## Gary Wysin, Condensed Matter Physics Condensed Matter Theory and Simulation

309 Cardwell Hall, Department of Physics, Kansas State University, Manhattan KS 66506-2601, USA.
(785) 532-1628 / FAX: (785) 532-6806
wysin@phys.ksu.edu

## Model for optical cavity modes

The graphs shown here represent the oscillation modes of the electromagnetic fields in a two-dimensional dielectric cavity. They could also represent the modes of vibration of an elastic membrane mounted on a special shaped form. That would be a triangular or hexagonal drum!

The sizes of the symbols represent the magnitude of one component of the electric field (usually, E_z). These wavefunctions represent solutions of the two-dimensional Helmholtz equation, which is the wave equation under the assumption of exponential time dependence like

Psi(x,y,t) = psi(x,y) * exp(-i*omega*t)

The wave equation being solved looks like

Laplacian * Psi - (n/c)^2 * Psi = 0

where the Laplacian is the sum of squared partial derivatives w.r.t. x and y, and n is the index of refraction, c is the speed of light in vacuum. It is assumed that n is much larger than the index of refraction n' existing outside the cavity being studied, which leads to the possibility of modes being confined in the cavity by total internal reflection.

These solutions assume the so-called Dirichlet boundary conditions, meaning, psi = 0 on the geometrical boundary being used. This is a reasonable approximation so long as the mode is undergoing total internal reflection, which may or may not be the case! TIR confinement depends on the ratio of index of refraction inside the cavity compared to outside the cavity, (n/n'); larger ratio improves the possibility for TIR for some of the modes.

The modes were tested for TIR confinement by looking at their Fourier transforms. Modes whose Fourier transforms only have intensity above a non-zero incident angle to the faces, can be TIR confined.

Modes in an Equilateral Triangular Cavity
These are the wavefunctions and their Fourier Transforms for the 24 lowest modes of an equilateral triangular cavity. They were found from an exact formula for the analytic solutions.
Modes in an Equilateral Hexagonal Cavity
These are the wavefunctions and Fourier Transforms for the 42 lowest modes of an equilateral hexagonal cavity, plus a few higher interesting states. They were found by a numerical relaxation procedure (Gauss-Seidel iteration) applied to the two-dimensional Helmholtz (wave) equation.