# Development of a Holographic Waveguide with Thermal Compensation for Augmented Reality Devices
## 2 Methods  
For correct reproduction of the augmented reality image in the observation plane, it is necessary that the parameters of the waveguide (refractive index, thickness, etc.) be consistent with the DOE structure (period, orientation) [17; 18; 19]. One of the main conditions for the correctness of image reproduction is the requirement to overlap the projections of the entrance pupils in the observation plane. When the radiation is introduced, the <<0>> diffraction order passes through the substrate without being refracted or reflected from the waveguide. In turn, radiation of the +1 diffraction order begins to propagate inside the substrate if the angle of refraction is greater than the angle of total internal reflection. Falling on DOE3, the radiation diffracts and is output to the observer at the same angle at which it was introduced into DOE1 (Figure 2). At the same time, part of the radiation continues to propagate in the waveguide under the action of TIR, and then it is withdrawn from it again, realizing an increase in the size of the exit pupil. This implementation makes it possible to minimize the size of the optical system, as noted above, due to the effect of multiplexing the input pupil size Din to the size of the output, for example, Dout = 20 mm. In order to ensure the propagation of radiation inside the waveguide, it is necessary to select its parameters so that the beam, after hitting the substrate through the input DOE, is refracted at an angle greater than or equal to the angle of total internal reflection [20]. \[\alpha_{TIR}\ =\ arcsin\left(\frac{1}{n_{g}}\right), \tag{1}\]  
where \(n_{g}\) represent the refractive index of the substrate glass. \[L\ =\ 2d\cdot tg(\alpha_{TIR}), \tag{2}\]  
where d is the thickness of the substrate. Figure 2: Geometry and ray propagation in the substrate: D\({}_{\text{in}}\) is the diameter of the entrance pupil of the substrate, D\({}_{\text{out}}\) is the diameter of the exit pupil of the substrate, L is the beam stoke length. To achieve the overlap of the projections of the entrance pupil in the observation plane, it is necessary that the distance that the beam passes with one reflection is less than or equal to the diameter of the entrance pupil (3). \[L\leqslant\ D_{in}, \tag{3}\]  
Based on Formulas (1) and (2), it is possible to plot the dependence of the distance that the beam will travel with one reflection on the thickness of the substrate for glasses with different refractive indices (Figure 3), on which the \(\alpha_{TIR}\) depends. In case of non-compliance with condition (3), \"blind spots\" will appear in the observer's field of view, into which the image is not transmitted, as shown in Figure 4. To study the parameters of the waveguide, a material with the most typical refractive index was selected (\(n_{g}=1.5139\)). As can be seen from the graph for this material (\(\lambda=0.65\)\(\mu\)m), the substrate thickness is d = 2 mm, the angle of total internal reflection from (1) is \(\alpha_{TIR}\ =41^{\circ}20^{\prime}\), and the beam stoke length from (2) L = 3.52 mm (Figure 4a). At the same time, if a substrate with a thickness of d = 2.5 mm is used (Figure 4b), \"blind spots\" will also occur in the observation plane, despite compliance with condition (3). Figure 4: Ray tracing in a substrate with a refractive index of the material \(n_{g}=1.5139\). (**a**) Substrate thickness d = 2 mm. (**b**) Substrate thickness d = 2.5 mm. Figure 3: The dependence of the beam stoke length on the thickness of the glass for different materials. To eliminate this effect, it is necessary to adjust condition (3) to the form \(L\leqslant\ D_{in}/\sqrt{2}\), i.e., the distance, the distance that the ray passes with one reflection is less than the side of the square with a diagonal equal to \(D_{in}\). In order to calculate the grating period, we use the formula:  
\[d_{gr}(n_{1}\sin(\alpha_{1})+n_{g}\sin(\alpha_{TIR}))\ =\ m\lambda, \tag{4}\]  
The paper considers the first order of diffraction (\(m=1\)), \(n_{1}=1\) (refractive index of air), \(\alpha_{1}=0\).