Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects

Abstract

We present the first rigorous electromagnetic theory of diffraction in nonlinear optics. This theory allows the study of any type of nonlinear grating: bare or coated, whatever the groove depth and the profile of grating and coatings may be. The formalism developed here is derived from Maxwell's equations. The existence of the excitation and its nonlinear feature on the one hand, and the diffraction of the pump beams and of the signal on the other hand, are fully taken into account. The calculation reported here is valid for all cases of polarization (TM or TE) of the pump beams and of the signal. Two expressions of the nonlinear polarization at the signal frequency are derived. One is valid below the modulated region; the other one, inside this region. These two expressions take into account all the diffracted orders at the pump frequencies: propagating and evanescent. We then get the expression of the electromagnetic field at the signal frequency everywhere: not only outside the modulated region, but also inside it. The results thus obtained show that this electromagnetic field is a superposition of a diffracted field, with radiated and evanescent orders, and an infinite number of elementary driven waves. We also derive the nonlinear grating equation which allows the determination of the directions of propagation of the radiated diffracted orders. This is achieved using a new geometrical construction. It is shown that the evanescent diffracted orders at the signal frequency and at the pump frequencies can be resonantly excited. The regorous feature of the electromagnetic theory developed here allows us to get the following new and important result: There exists an optimal groove depth for which the electromagnetic resonance contribution to the enhancement of the nonlinear optical process is the strongest. These results can be applied to the study of different nonlinear optical processes, such as enhanced second-harmonic generation, surface-enhanced Raman scattering, Pockels effect, and optical rectification.