Abstract

A two-dimensional analysis of a double-exposed speckle pattern as created by illuminating a plane surface with a beam of laser light is presented with the role played by the aperture illumination function described in detail. Between exposures, the surface is to undergo in-plane general deformation. Two approaches (one whole field and one pointwise) are presented to delineate from the speckle pattern the three Cartesian components of strain at each point in the field. An analogy is drawn between the whole field approach and the classical in-plane moiré method. the methods.

© 1976 Optical Society of America

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References

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  1. D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
    [CrossRef]
  2. L. T. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
    [CrossRef]
  3. J. D. Rigden, E. I. Gordon, Proc. IRE 50, 2367 (1962).
  4. R. A. Sprague, Appl. Opt. 11, 2811 (1972).
    [CrossRef] [PubMed]
  5. J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).
  6. E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
    [CrossRef]
  7. E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
    [CrossRef]
  8. D. E. Duffy, Exp. Mech. 14, 378 (1974).
    [CrossRef]
  9. P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, New York, 1967), Chapter 1.
  10. A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice Hall, New York, 1970), Chaps. 3 and 4.
  11. F. P. Chiang, Proc. Am. Soc. Civ. Eng. 96, 1285(1970).
  12. F. P. Chiang, Proc. Am. Soc. Civ. Eng. 91, 137 (1965).
  13. F. P. Chiang, Proc. Am. Soc. Civ. Eng. 95, 1379 (1969).

1974 (1)

D. E. Duffy, Exp. Mech. 14, 378 (1974).
[CrossRef]

1972 (2)

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

R. A. Sprague, Appl. Opt. 11, 2811 (1972).
[CrossRef] [PubMed]

1970 (3)

D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
[CrossRef]

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

F. P. Chiang, Proc. Am. Soc. Civ. Eng. 96, 1285(1970).

1969 (1)

F. P. Chiang, Proc. Am. Soc. Civ. Eng. 95, 1379 (1969).

1968 (1)

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

1965 (2)

L. T. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
[CrossRef]

F. P. Chiang, Proc. Am. Soc. Civ. Eng. 91, 137 (1965).

1962 (1)

J. D. Rigden, E. I. Gordon, Proc. IRE 50, 2367 (1962).

Archbold, E.

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

Burch, J. M.

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

Chiang, F. P.

F. P. Chiang, Proc. Am. Soc. Civ. Eng. 96, 1285(1970).

F. P. Chiang, Proc. Am. Soc. Civ. Eng. 95, 1379 (1969).

F. P. Chiang, Proc. Am. Soc. Civ. Eng. 91, 137 (1965).

Duffy, D. E.

D. E. Duffy, Exp. Mech. 14, 378 (1974).
[CrossRef]

Durelli, A. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice Hall, New York, 1970), Chaps. 3 and 4.

Ennos, A. E.

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

Gabor, D.

D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
[CrossRef]

Goldfischer, L. T.

Gordon, E. I.

J. D. Rigden, E. I. Gordon, Proc. IRE 50, 2367 (1962).

Parks, V. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice Hall, New York, 1970), Chaps. 3 and 4.

Rigden, J. D.

J. D. Rigden, E. I. Gordon, Proc. IRE 50, 2367 (1962).

Sprague, R. A.

Theocaris, P. S.

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, New York, 1967), Chapter 1.

Tokarski, J. M. J.

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

Appl. Opt. (1)

Exp. Mech. (1)

D. E. Duffy, Exp. Mech. 14, 378 (1974).
[CrossRef]

IBM J. Res. Dev. (1)

D. Gabor, IBM J. Res. Dev. 14, 509 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (3)

J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
[CrossRef]

E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

Proc. Am. Soc. Civ. Eng. (3)

F. P. Chiang, Proc. Am. Soc. Civ. Eng. 96, 1285(1970).

F. P. Chiang, Proc. Am. Soc. Civ. Eng. 91, 137 (1965).

F. P. Chiang, Proc. Am. Soc. Civ. Eng. 95, 1379 (1969).

Proc. IRE (1)

J. D. Rigden, E. I. Gordon, Proc. IRE 50, 2367 (1962).

Other (2)

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, New York, 1967), Chapter 1.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice Hall, New York, 1970), Chaps. 3 and 4.

