Optics Communications 249 (2005) 117–128
www.elsevier.com/locate/optcom
Coupled periodic waves with opposite dispersions
in a nonlinear optical fiber
S.C. Tsang a, K. Nakkeeran
b
b,*
, Boris A. Malomed c, K.W. Chow
a
a
Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong
Photonics Research Center, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong
c
Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University,
Tel Aviv 69978, Israel
Received 13 September 2004; received in revised form 17 December 2004; accepted 20 December 2004
Abstract
Using the HirotaÕs method and elliptic h-functions, we obtain three families of exact periodic (cnoidal) wave solutions for two nonlinear Schrödinger (NLS) equations coupled by XPM (cross-phase-modulation) terms, with a ratio
r of the XPM and SPM (self-phase-modulation) coefficients. Unlike the previous works, we obtain the solutions for
the case when the coefficients of the group-velocity-dispersion (GVD) in the coupled equations have opposite signs.
In the limit of the infinite period, the solutions with r > 1 carry over into inverted bound states of bright and dark solitons in the normal- and anomalous-GVD modes (known as ‘‘symbiotic solitons’’), while the infinite-period solution
with r < 1 is an uninverted bound state (also an unstable one). The case of r = 2 is of direct interest to fiber-optic telecommunications, as it corresponds to a scheme with a pulse stream in an anomalous-GVD payload channel stabilized
by a concomitant strong periodic signal in a mate normal-GVD channel. The case of arbitrary r may be implemented in
a dual-core waveguide. To understand the stability of the coupled waves, we first analytically explore the modulational
stability of CW (constant-amplitude) solutions, concluding that they may be completely stable for r P 1, provided that
the absolute value of the GVD coefficient is smaller in the anomalous-GVD mode than in the normal-GVD one, and
certain auxiliary conditions on the amplitudes are met. The stability of the exact cnoidal-wave solutions is tested in
direct simulations. We infer that, while, strictly speaking, in the practically significant case of r = 2 all the solutions
are unstable, in many cases the instability may be strongly attenuated, rendering the above-mentioned paired channels
scheme usable. In particular, the instability is milder for a smaller period of the wave pattern, and/or if the anomalous
*
Corresponding author. Tel.: +852 2766 6197; fax: +852 2362 8439.
E-mail addresses: [email protected] (K. Nakkeeran), [email protected] (B.A. Malomed), [email protected]
(K.W. Chow).
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2004.12.042
118
S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
GVD is weaker than the normal GVD in the mate channel. When the instability sets in, it first initiates quasi-reversible
modulations, and only at a later stage the wave pattern decays into a ‘‘turbulent’’ state.
Ó 2004 Elsevier B.V. All rights reserved.
PACS: 42.81.Dp; 42.65.Tg; 05.45.Yv
Keywords: Optical fiber; Coupled nonlinear Schrödinger (NLS) equations; Periodic solutions; Hirota method
1. Introduction
Exact solutions to the nonlinear Schrödinger
(NLS) equation, and coupled systems of such
equations, in the form of periodic arrays of pulses,
which are frequently called ‘‘cnoidal waves’’ (because they are based on elliptic functions like
cn), play an important role in the analysis of the
data transmission in fiber-optic telecommunications links. It has been demonstrated, theoretically and experimentally, that such arrays may
be created using the modulational instability of
the continuous-wave (CW) signal [1–3], or from
a dual-frequency pump [4,5], or directly (without
the use of a CW input) by a high-repetition-rate
pulse-generating fiber-ring laser (see, e.g., [6]).
Search for exact solutions to coupled (generally
speaking, nonintegrable) systems of NLS equations is also a mathematical problem of considerable interest in its own right [7–11].
A crucially important parameter of a datatransmitting channel in the optical fiber is its
group-velocity-dispersion (GVD) coefficient, which
depends on the carrier wavelength k [12,13]. In
particular, in the dispersion-shifted fiber, the transparency window (which is centered at k = 1540 nm
in all silica fibers) includes a zero-dispersion point
(ZDP) k0 (in fact, k0 in the dispersion-shifted fiber
is very close to 1540 nm), at which the GVD coefficient changes its sign from positive (normal
GVD) at k < k0 to negative (anomalous GVD) at
k > k0 (in fibers of other types, the ZDP is shifted
to smaller values of k, usually falling between 1300
and 1500 nm). The negative-GVD region, k > k0, is
employed for the data transmission in the returnto-zero (RZ), or quasi-soliton, format, in which a
bit of data is carried by an isolated light pulse
(with the zero field between them, hence the term
RZ). In this case, which is of major importance
to the applications [12,13], the bandpass in the
normal-GVD region remains idle, as normal
GVD cannot support quasi-soliton pulses in nonlinear optical fibers.
