Fractional Fourier transform of spherical refracting system and paraboloid refracting system

In the case of non-paraxial situation, the influence of the primary spherical aberration factor on fractional Fourier transform of single spherical refracting system was numerically and analytically studied and the difference between paraxial approximation and non-paraxial approximation was graphically presented. For a specific system, the numerical criterion in distinguishing paraxial and non-paraxial situation is given. The quadric surface was introduced to eliminate the influence of spherical aberration on fractional Fourier transform. It is shown that, with lax paraxial condition, the paraboloid which has the same curvature radius at the vertex as spherical refracting surface is capable of performing the same strict fractional Fourier transform as spherical refracting surface in the case of paraxial approximation.