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Rings as annular billiards
Ring resonators have found widespread use in nanophotonics because
their operating principle is extremely easy to understand, and many of
their properties can be described with the same concepts that apply to
one-dimensional optical waveguides, where light is allowed to
propagate freely in one direction and confined in the other
(transverse) directions. The ring is a waveguide bent back on itself
to form a loop. One then can characterize the propagation loss, bend
loss and coupling to adjacent waveguides. In a perfectly circular
ring, the curvature introduces a centrifugal potential that can
effectively be modelled as a slanted index-of-refraction profile in
the waveguide cross section.
An alternative but equivalent view is to describe a circular ring as a
layered dielectric with rotational symmetry, and apply generalizations
of Mie theory to it. See also the
hitchhiker's guide to dielectric cavities. The idea is simply that
a ring is nothing but a disk with a hole, and exact solutions can be
written down both for disks and holes.
But what about non-ideal, deformed rings?
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In a wide enough ring waveguide, the inner hole doesn't really
have any effect when the circulating modes are pushed to the outer
edge by the centrifugal potential. Then, even a non-concentric hole
shouldn't worry us at all. This is shown here for a whispering-gallery
mode that is just far enough removed from the inner edge to remain
essentially undisturbed by the off-center hole.
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Ray pictue
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The ray pattern shown here for a non-concentric, closed cavity is
chaotic. This can be seen clearly in a Poincaré surface of
section, where a
two-dimensional cloud of points is generated by a single
trajectory.
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The solid lines near the top of this surface of section
are whispering-gallery trajectories which never feel the inner
boundary, and thus behave as if the hole were not there.
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The off-center geometry has preferred coupling directions. This is one
of the main reasons why one should be interested in non-circularly
symmetric resonator designs. The red arrow indicates the direction of
an incoming wave, and the two plots differ only in this choice of
direction.
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Although the excitation beam is tuned to the frequency of a
cavity resonance, there is no coupling into that resonance in the
righthand picture. This is because the input
beam and cavity mode have opposite parity with respect to the
horizontal axis. To learn more about the treatment of discrete
symmetries in waveguide and resonator structures, you may want to look
at my work on Fano resonances and scattering theory in low-dimensional
structures.
Relation to quantum chaos
Rings are not just of great applied interest, but also useful as a
model system in quantum chaos. The first paper exploring this
connection is discussed here:
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This page © Copyright Jens Uwe
Nöckel, 10/2004
Last modified: Sat Jul 19 23:49:18 PDT 2008
