Abstract

The results of a Monte Carlo study on light propagation in dense turbid media are presented. The calculations refer to the radiation emerging from a spherical scattering cell containing the diffusing medium (no diffusers outside the cell are considered) in whose center a point source is placed. Both the total scattered power emerging from the sphere and the impulse response were evaluated for a large range of optical depths and different types of diffuser. The results pertaining to both the radiance at the surface of the scattering cell and the impulse response are described by simple empirical relations. The results suggest a method of measuring the albedo and the asymmetry factor of the diffusing medium. A comparison with some results of the diffusion approximation is also presented.

© 1991 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978).
  2. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [Crossref] [PubMed]
  3. P. Bruscaglioni, G. Zaccanti, “Multiple Scattering in Dense Media,” in Scattering in Volumes and Surfaces, M. Nieto Vesperinas, J. C. Dainty, Eds. (Elsevier, New York, 1990), pp. 53–71.
  4. J. F. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, MA, 1972), Sec. 2–9.
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  6. W. M. Star, J. P. A. Marijnissen, “Calculating the Response of Isotropic Light Dosimetry Probes as a Function of the Tissue Refractive Index,” Appl. Opt. 28, 2288–2291 (1989).
    [Crossref] [PubMed]
  7. G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
    [Crossref]
  8. S. Ito, “Comparison of Diffusion Theories for Optical Pulse Waves Propagated in Discrete Random Media,” J. Opt. Soc. Am. A 1, 502–505 (1984).
    [Crossref]

1989 (1)

1988 (1)

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
[Crossref]

1987 (1)

1984 (1)

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Bonner, R. F.

Bruscaglioni, P.

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
[Crossref]

P. Bruscaglioni, G. Zaccanti, “Multiple Scattering in Dense Media,” in Scattering in Volumes and Surfaces, M. Nieto Vesperinas, J. C. Dainty, Eds. (Elsevier, New York, 1990), pp. 53–71.

Havlin, S.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978).

Ito, S.

Lamarsh, J. F.

J. F. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, MA, 1972), Sec. 2–9.

Marijnissen, J. P. A.

Nossal, R.

Star, W. M.

Weiss, G. H.

Zaccanti, G.

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
[Crossref]

P. Bruscaglioni, G. Zaccanti, “Multiple Scattering in Dense Media,” in Scattering in Volumes and Surfaces, M. Nieto Vesperinas, J. C. Dainty, Eds. (Elsevier, New York, 1990), pp. 53–71.

Appl. Opt. (1)

J. Mod. Opt. (1)

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer Law in the Transmittance of a Light Beam Through Diffusing Media: Experimental Results,” J. Mod. Opt. 35, 229–242 (1988).
[Crossref]

J. Opt. Soc. Am. A (2)

Other (4)

P. Bruscaglioni, G. Zaccanti, “Multiple Scattering in Dense Media,” in Scattering in Volumes and Surfaces, M. Nieto Vesperinas, J. C. Dainty, Eds. (Elsevier, New York, 1990), pp. 53–71.

J. F. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, MA, 1972), Sec. 2–9.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1 (Academic, New York, 1978).

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Figures (14)

Fig. 1
Fig. 1

Phase functions used in Monte Carlo simulations. The curves pertain to polystyrene spheres suspended in water at the He–Ne wavelength with sizes 0.002, 0.33, and 15.8 μm (curves a, b, and c, respectively).

Fig. 2
Fig. 2

Monte Carlo results for β(ϕ) vs ϕ (continuous lines) pertaining to different values of τd. For any value of τd, the curves β ¯ (ϕ) [Eq. (11)] with n = 1.5 (dashed lines) and n = 2 (dotted lines) are also shown. Results pertain to 0.33-μm spheres and w0 = 1.

Fig. 3
Fig. 3

Monte Carlo results for the functions f1(l) and f2(K) pertaining to 0.002-μm spheres and to different values of τd are reported. The functions represent respectively the probability density for the optical length l followed by the photons, and for the number of scattering K undergone inside the spherical cell of radius R = τdld. w0 = 1.

