## 2 Operation Principle/System Design
### In-Coupling of Radiation of a Point Source into a Curved Waveguide
To optimize such an in-coupling DOE, a program was written using the MatLAB software, which searched inside an embedded loop all possible combinations of elements from the array of ray positions in the projector's exit pupil, \(y_{\text{pupil}}\), and the array of angular components of the FOV, \(\alpha_{\text{pfp}}\). For each combination, according to the law (2), the angle of ray in-coupling into the waveguide, \(\beta\), was calculated for the law (11) of the variation of the period of the in-coupling DOE, \(d^{IN}(L)\). Then, the differences (\(\Delta\beta=\beta-\beta_{0}\)) between the in-coupling angle of each ray (\(\beta\)) and the in-coupling angle \(\beta_{0}\) for the paraxial angular component of the FOV (\(\alpha_{\text{pfp}}=0^{\circ}\)) were found. Thus, a two-dimensional array of angular errors of the in-coupling ray (\(\Delta\beta\)) was built as a function of the co  
\[y_{\rm diff}=R_{1}\sin(\frac{L}{R_{1}}),z_{\rm diff}=R_{1}(\cos(L/R_{1})-1). \tag{3}\]4 Modeling of in-coupling of rays (blue lines) into a curved waveguide from the projector with a finite size of the exit pupil. 1—the inner surface of the curved waveguide in the meridional plane; 2—the plane of the projector’s exit pupil; 3—the center of the curvature radius of the waveguide surface. exit pupil (\(y_{\text{pupil}}\)) and the FOV angle (\(\alpha_{\text{prj}}\)). Then, the optimization procedure corrected the coefficients of the polynomial, by which the law (11) was presented in this optimization procedure to minimize the angular errors of ray in-coupling (\(\Delta\beta\)). 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 (11). An example of such an optimization procedure for the law of variation of the period of the in-coupling grating is shown in \[y_{\rm diff}=R_{1}\sin(\frac{L}{R_{1}}),z_{\rm diff}=R_{1}(\cos(L/R_{1})-1). \tag{3}\]5 The curved waveguide and the projector had the following parameters:  
* the inner curvature radius of the waveguide, \(R_{1}=150\) mm;
* the distance from the plane of the projector's exit pupil to the waveguide, \(z_{\text{pupil}}=15\) mm;
* the refractive index of the waveguide material, \(n_{w}=\)1.49;
* the projector's wavelength, \(\lambda=625\) nm;
* the size of the projector's exit pupil, 4 mm, i.e., \(y_{\text{pupil,max}}=2\) mm; \(y_{\text{pupil,min}}=-2\) mm;
* the angular size of the projector's FOV, \(2\alpha_{\text{prj,max}}=24^{\circ}\). The value of the angular error for in-coupling of radiation is shown by a colored histogram in this figure. From Figure 7, it can be seen that near \(\alpha_{\text{prj}}=0^{\circ}\) the error of ray in-coupling is zero (no aberrations) for any \(y_{\text{pupil}}\). However, for some rays with FOV directions of 10-12\({}^{\circ}\), this error is not equal to zero and reaches a value of about 3.5-4 arcmin. As a result of the optimization (Figure 7b), there was a significant (3-5 times) decrease in the angular errors of in-coupling angles for those rays. However, the error for angles, close to the normal, remained the same or increased insignificantly and did not exceed \(\sim\)0.8 arcmin. At the same time, it is known from the literature [12] that human eye resolution is \(\sim\)1-2 arcmin, which allows us to evaluate the optimization result as successful.