Abstract

New phase space distributions are proposed for describing pulse propagation in dispersive media for one spatial dimension. These distributions depend on time, position, and velocity, so that the pulse’s spatial propagation or temporal evolution is described by a free-particle-like transformation followed by integration over velocity. Examples are considered for approximate Lorentz-model dielectrics and metallic waveguides.

© 2010 Optical Society of America

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  1. P. Loughlin and L. Cohen, “A Wigner approximation method for wave propagation,” J. Acoust. Soc. Am. 118, 1268–1271 (2005).
    [CrossRef]
  2. J. Ojeda-Castañeda, J. Lancis, C. M. Gómez-Sarabia, V. Torres-Company, and P. Andrés, “Ambiguity function analysis of pulse train propagation: applications to temporal Lau filtering,” J. Opt. Soc. Am. A 24, 2268–2273 (2007).
    [CrossRef]
  3. P. Loughlin and L. Cohen, “Approximate wave function from approximate non-representable Wigner distributions,” J. Mod. Opt. 55, 3379–3387 (2008).
    [CrossRef]
  4. L. Cohen, P. Loughlin, and G. Okopal, “Exact and approximate moments of a propagating pulse,” J. Mod. Opt. 55, 3349–3358 (2008).
    [CrossRef]
  5. A. Papandreou-Suppappola, F. Hlawatsch, and G. Boudreaux-Bartels, “Quadratic time-frequency representations with scale covariance and generalized time-shift covariance: A unified framework for the affine, hyperbolic, and power classes,” Digit. Signal Process. 8, 3–48 (1998).
    [CrossRef]
  6. F. Hlawatsch, A. Papandreou-Suppappola, and G. Boudreaux-Bartels, “The power classes-quadratic time-frequency representations with scale covariance and dispersive time-shift covariance,” IEEE Trans. Signal Process. on 47, 3067–3083 (1999).
    [CrossRef]
  7. A. Papandreou-Suppappola, R. Murray, B. Iem, and G. Boudreaux-Bartels, “Group delay shift covariant quadratic time-frequency representations,” IEEE Trans. Signal Process. 49, 2549–2564 (2001).
    [CrossRef]
  8. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 92–102.
  9. G. Okopal, P. Loughlin, and L. Cohen, “Dispersion-invariant features for classification,” J. Acoust. Soc. Am. 123, 832–841 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas (Cambridge Univ. Press, 1995), p. 209.
  12. L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559–562 (1964).
  13. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  14. K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]
  15. M. A. Alonso, “Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2001).
    [CrossRef]
  16. M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).
    [CrossRef]
  17. S. Cho, J. C. Petruccelli, and M. A. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Opt. 56, 1843–1852 (2009).
    [CrossRef]
  18. L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101–112 (2002).
    [CrossRef]
  19. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
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    [CrossRef]

2009 (1)

S. Cho, J. C. Petruccelli, and M. A. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Opt. 56, 1843–1852 (2009).
[CrossRef]

2008 (3)

P. Loughlin and L. Cohen, “Approximate wave function from approximate non-representable Wigner distributions,” J. Mod. Opt. 55, 3379–3387 (2008).
[CrossRef]

L. Cohen, P. Loughlin, and G. Okopal, “Exact and approximate moments of a propagating pulse,” J. Mod. Opt. 55, 3349–3358 (2008).
[CrossRef]

G. Okopal, P. Loughlin, and L. Cohen, “Dispersion-invariant features for classification,” J. Acoust. Soc. Am. 123, 832–841 (2008).
[CrossRef] [PubMed]

2007 (2)

2005 (1)

P. Loughlin and L. Cohen, “A Wigner approximation method for wave propagation,” J. Acoust. Soc. Am. 118, 1268–1271 (2005).
[CrossRef]

2003 (1)

2002 (2)

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101–112 (2002).
[CrossRef]

P. Loughlin and L. Cohen, “Wigner distributions local properties of dispersive pulses,” J. Mod. Opt. 49, 2645–2655 (2002).
[CrossRef]

2001 (3)

1999 (2)

