Abstract

Ultrasound modulated light for optical tomography is very useful, since it can provide three-dimensional data with minimal mathematical processing. Although several experimental studies have shown the potential of this method, the link between the ultrasound location and the modulated signal intensity at the detector is not yet fully understood. We derive an analytical formula relating the position of the ultrasound transducer and the optical signal at the detector. We also derive an expression for the signal-to-shot-noise ratio as a function of the transducer position. We show that in certain conditions this ratio is only slowly decreasing as a function of the light penetration depth, which makes this technique attractive for optical tomography.

© 2001 Optical Society of America

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References

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  1. F. F. Jobsis, “Noninvasive infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
    [CrossRef]
  2. S. J. Matcher, P. Kirpatrick, K. Nahid, M. Cope, D. T. Delpy, “Absolute quantification methods in tissue near infrared spectroscopy,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, ed., Proc. SPIE2389, 486–495 (1995).
  3. M. A. Franceschini, E. Gratton, S. Fantini, “Noninvasive optical method of measuring tissue and arterial saturation: an application to absolute pulse oximetry of the brain,” Opt. Lett. 24, 829–831 (1999).
    [CrossRef]
  4. B. Chance, S. Nokia, J. Tracey, “Brain functional imaging in problem solving using NIR,” in Physics of Medical Imaging, J. T. Dobbins, J. M. Boone, eds., Proc. SPIE (to be published).
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    [CrossRef]
  6. D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 256, 1463–1466 (1993).
    [CrossRef]
  7. J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
    [CrossRef] [PubMed]
  8. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
    [CrossRef] [PubMed]
  9. J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
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    [CrossRef] [PubMed]
  11. H. Liu, D. A. Boas, Y. Zhang, A. G. Yodh, B. Chance, “A simplified approach to characterize optical properties and blood oxygenation in tissues using continuous near infrared light,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir eds., Proc. SPIE2389, 496–502 (1995).
  12. W. Leutz, G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B 204, 14–19 (1995).
    [CrossRef]
  13. H. W. Tomlinson, J. Tiemann, “Light imaging in a scattering medium, using ultrasonic probing and speckle image differencing,” U.S. patent5,212,667 (May18, 1993).
  14. M. Kempe, M. Larionov, D. Zaslavski, A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am. A 14, 1151–1158 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. A. Lev, Z. Kotler, B. Sfez, “Ultrasound tagged light imaging in turbid media in a reflectance geometry,” Opt. Lett. 25, 378–380 (2000).
    [CrossRef]
  18. G. D. Mahan, W. E. Engler, J. J. Tiemann, E. Uzgiris, “Ultrasonic tagging of light:  theory,” Proc. Natl. Acad. Sci. U.S.A. 95, 14015–14019 (1998).
    [CrossRef]
  19. G. Yao, L. V. Wang, “Theoretical and experimental studies of ultrasound-modulated optical tomography in biological tissues,” Appl. Opt. 39, 659–664 (2000).
    [CrossRef]
  20. G. H. Weiss, R. Nossal, R. F. Bonner, J. E. Kiefer, H. Taitelbaum, S. Havlin, “Random walk theory of photon migration in a turbid medium,” in Dynamical Processes in Condensed Molecular Systems, J. Klafter, J. Jortner, A. Blumen, eds. (World Scientific, Singapore, 1989), pp. 147–174.
  21. H. Taitelbaum, “Diagnosis using photon diffusion: from brain oxygenation to the fat of the atlantic salmon,” in Lecture Notes in Physics (Springer-Verlag, Berlin, 1998), Vol. 519, pp. 160–174.
  22. H. Taitelbaum, S. Havlin, G. H. Weiss, “Approximate theory of photon migration in a two-layer medium,” Appl. Opt. 28, 2245–2249 (1989).
    [CrossRef] [PubMed]
  23. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  24. S. C. Feng, F. A. Zeng, B. Chance, “Analytical perturbation theory of photon migration in the presence of a single absorbing or scattering defect sphere,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 54–63 (1995).

2000

1999

1998

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. Uzgiris, “Ultrasonic tagging of light:  theory,” Proc. Natl. Acad. Sci. U.S.A. 95, 14015–14019 (1998).
[CrossRef]

1997

1995

1994

1993

1992

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

1990

1989

1977

F. F. Jobsis, “Noninvasive infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[CrossRef]

Andersson-Engels, S. R.

Benaron, D. A.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 256, 1463–1466 (1993).
[CrossRef]

Berg, R.

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

H. Liu, D. A. Boas, Y. Zhang, A. G. Yodh, B. Chance, “A simplified approach to characterize optical properties and blood oxygenation in tissues using continuous near infrared light,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir eds., Proc. SPIE2389, 496–502 (1995).

Boccara, A. C.

Bonner, R. F.

