Abstract

The van Cittert-Zernike theorem, well known for the scalar optical fields, is generalized for the case of vector electromagnetic fields. The deduced theorem shows that the degree of coherence of the electromagnetic field produced by the completely incoherent vector source increases on propagation whereas the degree of polarization remains unchanged. The possible application of the deduced theorem is illustrated by an example of optical simulation of partially coherent and partially polarized secondary source with the controlled statistical properties.

© 2009 Optical Society of America

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References

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  1. P. H. van Cittert, "Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene," Physica 1, 201-210 (1934).
    [CrossRef]
  2. F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica 5, 785-795 (1938).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, UK, 1997).
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  5. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, UK, 2007).
  6. A. S. Ostrovsky, P. Martínez-Vara, M. A. Olvera-Santamaría, and G. Martínez-Niconoff, "Vector coherence theory: An overview of Basic concepts and definitions," in Recent Research Developments in Optics, S.G. Pandalai, ed. (Research Signpost, Trivandrum, Kerala, India, to be published).
  7. J. Tervo, T. Setälä, and A. T. Friberg, "Degree of coherence for electromagnetic fields," Opt. Express 11, 1137-1143 (2003).
    [CrossRef] [PubMed]
  8. J. Tervo, T. Setälä, and A. T. Friberg, "Theory of partially coherent electromagnetic fields in the space-frequency domain," J. Opt. Soc. Am. A 21, 2205-2215 (2004).
    [CrossRef]
  9. E. Wolf, "Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation," Opt. Lett. 28, 1078-1080 (2003).
    [CrossRef] [PubMed]

2004

2003

1938

F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica 5, 785-795 (1938).
[CrossRef]

1934

P. H. van Cittert, "Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene," Physica 1, 201-210 (1934).
[CrossRef]

Friberg, A. T.

Setälä, T.

Tervo, J.

van Cittert, P. H.

P. H. van Cittert, "Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene," Physica 1, 201-210 (1934).
[CrossRef]

Wolf, E.

Zernike, F.

F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica 5, 785-795 (1938).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Physica

P. H. van Cittert, "Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene," Physica 1, 201-210 (1934).
[CrossRef]

F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica 5, 785-795 (1938).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, UK, 1997).

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

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

A. S. Ostrovsky, P. Martínez-Vara, M. A. Olvera-Santamaría, and G. Martínez-Niconoff, "Vector coherence theory: An overview of Basic concepts and definitions," in Recent Research Developments in Optics, S.G. Pandalai, ed. (Research Signpost, Trivandrum, Kerala, India, to be published).

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Figures (1)

Fig. 1
Fig. 1

Modified Mach-Zehnder interferometer: BS - beam splitter; M - mirror; P - polarizer; OW - optical wedge.

Equations (29)

