# Addressable Refraction and Curved Soliton Waveguides Using Electric Interfaces
## 2 Materials and Methods  
We performed numerical experiments of propagation of soliton beams in biased media using a well-tested FDTD numerical code [19]. The experimental configuration considers a visible beam at 532 nm entering a material with saturable electro-optic nonlinearity, to which a static electric bias is applied. The experimental conditions are such that this beam is self-confined and eventually reaches the spatial soliton regime. In what follows, we will call this visible beam the writing beam, and we will indicate its electric field amplitude as \(A_{W}\). We consider it the writing beam because it is the one interacting with the nonlinear host and, as a consequence, writing the nonlinear modification of the refractive index (the solitonic waveguide). Overlapped with the writing beam, a second IR beam (at 800 nm) is injected in the experiment. This second beam will be called the signal beam, and its electric field amplitude \(A_{S}\) is as follows: it represents any information or data that can be transported within the soliton waveguide. Its IR wavelength ensures that it is not able to interact with the host material and to modify or destroy the waveguide. The nonlinear propagation of these two beams (\(A_{W}\) and \(A_{S}\)) is described by the following set of Helmholtz equations for an electro-optic saturable nonlinearity:  
\[\
abla^{2}A_{i}=-k_{0}^{2}\varepsilon_{NL}\frac{E_{bias}}{1+\frac{|A_{W}|^{2 }}{|A_{sat}|^{2}}}A_{i} \tag{1}\]  
where \(A_{i}\) represents either \(A_{W}\) or \(A_{S}\), while \(\varepsilon_{NL}E_{bias}\) is the nonlinear electro-optic dielectric constant, \(E_{bias}\) is an external electrical bias (a static electric field induced along the sample thickness by an applied static voltage) necessary for photorefractive screening solitons [6] and \(\mid A_{sat}\mid^{2}\) is the saturation intensity of the nonlinearity. Please note that only \(A_{W}\) contributes to the nonlinear variation of the refractive index (the denominator of the last term in the equation). We have used the numerical data corresponding to an intrinsically pure z-cut Lithium Niobate single crystal. We have neglected the photovoltaic effect from the calculation. For this material and for e-polarized light, \(\varepsilon_{NL}\) gets the form  
\[\varepsilon_{NL}=n_{e}^{4}r_{33} \tag{2}\]where the extraordinary refractive index is equal to \(n_{e}\) = 2.2332, and the electro-optic coefficient \(r_{33}\) = 32 pm/V. With respect to the beam amplitude, typical values of \(\frac{|A_{W}|^{2}}{|A_{sat}|^{2}}\) are between 1 and 10, i.e., in a regime of efficient saturation of the optical nonlinearity. To begin with, self-focusing, self-confinement and solitonic regimes were identified, as shown in Figure 1.  
Without bias, the light beam diffracts. Increasing the electric bias (5 kV/cm) causes self-focusing to occur, forcing the beam not to diverge. At about 10 kV/cm, the light becomes self-confined, even if a strong pulsing takes place as a consequence of continuous focusing and diffraction processes [20]. This pulsing regime continues for higher biases, up to 20-25 kV/cm; at these values, pulsing is so rapid that the oscillations almost disappear (the analysis of the signal beam shall reveal that pulsing is always present at every bias, with different oscillation period that depends on the value of the applied voltage). At 27 kV/cm the writing beam reaches the solitonic regime: input and output transverse profiles are similar, and no evident oscillations are detectable. At higher biases, the beam dynamics follows the solitonic regime. The soliton formation can be easy identified if the attention is moved towards the propagating signal \(A_{S}\) instead of the writing beam \(A_{W}\). In fact, the linear propagation inside a nonlinear waveguide is much more sensitive to any non-uniform refractive distribution. In Figure 2, we have reported the signal propagation inside self-confined channels as a function of the applied electric bias. Below 20 kV/cm, the strong pulsation makes the confinement partial or inefficient.