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Off-axis conics as base surfaces for freeform optics enable null testability

Nick Takaki, Jonathan C. Papa, Aaron Bauer, and Jannick P. Rolland

Open Access Open Access

Abstract

When conducting interferometric tests of freeform optical surfaces, additional optical components, such as computer-generated holograms or deformable mirrors, are often necessary to achieve a null or quasi-null. These additional optical components increase both the cost and the difficulty of interferometric tests of freeform optical surfaces. In this paper, designs using off-axis segments of conics as base surfaces for freeforms are explored. These off-axis conics are more complex base surfaces than typically-used base spheres but remain null-testable. By leveraging off-axis conics in conjunction with additional orthogonal polynomial departures, designs were found with up to an order-of-magnitude of improvement in testability estimates relative to designs that use base spheres. Two design studies, a three-mirror telescope and a wide field-of-view four-mirror telescope, demonstrate the impact of using off-axis conics as the base surface.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

One of the major sources of difficulty and cost when measuring the form of freeform optical surfaces is the task of achieving interferometric testability. Interferometric null tests, in which perfect test surfaces return perfectly spherical wavefronts, are ubiquitous in optics, providing non-contact, high precision and, ideally, economical measurements of surface form. Quasi-null tests, in which a perfect test surface returns a specific but non-spherical wavefront, offer similar test precision without the need for a complete null (assuming that retrace errors are negligible [1,2]). However, for all but mild freeform surfaces, the departures from the best-fit sphere are too extreme to achieve a suitable interferometric null (or quasi-null) without the use of additional nulling components, such as computer-generated holograms (CGHs) [3,4], spatial light modulators [5], or specific optics chosen to eliminate aberrations [6]. These additional components increase the cost and complexity of the interferometric test and are sometimes specific to the freeform surface being tested [7–9]. Since both the difficulty and the inflexibility of metrology negatively impact the broad adoption of freeform optical surfaces, it is critical to consider interferometric testability when specifying and designing freeform surfaces.

When specifying a freeform surface with a quasi-null test in mind, it is necessary to consider which surface is being nulled. One straightforward option is to use a base sphere, in which case the freeform surface sag $z$ is

$$z = f(\rho ,\theta ) = \frac{{c{\rho ^2}}}{{1 + \sqrt {1 - {c^2}{\rho ^2}} }} + \sum\limits_{n = 0}^N {{s_n}{P_n}(u,\theta ).}$$
Here, the sag departure is described by arbitrary polynomials $\{ {P_n}\} $ expressed in polar coordinates, ${s_n}$ is the weight for ${P_n},$ $u = \rho /{\rho _{\textrm{max}}}$ is the normalized radial coordinate, and $c$ is the curvature of the base surface.

If a non-zero conic constant is introduced into Eq. (1.1), an on-axis conic can also be used as a base surface [7,10]. It has been well established in the literature that freeform optics with base spheres or on-axis conics can be used successfully in design [11–13]. Descriptions like Eq. (1.1) above make it straightforward to separate the surface being nulled from the sag departures, provided that the base surface is chosen to best fit the overall freeform surface. For example, the interferometric testability estimates used in this paper – namely peak-to-valley (PV) sag departure, which corresponds to the number of fringes in an interferometric test, and maximum gradient normal departure, which corresponds to the fringe density – are both based on departures from the base surface that is intended to be nulled in the test [7,14].

In practice, achieving suitable optical performance often requires surface departures from a base sphere whose magnitudes exceed the limits of interferometric testability without additional null optics. Consequently, metrologists adopt other testing regimes or use additional optical components, such as those mentioned above [3–9].

Alternatively, more complex but still null-testable base surfaces can be used to reduce freeform surface departures and thereby improve testability. Off-axis segments of conical optical surfaces facilitate increased complexity in the base surface while remaining interferometrically testable [15–18]. Freeform surfaces using off-axis conics as base surfaces are also being used in design, especially for bilaterally symmetric systems and confocal designs that form sharp images after each surface [19–25]. However, design methods for bilaterally symmetric systems typically focus on aberration correction or starting point generation, rather than on leveraging the off-axis conic variables to improve the interferometric quasi-null testability of the surfaces in the final design. Further, literature using off-axis conics in freeform designs often use XY polynomials to specify sag departure [23–26], and thus does not explore the impact of design techniques that leverage orthogonal polynomials [27–33].

