Abstract

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

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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]

2007 (2)

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]

A. C. Fannjiang, "Self-averaging scaling limits of two-frequency Wigner distribution for random paraxial waves," J. Phys. A 40, 5025-5044 (2007).
[CrossRef]

2006 (2)

A. C. Fannjiang, "Information transfer in disordered media by broadband time reversal: stability, resolution and capacity," Nonlinearity 19, 2425-2439 (2006).
[CrossRef]

A. C. Fannjiang, "Self-averaging radiative transfer for parabolic waves," C. R. Math. 342(22), 109-114 (2006).
[CrossRef]

2005 (2)

A. C. Fannjiang, "Self-averaging scaling limits for random parabolic waves," Arch. Ration. Mech. Anal. 175, 343-387 (2005).
[CrossRef]

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]

2004 (1)

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]

2002 (2)

G. Samelsohn and V. Freilikher, "Two-frequency mutual coherence function and pulse propagation in random media," Phys. Rev. E 65, 046617 (2002).
[CrossRef]

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]

1999 (1)

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]

1996 (1)

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]

1994 (1)

R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994).
[CrossRef]

1987 (1)

A. Z. Genack, "Optical transmission in disordered media," Phys. Rev. Lett. 58, 2043-2046 (1987).
[CrossRef] [PubMed]

1986 (1)

B. Shapiro, "Large intensity fluctuations for wave propagation in random media," Phys. Rev. Lett. 57, 2168-2171 (1986).
[CrossRef] [PubMed]

1980 (1)

H. Spohn, "Kinetic equations from Hamiltonian dynamics: Markovian limits," Rev. Mod. Phys. 53, 569-615 (1980).
[CrossRef]

1905 (1)

A. Schuster, "Radiation through a foggy atmosphere," Astrophys. J. 21, 1-22 (1905).
[CrossRef]

Bensoussan, A.

A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, 1978).

Berkovits, R.

R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994).
[CrossRef]

Born, M.

M. Born and W. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Bronshtein, A.

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]

Chandrasekhar, S.

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

Dragoman, D.

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]

Fannjiang, A. C.

A. C. Fannjiang, "Self-averaging scaling limits of two-frequency Wigner distribution for random paraxial waves," J. Phys. A 40, 5025-5044 (2007).
[CrossRef]

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]

A. C. Fannjiang, "Self-averaging radiative transfer for parabolic waves," C. R. Math. 342(22), 109-114 (2006).
[CrossRef]

A. C. Fannjiang, "Information transfer in disordered media by broadband time reversal: stability, resolution and capacity," Nonlinearity 19, 2425-2439 (2006).
[CrossRef]

A. C. Fannjiang, "Self-averaging scaling limits for random parabolic waves," Arch. Ration. Mech. Anal. 175, 343-387 (2005).
[CrossRef]

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]

A. C. Fannjiang, "Space-frequency correlation of classical waves in disordered media: high-frequency asymptotics," submitted to Europhys. Lett.

Feng, S.

R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994).
[CrossRef]

Freilikher, V.

G. Samelsohn and V. Freilikher, "Two-frequency mutual coherence function and pulse propagation in random media," Phys. Rev. E 65, 046617 (2002).
[CrossRef]

Genack, A. Z.

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]

A. Z. Genack, "Optical transmission in disordered media," Phys. Rev. Lett. 58, 2043-2046 (1987).
[CrossRef] [PubMed]

Hopf, E.

E. Hopf, Mathematical Problems of Radiative Equilibrium (Cambridge U. Press, 1934).

Hu, B.

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]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. 1 and 2.

Keller, J. B.

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]

Lacis, A.

M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

Lions, J. L.

A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, 1978).

Lu, I. T.

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]

Mandel, L.

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

Mazar, R.

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]

Mishchenko, M.

M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

Nieuwenhuizen, Th. M.

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]

Papanicolaou, G.

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]

Papanicolaou, G. C.

A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, 1978).

Pnini, R.

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]

Ryzhik, L.

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]

Samelsohn, G.

G. Samelsohn and V. Freilikher, "Two-frequency mutual coherence function and pulse propagation in random media," Phys. Rev. E 65, 046617 (2002).
[CrossRef]

Schuster, A.

A. Schuster, "Radiation through a foggy atmosphere," Astrophys. J. 21, 1-22 (1905).
[CrossRef]

Sebbah, P.

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]

Shapiro, B.

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]

B. Shapiro, "Large intensity fluctuations for wave propagation in random media," Phys. Rev. Lett. 57, 2168-2171 (1986).
[CrossRef] [PubMed]

Spohn, H.

