Home
Search
Collections
Journals
About
Contact us
My IOPscience
Dispersive pulse dynamics and associated pulse velocity measures
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2002 J. Opt. A: Pure Appl. Opt. 4 S125
(http://iopscience.iop.org/1464-4258/4/5/359)
View the table of contents for this issue, or go to the journal homepage for more
Download details:
IP Address: 132.198.11.131
The article was downloaded on 23/02/2012 at 18:54
Please note that terms and conditions apply.
INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS
J. Opt. A: Pure Appl. Opt. 4 (2002) S125–S134
PII: S1464-4258(02)32839-3
Dispersive pulse dynamics and associated
pulse velocity measures
Kurt Edmund Oughstun and Natalie Anne Cartwright
College of Engineering and Mathematics, University of Vermont, Burlington, VM 05405-0156,
USA
E-mail: oughstun@emba.uvm.edu and ncartwri@emba.uvm.edu
Received 17 January 2002
Published 14 August 2002
Online at stacks.iop.org/JOptA/4/S125
Abstract
The classical theory of dispersive signal propagation by Sommerfeld and
Brillouin proposed in 1914 provided the first proof within the classical
Maxwell–Lorentz theory that an electromagnetic signal could not propagate
faster than the vacuum speed of light in a causal dielectric, also introducing
the signal velocity and the precursor fields (or forerunners). Modern
asymptotic theory has extended these results, providing a physically
meaningful definition of the signal velocity while showing the critical role
that the precursor fields play in the observed pulse dynamics. Their role in
defining the observed pulse velocity is presented and compared with the
classical group velocity as well as with other pulse velocity measures that
yield superluminal results.
Keywords: Electromagnetic pulse propagation, dispersion, ultrashort pulse
dynamics, ultrashort pulse velocity measures
1. Introduction
Ever since the advent of the special theory of relativity [1] in
1905, the velocity of light in a dispersive, absorptive medium
has presented a troublesome dilemma to the electromagnetic
and optical physics community. Although Lorentz’s classical
model [2] of dielectric dispersion is causal, the group velocity
was found to yield noncausal results in regions of anomalous
dispersion described by that model. With Sommerfeld’s
proof [3] within the classical Maxwell–Lorentz theory that
an electromagnetic signal could not propagate faster than
the vacuum speed of light in a Lorentz model dielectric,
Brillouin [4, 5] then introduced the signal and energy velocities
within that medium as a replacement for the group velocity.
As was pointed out by Brillouin [5], a group of waves
had been originally defined by Rayleigh [6, 7] ‘as moving
beats . . . following each other in a regular pattern. A signal
is a short isolated succession of wavelets, with the system
at rest before the signal arrived and also after it has passed.
. . . In general, the signal velocity will differ from the group
velocity, especially if the phase velocity is strongly frequency
dependent and if the absorption cannot be ignored.’ These
early results, including the later work by Baerwald [8],
have been borne out and improved upon by the modern
asymptotic theory [9–11]. Together with Loudon’s energy
velocity [12], a correct physical model of dispersive pulse
propagation has been presented [13, 14]. In spite of these
results, the group velocity has become one of the most
misused concepts in modern physics, engineering and applied
mathematics. Because of its inherent simplicity, the group
velocity description of dispersive pulse propagation, which is
based, in part, on the slowly-varying-envelope approximation,
originally introduced by Born and Wolf [15] in the context
of partial coherence theory, is widely accepted and employed
throughout physics [16] and applied mathematics [17] with
central importance in electromagnetics [18], acoustics [19]
and optics [20, 21]. Unfortunately, the approximations that
are introduced by this description have largely been ignored,
resulting in the current controversy regarding superluminal
velocities in dispersive materials [22].
The purpose of this paper is to briefly review past
and current research as well as to provide a solid
theoretical foundation on which to excuse experimental
measurements claiming superluminal pulse velocities. The
description is based upon the exact Fourier–Laplace integral
representation [11] of a linearly polarized, plane wave
electromagnetic pulse propagating in the positive z-direction
1464-4258/02/050125+10$30.00 © 2002 IOP Publishing Ltd Printed in the UK
S125
with electric and magnetic field vectors
1
E y (z, t ) =
f˜(ω) exp{i[k̃(ω)z − ωt]}dω
E0
2π
C
c
E 0 n(ω) f˜(ω) exp{i[k̃(ω)z − ωt]}dω
Bx (z, t ) = −
2πc
C
(1)
respectively, where E (z, t) = 1̂ y E y (z, t) and B (z, t) =
1̂x Bx (z, t). Here z = z −z 0 denotes the propagation distance
into the positive half-space z z 0 from the input plane at
z = z 0 where the initial field behaviour E y (z 0 , t ) = E 0 f (t) is
specified with electric field strength E 0 . The nondimensional
function f (t) describes the initial pulse structure with temporal
frequency spectrum
∞
˜f (ω) =
f (t) exp(iωt) dt
(2)
Real & Imaginary Parts of the Complex Index of Refraction
K E Oughstun and N A Cartwright
100
10–1
10–2
1015
1016
1017
1018
−∞
where ω is the angular frequency. The contour C appearing in
equation (1) is the Bromwich contour ω = ω + ia, where ω
varies from −∞ to +∞ and where the quantity a is greater or
equal to the abscissa of absolute convergence for the particular
pulse shape considered [11, 23]. The quantity
k̃(ω) ≡ β(ω) + iα(ω) = (ω/c)n(ω)
(3)
denotes the complex wavenumber in the dispersive medium
with complex index of refraction
n(ω) = n r (ω) + in i (ω) = (µε(ω)/(µ0 ε0 ))1/2
(4)
where n r (ω) = Re {n(ω)} is the real index of refraction,
n i (ω) = Im {n(ω)} is the imaginary part of the complex index
of refraction and β(ω) = Re {k̃(ω)} is the propagation factor
and α(ω) = Im {k̃(ω)} the attenuation factor for a timeharmonic plane wave with fixed angular frequency ω in the
dispersive material. Here Re {∗} denotes the real part, and
Im {∗} denotes the imaginary part of the quantity inside the
brackets. The exponential factor appearing in equation (1)
may then be rewritten as
exp{i[k̃(ω)z − ωt]}= exp[−α(ω)z] exp{i[β(ω)z − ωt]}
(5)
where the first exponential term on the right-hand side of
this expression describes the attenuation and the second term
describes the phase change upon propagation. Finally, notice
that both cgs and MKS units are employed here through the use
of a conversion factor that appears in double brackets ∗ in
the equation affected. If this factor is included in the equation
it is then in cgs units provided also that ε0 = µ0 = 1, while the
equation is in MKS units if this factor is omitted (i.e. replaced
by unity). If no such factor appears, then the equation is proper
in both systems of units.