This paper introduces freeform design methods that improve interferometric testability by leveraging base off-axis conics and orthogonal polynomial sag departure descriptions. The parameterization of the off-axis conic is initially discussed. Two design studies, a three-mirror telescope [12] and a wide field-of-view four-mirror telescope [34], are then conducted. Each design study is analyzed in terms of both optical performance and interferometric testability estimates.

The interferometric testability of the surfaces is estimated in terms of PV sag departure and maximum gradient normal departure, with both best-fit spheres and best-fit off-axis conics considered as base surfaces. While both metrics are only estimates and, in general, interferometric testability depends on a multitude of additional factors, it is nevertheless considered that improvements in these metrics are useful as estimates of improvement in surface interferometric testability when the differences between two designs lie primarily in the shape of the freeform surfaces.

2. Parameterizing the off-axis conic

The off-axis conic is parameterized via the radius of curvature R and the conic constant k of the parent, along with the off-axis angle $\omega$ defined to be the angle formed by the surface normal of the off-axis segment at the origin of its coordinate system with the parent axis of the conic, as shown in Fig. 1. This parameterization was discussed by Cardona-Nunez et al., and has been recently implemented in CODE V by Synopsys [35–37]. In this paper, only those off-axis conics that have two stigmatically-imaged foci are used, which corresponds to the requirement that the conic constant k is non-positive.

 figure: Fig. 1.

Fig. 1. The offset angle $\omega $ is the angle formed by the surface normal at the center of the off-axis segment with the axis of the parent.

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With this parameterization, the sag ${\tilde z_{\textrm{OAC}}}$ of the off-axis conic is calculated as [35]:

$${\tilde z_{\textrm{OAC}}}(\tilde x,\tilde y) = \frac{\gamma }{{\beta + \sqrt {{\beta ^2} - \alpha \gamma } }},$$
where the terms $\alpha = \frac{1}{R}({1 + k{{\cos }^2}\omega } ),\,\ \beta = \frac{1}{{\sqrt {1 + k{{\sin }^2}\omega } }} - \frac{{k\sin \omega \,\cos \omega }}{R}\tilde y$ and $\gamma = \frac{1}{R}({{{\tilde x}^2} + (1 + k{{\sin }^2}\omega ){{\tilde y}^2}} )$ are calculated using the approach of Cardona-Nunez et al., but with a YZ-plane symmetric conic rather than the XZ-plane symmetric conic originally considered.

Using Eq. (1.2) to specify the off-axis conic, the sag $\tilde z$ of a Zernike freeform surface with a base off-axis conic is specified as

$$\begin{array}{l} \tilde z = f(\tilde \rho ,\tilde \theta ) = {\tilde z_{\textrm{OAC}}}(\tilde x,\tilde y) + D(\tilde u,\tilde \theta ),\\ D(\tilde u,\tilde \theta ): = \sum\limits_{n = 0}^N {\sum\limits_{m ={-} n}^n {C_n^mZ_n^m(\tilde u,\tilde \theta )} } , \end{array}$$
where $\tilde \rho$ and $\tilde \theta$ are the polar coordinates of the off-axis segment, $\tilde u = \tilde \rho /{\rho _{\textrm{max}}}$ is the off-axis normalized radial coordinate and $C_n^m$ is the coefficient of the Zernike polynomial $Z_n^m.$

