1. Introduction

When conceiving a new lens design, the field-of-view (FOV) and aperture are two specifications that drive the choice of the starting design form. Shown in Fig. 1 is a commonly referenced diagram that was constructed in the prior art to provide a roadmap for the range of typical FOVs and speeds for an array of common design forms [1]. This diagram provides an optical designer with upfront knowledge about the most promising design forms to investigate for a given set of specifications. The advent of ultra-precision computer numerical control manufacturing of optics has opened a new frontier of realizable optical systems that leverage freeform optics [2]. When implemented in optical systems, freeform optics provide the capability for better optical performance [3–7], increased compactness [8,9], or increased etendue compared to similar systems using conventional optics. Consequently, the diagram shown in Fig. 1, which was historically constructed assuming the use of conventional optics, does not reflect the capabilities of systems using freeform optics, nor does it incorporate information related to the size of the system, which is often a driving factor in the decision to explore a freeform solution. To this date, over ten years after the start of the freeform revolution [10], we remain in the dark regarding the acute navigation of the full design space of freeform optical systems.

 

Fig. 1. Common design forms populate this map of achievable FOV and aperture specifications. Designers can reference this map to understand which design forms will most likely satisfy a set of design specifications (Adapted from [1]).

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One prevalent system type shown to have significant improvement through the inclusion of freeform optics is an unobscured three-mirror imager [9,11–13]. Three mirrors offer sufficient degrees of freedom to achieve diffraction-limited optical performance over a wide range of specifications while minimizing the number of surfaces to limit the system volume and cost. However, there is a myriad of different combinations of FOV, entrance pupil diameter (EPD), F-number, and volume specifications that an application may require. As such, it is not readily apparent if a freeform three-mirror solution is suitable for any given combination. This point is empirically corroborated by the fact that the authors frequently receive inquiries regarding the feasibility of designing a three-mirror freeform system with a specific FOV and F-number within a particular volume constraint. Without a roadmap for the three-mirror freeform imagers, the only way to determine if a combination of specifications is achievable in a three-mirror form is to invest time in designing it. This paper aims to uncover the possibilities for a three-mirror freeform imager by mapping out the design space. Specifically, we designed a plethora of systems that sample the specification space – full FOVs up to 50 degrees, EPDs up to 250 mm, and F-numbers ranging from F/4 down to F/1. In each design, diffraction-limited performance, defined as having a root-mean-square (RMS) wavefront error (WFE) of λ/14 or less in the visible spectrum, was sought at the minimum possible system volume. As a result, the question of whether a three-mirror freeform imager can be designed with a given set of specifications can be definitively answered using the proposed roadmap.

2. Unobscured three-mirror design forms

In the context of unobscured three-mirror imagers, solutions dating back to the 1970s employed off-axis sections of rotationally-symmetric conics or aspheres to combat the rotationally-variant aberrations. These early design forms became well-known in optical design, earning monikers such as the three-mirror anastigmat (TMA) [14], reflective triplet (or three-mirror compact) [15], three-mirror long [16], and WALRUS [17]. While off-axis conics or off-axis aspheres provide the ability to correct for rotationally variant aberrations, they do not match the ability for independent aberration control provided by freeform optics [8,18].

Freeform optics are optics whose surface shapes have no axes of rotational symmetry either within or outside the part aperture. The rotational variance of freeform surface shapes provides them with the ability to correct aberrations with rotational variance across the field that are typical of unobscured reflective systems. For three-mirror imagers using off-axis conics or aspheres, the predominant and best-performing folding geometry is the “zig-zag” geometry, where each successive mirror directs the rays further in the same direction. Bauer et al. explored the possibility that the enhanced aberration control of freeform optics would enable other geometries to provide a greater benefit [19]. However, while each explored geometry showed substantial improvement over their non-freeform equivalents, the “zig-zag” geometry was similarly found to be the most performant and compact. Thus, the “zig-zag” geometry will be the focus of this work.

Within the “zig-zag” geometry, there are variations that support increasing ranges of FOVs. The TMA is a reimaging form and supports the smallest FOV; the reflective triplet is the most compact and supports the next largest FOV; and the three-mirror long replaces the positively-powered primary mirror of the reflective triplet with a negatively-powered mirror that decreases the ray angles at the aperture stop, allowing for the largest FOV in exchange for occupying the largest volume.

