys. Rep. **223**, 45 (1993). [CrossRef]

41. S. D. Frischat and E. Doron, “Semiclassical description of tunneling in mixed systems: case of the annular billiard,” Phys. Rev. Lett. **75**, 3661 (1995). [CrossRef] [PubMed]

*classical*propagation in the chaotic portion of the phase space until the neighborhood of the other periodic orbit is reached, (iii) tunneling from the chaotic sea to the other periodic orbit. Note that the chaos-assisted processes are formally of higher order in the perturbation theory. However the corresponding matrix elements are much larger than those of the direct process. This can be understood intuitively as the tunneling from the periodic orbit to the chaotic sea typically involves a much smaller “violation” of classical mechanics and therefore has an exponentially larger amplitude.

40. O. Bohigas, S. Tomsovic, and D. Ullmo, “Manifestations of classical phase space structures in quantum mechanics,” Phys. Rep. **223**, 45 (1993). [CrossRef]

42. F. Leyvraz and D. Ullmo, “The level splitting distribution in chaos-assisted tunneling,” J. Phys. A **29**, 2529 (1996). [CrossRef]

*E*

_{C}with known statistical properties. The straightforward diagonalization of the resulting matrix yields[42

42. F. Leyvraz and D. Ullmo, “The level splitting distribution in chaos-assisted tunneling,” J. Phys. A **29**, 2529 (1996). [CrossRef]

44. M. V. Berry, “Regular and irregular semiclassical wavefunctions,” J. Phys. A **10**, 2083 (1977). [CrossRef]

*on the average*the Wigner function of a chaotic state is equally distributed across the chaotic portion of the phase space, and using the analytical expressions for the regular eigenstates calculated earlier, we find

*ϕ*, sin

*χ*) coordinates) occupied by the stable island supporting the regular eigenstate. Note that Eq. (60) holds only on average, since chaos-assisted tunneling always leads to strong fluctuations of the splittings which are of the same order as the average [40

40. O. Bohigas, S. Tomsovic, and D. Ullmo, “Manifestations of classical phase space structures in quantum mechanics,” Phys. Rep. **223**, 45 (1993). [CrossRef]

*ε*= 0.14, for different values of

*kR*. Note that although there are large fluctuations in the numerical data (as previously noted), the data are consistent with Eq. (60), while a calculation based on the “direct” coupling severely underestimates the splittings. Unfortunately, due to the large fluctuations in the splittings, knowledge of the average splitting size does not accurately predict the splitting of a specific doublet. Note also that small violations of symmetry in the fabrication of the resonator may lead to much larger splittings than these tunnel splittings; such an effect was recently observed for triangle-based modes of GaAs ARC micro-lasers [19

19. C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, “Kolmogorov-Arnold-Moser transition and laser action on scar modes in semiconductor diode lasers with deformed resonators,” Opt. Lett. **27**, 824–826 (2002). [CrossRef]

39. M. J. Davis and E. J. Heller, “Multidimensional wave functions from classical trajectories,” J. Chem. Phys. **75**, 246 (1981). [CrossRef]

## 5 Conclusions

*k*localized around the PO with mode spacings given by Δ

*k*

_{long}= 2

*π*/

*L*, Δ

*k*

_{trans}=

*φ*/

*L*where

*L*is the length of the PO and

*φ*is the Floquet phase associated with the eigenvalues of the monodromy matrix (round-trip ABCD matrix). For a dielectric cavity one finds similarly localized quasi-bound solutions at quantized complex values of

*nk*; in this case the imaginary part of

*nk*is determined by the Fresnel refractive loss at each bounce of the PO. Within this approximation the mode spacings are unchanged from the closed case (except for the trivial factor of

*n*). These regular modes coexist in a generic resonator with more complicated modes associated with the chaotic regions of phase space. Generalization of our results to the three-dimensional case appears straightforward for the scalar case and one expects only to have three-dimensional versions of the ABCD matrices enter the theory leading to some difference in details. More interesting would be the inclusion of the polarization degree of freedom, which seems possible in principle, but which we haven’t explored as yet.