G. H. Weiss, R. Nossal, R. F. Bonner, J. E. Kiefer, H. Taitelbaum, S. Havlin, “Random walk theory of photon migration in a turbid medium,” in Dynamical Processes in Condensed Molecular Systems, J. Klafter, J. Jortner, A. Blumen, eds. (World Scientific, Singapore, 1989), pp. 147–174.

Chance, B.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

H. Liu, D. A. Boas, Y. Zhang, A. G. Yodh, B. Chance, “A simplified approach to characterize optical properties and blood oxygenation in tissues using continuous near infrared light,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir eds., Proc. SPIE2389, 496–502 (1995).

B. Chance, S. Nokia, J. Tracey, “Brain functional imaging in problem solving using NIR,” in Physics of Medical Imaging, J. T. Dobbins, J. M. Boone, eds., Proc. SPIE (to be published).

S. C. Feng, F. A. Zeng, B. Chance, “Analytical perturbation theory of photon migration in the presence of a single absorbing or scattering defect sphere,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 54–63 (1995).

Cope, M.

S. J. Matcher, P. Kirpatrick, K. Nahid, M. Cope, D. T. Delpy, “Absolute quantification methods in tissue near infrared spectroscopy,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, ed., Proc. SPIE2389, 486–495 (1995).

Delpy, D. T.

J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
[CrossRef] [PubMed]

S. J. Matcher, P. Kirpatrick, K. Nahid, M. Cope, D. T. Delpy, “Absolute quantification methods in tissue near infrared spectroscopy,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, ed., Proc. SPIE2389, 486–495 (1995).

Engler, W. E.

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. Uzgiris, “Ultrasonic tagging of light:  theory,” Proc. Natl. Acad. Sci. U.S.A. 95, 14015–14019 (1998).
[CrossRef]

Fantini, S.

Feng, S. C.

S. C. Feng, F. A. Zeng, B. Chance, “Analytical perturbation theory of photon migration in the presence of a single absorbing or scattering defect sphere,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 54–63 (1995).

Fishkin, J. B.

Franceschini, M. A.

Genack, A. Z.

Gratton, E.

Havlin, S.

H. Taitelbaum, S. Havlin, G. H. Weiss, “Approximate theory of photon migration in a two-layer medium,” Appl. Opt. 28, 2245–2249 (1989).
[CrossRef] [PubMed]

G. H. Weiss, R. Nossal, R. F. Bonner, J. E. Kiefer, H. Taitelbaum, S. Havlin, “Random walk theory of photon migration in a turbid medium,” in Dynamical Processes in Condensed Molecular Systems, J. Klafter, J. Jortner, A. Blumen, eds. (World Scientific, Singapore, 1989), pp. 147–174.

Hebden, J. C.

Ishimaru, A.

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

Jacques, S. L.

Jarlman, O.

Jobsis, F. F.

F. F. Jobsis, “Noninvasive infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[CrossRef]

Kempe, M.

Kiefer, J. E.

G. H. Weiss, R. Nossal, R. F. Bonner, J. E. Kiefer, H. Taitelbaum, S. Havlin, “Random walk theory of photon migration in a turbid medium,” in Dynamical Processes in Condensed Molecular Systems, J. Klafter, J. Jortner, A. Blumen, eds. (World Scientific, Singapore, 1989), pp. 147–174.

Kirpatrick, P.

S. J. Matcher, P. Kirpatrick, K. Nahid, M. Cope, D. T. Delpy, “Absolute quantification methods in tissue near infrared spectroscopy,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, ed., Proc. SPIE2389, 486–495 (1995).

Kotler, Z.

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.

Liu, H.

H. Liu, D. A. Boas, Y. Zhang, A. G. Yodh, B. Chance, “A simplified approach to characterize optical properties and blood oxygenation in tissues using continuous near infrared light,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir eds., Proc. SPIE2389, 496–502 (1995).

Mahan, G. D.

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. Uzgiris, “Ultrasonic tagging of light:  theory,” Proc. Natl. Acad. Sci. U.S.A. 95, 14015–14019 (1998).
[CrossRef]

Maret, G.

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

Matcher, S. J.

S. J. Matcher, P. Kirpatrick, K. Nahid, M. Cope, D. T. Delpy, “Absolute quantification methods in tissue near infrared spectroscopy,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, ed., Proc. SPIE2389, 486–495 (1995).

Nahid, K.

S. J. Matcher, P. Kirpatrick, K. Nahid, M. Cope, D. T. Delpy, “Absolute quantification methods in tissue near infrared spectroscopy,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, ed., Proc. SPIE2389, 486–495 (1995).

Nokia, S.

B. Chance, S. Nokia, J. Tracey, “Brain functional imaging in problem solving using NIR,” in Physics of Medical Imaging, J. T. Dobbins, J. M. Boone, eds., Proc. SPIE (to be published).

Nossal, R.

