# Development of a Holographic Waveguide with Thermal Compensation for Augmented Reality Devices
## 2 Methods
We described relief-phase gratings that form several symmetric diffraction orders at once (\"\(-1\)\" and \"\(+1\)\", \"\(-2\)\" and \"\(+2\)\", etc.). This fact is a feature of this type of diffraction gratings that cannot be eliminated. It should be noted that they have practically no angular and spectral selectivity, so they can work with polychromatic radiation propagating in a wide range of angles, which is important for the formation of an angular field of view in augmented reality devices. Therefore, it is necessary to compromise between the color of the image, the size of the angular field of view and the presence of \"parasitic\" diffraction orders, which can also form an image or parts of it. At the same time, the diffraction angles of higher orders are quite large (2 times higher than the diffraction angle for \"\(+1\)\" and \"\(-1\)\"); therefore, when diffraction on the introductory grating (DOE 1), the radiation of these orders literally \"slides\" over the surface of the substrate, almost without undergoing re-reflections, i.e., practically does not leave the substrates of the waveguide to the observer. Another factor is the diffraction efficiency, which is lower in higher orders than in the considered ones, namely: \"\(+1\)\" and \"\(-1\)\"--34%, \"\(+2\)\" and \"\(-2\)\"--10%, \"\(+3\)\" and \"\(-3\)\"--1% [21]. Considering the fact that the radiation diffracts on at least two diffraction gratings during the input and output of radiation (DOE 1 and DOE 3), the total efficiency of the holographic waveguide for such orders does not exceed 1%; therefore, \"parasitic\" effects from such orders are simply not observed. The required period of the diffraction grating, which ensures the deviation of the first-order diffraction rays with a wavelength \(\lambda=0.65\) um by the angle of total internal reflection of the material with a refractive index \(n_{g}=1.5139\), will be equal to:  
\[d_{gr1,3}\ =\ \frac{\lambda}{n_{g}\sin(\alpha_{TIR})}\ =\ 0.65\ \upmu\text{m}, \tag{5}\]  
We used glass with a refractive index of 1.51 (analogous to BK7 glass). This brand of glass is quite popular, so it was used during research and for recording holographic waveguides. At the same time, it should be noted that the refractive index of glass determines the angle of total internal reflection in the material, i.e., the minimum angle at which radiation can propagate inside the glass substrate without leaving it, so if the task is to increase the size of the angular field of view (i.e., increasing the range of radiation propagation angles in the waveguide), it is recommended to choose glasses with a refractive index of 1.76 and higher. It is important to consider that the image expands along two coordinates [6]. For the relay grating DOE 2, the expression is valid:  
\[d_{gr2}\ =\ \frac{d_{gr1,3}}{2 cos\beta}, \tag{6}\]  
where \(\beta\) is the angle of inclination of the relay grating. It follows from (6) that the period of DOE2 will decrease by a factor of \(\sqrt{2}\) due to a change in the trajectory of radiation propagation by \(45^{\circ}\), and therefore the period of the relay grating will be equal to \(d_{gr2}=0\),\(46\ \upmu\)m.  
Another feature of the development of some optical systems based on a holographic waveguide is the need to consider the range of possible wavelength shift of an LED or laser radiation source within \(\pm 10\) nm, currently this issue remains unresolved in full [22; 23; 24]. The DOE design methodology should consider such features, since the radiation wavelength of semiconductor laser diodes and the dominant wavelength of the LED changes when its temperature or ambient temperature changes. Therefore, in the previously considered scheme, due to diffraction phenomena, according to (4), an angular displacement of the image on the holographic waveguide occurs, which, accordingly, leads to an inaccuracy in reproducing the line of sight and a shift in the angular position of the output image.