Mode degeneracies and the petermann excess-noise factor for unstable lasers

The linewidth of an unstable laser exceeds the quantum minimum by the Petermann factor K, which depends on the overlap between left and right eigenvectors of the (non-unitary) round-trip wave operator. When K is plotted as a function of the Fresnel number N, strong resonances occur, which are associated with degeneracies N c lying close to the real axis in the complex N plane. For certain values of the magnification, degeneracies can lie on the real axis, and K is infinite. The Horwitz-Southwell asymptotic theory of the spectrum, presented here in a very accurate form and with a streamlined derivation, is used to show that the peaks occur near N = s - 1/8 (s integer), and to give reliable formulae for the resonance widths Im N c and other features of the degeneracies. Low-lying resonances have discontinuous profiles associated with mode switching where the absolute values of the eigenvalues cross (these are not degeneracies).