Abstract

We demonstrate the phase fluctuation introduced by oscillation of scattering centers in the focal volume of an ultrasound transducer in an optical tomography experiment has a nonzero mean. The conditions to be met for the above are: (i) the frequency of the ultrasound should be in the vicinity of the most dominant natural frequency of vibration of the ultrasound focal volume, (ii) the corresponding acoustic wavelength should be much larger than n*, a modified transport mean-free-path applicable for phase decorrelation and (iii) the focal volume of the ultrasound transducer should not be larger than 4 – 5 times (n*)3. We demonstrate through simulations that as the ratio of the ultrasound focal volume to (n*)3 increases, the average of the phase fluctuation decreases and becomes zero when the focal volume becomes greater than around 4(n*)3; and through simulations and experiments that as the acoustic frequency increases from 100 Hz to 1 MHz, the average phase decreases to zero. Through experiments done in chicken breast we show that the average phase increases from around 110° to 130° when the background medium is changed from water to glycerol, indicating that the average of the phase fluctuation can be used to sense changes in refractive index deep within tissue.

© 2012 OSA

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References

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  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,” Proc. SPIE1888, 500–510 (1993).
    [CrossRef]
  2. 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(6), 629–631 (1995).
    [CrossRef] [PubMed]
  3. S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett.96, 163902 (2006).
    [CrossRef] [PubMed]
  4. S. Sakadzic and L. V. Wang, “Modulation of multiply scattered coherent light by ultrasonic pulses: an analytical model,” Phy. Rev. E.72, 036620 (2005).
    [CrossRef]
  5. M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am.14(5), 1151–1158 (1997).
    [CrossRef]
  6. R. S. Chandran, D. Roy, R. Kanhirodan, R. M. Vasu, and C. U. Devi, “Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration,” Opt. Exp.19(23), 22837–22850 (2011).
    [CrossRef]
  7. T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
    [CrossRef] [PubMed]
  8. C. U. Devi, R. M. Vasu, and A. K. Sood, “Design, fabrication, and characterization of a tissue-equivalent phantom for optical elastography,” J. Biomed. Opt.10(4), 044020 (2005).
    [CrossRef]
  9. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Elect.26(12), 2166–2185 (1990).
    [CrossRef]
  10. T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am.107(6), 3035–3046 (2000).
    [CrossRef] [PubMed]
  11. M. Suheshkumar Singh, P. K. Yalavarthy, K. Rajan, and R. M. Vasu, “Assessment of the effect of ultrasound modulation of near infrared light on the quantification of scattering coefficient,” Med. Phys.37(7), 3744–3751 (2010).
    [CrossRef] [PubMed]
  12. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed.47(2), 131–146 (1995).
    [CrossRef]
  13. S. Leveque-Fort, A. C. Boccara, M. Lebec, and H. Saint-Jalmes, “Ultrasonic tagging of photon paths in scattering media: parallel speckle modulation processing,” Opt. Lett.24(3), 181–183 (1999).
    [CrossRef]
  14. J. Li and L. V. Wang, “Methods for parallel-detection-based ultrasound modulated optical tomography,” Appl. Opt.41(10), 2079–2084 (2002).
    [CrossRef] [PubMed]
  15. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt.24 (18), 3053–3058 (1985).
    [CrossRef] [PubMed]

2011 (1)

R. S. Chandran, D. Roy, R. Kanhirodan, R. M. Vasu, and C. U. Devi, “Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration,” Opt. Exp.19(23), 22837–22850 (2011).
[CrossRef]

2010 (1)

M. Suheshkumar Singh, P. K. Yalavarthy, K. Rajan, and R. M. Vasu, “Assessment of the effect of ultrasound modulation of near infrared light on the quantification of scattering coefficient,” Med. Phys.37(7), 3744–3751 (2010).
[CrossRef] [PubMed]

2006 (1)

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

2005 (2)

S. Sakadzic and L. V. Wang, “Modulation of multiply scattered coherent light by ultrasonic pulses: an analytical model,” Phy. Rev. E.72, 036620 (2005).
[CrossRef]

C. U. Devi, R. M. Vasu, and A. K. Sood, “Design, fabrication, and characterization of a tissue-equivalent phantom for optical elastography,” J. Biomed. Opt.10(4), 044020 (2005).
[CrossRef]

2002 (2)

J. Li and L. V. Wang, “Methods for parallel-detection-based ultrasound modulated optical tomography,” Appl. Opt.41(10), 2079–2084 (2002).
[CrossRef] [PubMed]

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

2000 (1)

T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am.107(6), 3035–3046 (2000).
[CrossRef] [PubMed]

1999 (1)

1997 (1)

M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am.14(5), 1151–1158 (1997).
[CrossRef]

1995 (2)

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed.47(2), 131–146 (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(6), 629–631 (1995).
[CrossRef] [PubMed]

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,” Proc. SPIE1888, 500–510 (1993).
[CrossRef]

1990 (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Elect.26(12), 2166–2185 (1990).
[CrossRef]

1985 (1)

Boccara, A. C.

