Abstract

We present a study of the image blurring and depolarization resulting from the transmission of a narrow beam of light through a continuous random medium. We investigate the dependence of image quality degradation and of depolarization on optical thickness, correlation length of the inhomogeneities, and incident polarization state. This is done numerically with a Monte Carlo method based on a transport equation that takes into account polarization of light. We compare our results with those for transport in media with discrete spherical scatterers. We show that depolarization effects are different in these two models of biological tissue.

© 2001 Optical Society of America

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2000 (2)

S. L. Jacques, J. R. Roman, K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26, 119–129 (2000).
[CrossRef] [PubMed]

G. Bal, G. Papanicolaou, L. Ryzhik, “Probabilistic theory of transport processes with polarization,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 60, 1639–1666 (2000).
[CrossRef]

1999 (2)

1998 (3)

J. Przeslawski, K. Michielsen, H. DeRaedt, N. Garcia, “Computer simulation of time-resolved optical imaging of objects hidden in turbid media,” Phys. Rep. 304, 90–144 (1998).

O. Dorn, “A transport–backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[CrossRef]

J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. 37, 3586–3593 (1998).
[CrossRef]

1997 (1)

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

1996 (4)

1995 (1)

1994 (2)

D. Bicout, C. Brosseau, A. S. Martinez, J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[CrossRef]

B. Beauvoit, T. Kitai, B. Chance, “Contribution to the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (5)

J. M. Schmitt, A. H. Gandjbakche, 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]

K. M. Yoo, B. B. Das, R. R. Alfano, “Imaging of translucent object hidden in highly scattering medium from the early portion of the diffuse component of a transmitted ultrafast laser pulse,” Opt. Lett. 17, 958–960 (1992).
[CrossRef] [PubMed]

L. R. Bissonnette, “Imaging through fog and rain,” Opt. Eng. 31, 1045–1052 (1992).
[CrossRef]

C. Werner, J. Streicher, H. Herrmann, H. G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

R. F. Tusting, D. L. Davis, “Laser systems and structured illumination for quantitative undersea imaging,” Mar. Technol. Soc. J. 26, 5–12 (1992).

1991 (2)

1989 (2)

F. C. Mackintosh, S. John, “Diffusing-wave spectros-copy and multiple-scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

F. C. Mackintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

1988 (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]

1985 (1)

1983 (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824 (1983).
[CrossRef] [PubMed]

1981 (1)

1980 (1)

H. Iida, Y. Seki, “Simple method of eliminating infinite variance in point detector problem of Monte Carlo calculation,” J. Nucl. Sci. Technol. 17, 315–317 (1980).
[CrossRef]

1978 (2)

1977 (1)

1971 (1)

H. A. Steinberg, M. H. Kalos, “Bounded estimators for flux at a point in Monte Carlo,” Nucl. Sci. Eng. 44, 406–412 (1971).

1968 (2)

1963 (1)

M. H. Kalos, “On the estimation of flux at a point by Monte Carlo,” Nucl. Sci. Eng. 16, 111–117 (1963).

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824 (1983).
[CrossRef] [PubMed]

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]

Arridge, S. R.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Aruga, T.

Asselin, D.

Avrillier, S.

Bal, G.

G. Bal, G. Papanicolaou, L. Ryzhik, “Probabilistic theory of transport processes with polarization,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 60, 1639–1666 (2000).
[CrossRef]

G. Bal, M. Moscoso, “Theoretical and numerical analysis of polarization for time dependent radiative transfer equations,” J. Quant. Spectrosc. Radiat. Transf. (to be published).

G. Bal, M. Moscoso, “Polarization effects of seismic waves on the basis of radiative transport theory,” Geophys. J. Int. (to be published).

Beaudry, P.

Beauvoit, B.

B. Beauvoit, T. Kitai, B. Chance, “Contribution to the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

Bicout, D.

D. Bicout, C. Brosseau, A. S. Martinez, J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[CrossRef]

Bissonnette, L. R.

L. R. Bissonnette, “Imaging through fog and rain,” Opt. Eng. 31, 1045–1052 (1992).
[CrossRef]

Bonner, R. F.

Brosseau, C.

D. Bicout, C. Brosseau, A. S. Martinez, J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[CrossRef]

Bruscaglioni, P.

Carswell, A. I.

Chance, B.

B. Beauvoit, T. Kitai, B. Chance, “Contribution to the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, Oxford, UK, 1960).

Chatigny, S.

Cheong, W. F.

W. F. Cheong, “Summary of optical properties,” in Optical–Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 275–303.

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]

Dahn, H. G.

C. Werner, J. Streicher, H. Herrmann, H. G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

Das, B. B.

Davis, D. L.

R. F. Tusting, D. L. Davis, “Laser systems and structured illumination for quantitative undersea imaging,” Mar. Technol. Soc. J. 26, 5–12 (1992).