In Ref. [14] it was proposed to employ the normal-GVD band, k < k0, for service channels, which
carry strictly periodic signals. Such a signal cannot
transmit any information; instead, it is intended to
stabilize the concomitant array of RZ pulses in the
payload channels set at k > k0. In the usual on-off
realization of the binary code with the RZ signals,
the most important stability requirement is to prevent the pulse from leaving a prescribed temporal
slot (which would compromise ‘‘ons’’ and ‘‘offs’’).
A trend of the pulse to jitter, i.e., random walk off
the prescribed temporal position, is due to its
interaction with the optical noise created by amplifiers (the Gordon–Haus effect), as well as the nonlinearity-mediated interaction of the pulse with
adjacent ones in its own channel, and collisions
with pulses belonging to channels carried by other
wavelengths, in the wavelength-division-multiplexed (WDM) regime [15].
Suppression of the random walk is thus the
most important condition securing the use of the
fiber-optic links for long-haul telecommunications
in the quasi-soliton regime. A new approach proposed for this purpose in Ref. [14] was based on
the fact that the idle channel servicing its payload
mate may be chosen so that both have equal group
velocities (this is possible just because both channels are chosen on the opposite sides of the
ZDP), hence the pulses in the latter channel must
be immobile relative to the periodic signal in the
support channel. Further, if the periodic signal is
stable by itself and strong enough, it induces,
through the XPM (cross-phase-modulation) nonlinear interaction, an effective potential acting on
the pulse in the payload channel. Accordingly,
the pulse tends to be trapped in a local well belonging to the periodic potential. This should help one
S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
to suppress the pulseÕs random walk, provided that
the trapping potential well is deep enough. In fact,
this stabilization scheme may turn out to be simpler and more efficient than other known ones
(such as the use of guiding bandpass filters [12,15]).
Besides the applications, a possibility of direct
experimental observation of stabilized cnoidal
waves in the anomalous-GVD channel, as suggested by the results reported in this work, is a
physically interesting result in its own right. An
important circumstance which makes the use of
the proposed scheme relevant is that the cnoidal
wave in the normal-GVD NLS equation may indeed be dynamically stable, unlike the anomalous-GVD one [16,17], hence it can be employed
as a support structure (the stability of cnoidalwave solutions in the single-channel models with
various nonlinearities was recently reviewed in
[18], and a relevant issue of the cnoidal-wave stability in the dispersion-management scheme was
considered in Ref. [19]).
The same concept of pairing a payload channel
with an idle one carrying a periodic support structure may, in principle, be realized in terms of two
waves with one wavelength and orthogonal polarizations. In this context, periodic coupled-cnoidalwave solutions were studied in Ref. [11]. However,
the realization based on the pair of different wavelengths with equal group velocities, taken on the
opposite sides of the ZDP, is much more practical,
therefore in the present work we chiefly focus on
the latter case.
The actual difference between the two models
is in the size of the XPM/SPM ratio r, see Eqs.
(1) and (2) below: it is 2/3 for the orthogonal linear polarizations, and 2 for the different carrier
wavelengths (or orthogonal circular polarizations). In fact, any value of r may be realized
not in ordinary fibers, but rather in a dual-core
waveguide with different carrier wavelengths in
the two cores: obviously, r depends on the proximity between them. In particular, a promising
medium to realize such a structure is a photonic-crystal fiber, with two parallel holes made
in it (fabrication of such a system has already
been reported [20]).
As it was already mentioned, a strictly periodic signal in the payload channel cannot trans-
119
mit information. Nevertheless, the promising
application outlined above makes it a relevant
problem to find the broadest possible family of
exact solutions for coupled periodic (cnoidal)
waves in the XPM-coupled NLS equations
describing the pair of the payload and service
channels, with opposite signs of the GVD coefficients in them, and to test stability of such solutions. This problem, which is the subject of the
present work, is practically important, as it helps
to understand whether the scheme of the stabilization of the payload channel by the concomitant periodic signal in the service channel is
itself completely stable. On the other hand, this
problem is also novel in terms of the mathematical analysis of the coupled NLS equations, as
all the previously reported families of exact periodic solutions were only found for systems of
equations with the equal signs of the GVD
coefficients.