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,” Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Chance, B.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

Chandran, R. S.

R. S. Chandran, D. Roy, R. Kanhirodan, R. M. Vasu, and C. U. Devi, “Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration,” Opt. Exp.19(23), 22837–22850 (2011).
[CrossRef]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Elect.26(12), 2166–2185 (1990).
[CrossRef]

Choe, R.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

Creath, K.

Culver, J. P.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

Devi, C. U.

R. S. Chandran, D. Roy, R. Kanhirodan, R. M. Vasu, and C. U. Devi, “Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration,” Opt. Exp.19(23), 22837–22850 (2011).
[CrossRef]

C. U. Devi, R. M. Vasu, and A. K. Sood, “Design, fabrication, and characterization of a tissue-equivalent phantom for optical elastography,” J. Biomed. Opt.10(4), 044020 (2005).
[CrossRef]

Durduran, T.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

Genack, A. Z.

M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am.14(5), 1151–1158 (1997).
[CrossRef]

Giammarco, J.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

Holboke, M. J.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

Ishiwata, T.

T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am.107(6), 3035–3046 (2000).
[CrossRef] [PubMed]

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed.47(2), 131–146 (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(6), 629–631 (1995).
[CrossRef] [PubMed]

Kamakura, T.

T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am.107(6), 3035–3046 (2000).
[CrossRef] [PubMed]

Kanhirodan, R.

R. S. Chandran, D. Roy, R. Kanhirodan, R. M. Vasu, and C. U. Devi, “Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration,” Opt. Exp.19(23), 22837–22850 (2011).
[CrossRef]

Kempe, M.

M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am.14(5), 1151–1158 (1997).
[CrossRef]

Larionov, M.

M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am.14(5), 1151–1158 (1997).
[CrossRef]

Lebec, M.

Leveque-Fort, S.

Li, J.

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,” Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Matsuda, K.

T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am.107(6), 3035–3046 (2000).
[CrossRef] [PubMed]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Elect.26(12), 2166–2185 (1990).
[CrossRef]

Rajan, K.

M. Suheshkumar Singh, P. K. Yalavarthy, K. Rajan, and R. M. Vasu, “Assessment of the effect of ultrasound modulation of near infrared light on the quantification of scattering coefficient,” Med. Phys.37(7), 3744–3751 (2010).
[CrossRef] [PubMed]

Roy, D.

R. S. Chandran, D. Roy, R. Kanhirodan, R. M. Vasu, and C. U. Devi, “Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration,” Opt. Exp.19(23), 22837–22850 (2011).
[CrossRef]

Saint-Jalmes, H.

Sakadzic, S.

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

S. Sakadzic and L. V. Wang, “Modulation of multiply scattered coherent light by ultrasonic pulses: an analytical model,” Phy. Rev. E.72, 036620 (2005).
[CrossRef]

Sood, A. K.

C. U. Devi, R. M. Vasu, and A. K. Sood, “Design, fabrication, and characterization of a tissue-equivalent phantom for optical elastography,” J. Biomed. Opt.10(4), 044020 (2005).
[CrossRef]

Suheshkumar Singh, M.

M. Suheshkumar Singh, P. K. Yalavarthy, K. Rajan, and R. M. Vasu, “Assessment of the effect of ultrasound modulation of near infrared light on the quantification of scattering coefficient,” Med. Phys.37(7), 3744–3751 (2010).
[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,” Proc. SPIE1888, 500–510 (1993).
[CrossRef]

Vasu, R. M.

R. S. Chandran, D. Roy, R. Kanhirodan, R. M. Vasu, and C. U. Devi, “Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration,” Opt. Exp.19(23), 22837–22850 (2011).
[CrossRef]

M. Suheshkumar Singh, P. K. Yalavarthy, K. Rajan, and R. M. Vasu, “Assessment of the effect of ultrasound modulation of near infrared light on the quantification of scattering coefficient,” Med. Phys.37(7), 3744–3751 (2010).
[CrossRef] [PubMed]

C. U. Devi, R. M. Vasu, and A. K. Sood, “Design, fabrication, and characterization of a tissue-equivalent phantom for optical elastography,” J. Biomed. Opt.10(4), 044020 (2005).
[CrossRef]

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed.47(2), 131–146 (1995).
[CrossRef]

Wang, L. V.

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

S. Sakadzic and L. V. Wang, “Modulation of multiply scattered coherent light by ultrasonic pulses: an analytical model,” Phy. Rev. E.72, 036620 (2005).
[CrossRef]

J. Li and L. V. Wang, “Methods for parallel-detection-based ultrasound modulated optical tomography,” Appl. Opt.41(10), 2079–2084 (2002).
[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(6), 629–631 (1995).
[CrossRef] [PubMed]

Welch, A. J.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Elect.26(12), 2166–2185 (1990).
[CrossRef]

Yalavarthy, P. K.

