## 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

There is currently a great deal of interest in dielectric micro-cavities which can
serve as high-Q resonators by confining light on the basis of multiple reflections
from a boundary of dielectric mismatch[1, 2, 3, 4, 5]. Such resonators have been used to study fundamental optical
physics such as cavity quantum electrodynamics[6, 7] and have been proposed and demonstrated as the basis of both
active and passive optical components[8, 9]. Of particular interest in this work are dielectric
micro-cavity lasers which have already been demonstrated for a wide variety of
shapes: spheres[10], cylinders[1], squares[11], hexagons[12], and deformed cylinders and spheres (asymmetric resonant cavities[3] -ARCs)[4, 13, 14, 15, 16, 17]. The work on ARC micro-cavity lasers has shown the
possibility of producing high power directional emission from such lasers, which
lase in different spatial mode patterns depending on the index of refraction and
precise shape of the boundary. For example modes based on stable periodic ray orbits
with bow-tie[4] and triangular geometry[18] have been observed, as well as whispering gallery-like modes[16, 18] which are not obviously related to any periodic ray orbit.
In addition a periodic orbit mode can be selected for lasing even if it is unstable[17, 19, 20]; due to the analogy to quantum wavefunctions based on
unstable classical periodic orbits, such modes have been termed “scarred”[21]. There is at present no quantitative understanding of the
mode selection mechanism in ARCs and theory has tended to work backwards from
experimental observations; however the passive cavity solutions have been found to
explain quite well the observed lasing emission patterns. The formal analogy between
ARCs and the problem of classical and quantum billiards has given much insight into
their emission properties [1, 3]. For example it was predicted and recently observed that
polymer ARCs of elliptical shape (index *n* = 1.49) have dramatically
different emission patterns from quadrupolar shaped ARCs with the same major to
minor axis ratio[22]. The difference can be fully understood by the different
structure of the phase space for ray motion in the two cases, the ellipse giving
rise to integrable ray motion and the quadrupole to partially chaotic (mixed)
dynamics. Resonators are of course open systems for which radiation can leak out to
infinity. In the discussion of the ray-wave correspondence immediately following we
neglect this leakage assuming only perfect specular reflection of light rays at the
dielectric boundary. However the theory of ARCs[3, 5] includes these effects and they will be treated when
relevant below.

As is well-known, the geometric optics of a uniform dielectric region with perfectly
reflecting boundaries is formally analogous to the problem of a point mass moving in
a billiard and hence we may use the terminology of Hamiltonian dynamics to describe
ray motion in the resonator. As discussed below, we shall specialize to cylindrical
geometries in which the relevant motion will be in the plane transverse to the axis,
hence we focus on two-dimensional billiards. The quadrupole billiard is an example
of a generic deformation of the circle or cylinder in that it is smooth and analytic
and does not preserve any constant of motion. (Motion in the circle of course
conserves angular momentum and motion in a closed elliptical cavity conserves a
generalized angular momentum, the product of the instantaneous angular momenta with
respect to each focus[23]). A generic deformation such as the quadrupole will lead to
a phase space for ray motion which has three types of possible motion (depending on
the choice of initial conditions): oscillatory motion in the vicinity of a stable
periodic ray orbit, chaotic motion in regions associated with unstable periodic ray
orbits, and marginally stable motion associated with families of quasi-periodic
orbits. To elucidate the structure of phase space it is conventional to plot a
number of representative trajectories in a two-dimensional cut through phase space,
called the surface of section (Fig. 1). For our system the surface of section corresponds to
the boundary of the billiard and the coordinates of the ray are the polar angle
*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.

It has been shown[24] that the solutions of the wave equation for a generic shape
such as the quadrupole can be classified by their association with these three
different kinds of motion. The ray-mode (or wave-particle) correspondence becomes
stronger as we approach the short-wavelength (semi-classical) limit, which in this
work is defined by *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], and referred to in its most general form as EBK
(Einstein-Brillouin-Keller) quantization. The individual modes associated with
unstable periodic orbits and chaotic motion cannot be treated by any current
analytic methods (although the density of states for a chaotic system can be found
by a sophisticated analytic method based on Gutzwiller’s Trace Formula [26]). Finally, the modes associated with stable periodic orbits
can be treated by generalizations of gaussian optics and will be the focus of the
current work. It should be pointed out that modes associated with stable periodic
orbits can also be treated by the periodic orbit approach mentioned above and in the
*stable* case explicit quantization rules can be found which are
equivalent to our results for the closed cavity derived below [27].