The analysis is presented for the case of a single-resonance
Lorentz model dielectric with (relative) dielectric permittivity
ε(ω)/ε0 = 1 −
b2
ω2 − ω02 + 2iδω
charge and m e the mass of an electron, and where δ is the
phenomenological damping constant. With relative magnetic
permeability µ/µ0 = 1, the complex index of refraction is
given by
1/2
b2
.
(7)
n(ω) = 1 − 2
ω − ω02 + 2iδω
Brillouin’s choice
√ of the material parameters (ω0 = 4 ×
1016 r s−1 , b = 20 × 1016 r s−1 and δ = 0.28 × 1016 r s−1 )
which describe a highly absorptive material, are used in the
numerical examples presented here; however, the results are
not restricted to this highly absorptive case. The general results
obtained here are also applicable to more general models of the
dispersion relation than that given in equation (6), such as that
given by a multiple-resonance Lorentz model dielectric.
The frequency dispersion of the real and imaginary
parts of the complex index of refraction described by
equation (7) for Brillouin’s choice of the medium parameters
are illustrated in figure 1. The material absorption band
isdefined over
domain
the approximate angular frequency
[ ω02 − δ 2 , ω12 − δ 2 ], where ω1 =
ω02 + b2 .
The
material dispersion is normal (n r (ω) increases with increasing
frequency)
over the approximate
angular frequency domains
[0, ω02 − δ 2 ) and ( ω12 − δ 2 , ∞) above and below the
absorption band, respectively, while it is anomalous (n r (ω)
decreases with increasing frequency)
over the approximate
angular frequency domain [ ω02 − δ 2 , ω12 − δ 2 ] containing
the absorption band. The corresponding frequency dispersion
of the attenuation and propagation factors α(ω) and β(ω),
respectively, are depicted by the solid curves in figure 2.
(6)
where ω0 is the undamped resonance frequency, b =
[(4π/ε0 )N qe2 /m e ]1/2 the plasma frequency with number
density N of Lorentz oscillators, where qe denotes the absolute
S126
Figure 1. Angular frequency dependence of the real (solid curve)
and imaginary (dashed curve) parts of the complex index of
refraction for a single-resonance Lorentz model dielectric with
undamped resonance
frequency ω0 = 4 × 1016 r s−1 , plasma
√
frequency b = 20 × 1016 r s−1 and phenomenological damping
constant δ = 0.28 × 1016 r s−1 .
2. The phase velocity
The phase velocity describes the rate at which the phase fronts
appearing in the Fourier–Laplace integral representation of
Real & Imaginary Parts of the Complex Wavenumber (r/m)
Dispersive pulse dynamics and associated pulse velocity measures
frequency of the initial pulse, and the propagation factor is
given by
β(ω) ≈ (ω/c)n r (ωc ).
(9)
109
With the further usual approximation that the attenuation factor
is nondispersive so that α(ω) ≈ α(ωc ), the propagated field
vectors given in equation (1) may be directly evaluated as
108
E y (z, t) ≈ E 0 f [t − (β(ωc )/ωc )z] exp[−α(ωc )z]
c
Bx (z, t) ≈ −
n r (ωc ) f [t − (β(ωc )/ωc )z]
c
× exp[−α(ωc )z].
107
106
105
1015
1016
1017
1018
(10)
The electromagnetic pulse then propagates undistorted in
shape, but attenuated in amplitude, at the phase velocity
v p (ωc ) = ωc /β(ωc ).
3. The group velocity
Relative Phase, Group & Energy Velocity
Figure 2. Angular frequency dependence of the real (solid curve)
and imaginary (dashed curve) parts of the complex wavenumber
k̃(ω) = β(ω) + iα(ω) for the single-resonance Lorentz model
dielectric considered in figure 1. The dotted curves illustrate the
frequency dependence of the linear β (1) (ω) and quadratic β (2) (ω)
approximations to the real part of the complex wavenumber about
the angular frequency value ωc = 0.75ω0 .
10 0
β (1)(ω) = β(ωc ) + β (ωc )(ω − ωc )
1016
1017
1018
Figure 3. Angular frequency dependence of the relative phase
velocity ν p (ω)/c (dashed curve), relative group velocity νg (ω)/c
(alternating dashed-dotted curve) and relative energy velocity
ν E (ω)/c (solid curve) for the single-resonance Lorentz model
dielectric considered in figure 1.
equation (1) propagate through the dispersive medium. The
phase velocity for this electromagnetic wave field is obtained
from the second term on the right-hand side of equation (5) as
v p (ω) =
ω
.
β(ω)
(8)
The angular frequency dispersion of the relative phase velocity
v p /c is illustrated by the dashed curve in figure 3.
The phase velocity describes the pulse velocity only in the
simplest nonphysical situation when the index of refraction of
the medium is nondispersive (excluding the trivial, physically
realizable case when the medium is the ideal vacuum). In that
case, n(ω) ≈ n(ωc ), where ωc is a characteristic oscillation
(11)
with α(ω) ≈ α(ωc ), where β (ω) ≡ ∂β(ω)/∂ω. Here ωc
describes a characteristic oscillation frequency of the initial
pulse; for example, for a pulse envelope modulated sine wave,
f (t) = u(t) sin(ωc t + ψ) where u(t) is the envelope function
and ψ is a phase constant, ωc = 2π fc is the fixed carrier
frequency of the initial pulse. With this substitution, the
propagated field vectors given in equation (1) may be directly
evaluated as
E y (z, t) ≈ E 0 f (t − β (ωc )z) exp[−α(ωc )z]
× cos[β(ωc )z − ωc t]
c
Bx (z, t) ≈ −
E 0 f (t − β (ωc )z) exp[−α(ωc )z]
c
× [n r (ωc ) cos(β(ωc )z − ωc t)
− n i (ωc ) sin(β(ωc )z − ωc t)].
10 –1
1015
A slightly more accurate description of the material dispersion
is provided by the linear dispersion approximation
(12)
The pulse phase then travels with the phase velocity given in
equation (8) while the pulse structure propagates undistorted
in shape, attenuated with propagation distance, at the group
velocity
1
vg (ω) = .
(13)
β (ω)
The angular frequency dispersion of the group velocity is
illustrated by the alternating dashed-dotted curve presented in
figure 3. Note that the group velocity becomes negative in the
region of anomalous dispersion; this negative branch of the
group velocity curve is not depicted in this graph.
