Abstract

Phase evaluation methods based on the 2D spatial Fourier transform of a speckle interferogram with spatial carrier usually assume that the Fourier spectrum of the interferogram has a trimodal distribution, i. e. that the side lobes corresponding to the interferential terms do not overlap the other two spectral terms, which are related to the intensity of the object and reference beams, respectively. Otherwise, part of the spectrum of the object beam is inside the inverse-transform window of the selected interference lobe and induces an error in the resultant phase map. We present a technique for the acquisition and processing of speckle interferogram sequences that separates the interference lobes from the other spectral terms when the aforementioned assumption does not apply and regardless of the temporal bandwidth of the phase signal. It requires the recording of a sequence of interferograms with spatial and temporal carriers, and their processing with a 3D Fourier transform. In the resultant 3D spectrum, the spatial and temporal carriers separate the conjugate interferential terms from each other and from the term related to the object beam. Experimental corroboration is provided through the measurement of the amplitude of surface acoustic waves in plates with a double-pulsed TV holography setup. The results obtained with the proposed method are compared to those obtained with the processing of individual interferograms with the regular spatial-carrier 2D Fourier transform method.

© 2010 OSA

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References

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  1. D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25(10), 1653–1660 (1986).
    [CrossRef] [PubMed]
  2. H. O. Saldner, N.-E. Molin, and K. A. Stetson, “Fourier-transform evaluation of phase data in spatially phase-biased TV holograms,” Appl. Opt. 35(2), 332–336 (1996).
    [CrossRef] [PubMed]
  3. C. Trillo, and A. F. Doval, “Spatiotemporal Fourier transform method for the measurement of narrowband ultrasonic surface acoustic waves with TV holography,” Proc. SPIE 6341, 63410M–1-6 (2006).
  4. H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier Fringe Analysis,” Opt. Lasers Eng. 46(6), 446–455 (2008).
    [CrossRef]
  5. K. Qian, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42(6), 1721–1724 (2003).
    [CrossRef]
  6. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, (Cambridge University Press, 1988), Chap.12.
  7. C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
    [CrossRef]
  8. J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, New York, 1992).
  9. Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33(11), 3709–3714 (1994).
    [CrossRef]

2008 (1)

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier Fringe Analysis,” Opt. Lasers Eng. 46(6), 446–455 (2008).
[CrossRef]

2003 (2)

K. Qian, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42(6), 1721–1724 (2003).
[CrossRef]

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

1996 (1)

1994 (1)

Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33(11), 3709–3714 (1994).
[CrossRef]

1986 (1)

Abdul-Rahman, H. S.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier Fringe Analysis,” Opt. Lasers Eng. 46(6), 446–455 (2008).
[CrossRef]

Abid, A.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier Fringe Analysis,” Opt. Lasers Eng. 46(6), 446–455 (2008).
[CrossRef]

Asundi, A. K.

K. Qian, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42(6), 1721–1724 (2003).
[CrossRef]

Bachor, H.-A.

Bone, D. J.

Burton, D. R.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier Fringe Analysis,” Opt. Lasers Eng. 46(6), 446–455 (2008).
[CrossRef]

Cernadas, D.

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

Dorrío, B. V.

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

Doval, A. F.

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

Fernández, J. L.

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

Fujisawa, M.

Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33(11), 3709–3714 (1994).
[CrossRef]

Gdeisat, M. A.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier Fringe Analysis,” Opt. Lasers Eng. 46(6), 446–455 (2008).
[CrossRef]

Lalor, M. J.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier Fringe Analysis,” Opt. Lasers Eng. 46(6), 446–455 (2008).
[CrossRef]

Lilley, F.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier Fringe Analysis,” Opt. Lasers Eng. 46(6), 446–455 (2008).
[CrossRef]

López, J. C.

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

López, O.

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

Molin, N.-E.

Morimoto, Y.

Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33(11), 3709–3714 (1994).
[CrossRef]

Pérez-Amor, M.

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

Qian, K.

K. Qian, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42(6), 1721–1724 (2003).
[CrossRef]

Saldner, H. O.

Sandeman, R. J.

Seah, H. S.

K. Qian, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42(6), 1721–1724 (2003).
[CrossRef]

Stetson, K. A.

Trillo, C.

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

Appl. Opt. (2)

Meas. Sci. Technol. (1)

C. Trillo, A. F. Doval, D. Cernadas, O. López, J. C. López, B. V. Dorrío, J. L. Fernández, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14(12), 2127–2134 (2003).
[CrossRef]

Opt. Eng. (2)

Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33(11), 3709–3714 (1994).
[CrossRef]

K. Qian, H. S. Seah, and A. K. Asundi, “Algorithm for directly retrieving the phase difference: a generalization,” Opt. Eng. 42(6), 1721–1724 (2003).
[CrossRef]

Opt. Lasers Eng. (1)

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier Fringe Analysis,” Opt. Lasers Eng. 46(6), 446–455 (2008).
[CrossRef]

Other (3)

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, New York, 1992).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, (Cambridge University Press, 1988), Chap.12.

