Abstract

The nonnull interferometric method that employs a partial compensation system to compensate for the longitude aberration of the aspheric under test and a reverse optimization procedure to correct retrace errors is a useful technique for general aspheric testing. However, accurate system modeling and simulation are required to correct retrace errors and reconstruct fabrication error of the aspheric. Here, we propose a ray-tracing-based method to simulate the nonnull interferometer, which calculates the optical path difference by tracing rays through the reference path and the test path. To model a nonrotationally symmetric fabrication error, we mathematically represent it with a set of Zernike polynomials (i.e., Zernike deformation) and derive ray-tracing formulas for the deformed surface, which can also deal with misalignment situations (i.e., a surface with tilts and/or decenters) and thus facilitates system modeling extremely. Simulation results of systems with (relatively) large and small Zernike deformations and their comparisons with the lens design program Zemax have demonstrated the correctness and effectiveness of the method.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  8. C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” Proc. SPIE 7656, 765612–765616 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]

2010 (3)

2009 (4)

Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, and Y. Zhuo, “Research of precision interference locating method for a partial null compensator at aspheric testing,” Proc. SPIE 7426, 74260R (2009).
[CrossRef]

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[CrossRef]

B. D. Stone and K. P. Thompson, “Modeling interferometers with lens design software: beyond ray-based approaches,” Proc. SPIE 7427, 74270A (2009).
[CrossRef]

M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916–738934 (2009).
[CrossRef]

2008 (1)

G. M. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
[CrossRef]

2006 (2)

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[CrossRef]

J.-M. Asfour and A. G. Poleshchuk, “Asphere testing with a Fizeau interferometer based on a combined computer-generated hologram,” J. Opt. Soc. Am. A 23, 172–178 (2006).
[CrossRef]

2005 (1)

Zemax Development Corporation, Optical Design Program User’s Guide (Zemax Development Corporation, 2005).

2004 (4)

2003 (1)

2002 (1)

P. Scott, “Recent developments in the measurement of aspheric surfaces by contact stylus instrumentation,” Proc. SPIE 4927, 199–207 (2002).

2000 (4)

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39, 2696–2699 (2000).
[CrossRef]

P. E. Murphy, T. G. Brown, and D. T. Moore, “Interference imaging for aspheric surface testing,” Appl. Opt. 39, 2122–2129 (2000).
[CrossRef]

B. D. Stone, “Modeling interferometers with lens design software,” Opt. Eng. 39, 1748–1759 (2000).
[CrossRef]

W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, 3rd ed. (McGraw-Hill, 2000).

1999 (1)

H. P. Stahl, “Aspheric surface testing techniques,” in OSA Trends in Optics and Photonics (Optical Society of America, 1999), paper T2.

1995 (1)

A. E. Lowman and J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” Proc. SPIE 2536, 139–147 (1995).
[CrossRef]

1992 (1)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

1988 (1)

1987 (1)

1985 (2)

B. Dörband and H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors,” Appl. Opt. 24, 2604–2611 (1985).
[CrossRef] [PubMed]

K. Creath, Y.-Y. Cheng, and J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

1980 (1)

1976 (1)

1972 (1)

1963 (1)

1962 (1)

1951 (1)

Asfour, J.-M.

Bennett, V. P.

Brown, T. G.

Cheng, Y.-Y.

K. Creath, Y.-Y. Cheng, and J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

Creath, K.

K. Creath, Y.-Y. Cheng, and J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

Dai, G. M.

G. M. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
[CrossRef]

DeVries, G.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[CrossRef]

Dörband, B.

Feder, D. P.

Fleig, J.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[CrossRef]

Forbes, G.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[CrossRef]

Gappinger, R. O.

Greivenkamp, J. E.

Hayslett, C. R.

Kohno, T.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39, 2696–2699 (2000).
[CrossRef]

Koliopoulos, C. L.

Küchel, M. F.

M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916–738934 (2009).
[CrossRef]

Kwon, O.

Lawrence, G. N.

Liu, D.

