Spectrum of the parametric down converted radiation calculated in the Wigner function formalism

We continue the study of parametric down conversion within the framework of the Wigner representation, by using a Maxwellian approach developed in a r ecent paper [A. Casado et al., fur. Phys. J. D 11; 465 (2000)]. This gives a mechanism, inside the crystal, for the production of the down-converted r adiation. We obtain the electric field to second order in the coupling cons tant by using the Green's function method, and compare our treatment with t he standard Hamiltonian approach. The spectrum of the down-converted radiat ion is calculated as a function of the parameters of the nonlinear crystal tin particular the length) and the radius of the pumping beam.