Time Harmonic Two-Dimensional Cavity Scar Statistics: Convex Mirrors and Bowtie

Abstract

This article examines the localization of time harmonic high-frequency modal fields in two-dimensional cavities along periodic paths between opposing sides of the cavity. The cases where these orbits lead to unstable localized modes are known as scars. This article examines the enhancements for these unstable orbits when the opposing mirrors are convex, constructing the high-frequency field in the scar region using elliptic cylinder coordinates in combination with a random reflection phase from the outer chaotic region. The enhancements when the cavity is symmetric as well as asymmetric about the orbit are examined.

Three-Dimensional Electromagnetic High Frequency Axisymmetric Cavity Scars.

“…To understand this enhancement in value due to the axisymmetric nature of the cavity scar field we next attempt to construct the analog of the axisymmetric random plane wave field. We should also note that this enhancement in value represents the enhancement due to the axisymmetry of the cavity field, analogous to the factor of two due to the even symmetry enhancement perpendicular to the orbit, observed previously in 2D [2], [6]. This factor of two enhancement persisted when the symmetry of the outer wall of the cavity was broken but the local even symmetry was maintained on the mirrors at the ends of the high frequency ray trajectory, as observed in the case of the asymmetric bowtie cavity [2], [6].…”

“…We should also note that this enhancement in value represents the enhancement due to the axisymmetry of the cavity field, analogous to the factor of two due to the even symmetry enhancement perpendicular to the orbit, observed previously in 2D [2], [6]. This factor of two enhancement persisted when the symmetry of the outer wall of the cavity was broken but the local even symmetry was maintained on the mirrors at the ends of the high frequency ray trajectory, as observed in the case of the asymmetric bowtie cavity [2], [6]. We might conjecture here that the key in the axisymmetric case is the local symmetry of the mirrors at the ends of the orbital ray trajectory.…”

“…(For purposes of simplification this figure does not include the vertical evenness of the cavity, which confines the wave leaving and the wave reflected to either the upper half or lower half of the cavity.) Thus   =   (     ) and   =   (     ) are elliptic cylinder functions like in the two-dimensional case [6], [1].…”

“…Projections of the scarred eigenfunction along the orbit will now be discussed. In two dimensions [2], [6] we considered a trigonometric (or Fourier) projection as well as a scar (or Galerkin) projection of the eigenfunction. The first was useful to exhibit the Fourier decomposition of the eigenfunction and the second was useful because it restores certain approximate orthogonality properties among the scarred eigenfunctions.…”

“…The approach used by Antonsen [1], on convex mirror geometries in two dimensions, is generalized [2], [3], [4] by introducing the curved ray path formalism, used previously by Vaynshteyn [5] on stable orbits. This combined approach was used recently to investigate scars in two-dimensional geometry [6], [7]. This report explores both the scalar (acoustic) [8], [9] and vector (electromagnetic) problems in three-dimensional axisymmetric geometry, first with convex walls, and second with concave walls supporting interior foci.…”