# Holographic Lens Array for Solar Collector with Large Angle and Expanded Spectral Width
## 2 Principle
### Spectral Width Expansion Within a Large Angle  
Due to the fact that each OHL in the HLASC is identical, the following analysis within the scope of an OHL unit is applicable to the entire HLASC. As shown in Figure 2a, OHL is recorded by two beams. One is a spherical light formed by parallel light entering the lens array, which corresponds to the incident light. Another is an oblique parallel light to provide an off-axis angle, which corresponds to the diffracted light. Generally, OHL is considered as a whole for reconstruction analysis. Due to the different reconstruction results of different regions, OHL is considered as many tightly arranged points in the following analysis. Firstly, we only consider a certain point \"A\" on the OHL, as shown in Figure 2b. The recording light is green light, as shown by the two wave vectors indicated by the green arrows. In the reconstruction process, when green light is incident at \(\theta_{A-g}\), it can be efficiently diffracted, while light with other wavelengths that are incident at this angle experience a sharp decrease in diffraction efficiency due to the mismatch of Bragg conditions. However, this wavelength shift can be compensated by angular deviation. As shown by the the red and blue arrows in Figure 2b, when red and blue light are  
Figure 1: Schematic diagram of the working principle of the HLASC. incident at angles \(\theta_{A-r}\) and \(\theta_{A-b}\), which deviate from \(\theta_{A-g}\), respectively, the grating vector, reconstructed vector, and diffracted vector can still form a closed triangle on the vector circle. Therefore, red and blue light can also be efficiently diffracted by point \"A\". Similarly, every point on OHL conforms to the above analysis, as shown in Figure 2c,d. In summary, every point on OHL can efficiently diffract a wide spectrum of light, but different diffraction wavelengths correspond to different incident angles. Based on the wide spectrum reconstruction characteristics mentioned above, due to the different recording angles of the spherical light at each point of OHL, when a wide-spectrum light is incident on OHL at a certain angle, different points on OHL will diffract different wavelengths, as shown in Figure 2e-g. Typically, as shown in Figure 2f, when a wide-spectrum light is vertically incident on OHL, points \"A\", \"B\", and \"C\" will efficiently diffract blue, green, and red light, respectively, which is consistent with the analysis in Figure 2b-d. Therefore, the spatial allocation for the collection spectrum is achieved. Similarly, OHL can also diffract wide-spectrum incident light at other incident angles, but the results of spectrum spatial allocations are different, as shown by the brown dashed boxes in Figure 2e-g. Based on the above analysis, OHL achieves the spectral width expansion of collected sunlight over a large angle range by using a single wavelength recording of a spherical wave and a plane wave. Moreover, we can flexibly design the wavelength and angle of recording light according to different demands, which will be analyzed in detail below. Figure 2: Recording and reconstruction analysis of OHL. **(a)** Schematic diagram of the recording. (**b–d**) are the multi-wavelength reconstruction analyses of point “A”, “B”, and “C” of OHL, based on k-vector analysis, respectively. The red, green, and blue arrows represent the wave vectors of the incident light and diffracted light of the corresponding colors, respectively. (**e–g**) are the spectrum spatial allocation results for different incident angle \(\theta_{A-g}\), \(\theta_{B-g}\), and \(\theta_{C-g}\). Obviously, the collection spectrum of the HLASC is the same as the diffraction spectrum of OHL. The diffraction spectrum of OHL is related to the recording conditions, and the specific analysis is as follows. It is worth mentioning that the below angles are all in the holographic material, not in the air.