Abstract

The classical theory of Sommerfeld and Brillouin of pulse propagation in a Lorentz medium is reexamined. We show by numerical techniques that Brillouin’s approximations for the saddle-point locations break down in certain space–time regions. Analytic approximations that describe the correct saddle-point behavior are derived and applied to obtain improved asymptotic expressions. Qualitatively, the resulting pulse behavior is similar to that predicted by Brillouin. The quantitative improvements are significant, however, and have led to a simple mathematical procedure for determining the pulse dynamics in addition to a clear physical interpretation.

© 1988 Optical Society of America

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References

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  1. G. R. Baldock, T. Bridgeman, Mathematical Theory of Wave Motion (Halsted, New York, 1981), Chap.5.
  2. L. A. Segel, G. H. Handelman, Mathematics Applied to Continuum Mechanics (Macmillan, New York, 1977), Chap.9.
  3. I. Tolstoy, Wave Propagation (McGraw-Hill, New York, 1973), Chaps.1–2.
  4. L. B. Felsen, in Transient Electromagnetic Fields, L. B. Felsen, ed.(Springer-Verlag, New York, 1976), Chap. 1, p. 65.
  5. K. A. Connor, L. B. Felsen, “Complex space-time rays and their application to pulse propagation in lossy dispersive media,” Proc. IEEE 62, 1586–1598 (1974).
    [CrossRef]
  6. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Secs. 5.12 and 5.18.
  8. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 7.11.
  9. N. S. Shiren, “Measurement of signal velocity in a region of resonant absorption by ultrasonic paramagnetic resonance,” Phys. Rev. 128, 2103–2112 (1962).
    [CrossRef]
  10. N. S. Shiren, “Signal velocity in a region of resonant stimulated emission,” Phys. Rev. Lett. 15, 341–343 (1965);errata, Phys. Rev. Lett. 15, 597 (1965).
    [CrossRef]
  11. P. Pleshko, I. Palocz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett. 22, 1201–1204 (1969).
    [CrossRef]
  12. S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
    [CrossRef]
  13. C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
    [CrossRef]
  14. D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
    [CrossRef]
  15. E. Gitterman, M. Gitterman, “Transient processes for incidence of a light signal on a vacuum–medium interface,” Phys. Rev. A 13, 763–776 (1976).
    [CrossRef]
  16. M. J. Frankel, J. L. Birman, “Transient optical response of a spatially dispersive medium,” Phys. Rev. A 15, 2000–2008 (1977).
    [CrossRef]
  17. A. Puri, J. L. Birman, “Energy-transport, group, and signal velocities near resonance in spatially dispersive media,” Phys. Rev. Lett. 47, 173–177 (1981).
    [CrossRef]
  18. R. J. Vidmar, F. W. Crawford, K. J. Harker, “Dispersion characteristics for decaying or amplifying waves, parts I and II,” Radio Sci. (in press)
  19. K. E. Oughstun, Ph.D. dissertation, University of Rochester (University Microfilms, Ann Arbor, Mich., 1978).
  20. G. C. Sherman, 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]
  21. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. I (McGraw-Hill, New York, 1953).
  22. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford U. Press, London, 1972).
  23. M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge U. Press, London, 1970).
  24. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
    [CrossRef]
  25. L. Brillouin, “Über die Fortpflanzung des Licht in dispeerdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
    [CrossRef]
  26. H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952).
  27. L. Rosenfeld, Theory of Electrons (Dover, New York, 1965).
  28. F. W. J. Olver, “Why steepest descents?”Stud. Appl. Math. Rev. 12, 228–247 (1970).
  29. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, London, 1963).
  30. L. Mandel, “Interpretation of instantaneous frequencies,” Am. J. Phys. 42, 840–846 (1974).
    [CrossRef]
  31. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
    [CrossRef]

1982

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

1981

A. Puri, J. L. Birman, “Energy-transport, group, and signal velocities near resonance in spatially dispersive media,” Phys. Rev. Lett. 47, 173–177 (1981).
[CrossRef]

G. C. Sherman, 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]

1977

M. J. Frankel, J. L. Birman, “Transient optical response of a spatially dispersive medium,” Phys. Rev. A 15, 2000–2008 (1977).
[CrossRef]

1976

E. Gitterman, M. Gitterman, “Transient processes for incidence of a light signal on a vacuum–medium interface,” Phys. Rev. A 13, 763–776 (1976).
[CrossRef]

1974

K. A. Connor, L. B. Felsen, “Complex space-time rays and their application to pulse propagation in lossy dispersive media,” Proc. IEEE 62, 1586–1598 (1974).
[CrossRef]

L. Mandel, “Interpretation of instantaneous frequencies,” Am. J. Phys. 42, 840–846 (1974).
[CrossRef]

1970

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

F. W. J. Olver, “Why steepest descents?”Stud. Appl. Math. Rev. 12, 228–247 (1970).

C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

1969

P. Pleshko, I. Palocz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett. 22, 1201–1204 (1969).
[CrossRef]

1965

N. S. Shiren, “Signal velocity in a region of resonant stimulated emission,” Phys. Rev. Lett. 15, 341–343 (1965);errata, Phys. Rev. Lett. 15, 597 (1965).
[CrossRef]

1962

N. S. Shiren, “Measurement of signal velocity in a region of resonant absorption by ultrasonic paramagnetic resonance,” Phys. Rev. 128, 2103–2112 (1962).
[CrossRef]

1914

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

L. Brillouin, “Über die Fortpflanzung des Licht in dispeerdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

Baldock, G. R.

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

Birman, J. L.

A. Puri, J. L. Birman, “Energy-transport, group, and signal velocities near resonance in spatially dispersive media,” Phys. Rev. Lett. 47, 173–177 (1981).
[CrossRef]

M. J. Frankel, J. L. Birman, “Transient optical response of a spatially dispersive medium,” Phys. Rev. A 15, 2000–2008 (1977).
[CrossRef]

Bridgeman, T.

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

Brillouin, L.

L. Brillouin, “Über die Fortpflanzung des Licht in dispeerdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

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

Chu, S.

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

Connor, K. A.

K. A. Connor, L. B. Felsen, “Complex space-time rays and their application to pulse propagation in lossy dispersive media,” Proc. IEEE 62, 1586–1598 (1974).
[CrossRef]

Copson, E. T.

E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford U. Press, London, 1972).

Crawford, F. W.

R. J. Vidmar, F. W. Crawford, K. J. Harker, “Dispersion characteristics for decaying or amplifying waves, parts I and II,” Radio Sci. (in press)

Felsen, L. B.

K. A. Connor, L. B. Felsen, “Complex space-time rays and their application to pulse propagation in lossy dispersive media,” Proc. IEEE 62, 1586–1598 (1974).
[CrossRef]

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

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. I (McGraw-Hill, New York, 1953).

Frankel, M. J.

M. J. Frankel, J. L. Birman, “Transient optical response of a spatially dispersive medium,” Phys. Rev. A 15, 2000–2008 (1977).
[CrossRef]

Garrett, C. G. B.

C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

Gitterman, E.

E. Gitterman, M. Gitterman, “Transient processes for incidence of a light signal on a vacuum–medium interface,” Phys. Rev. A 13, 763–776 (1976).
[CrossRef]

Gitterman, M.

E. Gitterman, M. Gitterman, “Transient processes for incidence of a light signal on a vacuum–medium interface,” Phys. Rev. A 13, 763–776 (1976).
[CrossRef]

Handelman, G. H.

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

Harker, K. J.

R. J. Vidmar, F. W. Crawford, K. J. Harker, “Dispersion characteristics for decaying or amplifying waves, parts I and II,” Radio Sci. (in press)

Jackson, J. D.

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

Lighthill, M. J.

M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge U. Press, London, 1970).

Lorentz, H. A.

H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952).

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

Mandel, L.

L. Mandel, “Interpretation of instantaneous frequencies,” Am. J. Phys. 42, 840–846 (1974).
[CrossRef]

McCumber, D. E.

C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. I (McGraw-Hill, New York, 1953).

