Abstract

We develop a temporal correlation transfer equation (CTE) and a Monte Carlo algorithm (MC) for multiply scattered light modulated by an ultrasonic pulse propagating in an optically scattering medium, where the ultrasound field can be nonuniform and the medium can have spatially heterogeneous distribution of optical parameters. The CTE and MC can be used to obtain the time-varying specific intensity and the spatial distribution of the time-dependent power spectral density, respectively, of ultrasound-modulated light. We expect the CTE and MC to be applicable for estimation of contrast and resolution in a wide spectrum of conditions in ultrasound-modulated optical tomography of soft biological tissues.

© 2007 Optical Society of America

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  1. A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
    [CrossRef] [PubMed]
  2. F. A. Marks, H. W. Tomlinson, and G. W. Brooksby, "A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination," in Proc. SPIE 1888, 500-510 (1993).
    [CrossRef]
  3. L. V. Wang, S. L. Jacques, and X. Zhao, "Continuous-wave ultrasonic modulation of scattered laser light to image objects in turbid media," Opt. Lett. 20, 629-631 (1995).
    [CrossRef] [PubMed]
  4. S. Sakadzic and L. V. Wang, "High-resolution ultrasound-modulated optical tomography in biological tissues," Opt. Lett. 29, 2770-2772 (2004).
    [CrossRef] [PubMed]
  5. C. Kim, R. J. Zemp, and L. V. Wang, "Intense acoustic bursts as a signal-enhancement mechanism in ultrasound-modulated optical tomography," Opt. Lett. 31, 2423-2425 (2006).
    [CrossRef] [PubMed]
  6. M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, "Acousto-optic tomography with multiply scattered light," J. Opt. Soc. Am. A 14, 1151-1158 (1997).
    [CrossRef]
  7. L. V. Wang and G. Ku, "Frequency-swept ultrasound-modulated optical tomography of scattering media," Opt. Lett. 23, 975-977 (1998).
    [CrossRef]
  8. S. Leveque, A. C. Boccara, M. Lebec, and H. Saint-Jalmes, "Ultrasonic tagging of photon paths in scattering media: parallel speckle modulation processing," Opt. Lett. 24, 181-183 (1999).
    [CrossRef]
  9. G. Yao, S.-L. Jiao, and L. V. Wang, "Frequency-swept ultrasound-modulated optical tomography in biological tissue by use of parallel detection," Opt. Lett. 25, 734-736 (2000).
    [CrossRef]
  10. A. Lev, Z. Kotler, and B. G. Sfez, "Ultrasound tagged light imaging in turbid media in a reflectance geometry," Opt. Lett. 25, 378-380 (2000).
    [CrossRef]
  11. M. Hisaka, T. Sugiura, and S. Kawata, "Optical cross-sectional imaging with pulse ultrasound wave assistance," J. Opt. Soc. Am. A 18, 1531-1534 (2001).
    [CrossRef]
  12. J. Li, G. Ku, and L. V. Wang, "Ultrasound-modulated optical tomography of biological tissue by use of contrast of laser speckles," Appl. Opt. 41, 6030-6035 (2002).
    [CrossRef] [PubMed]
  13. A. Lev and B. G. Sfez, "Pulsed ultrasound-modulated light tomography," Opt. Lett. 28, 1549-1551 (2003).
    [CrossRef] [PubMed]
  14. M. Gross, P. Goy, and M. Al-Koussa, "Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging," Opt. Lett. 28, 2482-2484 (2003).
    [CrossRef] [PubMed]
  15. T. W. Murray, L. Sui, G. Maguluri, R. A. Roy, A. Nieva, F. Blonigen, and C. A. DiMarzio, "Detection of ultrasound modulated photons in diffuse media using the photorefractive effect," Opt. Lett. 29, 2509-2511 (2004).
    [CrossRef] [PubMed]
  16. A. Lev, E. Rubanov, B. Sfez, S. Shany, and A. J. Foldes, "Ultrasound-modulated light tomography assessment of osteoporosis," Opt. Lett. 30, 1692-1694 (2005).
    [CrossRef] [PubMed]
  17. E. Bossy, L. Sui, T. W. Murray, and R. A. Roy, "Fusion of conventional ultrasound imaging and acousto-optic sensing by use of a standard pulsed-ultrasound scanner," Opt. Lett. 30, 744-746 (2005).
    [CrossRef] [PubMed]
  18. W. Leutz and G. Maret, "Ultrasonic modulation of multiply scattered-light," Physica B 204, 14-19 (1995).
    [CrossRef]
  19. G. Maret and P. E. Wolf, "Multiple light-scattering from disordered media--the effect of Brownian-motion of scatterers," Z. Phys. B: Condens. Matter 65, 409-413 (1987).
    [CrossRef]
  20. G. D. Mahan, W. E. Engler, J. J. Tiemann, and E. G. Uzgiris, "Ultrasonic tagging of light: Theory," in Proc. Natl. Acad. Sci. U.S.A. 95, 14015-14019 (1998).
    [CrossRef] [PubMed]
  21. L. V. Wang, "Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model," Phys. Rev. Lett. 87, 043903-(1-4) (2001).
    [CrossRef] [PubMed]
  22. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, "Diffusing-wave spectroscopy," Phys. Rev. Lett. 60, 1134-1137 (1988).
    [CrossRef] [PubMed]
  23. S. Sakadzic and L. V. Wang, "Ultrasonic modulation of multiply scattered coherent light: an analytical model for anisotropically scattering media," Phys. Rev. E 66, 026603-(1-19) (2002).
    [CrossRef]
  24. A. Lev and B. Sfez, "In vivo demonstration of the ultrasound modulated light technique," J. Opt. Soc. Am. A 20, 2347-2354 (2003).
    [CrossRef]
  25. S. Sakadzic and L. V. Wang, "Modulation of multiply scattered coherent light by ultrasonic pulses: An analytical model," Phys. Rev. E 72, 036620-(1-12) (2005).
    [CrossRef]
  26. U. Frisch, Wave Propagation in Random Media, Vol. 1, Probabilistic Methods in Applied Mathematics (Academic, 1968), pp. 75-198.
  27. Y. N. Barabanenkov, A. G. Vinogradov, Yu. A. Kravtsov, and V. I. Tatarskii, "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1420-1425 (1972).
    [CrossRef]
  28. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  29. M. C. W. van Rossum, and Th. M. Nieuwenhuizen, "Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion," Rev. Mod. Phys. 71, 313-371 (1999).
    [CrossRef]
  30. Yu. A. Kravtsov and L. A. Apresyan, "Radiative transfer: new aspects of the old theory," Vol. 36, Progress in Optics, E.Wolf, ed. (Elsevier, 1996), pp. 179-244.
    [CrossRef]
  31. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation through Random Media, 1st ed. (Springer Verlag, 1989).
  32. V. L. Kuzmin and V. P. Romanov, "Coherent phenomena in light scattering from disordered systems," Phys. Usp. 39, 231-260 (1996).
    [CrossRef]
  33. S. Sakadzic and L. V. Wang, "Correlation transfer and diffusion of ultrasound-modulated multiply scattered light," Phys. Rev. Lett. 96, 163902-(1-4) (2006).
    [CrossRef] [PubMed]
  34. S. Sakadzic and L. V. Wang, "Correlation transfer equation for ultrasound-modulated multiply scattered light," Phys. Rev. E 74, 036618-(1-10) (2006).
    [CrossRef]
  35. A. Ishimaru, "Correlation functions of a wave in a random distribution of stationary and moving scatterers," Radio Sci. 10, 45-52 (1975).
    [CrossRef]
  36. M. J. Stephen, "Temporal fluctuations in wave propagation in random media," Phys. Rev. B 37, 1-5 (1988).
    [CrossRef]
  37. F. C. MacKintosh and J. Sajeev, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40, 2383-2406 (1989).
    [CrossRef]
  38. R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorri-Nowkoorani, and U. Nobbmann, "Correlation transfer: development and application," J. Quant. Spectrosc. Radiat. Transf. 52, 713-727 (1994).
    [CrossRef]
  39. L. V. Wang, "Mechanisms of ultrasonic modulation of multiply scattered coherent light: a Monte Carlo model," Opt. Lett. 26, 1191-1193 (2001).
    [CrossRef]
  40. G. Yao and L. Wang, "Signal dependence and noise source in ultrasound-modulated optical tomography," Am. J. Optom. Physiol. Opt. 43, 1320-1326 (2004).
  41. Y. N. Barabanenkov, "Application of the smooth-perturbation method to the solution of general equations of multiple wave-scattering theory," Sov. Phys. JETP 27, 954-959 (1968).
  42. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 3: Elements of Random Fields, 1st ed. (Springer Verlag, 1989).
    [CrossRef]
  43. L. Cohen, "Time-frequency distributions--A review," Proc. IEEE 77, 941-981 (1989).
    [CrossRef]
  44. L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941).
    [CrossRef]
  45. L. V. Wang, S. L. Jacques, and L.-Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
    [CrossRef] [PubMed]
  46. J. A. Jensen and N. B. Svendsen, "Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 262-267 (1992).
    [CrossRef] [PubMed]

