Download Energy-velocity description of pulse propagation

G. C. Sherman and K. E. Oughstun
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
229
Energy-velocity description of pulse
propagation in absorbing, dispersive dielectrics
George C. Sherman
Department of Physics, Santa Barbara City College, Santa Barbara, California 93109-2394
Kurt Edmund Oughstun
Department of Computer Science and Electrical Engineering and Department of Mathematics and Statistics,
University of Vermont, Burlington, Vermont 05405-0156
Received January 19, 1994; revised manuscript received July 28, 1994
The evolution of an electromagnetic pulse propagating through a linear, dispersive, and absorbing dielectric (as
predicted by the modern asymptotic extension of the classic theory of Sommerfeld and Brillouin) is described
in physical terms. The description is similar to the group-velocity description known for plane-wave pulses
propagating through lossless, gainless, dispersive media but with two modifications: (1) the group velocity
is replaced by the velocity of energy in time-harmonic waves, and (2) a nonoscillatory component is added
that consists of a wave that grows exponentially with time with a time-dependent growth rate. In the
nonoscillatory component the growth rate at each space – time point is determined by the velocity of energy
in exponentially growing waves in the medium. The new description provides, for the first time to our
knowledge, a physical explanation of the localized details of pulse dynamics in dispersive and absorbing
dielectric media and a simple mathematical algorithm for quantitative predictions. Numerical comparisons
of the results of the algorithm with the exact integral solution are presented for a highly transparent dielectric
and for a highly absorbing dielectric. In both cases the agreement is excellent.
1.
INTRODUCTION
The effects of dispersion and absorption on the evolution
of an electromagnetic pulse as it propagates through a homogeneous linear dielectric are predicted quantitatively
by a classic theory developed originally by Sommerfeld1
and Brillouin2,3 and described currently in advanced textbooks on electrodynamics.4,5 The results show that after
the pulse has propagated sufficiently far in the material
its dynamics settle into a relatively simple regime (known
as mature dispersion) for the rest of the propagation. In
this regime the field becomes locally quasi monochromatic
with fixed local frequency, wavelength, and attenuation in
small regions of space that move with their own characteristic constant velocity. The theory provides approximate
analytic expressions for the local wave properties to be expected at a given space – time point of observation. The
expressions are complicated, however, and neither the results nor their derivations have provided insight into the
physical reasons for the field’s having those particular local properties at the given observation point.
The mature-dispersion regime is well known in the
theory of propagation of rather general linear waves in
homogeneous dispersive media in which there is no absorption or gain for any frequency. It is exhibited by all
waves whose monochromatic components are described by
the Helmholtz equation with a real propagation constant,
such as electromagnetic, acoustic, elastic, and gravity
waves in lossless, gainless, linear systems. Furthermore, a physical explanation is available for the local
properties of all these waves based on the concept of the
group velocity of time-harmonic waves.6–8 When absorp0740-3224/95/020229-19$06.00
tion or gain is present in the medium for some range
of frequencies, however, the group-velocity description
breaks down, and no general physical description of pulse
dynamics is known.9 This is a severe limitation of a
basic kind, because a dispersive system that is lossless
and gainless for all frequencies is noncausal.10 Hence
we are left in an awkward position. Our general physical understanding of the details of pulse propagation in
dispersive systems is confined to the case of no absorption
or gain, which is itself fundamentally unphysical.
In 1981 we published a physical explanation of the local
wave properties of electromagnetic pulses propagating in a dispersive, absorbing dielectric in the maturedispersion regime.11 The explanation is similar to the
group-velocity description valid for lossless, gainless media except that the group velocity is replaced by the
velocity of energy in time-harmonic waves and the attenuation of the waves is included. To our knowledge
this is the only such physical description of the details of
pulse dynamics that is known for any dispersive system
that includes absorption or that is causal.
Our energy-velocity description referred to above was
published previously without proof or derivation. The
derivation had to await the publishing of extensions we
had made to Sommerfeld’s and Brillouin’s classic theory.
The required extensions are now available.12,13 Hence
we derive the energy-velocity description in the present
paper and add an additional physical contribution to the
description to extend its validity into the small portion of
the pulse that was not included in the original formulation
(the beginning of the Brillouin precursor).
In addition to explaining the details of the local behav1995 Optical Society of America
230
J. Opt. Soc. Am. B / Vol. 12, No. 2 / February 1995
G. C. Sherman and K. E. Oughstun
ior of the pulse in physical terms, the energy-velocity description provides a simple mathematical algorithm for
predicting those details quantitatively. We implement
that algorithm numerically in this paper for both a highly
absorbing dielectric and a highly transparent dielectric
and compare the results with the exact integral solution
for the field.
2. EXACT PROPAGATION INTEGRAL AND
ITS ASYMPTOTIC APPROXIMATION
In this section we specify the physical problem of interest
and present the exact integral solution and its asymptotic
approximation. The derivations of these results can be
found in Refs. 3 and 12. For convenience we apply the
notation used in Ref. 11 with two minor modifications.14
Since we are interested in the effects of dispersion
and absorption on the propagation of the pulse (rather
than in the effects of other phenomena such as diffraction and scattering), we consider the propagation of a
one-dimensional wave in a homogeneous, isotropic, linear
dielectric. Consider a plane electromagnetic wave with
real electric field Esz, td linearly polarized along the x axis
and traveling in the positive z direction in the dielectric,
where the dielectric occupies the half-space z . 0. The
field is taken to be zero for t , 0 for all z $ 0 and therefore
can be expressed in a Laplace representation as
1 Z ia1`
Esz, td ­
Esz, vdexps2ivtddv ,
2p ia2`
(2.2)
in the half-space z . 0.
The complex propagation constant ksvd is given in
terms of the complex index of refraction nsvd by ksvd ­
vnsvdyc, where c is the speed of light in vacuum. We
follow the classic theory in applying the Lorentz model,
which models the dielectric as consisting of many oscillating point dipoles with a single resonance frequency v0
and phenomenological damping constant d. According to
this model, the complex index of refraction is given by
√
nsvd ­ 1 2
2
b
v 2 2 v0 2 1 2dvi
!1/2
d ­ 2.8 3 10
15
21
s
,
(2.3)
v0 ­ 4 3 1016 s21 .
(2.4a)
,
(2.5)
where f std is a real function that satisfies f std ­ 0 for
t , 0. The case treated by Sommerfeld and Brillouin
was the special case f std ­ sinsvc td for t . 0. We are
concerned here with f std that is zero for all times after
some finite positive time. The case of primary interest
is the delta-function pulse having f std ­ Adstd, where A
is a constant and dstd is the Dirac delta function (not to
be confused with the damping constant d). In that case
Esz, tdyA is the impulse response of the medium.
The exact integral solution to this boundary-value problem can be written as
∑
∏
1 Z ai1` ˜
z
Esz, td ­
f svdexp
fsvd dv ,
2p ai2`
c
(2.6)
where
f˜svd ­
Z
`
2`
f stdexpsivtddt ,
fsvd ­ ivfnsvd 2 ug ,
(2.7)
(2.8)
u ­ ctyz .
(2.9)
For the delta-function pulse, the pulse spectrum f˜svd is
simply A.
Sommerfeld showed that it follows from Eq. (2.6) that
Esz, td ­ 0 for u , 1, thereby proving that the field is
relativistically causal3 (i.e., it does not travel faster than
c). Brillouin showed that as z tends to infinity with fixed
u . 1 the asymptotic approximation of the field is15
Esz, td , Es sz, td 1 Eb sz, td ,
(2.10)
where
Es sz, td ­
∑
∏
p ˜
c f svs d
z
fsv
exp
d
s
f22pzfs2d svs dg1/2
c
∑
∏
p ˜
z
c f s2vs p d
pd ,
fs2v
exp
1
s
f22pzf s2d s2vs p dg1/2
c
(2.11)
∑
∏
c f˜svb d
z
Eb sz, td ­
fsv
exp
d
b
f22pzf s2d svb dg1/2
c
p
where b is the plasma frequency of the medium. For
most of our numerical computations we apply the same
numerical values for the medium parameters v0 , d, and
b as used by Brillouin:
b2 ­ 2 3 1033 s22 ,
Es0, td ­ f std ,
(2.1)
where a is a positive constant. The contour of integration
for the integral in Eq. (2.1) is a straight horizontal line
in the complex v plane at a distance a above the real
axis. The spectral amplitude Esz, vd satisfies the scalar
Helmholtz equation
=2 Esz, vd 1 k2 svdEsz, vd ­ 0
the same parameter values, except that d is taken to
be nearly zero, corresponding to a highly transparent
medium.
Let the field satisfy the boundary value
(2.4b)
(2.4c)
These values correspond to a highly absorbing medium.
For comparison we also include some computations with
for 1 , u , u1 , (2.12a)
∑
∏
c f˜svb d
z
fsv
exp
d
Eb sz, td ­
b
f22pzf s2d svb dg1/2
c
∑
∏
p ˜
z
c f s2vb p d
pd
fs2v
exp
1
b
f22pzf s2ds2vb p dg1/2
c
p
for u . u1 . (2.12b)
Here and throughout the paper we apply the notation
gsnd svd to indicate the nth derivative of an arbitrary function gsvd with respect to v evaluated at va and the superscript * to indicate the complex-conjugate operation.
G. C. Sherman and K. E. Oughstun
Fig. 1.
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
Path traced out by the saddle points with increasing u in the complex v plane.
The quantities vs and vb are saddle points that are complex solutions to
fs1d svd ­ 0 .
(2.13)
These saddle points are functions of u. It is shown in
Ref. 12 that they trace out the paths in the complex v
plane that are indicated in Fig. 1 as u increases from
1. The saddle point vb moves down the positive imaginary axis until u reaches a value u1 , which depends on
the medium parameters. For u increasing beyond u1 , vb
moves into the fourth quadrant as shown in the figure.
The above results are referred to as nonuniform asymptotic results because they break down at two critical values of the parameter u. In particular, the right-hand
sides of Eqs. (2.12) become infinite when u ­ u1 and give
different asymptotic behaviors on opposite sides of those
values because f s1d svb d ­ 0 there. Uniform asymptotic
results that do not exhibit this behavior are derived in
Ref. 13. We employ the nonuniform results for our initial analysis here because they are much simpler to understand. The uniform results are applied in the final
analysis at the end of the paper, however, to yield an
energy-velocity description that is valid for all u.
It is shown in Appendix A that the second terms on the
right-hand sides of Eqs. (2.11) and (2.12b) are the complex
conjugates of the corresponding first terms. Also, it is
easily verified that the quantity on the right-hand side
of Eq. (2.12a) is real, since fsvd is real for imaginary v.
