RESUMEN
Light propagation on a two-dimensional curved surface embedded in a three-dimensional space has attracted increasing attention as an analog model of four-dimensional curved spacetime in the laboratory. Despite recent developments in modern cosmology on the dynamics and evolution of the universe, investigation of nonlinear dynamics of light on non-Euclidean geometry is still scarce, with fundamental questions, such as the effect of curvature on deterministic chaos, challenging to address. Here, we study classical and wave chaotic dynamics on a family of surfaces of revolution by considering its equivalent conformally transformed flat billiard, with nonuniform distribution of the refractive index. We prove rigorously that these two systems share the same dynamics. By exploring the Poincaré surface of section, the Lyapunov exponent, and the statistics of eigenmodes and eigenfrequency spectrum in the transformed inhomogeneous table billiard, we find that the degree of chaos is fully controlled by a single, curvature-related geometric parameter of the curved surface. A simple interpretation of our findings in transformed billiards, the "fictitious force," allows us to extend our prediction to other classes of curved surfaces. This powerful analogy between two a priori unrelated systems not only brings forward an approach to control the degree of chaos, but also provides potentialities for further studies and applications in various fields, such as billiards design, optical fibers, or laser microcavities.
RESUMEN
Investigation of physics on two-dimensional curved surface has significant meaning in study of general relativity, inasmuch as its realizability in experimental analogy and verification of faint gravitational effects in laboratory. Several phenomena about dynamics of particles and electromagnetic waves have been explored on curved surfaces. Here we consider Wolf effect, a phenomenon of spectral shift due to the fluctuating nature of light fields, on an arbitrary surface of revolution (SOR). The general expression of the propagation of partially coherent beams propagating on arbitrary SOR is derived and the corresponding evolution of light spectrum is also obtained. We investigate the extra influence of surface topology on spectral shift by defining two quantities, effective propagation distance and effective transverse distance, and compare them with longitudinal and transverse proper lengths. Spectral shift is accelerated when the defined effective quantities are greater than real proper lengths, and vice versa. We also employ some typical SORs, cylindrical surfaces, conical surfaces, SORs generated by power function and periodic peanut-shell shapes, as examples to provide concrete analyses. This work generalizes the research of Wolf effect to arbitrary SORs, and provides a universal method for analyzing properties of propagation compared with that in flat space for any SOR whose topology is known.
RESUMEN
In this work, a new class of partially coherent Schell-type sources is introduced by modifying its degree of coherence, which is a product of a parabolic function and a Gaussian function. Such sources are confirmed to be physically genuine and may be called parabolic-Gaussian Schell-model (PGSM) sources. The propagating expression of the cross-spectral density function of such PGSM sources is derived in a general linear optical system. In particular, the propagation properties of such PGSM beams in both free space and lens systems are discussed. The results show that such PGSM sources can produce dark-hollow intensity profiles in the regions of free space very near as well as far away from the source plane, or in the focal region of the lens system.