Abstract

We present a new nonperturbative theoretical method for the analytical description of light propagation in random multiscattering media. The method is illustrated through the calculation of an expression that describes optical backscattering from a semi-infinite disordered medium. A companion paper [J. Opt. Soc. Am. A 17, 2034 (2000)] compares the theoretical expression with experimental data.

© 2000 Optical Society of America

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References

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  1. M. van Rossum, Th. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 314–371 (1999).
    [CrossRef]
  2. I. Lifshitz, S. Gredescul, L. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).
  3. S. Rytov, V. Kravtsov, V. Tatarskii, Principles of Statistical Radiophysics (Springer, Berlin1988).
  4. M. Nieto-Vesperinas, J. Dainty, eds., Scattering in Volumes and Surfaces (North-Holland, Amsterdam, 1990); V. Tatarsky, A. Ishimaru, V. Zavorotny, eds., Wave Propagation in Random Media (SPIE Press, Bellingham, Wash., 1993); Ping Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic, New York, 1995).
  5. E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
    [CrossRef]
  6. Yu. Barabanenkov, Yu. Kravtsov, V. Ozvin, A. Saichev, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1991), Vol. 29, p. 66; V. Shalaev, “Electromagnetic properties of small-particle composites,” Phys. Rep. 272, 61–137 (1996).
    [CrossRef]
  7. F. Scheffold, G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800–5803 (1998).
    [CrossRef]
  8. R. H. Kop, P. de Vries, R. Sprik, Ad. Langendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997);P. de Vries, D. van Coevorden, Ad. Langendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998). These works demonstrate the breakdown of radiative transport theory in the description of light propagation through thin disordered slabs examined in LCI with widths even larger than distances l examined in LCI measurements.
    [CrossRef]
  9. P. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998).
    [CrossRef]
  10. A. Brodsky, P. Shelley, S. Thurber, L. Burgess, “Low-coherence interferometry of particles distributed in dielectric medium,” J. Opt. Soc. Am. A 14, 2263–2268 (1997). See also C. Popescu, A. Dogariu, “Optical path-length spectroscopy of wave propagation in random media,” Opt. Lett. 24, 442–444 (1998).
    [CrossRef]
  11. B. L. Danielson, C. D. Whittenberg, “Guided-wave reflectometry with micrometer resolution,” Appl. Opt. 26, 2836–2842 (1987).
    [CrossRef] [PubMed]
  12. S. Thurber, L. Burgess, A. Brodsky, P. Shelley, “Low coherence interferometry in random media. II. Experiment,” J. Opt. Soc. Am. A 17, 2034–2039 (2000).
    [CrossRef]
  13. K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge U. Press, New York, 1997).
  14. The solution of the Bloch–Nordsiek model is reviewed in W. Bogolubov, D. Shirkov, Introduction to the Theory of Quantized Fields (Wiley, New York, 1980).
  15. G. Samelsohn, R. Mazar, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E 54, 5697–5706 (1996).
    [CrossRef]
  16. In the coherent phase approximation, κ(ω)=2πcn(ω)2∑α〈Nα〉Aα(0),where 〈Nα〉is the mean density of nonuniformities (particles) of type α and Aα(0)are the complex amplitudes of forward scattering caused by these nonuniformities (Tmatrices). The imaginary component of Aα(0)includes an effect of coherence loss during scattering events. See Ref. 17for its applications of coherent phase approximation in optics.
  17. R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. I; H. van De Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  18. L. Landau, E. Lifshitz, Electrodynamics of Condensed Media (Pergamon, London, 1980), Chap. 10.
  19. The long tail of a distribution of waves scattered by nonuniformities is dominated by speckle effects. See Refs. 2 and 7 and J. F. de Boer, M. van Rossum, M. van Albada, Th. Nieuwenhuizen, Ad. Langendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett. 73, 2567–2570 (1994).
    [CrossRef] [PubMed]

2000 (1)

1999 (1)

M. van Rossum, Th. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 314–371 (1999).
[CrossRef]

1998 (2)

F. Scheffold, G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800–5803 (1998).
[CrossRef]

P. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998).
[CrossRef]

1997 (2)

