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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 22 — Oct. 24, 2011
  • pp: 21919–21928
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A new family of optical systems employing φ-polynomial surfaces

Kyle Fuerschbach, Jannick P. Rolland, and Kevin P. Thompson  »View Author Affiliations


Optics Express, Vol. 19, Issue 22, pp. 21919-21928 (2011)
http://dx.doi.org/10.1364/OE.19.021919


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Abstract

Unobscured optical systems have been in production since the 1960s. In each case, the unobscured system is an intrinsically rotationally symmetric optical system with an offset aperture stop, a biased input field, or both. This paper presents a new family of truly nonsymmetric optical systems that exploit a new fabrication degree of freedom enabled by the introduction of slow-servos to diamond machining; surfaces whose departure from a sphere varies both radially and azimuthally in the aperture. The benefit of this surface representation is demonstrated by designing a compact, long wave infrared (LWIR) reflective imager using nodal aberration theory. The resulting optical system operates at F/1.9 with a thirty millimeter pupil and a ten degree diagonal full field of view representing an order of magnitude increase in both speed and field area coverage when compared to the same design form with only conic mirror surfaces.

© 2011 OSA

1. Introduction

Historically, optical designers have a reputation for designing optical systems that exceed the industry capabilities for fabrication and/or assembly. However, in general, these systems are intrinsically rotationally symmetric using spheres, aspheres, or off-axis segments of a rotationally symmetric surface (other than the occasional use of cylindrical or toric surfaces for special case anamorphic systems). Recently, the optical fabrication industry has changed this paradigm by implementing a capability to fabricate diamond turned, optical quality surfaces in the Long Wave InfraRed (LWIR, 8-12 µm) that are not rotationally symmetric. In particular, it is now possible to fabricate an optical surface that is defined as a conic plus the lower order terms of a Zernike polynomial (< FRINGE term 16). This emerging fabrication capability has also been extended to the fabrication of surfaces designed with radial basis functions, a meshless surface description first applied to optical system design by Cakmakci et al. [1

1. O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16(3), 1583–1589 (2008). [CrossRef] [PubMed]

].

In this paper, an optical system design that is made up of tilted mirrors with a Zernike polynomial surface representation is optimized to dramatically increase the field of view (FOV) and speed of an LWIR optical system, facilitating the use of an uncooled microbolometer detector. The optimized system represents an order of magnitude increase in field area coverage when compared to current state-of-the-art. The optimization strategies that are employed during the optical design of this nonsymmetric system use nodal aberration theory developed by Shack and Thompson [2

2. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).

] as further detailed in Section 3. When the symmetry constraint is removed, the traditional aberrations (spherical, coma, and astigmatism) develop a multi-nodal field dependence where there may now be multiple points in the FOV where a specific aberration type may go to zero. The seminal example is binodal astigmatism, first recognized by Shack [2

2. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).

]. However, due to the fact that any tilted and decentered optical system with rotationally symmetric parent surfaces could not be corrected for axial coma, this theory has previously only been useful during the optical design of offset aperture and/or field biased optical systems. With this revolution in fabrication, it is now possible to apply a nodal based optical analysis approach to optimization in order to explore a new design space and create a new optical system form.

Thompson, in 2005, described the new aberration field dependencies that arise in nodal aberration theory using a new display, the full field aberration display [3

3. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). [CrossRef] [PubMed]

]. That analysis tool is applied here to develop a truly nonsymmetric optical system design. The theory developed by Thompson [3

3. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). [CrossRef] [PubMed]

] is limited to tilted and decentered optical imaging systems made up of rotationally symmetric components, or offset aperture portions thereof. Recently, Schmid et al. [4

4. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010). [CrossRef] [PubMed]

] combined a nonsymmetric surface placed at an aperture stop with nodal aberration theory. With this new result and new fabrication methods, the optical designer is now able to target the third order aberrations (spherical, coma, astigmatism) and their nodal behavior during optical design using tilted φ-polynomial surfaces to create high performance imaging systems with no particular symmetry constraints. While this paper explores only the design space of tilted φ-polynomial surfaces, tilted and decentered surfaces are equally valid, and remain to be explored in future work.

2. Optical surface representation with φ-polynomials

A φ-polynomial surface takes the form
z=F(ρ,φ),
(1)
where the sag, z, is represented by a function that depends on the radial component, ρ, and the azimuthal component, φ, within the aperture of the part. Until recently, methods of fabrication have constrained the shape of optical surfaces to depend on the radial component only, that is, z = F(ρ). This limitation has been a severe constraint in the optical design of unobscured optical systems. It is well known that when any powered optical surface in an optical system is tilted or decentered with respect to the optical axis, third order coma will appear on-axis. While there are some special configurations where axial coma is eliminated (1:1 systems and systems that use the coma free pivot design principle), in general, the ubiquitous presence of axial coma in tilted and decentered systems has prevented access to a substantial family of optical design forms.

