Fractional Fourier transform optimization approach for analyzing optical beam propagation between two spherical surfaces

A fractional Fourier transform (FrFT) relation exists between the light amplitude distributions on two spherical surfaces when the optical beam propagation in the optical surface resonators. Although the FrFT approach is appealing in that it describes the continuous evolution of a wave as it propagates through a system, the pattern scaling error still exists; i.e., the Fresnel diffraction pattern caused by an object is just an approximate enlarged pattern of the FrFT. This paper presents a precise approach to the light amplitude distributions on two spherical surfaces resonators for FrFT applications and provides a rigorous analysis to reduce the error in the pattern scale between the conventional Fourier transform and FrFT.