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P H Y S I C A L R E V I E W L E T T E R S

**Eliminating the Transverse Instabilities of Kerr Solitons**
Charalambos Anastassiou,1,2 Marin Soljaˇcic´,1,3 Mordechai Segev,1,2 Eugenia D. Eugenieva,4
Demetrios N. Christodoulides,4 Detlef Kip,1,5 Ziad H. Musslimani,6 and Juan P. Torres1,7
1

*Physics Department and the Solid State Institute, Technion, Haifa 32000, Israel*
2

*Electrical Engineering Department, Princeton University, Princeton, New Jersey 08544*
3

*Physics Department, Princeton University, Princeton, New Jersey 08544*
4

*Electrical Engineering and Computer Science Department, Lehigh University, Bethlehem, Pennsylvania 18015*
5

*Physics Department, Universität Osnabrück, 49069 Osnabrück, Germany*
6

*Mathematics Department, Technion, Haifa 32000, Israel*
7

*Department of Signal Theory and Communications, Polytechnic University of Catalonia, Jordi Girona 1-3,*
*Campus Nord D3, 08034 Barcelona, Spain*
We show analytically, numerically, and experimentally that a transversely stable one-dimensional
͓͑1 1 1͒

*D*͔ bright Kerr soliton can exist in a 3D bulk medium. The transverse instability of the solitonis completely eliminated if it is made sufficiently incoherent along the transverse dimension. We derivea criterion for the threshold of transverse instability that links the nonlinearity to the largest transversecorrelation distance for which the 1D soliton is stable.

Research on optical spatial solitons has made much
Thus far, in order to avoid TI, experiments with 1D
progress during the past decade: new systems that support
solitons were conducted in either planar waveguides [5,6]
solitons have been identified, solitons of more than one
or nonlinearities in which TI was greatly suppressed [8].

transverse dimension have been demonstrated, and a whole
Here we demonstrate how to produce a truly stable stripe
range of soliton interactions was explored [1]. Despite
Kerr soliton propagating in a 3D bulk medium without
the diversity of the physical systems that support them,
suffering from transverse instability. We show that if the
solitons are a universal phenomenon and share many com-
soliton is made “sufficiently” incoherent in its transverse
mon features [1], one of which is transverse instability (TI)
dimension

*y*, then TI is completely eliminated.

[2 – 10]. TI is a symmetry breaking instability: almost all
First, recall incoherent solitons made of partially inco-
solitons [11] of a particular dimension that propagate in a
herent light [12]. They are multimode (speckled) beams of
higher dimension system [by having a uniform wave func-
which the instantaneous amplitude varies randomly with
tion in the additional dimension(s)] are unstable to pertur-
time. If such beams are launched into a noninstantaneous
bations in the dimension(s) in which they are uniform. TI
self-focusing medium, so that the response time of the non-
occurs because perturbations in the dimension of unifor-
linearity greatly exceeds the fluctuation time, then self-
mity have nothing to restrain them from growing (driven
focusing is driven solely by the average intensity. Then,
by the nonlinearity) and breaking the soliton up.

the incoherent beam induces a multimode waveguide and
In the particular case of a spatial optical ͑1 1 1͒

*D *soli-
guides itself in it by properly populating the guided modes,
ton that is self-trapped in one dimension

*x*, is uniform
thus forming an incoherent soliton [12 – 18].

in the transverse dimension

*y*, and is propagating along
A clue that TI could be completely eliminated for soli-

*z*, TI causes the soliton to break up along

*y *into an ar-
tons was given by two recent discoveries: modulation
ray of 2D filaments [2 – 10]. The transverse wavelength
instability (MI) of incoherent light [19] and elliptical in-
of these perturbations is usually much larger than the soli-
coherent solitons [14,20]. MI belongs to the same family
ton width [2 – 4]. Transverse instability is especially severe
of symmetry breaking instabilities as TI does, and it oc-
for Kerr nonlinearities and prohibits spatial 1D Kerr soli-
curs when a plane wave (or a very broad beam or pulse) is
tons in a bulk medium. This is why spatial ͑1 1 1͒