A. Brodsky, P. Shelley, S. Thurber, L. Burgess, “Low-coherence interferometry of particles distributed in dielectric medium,” J. Opt. Soc. Am. A 14, 2263–2268 (1997). See also C. Popescu, A. Dogariu, “Optical path-length spectroscopy of wave propagation in random media,” Opt. Lett. 24, 442–444 (1998).
[CrossRef]

R. H. Kop, P. de Vries, R. Sprik, Ad. Langendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997);P. de Vries, D. van Coevorden, Ad. Langendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998). These works demonstrate the breakdown of radiative transport theory in the description of light propagation through thin disordered slabs examined in LCI with widths even larger than distances l examined in LCI measurements.
[CrossRef]

1996 (1)

G. Samelsohn, R. Mazar, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E 54, 5697–5706 (1996).
[CrossRef]

1994 (1)

The long tail of a distribution of waves scattered by nonuniformities is dominated by speckle effects. See Refs. 2 and 7 and J. F. de Boer, M. van Rossum, M. van Albada, Th. Nieuwenhuizen, Ad. Langendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett. 73, 2567–2570 (1994).
[CrossRef] [PubMed]

1988 (1)

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

1987 (1)

Akkermans, E.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

Barabanenkov, Yu.

Yu. Barabanenkov, Yu. Kravtsov, V. Ozvin, A. Saichev, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1991), Vol. 29, p. 66; V. Shalaev, “Electromagnetic properties of small-particle composites,” Phys. Rep. 272, 61–137 (1996).
[CrossRef]

Bogolubov, W.

The solution of the Bloch–Nordsiek model is reviewed in W. Bogolubov, D. Shirkov, Introduction to the Theory of Quantized Fields (Wiley, New York, 1980).

Brodsky, A.

Brouwer, P.

P. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998).
[CrossRef]

Burgess, L.

Danielson, B. L.

de Boer, J. F.

The long tail of a distribution of waves scattered by nonuniformities is dominated by speckle effects. See Refs. 2 and 7 and J. F. de Boer, M. van Rossum, M. van Albada, Th. Nieuwenhuizen, Ad. Langendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett. 73, 2567–2570 (1994).
[CrossRef] [PubMed]

de Vries, P.

R. H. Kop, P. de Vries, R. Sprik, Ad. Langendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997);P. de Vries, D. van Coevorden, Ad. Langendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998). These works demonstrate the breakdown of radiative transport theory in the description of light propagation through thin disordered slabs examined in LCI with widths even larger than distances l examined in LCI measurements.
[CrossRef]

Efetov, K.

K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge U. Press, New York, 1997).

Gredescul, S.

I. Lifshitz, S. Gredescul, L. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).

Kop, R. H.

R. H. Kop, P. de Vries, R. Sprik, Ad. Langendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997);P. de Vries, D. van Coevorden, Ad. Langendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998). These works demonstrate the breakdown of radiative transport theory in the description of light propagation through thin disordered slabs examined in LCI with widths even larger than distances l examined in LCI measurements.
[CrossRef]

Kravtsov, V.

S. Rytov, V. Kravtsov, V. Tatarskii, Principles of Statistical Radiophysics (Springer, Berlin1988).

Kravtsov, Yu.

Yu. Barabanenkov, Yu. Kravtsov, V. Ozvin, A. Saichev, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1991), Vol. 29, p. 66; V. Shalaev, “Electromagnetic properties of small-particle composites,” Phys. Rep. 272, 61–137 (1996).
[CrossRef]

Landau, L.

L. Landau, E. Lifshitz, Electrodynamics of Condensed Media (Pergamon, London, 1980), Chap. 10.

Langendijk, Ad.

R. H. Kop, P. de Vries, R. Sprik, Ad. Langendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997);P. de Vries, D. van Coevorden, Ad. Langendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998). These works demonstrate the breakdown of radiative transport theory in the description of light propagation through thin disordered slabs examined in LCI with widths even larger than distances l examined in LCI measurements.
[CrossRef]

The long tail of a distribution of waves scattered by nonuniformities is dominated by speckle effects. See Refs. 2 and 7 and J. F. de Boer, M. van Rossum, M. van Albada, Th. Nieuwenhuizen, Ad. Langendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett. 73, 2567–2570 (1994).
[CrossRef] [PubMed]

Lifshitz, E.

