Abstract

The simultaneous quantitative measurement of out-of-plane displacement and slope using the fast Fourier transform method with a single three-aperture digital speckle pattern interferometry (DSPI) arrangement is demonstrated. The method coherently combines two sheared object waves with a smooth reference wave at the CCD placed at the image plane of an imaging lens with a three-aperture mask placed in front of it. The apertures also introduce multiple spatial carrier fringes within the speckle. A fast Fourier transform of the image generates seven distinct diffraction halos in the spectrum. By selecting the appropriate halos, one can directly obtain two independent out-of-plane displacement phase maps and a slope phase map from the two speckle images, one before and the second after loading the object. It is also demonstrated that by subtracting the out-of-plane displacement phase maps one can generate the same slope phase map. Experimental results are presented for a circular diaphragm clamped along the edges and loaded at the center.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2006

B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, "A dual-function ESPI system for the measurement of out-of-plane displacement and slope," Opt. Lasers Eng. 44, 637-644 (2006).
[CrossRef]

B. Bhaduri, N. Krishna Mohan, M. P. Kothiyal, and R. S. Sirohi, "Use of spatial phase shifting technique in digital speckle pattern interferometry (DSPI) and digital shearography (DS)," Opt. Express 14, 11598-11607 (2006).
[CrossRef] [PubMed]

2001

2000

1997

P. A. Fomitchov and S. Krishnaswamy, "A compact dual-purpose camera for shearography and electronic speckle-pattern interferometry," Meas. Sci. Technol. 8, 581-583 (1997).
[CrossRef]

J. M. Huntley, "Random phase measurement errors in digital speckle interferometry," Opt. Lasers Eng. 26, 131-150 (1997).
[CrossRef]

R. S. Sirohi, J. Burke, H. Helmers, and K. D. Hinsch, "Spatial phase shifting for pure in-plane displacement and displacement-derivative measurements in electronic speckle pattern interferometry (ESPI)," Appl. Opt. 36, 5787-5791 (1997).
[CrossRef] [PubMed]

1996

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, 332-336 (1996).
[CrossRef] [PubMed]

G. Pedrini, Y. L. Zou, and H. Tiziani, "Quantitative evaluation of digital shearing interferogram using the spatial carrier method," Pure Appl. Opt. 5, 313-321 (1996).
[CrossRef]

1995

T. W. Ng, "Digital speckle pattern interferometer for combined measurements of out-of-plane displacement and slope," Opt. Commun. 116, 31-35 (1995).
[CrossRef]

1994

1993

1985

1982

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

P. A. Fomitchov and S. Krishnaswamy, "A compact dual-purpose camera for shearography and electronic speckle-pattern interferometry," Meas. Sci. Technol. 8, 581-583 (1997).
[CrossRef]

Opt. Commun.

T. W. Ng, "Digital speckle pattern interferometer for combined measurements of out-of-plane displacement and slope," Opt. Commun. 116, 31-35 (1995).
[CrossRef]

Opt. Express

B. Bhaduri, N. Krishna Mohan, M. P. Kothiyal, and R. S. Sirohi, "Use of spatial phase shifting technique in digital speckle pattern interferometry (DSPI) and digital shearography (DS)," Opt. Express 14, 11598-11607 (2006).
[CrossRef] [PubMed]

Opt. Lasers Eng.

B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, "A dual-function ESPI system for the measurement of out-of-plane displacement and slope," Opt. Lasers Eng. 44, 637-644 (2006).
[CrossRef]

J. M. Huntley, "Random phase measurement errors in digital speckle interferometry," Opt. Lasers Eng. 26, 131-150 (1997).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

G. Pedrini, Y. L. Zou, and H. Tiziani, "Quantitative evaluation of digital shearing interferogram using the spatial carrier method," Pure Appl. Opt. 5, 313-321 (1996).
[CrossRef]

Other

R. S. Sirohi, ed., Speckle Metrology (Marcel Dekker, 1993).

J. Burke, "Application and optimization of the spatial phase shifting technique in digital speckle interferometry," Ph.D. dissertation (Carl von Ossietzky University, Oldenburg, Germany, 2000).

