\documentstyle[preprint,aps]{revtex}
\newcommand{\bq}{\begin{equation}}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\eq}{\end{equation}}
\newcommand{\ea}{\end{eqnarray}}
\begin{document}
\date{\today}
\title{Spectral Statistics for Quantum Chaos with Ray Splitting}
\author{R. N. Oerter \cite{byline}, E. Ott, T. M. Antonsen, Jr., and P. So}
\address{University of Maryland\\
College Park, MD 20742}
\maketitle
\begin{abstract}
We investigate the behavior of ray trajectories and
solutions of the wave equation of two dimensional billiard-like
systems with ray splitting. By ``ray splitting'' we mean the phenomenon
whereby a ray incident on a sharp boundary leads to two or more rays
traveling away from the boundary (e.g. a transmitted ray and a reflected ray).
Billiard systems with the same overall shape, but with and without ray
splitting boundaries present are examined and compared. It is found that,
for the configurations considered,
the level spacing distribution and the spectral rigidity for the case without
ray splitting are intermediate between Poisson and Gaussian Orthogonal
Ensemble (GOE) statistics, while the behavior with ray splitting is
very close to GOE.
\end{abstract}
\par
\noindent
{\it PACS numbers:} 05.45.+b \\
\noindent
{\it Keywords:} resonance spectra, inhomogeneous media, semiclassical
dynamics
\pagebreak
\section{Introduction}
Quantum chaos focuses on the semi-classical,
or short-wavelength, limit of a system described by wave mechanics
\cite{Gutz}\cite{Reichl}\cite{Ed}.
In this limit the ray equations
are just the equations of classical mechanics for the motion of a particle.
When the particle motion is chaotic, some very general results have been
conjectured to hold,
and the behavior of the system should display universal properties. In
particular, the energy level distribution is expected to be given by the
gaussian orthogonal ensemble (GOE) if the system obeys an antiunitary
symmetry \cite{Gutz}\cite{Berry}. When the phase space consists of a mixture
of chaotic orbits and KAM tori, the level distribution is intermediate
between the integrable case of Poisson statistics and the GOE case
\cite{Intermediate}. At sufficiently short wavelengths, this
transition is expected to be universal \cite{Intermediate}.
In this paper we consider
systems for which the ray approximation may be considered to hold everywhere
except at an interface where there is an abrupt change of some physical
parameter. Examples of this type of system
include a quantum particle in a box where the potential
has a constant value in one region and jumps discontinuously to a different
constant value in a second region \cite{Blumel},
an optical system with two regions
having different indices of refraction, a bounded elastic medium supporting
both shear and compression waves \cite{Couchman}\cite{Weaver},
and a thin microwave cavity with a
sudden change in the cavity height. In this paper we use the thin microwave
cavity (Fig.\ \ref{fig:cavity})
as our model, but the other systems will have very similar behavior.
For these systems the usual semi-classical approach is insufficient in that
ray trajectories are complicated by the need to take into account the
reflection and transmission which occurs at the interface between the
regions.
Systems satisfying a wave equation and having hard boundaries exhibit
a type of ray splitting when a ray incident on the boundary excites a
so-called ``creeping wave'' \cite{Watson}. However, these
creeping waves appear at higher order in the semiclassical expansion
(higher order in $(kd)^{-1}$, where $k$ is the wavenumber and $d$ is a
typical length scale of the system), while ray splitting at an
interface occurs at lowest order in
$(kd)^{-1}$. In the present work we confine our attention to ray
splitting at interfaces.
Previous work \cite{Couchman} has shown that the presence of a ray splitting
interface can increase the degree of chaotic behavior of suitably defined
ray trajectories
in cases where there is a mixed (chaotic and KAM) phase space in the absence
of ray splitting.
