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Role of optical nonlinearity on transverse localization of light in a disordered
one-dimensional optical waveguide lattice
Somnath Ghosh, Bishnu P. Pal ⁎, R.K. Varshney
Physics Department, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India
abstractarticle info
Article history:
Received 27 September 2011
Received in revised form 7 February 2012
Accepted 9 February 2012
Available online 25 February 2012
Keyword:
Transverse localization of light
Disordered waveguide lattice
Focusing and defocusing nonlinearities
We report a detailed numerical investigation on transverse localization of light in a 1D disordered lattice con-
sisting of a large array of coupled waveguides in the presence of nonlinearity in the medium. Our study re-
veals that the presence of a focusing type of nonlinearity favors faster localization of light while a
defocusing type of nonlinearity degrades the quality of localization. It is shown that presence of either of
these could over-shadow localization of light unless the strength of disorder is sufficiently strong. Influence
of the input beam width on propagation of light in such a disordered nonlinear medium has also been dis-
cussed. The present study should be useful in potential applications, in which one could exploit dominance
of focusing nonlinearity on transverse localization of light in a disordered medium.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
The classic idea of localization of wavefunction in a disordered po-
tential, which was originally proposed by Anderson [1] from the per-
spective of quantum and solid state physics, is now accepted to be
ubiquitous in wave physics [2–4], which includes localization of
light in a disordered dielectric lattice [4]. Raedt et al. [3] revisited
the concept of Anderson localization and introduced the concept of
transverse localization of light in a semi-infinite disordered medium,
in which confinement occurs only in the plane perpendicular to the
direction of transport of light. With the advent of photonic bandgap
devices, and discrete photonic structures and intense research on
this particular area transverse localization of light in disordered peri-
odic structures has attracted considerable attention in recent years
[4–6]. Though the technique of laser-induced temporarily realized
disordered photonic lattices has been reported [5], the evanescently
coupled optical waveguides with controlled disorder in 1D (or 2D)
offer a versatile platform for modeling and studying the linear and
nonlinear dynamics of wave packets in a disordered medium [6,7].
This is due to the fact that such lattice geometries could be exploited
as an alternative test bed to visualize transverse localization of light
[8]. These waveguiding structures could be conveniently designed in
order to manipulate flow of light in them that is not otherwise easily
realizable through existing common routes like bulk or fiber optical
systems. This platform could also be exploited to demonstrate the
concept of random lasing [9] in localized regime by choosing a gain
medium, which is sufficiently disordered (for realizing transverse
localization of light) through selective pumping of the localized
modes with varying spatial extents and locations. Hence the med-
ium's nonlinearity naturally becomes an important issue. Some stud-
ies on the interplay between the nonlinear nature of the medium and
the scattering event responsible for localization have been reported in
the literature [5,6,10–12].
However, none of these studies has reported a detailed investiga-
tion on the effect of focusing or defocusing nonlinearity of various
strengths on a localized state in a given disordered lattice. Further, re-
search on the impact of one of the crucial parameters known as trans-
verse wavenumber (k
⊥
) on the study of transverse localization of
light is missing in the literature [5]. This motivated us to investigate
the role of nonlinearity on localization of light in a disordered non-
linear medium. This study should be important in the context of
ever-growing literature on the formation of bright and dark discrete
solitons and the possibility of random lasing that exploit transverse
localization of light [13]. In this paper, we report the dependence of
a localized state on various strengths of nonlinearities in the disor-
dered medium. Moreover, our study opens up the possibility of syn-
thesizing the three different routes of engineering light flow for
novel applications such as formation of discrete solitons in perfectly
ordered waveguide lattices under nonlinear conditions [14,15], light
confinement in custom fabricated waveguide arrays that could engi-
neer the coupling constant map [16], and transverse localization of
light [5,6,8].
2. Modeling the waveguide lattice
In order to study these effects, we have chosen a 1D coupled
waveguide lattice consisting of an array of 100 coupled waveguides
as a sample. Such a lattice can be fabricated either by lithography
Optics Communications 285 (2012) 2785–2789
⁎Corresponding author.
E-mail address: bppal@physics.iitd.ac.in (B.P. Pal).
