Three Wave Mixing for Multimode Beams

Abstract

Here we consider the dynamics of three wave mixing (3WM) processes involving multi-temporal mode time-dependent fields appropriate to the description of 3WM of typical laser beams. We set up and solve the dynamics and examine its consequences on the spectral character of the interacting fields. While previous studies have dealt with the spectral consequences of 3WM in the weak field limit, here we show that analytic expressions are easy to derive for the 3WM of fields of arbitrary strength. Specifically, we show that the spectrum of the output fields generated using sum frequency generation (SFG), difference frequency generation (DFG), and second harmonic generation (SHG) changes in a nontrivial manner under conditions of strong coupling. In previous studies, authors have claimed that the bandwidth of SHG equals $\sqrt 2$ times the bandwidth of the fundamental, and the bandwdith of SFG is the square root of the sum of the squares of the bandwdith of the input beams. We shall show that these estimates are not correct. The bandwidth of these processes depend upon the degree of conversion. The output spectrum for SFG and DFG can be dense and wide, even for input beams containing only two modes. For large nonlinear coupling the output spectrum broadens, becoming quasicontinuous and exhibiting a bandwidth that approaches the phasematch bandwidth of the nonlinear medium. In contrast, the SHG spectrum remains discrete for an input beam containing only two modes but its spectral bandwidth can still broaden to the phasematch limit.