The aperture stop location also plays an important role in determining the imaging capabilities of each “zig-zag” geometry variation. In the TMA, the aperture stop is ideally located at the primary mirror or in object space to allow for the formation of an intermediate image near the secondary mirror and an accessible pupil between the final image and the tertiary mirror. In the reflective triplet, the symmetry about the aperture stop provided by a secondary mirror located at the stop affords the best aberration correction. Still, there is a compactness advantage to an object-space (or primary mirror) located stop in the lower range of its supported FOV where the mirror aperture sizes are under control. The three-mirror long operates best with the stop at the secondary mirror, which also minimizes the aperture sizes of the three mirrors for the large FOVs that it supports.

It is essential to understand the differences in the unobscured three-mirror design landscape because they dictate how that landscape is populated. This paper aims to become a reference to those seeking a freeform three-mirror design and to determine the most compact volume into which the system can be encapsulated. To that end, the TMA form will not be a focus of this exploration – its intermediate image requirement prohibits it from occupying a smaller volume than an equivalently specified reflective triplet. However, if the ultimate control of stray light or an accessible exit pupil is essential for a given application, the TMA should undoubtedly be considered.

3. Design optimization

In pursuit of mapping out the unobscured three-mirror design space utilizing freeform optics, specification boundaries for EPD, FOV, and F-number were chosen that represent a wide range of applicability and feasibility. For each parameter, intervals were selected that span the chosen range. Designs were optimized for each combination of EPD, FOV, and F-number, where the performance target was diffraction-limited imaging (λ/14 RMS WFE) over the full FOV at 550 nm. The volume of each design was decreased until diffraction-limited performance was no longer achieved. The range of full FOVs explored was 2° – 50°, with designs at 2° intervals up to 30° so that the transitions between solution spaces (SS) were captured, and at 5° intervals beyond 30° up to each design’s maximum diffraction-limited FOV. The final interval may be less than 5° in cases where the maximum diffraction-limited FOV for a given design is not a multiple of 5. The range of EPDs studied was 50–250 mm, with designs at every 50 mm. Finally, the F-number range was F/4 down to F/1, with designs at each whole number focal ratio. These intervals result in a potential total of 250+ optimized designs.

3.1 Optimization procedure

The method used in this paper builds on a method reported in Bauer et al. (2018). One starts with choosing a geometry with the best potential for aberration correction based on its intrinsic low-order aberration contributions and concludes with a step-by-step optimization procedure that targets the limiting aberrations with selected freeform shapes [19]. In addition, in the current methodology, we ensured various constraints in the optimization process, as further described in Section 3.2. The “zig-zag” geometry, which is the ideal starting geometry for the implementation of freeform optics, was used as the starting geometry for the systems designed herein.

The three mirrors in each design were tilted about a single axis to create the ray clearances needed to remove mirror obstructions. As a result, each system maintained a symmetry plane that allowed field points to be defined and clustered closely on one half of the full FOV. The surfaces themselves were described by Fringe-ordered Zernike polynomials up to Z28 (secondary tetrafoil) atop a base conic. Zernike polynomials have been shown to perform equivalently to other sets of orthogonal polynomials during freeform design studies [20]; thus, the results found here broadly apply. Zernike terms higher than Z28 can be included in each surface, though, for the class of designs presented here, there was not a substantial change in performance that justified the additional optimization computation time and surface complexity.

With these tools, we started with the smallest volume geometry variation (the reflective triplet with the stop at the primary mirror) and optimized the systems at the low end of the FOV range. Inevitably, at some FOV, the reflective triplet can no longer support a diffraction-limited system. At this point, other variations, which are larger in volume but support a larger FOV, were explored. This process was repeated for all combinations of FOV, EPD, and F-number, with the all-spherical starting points for a given F-number with different EPDs being generated by scaling up the systems from the smallest EPD design, prior to optimization. In some cases, specifically for the faster F-numbers, the diffraction limit could not be reached for large FOVs and EPDs, so these combinations set the limits of solutions on the roadmap.

3.2 Optimization constraints

Various optimization constraints and software features were utilized to aid in the optimization and lessen the designer's cognitive load. Faster convergence towards the optimal SS was observed when using Step Optimization in CODE V, an optimization algorithm designed to avoid getting trapped in shallow local minima. In some cases, the best SS was not found until Step Optimization was activated. The clearances between mirror apertures and rays were managed using the CODE V-supplied JMRCC macro and associated optimization constraints to keep the ray-to-surface clearances greater than 10 mm. In a scenario where stray light control is essential, the clearances between rays and mirrors can be adjusted from the minimum required value to allow for baffle placement. Stray light constraints were not considered for this work as the specific requirements are application-based. If these constraints were implemented, the reported minimum diffraction-limited volumes for each specification combination would necessarily increase.

