Abstract

A detailed analysis of using multiple apertures to record laser speckles for strain analysis is presented. It is shown that the basic fringe-forming mechanism is no different from that of single-aperture recording, except that at the Fourier filtering stage the diffracted light energy is concentrated at the spatial frequencies admitted by the apertures. As a result, better isothetic fringes at higher frequencies are obtainable. The concept of moiré is not utilized in the analysis. Indeed, it is shown that some of the observed phenomena cannot be explained using the moiré concept. Multiaperture arrangements studied include two, three, and four apertures.

© 1979 Optical Society of America

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References

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  1. J. M. Burch, J. M. J. Tokarski, Opt. Acta 15, 101 (1968).
  2. E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 17, 883 (1970).
    [CrossRef]
  3. E. Archbold, A. E. Ennos, Opt. Acta 19, 253 (1972).
    [CrossRef]
  4. R. P. Khetan, F. P. Chiang, Appl. Opt. 15, 2205 (1976).
    [CrossRef] [PubMed]
  5. F. P. Chiang, R. M. Juang, Appl. Opt. 15, 2199 (1976).
    [CrossRef] [PubMed]
  6. F. P. Chiang, C. H. Lee, Appl. Opt. 19, 3085 (1977).
    [CrossRef]
  7. F. P. Chiang, R. M. Juang, Opt. Acta 23, 997 (1976).
    [CrossRef]
  8. F. P. Chiang, in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U. P., London, 1977, p. 249.
  9. D. B. Baker, M. J. Fourney, Exp. Mech. 16, 209 (1977).
    [CrossRef]
  10. D. E. Duffy, Appl. Opt. 11, 1787 (1972).
    [CrossRef]
  11. Y. Y. Hung, C. E. Taylor, Exp. Mech. 14, 281 (1974).
    [CrossRef]
  12. F. P. Chiang, in Proceedings of Conference on Speckle Phenomena and Their Applications, Loughborough University, 27, 28 March 1974 (Summary).
  13. Y. Y. Hung, C. E. Taylor, C. P. Hu, in Proceedings, Seventh Southeastern Conference on Theoretical and Applied Mechanics (1974), p. 497.
  14. J. M. Burch, C. Forno, Opt. Eng. 15, 178 (1974).
  15. A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).
  16. F. P. Chiang, Manual on Experimental Stress Analysis, A. S. Kobayashi, Ed. (Society for Experimental Stress Analysis, Westport, Conn., 1978), Chap. 6, p. 51.
  17. F. P. Chiang, J. Eng. Mech. Div. Proc. ASCE 96 (EM6), 1285 (1970).
  18. F. P. Chiang, J. Eng. Mech. Div. Proc. ASCE 95 (EM6), 1379 (1969).
  19. F. P. Chiang, G. Jaisingh, paper presented at SESA Spring Meeting, San Francisco, 20–25 May 1979 (Paper R79-1311).
  20. F. P. Chiang, T. Y. Kao, Mech. Res. Commun. 5(3), 133 (1978).
    [CrossRef]
  21. F. P. Chiang, R. P. Khetan, “Strain Analysis by One Beam Laser Speckle Interferometry Part II: Multi-Aperture Method,” SUNY Stony Brook, College of Engineering Technical Report No. 253, December1974 (revised November 1978).

1978 (1)

F. P. Chiang, T. Y. Kao, Mech. Res. Commun. 5(3), 133 (1978).
[CrossRef]

1977 (2)

F. P. Chiang, C. H. Lee, Appl. Opt. 19, 3085 (1977).
[CrossRef]

D. B. Baker, M. J. Fourney, Exp. Mech. 16, 209 (1977).
[CrossRef]

1976 (3)

1974 (2)

Y. Y. Hung, C. E. Taylor, Exp. Mech. 14, 281 (1974).
[CrossRef]

J. M. Burch, C. Forno, Opt. Eng. 15, 178 (1974).

1972 (2)

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

D. E. Duffy, Appl. Opt. 11, 1787 (1972).
[CrossRef]

1970 (2)

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

F. P. Chiang, J. Eng. Mech. Div. Proc. ASCE 96 (EM6), 1285 (1970).

1969 (1)

F. P. Chiang, J. Eng. Mech. Div. Proc. ASCE 95 (EM6), 1379 (1969).

1968 (1)

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

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]

Baker, D. B.

D. B. Baker, M. J. Fourney, Exp. Mech. 16, 209 (1977).
[CrossRef]

Burch, J. M.

