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)]. Fits to the data of Kadowaki {\it et al.} [J. Phys. Soc. Jpn. {\bf 79}, 02373 (2010)] using both a uniform $ac$ Josephson current source and a non-uniform cavity mode (or magnetic surface current) source suggest that both scenarios are equally probable. However, when $n\ell/2w$ is integral, where $n$ is the index of the rectangular TM$^z(n,0)$ mode, standing cavity wave modes 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.