Multi-stable regime and intermediate solutions in a nonlinear saturable coupler

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by  Diego Guzmán-Silva

Multi-stable regime and intermediate solutions in a nonlinear saturable coupler
Diego Guzma´n-Silva1, Cibo Lou2, Uta Naether1, Christian E. Ru¨ter2, Detlef Kip2 and Rodrigo A. Vicencio1
1Departamento de F´ısica, MSI-Nucleus on Advanced Optics,
and Center for Optics and Photonics (CEFOP),
Facultad de Ciencias, Universidad de Chile, Santiago, Chile and
2Faculty of Electrical Engineering, Helmut Schmidt University, 22043 Hamburg, Germany
(Dated: January 31, 2013)
We show, theoretically and experimentally, the existence of a multi-stable regime in a nonlinear
saturablecoupler. Inspiteofitssimplicity,wefoundthatthismodelshowsgenericandfundamental
properties of extended saturable lattices. The study of this basic unit becomes crucial to under-
3 stand localization mechanisms and dynamical properties of extended discrete nonlinear saturable
1
systems. We theoretically predict the regions of existence of intermediate solutions, and experi-
0
mentally confirm it byobserving amulti-stable propagation regime in a LiNbO3 saturable coupler.
2
Thisconstitutesthefirstexperimentalevidenceoftheexistenceoftheseunstablesymmetry-broken

n stationary solutions.
a
J PACSnumbers: 42.65.Wi,63.20.Pw,63.20.Ry,05.45.Yv

0
3
I. INTRODUCTION system; i.e., a periodic exchange of light for small power

] andhightransmission(localization)forlargerpowers. In
s
Ref. [16] authors explored the dynamics and stationary
c Nonlinear discrete systems appear in several branches
i behavior of a cubic coupler which presents only one bi-
t of science and have found a fruitful field of development
p furcation point for stationary solutions and no exchange
and realistic implementation during the last years [1–4].
o of stability properties [similar to larger one-dimensional
Many results obtained in very different physical settings
s. canbeextrapolatedtootherareasofresearchgenerating (1D) cubic systems]. On the other hand, Ref. [17] shows
c the appearance of an extra bifurcation point, change of
i a broader and deeper scientific impact. Different tech-
s stability properties and richer dynamics when consider-
niques and methods to study such systems experimen-
y ingasaturablenonlinearity. However,onlyveryrecently,
h tally,forexampleinthecontextofphotoniclattices,have
the concept of “intermediate solutions” (IS) was intro-
p been developed with many possibilities to change and
ducedtoexplainthepropertiesofextendedsaturable1D
[ control the key parameters. Here, merely Kerr-like (cu-
and 2D lattices [7–9] (this concept was introduced be-
bic) nonlinear systems have been a main subject of the-
3 fore for other models [18, 19]). These kind of “unstable”
oretical and experimental research. As a result, the cor-
v symmetry-broken solutions appear when two fundamen-
9 roboration of several former theoretical predictions and
tal modes are simultaneously stable (in other settings,
7 different new findings have been performed. However, a
the IS can also be stable [19–21]). Therefore,it becomes
3 different type of nonlinearity has opened new challenges
natural to formulate the question about the minimum
3 with new interesting dynamical properties, the so-called
. number of sites - in a saturable array - for which this
1 “saturable nonlinearity”. On one hand, from a dynam-
phenomenology emerges. In the present work, we will
1 ical point of view, this type of nonlinearity allows for a
show that the fundamental saturable properties are al-
2 morecomplexandricherphenomenologythantypicalcu-
1 ready present in a system of just two waveguides. More-
bic systems [5–9]. For example, in a saturable nonlinear
: over,for the first time to our knowledge, we show an ev-
v regimeanexchangeofstabilitypropertiesbetweenfunda-
idence for the existence of ISby observingamulti-stable
i mental solutions is allowed, promoting improved mobil-
X propagation in an experiment performed in iron doped
ity for high power solitons. This is certainly opposite to
r LiNbO3 samples.
the phenomenology observed for cubic systems [10, 11],
a
where an increment in power induces localization only.
On the other hand, only very little experimental realiza-
II. MODEL
tions showing the specific behavior of saturable systems
have been performed. So far, mainly gradual changes
The propagationof light in a system composed of two
from the behavior of Kerr-like systems were observed;
identical weakly coupled waveguides, with a defocusing
for example, suppression of modulation instability [12],
saturable nonlinearity, can be described as follows
stabilization of discrete vector solitons [13], and higher-
order gap solitons [14].
