Trace formula for dielectric cavities: General properties

Abstract

The construction of the trace formula for open dielectric cavities is examined in detail. Using the Krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside the cavity. The contribution of each periodic orbit is the product of the two factors. The first is the same as in the standard trace formula and the second is connected with the product of reflection coefficients for all points of reflection with the cavity boundary. Two asymptotic terms of the smooth resonance counting function related with the area and the perimeter of a convex cavity are derived. The coefficient of the perimeter term differs from the one for closed cavities due to unusual high-energy asymptotics of the S matrix for the scattering on the cavity. Corrections to the leading semi-classical formula are briefly discussed. Obtained formulas agree well with numerical calculations for circular dielectric cavities.

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

“…Therefore, "no mode can disappear to infinity" along the imaginary direction in complex frequency space. This implies that the total number of internal modes of a dielectric cavity fulfills the conventional k 2 law as it was pointed out by Bogomolny et al (2008). This, however, is not in contradiction with the fractal Weyl law which applies to the long-lived modes within the set of internal modes.…”

“…However, the authors emphasized the need of higher-order corrections of the trace formula and showed that the application of curvature corrections to the Fresnel reflection coefficients improves the agreement (Bittner et al, 2012b). Bogomolny et al (2008) have shown that higher-order corrections lead to a small shift of the peak positions in the length spectrum which can be interpreted as an analogue to the Goos-Hänchen shift. Hales et al (2011) extended the trace formula to the case of a dielectric disk with a point scatterer.…”

“…Such a formula relates the density of states (DOS) to the sum of a smooth part given by a series of Weyl terms and an oscillating part given in the leading order by a sum over classical periodic orbits. Bogomolny et al (2008) extended this concept to open dielectric cavities by using the Krein formula. Their result can be written in the form…”

Outer resonances and effective potential analogy in two-dimensional dielectric cavities

“…The outer resonances constitute extremely broad background in the density of states because of their quite high leakage and they should be take into account in the study of the trace formula in open cavity [22,33].…”

Ray model and ray-wave correspondence in coupled optical microdisks

“…Although the results presented are for the TM case, there is generally no lower limit of imaginary values because the mode coupling to the environment is not intrinsically limited. Rather, the coupling between internal and external (or outer or shape [49][50][51][52]) resonances, i.e. modes localized inside and outside the cavity respectively, causes as broad distribution of imaginary parts.…”