## 3 Analysis of the Influence of the Diffraction Effect  
As can be observed from Figure 1, BPS-FTS is a fusion of dispersive and Fourier transform spectroscopy. When the moving mirror is stationary and in the zero optical path difference position, the system is equivalent to a low-resolution grating spectrometer, where each sensor pixel outputs an integrated value of the spectrum with its narrowband. When the mirror moves back and forth, each pixel outputs a narrowband interferogram, and a higher-resolution spectrum is obtained by spectral recovery. However, after grating splitting, the spectral range of the narrowband corresponding to each pixel in the image plane is frequently unknown, and the main reason for this stems from the optical system-diffraction effect. Therefore, we first analyze the diffraction effect from the perspective of grating splitting [12; 13; 14], and the system-equivalent optical path is shown in Figure 2.  
Assuming that the focal length of the imaging mirror is \(f\), \(d\) is the distance between grating groves, and light of wavelength \(\lambda\) is incident on the grating at an angle of incidence \(\alpha\), the resulting diffraction angle is \(\beta\). Then, for the full broadband spectrum, the range of \(\beta\) is \([\beta_{\min},\beta_{\max}]\). We assume that the central wavelength \(\lambda_{c}\) has a diffraction angle \(\beta_{c}\) and is imaged at the origin of the image plane coordinates, with the dispersion direction  
Figure 2: System equivalent optical path schematic. coinciding with the y-axis. According to the geometric relations, the position \(y\) of any angle \(\beta\) in the image plane can be obtained as follows:  
\[\frac{y}{f}=\tan(\beta-\beta_{c}) \tag{7}\]  
Then  
\[\beta=\arctan(\frac{y}{f})+\beta_{c} \tag{8}\]  
According to the grating equation [12; 13; 14; 15; 16], the relationship between the incident angle, the diffraction angle, and the wavelength can be obtained as follows:  
\[d(sin\pm sin\beta)=n\lambda,n=0,\pm 1,\pm 2,\cdots \tag{9}\]  
Taking \(n=+1\) as the order of diffraction, we obtain:  
\[\lambda=d(sin\alpha\pm sin\beta) \tag{10}\]  
Combining Equations (7) and (10), we can obtain the distribution of wavelength \(\lambda\) within the detected spectral range \(y\) is:  
\[y=f\tan\biggl{\{}\arcsin[\pm(\frac{\lambda}{d}-\sin\alpha)]-\beta_{c}\biggr{\}} \tag{11}\]  
Obviously, Equation (11) represents the distribution of the spectrum over the sensor without considering optical diffraction. When considering the optical diffraction effect, we assume that the system aperture diaphragm is circular with a radius of \(a\). According to the Fraunhofer circular aperture diffraction theory, at this stage, for any point \(y\) on the sensor, the corresponding light of wavelength \(\lambda\) forms a diffraction spot on the image plane (typically its central bright spot is called an Airy disk) with the center point at \(y\). Then, the intensity of light at a distance \(r\) from the center can be expressed as follows [12; 13]:  
\[S(r)=S_{0}\left[\frac{2J_{1}\pi\frac{2\pi\alpha r}{\lambda f}\pi}{\frac{2\pi \alpha r}{\lambda f}}\right]^{2} \tag{12}\]  
where \(J_{1}\) is a first-order Bessel function and \(S_{0}\) is the light intensity at the center of the Airy disk. The intensity distribution of the diffraction pattern is shown in Figure 3, where \(r=\sqrt{x^{2}+y^{2}}\). It can be observed that the intensity of the diffraction pattern decays with the increase in \(r\) in a sharp oscillation, showing a ring-like distribution between light and dark. Table 1 provides the \(r\) values corresponding to different light and dark rings and the ratio of their intensity values to the total incident light intensity [11; 12; 13; 17; 18].