A class of optical spatial solitons exhibiting propagation in a closed-loop orbit in a two-dimensional plane is presented. A closed-form particlelike model is derived, indicating that the quasi-centrifugal force acting on these solitons can be balanced by an inhomogeneity in the nonlinear index of refraction. Specifically, a circular-shaped nonlinear interface is shown to facilitate stable orbital propagation of solitons that carve their own circular cavity for a wide range of nonlinearity parameters

Feigenbaum, Eyal

Orenstein, Meir

Scheuer, Jacob

English

Caltech Authors - Main

Circulating spatial solitons
Eyal Feigenbaum and Meir Orenstein
Department of Electrical Engineering, Technion, Haifa 32000, Israel
Jacob Scheuer
Center for the Physics of Information, California Institute of Technology, Pasadena, California 91125
Received November 3, 2005; accepted November 9, 2005; posted November 22, 2005 (Doc. ID 65495)
A class of optical spatial solitons exhibiting propagation in a closed-loop orbit in a two-dimensional plane is
presented. A closed-form particlelike model is derived, indicating that the quasi-centrifugal force acting on
these solitons can be balanced by an inhomogeneity in the nonlinear index of refraction. Specifically, a
circular-shaped nonlinear interface is shown to facilitate stable orbital propagation of solitons that carve
their own circular cavity for a wide range of nonlinearity parameters. © 2006 Optical Society of America
OCIS codes: 190.5530, 190.3270.
One-dimensional (1D) spatial optical solitons (i.e.,
nonlinear confinement in one dimension) in Kerr
media were discussed and measured in many
publications.1–3 The dynamics of such solitons
is characterized by linear trajectories within a two-
dimensional (2D) spatial plane (usually implemented
as a slab waveguide). Introduction of material
inhomogeneity (e.g., nonlinear interfaces) yields an
equivalent force that modifies the soliton’s
trajectory.4–6 Here, by applying a circular interface
(i.e., a disk) in a 2D plane, the respective soliton
trajectories are bent to accommodate with a
Keplerian-like orbit about the circumference of the
nonlinear interface. Such solitons, upon completing a
closed-loop trajectory, carve not only their self-
waveguides but also their self-circular ring cavity,
with the potential for realization with relatively
weak light sources.
Complex soliton trajectories, e.g., dipolelike propel-
ler 2D+1 incoherent spatial solitons (which self-
trap in both transverse dimensions) were reported
previously.1 These solitons exhibited a helical trajec-
tory in a nonlinear medium. However, self-generation
of a closed-loop trajectory, namely, self-carving of an
optical cavity by a soliton, is novel to the best of our
knowledge.
In the out-of-plane coordinate, referred to as z, the
simplified 1D soliton is invariant, z=0. In the more
general case the linear confinement along the z axis
(e.g., the slab waveguide mode) introduces a nonvan-
ishing wavenumber kz. This value of kz modifies the
effective linear index of refraction and may impede
the soliton stability.3,7 In the current analysis we em-
ploy a vanishing kz value and focus on the steady-
state dynamics leading to the soliton capture into or-
bit.
Starting with the wave equation in polar coordi-
nates r ,, we employ the slowly varying envelope
amplitude A under the paraxial approximation. An
important distinction from the conventional ap-
proach is that we seek an angularly propagating non-
linear wave packet with a radial envelope, and thus
the paraxial approximation is performed about the
angular coordinate:
2A
r2
+
1
r
A
r
+
2jk
r2
A