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{ E ( x , v ) } = { E x ( x , v ) E y ( x , v ) } ,
W ( x 1 , x 2 ) = [ W ij ( x 1 , x 2 ) ] = E i * ( x 1 ) E j ( x 2 ) ( i , j = x , y ) ,
S i ( x ) = W ii ( x , x ) = E i ( x ) 2 ,
S ( x ) = E ( x ) E ( x ) = S i ( x ) + S j ( x ) = Tr W ( x , x ) ,
W ij ( x 1 , x 2 ) S i ( x 1 ) S j ( x 2 ) .
μ ij ( x 1 , x 2 ) = W ij ( x 1 , x 2 ) S i ( x 1 ) S j ( x 2 ) ,
0 μ ij ( x 1 , x 2 ) 1 .
μ ˜ ( x 1 , x 2 ) = ( i , j W ij ( x 1 , x 2 ) 2 i , j S i ( x 1 ) S j ( x 2 ) ) 1 / 2 = ( Tr [ W ( x 1 , x 2 ) W ( x 1 , x 2 ) ] Tr W ( x 1 , x 1 ) Tr W ( x 2 , x 2 ) ) 1 / 2 .
μ ˜ ( x 1 , x 2 ) = ( i , j μ ij ( x 1 , x 2 ) 2 S i ( x 1 ) S j ( x 2 ) i , j S i ( x 1 ) S j ( x 2 ) ) 1 / 2 .
W ( x , x ) = W xx ( x , x ) [ 1 0 0 1 ] ,
Det W ( x , x ) = 0 ,
P ( x ) = ( 1 4 Det W ( x , x ) [ Tr W ( x , x ) ] 2 ) 1 / 2 .
2 Det W ( x , x ) = [ Tr W ( x , x ) ] 2 Tr [ W ( x , x ) W ( x , x ) ] ,
P ( x ) = 2 μ ˜ 2 ( x , x ) 1 .
E i ( z ) ( x ) = 1 i λz exp ( i 2 π λ z ) ( z = 0 ) E i ( 0 ) ( x ) exp [ i π λz ( x x ) ] d x ,
W ( z ) ( x 1 , x 2 ) = 1 ( λz ) 2 ( z = 0 ) W ( 0 ) ( x 1 , x 2 ) exp { i π λz [ ( x 2 x 2 ) 2 ( x 1 x 1 ) 2 ] } d x 1 d x 2 .
W ii ( 0 ) ( x 1 , x 2 ) = E i ( 0 ) * ( x 1 ) E i ( 0 ) ( x 2 ) = η S i ( 0 ) ( x 1 ) S i ( 0 ) ( x 2 ) δ ( x 1 x 2 )
W ij ( 0 ) ( x 1 , x 2 ) = E i ( 0 ) * ( x 1 ) E j ( 0 ) ( x 2 ) = E i ( 0 ) * ( x 1 ) E j ( 0 ) ( x 2 ) = 0 ,
W ii ( z ) ( x 1 , x 2 ) = η ( λ z ) 2 exp [ i π λ z ( x 1 2 x 2 2 ) ] ( z = 0 ) S i ( 0 ) ( x ) exp ( i 2 π λz x · Δ x ) d x
W ij ( z ) ( x 1 , x 2 ) = 0 ,
μ ii ( z ) ( x 1 , x 2 ) = ( z = 0 ) S i ( 0 ) ( x ) exp ( i 2 π λz x · Δ x ) d x ( z = 0 ) S i ( 0 ) ( x ) d x .
μ ˜ ( z ) ( x 1 , x 2 ) = ( i ( z = 0 ) S i ( 0 ) ( x ) exp ( i 2 π λz x · Δ x ) d x 2 i , j ( z = 0 ) S i ( 0 ) ( x ) d x ( z = 0 ) S j ( 0 ) ( x ) d x ) 1 / 2 .
i , j ( z = 0 ) S i ( 0 ) ( x ) d x ( z = 0 ) S i ( 0 ) ( x ) d x = [ ( z = 0 ) S ( 0 ) ( x ) d x ] 2 ,
μ ˜ ( z ) ( x 1 , x 2 ) = ( i ( z = 0 ) S i ( 0 ) ( x ) exp ( i 2 π λz x · Δ x ) d x 2 ) 1 / 2 ( z = 0 ) S ( 0 ) ( x ) d x .
P ( z ) ( x ) = 4 α 2 4 α + 1 ,
α = ( z = 0 ) S x ( 0 ) ( x ) d x ( z = 0 ) S ( 0 ) ( x ) d x .
μ ˜ ( z ) ( x 1 , x 2 ) = α 2 + ( 1 α ) 2 π R 2 ( x R ) exp ( i 2 π λz x · Δ x ) d x .
μ ˜ ( z ) ( x 1 , x 2 ) = 2 α 2 2 α + 1 2 J 1 ( 2 π R λ z Δ x ) ( 2 π R λ z Δ x ) ,
r coh = 0.16 λz 2 R .

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