K. E. Oughstun and G. C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
817
Propagation of electromagnetic pulses in a linear dispersive
medium with absorption (the Lorentz medium)
Kurt Edmund Oughstun
Department of Computer Science and Electrical Engineering, University of Vermont, Burlington, Vermont
05405
George C.Sherman
Mission Research Corporation,P.O. Box Drawer719, Santa Barbara,California93101
Received June 10, 1987; accepted October 12, 1987
The classical theory of Sommerfeld and Brillouin of pulse propagation in a Lorentz medium is reexamined. We
show by numerical techniques that Brillouin's approximations for the saddle-point locations break down in certain
space-time regions. Analytic approximations that describe the correct saddle-point behavior are derived and
applied to obtain improved asymptotic expressions. Qualitatively, the resulting pulse behavior is similar to that
predicted by Brillouin. The quantitative improvements are significant, however, and have led to a simple mathematical procedure for determining the pulse dynamics in addition to a clear physical interpretation.
1. INTRODUCTION
As a wave pulse propagates through a linear dispersive medium its properties can change in complicated ways that are
difficult to systematize. Eventually, however, after the
pulse has traveled far enough, its dynamics can settle into a
systematic pattern that remains intact for the rest of the
flight. Typically, the pulse becomes locally quasi-monochromatic with different local amplitudes, frequencies, and
wave numbers at different locations within the pulse. Each
quasi-monochromatic portion of the pulse propagates with
its own characteristic velocity, which remains constant as
the propagation continues. When this condition prevails,
the dispersion is said to be mature.'
For the special case of lossless media, but under otherwise
rather general conditions, a simple mathematical procedure
and its associated physical interpretation specify the pulse
dynamics when the dispersion is mature.1-3 To implement
this procedure we need only a knowledge of the spectral
content of the pulse and the group velocity of the medium
for all frequencies.
When the medium becomes absorbing, however, the above
mathematical procedure and physical interpretation break
down. Although analytical expressions for pulse dynamics
can sometimes be obtained under the conditions of mature
dispersion by using asymptotic techniques, the results have
not yielded a simple procedure or physical description similar to the group-velocity description.4 One promising approach that appears to have rather general validity has been
developed by Connor and Felsen 5 by introducing complex
space-time rays. However, the mathematical procedure is
much more complicated than the group-velocity procedure
for lossless media, and the resultant physical interpretation
is obscure.
The lossy dispersive system that is best understood is that
0740-3224/88/040817-33$02.00
of an electromagnetic pulse propagating through a Lorentz
medium. Sommerfeld and Brillouin 6 developed a theory of
that system in the early 1900's, which is now classic. 7 8 This
theory is valuable because it provides analytic expressions
for the dynamics of pulses under mature dispersion conditions in a medium that accurately models several real lossy
dispersive materials. It is a canonical theory that continues
to serve as the basis for comparison with experimental results9- 2 and the predictions of other theoretical approaches.' 3"14 This theory has also been useful as the foundation
for extensions to more-complicated geometries 5 and materials.' 6- 8 The value of the classical theory for such extensions, however, has been primarily as a guide for the complicated asymptotic analysis required. The possibility of extrapolating the known results for the Lorentz medium to
provide insight into other dispersive systems (without performing the detailed asymptotic analysis) has been limited
because the classical theory did not reveal a simple mathematical procedure or physical interpretation (such as the
group-velocity description).
We have reexamined' 9 the classical theory and have discovered a simple mathematical procedure for determining
the pulse dynamics and an associated physical interpretation that is similar to (and an extrapolation of) the groupvelocity description. We have described these results without proof or derivation in Ref. 20. The derivation consists of
two parts. In the first part, which is the subject of the
present paper, we improve substantially the accuracy of
Brillouin's results. These improved results are useful not
only for our own purpose but for all studies that employ the
classical theory. In the second part of the derivation, we
establish a connection between the improved asymptotic
results and the physical concept of the flow of energy in
monochromatic fields propagating through the Lorentz medium. This connection provides both the simple mathemat© 1988 Optical Society of America
818
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
ical procedure for determining the asymptotic results and
the resultant physical interpretation. This second part of
the derivation will be presented in a subsequent paper.
2. FORMULATION OF THE CANONICAL
PROBLEM
A(z, t) =
Consider an arbitrary plane electromagnetic wave that is
propagating in the positive z direction through a linear,
homogeneous, isotropic, temporally dispersive medium occupying the half space z > 0. Let A(z, t) represent either the
scalar potential or any scalar component of the electric field,
magnetic field, Hertz vector, or vector potential field. The
spectral amplitude A(z, w)appearing in either the Fourier or
the Laplace representation of the scalar wave field, namely,
A
1(.)exp{i[k(o)z - wt]}dw
(2.9)
for z 2 0. With the dispersion relation [Eq. (2.3)], Eq. (2.9)
may be written in the following form, which is suitable for
asymptotic methods of approximation:
j l(w)exp[z 0(w, 0)]dw,
A(z, t) =
(2.10)
where the complex phase function O(w, 0) is given by
O(W, 0) = i[n(w) - 0],
(2.1)
A(z, w)e-iWtdw,
A(z, t) =
where f(w) is the Laplace transform of the initial time behavior of the pulse at the plane boundary z = 0. Therefore the
exact integral formulation describing the propagation of an
arbitrary plane-wave pulse through a dispersive medium is
given by
(2.11)
and where
then satisfies the scalar Helmholtz equation
2
2
= ct
where the complex wave number fi(w) is given by
=
(2.3)
n(w),
with n(w) being the complex index of refraction of the dispersive medium.
Consider now the situation in which the field A(z, t) on the
plane z = 0 is known for all time t and is nonzero only after t
= 0, so that
(2.4)
A(0, t) = f(t)
for all t, where the function f(t) vanishes identically for t < 0.
The integral expression [Eq. (2.1)] for the-scalar field A(z, t)
is then taken to be a Laplace representation in which, for
purposes of convergence, 212 2 the contour of integration C in
the complex w plane is the straight line .o = w' + ia, with a
being a fixed positive constant that is 'greater than the abscissa of absolute convergence 7 for the function f(t) and
where w' Re(w) ranges from negative to positive infinity.
The general solution of the ordinary differential equation
[Eq. (2.2)] is given by
A(z, w) = A+(w)exp(ihz) + A.(w)exp(-ikz),
(2.5)
where the first term on the right-hand side represents a
plane-wave disturbance propagating, in the positive z direction and the second term represents propagation in the
negative z direction. For propagation in the positive z direction A_(X) must be zero, and Eqs. (2.1) and (2.5) give
A(z, t)
=
J
A+(w)exp[i(z
-
wt)]dw.
(2.6)
Evaluation of this expression at the plane boundary z = 0
and application of the boundary condition (2.4) then yields
f(t) =
J
t
(2.7)
A+(w)e-iwdw.
The inverse transform of this result then gives
A+(w)
=
2
J
f(t)ei" tdt =
(2.12)
(2.2)
[V + ) (w)JA(z, co) = 0,
(W),
(2.8)
is a dimensionless parameter that characterizes a spacetime point in the field.
From the constitutive relations for electromagnetic radiation in a linear, homogeneous, isotropic, temporally dispersive medium, admissible models that describe the behavior
of the complex index of refraction in the complex w plane
must obey the relation
(2.13)
n(-w) = n*(w*),
where the asterisk denotes the complex conjugate, and
hence, from Eq. (2.11),
0)
A*(w*,
q0(-o, 0) =
(2.14)
Furthermore, for a real-valued initial pulse f(t) it follows
directly that its Laplace transform satisfies the relation
A-c) = f*(w*).
(2.15)
With these three relations we can show that
A(z, t =
Re
Xo)exp -I(c, 0) dw
(2.16)
for z 2 0, that is, given that the initial field behavior is real,
the propagated field behavior must also be real.
A case of special interest is that of a pulse-modulated sine
wave of applied signal frequency w, namely,
f(t) = u(t)sin(wct),
(2.17)
where u(t) is the real-valued initial envelope function of the
input pulse that is zero for t < 0. The propagated field is
then given by
( ia+I
a(w - wo)expL- (w, 0)jdwo (2.18)
Re i
A(z, t) =
for z 2 0, where a(w) denotes the Laplace transform of u(t).
The integral representations of Eqs. (2.10) and (2.18) form
the basis of the problem treated in this paper. The same
representations also apply even when the initial pulse shape
does not vanish identically for times t < 0 but rather possesses a sufficiently large exponential decay as t approaches
negative infinity to ensure the existence of the spectrum
K. E. Oughstun and G.C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc Am. B
f(w), which is now given by the Fourier transform of the
initial pulse shape. Furthermore, with the use of generalized function theory, 23 these results may be extended further
to include, for example, the behavior of strictly monochromatic fields.
The analysis of the present paper focuses on the dispersive
field evolution of the following two fundamental input
fields:
A. First Canonical Problem: Delta-Function Pulse and
the Impulse Response of the Model Medium
The first canonical problem treats the propagation of an
input delta-function pulse at time t = to > 0,
f(t) = Ot - to),
(2.19)
whose Laplace transform is
f(W) = J 6(t - to)ei'tdt = exp(iwto).
where
(t
-
to)].
(2.22)
This field yields the impulse response of the model medium.
B. Second Canonical Problem: Unit Step-FunctionModulated Signal
For a unit step-modulated signal, the initial envelope of the
pulse is given by the Heaviside unit step function,
U()
1,
fort > 0
(2.23)
that is, the external current source for the field abruptly
begins to radiate harmonically in time at t = 0 and continues
indefinitely with a constant amplitude and frequency. The
Laplace transform of this initial pulse envelope is then given
by
1(w)=
J
e"'tdt= ±
(2.24)
for Im(co) > 0, and the integral representation of the propagated disturbance is
1
ia+ 1
Iz
A(z,t = -- Re
expi - (W, )d
(2.25)
27r
-ia
-co,
Ic
raphy of the real part X(w, 0) of the complex phase function
0(w, 6) in the complex plane. In particular, the location of
the saddle points of 0(w, 0), the value of 0(w, 0) = X(w, 0) +
iY(w, ) at these points, and the regions of the complex w
plane wherein X(w, 0) is less than the value of X(w, ) at the
dominant saddle point for a given value of 0 are all required.
Through an investigation of the structure of 0k(o, 0) for a
Lorentz model medium2 6 27 in special regions of the complex
co plane where its behavior is relatively simple, Brillouin 6 25
was able to determine a rough general picture of the topography of X(w, 0). He could not provide that topography with
much detail, however, as it was necessary to resort to numerical methods without the aid of modern computers. A
brief review of Brillouin's analysis is provided in Appendix
A. This review provides the framework for the purely numerical results that follow. The results of this analysis are
summarized in Figs. 1-3.
(2.20)
The propagated scalar wave disturbance is then given by the
integral representation
1
ia+"
rzdl
A(z, t) =Re
expi - ot 0(wl ) do
(2.21)
27r
fiaIC
0t,0(w, ) = i n(w) -
819
I I
for t > 0 and is zero for t <0. This signal is precisely the one
treated by Sommerfeld 2 4 and Brillouin 6' 25 in their early considerations of this problem to give a more precise definition
of the signal velocity in a causally dispersive medium.
3. ANALYSIS OF THE PHASE FUNCTION
AND ITS SADDLE POINTS
In preparation for the asymptotic analysis of the integral
representation for the propagated field in a temporally dispersive medium, it is necessary first to determine the topog-
A. Numerical Results of the Topography of X(w, )
These basic results concerning the behavior of X(w, ) =
Re[kO(w, 0)1 in specific regions of the complex co plane are now
supplemented with computer-generated contour plots of
X(w, ) in the right half-plane. These plots are needed to
give a more complete picture of the topography of X(w, 0) in
order to determine the number of saddle points of 0(w, ),
their locations and relative importance, the approximate
locations of the deformed contour of integration that passes
through these saddle points (necessary for the subsequent
asymptotic analysis), and the way they move about in the
complex plane with changing 0. With Brillouin's choice of
the medium parameters, that is,
coo=
4.0 X 1016 sec-1,
b2 = 20.0 X 1032 sec 2 ,
= 0.28 X 1016 sec-1,
specific contours of the real phase function X(w, 0) in the
complex plane [including the contours that pass through
the saddle points of (w, 0)] have been plotted by computer
techniques and are reproduced in Fig. 4 for several values of
the parameter increasing away from unity. The dominant
saddle point is taken here as the saddle point that exhibits
the least exponential decay. Those contours of X(w, ) that
are greater than that at the dominant saddle point are indicated by solid curves, while those with values of X(w, ) that
are less than that at the dominant saddle point are indicated
by short-dashed curves in Fig. 4.
As Brillouin observed, 6 25 for all values of except one
there are four first-order saddle points of the phase function
kO(a, ) symmetrically located with respect to the imaginary
axis. Two of the saddle points lie in the region near the
origin, and the other two lie in the region removed from the
origin. The two saddle points near the origin lie along the
imaginary axis at = 1 and approach each other along the
imaginary axis as increases [Figs. 4(a)-4(d)], coalescing
into a single second-order saddle point when 0 = 1, at which
point the value of Al for Brillouin's choice of the medium
parameters is just slightly larger than 1.501 [Fig. 4(d)]. For
larger values of , the near saddle points separate symmetrically with respect to the imaginary axis and approach the
branch points + and w- as 0 approaches infinity [Figs 4(e)
820
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
- -W
--e
-TWT -e-,"
: Bra nc h. ......................................
C&
C
Co.
W-
w
l
E
.
-,--sCPO
t.
. nch:
.................................. Bra
G)+F C Z
_W
-
C,:
c,
G
Fig. 1. Branch cuts for both the complex index of refraction n(w) and the complex phase function (c, 0) in the complex w plane for w > .
3
0
CO'( d"/sec)
0
Gmin
-Z
-4
X (d)
-6
-8
-10 J
(X e)
Fig. 2. Behavior of the real and imaginary parts of the complex index of refraction n(w') = n,(w') + ini(w') (upper curves) and the real part
real w'axis. The medium parameters are those chosen by Brillouin in his
function (lower curve) along the positive
X(') of the complex phase
6
2
32
2
analysis: wo = 4.0 X 1016 /sec, b = 20.0 X 10 /sec , 6 = 0.28 X 101 /sec.
K. E. Oughstun and G. C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
it
821
saddle points of the complex phase function 0(w, 0) must be
located in the complex co plane, and the behavior of k(W, 0)at
these points must be determined. The condition that (c,
0) be stationary at a saddle point is simply that 0'(,6) = 0,
where the prime here denotes differentiation with respect to
c, so that
l
n(c) + Wn'()-0 = 0.
