The range of validity of the Rayleigh hypothesis

Abstract

Summary form only given. The parameter range over which the Rayleigh hypothesis (RH) for optical gratings might be validly applied to analysis of high power backward wave oscillators (BWOs) has been investigated numerically. A widely used method to analyze high power backward wave oscillators (BWO's) is to represent the electromagnetic (EM) fields in axisymmetric slow wave structure (SWS) in terms of a Floquet harmonic expansion (FHE). EM fields, E and B, are expressed in a form as, [E B]=/spl Sigma//sub n=-N//sup N/[E/sub n/(r) B/sub n/(r)]exp i(k/sub zn/z+l/spl theta/-/spl omega/t) where k/sub zn/=k/sub z/+nK/sub 0/, K/sub 0/ is wavenumber of SWS periodicity, and n=0, /spl plusmn/1, /spl plusmn/2,.../spl plusmn/N is the Floquet harmonic number. Expansion similar to (1) was firstly introduced by Lord Rayleigh for diffraction of waves from planar grating. He assumed that the expansion was applicable both outside and inside the corrugation, and this assumption is called as RH. It was argued by some mathematician that our numerical analysis was applied to deep corrugation of hK/sub 0/=1.67 which was beyond the limiting value hK/sub 0/=0.448 for validity in RH for planar sinusoidal grating, consequently the results were invalid. Here, h is the amplitude of corrugation. To respond the doubt, EM fields and dispersion relation in the SWS are numerically analyzed with and without RH for a given set of size parameters. The field patterns and eigen frequency for the SWS are solved for a given k/sub z/ numerically by using finite difference code HIDM (Higher Order Implicit Difference Method) that is free from RH. The results are compared with those using (1). For a deep corrugation, hK/sub 0/=5/spl times/0.448, using RH is still valid for obtaining the dispersion relation, although the Floquet Harmonic Expansion (FHE) fails to correctly represent the field patterns inside the corrugation. Accordingly, there exists a discrepancy between the validity of using RH for obtaining dispersion relations and for an exact convergence of FHE everywhere in the SWS.