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” Proc. SPIE 7656, 765612–765616 (2010).
[CrossRef]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49, 170–179 (2010).
[CrossRef] [PubMed]

C. Tian, Y. Yang, S. Zhang, D. Liu, Y. Luo, and Y. Zhuo, “Regularized frequency-stabilizing method for single closed-fringe interferogram demodulation,” Opt. Lett. 35, 1837–1839(2010).
[CrossRef] [PubMed]

Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, and Y. Zhuo, “Research of precision interference locating method for a partial null compensator at aspheric testing,” Proc. SPIE 7426, 74260R (2009).
[CrossRef]

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[CrossRef]

Liu, Y.-M.

Lowman, A. E.

A. E. Lowman and J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” Proc. SPIE 2536, 139–147 (1995).
[CrossRef]

Luo, Y.

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” Proc. SPIE 7656, 765612–765616 (2010).
[CrossRef]

C. Tian, Y. Yang, S. Zhang, D. Liu, Y. Luo, and Y. Zhuo, “Regularized frequency-stabilizing method for single closed-fringe interferogram demodulation,” Opt. Lett. 35, 1837–1839(2010).
[CrossRef] [PubMed]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49, 170–179 (2010).
[CrossRef] [PubMed]

Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, and Y. Zhuo, “Research of precision interference locating method for a partial null compensator at aspheric testing,” Proc. SPIE 7426, 74260R (2009).
[CrossRef]

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[CrossRef]

Malacara, D.

D. Malacara and Z. Malacara, Handbook of Optical Design (Marcel Dekker, 2004).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Malacara, Z.

D. Malacara and Z. Malacara, Handbook of Optical Design (Marcel Dekker, 2004).

Matsumoto, D.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39, 2696–2699 (2000).
[CrossRef]

Miladinovic, D.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[CrossRef]

Moore, D. T.

Murphy, P.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[CrossRef]

Murphy, P. E.

Murty, M. V. R. K.

Noll, R. J.

O’Donohue, S.

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[CrossRef]

Offner, A.

Osten, W.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Opt. Eng. 43, 2534–2540 (2004).
[CrossRef]

Poleshchuk, A. G.

Pruss, C.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Opt. Eng. 43, 2534–2540 (2004).
[CrossRef]

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute interferometric test of aspheres by use of twin computer-generated holograms,” Appl. Opt. 42, 4468–4479 (2003).
[CrossRef] [PubMed]

Reichelt, S.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Opt. Eng. 43, 2534–2540 (2004).
[CrossRef]

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute interferometric test of aspheres by use of twin computer-generated holograms,” Appl. Opt. 42, 4468–4479 (2003).
[CrossRef] [PubMed]

Scott, P.

P. Scott, “Recent developments in the measurement of aspheric surfaces by contact stylus instrumentation,” Proc. SPIE 4927, 199–207 (2002).

Shen, Y.

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[CrossRef]

Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, and Y. Zhuo, “Research of precision interference locating method for a partial null compensator at aspheric testing,” Proc. SPIE 7426, 74260R (2009).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, 3rd ed. (McGraw-Hill, 2000).

Spencer, G. H.

Stahl, H. P.

H. P. Stahl, “Aspheric surface testing techniques,” in OSA Trends in Optics and Photonics (Optical Society of America, 1999), paper T2.

Stone, B. D.

B. D. Stone and K. P. Thompson, “Modeling interferometers with lens design software: beyond ray-based approaches,” Proc. SPIE 7427, 74270A (2009).
[CrossRef]

B. D. Stone, “Modeling interferometers with lens design software,” Opt. Eng. 39, 1748–1759 (2000).
[CrossRef]

Thompson, K. P.

B. D. Stone and K. P. Thompson, “Modeling interferometers with lens design software: beyond ray-based approaches,” Proc. SPIE 7427, 74270A (2009).
[CrossRef]

Tian, C.

