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Generalized Nonlinear Schro
¨
dinger Equation for Dispersive Susceptibility and Permeability:
Application to Negative Index Materials
Michael Scalora,
1
Maxim S. Syrchin,
2
Neset Akozbek,
3
Evgeni Y. Poliakov,
1
Giuseppe D’Aguanno,
3
Nadia Mattiucci,
3,4
Mark J. Bloemer,
1
and Aleksei M. Zheltikov
2
1
Charles M. Bowden Research Center, AMSRD-AMR-WS-ST, RDECOM, Redstone Arsenal, Alabama 35898-5000, USA
2
Physics Department and International Laser Center, M V Lomonosov Moscow State University,
Vorob’evy Gory, 119899 Moscow, Russian Federation
3
Time Domain Corporation, Cummings Research Park, 7057 Old Madison Pike, Huntsville, Alabama 35806, USA
4
Universita
`
‘‘Roma Tre,’’ Dipartimento di Fisica ‘‘E. Amaldi,’’ Via Della Vasca Navale 84, I-00146 Rome, Italy
(Received 3 January 2005; published 27 June 2005)
A new generalized nonlinear Schro
¨
dinger equation describing the propagation of ultrashort pulses in
bulk media exhibiting frequency dependent dielectric susceptibility and magnetic permeability is derived
and used to characterize wave propagation in a negative index material. The equation has new features that
are distinct from ordinary materials ( 1): the linear and nonlinear coefficients can be tailored through
the linear properties of the medium to attain any combination of signs unachievable in ordinary matter,
with significant potential to realize a wide class of solitary waves.
DOI: 10.1103/PhysRevLett.95.013902 PACS numbers: 42.25.Bs, 42.25.Gy, 78.20.Ci
Nonlinear wave propagation in optics has been widely
studied in the framework of the nonlinear Schro
¨
dinger
equation (NLSE). Its use in describing the propagation of
picosecond pulses has led to innumerable innovations in
many fields, the most notable of which is perhaps fiber
optics [1]. The NLSE describes the evolution of an enve-
lope function, which is assumed to vary slowly over an
optical cycle. Typical pulse durations are now routinely
below 50 femtoseconds (fs), a regime where pulse enve-
lopes can no longer be assumed to always vary slowly in
space and/or time, especially near resonance or unusual
dispersive conditions [2]. The advent of fs lasers and recent
demonstrations of attosecond pulses [3] highlight a need
for a tailored, more measured theoretical approach to study
specific problems, to address new, unusual materials, and
to go beyond well-established approximations. The fila-
mentation of intense fs pulses in air and supercontinuum
(SC) generation can be described by a properly managed,
modified NLSE [4]. The same is true for the case of
photonic crystal fibers [5], where a SC is generated under
conditions of transverse light confinement [6]. It has been
shown that the slowly varying envelope approximation
(SVEA) breaks down even for initial pulses that are
many optical cycles long [7]. In this respect, a modified
NLSE is used that includes correction terms that go beyond
the SVEA, such as the shock term [8] and coupled tempo-
ral and transverse spatial derivatives [7,9]. Other correc-
tions are derived from including the second order spatial
derivatives, leading to a pseudo-
5
-like effect and mod-
ifications of the shock term [10].
In this Letter we discuss the propagation of pulses at
least a few tens of optical cycles in duration, within the
context of a new NLSE derived for the general case of
dispersive dielectric susceptibility " and magnetic perme-
ability , and focus our attention on the unusual character-
istics of uniform bulk, negative index materials [11], in the
absence of feedback. When " and are simultaneously
dispersive and negative, the index of refraction n
"
p
allows the negative root as its solution [11], leading
to unusual refraction of the beam, as if the index of
refraction were negative. Our approach evolves through
the narrow-band constraints imposed by the SVEA [2], and
leads to an equation of motion where group velocity and
group velocity dispersion (GVD) can easily be identified
and quantified without ambiguity when the medium is
relatively transparent. Then, eliminating the magnetic field
and introducing a nonlinear polarization or magnetization
leads to a NLSE similar to the usual NLSE, but different in
some important aspects. We find that while the group ve-
locity is always positive, GVD, the transverse Laplacian,
and non-SVEA corrections can all have the same sign, a
positive or negative sign, or a variety of combinations,
depending on the specifics of the linear dispersion curves
and the sign of the nonlinear coefficient. The available
combination of signs leads to new, richer dynamical char-
acteristics compared to the case of ordinary nonlinear dy-
namics [1], and immediately suggests the existence of tem-
poral and spatial, bright and dark solitons in bulk NIMs.