In Section 2 of this paper, we produce three
families of such exact solutions in the system
with opposite signs of the GVD in the XPMcoupled NLS equations. Two families are found
for r > 1, and one for r < 1. In Section 3, we address stability of the solutions. To get a general
idea of the stability in this type of the model, we
first consider the modulational stability of CW
(constant-amplitude) solutions, which can be
done in a complete analytical form. We find that
they may be stable in the case of r P 1, provided that the absolute value of the dispersion
coefficient is larger in the normal-GVD channel
than in the anomalous-GVD one, and certain
auxiliary conditions on the amplitudes are met.
Then, we summarize results for the stability of
the cnoidal-wave solutions, obtained by dint of
direct simulations. It is concluded that, in the
physically important case of r = 2 (as well as
for r < 1) the cnoidal waves are never strictly
stable, which is not surprising, as all the cnoidal
waves are unstable in the single NLS equation
with anomalous GVD. Nevertheless, it is possible to identify a parameter region where the
instability is very weak, so that the region is of
direct interest to applications. A general trend
is that the coupled cnoidal waves are less unstable when the absolute value of the dispersion
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S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
coefficient in the anomalous-GVD channel is
small in comparison with the respective coefficient in its normal-GVD counterpart, and the
period of the cnoidal waves is smaller (in the
limit case of the infinite period, the coupled cnoidal waves with r > 1 carry over into a known
exact solution for coupled ‘‘symbiotic’’ solitons,
which is strongly unstable; the infinite-period
limit for r < 1 is another unstable bound state
of bright and dark solitons). In the case when
the instability is tangible, it may be of two different types. Moderate instability manifests itself as
onset of quasi-regular modulations of the coupled waves; on the other hand, strong instability
results in direct transition to a spatiotemporal
chaos. Results obtained from these simulations
are summarized in Section 3.
2. Exact solutions for the coupled waves
2.1. Cnoidal waves
The system of the normalized XPM-coupled
NLS equations describing the interaction of electromagnetic waves with local amplitudes A and
B in the idle and payload channels in a fiber-optic
telecommunications link, or in a dual-core waveguide (in both cases, two different carrier wavelengths are implied), is
2
2
iAz d1 Att þ jAj þ rjBj A ¼ 0;
ð1Þ
2
2
iBz þ d2 Btt þ jBj þ rjAj B ¼ 0;
ð2Þ
where z and t are, as usual, the propagation distance and ‘‘local time’’ [12,13], 2d1 and 2d2 are
the GVD coefficients (in accordance with what
said above, we consider the case when both d1
and d2, defined as in Eqs. (1) and (2), are positive),
and the SPM (self-phase-modulation) nonlinearity
coefficient is normalized to be 1, then r > 0 is the
relative XPM coefficient. As said above, the case
of r = 2 is of major interest to applications, but
we will display exact solutions for arbitrary
r > 0. We also assume, as explained above, that
the channels governed by Eqs. (1) and (2) have
the same group velocity, therefore group-veloc-
ity-difference terms do not appear in Eqs. (1) and
(2). The latter condition implies the symmetric
choice of the two carrier waves relative to the
ZDP, then the case of d1 = d2 is the most natural
one, although unequal values of d1 and d2 are
relevant too.
The system (1) and (2) is definitely nonintegrable, except for the ManakovÕs case [21], r = 1
(which does not apply to optical fibers). Nevertheless, a vast class of exact cnoidal-wave solutions
can be found, using the HirotaÕs method [22] and
special relations between the elliptic h-functions,
as explained in detail in Refs. [23,24]. Note that,
in all the previous works, this method (as well as
other techniques employed to generate exact periodic-wave solutions [7,8]) was applied to the case
when the GVD coefficients in the coupled NLS
equations had the same sign. Here, we apply the
HirotaÕs method to Eqs. (1) and (2), where these
signs are opposite.