45. H. E. Tureci and A. D. Stone, “Deviation from Snell’s law for beams transmitted near the critical angle: application to microcavity lasers,” Opt. Lett. **27**, 7–9 (2002). [CrossRef]

46. E. E. Narimanov, G. Hackenbroich, P. Jacquod, and A. D. Stone, “Semiclassical theory of the emission properties of wave-chaotic resonant cavities,” Phys. Rev. Lett. **83**, 4991–4994 (1999). [CrossRef]

*kl*. One possibility we are exploring is that a generalized ray optics with non-specular effects included can describe the open resonator and its emission pattern. Both experiments[4

4. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science **280**, 1556–1564 (1998). [CrossRef] [PubMed]

17. N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” Phys. Rev. Lett. **88**, art. no.094 102 (2002). [CrossRef]

17. N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” Phys. Rev. Lett. **88**, art. no.094 102 (2002). [CrossRef]

45. H. E. Tureci and A. D. Stone, “Deviation from Snell’s law for beams transmitted near the critical angle: application to microcavity lasers,” Opt. Lett. **27**, 7–9 (2002). [CrossRef]

*kl*not too large (~ 50 – 100) these higher order effects must be taken into account.

## References and links

1. | R. K. Chang and A. K. Campillo, eds., |

2. | H. Yokoyama, “Physics and device applications of optical microcavities,” Science |

3. | A. D. Stone, “Wave-chaotic optical resonators and lasers,” Phys. Scr. |

4. | C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science |

5. | J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature |

6. | A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated-emission in microdroplets,” Phys. Rev. Lett. |

7. | H. B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B |

8. | S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature |

9. | B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. |

10. | S. X. Qian, J. B. Snow, H. M. Tzeng, and R. K. Chang, “Lasing droplets - highlighting the liquid-air interface by laser-emission,” Science |

11. | A. W. Poon, F. Courvoisier, and R. K. Chang, “Multimode resonances in square-shaped optical microcavities,” Opt. Lett. |

12. | I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, “Hexagonal microlasers based on organic dyes in nanoporous crystals,” Appl. Phys. B-Lasers Opt. |

13. | A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and q spoiling in lasing droplets,” Phys. Rev. Lett. |

14. | J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. |

15. | S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength λ = 10 |

16. | S. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, “Observation of emission from chaotic lasing modes in deformed microspheres: displacement by the stable-orbit modes,” J. Opt. Soc. Am. B-Opt. Phys. |

17. | N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” Phys. Rev. Lett. |

18. | N. B. Rex, |

19. | C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, “Kolmogorov-Arnold-Moser transition and laser action on scar modes in semiconductor diode lasers with deformed resonators,” Opt. Lett. |

20. | S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. |

21. | E. J. Heller, “Bound-state eigenfunctions of classically chaotic hamiltonian-systems - Scars of periodic orbits,” Phys. Rev. Lett. |

22. | H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, and A. D. Stone, “Dramatic shape sensitivity of emission patterns for similarly deformed cylindrical polymer lasers,” in |

23. | M. V. Berry, “Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard,” Eur. J. Phys. |

24. | B. Li and M. Robnik, “Geometry of high-lying eigenfunctions in a plane billiard system having mixed type classical dynamics,” J. Phys. A |

25. | J. B. Keller and S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. |

26. | M. C. Gutzwiller, |

27. | W. H. Miller, “Semiclassical quantization of nonseparable systems: A new look at periodic orbit theory,” J. Chem. Phys. |

28. | S. D. Frischat and E. Doron, “Quantum phase-space structures in classically mixed systems: A scattering approach,” J. Phys. A-Math. Gen. |

29. | O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, “Signature of dynamical localization in the resonance width distribution of wave-chaotic dielectric cavities,” Phys. Rev. E |

30. | J. U. Nöckel, “Angular momentum localization in oval billiards,” Phys. Scr. |

31. | V. M. BabiČ and V. S. Buldyrev, |

32. | F. Laeri and J. U. Nöckel, Nanoporous compound materials for optical applications - Microlasers and microresonators, in |

33. | A. E. Siegman, |

34. | V. P. Maslov and M. V. Fedoriuk, |

35. | N. A. Chernikov, “System whose hamiltonian is a time-dependent quadratic form in x and p,” Sov Phys-Jetp Engl Trans |

36. | E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, T. S. S., and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. |

37. | J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math |

38. | J. M. Robbins, “Discrete symmetries in periodic-orbit theory,” Phys. Rev. A. |

39. | M. J. Davis and E. J. Heller, “Multidimensional wave functions from classical trajectories,” J. Chem. Phys. |

40. | O. Bohigas, S. Tomsovic, and D. Ullmo, “Manifestations of classical phase space structures in quantum mechanics,” Phys. Rep. |

41. | S. D. Frischat and E. Doron, “Semiclassical description of tunneling in mixed systems: case of the annular billiard,” Phys. Rev. Lett. |

42. | F. Leyvraz and D. Ullmo, “The level splitting distribution in chaos-assisted tunneling,” J. Phys. A |

43. | E. E. Narimanov, unpublished. |

44. | M. V. Berry, “Regular and irregular semiclassical wavefunctions,” J. Phys. A |