If a mode is found by numerical solution, its interpretation in terms of the ray
phase space can be determined with reasonable accuracy by means of the Husimi
projection onto the phase space (see Fig. 2), although it is well-known that for
*kl* not much greater than unity the exact solutions tend to smear
out in the phase space over regions of order 1/*kl* and do not
correspond very closely to specific classical structures. For a closed generic ARC
the full spectrum will look highly irregular (see Fig. 3(a)), but contained in the full spectrum will be
regular sequences associated with tori and stable periodic orbits (Fig. 3(b)). The stable periodic orbit modes will give the
simplest such sequences consisting of two different constant spacings, one
associated with the longitudinal quantization of the orbit (free spectral range) and
the other associated with transverse excitations. In the example of Fig. 3 the imbedded regular spectrum is due to the stable
“bow-tie” orbit. The regular portion of the spectrum is
extracted by weighting each level by the overlap of its Husimi function with the
islands corresponding to the stable periodic orbit in the surface of section.
Clearly, hidden within this complex spectrum are simple regular mode sequences of
the type familiar from Gaussian optics. In the current work we show how to calculate
the resonant energies and spatial intensity patterns of such modes associated with
arbitrary stable periodic ray orbits for both the ideal closed resonator and a
dielectric resonator of the same shape with arbitrary dielectric mismatch
*n*. We shall refer to these as periodic orbit modes or PO modes.

In the case of the open resonator, the modes have a width which can be expressed as a
negative imaginary part of k, and some of the PO modes may be so broad (short-lived)
that they would not appear as sharp spectral lines. Conversely there can be many
long-lived modes associated with the chaotic regions, due to scarred states [17, 19, 20, 21] or dynamically localized states [29, 30] which are localized in the region of phase space above the
critical angle for total internal reflection. Therefore it is not the case that all
the high-Q modes are based on stable POs and describable by the theory below. Within
the gaussian-optical theory we present below the width of the PO modes is entirely
determined by Fresnel reflection at the interface and would be zero for a periodic
orbit which has all bounces above the total internal reflection condition, but would
be quite large for a periodic orbit, such as the two-bounce Fabry-Perot orbit, which
has normal incidence on the boundary. An exact solution must find a non-zero width
for *all* PO modes, due to evanescent leakage across a curved
interface, even if all the bounces satisfy the total internal reflection condition.

Another limitation of the gaussian theory of stable PO modes is that it predicts
exactly degenerate modes when the associated orbit has discrete symmetries, even in
cases for which a group-theoretic analysis shows that there can be no exact
symmetries (this is the case, for example in the quadrupole). Instead the exact
solutions will have some integer *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

The quantization of electromagnetic modes within dielectric bodies enclosed by
metallic boundaries, including the case of arbitrary index variation inside the
resonator and three-dimensions has been treated before in the literature [31]. However the generality of the treatment makes it difficult
to extract simple results of use to researchers working with uniform dielectric
optical micro-cavities; more importantly all the work of which we are aware focuses
on the case of Dirichlet boundary conditions corresponding to perfect reflection at
the boundary. Perfect reflection of course leads to true bound states. In the next
section we show how to generalize these results to the correct boundary conditions
at a dielectric interface and hence for the case of quasi-bound as opposed to bound
states. However, first, in this section we develop the formalism for the closed case
which we will generalize to the case of interest. In all of this work we will
specialize to the case of two dimensions, corresponding to an infinite dielectric
cylinder with an arbitrary cross-section in the transverse plane (see Fig. 4) with the condition
*k*
_{∥} = 0. A recent review by Laeri and Nöckel[32] has treated this two-dimensional case for the closed cavity
and has shown how solutions related to stable periodic orbits can be found using the
parabolic equation method, although their results are less explicit than those we
present below.

To conform with conventions introduced below we will refer to the two-dimensional
coordinate system as (*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.

