Abstract

Time-averaged TV holography has been shown to be a useful technique for investigating acoustic fields in transparent media. The theory of time-averaged TV-holography measurements of ultrasonic fields in water is described. Projections of the phase and the amplitude of a 3.25-MHz ultrasonic field from an annular ultrasound probe operated in cw mode are presented. Quantitative measurements with a spatial resolution of better than 100 μm have been obtained. A set of such projections may be processed into a three-dimensional mapping of the phase and the amplitude of the acoustic field by tomographic techniques. This process is described, and an example of a tomographic reconstruction of the same ultrasonic field is presented.

© 1998 Optical Society of America

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References

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  1. B. R. Barnes, C. J. Burton, “Visual methods for studying ultrasonic phenomena,” J. Appl. Phys. 20, 286–294 (1949).
    [CrossRef]
  2. A. Korpel, L. W. Kessler, M. Ahmed, “Bragg diffraction sampling of a sound field,” J. Acoust. Soc. Am. 51, 1582–1592 (1972).
    [CrossRef]
  3. C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves, part I,” Proc. Indian Acad. Sci. Sec. A, 406–412 (1936).
  4. R. Reibold, W. Molkenstruck, “Light diffraction tomography applied to the investigation of ultrasonic fields. I: continuous waves,” Acoustica 56, 180–192 (1984).
  5. W. K. Fischer, M. Zambuto, “Optical holographic detection of ultrasonic waves,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 349–362.
    [CrossRef]
  6. P. Kwiek, R. Reibold, “Holographic investigation of transient ultrasonic fields,” Acoust. Lett. 7, 167–172 (1984).
  7. O. J. Løkberg, “Sound in flight: measurement of sound fields by use of TV holography,” Appl. Opt. 33, 2574–2584 (1994).
    [CrossRef] [PubMed]
  8. M. Espeland, O. J. Løkberg, R. Rustad, “Full field tomographic reconstruction of sound fields using TV holography,” J. Acoust. Soc. Am. 98, 280–287 (1995).
    [CrossRef]
  9. O. J. Løkberg, M. Espeland, H. M. Pedersen, “Tomographic reconstruction of sound fields using TV holography,” Appl. Opt. 34, 1640–1645 (1995).
    [CrossRef] [PubMed]
  10. R. Rustad, O. J. Løkberg, H. M. Pedersen, K. Klepsvik, T. Støren, “TV holography measurements of underwater acoustic fields,” J. Acoust. Soc. Am. 102, 1904–1906 (1997).
    [CrossRef]
  11. VINGMED Sound 3.25 MHz Annular Phased Array Transducer, TK100104.
  12. S. Ellingsrud, G. O. Rosvold, “Analysis of data-based TV holography system used to measure small vibration amplitudes,” J. Opt. Soc. Am. 9, 237–251 (1992).
    [CrossRef]
  13. W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
    [CrossRef]
  14. K. Høgmoen, O. J. Løkberg, “Detection and measurement of small vibrations using electronic speckle pattern interferometry,” Appl. Opt. 16, 1869–1875 (1977).
    [CrossRef] [PubMed]
  15. C. C. Aleksoff, “Time-averaged holography extended,” Appl. Phys. Lett. 14, 23 (1969).
    [CrossRef]
  16. W. A. Riley, W. R. Klein, “Piezo-optic coefficients of liquids,” J. Acoust. Soc. Am. 42, 1258–1261 (1967).
    [CrossRef]
  17. J. Radon, “Über die bestimmung von Funktionen durch ihre Intergralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Phys. Kl. 69 (1917).
  18. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).
  19. K. Høgmoen, H. M. Pedersen, “Measurement of small vibrations using electronic speckle pattern interferometry: theory,” J. Opt. Soc. Am. 67, 1578–1583 (1977).
    [CrossRef]
  20. B. A. J. Angelsen, Waves, Signals and Signal Processing in Medical Ultrasonics (Department of Physiology and Biomedical Engineering, Norwegian University of Science and Technology, Trondheim, Norway, 1996), Vol. I, pp. 6.2–6.5.