Olver, F. W. J.

F. W. J. Olver, “Why steepest descents?”Stud. Appl. Math. Rev. 12, 228–247 (1970).

Oughstun, K. E.

G. C. Sherman, 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, Ph.D. dissertation, University of Rochester (University Microfilms, Ann Arbor, Mich., 1978).

Palocz, I.

P. Pleshko, I. Palocz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett. 22, 1201–1204 (1969).
[CrossRef]

Pleshko, P.

P. Pleshko, I. Palocz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett. 22, 1201–1204 (1969).
[CrossRef]

Puri, A.

A. Puri, J. L. Birman, “Energy-transport, group, and signal velocities near resonance in spatially dispersive media,” Phys. Rev. Lett. 47, 173–177 (1981).
[CrossRef]

Rosenfeld, L.

L. Rosenfeld, Theory of Electrons (Dover, New York, 1965).

Segel, L. A.

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

Sherman, G. C.

G. C. Sherman, 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]

Shiren, N. S.

N. S. Shiren, “Signal velocity in a region of resonant stimulated emission,” Phys. Rev. Lett. 15, 341–343 (1965);errata, Phys. Rev. Lett. 15, 597 (1965).
[CrossRef]

N. S. Shiren, “Measurement of signal velocity in a region of resonant absorption by ultrasonic paramagnetic resonance,” Phys. Rev. 128, 2103–2112 (1962).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Über 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.1–2.

Trizna, D. B.

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

Vidmar, R. J.

R. J. Vidmar, F. W. Crawford, K. J. Harker, “Dispersion characteristics for decaying or amplifying waves, parts I and II,” Radio Sci. (in press)

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, London, 1963).

Weber, T. A.

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, London, 1963).

Wong, S.

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

Am. J. Phys.

L. Mandel, “Interpretation of instantaneous frequencies,” Am. J. Phys. 42, 840–846 (1974).
[CrossRef]

Ann. Phys.

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

L. Brillouin, “Über die Fortpflanzung des Licht in dispeerdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

J. Phys. A

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3, 233–245 (1970).
[CrossRef]

Phys. Rev.

N. S. Shiren, “Measurement of signal velocity in a region of resonant absorption by ultrasonic paramagnetic resonance,” Phys. Rev. 128, 2103–2112 (1962).
[CrossRef]

Phys. Rev. A

E. Gitterman, M. Gitterman, “Transient processes for incidence of a light signal on a vacuum–medium interface,” Phys. Rev. A 13, 763–776 (1976).
[CrossRef]

M. J. Frankel, J. L. Birman, “Transient optical response of a spatially dispersive medium,” Phys. Rev. A 15, 2000–2008 (1977).
[CrossRef]

C. G. B. Garrett, D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

Phys. Rev. Lett.

G. C. Sherman, 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]

A. Puri, J. L. Birman, “Energy-transport, group, and signal velocities near resonance in spatially dispersive media,” Phys. Rev. Lett. 47, 173–177 (1981).
[CrossRef]

N. S. Shiren, “Signal velocity in a region of resonant stimulated emission,” Phys. Rev. Lett. 15, 341–343 (1965);errata, Phys. Rev. Lett. 15, 597 (1965).
[CrossRef]

P. Pleshko, I. Palocz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett. 22, 1201–1204 (1969).
[CrossRef]

S. Chu, S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982).
[CrossRef]

Proc. IEEE

K. A. Connor, L. B. Felsen, “Complex space-time rays and their application to pulse propagation in lossy dispersive media,” Proc. IEEE 62, 1586–1598 (1974).
[CrossRef]

Radio Sci.

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

Stud. Appl. Math. Rev.

F. W. J. Olver, “Why steepest descents?”Stud. Appl. Math. Rev. 12, 228–247 (1970).

Other

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, London, 1963).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. I (McGraw-Hill, New York, 1953).

E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford U. Press, London, 1972).

M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge U. Press, London, 1970).

H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952).

L. Rosenfeld, Theory of Electrons (Dover, New York, 1965).

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.

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

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

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

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

R. J. Vidmar, F. W. Crawford, K. J. Harker, “Dispersion characteristics for decaying or amplifying waves, parts I and II,” Radio Sci. (in press)

K. E. Oughstun, Ph.D. dissertation, University of Rochester (University Microfilms, Ann Arbor, Mich., 1978).

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Figures (19)

Fig. 1
Fig. 1

Branch cuts for both the complex index of refraction n(ω) and the complex phase function ϕ(ω, θ) in the complex ω plane for ω0 > δ.

Fig. 2
Fig. 2

Behavior of the real and imaginary parts of the complex index of refraction n(ω′) = nr(ω′) + ini(ω′) (upper curves) and the real part X(ω′) of the complex phase function (lower curve) along the positive real ω′ axis. The medium parameters are those chosen by Brillouin in his analysis: ω0 = 4.0 × 1016/sec, b2 = 20.0 × 1032/sec2, δ = 0.28 × 1016/sec.

Fig. 3
Fig. 3

Behavior of n(ω) and X(ω, θ) in the immediate neighborhoods about the branch points ω+ and ω+′. The dashed curve indicates the approximate location of the contour X(ω, θ) = 0 for θ > 0.

Fig. 4
Fig. 4

(a) Isotimic contours of the real phase behavior X(ω, θ) in the right half of the complex ω plane for θ = 1. X(ω, 1) < 0 in the upper half-plane, and the dominant distant saddle points are located in the lower half-plane at ωSPD± = ±∞ − 2δi and X(ωSPD±, 1) = 0. (b) Isotimic contours of the real phase behavior X(ω, θ,) in the right half of the complex ω plane for θ = 1.25. The distant saddle point has moved in from infinity and is still dominant over the upper near saddle point SP1. (c) Isotimic contours of the real phase behavior X(ω, θ) in the right half of the complex ω plane for θ = θSB = 1.33425. At this value the distant saddle points and the upper near saddle point SP1 are all of equal importance. (d) Isotimic contours of the real phase behavior X(ω, θ) in the right half of the complex ω plane for θ = 1.501, which is just before the coalescence of the two near first-order saddle points SP1 and SP2 into a single second-order saddle point. The upper near saddle point SP1 is dominant over the distant saddle points. (e) Isotimic contours of the real phase behavior X(ω, θ) in the right half of the complex ω plane for θ = 1.65 > θ1. The dominant near saddle points have now moved off of the imaginary axis into the lower half of the complex plane. (f) Isotimic contours of the real phase behavior X(ω, θ) in the right half of the complex ω plane for θ = 5.0. Notice the approach of the dominant near saddle point toward the branch point ω+ and the approach of the distance saddle point toward the branch point ω+′.

Fig. 5
Fig. 5

The behavior of the saddle points in the region removed from the origin. The dotted curves indicate the trajectories followed by the saddle points as θ varies.

Fig. 6
Fig. 6

The behavior of the saddle points in the region near the origin for 1 ≤ θ < θ1. As θ increases over this range the two saddle points steadily approach each other.

Fig. 7
Fig. 7

The two saddle points in the region near the origin have coalesced into a single saddle point of second order at θ = θ1.

Fig. 8
Fig. 8

The behavior of the saddle points in the region near the origin for θ > θ1. The dotted curves indicate the trajectories followed by the saddle points as θ varies.

Fig. 9
Fig. 9

Behavior of the real part of the phase, X(ω, θ) = Re[ϕ(ω, θ)], at the near and distant saddle points as a function of θ for 1 ≤ θ ≤ 2.2. The solid curves represent the exact (numerically determined) behavior, the dashed curves represent the second approximation, and the dotted curves represent the first approximation.

Fig. 10
Fig. 10

Behavior of the real part of the phase, X(ω,θ) = Re[ϕ(ω, θ)], at the near and distant saddle points as a function of 0 for 2 ≤ θ ≤ 14. The solid curves represent the exact (numerically determined) behavior, the dashed curves the second approximation, and the dotted curves the first approximation.

