Self-Fourier functions and fractional Fourier transforms

The Fourier transform is perhaps the most important analytical tool in wave optics. Hence Fourier-related concepts are likely to have an important on optics. We will likely recall two novel concepts and then show how they are interrelated. A self-Fourier function (SFF) [1,2] is a function whose Fourier transform is identical to itself. Another issue that has been recently investigated is the fractional Fourier transform. Two distinct definitions of the fractional Fourier transform have been given. In the first one [1], the fractional Fourier transform was defined physically, based on propagation in quadratic graded index (GRIN) media. The second definition [2] is based on Wigner distribution functions (WDF). Here the fractional Fourier transform is calculated by finding the WDF of the input image, rotating it by an angle α = aπ/2, and performing the inverse Wigner transform.