## 2 Operation Principle/System Design
### Propagation of Optical Radiation Inside Planar and Cylindrical Waveguides
A schematic representation of propagation of rays in the meridional plane is shown in Figure 3b)  
The case of propagation of a ray in the sagittal plane fully corresponds to the case of propagation of a ray in a plane-parallel waveguide described above. Due to the parallelism of the normals to the waveguide surface at all points, the waveguide transmits a virtual image without aberrations and forms an image at infinity. Figure 3: Propagation of a ray inside (a)—a flat waveguide; (b)—a concentric waveguide. In turn, in the meridional plane, the angles of incidence and reflection of a ray relative to the normals to the surfaces of the waveguide will be equal and constant in pairs for its outer and inner surfaces [10]. Thus, we can write the following relations:  
\[\beta_{1}=\beta_{2}=\beta_{4k+1}=\beta_{4k+2};\,\beta_{3}=\beta_{4}=\beta_{4k+3} =\beta_{4k+4}\text{ for }k=1,2,3,4\,\ldots\,,\]  
\[\beta_{3}=-\text{arcsin}(\sin(\beta_{1})R_{2}/R_{1}). \tag{1}\]  
Due to the preservation of the angle of incidence of a ray on each of the surfaces, each interaction of a ray with the out-coupling DOE, having a constant period, forms a diffraction response in one direction relative to the normal to the waveguide surface. In this case, the normals to the waveguide surface are not parallel to each other, but intersect on the axis of the cylindrical surfaces of the waveguide. Thus, in the meridional plane, the described waveguide focuses the rays. Slanted rays that lie neither in the sagittal nor in the meridional planes will propagate along such a trajectory that its projections on these planes will correspond to those depicted in Figure 3a,b. Thus, for a slanted ray, the angles of incidence and reflection of a ray will also be equal relative to the normal to the waveguide surface at each point of incidence for the inner and outer surfaces of the waveguide. We used this property of a concentric cylindrical meniscus in the present paper for designing a curved combiner. Thus, astigmatic aberration is observed when operating the waveguide--the meridional components of all rays coupled out of the waveguide are not equal, while their sagittal components are equal. Moreover, this aberration manifests itself both when a ray is coupled out of the waveguide, and, in the opposite direction, when several parallel rays are coupled into the waveguide--their angles of in-coupling are not equal. The suppression of this aberration is the main problem being solved in the present paper. Now, let us consider the propagation of a beam of rays, which forms a certain component of a FOV inside a waveguide. This case seems to be the most relevant to practice, since micro-projectors used in wearable AR devices, as a rule, form each component of a FOV not by a single ray, but by a beam of rays with a non-zero aperture. The astigmatism described above leads to aberrations in a beam, both when coupling into a waveguide and out of waveguide. Due to this, when an image is transmitted through the described concentric meniscus, blurring occurs and image doubling appears in the meridional plane, shown in Figure 2b. Then, in order to transmit a beam of rays through a cylindrical waveguide without aberration, it is necessary to ensure the equality of the angles of incidence relative to the normals to the waveguide surface at the points of incidence, for all rays in the beam inside the waveguide. This will allow to uniquely identify the belonging of a ray to a specific beam when it is coupled out of the waveguide. It is important to note that in this case, each such beam will not be homocentric--the extensions of all rays will not intersect at one point. Moreover, the areas of intersection of rays associated with different parts of an angular field will be located at different distances. The formation of a beam of such a structure is a nontrivial task for optics.