Fig. 11
Fig. 11

Behavior of the imaginary part of the phase, Y(ω, θ) = Im[ϕ(ω, θ)], at the near and distant saddle points in the right half-plane as a function of θ. The solid curves represent the exact (numerically determined) behavior, the dashed curves represent the second approximation, and the dotted curves represent the first approximation.

Fig. 12
Fig. 12

The deformed contour of integration P(θ) through the relevant saddle points of ϕ(ω, θ). The dashed contours indicate the isotimic contours through the saddle points, and the shaded areas indicate the regions of the complex ω plane wherein X(ω, θ) is less than that at the relevant saddle point.

Fig. 13
Fig. 13

The instantaneous angular frequency of oscillation ωs of the first precursor field and the real coordinate position ξ(θ) ≅ Re(ωSPD+) of the distance saddle-point location in the right half of the complex ω plane as a function of θ for ω0 = 4.0 × 1016/sec, δ = 0.28 × 1016/sec, and b2 = 20 × 1032/sec2. The solid curve represents the behavior obtained in the second approximation, and the dotted curve is the behavior obtained in the first approximation owing to Brillouin.

Fig. 14
Fig. 14

The instantaneous angular frequency of oscillation ωB of the second precursor field and the real coordinate position ψ(θ) ≅ Re(ωSPN+) of the near saddle-point location in the right half of the complex ω plane as a function of θθ1 for ω0 = 4.0 × 1016/sec, δ = 0.28 × 1016 sec, and b2 = 20 × 1032/sec2. The solid curve represents the behavior obtained in the second approximation, and the dotted curve is that obtained in the first approximation due to Brillouin.

Fig. 15
Fig. 15

Nonuniform evolution of the Sommerfeld precursor As(z, t), the Brillouin precursor AB(z, t), and the total field A(z, t) = As(z, t) + AB(z, t) for an input delta-function pulse for Brillouin’s choice of the medium parameters at a propagation distance of z = 1 × 10−4 cm.

Fig. 16
Fig. 16

A deformed contour of integration P(θ) through both the near and distant saddle points for a fixed value of θ. This contour is an Olver-type path with respect to the near saddle point SPN+ in the right half of the complex ω plane and is an Olver-type path with respect to the near saddle point SPN in the left half of the complex ω plane. The hatched areas indicate the regions of the complex ω plane wherein X(ω) is less than that at the respective saddle points.

Fig. 17
Fig. 17

Dynamic behavior of the propagated field due to an input unit step-function-modulated signal with applied signal frequency ωc < ωSB.

Fig. 18
Fig. 18

Dynamic behavior of the propagated field due to an input unit step-function-modulated signal with applied signal frequency ωc > ωSB.

Fig. 19
Fig. 19

Frequency dependence of the main signal, anterior presignal, and posterior presignal velocities for a Lorentz-model medium with ω0 = 4.0 × 1016/sec, b2 = 20 × 1032/sec2, and δ = 0.28 × 1016/sec. The dashed curve depicts the behavior of the energy-transport velocity for a strictly monochromatic field in the medium.

Equations (209)