+ k02n02 + 2k02n0n2A2 − k2r2A
= 0, 1
where k0 and k are the free space and the angular
wavenumbers, respectively, and n0 ,n2 are the respec-
tive linear and Kerr indices of refraction. By use of
the conformal transformation U=R lnr /R ,V=R,8
the wave equation is transformed into
A
V
=
jR
2k
2A
U2
+
jk0
2n0n2R
k
exp2U
R
A2A
+
j
2k02n02Rk exp2UR − kR 	A, 2
where R is an arbitrary scaling factor. Equation (2)
may have general nonlinear wave solutions, but we
seek solutions similar to the classical solitons of the
nonlinear Schrödinger equation (NLSE).9 A formal
NLSE can be obtained from Eq. (2) by employing a
radially inhomogeneous nonlinear medium satisfying
n0 

k
k0
1
r
, n2 
1
r
. 3
For such index profiles, Eq. (2) exhibits an exact or-
biting soliton solution with a radial envelope propor-
tional to sechlogr /R.
Although the index profile of Eq. (3) supports an
exact soliton solution, it is not practically realizable.
Therefore, we seek circulating soliton solutions in a
simpler configuration comprising two homogeneous
nonlinear media that yield a stepwise inhomogeneity
and focus our analysis on the interface as a source of
soliton perturbation. To understand the required in-
terface characteristics, we first examine Eq. (2) in a
homogeneous medium, as a perturbed NLSE for a
wave packet with a lateral width (U coordinate) sig-
nificantly smaller than its orbital radius. For conve-
nience we choose R as the orbit radius. Equation (2)
in normalized coordinates becomes
486 OPTICS LETTERS / Vol. 31, No. 4 / February 15, 2006
0146-9592/06/040486-3/$15.00 © 2006 Optical Society of America
v
E =
j
2
2E
u2
+ jE2E + jn0
2 + 2E2
u
k0R
E, 4a
where
u 
 k0U, v 