∂u1 γu1 ∂u2 γu2
A nonlinear coupler (dimer) is the simplest discrete −i ∂z =u2+ 1+ u1 2 , −i ∂z =u1+ 1+ u2 2 (1)
system where there are just two identical nonlinear | | | |
waveguides (sites) - placed in close proximity - which where u represents the light amplitude at site n , γ
n
≡
evanescently interact. Thirty years ago, Jensen [15] γ¯/V > 0 corresponds to the strength of the defocusing
showed the main features for the corresponding cubic nonlinearity (γ¯) with respect to the coupling coefficient
2
(V)betweenthetwosites,andzdescribesthenormalized
"’ α =−0.15
propagation distance along the waveguides. Model (1) asy
αIS=−0.41
possesses two conserved quantities, the Power
"!
!
!$#
2 2
P u1 + u2 (2)
≡| | | | &’
(
and the Hamiltonian
&! !
H (u2u∗1+u∗2u1)+γln (1+ u1 2)(1+ u2 2) . (3) αasy=−0.13 ! " ! "
≡ | | | |
(cid:2) (cid:3) ’ αsym=1 αant=−1
Stationary solutions of model (1) have the form u (z)=
n
!"#
unexp(iλz), where λ represents the spatial frequency. !
Firstofall,welookforlinearsolutionsofmodel(1)that, ! " # $ % &!
duetothesaturation,existintwodifferentregions[7–9]. Λ
In a low power regime, we find that there are two solu-
tions with frequencies “γ +1” and “γ 1”. For higher
− Figure 1: (Color Online) (a) P vs λ diagram for γ = 10.
powers, the nonlinear response vanishes and frequencies
The symmetric, stable and unstable antisymmetric, asym-
become “+1” and “ 1”. The spatial profiles for these
metric, and intermediate solutions are plotted in black, full
−
two modes are equal in both regimes: the symmetric and dashed blue lines, red, and red-dashed line, respectively.
(u1 = u2) and the antisymmetric (u1 = u2) modes, Vertical lines indicate linear frequencies. (b) Antisymmetric,
−
respectively. asymmetric, and intermediate mode profiles, for P =20.
III. NONLINEAR MODES the sign “ ” in Eq. (4)]. For saturable systems, this
−
always unstable non-symmetric solution is called inter-
Now, we look for general nonlinear solutions of the mediate solution [7–9]. The case where two fundamen-
form: u1 = A and u2 = αA, where A is a positive am- tal solutions are simultaneously stable, sharing the same
plitude and α describes the ratio between these two site Hamiltonian value, was initially suggested [5] as a van-
amplitudes. We found that the symmetric (α = 1) and ishing Peierls-Nabarro barrier [23, 24]. However, recent
the antisymmetric (α = 1) modes are also solutions in works [7–9, 19] have shown that there is a nonzero effec-
thenonlinearregime. Th−eybifurcatefromthelowpower tive energy barrier, which also considers the IS. Surpris-
linear solutions and diverge when approaching the high ingly, the dimer model also shows this phenomenology
power linear modes: which is fundamental to understand deeply the proper-
ties of nonlinear saturable arrays. Examples for some
(γ+1) λ (γ 1) λ profiles are sketched in Fig. 1(b). The intermediate and
P =2 − and P =2 − − .
sym ant
(cid:20) λ 1 (cid:21) (cid:20) λ+1 (cid:21) asymmetricmodescorrespondbothtonon-symmetricso-
−
lutions of model (1); they have quite similar profiles and
In Fig. 1(a) these two families are plotted with thick- would be only identified by directly observing their un-
black and blue lines, respectively, including their corre- stable/stable dynamical propagation.
sponding profiles (to visualize them, we used a Gaussian
profile with amplitude u at site n). In addition, we
n
found two non-symmetric solutions
A. Effective potential
γA2 γ2A4 4A2(1+A2)2
α(γ,A)= − ±p2A2(1+−A2) . (4) Now,wegodeeperintothedynamicalpropertiesofthis
model by computing an effective potential [7–9, 25, 26].
Withoutlossofgenerality,werestricttothecase α <1. Model (1) is integrable, therefore the effective potential
The sign “+” in (4) corresponds to the asymme|tr|ic so- can be obtained analytically. First of all, by fixing the
cluattiionng (f|rαoamsy|th6=e a1n)titshyamtmeextirsitcs monoldyefo[sreeγF>ig.4,1(bai)fuart- oProw1erPP,w=edue21finoretuh22e,caenndtexro=fm0.a5ssasux21≡=uu2222/P=(xP/=2)0.
