## 4 Analysis
### Analysis of Luminous Flux Utilization
#### 4.2.2 Effect of the Air Gap  
The SBFTIS relies on adjusting \(s\) to achieve the desired spectral resolution. However, it is important to note that changes in the air gap thickness also influence the wave's propagation trajectory, which, consequently, affects the aperture stop and luminous flux of the system. Hence, it is crucial to maintain a balance between achieving the desired spectral resolution and optimizing the luminous flux.  
To determine the aperture stop size and luminous flux utilization for a different \(s\), calculations were performed using a point source of spherical waves with a fixed FOV of 10\({}^{\circ}\). Using spherical waves to calculate under different air gap thicknesses (s = 0 mm, 2.5 mm, 5 mm, 7.5 mm, 10 mm), the aperture stop and luminous flux utilization ratio of the prism at different spectral resolutions are shown in Figure 17:  
The meanings of the rectangles in Figure 17 are the same as those in Figure 15. As the air gap \(s\) increases, the aperture stop of the system (the gray part in the figure) shrinks, and  
Figure 16: Captured interferograms (**a**) FOV = 5\({}^{\circ}\), (**b**) FOV = 7.5\({}^{\circ}\), (**c**) FOV = 10\({}^{\circ}\).  
Figure 17: When FOV = 10\({}^{\circ}\), \(s\) = 0–10 mm: (**a–e**) Aperture stop of DWP at different \(s\); (**f**) Variation in luminous flux utilization with s.  
\(\eta\) also decreases. When FOV = 10\({}^{\circ}\), as \(s\) increases from 0 mm to 10 mm, the aperture stop area decreases from 293.44 mm\({}^{2}\) to 207.68 mm\({}^{2}\), as Figure 17a-e shows, and \(\eta\) decreases linearly from 73.36% to 51.92%, as Figure 17f shows. Combining Figures 15f and 17f, the change rate of \(\eta\) with \(s\) is greater than that with FOV, indicating that the luminous flux is more affected by the air gap thickness.  
To experimentally verify the results, interferograms of various air gap thickness were captured, and \(s\) was limited to a range of 0-10 mm due to the sampling law. Interferograms were captured for s values of 0, 2.5, 5, 7.5, and 10 mm with an FOV = 10\({}^{\circ}\), as shown in Figure 18.  
It should be noted that the total energy of light entering the system remained constant as the FOV was kept constant. Thus, as \(s\) increased from 0 mm to 10 mm, the aperture utilization of the system decreased by 21.44%, resulting in a decrease in the luminous flux. Therefore, for the same exposure time, a larger air gap thickness resulted in darker interferograms. The minimum aperture (\(S_{min}\) = 207.68 mm\({}^{2}\)) was still larger than the size of the camera photosensitive chip (\(S_{chip}\) = 145.41 mm\({}^{2}\)), and the interferogram had no error. While the spectral resolution of the system improved with increasing \(s\), the brightness of the whole picture was gradually dimmed due to the decrease in luminous flux, and thus the signal-to-noise ratio of the interference fringes decreased. As a result, the system noise increased, which directly affected the accuracy of the system's restored spectrum, reduced the availability of the system in complex environments, and limited the scope of the application of the system. Therefore, a balance needed to be maintained between spectral resolution and luminous flux in complex environments.