# Holographic Lens Array for Solar Collector with Large Angle and Expanded Spectral Width
## 2 Principle
### Spectral Width Expansion Within a Large Angle
As shown in Figure 3a, when recording OHL with wavelength \(\lambda_{1}\) and reconstructing it with wavelength \(\lambda_{2}\), the deviation angle \(\Delta\omega\) between the reconstruction and recording vectors can be expressed as follows:  
\[\Delta\omega=\frac{\omega_{2}-\omega_{1}}{2} \tag{2}\]  
where \(\omega_{1}\) and \(\omega_{2}\) are the angles between the two recording wave vectors and two reconstruction wave vectors, respectively. When \(\lambda_{2}\geq\lambda_{1}\), \(\Delta\omega\geq 0\); when \(\lambda_{2}<\lambda_{1}\), \(\Delta\omega<0\). According to geometric relationships, there exists the following relationship between two angles:  
\[\frac{\sin(\omega_{1}/2)}{\sin(\omega_{2}/2)}=\frac{|k_{r2}|}{|k_{r1}|} \tag{3}\]  
where \(k_{r1}\) and \(k_{r2}\) are the wave vectors of the incident light during recording and reconstruction. Additionally, the relationship between \(k_{ri}\) (\(i=1\), 2) and wavelength \(\lambda_{i}\) is as follows:  
\[k_{ri}=\frac{2\pi}{\lambda_{i}} \tag{4}\]  
Combining Equations (2)-(4), there exists the following relationship:  
\[\Delta\omega=\arcsin\biggl{(}\frac{\sin(\omega_{1}/2)\lambda_{2}}{\lambda_{1} }\biggr{)}-\frac{\omega_{1}}{2} \tag{5}\]  
\[\lambda_{2}=\frac{\lambda_{1}\sin(\Delta\omega+\omega_{1}/2)}{\sin(\omega_{1} /2)} \tag{6}\]  
In this case, the reconstruction wavelength \(\lambda_{2}\) can be calculated using Equation (6) based on the recording wavelength \(\lambda_{1}\), the recording angle \(\omega_{1}\), and the reconstruction deviation angle \(\Delta\omega\). Due to the non-uniform grating structure of OHL, the above process is used to analyze the reconstruction situation of a certain point. The diffraction wavelength of OHL varies monotonically within the diameter, so the diffraction wavelength on the two  
Figure 3: The model for analyzing the diffraction spectrum of OHL. (**a**) K-vector analysis model for the reconstruction with a variable wavelength. The green and red arrows represent the wave vectors during recording and reconstruction, respectively. (**b**) Multi-wavelength reconstruction model of OHL with spectrum spatial allocation. edge recording points can be analyzed to determine the overall diffraction spectrum. As shown in Figure 3b, the recording light is indicated by the green dashed arrow. For vertically incident wide-spectrum light, the diffraction wavelength \(\lambda_{o}\) at point \"o\" is the same as the recording wavelength \(\lambda_{1}\), and according to Equation (6), the diffraction wavelengths \(\lambda_{p}\) and \(\lambda_{q}\) at the two edge points \"\(p\)\" and \"\(q\)\" can be expressed as follows:  
\[\lambda_{p}=\frac{\lambda_{1}\text{sin}\big{[}\big{(}\omega_{o}+\Delta\omega_{ p}\big{)}\,/2\big{]}}{\text{sin}\big{[}\big{(}\omega_{o}-\Delta\omega_{p}\big{)}\,/2 \big{]}} \tag{7}\]  
\[\lambda_{q}=\frac{\lambda_{1}\text{sin}\big{[}\big{(}\omega_{o}-\Delta\omega_{ q}\big{)}\,/2\big{]}}{\text{sin}\big{[}\big{(}\omega_{o}+\Delta\omega_{q}\big{)}\,/2 \big{]}} \tag{8}\]  
where \(\omega_{o}\) is the angle between the two recording wave vectors at the center point \"\(o\)\", and where \(\Delta\omega_{p}\) and \(\Delta\omega_{q}\) are the deviation angles between the incident light at points \"\(p\)\" and \"\(q\)\" and the incident light at the center point \"\(o\)\" during recording, respectively. It should be noted that \(\Delta\omega\) and \(\Delta\omega_{p/q}\) do not represent the same angle difference. The former represents the difference between the reconstructed angle and the recorded angle at a certain point of OHL during the variable wavelength reconstruction process, while the latter represents the difference in the incident angle of the spherical wave at different points of OHL during the recording process. Therefore, the diffraction spectrum of the entire OHL is \(\lambda_{q}\)-\(\lambda_{p}\). In different application scenarios, the collection angle and spectrum required for the HLASC may be different.