P. K. Rastogi, ed., Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).

W. Steinchen and L. Yang, Digital Shearography: Theory and Application of Digital Speckle Pattern Shearing Interferometry (SPIE Press, 2003).

M. Kujawinska, "Spatial phase measurement methods," in Interferogram Analysis-Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), Chap. 5.

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Figures (9)

Fig. 1
Fig. 1

Schematic of the three-aperture arrangement: A, apertures; WP, wedge plate.

Fig. 2
Fig. 2

Schematic of the halos in the spatial frequency domain obtained with a three-aperture mask.

Fig. 3
Fig. 3

(Color online) Schematic of a three-aperture digital speckle pattern interferometric arrangement: O, object; RM, reference mirror; BS, beam splitter; M, mirrors; P, front-surfaces-coated right angle prism; A, three-aperture mask; NDF, neutral density filter; L 1 , lens; L 2 , imaging lens.

Fig. 4
Fig. 4

Magnified portion of the speckle pattern obtained using a multi-aperture arrangement revealing the multiple spatial carrier fringes within the speckle.

Fig. 5
Fig. 5

Moiré correlation fringes obtained as the real-time subtraction of the deformed frame from the initial frame.

Fig. 6
Fig. 6

Spectrum of the recorded interferogram obtained with fast Fourier transform. Halos 1 and 2 are used for inverse Fourier transformation.

Fig. 7
Fig. 7

(Color online) Out-of-plane displacement evaluation: (a) filtered phase map, (b) unwrapped 2D and (c) 3D plots.

Fig. 8
Fig. 8

(Color online) Slope evaluation using halo 2 in Fig. 6: (a) filtered phase map, (b) unwrapped 2D and (c) 3D plots.

Fig. 9
Fig. 9

(Color online) Slope evaluation by subtracting the out-of-plane displacement phase maps obtained from halos 1 and 3 in Fig. 6: (a) filtered phase map, (b) unwrapped 2D and (c) 3D plots.

Equations (102)