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]

Demos, S. G.

DeRaedt, H.

J. Przeslawski, K. Michielsen, H. DeRaedt, N. Garcia, “Computer simulation of time-resolved optical imaging of objects hidden in turbid media,” Phys. Rep. 304, 90–144 (1998).

Dorn, O.

O. Dorn, “A transport–backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[CrossRef]

Egan, W. G.

Eick, A. A.

Engheta, N.

Freyer, J. P.

Gandjbakche, A. H.

Garcia, N.

J. Przeslawski, K. Michielsen, H. DeRaedt, N. Garcia, “Computer simulation of time-resolved optical imaging of objects hidden in turbid media,” Phys. Rep. 304, 90–144 (1998).

Hebden, J. C.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine. II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Herrmann, H.

C. Werner, J. Streicher, H. Herrmann, H. G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

Hielscher, A. H.

Igarashi, T.

Iida, H.

H. Iida, Y. Seki, “Simple method of eliminating infinite variance in point detector problem of Monte Carlo calculation,” J. Nucl. Sci. Technol. 17, 315–317 (1980).
[CrossRef]

Ishimaru, A.

Jacques, S. L.

S. L. Jacques, J. R. Roman, K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26, 119–129 (2000).
[CrossRef] [PubMed]

John, S.

F. C. Mackintosh, S. John, “Diffusing-wave spectros-copy and multiple-scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

Johnson, T. M.

Johnson, W. R.

Kalos, M. H.

H. A. Steinberg, M. H. Kalos, “Bounded estimators for flux at a point in Monte Carlo,” Nucl. Sci. Eng. 44, 406–412 (1971).

M. H. Kalos, “On the estimation of flux at a point by Monte Carlo,” Nucl. Sci. Eng. 16, 111–117 (1963).

M. H. Kalos, P. A. Whitlock, Monte Carlo Methods (Wiley, New York, 1986).

Kattawar, G. W.

Keller, J. B.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Kitai, T.

B. Beauvoit, T. Kitai, B. Chance, “Contribution to the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

Kuga, Y.

Kumar, G.

Lee, K.

S. L. Jacques, J. R. Roman, K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26, 119–129 (2000).
[CrossRef] [PubMed]

Lewis, E. E.

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (Wiley, New York, 1984).

Mackintosh, F. C.

F. C. Mackintosh, S. John, “Diffusing-wave spectros-copy and multiple-scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

F. C. Mackintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Maitland, D. J.

Martinez, A. S.

D. Bicout, C. Brosseau, A. S. Martinez, J. M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E 49, 1767–1770 (1994).
[CrossRef]

McCartney, E. J.

E. J. McCartney, Optics of the Atmosphere (Wiley, New York, 1976).

Meier, R. R.

Mertens, L. E.

Michielsen, K.

J. Przeslawski, K. Michielsen, H. DeRaedt, N. Garcia, “Computer simulation of time-resolved optical imaging of objects hidden in turbid media,” Phys. Rep. 304, 90–144 (1998).

Miller, W. F.

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (Wiley, New York, 1984).

Morin, M.

Moscoso, M.

G. Bal, M. Moscoso, “Polarization effects of seismic waves on the basis of radiative transport theory,” Geophys. J. Int. (to be published).

G. Bal, M. Moscoso, “Theoretical and numerical analysis of polarization for time dependent radiative transfer equations,” J. Quant. Spectrosc. Radiat. Transf. (to be published).

Mourant, J. R.

Painchaud, Y.

Papanicolaou, G.

G. Bal, G. Papanicolaou, L. Ryzhik, “Probabilistic theory of transport processes with polarization,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 60, 1639–1666 (2000).
[CrossRef]

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Pine, D. J.

F. C. Mackintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Plass, G. N.

Przeslawski, J.

J. Przeslawski, K. Michielsen, H. DeRaedt, N. Garcia, “Computer simulation of time-resolved optical imaging of objects hidden in turbid media,” Phys. Rep. 304, 90–144 (1998).

Pungh, E. N.

Replogle, F. S.

Roman, J. R.

S. L. Jacques, J. R. Roman, K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26, 119–129 (2000).
[CrossRef] [PubMed]

Rowe, M. P.

Ryan, J. S.

Ryzhik, L.

G. Bal, G. Papanicolaou, L. Ryzhik, “Probabilistic theory of transport processes with polarization,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 60, 1639–1666 (2000).
[CrossRef]

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Sankaran, V.

Schmitt, J. M.

Schönenberger, K.

Seki, Y.

H. Iida, Y. Seki, “Simple method of eliminating infinite variance in point detector problem of Monte Carlo calculation,” J. Nucl. Sci. Technol. 17, 315–317 (1980).
[CrossRef]

Shen, D.

Steinberg, H. A.

