Abstract

A rapid algorithm for phase and amplitude reconstruction from a single spatial-carrier interferogram is proposed by bringing a phase-shifting mechanism into reconstruction of a carrier-frequency interferogram. The algorithm reconstructs phase through directly obtaining and integrating its real-value derivatives, avoiding a phase unwrapping process. The proposed method is rapid and easy to implement and is made insensitive to the profile of the interferogram boundaries by choosing a suitable integrating path. Moreover, the algorithm can also be used to reconstruct the amplitude of the object wave expediently without retrieving the phase profile in advance. The feasibility of this algorithm is demonstrated by both numerical simulation and experiment.

© 2008 Optical Society of America

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References

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    [CrossRef]
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  15. G. Páez and M. Strojnik, “Fringe analysis and phase reconstruction from modulated intensity patterns,” Opt. Lett. 22, 1669-1671 (1997).
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2007

2000

J. Yanez-Mendiola, S. Manuel, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-396(2000).
[CrossRef]

1999

G. Páez and M. Strojnik, “Phase-shifted interferometry with out phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc, Am. A 16, 475-480 (1999).
[CrossRef]

J. A. Ferrari, E. M. Frins, D. Perciante, and A. Dubra, “Robust one-beam interferometer with phase-delay control,” Opt. Lett. 24, 1272-1274 (1999).
[CrossRef]

1997

1994

1992

1986

1985

1984

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

D. Newman and J. C. Dainty, “Detection of gratings hidden by diffusers using intensity interferometry,” J. Opt. Soc. Am. A 1, 403-411 (1984).
[CrossRef]

1983

1982

Abdul-Rahman, H. S.

Bachor, H.-A.

Bone, D. J.

Burton, D. R.

Creath, K.

Dainty, J. C.

Dubra, A.

Ferrari, J. A.

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogarm with linear carrier,” Opt. Commun. 271, 59-64 (2007)
[CrossRef]

J. A. Ferrari, E. M. Frins, D. Perciante, and A. Dubra, “Robust one-beam interferometer with phase-delay control,” Opt. Lett. 24, 1272-1274 (1999).
[CrossRef]

Frins, E. M.

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogarm with linear carrier,” Opt. Commun. 271, 59-64 (2007)
[CrossRef]

J. A. Ferrari, E. M. Frins, D. Perciante, and A. Dubra, “Robust one-beam interferometer with phase-delay control,” Opt. Lett. 24, 1272-1274 (1999).
[CrossRef]

Gdeisat, M. A.

Ghiglia, D. C.

Ina, H.

Itoh, K.

Kobayashi, S.

Lalor, M. J.

Lilley, F.

Malacara-Hernandez, D.

J. Yanez-Mendiola, S. Manuel, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-396(2000).
[CrossRef]

Manuel, S.

J. Yanez-Mendiola, S. Manuel, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-396(2000).
[CrossRef]

Mertz, L.

Moore, C. J.

Newman, D.

Páez, G.

G. Páez and M. Strojnik, “Phase-shifted interferometry with out phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc, Am. A 16, 475-480 (1999).
[CrossRef]

G. Páez and M. Strojnik, “Fringe analysis and phase reconstruction from modulated intensity patterns,” Opt. Lett. 22, 1669-1671 (1997).
[CrossRef]

Perciante, D.

Romero, L. A.

Sandeman, R. J.

Shi, Y. Q.

Stetson, K. A.

Strojnik, M.

G. Páez and M. Strojnik, “Phase-shifted interferometry with out phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc, Am. A 16, 475-480 (1999).
[CrossRef]

G. Páez and M. Strojnik, “Fringe analysis and phase reconstruction from modulated intensity patterns,” Opt. Lett. 22, 1669-1671 (1997).
[CrossRef]

Takeda, M.

Womack, K. H.

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

Yanez-Mendiola, J.

J. Yanez-Mendiola, S. Manuel, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-396(2000).
[CrossRef]

Appl. Opt.

J. Opt. Soc, Am. A

G. Páez and M. Strojnik, “Phase-shifted interferometry with out phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc, Am. A 16, 475-480 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

J. Yanez-Mendiola, S. Manuel, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-396(2000).
[CrossRef]

J. A. Ferrari and E. M. Frins, “Multiple phase-shifted interferograms obtained from a single interferogarm with linear carrier,” Opt. Commun. 271, 59-64 (2007)
[CrossRef]

Opt. Eng.

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

Opt. Express

Opt. Lett.

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Figures (5)

Fig. 1
Fig. 1

Simulated object wave and the corresponding interferogram. (a) Simulated wavefront of the object wave, (b) simulated amplitude of the object wave, (c) simulated interferogram with a linear carrier-frequency in the x direction.

Fig. 2
Fig. 2

Phase and amplitude reconstruction. (a) Derivative of the simulated phase in the x direction φ x ,(b) derivative of the simulated phase in the y direction φ y ,(c) reconstructed phase φ r ,(d) reconstructed amplitude | O r | .