2006 (3)

S. Sakadzic and L. V. Wang, "Correlation transfer and diffusion of ultrasound-modulated multiply scattered light," Phys. Rev. Lett. 96, 163902-(1-4) (2006).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, "Correlation transfer equation for ultrasound-modulated multiply scattered light," Phys. Rev. E 74, 036618-(1-10) (2006).
[CrossRef]

C. Kim, R. J. Zemp, and L. V. Wang, "Intense acoustic bursts as a signal-enhancement mechanism in ultrasound-modulated optical tomography," Opt. Lett. 31, 2423-2425 (2006).
[CrossRef] [PubMed]

2005 (4)

E. Bossy, L. Sui, T. W. Murray, and R. A. Roy, "Fusion of conventional ultrasound imaging and acousto-optic sensing by use of a standard pulsed-ultrasound scanner," Opt. Lett. 30, 744-746 (2005).
[CrossRef] [PubMed]

A. Lev, E. Rubanov, B. Sfez, S. Shany, and A. J. Foldes, "Ultrasound-modulated light tomography assessment of osteoporosis," Opt. Lett. 30, 1692-1694 (2005).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, "Modulation of multiply scattered coherent light by ultrasonic pulses: An analytical model," Phys. Rev. E 72, 036620-(1-12) (2005).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

2004 (3)

2003 (3)

2002 (2)

J. Li, G. Ku, and L. V. Wang, "Ultrasound-modulated optical tomography of biological tissue by use of contrast of laser speckles," Appl. Opt. 41, 6030-6035 (2002).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, "Ultrasonic modulation of multiply scattered coherent light: an analytical model for anisotropically scattering media," Phys. Rev. E 66, 026603-(1-19) (2002).
[CrossRef]

2001 (3)

2000 (2)

1999 (2)

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

M. C. W. van Rossum, and Th. M. Nieuwenhuizen, "Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion," Rev. Mod. Phys. 71, 313-371 (1999).
[CrossRef]

1998 (2)

G. D. Mahan, W. E. Engler, J. J. Tiemann, and E. G. Uzgiris, "Ultrasonic tagging of light: Theory," in Proc. Natl. Acad. Sci. U.S.A. 95, 14015-14019 (1998).
[CrossRef] [PubMed]

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

1997 (1)

1996 (1)

V. L. Kuzmin and V. P. Romanov, "Coherent phenomena in light scattering from disordered systems," Phys. Usp. 39, 231-260 (1996).
[CrossRef]

1995 (3)

W. Leutz and G. Maret, "Ultrasonic modulation of multiply scattered-light," Physica B 204, 14-19 (1995).
[CrossRef]

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

L. V. Wang, S. L. Jacques, and L.-Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

1994 (1)

