Lasing threshold and mode competition in chaotic cavities

The lasing threshold of a multimode chaotic cavity (linear size $D$ $\gg$ wavelength $\lambda )$ coupled to the outside through a small hole (linear size $d\ll \lambda )$ is studied. For sufficiently weak absorption by the boundaries, the statistical distribution of the threshold is wide, its mean value being much less than the pumping rate needed to compensate the average loss. The average number $〈N_{\mathrm{nc}}〉\gg 1$ of noncompeting excited modes is proportional to the square root of the pumping rate. We use the classical model of spatial hole burning to account for mode competition and find a reduction in the average number of excited modes to $〈N〉{=3\mathrm{}}^{1/3}〈N_{\mathrm{nc}}{〉}^{2/3}$.