Open Access
Add to BibSonomy
Abstract
A freeform pushbroom hyperspectral imager design was investigated as a combination of a freeform reflective triplet imager and a freeform reflective triplet spectrometer used in double-pass. The design operates at about F/2 with a 15-degree cross-track field-of-view and a 30 mm entrance pupil diameter. The design process led to achieving a small volume of less than 2 liters that fits comfortably within a 3U CubeSat geometry, exemplifying the compactness of this hyperspectral imager. We report the freeform sag departures and maximum slopes of the freeform surfaces, as well as the manufacturing tolerances together with an evaluation of the system stray light, all of which highlight the feasibility of a design in this class to be manufactured. This design uniquely positions itself on the landscape of compact hyperspectral imagers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Hyperspectral imaging is an imaging method used to acquire both spatial and spectral information of a scene and has been widely used in chemical analysis and environmental monitoring [1]. Utilizing a hyperspectral imager in an aircraft or a satellite enables hyperspectral imaging over a large area using only a strip-field imager due to the relative movement between the ground and the aircraft or satellite. The strip image can then be fed into a spectrometer through a slit where it is dispersed into the spectral information of the strip image. This method of hyperspectral imaging is called pushbroom, which commenced with the development of the Airborne Imaging Spectrometer [2] and has since been used in many hyperspectral imager designs and missions [3–6].
In recent decades, standardized and miniaturized satellites such as CubeSats have gained popularity because they lower the cost of manufacturing and deployment. The size of a CubeSat is standardized to be multiples of a 10×10×10 cm3 cubic unit, referred to as a “U”. The standardized size of CubeSats contributes to the reduced cost of these satellites because standardized components can be leveraged for solar cells, batteries, and other electronic components [7]. A specialized deployment method called the Poly-PicoSatellite Orbital Deployer (P-POD) was also developed for CubeSats [8] which enables CubeSats to be launched as a secondary payload, further reducing the cost. The low-cost characteristic of CubeSats is ideal for industrial and educational research and experimentation, and the unique cost-performance relation also enables CubeSats to fill key gaps in astronomical research [9] and earth imaging [10]. Hyperspectral imagers compatible with CubeSats have been designed and deployed in past space missions [11–14].
CubeSats significantly benefit from the advancements in compact electronics with low power consumption. Similarly in optical systems, the development of freeform optics is enabling compact systems with high performance [15]. It was previously shown that freeform surfaces can effectively reduce the volume of both spectrometers and imagers while maintaining high performance [16,17]. The combination of freeform optics and the CubeSat format has the potential to enable small satellites with high-performing optical payloads. This paper investigates the design for manufacturing of a pushbroom hyperspectral imager within the CubeSat format with the goal to be less than 4U. In Section 2, the first-order specifications of the hyperspectral imager design are listed and discussed. In Section 3, we describe the design process of the hyperspectral imager. In Section 4, detailed information about the final design and its performance is provided. In Section 5, the process and results of the sensitivity analysis of the final design are discussed. In Section 6, a stray light analysis conducted on the final design is presented.
2. First-order specifications for the hyperspectral imager
Recent hyperspectral imager designs have shown a preference for concentric forms for the spectrometer (e.g., Dyson or Offner type), which have low distortion while being compact and fast [5,11,18–21]. However, curved gratings or customized prisms are needed in these designs. Curved gratings are effective at reducing the number of optical elements needed [22], but they are challenging to manufacture. Also, while Offner-Chrisp designs have been shown to provide excellent spectral-spatial uniformity, some limitations have been discussed by Cook and Silny [23]. Chrisp presented a compact spectrometer design that accommodated a planar grating in a configuration similar to an Offner type by adding a catadioptric lens [24]. This creative design with a planar grating is attractive, yet a limitation is the low dispersion that limits the tradeoff between the final wavelength range and the spectral sampling rate. For the design we present in this paper, we aimed to provide a solution with high dispersion and also a planar grating to ease the manufacturing process of the dispersive element as well as leverage the reported advantages of the reflective triplet [23,25,26]. The specifications reported in Table 1 were chosen in discussion with our industry partners to provide well-rounded performance for earth imaging accounting for prior art, also listed in Table 1. The design covers a wide spectral range, spanning from the visible spectrum to the short-wave infrared while maintaining a high spectral sampling rate compared with the examples shown.
