  Resonant electromagnetic modes are analyzed inside a dielectric cavity of
equilateral triangular cross section and refractive index n, surrounded by a
uniform medium of refractive index n'. The field confinement is determined only
under the requirements needed to maintain total internal reflection of the
internal electromagnetic fields, matched to exponentially decaying evanescent
waves outside the cavity. Two-dimensional electromagnetics is considered, with
no dependence on the coordinate perpendicular to the cross section; hence,
independent transverse electric (TE) and transverse magnetic (TM) polarizations
are described separately. A linear combination of six plane waves is sufficient
within the cavity, whose wavevectors are related by 120-degree rotations and
whose phases are related by Fresnel reflection coefficients. Generally, the
mode spectrum becomes sparse and the minimum mode frequency increases rapidly
as the index ratio N=n/n' approaches 2. For specified quantum numbers and N,
the TM modes are lower in frequency than the TE modes. Assuming the evanescent
boundary waves escape at the triangle vertices, TE modes generally are found to
have greater confinement of the fields inside the cavity and much higher
quality factors than TM modes.
