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Abstract
Current imaging spectrometers with conventional optical elements face major challenges in achieving a large field of view (FOV), broadband and compact structure simultaneously. In this paper, a compact freeform-surface-based Offner imaging spectrometer with both a long-slit and a broadband (CISLS) is proposed. To keep a long slit and an anastigmatic imaging, the slit off-axis amount of the initial system is within a specific range theoretically. While to achieve a compact structure, the slit off-axis amount should be away from the specific range and as small as possible. Based on the vector aberration theory and the analytical study, Zernike polynomial terms Z5 and Z6 introduce the astigmatism independent of FOV. They are utilized to well balance the astigmatism when the slit off-axis amount is away from the specific range, helping a miniaturization of the system. Other Zernike polynomial terms below the eighth order introduce the astigmatism related to FOV. They contribute to balancing the astigmatism that produced with the increasing of the FOV, thus achieving a wide FOV. The design results show that the proposed CISLS with a high spectral resolution of 2.7 nm achieves a long slit of 30 mm in length but a small size of only 60 mm × 64 mm × 90 mm in volume under a broadband from 400 nm to 1000 nm.
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1. Introduction
Imaging spectrometers are used to obtain the two-dimensional spatial information and the one-dimensional spectral information of the targets simultaneously [1–6], and play an important role in the areas of object recognition and detection, and remote sensing [6–13]. With the increasing demand for wide detecting coverage, imaging spectrometers are developing towards large FOV and long slit. However, widening the FOV increases the system volume and reduces the imaging quality, which restricts the application of the imaging spectrometers. The Offner imaging spectrometers with conventional spherical elements usually utilize the methods such as increasing the size or complicating the system to obtain a high image quality, which significantly increase the volume and the weight, and do not meet the requirements of instrument volume and weight limitations.
Optical freeform surfaces without rotational symmetry offer more freedom of optical design [14]. They are capable of obtaining a better system performance as well as achieving a much more compact configuration. With the development of precision machining and detection technology [15–16], freeform surfaces have been applied to optical imaging systems in many areas such as astronomical research, spectral analysis, remote sensing detection and virtual reality [17–18]. Astigmatism is the dominant aberration in the imaging spectrometers. In recent years, due to the restriction of the astigmatism, the imaging spectrometers face a stern challenge in achieving a large FOV, broadband and compact structure, simultaneously. Freeform surfaces have been applied to the system to break through the limitations. In 2017, Reimers et al. designed an Offner-Chrisp imaging spectrometer with a slit of 20 mm in length introducing freeform surfaces onto the primary mirror, grating and tertiary mirror. It showed about 5 times smaller in volume than the classical all-spherical designs, and illustrated that freeform surfaces have a great potential in reducing the volume of system [19]. In 2018, Yang et al. presented a point-by-point design method to obtain a freeform imaging spectrometer, which was composed of three freeform surface mirrors. The system worked at the wavelength range from 450 nm to 950 nm with a volume of 18 mm × 58 mm × 39 mm but a slit of only 4 mm in length [20]. In 2020, Liu et al. applied freeform surfaces to the Schwarzschild imaging spectrometer, and proposed a design method of splitting the secondary mirror into two parts, which eliminated the astigmatism of the collimation system and the focusing system in the initial system. Finally, a system with a slit of 32 mm in length but a larger volume of 230 mm × 220 mm × 200 mm was obtained [21]. In 2020, Zhang et al. proposed the design method of prism box and partial anastigmatism in the Offner imaging spectrometer, and the aspherical and freeform surfaces were introduced onto the primary and tertiary mirrors, respectively. The spectrometer had a volume of 190 mm × 118 mm × 107 mm and a slit of 70 mm in length. But it had a low dispersion linearity [22]. Further, after studying the Offner imaging spectrometers based on the vector aberration theory and designing the tertiary mirror as a freeform surface, the system with a volume of 85 mm × 64 mm × 56 mm but a slit of only 22.5 mm in length was obtained [23].