H. Spohn, "Kinetic equations from Hamiltonian dynamics: Markovian limits," Rev. Mod. Phys. 53, 569-615 (1980).
[CrossRef]

Travis, L.

M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

van Rossum, M. C. W.

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]

Wolf, E.

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

Wolf, W.

M. Born and W. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Arch. Ration. Mech. Anal. (1)

A. C. Fannjiang, "Self-averaging scaling limits for random parabolic waves," Arch. Ration. Mech. Anal. 175, 343-387 (2005).
[CrossRef]

Astrophys. J. (1)

A. Schuster, "Radiation through a foggy atmosphere," Astrophys. J. 21, 1-22 (1905).
[CrossRef]

C. R. Math. (1)

A. C. Fannjiang, "Self-averaging radiative transfer for parabolic waves," C. R. Math. 342(22), 109-114 (2006).
[CrossRef]

C. R. Phys. (1)

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]

J. Phys. A (1)

A. C. Fannjiang, "Self-averaging scaling limits of two-frequency Wigner distribution for random paraxial waves," J. Phys. A 40, 5025-5044 (2007).
[CrossRef]

J. Stat. Phys. (1)

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]

Nonlinearity (1)

A. C. Fannjiang, "Information transfer in disordered media by broadband time reversal: stability, resolution and capacity," Nonlinearity 19, 2425-2439 (2006).
[CrossRef]

Phys. Rep. (1)

R. Berkovits and S. Feng, "Correlations in coherent multiple scattering," Phys. Rep. 238, 135-172 (1994).
[CrossRef]

Phys. Rev. E (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]

G. Samelsohn and V. Freilikher, "Two-frequency mutual coherence function and pulse propagation in random media," Phys. Rev. E 65, 046617 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

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]

B. Shapiro, "Large intensity fluctuations for wave propagation in random media," Phys. Rev. Lett. 57, 2168-2171 (1986).
[CrossRef] [PubMed]

A. Z. Genack, "Optical transmission in disordered media," Phys. Rev. Lett. 58, 2043-2046 (1987).
[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

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]

H. Spohn, "Kinetic equations from Hamiltonian dynamics: Markovian limits," Rev. Mod. Phys. 53, 569-615 (1980).
[CrossRef]

Wave Motion (1)

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]

Other (10)

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

E. Hopf, Mathematical Problems of Radiative Equilibrium (Cambridge U. Press, 1934).

A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, 1978).

A. C. Fannjiang, "Space-frequency correlation of classical waves in disordered media: high-frequency asymptotics," submitted to Europhys. Lett.

M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

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]

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).

M. Born and W. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vols. 1 and 2.

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

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Equations (78)

