Abstract

The application of differential interferometry to bulk wave measurements can be analyzed using the kinematic formulation of the photoelastic effect. For isotropic linearly elastic materials, a standing longitudinal wave changes the indicatrix from a sphere to an ellipsoid of revolution with axes parallel and perpendicular to the direction of propagation of the acoustic wave. If the probing light is linearly polarized, differential interferometry produces a maximum signal when the polarization of the light is perpendicular to the direction of propagation. This prediction is confirmed by experiment. The case of standing transverse waves can be analyzed in a similar manner.

© 1983 Optical Society of America

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References

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  1. A. Toepler, “Optische Studien nach der Methode der Sclierenbeobachtung,” Pogg. Ann. Phys. 131, 33 (1867).
  2. L. Bergmann, Der Ultraschall (Berlin, 1942;Edwards Brothers Lithoprint, Ann Arbor, 1944).
  3. R. L. Whitman, A. Korpel, Appl. Opt. 8, 1567 (1969).
    [CrossRef] [PubMed]
  4. R. M. White, Proc. IEEE 58, 1238 (1970).
    [CrossRef]
  5. G. I. Stegeman, IEEE Trans. Sonics Ultrason. SU-23, 33 (1976).
    [CrossRef]
  6. R. Adler, A. Korpel, P. Desmares, IEEE Trans. Sonics Ultrason. SU-15, 157 (1968).
    [CrossRef]
  7. E. P. Ippen, Proc. IEEE (Lett. 55, 248 (1967);D. C. Auth, W. G. Mayer, J. Appl. Phys. 38, 5138 (1967); and others.
    [CrossRef]
  8. R. L. Whitman, L. J. Laub, W. Bates, IEEE Trans. Sonics Ultrason SU-15, 186 (1968).
    [CrossRef]
  9. C. H. Palmer, H. M. South, T. H. Mak, Ultrasonics 11, 106 (1974).
    [CrossRef]
  10. C. H. Palmer, R. O. Claus, S. E. Fick, Appl. Opt. 16, 1849 (1977).
    [CrossRef] [PubMed]
  11. C. H. Palmer, R. E. Green, “Optical Probing of Acoustic Emission Waves,” in Nondestructive Evaluation of Materials, J. J. Burke, V. Weiss, Eds. (Plenum, New York, 1979, pp. 347–378.
    [CrossRef]
  12. R. L. Whitman, A. Korpel, Appl. Opt. 8, 1567 (1969).
    [CrossRef] [PubMed]
  13. H. M. South, “Investigations of the Reflection, Transmission, and Conversion of Ultrasonic Waves,” Doctoral Dissertation, Electrical Engineering Department, The Johns Hopkins U., Baltimore (1981).
  14. J. T. Nye, Physical Properties of Crystals(Oxford U. P, London, 1972), pp. 235–238.
  15. Ref. 14, pp. 243–256.
  16. R. E. Green, Ultrasonic Investigation of Mechanical Properties (Academic, New York, p. 1973, 6.
  17. R. W. Dixon, J. Appl. Phys. 38, 5149 (1967).
    [CrossRef]

1977 (1)

1976 (1)

G. I. Stegeman, IEEE Trans. Sonics Ultrason. SU-23, 33 (1976).
[CrossRef]

1974 (1)

C. H. Palmer, H. M. South, T. H. Mak, Ultrasonics 11, 106 (1974).
[CrossRef]

1970 (1)

R. M. White, Proc. IEEE 58, 1238 (1970).
[CrossRef]

1969 (2)

1968 (2)

R. Adler, A. Korpel, P. Desmares, IEEE Trans. Sonics Ultrason. SU-15, 157 (1968).
[CrossRef]

R. L. Whitman, L. J. Laub, W. Bates, IEEE Trans. Sonics Ultrason SU-15, 186 (1968).
[CrossRef]

1967 (2)

R. W. Dixon, J. Appl. Phys. 38, 5149 (1967).
[CrossRef]

E. P. Ippen, Proc. IEEE (Lett. 55, 248 (1967);D. C. Auth, W. G. Mayer, J. Appl. Phys. 38, 5138 (1967); and others.
[CrossRef]

1867 (1)

A. Toepler, “Optische Studien nach der Methode der Sclierenbeobachtung,” Pogg. Ann. Phys. 131, 33 (1867).

Adler, R.

R. Adler, A. Korpel, P. Desmares, IEEE Trans. Sonics Ultrason. SU-15, 157 (1968).
[CrossRef]

Bates, W.

