Abstract

A generalized van Cittert–Zernike theorem for the cross-spectral density matrix of quasi-homogeneous planar electromagnetic sources is introduced. We present theoretical examples of using this theorem to generate fields with interesting polarization and spatial coherence properties by choosing the appropriate spectral density distribution of the source. We found that under certain conditions, a quasi-homogeneous, polarized source may produce a beam in the far field that is unpolarized in the typical one-point sense but polarized in the two-point, mutual polarization sense.

© 2012 Optical Society of America

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References

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  1. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. 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]
  5. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
    [CrossRef]
  6. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
    [CrossRef]
  7. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  8. J. W. Goodman, Statistical Optics (Wiley, 2000).
  9. E. Baleine and A. Dogariu, “Variable coherence tomography,” Opt. Lett. 29, 1233–1235 (2004).
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  12. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
    [CrossRef]
  13. O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22, 2547–2556 (2005).
    [CrossRef]
  14. 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).
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  15. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
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  16. 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]
  17. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
    [CrossRef]

2008

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

2004

2003

2000

1998

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

1994

Agrawal, G. P.

Baleine, E.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Dogariu, A.

Ellis, J.

Friberg, A. T.

Gamiz, V. L.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Hoover, B. G.

James, D. F. V.

Korotkova, O.

Piquero, G.

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Setälä, T.

Shirai, T.

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.

Turner, T. S.

Tyo, J. S.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Wolf, E.

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. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[CrossRef]

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22, 2547–2556 (2005).
[CrossRef]

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

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

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
[CrossRef]

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

J. Opt. Soc. Am. A

Opt. Commun.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[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. Lett.

Phys. Lett. A

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

Pure Appl. Opt.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Other

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

J. W. Goodman, Statistical Optics (Wiley, 2000).

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Figures (5)

Fig. 1.
Fig. 1.

Diagram with the geometry for the calculation of the generalized van Cittert–Zernike theorem for the cross-spectral density matrix.

Fig. 2.
Fig. 2.

Spectral density distribution produced by (a) the condition in Eq. (32) with α=0.25R, β=0, and x0=0 and (b) the condition in Eq. (39) with α=0.5R, β=0, x0=0.5R, and y0=0. The x-polarized and y-polarized components have the same spectral density distribution (ϵ=0).

Fig. 3.
Fig. 3.

Cross-spectral density matrix for y-oriented linear fringes with absolute phase difference ϵ=0.

Fig. 4.
Fig. 4.

Spectral density distribution produced by the condition in Eq. (32). (a) x-polarized component and (b) y-polarized component with α=0.25R, β=0, and x0=0. The y-polarized fringes are spatially shifted with respect to the x-polarized fringes due to the absolute phase difference ϵ=π/2.

Fig. 5.
Fig. 5.

Cross-spectral density matrix for y-oriented linear fringes with absolute phase difference ϵ=π/2.

Equations (48)

