Abstract

It is shown that, for a 3×3 correlation matrix Wijr,r,ω,i,j=x,y,z of the electric vector of a random, stationary electromagnetic field to represent a field that is completely polarized at a point r and frequency ω, each element of the matrix must factorize. More precisely, a necessary and sufficient condition for the correlation matrix to represent a fully polarized field at a point r is that the matrix has the form Wijr,r,ω=Ei*r,ωEjr,ω, where Eir,ω i=x,y,z are deterministic functions, i.e., that all pairs of the Cartesian components of the electric field at a point r and frequency ω are completely correlated.

© 2004 Optical Society of America

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References

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  1. T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  2. C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).
  3. T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000).
    [CrossRef]
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
    [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).
    [CrossRef]
  6. A referee of this paper pointed out that, in Ref. 1, a definition of the degree of polarization of three-dimensional fields was put forward that gives the value unity in this limiting case. This is, however, fortuitous, because the definition proposed in Ref. 1 is purely formal and does not have the meaning of the ratio of the intensity of a fully polarized field to the total intensity at a point.

2002 (1)

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

2000 (1)

T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000).
[CrossRef]

Bergman, J.

T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Brosseau, C.

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

Carozzi, T.

T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000).
[CrossRef]

Friberg, A. T.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Kaivola, M.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Karlsson, R.

T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Setala, T.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Shevchenko, A.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Phys. Rev. E (2)

T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000).
[CrossRef]

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Other (4)

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

A referee of this paper pointed out that, in Ref. 1, a definition of the degree of polarization of three-dimensional fields was put forward that gives the value unity in this limiting case. This is, however, fortuitous, because the definition proposed in Ref. 1 is purely formal and does not have the meaning of the ratio of the intensity of a fully polarized field to the total intensity at a point.

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Equations (19)

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μxyr,ω=Ex*r,ωEyr,ωEx*r,ωExr,ω1/2Ey*r,ωEyr,ω1/2,
Eir,ω=eir,ωUr,ω,
Eir,ωEjr,ω=eir,ωeir,ω
Eir,ωexpiωt=eir,ωUr,ω21/2 exp-iωt,
Wijr1,r2,ω=Ei*r1,ωEjr2,ω,
Wijr,r,ω=Ei*r,ωEjr,ω.
Wijr,r,ωEi*r1,ωEjr2,ω=ei*r,ωU*r,ωejr,ωUr,ω=ei*r,ωejr,ωU*r,ωUr,ω=Ei*r,ωEjr,ω,
detWr,r,ω-λI=0,
Ex2-λEy2-λEz2-λ-Ey2Ez2-Ex*EyEy*ExEz2-λ-Ez2Ey*Ex+Ex*EzEy2Ez*Ex-Ez*ExEy2-λ=0,
λ2λ-Tr Wr,rω=0,
λ1=Tr Wr,r,ω=Ex*r,ωExr,ω+Ey*r,ωEyr,ω+Ez*r,ωEzr,ω.
R=cosθ0sinθ010-sinθ0-cosθcosαsinα0-sinαcosα0001,
θr,ω=arctanEzr,ωExr,ωcosα+Eyr,ωsinα,
αr,ω=arctanExr,ωsinϕzr,ω-ϕxr,ωEyr,ωsinϕyr,ω-ϕzr,ω.
W˜r,r,ω=RWr,r,ωR-1=Ex*r,ωExr,ωEx*r,ωEyr,ωEx*r,ωEzr,ωEy*r,ωExr,ωEy*r,ωEyr,ωEy*r,ωEzr,ωEz*r,ωExr,ωEz*r,ωEyr,ωEz*r,ωEzr,ω,
Exr,ω=Exr,ωcosθcosα+Eyr,ωcosθsinα-Ezr,ωsinθ,
Eyr,ω=-Exr,ωsinα+Eyr,ωcosα,
Ezr,ω=-Exr,ωsinθcosα-Eyr,ωsinθsinα+Ezr,ωcosθ.
W˜r,r,ω=Ex*r,ωExr,ωEx*r,ωEyr,ω0Ey*r,ωExr,ωEy*r,ωEyr,ω0000,

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