Abstract

We study the spatial coherence of an optical beam in a strongly scattering medium confined in a slab geometry. Using the radiative transfer equation, we study numerically the behavior of the transverse spatial coherence length in the different transport regimes. Transitions from the ballistic to the diffusive regimes are clearly identified.

© 2005 Optical Society of America

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References

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  1. P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).
    [CrossRef]
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  3. L. A. Apresyan, Y. A. Kravtsov, Radiation Transfer—Statistical and Wave Aspects (Gordon & Breach, 1996).
  4. F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
    [CrossRef]
  5. V. I. Tatarskii, Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, 1971).
  6. L. C. Andrews, R. L. Philips, Laser Beam Propagation through Random Media (SPIE, 1998).
  7. S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
    [CrossRef]
  8. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  9. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  10. H. M. Pedersen, “Exact geometrical theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991).
    [CrossRef]
  11. H. M. Pedersen, “Exact geometrical theory of free-space radiative energy transfer: errata,” J. Opt. Soc. Am. A 8, 1518–1518 (1991).
    [CrossRef]
  12. E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19, 2024–2026 (1994).
    [CrossRef] [PubMed]
  13. M. Alonso, “Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2001).
    [CrossRef]
  14. M. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).
    [CrossRef]
  15. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
    [CrossRef]
  16. S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.
  17. Y. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).
  18. L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
    [CrossRef]
  19. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  20. R. G. Littlejohn, R. Winston, “Generalized radiance and measurement,” J. Opt. Soc. Am. A 12, 2736–2743 (1995).
    [CrossRef]
  21. R. Winston, R. G. Littlejohn, “Measuring the instrument function of radiometers,” J. Opt. Soc. Am. A 14, 3099–3101 (1997).
    [CrossRef]
  22. L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 18, 374–384 (2001).
    [CrossRef]
  23. G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
    [CrossRef]
  24. R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).
    [CrossRef]
  25. R. Elaloufi, R. Carminati, J.-J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am. A 21, 1430–1437 (2004).
    [CrossRef]
  26. R. Carminati, R. Elaloufi, J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
    [CrossRef]

2004 (2)

R. Elaloufi, R. Carminati, J.-J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am. A 21, 1430–1437 (2004).
[CrossRef]

R. Carminati, R. Elaloufi, J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

2002 (2)

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).
[CrossRef]

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

2001 (3)

1997 (1)

1996 (1)

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

1995 (1)

1994 (1)

1991 (3)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

H. M. Pedersen, “Exact geometrical theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991).
[CrossRef]

H. M. Pedersen, “Exact geometrical theory of free-space radiative energy transfer: errata,” J. Opt. Soc. Am. A 8, 1518–1518 (1991).
[CrossRef]

1989 (1)

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

1969 (1)

Y. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).

1968 (1)

Alonso, M.

Andrews, L. C.

L. C. Andrews, R. L. Philips, Laser Beam Propagation through Random Media (SPIE, 1998).

Apresyan, L. A.

L. A. Apresyan, Y. A. Kravtsov, Radiation Transfer—Statistical and Wave Aspects (Gordon & Breach, 1996).

Barabanenkov, Y. N.

Y. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).

Carminati, R.

R. Elaloufi, R. Carminati, J.-J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am. A 21, 1430–1437 (2004).
[CrossRef]

R. Carminati, R. Elaloufi, J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Elaloufi, R.

R. Elaloufi, R. Carminati, J.-J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am. A 21, 1430–1437 (2004).
[CrossRef]

R. Carminati, R. Elaloufi, J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).
[CrossRef]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Greffet, J.-J.

R. Carminati, R. Elaloufi, J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

R. Elaloufi, R. Carminati, J.-J. Greffet, “Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach,” J. Opt. Soc. Am. A 21, 1430–1437 (2004).
[CrossRef]

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).
[CrossRef]

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

L. Roux, P. Mareschal, N. Vukadinovic, J.-B. Thibaud, J.-J. Greffet, “Scattering by a slab containing randomly located cylinders: comparison between radiative transfer and electromagnetic simulation,” J. Opt. Soc. Am. A 18, 374–384 (2001).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

John, S.

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

Keller, J. B.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Kravtsov, Y. A.

L. A. Apresyan, Y. A. Kravtsov, Radiation Transfer—Statistical and Wave Aspects (Gordon & Breach, 1996).

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Littlejohn, R. G.

MacKintosh, F. C.

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

Mareschal, P.

Papanicolaou, G.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Pedersen, H. M.

Philips, R. L.

L. C. Andrews, R. L. Philips, Laser Beam Propagation through Random Media (SPIE, 1998).

Ponomarenko, S. A.