Hence it follows from relations (2.10) – (2.12) that Esz, td
can be expressed as the real part of a complex function
Asz, td:
Esz, td ­ RefAsz, tdg ,
(2.14)
with the asymptotic approximation given by
Asz, td , As sz, td 1 Ab sz, td ,
(2.15)
with
As sz, td ­
∑
∏
p
2 c f˜svs d
z
fsv
exp
d
,
s
f22pzf s2d svs dg1/2
c
(2.16)
231
Dashed curve, vs ; dotted curve, vb .
∑
∏
p ˜
z
c f svb d
fsvb d
exp
Ab sz, td ­
f22pzf s2d svb dg1/2
c
for 1 , u , u1 , (2.17a)
∑
∏
p
2 c f˜svb d
z
fsv
Ab sz, td ­
exp
d
b
f22pzf s2d svb dg1/2
c
for u . u1 . (2.17b)
Equations (2.14) –(2.17) predict the behavior of the
pulse after it has traveled far enough to enter the maturedispersion regime. The details of the dynamics are determined by the motion of the saddle points in the complex
v plane with increasing u. Crude analytic approximations for the saddle-point locations as functions of u are
derived in Ref. 3, with substantial improvements derived
in Ref. 12. Unfortunately, the formulas are complicated,
and neither the results nor their derivations provide insight into the physical reasons behind the pulse behavior.
3. APPROXIMATIONS HAVING
PHYSICAL INTERPRETATIONS
In this section we obtain approximations of Eqs. (2.16)
and (2.17) that do have physical interpretations and that
provide a simple physical model of pulse dynamics in
the mature-dispersion regime. The approximations are
valid for d much smaller than v0 and b. This requires
that the medium not be too highly absorbing. We show
in Section 4 that the requirement is not overly restrictive, however, by demonstrating numerically that the approximations are sufficiently accurate for a medium (the
Lorentz medium with the parameter values studied by
Brillouin) that is so absorbing that the material would be
considered to be opaque.
To obtain the desired simplifications, we replace the
saddle points in the asymptotic expressions by other frequencies that yield approximately the same results but
that have clearer physical interpretations. In particular, we replace the saddle points in Eqs. (2.16) and
(2.17b) with specific real frequencies, leading to quasitime-harmonic (quasi-monochromatic) waves with a local
frequency, phase, and amplitude that are easily understood in physical terms. Similarly, we replace the saddle
232
J. Opt. Soc. Am. B / Vol. 12, No. 2 / February 1995
G. C. Sherman and K. E. Oughstun
point in Eq. (2.17a) with a specific purely imaginary frequency, leading to a nonoscillatory field with a local
amplitude and growth rate that are easily understood in
physical terms.
A. Quasi-Monochromatic Contribution
To identify the real frequencies of interest, we focus our
attention on the attenuation of the field with increasing z.
It is important that our approximation have the correct
attenuation, since we are interested in the properties of
an exponentially decaying field after it has propagated
a large distance. Hence we search for time-harmonic
waves (with real frequencies) that are attenuated in the
medium at the same rate as the fields given in Eqs. (2.16)
and (2.17b). First let us define the notation that gr and
gi , respectively, represent the real and the imaginary
parts of the arbitrary complex quantity g. Then, for fixed
u, the attenuation with increasing z of a wave of the
form expfzfsvdycg for complex v is determined by f r svd.
For a given u, therefore, we define vEj to be the real
frequencies nearest the saddle points vj that satisfy
f r svEj d ­ f r svj d
(3.1)
with j ­ s, b. Then a time-harmonic plane wave with
real frequency vEj has the same attenuation as the pulse
in the mature-dispersion regime given in Eqs. (2.16) and
(2.17b). The locations of these frequencies in the complex
v plane are indicated in Fig. 2 (for some value of u greater
than u1 ) as the intersections with the real axis of the
contours of constant f r svd that pass through the saddle
points.
The frequencies vEj so defined are intimately connected to the physics of the propagation of time-harmonic
waves in the medium. It is shown in Appendix B that,
to a good approximation, they satisfy
VE svEj d ­ zyt
(3.2)
for d much smaller than v0 and b, where VE svd is the
velocity of energy in time-harmonic waves (with real frequencies) defined by
VE svd ­ Ssvdyusvd .
(3.3)
Here Ssvd is the time-averaged magnitude of the
Poynting vector and usvd is the time-averaged energy
density. Loudon16 has shown that VE svd is given by
VE svd ­
c
.
nr svd 1 ni svdvyd
(3.4)
We have now established, for what is the first time to
our knowledge, a connection between the attenuation of
the pulse and the physics of the problem. In particular
we have shown that in the mature-dispersion regime the
attenuation of As sz, td as the point of observation moves
with fixed velocity (i.e., for fixed u) is approximately the
same as the attenuation of a time-harmonic wave with
real frequency that has an energy velocity equal to the
velocity of the point of observation. The same is true
of Ab sz, td for u . u1 . Although this result does not tell
us anything about the phase of the field or the amplitude of the exponential term, it does give a description of
the main dynamics of the energy of the pulse in physical
terms. Moreover, it provides a simpler mathematical algorithm for calculating those dynamics than was available
previously.
We now extend the result to include the rest of the
dynamics of the pulse. First we deal with the amplitude of the exponential term. If the damping constant
d is not too large, the slowly varying functions of v in
Eqs. (2.16) and (2.17b) can be approximated by replacement of vj with vEj . We can see this by applying, as
follows, some general properties of the saddle-point locations (see Refs. 3 and 12). The saddle point vs is less
than 2d below the real axis, and its real part is greater
than fv0 2 1 b2 2 d 2 g1/2 . Even for the highly absorbing
medium treated by Brillouin, the former value is much
smaller than the latter. Hence the imaginary part of the
saddle point can be neglected compared with its real part.
The saddle point vb is less than d below the real axis,
and its real part starts at zero for u ­ u1 and increases
rapidly toward fv0 2 2 d 2 g1/2 for increasing u. Again the
numbers are such that even for Brillouin’s parameter values the imaginary part of vb can be neglected compared
with the real part in slowly varying functions for u sufficiently large. These approximations are found to be rea-
Fig. 2. Location in the complex v plane of the real frequencies vEj relative to the locations of the saddle points vj for a fixed value
of u greater than u1 . Dashed curves, the contours of constant f r svd that pass through the saddle points and cross the real axis.
G. C. Sherman and K. E. Oughstun
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
sonably accurate provided that the inequality dyv0 & 0.1
is satisfied (for the highly absorptive material that is described by Brillouin’s choice of the medium parameters,
dyv0 ­ 0.07).
Next we note that the relevant contour of constant
fr svd through each saddle point is vertical in the vicinity of the saddle point. This means that the real part of
the saddle point vj can be approximated by vEj for d not
too large. Again this approximation is found to be valid
for the highly absorbing case treated by Brillouin. Combining these two approximations for each saddle point,
we conclude that the slowly varying functions of v in
Eqs. (2.16) and (2.17b) can be approximated by replacement of vj with vEj [provided that u is not too close to
u1 in the case of Eq. (2.17b)]. Hence, we approximate
Eqs. (2.16) and (2.17b) by
∑
∏
p
2 c f˜svEs d
z
As sz, td >
fsvs d ,
exp
(3.5)
f22pzfs2d svEs dg1/2
c
∑
∏
p
2 c f˜svEb d
z
fsvb d
Ab sz, td >
exp
f22pzf s2d svEb dg1/2
c
for u .. u1 . (3.6)
We must be more careful about approximating the exponential function in the above relations because it is a
more rapidly varying function of v. The attenuation has
already been expressed in terms of vEj in Eq. (3.1). We
can express that expression in terms of the index of refraction by noting that for real frequencies we have
f r svd ­ 2vni svd .
(3.7)
Equation (3.1) therefore can be written as
f r svj d ­ 2vEj ni svEj d ­ 2cki svEj d .
izf i svdyc ­ if2v r t 1 znr svdv ryc 2 zni svdv iycg (3.9)
evaluated at the relevant saddle point v ­ vj . If we
replace vj by vEj in the index of refraction terms and
set vj r ­ vEj elsewhere in this expression, it becomes
izf svj dyc > if2vEj t 1 zk svEj d 2 zn svEj dvj ycg .
r
i
sorption band. In the same spirit we apply the approximations given in relations (3.6) and (3.11) with j ­ b for
all u . u1 with the understanding that they become inaccurate for u too close to u1 . The region in which this
problem exists becomes smaller as d decreases compared
with v0 . Our numerical computations of the field with
the physical model below display the effects of these simplifications for highly absorbing and highly transparent
cases.
B. Nonoscillatory Contribution
We now turn our attention to the contribution to the field
given in Eq. (2.17a). This contribution is nonoscillatory
and is important only for points of observation near the
region given by
zyt ­ VE s0d .
(3.12)
The group velocity or phase velocity for time-harmonic
waves with zero frequency could be applied in Eq. (3.12)
equally well, since all three velocities are equal for zero
frequency. Physically the contribution can be thought of
as a quasi-static field that propagates with the velocity of
zero-frequency fields in the medium. Hence we consider
nonoscillatory electromagnetic fields of the form
wsz, t, ṽd ­ expfṽt 2 k̃sṽdzg ,
(3.13)
where the growth rate ṽ is a real constant. Then it is
easy to show that the field is a solution to Maxwell’s
equations in a Lorentz medium if k̃sṽd is given by
√
!1/2
ṽ
b2
,
k̃sṽd ­ 2iksiṽd ­
11 2
c
ṽ 1 v0 2 1 2dṽ
(3.14)
(3.8)
The oscillations are determined by
i
233
which is real.
Since vb is purely imaginary for 1 , u , u1 , the exponential term in Eq. (2.17a) is a field of the form given
in Eqs. (3.13) and (3.14) except that the growth rate
is a function of position and time. The equation for
saddle point vb [Eq. (2.13)] can be written in the form
vb ­ i ṽb ,
(3.15)
i
where ṽb is the largest real solution to
(3.10)
With the combination of relations (3.8) and (3.10), the
quantity occurring in the exponential in relations (3.5)
and (3.6) can be written as
zfsvj dyc > if2vEj t 1 zksvEj dg 2 izn svEj dvj yc .
i
i
(3.11)
In the case j ­ b relation (3.11) applies when u . u1 and
is accurate only when u is not too close to u1 .
