Abstract

We demonstrate that coherence vortices, singularities of the correlation function, generally occur in partially coherent electromagnetic beams. In successive cross sections of Gaussian Schell-model beams, their locus is found to be a closed string. These coherence singularities have implications for both interference experiments and correlation of intensity fluctuation measurements performed with such beams.

© 2012 Optical Society of America

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References

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  1. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, 1999).
  2. M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 83–110.
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  4. G. Gbur and T. D. Visser, in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.
  5. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, Opt. Lett. 28, 968 (2003).
    [CrossRef]
  6. G. Gbur and T. D. Visser, Opt. Commun. 222, 117 (2003).
    [CrossRef]
  7. D. G. Fischer and T. D. Visser, J. Opt. Soc. Am. A 21, 2097 (2004).
    [CrossRef]
  8. M. L. Marasinghe, M. Premaratne, and D. M. Paganin, Opt. Express 18, 6628 (2010).
    [CrossRef]
  9. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  10. R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956).
    [CrossRef]
  11. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
    [CrossRef]
  12. T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, Opt. Express 19, 15188 (2011).
    [CrossRef]
  13. D. W. Diehl and T. D. Visser, J. Opt. Soc. Am. A 21, 2103 (2004).
    [CrossRef]

2011 (1)

2010 (1)

2008 (1)

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

2004 (2)

2003 (2)

1956 (1)

R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956).
[CrossRef]

Diehl, D. W.

Fischer, D. G.

Friberg, A. T.

Gbur, G.

G. Gbur and T. D. Visser, Opt. Commun. 222, 117 (2003).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, Opt. Lett. 28, 968 (2003).
[CrossRef]

G. Gbur and T. D. Visser, in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956).
[CrossRef]

Hassinen, T.

James, D. F. V.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

Mandel, L.

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

Marasinghe, M. L.

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, 1999).

Paganin, D. M.

Premaratne, M.

Schouten, H. F.

Setälä, T.

Shirai, T.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 83–110.

Tervo, J.

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 83–110.

Visser, T. D.

Volkov, S. N.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

Wolf, E.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, Opt. Lett. 28, 968 (2003).
[CrossRef]

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

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

J. Opt. A (1)

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A 10, 055001 (2008).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nature (1)

R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956).
[CrossRef]

Opt. Commun. (1)

G. Gbur and T. D. Visser, Opt. Commun. 222, 117 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Other (5)

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

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, 1999).

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 83–110.

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

G. Gbur and T. D. Visser, in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.

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Figures (6)

Fig. 1.
Fig. 1.

Illustrating the notation. The vector ρ = ( x , y ) indicates a transverse position.

Fig. 2.
Fig. 2.

Locus of equal modulus of W x x and W y y (red curve), and the contours of Arg [ W x x ] Arg [ W y y ] = π ( mod 2 π ). Their intersections, ρ A and ρ B , are correlation singularities. In this example A x = 1 , A y = 3 , λ = 632.8 nm , σ = 1 mm , δ x x = 0.2 mm , δ y y = 0.09 mm , z = 1.4 m , and ρ 1 = ( 2.5 , 0 ) mm .

Fig. 3.
Fig. 3.

Color-coded phase plot of the degree of coherence η ( ρ 1 , ρ 2 , z ) in the plane z = 1.4 m . The singularities at ρ A and ρ B have opposite topological charge.

Fig. 4.
Fig. 4.

Normalized spectral density of the beam in the cross section z = 1.4 m . The points ρ 1 , ρ A , and ρ B are indicated by the three white dots.

Fig. 5.
Fig. 5.

Intersection of a surface of equal amplitude (green) and a surface of opposite phase (red) constitutes a string of correlation singularities.

Fig. 6.
Fig. 6.

Two strings of correlation singularities in a partially coherent beam. The larger string (blue) is for δ y y = 0.09 mm , and the shorter string (black) is for the case δ y y = 0.12 mm .

Equations (12)

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W ( r 1 , r 2 , ω ) = ( W x x ( r 1 , r 2 , ω ) W x y ( r 1 , r 2 , ω ) W y x ( r 1 , r 2 , ω ) W y y ( r 1 , r 2 , ω ) ) ,
W i j ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω ) , ( i , j = x , y ) .
η ( r 1 , r 2 , ω ) = Tr W ( r 1 , r 2 , ω ) [ Tr W ( r 1 , r 1 , ω ) Tr W ( r 2 , r 2 , ω ) ] 1 / 2 ,
| W x x ( ρ 1 , ρ 2 , z ) | = | W y y ( ρ 1 , ρ 2 , z ) | ,
Arg [ W x x ( ρ 1 , ρ 2 , z ) ] Arg [ W y y ( ρ 1 , ρ 2 , z ) ] = π ( mod 2 π ) .
W i j ( ρ 1 , ρ 2 , z = 0 ) = S i ( ρ 1 ) S j ( ρ 2 ) μ i j ( ρ 2 ρ 1 ) ,
S i ( ρ ) = A i 2 exp ( ρ 2 / 2 σ i 2 ) ,
μ i j ( ρ 2 ρ 1 ) = B i j exp [ ( ρ 2 ρ 1 ) 2 / 2 δ i j 2 ] .
W i j ( ρ 1 , ρ 2 , z ) = A i A j B i j Δ i j 2 ( z ) exp [ ( ρ 1 + ρ 2 ) 2 8 σ 2 Δ i j 2 ( z ) ] × exp [ ( ρ 2 ρ 1 ) 2 2 Ω i j 2 Δ i j 2 ( z ) ] exp [ i k ( ρ 2 2 ρ 1 2 ) 2 R i j ( z ) ] ,
Δ i j 2 ( z ) = 1 + ( z / k σ Ω i j ) 2 ,
1 Ω i j 2 = 1 4 σ 2 + 1 δ i j 2 ,
R i j ( z ) = [ 1 + ( k σ Ω i j / z ) 2 ] z .

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