Abstract

Different types of random vector fields can be depolarized in a global sense. They can be physically discriminated by probing the statistical invariance when they are subject to operations such as a change of the reference frame or the introduction of an arbitrary retardance. With use of the observable polarization sphere as a visualization tool, a set of measurements is capable of discriminating between certain types of globally depolarized light, and we discuss the geometric interpretation of the physical invariances.

© 2004 Optical Society of America

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References

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  1. E. Collett, Polarized Light in Fiber Optics (PolaWave Group, Lincroft, N.J., 2003).
  2. C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).
  3. G. S. Agarwal, J. Lehner, H. Paul, “Invariances for states of light and their quasi-distributions,” Opt. Commun. 129, 369–372 (1996).
    [CrossRef]
  4. J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
    [CrossRef]
  5. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  6. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  7. G. Korn, T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).
  8. S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
    [CrossRef] [PubMed]
  9. A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
    [CrossRef]
  10. J. Ellis, A. Dogariu, data available from the authors.

2002

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

1997

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

1996

G. S. Agarwal, J. Lehner, H. Paul, “Invariances for states of light and their quasi-distributions,” Opt. Commun. 129, 369–372 (1996).
[CrossRef]

1991

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

Agarwal, G. S.

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

G. S. Agarwal, J. Lehner, H. Paul, “Invariances for states of light and their quasi-distributions,” Opt. Commun. 129, 369–372 (1996).
[CrossRef]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

Cohen, S. M.

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

Collett, E.

E. Collett, Polarized Light in Fiber Optics (PolaWave Group, Lincroft, N.J., 2003).

Dogariu, A.

J. Ellis, A. Dogariu, data available from the authors.

Eliyahu, D.

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

Ellis, J.

J. Ellis, A. Dogariu, data available from the authors.

Freund, I.

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Kaveh, M.

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

Korn, G.

G. Korn, T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Korn, T.

G. Korn, T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Lehner, J.

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

G. S. Agarwal, J. Lehner, H. Paul, “Invariances for states of light and their quasi-distributions,” Opt. Commun. 129, 369–372 (1996).
[CrossRef]

Luis, A.

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Paul, H.

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

G. S. Agarwal, J. Lehner, H. Paul, “Invariances for states of light and their quasi-distributions,” Opt. Commun. 129, 369–372 (1996).
[CrossRef]

Opt. Commun.

G. S. Agarwal, J. Lehner, H. Paul, “Invariances for states of light and their quasi-distributions,” Opt. Commun. 129, 369–372 (1996).
[CrossRef]

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

Phys. Rev. A

S. M. Cohen, D. Eliyahu, I. Freund, M. Kaveh, “Vector statistics of multiply scattered waves in random systems,” Phys. Rev. A 43, 5748–5751 (1991).
[CrossRef] [PubMed]

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

Other

J. Ellis, A. Dogariu, data available from the authors.

E. Collett, Polarized Light in Fiber Optics (PolaWave Group, Lincroft, N.J., 2003).

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

G. Korn, T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

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Figures (4)

Fig. 1
Fig. 1

Globally unpolarized light obtained from a uniform distribution of all polarization states resulting from underlying independent Gaussian-distributed random fields. This is type I unpolarized light; the distribution is not affected by any change of coordinates for the OPS (equivalently, the introduction of an arbitrary retardance or the reversal of the direction of propagation).

Fig. 2
Fig. 2

Globally unpolarized light obtained from a uniform distribution of all linear states of polarization. This is an example of type II unpolarized light; the distribution is not affected by the choice of reference frames or the introduction of a reflection that serves to reverse right- and left-circular polarizations.

Fig. 3
Fig. 3

Globally unpolarized light obtained from a uniform-banded distribution of polarization states about s1=0 that results from specific coupling in the underlying complex fields. This is an example of type III unpolarized light; the distribution is independent of the reversal of the direction of propagation. However, an arbitrary retardance (including the introduction of a half-wave plate at any angle, equivalent to a change in reference frame) will alter the distribution and the resulting correlations.

Fig. 4
Fig. 4

Globally unpolarized light obtained from a Gaussian-banded distribution of polarization states about s1=0 that results from specific coupling in the underlying complex fields. This is another example of type III unpolarized light.

