Abstract

This first part of a two-part investigation is concerned with the effects of dispersion on the propagation characteristics of the scalar field associated with a highly localized pulsed-beam (PB) wave packet in a lossless homogeneous medium described by the generic wave-number profile k(ω)=ω/c(ω), where c(ω) is the frequency-dependent wave propagation speed. While comprehensive studies have been performed for the one-dimensional problem of pulsed plane-wave propagation in dispersive media, particularly for specific c(ω) profiles of the Lorentz or Debye type, even relatively crude measures tied to generic k(ω) profiles do not appear to have been obtained for the three-dimensional problem associated with a PB wave packet with complex frequency and wave-number spectral constituents. Such wave packets have been well explored in nondispersive media, and simple asymptotic expressions have been obtained in the paraxial range surrounding the beam axis. These paraxially approximated wave objects are now used to formulate the initial conditions for the lossless generic k(ω) dispersive case. The resulting frequency inversion integral is reduced by simple saddle-point asymptotics to extract the PB phenomenology in the well-developed dispersive regime. The phenomenology of the transient field is parameterized in terms of the space–time evolution of the PB wave-front curvature, spatial and temporal beam width, etc., as well as in terms of the corresponding space–time-dependent frequencies of the signal, which are related to the local geometrical properties of the k(ω) dispersion surface. These individual parameters are then combined to form nondimensional critical parameters that quantify the effect of dispersion within the space–time range of validity of the paraxial PB. One does this by performing higher-order asymptotic expansions beyond the paraxial range and then ascertaining the conditions for which the higher-order terms can be neglected. In Part II [J. Opt. Soc. Am. A 15, 1276 (1998)], these studies are extended to include the transitional regime at those early observation times for which dispersion is not yet fully developed. Also included in Part II are analytical and numerical results for a simple Lorentz model that permit assessment of the performance of various nondimensional critical estimators.

© 1998 Optical Society of America

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References

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  1. K. A. Connor, L. B. Felsen, “Gaussian pulses as complex-source-point solutions in dispersive media,” Proc. IEEE 62, 1614–1615 (1974).
    [CrossRef]
  2. K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, Vol. 16 of Springer Series on Wave Phenomena (Springer, New York, 1997).
  3. K. E. Oughstun, “Dynamical structure of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 257–272.
  4. C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a linear, causally dispersive medium,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 273–283.
  5. J. G. Blaschak, J. Franzen, “Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence,” J. Opt. Soc. Am. A 12, 1501–1512 (1995).
    [CrossRef]
  6. P. G. Petropoulos, “Wave hierarchies for propagation in dispersive electromagnetic media,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 351–354.
  7. B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase-space beam summation for time-dependent radiation from large apertures: continuous parameterization,” J. Opt. Soc. Am. A 8, 943–958 (1991).
    [CrossRef]
  8. T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
    [CrossRef]
  9. E. Heyman, L. B. Felsen, “Complex source pulsed beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
    [CrossRef]
  10. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
    [CrossRef]
  11. T. Melamed, E. Heyman, “Spectral analysis of time-domain diffraction tomography,” invited paper for the special issue on the International Union of Radio Science 1995 Electromagnetic Theory Symposium, Radio Sci. 32, 593–604 (1997).
    [CrossRef]
  12. T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part I. Forward scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.
  13. T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part II. Inverse scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.
  14. E. Heyman, T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
    [CrossRef]
  15. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE, Piscataway, N.J., 1994). Classic reissue.
  16. T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. II. Applications,” J. Opt. Soc. Am. A 15, 1277–1284 (1998).
    [CrossRef]

1998 (1)

1997 (2)

T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
[CrossRef]

T. Melamed, E. Heyman, “Spectral analysis of time-domain diffraction tomography,” invited paper for the special issue on the International Union of Radio Science 1995 Electromagnetic Theory Symposium, Radio Sci. 32, 593–604 (1997).
[CrossRef]

1995 (1)

1994 (2)

E. Heyman, T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[CrossRef]

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

1991 (1)

1989 (1)

1974 (1)

K. A. Connor, L. B. Felsen, “Gaussian pulses as complex-source-point solutions in dispersive media,” Proc. IEEE 62, 1614–1615 (1974).
[CrossRef]

Balictsis, C. M.

C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a linear, causally dispersive medium,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 273–283.

Blaschak, J. G.

Connor, K. A.