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

A ( z , t ) = c A ( z , ω ) e i ω t d ω ,
[ 2 + k 2 ( ω ) ] A ( z , ω ) = 0 ,
k ( ω ) = ω c n ( ω ) ,
A ( 0 , t ) = f ( t )
A ( z , ω ) = A + ( ω ) exp ( i k z ) + A ( ω ) exp ( i k z ) ,
A ( z , t ) = c A + ( ω ) exp [ i ( k z ω t ) ] d ω .
f ( t ) = c A + ( ω ) e i ω t d ω .
A + ( ω ) = 1 2 π 0 f ( t ) e i ω t d t = 1 2 π f ( ω ) ,
A ( z , t ) = 1 2 π c f ( ω ) exp { i [ k ( ω ) z ω t ] } d ω
A ( z , t ) = 1 2 π c f ( ω ) exp [ z c ϕ ( ω , θ ) ] d ω ,
ϕ ( ω , θ ) = i ω [ n ( ω ) θ ] ,
θ = c t z
n ( ω ) = n * ( ω * ) ,
ϕ ( ω , θ ) = ϕ * ( ω * , θ ) .
f ( ω ) = f * ( ω * ) .
A ( z , t ) = 1 2 π Re { i a i a + f ( ω ) exp [ z c ϕ ( ω , θ ) ] d ω }
f ( t ) = u ( t ) sin ( ω c t ) ,
A ( z , t ) = 1 2 π Re { i i a i a + ũ ( ω ω c ) exp [ z c ϕ ( ω , θ ) ] d ω }
f ( t ) = δ ( t t 0 ) ,
f ( ω ) = 0 δ ( t t 0 ) e i ω t d t = exp ( i ω t 0 ) .
A ( z , t ) = 1 2 π Re { i a i a + exp [ z c ϕ t 0 ( ω , θ ) ] d ω } ,
ϕ t 0 ( ω , θ ) = i ω [ n ( ω ) c z ( t t 0 ) ] .
u ( t ) = { 0 , for t < 0 1 , for t > 0 ,
ũ ( ω ) = c e i ω t d t = i ω
A ( z , t ) = 1 2 π Re { i a i a + 1 ω ω c exp [ z c ϕ ( ω , θ ) ] d ω }
ω 0 = 4.0 × 10 16 sec 1 , b 2 = 20.0 × 10 32 sec 2 , δ = 0.28 × 10 16 sec 1 ,
n ( ω ) + ω n ( ω ) θ = 0.
[ ω 2 ω 1 2 + 2 δ i ω + b 2 ω ( ω + δ i ) ω 2 ω 0 2 + 2 δ i ω ] 2 = θ 2 ( ω 2 ω 1 2 + 2 δ i ω ) ( ω 2 ω 0 2 + 2 δ i ω ) ,
n ( ω ) 1 b 2 2 ω ( ω + 2 δ i ) ,
ω SP D ± ( θ ) ± b [ 2 ( θ 1 ) ] 1 / 2 2 δ i .
n ( ω ) ω 1 ω 2 + b 2 2 ω 1 ω 0 3 ω ( ω + 2 δ i ) δ 2 b 2 ( 4 ω 1 2 b 2 ) 2 ω 1 2 ω 0 5 ω 2 ,
ω SP N ± ± 1 3 [ 6 θ 0 ω 0 4 α b 2 ( θ θ 0 ) 4 δ 2 α 2 ] 1 / 2 i 2 δ 3 α ,
θ 0 = ω 1 ω 0 = ( 1 + b 2 / ω 0 2 ) 1 / 2 = n ( 0 ) ,
α = 1 δ 2 ( 4 ω 1 2 b 2 ) ω 0 2 ω 1 2 .
θ 1 = θ 0 + 2 δ 2 b 2 3 α θ 0 ω 0 4 ,
ω SP N ± ( ) = ± ( ω 0 2 δ 2 ) 1 / 2 δ i ,
ω SP D ± ( ) = ± ( ω 1 2 δ 2 ) 1 / 2 δ i .
1 θ 2 ( ω 2 ω 0 2 + 2 δ i ω ) [ ω 2 ω 1 2 + 2 δ i ω + b 2 ω ( ω + δ i ) ω 2 ω 0 2 + 2 δ i ω ] 2 = ω 2 ω 1 2 + 2 δ i ω .
b 2 ω ( ω + δ i ) ω 2 ω 0 2 + 2 δ i ω b 2 ω + δ i ω + 2 δ i b 2 ( 1 δ i ω ) ,
ω 3 + 2 δ i ω 2 ( ω 0 2 + b 2 θ 2 θ 2 1 ) ω + 2 i δ b 2 θ 2 1 = 0 ,
ω SP D ± ( θ ) ± 3 2 [ ( β 1 + β 2 ) 1 / 3 ( β 1 + β 2 ) 1 / 3 ] i { 2 3 δ + 1 2 [ ( β 1 + β 2 ) 1 / 3 + ( β 1 β 2 ) 1 / 3 ] } ,
β 1 = δ 3 [ ω 0 2 8 9 δ 2 + b 2 ( θ 2 + 3 ) θ 2 1 ] ,
β 2 = 1 3 3 { ω 0 4 ( ω 0 2 δ 2 ) + b 2 θ 2 1 × [ ( 3 ω 0 2 2 δ 2 ) ω 0 2 θ 2 + 2 δ 2 ( 9 ω 0 2 8 δ 2 ) ] + b 4 ( θ 2 1 ) 2 [ ( 3 ω 0 2 δ 2 ) θ 4 + 9 δ 2 ( 2 θ 2 + 3 ) ] + b 6 θ 6 ( θ 2 1 ) 3 1 / 2 } .
( β 1 + β 2 ) 1 / 3 β 2 1 / 3 + β 1 3 β 2 2 / 3 , ( β 1 β 2 ) 1 / 3 β 2 1 / 3 + β 1 3 β 2 2 / 3 .
ω SP D ± ( θ ) ± 3 β 2 1 / 3 i ( 2 3 δ + β 1 3 β 2 2 / 3 ) .
β 2 1 / 3 1 3 ( ω 0 2 δ 2 + b 2 θ 2 θ 2 1 ) 1 / 2 .
ω SP D ± ( θ ) ± ξ ( θ ) δ i [ 1 + η ( θ ) ] ,
ξ ( θ ) = ( ω 0 2 δ 2 + b 2 θ 2 θ 2 1 ) 1 / 2 ,
η ( θ ) = δ 2 / 27 + b 2 / ( θ 2 1 ) ξ 2 ( θ ) .
lim θ [ ω SP D ± ( θ ) ] ± ( ω 1 2 δ 2 ) 1 / 2 δ i = ω ± ,
ϕ ( ω , θ ) i ω ( 1 θ ) i b 2 2 ( ω + 2 δ i ) .
ω = ω SP D + ( θ ) + r e i ψ ξ ( θ ) δ i [ 1 + η ( θ ) ] + r e i ψ .
ϕ ( θ , r , ψ ) ( 1 θ ) { i ξ ( θ ) + δ [ 1 + η ( θ ) ] + i r e i ψ } i b 2 2 ξ ( θ ) δ i [ 1 η ( θ ) ] + r e i ψ ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 + 2 ξ ( θ ) r cos ( ψ ) + 2 δ [ 1 η ( θ ) ] r sin ( ψ ) + r 2 ,
X ( θ , r , ψ ) ( 1 θ ) { δ [ 1 + η ( θ ) ] r sin ψ } b 2 2 δ [ 1 η ( θ ) ] + r sin ψ ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 + 2 ξ ( θ ) r cos ( ψ ) + 2 δ [ 1 η ( θ ) ] r sin ( ψ ) + r 2 .
θ 2 ( ω 2 ω 0 2 + 2 δ i ω ) = ω 2 ω 1 2 + 2 δ i ω + 2 b 2 ω ( ω + δ i ) ω 2 ω 0 2 + 2 δ i ω + b 4 ω 2 ( ω + δ i ) 2 ( ω 2 ω 1 2 + 2 δ i ω ) ( ω 2 ω 0 2 + 2 δ i ω ) 2 .
ω ( ω + δ i ) ω 2 ω 0 2 + 2 δ i ω 1 ω 0 2 [ δ i ω + ( 1 2 δ 2 ω 0 2 ) ω 2 + δ i ω 0 2 ( 3 4 δ 2 ω 0 2 ) ω 3 ]
ω 2 ( ω + δ i ) 2 ( ω 2 ω 1 2 + 2 δ i ω ) ( ω 2 ω 0 2 + 2 δ i ω ) 2 ω 2 ω 1 2 ω 0 4 [ δ 2 + 2 δ i ω 2 δ 3 i ω 1 2 ω 0 2 ( 2 ω 1 2 + ω 0 2 ) ω ] .
