Abstract

The uniform asymptotic description of electromagnetic pulse propagation in a single-resonance Lorentz medium is presented. The modern asymptotic theory used here relies on Olver’s saddle-point method [ Stud. Appl. Math. Rev. 12, 228 ( 1970)] together with the uniform asymptotic theory of Handelsman and Bleistein [ Arch. Ration. Mech. Anal. 35, 267 ( 1969)] when two saddle points are at infinity (for the Sommerfeld precursor), the uniform asymptotic theory of Chester et al. [ Proc. Cambridge Philos. Soc. 53, 599 ( 1957)] for two neighboring saddle points (for the Brillouin precursor), and the uniform asymptotic theory of Bleistein [ Commun. Pure Appl. Math. 19, 353 ( 1966)] for a saddle point and nearby pole singularity (for the signal arrival). Together with the recently derived approximations for the dynamical saddle-point evolution, which are accurate over the entire space–time domain of interest, the resultant asymptotic expressions provide a complete, uniformly valid description of the entire dynamic field evolution in the mature dispersion limit. Specific examples of the delta-function pulse and the unit-step-function-modulated signal are considered.

© 1989 Optical Society of America

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References

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  1. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
    [Crossref]
  2. L. Brillouin, “Über die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
    [Crossref]
  3. L. Brillouin, Wave Propagation and Group Velicity (Academic, New York, 1960).
  4. K. E. Oughstun, “Propagation of optical pulses in dispersive media,” doctoral dissertation (University of Rochester, Rochester, N.Y., 1978); University Microfilms, Ann Arbor, Mich. (1978).
  5. K. E. Oughstun, 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]
  6. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 5.18.
  7. H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952).
  8. L. Rosenfeld, Theory of Electrons (Dover, New York, 1965).
  9. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. I.
  10. M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge U. Press, London, 1970).
  11. F. W. J. Olver, “Why steepest descents?” Stud. Appl. Math. Rev. 12, 228–247 (1970).
  12. R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
    [Crossref]
  13. C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
    [Crossref]
  14. N. Bleistein, “Uniform asymptotic expansion of integrals with stationary point near algebraic singularity,” Commun. Pure Appl. Math. 19, 353–370 (1966).
    [Crossref]
  15. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,”J. Math. Mech. 17, 533–559 (1967).
  16. H. Baerwald, “Über die Fortpflanzung von Signalen in disperdierenden Medien,” Ann. Phys. 7, 731–760 (1930).
    [Crossref]
  17. 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]
  18. 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]

1988 (1)

1982 (1)

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 (1)

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]

1970 (1)

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

1969 (1)

R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
[Crossref]

1967 (1)

N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,”J. Math. Mech. 17, 533–559 (1967).

1966 (1)

N. Bleistein, “Uniform asymptotic expansion of integrals with stationary point near algebraic singularity,” Commun. Pure Appl. Math. 19, 353–370 (1966).
[Crossref]

1957 (1)

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[Crossref]

1930 (1)

H. Baerwald, “Über die Fortpflanzung von Signalen in disperdierenden Medien,” Ann. Phys. 7, 731–760 (1930).
[Crossref]

1914 (2)

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 disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

Baerwald, H.

H. Baerwald, “Über die Fortpflanzung von Signalen in disperdierenden Medien,” Ann. Phys. 7, 731–760 (1930).
[Crossref]

Bleistein, N.

R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
[Crossref]

N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,”J. Math. Mech. 17, 533–559 (1967).

N. Bleistein, “Uniform asymptotic expansion of integrals with stationary point near algebraic singularity,” Commun. Pure Appl. Math. 19, 353–370 (1966).
[Crossref]

Brillouin, L.

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

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

Chester, C.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[Crossref]

Friedman, B.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[Crossref]

Handelsman, R. A.

R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
[Crossref]

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).

Nussenzveig, H. M.

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

Olver, F. W. J.

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

Oughstun, K. E.

K. E. Oughstun, 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, 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, “Propagation of optical pulses in dispersive media,” doctoral dissertation (University of Rochester, Rochester, N.Y., 1978); University Microfilms, Ann Arbor, Mich. (1978).

Rosenfeld, L.

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

Sherman, G. C.

K. E. Oughstun, 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, 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]

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), Sec. 5.18.

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]

Ursell, F.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[Crossref]

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]

Ann. Phys. (3)

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 disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

H. Baerwald, “Über die Fortpflanzung von Signalen in disperdierenden Medien,” Ann. Phys. 7, 731–760 (1930).
[Crossref]

Arch. Ration. Mech. Anal. (1)

R. A. Handelsman, N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
[Crossref]

Commun. Pure Appl. Math. (1)

N. Bleistein, “Uniform asymptotic expansion of integrals with stationary point near algebraic singularity,” Commun. Pure Appl. Math. 19, 353–370 (1966).
[Crossref]

J. Math. Mech. (1)

N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,”J. Math. Mech. 17, 533–559 (1967).

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

Phys. Rev. Lett. (1)

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]

Proc. Cambridge Philos. Soc. (1)

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[Crossref]

Radio Sci. (1)

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. (1)

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

Other (7)

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

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

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

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

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

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

K. E. Oughstun, “Propagation of optical pulses in dispersive media,” doctoral dissertation (University of Rochester, Rochester, N.Y., 1978); University Microfilms, Ann Arbor, Mich. (1978).

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Figures (12)

Fig. 1
Fig. 1

Deformed contour of integration P(θ) through the relevant saddle points of ϕ(ω, θ). The dashed curves indicate the isotimic contours of X(ω, θ) = Re[ϕ(ω, θ)] 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. 2
Fig. 2

Uniform asymptotic description of the dynamic evolution of the Sommerfeld precursor field As(z, t) for an input delta-function pulse at a propagation distance of z = 1 × 10−4 cm in a highly absorptive and dispersive medium.

Fig. 3
Fig. 3

Uniform asymptotic description of the dynamic evolution of the Sommerfeld precursor field As(z, t) for an input unit-step-function-modulated signal with carrier frequency ωc = 1 × 1016/sec at a propagation distance of z = 1 × 10−4 cm in a highly absorptive and dispersive medium.

Fig. 4
Fig. 4

Uniform asymptotic description of the dynamic evolution of the Brillouin precursor field AB(z, t) for an input delta-function pulse at a propagation distance of z = 1 × 10−4 cm in a highly absorptive and dispersive medium.

Fig. 5
Fig. 5

Uniform asymptotic description of the dynamic evolution of the Brillouin precursor field AB(z, t) for an input unit-step-function-modulated signal with carrier frequency ωc = 1 × 1016/sec at a propagation distance of z = 1 × 10−3 cm in a highly absorptive and dispersive medium.

Fig. 6
Fig. 6

Interaction of the near saddle point with a simple pole singularity at ω = ωp >0. The shaded area indicates the region of the complex ω plane wherein X(ωSP, θ) > X(ω, θ), where ωSP = ωSP1 for 1 < θ < θ1, ωSP = ωSPN at θ = θ, or ωSPN+ for θ > θ1.