Consider the solutions *E*(*X*, *Z*) of
the Helmholtz equation in two dimensions:

This solution is assumed to be defined in a bounded two dimensional domain
*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

The “N-bounce PO”’s corresponding to a boundary
*∂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}
= ${\sum}_{m=1}^{M}$
*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 (-∞, +∞).

We write the general solution as:

where the “local set” (*x*_{m}
,
*z*_{m}
) and the fixed set (*X*,
*Z*) are related by shifts and rotations (see Fig. 5). Next, in accordance with the parabolic equation approximation[31], we assume that the main variation of the phase in
z-direction is linear (“slowly varying envelope
approximation”) and factor it out:

This definition suggests defining the origins of the local coordinates such that
*z*_{m}
= *z* along the PO, and we
do so (see Fig. 5).

Next, we insert the solution Eq. (2) into Eq. (1) and using the invariance of the Laplacian, we obtain:

and the boundary condition translates into

Here ∇_{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.

Dropping for the moment the arm index, we will focus on Eq. (4). Inserting Eq. (3), we arrive at

The basic assumption of the method is that after removing phase factor
exp[*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 $\sqrt{l\lambda}$, intermediate between the wavelength and the cavity 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

We can now neglect the *u*_{zz}
term due to the condition
*kl* ≫ 1, and obtain a partial differential
equation of parabolic type:

where ℒ =
${\mathit{\partial}}_{\stackrel{\mathit{~}}{x}}^{2}$ +
2*i∂*_{z}
. Next, we make the ansatz

Inserting Eq. (9) into Eq. (8) and requiring it to be satisfied for all
*x*, we obtain the relations

Here and in the rest of the text we will use primes as a shorthand notation for a
*z*-derivative. Next, making the substitution

we obtain

Note that Eq. (13) is the Euler equation for ray propagation in a
homogeneous medium with general solution *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

Having found the general solution of Eq. (8) along one segment of the periodic orbit we must impose the boundary condition Eq. (5) in order to connect solutions in successive segments. Writing out this condition:

$$\phantom{\rule{10.2em}{0ex}}+\frac{{c}_{m+1}}{\sqrt{{Q}_{m+1}\left({z}_{m+1}\right)}}\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(\mathit{ik}{z}_{m+1}+\frac{i}{2}\Omega \left({z}_{m+1}\right){\tilde{x}}_{m+1}^{2}\right){|}_{\partial D}=0$$

This equation must be satisfied on an arc of the boundary of length $~\sqrt{\lambda l}$ around the reflection point. Since this length is much smaller
than *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,

$$k\left({l}_{m}+\frac{1}{\sqrt{k}}{\tilde{\xi}}_{1}\phantom{\rule{.2em}{0ex}}\mathrm{sin}{\chi}_{m}+\frac{1}{k}\frac{{\tilde{\xi}}_{1}^{2}}{2{\rho}_{m}}\phantom{\rule{.2em}{0ex}}\mathrm{cos}{\chi}_{m}\right)+\frac{1}{2}\frac{{Q}_{m+1}^{\prime}}{{Q}_{m+1}}{\left({\tilde{\xi}}_{1}\mathrm{cos}{\chi}_{m}+\frac{1}{k}\frac{{\tilde{\xi}}_{1}^{2}}{2{\rho}_{m}}\phantom{\rule{.2em}{0ex}}\mathrm{sin}{\chi}_{m}\right)}^{2}$$

and there is a amplitude condition as well,

Here, we assume that *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:

and

Here we always choose the principal branch of √*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

To formalize the relation to ray propagation in the paraxial limit we define the
ray position coordinate to be *Q* and the conjugate momentum
*P* = *Q*′, with *z*
playing the role of the time. Let’s introduce the column-vector $\left(\begin{array}{c}Q\\ P\end{array}\right)$. The “fundamental matrix”? ∏
is the matrix obtained by two such linearly independent vectors

where the linear independence, as expressed by the Wronskian condition
*W*(*Q*′,*Q*)
≠ 0 reduces to det ∏ ≠ 0. Then, for example,
the Euler equation Eq. (13) in each arm can be expressed as

where $\ud6ae=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ and $\mathit{\Im}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$. It’s straightforward to show that det∏
is a constant of motion. To take into account the discreteness of the dynamics
in a natural manner, we will introduce “coordinates” for
the ray, represented by a *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}