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Γ̳(r1,r2,τ)=[Ex*(r1,t)Ex(r2,t+τ)Ex*(r1,t)Ey(r2,t+τ)Ey*(r1,t)Ex(r2,t+τ)Ey*(r1,t)Ey(r2,t+τ)],
Ei(Q,t)=Σ1jλ¯rEi(P,trc)χ(θ)dS,
Γij(Q1,Q2,τ)=Σ(1)jλ¯r1Ei*(P1,tr1c)χ(θ1)dS1Σ1jλ¯r2Ej(P2,t+τr2c)χ(θ2)dS2.
Γij(Q1,Q2,τ)=ΣΣEi*(P1,t)Ej(P2,t+τ(r2r1)c)χ(θ1)λ¯r1χ(θ2)λ¯r2dS1dS2.
Γij(Q1,Q2,τ)Γij(Q1,Q2,0)ej2πν¯τ,
Γij(Q1,Q2,τ)ΣΣΓij(0)(P1,P2,τ)exp[j2πλ¯(r2r1)]χ(θ1)λ¯r1χ(θ2)λ¯r2dS1dS2,
(Q1,Q2;ω)=12πΓ̳(Q1,Q2,τ)ejωτdτ.
Wij(Q1,Q2;ω)ΣΣWij(0)(P1,P2;ω)exp[j2πλ¯(r2r1)]×χ(θ1)λ¯r1χ(θ2)λ¯r2dS1dS2.
(Q1,Q2;ω)ΣΣ(0)(P1,P2;ω)exp[j2πλ¯(r2r1)]×χ(θ1)λ¯r1χ(θ2)λ¯r2dS1dS2.
(Q1,Q2,ω)=[Ex*(Q1,ω)Ex(Q2,ω)Ex*(Q1,ω)Ey(Q2,ω)Ey*(Q1,ω)Ex(Q2,ω)Ey*(Q1,ω)Ey(Q2,ω)].
χ(θ1)χ(θ2)1.
r2r1ξ¯Δξ+η¯Δη+x¯Δx+y¯Δyξ¯Δxx¯Δξη¯Δyy¯Δηz,
ξ¯=ξ1+ξ22,Δξ=ξ2ξ1,η¯=η1+η22,Δη=η2η1,x¯=x1+x22,Δx=x2x1,y¯=y1+y22,Δy=y2y1,
z>4ξ¯Δξλ¯andz>4η¯Δηλ¯,
Wij(x1,y1,x2,y2,z;ω)=ejψ(λ¯z)2ΣΣWij(0)(ξ1,η1,ξ2,η2;ω)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]×exp[j2πλ¯z(x¯Δξ+y¯Δη)]dS1dS2,
ψ=2πλ¯z(x¯Δx+y¯Δy).
z>2Ddcλ¯,
(x1,y1,x2,y2,z;ω)=ejψ(λ¯z)2ΣΣ(0)(ξ1,η1,ξ2,η2;ω)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]×exp[j2πλ¯z(x¯Δξ+y¯Δη)]dS1dS2.
(0)(ξ1,η1,ξ2,η2;ω)=1/2(ξ¯,η¯;ω)μ̳(Δξ,Δη;ω)1/2(ξ¯,η¯;ω),
1/2(ξ¯,η¯;ω)=[[Ix(ξ¯,η¯;ω)]1/200[Iy(ξ¯,η¯;ω)]1/2],
(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2ΣΣ1/2(ξ¯,η¯;ω)μ̳(Δξ,Δη;ω)1/2(ξ¯,η¯;ω)×exp[j2πλ¯z(ξ¯Δx+η¯Δy)]exp[j2πλ¯z(x¯Δξ+y¯Δη)]dξ¯dη¯dΔξdΔη.
(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)21/2(ξ¯,η¯;ω)κ̳(x¯,y¯,z;ω)1/2(ξ¯,η¯;ω)×exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯,
κ̳(x¯,y¯,z;ω)=μ̳(Δξ,Δη;ω)exp[j2πλ¯z(x¯Δξ+y¯Δη)]dΔξdΔη.
(0)(ξ1,η1,ξ2,η2;ω)=1/2(ξ¯,η¯;ω)δ(Δξ,Δη)[1ejΔϕejΔϕ1]1/2(ξ¯,η¯;ω),
(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)21/2(ξ¯,η¯;ω)[1ejΔϕejΔϕ1]1/2(ξ¯,η¯;ω)×exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯.
1/2(ξ¯,η¯;ω)[1ejΔϕejΔϕ1]1/2(ξ¯,η¯;ω)=[Ix(ξ¯,η¯;ω)[Ix(ξ¯,η¯;ω)Iy(ξ¯,η¯;ω)]1/2ejΔϕ[Ix(ξ¯,η¯;ω)Iy(ξ¯,η¯;ω)]1/2ejΔϕIy(ξ¯,η¯;ω)].
(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2[Ix(ξ¯,η¯;ω)[Ix(ξ¯,η¯;ω)Iy(ξ¯,η¯;ω)]1/2ejΔϕ[Ix(ξ¯,η¯;ω)Iy(ξ¯,η¯;ω)]1/2ejΔϕIy(ξ¯,η¯;ω)]exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯.
1/2(ξ¯,η¯;ω)=[E0xcos(ϱ(ξ¯,η¯;ω))00E0ycos(ϱ(ξ¯,η¯;ω)+ϵ)],
Wxx(0)(ξ¯,η¯,Δξ,Δη;ω)=E0x22μxx(1+cos(2ϱ(ξ¯,η¯;ω))),Wxy(0)(ξ¯,η¯,Δξ,Δη;ω)=E0xE0y2μxy(cosϵ+cos(2ϱ(ξ¯,η¯;ω)+ϵ)),Wyx(0)(ξ¯,η¯,Δξ,Δη;ω)=E0xE0y2μyx(cosϵ+cos(2ϱ(ξ¯,η¯;ω)+ϵ)),Wyy(0)(ξ¯,η¯,Δξ,Δη;ω)=E0y22μyy(1+cos(2[ϱ(ξ¯,η¯;ω)+ϵ])),
