Abstract

We derive expressions that allow us to examine the influence of different source parameters on the correlation of intensity fluctuations (the Hanbury Brown–Twiss effect) at two points in the same cross section of a random electromagnetic beam. It is found that these higher-order correlations behave quite differently than the lower-order amplitude-phase correlations that are described by the spectral degree of coherence.

© 2014 Optical Society of America

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  1. R. Hanbury Brown and R. Q. Twiss, Philos. Mag. 45(366), 663 (1954).
  2. R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956).
    [CrossRef]
  3. G. Baym, Acta Phys. Pol. B 29, 1839 (1998).
  4. M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
    [CrossRef]
  5. A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 95, 090404 (2005).
    [CrossRef]
  6. T. D. Gutierrez, Phys. Rev. Lett. 96, 121802 (2006).
    [CrossRef]
  7. R. J. Glauber, Phys. Rev. 130, 2529 (1963).
    [CrossRef]
  8. D. Kuebel, T. D. Visser, and E. Wolf, Opt. Commun. 294, 43 (2013).
    [CrossRef]
  9. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  10. E. Wolf, Phys. Lett. A 312, 263 (2003).
    [CrossRef]
  11. E. Wolf, Opt. Lett. 28, 1078 (2003).
    [CrossRef]
  12. H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
    [CrossRef]
  13. T. Shirai and E. Wolf, Opt. Commun. 272, 289 (2007).
    [CrossRef]
  14. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A Pure Appl. Opt. 10, 055001 (2008).
    [CrossRef]
  15. A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, Opt. Express 18, 17124 (2010).
    [CrossRef]
  16. T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, Opt. Express 19, 15188 (2011).
    [CrossRef]
  17. Y. J. Li, Opt. Commun. 316, 67 (2014).
    [CrossRef]
  18. D. F. V. James, J. Opt. Soc. Am. A 11, 1641 (1994).
    [CrossRef]
  19. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, J. Opt. A Pure Appl. Opt. 7, 941 (1998).
    [CrossRef]
  20. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A Pure Appl. Opt. 3, 1 (2001).
    [CrossRef]
  21. T. Shirai and E. Wolf, J. Opt. Soc. Am. A 21, 1907 (2004).
    [CrossRef]
  22. O. Korotkova and E. Wolf, Opt. Commun. 246, 35 (2005).
    [CrossRef]
  23. S. B. Raghunathan, H. F. Schouten, and T. D. Visser, Opt. Lett. 37, 4179 (2012).
    [CrossRef]
  24. S. B. Raghunathan, H. F. Schouten, and T. D. Visser, J. Opt. Soc. Am. A 30, 582 (2013).
    [CrossRef]
  25. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  26. O. Korotkova, M. Salem, and E. Wolf, Opt. Lett. 29, 1173 (2004).
    [CrossRef]
  27. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, J. Opt. Soc. Am. A 25, 1016 (2008).
    [CrossRef]
  28. J. Tervo, T. Setälä, and A. T. Friberg, Opt. Express 11, 1137 (2003).
    [CrossRef]

2014 (1)

Y. J. Li, Opt. Commun. 316, 67 (2014).
[CrossRef]

2013 (2)

2012 (1)

2011 (1)

2010 (1)

2008 (2)

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, J. Opt. Soc. Am. A 25, 1016 (2008).
[CrossRef]

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

2007 (1)

T. Shirai and E. Wolf, Opt. Commun. 272, 289 (2007).
[CrossRef]

2006 (1)

T. D. Gutierrez, Phys. Rev. Lett. 96, 121802 (2006).
[CrossRef]

2005 (3)

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
[CrossRef]

A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef]

O. Korotkova and E. Wolf, Opt. Commun. 246, 35 (2005).
[CrossRef]

2004 (2)

2003 (4)

J. Tervo, T. Setälä, and A. T. Friberg, Opt. Express 11, 1137 (2003).
[CrossRef]

E. Wolf, Opt. Lett. 28, 1078 (2003).
[CrossRef]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

2001 (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A Pure Appl. Opt. 3, 1 (2001).
[CrossRef]

1998 (2)

G. Baym, Acta Phys. Pol. B 29, 1839 (1998).

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, J. Opt. A Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

1994 (1)

1963 (1)

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

1956 (1)

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

1954 (1)

R. Hanbury Brown and R. Q. Twiss, Philos. Mag. 45(366), 663 (1954).

Al-Qasimi, A.

