Abstract

Typical eddy-current system test data consist of the values of the system’s probe impedance. The wave theory that links the phase response of the eddy-current probe impedance to the defect location relative to the probe is presented. The technique of phase multiplying the diffraction-limited hologram generated from the probe impedance is discussed. The effects and limitations of this technique are illustrated with a mathematical model of the eddy-current probe. Experimental data are presented that confirm the theoretical analysis and illustrate the ability to focus eddy-current holographic data by using backward wave propagation.

© 1993 Optical Society of America

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References

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  1. P. McIntire, Nondestructive Testing Handbook, 2nd ed. (American Society for Nondestructive Testing, Columbus, Ohio, 1986), Vol. 4, Sec. 1, p. 59.
  2. H. D. Collins, T. J. Davis, L. J. Busse, R. P. Gribble, “Eddy current phasography,” in Acoustical Imaging, J. P. Powers, ed. (Plenum, New York, 1981), Vol. 11, pp. 609–623.
    [CrossRef]
  3. B. P. Hildebrand, A. J. Boland, T. J. Davis, “Holographic principles applied to low frequency electromagnetic imaging in conductors,” in Proceedings of the Tenth International Optical Computing Conference (Institute of Electrical and Electronics Engineers, 1983), pp. 59–66.
  4. E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems, 2nd ed. (Prentice-Hall, Englewoods Cliffs, N.J., 1968), pp. 129–130.
  5. B. A. Auld, F. G. Muennemann, M. Riaziat, “Quantitative modelling of flaw responses in eddy current testing,” in Research Techniques in Nondestructive Testing, R. S. Sharpe, ed. (Academic, London, 1984), Vol. 7, Chap. 2.
  6. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989), pp. 327–328.
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 9, pp. 563–565.
  8. C. V. Dodd, W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems,” J. Appl. Phys. 39, 2829–2838 (1968).
    [CrossRef]
  9. A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of ultrasonic images by backward propagation,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 333–348.
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–49, 129–131.
  11. B. P. Hildebrand, K. A. Haines, “Holography by scanning,” J. Opt. Soc. Am. 59, 1–6 (1969).
    [CrossRef]
  12. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986), pp. 209–211.

1969

1968

C. V. Dodd, W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems,” J. Appl. Phys. 39, 2829–2838 (1968).
[CrossRef]

Auld, B. A.

B. A. Auld, F. G. Muennemann, M. Riaziat, “Quantitative modelling of flaw responses in eddy current testing,” in Research Techniques in Nondestructive Testing, R. S. Sharpe, ed. (Academic, London, 1984), Vol. 7, Chap. 2.

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989), pp. 327–328.

Balmain, K. G.

E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems, 2nd ed. (Prentice-Hall, Englewoods Cliffs, N.J., 1968), pp. 129–130.

Boland, A. J.

B. P. Hildebrand, A. J. Boland, T. J. Davis, “Holographic principles applied to low frequency electromagnetic imaging in conductors,” in Proceedings of the Tenth International Optical Computing Conference (Institute of Electrical and Electronics Engineers, 1983), pp. 59–66.

Boyer, A. L.

A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of ultrasonic images by backward propagation,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 333–348.
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986), pp. 209–211.

Busse, L. J.

H. D. Collins, T. J. Davis, L. J. Busse, R. P. Gribble, “Eddy current phasography,” in Acoustical Imaging, J. P. Powers, ed. (Plenum, New York, 1981), Vol. 11, pp. 609–623.
[CrossRef]

Collins, H. D.

H. D. Collins, T. J. Davis, L. J. Busse, R. P. Gribble, “Eddy current phasography,” in Acoustical Imaging, J. P. Powers, ed. (Plenum, New York, 1981), Vol. 11, pp. 609–623.
[CrossRef]

Davis, T. J.

H. D. Collins, T. J. Davis, L. J. Busse, R. P. Gribble, “Eddy current phasography,” in Acoustical Imaging, J. P. Powers, ed. (Plenum, New York, 1981), Vol. 11, pp. 609–623.
[CrossRef]

B. P. Hildebrand, A. J. Boland, T. J. Davis, “Holographic principles applied to low frequency electromagnetic imaging in conductors,” in Proceedings of the Tenth International Optical Computing Conference (Institute of Electrical and Electronics Engineers, 1983), pp. 59–66.

