Abstract

The evolution of an electromagnetic pulse propagating through a linear, dispersive, and absorbing dielectric (as predicted by the modern asymptotic extension of the classic theory of Sommerfeld and Brillouin) is described in physical terms. The description is similar to the group-velocity description known for plane-wave pulses propagating through lossless, gainless, dispersive media but with two modifications: (1) the group velocity is replaced by the velocity of energy in time-harmonic waves, and (2) a nonoscillatory component is added that consists of a wave that grows exponentially with time with a time-dependent growth rate. In the nonoscillatory component the growth rate at each space–time point is determined by the velocity of energy in exponentially growing waves in the medium. The new description provides, for the first time to our knowledge, a physical explanation of the localized details of pulse dynamics in dispersive and absorbing dielectric media and a simple mathematical algorithm for quantitative predictions. Numerical comparisons of the results of the algorithm with the exact integral solution are presented for a highly transparent dielectric and for a highly absorbing dielectric. In both cases the agreement is excellent.

© 1995 Optical Society of America

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  1. A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44,177–202 (1914).
    [Crossref]
  2. L. Brillouin, “Uber die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
    [Crossref]
  3. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  4. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Secs. 5.12 and 5.18.
  5. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 7.11.
  6. B. R. Baldock and T. Bridgeman, Mathematical Theory of Wave Motion (Halsted, New York, 1981), Chap. 5.
  7. L. A. Segel and G. H. Handelman, Mathematics Applied to Continuum Mechanics (Macmillan, New York, 1977), Chap. 9.
  8. I. Tolstoy, Wave Propagation (McGraw-Hill, New York, 1973), Chaps. 1 and 2.
  9. L. B. Felsen, in Transient Electromagnetic Fields, L. B. Felsen, ed. (Springer-Verlag, New York, 1976), Chap. 1, p. 65.
  10. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.
  11. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
    [Crossref]
  12. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [Crossref]
  13. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1420 (1989).
    [Crossref]
  14. The first modification is that we apply the definition of the Fourier transform used in Refs. 12 and 13 rather than that used in Ref. 11. The second modification is that we use gr and gi, respectively, to denote the real and the imaginary parts of the arbitrary complex quantity g. The notation differs from that applied in Refs. 12 and 13 by the same modification plus the change that the electric field is denoted E(z, t) instead of A(z, t). The latter symbol is used to denote a complex quantity with the property that E(z, t) is the real part of A(z, t).
  15. See Ref. 3, Chap. 3. A more comprehensive derivation that uses modern asymptotic techniques is given in Ref. 12.
  16. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 223 (1970).
    [Crossref]
  17. These conditions are sufficient but are far from necessary.
  18. Note that we have ruled out a nonoscillatory contribution with ω∼E=0 at θ= θ0by including only positive solutions to Eq. (4.5). We have done this to avoid including the zero-frequency solution twice. It is included in the time-harmonic contribution, since we include all nonnegative solutions to Eq. (4.4).
  19. We applied the zreal routine of the IMSL Math/Library, available from IMSL, 2500 ParkWest Tower One, 2500 City-West Blvd., Houston, Tex. 77042.
  20. Note that we solve the exact saddle-point equation numerically rather than evaluating numerically the approximate analytical expressions for the saddle points given in Ref. 3 or 12. We applied the routine zanlyof the IMSL Math/Library cited in Ref. 19.
  21. The plot begins at θ= 1.00055 rather than at θ= 1.00000 because the second derivative of ϕ(ω) evaluated at ωE or at the saddle point tends to zero as θ tends to 1 from above, causing the approximations to tend to infinity. The uniform approximation presented in Subsection 4.B does not have this problem.
  22. P. Wyns, D. P. Foty, and K. E. Oughstun, “Numerical analysis of the precursor fields in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1421–1429 (1989).
    [Crossref]
  23. This is a consequence of the fact that the integral is divergent at θ= 1.
  24. The algorithm parameters used to generate the plot in Fig. 9 were k = 5000 and m = 250. To generate the dotted curves in Figs. 10 and 11, the same parameters were taken to be k = 500 and m = 250.
  25. Equation (4.8) follows from Eqs. (4.3) and (4.16) of Ref. 13 with Re ũ(ω) replaced by −if∼(ω). This substitution follows from a comparison of the integral in Eq. (2.6) of this paper with that in Eq. (1.7) of Ref. 13. In addition, we have made use of the discussion following Eqs. (4.3) and (4.16) of Ref. 13 about the argument of a1½.
  26. See Appendix B of Ref. 13. The right-hand side of Eq. (4.23) differs from the right-hand side of Eq. (B9) of Ref. 13 by a factor of 3/4 because the latter equation is incorrect owing to a typographical error.
  27. For a brief discussion of this effect in lossless, gainless media, see Sec. 5.7.4 of Ref. 6. The results there are in a slightly different form from ours because the integral being treated is over wave number k instead of frequency ω. They can be placed in our form by appropriate change of integration variable.
  28. M. A. Biot, “General theorems on the equivalence of group velocity and energy transport,” Phys. Rev. 105, 1129–1137 (1957);M. J. Lighthill, “Group velocity,” J. Inst. Math. Appl. 1, 1–28(1965).
    [Crossref]
  29. K. E. Oughstun and G. C. Sherman, “Asymptotic theory of pulse propagation in absorbing and dispersive dielectrics,” Review of Radio Science 1990–1992 (Oxford U. Press, Oxford, 1993), pp. 75–105.
  30. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
    [Crossref]
  31. F. W. J. Olver, “Why steepest descents,” SIAM Rev. 12, 228–247 (1970).
    [Crossref]

