Abstract

An heuristic model for ballistic photon detection in continuous-wave measurements of collimated transmittance through a slab is presented. The model is based on the small angle approximation and the diffusion equation and covers all the ranges of optical thicknesses of the slab from the ballistic to the diffusive regime. The performances of the model have been studied by means of comparisons with the results of gold standard Monte Carlo simulations for a wide range of optical thicknesses and two types of scattering functions. For a non-absorbing slab and field of view of the receiver less than 3° the model shows errors less than 15% for any value of the optical thickness. Even for an albedo value of 0.9, and field of view of the receiver less than 3° the model shows errors less than 20%. These results have been verified for a large set of scattering functions based on the Henyey-Greenstein model and Mie theory for spherical scatterers. The latter has also been used to simulate the scattering function of Intralipid, a diffusive material widely used as reference standard for tissue simulating phantoms. The proposed model represents an effective improvement compared to the existing literature.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  3. L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253(5021), 769–771 (1991).
    [Crossref] [PubMed]
  4. R. R. Alfano, X. Liang, and L. Wang, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264(5157), 1913–1915 (1994).
    [Crossref] [PubMed]
  5. A. Bassi, D. Brida, C. D’Andrea, G. Valentini, R. Cubeddu, S. De Silvestri, and G. Cerullo, “Time-gated optical projection tomography,” Opt. Lett. 35(16), 2732–2734 (2010).
    [Crossref] [PubMed]
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    [Crossref]
  7. L. Sinha, J. G. Brankov, and K. M. Tichaue, “Modeling subdiffusive light scattering by incorporating the tissue phase function and detector numerical aperture,” Opt. Lett. 41(14), 3225–3228 (2016).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  13. I. Ben, Y. Y. Layosh, and E. Granot, “Study of a simple model for the transition between the ballistic and the diffusive regimes in diffusive media,” J. Biom. Opt. 21, 066004 (2016).
    [Crossref]
  14. L. Wind and W. W. Szymanski, “Quantification of scattering corrections to the Beer-Lambert law for transmittance measurements in turbid media,” Meas. Sci. Technol. 13(3), 270–275 (2002).
    [Crossref]
  15. S. N. Thennadil and Y. Chen, “Alternative Measurement Configurations for Extracting Bulk Optical Properties Using an Integrating Sphere Setup,” Appl. Spectrosc. 71(2), 224–237 (2017).
    [Crossref]
  16. G. Zaccanti and P. Bruscaglioni, “Deviation from the Lambert-Beer law in the transmittance of a light beam through diffusing media: Experimental results,” J. Mod. Opt. 35(2), 229–242 (1988).
    [Crossref]
  17. A. Taddeucci, F. Martelli, M. Barilli, M. Ferrari, and G. Zaccanti, “Optical properties of brain tissue,” J. Biomed. Opt. 1(1), 117–123 (1996).
    [Crossref] [PubMed]
  18. A. Ishimaru, Waves propagation and scattering in random media (Academic Press, 1978).
  19. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2009).
  20. X.C. Li, J. M. Zhao, C. C. Wang, and L. H. Liu, “Improved transmission method for measuring the optical extinction coefficient of micro/nano particle suspensions,” Appl. Opt. 55(29), 8171–8179 (2016).
    [Crossref] [PubMed]
  21. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36(19), 4587–4599 (1997).
    [Crossref] [PubMed]
  22. F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44(5), 1257–1275 (1999).
    [Crossref] [PubMed]
  23. D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50(2), 857–866 (1994).
    [Crossref]
  24. F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results,” Appl. Opt. 36(19), 4600–4612 (1997).
    [Crossref] [PubMed]
  25. H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in lntralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991).
    [Crossref] [PubMed]
  26. P. Di Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. 56(2), N21–N28 (2011).
    [Crossref]
  27. G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
    [Crossref]
  28. S. Del Bianco, F. Martelli, F. Cignini, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Liquid phantom for investigating light propagation through layered diffusive media,” Opt. Express 12(10), 2102–2111 (2004).
    [Crossref] [PubMed]
  29. M. P. Shepilov, O. S. Dymshits, A. A. Zhilin, and S. S. Zapalova, “On the measurements of scattering coefficient of nanostructured glass-ceramics by a serial spectrophotometer,” Measurement 95(10), 306–316 (2017).
    [Crossref]
  30. M. P. Shepilov, “Asymmetry parameter for anomalous scattering of light in nanostructured glasses,” Opt. Lett. 42(21), 4513–4516 (2017).
    [Crossref] [PubMed]

