# Image Quality for Near-Eye Display Based on Holographic Waveguides
## 2 Materials and Methods  
### General Principles for the Holographic Waveguide Design  
The optical systems of HMD devices contain two main functional parts: the optical engine or image generation module, which is a compact projector; and the so-called transparent display, i.e., the optical component that transmits images from the projector to the observer's eye. AR display based on the holographic waveguide is notable for its field of view above 50\({}^{\circ}\)[11; 12; 26]. The angular field is provided by the high refractive index of the glass substrate, which is responsible for the cone of TIR beams in the substrate, and the presence of several waveguide plates for different spectral channels (e.g., RGB) needed to form a color image. For the light to be transmitted effectively through the plate to the eye, the field of view must be divided by angle and spectrum across several plates [27; 28]. Each of the plates propagates a certain portion of the light, as shown in Figure 1.  
Figure 1 illustrates a three-layer holographic waveguide. For each spectral channel, the collimated beams from the projector diffract into the substrate (+1 order of diffraction) and propagate inside by TIR means. For this, the diffraction angle must be larger than the TIR angle. Consequently, the greater the refractive index of the plate, the greater the potential angular field of view to be transmitted. The second condition applied to the diffraction angle inside the substrate is related to the pupil size and substrate thickness, and hence to the maximum pupil spacing during pupil replication. The light beams for each of the spectral channels, move along the corresponding direction in the light guide according to the grating equation  
\[n_{\rm tr}\sin(\varphi_{m}^{\rm tr})=n_{\rm inc}\sin(\theta)-m\frac{\lambda}{d}, \tag{1}\]  
Figure 1: Scheme of the full color transparent display based on planar holographic waveguides. where \(n_{\text{tr}}\) is the refractive index of the waveguide, \(\varphi_{m}\) is the diffraction angle (from the normal), \(n_{\text{inc}}\) is the refractive index of the media of the incident light (air), \(\theta\) is the incident angle, \(m\) is the order of diffraction, \(\lambda\) is the wavelength, \(d\) is the period of the diffraction grating. The output DOE (Figure 1) which breaks the TIR condition to output the light, is located near the eye. The light beams leave the waveguide in the same direction as the incident beams. Using a symmetrical configuration of the input/output DOE is a simple method of compensating for the spectral misalignment in the image. Each of the transmitting diffraction gratings in the three-layer holographic waveguide satisfies the coordinated propagation angles in the waveguide for red, green, and blue radiation. Figure 1 is only a schematic representation of the principle of radiation input and output relative to the plate. Since the surface diffraction gratings are non-selective and operate in the Raman-Nath diffraction mode according to equation (1), light with any incident angle for one wavelength can partially diffract on a grating designed for another wavelength, as described in [29]. The color channels may be partially mixed, but the final image is unaffected by that spectral overlapping. The principles of diffraction grating calculation are described in detail [2; 30]. Therefore, there can be two or three plates depending on the spectrally transmitted angular composition. This effect is visualized in more detail in Section 2.2. To provide the size of the exit pupil, which is many times larger than the entrance pupil, an intermediate DOE, relay grating, is placed between the input and output DOEs. DOE configurations that help to achieve two-dimensional eye box expansion, required for most AR displays, are described in [29] and presented in Figure 2. It is often recommended to make the output element DOE3 with variable diffraction efficiency increase in the direction of propagation.