It is natural to ask whether this increase of ray chaos shows up in
the quantum spectrum as a shift toward GOE and away from integrable (Poisson)
statistics. In fact, Schuetz \cite{Schuetz}, following Berry
\cite{Berry}, shows that the spectral rigidity, $\Delta (l)$, (to be
defined below), should have the leading order
behavior for the ray splitting case:
\bq
\Delta (l) = {\alpha \over 2 \pi^2} \ln (l) + O(1),
\eq
where $\alpha$ is a constant of order unity which the analysis
in \cite{Schuetz} did not pin down. In the non-ray splitting
case considered by Berry, $\alpha$ is exactly unity, which is the
correct leading order behavior for the GOE. The numerical results
of the present work indicate that for systems with ray splitting
the spectral statistics are very close to GOE.
Thus,
when ray splitting is present, GOE statistics
can be anticipated in a much wider class of system
shapes than has been considered
previously.
\section{Description of the Model}
The model system we consider for numerical study
is a microwave cavity made of a perfectly
conducting material. The cavity is large compared to the typical wavelength
in two dimensions, but small enough in the third dimension (which we take
as the {\it z} direction) so that the electric field is constant in that
direction. These waves are described by the {\it z}-component
of the electric
field (all other electric field
components are zero), which satisfies the Helmholtz
equation in two dimensions:
\bq
(\nabla^2 + k^2)E = 0, \label{helmholtz}
\eq
where the wavenumber $k$ is related to the resonant frequency $\omega$
of the
cavity by $k = \omega /c$, and $c$ is the speed of light.
The boundary conditions are that $E=0$ at the sides of the cavity.
Furthermore, we assume that the cavity consists of two regions of
different thicknesses $h_1$ and $h_2$
in the {\it z} direction (Fig.\ \ref{fig:cavity}), with $kh_{1,2} \gg
1$.
The boundary conditions at the interface between the two regions are:
\bq
h_1 E_1 = h_2 E_2, \label{bcint1}
\eq
\bq
{\bf n}\cdot\nabla E_1 = {\bf n}\cdot\nabla E_2, \label{bcint2}
\eq
where ${\bf n}$ denotes the unit normal to the boundary.
When constructing the semiclassical (ray) solutions of Eq. (1) one must
consider the reflection and transmission of plane waves incident on a boundary
such as that depicted in Fig.\ \ref{fig:cavity}
which is approximated as being locally flat.
Using the boundary conditions\ (\ref{bcint1}) and\ (\ref{bcint2}),
the power reflection and transmission
coefficients for a ray incident from region 1 to region 2 are
\ba
R &=& {(1-r)^2 \over (1+r)^2 } \label{rtcoeff1} \\
T &=& {4 r \over (1+r)^2}, \label{rtcoeff2}
\ea
where $r= h_1/h_2$. (If the incident wave is in region 2, $r$ should be
replaced by $1/r$.)
Note that, since Eq.\ (\ref{helmholtz}) applies in
both regions 1 and 2, with the same value of $k = \omega /c$ in both
regions,
the angle of incidence and the angle of transmission are the same. Thus,
in contrast to a quantum particle in a
box with different constant potentials in the two regions,
there is no refraction of the transmitted wave. Furthermore, the reflection and
transmission coefficients are independent of the angle of incidence, so
there is no total internal reflection for this system. This situation may
be thought to represent the simplest possible case where ray splitting
occurs.
Another situation which has similar characteristics (i.e.
absence of refraction at the interface), although with different
boundary conditions on the ray splitting surface, occurs for Schr\"odinger's
equation with a potential that is a delta function of strength $V_0$ on the ray
splitting surface. Here the boundary conditions on the wave function become
$\psi _1 = \psi _2$ and ${\bf n} \cdot \nabla \psi _1 - {\bf n} \cdot
\nabla \psi _2 = V_0\psi _{1,2}$. In what follows, with experimental
realizability in mind, we shall use the microwave boundary conditions (2) and
(3) rather than these quantum boundary conditions.
