Abstract

A method for determining the directions of displacement in double exposure holographic interferometry is described. The technique requires the use of different frequencies in each of the two holographic exposures and thus can be thought of as a combination of conventional multifrequency contouring and holographic interferometry. The necessary equations to describe the resulting fringe pattern are developed. Two deformation fields, one theoretical and one experimental, are used to illustrate the phenomena. The experimental deformation was the result of a Rayleigh surface wave propagating in a rock specimen. The holograms were obtained with a Q-switched, multifrequency pulsed ruby laser.

© 1978 Optical Society of America

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References

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  1. D. Gabor, Nature 161, 777 (1948).
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  2. E. N. Leith, J. Upatneiks, J. Opt. Soc. Am. 52, 1123 (1962).
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  3. R. E. Brooks, L. O. Heflinger, P. F. Wuerker, Appl. Phys. Lett. 7, 248 (1965).
    [CrossRef]
  4. R. L. Powell, K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
    [CrossRef]
  5. K. A. Haines, B. P. Hildebrand, Phys. Lett. 19, 10 (1965).
    [CrossRef]
  6. J. E. Sollid, Appl. Opt. 8, 1587 (1968).
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  7. P. C. Gupta, A. K. Aggarwal, Appl. Opt. 15, 2961 (1976).
    [CrossRef] [PubMed]
  8. F. Gori, G. Guattari, Opt Commun. 5, 239 (1972).
    [CrossRef]
  9. K. Haines, B. P. Hildebrand, J. Opt. Soc. Am. 52, 155 (1967).
  10. J. S. Zelenka, J. R. Varner, Appl. Opt. 8, 1431 (1969).
    [CrossRef] [PubMed]
  11. J. R. Varner, Appl. Opt. 9, 2098 (1970).
    [CrossRef] [PubMed]
  12. M. V. Klein, Optics (Wiley, New York, 1970).
  13. C. L. Pekeris, H. Lifson, J. Acoust. Soc. Am. 20, 11 (1957).
  14. J. A. Viecelli, Applications of the TENSOR Code to the Calculation of Rayleigh Waves, Lawrence Livermore Laboratory Report UCRL-50992 (1971).
    [CrossRef]

1976 (1)

1972 (1)

F. Gori, G. Guattari, Opt Commun. 5, 239 (1972).
[CrossRef]

1970 (1)

1969 (1)

1968 (1)

1967 (1)

K. Haines, B. P. Hildebrand, J. Opt. Soc. Am. 52, 155 (1967).

1965 (3)

K. A. Haines, B. P. Hildebrand, Phys. Lett. 19, 10 (1965).
[CrossRef]

R. E. Brooks, L. O. Heflinger, P. F. Wuerker, Appl. Phys. Lett. 7, 248 (1965).
[CrossRef]

R. L. Powell, K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
[CrossRef]

1962 (1)

1957 (1)

C. L. Pekeris, H. Lifson, J. Acoust. Soc. Am. 20, 11 (1957).

1948 (1)

D. Gabor, Nature 161, 777 (1948).
[CrossRef] [PubMed]

Aggarwal, A. K.

Brooks, R. E.

R. E. Brooks, L. O. Heflinger, P. F. Wuerker, Appl. Phys. Lett. 7, 248 (1965).
[CrossRef]

Gabor, D.

D. Gabor, Nature 161, 777 (1948).
[CrossRef] [PubMed]

Gori, F.

F. Gori, G. Guattari, Opt Commun. 5, 239 (1972).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, Opt Commun. 5, 239 (1972).
[CrossRef]

Gupta, P. C.

Haines, K.

K. Haines, B. P. Hildebrand, J. Opt. Soc. Am. 52, 155 (1967).

Haines, K. A.

K. A. Haines, B. P. Hildebrand, Phys. Lett. 19, 10 (1965).
[CrossRef]

Heflinger, L. O.

R. E. Brooks, L. O. Heflinger, P. F. Wuerker, Appl. Phys. Lett. 7, 248 (1965).
[CrossRef]

Hildebrand, B. P.

K. Haines, B. P. Hildebrand, J. Opt. Soc. Am. 52, 155 (1967).

K. A. Haines, B. P. Hildebrand, Phys. Lett. 19, 10 (1965).
[CrossRef]

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970).

Leith, E. N.

Lifson, H.

C. L. Pekeris, H. Lifson, J. Acoust. Soc. Am. 20, 11 (1957).

Pekeris, C. L.

C. L. Pekeris, H. Lifson, J. Acoust. Soc. Am. 20, 11 (1957).

Powell, R. L.

Sollid, J. E.

Stetson, K. A.

Upatneiks, J.

Varner, J. R.

Viecelli, J. A.

J. A. Viecelli, Applications of the TENSOR Code to the Calculation of Rayleigh Waves, Lawrence Livermore Laboratory Report UCRL-50992 (1971).
[CrossRef]

Wuerker, P. F.