The φ-polynomial surface type that is used for the optical design presented in this paper is a Zernike polynomial surface described by the following equation
z=cρ21+1(1+k)c2p2+j=116CjZj,
(2)
where z is the sag of the surface, c is the vertex curvature, k is the conic constant, ρ is the radial component in the aperture, Zj is the jth FRINGE Zernike polynomial, and Cj is the magnitude coefficient of Zj. Each Zj is a polynomial in polar coordinates (R and φ) where R is a quantity normalized to a radius Rnorm, that is, R = ρ/Rnorm. A table summarizing the FRINGE Zernike polynomials can be found in [5

5. Synopsys Inc, “Zernike polynomials,” in CODE V Reference Manual, (2011), Volume IV, Appendix C.

]. Figure 1
Fig. 1 The sag components of a powered FRINGE Zernike polynomial surface (right, top) when the base conic surface (left, top) is superimposed with contributions of, top to bottom: spherical aberration (Z9), coma (Z7, Z8), and astigmatism (Z5, Z6). When the sag is evaluated with respect to the base conic (right, bottom), the Zernike overlay on the surface can be seen directly.
illustrates the sag of a powered Zernike polynomial surface with spherical aberration (Z9), coma (Z7, Z8), and astigmatism (Z5, Z6), (in order of increasing field dependence). As can be seen from the resulting sag, the surface is asymmetric due to the comatic contribution and anamorphic due to the astigmatic contribution. When the sag is evaluated with respect to the base conic, as illustrated in the lower series of Fig. 1, the dominance of the comatic Zernike term appears, as is typical for this new design family.

The influence of a φ-polynomial surface in an optical system is highly dependent on its position relative to the stop surface. In any optical system, there are apertures that limit the light that can pass through the optical system. The aperture that determines the cone of light that can be accepted by the optical system, thereby defining the limiting f/number, is the aperture stop. At this surface all field points will fill the entire aperture in a system without vignetting, which we assume here. This property combined with the constraint of a rotationally symmetric surface meant that in prior designs only spherical aberration is present on-axis and its contribution throughout the field is constant. In the new fabrication paradigm, when φ-polynomial surfaces are placed at or near the stop surface, their contribution to the aberration function will also be field constant. The optical designer now has the ability to introduce field dependent aberrations on-axis and to remove on-axis coma introduced by a tilted surface.

For surfaces located away from the stop, the active area of the surface will be different for each field point, shrinking relative to the clear aperture and moving off center. Figure 2
Fig. 2 Demonstration of a beam footprint (grey dashed circle) from an off-axis field point striking a plane surface with 1λ Zernike spherical (Z9). It can be seen that the Zernike composition of the beam footprint sub-region contains components of Zernike coma and astigmatism by sequentially subtracting Zernike spherical and coma.
illustrates how a mix of spherical, coma, and astigmatism contribute to the wavefront of an off-axis field point as that field’s beam footprint (dashed circle) is shifted and scaled on a surface with traditional rotationally symmetric departure, in this case a plane surface with 1λ of Zernike spherical (FRINGE term Z9) [6

6. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24(3), 569–577 (2007). [CrossRef] [PubMed]

]. The ratio of each contribution is proportional to the initial full aperture amount and then scales with the offset. Similarly, for a φ-polynomial surface away from the stop, a mix of aberrations of lower radial order than the surface itself will contribute to the wavefront for each off-axis field point. While the exact mix of aberration content has not yet been isolated theoretically, Zernike decomposition of a real ray based wavefront fit over a modest grid of points in the field of view demonstrates an overall field dependence of the aberration types when using a φ-polynomial surface away from the stop.

3. Optical design with tilted φ-polynomial surfaces: a new optical system family

3.1 The new optical design

In the 1960s, the first optical designs that involved three or more mirrors in an unobscured configuration started to be unclassified and began to appear in limited distribution government reports [7

7. Air Force Avionics Laboratory, “Three mirror objective,” in Aerial Camera Lenses, Report 027000 from RECON Central, the Reconnaissance Division/Reconnaissance Applications Branch, (1967), pp. 2–109.

