Abstract

We show, by an example, that the knowledge of the degree of coherence and of the degree of polarization of a light beam incident on two photo detectors is not adequate to predict correlations in the fluctuations of the currents generated in the detectors (the Hanbury Brown-Twiss effect). The knowledge of the so-called degree of cross-polarization, introduced not long ago, is also needed.

© 2010 Optical Society of America

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  1. R. H. Brown, and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).
  2. R. H. Brown, and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
    [CrossRef]
  3. R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, I: basic theory: the correlation between photons in coherent beams of radiation,” Proc. Roy. Soc. (London) Sec. A 242, 300–324 (1957).
    [CrossRef]
  4. R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, II: an experimental test of the theory for partially coherent light,” Proc. Roy. Soc. (London) Sec. A 243, 291–319 (1957).
    [CrossRef]
  5. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, Cambridge University Press, 1995).
  6. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).
  7. G. Baym, “The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions,” Acta Phys. Poln. B 29, 1839–1884 (1998).
  8. D. Kleppner, “Hanbury Brown’s steamroller,” Phys. Today 61, 8–9 (2008).
    [CrossRef]
  9. T. Shirai, and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007).
    [CrossRef]
  10. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10, 055001 (2008).
    [CrossRef]
  11. In Refs. [9] and [10] the degree of cross-polarization was defined in the space-frequency domain. A definition of the degree of the degree of cross-polarization in the space-time domain was introduced in Ref. [12]. Another two point generalization of the degree of polarization, called complex degree of mutual polarization was introduced in Ref. [13].
  12. D. Kuebel, “Properties of the degree of cross-polarization in the space-time domain,” Opt. Commun. 282, 3397–3401 (2009).
    [CrossRef]
  13. J. Ellis, and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
    [CrossRef] [PubMed]
  14. 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]
  15. H. Roychowdhury, and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
    [CrossRef]
  16. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
    [CrossRef]

2009

D. Kuebel, “Properties of the degree of cross-polarization in the space-time domain,” Opt. Commun. 282, 3397–3401 (2009).
[CrossRef]

2008

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008).
[CrossRef]

D. Kleppner, “Hanbury Brown’s steamroller,” Phys. Today 61, 8–9 (2008).
[CrossRef]

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10, 055001 (2008).
[CrossRef]

2007

T. Shirai, and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007).
[CrossRef]

2005

H. Roychowdhury, and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

2004

2003

1998

G. Baym, “The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions,” Acta Phys. Poln. B 29, 1839–1884 (1998).

1957

R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, I: basic theory: the correlation between photons in coherent beams of radiation,” Proc. Roy. Soc. (London) Sec. A 242, 300–324 (1957).
[CrossRef]

R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, II: an experimental test of the theory for partially coherent light,” Proc. Roy. Soc. (London) Sec. A 243, 291–319 (1957).
[CrossRef]

1956

R. H. Brown, and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

1954

R. H. Brown, and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

Baym, G.

G. Baym, “The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions,” Acta Phys. Poln. B 29, 1839–1884 (1998).

Borghi, R.

Brown, R. H.

R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, I: basic theory: the correlation between photons in coherent beams of radiation,” Proc. Roy. Soc. (London) Sec. A 242, 300–324 (1957).
[CrossRef]

R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, II: an experimental test of the theory for partially coherent light,” Proc. Roy. Soc. (London) Sec. A 243, 291–319 (1957).
[CrossRef]

R. H. Brown, and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

R. H. Brown, and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

Dogariu, A.

Ellis, J.

Gori, F.

James, D. F. V.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10, 055001 (2008).
[CrossRef]

Kleppner, D.

D. Kleppner, “Hanbury Brown’s steamroller,” Phys. Today 61, 8–9 (2008).
[CrossRef]

Korotkova, O.

H. Roychowdhury, and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

Kuebel, D.

