OPTICS LETTERS / Vol. 23, No. 9 / May 1, 1998
Stabilization of dark and vortex parametric spatial solitons
Tristram J. Alexander
Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University,
Canberra, ACT 0200, Australia
Alexander V. Buryak
School of Mathematics and Statistics, Australian Defence Force Academy, Canberra, ACT 2600, Australia, and Optical Sciences Centre,
Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia
Yuri S. Kivshar
Australian Photonics Cooperative Research Centre, Research School of Physical Sciences and Engineering, Optical Sciences Centre,
Australian National University, Canberra, ACT 0200, Australia
Received December 22, 1997
We demonstrate that a weak defocusing Kerr effect in an optical medium with predominantly quadratic [or
x?2?] nonlinear response can eliminate the parametric modulational instability of plane waves, leading to the
existence of stable two-wave dark and vortex spatial solitons.
190.7070, 190.5530, 190.3270.
1998 Optical Society of America
Dark spatial optical solitons, or self-trapped station-
ary dark notches existing on an infinite plane-wave
background or embedded in wide diffracting beams,
have been investigated for a number of years (see, e.g.,
the recent review in Ref. 1).
is due to potential applications of spatial solitons in
photonics as steerable and reconfigurable self-induced
waveguides.Up to now, the experiments were limited
by materials with relatively weak nonlinear response,
and therefore successful demonstrations of dark soli-
tons and the waveguides that they induce was achieved
for just a few optical materials.2,3
in recent years remarkable progress was made in
the field of so-called cascaded nonlinearities of opti-
cal materials with a quadratic [or x?2?] nonlinear re-
In particular, it was shown that both bright
and dark parametric solitary waves can exist as mutu-
ally trapped two-color beams of the fundamental har-
monic (FH) and the second harmonic (SH).5
straightforward extension of the concept of spatial
dark solitons to the case of quadratic optical materials
fails, because all finite-amplitude plane waves (PW’s)
in a pure x?2?medium suffer from parametric modu-
As a result, no stable spatial
dark solitons have been found to exist in pure qua-
In this Letter we suggest a simple physical mecha-
nism leading to stable dark solitons in optical me-
dia with quadratic nonlinearities.
the stabilizing action of the defocusing cubic (or Kerr)
nonlinearity, which can eliminate parametric modula-
tional instability in two-wave mixing.
the instability, in turn, allows the existence of dark
and vortex spatial solitons stabilized by the presence
of a weak cubic nonlinearity in an optical medium with
a predominantly quadratic nonlinear response.
cubic nonlinearity may be inherent, i.e., due to the
next-order nonlinear contribution of a noncentrosym-
metric optical material,8or induced, e.g., generated by
an incoherent coupling of FH and SH beams with other
Interest in this research
On the other hand,
First, we reveal
modes or higher-order spatial frequencies in quasi-
Physical equations that describe the interaction of
the FH ?v1? v? and the SH ?v2? 2v? in a medium
with both x?2?and x?3?nonlinearities are well known
and can be found in earlier papers.8
waveguide geometry, we measure the transverse coor-
dinate in the units of the beam size R0and the propa-
gation coordinate in the units of the diffraction length
Rd ? 2R02k1, where k1? k1?v? is the wave number
of the FH. Then, making the scaling transformations
similar to those in Ref. 8, we obtain the system of two
coupled equations for the normalized envelopes of the
FH ?u? and the SH ?w? in the form
Assuming a slab
≠z1 r=?2u 2 u 1 wu?1 x
u ? 0,
≠z1 r=?2w 2 aw 11
w ? 0,
where r ? sign?b?, a ? ?4b 1 2DkRd??b, b is the
nonlinear contribution to the propagation constant
of the fundamental wave, Dk ? ?2k1 2 k2? is the
wave-vector mismatch between the harmonics, r is
the cross-phase modulation coefficient, the coefficients
x?2?and x?3?are proportional to the relevant ele-
ments of the corresponding nonlinear susceptibility
tensors, and, for the ?1 1 1?-dimensional geometry
case, we should take =?2? ≠2?≠x2.
ter x ? 3bc2x?3???16pv12R02?x?2??2? characterizes the
relative contribution of the cubic nonlinearity, and its
sign is defined by that of the product bx?3?.
we consider both r ? 11 and r ? 21, so Eqs. (1) have
only two continuous parameters, a and x, except for r.
