Abstract

First, the analytic expressions are derived that describe some properties of the Poincaré sphere that are related to the coherent sum of two waves having different polarizations. Then these expressions are used to obtain the differential equation that describes the evolution of the state of polarization of radiation propagating in a medium that is nonuniform, birefringent, optically active, and dichroic. Finally, an important example is presented: that of a magnetized plasma in which particle collisions are not negligible, such as the earth’s ionosphere or certain laboratory plasmas. The evolution equation is the basis of plasma polarimetry.

© 2000 Optical Society of America

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References

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  1. S. Huard, Polarization of Light (Wiley, New York, 1997).
  2. H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).
  3. M. A. Heald, C. B. Wharton, Plasma Diagnostics with Microwaves (Wiley, New York, 1965).
  4. K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, Cambridge, UK, 1961).
  5. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).
  6. S. E. Segre, “Plasma polarimetry for large Cotton–Mouton and Faraday effects,” Phys. Plasmas 2, 2908–2914 (1995).
    [CrossRef]
  7. G. N. Ramachandran, S. Ramaseshan, “Crystal optics,” in Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. XXV/1.
  8. S. E. Segre, “A review of plasma polarimetry,” Plasma Phys. Controlled Fusion 41, R57–R100 (1999).
    [CrossRef]
  9. F. De Marco, S. E. Segre, “The polarization of an em wave propagating in a plasma with magnetic shear,” Plasma Phys. 14, 245–252 (1972).

1999 (1)

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

1995 (1)

S. E. Segre, “Plasma polarimetry for large Cotton–Mouton and Faraday effects,” Phys. Plasmas 2, 2908–2914 (1995).
[CrossRef]

1972 (1)

F. De Marco, S. E. Segre, “The polarization of an em wave propagating in a plasma with magnetic shear,” Plasma Phys. 14, 245–252 (1972).

Aben, H.

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

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).

Budden, K. G.

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

De Marco, F.

F. De Marco, S. E. Segre, “The polarization of an em wave propagating in a plasma with magnetic shear,” Plasma Phys. 14, 245–252 (1972).

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).

Ramachandran, G. N.

G. N. Ramachandran, S. Ramaseshan, “Crystal optics,” in Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. XXV/1.

Ramaseshan, S.

G. N. Ramachandran, S. Ramaseshan, “Crystal optics,” in Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. XXV/1.

Segre, S. E.

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

S. E. Segre, “Plasma polarimetry for large Cotton–Mouton and Faraday effects,” Phys. Plasmas 2, 2908–2914 (1995).
[CrossRef]

F. De Marco, S. E. Segre, “The polarization of an em wave propagating in a plasma with magnetic shear,” Plasma Phys. 14, 245–252 (1972).

Wharton, C. B.

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

Phys. Plasmas (1)

S. E. Segre, “Plasma polarimetry for large Cotton–Mouton and Faraday effects,” Phys. Plasmas 2, 2908–2914 (1995).
[CrossRef]

Plasma Phys. (1)

F. De Marco, S. E. Segre, “The polarization of an em wave propagating in a plasma with magnetic shear,” Plasma Phys. 14, 245–252 (1972).

Plasma Phys. Controlled Fusion (1)

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

Other (6)

G. N. Ramachandran, S. Ramaseshan, “Crystal optics,” in Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. XXV/1.

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

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

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).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).

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Figures (7)

Fig. 1
Fig. 1

Poincaré sphere in the p,q,r reference system. S1 and S2 are the points at the tip of the characteristic Stokes vectors s1 and s2.

Fig. 2
Fig. 2

Circle u·s equals constant (a meridian).

Fig. 3
Fig. 3

Meridians on the Poincaré sphere.

Fig. 4
Fig. 4

Meridians observed in the direction opposite to r.

Fig. 5
Fig. 5

Circle w·s equals constant (a pseudoparallel).

Fig. 6
Fig. 6

Pseudoparallels observed in the direction of q.

Fig. 7
Fig. 7

Pseudoparallels on the Poincaré sphere.

Equations (66)

