Abstract

We present numerical results on the change in polarization state of light pulses transmitted through thick turbid media. These results were obtained with a modified version of a previous Monte Carlo code that takes into account depolarization introduced by multiple scattering. The results have shown that for scattered received power pulse shape, polarization and total received power mainly depend on the transport cross section, σd, of the medium. The effect of the angular field of view of the receiver or of the distance between the diffusing medium and the receiver is shown, whereas the effect of the lateral displacement of the receiver elements proves to be of minor importance. An example of measurements showed a good agreement with numerical results, indicating the adequacy of our numerical code.

© 1993 Optical Society of America

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  1. E. A. Bucher, R. M. Lerner, “Experiments on light pulse communication and propagation through atmospheric clouds,” Appl. Opt. 12, 2401–2414 (1973).
    [CrossRef] [PubMed]
  2. G. C. Mooradian, M. Geller, “Temporal and angular spreading of blue-green pulses in clouds,” Appl. Opt. 21, 1572–1577(1982).
    [CrossRef] [PubMed]
  3. R. Lerner, J. D. Summers, “Monte Carlo description of time- and space-resolved multiple forward scatter in natural water,” Appl. Opt. 21, 861–869 (1982).
    [CrossRef] [PubMed]
  4. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  5. B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. Jacques, “The potential of time-resolved reflectance measurements for the noninvasive determination of tissue optical properties,” in Thermal and Optical Interactions With Biological and Related Composite Material, M. J. Berry, G. M. Harpole, eds., Proc. Soc. Photo-Opt. Instrum.1064, 97–106 (1989).
  6. D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
    [CrossRef] [PubMed]
  7. S. T. Hong, I. Sreenivasiah, A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag., AP-25, 822–828 (1977).
    [CrossRef]
  8. S. Ito, K. Furutsu, “Theory of light pulse propagation through thick clouds,” J. Opt. Soc. Am. 70, 366–374 (1980).
    [CrossRef]
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    [CrossRef] [PubMed]
  10. A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
    [CrossRef]
  11. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  12. E. A. Bucher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391–2400 (1973).
    [CrossRef] [PubMed]
  13. J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
    [CrossRef] [PubMed]
  14. G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point-source case,” Appl. Opt. 30, 2031–2041 (1991).
    [CrossRef] [PubMed]
  15. Y. Kuga, A. Ishimaru, A. P. Bruckner, “Experiments on picosecond pulse propagation in a diffuse medium,” J. Opt. Soc. Am. 73, 1812–1815 (1983).
    [CrossRef]
  16. R. A. Elliott, “Multiple scattering of optical pulses in scale model clouds,” Appl. Opt. 22, 2670–2681 (1983).
    [CrossRef] [PubMed]
  17. K. M. Yoo, R. R. Alfano, “Time-resolved coherent and incoherent components of forward light scattering in random media,” Opt. Lett. 15, 320–322 (1990).
    [CrossRef] [PubMed]
  18. G. Zaccanti, P. Bruscaglioni, A. Ismaelli, L. Carraresi, M. Gurioli, Q. Wei, “Transmission of a pulsed light beam through thick turbid media: experimental results,” Appl. Opt. 31, 2141–2147 (1992).
    [CrossRef] [PubMed]
  19. P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nieto Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1990), pp. 53–71.
  20. K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis multiple scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
    [CrossRef]
  21. L. R. Poole, D. D. Venable, J. W. Campbell, “Semianalytic Monte Carlo radiative transfer model for onceanografic lidar system,” Appl. Opt. 20, 3653–3656 (1981).
    [CrossRef] [PubMed]
  22. A. Ishimaru, Propagation and Scattering of Radiation in Random Media (Academic, New York, 1982).
  23. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  24. G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer law in the transmittance of a light beam through diffusing media: experimental results,” J. Mod. Opt. 35, 229–242 (1988).
    [CrossRef]
  25. G. Zaccanti, P. Bruscaglioni, M. Dami, “Simple inexpensive method of measuring the temporal spreading of a light pulse propagating in a turbid medium,” Appl. Opt. 29, 3938–3944 (1991).
    [CrossRef]

1992 (1)

1991 (2)

1990 (2)

J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
[CrossRef] [PubMed]

K. M. Yoo, R. R. Alfano, “Time-resolved coherent and incoherent components of forward light scattering in random media,” Opt. Lett. 15, 320–322 (1990).
[CrossRef] [PubMed]

1989 (1)

1988 (3)

H. Leelavathi, J. P. Pichamuthu, “Propagation of optical pulses through dense scattering media,” Appl. Opt. 27, 2461–2468 (1988).
[CrossRef] [PubMed]

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer law in the transmittance of a light beam through diffusing media: experimental results,” J. Mod. Opt. 35, 229–242 (1988).
[CrossRef]

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

1987 (1)

1983 (2)

1982 (2)

1981 (1)

1980 (1)

1978 (1)

1977 (1)

S. T. Hong, I. Sreenivasiah, A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag., AP-25, 822–828 (1977).
[CrossRef]

1976 (1)

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis multiple scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

1973 (2)

Alfano, R. R.