\section{Classical Ray Mechanics}
\label{sec:classical}
When there is no ray splitting, the ray approximation to
Eq.\ (\ref{helmholtz}) gives a billiard in two dimensions, with specular
reflection at the boundary. For the ray splitting case, we can think
about the classical dynamics in the following way: when the particle
strikes one of the outer edges it is specularly reflected, but when it
strikes an interface
between regions with different physical properties
it has a probability $R$ of being reflected and
a probability $T$ of being transmitted, where $R$ and $T$ are the
reflection and transmission coefficients from Eqs.\ (\ref{rtcoeff1},
\ref{rtcoeff2}).
When the ray hits the interface, we randomly choose whether it is
reflected or transmitted, according to the probabilities $R$ and $T$.
Between bounces, the particle moves in a straight line. Thus a ray
trajectory can be specified by listing the successive points where the ray
hits the {\it side}, and the angle at which it hits each time.
(By a {\it side} we mean {\it either} an interface between $h_1$ and
$h_2$ regions {\it or} a perfectly reflecting billiard wall.)
The resulting map preserves phase space area if we use as phase space
coordinates the variables $(\sigma,\tau)$, defined as follows:
$\sigma$ is the distance to the bounce point measured along the boundary
from some arbitrary fixed reference point,
normalized to one, and $\tau$ is the cosine of
the angle between the momentum vector {\it after} the bounce
and the counterclockwise tangent to the
boundary (see Fig.\ \ref{fig:coords}). In the ray splitting case, there are two
pieces of phase space corresponding to the two regions, and a ray
hitting the interface is plotted in the region of phase space where
it ends up after the random choice is made.
Calculations were done for the particular billiard shown in
Fig.\ \ref{fig:shape}. The right (top) side
of the cavity is an arc of a circle of radius $R_1$ ($R_2$) whose center
lies on the $x$ ($y$) axis. The circles meet at the point $(a,b)$.
The ray splitting interface is the dashed line from the origin to $(a,b)$.
For the results reported in this paper
the parameters are
$R_1 = 20$, $R_2 = 6.2$, $(a,b) = (1.6,1)$,
$r = h_1 / h_2 = (\protect\sqrt{2} - 1)^{2} $.
For this configuration, if billiards with perfectly reflecting walls
were formed for the shapes of
each of the subregions, then there is mixed
chaotic and KAM behavior, with a significant fraction of the phase
space being occupied by KAM tori. This is exhibited in
Fig.\ \ref{fig:separate}, where Fig.\ \ref{fig:separate} (a) corresponds
to region 1 of Fig.\ \ref{fig:shape} and Fig.\ \ref{fig:separate} (b)
corresponds to region 2 of Fig.\ \ref{fig:shape}.
Similarly, with ray splitting removed,
Fig.\ \ref{fig:whole} (a) shows that the resulting billiard for the entire
region also displays mixed KAM/chaotic behavior.
Figure\ \ref{fig:whole} (b) demonstrates the effect of introducing ray
splitting.
The entire phase space becomes chaotic
in the sense that the ray trajectory eventually visits every region of
the phase space. Thus all of the KAM tori are destroyed by the ray splitting
interface, in contrast to the case that was studied in \cite{Couchman}.
The mechanism for the destruction of KAM tori is quite simple.
Suppose a ray
starts out on
what would be a KAM torus if the ray splitting interface were absent.
If it hits the ray splitting
interface, it can be transmitted, staying on the original torus, or be
reflected into some new orbit. The new orbit could correspond to another
KAM torus of the original billiard or to a chaotic orbit. A KAM torus
which retains orbits forever is thus only possible in the following
circumstances:
if the torus never
intersects the ray splitting surface, or if the portions of the KAM
tori defined by the two subregions (assuming reflection) coincide for
the part of phase space corresponding to the common boundary (e.g.,
when the region is a rectangle and the interface is parallel to one of
the sides). For the case of Fig.\ \ref{fig:shape}
there are no such tori, and therefore
all orbits eventually feel the effects of the chaotic
fraction of phase space.