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” Proc. SPIE 7656, 765612–765616 (2010).
[CrossRef]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49, 170–179 (2010).
[CrossRef] [PubMed]

C. Tian, Y. Yang, S. Zhang, D. Liu, Y. Luo, and Y. Zhuo, “Regularized frequency-stabilizing method for single closed-fringe interferogram demodulation,” Opt. Lett. 35, 1837–1839(2010).
[CrossRef] [PubMed]

Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, and Y. Zhuo, “Research of precision interference locating method for a partial null compensator at aspheric testing,” Proc. SPIE 7426, 74260R (2009).
[CrossRef]

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[CrossRef]

Tiziani, H. J.

Uda, Y.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39, 2696–2699 (2000).
[CrossRef]

Wyant, J. C.

Xin, G.

Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, and Y. Zhuo, “Research of precision interference locating method for a partial null compensator at aspheric testing,” Proc. SPIE 7426, 74260R (2009).
[CrossRef]

Yang, Y.

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” Proc. SPIE 7656, 765612–765616 (2010).
[CrossRef]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49, 170–179 (2010).
[CrossRef] [PubMed]

C. Tian, Y. Yang, S. Zhang, D. Liu, Y. Luo, and Y. Zhuo, “Regularized frequency-stabilizing method for single closed-fringe interferogram demodulation,” Opt. Lett. 35, 1837–1839(2010).
[CrossRef] [PubMed]

Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, and Y. Zhuo, “Research of precision interference locating method for a partial null compensator at aspheric testing,” Proc. SPIE 7426, 74260R (2009).
[CrossRef]

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[CrossRef]

Yazawa, T.

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39, 2696–2699 (2000).
[CrossRef]

Zhang, S.

Zhuo, Y.

C. Tian, Y. Yang, S. Zhang, D. Liu, Y. Luo, and Y. Zhuo, “Regularized frequency-stabilizing method for single closed-fringe interferogram demodulation,” Opt. Lett. 35, 1837–1839(2010).
[CrossRef] [PubMed]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49, 170–179 (2010).
[CrossRef] [PubMed]

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” Proc. SPIE 7656, 765612–765616 (2010).
[CrossRef]

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[CrossRef]

Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, and Y. Zhuo, “Research of precision interference locating method for a partial null compensator at aspheric testing,” Proc. SPIE 7426, 74260R (2009).
[CrossRef]

Appl. Opt. (11)

Y.-M. Liu, G. N. Lawrence, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27, 4504–4513 (1988).
[CrossRef] [PubMed]

A. Offner, “A null corrector for paraboloidal mirrors,” Appl. Opt. 2, 153–155 (1963).
[CrossRef]

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

O. Kwon, J. C. Wyant, and C. R. Hayslett, “Rough surface interferometry at 10.6 μm,” Appl. Opt. 19, 1862–1869 (1980).
[CrossRef] [PubMed]

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
[CrossRef] [PubMed]

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute interferometric test of aspheres by use of twin computer-generated holograms,” Appl. Opt. 42, 4468–4479 (2003).
[CrossRef] [PubMed]

B. Dörband and H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors,” Appl. Opt. 24, 2604–2611 (1985).
[CrossRef] [PubMed]

P. E. Murphy, T. G. Brown, and D. T. Moore, “Interference imaging for aspheric surface testing,” Appl. Opt. 39, 2122–2129 (2000).
[CrossRef]

R. O. Gappinger and J. E. Greivenkamp, “Iterative reverse optimization procedure for calibration of aspheric wave-front measurements on a nonnull interferometer,” Appl. Opt. 43, 5152–5161 (2004).
[CrossRef] [PubMed]

J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43, 5143–5151 (2004).
[CrossRef] [PubMed]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49, 170–179 (2010).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

K. Creath, Y.-Y. Cheng, and J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

Opt. Eng. (3)

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Opt. Eng. 43, 2534–2540 (2004).
[CrossRef]

T. Kohno, D. Matsumoto, T. Yazawa, and Y. Uda, “Radial shearing interferometer for in-process measurement of diamond turning,” Opt. Eng. 39, 2696–2699 (2000).
[CrossRef]

B. D. Stone, “Modeling interferometers with lens design software,” Opt. Eng. 39, 1748–1759 (2000).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (7)

