## 3 Optical Fabrication of the Designed System  
As shown in the previous parts of the paper, in order to obtain non-aberration image transmission through a cylindrical waveguide, DOEs/HOEs with a period variation along the curved coordinate \(L\) are required. In other words, the described DOEs should perform as holographic lenses that couple two wavefronts: flat and cylindrical ones. Considering an HOE located on the surface of a curved waveguide, in the global coordinate system, radiation propagating in the air is characterized by a flat wavefront, while radiation propagating in the waveguide is characterized by a cylindrical wavefront. If we pass to a coordinate system associated with the waveguide--to such a system in which one of the coordinate axes at each point is directed along the normal to the waveguide surface at this point--a flat wavefront, on the contrary, characterizes the radiation propagating in the waveguide, and a cylindrical wavefront characterizes the radiation propagating in the air. Thus, one of the methods for recording the described HOEs is the formation of a two-ray interferometer with one plane wavefront and one cylindrical wavefront. In this case, in order that the formed period of the HOE be equal to the required one, the recording and the readout wavelengths should be as close as possible. Since the HOE was recorded on the flat surface of a glass prism, the reference beam associated with the radiation propagating in the waveguide should be characterized by a flat wavefront. An object beam associated with radiation propagating in the air should be characterized by a cylindrical wavefront. Based on the prerequisites described above and using the obtained law of variation of the periods of the in-coupling and out-coupling HOEs, the parameters of the optical setup for recording the HOEs were calculated: the focal length and position of the cylindrical lens, as well as the convergence angle of the recording beams. Figure 11 shows the path of rays in a prism (Figure 11a), and a set of symbols used in the calculations (Figure 11b). Figure 10: Angular errors for rays out-coupling from the combiner for different combinations of coordinates in the EMB and components of the angular FOV for out-coupling DOE with a variable period: (**a**)—according to analytical dependence (17); (**b**)—after the optimization procedure. The diffraction grating Formula (2) was used to calculate the position of the virtual waist of the object beam. The problem was solved discretely for one hundred values of the coordinate \(L\) located along the \(OY\) axis in Figure 11b. For each coordinate \(L\), the angle of incidence of the object ray (\(\alpha_{f}\)) was calculated using Formula (18):  
\[\alpha_{f}(L)=\arcsin\left(\frac{\lambda}{n_{f}d(L)}+\sin\left(\beta_{f}\right) \right), \tag{18}\]  
where \(\lambda\)--the recording wavelength, \(n_{f}\)--the refractive index of the holographic medium, \(d\)--the surface period of the grating, \(\beta_{f}\)--the angle of incidence of the reference beam, which is a constant for all points of an HOE. Although a volume holographic grating is usually characterized by a normal period (i.e., the distance between holographic planes along the normal to them), it must be emphasized that in this work, holograms are characterized by a surface period (i.e., the distance between holographic planes along the surface of the hologram, as shown on the inset of Figure 11b). This way of characterizing a hologram is due to the fact that in the theoretical part of the work DOEs were regarded as surface ones. The angle \(\beta_{f}\) was chosen to be equal to the angle of propagation for the central ray of the FOV in the waveguide for the recording wavelength. The equivalent ray propagation angle in the prism (\(\alpha_{p}\)) and in the air (\(\alpha_{a}\)) were found using [PERSON]'s formula for each propagation angle of a ray of the object beam in the medium (\(\alpha_{f}\)).