In cases where there is a critical angle for total internal reflection,
such as the Schr\"{o}dinger equation with regions of different constant
potential, or the elastic medium considered in \cite{Couchman}, a
particular periodic orbit for one of
the subregions may only hit the interface
at angles greater than the critical angle. In such cases the orbit is not
split at the interface. Thus, there may be regions of phase space that
retain KAM tori even when ray splitting is present.
\section{Chaotic Case}
\label{sec:chaotic}
The wave properties of the billiards such as the one depicted in Fig.\
\ref{fig:shape} may be discussed in terms of several statistical
measures of the spectrum.
These
statistical measures are defined in terms of the
level counting function $N(k^2)$, the number of resonant modes with
wavenumber less than
a given value $k$.
Define an
``unfolded'' level counting function $\hat{N} (e)$
by fitting $N(k^2)$
with a quadratic function and then let the unfolded
``energies'' $e_i$ be defined by
\bq
e_i = a k_i^2 + b k_i +c, \label{unfold}
\eq
where $a$, $b$, and $c$ are the fitting parameters. This gives an unfolded
spectrum with unit level density if the resonant levels follow the expected
Weyl distribution:
\bq
N(k^2) \approx {A \over 4\pi}k^2 - {P \over 4\pi} k, \label{weyl}
\eq
where $A$ is the area and $P$ the perimeter of the region.
$N(k^2)$ is called the (smoothed) level counting function.
(It is shown in \cite{Weylnew} that this is the correct form of the Weyl
formula for a region with Dirichlet boundary conditions on the outer
boundary and boundary conditions of the form of Eqs.\ (\ref{bcint1})
and\ (\ref{bcint2}) on the ray splitting interface.)
Numerical solution of Eq.\ (\ref{helmholtz}) for the lowest 500
resonant values
of $k^2$ was accomplished using a modified boundary element technique
that we have developed for this problem, described in detail in the
appendix.
The probability distribution of level spacings $P(s)$ is defined so that
$P(s) ds$ is the probability
that $s$,
the normalized separation between neighboring values of $k^2$, lies between
$s$ and $s+ds$. To obtain $s$, the separation of neighboring values of $k^2$
is divided by $dN(k^2)/dk^2$, so that the
average value of $s$ is one.
For integrable systems a Poisson distribution is
expected \cite{Tabor}:
\bq
P(s) = \exp (-s).
\eq
The Brody distribution \cite{Brody} is a one-parameter family
of distributions given by
\bq
P_{\beta} (s) = A_{\beta} s^{\beta} \exp \left( -\alpha s^{1+\beta}\right) ,
\eq
where the normalization is $A_{\beta} = (1+\beta)\alpha$,
\bq
\alpha = \left( \Gamma \left( {2+\beta \over 1+\beta}
\right) \right)^{1+\beta} ,
\eq
$\Gamma $ is the gamma function, and $\beta$ is the Brody parameter.
The Brody distribution interpolates between the Poisson distribution,
$ \beta = 0 $,
and the Wigner distribution, $\beta = 1$ (which one expects to be valid for a
completely chaotic system),
so that $\beta$ can be used as a measure of how close a
given distribution is to the two extremes: fully integrable or fully
chaotic.
We will use the
Brody distribution as a convenient measure of the degree of chaoticity of
the system.
In the absence of ray splitting we expect to see statistics intermediate
between Poisson and GOE \cite{Intermediate},
because of the mixed phase
space evident in Fig.\ \ref{fig:whole} (a). When ray splitting is present
we expect to see purely GOE statistics since the phase space has become
completely chaotic (Fig.\ \ref{fig:whole} (b)). The
level spacing distribution in the absence of ray splitting is best fit
with a Brody parameter of $0.42$, clearly showing the effect of having
a mixed phase space
(Fig.\ \ref{fig:brody} (a)). When ray splitting is introduced
(Fig.\ \ref{fig:brody} (b)) the Brody parameter becomes $0.97$; very
close to the expected value of $1.0$ for the GOE spectrum.
