Modes in square resonators

Modes in square resonators are analyzed and classified according to the irreducible representations of the point group C_4v. If the mode numbers p and q that denote the number of wave nodes in the directions of two orthogonal square sides are unequal and have the same even-odd characteristics, the corresponding double modes are accidentally degenerate and can be combined into two new distributions with definite parities relative to the square diagonal mirror planes. The distributions with odd parities belong to the whispering-gallery-like modes in square resonators. The mode frequencies and quality factors are also calculated by the finite-difference time-domain technique and Pade´ approximation method. The numerically calculated mode frequencies agree with the theoretical ones very well and the whispering-gallery-like modes have quality factors much higher than other modes, including their accidentally degenerate counterparts in square resonators.