Abstract

In Part I of this two-part investigation we presented a theory for propagation of pulsed-beam wave packets in a homogeneous lossless dispersive medium with the generic dispersion relation k(ω). Emphasis was placed on the paraxial regime, and detailed studies were performed to parameterize the effect of dispersion in terms of specific physical footprints associated with the PB field and with properties of the k(ω) dispersion surface. Moreover, critical nondimensional combinations of these footprints were defined to ascertain the space–time range of applicability of the paraxial approximation. This was done by recourse to simple saddle-point asymptotics in the Fourier inversion integral from the frequency domain, with restrictions to the fully dispersive regime sufficiently far behind the wave front. Here we extend these studies by addressing the dispersive-to-nondispersive transition as the observer moves toward the wave front. It is now necessary to adopt a model for the dispersive properties to correct the nondispersive high-frequency limit k(ω)=ω/c with higher-order terms in (1/ω). A simple Lorentz model has been chosen for this purpose that allows construction of a simple uniform transition function which connects smoothly onto the near-wave-front-reduced generic k(ω) profile. This model is also used for assessing the accuracy of the various analytic parameterizations and estimates in part I through comparison with numerically generated reference solutions. It is found that both the asymptotics for the pulsed-beam field and the nondimensional estimators perform remarkably well, thereby lending confidence to the notion that the critical parameter combinations are well matched to the space–time wave dynamics.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
    [CrossRef]
  2. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE, Piscataway, N.J., 1994). Classic reissue.
  3. K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, Vol. 16 of Springer Series on Wave Phenomena (Springer, New York, 1997).
  4. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).

1998

Felsen, L. B.

T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
[CrossRef]

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

Magnus, W.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).

Marcuvitz, N.

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

Melamed, T.

Oberhettinger, F.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).

Oughstun, K. E.

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

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).

Soni, R. P.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).

J. Opt. Soc. Am. A

Other

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

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

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Off-axis PB field for the cold plasma dispersion. Solid curves, fast-Fourier-transform reference solution; dashed curves, paraxial approximation in relation (I.57) (both sets coincide on the scale of the plots). Problem parameters: z/c=2, T=0.005, β/c=5, ω0=402.

Fig. 2
Fig. 2

Envelope function E(t, z) (dashed curve) with constant z for the cold plasma dispersion. The thick line describes the approximated temporal width at 1/Qd=1/5.4 of the envelope maximum. The horizontal axis is in units of time, and all other parameters bear the same normalization.

Fig. 3
Fig. 3

Envelope function E(t, z) (dashed curve) with constant t for the cold plasma dispersion. The thick line describes the approximated on-axis spatial width at 1/Qd=1/5.4 of the envelope maximum. All quantities are normalized so that they bear dimensionality of length; i.e., cT=0.0005, β=5, ω0/c=202.

Fig. 4
Fig. 4

Spatial (off-axis) width of the PB field for the cold plasma dispersion at various observation times.

Fig. 5
Fig. 5

PB field in the dispersive regime Eqs. (I.57) for the cold plasma dispersion relation with ω0=202. The field parameters are as in Fig. I.1). (a) PB in (ρ, z) plane, (b) contour plot of the field magnitude. The wave-front radii of curvature Rd [Eq. (I.70)] for various z values corresponding to the field maxima are also shown (dashed curves).

Fig. 6
Fig. 6

Bilateral matching for the on-axis PB field of relation (33b) in the medium of relation (26a): z/c=2; T=0.005; β/c=5, with Qγ=0.071, 0.71, 2.8 for (a), (b), and (c), respectively.