Although freeform surfaces have been applied to imaging spectrometers to improve the system performance in a certain extent, achieving a large FOV, broadband and compact structure simultaneously is still a stern challenge. In this paper, by studying the astigmatism introduced by Fringe Zernike polynomial terms based on vector aberration theory, Z5 and Z6 Zernike polynomial terms introduce the astigmatism independent of the FOV and other Zernike polynomial terms below the eighth order introduce the astigmatism related to the FOV. To keep a long slit and an anastigmatic imaging, the slit off-axis amount of the initial system is within a specific range theoretically. While to achieve a compact structure, the slit off-axis amount should be away from the specific range and as small as possible. Z5 and Z6 Zernike polynomial terms are utilized to well balance the astigmatism when the slit off-axis amount is away from the specific range, helping a miniaturization of the system. Other Zernike polynomial terms below the eighth order contribute to balancing the astigmatism produced with the increasing of the FOV, achieving a wide FOV. Finally, CISLS with a volume of only 60 mm × 64 mm × 90 mm, a slit of 30 mm in length, a broadband from 400 nm to 1000 nm, and a spectral resolution of 2.7 nm is obtained. The system provides a good correction for the astigmatism and verifies the correctness of balancing the residual astigmatism in the system using the additional astigmatism introduced by the freeform surfaces, which provides a reference for the design of imaging spectrometers with freeform surfaces.
2. Analytical design
Astigmatism is the dominant aberration in the imaging spectrometers, which makes it difficult to achieve a large FOV, broadband and compact structure simultaneously. Through the vector aberration theory and the analytical study, how the Zernike polynomial terms Z5, Z6 and the other Zernike polynomial terms below the eighth order affect the system astigmatism are studied respectively. Thus, an anastigmatic Offner imaging spectrometer within the full FOV under a compact structure is obtained, providing a reference for the design of CISLS.
2.1 How Z5 and Z6 Zernike polynomial terms help to balance the astigmatism in the system with a smaller slit off-axis amount
Schematic of the chief ray in the Offner imaging spectrometer is shown in Fig. 1. It contains three main components, including a primary mirror M1, a grating G and a tertiary mirror M3, which satisfy the relationship of Roland circle. Figure 1(a) shows the path of the chief ray tracing from an object point A, forming a meridianal image IM and a sagittal image Is. H is the common spherical center of the primary mirror M1, the grating G and the tertiary mirror M3. Y0 HZ is the incident plane. ${\theta _1}$, ${\theta _2}$ and ${\theta _3}$ are the incident angle of the chief ray on the primary mirror M1, the grating G and the tertiary mirror M3, respectively. ${\theta _2}^{\prime}$ is the diffraction angle on the grating G. $\varphi$ is the deviation angle in object space. ${\varphi _M}$ and ${\varphi _s}$ are deviation angles in the meridianal and the sagittal image space, respectively. R1, R2, R3 are radii of three main components, respectively. Figure 1(b) shows the view along the Z-axis. $\gamma$ and $\gamma ^{\prime}$ are the incident azimuth and the diffraction azimuth on the grating, respectively. Astigmatism is defined as the distance between the meridianal and sagittal images [24]:
The anastigmatic solution can be derived from the Eq. (1):
Both the chief ray in the object plane and image plane are parallel to the optical axis, and the system is not only objective telecentric but also imagery telecentric. A(x,y) is coordinate of any object point, K is defined as $\sqrt {{x^2} + {y^2}} /{R_2}$, and K1 is the radius ratio defined as R1/R2. The three parameters in Eq. (1) can be calculated as:
Fig. 1. Schematic of the chief ray in the Offner imaging spectrometer (a) View along the X-axis; (b) View along the Z-axis.
In the initial structure of the Offner imaging spectrometer, both the R1 and R3 are 79 mm, the R2 is 40 mm, the grating groove density is 100 lp/mm, the diffraction order of the grating is −1, and the center of the working waveband is 700 nm. Figure 2(a) is the astigmatism distribution of the system, which shows that the system is anastigmatic when the object point is in the white anastigmatic areas S1 and S2. The astigmatism increases and the system is not anastigmatic when the object point is away from the areas. To keep a long slit and an anastigmatic imaging, the slit off-axis amount is within a specific range in the anastigmatic area S2. Further, the slit off-axis amount is chosen to be 20 mm at the point A1 in the initial structure. Figure 2(b) is the variation curve of the system astigmatism changing with the slit half-length. Astigmatism increases as the slit half-length increases and the corresponding anastigmatic threshold is 0.002. The slit length in the initial structure is 13.2 mm.
Fig. 2. (a)Astigmatism distribution; (b)Variation curve of system astigmatism changing with slit half-length.
Freeform surfaces offer more freedom of optical design and provide enhanced aberration correction than the spherical surfaces. In optical design, the mathematical description of a freeform surface which characterized by the even aspheric surface with the additional polynomial terms defined by the Zernike Fringe coefficients is as follow [25]:
Further, the aberration generated by the Zernike polynomial terms when the freeform surface is introduced at the stop and away from the stop has been studied. When introducing freeform surfaces into the off-axis system, the pupil off-axis can be taken as the surface deviation of the co-axial system, with the deviation vector written as $\vec{\mu }$. The grating is usually designed at the stop in the Offner imaging spectrometer.