First, we address the case of r > 1, which is
most important for applications (because it includes r = 2, i.e., the value relevant to fiber optics).
Skipping details of manipulations with the h-functions, which follow the pattern of [11], we display a
final form of the solutions found in this case. Two
families of the exact periodic waves correspond to
the signs + and in the following expressions:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðrd2 þ d1 Þ
ð Þ
2 1=4
ðÞ
A ðz; tÞ ¼ exp iQ1 z r 1 k
r2 1
"
#
2 1=4
1k
dnðrt; k Þ
ð3Þ
1=4 ;
dnðrt; k Þ
1 k2
B
ðÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðd2 þ rd1 Þ
ðz; tÞ ¼ exp
r
r2 1
2
k snðrt; k Þcnðrt; k Þ
;
dnðrt; k Þ
ð Þ
iQ2 z
ð4Þ
where the propagation constants are
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi 2r2 1 k 2
ð Þ
2
2
2
Q1 ¼ r d1 2 k 2 1 k þ
2
pffiffiffiffiffiffiffiffiffiffiffiffiffi
r 1
1
rd2
1 k 2 þ pffiffiffiffiffiffiffiffiffiffiffiffi2ffi 2
1k pffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
2
ffiffiffiffiffiffiffiffiffiffiffiffi
ffi
p
þ d1 r
1k þ
2 ; ð5Þ
1 k2
S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
ð Þ
Q2
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi
2r2 1 k 2
1
2
2
d2
¼
1 k þ pffiffiffiffiffiffiffiffiffiffiffiffi2ffi 2r
r2 1
1
k
pffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ rd1
1 k 2 þ pffiffiffiffiffiffiffiffiffiffiffiffi2ffi 2 ;
ð6Þ
1k
sn, cn and dn being the standard JacobiÕs elliptic
functions with the modulus k (0 < k < 1). The
solutions depend on two arbitrary constants, k
and r, but the latter one is actually a trivial parameter, as it accounts for the obvious scaling invariance of the underlying Eqs. (1) and (2). The
period of the solutions of both types is
T ¼ 2KðkÞ=r;
ð7Þ
where K is the complete elliptic integral of the first
kind [23].
These solutions disappear at the point r = 1,
which corresponds to the integrability in the ManakovÕs sense [21]. In the case of r < 1, the above
expressions are meaningless, as they are imaginary,
while, by definition, the magnitude of the amplitude must be real. However, another exact solution family can be found in this case:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðrd2 þ d1 Þ
A ¼ rk expðiQ1 zÞ
snðrt; kÞ;
ð8Þ
1 r2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðrd1 þ d2 Þ
B ¼ r expðiQ2 zÞ
dnðrt; kÞ;
ð9Þ
1 r2
Q1 ¼ r2 d1 ð1 k 2 Þ þ
2r2 ðd1 þ rd2 Þ
;
1 r2
ð10Þ
2rr2 ðd1 þ rd2 Þ
:
ð11Þ
1 r2
We note that, for both cases r 6 1, the above
solutions remain valid in the case when either d1
or d2 vanishes, i.e., one of the two channels is chosen exactly at the ZDP. This special case was not
considered in previous works [7–11].
Q2 ¼ r2 d2 ð2 k 2 Þ þ
2.2. The limit case: solitons
In the case of k = 0, which corresponds to the
smallest period (7), T = p/r, the solutions take a
trivial CW form: Eqs. (3) and (4) yield A = const,
B = 0, and Eqs. (8) and (9) yield A = 0, B = const.
It is obvious that the former CW state is stable
121
(any CW solution is modulationally stable in the
single-component normal-GVD model), and latter
is not, as it has a CW state in the anomalous-GVD
subsystem.
In the opposite limit, k = 1, the period (7)
diverges, and the solutions (3) and (4) turn into
an inverted bound pair of the bright and dark solitons in the normal-GVD and anomalous-GVD
channels, respectively
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðrd2 þ d1 Þ
ð Þ
iQ1 z
sechðrtÞ;
A ¼ e r
r2 1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðd2 þ rd1 Þ
tanhðrtÞ;
ð12Þ
BðÞ ¼ eiQ2 z r
r2 1
r2 2rd2 þ ð1 þ r2 Þd1 ;
1
2r2
Q2 ¼ 2
ðd2 þ rd1 Þ:
r 1
Q1 ¼
r2
ð13Þ
In fact, it is a known solution in the form of the socalled ‘‘symbiotic soliton’’, which has bright and
dark components in the subsystems with the positive and negative GVD, despite the fact that such
solitons cannot exist in isolation [25] (uninverted
bound states, with the bright and dark solitons
in the anomalous-GVD and normal-GVD components, do not exist for r > 1). The symbiotic soliton is strongly unstable, as the finite-amplitude
background of the dark soliton in the anomalous-GVD equation is subject to the modulational
instability.
