Parametric interaction of an arbitrary incident signal

We have considered the process of frequency conversion when the signal mode has an arbitrary field distribution. The set of coupled partial differential equations is solved employing the angular spectrum representation for the signal mode as well as for the idler mode. Explicit expressions for the signal and the idler field are obtained in the far-field approximation. An expression for the signal mode intensity is obtained which takes into account the effect of partial coherence as well as the effect of the bounded beam aperture. As an illustration we have considered the case of a Gaussian beam in detail.