An analysis of array illumination generalized by a kind of multilevel phase grating and a fractional Fourier transformation

We study how to use a special kind of multilevel phase grating and a fractional Fourier transformation to fulfil array illumination. Using the Fresnel diffraction theory and fractional Fourier transformation principle, we give out an analytical equation to calculate the diffraction field of the optical system. By comparing with a single multilevel phase grating, the optical system diffraction is equivalent to that of a single effective multilevel grating, nevertheless the period and the diffraction distance of the effective grating are changed, which is the main characteristic of our analysis. As an example we study the diffraction of the optical system of a multilevel phase grating. We analysis the array illumination condition of the optical system and its characteristic. We find that the array illumination relates to the structure of the multilevel phase grating and the parameters of the fractional Fourier transformation. Therefore, the spatial structure of array illumination of the optical system can be conveniently adjusted by the multilevel phase grating and the parameters of fractional Fourier transformation.