# Addressable Refraction and Curved Soliton Waveguides Using Electric Interfaces
## 3 Results  
### Critical Angle and Generalized Snell Law  
The Generalized Snell Law (GSL) [16] for soliton reflection has previously been introduced for high-order nonlinearities. In the critical angle regime, it can be written in terms of the complementary angle \(\theta\) as:  
\[\gamma\ n_{1}\cos\theta_{1c}=\ n_{2} \tag{3}\]  
where \(\gamma\) is a correction coefficient introduced to account for the refraction of beams instead of rays. Introducing the electro-optic interface, the GLS becomes:  
\[\gamma\ (n_{0}-\alpha E_{1})\cos\theta_{1c}=\ n_{0}-\alpha E_{2} \tag{4}\]  
which, solved by \(E_{2}\), takes the linear form of the equation of a line:  
\[E_{2}=\gamma\cos\theta_{1c}E_{1}+\frac{n_{0}}{\alpha}(1-\gamma\cos\theta_{1c} )=A\ E_{1}+B \tag{5}\]  
In Figure 4, we have reported all the experimental electric biases corresponding to the critical angles as function of the input complementary angles. All experimental data has been fitted with a straight line equation, obtaining in all cases a coefficient of determination R\({}^{2}\) of 1. From the A and B coefficients of the linear regressions (Equation (5)), we inverted the expressions to obtain the \(\gamma\) and \(\alpha\) coefficients:  
\[\gamma=\frac{A}{\cos\theta_{1c}} \tag{6}\]  
\[\alpha=\frac{n_{0}}{B}(1-A) \tag{7}\]  
Figure 3: Reflection and refraction dynamics of a soliton beam at an ideal electric interface. The dotted line identifies the position of the electric wall. The applied biases at the two volumes before and after the wall are reported in each case.  
In Figure 5 we have reported the \(\gamma\), B and \(\alpha\) coefficients calculated from Figure 3. The \(\gamma\) correction factor for the GSL still follows a linear trend (Figure 5a) with respect to the input complementary angle. The B intercept instead has a parabolic trend (Figure 5b). This discrepancy leads, as a consequence, that the nonlinear \(\alpha\) coefficient, calculated through Equation (7), receiving a parabolic trend as well, as shown in Figure 5c. In particular, this fit points out that the highest nonlinear efficiency for the \(\pi/2\) refraction of a solitonic beam occurs at a complementary incidence angle of \(0.6^{\circ}\). This result occurs because of the mutual effects of the beam refraction and the effective birefringence induced at the electric interface. Thus, we shall use a complementary input beam of \(0.6^{\circ}\) as the reference value for the following waveguide analysis.  
Figure 4: This figure reports the obtained values of the applied voltages that generated an EO interface for which a critical angle of the soliton is experimentally examined. The curves correspond to different input angles, whose values are reported on the right-hand side. All straight-line regression fits have a coefficient of determination \(\mathrm{R}^{2}=1\).  
Figure 5: (**a**) From the slopes of the regressions in Figure 4, the \(\gamma\) coefficients of the Generalized Snell Law are calculated. They still follow a linear trend as the function of the input angle. (**b**) The interceptions of the regressions in Figure 4 scale quadratically with the input angle. (**c**) The discrepancy between the trends in Figure 5a,b gives the most efficient EO nonlinearity at a \(0.6^{\circ}\) input complementary angle.