S Y N O P S I S    O F
T H E S I S    W O R K

This manuscript describes the research for which I won the first prize in the 1999 Outstanding Thesis Competition of American Physical Society, Division of Atomic, Molecular and Optical Physics.
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 109 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 $\phi$) and angles of incidence $\chi$ 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 $\sin\chi\approx 1$, 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 $\sin\chi$ fluctuates only weakly over many rotations in $\phi$. This slow diffusion in $\sin\chi$ 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 $\sin\chi$ 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 $\sin\chi$ 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 $R\approx(1-n)^2/(1+n)^2$ when $\sin\chi$ drops below $\sin\chi_c\equiv 1/n$, 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 $\sin\chi_c$, 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 $\sin\chi$ 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 $\sin\chi_c$ causes rays to escape preferentially at certain $\phi$ with $\sin\chi$ only slightly below $\sin\chi_c$ [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 $\sin\chi_c$ 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].
Fig. 4

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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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 $\chi$ with respect to the surface normal. The position along the rim can be parametrized by the polar angle $\phi$, and each trajectory then generates a set of points in a graph of $\sin\chi$ versus $\phi$.
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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 $\sin\chi$, and it applies to resonators whose refractive index is sufficiently small to place the total-internal-reflection condition $\sin\chi=1/n$ 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 J. U. Nöckel, A. D. Stone, G. Chen, H. Grossman and R. K. Chang, Opt. Lett. 21, 1609 (1996)
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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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