Firstly, when the freeform surface is introduced at the stop, the light beams of different fields will use the same area on the surface. The Zernike polynomial terms added on the surface can be regarded as a figure error or deformation with respect to the substrate. As shown in Fig. 3(a), the red dashed circle and the red solid circle indicate the footprint of the pupil when the optical system is co-axial and off-axis, respectively. The contribution of the Zernike polynomial terms to the aberration function can be expressed as [26]:
where the parameter $\vec{\rho }$ is the vector form of the pupil coordinate. n denotes the refractive index of the medium. $\lambda$ denotes the wavelength of the light. The general vector coefficient $\vec{M}$ is used to represent the coefficient $- 2n\vec{C}/\lambda$ in the above aberration equation. Then Eq. (7) can be transformed into:Fig. 3. The footprint of the pupil. (a)Freeform surface is introduced at the stop; (b) Freeform surface is away from the stop.
Secondly, when the freeform surface is away from the stop, the light beams of different fields will use the different area on the surface. The area used for the noncentral field displaces from the central field. A relative beam displacement vector $\mathrm{\Delta }\vec{h}$ can be used to account for this effect. As shown in Fig. 3(b), the red dashed circle and the red solid circle indicate the footprints of the pupil at the central field when the optical system is co-axial and off-axis, respectively. And the green solid circle indicates the footprint of the pupil at the noncentral field when the optical system is off-axis. The aberration equation when the freeform surface is away from the stop can be written as:
The $\mathrm{\Delta }\vec{h}$ can be expressed as $({{\bar{y}} / y})\vec{H}$. Where y is the marginal ray height of the on-axis field on the freeform surface, $\bar{y}$ is the chief ray height of the marginal field on the freeform surface, and $\vec{H}$ is the normalized FOV.
Then the aberration expression introduced by Z5 and Z6 Zernike polynomial terms when the freeform surface is away from the stop can be deduced. Astigmatism is the dominant aberration in the imaging spectrometers. Extracting the astigmatism term in the aberration expression, the astigmatism expression can be expressed as:
It shows that the astigmatism expression introduced by Z5 and Z6 Zernike polynomial terms is not related to the normalized FOV $\vec{H}$. Then, the astigmatism introduced by Zernike polynomial Z5 and Z6 terms independent of FOV can be deduced.
The slit off-axis amount is chosen to be 20 mm in the initial structure. While to achieve a compact structure, the slit off-axis amount should be away from the anastigmatic area and as small as possible. As shown in Fig. 4(a), the cyan variation curve represents the astigmatism changing with slit half-length in the initial structure. When the slit off-axis amount is reduced and away from anastigmatic area, the cyan curve shifts down to the blue curve. The system with a smaller slit off-axis amount is not anastigmatic. As shown in Fig. 4(b), through the astigmatism introduced by Z5 and Z6 Zernike polynomial terms, the blue curve shifts up the red curve and the system is anastigmatic. Therefore, the astigmatism in the system with a smaller slit off-axis amount can be balanced by the astigmatism introduced by Zernike polynomial terms Z5 and Z6, leading to a miniaturization of the system volume.
Fig. 4. Variation curves of astigmatism changing with slit half-length; (a) Reducing the slit off-axis amount; (b) Adding Z5 and Z6 Zernike polynomial terms.
2.2 How other Zernike polynomial terms balance the astigmatism produced with the increasing of the FOV
As mentioned in Section 2.1, Z5 and Z6 Zernike polynomial terms are helpful to balance the astigmatism in the system with a smaller slit off-axis amount, leading to a miniaturization of the system volume. Further, how to balance the astigmatism produced with the increasing of the FOV is a stern challenge, which makes it difficult to achieve a wide FOV in the Offner imaging spectrometer.
Z1 to Z4 Zernike polynomial terms are actually related to the position of the reference sphere and do not affect the image quality, and they are generally not applied in fabrication, so these Zernike polynomial terms are not discussed. Substituting other Zernike polynomial terms below the eighth order into Eq. (9) and extracting the astigmatism terms, then the astigmatism expressions can be deduced and are presented in Supplement 1, which shows that the astigmatism expressions introduced by the other Zernike polynomial terms below the eighth order are related to the normalized FOV $\vec{H}$. Then, the astigmatism introduced by other Zernike polynomial terms below the eighth order related to the FOV can be deduced.
