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Discrete solitons in photorefractive optically induced photonic lattices
Nikos K. Efremidis, Suzanne Sears, and Demetrios N. Christodoulides
Department of Electrical and Computer Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
Jason W. Fleischer and Mordechai Segev
Physics Department, Technion-Israel Institute of Technology, Haifa 32000, Israel
and Electrical Engineering Department, Princeton University, Princeton, New Jersey 08544
共Received 19 February 2002; published 2 October 2002兲
We demonstrate that optical discrete solitons are possible in appropriately oriented biased photorefractive
crystals. This can be accomplished in optically induced periodic waveguide lattices that are created via plane-
wave interference. Our method paves the way towards the observation of entirely new families of discrete
solitons. These include, for example, discrete solitons in two-dimensional self-focusing and defocusing lattices
of different group symmetries, incoherently coupled vector discrete solitons, discrete soliton states in optical
diatomic chains, as well as their associated collision properties and interactions. We also present results
concerning transport anomalies of discrete solitons that depend on their initial momentum within the Brillouin
zone.
DOI: 10.1103/PhysRevE.66.046602 PACS number共s兲: 42.65.Tg, 42.65.Hw
Wave propagation in nonlinear periodic lattices is associ-
ated with a host of exciting phenomena that have no coun-
terpart whatsoever in bulk media. Perhaps, the most intrigu-
ing entities that can exist in such systems are discrete self-
localized states—better known as discrete solitons 共DS兲
关1–4兴. By their very nature, these intrinsically localized
modes represent collective excitations of the chain as a
whole, and are the outcome of the balance between nonlin-
earity and linear coupling effects. Over the years, discrete
solitons have been a topic of intense investigation in several
branches of science such as biological physics 关1兴, nonlinear
optics 关2兴, Bose-Einstein condensates 关3兴, and solid state
physics 关4兴.
In optics, discrete solitons have been predicted in nonlin-
ear waveguide arrays 关2兴 and most recently in chains of
coupled microcavities embedded in photonic crystals 关5兴.
Thus far, nonlinear optical waveguide arrays have provided a
fertile ground for the experimental observation and study of
discrete solitons 关6–9兴. Both in-phase bright 关6兴 as well as
staggered (
out-of-phase兲 darklike DS 关9兴 have been suc-
cessfully demonstrated in one-dimensional 共1D兲 self-
focusing Al
x
Ga
1⫺ x
As arrays. Along these lines, DS transport
under the action of Peierls-Nabbaro effects 关7兴 and diffrac-
tion management 关8兴 has been investigated in such systems.
In addition, several other exciting theoretical predictions
have been made. These include among others, soliton inter-
actions and beam steering 关10兴, out-of-phase bright discrete
solitons 关11兴, discrete solitons in two-dimensional lattices
关12兴, vector-discrete solitons 关13兴,DSin
(2)
arrays 关14兴, and
diffraction managed solitons 关15兴. Furthermore, it has been
shown that discrete solitons hold great promise in terms of
realizing intelligent functional operations such as blocking,
routing, logic functions, and time gating in two-dimensional
DS array optical networks 关16,17兴. Yet, to date, only a small
subset of the plethora of such interesting predictions has ac-
tually been demonstrated at the experimental level. This is
partly due to the fact that such arrays have only been imple-
mented in single-row topologies 共on the surface of a wafer兲
using a particular self-focusing material system. Establishing
two-dimensional waveguide array lattices in the bulk is an
even more complicated task. It is, therefore, important to
identify highly versatile nonlinear lattice systems where such
DS entities can be observed, especially at low power levels.
In this paper, we show that optical spatial discrete solitons
are possible in appropriately oriented biased photorefractive
crystals. This can be accomplished through the screening
nonlinearity 关18兴 in optically induced waveguide periodic
lattices that are established via plane wave interference. To
do so, we exploit the large electro-optic anisotropy that is
possible in certain families of crystals that, in turn, allows
invariant propagation of 1D and 2D periodic intensity pat-
terns. Our method offers exciting possibilities towards the
observation of entirely new families of spatial discrete soli-
tons at milliwatt power levels. These include, for example,
discrete solitons in two-dimensional self-focusing and defo-
cusing lattices of different group symmetries 关19兴共i.e.,
square, rectangular, hexagonal, etc.兲, incoherently coupled
vector discrete solitons, discrete soliton states in optical di-
atomic chains, as well as their associated collision properties
and interactions 关16兴. Moreover we note, that our scheme
offers considerable flexibility in the sense that the same pho-
torefractive waveguide array 共1D or 2D兲 can be of the self-
focusing or defocusing type 共depending on the polarity of the
external bias 关18兴兲 with adjustable lattice parameters. In ad-
dition, we present results concerning transport anomalies of
DS that depend on the initial momentum within the Brillouin
zone. These transport properties can only be identified in
semidiscrete systems, such as the one presented here, and are
not encountered in fully discrete systems described by the
tight-binding approximation. This occurs, whenever, DS
waves exhibit nonzero transverse momentum, as a result of
radiation modes.
