# Optimal F-Number of Ritchey-Chretien Telescope Based on Tolerance Analysis of Mirror Components
## 2 Allowable Range of F-Numbers and Design Cases of the Telescope  
The optical specifications of the Ritchey-Chretien telescope is shown in Table 1.  
Considering the effect of central obstruction on small aberrations for the Ritchey-Chretien telescope, Figure 1 shows the MTF versus normalized spatial frequency \(\xi\)/\(\xi_{c}\) (\(\xi_{c}\) is the cut-off frequency) for various obstruction ratios \(\varepsilon\)[11; 12], in which only primary and secondary mirrors are involved. Based on the required specification of as-built MTF \(\geq\) 0.25, the obstruction ratio \(\varepsilon\) should not be more than 0.5. The colored region between the curves of \(\varepsilon\) = 0.3 and \(\varepsilon\) = 0.4 in Figure 1 can be divided into three spatial frequency bands, marked as I, II and III in Figure 1. It is desired that the sensor's Nyquist frequency 71.429 lp/mm used for specifying required MTF can be located in region II, because it tends to over design and induce higher cost than required if this spatial frequency is located in region I. In addition, the MTF started to drop quickly in region III and therefore the image performance tended to be poor if the spatial frequency was located in it. Similarly, the best obstruction ratio should be in the range between \(\varepsilon\) = 0.3 and \(\varepsilon\) = 0.4.  
Based on the above argument, normalized spatial frequency \(\xi\)/\(\xi_{c}\) was scanned from 0.35 to 0.45, which was in region II as shown in Figure 1, with a step of 0.01 to evaluate the allowable range of the f-number. Table 2 shows the f-numbers at various \(\xi\)/\(\xi_{c}\), where \(\xi\) = 71.429 lp/mm, \(\lambda\) = 632.8 nm and the relationship between cut-off frequency \(\xi_{c}\) and f-number \(f_{n}\) is shown in Equation (1) [13].  
\[\xi_{c}=\frac{1}{\lambda\cdot\left(f_{n}\right)}\, \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Description** \\ \hline Spectral Range & 400–900 nm \\ Filter Bands & 2 Panchromatic + 6 Multi-Spectral \\ Sensor Pixel Pitch/Resolution & Pan.: 7 \(\mu\)m/12,000 pixel, M.S.: 28 \(\mu\)m/3000 pixel \\ Ground Sampling Distance & 0.7 m/pixel @ 600 km of Altitude; Sun-Synchronous Orbit \\ Field of View & \(>\)1\({}^{\circ}\) (Diagonal) \\ Swath Width & \(>\)8.4 km \\ Optical MTF & As–Built \(\geq\) 0.25 @ 71.429 lp/mm, Panchromatic; 18 lp/mm, Multi-Spectral \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of the optical system.  
The optical design program, Synopsys CODE V(r) (Synopsys Inc., Mountain View, CA, USA), was used for the optimization of 9 cases with a different f-number [14, 15]. The merit functions used in the optimization process were similar in different cases, and the target was to achieve a minimal MTF of 0.3 at 71.429 lp/mm. The higher MTF than what is specified (0.25) must leave a sufficient error budget for tolerancing. Table 3 shows the optical performance and system parameter of those 9 design cases, including minimal MTF, distortion, chief ray angle and obstruction ratio. All the 9 cases meet the same system performance specification shown in Table 1 with same optical layout shown in Figure 2, including the lens material. The only difference made with a different f-number is the detail parameters, such as obstruction ratio, curvature and spacing/thickness. The lens data of the all 9 design cases are as listed in Appendix A, Table A1.