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorri-Nowkoorani, and U. Nobbmann, "Correlation transfer: development and application," J. Quant. Spectrosc. Radiat. Transf. 52, 713-727 (1994).
[CrossRef]

1993 (1)

F. A. Marks, H. W. Tomlinson, and G. W. Brooksby, "A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination," in Proc. SPIE 1888, 500-510 (1993).
[CrossRef]

1992 (1)

J. A. Jensen and N. B. Svendsen, "Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 262-267 (1992).
[CrossRef] [PubMed]

1989 (2)

L. Cohen, "Time-frequency distributions--A review," Proc. IEEE 77, 941-981 (1989).
[CrossRef]

F. C. MacKintosh and J. Sajeev, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40, 2383-2406 (1989).
[CrossRef]

1988 (2)

M. J. Stephen, "Temporal fluctuations in wave propagation in random media," Phys. Rev. B 37, 1-5 (1988).
[CrossRef]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, "Diffusing-wave spectroscopy," Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

1987 (1)

G. Maret and P. E. Wolf, "Multiple light-scattering from disordered media--the effect of Brownian-motion of scatterers," Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

1975 (1)

A. Ishimaru, "Correlation functions of a wave in a random distribution of stationary and moving scatterers," Radio Sci. 10, 45-52 (1975).
[CrossRef]

1972 (1)

Y. N. Barabanenkov, A. G. Vinogradov, Yu. A. Kravtsov, and V. I. Tatarskii, "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1420-1425 (1972).
[CrossRef]

1968 (1)

Y. N. Barabanenkov, "Application of the smooth-perturbation method to the solution of general equations of multiple wave-scattering theory," Sov. Phys. JETP 27, 954-959 (1968).

1941 (1)

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Ackerson, B. J.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorri-Nowkoorani, and U. Nobbmann, "Correlation transfer: development and application," J. Quant. Spectrosc. Radiat. Transf. 52, 713-727 (1994).
[CrossRef]

Al-Koussa, M.

Apresyan, L. A.

Yu. A. Kravtsov and L. A. Apresyan, "Radiative transfer: new aspects of the old theory," Vol. 36, Progress in Optics, E.Wolf, ed. (Elsevier, 1996), pp. 179-244.
[CrossRef]

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Barabanenkov, Y. N.

Y. N. Barabanenkov, A. G. Vinogradov, Yu. A. Kravtsov, and V. I. Tatarskii, "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1420-1425 (1972).
[CrossRef]

Y. N. Barabanenkov, "Application of the smooth-perturbation method to the solution of general equations of multiple wave-scattering theory," Sov. Phys. JETP 27, 954-959 (1968).

Blonigen, F.

Boccara, A. C.

Bossy, E.

Brooksby, G. W.

F. A. Marks, H. W. Tomlinson, and G. W. Brooksby, "A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination," in Proc. SPIE 1888, 500-510 (1993).
[CrossRef]

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, "Diffusing-wave spectroscopy," Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Cohen, L.

L. Cohen, "Time-frequency distributions--A review," Proc. IEEE 77, 941-981 (1989).
[CrossRef]

DiMarzio, C. A.

Dorri-Nowkoorani, F.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorri-Nowkoorani, and U. Nobbmann, "Correlation transfer: development and application," J. Quant. Spectrosc. Radiat. Transf. 52, 713-727 (1994).
[CrossRef]

Dougherty, R. L.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorri-Nowkoorani, and U. Nobbmann, "Correlation transfer: development and application," J. Quant. Spectrosc. Radiat. Transf. 52, 713-727 (1994).
[CrossRef]

Engler, W. E.

G. D. Mahan, W. E. Engler, J. J. Tiemann, and E. G. Uzgiris, "Ultrasonic tagging of light: Theory," in Proc. Natl. Acad. Sci. U.S.A. 95, 14015-14019 (1998).
[CrossRef] [PubMed]

Foldes, A. J.

Frisch, U.

U. Frisch, Wave Propagation in Random Media, Vol. 1, Probabilistic Methods in Applied Mathematics (Academic, 1968), pp. 75-198.

Genack, A. Z.

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Goy, P.

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Gross, M.

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, "Diffusing-wave spectroscopy," Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Hisaka, M.

Ishimaru, A.

A. Ishimaru, "Correlation functions of a wave in a random distribution of stationary and moving scatterers," Radio Sci. 10, 45-52 (1975).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Jacques, S. L.

L. V. Wang, S. L. Jacques, and L.-Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

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

Jensen, J. A.

J. A. Jensen and N. B. Svendsen, "Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 262-267 (1992).
[CrossRef] [PubMed]

Jiao, S.-L.

Kawata, S.

Kempe, M.

Kim, C.

Kotler, Z.

Kravtsov, Yu. A.

Y. N. Barabanenkov, A. G. Vinogradov, Yu. A. Kravtsov, and V. I. Tatarskii, "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1420-1425 (1972).
[CrossRef]

Yu. A. Kravtsov and L. A. Apresyan, "Radiative transfer: new aspects of the old theory," Vol. 36, Progress in Optics, E.Wolf, ed. (Elsevier, 1996), pp. 179-244.
[CrossRef]

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 3: Elements of Random Fields, 1st ed. (Springer Verlag, 1989).
[CrossRef]

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation through Random Media, 1st ed. (Springer Verlag, 1989).

Ku, G.

Kuzmin, V. L.

V. L. Kuzmin and V. P. Romanov, "Coherent phenomena in light scattering from disordered systems," Phys. Usp. 39, 231-260 (1996).
[CrossRef]

Larionov, M.

Lebec, M.

Leutz, W.

W. Leutz and G. Maret, "Ultrasonic modulation of multiply scattered-light," Physica B 204, 14-19 (1995).
[CrossRef]

Lev, A.

Leveque, S.

Li, J.

