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Abstract
Afocal telescopes are often used as foreoptics to existing imaging systems to allow for application flexibility. To properly combine an afocal telescope with an existing imaging system, the exit pupil of the afocal telescope and the entrance pupil of the imaging system must be coincident. Additionally, the exit pupil of the afocal telescope must be well-formed; that is, it must be the correct size and shape to mitigate pupil-matching challenges. This work introduces processes for designing freeform afocal telescopes with an emphasis on understanding how to analyze and control the exit pupil quality of such systems. The included 3-mirror design examples demonstrate the advantages of using freeform surfaces in afocal systems and quantify the tradeoffs required to improve the exit pupil quality.
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1. Introduction
The introduction of freeform surfaces, or surfaces with no axis of rotational invariance within or beyond the optical part [1], into imaging systems that lack rotational symmetry can provide significant improvements to imaging performance [2], system packaging [2,3], and functionality [4,5]. Some relevant works in freeform design that demonstrate these improvements include an infrared imager [6,7], a five-mirror electronic viewfinder [8], a spectrometer design survey [9], and a dual-FOV imager [10], among others [11–17]. All of these cited works explore using freeform surfaces in a focal system where the light is focused to an image plane (typically, a sensor). There are limited examples in the literature that apply freeform surfaces to afocal systems, or systems whose focal length is infinite [18]. This work intends to fill that gap in the published literature by detailing the processes involved in freeform design for afocal systems [19].
A property of afocal systems is that collimated light entering the system will exit the system collimated. The angle and cross-sectional diameter of a beam exiting the system relative to the beam entering the system are determined by the afocal magnification of the system. Adhering to the Lagrange invariant, a beam of light whose cross-sectional diameter decreases after propagating through an afocal system will see its angle become steeper and vice-versa. Afocal systems are often used in conjunction with existing focal imaging systems, where the afocal system has a large aperture for light gathering and fine object-space resolution. To properly combine an afocal system and an auxiliary imager, the exit pupil of the afocal system must be pupil-matched to the entrance pupil of the imager. The auxiliary imager is typically much smaller but can handle a wider field-of-view (FOV); thus, the large entrance pupil of the afocal system is demagnified to match a smaller entrance pupil.
While the focal length of an afocal system is infinite, that does not mean that it can only take collimated light as an input. Afocal systems can image finite objects to form finite images, but if the afocal system is only optimized to minimize the aberrations for infinite conjugates, the ability to image finite objects without degrading blur will be limited. Imaging a finite entrance pupil to a finite exit pupil is precisely that case. An aberrated exit pupil may have a field-dependent size, shape, and location, which can cause vignetting and image quality challenges when combining it with another optical system.
Interpreting and quantifying the pupil-to-pupil blur, referred to as pupil aberrations [20], is beyond the scope of this work, but evaluating the degree to which the pupil is aberrated and consequently adding the pupil quality to the optimization merit function will be addressed. Moreover, discussing the features of and processes for designing freeform afocal telescopes, including quantifying and controlling the pupil quality in optimization, is the primary goal of this work. To that end, various three-mirror freeform afocal telescope design studies will demonstrate the concepts and elucidate the advantages of using freeform surfaces in this regime. System improvements in optical design rarely come without a tradeoff, so it is critical to know the tradeoffs involved when controlling the pupil quality.
2. Design description
The application that underlies the specification choices for the design studies to be presented here is a remote-sensing satellite in low-earth orbit (700 km altitude) operating in the visible spectrum. The baseline satellite consists of afocal telescope foreoptics with an entrance pupil diameter (EPD) of 250 mm and a full FOV of 1 × 1 degree. The auxiliary imager (not part of the design study) has an EPD of 25 mm, thus requiring an afocal magnification for the telescope of 10x. The full FOV as seen by the auxiliary system is then 10 × 10 degrees. Diffraction-limited performance at 587 nm is required using three mirrors (i.e., RMS wavefront error < 0.07 waves). While the specifications for this application are suited for and will be demonstrated in a three-mirror system, the exit pupil analysis and optimization methods that are discussed are applicable to any afocal system with any number of surfaces.
