# Germanium Single Crystals for Photonics
## 7 Other Nonlinear Effects in Germanium  
When exposed to intense laser radiation, other nonlinear effects are also observed in Ge [2]. The optical constants of the materials are determined by the well-known equation:  
\[P=\varepsilon_{0}(\chi(1)E+\chi(2)E+\chi(3)E+\ldots), \tag{3}\]  
where \(P\) is the electric polarization induced by the electromagnetic wave; \(E\) is the field strength; \(\chi\) is the susceptibility tensor of the medium.  
In linear optics, only the first term of this equation is taken into account, but when working with powerful laser radiation, it is no longer possible to neglect nonlinear effects determined by other terms of the equation. For materials with a diamond structural type (in particular, for Ge), the coefficient \(\chi(2)=0\). However, the next coefficient \(\chi(3)=1.5\cdot 10^{-10}\) CGSE units (for germanium). Therefore, it was possible to use Ge single crystals for efficient wave front reversal [87]. In a parallel-plane plate made of Ge, installed inside the resonator of a CO\({}_{2}\) laser, phase-conjugate reflection was recorded at \(\lambda=10.6\)\(\upmu\)m from phase gratings arising in counter-propagating waves. The magnitude of the reflection coefficient reached 20%. The laser beam reflected by such a mirror returns along the same optical path. The use of wave front reversal allows one to obtain diffraction divergence of high-power laser radiation, despite the presence of inhomogeneities in the active medium, in optical elements and in the atmosphere.  
One of the possible applications of germanium single crystals in photonics at present is its use for stabilizing the emission frequency of IR laser diodes by generating optical frequency combs (OFCs) based on compact optical microresonators. Optical frequency combs are a set of narrow equidistant spectral lines; generation is possible using both complex setups based on mode-locked femtosecond lasers [88] and on spherical or disk optical microresonators with <<whispering gallery>> modes (WGM resonators) [89]. Such WGM resonators have small dimensions and a high-quality factor. The created optical frequency combs have a free dispersion region from units to hundreds of gigahertz and a fixed distance between the lines due to phase synchronization. Microresonators are used to stabilize laser radiation and generate photon pairs for quantum cryptography and communications. Based on microresonators, we can create devices and instruments operating in various fields of science and technology: spectroscopy, communications, telecommunications, navigation, and metrology (creation of reference sources of optical and radio frequencies), etc. [90; 91]. The OFC generation effect occurs in optically nonlinear  
Figure 9: Dependence of the concentration of nonequilibrium free carriers generated in the process of two-photon absorption on the depth (calculated at the time: t–200 ns).  
materials when pumped by a continuous laser. In this case, due to the Kerr effect, so-called four-wave mixing occurs, because of which the photons are redistributed between the modes, and an OFC is generated.  
Microresonators (1.35 and 1.5 mm-thick) were fabricated based on high-purity germanium (HPGe) [92]. Experimental setups were assembled, which provided excitation of \"whispering gallery\" modes in microresonators at a wavelength of 2.68 \(\upmu\)m using various coupling elements, such as rectangular prisms and hemispheres. A record quality factor was measured, which is (2.2 \(\pm\) 0.3)\(\cdot\)10\({}^{7}\) and is determined by radiation losses in germanium. It was found that high-quality modes in germanium microresonators provide impulse tightening of a laser diode with a tightening range width up to 100 MHz. The high-quality factor of germanium microresonators indicates the possible use of such devices for stabilizing the frequency of lasers in the IR region of the spectrum.