Fig. 7
Fig. 7

Uniform asymptotic description of the dynamic evolution of the pole contribution Ac(z, t) for the unit-step-function-modulated signal with signal frequency ωc = 1 × 1016/sec at a propagation distance of z = 1 × 10−3 cm in a highly absorptive and dispersive medium.

Fig. 8
Fig. 8

Interaction of the distant saddle point with a simple pole singularity at ω = ωp ≥ (ω12δ2)1/2 with finite ωp. The shaded area indicates the region of the complex ω plane wherein the inequality X(ωSPD+, θ) > X(ω, θ) is satisfied.

Fig. 9
Fig. 9

Uniform asymptotic description of the dynamic evolution of (a) the Brillouin precursor field AB(z, t), (b) the pole contribution Ac(z, t), and (c) the total propagated field A(z, t) = As(z, t) + AB(z, t) + Ac(z, t) for an input unit-step-function-modulated signal with carrier frequency ωc = 1.0 × 1016/sec at an observation distance of z = 100 cm in a weakly dispersive medium. Notice that the resonance peaks in AB(z, t) and Ac(z, t) cancel each other when added together to form the total field A(z, t).

Fig. 10
Fig. 10

Uniform asymptotic description of the dynamic evolution of the total propagated field A(z, t) for an input unit-step-function-modulated signal with carrier frequency ωc = 1.0 × 1016/sec at a propagation distance of z = 1 × 10−3 cm in a highly absorptive and dispersive medium.

Fig. 11
Fig. 11

Contours of integration ij to which the path P(θ) may be mapped.

Fig. 12
Fig. 12

Interaction of a first-order saddle point with a simple pole singularity of the integrand.

Equations (234)

