Ray chaos, i.e., exponential sensitivity of ray trajectories to launching conditions, could intuitively be expected in very complex and cluttered propagation scenarios (e.g., urban areas). Remarkably, it can also be observed in relatively simple (but coordinate-non-separable) structures that give rise to multiple reflections, focusing, and defocusing, such as homogeneously filled “billiard-shaped” enclosures.
In [1], by revisiting a body of results from quantum physics (classical vs. quantum chaos), we emphasized possible implications of internal ray chaos for high-frequency wave dynamics (“ray-chaotic footprints”). In the time-harmonic high-frequency limit, ray-chaotically-inclined systems tend to exhibit behaviors that differ fundamentally from those observed in ray-regular geometries. For instance, in connection with the eigenvalues, in regular billiards the normalized neighboring spacing statistics follow a Poisson probability density function (PDF), whereas in completely ray-chaotic billiards, they turn out to be accurately modeled by a Wigner-Rayleigh PDF. In connection with the eigenfunctions, in regular billiards, the wavefield at each point results from the superposition of plane waves (rays) from a finite number of possible directions, whereas in completely ray-chaotic billiards it is expected to be a superposition of a multitude of plane waves, with fixed wavevector amplitude, and uniform direction and phase distribution.
In [2], we carried out a full-wave investigation of short-pulse electromagnetic wavepacket propagation in ray-chaotic enclosures, via a numerical approach based on the finite difference time domain (FDTD) method.
The figure top panel shows the quarter-Sinai-stadium configuration considered, with a typical space-filling ray trajectory (after 250 reflections), and a map illustrating the uniform filling of the phase-space (reflection position vs. direction).
The bottom panels show instead the temporal evolution of a short-pulsed wavepacket launched in the cavity. While bouncing around the walls, along the ray path skeleton, the wavepacket undergoes focusing at the concave curved wall and natural spreading elsewhere (including straight-wall reflection), progressively losing its initial space-time localization. At a sufficiently long time, the wavepacket has already uniformly covered the entire enclosure, and the initially localized energy is spread across the mode spectrum.
Results from statistical analysis of the late-time spatial field distributions are consistent with the assumption of a random pulsed plane-wave model.
These results may find important applications to wideband electromagnetic interference and/or compatibility testbeds (whereby an equipment under test may be subjected to a “pulse shower” illumination with characteristics largely independent of its location, orientation, shape and constitutive properties), as well as the synthetic emulation of multi-path wireless channels.
Ray chaos, characterized by eventual exponential divergence of originally nearby multi-bounce ray trajectories, is an intriguing phenomenon. It can be observed in several electromagnetic wave propagation scenarios: both very complex (e.g., urban areas) and very simple (e.g., a stadium-shaped cavity) scenarios. This paper contains a compact review of known results on wave propagation in ray-chaotic scenarios. Attention is focused principally on two-dimensional simple paradigms of internal ray chaos (“ray-chaotic billiards”), with emphasis on possible implications for high-frequency wave dynamics (“ray-chaotic footprints”). General concepts, tools, and numerical examples are discussed, and their potential relevance to current challenges in electromagnetic engineering is noted.
@article{IJ30_IEEE_APMAG_47_62_2005, author = {Galdi, V. and Pinto, I. M. and Felsen, L. B.}, journal = {IEEE Antennas and Propagation Magazine}, title = {Wave propagation in ray-chaotic enclosures: paradigms, oddities and examples}, year = {2005}, volume = {47}, number = {1}, pages = {62--81}, keywords = {chaos;electromagnetic wave propagation;electromagnetic engineering;electromagnetic wave propagation;exponential divergence;high-frequency wave dynamics;internal ray chaos;multi-bounce ray trajectory;ray-chaotic enclosure;wave propagation;Aerodynamics;Aerospace engineering;Chaos;Electromagnetic propagation;Fluid dynamics;Geometrical optics;Laser cavity resonators;Mechanical engineering;Optical propagation;Urban areas}, doi = {10.1109/MAP.2005.1436220}, issn = {1045-9243}, month = feb }
Wave propagation in ray-chaotic scenarios, characterized by exponential sensitivity to ray-launching conditions, is a topic of significant interest, with deep phenomenological implications and important applications, ranging from optical components and devices to time-reversal focusing/sensing schemes. Against a background of available results that are largely focused on the time-harmonic regime, we deal here with short-pulsed wavepacket propagation in a ray-chaotic enclosure. For this regime, we propose a rigorous analytical framework based on a short-pulsed random-plane-wave statistical representation, and check its predictions against the results from finite-difference-time-domain numerical simulations.
@article{IJ100_IEEE_TAP_60_3827_2012, author = {Castaldi, G. and Galdi, V. and Pinto, I. M.}, journal = {IEEE Transactions on Antennas and Propagation}, title = {Short-pulsed wavepacket propagation in ray-chaotic enclosures}, year = {2012}, volume = {60}, number = {8}, pages = {3827--3837}, keywords = {chaos;electromagnetic wave propagation;statistical analysis;ray launching condition;ray-chaotic enclosures;short pulsed random plane wave statistical representation;short pulsed wavepacket propagation;time reversal focusing;time reversal sensing;time-harmonic regime;Apertures;Finite difference methods;Geometry;Numerical models;Time domain analysis;Trajectory;Vectors;Plane waves;random fields;ray chaos;short pulses}, doi = {10.1109/TAP.2012.2201126}, issn = {0018-926X}, month = aug }