## 2 Transmission Model of Stray Radiation  
Usually, due to the diversity of stray radiation sources and system design, it is difficult to obtain accurate mathematical expressions of stray radiation using theoretical methods. It is generally considered that stray radiation is transmitted from the previous stray radiation source to the next stray radiation source. The propagation process follows the law of energy conservation and satisfies the general law of light radiation propagation. Figure 1 shows the radiation transmission between two micro-facets. According to the theory of radiative energy transmission, it is assumed that there are two Lambert micro surface elements \(dA_{s}\) and \(dA_{r}\); the distance between them is \(l\) and the radiance is \(L_{s}\) and \(L_{r}\), respectively. The angles between the normal of the two facets and \(l\) are \(\theta_{s}\) and \(\theta_{r}\), respectively. According to the definition of radiance, the radiation power emitted from \(dA_{s}\) to \(dA_{r}\) is as follows:  
\[dP_{s\to r}=L_{s}\cos\theta_{s}dA_{s}d\Omega_{r\to s}, \tag{1}\]  
where the solid angle of \(dA_{r}\) to \(dA_{s}\) can be expressed as follows:  
\[d\Omega_{sr}=\frac{dA_{s}\cos\theta_{s}}{l^{2}}. \tag{2}\]  
By substituting Equation (2) into Equation (1), one can get  
\[dP_{s\to r}=\frac{L_{s}\cos\theta_{s}\cos\theta_{r}dA_{s}dA_{r}}{l^{2}}. \tag{3}\]  
Figure 1: Schematic diagram of radiation transmission between two micro-facets. It is assumed that the radiation energy incident on the micro-facet \(dA_{s}\) is \(d\Phi_{s}\), and the energy reflected on the other surface is \(d\Phi_{r}\). According to the definition of bidirectional reflectance distribution function (BRDF) for the Lambert radiation surface, the \(BRDF_{s}\) of the micro-facet \(dA_{s}\) can be expressed as follows:  
\[BRDF_{s}=\frac{L_{s}}{E_{s}}, \tag{4}\]  
where \(E_{s}\) is the irradiance on the micro-facet \(dA_{s}\). By substituting Equation (4) into Equation (3), one can get  
\[dP_{s\to r}=BRDF_{s}\cdot E_{s}\cdot\cos\theta_{s}\cdot dA_{s}\frac{dA_{r} \cdot\cos\theta_{r}}{l^{2}}. \tag{5}\]  
The angular coefficient, also known as the shape factor, represents the geometric relationship between the source object and the receiving object. According to the definition, the angle coefficient \(dF_{s\to r}\) from micro-facet \(dA_{s}\) to \(dA_{r}\) can be described by the following expression:  
\[dF_{s\to r}=\frac{dP_{s\to r}}{M_{s}dA_{s}}=\frac{\cos\theta_{s}\cos \theta_{r}}{\pi l^{2}}\cdot dA_{r}, \tag{6}\]  
where the \(M_{S}\) is the radiant emittance of \(dA_{s}\). The physical meaning of angular coefficient \(dF_{s\to r}\) is the ratio of the radiant power emitting from surface \(dA_{s}\) and received by surface \(dA_{r}\) to the total radiant power emitted from the surface. Considering  
\[d\Phi_{s}=E_{s}\cdot dA_{s}, \tag{7}\]  
\[d\Phi_{r}=dP_{s\to r}, \tag{8}\]  
one can get the following result by substituting Equations (6)-(8) into Equation (5):  
\[d\Phi_{r}=d\Phi_{s}\cdot BRDF_{s}\cdot dF_{s\to r}\cdot\pi. \tag{9}\]  
For the two finite surfaces, the result is as follows:  
\[d\Omega_{sr}=\frac{dA_{s}\cos\theta_{s}}{l^{2}}. \tag{2}\]0  
It can be seen from Equation (10) that the energy on the receiving surface is the product of three factors: the radiation power from the source surface, the BRDF of the source surface, and the angle coefficient \(F_{s\to r}\) from the source surface to the receiving surface. Therefore, in order to suppress the stray radiation in the infrared imaging system, the following methods are usually used: reducing the angle coefficient and the radiation power from the source surface as far as possible, and using the material with as low a BRDF as possible. There have been many studies on reducing the angle coefficient and the source surface emission power, but there are no reports on using a high-absorptivity ceramic coating to increase the absorption for stray radiation on the material surface, thereby reducing the BRDF.