Frobenius–Perron eigenstates in deformed microdisk cavities: non-Hermitian physics and asymmetric backscattering in ray dynamics

Abstract

In optical microdisk cavities with boundary deformations the backscattering between clockwise and counter-clockwise propagating waves is in general asymmetric. The striking consequence of this asymmetry is that these apparently weakly open systems show pronounced non-Hermitian phenomena. The optical modes appear in non-orthogonal pairs, where both modes copropagate in a preferred sense of rotation, i.e. the modes exhibit a finite chirality. Full asymmetry in the backscattering results in a non-Hermitian degeneracy (exceptional point) where the deviation from closed system evolution is strongest. We study the effects of asymmetric backscattering in ray dynamics. For this purpose, we construct a finite approximation of the Frobenius-Perron operator for deformed microdisk cavities, which describes the dynamics of intensities in phase space. Eigenstates of the Frobenius-Perron operator show nice analogies to optical modes: they come in non-orthogonal copropagating pairs and have a finite chirality. We introduce a new cavity system with a smooth asymmetric boundary deformation where we demonstrate our results and we illustrate the main aspects with the help of a simple analytically solvable 1D model.

Modeling propagation in large deformed step-index fibers using a finite operator method

“…In this section, we establish the underlying principles and notation that will later be used to propagate light in deformed optical fibers by the finite operator method. For most of the discussion in this section, it suffices to consider ray dynamics on the single boundary of an unstructured fiber (see also [28]), but for the more detailed applications in later sections we will treat step-index fibers (SIFs) and here define our notation accordingly.…”

“…Coarse-grained or discretized representations of PFOs have been extensively investigated in the past (see, for example, [25][26][27]). Of particular relevance to this paper is the simulation using similarly pixelated PFOs of the fully-chaotic SOS of deformed micro-resonant optical cavities in [28,29]. The calculations in this paper are distinct in allowing coupling between multiple domains (core and cladding) and explicitly accounting for the third dimension.…”

Lagrangian descriptors for open maps

“…For long times, this leaves just a repeller, which is a fractal invariant set. Nevertheless, reflection mechanisms at the boundaries are usually more complicated than a complete opening [11], and they may exhibit many interesting mathematical consequences [12]. This leads us to consider in this work a function depending on a reflectivity R, which rules the way in which the classical trajectories arriving at the opening are only partially reflected.…”

Role of short periodic orbits in quantum maps with continuous openings

“…In many experimental situations like in the case of optical cavities [1][2][3], the knowledge of properties of an open system becomes crucial. This is also a very interesting theoretical problem, even from a pure mathematical point of view [4].…”