The accuracy of the linear dispersion approximation
β (1)(ω) given in equation (11) is illustrated in figure 2 when
ωc = 0.75ω0 . As can be seen, this approximation is accurate
only over a very small neighbourhood about ωc . The local
approximation does improve as ωc is decreased towards zero
frequency while it gets worse as ωc is increased into the
region of anomalous dispersion. The local approximation also
improves as ωc increases above the absorption band.
A somewhat more accurate description of the material
dispersion is provided by the widely used quadratic dispersion
approximation
β (2) (ω) = β(ωc )+β (ωc )(ω − ωc )+ 12 β (ωc )(ω − ωc )2 (14)
S127
K E Oughstun and N A Cartwright
with α(ω) ≈ α(ωc ), where β (ω) ≡ ∂ 2 β(ω)/∂ω2 describes
the so-called group velocity dispersion (GVD). With this
substitution, the propagated field vectors given in equation (1)
become
E0
exp[−α(ωc )z]
E y (z, t) ≈
[2πβ (ωc )z]1/2
∞
f (t )
× Re exp[i(β(ωc )z − ωc t + 3π/4)]
−∞
(β (ωc )z + t − t)
dt × exp −i
(15)
2β (ωc )z
cE 0
exp[−α(ωc )z]
Bx (z, t) ≈ −
c[2πβ (ωc )z]1/2
∞
g(t )
× Re exp[i(β(ωc )z − ωc t + 3π/4)]
−∞
(β (ωc )z + t − t)
dt × exp −i
2β (ωc )z
where
∞
1
n(ω) f (ω) exp(−iωt) dω.
(16)
g(t) =
2π −∞
The pulse phase then propagates with the phase velocity
v p (ωc ) while the pulse itself propagates with the group velocity
vg (ωc ), where the pulse shape is proportional to the Fresnel
transform of the initial pulse shape. The propagated pulse
structure then depends upon the timescale parameter [24]
TF ≡ |2πβ (ωc )z|1/2
(17)
which depends upon the value of the GVD and corresponds
to the principal Fresnel zone in the analogous slit diffraction
problem. If T > 0 denotes the initial temporal pulse width,
then for sufficiently small propagation distances z 0, the
inequality T > TF will be satisfied and the pulse shape evolves
in the same fashion as the diffracted field in the near field of the
diffracting slit aperture, while for sufficiently large propagation
distance, the inequality T < TF will be satisfied and the pulse
shape approaches that given by the Fourier transform of the
initial pulse shape in the same manner as the diffracted field in
the Fraunhofer region.
Similar results are obtained if, instead of just the
propagation factor, the complex wavenumber k̃(ω) is
approximated by a finite number of terms in its Taylor series
expansion about ωc . The quadratic dispersion approximation
given in equation (14) is then replaced by
k̃ (2) (ω) = k̃(ωc ) + k̃ (ωc )(ω − ωc ) + 12 k̃ (ωc )(ω − ωc )2 . (18)
This form of the approximation results in a frequencydependent behaviour in the absorption factor. However, it also
results in a complex group velocity
ṽg (ω) =
1
k̃ (ω)
(19)
that is complex valued almost everywhere along the entire real
frequency axis, being real valued at ω = 0 and ω = ±∞. The
physical meaning of a complex-valued velocity remains to be
given.
The accuracy of the quadratic dispersion approximation
β (2) (ω) given in equation (14) is also illustrated in figure 2
S128
when ωc = 0.75ω0 . It is seen to be accurate only over
a slightly larger neighbourhood about ωc than is the linear
approximation given in equation (11). It has long been
assumed that increased accuracy over a larger frequency
domain about the characteristic pulse frequency ωc may be
obtained by using higher-order approximations to describe
the material dispersion. In 1975 Anderson et al stated in
the abstract of their paper [25] that ‘the evolution of slowly
varying wave pulses in strongly dispersive and absorptive
media is studied by a recursive method. It is shown that
the resulting envelope function may be obtained by including
correction terms of arbitrary dispersive and absorptive orders’.
This sentiment is extended in the 1990 text by Butcher and
Cotter [26] which states that ‘to describe pulse propagation
in dispersive media, in general, we must retain the secondorder dispersion, and for ultrashort pulses or those with a wide
frequency spectrum it may sometimes be necessary to also
include higher-order terms’. Continuing on in this tradition,
Akhmanov et al state in their 1992 text [27] that ‘one can
analyse how the dispersion of a medium affects a propagating
pulse for any higher-order approximation of the dispersion
theory. Naturally, the higher-order approximations make
the quantitative picture of dispersive spreading more precise
although its basic features obtained for the second- and thirdorder approximations remain unchanged.’ Unfortunately,
these assertions, although physically appealing, are not valid
in the ultrashort pulse, ultrawideband signal regime. As
first proved by Oughstun and Xiao [28, 29], ‘with the
exception of a small neighbourhood about some characteristic
frequency of the initial pulse, the inclusion of higherorder terms in the Taylor series approximation of the
complex wavenumber in a causally dispersive, attenuative
medium beyond the quadratic approximation is practically
meaningless from both the physical and mathematical points
of view’.
4. The energy velocity
A quantity of fundamental importance to both the analysis
and interpretation of propagation phenomena in a causally
dispersive medium is the velocity of energy transport (or the
energy velocity) of a monochromatic field. This physical
velocity is defined as the ratio of the time-average value of
the Poynting vector to the total time-average electromagnetic
energy density stored in both the field and the medium. The
original derivation of this quantity for a single-resonance
Lorentz model dielectric by Brillouin [4, 5] neglected to
include that portion of the electromagnetic energy that is
stored in the excited Lorentz oscillators of the medium and
consequently was in error. Loudon [12] was the first to derive a
correct expression for the energy velocity in a single-resonance
Lorentz medium. This momentous research would later prove
to be critical in providing a correct physical description of
dispersive pulse dynamics.
Consider a multiple-resonance Lorentz model dielectric
with complex index of refraction
n(ω) = 1 −
j
b2j
ω2 − ω2j + 2iδ j ω
1/2
(20)
Dispersive pulse dynamics and associated pulse velocity measures
where each separate resonance line is described by
the parameter triple (ω j , δ j , b j ) of undamped resonance
frequency, phenomenological damping constant and plasma
frequency, respectively, with summation index j extending
over the set of resonance lines present in the dielectric material.
The time-average value of the electromagnetic energy density
stored in the Lorentz oscillators is then found to be given
by [11, 30]
1 ε0
b2j (ω2 + ω2j )
2
Uosc = 4π 4 |E|
(ω2 − ω2 )2 + 4δ 2 ω2
j
j
(21)
j
where ω is the angular frequency of oscillation of the timeharmonic plane wave field with electric field strength E. The
time-average value of the energy density stored in the timeharmonic plane wave electromagnetic field is given by
1 ε0 2
2
2
(22)
U f = 4π 4 (n r (ω) + n i (ω) + 1)|E| .
