Content uploaded by Gregory Salamo
Author content
All content in this area was uploaded by Gregory Salamo on May 19, 2015
Content may be subject to copyright.
VOLUME 78, NUMBER 13 PHYSICAL REVIEW LETTERS 31MARCH 1997
Three-Dimensional Spiraling of Interacting Spatial Solitons
Ming-feng Shih and Mordechai Segev
Department of Electrical Engineering and Center for Photonics and Optoelectronic Materials (POEM),
Princeton University, Princeton, New Jersey 08544
Greg Salamo
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701
(Received 15 January 1997)
We report the observation of three-dimensional spiraling collision of interacting two-dimensional
spatial solitons. The solitons are photorefractive screening solitons and are phase incoherent to each
other at all times. The collision provides the solitons with angular momentum which is manifested in a
centrifugal repulsion force. When it is balanced by attraction, the solitons spiral about each other in a
DNA-like structure. [S0031-9007(97)02853-6]
PACS numbers: 42.65.Tg, 42.65.Hw
Optical spatial solitons [1] have attracted a substantial
research interest in the last three decades. Several types
of one-dimensional (1D) or two-dimensional (2D) bright
or dark (vortex in two dimensions) spatial solitons have
been demonstrated experimentally, including Kerr-type
solitons [2], photorefractive solitons [3], x
s2d
quadratic
solitons [4], and solitons in a saturable medium [5]. All
these solitons occur when the diffraction of a light beam
is exactly balanced by the nonlinear self-focusing effect
(bright solitons) or self-defocusing effect (dark solitons).
Collisions between solitons are perhaps the most fas-
cinating features of self-trapped beams, since, in many
aspects, solitons interact like particles: being able to main-
tain their separate identities (in some cases), fuse (in oth-
ers), or generate entirely new soliton beams [6]. Each
possibility is fully determined by the initial trajectories of
the colliding solitons and the interaction force they exert
on each other (resulting from the nonlinear change in the
refractive index induced by both solitons). For example,
if two bright spatial solitons are mutually coherent and in
phase, they constructively interfere, giving rise to an in-
crease in the optical intensity in the region between them.
This leads to an increase in the refractive index in their
central region. As a result, more light is attracted toward
the central region and is self-guided there. The net re-
sult is that the solitons appear to attract each other during
their propagation in the nonlinear medium. On the other
hand, when the initial relative phase between the colliding
solitons is equal to p, the solitons destructively interfere
in the central region and they appear to repel each other.
However, if the solitons are mutually phase incoherent
(i.e., the relative phase between the soliton beams varies
much faster than the response time of the medium) [7,8],
their intensities, rather than their amplitudes, are super-
imposed and this makes the interaction phase insensitive.
Obviously, the total intensity in the central region can-
not be lowered now; thus mutually incoherent bright soli-
tons always attract each other. In spite of the complexity
regarding soliton interactions, most of the soliton colli-
sion properties can be described using linear waveguide
theory [9,10]. Whether 1D or 2D bright spatial solitons
are involved in a collision, the attraction and repulsion
forces are key factors determining the result of the soliton
interaction.
Before stable 2D bright spatial solitons were observed,
the study of soliton interaction was limited to the uncon-
strained transverse dimension of a 1D waveguide and the
longitudinal dimension [2,11]. The recent observations
of stable 2D spatial solitons has enabled observations of
full s2 1 1dD (two transverse plus one longitudinal di-
mensions) soliton interactions. In particular, collisions of
2D solitons were observed in photorefractive media [7]
and in saturable nonlinear atomic media [5]. The lat-
ter has also reported 3D spiraling of bright spatial soli-
tons, when the solitons were generated from the breakup
of an input vortex beam. When this input “bright ring”
was launched into a self-focusing medium, it exhibited in-
stability and fragmented into two 2D solitonlike beams.
