Abstract

Pulsed beams (PB’s) are time-dependent wave fields that are confined in beamlike fashion in transverse planes perpendicular to the propagation axis, whereas confinement along the axis is due to temporal windowing. Because they have these properties, pulsed beams are useful wave objects for generating and synthesizing highly focused transient fields. The PB problem is addressed here within the context of fundamental Green’s-function propagators for the time-dependent field equations. In a departure from known results in the frequency domain, by which beam solutions can be generated from point-source solutions by displacing the source coordinate location into a complex coordinate space, the complex extension is applied here as well to the source initiation time. This procedure converts the conventional causal impulsive-source Green’s-function propagator into a noncausal PB propagator, which must be defined as an analytic signal because, owing to causality, the analytic continuation into the complex domain cannot be performed by direct substitution. This being done, PB’s can be manipulated as conventional Green’s functions. Some previous results obtained by similar methods are viewed here from a sharper perspective, and new results, both analytical and numerical, are presented that grant basic insight into the PB behavior, including the ability to excite these fields by finite-causal-aperture-source distributions. Besides the basic (analytic Green’s-function) PB, examples include PB’s with frequency spectra of special interest. Particular attention is paid to the PB synthesis of focus-wave modes, which are source-free solutions of the time-dependent wave equation, and to the compact PB formulation of wave fields synthesized by focus-wave-mode spectral superposition.

© 1989 Optical Society of America

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References

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  1. G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  2. E. Gowan, G. A. Deschamps, “Quasi-optical approaches to the diffraction and scattering of Gaussian beams,” (University of Illinois at Urbana-Champaign, Urbana, Ill., 1970).
  3. J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–412 (1973).
    [CrossRef]
  4. L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” in Istituto Nazionale di Alta Matematica, Symposia Matematica (Academic, London, 1976), Vol. 18, pp. 40–56.
  5. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 251–252.
  6. E. Heyman, L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,”IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
    [CrossRef]
  7. P. D. Einziger, S. Raz, “Wave solutions under complex space–time shifts,” J. Opt. Soc. Am. A 4, 3–10 (1987).
    [CrossRef]
  8. E. Heyman, B. Z. Steinberg, “Spectral analysis of complex-source pulsed beams,” J. Opt. Soc. Am. A 4, 473–480 (1987).
    [CrossRef]
  9. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
    [CrossRef]
  10. P. A. Belanger, “Packetlike solutions of the homogeneous wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984).
    [CrossRef]
  11. A. Sezginer, “A general formulation of focused wave modes,” J. Appl. Phys. 57, 678–683 (1985).
    [CrossRef]
  12. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,”J. Math. Phys. 26, 861–863 (1985).
    [CrossRef]
  13. E. Heyman, B. Z. Steinberg, L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
    [CrossRef]
  14. E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. (to be published).
  15. R. Ziolkowski, “New electromagnetic directed energy pulses,” in Microwave Particle and Beam Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 312–319 (1988).
    [CrossRef]
  16. L. B. Felsen, “Evanescent waves,”J. Opt. Soc. Am. 66, 751–760 (1976).
    [CrossRef]
  17. P. D. Einziger, L. B. Felsen, “Evanescent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–604 (1982).
    [CrossRef]
  18. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 8.1.
  19. R. W. Ziolkowski, D. K. Lewis, “Propagation of energy in nonspreading wave packets,” J. Acoust. Soc. Am. 84, S209 (1988).
    [CrossRef]
  20. R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “An experimental reach for the localized wave packet,”J. Acoust. Soc. Am. 84, S209 (1988).
    [CrossRef]
  21. E. Heyman, R. Ianconescu, L. B. Felsen, “Pulsed beam interaction with propagation environments: canonical example of reflection and diffraction,” in Microwave Particle and Beam Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

1988

R. W. Ziolkowski, D. K. Lewis, “Propagation of energy in nonspreading wave packets,” J. Acoust. Soc. Am. 84, S209 (1988).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “An experimental reach for the localized wave packet,”J. Acoust. Soc. Am. 84, S209 (1988).
[CrossRef]

1987

1986

E. Heyman, L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,”IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
[CrossRef]

1985

A. Sezginer, “A general formulation of focused wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,”J. Math. Phys. 26, 861–863 (1985).
[CrossRef]

1984

1983

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[CrossRef]

1982

P. D. Einziger, L. B. Felsen, “Evanescent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–604 (1982).
[CrossRef]

1976

1973

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

1971

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Belanger, P. A.