]. Motivated by the advance in LWIR detectors and the accompanying need for straylight control, a number of systems were designed, particularly, as concept designs for missile defense. While many of these systems appear to lack rotational symmetry, detailed analysis reveals that any successful design with a significant field of view was in fact based on a rotationally symmetric design with an offset aperture, a biased field, or both. Analysis shows that this fact could be anticipated, as many systems that depart from rotational symmetry immediately display on-axis coma, where the axis for a nonsymmetric system is defined by the optical axis ray (OAR) [8

8. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009). [CrossRef] [PubMed]

]. While there are special configurations that eliminate axial coma, there are very few practical forms that do not reduce to a rotational symmetric form.

In 1994, an optical system designed by Rodgers was patented that had the property of providing the largest planar, circular input aperture in the smallest overall spherical volume [9

9. J. M. Rodgers, “Catoptric optical system including concave and convex reflectors,” Optical Research Associates, US Patent 5,309,276 (1994).

]. A similar starting point can be found in 2005 by Nakano [10

10. T. Nakano and Y. Tamagawa, “Configuration of an off-axis three-mirror system focused on compactness and brightness,” Appl. Opt. 44(5), 776–783 (2005). [CrossRef] [PubMed]

]. The particular form embodied in the patent of [9

9. J. M. Rodgers, “Catoptric optical system including concave and convex reflectors,” Optical Research Associates, US Patent 5,309,276 (1994).

] is shown in Fig. 3(a)
Fig. 3 (a) Layout of U.S. Patent 5,309,276 consisting of three off-axis sections of rotationally symmetric mirrors and a fourth fold mirror (mirror 3). The optical system had, at the time of its design, the unique property of providing the largest planar, circular input aperture in the smallest overall spherical volume for a gimbaled application. (b) The new optical design based on tilted φ-polynomial surfaces to be coupled to an uncooled microbolometer.
. This optical design is a 9:1 afocal relay that operates over a 3° full FOV using four mirrors and it provides a real, accessible exit pupil that is often a requirement in earlier infrared systems requiring cooled detectors. In use, it is coupled with a fast f/number refractive component in a dewar near the detector. It is based on using off-axis sections of rotationally symmetric conic mirrors that are folded into the spherical volume by using one fold mirror (mirror 3).

As is often the case, many applications would exploit a larger FOV if it were available with usable performance. In addition, if an optical form could be developed at a fast enough f/number, it becomes feasible to transition to an uncooled detector thereby abandoning the need for the reimaging configuration, the external exit pupil, and the refractive component in the dewar. Using the new paradigm of tilted φ-polynomial optical surfaces, a three mirror, F/1.9 form with a 10° diagonal full FOV has been developed using the methods of nodal aberration theory for the optimization. The nominal optical design is shown in Fig. 3(b) and has an overall RMS wavefront error (RMS WFE) of less than < λ/100 at 10 µm over a 10° full FOV where the overall RMS WFE is computed as the average plus one standard deviation RMS WFE for all field points. The remainder of this paper will detail how this solution was developed using the tools and concepts of nodal aberration theory applied to tilted φ-polynomial surfaces.

3.2 Creating the starting form

3.3 Creating the unobscured form

Tilting the on-axis solution will break the rotational symmetry of the system and will change where the aberration field zeros (nodes) are located for each aberration type. The shift of the aberration fields will drastically degrade the overall performance of the system. A strategy for tracking the evolution of the nodal structure as the unobscured design form is created is to oversize the field of view to many times the intended field of view. As an example of this strategy, Figs. 5(a)
Fig. 5 The lens layout, Zernike coma (Z7, Z8) and astigmatism (Z5, Z6) full field displays for a ±40° FOV for the (a) on-axis optical system, (b) halfway tilted, 50% obscured system, and (c) fully tilted, 100% unobscured system. The region in red shows the field of interest, a 10° diagonal FOV.
-5(c) shows the design form at 0%, 50% and 100% unobscured accompanied by an evaluation of Zernike coma (Z7, Z8) and Zernike astigmatism (Z5, Z6) across a ±40° field (note there is a 12X scale change between Fig. 5(a) and Figs. 5(b)-5(c) so the nodal behavior can be seen for each tilt position). As can be seen from Fig. 5(a), the on-axis solution is well corrected for astigmatism and coma within the 10° diagonal full FOV (sub-region in red) and the nodes (blue star and green dot) are centered on the optical axis (zero field). As the system is tilted halfway to an unobscured solution, Fig. 5(b), the node for coma has moved immediately beyond the field being evaluated resulting in what is a field constant coma. For this intermediate tilt, one of the two astigmatic nodes remains within the extended analysis field moving linearly with tilt. When the system is tilted to an unobscured solution, Fig. 5(c), field constant coma is increased while the astigmatic node also moves out of the 8X oversized analysis field leaving the appearance of a field constant astigmatism. The first significant observation regarding formulating a strategy for correction is that in the unobscured configuration the nodes have moved so far out in the field that the astigmatism and coma contributions within the region of interest, a 10° full FOV, are nearly constant.