We will choose $\prod \left(z\right)=\left(\begin{array}{cc}1& z\\ 0& 1\end{array}\right)$, so that ∏^{-1}(*z*) =
∏(-*z*). Then if
*Q*_{m}
(*z*) =
*α*_{m}*z* +
*β*_{m}
, we have ${h}_{m}=\left(\begin{array}{c}{\beta}_{m}\\ {\alpha}_{m}\end{array}\right)$. In this notation, propagation within each arm (i.e.
*z* + 1 < *L*_{m}
where *L*
_{m-1} <
*z* < *L*_{m}
) is induced by
the ‘propagator’ ∏

Note that ∏ is the “ABCD” matrix for free propagation[33]. Thus, for example *z* →
*z*′ propagation for
*L*
_{m-1} <
*z* < *L*_{m}
<
*z*′ <
*L*
_{m+1} is

Of special importance is the monodromy matrix,
*𝚳*(*z*), which propagates rays a
full round-trip, i.e. by the length *L* of the corresponding PO
and is given by

for *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

Having determined how to propagate an initial solution *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

We will suppress again the reference to arm index and use the notation
*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

This periodicity condition will only be solvable for discrete values of
*k* and will lead to our quantization rule for the PO modes.

From the form of *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*.

On the other hand, for the stable case the eigenvalues come in complex conjugate
pairs and the reality of the monodromy matrix then implies that the
*eigenvectors* cannot be purely real and are related by
complex conjugation. Therefore we can always construct a gaussian solution based
on the eigenvector with *ImΩ* > 0 so that
there is a negative real part of the gaussian exponent and the resulting
solution decays away from the PO. It is easy to check that the range of the
decay is $~\sqrt{\lambda l}$ as assumed above.

The eigenvectors of the monodromy matrix are sometimes referred to as Floquet
rays of the corresponding Euler equation. In a more formal discussion which can
be straightforwardly generalized to arbitrary index variation and three
dimensions [31] our condition on the choice of Ω can be
expressed by recalling that the two eigenvectors must generate a constant
Wronskian, and we will choose the eigenvector
(*q*(*z*), *p*(*z*))
such that

This condition implies *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.

Having made this uniquely determined choice (up to a scale factor) and using $\left(\begin{array}{c}q\left(z+L\right)\\ p\left(z+L\right)\end{array}\right)={e}^{i\phi}\left(\begin{array}{c}q\left(z\right)\\ p\left(z\right)\end{array}\right)$ and
*c*
_{m+N}
=
*e*
^{-iπN}
*c*_{m}
we obtain

From Eq. (23) and Eq. (3), we finally obtain the quantization rule for the wavevectors of the bound states of the closed cavity:

where *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]. It arises in the following manner. As already noted,
the eigenvalues of the monodromy matrix for stable POs are complex conjugate
numbers of modulus unity whose phase is the Floquet phase. We can define $\phi =\mathrm{Arg}\left[\sqrt{\frac{q\left(L\right)}{q\left(0\right)}}\right]$, where Arg[∙] denotes the principal argument; hence
the Floquet phase depends only on *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*):

where [∙] denotes the integer part. There is no simple rule for reading off the Maslov index from the geometry of the PO; however the Maslov index doesn’t affect the free spectral range or the transverse mode spacing.

Except for the Maslov index, the quantization rule Eq. (27) is familiar from Fabry-Perot resonators. The
longitudinal mode index *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

As is well known, the solution Eq. (9) is only one family of the possible solutions of the
parabolic equation Eq. (8) satisfying the boundary conditions Eq. (5) and the periodicity condition Eq. (23), the one corresponding to the ground state for
transverse oscillations. Further solutions can be generated with algebraic
techniques which were originally developed in the context of quantum oscillators[35]. If we refer to Eq. (8) we see that we are looking for solutions of
*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.

We can easily show that the operators (Λ,
Λ^{†}, *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]. We can find the wavevector quantization condition for
this solution by calculating the additional phase acquired by
(Λ^{†}
*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

will satisfy the wavevector quantization rule:

This result has the well-known interpretation of adding *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)).