1989 (2)

1988 (1)

1981 (1)

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[Crossref]

1970 (2)

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 223 (1970).
[Crossref]

F. W. J. Olver, “Why steepest descents,” SIAM Rev. 12, 228–247 (1970).
[Crossref]

1957 (1)

M. A. Biot, “General theorems on the equivalence of group velocity and energy transport,” Phys. Rev. 105, 1129–1137 (1957);M. J. Lighthill, “Group velocity,” J. Inst. Math. Appl. 1, 1–28(1965).
[Crossref]

1914 (2)

A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44,177–202 (1914).
[Crossref]

L. Brillouin, “Uber die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

Baldock, B. R.

B. R. Baldock and T. Bridgeman, Mathematical Theory of Wave Motion (Halsted, New York, 1981), Chap. 5.

Biot, M. A.

M. A. Biot, “General theorems on the equivalence of group velocity and energy transport,” Phys. Rev. 105, 1129–1137 (1957);M. J. Lighthill, “Group velocity,” J. Inst. Math. Appl. 1, 1–28(1965).
[Crossref]

Bridgeman, T.

B. R. Baldock and T. Bridgeman, Mathematical Theory of Wave Motion (Halsted, New York, 1981), Chap. 5.

Brillouin, L.

L. Brillouin, “Uber die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

Felsen, L. B.

L. B. Felsen, in Transient Electromagnetic Fields, L. B. Felsen, ed. (Springer-Verlag, New York, 1976), Chap. 1, p. 65.

Foty, D. P.

Handelman, G. H.

L. A. Segel and G. H. Handelman, Mathematics Applied to Continuum Mechanics (Macmillan, New York, 1977), Chap. 9.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 7.11.

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 223 (1970).
[Crossref]

Nussenzveig, H. M.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.

Olver, F. W. J.

F. W. J. Olver, “Why steepest descents,” SIAM Rev. 12, 228–247 (1970).
[Crossref]

Oughstun, K. E.

K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1420 (1989).
[Crossref]

P. Wyns, D. P. Foty, and K. E. Oughstun, “Numerical analysis of the precursor fields in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1421–1429 (1989).
[Crossref]

K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[Crossref]

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[Crossref]

K. E. Oughstun and G. C. Sherman, “Asymptotic theory of pulse propagation in absorbing and dispersive dielectrics,” Review of Radio Science 1990–1992 (Oxford U. Press, Oxford, 1993), pp. 75–105.

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
[Crossref]

Segel, L. A.

L. A. Segel and G. H. Handelman, Mathematics Applied to Continuum Mechanics (Macmillan, New York, 1977), Chap. 9.

Sherman, G. C.

K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1420 (1989).
[Crossref]

K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[Crossref]

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[Crossref]

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
[Crossref]

K. E. Oughstun and G. C. Sherman, “Asymptotic theory of pulse propagation in absorbing and dispersive dielectrics,” Review of Radio Science 1990–1992 (Oxford U. Press, Oxford, 1993), pp. 75–105.

Sommerfeld, A.

A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44,177–202 (1914).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Secs. 5.12 and 5.18.

Tolstoy, I.

I. Tolstoy, Wave Propagation (McGraw-Hill, New York, 1973), Chaps. 1 and 2.

Wyns, P.