Another statistical measure of the resonance spectrum is the spectral
rigidity $\Delta (l)$ \cite{Dyson}\cite{Reichl}.
The spectral rigidity $\Delta (l)$ is
defined as the squared deviation of $\hat{N} (e)$
from the best-fitting straight line, integrated over an interval
in $e$ of length $l$, and then averaged over a number of intervals of the
same length.
The numerical results for the
spectral rigidity show a clear shift from intermediate statistics
(Fig.\ \ref{fig:d3chao} (a))
to GOE statistics (Fig.\ \ref{fig:d3chao} (b)).
Since many physical systems have some amount of ray splitting, and since
ray splitting always tends to increase the amount of chaos in the system,
our results suggest that GOE-like statistics should be found much more
commonly (i.e., for a much less restricted class of shapes) than would be the
case in the absence of ray splitting.
\section{Conclusions}
\label{sec:conclu}
The model of ray splitting presented in this paper is a particularly
straightforward one, in that there is no refraction or critical reflection, and
the classical orbits are independent of energy. The model may be realized
experimentally by a thin microwave cavity and thus is experimentally testable.
(In quantum mechanics somewhat similar behavior is obtained with a delta
function potential on the ray splitting surface.)
We have shown how ray splitting causes the destruction of KAM tori in the
semi-classical picture. In the full wave solution we have demonstrated a
corresponding transition in the spectral statistics. In the absence of ray
splitting, the classical phase space is mixed (KAM tori and chaotic regions)
and the spectral statistics of the wave solution are intermediate between the
statistics expected for an integrable system and those expected for a chaotic
system. When ray splitting is introduced the KAM tori are destroyed and the
spectral statistics become very close to the GOE statistics expected for a
classically chaotic system. Thus with ray splitting GOE statistics is to be
expected in a much broader class of billiard shapes.
This research was supported in part by the Office of Naval Research. R.N.O.
was also partially supported by Department of Energy grant DE-FG02-94ER-40854.
\appendix
\section*{Numerical Procedure}
The solution of the Helmholtz equation \ (\ref{helmholtz}) in the presence
of ray splitting introduces some difficulties which are not present when
only a single region is considered. The approach used here is based on
the boundary element method, which has been discussed in textbooks on
numerical techniques for partial differential equations \cite{Brebbia}.
Some of these texts discuss the problem of subregions, but we found the
standard procedure inadequate for our purposes. In this appendix we
outline the modified procedure we used to obtain the results reported
in this paper.
An integral representation for the solution to the Helmholtz equation
is obtained in the usual way by multiplying Eq.\ (\ref{helmholtz})
by the appropriate Green function and integrating the result over the
region. This is done for each of the two (or more) regions separately.
In our case the Green function is the same for both regions, and the
reflection comes about solely from the boundary conditions at the interface,
Eqs.\ (\ref{bcint1}),
and\ (\ref{bcint2}).
For the derivation that follows we allow the two Green functions,
$G_1$ and $G_2$, to be different. If $E_1$ and $E_2$ represent the
solution to the Helmholtz equation in region $1$ and $2$, respectively,
then the boundary integrals are
\ba
E_1(x) &=& \int_{S_1} ( E_1(y) \nabla_{\hat{n}(y)} G_1 (x,y) - G_1 (x,y) \nabla_{\hat{n}(y)} E_1 (y) ) dy \\
E_2(x) &=& \int_{S_2} ( E_2(y) \nabla_{\hat{n}(y)} G_2 (x,y) - G_2 (x,y) \nabla_{\hat{n}(y)} E_2 (y) ) dy ,
\label{boundint}
\ea
where $S_{1,2}$ are the boundaries to the two regions, including the
interface, and $\nabla_{\hat{n}(y)}$ is the normal derivative evaluated at the point $y$.