In order to obtain explicit forms for the excited state solutions found here one may use the theory of orthogonal polynomials to find

where *H*_{l}
are the Hermite polynomials[32].

As noted in the introduction, the solutions we have found by the parabolic equation method do not reflect correctly the discrete symmetries of the cavity which may be present. The theory of the symmetrized solutions and the breaking of degeneracy is essentially the same for both the closed and open cavity and will be presented in section 4 after treating the open case in the next section.

## 3 Opening the cavity - The dielectric resonator

We now consider the wave equation for a uniform dielectric rod of arbitrary cross-section surrounded by vacuum. As noted in the introduction, Maxwell’s equations separate into TM and TE polarized waves satisfying

Maxwell boundary conditions translate into continuity of Ψ and its normal
derivative across the boundary *∂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]. We now show how such quasi-bound modes can be incorporated
into the gaussian optical resonance theory just described.

As before we look for solutions which, within the cavity, are localized around
periodic orbits based on specular reflection within the cavity. In order to satisfy
the boundary conditions the solutions outside the cavity will be localized around
rays extending out to infinity in the directions determined by Snell’s
law at the bounce point of the PO. The quasi-bound state condition is imposed by
insisting that the solutions along those rays to infinity are only outgoing. This
can only be achieved by making *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]. It is interesting to attempt to incorporate such effects
into our approach at higher order, however we do not do so here and confine
ourselves to obtaining a consistent solution to lowest order in *kl*.

As before we will define the solution as the sum of solutions
*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*, ${\tilde{x}}_{\mathit{mt}}=\sqrt{k}x,\phantom{\rule{.2em}{0ex}}{\tilde{x}}_{m}=\sqrt{\mathit{nk}}x$, we get the parabolic equation Eq. (8) for each component, at lowest order in *k*.
The boundary conditions close to the *m*
^{th} bounce point
will take the form

and

Here *∂*_{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 ${\Phi}_{M}={n}_{M}\mathit{kz}+\frac{i}{2}{Q}_{M}^{\prime}{Q}_{M}^{-1}{\tilde{x}}_{M}^{2}$ and ${A}_{M}=\frac{{c}_{m}}{\sqrt{{Q}_{M}}}$. Here *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*.

Similarly to the closed case, the first continuity condition Eq. (35) must be satisfied on an arc of size $~\sqrt{\lambda l}$ on the boundary around each bounce point and that implies that the
phases of the incident, transmitted and reflected waves must be equal. This equality
will be implemented in the coordinate system
(*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.

The analysis of the phase equality on the boundary for the incident and reflected waves is exactly the same as in the closed case and again leads to Eq. (15) describing specular reflection. Equating the phases of the incident and transmitted wave leads to:

where we have defined *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*)

where *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].

Using the phase equality on the boundary the continuity of the field gives the general transport equation:

which becomes (using the conventions *Q*_{r}
=
*Q*_{i}
, *Q*_{t}
=
*Q*_{i}
/*μ*)

In order to find the quantization condition we need a direct relation between
*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:

At the level of approximation needed one finds the simple results
*∂*_{n}
Φ_{i} = *nk* cos *χ*,
*∂*_{n}
Φ_{r} = -*nk* cos *χ*,
*∂*_{n}
Φ_{t} = *k* cos *χ*_{t}
, leading
to

and

Note the key result that

which is precisely the Fresnel reflection law at a (flat) dielectric interface.

Now we impose the single-valuedness or periodicity condition to obtain the
quantization rule for *k*.