Arridge, S.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Bohren, C. F.

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

Bonner, R. F.

Bruckner, A. P.

Bruscaglioni, P.

G. Zaccanti, P. Bruscaglioni, A. Ismaelli, L. Carraresi, M. Gurioli, Q. Wei, “Transmission of a pulsed light beam through thick turbid media: experimental results,” Appl. Opt. 31, 2141–2147 (1992).
[CrossRef] [PubMed]

G. Zaccanti, P. Bruscaglioni, M. Dami, “Simple inexpensive method of measuring the temporal spreading of a light pulse propagating in a turbid medium,” Appl. Opt. 29, 3938–3944 (1991).
[CrossRef]

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer law in the transmittance of a light beam through diffusing media: experimental results,” J. Mod. Opt. 35, 229–242 (1988).
[CrossRef]

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nieto Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1990), pp. 53–71.

Bucher, E. A.

Campbell, J. W.

Carraresi, L.

Chance, B.

Cope, M.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Dami, M.

Delpy, D. T.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Elliott, R. A.

Furutsu, K.

Geller, M.

Gurioli, M.

Havlin, S.

Hebden, J. C.

J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
[CrossRef] [PubMed]

Hefetz, Y.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. Jacques, “The potential of time-resolved reflectance measurements for the noninvasive determination of tissue optical properties,” in Thermal and Optical Interactions With Biological and Related Composite Material, M. J. Berry, G. M. Harpole, eds., Proc. Soc. Photo-Opt. Instrum.1064, 97–106 (1989).

Hong, S. T.

S. T. Hong, I. Sreenivasiah, A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag., AP-25, 822–828 (1977).
[CrossRef]

Huffman, D. R.

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

Ishimaru, A.

Y. Kuga, A. Ishimaru, A. P. Bruckner, “Experiments on picosecond pulse propagation in a diffuse medium,” J. Opt. Soc. Am. 73, 1812–1815 (1983).
[CrossRef]

A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
[CrossRef]

S. T. Hong, I. Sreenivasiah, A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag., AP-25, 822–828 (1977).
[CrossRef]

A. Ishimaru, Propagation and Scattering of Radiation in Random Media (Academic, New York, 1982).

Ismaelli, A.

Ito, S.

Jacques, S.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. Jacques, “The potential of time-resolved reflectance measurements for the noninvasive determination of tissue optical properties,” in Thermal and Optical Interactions With Biological and Related Composite Material, M. J. Berry, G. M. Harpole, eds., Proc. Soc. Photo-Opt. Instrum.1064, 97–106 (1989).

Kruger, R. A.

J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
[CrossRef] [PubMed]

Kuga, Y.

Kunkel, K. E.

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis multiple scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

Leelavathi, H.

Lerner, R.

Lerner, R. M.

Madsen, S.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. Jacques, “The potential of time-resolved reflectance measurements for the noninvasive determination of tissue optical properties,” in Thermal and Optical Interactions With Biological and Related Composite Material, M. J. Berry, G. M. Harpole, eds., Proc. Soc. Photo-Opt. Instrum.1064, 97–106 (1989).

Mooradian, G. C.

Nossal, R.

Park, Y.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. Jacques, “The potential of time-resolved reflectance measurements for the noninvasive determination of tissue optical properties,” in Thermal and Optical Interactions With Biological and Related Composite Material, M. J. Berry, G. M. Harpole, eds., Proc. Soc. Photo-Opt. Instrum.1064, 97–106 (1989).

Patterson, M.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. Jacques, “The potential of time-resolved reflectance measurements for the noninvasive determination of tissue optical properties,” in Thermal and Optical Interactions With Biological and Related Composite Material, M. J. Berry, G. M. Harpole, eds., Proc. Soc. Photo-Opt. Instrum.1064, 97–106 (1989).

Patterson, M. S.

Pichamuthu, J. P.

Poole, L. R.

Sreenivasiah, I.

S. T. Hong, I. Sreenivasiah, A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag., AP-25, 822–828 (1977).
[CrossRef]

Summers, J. D.

van der Zee, P.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Venable, D. D.

