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October 1, 1995 / Vol. 20, No. 19 / OPTICS LETTERS
1961
Optical solitons supported by competing nonlinearities
Alexander V. Buryak and Yuri S. Kivshar
Optical Sciences Centre, Australian National University, ACT 0200, Canberra, Australia
Stefano Trillo
Fondazione Ugo Bordoni, Via B. Castiglione 59, 00142 Rome, Italy
Received May 17, 1995
It is demonstrated that optical solitons can propagate in a dispersive (or diffractive) medium with competing
quadratic [i.e., x s2d ] and cubic [i.e., x s3d ] nonlinearities. Strong interplay between the nonlinearities leads to
novel effects, in particular the following: (i) stable bright solitons can still exist in a self-defocusing (owing to
cubic nonlinearity) medium supported by quadratic parametric interactions and (ii) x s2d nonlinearity can lead
to instabilities of x s3d solitons.  1995 Optical Society of America
Recently it was demonstrated experimentally that
nonlinearity-induced phase shifts can be achieved
in x s2d materials (i.e., optical materials without
the inversion symmetry) as the result of cascaded
effects.1 This result has stimulated many efforts to
analyze a variety of x s2d nonlinear effects [previously
well understood only for x s3d materials] such as modulational instability, self-focusing or self-defocusing,
and optical solitons. As has been already demonstrated (see, e.g., Refs. 2–4), bright solitons can
exist in x s2d materials in the form of two-wave
localized modes of the strongly coupled fundamental
and second-harmonic fields. However, in x s2d materials there always exists next-order, x s3d nonlinearity, which, under certain conditions, might become
important and strongly compete with x s2d nonlinearity
(see, e.g., Refs. 5 and 6). In this Letter we discuss
how the interplay of nonlinearities inf luences the
existence and stability properties of bright (temporal
or spatial) optical solitons. We find and investigate
novel families of soliton solutions and bifurcations and
also analyze the soliton stability. Physical estimates
show that our results can be important for any x s2d
optical material, depending on the quality of phase
matching between the interacting harmonics.
Considering the interaction of the fundamental
sv1 ­ vd and second sv2 ­ 2vd harmonics in a medium
with both x s2d and x s3d nonlinearities described by the
amplitude envelopes E1 and E2 , we obtain coupled
equations in the form
i
≠E1
≠2 E1
≠E1
1 x2 E1p E2 exps2iDkzd
1 id1
1 g1
≠z
≠j
≠j 2
1 x3 sjE1 j2 1 rjE2 j2 E1 ­ 0 ,
i
≠E2
≠E2
≠2 E2
1 id2
1 g2
1 x2 E12 expsiDkzd
≠z
≠j
≠j 2
1 2x3 sjE1 j2 1 rjE1 j2 dE2 ­ 0 ,
(1)
where x2 and x3 are proportional to the elements x s2d
and x s3d of the quadratic and cubic susceptibility tensors, respectively; z is the propagation distance; Dk ;
s2k1 2 k2 d ; 2fnsvd 2 ns2vdgvyc is the wave-vector
0146-9592/95/191961-03$6.00/0
mismatch between the harmonics fnsvd and ns2vd are
the linear refractive indices of an optical medium at
v and 2v]; r is the parameter of the cross-phase
modulation; and the coeff icients gj s j ­ 1, 2d characterize either the mode dispersion, for pulse propagation
(temporal solitons), or the mode diffraction, gj ­ 1y2kj ,
for the case of self-localized beams (spatial solitons).
In the former case dj are the modal group velocities,
and in the latter case sd1 2 d2 d describes the spatial
walk-off effect.
We are interested in stationary phase-locked wave
propagation and applyp the following exact transformation: E1 ­ sky 2sx22 dw expsibz 1 iVjd
and E2 ­ skyx2 dv expsib2 z 1 2iVjd, where b
and b2 ; 2b 1 Dk are the nonlinearity-induced
shifts of the propagation constant, s ; jg1 jyjg2 j,
k ; b 1 d1 V 1 g1 V 2 , and V ; sd1 2 d2 dys4g2 2 2g1 d.
