Cavity mode waves during terahertz radiation from rectangular Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ mesas

We re-examined the angular dependence of the radiation from the intrinsic Josephson junctions in rectangular mesas of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, in order to determine if the cavity mode part of the radiation arises from waves across the width $w$ or along the length $\ell$ of the mesas, associated with ``hot spots'' [Wang {\it et al.}, Phys. Rev. Lett. {\bf 105}, 057002 (2010)]. We derived analytic forms for the angular dependence expected in both cases for a general cavity mode in which the width of the mesa corresponds to an integer multiple of one-half the wavelength of the radiation. Assuming the coherent radiation from the $ac$ Josephson current source and the cavity magnetic surface current density source combine incoherently, fits to the data of Kadowaki {\it et al.} [J. Phys. Soc. Jpn. {\bf 79}, 023703 (2010)] on a mesa with mean $\ell/w=5.17$ for both wave directions using two models for the incoherent combination were made, which correspond to standing and traveling waves, respectively. The results suggest that the combined output from the uniform $ac$ Josephson current source plus a cavity wave forming along the rectangle length is equally probable as that of the combined output from the uniform $ac$ Josephson current plus a cavity wave across the width. However, for mesas in which $n\ell/2w$ is integral, where $n$ is the index of the rectangular TM$^z_{n,0}$ mode, it is shown that standing cavity mode waves along the length of the mesa do not radiate in the $xz$ plane perpendicular to the length of the mesa, suggesting experiments on such mesas could help to resolve the question.