Fig. 4
Fig. 4

Same as Fig. 3 but for 15.8-μm spheres.

Fig. 5
Fig. 5

Probability density functions q K ( l / l K ) for a photon moving in an infinitely extended nonabsorbing medium for different values of K. The curves show that the distributions become narrower and narrower around the peak value l/K ≈ 1, when K increases.

Fig. 6
Fig. 6

Mean value 〈K〉 of the number of scattering events undergone by the photons inside the spherical cell of radius R = τdld vs τd for a nonabsorbing medium. Marks refer to the Monte Carlo results whereas continuous curves were obtained by using the empirical relation given by Eq. (18). Similar results hold for 〈l〉.

Fig. 7
Fig. 7

Monte Carlo results for the probability density functions f1(l) and f 1 ( l ). Where the function f1(l) refers to the optical lengths followed by all the photons before leaving the sphere, f 1 ( l ) refers to the photons exiting with angles Φ < 15° only. Spheres with Φ = 0.33 μm; w0 = 1.

Fig. 8
Fig. 8

Monte Carlo results for the impulse response g1(t/t0) referring to the whole diffused power leaving the sphere of radius R = τdld. The transit time t0 for the unscattered power is taken as the unit to measure t. Curves refer to 0.002-, 0.33-, and 15.8-μm spheres and to the values of τd indicated near the curves; w0 = 1.

Fig. 9
Fig. 9

Monte Carlo results for the probability density h(ξ) of ξ = l/3τs(τd + 1) for various values of τd. Curves referring to different values of τd are almost indistinguishable apart from the smaller values of ξ. The figure refers to 0.002-μm spheres and w0 = 1. Almost indistinguishable results were obtained for 0.33- and 15.8-μm spheres.

Fig. 10
Fig. 10

Comparison between the impulse response g1(t/t0) obtained by Monte Carlo simulations and by the empirical relation given by Eq. (21). The curves refer to 0.002-μm spheres for different values of τd and w0 = 1. Virtually identical results were obtained for 0.33- and 15.8-μm spheres (see Fig. 8).

Fig. 11
Fig. 11

Impulse response g1(t/t0) referring to the whole diffused power leaving the sphere when a nonunitary albedo is considered. Curves refer to 15.8-μm spheres. The figure shows how an albedo slightly <1 is sufficient to cause a large attenuation and distortion in the impulse response with respect to w0 = 1 when τd is large.

Fig. 12
Fig. 12

Logarithm of the attenuation of the total power leaving the sphere (apparent optical depth) vs τa for different values of τd and the three types of sphere considered. Dotted, dashed, and dotted–dashed lines were obtained by using the Monte Carlo results pertaining to 0.002-, 0.33-, and 15.8-μm spheres, respectively, for f1(l) (1) in Eq. (23). Continuous curves were obtained by using for f1(l) the function derived by the empirical relation for h ¯ (ξ) [Eq. (20)].

Fig. 13
Fig. 13

Comparison between the impulse response g1(t/t0) pertaining to the total diffused power leaving the sphere (radius R = 10 cm) when an isotropic point source is placed in the center (dotted curves) and when an infinitely thin beam source is considered (continuous lines) with a receiver of area 1 cm2 and field of view semiaperture of 9° in front of the beam source. Spheres with Φ = 0.33 μm; w0 = 1. To compare the results, the area of the curves was normalized to one.

Fig. 14
Fig. 14

Comparison between the attenuation of the total power leaving the sphere, plotted vs τa, obtained by using both the diffusion approximation [Eq. (9.54) of Ref. 1 and the Monte Carlo results. Results for the diffusion approximation refer to the attenuation of the total flux across the spherical surface of radius R = τdld centered on the point source placed in an infinitely extended diffusing medium [Eq. (9.54) of Ref. 1]. Results referred to as Monte Carlo ones were obtained by using in Eq. (24) the empirical formula for f1(l) derived from Eq. (20).