Since the input vortex had carried initial angular momen-
tum, the bright solitonlike beams were forced (by con-
servation of angular momentum) to spiral while moving
away from each other [5].
However, in principle, two 2D solitons should be able
to spiral about each other even when they individually
do not carry initial angular momentum, as predicted
by Snyder’s group in 1991 [12]. This should occur
when two 2D solitons collide with trajectories that are
not lying in a single plane, and at the same time,
they attract each other just enough to “capture” each
other [12]. Then, the solitons orbit about each other in
a DNA-like structure, as illustrated in Fig. 1. In this
Letter, we demonstrate just that: collision of two 2D
mutually incoherent photorefractive solitons that are fully
controllable in three dimensions. The solitons fuse, spiral
about each other, or bypass each other depending on the
distance between them and their trajectories. When each
input beam is individually launched (the other beam is
absent) it possesses no angular momentum. Nevertheless,
0031-9007y97y78(13)y2551(4)$10.00 © 1997 The American Physical Society 2551
VOLUME 78, NUMBER 13 PHYSICAL REVIEW LETTERS 31MARCH 1997
FIG. 1. An illustration of the soliton spiraling process. The
arrows indicate the initial direction of the two soliton beams.
the collision process provides the soliton pair with angular
momentum as the simultaneously launched solitons form
a two-body system, and this drives the solitons to spiral
about each other. The angular momentum is manifested
in a mutual repulsion (centrifugal) force. When repulsion
is exactly balanced by attraction due to the soliton
interaction, the solitons capture each other (as celestial
objects do) and spiral about each other in a DNA-like
structure. We find that this process is most easily realized
when the solitons are mutually incoherent with a very
small angular separation between their initial propagation
directions. When the initial distance between the two
solitons is too large providing not enough attraction force,
they move away from each other. When the distance is
too small, the solitons fuse into one beam.
The choice of two mutually incoherent solitons merits
further discussion. In principle, soliton spiraling should
be realizable with either mutually coherent or mutually
incoherent solitons. A necessary condition is, of course,
that the mutual repulsion due to the centrifugal force
will be balanced by attraction. However, while the
force between mutually incoherent solitons is always
attraction, the force between mutually coherent solitons
depends on their relative phase and can be manifested in
either attraction or repulsion. We find it much simpler,
therefore, to realize a system of spiraling solitons with
mutually incoherent solitons, for which the mutual force
is not subject to phase variations. Furthermore, it is
known that the soliton parameters, such as the propagation
constant, the beam shape, and the maximum amplitude,
are all interrelated. To make the delicate spiraling soliton
pair stable and accessible, the attraction force between
them throughout the entire propagation distance should be
maintained as constant as possible. For a coherent soliton
pair, this implies that the propagation constants must be
indentical (in some nonlinear medium, this also implies
that the two solitons must be identical). Otherwise, after
a certain propagation length, the soliton pair becomes out
of phase (due to their different propagation constants)
and then the solitons start to repel each other. Once the
solitons move away from each other, the interaction force
(which decays with the transverse distance between the
solitons) can no longer balance the centrifugal force and
the solitons move further apart. Thus, using a mutually
incoherent pair to observe the spiraling process can avoid
the stability problem since the interaction force between
the solitons is always attraction.
