Abstract

We show, for the first time to our knowledge, how wavelength-scanning interferometry can be used to measure depth-resolved displacement fields through semitransparent scattering surfaces. Temporal sequences of speckle interferograms are recorded while the wavelength of the laser is tuned at a constant rate. Fourier transformation of the resultant three-dimensional (3-D) intensity distribution along the time axis reconstructs the scattering potential within the medium, and changes in the 3-D phase distribution measured between two separate scans provide the out-of-plane component of the 3-D displacement field. The principle of the technique is explained in detail and illustrated with a proof-of-principle experiment involving two independently tilted semitransparent scattering surfaces. Results are validated by standard two-beam electronic speckle pattern interferometry.

© 2005 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1959).
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    [CrossRef]
  6. A. F. Fercher, W. Drexler, C. K. Hitzenberger, T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
    [CrossRef]
  7. A. F. Fercher, C. K. Hitzenberger, G. Kamp, S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43–48 (1995).
    [CrossRef]
  8. F. Lexer, C. K. Hitzenberger, A. F. Fercher, M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36, 6548–6553 (1997).
    [CrossRef]
  9. G. Gülker, A. Kraft, “Low-coherence ESPI in the investigation of ancient terracotta warriors,” in Speckle Metrology 2003, K. Gastinger, O. J. Løkberg, S. Winther, eds., Proc. SPIE4933, 53–58 (2003).
    [CrossRef]
  10. K. Gastinger, S. Winther, K. D. Hinsch, “Low-coherence speckle interferometer (LCSI) for characterization of adhesion in adhesive-bonded joints,” in Speckle Metrology 2003, K. Gastinger, O. J. Løkberg, S. Winther, eds., Proc. SPIE4933, 59–65 (2003).
    [CrossRef]
  11. P. D. Ruiz, Y. Zhou, J. M. Huntley, R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A Pure Appl. Opt. 6, 679–683 (2004).
    [CrossRef]

2004 (1)

P. D. Ruiz, Y. Zhou, J. M. Huntley, R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A Pure Appl. Opt. 6, 679–683 (2004).
[CrossRef]

2003 (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

2000 (1)

1997 (2)

1995 (1)

A. F. Fercher, C. K. Hitzenberger, G. Kamp, S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43–48 (1995).
[CrossRef]

1994 (1)

1992 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1959).

de Groot, P.

Dresel, T.

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

El-Zaiat, S. Y.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43–48 (1995).
[CrossRef]

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

F. Lexer, C. K. Hitzenberger, A. F. Fercher, M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36, 6548–6553 (1997).
[CrossRef]

A. F. Fercher, C. K. Hitzenberger, G. Kamp, S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43–48 (1995).
[CrossRef]

Gastinger, K.

K. Gastinger, S. Winther, K. D. Hinsch, “Low-coherence speckle interferometer (LCSI) for characterization of adhesion in adhesive-bonded joints,” in Speckle Metrology 2003, K. Gastinger, O. J. Løkberg, S. Winther, eds., Proc. SPIE4933, 59–65 (2003).
[CrossRef]

Gülker, G.

G. Gülker, A. Kraft, “Low-coherence ESPI in the investigation of ancient terracotta warriors,” in Speckle Metrology 2003, K. Gastinger, O. J. Løkberg, S. Winther, eds., Proc. SPIE4933, 53–58 (2003).
[CrossRef]

Häusler, G.

Hinsch, K. D.

K. Gastinger, S. Winther, K. D. Hinsch, “Low-coherence speckle interferometer (LCSI) for characterization of adhesion in adhesive-bonded joints,” in Speckle Metrology 2003, K. Gastinger, O. J. Løkberg, S. Winther, eds., Proc. SPIE4933, 59–65 (2003).
[CrossRef]

Hitzenberger, C. K.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

F. Lexer, C. K. Hitzenberger, A. F. Fercher, M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36, 6548–6553 (1997).
[CrossRef]

A. F. Fercher, C. K. Hitzenberger, G. Kamp, S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43–48 (1995).
[CrossRef]

Huntley, J. M.

P. D. Ruiz, Y. Zhou, J. M. Huntley, R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A Pure Appl. Opt. 6, 679–683 (2004).
[CrossRef]

Kamp, G.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43–48 (1995).
[CrossRef]

Kraft, A.

G. Gülker, A. Kraft, “Low-coherence ESPI in the investigation of ancient terracotta warriors,” in Speckle Metrology 2003, K. Gastinger, O. J. Løkberg, S. Winther, eds., Proc. SPIE4933, 53–58 (2003).
[CrossRef]

Kulhavy, M.

