Abstract

The propagation of a polarized pulse in random media is investigated using the discrete-ordinates method to solve the transient vector radiative transfer. The angular analysis of the transient polarized features of scattering fluxes makes it possible to investigate subtle details of the polarization flip encountered for circularly polarized waves. We found that, depending on the geometry, the state of polarization, and the angle of detection, the degree of polarization decays at either a slower or faster rate when the beam is impinging at an angle far from the normal incidence. At normal incidence, our results confirm that, for large particles, the circular polarization maintains a greater degree of polarization.

© 2006 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  2. A. Ishimaru, 'Diffusion of light in turbid material,' Appl. Opt. 28, 2210-2215 (1989).
    [CrossRef] [PubMed]
  3. M. Q. Brewster and Y. Yamada, 'Optical properties of thick turbid media from picosecond time-resolved light scattering measurement,' Int. J. Heat Mass Transfer 38, 2569-2581 (1995).
    [CrossRef]
  4. N. Xiaohui and R. R. Alfano, 'Time-resolved backscattering of circularly and linearly polarized light in a turbid medium,' Opt. Lett. 29, 2773-2775 (2004).
    [CrossRef]
  5. R. Elaloufi, R. Carminati, and J. J. Greffet, 'Time-dependent transport through scattering media: from radiative transfer to diffusion,' J. Opt. A, Pure Appl. Opt. 4, S103-S108 (2002).
    [CrossRef]
  6. K. Mitra and S. Kumar, 'Development and comparison of models for light-pulse transport through scattering-absorbing media,' Appl. Opt. 38, 188-196 (1999).
    [CrossRef]
  7. Z. M. Tan and P.-F. Hsu, 'An integral formulation of transient radiative transfer,' J. Heat Transfer 123, 466-475 (2001).
    [CrossRef]
  8. M. Sakami, K. Mitra, and P.-F. Hsu, 'Analysis of light-pulse transport through two-dimensional scattering and absorbing media,' J. Quant. Spectrosc. Radiat. Transf. 73, 169-179 (2002).
    [CrossRef]
  9. R.L.-T. Cheung and A. Ishimaru, 'Transmission, backscattering, and depolarization of waves in randomly distributed spherical particles,' Appl. Opt. 21, 3792-3798 (1982).
    [CrossRef] [PubMed]
  10. A. D. Kim, S. Jaruwatanadilok, A. Ishimaru, and Y. Kuga, 'Polarized light propagation and scattering in random media,' in Proc. SPIE 4257, 90-100 (2001).
    [CrossRef]
  11. R. Vaillon, B. T. Wong, and M. P. Mengüç, 'Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,' J. Quant. Spectrosc. Radiat. Transf. 84, 383-394 (2004).
    [CrossRef]
  12. Y. Jiang, Y. L. Yung, S. P. Sander, and L. D. Travis, 'Modeling of atmospheric radiative transfer with polarization and its application to the remote sensing of tropospheric ozone,' J. Quant. Spectrosc. Radiat. Transf. 84, 169-179 (2004).
    [CrossRef]
  13. Q. Ma and A. Ishimaru, 'Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding media,' Radio Sci. 25, 419-426 (1990).
    [CrossRef]
  14. A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, 'Polarized pulse waves in random discrete scatterers,' Appl. Opt. 40, 5495-5502 (2001).
    [CrossRef]
  15. G. Strang, 'On the construction and comparison of difference schemes,' SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 506-517 (1968).
  16. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  17. A. Harten, 'High resolution schemes for hyperbolic conservation laws,' J. Comput. Phys. 49, 357-393 (1983).
    [CrossRef]
  18. P. Colella and P. R. Woodward, 'The piecewise parabolic method (PPM) for gas-dynamical simulations,' J. Comput. Phys. 54, 174-201 (1984).
    [CrossRef]
  19. J. M. Stone and D. Mihalas, 'Upwind monotonic interpolation methods for the solution of the time dependent radiative transfer equation,' J. Comput. Phys. 100, 402-408 (1992).
    [CrossRef]
  20. R. L. Carpenter, Jr., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, 'Application of the piecewise parabolic method (PPM) to meteorological modeling,' Mon. Weather Rev. 118, 586-612 (1990).
    [CrossRef]
  21. C. P. Thurgood, 'A critical evaluation of the discrete ordinates method using HEART and Tn quadrature,' Ph.D. thesis, (Queen's University, Kingston, Ontario, Canada, 1992).
  22. A. Kim and M. Moscoso, 'Backscattering of circularly polarized pulses,' Opt. Lett. 27, 1589-1591 (2002).
    [CrossRef]
  23. J. Ellis, P. Caillard, and A. Dogariu, 'Off-diagonal Mueller matrix elements in backscattering from highly diffusive media,' J. Opt. Soc. Am. A 19, 43-48 (2002).
    [CrossRef]