Bertoni, H. L.

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

Brittingham, J. N.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[CrossRef]

Cook, B. D.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “An experimental reach for the localized wave packet,”J. Acoust. Soc. Am. 84, S209 (1988).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

E. Gowan, G. A. Deschamps, “Quasi-optical approaches to the diffraction and scattering of Gaussian beams,” (University of Illinois at Urbana-Champaign, Urbana, Ill., 1970).

Einziger, P. D.

P. D. Einziger, S. Raz, “Wave solutions under complex space–time shifts,” J. Opt. Soc. Am. A 4, 3–10 (1987).
[CrossRef]

P. D. Einziger, L. B. Felsen, “Evanescent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–604 (1982).
[CrossRef]

Felsen, L. B.

E. Heyman, B. Z. Steinberg, L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
[CrossRef]

E. Heyman, L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,”IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
[CrossRef]

P. D. Einziger, L. B. Felsen, “Evanescent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–604 (1982).
[CrossRef]

L. B. Felsen, “Evanescent waves,”J. Opt. Soc. Am. 66, 751–760 (1976).
[CrossRef]

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

E. Heyman, R. Ianconescu, L. B. Felsen, “Pulsed beam interaction with propagation environments: canonical example of reflection and diffraction,” in Microwave Particle and Beam Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” in Istituto Nazionale di Alta Matematica, Symposia Matematica (Academic, London, 1976), Vol. 18, pp. 40–56.

Gowan, E.

E. Gowan, G. A. Deschamps, “Quasi-optical approaches to the diffraction and scattering of Gaussian beams,” (University of Illinois at Urbana-Champaign, Urbana, Ill., 1970).

Heyman, E.

E. Heyman, B. Z. Steinberg, L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
[CrossRef]

E. Heyman, B. Z. Steinberg, “Spectral analysis of complex-source pulsed beams,” J. Opt. Soc. Am. A 4, 473–480 (1987).
[CrossRef]

E. Heyman, L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,”IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
[CrossRef]

E. Heyman, R. Ianconescu, L. B. Felsen, “Pulsed beam interaction with propagation environments: canonical example of reflection and diffraction,” in Microwave Particle and Beam Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. (to be published).

Ianconescu, R.

E. Heyman, R. Ianconescu, L. B. Felsen, “Pulsed beam interaction with propagation environments: canonical example of reflection and diffraction,” in Microwave Particle and Beam Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

Lewis, D. K.

R. W. Ziolkowski, D. K. Lewis, “Propagation of energy in nonspreading wave packets,” J. Acoust. Soc. Am. 84, S209 (1988).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “An experimental reach for the localized wave packet,”J. Acoust. Soc. Am. 84, S209 (1988).
[CrossRef]

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 251–252.

Ra, J. W.

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

Raz, S.

Sezginer, A.

A. Sezginer, “A general formulation of focused wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

Steinberg, B. Z.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 8.1.

Ziolkowski, R.

R. Ziolkowski, “New electromagnetic directed energy pulses,” in Microwave Particle and Beam Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 312–319 (1988).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, D. K. Lewis, “Propagation of energy in nonspreading wave packets,” J. Acoust. Soc. Am. 84, S209 (1988).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “An experimental reach for the localized wave packet,”J. Acoust. Soc. Am. 84, S209 (1988).
[CrossRef]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,”J. Math. Phys. 26, 861–863 (1985).
[CrossRef]

Electron. Lett.

G. A. Deschamps, “Gaussian beams as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

IEEE Trans. Antennas Propag.

E. Heyman, L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,”IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
[CrossRef]

P. D. Einziger, L. B. Felsen, “Evanescent waves and complex rays,”IEEE Trans. Antennas Propag. AP-30, 594–604 (1982).
[CrossRef]

J. Acoust. Soc. Am.

R. W. Ziolkowski, D. K. Lewis, “Propagation of energy in nonspreading wave packets,” J. Acoust. Soc. Am. 84, S209 (1988).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “An experimental reach for the localized wave packet,”J. Acoust. Soc. Am. 84, S209 (1988).
[CrossRef]

J. Appl. Phys.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[CrossRef]

A. Sezginer, “A general formulation of focused wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

J. Math. Phys.

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,”J. Math. Phys. 26, 861–863 (1985).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

SIAM J. Appl. Math.

