## 2 Theory
In LN-based setups, a viable alternative consists of restoring a quasi-phase matching condition after each coherence length (only few tens of \(\mu\)m) by introducing a periodic modulation of nonlinear susceptibility. The achievement of collineargeneration schemes by means of periodically-poled (PPLN) crystals is a fundamental step in order to accomplish optical parametric oscillators [44; 45] capable of THz emission with power levels in the order of tens of \(\upmu\)W. However, the strong THz absorption of LN [46] and the quasi-PM bandwidth limits the spectral coverage of these sources to few hundreds of GHz around a specific frequency fixed by the PPLN crystal design. This serious drawback can be completely overcome by realizing the PM in a non-collinear emission scheme within a LN surface waveguide. Indeed, since IR light propagates faster than the THz waves (\(n_{pump}<n_{THz}\)) [47; 48] in such a material, a perfect PM condition can be fulfilled exactly, and a THz wave with a conical wave-front, defined by the relation:  
\[\theta_{C}=\arccos\left(\frac{n_{pump}}{n_{THz}}\right), \tag{5}\]  
is coherently generated along the whole crystal length. In LN, the angle \(\theta_{C}\) (called the Cherenkov angle because of the analogy with the light emission by ultra-relativistic particles) is of about 64\({}^{\circ}\), requiring a cladding material with a proper prism shape to be held in optical contact with the emitting surface, in order to avoid total internal reflection of the produced THz radiation. Since the Cherenkov PM results in an infinite coherence length, it entails the experimental need to keep the IR pump beams collimated along the whole crystal length and, at the same time, overlapped on a small interaction area (\(\eta\propto L^{2}/r_{0}^{2}\)). To this purpose, the most suitable choice is to couple the IR light with the fundamental mode of a tiny channel waveguide. Furthermore, as the waveguide is ion-planted, it lies just beneath the surface of the LN substrate, and the THz radiation can be immediately extracted, minimizing the absorption losses. Another important advantage rising from the waveguided approach, is the reduction of the area of the interacting wave fronts. This leads to a dramatic increase in the generation bandwidth. Indeed, looking at the frequency dependence of the nonlinear efficiency entirely contained into the \(g\) function, we can see that while at the low frequencies, the efficiency scales as \(\omega^{2}\), the high-frequency behavior is described by a Gaussian profile with full-width-at-half-maximum \(\propto 1/r_{0}\). From a physical point of view, the high-frequency cut-off can be seen as a consequence of the destructive interference between THz waves generated on the crystal surface and the ones generated at a depth of about \(\lambda_{THz}/2\). Since LN possesses a refractive index of about 5.2 in the THz range, only the waves generated within a few microns below the surface will experience coherent interference. In Figure 1b, the function \(g(\omega_{THz},r_{0})\) has been plotted for different mode radii. As it can be seen, a broadening of the achievable bandwidth and a high-frequency shift of the maximum available  
Figure 1: (**a**) Polar coordinates reference system with respect to the waveguide input facet. (**b**) Accessible THz bandwidth for a continuous wave (CW) waveguided difference frequency generation (DFG) process in MgO-doped Lithium Niobate (LN). Different lines correspond to different guided mode radii between 4 \(\upmu\)m and 10 \(\upmu\)m.  
efficiency rapidly occurs once the mode size is reduced. For example, radiation beyond 4.5 THz, representing the current frequency limitation in CW THz QCLs, can hardly be obtained with mode dimensions exceeding 10 \(\upmu\)m. Conversely, a beam radius smaller than 5\(-\)6 \(\upmu\)m should, in principle, be sufficient in order to achieve nonlinear generation along the whole THz bandwidth.