2.1 Starting point construction
Our previous work in focal freeform design [11] informed our methodology for choosing the starting design. The starting design must, first and foremost, meet the first-order properties, which, in this case, are the afocal magnification and having a real, accessible exit pupil. The starting design is constructed with the full EPD and FOV – we have found no benefit in these types of systems to slowly increase the EPD or FOV from a small value towards the full specification as is commonly practiced for rotationally symmetric systems. Based on prior art with similar specifications (albeit in the focal regime) [11], we estimated a reasonable volume for the starting design to be 70 L. If we assume the Y and Z dimensions are equal and that the X-dimension is roughly equivalent to the EPD, then we can solve for the airspaces between the mirrors to be ∼600 mm. We also know from the prior art that to get a real, accessible exit pupil with the aperture stop in object space, an intermediate image is required with an ideal location shortly after the secondary mirror. With those constraints, a power range for the primary mirror can be established. Then, the powers of the secondary mirror and the tertiary mirror can be solved to give the desired afocal magnification of 10x while satisfying the flat-field condition where the sum of the mirror powers equals 0. The starting design using all spherical surfaces is shown in Fig. 1. It is important to note here that the specific values of the first-order layout parameters (tilt angles, airspaces, mirror radii, etc.) of the system and the magnitude of the aberrations present at this stage of the design are not critical.
Fig. 1. Layout of the starting design for the design studies in this work using all spherical surfaces.
The telescope is unobscured to allow maximum light throughput using tilts about the mirror vertices to create clearance between the rays and the optical surfaces. Only spherical surfaces are used at this point, so these tilts generate massive rotationally variant aberrations. Though the magnitude of these aberrations may seem problematic, they are precisely the types of aberrations that freeform surfaces are designed to correct.
2.2 Design optimization
Each design was optimized in CODE V with a common set of optimization constraints to meet the system requirements, as now detailed. First, the afocal magnification is calculated by tracing quasi-paraxial chief rays that are not significantly influenced by the system aberrations. The ratio of the output angle, θout, to the input angle, θin, for these chief rays, as illustrated in Fig. 2, gives the afocal magnification. The afocal magnification is calculated in this manner for both the X-Z and Y-Z planes since freeform systems can be anamorphic. Next, the ray-surface clearances are maintained using the @JMRCC macro supplied with CODE V. The exit pupil is included in the clearance constraints to ensure its accessibility. Then, to keep the size of the system in check and quantified, volume constraints, calculated as the smallest box to encompass the system with one face normal to the incoming on-axis beam, were used. Finally, constraints on the quality of the exit pupil were toggled on and off in separate design studies to determine the required tradeoffs. More details on the exit pupil quality constraint follow in Section 4.
Fig. 2. A simple afocal system showing a collimated input beam yielding an output collimated beam with some afocal magnification. The Lagrange invariant dictates that if the exit pupil diameter is smaller than the entrance pupil diameter (the scenario shown above), the chief ray angle at the exit pupil is larger than at the entrance pupil.
The freeform surfaces used in the design studies were mathematically defined by a base conic plus Zernike terms up to the 28th term in Fringe ordering. We found only minor improvements going beyond this term for all the studies performed in this work, including when exit pupil constraints are added. Additionally, since the system has plane-symmetry, only the Zernike terms that are symmetric about the Y-axis were used. Takaki et al. showed that there is a negligible performance difference between systems optimized with different polynomial types [16]. Zernike polynomials can exhibit piston, power, and tilt degeneracy when used with a base conic, so they were managed using constraints similar to those found in Takaki et al. [21]. One critical aspect of optimizing with orthogonal polynomials is handling the normalization radius. Our preferred method is to let the normalization radius vary in optimization and constrain its value always to be 5% larger than the semi-diameter of the optical surface. This technique allows the system to continuously change form without encountering scenarios where the surface semi-diameters become larger than the normalization radius, which can cause raytrace errors.
The aberration correction process follows the progression described in Bauer et al. [11]. Though the original method was done for a focal system, the principles from Fuerschbach et al. [22] apply independently of the focal length and work well for the afocal systems being explored here. The Step Optimization feature in CODE V was found to be particularly useful for timely convergence and avoiding local minima. To maximize the impact of the freeform surfaces, all mirrors in the freeform systems are freeform surfaces.