Kuwamura, S.

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

Lexer, F.

Ruiz, P. D.

P. D. Ruiz, Y. Zhou, J. M. Huntley, R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A Pure Appl. Opt. 6, 679–683 (2004).
[CrossRef]

Takeda, M.

Venzke, H.

Wildman, R. D.

P. D. Ruiz, Y. Zhou, J. M. Huntley, R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A Pure Appl. Opt. 6, 679–683 (2004).
[CrossRef]

Winther, S.

K. Gastinger, S. Winther, K. D. Hinsch, “Low-coherence speckle interferometer (LCSI) for characterization of adhesion in adhesive-bonded joints,” in Speckle Metrology 2003, K. Gastinger, O. J. Løkberg, S. Winther, eds., Proc. SPIE4933, 59–65 (2003).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1959).

Yamaguchi, I.

Yamamoto, H.

Zhou, Y.

P. D. Ruiz, Y. Zhou, J. M. Huntley, R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A Pure Appl. Opt. 6, 679–683 (2004).
[CrossRef]

Appl. Opt. (5)

J. Opt. A Pure Appl. Opt. (1)

P. D. Ruiz, Y. Zhou, J. M. Huntley, R. D. Wildman, “Depth-resolved whole-field displacement measurement using wavelength scanning interferometry,” J. Opt. A Pure Appl. Opt. 6, 679–683 (2004).
[CrossRef]

Opt. Commun. (1)

A. F. Fercher, C. K. Hitzenberger, G. Kamp, S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43–48 (1995).
[CrossRef]

Rep. Prog. Phys. (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. 66, 239–303 (2003).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1959).

G. Gülker, A. Kraft, “Low-coherence ESPI in the investigation of ancient terracotta warriors,” in Speckle Metrology 2003, K. Gastinger, O. J. Løkberg, S. Winther, eds., Proc. SPIE4933, 53–58 (2003).
[CrossRef]

K. Gastinger, S. Winther, K. D. Hinsch, “Low-coherence speckle interferometer (LCSI) for characterization of adhesion in adhesive-bonded joints,” in Speckle Metrology 2003, K. Gastinger, O. J. Løkberg, S. Winther, eds., Proc. SPIE4933, 59–65 (2003).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Layout for measuring depth-resolved out-of-plane displacements by WSI. Displacements w1(x, y) and w2(x, y) of scattering centers at surfaces S1 and S2 are measured between reference state a (solid lines) and loaded state b (dotted curves).

Fig. 2
Fig. 2

Schematic spectrum of a three-beam interference signal that is due to reference surface R and two test surfaces, S1 and S2.

Fig. 3
Fig. 3

Slice through the gauge volume in the WSI system, showing depth resolution δz, lateral resolution δx, and average displacement d of scatterers.

Fig. 4
Fig. 4

Wavelength-scanning speckle pattern interferometer. PC, personal computer; other abbreviations defined in text.

Fig. 5
Fig. 5

Experimental procedure for measuring displacement of surfaces S1 and S2 by using two-beam speckle pattern interferometry (2B) for validation purposes, and WSI. Subscripts a and b mean reference state and deformed state, respectively. See further description in the text.

Fig. 6
Fig. 6

(a) Intensity signal I3(x0, y0, t) recorded at a pixel (x0, y0) in the region of interest. (b) Average frequency spectrum of intensity signal I3(x, y, t) recorded at all the pixels from one column in the region of interest for x = 100 and y = 1, 2, …, 200. Peaks S1, S2, and S1S2 correspond to the interference signals for surfaces R and S1, R and S2, and S1 and S2, respectively.

Fig. 7
Fig. 7

Wrapped phase difference maps of (a) S1 and (b) S2 obtained by WSI, showing the tilt introduced for each surface. Black represents −π rad; white; +π rad.

Fig. 8
Fig. 8

Correlation fringes obtained for surfaces (a) S1 and (b) S2 by use of two-beam speckle pattern interferometry.

Fig. 9
Fig. 9

When there are two scattering slices, multiply scattered photons (dashed line) travel longer paths and therefore modulate the intensity signal with higher frequencies than direct photons (solid and dotted lines that indicate reflection at S2, S1, and R).