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Roux, L.

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Ryzhik, L.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sebbah, P.

P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).
[CrossRef]

Stamnes, K.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tatarskii, V. I.

V. I. Tatarskii, Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, 1971).

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Thibaud, J.-B.

Thomas, G. E.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
[CrossRef]

Vukadinovic, N.

Walther, A.

Winston, R.

Wolf, E.

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19, 2024–2026 (1994).
[CrossRef] [PubMed]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

R. Elaloufi, R. Carminati, J.-J. Greffet, “Time-dependent transport through scattering media: from radiative transfer to diffusion,” J. Opt. A, Pure Appl. Opt. 4, S103–S108 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Opt. Commun. (1)

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

F. C. MacKintosh, S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

Phys. Rev. Lett. (1)

R. Carminati, R. Elaloufi, J.-J. Greffet, “Beyond the diffusion-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

Y. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).

Wave Motion (1)

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Other (9)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

V. I. Tatarskii, Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, 1971).

L. C. Andrews, R. L. Philips, Laser Beam Propagation through Random Media (SPIE, 1998).

P. Sebbah, Waves and Imaging through Complex Media (Kluwer Academic, 2001).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

L. A. Apresyan, Y. A. Kravtsov, Radiation Transfer—Statistical and Wave Aspects (Gordon & Breach, 1996).

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry of the system. Medium 2 is a scattering slab of thickness L illuminated from the left by a plane wave at normal incidence. The notations used to decompose the specific intensity into ballistic and diffuse components propagating in the forward and backward directions are indicated.

Fig. 2
Fig. 2

Degree of spatial coherence w ( τ , ρ ) inside a semi-infinite scattering medium versus the normalized distance from the medium interface τ and the distance ρ between the two observation points. The scattering medium is described by a Henyey–Greenstein phase function with g = 0.90 and an albedo a = 0.98 . (a) Circles correspond to α c ( τ ) w c , and the arrow indicates an example of FWHM for the diffuse component only and is reported on Fig. 3.

Fig. 3
Fig. 3

Spatial coherence length (normalized by the wavelength) of (a) the diffuse radiation and (b) the diffuse and ballistic radiation propagating in the forward direction inside a semi-infinite scattering medium versus the normalized distance z l * from the slab interface, where l * is the transport mean-free path. The scattering medium is described by a Henyey–Greenstein phase function with different values of the anisotropy factor g, an albedo a = 0.98 , and a refractive index n 2 = 1 . The arrow indicates an example of FWHM for the scattered part only and is reported on Fig. 2.

Fig. 4
Fig. 4

Same as Fig. 3 with n 2 = 1.33 .

Fig. 5
Fig. 5

Plot of the variations of the specific intensity at the entrance of the slab. Three zones can be identified: For μ 2 < 0 , the specific intensity is quasi isotropic owing to multiple scattering of light in a semi-infinite slab; for 0 < μ 2 < μ l , the specific intensity is also quasi isotropic owing to the whole reflection; and, for μ 2 > μ l , there are two contributions, which are the partial reflection of I d and the single scattering of the incident specific intensity.

Fig. 6
Fig. 6

Comparison between the single-scattering model (solid curve) and the whole numerical computation (dashed curve) for the diffuse spatial coherence length in the forward direction normalized by the wavelength versus the normalized distance z l s for two anisotropy factors [(a) g = 0.16 and (b) g = 0.80 ], an albedo a = 1 , and a refractive index n 2 = 1.33 . We used a Henyey–Greenstein phase function. The scale for l coh d + is very small. The relative errors are about 2% for z < 8 l * 10 l * .

Fig. 7
Fig. 7

Geometrical filtering effect on the coherence length.

Equations (40)