K. A. Connor, L. B. Felsen, “Gaussian pulses as complex-source-point solutions in dispersive media,” Proc. IEEE 62, 1614–1615 (1974).
[CrossRef]

Felsen, L. B.

T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. II. Applications,” J. Opt. Soc. Am. A 15, 1277–1284 (1998).
[CrossRef]

B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase-space beam summation for time-dependent radiation from large apertures: continuous parameterization,” J. Opt. Soc. Am. A 8, 943–958 (1991).
[CrossRef]

E. Heyman, L. B. Felsen, “Complex source pulsed beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
[CrossRef]

K. A. Connor, L. B. Felsen, “Gaussian pulses as complex-source-point solutions in dispersive media,” Proc. IEEE 62, 1614–1615 (1974).
[CrossRef]

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE, Piscataway, N.J., 1994). Classic reissue.

T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part I. Forward scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.

T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part II. Inverse scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.

Franzen, J.

Heyman, E.

T. Melamed, E. Heyman, “Spectral analysis of time-domain diffraction tomography,” invited paper for the special issue on the International Union of Radio Science 1995 Electromagnetic Theory Symposium, Radio Sci. 32, 593–604 (1997).
[CrossRef]

E. Heyman, T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[CrossRef]

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase-space beam summation for time-dependent radiation from large apertures: continuous parameterization,” J. Opt. Soc. Am. A 8, 943–958 (1991).
[CrossRef]

E. Heyman, L. B. Felsen, “Complex source pulsed beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
[CrossRef]

T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part I. Forward scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.

T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part II. Inverse scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE, Piscataway, N.J., 1994). Classic reissue.

Melamed, T.

T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. II. Applications,” J. Opt. Soc. Am. A 15, 1277–1284 (1998).
[CrossRef]

T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
[CrossRef]

T. Melamed, E. Heyman, “Spectral analysis of time-domain diffraction tomography,” invited paper for the special issue on the International Union of Radio Science 1995 Electromagnetic Theory Symposium, Radio Sci. 32, 593–604 (1997).
[CrossRef]

E. Heyman, T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[CrossRef]

T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part II. Inverse scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.

T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part I. Forward scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.

Oughstun, K. E.

K. E. Oughstun, “Dynamical structure of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 257–272.

C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a linear, causally dispersive medium,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 273–283.

K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, Vol. 16 of Springer Series on Wave Phenomena (Springer, New York, 1997).

Petropoulos, P. G.

P. G. Petropoulos, “Wave hierarchies for propagation in dispersive electromagnetic media,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 351–354.

Sherman, G. C.

K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, Vol. 16 of Springer Series on Wave Phenomena (Springer, New York, 1997).

Steinberg, B. Z.

IEEE Trans. Antennas Propag. (2)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

E. Heyman, T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[CrossRef]

J. Electromagn. Waves Appl. (1)

T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
[CrossRef]

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

Proc. IEEE (1)

K. A. Connor, L. B. Felsen, “Gaussian pulses as complex-source-point solutions in dispersive media,” Proc. IEEE 62, 1614–1615 (1974).
[CrossRef]

Radio Sci. (1)

T. Melamed, E. Heyman, “Spectral analysis of time-domain diffraction tomography,” invited paper for the special issue on the International Union of Radio Science 1995 Electromagnetic Theory Symposium, Radio Sci. 32, 593–604 (1997).
[CrossRef]

Other (7)

T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part I. Forward scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.

T. Melamed, E. Heyman, L. B. Felsen, “Local spectral analysis of short-pulse-excited scattering from weakly inhomogeneous media: Part II. Inverse scattering,” Department of Aerospace and Mechanical Engineering, Boston University, Boston, Mass. 02215, , 1997.

K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, Vol. 16 of Springer Series on Wave Phenomena (Springer, New York, 1997).

K. E. Oughstun, “Dynamical structure of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 257–272.

C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a linear, causally dispersive medium,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 273–283.

P. G. Petropoulos, “Wave hierarchies for propagation in dispersive electromagnetic media,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 351–354.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE, Piscataway, N.J., 1994). Classic reissue.

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Figures (2)

Fig. 1
Fig. 1

Nondispersive PB field [Eq. (22)] for cT=0.005, β=5, ct=2. At the wave front z=ct=2, this set of parameters yields a spatial beam width D/2=0.16 in Eq. (23), a wave-front radius of curvature R=14.5, and a temporal on-axis width Tp(0)=T=0.005. (a) u versus (ρ, z); (b) (ρ, z) contour plot.

