1,414 research outputs found
Eigenfunction statistics for a point scatterer on a three-dimensional torus
In this paper we study eigenfunction statistics for a point scatterer (the
Laplacian perturbed by a delta-potential) on a three-dimensional flat torus.
The eigenfunctions of this operator are the eigenfunctions of the Laplacian
which vanish at the scatterer, together with a set of new eigenfunctions
(perturbed eigenfunctions). We first show that for a point scatterer on the
standard torus all of the perturbed eigenfunctions are uniformly distributed in
configuration space. Then we investigate the same problem for a point scatterer
on a flat torus with some irrationality conditions, and show uniform
distribution in configuration space for almost all of the perturbed
eigenfunctions.Comment: Revised according to referee's comments. Accepted for publication in
Annales Henri Poincar
Trace formula for a dielectric microdisk with a point scatterer
Two-dimensional dielectric microcavities are of widespread use in microoptics
applications. Recently, a trace formula has been established for dielectric
cavities which relates their resonance spectrum to the periodic rays inside the
cavity. In the present paper we extend this trace formula to a dielectric disk
with a small scatterer. This system has been introduced for microlaser
applications, because it has long-lived resonances with strongly directional
far field. We show that its resonance spectrum contains signatures not only of
periodic rays, but also of diffractive rays that occur in Keller's geometrical
theory of diffraction. We compare our results with those for a closed cavity
with Dirichlet boundary conditions.Comment: 39 pages, 18 figures, pdflate
Microdisk Resonators with Two Point Scatterers
Optical microdisk resonators exhibit modes with extremely high Q-factors. Their low lasing thresholds make circular microresonators good candidates for the realization of miniature laser sources. They have, however, the serious drawback that their light emission is isotropic, which is inconvenient for many applications. In our previous work, we showed that the presence of a point scatterer inside the disk can lead to highly directional modes in various frequency ranges while preserving the high Q-factors. In the present paper we generalize this idea to two point scatterers. The motivation for this work is that the strength of a point scatterer is difficult to control in experiments, and the presence of a second scatterer leads to a higher dimensional parameter space which permits to compensate this deficiency. Similar to the case of a single scatterer in a circular disk, the problem of finding the resonance modes in the presence of two scatterers is to a large extent analytically tractable.
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