45. | H. E. Tureci and A. D. Stone, “Deviation from Snell’s law for beams transmitted near the critical angle: application to microcavity lasers,” Opt. Lett. |

46. | E. E. Narimanov, G. Hackenbroich, P. Jacquod, and A. D. Stone, “Semiclassical theory of the emission properties of wave-chaotic resonant cavities,” Phys. Rev. Lett. |

**OCIS Codes**

(080.2720) Geometric optics : Mathematical methods (general)

(140.3410) Lasers and laser optics : Laser resonators

(140.4780) Lasers and laser optics : Optical resonators

**ToC Category:**

Focus Issue: Rays in wave theory

**History**

Original Manuscript: June 26, 2002

Revised Manuscript: July 10, 2002

Published: August 12, 2002

**Citation**

Hakan Tureci, H. Schwefel, A. Stone, and E. Narimanov, "Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities," Opt. Express **10**, 752-776 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-16-752

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### References

- R. K. Chang, A. K. Campillo, eds., Optical Processes in Microcavities (World Scienti.c, Singapore, 1996).
- H. Yokoyama, �??Physics and device applications of optical microcavities,�?? Science 256, 66�??70 (1992). [CrossRef] [PubMed]
- A. D. Stone, �??Wave-chaotic optical resonators and lasers,�?? Phys. Scr. T90, 248�??262 (2001). [CrossRef]
- C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, �??High-power directional emission from microlasers with chaotic resonators,�?? Science 280, 1556�??1564 (1998). [CrossRef] [PubMed]
- J. U. Nockel and A. D. Stone, �??Ray and wave chaos in asymmetric resonant optical cavities,�?? Nature 385, 45�??47 (1997). [CrossRef]
- A. J. Campillo, J. D. Eversole, and H. B. Lin, �??Cavity quantum electrodynamic enhancement of stimulated-emission in microdroplets,�?? Phys. Rev. Lett. 67, 437�??440 (1991). [CrossRef] [PubMed]
- H. B. Lin, J. D. Eversole, and A. J. Campillo, �??Spectral properties of lasing microdroplets,�?? J. Opt. Soc. Am. B 9, 43�??50 (1992). [CrossRef]
- S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, �??Ultralow-threshold Raman laser using a spherical dielectric microcavity,�?? Nature 415, 621�??623 (2002). [CrossRef] [PubMed]
- B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, �??Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,�?? IEEE Photonics Technol. Lett. 10, 549�??551 (1998). [CrossRef]
- S. X. Qian, J. B. Snow, H. M. Tzeng, and R. K. Chang, �??Lasing droplets - highlighting the liquid-air interface by laser-emission,�?? Science 231, 486�??488 (1986). [CrossRef] [PubMed]
- A. W. Poon, F. Courvoisier, and R. K. Chang, �??Multimode resonances in square-shaped optical microcavities,�?? Opt. Lett. 26, 632�??634 (2001). [CrossRef]
- I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Eklo., F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, �??Hexagonal microlasers based on organic dyes in nanoporous crystals,�?? Appl. Phys. B-Lasers Opt. 70, 335�??343 (2000). [CrossRef]
- A. Mekis, J. U. Nockel, G. Chen, A. D. Stone, and R. K. Chang, �??Ray chaos and q spoiling in lasing droplets,�?? Phys. Rev. Lett. 75, 2682�??2685 (1995). [CrossRef] [PubMed]
- J. U. Nockel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, �??Directional emission from asymmetric resonant cavities,�?? Opt. Lett. 21, 1609�??1611 (1996). [CrossRef]
- S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, �??Long-wavelength λ = 10μm quadrupolar-shaped GaAs-AlGaAs microlasers,�?? IEEE J. Quantum Electron. 36, 458�??464 (2000). [CrossRef]
- S. Chang, R. K. Chang, A. D. Stone, and J. U. Nockel, �??Observation of emission from chaotic lasing modes in deformed microspheres: displacement by the stable-orbit modes,�?? J. Opt. Soc.Am. B 17, 1828�??1834 (2000). [CrossRef]
- N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, �??Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,�?? Phys. Rev. Lett. 88, art. no.094 102 (2002). [CrossRef]
- N. B. Rex, Regular and chaotic orbit Gallium Nitride microcavity lasers, Ph.D. thesis, Yale University (2001).
- C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, �??Kolmogorov-Arnold-Moser transition and laser action on scar modes in semiconductor diode lasers with deformed resonators,�?? Opt. Lett. 27, 824�??826 (2002). [CrossRef]