Equations (33)

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l d = 1 σ s ( 1 - g ) ,
β ( ϕ i ϕ ϕ i + 1 ) = 1 2 π ( cos ϕ i - cos ϕ i + 1 ) L i i = 1 I L i ,
f 1 ( l i l < l i + 1 ) = 1 l i + 1 - l i M i i = 1 I M i ,
f 2 ( K i K < K i + 1 ) = 1 K i + 1 - K i N i i = 1 I N i ,
P d r ( τ s ) = P e [ 1 - exp ( - τ s ) ] A 4 π R 2 2 π 0 α β ( ϕ ) sin ϕ d ϕ ,
P o r ( τ s ) = P e exp ( - τ s ) A 4 π R 2 ,
g ( t ) = [ 1 - exp ( - τ s ) ] τ s R c f 1 ( l ) ,             l = t c τ s / R .
P d ( t ) = - P e ( t ) g ( t - t ) d t ,
P 0 ( t ) = P e ( t - t 0 ) exp ( - τ s ) ,
g 1 ( t t 0 ) = [ 1 - exp ( - τ s ) ] τ s f 1 ( l ) ,             l = τ s t t 0 .
β ¯ ( ϕ ) = n + 1 2 π cos n ϕ ,
q 1 ( l ) = exp ( - l ) .
q 2 ( l ) = 0 l q 1 ( l ) q 1 ( l - l ) d l = l exp ( - l ) .
q K ( l ) = 0 l q K - 1 ( l ) q 1 ( l - l ) d l = l K - 1 ( K - 1 ) ! exp ( - l ) .
l m K = 0 l m q K ( l ) d l = K ( K + 1 ) ( K + m - 1 ) ,
l max K = K - 1.
q K ( l K ) = K q K ( l )
K = 0.50 τ s τ d .
h ( ξ ) = 3 τ s ( τ d + 1 ) f 1 ( l ) ,             l = 3 τ s ( τ d + 1 ) ξ ,
h ¯ ( ξ ) = C ξ - 3.75 exp ( - 0.3465 ξ ) if ξ > ξ 0 , = 0 if ξ ξ 0 ,
ξ 0 h ¯ ( ξ ) d ξ = [ 1 - exp ( - τ s ) ] .
g ¯ 1 ( t / t 0 ) = [ 1 - exp ( - τ s ) ] C 3 ( τ d + 1 ) [ t 0 t 3 ( τ d + 1 ) ] 3.75 × exp [ - 1.04 ( τ d + 1 ) t 0 t ] ,             t > t 0 , = 0 ,             t t 0 .
t max t 0 = 1.04 3.75 ( τ d + 1 ) ,
t t 0 = 0.58 ( τ d + 1 ) ,
Δ t 1 / 10 t 0 = 0.97 ( τ d + 1 ) ,
σ a ρ = τ a l τ s .
g 1 ( t / t 0 ) = [ 1 - exp ( - τ s ) ] τ s f 1 ( l ) exp ( - τ a l / τ s ) ,             l = τ s t t 0 ,
P t ( τ s , τ a ) = P c { exp - ( τ s + τ a ) + [ 1 - exp ( - τ s ) ] τ s f 1 ( l ) exp - ( τ a l τ s ) d l } ,
P t = 4 π R 2 F d = P e { 1 + [ 3 τ a ( τ d + τ a ) ] 0.5 } exp { - [ 3 τ a ( τ d + τ a ) ] 0.5 } ,
G ( R , t t 0 ) = 2 π ( 4 π R ) 2 ( 3 τ d t 0 t ) 3 / 2 exp ( - τ a t t 0 - 3 4 τ d t 0 t ) , t > 0 , = 0 , t < 0 ,
P K ( r ) ( 3 2 π K ) 3 / 2 exp [ - 3 2 K ( x 2 + y 2 + z 2 ) ] ,
x 2 + y 2 + z 2 = R 2 σ s 2 2
P ( l , τ s ) A ( 1 l ) 3 / 2 exp ( - 3 4 τ s 2 l ) ,

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