2017 (4)

A. L. Post, S. L. Jacques, H. J. C. M. Sterenborg, D. J. Faber, and T. G. van Leeuwen, “Modeling subdiffusive light scattering by incorporating the tissue phase function and detector numerical aperture,” J. Biom. Opt. 22, 050501 (2017).
[Crossref]

S. N. Thennadil and Y. Chen, “Alternative Measurement Configurations for Extracting Bulk Optical Properties Using an Integrating Sphere Setup,” Appl. Spectrosc. 71(2), 224–237 (2017).
[Crossref]

M. P. Shepilov, O. S. Dymshits, A. A. Zhilin, and S. S. Zapalova, “On the measurements of scattering coefficient of nanostructured glass-ceramics by a serial spectrophotometer,” Measurement 95(10), 306–316 (2017).
[Crossref]

M. P. Shepilov, “Asymmetry parameter for anomalous scattering of light in nanostructured glasses,” Opt. Lett. 42(21), 4513–4516 (2017).
[Crossref] [PubMed]

2016 (3)

2015 (1)

B. Brezner, S. Cahen, Z. Glasser, S. Sternklar, and E. Granot, “Ballistic imaging of biological media with collimated illumination and focal plane detection,” J. Biom. Opt. 20, 076006 (2015).
[Crossref]

2013 (2)

A. Yaroshevsky, Z. Glasser, E. Granot, and S. Sternklar, “Transition from the ballistic to the diffusive regime in a turbid medium,” Opt. Lett. 36(8), 1395–1397 (2013).
[Crossref]

Z. Glasser, A. Yaroshevsky, B. Barak, E. Granot, and S. Sternklar, “Effect of measurement on the ballistic-diffusive transition in turbid media,” J. Biomed. Opt. 18, 106006 (2013).
[Crossref] [PubMed]

2011 (1)

P. Di Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. 56(2), N21–N28 (2011).
[Crossref]

2010 (1)

2004 (2)

2002 (1)

L. Wind and W. W. Szymanski, “Quantification of scattering corrections to the Beer-Lambert law for transmittance measurements in turbid media,” Meas. Sci. Technol. 13(3), 270–275 (2002).
[Crossref]

1999 (2)

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60(4), 4843–4850 (1999).
[Crossref]

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44(5), 1257–1275 (1999).
[Crossref] [PubMed]

1997 (2)

1996 (1)

A. Taddeucci, F. Martelli, M. Barilli, M. Ferrari, and G. Zaccanti, “Optical properties of brain tissue,” J. Biomed. Opt. 1(1), 117–123 (1996).
[Crossref] [PubMed]

1994 (3)

R. R. Alfano, X. Liang, and L. Wang, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264(5157), 1913–1915 (1994).
[Crossref] [PubMed]

D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50(2), 857–866 (1994).
[Crossref]

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[Crossref]

1991 (2)

H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in lntralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991).
[Crossref] [PubMed]

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253(5021), 769–771 (1991).
[Crossref] [PubMed]

1988 (1)

G. Zaccanti and P. Bruscaglioni, “Deviation from the Lambert-Beer law in the transmittance of a light beam through diffusing media: Experimental results,” J. Mod. Opt. 35(2), 229–242 (1988).
[Crossref]

Alfano, R. R.

R. R. Alfano, X. Liang, and L. Wang, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264(5157), 1913–1915 (1994).
[Crossref] [PubMed]

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253(5021), 769–771 (1991).
[Crossref] [PubMed]

R. R. Alfano, W. B. Wang, L. Wang, and S. K. Gayen, “Light propagation in highly scattering turbid media: concepts, techniques, and biomedical applications,” in Photonics, Vol. 4: Biomedical Photonics, Spectroscopy, and Microscopy, D. L. Andrews, ed. (John Wiley & Sons, Inc., 2015) Chap. 9.

Barak, B.