0030-4018/$ –see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2012.02.016
Contents lists available at SciVerse ScienceDirect
Optics Communications
journal homepage: www.elsevier.com/locate/optcom
technique [6] in semiconductors or ultrafast laser inscription [17] in
glass. For our simulation, we have considered propagation of a
continuous-wave (CW) beam in a lossless, disordered lattice. All the
waveguides in the lattice are embedded in a medium of constant re-
fractive index n
0
. The refractive index profile of the lattice is deliber-
ately perturbed to introduce the influence of transverse disorder,
while the spatial periodicity remains unaltered. The transverse refrac-
tive index variation Δn(x) (over the uniform background refractive
index (n
0
)) of this 1D waveguide lattice was assumed to be of the fol-
lowing form:
Δnx
ðÞ
¼ΔnpHx
ðÞ
þCδx
ðÞðÞ ð1Þ
where Cis a dimensionless constant whose value governs the level of
disorder; the periodic function H(x) takes the value 1 inside the
higher-index regions and is zero elsewhere; Δn(x) consists of a deter-
ministic periodic part Δn
p
of spatial period Λand a spatially periodic
random component δ(uniformly distributed over a specified range
varying from 0 to 1). This particular choice of randomly perturbed re-
fractive index profile enables us to mimic the simultaneous presence
of diagonal and off-diagonal disorders to study the optics of light lo-
calization [1,18]. The chosen waveguide lattice consists of an array
of periodic waveguides, each of 7 μm width and spatially separated
by 7 μm (see Fig. 1). The unperturbed lattice has a periodic index
modulation Δn
p
; on top of which the controlled disorder is intro-
duced following Eq. (1) for different values of the Cparameter. The
value of Δn
p
is chosen to be 0.001 over and above the background
material of refractive index n
0
=1.46.
3. Methodology
The dynamics of wave propagation through the chosen lattice ge-
ometry is governed by standard scalar Helmholtz equation [8], which
in the simultaneous presence of disorder and nonlinearity can be
written as [5]
i∂A
∂zþ1
2k
∂2A
∂x2
!
þk
n0
ΔnxðÞAþσk
n0
n2A
jj
2A¼0ð2Þ
where A(x,z) is the amplitude of an input CW optical beam having its
electric field asE(x,z,t)=Re[A(x,z)e
i(kz −ωt)
]; k=n
0
ω/c;n
2
represents
the nonlinear coefficient of the medium, and σcharacterizes the fo-
cusing (σ=1) or defocusing (σ=−1) nature of the medium. We
solve Eq. (2) numerically through the scalar beam propagation meth-
od, which we implemented in Matlab. We deliberately choose a rela-
tively small refractive index contrast along with a relatively large
period for the unit cell (Λ≫λ) in order to ensure that the photonic
bandgap effect is negligible at the operating wavelength. Fig. 2 de-
picts the evolution of an input CW beam having a plane phase front
with propagation at the operating wavelength of 980 nm. Fig. 2(a)
depicts the dynamics of a light beam as it propagates through a lattice
whose refractive index profile follows Eq. (1) and corresponds to
C=0.2. This figure evidently demonstrates transition to a localized
state after an initial ballistic mode of propagation. The input beam
width ω
0
(FWHM) was 10 μm.
4. Effect of nonlinearity
In order to investigate quantitatively the effect of optical nonli-
nearity on light localization in a disordered lattice, we have studied
the beam dynamics by solving Eq. (2) for different lengths of the lat-
tice. The various strengths of the medium's nonlinearity has been
taken into account through different values of the C
NL
-parameter de-
fined as
CNL ¼n2Ajj
2
Δnp
:ð3Þ
A measure of the localization is quantifiable through a decrease in
the average effective width (ω
eff
)defined as [8]
ωeff ¼P
hi
−1ð4Þ
of the propagating beam after including the statistical nature of the
localization phenomenon in our chosen lattice of finite length;
where the sign b..> represents statistical averaging over several real-
izations of the same level of disorder. After a certain propagation dis-
tance, depending upon the strength of disorder and lattice aspect
ratio, ω
eff
remains almost unchanged with a characteristic statistical
fluctuation [5], which is recognized in the literature as a signature
of transverse localization of light [3]. In addition, the so called charac-
teristic localization length (l
c
)[5,8] is a figure of merit; the smaller the
l
c
, the better is the quality of localization. Thus a larger value of ω
eff
in
the localized regime and an increase in l
c
of a localized eigen-state in a
disordered medium would imply degradation in the quality of locali-
zation, which is also commonly referred to in the literature as delocal-
ization. Influence of either a 10% focusing or a 10% defocusing
nonlinearity of the medium on localization is pictorially illustrated
in Fig. 2(b) and (c), respectively. It is evident from these plots that a
focusing nonlinearity favors localization to occur within a shorter
L
N = 100
y
x
z
7 µm
7 µm
Fig. 1. Schematic of the chosen lattice consisting of evanescently coupled large number
of waveguides (i.e. 100 unit cells (N)), each of equal width, and which are equally
spaced; different shades of color signify different refractive indices over an average re-
fractive index.
