Abstract

Within the framework of the phase-space representation of random electromagnetic fields provided by electromagnetic spatial coherence wavelets, and by using the Fresnel–Arago laws for interference and polarization as an analysis tool, the meaning of the spatial coherence-polarization tensor and its invariance under transformations is studied. The results give new insight into the definition and properties of the complex degree of spatial coherence by showing that its invariance is not required for properly describing the behavior of random electromagnetic fields within the scope of physically measurable quantities.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).
  18. D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles (Wiley Interscience, 1975).
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  20. R. Shankar, Principles of Quantum Mechanics (Plenum, 1994).

2009

2008

2007

M. R. Dennis, “A three-dimensional degree of polarization based on Rayleigh scattering,” J. Opt. Soc. Am. A 24, 2065-2069 (2007).
[CrossRef]

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J.: Appl. Phys. 40, 1-47 (2007).
[CrossRef]

2006

2005

2004

2003

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137-1143 (2003).
[CrossRef] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Agarwal, G. S.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Betancur, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

Carrasquilla, J.

Castañeda, R.

Dennis, M. R.

Dogariu, A.

Friberg, A. T.

Garcia, J.

Garcia-Sucerquia, J.

Gil, J. J.

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J.: Appl. Phys. 40, 1-47 (2007).
[CrossRef]

Goudail, F.

Herrera, J.

Korotkova, O.

Lovelock, D.

D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles (Wiley Interscience, 1975).

Mandel, L.

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

Mujat, M.

Réfrégier, P.

Rund, H.

D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles (Wiley Interscience, 1975).

Setälä, T.

Shankar, R.

R. Shankar, Principles of Quantum Mechanics (Plenum, 1994).

Tervo, J,. .

Tervo, J.

Visser, T. D.

Wolf, E.

Appl. Opt.

Eur. Phys. J.: Appl. Phys.

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J.: Appl. Phys. 40, 1-47 (2007).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

R. Castañeda, “Electromagnetic spatial coherence wavelets and the classical laws on polarization,” Opt. Commun. 267, 4-13 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Other

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles (Wiley Interscience, 1975).

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

M. Born and E. Wolf, Principles of Optics (Pergamon, 1993).

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

R. Shankar, Principles of Quantum Mechanics (Plenum, 1994).

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Figures (2)

Fig. 1
Fig. 1

Cartesian and center and difference coordinate systems for the propagation of a stationary random electromagnetic field.

Fig. 2
Fig. 2

Young’s experiment with random electromagnetic fields.

Equations (21)

