  A recent S matrix-based theory of the quantum-limited linewidth, which is
applicable to general lasers, including spatially non-uniform laser cavities
operating above threshold, is analyzed in various limits. For broadband gain, a
simple interpretation of the Petermann and bad-cavity factors is presented in
terms of geometric relations between the zeros and poles of the S matrix. When
there is substantial dispersion, on the frequency scale of the cavity lifetime,
the theory yields a generalization of the bad-cavity factor, which was
previously derived for spatially uniform one-dimensional lasers. This effect
can lead to sub-Schawlow-Townes linewidths in lasers with very narrow gain
widths. We derive a formula for the linewidth in terms of the lasing mode
functions, which has accuracy comparable to the previous formula involving the
residue of the lasing pole. These results for the quantum-limited linewidth are
valid even in the regime of strong line-pulling and spatial hole-burning, where
the linewidth cannot be factorized into independent Petermann and bad-cavity
factors.
