Abstract

We use a Mach-Zehnder interferometer with two afocal systems to generate the interference between an image and a diffracted copy of a test beam. The interference data is inserted in an iterative algorithm for phase retrieval. We carry out a numerical study of the performance of this wavefront reconstruction technique with simulated data and we present also experimental results obtained using several lenses as test objects.

© 2007 Optical Society of America

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References

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  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical testing. (Marcel Dekker, New York, 1998).
  2. W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. 59, 940-650 (1947).
    [CrossRef]
  3. J. C. Wyant, "Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems," Appl. Opt. 14, 2622-2626 (1975).
    [CrossRef] [PubMed]
  4. J. C. Wyant and F. D. Smith, "Interferometer for measuring power distribution of ophthalmic lenses," Appl. Opt. 14, 1607-1612 (1975).
    [CrossRef] [PubMed]
  5. P. Liang, J. Ding, Z. Jin, C. -S. Guo, and H. -T. Wang, "Two-dimensional wave-front reconstruction from lateral shearing interferograms," Opt. Express 14, 625-634 (2006).
    [CrossRef] [PubMed]
  6. A. Dubra, C. Paterson, and C. Dainty, "Study of the tear topography dynamics using a lateral shearing interferometer," Opt. Express 12, 6278-6288 (2004).
    [CrossRef] [PubMed]
  7. P. Hariharan and D. Sen, "Radial shearing interferometer," J. Sci. Instrum. 11, 428-432 (1961).
    [CrossRef]
  8. D. S. Brown, "Radial shear interferometry" J. Sci. Instrum. 39, 71-72(1962).
    [CrossRef]
  9. W. H. Steel, "A radial shear interferometer for testing microscope objectives," J. Sci. Instrum. 42, 102-104 (1965).
    [CrossRef]
  10. D. Li, H. Chen, and Z. Chen, "Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer," Opt. Eng. 41, 1893-1898 (2002).
    [CrossRef]
  11. M. Li, P. Wang, X. Li, H. Yang, and H. Chen, "Algorithm for near-field reconstruction based on radial-shearing interferometry," Opt. Lett. 30, 492-494 (2005).
    [CrossRef] [PubMed]
  12. C. -Y. Chung, K. -C. Cho, C. -C. Chang, C. -H. Lin, W. -C. Yen, and S. -J. Chen, "Adaptive-optics system with liquid-crystal phase-shift interferometer," Appl. Opt. 45, 3409-3414 (2006).
    [CrossRef] [PubMed]
  13. E. López-Lago and R. de la Fuente, "Wavefront sensing by diffracted beam interferometry," J. Opt. A: Pure Appl. Opt. 4, 299-302 (2002).
    [CrossRef]
  14. D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. of Mod. Opt. 44, 407-414 (1997).
    [CrossRef]
  15. H. Z. Hu, "Polarization heterodyne interferometry using a simple rotating analyzer: 1. theory and error analysis," Appl. Opt. 22, 2052-2056 (1983).
    [CrossRef] [PubMed]
  16. E. M Frins, W. Dultz, and J. A. Ferrari, "Polarization shifting method for step interferometry," Pure Appl. Opt 7, 53-60 (1998).
    [CrossRef]
  17. K. Creath, ‘Phase measurement interferometry techniques’ in Progress in Optics XXVI, E. Wolf Ed., 349-393 (Elsevier Science, 1988).
  18. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  19. R. A. Gonsalves, "Phase retrieval and diversity in adaptative optics," Opt. Eng. 21, 829-832 (1982).
  20. F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. Soc. Photo-Opt.Instrum. Eng. 976, 203-209 (1988).
  21. G. R. Brady and J. R. Fienup, "Nonlinear optimization algorithm for retrieving the full complex pupil function," Opt. Express 14, 474-486 (2006).
    [CrossRef] [PubMed]

2006 (3)

2005 (1)

2004 (1)

2002 (2)

D. Li, H. Chen, and Z. Chen, "Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer," Opt. Eng. 41, 1893-1898 (2002).
[CrossRef]

E. López-Lago and R. de la Fuente, "Wavefront sensing by diffracted beam interferometry," J. Opt. A: Pure Appl. Opt. 4, 299-302 (2002).
[CrossRef]

1998 (1)

E. M Frins, W. Dultz, and J. A. Ferrari, "Polarization shifting method for step interferometry," Pure Appl. Opt 7, 53-60 (1998).
[CrossRef]

1997 (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. of Mod. Opt. 44, 407-414 (1997).
[CrossRef]

1988 (1)

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. Soc. Photo-Opt.Instrum. Eng. 976, 203-209 (1988).

