  The quantum-limited linewidth of a laser cavity is enhanced above the
Schawlow-Townes value by the Petermann factor K, due to the non-orthogonality
of the cavity modes. The average Petermann factor $<K>$ in an ensemble of
cavities with chaotic scattering and broken time-reversal symmetry is
calculated non-perturbatively using random-matrix theory and the supersymmetry
technique, as a function of the decay rate $\Gamma$ of the lasing mode and the
number of scattering channels N. We find for $N\gg 1$ that for typical values
of $\Gamma$ the average Petermann factor $<K>\propto \sqrt{N}\gg 1$ is
parametrically larger than unity.