Z. Glasser, A. Yaroshevsky, B. Barak, E. Granot, and S. Sternklar, “Effect of measurement on the ballistic-diffusive transition in turbid media,” J. Biomed. Opt. 18, 106006 (2013).
[Crossref] [PubMed]

Barilli, M.

A. Taddeucci, F. Martelli, M. Barilli, M. Ferrari, and G. Zaccanti, “Optical properties of brain tissue,” J. Biomed. Opt. 1(1), 117–123 (1996).
[Crossref] [PubMed]

Bassi, A.

Battistelli, E.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[Crossref]

Ben, I.

I. Ben, Y. Y. Layosh, and E. Granot, “Study of a simple model for the transition between the ballistic and the diffusive regimes in diffusive media,” J. Biom. Opt. 21, 066004 (2016).
[Crossref]

Brankov, J. G.

Brezner, B.

B. Brezner, S. Cahen, Z. Glasser, S. Sternklar, and E. Granot, “Ballistic imaging of biological media with collimated illumination and focal plane detection,” J. Biom. Opt. 20, 076006 (2015).
[Crossref]

Brida, D.

Bruscaglioni, P.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[Crossref]

G. Zaccanti and P. Bruscaglioni, “Deviation from the Lambert-Beer law in the transmittance of a light beam through diffusing media: Experimental results,” J. Mod. Opt. 35(2), 229–242 (1988).
[Crossref]

Cahen, S.

B. Brezner, S. Cahen, Z. Glasser, S. Sternklar, and E. Granot, “Ballistic imaging of biological media with collimated illumination and focal plane detection,” J. Biom. Opt. 20, 076006 (2015).
[Crossref]

Carminati, R.

Cerullo, G.

Chen, Y.

Cignini, F.

Contini, D.

Cubeddu, R.

D’Andrea, C.

De Silvestri, S.

Del Bianco, S.

Di Ninni, P.

P. Di Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. 56(2), N21–N28 (2011).
[Crossref]

Durian, D. J.

D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50(2), 857–866 (1994).
[Crossref]

Dymshits, O. S.

M. P. Shepilov, O. S. Dymshits, A. A. Zhilin, and S. S. Zapalova, “On the measurements of scattering coefficient of nanostructured glass-ceramics by a serial spectrophotometer,” Measurement 95(10), 306–316 (2017).
[Crossref]

Elaloufi, R.

Faber, D. J.

A. L. Post, S. L. Jacques, H. J. C. M. Sterenborg, D. J. Faber, and T. G. van Leeuwen, “Modeling subdiffusive light scattering by incorporating the tissue phase function and detector numerical aperture,” J. Biom. Opt. 22, 050501 (2017).
[Crossref]

Ferrari, M.

A. Taddeucci, F. Martelli, M. Barilli, M. Ferrari, and G. Zaccanti, “Optical properties of brain tissue,” J. Biomed. Opt. 1(1), 117–123 (1996).
[Crossref] [PubMed]

Gayen, S. K.

R. R. Alfano, W. B. Wang, L. Wang, and S. K. Gayen, “Light propagation in highly scattering turbid media: concepts, techniques, and biomedical applications,” in Photonics, Vol. 4: Biomedical Photonics, Spectroscopy, and Microscopy, D. L. Andrews, ed. (John Wiley & Sons, Inc., 2015) Chap. 9.

Glasser, Z.

B. Brezner, S. Cahen, Z. Glasser, S. Sternklar, and E. Granot, “Ballistic imaging of biological media with collimated illumination and focal plane detection,” J. Biom. Opt. 20, 076006 (2015).
[Crossref]

A. Yaroshevsky, Z. Glasser, E. Granot, and S. Sternklar, “Transition from the ballistic to the diffusive regime in a turbid medium,” Opt. Lett. 36(8), 1395–1397 (2013).
[Crossref]

Z. Glasser, A. Yaroshevsky, B. Barak, E. Granot, and S. Sternklar, “Effect of measurement on the ballistic-diffusive transition in turbid media,” J. Biomed. Opt. 18, 106006 (2013).
[Crossref] [PubMed]

Granot, E.