Nonlinear pulse propagation in NIMs of both finite and
infinite lengths remains essentially unexplored. In the lin-
ear regime authors have investigated group delay and
superluminal propagation [12–14], and the anomalous
refraction process for wave packets of finite spatial and
temporal extent [15–17]. For structures of finite length
bright and dark gap solitonlike solutions can be dynami-
cally excited and made to propagate within a single layer of
NIM [18]. Here we use the same formal approach used in
Ref. [17] to derive a new wave equation that can be used to
study pulse propagation in uniform, bulk materials under
the conditions of a nonlinear polarization and/or magneti-
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zation [19]. We go beyond the usual SVEA to obtain only
qualitative understanding of how higher order terms con-
tribute to the dynamics when the material is magnetically
active, and compare with the dynamics that ensues in
ordinary materials. For simplicity we first consider one
longitudinal spatial coordinate and time, and define the
fields as follows:
D z; t
^
i
Z
1
1
"!E
x
z; !e
i!t
d! P
nl
z; t
; (1)
B z; t
^
j
Z
1
1
!H
y
z; !e
i!t
d!: (2)
P
nl
is the nonlinear polarization. Substitution into Maxwell
equations yields
@E
x
z; t
@z
i
c
Z
1
1
!!H
y
z; !e
i!t
d!; (3)
@H
y
z; t
@z
i
c
Z
1
1
!"!E
x
z; !e
i!t
d!
1
c
@P
nl
z; t
@t
: (4)
The symmetry of the equations suggests that a nonlinear
magnetization produces qualitatively similar effects. In
Eqs. (3) and (4) we have expanded " and as follows:
!"!
X
1
n0
@
n
!"!
@!
n
!!
0
! !
0
n
n!
; (5)
!! is written in a similar way. !
0
is the carrier
frequency of the incident pulse. Substituting the expan-
sions for !"! and !! into Eqs. (3) and (4) yields
@E
x
@z
i
c
e
ikzi!
0
t
X
1
n0
i
n
@
n
!
@!
n
!!
0
1
n!
@
n
H
y
@t
n
; (6)
@H
y
@z
i
c
e
ikzi!
0
t
X
1
n0
i
n
@
n
!"
@!
n
!!
0
1
n!
@
n
E
x
@t
n
1
c
@P
nl
@t
:
(7)
where all the fields are understood to be explicit functions
of z and t. Equations (6) and (7) are very general because
they include dispersion effects up to any desired order. For
ultrashort pulse propagation one requires an expansion of "
and at least up to second order to account for GVD
effects. Accordingly, we decompose the fields as a general
envelope function (not necessarily slowly varying), multi-
plied by carrier wave vector and frequency: Ez; t;
Hz; te
ikz!t
, and substitute in Eqs. (6) and (7) to obtain
@E
@
i
0
4
@
2
E
@
2
i"E inH
@H
@
i
3
jEj
2
E
3
@
@
jEj
2
E; (8)
@H
@
i
0
4
@
2
H
@
2
iH inE
@E
@
; (9)
where
@~!"~!
@ ~!
,
0
@
2
~!"~!
@ ~!
2
and
@~!~!
@ ~!
,
0
@
2
~!~!
@ ~!
2
. We have adopted the following scaling:
z=
p
; ct=
p
, 2 ~! 2!=!
p
; n
"
p
n~!, !
p
is the plasma frequency and
p
its corresponding
wavelength. We now combine Eqs. (8) and (9) and elimi-
nate the magnetic field. For relatively transparent materi-
als, the fact that energy should always be positive [20]
imposes the following conditions:
@~!"~!
@ ~!
> 0 and
@~!~!
