Abstract

The classic Hanbury Brown–Twiss experiment is analyzed in the space–frequency domain by taking into account the vectorial nature of the radiation. We show that as in scalar theory, the degree of electromagnetic coherence fully characterizes the fluctuations of the photoelectron currents when a random vector field with Gaussian statistics is incident onto the detectors. Interpretation of this result in terms of the modulations of optical intensity and polarization state in two-beam interference is discussed. We demonstrate that the degree of cross-polarization may generally diverge. We also evaluate the effects of the state of polarization on the correlations of intensity fluctuations in various circumstances.

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  1. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
    [CrossRef]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  3. G. Baym, “The physics of Hanbury Brown–Twiss intensity interferometer: from stars to nuclear collisions,” Acta Phys. Polonica B 29, 1839–1884 (1998).
  4. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
    [CrossRef]
  5. A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics , A. T. Friberg and R. Dändliker, eds. (SPIE Press, 2008), Chap. 9.
    [CrossRef]
  6. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic waves,” Opt. Express 11, 1137–1143 (2003).
    [CrossRef] [PubMed]
  7. A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. 20, 623–625 (1995).
    [CrossRef] [PubMed]
  8. T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. 34, 2924–2926 (2009).
    [CrossRef] [PubMed]
  9. M. Lahiri, “Polarization properties of stochastic light beams in the space–time and space–frequency domains,” Opt. Lett. 34, 2936–2938 (2009).
    [CrossRef] [PubMed]
  10. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [CrossRef] [PubMed]
  11. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
    [CrossRef] [PubMed]
  12. 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]
  13. M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (2008).
    [CrossRef]
  14. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
    [CrossRef] [PubMed]
  15. O. Korotokova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
    [CrossRef]
  16. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
    [CrossRef] [PubMed]
  17. J. Tervo, T. Setälä, A. Roueff, Ph. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
    [CrossRef] [PubMed]
  18. 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]
  19. 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]
  20. A. Al-Quasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown–Twiss effect,” Opt. Express 18, 17124–17129 (2010).
    [CrossRef]
  21. L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
    [CrossRef]
  22. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  23. T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34, 3866–3868 (2009).
    [CrossRef] [PubMed]
  24. F. Gori, J. Tervo, and J. Turunen, “Correlation matrices for completely unpolarized beams,” Opt. Lett. 34, 1447–1449 (2009).
    [CrossRef] [PubMed]

2010 (1)

2009 (5)

2008 (2)

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (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 (1)

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]

2006 (2)

2005 (1)

2004 (3)

2003 (1)

1998 (1)

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

1995 (1)

1965 (1)

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

1956 (1)

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

Alonso, M. A.

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (2008).
[CrossRef]

Al-Quasimi, A.

Baym, G.

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

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Dogariu, A.

Ellis, J.

Friberg, A. T.

T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. 34, 2924–2926 (2009).
[CrossRef] [PubMed]

T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34, 3866–3868 (2009).
[CrossRef] [PubMed]

J. Tervo, T. Setälä, A. Roueff, Ph. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
[CrossRef] [PubMed]

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]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
[CrossRef] [PubMed]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic waves,” Opt. Express 11, 1137–1143 (2003).
[CrossRef] [PubMed]

A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. 20, 623–625 (1995).
[CrossRef] [PubMed]

Gori, F.

Hanbury Brown, R.

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

Hassinen, T.

James, D. F. V.

A. Al-Quasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown–Twiss effect,” Opt. Express 18, 17124–17129 (2010).
[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]

Korotokova, O.

Kuebel, D.

Lahiri, M.

Luis, A.

A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics , A. T. Friberg and R. Dändliker, eds. (SPIE Press, 2008), Chap. 9.
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

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

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Nunziata, F.

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Réfrégier, Ph.

Roueff, A.

Setälä, T.

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]

Tervo, J.

Turunen, J.

Twiss, R. Q.

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

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.