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Figures (12)

Fig. 1
Fig. 1

Schematic for recording of speckle pattern.

Fig. 2
Fig. 2

Optical arrangement for whole field filtering.

Fig. 3
Fig. 3

Aperture illumination function: (a) in the image plane, (b) in the transform plane.

Fig. 4
Fig. 4

Graphical representation of: (a) F(v1,v2), (b) its autocorrelation and intensity I1(u1,u2) due to single exposure speckle pattern.

Fig. 5
Fig. 5

Optical arrangement for pointwise filtering.

Fig. 6
Fig. 6

Isothetics from whole field filtering for disk under rigid body in-plane rotation for various sensitivities. Fringes represent horizontal displacement component.

Fig. 7
Fig. 7

Fringe patterns from pointwise filtering for disk under rigid body in plane rotation.

Fig. 8
Fig. 8

Isothetics from whole field filtering for cantilever beam for various sensitivities.

Fig. 9
Fig. 9

Fringe patterns from pointwise filtering for cantilever beam.

Fig. 10
Fig. 10

Theoretical solution and calculated results from whole field and pointwise filtering for cantilever beam.

Fig. 11
Fig. 11

Isothetics for disk under three-point load: (a) u field, (b) v field.

Fig. 12
Fig. 12

Geometrical construction for derivation of G1(u1,u2).

Equations (55)