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

Other

L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” in Istituto Nazionale di Alta Matematica, Symposia Matematica (Academic, London, 1976), Vol. 18, pp. 40–56.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 251–252.

E. Gowan, G. A. Deschamps, “Quasi-optical approaches to the diffraction and scattering of Gaussian beams,” (University of Illinois at Urbana-Champaign, Urbana, Ill., 1970).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 8.1.

E. Heyman, R. Ianconescu, L. B. Felsen, “Pulsed beam interaction with propagation environments: canonical example of reflection and diffraction,” in Microwave Particle and Beam Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. (to be published).

R. Ziolkowski, “New electromagnetic directed energy pulses,” in Microwave Particle and Beam Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 312–319 (1988).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Confocal elliptical coordinates that define the beam oblate spheroidal coordinate system. (a) Source problem in relations (5a) and (6a). (b) Beam problem in relations (5b) and (6b).

Fig. 2
Fig. 2

Complex-source PB field Re G+ with beam-type radiation condition [relation (5b)]. The plots depict an axial cross-sectional cut (along the x coordinate) through the rotationally symmetric 3-space. Beam parameters [see Eq. (17)]: vt1 = 1.0005b (i.e., = 0.0005b). Observation times: (a) vt/b = 0.1, (b) vt/b = 1.0, and (c) vt/b = 2.5. All axes are normalized with respect to b and v as indicated. The propagating wave packet is localized around z = vt, ρ = 0; the field spikes at z = 0, ρ = b (i.e., x/b = ±1) are stationary and are attributable to singularities of the equivalent real-space source distribution corresponding to the complex-source location. The pulse profile away from the central peak is smooth. The lobe structure on the plots is due to the limits of resolution associated with the three-dimensional graphics.

Fig. 3
Fig. 3

Complex-source PB field Re G+ with beam parameters vt1 = 1.005b (i.e., β = 0.005b) observed at vt/b= 2.5. Note the differences in the beam width and peak height as compared with those in Fig. 2(c). The comment in the caption to Fig. 2, concerning the graphics, applies here also.

Fig. 4
Fig. 4

Locus of the stationary points ωs± for the Gaussian PB in the complex frequency plane as a function of time ( τ ¯). The field is dominated by the stationary point ωs± when ωs± is in the unshaded region and by the end-point contributions when it is in shaded region.

Fig. 5
Fig. 5

The complex source PB field Re G ¯ +. The beam parameters and the observation time are the same as in Fig. 2(c). Note that G ¯ + is regular in the source plane.

Equations (108)

