Abstract

A recently theoretically predicted backscattering halo appearing around the pencil beam in the scattering medium is studied. The criterion of seven-thirds, which is to be met in the experiment for the halo observation, is established. The possibility to register the effect in realistic media is discussed.

© 2012 Optical Society of America

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References

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  3. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
    [CrossRef]
  4. V. V. Marinyuk and D. B. Rogozkin, “Effects of nondiffusive wave propagation upon coherent backscattering by turbid media,” Laser Phy. 19, 176–184 (2009).
    [CrossRef]
  5. V. V. Marinyuk and D. B. Rogozkin, “Wings of coherent backscattering from a disordered medium with large inhomogeneities,” Phys. Rev. E 83, 066604 (2011).
    [CrossRef]
  6. Ya. A. Ilyushin, “Coherent backscattering enhancement in highly anisotropically scattering media: numerical solution,” J. Quant. Spectrosc. Radiat. Transfer 113, 348–354 (2012).
    [CrossRef]
  7. D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
    [CrossRef]
  8. D. Rogozkin, “Coherent backscattering of waves from disordered systems with large-scale inhomogeneities,” Laser Phys. 5, 787–792 (1995).
  9. C. E. Siewert, “On the singular components of the solution to the searchlight problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 33, 551–554 (1985).
    [CrossRef]
  10. R. Sanchez, “On the singular structure of the uncollided and first-collided components of the Green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
    [CrossRef]
  11. Y. A. Ilyushin and V. P. Budak, “Narrow beams in scattering media: the advanced small-angle approximation,” J. Opt. Soc. Am. A 28, 1358–1363 (2011).
    [CrossRef]
  12. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
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  13. D. Arnush, “Underwater light-beam propagation in the small-angle-scattering approximation,” J. Opt. Soc. Am. 62, 1109–1111(1972).
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  14. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

2012

Ya. A. Ilyushin, “Coherent backscattering enhancement in highly anisotropically scattering media: numerical solution,” J. Quant. Spectrosc. Radiat. Transfer 113, 348–354 (2012).
[CrossRef]

2011

V. V. Marinyuk and D. B. Rogozkin, “Wings of coherent backscattering from a disordered medium with large inhomogeneities,” Phys. Rev. E 83, 066604 (2011).
[CrossRef]

Y. A. Ilyushin and V. P. Budak, “Narrow beams in scattering media: the advanced small-angle approximation,” J. Opt. Soc. Am. A 28, 1358–1363 (2011).
[CrossRef]

2009

V. V. Marinyuk and D. B. Rogozkin, “Effects of nondiffusive wave propagation upon coherent backscattering by turbid media,” Laser Phy. 19, 176–184 (2009).
[CrossRef]

2007

2004

2000

R. Sanchez, “On the singular structure of the uncollided and first-collided components of the Green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
[CrossRef]

1995

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef]

D. Rogozkin, “Coherent backscattering of waves from disordered systems with large-scale inhomogeneities,” Laser Phys. 5, 787–792 (1995).

1986

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

1985

C. E. Siewert, “On the singular components of the solution to the searchlight problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 33, 551–554 (1985).
[CrossRef]

1972

1941

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

Arnush, D.

Bohren, C. F.

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

Budak, V. P.

Campbell, S. D.

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Grobe, R.

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Huffman, D. R.

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

Ilyushin, Y. A.

Ilyushin, Ya. A.

Ya. A. Ilyushin, “Coherent backscattering enhancement in highly anisotropically scattering media: numerical solution,” J. Quant. Spectrosc. Radiat. Transfer 113, 348–354 (2012).
[CrossRef]

Kim, A. D.

Lagendijk, A.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef]

Marinyuk, V. V.

V. V. Marinyuk and D. B. Rogozkin, “Wings of coherent backscattering from a disordered medium with large inhomogeneities,” Phys. Rev. E 83, 066604 (2011).
[CrossRef]

V. V. Marinyuk and D. B. Rogozkin, “Effects of nondiffusive wave propagation upon coherent backscattering by turbid media,” Laser Phy. 19, 176–184 (2009).
[CrossRef]

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

Moscoso, M.

O’Connell, A. K.

