T H E S I S W O R K

The first prize was shared between myself and Brett Esry.

More information on the prize, and a short abstract can be found here.

To find out more about the other finalists in that competition, take a look at the abstracts of the special session in Atlanta for the DAMOP Outstanding Thesis Award.

**The participants were**

*Michael Andrews (MIT), advisor: Wolfgang Ketterle
Brett Esry (JILA Colorado), advisor: Chris Greene
Lisa Wiese (Nebraska), advisor: Duane Jaecks
Fredrick Fatemi (Virginia), advisor: Louis Bloomfield
Jens Nöckel (Yale), advisor: Douglas Stone.*

**The
Award Committee consisted of:**

*Hossein Sadeghpour* ITAMP, Harvard University

*Eric Cornell*, JILA

*Wendell Hill* , University of Maryland

*Barbara Levi*, American Institute of Physics

*Michael Morrison*, University of Oklahoma

Taken to the extreme, total internal reflection makes it
possible to fabricate lasers with resonator dimensions on the
order of the optical wavelength [2,5]. In particular,
ultralow-threshold lasing without the need for Bragg mirrors
has been observed in thin semiconductor disk resonators with a
circular cross section. These *microdisk lasers* rely on
ring-shaped modes localized near the dielectric interface with
the surrounding lower-index medium (e.g. air). In the ray
picture, the light circulates around the disk in a series of
total internal reflections (see Fig. 1), forming a
trajectory that is known from acoustics as a
``*whispering-gallery*'' (WG) pattern.

For the analogous WG modes in fused-silica microspheres,
record Q-factors well above 10^{9} have been measured
at optical wavelengths [6,7]. *Infinite* lifetimes are unattainable
even in the absence of material losses, because total internal
reflection is frustrated by the finite curvature of the
surface. This unavoidable outcoupling loss is analogous to
quantum tunneling and its rate is correspondingly small. The
discrete wavelengths at which WG resonances occur in thin disks
or perfect spheres, as well as the associated linewidths, can
be calculated straightforwardly because for a rotationally
symmetric cross section, the conservation of angular momentum
(*L*) renders Maxwell's wave equations separable [8,9]. Besides
microlasers with different active materials [10,11], there is a
wide range of other applications that can benefit from the
long-lived (metastable) states in such dielectric resonators
[12,13].

In most of these applications, however, rotationally
symmetric microcavites have a shortcoming: they lack a
preferred emission direction, and the coupling between the
resonator and adjacent optical devices (such as fibers) poses
major technical challenges, requiring accurate placement of
coupling components in the exponentially decaying nearfield of
the resonator [7,12]. While it seems clear that rotational
symmetry should be broken to achieve directionality, it is far
from obvious how to chose the proper deformation. This is
because the wave equation is not separable when no good quantum
numbers such as angular momentum are preserved, and a typical
mode then contains admixtures of many different *L*.

One way to explore the modes of asymmetric dielectric
resonators are exact solutions of the wave equation. We have
carried out such calculations for dielectric microcylinders (or
disks) [14] at deformations well beyond the
reach of a shape perturbation theory [15,16] that had been
previously applied to microdroplets. However, in order to
decide for which shapes it is worthwhile to perform numerical
or real experiments in the first place, and what phenomena to
expect, we resort to the explanatory and predictive power of
the *ray picture*.

A nonseparable wave equation means that rays move along
*chaotic* trajectories. One therefore has to address the
implications of chaotic ray dynamics for the wave solutions: a
typical question in the field of quantum chaos [17]. This defines our agenda: **Analyze the
ray dynamics of asymmetric resonant cavities in order to draw
conclusions about the intrinsic emission properties of the
individual metastable states under consideration, i. e.,
their lifetimes and emission directionality. This requires
semiclassical methods to connect resonances with rays.**
Semiclassical quantization of course yields the resonance
frequencies, too - this has historically been its main
objective [17,18,19,20].

Semiclassical theory for the *lifetimes* of
metastable states has received strong impulses from chemical
physics, in particular from the study of reactive collisions
[20,21,22,23]. Completely regular
(i.e., non-chaotic) systems can be semiclassically quantized
using some version of the WKB or EBK approximation [18,19]; for fully
chaotic systems, spectrum and lifetimes can be obtained from
the periodic-orbit theory pioneered by Gutzwiller [17,23]. The deformed
dielectric resonator, however, generically belongs to the far
larger class of systems for which chaos and regularity
*coexist* because the break-up of regular structure as a
function of deformation is gradual. Semiclassical theory for
such ``mixed systems'' is as yet incomplete.

In a microdisk that has been deformed in a smooth and everywhere convex manner, the WG orbits closest to the edge remain intact because they skip along the interface in short, almost grazing line segments, across which the surface curvature changes only infinitesimally [24]. Chaos first appears in the neighborhood of certain short periodic orbits which become unstable under slight variations of their initial conditions. However, other periodic orbits are stabilized by the perturbation, and the resulting mixture of chaotic and regular motion is difficult to disentangle when conventional ray traces are plotted in real-space.

