Asymptotic solution of the Huygens–Fresnel integral in circular coordinates

Complex contour integration techniques have been used to derive an asymptotic solution of the Huygens–Fresnel integral in circular coordinates. The solution (which is valid everywhere, including across the shadow boundary) provides a computationally efficient means to propagate uniform plane waves with high Fresnel numbers or to permit the evaluation of high-Fresnel-number unstable resonator modes. The results are interpreted as consisting of a geometrical term plus diffraction terms originating from two points on the circular aperture.