A Study of Talbot Illuminator with a Binary Phase Grating and a Fractional Fourier Transformation under a Spherical Wave Illumination

In this paper using an optical system of a binary phase grating and a fractional Fourier transformation under a spherical wave illumination to fulfil array illumination is studied. Based on the Fresnel diffraction theory, an analytical equation to calculate the diffraction field of the optical system is present. Through comparison we know that the optical system diffraction is equivalent to that of a single effective binary phase grating and formulae to calculate the effective period and effective distance are given. As an example we analyse the diffraction of the optical system of a binary phase grating and study the array illumination condition of the optical system of the binary phase grating. Analysed results reveal that the array illumination of the optical system relates to the structure of the binary phase grating and the parameters of the fractional Fourier transformation. Therefore by changing the parameters of the fractional Fourier transformation and binary phase grating, we can conveniently adjust the array illumination period and realizable condition. Our analysis will be helpful for the measurement of distance and a particle´s velocities.