L. Landau, E. Lifshitz, Electrodynamics of Condensed Media (Pergamon, London, 1980), Chap. 10.

Lifshitz, I.

I. Lifshitz, S. Gredescul, L. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).

Maret, G.

F. Scheffold, G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800–5803 (1998).
[CrossRef]

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

Maynard, R.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

Mazar, R.

G. Samelsohn, R. Mazar, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E 54, 5697–5706 (1996).
[CrossRef]

Newton, R.

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. I; H. van De Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Nieuwenhuizen, Th.

M. van Rossum, Th. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 314–371 (1999).
[CrossRef]

The long tail of a distribution of waves scattered by nonuniformities is dominated by speckle effects. See Refs. 2 and 7 and J. F. de Boer, M. van Rossum, M. van Albada, Th. Nieuwenhuizen, Ad. Langendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett. 73, 2567–2570 (1994).
[CrossRef] [PubMed]

Ozvin, V.

Yu. Barabanenkov, Yu. Kravtsov, V. Ozvin, A. Saichev, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1991), Vol. 29, p. 66; V. Shalaev, “Electromagnetic properties of small-particle composites,” Phys. Rep. 272, 61–137 (1996).
[CrossRef]

Pastur, L.

I. Lifshitz, S. Gredescul, L. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).

Rytov, S.

S. Rytov, V. Kravtsov, V. Tatarskii, Principles of Statistical Radiophysics (Springer, Berlin1988).

Saichev, A.

Yu. Barabanenkov, Yu. Kravtsov, V. Ozvin, A. Saichev, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1991), Vol. 29, p. 66; V. Shalaev, “Electromagnetic properties of small-particle composites,” Phys. Rep. 272, 61–137 (1996).
[CrossRef]

Samelsohn, G.

G. Samelsohn, R. Mazar, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E 54, 5697–5706 (1996).
[CrossRef]

Scheffold, F.

F. Scheffold, G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800–5803 (1998).
[CrossRef]

Shelley, P.

Shirkov, D.

The solution of the Bloch–Nordsiek model is reviewed in W. Bogolubov, D. Shirkov, Introduction to the Theory of Quantized Fields (Wiley, New York, 1980).

Sprik, R.

R. H. Kop, P. de Vries, R. Sprik, Ad. Langendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997);P. de Vries, D. van Coevorden, Ad. Langendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998). These works demonstrate the breakdown of radiative transport theory in the description of light propagation through thin disordered slabs examined in LCI with widths even larger than distances l examined in LCI measurements.
[CrossRef]

Tatarskii, V.

S. Rytov, V. Kravtsov, V. Tatarskii, Principles of Statistical Radiophysics (Springer, Berlin1988).

Thurber, S.

van Albada, M.

The long tail of a distribution of waves scattered by nonuniformities is dominated by speckle effects. See Refs. 2 and 7 and J. F. de Boer, M. van Rossum, M. van Albada, Th. Nieuwenhuizen, Ad. Langendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett. 73, 2567–2570 (1994).
[CrossRef] [PubMed]

van Rossum, M.

M. van Rossum, Th. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 314–371 (1999).
[CrossRef]

The long tail of a distribution of waves scattered by nonuniformities is dominated by speckle effects. See Refs. 2 and 7 and J. F. de Boer, M. van Rossum, M. van Albada, Th. Nieuwenhuizen, Ad. Langendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett. 73, 2567–2570 (1994).
[CrossRef] [PubMed]

Whittenberg, C. D.

Wolf, P. E.

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

J. Phys. (Paris) (1)

E. Akkermans, P. E. Wolf, R. Maynard, G. Maret, “Theoretical study of the coherent backscattering of light by disordered media,” J. Phys. (Paris) 49, 77–98 (1988).
[CrossRef]

Phys. Rev. B (1)

P. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526–10536 (1998).
[CrossRef]

Phys. Rev. E (1)

G. Samelsohn, R. Mazar, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E 54, 5697–5706 (1996).
[CrossRef]

Phys. Rev. Lett. (3)