@ ~!
> 0 simultaneously. Although absorption is
an issue one must deal with in any bulk medium and at
any wavelength, including the visible range [21], we are
mainly interested in seeing how magnetic activity mani-
fests itself in the dynamics. Therefore, we assume that
linear absorption remains small [22]. Substituting and
retaining linear derivatives up to second order, and neglect-
ing nonlinear second order temporal derivatives, we can
write
@E
@
"
2n
@E
@
i
2n
@
2
E
@
2
@
2
E
@
2
i
3
2n
jEj
2
E
3
3
2n
@
@
jEj
2
E
i
"
0
0
8n
@
2
E
@
2
: (10)
Ordinarily, propagation takes place in low density gases or
other materials such that the index of refraction is greater
than one. Our system displays metallic behavior, and so
one has to be careful about extending the range of validity
of Eq. (10) to regions where n ! 0 [18]. In that case, one
should solve Eqs. (8) and (9) simultaneously.
The form of Eq. (10) suggests a group velocity: V
g
2n
"
. It can easily be shown that given n
2
", then
V
g
1
n!
@n
@!
@k
@!
1
, which is the usual expression for the
group velocity in units of c. Because n is negative when
both " and are simultaneously negative, and given >0
and >0 [20], it follows that the group velocity is always
positive without ambiguity.
Transforming to the retarded coordinates:
@
@z
@
@
1
V
g
@
@
, and
@
@t
@
@
and substituting into Eq. (10) yields
@E
@z
i
2n
1
V
2
g
"
0
0
2
@
2
E
@t
2
i
2n
@
2
E
@z
2
2
V
g
@
@z
@E
@t
i
3
2n
jEj
2
E
3
3
2n
@
@
jEj
2
E: (11)
The change of coordinates immediately reveals a GVD co-
efficient k
00
[1]. Although it is not obvious from its form,
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one can show that k
00
@
@!
1
V
g
1
n
1
V
2
g
"
0
0
2
.
Because we are assuming propagation in bulk media and
no feedback, and pulses at least a few tens of wave cycles
in duration, all higher order derivatives should give negli-
gible contributions. We now calculate the first order non-
SVEA correction terms by using Eq. (11) to evaluate
@
2
E
@z
2
and
@
@z
@E
@t
[10]. Neglecting higher order derivatives, we
estimate
@
2
E
@z
2
i
3
2n
@
@z
jEj
2
E and
@
2
E
@z@t
i
3
2n
@
@t
jEj
2
E. Therefore,
@E
@z
ik
00
2
@
2
E
@t
2
i
3
2n
1
3
4n
2
jEj
2
jEj
2
E
3
2V
g
n
2
2n
@jEj
2
E
@t
: (12)
Equation (12) is the generalized NLSE we sought. It can be
solved using any of a number of equivalent numerical
techniques [1].
There are some qualitative aspects of this modified
NLSE that are remarkable. First, the sign of the GVD
can be positive or negative, depending on the particular
choice of parameters. Second, assuming a positive
3
, the
sign of the leading nonlinear coefficient is always positive
because the ratio =n is positive. On the other hand, the
pseudo-
5
correction term is proportional to (
2
=n
3
),
which makes the coefficient positive, and its effect is to
enhance the nonlinearity. For ordinary materials the effect
of this term is to quench the nonlinearity. Finally, magnetic
contributions to the shock term are also evident. Usually
( 1 and n>1) this coefficient is negative [1], but its
form in Eq. (12) and the specific model used may allow for
it to be positive in a frequency range where 0 <n<1.In
optical fibers the shock term causes the pulse to steepen
along its trailing edge and the spectrum to split asymmet-
rically with larger identifiable redshifted peaks [1]. In
NIMs the opposite occurs, with self-steepening character-
izing the front of the pulse (see Fig. 3 below).
We now illustrate the dynamics with a representative
example. We use a Drude model described by "~!1
1
~!
2
i ~!
and ~!1
!
2
m
=!
2
p
~!
2
i ~!
. In our scaled units, we
choose 5 10
4
, which results in negligible absorp-
tion [12]. In Fig. 1 we show ", , and the index of
refraction n when !
