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Two-Dimensional Optical Lattice Solitons
Nikolaos K. Efremidis,1Jared Hudock,1Demetrios N. Christodoulides,1Jason W. Fleischer,1,2
Oren Cohen,2and Mordechai Segev2
1School of Optics/CREOL, University of Central Florida, Florida 32816-2700, USA
2Physics Department, Technion-Israel Institute of Technology, Haifa 32000, Israel
(Received 16 April 2003; published 21 November 2003)
We study various families of two-dimensional discrete or lattice solitons, and show that they are
possible only when their power level exceeds a critical threshold. In addition, we show that gap-lattice
solitons exist only when the lattice possesses a complete 2D band gap. Our results suggest that these
conditions are universally valid, irrespective of the nature of the nonlinearity or the specific structure
of the index lattice.The analysis explains fundamental aspects of behavior of two-dimensional discrete
solitons that have been very recently observed in photosensitive optical crystals.
DOI: 10.1103/PhysRevLett.91.213906PACS numbers: 42.65.Tg
Wave propagation in periodic lattices is known to ex-
hibit several fundamental features that arise from the
presence of allowed bands and forbidden gaps. One of
the most intriguing outcomes of nonlinearity in such
periodic systems is the existence of self-localized entities
better known as discrete or lattice solitons (LS). This
family of self-localized modes has recently attracted
considerable attention in diverse branches of science
such as biological physics [1], nonlinear optics [2], solid
state physics [3], and Bose-Einstein condensates [4].Thus
far, nonlinear waveguide lattices [2,5] have provided the
only fertile environment for the direct experimental ob-
servation and study of one-dimensional LS [6–8]. Yet, in
spite of the progress made in the past few years in 1D
topologies [6,9], very little is known regarding lattice
solitons in higher dimensions, where they are expected
to exhibit a much richer behavior [10].
Earlier this year, two-dimensional LS were reported
for the first time [11]. In this experiment, two basic
families of discrete solitons were demonstrated: in-phase
lattice solitons under self-focusing conditions and ? out-
of-phase gap soliton states in a defocusing environment
[11]. Surprisingly, in this experiment it was found that 2D
gap LSare possible even in‘‘backbone’’lattices [shown in
Fig. 1(b)] that totally lack an absolute potential minimum
on site. These observations indicate that the transition
from solitons in 1D lattices to those in higher dimensions
is far from being simple. In view of this, several funda-
mental issues must be addressed. For example, what are
the basic features of 2D LS and how do they differ from
their 1D counterparts? What are the required band gap
properties (partial or complete) for LS to exist? What
lattices possess the appropriate band structure to support
2D LS?
In this Letter, we study various families of solitons in
2D square lattices and show that they are possible only
when their power level exceeds a critical threshold. This
is in sharp contradistinction to 1D LS for which no such
threshold exists [12,13] and is closely associated with the
stabilityofsolitonsin 2Dlattices. Inaddition,wefindthat
a full two-dimensional photonic band gap is required for
a gap LS soliton to exist. Thus, only potentials that are
deep enough can support LS.The properties as well as the
parameters of the index potential lattice necessary to
establish a full band gap are presented. Our results are
in good agreement with the experimental observations of
Ref. [11]. Our analysis provides valuable information for
the experimental realization of discrete and gap LS in
other types of 2D photonic lattices (waveguide arrays and
arrays embedded in photonic crystal fibers) and in other
physical settings such as Bose-Einstein condensates [14].
The results presented here are applicable to other types of
lattices, such as hexagonal, etc.
We analyze two generic, experimentally feasible, types
of lattices: a sinusoidal lattice and a backbone lattice.We
study the linear properties (band structure) of these lat-
tices and consider the two most common types of optical
nonlinearities; namely, the (i) Kerr and the (ii) saturable
nonlinearity. In the Kerr case, the evolution of the slowly
varying amplitude of the light in a periodic array is
described by the nonlinear Schro ¨dinger equation:
i@
2r2
@z?1
? ? V?x;y? ? ?j j2 ? 0;
(1)
FIG. 1.
defocusing backbone lattice when A2? 1:21.
