## 3 Phase Modulation Effects of the Parameters and Parameter Selection
### Phase Modulation Effects of the Parameters  
#### 3.2.1 Phase Modulation by Varying Only \(\delta_{k}\)  
Set the fringe density to a constant value of \(f_{k}=f_{0}=n_{g0}-n_{g1}\)\(W_{g}/2\). It is equal to the desired density at \(x=0\). The step size is an integral multiple of the fringe period as that in a traditional EP-SBIL, which implies that \(S_{k}=N_{steps}/f_{0}\) and \(X_{k}=kN_{steps}/f_{0}\). \(N_{steps}\) influences the dose uniformity and exposure contrast. For general VLS gratings with a small coefficient \(n_{g1}\), the grating density is mainly determined by \(n_{g0}\). Therefore, \(N_{steps}\) is chosen as a constant expressed as  
\[N_{steps}=round\big{(}\omega\cdot StepRatio\cdot n_{g0}\big{)} \tag{16}\]  
Under these conditions, Equation (8) changes to a simple form \(\phi_{k}(x)=\delta_{k}\). If \(\delta_{k}\) is used for phase modulation, all fringes in the image have identical phase variations, which implies that \(\delta_{k}\) cannot be expressed as a function of the \(x\) coordinate. 1. Intuitively varying \(\delta_{k}\) according to \(\phi_{g}(x)\) Intuitively, changing the \(\delta_{k}\) value with the desired \(\phi_{g}\) phase in the \((k+1)\)-th scan appears to be feasible, which can be written as \[\delta_{k}=\Phi_{g}(X_{k})=2\pi\Big{[}\big{(}n_{g1}/2\big{)}X_{k}{}^{2}+\big{(} n_{g0}-n_{g1}W_{g}/2\big{)}X_{k}\Big{]}\] (17)  
\(\phi_{e}(x)\) is shown in Figure 2, and it is basically piecewise linear with a transition region between the two adjacent linear segments. The exposure contrast is shown in Figure 3. The troughs of the exposure are in contrast with an amplitude close to zero. We performed a differential operation to \(\phi_{e}(x)\). The differences in \(\phi_{e}(x)\) are also shown in Figure 3. The \(x\) coordinates of the cusps in \(\phi_{e}(x)\) coincide with those of the lowest contrasts. The frequency components of the exposure contrast are extracted using FFT (Fast Fourier Transform). Two main frequency components are approximately located at \(F_{\rm p1}=10.5n_{g1}\,1/(2\pi)\) and \(F_{\rm p2}=1/S_{k}\). The \(F_{\rm p1}\) component is influenced by the linear variation of the grating density. The \(F_{\rm p2}\) component is caused by the phase inconsistency in adjacent scans, which also exists in \(\phi_{e}(x)\), as shown in Figure 3. \(\phi_{e}(x)\) and \(\gamma(x)\) cannot satisfy the demands of grating fabrication. 2. Phase modulation effect by only varying \(\delta_{k}\) according to \(X_{k}\) To modulate the phase by only varying \(\delta_{k}\), \(\delta_{k}\) is different at different \(X_{k}\). As shown in Equation (2), the desired phase variation is a quadratic polynomial. More generally, \(\delta_{k}\) can be written in an analogous form expressed as \[\delta_{k}=2\pi\Big{(}a_{g2}X_{k}{}^{2}+a_{g1}X_{k}+a_{g0}\Big{)}\] (18)Through numerical calculation and theoretical analysis, the modulation effects of \(\delta_{k}\) are as follows. (1) \(a_{\delta 2}=a_{\delta 1}=0\), and \(\varphi_{k}=\delta_{c}=2\pi a_{\delta 0}\). The phase increments are constant for all scans. From Equation (13), \(\Psi_{T}(x)\) can be expressed as  
\[\Psi_{T}(x)=2\pi f_{0}x+\delta_{c} \tag{19}\]  
The patterned grating is an equal-period grating with density \(f_{0}\). The constant phase shift introduced by \(\delta_{c}\) does not affect the grating characteristics. From Equation (15), \(\gamma(x)=1\). (2) \(a_{\delta 2}=0\), \(\varphi_{k}=\delta_{k}=2\pi\left(a_{\delta 1}X_{k}+a_{\delta 0}\right)\). \(\Psi_{T}(x)\) is a linear function with the \(F_{\text{p2}}\) frequency component. The slope is influenced by \(a_{\delta 1}\), and the constant term and \(\gamma(x)\) are determined by \(a_{\delta 1}\) and \(a_{\delta 0}\). If the absolute value of \(a_{\delta 1}\) is small, the slope of \(\Psi_{T}(x)\) is approximately  
Figure 3: Exposure contrast and difference in \(\phi_{c}(x)\) under the condition \(\delta_{k}=\phi_{g}(X_{k})\). Figure 2: Phase error under the condition \(\delta_{k}=\phi_{g}(X_{k})\).