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Γ 12 ( x , y ) = U 1 ( x k 1 + y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) ,
Δ U j ( r ) + k j 2 ( ν j + V j ( r ) ) U j ( r ) = f j ( r ) , r R 3 , j = 1 , 2 ,
W ( x , p ) = 1 ( 2 π ) 3 e i p y U 1 ( x k 1 + y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) d y .
W 2 ( x , p ) d x d p = ( k 1 k 2 2 π ) 3 U 1 2 ( x ) d x U 2 2 ( x ) d x ,
W ( x , p ) e i p y d p = U 1 ( x k 1 + y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) ,
W ( x , p ) e i x q d x = ( π 2 k 1 k 2 ) 3 U ̂ 1 ( k 1 p 4 + k 1 q 2 ) U ̂ 2 * ( k 2 p 4 k 2 q 2 ) ,
p W ( x , p ) d p = i [ 1 2 k 1 U 1 ( x k 1 ) U 2 * ( x k 2 ) 1 2 k 2 U 1 ( x k 1 ) U 2 * ( x k 2 ) ] ,
p W = i 2 ( 2 π ) 3 e i p y U 1 ( x k 1 + y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) V 1 ( x k 1 + y 2 k 1 ) d y i 2 ( 2 π ) 3 e i p y U 1 ( x k 1 y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) V 2 * ( x k 2 y 2 k 2 ) d y + i 2 ( ν 1 ν 2 * ) W + F ,
F = i 2 ( 2 π ) 3 e i p y f 1 ( x k 1 + y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) d y + i 2 ( 2 π ) 3 e i p y U 1 ( x k 1 + y 2 k 1 ) f 2 * ( x k 2 y 2 k 2 ) d y .
V j ( x ) = e i q x V ̂ j ( d q ) ,
p W i 2 ( ν 1 ν 2 * ) W F = i 2 V ̂ 1 ( d q ) e i q x k 1 W ( x , p q 2 k 1 ) i 2 V ̂ 2 * ( d q ) e i q x k 2 W ( x , p q 2 k 2 ) .
2 2 Δ Ψ j + [ ν j + V j ( x ) ] Ψ j = ω j Ψ j + f j , j = 1 , 2 ,
W ( x , p ) = 1 ( 2 π ) 3 e i p y Ψ 1 ( x + y 2 ) Ψ 2 * ( x y 2 ) d y ,
p W + i ( ω 2 ω 1 ) W + i ( ν 2 * ν 1 ) W = i V ̂ 1 ( d q ) e i q x W ( x , p q 2 ) i V ̂ 2 * ( d q ) e i q x W ( x , p q 2 ) + F ,
V ̂ j ( d p ) V ̂ j * ( d q ) = δ ( p q ) Φ j ( p ) d p d q ,
V ̂ 1 ( d p ) V ̂ 2 * ( d q ) = δ ( p q ) Φ 12 ( p ) d p d q .
1 γ 2 ϵ 2 ( ν j + ϵ V j ( r ϵ ) ) , γ > 0 , ϵ 1 ,
W ( x , p ) = 1 ( 2 π ) 3 e i p y U 1 ( x k 1 + γ ϵ y 2 k 1 ) U 2 * ( x k 2 γ ϵ y 2 k 2 ) d y .
p W F = i 2 ϵ γ ( ν 1 ν 2 * ) W + 1 ϵ L W ,
L W ( x , p ) = i 2 γ V ̂ 1 ( d q ) exp ( i q x ϵ k 1 ) W ( x , p γ q 2 k 1 ) i 2 γ V ̂ 2 * ( d q ) exp ( i q x ϵ k 2 ) W ( x , p γ q 2 k 2 ) .
lim ϵ 0 k 1 = lim ϵ 0 k 2 = k , k 2 k 1 ϵ γ k = β ,
ν 2 * ν 1 2 ϵ γ = ν ,
x ̂ = x ϵ
p = p x + ϵ 1 p x ̃ .
W ( x , p ) = W ¯ ( x , x ̃ , p ) + ϵ W 1 ( x , x ̃ , p ) + ϵ W 2 ( x , x ̃ , p ) + O ( ϵ 3 2 ) , x ̃ = x ϵ 1 ,
p x ̃ W ¯ = 0 ,
p x ̃ W 1 = L W ¯ .
ϵ W 1 ϵ + p x ̃ W 1 ϵ = L W ¯ ,
W 1 ϵ ( x , x ̃ , p ) = i 2 γ V ̂ 1 ( d q ) exp ( i q x ̃ k 1 ) ϵ + i q p k 1 W ¯ ( x , p γ q 2 k 1 ) i 2 γ V ̂ 2 * ( d q ) exp ( i q x ̃ k 2 ) ϵ i q p k 2 W ¯ ( x , p γ q 2 k 2 ) .