Wxx(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E0x22κxx(1+12(ej2ϱ(ξ¯,η¯;ω)+ej2ϱ(ξ¯,η¯;ω)))Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯,Wxy(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E0xE0y2κxy(cosϵ+12(ej(2ϱ(ξ¯,η¯;ω)+ϵ)+ej(2ϱ(ξ¯,η¯;ω)+ϵ)))Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯,Wyx(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E0xE0y2κyx(cosϵ+12(ej(2ϱ(ξ¯,η¯;ω)+ϵ)+ej(2ϱ(ξ¯,η¯;ω)+ϵ)))Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯,Wyy(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E0y22κyy(1+12(ej2(ϱ(ξ¯,η¯;ω)+ϵ)+ej2(ϱ(ξ¯,η¯;ω)+ϵ)))Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯,
Disk(ξ¯2+η¯2R)={1,forξ¯2+η¯2<R0,otherwise.
2ϱ(ξ¯,η¯;ω)=2π(ξ¯x0)αβ.
(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022[1ejΔϕejΔϕ1]×{g(Δr)+12ejϕ(Δr0)g(Δr+Δr0)+12ejϕ(Δr0)g(ΔrΔr0)},
ϕ(Δr0)=2παx0+β,
g(Δr)=Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯,
g(Δr±Δr0)=Disk(ξ¯2+η¯2R)exp[j2πλ¯z[ξ¯(Δx±Δx0)+η¯Δy]]dξ¯dη¯,
Δr0=Δx0x^+Δy0y^+Δz0z^=λ¯zαx^.
2ϱ(ξ¯,η¯;ω)=2π(η¯y0)αβ.
2ϱ(ξ¯,η¯;ω)=2π(ξ¯x0)2+(η¯y0)2α2β,
ϕ(Δr0)=2πα2[x02+y02]β
g(Δr±Δr0)=Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯(Δx±Δx0)+η¯(Δy±Δy0))]×exp[j2πα2(ξ¯2+η¯2)]dξ¯dη¯,
Δr0=2λ¯zα2[x0x^+y0y^].
1/2(ξ¯,η¯;ω)=E0[cos(ϱ(ξ¯,η¯;ω))00sin(ϱ(ξ¯,η¯;ω))].
μ̳(Δξ,Δη;ω)=[1ejΔϕ(Δξ,Δη;ω)ejΔϕ(Δξ,Δη;ω)1],
μij(Δξ,Δη;ω)=Ei*(Δξ,Δη;ω)Ej(Δξ,Δη;ω)|Ei(Δξ,Δη;ω)||Ej(Δξ,Δη;ω)|.
(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]×[κxx(1+cos(2ϱ(ξ¯,η¯;ω)))κxysin(2ϱ(ξ¯,η¯;ω))κyxsin(2ϱ(ξ¯,η¯;ω))κyy(1cos(2ϱ(ξ¯,η¯;ω)))]dξ¯dη¯,
Wxx(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022κxx(1+12(ej2ϱ(ξ¯,η¯;ω)+ej2ϱ(ξ¯,η¯;ω)))Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯,Wxy(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022j2κxy(ej2ϱ(ξ¯,η¯;ω)ej2ϱ(ξ¯,η¯;ω))Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯,Wyx(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022j2κyx(ej2ϱ(ξ¯,η¯;ω)ej2ϱ(ξ¯,η¯;ω))Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯,Wyy(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022κyy(112(ej2ϱ(ξ¯,η¯;ω)+ej2ϱ(ξ¯,η¯;ω)))Disk(ξ¯2+η¯2R)exp[j2πλ¯z(ξ¯Δx+η¯Δy)]dξ¯dη¯.
Wxx(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022κxx(g(Δr)+12ejϕ(Δr0)g(Δr+Δr0)+12ejϕ(Δr0)g(ΔrΔr0)),Wxy(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022κxy(12ej(ϕ(Δr0)+π2)g(Δr+Δr0)+12ej(ϕ(Δr0)+π2)g(ΔrΔr0)),Wyx(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022κyx(12ej(ϕ(Δr0)+π2)g(Δr+Δr0)+12ej(ϕ(Δr0)+π2)g(ΔrΔr0)),Wyy(x¯,y¯,Δx,Δy,z;ω)=ejψ(λ¯z)2E022κyy(g(Δr)12ejϕ(Δr0)g(Δr+Δr0)12ejϕ(Δr0)g(ΔrΔr0)),

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