Y. Yang, D. Liu, G. Xin, C. Tian, Y. Luo, Y. Shen, and Y. Zhuo, “Research of precision interference locating method for a partial null compensator at aspheric testing,” Proc. SPIE 7426, 74260R (2009).
[CrossRef]

B. D. Stone and K. P. Thompson, “Modeling interferometers with lens design software: beyond ray-based approaches,” Proc. SPIE 7427, 74270A (2009).
[CrossRef]

A. E. Lowman and J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” Proc. SPIE 2536, 139–147 (1995).
[CrossRef]

D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[CrossRef]

C. Tian, Y. Yang, Y. Luo, D. Liu, and Y. Zhuo, “Study on phase retrieval of a single closed fringe interferogram in radial shearing interferometer for aspheric test,” Proc. SPIE 7656, 765612–765616 (2010).
[CrossRef]

M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916–738934 (2009).
[CrossRef]

P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).
[CrossRef]

Other (7)

P. Scott, “Recent developments in the measurement of aspheric surfaces by contact stylus instrumentation,” Proc. SPIE 4927, 199–207 (2002).

H. P. Stahl, “Aspheric surface testing techniques,” in OSA Trends in Optics and Photonics (Optical Society of America, 1999), paper T2.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, 3rd ed. (McGraw-Hill, 2000).

D. Malacara and Z. Malacara, Handbook of Optical Design (Marcel Dekker, 2004).

G. M. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
[CrossRef]

Zemax Development Corporation, Optical Design Program User’s Guide (Zemax Development Corporation, 2005).

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Figures (8)

Fig. 1
Fig. 1

Principle of the nonnull testing method. BS, beam splitter; PCS, partial compensation system. Note that surfaces in the test arm are sequentially numbered.

Fig. 2
Fig. 2

Partial compensation for the aberration of normals of the aspheric. (a) Spherical aberration F 0 F 1 of the PCS; (b) aberration of normals C 0 C 1 of the aspheric.

Fig. 3
Fig. 3

Determination of the intersection point of a ray and a nonrotationally symmetric aspheric using an iterative scheme.

Fig. 4
Fig. 4

Definition of decenters and tilts of a surface: (a) decenter along x and y axis, (b) tilt about x axis, (c) tilt about y axis, (d) tilt about z axis. Note that the rotation is defined as left handed about the positive axes.

Fig. 5
Fig. 5

Intermediate profile of the synthetic Zernike deformation (dashed curve) and the departure (solid curve) of the paraboloid from its vertex sphere: (a) large Zernike deformation, (b) small Zernike deformation.

Fig. 6
Fig. 6

Ray tracing results for the paraboloid with large Zernike deformation: (a) OPD calculated by the formulas in the paper. PV, 1219.93 λ ; rms, 769.15 λ ; (b) OPD from Zemax. PV, 1219.58 λ ; rms, 769.52 λ .

Fig. 7
Fig. 7

Ray-tracing results for the paraboloid with small Zernike deformation: (a) distribution of the deformation, (b) OPD calculated by the method in the paper. PV, 17.93 λ ; rms, 12.71 λ . (c) OPD from Zemax. PV, 17.94 λ ; rms, 12.70 λ .

Fig. 8
Fig. 8

Four-step phase-shifting interferograms in the nonnull interferometer.

Tables (4)

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Table 1 Zernike Polynomials Z n m ( ρ , θ ) and Their Derivatives up to the Fourth Order a

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Table 2 Surface Data of the Test Path (Dashed Box in Fig. 1) of the Nonnull Interferometer a

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Table 3 Synthetic 37 Zernike Coefficients to Simulate (Relatively) Large Zernike Deformation with a Normalization Radius R = 79 mm

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Table 4 Tracing a Specified Ray through the Test Path of the Nonnull Interferometer

Equations (32)