Table 1. Specifications of Pushbroom Hyperspectral Designs.
3. Design process
The design process started with selecting the design form for the imager and spectrometer, which were then separately designed according to the specifications. The two designs were adjusted for pupil matching and then combined into one system. A systemwide optimization was then conducted to reach the final design.
3.1 Imager design
The structure of the imager is a freeform reflective triplet with the stop at the first mirror. The imager Mirror 1 (IM1) and Mirror 3 (IM3) are positive, while Mirror 2 (IM2) is negative, referred to as a PNP power distribution. PNP has been shown to yield smaller volumes than NPP solutions with equivalent specifications [28,29]. The presence of a negative mirror is critical for minimizing the Petzval sum to meet overall specifications. Within PNP, the reflective triplet has also been shown to be the best folding geometry for aberration correction among all unobscured PNP three-mirror geometries [29].
The design was started by setting up a spherical three-mirror starting point that is unobscured and satisfies the aperture and field requirements, which occupies a volume of approximately 1U. This design started with spherical surfaces aligned coaxially. The coaxial design roughly met the first-order specifications and had a PNP power distribution. The design was then made unobscured by tilting the mirrors. After re-optimization, we then changed the surface type to Extended Fringe Zernike Polynomial in CODE V, which combines an off-axis conic base surface with a Fringe Zernike overlay and can be strategically used to improve the testability of an optical surface [30]. This surface type was investigated in both the imager and the spectrometer components. To fully leverage off-axis conics to correct aberrations, only the conic constant and the off-axis angle were optimized initially sans freeform departure, including using CODE V’s Global Synthesis (GS). After selecting the best-performing off-axis conic solution, the freeform departures were implemented to further improve the performance. The imager is symmetric about a plane that is perpendicular to the slit. Therefore, during the optimization, only the Fringe Zernike terms that are symmetric about the same plane were used, up to the 25th term. We applied degeneracy constraints to avoid tilt, power, and despace degeneracies [31]. A square sum penalty on the Fringe Zernike coefficients was used during the optimization process to minimize the freeform departure to improve the manufacturability and testability of each surface, which has been shown to be effective [32]. The resulting imager design has a compact size due to the minimum field size in the tangential direction. The final design layout is shown in Fig. 1, which has an average RMS spot size of 7 µm and a smile distortion of less than 2 µm.
3.2 Spectrometer design
The design form of the spectrometer is a freeform reflective triplet used in double-pass. The spectrometer utilizes a planar reflective grating as the dispersive element, which also serves as the aperture stop. The light coming from the slit interacts with each powered mirror before being dispersed by the grating, then interacts again with each mirror on the return trip after diffraction. Like the imager design form, the spectrometer leverages the PNP power distribution with the spectrometer Mirror 1 (SM1), Mirrror 2 (SM2), and Mirror 3 (SM3).
Similar to the design process of the imager, the spectrometer design was started by setting up a three-mirror unobscured imager design with sagittal fields only. The double-pass was modeled by placing the slit at the focal point of the imager and using a plane mirror to reflect the outgoing collimated light back into the imager to form an image at the slit location. Then, by changing the plane mirror into a linear grating, the image becomes a spectrum. The reflective grating operates at the -1 order with a constant line spacing.
Similar to the imager design process, the surface type was changed to off-axis conic, and the system was optimized without freeform departure to reach F/2 and 15-mm slit length. The spectrum length specification was met by varying the grating groove density. Additional fields were added at this stage to fully simulate the slit width. Since the incoming beam and the diffracted beam hit different parts of the surfaces, the vertices of the surfaces were kept near the center of the illuminated area for each surface. As the design stabilized (i.e., the design met all constraints and generally remained the same with consecutive optimization cycles), freeform departure was added.
The optimization of the system was performed in CODE V without Global Synthesis since the structure of the design cannot be drastically changed. The error function consisted of the default CODE V error function for imaging quality optimization, combined with weighted constraints on distortion and a square-sum penalty on the surface freeform coefficients. The first-order specifications such as volume, F-number, and image size were prioritized, and constraints such as for distortion, square-sum penalty, and telecentricity were added and tightened gradually.