The ``direct'' method of solving these equations \cite{Brebbia}
is to discretize them, impose the boundary conditions (Eqs.\ (\ref{bcint1}),
and\ (\ref{bcint2})
together with $E=0$ on the exterior boundaries), and then search for
values of $k$ for which the determinant of the resulting matrix is zero.
In this procedure the unknowns are the values of the normal derivative
of $E_{1,2}$ on the exterior boundaries and the value of $E_1$ and its
normal derivative on the interface. Experience with the case of a single
region indicates that it is better to introduce an auxiliary solution
which allows all terms involving the Green function itself to be eliminated,
leaving only terms involving its normal derivative. This is preferable
because the normal derivative is better behaved than the Green function
in the coincidence limit $y \rightarrow x$. In fact, our experience indicates
that this ``indirect'' method is more accurate
than the direct method by several orders of magnitude. It seems worthwhile,
then, to develop a similar scheme for the case of two regions.
Introduce auxiliary exterior solutions $\tilde{E}_{1,2}$ which are taken to be
zero inside the respective region, solve Eq.\ (\ref{helmholtz}) in the
exterior of the region, and have boundary conditions which we are free
to choose as we like. These exterior solutions satisfy the same
equations (\ref{boundint}) except that the left-hand side is zero
when $x$ is a point in the interior of the respective region.
Subtract the equations for $\tilde{E}_{1,2}$ from those for $E_{1,2}$.
Defining
\ba
D_1(y) &=& E_1(y) - \tilde{E}_1 (y), \text{ and}\\
P_1(y) &=& \nabla_{\hat{n}(y)} E_1 (y) - \nabla_{\hat{n}(y)} \tilde{E}_1 (y),
\ea
we find
\ba
\int_{S_1} (D_1 (y) \nabla_{\hat{n}(y)} G_1 (x,y) - P_1 (y) G_1 (x,y) ) dy &=& E_1 (x) \\
\int_{S_2} (D_2 (y) \nabla_{\hat{n}(y)} G_2 (x,y) - P_2 (y) G_2 (x,y) ) dy &=& E_2 (x) .
\label{geneqs}
\ea
Up to this point we have not imposed any boundary conditions. Let us
denote the ray splitting interface by $S_I$, and the remaining boundary of
each region,
excluding the interface, by $\hat{S}_{1,2}$. For $E_{1,2}$ we have
\ba
E_1(x) &=& 0 ; x \in \hat{S}_1, \\
E_2(x) &=& 0 ; x \in \hat{S}_2,
\ea
and Eqs.\ (\ref{bcint1}),
and\ (\ref{bcint2}) for $x$ on the interface. Choose boundary
conditions on $\tilde{E}_{1,2}$ by setting
\ba
P_1(x) &=& 0; x \in \hat{S}_1, \\
P_2(x) &=& 0; x \in \hat{S}_2, \\
D_1(x) &=& 0; x \in S_I, \\
D_2(x) &=& 0; x \in S_I. \label{bctilde}
\ea
A count of the degrees of freedom reveals that we are missing some equations.
