The diffraction theory of optical aberrations: Part II: Diffraction pattern in the presence of small aberrations

Abstract In this paper the diffraction theory for arbitrary aberrations of a symmetrical optical system is developed for the case that the amount of aberration is small. The aberration function, which measures the deviation of the actual wave-front from a sphere, is expanded in a series of the so-called circle polynomials, which were introduced by Zernike in a problem closely related to the one treated here. This new expansion appears to have considerable advantages as compared to the customary one. So, for instance, the problem of the counter-balancing of aberrations of various orders can be solved now completely. Formulae are given, from which the intensity distribution of the diffraction pattern in a receiving plane perpendicular to the principal ray in the neighbourhood of the Gaussian image can be numerically determined without much labour. The formulae are applied to the diffraction patterns of astigmatism and coma. The results of the numerical computations are given in diagrams showing the lines of equal intensity.