MacKintosh, F. C.

F. C. MacKintosh and J. Sajeev, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40, 2383-2406 (1989).
[CrossRef]

Maguluri, G.

Mahan, G. D.

G. D. Mahan, W. E. Engler, J. J. Tiemann, and E. G. Uzgiris, "Ultrasonic tagging of light: Theory," in Proc. Natl. Acad. Sci. U.S.A. 95, 14015-14019 (1998).
[CrossRef] [PubMed]

Maret, G.

W. Leutz and G. Maret, "Ultrasonic modulation of multiply scattered-light," Physica B 204, 14-19 (1995).
[CrossRef]

G. Maret and P. E. Wolf, "Multiple light-scattering from disordered media--the effect of Brownian-motion of scatterers," Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

Marks, F. A.

F. A. Marks, H. W. Tomlinson, and G. W. Brooksby, "A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination," in Proc. SPIE 1888, 500-510 (1993).
[CrossRef]

Murray, T. W.

Nieuwenhuizen, Th. M.

M. C. W. van Rossum, and Th. M. Nieuwenhuizen, "Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion," Rev. Mod. Phys. 71, 313-371 (1999).
[CrossRef]

Nieva, A.

Nobbmann, U.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorri-Nowkoorani, and U. Nobbmann, "Correlation transfer: development and application," J. Quant. Spectrosc. Radiat. Transf. 52, 713-727 (1994).
[CrossRef]

Pine, D. J.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, "Diffusing-wave spectroscopy," Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Reguigui, N. M.

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorri-Nowkoorani, and U. Nobbmann, "Correlation transfer: development and application," J. Quant. Spectrosc. Radiat. Transf. 52, 713-727 (1994).
[CrossRef]

Romanov, V. P.

V. L. Kuzmin and V. P. Romanov, "Coherent phenomena in light scattering from disordered systems," Phys. Usp. 39, 231-260 (1996).
[CrossRef]

Roy, R. A.

Rubanov, E.

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation through Random Media, 1st ed. (Springer Verlag, 1989).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 3: Elements of Random Fields, 1st ed. (Springer Verlag, 1989).
[CrossRef]

Saint-Jalmes, H.

Sajeev, J.

F. C. MacKintosh and J. Sajeev, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40, 2383-2406 (1989).
[CrossRef]

Sakadzic, S.

S. Sakadzic and L. V. Wang, "Correlation transfer equation for ultrasound-modulated multiply scattered light," Phys. Rev. E 74, 036618-(1-10) (2006).
[CrossRef]

S. Sakadzic and L. V. Wang, "Correlation transfer and diffusion of ultrasound-modulated multiply scattered light," Phys. Rev. Lett. 96, 163902-(1-4) (2006).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, "Modulation of multiply scattered coherent light by ultrasonic pulses: An analytical model," Phys. Rev. E 72, 036620-(1-12) (2005).
[CrossRef]

S. Sakadzic and L. V. Wang, "High-resolution ultrasound-modulated optical tomography in biological tissues," Opt. Lett. 29, 2770-2772 (2004).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, "Ultrasonic modulation of multiply scattered coherent light: an analytical model for anisotropically scattering media," Phys. Rev. E 66, 026603-(1-19) (2002).
[CrossRef]

Sfez, B.

Sfez, B. G.

Shany, S.

Stephen, M. J.

M. J. Stephen, "Temporal fluctuations in wave propagation in random media," Phys. Rev. B 37, 1-5 (1988).
[CrossRef]

Sugiura, T.

Sui, L.

Svendsen, N. B.

J. A. Jensen and N. B. Svendsen, "Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 262-267 (1992).
[CrossRef] [PubMed]

Tatarskii, V. I.

Y. N. Barabanenkov, A. G. Vinogradov, Yu. A. Kravtsov, and V. I. Tatarskii, "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1420-1425 (1972).
[CrossRef]

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation through Random Media, 1st ed. (Springer Verlag, 1989).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 3: Elements of Random Fields, 1st ed. (Springer Verlag, 1989).
[CrossRef]

Tiemann, J. J.

G. D. Mahan, W. E. Engler, J. J. Tiemann, and E. G. Uzgiris, "Ultrasonic tagging of light: Theory," in Proc. Natl. Acad. Sci. U.S.A. 95, 14015-14019 (1998).
[CrossRef] [PubMed]

Tomlinson, H. W.

F. A. Marks, H. W. Tomlinson, and G. W. Brooksby, "A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination," in Proc. SPIE 1888, 500-510 (1993).
[CrossRef]

Uzgiris, E. G.

G. D. Mahan, W. E. Engler, J. J. Tiemann, and E. G. Uzgiris, "Ultrasonic tagging of light: Theory," in Proc. Natl. Acad. Sci. U.S.A. 95, 14015-14019 (1998).
[CrossRef] [PubMed]

van Rossum, M. C. W.

M. C. W. van Rossum, and Th. M. Nieuwenhuizen, "Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion," Rev. Mod. Phys. 71, 313-371 (1999).
[CrossRef]

Vinogradov, A. G.

Y. N. Barabanenkov, A. G. Vinogradov, Yu. A. Kravtsov, and V. I. Tatarskii, "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1420-1425 (1972).
[CrossRef]

Wang, L.

G. Yao and L. Wang, "Signal dependence and noise source in ultrasound-modulated optical tomography," Am. J. Optom. Physiol. Opt. 43, 1320-1326 (2004).

Wang, L. V.

