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

Full Article  |  PDF Article
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References

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2019 (2)

2018 (6)

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]

N. Takaki, A. Bauer, and J. P. Rolland, “Degeneracy in freeform surfaces described with orthogonal polynomials,” Appl. Opt. 57(35), 10348–10354 (2018).
[Crossref]

E. M. Schiesser, S.-W. Bahk, J. Bromage, and J. P. Rolland, “Gaussian curvature and stigmatic imaging relations for the design of an unobscured reflective relay,” Opt. Lett. 43(20), 4855–4858 (2018).
[Crossref]

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using freeform surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57, 101705 (2018).
[Crossref]

J. Sasian, “The method of confocal mirror design,” Proc. SPIE 10690, 1069015 (2018).
[Crossref]

2017 (1)

M. P. Chrisp, “Wide Angle Reflective Telescopes with NURBS Freeform Surfaces,” Proc. SPIE 10590, 1059011 (2017).
[Crossref]

2015 (3)

S. Chang, “Linear astigmatism of confocal off-axis reflective imaging systems with N-conic mirrors and its elimination,” J. Opt. Soc. Am. A 32(5), 852–859 (2015).
[Crossref]

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” Proc. SPIE 9626, 962612 (2015).
[Crossref]

T. Blalock, K. Medicus, and J. D. Nelson, “Fabrication of freeform optics,” Proc. SPIE 9575, 95750H (2015).
[Crossref]

2014 (2)

2013 (3)

C. B. Kreischer, “Retrace error: interferometry's dark little secret,” Proc. SPIE 8884, 88840X (2013).
[Crossref]

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Advanced Optical Testing 2, 97–109 (2013).
[Crossref]

2012 (1)

2011 (5)

2009 (1)

R. N. Youngworth, “Tolerancing Forbes aspheres: advantages of an orthogonal basis,” Proc. SPIE 7433, 74330H (2009).
[Crossref]

2005 (2)

2004 (1)

J. H. Burge and J. C. Wyant, “Use of computer generated holograms for testing aspheric surfaces,” Proc. SPIE 5494, 1–31 (2004).
[Crossref]

1995 (1)

1994 (2)

1992 (1)

J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” Applied Optics and Optical Engineering Xl, 1–13 (1992).

1986 (1)

1972 (1)

1971 (1)

1934 (1)

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[Crossref]

Bahk, S.-W.

Bauer, A.

Bennett, V. P.

Blalock, T.

T. Blalock, K. Medicus, and J. D. Nelson, “Fabrication of freeform optics,” Proc. SPIE 9575, 95750H (2015).
[Crossref]

Bond, C.

C. Bond and C. Pipan, “How to align an off-axis parabolic mirror,” in SPIE 1989 Technical Symposium on Aerospace Sensing (SPIE, Orlando, FL, 1989).

Bromage, J.

Burge, J. H.

J. H. Burge and J. C. Wyant, “Use of computer generated holograms for testing aspheric surfaces,” Proc. SPIE 5494, 1–31 (2004).
[Crossref]

J. H. Burge, “Null test for null correctors: error analysis,” in SPIE's 1993 International Symposium on Optics, Imaging, and Instrumentation (SPIE1993).

Cardona-Nunez, O.

Chang, S.

S. Chang, “Linear astigmatism of confocal off-axis reflective imaging systems with N-conic mirrors and its elimination,” J. Opt. Soc. Am. A 32(5), 852–859 (2015).
[Crossref]

S. Chang, “A design method of linear-astigmatism-free three-mirror freeform imaging systems,” in Design and Fabrication Congress (Freeform, OFT) (Optical Society of America, OSA Technical Digest, 2019).

Chaudhuri, R.

Chrisp, M. P.

M. P. Chrisp, “Wide Angle Reflective Telescopes with NURBS Freeform Surfaces,” Proc. SPIE 10590, 1059011 (2017).
[Crossref]

Cordero-Davila, A.

Cornejo-Rodriguez, A.

Creath, K.

J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” Applied Optics and Optical Engineering Xl, 1–13 (1992).

D. Malacara, J. C. Wyant, K. Creath, and J. Schmit, “Testing of Aspheric Wavefronts and Surfaces,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, Inc, 2007), pp. 435–497.

de Groot, P.

