Cavity mode enhancement of terahertz emission from equilateral triangular microstrip antennas of the high-$T_c$ superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8 + δ}$

We study the transverse magnetic (TM) electromagnetic cavity mode wave functions for an ideal equilateral triangular microstrip antenna (MSA) exhibiting $C_{3v}$ point group symmetry. When the $C_{3v}$ operations are imposed upon the antenna, the TM($m,n$) modes with wave vectors $\alpha \sqrt[ ]{m^2 + nm + n^2}$ are much less dense than commonly thought. The $R_3$ operations restrict the integral $n$ and $m$ to satisfy $\mid m - n \mid$ = 3$p$, where $p \geq$ 0 and $p \geq$ 1 for the modes even and odd under reflections about the three mirror planes, respectively. We calculate the forms of representative wave functions and the angular dependence of the output power when these modes are excited by the uniform and non-uniform ac Josephson current sources in thin, ideally equilateral triangular MSAs employing the intrinsic Josephson junctions in the high transition temperature $T_c$ superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8 + \delta}$, and fit the emissions data from an earlier sample for which the $C_{3v}$ symmetry was apparently broken.