Ann. Phys. (2)

A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44,177–202 (1914).
[Crossref]

L. Brillouin, “Uber die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

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

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 223 (1970).
[Crossref]

Phys. Rev. (1)

M. A. Biot, “General theorems on the equivalence of group velocity and energy transport,” Phys. Rev. 105, 1129–1137 (1957);M. J. Lighthill, “Group velocity,” J. Inst. Math. Appl. 1, 1–28(1965).
[Crossref]

Phys. Rev. Lett. (1)

G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[Crossref]

SIAM Rev. (1)

F. W. J. Olver, “Why steepest descents,” SIAM Rev. 12, 228–247 (1970).
[Crossref]

Other (22)

K. E. Oughstun and G. C. Sherman, “Asymptotic theory of pulse propagation in absorbing and dispersive dielectrics,” Review of Radio Science 1990–1992 (Oxford U. Press, Oxford, 1993), pp. 75–105.

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
[Crossref]

This is a consequence of the fact that the integral is divergent at θ= 1.

The algorithm parameters used to generate the plot in Fig. 9 were k = 5000 and m = 250. To generate the dotted curves in Figs. 10 and 11, the same parameters were taken to be k = 500 and m = 250.

Equation (4.8) follows from Eqs. (4.3) and (4.16) of Ref. 13 with Re ũ(ω) replaced by −if∼(ω). This substitution follows from a comparison of the integral in Eq. (2.6) of this paper with that in Eq. (1.7) of Ref. 13. In addition, we have made use of the discussion following Eqs. (4.3) and (4.16) of Ref. 13 about the argument of a1½.

See Appendix B of Ref. 13. The right-hand side of Eq. (4.23) differs from the right-hand side of Eq. (B9) of Ref. 13 by a factor of 3/4 because the latter equation is incorrect owing to a typographical error.

For a brief discussion of this effect in lossless, gainless media, see Sec. 5.7.4 of Ref. 6. The results there are in a slightly different form from ours because the integral being treated is over wave number k instead of frequency ω. They can be placed in our form by appropriate change of integration variable.

The first modification is that we apply the definition of the Fourier transform used in Refs. 12 and 13 rather than that used in Ref. 11. The second modification is that we use gr and gi, respectively, to denote the real and the imaginary parts of the arbitrary complex quantity g. The notation differs from that applied in Refs. 12 and 13 by the same modification plus the change that the electric field is denoted E(z, t) instead of A(z, t). The latter symbol is used to denote a complex quantity with the property that E(z, t) is the real part of A(z, t).

See Ref. 3, Chap. 3. A more comprehensive derivation that uses modern asymptotic techniques is given in Ref. 12.

These conditions are sufficient but are far from necessary.

Note that we have ruled out a nonoscillatory contribution with ω∼E=0 at θ= θ0by including only positive solutions to Eq. (4.5). We have done this to avoid including the zero-frequency solution twice. It is included in the time-harmonic contribution, since we include all nonnegative solutions to Eq. (4.4).

We applied the zreal routine of the IMSL Math/Library, available from IMSL, 2500 ParkWest Tower One, 2500 City-West Blvd., Houston, Tex. 77042.

Note that we solve the exact saddle-point equation numerically rather than evaluating numerically the approximate analytical expressions for the saddle points given in Ref. 3 or 12. We applied the routine zanlyof the IMSL Math/Library cited in Ref. 19.

The plot begins at θ= 1.00055 rather than at θ= 1.00000 because the second derivative of ϕ(ω) evaluated at ωE or at the saddle point tends to zero as θ tends to 1 from above, causing the approximations to tend to infinity. The uniform approximation presented in Subsection 4.B does not have this problem.

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Secs. 5.12 and 5.18.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 7.11.

B. R. Baldock and T. Bridgeman, Mathematical Theory of Wave Motion (Halsted, New York, 1981), Chap. 5.

L. A. Segel and G. H. Handelman, Mathematics Applied to Continuum Mechanics (Macmillan, New York, 1977), Chap. 9.

I. Tolstoy, Wave Propagation (McGraw-Hill, New York, 1973), Chaps. 1 and 2.

L. B. Felsen, in Transient Electromagnetic Fields, L. B. Felsen, ed. (Springer-Verlag, New York, 1976), Chap. 1, p. 65.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.

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Figures (12)

Fig. 1
Fig. 1

Path traced out by the saddle points with increasing θ in the complex ω plane. Dashed curve, ωs; dotted curve, ωb.

