## 2 Operation Principle/System Design
### Out-Coupling of a Zero FOV from a Curved Waveguide in an EMB of Finite Dimensions  
Let us calculate the law of variation of the period of the out-coupling DOE in such a way as to form an image of one point of the virtual image in the meridional plane at a finite distance from the waveguide. The rays out-coupled by such a DOE must converge at a point behind the combiner at a distance of \(z_{\text{img}}\) (Figure 8). In addition, we must ensure that this image can be observed within an eye-motion box, so we need to form a set of rays for multiplication of pupils in the meridional plane. We will assume that all rays  
Figure 7: Angular errors for rays with different coordinates on the exit pupil for different components of the angular FOV for the in-coupling DOE with a variable period: (**a**)—according to the analytical dependence (11); (**b**)—after the optimization procedure. from the radiation source were in-coupled into the waveguide and propagate in it without aberrations. The out-coupling angle \(\alpha\) (the angle between the out-coupled ray and the normal to the waveguide surface at the out-coupling point) for different out-coupling points will vary depending on the coordinate \(L\) on the curved surface of the waveguide. Let us find this dependence. Let us consider the case of out-coupling of a beam of rays propagating inside the waveguide at the angle \(\beta_{0 out}\) relative to the normal to the inner surface of the waveguide (Figure 8). Plus one diffraction order (\(m=1\)) of the out-coupling DOE is used to form the out-coupling image here and beyond. Let the period of the out-coupling DOE at the point where the ray incident on the out-coupling DOE at the angle \(\beta_{0 out}\) is out-coupled towards the curvature center of the waveguide (\(\alpha=0\)) have the value \(d_{0}^{OUT}\). Let us choose this point as the origin of coordinates. Let the virtual image be located at the point with the coordinate \(z_{\text{img}}\) on the \(OZ\) axis. In this case, for the slope angle of a ray incident on the out-coupling DOE at the point with coordinate \(L\) on the waveguide surface, from (3) we obtain the following relationship:  
\[\varepsilon=-\text{arctg}\left(\frac{y_{\text{diff}}}{z_{\text{img}}-z_{\text {diff}}}\right)=-\text{arctg}\left(\frac{R_{1}\sin(L/R_{1})}{z_{\text{img}}-R _{1}\cos(\frac{L}{R_{1}})+R_{1}}\right). \tag{12}\]  
The angle for out-coupling a ray from the out-coupling DOE is determined as follows, as can be seen from Figure 8:  
\[-\alpha=\Omega-\varepsilon=\frac{L}{R_{1}}+\text{arctg}\left(\frac{R_{1}\sin( L/R_{1})}{z_{\text{img}}-R_{1}\cos(\frac{L}{R1})+R_{1}}\right). \tag{13}\]  
Then, using (2), we obtain the following relationship for the law of variation of the period of the out-coupling DOE:  
\[d_{mtd}^{OUT}(L)=\frac{\lambda}{\left(\sin(-\frac{L}{R_{1}}-\text{arctg}\left( \frac{R_{1}\sin(L/R_{1})}{Z_{\text{img}}-R_{1}\cos(\frac{L}{R1})+R_{1}}\right) \right)-n_{w}\sin(\beta)\right)}. \tag{14}\]  
As shown in Section 2.1, in the present paper we are designing a system with a common optical axis. Therefore, the period of the in-coupling DOE at the point of intersection with  
Figure 8: Formation of a virtual image (blue lines) of a point at a finite distance by using an out-coupling DOE. 1—the inner surface of the curved waveguide in the meridional plane; 2—the position of the virtual image; 3—the center of the curvature radius of the waveguide surface. the projector's axis must be chosen equal to the period of the out-coupling DOE at the point of intersection with the axis of the user's eye (\(d_{0}^{OUT}=d_{0}^{IN}\)). In such a system, a ray incident along the normal to the waveguide surface on the in-coupling DOE will also out-couple along the normal on the out-coupling DOE as well (\(\beta_{0 out}=-\beta_{0 in}\)).