2004 (3)

R. Vaillon, B. T. Wong, and M. P. Mengüç, 'Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,' J. Quant. Spectrosc. Radiat. Transf. 84, 383-394 (2004).
[CrossRef]

Y. Jiang, Y. L. Yung, S. P. Sander, and L. D. Travis, 'Modeling of atmospheric radiative transfer with polarization and its application to the remote sensing of tropospheric ozone,' J. Quant. Spectrosc. Radiat. Transf. 84, 169-179 (2004).
[CrossRef]

N. Xiaohui and R. R. Alfano, 'Time-resolved backscattering of circularly and linearly polarized light in a turbid medium,' Opt. Lett. 29, 2773-2775 (2004).
[CrossRef]

2002 (4)

J. Ellis, P. Caillard, and A. Dogariu, 'Off-diagonal Mueller matrix elements in backscattering from highly diffusive media,' J. Opt. Soc. Am. A 19, 43-48 (2002).
[CrossRef]

A. Kim and M. Moscoso, 'Backscattering of circularly polarized pulses,' Opt. Lett. 27, 1589-1591 (2002).
[CrossRef]

R. Elaloufi, R. Carminati, and J. J. Greffet, 'Time-dependent transport through scattering media: from radiative transfer to diffusion,' J. Opt. A, Pure Appl. Opt. 4, S103-S108 (2002).
[CrossRef]

M. Sakami, K. Mitra, and P.-F. Hsu, 'Analysis of light-pulse transport through two-dimensional scattering and absorbing media,' J. Quant. Spectrosc. Radiat. Transf. 73, 169-179 (2002).
[CrossRef]

2001 (3)

Z. M. Tan and P.-F. Hsu, 'An integral formulation of transient radiative transfer,' J. Heat Transfer 123, 466-475 (2001).
[CrossRef]

A. D. Kim, S. Jaruwatanadilok, A. Ishimaru, and Y. Kuga, 'Polarized light propagation and scattering in random media,' in Proc. SPIE 4257, 90-100 (2001).
[CrossRef]

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, 'Polarized pulse waves in random discrete scatterers,' Appl. Opt. 40, 5495-5502 (2001).
[CrossRef]

1999 (1)

1995 (1)

M. Q. Brewster and Y. Yamada, 'Optical properties of thick turbid media from picosecond time-resolved light scattering measurement,' Int. J. Heat Mass Transfer 38, 2569-2581 (1995).
[CrossRef]

1992 (1)

J. M. Stone and D. Mihalas, 'Upwind monotonic interpolation methods for the solution of the time dependent radiative transfer equation,' J. Comput. Phys. 100, 402-408 (1992).
[CrossRef]

1990 (2)

R. L. Carpenter, Jr., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, 'Application of the piecewise parabolic method (PPM) to meteorological modeling,' Mon. Weather Rev. 118, 586-612 (1990).
[CrossRef]

Q. Ma and A. Ishimaru, 'Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding media,' Radio Sci. 25, 419-426 (1990).
[CrossRef]

1989 (1)

1984 (1)

P. Colella and P. R. Woodward, 'The piecewise parabolic method (PPM) for gas-dynamical simulations,' J. Comput. Phys. 54, 174-201 (1984).
[CrossRef]

1983 (1)

A. Harten, 'High resolution schemes for hyperbolic conservation laws,' J. Comput. Phys. 49, 357-393 (1983).
[CrossRef]

1982 (1)

1968 (1)

G. Strang, 'On the construction and comparison of difference schemes,' SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 506-517 (1968).

Alfano, R. R.

Brewster, M. Q.

M. Q. Brewster and Y. Yamada, 'Optical properties of thick turbid media from picosecond time-resolved light scattering measurement,' Int. J. Heat Mass Transfer 38, 2569-2581 (1995).
[CrossRef]

Caillard, P.

Carminati, R.

R. Elaloufi, R. Carminati, and J. J. Greffet, 'Time-dependent transport through scattering media: from radiative transfer to diffusion,' J. Opt. A, Pure Appl. Opt. 4, S103-S108 (2002).
[CrossRef]

Carpenter, R. L.

