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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC, 2005).
    [CrossRef]
  2. P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26(13), 2504-2506 (1987), http://ao.osa.org/abstract.cfm?URI=ao-26-13-2504.
    [CrossRef] [PubMed]
  3. 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]
  4. Y. Surrel, "Phase stepping: a new self-calibrating algorithm," Appl. Opt. 32(19), 3598-3600 (1993), http://ao.osa.org/abstract.cfm?URI=ao-32-19-3598.
    [CrossRef] [PubMed]
  5. 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]
  6. J. L. Marroquin, M. Servin, and R. R. Vera, "Adaptive quadrature filters for multiple phase-stepping images," Opt. Lett. 23(4), 238-240 (1998), http://ol.osa.org/abstract.cfm?URI=ol-23-4-238.
    [CrossRef]
  7. K. G. Larkin, "A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns," Opt. Express 9(5), 236-253 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-5-236.
    [CrossRef] [PubMed]
  8. Z. Y. Wang and B. T. Han, "Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms," Opt. Lett. 29(14), 1671-1673 (2004), http://ol.osa.org/abstract.cfm?URI=ol-29-14-1671.
    [CrossRef] [PubMed]
  9. C. J. Morgan, "Least-squares estimation in phase-measurement interferometry," Opt. Lett. 7(8), 368 (1982), http://www.opticsinfobase.org/abstract.cfm?id=59997.
    [CrossRef] [PubMed]
  10. 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]
  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)

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]

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. 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]

2004 (1)

Z. Y. Wang and B. T. Han, "Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms," Opt. Lett. 29(14), 1671-1673 (2004), http://ol.osa.org/abstract.cfm?URI=ol-29-14-1671.
[CrossRef] [PubMed]

2001 (1)

K. G. Larkin, "A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns," Opt. Express 9(5), 236-253 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-5-236.
[CrossRef] [PubMed]

1998 (1)

J. L. Marroquin, M. Servin, and R. R. Vera, "Adaptive quadrature filters for multiple phase-stepping images," Opt. Lett. 23(4), 238-240 (1998), http://ol.osa.org/abstract.cfm?URI=ol-23-4-238.
[CrossRef]

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)

Y. Surrel, "Phase stepping: a new self-calibrating algorithm," Appl. Opt. 32(19), 3598-3600 (1993), http://ao.osa.org/abstract.cfm?URI=ao-32-19-3598.
[CrossRef] [PubMed]

1987 (1)

P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26(13), 2504-2506 (1987), http://ao.osa.org/abstract.cfm?URI=ao-26-13-2504.
[CrossRef] [PubMed]

1982 (1)

C. J. Morgan, "Least-squares estimation in phase-measurement interferometry," Opt. Lett. 7(8), 368 (1982), http://www.opticsinfobase.org/abstract.cfm?id=59997.
[CrossRef] [PubMed]

Eiju, T.

P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26(13), 2504-2506 (1987), http://ao.osa.org/abstract.cfm?URI=ao-26-13-2504.
[CrossRef] [PubMed]

Estrada, J.

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]

Estrada, J. C.

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]

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]

Han, B. T.

Z. Y. Wang and B. T. Han, "Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms," Opt. Lett. 29(14), 1671-1673 (2004), http://ol.osa.org/abstract.cfm?URI=ol-29-14-1671.
[CrossRef] [PubMed]

Hariharan, P.

P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26(13), 2504-2506 (1987), http://ao.osa.org/abstract.cfm?URI=ao-26-13-2504.
[CrossRef] [PubMed]

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.

K. G. Larkin, "A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns," Opt. Express 9(5), 236-253 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-5-236.
[CrossRef] [PubMed]

Marroquin, J. L.

J. L. Marroquin, M. Servin, and R. R. Vera, "Adaptive quadrature filters for multiple phase-stepping images," Opt. Lett. 23(4), 238-240 (1998), http://ol.osa.org/abstract.cfm?URI=ol-23-4-238.
[CrossRef]

Morgan, C. J.

C. J. Morgan, "Least-squares estimation in phase-measurement interferometry," Opt. Lett. 7(8), 368 (1982), http://www.opticsinfobase.org/abstract.cfm?id=59997.
[CrossRef] [PubMed]

Mosino, J. F.

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]

Oreb, B. F.

P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26(13), 2504-2506 (1987), http://ao.osa.org/abstract.cfm?URI=ao-26-13-2504.
[CrossRef] [PubMed]

Quiroga, J.

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]

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.

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]

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]

J. L. Marroquin, M. Servin, and R. R. Vera, "Adaptive quadrature filters for multiple phase-stepping images," Opt. Lett. 23(4), 238-240 (1998), http://ol.osa.org/abstract.cfm?URI=ol-23-4-238.
[CrossRef]

Surrel, Y.

Y. Surrel, "Phase stepping: a new self-calibrating algorithm," Appl. Opt. 32(19), 3598-3600 (1993), http://ao.osa.org/abstract.cfm?URI=ao-32-19-3598.
[CrossRef] [PubMed]

Vera, R. R.

J. L. Marroquin, M. Servin, and R. R. Vera, "Adaptive quadrature filters for multiple phase-stepping images," Opt. Lett. 23(4), 238-240 (1998), http://ol.osa.org/abstract.cfm?URI=ol-23-4-238.
[CrossRef]

Wang, Z. Y.

Z. Y. Wang and B. T. Han, "Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms," Opt. Lett. 29(14), 1671-1673 (2004), http://ol.osa.org/abstract.cfm?URI=ol-29-14-1671.
[CrossRef] [PubMed]

Appl. Opt. (2)

P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26(13), 2504-2506 (1987), http://ao.osa.org/abstract.cfm?URI=ao-26-13-2504.
[CrossRef] [PubMed]

Y. Surrel, "Phase stepping: a new self-calibrating algorithm," Appl. Opt. 32(19), 3598-3600 (1993), http://ao.osa.org/abstract.cfm?URI=ao-32-19-3598.
[CrossRef] [PubMed]

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)

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]

K. G. Larkin, "A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns," Opt. Express 9(5), 236-253 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-5-236.
[CrossRef] [PubMed]

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]

Opt. Lett. (4)

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]

Z. Y. Wang and B. T. Han, "Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms," Opt. Lett. 29(14), 1671-1673 (2004), http://ol.osa.org/abstract.cfm?URI=ol-29-14-1671.
[CrossRef] [PubMed]

C. J. Morgan, "Least-squares estimation in phase-measurement interferometry," Opt. Lett. 7(8), 368 (1982), http://www.opticsinfobase.org/abstract.cfm?id=59997.
[CrossRef] [PubMed]

J. L. Marroquin, M. Servin, and R. R. Vera, "Adaptive quadrature filters for multiple phase-stepping images," Opt. Lett. 23(4), 238-240 (1998), http://ol.osa.org/abstract.cfm?URI=ol-23-4-238.
[CrossRef]

Other (1)

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

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

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 ) ] ] ,

Metrics