Abstract

Standard phase-shifting interferometry (PSI) generally requires collecting at least three phase-shifted interferograms to extract the physical quantity being measured. Here, we propose the application of a simple two-frame PSI for the testing of a range of optical surfaces, including flats, spheres, and aspheres. The two-frame PSI extracts modulated phase from two randomly phase-shifted interferograms using a Gram-Schmidt algorithm, and can work in either null testing or non-null testing modes. Since only two interferograms are used for phase demodulation and the phase shift amount can be random, requirements on environmental conditions and phase shifter calibration are greatly relaxed. Experimental results of three different mirrors suggest that the two-frame PSI can achieve comparable measurement precision with conventional multi-frame PSI, but has faster data acquisition speed and less stringent hardware requirements. The proposed two-frame PSI expands the flexibility of PSI and holds great potential in many applications.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2016 (5)

2015 (1)

2014 (3)

2013 (4)

2012 (3)

2011 (4)

2010 (1)

2009 (1)

M. B. Dubin, P. Su, and J. H. Burge, “Fizeau interferometer with spherical reference and CGH correction for measuring large convex aspheres,” Proc. SPIE 7426, 74260S (2009).
[Crossref]

2007 (2)

2004 (4)

2001 (1)

2000 (1)

1997 (2)

1993 (1)

1992 (1)

T. M. Kreis and W. P. Jueptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

1974 (1)

1972 (1)

Auerbach, J. M.

Baer, G.

Belenguer, T.

Bennett, V. P.

Bliss, E. S.

Bone, D. J.

Bowers, M. W.

Brangaccio, D. J.

Bruning, J. H.

Burge, J. H.

C. Zhao and J. H. Burge, “Optical testing with computer generated holograms: comprehensive error analysis,” Proc. SPIE 8838, 88380H (2013).
[Crossref]

M. B. Dubin, P. Su, and J. H. Burge, “Fizeau interferometer with spherical reference and CGH correction for measuring large convex aspheres,” Proc. SPIE 7426, 74260S (2009).
[Crossref]

Carazo, J. M.

Chen, X.

Cuevas, F. J.

Dailey, M. J.

Dixit, S. N.

Dubin, M. B.

M. B. Dubin, P. Su, and J. H. Burge, “Fizeau interferometer with spherical reference and CGH correction for measuring large convex aspheres,” Proc. SPIE 7426, 74260S (2009).
[Crossref]

Dubos, T.

Elster, C.

Erbert, G. V.

Estrada, J. C.

Feldman, M.

Fortmeier, I.

Furey, L.

Gallagher, J. E.

Gappinger, R. O.

Greivenkamp, J. E.

Grey, A. A.

Gu, H.

Han, B.

Hansen, D.

Haynam, C. A.

Heestand, G. M.

Henesian, M. A.

Hermann, M. R.

Herriott, D. R.

Holdener, F. R.

Ishiyama, T.

T. Matsuyama, T. Ishiyama, and Y. Omura, “Nikon projection lens update,” Proc. SPIE 5377, 730–741 (2004).
[Crossref]

Jancaitis, K. S.

Jin, G.

Jueptner, W. P.

T. M. Kreis and W. P. Jueptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

Kemao, Q.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[Crossref]

Koch, J. A.

Kreis, T. M.

T. M. Kreis and W. P. Jueptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

Larkin, K. G.

Li, K.

X. Zhu, K. Li, P. Zhang, J. Zhu, J. Zhang, C. Tian, and S. Liu, “Implementation of dispersion-free slow acoustic wave propagation and phase engineering with helical-structured metamaterials,” Nat. Commun. 7, 11731 (2016).
[Crossref] [PubMed]

Ling, T.

Liu, D.

Liu, S.

Luo, Y.

Ma, J.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55, 205–211 (2014).
[Crossref]

Manes, K. R.

Marroquin, J. L.

Marshall, C. D.

Matsuyama, T.

T. Matsuyama, T. Ishiyama, and Y. Omura, “Nikon projection lens update,” Proc. SPIE 5377, 730–741 (2004).
[Crossref]

Mehta, N. C.

Menapace, J.

Moses, E.

Murray, J. R.

Nostrand, M. C.

Oldfield, M. A.

Omura, Y.

T. Matsuyama, T. Ishiyama, and Y. Omura, “Nikon projection lens update,” Proc. SPIE 5377, 730–741 (2004).
[Crossref]

Orth, C. D.

Osten, W.

Pan, T.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55, 205–211 (2014).
[Crossref]

Patorski, K.

Patterson, R.