D. M. Sykora and P. de Groot, “Instantaneous measurement Fizeau interferometer with high spatial resolution,” Proc. SPIE 8126, 812610 (2011).
[Crossref]

Diaz-Uribe, R.

Evans, C.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Evans, C. J.

Fang, F. Z.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Forbes, G. W.

Fuerschbach, K.

Howard, J. M.

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57, 101705 (2018).
[Crossref]

J. M. Howard and S. Wolbach, “Improving the performance of three-mirror imaging systems with Freeform Optics,” in Renewable Energy and the Environment (Optical Society of America, OSA Technical Digest (online), 2013).

Kreischer, C. B.

C. B. Kreischer, “Retrace error: interferometry's dark little secret,” Proc. SPIE 8884, 88840X (2013).
[Crossref]

Kumler, J.

J. Kumler, “Designing and specifying aspheres for manufacturability,” Proc. SPIE 5874, 58740C (2005).
[Crossref]

Li, L.

Ma, B.

MacGovern, A. J.

Malacara, D.

D. Malacara, J. C. Wyant, K. Creath, and J. Schmit, “Testing of Aspheric Wavefronts and Surfaces,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, Inc, 2007), pp. 435–497.

Medicus, K.

T. Blalock, K. Medicus, and J. D. Nelson, “Fabrication of freeform optics,” Proc. SPIE 9575, 95750H (2015).
[Crossref]

Menke, C.

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Advanced Optical Testing 2, 97–109 (2013).
[Crossref]

Nakano, T.

Nelson, J. D.

T. Blalock, K. Medicus, and J. D. Nelson, “Fabrication of freeform optics,” Proc. SPIE 9575, 95750H (2015).
[Crossref]

Ochse, D.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” Proc. SPIE 9626, 962612 (2015).
[Crossref]

Papa, J.

Papa, J. C.

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57, 101705 (2018).
[Crossref]

Parks, R. E.

R. E. Parks, C. J. Evans, and L. Shao, “Test of a slow off-axis parabola at its center of curvature,” Appl. Opt. 34(31), 7174–7178 (1995).
[Crossref]

R. E. Parks, “Alignment of off-axis conic mirrors,” in Optical Fabrication and Testing (Optical Society of America, Washington, D.C., 1980).

Pedraza-Contreras, J.

Pipan, C.

C. Bond and C. Pipan, “How to align an off-axis parabolic mirror,” in SPIE 1989 Technical Symposium on Aerospace Sensing (SPIE, Orlando, FL, 1989).

Reichmann, L.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” Proc. SPIE 9626, 962612 (2015).
[Crossref]

Reshidko, D.

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using freeform surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

Rogers, J. R.

J. R. Rogers, “Orthogonal polynomials and tolerancing,” Proc. SPIE 8131, 81310D (2011).
[Crossref]

Rolland, J. P.

R. Chaudhuri, J. Papa, and J. P. Rolland, “System design of a single-shot reconfigurable null test using a spatial light modulator for freeform metrology,” Opt. Lett. 44(8), 2000–2003 (2019).
[Crossref]

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]

N. Takaki, A. Bauer, and J. P. Rolland, “Degeneracy in freeform surfaces described with orthogonal polynomials,” Appl. Opt. 57(35), 10348–10354 (2018).
[Crossref]

E. M. Schiesser, S.-W. Bahk, J. Bromage, and J. P. Rolland, “Gaussian curvature and stigmatic imaging relations for the design of an unobscured reflective relay,” Opt. Lett. 43(20), 4855–4858 (2018).
[Crossref]

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57, 101705 (2018).
[Crossref]

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]

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]

K. Fuerschbach, K. P. Thompson, and J. P. Rolland, “Interferometric measurement of a concave, phi-polynomial Zernike mirror,” Opt. Lett. 39(1), 18–21 (2014).
[Crossref]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing phi-type polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
[Crossref]

B. Ma, L. Li, K. P. Thompson, and J. P. Rolland, “Applying slope constrained Q-type aspheres to develop higher performance lenses,” Opt. Express 19(22), 21174–21179 (2011).
[Crossref]

Sasian, J.

J. Sasian, “The method of confocal mirror design,” Proc. SPIE 10690, 1069015 (2018).
[Crossref]

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using freeform surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

J. Sasian, “How to approach the design of bilateral symmetric optical systems,” Opt. Eng. 33(6), 2045–2061 (1994).
[Crossref]

Schiesser, E. M.