Index potential of (a) a sinusoidal lattice and (b) of a
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where r2
tential, ? ? ?1 determines the nature of the medium
nonlinearity (focusing/defocusing), and the field and co-
ordinates in Eq. (1) have been normalized for conve-
nience. For the purposes of this study, we assume that
the 2D index potential is sinusoidal, e.g.,
V?x;y? ? ??V0=2??sin2??x? ? sin2??y??;
where the period of the square lattice is 1 and V0is the
index depression depth [see Fig. 1(a)]. Note that the linear
eigenvalue problem associated with this potential is ex-
actly solvable in terms of Mathieu functions [15].
For the saturable case, we assume a nonlinearity of the
form ?1=?1 ? j j2? that better suits the description of
photorefractive crystals where 2D LS have been recently
observed [11]. In this case, the waveguide lattice is opti-
cally induced and the nonlinear evolution equation for
the optical soliton field obeys a saturable nonlinear
Schro ¨dinger equation (see Ref. [16] for details):
?? @2
x? @2
y, V?x;y? is a periodic index po-
(2)
i z?1
2r2
? ?
V0
1 ? I?x;y? ? j j2? 0;
intensity
(3)
where
A2cos2??x?cos2??y? is created through the interfer-
ence of two pairs of laser beams that are coherently
superimposed in order to establish an optical lattice.
This intensity pattern is a 45?rotation of the expres-
sion I?x;y? ? ?A2=4?fcos???x ? y?? ? cos???x ? y??g2of
Ref. [11]. The potential depth as well as the sign and
magnitude of the nonlinearity can be controlled by V0,
which is proportional to the voltage applied to the photo-
refractive crystal. In Eq. (3), under linear conditions, the
index potential V?x;y? is given by [17]
V?x;y? ? ?V0=?1 ? I?x;y??:
Figure 1 depicts the lattice potentials of interest.
Figure 1(a) shows the potential of Eq. (2) and has the
same structure for both the focusing and defocusing Kerr
nonlinearity. The focusing case for the photorefractive
nonlinearity has a similarly modulated potential and is
not shown. The potential for the defocusing saturable
nonlinearity of Eq. (4) is shown in Fig. 1(b). Note that,
unlike the lattice of Fig. 1(a), which exhibits absolute
index maxima on site, the profile of Fig. 1(b) is consid-
erably different. In this latter case, the index maxima are
not isolated but instead are located along crossed ridges,
thus forming a backbone structure, as clearly shown in
Fig. 1(b).
To find the associated band structure, we assume that
the linear versions of Eqs. (1) and (3) admit solutions
of the form ? exp??iqz?u?x;y?. In this case, the fol-
lowing eigenvalue problem results: qu ? ?1=2?r2
V?x;y?u ? 0. Following Bloch’s theorem, the eigenfunc-
tion, uk?x;y? can be written as uk?x;y? ? Uk?x;y? ?
exp?ik ? r?, where Uk?x;y? is periodic (in r) having the
period of the lattice, k ? ^ x xkx? ^ y ykyis its Bloch momen-
thenormalized pattern
I?x;y? ?
(4)
?u ?
tum, and r ? ^ x xx ? ^ y yy. This eigenvalue problem is then
solved numerically.
Figure 2(a) [2(b)] corresponds to the band structure of
the potential shown in Fig. 1(a) [1(b)]. As can be seen,
different bands in these structures can overlap with each
other, restricting (or even eliminating altogether) the
number of complete band gaps. This is a general property
of 2D lattices and is in direct contrast to1D arrays, which
are generically associated with an infinite number of
complete Bragg resonance band gaps. In two dimensions,
the number of full band gaps depends on the properties of
the specific potential (depth, period, and form).
Figure 3(a) shows the eigenvalue (energy) bands (de-
picted in shaded regions) as a function of the potential
depth V0of Eq. (2). In this case, when the potential V0is
less than 13.8, we find that no complete band gap exists.