I. Ben, Y. Y. Layosh, and E. Granot, “Study of a simple model for the transition between the ballistic and the diffusive regimes in diffusive media,” J. Biom. Opt. 21, 066004 (2016).
[Crossref]

B. Brezner, S. Cahen, Z. Glasser, S. Sternklar, and E. Granot, “Ballistic imaging of biological media with collimated illumination and focal plane detection,” J. Biom. Opt. 20, 076006 (2015).
[Crossref]

A. Yaroshevsky, Z. Glasser, E. Granot, and S. Sternklar, “Transition from the ballistic to the diffusive regime in a turbid medium,” Opt. Lett. 36(8), 1395–1397 (2013).
[Crossref]

Z. Glasser, A. Yaroshevsky, B. Barak, E. Granot, and S. Sternklar, “Effect of measurement on the ballistic-diffusive transition in turbid media,” J. Biomed. Opt. 18, 106006 (2013).
[Crossref] [PubMed]

Greffet, J.-J.

Ho, P. P.

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253(5021), 769–771 (1991).
[Crossref] [PubMed]

Ishimaru, A.

A. Ishimaru, Waves propagation and scattering in random media (Academic Press, 1978).

Ismaelli, A.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2009).

Jacques, S. L.

A. L. Post, S. L. Jacques, H. J. C. M. Sterenborg, D. J. Faber, and T. G. van Leeuwen, “Modeling subdiffusive light scattering by incorporating the tissue phase function and detector numerical aperture,” J. Biom. Opt. 22, 050501 (2017).
[Crossref]

Jones, I. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60(4), 4843–4850 (1999).
[Crossref]

Layosh, Y. Y.

I. Ben, Y. Y. Layosh, and E. Granot, “Study of a simple model for the transition between the ballistic and the diffusive regimes in diffusive media,” J. Biom. Opt. 21, 066004 (2016).
[Crossref]

Li, X.C.

Liang, X.

R. R. Alfano, X. Liang, and L. Wang, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264(5157), 1913–1915 (1994).
[Crossref] [PubMed]

Liu, C.

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253(5021), 769–771 (1991).
[Crossref] [PubMed]

Liu, L. H.

Martelli, F.

P. Di Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. 56(2), N21–N28 (2011).
[Crossref]

S. Del Bianco, F. Martelli, F. Cignini, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Liquid phantom for investigating light propagation through layered diffusive media,” Opt. Express 12(10), 2102–2111 (2004).
[Crossref] [PubMed]

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44(5), 1257–1275 (1999).
[Crossref] [PubMed]

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36(19), 4587–4599 (1997).
[Crossref] [PubMed]

F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results,” Appl. Opt. 36(19), 4600–4612 (1997).
[Crossref] [PubMed]

A. Taddeucci, F. Martelli, M. Barilli, M. Ferrari, and G. Zaccanti, “Optical properties of brain tissue,” J. Biomed. Opt. 1(1), 117–123 (1996).
[Crossref] [PubMed]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2009).

Moes, C. J. M.

Page, J. H.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60(4), 4843–4850 (1999).
[Crossref]

Pifferi, A.

Post, A. L.

A. L. Post, S. L. Jacques, H. J. C. M. Sterenborg, D. J. Faber, and T. G. van Leeuwen, “Modeling subdiffusive light scattering by incorporating the tissue phase function and detector numerical aperture,” J. Biom. Opt. 22, 050501 (2017).
[Crossref]

Prahl, S. A.

Sassaroli, A.

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44(5), 1257–1275 (1999).
[Crossref] [PubMed]

Schriemer, H. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60(4), 4843–4850 (1999).
[Crossref]

Sheng, P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60(4), 4843–4850 (1999).
[Crossref]

Shepilov, M. P.

M. P. Shepilov, O. S. Dymshits, A. A. Zhilin, and S. S. Zapalova, “On the measurements of scattering coefficient of nanostructured glass-ceramics by a serial spectrophotometer,” Measurement 95(10), 306–316 (2017).
[Crossref]

M. P. Shepilov, “Asymmetry parameter for anomalous scattering of light in nanostructured glasses,” Opt. Lett. 42(21), 4513–4516 (2017).
[Crossref] [PubMed]

Sinha, L.

Sterenborg, H. J. C. M.

A. L. Post, S. L. Jacques, H. J. C. M. Sterenborg, D. J. Faber, and T. G. van Leeuwen, “Modeling subdiffusive light scattering by incorporating the tissue phase function and detector numerical aperture,” J. Biom. Opt. 22, 050501 (2017).
[Crossref]

Sternklar, S.