The final design is shown in Fig. 2, which has an average RMS spot size of 9.3 µm, smile distortion less than 2.5 µm, and keystone distortion less than 3 µm.
3.3 System matching
To ensure a smooth combination of the imager and spectrometer, the ideal case is when the output rays of the first system can perfectly match the input rays of the second system at the plane of connection. In the ideal case, as illustrated in Fig. 3(a), the two systems can be combined perfectly without any change to the image quality of each individual design. However, it is not practical to reach the ideal case. In practice, certain first-order quantities are matched as an approximation of a perfect match. For example, if the plane of connection is located in a space where the light beam is collimated in both systems, the two systems can be matched in the first-order by matching the exit pupil of the first system with the entrance pupil of the second system, as shown in Fig. 3(b). If the plane of connection is located at an internal image plane of the total system and is also telecentric in both systems, the two systems can also be matched in the first-order by matching the F-number or numerical aperture of the two systems, as shown in Fig. 3(c).
Fig. 3. Types of optical system matching: (a) perfect match for every ray, (b) pupil match in the space of collimation, (c) F/# match at an internal image in telecentric systems, and (d) F/# and chief ray match in a general internal image.
The first-order quantities can be determined by tracing the marginal and chief rays through the system. Therefore, in general, it can be seen that in any plane of connection, two systems can be matched in the first order by matching the marginal and chief rays of two systems. In the case of the hyperspectral imager design in this paper, the plane of connection is at the slit, which is an internal image plane of the hyperspectral imager. The match of the image planes ensures that the marginal ray and chief ray heights are matched at the plane of connection. In order to fully match the marginal and chief rays, their ray angles also need to be matched. This can be done by matching the F-number and the chief ray angle of the two systems. Because the slit only has a significant length in the sagittal direction, only the sagittal chief ray angle needs to be matched. This matching method is illustrated in Fig. 3(d). Both the imager and spectrometer were designed at F/2 for a matched F-number. The tangent of the sagittal chief ray angle was adjusted to 0.044 for the two components, which is close to being telecentric.
In addition to matching the rays, the imager can also be combined with the spectrometer in two orientations, as shown in Fig. 4. The aberration balance is different in each orientation resulting in the variation in the spectrometer geometry shown in Fig. 4. Results show that both orientations have similar image quality and distortion performance, yet orientation 2 is more compact. Orientation 2 was chosen for the final design due to the larger clearance between the imager and the detector. The compactness gain also supports the choice for this geometry.
3.4 Combined system optimization
After matching the two systems, a systemwide optimization was conducted to reach the final performance. The system stop location was kept at the first mirror to minimize the stray light level. In the optimization process, the grating was first constrained to be conjugate to the stop at the primary mirror. The conjugate constraint was then relaxed in the final optimization. The telecentricity was constrained to be less than 5 degrees in the image space to lower the incident angle on the detector. During the systemwide optimization, the imaging performance at the slit was also part of the error function besides that at the final spectrum. Therefore, the imager maintained an average RMS spot size of 9.8 µm and smile distortion of less than 2 µm.
4. Final design and performance of the combined system
4.1 Design layout and performance
The final optical design layout is shown in Fig. 5, with a volume of 91×161×116 mm3, which can be fully contained in a 3U format. In a 4U CubeSat, the system can fit comfortably with ample space for additional mechanics or electronics. At the final spectrum, the spot size is generally uniform across all fields and wavelengths, as shown in Fig. 6, and the average RMS spot size is 5.5 µm, about 28% of the pixel size. The smile distortion is kept under 25% of the pixel, and the keystone distortion is under 15% of the pixel size. The final grating dimension is 90.4 mm along the groove direction and 71 mm perpendicular to the groove direction. The groove density of the grating is 43.67 line/mm.
Fig. 6. The spectral full-field display for RMS spot size showing the performance over the slit and full spectrum for the combined system.