E. M. Schiesser, S.-W. Bahk, J. Bromage, and J. P. Rolland, “Gaussian curvature and stigmatic imaging relations for the design of an unobscured reflective relay,” Opt. Lett. 43(20), 4855–4858 (2018).
[Crossref]

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]

Schmit, J.

D. Malacara, J. C. Wyant, K. Creath, and J. Schmit, “Testing of Aspheric Wavefronts and Surfaces,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, Inc, 2007), pp. 435–497.

Shao, L.

Stone, B. D.

Sykora, D. M.

D. M. Sykora and P. de Groot, “Instantaneous measurement Fizeau interferometer with high spatial resolution,” Proc. SPIE 8126, 812610 (2011).
[Crossref]

Takaki, N.

Tamagawa, Y.

Thompson, K. P.

Uhlendorf, K.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” Proc. SPIE 9626, 962612 (2015).
[Crossref]

Weckenmann, A.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Wolbach, S.

J. M. Howard and S. Wolbach, “Improving the performance of three-mirror imaging systems with Freeform Optics,” in Renewable Energy and the Environment (Optical Society of America, OSA Technical Digest (online), 2013).

Wyant, J. C.

J. H. Burge and J. C. Wyant, “Use of computer generated holograms for testing aspheric surfaces,” Proc. SPIE 5494, 1–31 (2004).
[Crossref]

J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” Applied Optics and Optical Engineering Xl, 1–13 (1992).

J. C. Wyant and V. P. Bennett, “Using computer generated holograms to test aspheric wavefronts,” Appl. Opt. 11(12), 2833–2839 (1972).
[Crossref]

A. J. MacGovern and J. C. Wyant, “Computer generated holograms for testing optical elements,” Appl. Opt. 10(3), 619–624 (1971).
[Crossref]

D. Malacara, J. C. Wyant, K. Creath, and J. Schmit, “Testing of Aspheric Wavefronts and Surfaces,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, Inc, 2007), pp. 435–497.

Youngworth, R. N.

R. N. Youngworth, “Tolerancing Forbes aspheres: advantages of an orthogonal basis,” Proc. SPIE 7433, 74330H (2009).
[Crossref]

Zernike, F.

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[Crossref]

Zhang, G. X.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Zhang, X. D.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Advanced Optical Testing (1)

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Advanced Optical Testing 2, 97–109 (2013).
[Crossref]

Appl. Opt. (6)

Applied Optics and Optical Engineering (1)

J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” Applied Optics and Optical Engineering Xl, 1–13 (1992).

CIRP Ann. (1)

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Commun. (1)

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]

Opt. Eng. (3)

J. Sasian, “How to approach the design of bilateral symmetric optical systems,” Opt. Eng. 33(6), 2045–2061 (1994).
[Crossref]

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using freeform surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

J. C. Papa, J. M. Howard, and J. P. Rolland, “Starting point designs for freeform four-mirror systems,” Opt. Eng. 57, 101705 (2018).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Physica (1)

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[Crossref]

Proc. SPIE (10)

J. R. Rogers, “Orthogonal polynomials and tolerancing,” Proc. SPIE 8131, 81310D (2011).
[Crossref]

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” Proc. SPIE 9626, 962612 (2015).
[Crossref]

M. P. Chrisp, “Wide Angle Reflective Telescopes with NURBS Freeform Surfaces,” Proc. SPIE 10590, 1059011 (2017).
[Crossref]

J. Sasian, “The method of confocal mirror design,” Proc. SPIE 10690, 1069015 (2018).
[Crossref]

R. N. Youngworth, “Tolerancing Forbes aspheres: advantages of an orthogonal basis,” Proc. SPIE 7433, 74330H (2009).
[Crossref]

J. Kumler, “Designing and specifying aspheres for manufacturability,” Proc. SPIE 5874, 58740C (2005).
[Crossref]

T. Blalock, K. Medicus, and J. D. Nelson, “Fabrication of freeform optics,” Proc. SPIE 9575, 95750H (2015).
[Crossref]

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[Crossref]

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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.
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.
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.
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.
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.
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.
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.

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.

Equations (5)

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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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