G. H. Weiss, R. Nossal, R. F. Bonner, J. E. Kiefer, H. Taitelbaum, S. Havlin, “Random walk theory of photon migration in a turbid medium,” in Dynamical Processes in Condensed Molecular Systems, J. Klafter, J. Jortner, A. Blumen, eds. (World Scientific, Singapore, 1989), pp. 147–174.

O’Leary, M. A.

Saint-Jalmes, H.

Sfez, B.

Stevenson, D. K.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 256, 1463–1466 (1993).
[CrossRef]

Svanberg, S.

Taitelbaum, H.

H. Taitelbaum, S. Havlin, G. H. Weiss, “Approximate theory of photon migration in a two-layer medium,” Appl. Opt. 28, 2245–2249 (1989).
[CrossRef] [PubMed]

H. Taitelbaum, “Diagnosis using photon diffusion: from brain oxygenation to the fat of the atlantic salmon,” in Lecture Notes in Physics (Springer-Verlag, Berlin, 1998), Vol. 519, pp. 160–174.

G. H. Weiss, R. Nossal, R. F. Bonner, J. E. Kiefer, H. Taitelbaum, S. Havlin, “Random walk theory of photon migration in a turbid medium,” in Dynamical Processes in Condensed Molecular Systems, J. Klafter, J. Jortner, A. Blumen, eds. (World Scientific, Singapore, 1989), pp. 147–174.

Tiemann, J.

H. W. Tomlinson, J. Tiemann, “Light imaging in a scattering medium, using ultrasonic probing and speckle image differencing,” U.S. patent5,212,667 (May18, 1993).

Tiemann, J. J.

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. Uzgiris, “Ultrasonic tagging of light:  theory,” Proc. Natl. Acad. Sci. U.S.A. 95, 14015–14019 (1998).
[CrossRef]

Tomlinson, H. W.

H. W. Tomlinson, J. Tiemann, “Light imaging in a scattering medium, using ultrasonic probing and speckle image differencing,” U.S. patent5,212,667 (May18, 1993).

Tracey, J.

B. Chance, S. Nokia, J. Tracey, “Brain functional imaging in problem solving using NIR,” in Physics of Medical Imaging, J. T. Dobbins, J. M. Boone, eds., Proc. SPIE (to be published).

Uzgiris, E.

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. Uzgiris, “Ultrasonic tagging of light:  theory,” Proc. Natl. Acad. Sci. U.S.A. 95, 14015–14019 (1998).
[CrossRef]

Wang, L.

Wang, L. V.

Weiss, G. H.

H. Taitelbaum, S. Havlin, G. H. Weiss, “Approximate theory of photon migration in a two-layer medium,” Appl. Opt. 28, 2245–2249 (1989).
[CrossRef] [PubMed]

G. H. Weiss, R. Nossal, R. F. Bonner, J. E. Kiefer, H. Taitelbaum, S. Havlin, “Random walk theory of photon migration in a turbid medium,” in Dynamical Processes in Condensed Molecular Systems, J. Klafter, J. Jortner, A. Blumen, eds. (World Scientific, Singapore, 1989), pp. 147–174.

Yao, G.

Yodh, A. G.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

H. Liu, D. A. Boas, Y. Zhang, A. G. Yodh, B. Chance, “A simplified approach to characterize optical properties and blood oxygenation in tissues using continuous near infrared light,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir eds., Proc. SPIE2389, 496–502 (1995).

Zaslavski, D.

Zeng, F. A.

S. C. Feng, F. A. Zeng, B. Chance, “Analytical perturbation theory of photon migration in the presence of a single absorbing or scattering defect sphere,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 54–63 (1995).

Zhang, Y.

H. Liu, D. A. Boas, Y. Zhang, A. G. Yodh, B. Chance, “A simplified approach to characterize optical properties and blood oxygenation in tissues using continuous near infrared light,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir eds., Proc. SPIE2389, 496–502 (1995).

Zhao, X.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. Lett.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Physica B

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

Proc. Natl. Acad. Sci. U.S.A.

G. D. Mahan, W. E. Engler, J. J. Tiemann, E. Uzgiris, “Ultrasonic tagging of light:  theory,” Proc. Natl. Acad. Sci. U.S.A. 95, 14015–14019 (1998).
[CrossRef]

Science

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 256, 1463–1466 (1993).
[CrossRef]

F. F. Jobsis, “Noninvasive infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[CrossRef]

Other

S. J. Matcher, P. Kirpatrick, K. Nahid, M. Cope, D. T. Delpy, “Absolute quantification methods in tissue near infrared spectroscopy,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir, ed., Proc. SPIE2389, 486–495 (1995).

B. Chance, S. Nokia, J. Tracey, “Brain functional imaging in problem solving using NIR,” in Physics of Medical Imaging, J. T. Dobbins, J. M. Boone, eds., Proc. SPIE (to be published).