Similarly, the sag of a 2D-Q freeform surface with a base off-axis conic is

$$\begin{array}{l} z = f(\tilde \rho ,\tilde \theta ) = {\tilde z_{\textrm{OAC}}}(\tilde x,\tilde y) + \frac{{\delta (\tilde u,\tilde \theta )}}{{\sigma ({\tilde x,\tilde y} )}},\\ \delta (\tilde u,\tilde \theta ){\kern 1pt} : = {\kern 1pt} {\kern 1pt} {\tilde u^2}(1 - {\kern 1pt} {\tilde u^2})\sum\limits_{n = 0}^N {a_n^0Q_n^0({{\tilde u}^2} ){\kern 1pt} + \sum\limits_{m = 1}^M {{{\tilde u}^m}} } \sum\limits_{n = 0}^N {[a_n^m\cos m\tilde \theta + b_n^m\sin m\tilde \theta ]Q_n^m({{\tilde u}^2})} , \end{array}$$
where $a_n^m$ and $b_n^m$ are the 2D-Q coefficients and $\sigma ({\tilde x,\tilde y} )$ is the cosine factor which scales the (approximate) normal departure to sag departure. For an off-axis conic, $\sigma ({\tilde x,\tilde y} )$ is [37]
$$\sigma ({\tilde x,\tilde y} )= {\left[ {1 + \frac{{{c^2}{k_1}({k_3}^2 + {{\tilde x}^2})}}{{{k_4}}}} \right]^{ - \frac{1}{2}}}.$$
The factors ${k_1} = 1 + k{\sin ^2}\omega$, ${k_2} = ck\sqrt {{k_1}} \sin \omega \,{\kern 1pt} \cos \omega$, ${k_3} = {k_1}\tilde y + k\sin \omega \,{\kern 1pt} \cos \omega \;{\tilde z_{\textrm{OAC}}}(\tilde x,\tilde y)$ and ${k_4} = 1 - 2{k_2}\tilde y - {c^2}{k_1}[{(1 + k{{\cos }^2}\omega ){{\tilde x}^2} + (1 + k){{\tilde y}^2}}] $ are used for consistency with [37].

The Zernike polynomials (“Zernikes”), specifically in Fringe ordering, and the Forbes 2D-Q polynomials (“2D-Qs”) are used as representative examples of orthogonal polynomials over circular apertures in design [38–40]. All freeform surfaces discussed in this paper have approximately circular effective apertures, which is consistent with the circular domain of orthogonality of the Zernikes and the 2D-Qs.

3. Analysis of base off-axis conics plus orthogonal polynomials in design

3.1 Overview of the design studies

Two design case studies were selected to compare off-axis conics as base surfaces with base spheres. For each design case study, the starting point satisfied the same geometry constraints and had the same layout as the final design but used only spherical base surfaces and included no freeform terms. Design variables and parameters were then introduced based on their ability to correct the aberrations as observed via full-field display, including use of off-axis conic parameters, system layout parameters, and up to the 36th Fringe Zernike term or corresponding 2D-Q term [31,32]. In conjunction with the freeform variables, the square-sum penalty, as described in Takaki et al., was used to minimize freeform departures while achieving the specified optical performance [33].

Prior to implementing designs using base off-axis conics, benchmark designs were created using a base sphere plus orthogonal polynomial sag departures. These benchmark designs are analyzed in terms of optical performance and testability estimates, namely PV sag departure and maximum gradient normal departure. To facilitate comparison with off-axis conic designs, two sets of testability estimates are shown: the first reports departure from the base sphere, while the second reports the departures from the off-axis conic that best fits (in terms of RMS sag departure) the surfaces in the benchmark designs.

Designs with base off-axis conics plus orthogonal polynomial sag departures were then created, allowing use of polynomials of up to the same order as those in the benchmark designs and again leveraging the square-sum penalty. In both design studies, the initial limiting aberrations were field-constant astigmatism and field-constant coma. Consequently, the off-axis conic parameters were leveraged to correct these aberrations. Orthogonal polynomials corresponding to surface coma and surface astigmatism were not used, since doing so would potentially yield negative interactions between orthogonal polynomial variables and off-axis conic parameters, such as degeneracy [41]. The effective curvature, defined by Schiesser et al., as the (signed) square-root of the Gaussian curvature, was used during this design process to control the power of the base off-axis conic when required by the design method [42]. These designs were then analyzed in terms of optical performance and testability estimates. In both design studies, designs that use base off-axis conics from the start have improved testability estimates relative to the benchmark designs while achieving equivalent optical performance.

Both design studies were explored in terms of Zernikes vs. 2D-Qs to describe sag departure. While direct comparison of Zernikes vs. 2D-Qs is not the primary focus of this paper, the results are not specifically related to just the Zernikes or just the 2D-Qs, and instead are shared by both orthogonal bases. While there is an abundance of orthogonal polynomial surface descriptions and investigating just two sets does not comprehensively cover all of them, it is important to note that the results in this paper generalize beyond just one orthogonal polynomial set. However, because the optical performance and testability estimates are similar between Zernikes and 2D-Qs, only the 2D-Q design is shown for the first design study and only the Zernike design is shown for the second design study. For each design study, the results using the polynomial set not shown in the paper are included in Appendix A.