F. Hlawatsch, A. Papandreou-Suppappola, and G. Boudreaux-Bartels, “The power classes-quadratic time-frequency representations with scale covariance and dispersive time-shift covariance,” IEEE Trans. Signal Process. on 47, 3067–3083 (1999).
[CrossRef]

K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

1998 (1)

A. Papandreou-Suppappola, F. Hlawatsch, and G. Boudreaux-Bartels, “Quadratic time-frequency representations with scale covariance and generalized time-shift covariance: A unified framework for the affine, hyperbolic, and power classes,” Digit. Signal Process. 8, 3–48 (1998).
[CrossRef]

1995 (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 92–102.

1968 (1)

1964 (1)

L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559–562 (1964).

Alonso, M. A.

Andrés, P.

Boudreaux-Bartels, G.

A. Papandreou-Suppappola, R. Murray, B. Iem, and G. Boudreaux-Bartels, “Group delay shift covariant quadratic time-frequency representations,” IEEE Trans. Signal Process. 49, 2549–2564 (2001).
[CrossRef]

F. Hlawatsch, A. Papandreou-Suppappola, and G. Boudreaux-Bartels, “The power classes-quadratic time-frequency representations with scale covariance and dispersive time-shift covariance,” IEEE Trans. Signal Process. on 47, 3067–3083 (1999).
[CrossRef]

A. Papandreou-Suppappola, F. Hlawatsch, and G. Boudreaux-Bartels, “Quadratic time-frequency representations with scale covariance and generalized time-shift covariance: A unified framework for the affine, hyperbolic, and power classes,” Digit. Signal Process. 8, 3–48 (1998).
[CrossRef]

Boyd, T. J. M.

T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas (Cambridge Univ. Press, 1995), p. 209.

Cho, S.

S. Cho, J. C. Petruccelli, and M. A. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Opt. 56, 1843–1852 (2009).
[CrossRef]

Cohen, L.

G. Okopal, P. Loughlin, and L. Cohen, “Dispersion-invariant features for classification,” J. Acoust. Soc. Am. 123, 832–841 (2008).
[CrossRef] [PubMed]

P. Loughlin and L. Cohen, “Approximate wave function from approximate non-representable Wigner distributions,” J. Mod. Opt. 55, 3379–3387 (2008).
[CrossRef]

L. Cohen, P. Loughlin, and G. Okopal, “Exact and approximate moments of a propagating pulse,” J. Mod. Opt. 55, 3349–3358 (2008).
[CrossRef]

P. Loughlin and L. Cohen, “A Wigner approximation method for wave propagation,” J. Acoust. Soc. Am. 118, 1268–1271 (2005).
[CrossRef]

P. Loughlin and L. Cohen, “Wigner distributions local properties of dispersive pulses,” J. Mod. Opt. 49, 2645–2655 (2002).
[CrossRef]

Davis, B. J.

B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[CrossRef]

Dolin, L.

L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559–562 (1964).

Forbes, G. W.

Gómez-Sarabia, C. M.

Hlawatsch, F.

F. Hlawatsch, A. Papandreou-Suppappola, and G. Boudreaux-Bartels, “The power classes-quadratic time-frequency representations with scale covariance and dispersive time-shift covariance,” IEEE Trans. Signal Process. on 47, 3067–3083 (1999).
[CrossRef]

A. Papandreou-Suppappola, F. Hlawatsch, and G. Boudreaux-Bartels, “Quadratic time-frequency representations with scale covariance and generalized time-shift covariance: A unified framework for the affine, hyperbolic, and power classes,” Digit. Signal Process. 8, 3–48 (1998).
[CrossRef]

Iem, B.

A. Papandreou-Suppappola, R. Murray, B. Iem, and G. Boudreaux-Bartels, “Group delay shift covariant quadratic time-frequency representations,” IEEE Trans. Signal Process. 49, 2549–2564 (2001).
[CrossRef]

Lajunen, H.

Lancis, J.

Loughlin, P.

