Abstract

Intensity fluctuations of partially polarized light with Gaussian statistics are investigated using a field decomposition approach. These developments provide an enlightening interpretation of the Hanbury Brown–Twiss effect of partially polarized Gaussian light. In particular, the behavior of the intensity fluctuation correlations can be interpreted as resulting from the mixing of two incoherent lights between themselves.

© 2012 Optical Society of America

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References

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  1. R. H. 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, “The Hanbury Brown–Twiss effect,” in Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 458–460.
  3. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express 11, 1137–1142(2003).
    [CrossRef]
  4. 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]
  5. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beam,” J. Opt. A 10, 055001 (2008).
    [CrossRef]
  6. D. Kuebel, “Properties of the degree of cross-polarization in the space time domain,” Opt. Commun. 282, 3397–3401(2009).
    [CrossRef]
  7. A. Al-Qasimi, 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]
  8. T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown–Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
    [CrossRef]
  9. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267(2003).
    [CrossRef]
  10. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84(2003).
    [CrossRef]
  11. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
    [CrossRef]
  12. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063–1068 (2007).
    [CrossRef]
  13. R. Martinez-Herrero and P. M. Mejias, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504–1506 (2007).
    [CrossRef]
  14. R. Martinez-Herrero, P. M. Mejias, and G. Piquero, “Polarization and coherence of random electromagnetic fields,” in Characterization of Partially Polarized Light Fields (Springer-Verlag, 2009), pp. 93–124.
  15. Ph. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051–6060(2005).
    [CrossRef]
  16. L. Mandel and E. Wolf, “Random (or stochastic) processes,” in Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 41–88.
  17. J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, 1985), pp. 116–156.
  18. 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]
  19. A. Roueff and Ph. Réfrégier, “Separation technique of a mixing of two uncorrelated and perfectly polarized lights with different coherence and polarization properties,” J. Opt. Soc. Am. A 25, 838–845 (2008).
    [CrossRef]
  20. Ph. Réfrégier, M. Zerrad, and C. Amra, “Coherence and polarization properties in speckle of totally depolarized light scattered by totally depolarizing media,” Opt. Lett. 37, 205–207(2012).
    [CrossRef]
  21. Ph. Réfrégier, J. Tervo, and A. Roueff, “A temporal-coherence anisotropy of unpolarized light,” Opt. Commun. 282, 1069–1073 (2009).
    [CrossRef]
  22. J. Tervo, Ph. Réfrégier, and A. Roueff, “Minimum number of modulated Stokes parameters in Young’s interference experiment,” J. Opt. A 10, 3074–3076 (2009).

2012

2011

2010

2009

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

Ph. Réfrégier, J. Tervo, and A. Roueff, “A temporal-coherence anisotropy of unpolarized light,” Opt. Commun. 282, 1069–1073 (2009).
[CrossRef]

J. Tervo, Ph. Réfrégier, and A. Roueff, “Minimum number of modulated Stokes parameters in Young’s interference experiment,” J. Opt. A 10, 3074–3076 (2009).

2008

A. Roueff and Ph. Réfrégier, “Separation technique of a mixing of two uncorrelated and perfectly polarized lights with different coherence and polarization properties,” J. Opt. Soc. Am. A 25, 838–845 (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 beam,” J. Opt. A 10, 055001 (2008).
[CrossRef]

2007

2005

2004

2003

1956

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

Al-Qasimi, A.

Amra, C.

Borghi, R.

Brown, R. H.

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

Dogariu, A.

Ellis, J.

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, 1985), pp. 116–156.

Gori, F.

Goudail, F.

Guattari, G.

Hassinen, T.

James, D. F. V.

A. Al-Qasimi, 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 beam,” J. Opt. A 10, 055001 (2008).
[CrossRef]

Kuebel, D.

Lahiri, M.

Luis, A.

Mandel, L.

L. Mandel and E. Wolf, “The Hanbury Brown–Twiss effect,” in Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 458–460.

L. Mandel and E. Wolf, “Random (or stochastic) processes,” in Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 41–88.

Martinez-Herrero, R.

R. Martinez-Herrero and P. M. Mejias, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504–1506 (2007).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, and G. Piquero, “Polarization and coherence of random electromagnetic fields,” in Characterization of Partially Polarized Light Fields (Springer-Verlag, 2009), pp. 93–124.