In the same limit of k ! 1, the solution (3) and
(4) corresponding to r < 1 goes over into a ‘‘normal’’ (uninverted) bound state of dark and bright
solitons in the normal- and anomalous-GVD components, respectively (dn ! sech, sn ! tanh):
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðrd2 þ d1 Þ
A ¼ r expðiQ1 zÞ
tanhðrtÞ;
1 r2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðrd1 þ d2 Þ
B ¼ r expðiQ2 zÞ
sechðrtÞ;
ð14Þ
1 r2
Q1 ¼
2r2 ðd1 þ rd2 Þ
;
1 r2
Q2 ¼ r2 d2 þ
2rr2 ðd1 þ rd2 Þ
:
1 r2
ð15Þ
It is obvious that this bound state is also unstable, although for a reason completely different
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S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
from that in the case of the symbiotic soliton
(12). Indeed, while both bright and dark uninverted solitons are stable in isolation, their interaction is repulsive (as the XPM-mediated
interaction between the two fields is attractive,
the bright soliton is effectively repelled by the
hole representing the dark soliton in the mate
channel). Obviously, a bound state of the solitons may only be unstable in the case of repulsion between them.
resulting dispersion equation for the perturbation
propagation constant X is
½X2 d1 p2 ðd1 p2 þ 2A20 Þ½X2 d2 p2 ðd2 p2 2B20 Þ
þ 4r2 d1 d2 A20 B20 p4 ¼ 0:
The stability condition is that both roots for X2
determined by Eq. (18) must be real and positive
for all real p. For the biquadratic equation in the
form of (X2)2 + b(X2) + c = 0, the latter condition
amounts to the set of inequalities
b2 4c > 0;
3. Stability of the solutions
In the previous section, it was concluded that
the exact solution (3) and (4) for r > 1 is unstable
in the limiting case of k ! 1, and stable in the limit
of k ! 0. The solution (8) and (9) for r < 1 is
unstable in both limits. In the general case,
0 < k < 1, the stability can be tested by direct
simulations of perturbed periodic-wave solutions.
Results of the test are reported in the next
subsection.
Before proceeding to that, it is relevant to study
the modulational stability of the full family of CW
solutions in the present model, as this issue has
never been considered before for XPM-coupled
equations with opposite signs of the dispersion,
and exact analytical results for this relatively simple
problem may be a clue to the understanding of
stability of the cnoidal waves in the same system.
The CW solutions with arbitrary amplitudes A0
and B0 are
Að0Þ ¼ A0 exp½iðA20 þ rB20 Þz;
ð16Þ
A perturbed solution is taken in the form
A ¼ Að0Þ ðzÞð1 þ ar þ iai Þ;
B ¼ Bð0Þ ðzÞð1 þ br þ ibi Þ;
b < 0;
c > 0:
ð19Þ
For Eq. (18), these conditions take the form,
respectively,
pffiffiffiffiffiffiffiffiffi
2 A20 d1 þ B20 d2 2A0 B0 r d1 d2 þ ðd21 d22 Þp2 > 0;
3.1. Continuous-wave states
Bð0Þ ¼ B0 exp½iðrA20 þ B20 Þz:
ð18Þ
ð17Þ
where ar,i(z,t) and br,i(z,t) are infinitesimal real and
imaginary parts of the perturbation. After the substitution of Eqs. (17) and (16) in Eqs. (1), (2) and
subsequent linearization, eigenmodes are looked
for as exp[i(pt Xz)] with arbitrary real p. The
ð20Þ
2ðd1 A20 d2 B20 Þ þ ðd21 þ d22 Þp2 > 0;
ð21Þ
4A20 B20 ðr2 1Þ þ 2ðd2 A20 d1 B20 Þp2 þ d1 d2 p4 > 0:
ð22Þ
Straightforward analysis of these inequalities demonstrates that the CW solutions with A0B0 6¼ 0
may be stable only if r P 1 and d1 P d2. If the latter conditions hold, the stability region of the CW
solutions is
A0
P max
B0
sffiffiffiffiffi
(sffiffiffiffiffi
)
pffiffiffiffiffiffiffiffiffiffiffiffiffi d2 pffiffiffiffiffiffiffiffiffiffiffiffiffi
d1 2
2
r r 1 ;
rþ r 1 :
d2
d1
ð23Þ
The purport of this result is that, if the amplitude A0 of the solution in the normal-GVD channel, which by itself is modulationally stable, is
sufficiently large as compared to the amplitude
B0 in the anomalous-GVD channel, the CW as a
whole is stable. Further, consideration of Eq.