3.4 Creating field constant aberration correction

It is possible to correct the field constant aberrations shown in Fig. 6 by using the fact that the stop location for this optical system is the secondary mirror. When Zernike polynomials for coma and astigmatism are added as variables to the secondary conic surface, they will introduce, when optimized, the opposite amount of field constant coma and astigmatism independently – a completely new optical design degree of freedom. The effect of optimizing the optical system with these new variables is shown in Fig. 7
Fig. 7 The lower order spherical (Z9), coma (Z7, Z8), and astigmatism (Z5, Z6) and one higher order, elliptical coma (Z10,11) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree field of view for the optimized system where Zernike astigmatism and coma were used as variables on the secondary (stop) surface. When the system is optimized, the field constant contribution to astigmatism and coma are greatly reduced improving the RMS WFE from ~12λ to ~0.75λ.
where the field constant coma and astigmatism have been removed. The RMS WFE has gone from ~12λ for the tilted system without φ-polynomials to ~0.75λ for the tilted system with Zernike coma and astigmatism on the secondary surface (note that there is a 10X scale change from Fig. 6 to Fig. 7 to show the residual terms in further detail).

3.5 Creating field dependent aberration correction

By studying the residual behavior of the optical system after optimization of Zernike coma and astigmatism on the secondary surface, it can be seen from the displays, Fig. 7, that the dominant aberration contribution is Zernike astigmatism and it is the largest contributor to the RMS WFE of ~0.75λ. Moreover, the astigmatism has taken the form of field linear, field asymmetric astigmatism first described by Thompson et al. in 2008 [12

12. K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008). [CrossRef] [PubMed]

]. Since in this case the astigmatism contribution is one of the new asymmetric forms, which is a characteristic identified by nodal aberration theory, it is necessary to apply a new type of variable to reduce or eliminate its impact. A second design strategy uses φ-polynomials away from the stop location, that is, the primary and tertiary surfaces, to counteract field dependent aberration contributions with degrees of freedom previously not available. By adding Zernike terms to these mirrors, the designer can break the relationship between spherical aberration, coma, and astigmatism as a function of the conic distributions on the mirrors, as illustrated in Fig. 2. For example, by using a Zernike polynomial of radial order higher than Zernike astigmatism as a variable at a surface away from the stop, it will create a linear field dependent contribution to astigmatism that will reduce (and in some cases eliminate) the residual field linear, field asymmetric astigmatism. The effectiveness of this strategy is demonstrated in Fig. 8
Fig. 8 The lower order spherical (Z9), coma (Z7, Z8), and astigmatism (Z5, Z6) and one higher order, elliptical coma (Z10, Z11) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree field of view for the optimized system where Zernike coma is added as an additional variable to the tertiary surface. The RMS WFE has been reduced from ~0.75λ to ~0.25λ.
where the relevant aberration contributions after optimization with Zernike coma on the tertiary mirror as an additional variable are shown. As can be seen from Fig. 8, which is on the same scale as Fig. 7, the astigmatism contribution has been reduced and the RMS WFE has been improved by another factor of 3X going from ~0.75λ to ~0.25λ.

With the successful creation of a nearly compliant unobscured form, the remaining optimization proceeds with additional use of low order Zernike coefficients resulting in the system shown in Fig. 9
Fig. 9 (a) Layout of LWIR imaging system optimized with φ-polynomial surfaces and (b) the RMS WFE of the final, optimized system, which is < λ/100 (0.01λ) at 10 µm over a 10° diagonal full FOV.
. The overall RMS WFE over the 10° full FOV, as displayed in Fig. 9(b) is, less than λ/100 (0.01λ) at 10 µm, well within the diffraction limit (0.07λ). As a point of comparison, if the field and f/number of the unobscured, conic only solution presented in Fig. 5(c) are reduced to produce a diffraction limited system, the field must be reduced to a 3° diagonal full FOV and the system speed must be reduced to F/22. Thus with the φ-polynomial surface, there is a substantial advance in usable field of view in this design space, and a 3X increase when compared to the motivating afocal design form. In addition, the light collection capability is extended from F/22 to F/1.9, improving signal to noise by two orders of magnitude thereby enabling the transition to the use of an uncooled microbolometer.