k0
2R
k
V,
4b
Au,v 

Eu,v
n0n2
exp− j k22k02R2 − n0
2
2 v	 ,
Equation (4a) is a perturbed NLSE, which admits a
perturbed soliton solution with intensity profile A2
=2 sech2 , =u−u0, where  is the soliton peak
amplitude and u0 is the soliton center. By use of soli-
ton perturbation theory,10 the perturbation source
[the third term on the right-hand side of Eq. (4a)] in-
troduces an effective force that accelerates the soliton
outward:
Fr =
1
k0R
n02 + 232. 5
The existence of this (fictitious) centrifugal force, Fr,
in a homogeneous medium, is a manifestation of the
fact that we imposed a circular motion on the soliton,
which is following a linear trajectory in homogeneous
media.
Thus, maneuvering the soliton into orbit requires
an opposite force to compensate for Fr. Figure 1(a) de-
scribes the perturbation introduced by circular inter-
face, with general variations in both linear and non-
linear refractive indices. When one applies the step
index profile to the Kerr coefficient in Eq. (2), an ad-
ditional perturbation source is formed in the modified
perturbed nonlinear Schrödinger equation (4a):
sn = j
n0n2
n0n2
1u − uE2E, 1u − u 
 1 u u0 u u,
6
where n0n2
n0+n0n2−n2−n0n2. The
equivalent force acting on the soliton due to this disk
interface is a function of the soliton displacement
from this interface:
Fn =
3n0n2
4n0n2
sech4u − u0. 7
To set the soliton in a stable orbit, the total applied
force must vanish at a certain radius, around which
an equivalent mechanical potential well is formed.
Figure 1(b) illustrates the potential well shape,
W=−Fdu, and its two constituents, Wr and Wn, re-
lated to Fr and Fn, respectively. Since the potential
minimum is local, the capture of a soliton in orbit is
possible if the initial position of a stationary soliton is
within this local well. Solitons that are initially out-
side will be swept away from the interface and escape
unless sufficient equivalent kinetic energy that is re-
lated to their initial radial velocity ku is provided to
them. The attracting force in real space is scaled by
1/k
2, thus higher angular velocity (which is propor-
tional to k) reduces the depth of the potential well.
The necessary conditions for obtaining closed-loop
orbital motion are found by equating the two radial
forces acting on the soliton and using the fact that
sech4 is bounded by 0 and 1:
0
n0
2 +
2
3
2
k0R
4n0n2
3n0n2
 1. 8
The left inequality corresponds to n0n2 being posi-
tive. A particularly interesting case is the one of zero
(no interface for low intensity) or positive n0 for
which there is no linearly confined mode in the disk.
For this case, the Kerr coefficient of the inner (disk)
material should be higher than that of the cladding
material. From the right inequality we extract a
threshold value for n2, which is a necessary condi-
tion for supporting an orbital soliton:
minn2 =
n2
n0
n0 +
4n2n02 + 232
k0R
3 . 9
We compare our closed-form results with angular
beam propagation simulations based on the NLSE.9
For a linear positive index step height n0 and
without nonlinear step n2, the soliton escapes.
Fig. 1. (a) Refractive index profile (n0, linear; n2, Kerr). (b)
Potential well formation. =1, 2R /	0=200, u0=u=5, n0
=1.45, n2=0.1, n0=0.1, n2=0.05.
Fig. 2. Simulation trajectories. =1, R=2
100	0 , u
=0, n0=1.45, n2=0.1, k=1000. (a) ku=0, u0=0, n0=0.05.
(b)–(d) n0=0.01, u0=10, n2=0.03. SP, starting point.
February 15, 2006 / Vol. 31, No. 4 / OPTICS LETTERS 487
When the nonlinear step exceeds a threshold level,
the soliton is captured into a circular trajectory.
Equation (9) predicts quite accurately the threshold
value of the index step obtained by simulations, and
the agreement improves for larger k0R values. The
threshold value depends inversely on the radius be-
cause the centrifugal force Fr decreases for a larger
orbital radius, and a smaller balancing force Fn is
required. Increasing the negative linear index step
value necessitates a corresponding increase of the
nonlinear step to keep the soliton in orbit.
Figure 2(a) depicts the formation of the orbiting
soliton in the vicinity of an index step. Soliton trajec-
tories extracted from several beam propagation simu-
lations are shown. They verify that orbital solitons
can evolve even for an opposite (repelling) linear in-
dex of refraction [shown schematically in Fig. 1(a)].
Contrary to regular linear wave packets, which
would definitely escape orbit for such a linear refrac-
tive index profile, the soliton self-carves a circular
ring cavity as a result of nonlinear effects. Decreas-
ing the Kerr index difference would eventually elimi-
nate the potential well in the vicinity of the interface
and cause the soliton to escape. In Figs. 2(b)–2(d) the
soliton is initially displaced from the interface but is
provided with initial “kinetic energy,” namely, a ra-
dial velocity (wavenumber in the radial direction, ku).
Three trajectories are depicted for three different ini-
tial ku starting at the same initial spatial location.
For ku=0.69 the soliton is trapped in orbit.
The potential well analogy allows the identification
of more-complex trajectories than cyclic ones. Any
initial deviation of the soliton from the center of the
well results in oscillations around the orbit. To fulfill
the requirement of a closed-loop trajectory, the oscil-
lations must complete an integer number of periods
in a single round trip. These oscillations, similar to
the light-ray trajectories of geometrical optics, were
revealed in our simulation but exhibited a relatively
small amplitude compared with orbital radius (see
Fig. 3). Figure 3 depicts two examples for oscillating
orbital solitons. Because the induced potential well is
inharmonic, the oscillation period depends on the os-
cillation amplitude (i.e., the initial shift from the
minimum of the potential well), and as a result only
discrete values of the initial shift can satisfy the os-
cillation condition.
To transform the simulation results back to polar
coordinates, r , k should be determined:
k0r = k0Rexp uk0R,  = kk0R2v. 10
The effect of the self-cavity, manifested by the period-
icity of the angular trajectory, is obscured by the co-
ordinate transformation. Transforming the soliton
solution back to the conventional cylindrical coordi-
nates requires direct application of these cyclic
boundary conditions, resulting in an integer-valued
angular k-number k and a requirement for an in-
teger number of amplitude periods, per (correspond-
ing to vper), in one cycle 2. This leads to discrete
values of k0R that basically determine the resonance
wavelength of the nonlinear cavity:
mk = l =
2k0R2
vper
, 11
where m and l are integers and vper is obtained from
the simulation for each k0R.
To conclude, we have presented a new class of soli-
tons circulating in a closed-loop orbit. The centrifugal
force of the solitons is balanced by an attractive cen-
tral force induced by a nonlinear circular interface.
Such orbital solitons exist for a vanishing and even
for a repelling linear step index profile, given that the
nonlinear index difference is high enough. The
threshold value of the onset of orbital motion was cal-
culated and confirmed by numerical simulation. This
threshold can be exploited to achieve a soliton inten-
sity discriminator that deflects high-intensity beams
into a locked circular motion while preserving the
cross-field propagation of a lower-intensity beam.
The stability of the soliton circular motion and addi-
tional interesting cavity related dynamics (possible
bistability due to soliton energy storage) are subjects
for further research.
M. Orenstein’s e-mail address is meiro
@ee.technion.ac.il.
References
1. G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
2. Y. S. Kivshar and G. I. Stegeman, Opt. Photonics News
13, 59 (2002).
3. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner,
J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005).
4. J. Scheuer and M. Orenstein, Opt. Lett. 24, 1735
(1999).
5. E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-
Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh,
Opt. Quantum Electron. 30, 687 (1998).
6. A. B. Aceves, P. Varatharajah, A. C. Newell, E. M.
Wright, G. I. Stegeman, D. R. Heatley, J. V. Moloney,
and H. Adachihara, J. Opt. Soc. Am. B 7, 963 (1990).
7. L. Djaloshinski and M. Orenstein, IEEE J. Quantum
Electron. 35, 737 (1999).
8. M. Heiblum and J. H. Harris, IEEE J. Quantum
Electron. 11, 75 (1975).
9. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed.
(Academic, New York, 1995).
10. H. A. Haus and W. S. Wong, Rev. Mod. Phys. 68, 423
(1996).
Fig. 3. (a) Closed-loop oscillating soliton trajectories for
initial shifts of 5 (dashed curve) and 8.6 (solid curve). (b)
Polar view of the latter. The structure parameters are
Rk0=500, n01=1.5, n02=1.45, and n21=n22=0.0625.
488 OPTICS LETTERS / Vol. 31, No. 4 / February 15, 2006