Presta, R. W.

Pruss, C.

Quiroga, J. A.

Rajshekhar, G.

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50(8), iii–x (2012).
[Crossref]

Rastogi, P.

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50(8), iii–x (2012).
[Crossref]

Rosenfeld, D. P.

Sacks, R. A.

Salmon, J. T.

Samuels-Schwartz, J.

Schindler, J.

Schulz, M.

Seppala, L. G.

Servin, M.

Servín, M.

Shaw, M. J.

Siepmann, J.

Sluzewski, L.

Sorzano, C. O.

Spaeth, M.

Stavridis, M.

Su, P.

M. B. Dubin, P. Su, and J. H. Burge, “Fizeau interferometer with spherical reference and CGH correction for measuring large convex aspheres,” Proc. SPIE 7426, 74260S (2009).
[Crossref]

Sutton, S. B.

Tan, Q.

Tian, C.

Tkaczyk, T.

Toeppen, J. S.

Trusiak, M.

Van Atta, L.

Van Wonterghem, B. M.

Vargas, J.

Wang, Z.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55, 205–211 (2014).
[Crossref]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[Crossref] [PubMed]

Wegner, P. J.

Wei, T.

Whistler, W. T.

White, A. D.

White, R. K.

Whitehouse, D. J.

D. J. Whitehouse, “Surface metrology,” Meas. Sci. Technol. 8(9), 955–972 (1997).
[Crossref]

Widmayer, C. C.

Wiegmann, A.

Williams, W. H.

Winters, S. E.

Woods, B. W.

Wyant, J. C.

Yang, S. T.

Yang, Y.

Zacharias, R. A.

Zeng, F.

Zhang, J.

X. Zhu, K. Li, P. Zhang, J. Zhu, J. Zhang, C. Tian, and S. Liu, “Implementation of dispersion-free slow acoustic wave propagation and phase engineering with helical-structured metamaterials,” Nat. Commun. 7, 11731 (2016).
[Crossref] [PubMed]

Zhang, P.

X. Zhu, K. Li, P. Zhang, J. Zhu, J. Zhang, C. Tian, and S. Liu, “Implementation of dispersion-free slow acoustic wave propagation and phase engineering with helical-structured metamaterials,” Nat. Commun. 7, 11731 (2016).
[Crossref] [PubMed]

Zhang, S.

Zhao, C.

C. Zhao and J. H. Burge, “Optical testing with computer generated holograms: comprehensive error analysis,” Proc. SPIE 8838, 88380H (2013).
[Crossref]

Zhu, J.

X. Zhu, K. Li, P. Zhang, J. Zhu, J. Zhang, C. Tian, and S. Liu, “Implementation of dispersion-free slow acoustic wave propagation and phase engineering with helical-structured metamaterials,” Nat. Commun. 7, 11731 (2016).
[Crossref] [PubMed]

Zhu, X.

X. Zhu, K. Li, P. Zhang, J. Zhu, J. Zhang, C. Tian, and S. Liu, “Implementation of dispersion-free slow acoustic wave propagation and phase engineering with helical-structured metamaterials,” Nat. Commun. 7, 11731 (2016).
[Crossref] [PubMed]

Zhuo, Y.

Appl. Opt. (11)

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

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
[Crossref] [PubMed]

L. Furey, T. Dubos, D. Hansen, and J. Samuels-Schwartz, “Hubble Space Telescope primary-mirror characterization by measurement of the reflective null corrector,” Appl. Opt. 32(10), 1703–1714 (1993).
[Crossref] [PubMed]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36(19), 4540–4548 (1997).
[Crossref] [PubMed]

J. A. Koch, R. W. Presta, R. A. Sacks, R. A. Zacharias, E. S. Bliss, M. J. Dailey, M. Feldman, A. A. Grey, F. R. Holdener, J. T. Salmon, L. G. Seppala, J. S. Toeppen, L. Van Atta, B. M. Van Wonterghem, W. T. Whistler, S. E. Winters, and B. W. Woods, “Experimental comparison of a Shack-Hartmann sensor and a phase-shifting interferometer for large-optics metrology applications,” Appl. Opt. 39(25), 4540–4546 (2000).
[Crossref] [PubMed]

J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43(27), 5143–5151 (2004).
[Crossref] [PubMed]

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(27), 5152–5161 (2004).
[Crossref] [PubMed]

C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007).
[Crossref] [PubMed]

C. Tian, Y. Yang, and Y. Zhuo, “Generalized data reduction approach for aspheric testing in a non-null interferometer,” Appl. Opt. 51(10), 1598–1604 (2012).
[Crossref] [PubMed]

