Abstract

In a classic paper that may be regarded as the starting point of polarization optics, G. G. Stokes [Trans. Cambridge Philos. Soc. 9, 399 (1852)] presented a theorem according to which any light beam is equivalent to the sum of two light beams, one of which is completely polarized and the other completely unpolarized. We show that Stokes’ proof of this theorem is flawed. We present a condition for the theorem to be valid.

© 2008 Optical Society of America

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References

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  1. G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399-416 (1852), reprinted in W. Swindell, Polarized Light (Dowden, Hutchinson, P. Ross, 1975), pp. 124-141.
  2. E. Wolf, Phys. Lett. A 312, 263 (2003).
    [CrossRef]
  3. E. Wolf, Opt. Lett. 28, 1078 (2003).
    [CrossRef] [PubMed]
  4. H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
    [CrossRef]
  5. E. Wolf, Introduction to the Theory of Coherence of Polarization of Light (Cambridge U. Press, 2007).
  6. J. Tervo, T. Setälä, and A. T. Friberg, J. Opt. Soc. Am. A 21, 2205 (2004).
    [CrossRef]
  7. M. Alonso and E. Wolf, “A new representation of the cross-spectral density matrix of a planar, stochastic source,” Opt. Commun. (to be published).
  8. Usual treatments involving the Stokes parameters are carried out in the space-time rather than in the space-frequency domain. The two approaches are equivalent. However, the space-frequency representation has considerable advantages in treatments of problems involving propagation, because the propagation laws in the space-frequency domain are appreciably simpler than those in the space-time domain.
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  10. M. Lahiri and E. Wolf, “Cross-spectral density matrix of the far field generated by blackbody sources,” Opt. Commun. (to be published).

2007 (1)

E. Wolf, Introduction to the Theory of Coherence of Polarization of Light (Cambridge U. Press, 2007).

2004 (1)

2003 (3)

E. Wolf, Opt. Lett. 28, 1078 (2003).
[CrossRef] [PubMed]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

1995 (1)

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

1852 (1)

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399-416 (1852), reprinted in W. Swindell, Polarized Light (Dowden, Hutchinson, P. Ross, 1975), pp. 124-141.

Alonso, M.

M. Alonso and E. Wolf, “A new representation of the cross-spectral density matrix of a planar, stochastic source,” Opt. Commun. (to be published).

Friberg, A. T.

Lahiri, M.

M. Lahiri and E. Wolf, “Cross-spectral density matrix of the far field generated by blackbody sources,” Opt. Commun. (to be published).

Mandel, L.

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

Roychowdhury, H.

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

Setälä, T.

Stokes, G. G.

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399-416 (1852), reprinted in W. Swindell, Polarized Light (Dowden, Hutchinson, P. Ross, 1975), pp. 124-141.

Tervo, J.

Wolf, E.

E. Wolf, Introduction to the Theory of Coherence of Polarization of Light (Cambridge U. Press, 2007).

E. Wolf, Opt. Lett. 28, 1078 (2003).
[CrossRef] [PubMed]

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

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

M. Alonso and E. Wolf, “A new representation of the cross-spectral density matrix of a planar, stochastic source,” Opt. Commun. (to be published).

M. Lahiri and E. Wolf, “Cross-spectral density matrix of the far field generated by blackbody sources,” Opt. Commun. (to be published).

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399-416 (1852), reprinted in W. Swindell, Polarized Light (Dowden, Hutchinson, P. Ross, 1975), pp. 124-141.

Other (5)

E. Wolf, Introduction to the Theory of Coherence of Polarization of Light (Cambridge U. Press, 2007).

M. Alonso and E. Wolf, “A new representation of the cross-spectral density matrix of a planar, stochastic source,” Opt. Commun. (to be published).

Usual treatments involving the Stokes parameters are carried out in the space-time rather than in the space-frequency domain. The two approaches are equivalent. However, the space-frequency representation has considerable advantages in treatments of problems involving propagation, because the propagation laws in the space-frequency domain are appreciably simpler than those in the space-time domain.

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

M. Lahiri and E. Wolf, “Cross-spectral density matrix of the far field generated by blackbody sources,” Opt. Commun. (to be published).