Circulating spatial solitons

Circulating spatial solitons
Eyal Feigenbaum and Meir Orenstein
Department of Electrical Engineering, Technion, Haifa 32000, Israel
Jacob Scheuer
Center for the Physics of Information, California Institute of Technology, Pasadena, California 91125
Received November 3, 2005; accepted November 9, 2005; posted November 22, 2005 (Doc. ID 65495)
A class of optical spatial solitons exhibiting propagation in a closed-loop orbit in a two-dimensional plane is
presented. A closed-form particlelike model is derived, indicating that the quasi-centrifugal force acting on
these solitons can be balanced by an inhomogeneity in the nonlinear index of refraction. Specifically, a
circular-shaped nonlinear interface is shown to facilitate stable orbital propagation of solitons that carve
their own circular cavity for a wide range of nonlinearity parameters. © 2006 Optical Society of America
OCIS codes: 190.5530, 190.3270.
One-dimensional (1D) spatial optical solitons (i.e.,
nonlinear confinement in one dimension) in Kerr
media were discussed and measured in many
publications.1–3 The dynamics of such solitons
is characterized by linear trajectories within a two-
dimensional (2D) spatial plane (usually implemented
as a slab waveguide). Introduction of material
inhomogeneity (e.g., nonlinear interfaces) yields an
equivalent force that modifies the soliton’s
trajectory.4–6 Here, by applying a circular interface
(i.e., a disk) in a 2D plane, the respective soliton
trajectories are bent to accommodate with a
Keplerian-like orbit about the circumference of the
nonlinear interface. Such solitons, upon completing a
closed-loop trajectory, carve not only their self-
waveguides but also their self-circular ring cavity,
with the potential for realization with relatively
weak light sources.
Complex soliton trajectories, e.g., dipolelike propel-
ler 2D+1 incoherent spatial solitons (which self-
trap in both transverse dimensions) were reported
previously.1 These solitons exhibited a helical trajec-
tory in a nonlinear medium. However, self-generation
of a closed-loop trajectory, namely, self-carving of an
optical cavity by a soliton, is novel to the best of our
knowledge.
In the out-of-plane coordinate, referred to as z, the
simplified 1D soliton is invariant, z=0. In the more
general case the linear confinement along the z axis
(e.g., the slab waveguide mode) introduces a nonvan-
ishing wavenumber kz. This value of kz modifies the
effective linear index of refraction and may impede
the soliton stability.3,7 In the current analysis we em-
ploy a vanishing kz value and focus on the steady-
state dynamics leading to the soliton capture into or-
bit.
Starting with the wave equation in polar coordi-
nates r ,, we employ the slowly varying envelope
amplitude A under the paraxial approximation. An
important distinction from the conventional ap-
proach is that we seek an angularly propagating non-
linear wave packet with a radial envelope, and thus
the paraxial approximation is performed about the
angular coordinate:
2A
r2
+
1
r
A
r
+
2jk
r2
A

