THE EXTRAPOLATING PUPIL, IMAGE SYNTHESIS, AND APPLICATIONS FOR THE FUTURE

Abstract

A function PvN(ß) exists whose finite Fourier transform over a specified range of its argument is asymptotic (with N) to an Airy distribution with arbitrary scale compression. Consequently, when the function is applied as a passive coating to a diffraction -limited lens of fixed aperture,the point amplitude response collapses inward as if the lens were physically replaced by a diffraction-limited lens of greater aperture. Investigating the implications of coating PN(ß) to image theory, we find the following: (1) The scalar wave equation has intrinsically a particlelike solution. (2) A modification of PN(ß) causes an arbitrarily narrow depth of focus. (3) An arbitrary point amplitude response may be optically produced. (Suppose g(x) to be a required, and arbitrary, point response function with G(ß) its finite Fourier transform. Then pupil PN(ß)G(ß) produces g(x), asymptotic with N.) (4) When applied onto any band -limited pupil G(ß), coating PN(ß) effectively extrapolates G(ß) arbitrarily beyond the bounds of the aperture. Some amusing analog devices, based on the extrapolating property (No. 4 above), are next developed. These are an optical analog signal extrapolator, a picture extrapolator, and an analog method of band -unlimited image processing. We also suggest the existence of a laser "superposition mode" whose out- put would be arbitrarily directive, and the possibility of using an acoustical pupil NO) to resolve these long wavelengths with near-optical quality. The ultimate limitations on the practical use and fabrication of pupil PN(ß) are discussed.