B. Brezner, S. Cahen, Z. Glasser, S. Sternklar, and E. Granot, “Ballistic imaging of biological media with collimated illumination and focal plane detection,” J. Biom. Opt. 20, 076006 (2015).
[Crossref]

Z. Glasser, A. Yaroshevsky, B. Barak, E. Granot, and S. Sternklar, “Effect of measurement on the ballistic-diffusive transition in turbid media,” J. Biomed. Opt. 18, 106006 (2013).
[Crossref] [PubMed]

A. Yaroshevsky, Z. Glasser, E. Granot, and S. Sternklar, “Transition from the ballistic to the diffusive regime in a turbid medium,” Opt. Lett. 36(8), 1395–1397 (2013).
[Crossref]

Szymanski, W. W.

L. Wind and W. W. Szymanski, “Quantification of scattering corrections to the Beer-Lambert law for transmittance measurements in turbid media,” Meas. Sci. Technol. 13(3), 270–275 (2002).
[Crossref]

Taddeucci, A.

Taroni, P.

Thennadil, S. N.

Tichaue, K. M.

Torricelli, A.

Valentini, G.

van Gemert, M. J. C.

van Leeuwen, T. G.

A. L. Post, S. L. Jacques, H. J. C. M. Sterenborg, D. J. Faber, and T. G. van Leeuwen, “Modeling subdiffusive light scattering by incorporating the tissue phase function and detector numerical aperture,” J. Biom. Opt. 22, 050501 (2017).
[Crossref]

van Marie, J.

van Staveren, H. J.

Wang, C. C.

Wang, L.

R. R. Alfano, X. Liang, and L. Wang, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264(5157), 1913–1915 (1994).
[Crossref] [PubMed]

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253(5021), 769–771 (1991).
[Crossref] [PubMed]

R. R. Alfano, W. B. Wang, L. Wang, and S. K. Gayen, “Light propagation in highly scattering turbid media: concepts, techniques, and biomedical applications,” in Photonics, Vol. 4: Biomedical Photonics, Spectroscopy, and Microscopy, D. L. Andrews, ed. (John Wiley & Sons, Inc., 2015) Chap. 9.

Wang, L. V.

L. V. Wang and H. I. Wu, Biomedical Optics, Principles and Imaging (John Wiley & Sons, Inc., 2007) Chap. 7–8.

Wang, W. B.

R. R. Alfano, W. B. Wang, L. Wang, and S. K. Gayen, “Light propagation in highly scattering turbid media: concepts, techniques, and biomedical applications,” in Photonics, Vol. 4: Biomedical Photonics, Spectroscopy, and Microscopy, D. L. Andrews, ed. (John Wiley & Sons, Inc., 2015) Chap. 9.

Wei, Q. N.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[Crossref]

Weitz, D. A.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60(4), 4843–4850 (1999).
[Crossref]

Wind, L.

L. Wind and W. W. Szymanski, “Quantification of scattering corrections to the Beer-Lambert law for transmittance measurements in turbid media,” Meas. Sci. Technol. 13(3), 270–275 (2002).
[Crossref]

Wu, H. I.

L. V. Wang and H. I. Wu, Biomedical Optics, Principles and Imaging (John Wiley & Sons, Inc., 2007) Chap. 7–8.

Yamada, Y.

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44(5), 1257–1275 (1999).
[Crossref] [PubMed]

Yaroshevsky, A.

A. Yaroshevsky, Z. Glasser, E. Granot, and S. Sternklar, “Transition from the ballistic to the diffusive regime in a turbid medium,” Opt. Lett. 36(8), 1395–1397 (2013).
[Crossref]

Z. Glasser, A. Yaroshevsky, B. Barak, E. Granot, and S. Sternklar, “Effect of measurement on the ballistic-diffusive transition in turbid media,” J. Biomed. Opt. 18, 106006 (2013).
[Crossref] [PubMed]

Zaccanti, G.

