# Organic Crystals for THz Photonics
## 2 Second-Order Organic Nonlinear Optical (NLO) Crystals
### NLO Effects in Organic Materials  
NLO organic materials are based on the nonlinearity of the constituting molecules. An electric field \(\mathbf{E}\) induces a charge redistribution in an organic molecule. This induces a dipole moment \(\mathbf{p}\) (in addition to the permanent dipole moment \(\mathbf{\mu}\) of a molecule), which is most commonly described by the power series as a function of the electric field \(\mathbf{E}\)  
\[p_{i}=\epsilon_{0}\alpha_{ij}E_{j}+\epsilon_{0}\beta_{ijk}E_{j}E_{k}+\epsilon _{0}\gamma_{ijkl}E_{j}E_{k}E_{l}+\ldots, \tag{1}\]  
where the usual [PERSON] summation convention holds and where \(\epsilon_{0}\) is the electric constant, \(\alpha_{ij}\) the linear polarizability, \(\beta_{ijk}\) the first hyperpolarizability and \(\gamma_{ijkl}\) the second hyperpolarizability of the molecule. Note that, in this review, we use, unless explicitly stated, SI units and the above defined power series expansion, which is the same as in Ref. [3]. Note that several different formulations are used for the above equation in both SI and cgs unit systems, so care must be taken when comparing the results from different sources [3; 4; 5]. While all molecules exhibit \(\alpha_{ij}\) and \(\gamma_{ijkl}\), the first hyperpolarizability \(\beta_{ijk}\) is responsible for the asymmetric response to an external field and is only possible in asymmetric molecules. For molecules with non-zero \(\beta_{ijk}\), the second-order NLO effects such as second-harmonicgeneration, electro-optic effect, optical rectification, and THz-wave generation are possible and are of interest in this review. The conventional design of organic molecules to achieve a high first hyperpolarizability \(\beta_{ijk}\) is based on a \(\pi\)-conjugated bridge with strong donor and acceptor substituents at the end of the molecule to promote the asymmetric response and has been extensively studied [6; 7; 8; 9; 10; 11; 12; 13]. Figure 1a illustrates the simple physical mechanism responsible for the NLO response of an asymmetric molecule (\(\beta_{ijk}\
eq 0\)) to an external electric field: the electronic cloud favors the acceptor A over the donor D. As a consequence, the amplitude of the induced dipole moment depends on the polarity of the applied electric field. Note that the NLO response is usually significant only for very strong electric fields E.  
Many of the state-of-the-art NLO molecules only possess one dominant component of the \(\beta_{ijk}\) tensor along the main charge-transfer axis \(z\), the component \(\beta_{zzzz}\), while all other components are close to zero, which is the so-called 1D-NLO-chromophore approximation [14]. Note that other molecular-design approaches such as the octupolar approach are also being investigated, so that in  
Figure 1: Schematics of: (**a**) induced dipole moment \(p(t)\) of a donor(D)-acceptor(A) substituted \(\pi\)-conjugated nonlinear optical (NLO) molecule (chromophore) upon application of an external sinusoidal electric field \(E(t)\); (**b**) NLO molecules in the simple oriented-gas model. The molecular-ordering angle \(\theta_{kz}\) is the angle between the charge-transfer axis \(z\) of the molecule and the polar axis \(k\) of the crystal; (**c**) an example of a centrosymmetric crystal structure with \(\theta_{kz}=90^{\circ}\), resulting in zero macroscopic second-order NLO susceptibility \(\chi^{(2)}\). Examples of non-centrosymmetric structures with (**d**) perfectly parallel chromophores with \(\theta_{kz}=0^{\circ}\), maximizing the diagonal component of the \(\chi^{(2)}\) tensor and with (**e**) zig-zag-chromophore orientation with \(0<\theta_{kz}<90^{\circ}\) with both diagonal and non-diagonal components of \(\chi^{(2)}\); (**f**) the acentric order parameter for the diagonal (\(\cos^{3}\theta_{kz}\)) and non-diagonal (\(\cos\theta_{kz}\sin^{2}\theta_{kz}\)) components of \(\chi^{(2)}\) as a function of \(\theta_{kz}\) for simplest 1D-chromophores with one dominant first hyperpolarizability component \(\beta_{zzzz}\).