(1+)0
(3.1)
+Lc0
-
-
cL4t-eo
(i-L)co
+Lo
-Lo t W,
The roots of this equation then give the desired saddle-point
locations. With the complex index of refraction given by
Eq. (Al), the saddle-point locations are given exactly by the
equation
0
(-;)0
(a)
2
WI2
+ 2i +
b2W (co2
12
W2Wo + 26ik,
co)'
2
= 02 (W
_
12
+ 2bi)(W2 -
2
+ 2biw),
(3.2)
2
_
.
_
I
I
_
-_
_
-
_
~
-e
+m
-a:
"Iook-0
- 11
"I
CL)'
-Se
\AtJ _
_+
.
_
_
c
+co
+cX
c-Se
X(ca,e) = 0
(b)
Fig. 3. Behavior of n(w) and X(w, 0) in the immediate neighborhoods about the branch points + and +'. The dashed curve
indicates the approximate location of the contour X(co, 0) = 0 for >
0.
where W1 = L02 + b2 .
Since this expression is a complicated function of the
complex variable a, it is difficult (if not impossible) to determine exact general analytic expressions for the locations of
the saddle points as a function of . However, from the
computer plots of the topography of X(w, 0)in the complex co
plane, given in Figs. 4(a)-4(f), we find that there are, in
general, two saddle points in the regions near the origin and
two saddle points in the region well removed from the origin,
just as Brillouin found. 6 25 These two regions may then be
considered separately to determine the approximate locations of the saddle points.
In the region removed from the origin, Brillouin6 25 approximated the complex index of refraction as
n(w -
and 4(f)]. On the other hand, the two distant saddle points
are situated in the lower half of the complex plane for all
> 1 and are located at A- - 2i at 0 = 1. As increases,
these saddle points move symmetrically in from infinity and
approach the branch points +' and Wa' as approaches
infinity. Initially, the distant saddle points have less exponential decay associated with them than does the upper near
saddle point labeled SP 1 in Figs. 4(a) and 4(b). Since the
original path of integration for the integral of interest in this
paper is not deformable through the lower near saddle point
SP 2, this saddle point is irrelevant for the present analysis
for values of 6below and bounded away from = 1. At 0 =
OSB
1.33425, illustrated in Fig. 4(c), the upper near saddle
point SP 1 has precisely the same exponential decay associated with it as do the two distant saddle points. Consequently, at = SB those three saddle points are of equal importance in the asymptotic analysis of the propagated field.
Hence for values of 6within the range 1 < < OSB, the two
distant saddle points are dominant over the upper near
saddle point SP 1 ; for values of s < < 1, the upper near
saddle point SP 1 is dominant over the distant saddle points;
and for all later values of the parameter , the two near
saddle points are dominant over the two distant saddle
points.
B. Location of the Saddle Points and Approximation of
the Phase
To obtain the asymptotic expansion of the propagated field
A(z, t) for large values of the propagation distance z, the
-
b2
2w(w + 26i)
(3.3)
which is valid provided that 1w12
>> Co2 and JW12 >> b2 . With
this approximation the solutions of the saddle-point equation [Eq. (3.1)] are found as
6
5p
(00)~Cosp,
[2(0( -b 1)]112
-
25i.
(3.4)
At 0 = 1 these two saddle points are at infinity, and as 6
increases they move in symmetrically along the line " =
-26 toward the imaginary ." axis. However, as increases
and the saddle points come into the vicinity of the branch
points o' and +',as Brillouin stated, approximation (3.3) is
no longer valid, and expression (3.4) becomes inaccurate.
In the region about the origin, Brillouin 6' 25 approximated
the complex index of refraction as
n(w)
t
-C2
+
b 3 (o + 22i) _ 62b2(4w31 2
2w 1W
2 wO
0
1
5
b2)
2
W)
(3.5)
which is valid provided that JW12 << Wo2 [the final term in
expression (3.5) is missing in Brillouin's analysis and is included here so that this expression be correct to 0(W2 )].
With this approximation the solutions of the saddle-point
equation [Eq. (3.1)] are found to be
3
where
ab 2
(-&4a2
- 3
a
(3.6)
822
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
e
WI(lob/sec)
=.
N
ok
(a)~~~~~~~~(
X
F
lo
/
A((SV
/
I
N
e,-
eo=
0' (Sxec
A
40~~~~
(b)
e
i
e,,
133425
EO
Ad
4b
(C)
(xei)/sec)
K. E. Oughstun and G. C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
e=
823
1.501
H________
,_
-- --
________
_
-_
- - - - - - =
C/K
(X_
_/Sec)
= …
…_
===
1,0-
_
0
)
e = 1.bs
W"i
(Xlo`/c)
(l.ox1o")
-…
~~~
_
_ _ _,Q
_ * _'L _ _ _ __ _
(e)
e=
co 1 (xlo
s.o
Ise )
- - - - - - -…---"
- -- --
…
C (lo /se)
*
Fig. 4. (a) Isotimic contours of the real phase behavior X(w, 0) in the right half of the complex w plane for 0
= 1. X(co, 1) < 0 in the upper half-plane, and the
dominant distant saddle points are located in the lower half-plane at wSPD = X- 2i and X(wspDA, 1) = 0.
(b) Isotimic contours of the real phase behavior X(w,
0) in the right half of the complex plane for 0 = 1.25.
The distant saddle point has moved in from infinity
and is still dominant over the upper near saddle point
SP,. (c) Isotimic contours of the real phase behavior
X(w, 0) in the right half of the complex plane for 0 =
OSB = 1.33425. At this value the distant saddle points
and the upper near saddle point SP, are all of equal
importance. (d) Isotimic contours of the real phase
behavior X(w, 0) in the right half of the complex
plane for 0 = 1.501, which is just before the coalescence
of the two near first-order saddle points SP1 and P2
into a single second-order saddle point. The upper
near saddle point SP is dominant over the distant
saddle points. (e) Isotimic contours of the real phase
behavior X(w, 0) in the right half of the complex
plane for 0 = 1.65 > 1. The dominant near saddle
- -
~ ~ ~ ~ ~ ~ ~ ~
1--;~~(f
points have now moved off of the imaginary axis into
the lower half of the complex plane. (f) Isotimic contours of the real phase behavior X(w, 0) in the right
half of the complex plane for 0 = 5.0. Notice the
approach of the dominant near saddle point toward
the branch point + and the approach of the distance
saddle point toward the branch point wo+'.
824
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
00 =
l = (1 + b2/WO2) 1 /2 =
(3.7)
(0)
coo
2
6 (4
a:= 1-
2
2
- b)
(3.8)
2
2
Co W1
2 2
the two near saddle points lie along the imaginary axis,
symmetrically situated about the point w = -i2b/3a. At 0 =
01 the two near saddle points coalesce into a single saddle
point of higher order at w = -i2b/3a, and for 0 > 01 the two
saddle points move off of the imaginary axis and into the
complex w plane along the line w" = -26/3a, approaching
it- along this line as 0 goes to infinity. However, as 0
increases above 01 and the saddle points come into the vicinity of the branch points w_ and w+, approximation (3.5) is no
longer valid, and expression (3.6) becomes inaccurate.
The exact locations of the near and distant saddle points
as 0 approaches infinity may be obtained readily from the
exact saddle-point equation [Eq. (3.2)]. In that limit, either
2
2
(c - W + 26iw) = 0, yielding the root
OSPN (_) =
or n(w)
=
(Wo
- 62)1/2 -
Thus in the limit as 0 approaches infinity, the two near
saddle points move onto the branch points w+and co-, and
the two distant saddle points move onto the branch points
co+' and ca-'. An approximate solution of the saddle-point
equation that accurately models the evolution of the saddle
points for all 0 2 1 is now given.
1. The Region Removed from the Origin
To obtain a more accurate description of the distant saddlepoint locations over the entire range of 0 values of interest,
we begin with the exact saddle-point equation [Eq. (3.2)],
which may be rewritten as
02(-
1
- 2
wo2 + 2iw)
1 2 + 2ic +
b2W(W2 + i) 12
, 2 W_ O + 2iw
2
w2-W1 + 2'3iw.
=
(3.12)
In this particular form we can see that as 0 approaches
infinity the right-hand side of the equation must approach
zero, and Eq. (3.11) is obtained. With this limiting behavior
in mind, the rational function in the squared term of Eq.
(3.12) may be approximated as
2
b W(W + i)
2
2
W -o
+ 23ic
-b
2
22ibb
)
(3.13)
=
[(131+
(1
102)'/3 -
- 12)3]
i23 + 2 [(I1+ 00)1"3 + (1
-
(2)314
(3.14)
1
32
2
(3.15a)
= 3
o2 _
=1
J4(W2 - P2) + b
62 +
(0+3
2
2
2 2
2
2
X [(3wO
0 - 2 )WO0 + 2' (9ccO - 862)]
+
(02
[(3W0 2 -
-1)2
b606
1/21
(02 -1)
j
62)04 + 962(202 +
3)]
(3.15b)
This complicated expression is rather formidable to work
with, and a more simplified but accurate version is sought.
Since 12 >> 01 for 0 > 1, then
(3.11)
= ±(O2 - 62)1/2 - bi.
+
where
0, yielding the root
SPD ()
i-2
-
(3.10)
i,
- (@2
'sPD (0) (3.9)
234b
3a00 w0 4
2
2biW2
2
has been neglected. The appropriate
where the term 62b 4/wO
solutions of this cubic equation that give the locations of the
two distant saddle points are
0 < 01, where
For values of 0 in the range 1
a = 00 +
t3 +
L
+ 0201'
~~~(01
2/3'
= 021'3 +
132/3
+
(01-2)1/'3 =-1'3
With this approximation, expression (3.14) becomes
WsP,1_(0
=+ OS02 / i_(3-3
13
332 2/3/
-
(3.16)
Furthermore, the quantity 1321/3 may be approximated as
131/3 -
4
(1 2
-
2 +
)
b
(3.17)
With these approximate results, the second approximation
to the locations of the distant saddle points is then given by
(3.18)
where, for convenience, the following two functions have
been defined:
SP (0)
() - '3i[1 + q(0)],
t(0) = (Wo2 - 62 +
b2
1)
(3.19)
w+'3i
co + 2i
_b(1-
which is valid provided that 12>> wo2 and Iw>>I. With this
approximation saddle-point equation (3.12) becomes
(0) =
2
2
2/27 + b /(0 - 1)
~2(0)
(3.20)
We can see that the second approximation [expression
(3.18)] properly describes the evolution of the distant saddle
points. First, for values of 0 close to unity, t(0) _ b/[2(0 1)]1/2, n() _ 1, and Eq. (3.18) reduces to the first approxima-
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
K. E. Oughstun and G. C. Sherman
825
oI/
-
7-
cX_
co_
................
I
"',
1--l"",
-
Ca
-
Cl)/
CZ.
An.................
-26
Fig. 5. The behavior of the saddle points in the region removed from the origin. The dotted curves indicate the trajectories followed by the
saddle points as 0 varies.
We can see that X(6, r, ) attains its maximum variation
tion [expression (3.4)] for the location of the distant saddle
points. On the other hand, for very large values of 0, t(O)
2
2
(WZ12 - 62)1/2, ij(0) _ 6 /[27 W1 - 62)]
0 and
lim
[SPD'(6)] _
-(w2
62)1/2
with r about the distant saddle point wSpL+ for
ai = any,
in agreement with Eq. (3.11). Exact agreement is not obtained because 77(0) does not vanish identically as approaches infinity. However, for realistic values of the medium parameters Co, b2, and 6, this discrepancy is exceedingly
small.
With Eq. (3.3) the phase behavior in the region of the
complex w plane removed from the origin may be approximated as
6(,
0) i(l
2( + 2 i)
-
(3.21)
This approximation is valid provided that both I61 >> so and
1A >> b. Both of these inequalities are satisfied (at least in
the weak sense) for all values of 0 > 1 when the second
2. The Region Near the Origin
To obtain a more accurate description of the near saddlepoint locations, particularly for values of 0 greater than 01,
we begin with the exact saddle-point equation [Eq. (3.2)],
which may be rewritten as
approximation [expression (3.18)] for the distant saddlepoints locations is employed; however, this is not the case
when the first approximation [expression (3.4)] is employed
for increasing values of 0 >> 1. To determine the approximate behavior of X(w, 0) = Re[O(w, 0)] in the vicinity of the
distant saddle point in the right half of the complex w plane,
let
W= CSPD +()
2
-w
0 + 26ia)
= C2
+ b4
+ rei#
(W2
(O) - i[1 + n(0)] + rei.
)1i(O) + 6[1 + (0)] + ire
- i b2
- 2
_
2
-
1
2+
2
biw + 2b 2
2
W
(W,+ 6i)2
+ 25i*)(w 2
-
,O2
22(
2+ i)i
o -W
- 0 + 26iw
+ 2biw)2
(.4
(324)
For any value of col small in comparison to wo, we can reason
that
With this substitution, expression (3.21) becomes
0(0, r, ) - (1-
= 7r/4, 37r/4,
57r/4, and 7ir/4. Consequently, in the right half-plane the
lines of steepest descent through the distant saddle point are
at the angles 4' = 37r/4 and 7r/4, and the lines of steepest
ascent are at A = 7r/4 and 5r/4. Owing to the symmetric
behavior of X(w, ) about the imaginary axis, the reverse
holds true for the distant saddle point in the left half-plane.
These two saddle points are illustrated in Fig. 5, in which the
arrows indicate the direction of ascent along the lines of
steepest descent and ascent through the saddle points that
are at angles of 45° to the coordinate axes in the immediate
vicinity of the saddle points. The shaded areas in the figure
indicate the regions about the saddle points wherein the
inequality X(X) < X(wspDb, 0) is satisfied and in which the
path of descent from each saddle point lies. These general
features are in complete agreement with the numerical results presented in Figs. 4(a)-4(f).
2
t(O)
- q(6)]
2
-b
i[l - q(O)] + re-"P
+ 20(6)r cos(4) + 26[1 - (6)]r sin(4') + r2
(3.22)
and hence
X(6, r, 4')
(
-
)16[1 + 7(0)]
-
X0
b2 (1 [+ )- sn}
6[1 - x(0)] + r sin
r sin 4'j - 2 2(0) + 62[1 - 1(0)] 2 + 2(6)r cos(4) + 26[1 - f(O)]r sin(4) + r2
-
(3.23)
826
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
262
_
2 +6i
02 -.
K. E. Oughstun and G. C. Sherman
-2co
-
+
2b 2
0)2
(3.29)
02
2) 31
4
0o +
3
)02/
(3-
)2
2
02
co2
0o2 +
2a
Wo
where the positive branch of the square root is to be taken in
Eq. (3.28).