Fig. 2
Fig. 2

Location in the complex ω plane of the real frequencies ωEj relative to the locations of the saddle points ωj for a fixed value of θ greater than θ1. Dashed curves, the contours of constant ϕr(ω) that pass through the saddle points and cross the real axis.

Fig. 3
Fig. 3

Normalized energy velocity of monochromatic waves.

Fig. 4
Fig. 4

Attenuation coefficient of monochromatic waves.

Fig. 5
Fig. 5

Normalized energy velocity of nonoscillatory waves.

Fig. 6
Fig. 6

Attenuation coefficient of nonoscillatory waves.

Fig. 7
Fig. 7

Nonuniform results for the electric field of a delta-function pulse. Solid curve, physical model; dashed curve, asymptotic results.

Fig. 8
Fig. 8

Nonuniform results for the electric field of a delta-function pulse for θ near 1. Solid curve, physical model; dashed curve, asymptotic results. The dashed curve is not visible because it falls almost exactly on the solid curve.

Fig. 9
Fig. 9

Comparison of the uniform physical model with the numerical integration of the exact integral solution for the electric field of a delta-function pulse for θ near 1. Solid curve, uniform physical model; dashed curve, numerical evaluation of the exact integral solution.

Fig. 10
Fig. 10

Comparison of the uniform physical model with the numerical integration of the exact integral solution for the electric field of a step-modulated sinusoidal pulse for θ near 1. Solid curve, uniform physical model; dashed curve, numerical evaluation of the exact integral solution.

Fig. 11
Fig. 11

Comparison of the uniform physical model with the numerical integration of the exact integral solution for the electric field of a delta-function pulse in the highly absorbing case. Solid curve, physical model; dashed curve, exact integral solution.

Fig. 12
Fig. 12

Comparison of the uniform physical model with the numerical integration of the exact integral solution for the electric field of a delta-function pulse in the highly transparent case. Solid curve, physical model; dashed curve, exact integral solution. The dashed curve is not visible because it falls almost exactly on the solid curve.

Equations (177)