P. Di Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. 56(2), N21–N28 (2011).
[Crossref]

S. Del Bianco, F. Martelli, F. Cignini, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Liquid phantom for investigating light propagation through layered diffusive media,” Opt. Express 12(10), 2102–2111 (2004).
[Crossref] [PubMed]

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44(5), 1257–1275 (1999).
[Crossref] [PubMed]

D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36(19), 4587–4599 (1997).
[Crossref] [PubMed]

F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results,” Appl. Opt. 36(19), 4600–4612 (1997).
[Crossref] [PubMed]

A. Taddeucci, F. Martelli, M. Barilli, M. Ferrari, and G. Zaccanti, “Optical properties of brain tissue,” J. Biomed. Opt. 1(1), 117–123 (1996).
[Crossref] [PubMed]

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[Crossref]

G. Zaccanti and P. Bruscaglioni, “Deviation from the Lambert-Beer law in the transmittance of a light beam through diffusing media: Experimental results,” J. Mod. Opt. 35(2), 229–242 (1988).
[Crossref]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2009).

Zapalova, S. S.

M. P. Shepilov, O. S. Dymshits, A. A. Zhilin, and S. S. Zapalova, “On the measurements of scattering coefficient of nanostructured glass-ceramics by a serial spectrophotometer,” Measurement 95(10), 306–316 (2017).
[Crossref]

Zhang, G.

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253(5021), 769–771 (1991).
[Crossref] [PubMed]

Zhang, Z. Q.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60(4), 4843–4850 (1999).
[Crossref]

Zhao, J. M.

Zhilin, A. A.

M. P. Shepilov, O. S. Dymshits, A. A. Zhilin, and S. S. Zapalova, “On the measurements of scattering coefficient of nanostructured glass-ceramics by a serial spectrophotometer,” Measurement 95(10), 306–316 (2017).
[Crossref]

Appl. Opt. (4)

Appl. Spectrosc. (1)

J. Biom. Opt. (3)

I. Ben, Y. Y. Layosh, and E. Granot, “Study of a simple model for the transition between the ballistic and the diffusive regimes in diffusive media,” J. Biom. Opt. 21, 066004 (2016).
[Crossref]

B. Brezner, S. Cahen, Z. Glasser, S. Sternklar, and E. Granot, “Ballistic imaging of biological media with collimated illumination and focal plane detection,” J. Biom. Opt. 20, 076006 (2015).
[Crossref]

A. L. Post, S. L. Jacques, H. J. C. M. Sterenborg, D. J. Faber, and T. G. van Leeuwen, “Modeling subdiffusive light scattering by incorporating the tissue phase function and detector numerical aperture,” J. Biom. Opt. 22, 050501 (2017).
[Crossref]

J. Biomed. Opt. (2)

A. Taddeucci, F. Martelli, M. Barilli, M. Ferrari, and G. Zaccanti, “Optical properties of brain tissue,” J. Biomed. Opt. 1(1), 117–123 (1996).
[Crossref] [PubMed]

Z. Glasser, A. Yaroshevsky, B. Barak, E. Granot, and S. Sternklar, “Effect of measurement on the ballistic-diffusive transition in turbid media,” J. Biomed. Opt. 18, 106006 (2013).
[Crossref] [PubMed]

J. Mod. Opt. (1)

G. Zaccanti and P. Bruscaglioni, “Deviation from the Lambert-Beer law in the transmittance of a light beam through diffusing media: Experimental results,” J. Mod. Opt. 35(2), 229–242 (1988).
[Crossref]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

L. Wind and W. W. Szymanski, “Quantification of scattering corrections to the Beer-Lambert law for transmittance measurements in turbid media,” Meas. Sci. Technol. 13(3), 270–275 (2002).
[Crossref]

Measurement (1)

M. P. Shepilov, O. S. Dymshits, A. A. Zhilin, and S. S. Zapalova, “On the measurements of scattering coefficient of nanostructured glass-ceramics by a serial spectrophotometer,” Measurement 95(10), 306–316 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Med. Biol. (2)

P. Di Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. 56(2), N21–N28 (2011).
[Crossref]

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44(5), 1257–1275 (1999).
[Crossref] [PubMed]

Phys. Rev. E (2)

D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50(2), 857–866 (1994).
[Crossref]

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E 60(4), 4843–4850 (1999).
[Crossref]

Pure Appl. Opt. (1)

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
[Crossref]

Science (2)

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-d imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253(5021), 769–771 (1991).
[Crossref] [PubMed]

R. R. Alfano, X. Liang, and L. Wang, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264(5157), 1913–1915 (1994).
[Crossref] [PubMed]

Other (4)

L. V. Wang and H. I. Wu, Biomedical Optics, Principles and Imaging (John Wiley & Sons, Inc., 2007) Chap. 7–8.

R. R. Alfano, W. B. Wang, L. Wang, and S. K. Gayen, “Light propagation in highly scattering turbid media: concepts, techniques, and biomedical applications,” in Photonics, Vol. 4: Biomedical Photonics, Spectroscopy, and Microscopy, D. L. Andrews, ed. (John Wiley & Sons, Inc., 2015) Chap. 9.