H. W. Tomlinson, J. Tiemann, “Light imaging in a scattering medium, using ultrasonic probing and speckle image differencing,” U.S. patent5,212,667 (May18, 1993).

H. Liu, D. A. Boas, Y. Zhang, A. G. Yodh, B. Chance, “A simplified approach to characterize optical properties and blood oxygenation in tissues using continuous near infrared light,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies and Instrumentation, B. Chance, R. R. Alfano, A. Katzir eds., Proc. SPIE2389, 496–502 (1995).

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

S. C. Feng, F. A. Zeng, B. Chance, “Analytical perturbation theory of photon migration in the presence of a single absorbing or scattering defect sphere,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 54–63 (1995).

G. H. Weiss, R. Nossal, R. F. Bonner, J. E. Kiefer, H. Taitelbaum, S. Havlin, “Random walk theory of photon migration in a turbid medium,” in Dynamical Processes in Condensed Molecular Systems, J. Klafter, J. Jortner, A. Blumen, eds. (World Scientific, Singapore, 1989), pp. 147–174.

H. Taitelbaum, “Diagnosis using photon diffusion: from brain oxygenation to the fat of the atlantic salmon,” in Lecture Notes in Physics (Springer-Verlag, Berlin, 1998), Vol. 519, pp. 160–174.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of the geometry of the calculations in the case of a homogeneous turbid medium.

Fig. 2
Fig. 2

Comparison between the simulation (circles) and the analytical expression (24) (solid line) for the following configuration: The optodes are located at the surface (z=0), and the tagging point is located at zt=-15. The attenuation coefficient is μ=0.2. The x axis is the source-detector distance 2ρ.

Fig. 3
Fig. 3

Tagged signal as a function of the ultrasound transducer position for a given optode separation and for μ=0.2.

Fig. 4
Fig. 4

Schematic representation of the calculation geometry in the case of an absorbing object inside the diffusive medium.

Fig. 5
Fig. 5

Signal-to-noise ratio (without the prefactor (2πj0)1/2q0Δμd2 and in a log–log scale) as a function of the distance z inside the medium for different optodes separation (2ρ). The maximum of the signal is obtained for ρ=z/3. This maximum follows a polynomial decrease in z-5/2, as indicated by the straight line.

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

Qt=D2Q-αQ,
Q(planez=0,t)=0,
Qn=16 2Q-μQ.
μab=α/c,
μsc=c/3D
τ=L2μsc2c,μ=L2μscμab/2.
16 2Q(r)-μQ(r)=jext(r).
jext(r)=j0f(r-r0).
jext=τQext/t.
Q(r)=drjext(r)G(|r-r|),
G(r-r)exp(-6μ|r-r|)4π|r-r|.
Q(r)=j0drf(r-zˆ)G(|r-r|)-j0drf(r+zˆ)G(|r-r|).
Q(r)=2drd jext(r)dzz=0G(|r-r|).
Q(r)=Vtdrq(r)drd jext(r)dzz=0×G(|r-r|)G(|r-r|),
Q(r)=Vtdrq(r)drd jext(r)dzz=0×G(|r-r|)G(|r-r|)--Vtdrq(r)drd jext(r)dzz=0××G(|r-r|)G(|r-r|),
Q(r)=zVtdrq(r)drjext(r)zz=0×G(|r-r|)G(|r-r|)z=0
Pdet=dAQ(r).
jext(r)=4πj0δ(r-rs),
q(r)=4πq0δ(r-rt),
Q(rd)=(4π)2j0q0G(|rt-r|)zr=rs×G(|rt-r|)zr=rd,
Q(rd)=j0q0zt2(1+6μ|rt-rs|)(1+6μ|rt-rd|)|rt-rs|3|rt-rd|3×exp[-6μ(|rt-rs|+|rt-rd|)].
j0=jV0,
q0=qVt,
Q(r˜d)
 =μsc2c VsVtQextqtz˜s2(1+μt|r˜t-r˜s|)(1+μt|r^t-r˜d|)|r˜t-r˜s|3|r˜t-r˜d|3×exp[-μt(|r˜t-r˜s|+|r˜t-r˜d|)],
Q(ρ, z)=j0q0z2[1+6μ(z2+ρ2)1/2]2(z2+ρ2)3×exp[-26μ(z2+ρ2)]1/2.
Q=j0116πρ2 (6μ+1/ρ)exp(-26μρ).
QTAP=j0q0z2(1+6μz2+ρ2)2(z2+ρ2)3×exp(-26μz2+ρ2-Δμd2),
SNR(QT-QTAP)/Q,
SNR(2πj0)1/2q0Δμd2z2ρ3/2(z2+ρ2)3.
SNRexp{-2[6μ(ρ2+z2)]1/2+6μρ}Δμ.

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