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E ( z , t ) = 1 2 π i a i a + E ( z , ω ) exp ( i ω t ) d ω ,
2 E ( z , ω ) + k 2 ( ω ) E ( z , ω ) = 0
n ( ω ) = ( 1 b 2 ω 2 ω 0 2 + 2 δ ω i ) 1 / 2 ,
b 2 = 2 × 10 33 s 2 ,
δ = 2.8 × 10 15 s 1 ,
ω 0 = 4 × 10 16 s 1 .
E ( 0 , t ) = f ( t ) ,
E ( z , t ) = 1 2 π a i a i + f ( ω ) exp [ z c ϕ ( ω ) ] d ω ,
f ( ω ) = f ( t ) exp ( i ω t ) d t ,
ϕ ( ω ) = i ω [ n ( ω ) θ ] ,
θ = c t / z .
E ( z , t ) E s ( z , t ) + E b ( z , t ) ,
E s ( z , t ) = c f ( ω s ) [ 2 π z ϕ ( 2 ) ( ω s ) ] 1 / 2 exp [ z c ϕ ( ω s ) ] + c f ( ω s * ) [ 2 π z ϕ ( 2 ) ( ω s * ) ] 1 / 2 exp [ z c ϕ ( ω s * ) ] ,
E b ( z , t ) = c f ( ω b ) [ 2 π z ϕ ( 2 ) ( ω b ) ] 1 / 2 exp [ z c ϕ ( ω b ) ] for 1 < θ < θ 1 ,
E b ( z , t ) = c f ( ω b ) [ 2 π z ϕ ( 2 ) ( ω b ) ] 1 / 2 exp [ z c ϕ ( ω b ) ] + c f ( ω b * ) [ 2 π z ϕ ( 2 ) ( ω b * ) ] 1 / 2 exp [ z c ϕ ( ω b * ) ] for θ > θ 1 .
ϕ ( 1 ) ( ω ) = 0 .
E ( z , t ) = Re [ A ( z , t ) ] ,
A ( z , t ) A s ( z , t ) + A b ( z , t ) ,
A s ( z , t ) = c f ( ω s ) [ 2 π z ϕ ( 2 ) ( ω s ) ] 1 / 2 exp [ z c ϕ ( ω s ) ] ,
A b ( z , t ) = c f ( ω b ) [ 2 π z ϕ ( 2 ) ( ω b ) ] 1 / 2 exp [ z c ϕ ( ω b ) ] for 1 < θ < θ 1 ,
A b ( z , t ) = 2 c f ( ω b ) [ 2 π z ϕ ( 2 ) ( ω b ) ] 1 / 2 exp [ z c ϕ ( ω b ) ] for θ > θ 1 .
ϕ r ( ω E j ) = ϕ r ( ω j )
V E ( ω E j ) = z / t
V E ( ω ) = S ( ω ) ¯ / u ( ω ) ¯ .
V E ( ω ) = c n r ( ω ) + n i ( ω ) ω / δ .
A s ( z , t ) 2 c f ( ω E s ) [ 2 π z ϕ ( 2 ) ( ω E s ) ] 1 / 2 exp [ z c ϕ ( ω s ) ] ,
A b ( z , t ) 2 c f ( ω E b ) [ 2 π z ϕ ( 2 ) ( ω E b ) ] 1 / 2 exp [ z c ϕ ( ω b ) ] for θ θ 1 .
ϕ r ( ω ) = ω n i ( ω ) .
ϕ r ( ω j ) = ω E j n i ( ω E j ) = c k i ( ω E j ) .
i z ϕ i ( ω ) / c = i [ ω r t + z n r ( ω ) ω r / c z n i ( ω ) ω i / c ]
i z ϕ i ( ω j ) / c i [ ω E j t + z k r ( ω E j ) z n i ( ω E j ) ω j i / c ] .
z ϕ ( ω j ) / c i [ ω E j t + z k ( ω E j ) ] i z n i ( ω E j ) ω j i / c .
z / t = V E ( 0 ) .
w ( z , t , ω ) = exp [ ω t k ( ω ) z ] ,
k ( ω ) = i k ( i ω ) = ω c ( 1 + b 2 ω 2 + ω 0 2 + 2 δ ω ) 1 / 2 ,
ω b = i ω b ,
G ( ω b ) = z / t ,
G ( ω ) = 1 / d k ( ω ) d ω .
E ( ω ) = c n ( i ω ) Q 2 ( ω ) n 2 ( i ω ) Q 2 ( ω ) b 2 ω δ ,
Q ( ω ) = ω 2 + ω 0 2 + 2 δ ω ,
n ( i ω ) = [ 1 + b 2 Q ( ω ) ] 1 / 2 .
