# A High-Resolution MIR Echelle Grating Spectrometer with a Three-Mirror Anastigmatic System
## 4 Results and Discussion  
The working band range of the echelle grating spectrometer was mid-infrared. Based on the constraints between the components, a spectrometer optical system combining an immersion grating and concave grating was designed. In this design, an OAP mirror was selected as the collimation system instead of a lens; accordingly, the beam could be collimated with a relatively large aperture angle without unnecessary aberration. The off-axis angle was 15\({}^{\circ}\); the light was collimated after incidence on the main dispersion element, an immersed echelle grating with a groove spacing of 12.36 line/mm, an area of 70 mm \(\times\) 35 mm, a blaze angle of \(\theta_{B}\) = 63.5\({}^{\circ}\) (R = 2), and an auxiliary dispersion element composed of a concave grating with a typical radius of curvature of 0.5, 0.75, or 1 m, according to the size of the detector image plane and the focusing system. The off-axis three-mirror optical system was used as the condensing mirror to achieve aberration correction with high dispersion and a wide free spectrum, realize focal length compensation and high-quality imaging optimization across the whole band, and achieve relatively uniform imaging. In terms of the numerical spectral order separation of the number of scribe lines of the concave grating, to optimize the matching between the order separation and the detector, we modeled and analyzed the cross-dispersion in Zemax Optic Studio 19.4, sourced from Ansys Corporation;and tested different scribe lines of the concave grating. The grating order separation increased with increasing wavelength, as shown in Figure 9. We selected a concave mode grating of 75 lines/mm at an 8.8\({}^{\circ}\) angle of incidence; this parameter selection was ideal as the two gratings dispersed the incoming light from the immersion grating in the \(\sim\)3.7-4.8\(\upmu\)m wavelength range for a total of 36 orders (120 to 155). The software simulated the immersion grating working under the Littrow configurations, and the structure included multiple configurations. The ellipse dotted line represents the actual grating diffraction area. Figure 8: Flowchart of the design process in engineering. The optical performance of the optimally designed echelle spectrometer was analyzed from three aspects: a spot diagram, a point spread function, and an energy distribution curve. The two characteristic wavelengths at the edge of the spectral range and the central wavelength were selected for image quality evaluation. The echellogram in the left panel of Figure 10 shows the free spectral range of 36 orders on the 320 \(\times\) 256 pixel detector. To analyze the optical properties of the spectrometer throughout the entire wavelength range of \(\sim\)3.7-4.8\(\upmu\)m, multiple structures were established for the optical system, and multiple diffraction orders were selected for simulation. The diffraction orders corresponding to the two ends of the range were 155 (3.7 \(\upmu\)m) and 120 (4.8 \(\upmu\)m). Figure 10: The echellogram in (**a**) shows the free spectral range of 36 orders on the detector of 320 \(\times\) 256 pixels. The corresponding point spread function is plotted in (**b**). Figure 9: Optical configuration of the echelle spectrometer system: (**a**) Optical System; (**b**) Immersion Grating Simulation Model. The point spread function (PSF) is based on wave optics. Based on the deformed wavefront at the exit pupil of the optical system, the three-dimensional spatial energy distribution of the imaging beam on the image plane was determined, i.e., the light energy concentration on the spot was identified. The PSF represents the product of the Fourier transform of the pupil function and the conjugate of the Fourier transform of the pupil function, and, if the constant and high-order terms are removed, the pupil function represents the wave aberration. The right panel in Figure 10 shows that the diffraction pole was achieved in the entire working band range (Strehl ratio > 0.99).