Note however that we have a qualitatively different situation than for the closed
cavity; some amplitude can be lost at each reflection and it will in general be
impossible to make a loop around the PO and return to the same field amplitude
unless *k* is complex. We have the condition

which can be satisfied by choosing *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

Recall that the Fresnel reflection law has the property that it gives a pure phase for rays incident above total internal reflection. Thus the new term in the quantization law due to Fresnel reflection can be either purely real (all bounces of PO totally internally reflected), purely positive imaginary (all bounces below TIR) or complex (some TIR bounces, some refracted bounces). If we define: ${\phi}_{f}=\mathrm{Re}\left[-i{\sum}_{b}^{N}\mathrm{log}\left[\frac{n{\mu}_{b}-1}{n{\mu}_{b}+1}\right]\right]$ and ${\gamma}_{f}=\mathrm{Im}\left[-i{\sum}_{b}^{N}\mathrm{log}\left[\frac{n{\mu}_{b}-1}{n{\mu}_{b}+1}\right]\right]$ then the quantization rule gives

As noted above, this result is only in the leading order approximation, and it ignores both the effects of evanescent leakage at a curved interface and the momentum width of the gaussian “beam” which leads to violations of ray optics. These effects will give a non-zero imaginary part (width) to all resonances, even those with all bounce points above TIR.

In Table 3 we present a comparison between the numerically obtained quantized
wavevectors for a bow-tie resonance and the values for the real and imaginary part
of *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

As noted above, the gaussian theory we have just presented predicts exact
degeneracies if there exist several symmetry-related stable orbits (note that in
this context the same path traversed in the opposite sense is considered a distinct
symmetry-related orbit). However it is well-known that wave equations with discrete
symmetries cannot in general have a degeneracy which is larger than the largest
dimension of the irreducible representations of the symmetry group of the equation.
Here we are concerned with the point group (rotations and reflections) of the
dielectric resonator in two dimensions. Let G be a group which leaves the cavity
invariant. Then, if Ψ(x) is a solution to the Helmholtz equation, so is
Ψ(*g*x), where *g* ∊
*G* and x = (*X*, *Z*). The
symmetrized solutions, i.e. solutions which transform according to the irreducible
representations of *G* under the action of *G*, can be
obtained by the projection operators of the group:

Here *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

Let’s consider our canonical example, the quadrupole (see Fig. 5). The symmetry group of the quadrupole is
*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

$${E}_{\left(-+\right)}=\frac{1}{4}\left({e}_{1}-{e}_{2}+{e}_{3}-{e}_{4}\right),\phantom{\rule{.2em}{0ex}}{E}_{\left(--\right)}=\frac{1}{4}\left({e}_{1}-{e}_{2}-{e}_{3}+{e}_{4}\right)$$

where

In this case the symmetrized solutions are just the solutions with definite parity with respect to the symmetry axes of the quadrupole.

The key point here is that within the parabolic equation approximation these four
solutions are exactly degenerate, whereas our group-theoretic analysis for the
exact Helmholtz equation tells us that they cannot be so, although they will be
nearly degenerate. Moreover our original solution
*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.

Before discussing the general rule, we illustrate the basic procedure for the
case of a bow-tie PO. Let ℓ_{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*)$${e}_{4}=E\left(gx\right)={e}^{\frac{1}{2}i\pi}\frac{1}{\sqrt{{e}^{\frac{i\phi}{2}}q\left({\mathit{\ell}}_{1}-z\right)}}\phantom{\rule{.2em}{0ex}}\mathrm{exp}\phantom{\rule{.2em}{0ex}}\left[\mathit{ik}\left(\frac{L}{2}+{\mathit{\ell}}_{1}-z\right)+\frac{i}{2}\Omega \left({\mathit{\ell}}_{1}-z\right){x}^{2}\right]\equiv {e}^{i\zeta}{e}_{3}$$

where the phase factor *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 ${e}^{\frac{1}{2}i\pi}$, which is due to the specific choice of branch-cut for $\sqrt{q\left(z\right)}$. Putting these together, we obtain

$${E}_{\left(+-\right)}=\frac{1}{2}\left({e}_{1}-{e}_{3}\right){e}^{i\frac{\zeta}{2}}\phantom{\rule{.2em}{0ex}}\mathrm{cos}\frac{\zeta}{2}$$

$${E}_{\left(-+\right)}=\frac{1}{2i}\left({e}_{1}+{e}_{3}\right){e}^{i\frac{\zeta}{2}}\phantom{\rule{.2em}{0ex}}\mathrm{sin}\frac{\zeta}{2}$$

$${E}_{(--)}=\frac{1}{2i}\left({e}_{1}+{e}_{3}\right){e}^{i\frac{\zeta}{2}}\phantom{\rule{.2em}{0ex}}\mathrm{sin}\frac{\zeta}{2}$$

The solutions are *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

$${E}_{\left(-+\right)},{E}_{\left(--\right)}\propto \mathrm{sin}\frac{\zeta}{2}=0\phantom{\rule{.8em}{0ex}}m=\mathrm{0,2,4},.\phantom{\rule{.2em}{0ex}}.\phantom{\rule{.2em}{0ex}}.$$

Thus the quasi-degeneracy of the solutions is two for the bow-tie, the solutions
with identical parity under *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*.