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A ( z , t ) = 1 2 π C f ¯ ( ω ) exp [ z c ϕ ( ω , θ ) ] d ω ,
f ˜ ( ω ) = - f ( t ) e i ω t d t
ϕ ( ω , θ ) = i ω [ n ( ω ) - θ ] ,
θ = c t z
n ( ω ) = ( 1 - b 2 ω 2 - ω 0 2 + 2 i δ ω ) 1 / 2 .
f ( t ) = u ( t ) sin ( ω c t )
A ( z , t ) = 1 2 π Re { i i a - i a + u ˜ ( ω - ω c ) exp [ z c ϕ ( ω , θ ) ] d ω }
f ( t ) = δ ( t ) ,
f ˜ ( ω ) = 1
A ( z , t ) = 1 2 π Re { i a - i a + exp [ z c ϕ ( ω , θ ) ] d ω } .
u ( t ) = { 0 for t < 0 1 for t > 0 ,
u ˜ ( ω ) = 0 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 }
A ( z , t ) = I ( z , θ ) - Re [ 2 π i Λ ( θ ) ] ,
Λ ( θ ) = p Res ω = ω p { i 2 π u ˜ ( ω - ω c ) exp [ z c ϕ ( ω , θ ) ] }
I ( z , θ ) = 1 2 π Re { i P ( θ ) u ˜ ( ω - ω c ) exp [ z c ϕ ( ω , θ ) ] d ω } .
ω SP D ± ± 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 ) .
ω ± = ± ( ω 1 2 - δ 2 ) 1 / 2 - δ i
ω 1 = ( ω 0 2 + b 2 ) 1 / 2 .
ϕ ( ω SP D ± , θ ) - δ { [ 1 + η ( θ ) ] ( θ - 1 ) + b 2 [ 1 - η ( θ ) ] / 2 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 .
ω SP N ± ( θ ) i [ ± ψ ( θ ) - ( 2 / 3 ) δ ζ ( θ ) ] ,             1 θ < θ 1 ,
ω SP N ( θ 1 ) - ( 2 δ / 3 α ) i ,             θ = θ 1 ,
ω SP N ± ( θ ) ± ψ ( θ ) - ( 2 / 3 ) i δ ζ ( θ ) ,             θ > θ 1 ,
ψ ( θ ) = [ ω 0 2 ( θ 2 - θ 0 2 ) θ 2 - θ 0 2 + 3 α b 2 ω 0 2 - δ 2 ( θ 2 - θ 0 2 + 2 b 2 ω 0 2 θ 2 - θ 0 2 + 3 α b 2 ω 0 2 ) 2 ] 1 / 2 ,
ζ ( θ ) = 3 2 θ 2 - θ 0 2 + 2 b 2 ω 0 2 θ 2 - θ 0 2 + 3 α b 2 ω 0 2 ,
α = 1 - δ 2 2 ω 0 2 ω 1 2 ( 4 ω 1 2 + b 2 ) .
θ 0 = n ( 0 ) = ( 1 + b 2 ω 0 2 ) 1 / 2 ,
θ 1 θ 0 + 2 δ 2 b 2 θ 0 ω 0 2 ( 3 α ω 0 2 - 4 δ 2 ) .
ω ± = ± ( ω 0 2 - δ 2 ) 1 / 2 - δ i
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 ) }
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 SP N , θ 1 ) 3 i α b 2 θ 0 ω 0 4 ,
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 ) + / δ 2 ζ 2 ( θ ) [ α ζ ( θ ) - 1 ] } ) ± i ψ ( θ ) ( θ 0 - θ + b 2 2 θ 0 ω 0 4 × { / 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 ) }
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 ( z , θ ) ,
I 1 ( z , θ ) = A B ( z , t ) + R ( z , θ ) ,             1 θ θ 1 ,
I N - ( z , θ ) + I N + ( z , θ ) = A B ( z , t ) + R ( z , θ ) ,             θ > θ 1 ,
I D - ( z , θ ) + I D + ( z , θ ) = A s ( z , t ) + C D ± ( z , t ) + R ( z , θ ) ,
I 1 ( z , θ ) = A B ( z , t ) + C 1 ( z , t ) + R ( z , θ ) ,             1 θ θ 1 ,
I N - ( z , θ ) + I N + ( z , θ ) = A B ( z , t ) + C N + ( z , t ) + R ( 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 ) = - Re [ 2 π i Λ ( θ ) ] + C D - ( z , t ) + C D + ( z , t ) + C 1 ( z , t ) ,             1 θ θ 1 ,
A c ( z , t ) = - Re [ 2 π i Λ ( θ ) ] + C D - ( z , t ) + C D + ( z , t ) + C N - ( z , t ) + C N + ( z , t ) ,             θ > θ 1 .
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 ) .
ω S B ξ ( θ S B ) ω 0 ( 2 + b 2 ω 0 2 + 5 δ 2 3 ω 0 2 ) 1 / 2 .
A s ( z , t ) = Re ( exp [ - i z c β ( θ ) ] [ 2 α ( θ ) exp ( - i π 2 ) ] ν × { γ 0 J ν [ z c α ( θ ) ] + 2 α ( θ ) exp ( - i π 2 ) J ν + 1 [ z c α ( θ ) ] } ) + R ( z , θ )
α ( θ ) ξ ( θ ) { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ,
β ( θ ) - i δ { [ 1 + η ( θ ) ] ( θ - 1 ) + ( 1 / 2 ) b 2 [ 1 - η ( θ ) ] ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ,
γ 0 ( θ ) ( 1 2 ) 1 + ν ξ 1 / 2 ( θ ) b × ( ξ ( θ ) { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ) ( 1 / 2 ) - ν × ( u ˜ ( ω SP D + - ω c ) { ξ ( θ ) + ( 3 / 2 ) δ i [ 1 - η ( θ ) ] } + ( - 1 ) 1 + ν u ˜ ( ω SP D - - ω c ) { ξ ( θ ) - ( 3 / 2 ) δ i [ 1 - η ( θ ) ] } ) ,
γ 1 ( θ ) ( 1 2 ) 2 + ν ξ 1 / 2 ( θ ) b × ( ξ ( θ ) { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ) - [ ( 1 / 2 ) + ν ] × ( u ˜ ( ω SP D + - ω c ) { ξ ( θ ) + ( 3 / 2 ) δ i [ 1 - η ( θ ) ] } - ( - 1 ) 1 + ν u ˜ ( ω SP D - - ω c ) { ξ ( θ ) - ( 3 / 2 ) δ i [ 1 - η ( θ ) ] } ) .
A s ( z , t ) ~ ξ ( θ ) 2 b { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } 1 / 2 × exp ( - δ z c { [ 1 + η ( θ ) ] ( θ - 1 ) + ( 1 / 2 ) b 2 [ 1 - η ( θ ) ] ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ) × Re exp ( - i π 2 ν ) [ u ˜ ( ω SP D + - ω c ) { ξ ( θ ) + ( 3 / 2 ) δ i [ 1 - η ( θ ) } + ( - 1 ) 1 + ν u ˜ ( ω SP D - - ω c ) { ξ ( θ ) - ( 3 / 2 ) δ i [ 1 - η ( θ ) ] } ) × J ν ( z c ξ ( θ ) { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ) + exp ( - i π 2 ) ( u ˜ ( ω SP D + - ω c ) { ξ ( θ ) + ( 3 / 2 ) δ i [ 1 - η ( θ ) ] } - ( - 1 ) 1 + ν u ˜ ( ω SP D + - ω c ) { ξ ( θ ) + ( 3 / 2 ) δ i [ 1 - η ( θ ) ] } ) × J ν + 1 ( z c ξ ( θ ) { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] } ) ]