M. Suheshkumar Singh, P. K. Yalavarthy, K. Rajan, and R. M. Vasu, “Assessment of the effect of ultrasound modulation of near infrared light on the quantification of scattering coefficient,” Med. Phys.37(7), 3744–3751 (2010).
[CrossRef] [PubMed]

Yodh, A. G.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

Zaslavsky, D.

M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am.14(5), 1151–1158 (1997).
[CrossRef]

Zhao, X.

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed.47(2), 131–146 (1995).
[CrossRef]

Zubkov, L.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

Appl. Opt. (2)

Comput. Meth. Prog. Biomed. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed.47(2), 131–146 (1995).
[CrossRef]

IEEE J. Quant. Elect. (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quant. Elect.26(12), 2166–2185 (1990).
[CrossRef]

J. Acoust. Soc. Am. (1)

T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am.107(6), 3035–3046 (2000).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

C. U. Devi, R. M. Vasu, and A. K. Sood, “Design, fabrication, and characterization of a tissue-equivalent phantom for optical elastography,” J. Biomed. Opt.10(4), 044020 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am.14(5), 1151–1158 (1997).
[CrossRef]

Med. Phys. (1)

M. Suheshkumar Singh, P. K. Yalavarthy, K. Rajan, and R. M. Vasu, “Assessment of the effect of ultrasound modulation of near infrared light on the quantification of scattering coefficient,” Med. Phys.37(7), 3744–3751 (2010).
[CrossRef] [PubMed]

Opt. Exp. (1)

R. S. Chandran, D. Roy, R. Kanhirodan, R. M. Vasu, and C. U. Devi, “Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration,” Opt. Exp.19(23), 22837–22850 (2011).
[CrossRef]

Opt. Lett. (2)

Phy. Rev. E. (1)

S. Sakadzic and L. V. Wang, “Modulation of multiply scattered coherent light by ultrasonic pulses: an analytical model,” Phy. Rev. E.72, 036620 (2005).
[CrossRef]

Phys. Med. Biol. (1)

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol.47, 2847–2861 (2002).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett.96, 163902 (2006).
[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,” Proc. SPIE1888, 500–510 (1993).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Representation of a single scattering event from a point which suffered a maximum displacement of u⃗.

Fig. 2
Fig. 2

Variation of 〈Δϕ(τ)〉av (arrived at through MC simulation) with the volume of the ROI. Beyond V = 4 ( n * ) 3, 〈Δϕ(τ)〉av fluctuates rendering the average not reliable.

Fig. 3
Fig. 3

Variation of 〈Δϕ(τ)〉av with the ultrasound beat frequency. Beyond 1 kHz the variation in the mean is large and the measurement is inaccurate. The error in the experimental measurements is in the range of 4 – 6%.

Fig. 4
Fig. 4

Histogram of the phase fluctuation obtained from experiments and simulations. The object used is the PVA phantom. When the US frequency is (a) 1 MHz the mean is close to zero; when it is (b) 100 Hz the histogram has a clear bias with a peak around 140° (for experiment) and 145° (for simulations). The percentage error in phase extracted through measurements is 1 – 6% depending on intensity level.

Fig. 5
Fig. 5

Histogram of the measured phase fluctuation obtained from experiments and simulations. The object used is a slab of chicken breast which contains water in the ROI. When the US frequency is 1 MHz the phase fluctuation is random with the mean close to zero (a) and when it is 100 Hz the histogram has a clear bias with a peak around 110° (for experiment) and 105° (for simulations) (b). The percentage error in phase extracted through measurements is 1 – 8%.

Fig. 6
Fig. 6

Same as 5(b) but with glycerol injected around the ROI. It is seen that owing to the higher n of glycerol the average phase has increased to ≈ 130° (for experiment) and 120° (for simulations). The percentage error in phase extracted through measurements is 1 – 7%.

Fig. 7
Fig. 7

Schematic diagram of the experimental set-up. The CCD camera is focused onto the optical window. For a fuller description of the parts shown see Ref. [11].

Fig. 8
Fig. 8

Schematic diagram showing a typical cross section of the experimental set-up. Two focusing ultrasonic waves from the transducers are intersecting each other inside the tissue subtending an angle of 60°. A narrow laser beam (of diameter 2 mm) is incident onto the sample along the direction bisecting the subtending angle of the US focusing waves and the CCD faces the diode laser as shown to collect the transmitted light.

Equations (4)

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

Δ r j ( t ) = u ( r j ) sin ( k a r j ω a t ) .
δ ϕ p ( t ) = j = 1 N p q j Δ r j ( t ) .
Δ ϕ p ( τ ) = δ ϕ p ( t + τ ) δ ϕ p ( t ) lim T 1 T 0 T ( δ ϕ p ( t + τ ) δ ϕ p ( t ) ) d t .
Δ ϕ ( τ ) a v = s 1 s 2 p ( s ) Δ ϕ p ( τ ) d s .

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