With use of the identity [11]
n r2 (ω) − n 2i (ω) = 1 −
b2j (ω2 − ω2j )
j
(ω2 − ω2j )2 + 4δ 2j ω2
(23)
the total time-average electromagnetic energy density stored
in the coupled field–medium system is found to be
U = U f + Uosc 1 ε0
b2j ω2
|E|2 n 2 +
.
=
r
4π 2
2 2
2 2
2
j (ω − ω j ) + 4δ j ω
(24)
The time-average value of the magnitude of the Poynting vector
for the monochromatic plane wave field is given by [11]
2
c 1
2
(25)
|S| = 4π 2µ c n r (ω)|E| .
0
The time-average velocity of energy transport in a multiple
resonance Lorentz model dielectric is then given by the ratio
of these two quantities, so that [11, 30]
v E (ω) ≡
|S|
=
U n r (ω) +
1
nr (ω)
c
b2j ω2
j (ω2 −ω2j )2 +4δ 2j ω2
.
(26)
c
n r (ω) + ωn i (ω)/δ
5. The signal velocity
The signal velocity in a Lorentz model dielectric was originally
introduced by Brillouin [4, 5] and its description was later
improved upon by Baerwald [8]. Its original physical
interpretation has been criticized by Smith [33] as being
impractical from an experimental point of view since ‘the
difficulty of arriving at a workable definition for the signal
velocity is that a pulse of radiation is not a point, i.e. the
motion of the pulse cannot be equated to the motion of a point
associated with the pulse’. This criticism is not surprising
since the original definition of the signal velocity is based
upon the intricacies of the asymptotic method of steepest
descent, the signal arrival being defined by the instant the
path of steepest descent crossed the simple pole singularity
appearing in the spectrum of the initial signal. The modern
asymptotic theory [9–11] has redefined the signal velocity in
light of Olver’s saddle point method [34] which proved that
the path of steepest descent was irrelevant in determining the
signal velocity.
The signal velocity is defined for the fundamental
canonical problem of the step function modulated signal
f (t) = u(t) sin(ωc t)
(28)
where ωc = 2π fc is a fixed carrier frequency and where
the envelope function u(t) is the Heaviside unit step function
(u(t) = 0 for t < 0 and u(t) = 1 for t > 0). With this
substitution in equation (1), the integral representation for the
propagated signal becomes
E0
Re
ũ(ω − ωc ) exp[(z/c)φ(ω, θ )] dω
E y (z, t) =
2π
C
E0
n(ω − ωc )ũ(ω − ωc )
(29)
Bx (z, t) = −c
Re
2πc
C
× exp[(z/c)φ(ω, θ )] dω
with complex phase function
This result directly reduces to the expression
v E (ω) =
in v E (ω) corresponds to the angular frequency region where
the time-average electromagnetic energy density Uosc stored
in the Lorentz oscillators is near its local maximum value.
This linear result then complements the nonlinear result
of minimal propagation velocity observed in self-induced
transparency [31, 32].
(27)
given by Loudon [12] for the case of a single-resonance Lorentz
model dielectric.
The frequency dependence of the energy velocity in a
single-resonance dielectric is illustrated by the solid curve
presented in figure 3. Notice first that v E (ω) c for
all real ω with v E → c as ω → ∞. The energy
velocity is seen to attain a minimum value just above the
resonance frequency ω0 near to the frequency value where
n i (ω) attains its maximum value, and remains small through
the region of anomalous dispersion. Naturally, this occurs
over each absorption band in a dielectric material described
by multiple resonance frequencies. Each minimum region
φ(ω, θ ) ≡ i(c/z)(k̃(ω)z − ωt) = iω(n(ω) − θ)
(30)
where θ = ct/z is a nondimensional space-time value.
Although this analysis of the signal velocity is presented in
terms of the Heaviside unit step function signal, the expressions
appearing in equation (29) remain valid for any pulse that can
be expressed in the form given in equation (28) with fixed
carrier frequency ωc .
If the initial time behaviour E(z 0 , t ) = E 0 f (t) of the field
at the plane z = z 0 is zero for all time t < 0 and if the model
of the material dispersion is causal, then the propagated field
(29) identically vanishes for all θ < 1 with z > 0, so that
f (t) = 0 for t < 0 ⇒ E y (z, t ) = Bx (z, t ) = 0 for all θ < 1.
(31)
S129
K E Oughstun and N A Cartwright
A proof of this important result was first given by
Sommerfeld [3] for the Heaviside unit step function signal (28)
in a single-resonance Lorentz model dielectric, and was later
extended [9, 11] to an arbitrary plane wave pulse. The
dynamical field evolution then occurs for θ 1 and asymptotic
methods of analysis are typically required as z → ∞.
The first step in the asymptotic analysis of the integral
representation (29) of the propagated field for θ 1 is
to determine the set of saddle points of the complex phase
function where dφ/dω = 0, so that
n(ω) + ωn (ω) − θ = 0.
(32)
The roots of this saddle point equation then give the saddle
point locations in the complex ω-plane. Since the saddle
point equation depends upon the space-time parameter θ =
ct/z, the saddle points will then evolve with time at any
fixed propagation distance z. Because of the symmetry
relations [11] n(−ω) = n ∗ (ω∗ ) and φ(−ω, θ ) = φ ∗ (ω∗ , θ)
satisfied by a causal medium, if ω j (θ ) is a saddle point, then
so also is −ω∗j (θ ).
The physical significance of the saddle points can be
appreciated through consideration of the defining relation
(z/c)φ(ω, θ ) = i(k̃(ω)z − ωt). Upon differentiating this
expression with respect to ω one obtains (z/c)φ (ω, θ ) =
i((∂ k̃(ω)/∂ω)z − t). Since φ (ω, θ ) = 0 at the saddle
points of the complex phase function, then z/t =
(∂ k̃(ω)/∂ω)−1
ω=ω j = ṽg (ω j ) and the complex group velocity
is found to be real valued at the saddle points.
With the saddle point locations known for θ 1, the next
step in the analysis is to express the integral representation (29)
in terms of an integral I (z, θ ) with the same integrand but with
a new contour of integration P(θ ) to which the original contour
C may be deformed [9–11]. By Cauchy’s residue theorem, the
integral representation (29) of E y (z, t ) and the contour integral
I (z, θ ) are related by
E y (z, t ) = I (z, θ ) − Re {2πi(θ )}
(33)
where
1
ũ(ω − ωc ) exp[(z/c)φ(ω, θ )]
ω=ω p 2π
p
(34)
is the sum of the residues of the poles that were crossed in the
deformation from C to P(θ ), and where
1
ũ(ω − ωc ) exp[(z/c)φ(ω, θ )] dω.
