Abstract

We propose a new computer-controlled phase-shifting method based on computer-generated holograms (CGHs) displayed on a spatial light modulator (SLM). In this method the accurate phase shifts required in phase-shifting digital holography or interferometry are induced by a suitable transformation of the encoding patterns of the CGH displayed on a SLM. Both the theoretical analysis and the experimental results demonstrate the feasibility of this approach. We also discuss possible applications of this method in the field of interferometric null testing of aspheres.

© 2003 Optical Society of America

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  1. J. W. Goodman, R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
    [CrossRef]
  2. U. Schnars, W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  3. J. Pomarico, U. Schnars, H.-J. Hartmann, W. Juptner, “Digital recording and numerical reconstruction of holograms: a new method for displaying light in flight,” Appl. Opt. 34, 8095–8099 (1995).
    [CrossRef] [PubMed]
  4. C. Pedrini, P. Froing, H. J. Tiziani, M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method,” Appl. Opt. 38, 3460–3467 (1999).
    [CrossRef]
  5. C. Wagner, W. Osten, S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
    [CrossRef]
  6. I. Yamaguchi, T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [CrossRef] [PubMed]
  7. S. Lai, B. King, M. A. Neifeld, “Wavefront reconstruction by means of phase-shifting digital in-line holography,” Opt. Commun. 173, 155–160 (2000).
    [CrossRef]
  8. T. Zhang, I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998).
    [CrossRef]
  9. I. Yamaguchi, J. Kato, S. Ohta, J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177–6186 (2001).
    [CrossRef]
  10. C.-S. Guo, L. Zhang, H.-T. Wang, J. Liao, Y. Y. Zhu, “Phase-shifting error and its elimination in phase-shifting digital holography,” Opt. Lett. 27, 1–3 (2002).
    [CrossRef]
  11. Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  12. C. Jin, N. Bao, P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
    [CrossRef]
  13. M. B. North-Morris, J. VanDelden, J. C. Wyant, “Phase-shifting birefringent scatterplate interferometer,” Appl. Opt. 41, 668–677 (2002).
    [CrossRef] [PubMed]
  14. J. C. Wyant, V. P. Bennett, “Using computer generated holograms to test aspheric wave fronts,” Appl. Opt. 9, 2833–2839 (1972).
    [CrossRef]
  15. J. C. Wyant, P. K. O’Neill, “Computer generated hologram: null lens test of aspheric wave fronts,” Appl. Opt. 13, 2762–2765 (1974).
    [CrossRef] [PubMed]
  16. J. H. Burge, “Application of computer-generated holograms for interferometric measurement of large aspheric optics,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 258–269 (1995).
    [CrossRef]
  17. N. Lindlein, “Analysis of the disturbing diffraction orders of computer-generated holograms used for testing optical aspheres,” Appl. Opt. 40, 2698–2708 (2001).
    [CrossRef]
  18. M. Beyerlein, N. Lindlein, J. Schwider, “Dual-wave-front computer-generated holograms for quasi-absolute testing of aspheres,” Appl. Opt. 41, 2440–2447 (2002).
    [CrossRef] [PubMed]
  19. I. Yamaguchi, S. Ohta, J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
    [CrossRef]
  20. W. H. Lee, “Sampled Fourier-transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
    [CrossRef] [PubMed]
  21. N. Yoshikawa, M. Itoh, T. Yatagai, “Quantized phase optimization of two-dimensional Fourier kinoforms by a genetic algorithm,” Opt. Lett. 20, 752–754 (1995).
    [CrossRef] [PubMed]
  22. K. A. Goldberg, J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886–2894 (2001).
    [CrossRef]
  23. C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L. Z. Cai, Y.-R. Wang, “Determination of global phase shifts between interferograms using energy minimum algorithm,” Appl. Opt. 42, 6414–6519 (2003).
    [CrossRef]
  24. K. Creath, Y.-Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength phase-shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
    [CrossRef]

2003 (1)

C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L. Z. Cai, Y.-R. Wang, “Determination of global phase shifts between interferograms using energy minimum algorithm,” Appl. Opt. 42, 6414–6519 (2003).
[CrossRef]

2002 (3)

2001 (4)

2000 (2)

C. Wagner, W. Osten, S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

S. Lai, B. King, M. A. Neifeld, “Wavefront reconstruction by means of phase-shifting digital in-line holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

1999 (1)

1998 (1)

1997 (1)

1995 (2)

1994 (2)

U. Schnars, W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

C. Jin, N. Bao, P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

1985 (2)

Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

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

1974 (1)

1972 (1)

J. C. Wyant, V. P. Bennett, “Using computer generated holograms to test aspheric wave fronts,” Appl. Opt. 9, 2833–2839 (1972).
[CrossRef]

1970 (1)

1967 (1)

J. W. Goodman, R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Bao, N.

C. Jin, N. Bao, P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Bennett, V. P.

J. C. Wyant, V. P. Bennett, “Using computer generated holograms to test aspheric wave fronts,” Appl. Opt. 9, 2833–2839 (1972).
[CrossRef]

Beyerlein, M.

