## 3 Experimental Results and Discussion
### THz Generation Efficiency and Spectral Coverage  
The generation bandwidth achieved by our CW THz source has been investigated by tuning the wavelength (i.e., the frequency \(\omega_{1}\)) of one of the laser telecom diodes along the 1540-1565 nm operational bandwidth of the A1 amplifier, while the frequency \(\omega_{2}\) of the other laser is fixed within the A2 bandwidth. As previously discussed, two different combinations are possible, allowing us to span two different portions of the THz spectral window. Figure 4b shows two examples of these frequency scans, one for each amplifier's configuration, demonstrating a spectral coverage extended from 1 to 7.5 THz. The first half of the curve, spanning the 1 to 4.5 THz spectral window, has been obtained by fixing \(\lambda_{2}=2\pi c/\omega_{2}\) at 1575 nm and adjusting \(\lambda_{1}\) from 1541 nm to 1563.5 nm. In this range, a maximum THz power of 0.52 \(\mu\)W has been measured at 4.1 THz. This value, corresponding to 3.6 W of IR power confined in the waveguide (distributed in \(P_{1}=1.8\) W and \(P_{2}=1.8\) W), leads to a nonlinear efficiency of \(\eta=1.5\;\times\;10^{-7}\) W\({}^{-1}\). The second part of the blue trace in Figure 4b, from 5 to 7.5 THz, has been achieved with the same \(\lambda_{1}\) scan and setting \(\lambda_{2}=1605\) nm. Here, a maximum THz power of 0.2 \(\mu\)W (with \(P_{1}=1.8\) W and \(P_{2}=0.75\) W) has been measured at 5 THz, correspondingly, to the same nonlinear efficiency. The operation of a room-temperature setup, spanning the entire 1-7.5 THz frequency range (i.e., almost 3 octaves and well beyond the 4.5 THz current limit of THz-QCLs) with a fraction of \(\mu\)W generated power can represent a groundbreaking achievement for a number of applications. Nevertheless, some hints for further improvements can be found by comparing the experimental generation curve with the expected one. To this purpose, the nonlinear efficiency \(\eta\) has been calculated by numerical integration of Equation (3) over the emission angles. To this regard, given the asymmetric profile of the guided mode (\(\tau_{0y}>\tau_{0z}\)), Equation (4) has been generalized to:  
\[g(\omega_{THz}.\;r_{0y},r_{0z})=\omega_{THz}^{2}\;\exp(-\frac{\omega_{THz}^{2 }}{4c^{2}}\frac{n_{THz}^{2}}{(r_{0y}^{2}\sin^{2}\phi\sin^{2}\theta+r_{0z}^{2} \sin^{2}\theta\cos^{2}\phi)). \tag{10}\]  
Since for \(L=1\) cm, the term \(sinc^{2}(\Delta k\;L/2)\) in Equation (3) is sharply peaked around \(\Delta k=0\), \(\theta\) has simply been evaluated at the Cherenkov angle, while the integration on \(\phi\) has been performed  
Figure 4: (**a**) Infrared power coupled to the guided mode as a function of the incoming power for a channel waveguide (blue) and a planar waveguide (red). (**b**) The red curves represent the calculated nonlinear efficiency \(\eta\), with (solid) and without (dashed) the contribution of the transverse-optical phonon resonance at 7.5 THz. The blue curve is an example of the experimental nonlinear efficiency obtained with two different frequency scans. The low- and the high-frequency parts of the blue trace have been obtained by scanning the A1 amplifier from 1541 to 1563.5 nm and fixing the A2 wavelength at 1575 nm and 1605 nm, respectively. within the range allowed for transmission across the Si-air interface. The angular dependence has also been included to take into account the Fresnel losses at both the LN-Si and Si-air interfaces and in a multiplicative term \(\sin^{2}\phi\), in analogy with a linear dipole emission. In order to correctly reproduce the spectral profile of the measured efficiency, frequency dependence of \(n_{\text{{THz}}}\) and of the HR-Si have been considered. In particular, the dispersion curve for 5% MgO:LN has been obtained assuming the Sellmeier equation to be valid in the whole 1-8 THz spectral window, while the absorption spectrum of HR-Si has been retrieved from the THz-bridge spectral database [53]. The numerical result of such a calculation is represented by the red dashed line in Figure 4b.