Abstract

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

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  1. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238; C.-C. Cheng, M. G. Raymer, “Long-range saturation of spatial decoherence in wave-field transport in random multiple-scattering media,” Phys. Rev. Lett. 82, 4807–4810 (1999); M. G. Raymer, C.-C. Cheng, “Propagation of the optical Wigner function in random multiple-scattering media,” in Laser–Tissue Interaction XI: Photochemical, Photothermal, and Photomechanical, D. D. Duncan, J. O. Hollinger, S. L. Jacques, eds., Proc. SPIE3914, 376–380 (2000).
    [CrossRef]
  3. S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
    [CrossRef] [PubMed]
  4. A. Wax, J. E. Thomas, “Measurement of smoothed Wigner phase-space distributions for small-angle scattering in a turbid medium,” J. Opt. Soc. Am. A 15, 1896–1908 (1998).
    [CrossRef]
  5. R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  6. M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
    [CrossRef]
  7. It is straightforward to show that within the paraxial approximation, the specific radiance distribution N(P, θ)=k2W(P, kθ) for those cases where the Wigner phase-space distribution is positive definite.
  8. In random media where the scattering particles are large compared with the wavelength and the index of refraction ratio is near unity, the bulk backscattering efficiency is much smaller than the scattering efficiency. Moreover, the scattering is primarily in the forward direction, which is the basis of using the paraxial approximation. Therefore the bulk backscattering may be neglected when one is considering the light propagation problem, since its contribution is mall. An example of this is skin tissue (cell sizes of 5–10-µm diameter and index of refraction ratio of 1.45/1.4=1.04).
  9. V. A. Banakh, V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech House, Boston, Mass., 1987).
  10. L. Thrane, H. T. Yura, P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended Huygens–Fresnel principle,” J. Opt. Soc. Am. A 17, 484–490 (2000).
    [CrossRef]
  11. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  12. Because we are dealing with “real” ABCD optical systems, we tacitly assume that B≠0.
  13. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  14. H. T. Yura, S. G. Hanson, “Effects of receiver optics contamination on the performance of laser velocimeter systems,” J. Opt. Soc. Am. A 13, 1891–1902 (1996).
    [CrossRef]
  15. One can obtain corresponding results that are valid for propagation through the turbulent atmosphere by formally replacing the exponent in Eq. (4) by one-half the corresponding point-source wave structure function.13
  16. For completeness, the corresponding mutual coherence function is given by 〈U(P+p/2)U*(P+p/2)〉=(η/πB2)|Γpt(p)|2K(-p)exp(-ikDp·P/B).
  17. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  18. D. Arnush, “Underwater light-beam propagation in the small-angle-scattering approximation,” J. Opt. Soc. Am. 62, 1109–1111 (1972).
    [CrossRef]
  19. I. Dror, A. Sandrov, N. S. Kopeika, “Experimental investigation of the influence of the relative position of the scattering layer on image quality: the shower curtain effect,” Appl. Opt. 37, 6495–6499 (1998).
    [CrossRef]
  20. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  21. See, e.g., S. L. Jacques, L. Wang, “Monte Carlo modeling of light transport in tissues,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 4.
  22. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  23. It is straightforward to show that the quantity θrms appearing in Eq. (15) can be written as θrms=2(1-g).
  24. J. M. Schmitt, A. Knüttel, “Model of optical coherence tomography of heterogeneous tissue,” J. Opt. Soc. Am. A 14, 1231–1242 (1997).
    [CrossRef]
  25. M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
    [CrossRef] [PubMed]
  26. H. T. Yura, “A multiple scattering analysis of the propagation of radiance through the atmosphere,” in Proceedings of the Union Radio-Scientifique Internationale Open Symposium (Union Radio-Scientifique Internationale, Ghent, Belgium, 1977), pp. 65–69.
  27. A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986).
  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.

2000 (1)

1998 (2)

1997 (1)

1996 (2)

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

H. T. Yura, S. G. Hanson, “Effects of receiver optics contamination on the performance of laser velocimeter systems,” J. Opt. Soc. Am. A 13, 1891–1902 (1996).
[CrossRef]

1989 (1)

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

1987 (1)

1984 (1)

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

1972 (1)

1971 (1)

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Andersen, P. E.