A. Ishimaru, Waves propagation and scattering in random media (Academic Press, 1978).

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software (SPIE, 2009).

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Figures (11)

Fig. 1
Fig. 1 Basic schematic for measurements of ballistic light in a collimated transmittance configuration. Parameters are: r, radius of the entrance surface of the optical receiver; α, semi-aperture of the field of view of the optical receiver; s0, thickness of the scattering medium slab; d, distance between the scattering medium and the optical receiver; ni refractive index of the scattering medium; ne external refractive index (e.g. air).
Fig. 2
Fig. 2 Simulated MC experiments for a scattering medium constituted by a suspension of polystyrene latex spheres in water (〈ϕ〉 = 15.8 μm). The figure shows τA versus τ for three α values; s0 = 10 cm, d = 70 cm and r = 2 cm. The straight line representing the ideal measurement is also shown.
Fig. 3
Fig. 3 Schematic of multiple scattering interactions in a transmittance measurement for a pencil beam impinging onto the medium. Parameters are: lk, steps between scattering events; dlk, infinitesimal step before a scattering event; dΩk, solid angle; θk, scattering angle. This figure is a 2D projection, but must be imagined in 3D.
Fig. 4
Fig. 4 Examples of PsExt(r, α) obtained from the fitting of PDE(r)χ(α) for a slab 10 mm thick with μa = 0, ni = ne, α = 90° and for several values of r. PsExt(r, α) and PDE(r)χ(α) are normalized to the power Pe injected inside the slab and thus they have not units.
Fig. 5
Fig. 5 Examples of the scattering functions employed in the MC simulations.
Fig. 6
Fig. 6 Comparison heuristic model vs. MC results for τA versus τ in a transmittance measurement through a non-absorbing slab with s0 = 10 mm and ni = ne. The detector is coaxial to the pencil-beam source at d = 0. A HG model with g = 0 is used for p(θ). The straight line represents the ideal measurement.
Fig. 7
Fig. 7 As Fig. 6, but for a HG model with g = 0.8.
Fig. 8
Fig. 8 As Fig. 6, but for a HG model with g = 0.9.
Fig. 9
Fig. 9 As Fig. 6, but for a HG model with g = 0.95.
Fig. 10
Fig. 10 As Fig. 6, but for the scattering function of Intralipid at λ =632.8 nm.
Fig. 11
Fig. 11 As Fig. 6, but for a water suspension of polystyrene latex spheres having a Gaussian size distribution with 〈ϕ〉=15.8 μm and SD = 2.8μm at λ =632.8 nm.

Tables (2)

Tables Icon

Table 1 Coefficients a and b of Eq. (24) calculated for a slab 10 mm thick with ni = ne, μa = 0 and d = 0.

Tables Icon

Table 2 Accuracy of the heuristic model for a slab 10 mm thick, d = 0, μa = 0, and ni = ne. The percentage relative errors on τA calculated as percentage difference between the model and MC simulations are shown for several scattering functions p(θ). The values of the radius of the receiver considered are r = 0.5, 1, 2, 4, 10 and ∞ mm.