*D *Kerr
launched into a self-focusing medium. If this plane wave
solitons have to be launched in a planar waveguide con-
is fully coherent, it breaks up into a train of filaments due
figuration, in which the

*y *confinement is much narrower
to MI. Recently, it has been shown theoretically and ex-
than the self-trapped (soliton) width in

*x *[5,6]. TI actually
perimentally [20] that MI does exist also for incoherent
occurs for solitons in any nonlinearity, including, for ex-
light, but it occurs only if the nonlinearity exceeds a well-
ample, quadratic solitons [7] and photorefractive solitons
[8,9]. Interestingly, saturation arrests transverse instabil-
the coherence of the light. If the nonlinearity is below
ity [10] but never completely eliminates it. In fact, it is
threshold, then MI is eliminated and the wave is stable.

the suppression of TI due to saturation that facilitates the
This generic idea has enabled the observation of anti-
observation of stable 1D solitons in a bulk photorefractive
dark solitons [21], which were thought to be unstable in
crystal for more than ten diffraction lengths [8].

conservative nonlinear systems [22]. The new finding of
P H Y S I C A L R E V I E W L E T T E R S
incoherent elliptical solitons [14,20] is based on having
tion,

*E*͑

*x*,

*y*,

*z*,

*t*͒ being the slowly varying amplitude. The
different coherence function for the two transverse dimen-
͗ ͘ denote averaging over the response time of the medium
sions of self-trapping. Combing these ideas, one can gen-
t, which is much larger than tcoh. From the paraxial wave
erate a 1D soliton that is fully coherent in

*x *(direction of
trapping), partially incoherent but uniform in

*y*, and propa-
gating along

*z*. The intimate relation between TI and MI

*ik *͕d

*n*͑

*x*1,

*y*1,

*z*͒
suggests that TI of incoherent beams should also exhibit
a threshold for its existence. Therefore, if the degree of
2 d

*n*͑

*x*2,

*y*2,

*z*͖͒

*B *, (1)
coherence in

*y *is such that TI is below the threshold, alltransverse perturbations are suppressed and TI is elimi-
where

*z *is the propagation direction,

*k *is the carrier wave
nated. This is the core idea of our Letter. The idea of
number,

*n*0 is the bias refractive index, d

*n *is the nonlin-
using the threshold to eliminate TI applies to any type of
ear contribution to the refractive index,

*x * ͑

*x*1 1

*x*2͒͞2,
nonlinearity, yet we will concentrate on the Kerr nonlinear-

*y * ͑

*y*1 1

*y*2͒͞2 are the middle point coordinates, and
ity for two reasons. First, wave propagation in Kerr media
r

*x *

*x*1 2

*x*2, r

*y *

*y*1 2

*y*2 are the difference coordi-
is described by the cubic nonlinear Schrödinger equation
nates. When r

*x * r

*y * 0,

*B*͑

*x*,

*y*, r

*x*, r

*y*,

*z*͒ is the time-
(NLSE) which is one of the most general soliton equations
averaged intensity

*I*͑

*x*,

*y*,

*z*͒. Let

*BS*͑

*x*,

*y*, r

*x*, r

*y*͒

*u*͑

*x *1
[23]. Generally speaking, the NLSE describes envelope
r

*x*͞2͒

*u*ء͑

*x *2 r

*x*͞2͒

*An*͑r

*y*͒ be a

*z*-independent solution
solitons in dispersive wave systems with weak symmetric
It represents a 1D soliton stripe, which is
anharmonicity. Second, the effect of TI for Kerr solitons
self-trapped and fully coherent in

*x*, while being uni-
is very strong and we can demonstrate a convincing dif-
form and incoherent along

*y *with an angular spectrum of
ference between having TI and eliminating TI by making

*An*͑r

*y*͒.

