Abstract

We present both experimental measurements and Monte-Carlo-based simulations of the diffusely backscattered intensity patterns that arise from illuminating a turbid medium with a polarized laser beam. It is rigorously shown that, because of axial symmetry of the system, only seven elements of the effective backscattering Mueller matrix are independent. A new numerical method that allows simultaneous calculation of all 16 elements of the two-dimensional Mueller matrix is used. To validate our method we compared calculations to measurements from a turbid medium that consisted of polystyrene spheres of different sizes and concentrations in deionized water. The experimental and numerical results are in excellent agreement.

© 1999 Optical Society of America

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  1. A. H. Hielscher, J. R. Mourant, I. J. Bigio, “Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions,” Appl. Opt. 36, 125–135 (1997).
    [CrossRef] [PubMed]
  2. A. I. Carswell, S. R. Pal, “Polarization anisotropy in lidar multiple scattering from clouds,” Appl. Opt. 19, 4123–4126 (1980).
    [CrossRef] [PubMed]
  3. S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).
    [CrossRef] [PubMed]
  4. S. L. Jacques, M. Ostemeyer, L. Wang, D. Stephens, “Polarized light transmission through skin using video reflectometry: toward optical tomography of superficial tissue layers,” in Lasers in Surgery: Advanced Characterization, Therapeutics, and Systems VI, R. R. Anderson, ed., Proc. SPIE2671, 199–220 (1996).
  5. S. G. Demos, R. R. Alfano, “Optical polarization imaging,” Appl. Opt. 36, 150–155 (1997).
    [CrossRef] [PubMed]
  6. A. H. Hielscher, A. A. Eick, J. R. Mourant, I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” in Biomedical Sensing Imaging and Tracking Technologies II, T. Va-Dinh, R. A. Lieberman, G. G. Vuvek, eds. Proc. SPIE2976, 298–305 (1997).
    [CrossRef]
  7. M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
    [CrossRef]
  8. A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Exp. 1, 441–454 (1997).
    [CrossRef]
  9. W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
    [CrossRef]
  10. A. Ambirajan, D. C. Look, “A backward Monte Carlo study of the multiple scattering of a polarized laser beam,” J. Quantum Spectrosc. Radiat. Transfer 58, 171–192 (1997).
    [CrossRef]
  11. M. J. Raković, G. W. Kattawar, “Theoretical analysis of polarization patterns from incoherent backscattering of light,” Appl. Opt. 37, 3333–3338 (1998).
    [CrossRef]
  12. B. D. Cameron, M. J. Raković, M. Mehrubeoglu, G. Kattawar, S. Rastegar, L. V. Wang, G. Coté, “Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. 23, 485–487 (1998); erratum, 23, 1630 (1998).
  13. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  14. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).
  15. G. Bohren, D. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  16. J. W. Hovenier, J. F. de Haan, “Polarized light in planetary atmospheres for perpendicular directions,” Astron. Astrophys. 146, 185–191 (1985).

1998 (2)

1997 (4)

A. H. Hielscher, J. R. Mourant, I. J. Bigio, “Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions,” Appl. Opt. 36, 125–135 (1997).
[CrossRef] [PubMed]

S. G. Demos, R. R. Alfano, “Optical polarization imaging,” Appl. Opt. 36, 150–155 (1997).
[CrossRef] [PubMed]

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Exp. 1, 441–454 (1997).
[CrossRef]

A. Ambirajan, D. C. Look, “A backward Monte Carlo study of the multiple scattering of a polarized laser beam,” J. Quantum Spectrosc. Radiat. Transfer 58, 171–192 (1997).
[CrossRef]

1996 (1)

M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[CrossRef]

1985 (3)

S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).
[CrossRef] [PubMed]

J. W. Hovenier, J. F. de Haan, “Polarized light in planetary atmospheres for perpendicular directions,” Astron. Astrophys. 146, 185–191 (1985).

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

1980 (1)

Alfano, R. R.

Ambirajan, A.

A. Ambirajan, D. C. Look, “A backward Monte Carlo study of the multiple scattering of a polarized laser beam,” J. Quantum Spectrosc. Radiat. Transfer 58, 171–192 (1997).
[CrossRef]

Asakura, T.