Anderson, M.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238; C.-C. Cheng, M. G. Raymer, “Long-range saturation of spatial decoherence in wave-field transport in random multiple-scattering media,” Phys. Rev. Lett. 82, 4807–4810 (1999); M. G. Raymer, C.-C. Cheng, “Propagation of the optical Wigner function in random multiple-scattering media,” in Laser–Tissue Interaction XI: Photochemical, Photothermal, and Photomechanical, D. D. Duncan, J. O. Hollinger, S. L. Jacques, eds., Proc. SPIE3914, 376–380 (2000).
[CrossRef]

Arnush, D.

Banakh, V. A.

V. A. Banakh, V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech House, Boston, Mass., 1987).

Beck, M.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238; C.-C. Cheng, M. G. Raymer, “Long-range saturation of spatial decoherence in wave-field transport in random multiple-scattering media,” Phys. Rev. Lett. 82, 4807–4810 (1999); M. G. Raymer, C.-C. Cheng, “Propagation of the optical Wigner function in random multiple-scattering media,” in Laser–Tissue Interaction XI: Photochemical, Photothermal, and Photomechanical, D. D. Duncan, J. O. Hollinger, S. L. Jacques, eds., Proc. SPIE3914, 376–380 (2000).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Cheng, C.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238; C.-C. Cheng, M. G. Raymer, “Long-range saturation of spatial decoherence in wave-field transport in random multiple-scattering media,” Phys. Rev. Lett. 82, 4807–4810 (1999); M. G. Raymer, C.-C. Cheng, “Propagation of the optical Wigner function in random multiple-scattering media,” in Laser–Tissue Interaction XI: Photochemical, Photothermal, and Photomechanical, D. D. Duncan, J. O. Hollinger, S. L. Jacques, eds., Proc. SPIE3914, 376–380 (2000).
[CrossRef]

Dror, I.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hanson, S. G.

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hillery, M.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jacques, S. L.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

See, e.g., S. L. Jacques, L. Wang, “Monte Carlo modeling of light transport in tissues,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 4.

John, S.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

Knüttel, A.

Kopeika, N. S.

Lutomirski, R. F.

Mironov, V. L.

V. A. Banakh, V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech House, Boston, Mass., 1987).

O’Connell, R. F.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Pang, G.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

Raymer, M. G.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238; C.-C. Cheng, M. G. Raymer, “Long-range saturation of spatial decoherence in wave-field transport in random multiple-scattering media,” Phys. Rev. Lett. 82, 4807–4810 (1999); M. G. Raymer, C.-C. Cheng, “Propagation of the optical Wigner function in random multiple-scattering media,” in Laser–Tissue Interaction XI: Photochemical, Photothermal, and Photomechanical, D. D. Duncan, J. O. Hollinger, S. L. Jacques, eds., Proc. SPIE3914, 376–380 (2000).
[CrossRef]

Sandrov, A.

Schmitt, J. M.

Scully, M. O.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986).

Star, W. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Sterenborg, H. J. C. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Thomas, J. E.

Thrane, L.

Toloudis, D. M.

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238; C.-C. Cheng, M. G. Raymer, “Long-range saturation of spatial decoherence in wave-field transport in random multiple-scattering media,” Phys. Rev. Lett. 82, 4807–4810 (1999); M. G. Raymer, C.-C. Cheng, “Propagation of the optical Wigner function in random multiple-scattering media,” in Laser–Tissue Interaction XI: Photochemical, Photothermal, and Photomechanical, D. D. Duncan, J. O. Hollinger, S. L. Jacques, eds., Proc. SPIE3914, 376–380 (2000).
[CrossRef]

van Gemert, M. J. C.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Wang, L.

See, e.g., S. L. Jacques, L. Wang, “Monte Carlo modeling of light transport in tissues,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 4.

Wax, A.

Wigner, E. P.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Yang, Y.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

Yura, H. T.

Appl. Opt. (2)

Astrophys. J. (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Phys. Rep. (1)

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Phys. Rev. (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Other (20)

M. G. Raymer, C. Cheng, D. M. Toloudis, M. Anderson, M. Beck, “Propagation of Wigner coherence functions in multiple scattering media,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 236–238; C.-C. Cheng, M. G. Raymer, “Long-range saturation of spatial decoherence in wave-field transport in random multiple-scattering media,” Phys. Rev. Lett. 82, 4807–4810 (1999); M. G. Raymer, C.-C. Cheng, “Propagation of the optical Wigner function in random multiple-scattering media,” in Laser–Tissue Interaction XI: Photochemical, Photothermal, and Photomechanical, D. D. Duncan, J. O. Hollinger, S. L. Jacques, eds., Proc. SPIE3914, 376–380 (2000).
[CrossRef]

See, e.g., S. L. Jacques, L. Wang, “Monte Carlo modeling of light transport in tissues,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 4.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

It is straightforward to show that the quantity θrms appearing in Eq. (15) can be written as θrms=2(1-g).

