# A High-Resolution MIR Echelle Grating Spectrometer with a Three-Mirror Anastigmatic System
## 2 Dispersers and Methods  
An immersion grating is a plane diffraction grating combined with a prism (i.e., immersion medium) diffractive element, where the diffractive surface is immersed in a material with a high refractive index (\(n>2\)); relative to a non-immersion grating of the same face size, the immersion grating's resolution can be increased n times [6; 7]. The maximum spectral resolution of the immersion grating at wavelength \(\lambda\) is shown in Figure 1.  
The immersion grating equation is given in Equation (1). In this equation, m represents the diffraction order, \(\lambda\) represents the wavelength, \(n\) represents the refractive index of the light propagation medium, \(d\) represents the groove spacing, and \(\alpha\) and \(\beta\) represent the incident angle and diffraction angle, respectively. For gratings operating under Littrow conditions, the diffraction corresponds to the following equation [8]:  
\[m\lambda=nd(sin\alpha+sin\beta) \tag{1}\]  
The maximum spectral resolution, R, of an immersion grating at wavelength \(\lambda\) is given by:  
\[\text{R}=2 nL/\lambda \tag{2}\]  
\(L\) is the length of the bottom edge of the immersion grating. These equations demonstrate that immersion gratings can achieve improved spectral resolution by a factor of \(n\) and reduce the dispersive power of the additional dispersive element by a factor of \(n\) compared to those of a reflective echelle grating of the same aperture. In the infrared band, optical materials with a high refractive index (\(n>2\)) can be used for immersion gratings, which are primarily semiconductors with high refractive indexes. Due to the absorption of light inside the material, the greater the required volume, the greater the immersion, and the transmittance of the material will drop significantly. Studies by Lawrence Laboratory LLNL and Canon have shown infrared materials that can be applied to immersion gratings and have been successfully applied to both space-borne and ground-based high-spectral-resolution grating spectrometers. In terms of the commonly used materials shown in Table 1, single-crystal silicon is easy to process in mid-wave infrared and has relatively mature processing technology, thus, this study uses single-crystal silicon as a prism material [9]. As shown in Figure 2, the refractive index of Si decreases as the wavelength increases. However, changes in the refractive index are usually ignored or assumed to be constant for most optical applications due to the narrow range of operation. More detailed information on the optical properties of single-crystal silicon can be found in references [10; 11]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Materials** & **Usable Wavelength/\(\upmu\)m** & **Index** & **Crystal Type** \\ \hline Si & 1.2–6.9 & 3.4 & Single \\ Ge & 2.0–11 & 4.0 & Single \\ InP & 1.4–9.0 & 3.2 & Single \\ CdTe & 5.0–20 & 2.7 & Single \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optical properties of candidate materials for IR immersion gratings. Figure 1: Reflective grating and immersion grating. (**a**) Work mode of echelle grating: (**b**) Work mode of immersion grating. Accordingly, the dispersion of an immersion grating comprises two components: the refractive index and the diffraction angle. According to Equation (1), the angular dispersion of an immersion grating can be calculated as:  
\[\frac{d\beta}{d\lambda}=n(\lambda).\frac{m}{dcos\beta cos\gamma} \tag{3}\]  
\[\frac{dl}{d\lambda}=f.n(\lambda)\frac{m}{dcos\beta} \tag{4}\]  
Here, \(f\) represents the focal length of the imaging system and \(\gamma\) is the angle between the chief ray and the plane perpendicular to the groove. To avoid interaction between the incident light and the diffracted light, gratings can be employed slightly \"out of plane\", in which the incident and diffracted ray paths do not intersect but are separated by an angle \(\gamma\) (where \(\gamma<10\) degrees) [12]. Equation (1) assumes that the diffraction occurs within a plane oriented perpendicular to the grating grooves (see Figure 3).