A General Modal Theory for Reflection Gratings

The problem of diffraction from lossless reflection gratings is treated with a modal expansion formulation which incorporates an impedance condition on the mode eigenfunctions along the groove aperture. The theory for both fundamental polarizations is applied to three different grating profiles and is found to yield results which conform well with predictions of an established integral theory over a restricted range of groove depths. The upper bounds on these depths are approximately 0·2 periods for the symmetrical triangular profile and 0·5 periods for the semi-circular and sinusoidal profiles. Beyond these limits numerical instabilities occur and the results often fail to satisfy standard numerical tests.