P. Loughlin and L. Cohen, “Approximate wave function from approximate non-representable Wigner distributions,” J. Mod. Opt. 55, 3379–3387 (2008).
[CrossRef]

L. Cohen, P. Loughlin, and G. Okopal, “Exact and approximate moments of a propagating pulse,” J. Mod. Opt. 55, 3349–3358 (2008).
[CrossRef]

G. Okopal, P. Loughlin, and L. Cohen, “Dispersion-invariant features for classification,” J. Acoust. Soc. Am. 123, 832–841 (2008).
[CrossRef] [PubMed]

P. Loughlin and L. Cohen, “A Wigner approximation method for wave propagation,” J. Acoust. Soc. Am. 118, 1268–1271 (2005).
[CrossRef]

P. Loughlin and L. Cohen, “Wigner distributions local properties of dispersive pulses,” J. Mod. Opt. 49, 2645–2655 (2002).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 92–102.

Murray, R.

A. Papandreou-Suppappola, R. Murray, B. Iem, and G. Boudreaux-Bartels, “Group delay shift covariant quadratic time-frequency representations,” IEEE Trans. Signal Process. 49, 2549–2564 (2001).
[CrossRef]

Ojeda-Castañeda, J.

Okopal, G.

L. Cohen, P. Loughlin, and G. Okopal, “Exact and approximate moments of a propagating pulse,” J. Mod. Opt. 55, 3349–3358 (2008).
[CrossRef]

G. Okopal, P. Loughlin, and L. Cohen, “Dispersion-invariant features for classification,” J. Acoust. Soc. Am. 123, 832–841 (2008).
[CrossRef] [PubMed]

Papandreou-Suppappola, A.

A. Papandreou-Suppappola, R. Murray, B. Iem, and G. Boudreaux-Bartels, “Group delay shift covariant quadratic time-frequency representations,” IEEE Trans. Signal Process. 49, 2549–2564 (2001).
[CrossRef]

F. Hlawatsch, A. Papandreou-Suppappola, and G. Boudreaux-Bartels, “The power classes-quadratic time-frequency representations with scale covariance and dispersive time-shift covariance,” IEEE Trans. Signal Process. on 47, 3067–3083 (1999).
[CrossRef]

A. Papandreou-Suppappola, F. Hlawatsch, and G. Boudreaux-Bartels, “Quadratic time-frequency representations with scale covariance and generalized time-shift covariance: A unified framework for the affine, hyperbolic, and power classes,” Digit. Signal Process. 8, 3–48 (1998).
[CrossRef]

Petruccelli, J. C.

S. Cho, J. C. Petruccelli, and M. A. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Opt. 56, 1843–1852 (2009).
[CrossRef]

Sanderson, J. J.

T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas (Cambridge Univ. Press, 1995), p. 209.

Tervo, J.

Torres-Company, V.

Turunen, J.

Vahimaa, P.

Vicent, L. E.

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101–112 (2002).
[CrossRef]

Walther, A.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 92–102.

Wolf, K. B.

Wyrowski, F.

Digit. Signal Process. (1)

A. Papandreou-Suppappola, F. Hlawatsch, and G. Boudreaux-Bartels, “Quadratic time-frequency representations with scale covariance and generalized time-shift covariance: A unified framework for the affine, hyperbolic, and power classes,” Digit. Signal Process. 8, 3–48 (1998).
[CrossRef]

IEEE Trans. Signal Process. (2)

F. Hlawatsch, A. Papandreou-Suppappola, and G. Boudreaux-Bartels, “The power classes-quadratic time-frequency representations with scale covariance and dispersive time-shift covariance,” IEEE Trans. Signal Process. on 47, 3067–3083 (1999).
[CrossRef]

A. Papandreou-Suppappola, R. Murray, B. Iem, and G. Boudreaux-Bartels, “Group delay shift covariant quadratic time-frequency representations,” IEEE Trans. Signal Process. 49, 2549–2564 (2001).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 7, 559–562 (1964).