The last term in relation (3.11) involves the location of
the saddle point vj . That term is not important, however, because it is negligible except when vEj is in the
absorption band, and it contributes only a small phase
shift to the field even then. Hence we ignore that term
in our physical model of pulse dynamics with the understanding that the phase of the field that we obtain by
using the model may be a little off when vEj is in the ab-
ṼG sṽb d ­ zyt ,
(3.16)
with ṼG sṽd defined as
¡
dk̃sṽd .
ṼG sṽd ­ 1
dṽ
(3.17)
We see from the definition of ṼG sṽd that it can be taken
to be the group velocity of the nonoscillatory waves given
in Eq. (3.13). This identification does not provide much
physical insight, however, because the group velocity is
primarily a mathematical object rather than a physical
one. To connect ṼG sṽd with a more physical quantity,
we consider the velocity of energy in the nonoscillatory
waves.
We take the velocity of energy in fields of the form given
in Eq. (3.13) to be given by Eq. (3.3) with the change that
the Poynting vector and the energy density are not time
234
J. Opt. Soc. Am. B / Vol. 12, No. 2 / February 1995
G. C. Sherman and K. E. Oughstun
averaged (since the field is not oscillating). With this
definition it is shown in Appendix C that an electromagnetic field of the form given in Eqs. (3.13) and (3.14) has
energy velocity ṼE sṽd given by
ṼE sṽd ­
cnsi ṽdQ 2 sṽd
,
2 b2 ṽd
n2 siṽdQ 2 sṽd
(3.18)
Qsṽd ­ ṽ 2 1 v0 2 1 2dṽ ,
#1/2
"
b2
.
nsiṽd ­ 1 1
Qsṽd
(3.19)
where
(3.20)
It is shown also in Appendix C that the group velocity
of the nonoscillatory waves is given by
ṼG sṽd ­
cnsi ṽdQ 2 sṽd
.
2 b2 ṽd 2 b2 ṽ 2
n2 siṽdQ 2 sṽd
It is shown in Appendix C that the relative difference
between the two velocities is given by
ṼG sṽd 2 ṼE sṽd
b ṽ
.
­ 2
n siṽdQ 2 2 b2 dṽ 2 b2 ṽ 2
ṼE sṽd
2
(3.23)
Using this expression, we show in Appendix C that under the conditions17 d # 0.1v0 and b # 5v0 , which are
normally satisfied in the dielectrics of interest, the maximum value of the relative difference between the velocities with ṽ within the interval of interest specified in
inequality (3.22) is less than 1.15%. For most applications the agreement between the two velocities will be
much better than this.
To obtain a more physical result, therefore, we replace
ṼG sṽd with ṼE sṽd in Eq. (3.16). As result we obtain the
growth rate of the nonoscillatory contribution for u # u1
by finding the largest real solution to
ṼE sṽd ­ zyt ,
(3.24)
where ṼE sṽd is given by Eq. (3.18).
One more approximation is useful for the formulation
of our physical model. It follows from Eq. (3.9) of Ref. 12
that u1 can be approximated as u0 , defined as
u0 ­
c
c
­
­ ns0d ,
VE s0d
ṼE s0d
4. PHYSICAL MODEL OF
PULSE DYNAMICS
A. Nonuniform Model
We now present our physical model of pulse dynamics
based on the above results. As z tends to infinity with
fixed u, the electric field Esz, td can be expressed as the
real part of a complex quantity Asz, td, which, for d much
smaller than v and b, can be approximated by
(3.21)
Comparison of Eqs. (3.18) and (3.21) shows that the two
velocities differ only by an extra term 2b2 ṽ 2 in the denominator of Eq. (3.21). Hence the difference between
the two velocities is negligible for ṽ sufficiently close to
zero.
The above observation implies that we may be able to
apply the energy velocity in place of the group velocity
because we are interested only in nonoscillatory waves
with growth rates that satisfy (see Appendix C)
p
s2y3dd # ṽ # s2y3ds 3 2 1dd .
(3.22)
2
1.500 and u1 ­ 1.503.) As a result of this approximation
the nonoscillatory waves with negative growth rates are
not used in our physical model. The change in the second
precursor field Ab sz, td from the nonoscillatory form (with
growth rate equal to zero) to the time-harmonic form
(with frequency equal to zero) takes place at the point
of observation that is moving with velocity VE s0d ­ ṼE s0d.
Asz, td , ATH sz, td 1 AQS sz, td ,
(4.1)
where
ATH sz, td ­
X
j
p
2 c f˜svEj d
exphifzksvEj d 2 vEj tgj ,
f22pzfs2d svEj dg1/2
(4.2)
p ˜
c f si ṽE d
AQS sz, td ­
expfṽE t 2 k̃sṽE dzg . (4.3)
f22pzf s2dsiṽE dg1/2
Here vEj are defined to be the nonnegative, real solutions
to
c ,
z
VE svEj d ­
;
(4.4)
t
u
and ṽE is defined to be the positive solution to
ṼE sṽE d ­
c .
z
;
t
u
(4.5)
The quantity ATH sz, td given in Eq. (4.2) is the timeharmonic component discussed in Subsection 3.A. It is
shown in what follows that the sum in Eq. (4.2) includes
only one term, the Sommerfeld precursor As sz, td, for 1 ,
u , u0 and includes two terms, the Sommerfeld precursor
As sz, td and the Brillouin precursor Ab sz, td, for u $ u0 .
The quantity AQS sz, td is the nonoscillatory (quasi-static)
component discussed in Subsection 3.B.
We refer to the above results as a physical model because they can be used to describe the local dynamics
of the pulse in physical terms. They are similar to the
mathematical results that lead to the group-velocity description that is valid for lossless, gainless media but are
different in three respects:
a. The nonoscillatory contribution is included in addition to the time-harmonic contribution,
b. The energy velocity is used to determine pulse dynamics in place of the group velocity,
c. The pulse dynamics are strongly affected by the
relative attenuation of the various time-harmonic and
nonoscillatory contributions.
(3.25)
which is simply the index of refraction for static fields,
i.e., the square root of the static dielectric constant. The
validity of the approximation decreases with increasing
d but is still good even for the parameter values used
by Brillouin. (For those parameter values we have u0 ­
It follows from relations (4.1) – (4.5) that the primary
quantities that determine the local dynamics of the pulse
are the energy velocity and the attenuation coefficient
L of time-harmonic waves as functions of frequency and
the energy velocity and the attenuation coefficient L̃ of
nonoscillatory waves as a function of growth rate. Plots
G. C. Sherman and K. E. Oughstun
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
235
be eliminated in Eq. (4.6) to yield the formula used to
compute L̃ for Fig. 6:
L̃ ­ k̃sṽE d 2
ṽE ,
ṼE sṽE d
(4.7)
where k̃sṽd is given by Eq. (3.14) and ṼE sṽd is given by
Eq. (3.18).
We may obtain a qualitative description of all the main
features of pulse dynamics by studying these plots. We
first notice from Fig. 3 (by recalling that u0 ­ 1.5 in a
Lorentz medium with Brillouin’s parameter values) that
for 1 , u , u0 there is only one solution to Eq. (4.4), so
that the sum in Eq. (4.2) includes only one term. The
frequency vE of this term is large for u near 1 and de-
Fig. 3.
Normalized energy velocity of monochromatic waves.
Fig. 5.
Fig. 4.
Normalized energy velocity of nonoscillatory waves.
Attenuation coefficient of monochromatic waves.
of these functions for a medium with Brillouin’s parameter values [specified in Eqs. (2.4)] are given in Figs. 3 – 6.
The energy velocities were computed by application of
Eqs. (3.4) and (3.18). The attenuation coefficients of the
waves are the rates of exponential decay of the waves as
z increases with constant u. For time-harmonic waves
with real frequencies, L is equal to the imaginary part
of the propagation constant k [defined in the above text
between Eqs. (2.2) and (2.3)] with the index of refraction
given in Eq. (2.3). For the nonoscillatory waves, L̃ is
given by
L̃ ­ k̃sṽE d 2
ṽE u .
c
(4.6)
Since ṽE and u are related through Eqs. (4.5), u can
Fig. 6.
Attenuation coefficient of nonoscillatory waves.
236
J. Opt. Soc. Am. B / Vol. 12, No. 2 / February 1995
creases monotonically with increasing u. From Fig. 4
we see that, as the frequency decreases from a large
value, the attenuation increases. This means that this
high-frequency, quasi-time-harmonic term decreases in
frequency and amplitude as u increases. These features
of the Sommerfeld precursor As sz, td are already well
known; what we have obtained here that is new is an explanation of these features in terms of the physical properties of time-harmonic waves given in Figs. 3 and 4.
Continuing the physical description, we note from
Fig. 5 that there is one positive solution ṽE to Eq. (4.5)
with 1 , u , u0 . The growth rate ṽE decreases with increasing u, tending toward 0 as u approaches u0 . From
Fig. 6 we see that the attenuation with z of the nonoscillatory contribution is large for large growth rate but
gradually decreases to zero as ṽE approaches 0. Hence
the nonoscillatory contribution is negligible compared
with the Sommerfeld precursor for small u . 1 but gradually increases until it dominates Sommerfeld’s precursor
as u approaches u0 . This is the arrival of Brillouin’s
precursor. Finally, we note from Fig. 5 that there is
no positive solution to Eq. (4.5) for u $ u0 . Hence the
nonoscillatory part no longer contributes.18
Returning to Fig. 3, we note that for u $ u0 there are
now two nonnegative solutions to Eq. (4.4). The first is
a high-frequency solution, which is the continuation of
Sommerfeld’s precursor. The new solution is a lowfrequency solution, with frequency starting at zero for
u ­ u0 and increasing with increasing u. A glance at
Fig. 4 shows that the attenuation of this new wave is
much less than that of the high-frequency contribution.
Hence this wave dominates Sommerfeld’s precursor.
Figure 4 also reveals that the attenuation of this wave
increases with increasing v, causing the wave to decrease
in amplitude with increasing u. Hence this contribution
has increasing frequency and decreasing amplitude with
increasing u. These properties of Brillouin’s precursor
are well known, but we now have their physical basis for
the first time to our knowledge.
In addition to providing a description of the qualitative pulse behavior in physical terms, the physical model
gives approximate analytical formulas that predict pulse
dynamics quantitatively without requiring the evaluation
of the saddle-point locations in the complex plane. Of
course, the model does require the solution of Eqs. (4.4)
and (4.5), which are transcendental equations, but these
equations are simpler to deal with than the saddle-point
equations because they involve only real quantities.