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

d a
A 1
A 0
A 2
A 0
A 1
A 2
A 2
Δ x
Δ x = U ( n 1 ) α
A 0
A 1
A 2
u 0 ( x , y ) = | u 0 ( x , y ) | exp [ i { 2 π f 0 x x } ] ,
u 1 ( x , y ) = | u 1 ( x , y ) | exp [ i { ϕ ( x , y ) } ] ,
u 2 ( x , y ) = | u 1 ( x + Δ x , y ) | exp [ i { ϕ ( x + Δ x , y ) + 2 π f 0 y y } ] ,
| u | s
ϕ ( x , y )
ϕ ( x + Δ x , y )
f 0 x
f 0 y
f 0 x = f 0 y = f 0
f 0
f 0 = 1 / Δ
Δ = λ V / D
( d s )
d s = 1.22 λ V / d a
d a
I ( x , y ) = | u 0 ( x , y ) + u 1 ( x , y ) + u 2 ( x , y ) | 2 = u 0 u 0 * + u 1 u 1 * + u 2 u 2 * + u 0 u 1 * + u 1 u 0 * + u 0 u 2 * + u 2 u 0 * + u 1 u 2 * + u 2 u 1 * .
I ( x , y ) = | u 0 ( x , y ) | 2 + | u 1 ( x , y ) | 2 + | u 1 ( x + Δ x , y ) | 2 + | u 0 ( x , y ) | | u 1 ( x , y ) | exp [ i { 2 π f 0 x ϕ ( x , y ) } ] + | u 0 ( x , y ) | | u 1 ( x , y ) | exp [ i { 2 π f 0 x ϕ ( x , y ) } ] + | u 0 ( x , y ) | | u 1 ( x + Δ x , y ) | exp [ i { 2 π f 0 x 2 π f 0 y ϕ ( x + Δ x , y ) } ] + | u 0 ( x , y ) | | u 1 ( x + Δ x , y ) × | exp [ i { 2 π f 0 x 2 π f 0 y ϕ ( x + Δ x , y ) } ] + | u 1 ( x , y ) | | u 1 ( x + Δ x , y ) | exp [ i { ϕ ( x , y ) ϕ ( x + Δ x , y ) 2 π f 0 y } ] + | u 1 ( x , y ) | | u 1 ( x + Δ x , y ) × | exp [ i { ϕ ( x , y ) ϕ ( x + Δ x , y ) 2 π f 0 y } ] .
ϕ ( x , y )
ϕ ( x + Δ x , y )
[ ϕ ( x , y ) ϕ ( x + Δ x , y ) ]
i = F T [ I ] = U 0 U 0 * + U 1 U 1 * + U 2 U 2 * + U 0 U 1 * + U 1 U 0 * + U 0 U 2 * + U 2 U 0 * + U 1 U 2 * + U 2 U 1 * ,
U j = F T [ u j ] ; j = 0 , 1 , 2
f x
f y
A 1
A 2
A 0
A 1
u 1 ( x , y )
A 0
A 2
u 2 ( x , y )
A 1
A 2
d p
1 / d p
( f max )
1 / ( 2 d p )
2 f s = f max / 3 = 1 / ( 6 d p )
f s
U 1 U 0 *
u 0 u 1 *
[ ϕ ( x , y ) 2 π f 0 x ]
ϕ ( x , y ) 2 π f 0 x = arctan [ Im ( u 1 u 0 * ) Re ( u 1 u 0 * ) ] = arctan [ N B D B ] ,
U 1 U 2 *
u 1 u 2 *
[ ϕ ( x , y ) ϕ ( x + Δ x , y ) 2 π f 0 y ]
ϕ ( x , y ) ϕ ( x + Δ x , y ) 2 π f 0 y = arctan [ Im ( u 1 u 2 * ) Re ( u 1 u 2 * ) ]
= arctan [ N B D B ] ,
N
D
u 1 ( x , y ) = | u 1 ( x , y ) | exp [ i { ϕ ( x , y ) } ] ,
u 2 ( x , y ) = | u 1 ( x + Δ x , y ) | exp [ i { ϕ ( x + Δ x , y ) + 2 π f 0 y } ] .
[ ϕ ( x , y ) 2 π f 0 x ]
[ ϕ ( x , y ) ϕ ( x + Δ x , y ) 2 π f 0 y ]
Δ ϕ ( x , y )
Δ ϕ ( x , y ) = ϕ ( x , y ) ϕ ( x , y ) = [ ϕ ( x , y ) 2 π f 0 x ] [ ϕ ( x , y ) 2 π f 0 x ] = arctan ( N A D B N B D A N A N B + D A D B ) = arctan ( N D ) .
Δ ϕ ( x , y )
Δ ϕ ( x , y ) = 4 π λ w ( x , y ) .
Δ ϕ ( x , y )
Δ ϕ ( x , y ) = [ ϕ ( x , y ) ϕ ( x , y ) ] [ ϕ ( x + Δ x , y ) ϕ ( x + Δ x , y ) ] = [ ϕ ( x , y ) ϕ ( x + Δ x , y ) 2 π f 0 y ] [ ϕ ( x , y ) ϕ ( x + Δ x , y ) 2 π f 0 y ] = arctan ( N A D B N B D A N A N B + D A D B ) = arctan ( N D ) ,
Δ ϕ ( x , y )
w ( x , y ) / x
Δ ϕ ( x , y ) = 4 π λ w ( x , y ) x Δ x ,
Δ x
Δ ϕ ( x + Δ x , y )
Δ x
A 0
A 2
Δ ϕ ( x , y ) = Δ ϕ ( x + Δ x , y ) Δ ϕ ( x , y ) = 4 π λ [ w ( x + Δ x , y ) w ( x , y ) Δ x ] Δ x = 4 π λ w ( x , y ) x Δ x .
2 π
2.65   mm
4 .35   mm
2.8   mm
4 .2   mm
60   mm
20   mW
M 1
( 20
3   mm
( L 1 )
( M 2 )
L 2
1376 × 1035
4.65 × 4.65 μm 2
L 1
Δ x
L 1
L 2

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