## 5 Conclusions  
In this paper, the wave normal tracing method was used to accurately calculate the displacement and offset of light incident on the SBFTIS at any direction, and the spatial distribution of the displacement and offset was obtained. An experimental verification of the offset was conducted under different air gap thicknesses, and the obtained results were in good agreement with the theoretical results, thereby establishing the validity of the calculation. The aperture stop's size and luminous flux utilization of the SBFTIS under different FOVs and air gap thicknesses were analyzed. The proposed research filled the gap in the literature by providing a systematic study and analysis of the aperture and luminous flux in the SBFTIS, which had not been done before.  
The results indicated that the eooe light and oeeo light were mainly displaced in the \(z\) direction and had almost no displacement in the \(y\) direction. The displacement was verified by experiments, and the calculated results were in good agreement with the experimental results. As the FOV of the incident wave increased, the luminous flux utilization showed a linear decrease, ranging from 94.37% (FOV = 0\({}^{\circ}\)) to 73.36% (FOV = 10\({}^{\circ}\)) at \(s\) = 0 mm. Similarly, as the air gap thickness \(s\) increased, the luminous flux utilization linearly decreased, ranging from 73.36% (\(s\) = 0 mm) to 51.92% (\(s\) = 10 mm) at FOV = 10\({}^{\circ}\). The impact of the air gap thickness on the aperture stop size and luminous flux utilization was more pronounced. The experimental findings indicated that an increase in FOV from 5\({}^{\circ}\) to 10\({}^{\circ}\) led to a slight decrease in aperture but a doubling of the amount of light entering the system. Thus, a larger FOV resulted in higher luminous flux for the same exposure time. On the other hand, as s increased from 0 mm to 10 mm, the aperture utilization of the system decreased, leading to a reduction in luminous flux. As a result, a larger air gap thickness led to a darker interferogram, higher spectral resolution, and lower signal-to-noise ratio of the interference fringes for the same exposure time. Therefore, in complex environments, it is crucial to maintain a balance between spectral resolution and luminous flux.  
The aperture stop of the SBFTIS indirectly affects its spectral resolution adjustment range and FOV, while directly affecting its luminous flux utilization; hence, it is important to consider the aperture and maintain a balance between the various parameters. The application of the proposed research is to guide the design and development of a spatio-tempo hybrid modulated static birefringent crystal Fourier transform imaging spectrometer with adjustable spectral resolution. The obtained results, such as the relationship between aperture and luminous flux utilization, FOV and air gap thickness, and the restrictive relationship between aperture and luminous flux utilization, FOV and spectral resolution, provide meaningful guidance for the use of the system.  
Conceptualization, X.Z.; methodology, X.Z.; validation, X.Z.; formal analysis, X.Z.; writing--original draft preparation, X.Z.; writing--review and editing, X.Z., Y.Z., H.W., F.G. and L.H.; visualization, X.Z. and J.D.; supervision, X.Z.; project administration, J.Z.; funding acquisition, J.Z. All authors have read and agreed to the published version of the manuscript. This research was funded by National Natural Science Foundation of China, grant numbers 61890961, 62127813, 62001382, and 62201568; Key R&D project in the Shaanxi Province of China, grant number 2020 GY-274; Shaanxi Natural Science Basic Research Program, grant numbers 2018J6008 and 2022 JQ-693. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: All data are contained within the article. Conflicts of Interest: The authors declare no conflict of interest.