This paper deals with an integral equation method for analyzing the diffraction of a transverse magnetic (TM) plane wave by a perfectly conductive periodic surface. In the region below the periodic surface, the extinction theorem holds, and the total field vanishes if the field solution is determined exactly. For an approximate solution, the extinction theorem does not hold but an extinction error field appears. By use of an image Green's function, new formulae are given for the extinction error field and the mean square extinction error (MSEE), which may be useful as a validity criterion. Numerical examples are given to demonstrate that the formulae work practically even at a critical angle of incidence.

In the theory of diffraction gratings, the conventional integral method is considered as a powerful tool of numerical analysis. But it fails to work at a critical angle of incidence, because a periodic Green's function (integral kernel) diverges. This problem was resolved by the image integral equation in a previous paper. Newly introducing the reflection extinction theorem, this paper derives the image extinction theorem and the image integral equation. Then, it is concluded that the image integral equation is made up of two physical processes: the image surface radiates a reflected plane wave, whereas the periodic surface radiates the diffracted wave.