This illustrates a general procedure, valid for any stable PO. First, one finds
by the parabolic equation method a non-symmetrized approximate solution
*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

Although the construction just presented allows one to find the quasi-degeneracy and symmetry pairing, it is convenient when possible to have a simple rule to get the quasi-degeneracy and symmetry-pairing from the geometry of the orbit. The quasi-degeneracy is easily determined by the following rule:

*The quasi-degeneracy D is equal to the number of distinct classical
periodic orbits which are related by the spatial symmetry group and
time-reversal symmetry*.

In this rule “distinct” orbits are defined as orbits which cannot be mapped into one another by time translation. Therefore a self-retracing orbit such as the Fabry-Perot, two-bounce orbit, only counts as one orbit and is non-degenerate (see Table 2). In contrast for a circulating orbit like the diamond no translation in time will take the orbit into its time-reversed partner. This rule can be obtained from semiclassical methods similar to the Gutzwiller Trace formula[38]. The density of states can be expressed by a summation over periodic orbits and their repetitions; for stable periodic orbits the summation over repetitions yields a delta function at the semiclassical energies[27] (corresponding to the same wavevectors as we find from our quantization rule). This approach would give an alternative derivation of our results which is less familiar in optics than the parabolic equation method we have chosen. However the semiclassical method makes it clear that there will be a mode for every distinct symmetry-related PO using the definition we have just given (of course in this method, as in the parabolic equation method, one would predict an exact degeneracy instead of the quasi-degeneracy we have discussed).

Let us illustrate the application of this rule. The bow-tie orbit goes into itself under all the reflection symmetries and so spatial symmetry generates no new orbits; however time-reversal changes the sense of traversal of each leg of the orbit and does give a distinct orbit. Thus the predicted quasi-degeneracy is two, which we found to be correct by our explicit construction above. In contrast, the triangle orbit (see Table 2) has a symmetry related distinct orbit and a definite sense of circulation which is reversed by time-reversal, hence it should have a quasi-degeneracy 2 × 2 = 4 leading to quartets instead of doublets. A few different cases of this rule are illustrated in Table 2.

The rule we have just given tells one the quasi-degeneracy, D, but not the
symmetry-pairing. For the case of reflection symmetries one can state a second
rule which determines these pairings. First fold the PO of interest back into
the symmetry-reduced resonator[38] (see Table 2) using reflection until it completes *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.

We will illustrate this rule for the case of the bow-tie in the quadrupole. The
symmetry reduced PO is shown in the last column of Table 2. It has one corner bounce, one bounce on the
*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

The symmetry analysis above can only determine the existence of quasi-degenerate multiplets with small splittings, it cannot estimate the size of these splittings. In the phase space picture the splittings we are discussing come from tunneling between distinct periodic orbits, referred to as “dynamical tunneling” in the quantum chaos literature [39]. Techniques have been developed in that context for evaluating the splittings and we now apply those to the system we are considering.

Dynamical tunneling in integrable systems is essentially similar to the textbook
example of tunneling in one dimension (note that conservative dynamics in 1D is
always integrable), and the splittings in such a case come from first order
perturbation theory in the tunneling matrix element through an effective
barrier, just as they do for the one-dimensional double well potential. Systems
of the type we are considering, with mixed dynamics however show a very striking
difference. It has been found by both numerical simulations and analytic
arguments that the dynamical tunneling splittings in mixed systems are typically
many order of magnitude larger than found for similar but integrable systems[40, 41] (e.g in quadrupole vs. elliptical billiards). This
difference can be traced to the mechanism of “chaos-assisted
tunneling” (CAT). As opposed to the “direct”
processes when the particle (ray) “tunnels” directly from
one orbit to the other, the CAT corresponds to the following three-step process:
(i) tunneling from the periodic orbit to the nearest point of the chaotic
“sea”, (ii) *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.