A s ( z , t ) ~ ξ ( θ ) 2 b { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } 1 / 2 × exp ( - δ z c { [ 1 + η ( θ ) ] ( θ + 1 ) + ( 1 / 2 ) b 2 [ 1 - η ( θ ) ] ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ) × [ 2 ξ ( θ ) J - 1 ( z c ξ ( θ ) { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ) + 3 δ [ 1 - η ( θ ) ] J 0 ( z c ξ ( θ ) { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ) ] .
ω 0 = 4.0 × 10 16 / sec , b 2 = 20.0 × 10 32 / sec 2 , δ = 0.28 × 10 16 / sec ,
u ˜ ( ω SP D ± - ω c ) i ± ξ ( θ ) - ω c - δ i [ 1 + η ( θ ) ] .
A s ( z , t ) ~ ξ ( θ ) 2 b { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } 1 / 2 × exp ( - δ z c { [ 1 + η ( θ ) ] ( θ - 1 ) + ( 1 / 2 ) b 2 [ 1 - η ( θ ) ] ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ) × [ 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 } × J 0 ( z c ξ ( θ ) { θ - 1 + b 2 / 2 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 } × J 1 ( z c ξ ( θ ) { θ - 1 + b 2 / 2 ξ 2 ( θ ) + δ 2 [ 1 - η ( θ ) ] 2 } ) ]
ϕ ( ω SP 1 , θ ) [ 2 3 δ ζ ( θ ) - ψ ( θ ) ] ( θ 0 - θ ) - b 2 2 θ 0 ω 0 4 × [ - δ ζ ( θ ) + ψ ( θ ) ] 2 [ α δ ζ ( θ ) - α ψ ( θ ) - 2 δ ] ,
ϕ ( ω SP 2 , θ ) [ 2 3 δ ζ ( θ ) + ψ ( θ ) ] ( θ 0 - θ ) - b 2 2 θ 0 ω 0 4 × [ δ ζ ( θ ) + ψ ( θ ) ] 2 [ α δ ζ ( θ ) - α ψ ( θ ) - 2 δ ] ,
A B ( z , t ) = - Re [ exp [ z c α 0 ( θ ) ] ( ( c z ) 1 / 3 exp ( - i 2 π 3 ) × A i [ α 1 ( θ ) exp ( - i 2 π 3 ) ( z c ) 2 / 3 ] × { 1 2 [ u ˜ ( ω SP 1 - ω c ) h 1 ( θ ) + u ˜ ( ω SP 2 - ω c ) h 2 ( θ ) ] + O ( 1 z ) } + ( c z ) 2 / 3 exp ( - i 4 π 3 ) A i ( 1 ) [ α 1 ( θ ) exp ( - i 2 π 3 ) ( z c ) 2 / 3 ] × { 1 2 α 1 1 / 2 ( θ ) [ u ˜ ( ω SP 1 - ω c ) h 1 ( θ ) - u ˜ ( ω SP 2 - ω c ) h 2 ( θ ) ] + O ( 1 z ) } ) ]
α 0 ( θ ) - 2 3 δ ζ ( θ ) ( θ - θ 0 ) - δ b 2 θ 0 ω 0 4 { ψ ( θ ) 2 [ α ζ ( θ ) - 1 ] + 4 9 δ 2 ζ 2 ( θ ) [ 1 3 α ζ ( θ ) - 1 ] } ,
α 1 1 / 2 ( θ ) ψ ( θ ) ( 3 2 { ( θ - θ 0 ) + b 2 θ 0 ω 0 4 × [ ¾ α ψ ( θ ) 2 + α δ 2 ζ 2 ( θ ) - 2 δ 2 ζ ( θ ) ] } ) 1 / 3 ,
h 1 , 2 ( θ ) ( { 2 θ 0 ω 0 4 b 2 3 α ψ ( θ ) ± 2 δ [ 1 - α ζ ( θ ) ] } 3 ψ ( θ ) { 3 2 ( θ - θ 0 ) + b 2 θ 0 ω 0 4 [ 3 4 α ψ ( θ ) 2 + α δ 2 ζ 2 ( θ ) - 2 δ 2 ζ ( θ ) ] } ) 1 / 6
h ( θ 1 ) lim θ θ 1 - ( h 1 , 2 ( θ ) ) ( - 2 θ 0 ω 0 4 3 i α b 2 ) 1 / 3 .
lim θ θ 1 - { arg [ h 1 , 2 ( θ ) ] } = α ¯ + ,
arg [ h 1 , 2 ( θ ) ] = π / 6
lim θ θ 1 - { arg [ α 1 1 / 2 ( θ ) ] } = α ¯ 12 - α ¯ + ,
lim θ θ 1 - { arg [ α 1 1 / 2 ( θ ) ] } = π / 3.
arg [ α 1 1 / 2 ( θ ) ] = π / 3
A B ( z , t ) ~ exp [ z c α 0 ( θ ) ] ( 1 2 ( c z ) 1 / 3 Re { i [ u ˜ ( ω SP 1 - ω c ) h 1 ( θ ) + u ˜ ( ω SP 2 - ω c ) h 2 ( θ ) ] } A i [ α 1 ( θ ) ( z c ) 2 / 3 ] - 1 2 α 1 ( θ ) 1 / 2 ( c z ) 2 / 3 Re { i [ u ˜ ( ω SP 1 - ω c ) h 1 ( θ ) = u ˜ ( ω SP 2 - ω c ) h 2 ( θ ) ] } A i ( 1 ) [ α 1 ( θ ) ( z c ) 2 / 3 ] )
A B ( z , t 1 ) ~ ω 0 2 π 3 Γ ( 1 3 ) ( 2 θ 0 ω c c α b 2 z ) 1 / 3 Re [ i u ˜ ( ω SP N - ω c ) ] × exp [ 2 δ z 3 α c ( θ 0 - θ 1 + 4 δ 2 b 2 9 α θ 0 ω 0 4 ) ]
A B ( z , t ) ~ exp [ z c α 0 ( θ ) ] 1 2 ( c z ) 1 / 3 h ( θ 1 ) × Re { i [ u ˜ ( ω SP 1 - ω c ) + u ˜ ( ω SP 2 - ω c ) ] } A i [ α 1 ( θ ) ( z c ) 2 / 3 ]
A B ( z , t ) = - Re [ exp [ z c α 0 ( θ ) ] ( ( c z ) 1 / 3 exp ( - i 2 π 3 ) × A i [ α 1 ( θ ) exp ( - i 2 π 3 ) ( z c ) 2 / 3 ] { 1 2 [ u ˜ ( ω SP N + - ω c ) h + ( θ ) + u ˜ ( ω SP N - - ω c ) h - ( θ ) ] + O ( 1 z ) } + ( c z ) 2 / 3 exp ( - i 4 π 3 ) × A i ( 1 ) [ α 1 ( θ ) exp ( - i 2 π 3 ) ( z c ) 2 / 3 ] { 1 2 α 1 / 2 ( θ ) × [ u ˜ ( ω SP N + - ω c ) h + ( θ ) - u ˜ ( ω SP N - - ω c ) h - ( θ ) ] + O ( 1 z ) } ) ]
α 0 ( θ ) - δ ( 2 3 ζ ( θ ) ( θ - θ 0 ) + b 2 θ 0 ω 0 4 { [ 1 - α ζ ( θ ) ] ψ 2 ( θ ) + / δ 2 ζ 2 ( θ ) [ α ζ ( θ ) - 1 ] } ) ,
α 1 1 / 2 ( θ ) [ - 3 2 i ψ ( θ ) ( θ - θ 0 - b 2 2 θ 0 ω 0 4 × { / δ 2 ζ ( θ ) [ 2 - α ζ ( θ ) ] + α ψ 2 ( θ ) } ) ] 1 / 3 ,
h ± ( θ ) [ [ i 2 θ 0 ω 0 4 3 α b 2 ψ ( θ ) ] 3 [ - 3 2 i ψ ( θ ) ] ( θ - θ 0 - b 2 2 θ 0 ω 0 4 × { / δ 2 ζ ( θ ) [ 2 - α ζ ( θ ) ] + α ψ 2 ( θ ) } ) ] 1 / 6