(23) demonstrates that, for any r P 1, the minimum value of the ratio A0/B0 admitting the stability is exactly p(A
0/B0)min = 1, which is attained at
ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
d1 =d2 ¼ ðr þ r2 1Þ .
3.2. Cnoidal waves
For the numerical analysis of the stability of the
exact periodic-wave solutions, simulations of the
S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
perturbed solutions were performed by means of
the Hopscotch method, which amounts to marching forward in z using both implicit and explicit
schemes. It is known that the method itself is
definitely stable [26].
Initial conditions were taken, at z = 0, in the
form
prt
2prt
A0 ðtÞ ¼ Að0Þ ðz ¼ 0; tÞ þ 1 sin
þ 2 sin
;
KðkÞ
KðkÞ
B0 ðtÞ ¼ Bð0Þ ðz ¼ 0; tÞ;
ð24Þ
where A(0)(z,t) and B(0)(z,t) are the exact solution
(3) and (4) or (8) and (9), and 1,2 are small amplitudes of the perturbation. Recall that the period of
the unperturbed solutions is given by the expression (7), therefore the form of the initial perturbation in Eq. (24) implies that it is taken as a
combination of the fundamental and second harmonics of the underlying solution. Additional
numerical simulations demonstrate that inclusion
of higher-order harmonics in the perturbation
does not essentially alter the results. Note also
that the initial perturbation is added only to the
A-component of the solution; more extensive simulations show that a perturbation added to the Bcomponent does not lead to essentially new effects.
The results turn out to be nearly the same for the
perturbation of both solutions, (3), (4) and (8), (9),
123
therefore the conclusions are illustrated below by
plots obtained for the former one.
Extensive simulations have not revealed any
strictly stable solution, although it is difficult to
perform an exhaustive scan of the entire solution
family. However, the propagation distance after
which the instability sets in, as observed in the
direct simulations, strongly depends on the solutionÕs parameter k (i.e., on the period (7) of the
cnoidal solution): the instability of the cnoidal
waves (3) and (4) with r > 1 gets strongly suppressed with the decrease of k, making the solution
a ‘‘practically stable’’ one (see details below), and,
accordingly, rendering the system appropriate for
the applications to fiber-optic telecommunications.
This feature can be easily understood, as these exact solutions are definitely unstable if k = 1, and
stable if k = 0, see above. Note that, as the period
T decreases with k, see Eq. (7), while the bit-rate,
provided by the scheme in which the normalGVD channel is employed as the one supporting
the pulse stream in its anomalous-GVD channel,
is 1/T, the case of smaller k and, accordingly, smaller T is of major interest to the applications.
To illustrate the dependence of the instability
on the period, in Figs. 1(a) and (b) we compare
typical results of the simulations of Eqs. (1) and
(2) for the cases of k = 0.5 and k = 0.9, in the most
relevant model, with r = 2 and d1 = d2 = 1. Note
Fig. 1. Results of direct simulations of Eqs. (1) and (2) with r = 2, d1 = d2 = 1 and the initial condition (24), in which the unperturbed
solution is the one A(+), B(+) from Eqs. (3) and (4), and the perturbation amplitudes are 1 = 0.1 and 2 = 0. The other parameters are
r = 1, and k = 0.5 (a) or k = 0.9 (b). In this and subsequent figures, only the evolution of the field jA(z,t)j is displayed, as the field jB(z,t)j
demonstrates a very similar behavior.