R. L. Carpenter, Jr., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, 'Application of the piecewise parabolic method (PPM) to meteorological modeling,' Mon. Weather Rev. 118, 586-612 (1990).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Cheung, R. L.-T.

Colella, P.

P. Colella and P. R. Woodward, 'The piecewise parabolic method (PPM) for gas-dynamical simulations,' J. Comput. Phys. 54, 174-201 (1984).
[CrossRef]

Dogariu, A.

Droegemeier, K. K.

R. L. Carpenter, Jr., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, 'Application of the piecewise parabolic method (PPM) to meteorological modeling,' Mon. Weather Rev. 118, 586-612 (1990).
[CrossRef]

Elaloufi, R.

R. Elaloufi, R. Carminati, and J. J. Greffet, 'Time-dependent transport through scattering media: from radiative transfer to diffusion,' J. Opt. A, Pure Appl. Opt. 4, S103-S108 (2002).
[CrossRef]

Ellis, J.

Greffet, J. J.

R. Elaloufi, R. Carminati, and J. J. Greffet, 'Time-dependent transport through scattering media: from radiative transfer to diffusion,' J. Opt. A, Pure Appl. Opt. 4, S103-S108 (2002).
[CrossRef]

Hane, C. E.

R. L. Carpenter, Jr., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, 'Application of the piecewise parabolic method (PPM) to meteorological modeling,' Mon. Weather Rev. 118, 586-612 (1990).
[CrossRef]

Harten, A.

A. Harten, 'High resolution schemes for hyperbolic conservation laws,' J. Comput. Phys. 49, 357-393 (1983).
[CrossRef]

Hsu, P.-F.

M. Sakami, K. Mitra, and P.-F. Hsu, 'Analysis of light-pulse transport through two-dimensional scattering and absorbing media,' J. Quant. Spectrosc. Radiat. Transf. 73, 169-179 (2002).
[CrossRef]

Z. M. Tan and P.-F. Hsu, 'An integral formulation of transient radiative transfer,' J. Heat Transfer 123, 466-475 (2001).
[CrossRef]

Ishimaru, A.

A. D. Kim, S. Jaruwatanadilok, A. Ishimaru, and Y. Kuga, 'Polarized light propagation and scattering in random media,' in Proc. SPIE 4257, 90-100 (2001).
[CrossRef]

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, 'Polarized pulse waves in random discrete scatterers,' Appl. Opt. 40, 5495-5502 (2001).
[CrossRef]

Q. Ma and A. Ishimaru, 'Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding media,' Radio Sci. 25, 419-426 (1990).
[CrossRef]

A. Ishimaru, 'Diffusion of light in turbid material,' Appl. Opt. 28, 2210-2215 (1989).
[CrossRef] [PubMed]

R.L.-T. Cheung and A. Ishimaru, 'Transmission, backscattering, and depolarization of waves in randomly distributed spherical particles,' Appl. Opt. 21, 3792-3798 (1982).
[CrossRef] [PubMed]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Jaruwatanadilok, S.

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, 'Polarized pulse waves in random discrete scatterers,' Appl. Opt. 40, 5495-5502 (2001).
[CrossRef]

A. D. Kim, S. Jaruwatanadilok, A. Ishimaru, and Y. Kuga, 'Polarized light propagation and scattering in random media,' in Proc. SPIE 4257, 90-100 (2001).
[CrossRef]

Jiang, Y.

Y. Jiang, Y. L. Yung, S. P. Sander, and L. D. Travis, 'Modeling of atmospheric radiative transfer with polarization and its application to the remote sensing of tropospheric ozone,' J. Quant. Spectrosc. Radiat. Transf. 84, 169-179 (2004).
[CrossRef]

Kim, A.

Kim, A. D.

A. D. Kim, S. Jaruwatanadilok, A. Ishimaru, and Y. Kuga, 'Polarized light propagation and scattering in random media,' in Proc. SPIE 4257, 90-100 (2001).
[CrossRef]

Kuga, Y.

A. D. Kim, S. Jaruwatanadilok, A. Ishimaru, and Y. Kuga, 'Polarized light propagation and scattering in random media,' in Proc. SPIE 4257, 90-100 (2001).
[CrossRef]

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, 'Polarized pulse waves in random discrete scatterers,' Appl. Opt. 40, 5495-5502 (2001).
[CrossRef]

Kumar, S.

Ma, Q.