M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[CrossRef]

Bailey, W. M.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Bickel, W. S.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Bigio, I. J.

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Exp. 1, 441–454 (1997).
[CrossRef]

A. H. Hielscher, J. R. Mourant, I. J. Bigio, “Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions,” Appl. Opt. 36, 125–135 (1997).
[CrossRef] [PubMed]

A. H. Hielscher, A. A. Eick, J. R. Mourant, I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” in Biomedical Sensing Imaging and Tracking Technologies II, T. Va-Dinh, R. A. Lieberman, G. G. Vuvek, eds. Proc. SPIE2976, 298–305 (1997).
[CrossRef]

Bohren, G.

G. Bohren, D. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Cameron, B. D.

Carswell, A. I.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

Coté, G.

de Haan, J. F.

J. W. Hovenier, J. F. de Haan, “Polarized light in planetary atmospheres for perpendicular directions,” Astron. Astrophys. 146, 185–191 (1985).

Demos, S. G.

Dogariu, M.

M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[CrossRef]

Eick, A. A.

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Exp. 1, 441–454 (1997).
[CrossRef]

A. H. Hielscher, A. A. Eick, J. R. Mourant, I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” in Biomedical Sensing Imaging and Tracking Technologies II, T. Va-Dinh, R. A. Lieberman, G. G. Vuvek, eds. Proc. SPIE2976, 298–305 (1997).
[CrossRef]

Freyer, J. P.

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Exp. 1, 441–454 (1997).
[CrossRef]

Hielscher, A. H.

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Exp. 1, 441–454 (1997).
[CrossRef]

A. H. Hielscher, J. R. Mourant, I. J. Bigio, “Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions,” Appl. Opt. 36, 125–135 (1997).
[CrossRef] [PubMed]

A. H. Hielscher, A. A. Eick, J. R. Mourant, I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” in Biomedical Sensing Imaging and Tracking Technologies II, T. Va-Dinh, R. A. Lieberman, G. G. Vuvek, eds. Proc. SPIE2976, 298–305 (1997).
[CrossRef]

Hoffman, D.

G. Bohren, D. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Hovenier, J. W.

J. W. Hovenier, J. F. de Haan, “Polarized light in planetary atmospheres for perpendicular directions,” Astron. Astrophys. 146, 185–191 (1985).

Jacques, S. L.

S. L. Jacques, M. Ostemeyer, L. Wang, D. Stephens, “Polarized light transmission through skin using video reflectometry: toward optical tomography of superficial tissue layers,” in Lasers in Surgery: Advanced Characterization, Therapeutics, and Systems VI, R. R. Anderson, ed., Proc. SPIE2671, 199–220 (1996).

Kattawar, G.

Kattawar, G. W.

Look, D. C.

A. Ambirajan, D. C. Look, “A backward Monte Carlo study of the multiple scattering of a polarized laser beam,” J. Quantum Spectrosc. Radiat. Transfer 58, 171–192 (1997).
[CrossRef]

Mehrubeoglu, M.

Mourant, J. R.

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Exp. 1, 441–454 (1997).
[CrossRef]

A. H. Hielscher, J. R. Mourant, I. J. Bigio, “Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions,” Appl. Opt. 36, 125–135 (1997).
[CrossRef] [PubMed]

A. H. Hielscher, A. A. Eick, J. R. Mourant, I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” in Biomedical Sensing Imaging and Tracking Technologies II, T. Va-Dinh, R. A. Lieberman, G. G. Vuvek, eds. Proc. SPIE2976, 298–305 (1997).
[CrossRef]

Ostemeyer, M.

S. L. Jacques, M. Ostemeyer, L. Wang, D. Stephens, “Polarized light transmission through skin using video reflectometry: toward optical tomography of superficial tissue layers,” in Lasers in Surgery: Advanced Characterization, Therapeutics, and Systems VI, R. R. Anderson, ed., Proc. SPIE2671, 199–220 (1996).

Pal, S. R.

Rakovic, M. J.

