## 2 Theory  
DFG is a three-wave-mixing process, basically taking advantage of the anharmonic response of a nonlinear medium to an external dual frequency field. Let us consider a laser beam resulting on the superposition of two monochromatic waves at angular frequencies \(\omega_{1}\) and \(\omega_{2}\) (whose difference lies in the THz range \(\omega_{THz}=\omega_{1}-\omega_{2}\)), where their propagation direction is indicated as \(x\). When it passes through a non-centrosymmetric nonlinear medium, it excites a second-order polarization field \(P^{(2)}\), which acts as a source of new spectral components and, in particular, of a new field at angular frequency \(\omega_{THz}\). In the most general picture, the three waves \(E(\omega_{1})\), \(E(\omega_{2})\), and \(E(\omega_{THz})\) are coupled by the three nonlinear wave equations:  
\[\
abla^{2}E\left(\overset{\rightarrow}{r},\omega_{i}\right)+\frac{\omega_{i} ^{2}n^{2}(\omega_{i})}{c}\ E\left(\overset{\rightarrow}{r},\omega_{i}\right)= -\mu_{0}\omega_{i}^{2}P^{(DFG)}\left(\overset{\rightarrow}{r},\omega_{i} \right),\hskip 28.452756 pt(i=1,\ 2,\ THz), \tag{1}\]  
where \(\mu_{0}\) is the magnetic permeability of the vacuum, and \(c/n(\omega)\) is the phase velocity of the electric field Fourier component at the angular frequency \(\omega\). When the conversion efficiency is sufficiently low and the field amplitude does not vary appreciably within one wavelength (slow-varying-envelope approximation), one can assume the two optical waves to be independent on the THz one. Under these assumptions, the THz field can be described by the nonlinear equation:  
\[\frac{dE_{THz}}{dx}=-i\frac{\omega_{THz}d_{eff}}{n_{THz}c}E_{1}(y,z)E_{2}^{*} (y,z)\exp(-i\Delta kx), \tag{2}\]  
where \(d_{eff}\) is the effective nonlinear coefficient of the medium, and the wavevectors' mismatch \(\Delta k\) depends on the refractive indexes for the optical pump fields \(n_{pump}\overset{\mathrm{def}}{=}n_{1}=n_{2}\), the refractive index for the THz wave is \(n_{THz}\), and the angle \(\theta\) is formed by the emitted THz wave with respect to the \(x\) direction (see Figure 1a): \(\Delta k=\frac{\omega_{THz}}{c}\big{(}n_{pump}-n_{THz}cos\theta\big{)}\). Assuming the two pump beams have a Gaussian shape with a \(r_{0}\) radius at the 1/e level, the nonlinear wave equation can be easily integrated over the propagation length within the nonlinear medium \(L\), leading to a nonlinear efficiency \(\eta\) (defined by the ratio \(\frac{P_{THz}}{P_{1}P_{2}}\) between the THz power and the product of the pump beam powers). \[\eta=\frac{\mu_{0}L^{2}d_{eff}^{2}}{c\pi^{2}n_{THz}n_{pump}^{2}r_{0}^{2}}\cdot g (\omega_{THz},r_{0})\cdot sinc^{2}\left(\frac{\Delta k\,L}{2}\right), \tag{3}\]  
where:  
\[g(\omega_{THz},\ r_{0})=\omega_{THz}^{2}\exp\left(-\frac{\omega_{THz}^{2}r_{0 }^{2}}{4c^{2}}\Big{(}n_{THz}^{2}-n_{pump}^{2}\Big{)}\right). \tag{4}\]  
As it can be seen, unless a perfect phase-matching condition (\(\Delta k=0\)) is achieved, the THz generation process will take place only within a finite thickness of the nonlinear medium. It is given by the so-called coherence length \(L_{coh}=2\pi/\Delta k\), depending on the optical features of the material and on the wave vectors' geometry. In order to overcome this problem, different techniques have so far been developed. For example, perfect phase-matching in a collinear geometry (i.e., for \(\theta=0\)) can be achieved by taking advantage of the velocity mismatch experienced by non-parallel-polarized pump fields in birefringent materials [38]. To maximally exploit the nonlinear properties of certain materials, a parallel arrangement of the three involved polarizations is often required. This is the case for Lithium Niobate (LN), which is one of the best-suited materials for THz DFG because of its strong nonlinearities [39; 40], high photorefractive damage threshold [41; 42], and small absorption coefficient, in the order of \(10^{-4}\)cm\({}^{-1}\) in the infrared spectral region [43].