It is straightforward to show that within the paraxial approximation, the specific radiance distribution N(P, θ)=k2W(P, kθ) for those cases where the Wigner phase-space distribution is positive definite.

In random media where the scattering particles are large compared with the wavelength and the index of refraction ratio is near unity, the bulk backscattering efficiency is much smaller than the scattering efficiency. Moreover, the scattering is primarily in the forward direction, which is the basis of using the paraxial approximation. Therefore the bulk backscattering may be neglected when one is considering the light propagation problem, since its contribution is mall. An example of this is skin tissue (cell sizes of 5–10-µm diameter and index of refraction ratio of 1.45/1.4=1.04).

V. A. Banakh, V. L. Mironov, LIDAR in a Turbulent Atmosphere (Artech House, Boston, Mass., 1987).

One can obtain corresponding results that are valid for propagation through the turbulent atmosphere by formally replacing the exponent in Eq. (4) by one-half the corresponding point-source wave structure function.13

For completeness, the corresponding mutual coherence function is given by 〈U(P+p/2)U*(P+p/2)〉=(η/πB2)|Γpt(p)|2K(-p)exp(-ikDp·P/B).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Because we are dealing with “real” ABCD optical systems, we tacitly assume that B≠0.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

H. T. Yura, “A multiple scattering analysis of the propagation of radiance through the atmosphere,” in Proceedings of the Union Radio-Scientifique Internationale Open Symposium (Union Radio-Scientifique Internationale, Ghent, Belgium, 1977), pp. 65–69.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986).

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.

For axially symmetric scattering functions, the integral over scattering angles reduces to 2π∫0∞dθθσ(θ)J0kB(z′)Br+1-B(z′)Bp.

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).

Note that the limits on the z′ integration of the jth term in the summation are now from zj to zj+Δzj.

The corresponding Wigner function is obtained by replacing θ by q/km.

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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Figures (1)

Fig. 1
Fig. 1

Schematic illustration of the propagation geometry. For clarity, we show the standard OCT geometry, where the focal and discontinuity planes coincide. The discontinuity at a depth z gives rise to diffuse backscattering. In principle, the space between the lens plane and the scattering medium can contain optical elements that are characterized by ABCD ray matrices. Therefore to be as general as possible, we assume an arbitrary ABCD optical system between the lens plane and the scattering medium.

Equations (72)

Equations on this page are rendered with MathJax. Learn more.