h ( θ 1 ) lim θ θ 1 + [ h ± ( θ ) ] ( - 2 θ 0 ω 0 4 3 i α b 2 ) 1 / 3 .
lim θ 1 + { arg [ h ± ( θ ) ] } = α ¯ + ,
arg [ h ± ( θ ) ] = π / 6
lim θ θ 1 + { arg [ α 1 1 / 2 ( θ ) ] } = α ¯ 12 - α ¯ + ,
lim θ θ 1 + { arg [ α 1 1 / 2 ( θ ) ] } = - π / 6.
arg [ α 1 1 / 2 ( θ ) ] = - π / 6
A B ( z , t ) ~ exp [ z c α 0 ( θ ) ] ( 1 2 ( c z ) 1 / 3 Re { i [ u ˜ ( ω SP N + - ω c ) h + ( θ ) + u ˜ ( ω SP N - - ω c ) h - ( θ ) ] } A i [ - α 1 ( θ ) ( z c ) 2 / 3 ] + 1 2 α 1 ( θ ) 1 / 2 ( c z ) 2 / 3 × Re [ u ˜ ( ω SP N + - ω c ) h + ( θ ) - u ˜ ( ω SP N - - ω c ) h - ( θ ) ] × A i ( 1 ) [ - α 1 ( θ ) ( z c ) 2 / 3 ] )
A B ( z , t ) ~ exp [ z c α 0 ( θ ) ] 1 2 ( c z ) 1 / 3 h ( θ 1 ) × Re { i [ u ˜ ( ω SP N + - ω c ) + u ˜ ( ω SP N - - ω c ) ] } × A i [ - α 1 ( θ ) ( z c ) 2 / 3 ]
θ 2 = 2 θ 1 - θ 0 .
A B ( z , t ) ~ ω 0 2 2 b ( c z ) 1 / 3 exp [ z c α 0 ( θ ) ] × [ ( { 2 θ 0 3 α ψ ( θ ) + 2 δ [ 1 - α ζ ( θ ) ] } 1 / 2 + { 2 θ 0 3 α ψ ( θ ) - 2 δ [ 1 - α ζ ( θ ) ] } 1 / 2 ) × α 1 ( θ ) 1 / 4 A i [ α 1 ( θ ) ( z c ) 2 / 3 ] - ( { 2 θ 0 3 α ψ ( θ ) + 2 δ [ 1 - α ζ ( θ ) ] } 1 / 2 - { 2 θ 0 3 α ψ ( θ ) - 2 δ [ 1 - α ζ ( θ ) ] } 1 / 2 ) × ( c / z ) 1 / 3 α 1 ( θ ) 1 / 4 A i ( 1 ) [ α 1 ( θ ) ( z c ) 2 / 3 ] ]
A B ( z , t ) ~ ( 2 θ 0 ω 0 4 c 3 α b 2 z ) 1 / 3 A i [ α 1 ( θ ) ( z c ) 2 / 3 ] × exp [ 2 δ z 3 α c ( θ 0 + 4 δ 2 b 2 9 α θ 0 ω 0 4 - 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 ) ~ ( 2 θ 0 ω 0 4 c 3 α b 2 z ) 1 / 3 A i [ - α 1 ( θ ) ( z c ) 2 / 3 ] × exp [ - 2 δ z 3 α c ( θ - θ 0 - 4 δ 2 b 2 9 α θ 0 ω 0 4 ) ]
A B ( z , t ) ~ ω 0 2 b ( 2 θ 0 3 α ψ ( θ ) ) 1 / 2 ( c z ) 1 / 3 α 1 ( θ ) 1 / 4 × exp [ z c α 0 ( θ ) ] A i [ - α 1 ( θ ) ( z c ) 2 / 3 ]
u ˜ ( ω SP 1 - ω c ) ψ ( θ ) - ( 2 / 3 ) δ ζ ( θ ) - i ω c ω c 2 + [ ψ ( θ ) - ( 2 / 3 ) δ ζ ( θ ) ] 2 ,
u ˜ ( ω SP 2 - ω c ) ψ ( θ ) + ( 2 / 3 ) δ ζ ( θ ) - i ω c ω c 2 + [ ψ ( θ ) + ( 2 / 3 ) δ ζ ( θ ) ] 2
u ˜ ( ω SP N ± - ω c ) - ( 2 / 3 ) δ ζ ( θ ) + i [ ± ψ ( θ ) - ω c ] [ ± ψ ( θ ) - ω c ] 2 + ( 4 / 9 ) δ 2 ζ 2 ( θ )
A B ( z , t ) ~ ω 0 2 ω c 2 b ( c z ) 1 / 3 exp [ ( z c ) α 0 ( θ ) ] × [ ( 1 ω 2 + [ ψ ( θ ) - ( 2 / 3 ) δ ζ ( θ ) ] 2 { 2 θ 0 3 α ψ ( θ ) + 2 δ [ 1 - α ζ ( θ ) ] } 1 / 2 + 1 ω 2 + [ ψ ( θ ) + ( 2 / 3 ) δ ζ ( θ ) ] 2 { 2 θ 0 3 α ψ ( θ ) - 2 δ [ 1 - α ζ ( θ ) ] } 1 / 2 ) × α 1 ( θ ) 1 / 4 A i [ α 1 ( θ ) ( z c ) 2 / 3 ] - ( 1 ω c 2 + [ ψ ( θ ) - ( 2 / 3 ) δ ζ ( θ ) ] 2 × { 2 θ 0 3 α ψ ( θ ) + 2 δ [ 1 - α ζ ( θ ) ] } 1 / 2 - 1 ω c 2 + [ ψ ( θ ) + ( 2 / 3 ) δ ζ ( θ ) ] 2 × { 2 θ 0 3 α ψ ( θ ) - 2 δ [ 1 - α ζ ( θ ) ] } 1 / 2 ) ( c / z ) 1 / 3 α 1 ( θ ) 1 / 4 × A i ( 1 ) [ α 1 ( θ ) ( z c ) 2 / 3 ] ]
A B ( z , t ) ~ ω 0 ω c ω c 2 + 4 δ 2 9 α 2 ( 2 θ 0 ω 0 c 3 α b 2 z ) 1 / 3 A i [ α 1 ( θ ) ( z c ) 2 / 3 ] × exp [ 2 δ z 3 α c ( θ 0 + 4 δ 2 b 2 9 α θ 0 ω 0 4 - 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 0 - θ 1 + 4 δ 2 b 2 9 α θ 0 ω 0 4 ) ]
A B ( z , t ) ~ ω 0 ω c ω c 2 + 4 δ 2 9 α 2 ( 2 θ 0 ω 0 c 3 α b 2 z ) 1 / 3 A i [ - α 1 ( θ ) ( z c ) 2 / 3 ] × exp [ - 2 δ z 3 α c ( θ - θ 0 - 4 δ 2 b 2 9 α θ 0 ω 0 4 ) ]
A B ( z , t ) ~ - ω 0 2 b [ θ 0 6 α ψ ( θ ) ] 1 / 2 ( c z ) 1 / 3 exp [ z c α 0 ( θ ) ] × ( 2 3 ξ ζ ( θ ) α 1 ( θ ) 1 / 4 { 1 [ ψ ( θ ) + ω c ] 2 + ( 4 / 9 ) δ 2 ζ 2 ( θ ) - 1 [ ψ ( θ ) - ω c ] 2 + ( 4 / 9 ) δ 2 ζ 2 ( θ ) } A i [ - α 1 ( θ ) ( z c ) 2 / 3 ] - 1 α 1 ( θ ) 1 / 4 { ψ ( θ - ω c ) [ ψ ( θ ) - ω c ] 2 + ( 4 / 9 ) δ 2 ζ 2 ( θ ) - ψ ( θ ) + ω c [ ψ ( θ ) + ω c ] 2 + ( 4 / 9 ) δ 2 ζ 2 ( θ ) } ( c z ) 1 / 3 × A i ( 1 ) [ 1 - α 1 ( θ ) ( z c ) 2 / 3 ] )
ϕ ( w p , θ ) = - ω p n i ( ω p ) + i ω p [ n r ( ω p ) - θ ] ,
X ( ω p ) = - ω p n i ( ω p ) ,
Y ( ω p , θ ) = ω p [ n r ( ω p ) - θ ] ,
C ( z , t ) = 1 2 π Re ( i γ { ± i π erfc [ i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω p , θ ) ] + 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP , θ ) ] } ) , Im [ Δ ( θ ) ] 0 ,
C ( z , t ) = 1 2 π Re ( i γ { i π erfc [ - i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω p , θ ) ] - i π exp [ z c ϕ ( ω p , θ ) ] + 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP , θ ) ] } ) , Im [ Δ ( θ ) ] = 0 ,             Δ ( θ ) 0 ,
C ( z , t ) = - 1 2 π Re { i γ [ - 2 π c z ϕ ( 2 ) ( ω SP , θ ) ] 1 / 2 × [ 1 ω SP - ω c + ϕ ( 3 ) ( ω SP , θ ) 6 ϕ ( 2 ) ( ω SP , θ ) ] exp [ z c ϕ ( ω SP , θ ) ] } ,             Δ ( θ ) = 0