I (z, θ ) =
2π P(θ )
(35)
Similar expressions are obtained for the associated magnetic
field component. For the asymptotic evaluation of the contour
integral I (z, θ ) as z → ∞, the path P(θ ) is taken as a
union of Olver-type paths [9–11] with respect to a subset of
the set of saddle points of φ(ω, θ ) such that P(θ ) evolves
continuously for all θ 1. Not all saddle points in
this set may be appropriate in the asymptotic description
because the Olver-type paths with respect to them may not
be deformable to the original contour C owing, for example,
to the presence of the branch cuts of φ(ω, θ); such saddle
points are said to be inaccessible, otherwise they are said
(θ ) =
S130
Res
to be accessible. The dominant accessible saddle point (or
points) refers to the saddle point (or points) that has the
largest value of Re {φ(ω, θ )} at it, and hence, has the least
exponential attenuation associated with it. In comparison,
Brillouin’s interpretation [4, 5] of this asymptotic method
required that the contour of integration C be deformed so
that it lay along the entire path of steepest descent through
the accessible saddle points of the complex phase function.
Olver’s theorem [34] proved this requirement unnecessary with
important consequences regarding the physical significance
of whether or not a particular pole singularity is crossed in
deforming the contour C to P(θ ).
If ω j (θ ) and −ω∗j (θ ) are the dominant accessible firstorder saddle points at a particular value of θ and if they are
isolated from each other as well as from all other saddle points
of the complex phase function φ(ω, θ ) at that value of θ ,
then the nonuniform asymptotic approximation of I (z, θ ) as
z → ∞ is obtained from Olver’s theorem [34] as [9, 11]
1/2
c
ũ(ω j − ωc )
I (z, θ ) ∼ Re −
2πzφ (2) (ω j , θ )
× exp[(z/c)φ(ω j , θ )]
1/2
c
+ −
ũ(ω∗j − ωc )
2πzφ (2) (−ω∗j , θ )
(36)
× exp[(z/c)φ(ω∗j , θ )] .
The dynamical evolution of the saddle points then provides
a nearly complete description of the dynamical evolution of
the transient field behaviour associated with dispersive pulse
propagation.
The residue contribution (34) is nonzero only if ũ (ω − ωc )
has poles. For the Heaviside unit step function signal
(θ ) = 0
for θ < θ S
(θ ) = exp(−α(ωc )z) sin(β(ωc )z − ωc t)
for θ > θ S
(37)
where θ S denotes the space-time point at which the Olver-type
path P(θ ) crosses the simple pole singularity at ω = ωc . This
contribution to the asymptotic behaviour of the propagated
field describes the steady-state behaviour of the signal. The
arrival of this signal contribution is determined by the dynamics
of that dominant saddle point that becomes exponentially
negligible in comparison to this pole contribution.
In a single-resonance Lorentz model dielectric the
asymptotic theory [9–11] shows that the propagated field
described in equation (29) may be expressed in the form
E y (z, t ) = A S (z, t ) + A B (z, t ) + Ac (z, t)
(38)
as z → ∞, with a similar expression for Bx (z, t ). The
asymptotic behaviour of the component field A S (z, t ) is due to
the pair of distant saddle points [9, 11]
∼
ω±
S PD (θ ) = ±ξ(θ ) − iδ(1 + η(θ ))
(39)
with ξ(θ ) = (ω02 − δ 2 + b2 θ 2 /(θ 2 − 1))1/2 and η(θ ) =
(δ 2 /27 + b2 /(θ 2 − 1))/ξ 2 (θ ), and is referred to as the first or
Sommerfeld precursor field. The front of the Sommerfeld
precursor arrives at θ = 1 with zero amplitude and an
infinite instantaneous angular frequency. As θ increases from
Dispersive pulse dynamics and associated pulse velocity measures
unity the amplitude rapidly builds to a maximum value and
thereafter decays as the attenuation factor increases and the
instantaneous oscillation frequency chirps downward towards
the upper frequency edge of the material absorption band. The
asymptotic behaviour of the component field A B (z, t ) is due
to the near saddle points [9, 11]
ω+S PN (θ ) ∼
= i(|ψ(θ )| − (2/3)δζ (θ ))
(40a)
0.15
(a)
0.1
0.05
0
– 0.05
for 1 < θ θ1 , while for θ θ1 ,
∼
ω±
S PN (θ ) = ±ψ(θ ) − (2/3)iδζ (θ )
(40b)
where ζ (θ ) = (3/2)ς(θ ) and ψ(θ ) = (ω02 (θ 2 − θ02 )/(θ 2 − θ02 +
3αb2 /ω02 )−δ 2 ς 2 (θ ))1/2 with ς(θ ) = (θ 2 −θ02 +2b2 /ω02 )/(θ 2 −
θ02 + 3αb2 /ω02 ), and is referred to as the second or Brillouin
precursor field. Here
– 0.1
4
0.04
5
6
7
8
9
t (s)
10
11
12
13 14
x10–15
(b)
0.03
θ0 = n(0) = (1 + b2 /ω02 )1/2
(41)
θ1 ∼
= θ0 + 2δ 2 b2 /(θ0 ω02 (3αω02 − 4δ 2 ))
0.01
with α = 1 − δ 2 (4ω12 + b2 )/(3ω02 ω12 ). As the near saddle point
ω+S PN (θ ) moves down the imaginary axis for 1 < θ θ1 ,
as described by equation (40a), the Brillouin precursor is
nonoscillatory over this space-time domain and reaches a peak
amplitude near the space-time point θ = θ0 where there is no
exponential attenuation. As θ increases above θ1 the Brillouin
precursor becomes oscillatory with an instantaneous angular
frequency that chirps upward towards the lower frequency
edge of the absorption band with decreasing amplitude as
the attenuation factor monotonically increases. The final
contribution Ac (z, t ) appearing in equation (38) is due to
the simple pole singularity given by equation (37). This
contribution to the asymptotic behaviour of the propagated
field describes the steady state behaviour of the propagated
signal that oscillates at the input carrier frequency ω = ωc .
The Sommerfeld precursor arrives at the vacuum speed of
light c and dominates the initial evolution of the propagated
field.