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G ^ ( r , ω ; r ) = exp ( i k s ) / 4 π s ,             k = ω / v ,
s = [ ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 ] 1 / 2
r = ( x , y , z ) = i b r ^ ,
r ^ = z ^ ,             b > 0 ,
s = [ ρ 2 + ( z - i b ) 2 ] 1 / 2 ,             ρ = ( x 2 + y 2 ) 1 / 2 .
Re s 0
Im s 0 ,
( x , y , z ) = b ( cos ϕ sin η cosh ξ , sin ϕ sin η cosh ξ , cos η sinh ξ ) ,
0 ξ < ,             η < π ,
- < ξ < ,             η < π / 2 ,
s s R + i s I = b ( sinh ξ - i cos η ) .
G ^ ( r , ω ; r ) = exp ( - i k s ) / 4 π s ,
s ± [ z - i b + ½ ρ 2 / ( z - i b ) ] ,
G ^ 1 4 π ( z - i b ) exp [ i k ( z - i b ) + i k ρ 2 2 K ( z ) - ρ 2 2 W 2 ( z ; ω ) ] ,
W ( z ; ω ) = W ( 0 ; ω ) [ 1 + ( b / z ) 2 ] 1 / 2 ,             W ( 0 ; ω ) = ( b / k ) 1 / 2 ,
K ( z ) = z + b 2 / z .
D ( ω ) = k W 2 ( 0 ; ω ) = b ,
u + ( t ) = 1 π 0 d ω e - i ω t u ^ ( ω ) ,             Im t 0 ,
u + ( t ) = u ( t ) + i H u ( t )             for real t ,
u ^ ( ω ) = f ^ ( ω ) exp ( - i ω t ) G ^ ( r , ω ; r ) .
t = i t 1 ,             r - i b z ^             ( t 1 and b are real ) ,
0 < b v t 1 ,
u + ( r , t ; r , t ) = f + ( τ ) / 4 π s ,             τ = t - t - s / v ,
u + ( r , t ; r , t ) = f + ( τ ) / 4 π s ,             τ = t - t + s / v .
u + = f + [ t - s R / v - i ( t 1 + s I / v ) ] / 4 π ( s R + i s I ) ,
f + [ t - i Q ] f Q + ( t ) = f Q ( t ) + i H f Q ( t ) ,
- i ( t 1 + s I / v ) - i Q ,             Q > 0
u s I = const . = ( 4 π ) - 1 ( s R 2 + S I 2 ) - 1 ( s R + s I H ) f Q ( t - s R / v ) .
α = v t 1 + b ,             β = v t 1 - b ,             α > β > 0 ,
u + f + [ t - z / v - ½ ρ 2 / v ( z - i b ) ] / 4 π ( z - i b ) ,
G + ( r , t ; r , t ) = δ + ( τ ) / 4 π s ,
G + ( r , t ; r , t ) = δ + ( τ ) / 4 π s ,
δ + ( t ) = { 1 / π i t Im t < 0 δ ( t ) + P / π i t Im t = 0
G + ( v / 4 π 2 i ) ( z - i b ) - 1 [ v t - z - i β - ½ ρ 2 / ( z - b ) ] - 1 ,
G + ( v / 4 π 2 i ) ( z - i b ) - 1 [ v t + z - i α + ½ ρ 2 / ( z - i b ) ] - 1 ,
G + [ 4 π ( z - i b ) ] - 1 { δ ( t - z / v ) δ ( x ) δ ( y ) + ( P / π i ) [ t - z / v - ½ ρ 2 / v ( z - i b ) ] - 1 } .
W G ( z ) = W G ( 0 ) [ 1 + ( z / b ) 2 ] 1 / 2 ,             W G ( 0 ) = ( β b ) 1 / 2 [ 2 ( 2 - 1 ) ] 1 / 2
T G = 2 β / v ,             ω G = T G - 1 = v / 2 β
D G = b
