## 3 Optical Design and Analysis
### High-Fidelity Spectrometers  
The spectrometer is one of the most important parts of GeoFWHIS. In each band, four identical spectrometers are spliced to cover the image plane of the fore-optics completely. The slit of a single spectrometer is still quite long, especially in B1 and B2 bands, with a maximum slit length of 61.44 mm. The splicing diagram of four spectrometers is shown in Figure 4, and the slits of two adjacent spectrometers have some overlap. The total length of the spliced slit \(L\), the single slit length \(S\), and the overlap \(\delta_{S}\) have the following relationship:  
\[L=4S-3\delta_{S} \tag{1}\]  
To avoid the interference of two adjacent spectrometers at the same side, the spectrometer width \(W\) needs to satisfy the following inequality, which requires the spectrometer to have a small volume while having a long slit. \[W<2S-2\delta_{S} \tag{2}\]  
Figure 3: (**a**) Optical path of the afocal telescope and (**b**) its wavefront map. (**c**) Optical path of objective I and its (**d**) MTF curves. Besides the compact size, high-fidelity [39; 40] performance is also critical to spectrometers, and it is essential to realize quantitative hyperspectral remote sensing. The fidelity of a spectrometer is affected by many properties, such as MTF, spectral response function (SRF), spectral distortions (smile and keystone), stray light, SNR, etc. Various spectrometer configurations suitable for long slit have been reported, as shown in Figure 5. Including Offner configuration [41], Wynne-Offner configuration [42; 43], freeform Ofner configuration [44], immersed Offner configuration [45], Dyson configuration [46], Wynne-Dyson configuration [47], R-T configuration [25], the catadioptric system [48], and CCVIS [23] proposed by [PERSON]. These configurations have excellent spectral imaging performance in specific scenes, but not all of them are compact, high-fidelity, and fit for splicing. We designed these configurations to have the same wavelength range (0.55-1.01 \(\upmu\)m) but different slit lengths and F numbers. The various configurations are arranged in Figure 5 according to their normalized volume \(V\). Optical performance, including wavefront error (WFE) and spectral distortions of these configurations, are shown in Table 3, and \(\lambda\) is 632.8 nm in this table. Characteristics of different configurations are also compared in Table 3. The Wynne-Dyson and immersed Offner types have the smallest volume, only about 1/15 of the classic Offner type. However, the Wynne-Dyson configuration is unsuitable for splicing because of its short working distance and a small clearance between the slit and image plane. The immersed Offner configuration can be used for splicing, and we manufactured it in our previous work to solve the contradiction between large relative aperture, long slit, and small volume [45]. However, it has high cost and risk in the development process, mainly suitable for the system with large relative aperture. The R-T configuration and the freeform Ofner configuration also have relatively compact structures. However, the double-pass aspheric mirrors and freeform elements are challenging and costly to manufacture. Since spectrometers in GeoFWHIS have small relative apertures, the spectrometers in B1, B2, and  
Figure 4: Splicing of spectrometers. Figure 5: Optical path diagrams of different configurations and their volume. B3 bands are designed with Wynne-Offner configuration, and the spectrometers in B4 and B5 bands are designed with classic Offner configuration. There has been much research on the aberration theory and design methods of the Offner spectrometer [12; 49], and the authors have also made detailed analyses of the Wynne-Offner spectrometer in previous work [43; 50]. The designed spectrometers for five bands are shown in Figure 6, and their characteristics are shown in Table 4. There are two slits in B3 and B4 spectrometers because the detectors of B3 and B4 bands are made up of two chips with 1024 \(\times\) 256 pixels. The double slit assembly is shown in Figure 12c.