# See-Through Near-Eye Display with Built-in Prescription and Two-Dimensional Exit Pupil Expansion
## 3 Results and Discussion
### Diffraction Efficiency  
The algorithm for calculating the grating is the Fourier modal method [26]. The simulated annealing method [27] and Nelder-Mead or downhill simplex method [28] are in turn employed for the global and local optimizations. During the optimization, the slant angle, grating height and width are assigned as variables. In order to widen the spectral and angular bandwidths of a single slanted grating, both the ISG and OSG are multiplexed, which are formed through the fusion of plural sub-gratings, as itemized in Table 7. As for the spectral bandwidth, the DEs of all possible diffraction orders combined are calculated with respect to the wavelengths, as shown in Figure 8, where the average DEs of the ISG and OSG over the entire spectrum (460 to 660 nm) are 48.0% and 16.7%, respectively. As for the angular bandwidth, the DEs are calculated with respect to the out-coupling angles, at which the light is coupled out of the lens, the range of which equals the FOV, as shown in Figure 9. It can be seen that the average DEs of the ISG and OSG for the wavelength of 565 nm over the full horizontal/vertical FOV (\(\pm 12^{\circ}\)) are 14.40% and 10.53%, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Surface** & \begin{tabular}{c} **Y Radius** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Conic** \\ **Constant (K)** \\ \end{tabular} & \begin{tabular}{c} **4 th Order** \\ **Coefficient (A)** \\ \end{tabular} & \begin{tabular}{c} **6 th Order** \\ **Coefficient (B)** \\ \end{tabular} & \begin{tabular}{c} **8 th Order** \\ **Coefficient (C)** \\ \end{tabular} & \
\begin{tabular}{c} **10 th Order** \\ **Coefficient (D)** \\ \end{tabular} &
\begin{tabular}{c} **12 th Order** \\ **Coefficient (E)** \\ \end{tabular} \\ \hline
**1** & 4.1494 & \(-0.1154\) & 0.0003 & \(5.7069\times 10^{-7}\) & \(6.9614\times 10^{-6}\) & \(-5.4048\times 10^{-7}\) & \(2.1806\times 10^{-8}\) \\
**2** & 4.1379 & \(-0.6547\) & 0.0055 & 0.0003 & \(0.0003\) & \(-6.7023\times 10^{-5}\) & \(9.9782\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Detailed parameters of aspherical surfaces for the virtual image.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Object** & **Sub-Object** & **Parameter** & **Value** \\ \hline \multirow{8}{*}{ISG} & \multirow{4}{*}{Sub-grating A} & \(p\) & 511 nm \\  & & \(h\) & 1015.80 nm \\  & & \(\theta_{g}\) & 29.31\({}^{\circ}\) \\  & & \(w\) & 55.67 nm \\ \cline{2-4}  & & \(p\) & 587 nm \\  & & \(h\) & 637.79 nm \\  & & \(\theta_{g}\) & 29.31\({}^{\circ}\) \\  & & \(w\) & 272.30 nm \\ \hline \multirow{8}{*}{Sub-grating C} & \multirow{4}{*}{Sub-grating C} & \(p\) & 688 nm \\  & & \(h\) & 697.85 nm \\  & & \(\theta_{g}\) & 29.31\({}^{\circ}\) \\  & & \(w\) & 244.51 nm \\ \cline{2-4}  & Multiplexed grating & \(P_{m}\) & 19.95 μm \\ \hline \multirow{8}{*}{OSG} & \multirow{4}{*}{Sub-grating B} & \(p\) & 317 nm \\  & & \(h\) & 702.98 nm \\ \cline{1-1}  & & \(\theta_{g}\) & 42.413\({}^{\circ}\) \\ \cline{1-1}  & & \(w\) & 128.67 nm \\ \cline{1-1} \cline{2-4}  & & \(p\) & 364 nm \\ \cline{1-1} \cline{2-4}  & & \(h\) & 440.53 nm \\ \cline{1-1}  & & \(\theta_{g}\) & 42.413\({}^{\circ}\) \\ \cline{1-1}  & & \(w\) & 237.36 nm \\ \hline \hline \end{tabular}
\end{table}
Table 7: Optimized parameters of the ISG and OSG.