1997 (1)

R. Rustad, O. J. Løkberg, H. M. Pedersen, K. Klepsvik, T. Støren, “TV holography measurements of underwater acoustic fields,” J. Acoust. Soc. Am. 102, 1904–1906 (1997).
[CrossRef]

1995 (2)

M. Espeland, O. J. Løkberg, R. Rustad, “Full field tomographic reconstruction of sound fields using TV holography,” J. Acoust. Soc. Am. 98, 280–287 (1995).
[CrossRef]

O. J. Løkberg, M. Espeland, H. M. Pedersen, “Tomographic reconstruction of sound fields using TV holography,” Appl. Opt. 34, 1640–1645 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

1984 (2)

R. Reibold, W. Molkenstruck, “Light diffraction tomography applied to the investigation of ultrasonic fields. I: continuous waves,” Acoustica 56, 180–192 (1984).

P. Kwiek, R. Reibold, “Holographic investigation of transient ultrasonic fields,” Acoust. Lett. 7, 167–172 (1984).

1977 (2)

1972 (1)

A. Korpel, L. W. Kessler, M. Ahmed, “Bragg diffraction sampling of a sound field,” J. Acoust. Soc. Am. 51, 1582–1592 (1972).
[CrossRef]

1969 (1)

C. C. Aleksoff, “Time-averaged holography extended,” Appl. Phys. Lett. 14, 23 (1969).
[CrossRef]

1967 (2)

W. A. Riley, W. R. Klein, “Piezo-optic coefficients of liquids,” J. Acoust. Soc. Am. 42, 1258–1261 (1967).
[CrossRef]

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

1949 (1)

B. R. Barnes, C. J. Burton, “Visual methods for studying ultrasonic phenomena,” J. Appl. Phys. 20, 286–294 (1949).
[CrossRef]

1936 (1)

C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves, part I,” Proc. Indian Acad. Sci. Sec. A, 406–412 (1936).

1917 (1)

J. Radon, “Über die bestimmung von Funktionen durch ihre Intergralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Phys. Kl. 69 (1917).

Ahmed, M.

A. Korpel, L. W. Kessler, M. Ahmed, “Bragg diffraction sampling of a sound field,” J. Acoust. Soc. Am. 51, 1582–1592 (1972).
[CrossRef]

Aleksoff, C. C.

C. C. Aleksoff, “Time-averaged holography extended,” Appl. Phys. Lett. 14, 23 (1969).
[CrossRef]

Angelsen, B. A. J.

B. A. J. Angelsen, Waves, Signals and Signal Processing in Medical Ultrasonics (Department of Physiology and Biomedical Engineering, Norwegian University of Science and Technology, Trondheim, Norway, 1996), Vol. I, pp. 6.2–6.5.

Barnes, B. R.

B. R. Barnes, C. J. Burton, “Visual methods for studying ultrasonic phenomena,” J. Appl. Phys. 20, 286–294 (1949).
[CrossRef]

Burton, C. J.

B. R. Barnes, C. J. Burton, “Visual methods for studying ultrasonic phenomena,” J. Appl. Phys. 20, 286–294 (1949).
[CrossRef]

Cook, B. D.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Ellingsrud, S.

Espeland, M.

M. Espeland, O. J. Løkberg, R. Rustad, “Full field tomographic reconstruction of sound fields using TV holography,” J. Acoust. Soc. Am. 98, 280–287 (1995).
[CrossRef]

O. J. Løkberg, M. Espeland, H. M. Pedersen, “Tomographic reconstruction of sound fields using TV holography,” Appl. Opt. 34, 1640–1645 (1995).
[CrossRef] [PubMed]

Fischer, W. K.

W. K. Fischer, M. Zambuto, “Optical holographic detection of ultrasonic waves,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 349–362.
[CrossRef]

Høgmoen, K.

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).

Kessler, L. W.