3. Design studies without pupil constraints
The first set of designs explored the freeform advantage in afocal systems over a range of system volumes. Specifically, we compared the achievable root-mean-squared (RMS) wavefront error (WFE) of freeform designs to equivalently specified off-axis conic (OAC) designs from volumes of ∼20 L to 70 L. Off-axis conics were chosen to represent the group of more conventional optical surfaces because they can be both fabricated [23] and measured [23,24] using conventional techniques. Significantly, these initial design studies did not place any constraints on the quality of the exit pupil. While the results are interesting in themselves, they will also serve as a baseline for the subsequent design studies involving exit pupil quality constraints so we can understand the tradeoffs involved.
3.1 System volume reduction technique
The starting design was the system from Fig. 1 with a volume of 70 L. The constraints and processes described in Section 2 were used to optimize the starting design using freeform surfaces to obtain a freeform design at 70 L. One method to decrease the system volume is simply starting with a smaller all-spherical starting design and redoing the optimization. However, this method is time-consuming. Another approach is to reduce the target on the volume constraint for the already-optimized 70 L freeform design and reoptimize. However, we found that the latter method was not as stable for afocal systems as we previously found for focal systems. Specifically, we found that the volume constraint was effective at limiting the volume for a system already below the volume target, but it was not effective at decreasing the volume of a system that started larger than the target volume. To address the lack of stability using the volume constraint, we conceived an improved method that exploited the infinite focal length of afocal systems. To decrease the volume of an existing freeform system, one can scale the entire system by some factor less than 1. Due to its infinite focal length, scaling the system by a factor less than unity results in a smaller system volume while the system remains afocal since any factor times infinity is still infinity. Importantly, the aberration correction of the system is largely maintained. Conversely, for a focal system, scaling the system by a factor less than unity succeeds in making the system smaller, but it also decreases the focal length by the same factor and is not always easily recovered. In practice, we scaled the system to a slightly smaller value than the target volume, then optimized it with a volume constraint imposed to keep it below the target volume.
3.2 Freeform advantage for a range of system volumes
We implemented the optimization process outlined in this section for systems using freeform surfaces and systems using OAC surfaces for comparison. The layouts of the freeform designs for four selected volumes are shown in Fig. 3. The layouts of the OAC designs look quite similar to the freeform system of the same volume. The resulting performance of the freeform designs is plotted together with the performance of the OAC designs in Fig. 4. The main takeaway is that over the volume range studied, switching from OAC surfaces to freeform surfaces provides an RMS WFE improvement of 45–55%. We can also infer from the design data that, for a given level of RMS WFE, the freeform systems are 15 - 30% smaller in volume compared to the OAC systems, with the biggest volume decrease occuring for the lowest levels of wavefront error where the design difficulty is the greatest. Other off-axis sections of parent rotationally symmetric shapes, such as aspheres, can offer more degrees of freedom than conics and, thus, better performance. We found for the two points at the ends of the volume range (70 L and 25 L), that off-axis asphere designs had max RMS WFE values that would place them approximately halfway between the off-axis conic and freeform designs. However, since measuring off-axis aspheres cannot be done with conventional techniques, off-axis conics better represent the group of more conventional optics and will continue to be used here for comparison.
Fig. 3. 2D layouts of four selected freeform systems that span the volume range studied. The system at 21 L is the smallest volume where near diffraction-limited performance was achieved. The OAC design layouts look quite similar to the freeform designs of the same volume.
Fig. 4. The maximum RMS WFE over the 1° x 1° full FOV for the freeform and OAC designs are plotted over the studied volume range. The freeform advantage is shown on the right vertical axis as the percentage improvement of the maximum RMS WFE across the full FOV when using freeform surfaces (lower RMS WFE is better).
3.3 Freeform advantage quantified by increased diffraction-limited FOV
Improving the imaging performance of a system by using freeform surfaces is only one way to realize the benefit of freeform surfaces. Another way to leverage freeform surfaces is to enhance the system functionality by increasing the FOV. For this study, we optimized systems over a range of volumes for freeform and OAC systems. For each system, the FOV was increased until the system was no longer diffraction-limited (RMS WFE < 0.07 waves). All other parameters and specifications were equivalent between the freeform and OAC systems. In doing so, we were able to quantify the freeform advantage in terms of increased FOV, as illustrated in Fig. 5. The main takeaway is that over the volume range studied, the freeform systems were able to support a diffraction-limited FOV that was consistently between 25 - 30% greater than the OAC systems, which is equivalent to about 63% more etendue (averaged over the volume range).