For values of 0 approximately equal to 0o, the functions
given in Eqs. (3.28) and (3.29) become
and
6i) 2
02(0) +
(0)2 -
0)12
+ 26iw) (02
24
-
Co1 W0
+ 26ji)
0)2
-
2
[6
2
W1~W0
2(2)12 +
F200w
2)0)
With these approximations, Eq. (3.24) for the near saddle
points becomes
02(02 -
-=
25ib 2 F
b2
462
4[3 ++-- 2
L
0)2
I b 2 /2
)2
02)W
-
20
J
lim [SPj1(0)] = +(0)o2 -
2
Since 6 is typically much smaller than either b or wo, we can
see that the coefficient of the cubic term in the above equation is very small in comparison with the other terms.
Hence this term may be neglected to obtain the quadratic
equation
02 -
2
0 + 2i
02
O=oa
\12
6b
2
0)0 -b
2)1 3
w + Oco
i
4 2
41~
2 2
462
b2
+26i 1 -
62b2
-
~
02
-
o02 + 3b a
2
=
-i
/02~~~2
01
WO2(012
o2)
2 - o02 + 3b
2
0) 2(02 - o0 )
-
62)1/2
in agreement with Eq. (3.10).
To analyze the behavior of the near saddle points as specified by Eq. (3.27), we first need to determine the algebraic
sign of the argument of the square root in Eq. (3.28). This
process amounts to determining a more accurate value of 0
01 at which that argument will vanish, and hence
+ 2b2
WO262
2°
02 - 0o2 + 3b2 a
1/2
and Eq. (3.27) reduces to the first approximation [expression
(3.6)] for the location of the near saddle points. Furthermore, in the limit as 0 approaches infinity the second approximate solution [expression (3.27)] yields
+ 26iw)
0)2
Oo)
_
4
L03b2 a (0
=0
2
002
~~~
=0, (3.25)
02
~~621
~
\1
02
a
2
WO
o02 +3b2
Since 01 is greater than 00 for nonzero values of 6, the appropriate solution of this equation is
00
1/2
01
32e
b2
= (002
where the parameter a is now given by [see Eq. (3.8)]
a =1-
+ b2 )
2 2(4)
(3.26)
and is nearly unity.
The second approximation to the locations of the near
saddle points is then
0
SPN -
(0)
-
3 i6(0),
(3.27)
where, for convenience, the following two functions have
been defined:
2 1/2
'o2(2
62)
0
i (0) =
-
- 62
0o2 + 3ba°
0o2
(3.28)
+
-
2 02
462
0)021+
62'IkL
16
62(0 2 - 62) 11/2
2 462)2j
1 (3.30)
where the positive values of both square roots are to be taken
in this expression. The square of this expression may be
approximated as
02
-
02
2
462b2 2
COO(3a 0
- 462)'
thus
01 -00
+
60 C)O(3a0o
2
- 462)
(3.31)
which differs only slightly from the first approximation given in Eq. (3.9). The approximate expression (3.31) for 02 is
much simpler than that obtained without approximation
from Eq. (3.30) and greatly simplifies subsequent calculations regarding the behavior of the saddle points at that
critical value.
With Eqs. (3.5) and (3.26) the phase behavior in the region
of the complex 0) plane that is traversed by the near saddle
points may be approximated as
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
K. E. Oughstun and G. C. Sherman
and ascent are then parallel to the coordinate axes, as illustrated in Fig. 6. In this figure the arrows indicate ascent
along these lines, and the shaded areas indicate the regions
about the saddle points wherein X(Z) is below that value at
the respective saddle point. The saddle point labeled SP 1 is
located at WSPN+, and the saddle point SP 2 is located at
wSPN . For the upper near saddle point SP,, the paths of
steepest descent are at x = 0 and 7r; for the lower near saddle
point SP 2, the paths of steepest descent are at x = 7r/2 and
3ir/2. At 0 = 6o,WSPN+(0O) = 0 exactly, and both 0(WSPN+, 6 o)
and 0'(wSPN, 00) vanish identically.
At the critical value 0 = 01, expression (3.27) yields only
one saddle point, which is located at
c-oil
WA
4A
SP.
*m
lq
3
WSPN(Ol) -
NW_
_SP,
di E
Fig. 6. The behavior of the saddle points in the region near the
origin for 1 0 < 0. As 0 increases over this range the two saddle
points steadily approach each other.
0(w, 0) - i(6
0
w co(iaw - 26).
- 0) +
This approximation is valid provided that
(3.32)
Iw << wo. This
inequality is satisfied (at least in the weak sense) for all finite
values of 0 > 1 when the second approximation [expression
(3.27)] for the near saddle-point locations is employed; however, this is not the case when the first approximation (3.6) is
employed for increasing values of 0 >> 01.
For 1 < 0 < 01 the near saddle-point locations are given by
~23
i +I
SPN ()
-)l
(0)],
(3.33)
where the quantity inside the brackets is real. To determine
the behavior of X(w, 0) = Re[O(w, 0)] in the vicinity of either
near saddle point for 1 < 0 < 01, let
0
Co= USPN
= i
827
-
(3.36)
3.
At this critical value of 0 both ''(WSPN, 61)and 0"(WSPN, 01)
vanish, and the two near saddle points have coalesced into a
single higher-order saddle point. From expression (3.35),
with " = -26/3a, the behavior of X(co, 0) in the vicinity of
this saddle point is given by
X(01, r,X)
6
00
4
4( 6 2
27a
+
2
2
ar3 sin 3X)-
(3.37)
Hence X(01 r, X) attains its maximum variation for x = r/6,
7r/2, 57r/6, 77r/6, 37r/2, and 11r/6. The lines of steepest descent from the saddle point are at x = 7r/6, 57r/6, and 37r/2,
and the lines of steepest ascent are at x = ir/2, 77r/6, and li1r!
6, as illustrated in Fig. 7.
Finally, for 0 >01 the near saddle-point locations are given
by
USPNWSPN
(0) - +(0)
-2
(3.38)
WM(0)
~~3
where both AP(0) and (O)are real. As increases away from
01 the two near saddle points symmetrically move off the
imaginary axis into the lower half of the complex w plane,
and as 0 approaches infinity they approach the branch
(0) + reiX
+ reix,
where w = +Iql(6)l - 2/3 b(0). With this substitution, expression (3.32) becomes
40(0, r, x)
0) + 20owo4
(-a" + ire X)(00
-
X [acoz" 3 + 26e0w2
- i(3aw
,
-
2
2
(3acw" + 26)r ei x + ir
2
+ 46co')reiX
3 3
e X,
ca)
(3.34)
and hence
26 3CX
X(8, r, X)
(
+r
sin x)( - 0) +
200w 04
a
+ 2/2/
+ (3aw" 2 + 4w")r sin x - (3aw" + 2)r
X cos(2x)
-
ar3 sin 3 X].
2
(3.35)
We can see then that X(0, r, X) attains its maximum variation for x = 0, r/2, 7r, and 371r/2. The lines of steepest descent
Fig. 7. The two saddle points in the region near the origin have
coalesced into a single saddle point of second order at 0 = A.
828
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
K. E. Oughstun and G. C. Shermah
Q)e
Ca
26
.30(
CL)
co
Fig. 8. The behavior of the saddle points in the region near the origin for 0 > A. The dotted curves indicate the trajectories followed by the
saddle points as 0 varies.
points + and wa. To determine the behavior of X(cw, 0) in
the vicinity of the near saddle point in the right half-plane,
let
= WSPN+(O)
_CA() -
+ reix
2
i
WM~r8 + rei
3
With this substitution, expression (3.32) becomes
b(0, r, x) _ 2 6b(0) + i(0) + ireix](00- 0)
+ 20
+
[ 63
(0)] + 2a,2(0)[at(0)
62(0{1
4
- 1]
4 62(0)0(0)[2 - at(0)] + aci3(0)}
+ (4a6/(O)[ot(O) - 1] + i{3a02(9) +
+ 26[at(0) - 1] + 3ia0(0)}r 2 ei2 x
- 63(0)[2
-
+ iar3ei3x]
(0)I})reiX
(3.39)
and hence
X(, r, x)
[r sin(x)
-
2 M(0)](0 -
+ 20w 4 (9 63.2(0)[1
o)
+ 2if 2 (0)[Cz(0)
3
-
1]
+ 4(0)[a(O) - 1r cos(x)
_ {3C42(0) +
62t(0)[2 -
ce(0)]}r sin(x)
+ 25[c4(0) - 1hr 2 cos(2x)
3e(O)r2 sin(2x)
- cer
3
sin(3X))-
(3.40)
We can see then that X(0, r, x) attains its maximum variation for x = r/4, 3r/4, 57r/4, and 7r/4. In the right halfplane the lines of steepest descent through the near saddle
point wSPN+ are at x = r/4 and 57r/4, and the lines of steepest
ascent are at x = 37r/4 and 77r/4. In the left half-plane this
behavior is reversed because of the symmetry of X(w, 0)
about the imaginary axis. These two near saddle points are
illustrated in Fig. 8, in which the arrows indicate the direction of ascent along the lines of steepest descent and ascent
through each saddle point. The shaded areas about each
saddle point indicate the regions wherein X(w, 0) is less than
that at the saddle point.
C. Comparison with Numerical Results
The accuracy of these new approximate results for the saddle-point locations and the phase behavior at them is best
tested by comparison with the exact behavior of the complex
phase function O(w, 0) = X(w, 0) + iY(w, 0) at the exact
saddle-point locations. The exact locations of the saddle
points were obtained numerically from Eq. (3.2) by employing Newton's method. These purely numerical results were
then employed to calculate the real and imaginary parts of
the complex phase function p(w, ) = ico[n(w) - 0] at each of
the saddle points in the right half of the complex wplane as a
function of the parameter 0 1.
Consider first the real phase behavior at the saddle points
as a function of , given in Figs. 9 and 10. At the distant
saddle points, X(wspD,, 0) vanishes identically at 0 = 1 and
then decreases monotonically as 0 increases away from unity. The first approximation to X(wspD,, 0) is seen to diverge
rapidly away from the exact behavior over the entire range of
values of 0.
At the upper near saddle point. SP 1 (for 1 < 0 < O1), the
first and second approximations to X(wspN+, 0) lie close
together and are in fair agreement with the exact behavior,
the agreement becoming excellent for values of 0 roughly
greater than 1.3. However, for values of 0 up to OSB> 1.3, the
K. E. Oughstun and G. C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
829
Finally, for values of 0 >> 6 o, X(wsp, 6), at both the near and
distant saddle points, decreases steadily in a nearly linear
relationship to 0, with X(WSPN, 6) > X(oSpD", 6).
An approximate expression for the value of OSB at which
X(WSPD,, 6 SB) = X(COSPN+, OSB) may be obtained by using the
first approximate expressions
two distant saddle points are dominant over the upper near
saddle point SP 1 . From Fig. 9, the value of OSB at which
X(@, 6) at the upper near saddle point is equal to that at the
distant saddle points is SB = 1.334. Over the range of
values of 6 given by 1 < 6 < OSB, wherein the second approximation does not accurately describe the behavior of
X(coSpN+, ) at the upper near saddle point, the far saddle
points are dominant, and for these two distant saddle points
both the first and second approximations of X(wspD:, )
accurately describe the actual behavior. For 0 > OSB, during
which the near saddle point (or points for 0 > 01) remains
dominant over the distant saddle points, the second approximation of X(cospN+, 6) for 6SB < < 01 and X(ospN,, 6) for 6
> 01 accurately describes the exact behavior. For values of 6
in the region about 00 = 1.5, the first approximation is also
seen to describe accurately the actual behavior at the near
saddle points, as expected, but as 0 increases away from 00,
the accuracy of the first approximation steadily diminishes.
X(wsPD,
6)
-
2
X(Wp
(3.41)
25(0 - 1),
+ 6)
X
-
27
iLa
[
62
]2b2
26%2
4
00W
0
4-+6
La2
9 - (6o a
0
0004 (
ab
)
(o
_ 0)1/2
2
46b2
a26owo04
J
]
J
(3.42)
(x ob,1/sec)
LZ j1
1.0
0.8
0.4
0.2
e<
X@+p'e)
1.6.
0
-0.2
- 0.4
-0.8
- 1.0
Fig. 9. Behavior of the real part of the phase, X(w, 6) = Re[o(w, 6)], at the near and distant saddle points as a function of 6 for 1 < 0 < 2.2. The
solid curves represent the exact (numerically determined) behavior, the dashed curves represent the second approximation, and the dotted
curves represent the first approximation.
830
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
2
K. E. Oughstun and G. C. Sherman
4
6
e
8
10
12.
14
0
I
-2
-3
-4
X(c"-Pe)
1
-5
-6
-7
-8
'
(xI/sec)
Fig. 10. Behavioroftherealpartofthe phase, X(, 0) = Re[,(wO0)], atthe near and distant saddle points as a function of0for 2 <014. The
solid curves represent the exact (numerically determined) behavior, the dashed curves the second approximation, and the dotted curves the
first approximation.
with the result that (a being equal to unity)
4
2
2
b
-
{
[27a2 b (o0 - 1)2 1/3
27o(o 1)o ]
X{1+
f62b2
+
The frequency WSB is then the real coordinate value at which
the contour X(WSPD+, 0) at the distant saddle point in the
right half-plane crosses the positive real axis when 0 = OSB.
The solution of Eq. (3.44) for WSB is an extremely formidable,
+
12 - i})3 1
(3.43a)
2700(0 -1)wo42
N0
462b2
30 wX4
(3.44)
X(WSB) = X(WSPD y OSB)-
2
(3.43b)
When we use Brillouin's choice of the medium parameters,
expression (3.43a) yields the value OSB
1.295, which is in
good agreement with the actual value of 1.334.
A related quantity, of interest to the asymptotic evolution
of the field, is the value of the real frequency WSB > (W12 62)1/2 defined by
if not impossible, task. However, because the contour of
X(wSPD+, 0) through the distant saddle point at the angle 7/2
to the real axis remains at essentially this angle when it
intersects the real axis, as is evident in Figs. 4(b)-4(f), an
approximate expression for WSB is given by the real coordinate value of the distant saddle point wSPD+ at 0 = OsB.
Hence from Eq. (3.19) and expression (3.43b) we obtain
COSB t(
0
5B) = (O2-
(
62
b2
2
Wo
b2
+
OSB2
1/2
2
1
52
3WO2
/
1/2
(3.45a)
(3.45b)
K E. Oughstun and G. C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
where terms in 64 and higher have been neglected in the final
expression.
