Extremal characterization of band gaps in nonlinear gratings

In this paper we present an explicit extremal characterization of the edges of the lowest band gap in gratings; we restrict here to the case of TE-modes, but the TM case can be treated similarly. The characterization is valid for linear and Kerr-nonlinear gratings, for smooth as well as for step-variations of the index and the coefficient of nonlinearity. The variational characterization allows one to derive monotonicity properties by using simple comparison arguments. For instance, the monotonie shift of the band gap as a function of the angle of oblique incidence is proved, leading to a simple criterion for omni-directionally for the band gap. Hough estimates of the band gaps can be obtained by substituting suitable trial fields in the governing functional, while the same formulation can also be used to design a Finite Element scheme for accurate calculations.