## 2 Operation Principle/System Design
### In-Coupling of Radiation of a Point Source into a Curved Waveguide
The angle of in-coupling of this ray, \(\beta=\ \beta_{0 in}\), into the waveguide at this point will be equal to:  
\[\beta_{0 in}=\mathrm{arcsin}\left(\frac{\lambda}{n_{\omega}d_{0}^{IN}}\right). \tag{7}\]  
As mentioned above, in order to fulfill the condition of beam propagation inside a curved waveguide without aberration, all rays from a point source of radiation must be in-coupled at all points of the in-coupling DOE at the same angle \(\beta_{0 in}\) relative to the normal to the surface at the point of incidence, as shown in Figure 4.  
Then, from (2), (6) and (7), the law of variation of the period of the in-coupling DOE in the meridional plane, which ensures the in-coupling of rays with the required conditions from a point source of radiation located at a finite distance from the waveguide, will be as follows:  
\[d_{mrd}^{IN}(L)=\frac{\lambda}{\sin\bigg{(}\mathrm{arctg}\bigg{(}\frac{R_{1} \sin\left(\frac{L}{R1}\right)}{-Z_{LGT}+R_{1}\cos\left(\frac{L}{R1}\right)-R_{ 1}}\bigg{)}-\frac{L}{R_{1}}\bigg{)}-\frac{\lambda}{d_{0}^{IN}}}. \tag{8}\]  
It should be noted that such an in-coupling DOE will provide the required in-coupling of an image for only one position of the point source of radiation. In-coupling of an image from a source of non-zero size will be carried out by such a DOE with aberrations, the larger, the larger the source size. The relation (8) derived above is valid for the meridional plane. Let us consider the path of rays in the sagittal plane, in which the waveguide under consideration is a plane-parallel plate. To in-couple rays from the source into the waveguide without aberrations, we need the rays to enter the waveguide in the sagittal plane along the normal to the waveguide surface (Figure 5). To do this, it is possible, for example, to add a cylindrical positive lens between the radiation source and the in-coupling DOE. This lens should transfer the radiation source to infinity in the sagittal plane, i.e., the focus of the lens must be aligned with the position of the radiation source. In this case, the in-coupling DOE should be a linear structure with the grating vector parallel to the \(OY\) axis. This means that for all points of such an in-coupling DOE, the condition \(d_{gst}^{IN}(X)=\infty\) should be satisfied in the sagittal plane. The second option to arrange in-coupling of radiation without aberrations from a point source into a curved waveguide in the sagittal plane is to use a variable period of grating of the in-coupling DOE in this plane. Let us consider this case in more detail. For the angle of incidence, \(\alpha\), relative to the normal at each point of the waveguide in the sagittal plane, we can write  
\[\alpha=\varphi=\mathrm{arctg}(x/z_{LGT}). \tag{9}\]  
Then, the period of the in-coupling DOE in the sagittal plane from (2) for \(\beta=0\) will be determined as  
\[y_{\rm diff}=R_{1}\sin(\frac{L}{R_{1}}),z_{\rm diff}=R_{1}(\cos(L/R_{1})-1). \tag{3}\]0  
Now, let us return to the consideration of the meridional section and consider two important special cases of the location of the radiation source (see Figure 4). The first of them, when \(z_{LGT}=-R_{1}\), is considered in the publication [11]. In this case, the radiation source is located at the curvature center of the waveguide. According to expression (8), the in-coupling DOE with a constant period, \(d_{0}^{IN}\), will ensure the in-coupling of a beam of rays from the point source in that position into the waveguide without aberrations. However, this will be true for the only emitter point, which lies on the axis of the waveguide cylinder. Other points of the emitter of finite dimensions will be in-coupled by such a DOE with an angular error. Another important case is the position of the radiation source at infinity when \(z_{LGT}\rightarrow-\infty\). This case seems to be the most relevant to the practice, since micro-projectors, as a rule, form an image close to infinity.