Note that, while the systems in both studies are designed to have diffraction-limited performance in the infrared, it is expected that visible-light systems will follow the same trends as the systems in this paper when it comes to base spheres vs. base off-axis conics. When analyzing design performance, changes in wavelength matter because the diffraction limit, expressed in waves, is unique to the band and is linear with wavelength. These changes do not affect the proposed design methods. However, full-field displays are used to determine the field dependence of aberrations for visible-light systems, and surface description variables and parameters are introduced to correct these aberrations. Similarly, the method of analysis is independent of wavelength: for a fixed system, the departures from base sphere or base conic in microns stay the same if all that is changed is the wavelength at which the design is conducted.

3.2 Three-mirror telescope design study

The first design study is a return to the three-mirror, LWIR imaging telescope designed by Fuerschbach et al., [12] and recently revisited by Takaki et al. [33]. The design by Fuerschbach et al., maintained a ball geometry, prioritized low volume, and achieved less than λ/100 (λ=10 µm) of root-mean-square wavefront error (RMS WFE). In Takaki et al., it was shown that freeform departures could be reduced by as much as 40% while maintaining the same optical performance and geometry by leveraging the square-sum penalty with base spherical surfaces. These reduced-departure designs are used here as the benchmark. The specifications of this design are shown in Table 1 below. A YZ cross-section of the benchmark design is shown in Fig. 2 below.

 figure: Fig. 2.

Fig. 2. A YZ cross-section of the off-axis conic design of the three-mirror telescope. The benchmark design by Takaki et al., [27] which has base spheres, maintains a similar layout.

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Table 1. Three-mirror telescope specifications

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From a design perspective, the three-mirror telescope is useful to revisit due to its familiarity to designers. From a testability perspective, the three-mirror telescope design is also worth revisiting. Fuerschbach et al., document a procedure for testing the secondary mirror of the three-mirror telescope, using an Offner null for nulling spherical aberration, surface tilt for nulling astigmatism, and a deformable mirror for nulling coma and higher-order terms, as well as a software null [6]. The procedure of Fuerschbach et al., demonstrates both the capability and the increased complexity of quasi-null interferometric tests for freeform surfaces when additional null optics are included.

Table 2 shows the optical performance and testability estimates for the benchmark design with best-fit sphere, for the benchmark design fitted with best-fit off-axis conic, and for the system designed using base off-axis conics from the beginning of the design process. Fitting the base sphere design with the best-fit off-axis conic yields some reduction in the PV sag departure of Mirror 3 but does not reliably improve testability estimates for other surfaces. Further, as can be seen by examining the sag departure profiles in Fig. 3, surface coma dominates the departure profiles of Mirror 1 and Mirror 3 even after removal of the best-fit off-axis conic. This finding indicates that the best-fit off-axis conic may have some coma contributions but does not provide all the surface coma in the surface.

 figure: Fig. 3.

Fig. 3. Three-mirror telescope with 2D-Q surfaces for: (left) the benchmark designs with departures from best-fit sphere, (center) the benchmark designs with departures from best-fit off-axis conic, and (right) the off-axis conic designs with departures from base off-axis conic. The scales of the color bars are in units of microns.

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Table 2. Three-mirror telescope with 2D-Qs: optical performance and testability estimates of (left) the benchmark design with departures from best-fit sphere, (center) the benchmark design with departures from best-fit off-axis conic, and (right) the design with off-axis conics.

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On the other hand, the system designed from the beginning with base off-axis conics has significantly improved quasi-null testability estimates. In terms of PV sag departure, Mirror 1 improves from 34 µm to 4 µm, Mirror 2 improves from 9 µm to 4 µm, and Mirror 3 improves from 75 µm to 6 µm, an overall improvement of about an order-of-magnitude. The maximum gradient normal departures show similar improvement. Because of these improvements, the testability estimates of the off-axis conic designs may lie within the range of a standard Fizeau interferometer with no need for additional null optics.