D. Kuebel, “Properties of the degree of cross-polarization in the space-time domain,” Opt. Commun. 282, 3397–3401 (2009).
[CrossRef]

Ramírez-Sánchez, V.

Roychowdhury, H.

H. Roychowdhury, and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

Santarsiero, M.

Shirai, T.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10, 055001 (2008).
[CrossRef]

T. Shirai, and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007).
[CrossRef]

Twiss, R. Q.

R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, II: an experimental test of the theory for partially coherent light,” Proc. Roy. Soc. (London) Sec. A 243, 291–319 (1957).
[CrossRef]

R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, I: basic theory: the correlation between photons in coherent beams of radiation,” Proc. Roy. Soc. (London) Sec. A 242, 300–324 (1957).
[CrossRef]

R. H. Brown, and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

R. H. Brown, and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

Volkov, S. N.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10, 055001 (2008).
[CrossRef]

Wolf, E.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10, 055001 (2008).
[CrossRef]

T. Shirai, and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007).
[CrossRef]

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]

Acta Phys. Poln. B

G. Baym, “The Physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions,” Acta Phys. Poln. B 29, 1839–1884 (1998).

J. Opt. A, Pure Appl. Opt.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A, Pure Appl. Opt. 10, 055001 (2008).
[CrossRef]

J. Opt. Soc. Am. A

Nature

R. H. Brown, and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

Opt. Commun.

T. Shirai, and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007).
[CrossRef]

H. Roychowdhury, and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

D. Kuebel, “Properties of the degree of cross-polarization in the space-time domain,” Opt. Commun. 282, 3397–3401 (2009).
[CrossRef]

Opt. Lett.

Philos. Mag.

R. H. Brown, and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45, 663–682 (1954).

Phys. Today

D. Kleppner, “Hanbury Brown’s steamroller,” Phys. Today 61, 8–9 (2008).
[CrossRef]

Proc. Roy. Soc. (London) Sec. A

R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, I: basic theory: the correlation between photons in coherent beams of radiation,” Proc. Roy. Soc. (London) Sec. A 242, 300–324 (1957).
[CrossRef]

R. H. Brown, and R. Q. Twiss, ““Interferometry of the intensity fluctuations in light, II: an experimental test of the theory for partially coherent light,” Proc. Roy. Soc. (London) Sec. A 243, 291–319 (1957).
[CrossRef]

Other

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

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

In Refs. [9] and [10] the degree of cross-polarization was defined in the space-frequency domain. A definition of the degree of the degree of cross-polarization in the space-time domain was introduced in Ref. [12]. Another two point generalization of the degree of polarization, called complex degree of mutual polarization was introduced in Ref. [13].

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Figures (2)

Fig. 1.
Fig. 1.

Variation of the degree of cross-polarization ��( ρ ,− ρ , 0;ω) with ρ, at a pair of diametrically opposite points in two Gaussian Schell-model sources separated by distance 2ρ, for δ = 0.001m and σ = 0.01m. The labels (a) and (b) refer to the beams which are labeled in this way in the text.

Fig. 2.
Fig. 2.

Variation of the correlation between the intensity fluctuations C( ρ ,− ρ , z 0;ω) with ρ, at two diametrically opposite points separated by distance 2ρ, for the two beams, in a plane at distance z 0 = 10km from the sources. The labels (a) and (b) have the same meaning as in Fig. 1.

Equations (27)