A. Al-Quasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown–Twiss effect,” Opt. Express 18, 17124–17129 (2010).
[CrossRef]

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (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]

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]

O. Korotokova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[CrossRef]

A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. 20, 623–625 (1995).
[CrossRef] [PubMed]

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

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

Acta Phys. Polonica B (1)

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

J. Opt. A: Pure Appl. Opt. (1)

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 (1)

Nature (1)

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

Opt. Commun. (2)

M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393–2396 (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]

Opt. Express (2)

Opt. Lett. (11)

A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains,” Opt. Lett. 20, 623–625 (1995).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
[CrossRef] [PubMed]

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
[CrossRef] [PubMed]

O. Korotokova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[CrossRef]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[CrossRef] [PubMed]

F. Gori, J. Tervo, and J. Turunen, “Correlation matrices for completely unpolarized beams,” Opt. Lett. 34, 1447–1449 (2009).
[CrossRef] [PubMed]

T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. 34, 2924–2926 (2009).
[CrossRef] [PubMed]

M. Lahiri, “Polarization properties of stochastic light beams in the space–time and space–frequency domains,” Opt. Lett. 34, 2936–2938 (2009).
[CrossRef] [PubMed]

J. Tervo, T. Setälä, A. Roueff, Ph. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
[CrossRef] [PubMed]

T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34, 3866–3868 (2009).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Other (4)

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

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

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics , A. T. Friberg and R. Dändliker, eds. (SPIE Press, 2008), Chap. 9.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Square of the intensity fringe visibility |η 0(r 1, r 2, ω)|2 (dash, blue), square of the degree of cross-polarization �� 2(r 1, r 2, ω) (solid, green), and the normalized correlation of intensity fluctuations [left-hand side of Eq. (11)] (dash-dot, red), as a function of the polarization-plane rotation angle θ on a logarithmic scale.

Equations (30)

