Triangular cavity modes (G.M. Wysin at KSU)

Triangular cavity modes 1 -- 24

Triangular cavity of edge length = a.
Filled with a refractive material of index = n.
Surrounded by a refractive material of smaller index = n'.
Boundary condition of zero field at cavity edge.

These are the real-valued wavefunctions and complex-valued Fourier Transforms of the lowest 24 modes, obtained from the known analytic solutions. They are displayed on a grid for convenience; the wavefunction magnitude at each position is proportional to the symbol size there. Red is positive, blue is negative wavefunction value.

Modes are labelled by indexes m,n.
When m is zero, the mode is single or nondegenerate.
When m is non-zero, the mode is doubly degenerate (has the same frequency as another independent mode of vibration).

The dimensionless frequency numbers on the plots are actually omega*a*n/c, where n is the index of refraction inside the cavity, and c is the speed of light in vacuum.

For the Fourier Transforms, the arrows represent the magnitude and phase of the Fourier amplitude at a chosen wavevector. The spacing of the grid points in Fourier space is 4*pi/3a. The entire Brillouin zone is hexagonally shaped, as shown in the FT of the first mode. The wavevector (kx,ky)=(0,0) is the point at the center of the diagram. Only the important inner part of the BZ is shown for the other modes.

A mode can be confined by total internal reflection as long as there is no FT amplitude propagating normal to any boundary. For example, using the lower boundary, a mode cannot be confined if there is FT amplitude along the negative vertical axis of the FT diagram. Applying this type of analysis, only the modes with m=0 can never be confined by TIR. Conversely, all modes with nonzero m can be confined, at large enough refractive index ratio (n/n').

Click image for a postscript file of that figure.

Mode 1 (ground state, nondegenerate) and whole FT Brillouin zone: