## 3 Experimental Results and Discussion
### Waveguide Characterization and Power Handling
The transmitted intensity radius can be expressed as:  
\[\sigma^{2}=a+b\cdot\left(\Delta x\right)^{2}\]  
where the guided mode radius is related to the linear parameters \(a\) and \(b\) by:  
\[r_{0}=\frac{2}{\sqrt{b}}\sqrt{ab-\left(\frac{\lambda}{2\pi}\right)^{2}}. \tag{7}\]  
Such a broadening effect can be clearly observed in the different experimental traces of Figure 3b,c, representing the transmitted profiles in the y and \(z\) directions, respectively. It must be noted that, while the Gaussian shapes obtained in the \(z\) direction are symmetric for every value of \(\Delta x\), the LN-Silicon interface implies a visible decrease in guiding capability in the \(y>0\) half-space. Furthermore, since only the volume corresponding to \(y<0\) is involved in the generation process, the data corresponding to positive values of \(y\) have not been fitted. Figure 3: (**a**) Representation of the crystal plate containing the channel waveguides. (**b**,**c**) Transmission profiles in the \(z\) and \(y\) directions, respectively, measured for different waist displacement \(\Delta x\). (**d**,**e**) Mode Radii at the \(1/\mathrm{e}\) level as a function of \(\Delta x\). In Figure 3d,e, the \(\sigma^{2}\)-parameter has been plotted as a function of the squared waist displacement \(\Delta x^{2}\), while the measured waveguide parameters have been summarized in Table 1. The radii \(r_{0y}\) and \(r_{0z}\) of the guided mode have been obtained by substituting the fitting parameters into Equation (7), and the corresponding uncertainties have been calculated by means of the standard propagation formula. Knowledge of the intensity distribution \(I_{g}\) allows us to estimate the refractive index gap \(\Delta n\) between the guiding material and the substrate. It has been calculated on the basis of the measured geometrical parameters by applying the inverse WKB method [51]. In the weak guidance approximation (\(\Delta n\ll n_{pump}\)), the n-gap can be obtained by evaluating the expression:  
\[\Delta n(y,z)=-\frac{\
abla^{2}\sqrt{I_{g}(y,z)}}{2n_{pump}k_{0}^{2}\sqrt{I_{ g}(y,z)}} \tag{8}\]  
In the guiding core (\(y=z=0\)). In the previous expression, \(k_{0}\) represents the free-space IR wavevector, while for \(n_{pump}\) the value provided by the Sellmeier semi-empirical formula has been used [47]. The obtained result of \(\Delta n=0.004\) has been compared with the contribution, opposite in sign, due to the thermo-optic and photo-refractive effects. These quantities have been estimated from the relations [52]:  
\[\Delta n_{TOE}=-\left(\frac{dn}{dT}\right)\frac{\alpha\ r_{0}^{2}\ I}{k}\]  
and:  
\[\Delta n_{PRE}=-\left(n_{pump}\right)^{\frac{g}{2}}r_{33}\left(\frac{I}{2 ce_{ 0}}\right)^{1/2} \tag{9}\]  
providing a contribution in the order of \(10^{-6}\) and \(10^{-4}\), respectively. In the previous equations, \(\alpha\), \(k\) and \(r_{33}\) are the absorption coefficient, the thermal conductivity, and the electro-optic coefficient of the LN crystal, respectively, whose values used for calculations are summarized in \[T(y,z)=\int I_{f}(\widetilde{y},\widetilde{z})\ I_{g}(\widetilde{y}-y, \widetilde{z}-z)\ d\widetilde{y}d\widetilde{z} \tag{6}\]0 When compared with the previous ones, the measured value \(\Delta n\) predicts, for our waveguide, the capability to confine an IR intensity of up to \(10^{7}\) W/cm\({}^{2}\) in the guided mode. The experimental curve, printed in blue in Figure 4a, represents the measured values of the coupled vs. incoming IR power. For the sake of clarity, these data have been compared with an analogous measurement performed by confining light in the fundamental mode of a planar waveguide (red trace in Figure 4a). Such a waveguide is characterized by the same geometrical and optical properties of the channel one, except for the radius of the beam in the non-guiding direction \(z\), which, in this case, is \(r_{0z}=150\) um.