J. M. Burch, C. Forno, Opt. Eng. 15, 178 (1974).

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, T. Y. Kao, Mech. Res. Commun. 5(3), 133 (1978).
[CrossRef]

F. P. Chiang, C. H. Lee, Appl. Opt. 19, 3085 (1977).
[CrossRef]

F. P. Chiang, R. M. Juang, Opt. Acta 23, 997 (1976).
[CrossRef]

R. P. Khetan, F. P. Chiang, Appl. Opt. 15, 2205 (1976).
[CrossRef] [PubMed]

F. P. Chiang, R. M. Juang, Appl. Opt. 15, 2199 (1976).
[CrossRef] [PubMed]

F. P. Chiang, J. Eng. Mech. Div. Proc. ASCE 96 (EM6), 1285 (1970).

F. P. Chiang, J. Eng. Mech. Div. Proc. ASCE 95 (EM6), 1379 (1969).

F. P. Chiang, G. Jaisingh, paper presented at SESA Spring Meeting, San Francisco, 20–25 May 1979 (Paper R79-1311).

F. P. Chiang, R. P. Khetan, “Strain Analysis by One Beam Laser Speckle Interferometry Part II: Multi-Aperture Method,” SUNY Stony Brook, College of Engineering Technical Report No. 253, December1974 (revised November 1978).

F. P. Chiang, in Proceedings of Conference on Speckle Phenomena and Their Applications, Loughborough University, 27, 28 March 1974 (Summary).

F. P. Chiang, in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U. P., London, 1977, p. 249.

F. P. Chiang, Manual on Experimental Stress Analysis, A. S. Kobayashi, Ed. (Society for Experimental Stress Analysis, Westport, Conn., 1978), Chap. 6, p. 51.

Duffy, D. E.

D. E. Duffy, Appl. Opt. 11, 1787 (1972).
[CrossRef]

Durelli, A. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

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]

Forno, C.

J. M. Burch, C. Forno, Opt. Eng. 15, 178 (1974).

Fourney, M. J.

D. B. Baker, M. J. Fourney, Exp. Mech. 16, 209 (1977).
[CrossRef]

Hu, C. P.

Y. Y. Hung, C. E. Taylor, C. P. Hu, in Proceedings, Seventh Southeastern Conference on Theoretical and Applied Mechanics (1974), p. 497.

Hung, Y. Y.

Y. Y. Hung, C. E. Taylor, Exp. Mech. 14, 281 (1974).
[CrossRef]

Y. Y. Hung, C. E. Taylor, C. P. Hu, in Proceedings, Seventh Southeastern Conference on Theoretical and Applied Mechanics (1974), p. 497.

Jaisingh, G.

F. P. Chiang, G. Jaisingh, paper presented at SESA Spring Meeting, San Francisco, 20–25 May 1979 (Paper R79-1311).

Juang, R. M.

Kao, T. Y.

F. P. Chiang, T. Y. Kao, Mech. Res. Commun. 5(3), 133 (1978).
[CrossRef]

Khetan, R. P.

R. P. Khetan, F. P. Chiang, Appl. Opt. 15, 2205 (1976).
[CrossRef] [PubMed]

F. P. Chiang, R. P. Khetan, “Strain Analysis by One Beam Laser Speckle Interferometry Part II: Multi-Aperture Method,” SUNY Stony Brook, College of Engineering Technical Report No. 253, December1974 (revised November 1978).

Lee, C. H.

F. P. Chiang, C. H. Lee, Appl. Opt. 19, 3085 (1977).
[CrossRef]

Parks, V. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Taylor, C. E.

Y. Y. Hung, C. E. Taylor, Exp. Mech. 14, 281 (1974).
[CrossRef]

Y. Y. Hung, C. E. Taylor, C. P. Hu, in Proceedings, Seventh Southeastern Conference on Theoretical and Applied Mechanics (1974), p. 497.

Tokarski, J. M. J.

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

Appl. Opt. (4)

Exp. Mech. (2)

Y. Y. Hung, C. E. Taylor, Exp. Mech. 14, 281 (1974).
[CrossRef]

D. B. Baker, M. J. Fourney, Exp. Mech. 16, 209 (1977).
[CrossRef]

J. Eng. Mech. Div. Proc. ASCE (2)

F. P. Chiang, J. Eng. Mech. Div. Proc. ASCE 96 (EM6), 1285 (1970).

F. P. Chiang, J. Eng. Mech. Div. Proc. ASCE 95 (EM6), 1379 (1969).

Mech. Res. Commun. (1)

F. P. Chiang, T. Y. Kao, Mech. Res. Commun. 5(3), 133 (1978).
[CrossRef]

Opt. Acta (4)

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]

F. P. Chiang, R. M. Juang, Opt. Acta 23, 997 (1976).
[CrossRef]

Opt. Eng. (1)

J. M. Burch, C. Forno, Opt. Eng. 15, 178 (1974).

Other (7)

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

F. P. Chiang, Manual on Experimental Stress Analysis, A. S. Kobayashi, Ed. (Society for Experimental Stress Analysis, Westport, Conn., 1978), Chap. 6, p. 51.

