## 3 Optical Design and Analysis
### Fore-Optics  
The afocal telescope and imaging objectives constitute the fore-optics of GeoFWHIS. Optical path diagram of the designed fore-optics is shown in Figure 2. Three objectives are arranged at the back of the telescope's primary mirror, and they share an entrance pupil which is also the telescope's exit pupil. The scanning mirror is located on this pupil and reflects the light into the objectives. Dichroic mirrors in the image space of objectives I and II subdivide spectral bands, and a total of five image planes for five bands are formed.  
The telescope's entrance pupil diameter is 3.2 m, and its field of view (FOV) is 0.64\({}^{\circ}\)\(\times\) 0.64\({}^{\circ}\). Compression ratio, i.e., the ratio of the entrance pupil diameter to the exit pupil diameter, is a crucial parameter of the telescope. The larger the compression ratio is, the smaller the obtainable exit pupil. However, the FOV in image space will increase, which will also increase the difficulty of designing imaging objectives. The compression ratio of the telescope is 16, the exit pupil diameter is 200 mm, and the FOV in image space is 10.24\({}^{\circ}\)\(\times\) 10.24\({}^{\circ}\). Considering the requirements of a large aperture and high compression ratio, we designed an afocal three-mirror anastigmatic telescope with zigzag axis [38]. Based on a coaxial three-mirror system, the three mirrors are slightly decentered and tilted to avoid secondary obscuration, and thus the exit pupil is led out. The optical path of the telescope is shown in Figure 3a. Its primary mirror is a paraboloid, and its second and third mirrors are hyperboloids. The line obscuration coefficient of the telescope is 0.22. A folding plane mirror is added in front of the middle image to fold the optical path. Wavefront quality of the telescope is close to the diffraction limit, and its RMS wavefront error is 0.024 \(\lambda\) (\(\lambda\) = 632.8 nm) at the center of the FOV, shown in Figure 3b. Three imaging objectives are mounted behind the telescope, each with an entrance pupil diameter of 200 mm, a linear FOV of 10.24\({}^{\circ}\), and a focal length 16 times smaller than that of the GeoFWHIS, referring  
Figure 2: Optical path diagram of fore-optics.  
to the focal lengths contained in Table 1. The off-axis three-mirror system is suitable for medium FOV and long focal length objectives. In this paper, the design result of objective I is given as an example. Optical path of objective I is shown in Figure 3c, a folding mirror is added to reduce the total length, and the image plane is led to the back of the system to facilitate the docking with spectrometers. MTF curves of objective I are shown in Figure 3d. In the wavelength range of 0.3-1.01 \(\upmu\)m, the imaging quality is close to the diffraction limit, and its MTF value is above 0.77 at the Nyquist frequency. Its grid distortion is 0.29%. In addition, the objective is designed to be telecentric to match the pupil of spectrometers. At the objective's entrance pupil, light is scanned in a time-sharing manner to cover the FOV of 10.24\({}^{\circ}\)\(\times\) 10.24\({}^{\circ}\). Therefore, the scanning angle of the scanning mirror is 5.12\({}^{\circ}\) (\(\pm\)2.56\({}^{\circ}\)), corresponding to the FOV of 0.64\({}^{\circ}\) (\(\pm\)0.32\({}^{\circ}\)) for GeoFWHIS.