124
S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
that the respective periods of the unperturbed
solution are 3.37 and 4.56, according to Eq. (7).
In the case of k = 0.5, the instability, after it sets
in at z . 50, first gives rise to quasi-reversible
modulations of the wave pattern, and only at
z . 150 the pattern suffers destruction, which
leaves the system in an apparently chaotic state.
On the contrary to that, in the case of k = 0.9
the instability immediately destroys the wave patterns at a much earlier stage, z . 20.
To realize the actual meaning of these critical
values of the normalized transmission distance, it
is necessary to express them in units of the dispersion length of the periodic wave, which is
Z disp ’ T 2 =ð4pd2 Þ;
ð25Þ
(recall T is the waveÕs period). Thus, in the case
shown in Fig. 1(a), the instability commences at
z 50Zdisp, while in the case of Fig. 1(b) it starts
at z 10Zdisp. In the fiber-optic telecommunication networks, the dispersion length of RZ signals
may typically be in the range of 300 km [15], which
shows that the instability-free propagation distance suggested by Fig. 1(a) is quite appropriate
for the applications.
Next, we test how the stability is affected by variation of the relative strength of the normal and
anomalous GVD in the two channels, d1/d2. Generally, the cnoidal wave pattern gets more stable
with the increase of d1/d2, as is illustrated by Fig.
2. This is natural, as the cnoidal-wave solution in
the single NLS equation may be completely stable
only in the case of the normal GVD. Accordingly,
the decrease of d1/d2 makes the pattern less stable,
although not dramatically (which suggests the
robustness of the proposed scheme for the applications), as is shown in Fig. 3.
The trend to the stabilization of the solution
with the growth of d1/d2 is strongly suggested too
by the analytical conditions (23) for the CW solutions, as they demonstrate that the stability threshold gets lower with the increase of d1/d2, as long as
it
remains
than
ðd1 =d2 Þmax ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 smaller
p
ðr r2 1Þ . In particular, (d1/d2)max 14 for
r = 2, which includes the cases of d1/d2 = 2 and
d1/d2 = 0.5, that are displayed in Figs. 2 and 3.
The onset and development of the instability
should, obviously, depend on the amplitude of
the perturbation. To illustrate that dependence,
in Fig. 4 we display the situations which are observed if the initial amplitude of the perturbation
is taken, respectively, as (a) 10%, (b) 30%, and
(c) twice that in Fig. 1(a). From here, it is observed
that the weakest initial perturbation (1 = 0.01,
Fig. 4(a)) generates no visible instability within
the propagation distance simulated, and the perturbation which is somewhat stronger (1 = 0.03,
Fig. 4(b)) gives rise only to quasi-reversible modulations, that set in at the latest stage of the evolution. The really strong perturbation (1 = 0.2, Fig.
4(c)) initiates the instability at the value of z not
much smaller than that in the case which was
shown above in Fig. 1(a), but in this case, the
instability immediately causes complete destruc-
Fig. 2. The same as in Fig. 1, but with d2 = 0.5 (the anomalous GVD twice as weak as the normal GVD).
S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
125
Fig. 3. The same as in Fig. 1, but for d2 = 2 (the anomalous GVD twice as strong as the normal GVD).
Fig. 4. The same as in Fig. 1(a), but for different amplitudes of the initial perturbation: 1 = 0.01 (a), 1 = 0.03 (b), and 1 = 0.2 (c).
tion of the pattern and transition to a ‘‘turbulent’’
state.
We also studied the effect of adding the secondharmonic component to the initial perturbation,
i.e., 2 6¼ 0 in Eq. (24). If the perturbation is weak,
for instance, 1 = 2 = 0.01 or 0.03 (cf. Figs. 4(a)
and (b)), the extra disturbance does not produce
any noticeable effect. However, in the case of a
stronger initial disturbance, such as that with
1 = 2 = 0.1 or 0.2 (cf. Figs. 1(a) and 4(c)), the
126
S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
Fig. 5. The same as in Fig. 1(a), but for the initial perturbation (24) that includes the second-harmonic component, so that 1 = 2 = 0.1
(a), 1 = 2 = 0.2 (b).
additional component in the perturbation naturally makes the transition to chaos quicker, as is
shown in Fig. 5.