- S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and K. An, �??Observation of scarred modes in asymmetrically deformed microcylinder lasers,�?? Phys. Rev. Lett. 88, art. no.033903 (2002). [CrossRef] [PubMed]
- E. J. Heller, �??Bound-state eigenfunctions of classically chaotic hamiltonian-systems - Scars of periodic orbits,�?? Phys. Rev. Lett. 53, 1515�??1518 (1984). [CrossRef]
- H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, and A. D. Stone, �??Dramatic shape sensitivity of emission patterns for similarly deformed cylindrical polymer lasers,�?? in QELS Technical Digest (Optical Society of America, Washington, D.C., 2002), pp. 24�??25.
- M. V. Berry, �??Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard,�?? Eur. J. Phys. 2, 91�??102 (1981). [CrossRef]
- B. Li and M. Robnik, �??Geometry of high-lying eigenfunctions in a plane billiard system having mixed type classical dynamics,�?? J. Phys. A 28, 2799�??2818 (1995). [CrossRef]
- J. B. Keller and S. I. Rubinow, �??Asymptotic Solution of Eigenvalue Problems,�?? Ann. Phys. 9, 24�??75 (1960). [CrossRef]
- M. C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, New York, USA, 1990).
- W. H. Miller, �??Semiclassical quantization of nonseparable systems: A new look at periodic orbit theory,�?? J. Chem. Phys. 63, 996�??999 (1975). [CrossRef]
- S. D. Frischat and E. Doron, �??Quantum phase-space structures in classically mixed systems: A scattering approach,�?? J. Phys. A-Math. Gen. 30, 3613�??3634 (1997). [CrossRef]
- O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, �??Signature of dynamical localization in the resonance width distribution of wave-chaotic dielectric cavities,�?? Phys. Rev. E 62, 2078�??2084 (2000). [CrossRef]
- J. U. Nockel, �??Angular momentum localization in oval billiards,�?? Phys. Scr. T90, 263�??267 (2001). [CrossRef]
- V. M. Babi¡c and V. S. Buldyrev, Asymptotic Methods in Shortwave Di.raction Problems (Springer, New York, USA, 1991).
- F. Laeri and J. U. N¨ockel, �??Nanoporous compound materials for optical applications �?? Microlasers and microresonators,�?? in Handbook of Advanced Electronic and Photonic Materials, H. S. Nalwa, ed. (Academic Press, San Diego, 2001). [CrossRef]
- A. E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986).
- V. P. Maslov and M. V. Fedoriuk, Semiclassical Approximations in Quantum Mechanics (Reidel, Boston, USA, 1981). [CrossRef]
- N. A. Chernikov, �??System whose hamiltonian is a time-dependent quadratic form in x and p,�?? Sov. Phys.-Jetp Engl. Trans. 26, 603�??608 (1968).
- E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, T. S. S., and K. Young, �??Quasinormal-mode expansion for waves in open systems,�?? Rev. Mod. Phys. 70, 1545�??1554 (1998). [CrossRef]
- J. W. Ra, H. L. Bertoni, and L. B. Felsen, �??Reflection and transmission of beams at a dielectric interface,�?? SIAM J. Appl. Math 24, 396�??413 (1973). [CrossRef]
- J. M. Robbins, �??Discrete symmetries in periodic-orbit theory,�?? Phys. Rev. A 40, 2128�??2136 (1989). [CrossRef] [PubMed]
- M. J. Davis and E. J. Heller, �??Multidimensional wave functions from classical trajectories,�?? J. Chem. Phys. 75, 246 (1981). [CrossRef]
- O. Bohigas, S. Tomsovic, and D. Ullmo, �??Manifestations of classical phase space structures in quantum mechanics,�?? Phys. Rep. 223, 45 (1993). [CrossRef]
- S. D. Frischat and E. Doron, �??Semiclassical description of tunneling in mixed systems: case of the annular billiard,�?? Phys. Rev. Lett. 75, 3661 (1995). [CrossRef] [PubMed]
- F. Leyvraz and D. Ullmo, �??The level splitting distribution in chaos-assisted tunneling,�?? J. Phys. A 29, 2529 (1996). [CrossRef]
- E. E. Narimanov, unpublished.
- M. V. Berry, �??Regular and irregular semiclassical wavefunctions,�?? J. Phys. A 10, 2083 (1977). [CrossRef]
- H. E. Tureci and A. D. Stone, �??Deviation from Snell�??s law for beams transmitted near the critical angle: application to microcavity lasers,�?? Opt. Lett. 27, 7�??9 (2002). [CrossRef]
- E. E. Narimanov, G. Hackenbroich, P. Jacquod, and A. D. Stone, �??Semiclassical theory of the emission properties of wave-chaotic resonant cavities,�?? Phys. Rev. Lett. 83, 4991�??4994 (1999). [CrossRef]

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