Fig. 2
Fig. 2

On-axis PB asymptotics, dispersion surface, and space–time rays. (a) k(ω) dispersion surface. The normal to the surface [see Eqs. (45)] is parallel to the space–time ray to the observation point (z, ct). The construction determines the saddle-point values ω¯s(z, t) and k[ω¯s(z, t)]. The local on-axis radius of curvature R¯c of the dispersion curve is also shown [see Eq. (47)]. (b) Space–time ray to the observation point (z, ct).

Equations (88)

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uˆ(r, ω)=-u(r, t)exp(iωt)dt,
u(r, t)=12π-uˆ(r, ω)exp(-iωt)dω,
u+(r, t)=1π0uˆ(r, ω)exp(-iωt)dω,Im t0,
u(r, t)=Re u+(r, t).
u˜ˆ0(ξ, ω)=-d2xuˆ0(x, ω)exp[-ik(ω)ξ·x],
uˆ0(x, ω)=[k(ω)/2π]2d2ξu˜ˆ0(ξ, ω)exp[ik(ω)ξ·x].
uˆ(r, ω)=(k/2π)2d2ξu˜ˆ0(ξ)exp[ik(ξ·x+ζz)],
ζ=1-ξ2,ξ2ξ·ξ,Im ζ0.
u+(r, t)=1π0dωk(ω)2π2d2ξu˜ˆ0(ξ, ω)×exp[-iωt+ik(ω)(ξ·x+ζz)].
uˆ0(x, ω)=fˆ(ω)exp[-(1/2)k(ω)ρ2/β],
u˜ˆ0(ξ, ω)=(2πfˆβ/k)exp[-(1/2)kβξ2].
u0+(x, t)=0dωfˆ(ω)exp-iωt-12k(ω)ρ2/β.
uˆ(r, ω)=[βk(ω)fˆ(ω)/2π]d2ξ exp  -iωt+ik(ω)
×i2βξ2+ξ·x+ζz.
uˆ(r, ω)fˆ(ω) -iβz-iβexp[ik(ω)S(r)],
S(r)=z+½ρ2/(z-iβ)
S=z+½ρ2/(z-Z-iF)=z+½ρ2(1/R+i/I),I=kD2,
Z=-βi,F=βr,
D=F/k[1+(z-Z)2/F2]1/2,
R=(z-Z)+F2/(z-Z).
u+(r, t)=-iβz-iβ1π0dωfˆ(ω)exp[-iΦ(ω; r, t)],
Φ(ω; r, t)=ωt-k(ω)S(r),
u+(r, t)=-iβz-iβf+t-c-1z+12ρ2/(z-iβ).
fˆ(ω)=exp(-ωT/2),
f+(t)=δ+(t-iT/2),f(t)=Re f+=1πT/2t2+(T/2)2,
ωmaxT-1.
u+(r, t)=-iβz-iβδ+t-iT2-c-1z+12ρ2/(z-iβ).
D(z)=2cTI(z).
Θ=2cT/F.
tS(r)=dkdωωs.
zt=dkdω(ω¯s)-1=vg(ω¯s),
Φ(ω)=Φ0+Φ1(ω-ω¯s)+½Φ2(ω-ω¯s)2,
Φ0Φ(ω¯s)=ω¯st-k(ω¯s)S(r),
Φ1Φωω¯s=t-k(ω¯s)S=t1-Sz,
Φ22Φω2ω¯s=-k(ω¯s)S.
ωs=ω¯s-Φ1/Φ2.
B(ω; r, t)exp[-iΦ(ω; r, t)]dω
2πiΦ(ωs; r, t)1/2B(ωs; r, t)exp[-iΦ(ωs;r, t)]
Φ(ωs)=0,
u+(r, t)A(r, t)exp[-iΨ(r, t)],
Ψ(r, t)Φ(ωs; r, t)=Φ0-½Φ12/Φ2,
