Abstract

A method for directly simulating coherent backscattering of polarized light by a turbid medium has been developed based on the Electric field Monte Carlo (EMC) method. Electric fields of light traveling in a pair of time-reversed paths are added coherently to simulate their interference. An efficient approach for computing the electric field of light traveling along a time-reversed path is derived and implemented based on the time-reversal symmetry of electromagnetic waves. Coherent backscattering of linearly and circularly polarized light by a turbid medium containing Mie scatterers is then investigated using this method.

© 2008 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media, I and II (Academic, New York, 1978).
  2. A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 38-40 (1995).
    [CrossRef]
  3. S. K. Gayen and R. R. Alfano, "Emerging optical biomedical imaging techniques," Opt. Photon. News 7, 17-22 (1996).
  4. S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
    [CrossRef]
  5. P. Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
    [CrossRef] [PubMed]
  6. M. P. Van Albada and Ad Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett. 55, 2692-2695 (1985).
    [CrossRef] [PubMed]
  7. I. Lux and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).
  8. 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]
  9. 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]
  10. B. Kaplan, G. Ledanois, and B. Villon, "Mueller matrix of dense polystyrene latex sphere suspensions: Measurements and Monte Carlo simulation," Appl. Opt. 40, 2769-2777 (2001).
    [CrossRef]
  11. X. Wang and L. V. Wang, "Propagation of polarized light in birefringent turbid media: A Monte Carlo study," J. Biomed. Opt. 7, 279-290 (2002).
    [CrossRef] [PubMed]
  12. M. Xu, "Electric field Monte Carlo simulation of polarized light propagation through turbid media," Opt. Express 12, 6530-6539 (2004).
    [CrossRef] [PubMed]
  13. K. G. Philips, M. Xu, S. K. Gayen, and R. R. Alfano, "Time-resolved ring structures of circularly polarized beams backscattered from forward scattering media," Opt. Express 13, 7954-7969 (2005).
    [CrossRef]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  15. D. S. Saxon, "Tensor scattering matrix for the electromagnetic field," Phys. Rev. 100, 1771-1775 (1955).
    [CrossRef]
  16. M. Xu and R. R. Alfano, "Circular polarization memory of light," Phys. Rev. E 72, 065061(R) (2005).

2005 (2)

2004 (1)

2002 (1)

X. Wang and L. V. Wang, "Propagation of polarized light in birefringent turbid media: A Monte Carlo study," J. Biomed. Opt. 7, 279-290 (2002).
[CrossRef] [PubMed]

2001 (2)

2000 (1)

1999 (1)

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

1996 (1)

S. K. Gayen and R. R. Alfano, "Emerging optical biomedical imaging techniques," Opt. Photon. News 7, 17-22 (1996).

1995 (1)

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 38-40 (1995).
[CrossRef]

1985 (2)

P. Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[CrossRef] [PubMed]

M. P. Van Albada and Ad Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett. 55, 2692-2695 (1985).
[CrossRef] [PubMed]

1955 (1)

D. S. Saxon, "Tensor scattering matrix for the electromagnetic field," Phys. Rev. 100, 1771-1775 (1955).
[CrossRef]

Appl. Opt. (3)

Inverse Probl. (1)

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

J. Biomed. Opt. (1)

X. Wang and L. V. Wang, "Propagation of polarized light in birefringent turbid media: A Monte Carlo study," J. Biomed. Opt. 7, 279-290 (2002).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Photon. News (1)

S. K. Gayen and R. R. Alfano, "Emerging optical biomedical imaging techniques," Opt. Photon. News 7, 17-22 (1996).

Phys. Rev. (1)

D. S. Saxon, "Tensor scattering matrix for the electromagnetic field," Phys. Rev. 100, 1771-1775 (1955).
[CrossRef]

Phys. Rev. E (1)

M. Xu and R. R. Alfano, "Circular polarization memory of light," Phys. Rev. E 72, 065061(R) (2005).

Phys. Rev. Lett. (2)

P. Wolf and G. Maret, "Weak localization and coherent backscattering of photons in disordered media," Phys. Rev. Lett. 55, 2696-2699 (1985).
[CrossRef] [PubMed]

M. P. Van Albada and Ad Lagendijk, "Observation of weak localization of light in a random medium," Phys. Rev. Lett. 55, 2692-2695 (1985).
[CrossRef] [PubMed]

Phys. Today (1)

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 38-40 (1995).
[CrossRef]

Other (3)

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

I. Lux and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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Figures (6)

Fig. 1.
Fig. 1.

A pair of partial waves propagating along the forward and reverse paths. The numbered circles are the scatterers. In the forward direction, light enters the media in the direction S 0 and encounters the first scattering event at the site “1”. After being scattered with an angle of θ 1, light propagates in the S 1 direction. Light leaves in the S n direction after the last scattering at the site “n”. In the reverse path, light is incident in the direction S 0 and is first scattered by the scatterer “n”, the last scatterer in the forward path, with a scattering angle of θ n,R and to the -S n-1 direction. At the last scattering event in the reverse path, light is scattered at the site “1” with a scattering angle of θ1,R and escapes the medium in the same direction S n that light remits in the forward path.

Fig. 2.
Fig. 2.

The azimuthal rotations of the normal of the scattering plane along the forward and reverse paths. In the forward path, light is rotated for an azimuthal angle of ϕ i-1 at the i-th scattering event. In the reverse path, except for three special rotations of angles ϕ n n ′, and ϕ′1, all the other rotations in the reverse path use the same ϕ angles as those in the forward path. See the text for details.