Δ ( θ ) = [ ϕ ( ω SP 1 , θ ) - ϕ ( ω p , θ ] 1 / 2 ,             1 < θ < θ 1 ,
Δ ( θ ) = [ ϕ ( ω SP N , θ 1 ) - ϕ ( ω p , θ 1 ) ] 1 / 2 ,             θ = θ 1 ,
Δ ( θ ) = [ ϕ ( ω SP N + , θ ) - ϕ ( ω p , θ ) ] 1 / 2 ,             θ > θ 1 ,
arg [ Δ ( θ ) ] = - π / 2 ,             1 < θ < θ s ,
arg [ Δ ( θ ) ] = 0 ,             θ = θ s ,
arg [ Δ ( θ ) ] = 3 π / 4 ,             θ > θ s ,
A c ( z , t ) ~ 1 2 π Re ( i γ { - i π erfc [ i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω p , θ ) ] + 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP , θ ) ] } ) θ < θ s
A c ( z , t s ) ~ 1 2 π Re ( i γ { i π erfc [ - i Δ ( θ s ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω p , θ s ) ] + 1 Δ ( θ s ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP , θ s ) ] } ) + Re { γ exp [ z c ϕ ( ω p , θ s ) ] } ,             θ = θ s ,             ω p 0
A c ( z , t ) ~ 1 2 π Re ( i γ { i π erfc [ - i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω p , θ ) ] + 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP , θ ) ] } ) + Re { γ exp [ z c ϕ ( ω p , θ ) ] } , θ > θ s
Λ ( θ 0 ) = 1 2 1 2 π i γ exp [ z c ϕ ( 0 , θ 0 ) ] .
ϕ ( ω SP 1 , θ 0 ) = ϕ ( 0 , θ 0 ) = 0 , ϕ ( 2 ) ( ω SP 1 , θ 0 ) - 2 δ b 2 θ 0 ω 0 4 , ϕ ( 3 ) ( ω SP 1 , θ 0 ) - 3 i α b 2 θ 0 ω 0 4 ,
A c ( z , t 0 ) ~ ω 0 2 2 b ( θ 0 c π δ z ) 1 / 2 Re { i γ [ - 1 ω SP 1 ( θ 0 ) + i α 4 δ ] } + 1 2 Re ( γ ) ,             θ = θ s = θ 0 ,             ω p = 0
A c ( z , t ) ~ Re { γ exp [ z c ϕ ( ω p , θ ) ] } = exp [ - z α ( ω p ) ] { γ cos [ k ( ω p ) z - ω p t ] - γ sin [ k ( ω p ) z - ω p t ] }
α ( ω p ) = - 1 c X ( ω p ) = ω p c n i ( ω p ) ,
k ( ω p ) = ω p c n r ( ω p )
γ = lim ω ω c [ ( ω - ω c ) i ω - ω c ] = i .
A c ( z , t ) ~ 1 2 π Re { i π erfc [ i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω c , θ ) ] - 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP , θ ) ] } ,             θ < θ s ,
A c ( z , t s ) ~ 1 2 π Re { 2 π 1 / 2 exp [ z c ϕ ( ω SP , θ s ) ] F [ Δ ( θ ) s ( z c ) 1 / 2 ] - 1 Δ ( θ s ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP , θ s ) ] } - ( 1 / 2 ) exp [ - z α ( ω c ) ] sin [ k ( ω c ) z - ω c t s ] , θ = θ s = c t s / z ,             ω c 0 ,
A c ( z , t s ) ~ ω 0 2 2 b ( θ 0 c π δ z ) 1 / 2 1 ω SP 1 ( θ 0 ) ,             θ = θ s = θ 0 ,             ω c = 0 ,
A c ( z , t ) ~ 1 2 π Re { - i π erfc [ - i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω c , θ ) ] - 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP , θ ) ] } - exp [ - z α ( ω c ) ] sin [ k ( ω c ) z - ω c t ] ,             θ < θ s
F ( ζ ) = exp ( - ζ 2 ) 0 ζ exp ( t 2 ) d t
Δ ( θ ) = [ ϕ ( ω SP D + , θ ) - ϕ ( ω p , θ ) ] 1 / 2 ,
π / 4 arg [ Δ ( θ ) ] > 0 ,             1 θ < θ s ,
arg [ Δ ( θ s ) ] = 0 ,             θ = θ s ,
0 > arg [ Δ ( θ ) ] - 3 π / 4 ,             θ > θ s ,
A c ( z , t ) ~ 1 2 π Re ( i γ { i π erfc [ - i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω p , θ ) ] + 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP D + , θ ) ] } )
A c ( z , t s ) ~ 1 2 π Re ( i γ { i π erfc [ - i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω p , θ s ) ] + 1 Δ ( θ s ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP D + , θ s ) ] } ) + Re { γ exp [ z c ϕ ( ω p , θ s ) ] }
A c ( z , t ) ~ 1 2 π Re ( i γ { - i π erfc [ i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω p , θ ) ] + 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP D + , θ ) ] } ) + Re { γ exp [ z c ϕ ( ω p , θ ) ] }
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 ] }
A c ( z , t ) ~ 1 2 π Re { - i π erfc [ - i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω c , θ ) ] - 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP D + , θ ) ] } , 1 θ θ s ,
A c ( z , t s ) ~ 1 2 π Re { 2 π 1 / 2 exp [ z c ϕ ( ω SP D + , θ s ) ] F [ Δ ( θ s ) ( z c ) 1 / 2 ] - 1 Δ ( θ s ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP D + , θ s ) ] } - ( 1 / 2 ) exp [ - z α ( ω c ) ] sin [ k ( ω c ) z - ω c t s ] , θ = θ s = c t s / z ,
A c ( z , t ) ~ 1 2 π Re { i π erfc [ i Δ ( θ ) ( z c ) 1 / 2 ] exp [ z c ϕ ( ω c , θ ) ] - 1 Δ ( θ ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP D + , θ ) ] } - exp [ - z α ( ω c ) ] sin [ k ( ω c ) z - ω c t ] ,             θ > θ s