J. Acoust. Soc. Am. (2)

P. Loughlin and L. Cohen, “A Wigner approximation method for wave propagation,” J. Acoust. Soc. Am. 118, 1268–1271 (2005).
[CrossRef]

G. Okopal, P. Loughlin, and L. Cohen, “Dispersion-invariant features for classification,” J. Acoust. Soc. Am. 123, 832–841 (2008).
[CrossRef] [PubMed]

J. Mod. Opt. (4)

P. Loughlin and L. Cohen, “Wigner distributions local properties of dispersive pulses,” J. Mod. Opt. 49, 2645–2655 (2002).
[CrossRef]

P. Loughlin and L. Cohen, “Approximate wave function from approximate non-representable Wigner distributions,” J. Mod. Opt. 55, 3379–3387 (2008).
[CrossRef]

L. Cohen, P. Loughlin, and G. Okopal, “Exact and approximate moments of a propagating pulse,” J. Mod. Opt. 55, 3349–3358 (2008).
[CrossRef]

S. Cho, J. C. Petruccelli, and M. A. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Opt. 56, 1843–1852 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101–112 (2002).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (1)

B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[CrossRef]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 92–102.

T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas (Cambridge Univ. Press, 1995), p. 209.

Supplementary Material (6)

» Media 1: MOV (1031 KB)     
» Media 2: MOV (69 KB)     
» Media 3: MOV (730 KB)     
» Media 4: MOV (79 KB)     
» Media 5: MOV (1000 KB)     
» Media 6: MOV (99 KB)     

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Figures (5)

Fig. 1
Fig. 1

Illustration of the change of variables. (a) The integral in Eq. (13) is taken over all pairs [ ω 1 ( v , ξ ) , k 1 ( v , ξ ) ] and [ ω 2 ( v , ξ ) , k 2 ( v , ξ ) ] such that the line connecting the two points has slope 1 v . (b) For curves with inflection points, all pairs of intersection points between the dispersion curve and the straight lines of slope 1 v must be considered. (c) For pulses with multiple components, each pair of separate components contributes to an interference pattern in phase space, at velocities given by the inverse of the slope of the line joining the two corresponding segments of the dispersion curve.

Fig. 2
Fig. 2

Illustration of the change of variables for a dispersion relation in which k depends quadratically on ω. In this case, ω 1 ( v , 0 ) = ω 2 ( v , 0 ) = [ ω 1 ( v , ξ ) + ω 2 ( v , ξ ) ] 2 for any ξ.

Fig. 3
Fig. 3

(a) Illustration of the change of variables for the dispersion relation in Eq. (19). [Online: the movie illustrates the change of variables in (a) as v and ξ are varied independently. (Media 1)] (b) B v ( z , t = 0 , v ) for a Gaussian pulse in the analytic signal representation, traveling in a medium with dispersion described by Eq. (19). The dotted line indicates the group velocity associated with the pulse’s central frequency. Since B v can take on both positive and negative values, solid black lines indicate contours of B v = 0 . (c) The short-time-averaged intensity, I ( z , t ) , given by B v ( z v t , 0 , v ) d v , at three different times. [Online: the movie shows the variation of (b) and (c) with t (Media 2).]

Fig. 4
Fig. 4

(a) Illustration of the change of variables for the waveguide dispersion relation given in Eq. (24). [Online: the movie illustrates the change of variables in (a) as v and ξ are varied independently (Media 3).] (b) B v ( z , t , v ) for a pulse with square spectrum in the analytic signal representation, traveling in a medium with dispersion described by Eq. (24). Since B v can take on both positive and negative values, solid black lines indicate contours of B v = 0 . (c) The short-time-averaged intensity, I ( z , t ) , given by B v ( z v t , 0 , v ) d v , at three different times. [Online: the movie shows the variation of (b) and (c) with t (Media 4). Green indicates positive and red negative values of B.]

Fig. 5
Fig. 5

(a) Illustration of the change of variables for a medium with dispersion relation given in Eq. (27). [Online: the movie illustrates the change of variables in (a) as v and ξ are varied independently (Media 5).] (b) B v ( z , t = 0 , v ) for the instantaneous intensity of a Gaussian pulse. The dotted lines indicate the group, v g , and phase, v p , velocities associated with the pulse’s central frequency. The solid black lines indicate contours of B v = 0 . (c) The instantaneous intensity, I ( z , t ) , given by B v ( z v t , 0 , v ) d v , at three different times. [Online: the movie shows the variation (b) and (c) with t (Media 6). Green indicates positive and red negative values of B.]