We therefore image the internal ray dynamics in *phase
space*, using a Poincaré surface of section (SOS)
[25], cf. Fig. 2. The SOS
stroboscopically reveals the *combinations* of positions
(parametrized by polar angle ) and angles of
incidence at which different trajectories
impinge on the boundary. Regular orbits typically form
one-dimensional lines that are grouped into islands of
stability, while chaotic rays fill out two-dimensional clouds -
a consequence of the missing constraint of angular-momentum
conservation.

In the WG region corresponding to , chaotic motion is characterized by an approximate separation of time scales: although a chaotic ray launched in the WG region will eventually explore all accessible areas of the SOS, it can describe almost one-dimensional lines for intermediate times because fluctuates only weakly over many rotations in . This slow diffusion in is directed toward lower values, but the intermediate almost-regular motion can be characterized by an adiabatic invariant for which we then perform a semiclassical quantization in the spirit of the eikonal (EBK) approach [26].

Our eikonal theory not only provides the frequency shifts of
chaotic WG resonances [26], but also
relates each quantized WG mode to a particular adiabatic
invariant curve (AIC) in the SOS. While diffusion in
away from that curve can be
neglected in the constructive-interference argument leading to
the mode quantization [21], diffusion is
crucial for determining lifetime and emission directionality.
The reason is that determines the
reflectivity *R* of the interface: At refractive index
*n*, Fresnel's formula yields a jump (rounded by
tunneling) from *R*=1 to a lower value near when
drops below , which
delimits the total-internal-reflection condition.

If a *threshold deformation* [27] is exceeded such that a chaotic domain in
the SOS connects the quantized AIC with the critical value
, the average lifetime of rays
launched from this AIC is dominated *not* by the small
tunnel leakage known from the circle, but instead by classical
phase-space diffusion that allows to
reach values where refractive escape (following Snell's law)
can occur. This classically allowed escape mechanism is
wavelength-independent, and one thus expects a
*universal* resonace lifetime for all chaotic WG modes
supported by the same AIC. Comparison with exact wave solutions
(Fig. 3) confirms this prediction, showing moreover a
quantitative agreement between resonance lifetimes and
classical diffusion times at *large deformations* -
which is precisely the regime in which *wave*
calculations are especially difficult!

The classical picture proves even more useful when the
*emission directionality* of a metastable state is
required - a property that is directly observable in
microlasers, whereas it is typically averaged out in nuclear or
chemical decay processes and hence has not been studied in
detail prior to our work. The anisotropic structure of the SOS
in the neighborhood of causes
rays to escape preferentially at certain with only slightly below
[28]. As
a dramatic consequence, we predict that the emission profile of
an oval cylinder resonator can depend strongly on its
refractive index, even if all other parameters (i.e.
deformation and wavelength) are kept fixed (Fig. 4). This
is because a change in *n* may place in a part of the SOS where island structure
strongly modifies the chaotic diffusion of WG rays [29]. Applying an analogous phase-space analysis
to nonspherical lasing microdroplets, we explained their
experimentally observed anisotropic emission [30,31].

Finally, comparing with numerical wave solutions, we can
extract *wave corrections* to the ray theory, thereby
identifying the physics that is not contained in an essentially
classical model for the metastable states. At deformations too
small to permit chaotic diffusion from the AIC, decay rates are
found to be enhanced over the classical prediction by
*chaos-assited tunneling*. At large deformations, on the
other hand, chaotic diffusion toward the escape condition can
be suppressed by wave interference, leading to *dynamical
localization* [26]. These are
phenomena of great current interest in quantum chaos.
Micro-optics and quantum chaos, two fields with rather
different objectives, have thus found a promising connection
from which optical device design can benefit and new
theoretical questions emerge.

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4 |
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9 |
In the circular disk resonator, the out-of plane component of the wave equation can be approximately separated from the in-plane propagation for two limiting cases: if the disk's thickness permits exactly one vertical mode, or if its refractive index is large enough to render it effectively closed. In microdisk lasers, the former condition is usually satisfied. |

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13 |
R. K. Chang, J. U. Nöckel and A. D. Stone, U. S. Patent 5,742,633 (registered April 21, 1998) |

14 |
J. U. Nöckel and A. D. Stone, in Optical
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Campillo, eds. (World Scientific, Singapore, 1996) |

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25 |
The Poincaré surface of section is a central tool in nonlinear dynamics. In our case, it is obtained numerically by following a number of rays for several hundred reflections and recording not only the successive positions at which the surface is encountered, but also the angles of incidence with respect to the surface normal. The position along the rim can be parametrized by the polar angle , and each trajectory then generates a set of points in a graph of versus . |

26 |
J. U. Nöckel and A. D. Stone, Nature 385,
45 (1997) |

27 |
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Lett. 19, 1693 (1994). |

28 |
Often, the escape in fact occurs from the highest-curvature points as is intuitively expected. This is another consequence of the slow diffusion in , and it applies to resonators whose refractive index is sufficiently small to place the total-internal-reflection condition in the WG region of the SOS. However, stable islands always have to be avoided by chaotically diffusing rays, and this causes deviations from intuitive expectations. |

29 |
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and R. K. Chang, Opt. Lett. 21, 1609 (1996) |

30 |
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R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995) |

31 |
The microdroplet is one of the pioneering microcavity systems, but the emission anisotropy of such droplets in prolate and oblate phases of their natural shape oscillation had remained unexplained for approximately ten years prior to our work. |

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