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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 12 — Dec. 1, 2000
  • pp: 2464–2474

Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium

Harold T. Yura, Lars Thrane, and Peter E. Andersen  »View Author Affiliations

JOSA A, Vol. 17, Issue 12, pp. 2464-2474 (2000)

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Within the paraxial approximation, a closed-form solution for the Wigner phase-space distribution function is derived for diffuse reflection and small-angle scattering in a random medium. This solution is based on the extended Huygens–Fresnel principle for the optical field, which is widely used in studies of wave propagation through random media. The results are general in that they apply to both an arbitrary small-angle volume scattering function, and arbitrary (real) ABCD optical systems. Furthermore, they are valid in both the single- and multiple-scattering regimes. Some general features of the Wigner phase-space distribution function are discussed, and analytic results are obtained for various types of scattering functions in the asymptotic limit s1, where s is the optical depth. In particular, explicit results are presented for optical coherence tomography (OCT) systems. On this basis, a novel way of creating OCT images based on measurements of the momentum width of the Wigner phase-space distribution is suggested, and the advantage over conventional OCT images is discussed. Because all previous published studies regarding the Wigner function are carried out in the transmission geometry, it is important to note that the extended Huygens–Fresnel principle and the ABCD matrix formalism may be used successfully to describe this geometry (within the paraxial approximation). Therefore for completeness we present in an appendix the general closed-form solution for the Wigner phase-space distribution function in ABCD paraxial optical systems for direct propagation through random media, and in a second appendix absorption effects are included.

© 2000 Optical Society of America

OCIS Codes
(170.1650) Medical optics and biotechnology : Coherence imaging
(170.4500) Medical optics and biotechnology : Optical coherence tomography
(170.7050) Medical optics and biotechnology : Turbid media
(290.4210) Scattering : Multiple scattering

Original Manuscript: December 12, 1999
Revised Manuscript: April 21, 2000
Manuscript Accepted: May 1, 2000
Published: December 1, 2000

Harold T. Yura, Lars Thrane, and Peter E. Andersen, "Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium," J. Opt. Soc. Am. A 17, 2464-2474 (2000)

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  28. For axially symmetric scattering functions, the integral over scattering angles reduces to 2π∫0∞dθθσ(θ)J0kz′Zr+1-z′Zp, where J0(·) is the Bessel function of the first kind, of order zero.
  29. For axially symmetric scattering functions, the integral over scattering angles reduces to 2π∫0∞dθθσ(θ)J0kB(z′)Br+1-B(z′)Bp.
  30. For a spatially uniform medium of index of refraction n, we have A=D=1,B=Z/n,B(z′)=z′/n, and Eq. (A7) reduces to Eq. (A1).
  31. Note that the limits on the z′ integration of the jth term in the summation are now from zj to zj+Δzj.
  32. The corresponding Wigner function is obtained by replacing θ by q/km.
  33. As expected physically, the corresponding irradiance, obtained by integrating the radiance pattern over all solid angle, is given by exp(-μAZ)I0.

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