Abstract

We formulate the van Cittert–Zernike theorem for spatially incoherent partially polarized sources in terms of the traditional polarization (one-point) Stokes parameters and the coherence (two-point) Stokes parameters. This leads to a physically insightful connection between the source polarization and the field coherence. We show that the far-zone coherence is given by a Fourier relation of the source-plane polarization and apply the results to investigate the far-zone properties of a field emitted by an incoherent source limited by an annular aperture.

© 2013 Optical Society of America

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References

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  1. P. H. van Cittert, Physica 1, 201 (1934).
    [CrossRef]
  2. F. Zernike, Physica 5, 785 (1938).
    [CrossRef]
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  5. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
  6. J. C. Leader, J. Opt. Soc. Am. 68, 1332 (1978).
    [CrossRef]
  7. A. D. Jacobson, IEEE Trans. Antennas Propag. 15, 24 (1967).
    [CrossRef]
  8. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, Opt. Lett. 25, 1291 (2000).
    [CrossRef]
  9. M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
    [CrossRef]
  10. A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, Opt. Express 17, 1746 (2009).
    [CrossRef]
  11. T. Shirai, Opt. Lett. 34, 3761 (2009).
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  12. O. G. Rodríguez-Herrera and J. S. Tyo, J. Opt. Soc. Am. A 29, 1939 (2012).
    [CrossRef]
  13. O. Korotkova and E. Wolf, Opt. Lett. 30, 198 (2005).
    [CrossRef]
  14. J. Tervo, T. Setälä, and A. T. Friberg, Opt. Express 11, 1137 (2003).
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  15. T. Setälä, J. Tervo, and A. T. Friberg, Opt. Lett. 29, 328 (2004).
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  16. J. Tervo, T. Setälä, and A. T. Friberg, J. Opt. Soc. Am. A 21, 2205 (2004).
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  17. T. Setälä, J. Tervo, and A. T. Friberg, Opt. Lett. 31, 2208 (2006).
    [CrossRef]
  18. T. Setälä, J. Tervo, and A. T. Friberg, Opt. Lett. 31, 2669 (2006).
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  19. T. Setälä, F. Nunziata, and A. T. Friberg, Opt. Lett. 34, 2924 (2009).
    [CrossRef]
  20. M. Santarsiero, V. Ramírez-Sanchez, and R. Borghi, J. Opt. Soc. Am. A 27, 1450 (2010).
    [CrossRef]
  21. S. B. Saghunathan, T. van Dijk, E. J. G. Peterman, and T. D. Visser, Opt. Lett. 35, 4166 (2010).
    [CrossRef]

2012 (1)

2010 (2)

2009 (3)

2006 (3)

2005 (1)

2004 (2)

2003 (1)

2000 (1)

1978 (1)

1967 (1)

A. D. Jacobson, IEEE Trans. Antennas Propag. 15, 24 (1967).
[CrossRef]

1938 (1)

F. Zernike, Physica 5, 785 (1938).
[CrossRef]

1934 (1)

P. H. van Cittert, Physica 1, 201 (1934).
[CrossRef]

Alonso, M. A.

M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
[CrossRef]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).

Friberg, A. T.

Gori, F.

Jacobson, A. D.

A. D. Jacobson, IEEE Trans. Antennas Propag. 15, 24 (1967).
[CrossRef]

Korotkova, O.

M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
[CrossRef]

O. Korotkova and E. Wolf, Opt. Lett. 30, 198 (2005).
[CrossRef]

Leader, J. C.

Mandel, L.

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

Martínez-Niconoff, G.

Martínez-Vara, P.

Nunziata, F.

Olvera-Santamaría, M. A.

Ostrovsky, A. S.

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).

Peterman, E. J. G.

Piquero, G.

Ramírez-Sanchez, V.

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).

Rodríguez-Herrera, O. G.

Saghunathan, S. B.

Santarsiero, M.

Setälä, T.

Shirai, T.

Tervo, J.

Thompson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).

Tyo, J. S.

van Cittert, P. H.

P. H. van Cittert, Physica 1, 201 (1934).
[CrossRef]

van Dijk, T.

Visser, T. D.

Wolf, E.

