Theory of the four-photon parametric oscillator

The quantum statistics of parametric oscillation in a medium with cubic polarization is investigated in parametric approximation. Starting from an effective hamiltonian containing Bose operators of the signal, idler and pump mode and their coupling to heat baths, we obtain a c-number Fokker-Planck equation and the stochastically equivalent Langevin equations. These equations are solved, below threshold, well above threshold and near threshold. The linewidth of the signal and idler wave and their autocorrelation and crosscorrelation functions are calculated, characteristic differences between the four-photon and the three-photon parametric oscillators are discussed. The solution of the Fokker-Planck equation near threshold for a multimode pump laser is also obtained.