Objective Hyperspectral imaging is an imaging method to acquire spatial and spectral information of a scene. The satellite- based spectrometer can monitor moving targets in real time and is suitable for disaster emergency and target monitoring. In recent years, microsatellite spectrometers have caught increasing attention. Compact optical systems with small size, light weight, and high imaging quality should be designed to improve spatial resolution and reduce production costs. By comparing the characteristics of refractive imaging systems and reflective imaging systems, we find that reflective optical systems are easier to realize compact structures. The reflective imaging system has coaxial and off- axis types. The coaxial layout can fold the optical path, but the central obscuration blocks energy transfer. Meanwhile, the off- axis aberration correction is hard, but it can make full use of energy in off- axis systems. The determination of the initial optical structure is a challenge. A good initial structure greatly determines the efficiency and potential of subsequent optimizations, which will greatly reduce the optimization time and dependence on design experience. Nowadays, based on the generalized flowsheet, the general design of a reflective freeform system is to select a suitable system from a proprietary or existing structure as the starting point and then conduct optimization design in optical software. As special configurations are increasingly being employed, specific design structures are commonly limited for viable starting points. Zhang proposed to design an unobscured initial structure with a reflective freeform imaging system, in which a special algorithm is demonstrated to calculate the data points on the unknown freeform surfaces using the rays from multiple fields and different pupil coordinates, and thus construct multiple freeform surfaces in an imaging system. However, the above method is only suitable for the design of internal systems with less than three mirrors, but the stray light is difficult to suppress for the three- mirror system. Especially, it's complicated for reflective imaging systems with several mirrors. There is still a gap in the initial structure of the reflective spectrometer design method. Methods A novel method of reflective spectrometer structure design is proposed. It is an automatic design method for an off- axis, five- mirror spectrometer based on the Seidel aberration theory. Firstly, if the pitch of mirrors and the mirror curvature of the optical system are known, we can get the heights and paraxial angles on each surface by tracing the characteristic rays in the reflective system. In our study, the height, the angle on each surface, and the mirror curvature are three parameters for reflective optical systems. The optical system of the volumes can be calculated according to three parameters. Meanwhile, based on the aberration theory, we can characterize the primary Seidel aberration terms of the reflective spectrometer system by three parameters. This means that the volume and system aberration are represented by the same three parameters, with the relationship between the volume and the aberration established. Secondly, we consider the characteristics of the spectral system, in which the influence of the grating during light transmission in the system is considered, to ensure that the system meets the imaging conditions of different wavelengths. Then, we derive the mathematical relationships of the relevant parameters of the reflective system. Additionally, the evaluation criteria system is developed to narrow down the structure parameter ranges, thus obtaining the initial compensating optical system aberration under limited volume. Finally, we can import the initial structure into optical design software, which is the off- axis initial system with dispersive elements. As a result, optical systems of UV- visible imaging micro- spectrometers can be quickly optimized. The proposed scheme can satisfy the design requirements, including spectral resolution and spatial resolution. Results and Discussions To verify the feasibility of the method, we design a compact off- axis aspheric reflectance imaging spectrometer by this method. The working spectrum is 320-500 nm, and then the initial off- axis structure can be obtained. Then, freeform surfaces can further improve the imaging quality and expand the field of view. The design results show that the modulation transfer function of each wavelength at Nyquist frequency (12 lp/mm) is greater than 0.8 and the root mean square is better than 10 mu m (Fig. 11). The Keystone and Smile are smaller than one pixel in the system (Fig. 12), with the spectral resolution of 0.5 nm in Fig. 13. The system has high resolution and sound imaging quality, thus providing a new idea and method for the design of relevant off- axis reflective structures. Conclusions To meet the development trend of miniaturization of spaceborne spectrometers, we propose a design idea and method suitable for the automation of spectrometers structures under the volume requirement, which shortens the time to find the initial structure and provides a more appropriate initial structure design for reflective structure spectrometers. Given the system design parameters and indicators, an appropriate initial optical system can be generated to accelerate the optimization design in the later stage and greatly reduce the growth compared to traditional optical structure design. Finally, a compact off- axis optical system is obtained with meeting specifications, configurations, and the element number.
