General numerical methods for simulating second-order nonlinear interactions in birefringent media

Two computational methods are common for simulating the evolution of three beams propagating in a birefringent medium and interacting through a second-order nonlinearity: the split-step method and solution of the coupled equations for the amplitudes of the spatial frequency components of the beams (Fourier-space method). I (i) compare the accuracy and computational cost of both methods, (ii) investigate the effect of using a first-order expansion for the refractive index as a function of propagation direction, and (iii) generalize both methods to handle arbitrary propagation directions in biaxial crystals. It turns out that the Fourier-space method with a Runge–Kutta solver gives best accuracy, but a symmetrized split-step method may be faster when low accuracy is sufficient. The first-order expansion for the refractive index gives a very small error for well-collimated beams, but the approximation is not important for computational efficiency. Modeling of parametric amplification outside the principal planes of a biaxial crystal is demonstrated, and to the author's knowledge this process has not been modeled in such detail before.