Q. Ma and A. Ishimaru, 'Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding media,' Radio Sci. 25, 419-426 (1990).
[CrossRef]

Mengüç, M. P.

R. Vaillon, B. T. Wong, and M. P. Mengüç, 'Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,' J. Quant. Spectrosc. Radiat. Transf. 84, 383-394 (2004).
[CrossRef]

Mihalas, D.

J. M. Stone and D. Mihalas, 'Upwind monotonic interpolation methods for the solution of the time dependent radiative transfer equation,' J. Comput. Phys. 100, 402-408 (1992).
[CrossRef]

Mitra, K.

M. Sakami, K. Mitra, and P.-F. Hsu, 'Analysis of light-pulse transport through two-dimensional scattering and absorbing media,' J. Quant. Spectrosc. Radiat. Transf. 73, 169-179 (2002).
[CrossRef]

K. Mitra and S. Kumar, 'Development and comparison of models for light-pulse transport through scattering-absorbing media,' Appl. Opt. 38, 188-196 (1999).
[CrossRef]

Moscoso, M.

Sakami, M.

M. Sakami, K. Mitra, and P.-F. Hsu, 'Analysis of light-pulse transport through two-dimensional scattering and absorbing media,' J. Quant. Spectrosc. Radiat. Transf. 73, 169-179 (2002).
[CrossRef]

Sander, S. P.

Y. Jiang, Y. L. Yung, S. P. Sander, and L. D. Travis, 'Modeling of atmospheric radiative transfer with polarization and its application to the remote sensing of tropospheric ozone,' J. Quant. Spectrosc. Radiat. Transf. 84, 169-179 (2004).
[CrossRef]

Stone, J. M.

J. M. Stone and D. Mihalas, 'Upwind monotonic interpolation methods for the solution of the time dependent radiative transfer equation,' J. Comput. Phys. 100, 402-408 (1992).
[CrossRef]

Strang, G.

G. Strang, 'On the construction and comparison of difference schemes,' SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 506-517 (1968).

Tan, Z. M.

Z. M. Tan and P.-F. Hsu, 'An integral formulation of transient radiative transfer,' J. Heat Transfer 123, 466-475 (2001).
[CrossRef]

Thurgood, C. P.

C. P. Thurgood, 'A critical evaluation of the discrete ordinates method using HEART and Tn quadrature,' Ph.D. thesis, (Queen's University, Kingston, Ontario, Canada, 1992).

Travis, L. D.

Y. Jiang, Y. L. Yung, S. P. Sander, and L. D. Travis, 'Modeling of atmospheric radiative transfer with polarization and its application to the remote sensing of tropospheric ozone,' J. Quant. Spectrosc. Radiat. Transf. 84, 169-179 (2004).
[CrossRef]

Vaillon, R.

R. Vaillon, B. T. Wong, and M. P. Mengüç, 'Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,' J. Quant. Spectrosc. Radiat. Transf. 84, 383-394 (2004).
[CrossRef]

Wong, B. T.

R. Vaillon, B. T. Wong, and M. P. Mengüç, 'Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,' J. Quant. Spectrosc. Radiat. Transf. 84, 383-394 (2004).
[CrossRef]

Woodward, P. R.

R. L. Carpenter, Jr., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, 'Application of the piecewise parabolic method (PPM) to meteorological modeling,' Mon. Weather Rev. 118, 586-612 (1990).
[CrossRef]

P. Colella and P. R. Woodward, 'The piecewise parabolic method (PPM) for gas-dynamical simulations,' J. Comput. Phys. 54, 174-201 (1984).
[CrossRef]

Xiaohui, N.

Yamada, Y.

M. Q. Brewster and Y. Yamada, 'Optical properties of thick turbid media from picosecond time-resolved light scattering measurement,' Int. J. Heat Mass Transfer 38, 2569-2581 (1995).
[CrossRef]

Yung, Y. L.