θ 2 ( ω 2 ω 0 2 + 2 δ i ω ) 2 δ i b 2 ω 0 4 [ 3 + b 2 ω 1 2 4 δ 2 ω 0 2 δ 2 b 2 ω 1 4 ω 0 2 ( 2 ω 1 2 + ω 0 2 ) ] ω 3 + [ 1 b 2 ω 0 2 ( 2 4 δ 2 ω 0 2 δ 2 b 2 ω 1 2 ω 0 2 ) ] ω 2 + 2 δ i ( 1 b 2 ω 0 2 ) ω ω 0 2 b 2 .
ω 2 + 2 δ i θ 2 θ 0 2 + 2 b 2 ω 0 2 θ 2 θ 0 2 + 3 b 2 ω 0 2 α ω ω 0 2 ( θ 2 θ 0 2 ) θ 2 θ 0 2 + 3 b 2 ω 0 2 α = 0 ,
α = 1 δ 2 3 ω 0 2 ω 1 2 ( 4 ω 1 2 + b 2 )
ω SP N ± ± ψ ( θ ) 2 3 i δ ζ ( θ ) ,
ψ ( θ ) = [ ω 0 2 ( θ 2 θ 0 2 ) θ 2 θ 0 2 + 3 b 2 ω 0 2 α δ 2 ( θ 2 θ 0 2 + 2 b 2 ω 0 2 θ 2 θ 0 2 + 3 b 2 ω 0 2 α ) 2 ] 1 / 2 ,
ζ ( θ ) = 3 2 θ 2 θ 0 2 + 2 b 2 ω 0 2 θ 2 θ 0 2 + 3 b 2 ω 0 2 α ,
ψ ( θ ) | θ θ 0 [ 2 θ 0 ω 0 4 3 b 2 α ( θ θ 0 ) 4 δ 2 9 α 2 ] 1 / 2 , ζ ( θ ) | θ θ 0 1 α ,
lim θ [ ω SP N ± ( θ ) ] = ± ( ω 0 2 δ 2 ) 1 / 2 δ i = ω ± ,
ω 0 2 ( θ 1 2 θ 0 2 ) θ 1 2 θ 0 2 + 3 b 2 ω 0 2 α δ 2 ( θ 1 2 θ 0 2 + 2 b 2 ω 0 2 θ 1 2 θ 0 2 + 3 b 2 ω 0 2 α ) 2 = 0 .
θ 1 = ( θ 0 2 + b 2 2 3 α 4 δ 2 ω 0 2 ω 0 2 δ 2 { [ 1 + 16 δ 2 ( ω 0 2 δ 2 ) ( 3 α ω 0 2 4 δ 2 ) 2 ] 1 / 2 1 } ) 1 / 2 ,
θ 1 2 θ 0 2 4 δ 2 b 2 ω 0 2 ( 3 α ω 0 2 4 δ 2 ) ;
θ 1 θ 0 + 2 δ 2 b 2 θ 0 ω 0 2 ( 3 α ω 0 2 4 δ 2 ) ,
ϕ ( ω , θ ) i ω ( θ 0 θ ) + b 2 2 θ 0 ω 0 4 ω 2 ( i α ω 2 δ ) .
ω SP N ± ( θ ) i [ ± | ψ ( θ ) | 2 3 δ ζ ( θ ) ] ,
ω = ω SP N ± ( θ ) + r e i χ = i ω + r e i χ ,
ϕ ( θ , r , χ ) ( ω + i r e i χ ) ( θ 0 θ ) + b 2 2 θ 0 ω 0 4 × [ α ω 3 + 2 δ ω 2 i ( 3 α ω 2 + 4 δ ω ) r e i χ ( 3 α ω + 2 δ ) r 2 e i 2 χ + i α r 3 e i 3 χ ] ,
X ( θ , r , χ ) ( ω + r sin χ ) ( θ θ 0 ) + b 2 2 θ 0 ω 0 4 [ α ω 3 + 2 δ ω 2 + ( 3 α ω 2 + 4 δ ω ) r sin χ ( 3 α ω + 2 δ ) r 2 × cos ( 2 χ ) α r 3 sin 3 χ ] .
ω SP N ( θ 1 ) 2 3 i δ ζ ( θ 1 ) 2 δ 3 α i .
X ( θ 1 , r , χ ) b 2 θ 0 ω 0 4 ( 4 δ 3 27 α 2 + 1 2 α r 3 sin 3 χ ) .
ω SP N ± ( θ ) ± ψ ( θ ) 2 3 i δ ζ ( θ ) ,
ω = ω SP N + ( θ ) + r e i χ ψ ( θ ) 2 3 i δ ζ ( θ ) + r e i χ .
ϕ ( θ , r , χ ) [ 2 3 δ ζ ( θ ) + i ψ ( θ ) + i r e i χ ] ( θ 0 θ ) + b 2 2 θ 0 ω 0 4 [ 8 9 δ 3 ζ 2 ( θ ) [ 1 α 3 ζ ( θ ) ] + 2 δ ψ 2 ( θ ) [ α ζ ( θ ) 1 ] + i { 4 3 δ 2 ζ ( θ ) ψ ( θ ) [ 2 α ζ ( θ ) ] + α ψ 3 ( θ ) } + ( 4 δ ψ ( θ ) [ α ζ ( θ ) 1 ] + i { 3 α ψ 2 ( θ ) + 4 3 δ 3 ζ ( θ ) [ 2 α ζ ( θ ) ] } ) r e i χ + { 2 δ [ α ζ ( θ ) 1 ] + 3 i α ψ ( θ ) } r 2 e i 2 χ + i α r 3 e i 3 χ ] ,
X ( θ , r , χ ) [ r sin ( χ ) 2 3 δ ζ ( θ ) ] ( θ θ 0 ) + b 2 2 θ 0 ω 0 4 ( 8 9 δ 3 ζ 2 ( θ ) [ 1 α 3 ζ ( θ ) ] + 2 δ ψ 2 ( θ ) [ α ζ ( θ ) 1 ] + 4 δ ψ ( θ ) [ α ζ ( θ ) 1 ] r cos ( χ ) { 3 α ψ 2 ( θ ) + 4 3 δ 2 ζ ( θ ) [ 2 α ζ ( θ ) ] } r sin ( χ ) + 2 δ [ α ζ ( θ ) 1 ] r 2 cos ( 2 χ ) 3 α ψ ( θ ) r 2 sin ( 2 χ ) α r 3 sin ( 3 χ ) ) .
X ( ω SP D ± , θ ) 2 δ ( θ 1 ) ,
X ( ω SP N + , θ ) 2 27 { [ 2 δ 2 b 2 α θ 0 ω 0 4 + 3 ( θ 0 θ ) ] × [ 4 δ 2 α 2 + 6 θ 0 ω 0 4 α b 2 ( θ 0 θ ) ] 1 / 2 9 δ α ( θ 0 θ ) 4 δ 3 b 2 α 2 θ 0 ω 0 4 } ,
θ SB θ 0 4 δ 2 b 2 3 θ 0 ω 0 4 [ 27 δ 2 b 2 ( θ 0 1 ) 2 4 θ 0 ω 0 4 ] 1 / 3 × ( { [ 1 + δ 2 b 2 27 θ 0 ( θ 0 1 ) ω 0 4 ] 1 / 2 + 1 } 1 / 3 { 1 + [ δ 2 b 2 27 θ 0 ( θ 0 1 ) ω 0 4 ] 1 / 2 1 } 1 / 3 )
θ 0 4 δ 2 b 2 3 θ 0 ω 0 4 .
X ( ω SB ) = X ( ω SP D + , θ SB ) .
ω SB ξ ( θ SB ) = ( ω 0 2 δ 2 + b 2 θ SB 2 θ SB 2 1 ) 1 / 2
ω 0 ( 2 + b 2 ω 0 2 + 5 δ 2 3 ω 0 2 ) 1 / 2 ,
A ( z , t ) = I ( z , θ ) Re [ 2 π i Λ ( θ ) ] ,
Λ ( θ ) = P Res ω = ω p { i 2 π ũ ( ω ω c ) exp [ z c ϕ ( ω , θ ) ] }
I ( z , θ ) = 1 2 π Re { i P ( θ ) ũ ( ω ω c ) exp [ z c ϕ ( ω , θ ) ] d ω } .
I ( z , θ ) = I D ( z , θ ) + I 1 ( z , θ ) + I D + ( z , θ ) , 1 θ θ 1 ,
I ( z , θ ) = I D ( z , θ ) + I N ( z , θ ) + I N + ( z , θ ) + I D + ( z , θ ) , θ > θ 1 ,
I D ( z , θ ) + I D + ( z , θ ) = A s ( z , t ) + R 1 ( z , θ ) ,
I 1 ( z , θ ) = A B ( z , t ) + R 2 ( z , θ ) , 1 θ < θ 1 ,
I 1 ( z , θ 1 ) = A B ( z , t 1 ) + R 2 ( z , θ 1 ) , θ = θ 1 ,
I N ( z , θ ) + I N + ( z , θ ) = A B ( z , t ) + R 2 ( z , θ ) , θ > θ 1 .
A ( z , t ) = A s ( z , t ) + A B ( z , t ) + A c ( z , t ) + R ( z , θ ) .
A c ( z , t ) = R e [ 2 π i Λ ( θ ) ] ,
ũ ( ω ) = 0 u ( t ) e i ω t d t .
I ( z , θ , | Ω | ) = C ũ ( ω ω c ) exp [ z c ϕ ( ω , θ ) ] d ω ,
| I ( z , θ , | Ω | ) | C | | ũ ( ω ω c ) | exp [ z c X ( ω , θ ) ] | d ω | ,
X ( ω , θ ) = ω [ n r ( ω ) θ ] ω n i ( ω ) .
X ( ω , θ ) < ω [ n r ( ω ) θ ] .
X ( ω , θ ) ω ( 1 θ )