S. Sakadzic and L. V. Wang, "Correlation transfer and diffusion of ultrasound-modulated multiply scattered light," Phys. Rev. Lett. 96, 163902-(1-4) (2006).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, "Correlation transfer equation for ultrasound-modulated multiply scattered light," Phys. Rev. E 74, 036618-(1-10) (2006).
[CrossRef]

C. Kim, R. J. Zemp, and L. V. Wang, "Intense acoustic bursts as a signal-enhancement mechanism in ultrasound-modulated optical tomography," Opt. Lett. 31, 2423-2425 (2006).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, "Modulation of multiply scattered coherent light by ultrasonic pulses: An analytical model," Phys. Rev. E 72, 036620-(1-12) (2005).
[CrossRef]

S. Sakadzic and L. V. Wang, "High-resolution ultrasound-modulated optical tomography in biological tissues," Opt. Lett. 29, 2770-2772 (2004).
[CrossRef] [PubMed]

J. Li, G. Ku, and L. V. Wang, "Ultrasound-modulated optical tomography of biological tissue by use of contrast of laser speckles," Appl. Opt. 41, 6030-6035 (2002).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, "Ultrasonic modulation of multiply scattered coherent light: an analytical model for anisotropically scattering media," Phys. Rev. E 66, 026603-(1-19) (2002).
[CrossRef]

L. V. Wang, "Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model," Phys. Rev. Lett. 87, 043903-(1-4) (2001).
[CrossRef] [PubMed]

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

G. Yao, S.-L. Jiao, and L. V. Wang, "Frequency-swept ultrasound-modulated optical tomography in biological tissue by use of parallel detection," Opt. Lett. 25, 734-736 (2000).
[CrossRef]

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

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

L. V. Wang, S. L. Jacques, and L.-Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Weitz, D. A.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, "Diffusing-wave spectroscopy," Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Wolf, P. E.

G. Maret and P. E. Wolf, "Multiple light-scattering from disordered media--the effect of Brownian-motion of scatterers," Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

Yao, G.

G. Yao and L. Wang, "Signal dependence and noise source in ultrasound-modulated optical tomography," Am. J. Optom. Physiol. Opt. 43, 1320-1326 (2004).

G. Yao, S.-L. Jiao, and L. V. Wang, "Frequency-swept ultrasound-modulated optical tomography in biological tissue by use of parallel detection," Opt. Lett. 25, 734-736 (2000).
[CrossRef]

Zaslavsky, D.

Zemp, R. J.

Zhao, X.

Zheng, L.-Q.

L. V. Wang, S. L. Jacques, and L.-Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Am. J. Optom. Physiol. Opt. (1)

G. Yao and L. Wang, "Signal dependence and noise source in ultrasound-modulated optical tomography," Am. J. Optom. Physiol. Opt. 43, 1320-1326 (2004).

Appl. Opt. (1)

Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Comput. Methods Programs Biomed. (1)

L. V. Wang, S. L. Jacques, and L.-Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. A. Jensen and N. B. Svendsen, "Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 262-267 (1992).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

J. Quant. Spectrosc. Radiat. Transf. (1)

R. L. Dougherty, B. J. Ackerson, N. M. Reguigui, F. Dorri-Nowkoorani, and U. Nobbmann, "Correlation transfer: development and application," J. Quant. Spectrosc. Radiat. Transf. 52, 713-727 (1994).
[CrossRef]

Opt. Lett. (13)

A. Lev, Z. Kotler, and B. G. Sfez, "Ultrasound tagged light imaging in turbid media in a reflectance geometry," Opt. Lett. 25, 378-380 (2000).
[CrossRef]

G. Yao, S.-L. Jiao, and L. V. Wang, "Frequency-swept ultrasound-modulated optical tomography in biological tissue by use of parallel detection," Opt. Lett. 25, 734-736 (2000).
[CrossRef]

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

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

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

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

M. Gross, P. Goy, and M. Al-Koussa, "Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging," Opt. Lett. 28, 2482-2484 (2003).
[CrossRef] [PubMed]

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

S. Sakadzic and L. V. Wang, "High-resolution ultrasound-modulated optical tomography in biological tissues," Opt. Lett. 29, 2770-2772 (2004).
[CrossRef] [PubMed]

E. Bossy, L. Sui, T. W. Murray, and R. A. Roy, "Fusion of conventional ultrasound imaging and acousto-optic sensing by use of a standard pulsed-ultrasound scanner," Opt. Lett. 30, 744-746 (2005).
[CrossRef] [PubMed]

A. Lev, E. Rubanov, B. Sfez, S. Shany, and A. J. Foldes, "Ultrasound-modulated light tomography assessment of osteoporosis," Opt. Lett. 30, 1692-1694 (2005).
[CrossRef] [PubMed]

C. Kim, R. J. Zemp, and L. V. Wang, "Intense acoustic bursts as a signal-enhancement mechanism in ultrasound-modulated optical tomography," Opt. Lett. 31, 2423-2425 (2006).
[CrossRef] [PubMed]

A. Lev and B. G. Sfez, "Pulsed ultrasound-modulated light tomography," Opt. Lett. 28, 1549-1551 (2003).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Phys. Rev. B (2)

M. J. Stephen, "Temporal fluctuations in wave propagation in random media," Phys. Rev. B 37, 1-5 (1988).
[CrossRef]

F. C. MacKintosh and J. Sajeev, "Diffusing-wave spectroscopy and multiple scattering of light in correlated random media," Phys. Rev. B 40, 2383-2406 (1989).
[CrossRef]

Phys. Rev. E (3)

S. Sakadzic and L. V. Wang, "Ultrasonic modulation of multiply scattered coherent light: an analytical model for anisotropically scattering media," Phys. Rev. E 66, 026603-(1-19) (2002).
[CrossRef]

S. Sakadzic and L. V. Wang, "Modulation of multiply scattered coherent light by ultrasonic pulses: An analytical model," Phys. Rev. E 72, 036620-(1-12) (2005).
[CrossRef]

S. Sakadzic and L. V. Wang, "Correlation transfer equation for ultrasound-modulated multiply scattered light," Phys. Rev. E 74, 036618-(1-10) (2006).
[CrossRef]

Phys. Rev. Lett. (3)

L. V. Wang, "Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model," Phys. Rev. Lett. 87, 043903-(1-4) (2001).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, "Diffusing-wave spectroscopy," Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