Y. Jiang, Y. L. Yung, S. P. Sander, and L. D. Travis, 'Modeling of atmospheric radiative transfer with polarization and its application to the remote sensing of tropospheric ozone,' J. Quant. Spectrosc. Radiat. Transf. 84, 169-179 (2004).
[CrossRef]

Appl. Opt. (4)

Int. J. Heat Mass Transfer (1)

M. Q. Brewster and Y. Yamada, 'Optical properties of thick turbid media from picosecond time-resolved light scattering measurement,' Int. J. Heat Mass Transfer 38, 2569-2581 (1995).
[CrossRef]

J. Comput. Phys. (3)

A. Harten, 'High resolution schemes for hyperbolic conservation laws,' J. Comput. Phys. 49, 357-393 (1983).
[CrossRef]

P. Colella and P. R. Woodward, 'The piecewise parabolic method (PPM) for gas-dynamical simulations,' J. Comput. Phys. 54, 174-201 (1984).
[CrossRef]

J. M. Stone and D. Mihalas, 'Upwind monotonic interpolation methods for the solution of the time dependent radiative transfer equation,' J. Comput. Phys. 100, 402-408 (1992).
[CrossRef]

J. Heat Transfer (1)

Z. M. Tan and P.-F. Hsu, 'An integral formulation of transient radiative transfer,' J. Heat Transfer 123, 466-475 (2001).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

R. Elaloufi, R. Carminati, and J. J. Greffet, 'Time-dependent transport through scattering media: from radiative transfer to diffusion,' J. Opt. A, Pure Appl. Opt. 4, S103-S108 (2002).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transf. (3)

M. Sakami, K. Mitra, and P.-F. Hsu, 'Analysis of light-pulse transport through two-dimensional scattering and absorbing media,' J. Quant. Spectrosc. Radiat. Transf. 73, 169-179 (2002).
[CrossRef]

R. Vaillon, B. T. Wong, and M. P. Mengüç, 'Polarized radiative transfer in a particle-laden semi-transparent medium via a vector Monte Carlo method,' J. Quant. Spectrosc. Radiat. Transf. 84, 383-394 (2004).
[CrossRef]

Y. Jiang, Y. L. Yung, S. P. Sander, and L. D. Travis, 'Modeling of atmospheric radiative transfer with polarization and its application to the remote sensing of tropospheric ozone,' J. Quant. Spectrosc. Radiat. Transf. 84, 169-179 (2004).
[CrossRef]

Mon. Weather Rev. (1)

R. L. Carpenter, Jr., K. K. Droegemeier, P. R. Woodward, and C. E. Hane, 'Application of the piecewise parabolic method (PPM) to meteorological modeling,' Mon. Weather Rev. 118, 586-612 (1990).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

A. D. Kim, S. Jaruwatanadilok, A. Ishimaru, and Y. Kuga, 'Polarized light propagation and scattering in random media,' in Proc. SPIE 4257, 90-100 (2001).
[CrossRef]

Radio Sci. (1)

Q. Ma and A. Ishimaru, 'Scattering and depolarization of waves incident upon a slab of random medium with refractive index different from that of the surrounding media,' Radio Sci. 25, 419-426 (1990).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

G. Strang, 'On the construction and comparison of difference schemes,' SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 506-517 (1968).

Other (3)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

C. P. Thurgood, 'A critical evaluation of the discrete ordinates method using HEART and Tn quadrature,' Ph.D. thesis, (Queen's University, Kingston, Ontario, Canada, 1992).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

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Figures (10)

Fig. 1
Fig. 1

Physical model.

Fig. 2
Fig. 2

Time-resolved backscattered flux for LP input LP and for CP input calculated for a particle size a = 0.5 μ m and for optical thickness is τ 0 = 10 (X-Pol: cross polarized, Co-pol: co-polarized).

Fig. 3
Fig. 3

DOP in the backward direction for two different particle sizes as indicated.

Fig. 4
Fig. 4

DOP in the backward direction for an incident CP light and different particle sizes.

Fig. 5
Fig. 5

DOP in the backward direction for CP incident light and for different angles of incidence as indicated ( a = 0.5 μ m ) . The backscattered radiation is collected in nearly normal direction. Also indicated is the handedness of scattered fluxes at specific times.

Fig. 6
Fig. 6

Time-resolved backscattered flux for a CP incident wave at an angle of incidence μ = 0.6 and for a particle size a = 0.5 μ m and optical thickness τ 0 = 10 (X-Pol: cross polarized, Co-pol: co-polarized).

Fig. 7
Fig. 7

Sign of V with respect to the scattering angle, computed using the Mie theory.

Fig. 8
Fig. 8

DOP in the backward direction for an incident LP light and different angle of incidence ( a = 0.5 μ m ) . Light collected in nearly normal direction.

Fig. 9
Fig. 9

DOP in the backward direction for an incident LP light and different angles of incidence ( a = 0.5 μ m ) . The light is collected in the opposite direction of incidence. The inset shows the variation of the DOP with the angle of incidence of steady-state waves.