Further, as shown in Fig. 3, surface coma no longer dominates the sag departure profiles of the off-axis conic designs. By re-designing with base off-axis conics from the beginning, surfaces are found that account for coma and astigmatism via the off-axis conic parameters. The resulting design has the same optical performance as the benchmark designs but features significantly improved testability estimates.

The results presented here use the 2D-Q polynomials to describe sag departure. The designs using Zernike polynomials to describe sag departure are shown in Appendix A and show similar trends as the 2D-Q designs.

3.3 Second design study: wide field-of-view four-mirror telescope

In the second design study, a more aggressive specification is considered, in which the freeform departures required to achieve suitable optical performance – whether with base sphere or base off-axis conic – exceed what can be tested without the use of additional null optics. Even if additional null optics are necessary, improving testability estimates is advantageous because doing so reduces the complexity of these additional null optics. For example, if a CGH is used, larger wavefront errors to be nulled lead to denser fringe spacing on the CGH and, in turn, greater sensitivity to manufacturing and alignment errors of the CGH [43,44]. Further, by studying a design with a much wider field-of-view and larger volume, it is shown that the results from Section 3.2 are also applicable to design studies with more aggressive specifications.

Thus, the second design study is based on the wide field-of-view, large-volume four-mirror telescope for use in MWIR, based on a design by Chrisp [34]. Because of the wide field-of-view, large entrance pupil diameter, and relatively fast F/#, Chrisp’s design has significantly larger testability estimates than those of the three-mirror telescope design. Chrisp’s design features a 40° by 40° field-of-view, with a 100 mm entrance pupil diameter at F/2. Because of the speed, entrance pupil diameter and field-of-view, Chrisp’s design has dimensions 140 × 130 × 85 cm3 (i.e. ∼1500 L) and PV sag departures from best-fit sphere of up to 12 mm. The design study in this paper features equivalent field-of-view, F/#, focal length and optical performance while utilizing a modified geometry that supports a reduced volume of 700 L. A YZ cross-section of the off-axis conic design, including the bounding box which has dimensions 51 × 117 × 116 cm3, is shown in Fig. 4. Chrisp’s design does not report distortion; in this design study, distortion was constrained to be less than 10%. The full specifications of the current design are shown in Table 3.

 figure: Fig. 4.

Fig. 4. A YZ cross-section of the off-axis conic design of the four-mirror telescope. The benchmark design, which has base spheres, has a similar layout.

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Table 3. Wide-field-of-view, four-mirror telescope specifications

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As with the three-mirror telescope design, a benchmark study was created using a base sphere, and testability estimates are reported first using base spheres and second using best-fit off-axis conics. The results are shown in Table 4 below. As with the three-mirror telescope design, fitting the benchmark designs with the best-fit off-axis conic yields some improvement in testability estimates for some surfaces, but improvements are not present across all surfaces. Unlike the three-mirror design study, the residual sag departures after removing the best-fit off-axis conic, reported in Fig. 5, are not all dominated by terms that are expected to be provided by an off-axis conic, like surface coma and astigmatism. Instead, sag departure profiles of the base-sphere design include more complicated, higher-order terms. Consequently, fitting surfaces with the best-fit off-axis conic does not reliably improve testability estimates. Note that the results in the central column of Table 4 use the best-fit off-axis conic in the sense of RMS sag departure, but other definitions of best-fit conic (such as PV sag departure) were investigated and show similar trends.

 figure: Fig. 5.

Fig. 5. Four-mirror telescope with Zernike surfaces for: (left) the benchmark designs with departures from best-fit sphere, (center) the benchmark designs with departures from best-fit off-axis conic, and (right) the off-axis conic designs with departures from base off-axis conic. The scales of the color bars are in units of microns.

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Table 4. For the wide-field-of-view, four-mirror telescope with Zernikes, optical performance and testability estimates of (left) the benchmark design with departures from best-fit sphere, (center) the benchmark design with departures from best-fit off-axis conic, and (right) the design with off-axis conics.

