Time-resolved ring structure of circularly polarized beams backscattered from forward scattering media

Kevin G. Phillips; Min Xu; S.K. Gayen; R.R. Alfano

doi:10.1364/OPEX.13.007954

Abstract

The backscattering of circularly polarized light at normal incidence to a half-space of scattering particles is studied using the Electric Field Monte Carlo (EMC) method. The spatial distribution of the backscat-tered light intensity is examined for both the time-resolved and continuous wave cases for large particles with anisotropy factor, g, in the range 0.8 to 0.97. For the time-resolved case, the backscattered light with the same helicity as that of the incident beam (co-polarized) is found to form a ring centered on the point of incidence. The ring expands and simultaneously grows weak as time increases. The intensity of backscattered light with helicity opposite to that of the incident beam (cross-polarized) is found to exhibit a ring behavior for g ≥ 0.85, with significant backscattering at the point of incidence. For the continuous-wave case no such ring pattern is observed in backscattered light for either helicity. The present EMC study suggests that the ring behavior can only be observed in the time domain, in contrast to previous studies of light backscattered from forward scattering media based on the scalar time-independent Fokker-Planck approximation to the radiative transfer equation. The time-dependent ring structure of backscattered light may have potential use in subsurface imaging applications.

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G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt. 6, 741–746 (1967).
[Crossref] [PubMed]
G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38, 3937–3944 (1999).
[Crossref]
S. P. Morgan and M. E. Ridgway, “Polarization properties of light backscattered from a two layer scattering medium,” Opt. Express 7, 395–402 (2000).
[Crossref] [PubMed]
X. Ni and R. R. Alfano, “Time-resolved backscattering of circularly and linearly polarized light in a turbid medium,” Opt. Lett. 29, 2773–2775 (2004).
[Crossref] [PubMed]
M. Xu and R. R. Alfano, “Circular polarization memory of light,” Phys. Rev. E (submitted for publication)
F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, R9342–R9345 (1989).
[Crossref]
A. D. Kim and M. Moscoso, “Backscattering of beams by forward-peaked scattering media,” Opt. Lett. 29, 74–76 (2004).
[Crossref] [PubMed]
Min Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12, 6530–6539 (2004).
[Crossref] [PubMed]
A. Ishimaru, Wave propagation and scattering in random media, I and II (Academic, New York, 1978).
S. Chandrasekhar, Radiative transfer (Oxford University Press, Oxford, UK, 1960).
K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[Crossref]
A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. 23, 2074–2094 (2002).
[Crossref]
G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiply scattered light from haze clouds,” Appl. Opt. 7, 1519–1527 (1968).
[Crossref] [PubMed]
I. Lux and L. Koblinger, Monte Carlo particle transport methods: neutron and photon calculations (CRC Press, Boca Raton, Fla., 1991).
J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535–6546 (1992).
[Crossref] [PubMed]
P. Bruscaglioni, G. Zaccanit, and Q. Wei, “Transmission of a pulsed polarized light beam through thick turbid media: numerical results,” Appl. Opt. 32, 6142–6150 (1993).
[Crossref] [PubMed]
M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G.L. Cote, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[Crossref]
S. Bartel and A. H. Hielscher,“Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39, 1580–1588 (2000).
[Crossref]
M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[Crossref]
H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]
R. Y. Rubinstein, Simulation and the Monte Carlo method (John Wiley and Sons, 1981)
[Crossref]

2004 (3)

X. Ni and R. R. Alfano, “Time-resolved backscattering of circularly and linearly polarized light in a turbid medium,” Opt. Lett. 29, 2773–2775 (2004).
[Crossref] [PubMed]

A. D. Kim and M. Moscoso, “Backscattering of beams by forward-peaked scattering media,” Opt. Lett. 29, 74–76 (2004).
[Crossref] [PubMed]

Min Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12, 6530–6539 (2004).
[Crossref] [PubMed]

2002 (1)

A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. 23, 2074–2094 (2002).
[Crossref]

1991 (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[Crossref]

1989 (1)

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, R9342–R9345 (1989).
[Crossref]

1968 (1)

G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiply scattered light from haze clouds,” Appl. Opt. 7, 1519–1527 (1968).
[Crossref] [PubMed]

1967 (1)

G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt. 6, 741–746 (1967).
[Crossref] [PubMed]

Alfano, R. R.