Fig. 3
Fig. 3

Error in reconstruction. (a) Error in phase reconstruction, (b) error in amplitude reconstruction. Δ φ = | φ r ( x , y ) φ ( x , y )| , Δ A = | A r ( x , y ) A ( x , y )| .

Fig. 4
Fig. 4

Experimentally measured carrier-frequency interferogram of a spherical lens.

Fig. 5
Fig. 5

Results of reconstruction. (a) Derivative of the phase in the x direction φ x , (b)  derivative of the phase in the y direction φ y , (c)  reconstructed phase φ, (d)  reconstructed intensity distribution of the object wave.

Equations (19)

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I ( x , y ) = I ¯ 0 ( x , y ) { 1 + V ( x , y ) cos [ ϕ ( x , y ) + ω 0 x ] } ,
I ( x 2 , y ) = I ¯ 0 ( x , y ) × { 1 + V ( x , y ) × cos [ ϕ ( x 2 , y ) + ω 0 ( x 2 ) ] } ,
I ( x 1 , y ) = I ¯ 0 ( x , y ) × { 1 + V ( x , y ) × cos [ ϕ ( x 1 , y ) + ω 0 ( x 1 ) ] } ,
I ( x , y ) = I ¯ 0 ( x , y ) { 1 + V ( x , y ) cos [ ϕ ( x , y ) + ω 0 x ] } ,
I ( x + 1 , y ) = I ¯ 0 ( x , y ) × { 1 + V ( x , y ) × cos [ ϕ ( x + 1 , y ) + ω 0 ( x + 1 ) ] } ,
I ( x + 2 , y ) = I ¯ 0 ( x , y ) × { 1 + V ( x , y ) × cos [ ϕ ( x + 2 , y ) + ω 0 ( x + 2 ) ] } ,
I 1 + I 1 2 I 0 = 2 I ¯ 0 ( x , y ) V ( x , y ) [ cos ( ϕ x + ω 0 ) 1 ] × cos [ ϕ ( x , y ) + ω 0 x ] , I 1 I 1 = 2 I ¯ 0 ( x , y ) V ( x , y ) sin ( ϕ x + ω 0 ) × sin [ ϕ ( x , y ) + ω 0 x ] ,
tan [ ϕ ( x , y ) + ω 0 x ] = tan ( ω 0 + ϕ x 2 ) I 1 I 1 I 1 + I 1 2 I 0 .
tan [ ϕ ( x , y ) + ω 0 x ] = tan ( ω 0 + ϕ x ) I 2 I 2 I 2 + I 2 2 I 0 .
ϕ x ( x , y ) = 2 arctan [ 1 2 ( I 2 I - 2 ) ( I 1 + I 1 2 I 0 ) ( I 1 I 1 ) ( I 2 + I 2 2 I 0 ) ] ω 0 .
ϕ y ( x , y ) = 2 tan ( ϕ x + ω 0 2 ) ( I 1 I 1 y I 1 y I 1 + I 0 y I 1 I 0 I 1 y + I 1 y I 0 I 1 I 0 y ) ( I 1 + I 1 2 I 0 ) 2 + tan 2 ( ϕ x + ω 0 2 ) ( I 1 I 1 ) 2 .
ϕ r ( x , y ) = ϕ ( x 0 , y 0 ) + y 0 y ϕ y ( x 0 , y ) d y + x 0 x ϕ x ( x , y ) d y ,
I 1 = | O | 2 + | R ' | 2 + O R ' * exp [ i ( ϕ x + ω 0 ) ] + O * R ' exp [ i ( ϕ x + ω 0 ) ] ,
I 0 = | O | 2 + | R ' | 2 + O R ' * + O * R ' ,
I 1 = | O | 2 + | R ' | 2 + O R ' * exp [ i ( ϕ x + ω 0 ) ] + O * R ' × exp [ i ( ϕ x + ω 0 ) ] ,
O R ' * = i ( I 0 = I 1 ) { exp [ i ( ω 0 + ϕ x ) ] 1 } ( I 1 I 0 ) { 1 exp [ i ( ω 0 + ϕ x ) ] } 2 sin ( 2 ω 0 ) 4 sin ( ω 0 ) .
| O | = | ( I 0 I 1 ) { exp [ i ( ω 0 + ϕ x ) ] 1 } ( I 1 I 0 ) { 1 exp [ i ( ω 0 + ϕ x ) ] } | | 2 sin ( 2 ω 0 ) 4 sin ( ω 0 ) | | R ' | .
O = { 0.45 0.02 [ ( x / 100 3 ) 2 + ( y / 100 3 ) 2 ] } × exp { i π 2 [ 3 ( x / 100 3 ) 2 + ( y / 100 ) 2 ] } R ' , R ' = 0.4 exp ( i ω 0 x ) .
I ( x , y ) = | O ( x , y ) | 2 + | R ( x , y ) | 2 + O * ( x , y ) R ( x , y ) + O ( x , y ) R * ( x , y ) ,

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