|
The Fast Fourier Transform (FFT) algorithm makes up the backbone of fast physical optics modeling. Its nu- merical effort, approximately linear on the sample number of the function to be transformed, already constitutes a huge improvement on the original Discrete Fourier Transform (whose own numerical effort depends quadrati- cally on the sample number). However, even this orders-of-magnitude improvement in the number of operations required can turn out to 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 gigantic sample number. So much so, in fact, that even with the reduced effort of the FFT, the operation becomes impractica- ble. Finding a workaround that allows us to evade, at least in part, the stringent sampling requirements of the Nyquist-Shannon theorem 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.
|