A(r, t)=-iβz-iβfˆ(ω¯s)-2πik(ω¯s)S1/2.
Δωs(z, ρ, t)=t1-Szk(ω¯s)S=-12ρ2 1z-iβtk(ω¯s)zS.
Δωs(z, ρ, t)=-12ρ2 tk(ω¯s)zz2+½ ρ2+βiz+βr2z2z2+½ρ2+βiz-1+iβrz+(z2+½ρ2+βiz)2βrz-1.
Δωs(z, ρ, t)=-12ρ2 tk(ω¯s)zz2+βiz+βr2z2z2+βiz-1+iβrz+(z2+βiz)2βrz-1+O(ρ4),
Ψ(z, ρ, t)=ω¯st-k(ω¯s)S(z, ρ)+12t1-S(z, ρ)zΔωs(z, ρ, t).
Ψ(ρ)=Ψ¯+ΔΨ(ρ),
Ψ¯=ω¯st-k(ω¯s)z,
ΔΨ(ρ)=-12ρ2 1z(z-iβ)12Δωst+k(ω¯s)z.
ΔΨ=12ρ2 1z(z-iβ)14ρ2 1z-iβt2k(ω¯s)zS-k(ω¯s)z,
ΔΨ=12ρ2 1z(z-iβ)14ρ2 1z-iβt2k(ω¯s)z2-k(ω¯s)z+O(ρ6),
Ψp(r, t)=ω¯st-k(ω¯s)S(z, ρ),
c dkdωω=ω¯s=Ω¯,Ω¯ctz.
Rc(ω)={1+[ck(ω)]2}3/2/ck(ω),
R¯cRc(ω¯s)=(1+Ω¯2)3/2ck(ω¯s).
Ψ=Ψp+Ψd,
Ψd=12t1-S(z, ρ)zΔωs(z, ρ, t).
Ψd=18ρ4(z-iβ)2z2t2zk(ω¯s)+O(ρ6).
Ψd=18ρ4(z-iβ)2czΩ¯2(1+Ω¯2)3/2R¯c.
Qρ116πρ4|z-iβ|2czΩ¯2(1+Ω¯2)3/2|R¯c|1,
u+(r, t)=-iβz-iβ1π0dω exp[-iΦ(ω; r, t)],
Φ(ω; r, t)=ω[t-i(T/2)]-k(ω)S(r).
t-iT/2S=k(ωs).
ωs=ω¯s(z, t)-12ρ2tz(z-iβ)-iTk(ω¯s)S
Ψ(ρ)=ω¯st-i T2-k(ω¯s)S-18T2k(ω¯s)S+2iTtρ2z(z-iβ)k(ω¯s)S+Ψd,
u+p(r, t)=-iβz-iβ-2πik(ω¯s)S1/2 exp[-iΨp(r, t)],
Ψp(r, t)=ω¯st-i T2-k(ω¯s)S.
18T2k(ω¯s)S+2iTtρ2z(z-iβ)k(ω¯s)S2π.
QTc16πz(1+Ω¯2)-3/2 |R¯c|T21,
QpQρ+QT1
E(z, t)=|β||z-iβ|2zπ exp[Im Ψ(z, t)]/|k(ω¯s)|exp[-ω¯s(z, t)T/2]/|k(ω¯s)|.
dω¯sdtddω¯s[exp(-Tω¯s/2)/k(ω¯s)]|t=tmax=0,
z=const.,
[k(ω¯s)+Tk(ω¯s)]|t=tmax=0.
D(z, t)=I(z)k[ω¯s(z, t)]1/2.
Re Ψp(r, t)=ω¯st-k(ω¯s)z-½k(ω¯s)ρ2/R(z)=const.,
dω¯sdρ[t-k(ω¯s)z(ρ)]-k(ω¯s)z(ρ)-k(ω¯s)ρ/R(z)
+O(ρ2)=0.
dω¯sdρ[t-k(ω¯s)z]=0.
-k(ω¯s)z(ρ)-k(ω¯s)ρ/R(z)+O(ρ2)=0,
z(ρ)|ρ=0=-1R(z).
Rd(z)=-{1+[z(ρ=0)]2}3/2[z(ρ=0)]-1=R(z),
ωi=ω¯s+dω¯sdt[t-k(ω¯s)z]-dω¯sdtk(ω¯s) ½ρ2R(z)=ω¯s-dω¯sdttz½ρ2R(z),
ωi(r, t)=ω¯s(z, t)-tz2k(ω¯s)ρ2/2R(z).
ωi(r, t)=ω¯s(z, t)-12ρ2 R¯czR(z)Ω¯(1+Ω¯2)3/2,
Ω¯ctz.
Φ0=ω¯s[t-(i/2)T]-k(ω¯s)S(r),
Φ1=t[1-(S/z)]-(i/2)T,

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