Numerical solution of the Huygens–Fresnel–Kirchhoff diffraction of spherical waves by a circular aperture

Using the rigorous wave-front formulation for scalar wave diffraction of Kraus [ J. Opt. Soc. Am. A6, 1196 ( 1989); J. Opt. Soc. Am. A9, 1132 ( 1992)], it is shown that the two-dimensional integral used to calculate the diffraction of spherical waves by a circular aperture may be reduced to a one-dimensional integral by choosing an appropriate coordinate frame. Both the two-dimensional integral and the one-dimensional integral must be evaluated numerically, but because each dimension must be sampled at approximately N locations to calculate accurately the integral (where N is the number of wavelengths across the aperture) the two-dimensional integration will require of the order of N2 evaluations of the integrand, whereas the one-dimensional integration will require of the order of only N evaluations, a substantial decrease in computing time for apertures that are large compared with optical wavelengths.