## 2 Model of the Total Exposure Dose for VLS Grating Patterning  
The desired groove density function of the VLS plane grating without higher-order aberration compensation can be expressed as follows  
\[g(x)=n_{g0}+n_{g1}\big{(}x-W_{g}/2\big{)} \tag{1}\]  
The desired phase variation of the VLS grating can be expressed as  
\[ \begin{array}{ll}\Phi_{g}(x)&=2\pi\int\limits_{0}^{x}g(x)dx\\ &=2\pi\big{[}\big{(}n_{g1}/2\big{)}x^{2}+\big{(}n_{g0}-n_{g1}W_{g}/2\big{)}x \big{]}\end{array}  \tag{2}\]  
where the \(x\) coordinate is along the vector direction of the grating. It is assumed that \(x=0\) is at the margin of the grating substrate. \(W_{g}\) is the grating width along the \(x\)-axis; \(n_{g0}\) is the groove density in the center of the grating with units of L/mm; \(n_{g1}\) is the slope with units of L/mm\({}^{2}\); and \(n_{g0}\) and \(n_{g1}\) are determined according to the aberration correction principle of the VLS grating and the structure of the spectrometer or the device, respectively. In the VP-SBIL system, the step direction is along the \(x\)-axis, and the scanning direction is along the \(y\)-axis, which is parallel to the fringe direction. In this paper, the VLS grating was patterned with substrate parallel scanning. At the end of a scan, the stage stepped over and reversed direction for a new scan. The exposure phase was modulated during the step process, and the phase relative to the substrate remained unchanged during scanning. Other written modes such as Doppler scanning are not described here [16; 17]. Let us assume a perfect stage scan, fringe period regulation, and fringe phase locking. The initial scan is at \(x=X_{0}=0\). The fringe density in the initial scan is\(f_{0}\). The dose delivered to the resist can be mathematically written as  
\[D_{0}(x)=B(x)+A(x)\cos(2\pi x\cdot f_{0}+\delta_{0}) \tag{3}\]where \(B(x)\) is the background dose. \(\delta_{0}\) is the locked phase introduced by the fringe locking unit in the initial scan. \(A(x)\) is the normalized Gaussian dose amplitude in the \(x\)-direction and is expressed as  
\[A(x)=\exp(-2x^{2}/\omega^{2}) \tag{4}\]  
where \(\omega\) is the \(1/\mathrm{e}^{2}\) radius of the Gaussian intensity envelope [16; 17]. A uniform exposure dose is achieved by overlapping subsequent scans. After the first step with a step size \(S_{1}\), \(x=X_{1}=S_{1}\) and its serial number \(k=1\). The dose deposited by the second scan (\(k=1\)) is expressed as  
\[D_{1}(x)=B(x-X_{1})+A(x-X_{1})\cos[2\pi\cdot f_{1}\cdot(x-X_{1})+\delta_{1}] \tag{5}\]  
where \(f_{1}\) is the fringe density in the second scan, and \(\delta_{1}\) is the phase introduced by the fringe locking unit. Similarly, the exposure dose deposited by the (\(k+1\))-th scan after \(k\) steps (\(k\geq 1\)) is equal to the initial dose, but it has shifted in position due to the discrete stepping of the stage, which can be expressed with Equation (6). This is illustrated in Figure 1 (the fringe density is exaggerated for clarity). \[D_{k}(x)=B(x-X_{k})+A(x-X_{k})\cos[2\pi f_{k}\cdot(x-X_{k})+\delta_{k}] \tag{6}\]  
where \(S_{k}\) is the \(k\)-th step size. The (\(k+1\))-th scan is at \(x=X_{k}=\sum\limits_{i=1}^{k}S_{i}\) after \(k\) steps. \(f_{k}\) and \(\delta_{k}\) are the fringe density and locked phase introduced by the fringe locking unit in the (\(k+1\))-th scan, respectively. \(f_{k}\) (\(k\geq 1\)) can be written as \(f_{k}=f_{k-1}+\Delta_{k}=f_{0}+\sum\limits_{i=1}^{k}\Delta_{i}\), where \(\Delta_{k}\) is the density increment between the (\(k+1\))-th scan and the \(k\)-th scan.