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σ = 1.2 ( λ q ) / D ,
σ = 1.2 λ F
f ( x 1 , x 2 ) = K h ( ξ 1 , ξ 2 ) A ( r 1 , r 2 ) × exp { i k [ r 1 ( ξ 1 / p + x 1 / q ) + r 2 ( ξ 2 / p + x 2 / q ) ] } d ξ 1 d ξ 2 d r 1 d r 2 ,
e ( x 1 , x 2 ) = [ f ( x 1 , x 2 ) 2 + f ( x 1 + d 1 , x 2 + d 2 ) 2 ] t ,
g ( x 1 , x 2 ) = b - c e ( x 1 , x 2 ) ,
g ( x 1 , x 2 ) = b - c t [ f ( x 1 , x 2 ) 2 + f ( x 1 + d 1 , x 2 + d 2 ) 2 ] .
I ( u 1 , u 2 ) = g ( x 1 , x 2 ) exp [ i k ( x 1 u 1 + x 2 u 2 ) / L ] d x 1 d x 2 2 ,
I ( u 1 , u 2 ) = c 2 t 2 f ( x 1 , x 2 ) 2 exp [ i k ( x 1 u 1 + x 2 u 2 ) / L ] d x 1 d x 2 + f ( x 1 + d 1 , x 2 + d 2 ) 2 exp ( i k x 1 u 1 + x 2 u 2 / L ) d x 1 d x 2 2 ,
I ( u 1 , u 2 ) = c 2 t 2 f ( x 1 , x 2 ) 2 exp [ i k ( x 1 u 1 + x 2 u 2 ) / L ] d x 1 d x 2 × { 1 + exp [ - i k ( u 1 d 1 + u 2 d 2 ) / L ] } 2 ,
I ( u 1 , u 2 ) = 4 cos 2 k ( u 1 d 1 + u 2 d 2 L ) I 1 ( u 1 , u 2 ) ,
I 1 ( u 1 , u 2 ) = c 2 t 2 f ( x 1 , x 2 ) 2 exp [ i k ( x 1 u 1 + x 2 u 2 ) / L ] d x 1 d x 2 2 = c 2 t 2 G 1 ( u 1 , u 2 ) 2
G 1 ( u 1 , u 2 ) = F ( v 1 , v 2 ) F * ( v 1 - u 1 , v 2 - u 2 ) d v 1 d v 2 ,
F ( v 1 , v 2 ) = f ( x 1 , x 2 ) exp [ i k ( x 1 v 1 + x 2 v 2 ) / L ] d x 1 d x 2
F ( v 1 , v 2 ) = K h ( ξ 1 , ξ 2 ) A ( r 1 , r 2 ) exp [ i k ( r 1 ξ 1 + r 2 ξ 2 ) / p ] d ξ 1 d ξ 2 × exp [ i k x 1 ( r 1 / q + v 1 / L ) ] d x 1 exp [ i k x 2 ( r 2 / q + v 2 / L ) ] d x 2 d r 1 d r 2 = K h ( ξ 1 , ξ 2 ) A ( r 1 , r 2 ) exp [ i k ( r 1 ξ 1 + r 2 ξ 2 ) / p ] d ξ 1 d ξ 2 × δ ( r 1 q + v 1 L ) δ ( r 2 q + v 2 L ) d r 1 d r 2 = K h ( ξ 1 , ξ 2 ) A ( r 1 , r 2 ) exp [ i k ( r 1 ξ 1 ) / p ] δ ( r 1 q + v 1 L ) d r 1 × exp [ i k ( r 2 ξ 2 ) / p ] δ ( r 2 q + v 2 L ) d r 2 d ξ 1 d ξ 2 = K h ( ξ 1 , ξ 2 ) A ( - v 1 q L , - v 2 q L ) exp [ - i k ( v 1 q / L ) ( ξ 1 / p ) ] × exp [ - i k v 2 q / L ( ξ 2 / p ) ] d ξ 1 d ξ 2 = K A ( - v 1 q L , - v 2 q L ) h ( ξ 1 , ξ 2 ) × exp [ - i k ( q / L p ) ( v 1 ξ 1 + v 2 ξ 2 ) ] d ξ 1 d ξ 2 ,
F ( v 1 , v 2 ) = K A ( - v 1 q L , - v 2 q L ) H ( - q v 1 L p , - q v 2 L p ) ,
H ( - q v 1 L p , - q v 2 L p ) = h ( ξ 1 , ξ 2 ) × exp [ - i k ( q / L p ) ( v 1 ξ 1 + v 2 ξ 2 ) ] d ξ 1 d ξ 2
v max = ( v 1 max 2 + v 2 max 2 ) 1 / 2 .
A ( r 1 , r 2 ) = 1 for ( r 1 2 + r 2 2 ) 1 / 2 D / 2 = 0 for ( r 1 2 + r 2 2 ) 1 / 2 > D / 2 ,
A ( - v 1 q L , - v 2 q L ) = 1 for ( v 1 2 + v 2 2 ) 1 / 2 L D 2 q = 0 for ( v 1 2 + v 2 2 ) 1 / 2 > L D 2 q .
H ( - q v 1 L P , - q v 2 L P )
G 1 ( u 1 , u 2 ) = 2 π { cos - 1 u q L D - u q L D [ 1 - ( u q L D ) 2 ] 1 / 2 }
I 1 ( u 1 , u 2 ) = ( 2 π ) 2 { cos - 1 u q L D - u q L D [ 1 - ( u q L D ) 2 ] 1 / 2 } 2
cos k ( u 1 d 1 + u 2 d 2 ) 2 L = 0
