Lommel’s problem, the diffraction of spherical waves by a circular aperture and opaque disk, is extended to the case of a concentric array of ring-shaped apertures with arbitrary distribution of circle radii. The evaluation of the diffracted amplitude resulting from the entire array by the obvious process of numerical summation, applied to amplitude expressions pertaining to the sequence of circles involved, is found unsuitable and, instead, series expansions are investigated in which the location of the off-axis observation point and the interference effects of the ring openings are formally separated. The already available solutions of Lommel’s problem are examined from this viewpoint, and two new expansions in complex power series are derived. By means of two novel multiplication theorems for Lommel’s functions of two variables, convenient expansions in series of these functions are obtained for the amplitude diffracted by the annular array, thus affording the basis of a general theory of diffraction by ring systems, including the case of phase-reversal construction. Furthermore, equivalent general expressions are directly obtained from the power series formulas mentioned previously. A first application is made to the special case of the Soret zone plate. Finally Hopkins’ assumption of radially non-uniform amplitude is briefly dealt with in connection with the circular aperture and the annular systems.
© 1952 Optical Society of America
Equations on this page are rendered with MathJax. Learn more.