Wei, Q.

Weinman, J. A.

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis multiple scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

Weiss, G. H.

Wilson, B.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. Jacques, “The potential of time-resolved reflectance measurements for the noninvasive determination of tissue optical properties,” in Thermal and Optical Interactions With Biological and Related Composite Material, M. J. Berry, G. M. Harpole, eds., Proc. Soc. Photo-Opt. Instrum.1064, 97–106 (1989).

Wilson, B. C.

Wray, S.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyatt, J.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Yoo, K. M.

Zaccanti, G.

Appl. Opt. (11)

E. A. Bucher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391–2400 (1973).
[CrossRef] [PubMed]

E. A. Bucher, R. M. Lerner, “Experiments on light pulse communication and propagation through atmospheric clouds,” Appl. Opt. 12, 2401–2414 (1973).
[CrossRef] [PubMed]

L. R. Poole, D. D. Venable, J. W. Campbell, “Semianalytic Monte Carlo radiative transfer model for onceanografic lidar system,” Appl. Opt. 20, 3653–3656 (1981).
[CrossRef] [PubMed]

R. Lerner, J. D. Summers, “Monte Carlo description of time- and space-resolved multiple forward scatter in natural water,” Appl. Opt. 21, 861–869 (1982).
[CrossRef] [PubMed]

G. C. Mooradian, M. Geller, “Temporal and angular spreading of blue-green pulses in clouds,” Appl. Opt. 21, 1572–1577(1982).
[CrossRef] [PubMed]

R. A. Elliott, “Multiple scattering of optical pulses in scale model clouds,” Appl. Opt. 22, 2670–2681 (1983).
[CrossRef] [PubMed]

H. Leelavathi, J. P. Pichamuthu, “Propagation of optical pulses through dense scattering media,” Appl. Opt. 27, 2461–2468 (1988).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

G. Zaccanti, P. Bruscaglioni, M. Dami, “Simple inexpensive method of measuring the temporal spreading of a light pulse propagating in a turbid medium,” Appl. Opt. 29, 3938–3944 (1991).
[CrossRef]

G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point-source case,” Appl. Opt. 30, 2031–2041 (1991).
[CrossRef] [PubMed]

G. Zaccanti, P. Bruscaglioni, A. Ismaelli, L. Carraresi, M. Gurioli, Q. Wei, “Transmission of a pulsed light beam through thick turbid media: experimental results,” Appl. Opt. 31, 2141–2147 (1992).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (1)

S. T. Hong, I. Sreenivasiah, A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag., AP-25, 822–828 (1977).
[CrossRef]

J. Atmos. Sci. (1)

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis multiple scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

J. Mod. Opt. (1)

G. Zaccanti, P. Bruscaglioni, “Deviation from the Lambert-Beer law in the transmittance of a light beam through diffusing media: experimental results,” J. Mod. Opt. 35, 229–242 (1988).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Med. Phys. (1)

J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Med. Biol. (1)

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Other (4)

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. Jacques, “The potential of time-resolved reflectance measurements for the noninvasive determination of tissue optical properties,” in Thermal and Optical Interactions With Biological and Related Composite Material, M. J. Berry, G. M. Harpole, eds., Proc. Soc. Photo-Opt. Instrum.1064, 97–106 (1989).

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nieto Vesperinas, J. C. Dainty, eds. (Elsevier, New York, 1990), pp. 53–71.

A. Ishimaru, Propagation and Scattering of Radiation in Random Media (Academic, New York, 1982).

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

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Figures (12)

Fig. 1
Fig. 1

Geometrical schemes for Monte Carlo calculations: (a) example of photon path used to describe the Semi Monte Carlo procedure, (b) Geometrical situation considered for Monte Carlo simulations.

Fig. 2
Fig. 2

Scalar scattering functions used for Monte Carlo simulations. The curves pertain to polystyrene spheres suspended in water with ϕ = 0.155, 0.300, and 0.993 μm, respectively. λ = 0.633 μm.

Fig. 3
Fig. 3

Examples of impulse response: total received power g(x), shown by the continuous curve, versus x[x = (tt0)/t0] together with the parallel polarized g(x), shown by dotted curve, and cross polarized components g(x), shown by dashed curve. (a) ϕ = 0.3 and (b) ϕ = 0.993 μm. L = 4cm, d = 15 cm; D = 20; r = 0; α = 15°; λ = 0.633 μm.