Fig. 10
Fig. 10

DOP in the backward direction for an incident CP light and different angles of incidence ( a = 0.5 μ m ) . The light is collected in the opposite direction of incidence.

Equations (19)

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

1 c I ( x , Ω , t ) t + μ I ( x , Ω , t ) x + σ I ( x , Ω , t ) = 1 4 π 4 π P ( Ω , Ω ) I ( x , Ω , t ) d Ω
I = [ E 1 E 1 * E 2 E 2 * 2 Re E 1 E 2 * 2 Im E 1 E 2 * ] T = [ I I U V ] T ,
P = 1 σ s [ f 11 2 f 12 2 Re ( f 11 f 12 * ) Im ( f 11 f 12 * ) f 21 2 f 22 2 Re ( f 21 f 22 * ) Im ( f 21 f 22 * ) 2 Re ( f 11 f 21 * ) 2 Re ( f 12 f 22 * ) Re ( f 11 f 22 * + f 12 f 21 * ) Im ( f 11 f 22 * f 12 f 21 * ) 2 Im ( f 11 f 21 * ) 2 Im ( f 12 f 22 * ) Im ( f 11 f 22 * + f 12 f 21 * ) Re ( f 11 f 22 * f 12 f 21 * ) ] ,
I ( x , Ω , t ) = I c ( x , Ω , t ) + I d ( x , Ω , t ) ,
1 c I m ( x , t ) t + μ I m ( x , t ) x = σ I m ( x , Ω m , t ) + σ m = 1 M w m P m m I m ( x , t ) + S m i n c ( x , t ) ,
m = 1 , 2 , , M ,
S m i n c ( x , t ) = 1 4 π PI i n c e σ x F ( x , t , t p ) = [ S i n c S i n c S U i n c S V i n c ] ,
F ( x , t , t p ) = exp [ σ x ( t x c 2 ) T 2 ] δ ( μ μ 0 ) δ ( φ φ 0 )
1 c I i n + 1 I i n Δ t + μ I i + 1 2 n I i 1 2 n Δ x = σ I i n + 1 + σ m = 1 M w m P 1 1 , m , m I i , m n + 1 + S i n + 1 .
I i , m n + 1 = 1 ( 1 + c β Δ t ) [ I i , m n μ m c Δ t Δ x ( I i + 1 2 , m n I i 1 2 , m n ) + c Δ t ( β m = 1 M w m P 11 , m m I i , m n + 1 + S I i , m n + 1 ) ] ,
I i , m n + 1 = 1 ( 1 + c β Δ t ) [ I i , m n μ m c Δ t Δ x ( I i + 1 2 , m n I i 1 2 , m n ) + c Δ t ( β m = 1 M w m P 22 , m m I i , m n + 1 + S I i , m n + 1 ) ] ,
U i , m n + 1 = 1 ( 1 + c β Δ t ) [ U i , m n μ m c Δ t Δ x ( U i + 1 2 , m n U i 1 2 , m n ) + c Δ t ( β m = 1 M w m P 33 , m m U i , m n + 1 + S U i , m n + 1 ) ] ,
V i , m n + 1 = 1 ( 1 + c β Δ t ) [ V i , m n μ m c Δ t Δ x ( V i + 1 2 , m n V i 1 2 , m n ) + c Δ t ( β m = 1 M w m P 44 , m m V i , m n + 1 + S V i , m n + 1 ) ] ,
m = M , . . . . , 1 , 1 , , M ,
S I i , m n + 1 = c Δ t β [ m = 1 M w m ( P 1 2 , m m I i , m n + 1 + P 1 3 , m m U i , m n + 1 + P 1 4 , m m V i , m n + 1 ) + S m , i n c ] ,
S I i , m n + 1 = c Δ t β [ m = 1 M w m ( P 2 1 , m m I i , m n + 1 + P 2 3 , m m U i , m n + 1 + P 2 4 , m m V i , m n + 1 ) + S m , i n c ] ,
S U i , m n + 1 = c Δ t β [ m = 1 M w m ( P 3 1 , m m I i , m n + 1 + P 3 2 , m m I i , m n + 1 + P 3 4 , m m V i , m n + 1 ) + S m , U i n c ] ,
S V i , m n + 1 = c Δ t β [ m = 1 M w m ( P 4 1 , m m I i , m n + 1 + P 4 2 , m m I i , m n + 1 + P 4 3 , m m U i , m n + 1 ) + S m , V i n c ] ,
D O P = [ ( I I ) 2 + U 2 + V 2 ] 1 2 I + I .

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