⇒ ⇒
λ 8]. Oncethis asymmetricsolutionappears,the anti- Then, as we are considering a defocusing nonlinearity,
sy∼mmetric one becomes unstable [a standard linear sta- we study staggered solutions (α < 0) and express the
bility analysis [22] was performed and full (dashed) lines amplitudes as
indicate stable (unstable) solutions in Fig. 1]. Fig. 1(a)
shows a monotonic increment of the asymmetric power u1 = P(1 x) and u2 = √xP .
± − ∓
up to some maximum value. All this branch (full red p
line) is stable. However, after achieving this maximum, WiththeseexpressionsinsertedintotheHamiltonian(3),
the branch changes its curvature (power decreases) and we get
the solution gets unstable until it fuses with the anti-
symmetric branch [this unstable branch corresponds to H(x,P,γ)= 2P x x2+γln 1+P +P2(x x2) .
− − −
p (cid:2) (cid:3)
3
L
Γ
P,
x,
H
H
0.0 0.2 0.4 0.6 0.8 1.0
x
Figure 2: (Color Online) Effective potential for γ = 10 and
P =0.15(dotted),P =15(dashed),P =19(full)andP =26
(full thick). Filled circles correspond to stationary solutions.
The critical points ∂H/∂x = 0 represent different sta-
tionary solutions. A first one corresponds to x 0.5
ant
≡
(α=−1),asolutionexistingforalllevelofpowersP [see Figure 3: (Color Online) Density Plot of “xout” versus |α|
bluelineinFig.1(a)]. Therearefouradditionalsolutions and P for γ =10. Full and dashed white lines correspond to
theasymmetric and intermediate solutions, respectively.
1 1
x= P2 2γ2+4(P +1) 2γ γ2 4(P +1),
2±2Pq − ± −
p
(5)
ricmodeistheonlycriticalpointintheeffectivepotential
where asymmetric modes correspond to the sign “+” in
(full thick line). This is a direct consequence of the sat-
front to the inner square root. They are symmetrically
urable nature of the system, where the nonlinear term
located - to the right and to left - from the antisym-
vanishes for powers above a critical value (P >P ).
up
metric solution (x = 1/2). Asymmetric solutions ex-
ant
ist only for γ > 4, bifurcating from the antisymmetric
mode at power P γ 2 (γ 2)2 4. Unsta-
min
ble IS correspond to t≡he s−ign “−p” in−front −to the inner B. Numerical propagation
−
squarerootin(5). Theyexistabovethe powerthreshold
P γ 2+ (γ 2)2 4. The asymmetric and the Now, in order to test these stationary properties
th
≡ − − −
intermediate soplutions exist up to an upper power value andtheir dynamicalconsequences,we numerically study
2
given by P (γ/2) 1. (For γ = 10, P = 0.254, model (1) by considering the general initial condition
up min
≡ −
Pth = 15.746, and Pup = 24 [see Fig. 1(a)]). IS also ex- u1(0) = A and u2(0) = αA, with 1 6 α 6 0. This
−
ist for γ > 4; i.e, the intrinsic saturable phenomenology input condition allow us to excite all different staggered
would be only observedabove some critical nonlinearity. solutions. Fig.3showstheoutputcenterofmass,defined
as xout u2(zmax)2/P, measured after a given propa-
≡ | |
Fig. 2 shows the effective potential H(x,P,γ) versus gation distance z = z (similar density maps are ob-
max
the center of mass, for γ = 10 and for four different tained for different propagation distances). Purple-blue
level of power (curves have been normalized for compar- colors (x . 0.2) represent solutions localized close to
out
ison). Below the P (dotted line), the potential looks site n = 1; green colors (0.4 . x . 0.6) represent an-
min out
as a typical potential well with a minimum located at tisymmetric profiles; orange-red colors (x & 0.8) rep-
out
x . In the range P ,P the effective potential is resent solutions localized close to site n=2. For P .1,
ant min th
{ }
cubic-like (dashed line); i.e, the antisymmetric solution asymmetric profiles (α 0) do not correspond to any
| | ∼
is unstable (maximum) while the asymmetric solution is stationary solution and the light just oscillates between
stable (minimum). The saturable nature of this model the two sites (see the appearance of multiple colors as
manifests for powers above P (full line). The antisym- an indication of strong oscillation, i.e. switching [15]).
