The Fourier transform in optics: from continuous to discrete (II)

Not long ago we published an article dedicated to attempt the creation of a bridge between the continuous, physical optics and the discrete, mathematical optics [P. C. Logofatu and D. Apostol, "The Fourier transform in optics: from continuous to discrete or from analogous experiment to digital calculus," J. Optoelectron. Adv. M., 9(9), 2838-2846 (2007)1 Our motivation was that the connection between continuous and discrete is insufficiently investigated and the two formalisms stand alone for the most part. Our approach was one of the type top-down by enunciating the principles and then proving them, though we tried to be as user-friendly as possible and limit the inevitable mathematics to a minimum. In this article the theme is retaken from a different perspective, using a more bottom-up type of approach. Formalisms are built from one another. A great importance is accorded to the sampling theorem which is used to show that in the case of the functions with limited bandwidth the continuous and the discrete Fourier transform function coincide in the sample points if the sampling is properly made. The alteration of the output of the Fast Fourier Transform due to the shifting of the input is analyzed and ways to undo it are devised. We also found out an improved, more accurate form of the sinc interpolation function from the Nyquist-Shannon theorem.