Mejias, P. M.

R. Martinez-Herrero and P. M. Mejias, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504–1506 (2007).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, and G. Piquero, “Polarization and coherence of random electromagnetic fields,” in Characterization of Partially Polarized Light Fields (Springer-Verlag, 2009), pp. 93–124.

Piquero, G.

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84(2003).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, and G. Piquero, “Polarization and coherence of random electromagnetic fields,” in Characterization of Partially Polarized Light Fields (Springer-Verlag, 2009), pp. 93–124.

Réfrégier, Ph.

Roueff, A.

J. Tervo, Ph. Réfrégier, and A. Roueff, “Minimum number of modulated Stokes parameters in Young’s interference experiment,” J. Opt. A 10, 3074–3076 (2009).

Ph. Réfrégier, J. Tervo, and A. Roueff, “A temporal-coherence anisotropy of unpolarized light,” Opt. Commun. 282, 1069–1073 (2009).
[CrossRef]

A. Roueff and Ph. Réfrégier, “Separation technique of a mixing of two uncorrelated and perfectly polarized lights with different coherence and polarization properties,” J. Opt. Soc. Am. A 25, 838–845 (2008).
[CrossRef]

Santarsiero, M.

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 beam,” J. Opt. A 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]

Simon, R.

Tervo, J.

Twiss, R. Q.

R. H. 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 beam,” J. Opt. A 10, 055001 (2008).
[CrossRef]

Wolf, E.

A. Al-Qasimi, 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 beam,” J. Opt. A 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, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267(2003).
[CrossRef]

L. Mandel and E. Wolf, “The Hanbury Brown–Twiss effect,” in Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 458–460.

L. Mandel and E. Wolf, “Random (or stochastic) processes,” in Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 41–88.

Zerrad, M.

J. Opt. A

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

J. Tervo, Ph. Réfrégier, and A. Roueff, “Minimum number of modulated Stokes parameters in Young’s interference experiment,” J. Opt. A 10, 3074–3076 (2009).

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]

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

Ph. Réfrégier, J. Tervo, and A. Roueff, “A temporal-coherence anisotropy of unpolarized light,” Opt. Commun. 282, 1069–1073 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267(2003).
[CrossRef]

Other

L. Mandel and E. Wolf, “The Hanbury Brown–Twiss effect,” in Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 458–460.

R. Martinez-Herrero, P. M. Mejias, and G. Piquero, “Polarization and coherence of random electromagnetic fields,” in Characterization of Partially Polarized Light Fields (Springer-Verlag, 2009), pp. 93–124.

L. Mandel and E. Wolf, “Random (or stochastic) processes,” in Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 41–88.

J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, 1985), pp. 116–156.

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Figures (10)

Fig. 1.
Fig. 1.

Schematic illustration of the polarization states that appear in the field decomposition used for the analysis of the intensity fluctuation correlation.

Fig. 2.
Fig. 2.

(Left) Unitary transformations U1,j and U2,j are applied sequentially so that U1,j transforms uj and vj into uj and vj and U2,j transforms uj and vj into uj and vj. (Right) Schematic representation of the field decomposition obtained after unitary transformation.

Fig. 3.
Fig. 3.

Schematic representation of δI1δI2/I1I2 when the two fields of the decomposition have (left) orthogonal or (right) parallel polarization states.

Fig. 4.
Fig. 4.

Schematic representation of δI1δI2/I1I2 when the two fields of the decomposition have arbitrary polarization states. It is recalled that hθ1,θ2=cos(θ1)cos(θ2).

Fig. 5.
Fig. 5.

Sets of phase Δϕ, and angles θ1 and θ2 for which (left) g(Δϕ,θ1,θ2)=1/2 and (right) g(Δϕ,θ1,θ2)=1/20.

Fig. 6.
Fig. 6.

Representation of μTSF2/γO when the two fields of the decomposition have arbitrary polarization states, Δϕ=0 and K=1.

Fig. 7.
Fig. 7.

Representation of ρ12/μeq2 in the conditions of Example A when K=1 and Δϕ=0.

Fig. 8.
Fig. 8.

Same as Fig. 7 but when K=0.5.

Fig. 9.
Fig. 9.

Representation of ρ12/μeq2 in the conditions of Example B.

Fig. 10.
Fig. 10.

Representation of ρ12/μeq2 in the conditions of Example C.

