Abstract

Acoustophotonic imaging uses ultrasound-modulated scattered light to improve the quality of optical imaging in diffusive media. Experiments that use photorefractive-crystal-based detection have shown that there is a large dc shift in the acoustically modulated or ac optical signal, which could be utilized to further improve optical imaging resolution. We report that photon paths in a diffusive medium were generated by a Monte Carlo simulation, and the optical phase shifts of the various photons induced by the presence of a realistic focused ultrasound beam were calculated. Quantities that characterize the ac and dc signal components were evaluated by use of the calculated phase shifts. It was confirmed that the dc component dominates owing to coherent summation of the contributions from all the photons.

© 2005 Optical Society of America

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  31. V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
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  32. S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model for light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS5 of SPIE Institute Series (SPIE, 1989), pp. 102–111.
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    [CrossRef] [PubMed]

2004 (2)

2003 (2)

2002 (1)

S. Sakadžić, L.-H. V. Wang, “Ultrasonic modulation of multiply scattered coherent light: an analytical model for anisotropically scattering media,” Phys. Rev. E 66, 026603 (2002).
[CrossRef]

2001 (4)

L.-H. V. Wang, “Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model,” Phys. Rev. Lett. 87, 043903 (2001).
[CrossRef] [PubMed]

L.-H. V. Wang, “Mechanisms of ultrasonic modulation of multiply scattered coherent light: a Monte Carlo model,” Opt. Lett. 26, 1191–1193 (2001).
[CrossRef]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” in IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

S. Leveque, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

1999 (2)

S. Leveque, A. C. Boccara, M. Lebec, H. Saint-Jalmes, “Ultrasonic tagging of photon paths in scattering media: parallel speckle modulation processing,” Opt. Lett. 24, 181–183 (1999).
[CrossRef]

I. M. Hallaj, R. O. Cleveland, “FDTD simulation of finite amplitude pressure and temperature fields for biomedical ultrasound,” J. Acoust. Soc. Am. 105, L7–L12 (1999).
[CrossRef] [PubMed]

1998 (2)

L.-H. V. Wang, G. Ku, “Frequency-swept ultrasound-modulated optical tomography of scattering media,” Opt. Lett. 23, 975–977 (1998).
[CrossRef]

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. G. Uzgiris, “Ultrasonic tagging of light: theory,” Proc. Natl. Acad. Sci. USA 95, 14,015–14,019 (1998).
[CrossRef]

1997 (3)

1995 (3)

W. Leutz, G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204, 14–19 (1995).
[CrossRef]

P. Delaye, L.-A. de Montmorillon, G. Roosen, “Transmission of time modulated optical signals through an absorbing photorefractive crystal,” Opt. Commun. 118, 154–164 (1995).
[CrossRef]

L.-H. V. Wang, S. L. Jacques, X.-M. Zhao, “Continuous-wave ultrasonic modulation of scattered laser light to image objects in turbid media,” Opt. Lett. 20, 629–631 (1995).
[CrossRef] [PubMed]

1994 (1)

A. Blouin, J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by two-wave mixing in a photorefractive GaAs crystal,” Appl. Phys. Lett. 65, 932–934 (1994).
[CrossRef]

1993 (1)

1991 (1)

R. K. Ing, J. P. Monchalin, “Broadband optical detection of ultrasound by two-wave mixing in a photorefractive crystal,” Appl. Phys. Lett. 59, 3233–3235 (1991).
[CrossRef]

1990 (2)

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

1955 (1)

K. Yosioka, Y. Kawasima, “Acoustic radiation pressure on a compressible sphere,” Acustica 5, 167–173 (1955).

Blackstock, D.

M. Hamilton, D. Blackstock, Nonlinear Acoustics: Theory and Applications (Academic, 1998).

Blonigen, F.

Blouin, A.