To observe soliton spiraling, the most critical require-
ment is to find the condition for which the attraction force
can compensate exactly the centrifugal force caused by
the acquired angular momentum. In previous experiments
[13,14], we have studied the behaviors of waveguides in-
duced by photorefractive screening solitons. For 1D pho-
torefractive screening spatial solitons, the refractive index
profile of the soliton-induced waveguide is controlled by
the so-called intensity ratio, which is the ratio between the
peak soliton intensity and the sum of the dark and back-
ground irradiances [14]. It is found that, at a large inten-
sity ratio, the soliton-induced waveguide is multimode and
its index profile is wider and deeper than at low intensity
ratios (where the soliton resembles a Kerr soliton and the
waveguide it induces is a single mode waveguide). At
the same time, the index of the soliton-induced waveguide
drops more dramatically at the boundary of the soliton at
high intensity ratio, while at low intensity ratio (around
intensity ratio 3), the refractive index profile varies more
smoothly across the solitons. A similar trend has also been
found in 2D bright screening solitons [13]. As pointed out
in a recent theoretical Letter [15], the interaction force be-
tween two solitons is proportional to the gradient of the
index perturbation induced by the solitons. In our spiral-
ing experiment, we find the most suitable intensity ratio for
observation of the spiraling process is around 4 to 6. If the
intensity ratio is smaller, the attraction force is too weak
(the gradient of the refractive index change is too small)
to compensate the centrifugal force. On the other hand, at
high intensity ratios, the soliton-induced waveguide is mul-
timode, which means that nonfundamental guided modes
can be excited in the collision process [7], and this breaks
the 2D symmetry and deteriorates the solitons [7,14]. In
summary, to observe spiraling solitons, it is (1) necessary
to have a saturable nonlinearity (such as the photorefractive
nonlinearity) that stabilizes 2D solitons, and it is desirable
to have (2) mutually incoherent solitons to ensure a phase-
insensitive attraction force, and (3) the nonlinearity should
be operated at maximum saturation that still gives rise only
to single-mode soliton-induced waveguides. Only after
these conditions are satisfied, one can resort to the deli-
cate work of adjusting the initial trajectories of the collid-
ing solitons (that should not lie in the same plane) and the
distance between them.
The experiment setup is similar to that of Ref. [7] ex-
cept that the input beams are now launched with their tra-
jectories skewed with respect to each other, as illustrated
by the arrows in Fig. 1. We use a 14 3 13 3 6.5 mm
3
SBN:60 (Ba
0.4
Sr
0.6
Nb
2
O
6
) photorefractive crystal. The
two 12 mm wide (FWHM) beams are first launched into
the crystal with their minimum waists on the input face,
marked A and B in Fig. 2(a). The angular separation
and distance between the solitons are 7 3 10
23
rad and
2552
VOLUME 78, NUMBER 13 PHYSICAL REVIEW LETTERS 31MARCH 1997
FIG. 2. (a) Beams A and B at the input face of the crystal,
(b) the spiraling soliton pair after 6.5 mm of propagation, and
(c) after 13 mm of propagation. The centers of diffracting A
and B are marked by white triangles. The white cross indicates
the center of mass of the diffracting beams A and B in (b)
and (c).
14 mm, respectively. After 6.5 mm of propagation, the
beams diffract to about 55 mm and increase their sep-
aration, their centers being marked by the triangles in
Fig. 2(b). The diffracted beams were sampled immedi-
ately (0.1 s) after launching each individual input beam,
that is, long before fanning evolves. We define the cen-
ter between the diffracting beams as the “center of mass”
and mark it by a white cross shown in Figs. 2–4. After we
apply 6.1 kV between electrodes separated by 14 mm, two
solitons form and, at the same time, their relative position
rotates [16] by roughly 270
±
[Fig. 2(b)]. We distinguish
between the output beams by blocking (or modulating) one
of them at the input for a time “window” much shorter than
the response time of the photorefractive material and thus
being able to identify the modulation in one of the output
beams. The photorefractive nonlinearity is not affected by
such fast modulation as the nonlinear index change does
not have time to adjust; thus the beams are easily distin-
guishable from the other (this technique is used in [17]).
Then, we flip the crystal over and let the solitons propa-
gate along 13 mm. We find that the solitons now rotate an
additional 270
±
[Fig. 2(c)]. We also observe roughly 19%
of energy exchange [7] between A and B after 6.5 mm of
propagation and 30% of energy exchange after 13 mm of
propagation because the soliton-induced waveguides are
so close to each other, that energy from each soliton beam
can be coupled into the waveguide induced by the other.