Equations (21)

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ϕ j ( x , y = ϕ 0 j ( x , y ) + 4 π λ z j ( x , y ) ,
ϕ j ( x , y , t ) = ϕ 0 j ( x , y ) + 4 π λ c z j 4 π z j Δ λ T λ c 2 t .
f j = 2 Δ λ T λ c 2 z j .
I ( x , y , t ) = I R ( x , y ) + I 1 ( x , y ) + I 2 ( x , y ) + 2 [ I R ( x , y ) I 1 ( x , y ) ] 1 / 2 cos [ ϕ 1 ( x , y , t ) ] + 2 [ I R ( x , y ) I 2 ( x , y ) ] 1 / 2 cos [ ϕ 2 ( x , y , t ) ] + 2 [ I 1 ( x , y ) I 2 ( x , y ) ] 1 / 2 cos [ ϕ 12 ( x , y , t ) ] ,
Ĩ ( x , y , f ) = [ I ( x , y , t ) n = δ ( t n Δ t ) ] W ( t ) × exp ( i 2 π f t ) d t ,
Ĩ ( x , y , f ) = W ( f ) { I R + I 1 + I 2 Δ t n = δ ( f n Δ t ) + I R I 1 Δ t exp [ ± i ϕ 1 ( x , y , 0 ) ] n = δ ( f ± f 1 n Δ t ) + I R I 2 Δ t exp [ ± i ϕ 2 ( x , y , 0 ) ] × n = δ ( f ± f 2 n Δ t ) + I R I 2 Δ t × exp [ ± i ϕ 12 ( x , y , 0 ) ] × n = δ ( f ± f 12 n Δ t ) } ,
ϕ 1 ( x , y , 0 ) = ϕ 01 ( x , y ) + 4 π z 1 λ c , ϕ 2 ( x , y , 0 ) = ϕ 02 ( x , y ) + 4 π z 2 λ c , ϕ 21 ( x , y , 0 ) = ϕ 2 ( x , y , 0 ) ϕ 1 ( x , y , 0 ) ,
Ĩ ( x , y , f ) = I R + I 1 + I 2 Δ t W ( f ) + I R I 1 Δ t × exp [ ± i ϕ 1 ( x , y , 0 ) ] W ( f ± f 1 ) + I R I 2 Δ t × exp [ ± i ϕ 2 ( x , y , 0 ) ] W ( f ± f 2 ) + I 1 I 2 Δ t × exp [ ± i ϕ 12 ( x , y , 0 ) ] W ( f ± f 12 ) .
Ĩ ( x , y , f ) = 1 Δ t ( I R + j = 1 N s I j ) W ( f ) + 1 Δ t j = 1 N s I R I j × exp [ ± i ϕ j ( x , y , 0 ) ] W ( f ± f j ) + 1 Δ t j = 1 N s 1 k = j + 1 N s I j I k exp [ ± i ϕ j k ( x , y , 0 ) ] × W ( f ± f j k ) .
Ĩ ( x , y , f 1 ) = I R + I 1 + I 2 Δ t W ( f 1 ) + I R I 1 Δ t × exp [ ± i ϕ 1 ( x , y , 0 ) ] W ( 0 ) + I R I 2 Δ t × exp [ ± i ϕ 2 ( x , y , 0 ) ] W ( f 1 ± f 2 ) + I 1 I 2 Δ t × exp [ ± i ϕ 12 ( x , y , 0 ) ] W ( f 1 ± f 12 ) .
Ĩ ( x , y , f 1 ) I R I 1 Δ t exp [ ± i ϕ 1 ( x , y , 0 ) ] W ( 0 ) .
Δ ϕ 1 ( x , y ) = 4 π w 1 ( x , y ) / λ c .
Δ ϕ 1 ( a , b ) = tan 1 [ N ( b ) D ( a ) D ( b ) N ( a ) D ( b ) D ( a ) + N ( b ) N ( a ) ] ,
N ( a ) = Im [ Ĩ ( x , y , f 1 ) ] , D ( a ) = Re [ Ĩ ( x , y , f 1 ) ] .
Δ ϕ 1 u ( x , y ) = Δ ϕ 1 ( x , y ) + 2 π ν ( x , y ) .
w 1 ( x , y , z 1 ) = λ c Δ ϕ 1 u ( x , y , z 1 , 0 ) 4 π .
Δ ϕ ( x , y , z 1 ) = 4 π z 1 Δ λ λ 1 λ u ,
N f 4 z 1 Δ λ λ 1 λ 2 .
Δ z N f λ c 2 4 Δ λ .
δ z = γ λ c 2 Δ λ ,
D R = N f / 2 γ .

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