P. Delaye, A. Blouin, D. Drolet, L.-A. de Montmorillon, G. Roosen, J.-P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive InP:Fe under an applied dc field,” J. Opt. Soc. Am. B 14, 1723–1734 (1997).
[CrossRef]

A. Blouin, J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by two-wave mixing in a photorefractive GaAs crystal,” Appl. Phys. Lett. 65, 932–934 (1994).
[CrossRef]

Boas, D. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” in IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

A. Nieva, S. Manneville, D. A. Boas, R. Roy, C. A. DiMarzio, “Monte Carlo simulations in acousto-photonic imaging,” in Digest of Topical Meeting on Biomedical Optics (Optical Society of America, 2002).

Boccara, A. C.

S. Leveque, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

S. Leveque, A. C. Boccara, M. Lebec, H. Saint-Jalmes, “Ultrasonic tagging of photon paths in scattering media: parallel speckle modulation processing,” Opt. Lett. 24, 181–183 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1987).

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” in IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Brooksby, G. W.

F. A. Marks, H. W. Tomlinson, G. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. A. Alfano, eds., Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Cleveland, R. O.

I. M. Hallaj, R. O. Cleveland, “FDTD simulation of finite amplitude pressure and temperature fields for biomedical ultrasound,” J. Acoust. Soc. Am. 105, L7–L12 (1999).
[CrossRef] [PubMed]

de Montmorillon, L.-A.

P. Delaye, A. Blouin, D. Drolet, L.-A. de Montmorillon, G. Roosen, J.-P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive InP:Fe under an applied dc field,” J. Opt. Soc. Am. B 14, 1723–1734 (1997).
[CrossRef]

P. Delaye, L.-A. de Montmorillon, G. Roosen, “Transmission of time modulated optical signals through an absorbing photorefractive crystal,” Opt. Commun. 118, 154–164 (1995).
[CrossRef]

Delaye, P.

P. Delaye, A. Blouin, D. Drolet, L.-A. de Montmorillon, G. Roosen, J.-P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive InP:Fe under an applied dc field,” J. Opt. Soc. Am. B 14, 1723–1734 (1997).
[CrossRef]

P. Delaye, L.-A. de Montmorillon, G. Roosen, “Transmission of time modulated optical signals through an absorbing photorefractive crystal,” Opt. Commun. 118, 154–164 (1995).
[CrossRef]

DiMarzio, C. A.

T. W. Murray, L. Sui, G. Maguluri, R. A. Roy, A. Nieva, F. Blonigen, C. A. DiMarzio, “Detection of ultrasound-modulated photons in diffuse media using the photorefractive effect,” Opt. Lett. 29, 2509–2511 (2004).
[CrossRef] [PubMed]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” in IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

C. A. DiMarzio, R. J. Gaudette, T. J. Gaudette, “A new imaging technique combining diffusive photon density waves and focused ultrasound,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. A. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 376–384 (1999).
[CrossRef]

A. Nieva, S. Manneville, D. A. Boas, R. Roy, C. A. DiMarzio, “Monte Carlo simulations in acousto-photonic imaging,” in Digest of Topical Meeting on Biomedical Optics (Optical Society of America, 2002).

Dolfi, D.

D. Dolfi, F. Micheron, “Imaging process and system for transillumination with photon frequency marking,” U.S. patent, 5,174,298 (29December1992).

Drolet, D.

Duncan, M. D.

Engler, W. E.

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. G. Uzgiris, “Ultrasonic tagging of light: theory,” Proc. Natl. Acad. Sci. USA 95, 14,015–14,019 (1998).
[CrossRef]

Frank, G. L.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” in IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

C. A. DiMarzio, R. J. Gaudette, T. J. Gaudette, “A new imaging technique combining diffusive photon density waves and focused ultrasound,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. A. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 376–384 (1999).
[CrossRef]

Gaudette, T. J.

C. A. DiMarzio, R. J. Gaudette, T. J. Gaudette, “A new imaging technique combining diffusive photon density waves and focused ultrasound,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. A. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 376–384 (1999).
[CrossRef]

Genack, A. Z.

Guenther, R. D.

R. D. Guenther, Modern Optics (Wiley, 1990).

Hallaj, I. M.

