Raman-Nath nonlinear diffraction in second harmonic generation

In the linear optics, light beam propagating through a structure with periodic refractive index experiences either single order Bragg diffraction, when the full vectorial phase matching (PM) condition is satisfied, or multi-order Raman-Nath diffraction (RND), otherwise. Analogously, in parametric processes such as second harmonic (SH) generation light can experience nonlinear diffraction in periodically poled crystals with homogeneous linear refractive index. In this, the so called, nonlinear Bragg diffraction [1], the SH beam is emitted at a specific angle, determined by the vectorial phase-matching condition 2k 1 + G n = k 2 , where k 1 and k 2 are the respective wave-vectors of the fundamental and the second harmonic and G n denotes the grating vector of the periodic modulation of the sign of nonlinearity [see Fig. 1(a), left]. However, if this vectorial condition is not satisfied, a multi-order nonlinear diffraction of the SH should be seen instead [see Fig. 1(a), right], in analogy to the linear Raman-Nath diffraction.