G ( ω ) = c n ( i ω ) Q 2 ( ω ) n 2 ( i ω ) Q 2 ( ω ) b 2 ω δ b 2 ω 2 .
( 2 / 3 ) δ ω ( 2 / 3 ) ( 3 1 ) δ .
G ( ω ) E ( ω ) E ( ω ) = b 2 ω 2 n 2 ( i ω ) Q 2 b 2 δ ω b 2 ω 2 .
E ( ω ) = z / t ,
θ 0 = c V E ( 0 ) = c E ( 0 ) = n ( 0 ) ,
A ( z , t ) A TH ( z , t ) + A QS ( z , t ) ,
A TH ( z , t ) = j 2 c f ( ω E j ) [ 2 π z ϕ ( 2 ) ( ω E j ) ] 1 / 2 exp { i [ z k ( ω E j ) ω E j t } ,
A QS ( z , t ) = c f ( i ω E ) [ 2 π z ϕ ( 2 ) ( i ω E ) ] 1 / 2 exp [ ω E t k ( ω E ) z ] .
V E ( ω E j ) = z t c θ ,
E ( ω E ) = z t c θ .
L = k ( ω E ) ω E θ c .
L = k ( ω E ) ω E E ( ω E ) ,
f ( ω ) = ω ( 1 + υ ) F ( ω ) ,
E ( z , t ) i 2 α ( i ) υ exp ( z c X ) × [ Γ 0 J υ ( z c α ) i Γ 1 J υ + 1 ( z c α ) ] ,
α = Im [ ϕ ( ω E ) ] ,
X = Re [ ϕ ( ω E ) ] ,
Γ 0 = f ( ω E ) Λ ( 1 ) υ f ( ω E ) Λ ,
Γ 1 = f ( ω E ) Λ + ( 1 ) υ f ( ω E ) Λ ,
Λ = α 3 i ϕ ( 2 ) ( ω E ) .
E ( z , t ) 1 α exp ( z c X ) { Re [ f ( ω E ) Λ ] J 1 ( z c α ) + Im [ f ( ω E ) Λ ] J 0 ( z c α ) }
f ( ω ) = A ,
ω E b [ 2 ( θ 1 ) ] 1 / 2 .
E ( z , t ) A b 2 ( θ 1 ) exp [ 2 δ z c ( θ 1 ) ] × ( J 1 { z c b [ 2 ( θ 1 ) ] 1 / 2 } + 3 δ b [ 2 ( θ 1 ) ] 1 / 2 J 0 { z c b [ 2 ( θ 1 ) ] 1 / 2 } ) .
E ( z , t ) A b 2 2 z c exp [ 2 δ z c ( θ 1 ) ] ( 1 6 δ b 2 c z )
E ( z , t ) A b [ 2 ( θ 1 ) ] 1 / 2 J 1 { z c b [ 2 ( θ 1 ) ] 1 / 2 }
f ( ω ) = A 2 ( 1 ω + ω c 1 ω ω c ) .
E b ( z , t ) = i 2 exp ( z α 0 / c ) { ( c z ) 1 / 3 exp ( i 2 π / 3 ) × [ f ( ω + ) h + + f ( ω ) h ] Ai [ | α 1 | ( z c ) 2 / 3 ] + ( c z ) 2 / 3 exp ( i 4 π / 3 ) [ f ( ω + ) h + f ( ω ) h ] × Ai ( 1 ) [ | α 1 | ( z c ) 2 / 3 / α 1 1 / 2 ]
α 0 = 0.5 [ ϕ ( ω + ) + ϕ ( ω ) ] ,
α 1 1 / 2 = { 3 / 4 [ ϕ ( ω + ) ϕ ( ω ) ] } 1 / 3 ,
h ± = [ ( ± 1 ) 2 α 1 1 / 2 ϕ ( 2 ) ( ω ± ) ] 1 / 2
lim θ θ 1 h ± = [ 2 ϕ ( 3 ) ( ω 1 ) ] 1 / 3 h 1 ,
lim θ θ 1 [ f ( ω + ) h ( ω + ) + f ( ω ) h ( ω ) ] = 2 f ( ω 1 ) h 1 ,
lim θ θ 1 [ f ( ω + ) h ( ω + ) f ( ω ) h ( ω ) ] / α 1 1 / 2 = 2 f ( 1 ) ( ω 1 ) h 1 2 ,
exp { i [ k ( ω ) z ω t ] }
exp [ ω t k ( ω ) z ]
V E ( ω ) = z / t ,
E ( ω ) = z / t ,
V < c n ( 0 ) ,
A j = f ( ω j ) [ z ϕ ( 2 ) ( ω j ) ] 1 / 2 exp [ z c ϕ ( ω j ) ] ,
B j = f ( ω j * ) [ z ϕ ( 2 ) ( ω j * ) ] 1 / 2 exp [ z c ϕ ( ω j * ) ] ,
f ( ω j * ) = f * ( ω j ) ,
n 2 ( ω ) = 1 b 2 ω 2 ω 0 2 + 2 δ ω i ,