*u*͑

*x*͒ is determined by the nonlinearity and can be
the soliton incoherent along

*y*.

taken to be real without loss of generality. To study TI,
An incoherent beam can be represented as a series
we add a small perturbation

*B*1 to

*BS *where

*B*1 ø

*BS*.

of coherent speckles that change, on average, every
The nonlinear index change in Kerr media is d

*n*͑

*I*͒ g

*I*,
We define

*B*͑

*x*1,

*y*1

*x*2,

*y*2,

*z*͒
where g is the nonlinear coefficient ͑

*n*2͒. Linearizing
͗

*E*ء͑

*x*2,

*y*2,

*z*,

*t*͒

*E*͑

*x*1,

*y*1,

*z*,

*t*͒͘, the spatial correlation func-
, r

*x * 0, r

*y * 0,

*z*͒ 2

*B*1͑

*x *2
, r

*x * 0, r

*y * 0,

*z*͔͒

*BS*͑

*x*,

*y*, r

*x*, r

*y*͒

*x * 0, r

*y * 0͒ 2

*BS *͑

*x *2 2

*x * 0, r

*y * 0͔͒

*B*1͑

*x*,

*y*, r

*x *, r

*y *,

*z*͒
We seek solutions in the form

*B*1͑

*x*,

*y*, r

*x*, r

*y*,

*z*͒ exp͑

*gz*͒ exp͑

*i*a

*y*͒

*L*͑

*x*, r

*x*͒

*Af*͑r

*y*͒ 1 exp͑

*g*ء

*z*͒ exp͑2

*i*a

*y*͒ 3

*L*ء͑

*x*, 2r

*x*͒

*A*ء

*f*͑2r

*y*͒, where a is the transverse wave number,

*g *is the TI growth rate (gain), and

*Af*͑r

*y*͒ is the angularspectrum. The necessary condition

*B*1͑

*x*,

*y*, r

*x*, r

*y*,

*z*͒

*B*ء1͑

*x*,

*y*, 2r

*x*, 2r

*y*,

*z*͒ is satisfied [17,19]. Substituting

*B*1 intoEq. (2) gives

*gL*͑

*x*, r

*x*͒

*Af*͑r

*y*͒ 2
͓

*L*͑

*x *1

*x *, 0͒

*A*
*x *͒

*u*͑

*x *2

*x *͒

*A*
*f *͑0͒

*ei*ar

*y*͞2 2

*L*͑

*x *2 2

*f *͑0͒

*e*2

*i*ar

*y*͞2͔

*u*͑

*x *1 2
1 ͓

*u*͑

*x *1 r

*x*͞2͒2 2

*u*͑

*x *2 r

*x*͞2͒2͔

*L*͑

*x*, r

*x*͒

*Af*͑r

*y*͒

*An*͑0͒
We are interested in determining the threshold condi-
the “cutoff wavelength” [2 – 4]. This means that for a co-
tion: to find the conditions under which the growth rate
herent soliton, the growth rate is positive (and TI exists)

*g*͑a͒ goes from a positive value to a negative value for all
for a band of wave numbers a between zero and the cutoff
a. The procedure of determining the threshold applies to
wave number. For a soliton that is partially coherent in

*y*,
any form of spatial coherence (angular power spectrum),
we expect that for u0 small enough (a beam that is coherent
but for simplicity, we consider an initial Gaussian angu-
enough),

*g*͑a͒ will be positive in a band of wave numbers,
lar power spectrum

*An*͑r

*y*͒ exp͓2͑r

*y*u0

*k*͞2͒2͔, where
just as the coherent case. But, as u0 increases, this band
u0 defines the degree of coherence (correlation distance).

becomes narrower until it completely disappears at some
The higher u0 the more incoherent the soliton is. For a
value u0

*T *. If u0 is larger than this value, then TI is elimi-
fully coherent soliton, if we were to calculate the growth
nated. We therefore expect that, at the threshold u0 u0

*T *,
rate

*g *as a function of transverse wave number a, then

*g*
the two boundary points at which

*g*͑a͒ 0 (one at a 0
starts from 0 (at a 0), increases and reaches a maxi-
and the other at the cutoff wave number) coincide. Thus,
mum positive value (at the wave number with the largest
we seek the value of u0 at which (i)

*g*͑a 0͒ 0 and
growth rate), and then drops back to 0 at a associated with
(ii)

*g*0͑a 0͒ 0. We solve Eq. (3) by expansion while

P H Y S I C A L R E V I E W L E T T E R S
retaining up to O͑a͒. This eliminates the first term on theleft-hand side (LHS) because