Rogozkin, D.

D. Rogozkin, “Coherent backscattering of waves from disordered systems with large-scale inhomogeneities,” Laser Phys. 5, 787–792 (1995).

Rogozkin, D. B.

V. V. Marinyuk and D. B. Rogozkin, “Wings of coherent backscattering from a disordered medium with large inhomogeneities,” Phys. Rev. E 83, 066604 (2011).
[CrossRef]

V. V. Marinyuk and D. B. Rogozkin, “Effects of nondiffusive wave propagation upon coherent backscattering by turbid media,” Laser Phy. 19, 176–184 (2009).
[CrossRef]

Rutherford, G. H.

Sanchez, R.

R. Sanchez, “On the singular structure of the uncollided and first-collided components of the Green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
[CrossRef]

Siewert, C. E.

C. E. Siewert, “On the singular components of the solution to the searchlight problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 33, 551–554 (1985).
[CrossRef]

van Albada, M. P.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef]

van Tiggelen, B. A.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef]

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

Ann. Nucl. Energy

R. Sanchez, “On the singular structure of the uncollided and first-collided components of the Green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
[CrossRef]

Astrophys. J.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

C. E. Siewert, “On the singular components of the solution to the searchlight problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 33, 551–554 (1985).
[CrossRef]

Ya. A. Ilyushin, “Coherent backscattering enhancement in highly anisotropically scattering media: numerical solution,” J. Quant. Spectrosc. Radiat. Transfer 113, 348–354 (2012).
[CrossRef]

Laser Phy.

V. V. Marinyuk and D. B. Rogozkin, “Effects of nondiffusive wave propagation upon coherent backscattering by turbid media,” Laser Phy. 19, 176–184 (2009).
[CrossRef]

Laser Phys.

D. Rogozkin, “Coherent backscattering of waves from disordered systems with large-scale inhomogeneities,” Laser Phys. 5, 787–792 (1995).

Opt. Lett.

Phys. Rev. E

V. V. Marinyuk and D. B. Rogozkin, “Wings of coherent backscattering from a disordered medium with large inhomogeneities,” Phys. Rev. E 83, 066604 (2011).
[CrossRef]

Phys. Rev. Lett.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef]

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

Other

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

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Figures (5)

Fig. 1.
Fig. 1.

Sketch of the pencil beam backscattering halo.

Fig. 2.
Fig. 2.

Thick dashed curve, numerical estimates of the critical normalized backscattering cross sections σb/σ [Eq. (7)] for the exponential phase function (17); solid curve, analytical criterion of seven-thirds [Eq. (21)]; dotted curve, actual ratio for the Henyey–Greenstein phase function (9). Monte Carlo simulations: solid triangles, no halo observed; solid squares, weak halo (local maximum near ρltr) is observed; crosses, strong halo (global maximum near ρltr) is observed.

Fig. 3.
Fig. 3.

Reflected light distribution F(ρ) at the exit surface of the medium (Monte Carlo simulations, μ0=0.9, arbitrary units). Solid curve, σb/σ8·103 (no halo); dashed curve, σb/σ8·105 (weak halo); dotted curve, 8·1010 (strong halo).

Fig. 4.
Fig. 4.

Solid curve, normalized backscattering cross section σb/σ; dashed curve, mean scattering cosine g versus size parameter x (spherical latex particles in water, Mie theory [14]).

Fig. 5.
Fig. 5.

Geometry of the boundary problem for the slab medium.

Equations (46)

