# Organic Crystals for THz Photonics
## 2 Second-Order Organic Nonlinear Optical (NLO) Crystals
### Organic Single Crystalline Materials
For an orientation with \(0<\theta_{kz}<90^{\circ}\) and in 1D chromophore orientation as illustrated in Figure 1e, the maximal diagonal effective first hyperpolarizability component is \(\beta_{kkk}^{\text{eff}}=\beta_{zzz}\cos^{3}\theta_{kz}\) and the maximal off-diagonal effective first hyperpolarizability component is \(\beta_{ikk}^{\text{eff}}=\beta_{zzz}\cos\theta_{kz}\sin^{2}\theta_{kz}\). The dependence of maximal diagonal and maximal off-diagonal components of \(\beta_{ijk}^{\text{eff}}\) are illustrated in Figure 1f. Note that in case the chromophores are additionally distributed in the azimuthal angle around the polar axis, another projection needs to be considered for different off-diagonal effective first hyperpolarizability components [14]. As seen in Figure 1f, to maximize the diagonal NLO susceptibility \(\chi_{kkk}^{(2)}\) along the polar axis \(k\), the projection angles \(\theta_{kz}\) should be close to zero, i.e., the charge transfer axes of the molecules should be close to parallel. To maximize the off-diagonal susceptibility elements \(\chi_{ikk}^{(2)}\), the molecules should make an angle of about \(54.7^{\circ}\) with respect to the polar axis [14]. For electro-optic and THz photonics applications, for the majority of cases, the large diagonal NLO coefficient is desired and therefore the molecular orientation with small or zero \(\theta_{kz}\) is optimal. The dispersion of the macroscopic NLO susceptibilities is usually modeled using Equation (8), by assuming a simple two-level model [33] for the dispersion of first hyperpolarizabilities \(\beta_{zzz}\) and local field factors \(f_{\text{local}}\)[8; 9]. The dispersion depends on the frequencies of all involved interacting electric fields, therefore the NLO susceptibilities are different for different NLO processes. For the linear electro-optic effect, the dispersion of the dominant first hyperpolarizability component is in the two-level model given by  
\[\beta_{zzz}(-\omega;\omega,0)=\frac{\omega_{eg}^{2}(3\omega_{eg}^{2}-\omega^ {2})}{3(\omega_{eg}^{2}-\omega^{2})^{2}}\beta_{0} \tag{9}\]  
and for the second-harmonic generation by  
\[\beta_{zzz}(-2\omega;\omega,\omega)=\frac{\omega_{eg}^{4}}{(\omega_{eg}^{2}-4 \omega^{2})(\omega_{eg}^{2}-\omega^{2})}\beta_{0}, \tag{10}\]  
where \(\omega_{eg}\) denotes the resonant angular frequency of the electronic transition from the ground (\(g\)) to the excited (\(e\)) state and a dispersion-free first hyperpolarizability \(\beta_{0}\) presents the first hyperpolarizability extrapolated to infinite wavelengths (\(\omega\to 0\)), away from electronic resonances. The simple two-level model describes well the dispersion away from electronic resonances. More sophisticated models are available for modeling the dispersion closer or at electronic resonances such as the homogeneously damped two-level model [34; 35], the homogeneously broadened vibronic model [34; 36] and the inhomogeneously broadened vibronic model [36]. The dispersion-free first hyperpolarizability \(\beta_{0}\) is a very useful parameter when comparing the nonlinearity of molecules with different resonant frequencies \(\omega_{eg}\). In the conventional two-level model [33], the dispersion-free first hyperpolarizability \(\beta_{0}\) depends on \(\omega_{eg}\), the difference between the excited \(\mu_{e}\) and the ground-state \(\mu_{g}\) dipole moments and the transition dipole moment \(\mu_{cg}\) as  
\[\beta_{0}=\frac{3}{2\varepsilon_{0}\hbar^{2}}\frac{\Delta\mu\,\mu_{cg}^{2}}{ \omega_{cg}^{2}}. \tag{11}\]  
The inverse-square dependence of \(\beta_{0}\) on \(\omega_{cg}\) is often referred to as the nonlinearity-transparency trade-off: the chromophores with a red-shifted absorption maximum (lower transmission in the visible wavelength range, lower \(\omega_{cg}\)) typically present a higher \(\beta_{0}\). This is strongly related to the best phase-matching range for THz-wave generation with state-of-the-art organic materials, as discussed later on.