The long tail of a distribution of waves scattered by nonuniformities is dominated by speckle effects. See Refs. 2 and 7 and J. F. de Boer, M. van Rossum, M. van Albada, Th. Nieuwenhuizen, Ad. Langendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett. 73, 2567–2570 (1994).
[CrossRef] [PubMed]

F. Scheffold, G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800–5803 (1998).
[CrossRef]

R. H. Kop, P. de Vries, R. Sprik, Ad. Langendijk, “Observation of anomalous transport of strongly multiple scattered light in thin disordered slabs,” Phys. Rev. Lett. 79, 4369–4372 (1997);P. de Vries, D. van Coevorden, Ad. Langendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998). These works demonstrate the breakdown of radiative transport theory in the description of light propagation through thin disordered slabs examined in LCI with widths even larger than distances l examined in LCI measurements.
[CrossRef]

Rev. Mod. Phys. (1)

M. van Rossum, Th. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. 71, 314–371 (1999).
[CrossRef]

Other (9)

I. Lifshitz, S. Gredescul, L. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).

S. Rytov, V. Kravtsov, V. Tatarskii, Principles of Statistical Radiophysics (Springer, Berlin1988).

M. Nieto-Vesperinas, J. Dainty, eds., Scattering in Volumes and Surfaces (North-Holland, Amsterdam, 1990); V. Tatarsky, A. Ishimaru, V. Zavorotny, eds., Wave Propagation in Random Media (SPIE Press, Bellingham, Wash., 1993); Ping Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Academic, New York, 1995).

Yu. Barabanenkov, Yu. Kravtsov, V. Ozvin, A. Saichev, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1991), Vol. 29, p. 66; V. Shalaev, “Electromagnetic properties of small-particle composites,” Phys. Rep. 272, 61–137 (1996).
[CrossRef]

K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge U. Press, New York, 1997).

The solution of the Bloch–Nordsiek model is reviewed in W. Bogolubov, D. Shirkov, Introduction to the Theory of Quantized Fields (Wiley, New York, 1980).

In the coherent phase approximation, κ(ω)=2πcn(ω)2∑α〈Nα〉Aα(0),where 〈Nα〉is the mean density of nonuniformities (particles) of type α and Aα(0)are the complex amplitudes of forward scattering caused by these nonuniformities (Tmatrices). The imaginary component of Aα(0)includes an effect of coherence loss during scattering events. See Ref. 17for its applications of coherent phase approximation in optics.

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. I; H. van De Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

L. Landau, E. Lifshitz, Electrodynamics of Condensed Media (Pergamon, London, 1980), Chap. 10.

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Figures (1)

Fig. 1
Fig. 1

Block diagram of the LCI experiments.

Equations (83)