Consider finally the imaginary part of the complex phase
behavior at the saddle points in the right half of the complex
w plane as a function of the continuous parameter , as
illustrated in Fig. 11. Because Y(-cw' + i", ) = -Y(6)' +
iw", 6), just the negative of'this behavior is obtained in the
left half-plane. As can be seen from the figure, the first
approximate behavior at the distant saddle points diverges
rapidly away from the actual behavior as increases away
from unity, and the first approximate behavior at the near
saddle points diverges rapidly away from the actual behavior
as increases away from 1 [for 1 < ' 01, Y(wSPN-, 6) = 0].
However, the second approximations in both of these cases
provide a significant improvement over their respective first
approximations in depicting the actual behavior. The accuracy of this approximation is critical in obtaining an accurate
expression for the instantaneous frequency of oscillation for
the propagated field.
In conclusion, we have shown that the first approximation
fails to describe accurately the actual phase behavior at the
relevant saddle points over the range of values of 6 necessary
831
to describe the asymptotic behavior of optical pulse propagation in a temporally dispersive Lorentz medium. The
second approximation, however, accurately describes the actual behavior over the entire range of values of 0 needed.
4. ASYMPTOTIC DESCRIPTION OF THE
PROPAGATED FIELD
Equipped with the knowledge of the topography of the
phase function 0(w, 6) and the dynamics of its saddle points,
we can now perform the asymptotic analysis of the field A(z,
t) as given by either Eq. (2.16) or Eq. (2.18). This analysis
uses the modern asymptotic method of Olver, 28 which does
not require a knowledge of the path of steepest descent
through the saddle points.
A. Procedure for the Asymptotic Analysis of the Field
The first step in the asymptotic analysis of the propagated
field A(z, t) is to express the integral representation of A(z, t)
in terms of an integral I(z, ) with the same integrand but
with a new contour of integration P(O) to which the original
contour of integration may be deformed. In the present
E
z
'x
6
J
ID ,
3
4
\
s
e
¢ )
5
\
y
( )
e)
Fig. 11. Behavior of the imaginary part of the phase, Y(c, ) = Im[o(w, )], at the near and distant saddle points in the right half-plane as a
function of 0. The solid curves represent the exact (numerically determined) behavior, the dashed curves represent the second approximation,
and the dotted curves represent the first approximation.
832
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
application we found that any poles of the spectral function
(w - wc) that are crossed when the original contour is
deformed to P() are encircled in the process in the clockwise
sense [we assume here that the spectral function a(w - c) is
an analytic function of complex w, regular in the entire
complex w pane except at a countable number of isolated
points where it may exhibit poles]. Hence, according to the
residue theorem of Cauchy, 2 2 the integral representation
A(z, t) and the integral I(z, 0) are related by
A(z, t) = I(z, 0) - Re[27riA(0)],
(4.1)
where
A(0)
Res
u(w - wc)exp[Z O(,
0)11
(4.2)
is the sum of the residues of the poles that were crossed and
where I(z, 0) is defined by
I(z, 0) =
kR
ei
J
ii(wo- wc)exP[z k(W, 0)]dw}-
spect to the saddle points SPD' and SPN3, respectively, and
the subpath P 1 (0) is an Olver-type path with respect to the
saddle point SP 1. Provided that the path P(O) and its component subpaths satisfy the aforementioned constraints, it is
unimportant which particular paths are used; the asymptotic results are independent of the choice. Some particular
choices may be more convenient, however, in that they reduce the computation required.
The deformed contour employed by Brillouin 6 25 followed
along the entire paths of steepest descent through the distant saddle points and the entire steepest descent paths
through SP 1 for 1 < 0 < 01 and through both SPNe for 0 > 01
with the various paths connected along the branch cuts.
Although it is perfectly valid, that path is unnecessarily
complicated and is avoided in this analysis by applying the
method of Olver instead of the method of steepest descent.
The integral I(z, ) given by Eq. (4.3) may now be expressed as a sum of integrals with the same integrand over
the various subpaths, so that
(4.3)
Note that A(0) changes discontinuously with the parameter 0
as the path P(O) crosses over the poles of i(w - w,).
Because the distant saddle points are dominant for some
values of 0 and the near saddle points are dominant for other
values of 0, there is no path P(O) that is an Olver-type path2 8
with respect to a single saddle point and that evolves continuously with for all 0 > 1. The contour P(O) must evolve
continuously for all 0 2 1, and, in the vicinity of 0 = OSB
(when the saddle-point dominance changes), the path must
pass through both the dominant and nondominant saddle
points. Moreover, the path must be divisible into a sum of
subpaths, each of which is an Olver-type path with respect to
one of the saddle points.
For values of 0 in the range 1 < 0 < 01, during which the two
near saddle points are on the imaginary axis, the lower near
saddle point SP 2 is dominant over all the other saddle
points. That saddle point is not useful, however, because
the Olver-type paths with respect to it are not deformable to
the original contour (and vice versa) owing to the presence of
the branch points. For this reason the saddle point SP 2 is
not included in the subsequent discussion.
Many paths having the required properties pass through
both the upper near saddle points SP 1 and the two distant
saddle points SPD' for 1 < 0 < 01. There are also many
paths having the required properties that pass through both
of the near saddle points SPN' and the distant saddle points
SPD- for 0 > 0. Finally, many paths having the required
properties pass through the single higher-order near saddle
point SP1 = SP 2 and both of the distant saddle points SPD'
cWIl
I
e
S5
e05 <e < E,
for 0 = 01. As a result, the contour P(0) always can be chosen
so that it passes through the upper near saddle points SP 1
and the two distant saddle points SPD' for 1 < 0 < 01
(remembering that SP 1 = SP 2 at 0 = 01) and through all four
saddle points for 0 > 01 such that the path evolves in a
continuous fashion as 0 varies over 0 > 1 and can be divided
into the desired subpaths.
An example of such a path P(0) and its component subpaths is illustrated in Fig. 12. For 0 in the range 1 < 0 < 01,
the component subpaths arc PD-(O), PI(O), and PD+(0), and
for 0 > 01 they are PD-(O), PN-(O), PN+(O), and PD+(O). The
subpaths PD'(O) and PN'(0) are Olver-type paths with re-
p+
e, < e
Fig. 12. The deformed contour of integration P(O) through the
relevant saddle points of 0(w, 0). The dashed contours indicate the
isotinic contours through the saddle points, and the shaded areas
indicate the regions of the complex coplane wherein X(w, 0) is less
than that at the relevant saddle point.
K. E. Oughstun and G. C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
I(Z, 9) = ID-(Z, 9) + I(z, 0) + ID+(Z, ),
1 < ° 5 01,
(4.4a)
I(z, 9) = ID-(Z, 9) + IN-(Z, 9) + IN(Z, 9) + ID(Z, 9),
0 > 01,
(4.4b)
where IDA(Z, ) and Ie(Z, ) denote the contour integrals
taken over the paths PD(0) and P#(O), respectively, and
there I(z, ) denotes the contour integral taken over the
path P1 (9). To obtain an asymptotic approximation of the
integral representation of the field A(z, t) we still need to
obtain asymptotic approximations of the integrals appearing on the right-hand sides of Eqs. (4.4). Only the nonuniform asymptotic behavior of these integrals will be described
in the present paper; the uniform asymptotic description
will be presented in a subsequent paper.
Application of the method of Olver2 8 to the two distant
saddle points and adding the results yields an asymptotic
representation of the form
ID-(Z, ) + ID'(z, ) = A(z, t) + R(z, ),
(4.5)
which is valid for values of bounded away from unity from
above. (As 0 - 1+ the two distant saddle points move off to
infinity, and a uniform asymptotic procedure must be employed.) The component field A(z, t) is the contribution
due to the two distant saddle points alone and is referred to
as the first, or Sommerfeld, precursor field. The quantity
Ri(z, ) is an estimate of the remainder in the asymptotic
approximation as z .
Application of the method of Olver to the near saddle
points yields an asymptotic representation of the form
I(z, 9) = AB(Z, t) + R2 (Z, 9),
1 < O < 01,
Ij(z, 1) = AB(Z, t) + R2 (Z, 01),
IN (Z,9) + IN (Z, ) = AB(Z, t) + R2 (z, 9),
0 = 1,
0>
(4.6a)
(4.6b)
1. (4.6c)
Equation (4.6a) is valid for values of 9 bounded away from °1
from below, and Eq. (4.6c) is valid for values of bounded
away from 01 from above. (As the two near saddle
points approach each other and coalesce into a single saddle
point of second order at = 1; thus a uniform asymptotic
procedure must be employed about = 1 to obtain a continuous evolution of this contribution.) The component field
AB(Z, t) is the contribution due to the upper near saddle
point alone for 1 < < 1, the single near saddle point at =
°1, and the two near saddle points for > 1 and is referred to
as the second, or Brillouin, precursor field. The quantity
R2 (z, ) is an estimate of the remainder in the asymptotic
approximation as z
Combination of Eqs. (4.1)-(4.6) leads to the following general expression for the asymptotic approximation of the integral representation of the field A(z, t):
A(z, t) = A(z, t) + AB(Z t) + A(z, t) + R(z, 0).
(4.7)
The contribution A(z, t) is simply the residue contribution,
AC(z, t) = -Re[27riA(9)],
(4.8)
833
The dynamic behavior of A,(z, t) is determined by the
poles of the spectral function i(co - w) and the dynamics of
the saddle points that pass near these poles. The contributionA,(z, t) is nonzero only if 2(w - w) [or(w)] has poles. If
the envelope function u(t) of the field A(O, t) on the plane z =
0 is bounded for all time t, then z(Q- wc) can have poles only
if u(t) does not tend to zero too fast as t
-
x. Hence the
implication of nonzero A(z, t) is that the field A(z, t) oscillates with angular frequency wc for positive times t at the
plane z = 0 and will tend to do the same at larger values of z
for large enough t. As a result, the contribution Ac(z t)
describes the dynamic behavior of the main signal oscillating
with angular frequency wc, This contribution is negligible
during most of the evolution of the precursor fields.
For most values of only one of the terms appearing in Eq.
(4.7)-A(z, t), AB(Z, t), or A(z, t)-is important at a time.
There are short intervals of 0, however, during which two or
more of these terms are significant for fixed values of z.
These intervals mark the transition periods when the field is
changing its character from one form to another. When the
uniform asymptotic descriptions of As, AB, and AC are employed, the presence of both terms in Eq. (4.7) leads to a
continuous transition in the behavior of the field, and Eq.
(4.7) displays the entire evolution of the field through its
various forms in a completely continuous manner.
B. Field Behavior for < 1.
If the initial envelope u(t) of the field at the plane z = 0 is
zero for times t < 0, then the field A(z, t) can be nonzero for
values of = ct/z < 1 only if the field propagates with a
velocity greater than the vacuum speed of light c, which is in
violation of the relativistic principle of causality. Sommerfeld24 proved that for the step-function-modulated signal,
the propagated signal is identically zero for all < 1. For
completeness, his argument is extended here to show that
the same result holds for all fields with initial envelopes u(t),
at the plane z = 0, that are zero for all t < 0.
For fields with an envelope function u(t) that is zero for t
< 0, the spectral amplitude function appearing in the integrand of Eq. (2.18) is given by
t2(w) =
J
u(t)etwtdt.
(4.9)
If u(t) is bounded for all t, it then follows by direct differentiation of Eq. (4.9) that u(co) is an analytic function of complex
cofor wo > 0. In addition, if du/dt is bounded for all t, then
integration by parts of the integral in Eq. (4.9) shows that
a(w) tends to zero uniformly with respect to 4 = arg(Q) for 0
<'<7ras I-QIo,where Q='-iawitha>O. Because(o,
is real, the same results apply to (c - wc).
With the knowledge that the spectral amplitude function
u() satisfies the above conditions, it is now possible to
express A(z, t) by Eq. (2.18) with the change that the integration is taken over the closed contour that encircles the
region cw > a > 0 of the complex
plane. All that is
required is to show that I(z, , c) - 0 uniformly with
respect to z and 0 for z > Z and0 < 1 - A as IQI
for
arbitrary Z > 0 and A such that 0 < A < 1. Here
-
obtained when the original contour of integration is deformed to P(9). An estimate of the remainder term R(z, 9) as
z -is obtained by taking the largest estimate of the
remainder terms appearing in Eqs. (4.5) and (4.6).
I(Z ,IQI) =
t( - wc)exp[- 0(w, 0)]dw,
(4.10)
834
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
with C being the contour 0 § fi S xr for fixed Ill. The proof
makes use of the method of proof of Jordan's lemma. It
follows from Eq. (4.10) that
I(Z, O,kl)
li(wco
a
I
d)lexp[ X( ,
0)]ldw
(4.11)
where, from Eq. (All),
X(w, 6) = -o[nr(w)
-
6] - o'nj(w).
(4.12)
It follows from Eqs. (A9) and (A10) that a positive constant
Qo exists such that for QI> Qos w'ni(w) > O. Hence for ilgl>
Qo and 0 <
St7r,
6)
X(w,
<-w"[nfr(w)-
6].
(4.13)
In addition, it follows from Eqs. (A9) and (A1o) that there is
a positive constant such that for IQI larger than that constant, nr(w) 2 1. Henceforth go is taken to be larger than
that constant. It then follows that
X(w, 6) S -w"(l - 6)
(4.14)
for IQI > go with 0 S 4A 7r [this final inequality also follows
directly from expression (3.21)]. Consequently, for 6 S 1 A with 0 < A < 1, the combination of inequalities (4.11) and
(4.14) yields
I(Z, 6, IQI) S
J
|u(w - wc)lexp(-
W"A)ldw
(4.15)
for lQI> Qo. From here the proof follows exactly the proof of
Jordan's lemma as given in §6.222 of Whittaker and Watson. 29 Hence I(Z, 0, IQ) - 0 uniformly with respect to both
z and 0 for z 2 Z and 0 S 1 - A as IQI - - for arbitrary Z > 0
and A such that 0 < A < 1. Consequently, that contour
integral can be added to the integral in Eq. (2.18) to express
A(z, t) as an integral over a closed contour that encircles the
region w" > a of the complex wplane.
Because the integrand of the closed contour integral is a
regular analytic function of complex w for w" > a, with a
positive, it then follows from the residue theorem of
Cauchy22 '29 that the integral is identically zero for z > Z and
6 < 1 - A for arbitrary Z > 0 and arbitrarily small A > 0.
Hence A(z,. t) = 0 for z > 0 and 0 < 1 when the initial
envelope function u(t) is zero for t < 0. The same result
holds for the integral representation of Eq. (2.16) when f(t) is
zero for t < 0.
