# A High-Resolution MIR Echelle Grating Spectrometer with a Three-Mirror Anastigmatic System
## 3 Optical Design Method  
Typical plane grating spectrometer structures, such as Littrow [14], Ebert-Fastie, Chupp-Grantz, and Czerny-Turner [15], are unsuitable for the design of this work; accordingly, this paper follows the principle of ensuring that the overall resolution of the instrument is not affected by design but also that the volume is controlled and stray light of the entire system is kept to a minimum. To improve and simplify the optical structure of the echelle spectrometer, a quasi-Littrow structure [16] and a three-mirror anastigmatic (TMA) structure were used for cross design [17]. The Immersion Echelle Spectrometer with Three-Mirror Anastigmatic (IESTMA) system is shown in Figure 6. The light enters incident through a pinhole, which is located at the telescope focal plane, and is reflected by the off-axis parabolic mirror (OAP) before entering the immersion grating, where the angle between the chief ray and the plane perpendicular to the groove (\(\gamma\) angle) is \(4^{\circ}\). The dispersed light is separated into orders by the concave grating (first order used). The cross-dispersed spectra are focused on the detector by the TMA camera system. The detector covers a wavelength range from 3.7 to 4.8 \(\upmu\)m in a single exposure. By using the concave grating as the cross-dispersion element, the difficulties caused by using prisms for infrared immersion echelle grating spectrometers are resolved; the beam can be focused to a certain extent through the concave grating, and the beam aperture is also reduced relative to the use of prisms or plane gratings. In addition, an off-axis three-mirror anti-astigmatism optical system is used to replace the imaging system composed of refractive elements,which fundamentally reduces the aberration of the optical system. The proposed IESTMA solution also offers the possibility of developing compact immersion spectrometers. The design of the off-axis three-mirror reflective system is based on an initial structure of a coaxial three-mirror system as shown in Figure 7. To achieve the off-axis three-reflection optical system design, a coaxial three-reflection mirror structure must first be found that meets the requirements of the initial structure of the system; the center of the system can then be found using aperture off-axis or field of view off-axis methods. The system structure variables can then be further optimized to achieve the desired results. As the system does not form a real image in the center, the traditional method of determining the initial structural parameters is discussed below, including the value ranges of various structural parameter types. First, assuming that the object is infinitely distant, the entrance pupil is located on the primary mirror \(M_{1}\), and \(M_{2}\) and \(M_{3}\) are the secondary mirror and tertiary mirror, respectively. On this basis, the following equations can be derived:  
The blocking ratio of the secondary mirror to the primary mirror: \(\alpha_{1}=l_{2}/f_{1}^{\prime}\approx h_{2}/h_{1}\)  
The blocking ratio of the third mirror to the second mirror: \(\alpha_{2}=l_{3}/l_{2}^{\prime}\approx h_{3}/h_{2}\)  
Figure 6: Basic optical path of the IESTMA system. Figure 7: Sketch of the off-axis three-mirror reflective system. The magnification of the secondary mirror: \(\beta_{1}=l_{2}^{\prime}/l_{2}\)  
The magnification of the tertiary mirror: \(\beta_{2}=l_{3}^{\prime}/l_{3}\)  
For reflective systems, these are the following definitions: \(n_{1}=n_{2}^{\prime}=n_{3}=1\ n_{1}^{\prime}=n_{2}=n_{3}^{\prime}=-1\)  
The parameter \(d_{1}\) represents the interval between the primary mirror and the secondary mirror, \(d_{2}\) represents the interval between the secondary mirror and the tertiary mirror, and \(d_{3}\) is the back focal length. Logically, \(d_{1}\)\(<\)\(0\), \(d_{2}\)\(>\)\(0\), and \(d_{3}<0\), as follows:  
\[\left\{ \begin{array}{l}d_{1}=\frac{(1-\alpha_{1})f^{\prime}}{(\beta_{1}\beta _{2})}\\ d_{2}=\frac{\alpha_{1}(1-\alpha_{2})f^{\prime}}{\beta_{2}}\\ d_{3}=\alpha_{1}\alpha_{2}f^{\prime}<0\end{array}