  Frequency combs are light sources with coherent evenly-spaced lines. It has
been observed that in certain laser systems, combs can form whose output is
frequency-modulated (FM) in time. In this state, they produce an output whose
frequency sweeps linearly and periodically. These results have been replicated
numerically, but a thorough understanding of their core physics remains
elusive. Surprisingly, we have found that these lasers are described by a
nonlinear Schrodinger equation whose potential is proportional to the phase of
the electric field. This equation can be solved exactly and produces a field
whose phase is piecewise quadratic in time---an FM comb. These results can be
used to derive all of the salient features of FM combs, and our general theory
is applicable to any nonlinear optical system with large internal gain. More
generally, this result portends the development of new coherent states of light
governed by phase potentials rather than amplitude potentials.
