## 2 Operation Principle/System Design
### In-Coupling of Radiation of a Point Source into a Curved Waveguide
In this case, the function (8) of the period variation of the in-coupling DOE in the meridional section will take the form:  
\[y_{\rm diff}=R_{1}\sin(\frac{L}{R_{1}}),z_{\rm diff}=R_{1}(\cos(L/R_{1})-1). \tag{3}\]1  
In the sagittal plane, the period of such a DOE at all points should be infinite, \(d_{sgst}^{IN}(X)=\infty\). In this case, similarly to the case considered above, the in-coupling DOE should be a linear structure, with the grating vector lying in the YOZ plane. In-Coupling of a Wide Angular Field from a Source with a Pupil of a Finite Size into a Curved Waveguide  
In practice, a projector, consisting of a light emitting matrix and a collimator, is often used as a light source. The radiation of each pixel of the projector matrix is collimated, and thus an image of each pixel is formed at infinity. In this case, parallel beams from each individual pixel propagate at individual angles, thus, each pixel of the matrix (each  
\[y_{\rm diff}=R_{1}\sin(\frac{L}{R_{1}}),z_{\rm diff}=R_{1}(\cos(L/R_{1})-1). \tag{3}\]2 In-coupling of rays (blue lines) from a point source of radiation located at a finite distance, for the sagittal plane. \(I\)—the inner surface of the curved waveguide in the sagittal plane; \(2\)—a point source of radiation. point of the original image) is uniquely set by an angular direction, thus forming the angular field of the projector. It can be concluded that the described optical system is characterized by an exit pupil of a non-zero (finite) size and a non-zero angular field, which leads to an increase in the required in-coupling area on a waveguide. In turn, due to the curvature of the waveguide, the projector field is differently oriented with respect to the normals in different parts of the in-coupling DOE. At the same time, if the angular step between the beams of rays, associated with different parts of the image, is constant, then after in-coupling them into the waveguide the step between the beams will no longer be constant. Such a conclusion is obvious when considering Formula (2) for a set of initial angles. Thus, when considering the operation of the in-coupling DOE on a curved surface, the described image aberrations will be different for each part of the field (determined by the angle of propagation) in different parts of the in-coupling DOE. So, it is possible to fulfill the condition of radiation in-coupling into the waveguide without aberrations by using the in-coupling DOE with a variable period for a single angular component only or for a radiation source with a zero aperture. Otherwise, it is impossible to introduce radiation into the waveguide without aberrations. However, aberrations can be minimized by optimizing the law of variation of the grating period. In the optimization process, a law is selected that will provide the minimum in-coupling angular errors for all combinations of angles and points in the area of the projector's exit pupil. Let us consider the case of using a radiation source of finite dimensions that forms an image at infinity. As shown above, the in-coupling DOE, operating without aberrations for one direction of the FOV, is a grating with a variable period determined by expression (11). The grating vector lies in the _YOZ_ plane. We optimize the law of variation of the period of the in-coupling DOE in the meridional plane only. The mutual arrangement of the projector's pupil, the waveguide and the path of rays in the meridional plane are shown in \[y_{\rm diff}=R_{1}\sin(\frac{L}{R_{1}}),z_{\rm diff}=R_{1}(\cos(L/R_{1})-1). \tag{3}\]3 Let the projector's exit pupil be located at a known distance from the waveguide, \(z_{\text{pupil}}\), the pupil size is \(2y_{\text{pupil}\_{\text{max}}}\), and the angular field is \(2\alpha_{\text{pupil}\_{\text{max}}}\).