However, since both solitons induce single-mode wave-
guides, this energy exchange does not break the symmetry
and does not affect the interaction [7].
As we increase the initial distance between A and B
to 22 mm [Fig. 3(a)] while keeping their initial angular
FIG. 3. (a) Beams A and B at the input face of the crystal,
(b) the bypassing soliton pair after 6.5 mm of propagation. The
centers of diffracting A and B are marked by white triangles.
The white cross indicates the center of mass of the diffracting
beam A and B in (b).
separation unchanged, the attraction force decreases and
cannot balance the centrifugal force anymore. As a result,
the solitons now bypass and move away from each other.
The separation becomes 28 mm [Fig. 3(b)] after 6.5 mm
of propagation, although some attraction is still observed
when we compare the distance between solitons A and
B with the distance between diffracting beams A and
B. Finally, as we reduce the initial distance between the
solitons to 7 mm, [Fig. 4(a)] and also adjust the initial
angular separation to 6 3 10
23
rad, we observe that A
and B fuse into one beam [Figs. 4(b) and 4(c)] because
the attraction force is now larger than needed for spiraling
and the solitons coalesce into their common center of
mass. A similar fusion result has also been observed in
the previous experiments of planar incoherent collisions
(that is, when the trajectories are in a single plane) between
bright screening solitons [7,14].
A careful look at Fig. 2 reveals an interesting obser-
vation: the two identical solitons spiral about each other
in elliptical (rather than circular) orbits. The reason for
that is twofold. First, the trajectories of identical inter-
acting “particles” in an effective two-body problem are in
general always elliptical (circular trajectories can be ob-
tained only under special conditions) [18]. Second, the
photorefractive nonlinearity is anisotropic, and only un-
der specific conditions is one able to obtain even individ-
ual circular 2D solitons [19]. It is, therefore, expected
that the interaction force between solitons will depend
on the plane of collision: whether the space charge field
in the center region between the solitons is parallel (or
perpendicular) to the crystalline c axis, thus maximizing
(minimizing) the influence of the large r
33
electro-optic
coefficient of SBN. For both these reasons we expect that
the spiraling solitons will follow elliptical orbits. Indeed,
Figs. 2(b) and 2(c) show that the solitons move closer and
then break apart periodically.
In addition to the nonlinearity that gives rise to screening
solitons, photorefractive solitons also self-bend toward
the c axis as a result of asymmetric diffusion fields
[20]. Self-bending of individual and colliding solitons was
observed in Refs. [7], [14], and [18] respectively. Here,
the diffusion field acts as an additional force exerted on
both solitons. It is, therefore, expected that the center
of mass will bend toward the c axis. When the solitons
FIG. 4. (a) Beams A and B at the input face of the crystal,
(b) the fused soliton pair after 6.5 mm of propagation, and
(c) after 13 mm of propagation. The centers of diffracting A
and B are marked by white triangles. The white cross indicates
the center of mass of the diffracting beams A and B in (b)
and (c).
2553
VOLUME 78, NUMBER 13 PHYSICAL REVIEW LETTERS 31MARCH 1997
fuse or bypass (Figs. 3 and 4, respectively), their center
of mass is indeed always bent (shifted) toward the c axis
as shown in Figs. 3(b), 4(b), and 4(c). When the beams
spiral about each other, however, the center of mass seems
to wobble, as observed in Fig. 2: after 6.5 mm it appears to
shift toward the c axis [Fig. 2(b)], whereas after 13 mm it
shifts in the opposite direction [Fig. 2(c)]. This feature of
the spiraling soliton dynamics is certainly worthy of further
study in the future.
We emphasize that we have reproduced all the delicate
experiments described in this Letter, including the spiral-
ing, fusion, elliptic orbits, and wobbling, in two different
SBN crystals of different length and strength of nonlin-
earity sr
33
d. It thus confirms that these observations stem
from the generic nature of spatial solitons and depend very
little on the specific material properties.
