Theory of Kerr frequency combs in Fabry-Perot resonators

We derive a spatiotemporal equation describing nonlinear optical dynamics in Fabry-Perot (FP) cavities containing a Kerr medium. This equation is an extension of the equation that describes dynamics in Kerr-nonlinear ring resonators, referred to as the Lugiato-Lefever equation (LLE) due to its formulation by Lugiato and Lefever in 1987. We use the equation to study the generation and properties of Kerr frequency combs in FP resonators. The derivation of the equation starts from the set of Maxwell-Bloch equations that govern the dynamics of the forward and backward propagating envelopes of the electric field coupled to the atomic polarization and population difference variables in a FP cavity. The final equation is formulated in terms of an auxiliary field psi(z,t) that evolves over a slow time t on the domain -L <= z <= L with periodic boundary conditions, where L is the cavity length. We describe how the forward and backward propagating field envelopes can be obtained after solving the equation for psi. This formulation makes the comparison between the FP and ring geometries straightforward. The FP equation includes an additional nonlinear integral term relative to the LLE for the ring cavity, with the effect that the value of the detuning parameter a of the ring LLE is increased by an amount equal to twice the spatial average of vertical bar psi vertical bar(2). This feature establishes a general connection between the stationary phenomena in the two geometries. For the FP-LLE, we discuss the linear stability analysis of the flat stationary solutions, analytic approximations of solitons, Turing patterns, and nonstationary patterns. We note that Turing patterns with different numbers of rolls may exist for the same values of the system parameters. We then discuss some implications of the nonlinear integral term in the FP-LLE for the kind of experiments that have been conducted in Kerr-nonlinear ring resonators.