# Organic Crystals for THz Photonics
## 2 Second-Order Organic Nonlinear Optical (NLO) Crystals
### Organic Single Crystalline Materials
The commercial availability of these crystals has helped to considerably expand the research possibilities particularly for THz-wave generation with organic NLO crystals during the last couple of years, which has additionally motivated the continued research and development of novel and optimized organic NLO crystals for THz photonics applications. The relation between the macroscopic and the molecular NLO coefficients is complex because the interactions between the neighboring molecules influence the first hyperpolarizability \(\beta\) in the crystalline form [11; 31; 32]. However, often the macroscopic second-order nonlinearities of organic materials can be reasonably well explained by the nonlinearities of the constituent molecules by the so-called oriented-gas model [14]. This relation is in particular simple for the chromophores in the 1D approximation, for which only one dominant tensor element \(\beta=\beta_{zzz}\) can be considered, which is the case of most state-of-the-art organic NLO crystals. For example, the maximal diagonal element for the NLO susceptibility along the polar axis \(k\) of the crystal can be expressed as  
\[\chi^{(2)}_{kkk}=Nf_{\rm local}\ \beta^{\rm eff}_{kkk}=Nf_{\rm local}\langle \cos^{3}\theta_{kz}\rangle\ \beta_{zzz}. \tag{6}\]  
The above relation reveals the most important contributions to the NLO effect in organic materials: the chromophore number density \(N\), the local-field factor \(f_{\rm local}\), the molecular nonlinearity \(\beta_{zzz}\) along the main charge-transfer axis of the chromophore, and the orientational factor, the so-called acentric order parameter \(\langle\cos^{3}\theta_{kz}\rangle\), which is averaged over all molecules in a macroscopic system. For crystalline materials, the angle \(\theta_{kz}\) is the angle between the main charge transfer axis \(z\) of the chromophores and the polar axis \(k\) of the crystal, as illustrated in Figure 1b. Note that, for simplicity and practical relevance, we define here the polar axis \(k\) along the effective charge-transfer direction of the crystal, which may be, in low symmetry crystals (monoclinic and triclinic) slightly off (for highly nonlinear crystals, not more than a few degrees), the real polar axis defined by the effective permanent dipole moment of the crystal. The effective first hyperpolarizability coefficients \(\beta^{\rm eff}_{ijk}\) in the crystal can be introduced to consider the first hyperpolarizability tensor components \(\beta_{xyz}\) of all single chromophores in the crystalline unit cell projected to a suitable Cartesian coordinate system of the crystal based on the oriented-gas model as [14; 15]  
\[\beta^{\rm eff}_{ijk}=\frac{1}{n}\sum_{s}\sum_{xyz}\cos(\theta^{s}_{ix})\cos( \theta^{s}_{yy})\cos(\theta^{s}_{kz})\beta_{xyz}, \tag{7}\]where \(n\) denotes the number of chromophores in the unit cell, \(s\) the site in the unit cell and \(\theta_{im}^{s}\) are the angles between the dielectric axes \(i\) (\(i=1,2,3\)) of the crystal and the molecular axis \(m\) (\(m=x,y,z\)) of the chromophore. The macroscopic second-order susceptibility tensor components can be then evaluated as  
\[\chi_{ijk}^{(2)}=Nf_{\text{local}}\,\beta_{ijk}^{\text{eff}}. \tag{8}\]  
Note that both the effective first hyperpolarizability components \(\beta_{ijk}^{\text{eff}}\) as well as the local-field factors \(f_{\text{local}}\) strongly depend on the frequencies of the interacting electric fields. The effective first hyperpolarizability components vanish in case of a centrosymmetric crystalline arrangement (\(\beta_{ijk}^{\text{eff}}=0\)) like the alignment illustrated in Figure 1c. For perfectly parallel chromophore alignment along the \(k\)-axis as illustrated in Figure 1d and 1D chromophore approximation, the diagonal component along the polar axis is maximal \(\beta_{kkk}^{\text{eff}}=\beta_{zzz}\) and all other tensor components of \(\beta_{ijk}^{\text{eff}}\) are zero.