Abstract

A necessary and sufficient condition is derived for certain ad hoc expressions that are frequently used in the literature to represent correctly the degree of polarization of a light beam.

© 2007 Optical Society of America

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References

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  1. G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 339 (1952), Sec. 19. Reprinted in Polarized Light, W.Swindell, ed. (Dowden, Hutchinson and Ross, 1975), pp. 124-141.
  2. E. Wolf, Nuovo Cimento 13, 1165 (1959).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  5. S. Chandrasekhar, Radiative Transfer (Dover, 1960), p. 247, Eq. (100).
  6. W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University Press, 2000), p. 82, Eq. (3.71).
  7. F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Addison-Wesley, 1969, reprinted by Dover, 1992), p. 165, Theorem 4.20.
  8. In his classic paper on generalized harmonic analysis N. Wiener, Acta Math. 55, 117 (1930) defined 'percentage of polarization' in term of the coherency matrix (Eq. (9.5.1), p. 191). It is not difficult to show that his definition can be expressed in the form of Eq. . Wiener obtained the formula by use of a real unitary transformation. However, in general, the unitary transformation which diagonalizes the coherency matrix is a complex matrix. To perform the diagonalization physically one has, in general, to make use not only of a rotator (real transformer) but also of a phase plate (complex transformer).
    [CrossRef]

1959 (1)

E. Wolf, Nuovo Cimento 13, 1165 (1959).
[CrossRef]

1952 (1)

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 339 (1952), Sec. 19. Reprinted in Polarized Light, W.Swindell, ed. (Dowden, Hutchinson and Ross, 1975), pp. 124-141.

1930 (1)

In his classic paper on generalized harmonic analysis N. Wiener, Acta Math. 55, 117 (1930) defined 'percentage of polarization' in term of the coherency matrix (Eq. (9.5.1), p. 191). It is not difficult to show that his definition can be expressed in the form of Eq. . Wiener obtained the formula by use of a real unitary transformation. However, in general, the unitary transformation which diagonalizes the coherency matrix is a complex matrix. To perform the diagonalization physically one has, in general, to make use not only of a rotator (real transformer) but also of a phase plate (complex transformer).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

Byron, F. W.

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Addison-Wesley, 1969, reprinted by Dover, 1992), p. 165, Theorem 4.20.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960), p. 247, Eq. (100).

Fuller, R. W.

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Addison-Wesley, 1969, reprinted by Dover, 1992), p. 165, Theorem 4.20.

Grandy, W. T.

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University Press, 2000), p. 82, Eq. (3.71).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Stokes, G. G.

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 339 (1952), Sec. 19. Reprinted in Polarized Light, W.Swindell, ed. (Dowden, Hutchinson and Ross, 1975), pp. 124-141.

Wiener, N.

In his classic paper on generalized harmonic analysis N. Wiener, Acta Math. 55, 117 (1930) defined 'percentage of polarization' in term of the coherency matrix (Eq. (9.5.1), p. 191). It is not difficult to show that his definition can be expressed in the form of Eq. . Wiener obtained the formula by use of a real unitary transformation. However, in general, the unitary transformation which diagonalizes the coherency matrix is a complex matrix. To perform the diagonalization physically one has, in general, to make use not only of a rotator (real transformer) but also of a phase plate (complex transformer).
[CrossRef]

Wolf, E.

E. Wolf, Nuovo Cimento 13, 1165 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Acta Math. (1)

In his classic paper on generalized harmonic analysis N. Wiener, Acta Math. 55, 117 (1930) defined 'percentage of polarization' in term of the coherency matrix (Eq. (9.5.1), p. 191). It is not difficult to show that his definition can be expressed in the form of Eq. . Wiener obtained the formula by use of a real unitary transformation. However, in general, the unitary transformation which diagonalizes the coherency matrix is a complex matrix. To perform the diagonalization physically one has, in general, to make use not only of a rotator (real transformer) but also of a phase plate (complex transformer).
[CrossRef]

Nuovo Cimento (1)

E. Wolf, Nuovo Cimento 13, 1165 (1959).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 339 (1952), Sec. 19. Reprinted in Polarized Light, W.Swindell, ed. (Dowden, Hutchinson and Ross, 1975), pp. 124-141.

Other (5)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

S. Chandrasekhar, Radiative Transfer (Dover, 1960), p. 247, Eq. (100).

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University Press, 2000), p. 82, Eq. (3.71).

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Addison-Wesley, 1969, reprinted by Dover, 1992), p. 165, Theorem 4.20.

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Equations (16)

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Q ( r ) = I x ( r ) I y ( r ) I x ( r ) + I y ( r ) ,
J ( r ) = [ E x * ( r ) E x ( r ) E x * ( r ) E y ( r ) E y * ( r ) E x ( r ) E y * ( r ) E y ( r ) ] .
P ( r ) = I p ( r ) I ( r )
P ( r ) = 1 4 Det J ( r ) [ Tr J ( r ) ] 2 ,
Det J = J x x J y y J x y J y x ,
Tr J = J x x + J y y .
1 4 ( J x x J y y J x y J y x ) ( J x x + J y y ) 2 = ( J x x J y y ) 2 ( J x x + J y y ) 2 .
J x y J y x = 0
J x y = J y x * = 0 .
J x y = ( J y y J x x ) c s + J x y c 2 J y x s 2 ,
J y x = ( J y y J x x ) c s + J y x c 2 J x y s 2 ,
J x y = J y x = 0
( J y y J x x ) c s + J x y c 2 J y x s 2 = 0 ,
( J y y J x x ) c s + J y x c 2 J x y s 2 = 0 .
J x y J y x J x y J x y * = 0 ,
Im J x y = 0 ,

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