The above approach works by virtue of the fact that for 6 <
1 the real phase function X(w, 6) = Re[k(w, 6)] tends to
negative infinity in the limit as Iwl approaches infinity in the
upper half of the complex w plane. For values of 6 > 1,
however, X(w, 6) does not tend to negative infinity as Iwl - o
in the upper half-plane. From expression (3.21), we obtain
lim [X(w, 6)] = w"(O - 1).
1 were not successful. Beevaluating the integral for 0
cause all the features of pulse propagation of interest occur
in the regime 0 2 1, analytic approximations for A(z, t) are
required.
C. First Precursor Field (Sommerfeld's Precursor)
The contributions of the two distant saddle points to the
asymptotic behavior of the field A(z, t) for sufficiently large
values of the observation distance z yield the dynamic evolution of the first precusor field. This contribution to the total
asymptotic behavior of the field A(z, t) is denoted by A,(z t)
[see Eqs. (4.5) and (4.7)] and is dominant over the second
precursor field for all values of 0 in the range 1 < 0 < OSBBecause the two distant saddle points SPD' remain isolated
from each other and do not change their order, a straightforward asymptotic analysis based on the method of Olver is
applicable to obtain the first precursor evolution for 6,> 1.
However, as the parameter 6 approaches unity from above,
the two distant saddle points approach infinity, and Olver's
method fails; thus it is valid only for values of 0 bounded
away from unity.
For the example considered in this analysis, both the spectral functions Cz(w - wc) and 7(w) and the phase function o(w,
6) appearing in the integrand of A(z, t) are analytic about the
two distant saddle points. Olver's theorem2 8 then applies
for each of the two distant saddle points for 6 > 1 withy = 2
(since SPD1 represents first-order saddle points) and with X
= 1 (since either of the spectral functions C1or 7 is regular at
these saddle points). Hence
A,(z, t)
- Re(i{2 exp[c
=
6)] (
4(WSPD",
X [1 + 0(z-1)] + 2 exp(-
0GOSPD-, °)
IC
6)Ipj 2E
X ao(wsp,-)[l + O(Z-1)]3)
)
/
(4.17)
as z - - uniformly for 6 > 1 + A, where A > 0.
To evaluate the coefficients ao(wspDl) appearing in Eq.
(4.17), the first two coefficients in the Taylor series expansions of 0 (w, 6) about the distant saddle points are required.
From expression (3.21), the approximate phase behavior in
the region of the complex w plane far removed from the
origin yields, with expression (3.18),
4(WSPD, '6)
[1 + 7(0)](0
-
-
1)
V~~~~~~~~
2 42(0) + 62[1 - 7(0)]2
(4.16)
t
i~(6){6
1 +2(6)
-
1
+ 62[l_ (0)]2
11-a
Hence for 0 = 1, X(w, 1) is equal to zero everywhere at Iwl =
-, and for 0 > 1, X(w, 6) is equal to +- in the upper half of
the complex w plane, zero at the real w' axis, and -- in the
lower half-plane. Consequently the method used for 0 < 1
cannot be applied to evaluate the integral for A(z, t) for 0 2:
1. Moreover, attempts to find other methods for exactly
)o(wsp,+)
(4.18a)
)(WSPD4
0)
- i {+l(O) + bi[l -
()]3
(4.18b)
so that the second coefficient in the Taylor series expansion
of 0(w, )is
K. E. Oughstun and G. C. Sherman
Po(wsPD±, 0)
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
012 1(wsp
=
2!
,
W = d/z
- {t(O)0-1+ +
dt c
t
t2(0)
O)
835
b2/
2
b /2
+ 62[1
-
7(0)12J)
2
b /2
_ -i
= V(O) +
(4.19)
1+(0) + i[1 - q(o)]13
The proper value of o = arg[-Po(wspD,, 0)], as specified by
the convention set forth in Glver's analysis, 28 is - _ :7r/4.
The coefficients ao(cvspD,) appearing in expression (4.17) are
then given by
X
b20
2(O)(02 - 1)2
2 2(0) - 562[1
n(0)] 2
-
_
2(0
2
-
1))
\{ (0) + 62[1 - (o)]212
(4.22a)
=_VO)ao(wsp-)
Z( SPD
C)
[-Po(WsSP", a)O]
-
Z(WSPD
1(2 {:iW(C) +
2
-
- rZ(WSPD -
C)-
c~fb
X {4/2(0) +1
i[1
i[-n(0)1&3
exp(=Fi7r/4)
-
n(0)/I2(0)},
(4.20)
where we have used the inequalities 0 < 1 - tn(0) < 1 and (0)
>> 6for 1 < 0 < to obtain the final approximation. Substitution of expressions (4.18a) and (4.20) into expansion (4.17)
for the first precursor field then yields
j(0, t)
b
X
[
1-
Cf-,
[ +
c~
e
ep
27-Z
(0)
+
The behavior of the instantaneous angular frequency of oscillation of the first precursor field as a function of 0 as given
by Eq. (4.22a) is illustrated in Fig. 13 along with the corresponding behavior of (O) _ RewSpD+(0)] for Brillouin's
choice of the medium parameters. We can see that, for
values of 0 very close to unity, cw,is given by (0) to a high
degree of accuracy. Furthermore, for large values of 0, the
second term on the right-hand side of Eq. (4.22a) approaches
zero and w, approaches the value (-) = (i2 - 62)1/2 asymptotically from above. However, approximation (4.22b) becomes less accurate for intermediate values of 0, as seen in
Fig. 13. For comparison, the behavior of the instantaneous
angular frequency that is obtained from the first approximation [expression (3.4)] for the distant saddle-point locations
is illustrated by the dotted curve in Fig. 13. The accuracy of
2
30 -
1}
28
42(0) + 62[l _ n(ow21
X
i a(wSPD
-
X exp[-if t(0){0
WC){ (O) + 3 bi[
-
(4.22b)
(0)]}
-
24 -
1 + 2( + 62[1 + ?(0)]2} + 4)
zz-
a)
20 -
U
+ U(SP
2 641l - t1(0)]}
4~2() + 62[1 +
wC){t(0)
L \C t
-
a) ,
()2
4J
L
I O'
-a
X exp i
4()0
- 1+
b2(0
(012, + 7
14 -
(4.21)
a)
CD
<to4
(0E
as z
uniformly for 0
1 + , where
> . This
expression constitutes the second approximate expression of
the nonuniform asymptotic behavior of the Sommerfeld precursor field. Expression (4.21) also holds if one begins with
the integral representation of Eq. (2.16), but with (WspD WC) replaced with (wSPD).
It is convenient to define the instantaneous angular frequency of oscillation of the precursor field as the time derivative of the oscillatory phase. 6 Because dO/dt = c/z and
d4(0)
dt
dn(O)
dt
2
cb 0
Z~(0)(0 2 - 1)2
2b 2c[7(0) - 1]0
zt 2 (0)(0 2 - 1)2
then the instantaneous angular frequency of oscillation of
the first precursor field is given by
t-4
8-
lsi. apprUinmaiito
4-
+0 o5
20
1.0
l.o
1.1
_.112.I
1
1.3
1.4.
e
Fig. 13. The instantaneous angular frequency of oscillation W of
the first precursor field and the real coordinate position (O)
Re(wspD+) of the distance saddle-point location in the right half of
the complex plane as a function of 0 for o = 4.0 X 11 6/sec, 6 = 0.28
X 1016 /sec, and b2 = 20 X 10 32 /sec2 . The solid curve represents the
behavior obtained in the second approximation, and the dotted
curve is the behavior obtained in the first approximation owing to
Brillouin.
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
836
4( 2 )("Sp"
this approximation is seen to decrease monotonically as 0
increases and will yield a 10% error in the value of the
frequency of oscillation of the first precursor field at 0 = 1.2,
which is well within the space-time domain in which the first
precursor field is dominant. Finally, we must keep in mind
that the instantaneous angular frequency is only a heuristic
mathematical concept, which, at times, may yield completely erroneous results. 30 For the present situation, however,
the instantaneous frequency monotonically decreases with
increasing 0; thus it does yield proper results (i.e., results
that describe correctly the dynamical evolution of the field
in question).
D. Second Precursor Field (Brillouin's Precursor)
The contributions of the near saddle points to the asymptotic behavior of the field A(z, t) for sufficiently large values of
the observation distance z yield the dynamic evolution of the
second precursor field. This contribution to the total asymptotic behavior of the field A(z, t) is denoted by AB(Z, t)
[see Eqs. (4.6) and (4.7)] and is dominant over the first
precursor field A,(z, t) for 0 > OSB- Since the two near firstorder saddle points, which are isolated initially from each
other at 0 = 1, approach each other along the imaginary axis
as 0 approaches 01 and coalesce into a single second-order
saddle point at 0 = 01, after which they separate into two
first-order saddle points again and move away from each
other in the complex w plane, a straightforward asymptotic
analysis based on the method of Olver yields the second
precursor evolution for values of 0 in three separate domains
1 • 0 < 01, 0 = 01, and 0 > 01. These results are nonuniform
in a neighborhood about the critical value 0 = 01.
For values of 0 within the range 1 S 0 < 01, the conditions
of Olver's theorem 28 are satisfied at the upper near saddle
point SP1 with A = 2 (since SP1 is a first-order saddle point)
and X = 1 (since either of the spectral functions C or f is
regular at this saddle point). Hence
AB(Z, t) = I rR,,(12 exP c
0(wsi 01)](7ZC)11
(4.23)
X ao(Wsp1)[1 + o(Z-1)]})s
as z
uniformly for 1 S 0 • Oi - A with 0 < A < 01 - 1.
-
To evaluate the coefficient ao(ospl) appearing in Eq.
(4.23), the first two coefficients in the Taylor series expansion of 0(w, 0) about the upper near saddle point SP, are
required. From Eq. (3.32) the approximate phase behavior
in the region of the complex w plane about the origin yields,
with expression (3.33),
,p(wsp,, 0)
1
- [26r(0) - 31'(0)1](0o - 0) +
b2_
X [263(0) - 31Y(0)] 2126[3 - ar(0)] + 3actl(0)1,
(4.24a)
2
k(
)(wSP
1
,
0)
_
126[1
-
a(0)] + 3a1l(0)1,
(4.24b)
so that the second coefficient in the Taylor series expansion
of O(W, 0) is
0)
2!
P0 (c,, 5 p1 , 0) =
-
126[1 - a(O)] + 3a1l(0)1.
20o
(4.25)
The proper value of = arg[-Po(sp,, 0)] as specified by the
convention in Olver's analysis 2 8 is ao = 0. The coefficient
ao(wsp,) appearing in Eq. (4.23) is then given by
dWSP,
aowp
WI
,-P 0o(wsp, 0)]XL/
O SP1
001/2
W2
U~a'Sp
1
46[1 - ac(0)] + 6a4(0)I
-XC) b
(4.26)
where the positive value of the square root in this expression
is to be taken. Substitution of expressions (4.24a) and
(4.26) into the expansion of Eq. (4.23) for the second precursor field then yields
O2/
AB(Z, t
0
b
7rz(45[1
X Re{[i(wSP
X
1/2
00
00c
ar(0)] + 6aji(0)1I
-
0 - 0 + 18
wc)]exp
1800W04
[25v(0)
[26(0)
-
-uniformly for 1 S 0 S 01-
3t(0)1]
31(0)1]
-
X 26[3 - a(0)] + 3acd(0)1)]}
asz
-
A with
(4.27)
<A < 01- 1.
Since the quantity [26r(0) - 314(0)1] is negative for 1 S 0 <
00, vanishes at 0 = 00, and is positive for 00 < 0 < Oi, and since
the quantity 126[3 - ao(O)] + 3a1l(0)i} is positive for 1 S 0 S
01, and since the inequality
100
-
01
8
1285(0) - 31A(0)1125[3
-
ac(0)] + 3a14(0)11
is satisfied for all 0 in the range 1 S 0 <Oi, with the equality
holding only at 0 = 00 when both sides vanish, then the
exponential argument in expression (4.27) is negative for 1 <
0 < 00, vanishes at 0 = 0o, and is again negative for 0o < 0 < 01.
Consequently, the second precursor field AB(Z, t) first grows
with 0 as the exponential argument decreases for increasing
values of 0 in the range 1 S 0 < 00 and becomes exponentially
dominant over the first precursor field for 0 > OSB (where 1 <
OSB < 00). At 0 = Oo the exponential argument vanishes, and
the field [expression (4.27)] varies only as z-1 12 , making this
particular case strikingly different from the field observed at
any other value of 0. Finally, for increasing values of 0 in the
range 00 < 0 < 01 the exponential argument decreases, and
the field is again exponentially attenuated. In this range of
values of 0, the second precursor field is nonoscillatory.
At 0 = 01 the conditions of Olver's theorem are satisfied at
the second-order near saddle point SPN with ,u= 3 and X= 1.
From expression (3.32) the approximate phase behavior in
the region about the origin yields, with expression (3.36),
K. E. Oughstun and G. C. Sherman
(WSPN
01) -
P(w,
(3
0)
(0
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
(4.28a)
01 + 940)'
-
3i ab2
AB(Z, t) = 2
Re(i{2exP[
(WSPN+, 0)](e) ao(sp +)
X [1 + O(z-1)] + 2exp[
(4.28b)
837
)
P(WSPj-, 0) I
ao4
where 003 ) is independent of both 0 and the saddle-point
location (in this approximation). The second coefficient in
the Taylor, series expansion of O(w, 0) is then given by
3!
PO(wSPN, 01) =
_ iS
(4.29)
The proper value of oao = arg[-Po(cspN, 01)] _ arg(-i) as
specified by the convention of Over's analysis is oao+ = -7r/2
along the deformed contour leading away from SPN in the
right half-plane and iio- = -57r/2 along the deformed contour leading away from SPN in the left half of the complex
plane. The phase difference between the coefficients
ao+(wspN) and ao-(WspN) then yields the factor
exp(-i&0 +/3) - exp(-ii 0 -/3) =
X ao(wsp
0(wspN , 0)
(O)(0
ao+(WSP)
=
9
Thus asymptotically as z
-
w)Lab2
J
1r/
e~'.(3) (4.30)
second precursor field value is given by
AB(Z t)
1
(4[
-R,(ifxpL
o(wsp,,
2ir e
C
-
,](
3
r
X exp
L3 ac
01
~j
4a6b
a(O)] + ac2(0)})
-
(4.33a)
]
ih 01
(4.33b)
and the second coefficient in the Taylor series expansion of
I (-,
0)
is
PO(WSPNL'
,0(2)
(.SPN
-
ao = N a(ws SP
4b
4)]
0)
2!
3icaV(0)j. (4.34)
c)
-
[-PO((SPN4,
4
9a00 W,
0
(4.31)
<
4
Ow2
125[a~(O) - 1]
The coefficients a(wspNA) appearing in Eq. (4.32) are then
given by
-C)]
4
9c0 0w0
0
s~ {3
the convention set forth in Olver's analysis is co = Fr/4.