I. M. Hallaj, R. O. Cleveland, “FDTD simulation of finite amplitude pressure and temperature fields for biomedical ultrasound,” J. Acoust. Soc. Am. 105, L7–L12 (1999).
[CrossRef] [PubMed]

Hamilton, M.

M. Hamilton, D. Blackstock, Nonlinear Acoustics: Theory and Applications (Academic, 1998).

Ing, R. K.

R. K. Ing, J. P. Monchalin, “Broadband optical detection of ultrasound by two-wave mixing in a photorefractive crystal,” Appl. Phys. Lett. 59, 3233–3235 (1991).
[CrossRef]

Jacques, S. L.

L.-H. V. Wang, S. L. Jacques, X.-M. Zhao, “Continuous-wave ultrasonic modulation of scattered laser light to image objects in turbid media,” Opt. Lett. 20, 629–631 (1995).
[CrossRef] [PubMed]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model for light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS5 of SPIE Institute Series (SPIE, 1989), pp. 102–111.

Kawasima, Y.

K. Yosioka, Y. Kawasima, “Acoustic radiation pressure on a compressible sphere,” Acustica 5, 167–173 (1955).

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model for light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS5 of SPIE Institute Series (SPIE, 1989), pp. 102–111.

Kempe, M.

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” in IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Ku, G.

Larionov, M.

Lebec, M.

Leutz, W.

W. Leutz, G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204, 14–19 (1995).
[CrossRef]

Lev, A.

Leveque, S.

S. Leveque, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

S. Leveque, A. C. Boccara, M. Lebec, H. Saint-Jalmes, “Ultrasonic tagging of photon paths in scattering media: parallel speckle modulation processing,” Opt. Lett. 24, 181–183 (1999).
[CrossRef]

Maguluri, G.

Mahan, G. D.

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. G. Uzgiris, “Ultrasonic tagging of light: theory,” Proc. Natl. Acad. Sci. USA 95, 14,015–14,019 (1998).
[CrossRef]

Mahon, R.

Manneville, S.

A. Nieva, S. Manneville, D. A. Boas, R. Roy, C. A. DiMarzio, “Monte Carlo simulations in acousto-photonic imaging,” in Digest of Topical Meeting on Biomedical Optics (Optical Society of America, 2002).

Maret, G.

W. Leutz, G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204, 14–19 (1995).
[CrossRef]

Marks, F. A.

F. A. Marks, H. W. Tomlinson, G. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. A. Alfano, eds., Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Micheron, F.

D. Dolfi, F. Micheron, “Imaging process and system for transillumination with photon frequency marking,” U.S. patent, 5,174,298 (29December1992).

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, Q. Zhang, “Imaging the body with diffuse optical tomography,” in IEEE Signal Process. Mag. 18, 57–75 (2001).
[CrossRef]

Monchalin, J. P.

A. Blouin, J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by two-wave mixing in a photorefractive GaAs crystal,” Appl. Phys. Lett. 65, 932–934 (1994).
[CrossRef]

R. K. Ing, J. P. Monchalin, “Broadband optical detection of ultrasound by two-wave mixing in a photorefractive crystal,” Appl. Phys. Lett. 59, 3233–3235 (1991).
[CrossRef]

Monchalin, J.-P.

Moon, J. A.

Murray, T. W.

Nieva, A.

T. W. Murray, L. Sui, G. Maguluri, R. A. Roy, A. Nieva, F. Blonigen, C. A. DiMarzio, “Detection of ultrasound-modulated photons in diffuse media using the photorefractive effect,” Opt. Lett. 29, 2509–2511 (2004).
[CrossRef] [PubMed]

A. Nieva, S. Manneville, D. A. Boas, R. Roy, C. A. DiMarzio, “Monte Carlo simulations in acousto-photonic imaging,” in Digest of Topical Meeting on Biomedical Optics (Optical Society of America, 2002).

Patterson, M. S.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Peters, V. G.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Pottier, L.

S. Leveque, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model for light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS5 of SPIE Institute Series (SPIE, 1989), pp. 102–111.