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W ( r A + r D 2 , r A r D 2 , ξ A ; ν ) = S ( r A , ξ A ; ν ) exp ( i k z r D ξ A ) ,
S ( r A , ξ A ; ν ) = AP E ( + ) ( ν ) η ( + , ) ( ν ) E ( ) ( ν ) exp [ i k z ( ξ A r A ) ξ D ] d 2 ξ D
S ( r A ; ν ) = ( 1 λ z ) 2 AP tr [ S ( r A , ξ A ; ν ) ] d 2 ξ A ,
η ( + , ) ( ν ) = | η 0 ( + , ) ( ν ) | [ cos ϑ ( + ) cos ϑ ( ) cos 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 exp [ i β x x ( + , ) ( ν ) ] cos ϑ ( + ) sin ϑ ( ) cos 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 exp [ i β x y ( + , ) ( ν ) ] sin ϑ ( + ) cos ϑ ( ) sin 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 exp [ i β y x ( + , ) ( ν ) ] sin ϑ ( + ) sin ϑ ( ) sin 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 exp [ i β y y ( + , ) ( ν ) ] ] ,
η ( + , ) ( ν ) = | η 0 ( + , ) ( ν ) | 2 { [ cos ϑ ( + ) cos ϑ ( ) cos 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 exp [ i β x x ( + , ) ( ν ) ] + sin ϑ ( + ) sin ϑ ( ) sin 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 exp [ i β y y ( + , ) ( ν ) ] ] σ 0 + [ cos ϑ ( + ) cos ϑ ( ) cos 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 exp [ i β x x ( + , ) ( ν ) ] sin ϑ ( + ) sin ϑ ( ) sin 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 exp [ i β y y ( + , ) ( ν ) ] ] σ 1 + [ cos ϑ ( + ) sin ϑ ( ) cos 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 exp [ i β x y ( + , ) ( ν ) ] + sin ϑ ( + ) cos ϑ ( ) sin 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 exp [ i β y x ( + , ) ( ν ) ] ] σ 2 i [ cos ϑ ( + ) sin ϑ ( ) cos 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 exp [ i β x y ( + , ) ( ν ) ] sin ϑ ( + ) cos ϑ ( ) sin 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 exp [ i β y x ( + , ) ( ν ) ] ] σ 3 } .
E ( + ) ( ν ) η ( + , ) ( ν ) E ( ) ( ν ) = 1 2 | E 0 ( + ) ( ν ) | 2 1 2 | η 0 ( + , ) ( ν ) | | E 0 ( ) ( ν ) | 2 1 2 × { [ cos ϑ ( + ) cos ϑ ( ) exp [ i β x x ( + , ) ( ν ) ] + sin ϑ ( + ) sin ϑ ( ) exp [ i β y y ( + , ) ( ν ) ] ] σ 0 + [ cos ϑ ( + ) cos ϑ ( ) exp [ i β x x ( + , ) ( ν ) ] sin ϑ ( + ) sin ϑ ( ) exp [ i β y y ( + , ) ( ν ) ] ] σ 1 + [ cos ϑ ( + ) sin ϑ ( ) exp [ i β x y ( + , ) ( ν ) ] + sin ϑ ( + ) cos ϑ ( ) exp [ i β y x ( + , ) ( ν ) ] ] σ 2 i [ cos ϑ ( + ) sin ϑ ( ) exp [ i β x y ( + , ) ( ν ) ] sin ϑ ( + ) cos ϑ ( ) exp [ i β y x ( + , ) ( ν ) ] ] σ 3 } .
S ( r A ; ν ) = ( 1 λ z ) 2 AP AP { C δ ( ξ A a b 2 ) δ ( ξ D ) + C δ ( ξ A a + b 2 ) δ ( ξ D ) + δ ( ξ A a ) δ ( ξ D b ) } × { cos ϑ ( + ) cos ϑ ( ) exp [ i β x x ( + , ) ( ν ) ] + sin ϑ ( + ) sin ϑ ( ) exp [ i β y y ( + , ) ( ν ) ] } × E 0 ( + ) ( ν ) E 0 ( ) ( ν ) exp [ i k z ( ξ A r A ) ξ D ] d 2 ξ A d 2 ξ D .
S ( r A ; ν ) = ( 1 λ z ) 2 { C E 0 2 ( + ; ν ) [ cos 2 ϑ ( + ) + sin 2 ϑ ( + ) ] + C E 0 2 ( ; ν ) [ cos 2 ϑ ( ) + sin 2 ϑ ( ) ] + 2 | E 0 ( + ; ν ) | 2 1 2 | η 0 ( + , ; ν ) | | E 0 ( ; ν ) | 2 1 2 cos ϑ ( + ) cos ϑ ( ) cos ( k z b ( a r A ) + β x x ( + , ; ν ) ) + 2 | E 0 ( + ; ν ) | 2 1 2 | η 0 ( + , ; ν ) | | E 0 ( ; ν ) | 2 1 2 sin ϑ ( + ) sin ϑ ( ) cos ( k z b ( a r A ) + β y y ( + , ; ν ) ) } ,
S ( r A ; ν ) = ( 1 λ z ) 2 C { E 0 2 ( + ; ν ) + E 0 2 ( ; ν ) } ,
β x x ( + , ; ν ) = β y y ( + , ; ν ) = β 0 ( + , ; ν )
