Study of the differential theory of lamellar gratings made of highly conducting materials

Differential theory is said to be difficult to apply to surface-relief gratings made of metals with very high conductivity even though the formulation follows Li's Fourier factorization rules. Recently, Popov et al. [J. Opt. Soc. Am. 21, 199 (2004)] pointed out this difficulty and explained that its origin is related to the inversion of Toeplitz matrices constructed by the permittivity distribution inside the groove region. The current paper provides information about the differential theory for highly conducting gratings and considers the numerical instability problems. A stable calculation for lossless gratings is described, based on the extrapolation technique with the assumption of small losses.