FDTD modeling of conversion and separatrix crossing in second harmonic generation

Abstract

We investigate forward and backward second-harmonic generation by means of FDTD method. We show that numerical dispersion of the method can strongly affect the generation process. Nevertheless FDTD captures complex dynamical phenomena such as separatrix crossing.Introduction. Frequency conversion processes and in particular second-harmonic generation, both in forward (FSHG) and backward (BSHG, requiring a QPM grating) configuration, are usually described in the framework of the coupled-mode theory (CMT) [1]. Beyond the undepleted regime, CMT predicts complex phenomena such as controllable qualitative changes of the dynamical field behavior accompanied by quantitative changes of the conversion. These phenomena are associated with separatrix crossing in a suitable reduced phase-space, and can occur both in FSHG and BSHG, though driven by different mechanisms: competition between quadratic and unavoidable cubic nonlinearities in the former case [2], and due to intrinsic nonlinear feedback in the latter one [3]. However, no experimental evidence for these phenomena has been reported so far, and one naturally wonders whether they are expected to persist when full Maxwell equations are considered, thereby going beyond the intrinsic approximations of CMT (slowly-varying, rotating-wave, other generated frequencies neglected). To this end one can apply nonlinear and dispersive finite-difference time-domain (FDTD) methods [4], which imply no other approximation than discretization, and are suitable to describe nonlinear frequency conversion in complex (non-homogeneous and/or anisotropic) structures [5,6] as well as self-pulsing and hysteresis phenomena characteristic of nonlinear distributed feedback structures [7]. In order to emphasize basic limitations and advantages of the method, and to compare with CMT, we run FDTD simulations of FSHG and BSHG both in homogeneous media and QPM gratings, keeping complexity to the minimum by considering the 1D quasi-scalar case, and mainly continuous wave excitation.