System volume is a critical component of this work and is best controlled with a macro. Decreasing the volume can be accomplished by a reduction in any of the three linear system dimensions. However, it is leveraged to constrain only the volume and let the optimizer massage all three dimensions simultaneously to pursue a lower volume rather than the designer choosing which dimension(s) to decrease. In this macro, the volume is calculated by cycling through every reference ray for each field point and finding the ray intersection points on each surface. The extent of each dimension is found from the bounding rays, and the volume is the product of each dimension. The coordinate system axes, and therefore the dimensions, of the bounding box, are aligned with the object space coordinate system axes before any coordinate axis transformations from the tilting of the mirrors occur. The center field for all of the systems designed here is (x = 0°,y = 0°), thus the coordinate system for the bounding box used in the volume calculations is consistent across all of the designs. There are cases where a rotated bounding box or a polygon with additional faces could provide an optimized representation of the volume, but here we opted for a consistent approach for the volume calculation. Additionally, the aperture stop was restricted to one of the three mirror surfaces. While the stop could reside in an airspace between the mirrors, physically realizing such a stop would create ray obscuration issues that would require additional clearances, and therefore, a greater system volume.

The degeneracies of the Zernike polynomials with the base conic and layout parameters can hinder convergence and, thus, were controlled. Specifically, contributions to the mirror tilts and the airspaces between each surface were constrained to originate only from the corresponding physical variable as opposed to the degenerate Zernike tilt and piston, which are present in more than just the first couple of terms. The optical power was similarly constrained to come from the base surface [21]. Real-ray constraints were used at the image plane to constrain distortion to less than 5%. Finally, the image plane was allowed to have a slight tilt of fewer than 5° to help balance field-linear medial field curvature [22]. The design specifications for developing the roadmap are summarized in Table 1.

Table 1. Roadmap full design specifications

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4. Solution spaces

After completing the system optimizations, there were design forms that had similarities amongst the combinations of specifications and could be categorized into a small number of SS. There were five SS identified in this study, and one representative layout of each is shown on the map in Fig. 2. SS1, shown in the blue highlighted region of the F/4 plot, resembles the reflective triplet with the aperture stop located at the primary mirror. This SS has the smallest volume given its positive powered primary mirror but supports the lowest FOV. SS2 is shown in the green highlighted area for F/2. It is similar to SS1, except that the stop has been moved to the secondary mirror. Its volume is the second smallest and supports a larger FOV than SS1. Next is SS3, shown in the orange highlighted area for F/2, which moves into the territory of the three-mirror long with a negative primary mirror and substantially larger volume than the first two SS. However, it can support among the widest FOV. A variation of the three-mirror long is SS4, indicated in the red highlighted area for F/3, which was found for systems in the higher end of the EPD and FOV range. It is differentiated from SS3 by the significant separation between the tertiary and primary mirrors. Finally, SS5, shown in the purple highlighted area for F/1, can achieve the fastest speed. It is most similar to the reflective triplet with the stop at the secondary mirror but uses the primary mirror at a steep angle. These solution space characteristics are compiled in Table 2. We note that though the solution spaces are reminiscent of the classical design forms, our study was not biased towards any specific solution space.

 

Fig. 2. Each of the four plots above shows the minimum diffraction-limited volume achievable for a range of full FOVs at five different EPD values (in mm). The volume is plotted on a logarithmic axis. The diffraction-limit was calculated at λ = 550 nm. A representative optical layout for each SS is shown with an arrow indicating its SS. The precise location of the representative design for each SS is circled on the map in the corresponding SS region.

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Table 2. Characteristics of the five solution spaces identified in this study. The ranges shown for volume and FOV consider all EPDs and F-numbers for which each solution space is found.

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5. Design results and discussion

Here we will provide means to visualize the findings of this roadmap study.

5.1 Constant F-number plots

After optimizing 200+ designs to their minimum volumes while maintaining diffraction-limited performance at 550 nm, the design data was organized to create maps of the unobscured freeform three-mirror design space.

The plots in Fig. 2 correspond to a constant F-number with the various EPDs plotted as separate lines for F/4 down to F/1. The volume range on these plots is too extreme for visualization on a linearly scaled axis, thus the system volume is plotted with a logarithmic scale on the vertical axis. For each F-number, the specification area in which each SS can be found are indicated by various colored regions. The regions are not meant to be rigid. Applying constraints differently, adding new constraints, or using more Zernike terms are examples of factors that can gently morph the SS regions. The upper-right portions of each plot in Fig. 2 correspond to a design space with FOVs that are too great for systems to achieve diffraction-limited performance at 550 nm. Designs exist in these regions, but they will not be diffraction-limited.