1983 (1)

1982 (2)

J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
[CrossRef] [PubMed]

R. A. Gonsalves, "Phase retrieval and diversity in adaptative optics," Opt. Eng. 21, 829-832 (1982).

1975 (2)

1965 (1)

W. H. Steel, "A radial shear interferometer for testing microscope objectives," J. Sci. Instrum. 42, 102-104 (1965).
[CrossRef]

1962 (1)

D. S. Brown, "Radial shear interferometry" J. Sci. Instrum. 39, 71-72(1962).
[CrossRef]

1961 (1)

P. Hariharan and D. Sen, "Radial shearing interferometer," J. Sci. Instrum. 11, 428-432 (1961).
[CrossRef]

1947 (1)

W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. 59, 940-650 (1947).
[CrossRef]

Bates, W. J.

W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. 59, 940-650 (1947).
[CrossRef]

Brady, G. R.

Brown, D. S.

D. S. Brown, "Radial shear interferometry" J. Sci. Instrum. 39, 71-72(1962).
[CrossRef]

Chang, C. -C.

Chen, H.

M. Li, P. Wang, X. Li, H. Yang, and H. Chen, "Algorithm for near-field reconstruction based on radial-shearing interferometry," Opt. Lett. 30, 492-494 (2005).
[CrossRef] [PubMed]

D. Li, H. Chen, and Z. Chen, "Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer," Opt. Eng. 41, 1893-1898 (2002).
[CrossRef]

Chen, S. -J.

Chen, Z.

D. Li, H. Chen, and Z. Chen, "Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer," Opt. Eng. 41, 1893-1898 (2002).
[CrossRef]

Cho, K. -C.

Chung, C. -Y.

Dainty, C.

de la Fuente, R.

E. López-Lago and R. de la Fuente, "Wavefront sensing by diffracted beam interferometry," J. Opt. A: Pure Appl. Opt. 4, 299-302 (2002).
[CrossRef]

Ding, J.

Dubra, A.

Dultz, W.

E. M Frins, W. Dultz, and J. A. Ferrari, "Polarization shifting method for step interferometry," Pure Appl. Opt 7, 53-60 (1998).
[CrossRef]

Ferrari, J. A.

E. M Frins, W. Dultz, and J. A. Ferrari, "Polarization shifting method for step interferometry," Pure Appl. Opt 7, 53-60 (1998).
[CrossRef]

Fienup, J. R.

Frins, E. M

E. M Frins, W. Dultz, and J. A. Ferrari, "Polarization shifting method for step interferometry," Pure Appl. Opt 7, 53-60 (1998).
[CrossRef]

Gonsalves, R. A.

R. A. Gonsalves, "Phase retrieval and diversity in adaptative optics," Opt. Eng. 21, 829-832 (1982).

Guo, C. -S.

Hariharan, P.

P. Hariharan and D. Sen, "Radial shearing interferometer," J. Sci. Instrum. 11, 428-432 (1961).
[CrossRef]

Hu, H. Z.

Jin, Z.

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. of Mod. Opt. 44, 407-414 (1997).
[CrossRef]

Li, D.

D. Li, H. Chen, and Z. Chen, "Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer," Opt. Eng. 41, 1893-1898 (2002).
[CrossRef]

Li, M.

Li, X.

Liang, P.

Lin, C. -H.

López-Lago, E.

E. López-Lago and R. de la Fuente, "Wavefront sensing by diffracted beam interferometry," J. Opt. A: Pure Appl. Opt. 4, 299-302 (2002).
[CrossRef]

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. of Mod. Opt. 44, 407-414 (1997).
[CrossRef]

Paterson, C.

Roddier, C.

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. Soc. Photo-Opt.Instrum. Eng. 976, 203-209 (1988).

Roddier, F.

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. Soc. Photo-Opt.Instrum. Eng. 976, 203-209 (1988).

Roddier, N.

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. Soc. Photo-Opt.Instrum. Eng. 976, 203-209 (1988).

Sen, D.

P. Hariharan and D. Sen, "Radial shearing interferometer," J. Sci. Instrum. 11, 428-432 (1961).
[CrossRef]

Smith, F. D.

Steel, W. H.

W. H. Steel, "A radial shear interferometer for testing microscope objectives," J. Sci. Instrum. 42, 102-104 (1965).
[CrossRef]

Wang, H. -T.

Wang, P.

Wyant, J. C.

Yang, H.

Yen, W. -C.

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. of Mod. Opt. 44, 407-414 (1997).
[CrossRef]

Appl. Opt. (5)

Instrum. Eng. (1)

F. Roddier, C. Roddier, and N. Roddier, "Curvature sensing: a new wavefront sensing method," Proc. Soc. Photo-Opt.Instrum. Eng. 976, 203-209 (1988).

