# Organic Crystals for THz Photonics
## 2 Second-Order Organic Nonlinear Optical (NLO) Crystals
### NLO Effects in Organic Materials
general we need to consider the full tensorial nature of the first hyperpolarizability \(\beta_{ijk}\). The first hyperpolarizability of NLO chromophores can be evaluated theoretically using quantum chemical calculations [6,15] or measured experimentally, mostly using the Electric Field Induced Second Harmonic Generation (EFISH) [16] or the hyper-Rayleigh scattering (HRS) technique [17]. The macroscopic NLO material response is in general (also for inorganic materials) described using an analogous power series for the induced material polarization \(\mathbf{P}\) as a function of the applied electric field \(\mathbf{E}\)  
\[P_{i}=e_{0}\chi_{ij}^{(1)}E_{j}+e_{0}\chi_{ijk}^{(2)}E_{j}E_{k}+e_{0}\chi_{ijkl }^{(3)}E_{j}E_{k}E_{l}+\ldots, \tag{2}\]  
where \(\chi^{(1)}\) is the linear susceptibility, while \(\chi^{(2)}\) and \(\chi^{(3)}\) are the nonlinear susceptibilities of the second and the third order, respectively. The susceptibility tensor coefficients are all dependent on the frequency \(f\) (or the angular frequency \(\omega=2\pi f\)) of the involved electric fields \(\mathbf{E}\). As seen in Equation (2), the NLO effects of the second-order, which are of interest in this review, allow interaction of two electric fields \(\mathbf{E}(\omega_{1})\) and \(\mathbf{E}(\omega_{2})\) to induce a macroscopic polarization at a combination of their frequencies \(\omega_{3}\), which is the source of the generated electric field \(\mathbf{E}(\omega_{3})\). Figure 2 shows schematic illustrations of two basic second-order NLO processes: sum-frequency generation and difference frequency generation. In the sum-frequency generation process, the generated frequency is the sum of the two input frequencies \(\omega_{3}=\omega_{1}+\omega_{2}\). From the quantum optics point of view, an NLO material allows interactions between photons, so that the total photon energy is conserved \(\hbar\omega_{3}=\hbar\omega_{1}+\hbar\omega_{2}\) and the total photon momentum is conserved \(\hbar\mathbf{k}_{3}=\hbar\mathbf{k}_{1}+\hbar\mathbf{k}_{2}\) (\(\hbar\) is the reduced Planck constant and \(\mathbf{k}_{i}\) the wave vector of the corresponding optical wave). The momentum conservation is in the field of NLO commonly known as the phase-matching condition. In Figure 2c, a general non-collinear phase-matching geometry is presented, while practically often a collinear geometry is preferred, in which all wave vectors are parallel. Any phase mismatch decreases the efficiency of the NLO conversion process. Phase matching requirements relevant for THz-wave generation are discussed in Section 3.4, while the reader is referred to general books on NLO for phase-matching approaches for other processes (see e.g., Refs. [18, 19, 20]). The schematic energy-level diagram in Figure 2b illustrates the photon energy conservation with the electronic ground state indicated by the solid line and the virtual states indicated by the dashed lines. The corresponding \(\chi^{(2)}\) NLO susceptibility depends on the frequencies of the involved fields, depending on how close these virtual levels are to the real electronic levels. In case one of the virtual levels is close to a real level, the NLO susceptibility is said to be resonantly enhanced. The frequency dependence of the NLO susceptibility for sum-frequency generation is here denoted as \(\chi^{(2)}(-\omega_{3};\omega_{1},\omega_{2})\), where \(\omega_{3}\) is the frequency of the generated electric field. This description holds for the so-called parametric NLO processes, for which the photon energy is conserved and there is no electronic transition within the material involved. For non-parametric  
Figure 2: (**a**) schematics of the two basic parametric second-order NLO processes: sum-frequency generation and difference-frequency generation with (**b**) the corresponding energy-level diagram and (**c**) wavevectors \(\mathbf{k}_{i}\) that satisfy the phase-matching condition. processes such as, e.g., the two photon-absorption, the imaginary part of the NLO susceptibility should also be considered [19].