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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 22, Iss. 11 — Nov. 1, 2005
  • pp: 2547–2556

Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources

Olga Korotkova, Brian G. Hoover, Victor L. Gamiz, and Emil Wolf  »View Author Affiliations

JOSA A, Vol. 22, Issue 11, pp. 2547-2556 (2005)

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In studies of radiation from partially coherent sources the so-called quasi-homogeneous (QH) model sources have been very useful, for instance in elucidating the behavior of fields produced by thermal sources. The analysis of the fields generated by such sources has, however, been largely carried out in the framework of scalar wave theory. In this paper we generalize the concept of the QH source to the domain of the electromagnetic theory, and we derive expressions for the elements of the cross-spectral density matrix, for the spectral density, the spectral degree of coherence, the degree of polarization, and the Stokes parameters of the far field generated by planar QH sources of uniform states of polarization. We then derive reciprocity relations analogous to those familiar in connection with the QH scalar sources. We illustrate the results by determining the properties of the far field produced by transmission of an electromagnetic beam through a system of spatial light modulators.

© 2005 Optical Society of America

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(260.2110) Physical optics : Electromagnetic optics
(260.5430) Physical optics : Polarization

ToC Category:
Physical Optics

Original Manuscript: January 3, 2005
Manuscript Accepted: February 3, 2005
Published: November 1, 2005

Olga Korotkova, Brian G. Hoover, Victor L. Gamiz, and Emil Wolf, "Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources," J. Opt. Soc. Am. A 22, 2547-2556 (2005)

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995). [CrossRef]
  2. W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975). [CrossRef]
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  8. J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.
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  10. H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003). [CrossRef]
  11. 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] [PubMed]
  12. We use the symbol μij(0) rather than ηij(0) that was employed in Ref. [9] to avoid confusion with the scalar theory.
  13. M. Born, E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK 1999). [CrossRef]
  14. The Stokes parameter S0(0)(ρ,ω) that we use for normalization is just the spectral density S(0)(ρ,ω) of the source, given by Eq. (2.4).
  15. A source that is linearly polarized along the y direction, say, obviously corresponds to the limit α(ω)→∞.
  16. This result represents essentially a generalization to the far field generated by QH electromagnetic sources of the well-known van Cittert–Zernike theorem of the scalar theory.
  17. The Stokes parameters of the far-zone field can also be calculated from the normalized ones, because the Stokes parameter used for normalization, is just the spectral density S(∞)(rs,ω) of the far field, given by reciprocity relation (3.9).
  18. The results derived in this section were found for some special cases in G. Piquero, R. Borghi, A. Mondello, M. Santarsiero, “Far-field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339–350 (2001). [CrossRef]
  19. T. Shirai, O. Korotkova, E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” Pure Appl. Opt. 7, 232–237 (2005). [CrossRef]
  20. T. Shirai, E. Wolf, “Spatial coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907–1916 (2004). [CrossRef]
  21. The amplitude filters in the experiments described in Ref. [19] generally produce nonuniform polarization. To ensure that the polarization of the synthesized source is uniform, these filters must have the same rms widths, i.e., σF1=σF2 [see Eqs. (15) and (16) of Ref. [19]].
  22. H. Roychowdhury, O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005). [CrossRef]
  23. O. Korotkova, M. Salem, E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29, 1173–1175 (2004). [CrossRef] [PubMed]
  24. The parameter α(ω) may be shown to be related to the angle ψ(ω) between the x axis and the direction of polarization at the source plane by the formula α(ω)=tan−1∣ψ(ω)∣, −π∕2<ψ⩽π∕2. In practice, in order to control α(ω), one can replace the beam splitter BS1 introduced in Ref. [19] with a polarizing beam splitter and place a polarization rotator (half-wave plate) in front of it. The parameter α(ω) can then be determined by rotating the half-wave plate.
  25. The parameters characterizing the spectral polarization ellipse at each point can also be calculated directly from the elements of the cross-spectral density matrix [see O. Korotkova, E. Wolf, “Polarization state changes in random electromagnetic beams on propagation,” Opt. Commun. 246, 35–43 (2005)]. [CrossRef]

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