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A. referee has drawn the author's attention to an unpublished NDA Report (18-8) by B. A. Lippmann, in which the same Green's function has been derived in connection with his "variational formulation of a grating problem."
The application of the surface impedance (Leontovich) boundary condition would introduce an extra parameter, viz., the frequency-dependent surface impedance. Since the surface-impedance boundary condition holds only in a limited frequency domain, we cannot represent the field quantities as Fourier integrals.
Note that this Green's function is the same as the one used by B. A. Lippmann in NDA Report No. 18-8.
This phenomenon is just the "lateral wave" case of the "Wood anomalies" (Rayleigh wavelengths) observed with diffraction gratings [see J. E. Stewart and W. S. Gallaway, Appl. Opt. 1, 421 (1962)].
P. M. van den Berg, Thesis, Delft University of Technology, The Netherlands (1971).
P. M. van den Berg and O. J. Voorman, Appl. Sci. Res. 26, 175 (1972).
H. Blok and G. Mur, Appl. Sci. Res. 26, 389 (1972).