Equations (40)

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U ( z , t ) = 1 2 π A ( ω ) exp { i [ k ( ω ) z ω t ] } d ω ,
B k ( z , t , k ) = 1 2 π U * ( z z 2 , t ) U ( z + z 2 , t ) exp ( i k z ) d z .
I ( z , t ) = B k ( z , t , k ) d k .
B k ( z , t , k ) B k [ z v g ( ω ) t , 0 , k ] B k [ 0 , t z v g ( ω ) , k ] ,
B ω ( z , t , ω ) = 1 2 π U * ( z , t t 2 ) U ( z , t + t 2 ) exp ( i ω t ) d t .
B ω ( z , t , ω ) = 1 2 π A * ( ω ω 2 ) A ( ω + ω 2 ) exp { i [ k ( ω + ω 2 ) k ( ω ω 2 ) ] z } exp ( i ω t ) d ω .
I ( z , t ) = B ω ( z , t , k ) d ω .
B ω ( z , t , ω ) B ω [ z v g ( ω ) t , 0 , ω ] B ω [ 0 , t z v g ( ω ) , ω ] .
I ( z , t ) = B v ( z , t , v ) d v .
B v ( z , t , v ) = B v ( z v t , 0 , v ) = B v ( 0 , t z v , v ) .
( v z + t ) B v ( z , t , v ) = 0 .
I ( z , t ) = 1 2 π A * ( ω 1 ) A ( ω 2 ) exp ( i { [ k ( ω 2 ) k ( ω 1 ) ] z ( ω 2 ω 1 ) t } ) d ω 1 d ω 2 .
B v ( z , t , v ) = 1 2 π δ ( ω 1 , ω 2 ) δ ( v , ξ ) A * [ ω 1 ( v , ξ ) ] A [ ω 2 ( v , ξ ) ] exp ( i { [ k 2 ( v , ξ ) k 1 ( v , ξ ) ] z [ ω 2 ( v , ξ ) ω 1 ( v , ξ ) ] t } ) d ξ ,
k 2 ( v , ξ ) k 1 ( v , ξ ) ω 2 ( v , ξ ) ω 1 ( v , ξ ) = 1 v .
B v ( z , t , v ) = 1 2 π δ ( ω 1 , ω 2 ) δ ( v , ξ ) A * [ ω 1 ( v , ξ ) ] A [ ω 2 ( v , ξ ) ] × exp [ i ω 2 ( v , ξ ) ω 1 ( v , ξ ) v ( z v t ) ] d ξ .
B v ( z , t , v ) = 1 2 π δ ( k 1 , k 2 ) δ ( v , ξ ) U ̃ * [ k 1 ( v , ξ ) , 0 ] U ̃ [ k 2 ( v , ξ ) , 0 ] × exp { i [ k 2 ( v , ξ ) k 1 ( v , ξ ) ] ( z v t ) } d ξ .
B v [ z , t , v g ( ω ) ] = k 2 ( ω ) 2 π k ( ω ) A * ( ω ξ 2 ) A ( ω + ξ 2 ) exp { i ξ [ t z v g ( ω ) ] } d ξ = k 2 ( ω ) k ( ω ) B ω ( z , t , ω ) ,
B v { z , t , v g [ ω ( k ) ] } = 1 ω ( k ) B k ( z , t , k ) ,
k ( ω ) = ω c ( 1 β ω ω 0 ) ,
1 + β ω 0 [ ω 1 ( v , ξ ) ω 0 ] [ ω 2 ( v , ξ ) ω 0 ] = c v .
ω 1 , 2 ( v , ξ ) = ω 0 β ω 0 v c v exp ( ± ξ 2 ) .
