Abstract

Speckle photography offers a technique for measuring small strains on an object in the presence of large displacements. An analysis is presented of this technique, which accounts for the phenomena observed and predicts its capabilities and limitations. The analysis is sufficiently general to describe either of two related techniques, one that permits measurement of strains in the absence of large displacements and one that permits measurement of large displacements alone.

© 1974 Optical Society of America

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References

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  1. E. Archbold, J. M. Burch, and A. E. Ennos, Opt. Acta 17, 883 (1970).
    [CrossRef]
  2. J. A. Leendertz, J. Phys. 3, 214 (1970).
  3. J. M. Burch and J. M. J. Tokarski, Opt. Acta 15, 101 (1968).
  4. K. Biedermann, Optik 28, 160 (1968).
  5. H. J. Tiziani, Opt. Acta 18, 891 (1971).
    [CrossRef]
  6. H. J. Tiziani, Opt. Commun. 5, 217 (1972).
    [CrossRef]
  7. H. J. Tiziani, Optik 34, 442 (1972).
  8. H. J. Tiziani, Appl. Opt. 11, 2911 (1972).
    [CrossRef] [PubMed]
  9. U. Köpf, Optik 33, 517 (1971).
  10. E. Archbold and A. E. Ennos, Opt. Acta 19, 253 (1972).
    [CrossRef]
  11. K. A. Stetson, J. Opt. Soc. Am. 60, 1378 (1970).
    [CrossRef]

1972 (4)

H. J. Tiziani, Opt. Commun. 5, 217 (1972).
[CrossRef]

H. J. Tiziani, Optik 34, 442 (1972).

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

H. J. Tiziani, Appl. Opt. 11, 2911 (1972).
[CrossRef] [PubMed]

1971 (2)

U. Köpf, Optik 33, 517 (1971).

H. J. Tiziani, Opt. Acta 18, 891 (1971).
[CrossRef]

1970 (3)

K. A. Stetson, J. Opt. Soc. Am. 60, 1378 (1970).
[CrossRef]

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

J. A. Leendertz, J. Phys. 3, 214 (1970).

1968 (2)

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

K. Biedermann, Optik 28, 160 (1968).

Archbold, E.

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

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

Biedermann, K.

K. Biedermann, Optik 28, 160 (1968).

Burch, J. M.

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

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

Ennos, A. E.

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

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

Köpf, U.

U. Köpf, Optik 33, 517 (1971).

Leendertz, J. A.

J. A. Leendertz, J. Phys. 3, 214 (1970).

Stetson, K. A.

Tiziani, H. J.

H. J. Tiziani, Opt. Commun. 5, 217 (1972).
[CrossRef]

H. J. Tiziani, Optik 34, 442 (1972).

H. J. Tiziani, Appl. Opt. 11, 2911 (1972).
[CrossRef] [PubMed]

H. J. Tiziani, Opt. Acta 18, 891 (1971).
[CrossRef]

Tokarski, J. M. J.

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

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Phys. (1)

J. A. Leendertz, J. Phys. 3, 214 (1970).

Opt. Acta (4)

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

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

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

H. J. Tiziani, Opt. Acta 18, 891 (1971).
[CrossRef]

Opt. Commun. (1)

H. J. Tiziani, Opt. Commun. 5, 217 (1972).
[CrossRef]

Optik (3)

H. J. Tiziani, Optik 34, 442 (1972).

U. Köpf, Optik 33, 517 (1971).

K. Biedermann, Optik 28, 160 (1968).

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Figures (4)

Fig. 1
Fig. 1

System for recording a speckle photograph. The diffuse object O lies in the x, y plane, with the z axis passing through its center. It is illuminated by two mutually coherent beams of light whose propagation vectors are K1 and K2. The reflected light is imaged by lens L to form the image I.

Fig. 2
Fig. 2

Optical processor for speckle interference. The photographic image, I, obtained in Fig. 1, is placed in a converging beam of coherent light. An aperture is placed in the plane P, where that beam comes to focus and an image, I, of the transparency is formed from the light transmitted by the aperture. When the aperture is properly located, fringes are seen on the object, that can measure strains in the presence of large object displacements.

