# Image Quality for Near-Eye Display Based on Holographic Waveguides
## 2 Materials and Methods
### Diffraction Analysis for the Waveguide Gratings  
To analyze the correct profile height, analog grating diffraction was modeled using a Matlab numerical solution based on RCWA (rigorous coupled wave analysis). Figure 3 shows the resulting characteristics of diffraction efficiency via relief height. Figure 3: Dependences of diffraction efficiency (\(\eta_{-1}\)) on the relief height of holographic diffraction grating for three color channels according to Table 1. RGB diffraction analysis: for the red channel grating in TE polarization (**a**) and TM polarization (**b**); for the green channel grating in TE polarization (**c**) and TM polarization (**d**); for the blue channel grating in TE polarization (**e**) and TM polarization (**f**). Plots are given for angles of incidence 0 and 15 degrees combined for two perpendicular polarizations of incident light. RCWA is one of the most popular and efficient methods to solve the diffraction problem since the 1980s [31, 32]. The code calculates the interaction of a plane electromagnetic wave with the periodic structure providing the diffraction efficiencies of all existing reflected and transmitted diffraction orders. As initial data for the calculation, the incident wave has a unity amplitude of electric vector E and zero phases. Since the wave field is polarization-dependent, two conditions were analyzed: TE polarization, as Figure 4a illustrates, and TE polarization, as shown in Figure 4b. Unpolarized light is equivalent to the average of TE and TM diffraction efficiency. In this case, the \(x\)-axis corresponds to the grating vector, the \(y\)-axis corresponds to the lines of the diffraction grating, and the \(z\)-axis is perpendicular to the plane of the diffraction grating. The radiation wavelengths are \(\lambda_{\text{B}}\) = 0.459 \(\upmu\)m, \(\lambda_{\text{G}}\) = 0.53 \(\upmu\)m, and \(\lambda_{\text{R}}\) = 0.625 \(\upmu\)m.  
Table 1 combines not only the parameters for modeling but also the general parameters of the waveguide. The substrate material (F6) determines the minimum TIR angle for light propagation in the waveguide. The recording angle is angle \(\alpha\) in Figure 5a--the angle between the normal to the grating surface and the angle of incidence of the interfering wave, which will be described in more detail in the following section. The diffraction distribution in -1 diffraction order was calculated for two incidence conditions from cover to substrate: normal to the grating plane and oblique at the angle of \(15^{\circ}\). The dielectric permittivity of the media was specified as follows. The cover zone corresponds to air, substrate zone corresponds to F6. The structure zone is resin S1813 having dielectric permittivity distribution defined with sinusoidal profile functions. As for the calculation parameter, the number of harmonics in the expansion series is N = 16 (the total number \(2\text{N}+1=33\)) for every case. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Spectral Channel**} & \multicolumn{2}{c}{**Recording**} & \multirow{2}{*}{**Grating Period**} & **Substrate** & **Resin** \\  & **Angle \(\alpha\)** & & **Refractive** & **Refractive** \\  & (\(\alpha_{\text{rec}}\) = 441.6 nm) & & **Index n\({}_{\text{F6}}\)** & **Index n\({}_{\text{S1813}}\)** \\ \hline R (\(\lambda_{\text{R}}\) = 625 nm) & 25\({}^{\circ}\) & 516 nm & 1.6005 & 1.63 \\ \hline G (\(\lambda_{\text{G}}\) = 530 nm) & 31\({}^{\circ}\) & 421 nm & 1.6087 & 1.66 \\ \hline B (\(\lambda_{\text{B}}\) = 459 nm) & 38\({}^{\circ}\) & 357 nm & 1.6193 & 1.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the coupling gratings. Figure 4: Illumination geometry for TE polarization (**a**) and TM polarization (**b**): **E—vector of electric field (magnetic field is not shown), \(\theta\)—angle of incidence, \(d\)—diffraction grating period, \(h\)—relief height.