Fig. 3.
Fig. 3.

(Color) Backscattering of normally incident circularly polarized light. The first row displays I++I_, I+, and I- from left to right for incident coherent light; the bottom row displays the corresponding intensities for incident incoherent light. Each circle depicts the view around the exact backscattering direction such that the zenith angle (θb) is the circle’s radius, in degrees, and the azimuthal angle (ϕb) is the polar angle, in degrees. θb starts at zero degrees exactly in the center, the exact backscattering direction, and ends at the edge (75 degrees). ϕb is zero degrees in the direction of a standard Cartesian positive x-axis and increases counterclockwise until 360 degrees. Notice all I’s are not dependent on the azimuthal angle.

Fig. 4.
Fig. 4.

(Color) Angular profiles of backscattering of circularly polarized light versus the zenith detection angle. Coherent backscattered intensities, incoherent backscattered intensities, and the enhancement factor are displayed in (a,d), (b,e), and (c,f), respectively. Near exact backscattering direction θb~0, coherent backscattered I+ is greater than coherent backscattered I- and the enhancement factor for I+ is larger than I-. Incoherent backscattered I+ is less than incoherent backscattered I- for all θb. (Note that I- in (f) is raised for 1.75×104 to show how it compares with I+ within the angular range).

Fig. 5.
Fig. 5.

(Color) Backscattering of normally incident linearly polarized light. The incident beam is polarized linearly in the x-direction. Each circle depicts the view around the exact backscattering direction such that the zenith angle (θb) is the circle’s radius, in degrees, and the azimuthal angle (ϕb) is the polar angle, in degrees. θb starts at zero degrees exactly in the center, the exact backscattering direction, and ends at the edge at 75 degrees for the top and middle rows and ends at 2.25 degrees for the bottom row. ϕb is zero degrees in the direction of a standard Cartesian positive x-axis and increases counterclockwise until 360 degrees.

Fig. 6.
Fig. 6.

(Color) Angular profiles of backscattering of linearly polarized light along three directions corresponding to the azimuthal angle of 0, 45, and 90 degrees, respectively. Coherent backscattered intensities, incoherent backscattered intensities, and the enhancement factor are displayed in (a,d), (b,e), and (c,f), respectively. (Note that Iy in (e) is raised for 2.54×104 to show how it compares with Ix within the angular range).

Equations (16)

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I = ( I , Q , U , V ) T ,
I = | E x | 2 + | E y | 2 , Q = | E x | 2 | E y | 2 , U = E x * E y + E x E y * , V = i E x * E y E x E y *
E = SR E ,
( E 1 E 2 ) = [ S 2 ( θ ) 0 0 S 1 ( θ ) ] [ cos ϕ sin ϕ sin ϕ cos ϕ ] ( E 1 E 2 ) ,
E out = S ( n ) ( s n , s n 1 ) R ( ϕ n 1 ) S ( n 1 ) ( s n 1 , s n 2 ) R ( ϕ n 2 ) R ( ϕ 2 ) S ( 2 ) ( s 2 , s 1 ) R ( ϕ 1 ) S ( 1 ) ( s 1 , s 0 ) R ( ϕ 0 ) E in ,
E out Re v = S ( 1 ) ( s n , s 1 ) R ( ϕ 1 ) R ( ϕ 1 ) S ( 2 ) ( s 1 , s 2 ) R ( ϕ 2 ) R ( ϕ n 2 ) S ( n 1 ) ( s n 2 , s n 1 ) R ( ϕ n ) S ( n ) ( s n 1 , s 0 ) R ( ϕ n ) E in .
T = S ( n 1 ) ( s n 1 , s n 2 ) R ( ϕ n 2 ) R ( ϕ 2 ) S ( 2 ) ( s 2 , s 1 ) R ( ϕ 1 )
T Re v = R ( ϕ 1 ) S ( 2 ) ( s 1 , s 2 ) R ( ϕ 2 ) R ( ϕ n 2 ) S ( n 1 ) ( s n 2 , s n 1 ) .
T T = R T ( ϕ 1 ) S ( 2 ) T ( s 2 , s 1 ) R T ( ϕ 2 ) R T ( ϕ n 2 ) S ( n 1 ) T ( s n 1 , s n 2 ) .
R T ( ϕ ) = [ cos ϕ sin ϕ sin ϕ cos ϕ ] = Q [ cos ϕ sin ϕ sin ϕ cos ϕ ] Q = QR ( ϕ ) Q ,
S ( 2 ) T ( s 2 , s 1 ) = QS ( 2 ) ( s 1 , s 2 ) Q ,
T T = QR ( ϕ 1 ) QQS ( 2 ) ( s 1 , s 2 ) QQR ( ϕ 2 ) Q QR ( ϕ n 2 ) QQS ( n 1 ) ( s n 2 , s n 1 ) Q ,
T T = QT Re v Q ,
T Re v = QT T Q .
E out = S ( n ) ( s n , s n 1 ) R ( ϕ n 1 ) TS ( 1 ) ( s 1 , s 0 ) R ( ϕ 0 ) E in ,
E out Re v = S ( 1 ) ( s n , s 1 ) R ( ϕ 1 ) QT T QR ( ϕ n ) S ( n ) ( s n 1 , s 0 ) R ( ϕ n ) E in .

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