5.2 Example plot usage

As an example of how these plots can guide decision-making, imagine a scenario where a Company wants to design a new freeform imager, if possible, using a maximum of three mirrors. They specify that the system should ideally operate at F/2 with a 100 mm EPD with a volume no larger than 100 L, but they are not sure what FOV is possible in that space. The F/2 plot in Fig. 2 shows that the largest full FOV for a diffraction-limited system in the visible is about 16 degrees, and the form falls into the three-mirror long-type layout. Note that the inclusion of additional application-specific constraints to the system design may lower the FOV limit indicated by this work.

In another scenario, another Company’s application is to look at the earth from low-earth orbit (about 400 km) and resolve details as small as 1 meter on the surface with a 10° full FOV. The required EPD can be calculated as $({1.22} )({550\; \textrm{nm}} )({400\; \textrm{km}} )/({1\; \textrm{m}} )$, or approximately 250 mm. From Fig. 2, the F/4 and F/3 solutions are less than 1000 L in volume, while the F/2 solution is almost 2x the volume of the F/3 solution. The F/1 solution cannot reach the diffraction limit and, thus, cannot meet the resolution requirement. The likely choice would be between F/4 and F/3, depending on the other design specifications.

5.3 Extension to the long-wave infrared spectrum

The design results presented in Fig. 2 were predicated on operation in the visible spectrum. The level of aberration correction required for diffraction-limited performance in these systems is significantly higher than what would be necessary for diffraction-limited performance for longer wavelengths, such as the long-wave infrared (LWIR). It can be presumed, then, that for a given F-number, EPD, and FOV combination, the minimum diffraction-limited volume of a system operating in the LWIR is substantially lower than the same system operating with visible light.

To understand the impact on the Fig. 2 roadmap of moving from visible light to LWIR radiation, we optimized a sample set of systems in the LWIR. For the F/3, 250 mm EPD set of designs, we selected the largest FOV in each SS region, as well as the lowest overall FOV (2°) for optimization. An example of the design transformation from the visible spectrum to LWIR is shown in Fig. 3 for a 12° full FOV. Further design results are shown in Table 3. Note that only non-reimaging systems were considered here, which are appropriate for the use of uncooled focal plane arrays in the LWIR. Cooled sensors for LWIR imaging require a cold stop and a reimaging optical system, which could be part of a future study.

 

Fig. 3. Optical layouts for the minimum diffraction-limited volume systems when designed in the (a) visible spectrum (λ = 550 nm) and the (b) LWIR (λ = 10 µm) with a 250 mm EPD and a 12° full FOV. The visible spectrum system falls into SS4 and has a volume of 1870 L, while the LWIR system can achieve diffraction-limited performance in SS2 at a volume of 70 L.

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Table 3. Design results for selected FOVs with a 250 mm EPD operating at F/3.

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Because less aberration correction is needed for diffraction-limited performance in the LWIR, the system package can be squeezed tighter. Furthermore, the FOV range possible within a given SS is significantly expanded, leading to drastic volume reductions from the visible designs. However, order-of-magnitude reductions in the diffraction-limited volume are only observed when there is a new, more compact SS into which the design can jump. For example, the 12° FOV system jumped from SS4 in the visible to SS2 in the LWIR. The 8° and 2° FOV systems were already in the most compact SS in the visible and, therefore, do not experience as great of a volume reduction when moving to the LWIR. Similar drastic volume reductions are expected for other designs with specifications combinations that allow jumps to more compact SS when optimized for diffraction-limited performance in the LWIR.

6. Conclusion

As substantiated by the investigation and associated visual plots presented in this paper, unobscured three-mirror imagers that utilize freeform surfaces cover a wide range of EPD, FOV, F-number, and volume specifications. This complexity makes it difficult to know where to focus design efforts and how the final optical layout may look. Much like the popular diagram shown in Fig. 1 did for rotationally symmetric lens design, the plots shown in Fig. 2 provide quantitative bounds for the capabilities of a three-mirror freeform imager and identify five solution spaces into which the designs are categorized. In cases where achieving cost-effective designs is prioritized over compactness, we expect that the roadmap provided would enable designers to narrow-down the explorable solution space to solve their problem from which volume tradeoffs with metrics such as freeform departure and slope can be further studied and optimized. This roadmap is expected to divert the time and effort one may put into a design with no chance of success towards a plausible combination of design specifications. We envision this work may stimulate an expansion of this roadmap by others in the area of large-scale optics or entirely new roadmaps that explore system parameters such as reimaging, number of mirrors, floating stops (for telecentric systems), or the scalability to other spectral ranges.