F. P. Chiang, R. P. Khetan, “Strain Analysis by One Beam Laser Speckle Interferometry Part II: Multi-Aperture Method,” SUNY Stony Brook, College of Engineering Technical Report No. 253, December1974 (revised November 1978).

F. P. Chiang, G. Jaisingh, paper presented at SESA Spring Meeting, San Francisco, 20–25 May 1979 (Paper R79-1311).

F. P. Chiang, in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U. P., London, 1977, p. 249.

F. P. Chiang, in Proceedings of Conference on Speckle Phenomena and Their Applications, Loughborough University, 27, 28 March 1974 (Summary).

Y. Y. Hung, C. E. Taylor, C. P. Hu, in Proceedings, Seventh Southeastern Conference on Theoretical and Applied Mechanics (1974), p. 497.

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Figures (15)

Fig. 1
Fig. 1

Multiaperture arrangement for recording laser speckles.

Fig. 2
Fig. 2

Fourier filtering of recorded speckles.

Fig. 3
Fig. 3

Schematic representation of normalized A1(r1,r2) and F1(v1,v2).

Fig. 4
Fig. 4

Contour plot and sectional view of normalized G1(u1,u2).

Fig. 5
Fig. 5

Contour plot and sectional view of normalized I1(u1,u2).

Fig. 6
Fig. 6

Sectional view of normalized I(u1,u2) with a rigid body translation between exposures.

Fig. 7
Fig. 7

Two-aperture recording; numerals denote peak values of normalized functions: (a) A2(r1,r2); (b) F2(v1,v2)/C2; and (c) G2(u1,u2)/(2LA2C2)/q] or I2(u1,u2)/k2.

Fig. 8
Fig. 8

Three-aperture recording; numerals denote peak values of normalized functions: (a) A3(r1,r2); (b) F3(v1,v2)/C3; and (c) G3(u1,u2)/[(2LA2C3/q]2 or I3(u1,u2)/k3.

Fig. 9
Fig. 9

Four-aperture recording; numerals denote peak values of normalized functions: (a) A4(r1,r2); (b) F4(v1,v2)/C4; and (c) G4(u1,u2)/[(2LA4C4)/q]2 or I4(u1u2)/k4.

Fig. 10
Fig. 10

Diffraction spectra of double-exposure speckle patterns with horizontal rigid body displacement as recorded by (a) two-, (b) three-, and (c) four-aperture arrangements.

Fig. 11
Fig. 11

Isothetic fringes of a disk under rigid body rotation as recorded by a four-aperture arrangement: (a) d; (b) d90° and (c) d45°.

Fig. 12
Fig. 12

Postrecording sensitivity and orientation change of a four-aperture recorded speckle interferogram.

Fig. 13
Fig. 13

Typical fringe densities obtainable by four-aperture recording: (a) strain induced and (b) rigid body rotation.

Fig. 14
Fig. 14

Comparison between theory and experiment of a cantilever experiment.

Fig. 15
Fig. 15

Dynamic speckle isothetics showing propagating stress wave as obtained by a four-aperture arrangement.

Equations (45)

