## 3 Phase Modulation Effects of the Parameters and Parameter Selection
### Phase Modulation Effects of the Parameters
#### 3.2.2 Phase Modulation by Varying \(f_{k}\) and \(\delta_{k}\)  
Instinctively, the VLS grating can be patterned by directly varying the fringe period according to \(g(x)\). In the \((k+1)\)-th scan, \(f_{k}\) is set to the desired density \(g(X_{k})\) of this scan, and \(\Delta_{k}=f_{k}-f_{k-1}=n_{\mathrm{g}}1S_{k}\). \(S_{k}\) is not used to modulate the phase in this manner. We chose a constant value of \(S_{c}=\mathrm{StepRatio}\cdot\omega\), and \(X_{k}=kS_{c}\).  
The unequal fringe density in the adjacent scans introduces the phase inconsistency. As described in Section 3.2.1, \(\delta_{k}\) cannot compensate for the phase error varying with \(x\). However, with \(\delta_{k}\) in Equation (20), the fringe phase difference between the \((k+1)\)-th scan and the \(k\)-th scan at a certain \(x_{comb}\) coordinate can be compensated.  
\[\delta_{k}=\delta_{k-1}-\varphi_{kdx}(x_{comb})-\varphi_{kdc} \tag{20}\]  
With \(f_{0}=n_{g0}-n_{g1}W_{g}/2\), \(\phi_{e}(x)\) with different \(\delta_{k}\) are shown in Figure 4. \(\phi_{e}(x)\) is generally a piecewise quadratic polynomial with \(\delta_{k}=0\) or \(\delta_{k}=\delta_{k-1}-\varphi_{kdc}\). The \(x\) coordinates of the cusps in \(\phi_{e}(x)\) coincide with those of the lowest contrasts. \(\phi_{e}(x)\) with \(x_{comb}=(X_{k}+X_{k-1})/2\) is close to 0, and \(\phi_{e}(x)\) with \(x_{comb}=X_{k}\) and \(x_{comb}=X_{k-1}\) have a cumulative characteristic. For small VLS gratings, the cumulative effects can be neglected. However, with increasing the grating size, the cumulative \(\Phi_{e}(x)\) may have a great impact. The \(F_{\mathrm{P2}}\) frequency component in \(\phi_{e}(x)\) can be acquired by removing the linear polynomial from \(\phi_{e}(x)\), as shown in Figure 5. The \(F_{\mathrm{P2}}\) frequency components indicate that the phase inconsistencies in the adjacent scans are not fully compensated. The peak-to-valley amplitude of the phase fluctuation is close to \(2\times 10^{-4}\) rad, which is smaller than 1/30,000 groove period. The grating groove position error caused by the phase fluctuation is smaller than 0.028 nm for the grating example. Compared with the phase errors in nanometer scale caused by the substrate and resist roughness, development, and metrology frame, this small phase fluctuation can usually be neglected for general VLS gratings.  
We performed a quadratic polynomial fitting to \(\Psi_{T}(x)\). The fitting polynomial can be expressed as  
\[\Psi_{Tp}(x) =2\pi\left(a_{\Psi 2}x^{2}+a_{\Psi 1}x+a_{\Psi 0}\right) \tag{21}\] \[=2\pi\int\limits_{0}^{x}g_{p}(x)dx+\Psi_{Tp0}\]  
where \(g_{p}(x)=n_{g0}+n_{g1}(x-W_{g}/2)\) can be treated as the theoretical groove density of the patterned grating. \(\Psi_{Tp0}=2\pi a_{\Psi 0}\) is the constant component of \(\Psi_{Tp}(x)\), which has little effect on the characteristics of the grating. The groove density coefficients with different phase compensations are shown in Table 1. The theoretical groove densities \(g_{p}(x)\) are close to the desired value, especially \(g_{p}(x)\) with \(x_{\mathrm{comb}}=(X_{k}+X_{k-1})/2\).  
The exposure contrasts \(\gamma(x)\) are shown in Figure 6. The exposure contrasts are better than 0.99, and \(\gamma(x)\) with \(x_{comb}=(X_{k}+X_{k-1})/2\) is better than the others. The resist has low-frequency filtering characteristics, so the influence of the \(F_{\rm P2}\) fluctuation can be neglected [22; 23]. The contrast can satisfy the demands of the grating exposure.