Fractional Fourier transform and spherical refracting imaging system

Using paraxial Fresnel diffraction formula, the relation between input and output function of a single spherical refracting system was derived. It is shown that fractional Fourier transform can be implemented on such optical system. Based on the property of fractional Fourier transform and coordinate scale change of input and output plane, the Gaussian formula for perfect imaging was obtained. It is pointed out that, a family of thick lens systems composed of multiple spherical refracting surfaces is potentially capable of performing fractional Fourier transform and its imaging process could be analysed by means of fractional Fourier transform. A unique design of double spherical refracting imaging system with four media is also presented.