Fig. 5. The achievable diffraction-limited full FOV for the freeform and OAC designs is plotted over the studied volume range. The freeform advantage is shown on the right vertical axis as the percentage increase in the achievable FOV when using freeform surfaces in place of OACs.
4. Design studies with pupil constraints
The designs in Section 3 were all optimized without controlling the exit pupil quality. However, a visual inspection of the 2D layouts, especially for the systems with the largest FOV, can confirm that the exit pupil is not always well-formed. To better understand the exit pupil quality of a given design, we defined two pupil metrics – the RMS pupil size/shape error and the RMS pupil offset.
4.1 Exit pupil quality evaluation
The pupil size/shape error is precisely what it sounds like – the difference in size and shape between the real exit pupil for a given field point and the paraxial (perfect) exit pupil. The first step to calculate this metric is to determine which plane to call the “exit pupil plane.” To find this plane, a dummy surface is shifted along the gut ray to the point where the size disparity between the field points is minimized. Then, for a densely and evenly sampled selection of field points, we find the X and Y diameters of the beam footprints at this exit pupil plane. The RMS of the differences between the X and Y diameters of the real exit pupils and the paraxial exit pupil is found and reported as a percentage of the paraxial exit pupil diameter. Any difference in the size of the exit pupil in X or Y results in a corresponding difference in the chief ray angle in image space, as predicted by the Lagrange invariant.
The pupil offset is the variation in the location of the exit pupil with FOV. To calculate this value, the (x,y) intersection of the chief ray for a given field point is found at the exit pupil plane. Ideally, this value is zero, but pupil aberrations smear the real exit pupils along the gut ray, so, in practice, the value will likely be non-zero. The RMS of the distances to the chief ray intersection point is taken and reported as a percentage of the paraxial exit pupil diameter. Again, the FOV is densely and evenly sampled to not overweight any section of the FOV.
Now that we have a way of quantifying the quality of the exit pupil, we evaluate the two metrics for the systems we optimized without exit pupil constraints in Section 3. The results are shown in Fig. 6 for the design study of Section 3.2 and in Fig. 7 for the maximized diffraction-limited FOV study of Section 3.3. In Fig. 6, the exit pupil quality between the freeform and off-axis conic designs for these diffraction-limited systems with identical specifications are more-or-less equivalent, with the smallest volumes having slightly worse overall pupil quality. In Fig. 7, a clear trend is observed that the smaller volume systems have better exit pupil quality for both freeform and OAC designs. However, the main driver for the exit pupil quality is the magnitude of the FOV, which is not shown explicitly in Fig. 7. The diffraction-limited FOV increases with volume and the exit pupil quality decreases as the FOV increases, which together explains why the OAC designs have better pupil quality metrics than the freeform designs. Freeform surfaces allow for larger FOVs at the same volume, but larger FOVs are accompanied by worse pupil quality (when not constrained). This behavior is analogous to distortion in a focal system where the distorted image size can be thought of as a field-dependent magnification error.
Fig. 6. The pupil size difference and pupil offset of the freeform systems and OAC systems optimized without pupil constraints in Section 3.2.
Fig. 7. The pupil size difference and pupil offset of the freeform systems and OAC systems optimized without pupil constraints in Section 3.3.
4.2 Exit pupil quality optimization constraints
If the exit pupil quality in a system is worse than what is needed, the designer can add constraints to the optimizer merit function so designs with better exit pupil quality are favored. However, the improvement in exit pupil quality will likely not come for free – there will be a tradeoff.
Because the pupil quality is quantified in two ways, it is leveraged to constrain the pupil quality in two separate ways that align with the quantifications. To constrain the pupil size, we find the distance between the + X and -X marginal rays and the + Y and -Y marginal rays at the exit pupil plane for each field point. These distances are the X and Y pupil diameters, which are set equal to the desired exit pupil diameter as a weighted constraint. The merit function is penalized for values that differ from the desired value.
