# Analysis and Reduction of Nonlinear Effects in Optical Fiber Frequency Transfer
## 2 Fundamental Principle  
Figure 1 illustrates the system schematic of high-precision optical frequency signal transmission based on a fiber optic link. The optical frequency signal is generated by a laser that can output continuous laser light and is injected into the fiber optic link. At the transmitter end of the system, the optical signal is split into two paths, with one path serving as the reference light and the other path being modulated by Bi-Actuator 1 before being  
transmitted through the fiber optic link to the remote end, where it is further modulated by Bi-Actuator 2. At the remote end, a portion of the optical signal is reflected back to the transmitter end by a reflective mirror along the same path, while another portion is provided to the user. The light that is reflected back and the reference light at the transmitter end undergo beat frequency comparison to extract phase noise information introduced by the transmission system. This information is then subjected to noise suppression through a noise feedback control system to ensure that the transmitted light at the user end exhibits the same characteristics as the light source. As shown in Figure 2, laser signals transmitted in optical fibers can result in nonlinear effects due to non-elastic collisions [24,29,30]. Depending on the optical intensity, two different nonlinear effects can occur: stimulated Raman scattering and stimulated Brillouin scattering. Both of these nonlinear effects are intensity dependent, but the optical intensity threshold for stimulated Brillouin scattering is significantly lower compared to Raman scattering. Even at lower power levels, Brillouin scattering can occur spontaneously, as it is driven by the scattering of phonons generated thermally. Higher input optical power leads to the generation of a large number of phonons, and when two counter-propagating optical waves in the fiber intersect, they create a moving refractive index grating. The greater the reflected optical power, the stronger the grating, resulting in higher effective reflectivity. When the incident optical power exceeds a certain threshold, stimulated Brillouin phonons can scatter a significant portion of the incident optical power back to the input end of the optical fiber. For silica optical fibers, the material itself exhibits relatively weak nonlinear effects, but the combination of a small effective mode area and long propagation length in the optical fiber greatly enhances these nonlinear effects. The frequency of the reflected beam generated by stimulated Brillouin scattering is slightly lower than that of the incident beam. It corresponds to the frequency \(v_{B}\) of phonons, and the corresponding frequency shift is called the Brillouin frequency shift. The frequency  
Figure 1: Schematic of the basic principle of optical frequency transmission via fiber with the active phase-noise compensation system. Laser Source and Phase-Servo System represent continuous laser light and the active phase-noise compensation system, respectively. Bi-Actuator denotes bidirectional modulator. Figure 2: Schematic of the basic principle of nonlinear effects in optical fiber frequency transfer [31]. \(\lambda_{0}\) is the center wave length of the transmitted light. shift is determined by the phase matching condition, and it can be calculated using the refractive index \(n\), the speed of sound \(v_{a}=6\times 10^{3}\) m/s, and the vacuum wavelength \(\lambda_{0}\). The frequency \(v_{B}\) can be expressed as [24; 32; 33]:  
\[v_{B}=\frac{2 nv_{a}}{\lambda_{0}} \tag{1}\]  
The Brillouin frequency shift depends on the material composition and, to some extent, the temperature and pressure of the medium. For silica optical fibers, the Brillouin frequency shift is approximately 11 GHz. The intrinsic bandwidth of Brillouin gain is typically between 10 and 100 MHz, and the linewidth \(\Delta v_{B}\) is determined by the phonon lifetime \(t\), where \(\Delta v_{B}=1/2\pi t\). In theory, phonon absorption leads to a shorter phonon lifetime and thus a narrower linewidth.