## 2 Theoretical Background  
[PERSON]'s theory [17] is widely accepted for modelling the performance of volume holograms. This theory shows that high diffraction efficiency can be related to some physical characteristics, such as thickness, spatial frequency and refractive index modulation. According to [PERSON]'s theory, the diffraction efficiency (\(\eta\)) can be theoretically evaluated as: [18]  
\[\eta=\frac{\sin^{2}\sqrt{\xi^{2}+v^{2}}}{\left(1+\frac{\xi^{2}}{v^{2}}\right)}, \tag{1}\]  
where the parameters \(\xi\) and \(v\) are defined as:  
\[\xi=\Delta\theta\frac{\mid\stackrel{{\rightarrow}}{{G}}\mid d}{2}, \tag{2}\]  
\[\
u=\frac{\pi\Delta nd}{\lambda\cos\theta_{Bragg}}, \tag{3}\]  
In Equations (2) and (3) \(d\) is the holographic film thickness, \(\Delta n\) is the refractive index modulation; \(\lambda\) is the wavelength of the reconstructing beam; \(\theta_{Bragg}\) is Bragg diffraction angle; \(\Delta\theta\) is the deviation from the Bragg angle and \(\stackrel{{\rightarrow}}{{G}}\) is the grating vector, normal to the fringes with a magnitude \(\mid\stackrel{{\rightarrow}}{{G}}\mid=2\pi/\Lambda\), with \(\Lambda\) grating period:  
\[\Lambda=\lambda/2*\sin(\theta/2), \tag{4}\]  
here \(\lambda\) is the recording wavelength and \(\theta\) is the angle between the incident and diffracted angles inside the medium. When the Bragg condition is satisfied, that is, \(\Delta\theta\) = 0, the volume holographic grating (VHG) diffraction efficiency \(\eta\) can be theoretically evaluated as [17; 19]:  
\[\eta=\sin^{2}\left(\frac{\pi\Delta nd}{\lambda\cos\theta_{Bragg}}\right), \tag{5}\]  
Depending on the applications, the holographic optical element and in particular for solar applications the holographic lens, requires particular values of angular and/or wavelength selectivity. Due to the chromatic selectivity, hologram diffraction efficiency depends on wavelength; while regarding the angular selectivity, the hologram diffraction efficiency decreases very quickly when the direction of the incident radiation does not fulfil the Bragg condition in the recording plane [10]. It is important to point out that the main difference between conventional optics and holographic optics is that to determine diffracted ray direction, Snell's law is replaced by the grating equation [20]. As expected, the angle at which the diffraction intensity reaches its maximum is strictly related to the incident wavelength [21]. In particular, the angle increases as the wavelength increases. This behaviour is due to Bragg phase-matching condition for VHG, which (considering a non-slanted transmission grating, i.e., gratings with a grating period vector forming an angle of 90\({}^{\circ}\) with the perpendicular to the plane of incidence) is defined as:  
\[\stackrel{{\rightarrow}}{{k_{i}}}-\stackrel{{ \rightarrow}}{{k_{d}}}=\stackrel{{\rightarrow}}{{G}}, \tag{6}\]  
where \(\mid\stackrel{{\rightarrow}}{{k_{i}}}\mid=\mid\stackrel{{ \rightarrow}}{{k_{d}}}\mid=2\pi n/\lambda\) are the magnitude of the incident and diffracted wave vectors inside the medium and angles \(\theta_{i}\), \(\theta_{d}\) are the incident and diffracted angles inside the medium. \(\stackrel{{\rightarrow}}{{G}}\) is the grating vector defined above. Generally, two regimes in which phase gratings operate are defined: the Raman-Nath regime, in which several diffracted waves are produced and the Bragg regime, where basically only one diffracted wave is formed and this occurs only for near Bragg incidence. It has been usual to refer to gratings that work in the Raman-Nath or the Bragg regimes as thin and thick gratings, respectively. A confirmation that the recorded hologram is a volume and not a surface hologram can be obtained by evaluating a parameter called \(Q\) factor.