Abstract

The weak localization (coherent backscattering enhancement) phenomenon in media with graded refraction index is investigated within the diffusion approximation. The obtained analytic results are compared with numerical solutions by finite-difference and Monte Carlo calculations.

© 2013 Optical Society of America

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References

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  1. L. Tsang and A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–839 (1984).
    [CrossRef]
  2. 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]
  3. B. Hapke, “Coherent backscatter and the radar characteristics of outer planet satellites,” Icarus 88, 407–417 (1990).
    [CrossRef]
  4. V. V. Marinyuk and D. B. Rogozkin, “Effects of nondiffusive wave propagation upon coherent backscattering by turbid media,” Laser Physics 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. S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40, 831–848 (1949).
    [CrossRef]
  7. L. Liu, L. Zhang, and H. Tan, “Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems,” J. Quant. Spectrosc. Radiat. Transfer 97, 446–456 (2006).
    [CrossRef]
  8. L. Liu, H. Zhang, and H. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 84, 357–362 (2004).
    [CrossRef]
  9. D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
    [CrossRef]
  10. A. Afanasiev, “The energy spectrum of spacecraft radio signals in the caustic shadow zone of the Sun: a new diagnostic of the solar coronal plasma,” J. Atmos. Sol. Terr. Phys. 67, 1002–1013 (2005).
    [CrossRef]
  11. Y. A. Ilyushin, “Impact of the plasma fluctuations in the martian ionosphere on the performance of the synthetic aperture ground-penetrating radar,” Planet. Space Sci. 57, 1458–1466 (2009).
    [CrossRef]
  12. Y. A. Ilyushin, “Martian northern polar cap: layering and possible implications for radar sounding,” Planet. Space Sci. 52, 1195–1207 (2004).
    [CrossRef]
  13. Y. A. Ilyushin, “Coherent backscattering enhancement in highly anisotropically scattering media: numerical solution,” J. Quant. Spectrosc. Radiat. Transfer 113, 348–354 (2012).
    [CrossRef]
  14. Y. A. Ilyushin, “Coherent backscattering enhancement in medium with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 117, 133–139 (2013).
    [CrossRef]
  15. G. Bekefi, Radiation Processes in Plasmas (Wiley, 1966).
  16. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).
  17. K.-Y. Zhu, Y. Huang, and J. Wang, “Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index,” J. Quant. Spectrosc. Radiat. Transfer 112, 377–383 (2011).
    [CrossRef]
  18. A. Sassaroli and F. Martelli, “Equivalence of four Monte Carlo methods for photon migration in turbid media,” J. Opt. Soc. Am. A 29, 2110–2117 (2012).
    [CrossRef]
  19. M. L. Shendeleva, “Radiative transfer in a turbid medium with a varying refractive index: comment,” J. Opt. Soc. Am. A 21, 2464–2467 (2004).
    [CrossRef]
  20. A. Isimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  21. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  22. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy.” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]

2013 (1)

Y. A. Ilyushin, “Coherent backscattering enhancement in medium with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 117, 133–139 (2013).
[CrossRef]

2012 (2)

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

A. Sassaroli and F. Martelli, “Equivalence of four Monte Carlo methods for photon migration in turbid media,” J. Opt. Soc. Am. A 29, 2110–2117 (2012).
[CrossRef]

2011 (2)

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]

K.-Y. Zhu, Y. Huang, and J. Wang, “Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index,” J. Quant. Spectrosc. Radiat. Transfer 112, 377–383 (2011).
[CrossRef]

2009 (2)

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

Y. A. Ilyushin, “Impact of the plasma fluctuations in the martian ionosphere on the performance of the synthetic aperture ground-penetrating radar,” Planet. Space Sci. 57, 1458–1466 (2009).
[CrossRef]

2006 (1)

L. Liu, L. Zhang, and H. Tan, “Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems,” J. Quant. Spectrosc. Radiat. Transfer 97, 446–456 (2006).
[CrossRef]

2005 (1)

A. Afanasiev, “The energy spectrum of spacecraft radio signals in the caustic shadow zone of the Sun: a new diagnostic of the solar coronal plasma,” J. Atmos. Sol. Terr. Phys. 67, 1002–1013 (2005).
[CrossRef]

2004 (3)

Y. A. Ilyushin, “Martian northern polar cap: layering and possible implications for radar sounding,” Planet. Space Sci. 52, 1195–1207 (2004).
[CrossRef]

M. L. Shendeleva, “Radiative transfer in a turbid medium with a varying refractive index: comment,” J. Opt. Soc. Am. A 21, 2464–2467 (2004).
[CrossRef]

L. Liu, H. Zhang, and H. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 84, 357–362 (2004).
[CrossRef]

2002 (1)

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[CrossRef]

1994 (1)

1990 (1)

B. Hapke, “Coherent backscatter and the radar characteristics of outer planet satellites,” Icarus 88, 407–417 (1990).
[CrossRef]

1986 (1)

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]

1984 (1)

1949 (1)

S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40, 831–848 (1949).
[CrossRef]

1941 (1)

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

Afanasiev, A.