We begin our analysis by considering a biased photore-
fractive crystal as shown in Fig. 1. For demonstration pur-
poses, let the crystal be of the Strontium Barium Niobate
type 共SBN:75兲 having length L and width W in both trans-
PHYSICAL REVIEW E 66, 046602 共2002兲
1063-651X/2002/66共4兲/046602共5兲/$20.00 ©2002 The American Physical Society66 046602-1
verse dimensions. The SBN sample is externally biased
along its extraordinary x axis 共crystalline c axis兲 with voltage
V. The refractive index along the extraordinary axis is n
e
⫽ 2.299, whereas that along the ordinary (y axis兲 is n
0
⫽ 2.312. The relevant electro-optic coefficients of this crys-
tal are r
33
⫽ 1340 pm/V and r
13
⫽ 67 pm/V and the wave-
length of the lightwaves used is taken here to be
0
⫽ 0.5
m. In this case an x-polarized wave will see a refrac-
tive index n
e
⬘
2
⫽ n
e
2
⫺ n
e
4
r
33
E
sc
while the corresponding n
0
⬘
for
a y-polarized wavefront is given by n
0
⬘
2
⫽ n
0
2
⫺ n
0
4
r
13
E
sc
,
where E
sc
x
ˆ
is the external space-charge field under external
bias.
Next, we identify methods to establish optically induced
waveguide lattices in the bulk of the photorefractive crystal,
where discrete solitons are expected to occur. Such stationary
1D or 2D array lattices can be photoinduced by periodic
diffraction-free intensity patterns that result from plane-wave
superposition 共provided that the system is linear for the in-
terfering waves兲. In the suggested configuration of Fig. 1,
this is accomplished by linearly polarizing these plane waves
along the ordinary y axis 共since r
13
Ⰶ r
33
) and, therefore,
propagation along z is essentially linear. On the other hand,
it is important to note that these same induced waveguides
are highly nonlinear for extraordinary polarized waves be-
cause of the large value of r
33
. For example, a one-
dimensional periodic intensity pattern I⫽I
0
cos
2
关
k
2
sin(
)x
兴
can be generated from the interference of two plane waves
y
ˆ
E
0
exp
关
⫾ik
2
sin(
)x
兴
exp
关
ik
2
cos(
)z
兴
, where k
2
⫽ k
0
n
0
, k
0
⫽ 2
/
0
, and ⫾
is the angle at which these two plane
waves propagate with respect to the z axis. The spatial period
of this array lattice is D⫽
0
/(2n
0
sin
) and is, therefore,
highly adjustable with
or with the wavelength
0
. Using
two orthogonal mutually incoherent plane-wave pairs 2D
‘‘crystals’’ can be established from a diffraction-free inten-
sity pattern I⫽ I
0
兵
cos
2
关
k
2
sin(
1
)x
兴
⫹cos
2
关
k
2
sin(
2
)y)
兴
其
.Inad-
dition, such 2D structures can also be created by coherent
superposition of four plane waves in which case I
⫽ I
0
cos
2
关
k
2
sin(
1
)x
兴
cos
2
关
k
2
sin(
2
)y
兴
. These waveguide ar-
rays can be rectangular or square depending whether
1
⫽
2
or not. More complicated 共hexagonal, etc.兲 nonlinear
lattices can be generated by superimposing two or more mu-
tually incoherent plane-wave pairs at different angles. We
emphasize again that what makes this possible is the large
electro-optic anisotropy (r
33
vs r
13
) of the photorefractive
crystal. This allows almost diffraction-free propagation of
ordinary polarized periodic patterns and highly nonlinear
evolution for extraordinary polarized waves.
We first consider a one-dimensional array configuration.