J. of Mod. Opt. (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, "Computation considerations and fast algorithms for calculating the diffraction integral," J. of Mod. Opt. 44, 407-414 (1997).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

E. López-Lago and R. de la Fuente, "Wavefront sensing by diffracted beam interferometry," J. Opt. A: Pure Appl. Opt. 4, 299-302 (2002).
[CrossRef]

J. Sci. Instrum. (3)

P. Hariharan and D. Sen, "Radial shearing interferometer," J. Sci. Instrum. 11, 428-432 (1961).
[CrossRef]

D. S. Brown, "Radial shear interferometry" J. Sci. Instrum. 39, 71-72(1962).
[CrossRef]

W. H. Steel, "A radial shear interferometer for testing microscope objectives," J. Sci. Instrum. 42, 102-104 (1965).
[CrossRef]

Opt. Eng. (2)

D. Li, H. Chen, and Z. Chen, "Simple algorithms of wavefront reconstruction for cyclic radial shearing interferometer," Opt. Eng. 41, 1893-1898 (2002).
[CrossRef]

R. A. Gonsalves, "Phase retrieval and diversity in adaptative optics," Opt. Eng. 21, 829-832 (1982).

Opt. Express (3)

Opt. Lett. (1)

Proc. Phys. Soc. (1)

W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. 59, 940-650 (1947).
[CrossRef]

Pure Appl. Opt (1)

E. M Frins, W. Dultz, and J. A. Ferrari, "Polarization shifting method for step interferometry," Pure Appl. Opt 7, 53-60 (1998).
[CrossRef]

Other (2)

K. Creath, ‘Phase measurement interferometry techniques’ in Progress in Optics XXVI, E. Wolf Ed., 349-393 (Elsevier Science, 1988).

D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical testing. (Marcel Dekker, New York, 1998).

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Figures (11)

Fig. 1.
Fig. 1.

Scheme of a diffracted beam interferometer.

Fig. 2.
Fig. 2.

Mach-Zehnder configuration for a diffracted beam interferometer.

Fig. 3.
Fig. 3.

Typical interferograms obtained with the DBI sensor presented in the text. Left, for gaussian illumination and, right, for uniform illumination.

Fig. 4.
Fig. 4.

Flow chart of the DBI algorithm.

Fig. 5.
Fig. 5.

Results obtained with the numerical examples explained in the text.

Fig. 6.
Fig. 6.

Same as Fig. 5 but with noisy signals.

Fig. 7.
Fig. 7.

Experimental set-up. L0, L1, L2, achromatic doublets; PL1, PL2, PL3 linear polarizers; BS1, BS2 beam splitters, M1, M2 mirrors.

Fig. 8.
Fig. 8.

Experimental results for a spherical lens.

Fig. 9.
Fig. 9.

Comparison between measurements performed in plan P1 and P2 (see text).

Fig. 10.
Fig. 10.

(a). Continuous line: experimental irradiance of beam 1 measured at its image plane (P1); dotted line: irradiance of beam 1 at its image plane calculated from its numerical irradiance at plane P2. (b). Continuous line: experimental irradiance of beam 2 measured at its image plane (P2); dotted line: irradiance of the beam 2 at its image plane calculated from its numerical irradiance at plane P1. (c). Numerical phase of beam 1 at its image plane obtained from the measurement at this plane (continuous line) and from the measurement at plane P2 (dotted line). (d). Numerical phase of beam 2 at its image plane obtained from the measurement at this plane (continuous line) and from the measurement at plane P1 (dotted line).

Fig. 11.
Fig. 11.

Results for the cylindrical lenses along the axes of the cylinder and along the orthogonal direction. Continuous line: measured irradiances, dotted line: numerical irradiances, dashed line: numerical phases.

Equations (8)

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u i ( x , y ) = f 0 f i u 0 ( f 0 f i x , f 0 f i y ) ( i = 1 , 2 )
u 1 d ( x , y ) = C u 2 ( f 2 f 1 x ´ , f 2 f 1 y ´ ) × exp { ik [ ( x x ´ ) 2 + ( y y ´ ) 2 ] 2 d } dx ´ dy ´
I ( x , y ) = I 2 ( x , y ) + I 1 d ( x , y ) + 2 I 2 ( x , y ) I 1 d ( x , y ) cos ( Δ ϕ )
u 2 ( ie ) = I 2 exp ( 2 )
u 1 d ( e ) ( x , y ) = IFFT { FFT [ u 2 ( ie ) ( , ) ] exp [ iπλd ( η x 2 + η y 2 ) ] }
u 2 ( ne ) = u 1 d ( ie ) I f ( u 1 d ( ie ) 2 + α ) .
u 2 ( fe ) = βu 2 ( ie ) + ( 1 β ) u 2 ( ne )
Q = 1 N i , j = 1 N ( I d ( x i , y j ) u d ( e ) ( x i , y j ) 2 ) 2

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