S ( r A ; ν ) = ( 1 λ z ) 2 { C E 0 2 ( + ; ν ) + C E 0 2 ( ; ν ) + 2 E 0 2 ( + ; ν ) 1 2 | η 0 ( + , ; ν ) | E 0 2 ( ; ν ) 1 2 cos ( k z b ( a r A ) + β 0 ( + , ; ν ) ) }
η ( + , ) ( ν ) = 1 2 { [ cos ϑ ( + ) cos ϑ ( ) cos 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 + sin ϑ ( + ) sin ϑ ( ) sin 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 ] η 0 ( + , ) ( ν ) σ 0 + [ cos ϑ ( + ) cos ϑ ( ) cos 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 sin ϑ ( + ) sin ϑ ( ) sin 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 ] η 0 ( + , ) ( ν ) σ 1 + [ cos ϑ ( + ) sin ϑ ( ) cos 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 η x y ( + , ) ( ν ) + sin ϑ ( + ) cos ϑ ( ) sin 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 η y x ( + , ) ( ν ) ] σ 2 i [ cos ϑ ( + ) sin ϑ ( ) cos 2 ϑ ( + ) 1 2 sin 2 ϑ ( ) 1 2 η x y ( + , ) ( ν ) sin ϑ ( + ) cos ϑ ( ) sin 2 ϑ ( + ) 1 2 cos 2 ϑ ( ) 1 2 η y x ( + , ) ( ν ) ] σ 3 } ,
E ( + ) ( ν ) η ( + , ) ( ν ) E ( ) ( ν ) = 1 2 | E 0 ( + ) ( ν ) | 2 1 2 | E 0 ( ) ( ν ) | 2 1 2 × { [ cos ϑ ( + ) cos ϑ ( ) + sin ϑ ( + ) sin ϑ ( ) ] η 0 ( + , ) ( ν ) σ 0 + [ cos ϑ ( + ) cos ϑ ( ) sin ϑ ( + ) sin ϑ ( ) ] η 0 ( + , ) ( ν ) σ 1 + [ cos ϑ ( + ) sin ϑ ( ) η x y ( + , ) ( ν ) + sin ϑ ( + ) cos ϑ ( ) η y x ( + , ) ( ν ) ] σ 2 i [ cos ϑ ( + ) sin ϑ ( ) η x y ( + , ) ( ν ) sin ϑ ( + ) cos ϑ ( ) η y x ( + , ) ( ν ) ] σ 3 } ,
η ( + , ) ( ν ) = { σ 0 , ξ A = a ± b 2 , ξ D = 0 { η 0 ( + , ) ( ν ) σ 1 1 2 [ η x y ( + , ) ( ν ) η y x ( + , ) ( ν ) ] σ 2 + i [ η x y ( + , ) ( ν ) + η y x ( + , ) ( ν ) ] σ 3 } , ξ A = a , ξ D = b } ,
E ( + ) ( ν ) η ( + , ) ( ν ) E ( ) ( ν ) = { 1 2 | E 0 ( a ± b 2 ; ν ) | 2 σ 0 ± 1 2 | E 0 ( a ± b 2 ; ν ) | 2 cos 2 ( 2 ϑ 0 ) σ 1 , ξ A = a ± b 2 , ξ D = 0 1 2 | E 0 ( + ) ( ν ) | 2 1 2 | E 0 ( ) ( ν ) | 2 1 2 { 2 sin ϑ 0 cos ϑ 0 η 0 ( + , ) ( ν ) σ 1 { ξ A = a , ξ D = b { [ cos 2 ϑ 0 η x y ( + , ) ( ν ) sin 2 ϑ 0 η y x ( + , ) ( ν ) ] σ 2 + i [ cos 2 ϑ 0 η x y ( + , ) ( ν ) + sin 2 ϑ 0 η y x ( + , ) ( ν ) ] σ 3 } , } ,
η ( + , ) ( ν ) = { η 0 ( + , ) ( ν ) σ 0 + 1 2 ( [ η x y ( + , ) ( ν ) + η y x ( + , ) ( ν ) ] σ 2 i [ η x y ( + , ) ( ν ) η y x ( + , ) ( ν ) ] σ 3 ) } ,
E ( + ) ( ν ) η ( + , ) ( ν ) E ( ) ( ν ) = 1 2 | E 0 ( + ) ( ν ) | 2 1 2 | E 0 ( ) ( ν ) | 2 1 2 { η 0 ( + , ) ( ν ) σ 0 + cos 2 ϑ 0 η 0 ( + , ) ( ν ) σ 1 + sin ϑ 0 cos ϑ 0 ( [ η x y ( + , ) ( ν ) + η y x ( + , ) ( ν ) ] σ 2 i [ η x y ( + , ) ( ν ) η y x ( + , ) ( ν ) ] σ 3 ) } ,
E ( + ) ( ν ) η ( + , ) ( ν ) E ( ) ( ν ) = T E ( + ) ( ν ) T 1 T η ( + , ) ( ν ) T 1 T E ( ) ( ν ) T 1 = T E ( + ) ( ν ) η ( + , ) ( ν ) E ( ) ( ν ) T 1
T σ 1 T = ( 2 | T d | 2 1 ) σ 1 ( T x y T y y * + T x y * T y y ) σ 2 + i ( T x y T y y * T x y * T y y ) σ 3 ,
T σ 2 T = ( T y x T y y * + T y x * T y y ) σ 1 + 1 2 ( T x x T y y * + T x y T y x * + T x x * T y y + T x y * T y x ) σ 2 i 2 ( T x x T y y * + T x y T y x * T x x * T y y T x y * T y x ) σ 3 ,
T σ 3 T = i ( T x x T x y * T x x * T x y ) σ 1 + i 2 ( T x x T y y * T x y T y x * + T x y * T y x T x x * T y y ) σ 2 + 1 2 ( T x x T y y * T x y T y x * T x y * T y x + T x x * T y y ) σ 3 .

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