4.2 Sag and slope departure of the optical surfaces
To get a basic understanding of the manufacturability of the resulting optical surfaces, we look at the freeform sag and slope departures. The sag and slope departures were calculated by sampling the surface at a grid of points, the mathematical definitions of which can be found in Supplement 1. The spacing between points was chosen to be 0.05 mm, resulting in a slope value that was invariant within one digit of the value. Finer sampling gave refinement of less than one percent of the value.
Figure 7 and 8 show the sag and slope departure maps from the best fit off-axis conic of each mirror surface. The aperture of each mirror is circular, with portions cut out to display only the used area. The maximum peak-to-valley (PV) departure for all mirrors is about 250 µm, and the maximum slope departure is 4.0 degrees.
Table 2 shows the PV sag departure and the maximum slope departure for each mirror from the best-fit sphere and the best-fit off-axis conic. One might want to analyze the departure from best-fit off-axis conic because there is the potential to null out the off-axis conic during the surface metrology. The PV sag departure and the maximum slope departure are lower when measured from the best-fit off-axis conic than when measured from the best-fit sphere, especially for the mirrors in the spectrometer. For the current design, however, the residual freeform departure of all mirrors is too large to do a full-field interferometric test after nulling out the best-fit off-axis conic. To make the interferometric test viable, additional nulling optics can be used, such as computer-generated holograms (CGHs) or reconfigurable CGHs [33]. Additionally, other non-interferometric metrology methods can be used to measures the surfaces such as coordinate measuring machines (CMMs).
Table 2. The PV Sag Departure and the Maximum Slope Departure for Each Mirror.
5. Sensitivity analysis
A Monte Carlo process was used to evaluate the as-built performance after random perturbations that emulate fabrication errors. The values of the tolerances were determined so that the as-built performance has above 95% chance to have an average RMS spot size contained within one pixel (20 μm), smile distortion < 8 μm, and keystone distortion < 5 μm.
The tolerances that were investigated for the mirrors and the grating are the decenter, tilt, and the surface figure error modeled by 25 Fringe Zernike polynomials. Image plane tilt (-3.6–4.2 arcmin range), decenter (-50–50 μm range), and despace (-40–30 μm range) were used as compensators.
For the figure error, each Zernike term was assigned a tolerance value that can perturb the surface shape. In this tolerancing process, all Zernike terms were assigned the same value to model a random figure error. If a particular type of figure error is more significant than other types in a known manufacturing process, the tolerance value should be adjusted to reflect the figure error. The resulting tolerance values are summarized in Table 3. The tolerance value for each Zernike term is 10 nm, the accumulative effect of all 37 Fringe Zernike terms can reach a maximum PV sag departure of about 0.3 μm by setting all coefficients to be the maximum tolerance. To represent a lower-order figure error, 25 terms of Fringe Zernike terms were also used instead of all 37 terms. Figure 9 shows a histogram of PV values for 50000 simulated figure errors, from which it can be seen that the maximum PV value is about 0.12 μm.
Fig. 9. Histogram of PV values of 50000 simulated figure error cases using 25 Fringe Zernike polynomials.
Table 3. Tolerance Values to Reach the Target As-built Performance.
6. Stray light analysis and spectrum sorting
Two sources of unwanted light on the detector were investigated: the stray light that interacts with the surfaces in the incorrect order and the light from unwanted diffraction orders of the grating. Note that a full stray light analysis requires the inclusion of mechanical features of the design. Since the mechanical design is beyond the scope of this optical design study, a brief discussion of the stray light sources in the optical design was done to indicate that the parasitic rays coming from the optics themselves and that are amplified when the optical design is compacted, can be managed.
6.1 Stray light that interacts with the surfaces in the incorrect order
A stray light analysis was performed in LightTools on the solid model of the design, as shown in Fig. 10. A physical aperture bigger than the beam footprint was created in front of the first mirror, which all light entering the system must pass through. The location and the size of the physical aperture are illustrated in Fig. 11. It was also assumed that the imager housing and the spectrometer housing were separated and only connected with the slit, which essentially acts as a field-stop with regards to limiting the stray light. The analysis assumes smooth optical surfaces (i.e., scatter-free) and that the light missing optical surfaces will be absorbed by the housing. Therefore, any light that goes through the physical aperture, the slit, and lands on the detector in the incorrect optical surface order can be regarded as stray light. With 40 million rays traced backward from the detector with uniform angular distribution across the whole wavelength range, not a single stray light ray path was found.