(c)
L
Unit cell positions
(b)
(a)
L
Unit cell positions
020406080
0 20 40 60 80 0 20 40 60 80
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
Fig. 2. Beam dynamics showing a) transverse localization of the beam in the presence
of 20% refractive index disorder (i.e. C= 0.20) when the ballistic expansion has been
eliminated as it propagates along a 20 mm long lattice; b) enhancement in localization
in the presence of an additional 10% focusing nonlinearity (C
NL
); c) degradation in lo-
calization in the presence of a 10% defocusing nonlinearity.
2786 S. Ghosh et al. / Optics Communications 285 (2012) 2785–2789
distance of propagation, whereas a defocusing nonlinearity of the
same strength degrades localization.
To estimate the contribution of the focusing nonlinearity quantita-
tively in a disordered waveguide lattice under consideration here, we
have calculated the ω
eff
parameter over 100 realizations for different
sets of Cand C
NL
parameters. The results are plotted in Fig. 3, where
ω
0
denotes FWHM of the launched input beam. The black solid
curve through the circles depicts variation of ω
eff
with increase in
the value of Cparameter in the absence of any nonlinearity
(C
NL
=0) [8]. It is also evident that beyond the threshold value of
0.50 of the disorder parameter C, the beam gets localized after prop-
agating 15 mm through the lattice. If we now introduce the focusing
nonlinearity of the medium through the parameter C
NL
, the interplay
between the roles of Cand C
NL
gets revealed as shown in the other
three curves in Fig. 3. The curve corresponding to C
NL
=7.5% is repre-
sentative of an intermediate regime of the aforementioned interplay.
If the strength of nonlinearity is lower than this particular value, then
the ω
eff
variation is mainly dominated by the sole effect of disorder
(i.e. Cparameter). Whereas, if the chosen value of C
NL
is more than
7.5%, the beam dynamics is mainly determined by the effect of nonli-
nearity. It is well known that in a perfectly ordered structure, due to
sufficiently high C
NL
(more than 7.5% in our case), the broadening
due to discrete diffraction is balanced by the focusing effect of the
medium to form so-called discrete spatial solitons [14,15]. This is
often referred to as another form of light localization due to self-
trapping depending on the input power level. In this figure, the
curves corresponding to C
NL
parameters as 10% and 15% show two
distinct regimes for variation in ω
eff
with C, as a result of the interest-
ing interplay between the above-mentioned two light localization
mechanisms. For these two cases, initially there is a self-trapped
state due to the presence of a strong nonlinearity in a perfectly or-
dered lattice, and then as we increase the disorder, these states de-
grade their quality of localization with an increase in magnitude of
ω
eff
. Subsequently, as we further increase the level of disorder, be-
yond a certain level of disorder no further increase in ω
eff
occurs
but a crossover to the next regime of localization takes place. The fig-
ure shows that the higher the value of C
NL
, the smoother is the tran-
sition from the initial self-trapped state due to nonlinearity to the
final localized state. These results evidently show that the simulta-
neous presence of a finite magnitude of C
NL
and any level of disorder
(C) in refractive index in the medium always enhances the effect of
localization. In other words, the curves corresponding to C
NL
as 10%
and 15% clearly reveal that the presence of a finite focusing type of
nonlinearity in a disordered medium would help in realization of lo-
calized light even in the presence of a relatively weak level of disorder
(e.g. Cb40%). On the other hand, in the absence of nonlinearity, one
requires a higher strength of disorder (e.g. Cas ~ 50%). This essentially
reflects that the threshold level of disorder for transverse localization
of light can be sufficiently lowered by choosing a medium in which
disorder and focusing type of nonlinearity are simultaneously pre-
sent. To visualize this effect on the degree of localization, we have
plotted in Fig. 3(b) the ensemble averaged two output intensity pro-
files from the 15 mm long waveguide lattice in which Cwas chosen to
be 0.6. The two profiles one in black (solid) and the other in red
(dashed), respectively correspond to absence and simultaneous pres-
ence of 10% focusing nonlinearity. Thus this semi-log plot clearly
manifests that a focusing nonlinearity always favors localization. Un-
like the above mentioned beneficial effect of focusing nonlinearity, a
defocusing type (σ=−1) nonlinear disordered medium adversely
affects transverse localization of light as is shown in Fig. 4. In this fig-
ure, we have presented various curves for the variations of ω
eff
with C
with C
NL
as the labeling parameter. It can be clearly seen from this fig-
ure that the presence of a defocusing nonlinearity degrades the qual-
ity of localization as is evidenced by resultant increase in ω
eff
.