Rastegar, S.

Shen, D.

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Exp. 1, 441–454 (1997).
[CrossRef]

Stephens, D.

S. L. Jacques, M. Ostemeyer, L. Wang, D. Stephens, “Polarized light transmission through skin using video reflectometry: toward optical tomography of superficial tissue layers,” in Lasers in Surgery: Advanced Characterization, Therapeutics, and Systems VI, R. R. Anderson, ed., Proc. SPIE2671, 199–220 (1996).

van de Hulst, H. C.

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

Wang, L.

S. L. Jacques, M. Ostemeyer, L. Wang, D. Stephens, “Polarized light transmission through skin using video reflectometry: toward optical tomography of superficial tissue layers,” in Lasers in Surgery: Advanced Characterization, Therapeutics, and Systems VI, R. R. Anderson, ed., Proc. SPIE2671, 199–220 (1996).

Wang, L. V.

Am. J. Phys. (1)

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Appl. Opt. (5)

Astron. Astrophys. (1)

J. W. Hovenier, J. F. de Haan, “Polarized light in planetary atmospheres for perpendicular directions,” Astron. Astrophys. 146, 185–191 (1985).

J. Quantum Spectrosc. Radiat. Transfer (1)

A. Ambirajan, D. C. Look, “A backward Monte Carlo study of the multiple scattering of a polarized laser beam,” J. Quantum Spectrosc. Radiat. Transfer 58, 171–192 (1997).
[CrossRef]

Opt. Eng. (1)

M. Dogariu, T. Asakura, “Photon pathlength distribution from polarized backscattering in random media,” Opt. Eng. 35, 2234–2239 (1996).
[CrossRef]

Opt. Exp. (1)

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Exp. 1, 441–454 (1997).
[CrossRef]

Opt. Lett. (1)

Other (5)

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

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

G. Bohren, D. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

A. H. Hielscher, A. A. Eick, J. R. Mourant, I. J. Bigio, “Biomedical diagnostic with diffusely backscattered linearly and circularly polarized light,” in Biomedical Sensing Imaging and Tracking Technologies II, T. Va-Dinh, R. A. Lieberman, G. G. Vuvek, eds. Proc. SPIE2976, 298–305 (1997).
[CrossRef]

S. L. Jacques, M. Ostemeyer, L. Wang, D. Stephens, “Polarized light transmission through skin using video reflectometry: toward optical tomography of superficial tissue layers,” in Lasers in Surgery: Advanced Characterization, Therapeutics, and Systems VI, R. R. Anderson, ed., Proc. SPIE2671, 199–220 (1996).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the multiple scattering.

Fig. 2
Fig. 2

Multiple-scattering trajectory: scattering points, scattering and rotational angles.

Fig. 3
Fig. 3

Schematic diagram of the experimental setup.

Fig. 4
Fig. 4

Experimental and Monte Carlo backscattering Mueller matrix. The phantom was comprised of a 0.05-wt% suspension of polystyrene spheres (of diameter 2.02 µm) in deionized water. The approximate size of each image is 1.6 cm × 1.6 cm. Light wavelength was 632.8 nm. The normalized (see the text) absolute values corresponding to the contours of the theoretical matrix elements are S 11: 266, 111, 69, 49 × 10-4; S 12, S 21, S 13, S 31: 15, 6, 2.5, 1 × 10-4; S 22, S 33: 38, 20, 14, 10 × 10-4; S 23, S 32: 26, 15, 10, 6 × 10-4; S 24, S 34, S 42, S 43: 10, 5, 2, 1 × 10-4; S 44: 129, 60, 38, 28 × 10-4. The smaller values correspond to the contours located farther from the center of each plot.

Fig. 5
Fig. 5

Experimental (scattered symbols) and Monte Carlo (solid curves) results for the azimuthal dependence of crossed patterns C(ρ, ϕ), Eq. (33), for polystyrene spheres of diameter 2.02 µm and a concentration of 0.05%. The upper data correspond to 2-mm radial distance, whereas the lower data correspond to 4 mm. Light wavelength was 632.8 nm.