Aspect, A.

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
[CrossRef]

Baym, G.

G. Baym, Acta Phys. Pol. B 29, 1839 (1998).

Boiron, D.

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, J. Opt. Soc. Am. A 25, 1016 (2008).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A Pure Appl. Opt. 3, 1 (2001).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, J. Opt. A Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Esslinger, T.

A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef]

Friberg, A. T.

Glauber, R. J.

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, J. Opt. Soc. Am. A 25, 1016 (2008).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A Pure Appl. Opt. 3, 1 (2001).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, J. Opt. A Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, J. Opt. A Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Gutierrez, T. D.

T. D. Gutierrez, Phys. Rev. Lett. 96, 121802 (2006).
[CrossRef]

Hanbury Brown, R.

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

R. Hanbury Brown and R. Q. Twiss, Philos. Mag. 45(366), 663 (1954).

Hassinen, T.

Hoppeler, R.

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
[CrossRef]

James, D. F. V.

Köhl, M.

A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef]

Korotkova, O.

Kuebel, D.

Lahiri, M.

Li, Y. J.

Y. J. Li, Opt. Commun. 316, 67 (2014).
[CrossRef]

Mandel, L.

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

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A Pure Appl. Opt. 3, 1 (2001).
[CrossRef]

Öttl, A.

A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef]

Perrin, A.

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
[CrossRef]

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A Pure Appl. Opt. 3, 1 (2001).
[CrossRef]

Raghunathan, S. B.

Ramírez-Sánchez, V.

Ritter, S.

A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

Salem, M.

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, J. Opt. Soc. Am. A 25, 1016 (2008).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A Pure Appl. Opt. 3, 1 (2001).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, J. Opt. A Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Schellekens, M.

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
[CrossRef]

Schouten, H. F.

Setälä, T.

Shirai, T.

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

T. Shirai and E. Wolf, Opt. Commun. 272, 289 (2007).
[CrossRef]

T. Shirai and E. Wolf, J. Opt. Soc. Am. A 21, 1907 (2004).
[CrossRef]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A Pure Appl. Opt. 3, 1 (2001).
[CrossRef]

Tervo, J.

Twiss, R. Q.

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

R. Hanbury Brown and R. Q. Twiss, Philos. Mag. 45(366), 663 (1954).

Viana Gomes, J.

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, J. Opt. A Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Visser, T. D.

Volkov, S. N.

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

Westbrook, C. I.

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
[CrossRef]

Wolf, E.

D. Kuebel, T. D. Visser, and E. Wolf, Opt. Commun. 294, 43 (2013).
[CrossRef]

A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, Opt. Express 18, 17124 (2010).
[CrossRef]

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

T. Shirai and E. Wolf, Opt. Commun. 272, 289 (2007).
[CrossRef]

O. Korotkova and E. Wolf, Opt. Commun. 246, 35 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, Opt. Lett. 29, 1173 (2004).
[CrossRef]

T. Shirai and E. Wolf, J. Opt. Soc. Am. A 21, 1907 (2004).
[CrossRef]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

E. Wolf, Opt. Lett. 28, 1078 (2003).
[CrossRef]

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

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

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

Acta Phys. Pol. B (1)

G. Baym, Acta Phys. Pol. B 29, 1839 (1998).

J. Opt. A Pure Appl. Opt. (3)

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

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, J. Opt. A Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A Pure Appl. Opt. 3, 1 (2001).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nature (1)

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

Opt. Commun. (5)

D. Kuebel, T. D. Visser, and E. Wolf, Opt. Commun. 294, 43 (2013).
[CrossRef]

Y. J. Li, Opt. Commun. 316, 67 (2014).
[CrossRef]

H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57 (2003).
[CrossRef]

T. Shirai and E. Wolf, Opt. Commun. 272, 289 (2007).
[CrossRef]

O. Korotkova and E. Wolf, Opt. Commun. 246, 35 (2005).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Philos. Mag. (1)

R. Hanbury Brown and R. Q. Twiss, Philos. Mag. 45(366), 663 (1954).

Phys. Lett. A (1)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Phys. Rev. (1)

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

Phys. Rev. Lett. (2)

A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 95, 090404 (2005).
[CrossRef]

T. D. Gutierrez, Phys. Rev. Lett. 96, 121802 (2006).
[CrossRef]

Science (1)

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005).
[CrossRef]

Other (2)

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

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

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Figures (9)

Fig. 1.
Fig. 1.

