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
1Physics Department and the Solid State Institute, Technion, Haifa 32000, Israel
2Electrical Engineering Department, Princeton University, Princeton, New Jersey 08544
3Physics Department, Princeton University, Princeton, New Jersey 08544
4Electrical Engineering and Computer Science Department, Lehigh University, Bethlehem, Pennsylvania 18015
5Physics Department, Universität Osnabrück, 49069 Osnabrück, Germany
6Mathematics Department, Technion, Haifa 32000, Israel
7Department 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 1͒D͔ 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 1͒D 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 1͒D 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, E͑x, y, z, t͒ 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 ͕dn͑x1, y1, z͒
suggests that TI of incoherent beams should also exhibit
a threshold for its existence. Therefore, if the degree of
2 dn͑x2, y2, z͖͒B , (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, n0 is the bias refractive index, dn is the nonlin-
using the threshold to eliminate TI applies to any type of
ear contribution to the refractive index, x ͑x1 1 x2͒͞2,
nonlinearity, yet we will concentrate on the Kerr nonlinear-
y ͑ y1 1 y2͒͞2 are the middle point coordinates, and
ity for two reasons. First, wave propagation in Kerr media
rx x1 2 x2, ry y1 2 y2 are the difference coordi-
is described by the cubic nonlinear Schrödinger equation
nates. When rx ry 0, B͑x, y, rx, ry, z͒ is the time-
(NLSE) which is one of the most general soliton equations
averaged intensity I͑x, y, z͒. Let BS͑x, y, rx, ry͒ u͑x 1
[23]. Generally speaking, the NLSE describes envelope
rx͞2͒ uء͑x 2 rx͞2͒An͑ry͒ 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
An͑ry͒. u͑x͒ 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 B1 to BS where B1 ø BS.
of coherent speckles that change, on average, every
The nonlinear index change in Kerr media is dn͑I͒ gI,
We define B͑x1, y1x2, y2, z͒
where g is the nonlinear coefficient ͑n2͒. Linearizing
͗Eء͑x2, y2, z, t͒E͑x1, y1, z, t͒͘, the spatial correlation func-
, rx 0, ry 0, z͒ 2 B1͑x 2
, rx 0, ry 0, z͔͒BS͑x, y, rx, ry͒
x 0, ry 0͒ 2 BS ͑x 2 2
x 0, ry 0͔͒B1͑x, y, rx , ry , z͒
We seek solutions in the form B1͑x, y, rx, ry, z͒ exp͑gz͒ exp͑iay͒L͑x, rx͒Af͑ry͒ 1 exp͑gءz͒ exp͑2iay͒ 3
Lء͑x, 2rx͒Aءf͑2ry͒, where a is the transverse wave number, g is the TI growth rate (gain), and Af͑ry͒ is the angularspectrum. The necessary condition B1͑x, y, rx, ry, z͒ Bء1͑x, y, 2rx, 2ry, z͒ is satisfied [17,19]. Substituting B1 intoEq. (2) gives
gL͑x, rx͒Af͑ry͒ 2
͓L͑x 1 x , 0͒Ax ͒u͑x 2 x ͒Af ͑0͒eiary͞2 2 L͑x 2 2
f ͑0͒e2iary͞2͔u͑x 1 2
1 ͓u͑x 1 rx͞2͒2 2 u͑x 2 rx͞2͒2͔L͑x, rx͒Af͑ry͒An͑0͒
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)
g͑a͒ 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), g͑a͒ will be positive in a band of wave numbers,
lar power spectrum An͑ry͒ exp͓2͑ryu0k͞2͒2͔, 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 u0T . 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 u0T ,
rate g as a function of transverse wave number a, then g
the two boundary points at which g͑a͒ 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) g͑a 0͒ 0 and
growth rate), and then drops back to 0 at a associated with
(ii) g0͑a 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., atx rx 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 x1 and x2)that the second term of the LHS is zero for x1 x2, i.e.,for rx 0. Thus, from Eq. (3) we get
Simulations of a 1D Kerr soliton with a Gaussian spec-
3 exp͓2͑ku0T ry͞2͒2͔ , (4)
trum, for various degrees of coherence. The predicted thresh-old is u0 0.549±. (a) Input intensity. ( b) Output beam after
where I0 u͑0͒2 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 ͑ry͒ An͑ry͒. Equation (4)
which is below the threshold, after 4.5 cm of propagation. TI is
Dn0 gI0 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 g͑a͒ 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, A͑ ry͒ exp͓2͑ryu0k͞2͒2͔, for various values of
size in x, and it effectively creates a 1D beam that is
u0. In this example, n0 2.3, l 0.5 mm in vacuum,
narrow and coherent in x and “infinitely” long (uniform)
FWHM 9 mm, which yields a Dn0 0.000 105 6 and
and incoherent in y. The slit is then imaged to the input
an analytic prediction of the threshold of u0T 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
[2] For recent reviews on instabilities, see Yu. S. Kivshar and
D. E. Pelinovsky, Phys. Rep. 331, 117 (2000); E. Kuznetsov et al., Phys. Rep. 142, 103 (1986).
[3] V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38,
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[5] J. S. Aitchison et al., Opt. Lett. 15, 471 (1990).
