# Optimal F-Number of Ritchey-Chretien Telescope Based on Tolerance Analysis of Mirror Components
## 1 Introduction  
With the advent of the space age, the demand of the remote sensing instrument (RSI) is becoming higher. Current technology has been developed to launch remote sensing satellite constellation within the same period of the launch window for multiple Earth observation purposes [1]. Therefore, efficiency of RSI fabrication is becoming essential in order to meet the schedule of the launch window with a large amount of satellites. The most time-consuming procedure in making remote sensing satellites would be the manufacturing and the alignment of optical components in the telescope [2; 3; 4]. Although computer-added engineering has been developed for reducing the effort of manual work, it is still highly desired that the tolerance of those precision optics can be increased, so as to reduce the time consumed in the fabrication and assembly, even with automated process [5; 6].  
For a Ritchey-Chretien telescope, the major tolerance items would be a tilt and lateral shift, as well as the RMS random surface wavefront error (WFE) of the two mirror components [7; 8; 9]. Normally, tolerance is set based on the minimal requirement of image quality, for which the modulation transfer function (MTF) has been largely used as the major/reference index. It is possible to manipulate system layout for relaxing tolerance during the design phase, which in turn could benefit from the efficiency of fabrication and the cost of the product. However, the basic structure of the Ritchey-Chretien telescope has a compact layout with only two mirror components, regardless of those corrector lens elements close to the image sensor for aberration correction, and therefore has not much of this kind degree of freedom to control systematic errors in telescope engineering [10]. In this paper, a new concept ofdesign for tolerance is proposed, in which some specific system parameters could be chosen as the degree of freedom for trading the tolerance issue.  
System parameters, such as focal length, f-number, etc., are normally fixed in the beginning of the design phase, and all the paraxial layout and optical analysis should stick to these basic requirements. However, in many cases, those system parameters might not be as rigid as they should be when tracing back to the optical functionality of the system. For some specific system parameters, there could be a certain amount of allowable range while keeping the required optical performance. In developing the remote sensing satellite where the Ritchey-Chretien telescope architecture is adapted, the f-number of the telescope has been used as a tunable parameter in the tolerance analysis. The relationship between the f-number and the major tolerance items of two mirror components is investigated as the reference to choose the optimal f-number in favor of those tolerance items for the ease and efficiency of manufacturing and assembly process. Section 2 presents the procedure of finding the allowable range of f-number and nine design cases of the Ritchey-Chretien telescope, which all meet the required optical specification but with a different f-number within the allowable range. Section 3 gives the tolerance analysis on RMS random surface WFE and the alignment error of the primary and the secondary mirror versus nine different f-numbers from the design cases in Section 2. Section 4 gives the conclusion.