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Figures (1)

Fig. 1
Fig. 1

Illustrating the notation.

Equations (22)

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W ( r 1 , r 2 , ω ) = [ W i j ( r 1 , r 2 , ω ) ] = [ 1 2 π E i * ( r 1 , t ) E j ( r 2 , t + τ ) e i ω τ d τ ] , ( i = x , y ; j = x , y ) ,
W i j ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω ) ,
s 0 ( r , ω ) = E x * ( r , ω ) E x ( r , ω ) + E y * ( r , ω ) E y ( r , ω ) ,
s 1 ( r , ω ) = E x * ( r , ω ) E x ( r , ω ) E y * ( r , ω ) E y ( r , ω ) ,
s 2 ( r , ω ) = E x * ( r , ω ) E y ( r , ω ) + E y * ( r , ω ) E x ( r , ω ) ,
s 3 ( r , ω ) = i [ E y * ( r , ω ) E x ( r , ω ) E x * ( r , ω ) E y ( r , ω ) ] .
s 0 ( r , ω ) = W x x ( r , r , ω ) + W y y ( r , r , ω ) ,
s 1 ( r , ω ) = W x x ( r , r , ω ) W y y ( r , r , ω ) ,
s 2 ( r , ω ) = W x y ( r , r , ω ) + W y x ( r , r , ω ) ,
s 3 ( r , ω ) = i [ W y x ( r , r , ω ) W x y ( r , r , ω ) ] .
P ( r , ω ) = [ s 1 2 ( r , ω ) + s 2 2 ( r , ω ) + s 3 2 ( r , ω ) ] 1 2 s 0 ,
P ( r , ω ) = [ 1 4 Det W ( r , r , ω ) [ Tr W ( r , r , ω ) ] 2 ] 1 2 ,
W ( r 1 , r 2 , ω ) = W ( ρ 1 , ρ 2 ; z = 0 ; ω ) K ( ρ 1 ρ 1 ; ρ 2 ρ 2 ; z ; ω ) d 2 ρ 1 d 2 ρ 2 ,
K ( ρ 1 ρ 1 ; ρ 2 ρ 2 ; z ; ω ) = G * ( ρ 1 ρ 1 , z ; ω ) G ( ρ 2 ρ 2 , z ; ω ) ,
G ( ρ ρ ; z ; ω ) = i k 2 π z exp ( i k ρ ρ 2 2 z ) .
W ( p ) ( ρ 1 , ρ 2 ; z = 0 ; ω ) = [ e x * ( ρ 1 , ω ) e x ( ρ 2 , ω ) e x * ( ρ 1 , ω ) e y ( ρ 2 , ω ) e y * ( ρ 1 , ω ) e x ( ρ 2 , ω ) e y * ( ρ 1 , ω ) e y ( ρ 2 , ω ) ] .
W ( u ) ( ρ 1 , ρ 2 , z = 0 ; ω ) = a ( ρ 1 , ρ 2 ; ω ) [ 1 0 0 1 ] .
W ( ρ 1 , ρ 2 ; z = 0 ; ω ) = W ( p ) ( ρ 1 , ρ 2 ; z = 0 ; ω )
+ W ( u ) ( ρ 1 , ρ 2 ; z = 0 ; ω ) .
W ( ρ 1 , ρ 2 , z ; ω ) = W ( p ) ( ρ 1 , ρ 2 , z ; ω ) + W ( u ) ( ρ 1 , ρ 2 , z ; ω ) ,
W ( p ) ( ρ 1 , ρ 2 ; z ; ω ) = [ E x * ( ρ 1 , z ; ω ) E x ( ρ 2 , z ; ω ) E x * ( ρ 1 , z , ω ) E y ( ρ 2 , z ; ω ) E y * ( ρ 1 , z ; ω ) E x ( ρ 2 , z , ω ) E y * ( ρ 1 , z ; ω ) E y ( ρ 2 , z , ω ) ] ,
W ( u ) ( ρ 1 , ρ 2 , z ; ω ) = A ( ρ 1 , ρ 2 , z ; ω ) [ 1 0 0 1 ] .

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