[6] A. Barthelemy et al., Opt. Commun. 55, 201 (1985) have
suppressed TI, by employing self-induced-waveguiding iny, through interference.
Experiments in photorefractive SBN:75 in the Kerr
[7] R. A. Fuerst et al., Phys. Rev. Lett. 78, 2756 (1997).
regime (intensity ratio ϳ0.1). (a) Input 12 mm FWHM beam.
[8] K. Kos et al., Phys. Rev. E 53, R4330 (1996).
( b) Output beam after 6 mm of linear diffraction when nonlin-
[9] A. V. Mamaev et al., Phys. Rev. A 54, 870 (1996).
earity is off. (c), (d ), and (e) show the output beams with an ap-
[10] N. Akhmediev, V. Korneev, and R. Nabiev, Opt. Lett. 17,
plied field of 2.7 kV͞cm, for various degrees of coherence along
393 (1992); Z. H. Musslimani et al., Phys. Rev. E 60, y and all other parameters kept constant. (c) A fully coherent
soliton breaks up into filaments because of TI. (d ) The soliton is
[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.
0) and still displays a strong TI. (e) The speckle sizes
are reduced to ϳ5 mm: TI is completely eliminated.
M. Haelterman et al., Opt. Lett. 19, 96 (1994); A. Shep- hard et al., Opt. Lett. 19, 859 (1994).
TI threshold and the beam suffers from TI [Fig. 2(d )].
[12] M. Mitchell et al., Phys. Rev. Lett. 77, 490 (1996);
Finally, by decreasing the speckle sizes to ϳ5 mm,
M. Mitchell and M. Segev, Nature (London) 387, 880
TI is eliminated and we get a stable ͑1 1 1͒D soliton
(1997); Z. Chen et al., Science 280, 889 (1998).
[Fig. 2(e)]. Thus, we have shown that a stable ͑1 1 1͒D
[13] D. N. Christodoulides et al., Phys. Rev. Lett. 78, 646
soliton can propagate in a 3D bulk medium if it is made
partially incoherent in the “uniform” transverse dimen-
[14] D. N. Christodoulides et al., Phys. Rev. Lett. 80, 2310
sion. For the soliton to be stable, the degree of coherence
[15] M. Mitchell et al., Phys. Rev. Lett. 79, 4990 (1997).
in the “dimension of uniformity” must be such that the
[16] A. W. Snyder and D. J. Mitchell, Phys. Rev. Lett. 80, 1422
nonlinearity is below threshold for transverse instability.
In conclusion, we have derived the threshold for TI of
[17] V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81,
͑1 1 1͒D solitons that are fully coherent in their direction
of trapping yet are partially incoherent in their direction
[18] N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys.
of uniformity. We predicted that if the nonlinearity is be-
Rev. Lett. 81, 4632 (1998).
low a well-defined threshold, then transverse instability of
[19] M. Soljaˇcic´ et al., Phys. Rev. Lett. 84, 467 (2000);
such 1D solitons is completely eliminated. We proved our
D. Kip, M. Soljaˇcic´, M. Segev, E. Eugenieva, and D. N.
results analytically, numerically, and experimentally, and
Christodoulides, Science 290, 495 (2000).
showed that it is possible to generate stable 1D Kerr-like
[20] E. Eugenieva, D. N. Christodoulides, and M. Segev, Opt.
solitons in a 3D bulk material. This is the only method
Lett. 25, 972 (2000).
we know of for propagating truly stable 1D solitons in a
[21] T. Coskun et al., Phys. Rev. Lett. 84, 2374 (2000).
bulk material. Our method applies to all types of saturable
[22] Y. S. Kivshar, Phys. Rev. A 43, 1677 (1991); 44, R1446
nonlinearities and could be used to eliminate TI in them as
[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 L͑x, rx͒Af͑ry͒ 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
Dn 2I͑͞1 1 I͒ 21 1 1͑͞1 1 I͒. et al., Phys. Rev. Lett. 73, 3211 (1994); D. N. Christo- doulides and M. I. Carvalho, J. Opt. Soc. Am. B 12, 1628
[1] G. I. Stegeman and M. Segev, Science 286, 1518 (1999);
(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).

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