R. E. Brooks, L. O. Heflinger, P. F. Wuerker, Appl. Phys. Lett. 7, 248 (1965).
[CrossRef]

Zelenka, J. S.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

R. E. Brooks, L. O. Heflinger, P. F. Wuerker, Appl. Phys. Lett. 7, 248 (1965).
[CrossRef]

J. Acoust. Soc. Am. (1)

C. L. Pekeris, H. Lifson, J. Acoust. Soc. Am. 20, 11 (1957).

J. Opt. Soc. Am. (3)

Nature (1)

D. Gabor, Nature 161, 777 (1948).
[CrossRef] [PubMed]

Opt Commun. (1)

F. Gori, G. Guattari, Opt Commun. 5, 239 (1972).
[CrossRef]

Phys. Lett. (1)

K. A. Haines, B. P. Hildebrand, Phys. Lett. 19, 10 (1965).
[CrossRef]

Other (2)

M. V. Klein, Optics (Wiley, New York, 1970).

J. A. Viecelli, Applications of the TENSOR Code to the Calculation of Rayleigh Waves, Lawrence Livermore Laboratory Report UCRL-50992 (1971).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Definition of the vectors used in the analysis of multifrequency holographic interferometry.

Fig. 2
Fig. 2

Midplane of prolate spheroids generated by holding the phase constant in Eq. (10). The intersection of the object plane with the ellipses forms contour fringes in the object plane.

Fig. 3
Fig. 3

Mathematical model of a surface displacement: (a) source and viewing positions; (b) displacement profile.

Fig. 4
Fig. 4

Computer generated fringe patterns of the deformation field of Fig. 3, formed by various combinations of the terms in Eq. (15) with ν1 <ν2: a) all terms; (b) constants from Eq. (15a) along with (15f) yields eq (16) and ν1 in the first exposure and ν2 in the second exposure or B1 =A2 =0 and A1 =B2 =1; (c) constants from Eq. (15a) along with Eq. (15g) and ν2 in the first exposure and ν1 in the second exposure or A1 =B2 =0 and A2 =B1 =1; (d) all terms and B1 =0.25A1, A2 =0.25B2, A1 =B2 =1.

Fig. 5
Fig. 5

Schematic of the ruby laser.

Fig. 6
Fig. 6

Experimental arrangement used in multifrequency holographic interferometry. Numbers in parenthesis refer to positions in cm with respect to an origin at the explosive source.

Fig. 7
Fig. 7

Photograph of a propagating Rayleigh wave in a half space taken using multifrequency holographic interferometry.

Equations (28)

