## 3 Optical Design and Analysis
### High-Fidelity Spectrometers
If the detector sampling is taken into account, SRF of the instrument--i.e., \(SRF_{ims}\)--can be computed as the convolution of the \(SRF_{opt}\) and the detector sampling function (a rectangular function of width \(y_{d}\), \(y_{d}\) the sampling width). \[SRF_{opt}=rect(y_{s})*LSF(y)\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bands** & **B1** & **B2** & **B3** & **B4** & **B5** \\ \hline MTF & 0.86 & 0.77 & 0.72 & 0.41 & 0.28 \\ Smile/pixel & 0.06\% & 0.11\% & 0.26\% & 0.31\% & 0.95\% \\ Keystone/pixel & 0.33\% & 0.47\% & 0.79\% & 0.42\% & 1.00\% \\ Size/mm & 140 \(\times\) 110 \(\times\) 100 & 170 \(\times\) 111 \(\times\) 106 & 160 \(\times\) 132 \(\times\) 89 & 168 \(\times\) 165 \(\times\) 88 & 90 \(\times\) 92 \(\times\) 49 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: System parameters and performance of spectrometers. Figure 6: Profile map of spectrometers in GeoFWHIS. (**a–e**) from B1 band to B5 band. Figure 7: Spectral lines showing smile and keystone distortions. (**a**) Smile, (**b**) keystone. Figure 8a,b show the \(SRF_{opt}\) of B1 and B3 spectrometers at the middle wavelength of the band, respectively. Figure 8c,d show \(SRF_{ins}\) of B1 and B3 spectrometers with pixel-binning sampling, respectively. A high-fidelity spectrometer shall have uniform SRFs at different fields. It can be seen from the curves in Figure 8a-d that the SRFs at different fields coincide well. The full width at half maximum (FWHM) of \(SRF_{opt}\) accounts for 1 pixel (pixel size 15 \(\upmu\)m) in B1 band and 0.75 pixel (pixel size 32 \(\upmu\)m) in B3 band. The FWHM of \(SRF_{ins}\) accounts for 6 pixels in the B1 band and 2 pixels in the B3 band. SFFD displays the aberration distribution of the image plane by traversing the aberrations of multiple sampling points in spatial and spectral dimensions. Figure 8e,f show the SFFD of wavefront error of B1 and B3 spectrometers, respectively. The size of each circle represents the wavefront error at the corresponding position in the image plane. The aberration distribution in the spectral plane is uniform without drastic change. The maximum wavefront error is only 0.031 \(\lambda\) in the B1 band and 0.035 \(\lambda\) in the B3 band, close to the diffraction limit. Low stray light is another important property for realizing high-fidelity spectral imaging. We design multiple baffles to reduce stray light, as shown in Figure 9a, mainly to eliminate the reflected light from meniscus surface and the diffraction light in non-working orders. Using non-sequential ray tracing, the stray light coefficients of the spectrometer in spatial dimension and spectral dimension are analyzed, respectively. The simulated illuminance of the image plane is shown in Figure 9b-d. Figure 9c shows the illuminance map when the slit is blocked by 10%, and the average illuminance at the center of the blocked portion is \(I_{spa}\). Figure 9d shows the illuminance map when the wavelength range is notched by 10%, and the average illuminance at the center of the notched portion is \(I_{spa}\). The stray light coefficient in spatial dimension \(S_{spa}\) and in spectral dimension \(S_{spc}\) can be calculated by the following equations. \[S_{spa}=\frac{I_{spa}}{I_{nqr}} \tag{5}\]\[S_{spe}=\frac{I_{spe}}{I_{nor}} \tag{6}\]  
where \(I_{nor}\) is the average illuminance at the center of the image plane in Figure 9b. The analysis results show that the stray light coefficients are pretty small in the B1 band, only 0.05% in the spatial dimension, and 0.09% in the spectral dimension.