In this case one can show that the spatial evolution dynamics
of both the discrete soliton and the optically induced lattice
fields in a biased photorefractive SBN crystal is governed by
the following set of equations 关20兴:
iu
z
⫹
1
2k
1
u
xx
⫺
k
0
n
e
3
r
33
2
E
sc
u⫽ 0, 共1兲
i
v
z
⫹
1
2k
2
v
xx
⫺
k
0
n
0
3
r
13
2
E
sc
v
⫽ 0, 共2兲
where k
1
⫽ k
0
n
e
, E
sc
is the steady state space-charge field
given by 关18兴,
E
sc
⫽
E
0
1⫹ I
共
x
兲
⫺
K
B
T
e
I/
x
1⫹ I
共
x
兲
, 共3兲
and I⫽
兩
u
兩
2
⫹
兩
v
兩
2
is the normalized total intensity with re-
spect to the dark irradiance of the crystal I
d
关18兴.InEq.共3兲
the first term associated with E
0
describes the dominant 共un-
der appreciable external bias兲 screening nonlinearity of the
photorefractive crystal, whereas the second term accounts for
weak diffusion effects that have been incorporated for com-
pleteness in this discussion. K
B
is the Boltzmann constant, T
is the temperature, and e is the electron charge. u represents
the x-polarized discrete soliton field that is affected by the
strong r
33
nonlinearity, and
v
is the y-polarized periodic field
共evolving almost linearly兲 responsible for setting up the
waveguide lattice. In addition, under a constant bias V, the
following constraint holds true along z, V⫽
兰
0
W
E
sc
dx.
Using numerical relaxation methods, we obtained discrete
soliton solutions in this system. The dynamical evolution of
these states is then examined by exactly solving Eqs. 共1兲–共3兲
under a constant bias V. As an example, let the dimensions of
the SBN crystal be L⫽ W⫽ 6 mm. Let also the normalized
v
field at the input be
v
⫽
v
0
cos(
x/D), where here
兩
v
0
兩
2
⫽ 2.56 and D⫽ 9
m. The periodic
v
field is assumed to
cover the entire W⫻W input face of the crystal. The applied
voltage across W is taken to be 325 V, which corresponds to
an E
0
⯝(V/W)
冑
1⫹
v
0
2
⫽ 102 V/mm and leads to a self-
focusing nonlinearity. Under these conditions, the refractive
index change between the maxima and minima of the in-
duced waveguides is approximately 6⫻10
⫺ 4
. Figure 2共a兲
depicts the propagation dynamics of a well confined in-phase
DS when its normalized peak intensity
兩
u
0
兩
2
⫽ 0.36. As it is
illustrated in this figure, this DS state propagates in an in-
variant fashion along z. In addition, our simulations indicate
that the 1D waveguide lattice, as induced by the
v
field,
remains essentially undistorted over the length of this crystal
despite of the presence of small diffusion effects. Note that
the peaks of the DS reside on the maxima of the
兩
v
兩
2
inten-
sity pattern, since the system is of the self-focusing type. If,
on the other hand, the intensity of the same field pattern has
FIG. 1. A biased photorefractive crystal illuminated by a peri-
odic intensity pattern created through the interference of plain wave
pairs.
NIKOS K. EFREMIDIS et al. PHYSICAL REVIEW E 66, 046602 共2002兲
046602-2
been appreciably reduced, the beam expands considerably, as
shown in Fig. 3共a兲. In this case, light tends to oscillate in the
photorefractive waveguides as a result of beam self-bouncing
effects 关21兴. In addition to in-phase bright DS staggered dark
solitons are also possible in this self-focusing system pro-
vided that the phase shift among sites is
. We emphasize
again that these DS can be observed at low power levels
共milliwatts兲 because of the high nonlinearity that is offered
by the photorefractive crystal. By reversing the polarity of
the applied voltage, the nonlinearity of the lattice becomes
defocusing. In this regime, the induced waveguide sites are
located on the dark regions of the
兩
v
兩
2
periodic intensity
pattern. In such defocusing lattices, two families of DS exist.
These are in-phase dark solitons 共at the center of the Bril-
louin zone兲 and staggered (
out of phase兲 bright solitons at
the edge of the Brillouin zone 关11兴. Figure 2共b兲 depicts the
propagation dynamics of a staggered bright soliton. This DS
solution was obtained numerically for
兩
u
0
兩
2
⫽ 0.36,
兩
v
0
兩
2
⫽ 4, D⫽ 9
m, and by assuming again that the
v
field cov-
ers the entire crystal. The applied voltage in this case is
⫺ 182 V. Note that this particular type of DS solution can
not exist in the bulk and is purely the outcome of discrete-
ness. The diffraction dynamics of the field pattern shown in
Fig. 2共b兲, when the intensity is considerably reduced, is de-
picted in Fig. 3共b兲.
