arXiv:nlin/0309028v1 [nlin.PS] 9 Sep 2003
Partially incoherent optical vortices in self-focusing nonlinear media
Chien-Chung Jeng1, Ming-Feng Shih1, Kristian Motzek2,3, and Yuri Kivshar3
1Physics Department, National Taiwan University, Taipei, 106, Taiwan
2Institute of Applied Physics, Darmstadt University of Technology, D-64289 Darmstadt, Germany
3Nonlinear Physics Group, Research School of Physical Sciences and Engineering,
Australian National University, Canberra ACT 0200, Australia
We observe stable propagation of spatially localized single- and double-charge optical vortices in
a self-focusing nonlinear medium. The vortices are created by self-trapping of partially incoherent
light carrying a phase dislocation, and they are stabilized when the spatial incoherence of light
exceeds a certain threshold. We confirm the vortex stabilization effect by numerical simulations and
also show that the similar mechanism of stabilization applies to higher-order vortices.
Vortices are fundamental localized objects which ap-
pear in many branches of physics, ranging from liquid
crystals to superfluids and Bose-Einstein condensates .
In optics, vortices are associated with phase disloca-
tions (or phase singularities) carried by diffracting op-
tical beams . When such vortices propagate in self-
defocusing nonlinear media, the vortex core with a phase
dislocation becomes self-trapped, and the resulting sta-
tionary singular beam is known as an optical vortex soli-
ton . Such vortex solitons have been generated ex-
perimentally (e.g. by using a phase mask) within broad
diffracting beams in different types of defocusing nonlin-
ear media [4, 5, 6, 7, 8]. Optical vortex solitons demon-
strate many common properties with the vortices ob-
served in superfluids and, more recently, in Bose-Einstein
In contrast, optical vortices become highly unstable
in self-focusing nonlinear media. Indeed, when a ring-
like optical beam with zero intensity at the center and a
phase singularity  propagates in a self-focusing non-
linear medium, it always decays into several fundamental
solitons flying off the main ring . This effect has been
observed experimentally in different nonlinear systems,
including the saturable Kerr-like nonlinear media ,
biased photorefractive crystals , and quadratic nonlin-
ear media  in the self-focusing regime. This effect has
also been predicted to occur in many other physical sys-
tems including attractive Bose-Einstein condensates .
A number of recent theoretical papers , including
the rigorous studies of the vortex stability , suggest
that the stable propagation of spatial and spatiotempo-
ral vortex-like stationary structures may become possible
in the models with competing nonlinearities in the pres-
ence of a large higher-order defocusing nonlinearity, but
no realistic physical systems to support these theoretical
findings have been found so far. Thus, the main ques-
tion remains open: Can stable optical vortices readily be
observed in self-focusing nonlinear media?
The main purpose of this Letter is to demonstrate,
for the first time to our knowledge, that stable propaga-
tion of spatially localized optical vortices in a self-focusing
photorefractive crystal can indeed be observed provided
such vortices are created by partially incoherent light car-
rying a phase dislocation. In particular, we show, both
experimentally and theoretically, that single- and double-
charge optical vortices can be stabilized in self-focusing
nonlinear media when the value of the spatial incoherence
of light exceeds a certain threshold, and these vortices are
readily observed in experiment as stationary self-trapped
structures propagating for many diffraction lengths.
We should mention that the generation and proper-
ties of singular optical beams created by partially inco-
herent light is an important issue which is a subject of
a current active research even in linear optics (see, e.g,
Ref.  and references therein).
strate that partially incoherent singular beams can exist
in self-focusing nonlinear media being stabilized by the
light incoherence effect.
First, we present our experimental results. The main
purpose of our experiments conducted for a biased pho-
torefractive medium is to generate partially incoherent
vortices and vortex solitons, and then inspect their sta-
bility in such a self-focusing nonlinear medium. Being
driven by the earlier results of Anastassiou et al. 
who demonstrated the suppression of modulational insta-
bility for the stripe spatial solitons created by partially
incoherent light, we try to understand whether the other
type of the nonlinearity-driven instability, the so-called
azimuthal instability, can be suppressed by reducing the
coherent properties of optical vortex beams.
