## 4 Discussion  
Although \(S_{k}\) and \(\delta_{k}\) have similar phase compensation effects, the results show that parameter combination (a) performs better, and \(x_{comb}=(X_{k}+X_{k-1})/2\) is the optimal coordinate for phase compensation. The reason for the phenomenon will be discussed. First, let us focus on the performance differences with different parameter combinations. In parameter combination (b) and (c), \(S_{k}\) varies with \(f_{k-1}\), as expressed in Equation (22). For the VLS grating example in this paper, \(S_{k}\) increases with \(k\). With an increasing \(S_{k}\), the phase inconsistency from adjacent scans has a greater impact on the phase error and exposure contrast. Therefore, the overall performances with parameter combination (b) and (c) decrease with increasing \(S_{k}\), and a constant \(S_{k}=S_{c}\) maintains a consistent performance with combination (a). Second, the selection of \(x_{comb}\) and the cause of the cumulative characteristic in \(\Phi_{c}(x)\) with \(x_{comb}\
eq(X_{k}+X_{k-1})/2\) are discussed. As shown in Equation (9), after the phase difference at \(x_{comb}\) is compensated with \(\delta_{k}\) or \(S_{k}\), the residual phase in \(\varphi_{bklx}(x)\) is expressed as  
\[ \begin{array}{ll}\varphi_{bkl\_res}(x)&=\varphi_{bklx}(x)-\varphi_{bklx}(x_{ comb})\\ &=2\pi\Delta_{k}(x-x_{comb})\end{array}  \tag{26}\]  
As shown in Figure 1, the overall phase compensation performance in the \(x\) range between \(X_{k-1}\) and \(X_{k}\) is mainly determined by the \(k\)-th scan and \((k+1)\)-th scan. The overall phase compensation error can be estimated with the integral of \(\varphi_{bkl\_res}(x)\) in the range \([X_{k-1},\,X_{k}]\) as  
\[ \begin{array}{ll}I_{k}(x_{comb})&=\int\limits_{X_{k-1}}^{X_{k}}\varphi_{bkl \_res}(x)dx\\ &=2\pi\Delta_{k}S_{k}[(X_{k}+X_{k-1})/2-x_{comb}]\end{array}  \tag{27}\]  
Although Equation (27) is not a strict mathematical deduction based on the exposure model, it provides a tool for the analysis. The absolute value of \(I_{k}\) is smaller, and the phase compensation performance is better. If \(x_{comb}=(X_{k}+X_{k-1})/2\), the best phase compensation performance can be obtained with \(I_{k}=0\), and it is not affected by \(S_{k}\). Therefore, \((X_{k}+X_{k-1})/2\) is a good selection for \(x_{comb}\). If \(x_{comb}\
eq(X_{k}+X_{k-1})/2\), the phase compensa \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(x_{comb}\)} & \multicolumn{2}{c}{\(g_{p}\)**(x) Coefficients**} & \multicolumn{2}{c}{**Relative Errors of \(g_{p}\)(x) Coefficients**} \\ \cline{2-5}  & \(n_{gp0}\)**, L/mm** & \(n_{gp1}\)**, L/mm\({}^{2}\)** & \(\epsilon\)**(\(n_{g0}\))** & \(\epsilon\)**(\(n_{g1}\))** \\ \hline \(X_{k}\) & 1200.0311 & \(-\)0.7783 & 2.5943 \(\times\) 10\({}^{-5}\) & \(-\)2.5946 \(\times\) 10\({}^{-5}\) \\ \((X_{k}+X_{k-1})/2\) & 1200.0000 & \(-\)0.7783 & \(<\)1 \(\times\) 10\({}^{-8}\) & \(<\)1 \(\times\) 10\({}^{-8}\) \\ \(X_{k-1}\) & 1199.9689 & \(-\)0.7783 & \(-\)2.5943 \(\times\) 10\({}^{-5}\) & 2.5946 \(\times\) 10\({}^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(g_{p}(x)\) under the condition of \(f_{k}=g(X_{k})\) and phase compensation with \(S_{k}\). * [PERSON] (2015) [PERSON] The Research on Design Methods and Fabricating Technology of Varied Line-Space Holographic Plane Grating. Ph.D. Thesis, Chinese Academy of Sciences, Beijing China, 2015. * [PERSON] et al. (2017) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON] Electrically tunable two-dimensional holographic polymer-dispersed liquid crystal grating with variable period. _Opt. Commun._**2017**, _392_, 128-134. [CrossRef]
* [PERSON] et al. (2006) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Advanced interference lithography for Nanomanufacturing. _Int. Soc. Nanomufacturing_**2006**, _44_, 441-445. * [PERSON] et al. (2002) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Progress towards a general grating patterning technology using phase-locked scanning beams. _Proc. SPIE_**2002**, _4485_, 378-384. * [PERSON] et al.