We would like to emphasize that there are important dif-
ferences between the soliton families found in the system
examined here and the DS solutions as obtained from a dis-
crete nonlinear Schro
¨
dinger 共DNLS兲 equation 关1兴. One such
major difference appears in their respective transport proper-
ties 共steering兲 in these models. This is due to the fact that the
DNLS equation accounts only for bound states, whereas con-
tinuous models 关Eqs. 共1兲–共2兲兴 with periodic potentials 共semi-
discrete兲 also account for radiation modes. In the latter case,
a general excitation,
兩
典
can be described in terms of a com-
plete set of bound states,
兩
n
典
, and a radiation mode con-
tinuum,
兩
R(
␣
)
典
, i.e.,
兩
典
⫽
兺
n
c
n
兩
n
典
⫹
冕
共
␣
兲
兩
R
共
␣
兲
典
d
␣
. 共4兲
For example if, during excitation, the discrete soliton mo-
mentum is 0 or 2
within the Brillouin zone, the DNLS
model predicts exactly the same behavior since its solution
remains invariant. However, this is not the case in the system
described here. Figure 4共a兲 depicts the propagation of a DS
关of the same field profile as that of Fig. 2共a兲兴 when it is
excited at an angle corresponding to 2
in the Brillouin
zone. Evidently, the transport dynamics are totally different
from that of Fig. 2共a兲 and can not be explained from the
DNLS model: the DS is no longer immobile in the lattice and
tends to deteriorate very fast. These transport anomalies are
FIG. 2. Invariant propagation of a DS in a 1D photorefractive
optically induced potential 共a兲 for a bright in-phase DS; 共b兲 for a
staggered bright DS.
FIG. 3. Diffraction dynamics 共a兲 of the in-phase DS and 共b兲 of
the staggered DS shown in Figs. 2共a兲 and 2共b兲, respectively, when
their intensities are considerably reduced.
DISCRETE SOLITONS IN PHOTOREFRACTIVE... PHYSICAL REVIEW E 66, 046602 共2002兲
046602-3
due to the presence of a radiation continuum. In fact, for
single mode local potentials, the amount of power escaping
in the radiation modes is approximately given by r⫽ 1
⫺
兩
具
0
兩
0
典
兩
2
⬇1⫺ exp
关
⫺(qw/2D)
2
兴
, where w is the spatial
extend of the local Wannier function, and q is the initial
momentum. These estimates are in very good agreement
with the results of Fig. 4共a兲. Similarly Fig. 4共b兲 shows the
transport dynamics of a DS at 2
, when D is 14
m. In this
case, the transport anomalies are significantly reduced, since
w/D is now smaller by a factor 1.5. In addition, we have
found that there are also differences between these two mod-
els in connection to in-phase and staggered bright discrete
solitons. In the DNLS limit these two classes happen to be
fully identical, i.e., they share the same profile and properties
共since the one can be deduced from the other through a
trivial
phase transformation兲. On the other hand, in con-
tinuous periodic lattices 共as in photorefractives兲 we found
that the profile and behavior of staggered DS can not be
extracted from the in-phase family.
Similarly two-dimensional DS are also possible in opti-
cally induced photonic lattices in biased photorefractive
crystals. As previously mentioned, such lattices can be estab-
lished in the bulk by coherently superimposing two plane-
wave pairs. In this way tetragonal, hexagonal, etc, array
structures can be created. For example, Fig. 5 shows an in-
phase two-dimensional bright DS in a square and a hexago-
nal lattice with D⫽ 15
m, as obtained using relaxation
methods. This solution was obtained numerically by assum-
ing, for simplicity, an isotropic model for the photorefractive
nonlinearity
关
⌬n
NL
⬀⫺ 1/(1⫹ I(x,y))
兴
and by neglecting
small diffusion effects. As a result of the saturable photore-
fractive nonlinearity these 2D, DS happen to be stable. Other
more involved types of 2D DS, such as staggered states, are
also possible in these lattices. In addition, our scheme offers
unique opportunities to study diffraction management 关8兴 in
a two-dimensional environment.
In conclusion, we have shown that optical discrete soli-
tons are possible in appropriately oriented biased photore-
fractive crystals. This can be accomplished in optically in-
duced periodic waveguide lattices that are created via plane-
wave interference. Our method paves the way towards the
observation of entirely new families of discrete solitons, such
as discrete solitons in two-dimensional self-focusing and de-
focusing lattices of different group symmetries, incoherently
coupled vector discrete solitons, discrete soliton states in op-
tical diatomic chains. Before closing, we would like to note
that a possible observation of such families of DS may have
an impact in other areas of physics that share similar dynam-
ics, such as for example Bose-Einstein condensates in light-
induced periodic potentials 关3,22兴.
This work was supported by ARO MURI, the National
Science Foundation, and by a grant from the Pittsburgh Su-
percomputing Center.
FIG. 4. Transport dynamics of a DS when q⫽ 2
and 共a兲 D
⫽ 9
m, 共b兲 D⫽ 14
m.
FIG. 5. A 2D in-phase discrete soliton in a biased photorefrac-
tive crystal 共a兲 square lattice and 共b兲 in a hexagonal lattice.
NIKOS K. EFREMIDIS et al. PHYSICAL REVIEW E 66, 046602 共2002兲
046602-4
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DISCRETE SOLITONS IN PHOTOREFRACTIVE... PHYSICAL REVIEW E 66, 046602 共2002兲
046602-5