As shown in Fig. 5, the cyan variation curve represents the astigmatism changing with slit half-length in the initial structure. The blue curve shows that the maximum slit half-length can be set to 15 mm at the point A2 when continuing to add Zernike polynomial terms to eighth order. At this time, the astigmatism except for the marginal FOV is uniform. The range of the FOV doubles compared with the initial structure. Therefore, the astigmatism produced with the increasing of the FOV can be balanced using the additional astigmatism introduced by other Zernike polynomial terms below the eighth order, which improve the ability of astigmatism correction and widen the range of the FOV.
Fig. 5. Variation curve of astigmatism changing with slit half-length when adding Zernike polynomial terms to eighth order.
3. Optimum design
3.1 Design of CISLS
Based on the theoretical study in Section 2, when the freeform surfaces are introduced on the primary and tertiary mirrors, a CISLS is obtained and shown in Fig. 6.
The specifications of CISLS are shown in Table 1. The system with a broadband from 400 nm to 1000 nm, a slit of 30 mm in length and a volume of 345.6 cm3 is obtained, which shows that the CISLS can achieve a large FOV and a compact structure under a broadband.
Table 1. Specifications of the CISLS
The optical system is symmetric about the YOX plane. Only the even-order terms of Z in the polynomials are used. The lens data of the system and coefficients of the Zernike polynomial terms in M1 and M3 are shown in the Table 2 and Table 3, respectively.
Table 2. Lens data of the system
Table 3. Coefficients of the Zernike polynomial terms
3.2 Performance evaluation
The RMS spot radii within the full FOV of the CISLS are shown in Fig. 7(a), with the black line indicating the RMS spot radius of the Airy disk. The RMS spot radii of the system are less than 5 µm within the full FOV, which are smaller than a half of the pixel size.
Fig. 7. (a) Performance evaluation of RMS spot radii; Performance evaluation of MTFs at different wavelengths. (b) 400 nm; (c) 700 nm; (d) 1000 nm.
MTFs of wavelengths 400 nm, 700 nm, and 1000 nm are shown in Figs. 7(b)–7(d), respectively. MTFs within the full FOV are greater than 0.56 at the Nyquist frequency and the imaging quality is close to the diffraction limit.
The spectral smile and keystone are shown in the Fig. 8(a)–8(b), and both are less than 14% pixels.
3.3 Comparison with the system designed only of spherical mirrors
The Offner imaging spectrometer using the spherical mirrors with the same parameters is designed for comparison. Figure 9(a) is the schematic diagram of the system with the same volume as CISLS. The RMS spot radii shown in Fig. 10(a) are much larger than the RMS spot radius of the Airy disk, and a high imaging quality cannot be obtained. Figure 9(b) is the schematic diagram of the system with the magnifying volume. The system volume is 80 × 105 × 153 mm3, which is about 3.7x of CISLS. The curves in Fig. 10(b) show that the RMS spot radii of system are less than 5 µm within the full FOV, which are smaller than a half of the pixel size. The imaging quality reaches the diffraction limit.
Fig. 9. Schematic diagram of Offner imaging spectrometer designed only of spherical elements. (a) Spherical system with the same volume; (b) Spherical system with the magnifying volume.
Fig. 10. RMS spot radii of Offner imaging spectrometer designed only of spherical elements. (a) Spherical system with the same volume; (b) Spherical system with the magnifying volume.
4. Conclusion
In this study, we propose a CISLS, which achieves a large FOV and a compact structure under a broadband. Based on the vector aberration theory and the analytical study, we find that Z5 and Z6 Zernike polynomial terms introduce the astigmatism independent of the FOV and are helpful to balance the astigmatism in the system with a smaller slit off-axis amount, contributing to a compact structure. Other Zernike polynomial terms below the eighth order introduce the astigmatism related to the FOV and are utilized to balance the astigmatism that produced with the increasing of the FOV, achieving a wide FOV. Finally, CISLS with a slit of 30 mm in length, a broadband from 400 nm to 1000 nm, a numerical aperture of 0.167, and a spectral resolution of 2.7 nm is obtained. The MTFs within the full FOV are over 0.56 at the Nyquist frequency. Both the spectral smile and keystone are within 14% pixel. The volume of CISLS is only 60 mm × 64 mm × 90 mm, which is 3.7x smaller than the conventional Offner imaging spectrometer. The CISLS provides a reference for the design of imaging spectrometers with freeform surfaces.
Funding
National Natural Science Foundation of China (61340007, 61405134); National Defense Basic Scientific Research Program of China (JCKY2018414C013); Natural Science Foundation of Jiangsu Province (BK20161512); Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD); Suzhou Basic Research Project (SJC2023003).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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