Abstract

The evolution equation is obtained for the state of polarization of partially polarized radiation propagating in a medium that is nonuniform and fully anisotropic, i.e., that presents both linear and circular birefringence as well as dichroism. The treatment covers the general case in which the characteristic polarizations are not necessarily orthogonal, such as occurs for propagation in a magnetized, dissipative plasma. The differential Mueller matrix that appears in the evolution equation is obtained explicitly for two particular cases. The resulting formalism is convenient for numerical integration.

© 2001 Optical Society of America

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References

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  1. S. Huard, Polarization of Light (Wiley, New York, 1997).
  2. M. A. Heald, C. B. Wharton, Plasma Diagnostics with Microwaves (Wiley, New York, 1965).
  3. K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, Cambridge, UK, 1961).
  4. H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).
  5. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767 (1978).
    [CrossRef]
  6. D. B. Melrose, R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge U. Press, Cambridge, UK, 1991).
  7. S. E. Segre, “Evolution of the polarization state for radiation propagating in a nonuniform, birefringent, optically active and dichroic medium: the case of a magnetized plasma,” J. Opt. Soc. Am. A 17, 95–100 (2000).
    [CrossRef]
  8. E. Collett, Polarized Light: Fundamentals and Applications (Dekker, New York, 1992).
  9. S. E. Segre, “A review of plasma polarimetry,” Plasma Phys. Controlled Fusion 41, R57–R100 (1999).
    [CrossRef]
  10. S. E. Segre, “Effect of ray refraction on evolution of the polarization state of radiation propagating in a nonuniform, birefringent, optically active and dichroic medium,” J. Opt. Soc. Am. A 17, 1682–1683 (2000).
    [CrossRef]

2000 (2)

1999 (1)

S. E. Segre, “A review of plasma polarimetry,” Plasma Phys. Controlled Fusion 41, R57–R100 (1999).
[CrossRef]

1978 (1)

Aben, H.

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

Azzam, R. M. A.

Budden, K. G.

K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, Cambridge, UK, 1961).

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Dekker, New York, 1992).

Heald, M. A.

M. A. Heald, C. B. Wharton, Plasma Diagnostics with Microwaves (Wiley, New York, 1965).

Huard, S.

S. Huard, Polarization of Light (Wiley, New York, 1997).

McPhedran, R. C.

D. B. Melrose, R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge U. Press, Cambridge, UK, 1991).

Melrose, D. B.

D. B. Melrose, R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge U. Press, Cambridge, UK, 1991).

Segre, S. E.

Wharton, C. B.

M. A. Heald, C. B. Wharton, Plasma Diagnostics with Microwaves (Wiley, New York, 1965).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Plasma Phys. Controlled Fusion (1)

S. E. Segre, “A review of plasma polarimetry,” Plasma Phys. Controlled Fusion 41, R57–R100 (1999).
[CrossRef]

Other (6)

S. Huard, Polarization of Light (Wiley, New York, 1997).

M. A. Heald, C. B. Wharton, Plasma Diagnostics with Microwaves (Wiley, New York, 1965).

K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, Cambridge, UK, 1961).

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

D. B. Melrose, R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge U. Press, Cambridge, UK, 1991).

E. Collett, Polarized Light: Fundamentals and Applications (Dekker, New York, 1992).

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Equations (52)