+ k02n02 + 2k02n0n2A2 − k2r2A
= 0, 1
where k0 and k are the free space and the angular
wavenumbers, respectively, and n0 ,n2 are the respec-
tive linear and Kerr indices of refraction. By use of
the conformal transformation U=R lnr /R ,V=R,8
the wave equation is transformed into
A
V
=
jR
2k
2A
U2
+
jk0
2n0n2R
k
exp2U
R
A2A
+
j
2k02n02Rk exp2UR − kR 	A, 2
where R is an arbitrary scaling factor. Equation (2)
may have general nonlinear wave solutions, but we
seek solutions similar to the classical solitons of the
nonlinear Schrödinger equation (NLSE).9 A formal
NLSE can be obtained from Eq. (2) by employing a
radially inhomogeneous nonlinear medium satisfying
n0 

k
k0
1
r
, n2 
1
r
. 3
For such index profiles, Eq. (2) exhibits an exact or-
biting soliton solution with a radial envelope propor-
tional to sechlogr /R.
Although the index profile of Eq. (3) supports an
exact soliton solution, it is not practically realizable.
Therefore, we seek circulating soliton solutions in a
simpler configuration comprising two homogeneous
nonlinear media that yield a stepwise inhomogeneity
and focus our analysis on the interface as a source of
soliton perturbation. To understand the required in-
terface characteristics, we first examine Eq. (2) in a
homogeneous medium, as a perturbed NLSE for a
wave packet with a lateral width (U coordinate) sig-
nificantly smaller than its orbital radius. For conve-
nience we choose R as the orbit radius. Equation (2)
in normalized coordinates becomes
486 OPTICS LETTERS / Vol. 31, No. 4 / February 15, 2006
0146-9592/06/040486-3/$15.00 © 2006 Optical Society of America

