Joint transform correlator: expansion of the aberration function for a compact design

Abstract

A method to compute the aberrations of any order introduced into an optical system which performs the first Fourier transform in a joint transform correlator is described. The optical system considered is illuminated by a quasimonochromatic axial point source and it optically processes a structure placed between two centred optical systems, A and B. The structure acts as an aperture stop, its plane is perpendicular to the optical axis of A and B and it consists of two rectangular displays (object and sample) placed symmetrically with respect to this axis. Each display can be considered as a superposition of sinusoidal gratings of different spatial frequencies and orientations. The aberrations which correspond to each spatial frequency must be evaluated separately for the object (scene) and for the sample (target) and then the device is neither centred nor symmetric. Here it is shown that the aberration function for one of the displays and a given spatial frequency can be expanded in Zernike's polynomials considering both the usual symmetric terms and the non-symmetric ones and assuming that the aperture stop is the circle which contains the display. Moreover, if the optical system which performs the second Fourier transform in the correlator is free from aberrations, it is shown that a criterion which can be used to ensure that an accurate correlation signal is obtained is that the difference between the wavefront aberrations introduced in both displays is less than a quarter of a wavelength.