## 3 Experimental Results and Discussion
### Waveguide Characterization and Power Handling  
Knowledge of the geometrical parameters describing the intensity distribution of the guided fields, related to the refractive index gap \(\Delta n\) between the core of the nonlinear waveguide and the bulk LN crystal, is crucial for estimation of both the spectral coverage of the CW source (see Equations (3) and (4)) and the maximum IR power available for frequency down-conversion. Indeed, the generation of broadly tunable CW THz radiation, with power levels allowing room-temperature detection,  
Figure 2: Schematic of the experimental setup for waveguided CW THz DFG. implies the IR pump fields' intensity in the order of \(I\simeq 10^{7}\,\mathrm{W/cm^{2}}\). In these conditions, third-order nonlinear susceptibility and extrinsic defects in the material commonly trigger thermo-optic (TOE) and photo-refractive (PRE) effects, inducing self-defocusing of the pump beams and hampering the achievement of DFG phase-matching [35]. Since the self-defocusing is caused by an intensity-dependent negative modulation of the refractive index (\(\Delta n(I)<0\)) between the center and peripheral rays of the Gaussian IR mode, the waveguide refractive index gap has to be larger than \(|\Delta n(I)|\). In order to measure the size of the guided mode, the crystal plate containing the surface waveguides, held by a metallic support, is mounted on a 3 axis platform (Thorlabs MAX313D) allowing a 1 \(\upmu\)m accuracy in the positioning of the waveguide input facet with respect to the incoming beam axis. Referring to Figures 1 and 3, we will indicate the propagation direction as the \(x\) axis, coinciding with the side of the crystal that is 1 cm long. The plane containing the input facet will be indicated as \(yz\), being the vertical \(z\) axis parallel to the extraordinary crystallographic axis. The crystal is mechanically held between a copper plate (ensuring optimal heat dissipation) on one side, and the HR-Si prism on the side containing the waveguides. When the pump fields are coupled to the waveguide, the transmitted intensity can be measured as a function of the displacement of the center of the guiding core with respect to the optical axis. Hence, the resulting spatial distribution in the \(yz\) plane will be in the form of a convolution that is integral between the intensity profile of the field \(I_{f}\) and of the guided mode \(I_{g}\):  
\[T(y,z)=\int I_{f}(\widetilde{y},\widetilde{z})\ I_{g}(\widetilde{y}-y, \widetilde{z}-z)\ d\widetilde{y}d\widetilde{z} \tag{6}\]  
Generally, the guided mode intensity profile \(I_{g}\) can undergo an exponential or a Gaussian decay at the edge of the guiding core, depending on the fabrication process [49]. In the following, we will assume for it a Gaussian profile, since the convolution integral obtained assuming an exponential decay did not reproduce our experimental data. On the contrary, as it will be shown, they are correctly reproduced by assuming a Gaussian shape for both the field intensity \(I_{f}(y,z)=I_{f0}\exp\left(-\frac{2\left(y^{2}+z^{2}\right)}{w_{0}^{2}}\right)\) and the guided mode \(I_{g}(y,z)=I_{g0}\exp\left(-2\left(\frac{y^{2}}{r_{0}^{2}}+\frac{z^{2}}{r_{0 }^{2}z}\right)\right)\). In so doing, even the transmitted power profile will be described by a Gaussian with radii at the 1/e level:  
\[\sigma_{j}=\frac{1}{2}\sqrt{w_{0}^{2}+r_{0j}^{2}}\ (j=y,z).\]  
The measurement of the waist of the input beam \(w_{0}\) is non-trivial. Indeed, since an optimal mode coupling requires \(w_{0}\) to be of the same order of magnitude of the mode size (namely, a few microns), it cannot be measured by means of the standard knife-edge method [50]. Our approach consists in acquiring different transmission profiles for different positions of the waveguide facet with respect to the focal plane. According with the Gaussian beam propagation, the acquired profiles will be broadened, increasing the relative displacement \(\Delta x\) between the two planes.