These are obtained by taking the normal derivative of Eqs.\ (\ref{geneqs})
with respect to $x$, for $x \in S_I$. The resulting system of equations is
\ba
0 &= \int_{\hat{S}_1} D_1 (y) \nabla_{\hat{n}(y)} G_1 (x,y) dy
- \int_{S_I} P_1 (y) G_1 (x,y) dy , & x \in \hat{S}_1 \\
0 &= \int_{\hat{S}_2} D_2 (y) \nabla_{\hat{n}(y)} G_2 (x,y) dy
- \int_{S_I} P_2 (y) G_2 (x,y) dy , & x \in \hat{S}_2 \\
0 &= r \int_{\hat{S}_1} D_1 (y) \nabla_{\hat{n}(y)} G_1 (x,y) dy
- \int_{\hat{S}_2} D_2 (y) \nabla_{\hat{n}(y)} G_2 (x,y) dy\\
&+ \int_{S_I} ( P_2 (y) G_2 (x,y) - r P_1 (y) G_1 (x,y) ) dy , & x \in S_I \\
0 &= \int_{\hat{S}_1} D_1 (y) \nabla_{\hat{n}(x)} \nabla_{\hat{n}(y)} G_1 (x,y) dy
- \int_{\hat{S}_2} D_2 (y) \nabla_{\hat{n}(x)} \nabla_{\hat{n}(y)} G_2 (x,y) dy\\
&+ \int_{S_I} ( P_2 (y) \nabla_{\hat{n}(x)} G_2 (x,y) - P_1 (y) \nabla_{\hat{n}(x)}
G_1 (x,y) ) dy , & x \in S_I .
\label{fulleq}
\ea
This procedure has introduced the second derivative of the Green function,
but because of the particular choice of boundary conditions,
Eq.\ (\ref{bctilde}), the second derivative is never evaluated in the
coincidence limit. The price we pay for this is that the terms where the
Green function itself needs to be evaluated in the coincidence limit
have not been entirely eliminated. However, this procedure does yield
a significant improvement in accuracy over the direct method.
The solution follows by discretizing Eqs.\ (\ref{fulleq}) and
searching for the values of the wavenumber, $k$, for which the
determinant of the resulting matrix is zero. The numerical search is
facilitated by the fact that the matrix involved contains blocks of
zeroes, allowing a block LU decomposition.
This procedure yields
twice the actual number of eigenvalues, just as in the case of a
single homogeneous region. The excess eigenvalues are eliminated using
an auxiliary refractive index, in the same way as in that case
\cite{McDonald}.
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\end{references}
\begin{figure}
\caption{Microwave cavity with two regions of different thicknesses
$h_1$ and $h_2$.}
\label{fig:cavity}
\end{figure}
\begin{figure}
\caption{Coordinates for the ray problem. $s$ is the distance
along the perimeter measured from an arbitrary reference point $p_0$.
$\sigma \equiv {s \over
\text{perimeter}}$, and $\tau \equiv \cos \alpha$.}
\label{fig:coords}
\end{figure}
\begin{figure}
\caption{Cavity used in sections \protect\ref{sec:classical} and
\protect\ref{sec:chaotic}. }
\label{fig:shape}
\end{figure}
\begin{figure}
\caption{Classical ray trajectories for (a) region 1 and (b) region 2 of
the cavity shown in Fig.\ \protect\ref{fig:shape}, considered as
separate billiards.
In both cases two orbits are plotted, one lying on a KAM torus
(1000 iterations of a single initial condition)
and one chaotic
orbit (20,000 iterations of a single initial condition). The
discontinuous-looking behavior at certain values of $\sigma$ is caused
by the corners of the billiard, where the tangent to the perimeter jumps
discontinuously.}
\label{fig:separate}
\end{figure}
\begin{figure}
\caption{Classical ray trajectories for the whole cavity of Fig.\
\protect\ref{fig:shape}:
(a) without ray splitting and (b) with ray splitting. The
surface of section is taken to be the boundary of region 2.}
\label{fig:whole}
\end{figure}
\begin{figure}
\caption{Histogram of $P(s)$ and the best fitting Brody distribution
(dashed line) for the cavity of Fig.\ \protect\ref{fig:shape}: (a) without
ray splitting and (b) with ray splitting.}
\label{fig:brody}
\end{figure}
\begin{figure}
\caption{Spectral rigidity
for the cavity of Fig.\ \protect\ref{fig:shape} (a) without
ray splitting and (b) with ray splitting.}
\label{fig:d3chao}
\end{figure}
\end{document}