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

W ( r 1 , r 2 , ν ) = + u ( r 1 , t ) u * ( r 2 , t + τ ) exp [ 2 i π ν τ ] d τ ,
W ( r , ρ ) = 4 π I ν ( r , u ) exp ( i k u ρ ) d Ω ,
u r I ν ( r , u ) = ( μ a + μ s ) I ν ( r , u ) + μ s 4 π 4 π p ( u , u ) I ν ( r , u ) d Ω .
W ( z , ρ ) = I ν ( z , μ , φ ) exp ( i k ρ u ) d μ d φ .
J 0 ( x ) = 1 2 π 0 2 π exp ( i x cos φ ) d φ .
W ( z , ρ ) = 1 + 1 I ν ( z , μ ) J 0 ( 2 π λ ρ 1 μ 2 ) d μ ,
W ( z , ρ ) = 2 I ν ( z ) sinc ( 2 π ρ λ ) .
l coh 0.3 λ .
μ τ I ν ( τ , μ ) = I ν ( τ , μ ) + a 2 1 + 1 p ( μ , μ ) I ν ( τ , μ ) d μ .
I ( τ , μ ) = I c + ( τ ) δ ( μ 1 ) + I c ( τ ) δ ( μ + 1 ) + I d ( τ , μ ) ,
I d + ( τ , μ ) = I d ( τ , μ > 0 ) ,
I d ( τ , μ ) = I d ( τ , μ < 0 ) .
W c + ( τ ) = I c + ( τ ) ,
W c ( τ ) = I c ( τ ) ,
W c ( τ ) = W c + ( τ ) + W c ( τ ) ,
W d + ( τ , ρ ) = 0 + 1 I d + ( τ , μ ) J 0 ( 2 π λ ρ 1 μ 2 ) d μ ,
W d ( τ , ρ ) = 1 0 I d ( τ , μ ) J 0 ( 2 π λ ρ 1 μ 2 ) d μ ,
W d ( τ , ρ ) = W d + ( τ , ρ ) + W d ( τ , ρ ) .
W ( τ , ρ ) W ( τ , 0 ) = W c ( τ ) W c ( τ ) W c ( τ ) W c ( τ ) + W d ( τ , 0 ) + W d ( τ , ρ ) W d ( τ , 0 ) W d ( τ , 0 ) W c ( τ ) + W d ( τ , 0 ) .
w ( τ , ρ ) = w c α c ( τ ) + w d ( τ , ρ ) α d ( τ ) ,
W ± ( τ , ρ ) W ± ( τ , 0 ) = W c ± ( τ ) W c ± ( τ ) W c ± ( τ ) W c ± ( τ ) + W d ± ( τ , 0 ) + W d ± ( τ , ρ ) W d ± ( τ , 0 ) W d ± ( τ , 0 ) W c ± ( τ ) + W d ± ( τ , 0 ) .
w ± ( τ , ρ ) = w c ± α c ± ( τ ) + w d ± ( τ , ρ ) α d ± ( τ ) ,
d Ω = d u x d u y u z = d 2 u u z .
W ( r , ρ ) = exp ( i k ρ z u z ) u z I ν ( r , u ) exp ( i k ρ u ) d 2 u .
I ν ( r , u ) = ( k 2 π ) 2 μ exp ( i k ρ z u z ) W ( r , ρ ) exp ( i k ρ u ) d 2 ρ ,
I ν ( r , u ) = ( k 2 π ) 2 μ W ( r , ρ ) exp ( i k ρ u ) d 2 ρ ,
I ( τ , μ ) = I ( τ ) .
W ( τ , ρ ) = I ( τ ) 1 + 1 J 0 ( 2 π ρ λ 1 μ 2 ) d μ .
W + ( τ , X ) = I ( τ ) 0 + 1 J 0 ( X u ) u d u 1 u 2 = I ( τ ) sinc ( X ) .
W ( τ , X ) = I ( τ ) sinc ( X ) ,
W ( τ , X ) = 2 I ( τ ) sinc ( X ) .
T 21 d ( μ 1 ) = n 2 2 n 1 2 T 21 ( μ 1 ) .
I d ( τ , μ 2 ) = 1 R 12 ( 1 ) 0 1 T 21 d ( μ 1 ) μ 1 d μ 1 I inc = X I inc .
I d ( τ , μ 2 ) R 21 ( μ 2 ) = X I inc ,
a 2 1 1 p ( μ 2 , μ 2 ) I d ( τ , μ 2 ) d μ 2 a 2 1 μ l p ( μ 2 , μ 2 ) X L inc d μ 2 = a I inc J ( μ 2 ) ,
J ( μ 2 ) = X 2 1 μ l p ( μ 2 , μ 2 ) d μ 2 .
I c + ( τ ) = T 12 ( 1 ) exp ( τ ) .
μ I d + τ ( τ , μ 2 ) + I d + ( τ , μ 2 ) = a I inc J ( μ 2 ) + a 2 p ( μ 2 , 1 ) T 12 ( 1 ) I inc exp ( τ ) .
I d + ( 0 , μ 2 ) = 0 .
I d + ( τ , μ 2 ) = a I inc 2 ( 1 μ 2 ) { 2 I ( μ 2 ) ( 1 μ 2 ) + p ( μ 2 , 1 ) T 12 ( 1 ) exp ( τ ) [ 2 I ( μ 2 ) ( 1 μ 2 ) + p ( μ 2 , 1 ) T 12 ( 1 ) ] exp ( τ μ 2 ) } .

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