The experiment (the setup is shown in Fig. 1) is con-
ducted with a biased photorefractive SBN crystal (a x b
x c = 5 mm x 10mm x 5mm). First, a cw laser light
beam (at 488 nm) of the extraordinary polarization is
made partially incoherent by passing it through a lens
and then through a rotating diffuser. The rotating dif-
fuser introduces random-varying phase and amplitude on
the light beam every 1 µs, which is much shorter than
the response time (about 1 s) of the crystal under our
experimental light illumination. By adjusting the posi-
tion of the diffuser to near (away from) the focal point of
the lens in front the diffuser, we can increase (decrease)
the degree of the light coherence. We collect the light
after the rotating diffuser by a second lens and then pass
is through a computer-generated hologram to imprint a
vortex phase (with a single-, for m = 1, or double-, for
Our results demon-
FIG. 1: Schematic of the experimental setup for the observa-
tion of the partially incoherent optical vortices; SBN: Stron-
tium Barium Niobate crystal.
m = 2, charge) on the light beam. Since the partially in-
coherent light beam can be considered as a superposition
of many mutually-incoherent light beams, the first-order
diffracted light beam after the hologram becomes a su-
perposition of many mutually-incoherent vortex beams.
We then focus and launch the partially coherent vortex
beam into the SBN crystal along its a-axis. The crys-
tal is illuminated by a halogen lamp from its top side as
necessary for photorefractive screening solitons. The to-
tal power of the vortex beam is of 0.17µW, which results
in the nonlinearity of the photorefractive crystal falling
into the Kerr region for the peak intensity of the vor-
tex beam to the background intensity is much less than
unity. Then, a lens is used to project the images at the
input and output faces onto a CCD camera.
Before showing that reducing the degree of coherence
can stabilize the vortex beam propagating in a self-
focusing medium, we reproduce the experiment that the
coherent single-charge(m = 1) vortex light beam cannot
stably propagate in a self-focusing medium [7, 12]. We
first remove the diffuser from the experimental setup.
The vortex beam at the input face of the crystal is
shown as Fig. 2(a). With zero biasing voltage, Fig. 2(b)
shows the natural diffraction of the vortex light beam.
While a 2.5 kV biasing voltage is applied on the photore-
fractive crystal creating a Kerr-type self-focusing non-
linear medium, the vortex beam breaks up into two
pieces [Fig. 2(c)]. This vortex break-up observed in a
self-focusing medium is due to the azimuthal instability,
and it has been theoretically and experimentally demon-
strated previously [11, 12]. We then put back the rotating
diffuser and adjust it to a suitable position. The degree
of coherence of the vortex light beam can be estimated by
the speckle size at the input face of the crystal [Fig. 2(d)]
when we stop the diffuser from rotating. As a voltage of
2.5 kV is applied on the crystal, Fig. 2(g) clearly shows
that the vortex light beam is stabilized by the reduction
of the degree of coherence though two very unclear bright
spots still can be seen on the opposite sides (top and bot-
tom) of the ring-like intensity distribution. As we move
the rotating diffuser further away from the focal point of
the lens to make the light more incoherent [indicated by
Fig. 2(h)] and apply a voltage of 2.5 kV on the crystal,
Fig. 2(k) shows the generated stable partially incoherent
FIG. 2: Experimental results for the intensity distribution
of a single-charge vortex beam (m = 1) for different degrees
of coherence (top: coherent, middle: less coherent, bottom:
least coherent). At input face, (a) is coherent light, (e) and
(i) are partially incoherent, but (i) is more incoherent than
(e) by comparing the speckle pattern shown in (d) and (h).
At output face, (b) (f) and (j) show the diffraction when the
nonlinearity is off, and (c), (g) and (k) shows the incoherence
stabilize the vortex soliton when voltage of 2.5 kV is applied.
vortex soliton at the output face of the crystal.
We notice that the vortex stabilization by the light in-
coherence can possibly be explained by employing simple
physics. Indeed, after some propagation of the partially
incoherent beam with an imprinted phase, we observe
that the intensity at the center of the beam does not
vanish. This means that the incoherent vortex can be de-
composed into many mutually incoherent vortex beams
with not only their phases randomly distributed but also
their central positions offset from each others. In this
way, the core of a composite vortex beam will be filled by
some light, and the index change at the beam center will
become nonzero. We believe that this effect contributes
strongly to the vortex stabilization.
We continue with the experiments for the double-
charge vortex beams generated by a phase mask with
the m = 2 dislocation. Figures 3(a-c) show that a coher-
ent double-charge vortex light beam cannot stably prop-
agate in a self-focusing medium (we applied 3 kV here)
as well, and it breaks up into pieces as been observed
in a self-focusing atomic vapor . When we make the
light more incoherent, the double-charge vortex becomes
more stable, as shown in Figs. 3(g) and (k). However, we
could not make a double-charge vortex soliton here. The
double-charge vortex beam diffracts more than a single-
charge vortex beam and, therefore, it requires higher non-
FIG. 3: Experimental results for a double-charge vortex beam
(m = 2) for different degrees of coherence (top: coherent, mid-
dle: less coherent, bottom: least coherent). At input face, (a)
is coherent light, (e) and (i) are partially incoherent, but (i)
is more incoherent than (e) by comparing the speckle pattern
shown in (d) and (h). At output face, (b) (f) and (j) show
the diffraction when the nonlinearity is off, and (c), (g) and
(k) shows the incoherence stabilize the vortex soliton when
voltage of 3 kV is applied.