A c ( z , t s ) ~ 1 2 Re { γ exp [ z c ϕ ( ω p , θ s ) ] } = ( 1 / 2 ) 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 = c t s / z
A c ( z , t ) ~ Re { γ exp [ z c ϕ ( ω p , θ s ) ] } = 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 ) = 0 ,             θ < θ s ,
A c ( z , t s ) ~ - ( 1 / 2 ) exp [ - z α ( ω c ) ] sin [ k ( ω c ) z - ω c t s ] ,             θ = θ s = c t s / z ,
A c ( z , t ) ~ - exp [ - z α ( ω c ) ] sin [ k ( ω c ) z - ω c t ] ,             θ > θ s
A ( z , t ) = A s ( z , t ) + A B ( z , t ) + A c ( z , t )
A P ( z , t ) = A s ( z , t ) + A B ( z , t ) .
I ( z , t ) ~ q ( ω sp ) [ - 2 π z p ( 2 ) ( ω sp , θ ) ] 1 / 2 exp [ z p ( ω sp , θ ) ] ,
A ( z , t ) ~ q ( ω sp ) [ - 2 π z p ( 2 ) ( ω sp , θ ) ] 1 / 2 exp [ z p ( ω sp , θ ) ] + A c ( z , t ) .
A c ( z , t ) ~ γ Δ ( θ ) ( π z ) 1 / 2 exp [ z p ( ω sp , θ ) ] + f 0 ( ω p ) ,
p ( ω p , θ ) = p ( ω sp , θ ) + ( 1 / 2 ) p ( 2 ) ( ω sp , θ ) ( ω p - ω sp ) 2 +
Δ ( θ ) = [ - 1 2 p ( 2 ) ( ω sp , θ ) ] 1 / 2 ( ω p - ω sp ) + f 1 ( ω p ) ,
A c ( z , t ) ~ γ [ - ( 1 / 2 ) p ( 2 ) ( ω sp , θ ) ] 1 / 2 ( ω p - ω sp ) × ( π z ) 1 / 2 exp [ z p ( ω sp , θ ) ] + f 2 ( ω p ) ,
q ( ω sp ) [ - 2 π z p ( 2 ) ( ω sp , θ ) ] 1 / 2 exp [ z p ( ω sp , θ ) ] = γ ω sp - ω p [ - 2 π z p ( 2 ) ( ω sp , θ ) ] 1 / 2 exp [ z p ( ω sp , θ ) ] + f 3 ( ω p ) ,
A ( z , t ) ~ f 2 ( ω p ) + f 3 ( ω p ) .
ω 0 = 2.0 × 10 16 / sec , b 2 = 0.4 × 10 32 / sec , δ = 1.7 × 10 12 / sec
Y ( ω sp , θ s ) = Y ( ω p , θ s ) ,
X ( ω sp , θ c ) = X ( ω p ) ,
A c ( z , t ) ~ exp [ - z α ( ω p ) ] { γ p cos [ k ( ω p ) z - ω p t ] - γ p sin [ k ( ω p ) z - ω p t ] }
γ p = lim ω ω p [ ( ω - ω p ) u ˜ ( ω - ω p ) ]
arg [ - i Δ ( θ c ) ( z c ) 1 / 2 ] = - π 4 ,
erfc [ ρ exp ( ± i π 4 ) 1 / 2 ] = 1 - 2 1 / 2 exp ( ± i π 4 ) × { C [ ( 2 π ) 1 / 2 ρ ] i S [ ( 2 π ) 1 / 2 ρ ] } ,
A c ( z , t c ) ~ 1 2 π Re [ i γ ( i 2 1 / 2 π exp ( - i π 4 ) { C [ Δ ( θ c ) ( 2 z π c ) 1 / 2 ] + i S [ Δ ( θ c ) ( 2 z π c ) 1 / 2 ] } exp [ z c ϕ ( ω p , θ c ) ] + 1 Δ ( θ c ) × ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP N + , θ c ) ] ) ] + 3 2 Re { γ exp [ z c ϕ ( ω p , θ c ) ] } ,             θ = θ c = c t c z
A c ( z , t c ) ~ - 1 2 ( C [ Δ ( θ c ) ( 2 z π c ) 1 / 2 ] { cos [ k ( ω c ) z - ω c t c ] - sin [ k ( ω c ) z - ω c t c ] } - S [ Δ ( θ c ) ( 2 z π c ) 1 / 2 ] × { cos [ k ( ω c ) z - ω c t c ] + sin [ k ( ω c ) z - ω c t c ] } + 3 sin [ k ( ω c ) z - ω c t c ] ) exp [ - z α ( ω c ) ] - 1 2 π Re { 1 Δ ( θ c ) ( π c z ) 1 / 2 exp [ z c ϕ ( ω SP N + , θ c ) ] } , θ = θ c = c t c z
I ( z , θ ) = P ( θ ) q ( ω ) exp [ z p ( ω , θ ) ] d ω
ψ ( ω , θ ) - i p ( ω , θ )
ψ ( ω , θ ) ω ( 1 - θ ) + n = 0 a n ( θ ) ω - n
q ( ω ) = ω - ( 1 + ν ) q ˜ ( ω )
lim ω [ q ˜ ( ω ) ] 0.
I ( z , θ ) = - 2 π i exp [ - i z β ( θ ) ] [ 2 α ( θ ) exp ( - i π 2 ) ] ν × { γ 0 J ν [ α ( θ ) z ] + 2 α ( θ ) exp ( - i π 2 ) γ 1 J ν + 1 [ α ( θ ) z ] } + R ( z , θ ) ,
R ( z , θ ) K 2 α ( θ ) ν + 1 z { J ν + 1 [ α ( θ ) z ] + J ν + 2 [ α ( θ ) z ] }
α ( θ ) = - ( 1 / 2 ) [ ψ ( ω + , θ ) - ψ ( ω - , θ ) ] ,
β ( θ ) = - ( 1 / 2 ) [ ψ ( ω + , θ ) + ψ ( ω - , θ ) ] ,
γ 0 ( θ ) = 1 2 { q ( ω + ) ( 1 2 α ( θ ) ) 1 + ν [ - 4 α 3 ( θ ) ψ ( 2 ) ( ω + , θ ) ] 1 / 2 + q ( ω - ) [ - 1 2 α ( θ ) ] 1 + ν [ 4 α 3 ( θ ) ψ ( 2 ) ( ω - , θ ) ] 1 / 2 } ,
γ 1 ( θ ) = 1 4 α ( θ ) { q ( ω + ) [ 1 2 α ( θ ) ] 1 + ν [ - 4 α 3 ( θ ) ψ ( 2 ) ( ω + , θ ) ] 1 / 2 - q ( ω - ) [ - 1 2 α ( θ ) ] 1 + ν [ 4 α 3 ( θ ) ψ ( 2 ) ( ω - , θ ) ] 1 / 2 } .
α ¯ ± + Θ + ν α ¯ + π / 2 ,
α ( θ ) = - 2 [ - a 1 ( θ ) ( θ - 1 ) ] 1 / 2 + O [ ( θ - 1 ) 1 / 2 ] ,
β ( θ ) = a 0 ( θ ) + O [ ( θ - 1 ) 1 / 2 ] ,
[ 4 α 3 ( θ ) ψ ( 2 ) ( ω ± , θ ) ] 1 / 2 = 2 1 / 2 a 1 ( θ ) { 1 + O [ ( θ - 1 ) 1 / 2 ] }
I ( z , θ ) = P ( θ ) q ( ω ) exp [ z p ( ω , θ ) ] d ω
p ( 1 ) ( ω 1 ) = p ( 1 ) ( ω 2 ) = 0 , p ( 2 ) ( ω 1 ) 0 ,             p ( 2 ) ( ω 2 ) 0 ,
p ( 1 ) ( ω s ) = p ( 2 ) ( ω s ) = 0 , p ( 3 ) ( ω s ) 0.
I ( z , θ ) = exp [ α 0 ( θ ) z ] ( 2 π i z 1 / 3 C [ α 1 ( θ ) z 2 / 3 ] × { 1 2 [ q ( ω 1 ) h 1 ( θ ) + q ( ω 2 ) h 2 ( θ ) ] + O ( 1 z ) } + 2 π i z 2 / 3 C ( 1 ) [ α 1 ( θ ) z 2 / 3 ] { 1 2 α 1 1 / 2 ( θ ) × [ q ( ω 1 ) h 1 ( θ ) - q ( ω 2 ) h 2 ( θ ) ] + O ( 1 z ) } ) ,
