Abstract

Although the theories of polarization in the space–time and space–frequency domains are somewhat analogous, they have been developed independently, and there is no obvious connection between them. We investigate how they are related.

© 2009 Optical Society of America

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References

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  1. N. Wiener, J. Math. Phys. (Cambridge, Mass.) 7, 109 (1927-28).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  3. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  5. T. Setälä, F. Nunziata, and A. T. Friberg, presented at the Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization, May 24-27, 2009, Koli, Finland.
  6. H. Roychowdhury and E. Wolf, Opt. Commun. 248, 327 (2005).
    [CrossRef]
  7. E. Wolf, “Statistical similarity as a unifying concept of the theories of coherence and polarization,” Opt. Commun., submitted for publication in Special Issue on Electromagnetic Coherence and Polarization.
  8. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, Opt. Lett. 29, 1536 (2004).
    [CrossRef] [PubMed]
  9. This factorization property was also noted in However, no connection of this results with the factorization in the space-time domain was discussed.
  10. This formula can be derived by using the expression for the intensity at the output of an Michelson's interferometer, in the usual formula V=(Imax−Imin)/(Imax+Imin) for fringe visibility.
  11. B. Karczewski, Nuovo Ciemento 30, 906 (1963).
    [CrossRef]

2005 (1)

H. Roychowdhury and E. Wolf, Opt. Commun. 248, 327 (2005).
[CrossRef]

2004 (1)

1963 (1)

B. Karczewski, Nuovo Ciemento 30, 906 (1963).
[CrossRef]

Born, M.

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

Dogariu, A.

Ellis, J.

Friberg, A. T.

T. Setälä, F. Nunziata, and A. T. Friberg, presented at the Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization, May 24-27, 2009, Koli, Finland.

Karczewski, B.

B. Karczewski, Nuovo Ciemento 30, 906 (1963).
[CrossRef]

Mandel, L.

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

Nunziata, F.

T. Setälä, F. Nunziata, and A. T. Friberg, presented at the Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization, May 24-27, 2009, Koli, Finland.

Ponomarenko, S.

Roychowdhury, H.

H. Roychowdhury and E. Wolf, Opt. Commun. 248, 327 (2005).
[CrossRef]

Setälä, T.

T. Setälä, F. Nunziata, and A. T. Friberg, presented at the Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization, May 24-27, 2009, Koli, Finland.

Wiener, N.

N. Wiener, J. Math. Phys. (Cambridge, Mass.) 7, 109 (1927-28).

Wolf, E.

H. Roychowdhury and E. Wolf, Opt. Commun. 248, 327 (2005).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, Opt. Lett. 29, 1536 (2004).
[CrossRef] [PubMed]

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

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

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

E. Wolf, “Statistical similarity as a unifying concept of the theories of coherence and polarization,” Opt. Commun., submitted for publication in Special Issue on Electromagnetic Coherence and Polarization.

J. Math. Phys. (Cambridge, Mass.) (1)

N. Wiener, J. Math. Phys. (Cambridge, Mass.) 7, 109 (1927-28).

Nuovo Ciemento (1)

B. Karczewski, Nuovo Ciemento 30, 906 (1963).
[CrossRef]

Opt. Commun. (1)

H. Roychowdhury and E. Wolf, Opt. Commun. 248, 327 (2005).
[CrossRef]

Opt. Lett. (1)

Other (7)

E. Wolf, “Statistical similarity as a unifying concept of the theories of coherence and polarization,” Opt. Commun., submitted for publication in Special Issue on Electromagnetic Coherence and Polarization.

This factorization property was also noted in However, no connection of this results with the factorization in the space-time domain was discussed.

This formula can be derived by using the expression for the intensity at the output of an Michelson's interferometer, in the usual formula V=(Imax−Imin)/(Imax+Imin) for fringe visibility.

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

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

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

T. Setälä, F. Nunziata, and A. T. Friberg, presented at the Koli Workshop on Partial Electromagnetic Coherence and 3D Polarization, May 24-27, 2009, Koli, Finland.

