Imaging spectrometer can obtain spatial and spectral information of targets simultaneously, and it has been widely applied in ground object analysis, space remote sensing, target reconnaissance, and other aspects. As spectral imaging technology develops, designers hope that the imaging spectrometer can achieve a wider field of view (FOV) and wavelength band while meeting the structural compactness. In this aspect, the Offner imaging spectrometer shows excellent performance. However, with the expanding FOV and wavelength band, the aberration correction is more difficult. The Offner imaging spectrometer employing a traditional spherical mirror usually balances aberration by relaxing the size limits or adding lenses to the system, thereby leading to the increasing volume or complexity but failing to meet the requirements of lightweight and compactness. With the development of manufacturing and testing technology, the free-form surface has been widely employed in optical design. It is a kind of non-rotationally symmetric optical element, which can introduce more degrees of freedom into optical design. Introducing free-form surfaces into imaging spectrometers can improve the aberration correction ability of the system. However, too many free-form surfaces in optical systems or too many free-form terms on optical surfaces will cause a large deviation in the sag of surfaces, which will not only reduce the aberration correction ability of free-form surfaces in the system but also make the manufacturing of free-form surfaces more difficult. Therefore, we want to use a reasonable free-form surface in the imaging spectrometer design to achieve broad wavelength band and compact volume simultaneously.

The main aberration in the Offner imaging spectrometer is astigmatism. Firstly, the expression of the third-order astigmatism of the system is obtained based on the vector aberration theory. The analysis of the expression of third-order astigmatism shows that the astigmatism of the system increases with the increase of FOV and wavelength band. In the Offner imaging spectrometer, the design of the third mirror as a free-form surface allows for the correction of aberrations associated with both FOV and dispersion. Then we calculate the expression for astigmatism introduced in the Offner imaging spectrometer by Zernike polynomials of 4th^{order and below when the third mirror is designed as a free-form surface with conical surface adding fringe Zernike polynomials. In the calculation, since the diffraction grating is set as aperture stop and the third mirror is a surface away from the stop, the pupil employed in the non-central FOV is shifted relative to the central FOV, and an offset vector is introduced to describe the pupil utilized by the non-center field on the third mirror. Additionally, since the third mirror is set behind the grating in the system, the rays passing through the grating have been dispersed on the third mirror, and the other offset vector is introduced to describe the pupil region leveraged by the rays with different wavelengths on the third mirror at this time. Finally, the relationship among the introduced astigmatism, wavelength, and FOV of the system can be obtained by analyzing the calculated expressions. The results show that in the optical design of the Offner imaging spectrometer based on free-form surfaces when Zernike free-form terms of 4th order and below are selected, the eighth, eleventh, and twelfth terms of Zernike polynomial can be selected to correct astigmatism in the wide FOV and wavelength band.}

An imaging spectrometer with broad wavelength band is designed, and its structure is shown in Fig. 4. The groove density of grating is 100 lp/mm and the diffraction order of -1 is selected, with a wavelength band from 400 to 2500 nm. The system is designed for dual-band detection of visible-near-infrared (VNIR) and shortwave-infrared (SWIR). Two slits with different widths are adopted to ensure the independent spectral resolution of each band. After the rays from the dual slit pass through M₁, M₂, and M₃ successively, they are imaged to corresponding detectors respectively through a dichroic mirror as a beam splitter. The detectors have pixels of 14 μm for the VNIR band and 20 μm for the SWIR band. The system specifications are shown in Table 3 and the lens data of the system are shown in Table 4. M₃ is designed as a free-form surface. The coefficients of different Zernike terms in M₃ are shown in Table 5. The volume of the system is 42 mm×82 mm×100 mm, which is one-third of the spherical mirror system with the same specifications.

The imaging spectrometer based on the conventional optical element is difficult to meet the structural compactness and realize wide FOV and wavelength. Thus, the third mirror in the Offner imaging spectrometer is designed as a free-form surface expressed by a conical surface with fringe Zernike polynomial added. The relations among astigmatism introduced by polynomials of fourth order and below, wavelength, and FOV in the Offner imaging spectrometer based on vector aberration theory are analyzed. Then an imaging spectrometer with a wavelength range from 400 to 2500 nm and a volume of only 42 mm×82 mm×100 mm is proposed by selecting a reasonable polynomial adding on a conical surface. The system achieves dual-band detection. The rays from two slits with different widths are dispersed by the diffraction grating from the beam splitter to work in visible-near-infrared (400-1000 nm) and shortwave-infrared (1000-2500 nm) bands. The optimized design results indicate that the spectral resolution is 2.8 nm and 4 nm respectively with high imaging quality. Finally, theoretical reference implications are provided for the application of free-form surfaces in the design of imaging spectrometers.