H. A. Steinberg, M. H. Kalos, “Bounded estimators for flux at a point in Monte Carlo,” Nucl. Sci. Eng. 44, 406–412 (1971).

Streicher, J.

C. Werner, J. Streicher, H. Herrmann, H. G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

Tinet, E.

Tualle, M.

Tusting, R. F.

R. F. Tusting, D. L. Davis, “Laser systems and structured illumination for quantitative undersea imaging,” Mar. Technol. Soc. J. 26, 5–12 (1992).

Tyo, J. S.

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]

Walsh, J. T.

Wei, Q.

Weitz, D. A.

F. C. Mackintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Werner, C.

C. Werner, J. Streicher, H. Herrmann, H. G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

Whitehead, V. S.

Whitlock, P. A.

M. H. Kalos, P. A. Whitlock, Monte Carlo Methods (Wiley, New York, 1986).

Wilson, B. C.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824 (1983).
[CrossRef] [PubMed]

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.

Zhu, J. X.

F. C. Mackintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Appl. Opt. (11)

V. Sankaran, K. Schönenberger, J. T. Walsh, D. J. Maitland, “Polarization discrimination of coherently propagating light in turbid media,” Appl. Opt. 38, 4252–4261 (1999).
[CrossRef]

J. M. Schmitt, A. H. Gandjbakche, 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]

T. Aruga, T. Igarashi, “Narrow beam light transfer in small particles: image blurring and depolarization,” Appl. Opt. 20, 2698–2705 (1981).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

(a) Polarization ellipse for a transverse wave and (b) coordinate system for the polarization parameters of the incident and scattered waves in single scattering. The scattering plane is defined by the incident and scattered directions, and l and r stand for directions parallel and perpendicular to the plane of reference, respectively, in planes transverse to the incident and scattered waves.

Fig. 2
Fig. 2

Setup used in the numerical simulations. (a) The sample is an 8-mm cube, so L=Z=8 mm. It contains a 2-mm×8-mm opaque absorbing plate. The distance between two consecutive incident beam positions is 0.5 mm. The detectors are placed at a distance d=1 mm from the exit surface. The width of the incident laser beam, measured at 1/e of the maximum, is 0.5 mm. (b) Geometry used for the numerical computation of the point-spread functions. The incident beam, impinging on (x, y)=(0, 0), has zero width. The detector positions are 0.2 mm apart.

Fig. 3
Fig. 3

Numerical results for scan profiles with (a) τ=4 and (b) τ=9. The solid curves are the computed total intensities, and the dashed curves are the computed polarization differences. All curves are normalized to their maximum values. The semiaperture angle of the receiver is 16°, and the width of the incident beam is 0.5 mm. The thick line at the top of the plots represents the position of the absorbing strip.

Fig. 4
Fig. 4

Point-spread functions in the xz plane. The normally incident beam is 100% linearly polarized parallel to this plane. The solid curve, the dashed curve, and the dotted–dashed curve are the total intensity, the polarized difference intensity, and the polarized difference intensity divided by the total intensity, respectively. The spatial profiles are calculated at the bottom of the sample [see Fig. 2(b)]. The optical thickness τ=9, the semiaperture angle of the receiver is 16°, and kl1. All curves are normalized to their maximum values at x=0. The horizontal dotted line is the value 1/e.

Fig. 5
Fig. 5

Beam half-width at 1/e of the maximum (in millimeters) of the point-spread functions. Solid and open symbols are for circularly [(1, 0, 0, 1)] and linearly [(1, 1, 0, 0)] polarized incident light, respectively. Circles, diamonds, and squares are for total intensity, polarized difference intensity, and polarized difference intensity divided by the total intensity, respectively. kl=1. The semiaperture angle of the receiver is (a) 16° and (b) 90°.

Fig. 6
Fig. 6

Maximum degree of polarization for media with (a) kl=0, (b) kl=1, and (c) kl=2. Solid and open symbols are for circularly and linearly polarized incident light, respectively. Squares and circles are for semiapertures of the receiver equal to 16° and 90°, respectively. The relative standard error is less than 0.01.

Fig. 7
Fig. 7

Coordinate system for the Stokes vector. The shaded plane is the plane of reference, the meridian plane.

