Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation

The Brekhovskikh variant of the Kirchhoff approximation (KA), as it applies to the description of the scattered field at large distances from a perfectly conducting sinusoidal grating (of amplitude h and period d) exposed to a plane wave of wavelength λ and incidence angle θi, is evaluated by means of energy-balance error calculations and comparisons with quasi-rigorous reference results obtained from the extensive form of the Rayleigh theory (RT). The RT is theoretically exact for all d/λ provided that h/d ≲ 0.072 and can be made to yield accurate numerical solutions at all θi and for all polarizations roughly for h/d < 0.39 − 0.0177 (d/λ) when d/λ ≤ 15. The KA is found to yield sufficiently accurate solutions at normal incidence only for roughly h/d < 0.011 (d/λ). Furthermore, the quality of the KA diminishes rapidly with increasing angle of incidence, as is shown by the fact that unacceptable errors occur for h/d ≳ 0.1 above θi ≃ 6°, for h/d ≳ 0.05 above θi ≃ 37°, and for h/d ≳ 0.016 above θi ≃ 63° at the period/wavelength ratio of 10. At higher frequencies (d/λ ≃ 36) there is some improvement in that, for example, the error becomes excessive for h/d ≳ 0.1 only above θi ≃ 25.5°. The KA proves to be unsafe at any frequency and incident angle as soon as h/d exceeds 0.13. This threshold is shown to be related to the onset of multiple scattering in the sense of geometrical optics.