In conclusion, we have observed full three-dimensional
spiraling of a 2D bright soliton pair. For proper initial
conditions the solitons spiral about each other in elliptical
orbits. A deviation from these conditions leads to soliton
fusion (“impact”) or to their escape from mutual orbiting.
These experiments reveal the deep similarity between
the solitons in nature (not only in nonlinear optics) and
particles.
M. Segev gratefully acknowledges the support of a
Sloan Fellowship. This research was supported by the
U.S. Army Research Office and the National Science
Foundation. The authors are in debt to Allan Snyder and
Yuri Kivshar, both of the Optical Science Center, Aus-
tralia National University, and to Z. Chen and M. Mitchell
of Princeton University for many helpful discussions.
[1] R.Y. Chiao, E. Garmire, and C.H. Townes, Phys. Rev.
Lett. 13, 479 (1964).
[2] J.S. Aitchinson, A. M. Weiner, Y. Silberberg, M.K.
Oliver, J.L. Jackel, D.E. Leaird, E. M. Vogel, and P.W.
Smith, Opt. Lett. 15, 471 (1990).
[3] G. Duree, J.L. Shultz, G. Salamo, M. Segev, A. Yariv,
B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar,
Phys. Rev. Lett. 71, 533 (1993); Phys. Rev. Lett. 74, 1978
(1995).
[4] W. E. Torruellas, Z. Wang, D.J. Hagan, E. W. Van
Stryland, G.I. Stegeman, L. Torner, and C.R. Menyuk,
Phys. Rev. Lett. 74, 5036 (1995).
[5] V. Tikhonenko, J. Christou, and B. Luther-Davies, Opt.
Lett. 19, 1817 (1994); J. Opt. Soc. Am. B 12, 2046 (1995);
Phys. Rev. Lett. 76, 2698 (1996).
[6] A. W. Snyder and A.P. Sheppard, Opt. Lett. 18, 482
(1993).
[7] M. Shih and M. Segev, Opt. Lett. 21, 1538 (1996).
[8] D. N. Christodoulides, S. R. Singh, M.I. Carvalho, and
M. Segev, Appl. Phys. Lett. 68, 1763 (1996).
[9] Y. Silbergerg, in Anisotropic and Nonlinear Optical
Waveguides, edited by G.C. Someda and G.I. Stegeman
(Elsevier, New York, 1992).
[10] A.W. Snyder, D.J. Mitchell, and Y.S. Kivshar, Mod.
Phys. Lett. B 9, 1479 (1995).
[11] F. Reynaud and A. Barthelemy, Europhys. Lett. 12, 401
(1990).
[12] D.J. Mitchell, A.W. Snyder, and L. Poladian, Opt.
Commun. 85, 59 (1991).
[13] M. Shih, M. Segev, and G. Salamo, Opt. Lett. 21, 931
(1996).
[14] M. Shih, Z. Chen, M. Segev, T.H. Coskun, and
D.N. Christodoulides, Appl. Phys. Lett. 69, 4151
(1996).
[15] D.J. Mitchell, A. W. Snyder, and L. Poladian, Phys. Rev.
Lett. 77, 271 (1996).
[16] Unfortunately, since the two solitons are very close to
each other, they are indistinguishable when viewed from
the top of the crystal, and we are unable to take the top-
view photograph of the collision process, as we did in
previous experiments [7,13,14,19].
[17] Z. Chen, M. Segev, T.H. Coskun, and D.N.
Christodoulides, Opt. Lett. 21, 1436 (1996).
[18] L.D. Landau and E. M. Lifshitz, Mechanics (Pergamon
Press, Oxford, 1973), 3rd ed., Chap. 3, pp. 29–40.
[19] M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamo,
and G.C. Valley, Opt. Lett. 21, 324 (1996).
[20] S. R. Singh and D. N. Christodoulides, Opt. Commun. 118,
569 (1995).
2554