M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
[CrossRef]

O. Korotkova and E. Wolf, Opt. Lett. 30, 198 (2005).
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Zernike, F.

F. Zernike, Physica 5, 785 (1938).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. D. Jacobson, IEEE Trans. Antennas Propag. 15, 24 (1967).
[CrossRef]

J. Mod. Opt. (1)

M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Express (2)

Opt. Lett. (8)

Physica (2)

P. H. van Cittert, Physica 1, 201 (1934).
[CrossRef]

F. Zernike, Physica 5, 785 (1938).
[CrossRef]

Other (3)

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).

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Figures (2)

Fig. 1.
Fig. 1.

Geometry and notations. Here σ denotes the source at z=0, rm=rmu^m, m=(1,2), are the two observation points, u^m are the unit direction vectors, ρ is a position vector in the source plane, and Rm=|rmρ|.

Fig. 2.
Fig. 2.

Degree of coherence with ϱ=15λ, where λ is the wavelength, with w=0 (solid red), w=1/2 (dashed green), and w1 (dotted blue).

Equations (22)

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μ(r1,r2,ω)=(k2π)2[S(r1,ω)S(r2,ω)]1/2×σS(ρ,ω)exp[ik(R2R1)]R1R2d2ρ,
μ(r1,r2,ω)=exp[ik(r2r1)]×σs(ρ)exp(ikΔu^·ρ)d2ρ,
W(r1,r2,ω)=(k2π)2W(ρ1,ρ2,ω)×exp[ik(R2R1)]R1R2d2ρ1d2ρ2.
W(r1,r2,ω)=(k2π)2σJ(ρ,ω)exp[ik(R2R1)]R1R2d2ρ.
W(r1,r2,ω)=(k2π)2exp[ik(r2r1)]r1r2×σJ(ρ,ω)exp(ikΔu^·ρ)d2ρ,
S0(r1,r2,ω)=Wxx(r1,r2,ω)+Wyy(r1,r2,ω),
S1(r1,r2,ω)=Wxx(r1,r2,ω)Wyy(r1,r2,ω),
S2(r1,r2,ω)=Wxy(r1,r2,ω)+Wyx(r1,r2,ω),
S3(r1,r2,ω)=i[Wyx(r1,r2,ω)Wxy(r1,r2,ω)].
ηj(r1,r2,ω)=Sj(r1,r2,ω)[S0(r1,ω)S0(r2,ω)]1/2,j=0,,3
μE2(r1,r2,ω)=12j=03|ηj(r1,r2,ω)|2.
ηj(r1,r2,ω)=(k2π)2[S0(r1,ω)S0(r2,ω)]1/2×σSj(ρ,ω)exp[ik(R2R1)]R1R2d2ρ.
ηj(r1,r2,ω)=exp[ik(r2r1)]×σsj(ρ,ω)exp(ikΔu^·ρ)d2ρ,
Sj(r,ω)=(k2πr)2σSj(ρ,ω)d2ρ,
μE2(r1,r2,ω)=12σexp(ikΔu^·Δρ)×s(ρ1,ω)·s(ρ2,ω)d2ρ1d2ρ2,
ηj(r1,r2,ω)=exp[ik(r2r1)]wϱϱ02πsj(α,ϕ,ω)×exp[ikαsinϑcos(ϕψ)]αdϕdα.
ηj(r1,r2,ω)=2s¯j(ω)exp[ik(r2r1)]ϱ(1w2)ksinϑ×[J1(kϱsinϑ)wJ1(kwϱsinϑ)],
μE(r1,r2,ω)=21/2[1+Pσ2(ω)]1/2×|J1(kϱsinϑ)wJ1(kwϱsinϑ)ϱ(1w2)ksinϑ|,
ηj(r1,r2,ω)=2s¯j(ω)exp[ik(r2r1)]J1(kϱsinϑ)kϱsinϑ,
μE(r1,r2,ω)=21/2[1+Pσ2(ω)]1/2|J1(kϱsinϑ)kϱsinϑ|.
ηj(r1,r2,ω)=s¯j(ω)exp[ik(r2r1)]J0(kϱsinϑ),
μE(r1,r2,ω)=21/2[1+Pσ2(ω)]1/2|J0(kϱsinϑ)|.

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