Perturbation approximation for higher modes in nearly regular two-dimensional cavities

Abstract

A perturbation theory for weakly distorted regular cavity which has classical ray trajectories lying on invariant tori, is constructed to a higher perturbation order, than for the general case. This is possible because of a special structure of semi-classical eigenvalues for integrable Hamiltonians. The perturbation magnitude here has an order of a characteristic wavelength of a mode instead of usual wavelength square. The results are expressed in solutions of the Hill equation. The set includes modes localized along stable periodic ray trajectories; scar modes, corresponding to unstable periodic trajectories; weakly distorted modes of regular cavity, and intermediate cases. The application of the method to square, circular and elliptical cavities is outlined.

Keywords:

• cavity modes
• quantum billiards
• semi-classical approximation
• perturbation theory

Public Interest Statement

If a resonator does not have one of a small number of shapes, for which analytical solutions are known, its higher modes are quite complicated. By perturbation analysis, it is possible to obtain modes for a slightly deformed regular resonator, but the usual technique is efficient only for deformations of an order of the square of mode wavelength divided by a resonator length. Using the relation between wave and ray description, we identify special subsets of regular modes, for which the perturbation problem can be solved in special functions. In comparison with a general perturbation scheme, the approach of the paper permits to take into account larger distortions from regular shape (of an order of wavelength). The results can be useful for better understanding of relation between classical and quantum chaos, and for practical calculations in laser microcavities.

Additional information

Funding

The author received no direct funding for this research.

Notes on contributors

N. Korneev

N. Korneev obtained Ph.D. from A.F.Ioffe Physical-Technical Institute, St. Petersburg, Russia in 1994, and works at the National Institute of Astrophysics, Optics and Electronics (INAOE), Puebla, Mexico. He has co-authored around 70 papers, both theoretical and experimental. His interests include physics and applications of photorefractive crystals, nonlinear fiber optics, and chaos theory.