A rigorous analysis of the generalized Rayleigh-Gans approximation

Since its introduction by Schiffer and Thielheim [1979], the generalized Rayleigh-Gans (GRG) approximation has frequently been employed to determine the scattering from thin dielectric cylinders. In Schiffer and Thielheim, the validity of the approximation is demonstrated for a thin circular cylinder at both the long wavelength and the short wavelength cases by direct comparison with Rayleigh and high frequency models, respectively. Although this provides strong evidence as to the validity of the GRG approximation, it does not prove its general validity, particularly with regards to resonant length cylinders and cylinders with non-circular cross-sections. Thus, to provide a firm theoretical basis for the implementation of the GRG approximation, a rigorous analysis is required to demonstrate GRG validity for arbitrary values of cylinder electrical length, kl, in addition to arbitrary cross-sections. Beginning with a general scattering formulation, it is shown that the solution to the integral equation converges to the GRG approximation as the cylinder electrical radius converges to zero, for any arbitrary value of kl. In addition, the proof is independent of cylinder cross-section shape, thus demonstrating the validity of GRG for cylinders with non-circular cross-sections. Finally, the error associated with the approximation is examined for all values of kl by developing a moment method solution for thin dielectric cylinders.