| I ( z , θ , | Ω | ) | C | ũ ( ω ω c ) | exp ( z c ω Δ ) | d ω |
lim | ω | [ X ( ω , θ ) ] = ω ( θ 1 ) .
A s ( z , t ) = 1 2 π Re ( i { 2 exp [ z c ϕ ( ω SP D + , θ ) ] ( π c z ) 1 / 2 a 0 ( ω SP D + ) × [ 1 + 0 ( z 1 ) ] + 2 exp [ z c ϕ ( ω SP D , θ ) ] ( π c z ) 1 / 2 × a 0 ( ω SP D ) [ 1 + 0 ( z 1 ) ] } ) ,
ϕ ( ω SP D ± , θ ) δ { [ 1 + η ( θ ) ] ( θ 1 ) + b 2 2 1 η ( θ ) ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } i ξ ( θ ) { θ 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } ,
ϕ ( 2 ) ( ω SP D ± , θ ) i b 2 { ± ξ ( θ ) + δ i [ 1 η ( θ ) ] } 3 ,
P 0 ( ω SP D ± , θ ) = ϕ ( 2 ) ( ω SP D ± , θ ) 2 ! i b 2 / 2 { ± ξ ( θ ) + δ i [ 1 η ( θ ) ] } 3 .
a 0 ( ω SP D ± ) = ũ ( ω SP D ± ω c ) μ [ P 0 ( ω SP D ± , θ ) ] λ / μ ũ ( ω SP D ± ω c ) 1 2 ( 2 i b 2 { ± ξ ( θ ) + δ i [ 1 η ( θ ) ] } 3 ) 1 / 2 ũ ( ω SP D ± ω c ) 1 2 b exp ( i π / 4 ) × { ξ 3 / 2 ( θ ) ± 3 2 δ i [ 1 η ( θ ) ] ξ 1 / 2 ( θ ) } ,
A s ( z , t ) 1 b [ c ξ ( θ ) 2 π z ] 1 / 2 exp ( δ z c { [ 1 + η ( θ ) ] ( θ 1 ) + b 2 2 × 1 η ( θ ) ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } ) × Re ( i { ũ ( ω SP D + ω c ) { ξ ( θ ) + 3 2 δ i [ 1 η ( θ ) ] } × exp [ i ( z c ξ ( θ ) { θ 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 + η ( θ ) ] 2 } + π 4 ) ] + ũ ( ω SP D + ω c ) { ξ ( θ ) 3 2 δ i [ 1 η ( θ ) ] } × exp [ i ( z c ξ ( θ ) { θ 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 + η ( θ ) ] 2 } + π 4 ) ] } ) ,
d ξ ( θ ) d t = c b 2 θ z ξ ( θ ) ( θ 2 1 ) 2 , d η ( θ ) d t = 2 b 2 c [ η ( θ ) 1 ] θ z ξ 2 ( θ ) ( θ 2 1 ) 2 ,
ω s = d d t ( z c ξ ( θ ) { θ 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } ) = ξ ( θ ) + b 2 θ 2 ξ ( θ ) ( θ 2 1 ) 2 × ( b 2 ξ 2 ( θ ) 5 δ 2 [ 1 η ( θ ) ] 2 { ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } 2 2 ( θ 1 ) )
ξ ( θ ) .
A B ( z , t ) = 1 2 π Re ( i { 2 exp [ z c ϕ ( ω SP 1 , θ ) ] ( π c z ) 1 / 2 × a 0 ( ω SP 1 ) [ 1 + 0 ( z 1 ) ] } ) ,
ϕ ( ω SP 1 , θ ) 1 3 [ 2 δ ζ ( θ ) 3 | ψ ( θ ) | ] ( θ 0 θ ) + b 2 54 θ 0 ω 0 4 × [ 2 δ ζ ( θ ) 3 | ψ ( θ ) | ] 2 { 2 δ [ 3 α ζ ( θ ) ] + 3 α | ψ ( θ ) | } ,
ϕ ( 2 ) ( ω SP 1 , θ ) b 2 θ 0 ω 0 4 { 2 δ [ 1 α ζ ( θ ) ] + 3 α | ψ ( θ ) | } ,
P 0 ( ω SP 1 , θ ) = ϕ ( 2 ) ( ω SP 1 , θ ) 2 ! b 2 2 θ 0 ω 0 4 { 2 δ [ 1 α ζ ( θ ) ] + 3 α | ψ ( θ ) | } .
a 0 ( ω SP 1 ) = ũ ( ω SP 1 ω c ) μ [ P 0 ( ω SP 1 , θ ) ] λ / μ ũ ( ω SP 1 ω c ) ω 0 2 b { θ 0 4 δ [ 1 α ζ ( θ ) ] + 6 α | ψ ( θ ) | } 1 / 2 ,
A B ( z , t ) ω 0 2 b ( θ 0 c π z { 4 δ [ 1 α ζ ( θ ) ] + 6 α | ψ ( θ ) | } ) 1 / 2 × R e { [ i ũ ( ω SP 1 ω c ) ] exp [ z 3 c [ 2 δ ζ ( θ ) 3 | ψ ( θ ) | ] × ( θ 0 θ + b 2 18 θ 0 ω 0 4 [ 2 δ ζ ( θ ) 3 | ψ ( θ ) | ] × { 2 δ [ 3 α ζ ( θ ) ] + 3 α | ψ ( θ ) | } ) ] } ,
| θ 0 θ | b 2 18 θ 0 ω 0 4 | 2 δ ζ ( θ ) 3 | ψ ( θ ) { 2 δ [ 3 α ζ ( θ ) ] + 3 α | ψ ( θ ) | }
ϕ ( ω SP N , θ 1 ) 2 δ 3 α ( θ 0 θ 1 + 4 δ 2 b 2 9 α θ 0 ω 0 4 ) ,
ϕ ( 3 ) ( ω , θ ) 3 i α b 2 θ 0 ω 0 4 ,
P 0 ( ω SP N , θ 1 ) = ϕ ( 3 ) ( ω SP N , θ 1 ) 3 ! i α b 2 2 θ 0 ω 0 4 .
exp ( i 0 + / 3 ) exp ( i 0 / 3 ) = 3 ,
a 0 + ( ω SP N ) = ũ ( ω SP N ω c ) μ [ P 0 ( ω SP N , θ 1 ) ] λ / μ 1 3 ũ ( ω SP N ω c ) [ 2 θ 0 ω 0 4 α b 2 ] 1 / 3 e i π / 6 .
A B ( z , t 1 ) = 1 2 π R e ( i { exp [ z c ϕ ( ω SP N , θ 1 ) ] Γ ( 1 3 ) [ a 0 + ( ω SP N ) ā 0 ( ω SP N ) ] ( c z ) 1 / 3 [ 1 + 0 ( z 1 / 3 ) ] } ) ω 0 2 π 3 Γ ( 1 3 ) [ 2 θ 0 ω 0 c α b 2 z ] 1 / 3 × exp [ 2 δ z 3 α c ( θ 0 θ 1 + 4 δ 2 b 2 9 α θ 0 ω 0 4 ) ] × R e [ i ũ ( ω SP N ω c ) ] .
θ 0 θ 1 + 4 δ 2 b 2 9 α θ 0 ω 0 4 < 2 δ 2 b 2 9 α θ 0 ω 0 4 ,
A B ( z , t ) = 1 2 π R e ( i { 2 exp [ z c ϕ ( ω SP N + , θ ) ] ( π c z ) 1 / 2 a 0 ( ω SP N + ) × [ 1 + 0 ( z 1 ) ] + 2 exp [ z c ϕ ( ω SP N , θ ) ] ( π c z ) 1 / 2 × a 0 ( ω SP N ) [ 1 + 0 ( z 1 ) ] } ) ,
ϕ ( ω SP N ± , θ ) δ ( 2 3 ζ ( θ ) ( θ θ 0 ) + b 2 θ 0 ω 0 4 { [ 1 α ζ ( θ ) ] ψ 2 ( θ ) + 4 9 δ 2 ζ 2 ( θ ) [ 1 3 α ζ ( θ ) 1 ] } ) ± i ψ ( θ ) ( θ 0 θ + b 2 2 θ 0 ω 0 4 { 4 3 δ 2 ζ ( θ ) × [ 2 α ζ ( θ ) ] + α ψ 2 ( θ ) } ) ,
ϕ ( 2 ) ( ω SP N ± , θ ) b 2 θ 0 ω 0 4 { 2 δ [ α ζ ( θ ) 1 ] ± 3 i α ψ ( θ ) } ,