Equations (33)

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τ = ln P 0 P e ,
τ A = ln P R P e = ln P 0 + P s P e ,
P R = P 0 + P s = P 0 + P s 1 + P s 2 + + P sk +
d P sk = P e exp ( μ e l 1 ) μ s d l 1 p ( θ 1 ) d Ω 1 × exp ( μ e l 2 ) μ s d l 2 p ( θ 2 ) d Ω 2 exp ( μ e l k ) μ s d l k p ( θ k ) d Ω k exp ( μ e l k + 1 ) = P e μ s k exp ( μ e i = 1 k + 1 l i ) p ( θ 1 ) p ( θ 2 ) p ( θ k ) d Ω 1 d Ω 2 d Ω k d l 1 d l 2 d l k .
P sk P e τ s k exp ( τ ) K k ( p ( θ ) , s 0 , d , α , r ) ,
K k ( p ( θ ) , s 0 , d , α , r ) = p ( θ 1 ) p ( θ 2 ) p ( θ k ) d Ω 1 d Ω 2 d Ω k d l 1 d l 2 d l k
P R = P e exp ( τ ) [ 1 + k τ s k K k ( p ( θ ) , s 0 , d , α , r ) ] .
τ A = ln P R P e = τ ln [ 1 + k τ s k K k ( p ( θ ) , s 0 , d , α , r ) ] ,
ε = τ τ A τ = ln [ 1 + k τ s k K k ( p ( θ ) , s 0 , d , α , r ) ] τ .
lim τ s 0 ε ε S S = τ s K 1 ( p ( θ ) , s 0 , d , α , r ) τ = Λ K 1 ( p ( θ ) , s 0 , d , α , r ) ,
K 1 ( p ( θ ) , s 0 , d , α , r ) = p ( θ 1 ) d Ω 1 d l 1 = 2 π 1 s 0 0 s 0 0 θ ( l , s 0 , d , α , r ) p ( θ ) sin θ d θ d l ,
θ ( l , s 0 , d , α , r ) = min [ α , arctan ( r s 0 + d l ) ] .
K 1 ( p ( θ ) , s 0 , d , α , r ) = 2 π 0 α p ( θ ) sin θ d θ ,
τ A S S = τ ( 1 ε ) τ ( 1 ε S S ) = τ [ 1 Λ K 1 ( p ( θ ) , s 0 , d , α , r ) ] .
P R = P e exp ( τ A S S ) = P e exp [ μ e ( 1 ε S S ) s 0 ] ,
μ e A S S = μ e ( 1 ε S S ) ,
P DE ( r ) = P e 1 2 m = m = + { sgn ( z 1 m ) exp ( μ eff | z 1 m | ) sgn ( z 2 m ) exp ( μ eff | z 2 m | ) + z 1 m exp [ μ eff r 2 + z 1 m 2 ] r 2 + z 1 m 2 + z 2 m exp [ μ eff r 2 + z 2 m 2 ] r 2 + z 2 m 2 } ,
{ z 1 m = ( 1 2 m ) s 0 4 m z e z s z 2 m = ( 1 2 m ) s 0 ( 4 m 2 ) z e + z s ,
F ( θ e ) = 1 4 π ( n e n i ) 2 { 1 R F [ arcsin ( n e n i sin θ e ) ] } { 2 A + 3 cos [ arcsin ( n e n i sin θ e ) ] } ,
P R ( r , α ) P 0 + P DE ( r ) χ ( α ) ,
χ ( α ) = 0 α F ( θ e ) cos θ e 2 π sin θ e d θ e .
χ ( α ) = 1 1 2 ( cos 2 α + cos 3 α ) .
τ A DE α = ln P R ( r , α ) P e .
P s Ext ( r , α ) = a ( τ , α ) τ s b ( r , α ) , ( τ s 4 , τ s 1.5 ) ,
P R ( r , α ) = P 0 + P s Ext ( r , α ) ,
τ A Ext = ln P R ( r , α ) P e .
τ A = τ A S S for τ s < 4 ( Eq . ( 14 ) ) ,
τ A = min ( τ A S S , τ A Ext ) for τ s 4 and τ s 1.5 ( Eq . ( 26 ) ) ,
τ A = min ( τ A S S , τ A DE ) for τ s 4 and τ s 1.5 ( Eq . ( 23 ) ) .
r eqF = r , d eqF = n i n e d α eqF = n e n i α .
P s ( r , α , d > 0 ) P s ( r eqd , α eqd , d = 0 ) = P DE ( r eqd , d = 0 ) χ ( α eqd ) ,
r eqd ( α , r , d ) = max ( r , d tan α ) , and α eqd ( α , r , d ) = min ( α , arctan ( r / d ) ) .
r eq ( α , r , d , R d ) = min [ R d , max ( r , d , tan α ) ] ,

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