A. Korpel, L. W. Kessler, M. Ahmed, “Bragg diffraction sampling of a sound field,” J. Acoust. Soc. Am. 51, 1582–1592 (1972).
[CrossRef]

Klein, W. R.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

W. A. Riley, W. R. Klein, “Piezo-optic coefficients of liquids,” J. Acoust. Soc. Am. 42, 1258–1261 (1967).
[CrossRef]

Klepsvik, K.

R. Rustad, O. J. Løkberg, H. M. Pedersen, K. Klepsvik, T. Støren, “TV holography measurements of underwater acoustic fields,” J. Acoust. Soc. Am. 102, 1904–1906 (1997).
[CrossRef]

Korpel, A.

A. Korpel, L. W. Kessler, M. Ahmed, “Bragg diffraction sampling of a sound field,” J. Acoust. Soc. Am. 51, 1582–1592 (1972).
[CrossRef]

Kwiek, P.

P. Kwiek, R. Reibold, “Holographic investigation of transient ultrasonic fields,” Acoust. Lett. 7, 167–172 (1984).

Løkberg, O. J.

Molkenstruck, W.

R. Reibold, W. Molkenstruck, “Light diffraction tomography applied to the investigation of ultrasonic fields. I: continuous waves,” Acoustica 56, 180–192 (1984).

Nagendra Nath, N. S.

C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves, part I,” Proc. Indian Acad. Sci. Sec. A, 406–412 (1936).

Pedersen, H. M.

Radon, J.

J. Radon, “Über die bestimmung von Funktionen durch ihre Intergralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Phys. Kl. 69 (1917).

Raman, C. V.

C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves, part I,” Proc. Indian Acad. Sci. Sec. A, 406–412 (1936).

Reibold, R.

P. Kwiek, R. Reibold, “Holographic investigation of transient ultrasonic fields,” Acoust. Lett. 7, 167–172 (1984).

R. Reibold, W. Molkenstruck, “Light diffraction tomography applied to the investigation of ultrasonic fields. I: continuous waves,” Acoustica 56, 180–192 (1984).

Riley, W. A.

W. A. Riley, W. R. Klein, “Piezo-optic coefficients of liquids,” J. Acoust. Soc. Am. 42, 1258–1261 (1967).
[CrossRef]

Rosvold, G. O.

Rustad, R.

R. Rustad, O. J. Løkberg, H. M. Pedersen, K. Klepsvik, T. Støren, “TV holography measurements of underwater acoustic fields,” J. Acoust. Soc. Am. 102, 1904–1906 (1997).
[CrossRef]

M. Espeland, O. J. Løkberg, R. Rustad, “Full field tomographic reconstruction of sound fields using TV holography,” J. Acoust. Soc. Am. 98, 280–287 (1995).
[CrossRef]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).

Støren, T.

R. Rustad, O. J. Løkberg, H. M. Pedersen, K. Klepsvik, T. Støren, “TV holography measurements of underwater acoustic fields,” J. Acoust. Soc. Am. 102, 1904–1906 (1997).
[CrossRef]

Zambuto, M.

W. K. Fischer, M. Zambuto, “Optical holographic detection of ultrasonic waves,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 349–362.
[CrossRef]

Acoust. Lett. (1)

P. Kwiek, R. Reibold, “Holographic investigation of transient ultrasonic fields,” Acoust. Lett. 7, 167–172 (1984).

Acoustica (1)

R. Reibold, W. Molkenstruck, “Light diffraction tomography applied to the investigation of ultrasonic fields. I: continuous waves,” Acoustica 56, 180–192 (1984).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. C. Aleksoff, “Time-averaged holography extended,” Appl. Phys. Lett. 14, 23 (1969).
[CrossRef]

Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Phys. Kl. (1)

J. Radon, “Über die bestimmung von Funktionen durch ihre Intergralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Phys. Kl. 69 (1917).