P 0 ( ω SP N ± , θ ) = ϕ ( 2 ) ( ω SP N ± , θ ) 2 ! b 2 2 θ 0 ω 0 4 { 2 δ [ α ζ ( θ ) 1 ] ± 3 i α ψ ( θ ) } .
a 0 ( ω SP N ± ) = ũ ( ω SP N ± ω c ) μ [ P 0 ( ω SP N ± , θ ) ] λ / μ ũ ( ω SP N ± ω c ) ω 0 2 2 b { 2 θ 0 2 δ [ α ζ ( θ ) 1 ] ± 3 i α ψ ( θ ) } 1 / 2 .
a 0 ( ω SP N ± ) ũ ( ω SP N ± ω c ) ω 0 2 2 b [ 2 θ 0 3 α ψ ( θ ) ] 1 / 2 e ± i π / 4 .
A B ( z , t ) ω 0 2 2 b [ 2 θ 0 c 3 π α ψ ( θ ) z ] 1 / 2 × exp [ δ z c ( 2 3 ζ ( θ ) ( θ θ 0 ) + b 2 θ 0 ω 0 4 × { [ 1 α ζ ( θ ) ] ψ 2 ( θ ) + 4 9 δ 2 ζ 2 ( θ ) [ 1 3 α ζ ( θ ) 1 ] } ) ] × R e ( i { ũ ( ω SP N + ω c ) exp [ i ψ ( θ ) z c ( θ 0 θ + b 2 2 θ 0 ω 0 4 × { 4 3 δ 2 ζ ( θ ) [ 2 α ζ ( θ ) ] + α ψ 2 ( θ ) } ) + i π 4 ] + ũ ( ω SP N ω c ) exp [ i ψ ( θ ) z c ( θ 0 θ + b 2 2 θ 0 ω 0 4 × { 4 3 δ 2 ζ ( θ ) [ 2 α ζ ( θ ) ] + α ψ 2 ( θ ) } ) i π 4 ] } ) ,
ω B = 0
ω B = d d t [ z c ψ ( θ ) ( θ θ 0 b 2 2 θ 0 ω 0 4 { 4 3 δ 2 ζ ( θ ) [ 2 ζ ( θ ) ] + ψ 2 ( θ ) } ) + π 4 ] ψ ( θ ) [ 1 b 4 θ ω 0 4 θ 0 ( 9 2 + 8 δ 2 ω 0 2 ) ( θ 2 θ 0 2 + 3 b 2 ω 0 2 ) 15 δ 2 ω 0 2 ( θ 2 θ 0 2 + 2 b 2 ω 0 2 ) ( θ 2 θ 0 2 + 3 b 2 ω 0 2 ) 3 ] + b 2 θ ψ ( θ ) ( θ θ 1 ) 3 ( θ 2 θ 0 2 + 3 b 2 ω 0 2 ) 2 δ 2 ω 0 2 ( θ 2 θ 0 2 + 2 b 2 ω 0 2 ) ( θ 2 θ 0 2 + 3 b 2 ω 0 2 ) 3
ψ ( θ ) .
γ p = lim ω ω p [ ( ω ω p ) ũ ( ω ω p ) ] .
A c ( z , t ) = R e [ 2 π i Λ ( θ ) ] ,
Λ ( θ ) = p Res ω = ω p { 1 2 π i ũ ( ω ω c ) exp [ z c ϕ ( ω , θ ) ] } = i 2 π p γ p exp [ z c ϕ ( ω p , θ ) ]
Λ ( θ ) = 0 , θ < θ s ,
Λ ( θ s ) = i 4 π γ p exp [ z c ϕ ( ω p , θ s ) ] , θ = θ s ,
Λ ( θ ) = i 2 π γ p exp [ z c ϕ ( ω p , θ ) ] , θ > θ s .
A c ( z , t ) = 0 , θ < θ s ,
A c ( z , t s ) = ½ exp [ z c X ( ω p , θ s ) ] { γ p cos [ z c Y ( ω p , θ s ) ] γ p sin [ z c Y ( ω p , θ s ) ] } , θ = θ s ,
A c ( z , t ) = exp [ z c X ( ω p , θ ) ] { γ p cos [ z c Y ( ω p , θ ) ] γ p sin [ z c Y ( ω p , θ ) ] } , θ > θ s ,
X ( ω p ) = ω p n i ( ω p ) = c α ( ω p ) ,
Y ( ω p , θ ) = ω p [ n r ( ω p ) θ ] ,
z c Y ( ω p , θ ) = z c ω p [ n r ( ω p ) c t z ] = k ( ω p ) z ω p t ,
k ( ω p ) = ω p c n r ( ω p ) .
A c ( z , t ) = 0 , θ < θ s ,
A c ( z , t s ) = ½ exp [ z α ( ω p ) ] { γ cos [ k ( ω p ) z ω p t ] γ sin [ k ( ω p ) z ω p t ] } , θ = θ s ,
A c ( z , t ) = exp [ z α ( ω p ) ] { γ cos [ k ( ω p ) z ω p t ] γ sin [ k ( ω p ) z ω p t ] } θ > θ s ,
ũ ( ω ω c ) = i .
A ( z , t ) = A s ( z , t ) + A B ( z , t ) ,
A s ( z , t ) ~ 1 b [ c ξ ( θ ) 2 π z ] 1 / 2 exp ( δ z c { [ 1 + η ( θ ) ] ( θ 1 ) + b 2 2 1 η ( θ ) ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } ) × [ 2 ξ ( θ ) cos ( z c ξ ( θ ) { θ 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } + π 4 ) + 3 δ [ 1 η ( θ ) ] sin ( z c ξ ( θ ) { θ 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } + π 4 ) ] ,
A B ( z , t ) ~ ω 0 2 b ( θ 0 c 2 π z { 2 δ [ 1 α ζ ( θ ) ] + 3 α | ψ ( θ ) | } ) 1 / 2 × exp [ z 3 c [ 2 δ ζ ( θ ) 3 | ψ ( θ ) | ] × ( θ 0 θ + b 2 18 θ 0 ω 0 4 [ 2 δ ζ ( θ ) 3 | ψ ( θ ) | ] × { 2 δ [ 3 α ζ ( θ ) ] + 3 α | ψ ( θ ) | } ) ] ,
A B ( z , t 1 ) ~ Γ ( 1 3 ) 2 π 3 ( 2 θ 0 ω 0 4 c α b 2 z ) 1 / 3 × exp [ 2 δ z 3 α c ( θ 0 + 4 δ 2 b 2 9 α θ 0 ω 0 4 θ 1 ) ] ,
A B ( z , t ) ~ ω 0 2 b [ 2 θ 0 c 3 π α ψ ( θ ) z ] 1 / 2 × exp [ δ z c ( 2 3 ζ ( θ ) ( θ θ 0 ) + b 2 θ 0 ω 0 4 × { [ 1 α ζ ( θ ) ] ψ 2 ( θ ) + 4 9 δ 2 ζ 2 ( θ ) [ 1 3 α ζ ( θ ) 1 ] } ) ] × cos [ z c ψ ( θ ) ( θ 0 θ + b 2 2 θ 0 ω 0 4 × { 4 3 δ 2 ζ ( θ ) [ 2 α ζ ( θ ) ] + α ψ 2 ( θ ) } ) + π 4 ] ,
ũ ( ω ω c ) = i ω ω c .
A ( z , t ) = A s ( z , t ) + A B ( z , t ) + A c ( z , t ) ,
A s ( z , t ) ~ 1 b [ c ξ ( θ ) 2 π z ] 1 / 2 exp ( δ z c { [ 1 + η ( θ ) ] ( θ 1 ) + b 2 2 1 η ( θ ) ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } ) × ( { ξ ( θ ) [ ξ ( θ ) ω c ] 3 2 δ 2 [ 1 η 2 ( θ ) ] [ ξ ( θ ) ω c ] 2 + δ 2 [ 1 + η ( θ ) ] 2 ξ ( θ ) [ ξ ( θ ) + ω c ] 3 2 δ 2 [ 1 η 2 ( θ ) ] [ ξ ( θ ) + ω c ] 2 + δ 2 [ 1 + η ( θ ) ] 2 } × cos { z c ξ ( θ ) ( θ 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 ) + π 4 } + δ 2 { ξ ( θ ) [ 5 η ( θ ) ] 3 ω c [ 1 η ( θ ) ] [ ξ ( θ ) ω c ] 2 + δ 2 [ 1 + η ( θ ) ] 2 ξ ( θ ) [ 5 η ( θ ) ] + 3 ω c [ 1 η ( θ ) ] [ ξ ( θ ) ω c ] 2 + δ 2 [ 1 + η ( θ ) ] 2 } × sin [ z c ξ ( θ ) { θ 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 η ( θ ) ] 2 } + π 4 ] ) ,