Fig. 3
Fig. 3

The set up for recording photographs of a compressed disk. A laser beam is expanded and filtered by a lens and a pinhole, LP, and collimated by a lens, L. Half of the beam strikes the object, O, which is a circular disk, at +45° to its surface normal; the other half strikes O at −45°, after reflection from the mirror M. The object is photographed before and after loading, as a double exposure, by camera C.

Fig. 4
Fig. 4

Strain fringes due to the orthogonal expansion of a compressed disk. A Plexiglass disk, whose surface was painted white, was compressed in the vertical direction against a leaf spring. A double-exposure photograph was recorded before and after the loading with the set up of Fig. 3. The fringes show a combination of rotation about the axis of the disk and expansion in the horizontal direction.

Equations (27)

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S 1 = a 1 ( x , y ) exp [ i θ 1 ( x , y ) ]
S 2 = a 1 ( x , y ) exp [ i θ 2 ( x , y ) ] ,
L 0 ( x , y ) = î L x 0 ( x , y ) + j ˆ L y 0 ( x , y ) + k ˆ L z 0 ( x , y ) ,
L i ( x , y ) = î L x i + j ˆ L y i + k ˆ L z i
= î M x L x 0 + j ˆ M y L y 0 + k ˆ M z L z 0 ,
S 1 = a 1 ( x - L x i , y - L y i ) exp [ ( i θ 1 ( x - L x i , y - L y i ) ) ] × exp [ i K 1 · L 0 ] ,
S 2 = a 2 ( x - L x i , y - L y i ) exp [ ( i θ 2 ( x - L x i , y - L y i ) ) ] × exp [ i K 2 · L 0 ] .
E A = T { a 1 2 + a 2 2 + 2 a 1 a 2 cos [ θ ( x , y ) ] } ,
E B = T { a 1 2 ( x - L x i , y - L y i ) + a 2 2 ( x - L x i , y - L y i ) + 2 a 1 ( x - L x i , y - L y i ) a 2 ( x - L x i , y - L y i ) × cos [ θ ( x - L x i , y - L y i ) + K · L 0 ] } ,
τ amp = τ amp ( E av ) + 0.434 M ( ν ) α ( E av ) E A + E B - E av E av ,
T ω { E A + E B } = A 1 ( ω ) * A 1 ( ω ) [ 1 + exp ( - i ω · L i ) ] + A 2 ( ω ) * A 2 ( ω ) [ 1 + exp ( - i ω · L i ) ] + 2 A 1 ( ω ) * A 2 ( ω ) * C ( ω ) [ 1 + cos ( K · L 0 ) exp ( - i ω · L i ) ] - 2 A 1 ( ω ) * A 2 ( ω ) * S ( ω ) sin ( K · L 0 ) exp ( - i ω · L i ) .
C ( ω ) = T ω { cos [ θ ( x , y ) ] } , S ( ω ) = T ω { sin [ θ ( x , y ) ] } .
exp ( - i ω · L i ) = - 1 ,             i . e . ,             ω · L i = ( 2 m - 1 ) π ,
cos ( K · L 0 ) = 1 ,             i . e . ,             K · L 0 = 2 n π .
( 2 L x 0 / λ ) sin ϕ = x M x / d .
x = d L x 0 / d x = λ M x / d 2 sin ϕ .
L 0 = Θ × R
K · L = K · Θ × R = K × Θ · R ,
d y = λ M y / 2 Θ z sin ϕ .
d x , y = M x , y λ / 2 Θ y sin ϕ tan γ ,
f / N x M x / 6.1 λ = M x 12.2 x sin ϕ .
Δ ω · L i π / 2.
Δ ω x = 2 π / λ ( f / N x ) .
f / N x 4 M x / L x 0 / λ .
x λ / ( 50 sin ϕ L x 0 ) ,
T ω { E B } = [ A 1 * A 1 + A 2 * A 2 ] M ( - ω · L i ) + A 1 * A 2 * C [ M ( - ω · L i + K · L 0 ) + M ( - ω · L i - K · L 0 ) ] + i A 1 * A 2 * S [ M ( - ω · L i - K · L 0 ) - M ( - ω · L i + K · L 0 ) ] ,
M ( Ω ) = ( 1 / T ) 0 T exp [ i Ω f ( t ) ] d t