C. Trillo, and A. F. Doval, “Spatiotemporal Fourier transform method for the measurement of narrowband ultrasonic surface acoustic waves with TV holography,” Proc. SPIE 6341, 63410M–1-6 (2006).

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Figures (9)

Fig. 1
Fig. 1

Summary of the 2DFTM method–subfigures (a) to (d)– and the 3DFTM method –subfigures (e) to (h). (a) Single interferogram. (b) Modulus of the 2D Fourier transform of the data and bandpass filter. (c) Filtered spectrum. (d) Complex-valued map containing the optical phase of the interferogram. (e) Sequence of interferograms. (f) Modulus of the 3D Fourier transform of the data. A 3D bandpass filter is also shown. (g) Filtered spectrum. (h) Sequence of complex-valued maps containing the optical phases of the interferograms.

Fig. 2
Fig. 2

Modulus of the 2D Fourier transform of a speckle interferogram with spatial carrier.

Fig. 3
Fig. 3

(a) Experimental set-up; (b) phase modulation of the reference beam; (c) laser pulses; (d) out-of plane displacement u3 of the surface due to the ultrasonic wave; (e) and (f) several interferograms from sequences Ib and Ia respectively.

Fig. 4
Fig. 4

(a) Modulus of the 3DFTM of a sequence of eight interferograms. Black and white represent zero and the maximum modulus respectively. (b) Average modulus calculated in the region delimited by the dotted rectangle in map n'=2. (c) Result obtained for a sequence of sixteen interferograms.

Fig. 5
Fig. 5

Summary of the 3DFTM. Only 5 maps out of 16 are used to illustrate the different stages. (a) Sequences of temporally phase-shifted interferograms. (b) Modulus of the 3D Fourier transform (FT (3D)) of the sequences and the 3D filter used in the processing. (c) Sequence of optical phases Φ a and Φ b , obtained as the argument of the inverse 3D Fourier transform (FT−1 (3D)) of the filtered data (d) Sequence of optical phase-change maps calculated as indicated in Eq. (11). * stands for complex conjugation.

Fig. 6
Fig. 6

Calculation of a complex optical phase-change map Δ Φ ^ n from an optical phase-change map ΔΦ n . (a) ΔΦ23 obtained by applying Eq. (11) to the pair of complex-valued maps (I' a 23,I' b 23). (b) Modulus of the 2D Fourier transform of (a). The filter size is 25×25 pixel. An inverse 2D Fourier transform yields a complex optical phase-change map Δ Φ ^ n . (c) Acoustic amplitude u 3m, n =[λ/(4π)]mod( Δ Φ ^ n ). The average value of u 3m, n is computed in the region A1 defined by the dotted rectangle. Black and white represent, respectively (a) –π and π (b) minimum and maximum values of the modulus (c) minimum and maximum values of u 3m, n .

Fig. 7
Fig. 7

ΔΦ20 obtained with (a) 2DFTM and (b) 3DFTM. ΔΦ24 obtained with (c) 2DFTM and (d) 3DFTM.

Fig. 8
Fig. 8

Acoustic amplitude u 3m, n obtained with (i) 2DFTM and (ii) 3DFTM. Columns (a), (b) and (c) correspond to maps at n=17, n=20 and n=24, respectively.

Fig. 9
Fig. 9

Average of the acoustic amplitude u 3m, n in region A1 (see Fig. 6), calculated for a sequence of 16 maps with the 2DFTM (in black) and 3DFTM (in grey).

Equations (11)

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I n ( x ) = I o , n + I r , n + 2 I o , n I r , n cos ( ψ p , n + ϕ o , n ϕ r , n + 2 π f c x x )
f c t = α 2 π ( t n + 1 t n ) = α 2 π Δ t
I ( x , t ) = I o + I r + 2 I o I r cos ( ϕ + 2 π f c x x + 2 π f c t t )
I = I o + I r + 2 I o I r { 1 2 exp [ i ( ϕ + 2 π f c x x + 2 π f c t t ) ] + C * }
F ( I ) = F ( I o ) + F ( I r ) + F ( I o I r ) { F [ exp ( i ϕ ) ] δ ( f x f c x , f t f c t ) + F ( C * ) }
I ' ( x , t ) = I o I r exp ( i ϕ ) exp [ i ( 2 π f c x x + 2 π f c t t ) ]
Δ Φ ( x , t ) = Φ j Φ i = ϕ o , j ϕ o , i = arg [ I ' j ( I ' i ) ]
I ( x 1 p , x 2 q , t n ) = I ( x 10 + p Δ x 1 , x 20 + q Δ x 2 , t 0 + n Δ t )
f p ' = p ' P Δ x 1 , f q ' = q ' Q Δ x 2 , f n ' = n ' N Δ t
n ' = ± N Δ t f c t = ± N 4
Δ Φ n = Φ n b Φ n a = ϕ o , n b ϕ o , n a = arg [ I ' n b ( I ' n a ) ]

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