## 2 Operation Principle/System Design
### In-Coupling of Radiation of a Point Source into a Curved Waveguide  
Let us consider the in-coupling of radiation from a point source into a waveguide using a DOE (2) located on the inner surface of a concentric cylindrical waveguide (1) (Figure 1). Let us derive formulas describing variation of the period of the in-coupling DOE (2), which ensures the in-coupling of radiation into a waveguide with the conditions formulated in the previous part of the paper. Let us consider the path of rays in the meridional plane (Figure 4). In case of a curved waveguide, for diffraction with an order of minus one and refraction of a ray incident at an angle \(\alpha\) at an arbitrary point with a curvilinear coordinate \(L\) of such an in-couplingDOE on a curved waveguide surface, the diffraction grating equation can be written in the following form:  
\[d^{IN}(L)(\sin(\alpha)-n_{w}\sin(\beta))=m\lambda, \tag{2}\]  
where \(n_{w}\)'the refractive index of the waveguide, \(\lambda\)--the radiation wavelength, \(m\)--the diffraction order number (\(m=\pm 1\), \(\pm 2\),... ); \(\beta\)--the ray propagation angle inside the waveguide (with respect to the normal at the entry/incidence point), \(L\)--the curvilinear coordinate on the curved surface of the waveguide (its projection on the \(OY\) axis is marked as \(y_{\rm diff}\), and on the \(OZ\) axis, as \(z_{\rm diff}\)). By setting one or another dependence of the period of the in-coupling DOE on the coordinate \(d^{IN}(L)\), we can synthesize a DOE that will solve the above-described problem of in-coupling of radiation into a curved waveguide. We use in the calculations the curvilinear coordinate on the curved surface of the waveguide \(L\), because, as will be shown below, in practice (in the experiment) we used HOEs as DOEs. The HOEs were recorded on a flat surface of a technological prism. Therefore, to calculate the HOE recording scheme, we needed the curvilinear coordinate \(L\), which is converted into one of the coordinate axes of the Cartesian system in the HOE recording plane. First of all, let us consider the case when we need to in-couple radiation from a point source located at a finite distance from the waveguide, \(Z_{LGT}\) (Figure 4). In this figure and all the figures below, angles counted counterclockwise are considered positive; counted clockwise--negative. The dependences of linear coordinates of the ray incidence point on the in-coupling DOE in the meridional plane on \(L\) can be expressed as follows:  
\[y_{\rm diff}=R_{1}\sin(\frac{L}{R_{1}}),z_{\rm diff}=R_{1}(\cos(L/R_{1})-1). \tag{3}\]  
On the other hand, the angular coordinate \(\blacksquare\) of this point relative to the curvature center of the waveguide is defined as  
\[\blacksquare=\frac{L}{R_{1}}. \tag{4}\]  
Figure 4: Coupling of rays (blue lines) into a waveguide from a point source of radiation (for the meridional plane). 1—the inner surface of the curved waveguide in the meridional plane; 2—the point source of radiation; 3—the center of the curvature radius of the waveguide surface. The angle of incidence of an arbitrary ray relative to the OZ axis, \(\varphi\), from a point source of radiation to a point with the coordinate \(L\) on the in-coupling DOE can be expressed from (3):  
\[\varphi=\mathrm{arctg}\bigg{(}\frac{y_{\mathrm{dif}}}{-Z_{LGT}+z_{\mathrm{dif}}} \bigg{)}=\mathrm{arctg}\left(\frac{R_{1}\sin(L/R1)}{-Z_{LGT}+R_{1}\cos\left( \frac{L}{R_{1}}\right)-R_{1}}\right). \tag{5}\]  
From (4) and (5), the angle of incidence of the same ray relative to the normal at the point of incidence will be:  
\[\alpha=\varphi-\blacksquare=\mathrm{arctg}\left(\frac{R_{1}\sin\left(\frac{L}{R_{1 }}\right)}{-Z_{LGT}+R_{1}\cos\left(\frac{L}{R_{1}}\right)-R_{1}}\right)-\frac{ L}{R_{1}}. \tag{6}\]  
Let the period of the in-coupling DOE is equal to \(d_{0}^{IN}\) at the point of incident ray in-coupling, wherein the direction of this ray coincides with the direction of the curvature radius of the waveguide surface, R1 (i.e., \(\alpha=0\)). We choose this point as the origin of coordinates (\(L=0\)).