Abstract

We present a simple and novel algorithm for the phase extraction from a single interferogram based on the spatial processing of interference patterns. This new evaluation procedure is suitable for application in environments where the presence of vibrations impedes the use of a classical phase-shifting interferometry scheme with multiple exposures. The algorithm does not require the introduction of a linear carrier as required in Fourier transform techniques. The addition of a carrier can be a significant drawback, e.g. in the case of wavefronts with strong aberrations where the minimum required linear carrier is not even resolved by the detector. The basic idea relies on the spatial application of a temporal phase-shifting algorithm and an iterative correction process to obtain an accurate reconstruction of the wavefront. The validity and performance of the proposed method is shown with numerical and experimental results.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  27. X. Colonna de Lega and P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35, 5115-5121 (1996).
    [CrossRef] [PubMed]
  28. M. Küchel, “Some progress in phase measurement techniques,” Fringe 1997, Proceedings of the International Workshop on Automatic Processing of Fringe Patterns, (Akademie Verlag, 1997), pp. 27-44.
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2005

2004

2001

1998

1997

1996

1995

M. Melozzi, L. Pezzati, and A. Mazzoni, “Vibration insensitive interferometer for on-line measurements,” Appl. Opt. 34, 5595-5601 (1995).
[CrossRef] [PubMed]

M. Pirga and M. Kujawinska, “Two directional spatial carrier phase shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

A. J. Moore and F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

1992

R. Jozwicki, M. Kujawinska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422-433 (1992).
[CrossRef]

1991

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

1986

1984

O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9, 59-61 (1984).
[CrossRef] [PubMed]

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 2, 361-365 (1984).

1982

1974

1972

1969

1966

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mésures,” Metrologia 2, 13-23 (1966).
[CrossRef]

Appl. Opt.

H. J. Okoomian, “A two-beam polarization technique to measure optical phase,” Appl. Opt. 8, 2363-2365 (1969).
[CrossRef]

J. Bruning, J. Gallagher, D. Rosenfeld, D. White, D. Brangaccio, and D. Herriot, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693-2703 (1974).
[CrossRef] [PubMed]

H. Schreiber and J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321-5324 (1997).
[CrossRef] [PubMed]

A. L. Weijers, H. van Brug, and H. J. Frankena, “Polarization phase stepping with a savart element,” Appl. Opt. 37, 5150-5155 (1998).
[CrossRef]

J. Garcia-Marquez, D. Malacara-Hernandez, and M. Servin, “Analysis of interferograms with a spatial radial carrier or closed fringes and its holographic analogy,” Appl. Opt. 37, 7977-7982 (1998).
[CrossRef]

E. Robin and V. Valle, “Phase demodulation from a single fringe pattern based on a correlation technique,” Appl. Opt. 43, 4355-4361 (2004).
[CrossRef] [PubMed]

E. Robin, V. Valle, and F. Brémand, “Phase demodulation method from a single fringe patter based on correlation with a polynomial form,” Appl. Opt. 44, 7261-7269 (2005).
[CrossRef] [PubMed]

M. Melozzi, L. Pezzati, and A. Mazzoni, “Vibration insensitive interferometer for on-line measurements,” Appl. Opt. 34, 5595-5601 (1995).
[CrossRef] [PubMed]

X. Colonna de Lega and P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35, 5115-5121 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Metrologia

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mésures,” Metrologia 2, 13-23 (1966).
[CrossRef]

Opt. Eng.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 2, 361-365 (1984).

M. Pirga and M. Kujawinska, “Two directional spatial carrier phase shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

R. Jozwicki, M. Kujawinska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422-433 (1992).
[CrossRef]

Opt. Laser Technol.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase step interferometry: a simplified approach,” Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Opt. Lasers Eng.

A. J. Moore and F. Mendoza-Santoyo, “Phase demodulation in the space domain without a fringe carrier,” Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

Opt. Lett.

Other

R. Doloca and R. Tutsch, “Random phase shift interferometer,” Fringe 2005, Proceedings of the International Workshop on Automatic Processing of Fringe Patterns (Springer-Verlag, 2005), pp. 166-174.

D. Malacara, “Phase shifting interferometers” in Optical Shop Testing, 2nd ed. (Wiley, 1998), Chap. 14.

M. Küchel, “Some progress in phase measurement techniques,” Fringe 1997, Proceedings of the International Workshop on Automatic Processing of Fringe Patterns, (Akademie Verlag, 1997), pp. 27-44.

C. Koliopoulus, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, 1981).

D. W. Robinson, “Spatial phase measurement methods,” in Interferogram Analysis, D.Robinson and G.T.Reid, eds. (Institute of Physics, University of Reading, 1993), pp. 141-193.

J. E. Millerd and N. J. Brock, “Methods and apparatus for splitting, imaging, and measuring wavefronts in interferometry,” U. S. Patent 6,304,330 (16 October 2001).

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 1st ed. (Marcel Dekker, 1998), Chap. 8.