We keep r unfixed to present the exact soliton solution
1998 Optical Society of America
May 1, 1998 / Vol. 23, No. 9 / OPTICS LETTERS
[Eqs. (3), below] but set it as r ? 2 in the rest of the
We can find solutions of Eqs. (1) for PW’s by solving
the algebraic equations 12xwp31 12wp21 wp?a 2 8 1
2?x? 2 2?x ? 0 and up2? 4?1 2 wp??x 2 8wp2for real
upand wp.There exist as many as three branches of
PW’s for which both amplitudes upand wpare nonzero.
Performing standard parametric modulational insta-
bility analysis (see, e.g., Refs. 6 and 8) for each of the
three branches of PW solutions at r ? 61, we find that
there exists only one branch of modulationally stable
PW solutions.The parameter domains in which such
a solution exists are presented in Fig. 1.
sign?rx? ? signx?3?, and thus the modulationally stable
PW’s exist only for x?3?, 0.
Importantly, the amplitudes of the modulationally
stable PW’s diverge in the limit x ! 0, as is clearly
seen in Fig. 2, so the stable branches of the PW’s exist
exclusively because of the mutual action of quadratic
and cubic nonlinearities.Other PW’s are modulation-
ally unstable in the whole domain of their existence,
and they are not presented in Figs. 1 and 2.
tional stability of PW’s in the limit of large negative x?3?
[e.g., for r ? 21 and x . 0; see Figs. 1(b) and 2(b)] is
not surprising because stable dark solitons are known
to exist in a defocusing Kerr medium.
interested in the case in which the effective nonlinear-
ity is predominantly quadratic, i.e., jxupj ? jxwpj , ,1.
We found that this condition can be satisfied only for
r ? 11, where modulationally stable PW’s of moderate
amplitudes exist for relatively small values of x [see
Figs. 1(a) and 2(a)].
Modulational stability of PW’s is promising for the
existence of stable dark solitons.
solitons are described by the system of two second-
order ordinary differential equations for the real func-
tions u?x? and w?x?,
Here we are
For our model dark
dx22 u 1 wu 1 x
u ? 0,
dx22 av 1
with nonvanishing boundary conditions.
these stationary solutions numerically, applying the
relaxation technique.As a result, we find that dark
solitons exist in the whole domain of the modulation-
ally stable PW’s shown in Fig. 1, and characteristic ex-
amples of such solitons are presented in Fig. 3.
It is interesting to notice that at a ? 1??r2x? Eqs. (1)
have the exact solution
2u21 x?4w21 2u2?w ? 0,
w ? 21??2rx?,
which exists for x . 0 ?r ? 21? or 21??4r? , x , 0 ?r ?
11?.For r ? 11 and r ? 2 and close enough to the
value x ? 21?8, we have jxupj ? jxwpj ? 0.2.
As is shown in Fig. 4, on the parameter plane ?x, a?
the exact solution (3) separates two types of dark soli-
solitons with a dip-shaped SH; see Figs. 3(c) and 3(d)]
and the other for which the SH component has a maxi-
mum [i.e., dark solitons with a hump-shaped SH; see
In spite of the fact that the dark solitons that
we found here exist on a modulationally stable PW
background, they can become inherently unstable,
displaying so-called drift instability.10
tain the stability criterion for dark solitons described
by Eqs. (1) by generalizing the formalism of Ref. 10,
which, however, requires knowledge of more-general
gray solitons, i.e., dark solitons propagating under a
one for which the SH has a minimum [i.e., dark
We can ob-
Existence domains for the modulationally stable
(a) r ? 11, (b) r ? 21.
versus x at a ? 22.0.
