Abstract

Optical interferometers are very sensitive when environment perturbations affect its optical path. The wavefront under test is not static at all. In this paper, it is proposed a novel and robust phase-shifting demodulation method. This method estimates the interferogram’s phase-shifting locally, reducing detuning errors due to environment perturbations like vibrations and/or miscalibrations of the Phase-Shifting Interferometry setup. As we know, phase-shifting demodulation methods assume that the wavefront under test is static and there is a global phase-shifting for all pixels. The demodulation method presented here is based on local weighted least-squares, letting each pixel have its own phase-shifting. This is a different and better approach, considering that all previous works assume a global phase-shifting for all pixels of interferograms. Seeing this method like a black box, it receives an interferogram sequence of at least 3 interferograms and returns the modulating phase or wavefront under test. Here it is not necessary to know the phase shifts between the interferograms. It does not assume a global phase-shifting for the interferograms, is robust to the movements of the wavefront under test and tolerates miscalibrations of the optical setup.

© 2013 Optical Society of America

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References

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2010 (2)

2009 (3)

2006 (2)

2004 (1)

1987 (1)

1985 (1)

1983 (1)

1982 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Cai, L. Z.

Cheng, Y.-Y.

Cywiak, M.

Dong, G. Y.

Eiju, T.

Elssner, K. E.

Estrada, J. C.

Gallagher, J. E.

Grzanna, J.

Han, B.

Hariharan, P.

Herriott, D. R.

Kimbrough, B. T.

B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt.45, 4554–4562 (2006).
[CrossRef] [PubMed]

Medina, O.

Meng, X. F.

Merkel, K.

Morgan, C. J.

Mosino, J. F.

Mosiño, J. F.

Oreb, B. F.

Quiroga, J. A.

Rosenfeld, D. P.

Schwider, J.

Servin, M.

Shen, X. X.

Spolaczyk, R.

Wang, Z.

White, A. D.

Wyant, J. C.

Xu, X. F.

Appl. Opt. (1)

B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt.45, 4554–4562 (2006).
[CrossRef] [PubMed]

Appl. Opt. (4)

Opt. Express (4)

Opt. Lett. (4)

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Figures (4)

Fig. 1
Fig. 1

From (a) to (b) we show the wrapped wavefronts generated with Eq. (11). From (e) to (h) we show the estimated wavefronts using the RAPS algorithm presented here. From (i) to (l) we show the estimated wavefronts using the AIA. Above each estimated wavefront we show the estimation error.

Fig. 2
Fig. 2

Electronic Speckle Pattern Interferometry (ESPI) setup. The object under test is a metal plate perturbed with a horn. The reference mirror has attached a piezoelectric transducer (PZT) as phase shifter.

Fig. 3
Fig. 3

From (a) to (d) we show the interferogram sequence. From (e) to (h) we show the estimated wavefront using the AIA, and from (i) to (l) we show the estimated wavefront using the RAPS.

Fig. 4
Fig. 4

Phase difference with respect to x (partial approximated derivative). The dynamic range of the difference is quantized to the gray levels 1, 102, 153, 203 and 255, for a better appreciation of the detuning error. (a) shows the phase difference corresponding to the AIA and (b) shows the difference corresponding to RAPS method.

Equations (14)

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I k ( x , y ) = a ( x , y ) + b ( x , y ) cos [ θ 0 ( x , y ) + ω 0 k ] ,
I k ( x , y ) = a ( x , y ) + b ( x , y ) cos [ θ 0 ( x , y ) + ( ω 0 k + η k ( x , y ) ) ] .
I k ( x , y ) = a ( x , y ) + b ( x , y ) cos [ θ 0 ( x , y ) + β k ( x , y ) ] ,
E [ a ( x , y ) , f ( x , y ) ] = k = 0 K 1 [ a ( x , y ) + Re { f ( x , y ) e i β k ( x , y ) } I k ( x , y ) ] 2 .
θ ^ ( x , y ) = angle [ f ^ ( x , y ) ] ,
( K c k ( x , y ) s k ( x , y ) c k ( x , y ) c k ( x , y ) 2 c k ( x , y ) s k ( x , y ) s k ( x , y ) c k ( x , y ) s k ( x , y ) s k ( x , y ) 2 ) ( a ^ ( x , y ) ϕ ^ ( x , y ) ψ ^ ( x , y ) ) = ( I k ( x , y ) I k ( x , y ) c k ( x , y ) I k ( x , y ) s k ( x , y ) ) .
E [ a ( x , y ) , g k ( x , y ) ] = m = 0 M 1 n = 0 N 1 [ { a ( m , n ) + Re { g k ( m , n ) e i θ 0 ( m , n ) } I k ( n , m ) } Least-squares error h ( x m , y n ) ] 2 .
β ^ k ( x , y ) = angle [ g ^ k ( x , y ) ] ,
( [ 1 s * h ] ( x , y ) [ ϕ * h ] ( x , y ) [ ψ * h ] ( x , y ) [ ϕ * h ] ( x , y ) [ ϕ * h ] 2 ( x , y ) [ ϕ ψ * h ] ( x , y ) [ ψ * h ] ( x , y ) [ ϕ ψ * h ] ( x , y ) [ ψ * h ] 2 ( x , y ) ) ( a ^ ( x , y ) c ^ k ( x , y ) s ^ k ( x , y ) ) = ( [ I k * h ] ( x , y ) [ I k ϕ * h ] ( x , y ) [ I k ψ * h ] ( x , y ) ) ,
ε = x , y | θ ^ 0 + ( x , y ) θ ^ 0 ( x , y ) | ,
θ k ( x , y ) = A cos ( 2 π 256 x ) cos ( 2 π 256 y ) cos ( π 17 k )
ε = 1 M × N x = 0 M 1 y = 0 N 1 [ θ k ( x , y ) θ ^ k ( x , y ) ] 2 ,
θ = arctan ( I 0 I 2 I 1 I 3 ) .
θ ( x , y , t ) = θ ( x , y , t 0 ) + θ ( x , y , t 0 ) t ( t t 0 ) .

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