n 2 ( ω * ) = [ n 2 ( ω ) ] * = [ n * ( ω ) ] 2 .
n * ( 0 ) = n ( 0 ) .
n ( ω j * ) = n * ( ω j ) .
ϕ ( ω ) = i ω [ n ( ω ) θ ]
ϕ ( ω j * ) = ϕ * ( ω j ) .
ϕ ( 2 ) ( ω ) = i [ 2 n ( 1 ) ( ω ) + ω n ( 2 ) ( ω ) ] ,
n ( 1 ) ( ω ) = b 2 P ( ω ) n Q 2 ( ω ) ,
n ( 2 ) ( ω ) = b 2 n ( ω ) Q 2 ( ω ) [ n ( 1 ) ( ω ) n ( ω ) P ( ω ) + 1 4 P 2 ( ω ) Q ( ω ) ] ,
P ( ω ) = ω + i δ ,
Q ( ω ) = ω 2 ω 0 2 + 2 i δ ω .
P ( ω * ) = P * ( ω ) ,
Q ( ω * ) = Q * ( ω ) .
n ( 1 ) ( ω * ) = [ n ( 1 ) ( ω ) ] * ,
n ( 2 ) ( ω * ) = [ n ( 2 ) ( ω ) ] * .
ϕ ( 2 ) ( ω * ) = [ ϕ ( 2 ) ( ω ) ] * .
| α ¯ o + 2 α ¯ | π 2 ,
[ z ϕ ( 2 ) ( ω j * ) ] 1 / 2 = { [ z ϕ ( 2 ) ( ω j ) ] 1 / 2 } *
B j = f * ( ω j ) { [ z ϕ ( 2 ) ( ω j ) ] 1 / 2 } * exp [ z c ϕ * ( ω j ) ] = A j * ,
V E ( ω E j ) = z / t
θ E = n r ( ω E j ) + ω E j δ n i ( ω E j ) .
ϕ r ( ω j , θ E ) = ϕ r ( ω E j ) ,
ϕ r ( ω b , θ ) δ ( θ θ 0 ) θ 2 θ 0 2 + 2 b 2 / ω 0 2 θ 2 θ 0 2 + 3 b 2 / ω 0 2 ,
ϕ r ( ω E b ) δ ( θ E θ 0 ) θ E 2 θ 0 2 + 2 b 2 / ω 0 2 θ E 2 θ 0 2 + 3 b 2 / ω 0 2 .
θ E θ 0 3 ϕ r ( ω E b ) 2 δ , 0 ω E b ω 0 ,
θ E θ 0 ϕ r ( ω E b ) δ , ω 0 ω E b ω 1 ,
ϕ r ( ω s , θ ) 2 δ ( θ 1 ) ,
θ E 1 ϕ r ( ω E b ) / 2 δ , ω E s ω 1 .
n ( ω E b ) θ 0 + b 2 2 θ 0 ω 0 4 ω E b 2 + i δ b 2 θ 0 ω 0 4 ω E b ,
ϕ r ( ω E b ) = ω E b n i ( ω E b ) δ b 2 θ 0 ω 0 4 ω E b 2 .
θ E θ 0 + 3 b 2 2 θ 0 ω 0 4 ω E b 2 = θ 0 3 ϕ r ( ω E b ) 2 δ , 0 ω E b ω 0 ,
n ( ω E b ) 1 + i b 2 4 δ ω E b ,
ϕ r ( ω E b ) b 2 4 δ .
θ E 1 + b 2 4 δ 2 1 ϕ r ( ω E b ) δ , ω 0 ω E b ω 1 ,
n ( ω E s ) 1 b 2 2 ω E S 3 ( ω E s 2 δ i ) ,
ϕ r ( ω E s ) = ω E s n i ( ω E s ) δ b 2 ω E S 2 .
θ E 1 + b 2 2 ω E s 2 1 ϕ r ( ω E s ) 2 δ , ω E s ω 1 ,
E ( ω ) = S u ,
u = m N 2 ( r 2 + ω 0 2 r 2 ) + E 2 + H 2 8 π ,
r = e E m ( ω 2 ω 0 2 + 2 δ i ω ) ,
r = e E m Q ,
Q = ω 2 + ω 0 2 + 2 δ ω .
= ω r .
u = m N r 2 2 ( ω 2 + ω 0 2 ) + E 2 + H 2 8 π .
H = n ( i ω ) E ,
n ( i ω ) = ( 1 + b 2 Q ) 1 / 2 .
u = m N 2 ( ω 2 + ω 0 2 ) e 2 E 2 m 2 Q 2 + E 2 8 π [ 1 + n 2 ( i ω ) ] .
S = c n ( i ω ) 4 π E 2 .
b = ( 4 π N e 2 m ) 1 / 2 ,
u S = 1 2 c n ( i ω ) [ b 2 Q 2 ( ω 2 + ω 0 2 ) + 1 + n 2 ( i ω ) ] .
1 + n 2 ( i ω ) = 2 n 2 ( i ω ) b 2 Q .
u S = n 2 ( i ω ) Q 2 b 2 δ ω c n ( i ω ) Q 2 .
E ( ω ) = c n ( i ω ) Q 2 n 2 ( i ω ) Q 2 b 2 ω δ .