Fig. 10. The 3D model of the hyperspectral imager in LightTools (the blue cuboid indicates a 4U CubeSat).
Fig. 11. (a) The physical aperture size and location with respect to the first mirror in the imager, (b) the size comparison between the physical aperture and the beam footprint on the aperture.
The main factor for the low stray light from specular reflections is that the slit serves as an effective field stop to block unwanted light. In general, the aperture stop being at the first mirror of the imager generates a lower stray light level compared with the case where the stop is inside the spectrometer [5]. If the stop is set inside the spectrometer, some unwanted light can still enter the spectrometer before blocked by the stop, thus becoming seeds for stray light if not absorbed.
When expanding the stray light analysis to consider surface scatter, particular attention during fabrication should be given to reduce the surface roughness of the optics and the reflectivity of the housing interior. Furthermore, the residual particles inside the system can be minimized to mitigate scatter by assembling the system in a class 100 clean room.
6.2 Spectrum sorting
The hyperspectral imager was designed for a wavelength range of 400–1700 nm. The ratio of the upper to lower wavelength is larger than two, which leads to an overlap of higher-order spectra with the first-order spectrum. Since only the first-order spectrum is wanted, spectra of higher orders must be removed. There are two ways to avoid higher-order spectra. The first way is to reduce the efficiency of the grating on higher orders. The second way is to place filters on the detector to filter out the higher-order spectra.
Figure 12 illustrates how spectra of different orders are distributed on the detector. The green and orange lines represent two types of high-pass filters that can be used for spectrum sorting. The green step lines represent multiple high-pass filters used in conjunction. Because the wavelength of the second-order spectrum at the end of the detector (850 nm) is higher than the wavelength of the first-order spectrum at where the second-order spectrum starts (800 nm), the spectrum sorting cannot be achieved by using a single high-pass filter. If the upper wavelength was allowed to be adjusted from 1700 nm down to below 1600 nm, a single high-pass filter could be enough to filter out all higher-order spectra. The orange line in Fig. 12 represents a high-pass linear variable filter that can be used for spectrum sorting. The cutoff wavelength of a linear variable filter changes linearly with spatial positions. By placing the line of cutoff wavelength between the first-order spectrum and higher-order spectra, the spectrum sorting can be achieved with one high-pass linear variable filter. Important factors that affect the manufacturability of linear viable filters are the change rate of the cutoff wavelength and the blocked wavelength range [34].
Fig. 12. Illustration of the overlap of spectra of different orders and the potential spectrum sorting filters.
Spectrum sorting filters were not integrated during the design process since specific parameters of the filters have not been decided. To estimate the impact of these filters on the imaging qualities, a 2-mm thick NBK7 cover glass was inserted in the final design, 5 mm away from the image plane, to simulate the worst-case scenario of adding the filter after the design was complete. After refocusing, the resulting average RMS spot size increased from 5.5 μm to 6.3 μm, which is not a significant change compared to the 20 μm pixel size. Figure 13 shows a spectral full-field display for the RMS spot size of the design with the cover glass inserted. The distortion performance remained the same since the design is nearly telecentric at the image plane.
Fig. 13. The spectral full-field display for RMS spot size of the design with the cover glass inserted.
7. Conclusion
In this paper, we have presented a hyperspectral imager design compatible with the 3U optical CubeSat format that is 1.7 liter in volume. The design provides a solution with freeform mirrors and a planar grating. Prior art focused on off-axis conics for the highest complexity surface type and curved gratings, the latter being harder to manufacture than planar gratings. Within the 4U CubeSat, a prior art achieved a 3.5 liter volume. The hyperspectral imager presented in this paper works in pushbroom mode and consists of a reflective triplet imager and a double-pass reflective triplet spectrometer. The system has a 30 mm aperture and works at F/2 with a 15-degree cross-track field of view and a 400–1700 nm wavelength range. The nominal design reaches a 5.5 µm average RMS spot size and distortion less than 25% of a pixel. With misalignment tolerances at 10 µm, tilt tolerances at 0.35 arcmin, and figure error tolerances at 10 nm on the first 25 Fringe Zernike polynomials, the system has more than 95% chance to have the as-built average RMS spot size contained in a pixel, and distortion to be less than 40% of a pixel. This optical system also has a low level of stray light from specular reflections mainly due to the filtering effect of the slit. The stray light analysis in the optical design was done to indicate that the parasitic rays coming from the optics themselves and that are amplified when the optical design is compacted, can be managed. Future work towards manufacturing will include mechanical features of the design.