X. Ni and R. R. Alfano, “Time-resolved backscattering of circularly and linearly polarized light in a turbid medium,” Opt. Lett. 29, 2773–2775 (2004).
[Crossref] [PubMed]

M. Xu and R. R. Alfano, “Circular polarization memory of light,” Phys. Rev. E (submitted for publication)

Bartel, S.

S. Bartel and A. H. Hielscher,“Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39, 1580–1588 (2000).
[Crossref]

Bonner, R. F.

J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535–6546 (1992).
[Crossref] [PubMed]

Bruscaglioni, P.

P. Bruscaglioni, G. Zaccanit, and Q. Wei, “Transmission of a pulsed polarized light beam through thick turbid media: numerical results,” Appl. Opt. 32, 6142–6150 (1993).
[Crossref] [PubMed]

Cameron, B. D.

M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G.L. Cote, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[Crossref]

Chaikovskaya, L. I.

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative transfer (Oxford University Press, Oxford, UK, 1960).

Cote, G.L.

M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G.L. Cote, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[Crossref]

Evans, K. F.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[Crossref]

Gandjbakhche, A. H.

J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535–6546 (1992).
[Crossref] [PubMed]

Gilbert, G. D.

G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt. 6, 741–746 (1967).
[Crossref] [PubMed]

Hielscher, A. H.

S. Bartel and A. H. Hielscher,“Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39, 1580–1588 (2000).
[Crossref]

Ishimaru, A.

A. Ishimaru, Wave propagation and scattering in random media, I and II (Academic, New York, 1978).

Jordan, D. L.

G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38, 3937–3944 (1999).
[Crossref]

Katsev, I. L.

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]

Kattawar, G. W.

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]

M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G.L. Cote, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[Crossref]

G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiply scattered light from haze clouds,” Appl. Opt. 7, 1519–1527 (1968).
[Crossref] [PubMed]

Keller, J. B.

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[Crossref]

Kim, A. D.

A. D. Kim and M. Moscoso, “Backscattering of beams by forward-peaked scattering media,” Opt. Lett. 29, 74–76 (2004).
[Crossref] [PubMed]

A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. 23, 2074–2094 (2002).
[Crossref]

Koblinger, L.

I. Lux and L. Koblinger, Monte Carlo particle transport methods: neutron and photon calculations (CRC Press, Boca Raton, Fla., 1991).

Lewis, G. D.

G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38, 3937–3944 (1999).
[Crossref]

Lux, I.

I. Lux and L. Koblinger, Monte Carlo particle transport methods: neutron and photon calculations (CRC Press, Boca Raton, Fla., 1991).

MacKintosh, F. C.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, R9342–R9345 (1989).
[Crossref]

Mehrbeolu, M.

M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G.L. Cote, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[Crossref]

Morgan, S. P.

S. P. Morgan and M. E. Ridgway, “Polarization properties of light backscattered from a two layer scattering medium,” Opt. Express 7, 395–402 (2000).
[Crossref] [PubMed]

Moscoso, M.

A. D. Kim and M. Moscoso, “Backscattering of beams by forward-peaked scattering media,” Opt. Lett. 29, 74–76 (2004).
[Crossref] [PubMed]

A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. 23, 2074–2094 (2002).
[Crossref]

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[Crossref]

Ni, X.

X. Ni and R. R. Alfano, “Time-resolved backscattering of circularly and linearly polarized light in a turbid medium,” Opt. Lett. 29, 2773–2775 (2004).
[Crossref] [PubMed]

Papanicolaou, G.

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[Crossref]

Pernicka, J. C.

G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt. 6, 741–746 (1967).
[Crossref] [PubMed]

Pine, D. J.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, R9342–R9345 (1989).
[Crossref]

Plass, G. N.

G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiply scattered light from haze clouds,” Appl. Opt. 7, 1519–1527 (1968).
[Crossref] [PubMed]

Prikhach, A. S.

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]

Rakovic, M. J.

M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G.L. Cote, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[Crossref]

Rastegar, S.

M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G.L. Cote, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[Crossref]

Ridgway, M. E.

S. P. Morgan and M. E. Ridgway, “Polarization properties of light backscattered from a two layer scattering medium,” Opt. Express 7, 395–402 (2000).
[Crossref] [PubMed]

Roberts, P. J.

G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38, 3937–3944 (1999).
[Crossref]

Rubinstein, R. Y.

R. Y. Rubinstein, Simulation and the Monte Carlo method (John Wiley and Sons, 1981)
[Crossref]

Schmitt, J. M.