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U = A 1 exp j k 1 [ a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) ] + B 1 exp j k 2 [ a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) ] + A 2 exp j k 1 [ ( a ̅ 1 + Δ a ̅ 1 ) · r ̅ 2 + ( a ̅ 2 + Δ a ̅ 2 ) · ( R ̅ r ̅ 2 ) ] + B 2 exp j k 2 [ ( a ̅ 1 + Δ a ̅ 1 ) · r ̅ 2 + ( a ̅ 2 + Δ a ̅ 2 ) · ( R ̅ r ̅ 2 ) ] ,
UU * = A 1 A 1 * + B 1 B 1 * + A 2 A 2 * + B 2 B 2 * + A 1 A 2 * + A 2 A 1 * + B 1 B 2 * + B 2 B 1 * + A 1 B 1 * + B 1 A 1 * + A 2 B 2 * + B 2 A 2 * + A 1 B 2 * + B 2 A 1 * + B 1 A 2 * + A 2 B 1 * ,
A 1 A 1 * = A 1 2 exp j k 1 [ a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) a ̅ 1 · r ̅ 1 a ̅ 2 · ( R ̅ r ̅ 1 ) ] = A 1 2 ,
B 1 B 1 * = B 1 2 , A 2 A 2 * = A 2 2 , B 2 B 2 * = B 2 2 .
A 1 A 2 * = A 1 A 2 exp j k 1 [ a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) ( a ̅ 1 + Δ a ̅ 1 ) · r ̅ 2 ( a ̅ 2 + Δ a ̅ 2 ) · ( R ̅ r ̅ 2 ) ] .
δ = ( a ̅ 1 · r ̅ 1 + a ̅ 2 · R ̅ a ̅ 2 · r ̅ 1 a ̅ 1 r ̅ 2 Δ a ̅ 1 · r ̅ 2 a ̅ 2 · R ̅ + a ̅ 2 · r ̅ 2 a ̅ 2 · R ̅ + a ̅ 2 · r ̅ 2 ) ,
δ = ( a ̅ 1 a ̅ 2 ) · ( r ̅ 1 r ̅ 2 ) Δ a ̅ 2 · ( R ̅ r ̅ 2 ) Δ a ̅ 1 · r ̅ 2 .
A 1 A 2 * + A 2 A 1 * = 2 A 1 A 2 cos k 1 [ ( a ̅ 1 a ̅ 2 ) · ( r ̅ 1 r ̅ 2 ) ] .
B 1 B 2 * + B 2 B 1 * = 2 B 1 B 2 cos k 2 [ ( a ̅ 1 a ̅ 2 ) · ( r ̅ 1 r ̅ 2 ) ] .
A 1 B 1 * = A 1 B 1 exp j [ ( k 1 k 2 ) ( a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) ] , B 1 A 1 * = A 1 B 1 exp j [ ( k 1 k 2 ) ( a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) ] .
A 1 B 1 * + B 1 A 1 * = 2 A 1 B 1 cos ( k 1 k 2 ) [ a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) ] ,
A 2 B 2 * + B 2 A 2 * = 2 A 2 B 2 cos ( k 1 k 2 ) [ a ̅ 1 · r ̅ 2 + a ̅ 2 · ( R ̅ r ̅ 2 ) ] .
A 1 B 2 * = A 1 B 2 exp j { k 1 [ a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) ] k 2 [ ( a ̅ 1 + Δ a ̅ 1 ) · r ̅ 2 + ( a ̅ 2 + Δ a ̅ 2 ) · ( R ̅ r ̅ 2 ) ] } , B 2 A 1 * = A 1 B 2 exp j { k 1 [ a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) ] k 2 [ ( a ̅ 1 + Δ a ̅ 1 ) · r ̅ 2 + ( a ̅ 2 + Δ a ̅ 2 ) · ( R ̅ r ̅ 2 ) ] } .
A 1 B 2 * + B 2 A 1 * = 2 A 1 B 2 cos { ( k 1 r ̅ 1 k 2 r ̅ 2 ) · a ̅ 1 + [ k 1 ( R ̅ r ̅ 1 ) k 2 ( R ̅ r ̅ 2 ) ] · a ̅ 2 } .
B 1 A 2 * + A 2 B 1 * = 2 A 2 B 1 cos { ( k 1 r ̅ 2 k 2 r ̅ 1 ) · a ̅ 1 + [ k 1 ( R ̅ r ̅ 2 ) k 2 ( R ̅ r ̅ 1 ) ] · a ̅ 2 } .
UU * = A 1 2 + B 1 2 + A 2 2 + B 2 2 } Constant
+ 2 A 1 A 2 cos k 1 [ ( a ̅ 1 a ̅ 2 ) · ( r ̅ 1 r ̅ 2 ) ] + 2 B 1 B 2 cos k 2 [ ( a ̅ 1 a ̅ 2 ) · ( r ̅ 1 r ̅ 2 ) ] } Displacement
+ 2 A 1 B 1 cos [ ( k 1 k 2 ) ( a ̅ 1 · r ̅ 1 + a ̅ 2 · ( R ̅ r ̅ 1 ) ] + 2 A 2 B 2 cos [ ( k 1 k 2 ) ( a ̅ 1 · r ̅ 2 + a ̅ 2 · ( R ̅ r ̅ 2 ) ] } Contour
+ 2 A 1 B 2 cos { ( k 1 r ̅ 1 k 2 r ̅ 2 ) · a ̅ 1 + 2 A 2 B 1 cos { ( k 1 r ̅ 2 k 2 r ̅ 1 ) · a ̅ 1
+ [ k 1 ( R ̅ r ̅ 1 ) k 2 ( R ̅ r ̅ 2 ) ] · a ̅ 2 } + [ k 1 ( R ̅ r ̅ 2 ) k 2 ( R ̅ r ̅ 1 ) ] · a ̅ 2 } } Mixed .
UU * = A 1 2 + B 2 2 + 2 A 1 B 2 cos { ( k 1 r ̅ 1 k 2 r ̅ 2 ) · a ̅ 1 + [ k 1 ( R ̅ r ̅ 1 ) k 2 ( R ̅ r ̅ 2 ) ] · a ̅ 2 }
UU * = A 1 2 + B 2 2 + 2 A 1 B 2 cos { k 0 ( a ̅ 1 a ̅ 2 ) · 2 Δ r ̅ + 2 Δ k [ r ̅ 0 · ( a ̅ 1 a ̅ 2 ) + R ̅ · a ̅ 2 ] } .
Z = W sin [ π 3 ( r 1.0 ) ] 1.0 cm r 4.0 cm Z = 0 { r < 1.0 cm r > 4.0 cm ,
A 1 ( ν 1 ) B 1 ( ν 2 ) = B 2 ( ν 2 ) A 2 ( ν 1 ) ,
Δ ν = ν m + 1 ν m = ( c 0 ) / ( 2 n l ) ,
Δ R = ( c 0 ) / [ 2 ( ν 2 ν 1 ) ] .
Δ R = nl .
I = A 1 2 + B 2 2 + 2 A 1 B 2 cos { ( k 1 r ̅ 1 k 2 r ̅ 2 ) · a 1 + [ k 1 ( R ̅ r ̅ 1 ) k 2 ( R ̅ r ̅ 2 ) ] · a ̅ 2 } ,

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