Classical Ray Dynamics Research Articles

Impact of Cavity Geometry on Microlaser Dynamics.

We experimentally investigate spatiotemporal lasing dynamics in semiconductor microcavities with various geometries, featuring integrable or chaotic ray dynamics. The classical ray dynamics directly impacts the lasing dynamics, which is primarily determined by the local directionality of long-lived ray trajectories. The directionality of optical propagation dictates the characteristic length scales of intensity variations, which play a pivotal role in nonlinear light-matter interactions. While wavelength-scale intensity variations tend to stabilize lasing dynamics, modulation on much longer scales causes spatial filamentation and irregular pulsation. Our results will pave the way to control the lasing dynamics by engineering the cavity geometry and ray dynamical properties.

Resonance-assisted tunneling in deformed optical microdisks with a mixed phase space.

The lifetimes of optical modes in whispering-gallery cavities depend crucially on the underlying classical ray dynamics, and they may be spoiled by the presence of classical nonlinear resonances due to resonance-assisted tunneling. Here we present an intuitive semiclassical picture that allows for an accurate prediction of decay rates of optical modes in systems with a mixed phase space. We also extend the perturbative description from near-integrable systems to systems with a mixed phase space, and we find equally good agreement. Both approaches are based on the approximation of the actual ray dynamics by an integrable Hamiltonian, which enables us to perform a semiclassical quantization of the system and to introduce a ray-based description of the decay of optical modes. The coupling between them is determined either perturbatively or semiclassically in terms of complex paths.

Diffusive transport in highly corrugated channels

Diffusion in waveguides with spatially modulated profiles is an important topic in modern electromagnetics and optics. Wave dynamics in the high-frequency asymptotics are governed by classical ray dynamics which can be characterised by looking at the diffusion of particles throughout the channel. We study the transport of particles (rays) in a channel with a sinusoidal profile at different values of the corrugation amplitude. We find that below a certain corrugation level the transport is ballistic, beyond this threshold, a diffusion-like behaviour emerges in the asymptotic limit of large times. In this regime particle transport slows down due to the trapping mechanism in the corrugated regions of the channel. We use the analogy with correlated random walks to discuss the observed transport regimes.

Extremely high Q and unidirectional laser emission due to combination of the Kolmogorov-Arnold-Moser barrier and the chaotic sea in a dielectric microdisk.

Emission characteristics of an oval-shaped microcavity laser are studied. In experiments, modes localized on periodic orbits emit unidirectionally with a narrow in-plane divergence angle of about 12deg. The origin of high directionality is elucidated by means of classical ray dynamics. Wave calculations show that the Q-factors of the resonances are higher than 108. We explain this extraordinary high Q-factor in relation with a dynamical barrier region where Kolmogorov-Arnold-Moser curves significantly obstruct leakages of resonances.

Chaos-assisted tunneling in a deformed microcavity laser

We investigate the impact of local dynamics on chaos-assisted tunneling in a highly deformed microcavity whose classical ray dynamics exhibits a small measure of trapezoidal-shaped orbit (TSO) stability islands in a main chaotic sea. These two classically completely decomposed regions in phase space can support resonance modes of their own respectively. Using numerical ray and wave analyses, we show that the emission characteristics of the TSO resonance mode are determined by local ray dynamics near TSO islands. The emission characteristics of the other high-Q resonance modes, on the other hand, are governed by usual ray-wave correspondence. We experimentally demonstrate that the TSO emission mode can be lased without selective excitations by devising a half-moon shaped highly deformed cavity. And we also show that the emission characteristics of the TSO lasing modes are well explained by numerical ray and wave analyses.

Nodal domains of a non-separable problem—the right-angled isosceles triangle

We study the nodal set of eigenfunctions of the Laplace operator on the right-angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number νn of nodal domains for any eigenfunction. In addition, an exact recursive formula for the number of nodal domains is found to reproduce all existing data. Eventually, we use the recursion formula to analyse a large sequence of nodal counts statistically. Our analysis shows that the distribution of nodal counts for this triangular shape has a much richer structure than the known cases of regular separable shapes or completely irregular shapes. Furthermore, we demonstrate that the nodal count sequence contains information about the periodic orbits of the corresponding classical ray dynamics.

Quasieigenstate evolution in open chaotic billiards

We experimentally studied the evolution of quasieigenmodes as classical dynamics undergoing a transition from being regular to chaotic in open quantum billiards. In a deformation-variable microcavity, we traced all high-$Q$ cavity modes in a wide range of frequency as the cavity deformation increased. By employing an internal parameter, we were able to obtain a mode-dynamics diagram at a given deformation, showing avoided crossings between different mode groups, and could directly observe the coupling strengths induced by ray chaos among encountering modes. We also show that the observed mode-dynamics diagrams reflect the underlying classical ray dynamics in the phase space.

Quantum chaos and the double-slit experiment

We report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behaviour.

Quantum chaos, dynamical stability and decoherence

We discuss the stability of quantum motion under system's perturbations in the light of the corresponding classical behavior. In particular we focus our attention on the so called fidelity or Loschmidt echo, its relation with the decay of correlations, and discuss the quantum-classical correspondence. We then report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.

Chaotic waveguide-based resonators for microlasers

We propose the construction of highly directional emission microlasers using two-dimensional high-index semiconductor waveguides as {\it open} resonators. The prototype waveguide is formed by two collinear leads connected to a cavity of certain shape. The proposed lasing mechanism requires that the shape of the cavity yield mixed chaotic ray dynamics so as to have the appropiate (phase space) resonance islands. These islands allow, via Heisenberg's uncertainty principle, the appearance of quasi bound states (QBS) which, in turn, propitiate the lasing mechanism. The energy values of the QBS are found through the solution of the Helmholtz equation. We use classical ray dynamics to predict the direction and intensity of the lasing produced by such open resonators for typical values of the index of refraction.

Wave chaos in the elastic disk.

The relation between the elastic wave equation for plane, isotropic bodies and an underlying classical ray dynamics is investigated. We study, in particular, the eigenfrequencies of an elastic disk with free boundaries and their connection to periodic rays inside the circular domain. Even though the problem is separable, wave mixing between the shear and pressure component of the wave field at the boundary leads to an effective stochastic part in the ray dynamics. This introduces phenomena typically associated with classical chaos as, for example, an exponential increase in the number of periodic orbits. Classically, the problem can be decomposed into an integrable part and a simple binary Markov process. Similarly, the wave equation can, in the high-frequency limit, be mapped onto a quantum graph. Implications of this result for the level statistics are discussed. Furthermore, a periodic trace formula is derived from the scattering matrix based on the inside-outside duality between eigenmodes and scattering solutions and periodic orbits are identified by Fourier transforming the spectral density.