The pupil offset is the variation with field of the location of the exit pupil along the gut ray. At the defined exit pupil plane, the chief ray of a field point with an exit pupil not located at the plane will have a non-zero (x,y) intersection point. To constrain this parameter, the (x,y) intersection point of the chief ray at the exit pupil plane for each field point is found. The distances from the origin of the exit pupil plane are summed, and the sum is equal to 0 with a weighted constraint. The merit function is penalized for a non-zero sum of the distances, thus driving the chief rays to intersect that exit pupil plane closer to the origin.
4.3 RMS WFE tradeoff when using exit pupil constraints
We optimized systems with the same specifications as those shown in Section 3.2 but with the addition of the two aforementioned exit pupil quality constraints. The tradeoff to be quantified is the RMS WFE increase to meet a goal of < 1% RMS pupil offset and < 1% RMS pupil size. The performance results over volumes from 21–70 L are plotted in Fig. 8, both with the pupil quality constraints (dotted blue) and without the constraints (solid blue). The percent RMS WFE increase (worse performance) after adding the pupil constraints, also shown in Fig. 8, ranges from 80% at the smallest volume to near no change for the larger volumes.
Fig. 8. Performance tradeoff when optimizing the freeform systems with exit pupil quality constraints.
At 70 L, there is basically no performance difference between the system with the pupil constraints enforced and the system without the constraints. At this larger volume, there is no hardship presented by the volume constraint, and the nominal performance is quite good; therefore, there is the freedom to assume configurations with better exit pupil quality. However, at the other end of the volume range (21 L), the volume constraint is much more impactful, and changes in the configuration/surfaces to yield better pupil quality significantly affect the RMS WFE.
4.4 Maximizing the diffraction-limited FOV with exit pupil constraints
Here, we repeat the study from Section 3.3, but with the aforementioned exit pupil constraints enforced. The goal is to reach less than 1% RMS pupil offset and 2% RMS pupil size error (1% RMS pupil size error was too challenging to achieve with the larger FOVs in this study). The freeform designs are compared directly with OAC designs in Fig. 9 over volumes from 25–70 L with the pupil quality constraints enforced. The freeform designs can achieve about a 20% increase in FOV compared to the OAC designs, which translates to a 44% increase in etendue. The difference visibly widens as the system volumes increase. Then, to understand the tradeoff required to improve the exit pupil for the freeform and OAC designs separately, we plotted the maximum FOV achieved with and without the pupil constraints in Fig. 10. At small volumes, the achievable FOV is small, which intrinsically leads to better exit pupil quality; thus there is not a significant loss of FOV. As the volume increases, so does the achievable FOV. Larger FOVs have worse exit pupil quality, so there is a larger tradeoff to correct the pupil quality, seen as a 25% reduction in the achievable FOV at 70 L volume for the freeform systems.
Fig. 9. Freeform vs. OAC comparison after implementing exit pupil quality constraints and maximizing the diffraction-limited FOV.
Fig. 10. The tradeoff in achievable diffraction-limited full FOV when implementing exit pupil quality constraints is shown for the freeform designs and the OAC designs.
5. Conclusion
The design and analysis methods developed for optimizing afocal systems, evaluating the exit pupil quality, and constraining of the exit pupil quality during optimization enabled the afocal design studies performed in this work. Though performed for example three-mirror systems, the exit pupil analysis and optimization methods can be applied broadly to all afocal sytems. As evidenced by these studies, there is a significant benefit (in performance and FOV) offered by freeform optics in the context of afocal telescope systems, both with and without exit pupil quality constraints, when compared to the more conventional surface type of OACs. Using these techniques when designing afocal systems can provide immediate feedback when there is a specification on the quality of the exit pupil.
It was shown in the studies that when systems have a large FOV, constraining and controlling the exit pupil quality becomes a challenge, much like controlling image distortion of a focal system. In cases where the FOV is wide and the exit pupil quality is untenable, one could explore moving the physical stop to the exit pupil plane. In this case, the exit pupil is perfect, but the entrance pupil becomes aberrated. Determining the consequences and tradeoffs of this arrangement is beyond the scope of this work but remains an interesting aspect to investigate in further studies. For situations and applications whose cost has an upper bound exceeded by an all-freeform solution, each freeform surface can be analyzed to determine if it could be simplified to a lower cost off-axis conic surface with an acceptable trade-off in performance.