Bokor, J.

Burge, J. H.

J. H. Burge, “Application of computer-generated holograms for interferometric measurement of large aspheric optics,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 258–269 (1995).
[CrossRef]

Cai, L. Z.

C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L. Z. Cai, Y.-R. Wang, “Determination of global phase shifts between interferograms using energy minimum algorithm,” Appl. Opt. 42, 6414–6519 (2003).
[CrossRef]

Cheng, Y.-Y.

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

Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

Chung, P. S.

C. Jin, N. Bao, P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Creath, K.

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

Froing, P.

Goldberg, K. A.

Goodman, J. W.

J. W. Goodman, R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Guo, C.-S.

C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L. Z. Cai, Y.-R. Wang, “Determination of global phase shifts between interferograms using energy minimum algorithm,” Appl. Opt. 42, 6414–6519 (2003).
[CrossRef]

C.-S. Guo, L. Zhang, H.-T. Wang, J. Liao, Y. Y. Zhu, “Phase-shifting error and its elimination in phase-shifting digital holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

Gusev, M. E.

Hartmann, H.-J.

He, J.-L.

C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L. Z. Cai, Y.-R. Wang, “Determination of global phase shifts between interferograms using energy minimum algorithm,” Appl. Opt. 42, 6414–6519 (2003).
[CrossRef]

Itoh, M.

Jin, C.

C. Jin, N. Bao, P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Juptner, W.

Kato, J.

I. Yamaguchi, J. Kato, S. Ohta, J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177–6186 (2001).
[CrossRef]

I. Yamaguchi, S. Ohta, J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
[CrossRef]

King, B.

S. Lai, B. King, M. A. Neifeld, “Wavefront reconstruction by means of phase-shifting digital in-line holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

Lai, S.

S. Lai, B. King, M. A. Neifeld, “Wavefront reconstruction by means of phase-shifting digital in-line holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

Lawrence, R. W.

J. W. Goodman, R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Lee, W. H.

Liao, J.

C.-S. Guo, L. Zhang, H.-T. Wang, J. Liao, Y. Y. Zhu, “Phase-shifting error and its elimination in phase-shifting digital holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

Lindlein, N.

Mizuno, J.

Neifeld, M. A.

S. Lai, B. King, M. A. Neifeld, “Wavefront reconstruction by means of phase-shifting digital in-line holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

North-Morris, M. B.

O’Neill, P. K.

Ohta, S.

I. Yamaguchi, J. Kato, S. Ohta, J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177–6186 (2001).
[CrossRef]

I. Yamaguchi, S. Ohta, J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
[CrossRef]

Osten, W.

C. Wagner, W. Osten, S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Pedrini, C.

Pomarico, J.

Rong, Z.-Y.

C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L. Z. Cai, Y.-R. Wang, “Determination of global phase shifts between interferograms using energy minimum algorithm,” Appl. Opt. 42, 6414–6519 (2003).
[CrossRef]

Schnars, U.

Schwider, J.

Seebacher, S.

C. Wagner, W. Osten, S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Tiziani, H. J.

VanDelden, J.

Wagner, C.

C. Wagner, W. Osten, S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Wang, H.-T.

C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L. Z. Cai, Y.-R. Wang, “Determination of global phase shifts between interferograms using energy minimum algorithm,” Appl. Opt. 42, 6414–6519 (2003).
[CrossRef]

C.-S. Guo, L. Zhang, H.-T. Wang, J. Liao, Y. Y. Zhu, “Phase-shifting error and its elimination in phase-shifting digital holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

Wang, Y.-R.

C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L. Z. Cai, Y.-R. Wang, “Determination of global phase shifts between interferograms using energy minimum algorithm,” Appl. Opt. 42, 6414–6519 (2003).
[CrossRef]

Wyant, J. C.

Yamaguchi, I.

Yatagai, T.

Yoshikawa, N.

Zhang, L.

C.-S. Guo, L. Zhang, H.-T. Wang, J. Liao, Y. Y. Zhu, “Phase-shifting error and its elimination in phase-shifting digital holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

Zhang, T.

Zhu, Y. Y.