u + ( r , t ) = S d x ¯ d y ¯ 1 4 π R × { - z ¯ + ( z ^ · R ^ ) [ 1 R + 1 v t ¯ ] } u 0 + ( r ¯ , t ¯ ) ,
r ¯ = ( x ¯ , y ¯ , 0 + ) ,             t ¯ = t - R / v ,
R r - r R R ^ ,             R R .
lim r { r [ r + 1 v t ] u + ( r , t ) } = 0
f ^ ( ω ) = H ( ω - γ ) ( ω - γ ) μ / Γ ( μ + 1 ) ,             μ 0 ,             γ 0 ,
f + ( t ) = π - 1 ( i t ) - μ - 1 exp ( - i γ t ) ,
u + ( r , t ; r , t ) = ( 4 π 2 s ) - 1 ( i τ ) - μ - 1 exp ( - i γ τ )
= G + ( r , t ; r , t ) ( i τ ) - μ exp ( - i γ τ ) .
f ( t ) = exp ( - Ω 2 t 2 / 2 ) cos ω m t ,
f ^ ( ω ) = ( π / 2 Ω 2 ) 1 / 2 { exp [ - ω - ω m ) 2 / 2 Ω 2 ] + exp [ - ( ω + ω m ) 2 / 2 Ω 2 ] } .
u + = ( 8 π s ) - 1 { exp ( - i ω m τ - Ω 2 τ 2 / 2 ) erfc ( - ω s + / 2 Ω ) + exp ( i ω m τ - Ω 2 τ 2 / 2 ) erfc ( - ω s - / 2 Ω ) } ,
erfc ( σ ) = 2 π σ e - x 2 d x
ω s ± ( τ ) = ± ω m - i Ω 2 τ
ω m Ω ,
ω s ± = ω ¯ ± - i Ω τ ¯ ,
ω ¯ ± = ± ω m - Ω 2 Q ,             τ ¯ = t - s R / v ,
erfc ( σ ) ~ 2 H ( - Re σ ) + exp ( - σ 2 ) / σ π ,
u + = ( 4 π s ) - 1 [ exp ( - i ω ¯ + τ ¯ - Ω 2 τ ¯ 2 / 2 - ω m Q + Ω 2 Q 2 / 2 ) + ( 2 / π ) 1 / 2 e x p ( - ω m 2 / 2 Ω 2 ) Ω ( i τ ¯ + Q ) Ω 2 ( i τ ¯ + Q ) 2 - ( ω m / Ω ) 2 ] ,
arg ω s + π / 4             for τ ¯ ω ¯ + / Ω 2 ,
τ ¯ t - ( z + ρ 2 / 2 K ) / v ,
Q ω m - 1 ρ 2 / 2 W m 2 ,
Φ + ( r , t ; ω 0 ; z 0 ) = [ 4 π ( ζ + i z 0 ) ] - 1 exp { - i k 0 [ σ + ρ 2 / ( ζ + i z 0 ) ] } ,
ζ = z - v t ,             σ = z + v t .
W FWM ( ζ ; ω 0 ) = W FWM ( 0 ; ω 0 ) [ 1 + ( ζ / z 0 ) 2 ] 1 / 2 , W FWM ( 0 ; ω 0 ) = ( z 0 / 2 k 0 ) 1 / 2 ,
ζ ~ O ( z 0 ) .
z 0 k 0 1.
u + ( r , t ; z , t ; ω 0 ) = - 1 2 - d t 0 exp ( - i ω 0 t 0 ) G ¯ + ( r , t ; r ¯ , t ¯ ) ,
r ¯ = z ^ z ¯ ,             z ¯ z + v t 0 / 2 ,             t ¯ t + t 0 / 2.
G ¯ + ( r , t ; r , t ) = ½ [ G + ( r , t ; r , t ) - G + ( r , t ; r , t ) ] ,
z = i z 1 ,             t = i t 1 ,
- v t 1 z 1 v t 1 .
G ¯ + ( r , t ; r , t ) = ( i 4 π 2 v ) - 1 [ ( t - t ) 2 - ( s / v ) 2 ] - 1 ,
G ¯ + ( r , t ; r ¯ , t ¯ ) = ( - i 4 π 2 ) - 1 ( ζ + i β ) - 1 ( t 0 - t 0 p ) - 1 ,
α = v t 1 + z 1 > 0 ,             β = v t 1 - z 1 > 0.
t 0 p = v - 1 [ σ - i α + ρ 2 ( ζ + i β ) - 1 ]
u + = Φ + ( r , t ; ω 0 ; β ) exp ( - k 0 α ) ,
z = - i z 0 / 2 ,             t = i z 0 / 2 v ,             i . e . , α = 0 ,             β = z 0 .
Ψ + ( r , t ) = 1 π 0 f ^ ( ω 0 ) Φ + ( r , t ; ω 0 ; z 0 ) d ω 0 ,