IEEE Trans. Sonics Ultrason. (1)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

J. Acoust. Soc. Am. (4)

W. A. Riley, W. R. Klein, “Piezo-optic coefficients of liquids,” J. Acoust. Soc. Am. 42, 1258–1261 (1967).
[CrossRef]

A. Korpel, L. W. Kessler, M. Ahmed, “Bragg diffraction sampling of a sound field,” J. Acoust. Soc. Am. 51, 1582–1592 (1972).
[CrossRef]

M. Espeland, O. J. Løkberg, R. Rustad, “Full field tomographic reconstruction of sound fields using TV holography,” J. Acoust. Soc. Am. 98, 280–287 (1995).
[CrossRef]

R. Rustad, O. J. Løkberg, H. M. Pedersen, K. Klepsvik, T. Støren, “TV holography measurements of underwater acoustic fields,” J. Acoust. Soc. Am. 102, 1904–1906 (1997).
[CrossRef]

J. Appl. Phys. (1)

B. R. Barnes, C. J. Burton, “Visual methods for studying ultrasonic phenomena,” J. Appl. Phys. 20, 286–294 (1949).
[CrossRef]

J. Opt. Soc. Am. (2)

Proc. Indian Acad. Sci. Sec. A (1)

C. V. Raman, N. S. Nagendra Nath, “The diffraction of light by high frequency sound waves, part I,” Proc. Indian Acad. Sci. Sec. A, 406–412 (1936).

Other (4)

W. K. Fischer, M. Zambuto, “Optical holographic detection of ultrasonic waves,” in Acoustical Holography, A. F. Metherell, ed. (Plenum, New York, 1971), Vol. 3, pp. 349–362.
[CrossRef]

VINGMED Sound 3.25 MHz Annular Phased Array Transducer, TK100104.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).

B. A. J. Angelsen, Waves, Signals and Signal Processing in Medical Ultrasonics (Department of Physiology and Biomedical Engineering, Norwegian University of Science and Technology, Trondheim, Norway, 1996), Vol. I, pp. 6.2–6.5.

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Figures (15)

Fig. 1
Fig. 1

Pressure phasor P of magnitude P and phase Φ and its real and imaginary parts P Re and P Im, respectively.

Fig. 2
Fig. 2

Modulation phasor aψ of magnitude a ψ and phase ϕ, can its real and imaginary parts, the integrals p opRe and p opIm.

Fig. 3
Fig. 3

Recording geometry for tomography by TV holography. The projection of a function in the plane xy is recorded by a TV camera at an angle θ. The term t is the position along a line running through the origin at the same angle, and d is the distance between points on this line being imaged onto adjacent pixels on the TV target.

Fig. 4
Fig. 4

Experimental setup for TV-holography measurements of ultrasonic fields in water. See text for explanation of notation used.

Fig. 5
Fig. 5

Schematic drawing of the medical ultrasound probe.

Fig. 6
Fig. 6

Projection measurement of the ultrasonic field outside the tip of the probe: (left) a gray-scale image of the projection of the amplitude, (right) a gray-scale representation of the projection of the phase. The phase is wrapped so that black corresponds to -π and white to π. The cap of the probe is semitransparent, and the wave fronts inside the cap are visible in the phase image.

Fig. 7
Fig. 7

Plot of the amplitude projection a ψ in nanometers along the line A–A in Fig. 6.

Fig. 8
Fig. 8

Close-up of the projection of the phase of the field at the tip of the probe. The measured data have been filtered once with a 3 × 3 spatial median filter.

Fig. 9
Fig. 9

Plot of the phase along a vertical line in Fig. 8. The unfiltered data are shown.

Fig. 10
Fig. 10

Three light rays passing through an ultrasonic field of plane waves. One of the quadrature components of the amplitude phasor of the field is shown to the left. The lens system of the interferometer is focused at the center of the ultrasonic field, and all three rays will illuminate the same pixel in the TV camera. They all undergo approximately the same phase modulation as they pass through the field, except when they pass through the maxima or the minima of the acoustic wave.

Fig. 11
Fig. 11

Reconstruction of the ultrasonic field at 3.25 MHz in a plane normal to the direction of propagation, immediately below the tip of the transducer. The amplitude reconstruction is presented in gray scale so that lighter gray corresponds to higher pressure. The phase is presented in a wrapped gray-scale representation, so that black corresponds to -π and white to π.

