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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics


  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 9 — Sep. 26, 2007

Two-frequency radiative transfer and asymptotic solution

Albert C. Fannjiang  »View Author Affiliations

JOSA A, Vol. 24, Issue 8, pp. 2248-2256 (2007)

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Two-frequency radiative transfer (2f-RT) theory is developed for classical waves in random media. Depending on the ratio of the wavelength to the scale of medium fluctuation, the 2f-RT equation is either a Boltzmann-like integral equation with a complex-valued kernel or a Fokker–Planck-like differential equation with complex-valued coefficients in the phase space. The 2f-RT equation is used to estimate three physical parameters: the spatial spread, the coherence length, and the coherence bandwidth (Thouless frequency). A closed-form solution is given for the boundary layer behavior of geometrical radiative transfer and shows highly nontrivial dependence of mutual coherence on the spatial displacement and frequency difference. It is shown that the paraxial form of 2f-RT arises naturally in anisotropic media that fluctuate slowly in the longitudinal direction.

© 2007 Optical Society of America

OCIS Codes
(030.5620) Coherence and statistical optics : Radiative transfer
(290.4210) Scattering : Multiple scattering

ToC Category:
Coherence and Statistical Optics

Original Manuscript: October 20, 2006
Revised Manuscript: February 8, 2007
Manuscript Accepted: February 16, 2007
Published: July 11, 2007

Virtual Issues
Vol. 2, Iss. 9 Virtual Journal for Biomedical Optics

Albert C. Fannjiang, "Two-frequency radiative transfer and asymptotic solution," J. Opt. Soc. Am. A 24, 2248-2256 (2007)

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  1. M. Born and W. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).
  2. A. Bronshtein, I. T. Lu, and R. Mazar, "Reference-wave solution for the two-frequency propagator in a statistically homogeneous random medium," Phys. Rev. E 69, 016607 (2004). [CrossRef]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. 1 and 2.
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  5. G. Samelsohn and V. Freilikher, "Two-frequency mutual coherence function and pulse propagation in random media," Phys. Rev. E 65, 046617 (2002). [CrossRef]
  6. R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994). [CrossRef]
  7. P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, "Spatial-field correlation: the building block of mesoscopic fluctuations," Phys. Rev. Lett. 88, 123901 (2002). [CrossRef] [PubMed]
  8. M. C. W. van Rossum and Th. M. Nieuwenhuizen, "Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion," Rev. Mod. Phys. 71, 313-371 (1999). [CrossRef]
  9. B. Shapiro, "Large intensity fluctuations for wave propagation in random media," Phys. Rev. Lett. 57, 2168-2171 (1986). [CrossRef] [PubMed]
  10. D. Dragoman, "The Wigner distribution function in optics and optoelectronics," in Progress in Optics, Vol. 37, E.Wolf, ed. (Elsevier, 1997), , pp. 1-56. [CrossRef]
  11. G.W.Forbes, V.I.Man'ko, H.M.Ozaktas, R.Simon, and K.B.Wolf, eds., "Wigner Distributions and Phase Space in Optics," J. Opt. Soc. Am. A 17, 2274-2354 (2000) (feature issue).
  12. A. C. Fannjiang, "White-noise and geometrical optics limits of Wigner-Moyal equation for wave beams in turbulent media II. Two-frequency Wigner distribution formulation," J. Stat. Phys. 120, 543-586 (2005). [CrossRef]
  13. A. C. Fannjiang, "Radiative transfer limit of two-frequency Wigner distribution for random parabolic waves: an exact solution," C. R. Phys. 8, 267-271 (2007). [CrossRef]
  14. A. C. Fannjiang, "Self-averaging scaling limits of two-frequency Wigner distribution for random paraxial waves," J. Phys. A 40, 5025-5044 (2007). [CrossRef]
  15. A. C. Fannjiang, "Space-frequency correlation of classical waves in disordered media: high-frequency asymptotics," submitted to Europhys. Lett.
  16. M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).
  17. H. Spohn, "Kinetic equations from Hamiltonian dynamics: Markovian limits," Rev. Mod. Phys. 53, 569-615 (1980). [CrossRef]
  18. A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, 1978).
  19. L. Ryzhik, G. Papanicolaou, and J. B. Keller, "Transport equations for elastic and other waves in random media," Wave Motion 24, 327-370 (1996). [CrossRef]
  20. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  21. E. Hopf, Mathematical Problems of Radiative Equilibrium (Cambridge U. Press, 1934).
  22. A. Schuster, "Radiation through a foggy atmosphere," Astrophys. J. 21, 1-22 (1905). [CrossRef]
  23. A. Z. Genack, "Optical transmission in disordered media," Phys. Rev. Lett. 58, 2043-2046 (1987). [CrossRef] [PubMed]
  24. A. C. Fannjiang, "Information transfer in disordered media by broadband time reversal: stability, resolution and capacity," Nonlinearity 19, 2425-2439 (2006). [CrossRef]
  25. A. C. Fannjiang, "Self-averaging radiative transfer for parabolic waves," C. R. Math. 342(22), 109-114 (2006). [CrossRef]
  26. A. C. Fannjiang, "Self-averaging scaling limits for random parabolic waves," Arch. Ration. Mech. Anal. 175, 343-387 (2005). [CrossRef]

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