Funding
National Science Foundation (IIP-1338877, IIP-1338898, IIP-1822026, IIP-1822049).
Acknowledgment
This research was supported in part by the National Science Foundation I/UCRC Center for Freeform Optics (IIP-1338877, IIP-1338898, IIP-1822049 and IIP-1822026).
Disclosures
The authors declare that there are no conflicts of interest to this article.
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.
References
1. 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]
2. 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]
3. J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light: Sci. Appl. 6(7), e17026 (2017). [CrossRef]
4. C. Yoon, A. Bauer, D. Xu, C. Dorrer, and J. P. Rolland, “Absolute linear-in-k spectrometer designs enabled by freeform optics,” Opt. Express 27(24), 34593–34602 (2019). [CrossRef]
5. Z. Tang and H. Gross, “Improved correction by freeform surfaces in prism spectrometer concepts,” Appl. Opt. 60(2), 333–341 (2021). [CrossRef]
6. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing phi-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011). [CrossRef]
7. K. Fuerschbach, G. E. Davis, K. P. Thompson, and J. P. Rolland, “Assembly of a freeform off-axis optical system employing three phi-polynomial Zernike mirrors,” Opt. Lett. 39(10), 2896–2899 (2014). [CrossRef]
8. A. Bauer, M. Pesch, J. Muschaweck, F. Leupelt, and J. P. Rolland, “All-reflective electronic viewfinder enabled by freeform optics,” Opt. Express 27(21), 30597–30605 (2019). [CrossRef]
9. C. Liu, C. Straif, T. Flügel-Paul, U. D. Zeitner, and H. Gross, “Comparison of hyperspectral imaging spectrometer designs and the improvement of system performance with freeform surfaces,” Appl. Opt. 56(24), 6894–6901 (2017). [CrossRef]
10. C. Xu, D. Cheng, J. Chen, and Y. Wang, “Design of all-reflective dual-channel foveated imaging systems based on freeform optics,” Appl. Opt. 55(9), 2353–2362 (2016). [CrossRef]
11. 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]
12. Y. Liu, A. Bauer, T. Viard, and J. P. Rolland, “Freeform hyperspectral imager design in a CubeSat format,” Opt. Express 29(22), 35915–35928 (2021). [CrossRef]
13. 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]
14. B. Narasimhan, P. Benitez, J. C. Miñano, M. Nikolic, and D. Grabovickic, “Three surface freeform aplanatic systems,” Opt. Express 25(10), 10710–10715 (2017). [CrossRef]
15. 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]
16. 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]
17. T. Yang, L. Zhou, D. Cheng, and Y. Wang, “Designing reflective imaging systems with multiple-surfaces-integrated elements using a Gaussian function freeform surface,” Appl. Opt. 61(17), 5215–5225 (2022). [CrossRef]
18. T. Yang, J. Zhu, and G. Jin, “Starting configuration design method of freeform imaging and afocal systems with a real exit pupil,” Appl. Opt. 55(2), 345–353 (2016). [CrossRef]
19. W. B. Wetherell, “AFOCAL SYSTEMS,” in Handbook of Optics: Volume I - Geometrical and Physical Optics, Polarized Light, Components and Instruments, M. Bass, ed. (McGraw-Hill Education, 2010).
20. J. Sasian, “Interpretation of pupil aberrations in imaging systems,” Proc. SPIE 6342, 634208 (2006). [CrossRef]
21. N. Takaki, A. Bauer, and J. P. Rolland, “Degeneracy in freeform surfaces described with orthogonal polynomials,” Appl. Opt. 57(35), 10348–10354 (2018). [CrossRef]
22. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014). [CrossRef]
23. T. J. Magner, J. Zaniewski, S. Rice, and C. Fleetwood, “Fabrication and testing of off-axis parabolic mirrors,” Opt. Laser Technol. 19(2), 91–96 (1987). [CrossRef]
24. D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007).
