Abstract

In the traditional treatment of the spectral degree of coherence, the Cartesian coordinate system is deployed to describe the electromagnetic field. In the description of the far field scattered from random media, however, the spherical polar coordinates system is more suitably used due to the field’s outgoing spherical form. We hence derive the expression for the spectral degree of coherence in the spherical polar coordinates system. An example of one polychromatic plane wave scattered by a collection of identical particles is given.

© 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]
  2. 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]
  3. H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
    [CrossRef]
  4. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  5. O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
    [CrossRef]
  6. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
    [CrossRef]
  7. J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [CrossRef]
  8. T. Setala, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [CrossRef]
  9. T. Setala, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
    [CrossRef]
  10. T. Jouttenus, T. Setala, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
    [CrossRef]
  11. E. Wolf, “Comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712 (2004).
    [CrossRef]
  12. T. Setala, J. Tervo, and A. T. Friberg, “Reply to comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1713–1714 (2004).
    [CrossRef]
  13. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19, 5979–5992 (2011).
    [CrossRef]
  14. O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
    [CrossRef]
  15. T. Wang and D. Zhao, “Determination of pair-structure factor of scattering potential of a collection of particles,” Opt. Lett. 35, 318–320 (2010).
    [CrossRef]
  16. S. Serkan and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34, 1762–1764 (2009).
    [CrossRef]
  17. T. Wang and D. Zhao, “Scattering theory of stochastic electromagnetic light waves,” Opt. Lett. 35, 2412–2414 (2010).
    [CrossRef]
  18. G. Arfken and H. J. Weber, Mathematical Methods for physicists (Academic, 2001).

2011 (1)

2010 (3)

2009 (1)

2007 (1)

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

2005 (1)

T. Jouttenus, T. Setala, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
[CrossRef]

2004 (4)

2003 (5)

T. Setala, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

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

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

Arfken, G.

G. Arfken and H. J. Weber, Mathematical Methods for physicists (Academic, 2001).

Blomstedt, K.

T. Setala, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

Cai, Y.

Dong, Y.

Friberg, A. T.

T. Jouttenus, T. Setala, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
[CrossRef]

T. Setala, J. Tervo, and A. T. Friberg, “Reply to comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1713–1714 (2004).
[CrossRef]

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

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

T. Setala, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

Jouttenus, T.

T. Jouttenus, T. Setala, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
[CrossRef]

Kaivola, M.

T. Jouttenus, T. Setala, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
[CrossRef]

T. Setala, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

Korotkova, O.

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

S. Serkan and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34, 1762–1764 (2009).
[CrossRef]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

Serkan, S.

Setala, T.

T. Jouttenus, T. Setala, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
[CrossRef]

T. Setala, J. Tervo, and A. T. Friberg, “Reply to comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1713–1714 (2004).
[CrossRef]

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

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

T. Setala, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

Tervo, J.

Tong, Z.

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

Wang, T.

Weber, H. J.

G. Arfken and H. J. Weber, Mathematical Methods for physicists (Academic, 2001).

Wolf, E.

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
[CrossRef]

E. Wolf, “Comment on complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712 (2004).
[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]

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

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

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

Yao, M.

Zhao, C.

Zhao, D.

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (7)

Phys. Lett. A (1)

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

Phys. Rev. A (1)

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

Phys. Rev. E (3)

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

T. Setala, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Jouttenus, T. Setala, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
[CrossRef]

Other (2)

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

G. Arfken and H. J. Weber, Mathematical Methods for physicists (Academic, 2001).

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Figures (3)

Fig. 1.
Fig. 1.

Illustrating the spherical coordinate system and notation relating the scatterer and scattered field in the far zone.

Fig. 2.
Fig. 2.

Spectral degree of coherence of the far scattered field calculated from Eq. (28), in the sense of Wolf’s definition in Cartesian coordinates and in spherical polar coordinates as a function of Δθ with various δ. λ=0.6328μm, Ax=Ay=1, Bxy=Byx=0.2, σ=0.3λ, θ1=0, Δθ=θ2θ1=θ2, φ1=φ2=π/2, r1=r2.

Fig. 3.
Fig. 3.

Spectral degree of coherence of the far scattered field calculated from Eq. (28) in the sense of Wolf’s definition in spherical polar coordinates and in the sense of Friberg’s definition as a function of Δθ with various δ. The other parameters are the same as in Fig. 2.

Equations (46)