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W ( r 1 , r 2 , ω ) = E * ( r 1 , ω ) E T ( r 2 , ω ) ,
I ( r , ω ) = | E x ( r , ω ) | 2 + | E y ( r , ω ) | 2 = tr W ( r , r , ω ) ,
μ E 2 ( r 1 , r 2 , ω ) = tr  [ W ( r 1 , r 2 , ω ) W ( r 2 , r 1 , ω ) ] I ( r 1 , ω ) I ( r 2 , ω ) .
Δ I ( r , ω ) = I ( r , ω ) I ( r , ω ) = I ( r , ω ) tr W ( r , r , ω ) .
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) = i , j | W i j ( r 1 , r 2 , ω ) | 2 = tr [ W ( r 1 , r 2 , ω ) W ( r 2 , r 1 , ω ) ] ,
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) = I ( r 1 , ω ) I ( r 2 , ω ) μ E 2 ( r 1 , r 2 , ω ) .
μ E 2 ( r 1 , r 2 , ω ) = 1 2 j = 0 3 | η j ( r 1 , r 2 , ω ) | 2 ,
η j ( r 1 , r 2 , ω ) = 𝒮 j ( r 1 , r 2 , ω ) [ I ( r 1 , ω ) I ( r 2 , ω ) ] 1 / 2 , j = ( 0 , , 3 ) ,
𝒮 0 ( r 1 , r 2 , ω ) = W x x ( r 1 , r 2 , ω ) + W y y ( r 1 , r 2 , ω ) ,
𝒮 1 ( r 1 , r 2 , ω ) = W x x ( r 1 , r 2 , ω ) W y y ( r 1 , r 2 , ω ) ,
𝒮 2 ( r 1 , r 2 , ω ) = W y x ( r 1 , r 2 , ω ) + W x y ( r 1 , r 2 , ω ) ,
𝒮 3 ( r 1 , r 2 , ω ) = i [ W y x ( r 1 , r 2 , ω ) W x y ( r 1 , r 2 , ω ) ] .
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) I ( r 1 , ω ) I ( r 2 , ω ) = 1 2 j = 0 3 | η j ( r 1 , r 2 , ω ) | 2 .
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) I ( r 1 , ω ) I ( r 2 , ω ) = 1 2 [ 1 + 𝒫 2 ( r 1 , r 2 , ω ) ] | η 0 ( r 1 , r 2 , ω ) | 2 ,
𝒫 ( r 1 , r 2 , ω ) = Σ j = 1 3 | η j ( r 1 , r 2 , ω ) | 2 | η 0 ( r 1 , r 2 , ω ) |
W ( r 1 , r 2 , ω ) = F ( r 1 , r 2 , ω ) J ( r 1 , r 2 , ω ) ,
F ( r 1 , r 2 , ω ) = a * ( r 1 ) a ( r 2 )
J ( r 1 , r 2 , ω ) = e ^ * ( r 1 ) e ^ T ( r 2 ) .
μ i j ( r 1 , r 2 , ω ) = W i j ( r 1 , r 2 , ω ) [ | E i ( r 1 , ω ) | 2 | E j ( r 2 , ω ) | 2 1 / 2 , ( i , j ) = ( x , y ) ,
W ( r 1 , r 2 , ω ) = [ F ( r 1 , ω ) F ( r 2 , ω ) ] 1 / 2 × ( μ x x ( r 1 , r 2 , ω ) [ J x x ( r 1 , ω ) J x x ( r 2 , ω ) ] 1 / 2 μ x y ( r 1 , r 2 , ω ) [ J x x ( r 1 , ω ) J y y ( r 2 , ω ) ] 1 / 2 μ y x ( r 1 , r 2 , ω ) [ J y y ( r 1 , ω ) J x x ( r 2 , ω ) ] 1 / 2 μ y y ( r 1 , r 2 , ω ) [ J y y ( r 1 , ω ) J y y ( r 2 , ω ) ] 1 / 2 ) ,
μ x y ( r 1 , r 2 , ω ) = μ x y ( r 1 , r 1 , ω ) μ x x ( r 1 , r 2 , ω ) ,
μ x x ( r 1 , r 2 , ω ) = μ y y ( r 1 , r 2 , ω ) μ ( r 1 , r 2 , ω ) ,
W ( r 1 , r 2 , ω ) = μ ( r 1 , r 2 , ω ) [ F ( r 1 , ω ) F ( r 2 , ω ) ] 1 / 2 × ( [ J x x ( r 1 , ω ) J x x ( r 2 , ω ) ] 1 / 2 μ x y ( r 1 , r 1 , ω ) [ J x x ( r 1 , ω ) J y y ( r 2 , ω ) ] 1 / 2 μ y x ( r 2 , r 2 , ω ) [ J y y ( r 1 , ω ) J x x ( r 2 , ω ) ] 1 / 2 [ J y y ( r 1 , ω ) J y y ( r 2 , ω ) ] 1 / 2 ) .
P ( r , ω ) = [ 1 4 det J ( r , ω ) tr 2 J ( r , ω ) ] 1 / 2 .
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) I ( r 1 , ω ) I ( r 2 , ω ) = 1 2 [ 1 + P 2 ( r 1 , ω ) + P 2 ( r 2 , ω ) 2 ] | μ ( r 1 , r 2 , ω ) | 2 .
W ( r 1 , r 2 , ω ) = F ( r 1 , r 2 , ω ) U ( r 1 ) J ( ω ) U ( r 2 ) ,
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) I ( r 1 , ω ) I ( r 2 , ω ) = 1 2 [ 1 + P 2 ( ω ) ] | f ( r 1 , r 2 , ω ) | 2 ,
f ( r 1 , r 2 , ω ) = F ( r 1 , r 2 , ω ) [ F ( r 1 , r 1 , ω ) F ( r 2 , r 2 , ω ) ] 1 / 2
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) I ( r 1 , ω ) I ( r 2 , ω ) = 1 2 [ 1 + P 2 ( ω ) ] | μ ( r 1 , r 2 , ω ) | 2 ,
𝒫 2 ( r 1 , r 2 , ω ) = [ 1 + P 2 ( ω ) ] | f ( r 1 , r 2 , ω ) η 0 ( r 1 , r 2 , ω ) | 2 1.

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