Now the normalized equations for w and v take the
form
∂
µ
2
≠2 w
≠w
p 1 x jwj 1 rjvj2 w ­ 0 ,
1r
2
w
1
vw
≠z
≠t 2
2s
∂
µ
2
2
≠v
w
≠ v
is
1 s 2 2 ay 1
1 x 2sjvj2 1 rjwj2 v ­ 0 ,
≠z
≠t
2
i
(2)
where z ­ kz, t ­ sjkjyjg1 jd1/2 sj 2 nzd, n ­ s2g2 d1 2
g1 d2 dys2g2 2 g1 d, r ­ signskg1 d, s ­ signskg2 d, a ­
sb2 1 2d2 V 1 4g2 V 2 dsyk, and x ­ kx3 yx22 .
System (2) is rather complicated for analyzing all
possible localized solutions. Nevertheless, we recently
showed6 the existence of an exact analytical solution of
these equations in the case of some special relations
between the system parametersp (a ­ 1, v and w are
real, and the constraints w ­ v 2 and r ­ s 21 2 2s
are fulf illed). Stable propagation of this bright twowave soliton supported by both nonlinearities has been
shown in Ref. 6. The situation resembles the case of
pure x s2d solitons for which a similar exact solution is
known (see, e.g., Ref. 2).
To demonstrate the existence of more general
types of bright soliton in Eqs. (2) and analyze their
 1995 Optical Society of America
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OPTICS LETTERS / Vol. 20, No. 19 / October 1, 1995
properties and stability, we consider the case s ­ 2
and r ­ s ­ 11, which corresponds to spatial bright
solitons. For definiteness, we select the cross-phase
modulation coeff icient to be r ­ 2 and also a ­ 2. We
expect that the general features of the stationary solutions of Eqs. (2) and their stability properties will be
similar for other values of a, at least when a is not too
small. [For small a the problem is more complicated;
e.g., it is possible to show that, for a , acr ø 0.211
at s ­ 2, two-wave solitons are unstable in the case
of a pure x s2d nonlinearity. See Ref. 7.] We should
emphasize that the general case of Eqs. (2) cannot
be understood in all detail before the analysis of two
limiting cases, namely, solely x s2d or x s3d nonlinearities,
is completed.
To find the stationary solutions of Eqs. (2) for the
particular values of the parameters mentioned above
we omit the derivatives in z and therefore reduce the
problem to the analysis of localized solutions of the
system
µ
∂
1
d2 w
2
2
2
w
1
wv
1
x
1
2jvj
w ­ 0,
jwj
dt 2
4
∂
µ
d2 v
1 2
2
2
v ­ 0,
w
2
2v
1
1
x
4
jvj
1
2jwj
dt 2
2
(3)
which can be treated as equations of motion for
an effective particle in a two-dimensional potential
parameterized by x. Separatrix trajectories of system (3) correspond to soliton solutions of Eqs. (2).
The main types of localized solution that describe
bright solitons of the lowest orders (i.e., solitons with
the lowest energy) are shown in Fig. 1 by the change of
the scaled (by the factor jxj) soliton energy
P ­ jxj
Z
Note that these nontrivial two-wave solitons exist for
x ­ 0 [the case of pure x s2d solitons3] and even for
negative values of x larger than xthr ø 20.0616 [i.e.,
in the region of defocusing cubic nonlinearity where
the standard x s3d solitons do not exist], and Fig. 2(c)
presents an example of the latter solitons. The soliton
shown in Fig. 2(c) can be interpreted as composed of
a pair of kinks with the separation between the kinks
increasing as x ! xthr . For x , xthr self-defocusing
owing to x s3d nonlinearity dominates, and this can
allow stable dark solitons to exist in this region.
Additionally to solitons of the V and C types,
system (3) has other types of soliton that, in some
Fig. 1. Bifurcation diagram of the localized solutions of
Eqs. (3). Stable solitons are represented by solid curves,
and unstable ones are shown by dashed curves. Note the
different scales for positive and negative values of x.
1`
2`
sjwj2 1 4jvj2 ddt
(4)
with the parameter x that characterizes a balance of contributions that are due to x s2d and x s3d
nonlinearities.
To understand the physical origin of these localized solutions we should discuss them in more detail.