To investigate the accuracy of our approximate formulas, we have evaluated them numerically for the deltafunction pulse, in which case f˜svd ­ A. The results are
shown in a series of plots (Figs. 7 – 12) in which we set
z ­ 1024 cm and applied Brillouin’s values for the medium
parameters as given in Eqs. (2.4) (except in the lossless
case shown in Fig. 12, in which we set d equal to 1.0 s–1 ).
The plots in Figs. 7 – 12 are for the delta-function pulse
with A ­ 1 (statvolt s)ycm, except for Fig. 10, which is
for the step-modulated sinusoidal pulse. Equations (4.4)
and (4.5) were solved numerically by Mueller’s method.19
The numerical results of the nonuniform physical model
given in relations (4.1)– (4.5) are plotted as a solid curve
in Fig. 7, superimposed upon a dashed curve, which is
a plot of the nonuniform asymptotic result given by
G. C. Sherman and K. E. Oughstun
Eqs. (2.13) –(2.16) for the same parameter values. To
evaluate the asymptotic result, we applied Mueller’s
method to solve Eq. (2.13) to determine the saddle points
numerically.20 It is apparent from the figure that the
accuracy of the physical model is good. The main discrepancy is a minor shift in the phase of the Brillouin
precursor as discussed at the end of Subsection 3.A.
The scale in Fig. 7 was chosen so that the transition
between the two precursors can be seen clearly. Since
the curves are off the scale for small u in that figure,
the same two quantities are replotted for small u with an
Fig. 7. Nonuniform results for the electric field of a deltafunction pulse. Solid curve, physical model; dashed curve,
asymptotic results.
Fig. 8. Nonuniform results for the electric field of a deltafunction pulse for u near 1. Solid curve, physical model; dashed
curve, asymptotic results. The dashed curve is not visible because it falls almost exactly on the solid curve.
G. C. Sherman and K. E. Oughstun
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
237
asymptotic analysis that makes the results invalid in that
region. To obtain results that are useful there, it is necessary to employ the uniform asymptotic approximation
of the field. This is done Subsection 4.B.
B. Uniform Model
The asymptotic results that we have been using so far
are nonuniform in the vicinity of two values of u: (a)
Fig. 9. Comparison of the uniform physical model with the numerical integration of the exact integral solution for the electric
field of a delta-function pulse for u near 1. Solid curve, uniform
physical model; dashed curve, numerical evaluation of the exact
integral solution.
Fig. 11. Comparison of the uniform physical model with the numerical integration of the exact integral solution for the electric
field of a delta-function pulse in the highly absorbing case. Solid
curve, physical model; dashed curve, exact integral solution.
Fig. 10. Comparison of the uniform physical model with the
numerical integration of the exact integral solution for the electric field of a step-modulated sinusoidal pulse for u near 1.
Solid curve, uniform physical model; dashed curve, numerical
evaluation of the exact integral solution.
appropriate scale in Fig. 8.21 The results of the physical
model agree so well with the asymptotic results that the
two curves are indistinguishable, making it impossible to
see the dashed curve.
The large peak that occurs in both the physical-model
results and the asymptotic results in Fig. 7 for u near
u0 is a consequence of the nonuniform nature of the
results as explained in Section 2 in the paragraph between Eqs. (2.12) and (2.13). This is an artifact of the
Fig. 12. Comparison of the uniform physical model with the numerical integration of the exact integral solution for the electric
field of a delta-function pulse in the highly transparent case.
Solid curve, physical model; dashed curve, exact integral solution.
The dashed curve is not visible because it falls almost exactly on
the solid curve.
238
J. Opt. Soc. Am. B / Vol. 12, No. 2 / February 1995
G. C. Sherman and K. E. Oughstun
u ­ 1, which corresponds to the arrival of Sommerfeld’s
precursor; and (b) u ­ u1 , which occurs during the arrival
of Brillouin’s precursor. This means that, in order for
the results to be useful approximations for large z, the
parameter z must be taken to be larger and larger as
u approaches one of these critical values. Furthermore,
the functional forms of the results are different for u on
opposite sides of a critical value.
It is possible to obtain asymptotic approximations,
called uniform asymptotic approximations, that do not
have these difficulties. As the value of u tends away
from the critical values, the uniform results tend asymptotically to the same formulas as given in the nonuniform
results. We have presented such uniform asymptotic
results in Ref. 13 for both of the above critical points.
To use them in our physical model, we simply replace
the saddle points occurring in the uniform asymptotic
formulas with the approximations used in the nonuniform physical model, i.e., the corresponding real solutions to Eqs. (4.4) and (4.5). Because the results involve
Bessel functions and Airy functions instead of quasi-timeharmonic waves and nonoscillatory exponentially growing
waves, the physical interpretation of the results is not so
obvious as with the nonuniform results. Nevertheless,
in the uniform physical model that we are proposing,
the arguments of these functions are real, involving real
frequencies and growth rates that are clearly connected
with the physics of time-harmonic waves and nonoscillatory waves in the medium through Eqs. (4.4) and (4.5).
1. Beginning of Sommerfeld’s Precursor
We first address the critical point u ­ 1. The form of the
uniform asymptotic expansion depends on the behavior of
f˜svd as jvj tends to infinity. Assume that f˜svd is of the
form
f˜svd ­ v 2s11vd F̃ svd ,
(4.8)
with bounded, analytic F̃ svd for jvj larger than some positive constant. Then Ref. 13 gives an asymptotic approximation of the field that is uniformly valid at the critical
point if v is greater than zero. If v is less than or equal to
zero, the asymptotic approximation may not be uniformly
valid at the critical point, but it does give a valid approximation of the field in the region u $ 1 if it approaches a
finite limit as u approaches 1.
To obtain the uniform energy-velocity description, we
start with Eq. (2.6) and apply Eq. (A6) from Appendix A
of Ref. 13 with the saddle points v6 replaced by their
approximations 6vE , where vE is the nonnegative, real
solution to Eq. (4.4) for the appropriate value of u near
1. By employing Eq. (A9) of Appendix A of this paper,
we can simplify the result to
!
√
i
z
v
Esz, td , 2
s2id exp
X
2a
c
2
!
!3
√
√
z
z
3 4G0 Jv
a 2 iG1 Jv11
a 5,
c
c
(4.9)
where Jv sxd is the Bessel function of the first kind of order
v and
a ­ 2ImffsvE dg ,
X ­ ReffsvE dg ,
p
p
G0 ­ f˜svE d L 2 s21dv f˜s2vE d 2L ,
p
p
G1 ­ f˜svE d L 1 s21dv f˜s2vE d 2L ,
L­
a3
.
if s2d svE d
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
The branches of the multiple-valued functions in the
above expressions are specified in Appendix A of Ref. 13.
Relation (4.9) asymptotically approaches the nonuniform
approximation of the Sommerfeld precursor given in
Eq. (4.2) as u increases away from 1.
The examples discussed below involve the special cases
of v ­ 21 and v ­ 0. With the help of Eq. (A19) below,
relation (4.9) reduces to the simple form
√
!(
!
√
p
1
z
z
exp
X 2Ref f˜svE d LgJ1
a
Esz, td ,
a
c
c
!)
√
p
z
1 Imf f˜svE d LgJ0
a
(4.15)
c
for both of these cases.
We now show that relation (4.15) is valid for the special
case of the delta-function pulse. Since
f˜svd ­ A ,
(4.16a)
with constant A in this case, we have v ­ 21. Hence
relation (4.15) gives a valid approximation in the region
u $ 1, provided that it approaches a finite limit as u
approaches 1 from above. We investigate this limit by
using the approximation
vE >
b
.
f2su 2 1dg1/2
(4.16b)
It is shown in Appendix D that, for small, positive
u 2 1, relation (4.15) combined with Eq. (4.16a) and
relation (4.16b) can be approximated by
∑
∏
z
Ab
exp 22d su 2 1d
Esz, td , p
c
2su 2 1d
Ω
æ
µ
z
bf2su 2 1dg1/2
3 2J1
c
Ω
æ∂
d
z
1/2
1/2
.
13
f2su 2 1dg J0
bf2su 2 1dg
b
c
(4.17)
Since relation (4.17) approaches the finite limit
∑
∏µ
∂
Ab2 z
z
d c
Esz, td , 2
exp 22d su 2 1d 1 2 6 2
2 c
c
b z
(4.18)
as u approaches 1, relation (4.15) combined with
relations (4.16) is a valid approximation of the deltafunction pulse in the region u $ 1.
A plot of relation (4.15) using Eq. (4.16a) with A ­ 1
(statvolt s)ycm is shown as a solid curve from u ­ 1 to
u ­ 1.005 in Fig. 9. For u beyond 1.005 the nonuniform
G. C. Sherman and K. E. Oughstun
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
expression Eq. (4.2) was applied. The dashed curve in
Fig. 9 is the result of numerical evaluation of the exact
integral representation of the wave field of the deltafunction pulse as given in Eq. (2.6) for the same parameter values. The algorithm that we used to evaluate
the integral numerically for this and subsequent plots is
described elsewhere.22 As was discussed in Ref. 22, the
algorithm begins to produce artifacts in the results as u
approaches 1 from above for the delta-function pulse.23
In Fig. 9 we see that the dashed curve has the same
form as the solid curve but with a high-frequency ripple
superimposed. We have verified that the ripple is an
artifact of the numerical algorithm by showing that its
frequency changed when we changed the numerical value
of the algorithm parameter k, whereas the rest of the
curve remained unchanged.24 Apart from this spurious
ripple, the two curves agree well, except perhaps for very
small u 2 1 of the order of 0.0005 and less.
To investigate the behavior of the exact integral solution with u very close to 1 more carefully, we apply the
method used by Sommerfeld for the lossless case in Refs. 1
and 3 to approximate the integral in that region. It is
shown in Appendix E by direct approximation of the integral (without the asymptotic analysis) that, for the deltafunction pulse with d ­ 0, the integral in Eq. (2.6) is given
approximately by
Ω
æ
Ab
z
1/2
Esz, td , 2
bf2
su
2
1dg
J
1
f2su 2 1dg1/2
c
(4.19)
for small u 2 1. This provides a useful check on our
results in this region, since relation (4.17) reduces to
relation (4.19) when d is set equal to zero.
As a further check of our uniform physical model in
this region, we consider another pulse, the step-modulated
sinusoidal pulse, which was treated by Sommerfeld and
Brillouin in Refs. 1 – 3. We treat this pulse because its
integral solution is convergent at u ­ 1, and therefore
the algorithm that we use to evaluate the integral numerically does not produce the artifacts near u ­ 1 that
it produces for the delta-function pulse. For the stepmodulated sinusoid we have f std ­ 0 for t # 0 and f std ­
A sinsvc td for t $ 0. The spectrum is then given by
A
f˜svd ­
2
µ
1
1
2
v 1 vc
v 2 vc
∂
.
