# Organic Crystals for THz Photonics
## 3 THz-Wave Generation and Detection via NLO Processes
### Optical Rectification  
While for difference-frequency generation two pump-optical beams at closely spaced narrow single frequencies are used, THz-wave generation by optical rectification employs a single pump pulse of a very short pulse length. An ultrashort laser pulse in the femtosecond range induces a quasi-static polarization in \(\chi^{(2)}\)-NLO materials through optical rectification, which acts as a source of a THz field. A short laser pulse has an intrinsically broad bandwidth, i.e., a transform-limited laser beam with a pulse length of 10 fs or 200 fs has a bandwidth of roughly 40 THz or 2 THz, respectively. Different frequency components in such a pulse can mix with each other in an NLO material, producing a broadband THz pulse. In first approximation, considering non-depleted pump, negligible optical and THz absorption, plane-wave approximation and at perfect phase matching, the intensity of the generated THz wave is in frequency domain given by [6; 112]  
\[I(\omega_{\rm THz},L) = \frac{1}{2}\epsilon_{0}cn_{\rm THz}|E(\omega_{\rm THz},L)|^{2} \tag{19}\] \[\approx \frac{d_{\rm THz}^{2}I_{0}^{2}\omega_{\rm THz}^{2}L^{2}}{2 \epsilon_{0}c^{3}n_{o}^{2}n_{\rm THz}} \tag{20}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \(n_{o}\) & \(n_{g}\) & \(n_{\rm THz}\) & \(r\) **(pm/V)** & \(d_{\rm THz}\) **(pm/V)** & \({\rm FM}_{\rm THz}\) **(pm/V)\({}^{2}\)** & \(\lambda\) **(nm)** \\ \hline DAST & 2.13 & 2.26 & \(\sim\) 2.3 & 47 & 240 & 5600 & 1500 \\ DSTMS & 2.08 & 2.19 & \(\sim\) 2.2 & 50 & 230 & 5800 & 1500 \\ OH1 & 2.16 & 2.33 & \(\sim\) 2.3 & 52 & 280 & 7500 & 1300 \\ ZnTe & 2.85 & 3.2 & \(\sim\) 3.2 & 4 & 66 & 170 & 800 \\ GaP & 3.12 & 3.36 & \(\sim\) 3.35 & 1 & 24 & 17 & 1000 \\ LiNbO\({}_{3}\) & 2.16 & 2.22 & \(\sim\) 5.0 & 30 & 160 & 1100 & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Selection of organic and inorganic NLO materials that have been investigated for THz-wave generation and their most relevant parameters. Where possible, the parameters are given close to the optimal phase matching range: refractive index \(n_{o}\) and the group index \(n_{g}\) at the pump optical wavelength \(\lambda\); refractive index \(n_{\rm THz}\) in the THz range; the electro-optic coefficient \(r\); the NLO coefficient \(d_{\rm THz}\) for THz-wave generation according to Equation (17); the figure of merit FM\({}_{\rm THz}\) for THz-wave generation according to Equation (18). and the conversion efficiency by  
\[\eta=\frac{I(\omega_{\rm THz},L)}{I_{0}}=\frac{d_{\rm THz}^{2}I_{0}\omega_{\rm THz }^{2}L^{2}}{2\epsilon_{0}c^{3}n_{g}^{2}n_{\rm THz}}=\frac{1}{2\epsilon_{0}c^{3 }}{\rm FM}_{\rm THz}\omega_{\rm THz}^{2}I_{0}L^{2}, \tag{21}\]  
where the same figure of merit FM\({}_{\rm THz}\) as the one defined for difference-frequency generation in Equation (18) and listed in Table 2 is relevant. An analytical solution has been also derived for a more general case, considering both optical and THz-wave absorption and non-perfect phase matching, but still considering a non-depleted pump approximation [112; 113]. In the plane-wave approximation and neglecting surface contributions, the generated electric field \(E_{\rm THz}(\omega,\lambda,L)\) at the angular THz frequency \(\omega_{\rm THz}=\omega\) pumped by an optical pulse with the central wavelength \(\lambda\) is given by [112; 113]  
\[E_{\rm THz}(\omega,\lambda,L)=\frac{2d_{\rm THz}\omega^{2}I(\omega)}{\left( \frac{\omega}{c}(n_{\rm THz}+n_{g})+i(\alpha_{\rm THz}/2+\omega)\right)n \epsilon_{0}c^{3}}\cdot\frac{e^{i\left(\frac{\omega}{c}n_{\rm THz}+i\alpha_{ \rm THz}/2\right)L}-e^{i\left(\frac{\omega}{c}n_{g}+i\alpha\right)L}}{\frac{ \omega}{c}(n_{\rm THz}-n_{g})+i(\alpha_{\rm THz}/2-\alpha)}, \tag{22}\]  
where \(I(\omega)\) is the Fourier transform of the pump optical pulse \(I(t)\) and \(L\) the material thickness.