Abstract

The phase shift method, well known in holographic interferometry, is applied to the deformation of line gratings. The method uses the moire effect but it determines local displacement or strain from the grey values of three shifted images instead of the coordinates of the fringes. The basic equations are derived from the transmittance function. The theoretical error with respect to a given displacement field is calculated by simulation. Real errors are investigated by related plotter-generated line gratings. The phase shift method is especially suitable for automatic digital image processing.

© 1987 Optical Society of America

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References

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  1. R. Daendlicker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
    [CrossRef]
  2. B. Breuckmann, W. Thieme, “Ein rechnergestuetztes Holografie-System fuer industriellen Einsatz,” VDI Ber. (Ver. Dtsch. Ing.) 552, 27 (1985).
  3. K. Andresen, “The Phase Shift Method Applied to Moire Image Processing,” Optik 72, 115 (1986).
  4. K. Andresen, R. Ritter, “The Phase Shift Method Applied to Reflection Moire Pattern,” in Proceedings, Eighth International Conference on Experimental Stress Analysis351 (1986).
  5. K. Andresen, D. Klassen, “The Phase Shift Method Applied to Cross Grating Moire Measurement,” Opt. Laser Eng.7, 101 (1987), to be published.
    [CrossRef]

1986 (1)

K. Andresen, “The Phase Shift Method Applied to Moire Image Processing,” Optik 72, 115 (1986).

1985 (1)

B. Breuckmann, W. Thieme, “Ein rechnergestuetztes Holografie-System fuer industriellen Einsatz,” VDI Ber. (Ver. Dtsch. Ing.) 552, 27 (1985).

1982 (1)

R. Daendlicker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

Andresen, K.

K. Andresen, “The Phase Shift Method Applied to Moire Image Processing,” Optik 72, 115 (1986).

K. Andresen, D. Klassen, “The Phase Shift Method Applied to Cross Grating Moire Measurement,” Opt. Laser Eng.7, 101 (1987), to be published.
[CrossRef]

K. Andresen, R. Ritter, “The Phase Shift Method Applied to Reflection Moire Pattern,” in Proceedings, Eighth International Conference on Experimental Stress Analysis351 (1986).

Breuckmann, B.

B. Breuckmann, W. Thieme, “Ein rechnergestuetztes Holografie-System fuer industriellen Einsatz,” VDI Ber. (Ver. Dtsch. Ing.) 552, 27 (1985).

Daendlicker, R.

R. Daendlicker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

Klassen, D.

K. Andresen, D. Klassen, “The Phase Shift Method Applied to Cross Grating Moire Measurement,” Opt. Laser Eng.7, 101 (1987), to be published.
[CrossRef]

Ritter, R.

K. Andresen, R. Ritter, “The Phase Shift Method Applied to Reflection Moire Pattern,” in Proceedings, Eighth International Conference on Experimental Stress Analysis351 (1986).

Thalmann, R.

R. Daendlicker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

Thieme, W.

B. Breuckmann, W. Thieme, “Ein rechnergestuetztes Holografie-System fuer industriellen Einsatz,” VDI Ber. (Ver. Dtsch. Ing.) 552, 27 (1985).

Willemin, J. F.

R. Daendlicker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

Opt. Commun. (1)

R. Daendlicker, R. Thalmann, J. F. Willemin, “Fringe Interpolation by Two-Reference-Beam Holographic Interferometry: Reducing Sensitivity to Hologram Misalignment,” Opt. Commun. 42, 301 (1982).
[CrossRef]

Optik (1)

K. Andresen, “The Phase Shift Method Applied to Moire Image Processing,” Optik 72, 115 (1986).

VDI Ber. (Ver. Dtsch. Ing.) (1)

B. Breuckmann, W. Thieme, “Ein rechnergestuetztes Holografie-System fuer industriellen Einsatz,” VDI Ber. (Ver. Dtsch. Ing.) 552, 27 (1985).

Other (2)

K. Andresen, R. Ritter, “The Phase Shift Method Applied to Reflection Moire Pattern,” in Proceedings, Eighth International Conference on Experimental Stress Analysis351 (1986).

K. Andresen, D. Klassen, “The Phase Shift Method Applied to Cross Grating Moire Measurement,” Opt. Laser Eng.7, 101 (1987), to be published.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Calculation of the displacement function: (a) theoretical function û, (b) wrapped function u modp0, (c) unwrapped function u, (d) error uû.