(4.20)
Hence, for this case, v ­ 0 and relation (4.15) applies. A
plot of relation (4.15) for Eq. (4.20) with A ­ 1 statvoltycm
is shown as a solid curve from u ­ 1 to u ­ 1.005 in
Fig. 10. For u beyond 1.005 the nonuniform expression, Eq. (4.2), was applied. The result of the numerical
evaluation of the exact integral solution for the same
pulse is superimposed as a dashed curve. It is seen that
the agreement is almost perfect.
2. Beginning of Brillouin’s Precursor
We now proceed to make the physical model uniform in
the vicinity of the point u ­ u1 . We start with Eq. (2.6)
for the field and apply the results of Section 4 of Ref. 13
to obtain the uniform asymptotic expansion25
(√
i
expsza0ycd
Eb sz, td ­
2
c
z
!1/3
exps2i2py3d
"
√
3 f f˜sv1 dh1 1 f˜sv2 dh2 gAi ja1 j
√
1
c
z
239
!2/3
z
c
!2/3 #
exps2i4py3df f˜sv1 dh1 2 f˜sv2 dh2 g
"
√
3 Ais1d ja1 j
z
c
!2/3 #,
)
a1 1/2
(4.21)
in place of Eq. (2.12).
In Eq. (4.21), Aisxd represents the Airy function of
x. For 1 # u # u1 , v1 represents the saddle point vb ,
which moves down the positive real axis as u increases,
whereas v2 represents another saddle point that moves
up the negative imaginary axis as u increases. These two
saddle points coalesce when u reaches u1 . For u . u1 ,
v1 represents the saddle point vb as it moves into the
fourth quadrant of the v plane with increasing u, and v2
represents 2vb p . The other new quantities occurring in
Eq. (4.21) are defined by26
a0 ­ 0.5ffsv1 d 1 fsv2 dg ,
(4.22)
­ h3y4ffsv1 d 2 fsv2 dgj
# 1/2
"
2a1 1/2
h6 ­ 2s61d s2d
f sv6 d
a1
1/2
1/3
,
(4.23)
(4.24)
for u fi u1 . At the critical value u1 of u at which the
two saddle points coalesce, these coefficients take on the
limiting values
# 1/3
"
2
lim h6 ­ 2 s3d
; h1 ,
(4.25)
u!u1
f sv1 d
lim f f˜sv1 dhsv1 d 1 f˜sv2 dhsv2 dg ­ 2f˜sv1 dh1 ,
(4.26)
u!u1
lim f f˜sv1 dhsv1 d 2 f˜sv2 dhsv2 dg ya1 1/2 ­ 2f˜s1d sv1 dh1 2 ,
u!u1
(4.27)
and a0 at u1 is given by Eq. (4.22). See Appendix B of
Ref. 13 for the specification of the branches used to make
the multivalued functions that occur in the above equations single valued.
On obtains the uniform physical model from the above
uniform asymptotic approximation by making the same
approximations that were used in the nonuniform asymptotic approximation, to yield the nonuniform physical
model. The critical value u1 is approximated by u0 , and
the saddle points are approximated by use of the appropriate solutions to the energy velocity equations (4.4) and
(4.5). In particular, for 1 # u # u0 , v1 is approximated
by iṽ1 , and v2 is approximated by iṽ2 , where ṽ6 are
the real solutions to Eq. (4.5) such that ṽ1 $ ṽ2 . Similarly, for u $ u0 , v1 is approximated by the low-frequency
real solution to Eq. (4.4), and v2 is approximated by minus that value. As in the nonuniform physical model, vs
is approximated in Eq. (2.16) by the high-frequency real
solution to Eq. (4.4).
As u moves away from u0 in either direction, the uniform physical model approaches the nonuniform physical
model asymptotically. Hence we can apply the nonuniform model for all u, except in the vicinity of u0 , where
240
J. Opt. Soc. Am. B / Vol. 12, No. 2 / February 1995
we apply the uniform model. To examine the validity
of the uniform physical model, we repeated the computations that were used to obtain Fig. 7 with the change
that we applied the uniform physical model in the region
1.43 , u # 1.55. Since the expressions for some of the coefficients in Eq. (4.21) become indeterminate for u ­ u0 ,
the limiting expressions given in Eqs. (4.25) – (4.27) were
used for 1.45 # u # 1.503. The results are presented as
the solid curve superimposed upon the results of numerical integration of the exact integral solution [Eq. (2.6)],
shown as the dashed curve in Fig. 11. The blip in the
solid curve at u ­ u0 and its departure from the dashed
curve for u larger than but near u0 is a result of the
fact that our approximation of the saddle-point location
by a real solution to the energy velocity equation (4.4)
is not very good for u in this region, as is discussed in
Subsection 3.A. The small shift in phase between the
two curves for u larger than u0 has been discussed in
Subsection 4.A. Apart from these rather minor discrepancies, the results of the uniform physical model are in
excellent agreement with the exact integral solution for
the field for u varying over the entire range shown.
As we have already mentioned, the medium parameter values chosen for our numerical computations correspond to a medium with very high absorption, much too
high for the material to be considered to be transparent.
Since our approximations become better as the absorption decreases, these numerical results can be considered
to be a worst-case test of the validity of our physical model
for describing pulse propagation in absorbing dielectrics
of practical interest. To verify the utility of the physical model in the other extreme of a highly transparent
medium, we have repeated the same computations with
the same parameter values except that d was taken to
be 1.0 s–1 for both the uniform physical model computations and the numerical evaluation of the exact integral
solution. The results are shown in Fig. 12. The physical model agrees so well with the exact solution that the
two curves can be distinguished from each other only at
a few isolated points.
We might expect the group-velocity description to be
applicable for obtaining the results plotted in Fig. 12,
since that case is nearly lossless. Indeed, it can be shown
that the energy velocity given in Eq. (3.4) approaches the
group velocity for all frequencies for which the medium is
lossless as d approaches zero. The medium is not lossless
for all frequencies, however, even when d is identically
zero, because the index of refraction given by Eq. (2.3) is
purely imaginary for v 2 slightly larger than v0 2 . As a
result the group-velocity description is not strictly valid
for the Lorentz medium even when d is identically zero.
Nevertheless, we have applied the group-velocity description to that case to see what it would yield. The results
were identical to those shown in Fig. 12 except in the region 1.35 # u # 1.5, where the field failed to rise as it
does in Fig. 12. It is apparent that all that is missing in
the group-velocity description in this case is the nonoscillatory contribution.
5.
SUMMARY AND CONCLUSIONS
We have presented a model that accurately describes in
physical terms all the dynamics of an electromagnetic
G. C. Sherman and K. E. Oughstun
pulse as it propagates through a Lorentz medium in the
mature-dispersion regime. Its accuracy decreases as the
absorption increases, but we have shown that the accuracy is still excellent even for media that are too absorbing to be of practical interest as absorbing transparent
materials. According to the model, once the pulse propagates far enough to be in the mature-dispersion regime,
it separates into two distinct components at any given
space – time point. Each component is either a quasimonochromatic wave of the form
exphifksvdz 2 vtdgj
with real frequency v, which is a slowly varying function
of position and time, or a nonoscillatory wave of the form
expfṽt 2 k̃sṽdzg
with real growth rate ṽ, which is a slowly varying function of position and time. The propagation constant in
the quasi-monochromatic wave is ksvd ­ vnsvdyc, where
the index of refraction nsvd is given in Eq. (2.3). The
quantity k̃sṽd in the nonoscillatory waves is given in
Eq. (3.14). The frequencies of the quasi-monochromatic
components satisfy
VE svd ­ zyt ,
(5.1)
where VE svd is the velocity of energy in a monochromatic
wave with real frequency v. The growth rates of the
nonoscillatory components satisfy
ṼE sṽd ­ zyt ,
(5.2)
where ṼE sṽd is the velocity of energy in a nonoscillatory
wave with growth rate ṽ.
The easiest way to describe the components is to consider the point of observation to be moving with a fixed
velocity V ­ zyt. Then, according to Eqs. (5.1) and (5.2),
the components that contribute at a specific point of observation are the ones with energy velocities equal to the
velocity of that point. If V satisfies
V,
c ,
ns0d
(5.3)
where ns0d is the index of refraction at zero frequency
(i.e., the square root of the static dielectric constant),
then there is only one real solution to each of the above
two equations. Hence, in the region of space– time in
which condition (5.3) is satisfied, the pulse has one
quasi-monochromatic component (the Sommerfeld precursor) and one nonoscillatory component (the rise of
the Brillouin precursor). When V is larger than cyns0d,
then there is no real solution to Eq. (5.2) and there are
two real solutions to Eq. (5.1). Hence, in that region of
space – time, the pulse has two quasi-monochromatic components (the Sommerfeld and Brillouin precursors) and
no nonoscillatory component. In the transition region in
which V is approximately equal to cyns0d, the functional
form of the field is more complicated, but its dynamics
are still controlled by the solutions to Eqs. (5.1) and (5.2).
The variations of the precursor fields with changes in
V are determined by the variation of the energy veloci-
G. C. Sherman and K. E. Oughstun
ties and attenuation rates of monochromatic waves and
nonoscillatory waves as functions of frequency and growth
rate, respectively. These latter variations are the simple
smooth curves plotted in Figs. 3 – 6. They are determined by the physics of energy flow and absorption in the
medium through the interaction of the electromagnetic
field with the molecules according to the microscopic
model of the material.
After having described the field behavior as above, we
can now describe it from a different viewpoint. If we observe the pulse in a small region of space at fixed time in
the mature-dispersion regime, we find it to be made up
of the superposition of a quasi-monochromatic component
of some real frequency and another component that is either a quasi-monochromatic wave with a lower real frequency or a nonoscillatory wave with a real growth rate.
These two components will have the same energy velocity.
Moreover, if we then follow these components through
space as time progresses, they move together with that
velocity, and as each component propagates it will decay
exponentially with the corresponding attenuation coefficient. This will be true throughout the pulse, with the
components with higher energy velocities being ahead of
those with lower velocities, and the separation between
the various components will increase with time. As a
consequence of this spreading of the components, the energy in any given frequency (or growth-rate) interval will
spread out over an increasing region of space, causing the
component amplitudes to decrease (in addition to the exponential attenuation) by a factor of 1 over the square root
that occurs in the denominator of Eqs. (4.2) and (4.3).27
The accuracy of the physical model is illustrated in
Fig. 11 for a highly absorbing medium and in Fig. 12 for
a nearly nonabsorbing medium by superimposing of the
result of the model upon the results of numerical evaluation of the exact integral solution. In both cases the
agreement between the model and exact results is excellent. Since these computations were made for a propagation distance of only 10–4 cm, it follows that the pulse
reaches the mature-dispersion regime after a very short
propagation distance. Hence, the physical model is applicable nearly everywhere in the medium (i.e., at least
for all z greater than or equal to 10–4 cm).