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I(ρ)=0J0(qρ)Jc(q)qdq,
Jc(q)=σblσ0dzexp(2z/l)[exp(20zd(z/l)χ(qz))1],
χ(ω)=02πθdθJ0(ωθ)1σdσdΩ(θ),
I(0)=0q0σblσ0dzexp(2z/l)[exp(20zd(z/l)χ(qz))1]qdq.
Jc(q)=3a4πl1exp(qa)qa,
I(ρ)=34πltr2{111+α2}.
σbσ34πltr2{111+α2}/0q00d(z/l)exp(2z/l)[exp(20zd(z/l)χ(qz))1]qdq.
dσdΩ=σ4π1g2(1+g22gcosθ)3/2,
σbσ=4π1g(1+g)2π(1g),
χ(ω)=exp((1g)ω).
f(cosθ)=12πεexp((1cosθ)/ε)1exp(2/ε)12πεexp(θ22ε),
χ(ω)exp(εω22),
0zχ(qz)dz=1qπ2εerf(qzε2),
σbσ=exp(2/ε)2πε.
f(θ)=γ2πθexp(γθ),
χ(ω)=γγ2+ω2.
f(θ)=12πθ02exp(θθ0)
χ(ω)=1(1+ω2θ02)3/2,
0z/lχ(qz)dz/l=z/l(1+q2z2θ02)1/2z/lz/l2q2z2θ02.
I(0)=σbσ0q0[Γ(4/3)(qlθ0)2/312]qdqσbσ0q0Γ(4/3)(qlθ0)2/3qdq34Γ(4/3)q04/3(lθ0)2/3.
σbσ<(1g)7/3(α2+11)33π(q0l)4/3α2+1Γ(4/3)(1g)7/3
dσdΩ(μ)={σb,μμ0,σf,μ>μ0,
dσdΩdΩ=2π(1+μ0)σb+2π(1μ0)σf=1
σf=12π(1μ0)1+μ01μ0σb.
g=μdσdΩ(μ)dΩ=2πσb1μ0μdμ+2πσfμ01μdμ=π(1μ02)(σfσb)1+μ02,
F(ρ)=Ω^·z^<0(Ω^·(z^))I(Ω,ρ)|z=0dΩ
(Ω·)G(r,Ω,r,Ω)+G(r,Ω,r,Ω)Λ4πG(r,Ω,r,Ω)x(Ω·Ω)dΩ=δ(rr)δ(ΩΩ),
G=0atz=z0/2,μz>0
G=0atz=+z0/2,μz<0
G(r,Ω,r,Ω)=G(r,Ω,r,Ω).
L0(r,Ω)=exp(τ)δ(x)δ(y)δ(Ω),
(Ω·)L(r,Ω)+L(r,Ω)Λ4πL(r,Ω)x(Ω·Ω)dΩ=f(r,Ω),
f(r,Ω)=Λ4πL0(r,Ω)x(Ω·Ω)dΩ=Λ4πexp(τ)x(Ω)δ(x)δ(y).
L(r,Ω)=G(r,Ω,r,Ω)f(r,Ω)drdΩ=Λ4πG(r,Ω,r,Ω)exp(τ)x(Ω)δ(x)δ(y)drdΩ,
0<minΩx(Ω)<x(Ω)<maxΩx(Ω).
c1G(r,Ω,r,Ω)δ(x)δ(y)drdΩG(r,Ω,r,Ω)exp(τ)x(Ω)δ(x)δ(y)drdΩc2G(r,Ω,r,Ω)δ(x)δ(y)drdΩ,
L(r,Ω)=G(r,Ω,r,Ω)f(r,Ω)drdΩ=Λ4πG(r,Ω,r,Ω)exp(τ)x(Ω)δ(x)δ(y)drdΩ.
c1G(r,Ω,r,Ω)δ(x)δ(y)drdΩG(r,Ω,r,Ω)exp(τ)x(Ω)δ(x)δ(y)drdΩc2G(r,Ω,r,Ω)δ(x)δ(y)drdΩ.
G(r,Ω,r,Ω)exp(τ)x(Ω)δ(x)δ(y)drdΩC1G(r,Ω,r,Ω)exp(τ)x(Ω)δ(x)δ(y)drdΩ,
G(r,Ω,r,Ω)exp(τ)x(Ω)δ(x)δ(y)drdΩC2G(r,Ω,r,Ω)exp(τ)x(Ω)δ(x)δ(y)drdΩ,
L(r,Ω)C1L(r,Ω),
L(r,Ω)C2L(r,Ω),
L(r,Ω)L(r,Ω).
L0(r,Ω)=exp(τ/μ0)δ(ΩΩ0)
f(r,Ω)=Λ4πL0(r,Ω)x(Ω·Ω)dΩ=Λ4πexp(τ/μ0)x(Ω·Ω0),
c1f(r,Ω)c2.

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