Abstract

Abstract

We investigate the scattering and multiple scattering of a typical laser beam (λ=800 nm) in the intermediate scattering regime. The turbid media used in this work are homogeneous solutions of monodisperse polystyrene spheres in distilled water. The two-dimensional distribution of light intensity is recorded experimentally, and calculated via Monte Carlo simulation for both forward and side scattering. The contribution of each scattering order to the total detected light intensity is quantified for a range of different scattering phase functions, optical depths, and detection acceptance angles. The Lorentz-Mie scattering phase function for individual particles is varied by using different sphere diameters (D=1 and 5 µm). The optical depth of the turbid medium is varied (OD=2, 5, and 10) by employing different concentrations of polystyrene spheres. Detection angles of θa=1.5° and 8.5° are considered. A novel approach which realistically models the experimental laser source is employed in this paper, and very good agreement between the experimental and simulated results is demonstrated. The data presented here can be of use to validate other modern Monte Carlo models, which generate high resolution light intensity distributions. Finally, an extrapolation of the Beer-Lambert law to multiple scattering is proposed based on the Monte Carlo calculation of the ballistic photon contribution to the total detected light intensity.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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  18. X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003).
    [CrossRef]
  19. H. C. van de Hulst, Light scattering by small particles (Dover, N.Y., 1981).
  20. C. Bohren, D. Huffman, Absorption and scattering of light by small particles (Wiley, N.Y., 1983).
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    [CrossRef]

2006

2005

2004

2003

X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

2002

1995

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

1990

I. R. Abubakirov, A. A. Gusev, "Estimation of scattering properties of lithosphere of Kamchatka based on Monte-Carlo simulation of record envelope of a near earthquake," Phys. Earth Planet. Inter. 64, 52-67 (1990).
[CrossRef]

1941

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

Abubakirov, I. R.

I. R. Abubakirov, A. A. Gusev, "Estimation of scattering properties of lithosphere of Kamchatka based on Monte-Carlo simulation of record envelope of a near earthquake," Phys. Earth Planet. Inter. 64, 52-67 (1990).
[CrossRef]

Berrocal, E.

Boas, D.

Brocks, S.

X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Chachisvilis, M.

Churmakov, D. Y.

Côté, D.

Culver, J.

Dunn, A.

Esener, S. C.

Greenstein, J. L.

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

Gusev, A. A.

I. R. Abubakirov, A. A. Gusev, "Estimation of scattering properties of lithosphere of Kamchatka based on Monte-Carlo simulation of record envelope of a near earthquake," Phys. Earth Planet. Inter. 64, 52-67 (1990).
[CrossRef]

Henyey, L. G.

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

Hsieh, J. -C.

Jacob, K.

X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Jacques, S.

Jacques, S. L.

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

Jaffe, J. S.

Jermy, M. C.

Jiang, C. -P.

Lee, C. -K.

Lee, H. -C.

Lee, P. -L.

Lu, J.

X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Ma, X.

X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Meglinski, I. V.

Melville, W. K.

Piskozub, J.

Prahl, S.

Ramella-Roman, J.

Romanov, V. P.

Shao, B.

Stott, J.

Stramski, D.

Sun, C. -W.

Terrill, E.

Tong, Y. -P.

Vitkin, I.

Wang, L.

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

Xin, X.-H.

X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Yang, C.

Yang, P.

X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Yeh, T. -C.

Zheng, L.

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

Appl. Opt.

Astrophys. J.

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

Comput. Methods Programs Biomed.

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

Opt. Express

Phys. Earth Planet. Inter.

I. R. Abubakirov, A. A. Gusev, "Estimation of scattering properties of lithosphere of Kamchatka based on Monte-Carlo simulation of record envelope of a near earthquake," Phys. Earth Planet. Inter. 64, 52-67 (1990).
[CrossRef]

Phys. Med. Biol.

X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang and X.-H. Xin, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Other

H. C. van de Hulst, Light scattering by small particles (Dover, N.Y., 1981).

C. Bohren, D. Huffman, Absorption and scattering of light by small particles (Wiley, N.Y., 1983).

E. Berrocal, D. L. Sedarsky, M. E. Paciaroni, I. V. Meglinski and M. A. Linne, "Laser light scattering in turbid media Part II: Spatial and temporal analysis of individual scattering orders via Monte Carlo simulation," Opt. Express, manuscript in preparation (to be submitted).

D. Y. Churmakov, Multipurpose computational model for modern optical diagnostics and its biomedical applications (PhD Thesis, Cranfield University, 2005).

T. Girasole, C. Roze, B. Maheu, G. Grehan and J. Menard, "Visibility distances in a foggy atmosphere: Comparisons between lighting installations by Monte Carlo simulation," Int. Journal of Lighting Research and technology 30, 29-36 (1998).
[CrossRef]

R. M. Measures, Laser Remote Sensing: Fundamentals and applications (Krieger, Florida, 1992).

V. V. Tuchin (ed.), Handbook of Optical Biomedical Diagnostics, (SPIE Press, Bellingham, WA, 2002).

E. Berrocal, Multiple scattering of light in optical diagnostics of dense sprays and other complex turbid media (PhD Thesis, Cranfield University, 2006).

I. Sobol, The Monte Carlo method, (The University of Chicago Press, 1974).

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Figures (12)

Fig. 1.
Fig. 1.