Funding
National Science Foundation Industry-University Cooperative Research Centers Program Center for Freeform Optics (IIP-1338877, IIP-1338898, IIP-1822026, IIP-1822049).
Disclosures
The authors declare no conflicts of interests.
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.
References
1. M. B. Stuart, A. J. S. McGonigle, and J. R. Willmott, “Hyperspectral Imaging in Environmental Monitoring: A Review of Recent Developments and Technological Advances in Compact Field Deployable Systems,” Sensors 19(14), 3071 (2019). [CrossRef]
2. G. Vane, A. F. H. Goetz, and J. B. Wellman, “Airborne imaging spectrometer: A new tool for remote sensing,” IEEE Trans. Geosci. Remote Sensing GE-22(6), 546–549 (1984). [CrossRef]
3. J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sensing 41(6), 1160–1173 (2003). [CrossRef]
4. M. J. Barnsley, J. J. Settle, M. A. Cutter, D. R. Lobb, and F. Teston, “The PROBA/CHRIS mission: a low-cost smallsat for hyperspectral multiangle observations of the Earth surface and atmosphere,” IEEE Trans. Geosci. Remote Sensing 42(7), 1512–1520 (2004). [CrossRef]
5. P. Mouroulis and R. Green, “Review of high fidelity imaging spectrometer design for remote sensing,” Opt. Eng. 57(04), 1 (2018). [CrossRef]
6. Y. N. Liu, D. X. Sun, X. N. Hu, X. Ye, Y. D. Li, S. F. Liu, K. Q. Cao, M. Y. Chai, W. Y. N. Zhou, J. Zhang, Y. Zhang, W. W. Sun, and L. L. Jiao, “The Advanced Hyperspectral Imager: Aboard China's GaoFen-5 Satellite,” IEEE Geosci. Remote Sens. Mag. 7(4), 23–32 (2019). [CrossRef]
7. H. Heidt, J. Puig-Suari, A. Moore, S. Nakasuka, and R. Twiggs, “CubeSat: A new generation of picosatellite for education and industry low-cost space experimentation,” in 14th Annual/USU Conference on Small Satellites (2000).
8. A. Chin, R. Coelho, R. Nugent, R. Munakata, and J. Puig-Suari, “CubeSat: the pico-satellite standard for research and education,” in AIAA Space 2008 Conference & Exposition (2008), pp.7734.
9. E. L. Shkolnik, “On the verge of an astronomy CubeSat revolution,” Nat Astron 2(5), 374–378 (2018). [CrossRef]
10. C. Boshuizen, J. Mason, P. Klupar, and S. Spanhake, “Results from the planet labs flock constellation,” in 24th Annual AIAA/USU Conference on Small Satellites (2014).