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s1=cos 2χ cos 2ψ,s2=cos 2χ sin 2ψ,
s3=sin 2χ,
s1·s2=cos 4χo.
p=s1+s2|s1+s2|,r=s1-s2|s1-s2|,q=r×p,
s1=(cos 2χo, 0, sin 2χo),s2=(cos 2χo, 0, -sin 2χo)
s1=s·p,s2=s·q,s3=s·r.
s1=cos 2χo+α cos ϕ1+α cos ϕ cos 2χo,
s2=α sin ϕ sin 2χo1+α cos ϕ cos 2χo,
s3=β sin 2χo1+α cos ϕ cos 2χo,
ϕ=ϕ1-ϕ2,α=2I1I2I1+I2,
β=I1-I2I1+I2,α2+β2=1,
I=I1+I2+2I1I2 cos ϕ cos 2χo.
β=s3 sin 2χo1-s1 cos 2χo,α=1-β2,
cos ϕ=β sin 2χo-s3αs3 cos 2χo,sin ϕ=αβs2s3,
β=(s·r)sin 2χo1-(s·p)cos 2χo,α=1-β2,
cos ϕ=β sin 2χo-(s·r)α(s·r)cos 2χo,sin ϕ=αβ(s·q)(s·r).
ϕ=ϕ1-ϕ2=ωc(μ1-μ2)z,
I1/I2=(I1/I2)o exp[-(κ1-κ2)].
u=(s1-s)×(s2-s)|(s1-s)×(s2-s)|.
u·s=sin ϕ cos 2χo(1-cos2 ϕ cos2 2χo)1/2.
w=(h-s)×q|(h-s)×q|,
h=1cos 2χop=2(s1+s2)|s1+s2|2,
w·s=1-α21-α2 cos2 2χo1/2=β2sin2 2χo+β2 cos2 2χo1/2,
sβ=sβu×s|(u×s)|,sϕ=sϕw×s|(w×s)|.
sβ=sin 2χoα(1+α cos ϕ cos 2χo),
sϕ=α sin 2χo(1+α cos ϕ cos 2χo),
|(u×s)|=sin 2χo(1-cos2 ϕ cos2 2χo)1/2,
|(w×s)|=α sin 2χo(1-α2 cos2 2χo)1/2.
dϕ=ωc(μ1-μ2)dz,dβ=12(κ2-κ1)(1-β2)dz
ds=sϕdϕ+sβdβ,
dsdz=ωc(μ1-μ2)sϕ+12(κ2-κ1)(1-β2)sβ,
dsdz=(W+U)×s,
W=ωc(μ1-μ2)sin 2χo[(1-s1 cos 2χo)2+s32 cos2 2χo]1/2,
U=(κ2-κ1)2 sin 2χo[(s1-cos 2χo)2+s22]1/2,
w=cos 2χos3|s3|1-α21-α2 cos2 2χo1/2p+sin 2χo(1-α2 cos2 2χo)1/2r,
(N1,2)2=1-X1-iZ-(G±G2+1)Y cos θ,
G=sin2 θ2 cos θY(1-X-iZ),
ρ1,2=-i(G±G2+1),
tan 2χ=2q sin δ[(1-q2)2+4q2 cos2 δ]1/2,
tan 2ψ=2q cos δ(1-q2),
cos 2ψ=(1-q2)q2-1+[(1-q2)2+4q2 cos2 δ]1/2(1-q2)2+4q2 cos2 δ+(q2-1)[(1-q2)2+4q2 cos2 δ]1/2.
Ex=A cos(ωt-ϕo)-B sin(ωt-ϕo),
Ey=C cos(ωt-ϕo)-D sin(ωt-ϕo),
I=A2+B2+C2+D2,
A=cos ψ cos χ,B=-sin ψ sin χ,
C=sin ψ cos χ,D=cos ψ sin χ
s1=cos 2χ cos 2ψ=A2+B2-C2-D2A2+B2+C2+D2,
s2=cos 2χ sin 2ψ=2(AC+BD)A2+B2+C2+D2,
s3=sin 2χ=2(AD-BC)A2+B2+C2+D2.
ψ1=ψ2=0,χ1=-χ2=χo.
Ex=I1 cos χo cos(ωt-ϕ1)+I2 cos χo cos(ωt-ϕ2),
Ey=-I1 sin χo sin(ωt-ϕ1)+I2 sin χo sin(ωt-ϕ2),
Ex=(I1+cos ϕI2)cos χo cos(ωt-ϕ1)-sin ϕ cos χoI2 sin(ωt-ϕ1),
Ey=sin ϕ cos χoI2 cos(ωt-ϕ1)-(I1-cos ϕI2)sin χo sin(ωt-ϕ1).
A=(I1+cos ϕI2)cos χo,B=sin ϕ cos χoI2,
C=sin ϕ sin χoI2,D=(I1-cos ϕI2)sin χo,
I=I1+I2+2I1I2 cos ϕ cos 2χo,
s1=(I1+I2)cos 2χo+2I1I2 cos ϕI1+I2+2I1I2 cos ϕ cos 2χo,
s2=2I1I2 sin ϕ sin 2χoI1+I2+2I1I2 cos ϕ cos 2χo,
s3=(I1-I2)sin 2χoI1+I2+2I1I2 cos ϕ cos 2χo.
Ex=Eo cos ωt,Ey=qEo cos(ωt+δ).
Ex=Ex cos ψ+Ey sin ψ=a cos(ωt+ϕ),
Ey=-Ex sin ψ+Ey cos ψ=b sin(ωt+ϕ).
ab=-q sin δ,ba=-q cos ψ sin δq sin ψ cos δ+cos ψ=q cos ψ cos δ-sin ψq sin ψ sin δ.
tan ψ=12q cos δ{q2-1+[(1-q2)2+4q2 cos2 δ]1/2}
tan χ=12q sin δ{q2+1-[(1-q2)2+4q2 cos2 δ]1/2}

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