## 2 Operation Principle/System Design
### Out-Coupling of a Wide FOV from a Curved Waveguide in an EMB of Finite Dimensions  
The task of the combiner is to out-couple all the angular components, described in Section 2.3, within an EMB of a finite size. At the same time, for reasons similar to those described for the in-coupling DOE (Section 2.3), it is impossible to form a virtual image with a non-zero FOV in an extended EMB without aberrations by using the out-coupling DOE with a variable period. That is why, by analogy with the in-coupling DOE, it is necessary to optimize the law of variation of the period of the out-coupling DOE to reduce the aberrations introduced by the out-coupling DOE. The law of variation of the period of the out-coupling DOE was optimized for the case where a virtual image was located at infinity (17), since, from a practical point of view, this case is of the greatest interest. It was also taken into account that the size of the pupil of human eye is several times smaller than the size of an EMB. In this regard, all possible positions of the user's eye pupil inside an EMB were searched in the calculation, and for each such position, all rays entering the pupil from one or another direction of the FOV were calculated. Thus, the criterion for estimating the magnitude of aberrations in the optimization process was the angular error for all combinations of the position of the user's pupil in the eye-box and the angles of the FOV. It was also assumed in the calculation that the image from the source was in-coupled into the waveguide and transmitted without aberration through it. Optimization was carried out only for the meridional plane. A scheme representing a ray out-coupling in the meridional plane of the waveguide is shown in Figure 9. Let each ray under consideration falls on the out-coupling DOE at the point with the coordinate \(L\) at the angle \(\beta\), diffracts and leaves the waveguide at the angle \(\alpha_{z}\), after which it falls into the EMB at the point with the coordinate \(y_{\mathrm{eye\_box}}\). To optimize the period of the out-coupling DOE, a program was written using the MatLAB software. For each coordinate \(L\) on the out-coupling DOE and for each component of the angular field \(\beta\), the program calculated the diffraction angle \(\alpha_{z}\) relative to the \(OZ\) axis and the coordinate of the beam incidence on the plane of an eye-box, \(y_{\text{eye\_box}}\). Then, for each combination of the angular component of the field \(\beta\) and the position of the center of the pupil of the eye in the EMB, the maximum error (angle spread) was found for all rays of this angular component that fell into the pupil (\(\Delta\alpha=\ \alpha_{z\_max}-\ \alpha_{z\_min}\)). Thus, a two-dimensional array of maximum angular errors of the out-coupled ray (\(\Delta\alpha\)) was built as a function of the pupil center coordinate in EMB, \(y_{\text{pupil}}\), and the FOV angle \(\alpha_{z}\). Then, the optimization procedure corrected the coefficients of the polynomial describing the law of variation of the period of the out-coupling DOE in order to minimize the angular errors of ray out-coupling. The cycle was repeated until the optimization procedure had found the local minimum of the angular error function depending on the values of the polynomial coefficients. The initial values of the polynomial coefficients were taken from the Taylor's expansion of the function (17). An example of applying the optimization procedure for the law of variation of the period of the out-coupling grating is shown in Figure 10. The waveguide parameters are the same as for the in-coupling DOE, the eye pupil diameter is equal to 4 mm, and the distance from the pupil to the waveguide (eye-relief) \(z_{\text{eye\_box}}=15\) mm. Figure 10 shows a two-dimensional picture of the dependence of the angular error of the formed virtual image with a width of \(24^{\circ}\) in an eye-box with a width of 12 mm before (a) and after (b) the optimization procedure.