*g * O͑a2͒. The growth rateof the transverse instabilities,

*g*, is independent of

*x *[2 –4]even though the actual shape of the perturbations dependson

*x*. We can therefore seek solutions of Eq. (3) under theconditions (i) and (ii) at the center of the soliton, i.e., at

*x * r

*x * 0, and assume that the threshold we find is thesame everywhere on the soliton. It can be easily shown(by expanding into derivatives with respect to

*x*1 and

*x*2)that the second term of the LHS is zero for

*x*1

*x*2, i.e.,for r

*x * 0. Thus, from Eq. (3) we get
Simulations of a 1D Kerr soliton with a Gaussian spec-
3 exp͓2͑

*k*u0

*T *r

*y*͞2͒2͔ , (4)
trum, for various degrees of coherence. The predicted thresh-old is u0 0.549±. (a) Input intensity. ( b) Output beam after
where

*I*0

*u*͑0͒2 is the peak intensity of the soliton. It is
0.8 cm of propagation for a fully coherent beam u0 0±; the
unlikely that a small perturbation will alter the coherence
soliton is destroyed by TI. (c) Output beam close to the threshold
statistics of the soliton (especially here that propagation
(for u0 0.5±), after 4.5 cm of propagation: As the thresholdis approached, TI gain is reduced and it takes a longer propa-
effects, given by

*g*, are of the order of a2 and are ignored).

gation before TI is evident. (d ) Output beam for u0 0.56±,
Thus, we assume that

*Af *͑r

*y*͒

*An*͑r

*y*͒. Equation (4)
which is below the threshold, after 4.5 cm of propagation. TI is
D

*n*0 g

*I*0 is the maximum change in the refractive in-dex. One can actually calculate, using numerical methods
diffuser. Then, by moving the focal point of this lens closer
similar to [10], the function

*g*͑a͒ and from it obtain the
(farther away) from the diffuser, we increase (decrease)
threshold for any angular distribution function [24].

the coherence in

*y*. The

*x *coherence is not affected by
To verify the analytic predictions, we perform simula-
the translation of this lens. After the diffuser, the beam
tions using the coherent density approach [12]. We launch
is collimated (to ϳ2 cm) and passed through a narrow
a 1D Kerr soliton with a Gaussian angular power spec-
(along

*x*) slit. The slit is made narrower than the speckle
trum,

*A*͑ r

*y*͒ exp͓2͑r

*y*u0

*k*͞2͒2͔, for various values of
size in

*x*, and it effectively creates a 1D beam that is
u0. In this example,

*n*0 2.3, l 0.5 mm in vacuum,
narrow and coherent in

*x *and “infinitely” long (uniform)
FWHM 9 mm, which yields a D

*n*0 0.000 105 6 and
and incoherent in

*y*. The slit is then imaged to the input
an analytic prediction of the threshold of u0

*T * 0.55±. Our
face of the crystal. We get a reasonable estimate of the
results are displayed in Fig. 1, where we show images of
correlation distance by stopping the diffuser and measur-
the intensity distribution of the soliton and cross sections
ing the average speckle size at the crystal input plane.

of the intensity along

*y *for

*x * 0. Figure 1(a) shows the
Finally, we use an orthogonally polarized background
input soliton at

*z * 0, and Fig. 1(b) shows a fully coherent
beam that covers the crystal uniformly as necessary for
soliton ͑u0 0±͒ after 0.8 cm of propagation. As clearly
photorefractive screening solitons [8].

depicted there, TI breaks the soliton up into a train of 2D
output faces of the crystal are imaged onto a CCD camera.

filaments. As we approach the threshold, the TI gain is
The photorefractive nonlinearity is in general saturable
getting smaller: As we set u0 to 0.5±, it takes a 4.5 cm
but resembles the Kerr nonlinearity when the peak inten-
propagation to exhibit signs of TI [Fig. 1(c)]. To show that
sity of the soliton normalized to the background intensity
TI is completely eliminated when the nonlinearity is below
is much smaller than unity [8,25]. In our case, this ratio
threshold, we increase u0 to 0.56±. As shown in Fig. 1(d),
is ϳ0.1. At this normalized intensity, a soliton that is
after 4.5 cm of propagation there are absolutely no signs
fully coherent in both