From expression (3.31), the bracketed quantity in the exponential argument appearing in expression (4.31) satisfies the
inequality
0-01 +
(O)J2(0)
The proper value of cio
0 = arg[-PO(wspN+, 0)] as specified by
(00 - 01 +
X Re[i2(.SpN
13
20ogi~o f26[c(O) - 1]
A200owoc /3
3/L ab2z
27rW-
-
t(O) -1]}
_)
[,+.
o(wsPN)IQ) 1
W°O
, 0) -
~~
0(2)(iSPN±
with fixed 0 = 01 = ctllz, the
-
X4 {[1
,
01]
) F20 0w0 4
3
0o) +
W)
AlP0(wSPN
1 u
-
+ 4 6Yv(0) [1
X [2
-
(4.32)
as z - - uniformly for 0
0 + A for positive A that is
arbitrarily small.
To evaluate the coefficients ao(wspN-) appearing in Eq.
(4.32) the first two coefficients in the Taylor series expansions of O(w, 0) about the near saddle points are required.
From expressions (3.32) and (3.38), one obtains
where
U (SN
)[1 + O(z1)]}),
__262b2
9a00oco 4
and the second precursor field is exponentially attenuated at
0 = 01.
For values of 0 within the range 0 > 01, the conditions of
Olver's theorem are satisfied at the two near saddle points
SPAA with , = 2 (since each is a first-order saddle point) and
X = 1 (since either of the spectral functions or f is regular at
these points). Hence
_ U(SPN+ -
)]X/P
Cb
{
-200
1/2
1] + 3iav(0)f
-
(4.35)
For values of 0 slightly larger than 01, the term 26[ao(O) - 1]
becomes negligible in comparison with 3q/(0) and remains
so for all larger values of 0. Hence for values of 0 sufficiently
larger than 01, expression (4.35) may be simplified somewhat
to
ao(cospN) =_ cspN'
°
2 0 i()
2bWC)
)]
ee
/1/2
(4.36)
When 0 approaches 01 from above, this analysis must be
replaced by a uniform asymptotic procedure as the two saddle points coalesce, so that approximation (4.36) is valid for
the nonuniform asymptotic analysis considered here. Sub-
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
838
The behavior of the instantaneous angular frequency of oscillation of the second precursor field as a function of 0 as
given by expression (4.39a) is illustrated in Fig. 14 along with
Re(wspN+) for Brilthe corresponding behavior of ,(O)
louin's choice of the medium parameters. As 0 increases to
infinity, the instantaneous angular frequency WB approaches
stitution of expressions (4.33a) and (4.36) into Eq. (4.32) for
the second precursor field then yields the result
uo22
W
0
A ()
AB(Z, t)--
200c 1/21/
Fz/2
0)
(0)(0o)
X exp,[-5 Z
___(
+
asymptotically the value P(X) = (wo2
4
x {[1 - a(0)]02 (0) + 4 622(0)[I c4 (0)
X Reiju(wsPN
{X
+
P2(0)[2
-
wc)exp [i1(0) c
-
732
4
'() c
ac(0)]
+
°
(o~
0 wo
20
0+
al2(0)})- i
tude function a(w
-
I =im[W-p
- for 0 > 01. As can be seen, the second precursor
-
i&()I
-
-
c
-
-
62
200w 4 134
8 6
)(02
.(0) [2
-
0o2+
b20
-( ( - 0)
) _.
3b2)
2
3b2
3
(02-02+
'p
1
(4.42)
I
is the sum of the residues of the poles that are crossed when
the original contour of integration is deformed to the path
P(O) that is specified in Subsection 4.A. For simplicity, we
assume here that the deformed contour of integration P(0) is
near only one pole at a time, and attention is restricted to
obtaining the nonuniform asymptotic contribution that is
due to that single pole contribution alone. The results ob-
+
+
002 +
(02
302 -o02 +
~(O)]
C
2YZYE P exP[4(WP 0)]
(4.38)
~ ~
OA
c)exP[ - 0(&, 0)]}
p1 ,=P11
2
(9 +
+
Res {27 ia(w -
A(0) =
for 1 < 0 < 01. For 0 > 01 one obtains19
dt
(4.41)
where
The instantaneous angular frequency of oscillation of the
second precursor field is defined as minus the time derivative of the oscillatory phase. Because AB(Z, t) is not oscillatory for values of 0 prior and equal to 01, then
WB
(4.40)
p]
aW-
A,(z, t) = -Re[2iA(0)],
with f(WSPN) throughout.
0
~
From Eqs. (4.7) and (4.8), the nonuniform asymptotic approximation of A,(z, t) is simply the residue contribution
field for 0 > 01 is oscillatory and exponentially attenuated
with the propagation distance z. The attenuation coefficient increases monotonically with increasing 0 > 01; thus the
second precursor field is increasingly attenuated as 0 increases with fixed z.
Expressions (4.27), (4.31), and (4.37) constitute the nonuniform asymptotic description of the second precursor
field, valid in the intervals 1 < 0 < 01, 0 = 01 and 0 > 01,
respectively, but not continuous in a neighborhood about 0 =
01. These expressions also hold if one begins with the integral representation of Eq. (2.16) but replaces ii2(WSPN - Co')
WB =
from below.
wc), with residue -yp given by
(4.37)
as z
62)1/2
E. Spectral Pole Contributions
Consider finally the nonuniform asymptotic approximation
of the field A,(z, t). This field component is due to the
contribution of any pole singularities appearing in the integrand of the propagated field [Eq. (2.18)]. For that purpose,
let Up denote a simple pole singularity of the spectral ampli-
at(0)] + a1f2()}) + i 4
U(WSPN- - XC~)eXt
X 14 62r() [2 -
1
+ 2O+
(
-
For comparison, the behavior of the instantaneous angular
frequency that is obtained from the first approximation [expression (3.6)] for the near saddle-point locations is illustrated by the dotted curve in Fig. 14. The accuracy of this
approximation is seen to decrease monotonically as 0 increases and will yield a 10% error in the value of the frequency of oscillation of the second precursor field at 0 = 2.4.
2b2)
Wo2J
(4.39a)
(2 2+3b 2 3
02o2Ž
(4.39b)
K. E. Oughstun and G. C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
4
I
839
a.
C)
C
LL
3
(
Xa .0C
cJ
..
:
5?
11
..
- -
2
o
ro
(r)
c
I
0 '
,
2.0
el
2.5
3.0
Fig. i4. The instantaneous angular frequency of oscillation WBof the second precursor field and the real coordinate position A(0) Re(osp,+)
of the near saddle-point location in the right half of the complex. wplane as a function of > 1for wo = 4.0 X 1016 /sec, 6 = 0.28 X 1016 sec, and b2
= 20 X 103 2/sec 2. The solid curve represents the behavior obtained in the second approximation, and the dotted curve is that obtained in the
first approximation due to Brillouin.
tained may be generalized to account for several individual
pole contributions.
Assume then that only one single pole occurs at = p,
and let P(8) and the original contour of integration lie on the
same side of the pole for < and on opposite sides for 0 >
0. Then
A(O) =
S
.A(0)
0,
8 < 8O
47 -y
?P[C 0(P' OS)]
y exP[I
=
0(w'
=
8)]'
8 > os.
-Y((P'
= O.
<
,
(4.44a)
AC(Z, t) = /2 exp[ X(COP, Os)] {Yp
Cos[Z Y(WP O)]
P sin[C
(
8s)]}
A(z, t) = exp[ X(CoP, )] {YP' Cos[
- y;'
sin[- Y(COP,
IP
C
( P
0 = "
(4.44b)
'
L~~
~
(4.46)
k(p) =opnrp).
(4.47)
With these results the expression for the pole contribution
given in Eqs. (4.44) may now be written as
A,(z, t) = 0,
0 <Os,
y" sin[k(cop)z - cpt]),
A,(z, t) = exp[-za(wp)]j-y' cos[k(wp)z
- y" sin[k(op)z -
(4.44c)
)]
where yp' = Re(-yp) and yp" = Im(,yp).
A special case of interest is that for wl hich the simple pole
singularity of the spectral amplitude fu nction (-C y)
iS
real and positive. In that case Eqs. (Al:
X(cop) = -copnj(wp) = -ca(u,P)'
I
= cP[nr(Wp) -
~=~ c
(4.48a)
Aj(z, t ) = 1/2exp[-za(wp)] {y' cos[k(cop)z
7(cp) °)]
8 > O°,
8)
where the propagation constant k(wp) at the real frequency
cop is given by
-
c
= k(wp)z - Wpt,
(4.43b)
(4.43c)
(4.45b)
where a(cop) is the coefficient of absorption of the medium at
the real frequency wp [see Eq. (A8)]. Furthermore, the oscillatory phase term appearing in Eqs. (4.44) may be written as
(4.43a)
On substitution of Eqs. (4.43) into Eq. (4.41), the following
nonuniform expression for the pole contribution at co = op is
obtained:
AC(Z t
Y(wp, 8) = wp[n,(wp) - 8],
(4.45a)
-
0= s
-
pt]
(4.48b)
Cpt]
08pt]}
> Os,
(4.48c)
for wp real and positive. Hence in this special case the pole
contribution has an angular frequency of wp and is attenuated with propagation distance z with a constant attenuation
coefficient a(wp) that is independent of the time t.
The integral representation in Eq. (2.18) of the propagated field A(z, t) is a continuous function of 8 and, in particular, is continuous at 8 = S. However, the resulting asymptotic approximation of A(z, t) is a discontinuous function of
at = when the pole contribution is given by the nonuni-
840
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
form expression [Eqs. (4.44)]. Furthermore, the value of Li
depends on which Olver-type path is chosen for P(L). [If
that path is taken to lie along the path of steepest descent
through the saddle point nearest the pole, then the value of
0i is given by the expression Y(wsp, Li) = Y(wp, 0i).] However, the resultant discontinuity is of no consequence for fixed
values of z larger than some positive constant Z because the
contribution to the field from the dominant saddle point at
wsp varies exponentially as exp[z/c X(wsp, i)], which dominates the exponential behavior of the pole contribution given by Eq. (4.44b) at L = Oi[the dominance of the saddle-point
contribution over the pole contribution at L = 0i is guaranteed by the fact that P(O) is an Olver-type path]. Hence at 0
=e,, when the discontinuity in the asymptotic behavior of
the propagated field occurs, the pole contribution is negligible in comparison with the saddle-point contribution, and
consequently the discontinuous behavior itself is negligible.
For that reason, the particular value of As at which the pole
crossing occurs is of little or no consequence to the asymptotic behavior of the propagated field for sufficiently large z.
5. ASYMPTOTIC BEHAVIOR OF THE INPUT
DELTA-FUNCTION PULSE
-
A(z, t) = Aj(z, t) + AB(Z, t),
X 0o
)] + 3al(0)11)
°+ 180io& 4 [265(O)
31(0)1]
-
(5.4a)
for 1 < 0 < i with 0 bounded away from °1;
-
2r
AB(Z, t1 )
( 2bz
[3ac
as z
-
)/
(
9aoo0w0 4
(
-
at L = 01 = ctj/z; and
AB(Z, t)
(5.2)
11/2 exp(-5£{[1
[
L.c j/
0
B
b 3£rao(O)z L
X
exp[-6 -
X {[1
With Eq. (5.1) the nonuniform asymptotic behavior [ex-
-
(
X t4
L) +
4
a(O)]4' 2 ( 0) + 4 52.2(L)[I a'(L)
(0
X os[£ (Lo)
+ (0)] (L - 1)
-
2.()[2 -
-
0 +Le 1
at(O) +
a2(L)}) +-]
(5.4c)
1-J()
2 42 (0) + 62[1 -
n(L)] 2 J
- 1
X [2(L)cos(z (Li){L
- for 0 > L1 with 0 bounded away from L1.
The individual precursor and total propagated field behavior as a function of 0is illustrated in Fig. 15 for Brillouin's
choice of the medium parameters at a propagation distance
as z -
of z
2
0
X 2 43 - at(0)] + 3a&(Li)I)]
pression (4.21)] of the first precursor field becomes
+b2
-
X exp[ - [26(O) - 31i(0)]
where AB(Z, t) is negligible when the first precursor is predominant, A,(z, t) is negligible when the second precursor is
predominant, and both A8 (z, t) and AB(Z, t) are important at
the same time during the transition from the first to the
second precursor field for values of 0 about 0 = OSB-
[c(L)
2rz26[1
(5.1)
a) = -i.
Because there is no pole contribution in this case, the asymptotic behavior of the propagated field is given by
A)'z,
)
b
AB(Z, t)
as z
The integral representation of the propagated disturbance
that is due to an input delta-function pulse at the time t = to
is given by Eq. (2.21). We need consider only an input deltafunction pulse at to = 0 since other such pulses at arbitrary
times to can be constructed by a linear shift in the time
coordinate. The spectral amplitude function is then given
by
z(c
- for 0 > 1 + A with A > 0. The relative amplitude of
the sine term is negligible in comparison with the amplitude
of the cosine term for all values of 0 > 1 so that the field
oscillates nearly as a pure cosine with instantaneous angular
frequency w given by expression (4.22a).
From expressions (4.27), (4.31), and (4.37), the nonuniform asymptotic behavior of the second precursor field is
given by, with Eq. (5.1),
as z -
I()]2
[
+4
+ 4 (0) + 62 1
+ 3[1-q(0)]sin(-z(){
+ 2(0) + 62[1 -
0-1
1
(L)]2
+ 4(53)
=
1.0 X 10-4 cm.
(Although this propagation distance
seems small for asymptotic considerations, it is indeed large
when compared with the average absorption or penetration
depth c/6 = 1.07 X 10-5 cm of the medium.) The field
identically vanishes for all L < 1. At 0 = 1 the first precursor
field arrives and evolves up until 0 = OSB, at which point a
transition to the second precursor field occurs. The second
precursor then evolves uninterrupted for all 0 > OSB. At 0 =
Lo a maximum in amplitude is reached for the Brillouin
precursor, at which point there is no exponential decay in
K. E. Oughstun and G.C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
the field, and as 0 increases away from Oo the amplitude
steadily decreases because of the monotonically increasing
exponential decay with 0. During the evolution of the first
precursor field the angular frequency of oscillation decreases
monotonically with increasing 0,while during the evolution
of the second precursor field the angular frequency of oscillation increases monotonically with increasing > 01.
As pointed out earlier in Subsection 4.C, expression (6.3)
is not a valid approximation of the first precursor field in
the
limit as 0 - 1+ for fixed values of the observation distance
z.
