## 3 Optical Design and Analysis
### High-Fidelity Spectrometers
It can be seen from Table 4 that the MTF values of spectrometers at the Nyquist frequency are close to the diffraction limit. The smile and keystone are both spectral distortions of the spectrometer, which will affect the spectral and spatial uniformity, respectively. Spectral uniformity demands that the monochromatic image of the slit at any wavelength be straight [40]. Smile is the deviation of monochromatic slit image from a straight line, as \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Spectrometer Types** & **WFE/A** & **Smile/\(\mu\)m** & **Keystone/\(\mu\)m** & \(V\) & \multicolumn{2}{c}{**Characteristics**} \\ \hline Offner & 0.10 & 0.12 & 0.09 & 1 & \(\bullet\) & long working distance \\  & & & & & \(\bullet\) & large volume with long slit \\ \hline Wynne–Offner & 0.04 & 0.02 & 0.07 & 0.26 & \(\bullet\) & compact with long slit \\  & & & & & \(\bullet\) & anastigmatic \\ \hline Freeform Offner & 0.07 & 0.08 & 0.15 & 0.2 & \(\bullet\) & compact with long slit \\  & & & & & \(\bullet\) & extremely difficulty \\  & & & & & \(\bullet\) & high cost \\ \hline Immersed Offner & 0.03 & 0.03 & 0.08 & 0.06 & \(\bullet\) & ultra-compact \\  & & & & & \(\bullet\) & large relative aperture \\  & & & & & \(\bullet\) & short working distance \\ \hline Dyson & 0.18 & 0.35 & 0.22 & 0.29 & \(\bullet\) & short working distance \\  & & & & & \(\bullet\) & large relative aperture \\  & & & & & \(\bullet\) & short slit \\ \hline Wynne–Dyson & 0.06 & 0.12 & 0.10 & 0.07 & \(\bullet\) & ultra-compact \\  & & & & & \(\bullet\) & small slit-to-image space \\  & & & & & \(\bullet\) & long slit \\ \hline R-T & 0.09 & 1.55 & 1.03 & 0.23 & \(\bullet\) & compact with long slit \\  & & & & & \(\bullet\) & tight tolerance \\  & & & & & \(\bullet\) & small slit-to-image space \\  & & & & & \(\bullet\) & residual distortions \\ \hline Chrisp catadioptric type & 0.24 & 2.20 & 2.35 & 1.1 & \(\bullet\) & long working distance \\  & & & & & \(\bullet\) & residual aberration \\ \hline CCVIS & 0.11 & 0.34 & 0.25 & 0.61 & \(\bullet\) & long working distance \\  & & & & & \(\bullet\) & large volume with long slit \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optical performance of the designed spectrometers and comparison of their characteristics. shown in Figure 7a. Spatial uniformity demands that the magnification of the spectrometer be independent of wavelength so that the monochromatic slit images are of the same length at any wavelength. Deviation from this condition is often referred to as 'keystone', as shown in Figure 7b. The smile and keystone requirements for the GeoFWHIS spectrometers are both less than 10% of a pixel. Moreover, in addition to reducing smile and keystone in optical design, the spectral uniformity requires the slit image to be aligned with the detector array, and the spatial uniformity requires the dispersion line of any field point be aligned with the detector array. This requires high alignment accuracy of the spectrometer and detector, which will be discussed in Section 4.2. In this section, we evaluate the smile and keystone, as shown in Table 4; both of them are less than 1% of a pixel in each band, indicating extremely low spectral distortions. The spectral response function (SRF) [39] and the spectral full field display (SFFD) [51] of B1 and B3 bands are given in Figure 8. SRF describes the response of the spectrometer to the monochromatic light with unit intensity, and it is affected by the spectrometer optical system and the detector sampling. We consider that the slit is along the \(x\)-axis, and then the spectral direction is along the \(y\)-axis. If the detector sampling is not taken into account, SRF of the optical system--i.e., \(SRF_{opt}\)--can be computed as the convolution of the slit function (a rectangular function of width \(y_{s}\), \(y_{s}\) the slit width) and the y-line spread function (LSF) of the optical system.