Tables (1)

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Table 1 Typical Mean Free Paths of Visible Light in Different Media

Equations (66)

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I=IQUV,
I=ElEl*+Er Er*),
Q=ElEl*-Er Er*,
U=ElEr*+Er El*,
V=iElEr*-Er El*.
I(s)=F I(i).
R(r)=δ(r)δ(r+r).
Rˆ(k)=1(2π)3  exp(ik·r)R(r)dr.
F(Θ)=π2k4Rˆ2k sin Θ2S(Θ).
S(Θ)
=12 1+cos2 Θcos2 Θ-100cos2 Θ-11+cos2 Θ00002 cos Θ00002 cos Θ.
s(k)=π22k4-11Rˆ(k2-2 cos Θ)×(1+cos2 Θ)d(cos Θ).
M(Θ)=M11M1200M21M220000M33M3400M43M44.
R(r)=ε2 exp(-r/l).
Rˆ(k)=ε2l3π2(1+k2l2)2.
g(k)=π2k42s(k) -11(cos Θ)Rˆ(k2-2 cos Θ)×(1+cos2 Θ)d(cos Θ).
lcoll=-(ln ξ)/s,
r1=r0+kˆ0 lcoll,
P(kˆ1, kˆ0, Q0, U0)=πk42sRˆT112+T212+T122+T2222+T112+T212-T122-T2222Q0+(T11T12+T21T22)U0.
P(kˆ1, kˆ0, Q0, U0)dkˆ1=1,
I(1)=1sP(kˆ1, kˆ0, Q0, U0)F¯(kˆ1, kˆ0)I(0).
I=dI,
Il=dI [cos2(ϕd+χd)cos2 θd+sin2 θd],
Ir=dI [sin2(ϕd+χd)cos2 θd+sin2 θd].
I¯=1N n=1NI(n),I¯l=1N n=1NIl(n),
I¯r=1N n=1NIr(n),V¯=1N n=1NV(n).
P(Θ, P)dΩ exp-0Rs(l)d l,
P(Θ, P) cos θd exp-0Rs(l)dlR2 dA.
FIi(n)=Wi(n)P(Θ, P) cos θd exp(-sR)R2.
F¯I=1N n=1Ni=1SnFIi(n),F¯l=1N n=1Ni=1SnFli(n),
F¯r=1N n=1Ni=1SnFri(n),F¯V=1N n=1Ni=1SnFVi(n),
E(t, x)=e1E10 cos(ωt-kx+α)+e2E20 sin(ωt-kx+α).
E(t, x)=E exp[-i(ωt-kx+α)]+E* exp[i(ωt-kx+α)],
E=E0(e1 cos β+ie2 sin β).
E(t, x)=Elel+Erer,
El=E0(cos χ cos β-i sin χ sin β),
Er=E0(sin χ cos β+i cos χ sin β).
W(t, x, k)=|El|2ElEr*El*Er|Er|2.
I=ElEl*+ErEr*=Il+Ir,
Q=ElEl*-ErEr*=Il-Ir,
U=ElEr*+ErEl*=(Il-Ir)tan 2χ,
V=iElEr*-ErEl*=(Il-Ir)tan 2β sec 2χ.
W(t, x, k)=12 I+QU-iVU+iVI-Q.
1v W(t, x, k)t+kˆ·xW(t, x, k)=σ(k, k)[W(k)]dk-Σ(k)W(t, x, k).
W(0, ρ, z=0, k)=1000 12πρδ(ρ)δ(k-k0),
12 1-ii1.
σ(k, k)[W(k)]=σT¯(k, k)W(k)T*(k, k)δ(|k|-|k|),
σ¯=π2|k|2Rˆ(|k-k|)
Tij(k, k)=z(i)(k)·z(j)(k),i, j=1, 2.
kˆ=sin θ cos ϕsin θ sin ϕcos θ,
z(1)(k)=cos θ cos ϕcos θ sin ϕ-sin θ,
z(2)(k)=-sin ϕcos ϕ0,
3σ(k, k)[2]dk=(k)2,
(k)=π2|k|42 -11 Rˆ(|k|2-2 cos Θ)×(1+cos2 Θ)d(cos Θ).
I=IQUV
I(s)=FI(i),
TWT*=12TI(i)+Q(i)U(i)-iV(i)U(i)+iV(i)I(i)-Q(i)T*=12 AB-iCB+iCD,
A=(T112+T122)I(i)+(T112-T122)Q(i)+2T11T12U(i),
B=(T11T21+T12T22)I(i)+(T11T21-T12T22)Q(i)+(T11T22+T12T21)U(i),
C=(T11T22-T12T21)V(i),
D=(T212+T222)I(i)+(T212-T222)Q(i)+2T21T22U(i).
I(s)=σ¯12[(T112+T122+T212+T222)I(i)+(T112+T212-T122+T222)Q(i)+2(T11T12+T21T22)U(i)],
Q(s)=σ¯12[(T112+T122-T212-T222)I(i)+(T112+T222-T122-T212)Q(i)+2(T11T12-T21T22)U(i)],
U(s)=σ¯[T11T21+T12T22]I(i)+(T11T21-T12T22)Q(i)+(T11T22+T12T21)U(i),
V(s)=σ¯(T11T22-T12T21)V(i).
F˜=σ¯T112T122T11T120T212T222T22T2102T11T212T22T12T11T22+T12T210000T11T22-T12T21.

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