For greater values of V0, an indirect full band gap opens
up between the ??;?? k vector of the first Brillouin zone
and the ???;0?, ?0;??? kvectors of the second Brillouin
zone. By increasing the potential depth further, a second
band gap opens between the third and the fourth band
when V0? 40:7. For evengreater values of V0, more band
gaps start to form. For the Kerr optical lattice of Eq. (1),
the refractive index modulation, ?, is related to the nor-
malized potential depth via V0? ?2?n0a=?0?2?. As an
example, for wavelength ?0? 0:5 ?m, lattice spacing
a ? 11 ?m, and refractive index n0? 2:3, the minimum
value of ? required for a complete band gap is equal to
1:37 ? 10?4.
The backbone potential of Fig. 1(b) exhibits different
behavior. As we can see in Fig. 3(b), when the potential
depth is less than 28.4 (and A2? 1:21) all the bands
overlap, i.e., no complete band gap exists. Using the
same parameters as in the previous example, this poten-
tial depth corresponds to ? ? 1:05 ? 10?4. On the other
hand, for bigger values of V0, one gap opens up between
the first and the second bands. Unlike the sinusoidal
lattice case, we find that no other band gaps emerge for
even greater values of V0. This seems to be a general
π
0
ky
−ππ
−π
0
kx
10
20
30
40
50
60
ky
ky
ky
ky
ky
kx
kx
kx
kx
kx
(a)(a)(a) (a)(a)(a)
q q q q q q
π
0
ky
−π π
−π
0
kx
−30
−20
−10
0
10
ky
kx
(b)(b)
q q
FIG. 2.
(a) of Fig. 1(a) when V0? 21:6 (b) of Fig. 1(b), for A2? 1:21,
and V0? ?36:3.
Band structure corresponding to the lattice potential
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feature of backbone lattices and results from the single-
mode nature of this type of potential. The backbone
structure resembles an array of ridge waveguides, where
the effective refractive index at the center of every inter-
section is increased relative to other points on the grid.
Waveguiding is weak at such a junction, and simulations
of a single intersection (isolated ‘‘cross’’) show that the
corresponding potential can support only a single bound
mode and thus a single band gap. In principle, a gap can
open up between two radiation bands, as it does in 1D
structures [18], but we do not observe such a process in
any 2D square lattice.
To find the lattice soliton solutions of Eq. (1) [and
Eq. (3)], we first assume that ?x;y? ? u?x;y? ?
exp??iqz?. The resulting static or ‘‘time-independent’’
nonlinear Schro ¨dinger equation is then solved numeri-
cally using relaxation methods. Two families of bright
solitons were identified: self-focusing LS that exist in the
semi-infinite band gap as well as self-defocusing LS that
exist between the first and the second band.
Typical soliton structures for the Kerr nonlinearity and
their corresponding existence curves are shown in Figs. 4
and 5 [19]. Theoretically, self-focusing 2D LS were first
found in [16]. Subsequently, It was also shown that such
structures can be induced by modulational instability
[20]. The self-focusing (in-phase) soliton shown in
Fig. 4(a) exists in the semi-infinite band gap below the
first band. When its corresponding eigenvalue is close to
the lowest energy band, the LS are broad, with the width
of the soliton narrowing as the eigenvalue is lowered. In
the self-defocusing case, represented by the ? out-of-
phase gap soliton [between bands 1 and 2 in Fig. 3(b)],
this behavior is reversed. If its corresponding eigenvalue
is close to its lower limit (the first band), the solutions are
broad, whereas as the eigenvalue increases towards the
second band, the LS get narrower.When the LS gets very
close to the second band, its tails expand to occupy many
lattice sites, and the LS itself exhibits a cusplike behavior.
As in the focusing case, a complete band gap is always
required; i.e., shallow potentials do not support LS. If the
bandgap isonly partial(a situation not encountered in1D
or for focusing potentials),an input beam will radiate due
to interaction with the linear spectrum in the transverse
directions lacking a gap.
The power P ?R Rjuj2dxdy conveyed by the solitons
versus the eigenvalue q is shown in Figs. 4(b) and 5(b).
Note that there is a minimum power threshold required in
order to observe a lattice soliton intwo dimensions. Inthe
1D case, such a threshold does not exist [12,13].We would
like tomention that inthe case of a semi-infinite band gap
our results are in agreement with the discrete nonlinear
Schro ¨dinger case [21] as rigorously proven in [22].