S. Sakadzic and L. V. Wang, "Correlation transfer and diffusion of ultrasound-modulated multiply scattered light," Phys. Rev. Lett. 96, 163902-(1-4) (2006).
[CrossRef] [PubMed]

Phys. Usp. (1)

V. L. Kuzmin and V. P. Romanov, "Coherent phenomena in light scattering from disordered systems," Phys. Usp. 39, 231-260 (1996).
[CrossRef]

Physica B (1)

W. Leutz and G. Maret, "Ultrasonic modulation of multiply scattered-light," Physica B 204, 14-19 (1995).
[CrossRef]

Proc. IEEE (1)

L. Cohen, "Time-frequency distributions--A review," Proc. IEEE 77, 941-981 (1989).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

G. D. Mahan, W. E. Engler, J. J. Tiemann, and E. G. Uzgiris, "Ultrasonic tagging of light: Theory," in Proc. Natl. Acad. Sci. U.S.A. 95, 14015-14019 (1998).
[CrossRef] [PubMed]

Proc. SPIE (1)

F. A. Marks, H. W. Tomlinson, and G. W. Brooksby, "A comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination," in Proc. SPIE 1888, 500-510 (1993).
[CrossRef]

Radio Sci. (1)

A. Ishimaru, "Correlation functions of a wave in a random distribution of stationary and moving scatterers," Radio Sci. 10, 45-52 (1975).
[CrossRef]

Radiophys. Quantum Electron. (1)

Y. N. Barabanenkov, A. G. Vinogradov, Yu. A. Kravtsov, and V. I. Tatarskii, "Application of the theory of multiple scattering of waves to the derivation of the radiation transfer equation for a statistically inhomogeneous medium," Radiophys. Quantum Electron. 15, 1420-1425 (1972).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. W. van Rossum, and Th. M. Nieuwenhuizen, "Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion," Rev. Mod. Phys. 71, 313-371 (1999).
[CrossRef]

Sov. Phys. JETP (1)

Y. N. Barabanenkov, "Application of the smooth-perturbation method to the solution of general equations of multiple wave-scattering theory," Sov. Phys. JETP 27, 954-959 (1968).

Z. Phys. B: Condens. Matter (1)

G. Maret and P. E. Wolf, "Multiple light-scattering from disordered media--the effect of Brownian-motion of scatterers," Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

Other (5)

U. Frisch, Wave Propagation in Random Media, Vol. 1, Probabilistic Methods in Applied Mathematics (Academic, 1968), pp. 75-198.

Yu. A. Kravtsov and L. A. Apresyan, "Radiative transfer: new aspects of the old theory," Vol. 36, Progress in Optics, E.Wolf, ed. (Elsevier, 1996), pp. 179-244.
[CrossRef]

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation through Random Media, 1st ed. (Springer Verlag, 1989).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 3: Elements of Random Fields, 1st ed. (Springer Verlag, 1989).
[CrossRef]

Supplementary Material (1)

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Figures (4)

Fig. 1
Fig. 1

Configuration of the scattering sample. Dimensions in the X, Y, and Z directions are 20 mm , 100 mm , and 100 mm , respectively.

Fig. 2
Fig. 2

(a) Temporal profile of the ultrasound pulse. (b) Power spectrum of the ultrasound pressure (solid curve) calculated from (a) and time-varying power spectrum of the ultrasound-modulated light (dotted-dashed curve) simulated at the transmission side of the slab at { x , y , z } = { 20 mm , 0 mm , 0 mm } . The simulation results are shown for the ultrasound pulse at the focal point. (c) Contour plot of the spatial distributions of the ultrasonic power density (solid curve) and the time-varying power density (dashed curve) of the ultrasound-modulated light. The distributions are presented in the plane defined by y = 0 mm , for ultrasound pulse at the focal point. The contours are plotted at 25%, 50%, and 75% of the maximum power levels. In this simulation no absorbing objects are present in the scattering slab.

Fig. 3
Fig. 3

(Multimedia online; josaa.osa.org) Static frame of the time-varying power density of the ultrasound-modulated light simulated for the duration of time-of-flight of the ultrasound pulse in the scattering sample. The spatial distribution of the power density is presented in the plane defined by y = 0 mm . The values are presented in shades of gray, with the levels equally spaced from zero to the maximum value. The maximum value of distribution ( I m ) in each frame is given in the upper right corner.

Fig. 4
Fig. 4

Modulation depth simulated at the transmission plane of the scattering sample during the ultrasound pulse propagation. Solid curve, the case when the detection point is placed at { x , y , z } = { 20 mm , 0 mm , 0 mm } —symmetrically with respect to both the illumination beam and the two objects. Dashed curve, the case when the detection point is placed at { x , y , z } = { 20 mm , 0 mm , 3 mm } 3 mm away from the previous detection point.

Equations (59)

