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

Full Article  |  PDF Article
Related Articles
Propagation-induced polarization changes in partially coherent optical beams

Govind P. Agrawal and Emil Wolf
J. Opt. Soc. Am. A 17(11) 2019-2023 (2000)

Use of the van Cittert–Zernike theorem for partially polarized sources

Franco Gori, Massimo Santarsiero, Riccardo Borghi, and Gemma Piquero
Opt. Lett. 25(17) 1291-1293 (2000)

Generation of complete coherence in Young’s interference experiment with random mutually uncorrelated electromagnetic beams

G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf
Opt. Lett. 30(2) 120-122 (2005)

References

  • View by:
  • |
  • |
  • |

  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 (1)

2003 (2)

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

1934 (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]

Born, M.

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

Cittert, P.H. van

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.

Mandel, L.

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

Martínez-Niconoff, G.

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).

Martínez-Vara, P.

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).

Olvera-Santamaría, M. A.

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).

Ostrovsky, A. S.

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).

Setälä, T.

Tervo, J.

Wolf, E.

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]

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).

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

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 (1)

Opt. Express (1)

Opt. Lett. (1)

Physica (2)

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 (4)

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).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

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

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

{ 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 .

Metrics