Schematic illustration of attenuation (a) and multiple scattering (b). The incident radiation is attenuated when it traverses the turbid medium due to both scattering and absorption. Depending on the position along the laser line-of-sight, particles are not illuminated in the same conditions. Multiple scattering, so-called “Extraneous light”, is detected after being multiply scattered by a number of particles (b).

Fig. 2.
Fig. 2.

Configuration of the experimental set-up: Two optical paths are independently considered for either the forward or the side detection.

Fig. 3.
Fig. 3.

Simulation configuration: The laser source S is modeled from the experimental image matrix (200×200 pixels). Photons are sent from S into a scattering single cubic volume (L=10 mm) containing scattering polystyrene spheres suspended in distilled water.

Fig. 4.
Fig. 4.

Polar scattering phase function (logarithmic scale) for a single polystyrene sphere of refractive index n=1.578-0.0i suspended in a solution of distilled water of refractive index n=1.33-0.0i. In (a) D=1 µm and in (b) D=5 µm.

Fig. 5.
Fig. 5.

Comparison between the front face experimental and simulated images at detection acceptance angle θa =8.5° in (a) and θa =1.5° in (b). Solutions of polystyrene spheres of 1 µm diameter are considered at optical depths OD=2, OD=5 and OD=10 in (a) and for OD=10 only in (b). The intensity scale of the images corresponds to the final light intensity, If , detected per pixel divided by the maximum value of the incident light intensity Ii . A comparison of the intensity profile along the vertical axis at X=5 mm is also shown on the right side of the figure. The solid line corresponds to the experimental results and the circles are the results from simulation.

Fig. 6.
Fig. 6.

Probability distribution of scattering orders, P(n), at various optical depths, for the polystyrene spheres of 1 µm diameter. Recorded photons exit the scattering medium through the front face, within the indicated acceptance angle θa .

Fig. 7.
Fig. 7.

Comparison between the front face experimental and simulated images at detection acceptance angle θa =8.5° in (a) and θa =1.5° in (b). Solutions of polystyrene spheres of 5 µm diameter are considered at optical depths OD=2, OD=5 and OD=10 in (a) and for OD=10 only in (b). The intensity scale of the images corresponds to the final light intensity, If , detected per pixel divided by the maximum value of the incident light intensity Ii . A comparison of the intensity profile along the vertical axis at X=5 mm is also shown on the right side of the figure. The solid line corresponds to the experimental results and the circles are the results from simulation.

Fig. 8.
Fig. 8.

Probability distribution of scattering orders, P(n), at various optical depths, for the polystyrene spheres of 5 µm diameter. Recorded photons exit the scattering medium through the front face, within the indicated acceptance angle θa .

Fig. 9.
Fig. 9.

Factor 1/P(0) as a function of the optical depth. Each value of 1/P(0) is deduced from the MC simulation and is related to the amount of multiply scattered light intensity detected. The red and black lines are the best fit of the calculated data.

Fig. 10.
Fig. 10.

Comparison between the side face experimental and simulated images at detection acceptance angle θa =8.5° in (a) and θa =1.5° in (b). Solutions of polystyrene spheres of 1 µm diameter are considered at optical depths OD=2, OD=5 and OD=10 in (a) and for OD=10 only in (b). The intensity scale of the images corresponds to the final light intensity, If , detected per pixel divided by the maximum value of the incident light intensity Ii . A comparison of the intensity profile along the vertical axis at Y=5 mm is also shown on the right side of the figure. The solid line corresponds to the experimental results and the circles are the results from simulation.

Fig. 11.
Fig. 11.

Comparison between the side face experimental and simulated images at detection acceptance angle θa =8.5° in (a) and θa =1.5° in (b). Solutions of polystyrene spheres of 5 µm diameter are considered at optical depths OD=2, OD=5 and OD=10 in (a) and for OD=10 only in (b). The intensity scale of the images corresponds to the final light intensity, If , detected per pixel divided by the maximum value of the incident light intensity Ii . A comparison of the intensity profile along the vertical axis at Y=5 mm is also shown on the right side of the figure. The solid line corresponds to the experimental results and the circles are the results from simulation.

Fig. 12.
Fig. 12.

Probability distribution of scattering orders, P(n), at various optical depths. (a)–(b) correspond to the 1 µm polystyrene spheres and (c)–(d) are for the 5 µm particles. Recorded photons exit the scattering medium through the side face, within θa .

Tables (2)

Tables Icon

Table 1. Classification of the scattering regimes as a function of the optical depth. The intermediate scattering regime is the most complex. Neither the single scattering assumption nor the diffusion approximation applies in this regime.

Tables Icon

Table 2. Deduction of the factor 1/P(0) from the Monte Carlo calculations. The results from the extrapolated Beer-Lambert transmission, given in Eq.(8), are compared with both the simulated results and the experiment.

Equations (9)

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1 c I ( r , s , t ) t ( a ) = μ e I ( r , s , t ) ( b ) + μ s I ( r , s , t ) 4 π f ( s , s ) I ( r , s , t ) d Ω ( c )
I f = I b + I ms
I f = I b + k . I b
k = 1 P ( 0 ) P ( 0 )
I f = I b + 1 P ( 0 ) P ( 0 ) . I b
I b = I i · e O D
I f = I i · e O D + 1 P ( 0 ) P ( 0 ) . I i · e O D
I f = 1 P ( 0 ) · I i · e O D
I f = I i · e O D + α · O D β

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