Degenerate Optical Parametric Oscillator with a Saturable Absorber

We consider a degenerate optical parametric oscillator (DOPO). A coherent signal is injected with real amplitude ϵ and phase 2iω0t. The cavity has two resonant modes at frequencies ν1 ≃ ω0 and ν2 ≃ 2ω0. A nonlinear crystal converts photons of frequency ν2 into pairs of photons of frequency ν1 = v2/2. A saturable absorber is inserted in the cavity and can absorb photons at the frequency ν1. The absober is described by a collection of N homogeneously broadened two-level atoms with atomic frequency ω ≃ ω0.