p x ̃ W 2 ( x , x ̃ , p ) = p x W ¯ ( x , p ) i ν W ¯ + F + i 2 γ V ̂ 1 ( d q ) exp ( i q x ̃ k 1 ) W 1 ϵ ( x , x ̃ , p γ q 2 k 1 ) i 2 γ V ̂ 2 * ( d q ) exp ( i q x ̃ k 2 ) W 1 ϵ ( x , x ̃ , p γ q 2 k 2 ) ,
p x W ¯ ( x , p ) + i ν W ¯ F = k 1 3 2 γ 4 d q Φ 1 ( k 1 γ ( p q ) ) π δ ( p 2 q 2 ) W ¯ ( x , p ) + i k 1 3 2 γ 4 d q Φ 1 ( k 1 γ ( p q ) ) p 2 q 2 W ¯ ( x , p ) k 2 3 2 γ 4 d q Φ 2 ( k 2 γ ( p q ) ) π δ ( p 2 q 2 ) W ¯ ( x , p ) i k 2 3 2 γ 4 d q Φ 2 ( k 2 γ ( p q ) ) p 2 q 2 W ¯ ( x , p ) + 1 4 γ 2 d q Φ 12 ( q ) e i x ̃ q ( k 1 1 k 2 1 ) π δ ( q k 2 ( p γ q 2 k 1 ) ) W ¯ ( x , p γ q 2 k 1 γ q 2 k 2 ) + 1 4 γ 2 d q Φ 12 ( q ) e i x ̃ q ( k 1 1 k 2 1 ) π δ ( q k 1 ( p γ q 2 k 2 ) ) W ¯ ( x , p γ q 2 k 1 γ q 2 k 2 ) + i 4 γ 2 d q [ 1 q k 2 ( p γ q 2 k 1 ) 1 q k 1 ( p γ q 2 k 2 ) ] Φ 12 ( q ) e i x ̃ q ( k 1 1 k 2 1 ) W ¯ ( x , p γ q 2 k 1 γ q 2 k 2 ) ,
lim η 0 1 η + i ξ = π δ ( ξ ) i ξ ,
F = i 2 ( 2 π ) 3 e i p y f 1 ( x k 1 + y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) d y + i 2 ( 2 π ) 3 e i p y U 1 ( x k 1 + y 2 k 2 ) f 2 * ( x k 2 y 2 k 2 ) d y ,
W ( x , p ) = W ¯ ( x , p ) + ϵ W 1 ϵ ( x , x ̃ , p ) + ϵ W 2 ϵ ( x , x ̃ , p ) ,
( p 1 ϵ L ) W + i ν W F = ( i ν 1 ) ϵ W 1 ϵ + ϵ p x W 1 ϵ ϵ L W 2 ϵ + ( i ν 1 ) ϵ W 2 ϵ + ϵ p x W 2 ϵ .
lim ϵ 0 ϵ d x d p W 1 ϵ ( x , x ϵ , p ) ψ ( p ) 2 = 0 , ψ L 2
p x W ¯ + i ν W ¯ F = π k 3 γ 4 d q Φ ( k γ ( p q ) ) δ ( p 2 q 2 ) [ e i x ( p q ) β W ¯ ( x , q ) W ¯ ( x , p ) ] .
p x W ¯ + i ν W ¯ F = π k 3 γ 4 d q Φ ( k γ ( p q ) ) δ ( p 2 q 2 ) W ¯ ( x , p ) .
W ( x , p ) = 1 ( 2 π ) 3 e i p y Ψ 1 ( x + ϵ y 2 ) Ψ 2 * ( x ϵ y 2 ) d y
p x W ¯ + i ( ω 2 ω 1 ) W ¯ + 2 i ν W ¯ F = 4 π 4 d q Φ ( p q ) δ ( p 2 q 2 ) [ W ¯ ( x , q ) W ¯ ( x , p ) ] .
1 η Φ ( k η γ ( p q ) , k γ ( p q ) ) , η 1 ,
γ k δ ( p q ) d w Φ ( w , k γ ( p q ) ) .
p z W ¯ + p x W ¯ + i ν W ¯ F = π k 2 γ 3 d q d w Φ ( w , k γ ( p q ) ) δ ( p 2 q 2 ) × [ e i x ( p q ) β W ¯ ( z , x , p , q ) W ¯ ( z , x , p , p ) ] .
p x W ¯ + i ν W ¯ F = π k 2 γ 2 d q Φ ( q ) δ ( q ( p γ q 2 k ) ) [ e i x q β γ k W ¯ ( x , p γ q k ) W ¯ ( x , p ) ] ,
p x W ¯ + i ν W ¯ F = 1 4 k ( p i β x ) D ( p i β x ) W ¯
D ( p ) = π Φ ( q ) δ ( p q ) q q d q .
C = π 3 δ ( p p q q ) Φ ( q ) q d q
x = σ x k x ̃ , p = σ p p ̃ k , β = β c β ̃ ,
p ̃ x ̃ W ¯ + i ν W ¯ F = ( p ̃ i β x ̃ ) p ̃ 1 P ( p ̃ ) ( p ̃ i β x ̃ ) W ¯ .
σ x σ p 1 k 2 .
σ x σ p 1 β c ,
σ p k 2 3 C 1 3 ,
σ x k 4 3 C 1 3 , β c k 2 3 C 2 3 .
( z + p x ) W ¯ + i ν W ¯ F = C 4 k ( p i β x ) 2 W ¯ ,
C = π 2 R 2 Φ ( 0 , q ) q 2 d q .
