Abstract

Recently, Nussenzveig and Wiscombe, using complex angular momentum theory, derived an asymptotic expansion for the Mie theory extinction efficiency factor Q. From the relation between Q and the forward scattering amplitude S, plus the known analyticity properties of S (which is a complex function), we have derived the corresponding asymptotic expansion for S. If we then assume that, subject only to analyticity constraints, this result may be extended into the lower half of the complex plane, we may perform appropriate contour integrals around a semicircle of large radius. Since S has no poles in the lower half of the complex plane, we are thus able to find the value of desired integrals along the real axis–sum rules. In the case of a fixed real refractive index, we find explicit results for the three integrals: We have evaluated these integrals numerically for two different real refractive indices and found excellent agreement, indicating that the validity of our asymptotic expansion for S extends into the complex plane and also holds in the mean.

© 1985 Optical Society of America