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f 1 ( x 1 , x 2 ) = K h ( ξ 1 , ξ 2 ) A 1 ( 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 ,
I ( u 1 , u 2 ) = 4 cos 2 k ( d · u 2 L ) I 1 ( u 1 , u 2 ) ,
I 1 ( u 1 , u 2 ) = K 0 G 1 ( u 1 , u 2 ) 2 ,
G 1 ( u 1 , u 2 ) = F 1 ( v 1 , v 2 ) F 1 * ( v 1 - u 1 , v 2 - u 2 ) d v 1 d v 2 ,
F 1 ( v 1 , v 2 ) = f 1 ( x 1 , x 2 ) exp [ i k ( x 1 v 1 + x 2 v 2 ) / L ] d x 1 d x 2 .
F 1 ( v 1 , v 2 ) = c 1 A 1 ( - v 1 q L , - v 2 q L ) ,
A 1 ( r 1 , r 2 ) = { 1 ,             for r 1 , r 2 a , 0 ,             for r 1 , r 2 > a ;
F 1 ( v 1 , v 2 ) = { c 1 ,             for v 1 , v 2 L a / q , 0 ,             for v 1 , v 2 > L a / q ;
G 1 ( u 1 , u 2 ) = { ( 2 L a q c 1 ) 2 ( 1 - u 2 q 2 L a ) ( 1 - u 2 q 2 L a ) , for u 1 , u 2 2 L a / q , 0 ,             for u 1 , u 2 > 2 L a / q ;
I 1 ( u 1 , u 2 ) = { K 1 ( 1 - u 1 q 2 L a ) 2 ( 1 - u 2 q 2 L a ) 2 , for u 1 , u 2 2 L a / q , 0 ,             for u 1 , u 2 > 2 L a / q ;
( I 1 ) T = I 1 ( u 1 , u 2 ) d u 1 d u 2 = 4 K 1 0 2 L a / q 0 2 L a / q ( 1 - u 1 q 2 L a ) 2 ( 1 - u 2 q 2 L a ) 2 d u 1 d u 2 = 16 9 ( L a q ) 2 K 1 .
A 2 ( r 1 , r 2 ) = { 1 , for ( a 2 - a ) r 1 ( a 2 + a ) and r 2 a , 0 , otherwise .
F 2 ( v 1 , v 2 ) = { c 2 , for L ( a 2 - a ) q v 1 L ( a 2 + a ) q and v 2 L a q , 0 , otherwise ,
( I 2 ) T = 6 × 16 9 ( L a q ) 2 K 2 ,
( I 1 ) T = 16 9 [ L ( a + a 2 ) q ] 2 K 1 .
( I 2 ) T = ( I 1 ) T .
K 2 K 1 = 1 6 ( a + a 2 a ) 2 ,
I 2 ( 2 L a 2 q , 0 ) = K 2 .
I 1 ( 2 L a 2 q , 0 ) = K 1 ( a a + a 2 ) 2 .
m 2 = ( a + a 2 a ) 2 K 2 K 1             or             m 2 = 1 6 ( a + a 2 a ) 4 .
A 3 = { 1 , for ( a 3 - a ) r 1 ( a 3 + a ) , r 2 a , and r 1 a , ( 3 a 3 - a ) r 2 ( 3 a 3 + a ) , 0 , otherwise ;
F 3 = { c 3 , for L q ( a 3 - a ) v 1 L q ( a 3 + a ) , v 2 L a q , and v 1 L a q , L q ( 3 a 3 - a ) v 2 v 2 L q ( 3 a 3 + a ) , 0 , otherwise .
( I 3 ) T = 15 × 16 9 ( L a q ) 2 K 3 .
( I 1 ) T = 16 9 [ L ( a 3 + a ) q ] 2 K 1 .
I 3 ( 2 L a 3 q , 0 ) = K 3 .
I 1 ( 2 L a 3 q , 0 ) = K 1 ( a a + a 3 ) 2 .
m 3 = I 3 I 1 = ( a 3 + a a ) 2 K 3 K 1 = 1 15 ( a 3 + a a ) 4 .
I 3 ( L a 3 q , 3 L a 3 q ) = K 3 .
I 1 ( L a 3 q , 3 L a 3 q ) = K 1 [ 1 - a 3 2 ( a 3 + a ) ] 2 [ 1 - 3 a 3 2 ( a 3 + a ) ] 2 .
m 3 = 1 15 ( a 3 + a a ) 2 / { [ 1 - a 3 2 ( a 3 + a ) ] 2 [ 1 - 3 a 3 2 ( a 3 + a ) ] 2 } 318
A r ( r 1 , r 2 ) = { 1 , for ( a 4 - a ) r 1 ( a 4 + a ) , r 2 a or r 1 a , ( a 4 - a ) r 2 ( a 4 + a ) , 0 , otherwise ,
F 4 ( v 1 , v 2 ) = { c 4 , for L q ( a 4 - a ) v 1 L q ( a 4 + a ) , v 2 L a q or v 1 L a q , L q ( a 4 - a ) v 2 L q ( a 4 + a ) , 0 , otherwise .
( I 4 ) T = 36 × 16 9 ( L a q ) 2 K 4 .
( I 1 ) T = 16 9 L q ( a 4 + a ) 2 K 1 .
I 4 ( 2 L a 4 q , 0 ) = I 4 ( 0 , 2 L a 4 q ) = K 4 .
I 1 ( 2 L a 4 q , 0 ) = I 1 ( 0 , 2 L a 4 q ) = K 1 ( a a + a 4 ) 2 .
m 4 = 1 36 ( a 4 + a a ) 4 .
I 4 ( L a 4 q , L a 4 q ) = 4 K 4 .
I 1 ( L a 4 q , L a 4 q ) = K 1 | a 4 + 2 a 2 ( a 4 + a ) | 4 .
m 4 = 1 9 ( a 4 + a a ) 2 / [ a 4 + 2 a 2 ( a 4 + a ) ] 4 .
d θ = ( n λ L ) / u θ ,             n = 0 , ± 1 , ± 2 , ,
d θ = ( n + ½ ) [ ( λ L ) / u θ ] ,             n = 0 , ± 1 , ± 2 , ,
θ i = d θ i / s i ,
1 2 } = 1 3 ( θ 1 + θ 2 + θ 3 ) ± 2 3 [ ( θ 1 - θ 2 ) 2 + ( θ 2 - θ 3 ) 2 + ( θ 3 - θ 1 ) 2 ] 1 / 2 .
x = d x x ,             y = d y y x y = d x y + d y x } .

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