5. Conclusion

With the introduction of slow-servo diamond turning technology, we have shown in this paper that a new type of optical surface, φ-polynomials, have become available for the optical design of LWIR systems. As the fabrication technology improves, the design strategies shown here will continue to support solutions at increasingly shorter wavelengths. This is a paradigm shift in optical design, allowing for the first time, truly nonsymmetric optical systems to provide diffraction limited performance over large fields of view. Using the new optical design degrees of freedom, a three mirror system has been designed using tilted φ-polynomial surfaces that extend the usable field by an order of magnitude in area while enabling for the first time the use of an uncooled microbolometer, which requires an F/number faster than F/2.

By using the nonsymmetric aberration field analysis techniques enabled by full field displays, a strategy for the optical design based in nodal aberration theory was presented that resulted in an efficient path to a solution with minimum added complexity and testable surfaces. At this time one of the three mirrors has been fabricated and is about to enter interferometric null testing using a new configuration to be reported elsewhere. The complete system is on course to be assembled in the coming months.

Acknowledgments

We thank the Frank J. Horton Research Fellowship, The NYSTAR Foundation (C050070), the II-VI Foundation, and the National Science Foundation (EECS-1002179) for supporting this research as well as Synopsys Inc. for the student license of CODE V.

References and links

1.

O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16(3), 1583–1589 (2008). [CrossRef] [PubMed]

2.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).

3.

K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). [CrossRef] [PubMed]

4.

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010). [CrossRef] [PubMed]

5.

Synopsys Inc, “Zernike polynomials,” in CODE V Reference Manual, (2011), Volume IV, Appendix C.

6.

L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24(3), 569–577 (2007). [CrossRef] [PubMed]

7.

Air Force Avionics Laboratory, “Three mirror objective,” in Aerial Camera Lenses, Report 027000 from RECON Central, the Reconnaissance Division/Reconnaissance Applications Branch, (1967), pp. 2–109.

8.

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009). [CrossRef] [PubMed]

9.

J. M. Rodgers, “Catoptric optical system including concave and convex reflectors,” Optical Research Associates, US Patent 5,309,276 (1994).

10.

T. Nakano and Y. Tamagawa, “Configuration of an off-axis three-mirror system focused on compactness and brightness,” Appl. Opt. 44(5), 776–783 (2005). [CrossRef] [PubMed]

11.

J. W. Figoski, “Aberration characteristics of nonsymmetric systems,” in 1985 International Optical Design Conference, W.H. Taylor, and D.T. Moore, eds. (SPIE, 1985), pp. 104–111.

12.

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008). [CrossRef] [PubMed]

OCIS Codes
(220.1920) Optical design and fabrication : Diamond machining
(220.2740) Optical design and fabrication : Geometric optical design
(080.4035) Geometric optics : Mirror system design
(080.4228) Geometric optics : Nonspherical mirror surfaces

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: August 3, 2011
Revised Manuscript: September 27, 2011
Manuscript Accepted: September 28, 2011
Published: October 21, 2011

Citation
Kyle Fuerschbach, Jannick P. Rolland, and Kevin P. Thompson, "A new family of optical systems employing φ-polynomial surfaces," Opt. Express 19, 21919-21928 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21919


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References

  1. O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express16(3), 1583–1589 (2008). [CrossRef] [PubMed]
  2. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE251, 146–153 (1980).
  3. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A22(7), 1389–1401 (2005). [CrossRef] [PubMed]
  4. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express18(16), 17433–17447 (2010). [CrossRef] [PubMed]
  5. Synopsys Inc, “Zernike polynomials,” in CODE V Reference Manual, (2011), Volume IV, Appendix C.
  6. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A24(3), 569–577 (2007). [CrossRef] [PubMed]
  7. Air Force Avionics Laboratory, “Three mirror objective,” in Aerial Camera Lenses, Report 027000 from RECON Central, the Reconnaissance Division/Reconnaissance Applications Branch, (1967), pp. 2–109.
  8. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A26(6), 1503–1517 (2009). [CrossRef] [PubMed]
  9. J. M. Rodgers, “Catoptric optical system including concave and convex reflectors,” Optical Research Associates, US Patent 5,309,276 (1994).
  10. T. Nakano and Y. Tamagawa, “Configuration of an off-axis three-mirror system focused on compactness and brightness,” Appl. Opt.44(5), 776–783 (2005). [CrossRef] [PubMed]
  11. J. W. Figoski, “Aberration characteristics of nonsymmetric systems,” in 1985 International Optical Design Conference, W.H. Taylor, and D.T. Moore, eds. (SPIE, 1985), pp. 104–111.
  12. K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express16(25), 20345–20353 (2008). [CrossRef] [PubMed]

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