We can show, in the limit as 0 - 1+, that expression
(6.3)
simplifies to Brillouin's result (Ref. 6, p. 73). Brillouin
also
noted its invalidity for 0 close to unity. For values
of 0
approximately equal to but greater than unity, expression
(6.3) becomes
6. ASYMPTOTIC BEHAVIOR OF THE INPUT
UNIT STEP-FUNCTION-MODULATED SIGNAL
A,(z
(b,)12
(27rz)
For a unit step-function-modulated signal, which begins to
oscillate harmonically in time at t = 0 at the plane z = 0 and
continues indefinitely with unit amplitude and angular frequency c, the spectral amplitude function is
[2(0-l
For this canonical problem there is then a simple pole singularity located at = Calong the positive real frequency axis.
The asymptotic behavior of the propagated field is then
given by
A(z, t) = AS(z, t) + AB(Z, t) + A,(z, t),
1
[2(0 - 1)]3/4
LK
(6.1)
- co-
841
exp[-26
}2
-
(
-
j
C
++4X2
[2(0
- )1/2+1+462)
(6.2)
X cos{ c [2(0 -)1/2 +
_
where As(z, t) and AB(z, t) are the precursor fields and
A,(z,
t) is given by the pole contribution.
~
~~~
[2(0-
1)]1/2
WC
4
+
A. Precursor Fields
With Eq. (6.1), the nonuniform asymptotic behavior
[expression (4.21)] of the first precursor field becomes
A,(z, t)
as z
b- ct()1 exp(-6 z
+b2
{[2(0 -1)1/2 +
+ (0)](0 - 1)
1-()
CL}
+ 462)
l
2
2(0) + 62[1 - (0)1
J
2
X sint C [2(0 - 1)]1/2 +
)O()
V14
[4(0)
()[o)
- WI] - 2 62[1 -
]Wj2
+ 62[1 + (o)] 2
-
+
c] -
2
3 62[1 -
+ 6
2
4(0)(0
(C \
f(0)[5 [4(0) -
_(0)[5-
-
1 +
(0)]-
()]1
[4(0) + WC]2 +
[12[
4
4
A8 (z, t)
(0)]
2
/2
42(0) + 62[l _7 n(f)
l +
41'
42
(6.3)
-
for 0 2 1 + A with A > 0.
~ 2bc
/2 Cc,[2(0
cos{Ž
c[2(0 -
()]
+ 7(0)]
- 1 +
+
1)]"
exp[-26N Z (0 - 1)]
(0)] 2
C]2 + 62[ +
W()]+ 3
X sin[- 4(0){
C
)
b/2
4~2(0) + 62[1 _ n(0)]2
UC[1 -
as z a with 0 > 1 and 0 1. This equation is the result
that would be obtained by using approximation (3.4) for the
locations of the distant saddle points in the nonuniform
expansion. The expression can be simplified further
by
noting that for values of 0 close to unity any finite applied
signal frequency will be negligible in comparison with the
quantity b[2(0 - 1)1-1/2. Hence the above equation simplifies further to
-
[4(0) + W]2 + 62[1 + (o)]2
X cos{
},
2(o)]
1)]/2 +
as z and 0- 1+. This result is precisely the same result
obtained by Brillouin (Ref. 6, p. 73), which is seen to be an
approximation valid for 0 near 1 of an expression that is not
valid for 0 near 1. Consequently, Brillouin's expression for
the asymptotic behavior of the first precursor field for the
unit step-function-modulated signal is not applicable.
From expressions (4.27), (4.31), and (4.37), the nonuniform asymptotic behavior of the second precursor field is
given by, with Eq. (6.1),
842
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
0.01.
e
0-
-0.010.004
AB(z,t)
0
-0.004J
0.01
A(zt)
_
_ _
//X/\/~_
_
.
2.
2.z
_
_ _
ee
_
_ _
0
-0.01
6
5B
G.
11.2
- i.A
.
I
I
I
2.0
.~~~~~~~
2.
2.4
Fig. 15. Nonuniform evolution of the Sommerfeld precursor A(z, t), the Brillouin precursor AB(z, t), and the total field A(z, t) = A(z, t) +
AB(Z, t) for an input delta-function pulse for Brillouin's choice of the medium parameters at a propagation distance of z = 1 X 10-4 cm.
as z
Wo2W,
AB(Z,
-
for 1 < 0 < 01 with 0 bounded away from 01;
t)
b{wc2 + 9 [31(6) - 26?(0)I2}
r(3
X (wzttt~3~i
aOC
)
1/2
AB(Z, t1 )
'.
)
2rsJ3
600(W
2
(26owoC
1/3
2
+
ab Z
9a 2
X exp
[25t(0) - 310)1(0
-
+ -
4
kW0
x Xp e 25z
x (0
X [26r(O) - 314(0)1] 26[3 - ar(0)] - 3a1i(0)1})],
(6.4a)
as z
-
- for 0 = °1
= Ctl/2 ; and
+
I (64b)
a O wo0/ '4
K. R Oughstun and G. C. Sherman
AB(Z,
_
t)'-
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
_ _ __ _ _
-
o ,_
~~26
_ _ _ _
_ _ _co
_ _co
_
_ _ _ _
b{kt'(-) w + 4 62 (0)}{[t(6) + wj]2 +
2
- a(0)] 2 (0) + 4 2r2(O)
X {[1
)(
+,
{[
X sin[
[C
as z -X
for
(0
-
k )Cos[
9
]
(0)(60- 0 +
\
[ ac(O)
4
20oo( 413
(
B. Spectral Pole Contribution and Arrival and Evolution
of the Main Signal
The residue due to the simple pole singularity at = co of
the spectral amplitude function [Eq. (6.1)] is simply y = i,
and Eqs. (4.48) yield
A,(z,
(6.5a)
) = -1/2 exp[-za(co)]sin[k(wo)z
-
cts]
exp
F(6z
-
+ 20~w
as z
O. Hence at
=
-
cot], 0 > O,
4
{
~2-0[2
a~(O)] + a 4 (0 })+
-
] + 43MO()t(0)
})4
3
J
(6.4c)
figure, which is drawn for one fixed value of 0. With the
path P(O) shown, 0 < , if the pole at = C lies within the
domain
ranges 0
c < CIc4, 0
Wc2 <
>
,
if Oc lies within either of the
and 0 = As if co = C(c2 or c =
or (c, >
Wc4,
The pole contribution at
=
< Cc <
C2
Wc4-
is the dominant contribu-
tion to the asymptotic behavior of the field for >
where , is that value of that satisfies the relation
> Os,
X(WSP 6) = X(UCd
(6.7)
where wsp denotes the dominant saddle point at that value of
6. From Eq. (A15), X(wcc) is independent of when c is
along the real axis. For values of <
such that the
inequality X(cosp, 0) > X(coc) is satisfied, the saddle point is
the dominant contribution to the asymptotic behavior of the
field, and the pole contribution is asymptotically negligible
by comparison. For values of > c such that the inequality
X(wsp, 0) < X(c,) is satisfied, the pole contribution is the
dominant contribution to the asymptotic behavior of the
field, and the saddle point is asymptotically negligible by
comparison. For example, for the situation illustrated in
= Os,
Cj/
(6.5c)
the field begins to oscillate in a
time-harmonic fashion at the applied signal frequency co,
.and with an amplitude that is determined by the medium
attenuation coefficient a(wc) and the propagation distance z.
The value of s depends on which Olver-type path is chosen
for P(6). If that path is taken to lie along the path of
steepest descent through the saddle point nearest the pole,
then Os is given by the expression
Y(cosP, A) = Y(co, 0),
8
200wo 4 {3
(6.5b)
Ac(z, t) = -exp-za(co)]sin[k(wc)z
(6 - 00) (O) + b2
-
L- c 3
3 ai( )z
/
>
01 with 0 bounded away from 01. Expression
0<
/
2(0)[2 - ar(0)] + a4A(6)} + 4
71lI
(6.4c) reduces to Brillouin's result (Ref. 6, p. 71) if (0) is
replaced by the absolute value of the real part of expression
(3.6), and if (O)is replaced by 1/a
1. However, these
approximations are valid only for values of 0 not to distant
from 0. . Hence Brillouin's expression for the second precursor field for 0 > 1is an approximation, valid for near 1, of
an expression that is not valid for 0 near 01. Consequently,
Brillouin's expression for the asymptotic behavior of the
second precursor field for the unit step-function-modulated
signal for 0 > 01 is not accurate in this space-time regime,
particularly with regard to the instantaneous angular frequency of oscillation of the propagated field.
A,(z, t) = 0,
62c2(O)
F1
- 1
V/(6)(00
[C
_ _
843
OriAinal
Contour
____
I;,
_
of
Inteoration
__________________
___
_
_~~~~~~~Ci
/~~~~~~~~~~~~W
(6.6)
where wsp(O) denotes the saddle point that interacts with the
pole singularity at
=
c. At 0 = Os, however, the pole
contribution is negligible in comparison with the saddlepoint contribution since P(O) is an Olver-type path. That is,
the value of as at which the pole crossing occurs is of little
importance to the nonuniform asymptotic behavior of the
propagated field as z
-
. An example of such an Olver-
type path when the two near saddle points SPNy are dominant over the two distant saddle points SPD' is illustrated in
Fig. 16. The path P(6) through the near saddle points SPNand SPN+ can lie anywhere within the hatched area of the
Fig. 16. A deformed contour of integration P(O) through both the
near and distant saddle points for a fixed value of 0. This contour is
an Olver-type path with respect to the near saddle point SPN+ in the
right half of the complex w plane and is an Olver-type path with
respect to the near saddle point SPN- in the left half of the complex
wplane. The hatched areas indicate the regions of the complex
plane wherein X(w) is less than that at the respective saddle points.
844
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
Fig. 16, the pole contribution is asymptotically negligible in
comparison with the saddle-point contribution to the field if
Wc lies within the domain Wcl < CO,< Wc6, whereas the pole
contribution is the dominant contribution to the asymptotic
behavior of the field if oC lies within either of the domains 0
• WC < WC, or Wc > Wc6.
On the basis of the above results, we can see that the
arrival of the main signal for a fixed value of Xc occurs at the
value of 0 = Oc, satisfying Eq. (6.7). This definition of the
signal arrival yields a signal velocity that depends only on
the properties of the medium and the value of the simple
pole singularity of the spectral amplitude function U(w - Wc).
From the discussion following Eq. (4.8), the signal arrival is
seen to be independent of the initial envelope of the field as
well as of the propagation distance z.
A general description of the arrival of the main signal and
its interaction with the Sommerfeld and Brillouin precursor
fields may now be given entirely on the basis of Eq. (6.7) and
the topography of X(w, 6) as a function of 0 in the complex c
plane presented in Figs. 4(a)-4(f). Consider first the range
of values of 6 during which the distant points are dominant
over the upper near saddle point SP 1 (i.e., 1 • 6 • OSB), as
depicted in Figs. 4(a)-4(c). From these three diagrams we
can see that for frequencies WC > W'SB signal arrival will occur
during the evolution of the first precursor field. At 0 = 6 sB
[Fig. 4(c)] the distant saddle points and the upper near
saddle point are of equal exponential importance in their
individual contributions to the asymptotic behavior of the
total field A(z, t). In that case, for WC > "SB, the signal has
already arrived and is the dominant contribution to A(z, t).
For Wc < COSB,however, the signal has yet to arrive at 0 = SB
and the precursor field is the dominant contribution to the
total field.
For all values of 0 larger than OSB, the upper near saddle
point SP 1 for OSB < 6 < 01 and the two near saddle points
SPN' for 0 > °1 are dominant over the distant saddle points.
As 6 increases over the range OSB <6 • ao, the contour X(w) =
X(wsp) through the dominant upper near saddle point recrosses any pole singularity at WC> WSB that had been crossed
previously by the contour X(w) = X(wspD+) through the
distant right saddle point when 0 varied over the range 1 • 0
< SB. However, any pole contribution at WC > "SB has not
been canceled by this phenomenon but rather has only be-
come less dominant than the evolving second precursor field.
At 0 = o, X(osp,) is identically zero at the dominant near
saddle point SP 1, and the contour X(w) = X(wsp,, 00) intersects the real axis only at w' = 0 and infinity and remains
above the real axis for all other positive values of w'. Consequently, at 0 = 0o the second precursor field is exponentially
dominant over all other contributions to the asymptotic
behavior of the propagated field A(z, t).
As 0 increases above 00 the contour X(w) = X(wsp,, 6) for 00
< 0 • 01 and the contour X(@) = X(WSPN+, 6) for > 1
crosses the positive real axis at both a low-frequency value
wpl, where 0 < Cpl • wmin and a high-frequency value Wp2 2
Wmin, where Upl monotonically increases to Wmin and wp2
monotonically decreases to Wmin as 6 increases, as seen in
Figs. 4(a)-4(f). A value of 0 = Or is reached at which the
relation
X(WSPN,
m) = X(Wmin)
(6.8)
is satisfied, where wmin is that value of w along the positive
real axis at which X(w) attains its minimum value [see expression (A16)]. Hence as 0increases from 00 to 6 m, any pole
singularity at w = WC along the real frequency axis will become the dominant contribution to the asymptotic behavior
of the field. For the frequency WC = Wmin the signal arrives at
Oc = m, which is larger than any other value of O, for co, That is, an input signal with applied frequency WC =
CWmin0
min has a main signal whose front propagates with a miniW
mum velocity in the medium. For all 0 > Gm, the main signal
at any frequency value WC has already arrived and is the
dominant contribution to the propagated field A(z, t).
It is then seen that the signal arrival separates naturally
into two distinct cases that depend on the value of the applied signal frequency WC For values of c in the domain 0 <
(c • "SB, the signal arrival is due to the crossing of the
contour X(w) = X(CwSpN) with the simple pole singularity at w
= c',, where SP, denotes the location of the upper near
saddle point SP 1 for 0 < 01, the second-order near saddle
point at 6 = A, and the near saddle point SPN+ for 0 > 01.
For such values of WC the main signal due to the pole contribution at X = Wc is preceded by the first and second precursor
fields, and the main signal evolves essentially undisturbed as
depicted in Fig. 17.
:
l
1
Fig. 17.
First Precursor
Second Precursor
Main Signal
Evolu+ion
Evolution
Vvolution
Dynamic behavior of the propagated field due to an input unit step-function-modulated signal with applied signal frequency wC <WSB
K. E. Oughstun and G. C. Sherman
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
845
11
"
0-
':S
I
I
I,
T-.
First Precursor
Evolution
Fig. 18.