The agreement between the physical model and the
exact solution is especially striking in Fig. 12, where the
plots describe a highly complicated and strange-looking
curve. If we had only the results of the numerical integration, the strange appearance of the curve might lead us
to question the validity of our numerical algorithm or computer code. But since both the physical model and the
numerical integration give precisely the same curve, even
though they use different mathematical algorithms and
computer codes, there can be no doubt about the validity of both approaches. Moreover, without the physical
model, we would be at a loss to explain the peculiar features of the curve. The physical model, however, reveals
the source of this behavior. Because the attenuation of
the monochromatic waves is so low in this case, the highfrequency component does not decay rapidly as it does
in Fig. 11. Hence it interferes first with the nonoscillatory component, and later with the low-frequency
quasi-monochromatic component, to give the complicated
behavior.
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
241
The physical model that we have presented is intuitively satisfying from a physical point of view. It seems
to be the natural extension of the group-velocity description that is known to be valid for lossless, gainless, dispersive media. In fact the group-velocity description can be
considered to be an energy-velocity description, since, under general conditions, it has been shown that the energy
and the group velocities are identical in lossless, gainless
media.28 Hence an energy-velocity description of pulse
dynamics is now available for all lossless, gainless, dispersive media and for one dispersive medium with loss,
the Lorentz medium. As was stated in Section 1, we are
not aware of a physical description of pulse dynamics in
any other dispersive system that includes loss or gain.
But it is tempting to speculate that the energy-velocity
model is valid for general dispersive media with small
(or zero) loss or gain, because the model appears to account for the effects of small loss or gain in terms of physical principles that transcend the details of the particular
dispersion relation of the medium. With this in mind,
it may be possible to avoid a lengthy and complicated
asymptotic analysis (such as those given in Refs. 1 – 3, 12,
and 13) when we are confronted with the need to determine the functional behavior of the pulse in the maturedispersion regime for a new dispersive system. If the
velocity of energy in monochromatic and nonoscillatory
exponentially growing waves in that system can be determined, then one could try the energy-velocity model
given here and compare it with numerical integration of
the exact integral solution [Eqs. (2.6) –(2.9)] with the appropriate dispersion relation. If the agreement were to
be good, then one would have obtained a more useful
result.
A concise overview of the modern asymptotic description of dispersive pulse propagation may be found in the
authors’ review paper.29 A complete, detailed development of the modern asymptotic theory of dispersive pulse
propagation, which leads to the energy-velocity model,
may be found in the research monograph30 recently published by the authors.
APPENDIX A
In this appendix we show that the terms
∑
∏
f˜svj d
z
fsv
exp
d
,
j
f2zfs2d s2vj dg1/2
c
∑
∏
f˜s2vj p d
z
pd
fsv
exp
Bj ­
j
f2zf s2d s2vj p dg1/2
c
Aj ­
(A1)
(A2)
[occurring in Eqs. (2.11) and (2.12)] are complex conjugates of each other, where vj are the saddle points with
j ­ s or j ­ b. First, we note that
f˜s2vj p d ­ f˜ p svj d ,
(A3)
because f˜svd is the Fourier transform of a real function
f std. Next, we note that, since
n2 svd ­ 1 2
b2
v2
2 v0 2 1 2dvi
,
(A4)
242
J. Opt. Soc. Am. B / Vol. 12, No. 2 / February 1995
with real b2 , v0 , and d, we have
n2 s2v p d ­ fn2 svdgp ­ fnp svdg2 .
G. C. Sherman and K. E. Oughstun
(A5)
We determine the branch of the square root of n svd by
using the branch cuts shown in Fig. 1 and taking ns0d
(which is real) to be positive. Then, nsvd is single valued
and satisfies
np s0d ­ ns0d .
(A6)
2
Hence, when we take the square root of Eq. (A5), requiring Eq. (A6) to be satisfied, we obtain
(A7)
ns2vj p d ­ np svj d .
Since fsvd is given by
with real u, Eq. (A7) implies that
fs2v p d ­ f p sv d .
j
j
(A8)
fs2d svd ­ if2ns1d svd 1 vns2d svdg ,
(A9)
for j ­ s, b.
Finally, we substitute Eqs. (A3), (A9), and (A21) into
Eq. (A2) to obtain
(A11)
"
#
ns1d svd
4P 2 svd ,
b2
ns2d svd ­
2
P
svd
1
1
2
nsvdQ 2 svd
nsvd
Qsvd
(A12)
where
(A13)
Qsvd ­ v 2 v0 1 2idv .
2
We see from Eqs. (A13) and (A14) that
P s2v p d ­ 2P p svd ,
Qs2v p d ­ Q p svd .
∏
∑
f˜ p svj d
z p
f
sv
d
­ Aj p ,
exp
j
hf2zf s2d svj dg1/2 jp
c
(A22)
APPENDIX B
In this appendix we show that, to a good approximation,
VE svEj d ­ zyt
(A10)
b2 P svd ,
nQ 2 svd
P svd ­ v 1 id ,
(A21)
which is the result to be proved.
Next we consider the square roots in the denominators
of Eqs. (A1) and (A2). By differentiation we find that
2
p
f2zf s2d s2vj p dg1/2 ­ hf 2 zf s2d svj dg1/2 jp
Bj ­
fsvd ­ ivfnsvd 2 ug
ns1d svd ­
therefore, that 2p # a o , 0 for the saddle points vs and
2vb p , whereas 0 # a o , p for the saddle points 2vs p
and vb . This means that the square root in Eq. (A2) is
in the upper half of the complex plane if the square root
in Eq. (A1) is in the lower half-plane and vice versa. It
then follows from Eq. (A19) that
(A14)
(A15)
(A16)
It therefore follows from Eqs. (A11) and (A12) that
(A17)
ns1d s2v p d ­ 2fns1d svdgp ,
p
p
s2d
s2d
n s2v d ­ fn svdg .
(A18)
With these relations applied to Eq. (A9), we have
(A19)
f s2d s2v p d ­ ff s2d svdgp .
In our derivation of Eqs. (2.11) and (2.12) of Ref. 12 we
applied the method of Olver31 for the asymptotic analysis.
For second-order saddle points and real asymptotic parameter zyc, as we have here, Olver specifies the branch
of the square roots occurring in Eqs. (A1) and (A2) to be
given by
p ,
ja o 1 2aj #
(A20)
2
where a is the angle between the integration path and
a line parallel to the real axis at the saddle point as the
integration path leaves the saddle point and a o is the
argument of the complex square root in question. Choosing our path of integration to be along the path of steepest descent at each saddle point, we find that a ­ 2py4
at 2vb p (for u . u1 ) and at vs and a ­ py4 at vb (for
u . u1 ) and at 2vs p . It follows from inequality (A20),
(B1)
for d much smaller than v0 and b, where VE svd is the
time-harmonic energy velocity given in Eq. (3.4). With
uE ­ cyVE svd, Eq. (B1) can be rewritten as
uE ­ nr svEj d 1
vEj i
n svEj d .
d
(B2)
The quantities uE and vEj are defined through Eq. (3.1)
as
f r svj , uE d ­ f r svEj d ,
(B3)
with j ­ s, b. Notice that f r is independent of u along
the real frequency axis, as is accounted for on the righthand side of Eq. (B3).
Consider first using the defining relation given in
Eq. (B3) to obtain explicit, albeit approximate, expressions for uE in terms of vEj and the medium parameters.
For u $ u0 , f r svb , ud may be approximated as30
f r svb , ud > 2dsu 2 u0 d
u 2 2 u0 2 1 2b2yv0 2 ,
u 2 2 u0 2 1 3b2yv0 2
(B4)
where terms containing factors of d 3 or higher have been
neglected. With this approximation Eq. (B3) becomes
f r svEb d > 2dsuE 2 u0 d
uE 2 2 u0 2 1 2b2yv0 2 .
uE 2 2 u0 2 1 3b2yv0 2
(B5)
For values of vEb that satisfy the inequality 0 # vEb ,,
v0 , the right-hand side of relation (B5) may be approximated by 2s2y3ddsuE 2 u0 d, so that
uE > u0 2
3f r svEb d ,
2d
0 # vEb ,, v0 ,
(B6)
whereas, if v0 # vEb # v1 , the right-hand side of
relation (B5) may be approximated by 2dsuE 2 u0 d, so
that
uE > u0 2
f r svEb d ,
d
v0 # vEb # v1 ,
(B7)
G. C. Sherman and K. E. Oughstun
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
where v1 ­ sv0 2 1 b2 2 d 2 d1/2 . Finally, for u $ 1,
fr svs , ud may be approximated as29
f r svs , ud > 22dsu 2 1d ,
(B8)
which is valid provided that jvs j .. v1 . Substitution of
this expression into Eq. (B3) then gives
uE > 1 2 f r svEs dy2d,
vEs .. v1 .
(B9)
Relations (B6), (B7), and (B9) provide the desired approximations of the solution of the defining relation given
in Eq. (B3) in the below-resonance, absorption-band, and
above-resonance frequency domains of a single-resonance
Lorentz-model dielectric.
Attention is now turned to the expression for uE given
in Eq. (B2). For values of vEb in the below-resonance
domain 0 # vEb ,, v0 , the complex index of refraction
given in Eq. (2.3) may be approximated as30
nsvEb d > u0 1
b2
db2
vEb 2 1 i
vEb ,
4
2u0 v0
u 0 v0 4
(B10)
so that (since vEb is real valued)
f r svEb d ­ 2vEb ni svEb d > 2
db2
vEb 2 .
u 0 v0 4
(B11)
Substitution of the real and the imaginary parts of
relation (B10) into Eq. (B2) then yields
3b2
3f r svEb d ,
2
v
­
u
2
uE > u0 1
Eb
0
2u0 v0 4
2d
0 # vEb ,, v0 , (B12)
which is precisely the expression in relation (B6). For
values of vEb in the absorption band, v0 # vEb # v1 , the
complex index of refraction may be approximated as30
nsvEb d > 1 1 i
b2 ,
4dvEb
2
b .