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Figures (1)

Fig. 1
Fig. 1

Illustrating the notations.

Equations (25)

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J i j ( r ) E i * ( r , t ) E j ( r , t ) , ( i = x , y ; j = x , y ) ,
P ( r ) 1 4 Det J ( r ) [ Tr J ( r ) ] 2 ,
W ( r 1 , r 2 , ω ) Γ ( r 1 , r 2 , τ ) exp [ i ω τ ] d τ ,
Γ ( r 1 , r 2 , τ ) [ Γ i j ( r 1 , r 2 , τ ) ] [ E i * ( r 1 , t ) E j ( r 2 , t + τ ) ] ,
P ( r ; ω ) 1 4 Det W ( r , r ; ω ) [ Tr W ( r , r ; ω ) ] 2 .
J i j ( p ) ( r ) = E i * ( r ) E j ( r ) , ( i = x , y ; j = x , y ) ,
E y ( r , t ) = A ( r ) E x ( r , t ) ,
J ( p ) ( r ) = J x x ( p ) ( r ) ( 1 A ( r ) A * ( r ) | A ( r ) | 2 ) .
E x ( r ) = J x x ( p ) ( r ) exp [ i φ x ( r ) ] ,
E y ( r ) = | A ( r ) | J x x ( p ) ( r ) exp [ i φ y ( r ) ] ,
Γ ( p ) ( r , r , τ ) [ E i * ( r , t ) E j ( r , t + τ ) ] = Γ x x ( p ) ( r , r , τ ) ( 1 A ( r ) A * ( r ) | A ( r ) | 2 ) .
W ( p ) ( r , r , ω ) = W x x ( p ) ( r , r , ω ) ( 1 A ( r ) A * ( r ) | A ( r ) | 2 ) .
W i j ( r , r ; ω ) = E ̃ i * ( r ; ω ) E ̃ j ( r ; ω ) ,
E ̃ x ( r ; ω ) = W x x ( p ) ( r , r ; ω ) exp [ i φ x ( r ) ] ,
E ̃ y ( r ; ω ) = | A ( r ) | W x x ( p ) ( r , r ; ω ) exp [ i φ y ( r ) ] ,
I θ , ε = J x x cos 2 θ + J y y sin 2 θ + 2 J x x J y y cos θ sin θ | j x y | cos ( β x y ε ) ,
V = γ ( + ) ( r , r ; τ ) = | Tr Γ ( + ) ( r , r ; τ ) | Tr Γ ( + ) ( r , r ; 0 ) ,
Tr Γ ( + ) ( r , r ; τ ) = γ θ , ε ( + ) I θ , ε α ( θ , ε ) , ( say ) .
Tr Γ ( + ) ( r , r ; τ ) = Γ x x ( r , r ; τ ) cos 2 θ + Γ y y ( r , r ; τ ) sin 2 θ + [ Γ x y ( r , r ; τ ) e i ε + Γ y x ( r , r ; τ ) e i ε ] sin θ cos θ .
α ( θ , ε ) = Γ x x ( r , r ; τ ) cos 2 θ + Γ y y ( r , r ; τ ) sin 2 θ + [ Γ x y ( r , r ; τ ) e i ε + Γ y x ( r , r ; τ ) e i ε ] sin θ cos θ .
Γ x x ( r , r ; τ ) = α ( 0 , 0 ) ,
Γ y y ( r , r ; τ ) = α ( π 2 , 0 ) ,
Γ x y ( r , r ; τ ) = 1 2 [ α ( π 4 , 0 ) α ( 3 π 4 , 0 ) ] + i 2 [ { α ( π 4 , π 2 ) α ( 3 π 4 , π 2 ) } ] ,
Γ y x ( r , r ; τ ) = 1 2 [ α ( π 4 , 0 ) α ( 3 π 4 , 0 ) ] i 2 [ { α ( π 4 , π 2 ) α ( 3 π 4 , π 2 ) } ] .
Γ ( r , r ; τ ) = Γ x x ( r , r ; τ ) ( 1 0 0 1 ) .

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