Tables (1)

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Table 1 Statistical Characteristics of Globally Depolarized Light a

Equations (54)

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p(x1, x2,,xN)=p(y1, y2,,yN)(y1, y2,,yN)(x1, x2,,xN),
p(|Ex|, |Ey|, θx, θy, D)=p(Ex, Ex, Ey, Ey, D) ×(Ex, Ex, Ey, Ey, D)(|Ex|, |Ey|, θx, θy, D),
p(|Ex|, |Ey|, θ, D)=p(|Ex|, |Ey|,θx-θy, θy, D)dθy,
p(It, s1, s2, s3)=p(|Ex|, |Ey|, θ, D)×(|Ex|, |Ey|, θ, D)(It, s1, s2, s3).
p(s1, s2, s3)=p(It, s1, s2, s3)dIt,
s1=ExEx*-EyEy*ExEx*+EyEy*,
s2=ExEy*+EyEx*ExEx*+EyEy*,
s3=iExEy*-EyEx*ExEx*+EyEy*,
p(r, 2α, Δ)=r2sin(2α)p(s1, s2, s3, D),
s1=r cos(2α),
s2=r sin(2α)cos(Δ),
s3=r sin(2α)sin(Δ),
p(θx, θy)=2π-|θx-θy|2π2 δ(θx+θy),
θi[0, 2π),
p(Ex, Ex, Ey, Ey, D)
=δ(D-1) 1σ4exp-(Ex)2+(Ex)2+(Ey)2+(Ey)22σ2× 2π-arctanExEx-arctanEyEy2π2
× δarctanExEx+arctanEyEy.
p(Ex, Ex, Ey, Ey, D)
=14π2σ4exp-(Ex)2+(Ex)2+(Ey)2+(Ey)22σ2δ(D-1),
 
p(Ix, Iy, θ, D)=14σ4exp-Ix+Iy2σ22π-|θ|4π2×δ(D-1),
|θ|2π,Ix,y0,D[0, 1],
p(r, 2α, Δ)=δ(r-1) 12πsin(2α)2,
Δ[0, 2π),2α[0, π].
p(r, 2α, Δ)=δ(r-1) sin(2α)212 [δ(Δ)+δ(Δ-π)],
2α[0, π],Δ[0, 2π],
p(r, 2α, Δ)=δ(r-1)δ2α-π212δΔ-π2+δΔ-3π2,
2α[0, π],Δ[0, 2π],
p(r, 2α, Δ)=δ(r-1) 12πsin(2α)2 δ2α-π2,
2α[0, π],Δ[0, 2π].
p(r, 2α, Δ)=δ(r-1) 12π1arect1a2α-π2,
2α[0, π],Δ[0, 2π],
p(r, 2α, Δ)=δ(r-1)exp(2α-π/2)2σ2×12π1erf(π2/4σ)2πσ2,2α[0, π],Δ[0, 2π],
p(It)=Itα2exp-Itα;
pI(Ex, Ex, Ey, Ey, D)
=δ(D-1)4π2σ4exp-(Ex)2+(Ex)2+(Ey)2+(Ey)22σ2,
pII(Ex, Ex, Ey, Ey, D)
=2pI(Ex, Ex, Ey, Ey, D)×δarctanExEx-arctanEyEy,
pIII(Ex, Ex, Ey, Ey, D)
=pI(Ex, Ex, Ey, Ey, D)1-(Ex)2+(Ex)2-(Ey)2-(Ey)2(Ex)2+(Ex)2+(Ey)2+(Ey)221/2rect 12 sin(1)(Ex)2+(Ex)2-(Ey)2-(Ey)2(Ex)2+(Ex)2+(Ey)2+(Ey)2,
pI(It, s1, s2, s3)
=It4σ4exp-It2σ214πδ[(s12+s22+s32)1/2-1]s12+s22+s32,
pII(It, s1, s2, s3)
=It4σ4exp-It2σ212π×δ[(s12+s22+s32)1/2-1]δ[arctan(s3/s2)]s12+s22+s32,
pIII(It, s1, s2, s3)
=It4σ4exp-It2σ214πδ[(s12+s22+s32)1/2-1]s12+s22+s32×rect{s1/[2 sin(1)]}[1-s12/(s12+s22+s32)]1/2.
It=Itp(It, s1, s2, s3)ds1ds2ds3dIt=4σ2
sisj=sisp(It, s1, s2, s3)ds1ds2ds3dIt,
sisj=si(r, 2α, Δ)sj(r, 2α, Δ)p(r, 2α, Δ)×drd(2α)dΔ,
s1(r, 2α, Δ)=r cos(2α),
s2(r, 2α, Δ)=r sin(2α)cos(Δ),
s3(r, 2α, Δ)=r sin(2α)sin(Δ).
s12I=01r2δ(r-1)dr02π12πdΔ×0πcos2(2α) sin(2α)2d(2α)=13,
s1s2I=01r2δ(r-1)dr02πcos(Δ) 12πdΔ×0πcos(2α) sin(2α)2d(2α)=0,

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