R. L. Whitman, L. J. Laub, W. Bates, IEEE Trans. Sonics Ultrason SU-15, 186 (1968).
[CrossRef]

Bergmann, L.

L. Bergmann, Der Ultraschall (Berlin, 1942;Edwards Brothers Lithoprint, Ann Arbor, 1944).

Claus, R. O.

Desmares, P.

R. Adler, A. Korpel, P. Desmares, IEEE Trans. Sonics Ultrason. SU-15, 157 (1968).
[CrossRef]

Dixon, R. W.

R. W. Dixon, J. Appl. Phys. 38, 5149 (1967).
[CrossRef]

Fick, S. E.

Green, R. E.

R. E. Green, Ultrasonic Investigation of Mechanical Properties (Academic, New York, p. 1973, 6.

C. H. Palmer, R. E. Green, “Optical Probing of Acoustic Emission Waves,” in Nondestructive Evaluation of Materials, J. J. Burke, V. Weiss, Eds. (Plenum, New York, 1979, pp. 347–378.
[CrossRef]

Ippen, E. P.

E. P. Ippen, Proc. IEEE (Lett. 55, 248 (1967);D. C. Auth, W. G. Mayer, J. Appl. Phys. 38, 5138 (1967); and others.
[CrossRef]

Korpel, A.

Laub, L. J.

R. L. Whitman, L. J. Laub, W. Bates, IEEE Trans. Sonics Ultrason SU-15, 186 (1968).
[CrossRef]

Mak, T. H.

C. H. Palmer, H. M. South, T. H. Mak, Ultrasonics 11, 106 (1974).
[CrossRef]

Nye, J. T.

J. T. Nye, Physical Properties of Crystals(Oxford U. P, London, 1972), pp. 235–238.

Palmer, C. H.

C. H. Palmer, R. O. Claus, S. E. Fick, Appl. Opt. 16, 1849 (1977).
[CrossRef] [PubMed]

C. H. Palmer, H. M. South, T. H. Mak, Ultrasonics 11, 106 (1974).
[CrossRef]

C. H. Palmer, R. E. Green, “Optical Probing of Acoustic Emission Waves,” in Nondestructive Evaluation of Materials, J. J. Burke, V. Weiss, Eds. (Plenum, New York, 1979, pp. 347–378.
[CrossRef]

South, H. M.

C. H. Palmer, H. M. South, T. H. Mak, Ultrasonics 11, 106 (1974).
[CrossRef]

H. M. South, “Investigations of the Reflection, Transmission, and Conversion of Ultrasonic Waves,” Doctoral Dissertation, Electrical Engineering Department, The Johns Hopkins U., Baltimore (1981).

Stegeman, G. I.

G. I. Stegeman, IEEE Trans. Sonics Ultrason. SU-23, 33 (1976).
[CrossRef]

Toepler, A.

A. Toepler, “Optische Studien nach der Methode der Sclierenbeobachtung,” Pogg. Ann. Phys. 131, 33 (1867).

White, R. M.

R. M. White, Proc. IEEE 58, 1238 (1970).
[CrossRef]

Whitman, R. L.

Appl. Opt. (3)

IEEE Trans. Sonics Ultrason (1)

R. L. Whitman, L. J. Laub, W. Bates, IEEE Trans. Sonics Ultrason SU-15, 186 (1968).
[CrossRef]

IEEE Trans. Sonics Ultrason. (2)

G. I. Stegeman, IEEE Trans. Sonics Ultrason. SU-23, 33 (1976).
[CrossRef]

R. Adler, A. Korpel, P. Desmares, IEEE Trans. Sonics Ultrason. SU-15, 157 (1968).
[CrossRef]

J. Appl. Phys. (1)

R. W. Dixon, J. Appl. Phys. 38, 5149 (1967).
[CrossRef]

Pogg. Ann. Phys. (1)

A. Toepler, “Optische Studien nach der Methode der Sclierenbeobachtung,” Pogg. Ann. Phys. 131, 33 (1867).

Proc. IEEE (1)

R. M. White, Proc. IEEE 58, 1238 (1970).
[CrossRef]

Proc. IEEE (Lett. (1)

E. P. Ippen, Proc. IEEE (Lett. 55, 248 (1967);D. C. Auth, W. G. Mayer, J. Appl. Phys. 38, 5138 (1967); and others.
[CrossRef]

Ultrasonics (1)

C. H. Palmer, H. M. South, T. H. Mak, Ultrasonics 11, 106 (1974).
[CrossRef]

Other (6)

L. Bergmann, Der Ultraschall (Berlin, 1942;Edwards Brothers Lithoprint, Ann Arbor, 1944).

C. H. Palmer, R. E. Green, “Optical Probing of Acoustic Emission Waves,” in Nondestructive Evaluation of Materials, J. J. Burke, V. Weiss, Eds. (Plenum, New York, 1979, pp. 347–378.
[CrossRef]

H. M. South, “Investigations of the Reflection, Transmission, and Conversion of Ultrasonic Waves,” Doctoral Dissertation, Electrical Engineering Department, The Johns Hopkins U., Baltimore (1981).