Ψ + ( r , t ) = - 1 2 - d t 0 f + ( t 0 ) G ¯ + ( r , t ; r ¯ , t ¯ ) ,
f + ( t 0 ) = 1 π 0 d ω 0 exp ( - i ω 0 t 0 ) f ^ ( ω 0 ) ,
Ψ + ( r , t ) = f + ( t 0 p ) / 4 π ( ζ + i z 0 ) ,
t 0 p = v - 1 [ σ + ρ 2 ( ζ + i z 0 ) ] ,
f ^ ( ω 0 ) = exp [ - ( ω 0 - γ 0 ) α 0 / v ] H ( ω 0 - γ 0 ) ( ω 0 - γ 0 ) μ / Γ ( μ + 1 ) ,             μ 0 ,             γ 0 0 ,
f + ( t 0 ) = π - 1 ( i t 0 + α 0 / v ) - μ - 1 exp ( - i γ 0 t 0 ) .
Ψ + ( r , t ) = π - 1 Φ + ( r . t ; γ 0 ; z 0 ) [ i v - 1 ( σ + ρ 2 / ( ζ + i z 0 ) - i α 0 ) ] - μ - 1
Ψ + ( r , t ) = - G ¯ + ( r , t ; r , t ) { i v - 1 [ σ + ρ 2 / ( ζ + i z 0 ) - i α 0 ] } - μ × exp { - i γ 0 / v ) [ σ + ρ 2 / ( ζ + i z 0 ) ] } ,
t 1 = ( α 0 + z 0 ) / 2 ,             z 1 = ( α 0 - z 0 ) / 2.
a 0 z 0
T Ψ = 2 z 0 / v ,             D Ψ = z 1 α 0 / 2.
2 α 0 γ 0 / v 1 ,
W Ψ ( 0 ) W FMW ( 0 ; γ 0 ) .
D Ψ k Ψ W Ψ 2 ,
W G ( 0 ) L z 1 .
Δ t ~ O ( T G ) .
z ¯ G + ( r ¯ , t ¯ ) z ¯ = 0 = G + ( r ¯ , t ¯ ) { i / b + ( 1 + ρ ¯ 2 / 2 b 2 ) [ v t ¯ - i β - i ρ ¯ 2 / 2 b ] - 1 } ,
1 v t ¯ G + ( r ¯ , t ¯ ) z ¯ = 0 = - G + ( r ¯ , t ¯ ) [ v t ¯ - i β - i ρ ¯ 2 / 2 b ] - 1 .
u + L ( r , t ) = ρ ¯ < L z ¯ = 0 d x ¯ d y - 1 4 π R G + ( r ¯ , t ) × ( 1 + ^ z ^ · R ^ ) [ v t ¯ - i β - i ρ ¯ 2 / 2 b ] - 1 ,
R z + ρ 2 / 2 z + ρ ¯ 2 / 2 z + ρ ¯ 2 / 2 z - ( ρ ρ ¯ / z ) cos ( ϕ - ϕ ¯ ) ,
u + L ( r , t ) = - v 8 π 3 b z ρ ¯ = 0 L ρ ¯ d ρ ¯ ϕ ¯ = π ¯ π d ϕ ¯ [ v t - z - i β - ρ 2 / 2 z - ( z - 1 + i b - 1 ) ρ ¯ 2 / 2 + ( ρ ρ ¯ / z ) cos ϕ ¯ ] - 2 .
I ϕ ¯ = 2 0 π ( a 1 + a 2 cos ϕ ¯ ) - 2 d ϕ ¯ ,
a 1 = a 3 + a 4 ρ ¯ 2 ,             a 2 = ρ ρ ¯ / z , a 3 = v t - z - i β - ρ 2 / 2 z ,             a 4 = - ( z - 1 + i b - 1 ) / 2
I ϕ ¯ = - 2 π i a 1 ( a 2 2 - a 1 2 ) - 3 / 2 .
I ρ ¯ = - 2 π i 0 L ρ ¯ d ρ ¯ ( a 3 + a 4 ρ ¯ 2 ) ( - a 3 2 + a 5 ρ ¯ 2 - a 4 2 ρ ¯ 4 ) - 3 / 2 ,
a 5 = - 2 a 3 a 4 + ( ρ / z ) 2 ,
I ρ ¯ = - 2 π i ( 2 a 3 a 4 - a 5 ) - 1 ( a 3 - a 4 ρ ¯ 2 ) ( - a 3 2 + a 5 ρ ¯ 2 - a 4 2 ρ ¯ 4 ) - 1 / 2 0 L .
I ρ = 4 π / ( 2 a 3 a 4 - a 5 ) ,
u + L ( r , t ¯ ) = ½ u + [ 1 - i ( a 3 - a 4 L 2 ) ( - a 3 2 + a 5 L 2 - a 4 2 L 4 ) - 1 / 2 ] ,
( - a 3 2 + a 5 L 2 - a 4 2 L 4 ) - 1 / 2 ( i a 4 L 2 ) - 1 ( 1 - ½ ( a 3 2 - a 5 L 2 ) / a 4 2 L 4 ) ,
u + L u + [ 1 + ( a 5 - 2 a 4 a 3 - a 3 L - 2 ) / 4 a 4 2 L 2 ] u + + v 2 π 2 L 2 z / b ( 1 + i z / b ) 2 .

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