Amplitudes of the modulationally stable PW’s
(a), (b), Figs. 1(a) and 1(b),
The limit x ! 0 is singular.
x ? 20.1, (c) a ? 21.0 and x ? 20.1, (d) a ? 4.0 and
x ? 20.05.
Examples of dark solitons of Eqs. (1) at r ?
(a) a ? 22.5 and x ? 20.1, (b) a ? 24.0 and
OPTICS LETTERS / Vol. 23, No. 9 / May 1, 1998
r ? 11.
solution [Eq. (3)].
dark solitons presented in Figs. 3(a), 3(b), 3(c), and 3(d),
Existence domains for dark solitons of Eqs. (1) at
The curve x ? 1??4a? is defined by the exact
Points A, B, C, and D correspond to the
x ? 20.1, a ? 22.5, and r ? 11.
for the FH. (b) Radial dependence for the FH and
(a) Intensity profile
certain angle to the background wave.
of gray soliton families is still an open problem even
for the case of pure x?2?nonlinearity.
confirmed the soliton stability directly by modeling
the propagation of dark solitons numerically for suf-
ficiently large distances.
The analysis presented above can be readily ex-
tended to the case of higher dimensions, i.e., for dark
solitons of circular symmetry in a bulk (vortex soli-
tons) associated with a phase singularity embedded
in a background wave (see, e.g., Ref. 1).
structure of the vortex solitons, we look for radially
symmetric solutions of the ?2 1 1?-dimensional version
of Eqs. (1) with =?2? ≠2?≠x21 ≠2?≠y2, in the form
u?R, w? ? U?R?exp?imw?, w?R, w? ? W?R?exp?2imw?,
where R ? ?x21 y2?1/2, w ? tan21?x?y?, and m is
an integer number (the vortex charge).
the lowest possible charge ?m ? 61? corresponds to a
2p phase twist of the FH and a 4p phase twist of
the SH. Solutions of the corresponding equations for
the amplitudes U?R? and W?R? satisfy the conditions
U?0? ? W?0? ? 0 and dU?dR ? dW?dR ? 0 at R ! `.
Figure 5 shows an example of a stable two-wave vortex
soliton of the first order, jmj ? 1.
for the parameters x and a shown by point A in Fig. 4.
We expect that stable dark and vortex solitons in
quadratic media can be observed in typical upconver-
sion experiments when a high-intensity beam under-
goes frequency doubling simultaneously with creation
of a phase singularity produced by a phase mask at
the input, similar to recent experiments performed at
At higher input powers, the ap-
To find the
A vortex of
The soliton exists
propriately chosen nonzero phase matching will lead to
the formation of a dark (or vortex) soliton stabilized by
the effect of incoherent coupling between the harmon-
ics owing to the cubic nonlinearity.
note that optical materials used in these experiments
should demonstrate the defocusing Kerr effect for both
harmonics. The most-likely candidates for such ma-
terials are poled optical polymers (see, e.g., Ref. 12),
which often have negative x?3?in a large range of op-
tical frequencies.The minimum light intensity that
is required for generation of a stable dark soliton in a
predominantly quadratic medium can be estimated as
Imin? ?x?2??x?3??23 109, where x?2?and x?3?should be
inserted in picometers per volt and picometers squared
per volt squared, respectively, whereas Iminis given in
megawatts per square centimeter.
In conclusion, we have demonstrated the existence
of modulationally stable plane waves for the process of
two-wave mixing in a parametric medium with both
a quadratic and a cubic nonlinear response and found
the families of dark and vortex spatial solitons with
a nonvanishing background field.
that these dark solitons can exist in the region of
parameters that correspond to a predominantly x?2?
medium and are therefore stabilized by a weak cubic
nonlinearity. This result gives us what we believe
to be the first example of stable spatial dark solitons
supported by quadratic nonlinearities.
It is important to
We have shown
The authors thank Ole Bang and Stefano Trillo
for useful discussions.A. V. Buryak acknowledges
support from the Australian Research Council.
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