4d
(B14)
Substitution of the real and the imaginary parts of
relation (B12) into Eq. (B2) then yields
b2
f r svEb d ,
uE > 1 1
>12
2
4d
d
b2
f r svEs d ,
>
1
2
2vEs 2
2d
v0 # vEb # v1 ,
APPENDIX C
In this appendix we derive an expression for the energy
and the group velocities of nonoscillatory electromagnetic
fields of the form given in Eq. (3.13) in a Lorentz medium.
We also derive an expression for the relative difference
between the two velocities.
We begin with the energy velocity, which is defined by
ṼE sṽd ­
nsvEs d > 1 2
b2
svEs 2 2did ,
2vEs 3
(B16)
u­
f r svEs d ­ 2vEs ni svEs d > 2
2
db .
vEs 2
(B17)
Substitution of the real and the imaginary parts of
relation (B16) into Eq. (B2) then yields
(C1)
mN 2
E2 1 H 2 ,
srÙ 1 v0 2 r 2 d 1
2
8p
(C2)
where r and N are, respectively, the displacement and the
number density of the molecular dipoles in the medium.
The dot over a symbol indicates the time derivative.
For a monochromatic field with frequency v, the complex phasor representation of the displacement is given by
r­
eE
,
msv 2 2 v0 2 1 2divd
(C3)
where e and m are, respectively, the charge and the
mass of the electron. The nonoscillatory field given in
Eq. (3.13) has the same form as a monochromatic field
with v ­ i ṽ. For such fields Eq. (C3) becomes
r­2
eE ,
mQ
(C4)
where
Q ­ ṽ 2 1 v0 2 1 2dṽ .
(C5)
Since the time dependency of both r and E is given by
Eq. (3.13), we have
rÙ ­ ṽr .
(C6)
Hence, Eq. (C2) can be written as
u­
so that
S ,
u
where the energy density u and the magnitude S of the
Poynting vector are not time averaged (because the field
is not oscillating).
To calculate an expression for usṽd, we follow the
approach used by Loudon for time-harmonic fields in
Section 3 of Ref. 16; see also Refs. 4 and 5 for the theory
of time-harmonic fields in Lorentz media. The definition
of the energy density in a Lorentz medium of an electromagnetic wave with arbitrary time dependence is
(B15)
which is precisely the expression in relation (B7). Finally, for values of vEs in the above-resonance domain
vEs .. v1 , the complex index of refraction given in
Eq. (2.3) may be approximated as30
vEs .. v1 , (B18)
which is precisely the expression in relation (B9). This
then establishes the validity of Eq. (B2) and hence of
Eq. (B1).
(B13)
so that
f r svEb d > 2
uE > 1 1
243
mNr 2
E2 1 H 2 .
sṽ 2 1 v0 2 d 1
2
8p
(C7)
For fields of the form given in Eq. (3.13) in a Lorentz
medium, we have
H ­ nsiṽdE ,
(C8)
244
J. Opt. Soc. Am. B / Vol. 12, No. 2 / February 1995
with
√
nsiṽd ­ 1 1
b2
Q
G. C. Sherman and K. E. Oughstun
Combination of Eqs. (C19) and (C20) then yields
!1/2
.
(C9)
With the application of Eqs. (C8) and (C4), Eq. (C7) becomes
u­
e2 E 2
E2
mN
sṽ 2 1 v0 2 d 2 2 1
f1 1 n2 siṽdg .
2
m Q
8p
cnsi ṽd 2
E .
4p
ṼG sṽd ­
(C11)
b­
4pNe2
m
!1/2
(C12)
"
#
1
b2
u
2
2
2
­
s
ṽ
1
v
d
1
1
1
n
si
ṽd
.
0
S
2cnsiṽd Q 2
(C13)
1 1 n2 siṽd ­ 2n2 siṽd 2
b2 .
Q
(C14)
Substitution of Eq. (C14) into Eq. (C13) and rearrangement of terms with the help of Eq. (C5) yields
n2 siṽdQ 2 2 b2 dṽ .
u
­
S
cnsiṽdQ 2
(C15)
The final expression for the energy velocity follows
when Eq. (C15) is substituted into Eq. (C1) to yield
cnsi ṽdQ 2
.
2 b2 ṽd
n2 siṽdQ 2
¡
dk̃sṽd ,
ṼG sṽd ­ 1
dṽ
(C24)
­
ṽ
nsi ṽd .
c
ṽn0 siṽd ,
nsiṽd
k̃ 0 sṽd ­
1
c
c
b2
b2 sṽ 1 dd .
­2
n siṽd ­ 2
2
2nsiṽdQ
nsiṽdQ 2
4d 2 b2 ,
3u0 v0 4
(C17)
(C20)
(C27)
"
#1/2
u 0 v0 4
1
2
2
d , (C28)
6
su
2
u
d
2
4d
2i
SB
0
3
b2
3
where we have approximated a as 1. Substitution of
relation (C26) into relation (C28) then yields
v̂SB >
(C19)
(C26)
The location v̂SB of the saddle point of interest for u ­ uSB
is given approximately by [see Eq. (3.6) of Ref. 12]
v̂SB >
(C18)
(C25)
where
u0 ­ ns0d .
In this appendix we use the prime to indicate differentiation with respect to ṽ. We have
0
ṼG sṽd
b2 ṽ 2 .
­11
F
ṼE sṽd
uSB > u0 2
where k̃sṽd is defined by Eq. (3.14) as
ṽ
b2
11
c
Q
(C23)
Since F is nonzero for ṽ ­ 0, we see that the relative
difference between the two velocities is small for small
jṽj. This is significant because we are interested in these
nonoscillatory waves only when they correspond to the
contribution of the saddle point vb when it is the dominant saddle point and is located in the imaginary axis.
This means that we are interested in these waves only
when vb and hence ṽ are near the origin.
To see this, recall that vb becomes dominant when
u reaches a value uSB given approximately by [see
Eq. (3.43b) of Ref. 12]
(C16)
We now derive the expression for the group velocity of
the same waves. It is defined by Eq. (3.17) as
k̃sṽd ­
F ­ n2 siṽdQ 2 2 b2 dṽ 2 b2 ṽ 2 ,
ṼG sṽd 2 ṼE sṽd
b2 ṽ 2 .
­
F
ṼE sṽd
!1/2
(C22)
Hence the relative difference between the two velocities
is given by
It follows from Eq. (C9) that
√
cnsiṽdQ 2
.
2 b2 dṽ 2 b2 ṽ 2
n2 siṽdQ 2
then
,
to obtain
ṼE sṽd ­
(C21)
Comparison of Eqs. (C16) and (C22) reveals that the
energy and the group velocities differ only by the extra
term 2b2 ṽ 2 in the denominator of the group velocity. If
we define the function
We now combine Eqs. (C10) and (C11) with the definition of the plasma frequency,
√
n2 siṽdQ 2 2 b2 ṽsṽ 1 dd .
cnsi ṽdQ 2
It follows from Eqs. (C17) and (C21) that the group velocity of these waves is given by
(C10)
Since H and E are related by Eq. (C8), the magnitude of
Poynting’s vector is given by
S­
k̃ 0 sṽd ­
2 p
ds 3 2 1di ­ 1.07di .
3
(C29)
As u increases beyond uSB , the saddle point moves down
the imaginary axis until it reaches the point 22diy3,
at which time (with the approximation a ­ 1) it moves
G. C. Sherman and K. E. Oughstun
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
off the imaginary axis. Hence we are interested in the
behavior of the nonoscillatory waves for ṽ in the region
2
2
d # ṽ # 1.07d .
3
(C30)
Therefore jṽj is small for all waves of interest if d is small.
To investigate the relative difference between the
velocities over this range of frequencies, we return to
Eq. (C25). We begin by applying Eqs. (C5) and (C9) in
Eq. (C23) to expand F, with the result
F ­ ṽ 4 1 4d ṽ 3 1 s2v0 2 1 4d 2 dṽ 2 1 dsb2 1 4v0 2 dṽ
1 v0 4 1 b2 v0 2 .
(C31)
It is easy to obtain a rough lower limit on F and hence
a rough upper limit on the relative difference between
the two velocities when ṽ $ 0. Then all the terms in
Eq. (C31) are nonnegative, so that dropping any of the
terms does not decrease the value of the right-hand side.
By dropping all but the last term, we have
F . b 2 v0 2
(C32)
for 0 # ṽ. This is a crude upper limit, which can be
improved if needed. Application of this inequality to
Eq. (C25) then yields
ṼG sṽd 2 ṼE sṽd
ṽ 2
,
v0 2
ṼE sṽd
is satisfied. This inequality is satisfied if the equivalent
inequality with the positive terms omitted from the lefthand side, i.e.,
(C35)
is satisfied. Define the quantity on the left-hand side of
inequality (C35) to be
Gsṽd ­ 2f4d ṽ 3 1 dsb2 1 4v0 2 dṽg .
APPENDIX D
In this appendix we obtain relation (4.17), which is the
uniform physical model for the delta-function pulse that
is uniformly valid at the beginning of Sommerfeld’s precursor. It contains explicit approximate expressions for
vE that are valid for small positive values of u 2 1.
The simplest way to proceed is to employ Eq. (37) of
Ref. 13. That equation is the uniform asymptotic result
for the delta-function pulse containing the approximate
expression
vsp > 6j 2 dis1 1 hd
(C36)
b
f2su 2 1dg1/2
Gs22dy3d # 0.193v0 4 .
Hence, inequality (C35) is satisfied at this limit of the
range of interest. It is obviously satisfied at ṽ ­ 0 as
well. Therefore, it is satisfied for all 0 $ ṽ $ 22dy3 if
Gsṽd has no maxima or minima in that interval. It is
easy to see that no such extrema exist in the interval
(D2)
for small u 2 1, we can obtain the desired result by setting
h ­ 21 and setting j ­ vE in Eq. (37) of Ref. 13. This
works because h is not used in any way other than in the
approximation of the saddle-point location. The result is
!1/2
√
AvE
b2y2
Esz, td ,
u211
2b
vE 2 1 4d 2
√
!
b2
z
3 exp 2d
c vE 2 1 4d 2
(
"
√
!#
z
b2y2
3 22vE J1
vE u 2 1 1
c
vE 2 1 4d 2
"
√
!#)
b2y2
z
. (D3)
1 6dJ0
vE u 2 1 1
c
vE 2 1 4d 2
Since vE tends to ` as u tends to 1, we neglect 4d 2 compared with vE 2 . Then, employing approximation (D2),
we have
b2y2
> u 2 1.
vE 1 4d 2
2
Let us evaluate Gsṽd, assuming that ṽ has its most negative value of interest (i.e., 22dy3). Assume also that
d # 0.1v0 and that b # 5v0 . We then obtain
(D1)
for the saddle points. We want the same result except
with vE inserted in place of vsp . Since we are using the
approximation
vE >
2fṽ 4 1 4dṽ 3 1 s2v0 2 1 4d 2 dṽ 2 1 dsb2 1 4v0 2 dṽg , v0 4
(C34)
2f4d ṽ 3 1 dsb2 1 4v0 2 dṽg , v0 4 ,
by taking the first derivative of Gsṽd, setting it equal to
zero, and solving for its roots. It is found that no real
roots exist in the interval. Hence inequalities (C35) and
(C34) are satisfied for ṽ in the interval 0 $ ṽ $ 22y3d.