FIG. 4: Numerical results showing the comparison between
the evolution of the vortex created by a coherent (upper row)
and partially incoherent (lower row) light (θ0 = 0.38). The
vortex stabilization by partial incoherence is clearly seen.
linearity (or higher voltage) to form the vortex soliton,
but our crystal begins to arc at 3.5 kV. Nevertheless,
these two experiments show that the reduction of the
degree of coherence of the light beam can indeed stabi-
lize the single- or double-charge (or even higher-charge)
vortex solitons propagating in a self-focusing medium.
We have also studied numerically the propagation
of partially incoherent optical vortices in a photore-
fractive medium, employing the coherent density ap-
FIG. 5: Numerical results showing the stabilization of the
vortex with growing incoherence: (a) input intensity, (b) vor-
tex after 9mm of propagation for the coherent case, (c) vortex
after 9mm for the partially incoherent case, θ0 = 0.14, and (d)
vortex after 9mm for the partially incoherent case, θ0 = 0.29.
proach [18, 19]. The coherent density approach is based
on the fact that partially incoherent light can be de-
scribed by a superposition of mutually incoherent light
beams that are tilted with respect to the z-axis at differ-
ent angles. One thus makes the ansatz that the partially
incoherent light consists of many coherent, but mutually
incoherent light beams Ej: I =
|Ej|2= G(jϑ)I, where
j|Ej|2. By setting
is the angular power spectrum, one obtains a partially
incoherent light beam whose coherence is determined by
the parameter θ0, i.e. less coherence means bigger θ0.
Here, jϑ is the angle at which the j-th beam is tilted
with respect to the z-axis. For our simulations we used
a set of 1681 mutually incoherent vortices, all initially
tilted at different angles. To simulate the photorefrac-
tive nonlinearity we use a simple model which predicts
that the refractive index change is approximately given
by I/(1 + I).
Figure 4 shows a comparison between the propagation
of the vortices generated by coherent and partially inco-
herent light (for θ0= 0.38). Increasing the incoherence
further leads to the case where the vortex beam radiates
off a lot of its intensity before the azimuthal instability
can set in. Larger values of incoherence (i.e. larger values
of θ0) correspond to a complete suppression of the vor-
tex instability, and this confirms the experimental results
Figures 5(a-d) show our numerical results for the prop-
agation of an input Gaussian beam carrying a phase dis-
location [(a)] after the total propagation (9 mm) in a
nonlinear medium for the coherent light [(b)] and two
different partially incoherent beams [(c,d)], correspond-
ing to the values θ0= 0.14 and θ0= 0.29, respectively.
The most obvious difference to the scenario of the prop-
agation the a coherent vortex is that the vortex decay
undergoes a visible delay when the degree of incoherence
grows. Furthermore, in the incoherent case the vortex
changes its profile only very slowly as it propagates and
thus can be considered as being in a transition stage be-
tween the decay and stabilization.
Finally, we have studied numerically the propagation
of a double-charge vortex beam in both coherent and
FIG. 6: Propagation of a double-charge vortex in (top row)
the coherent case at z =0, 8.6, and 14.5 mm, and (bottom
row) in the partially incoherent case with θ0 = 0.35 at z =0,
8.6, 14.7, and 18.8 mm.
partially incoherent cases, as shown in Fig. 6. The initial
perturbation in both cases have been chosen to be very
small in order to obtain a clearer picture of the insta-
bility, therefore the propagation distances are quite long.
In numerics, since we have a possibility to observe long
propagation distances, we did not find that the vortices
can be completely stabilized by increasing incoherence.
When using the values of θ0far above the value θ0= 0.35
used in Fig. 6, we observe that the vortex just radiates
off a lot of its intensity and then decays. The problem is
obviously that the vortices are not only incoherent along
the angular, but also the radial direction. We believe
that the experiment would confirm this observation if a
longer crystal was available.
In conclusion, for the first time to our knowledge,
we have observed the stable propagation of single- and
double-charge optical vortices in a self-focusing nonlin-
ear medium. The vortices have been created by partially
incoherent light beams carrying a phase dislocation and
propagating in a photorefractive nonlinear medium. The
vortex azimuthal instability in a self-focusing nonlinear
medium was suppressed for the light incoherence above
a critical value. The experimental results have been con-
firmed by numerical simulations which also provide an
insight into the physical mechanisms of the vortex stabi-
Yuri Kivshar thanks the Physics Department of the
Taiwan University for hospitality. This work was sup-
ported by a collaboration program between the Aus-
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