Abstract

In Phase Stepping Interferometry (PSI) an interferogram sequence having a known, and constant phase shift between the interferograms is required. Here we take the case where this constant phase shift is unknown and the only assumption is that the interferograms do have a temporal carrier. To recover the modulating phase from the interferograms, we propose a self-tuning phase-shifting algorithm. Our algorithm estimates the temporal frequency first, and then this knowledge is used to estimate the interesting modulating phase. There are several well known iterative schemes published before, but our approach has the unique advantage of being very fast. Our new temporal carrier, and phase estimator is capable of obtaining a very good approximation of their temporal carrier in a single iteration. Numerical experiments are given to show the performance of this simple yet powerful self-tuning phase shifting algorithm.

© 2010 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, 2nd ed. (CRC, 2005).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  11. J. Estrada, M. Servin, and J. Quiroga, “Easy and straightforward construction of wideband phase-shifting algorithms for interferometry,” Opt. Lett. 34(4), 413–415 (2009). http://ol.osa.org/abstract.cfm?URI=ol-34-4-413.
    [Crossref] [PubMed]

2009 (3)

2004 (1)

2001 (1)

1998 (1)

1995 (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995). http://link.aip.org/link/?JOE/34/183/1.
[Crossref]

1993 (1)

1987 (1)

1982 (1)

Eiju, T.

Estrada, J.

Estrada, J. C.

M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase?shifting algorithms,” Opt. Express 17(19), 16,423–16,428 (2009). http://www.opticsexpress.org/abstract.cfm?URI=oe-17-19-16423.
[Crossref]

J. F. Mosino, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009). http://www.opticsexpress.org/abstract.cfm?URI=oe-17-7-5618.
[Crossref] [PubMed]

Han, B. T.

Hariharan, P.

Kim, S.-W.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995). http://link.aip.org/link/?JOE/34/183/1.
[Crossref]

Kong, I.-B.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995). http://link.aip.org/link/?JOE/34/183/1.
[Crossref]

Larkin, K. G.

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC, 2005).
[Crossref]

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC, 2005).
[Crossref]

Marroquin, J. L.

Morgan, C. J.

Mosino, J. F.

Oreb, B. F.

Quiroga, J.

Quiroga, J. A.

M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase?shifting algorithms,” Opt. Express 17(19), 16,423–16,428 (2009). http://www.opticsexpress.org/abstract.cfm?URI=oe-17-19-16423.
[Crossref]

J. F. Mosino, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009). http://www.opticsexpress.org/abstract.cfm?URI=oe-17-7-5618.
[Crossref] [PubMed]

Servin, M.

Surrel, Y.

Vera, R. R.

Wang, Z. Y.

Appl. Opt. (2)

Opt. Eng. (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995). http://link.aip.org/link/?JOE/34/183/1.
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Other (1)

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC, 2005).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1.

A sample of an interferogram sequence used for testing the algorithm.

Table 1.
Table 1.

Numerical results.

Fig. 2.
Fig. 2.

Phase maps obtained from image sequence of Fig. 1. The left one is obtained using the method as is shown in Ref [8], while the right one is obtained using the method as depicted in this paper.

Equations (23)

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I t ( x , y ) = a ( x , y ) + b ( x , y ) cos ( ϕ ( x , y ) + ω 0 t ) ,
h ( t ) = [ 2 δ ( t ) δ ( t 1 ) δ ( t + 1 ) ] cos ( ω 0 2 ) + i [ δ ( t 1 ) + δ ( t + 1 ) ] sin ( ω 0 2 ) ,
ϕ ̂ = arctan [ Im { [ h * I ] ( 0 ) } Re { [ h * I ] ( 0 ) } ] = arctan [ I 1 I 1 2 I 0 I 1 I 1 tan ( ω 0 2 ) ] ,
H ( ω ) = [ h ( t ) ] = 4 sin ( ω 2 ) sin ( ω ω 0 2 ) ,
H 1 ( ω ) = sin ( ω ) ,
H 2 ( ω ) = 1 cos ( ω ω 0 ) .
H ( ω ) = H 1 ( ω ) H 2 ( ω )
= sin ( ω ) [ 1 cos ( ω ω 0 ) ] .
h ( t ) = [ 2 δ ( t ) δ ( t 2 ) δ ( t + 2 ) ] sin ( ω 0 ) / 2
+ i [ 2 δ ( t 1 ) 2 δ ( t + 1 ) ] / 2 i [ δ ( t 2 ) δ ( t + 2 ) ] cos ( ω 0 ) / 2 .
I ̂ t = h ( t ) * I t
= [ 2 I t I t 2 I t + 2 ] sin ( ω 0 ) / 2 + i ( 2 I t 1 2 I t + 1 ) / 2 i ( I t 2 I t + 2 ) cos ( ω 0 ) / 2 ] ,
ϕ ̂ = arctan [ Im { ( I ̂ 0 ) } Re { ( I ̂ 0 ) } ]
= arctan [ 2 I 1 2 I 1 [ I 2 I 2 ] cos ( ω 0 ) [ 2 I 0 I 2 I 2 ] sin ( ω 0 ) ] ,
ϕ ̂ 0 = ϕ + ε 0 , and
ϕ ̂ 1 = ϕ + ω 0 + ε 1 ,
ε 0 H ( ω 0 ) H ( ω 0 ) sin ( 2 ϕ ) , ε 1 H ( ω 0 ) H ( ω 0 ) sin [ 2 ( ϕ + ω 0 ) ] .
ϕ ̂ 1 ϕ ̂ 0 ω 0 2 H ( ω 0 ) H ( ω 0 ) [ cos ( 2 ϕ ω 0 ) sin ( ω 0 ) ] .
ω ̂ 0 = 1 MN x N y M W [ ϕ ̂ 1 ( x , y ) ϕ ̂ 0 ( x , y ) ] ,
I t ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + ω 0 t ] + η t ( x , y ) ,
ϕ ( x , y ) = 4 π N x + 4 π M y ,
a ( x , y ) = 5 · e ( x 128 ) 2 + ( y 128 ) 2 60 2 , and b ( x , y ) = e ( x 128 ) 2 + ( y 128 ) 2 95 2 ,
Error = 1 MN x N y M W [ ϕ ̂ ( x , y ) W [ ϕ ( x , y ) ] ] ,

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