Equations (52)

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

k(ω)=1cω2-ω02,
ωs=ω01-Sct2,
k(ωs)=ω0Sc2t11-Sct2,
k(ωs)=-1cω0ctS2-13/2.
ω¯s=ω0Ω¯/Ω¯2-1,Ω¯=ctz.
k(ω¯s)=ω0cΩ¯2-1,k(ω¯s)=-1cω0(Ω¯2-1)3/2.
Δωs(z, ρ, t)ωs(z, ρ, t)-ω¯s(z, t)=ω0zρ22c2t2(z-iβ)11-zct23+O(ρ4).
Δωs(z, ρ, t)=-12ρ2 1z-iβtk(ω¯s)z2+O(ρ4).
Ψ(r, t)=Φ(ωs)=ωst-k(ωs)S(r),
A(r, t)=-iβz-iβ-2πik(ωs)S1/2.
Ψ(r, t)=ω0t1-Sct2.
Ψ-Ψ(ρ=0)ΔΨ
=12ρ2 1z(z-iβ)14ρ2 1z-iβt2k(ω¯s)z2-k(ω¯s)z+O(ρ6).
E(z, t)=|β||z-iβ|2cω0πz1/2 ×exp(-ω0TΩ¯/2Ω¯2-1)/(Ω¯2-1)3/4.
Ωm=12+121+4ω0T321/21/21+12ω0T32,
Emax=|β||z-iβ|2cω0πz1/2(eω0T/3)-3/2.
E(Ωd)=Emax/d,
exp(-ω0TΩ¯/2Ω¯2-1)>0.9 exp(-ω0T).
Ω¯>1+121+15ω0T-2.
Ωd=1+d4/3ω0Te32 exp-23Tω01/21+12d4/3ω0Te32 exp-23Tω0,
dω0Te3-3/2 exp12Tω0.
Δtzc(Ωd-1)=12zcd4/3ω0Te32 exp-23Tω0.
d>Qdexp-23Tω0ω0Te321+15ω0T2-3/4,
Emax=βzm2+β22cω0πzm1/2(eω0T/3)-3/2,zm=ctΩm.
Δzct1-1Ωd,
D(z, t)=I(z)cω0 (Ω¯2-1)1/4.
ωi(r, t)=ω¯s(z, t)-tz2k(ω¯s)½ ρ2R(z).
ωi(r, t)=ω0Ω¯Ω¯2-11+12ρ2z1R(z)(Ω¯2-1).
k(ω)ωc-ωα2cω,ω
ω¯s=ωα/Ω¯-1.
R¯c=-ωα2Ω¯2+1Ω¯-13/2.
ρmax=32πcz|z-iβ|2ωα(Ω¯-1)3/2Ω¯21/4.
fˆ(ω)exp[-(½)Tω],ω,
fˆ(ω)exp[-(½Tω+a/ω)],ω,a>0,
u+(r, t)=1π-iβz-iβ0dω exp[-iωτ-ib¯(1/ω)],
τ=t-i T2-Sc,b¯=ωα2S(r)c-ia.
a12ρ2 ωα2cβr1+z+βiβr2-1.
u+(r, t)=iβz-iβb¯τ H1(2)(2bτ¯),
u+(r, t)iβz-iβbτ H1(2)(2bτ),b=ωα2Sc.
u(z, t)=-b/τJ1(2bτ),
|2bτ|=2t-i T2-Sc1/2ωα2Sc1/21.
u+(r, t)=-iβz-iβ1πi1τ=-iβz-iβδ+t-i T2-c-1z+12ρ2/(z-iβ).
QΩ2ωα2zct-zc2+T221/21/21.
QΩ2 ωαzc(Ω¯-1)2+(Ω¯T/2)21/41,
Ω¯lim=1+c2zωα4-(Ω¯T/2)21/2.
QγQΩ|Ω=1=ωα2Tzc1.
u+(r, t)--iβz-iβbτ1πbτ exp(-i2bτ+¾ πi).
ωsωs(r, t)=ωαc(t-iT/2)S-1-1/2=bτ,
2bτ=(t-iT/2)ωs(r, t)-k(ωs)S,
k(ωs)=-2ωα2cωs-3=-2ωα2cbτ-3/2.
u+(r, t)-iβz-iβ1π-2πik(ωs)S1/2 exp(-iΨ),
Ψ=(t-iT/2)ωs(r, t)+ik(ωs)S,

Metrics