Reintjes, J.

Roosen, G.

P. Delaye, A. Blouin, D. Drolet, L.-A. de Montmorillon, G. Roosen, J.-P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive InP:Fe under an applied dc field,” J. Opt. Soc. Am. B 14, 1723–1734 (1997).
[CrossRef]

P. Delaye, L.-A. de Montmorillon, G. Roosen, “Transmission of time modulated optical signals through an absorbing photorefractive crystal,” Opt. Commun. 118, 154–164 (1995).
[CrossRef]

Roy, R.

A. Nieva, S. Manneville, D. A. Boas, R. Roy, C. A. DiMarzio, “Monte Carlo simulations in acousto-photonic imaging,” in Digest of Topical Meeting on Biomedical Optics (Optical Society of America, 2002).

Roy, R. A.

Saint-Jalmes, H.

Sakadžic, S.

S. Sakadžić, L.-H. V. Wang, “Ultrasonic modulation of multiply scattered coherent light: an analytical model for anisotropically scattering media,” Phys. Rev. E 66, 026603 (2002).
[CrossRef]

Selb, J.

S. Leveque, J. Selb, L. Pottier, A. C. Boccara, “In situ local tissue characterization and imaging by backscattering acousto-optic imaging,” Opt. Commun. 196, 127–131 (2001).
[CrossRef]

Sfez, B.

Sfez, B. G.

Sui, L.

Tiemann, J. J.

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. G. Uzgiris, “Ultrasonic tagging of light: theory,” Proc. Natl. Acad. Sci. USA 95, 14,015–14,019 (1998).
[CrossRef]

Tomlinson, H. W.

F. A. Marks, H. W. Tomlinson, G. W. Brooksby, “A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. A. Alfano, eds., Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Uzgiris, E. G.

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. G. Uzgiris, “Ultrasonic tagging of light: theory,” Proc. Natl. Acad. Sci. USA 95, 14,015–14,019 (1998).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, 1980), Vol. 2.

Wang, L.-H. V.

Welch, A. J.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model for light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, Vol. IS5 of SPIE Institute Series (SPIE, 1989), pp. 102–111.

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Figures (8)

Fig. 1
Fig. 1

Plot of [J0(x) − 1] (solidcurve) and J1(x) (dashed curve). The function J1(x) characterizes a contribution of a photon to the ac component amplitude of the API signal when the magnitude of the photon’s acoustically induced phase shift is substituted for x, while the function [J0(x) − 1] characterizes the dc shift. When x is small, [J0(x) − 1] is dominated by J1(x). As x becomes large, [J0(x) − 1] asymptotes to —1 while J1(x) asymptotes to zero, and [J0(x) − 1] dominates in this case.

Fig. 2
Fig. 2

Intensity plot of the pressure in water generated by a circular focusing transducer with an aperture of 7 cm diameter and a focal length of 6.26 cm operating at 1 MHz. The pressure was computed with a finite-difference time-domain code to solve the Westervelt equation [Eq. (11)]. The center of the transducer is at r = z = 0. There is a high-pressure focal region along the axis near z = 6 cm.

Fig. 3
Fig. 3

View down the ultrasound axis, indicating the setup for the optical modeling of the API system as described in Section 4. For the coordinate system indicated, this figure shows the x–y plane with the origin at the ultrasound axis. All photons are launched toward the focus from 1 cm away, with the launch direction along the y axis. The detection region has a cross-sectional shape of a half-annulus of 0.5 cm in thickness. The detection region extends out of the page for 12 cm parallel to the ultrasound beam.

Fig. 4
Fig. 4

Side view of four hollowed cylindrical regions of interest in the diffusive medium, with their perimeters shown in black. They are superimposed upon an intensity plot of the pressure magnitude of the numerically calculated ultrasound beam. The horizontal axis is the acoustic axis. The centers of all four hollowed cylinders are at the ultrasound focus, which is at coordinates (0,0) in this plot. Region 1 is not hollowed and is almost entirely contained in the ultrasound focus. Other regions do not contain the smaller regions and thus tend to lie outside the focal region.

