Other intriguing examples of ray chaos can be found in *open* configurations such as (n>3)-disk “pinballs”,
or appropriately configured inhomogeneous refractive media with ray-trapping properties.

In [1], in collaboration with the late Prof. Leo Felsen (Boston University), we studied a two-dimensional test example of such an environment which embodies intimately coupled refractive wave-trapping and periodicity-induced multiple scattering phenomenologies, and which is amenable to explicit full-wave analysis. Though strictly nonchaotic, we demonstrated that under appropriate conditions the high-frequency wave dynamics exhibits trends toward irregularity and other peculiar characteristics; these features can be interpreted as “ray-chaotic footprints”, and they are usually not observed in geometries characterized by “regular” ray behavior.

In [2], we extended this study to a cylindrical scatterer made of a perfectly electric-conducting azimuthally corrugated boundary coated by a radially inhomogeneous (ray-trapping) dielectric layer, as shown in the figure (top panel). We showed that, for such configuration, ray dynamics is generally of “mixed type,” with the presence of both regular and chaotic regions in the phase space. It is possible to “tune” the parameter configuration (e.g., acting on the refractive index gradient) so as to render the dynamics strongly chaotic.

By relying on a full-wave solution (via a problem-matched semi-analytic technique), we studied
the monostatic (i.e., observed at backscattering direction) radar-cross-section (RCS) response,
for ﬁxed incidence angle and various parametric conﬁgurations ranging from regular to chaotic ray dynamics,
for various representative frequency and incidence conditions.
As illustrated in the figure bottom panels, starting from the fairly regular response of a plain circular cylinder,
it is observed how introducing either corrugations or graded-index-coating alone
(i.e., preserving the ray-regular behavior) produces only smooth changes or the appearance of isolated peaks.
Conversely, introducing both corrugation and graded-index-coating (which are both required for the onset of ray-chaos)
produces increasingly complex RCS responses. In particular, the conﬁguration corresponding to ray-chaotic dynamics

exhibits a monostatic RCS response with very rapid and irregular frequency variations, and
wide (up to three orders of magnitude) dynamic ranges.

Our results clearly indicate that electrically large ray-chaotically inclined conﬁgurations exhibit fairly complex monostatic and bistatic RCS responses, characterized by wide dynamic ranges and strong sensitivity on the frequency and incidence direction, which are fairly different from those observed in ray-regular conﬁgurations. Particularly intriguing from the application viewpoint is the possible exploitation of the inherent richness and complexity of the RCS signatures in radar-counter-measure scenarios.

## Relevant papers

- Castaldi, G., Fiumara, V., Galdi, V., Pierro, V., Pinto, I. M., & Felsen, L. B. (2005). Ray-chaotic footprints in deterministic wave dynamics: a test model with coupled Floquet-type and ducted-type mode characteristics.
*IEEE Transactions on Antennas and Propagation**53*(2), 753–765.Ray chaos, manifested by the exponential divergence of trajectories in an originally thin ray bundle, can occur even in linear electromagnetic propagation environments, due to the inherent nonlinearity of ray-tracing maps. In this paper, we present a novel (two-dimensional) test example of such an environment which embodies intimately coupled refractive wave-trapping and periodicity-induced multiple scattering phenomenologies, and which is amenable to explicit full-wave analysis. Though strictly nonchaotic, it is demonstrated that under appropriate conditions which are inferred from a comprehensive parametric database generated via the above-noted rigorous reference solution, the high-frequency wave dynamics exhibits trends toward irregularity and other peculiar characteristics; these features can be interpreted as “ray-chaotic footprints”, and they are usually not observed in geometries characterized by “regular” ray behavior. In this connection, known analogies from other disciplines (particularly quantum physics) are briefly reviewed and related to the proposed test configuration. Moreover, theoretical implications and open issues are discussed, and potential applications are conjectured.

@article{IJ29_IEEE_TAP_53_753_2005, author = {Castaldi, G. and Fiumara, V. and Galdi, V. and Pierro, V. and Pinto, I. M. and Felsen, L. B.}, journal = {IEEE Transactions on Antennas and Propagation}, title = {Ray-chaotic footprints in deterministic wave dynamics: a test model with coupled Floquet-type and ducted-type mode characteristics}, year = {2005}, volume = {53}, number = {2}, pages = {753--765}, keywords = {chaos;electromagnetic wave propagation;electromagnetic wave scattering;nonlinear dynamical systems;ray tracing;coupled Floquet-type characteristics;coupled refractive wave-trapping;deterministic wave dynamics;ducted-type mode characteristics;exponential trajectory divergence;full-wave analysis;linear electromagnetic propagation;multiple scattering phenomenology;parametric database generation;periodicity;ray tracing map;ray-chaotic footprint;Chaos;Character generation;Electromagnetic propagation;Electromagnetic refraction;Electromagnetic scattering;Particle scattering;Ray tracing;Spatial databases;Testing;Trajectory;Ducted-type modes;Floquet theory;ray chaos}, doi = {10.1109/TAP.2004.841296}, issn = {0018-926X}, month = feb }

- Castaldi, G., Galdi, V., & Pinto, I. M. (2008). A study of ray-chaotic cylindrical scatterers.
*IEEE Transactions on Antennas and Propagation**56*(8), 2638–2648.Ray chaos, manifested by the exponential divergence of trajectories originating from an originally thin ray bundle, can occur even in linear electromagnetic propagation environments, due to the inherent nonlinearity of ray-tracing (eikonal) maps. In this paper, extending our previous study of a two-dimensional planar ray-chaotic prototype scenario, we consider a cylindrical scatterer made of a perfectly electric-conducting azimuthally corrugated boundary coated by a radially inhomogeneous (ray-trapping) dielectric layer. For this configuration, we carry out a comprehensive parametric study of the ray-dynamical and full-wave scattering (monostatic and bistatic radar-cross-section) signatures, with emphasis on possible implications for high-frequency wave asymptotics (“ray-chaotic footprints”).

@article{IJ57_IEEE_TAP_56_2638_2008, author = {Castaldi, G. and Galdi, V. and Pinto, I. M.}, journal = {IEEE Transactions on Antennas and Propagation}, title = {A study of ray-chaotic cylindrical scatterers}, year = {2008}, volume = {56}, number = {8}, pages = {2638--2648}, keywords = {electromagnetic wave propagation;electromagnetic wave scattering;ray tracing;dielectric layer;electric-conducting azimuthally corrugated boundary;full-wave scattering;high-frequency wave asymptotics;linear electromagnetic propagation environments;ray-chaotic cylindrical scatterers;ray-tracing maps;two-dimensional planar ray-chaotic prototype scenario;Chaos;Dielectrics;Electromagnetic propagation;Electromagnetic scattering;Hafnium;Particle scattering;Radar cross section;Radar scattering;Ray tracing;Trajectory;High-frequency;radar cross-section;ray chaos}, doi = {10.1109/TAP.2008.927568}, issn = {0018-926X}, month = aug }