Application of the semi-analytical Fourier transform to electromagnetic modeling

The Fast Fourier Transform (FFT) algorithm makes up the backbone of fast physical optics modeling. Its numerical effort, approximately linear on the sample number of the function to be transformed, already constitutes a huge improvement on the original Discrete Fourier Transform. However, even this orders-of-magnitude improvement in the number of operations required can fall short in optics, where the tendency is to work with field components that present strong wavefront phases: this translates, as per the Nyquist-Shannon sampling theorem, into a huge sample number. So much so, in fact, that even with the reduced effort of the FFT, the operation becomes impracticable. Finding a workaround that allows us to evade, at least in part, these stringent sampling requirements is then fundamental for the practical feasibility of the Fourier transform in optics. In this work we propose, precisely, a way to tackle the Fourier transform that eschews the sampling of second-order polynomial phase terms, handling them analytically instead: it is for this reason that we refer to this method as the "semi-analytical Fourier transform". We present here the theory behind this concept and show the algorithm in action at several examples which serve to illustrate the vast potential of this approach.