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[ 2 + k 0 2 n 0 2 ( 1 + 2 η P ( r , t ) ρ v a 2 ) ] G a ( r , r 0 , t ) = δ ( r r 0 ) ,
G a ( r , r 0 , t ) = exp ( i k 0 n 0 r r 0 [ 1 + ξ ( r , r 0 , t ) ] ) 4 π r r 0 ,
ξ ( r , r 0 , t ) = η ρ v a 2 r r 0 r 0 r P ( r , t ) d r ,
ξ ( r , r 0 , t ) = 1 2 M r 0 r h ( t v a Ω ̂ a r + ϕ ) d r ,
G s ( r b , r a , t ) = G a ( r b , r a , t ) 4 π ρ s G a ( r b , r s , t ) f ( Ω ̂ s b , Ω ̂ a s ) × exp [ i k 0 n 0 e s ( t ) ( Ω ̂ a s Ω ̂ s b ) ] G s ( r s , r a , t ) d r s .
G s ( r b , r a , t ) = exp [ i K ( r b , r a , t ) r b r a ] 4 π r b r a ,
K ( r b , r a , t ) = k 0 n 0 [ 1 + ξ ( r b , r a , t ) ] + 2 π ρ s f ( Ω ̂ , Ω ̂ ) ( k 0 n 0 ) .
Γ ( r a , r b , t , τ ) = E ( r a , t τ 2 ) E * ( r b , t + τ 2 ) ,
Γ ( r a , r b , t , τ ) = Γ 0 ( r a , r b , t , τ ) + v s a ( t τ 2 ) v s b * ( t + τ 2 ) Γ ( r s , r s , t , τ ) ρ ( r s , t τ 2 ; r s , t + τ 2 ) d r s d r s .
Γ ̃ ( r c s , q , t , τ ) = ( 2 π ) 3 Γ ( r c s , r d s , t , τ ) exp ( i q r d s ) d r d s ,
v s a ( t τ 2 ) v s b * ( t + τ 2 ) Γ ( r s , r s , t , τ ) f ( Ω ̂ s a , Ω ̂ ) f * ( Ω ̂ s b , Ω ̂ ) r a r s r b r s Γ ̃ ( r c s , q , t , τ ) exp [ i K ( r a , r s , t τ 2 ) r a r s ] exp [ i K * ( r b , r s , t + τ 2 ) r b r s ] exp ( i q r d s ) d q ,
r a r s r c r c s + ( r d r d s ) Ω ̂ 2 ,
r b r s r c r c s ( r d r d s ) Ω ̂ 2 ,
( r a r s r b r s ) 1 r c r c s 2 ,
f ( Ω ̂ s a , Ω ̂ ) f * ( Ω ̂ s b , Ω ̂ ) r a r s r b r s exp [ i K ( r a , r s , t τ 2 ) r a r s i K * ( r b , r s , t + τ 2 ) r b r s ] = σ s p ( Ω ̂ , Ω ̂ ) r c r c s 2 exp [ i K r ( r d r d s ) Ω ̂ μ t r c r c s ] exp [ i Ψ n ( r a , r b , r s , r s , t , τ ) ] ,
Ψ n ( r a , r b , r s , r s , t , τ ) = k 0 n 0 r a r s ξ ( r a , r s , t τ 2 ) k 0 n 0 r b r s ξ ( r b , r s , t + τ 2 ) ,
Ψ n ( r c , r c s , t , τ ) = 1 2 k 0 n 0 M r c s r c [ h ( t τ 2 v a Ω ̂ a r + ϕ ) h ( t + τ 2 v a Ω ̂ a r + ϕ ) ] d r .
Γ ( r c , r d , t , τ ) = Γ 0 ( r c , r d , t , τ ) + μ s p ( Ω ̂ , Ω ̂ ) exp ( i K r r d Ω ̂ ) exp ( μ t r c r c s ) exp [ i ( q K r Ω ̂ ) Δ e ( r s , t , τ ) ] exp [ i Ψ n ( r c , r c s , t , τ ) ] Γ ̃ ( r c s , q , t , τ ) d r c r c s d Ω d q .
Γ ̃ ( r c s , q , t , τ ) δ ( q K r ) I ( r c s , Ω ̂ , t , τ ) K r 2 .
Γ ( r c , r d , t , τ ) = I ( r c , Ω ̂ , t , τ ) exp ( i K r Ω ̂ r d ) d Ω ,
Γ 0 ( r c , r d , t , τ ) = I 0 ( r c , Ω ̂ , t , τ ) exp ( i K r Ω ̂ r d ) d Ω ,
P ( r c , t , ω ) = 1 2 π + I Ω 0 ( r c , t , τ ) exp ( i ω τ ) d τ ,
I ( r , Ω ̂ , t , τ ) = I 0 ( r , Ω ̂ , t , τ ) + μ s p ( Ω ̂ , Ω ̂ ) exp ( μ t r r s ) I ( r s , Ω ̂ , t , τ ) × Φ ( r , r s , Ω ̂ , Ω ̂ , t , τ ) d r r s d Ω .
Φ ( r , r s , Ω ̂ , Ω ̂ , t , τ ) = exp [ i Ψ d ( r s , Ω ̂ , Ω ̂ , t , τ ) ] exp [ i Ψ n ( r , r s , t , τ ) ]
Ψ d ( r s , Ω ̂ , Ω ̂ , t , τ ) = K r ( Ω ̂ Ω ̂ ) Δ e ( r s , t , τ )
Ψ d ( r s , Ω ̂ , Ω ̂ , t , τ ) = K r P 0 ρ v a [ ( Ω ̂ Ω ̂ ) Ω ̂ a ] t τ 2 t + τ 2 h ( t v a Ω ̂ a r s + ϕ ) d t ,
Δ φ d , j ( t ) = k 0 n 0 [ ( Ω ̂ inc , j Ω ̂ sc , j ) Ω ̂ a , n j ] A n j ( t ) ,
W ph P ( t , ω ) = W ph ( 2 π ) 1 + W ( t , τ ) exp ( i ω τ ) d τ ,
W ( t , τ ) = exp [ i Δ φ ( t , τ ) ] ,
Δ φ ( t , τ ) = Δ φ ( t + τ 2 ) Δ φ ( t τ 2 ) .
P ( t , ω ) ( 2 π ) 1 + [ 1 + i Δ φ ( t , τ ) 1 2 Δ φ 2 ( t , τ ) ] exp ( i ω τ ) d τ .