Fig. 12
Fig. 12

Same reconstruction as in Fig. 11 in a plane parallel to the direction of propagation intersecting the center of the probe. In the region above the line A–A the probe intersects some of the projections. The reconstruction is not reliable in this region. The arrow in the phase image points at a disturbance in the phase.

Fig. 13
Fig. 13

Projection measurements of the field at different distances from the probe. All the amplitude plots use the same gray scale. In the topmost projection, the probe is visible. The second projection shows the focus of the field, at the white markers, 72 mm below the tip of the probe, while the last projection shows the field as it hits a layer of foam rubber 150 mm from the tip of the probe. The horizontal lines are distance markers, 10 mm apart.

Fig. 14
Fig. 14

Plot of the amplitude projection a ψ in nanometers at the focus. The bar shows the estimated half-width of the beam at the focus.

Fig. 15
Fig. 15

Projection of the field as it strikes a layer of foam rubber at the bottom of the tank.

Equations (19)

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ψ t = k   S   μ r cos Ω t + Φ r d s = kp op S   P r cos Ω t + Φ r d s .
υ = k μ L ,
Q = K 2 L k ,
Q υ     1 .
ψ t = kp op cos Ω t S   P r cos Φ r d s - sin Ω t S   P r sin Φ r d s kp op cos Ω t S   P Re r d s - sin Ω t S   P Im r d s .
Π Re = S   P Re r d s , Π Im = S   P Im r d s .
a ψ = p op Π Re 2 + Π Im 2 1 / 2 , ϕ = tan - 1 Π Im Π Re mod   2 π ,
ψ t = ka ψ   cos Ω t + ϕ .
ψ t = kp op   - 1 / 2 1 / 2   P r d x cos Ω t cos Φ z - sin Ω t sin Φ z .
α ψ = p op - 1 / 2 1 / 2   P r d x = p op P ¯ z L , ϕ = Φ z ,
P ¯ z = 1 L - 1 / 2 1 / 2   P r d x
I x ,   y     J 0 2 k | a r - a ψ x ,   y | = J 0 2 ( k a ψ 2 x ,   y + a r 2 - 2 a ψ x ,   y a r   cos ϕ x ,   y - θ 1 / 2 ) .
I x ,   y     J 0 2 ( k a ψ 2 x ,   y + a r 2 - 2 a ψ x ,   y a r × cos ϕ x ,   y - θ 1 / 2 ) I b x ,   y - c x ,   y a ψ x ,   y cos ϕ x ,   y - θ .
I θ = 0 x ,   y = I b x ,   y - c x ,   y a ψ x ,   y cos ϕ x ,   y , I θ = 90 x ,   y = I b x ,   y - c x ,   y a ψ x ,   y cos ϕ x ,   y + 90 = I b x ,   y + c x ,   y a ψ x ,   y sin ϕ x ,   y , I θ = 180 x ,   y = I b x ,   y - c x ,   y a ψ x ,   y cos ϕ x ,   y + 180 = I b x ,   y + c x ,   y a ψ x ,   y cos ϕ x ,   y , I θ = 270 x ,   y = I b x ,   y - c x ,   y a ψ x ,   y cos ϕ x ,   y + 270 = I b x ,   y - c x ,   y a ψ x ,   y sin ϕ x ,   y .
Π Re x ,   y = I θ = 180 x ,   y - I θ = 0 x ,   y p op c x ,   y , Π Im x ,   y = I θ = 90 x ,   y - I θ = 270 x ,   y p op c x ,   y .
t = x   cos θ + y   sin θ ,
Π Re , θ t = - -   P Re x ,   y δ x   cos θ + y   sin θ - t d x d y , Π Im , θ t = - -   P Im x ,   y δ x   cos θ + y   sin θ - t d x d y .
P Re i ,   j = P   cos Φ d , P Im i ,   j = P   sin Φ d .
P x ,   y = 1 d P Re 2 i ,   j + P Im 2 i ,   j 1 / 2 , Φ x ,   y = tan - 1 P Im i ,   j P Re i ,   j mod   2 π .

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