First, we note that Eqs. (2) [and (3)] always have the
simple solutions with w ­ 0. For this type of solution
solitons p
of the
the x s3d nonlinearity itself can support
p
ayssxd sechs a td.
second
harmonic: vs std ­
These p one-wave
solitons
have
the
energy
PV ­ 4 2 saysd, and we denote them in Fig. 1 as solitons of the V type. The analysis indicates that there
are many bifurcations from the V -type solitons, but the
most important one, which gives birth to the branch
with the lowest value of energy [Eq. (4)], starts at
x ­ xcr ø 8.76, where a nontrivial two-wave soliton appears (bifurcation point B in Fig. 1). These nontrivial
solitons exist because of strong interaction between
the harmonics, and we call them the solitons of the
combined, or C, type. For small values of x, when
x s2d nonlinearity dominates, these solitons look similar
to those of a pure x s2d medium. In Fig. 2 we show
the characteristic prof iles of these C-type solitons at
various points of the bifurcation curve shown in Fig. 1.
Fig. 2. Characteristic profiles of the two-wave solitons of
the C type at (a) x ­ 0.2 (point L in Fig. 1), (b) x ­ 8.0
(point M in Fig. 1), (c) x ­ 20.0615 (not shown in Fig. 1).
October 1, 1995 / Vol. 20, No. 19 / OPTICS LETTERS
Fig. 3. Example of the two-wave solitons of the W type at
x ­ 8.0 (point N in Fig. 1).
limits, are close to the one-wave solitons of a cubic
medium. An example of such solitons is shown by the
two-wave solitons of the W type (see Fig. 1). The W
solitons are supported mainly by the x s3d nonlinearity
acting on the fundamental harmonic. This can be
seen in the limit of large x when the
p W solitons
are described approximately as ws ­ 2 syx sech t 1
Osx 21 d and vs ­ Osx 21 d, so that limx!` PW ­ 8s.
However, because of parametric coupling, the second
harmonic is also generated, and it has a two-hump
prof ile as shown in Fig. 3.
Note that Fig. 1 does not display all possible soliton
families and bifurcations that exist in this model.
We show only three of the most physically important
soliton families of low energy P . There are many
other localized solutions that describe bound states
of single solitons and more complicated higher-order
solitons. All these solitons are likely always to be
unstable.
We also carried out a numerical analysis to investigate the stability of the solitons described above. This
analysis shows that the W -type solitons are unstable
for any value of the parameter x of Fig. 1. The solitons of the V type are stable for x $ xcr but unstable for
0 , x , xcr . This demonstrates how the inf luence of
parametric x s2d interactions leads to instabilities of the
solitons that are supported mainly by x s3d nonlinearity. On the other hand, the solitons of the C type are
stable in all domain of the parameters where they exist (i.e., for xthr , x , xcr ). We should note, however,
that a rigorous analysis of the soliton stability is still
an open problem. [For the case of a pure x s2d medium
it was recently shown that the stability of bright and
dark solitons depends strongly on the values of the
parameters s and a (Refs. 7 and 8).]
Finally, we make some physical estimates. The key
parameter of our analysis is x , skcyvdx s3d yfx s2d g2 ,
which depends on two dimensionless (for the esu
system) factors, m ; x s3d yfx s2d g2 , determined by the
type of material, and h ; skcyvd ­ lys2pzc d, a ratio of
1963
the fundamental harmonic wavelength in vacuum and
the characteristic propagation scale. Typical values
of x s2d and x s3d nonlinearities in crystals without
the inversion symmetry are x s2d , 1027 esu units
(10222 AsyV2 in mks units) and x s3d , 10212 esu units
(10232 AsmyV2 in mks units). This gives the estimate
m , 100. The value of h determines the importance
of the competition between the nonlinearities. For
the case of fixed a considered in this Letter k , Dk,
which gives h , Dn ; jns2vd 2 nsvdj. As follows from
our results, strong competition between x s2d and x s3d
nonlinearities is expected for jxj . 1022 , which (for
materials with m , 100) gives Dn . 1024 . In the
opposite case, i.e., for jxj # 1022 , the inf luence of x s3d
nonlinearity on solitons can be neglected, and one can
readily use the standard model for x s2d solitons (see,
e.g., Ref. 3).
In conclusion, we have shown that stable spatial (or
temporal) optical solitons can exist in a diffractive (or
dispersive) medium supported by competing quadratic
and cubic nonlinearities. We have found three main
classes of two-wave bright soliton and have demonstrated that competition between the nonlinearities
leads to novel effects. In particular, on the one hand,
stable bright solitons can exist because of x s2d nonlinearity in the region where the x s3d nonlinearity is selfdefocusing and, on the other hand, the x s2d nonlinearity
can make certain types of x s3d soliton unstable. We
have also discussed the applicability limits of the pure
x s2d parametric interaction model.
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