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Despite the complexity of the residual sag departures of the base sphere designs, designing with the base off-axis conic from the beginning nevertheless yields significant improvements in testability estimates while preserving optical performance. As Table 4 below shows, PV sag departure and max gradient normal departure improve by as much as an order-of-magnitude, from 3.66 mm and 1.93° to 0.34 mm and 0.12° for Mirror 1. These improvements show that use of off-axis conics as base surfaces enables improvement in testability estimates even with a wider field-of-view and larger volume.

4. Conclusion

In this paper, the design of systems using freeform surfaces that are specified with base off-axis conics and orthogonal polynomials sag departures is discussed. Fitting an off-axis conic to freeform surfaces that have been designed with base spheres can sometimes improve testability estimates, but these improvements are not reliable and do not completely remove terms like surface coma from the residual sag departure profiles. In two design studies – a three-mirror ball geometry telescope and a wide field-of-view four-mirror telescope – designing with base off-axis conics from the start of the design process yields new surface descriptions that make better use of the off-axis conic parameters to reduce freeform surface departures and interferometric testability estimates. In both design studies, base off-axis conic designs maintain the same optical performance but have significantly improved testability estimates relative both to the base sphere designs using best-fit sphere and to the base-sphere designs fit with the best-fit off-axis conic.

Appendix A

The optical performance and testability estimates of the three-mirror telescope design study using Zernike polynomials are reported in Table 5. These results show similar trends as those shown in Table 2 for the three-mirror telescope design study with 2D-Qs. Removing the best-fit off-axis conic from the benchmark design, which was designed using a base sphere plus Zernikes, shows improvement in the testability estimates of Mirror 3, but neither Mirror 1 nor Mirror 2 show significant improvement. Further, as shown in Fig. 6, the residual sag departure profile for Mirror 3 still retains coma. On the other hand, designing with base off-axis conics from the start improves testability estimates while preserving optical performance relative to the benchmark design, a trend which was observed for the 2D-Qs in Section 3.2 and is shown for the Zernikes in Table 5.

 figure: Fig. 6.

Fig. 6. Three-mirror telescope with Zernike surfaces for: (left) the benchmark designs with departures from best-fit sphere, (center) the benchmark designs with departures from best-fit off-axis conic, and (right) the off-axis conic designs with departures from base off-axis conic. The scales of the color bars are in units of microns.

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Table 5. For the three-mirror telescope with Zernikes, optical performance and testability estimates of (left) the benchmark design with departures from best-fit sphere, (center) the benchmark design with departures from best-fit off-axis conic, and (right) the design with off-axis conics.

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The optical performance and testability estimates of the wide field-of-view four-mirror telescope design study using 2D-Qs polynomials are reported in Table 6. As with the three-mirror telescope design study, the trends of the designs with 2D-Qs are like those of the designs with Zernikes. Removing the best-fit off-axis conic improves the testability estimates of some surfaces in the benchmark design, whereas designing with base off-axis conics plus 2D-Qs yields significant improvement on all surfaces. The sag departure profiles shown in Fig. 7 indicate residual coma even after the removal of the best-fit off axis conic, which is not present when base off-axis conics are used from the beginning of the design process.

 figure: Fig. 7.

Fig. 7. Four-mirror telescope with 2D-Q surfaces for: (left) the benchmark designs with departures from best-fit sphere, (center) the benchmark designs with departures from best-fit off-axis conic, and (right) the off-axis conic designs with departures from base off-axis conic. The scales of the color bars are in units of microns.

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Tables Icon

Table 6. For the wide-field-of-view, four-mirror telescope with 2D-Qs, optical performance and testability estimates of (left) the benchmark design with departures from best-fit sphere, (center) the benchmark design with departures from best-fit off-axis conic, and (right) the design with off-axis conics.

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Funding

National Science Foundation (IIP-1338877, IIP-1338898, IIP-1822026, IIP-1822049).

Acknowledgements

We thank Eric Schiesser for insight about designing with Zernike freeform surfaces and with off-axis conics. We thank Bryan Stone for discussions of off-axis conic parameterizations. We thank Greg Forbes for insight learned from discussions of freeform surface descriptions with both Zernikes and 2D-Qs, and about degeneracy. We thank Synopsys for the student license of CODE V.

Disclosures

The authors declare no conflicts of interest.