Furthermore, unless the strength of nonlinearity is relatively large,
the effect of disorder in a strongly (C~ 0.5) disordered lattice may
dominate over this adverse effect of defocusing nonlinearity. To ana-
lyze this underlying competition in more detail, we plot ω
eff
as a func-
tion of C
NL
for both cases of σ=±1 in Fig. 5(a) for three different
values of C. For a Cas 0.2, the variation plotted in black-solid (circles)
clearly manifests the adverse effect of defocusing on localization,
whereas the blue-dashed curve (filled circles) confirms enhancement
in localization. Other curves corresponding to larger Cof 0.3 and 0.5
reaffirm the similar trend. However, as Cincreases for a fixed C
NL
,
the difference between the values of ω
eff
corresponding to strongly
localized and a delocalized state decreases. Hence it can be concluded
that a medium's nonlinearity plays a key role in determining the
5
10
15
20
25
30
35
40
ωeff /ω0
C (%)
CNL 15 %
10 %
7.5 %
0 %
a)
0 102030405060
15 30 45 60 75 90
0.01
0.02
0.05
0.14
0.37
1.00
Normalized intensity (a.u.)
Unit cells
b)
Fig. 3. a) Variation of the effective beam width (averaged over 100 realizations) with
levels of disorder for different values of C
NL
of focusing nonlinearity. b) The ensemble
averaged localized profiles achieved for a 15 mm long lattice with C= 0.60; the black
solid curve has no nonlinearity while the red dashed curve corresponds to 10% focusing
nonlinearity (C
NL
). An enhancement in the quality of localization is evident from this
plot.
5
10
15
20
25
30
35
40
ωeff /ω0
C (%)
0 102030405060
CNL 15 %
10 %
5 %
0 %
Fig. 4. The effective beam width (averaged over 100 realizations) is plotted as a func-
tion of various levels of disorder for different values of the strength (C
NL
) of defocusing
nonlinearity. A lattice of 15 mm length was chosen for this simulation.
2787S. Ghosh et al. / Optics Communications 285 (2012) 2785–2789
extent of increase or decrease in ω
eff
in a disordered optical wave-
guide lattice.
For a better appreciation of this effect on the degree of localiza-
tion, we have also estimated the localization length (l
C
), which is an-
other important characteristic of a localized state in a disordered
lattice [5]. To obtain l
C
, we have averaged 100 output intensity pro-
files for a given value of Cand then performed a three-point moving
average to smoothen further the resulting profile as is explained in
[8]. The corresponding variation of l
C
with Cfor three different cases
namely medium without nonlinearity, with 10% focusing nonlinearity
and 10% defocusing nonlinearity, respectively are shown in Fig. 5(b).