The contribution of chaos-assisted tunneling can be evaluated both qualitatively
and quantitatively using the so called “three-level model”[40, 42], where the chaotic energy levels (the eigenstates
localized in the chaotic portion of the phase space) are represented by a single
state *E*_{C}
with known statistical properties. The
straightforward diagonalization of the resulting matrix yields[42]

where *E*_{R}
is the semiclassical energy of the
“regular” states (localized at the periodic orbits) which
does not include the tunneling contribution, and *V*_{RC}
is the corresponding coupling matrix element with the chaotic state. The
resonant denominator in Eq. (58) leads to strong fluctuations of the CAT-related
doublets, as is found in numerical simulations[42]. The average behavior of the splittings however is
determined by the matrix element *V*_{RC}
. By virtue of
the Wigner transformation[26] it can be shown [43] that |*V*_{RC}
|^{2} is
proportional to the overlap of the Wigner transforms of the
“regular” and “chaotic” states:

Assuming that, as required by Berry’s conjecture[44], *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

where 𝜜 is the area in the Poincare Surface of Section (in
(*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].

In Fig. 8 we show the numerically calculated splittings for
the bow-tie resonances in a resonant cavity with a fixed quadrupolar deformation
*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].

For the bow-tie orbit the quasi-degeneracy is associated with time-reversal symmetry, but as discussed above, it can equally well be associated with spatial symmetries and indeed this was the situation in the first work on dynamical tunneling [39]. Splittings of this type can also be adequately described using the framework developed in the present section.

## 5 Conclusions

We have generalized gaussian optical resonator theory to describe the resonances
associated with stable periodic orbits of arbitrary shaped two-dimensional
dielectric resonators using the parabolic equation method. The correspondence to ray
optics emerges naturally when imposing the boundary conditions at the dielectric
interface which leads to the appropriate ABCD matrices for reflection, transmission
and propagation. For a perfectly-reflecting cavity one gets quantized solutions at
real values of *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.

We noted that for a cavity with discrete symmetries one will typically be able to construct several symmetry-related, nominally degenerate solutions of the wave equation for each PO. However group-theoretic arguments indicate that these solutions cannot be exactly degenerate and lead us to construct symmetrized solutions which form quasi-degenerate multiplets. We presented a construction and then two simple rules for calculating the quasi-degeneracy of these multiplets and their symmetry quantum numbers. In the final section of the paper we show that the splittings of these multiplets are much larger than expected, due to the phenomenon of “chaos-assisted” tunneling, and estimate the average splitting in terms of classical quantities.

There are several limitations of this work which we hope to address in future work.
One obvious shortcoming is the prediction of zero width modes for the dielectric
cavity if the underlying PO has all of its bounces above total internal reflection.
Internal reflection is not perfect for these systems at any angle of incidence for
two reasons. First, as these solutions describe gaussian beams with some momentum
spread, every solution should have some plane-wave amplitude at an angle of
incidence for which it can be partially transmitted. This type of correction exists
even for a gaussian beam incident on an infinite planar surface and leads to an
outgoing beam direction which can be significantly different from that predicted
from Snell’s law; we have analyzed the beam deflection for this case
recently [45]. For the case of the bow-tie modes of the dielectric
resonator the effect of the momentum spread in inducing a finite width was also
evaluated using a semiclassical method in Ref. [46]. Second, due to the curvature of the interface in such
resonators, there will always be some evanescent leakage, which we can think of as
due to direct tunneling through the angular momentum barrier as is known to occur
even for perfectly circular resonators. We have evaluated this type of correction
also recently[43]. It is still not clear to what extent these effects can be
accounted for systematically within a generalization of our approach here, e.g. to
higher order in *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, 17] and numerical studies[17, 45] demonstrate that for *kl* not too large
(~ 50 – 100) these higher order effects must be taken into
account.

We acknowledge discussions with J. U. Nöckel that stimulated this work as well as helpful discussions with M. Gutzwiller. This work was partially supported by NSF grants DMR-0084501 and DMR-0134736

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