## 2 Operation Principle/System Design
### Propagation of Optical Radiation Inside Planar and Cylindrical Waveguides  
In order to formulate the conditions that provide propagation of a beam of rays inside a cylindrical waveguide without aberration, at first, we should analyze the propagation of rays inside a plane-parallel waveguide. To do this, let us consider a flat, equal thickness waveguide with in-coupling and out-coupling DOEs having constant and equal periods of gratings located on the inner surface of the waveguide (Figure 2a). The figure shows the initial image of the test template introduced into the waveguide by the in-coupling DOE, and the image emerged from the waveguide by the out-coupling DOE. As we can see, such a waveguide transmits the test image in a certain angular range (between the angle of TIR and the glancing angle of \(\sim\)90\({}^{\circ}\)), and the image is transmitted without aberration. Figure 1: A scheme of the developed system. A 3D view (a); a section of a waveguide by the _YOZ_ plane and the path of rays (b). 1—a cylindrical concentric waveguide; 2—an in-coupling DOE; 3—an out-coupling DOE; 4—the direction of the projector’s beam; 5—the direction of the out-coupling beams from the waveguide to the user’s eye; 6—the axis of the cylinder, passing through the curvature center of the waveguide. Figure 2: An image transmitted by equal thickness waveguides with in-coupling and out-coupling DOEs having constant periods equal to each other: (a)—a flat shape of a waveguide; (b)—a cylindrical shape. Now, let us bend the same waveguide and fix its shape. Since the initial shape of the waveguide is plane-parallel, we obtain the waveguide in the form of a cylindrical concentric meniscus as a result of bending (Figure 2b). Such a waveguide is formed by two cylindrical surfaces, whose curvature centers coincide. If we now introduce an image of the same template into this waveguide through the in-coupling DOE and then observe the image out-coupled from the out-coupling DOE, then we will obtain an out-coupled image with significant aberration, which is illustrated in Figure 2b. In this case, aberration of the image is present only along the axis perpendicular to the axis of the cylinder. To explain the observed quality of the transmitted image, let us consider the path of rays inside plane-parallel and concentric waveguides. In case of TIR, a ray inside a plane-parallel waveguide propagates with the angles of incidence and reflection \(\beta\) (Figure 3a), equal in absolute value relative to the normal to the surface for both surfaces of the waveguide:  
\[\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=\beta_{4k+1}=\beta_{4k+2}=\beta_{4k+3} =\beta_{4k+4}\text{ for }k=1,2,3,4\dots\]  
In a plane-parallel waveguide, each interaction of a ray with an in-coupling/out-coupling DOE having a constant period leads to the occurrence of a diffraction response making the same angle with normals to the waveguide surface. Since the normals to the waveguide surfaces at each interaction point are parallel to each other, it can be concluded that a beam of parallel rays incident on an in-coupling DOE will be transferred by the waveguide and out-coupled using an out-coupling DOE, also in the form of a parallel beam. Thus, a virtual image will be formed at infinity. This transformation of beams by a waveguide allows the user to observe a sharp virtual image. Now, let us consider the path of a ray inside a cylindrical concentric waveguide. For rays propagating in a plane parallel to the axis of the cylindrical surfaces of the waveguide, such a waveguide will operate as a plane-parallel one (Figure 3a). Let us take this plane as the sagittal _XOZ_ plane. In a plane perpendicular to the axis of the cylindrical surfaces, the waveguide is a concentric meniscus. For further study, let us take this plane as the meridional plane _YOZ_.