# An Integration Model for Flux Density Distribution Formed by a Heliostat
## 2 Methodology  
### The Energy Density Ratios of the Reflected Ray  
Solar rays from various positions will reach different positions on the receiver surface after being reflected by a certain point of the heliostat. Optical errors at different points of the heliostat will cause the reflected ray to deviate from the ideal direction, as shown in Figure 1. We calculated the flux density contribution of the reflected solar ray to a certain point on the receiver surface by integrating all of the reflection points on the heliostat [9]. Therefore, the distribution of the flux density on the whole receiver surface could be computed. The rays reflected by the heliostat were divided into three main parts [11]: \(I_{1}\)(4%), \(I_{2}\)(89.5%), and \(I_{3}\)(3.5%), as shown in Figure 2. The incident rays first reach the upper surface of the glass, and part \(I_{1}\) is directly reflected, accounting for approximately 5%. The rest of the rays are transmitted through glass, reflected by the metal reflecting surface, and transmitted out of the upper surface. These are the main reflected rays, namely part \(I_{2}\), accounting for approximately 91.5% when the incidence angle is small. The remaining part is reflected again from the upper surface, and again from the reflective metal surface. When it reaches the upper surface, it transmits out part \(I_{3}\), accounting for approximately 3%. The rest of the rays still travel inside the glass, but the fraction is negligible. The focuses of the three parts of the beams are different, but the part \(I_{2}\) mainly determines the distribution of the flux density on the receiver plane, so, in practice, the rays of part \(I_{2}\) are aligned with the center of the receiver plane. When the distance between the receiver plane and the heliostat is large enough, the difference in the focal positions of the three parts can be ignored. Meanwhile, the contribution of part \(I_{1}\) and part \(I_{3}\) to the flux density is not tiny. When a more accurate flux density distribution is required, the optical errors of these two parts should be taken into account, and the flux density distribution is different from that of part \(I_{2}\). In particular, part \(I_{1}\) and \(I_{3}\) are not taken into account when calculating the intercept factor, and the curvature of the intercept curve will differ significantly from the actual situation, especially when the intercept factor reaches 90%. The proportion of \(I_{1}\), \(I_{2}\), and \(I_{3}\) is related to the incident angle. There are also polarization changes in reflection and refraction on the glass surface. We first decomposed the amplitude vector of the ray into two directions. The one perpendicular to the vibration of the incident plane is called the S component, and the one parallel to the vibration of the incident plane is called the P component. They have identical parts in solar rays. Figure 1: Slope error \(\sigma_{slope}\) and reflection cone \(\theta\). Figure 2: Composition of reflected rays. The incident angle is \(i_{1}\) and the ratios of the refractive index of the two sides of the reflected glass interface are \(n_{1}\) and \(n_{2}\), respectively. According to the refraction law, the refraction angle inside the glass can be calculated as \(i_{2}\).