11. P. Mouroulis, B. Van Gorp, R. Green, and D. Wilson, “Optical design of a CubeSat-compatible imaging spectrometer,” Proc. SPIE 9222, 92220D (2014). [CrossRef]
12. M. Beasley, H. Goldberg, C. Voorhees, and P. Illsley, “Big capabilities in small packages: hyperspectral imaging from a compact platform,” Proc. SPIE 9978, 99780G (2016). [CrossRef]
13. S. Grabarnik, M. Taccola, L. Maresi, V. Moreau, L. de Vos, J. Versluys, and G. Gubbels, “Compact multispectral and hyperspectral imagers based on a wide field of view TMA,” Proc. SPIE 10565, 1056505 (2017). [CrossRef]
14. H. Bender, P. Mouroulis, H. Dierssen, T. Painter, D. Thompson, C. Smith, J. Gross, R. Green, J. Haag, B. Van Gorp, and E. Diaz, “Snow and Water Imaging Spectrometer: mission and instrument concepts for earth-orbiting CubeSats,” J. Appl. Rem. Sens. 12(04), 1 (2018). [CrossRef]
15. J. P. Rolland, M. A. Davies, T. J. Suleski, C. Evans, A. Bauer, J. C. Lambropoulos, and K. Falaggis, “Freeform optics for imaging,” Optica 8(2), 161–176 (2021). [CrossRef]
16. J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci Appl 6(7), e17026 (2017). [CrossRef]
17. E. M. Schiesser, A. Bauer, and J. P. Rolland, “Effect of freeform surfaces on the volume and performance of unobscured three mirror imagers in comparison with off-axis rotationally symmetric polynomials,” Opt. Express 27(15), 21750–21765 (2019). [CrossRef]
18. J.-H. Lee, T.-S. Jang, H.-S. Yang, and S.-W. Rhee, “Optical Design of A Compact Imaging Spectrometer for STSAT3,” J. Opt. Soc. Korea 12(4), 262–268 (2008). [CrossRef]
19. P. Mouroulis, R. O. Green, and D. W. Wilson, “Optical design of a coastal ocean imaging spectrometer,” Opt. Express 16(12), 9087–9096 (2008). [CrossRef]
20. J. Zhu, X. Chen, Z. Zhao, and W. Shen, “Design and manufacture of miniaturized immersed imaging spectrometer for remote sensing,” Opt. Express 29(14), 22603–22613 (2021). [CrossRef]
21. C. Graham, J. M. Girkin, and C. Bourgenot, “Freeform based hYperspectral imager for MOisture Sensing (FYMOS),” Opt. Express 29(11), 16007–16018 (2021). [CrossRef]
22. B. Zhang, Y. Tan, G. Jin, and J. Zhu, “Imaging spectrometer with single component of freeform concave grating,” Opt. Lett. 46(14), 3412–3415 (2021). [CrossRef]
23. L. Cook and J. Silny, “Imaging spectrometer trade studies: a detailed comparison of the Offner-Chrisp and reflective triplet optical design forms,” Proc. SPIE 7813, 78130F (2010). [CrossRef]
24. M. P. Chrisp, R. B. Lockwood, M. A. Smith, G. Balonek, C. Holtsberg, K. J. Thome, K. E. Murray, and P. Ghuman, “Development of a compact imaging spectrometer form for the solarreflective spectral region,” Appl. Opt. 59(32), 10007–10017 (2020). [CrossRef]
25. L. G. Cook, “Compact all-reflective imaging spectrometer,” U.S. patent 5,260,767 (9 November 1993).
26. L. G. Cook, “High-resolution, all-reflective imaging spectrometer,” U.S. patent 7,080,912B2 (3 May 2005).
27. J. Huang, Y. Wang, D. Zhang, L. Yang, M. Xu, D. He, X. Zhuang, Y. Yao, and J. Hou, “Design and demonstration of airborne imaging system for target detection based on area-array camera and push-broom hyperspectral imager,” Infrared Phys. Technol. 116, 103794 (2021). [CrossRef]
28. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011). [CrossRef]
29. A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018). [CrossRef]
30. N. Takaki, J. C. Papa, A. Bauer, and J. P. Rolland, “Off-axis conics as base surfaces for freeform optics enable null testability,” Opt. Express 28(8), 10859–10872 (2020). [CrossRef]
31. N. Takaki, A. Bauer, and J. P. Rolland, “Degeneracy in freeform surfaces described with orthogonal polynomials,” Appl. Opt. 57(35), 10348–10354 (2018). [CrossRef]
32. N. Takaki, A. Bauer, and J. P. Rolland, “On-the-fly surface manufacturability constraints for freeform optical design enabled by orthogonal polynomials,” Opt. Express 27(5), 6129–6146 (2019). [CrossRef]
33. R. Chaudhuri, J. Papa, and J. P. Rolland, “System design of a single-shot reconfigurable null test using a spatial light modulator for freeform metrology,” Opt. Lett. 44(8), 2000–2003 (2019). [CrossRef]
34. P. Svensgaard, Delta Optical Thin Film (personal communication, 2021).