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p=(S12+S22+S32)1/2/S0,
s1=cos 2χ cos 2ψ,s2=cos 2χ sin 2ψ, s3=sin 2χ,
S=S0(1, ps1, ps2, ps3)=U+V+W.
Ex=A cos(ωt-ϕ)-B sin(ωt-ϕ),
Ey=C cos(ωt-ϕ)-D sin(ωt-ϕ),
2S0=A2+B2+C2+D2,
2S1=A2+B2-C2-D2,
2S2=2(AC+BD),
2S3=2(AD-BC).
A=2I cos ψ cos χ,B=-2I sin ψ sin χ,
C=2I sin ψ cos χ,D=2I cos ψ sin χ.
Ex=E1x+E2x=A1 cos(ωt-ϕ1)-B1 sin(ωt-ϕ1)+A2 cos(ωt-ϕ2)-B2 sin(ωt-ϕ2)=(A1+A2 cos ϕ-B2 sin ϕ)cos ωt-(B1+A2 sin ϕ+B2 cos ϕ)sin ωt,
Ey=E1y+E2y=C1 cos(ωt-ϕ1)-D1 sin(ωt-ϕ1)+C2 cos(ωt-ϕ2)-D2 sin(ωt-ϕ2)=(C1+C2 cos ϕ-D2 sin ϕ)cos ωt-(D1+C2 sin ϕ+D2 cos ϕ)sin ωt,
A=A1+A2 cos ϕ-B2 sin ϕ,
B=B1+A2 sin ϕ+B2 cos ϕ,
C=C1+C2 cos ϕ-D2 sin ϕ,
D=D1+C2 sin ϕ+D2 cos ϕ.
S0I=I1+I2+2I1I2 cos ϕ cos(ψ1-ψ2)×cos(χ1-χ2)-2I1I2 sin ϕ sin(ψ1-ψ2)×sin(χ1+χ2),
S1=I1 cos 2χ1 cos 2ψ1+I2 cos 2χ2 cos 2ψ2+2I1I2 cos ϕ cos(ψ1+ψ2)cos(χ1+χ2)-2I1I2 sin ϕ sin(ψ1+ψ2)sin(χ1-χ2),
S2=I1 cos 2χ1 sin 2ψ1+I2 cos 2χ2 sin 2ψ2+2I1I2 cos ϕ sin(ψ1+ψ2)cos(χ1+χ2)+2I1I2 sin ϕ cos(ψ1+ψ2)sin(χ1-χ2),
S3=I1 sin 2χ1+I2 sin 2χ2+2I1I2 cos ϕ×cos(ψ1-ψ2)sin(χ1+χ2)-2I1I2 sin ϕ sin(ψ1-ψ2)cos(χ1-χ2).
X=(I1, I2, 2I1I2 cos ϕ, 2I1I2 sin ϕ),
S=A·X,
A=11cos(ψ1-ψ2)cos(χ1-χ2)-sin(ψ1-ψ2)sin(χ1+χ2)cos 2χ1 cos 2ψ1cos 2χ2 cos 2ψ2cos(ψ1+ψ2)cos(χ1+χ2)-sin(ψ1+ψ2)sin(χ1-χ2)cos 2χ1 sin 2ψ1cos 2χ2 sin 2ψ2sin(ψ1+ψ2)cos(χ1+χ2)cos(ψ1+ψ2)sin(χ1-χ2)sin 2χ1sin 2χ2cos(ψ1-ψ2)sin(χ1+χ2)-sin(ψ1-ψ2)cos(χ1-χ2)
dS(z)dz=A·dX(z)dz.
I1(z)=I1(0)exp(-κ1z),
I2(z)=I2(0)exp(-κ2z),
ϕ(z)=kΔμz,
dX(z)dz=D·X(z),
D=-κ10000-κ20000-(κ1+κ2)/2-kΔμ00kΔμ-(κ1+κ2)/2,
D=-(κ1+κ2)2 1+(κ2-κ1)2 Dd+kΔμDb,
Dd=10000-10000000000;Db=00000000000-10010
dS(z)dz=M·S(z),
M=A·D·A-1,
M=-(κ1+κ2)2 1+(κ2-κ1)2 Md+kΔμMb,
Md=A·Dd·A-1,Mb=A·Db·A-1,
A=1100cos 2χ1 cos 2ψ1-cos 2χ1 cos 2ψ1-sin 2ψ1-sin 2χ1 cos 2ψ1cos 2χ1 sin 2ψ1-cos 2χ1 sin 2ψ1cos 2ψ1-sin 2χ1 sin 2ψ1sin 2χ1-sin 2χ10cos 2χ1,
A-1=12 1cos 2χ1 cos 2ψ1cos 2χ1 sin 2ψ1sin 2χ11-cos 2χ1 cos 2ψ1-cos 2χ1 sin 2ψ1-sin 2χ10-2 sin 2ψ12 cos 2ψ100-2 sin 2χ1 cos 2ψ1-2 sin 2χ1 sin 2ψ12 cos 2χ1,
Md=0g1g2g3g1000g2000g3000,
Mb=000000-g3g20g30-g10-g2g10,
Mb·S=S0(0, s1×s).
A=11cos 2χ00cos 2χ0cos 2χ010000sin 2χ0sin 2χ0-sin 2χ000,
A-1=12 sin2 2χ0×1-cos 2χ00sin 2χ01-cos 2χ00-sin 2χ0-2 cos 2χ0200002 sin 2χ00,
Md=1sin 2χ0 0001000cos 2χ000001-cos 2χ000,
Mb=1sin 2χ0 00-cos 2χ0000-10-cos 2χ01000000,
ds(z)dz=Ω×s(z),
X(z)=exp(-κz){I1(0)exp(-δz), I2(0)×exp(δz), 2[I1(0)I2(0)]1/2×cos ϕ(z), 2[I1(0)I2(0)]1/2 sin ϕ(z)},
S(z)=A·X(z)=A·Q(z)·X(0)=A·Q(z)·A-1·S(0),
Q(z)=exp(-κz)exp(-δz)0000exp(δz)0000cos ϕ(z)/cos ϕ(0)0000sin ϕ(z)/sin ϕ(0).
dS(z)dz=M(z)·S(z),
S(z)=P(z)·S(0),
dP(z)dz=M(z)·P(z).

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