# Organic Crystals for THz Photonics
## 3 THz-Wave Generation and Detection via NLO Processes
### Difference-Frequency Generation  
To induce a nonlinear polarization oscillating at a THz (angular) frequency \(\omega_{\rm THz}\) based on difference-frequency generation, the interacting electric fields \(\mathbf{E}(\omega_{1})\) and \(\mathbf{E}(\omega_{2})\) should oscillate at closely spaced frequencies \(\omega_{1}\) and \(\omega_{2}\), so that their difference \(\omega_{3}=\omega_{\rm THz}=\omega_{1}-\omega_{2}\) is in the THzfrequency range. In order to efficiently generate \(\omega_{\rm THz}\), the phase matching condition \(\Delta\mathbf{k}\approx 0\) should be possibly well satisfied  
\[\Delta\mathbf{k}=\mathbf{k}_{\rm THz}-(\mathbf{k}_{1}-\mathbf{k}_{2}), \tag{12}\]  
where \(\mathbf{k}_{\rm THz}\) is the wave vector of the THz-wave with the magnitude of \(|\mathbf{k}_{\rm THz}|=2\pi\,n_{\rm THz}/\lambda_{\rm THz}\) and \(\mathbf{k}_{i}\) (\(i=1,2\)) are the wave vectors of the interacting optical fields with the magnitude of \(|\mathbf{k}_{i}|=2\pi\,n_{i}/\lambda_{i}\). \(n_{\rm THz}\) is the refractive index of the NLO material at the wavelength \(\lambda_{\rm THz}\) of the generated THz field, while \(n_{i}\) (\(i=1,2\)) are the refractive indices of the NLO material at the wavelengths \(\lambda_{i}\) (\(i=1,2\)) of the two interacting fields \(\mathbf{E}(\omega_{1})\) and \(\mathbf{E}(\omega_{2})\), respectively. The phase matching condition is often quantified by the so-called coherence length \(l_{c}\), which defines the maximum length of the NLO material, over which the generated field increases with the thickness  
\[l_{c}=\frac{\pi}{|\Delta\mathbf{k}|}. \tag{13}\]  
In the simplest case of the collinear difference-frequency generation, the three interacting waves propagate in the same direction and the corresponding wave vectors are all parallel. The resulting coherence length is then given by  
\[l_{c}=\frac{1}{2}\left(\frac{n_{\rm THz}}{\lambda_{\rm THz}}-\frac{n_{1}}{ \lambda_{1}}+\frac{n_{2}}{\lambda_{2}}\right)^{-1}. \tag{14}\]  
This expression can be additionally simplified for relatively small differences in frequencies of the two pump fields (up to several THz) and in case the dispersion of the refractive index is relatively small, e.g., like in the infrared range for the organic crystals shown in Figure 5a. In this case, the dispersion in the optical range can be considered as \(n_{2}=n_{1}+(\partial n/\partial\lambda)(\lambda_{2}-\lambda_{1})\). This leads to the following coherence length for THz-wave generation  
\[l_{c}=\frac{\lambda_{\rm THz}}{2(n_{\rm THz}-n_{g})}, \tag{15}\]  
where \(n_{g}=n_{1}-(\partial n/\partial\lambda)\lambda_{1}\) is the group index of the optical wave. For efficient generation of THz waves by difference-frequency generation, besides a high \(\chi^{(2)}\) material susceptibility, the most important parameter is the low refractive index mismatch \(\Delta n=n_{\rm THz}-n_{g}\) between the generated THz and the pump optical waves. Organic materials have an important advantage compared to standard inorganic NLO materials such as LiNbO\({}_{3}\) because of the relatively weak dispersion of the refractive index between the optical and the THz frequency range, as discussed in Section 2.1. The phase-matching condition for THz-wave generation is in organics almost naturally satisfied, while for inorganic NLO crystals such as LiNbO\({}_{3}\), more complex interacting-beam orientations are needed, e.g., tilted-pulse-front configuration [111]. The particular coherence lengths \(l_{c}\) for organic NLO crystals are discussed in Section 3.4.