## 3 Phase Modulation Effects of the Parameters and Parameter Selection  
### General Parameters and Numerical Calculation  
As shown in Equations (8) and (13), \(\Psi_{T}(x)\) is mainly determined by the locked phase \(\delta_{k}\), fringe density \(f_{k}\), and step size \(S_{k}\) from all of the scans. Fortunately, all of these parameters can be adjusted in a VP-SBIL system. However, \(\Psi_{T}(x)\) cannot be expressed with a simple mathematical form. It is difficult to directly solve the equation \(\Phi_{e}(x)=0\). The numerical calculation based on the exposure model is an effective means of analysis.  
For simplicity, assume that the contrast in a single scan is 1, which implies that \(A\) (\(x\)) = \(B\) (\(x\)) in Equation (3). A grating in a surface physics beamline monochromator is taken as an example. The parameters of the grating are \(W_{g}\) = 30 mm, \(n_{g0}\) = 1200 L/mm, and \(n_{g1}\) = \(-0.7783\) L/mm\({}^{2}\)[20]. To avoid fringe smearing in VP-SBIL, a spot diameter in the range of 10-100 microns is optimal [11]. We chose \(\omega\) = 100 \(\mu\)m for the numerical calculation and analysis. To achieve a uniform exposure dose, the adjacent scans overlap. As described in previous research on EP-SBIL, the step size is smaller, the uniformity and contrast decrease caused by phase error are better, but the exposure is slower. If the fringe period is constant in all scans, a step size of 0.9 times the 1/e\({}^{2}\) radius (StepRatio = \(S_{k}/\omega\) = 0.9) produces a better dose uniformity than 1% [18; 19]. VLS grating fabrication requires fringe phase modulation in each step process, which implies that phase inconsistency in the adjacent scans is inevitable. The step size should decrease with increasing the coefficient \(n_{g1}\) to satisfy the exposure contrast demand. As StepRatio is not a crucial parameter for phase modulation, we arbitrarily chose StepRatio = 0.8 for the small \(n_{g1}\) value in this paper.  
With the above general parameters, the numerical calculation was developed with Matlab. To minimize the numerical artifact that may arise from an abrupt data cutoff, the total step-scan width \(W_{t}\) is larger than \(W_{g}\) (\(W_{t}=W_{g}+1\) mm), and a sufficiently wide \(x\) range [\(-5\) mm, \(W_{t}+5\) mm] is chosen for calculation. To guarantee calculation precision, the calculation step is taken to be \(1/100\) times \(\omega\). Based on the exposure model, the exposure doses are calculated and accumulated step by step in the same way as those in a real exposure process, and \(\phi_{e}(x)\) and \(\gamma(x)\) are calculated according to Equations (14) and (15). Because of the lack of sufficient overlap at the start and end scan segments, the data from \(x=0\) to \(x=W_{g}\) are the focus. To observe the effect of \(\delta_{k},f_{k}\), and \(S_{k}\) on \(\Psi_{T}(x)\), three phase modulation manners by varying these parameters in an intuitive way with other parameters remaining unchanged are demonstrated. These approaches appeared to be feasible and needed to be verified, and the modulation effects of the parameters were analyzed. According to the numerical calculation results, the parameters for phase modulation were chosen.