B v ( z , t , v ) = c β ω 0 4 π ( c v ) 2 A * [ ω 0 β ω 0 v c v exp ( ξ 2 ) ] A [ ω 0 β ω 0 v c v exp ( ξ 2 ) ] exp [ 2 i β ω 0 v c v z v t v sinh ( ξ 2 ) ] d ξ .
A ( ω ) = 1 π exp [ ( | ω | ω c ) 2 2 σ 2 ]
k = ω 2 ω co 2 c ,
ω 1 , 2 ( v , ξ ) = ω co cosh ( ξ ¯ ξ 2 ) ,
B v ( z , t , v ) = c ω co 2 2 π ( c 2 v 2 ) 2 ξ ¯ 2 ξ ¯ sinh ( ξ ¯ ξ 2 ) sinh ( ξ ¯ + ξ 2 ) A * [ ω co cosh ( ξ ¯ ξ 2 ) ] A [ ω co cosh ( ξ ¯ + ξ 2 ) ] exp [ 2 i ω co ( z v t ) c 2 v 2 sinh ( ξ 2 ) ] d ξ .
k ( ω ) = ω c ( 1 γ ω 2 ω 0 2 ) ,
c v v γ = 1 2 ( ω 0 ω 1 ) ( ω 0 ω 2 ) + 1 2 ( ω 0 + ω 1 ) ( ω 0 + ω 2 ) .
1 ( ω 0 ± ω 1 ) ( ω 0 ± ω 2 ) = 1 [ ω 0 ± ω ¯ ( v ) ] 2 ± f ( v , ξ ) ,
ω ¯ ( v ) = { ω 0 2 γ v 2 ( c v ) [ 1 + 8 ω 0 2 ( c v ) γ v 1 ] } 1 2 .
f ( ξ ) = 4 ω ¯ ω 0 ( ω ¯ 2 ω 0 2 ) 2 sin 2 ( ξ 2 ) ,
ω 1 , 2 = ω ¯ ω 0 4 2 ω 0 2 ω ¯ 2 cos 2 ξ + ω ¯ 4 [ ( ω 0 2 ω ¯ 2 ) 2 cos ξ ± sin ξ 3 ( ω 0 4 + ω ¯ 4 ) 2 4 ω ¯ 2 ( ω ¯ 6 + 2 ω 0 6 cos 2 ξ ) ] .
A * [ ω 1 ( v , ξ ) ] A [ ω 2 ( v , ξ ) ] = [ δ ( ω 1 , ω 2 ) δ ( v , ξ ) ] 1 [ ω 2 ( v , ξ ) ω 1 ( v , ξ ) ] ξ B v ( z , t , v ) exp [ i ω 2 ( v , ξ ) ω 1 ( v , ξ ) v ( z v t ) ] d τ ,
| A ( ω ) | 2 = k ( ω ) k ( ω ) B v [ z , t , v g ( ω ) ] d z = k ( ω ) k 2 ( ω ) B v [ z , t , v g ( ω ) ] d t .
Φ D = | A ( D ) * ( ω ) A ( ω ) d ω | 2 ,
Φ D = 2 π B ¯ v ( D ) ( z , t , v ) B v ( z , t , v ) d τ d v ,
B ¯ v ( z , t , v ) = 1 2 π [ ω 2 ( v , ξ ) ω 1 ( v , ξ ) ] ξ A * [ ω 1 ( v , ξ ) ] A [ ω 2 ( v , ξ ) ] × exp [ i ω 2 ( v , ξ ) ω 1 ( v , ξ ) v ( z v t ) ] d ξ .
B v ( z , t , v ) = ω 2 v ( v , 0 ) | { [ 1 + a 2 , 0 2 t 2 + a 2 , 1 3 t 2 ω + ] × B ω ( 0 , t z v , ω ) } | ω = ω 2 ( v , 0 ) ,
a 2 , 0 ( v ) = 1 8 | { ( ω 2 v ) 1 ξ [ ( ω 2 ξ ) 2 2 ω 2 v ξ ] } | ξ = 0 ,
a 2 , 2 ( v ) = 1 8 | [ ( ω 2 ξ ) 2 2 ω 2 ξ 2 ] | ξ = 0 .

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