Fig. 2
Fig. 2

Calculation of the strain function: (a) theoretical function û′, (b) wrapped function u′ modp0/d, (c) unwrapped function u′ (d) error u′ − û′.

Fig. 3
Fig. 3

Maximum error u′ − û′ related to global shift d/P.

Fig. 4
Fig. 4

Moire image by superimposition of (a) deformed and undeformed gratings, and (b) deformed and deformed gratings displaced by d.

Fig. 5
Fig. 5

Displacement function calculated from plotter-generated line gratings: (a) wrapped function u mod p0, (b) unwrapped function u.

Fig. 6
Fig. 6

Strain function calculated from plotter-generated line gratings: (a) wrapped function u′ mod p0/d, (b) unwrapped function u′.

Equations (29)

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T 0 = T 0 + T a cos 2 π ξ p 0 ,
x = ξ + v ( ξ ) = g ( ξ ) .
ξ = g - 1 ( x ) ,
u ( x ) = v ( ξ ) = v [ g - 1 ( x ) ]
x = d u / d x .
T 1 = T 0 + T a cos 2 π [ x - u ( x ) ] p 0 .
T d = T 0 T 1 = T 0 2 + T 0 T a { cos 2 π x p 0 + cos 2 π [ x - u ( x ) ] p 0 } + T a 2 cos 2 π x p 0 cos 2 π [ x - u ( x ) ] p 0 .
I ¯ = T ¯ d I e ,
T ¯ d = 1 p 0 p 0 T d d x .
u i ( x ) = u i 0 + x i x ( x ) d x = u i 0 + i ( x - x i ) .
T ¯ d i = 1 p 0 p 0 ( T 0 2 + T 0 T a { cos 2 π x p 0 + cos 2 π p 0 [ x - u i 0 - i ( x - x i ) ] } + T a 2 cos 2 π x p 0 cos 2 π p 0 [ x - u i 0 - i ( x - x i ) ] } d x .
T ¯ d i = T 0 2 + T a 2 2 cos 2 π u i p 0 + i R d ,
I 0 = T 0 2 I e ;             I a = T a 2 2 I e ,
I ¯ i = I 0 + I a cos 2 π u i p 0 .
I ¯ i k = I 0 + I a cos ( 2 π u i p 0 + k 2 π 3 ) ,             k = 0 , 1 , 2.
u i = p 0 2 π arctan [ ( I ¯ i 2 - I ¯ i 1 ) 3 2 I ¯ i 0 - I ¯ i 1 - I ¯ i 2 ] .
T 2 = T 0 + T a cos 2 π p 0 [ x - d - u ( x - d ) ] .
T s = T 1 T 2 = T 0 2 + T 0 T a ( cos 2 π p 0 [ x - u ( x ) ] + cos 2 π p 0 [ x - d - u ( x - d ) ] ) + T a 2 cos 2 π p 0 [ x - u ( x ) ] cos 2 π p 0 [ x - d - u ( x - d ) ] .
u ( x - d ) = u 0 i + x i x - d x ( η ) d η = u 0 i + x i x k x ( η ) d η + x k x - d x ( η ) d η = u 0 i - d ¯ i + k ( x - x i ) ,
¯ i = - 1 d x i x k x ( η ) d η .
T ¯ s i = T 0 2 + T a 2 2 cos 2 π p 0 d ( 1 - ¯ i ) + k R s .
I 0 = T 0 2 I e ,             I a = T a 2 2 I e ,
I ¯ s i = T ¯ s i I e = I 0 + I a cos 2 π p 0 d ( 1 - ¯ i ) .
d = ( n + x ) p 0 ,
I ¯ s i = I 0 + I a cos 2 π ( x - d p 0 ¯ i ) .
¯ i = [ x - 1 2 π arctan ( I ¯ s i 2 - I ¯ s i 1 ) 3 2 I ¯ s i 0 - I ¯ s i 1 - I ¯ s i 2 ] p 0 d .
ξ = x 1 - x .
v ( ξ ) = v 0 sin 2 π f ξ ,             v 0 = 16.7 ,             f = 1 / 2100 ,
T 1 = T 0 + T a cos 2 π p 0 [ x - u ( x ) ] ,

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