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Figures (7)
Fig. 1.
Fig. 1. The offset angle $\omega $ is the angle formed by the surface normal at the center of the off-axis segment with the axis of the parent.

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Fig. 2.
Fig. 2. A YZ cross-section of the off-axis conic design of the three-mirror telescope. The benchmark design by Takaki et al., [27] which has base spheres, maintains a similar layout.

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Fig. 3.
Fig. 3. Three-mirror telescope with 2D-Q surfaces for: (left) the benchmark designs with departures from best-fit sphere, (center) the benchmark designs with departures from best-fit off-axis conic, and (right) the off-axis conic designs with departures from base off-axis conic. The scales of the color bars are in units of microns.

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Fig. 4.
Fig. 4. A YZ cross-section of the off-axis conic design of the four-mirror telescope. The benchmark design, which has base spheres, has a similar layout.

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Fig. 5.
Fig. 5. Four-mirror telescope with Zernike surfaces for: (left) the benchmark designs with departures from best-fit sphere, (center) the benchmark designs with departures from best-fit off-axis conic, and (right) the off-axis conic designs with departures from base off-axis conic. The scales of the color bars are in units of microns.

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Fig. 6.
Fig. 6. Three-mirror telescope with Zernike surfaces for: (left) the benchmark designs with departures from best-fit sphere, (center) the benchmark designs with departures from best-fit off-axis conic, and (right) the off-axis conic designs with departures from base off-axis conic. The scales of the color bars are in units of microns.

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Fig. 7.
Fig. 7. Four-mirror telescope with 2D-Q surfaces for: (left) the benchmark designs with departures from best-fit sphere, (center) the benchmark designs with departures from best-fit off-axis conic, and (right) the off-axis conic designs with departures from base off-axis conic. The scales of the color bars are in units of microns.

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Tables (6)

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Table 1. Three-mirror telescope specifications

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Table 2. Three-mirror telescope with 2D-Qs: optical performance and testability estimates of (left) the benchmark design with departures from best-fit sphere, (center) the benchmark design with departures from best-fit off-axis conic, and (right) the design with off-axis conics.

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Table 3. Wide-field-of-view, four-mirror telescope specifications

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Table 4. For the wide-field-of-view, four-mirror telescope with Zernikes, optical performance and testability estimates of (left) the benchmark design with departures from best-fit sphere, (center) the benchmark design with departures from best-fit off-axis conic, and (right) the design with off-axis conics.

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Table 5. For the three-mirror telescope with Zernikes, optical performance and testability estimates of (left) the benchmark design with departures from best-fit sphere, (center) the benchmark design with departures from best-fit off-axis conic, and (right) the design with off-axis conics.

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Table 6. For the wide-field-of-view, four-mirror telescope with 2D-Qs, optical performance and testability estimates of (left) the benchmark design with departures from best-fit sphere, (center) the benchmark design with departures from best-fit off-axis conic, and (right) the design with off-axis conics.

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Equations (5)

Equations on this page are rendered with MathJax. Learn more.

z = f ( ρ , θ ) = c ρ 2 1 + 1 − c 2 ρ 2 + ∑ n = 0 N s n P n ( u , θ ) .
z ~ OAC ( x ~ , y ~ ) = γ β + β 2 − α γ ,
z ~ = f ( ρ ~ , θ ~ ) = z ~ OAC ( x ~ , y ~ ) + D ( u ~ , θ ~ ) , D ( u ~ , θ ~ ) := ∑ n = 0 N ∑ m = − n n C n m Z n m ( u ~ , θ ~ ) ,
z = f ( ρ ~ , θ ~ ) = z ~ OAC ( x ~ , y ~ ) + δ ( u ~ , θ ~ ) σ ( x ~ , y ~ ) , δ ( u ~ , θ ~ ) := u ~ 2 ( 1 − u ~ 2 ) ∑ n = 0 N a n 0 Q n 0 ( u ~ 2 ) + ∑ m = 1 M u ~ m ∑ n = 0 N [ a n m cos ⁡ m θ ~ + b n m sin ⁡ m θ ~ ] Q n m ( u ~ 2 ) ,
σ ( x ~ , y ~ ) = [ 1 + c 2 k 1 ( k 3 2 + x ~ 2 ) k 4 ] − 1 2 .
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