W(P, q)=dp(2π)2U(P+p/2)U*(P-p/2)×exp(iq·p),
U(P+p/2)U*(P-p/2)
=4πk2drIB(r)G0(r, P+p/2)
×G0*(r, P-p/2)Γpt(p).
G0(r, p)=-ik2πBexp-ik2B(Ar2-2r·p+Dp2),
Γpt(p)=exp{-s[1-bϕ(p)]},
bϕ(p)=0z dz0σ(θ; z)J0(kpSθ)θdθ0z dz0σ(θ; z)θdθ,
pS=B(z)Bp,
W(P, q)=1πB2dp(2π)2×expip·q-kDBPΓpt(p)H(p),
H(p)=drIB(r)expikBp·r
H(p)=ηΓpt(-p)K(-p)=ηΓpt*(p)K(-p),
K(r)=dRU0(R+r/2)U0*(R-r/2)×exp-ikABr·R,
W(P, q)=ηπB2dp(2π)2|Γpt(p)|2K(-p)×expip·q-kDBP=η2π2B20dpp|Γpt(p)|2K(p)×J0q-kDBPp (for axially symmetric U0).
Γpt(p)=exp(-s)+exp(-s){exp[sbϕ(p)]-1},
Γpt(p)=exp(-p2/ρ02)(s1),
ρ0=p0/s.
ρ0=3sλπθrmsnBz,
θrms=0dθθσ(θ)θ20dθθσ(θ)1/2.
Γpt(p)exp(-s)+[1-exp(-s)]exp(-p2/ρ02).
W(P, q)=P0ηρ022π(2πB)2exp-ρ028q-kDBP2.
σ(θ)=14π1-g2(1+g2-2g cos θ)3/2,
g=0πσ(θ)cos θ sin θdθ0πσ(θ)sin θdθ.
σ(θ)14π1-g2(θHG2+θ2)3/2,θ1,
θHG=(1-g)/g.
bϕ(p)=1-exp(-p/r0)p/r0,
r0=1kθHGnBz=λg2π(1-g)nBz.
Γpt(p)=exp[-sp/2r0]fors1.
Γpt(p)exp(-s)+[1-exp(-s)]exp(-sp/2r0).
W(P, q)=P0ηπ(2πB)2r0s211+r0s2q-kDBP23/2
fors1.
SNR=constant×ReΓR(P+p/2, P-p/2)×ΓS(P+p/2, P-p/2)dPdp,
SNR=constant×ReWR(P, -q)WS(P, q)dPdq,
U0(r)=P0πw02 exp-r221w02+ikf,
K(r)=P0 exp-r24w02.
W(P, q)ηP04w02(2πf)2exp(-2s)[exp(-Q2w02)]+exp(-s)[1-exp(-s)]ρ˜024w02×exp-Q2ρ˜024+[1-exp(-s)]2×ρ˜028w02exp-Q2ρ˜028,
1ρ˜02=1ρ02+14w02,
Q=q-kfP.
W(P, q)ηP0(2πf)2(1-2s)exp(-Q2w02)fors1
W(P, q)P0ηπ(2πf)2r0s211+r0s2q-kfP23/2
fors1,
W(P, q; Z)=dRdqH(P-R, q-q; Z)×W(R-Zq/k, q; 0),
W(R, q; 0)=dr(2π)2U0(R+r/2)U0*(R-r/2)×exp(iq·r),
H(P-R, q-q; Z)=km4π2Z2dpdrF(p, r)×expikmZ(r-p)·(P-R)+ip·(q-q),
F(p, r)=exp{-s[1-f(p, r)]},
s=0Zdzdθσ(θ)=μsZ
f(p, r)=1s0Zdzdθσ(θ)×expikθ·zZr+1-zZp.
W(P, q; Z)=dRdqH(P, R, q, q; Z)×W(R-Bq/k, q; 0),
H(P, R, q, q; Z)=k4π2B2dpdrF(p, r)×expikB[R·(p-Ar)-P·(Dp-r)]+ip·(q-q)+iq·r(A-1),
F(p, r)=exp{-s[1-f(p, r)]},
f(p, r)=1szz+Δzdzdθσ(θ; z)×expikθ·B(z)Br+1-B(z)Bp,
exp-j=1Nsj[1-fj(p, r)],
Γ(P, p; Z)U(P+p/2; Z)U*(P-p/2; Z),
Γ(P, p; Z)=dRdrG(P, p, R, r; Z)Γ(R, r; 0),
G(P, p, R, r; Z)=k2πB2F(p, r)×expikB[R·(p-Ar)-P·(Dp-r)],
I(P; Z)=dRdrG(P, 0, R, r; Z)Γ(R, r; 0)=drK(r)Γpt(r)expikBP·r,
K(r)=k2πB2dR exp-ikBAR·r×U(R+r/2; 0)U*(R-r/2; 0)
GA(R, P)=exp(-ikmZ-μAZ/2)Z×exp-ikm2Z(R-P)2-μA2Z(R-P)2,
WA(P, q; Z)=dRdqHA(P-R, q-q; Z)×W(R-Zq/k, q; 0),
HA(P-R, q-q; Z)=exp(-μAZ)exp-μA8Z[(p-r)2+4(P-R)2]×H(P-R, q-q; Z).
N(θ; Z)=exp(-μAZ)I0πθ0A2exp(-θ2/θ0A2),
θ0A2=θ021+(μAZθ02)/6,
θ02=sθrms2=μsZθrms2.
HA(P, R, q, q; Z)=exp(-μAΔz)exp-μA8B[A(p2+4P2)-2p·r+Dr2-8P·R+4DR2]
×H(P, R, q, q; Z),
F(q)=drIB(r)exp(iq·r),
IB(r)=ηk2πB2du expikBr·uK(u)Γpt(u),
F(q)=ηk2πB2drK(r)Γpt(r)(2π)2δ(q+kr/B)=ηΓpt-BkqK-Bkq,
H(p)=F(kp/B)=ηΓpt(-p)K(-p)=ηΓpt*(p)K(-p).
U(P+p/2)U*(P+p/2)
=(η/πB2)|Γpt(p)|2K(-p)exp(-ikDp·P/B).
2π0dθθσ(θ)J0kzZr+1-zZp,
2π0dθθσ(θ)J0kB(z)Br+1-B(z)Bp.

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