C ( ζ ) = 1 2 π i L exp ( ζ v - 1 3 v 3 ) d v .
v 3 - α 1 ( θ ) v - α 0 ( θ ) + p ( ω , θ ) = 0
( d v d ω ) ω = ω s = 1 h s ( θ s )
α 0 ( θ ) = ( 1 / 2 ) [ p ( ω 1 , θ ) + p ( ω 2 , θ ) ] ,
α 1 1 / 2 ( θ ) = ¾ { ¾ [ p ( ω 1 , θ ) - p ( ω 2 , 0 ) ] } 1 / 3 ,
h 1 ( θ ) = [ - 2 α 1 1 / 2 ( θ ) p ( 2 ) ( ω 1 , θ ) ] 1 / 2 ,
h 2 ( θ ) = [ 2 α 1 1 / 2 ( θ ) p ( 2 ) ( ω 2 , θ ) ] 1 / 2
lim θ θ s [ h i ( θ ) ] = [ - 2 p ( 3 ) ( ω s , θ ) ] 1 / 3 h s ( θ s ) ,             i = 1 , 2 ,
lim θ θ s ( 1 / 2 ) [ q ( ω 1 ) h 1 ( θ ) + q ( ω 2 ) h 2 ( θ ) ] = q ( ω s ) h s ( θ s ) ,
lim θ θ s 1 2 α 1 1 / 2 ( θ ) [ q ( ω 1 ) h 1 ( θ ) - q ( ω 2 ) h 2 ( θ ) ] = h s 2 ( θ s ) q ( 1 ) ( ω s ) ,
arg [ h s ( θ s ) ] = α ¯ s ,
lim θ θ s [ α 1 1 / 2 ( θ ) ω 1 ( θ ) - ω 2 ( θ ) ] = 1 2 h s ( θ s ) .
lim θ θ s arg [ α 1 1 / 2 ( θ ) ] = α ¯ 12 - α ¯ s + 2 π n ,
lim θ θ s [ - p ( 2 ) ( ω 1 ) ω 1 - ω 2 ] = lim θ θ s [ p ( 2 ) ( ω 2 ) ω 1 - ω 2 ] = - 1 2 p ( 3 ) ( ω s ) .
lim θ θ s arg ( - ω 1 - ω 2 p ( 2 ) ( ω 1 ) ) = lim θ θ s arg [ ω 1 - ω 2 p ( 2 ) ( ω 2 ) ] = 3 α ¯ s .
lim θ θ s arg [ h i 2 ( θ ) ] = 2 α ¯ s + 2 π n ,             i = 1 , 2 ,
arg [ h i ( θ ) ] = ( 1 / 2 ) arg [ h i 2 ( θ ) ] ,             i = 1 , 2.
( d v d ω ) ω = ω s = - α ¯ s
Region 1 :             - π / 6 < arg ( v ) < π / 6 ,
Region 2 :             π / 2 < arg ( v ) < 5 π / 6 ,
Region 3 :             - 5 π / 6 < arg ( v ) < - π / 2 ,
C ( ζ ) = A i ( ζ ) ,             i = 3 ,             j = 2 ,
C ( ζ ) = exp ( - i 2 π 3 ) A i [ ζ exp ( - i 2 π 3 ) ] ,             i = 2 ,             j = 1 ,
C ( ζ ) = exp ( - i π 3 ) A i [ ζ exp ( i 2 π 3 ) ] ,             i = 3 ,             j = 1 ,
C ( ζ ) = 0 ,             i = j ,
arg ( Δ v ) = - α ¯ s + arg ( Δ ω )
I ( z , θ ) = P q ( ω ) exp [ z p ( ω , θ ) ] d ω ,
I sp ( z , θ ) = P ( θ ) q ( ω ) exp [ z p ( ω , θ ) ] d ω .
Re [ p ( ω , θ ) ] < Re { p [ ω sp ( θ ) , θ ] }
I ( z , θ ) = I sp ( z , θ ) ,             θ < θ s ,
I ( z , θ ) = I sp ( z , θ ) - π i γ exp [ z p ( ω c , θ s ) ] ,             θ = θ s ,
I ( z , θ ) = I sp ( z , θ ) - 2 π i γ exp [ z p ( ω c , θ ) ] ,             θ > θ s ,
γ = lim ω ω c [ ( ω - ω c ) q ( ω ) ]
I sp ( z , θ ) = q ( ω sp ) [ - 2 π z p ( 2 ) ( ω sp , θ ) ] 1 / 2 exp [ z p ( ω sp , θ ) ] + γ { ± i π erfc [ i Δ ( θ ) z 1 / 2 ] exp [ z p ( ω c , θ ) ] + 1 Δ ( θ ) ( π z ) 1 / 2 exp [ z p ( ω sp , θ ) ] } + R 1 exp [ z p ( ω sp , θ ) ] , Im [ Δ ( θ ) ] 0 ,
I sp ( z , θ ) = q ( ω sp ) [ - 2 π z p ( 2 ) ( ω sp , θ ) ] 1 / 2 exp [ z p ( ω sp , θ ) ] + γ { i π erfc [ - i Δ ( θ ) z 1 / 2 ] exp [ z p ( ω c , θ ) ] + 1 Δ ( θ ) ( π z ) 1 / 2 exp [ z p ( ω sp , θ ) ] } - i π γ exp [ z p ( ω c , θ ) ] + R 1 exp [ z p ( ω sp , θ ) ] , Im [ Δ ( θ ) ] = 0 ,             Δ ( θ ) 0 ,
I sp ( z , θ ) = [ - 2 π z p ( 2 ) ( ω sp , θ ) ] 1 / 2 exp [ z p ( ω sp , θ ) ] { [ q ( ω sp ) - γ ω sp - ω c ] - γ p ( 3 ) ( ω sp , θ ) 6 p ( 2 ) ( ω sp , θ ) } + R 1 exp [ z p ( ω sp , θ ) ] ,             Δ ( θ ) = 0 ,
R 1 = O ( z - 3 / 2 )
Δ ( θ ) { p [ ω sp ( θ ) , θ ] - p ( ω c , θ ) } 1 / 2 .
lim ω c ω sp ( θ ) [ Δ ( θ ) ] = [ ω c - ω sp ( θ ) ] { - 1 2 p ( 2 ) [ ω sp ( θ ) , θ ] } 1 / 2 .
erfc ( ζ ) 2 π 1 / 2 ζ exp ( ξ 2 ) d ξ .
lim ω c ω s p ( θ ) { arg [ Δ ( θ ) ] } = α ¯ c + arg ( { - p ( 2 ) [ ω sp ( θ ) , θ ] } 1 / 2 ) + 2 π n ,
arg ( { - p ( 2 ) [ ω sp ( θ ) , θ ] } 1 / 2 ) = - α ¯ SD ,
lim ω c ω s p ( θ ) { arg [ Δ ( θ ) ] } = α ¯ c - α ¯ SD + 2 π n .
I ( z ) = P q ( ω ) exp [ z p ( ω , θ ) ] d ω
Re [ p ( ω 1 , θ ) ] > Re [ p ( ω 2 , θ ) ] ,             θ < θ s ,
Re [ p ( ω 1 , θ s ) ] = Re [ p ( ω 2 , θ s ) ] ,             θ = θ s ,
Re [ p ( ω 1 , θ ) ] < Re [ p ( ω 2 , θ ) ] ,             θ > θ s .
I ( z , θ ) = I 1 ( z , θ ) + I 2 ( z , θ ) ,
I i ( z , θ ) = P i q ( ω ) exp [ z p ( ω , θ ) ] d ω ,             i = 1 , 2.
I ( z , θ ) ~ 2 exp [ z p ( ω i , θ ) ] s = 0 Γ ( s + λ 2 ) a 2 s ( i ) z s + λ / 2 ,
I ( z , θ ) = 2 exp [ z p ( ω 1 , θ ) ] { s = 0 N - 1 Γ ( s + λ 2 ) a 2 s ( 1 ) z s + λ / 2 + O [ z - ( N + λ / 2 ) ] } + 2 exp [ z p ( ω 2 , θ ) ] { s = 0 M - 1 Γ ( s + λ 2 ) a 2 s ( 2 ) z s + λ / 2 + O [ z - ( M + λ / 2 ) ] }

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