J. C. Wyant, “Computerized interferometric surface measurements [Invited],” Appl. Opt. 52(1), 1–8 (2013).
[Crossref] [PubMed]

C. Tian, Y. Yang, T. Wei, and Y. Zhuo, “Nonnull interferometer simulation for aspheric testing based on ray tracing,” Appl. Opt. 50(20), 3559–3569 (2011).
[Crossref] [PubMed]

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

Meas. Sci. Technol. (1)

D. J. Whitehouse, “Surface metrology,” Meas. Sci. Technol. 8(9), 955–972 (1997).
[Crossref]

Nat. Commun. (1)

X. Zhu, K. Li, P. Zhang, J. Zhu, J. Zhang, C. Tian, and S. Liu, “Implementation of dispersion-free slow acoustic wave propagation and phase engineering with helical-structured metamaterials,” Nat. Commun. 7, 11731 (2016).
[Crossref] [PubMed]

Opt. Express (10)

J. Vargas, J. A. Quiroga, T. Belenguer, M. Servín, and J. C. Estrada, “Two-step self-tuning phase-shifting interferometry,” Opt. Express 19(2), 638–648 (2011).
[Crossref] [PubMed]

K. Patorski and M. Trusiak, “Highly contrasted Bessel fringe minima visualization for time-averaged vibration profilometry using Hilbert transform two-frame processing,” Opt. Express 21(14), 16863–16881 (2013).
[Crossref] [PubMed]

F. Zeng, Q. Tan, H. Gu, and G. Jin, “Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method,” Opt. Express 21(14), 17234–17248 (2013).
[Crossref] [PubMed]

K. Patorski, M. Trusiak, and T. Tkaczyk, “Optically-sectioned two-shot structured illumination microscopy with Hilbert-Huang processing,” Opt. Express 22(8), 9517–9527 (2014).
[Crossref] [PubMed]

G. Baer, J. Schindler, C. Pruss, J. Siepmann, and W. Osten, “Calibration of a non-null test interferometer for the measurement of aspheres and free-form surfaces,” Opt. Express 22(25), 31200–31211 (2014).
[Crossref] [PubMed]

M. Trusiak and K. Patorski, “Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering,” Opt. Express 23(4), 4672–4690 (2015).
[Crossref] [PubMed]

C. Tian and S. Liu, “Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization,” Opt. Express 24(4), 3202–3215 (2016).
[Crossref] [PubMed]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, W. Osten, and C. Elster, “Evaluation of absolute form measurements using a tilted-wave interferometer,” Opt. Express 24(4), 3393–3404 (2016).
[Crossref] [PubMed]

C. Tian, X. Chen, and S. Liu, “Modal wavefront reconstruction in radial shearing interferometry with general aperture shapes,” Opt. Express 24(4), 3572–3583 (2016).
[Crossref] [PubMed]

M. Trusiak, Ł. Służewski, and K. Patorski, “Single shot fringe pattern phase demodulation using Hilbert-Huang transform aided by the principal component analysis,” Opt. Express 24(4), 4221–4238 (2016).
[Crossref] [PubMed]

Opt. Lasers Eng. (3)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[Crossref]

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Lasers Eng. 50(8), iii–x (2012).
[Crossref]

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55, 205–211 (2014).
[Crossref]

Opt. Lett. (5)

Proc. SPIE (4)

M. B. Dubin, P. Su, and J. H. Burge, “Fizeau interferometer with spherical reference and CGH correction for measuring large convex aspheres,” Proc. SPIE 7426, 74260S (2009).
[Crossref]

C. Zhao and J. H. Burge, “Optical testing with computer generated holograms: comprehensive error analysis,” Proc. SPIE 8838, 88380H (2013).
[Crossref]

T. M. Kreis and W. P. Jueptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

T. Matsuyama, T. Ishiyama, and Y. Omura, “Nikon projection lens update,” Proc. SPIE 5377, 730–741 (2004).
[Crossref]

Other (4)

H. Schreiber and J. H. Bruning, “Chapter 14. Phase Shifting Interferometry,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, 2007).

C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2000).

V. N. Mahajan, “Chapter 13. Zernike Polynomial and Wavefront Fitting,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, 2007).

D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007).