Fig. 6
Fig. 6

Experimental (scattered symbols) and Monte Carlo (solid curves) results for the azimuthal dependence of crossed patterns C(ρ, ϕ), Eq. (33), for polystyrene spheres of diameter 0.482 µm and three different concentrations: (a) 0.1%, (b) 0.05%, and (c) 0.025%. In (a)–(c), data at the top of the plots correspond to 2-mm radial distance, whereas data on the bottom of the plots correspond to 4 mm. In (a) and (b), the other two data correspond to radial distances of 3 and 4 mm. Light wavelength was 632.8 nm.

Fig. 7
Fig. 7

Experimental and theoretical (Monte Carlo) backscattering crossed-polarization patterns for the suspension from Fig. 6 and three concentrations: (a) 0.1%, (b) 0.05%, and (c) 0.025%. Inner and outer patterns in each image are the theoretical results, whereas the middle pattern in each image is the experimental result.

Equations (38)

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Ibsρ, ϕ=μs2Sρ, ϕ; μs, μTP0,
Ibsρ, ϕ=μs2Sρ, ϕ; μs, μTP0,
Sρ, ϕ; μs, μTS˜ρ; μs, μT,
Sρ, ϕ; μs, μT=R-ϕS˜ρ; μs, μTR-ϕ,
Rϕ=10000cos 2ϕsin 2ϕ00-sin 2ϕcos 2ϕ00001
S˜ρ; μs, μT=Sρ, ϕ=0; μs, μT.
S˜ρ; μs, μT=n=2S˜nρ; μs, μT,
S˜n0ρ; μs, μT=μsn-2-0dz1z20dr2z30dr3×zn-10drn-1-0dzn exp-μT|z1|×i=1n-1exp-μT|ri+1-ri||ri+1-ri|2×exp-μT|zn|RϕnMθnRϕn-1 n×Mθ2Rϕ12Mθ1Rϕ1,
Tθin, n12=1000010000cos δ-sin δ00sin δcos δ,
Tθin, n12=c12+c222c12-c22200c12-c222c12+c2220000c1c20000c1c2,
rs=μTr.
S˜n0ρ; μs, μT=ω¯n-2Ln0ρs=ω¯n-2-0dz1z20dr2z30dr3×zn-10drn-1-0dzn exp-|z1|×i=1n-1exp-|ri+1-ri||ri+1-ri|2×exp-|zn|RϕnMθnRϕn-1 n×Mθ2Rϕ12Mθ1Rϕ1,
S˜ρ; μs, μT=n=2 ω¯n-2Lnρs.
L20ρs=00dz1dz2r2exp-z1+z2+rMπ-θMθ,r=ρs2+z1-z221/2, tan θ=ρs/z2-z1,L21ρs=00dz1dz2r2exp-z1+z2+rMπ-θTθMπ-θ,r=ρs2+z1+z221/2, tan θ=ρs/z1+z2.
Mθ=M11M12M13M14M12M22M23M24-M13-M23M33M34M14M24-M34M44.
P=diag1, 1, -1, 1, P2=E,