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Figures (6)

Fig. 1
Fig. 1

Phase shifting with a set of adjacent pixels of the detector array. Choosing the pixels with the correct separations, we obtain the appropriate phase shifts in the wavefront in order to apply a temporal phase-shifting algorithm. The Δ Φ n phase shift caused by a deviation of n p pixels forms the central position.

Fig. 2
Fig. 2

(a) Two-dimensional local pixel window to determine the direction of the maximal intensity gradient and apply the phase-shifting algorithm. (b) Experimental interferogram showing the local fringe distribution. The arrow indicates the direction where the local gradient of the interference pattern is maximal and, thus, where the phase-shifting algorithm is to be applied.

Fig. 3
Fig. 3

Two-dimensional set of pixels for application of the algorithm. The dark bands represent the local interference fringe distribution in the working window centered at pixel ( x 0 , y 0 )

Fig. 4
Fig. 4

(a) Computer-simulated interference pattern. The signal presents variable fringe density and several points where the gradient of the phase reaches a null value. (b) The rms value of the difference between the calculated and the original profiles amounts to approximately 0.014 λ .

Fig. 5
Fig. 5

Experimental results. (a) Measurement of the aberrations of a lens measured in double-pass configuration with a Fisba μ Phase HR2 Twyman–Green interferometer. (b) Calculated phase profile (modulo- 2 π ) from the previous interferogram applying the proposed procedure. At the edge of the phase map it is possible to see the influence of pupil diffraction effects of the interferometer on the calculated phase.

Fig. 6
Fig. 6

Difference between the wavefronts calculated with the Schwider–Hariharan five-frame algorithm and the proposed one. The peak-to-valley difference amounts to 0.064 λ mostly due to the spurious intensity variations that remain static for the temporal phase shifting (five-frame algorithm) and, thus, do not influence the resulting phase calculated through this method.

Equations (14)

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I ( x , y ) = I 0 ( x , y ) · { 1 + V ( x , y ) · cos [ Φ ( x , y ) ] } ,
I 0 ( x 0 , y 0 ) I 0 ( x + i n , y + j m ) , | i n | < M , | j m | < N , V ( x 0 , y 0 ) V ( x + i n , y + j m ) , | i n | < M , | j m | < N .
Φ ( x , y ) = Φ ( x 0 , y 0 ) + Φ x ( x 0 , y 0 ) · ( x x 0 ) + 1 2 2 Φ x 2 ( x 0 , y 0 ) · ( x x 0 ) 2 + 1 3 ! 3 Φ x 3 ( x 0 , y 0 ) · ( x x 0 ) 3 +
Φ ( x , y ) Φ ( x 0 , y 0 ) + Φ x ( x 0 , y 0 ) · ( x x 0 ) .
I ( x , y ) = A + B cos [ Φ ( x , y ) ] A + B cos [ Φ ( x 0 , y 0 ) + Φ x ( x 0 , y 0 ) · ( x x 0 ) ] ,
Φ x ( x 0 , y 0 ) · ( x 1 x 0 ) = 3 · p · Φ x ( x 0 , y 0 ) = 3 α , Φ x ( x 0 , y 0 ) · ( x 2 x 0 ) = p · Φ x ( x 0 , y 0 ) = α , Φ x ( x 0 , y 0 ) · ( x 3 x 0 ) = p · Φ x ( x 0 , y 0 ) = α , Φ x ( x 0 , y 0 ) · ( x 4 x 0 ) = 3 · p · Φ x ( x 0 , y 0 ) = 3 α ,
I 1 A + B cos [ Φ ( x 0 , y 0 ) 3 α ] , I 2 A + B cos [ Φ ( x 0 , y 0 ) α ] , I 3 A + B cos [ Φ ( x 0 , y 0 ) + α ] , I 4 A + B cos [ Φ ( x 0 , y 0 ) + 3 α ] .
tan 2 α = 3 ( I 2 I 3 ) ( I 1 I 4 ) ( I 1 I 4 ) + ( I 2 I 3 ) ,
Φ ( x 0 , y 0 ) = tan 1 { { | [ 3 ( I 2 I 3 ) ( I 1 I 4 ) ] · [ ( I 2 I 3 ) + ( I 1 I 4 ) ] | } 1 2 [ ( I 2 + I 3 ) ( I 1 + I 4 ) ] } .
max { ϕ } ( x 0 , y 0 ) = max { I } ( x 0 , y 0 ) B sin [ ϕ ( x 0 , y 0 ) ] ,
I e ( x , y ) = I 0 e { 1 + V e cos [ Φ e ( x , y ) ] } , I ( x , y ) = I 0 { 1 + V cos [ Φ ( x , y ) ] } , δ = I I e , ε = Φ Φ e ,
δ = I I e = I 0 e V e ( cos Φ cos Φ e ) = 2 I 0 e V e sin ( Φ + Φ e 2 ) sin ( Φ Φ e 2 ) 2 I 0 e V e sin ( Φ e ) sin ( ε 2 ) ,
I 0 I 0 e , V V e , Φ + Φ e = ε + Φ e + Φ e 2 Φ e ( ε Φ e ) .
ε 2 sin 1 ( δ 2 I 0 e V e sin Φ e ) ,

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