Fig. 5
Fig. 5

Distribution of the complex phase shifts, Φj, for the 3603 detected photons in the simulation of the API system by use of the numerical model for the ultrasound as described in Section 3. (a) The magnitude of the phase shifts. The histogram bins have a width of 2°. The average magnitude is 12.2°. The standard deviation is 15.2°. (b) The arguments of the phase shifts to within a global phase factor. The average is —5.4° and the standard deviation is 107°. There are 100 histogram bins equally spaced from + 180° to −180°.

Fig. 6
Fig. 6

Distribution of the complex phase shifts, Φj, for the 3603 detected photons in the simulation of the API system by use of the analytical Gaussian beam model for the ultrasound. (a) The magnitude of the phase shifts. The histogram bins have a width of ∼3.5°. The average magnitude is 8.8°. The standard deviation is 31°. (b) The arguments of the phase shifts to within a global phase factor. The average is 0.86° and the standard deviation is 99.9°. There are 100 histogram bins equally spaced from + 180° to −180°.

Fig. 7
Fig. 7

Average phase-shift magnitudes of four subsets of the detected photons grouped according to the innermost region through which they passed; the regions are those indicated in Fig. 4. These magnitudes were calculated with the numerical model for the ultrasound. Photons have larger phase shifts the closer they pass by the focus. Thus the largest average phase-shift magnitude is for those photons that passed through region 1.

Fig. 8
Fig. 8

Magnitude of the average value of Aj = J1(|Φj)|)exp(iδj) (dashed curve) and Dj = [J0(x) − 1] (solid curve) for the four subsets of the detected photons.

Equations (20)

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Φ j = m = 1 N j ϕ j m
ϕ j m ( d ) = Δ k j m · x j m ( t ) ,
ϕ j m ( n ) = r j , m 1 r j m k Δ n ( r , t ) ( l ̂ j m · d r ) ,
ϕ j m = ϕ j m ( d ) + ϕ j m ( n ) .
Φ j = | Φ j | sin ( ω a t δ j ) ,
| E S | 2 = exp ( α L ) Σ j | E j | 2 ( | exp ( γ L ) | 2 + 2 Re { [ exp ( γ L ) 1 ] * [ exp ( i Φ j ) 1 ] } ) ,
| E S | 2 dc = 2 exp ( γ R L α L ) cos ( γ I L ) × j | E j | 2 [ J 0 ( | Φ j | ) 1 ] .
| E S | 2 ac = 4 exp ( γ R L α L ) sin ( γ 1 L ) × j | E j | 2 J 1 ( | Φ j | ) sin ( ω a t δ j ) .
A j = J 1 ( | Φ j | ) exp ( i δ j ) ,
D j = J 0 ( | Φ j | ) 1 .
2 p 1 c a 2 2 p t 2 + δ c a 4 3 p t 3 + β ρ c a 4 2 p 2 t 2 = 0 .
A ( r ) = p ( r , t o ) + i p ( r , t o + 1 4 f a ) .
k a ( r ) = ε A ( r ) i A ( r ) ,
m 2 x 2 t = S p n d s 6 π ρ ν a ( x t y ) .
S p i n d s = iVA ( r o ) k a ( r o ) exp [ i ω a ( t t o ) ] ,
S p s n d s = i ρ ρ 2 ρ + ρ V A ( r o ) k a ( r o ) × exp [ i ω a ( t t o ) ] ,
x ( t ) = i χ A ( r o ) k a ( r o ) ρ ω a 2 exp [ ω a ( t t o ) ] .
χ = ( 1 3 ρ 2 ρ + ρ i ω a τ ) / ( 1 i ω a τ ) ,
ϕ j m ( n ) = k η r j , m 1 r j m A ( r j m ) exp [ i k a ( r j m ) ( r r j m ) ( l ̂ jm d r ) .
ϕ j m ( n ) = i k η A ( r j m ) | l j m | [ 1 exp ( i θ i m ) ] θ j m

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