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η(r1,r2,ω)=TrW(r1,r2,ω)TrW(r1,r1,ω)TrW(r2,r2,ω).
μ(r1,r2,ω)=Tr[W(r1,r2,ω)·W(r2,r1,ω)]TrW(r1,r1,ω)TrW(r2,r2,ω),
Exk=Eqlhlxkql,
W(r1,r2,ω)=[Ei*(r1,ω)Ej(r2,ω)],
Ex=Ersinθcosφ+EθcosθcosφEφsinφ,
Ey=Ersinθsinφ+Eθcosθsinφ+Eφcosφ,
Ez=ErcosθEθsinθ,
Tr(C)Wsp(r1,r2,ω)=TrW(r1,r2,ω)=Wr1r2(r1,r2,ω)[sinθ1sinθ2cos(φ1φ2)+cosθ1cosθ2]+Wθ1θ2(r1,r2,ω)[cosθ1cosθ2cos(φ1φ2)+sinθ1sinθ2]+Wφ1φ2(r1,r2,ω)cos(φ1φ2)+Wr1θ2(r1,r2,ω)[sinθ1cosθ2cos(φ1φ2)cosθ1sinθ2]+Wθ1r2(r1,r2,ω)[cosθ1sinθ2cos(φ1φ2)sinθ1cosθ2]+Wr1φ2(r1,r2,ω)sinθ1sin(φ1φ2)+Wφ1r2(r1,r2,ω)sinθ2sin(φ2φ1)+Wθ1φ2(r1,r2,ω)cosθ1sin(φ1φ2)+Wφ1θ2(r1,r2,ω)cosθ2sin(φ2φ1),
Tr(C)Wsp(r1,r2,ω)=TrWsp(r1,r2,ω),
ηsp(r1,r2,ω)=Tr(C)Wsp(r1,r2,ω)TrWsp(r1,r1,ω)TrWsp(r2,r2,ω).
Tr(C)Wcc(r1,r2,ω)=TrW(r1,r2,ω)=[Wρ1ρ2(r1,r2,ω)+Wφ1φ2(r1,r2,ω)]cos(φ1φ2)+Wz1z2(r1,r2,ω)+[Wρ1φ2(r1,r2,ω)Wφ1ρ2(r1,r2,ω)]sin(φ1φ2),
Tr(C)Wcc(r1,r2,ω)=TrWcc(r1,r2,ω).
ηcc(r1,r2,ω)=Tr(C)Wcc(r1,r2,ω)TrWcc(r1,r1,ω)TrWcc(r2,r2,ω),
Tr[W(r1,r2,ω)·W(r2,r1,ω)]=Tr[W(r1,r2,ω)·W(r1,r2,ω)]
[Ex,Ey,Ez]=[Er,Eθ,Eφ]·M(θ,φ),
M(θ,φ)=(sinθcosφsinθsinφcosθcosθcosφcosθsinφsinθsinφcosφ0),
W(r1,r2,ω)=M(θ1,φ1)·Wsp(r1,r2,ω)·M(θ2,φ2).
Tr[W(r1,r2,ω)·W(r1,r2,ω)]=Tr[M(θ1,φ1)·Wsp(r1,r2,ω)·Wsp(r1,r2,ω)·M(θ1,φ1)]=Tr[Wsp(r1,r2,ω)·Wsp(r1,r2,ω)],
W(i)(r1,r2,ω)=[Ei(i)*(r1,ω)Ej(i)(r2,ω)],
Ei(i)(r,ω)=ai(i)(s,ω)eik(s·r+szz)d2s,
W(i)(r1,r2,ω)=A(i)(s1,s2,ω)eik(s2·r2s1·r1)d2s1d2s2,
E(s)(rs,ω)=eikrrVF(r,ω)[E(i)(r,ω)(s·E(i)(r,ω))s]eiks·rd3r.
E(s)(rs,ω)=eikrrVF(r,ω)E(i)(r,ω)eiks·rd3r·S(θ,φ),
S(θ,φ)=(cosθcosφsinφcosθsinφcosφsinθ0).
Wsp(r1s1,r2s2,ω)=[Eα(s)*(r1s1,ω)Eβ(s)(r2s2,ω)],
Wsp(r1s1,r2s2,ω)=eik(r2r1)r1r2ST(θ1,φ1)·C˜F(K1,K2,ω)A(i)(s1,s2,ω)d2s1d2s2·S(θ2,φ2),
C˜F(K1,K2,ω)=VVCF(r1,r2,ω)ei(K2·r2K1·r1)d3r1d3r2
CF(K1,K2,ω)=f*(K1,ω)f(K2,ω)M(K1,K2,ω),
M(K1,K2,ω)=M(K1,K1)M(K2,K2)m(|K1K2|),
W(r1s1,r2s2,ω)=eik(r2r1)S(i)(ω)r1r2f*(K1,ω)f(K2,ω)M(K1,K2,ω)ST(θ1,φ1)·B·S(θ2,φ2),
B=(Ax2AxAyBxy0AyAxByxAy20000).
TrWsp(r1,r2,ω)=Wθ1θ2(r1,r2,ω)+Wφ1φ2(r1,r2,ω),
Tr(C)Wsp(r1,r2,ω)=Wθ1θ2(r1,r2,ω)[cosθ1cosθ2cos(φ1φ2)+sinθ1sinθ2]+Wφ1φ2(r1,r2,ω)cos(φ1φ2)+Wθ1φ2(r1,r2,ω)cosθ1sin(φ1φ2)+Wφ1θ2(r1,r2,ω)cosθ2sin(φ2φ1),
Tr[W(r1,r2,ω)·W(r2,r1,ω)]=|Wθ1θ2(r1,r2,ω)|2+|Wφ1φ2(r1,r2,ω)|2+|Wθ1φ2(r1,r2,ω)|2+|Wφ1θ2(r1,r2,ω)|2.
ST(θ1,φ1)·B·S(θ2,φ2)=(cosθ20.20.2cosθ21),
TrWsp(r1,r2,ω)=(1+cosθ2)A(r1,r2,ω),
Tr(C)Wsp(r1,r2,ω)=(1+cos2θ2)A(r1,r2,ω),
Tr[W(r1,r2,ω)·W(r2,r1,ω)]=1.04+1.04cos2θ2A(r1,r2,ω),
ηspη=1+cos2θ21+cosθ2,
ηspμ=1+cos2θ21.04.
E=Eq1qˆ1+Eq2qˆ2+Eq3qˆ3.
hi2=xqi·xqi+yqi·yqi+zqi·zqi=rqi·rqi,
qˆi=1hirqi.
E=Eqlhlxqleˆ1+Eqlhlyqleˆ2+Eqlhlzqleˆ3=Eqlhlrql,
E=Exeˆ1+Eyeˆ2+Ezeˆ3.
Exk=Eqlhlxkql,

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