Pre pulse
Evo luflon
Second Precursor
Evolution
Dynamic behavior of the propagated field due to an input unit step-function-modulated signal with applied signal frequency c
For values of the frequency co, in the domain
> SB, the
signal arrival first occurs because of the crossing of the contour X(w) = X(WSPD) with the simple pole singularity at =
w. The arrival occurs for a value of = 1 in the range 1 <
0c1 < OSB for finite c,. However, for some value = C2 within
the range SB < ,2 < o, the pole is again crossed in the
opposite direction by the contour X(@) = X(wsp,) so that the
pole contribution becomes less dominant than the second
precursor field. Then, for some value of = 0 > 00, the pole
is again crossed by the contour X(cw) = X(wspN) so that
finally it becomes dominant over all other contributions to
the asymptotic behavior of A(z, t) for all remaining values of
0. Consequently, for sources with co > SB, there is the
existence of a prepulse that is due to the interruption of the
signal evolution by the second precursor field, which becomes dominant over the pole contribution for some finite
interval of . This prepulse formation is an integral part of
the dynamic evolution of the second precursor field that is
superimposed upon the evolution of the main signal. The
signal evolution for such a high signal frequency, c > CLSB,
may then be considered to be separated into three parts, as
depicted in Fig. 18: the so-called prepulse, which is preceded by the first precursor field and is followed by the dominant second precursor field superimposed upon the signal,
which is then followed by the main signal, which remains
dominant for all later times. It is important to remember
that the prepulse is not independent of the main signal
evolution.
7.
~~~...
y
Main S nal
Evoktion
coSB.
at which the pole contribution becomes the dominant contribution to the asymptotic behavior of the total field A(z, t).
The velocity at which this point in the field propagates
through the dispersive medium is the main signal velocity,
given by
C
C,
(7.2)
-,
where c is the vacuum speed of light. Furthermore, for
frequencies
> SB a prepulse exists, the front of which
arrives at that value of 6 = ac, satisfying the relation
X(COSPD, act) = X(W@),
1 <
C1 < OSB-
(7.3)
At this value the pole contribution becomes the dominant
contribution to the asymptotic behavior of the total field.
The back of the prepulse arrives at the value of = c2
satisfying the relation
X(COSP, 0c2) = X(wc),
OSB <
2 <
(7.4)
at which the second precursor field becomes the dominant
contribution to the asymptotic behavior of the total field.
The velocity at which the front of the prepulse propagates
through the dispersive medium will be called the anterior
presignal velocity and is given by
Vc =-
C
0ci
Wc >
(7.5)
WSB-
The velocity at which the back of the prepulse propagates
will be called the posterior presignal velocity and is given by
SIGNAL VELOCITY
Physically, the first precursor field is due to the high-frequency components that are present in the frequency spectrum of the initial pulse envelope, and the second precursor
field is due to the low-frequency components present in the
initial envelope spectrum. The pole contribution is due to
the frequency component at the applied signal frequency of
the source. Consequently a well-defined signal velocity may
be defined in the following manner.
The main signal arrival is defined to occur at the value of
= 0, satisfying the relation
X(WSPN, OC) = X(WC),
>
0C >
(.
(7.1)
Vc2 =
C
(0c >
(7.6)
SB-
0c2
According to Eqs. (7.1)-(7.6), these three velocities satisfy
the inequality
C>
V
>-C
OSB
> V2 >C>
(7.7)
Vc.
0
This branching character of the signal velocity is a direct
consequence of the asymptotic dominance of the second
precursor field for c2 < 0 < Oc when
c>
SB-
846
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
///C
/C
/
/:
VI
C/
/
0.4
0.2
\
0
0
2
//
Ba12.
SAo
I,
1
12
l
14
(X10D/sec)
C
Fig. 19. Frequency dependence of the main signal, anterior presignal, and posterior presignal velocities for a Lorentz-model medium with =
16
4.0 X 10 16 /sec, b2 = 20 X 1032 /sec2 , and = 0.28 X 10 /sec. The dashed curve depicts the behavior of the energy-transport velocity for a strictly
monochromatic field in the medium.
When we know the behavior of X(w, 0) along the positive
w' axis and at the near and distant saddle points, these three
velocities may be determined directly at any applied signal
frequency w,. By using the behavior depicted in Figs. 2, 9,
and 10, which were all determined for Brillouin's choice of
the medium parameters, we can obtain the frequency dependence depicted in Fig. 19. As can be seen, the main signal
wo [see Eq.
velocity attains a minimum value at wc = Wmin
(A16)], at which value X(w,) attains its minimum value
along the positive real axis. Consequently the signal velocity does not peak to the vacuum speed of light c near resonance, as was indicated by Brillouin, 6 but rather it attains a
minimum value near that point. The dashed curve in the
figure connects the main signal velocity curve below resonance with the anterior presignal velocity and depicts the
behavior of the velocity of energy transport for a strictly
monochromatic field in the medium, as given by Loudon. 31
The close agreement between these velocities will be explained in subsequent papers.
8. DISCUSSION
This rather lengthy paper serves to lay the foundation for a
clear, detailed physical description of optical pulse propagation in a dispersive Lorentz-model medium. First, a detailed, accurate approximation of the relevant saddle-point
dynamics and the associated complex phase behavior at
them was developed fully. These results yielded an accurate representation of the dynamic evolution of the Sommerfeld and Brillouin precursor fields over the entire range of
values of 0 = ctlz for which they are the dominant contributions to the asymptotic behavior of the propagated field. Of
particular importance here is the improved accuracy in the
value of the instantaneous angular frequency of oscillation
of the dominant precursor field at a given space-time point.
This matter is of critical importance to the accurate prediction of the predominant frequency components that are pro-
duced by rapid rise-time pulses in dispersive media and
systems. The overall improved description is critical both
to the new physical description of dispersive pulse dynamics
and to the uniform asymptotic representation of the propagated field (both topics are to be presented in a subsequent
paper).
Finally, a correct description of the signal velocity of the
field propagating in a dispersive Lorentz medium was provided. This description accounts for the breakup of the
signal into a prepulse and the main signal by the second
precursor field for applied signal frequencies wC satisfying wc
> WSB- In addition, this analysis showed that the signal
velocity is a minimum near the resonance frequency of the
dispersive medium and does not peak to the vacuum speed
of light there, as was indicated by Brillouin. 6
APPENDIX A: ANALYTIC STRUCTURE OF
i4w, 0) IN THE COMPLEX w PLANE
The behavior of the complex phase function X(, 0) =
iw[n(w) - 0] is dictated by the analytic form of the complex
index of refraction n(w). This form is taken here to be given
by the Lorentz model26' 27 for a homogeneous, isotropic, locally linear, temporally dispersive medium characterized by
a single resonance frequency wo and given by
-=
)
/2
(Al)
Here, b2 = 4rNe 2/m is the square of the plasma frequency of
the medium, N is the number density of electrons of charge e
and mass m bound with the resonant frequency wo, and 3is
the associated phenomenological damping constant of these
harmonically bound electrons.
The branch points for n(), and hence for o(w, 0), can be
determined directly by rewriting the complex index of refraction as
K. E. Oughstun and G. C. Sherman
n
w2' -
W12
2 -
2
=
co
+ 26iw
Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. B
1/2
(a' -
-
26ho
coo
a+')
(co -
a+)(co
(a' -
-
/2
a-')
a')
(A2)
Y(a',
6)
=
[fnr()
"2 ) 1/4
b4 + 2b2 ( a' 0 2 2 =
2
W
2
+ b.
(A3)
nr(a)
2
= [1 + (a" -
The branch point locations are then given by
2
04 = +(W'
W+
= +(Wo2
62)1/2 -
i,
(A4a)
-
62)1/2 -
i
(A4b)
and lie along the line a" = Im(w) = -6 symmetrically situat-
ed about the imaginary axis (it is assumed here that w > ).
The branch cuts chosen here are the line segments a'.'w- and
w+w+', as illustrated in Fig. 1. The complex index of refraction n(w) and the complex phase function (a', ) are then
analytic in the complex plane with the exception of the
branch points wa' and w±.
The remaining analysis is simplified by the observation
that since Eq. (2.14) holds, then the real part of the phase
function satisfies
) = X(a" + i" 1 , ),
X(-a' + i",
(A5)
whereas the imaginary part of (a', ) satisfies
Y(-a' + i",
where
),
) = -Y(W' + i",
2 2
aO'
4
-
(A6)
= Re(a) and " = Im(a). Consequently, X(a,
) =
Re[o(w, )] is symmetric about the imaginary axis, whereas
Y(a', ) = I[b(a', )] is antisymmetric about the imaginary
ni(a') = [1 +
(A7)
where nr() is the real-valued index of refraction of the
medium and na) is related to the coefficient of absorption
a(a) of the medium, given by
a(a)= C ni(w)
(a/
) + 46 a,
2
2
2
2
-
2
4-2b
(a/
2
-
2
(a"
W2-
2
-
2
o -
a"'2
- a'
2ba"') 2 +
2
_
2
2
4a, (a"' +
(A9)
where
1
(a")j,
(A14)
(A15)
and hence is proportional in magnitude to the coefficient of
absorption a(w"). Since ni(') > 0 for ' >-O and ni(')
for ' < 0, then X(a") < 0 for all values of a',as seen in Fig. 2.
X(a') is independent of the parameter and hence is independent of both z and t. The value of
=
min along the
real axis at which X(a') attains its minimum value and a(w')
attains its maximum value is given approximately by
[CO1+
(
1
-
)]'
(A16)
which is slightly above the resonance frequency along the
positive real axis.
At I1a
= -, n(a) is real and equal to unity. Hence in the
limit as co approaches infinity in an arbitrary direction, the
real part of the phase function becomes
lim [X(a',
)] =
"(0 - 1).
(A17)
The following behavior for X(a, 0) at ja' =
is then ob-
tained: for 0 < 1, X(w, 6) is equal to - in the upper half of
the complex plane, zero at the real axis, and + - in the
everywhere at
)2]
Fl
sin[-
4652a,2
lower half of the plane; at
1/4
6/"')
2
(A13)
1(a)j
X(a') = -o'ni(co')
11w-
1+
1
r1
cos[2
2
a" ) - 1/2
-
+
ao' )
2
respectively. The spectral regions wherein the real index of
refraction nr(a") increases with co' (i.e., of positive slope) are
termed normally dispersive, and the region wherein nr(a')
decreases with increasing a"(i.e., of negative slope) is said to
exhibit anomolous dispersion. The real index of refraction
nr(')varies rapidly within the region of anomolous dispersion, and this region essentially coincides with the region of
strong absorption of the medium, as seen in Fig. 2. From
Eq. (All) the behavior of X(a') along the real axis is given by
(AS)
along the real axis. The complex index of refraction may be
written as
2
b + 2b (a' 0
axis so that one need only examine the phase behavior in the
right half of the complex a' plane.
The complex index of refraction may be written as
n(a) = nr(a) + ini(a),
(A12)
"'n(w).
-
Along the real axis, w" = 0, and the real and imaginary parts
of the complex index of refraction are given by
where
=
]
-
847
a'!
= 1, X(a, 1) is equal to zero
= -; and for
> 1, X(co, ) is equal to +
the upper half of the plane, zero at the real axis, and the lower half of the complex plane.
Along the line
a'
=
in
in
'- i, the complex index of refraction
is given by
t(w) = arg[nl(a)]
= tan
1
[(/,2
- W,
-
-
-
2
_n2W126)b
b22(a"' -+
b2(w
o2 _ao2
with -7r < < ir. The real and imaginary parts of the phase
function are then given, respectively, by
X(W, 0 = -jW'[n,(w
- 0] + W'nj(w)j,
(All)
+)2J
26w") + 4W, 2 (a"' +
n(W' - i)
(lO
=
[1 +
2
I
(A18)
which is real and positive except for along the two branch
848
K. E. Oughstun and G. C. Sherman
J. Opt. Soc. Am. B/Vol. 5, No. 4/April 1988
cuts ax.w._ and w+wo+', wherein it is purely imaginary. Furthermore, X(w' - i, 0) is given by
X( ' - i, 0) =
[1 +
LkCoo
2
_
-
2
0
(Al9a)
2 - W7,j
-
when w' < Re(c_), Re(w-) < co' < Re(w+), or Re(w+') <
and
XW- ib, 0) = -60-II
i-(,
-
12
',
Near w+, however, X(o, 0) is negative on the upper side of the
branch cut and positive on the lower side for all 0. We can
then see that the contour X(w, 0) = 0 must pass through the
branch point w+ and continue on from the lower side of the
branch cut between w+ and w+' for 0 > 0, as illustrated in Fig.
3(b). For 0 < 0 the contour X(@, ) = 0 continues on from
the upper side of the branch cut between co+ and w+', and for
0 = 0 the contour continues on from the branch point w+'.
(Al9b)
1)/
REFERENCES
when either Re(w-') • ' < Re(w-) or Re(w+) C co < Re(w+').
Hence, X(@' - i, 0) is real valued except for along the two
branch cuts a'@'_ and w+w+', wherein it is generally complex
valued
In the region about w+', c, may be written in polar-coordi-
nate form as
2
w = (
1
-
+ rei#.
62)1/2 -iS
In the limit of small r the complex index of refraction in the
region about the branch point c,+' is given by
[2(a'
n(r, ') =
2
1
-
62) /2r]1/2
b
Similarly, in the region about w+,
coordinate form as
W=
(ao
2
-
may be written in polar-
a
iS + Re'i.
62)1/2 -
In the limit of small R the complex index of refraction in the
region about the branch point w+ is given by
[2(n( 2 _ S2 )' 12 R]' 12 exp[i(r - 4)/2].
(A21)
The behavior of n(a') at the two branch points w+' and w+, as
described by expressions (A20) and (A21), respectively, is
illustrated in Fig. 3(a). Similar results hold for the behavior
of the complex index of refraction about the branch points
w-' and
-. in the left half of the complex w plane.
From these results the behavior of X(@, 0) about the
branch points w+' and co+ is determined by using Eq. (All)
with " = -.
there results
X(r,
ik,
In the neighborhood of the branch point w+'
~ {[2(a'
0)
2
1
(W, 2 -
( )- 0
1
2) /r]2 Ccos
-
2)12[2(a 1
2
-
2
)12 r]1
2
i
b~~~~~~()
and in the neighborhood of the branch point + one obtains
X(R , 4,0)
-
[2('
(O
2-
0
2
-
2 12
6)7 R
] 2 cos ( 2
2)1/2
[2(a'o
2
-
b2(C>
71/~2R/
sin
112_
52)
R J"
)
2 )-
\ 2
/
Hence X(@, ) is negative on both sides of the branch cut
near
+' for 0 > 0, is zero at 0 = 0, and is positive for 0 < 0.
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