# Addressable Refraction and Curved Soliton Waveguides Using Electric Interfaces
## 3 Results
### Waveguiding and Propagation Losses
#### 3.2.2 Curved Propagation  
A similar procedure has been applied for curved self-confined structures, i.e., for signal beams propagating inside those written in presence of an electric interface. First of all, the radii of curvature of the osculating circles of the curved trajectories have been calculated from the experimental maps (Figure 7). The obtained values are reported in Figure 8a versus the refractive index contrast \(\delta n\) defined as:  
\[\delta n=n_{1}-n_{2}=\alpha(E_{2}-E_{1}) \tag{8}\]  
For negative contrasts, as well as for positive contrasts until the critical angles, the curvatures are always the same, all having radii of about 200 \(\upmu\)m. The Giant Goos-Hanchen regime occurs [16] above the critical angle for a \(\delta n\) of up to 0.5 above the critical value.  
Just above the critical angle, the refracted beam experiences a transverse instability that forces it to be reflected back within the input volume 1. Such regimes are well identified by monitoring the propagation losses of the signal beam, as shown in Figure 8b.  
Figure 6: (a) Input (red) and output (black) profiles of the signal beam for different applied biases; (b) propagation losses of solitonic waveguides for different applied biases.  
A single rapid curve of the solitonic waveguide introduces a signal loss as high as 0.2 dB. This is deduced by Figure 8b, where the straight propagation losses (red square) are reported together with the curved waveguide ones (white squares). Until the critical angle \(\left(\delta n\approx 1.5\times 10^{-4}\right)\), the total losses are about 0.2 dB above the straight propagation ones. With \(1.7\times 10^{-4}\geq\delta n>1.5\times 10^{-4}\), the Giant Goos-Hanchen occurs; this regime is governed by a double refraction, one towards the \(\pi/2\) refraction and one towards the back reflection (see for example \(\delta n>1.6\times 10^{-4}\) in Figure 6), connected together by a portion of the trajectory with low curvature (long radius). In Figure 8a such regime is reported by hyperbolic portion of the curve with a vertical asymptote in correspondence of the critical angle. Such double refraction doubles the losses, as well. In fact, the losses' curve gets a peak up to about 0.6 dB, as the sum of the straight propagation losses (0.07 dB/cm) and two rapid curves, each of which is about 0.2 dB. Above \(\delta n=1.7\times 10^{-4}\), the two sharp curves homogenize in a larger single curve. An optimal smooth curvature was obtained at about \(\delta n=2.0\times 10^{-4}\) (Figure 6) for which the propagation losses of the signal beam drop down to almost the straight-propagation ones. Above this value, the reflection curve gets sharper and sharper, increasing again the propagation losses up to the single-sharp curve value (see, for example, the 25/32 case in Figure 3).  
Figure 8: Radius of curvature and propagation losses calculated from numerical experiments: (**a**) Radius of curvature as a function of the refractive index contrast; (**b**) Propagation losses as a function of the refractive index contrast.  
Figure 7: Determination of the radius of curvature of the osculating circles for different trajectories of the solitonic waveguides.