## 4 Experimental Verification and Discussion
### Wavefront Aberrations Evaluation  
To specify the errors of the gratings, an absolute interferometric test method based on a Fizeau interferometer was introduced. The He-Ne laser was operated at a wavelength  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Points**} & \multicolumn{4}{c}{\(S\) = 2 \(\upmu\)m} & \multicolumn{4}{c}{\(S\) = 4 \(\upmu\)m} \\ \cline{2-9}  & \(S\) [\(\upmu\)m] & \(d\) [\(\upmu\)m] & \(h\) [nm] &  
of 632.8 nm in the interferometer. The gratings were measured in reflection as shown in Figure 9, where the incident angle \(\alpha\) and diffraction angle \(\theta\) conform to:  
\[S(\sin\alpha+\sin\theta)=m\lambda \tag{9}\]  
If \(\alpha\) is equal to \(\theta\), the tested grating and interferometer would be collimated, and the incident wave diffracted back along the same optical path. The incident angle \(\alpha\) of the plane wave satisfies the following equation:  
\[\alpha=\arcsin(\frac{m\lambda}{2S}) \tag{10}\]  
The light path for testing realized self-collimation. As shown in Figure 10, the wavefronts of the +1 st (\(W_{+1}\)) and \(-\)1 st (\(W_{-1}\)) diffracted orders were measured to determine the effects of fabrication errors. The direct measurement interferometry \(W_{+1}\) or \(W_{-1}\) contained aberrations from the interferometer system \(W_{\textit{inf}}\), the substrate surface figure error \(W_{\textit{sub}}\), and the fabrication error of elements \(W_{\textit{fib}}\). Thus, the wavefront of tests \(W_{+1}\) and \(W_{-1}\) can be respectively expressed by the following two equations:  
\[W_{+1}=W_{\textit{inf}}+W_{\textit{sub}}+W_{\textit{fab}+1} \tag{11}\]  
and,  
\[W_{-1}=W_{\textit{inf}}+W_{\textit{sub}}+W_{\textit{fab}-1} \tag{12}\]  
Figure 10: Schematics of the absolute test method for linear gratings. Optical configuration includes (**a**) normal incidence, (**b**) +1 st diffracted wavefront and (**c**) \(-\)1 st diffracted wavefront. Figure 9: Linear grating in the case of reflection. When the wavefront was generated in the \(-1\)st diffracted order, the aberration of the fabrication errors was complementary to the \(+1\)st diffracted order. That is, as follows:  
\[W_{fab-1}=-W_{fab+1} \tag{13}\]  
Thus, the aberrations in \(W_{inf}\) and \(W_{sub}\) could be eliminated by subtracting Equation (12) from Equation (11), and the effect of the fabrication errors was obtained as follows:  
\[W_{fab+1}=\frac{1}{2}(W_{+1}-W_{-1}) \tag{14}\]  
Based on this absolute testing method, the wavefront quality evaluation of the two gratings was executed. The test results were recorded in Table 2. As shown in Figure 11, the wavefront aberration of the \(4\)\(\upmu\)m grating had a RMS value of \(0.005\lambda\), while that of the \(2\)\(\upmu\)m grating had a RMS value of \(0.01\lambda\). The wavefront of small period grating appears more sensitive to fabrication errors. The wavefront aberrations of two gratings are distinguished, but the scalar approximation indicates that the wavefront aberrations are almost the same. Therefore, experimental \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **+1 st Order/\(\Lambda\)** & **\(-1\)st Order/\(\Lambda\)** & **Final Result/\(\Lambda\)** \\ \hline S = \(2\)\(\upmu\)m & 0.016 & 0.017 & 0.010 \\ S = \(4\)\(\upmu\)m & 0.011 & 0.011 & 0.005 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The RMS values of test results in Figure 11.  
Figure 11: Wavefront aberration of \(\pm 1\) diffraction orders and final data for \(S\) = \(2\) and \(4\)\(\upmu\)m grating elements caused by fabrication errors. results show the inaccuracy of scalar approximation in measuring CGHs with small periods, and the imperfections cannot be ignored. Conversely, the wavefront aberrations caused by duty cycle errors, which are not considered in scalar approximation, cannot be ignored according to our rigorous analysis in Section 3. The aberration of the 2 \(\upmu\)m grating in Figure 11 is more serious than that of the 4 \(\upmu\)m grating and proves the imperfections of scalar approximation for structures with small periods.