A. Afanasiev, “The energy spectrum of spacecraft radio signals in the caustic shadow zone of the Sun: a new diagnostic of the solar coronal plasma,” J. Atmos. Sol. Terr. Phys. 67, 1002–1013 (2005).
[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]

Bekefi, G.

G. Bekefi, Radiation Processes in Plasmas (Wiley, 1966).

Dez, V. L.

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[CrossRef]

Feng, T.-C.

Greenstein, J. L.

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

Hapke, B.

B. Hapke, “Coherent backscatter and the radar characteristics of outer planet satellites,” Icarus 88, 407–417 (1990).
[CrossRef]

Haskell, R. C.

Henyey, L. G.

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

Huang, Y.

K.-Y. Zhu, Y. Huang, and J. Wang, “Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index,” J. Quant. Spectrosc. Radiat. Transfer 112, 377–383 (2011).
[CrossRef]

Ilyushin, Y. A.

Y. A. Ilyushin, “Coherent backscattering enhancement in medium with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 117, 133–139 (2013).
[CrossRef]

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

Y. A. Ilyushin, “Impact of the plasma fluctuations in the martian ionosphere on the performance of the synthetic aperture ground-penetrating radar,” Planet. Space Sci. 57, 1458–1466 (2009).
[CrossRef]

Y. A. Ilyushin, “Martian northern polar cap: layering and possible implications for radar sounding,” Planet. Space Sci. 52, 1195–1207 (2004).
[CrossRef]

Ishimaru, A.

Isimaru, A.

A. Isimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Lemonnier, D.

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[CrossRef]

Liu, L.

L. Liu, L. Zhang, and H. Tan, “Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems,” J. Quant. Spectrosc. Radiat. Transfer 97, 446–456 (2006).
[CrossRef]

L. Liu, H. Zhang, and H. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 84, 357–362 (2004).
[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 Physics 19, 176–184 (2009).
[CrossRef]

Martelli, F.

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]

McAdams, M. S.

Morton, K. W.

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).

Richtmyer, R. D.

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).

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 Physics 19, 176–184 (2009).
[CrossRef]

Sassaroli, A.

Shendeleva, M. L.

Smerd, S. F.

S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40, 831–848 (1949).
[CrossRef]

Svaasand, L. O.

Tan, H.

L. Liu, L. Zhang, and H. Tan, “Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems,” J. Quant. Spectrosc. Radiat. Transfer 97, 446–456 (2006).
[CrossRef]

L. Liu, H. Zhang, and H. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 84, 357–362 (2004).
[CrossRef]

Tromberg, B. J.

Tsang, L.

Tsay, T.-T.

Wang, J.

K.-Y. Zhu, Y. Huang, and J. Wang, “Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index,” J. Quant. Spectrosc. Radiat. Transfer 112, 377–383 (2011).
[CrossRef]

Westfold, K. C.

S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40, 831–848 (1949).
[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]

Zhang, H.

L. Liu, H. Zhang, and H. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 84, 357–362 (2004).
[CrossRef]

Zhang, L.

L. Liu, L. Zhang, and H. Tan, “Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems,” J. Quant. Spectrosc. Radiat. Transfer 97, 446–456 (2006).
[CrossRef]

Zhu, K.-Y.

K.-Y. Zhu, Y. Huang, and J. Wang, “Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index,” J. Quant. Spectrosc. Radiat. Transfer 112, 377–383 (2011).
[CrossRef]

Astrophys. J. (1)

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

Icarus (1)

B. Hapke, “Coherent backscatter and the radar characteristics of outer planet satellites,” Icarus 88, 407–417 (1990).
[CrossRef]

J. Atmos. Sol. Terr. Phys. (1)

A. Afanasiev, “The energy spectrum of spacecraft radio signals in the caustic shadow zone of the Sun: a new diagnostic of the solar coronal plasma,” J. Atmos. Sol. Terr. Phys. 67, 1002–1013 (2005).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Quant. Spectrosc. Radiat. Transfer (6)

K.-Y. Zhu, Y. Huang, and J. Wang, “Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index,” J. Quant. Spectrosc. Radiat. Transfer 112, 377–383 (2011).
[CrossRef]

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

Y. A. Ilyushin, “Coherent backscattering enhancement in medium with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 117, 133–139 (2013).
[CrossRef]