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W ( ρ 1 , ρ 2 , z ; ω ) [ W ij ( ρ 1 , ρ 2 , z ; ω ) ] = E i * ( ρ 1 , z ; ω ) E j ( ρ 2 , z ; ω ) , ( i = x , y ; j = x , y ) .
S ( ρ , z ; ω ) Tr W ( ρ , ρ , z ; ω ) ,
μ ( ρ 1 , ρ 2 , z ; ω ) Tr W ( ρ 1 , ρ 2 , z ; ω ) S ( ρ 1 , z ; ω ) S ( ρ 2 , z ; ω ) ,
𝒫 ( ρ , z ; ω ) 1 4 Det W ( ρ , ρ , z ; ω ) { Tr W ( ρ , ρ , z ; ω ) } 2 ,
C ( ρ 1 , ρ 2 , z ; ω ) Δ I ( ρ 1 , z ; ω ) Δ I ( ρ 2 , z ; ω )
= 1 2 [ 1 + 𝒬 ( ρ 1 , ρ 2 , z ; ω ) ] μ ( ρ 1 , ρ 2 , z ; ω ) 2 S ( ρ 1 , z ; ω ) S ( ρ 2 , z ; ω ) ,
𝒬 ( ρ 1 , ρ 2 , z ; ω ) = 2 Tr [ W ( ρ 1 , ρ 2 , z , ω ) · W ( ρ 1 , ρ 2 , z ; ω ) ] Tr W ( ρ 1 , ρ 2 , z ; ω ) 2 1
W ( ρ 1 , ρ 2 , z ; ω ) = W ( ρ 1 , ρ 2 , 0 ; ω ) G * ( ρ 1 ρ 1 , z ; ω ) G ( ρ 2 ρ 2 , z ; ω ) d 2 ρ 1 d 2 ρ 2 ,
G ( ρ ρ , z ; ω ) = ik 2 π z exp [ ik ( ρ ρ ) 2 2 z ]
W ij ( ρ 1 , ρ 2 , 0 ; ω ) = A i A j B ij exp [ ( ρ 1 2 4 σ i 2 + ρ 2 2 4 σ j 2 ) ] exp [ ( ρ 2 ρ 1 ) 2 2 δ ij 2 ] , i = x , y ; j = x , y ,
B ij = 1 when i = j ,
B ij < 1 when i j ,
B ij = B ij * ,
δ ij = δ ji .
max { δ xx , δ yy } δ xy min { δ xx B xy , δ yy B xy } .
W ij ( ρ 1 , ρ 2 , z ; ω ) = A i A j B ij Δ ij 2 ( z ) exp [ ( ρ 1 + ρ 2 ) 2 8 σ 2 Δ ij 2 ( z ) ] exp [ ( ρ 2 ρ 1 ) 2 2 Ω ij Δ ij 2 ( z ) ] exp [ ik ( ρ 2 2 ρ 1 2 ) 2 R ij ( z ) ] ,
R ij = z [ 1 + ( k σ Ω ij z ) 2 ] ,
1 Ω ij 2 = 1 4 σ 2 + 1 δ ij 2 ,
Δ ij ( z ) = 1 + ( z k σ Ω ij ) 2 .
W ( a ) ( ρ 1 , ρ 2 , 0 ; ω ) = exp [ ρ 1 2 + ρ 2 2 4 σ 2 ] ( 𝒜 𝓑 𝓑 𝒜 ) ,
𝒜 = exp [ ( ρ 2 ρ 1 ) 2 2 δ 2 ] ,
𝓑 = 9 16 exp [ 9 ( ρ 2 ρ 1 ) 2 32 δ 2 ] .
W ( b ) ( ρ 1 , ρ 2 , 0 ; ω ) = exp [ ρ 1 2 + ρ 2 2 4 σ 2 ] ( 𝒜 𝒞 𝒞 𝒜 ) ,
𝒞 = 9 16 exp [ ( ρ 2 ρ 1 ) 2 2 δ 2 ] .
S ( a ) ( ρ , 0 ; ω ) = S ( b ) ( ρ , 0 ; ω ) = 2 exp [ ρ 2 2 σ 2 ] ,
μ ( a ) ( ρ 1 , ρ 2 , 0 ; ω ) = μ ( b ) ( ρ 1 , ρ 2 , 0 ; ω ) = exp [ ( ρ 2 ρ 1 ) 2 δ 2 2 ] ,
𝒫 ( a ) ( ρ , 0 ; ω ) = 𝒫 ( b ) ( ρ , 0 ; ω ) = 9 16 .

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