We conclude, then, that inequality (C33) holds for all
ṽ in the range of interest specified in inequality (C30) if
d # 0.1v0 and b # 5.0v0 . Hence, under these conditions,
the relative difference between the group and the energy
velocities is less than 1.15% for all ṽ in the range of
interest.
(C33)
for 0 # ṽ. This shows that for ṽ in the nonnegative
part of the range given in inequality (C30) the relative
difference between the two velocities is less than 1.15%
if d # 0.1v0 .
The situation is more complicated for negative values of
ṽ, because some of the terms in the expression for F are
negative. For negative ṽ the same logic applied above
for nonnegative ṽ is valid [and so are inequalities (C32)
and (C33)] if the inequality
245
(D4)
Thus relation (D3) becomes
#
"
q
AvE 2
z
Esz, td ,
2su 2 1d exp 22d su 2 1d
b
c
"
#
(
z
vE 2su 2 1d
3 2J1
c
"
#)
z
3d
.
1
vE 2su 2 1d
J0
vE
c
(D5)
246
J. Opt. Soc. Am. B / Vol. 12, No. 2 / February 1995
G. C. Sherman and K. E. Oughstun
Finally, we use approximation (D2) to substitute for vE
in relation (D5) to obtain relation (4.17):
#
"
Ab
z
Esz, td ,
exp 22d su 2 1d
f2su 2 1dg1/2
c
(
)
√
z
1/2
bf2su 2 1dg
3 2J1
c
(
)!
q
3d
z
. (D6)
1
bf2su 2 1dg1/2
2su 2 1d J0
b
c
delta function needs to be included. Hence we multiply
the integrand in Eq. (E4) by the quantity
2A
(D7)
as u tends to 1 from above.
becomes
In the case with d ­ 0, this
A
Esz, td > 2
2p
vc Z
z
exp 2ivt 2 i
2p C
v
!
Z
2p
0
expf22istz d1/2 cos ug
(E6)
3 expsiudidu .
Since
J1 szd ­
1 Z 2p
exps2iz cos adexpsiadida ,
2p 0
dv
,
v 2 2 vc 2
b2
zt ­
2
(E1)
√
z
c
su 2 1d ,
(
)
Ab
z
1/2
f2su
2
1dg
J
b
1
f2su 2 1dg1/2
c
(E2a)
Esz, td > 2
t­
z
su 2 1d .
c
(E2b)
as u tends to 1 from above.
to obtain
vc
2p
(E9b)
Substitution of Eqs. (E9) into Eq. (E7) then yields
b2 z ,
2c
(E3)
(E9a)
b2
z
.
­
t
2su 2 1d
z ­
For small positive values of u 2 1, he approximated the
integral by taking the contour C of integration to be a
closed circle of large jvj and neglecting vc 2 compared to
v 2 in the denominator. He then changed the variable of
integration to the variable u given by
r
(E8)
!2
for small positive values of u 2 1.
proaches the limit
expsiud ­ vstyz d1/2
(E7)
we have
Esz, td > 2
with
Esz, td ­
!1/2
It follows from Eqs. (E2) that
In this appendix we approximate the integral expression
for the delta-function pulse for u very close to 1 and d ­ 0
without applying the asymptotic analysis. The integral
expression is given in Eq. (2.6) with f˜svd ­ A and d ­ 0.
Sommerfeld approximated the step-function pulse under
the same conditions (see Ref. 2). He wrote the integral
expression for that field in the form
Esz, td ­
z
t
(D8)
APPENDIX E
√
√
Esz, td > 2Aszytd1/2 J1 f2stz d1/2 g .
Ab2 z .
Esz, td , 2
2c
(E5)
to obtain the corresponding approximation for the deltafunction pulse:
Since J1 sxd approaches xy2 and J0 sxd approaches 1 as x
approaches 0, Esz, td given by relation (D6) approaches
"
#√
!
z
dc
Ab2 z
exp 22d su 2 1d
126 2
Esz, td , 2
2c
c
b z
z
v2
exps2iud
­ 2A
vc
vc t
Ab2 z
2c
(E10)
This expression ap-
(E11)
ACKNOWLEDGMENTS
The research presented in this paper was, in part, supported by the Applied Mathematics Group of the U.S. Air
Force Office of Scientific Research under grant F4962092-J-0206. The writing of this paper was supported in
part by the Rocketdyne Division of Rockwell International
Corporation.
REFERENCES AND NOTES
t Z 2p
expf22istz d1/2 cos ugexps2iudidu .
z 0
(E4)
The integral for the delta-function pulse is the same as
the integral in Eq. (E1) except that it does not contain the
term vcysv 2 2 vc 2 d > vcyv 2 . Also, the path of integration is in the opposite direction and the amplitude A of the
1. A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177 – 202 (1914).
2. L. Brillouin, “Uber die Fortpflanzung des Licht in disperdierenden Medien,” Ann. Phys. 44, 203 – 240 (1914).
3. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
4. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New
York, 1941), Secs. 5.12 and 5.18.
5. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley,
New York, 1975), Sec. 7.11.
G. C. Sherman and K. E. Oughstun
6. B. R. Baldock and T. Bridgeman, Mathematical Theory of
Wave Motion (Halsted, New York, 1981), Chap. 5.
7. L. A. Segel and G. H. Handelman, Mathematics Applied
to Continuum Mechanics (Macmillan, New York, 1977),
Chap. 9.
8. I. Tolstoy, Wave Propagation (McGraw-Hill, New York,
1973), Chaps. 1 and 2.
9. L. B. Felsen, in Transient Electromagnetic Fields, L. B.
Felsen, ed. (Springer-Verlag, New York, 1976), Chap. 1,
p. 65.
10. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.
11. G. C. Sherman and K. E. Oughstun, “Description of pulse
dynamics in Lorentz media in terms of the energy velocity
and attenuation of time-harmonic waves,” Phys. Rev. Lett.
47, 1451 – 1454 (1981).
12. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817 – 849
(1988).
13. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic
description of electromagnetic pulse propagation in a linear
dispersive medium with absorption (the Lorentz medium),”
J. Opt. Soc. Am. A 6, 1394 – 1420 (1989).
14. The first modification is that we apply the definition of the
Fourier transform used in Refs. 12 and 13 rather than that
used in Ref. 11. The second modification is that we use g r
and g i , respectively, to denote the real and the imaginary
parts of the arbitrary complex quantity g. The notation differs from that applied in Refs. 12 and 13 by the same modification plus the change that the electric field is denoted
Esz, td instead of Asz, td. The latter symbol is used to denote a complex quantity with the property that Esz, td is
the real part of Asz, td.
15. See Ref. 3, Chap. 3. A more comprehensive derivation that
uses modern asymptotic techniques is given in Ref. 12.
16. R. Loudon, “The propagation of electromagnetic energy
through an absorbing dielectric,” J. Phys. A 3, 223 (1970).
17. These conditions are sufficient but are far from necessary.
18. Note that we have ruled out a nonoscillatory contribution
with ṽE ­ 0 at u ­ u0 by including only positive solutions to Eq. (4.5). We have done this to avoid including the
zero-frequency solution twice. It is included in the timeharmonic contribution, since we include all nonnegative solutions to Eq. (4.4).
19. We applied the ZREAL routine of the IMSL Math/Library,
available from IMSL, 2500 ParkWest Tower One, 2500 CityWest Blvd., Houston, Tex. 77042.
20. Note that we solve the exact saddle-point equation numeri-
Vol. 12, No. 2 / February 1995 / J. Opt. Soc. Am. B
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
247
cally rather than evaluating numerically the approximate
analytical expressions for the saddle points given in Ref. 3 or
12. We applied the routine ZANLY of the IMSL Math/Library
cited in Ref. 19.
The plot begins at u ­ 1.00055 rather than at u ­ 1.00000
because the second derivative of fsvd evaluated at vE or at
the saddle point tends to zero as u tends to 1 from above,
causing the approximations to tend to infinity. The uniform
approximation presented in Subsection 4.B does not have
this problem.
P. Wyns, D. P. Foty, and K. E. Oughstun, “Numerical analysis of the precursor fields in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1421 – 1429 (1989).
This is a consequence of the fact that the integral is divergent at u ­ 1.
The algorithm parameters used to generate the plot in Fig. 9
were k ­ 5000 and m ­ 250. To generate the dotted curves
in Figs. 10 and 11, the same parameters were taken to be
k ­ 500 and m ­ 250.
Equation (4.8) follows from Eqs. (4.3) and (4.16) of Ref. 13
with Re ũsvd replaced by 2i f˜svd. This substitution follows
from a comparison of the integral in Eq. (2.6) of this paper
with that in Eq. (1.7) of Ref. 13. In addition, we have made
use of the discussion following Eqs. (4.3) and (4.16) of Ref. 13
about the argument of a1 1/2 .
See Appendix B of Ref. 13. The right-hand side of Eq. (4.23)
differs from the right-hand side of Eq. (B9) of Ref. 13 by a
factor of 3y4 because the latter equation is incorrect owing
to a typographical error.
For a brief discussion of this effect in lossless, gainless media, see Sec. 5.7.4 of Ref. 6. The results there are in a
slightly different form from ours because the integral being treated is over wave number k instead of frequency v.
They can be placed in our form by appropriate change of
integration variable.
M. A. Biot, “General theorems on the equivalence of group
velocity and energy transport,” Phys. Rev. 105, 1129 – 1137
(1957); M. J. Lighthill, “Group velocity,” J. Inst. Math. Appl.
1, 1 – 28(1965).
K. E. Oughstun and G. C. Sherman, “Asymptotic theory of
pulse propagation in absorbing and dispersive dielectrics,”
Review of Radio Science 1990 – 1992 (Oxford U. Press,
Oxford, 1993), pp. 75 – 105.
K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse
Propagation in Causal Dielectrics (Springer-Verlag, Berlin,
1994).
F. W. J. Olver, “Why steepest descents,” SIAM Rev. 12,
228 – 247 (1970).