The stability of the exact cnoidal-wave solutions (8), (9) corresponding to r < 1 was explored
too. An inference is that these solutions are more
unstable than their counterparts with r > 1, which
is quite natural, as all the CW solutions with r < 1
are unstable, and the limiting case of the solutions
(8), (9) corresponding to k = 0 is unstable too. A
typical example is shown, for r = 2/3 (recall this
value corresponds to the interaction between
orthogonal linear polarizations of light), in Fig.
6. Comparing with the case displayed above in
Fig. 1 for the same parameters, except that r = 2,
one concludes that, in the present case, the initial
pattern gets completely destroyed and replaced
Fig. 6. The same as in Fig. 1(a), but with r = 2/3.
by a turbulent state already by z = 40, while in
the pattern corresponding to r = 2 survived unperturbed up to z = 50, and then persisted at least up
to z = 150 in a periodically modulated state.
4. Conclusion
In this work, using the technique based on the
HirotaÕs method and elliptic theta-functions, we
have constructed three families of exact coupled
periodic (cnoidal) waves for XPM-coupled NLS
equations, with the XPM/SPM ratio r. Two families were found for r > 1, and one for r < 1. Unlike
all the previous works on this topic, we have
obtained the solutions for the case when the groupvelocity-dispersion (GVD) coefficients in the two
equations have opposite signs (including the case
when the GVD vanishes in one equation). The
model with r = 2 has direct interest to fiber-optic
telecommunications, where it pertains to a scheme
with a pulse stream in an anomalous-GVD payload channel stabilized by a strong periodic signal
in a mate normal-GVD channel. The model with
smaller values of r may be realized as a dual-core
waveguide with different carrier wavelengths
launched into the cores, an especially promising
realization being the one with two parallel cores
embedded in a photonic-crystal fiber.
To attack the stability problem, we have first
explored, in an analytical form, the modulational
stability of the two-component CW solutions. A
S.C. Tsang et al. / Optics Communications 249 (2005) 117–128
stability domain was identified, with the main
inference that the CW state may be stable if its
normal-GVD amplitude is sufficiently large in
comparison with the anomalous-GVD one. In
addition, necessary stability conditions for the
CW states are r > 1, and a demand that the absolute value of the anomalous-GVD coefficient must
be smaller than its normal-GVD counterpart.
The stability of the exact periodic solutions was
tested by means of direct simulations. It was concluded that, although all the coupled-cnoidal-wave
patterns are, strictly speaking, unstable, a parameter region with a very weak instability can be
found in the model with r = 2, making the
above-mentioned dual-channel scheme appropriate for the applications to fiber-optics telecommunications. The instability is milder when the period
of the wave pattern is smaller, and/or the anomalous GVD is weaker than the normal GVD in
the mate channel. When the instability sets in, it
may first take the form of quasi-reversible modulations, and only at a later stage will the wave pattern be completely destroyed. In agreement with
what is suggested by the analytical results for the
stability of the CW solutions, the cnoidal waves
are much more unstable for r < 1.
The analysis presented in this work calls for an
extension, as the analytical cnoidal-wave solutions
are, obviously, not the most generic periodic ones.
In particular, it was shown above that, in the limit
of k ! 0 (i.e., for the smallest period), the solutions go over into particular CW solutions with
the vanishing amplitude in either of the two
XPM-coupled modes. On the other hand, it was
also shown, in the analytical form, that general
CW solutions have two nonzero amplitudes, and
a part of such solutions is modulationally stable.
Thus, a natural issue is to find the most generic
periodic-wave solutions that would abut, in the
limit similar to k ! 0, upon the general CW states;
in particular, the periodic solutions that carry
over, in this limit, into the stable CW solutions,
may themselves have a chance to be truly stable.
However, a difficulty is that no analytical method
is currently available to construct the most generic
periodic-wave solution, therefore the consideration may only rely on quite heavy numerical
computations.
127
Acknowledgements
K.N. acknowledges the support from The Hong
Kong Polytechnic University (PolyU5242/03E).
This author is grateful to P.K.A. Wai for valuable
help. B.A.M. appreciates hospitality of the Department of Electronics Engineering at the City University of Hong Kong, and Department of Mechanical
Engineering at the University of Hong Kong.
K.W.C. acknowledges a partial financial support
from RGC Contracts HKU 7184/04E and HKU
7006/02E.
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