## 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.

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Last update: Monday, 23-Aug-2004 13:46:42 CDT.

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