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W det f ( W asp , T asp ) ,
W asp W asp .
W det f ( W asp , T asp ) .
[ W asp , T asp ] T f 1 ( W det ) .
U = [ W det f ( W asp , T asp ) ] 2 + cons ,
f ( W asp , T asp ) = W test ( W asp , T asp ) W ref ,
z = c r 2 1 + 1 c 2 r 2 + j = 1 M A 2 j r 2 j + j = 1 N B j Z j ( ρ , θ ) ,
Z j = R n m ( ρ ) Θ m ( θ ) ,
R n m ( ρ ) = s = 0 ( n | m | ) / 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) / 2 s ] ! [ ( n m ) / 2 s ] ! ρ n 2 s ,
Θ m ( θ ) = { 2 cos | m | θ , m 0 , even     j 1 , m = 0 , 2 sin | m | θ , m 0 , odd     j .
d k = z k z k = z k [ c r k 2 1 + W k + j = 1 M A 2 j r k 2 j + j = 1 N B j Z j ( ρ , θ ) ] ,
F ( x , y , z ) = z [ c r 2 1 + W + j = 1 M A 2 j r 2 j + j = 1 N B j Z j ( ρ , θ ) ] = 0 .
( F x ) Q k ( x x k ) + ( F y ) Q k ( y y k ) + ( F z ) Q k ( z z k ) = 0 .
F x = x W c 2 x j = 1 M j A 2 j r 2 j 2 1 R j = 1 N B j Z j ( ρ , θ ) x N = U W ,
F y = y W c 2 y j = 1 M j A 2 j r 2 j 2 1 R j = 1 N B j Z j ( ρ , θ ) y N = V W ,
F z = 1 ,
U = [ c x + 2 W x j = 1 M j A 2 j r 2 j 2 + W 1 R j = 1 N B j Z j ( ρ , θ ) x N ] ,
V = [ c y + 2 W y j = 1 M j A 2 j r 2 j 2 + W 1 R j = 1 N B j Z j ( ρ , θ ) x N ] ,
Z j ( ρ , θ ) x N = n + 1 [ R n m ( ρ ) ρ Θ m ( θ ) cos θ R n m ( ρ ) ρ Θ m ( θ ) θ sin θ ] , Z j ( ρ , θ ) y N = n + 1 [ R n m ( ρ ) ρ Θ m ( θ ) sin θ + R n m ( ρ ) ρ Θ m ( θ ) θ cos θ ] ,
R n m ρ = n [ R n 1 m 1 + R n 1 m + 1 ] + R n 2 m ρ ,
( x x k ) U k + ( y y k ) V k + ( z z k ) W k = d k W k ,
( x k + 1 x k ) U k + ( y k + 1 y k ) V k + ( z k + 1 z k ) W k = d k W k .
x k + 1 = x k + Δ P k X 1 , y k + 1 = y k + Δ P k Y 1 , z k + 1 = z k + Δ P k Z 1 ,
Δ P k = d k W k X 1 U k + Y 1 V k + Z 1 W k .
cos I = ( X 1 U + Y 1 V + Z 1 W ) / G ,
H = G n 1 cos I = n 1 ( X 1 U + Y 1 V + Z 1 W ) ,
H = G n cos I = n G 2 ( 1 n 1 2 n 2 ) + ( 1 n ) 2 H 2 .
X = n 1 n X 1 + 1 n Γ U G , Y = n 1 n Y 1 + 1 n Γ V G , Z = n 1 n Z 1 + 1 n Γ W G ,
X = n 1 n X 1 + 1 n P U , Y = n 1 n Y 1 + 1 n P V , Z = n 1 n Z 1 + 1 n P W ,
T d = [ x d y d 0 ] , T x = [ 1 0 0 0 cos θ x sin θ x 0 sin θ x cos θ x ] , T y = [ cos θ y 0 sin θ y 0 1 0 sin θ y 0 cos θ y ] , T z = [ cos θ z sin θ z 0 sin θ z cos θ z 0 0 0 1 ] ,
[ x 1 y 1 z 1 ] = T z y x { [ x 1 y 1 z 1 ] T d } ,
[ X 1 Y 1 Z 1 ] = T z y x [ X 1 Y 1 Z 1 ] ,

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