Derivation of analytical expressions for anomalous reflection in the limit of zero thickness and weakly modulated dielectric grating.

It is known that dielectric gratings exhibit anomalous scattering behavior. At certain incident angles, which are not related to the grating's formula, 100% of the incident beam is reflected and, at other angles, 100% is transmitted. In this paper, analytical expressions are derived, for the first time, to the best of our knowledge, for these angles in the regime of narrow grating and weak modulation depth. In these expressions, the parameters emerge from basic principles. Moreover, in this weak modulation regime, a simple and analytically solvable model can be used to derive an analytical expression for the scattered electromagnetic field. Furthermore, it is shown that 100% reflection is achieved even when the grating layer shrinks to zero, the change in the layer's refractive index is zero, and even when the modulation depth is arbitrarily weak, in which case, the incident angle satisfies ⁡≅±(1-/), where is the grating spacing and is the beam's wavelength. This result is valid for any ratio /. Finally, it is shown that these anomalous transmission behaviors occur even when the modulation coefficient is imaginary and that these analytical expressions are still valid and can predict the corresponding angles.