Mt=PMP,
Rt=PRP, Tt=PTP.
S˜n0tρ; μs, μT=PS˜n0ρ; μs, μTP,
S˜n0ρ; μs, μT=μsn-2-0dz1z20dr2z30×dr3zn-10drn-1-0dzn×exp-μT|z1|×i=1n-1exp-μT|ri+1-ri||ri+1-ri|2×exp-μT|zn|Rϕ1Mθ1Rϕ12×Mθn-1Rϕn-1 nMθnRϕn.
S˜n0tρ; μs, μT=PS˜n0ρ; μs, μTP.
S˜tρ; μs, μT=PS˜ρ; μs, μTP.
S˜=S˜11S˜12S˜13S˜14S˜12S˜22S˜23S˜24-S˜13-S˜23S˜33S˜34S˜14S˜24-S˜34S˜44.
S11ρ, ϕ=S˜11ρ,S12ρ, ϕ=S˜12ρcos 2ϕ+S˜13ρsin 2ϕ,S13ρ, ϕ=-S˜12ρsin 2ϕ+S˜13ρcos 2ϕ,S14ρ, ϕ=S˜14ρ, S21ρ, ϕ=S12ρ, ϕ,S22ρ, ϕ=S˜22ρ-S˜33ρ2+S˜22ρ+S˜33ρ2cos 4ϕ+S˜23ρsin 4ϕ,S23ρ, ϕ=S˜23ρcos 4ϕ-S˜22ρ+S˜33ρ2sin 4ϕ,S24ρ, ϕ=S˜24ρcos 2ϕ-S˜34ρsin 2ϕ,S31ρ, ϕ=-S13ρ, ϕ, S32ρ, ϕ= -S23ρ, ϕ,S33ρ, ϕ=-S˜22ρ-S˜33ρ2+S˜22ρ+S˜33ρ2cos 4ϕ+S˜23ρsin 4ϕ,S34ρ, ϕ=S˜24ρsin 2ϕ+S˜34ρcos 2ϕ,S41ρ, ϕ=S14ρ, ϕ=S˜14ρ,S42ρ, ϕ=S24ρ, ϕ, S43ρ, ϕ=-S34ρ, ϕ,S44ρ, ϕ=S˜44ρ.
S13ρ, ϕ=S12ρ, ϕ+π/4, S21ρ, ϕ=S12ρ, ϕ,S31ρ, ϕ=-S13ρ, ϕ=S12ρ, ϕ-π/4,S32ρ, ϕ=-S23ρ, ϕ=S23ρ, ϕ±π/4,S33ρ, ϕ=-S22ρ, ϕ±π/4,S34ρ, ϕ=S24ρ, ϕ-π/4,S41ρ, ϕ=S41ρ=S14ρ, S42ρ, ϕ=S24ρ, ϕ,S43ρ, ϕ=-S34ρ, ϕ=S24ρ, ϕ+π/4,
Mθ=aθbθ00bθaθ0000dθ-eθ00eθdθ,
M=QMQ,
Q=diag1, 1, -1, -1, Q2=E.
Rt=QRQ, Tt=QTQ.
S˜ρ; μs, μT=QS˜ρ; μs, μTQ.
S˜=S˜11S˜1200S˜12S˜220000S˜33S˜3400-S˜34S˜44.
S11ρ, ϕ=S˜11ρ, S12ρ, ϕ=S˜12ρcos 2ϕ, S14ρ, ϕ=S˜14ρ=0, S22ρ, ϕ=S˜22ρ-S˜33ρ2+S˜22ρ+S˜33ρ2cos 4ϕ, S23ρ, ϕ=-S˜22ρ+S˜33ρ2sin 4ϕ, S24ρ, ϕ=-S˜34ρsin 2ϕ, S44ρ, ϕ=S˜44ρ,
S13ρ, ϕ=S12ρ, ϕ+π/4=-S˜12ρsin 2ϕ,S21ρ, ϕ=S12ρ, ϕ,S31ρ, ϕ=-S13ρ, ϕ, S32ρ, ϕ=-S23ρ, ϕ,S33ρ, ϕ=-S22ρ, ϕ±π/4=-S˜22ρ-S˜33ρ2+S˜22ρ+S˜33ρ2cos 4ϕ,S34ρ, ϕ=S24ρ, ϕ-π/4=S˜34ρcos 2ϕ,S41ρ, ϕ=S14ρ, ϕ=0,S42ρ, ϕ=S24ρ, ϕ, S43ρ, ϕ=-S34ρ, ϕ.
wi exp-μT|zi|RϕiMθiMi,
M1=0.820.055000.0550.820000-0.79-0.20000.20-0.79.
Cρ, ϕ=S11ρ, ϕ+S12ρ, ϕ-S21ρ, ϕ-S22ρ, ϕ.
Cρ, ϕ=S˜11ρ-S˜22ρ-S˜33ρ2-S˜22ρ+S˜33ρ2×cos 4ϕ-S˜23ρsin 4ϕ,
Cρ, ϕ=S˜11ρ-S˜22ρ-S˜33ρ2-S˜22ρ+S˜33ρ2cos 4ϕ.

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