*x *and

*y *exhibits strong transverse
instability [8]. We then gradually increase the incoherence
Our experiments are conducted in a photorefractive
in

*y *(decrease the speckle size) until the soliton becomes
SBN:75 crystal in a setup similar to that of [12]. The beam
transversely stable, while keeping all other parameters
is made spatially incoherent by passing it through a rotat-
(applied field, intensity) constant. Our results are shown in
ing diffuser. The rotating diffuser provides a new phase
Fig. 2. The 12 mm FWHM input beam [Fig. 2(a)] linearly
and amplitude distribution every tcoh ϳ 1 ms, which
diffracts to a 60 mm output after 6 mm propagation in the
is much shorter than the response time of the medium
crystal [Fig. 2( b)]. The nonlinearity is turned on with the
t ϳ 1 s. Unlike all previous experiments with incoherent
application of 2.7 kV͞cm and the beam self-traps forming
solitons, here we need to generate a beam which is very
a soliton in

*x*. When the beam is fully coherent, the soliton
narrow and fully coherent in

*x*, yet uniform and partially
suffers from TI and breaks up into filaments [Fig. 2(c)].

incoherent

*y*. To do that, we use a cylindrical lens which
When the beam is made incoherent in

*y*, but with a large
focuses the beam only in the

*y *direction onto the rotating
speckle size ͑ϳ100 mm͒, the nonlinearity is still above

P H Y S I C A L R E V I E W L E T T E R S
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[11] An exception is domain-wall solitons, whose stability
made incoherent along

*y *but with large speckle sizes ϳ100 mm(small u
is caused by an effect similar to surface tension.

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M. Haelterman

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hard

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*D *soliton
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*D*
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**78**, 646

soliton can propagate in a 3D bulk medium if it is made
partially incoherent in the “uniform” transverse dimen-
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*et al., *Phys. Rev. Lett.

**80**, 2310

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in the “dimension of uniformity” must be such that the
[16] A. W. Snyder and D. J. Mitchell, Phys. Rev. Lett.

**80**, 1422

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In conclusion, we have derived the threshold for TI of
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**81**,

͑1 1 1͒

*D *solitons that are fully coherent in their direction
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such 1D solitons is completely eliminated. We proved our
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**25**, 972 (2000).

we know of for propagating truly stable 1D solitons in a
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**44**, R1446

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[23] E. Infeld and G. Rowlands,

*Nonlinear Waves, Solitons and*
well. We believe that this work opens up a range of possi-

*Chaos *(Cambridge University Press, Cambridge, England,
bilities of eliminating transverse instabilities in many soli-
ton systems, for example, instabilities of 1D dark solitons
[24] The separability of the correlation function for the pertur-
in bulk media, instabilities of ring beams (with and with-
bation into

*L*͑

*x*, r

*x*͒

*Af*͑r

*y*͒ is correct only at the threshold
out topological charge) in self-focusing media, and more.

and close to it. We compared the threshold found here (an-
This work was supported by the Israeli Science Foun-
alytically) to the value we find using numerical methods
dation, the U.S. Army Research Office, the NSF, and
similar to [10], and to the value found from beam prop-
AFOSR. It is part of the MURI program on optical spa-
agation (Fig. 1). All three values are in excellent agree-
tial solitons. E. E. acknowledges support by NSF-NATO.

ment. For the purpose of calculating the threshold, the
This research was supported in part by the Pittsburgh
above ansatz is the most convenient means.

Supercomputing Center, Pittsburgh, PA.

[25] The photorefractive screening nonlinearity is given by
D

*n * 2

*I*͑͞1 1

*I*͒ 21 1 1͑͞1 1

*I*͒.

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**73**, 3211 (1994); D. N. Christo-

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**12**, 1628

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(1995); M. Segev, M. Shih, and G. C. Valley, J. Opt. Soc.

N. N. Akhemediev, Opt. Quantum Electron.

**30**, 535 (1998);

Am. B

**13**, 706 (1996).

Y. S. Kivshar,

*ibid. ***30**, 571 (1998).

Source: http://www.iept.tu-clausthal.de/fileadmin/homes/agkip/reprints/p35.pdf

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