G ( ω ) = 1 / d k ( ω ) d ω ,
k ( ω ) = ω c ( 1 + b 2 Q ) 1 / 2 = ω c n ( i ω ) .
k ( ω ) = n ( i ω ) c + ω n ( i ω ) c ,
n ( i ω ) = b 2 2 n ( i ω ) Q 2 = b 2 ( ω + δ ) n ( i ω ) Q 2 .
k ( ω ) = n 2 ( i ω ) Q 2 b 2 ω ( ω + δ ) c n ( i ω ) Q 2 .
G ( ω ) = c n ( i ω ) Q 2 n 2 ( i ω ) Q 2 b 2 δ ω b 2 ω 2 .
F = n 2 ( i ω ) Q 2 b 2 δ ω b 2 δ ω 2 ,
G ( ω ) E ( ω ) = 1 + b 2 ω 2 F .
G ( ω ) E ( ω ) E ( ω ) = b 2 ω 2 F .
θ SB θ 0 4 δ 2 b 2 3 θ 0 ω 0 4 ,
θ 0 = n ( 0 ) .
ω ̂ SB 1 3 [ 6 θ 0 ω 0 4 b 2 ( θ SB θ 0 ) 4 δ 2 ] 1 / 2 i 2 3 δ ,
ω ̂ SB 2 3 δ ( 3 1 ) i = 1.07 δ i .
2 3 δ ω 1.07 δ .
F = ω 4 + 4 δ ω 3 + ( 2 ω 0 2 + 4 δ 2 ) ω 2 + δ ( b 2 + 4 ω 0 2 ) ω + ω 0 4 + b 2 ω 0 2 .
F > b 2 ω 0 2
G ( ω ) E ( ω ) E ( ω ) < ω 2 ω 0 2
[ ω 4 + 4 δ ω 3 + ( 2 ω 0 2 + 4 δ 2 ) ω 2 + δ ( b 2 + 4 ω 0 2 ) ω ] < ω 0 4
[ 4 δ ω 3 + δ ( b 2 + 4 ω 0 2 ) ω ] < ω 0 4 ,
G ( ω ) = [ 4 δ ω 3 + δ ( b 2 + 4 ω 0 2 ) ω ] .
G ( 2 δ / 3 ) 0.193 ω 0 4 .
ω sp ± ξ δ i ( 1 + η )
ω E b [ 2 ( θ 1 ) ] 1 / 2
E ( z , t ) A ω E 2 b ( θ 1 + b 2 / 2 ω E 2 + 4 δ 2 ) 1 / 2 × exp ( δ z c b 2 ω E 2 + 4 δ 2 ) × { 2 ω E J 1 [ z c ω E ( θ 1 + b 2 / 2 ω E 2 + 4 δ 2 ) ] + 6 δ J 0 [ z c ω E ( θ 1 + b 2 / 2 ω E 2 + 4 δ 2 ) ] } .
b 2 / 2 ω E 2 + 4 δ 2 θ 1 .
E ( z , t ) A ω E 2 b 2 ( θ 1 ) exp [ 2 δ z c ( θ 1 ) ] × { J 1 [ z c ω E 2 ( θ 1 ) ] + 3 δ ω E J 0 [ z c ω E 2 ( θ 1 ) ] } .
E ( z , t ) A b [ 2 ( θ 1 ) ] 1 / 2 exp [ 2 δ z c ( θ 1 ) ] × ( J 1 { z c b [ 2 ( θ 1 ) ] 1 / 2 } + 3 δ b 2 ( θ 1 ) J 0 [ z c b [ 2 ( θ 1 ) ] 1 / 2 ] ) .
E ( z , t ) A b 2 z 2 c exp [ 2 δ z c ( θ 1 ) ] ( 1 6 δ c b 2 z )
E ( z , t ) A b 2 z 2 c .
E ( z , t ) = ω c 2 π C exp ( i ω τ i ζ ω ) d ω ω 2 ω c 2 ,
ζ = b 2 z 2 c ,
τ = z c ( θ 1 ) .
exp ( i u ) = ω ( τ / ζ ) 1 / 2
E ( z , t ) = ω c 2 π τ ζ 0 2 π exp [ 2 i ( τ ζ ) 1 / 2 cos u ] exp ( i u ) i d u .
A ω 2 ω c = A ζ ω c τ exp ( 2 i u )
E ( z , t ) = A 2 π ( ζ τ ) 1 / 2 0 2 π exp [ 2 i ( τ ζ ) 1 / 2 cos u ] exp ( i u ) i d u .
J 1 ( z ) = 1 2 π 0 2 π exp [ i z cos α ] exp ( i α ) i d α ,
E ( z , t ) A ( ζ / τ ) 1 / 2 J 1 [ 2 ( τ ζ ) 1 / 2 ] .
ζ τ = b 2 2 ( z c ) 2 ( θ 1 ) ,
ζ τ = b 2 2 ( θ 1 ) .
E ( z , t ) A b [ 2 ( θ 1 ) ] 1 / 2 J 1 { b z c [ 2 ( θ 1 ) ] 1 / 2 }
E ( z , t ) A b 2 z 2 c

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