Nonlinear Snell law for grazing incidence along interfaces with discontinuous second-order susceptibilities

In this article, we demonstrate that the angles at which second-harmonic (SH) waves are generated at ferroelectric domain walls satisfy the Snell law for nonlinear media. Nonlinear reflection and refraction are observed experimentally and the relation is found to be in good agreement with theoretical predictions. The ratio of the intensities of refracted and reflected waves has been measured. Under an anomalous-dispersion-like condition, the forbidden nonlinear reflection and refraction is analyzed and found to have a behavior similar to that of the total internal reflection in linear optics. In the periodic domain structure, the coherent superposition of SH waves has been observed, on the basis of which we have proposed a comprehensive theory to explain nonlinear effects in multilayered structures.

Figure 1

Schematic representation of a nonlinear medium with opposite polarization directions and nonlinear coefficients ${\chi }^{\left(2\right)}$ and $-{\chi }^{\left(2\right)}$, respectively.

Figure 2

(a) Phase-matching diagram of a parametric process at the domain wall. (b) Simulation results of the SH wave intensity as a function of the incident angle of the FW.

Figure 3

(a) Schematic of the experimental setup. (b) Variation of the recorded patterns with the incident angle of the FW. The internal incident angles, from top to bottom, are $74.1^{\circ }$, $75.4^{\circ }$, $79.6^{\circ }$, $82.2^{\circ }$, and ${90}^{\circ }$. (c) Reflected and refracted angles of the SH wave as functions of the incident angle of the FW. (d) Ratio of the intensities of the refracted and reflected waves as a function of the incident angle of the FW.

Figure 4

Reflected and refracted angles of the SH wave as a function of the incident angle $\alpha$ for an o-polarized FW with a wavelength of $1250\text{nm}$.

Figure 5

(a) Schematic representation of refracted SH waves generated at adjacent domain walls. (b) Intensity of the refracted SH wave as a function of the incident angle of the FW.