The Brillouin precursor becomes exponentially
dominant over the Sommerfeld precursor at the space-time
point θ = θ S B where [9, 11]
θ S B ≈ θ0 −
4δ 2 b2
3θ0 ω04
(42)
at which point the oscillation frequency of the Sommerfeld
precursor has decreased to the value [9, 11]
ωS B ∼
= ξ(θ S B ) ≈ ω0
2 1/2
b2 5δ
2+ 2 +
ω0 3ω02
0.02
.
(43)
The dynamical evolution of the propagated signal due to an
input Heaviside unit step function signal then separates into
two cases depending upon the value of the carrier frequency
ωc in comparison with the value ω S B that is a characteristic of
the dispersive material.
For 0 < ωc < ω S B the dynamical field evolution is first
dominated by the Sommerfeld precursor over the space-time
domain 1 θ < θ S B , is next dominated by the Brillouin
precursor over the space-time domain θ S B < θ < θc and is
finally dominated by the pole contribution for all θ > θc , as
0
– 0.01
– 0.02
– 0.03
– 0.04
5
6
7
8
t (s)
9
10
11
x10–15
Figure 4. Dynamical field evolution due to an input unit step
function modulated signal with (a) below the resonance carrier
frequency (ωc = ω0 /2) at five absorption depths (z = 5z d ), and
(b) above the resonance carrier frequency (ωc = 2.5ω0 > ω S B ) at
five absorption depths (z = 5z d ) in the single-resonance Lorentz
model dielectric considered in figure 1.
illustrated in figure 4(a), where θc denotes the space-time value
at which the exponential attenuation of the Brillouin precursor
first equals and thereafter remains greater than that at the carrier
frequency ωc . The propagated signal is then characterized
by several distinct features: the arrival of the Sommerfeld
precursor front at θ = 1 which travels at the velocity v S = c,
the peak of the Brillouin precursor at θ = θ0 which travels at
the velocity v B = c/θ0 = c/n(0), and the arrival of the main
signal at θ = θc which travels at the main signal velocity
vc (ωc ) =
c
θc
(44)
where 0 < vc v B .
For ωc > ω S B the dynamical field evolution is first
dominated by the Sommerfeld precursor over the space-time
domain 1 θ < θc1 , by the pole contribution over the spacetime domain θc1 < θ < θc2 , by the Brillouin precursor over
the space-time domain θc2 < θ < θc , and is finally dominated
by the pole contribution for all θ > θc , as illustrated in
figure 4(b). Here θc1 denotes the space-time value at which
the exponential attenuation of the Sommerfeld precursor first
equals and thereafter remains greater than that at the carrier
frequency ωc , and θc2 denotes the space-time value at which the
exponential attenuation of the pole contribution first equals and
S131
K E Oughstun and N A Cartwright
intensity centroid of the pulse. A variant of this velocity
measure which is defined in terms of the Poynting vector of
the pulse has recently been introduced by Peatross, Glasgow
and Ware [35, 36]. The temporal centre of the Poynting vector
S (z, t ) of a plane wave pulse propagating in the positive z
direction is defined as
∞
∞
t S (z, t) dt
ẑ ×
S (z, t) dt (47)
tz ≡ ẑ ×
Relative Signal & Energy Velocities
1
SB
0.9
0.8
0.7
0.6
0.5
0.4
−∞
0.3
0.2
0.1
0
0
5
10
15
−∞
where ẑ denotes the unit vector along the propagation
direction. If tz 0 denotes the temporal centroid of the Poynting
vector of the initial pulse at the input plane at z = z 0 , then the
pulse centro-velocity is given by [35]
vcentro =
Figure 5. Angular frequency dependence of the relative energy
velocity ν E (ωc )/c (solid curve) and the relative anterior presignal
velocity νc1 (ωc )/c, posterior presignal velocity νc2 (ωc )/c, and main
signal velocity νc (ωc )/c branches (open circles) for the
single-resonance Lorentz model dielectric considered in figure 1.
then becomes greater than that for the Brillouin precursor for
a finite θ interval. The propagated signal is now characterized
by several distinct features: the arrival of the Sommerfeld
precursor front at θ = 1 which travels at the velocity v S = c,
the first arrival of the signal at θ = θc1 which travels at the
anterior presignal velocity
c
vc1 (ωc ) =
(45)
θc1
where c > vc1 > c/θ S B , the arrival of the Brillouin precursor
at θ = θc2 which travels at the posterior presignal velocity
c
vc2 (ωc ) =
(46)
θc2
where c/θ S B > vc2 > c/θ0 , and finally the arrival of the
main signal which travels at the main signal velocity given
by equation (44).
The frequency dependence of the signal velocity in
a single-resonance Lorentz model dielectric is illustrated
in figure 5. Each sequence of open circles describes a
separate branch of the signal velocity obtained from a series
of numerical calculations given in table 9.1 of [11]: the
anterior presignal velocity vc1 (ωc ), the posterior presignal
velocity vc2 (ωc ) and the main signal velocity vc (ωc ). The
solid curve in the figure describes Loudon’s energy velocity
v E (ωc ), which is seen to form an upper envelope to the
signal velocity values [9, 11]. Based upon this important
result, a new physical description of dispersive pulse dynamics
has been given [11, 13, 14] in terms of the energy velocity
and attenuation of a time-harmonic plane wave in the
causally dispersive dielectric. This new description accurately
describes all of the features that occur in the evolution of
an ultrawideband pulse as it propagates through a dispersive
material, including the precursor fields, and reduces to the
group velocity description in the limit of zero loss.
6. The pulse centroid velocity
In 1970 Smith [33] proposed a new pulse velocity measure
called the centro-velocity which described the motion of the
S132
z − z0
.
tz − tz 0
(48)
The integrals appearing in equation (47) may be transformed
into the angular frequency domain upon application of the
Parseval–Plancherel theorem with the result that [35]
tz = T {Ẽ (z, ω)}
∞
≡ − i ẑ ×
∂ Ẽ (z, ω)/∂ω × H̃ ∗ (z, ω) dω
× ẑ ×
−∞
∞
−∞
S̃ (z, ω) dω
−1
.
(49)
The Poynting vector centroid delay may then be expressed in
the form
(50)
tz − tz 0 = G z + Rz 0
where
G z ≡ (∂β(ω)/∂ω)z
∞
S̃(z, ω)(∂β(ω)/∂ω) dω
= z
−∞
∞
−∞
S̃(z, ω) dω
(51)
is the group delay energy centroid, and where
Rz 0 ≡ T {Ẽ (z 0 , ω) exp(−α(ω)z)} − T {Ẽ (z 0 , ω)}
(52)
is called the pulse reshaping delay. As stated in the abstract
of their paper [35], Peatross, Glasgow and Ware state that this
‘result provides a context wherein group velocity is always
meaningful even for broad band pulses and when the group
velocity is superluminal or negative. The result imposes
superluminality on sharply defined pulses’.