2
m
=!
2
p
0:64. Using these dispersion
curves, we calculate the group velocity, GVD, and the
shock term and show them in Fig. 2. The figure shows
that the shock term can be zero, which means shorter
pulses can evolve into solitary waves, at least compared
to ordinary media. In general, the dispersion curves can be
engineered by choosing the size of split-ring resonator
circuit elements [21,23], which have been shown to exhibit
negative refraction under microwave illumination [23]. In
Fig. 3 we illustrate the effect of a positive shock coeffi-
cient. The figure shows that the pulse self-steepens along
its leading edge, and the peaks with the largest amplitude
are blueshifted (inset).
Magnetic nonlinearities can also play a role, and indeed
their contribution may be more pronounced than electric
nonlinearities [19]. A magnetic nonlinearity would pro-
duce additional terms, and Eq. (12) should be modified
accordingly. If only a magnetic nonlinearity is considered,
the qualitative aspects would remain the same thanks to the
symmetry properties of Eqs. (6) and (7). We note that a
negative 3may be used to make similar arguments, with
appropriate sign changes applied to the other coefficients.
We can gain further insight if we consider Eq. (12) in its
simplest form, but with the addition of transverse coordi-
nates. For the simple case of linearly polarized fields, and
to the extent that one can neglect the
~
r
~
rE in the vector
equations, the analysis yields the expected transverse
Laplacian that describes diffraction. The result is
-1.0
-0.5
0
0.5
0.6 0.7 0.8 0.9 1.0 1.1
Real(n)
Imag(n)
Real(ε)
Real(µ)
ω/ω
p
FIG. 1. Dispersion of ", , and resulting n. The region 0:8
~! 1 is characterized by metal-like reflections, as n becomes
almost purely imaginary. n<0 in the region ~! 0:8.
-2.5
0
2.5
5.0
0.4 0.5 0.6 0.7 0.8
Shock Term
GVD
V
g
ω/ω
p
FIG. 2. V
g
2n
"
, GVD k
00
@
@!
1
V
g
1
n
1
V
2
g
"
0
0
2
, and shock term S
2V
g
n
2
2n
in units
of
3
.
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@E
@z
ir
2
?
E
2n
ik
00
2
@
2
E
@t
2
i
3
2n
jEj
2
E: (14)
Although the sign of the nonlinear coefficient is unchanged
in NIMs, the GVD term can have the same or opposite sign.
Bright (dark) soliton solutions thus emerge when both
coefficients are positive (negative), with solutions that are
identical to those discussed for ordinary materials [1].
Because the temporal derivative and the transverse
Laplacian are on equal footing, the same arguments we
have made for temporal solitons may be made for trans-
verse or spatial solitons. Finally, the index of refraction
also rescales the Fresnel number (the coefficient in front of
the transverse Laplacian) and determines its sign. This
provides a simple dynamical explanation for the negative
refraction process.
In conclusion, a generalized nonlinear Schro
¨
dinger
equation for a dispersive dielectric susceptibility and per-
meability was derived from first principles, and was used to
describe pulse propagation in a NIM. We find that the
linear properties of the medium can be tailored to change
both the linear and nonlinear effective properties of the
medium leading to a new class of dynamic behavior and
solitary waves. We hope our findings will further stimulate
the study of nonlinear wave dynamics in NIMs, in the
visible range [21] in particular and in the field of solitary
waves in general.
We thank the National Research Council (E. Y. P.), the
President of Russian Federation Grant, and the Russian
Foundation for Basic Research (M. S. S. and A. M. Z.) for
partial financial support.
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0
0.4x10
9
0.8x10
9
-100 -50 0 50 100
Time (λ
p
/c)
|E|
2
0
1x10
7
-1 0 1
ω
FIG. 3. Field intensity profiles with ~! 0:707, where k
00
0,
and S 0:41. The pulse self-steepens with larger blueshifted
spectral components (inset). The input field is EtE
0
e
t
2
=
2
p
,
E
0
3 10
4
Statvolts=cm, and
p
20,
3
10
10
esu.
Propagation distance is 200
p
.
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