Fourier treatment of nonlinear optics

As the spatial angular bandwidth and temporal frequency bandwidth of nonlinear optical interactions is increased it becomes necessary to fully account for the effects of diffraction, anisotropy, and dispersion upon propagation through the volume of the nonlinear optical media. This increased bandwidth is required in order to increase the peak intensities through focusing or the use of ultrashort pulses and thereby increase the strength and efficiency of nonlinear interactions, or in order to increase the information capacity of the optical fields. The increased bandwidth appears to significantly complicate the simple plane wave coupled mode analytic formulation, however in the paper we show that in the Born regime of weak scattering a simple Fourier decomposition mechanism can be employed that fully accounts for the spatial, temporal, and polarization structure of the nonlinearly generated fields. This momentum space approach allows simple visualizations of the process, allows optimizations of the efficiency, resolution, and bandwidth of the nonlinearity, and enables the design and interpretation of novel interaction geometries. The momentum space solution begins with a 3-D spatio-temporal Fourier transformation of the vector field applied to the planar boundary of the nonlinear optical medium