## 2 Principle and System Structure  
### Principle of GVD Compensation  
GVD compensation technology is very mature; the commonly used schemes are mainly divided into two types, including chirped fiber grating compensation and solid grating-pair compensation techniques. Among the two compensation schemes, chirped fiber grating introduces a larger dispersion, and is usually used to compensate for the long transmission fibers (several kilometers) in communication systems. By contrast, parallel grating pairs introduce a smaller dispersion and are more suitable for the GVD compensation of short transmission fibers. The length of the transmission optical fiber used in the system is 50-200 m, the dispersion coefficient is 18.7 ps\({}^{2}\)/km, and the introduced dispersion is approximately 0.94-3.74 ps\({}^{2}\). Solid gratings are more stable and less susceptible to environmental influences than fiber gratings; therefore, the solid grating was chosen for the GVD compensation scheme. The principle of parallel grating-pair compensation is illustrated in Figure 1. The incident light was dispersed by the grating. By selecting an appropriate incident angle and grating parameters, the long-wavelength component takes longer to pass through the grating pair than the short-wavelength component. Thus, the long-wavelength component is output after the short-wavelength component on passing through the grating pair; in other words, a negative GVD is introduced to offset the positive GVD introduced by the transmission fiber to realize compensation. As shown in Figure 1, \(\gamma\) is the angle of incidence, \(\alpha\) is the angle of diffraction, and \(G\) is the vertical spacing of the grating pair. The path of a monochromatic light with a frequency component \(\omega\) in the grating pair is [23]  
\[\mathrm{x}(\omega)=\frac{G\{1+\cos[\gamma-\alpha(\omega)]\}}{\cos[\alpha( \omega)]} \tag{1}\]  
where \(\alpha(\omega)\) denotes the frequency functions satisfying the grating equation  
\[\sin\gamma+\sin[\alpha(\omega)]=\frac{2\pi c}{wd} \tag{2}\]  
When the signal light is diffracted at different positions of the grating, the phase delay satisfies:  
\[\varnothing(\omega)=\Big{(}\frac{\omega}{c}\Big{)}x(\omega)-\frac{2\pi G}{d} \tan[\alpha(\omega)] \tag{3}\]  
where \(c\) is the speed of light and d is the grating constant. For nanosecond pulses, the effect of higher-order dispersion can usually be neglected and only the second-order dispersion needs to be considered. By substituting Equation (1) into Equation (3) and performing derivation, the second-order dispersion \(D\) introduced by the parallel grating pairs can be obtained  
\[D=\frac{G\lambda}{c\cos[\alpha(\lambda)]^{3}d^{2}} \tag{4}\]  
where \(\lambda\) is the wavelength. The amount of dispersion can be tuned by varying the grating constant \(d\), grating-pair spacing \(G\), and angle of incidence \(\gamma\) for parallel grating pairs. For nanosecond pulses, although GVD does not cause significant pulse spreading, the strong GVD induced by long transmission fibers in a system can lead to severe amplitude and frequency modulation, resulting in waveform distortion. After long fiber transmission, the output optical power \(P_{2}\) can be expressed as:  
\[P_{2}=P_{1}\Big{\{}1-\beta_{2}L\Big{[}m_{1}{\omega_{1}}^{2}\sin(\omega_{1}T)+m _{2}{\omega_{2}}^{2}\sin(\omega_{2}T)\Big{]}\Big{\}} \tag{5}\]  
where \(P_{1}\) is the input optical power, \(L\) is the fiber length, \(m_{1}\) and \(m_{2}\) are the depths of the two levels of phase modulation; \(\omega_{1}\) and \(\omega_{2}\) are the modulation frequency of the two levels of phase modulation; \(\beta_{2}\) is the GVD; and T is the relative time of the pulse envelope  
Figure 1: Principle diagram of dispersion induced by parallel grating pairs. of the GVD. From Equation (5), the frequency modulation is partially converted into the amplitude modulation of the power. The amplitude modulation takes the same functional form as the frequency modulation, and the modulation depth increases with \(\beta_{2}L\).