Image formation with coherent light: The fractional Fourier transform approach

Abstract

The linear system theory of image formation tells us that, with coherent light, the image formed by a lens is obtained as the convolution product of the object amplitude with the scaled Fourier transform of the pupil function or, equivalently, that the lens transfer function is identical to the pupil function. This result is only true for spherical emitters centred on the entrance pupil of the lens. The use of the fractional-order Fourier transform and its relation to Fresnel diffraction provides a new way of considering image formation theory; the requirement on the curvature centre position is removed and the role of the pupil function as transfer function is generalized.