u ¯ · d ¯ = ( n + ½ ) λ L ,             n = 0 , ± 1 , ± 2 , .
u ¯ · d ¯ = n λ L             n = 0 , ± 1 , ± 2 , .
d cos θ = n L λ u ¯ for bright fringes = ( n + ½ ) L λ u ¯ for dark fringes ,
( d min ) i = λ q D = σ 1.2 .
l = { L 2 + [ u 1 - ( x 1 - x ¯ 1 ) ] 2 + [ u 2 - ( x 2 - x ¯ 2 ) ] 2 } 1 / 2 ,
l L + u 1 2 + u 2 2 + 2 u 1 x ¯ 1 + 2 u 2 x ¯ 2 2 L + ( x 1 - x ¯ 1 ) 2 + ( x 2 - x ¯ 2 ) 2 2 L - u 1 x 1 + u 2 x 2 L .
I ( u 1 , u 2 ) = g ( x 1 , x 2 ) exp ( - i k l ) d x 1 d x 2 2 .
( x 1 - x ¯ 1 ) 2 + ( x 2 - x ¯ 2 ) 2 2 L u 1 x 1 + u 2 x 2 L ,
I ( u 1 , u 2 ) = beam size g ( x 1 x 2 ) exp [ i k ( u 1 x 1 + u 2 x 2 ) / L ] d x 1 d x 2 2 .
α = D / q .
β = λ L d · 1 L = λ d .
α = ( N - 1 ) β ,
h ( ξ 1 , ξ 2 ) exp ( - i k S ) S ,
h ( ξ 1 , ξ 2 ) S exp ( - i k S ) d ξ 1 d ξ 2 .
f ( x 1 , x 2 ) = [ h ( ξ 1 , ξ 2 ) exp ( i k s ) S d ξ 1 d ξ 2 ] × A ( r 1 , r 2 ) exp [ i k ( r 1 2 + r 2 2 ) / 2 f ] exp ( - i k S ) S d r 1 d r 2 ,
f ( x 1 , x 2 ) = 1 p q h ( ξ 1 , ξ 2 ) A ( r 1 , r 2 ) exp [ - i k ( r 1 2 + r 2 2 ) / 2 f ] × exp ( - i k S ) exp ( - i k S ) d ξ 1 d ξ 2 d r 1 d r 2 .
S 2 = P 2 + ( ξ 1 - r 1 ) 2 + ( ξ 2 - r 2 ) 2 R 2 - 2 ( ξ 1 r 1 + ξ 2 r 2 ) + ( r 1 2 + r 2 2 ) ,
S = [ R 2 - 2 ( ξ 1 r 1 + ξ 2 r 2 ) + ( r 1 2 + r 2 2 ) ] 1 / 2 R - ( ξ 1 r 1 + ξ 2 r 2 ) R + r 1 2 + r 2 2 2 R .
S p - ξ 1 r 1 + ξ 2 r 2 p + r 1 2 + r 2 2 2 p .
S = q - r 1 x 1 + r 2 x 2 q + r 1 2 + r 2 2 2 q .
f ( x 1 , x 2 ) = 1 p q h ( ξ 1 , ξ 2 ) × exp [ i k ( - p + r 1 ξ 1 + r 2 ξ 2 p - r 1 2 + r 2 2 2 p ) ] × exp [ i k ( - q + r 1 x 1 + r 2 x 2 q - r 1 2 + r 2 2 2 q ) ] × A ( r 1 , r 2 ) exp [ - i k ( r 1 2 + r 2 2 ) 2 f ] d ξ 1 d ξ 2 d r 1 d r 2 = exp [ i k ( p + q ) ] p q h ( ξ 1 ξ 2 ) A ( r 1 , r 2 ) × exp { i k [ r 1 ( ξ 1 / p + x 1 / q ) + r 2 ( ξ 2 / p + x 2 / q ) ] } × exp [ - i k ( r 1 2 + r 2 2 ) / 2 ] ( 1 p + 1 q - 1 f ) d ξ 1 d ξ 2 d r 1 d r 2 .
1 p + 1 q - 1 f = 0 ,
f ( x 1 , x 2 ) = K h ( ξ 1 , ξ 2 ) A ( r 1 , r 2 ) × exp { i k [ r 1 ( ξ 1 p + ξ 2 q ) + r 2 ( ξ 2 p + ξ 2 q ) ] } d ξ 1 d ξ 2 d r 1 d r 2 ,
K = 1 p q exp [ - i k ( p + q ) ]
H ( - q v 1 L P , - q v 2 L P )
F ( v 1 , v 2 ) K = A ( - v 1 q L , - v 2 q L ) ,
F ( v 1 , v 2 ) K = 1 for ( v 1 2 + v 2 2 ) 1 / 2 L D 2 q = 0 for ( v 1 2 + v 2 2 ) 1 / 2 > L D 2 q ,
K = K H ( - q v 1 L P , q v 2 L P ) .
F * ( v 1 - u 1 , v 2 - u 2 ) = F ( v 1 - u 1 , v 2 - u 2 ) .
G ( u 1 , u 2 ) = 2 | ( L D 2 q ) 2 cos - 1 u / 2 L D / 2 q - u z [ ( L D / 2 q ) 2 - ( u / 2 ) 2 ] 1 / 2 | π ( L D 2 q ) 2 ,
G ( u 1 , u 2 ) = 2 π [ cos - 1 u q L D - u q L D ( 1 - ( u q / L D ) 2 ) 1 / 2 ]
I ( u 1 , u 2 ) = 4 π 2 { cos - 1 u q L D - u q L D [ 1 - ( u q / L D ) 2 ] 1 / 2 } 2 .

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