v
E =
j
2
2E
u2
+ jE2E + jn0
2 + 2E2
u
k0R
E, 4a
where
u 
 k0U, v 

k0
2R
k
V,
4b
Au,v 

Eu,v
n0n2
exp− j k22k02R2 − n0
2
2 v	 ,
Equation (4a) is a perturbed NLSE, which admits a
perturbed soliton solution with intensity profile A2
=2 sech2 , =u−u0, where  is the soliton peak
amplitude and u0 is the soliton center. By use of soli-
ton perturbation theory,10 the perturbation source
[the third term on the right-hand side of Eq. (4a)] in-
troduces an effective force that accelerates the soliton
outward:
Fr =
1
k0R
n02 + 232. 5
The existence of this (fictitious) centrifugal force, Fr,
in a homogeneous medium, is a manifestation of the
fact that we imposed a circular motion on the soliton,
which is following a linear trajectory in homogeneous
media.
Thus, maneuvering the soliton into orbit requires
an opposite force to compensate for Fr. Figure 1(a) de-
scribes the perturbation introduced by circular inter-
face, with general variations in both linear and non-
linear refractive indices. When one applies the step
index profile to the Kerr coefficient in Eq. (2), an ad-
ditional perturbation source is formed in the modified
perturbed nonlinear Schrödinger equation (4a):
sn = j
n0n2
n0n2
1u − uE2E, 1u − u 
 1 u  u0 u  u,
6
where n0n2
n0+n0n2−n2−n0n2. The
equivalent force acting on the soliton due to this disk
interface is a function of the soliton displacement
from this interface:
Fn =
3n0n2
4n0n2
sech4u − u0. 7
To set the soliton in a stable orbit, the total applied
force must vanish at a certain radius, around which
an equivalent mechanical potential well is formed.
Figure 1(b) illustrates the potential well shape,
W=−Fdu, and its two constituents, Wr and Wn, re-
lated to Fr and Fn, respectively. Since the potential
minimum is local, the capture of a soliton in orbit is
possible if the initial position of a stationary soliton is
within this local well. Solitons that are initially out-
side will be swept away from the interface and escape
unless sufficient equivalent kinetic energy that is re-
lated to their initial radial velocity ku is provided to
them. The attracting force in real space is scaled by
1/k
2, thus higher angular velocity (which is propor-
tional to k) reduces the depth of the potential well.
The necessary conditions for obtaining closed-loop
orbital motion are found by equating the two radial
forces acting on the soliton and using the fact that
sech4 is bounded by 0 and 1:
0 
n0
2 +
2
3
2
k0R
4n0n2
3n0n2
 1. 8
The left inequality corresponds to n0n2 being posi-
tive. A particularly interesting case is the one of zero
(no interface for low intensity) or positive n0 for
which there is no linearly confined mode in the disk.
For this case, the Kerr coefficient of the inner (disk)
material should be higher than that of the cladding
material. From the right inequality we extract a
threshold value for n2, which is a necessary condi-
tion for supporting an orbital soliton:
minn2 =
n2
n0
n0 +
4n2n02 + 232
k0R
3 . 9
We compare our closed-form results with angular
beam propagation simulations based on the NLSE.9
For a linear positive index step height n0 and
without nonlinear step n2, the soliton escapes.
Fig. 1. (a) Refractive index profile (n0, linear; n2, Kerr). (b)
Potential well formation. =1, 2R /	0=200, u0=u=5, n0
=1.45, n2=0.1, n0=0.1, n2=0.05.
Fig. 2. Simulation trajectories. =1, R=2
100	0 , u
=0, n0=1.45, n2=0.1, k=1000. (a) ku=0, u0=0, n0=0.05.
(b)–(d) n0=0.01, u0=10, n2=0.03. SP, starting point.
February 15, 2006 / Vol. 31, No. 4 / OPTICS LETTERS 487
When the nonlinear step exceeds a threshold level,
the soliton is captured into a circular trajectory.
Equation (9) predicts quite accurately the threshold
value of the index step obtained by simulations, and
the agreement improves for larger k0R values. The
threshold value depends inversely on the radius be-
cause the centrifugal force Fr decreases for a larger
orbital radius, and a smaller balancing force Fn is
required. Increasing the negative linear index step
value necessitates a corresponding increase of the
nonlinear step to keep the soliton in orbit.
Figure 2(a) depicts the formation of the orbiting
soliton in the vicinity of an index step. Soliton trajec-
tories extracted from several beam propagation simu-
lations are shown. They verify that orbital solitons
can evolve even for an opposite (repelling) linear in-
dex of refraction [shown schematically in Fig. 1(a)].
Contrary to regular linear wave packets, which
would definitely escape orbit for such a linear refrac-
tive index profile, the soliton self-carves a circular