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

n¯(ω, x)=θ(-x1)+n(ω, x)θ(x1),
θ(x1)=1forx1>012forx1=00forx1<0.
n(ω; x)=n(ω)+κ(ω) δρ(x)ρ¯,n(ω)=n(ω; x).
E1ref(2l+x; t)=E0ref2πσdω exp-iωt+ ωc (2l+x1)-(|ω|-ω0)22σ2,
JSUT(l)const.×[J(l)+Jdev],
J(l)=Re(E0ref)*2πσdω|x-x0|<Ld3x E(ω; x)×exp--iωc (2l+x1)+(|ω|-ω0)22σ2av,
F(1,2)(ω, x)=E(ω, x)±ih(ω; x)H(ω, x),
curl F(1,2)±ωc n¯(ω; x)F(1,2)
±[gradln n¯(ω; x)] F(1)-F(2)2=0.
i, α|I|j, β=δijδα,β,
i, α|Ikl|j, β=δikδjlδα, β,
i, α|Sk|j, β=-δα,βieijk,
i, α|σ1|j, β=δij(δα1δβ2+δα2δβ1),
i, α|σ3|j, β=δij(δα1δβ1-δα2δβ2),
ik=13Skxk F+σ3ωc n¯(ω; x)F
+I-τ12 ik=13Sk ln n¯(ω; x)xk F=0,
FF(1)(ω, x)F(2)(ω, x)F1(1)F2(1)F3(1)F1(2)F2(2)F3(2),
F*(ω, x)=σ1F(-ω, x).
k=13iSkxk+σ3ωc n¯(ω; x)+I-τ12k=13iSk ln n¯(ω; x)xk
×G(ω; x-x)=Iδ3(x, x).
S12|x-x0|<LG(ω; x; x)d2x
=S12exp-i ωc x12π|x-x0|<LG(ω; x, x)d2xx1=0
forx10,x10,x={x2, x3}.
J(l)=LI0(2πσ)2|x-x0|<Lδ(x1)d3xd3x×[F(0)]trdωexp-(|ω|-ω0)2σ2×exp-i ωc [2l-n(ω)x1]×G(ω; x, x)θ(x1)i ωc κ(ω) δρ(x)ρ¯ σ3+I-σ12n(ω; x)k=13Skxk θ(x1)κ(ω) δρ(x)ρ¯,
F(0)=σ1(F0)*=1401-i01i.
G(ω; x, x)=I+σ12+I-σ12n0(ω; x1)n¯(ω; x)×Gg(ω; x, x)×I+σ12+I-σ12n¯(ω; x)n0(ω; x1),
Gg(ω; x, x)=iσ30U(ν)dν,
U(ν)U(ν, ω; x, x)=exp(iσ3Kν)δ3(x-x),
K=i k=13Skxk+ωc σ3n¯(ω; x1)+I-δ12ωc δ3n2(ω; x)-n¯2(ω; x1)n¯(ω; x)
n0(ω; x1)=θ(x1)n(ω)+θ(-x1).
-i Uν=σ3KU
U(ν)=Iδ3(x-x)|ν=0.
U(ν)=1(2π)3/2d3p expiωc [1-n(ω)]×σ3S1(x1+ν)θ(-x1-ν)+(x1-ν)θ(-x1+ν)-2x1θ(-x1)2-S12νθ(-x1)-(x1+ν)θ(-x1-ν)+(ν-x1)θ(-x1+ν)2+Iνθ(-x1)
×expip(x-x)+νσ3k=13Skpk+νn(ω) ωcexp[iΛ(ω; ν; x)],
Λ(ω; 0; x)=0.
iI ν+iσ3k=13Skxk,exp(iΛ)
+κ(ω) ωc θ(x1) δρ(x)ρ¯exp(iΛ)=0,
[A, B]=AB-BA.
I ν+σ3k=13SkxkΛ+ωc κ(ω) δρ(x)θ(x1)ρ¯=0.
I Λν+σ3k=13SkΛxk, Λ=0.
Λ=ωκ(ω)c(2π)3/2d3q exp(iqx)×δρ(q)ρ¯iσ3k=13Skqk1-cosνqq2+[I-I˜(q)]ν-sinνqq+I sinνqq,
Λ*(ω; ν; x)=-σ1Λ(-ω; ν; x)σ1,Λσ3-σ3Λ=0,
αi|I˜(q)|βj=δαβδij-qiqjq2,q=(q)2
δρ(q)=1(2π)3/2d3xexp(-iqx)δρ(x)θ(x1).
k=13SkxkI2-k,k=13SkSk2xkxkF(x)0,
J(l)=-LI0(2π)σ2i|x|<Ldωd3pd3xd3x δ(x1)θ(x1)×0dν exp-2i ωc n1(ω)l×[F(0)]trexp-(|ω|-ω0)2σ2+iΛ(ω; ν; x)×exp-iωcν[1-n(ω)] σ3S12+iωcν n(1)+n(ω)2 I×expip(x-x)+νk=13σ3Skpk+ωc n(ω)x1iωc κ(ω) δρ(x)ρ¯+I+σ12n(ω)κ(ω)ρ¯ δρ(x)κ(ω) δρ(x)ρ¯+k=13iσ3Skpk+σ3n(ω) ωc S1ωcF(0).