C.-S. Guo, L. Zhang, H.-T. Wang, J. Liao, Y. Y. Zhu, “Phase-shifting error and its elimination in phase-shifting digital holography,” Opt. Lett. 27, 1–3 (2002).
[CrossRef]

Appl. Opt. (13)

U. Schnars, W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

J. Pomarico, U. Schnars, H.-J. Hartmann, W. Juptner, “Digital recording and numerical reconstruction of holograms: a new method for displaying light in flight,” Appl. Opt. 34, 8095–8099 (1995).
[CrossRef] [PubMed]

C. Pedrini, P. Froing, H. J. Tiziani, M. E. Gusev, “Pulsed digital holography for high-speed contouring that uses a two-wavelength method,” Appl. Opt. 38, 3460–3467 (1999).
[CrossRef]

I. Yamaguchi, J. Kato, S. Ohta, J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177–6186 (2001).
[CrossRef]

M. B. North-Morris, J. VanDelden, J. C. Wyant, “Phase-shifting birefringent scatterplate interferometer,” Appl. Opt. 41, 668–677 (2002).
[CrossRef] [PubMed]

J. C. Wyant, V. P. Bennett, “Using computer generated holograms to test aspheric wave fronts,” Appl. Opt. 9, 2833–2839 (1972).
[CrossRef]

J. C. Wyant, P. K. O’Neill, “Computer generated hologram: null lens test of aspheric wave fronts,” Appl. Opt. 13, 2762–2765 (1974).
[CrossRef] [PubMed]

Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

N. Lindlein, “Analysis of the disturbing diffraction orders of computer-generated holograms used for testing optical aspheres,” Appl. Opt. 40, 2698–2708 (2001).
[CrossRef]

M. Beyerlein, N. Lindlein, J. Schwider, “Dual-wave-front computer-generated holograms for quasi-absolute testing of aspheres,” Appl. Opt. 41, 2440–2447 (2002).
[CrossRef] [PubMed]

W. H. Lee, “Sampled Fourier-transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
[CrossRef] [PubMed]

K. A. Goldberg, J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886–2894 (2001).
[CrossRef]

C.-S. Guo, Z.-Y. Rong, J.-L. He, H.-T. Wang, L. Z. Cai, Y.-R. Wang, “Determination of global phase shifts between interferograms using energy minimum algorithm,” Appl. Opt. 42, 6414–6519 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

J. W. Goodman, R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Opt. Acta (1)

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

Opt. Commun. (1)

S. Lai, B. King, M. A. Neifeld, “Wavefront reconstruction by means of phase-shifting digital in-line holography,” Opt. Commun. 173, 155–160 (2000).
[CrossRef]

Opt. Eng. (2)

C. Wagner, W. Osten, S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

C. Jin, N. Bao, P. S. Chung, “Application of a novel phase-shift method using a computer-controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Opt. Lasers Eng. (1)

I. Yamaguchi, S. Ohta, J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
[CrossRef]

Opt. Lett. (4)

Other (1)

J. H. Burge, “Application of computer-generated holograms for interferometric measurement of large aspheric optics,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 258–269 (1995).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Experimental arrangement for in-line digital holography with CGHPS device. BS1, BS2, optical splitters; M1, M2, mirrors; LCD, liquid crystal display; CCD, camera; F, spatial filter; O, object; PC, computer.

Fig. 2
Fig. 2

(a) Structure of a typical cell in Lee’s phase-detour CGH; (b), (c), (d) show the changes of the cell of the CGH shown in (a) in order to introduce a phase shift of π/2, π, and 3π/2 rad, respectively, into the reconstructed wave. δx and δy are the size of one cell; m and n are the orders of the cell in the x and the y direction, respectively.

Fig. 3
Fig. 3

Enlarged drawings of the CGHs of a conical wave front with phase shifts of (a) 0 rad, (b) π/2 rad, (c) π rad, (d) 3π/2 rad, respectively.

Fig. 4
Fig. 4

Photos of the interferograms between a plane wave and four conical waves produced by the CGH patterns with phase shift of (a) 0 rad, (b) π/2 rad, (c) π rad, (d) 3π/2 rad, respectively.

Fig. 5
Fig. 5

Measured phase-shifting values of the CGHPS device versus display duration of the CGH displayed on the LCD.

Fig. 6
Fig. 6

(a) Photo of object O (a traditional Chinese character meaning “body”) used in the experiment. (b) One of four digital holograms recorded on the CCD camera. (c) Image directly reconstructed by the hologram shown in (b). (d) Final reconstructed image after the phase-shifting technique is used, in which the influence of the conjugate and autocorrelation parts is eliminated.

Fig. 7
Fig. 7

Typical configuration of the interferometric null testing of aspheres with a CGH null corrector.

Equations (3)

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

A expiφ=A cosφ+iA sinφ.
Uhx, y=14URI1x, y, 0-I3x, y, π+iI2x, y, π2-I4x, y, 3π2.
Uox, y, zo=expi πλzox2+y2Uhx, y.

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