x ̃ = x σ k , p ̃ = p k l c , z ̃ = z L k , β ̃ = β β c ,
z ̃ W ¯ + p ̃ x ̃ W ¯ + L k i ν W ¯ L k F = ( p ̃ i β ̃ x ̃ ) 2 W ¯ .
l c σ L k , σ l c 1 β c , l c k 1 L 1 2 C 1 2
σ L 3 2 C 1 2 , β c k 1 C 1 L 2 .
z ̃ Γ i y ̃ x ̃ Γ + L k i ν Γ L k F = y ̃ + β ̃ x ̃ 2 Γ ,
e i L k ν ( i 4 β ̃ ) 1 2 ( 2 π ) 2 z ̃ sinh [ ( i 4 β ̃ ) 1 2 z ̃ ] exp ( 1 i 4 β ̃ z ̃ y ̃ β ̃ x ̃ y + β ̃ x 2 ) × exp { coth [ ( i 4 β ̃ ) 1 2 z ̃ ] ( i 4 β ̃ ) 1 2 y ̃ + β ̃ x ̃ y + β ̃ x cosh [ ( i 4 β ̃ ) 1 2 z ̃ ] 2 } × exp { tanh [ ( i 4 β ̃ ) 1 2 z ̃ ] ( i 4 β ̃ ) 1 2 y + β ̃ x 2 } .
exp ( 2 2 β ̃ ) ,
δ ( q ) d w Φ ( w , q )
D ( p ) = π d q d w Φ ( w , q ) δ ( p q ) q q ,
C = π 2 δ ( p p q q ) Φ ( w , q ) q d w d q
( z + p x ) W ¯ + i ν W ¯ F = C 4 k ( p i β x ) p 1 P ( p ) ( p i β x ) W ¯ .
[ z ̃ + p ̃ x ̃ ] W ¯ + L k i ν W ¯ L k F = ( p ̃ i β ̃ x ̃ ) p ̃ 1 P ( p ̃ ) ( p ̃ i β ̃ x ̃ ) W ¯ ,
l c σ L k , σ l c 1 β c , l c C 1 3 L 1 3 k 1 ,
σ C 1 3 L 4 3 , β c C 2 3 L 5 3 k 1 .
W ̃ ( x , p ) = e i β x p W ¯ ( x , p )
p x W ̃ + i β p 2 W ̃ + i ν W ̃ e i β x p F = π k 3 γ 4 d q Φ ( k γ ( p q ) ) δ ( p 2 q 2 ) [ W ̃ ( x , q ) W ̃ ( x , p ) ] .
A W = p x W + π k 3 γ 4 d q Φ ( k γ ( p q ) ) δ ( p 2 q 2 ) [ W ( x , q ) W ( x , p ) ]
p x W = 1 ( 2 π ) 3 e i p y 2 p y U 1 ( x k 1 + y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) d y 1 ( 2 π ) 3 e i p y U 1 ( x k 1 + y 2 k 1 ) 2 p y U 2 * ( x k 2 y 2 k 2 ) d y = 2 i ( 2 π ) 3 ( y e i p y ) y U 1 ( x k 1 + y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) d y 2 i ( 2 π ) 3 ( y e i p y ) U 1 ( x k 1 y 2 k 1 ) y U 2 * ( x k 2 y 2 k 2 ) d y .
p x W = 2 i ( 2 π ) 3 e i p y y 2 U 1 ( x k 1 + y 2 k 1 ) U 2 * ( x k 2 y 2 k 2 ) d y + 2 i ( 2 π ) 3 e i p y U 1 ( x k 1 + y 2 k 1 ) y 2 U 2 * ( x k 2 y 2 k 2 ) d y ,
y 2 U j ( x k j + y 2 k j ) = 1 4 [ ν j + V j ( x k j + y 2 k j ) ] U j ( x k j + y 2 k j ) + 1 4 f j ( x k j + y 2 k j ) .
lim ϵ 0 1 4 d p d x d q Φ 1 ( q ) ϵ ϵ 2 + ( p q k 1 ) 2 W ¯ ( x , p q 2 k 1 ) 2 = π 4 d p d x d q Φ ( q ) δ ( p q k ) W ¯ ( x , p q 2 k ) 2 ,
lim ϵ 0 ϵ d x d p W 1 ϵ ( x , X ϵ , p ) ψ ( p ) 2 = 0 , ψ L 2 .
lim ϵ 0 ϵ 4 d p d p d x d q Φ 1 ( q ) ψ ( p ) ψ * ( p ) ( ϵ + i p q k 1 ) ( ϵ i p q k 1 ) W ¯ ( x , p q 2 k 1 ) W ¯ * ( x , p q 2 k 1 ) = lim ϵ 0 ϵ 4 d x d q Φ 1 ( q ) q 2 [ π d p δ ( p q ̂ k 1 ) ψ ( p ) W ¯ ( x , p q 2 k 1 ) d p i ψ ( p ) p q ̂ k 1 W ¯ ( x , p q 2 k 1 ) ] × [ π d p δ ( p q ̂ k 1 ) ψ * ( p ) W ¯ * ( x , p q 2 k 1 ) + d p i ψ * ( p ) p q ̂ k 1 W ¯ * ( x , p q 2 k 1 ) ] ,

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