# A High-Resolution MIR Echelle Grating Spectrometer with a Three-Mirror Anastigmatic System
## 2 Dispersers and Methods
Considering the out-of-plane scenario, a more complete version of the grating equation is given by:  
\[m\lambda=nd(sin\ \alpha+sin\ \beta)cos\ \gamma \tag{5}\]  
Figure 3: Schematic of echelle optics in quasi-Littrow configuration. Figure 2: Optical properties of single crystal silicon (**a**) Refractive index; (**b**) transmittance of silicon in the mid-infrared. where \(1/d\) is the groove spacing (line mm\({}^{-1}\)) and \(\gamma\) is the out-of-plane angle of incidence (degrees). Based on the diffraction equation of the grating, the angular dispersion, and the linear dispersion, the central wavelength corresponding to each diffraction order is calculated as:  
\[\lambda_{\text{cen}}\ =\frac{2 ndsin\ \beta}{m}cos\ \gamma\lambda_{high}\ (m)=\frac{2m+1}{2m} \lambda_{cen}\ \lambda_{low}\ (m)=\frac{2m-1}{2m}\lambda_{cen} \tag{6}\]  
Although the various diffraction orders of a grating overlap, for a given order, there is a subset of wavelengths that do not overlap the same wavelengths in a neighboring order. For each order, this set of wavelengths is known as the free spectral range. Equivalently, it can be defined as the change in wavelength between adjacent orders for a given angle of diffraction:  
\[\Delta\lambda=\frac{\lambda}{m} \tag{7}\]  
To make the free spectrum corresponding to each order, based on the constraint relationship between the detector and the grating dispersion, the following equation can be obtained, where the dispersion direction of the middle step is parallel to the shorter side of the detector:  
\[\Delta l_{m}=\frac{\lambda_{max}\cdot f}{ndcos\ \theta_{B}\cdot cos\ \gamma}\cdot H \tag{8}\]  
where \(H\) is the size of the main dispersion direction of the detector. As shown in Figure 4, at 4 um and R = 40,000, the resolvable spectral width is 0.05 cm\({}^{-1}\). If there are 320 pixels in a row (320 \(\times\) 256 array) and sampling theory requires two pixels per resolvable spectral element width, then the row will cover 50 cm\({}^{-1}\). Thus, the spacing between echelle orders should be 16 cm\({}^{-1}\). For an immersed germanium echelle with groove spacing d and blaze angle \(\theta\), the order spacing is 1/2 nd(sin \(\theta\)). Taking \(n\) = 4 and a 63.4\({}^{\circ}\) blaze angle, the corresponding groove spacing is 88.08 um. In echelle grating spectrometers, most low-dispersion elements use prisms, but gratings can also be used. However, in the infrared band, if prisms continue to be used, then larger sizes are required. Accordingly, concave gratings are being developed that can achieve convergence and improve imaging performance. In this paper, the concave grating was used in a Wadsworth imaging system, such that the concave grating was illuminated with a collimated beam, as shown in Figure 1:0 The spectral astigmatism in and near the  
Figure 1:1 The relationship between grating line density and diffraction order in (**a**); Plot showing the relationship between the grating angular dispersion rate and order, with the blaze angle shown in the upper panel and groove spacing as a function of wavelength and pixel format in (**b**). grating normal direction is greatly reduced, where O is the center of the grating (i.e., the origin of the coordinate system), the X-axis is along the normal to point O, and the Z-axis is along the dispersion direction. \(A(a,b,0)\) is the object point, \(P_{1}\) and \(P_{2}\) are the intersection points of the light emitted by it and the grating surface, and \(A^{\prime}(a^{\prime},b^{\prime},0)\) is the image point, such as \(P_{1}(x_{1},z_{1},y_{1})\), \(P_{2}(x_{2},z_{2}+d,y)\). Accordingly:  
For a concave grating surface, the ratio of the sag depth to its length and width is very small. Accordingly, it is assumed that \(x\approx 0\).