Equations (64)

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Γj=EjEj,
Pj2=14det(Γj)tr(Γj)2,
Pj2=2tr(Γj2)tr(Γj)21.
Ω12=E1E2.
μW=tr(Ω12)tr(Γ1)tr(Γ2),
μTSF2=tr(Ω12Ω12)tr(Γ1)tr(Γ2).
M12=Γ112Ω12Γ212.
M12=U1D12U2,
D12=(μS00μI).
ρ12=δI1δI2δI12δI22,
Q12=δI1δI2I1I2.
Q12=μTSF2.
δIj2Ij2=tr(Γj2)tr(Γj)2
Q12=12(1+P122)|μW|2,
P122=2tr(Ω12Ω12)|tr(Ω12)|21,
ρ12=2μTSF21+P121+P22.
Ej=UjΓj12Ej,
EjEj=IdandE1E2=D12,
Ej=Ej,uaj,u+Ej,vaj,v,
(aj,u,aj,v)=Γj12Uj,
Ej=(Ej,uEj,v).
Ej=Ej,uβj,uu˜j+Ej,vβj,vv˜j,
uj=eiϕj,uu˜j,vj=eiϕj,vv˜j,
Ej=Aj,uuj+Aj,vvj,
Aj,m=Ej,mβj,meiϕj,m,
Ω12=αuu1u2+αvv1v2
αm=A1,mA2,m*,
αm=μmeiδϕmβ1,mβ2,m
Aj=(Aj,u+Aj,vcj)x+Aj,vsjy,
μTSF2=γO+γCg(Δϕ,θ1,θ2),
g(Δϕ,θ1,θ2)=cos(Δϕ)cos(θ1)cos(θ2),
γO=μS2β1,uβ2,u+μI2β1,vβ2,vI1I2,
γC=2μSμIβ1,uβ2,uβ1,vβ2,vI1I2.
μTSF2=γO,
μTSF2=γO+γCcos(Δϕ)=γS.
μTSF2=γO[1hθ1,θ2]+γShθ1,θ2,
μTSF2=γO[1+Kg(Δϕ,θ1,θ2)],
μTSF2=γS=γO[1+Kcos(Δϕ)].
K=2μSβ1,uβ2,uμIβ1,vβ2,vμS2β1,uβ2,u+μI2β1,vβ2,v.
r=μSμIβ1,uβ2,uβ1,vβ2,v,
K=2r1+r2,
0K1.
γO=μ2R,
R=β1,uβ2,u+β1,vβ2,vI1I2,
μTSF2=μ2R[1+Kg(Δϕ,θ1,θ2)],
K=2β1,uβ2,uβ1,vβ2,vβ1,uβ2,u+β1,vβ2,v.
(aj,u,aj,v)=Ij2Uj,
μTSF2=γO=μS2+μI24,
ρ12=μeq21+Kcos(Δϕ)P1P21+P121+P22,
μeq2=μS2+μI22,K=2μI/μS1+(μI/μS)2.
ρ12=μeq21+Kcos(Δϕ)P21+P2,
ρ12=μeq21+Kcos(Δϕ)P221+P22.
A1A2=[(A1,u+A1,vc1)x+A1,vs1y][(A2,u+A2,vc2)*x+((A2,v)*s2y],
Ω12=ηxxxx+ηyyyy+ηxyxy+ηyxyx,
ηxx=αu+αvc1c2,ηyy=αvs1s2,ηxy=αvc1s2,ηyx=αvs1c2.
Ω12Ω12=[ηxxxx+ηyyyy+ηxyxy+ηyxyx]×[ηxx*xx+ηyy*yy+ηxy*yx+ηyx*xy].
tr(Ω12Ω12)=|ηxx|2+|ηyy|2+|ηxy|2+|ηyx|2.
tr(Ω12Ω12)=|αu|2+|αv|2+(αuαv*+αu*αv)c1c2,
Ej=Aj,u+Aj,v.
Aj,uAj,v*=0,A1,mA2,m*=μmeiδϕmI1,mI2,m,
μ=E1E2*I1I2,
E1E2*=A1,uA2,u*+A1,vA2,v*.
|E1E2*|2I1I2=κu2+κv2+2κuκvcos(δϕuδϕv),
δI1δI2I1I2=δI1δI2(δI1)2(δI2)2.

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