On the Inversion of an Integral Equation Relating Two Wavefunctions in Planes of an Optical System Suffering from an Arbitrary Number of Aberrations

If the wavefunction in the (not necessarily gaussian) image plane of an optical instrument is distorted by an arbitrary number of aberrations, the wavefunction in planes situated between the image plane and the plane of the specimen holder cannot be reconstructed by a Fourier series or a Fourier integral. This paper shows that the wavefunctions in those planes are uniquely determined by the values of the wavefunction in the image plane, and that an inversion formula, which in its structure is very similar to the well-known Fourier series, can be derived. Mathematically the problem concerns the inversion of the integral equation h (x 1, x 2) = Z exp iS (x 1, x 2, y 1, y 2)(y 1, y 1) dy 1 dy 2 if the eikonal S is a multinomial.