Deeds, W. E.

C. V. Dodd, W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems,” J. Appl. Phys. 39, 2829–2838 (1968).
[CrossRef]

Dodd, C. V.

C. V. Dodd, W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems,” J. Appl. Phys. 39, 2829–2838 (1968).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–49, 129–131.

Gribble, R. P.

H. D. Collins, T. J. Davis, L. J. Busse, R. P. Gribble, “Eddy current phasography,” in Acoustical Imaging, J. P. Powers, ed. (Plenum, New York, 1981), Vol. 11, pp. 609–623.
[CrossRef]

Haines, K. A.

Hildebrand, B. P.

B. P. Hildebrand, K. A. Haines, “Holography by scanning,” J. Opt. Soc. Am. 59, 1–6 (1969).
[CrossRef]

B. P. Hildebrand, A. J. Boland, T. J. Davis, “Holographic principles applied to low frequency electromagnetic imaging in conductors,” in Proceedings of the Tenth International Optical Computing Conference (Institute of Electrical and Electronics Engineers, 1983), pp. 59–66.

Hirsch, P. M.

A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of ultrasonic images by backward propagation,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 333–348.
[CrossRef]

Jordan, E. C.

E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems, 2nd ed. (Prentice-Hall, Englewoods Cliffs, N.J., 1968), pp. 129–130.

Jordan, J. A.

A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of ultrasonic images by backward propagation,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 333–348.
[CrossRef]

Lesem, L. B.

A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of ultrasonic images by backward propagation,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 333–348.
[CrossRef]

McIntire, P.

P. McIntire, Nondestructive Testing Handbook, 2nd ed. (American Society for Nondestructive Testing, Columbus, Ohio, 1986), Vol. 4, Sec. 1, p. 59.

Muennemann, F. G.

B. A. Auld, F. G. Muennemann, M. Riaziat, “Quantitative modelling of flaw responses in eddy current testing,” in Research Techniques in Nondestructive Testing, R. S. Sharpe, ed. (Academic, London, 1984), Vol. 7, Chap. 2.

Riaziat, M.

B. A. Auld, F. G. Muennemann, M. Riaziat, “Quantitative modelling of flaw responses in eddy current testing,” in Research Techniques in Nondestructive Testing, R. S. Sharpe, ed. (Academic, London, 1984), Vol. 7, Chap. 2.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 9, pp. 563–565.

Van Rooy, D. L.

A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of ultrasonic images by backward propagation,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 333–348.
[CrossRef]

J. Appl. Phys.

C. V. Dodd, W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems,” J. Appl. Phys. 39, 2829–2838 (1968).
[CrossRef]

J. Opt. Soc. Am.

Other

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986), pp. 209–211.

A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of ultrasonic images by backward propagation,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 333–348.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–49, 129–131.

P. McIntire, Nondestructive Testing Handbook, 2nd ed. (American Society for Nondestructive Testing, Columbus, Ohio, 1986), Vol. 4, Sec. 1, p. 59.

H. D. Collins, T. J. Davis, L. J. Busse, R. P. Gribble, “Eddy current phasography,” in Acoustical Imaging, J. P. Powers, ed. (Plenum, New York, 1981), Vol. 11, pp. 609–623.
[CrossRef]

B. P. Hildebrand, A. J. Boland, T. J. Davis, “Holographic principles applied to low frequency electromagnetic imaging in conductors,” in Proceedings of the Tenth International Optical Computing Conference (Institute of Electrical and Electronics Engineers, 1983), pp. 59–66.

E. C. Jordan, K. G. Balmain, Electromagnetic Waves and Radiating Systems, 2nd ed. (Prentice-Hall, Englewoods Cliffs, N.J., 1968), pp. 129–130.

B. A. Auld, F. G. Muennemann, M. Riaziat, “Quantitative modelling of flaw responses in eddy current testing,” in Research Techniques in Nondestructive Testing, R. S. Sharpe, ed. (Academic, London, 1984), Vol. 7, Chap. 2.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989), pp. 327–328.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 9, pp. 563–565.

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Figures (15)

Fig. 1
Fig. 1

One-dimensional holographic scanner.