Δ φ ( t , τ ) = k 0 n 0 η ρ v a 2 k l k P 0 , m k [ h ( t + τ 2 t m k ) h ( t τ 2 t m k ) ] + k 0 n 0 ρ v a j [ ( Ω ̂ inc , j Ω ̂ sc , j ) Ω ̂ a , n j ] P 0 , n j t τ 2 t + τ 2 h ( u t n j ) d u ,
( ρ v a ) 1 P 0 , n j t τ 2 t + τ 2 h ( u t n j ) d u = A n j ( t + τ 2 ) A n j ( t τ 2 ) .
P 1 ( t , ω ) = 4 k 0 n 0 η ρ v a 2 k l k P 0 , m k Im [ h ̃ ( 2 ω ) exp [ i 2 ω ( t t m k ) ] ] 4 k 0 n 0 ρ v a j P 0 , n j [ ( Ω ̂ inc , j Ω ̂ sc , j ) Ω ̂ a , n j ] Re [ h ̃ ( 2 ω ) 2 ω exp [ i 2 ω ( t t n j ) ] ] ,
P 2 ( t , ω ) = 1 2 1 2 π + Δ φ 2 ( t , τ ) exp ( i ω τ ) d τ .
Δ φ ( t , τ ) = k Δ φ n , k ( t , τ ) + j Δ φ d , j ( t , τ ) ,
Δ φ n , k ( t , τ ) = Δ φ n , k ( t + τ 2 ) Δ φ n , k ( t τ 2 )
Δ φ d , j ( t , τ ) = Δ φ d , j ( t + τ 2 ) Δ φ d , j ( t τ 2 ) ,
P 2 ( t , ω ) = P 2 , n n ( t , ω ) + P 2 , d d ( t , ω ) + 2 P 2 , n d ( t , ω ) ,
P 2 , a b ( t , ω ) = 1 2 1 2 π + k , j Δ φ a , k ( t , τ ) Δ φ b , j ( t , τ ) exp ( i ω τ ) d τ .
I a b , k , j ( t , ω ) = ( 2 π ) 1 Δ φ a , k ( t , τ ) Δ φ b , j ( t , τ ) exp ( i ω τ ) d τ
I n n , k , j ( t , ω ) = 4 Λ n 2 l k l j P 0 , m k P 0 , n j { Re [ exp [ i ω ( t m k t n j ) ] + h ̃ ( ω ω t ) h ̃ * ( ω + ω t ) exp [ i ω t ( 2 t t m k t n j ) ] d ω t ] Re [ exp [ i ω ( 2 t t m k t n j ) ] + h ̃ ( ω ω t ) h ̃ ( ω + ω t ) exp [ i ω t ( t m k t n j ) ] d ω t ] } ,
I d d , k , j ( t , ω )
= 4 Λ d 2 P 0 , m k P 0 , n j [ Ω ̂ a , m k ( Ω ̂ inc , k Ω ̂ sc , k ) ]
] [ Ω ̂ a , n j ( Ω ̂ inc , j Ω ̂ sc , j ) ] { Re [ exp [ i ω ( t m k t n j ) ] }
[ × + h ̃ ( ω ω t ) h ̃ * ( ω + ω t ) ( ω ω t ) ( ω + ω t ) exp [ i ω t ( 2 t t m k t n j ) ] d ω t ] + Re [ exp [ i ω ( 2 t t m k t n j ) ] { + h ̃ ( ω ω t ) h ̃ ( ω + ω t ) ( ω ω t ) ( ω + ω t ) exp [ i ω t ( t m k t n j ) ] d ω t ] } ,
I n d , k , j ( t , ω )
= 4 Λ n Λ d l k P 0 , m k P 0 , n j [ Ω ̂ a , n j ( Ω ̂ inc , j Ω ̂ sc , j ) ]
{ Im [ exp [ i ω ( t m k t n j ) ]
[ + h ̃ ( ω ω t ) h ̃ * ( ω + ω t ) ( ω + ω t ) exp [ i ω t ( 2 t t m k t n j ) ] d ω t ] + { Im [ exp [ i ω ( 2 t t m k t n j ) ] + h ̃ ( ω ω t ) h ̃ ( ω + ω t ) ω + ω t exp [ i ω t ( t m k t n j ) ] d ω t ] } ,
h ( t ) = exp ( t 2 2 ( σ T a ) 2 ) [ cos ( ω a t ) t ω a ( σ T a ) 2 sin ( ω a t ) ] ,
h ̃ ( ω ) = 1 2 ω ω a σ a 2 π [ exp ( ( ω + ω a ) 2 2 σ a 2 ) exp ( ( ω ω a ) 2 2 σ a 2 ) ] ,
I n n , k , j ( t , ω ) = Λ η 2 v a 2 l k l j P 0 , m k P 0 , n j ω 2 Ψ ( t , t m k , t n j ) Ξ ( ω ) cos [ ω ( t m k t n j ) ] ,
I d d , k , j ( t , ω ) = Λ P 0 , m k P 0 , n j [ Ω ̂ a , m k ( Ω ̂ inc , k Ω ̂ sc , k ) ] [ Ω ̂ a , n j ( Ω ̂ inc , j Ω ̂ sc , j ) ] × Ψ ( t , t m k , t n j ) Ξ ( ω ) cos [ ω ( t m k t n j ) ] ,
I n d , k , j ( t , ω ) = Λ η v a 1 l k P 0 , m k P 0 , n j [ Ω ̂ a , n j ( Ω ̂ inc , j Ω ̂ sc , j ) ] × Ψ ( t , t m k , t n j ) Ξ ( ω ) ( ω sin [ ω ( t m k t n j ) ] σ a 2 ( t t m k + t n j 2 ) cos [ ω ( t m k t n j ) ] ) ,
Λ = ( k 0 n 0 ) 2 [ 2 ( ρ v a ) 2 ω a 2 σ a π ] ,
Ψ ( t , t m k , t n j ) = exp [ σ a 2 ( t t m k 2 t n j 2 ) 2 ] ,
Ξ ( ω ) = exp [ ( ω + ω a ) 2 σ a 2 ] + exp [ ( ω ω a ) 2 σ a 2 ] .
P 2 ( t , ω ) = 1 2 k = 1 N M j = 1 N M I n n , k , j ( t , ω ) 1 2 k = 1 M j = 1 M I d d , k , j ( t , ω ) k = 1 N M j = 1 M I n d , k , j ( t , ω ) .

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