These plots also confirm the advantage of a focusing nonlinearity over
a defocusing nonlinearity of the medium from a localization point of
view. If we compare the values of l
C
s corresponding to the Cparame-
ter 0.4, it is evident that l
C
is reduced by ~50% in the presence of the
medium having a 10% focusing nonlinearity. Hence these results
could be advantageously exploited to optimize the lattice geometry
for light localization experiments. The results also indicate that in
the case of a strongly disordered lattice (C= 0.6), disorder overtakes
the effect of a focusing nonlinearity. Physically transverse localization
occurs due to elimination of diffraction broadening in the transverse
direction of the propagating beam in the presence of a randomly per-
turbed refractive index profile. Hence besides the scattering strength
of the disordered medium, the transverse wavenumber (k
⊥
) which is
inversely proportional to the width of the propagation beam (ω
0
),
plays a key role to dictate the characteristics (like ω
eff
,l
C
) of a local-
ized mode. In view of the above, to investigate the influence of the
width (ω
0
) of an input excitation while studying the effect of a med-
ium's nonlinearity in the context of transverse localization of light, we
have chosen three different sample beam widths that cover nearly
two (ω
0
=7.5 μm), three (ω
0
=10μm) and four sites (ω
0
=12 μm)
of the lattice, respectively. In Fig. 6, we have plotted the variation in
ω
eff
(normalized with respect to ω
0
) as a function of the nonlinear pa-
rameter C
NL
for both types of nonlinearities for the three different
input beam widths. It could be seen that in the absence of nonlinear-
ity (C
NL
=0) the beams while propagating through a 15 mm long dis-
ordered optical waveguide lattice with Cfixed at 0.5 evolve to
respective localized states with three different ω
eff
s. As we introduce
focusing nonlinearity all of the three propagating beams evolve to a
localized output, which is characterized by nearly the same value of
ω
eff
. Thus the combined effect of focusing nonlinearity and transverse
disorder may help to excite a localized state of light in a disordered
lattice, which would be nearly independent of ω
0
. These results
should be useful for selective excitation of localized modes in disor-
dered lattice with a finite gain coefficient for lasing applications.
5. Conclusions
Our results reveal that a focusing nonlinearity in a disordered me-
dium always favors transverse localization of light, whereas a defo-
cusing nonlinearity degrades the quality of localization. Thus the
medium's nonlinearity plays the role of a new degree of freedom to
control the localization and delocalization properties of a spatially lo-
calized state in a finite disordered lattice. Hence, our numerical find-
ings confirm that medium's nonlinearity can play a key role in
manipulating the flow of light in disordered discrete optical struc-
tures and these results should be of interest in investigations con-
cerned with light localization in waveguide lattices.
Acknowledgment
This work was partially supported by an ongoing collaborative
research project at IIT Delhi with Naval Research Laboratory,
Washington, D. C. under the ONRG grant N62909-10-1-7141 on
“Micro-Structured Fibers for Wavelength Translation and Super-
continuum Lasers”.
ωeff /ω0
(/c / 2ω0)
5
9
14
18
23
CNL(%)
A1
B1
A2
B2
A3
B3
C = 0.3
C = 0.5
C = 0.2
a)
02468101214
0.40 0.45 0.50 0.55 0.60
2
4
6
8
10
12
C (%)
no NL
10 % focusing NL
10 % defocusing NL
b)
Fig. 5. a) Variation of effective beam width of a propagating beam through a disorder
lattice of 15 mm as function of nonlinearity parameter C
NL
for the levels of disorder
C=0.20 (A1 and B1), 0.30 (A2 and B2) and 0.50 (A3 and B3), respectively. The curves
plotted in black (solid) correspond to defocusing case; whereas the blue (dashed)
curves show the variation due to a focusing nonlinearity. b) Variation of localization
lengths (l
c
) as a function of C-parameter for the same Gaussian input beam profile
along a 15 mm long lattice geometry. Solid, dashed, and dotted curves correspond to
the medium without nonlinearity, with 10% focusing and 10% defocusing nonlinearities
respectively.
ωeff /ω0
CNL(%)
024681012
4
6
8
10 7.5 μm with σ = +1
7.5 μm with σ = −1
10 μm with σ = +1
10 μm with σ = −1
12 μm with σ = +1
12 μm with σ = −1
Fig. 6. Variation of ω
eff
(normalized with respect to ω
0
) as a function of strength of
medium's nonlinearity for three different values of input width. The ω
eff
value of the
respective localized output state evolves to nearly identical states (ω
eff
is nearly the
same as the encircled values) when the strength of focusing nonlinearity (dashed
blue curves) of the medium reaches a certain level. The adverse effect of defocusing
nonlinearity (solid black curves) is still prominent in each case and leads to different
localized states (as the encircled values in the dashed line).
2788 S. Ghosh et al. / Optics Communications 285 (2012) 2785–2789
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