Abstract

We show that the cross-spectral density in the far zone of a homogeneous spherical source can be described as a low-pass filtered version of that existing across the source surface. We prove that, to an excellent approximation, the corresponding filter with respect to a (normalized) spatial frequency ξ has a functional structure of the form 1ξ2, for 0ξ1. The cases of spatially incoherent and Lambertian sources are treated as significant examples.

© 2012 Optical Society of America

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References

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  1. G. S. Agarwal, G. Gbur, and E. Wolf, Opt. Lett. 29, 459 (2004).
    [CrossRef]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. Digital Library of Mathematical Functions. Release date 2012-03-23. National Institute of Standards and Technology, from http://dlmf.nist.gov/ .
  4. F. Gori, Opt. Lett. 30, 2840 (2005).
    [CrossRef]
  5. F. Gori and O. Korotkova, Opt. Commun. 282, 3859 (2009).
    [CrossRef]
  6. I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, 1994).
  7. C. M. Bender and S. A. Orszag, Advanced Mathematical Method for Scientists and Engineers (McGraw-Hill, 1978).
  8. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).
  9. T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. Friberg, Phys. Rev. E 71, 036618 (2005).
    [CrossRef]

2009 (1)

F. Gori and O. Korotkova, Opt. Commun. 282, 3859 (2009).
[CrossRef]

2005 (2)

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. Friberg, Phys. Rev. E 71, 036618 (2005).
[CrossRef]

F. Gori, Opt. Lett. 30, 2840 (2005).
[CrossRef]

2004 (1)

Agarwal, G. S.

Bender, C. M.

C. M. Bender and S. A. Orszag, Advanced Mathematical Method for Scientists and Engineers (McGraw-Hill, 1978).

Blomstedt, K.

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. Friberg, Phys. Rev. E 71, 036618 (2005).
[CrossRef]

Friberg, A.

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. Friberg, Phys. Rev. E 71, 036618 (2005).
[CrossRef]

Gbur, G.

Gori, F.

F. Gori and O. Korotkova, Opt. Commun. 282, 3859 (2009).
[CrossRef]

F. Gori, Opt. Lett. 30, 2840 (2005).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, 1994).

Korotkova, O.

F. Gori and O. Korotkova, Opt. Commun. 282, 3859 (2009).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Lindberg, J.

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. Friberg, Phys. Rev. E 71, 036618 (2005).
[CrossRef]

Mandel, L.

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

Orszag, S. A.

C. M. Bender and S. A. Orszag, Advanced Mathematical Method for Scientists and Engineers (McGraw-Hill, 1978).

Rhyzhik, I. M.

I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, 1994).

Setälä, T.

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. Friberg, Phys. Rev. E 71, 036618 (2005).
[CrossRef]

Tervo, J.

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. Friberg, Phys. Rev. E 71, 036618 (2005).
[CrossRef]

Wolf, E.

G. S. Agarwal, G. Gbur, and E. Wolf, Opt. Lett. 29, 459 (2004).
[CrossRef]

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

Opt. Commun. (1)

F. Gori and O. Korotkova, Opt. Commun. 282, 3859 (2009).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E (1)

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. Friberg, Phys. Rev. E 71, 036618 (2005).
[CrossRef]

Other (5)

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

Digital Library of Mathematical Functions. Release date 2012-03-23. National Institute of Standards and Technology, from http://dlmf.nist.gov/ .

I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, 1994).

C. M. Bender and S. A. Orszag, Advanced Mathematical Method for Scientists and Engineers (McGraw-Hill, 1978).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

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Figures (2)

Fig. 1.
Fig. 1.

Behavior, as a function of l, of the l.h.s. (dots) and of the r.h.s. (solid curve) of Eq. (10), for ka=100.

Fig. 2.
Fig. 2.

Behavior, as a function of the kaφ, of the spectral degree of coherence for a δ-correlated (dots) and for a Lambertian (open circles) source, together with the theoretical estimates provided by Eqs. (15) (solid curve) and (17) (dotted curve), respectively. ka=100 and kr1=kr2=1000.

Equations (18)

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W0(r^1,r^2)=S0g0(cosφ),
g0(cosφ)=l=0(l+12)BlPl(cosφ),
Bl=11g0(ξ)Pl(ξ)dξ,l=0,1,,
W(r1,r2)=S0l=0(l+12)Blhl*(kr1)hl(kr2)|hl(ka)|2Pl(cosφ),
W(r1,r2)S0exp[i(kr2kr1)]r1r2g(cosφ),
g(cosφ)=l=0(l+12)Bl|hl(ka)|2Pl(cosφ).
Pl(cosφ)=2φπ01cos[(l+12)φt]2(cosφtcosφ)dt.
Pl(cosφ)J0[(l+12)φ],
g(cosφ)l=0(l+12)Bl|hl(ka)|2J0[(l+12)φ].
1(ka)2|hl(ka)|21(l+12ka)2,ka1,
g(cosφ)(ka)2×l=0[ka](l+12)Bl1(l+12ka)2J0[(l+12ka)kaφ],
g(cosφ)01B(ξ)1ξ2J0(kaφξ)ξdξ,
Bl=B(l+1/2ka).
F(ξ)=1ξ2circ(ξ),
g(cosφ)3j1(kaφ)kaφ,
W0(r^1,r^2)=S0sin(ka|r^1r^2|)ka|r^1r^2|.
g011sin(kaπξ2ka)2J0(kaφξ)ξdξ,
g(cosφ)2J1(kaφ)kaφ,

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