Fast diffraction computation algorithms based on FFT

The discovery of the Fast Fourier transform (FFT) algorithm by Cooley and Tukey meant for diffraction computation what the invention of computers meant for computation in general. The computation time reduction is more significant for large input data, but generally FFT reduces the computation time with several orders of magnitude. This was the beginning of an entire revolution in optical signal processing and resulted in an abundance of fast algorithms for diffraction computation in a variety of situations. The property that allowed the creation of these fast algorithms is that, as it turns out, most diffraction formulae contain at their core one or more Fourier transforms which may be rapidly calculated using the FFT. The key in discovering a new fast algorithm is to reformulate the diffraction formulae so that to identify and isolate the Fourier transforms it contains. In this way, the fast scaled transformation, the fast Fresnel transformation and the fast Rayleigh-Sommerfeld transform were designed. Remarkable improvements were the generalization of the DFT to scaled DFT which allowed freedom to choose the dimensions of the output window for the Fraunhofer-Fourier and Fresnel diffraction, the mathematical concept of linearized convolution which thwarts the circular character of the discrete Fourier transform and allows the use of the FFT, and last but not least the linearized discrete scaled convolution, a new concept of which we claim priority.