Illustrating the notation.

Fig. 2.
Fig. 2.

Contours of the normalized correlation of intensity fluctuations CN(0,ρ2,z) in the z ρ2 plane.

Fig. 3.
Fig. 3.

Contours of the modulus of the spectral degree of coherence |η(0,ρ2,z)| in the z ρ2 plane.

Fig. 4.
Fig. 4.

Evolution of (a) the normalized correlation of intensity fluctuations and (b) the modulus of the spectral degree of coherence when ρ2=0.65mm. The dashed lines are the asymptotic values given by Eqs. (25) and (27), respectively.

Fig. 5.
Fig. 5.

Evolution of the normalized correlation of intensity fluctuations as a function of z for different values of the parameter δxy. From bottom to top: δxy=2.3mm (blue), 2.6 mm (red), 2.9 mm (green), and 3.2 mm (purple). As in Fig. 4, ρ2=2mm.

Fig. 6.
Fig. 6.

Variation of (a) the normalized correlation of intensity fluctuations and (b) the modulus of the spectral degree of coherence as a function of δxx at the point ρ2=2mm, z=200m.

Fig. 7.
Fig. 7.

Variation of (a) the normalized correlation of intensity fluctuations and (b) the modulus of the spectral degree of coherence as a function of Ay at the point ρ2=2mm, z=200m.

Fig. 8.
Fig. 8.

Evolution of (a) the normalized correlation of intensity fluctuations and (b) the modulus of the spectral degree of coherence for different choices of ρ2. From bottom to top, ρ2=1.5mm (blue), 1 mm (red), 0.5 mm (green), and 0.2 mm (purple); the dashed lines are the asymptotic values given by Eqs. (24) and (26), respectively.

Fig. 9.
Fig. 9.

Variation of the far-zone value of CN(0,ρ2,z) as a function of the ratio Ay/Ax.

Equations (27)

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

I(ρ,z,ω)=|Ex(ρ,z,ω)|2+|Ey(ρ,z,ω)|2.
ΔI(ρ,z)=I(ρ,z)I(ρ,z),
Wij(ρ1,ρ2,z)=Ei*(ρ1,z)Ej(ρ2,z),(i,j=x,y).
I(ρ,z)=TrW(ρ,ρ,z),
C(ρ1,ρ2,z)=ΔI(ρ1,z)ΔI(ρ2,z).
C(ρ1,ρ2,z)=i,j|Wij(ρ1,ρ2,z)|2.
Wij(ρ1,ρ2,0)=Si(ρ1)Sj(ρ2)μij(ρ2ρ1),
Si(ρ)=Ai2exp(ρ2/2σi2),
μij(ρ2ρ1)=Bijexp[(ρ2ρ1)2/2δij2].
Bxx=Byy=1,
Bxy=Byx*,
|Bxy|,|Byx|1,
δxy=δyx.
14σ2+1δii22π2λ2,
δxx2+δyy22δxyδxxδyy|Bxy|,
|Bxy|2δyy/δxx+δxx/δyy.
Wij(ρ1,ρ2,z)=AiAjBijΔij2(z)exp[(ρ1+ρ2)28σ2Δij2(z)]×exp[(ρ1ρ2)22Ωij2Δij2(z)+ik(ρ22ρ12)2Rij(z)],
Δij2(z)=1+(z/σkΩij)2,
1Ωij2=14σ2+1δij2,
Rij(z)=[1+(σkΩij/z)2]z,
C(0,ρ2,z)=i,jAi2Aj2|Bij|2Δij4(z)×exp[ρ224σ2Δij2(z)ρ22Ωij2Δij2(z)].
CN(0,ρ2,z)=C(0,ρ2,z)I(0,z)I(ρ2,z),
I(0,z)=Ax2Δxx2(z)+Ay2Δyy2(z),
I(ρ2,z)=Ax2Δxx2(z)exp[ρ222σ2Δxx2(z)]+Ay2Δyy2(z)exp[ρ222σ2Δyy2(z)].
limzCN(0,ρ2,z)=i,jAi2Aj2|Bij|2Ωij4(Ax2Ωxx2+Ay2Ωyy2)2.
η(0,ρ2,z)=TrW(0,ρ2,z)I(0,z)I(ρ2,z),
limzη(0,ρ2,z)=1.

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