iτ1τ3I+σ12+I-σ12n¯(ω; x)n0(ω; x1)ωc σ3κ(ω)θ(x1) δρ(x)ρ¯
=σ11+I+σ12δρ(ω; x)n(ω)ρ¯i ωc κ(ω)θ(x1) δρ(x)ρ¯
I+σ12δρ(ω; x)n(ω)ρ¯ωc (ω)θ(x1) δρ(x)ρ¯.
exp[iΛ(ω, ν, x)exp-iνσ3k=13Skpk*
=exp[iσ1Λ(-ω, ν, x)σ1]exp-iνσ1σ3k=13Skpkσ1=σ1exp[iΛ(-ω, ν, x)]expiνσ3k=13Skpνσ1,
I+σ12 σ1=I+σ12,I-σ12 σ1=-I-σ12,
Sk*=-Sk,
δρ(x)
=δρ(q)=0,
δρ(x)ρ¯δρ(x)ρ¯
=Δ exp-(x-x)2a2 forx1, x10,
δρ(q)ρ¯δρ(q)ρ¯
=(aπ)3Δ exp-q2a24δ3(q+q)×δρ(q1)ρ¯  δρ(qn)ρ¯expiϕ(q) δρ(q)ρ¯d3q=(-i)nδnδϕ(q1)  δϕ(qn)×exp-12ϕ(q)D(q, q)ϕ(q)d3qd3q,
γ(z)=exp-z22Δ1forz1.
J(l)=Δa3L32σ22πidω0dνωc κ(ω)2×expiωcν 1+n2(ω)2-2l-(|ω|-ω0)2σ2×dp1p1-ωc n2(ω) [F(0)]trexp-Δ3π2ωac κ(ω)2×1+ν22a2-exp-νa2×exp-q2a24d3qiσ3k=13Skqk1-cosνqq2+[I-I˜(q)]ν-sinνqq+I sinνqq×expiωcν 1-n(ω)2 σ3S1×expiνσ3S1p1-iνk=13σ3Skqk×iσ3S1p1-ik=13σ3SkqkI-σ12 F(0)+O(Δ2).
dx1 θ(x)expiq1-p1+ωc n(ω)x
=i[q1-p1+ωc n(ω)]-1,
σ3S1F(0)=-F(0).
12πidp1p1mp1-n(ω)exp(iS1τ3p1ν)F(0)
=12πidp1p1mp1-n(ω)exp(-iνp1)F(0)=0,
τ12π1dp1p1mp1-n(ω)exp(iS1τ3p1ν)τ1F(0)
=12πidp1p1mp1-n(ω)exp(iνp1)F(0)=[n(ω)]mF(0),ν0,m=0, 1, 2 , .
J(l)=L3I0Δ3σ20dν0dωνκ(ω) ωc2exp-(ω-ω0)2σ2×12+exp-νa2×exp2iωc [n(ω)ν-l]-ωc2κ2(ω)gνa.
gνa=Δ3 2a21+12νa2-exp-νa2
=Δν2for νa0Δ3ν2for Reνa.
ωc n(ω)=ω0c n(ω0)+(ω-ω0) 1v,
ωc κ(ω)=ω0c κ(ω0)+(ω-ω0) 1v(0),
J(l)=-π2L3I(0)σ20dν ln[ν] ν ×11+σv2gν la1/2×12-v2κ(ω0) iω0c-σ2lv2vν-vc2σ21+σv2gν la×expσv2gν la1+σv2gν laσlv22ν-vc2-2iω0lc κ(ω0)vv2ν-vc-ω0vcσ κ(ω0)2×exp-σlv22ν-vc2-2iln2(ω0) ω0c ×ν-vc+2iω0lcdln n(ω)dln ω1+dln n(ω)dln ωω=ω0,
[ν]1+σν2gν la.
4νla2Δ3σv212+exp-vla21+σv2gν la
=νln1+σv2gν la.
νs=vc,
ω0σdln n(ω)dln ω1+dln n(ω)dln ωω=ω0<1,
σcvω0κ(ω0)=σ dlnωc κ(ω)dωω=ω0<1.
JSUT(l)=const.×κ2(ω0)Δl+i n(ω0)ω0vσ21+σν2gνsla5/212+exp-νsla2exp-ω0c κ(ω0)2gνSla[νS]×exp-2lω0cIm[d/(dln ω)][ln n(ω)]1+dln n(ω)dln ωω=ω0+Jdev,
ISUT(l)exp-lcτ22,τ2=1Δνsω0κ(ω0) forl<aνsΔexp-lcτ3,τ3=2ω0Imd/(dln ω)[ln n(ω)]1+dln n(ω)dln ωω=ω0-1 forlaνsΔ.
κ(ω)=2πcn(ω)2αNαAα(0),

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