J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535–6546 (1992).
[Crossref] [PubMed]

Stephens, G. L.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[Crossref]

Tynes, H. H.

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]

Wang, L. V.

M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G.L. Cote, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[Crossref]

Wei, Q.

P. Bruscaglioni, G. Zaccanit, and Q. Wei, “Transmission of a pulsed polarized light beam through thick turbid media: numerical results,” Appl. Opt. 32, 6142–6150 (1993).
[Crossref] [PubMed]

Weitz, D. A.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, R9342–R9345 (1989).
[Crossref]

Xu, M.

M. Xu and R. R. Alfano, “Circular polarization memory of light,” Phys. Rev. E (submitted for publication)

Xu, Min

Min Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12, 6530–6539 (2004).
[Crossref] [PubMed]

Zaccanit, G.

P. Bruscaglioni, G. Zaccanit, and Q. Wei, “Transmission of a pulsed polarized light beam through thick turbid media: numerical results,” Appl. Opt. 32, 6142–6150 (1993).
[Crossref] [PubMed]

Zege, E. P.

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]

Zhu, J. X.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, R9342–R9345 (1989).
[Crossref]

Appl. Opt. (8)

G. D. Gilbert and J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” Appl. Opt. 6, 741–746 (1967).
[Crossref] [PubMed]

G. D. Lewis, D. L. Jordan, and P. J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38, 3937–3944 (1999).
[Crossref]

G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiply scattered light from haze clouds,” Appl. Opt. 7, 1519–1527 (1968).
[Crossref] [PubMed]

J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535–6546 (1992).
[Crossref] [PubMed]

P. Bruscaglioni, G. Zaccanit, and Q. Wei, “Transmission of a pulsed polarized light beam through thick turbid media: numerical results,” Appl. Opt. 32, 6142–6150 (1993).
[Crossref] [PubMed]

M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G.L. Cote, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38, 3399–3408 (1999).
[Crossref]

S. Bartel and A. H. Hielscher,“Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39, 1580–1588 (2000).
[Crossref]

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]

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

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer 46, 413–423 (1991).
[Crossref]

Opt. Express (2)

S. P. Morgan and M. E. Ridgway, “Polarization properties of light backscattered from a two layer scattering medium,” Opt. Express 7, 395–402 (2000).
[Crossref] [PubMed]

Min Xu, “Electric field Monte Carlo simulation of polarized light propagation in turbid media,” Opt. Express 12, 6530–6539 (2004).
[Crossref] [PubMed]

Opt. Lett. (2)

A. D. Kim and M. Moscoso, “Backscattering of beams by forward-peaked scattering media,” Opt. Lett. 29, 74–76 (2004).
[Crossref] [PubMed]

X. Ni and R. R. Alfano, “Time-resolved backscattering of circularly and linearly polarized light in a turbid medium,” Opt. Lett. 29, 2773–2775 (2004).
[Crossref] [PubMed]

Phys. Rev. B (1)

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, R9342–R9345 (1989).
[Crossref]

SIAM J. Sci. Comput. (1)

A. D. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. 23, 2074–2094 (2002).
[Crossref]

Other (5)

I. Lux and L. Koblinger, Monte Carlo particle transport methods: neutron and photon calculations (CRC Press, Boca Raton, Fla., 1991).

A. Ishimaru, Wave propagation and scattering in random media, I and II (Academic, New York, 1978).

S. Chandrasekhar, Radiative transfer (Oxford University Press, Oxford, UK, 1960).

M. Xu and R. R. Alfano, “Circular polarization memory of light,” Phys. Rev. E (submitted for publication)

R. Y. Rubinstein, Simulation and the Monte Carlo method (John Wiley and Sons, 1981)
[Crossref]

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Fig. 1.

Backscattered time-resolved intensity of co-polarized light for g = 0.96 at times: (a.) $t = 1 [\frac{l_{s}}{c}]$ , (b.) $t = 2 [\frac{l_{s}}{c}]$ , (c.) $t = 3 [\frac{l_{s}}{c}]$ and (d.) $t = 4 [\frac{l_{s}}{c}]$ .

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Fig. 2.

Comparison of backscattered intensity of light co-polarized with the incident beam for different anisotropy factors at $t = 2 [\frac{l_{s}}{c}]$ .

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Fig. 3.