Fig. 2
Fig. 2

Contour plot of electric field phase.

Fig. 3
Fig. 3

One-dimensional real (top) and imaginary (bottom) components of the computed point defect hologram,.

Fig. 4
Fig. 4

Reconstruction of unmodified point defect hologram at depths of 0 to 25.6 mm.

Fig. 5
Fig. 5

Phase-multiplied real (top) and imaginary (bottom) components of the point defect hologram.

Fig. 6
Fig. 6

Point spread function of probe at pm = 16.

Fig. 7
Fig. 7

Lateral (top) and depth point (bottom) spread functions of probe at pm = 16.

Fig. 8
Fig. 8

Point spread functions of probe at pm = 8 (top) and pm = 32 (bottom).

Fig. 9
Fig. 9

Phase-multiplication-induced aberrations with no attenuation.

Fig. 10
Fig. 10

Effect of limited beamwidth on phase-multiplication-induced aberrations.

Fig. 11
Fig. 11

Effect of overlap on reconstructed phase-multiplied two-point holograms.

Fig. 12
Fig. 12

Flat-bottom hole two-point resolution test.

Fig. 13
Fig. 13

Thin-slot two-point resolution test.

Fig. 14
Fig. 14

Large square defect image.

Fig. 15
Fig. 15

Large letter ℱ defect image.

Equations (23)

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α = β = π f μ σ ,
Θ ( x ) = - 2 β R ,
Δ Z = - 1 I 2 V D E ¯ i · J ¯ D d v ,
J ¯ D = - σ E ¯ D ,
Δ Z = 3 2 σ V D I 2 ( E ¯ i · E ¯ i ) .
E ¯ i ( r , z ) = E ( r , z ) exp [ j Θ ( r , z ) ] ϕ ^ ,
Δ Z = 3 2 σ V D E 2 I 2 exp ( j 2 Θ ) .
A ¯ ( r , z , r 0 , l ) = μ I r 0 0 J 1 ( τ r 0 ) J 1 ( τ r ) × exp ( - τ l ) exp [ ( τ 2 + j ω μ σ ) 1 / 2 z ] τ + ( τ 2 + j ω μ σ ) 1 / 2 τ d τ ϕ ^ ,
E ¯ i ( r , z ) = - j ω n = 1 N A ¯ ( r , z , r n , l n ) ,
U ( x , y , 0 ) = g ( x , y ) exp [ j Θ ( x , y ) ] ,
A 0 ( u , v ) = F [ U ( x , y , 0 ) ] ,
U ( x , y , z ) = F - 1 { F [ U ( x , y , 0 ) ] P ( u , v ; z ) } ,
2 U ( x , y , z ) + k 2 U ( x , y , z ) = 0.
U ( x , y , z ) = - A ( u , v ; z ) exp [ j 2 π ( u x + u y ) ] d u d v .
A ( u , v ; z ) = c 1 exp { - j k [ 1 - ( 2 π u k ) 2 - ( 2 π v k ) 2 ] 1 / 2 z } .
A ( u , v ; z ) = A 0 ( u , v ) exp { - j k [ 1 - ( 2 π u k ) 2 - ( 2 π v k ) 2 ] 1 / 2 z } .
P ( u , v ; z ) = exp { - j k [ 1 - ( 2 π u k ) 2 - ( 2 π v k ) 2 ] 1 / 2 z } ,
U ( x , y , 0 ) = g ( x , y ) exp [ j p m Θ ( x , y ) ] ,
P ( u , v ; z ) = exp { - j 4 π pm λ 0 [ 1 - ( λ o u 2 pm ) 2 - ( λ o v 2 pm ) 2 ] 1 / 2 z } .
I ( x , y ) = U ( x , y , z ) 2 ,
Θ N = 46.2 pm deg peak .
U ( x ) = exp { - j 2 π [ ( x - x 1 ) 2 + z 0 2 ] 1 / 2 λ } [ ( x - x 1 ) 2 + z 0 2 ] 1 / 2 + exp { - j 2 π [ ( x - x 2 ) 2 + z 0 2 ] 1 / 2 λ } [ ( x - x 2 ) 2 + z 0 2 ] 1 / 2 ,
U ( x ) = g ( x ) exp [ j Θ ( x ) ] .

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