Abstract

Deflectometry is a well-known method to characterize pure phase objects by measuring the deformation of fringes. In principle, the retrieved magnitude is the partial derivative of the phase along the coordinate orthogonal to the fringes. In order to recover the phase it is necessary to know the derivatives in two orthogonal directions, which is usually achieved by rotating 90° the original fringes and acquiring a new deformed pattern. This “time-multiplexed” two-dimensional deflectometry is a time-consuming operation if the goal is to characterize phase objects in real time. In the present paper we propose a kind of two-dimensional deflectometry that allows acquisition of fringe patterns in two orthogonal directions in a single frame. The proposed procedure utilizes a two-dimensional (“additive”) fringe pattern that allows the application of Takeda’s method to each coordinate independently. The advantage of the method (with respect to the traditional one) is that it simplifies the setup and reduces the acquisition time. Validation experiments are presented.

© 2013 Optical Society of America

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References

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  3. G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
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2013

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A 710, 78–81 (2013).
[CrossRef]

2011

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50, 040501 (2011).
[CrossRef]

2007

R. Legarda-Saenz, “Robust wavefront estimation using multiple directional derivatives in moire deflectometry,” Opt. Lasers Eng. 45, 915–921 (2007).
[CrossRef]

2006

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[CrossRef]

2003

2002

G. García-Torales, M. Strojnik, and G. Paez, “Risley prisms to control wavefront tilt and displacement in a vectorial shearing interferometer,” Appl. Opt. 41, 1380–1384 (2002).
[CrossRef]

H. Canabal and J. Alonso, “Automatic wavefront measurement technique using a computer display and a charge-coupled device camera,” Opt. Eng. 41, 822–826 (2002).
[CrossRef]

2000

1999

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of Moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

1998

1996

1994

1991

1986

1984

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

1983

W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3899–3901 (1983).

1982

Alonso, J.

H. Canabal and J. Alonso, “Automatic wavefront measurement technique using a computer display and a charge-coupled device camera,” Opt. Eng. 41, 822–826 (2002).
[CrossRef]

Bernabeu, E.

Berujon, S.

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A 710, 78–81 (2013).
[CrossRef]

Blumel, T.

Bone, D. J.

Burow, R.

Canabal, H.

H. Canabal and J. Alonso, “Automatic wavefront measurement technique using a computer display and a charge-coupled device camera,” Opt. Eng. 41, 822–826 (2002).
[CrossRef]

H. Canabal, J. A. Quiroga, and E. Bernabeu, “Improved phase-shifting method for automatic processing of moiré deflectograms,” Appl. Opt. 37, 6227–6233 (1998).
[CrossRef]

Crespo, D.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of Moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

David, C.

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A 710, 78–81 (2013).
[CrossRef]

Elssner, K.-E.

Espinosa-Romero, A.

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50, 040501 (2011).
[CrossRef]

Ferrari, J. A.

Flores, J. L.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[CrossRef]

García-Torales, G.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[CrossRef]

G. García-Torales, M. Strojnik, and G. Paez, “Risley prisms to control wavefront tilt and displacement in a vectorial shearing interferometer,” Appl. Opt. 41, 1380–1384 (2002).
[CrossRef]

Ghiglia, D.

D. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

González-Álvarez, A.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[CrossRef]

Ina, H.

Kobayashi, S.

Kreis, T.

Legarda-Saenz, R.

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50, 040501 (2011).
[CrossRef]

R. Legarda-Saenz, “Robust wavefront estimation using multiple directional derivatives in moire deflectometry,” Opt. Lasers Eng. 45, 915–921 (2007).
[CrossRef]

Lindlein, N.

Macy, W. W.

W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3899–3901 (1983).

Marroquin, J. L.

Paez, G.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[CrossRef]

G. García-Torales, M. Strojnik, and G. Paez, “Risley prisms to control wavefront tilt and displacement in a vectorial shearing interferometer,” Appl. Opt. 41, 1380–1384 (2002).
[CrossRef]

Pape, I.

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A 710, 78–81 (2013).
[CrossRef]

Perciante, C. D.

Pfund, J.