J. T. Nye, Physical Properties of Crystals(Oxford U. P, London, 1972), pp. 235–238.

Ref. 14, pp. 243–256.

R. E. Green, Ultrasonic Investigation of Mechanical Properties (Academic, New York, p. 1973, 6.

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Figures (6)

Fig. 1
Fig. 1

Standing longitudinal wave pattern near a piezoelectric transducer mounted on a fused quartz prism.

Fig. 2
Fig. 2

Bulk wave probe using differential interferometry.

Fig. 3
Fig. 3

Standing longitudinal wave pattern in glass, measured using differential interferometry.

Fig. 4
Fig. 4

Indicatrix orientation.

Fig. 5
Fig. 5

Calculated phase shift as a function of polarization orientation, fused quartz specimen.

Fig. 6
Fig. 6

Measured phase shift as a function of polarization orientation, fused quartz specimen.

Tables (1)

Tables Icon

Table I Parameters for Phase Shift Calculation

Equations (22)

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u i = A α i exp [ j ( ω t k l j x j ) ] i , j = 1 , 2 , 3 ,
u 1 = A exp [ j ( ω t k x 1 ) ] + A exp [ j ( ω t + k x 1 ) ]
u 1 = 2 A cos ( k x 1 ) exp ( j ω t ) .
i j = 1 2 ( u i x j + u j x i ) .
11 = 2 k A sin ( k x 1 ) exp ( j ω t ) .
B 1 x 1 2 + B 2 x 2 2 + B 3 x 3 2 + 2 B 4 x 2 x 3 + 2 B 5 x 3 x 1 + 2 B 6 x 1 x 2 = 1 .
B i = B i 0 + Δ B i .
Δ B i = p i j j .
B 1 = B 0 + p 11 1 ; B 2 = B 3 = B 0 + p 21 1 ; B 4 = B 5 = B 6 = 0 .
d B i d n i = 2 n i 3 .
Δ B i 2 n i 3 Δ n i .
Δ n 1 = ½ n 0 3 p 11 1 , Δ n 2 = Δ n 3 = ½ n 0 3 p 21 1 .
Δ n 1 = n 0 3 p 11 A k sin ( k x 1 ) exp ( j ω t ) , Δ n 2 = Δ n 3 = n 0 3 p 21 A k sin ( k x 1 ) exp ( j ω t ) .
n i = n 0 + Δ n i , n i = n 0 + Δ n i , at x 1 = q Λ , at x 1 = ( q + 1 2 ) Λ .
n 2 = n 1 2 n 2 2 n 1 2 + ( n 2 2 n 1 2 ) cos 2 θ , n 2 = n 1 2 n 2 2 n 1 2 + ( n 2 2 n 1 2 ) cos 2 θ .
ϕ = ( n n ) W 2 π λ .
n = n 1 = n 0 + Δ n 1 n = n 1 = n 0 + Δ n 1 .
n n = n 0 3 p 11 A k exp ( j ω t ) [ sin k q Λ sin k ( q + ½ ) Λ ] , n n = n 0 3 p 11 A k exp ( j ω t ) [ sin 2 π q sin ( 2 π q + π ) ] , n n = n 0 3 p 11 A k exp ( j ω t ) 2 sin 2 π q .
u 3 = A exp [ ( j ω t k x 1 ) ] + A exp [ ( j ω t + k x 1 ) ] , u 3 = 2 A cos ( k x 1 ) exp ( j ω t ) .
B 0 ( x 1 2 + x 2 2 + x 3 2 ) + 2 B 5 x 3 x 1 = 1 .
x 1 = x 1 cos θ x 3 sin θ , x 2 = x 2 1 , x 3 = x 1 sin θ + x 3 cos θ .
( B 0 + 2 B 5 sin θ cos θ ) x 1 2 + B 0 x 2 2 + ( B 0 2 B 5 sin θ cos θ ) x 3 2 + 2 B 5 x 1 x 3 cos 2 θ = 1 .

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