A s ( z , t ) ~ ( b c 2 π z ) 1 / 2 1 [ 2 ( θ 1 ) ] 3 / 4 exp [ 2 δ z c ( θ 1 ) ] × [ ( b [ 2 ( θ 1 ) ] 1 / 2 ω c { b [ 2 ( θ 1 ) ] 1 / 2 ω c } 2 + 4 δ 2 b [ 2 ( θ 1 ) ] 1 / 2 + ω c { b [ 2 ( θ 1 ) ] 1 / 2 + ω c } 2 + 4 δ 2 ) × cos { b z c [ 2 ( θ 1 ) ] 1 / 2 + π 4 } + 2 δ 1 ( { b [ 2 ( θ 1 ) ] 1 / 2 ω c } 2 + 4 δ 2 1 { b [ 2 ( θ 1 ) ] 1 / 2 + ω c } 2 + 4 δ 2 ) × sin { b z c [ 2 ( θ 1 ) ] 1 / 2 + π 4 } ] ,
A s ( z , t ) ~ ( 2 b c π z ) 1 / 2 ω c [ 2 ( θ 1 ) ] 1 / 4 b 2 + 8 δ 2 ( θ 1 ) exp [ 2 δ z c ( θ 1 ) ] cos { b z c [ 2 ( θ 1 ) ] 1 / 2 + π 4 } ,
A B ( z , t ) ~ ω 0 2 ω c b { ω c 2 + 1 9 [ 3 | ψ ( θ ) | 2 δ ζ ( θ ) ] 2 } × ( θ 0 c π z { 4 δ [ 1 α ζ ( θ ) ] + 6 α | ψ ( θ ) | } ) 1 / 2 × exp [ z 3 c [ 2 δ ζ ( θ ) 3 | ψ ( θ ) | ] ( θ 0 θ + b 2 18 θ 0 ω 0 4 × [ 2 δ ζ ( θ ) 3 | ψ ( θ ) | ] { 2 δ [ 3 α ζ ( θ ) ] 3 α | ψ ( θ ) | } ) ] ,
A B ( z , t 1 ) ~ Γ ( 1 3 ) 2 π 3 ω 0 ω c ω c 2 + 4 δ 2 9 α 2 ( 2 θ 0 ω 0 c α b 2 z ) 1 / 3 × exp [ 2 δ z 3 α c ( θ θ 1 + 4 δ 2 b 2 9 α θ 0 ω 0 4 ) ] ,
A B ( z , t ) ~ ω 0 2 ω c b { [ ψ ( θ ) ω c ] 2 + 4 9 δ 2 ζ 2 ( θ ) } { [ ψ ( θ ) + ω c ] 2 + 4 9 δ 2 ζ 2 ( θ ) } [ 2 θ 0 c 3 π α ψ ( θ ) z ] 1 / 2 exp [ δ z c ( 2 3 ( θ θ 0 ) ζ ( θ ) + b 2 θ 0 ω 0 4 × { [ 1 α ζ ( θ ) ] ψ 2 ( θ ) + 4 9 δ 2 ζ 2 ( θ ) [ 1 3 α ζ ( θ ) 1 ] } ) ] × { [ ω c 2 + 4 9 δ 2 ζ 2 ( θ ) ψ 2 ( θ ) ] cos [ z c ψ ( θ ) ( θ 0 θ + b 2 2 θ 0 ω 0 4 { 4 3 δ 2 ζ ( θ ) [ 2 α ζ ( θ ) ] + α ψ 2 ( θ ) } ) + π 4 ] + 4 3 δ ζ ( θ ) ψ ( θ ) × sin [ z c ψ ( θ ) ( θ 0 θ + b 2 2 θ 0 ω 0 4 { 4 3 δ 2 ζ ( θ ) [ 2 α ζ ( θ ) ] + α ψ 2 ( θ ) } ) + π 4 ] } ,
A c ( z , t ) = 0 , θ < θ s ,
A c ( z , t s ) = ½ exp [ z α ( ω c ) ] sin [ k ( ω c ) z ω c t s ] , θ = θ s ,
A c ( z , t ) = exp [ z α ( ω c ) ] sin [ k ( ω c ) z ω c t ] , θ > θ s ,
Y ( ω SP , θ s ) = Y ( ω c , θ s ) ,
X ( ω SP , θ c ) = X ( ω c ) ,
X ( ω SP N + , θ m ) = X ( ω min )
X ( ω SP N , θ c ) = X ( ω c ) , θ c θ 0
υ c = c θ c ,
X ( ω SP D + , θ c 1 ) = X ( ω c ) , 1 < θ c 1 < θ SB .
X ( ω SP 1 , θ c 2 ) = X ( ω c ) , θ SB < θ c 2 < θ 0
υ c 1 = c θ c 1 , ω c > ω SB .
υ c 2 = c θ c 2 , ω c > ω SB .
c > υ c 1 > c θ SB > υ c 2 > c θ 0 υ c .
n ( ω ) = ( 1 b 2 ω 2 ω 0 2 + 2 δ i ω ) 1 / 2 .
n ( ω ) = ( ω 2 ω 1 2 + 2 δ i ω ω 2 ω 0 2 + 2 δ i ω ) 1 / 2 = [ ( ω ω + ) ( ω ω ) ( ω ω + ) ( ω ω ) ] 1 / 2 ,
ω 1 2 = ω 0 2 + b 2 .
ω ± = ± ( ω 1 2 δ 2 ) 1 / 2 δ i ,
ω ± = ± ( ω 0 2 δ 2 ) 1 / 2 δ i
X ( ω + i ω , θ ) = X ( ω + i ω , θ ) ,
Y ( ω + i ω , θ ) = Y ( ω + i ω , θ ) ,
n ( ω ) = n r ( ω ) + i n i ( ω ) ,
α ( ω ) = ω c n i ( ω )
n ( ω ) = [ 1 + b 4 2 b 2 ( ω 2 ω 2 ω 0 2 2 δ ω ) ( ω 2 ω 2 ω 0 2 2 δ ω ) 2 + 4 ω 2 ( ω + δ ) 2 ] 1 / 4 e i ζ / 2 ,
ζ ( ω ) = arg [ n 2 ( ω ) ] = tan 1 [ 2 ω ( ω + δ ) b 2 ( ω 2 ω 2 ω 0 2 2 δ ω ) 2 b 2 ( ω 2 ω 2 ω 0 2 2 δ ω ) + 4 ω 2 ( ω + δ ) 2 ] ,
X ( ω , θ ) = { ω [ n r ( ω ) θ ] + ω n i ( ω ) } ,
Y ( ω , θ ) = ω [ n r ( ω ) θ ] ω n i ( ω ) .
n r ( ω ) = [ 1 + b 4 + 2 b 2 ( ω 0 2 ω 2 ) ( ω 2 ω 0 2 ) 2 + 4 δ 2 ω 2 ] 1 / 4 cos [ 1 2 ζ ( ω ) ] ,
n i ( ω ) = [ 1 + b 4 + 2 b 2 ( ω 0 2 ω 2 ) ( ω 2 ω 0 2 ) 2 + 4 δ 2 ω 2 ] 1 / 2 sin [ 1 2 ζ ( ω ) ] ,
X ( ω ) = ω n i ( ω )
ω min ω 0 [ 1 + 2 δ 2 ω 0 2 ( 1 ω 0 2 b 2 ) ] ,
lim | ω | [ X ( ω , θ ) ] = ω ( θ 1 ) .
n ( ω i δ ) = [ 1 + b 2 ω 0 2 δ 2 ω 2 ] 1 / 2 ,
X ( ω i δ , θ ) = δ [ ( 1 + b 2 ω 0 2 δ 2 ω 2 ) 1 / 2 θ ]
X ( ω i δ , θ ) = δ θ i ω ( b 2 ω 2 + δ 2 ω 0 2 1 ) 1 / 2
ω = ( ω 1 2 δ 2 ) 1 / 2 i δ + r e i ψ .
n ( r , ψ ) [ 2 ( ω 1 2 δ 2 ) 1 / 2 r ] 1 / 2 b e i ψ / 2 .
ω = ( ω 0 2 δ 2 ) 1 / 2 i δ + R e i ξ .
n ( R , ξ ) b [ 2 ( ω 0 2 δ 2 ) 1 / 2 R ] 1 / 2 exp [ i ( π ξ ) / 2 ] .
X ( r , ψ , θ ) δ { [ 2 ( ω 1 2 δ 2 ) 1 / 2 r ] 1 / 2 b cos ( ψ 2 ) θ } ( ω 1 2 δ 2 ) 1 / 2 [ 2 ( ω 1 2 δ 2 ) 1 / 2 r ] 1 / 2 b sin ( ψ 2 ) ,
X ( R , ξ , θ ) δ { b [ 2 ( ω 0 2 δ 2 ) 1 / 2 R ] 1 / 2 cos ( π ξ 2 ) θ } ( ω 0 2 δ 2 ) 1 / 2 b [ 2 ( ω 0 2 δ 2 ) 1 / 2 R ] 1 / 2 sin ( π ξ 2 ) .

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