L. Liu, L. Zhang, and H. Tan, “Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems,” J. Quant. Spectrosc. Radiat. Transfer 97, 446–456 (2006).
[CrossRef]

L. Liu, H. Zhang, and H. Tan, “Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index,” J. Quant. Spectrosc. Radiat. Transfer 84, 357–362 (2004).
[CrossRef]

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[CrossRef]

Laser Physics (1)

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

Philos. Mag. (1)

S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40, 831–848 (1949).
[CrossRef]

Phys. Rev. E (1)

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. (1)

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]

Planet. Space Sci. (2)

Y. A. Ilyushin, “Impact of the plasma fluctuations in the martian ionosphere on the performance of the synthetic aperture ground-penetrating radar,” Planet. Space Sci. 57, 1458–1466 (2009).
[CrossRef]

Y. A. Ilyushin, “Martian northern polar cap: layering and possible implications for radar sounding,” Planet. Space Sci. 52, 1195–1207 (2004).
[CrossRef]

Other (3)

A. Isimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

G. Bekefi, Radiation Processes in Plasmas (Wiley, 1966).

R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems (Interscience, 1967).

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Figures (4)

Fig. 1.
Fig. 1.

Normalized coherent backscattering albedo of the semi-infinite conservative nonrefracting medium. Diffusion approximation Eqs. (27) and (28) (dashed and dotted curve, respectively). Solid curve, Monte Carlo simulation (Henyey–Greenstein phase scattering function, Λ=1, g=0.9).

Fig. 2.
Fig. 2.

Normalized coherent backscattering albedo of the conservative refracting slab medium. Finite difference scheme calculation. Henyey–Greenstein phase scattering function, total optical thickness of the slab ε(z2z1)=40, Λ=1, g=0.5. Solid curve, no refraction (γ=0); dotted and dashed curves, γ=+0.25 and γ=0.25, respectively.

Fig. 3.
Fig. 3.

Normalized coherent backscattering albedo of the conservative refracting slab medium. Monte Carlo simulation, 109 photons. Henyey–Greenstein phase scattering function, total optical thickness of the slab ε(z2z1)=40, Λ=1, g=0.5, Λ=1. Solid curve, no refraction (γ=0); dotted and dashed curves, γ=+0.25 and γ=0.25, respectively.

Fig. 4.
Fig. 4.

Normalized coherent backscattering albedo of the conservative refracting slab medium. Diffusion approximation. Solid curve, no refraction (γ=0); dashed curve, γ=±0.25, respectively. The slab thickness z2z1=20l.

Equations (28)

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n2dds(Ln2)+εL=S+F,
(Ω·)L+γ(1μz2)dLdμz=εL+2γμzL+Λε4πL(r,Ω)x(Ω,Ω)dΩ+f(r,Ω),
n(z)=n(z1)eγz,
n(z)sinθ=n0sinθ0=const.
dz=μzds,
dr=1μz2ds,
ϕ=const,
γ(zz0)=ln(1+μ02eγs+1μ02eγs),
γr=2tan1(eγs1+μ01μ0)2tan1(1+μ01μ0).
γs=ln(eγz±μ02+e2γ(zz0)1μ0+1),
(zturnz0)=ln(1μ02)2γ.
R=1γ1μ2.
α(ki,kf)=c4πl2dzdzd2ρexp(z/μ0lz/μl)Q(r,r){1+cos(q·ρ)},
(Dn2ϕ(r,r)n2)+μaϕ(r,r)=δ(r),
D=13(μa+(1g)μs),
ϕ(r,r)=0G(z,z,q)J0(qρ)qdq.
αc(ki,kf)=12l2exp(z/μ0lz/μl)G(z,z,q)dzdz.
(q2D+μa)G(z,z,q)+2γDddzG(z,z,q)Dd2dz2G(z,z,q)=δ(zz),
G(z1z0,z,q)=0,
G(z2+z0,z,q)=0,
G(z,z,q)=C1exp(γzzq2+γ2+μa/D)+C2exp(γz+zq2+γ2+μa/D).
G(z,z,q)={C1(z)exp(γzzq2+γ2+μa/D)+C2(z)exp(γz+zq2+γ2+μa/D),z1<z<z<z2C3(z)exp(γzzq2+γ2+μa/D)+C4(z)exp(γz+zq2+γ2+μa/D),z1<z<z<z2.
G(z,z,q)=G(z,z,q),
G(z,z,q)|zz+0G(z,z,q)|zz0=1D.
G(z,z,q)|γ=G(z,z,q)|γ,
αc(ki,kf)|γ=αc(ki,kf)|γ.
αc(ki,kf)=18πqD(1+ql)e2qz0(1+ql)2.
αc(ki,kf)=3a4πl1e2q(l+z0)2q(l+z0),

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