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Electromagnetic modes in dielectric equilateral triangle resonators

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.