## 4 Simulation Analysis and Verification  
To verify the influence of the optical diffraction effect on the BPS-FTS spectrum, we set the pixel size of the simulation analysis to 18\(\upmu\)m; the simulated spectral range is 2040 nm\(\sim\)2060 nm, and the corresponding wavenumber range is 4887 cm\({}^{-1}\)\(\sim\)4902 cm\({}^{-1}\). The spectral curve is shown in Figure 6.  
For the simulation process, we assume that the optical system is ideal and that no other error effect exists except for optical diffraction. To meet conventional optical design requirements, we assume that the Airy disk radius of the central wavelength spectrum is exactly half the pixel size; that is, \(r_{0}\)= \(P/2\). According to the data in Table 1, we can obtain the radius corresponding to the fourth dark ring in a straightforward manner as follows:  
\[r_{4}\text{= }(4.24/1.22)r_{0}\vartheta \tag{21}\]  
Because the spectral range is narrow, in the 20 nm band, the diffraction pattern distributions of different wavelength spectra on the image plane are fundamentally the same. To reduce simulation complexity, the intensity distribution of the diffraction pattern of the central wavelength spectrum is uniformly used to approximate the intensity distribution of all other wavelength spectra, and this approximation will not affect the judgment of the final result. Figure 5: Relationship between the energy distribution of the diffraction pattern and pixels. For BPS-FTS, a grating is typically used to subdivide the spectrum. It is assumed that under the effects of the grating, the 20 nm spectrum is uniformly subdivided into 10 narrowband spectra and sampled by 10 pixels, each of which corresponds to a sampling bandwidth of 2 nm. Using an interferometer, an interferogram corresponding to the spectrum is obtained on the image plane. We assume that the Fourier transform spectrometer can achieve a spectral resolution of 0.1 nm and the maximum optical path difference of the interferometer is 2.5 cm. Therefore, we obtain a set of reasonable optical system simulation parameters, as shown in Table 2.  
The simulation is divided into three parts and is based on the above set of optical parameters. Firstly, the influence of the diffraction effect on spectral distribution is simulated. The influence of the diffraction effect on the interferogram is simulated and analyzed on this basis. Finally, through the simulation of the recovery spectrum, we compare and analyze the influence of the diffraction effect on the recovery spectrum. In the data comparative analysis, some edge spectra are discarded. (1) Simulation of the diffraction effect on the spectral bandwidth of pixels  
In this section, we simulated the distribution of the original input spectrum on each pixel. The spectral bandwidth of each pixel is shown in Table 3; the last column shows the ratio of bandwidth both with and without considering the diffraction effect. It can be observed from the above calculation results that, owing to the diffraction effect influence, the actual spectral bandwidth corresponding to each pixel is wider than the ideal spectral bandwidth. Due to the boundary effect, the spectral bandwidth and energy corresponding to the four pixels on the edge are lost. Therefore, in this study, only spectral data from the middle six pixels were used in the data comparison analysis, which did not affect the conclusion. Based on the spectral bandwidth data of the middle six pixels, it can \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Radius of circular hole & 0.5706 cm \\ F-number & 3.5986 \\ Number of grating lines & 200 lines/mm \\ Focal length & 4.1064 cm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optical system parameters. Figure 6: Input spectrum. be observed that the real spectral bandwidth caused by the diffraction effect is 4.48 times the ideal spectral bandwidth.