Time evolution of co-polarized backscattered light for g = 0.80. Note at $t = 19 [\frac{l_{s}}{c}]$ plateauing of the ring-peak occurs. Inset: the radial profile at $t = 30 [\frac{l_{s}}{c}]$ , convergence to a Gaussian-like distribution.

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Fig. 4.

Schematic diagram illustrating light pathways contributing to ring formation with backscattered light (a.) co-polarized , and (b.) cross-polarized, with the incident beam.

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Fig. 5.

Time dependence of the radius of the ring-peak of co-polarized backscattered light for various values of g.

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Fig. 6.

Time-resolved intensity of cross-polarized backscattered light for g = 0.96 at times (a.) $t = 1 [\frac{l_{s}}{c}]$ , (b.) $t = 2 [\frac{l_{s}}{c}]$ , (c.) $t = 3 [\frac{l_{s}}{c}]$ and (d.) $t = 4 [\frac{l_{s}}{c}]$ .

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Fig. 7.

Time evolution of cross-polarized backscattered light for g = 0.85. Note at $t = 11 [\frac{l_{s}}{c}]$ plateauing of the ring-peak occurs. Inset: the radial profile at $t = 25 [\frac{l_{s}}{c}]$ , convergence to a Gaussian-like distribution.

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Fig. 8.

Comparison of cross-polarized backscattered light for various anisotropies, $t = 4 [\frac{l_{s}}{c}]$

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Fig. 9.

Comparison of backscattered light with opposite and preserved polarization at $t = 6.5 [\frac{l_{s}}{c}]$ and with g = 0.90.

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Fig. 10.

Backscattered continuous-wave intensity for (a) co-polarized and (b) cross-polarized light with the incident beam.

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Fig. 11.

Comparison of backscattered continuous-wave intensity for various anisotropies with the same (a) and opposite (b) helicity as the incident beam.

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Fig. 12.

Comparison of continuous-wave and time-resolved co-polarized backscattered light for g = 0.85.

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Fig. 13.

Comparison of continuous-wave and time-resolved cross-polarized backscattered light for g = 0.85.

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Fig. 14.

Time evolution of simultaneous collection of co-polarized and cross-polarized backscattered light for g = 0.85.

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Tables (1)

Table 1. The size parameters and the relative indices of refraction for scattering spheres of the specified anisotropy factors investigated.

View Table

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

(c_{o}^{- 1} \partial_{t} + \hat{s} ∙ \nabla_{r} + σ_{t}) I (\vec{r}, \hat{s}, t) = \int_{Ω} P (\hat{s} ∙ \hat{s}') ∙ I (\vec{r}, \hat{s}', t) d s'

s' = m \sin (θ) \cos (ϕ) + n \sin (θ) \sin (ϕ) + s \cos (θ)

e_{1} = m \cos (ϕ) + n \sin (ϕ)

e_{2} = - m \sin (ϕ) + n \cos (ϕ)

e_{1}^{'} = m \cos (θ) \cos (ϕ) + n \cos (θ) \sin (ϕ) - s \sin (θ) = m'

e_{2}^{'} = e_{2} = n'

E' = E_{1}^{'} m' + E_{2}^{'} n' = (S_{2} E ∙ e_{1}) m' + (S_{1} E ∙ e_{2}) n'

(\begin{matrix} m' \\ n' \\ s' \end{matrix}) = (\begin{matrix} \cos (θ) \cos (ϕ) & \cos (θ) \sin (ϕ) & - \sin (θ) \\ - \sin (ϕ) & \cos (ϕ) & 0 \\ \sin (θ) \cos (ϕ) & \sin (θ) \sin (ϕ) & \cos (θ) \end{matrix}) ∙ (\begin{matrix} m \\ n \\ s \end{matrix})

(\begin{matrix} E_{1}^{'} \\ E_{2}^{'} \end{matrix}) = \frac{1}{\sqrt{F (θ, ϕ)}} (\begin{matrix} S_{2} \cos (ϕ) & S_{2} \sin (ϕ) \\ - S_{1} \sin (ϕ) & S_{1} \cos (ϕ) \end{matrix}) ∙ (\begin{matrix} E_{1} \\ E_{2} \end{matrix})

Anisotropy	Size	Index of
Factor	Parameter	Refraction
0.80	3.15	1.1
0.85	3.61	1.1
0.90	4.88	1.1
0.95	7.8	1.1
0.96	10	1.1
0.97	12.1	1.1

Abstract

References

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