# Organic Crystals for THz Photonics
## 2 Second-Order Organic Nonlinear Optical (NLO) Crystals
### NLO Effects in Organic Materials
A special case of sum-frequency generation is the well-known second-harmonic generation process with \(\omega_{1}=\omega_{2}=\omega\), where the generated frequency is twice the pump optical frequency \(\omega_{3}=2\omega\) and only a single-frequency pump optical field is required. The difference-frequency generation process, also called an optical parametric amplification process, in which generated frequency \(\omega_{3}\) presents the difference of the pump frequencies \(\omega_{1}-\omega_{2}\), can be explained by the absorption of the first photon at \(\omega_{1}\) to a virtual state and then a two-photon emission process, stimulated by the presence of the field at \(\omega_{2}\) (see Figure 2b). The field at \(\omega_{2}\) can be also spontaneously generated; in this case, the process is referred to as the optical parametric generation. An appropriate optical resonator can be added to an optical parametric generator to enhance the emission; this case is referred to as the optical parametric oscillator. A special case of difference-frequency generation is the optical rectification process, for which \(\omega_{1}\approx\omega_{2}\approx\omega\) and the generated field is a (quasi-)static field at zero (or close to zero) frequency \(\omega_{3}\approx 0\). THz-wave generation by difference-frequency generation or optical rectification results in a field at the THz frequency \(\omega_{3}\ll\omega_{1}\), \(\omega_{2}\) as discussed in more detail in Section 3. For THz-wave generation and electro-optic detection, the relevant processes involve static or quasi-static fields (including THz fields), i.e., oscillating at frequencies much lower than the optical frequencies. The electro-optic effect is most often described as a field-induced change of the refractive index \(n\) at \(\omega\) as  
\[\Delta\left(\frac{1}{n^{2}}\right)_{ij}=r_{ijk}E_{k}, \tag{3}\]  
where the usual summation convention holds and where \(r_{ijk}\) is the electro-optic tensor and can be related to the second-order susceptibility as  
\[r_{ijk}(\omega)=-\frac{2\chi_{ijk}^{(2)}(-\omega;\omega,0)}{n_{i}^{2}(\omega )n_{j}^{2}(\omega)}, \tag{4}\]  
where \(n_{i}(\omega)\) are the frequency-dependent refractive indices along the corresponding dielectric axis \(i\). The linear susceptibility tensor \(\chi^{(1)}\) and the related relative permittivity tensor \(\epsilon=\chi^{(1)}+1\) describe the common linear polarization response, which depends on the frequency of the applied field \(\mathbf{E}(\omega)\). Figure 3 schematically shows the frequency dependence of the relative permittivity \(\epsilon(\omega)\). The relative permittivity at low frequencies \(\epsilon^{T}\) (zero-stress relative permittivity) contains different contributions from the material polarization response: from acoustic phonon vibrations \(\epsilon^{a}\), from optical phonon vibrations \(\epsilon^{o}\), which also includes molecular vibrations in organic materials, and from electronic vibrations \(\epsilon^{e}\)  
\[\epsilon^{T}=\epsilon^{a}+\epsilon^{o}+\epsilon^{e}=\epsilon^{a}+\epsilon^{S}, \tag{5}\]  
where \(\epsilon^{S}=\epsilon^{o}+\epsilon^{e}\) is the clamped (zero-strain) relative permittivity and \(\epsilon^{e}\) determines the refractive index in the optical range as \(\epsilon^{e}=n^{2}\). Analogously, the nonlinear susceptibilities \(\chi^{(2)}\) and \(\chi^{(3)}\) also contain contributions from different possible material polarization responses [6; 21; 22]. The electrons can follow the fastest electric-field oscillations, therefore the electronic polarizability contributes to the linear and the nonlinear susceptibility up until the band-to-band transitions (electronic excitations). The optical phonon vibrations and the molecular vibrations contribute to the polarization response below about the THz/far-infrared range. The acoustic lattice vibrations can only follow relatively slow electric field oscillations below about the MHz range.