Fig. 4
Fig. 4

Comparison between the normalized pulse shape g ¯ (x) pertaining to different sizes of spheres (total power). The area of the curves is normalized to 1. L = 4 cm, d = 15 cm; D = 20 cm; r = 0; α = 15°; λ = 0.633 μm.

Fig. 5
Fig. 5

Comparison between the ratios g(x)/g(x) pertaining to different sizes of spheres. Continuous, dashed, and dotted curves pertain to ϕ = 0.155, 0.3, and 0.993 μm. From upper to lower curves τd = 10, 4, and 2, respectively. L = 4 cm, d = 15 cm; D = 20 cm; r = 0; α = 15°; λ = 0.633 μm.

Fig. 6
Fig. 6

Comparison between the normalized pulse shape g ¯ (x) pertaining to different values of α. Curves a, b, and c pertain to α = 1°, 7°, and 15°, respectively. L = 4 cm, d = 15 cm, D = 20 cm, r = 0, ϕ = 0.3 μm, λ = 0.633 μm.

Fig. 7
Fig. 7

Comparison between g(x), shown by the continuous curves, and g(x), shown by the dashed curves, pertaining to different values of α. For curves a, b, c, and d, τd = 2; for curves a′, b′, c′, and d′, τd = 6. From upper to lower curves, α = 15°, 7°, 3.5°, and 1°, respectively. L = 4 cm, d = 15 cm, D = 20 cm, r = 0, ϕ = 0.3 μm, λ = 0.633 μm.

Fig. 8
Fig. 8

Comparison between the pulse shapes pertaining to different values of the scattering cell–receiver distance d. For curves a, b, and c, d = 2 cm, α = 7.5° and D = 8 cm; d = 15 cm, α = 1°, and D = 20 cm; and d = 30 cm, α = 0.5° and D = 60 cm, respectively. τd = 6, L = 4 cm, r = 0, ϕ = 0.3 μm, λ = 0.633 μm.

Fig. 9
Fig. 9

Comparison between the impulse response pertaining to different distances r of the receiver element from the beam axis. L = 4 cm, d = 2 cm, D = 8 cm, α = 7°, ϕ = 0.3 μm, λ = 0.633 μm.

Fig. 10
Fig. 10

Ratio between the scattered power received by a coaxial receiver (radius 0.5 cm, d = 15 cm, D = 20 cm) and the emitted power (continuous-beam case) is plotted versus τd for different values of α and different types of spheres. For any value of α, from upper to lower curves, ϕ = 0.155, 0.300, and 0.993, μm respectively. The curve pertaining to α = 1° and ϕ = 0.155 μm is not shown. λ = 0.633 μm.

Fig. 11
Fig. 11

Ratio between the cross-polarized and the parallel polarized components of the scattered power received on a coaxial receive r with radius 0.3 cm is plotted versus τd for different types of sphere, From upper to lower curves α = 15°, 7° and 1°, respectively. L = 4 cm, d = 15 cm, D = 20 cm, r = 0, and λ = 0.633 μm.

Fig. 12
Fig. 12

Comparison between the ratio Ps/Ps obtained by Monte Carlo simulations and by laboratory measurements. Both numerical and experimental results pertain to the scattered power received by a coaxial receiver with radius 1.25 cm. L = 10.8 cm, D = 14 cm, d = 17 cm, α = 6.9°, and λ = 0.633 μm.

Equations (9)

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

I = | I 1 I 2 U V | .
p ( θ ) = [ P 1 ( θ ) + P 2 ( θ ) ] / 2
M ( θ ) = [ P 1 ( θ ) 0 0 0 0 P 2 ( θ ) 0 0 0 0 P 3 ( θ ) P 4 ( θ ) 0 0 - P 4 ( θ ) P 3 ( θ ) ] ,
w ( φ ) = 2 I 1 R P 1 ( θ ) + I 2 R P 2 ( θ ) ( I 1 R + I 2 R ) [ P 1 ( θ ) + P 2 ( θ ) ] ,
I k i d = exp ( - ρ k i σ s ) Δ A ρ k i 2 F ( φ k i ) M ( θ k i ) F ( φ k i ) I k i cos β k i if β k i α I k i d = 0 if β k i > α ,
x = t - t o t o = l - L L ,
g ( x n x x n + 1 ) = 1 Δ A N ( x n + 1 - x n ) i = 1 N k I k i d ( x n x k i < x n + 1 ) ,
g ( x n x x n + 1 ) = 1 Δ A N ( x n + 1 - x n ) i = 1 N k I k i d ( x n x k i < x n + 1 ) ,
g ( x ) = g ( x ) + g ( x ) .

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