The asymptotic representation of the propagated field
given in equation (38) shows that, as an ultrawideband pulse
propagates into a Lorentz model dielectric, the field becomes
dominated by the Brillouin precursor whose peak amplitude
travels at the velocity
v B = c/θ0 = c/n(0).
(53)
Hence, as the propagation distance increases, the pulse centrovelocity asymptotically approaches this characteristic velocity
of the Brillouin precursor, so that
lim vcentro = v B .
z→∞
(54)
The evolution of the centro-velocity for an input ten-cycle
rectangular envelope pulse for a wide range of input pulse
Dispersive pulse dynamics and associated pulse velocity measures
1
relative centrovelocity
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
1
2
3
4
5
6
7
8
9
10
Figure 6. Behaviour of the relative centro-velocity νcentro (ωc )/c
with relative propagation distance z/z d for an input ten-cycle
rectangular envelope pulse for several values of the input pulse
carrier frequency ωc in the single-resonance Lorentz model
dielectric considered in figure 1.
frequencies in a single-resonance Lorentz model dielectric is
illustrated by the family of curves in figure 6. Each open circle
in the figure for z/z d 1 describes a data point for the
centro-velocity determined from a numerical experiment that
was based upon two independent techniques (an asymptotic
code and an FFT-based code). Each asterisk for z/z d = 0
describes the relative group velocity value vg (ωc )/c at that
particular frequency. For each carrier frequency considered,
the centro-velocity asymptotically approaches the subluminal
limiting value v B /c = 1/θ0 = 0.667 set by the peak velocity
of the Brillouin precursor as the propagation distance increases
in the mature dispersion regime (typically z/z d 1 outside
of the absorption band). Superluminal centroid velocity results
are only observed for carrier frequency values within the
absorption band for sufficiently small propagation distances.
7. Discussion
The velocity of an electromagnetic pulse as it propagates
through a causally dispersive material is a fundamental
problem in physics that has remained unresolved for nearly a
century. Nevertheless, a significant body of published research
has now been established that shows that:
(1) any study of the pulse velocity of an ultrawideband pulse
in a dispersive material must give careful consideration to
the material dispersion, including the attenuation, over the
entire frequency domain, and
(2) any conclusions regarding either ultrawideband or
ultrashort pulse dynamics, including the pulse velocity,
that are based upon the quadratic or any higher-order
dispersion approximation should be viewed with extreme
scepticism for propagation distances that exceed a single
absorption depth (at some characteristic frequency of the
initial pulse) in the material.
The detailed results presented here show that the only
physically meaningful velocity measures for dispersive pulse
propagation are the energy velocity, signal velocity and the
pulse centro-velocity. Although the energy velocity has only
been derived for a monochromatic plane wave in a Lorentz
model dielectric, it is an essential ingredient in the new physical
description of dispersive pulse dynamics [11, 13, 14]. This
description of the pulse evolution in terms of the attenuation
and energy velocity of each spectral component present in
the initial pulse spectrum provides a detailed representation
of each feature present in the propagated pulse, including the
precursor fields. The signal velocity describes the point at
which the propagated pulse begins to predominantly oscillate
at the input carrier frequency of the pulse [37]; consequently, it
can be measured in a carefully defined laboratory experiment.
This causal velocity measure is fundamental to the problem
of signal (and hence information) propagation through a
dispersive material or system.
Claims of superluminal pulse propagation [38–41] are
primarily based upon the group velocity approximation which
has been proven [28, 29] to be invalid in the mature
dispersion limit that is typically achieved when the propagation
distance exceeds a single absorption depth in the dispersive
material. Superluminal peak velocities can briefly occur for
such temporally unbounded pulse envelope functions as the
Gaussian envelope function [42–44]. For such an ultrashort
Gaussian envelope pulse, the peak velocity is found [43, 44]
to move along the group velocity curve in such a manner
that as the propagation distance increases, the peak velocity
approaches the energy velocity [43, 44]. Although the peak
velocity may briefly exceed the vacuum speed of light (or
may even be negative) during the initial evolution of the
pulse, this is primarily due to pulse reshaping and does
not entail the transmission of either energy or information
superluminally [22, 45]. That any energy or information is
not transmitted superluminally follows immediately through
consideration of a unit step function modulated Gaussian pulse.
In that case, the exact result expressed in equation (31) directly
applies and no information can have been transmitted faster
than the vacuum speed of light. Furthermore, as pointed out
by Landauer [22], ‘an emerging peak is not necessarily related
to the incident peak in a causative physical way’. Such is
the case in ultrashort Gaussian pulse propagation [44] where
the oscillation frequency at the peak amplitude point changes
continuously as the pulse evolves.
Finally, the pulse centro-velocity recently introduced
by Peatross, Glasgow and Ware [35] provides a convenient
measure of the pulse velocity for temporal pulses with a
sharply defined envelope function. However, their assertion
that superluminality is imposed on sharply defined pulses
requires further careful study. The delta function pulse is
certainly sharply defined, yet its propagated field identically
vanishes over the entire space-time domain ct/z < 1. The
same result holds for a rectangular envelope pulse of arbitrarily
short initial temporal width [46].
These results then establish that:
(1) superluminal peak velocities in causally dispersive
materials can momentarily occur for sufficiently small
propagation distances in the immature dispersion regime
when ω0 ωc ω1 ,
(2) however, as the propagation distance increases into the
mature dispersion regime, the pulse centro-velocity for an
ultrashort pulse becomes subluminal and approaches the
subluminal velocity of the dominant precursor field. As
S133
K E Oughstun and N A Cartwright
tempting as it may seem, pulse reshaping does not imply
either superluminal energy or superluminal information
transmission. In the light of these results,
(3) superluminal energy and information transfer is not
physically possible within the framework of the Maxwell–
Lorentz theory in linear, causally dispersive systems.
Acknowledgments
The research presented in this paper has been supported,
in part, by the United States Air Force Office of Scientific
Research under grant no 49620-01-1-0306. The first author
(KEO) would like to express his gratitude to Professor JeanJacques Greffet for his successful effort in organizing the
highly productive meeting on Electromagnetic Optics in Paris
this past summer that resulted in the preparation of this paper.
It was my great pleasure to finally meet both him and Professor
Rodney Loudon during this meeting.
References
[1] Einstein A 1905 Zur Elektrodynamik Bewegter Körper Ann.