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Figures (9)

Fig. 1
Fig. 1 Schematic illustration of the two-frame PSI. PZT: piezoelectric transducer; CCD: charge-coupled device.
Fig. 2
Fig. 2 Null testing and non-null testing modes of the two-frame PSI. (a) Null testing configuration and a typical interfergoram on the image plane. In this case, an aplanatic lens is employed as the compensation lens and its focal point coincides with the curvature center of the spherical surface under test. The incident rays and return rays go through the same optical paths, and no retrace error exists. (b) Non-null testing configuration and a typical interferogram on the image plane. In this case, a simple singlet lens is employed as the compensation lens partially compensating for the aspheric mirror under test. The incident rays and return rays follow different optical paths and retrace error exists. Blue: incident rays; red: return rays.
Fig. 3
Fig. 3 Workflow of the proposed two-frame PSI.
Fig. 4
Fig. 4 PV and RMS demodulation error with respect to the fringe number n.
Fig. 5
Fig. 5 Phase demodulation of two-frame interferograms when only one fringe (n = 1) is present. (a) and (b) two phase-shifted interferograms; (c) and (d) theoretical and demodulated phase; (e) demodulation error. Colorbar unit: rad.
Fig. 6
Fig. 6 Null testing of the spherical mirror using the proposed two-frame PSI. (a) and (b) Acquired two-frame phase-shifted fringe patterns; (c) retrieved phase using the GS algorithm; (d) reconstructed surface figure; (e) surface figure measured by a Zygo interferometer using 13-frame fringe patterns.
Fig. 7
Fig. 7 Non-null testing of the paraboloidal mirror with a 2.2 μm asphericity. (a) and (b) Acquired two-frame randomly phase-shifted fringe patterns, (c) computed retrace error. The retrace error was wrapped for comparison.
Fig. 8
Fig. 8 Non-null testing of the paraboloidal mirror with a 2.2 μm asphericity. 1st row: demodulated phases using 2-frame, 3-frame, and 13-frame interferograms. 2nd row: reconstructed surface figure maps. The 13-frame results were regarded as the reference. 3rd row: differences of the 2-frame and the 3-frame results with the reference results.
Fig. 9
Fig. 9 Non-null testing of the paraboloidal mirror with a 14.5 μm asphericity using two-frame interferograms. (a) and (b) Acquired two-frame randomly phase-shifted interferograms; (c) retrieved phase using the GS algorithm; (d) computed retrace error; (e) reconstructed surface figure map after removing the retrace error and the alignment error; (f) 1st comparison result: repeated measurement result using the same setup by rotating the mirror for 180 degrees; (g) 2nd comparison result: null testing result using an auto-collimation setup and a Zygo interferometer with 13-frame interferograms. The results show that the proposed two-frame PSI has excellent measurement repeatability and accuracy.

Tables (3)

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Table 1 Specifications of the three optical surfaces

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Table 2 Specifications of the test path

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Table 3 Summary of the PV and RMS values of all measurements

Equations (15)

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I 1 (x,y)=a(x,y)+b(x,y)cosϕ(x,y),
I 2 (x,y)=a(x,y)+b(x,y)cos[ ϕ(x,y)+δ ],
I 1 (x,y)=b(x,y)cosϕ(x,y),
I 2 (x,y)=b(x,y)cos[ ϕ(x,y)+δ ].
u 1 = I 1 =b(x,y)cosϕ(x,y).
u 2 = I 2 pro j u 1 ( I 2 )=b(x,y)cos[ ϕ(x,y)+δ ]pro j u 1 ( I 2 ),
pro j u 1 ( I 2 )= I 2 , u 1 u 1 , u 1 u 1 = (x,y) { b(x,y)cos[ ϕ(x,y)+δ ]b(x,y)cosϕ(x,y) } (x,y) { [ b(x,y)cosϕ(x,y) ] 2 } b(x,y)cosϕ(x,y),
pro j u 1 ( I 2 )b(x,y)cosϕ(x,y)cosδ.
u 2 b(x,y)sinϕ(x,y)sinδ.
e 1 = u 1 u 1 = b(x,y)cosϕ(x,y) { (x,y) [ b(x,y)cosϕ(x,y) ] 2 } 1/2 ,
e 2 = u 2 u 2 = b(x,y)sinϕ(x,y) { (x,y) [ b(x,y)sinϕ(x,y) ] 2 } 1/2 .
ϕ(x,y)arctan( e 2 e 1 ).
W= 1 2 { W CCD [ a+bx+cy+d( x 2 + y 2 ) ] },
W= 1 2 { ( W CCD W retrace )[ a+bx+cy+d( x 2 + y 2 ) ] }.
ϕ(x,y)=n( x 2 + y 2 )×2π,1x,y1,

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