The existence curves also give information on the
stability of the solitons. For the focusing case, Fig. 4(b),
the stability can be determined by a straightforward
application of the Vakhitov-Kolokolov criterion [23].
More specifically, when @P=@q < 0, the solutions are
stable, while close to the band @P=@q > 0 and the lattice
solitons become unstable.This analysis cannot be applied
directly to the defocusing case, Fig. 5(b), because gap LS
represent higher order modes. For behavior close to the
first band, the stability of these defocusing solutions can
FIG. 4.
curve for a Kerr self-focusing in-phase LS with V0? 28:8.
(a) The field profile and (b) the associated P ? q
FIG. 5.
curve for a Kerr self-defocusing gap LS with V0? 21:6.
(a) The field profile and (b) the associated P ? q
0204060
0
10
20
30
40
V0
q
(a)
02040
V0
6080
−70
−60
−50
−40
−30
q
−20
−10
0
(b)
FIG. 3.
potential depth V0for (a) the sinusoidal lattice of Fig. 1(a) and
(b) for the backbone lattice of Fig. 1(b).
Eigenvalue or ‘‘energy’’ bands as a function of the
FIG. 6.
existence curve for a photorefractive gap LS in a defocusing
lattice for V0? ?36:3.
(a) The field profile and (b) the associated P ? q
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be studied using a Bloch envelope approximation (which
cannot be applied close to the second band due to the
mixed structure of the nonlinear mode). Close to the edge
of the first band the LS are broad, so that they can be
expressed as a periodic Bloch mode times a slowly vary-
ing envelope. This envelope obeys an effective nonlinear
Schro ¨dinger equation, to which the Vakhitov-Kolokolov
criterion can be applied. As before, the defocusing non-
linearity gives us behavior that is opposite to the focusing
case: Close to the first band (lower limit), @P=@q < 0 and
the solution will be unstable, while increasing the eigen-
value to the regime where @P=@q > 0 will stabilize the
solution. However, as the eigenvalue further increases, the
soliton is affected by the second band and the solution
ultimately becomes unstable again. We would like to
mention that in a recent study gap LS in similar systems
have been identified [24]. However, in contrast to these
results [24], our analysis indicates that (i) a full 2D band
gap is necessary for gap LS to exist and (ii) LS always
exhibit power thresholds.
We find that the results obtained from the saturable
model of Eq. (3) are in qualitative agreement with those
obtained in the Kerr regime. In doing so, we have studied
the backbone structure of Fig. 1(b). Figure 6(a) demon-
strates a typical self-defocusing gap LS. In Fig. 6(b), the
required power of such a soliton is depicted as a function
of its eigenvalue. This curve has a minimum, although,
due to the saturable nature of the nonlinearity, it is not as
prominent asintheKerrcase.Ontheotherhand, Fig.7(a),
shows a typical in-phase self-focusing LS (which occurs
in a modulated lattice similar to the one in Fig. 1(a) [16])
and Fig. 7(b) depicts the total LS power as a function of q.
Finally, we would like to mention the connection be-
tween our results and Bose-Einstein condensates in an
optical lattice [14]. In this case, Eq. (1) describes the
evolution of the mean-field atomic condensate wave func-
tion in a potential V?x;y?. The potential of Eq. (2) can
then be created by the interference of two pairs of laser
beams with orthogonal polarizations. A lattice similar to
that of Fig. 1(b) can also be established when the two
polarizations are parallel in a dark lattice. Using the
normalizations of Ref. [13], one can show that for Rb87
atoms, and the lattice of Eq. (2) with 1:2 ?m spacing, the
potential depth required to create a band gap is approxi-
mately equal to 9.6 times the recoil energy. Also, in
contract with the 1D case, a minimum number of atoms
is always required to establish a LS.
This work is part of a MURI program on optical
solitons and is also supported by the Israeli Science
Foundation.
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FIG. 7.
existence curve for a photorefractive in-phase LS in a focusing
lattice for V0? 36:3.
(a) The field profile and (b) the associated P ? q
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