Phys., Lpz. 17 891–921
[2] Lorentz H A 1906 Versuch einer Theorie der Electrischen und
Optischen Erscheinungen in Bewegten Körpern (Leipzig:
Teubner) see also
Lorentz H A 1909 The Theory of Electrons (Leipzig: Teubner)
[3] Sommerfeld A 1914 Über die Fortpflanzung des Lichtes in
disperdierenden Medien Ann. Phys., Lpz. 44 177–202
[4] Brillouin L 1914 Über die Fortpflanzung des Licht in
disperdierenden Medien Ann. Phys., Lpz. 44 204–40
[5] Brillouin L 1960 Wave Propagation and Group Velocity (New
York: Academic)
[6] Lord Rayleigh 1877 On progressive waves Proc. London
Math. Soc. IX 21–6
[7] Lord Rayleigh 1881 On the velocity of light Nature XXIV
52–5
[8] Baerwald H G 1930 Über die fortpflanzung von signalen in
dispergierenden systemen Ann. Phys. 7 731–60
[9] Oughstun K E and Sherman G C 1988 Propagation of
electromagnetic pulses in a linear dispersive medium with
absorption (the Lorentz medium) J. Opt. Soc. Am. B 5
817–49
[10] Oughstun K E and Sherman G C 1989 Uniform asymptotic
description of electromagnetic pulse propagation in a linear
dispersive medium with absorption (the Lorentz medium) J.
Opt. Soc. Am. A 6 1394–420
[11] Oughstun K E and Sherman G C 1994 Electromagnetic Pulse
Propagation in Causal Dielectrics (Berlin:
Springer-Verlag)
[12] Loudon R 1970 The propagation of electromagnetic energy
through an absorbing dielectric J. Phys. A: Math. Gen. 3
233–45
[13] Sherman G C and Oughstun K E 1981 Description of pulse
dynamics in Lorentz media in terms of the energy velocity
and attenuation of time-harmonic waves Phys. Rev. Lett. 47
1451–4
[14] Sherman G C and Oughstun K E 1995 Energy velocity
description of pulse propagation in absorbing, dispersive
dielectrics J. Opt. Soc. Am. B 12 229–47
[15] Born M and Wolf E 1959 Principles of Optics (Oxford:
Pergamon)
[16] Eckart C 1948 The approximate solution of one-dimensional
wave equations Rev. Mod. Phys. 20 399–417
[17] Whitham G B 1974 Linear and Nonlinear Waves (New York:
Wiley)
[18] Jackson J D 1975 Classical Electrodynamics (New York:
Wiley)
S134
[19] Lindsay R B 1960 Mechanical Radiation (New York:
McGraw-Hill)
[20] Bloembergen N 1965 NonlinearOptics (New York: Benjamin)
[21] Akhmanov S A, Vysloukh V A and Chirkin A S 1992 Optics
of Femtosecond Laser Pulses (New York: AIP)
[22] Landauer R 1993 Light faster than light? Nature 365 692–3
[23] Stratton J A 1941 Electromagnetic Theory (New York:
McGraw-Hill)
[24] Jones J 1974 On the propagation of a pulse through a
dispersive medium Am. J. Phys. 42 43–6
[25] Anderson D, Askne J and Lisak M 1975 Wave packets in an
absorptive and strongly dispersive medium Phys. Rev. A 12
1546–52
[26] Butcher P N and Cotter D 1990 The Elements of Nonlinear
Optics (Cambridge: Cambridge University Press) ch 2
[27] Akhmanov S A, Vysloukh V A and Chirkin A S 1992 Optics
of Femtosecond Laser Pulses (New York: AIP) ch 1
[28] Oughstun K E and Xiao H 1997 Failure of the
quasimonochromatic approximation for ultrashort pulse
propagation in a dispersive, attenuative medium Phys. Rev.
Lett. 78 642–5
[29] Xiao H and Oughstun K E 1999 Failure of the group velocity
description for ultrawideband pulse propagation in a double
resonance Lorentz model dielectric J. Opt. Soc. Am. B 16
1773–85
[30] Oughstun K E and Shen S 1988 Velocity of energy transport
for a time-harmonic field in a multiple-resonance Lorentz
medium J. Opt. Soc. Am. B 5 2395–8
[31] McCall S L and Hahn E L 1969 Self-induced transparency
Phys. Rev. 183 457–85
[32] Allen L and Eberly J H 1975 Optical Resonance and
Two-Level Atoms (New York: Wiley) ch 4
[33] Smith R L 1970 The velocities of light Am. J. Phys. 38 978–84
[34] Olver F W J 1970 Why steepest descents? SIAM Rev. 12
228–47
[35] Peatross J, Glasgow S A and Ware M 2000 Average energy
flow of optical pulses in dispersive media Phys. Rev. Lett.
84 2370–3
[36] Ware M, Glasgow S A and Peatross J 2001 Role of group
velocity in tracking field energy in linear dielectrics Opt.
Express 9 506–18
[37] Oughstun K E, Wyns P and Foty D P 1989 Numerical
determination of the signal velocity in dispersive pulse
propagation J. Opt. Soc. Am. A 6 1430–40
[38] Chu S and Wong S 1982 Linear pulse propagation in an
absorbing medium Phys. Rev. Lett. 48 738–41
[39] Laude V and Tournois P 1999 Superluminal asymptotic
tunneling times through one-dimensional photonic
bandgaps in quarter-wave-stack dielectric mirrors J. Opt.
Soc. Am. B 16 194–8
[40] Milonni P W, Furuya K and Chiao R Y 2000 Quantum theory
of superluminal pulse propagation Opt. Express 8 59–65
[41] Dogariu A, Kuzmich A, Cao H and Wang L J 2001
Superluminal light pulse propagation via rephasing in a
transparent anomalously dispersive medium Opt. Express 8
344–50
[42] Garrett C G B and McCumber D E 1970 Propagation of a
gaussian light pulse through an anomalous dispersion
medium Phys. Rev. A 1 305–13
[43] Oughstun K E and Balictsis C M 1996 Gaussian pulse
propagation in a dispersive, absorbing dielectric Phys. Rev.
Lett. 77 2210–13
[44] Balictsis C M and Oughstun K E 1997 Generalized asymptotic
description of the propagated field dynamics in gaussian
pulse propagation in a linear, causally dispersive medium
Phys. Rev. E 55 1910–21
[45] Diener G 1996 Superluminal group velocities and information
transfer Phys. Lett. A 223 327–31
[46] Oughstun K E and Sherman G C 1990 Uniform asymptotic
description of ultrashort rectangular optical pulse
propagation in a linear, causally dispersive medium Phys.
Rev. A 41 6090–113