Pritt, M. D.

D. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

Quiroga, J. A.

Rutishauser, S.

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A 710, 78–81 (2013).
[CrossRef]

Sawhney, K.

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A 710, 78–81 (2013).
[CrossRef]

Schwider, J.

Servin, M.

Servín, M.

Ströbel, B.

Strojnik, M.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[CrossRef]

G. García-Torales, M. Strojnik, and G. Paez, “Risley prisms to control wavefront tilt and displacement in a vectorial shearing interferometer,” Appl. Opt. 41, 1380–1384 (2002).
[CrossRef]

Surrel, Y.

Takeda, M.

Villa, J.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[CrossRef]

J. Villa, J. A. Quiroga, and M. Servín, “Improved regularized phase-tracking technique for the processing of squared-grating deflectograms,” Appl. Opt. 39, 502–508 (2000).
[CrossRef]

Wang, H.

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A 710, 78–81 (2013).
[CrossRef]

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nucl. Instrum. Methods Phys. Res. A

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A 710, 78–81 (2013).
[CrossRef]

Opt. Commun.

G. García-Torales, G. Paez, M. Strojnik, J. Villa, J. L. Flores, and A. González-Álvarez, “Experimental intensity patterns obtained from a 2D shearing interferometer with adaptable sensitivity,” Opt. Commun. 257, 16–26 (2006).
[CrossRef]

Opt. Eng.

H. Canabal and J. Alonso, “Automatic wavefront measurement technique using a computer display and a charge-coupled device camera,” Opt. Eng. 41, 822–826 (2002).
[CrossRef]

R. Legarda-Saenz and A. Espinosa-Romero, “Wavefront reconstruction using multiple directional derivatives and Fourier transform,” Opt. Eng. 50, 040501 (2011).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of Moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Opt. Lasers Eng.

R. Legarda-Saenz, “Robust wavefront estimation using multiple directional derivatives in moire deflectometry,” Opt. Lasers Eng. 45, 915–921 (2007).
[CrossRef]

Opt. Lett.

Other

D. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).

D. Malacara, ed., Optical Shop Testing (Wiley, 2007).

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Figures (6)

Fig. 1.
Fig. 1.

One-dimensional deflectometry, P is a printed fringe pattern and T is a pure phase object.

Fig. 2.
Fig. 2.

(a) I(x,y)=I0[1+cos(2πx/p)cos(2πy/p)] and (b) I(x,y)=I0[2+cos(2πx/p)+cos(2πy/p)].

Fig. 3.
Fig. 3.

Progressive lens in its holder in front of the sinusoidal pattern. The circle indicates the effective lens size utilized for reconstructing the wavefront.

Fig. 4.
Fig. 4.

Frequency spectrum of the acquired pattern.

Fig. 5.
Fig. 5.

(a) Wrapped values of W(x,y)/x, (b) wrapped values of W(x,y)/y, (c) unwrapped map of W(x,y)/x, and (d) unwrapped map of W(x,y)/y.

Fig. 6.
Fig. 6.

Wavefront W(x,y) obtained by numerical integration.

Equations (9)

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

I(x,y)=I0[1+cos(2πxp)],
I(x,y)=I0[1+cos(2πxp+2πdpW(x,y)x)].
I(x,y)=I0[1+cos(2πxp+2πdpW(x,y)x)×cos(2πyp+2πdpW(x,y)y)].
I(x,y)=I0[2+cos(2πxp)+cos(2πyp)].
I(x,y)=I0[2+cos(2πxp+2πdpW(x,y)x)+cos(2πyp+2πdpW(x,y)y)].
I˜(ωx,ωy)=(I0/2){4δ(ωx)δ(ωy)+[Cx(ωxω0,ωy)+Cx*(ωx+ω0,ωy)]+[Cy(ωx,ωyω0)+Cy*(ωx,ωy+ω0)]},
2f(x,y)=g(x,y),
f(x,y)=(Lπ)2k=1Ln=1Ldkn(n2+k2)sin(πkxL)sin(πnyL),
dkn=1L202L02Lg(x,y)sin(πkxL)sin(πnyL)dxdy.

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