ring cavity as a result of nonlinear effects. Decreas-
ing the Kerr index difference would eventually elimi-
nate the potential well in the vicinity of the interface
and cause the soliton to escape. In Figs. 2(b)–2(d) the
soliton is initially displaced from the interface but is
provided with initial “kinetic energy,” namely, a ra-
dial velocity (wavenumber in the radial direction, ku).
Three trajectories are depicted for three different ini-
tial ku starting at the same initial spatial location.
For ku=0.69 the soliton is trapped in orbit.
The potential well analogy allows the identification
of more-complex trajectories than cyclic ones. Any
initial deviation of the soliton from the center of the
well results in oscillations around the orbit. To fulfill
the requirement of a closed-loop trajectory, the oscil-
lations must complete an integer number of periods
in a single round trip. These oscillations, similar to
the light-ray trajectories of geometrical optics, were
revealed in our simulation but exhibited a relatively
small amplitude compared with orbital radius (see
Fig. 3). Figure 3 depicts two examples for oscillating
orbital solitons. Because the induced potential well is
inharmonic, the oscillation period depends on the os-
cillation amplitude (i.e., the initial shift from the
minimum of the potential well), and as a result only
discrete values of the initial shift can satisfy the os-
cillation condition.
To transform the simulation results back to polar
coordinates, r , k should be determined:
k0r = k0Rexp uk0R,  = kk0R2v. 10
The effect of the self-cavity, manifested by the period-
icity of the angular trajectory, is obscured by the co-
ordinate transformation. Transforming the soliton
solution back to the conventional cylindrical coordi-
nates requires direct application of these cyclic
boundary conditions, resulting in an integer-valued
angular k-number k and a requirement for an in-
teger number of amplitude periods, per (correspond-
ing to vper), in one cycle 2. This leads to discrete
values of k0R that basically determine the resonance
wavelength of the nonlinear cavity:
mk = l =
2k0R2
vper
, 11
where m and l are integers and vper is obtained from
the simulation for each k0R.
To conclude, we have presented a new class of soli-
tons circulating in a closed-loop orbit. The centrifugal
force of the solitons is balanced by an attractive cen-
tral force induced by a nonlinear circular interface.
Such orbital solitons exist for a vanishing and even
for a repelling linear step index profile, given that the
nonlinear index difference is high enough. The
threshold value of the onset of orbital motion was cal-
culated and confirmed by numerical simulation. This
threshold can be exploited to achieve a soliton inten-
sity discriminator that deflects high-intensity beams
into a locked circular motion while preserving the
cross-field propagation of a lower-intensity beam.
The stability of the soliton circular motion and addi-
tional interesting cavity related dynamics (possible
bistability due to soliton energy storage) are subjects
for further research.
M. Orenstein’s e-mail address is meiro
@ee.technion.ac.il.
References
1. G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
2. Y. S. Kivshar and G. I. Stegeman, Opt. Photonics News
13, 59 (2002).
3. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner,
J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005).
4. J. Scheuer and M. Orenstein, Opt. Lett. 24, 1735
(1999).
5. E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-
Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh,
Opt. Quantum Electron. 30, 687 (1998).
6. A. B. Aceves, P. Varatharajah, A. C. Newell, E. M.
Wright, G. I. Stegeman, D. R. Heatley, J. V. Moloney,
and H. Adachihara, J. Opt. Soc. Am. B 7, 963 (1990).
7. L. Djaloshinski and M. Orenstein, IEEE J. Quantum
Electron. 35, 737 (1999).
8. M. Heiblum and J. H. Harris, IEEE J. Quantum
Electron. 11, 75 (1975).
9. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed.
(Academic, New York, 1995).
10. H. A. Haus and W. S. Wong, Rev. Mod. Phys. 68, 423
(1996).
Fig. 3. (a) Closed-loop oscillating soliton trajectories for
initial shifts of 5 (dashed curve) and 8.6 (solid curve). (b)
Polar view of the latter. The structure parameters are
Rk0=500, n01=1.5, n02=1.45, and n21=n22=0.0625.
488 OPTICS LETTERS / Vol. 31, No. 4 / February 15, 2006


Nonlinear Fiber Optics, 2nd ed.

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Circulating spatial solitons

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