Analytical astigmatism correction in an off-axis dual-Schwarzschild grating spectrometer

Conventional broadband spectrometers often suffer from complex system configurations, difficulties in aberration correction, and relatively large system volumes. To address these issues, this paper proposes a two-element concave-grating spectrometer based on a dual-Schwarzschild configuration. By coupling concave–convex mirror combinations with a multi-pass folded optical path, high-quality spectral imaging is achieved over a broad wavelength range. The system employs a curvature-coupled architecture with multiple optical working surfaces, in which the collimating mirror, focusing mirror, and concave grating are spherical surfaces with the same curvature, while the two convex mirrors are aspheric surfaces with identical curvature. This configuration integrates multiple optical working surfaces onto two types of concave and convex optical substrates, thereby reducing the number of independent structural parameters while enabling a compact optical layout. Based on geometrical aberration theory, the aberration characteristics of the off-axis reflective grating spectrometer are analyzed, with particular emphasis on the formation mechanism of astigmatism and its relationship with the system structural parameters. An astigmatism correction condition is derived to provide a theoretical basis for determining the parameters of the low-degree-of-freedom configuration. On this basis, a broadband concave-grating spectrometer operating from 400 to 700 nm is designed, and ray tracing and optimization are performed using optical design software. The results show that, over the entire 400–700 nm wavelength range, the modulation transfer function (MTF) exceeds 0.65 across the full field of view, and a spectral sampling interval of 0.6 nm is achieved with two-pixel sampling. System aberrations are effectively suppressed, while spectral line curvature and spectral smile are well corrected. These results demonstrate that the proposed two-element concave-grating spectrometer based on a dual-Schwarzschild configuration maintains high-quality broadband spectral imaging performance while reducing the number of independent design degrees of freedom, providing a new technical approach for the development of broadband, high-resolution, and compact spectrometers.