th
metric andthe asymmetric solutions become, both, a lo- On the other hand, for α 1 we see a greener color
| | ∼
cal minimum and, therefore, simultaneously stable. The that indicates a small oscillation in the vicinity of the
unstable symmetry-broken IS appears as a maximum in antisymmetric solution (x = 0.5), the only stationary
out
this potential, located in between the stable solutions. solution at this power regime. At the bifurcation point
Therefore, at P the IS has a center of mass x =x power(when H changesits shape fromdotted to dashed
th IS ant
that evolves in the direction of an asymmetric configu- in Fig. 2), both solutions are quite similar and the anti-
ration when approaching P . Finally, above P , both symmetric mode is slightly unstable. Then, in the range
up up
non-symmetricsolutionsdisappear,andtheantisymmet- P 1,15 for α .0.4, the light is well trapped at the
∈{ } | |
4
vicinity of site n = 1. This is an indication of the exci- in-diffusion on an x-cut lithium niobate substrate doped
tation of an asymmetric stationary state (see full white with iron. The end facet of the sample is monitored
line in Fig. 3). For α 1, there is an oscillation of the by a high-resolution CCD camera. The length of our
| |→
light in the interval x 0,0.5 , as expected for an sample along the propagationz-direction is 18 mm ( 5
out
∈ { } ∼
unstable antisymmetric configuration(see dashed line in diffractionlengths)wherethewaveguidechannelsare4.0
Fig. 2). In the region of power 16,24 , a very inter- µm wide with a separation of 2.2 µm. Our photovoltaic
≈ { }
esting behavior is observed: we found a simultaneously samples have a nonlinearity which grows exponentially
stabledynamicalpropagationofasymmetric(α 0.15) in time, γ(t) = γ(1 exp[ t/τ]), where τ is the dielec-
| |≈ − −
andantisymmetric(α 1)inputprofiles. Theconstant tric response time [27]. In order to reach a steady-state,
| |≈
value of x (constant color) when increasing P, indicates saturation of the photovoltaic nonlinearity is required,
a stable dynamicalevolution. This constitutes a dynam- which typically occurs, in our samples, for t 25 min-
∼
ical and indirect evidence of the existence of the inter- utes. Fig. 5 showssome examples of the evolutionof the
mediate solutions as a fundamental entity for saturable center of mass in time. We see how some input condi-
systems: if both staggered solutions are stable simultane- tions propagatesin a verystable wayfor the whole mea-
ously, an extra unstable intermediate solution must ex- surementperiod(black-full,orange-fullandblack-dashed
ist. The dashed white line in Fig. 3 corresponds to the lines)while the orange-dashedcurvetend to stabilize for
Intermediate solution and shows a clear connection in t&25.
parameter space between the asymmetric and the anti-
symmetric staggered solutions. For P & 24, we observe
0.6
an oscillation of energy between sites 1 and 2 for the
input condition 0 . α . 0.8, while an antisymmetric
| | 0.5
configuration is stable for α & 0.8. All the dynamical
| |
results are in perfect agreement with the stationary pic- 0.4
ture sketched in Figs. 1 and 2, including the region of
ut
multi-stability and the existence of the IS. xo0.3
0.2
IV. EXPERIMENTAL RESULTS
0.1
0.0
%& #$ 0 10 20 30 40
& Time@minD
-./01)23
Figure 5: (Color Online) Center of mass evolution at the
’( sample output for asymmetric (full lines) and antisymmet-
ric (dashed lines) input conditions, for input powers 100nW
(black) and 500nW (orange). The filled rectangular area de-
’) notestheregionwhereweexperimentallyobserve(define)an
5
antisymmetric configuration.
*&
6 4
Werepeattheexperimentforseveralinputpowersand
!!"
& fortheasymmetricandtheantisymmetricinputprofiles,
&+ $,! &+ #$ andaveragethecenterofmassvaluesforthelast10min-
utes of each experiment. Compiled results are shown in
Figure 4: (Color Online) Experimental setup for observing Fig.6with afairly goodagreementbetweenthe theoret-
themultistable regime in a saturable nonlinear coupler.
ically predicted phenomenology and what is observed in
direct experiments. The initial conditions are indicated
To verify our theoreticalandnumericalpredictions we bydotsconnectedwithdashedlines. Thedeviationofthe
usetheexperimentalsetupsketchedinFig.4. Acwlaser output values (dots connectedby full lines) fromthe ini-
withawavelengthof532nmpropagatesthroughaphase tialconditions canbe understoodasthe degreeofstabil-
mask (PM) covering half of the beam along the trans- ity. Inaddition,theerrorbars(obtainedintheaveraging
verse direction x (coinciding with the crystallographic process)alsoindicateshowstableistheprofile: asmaller
c-axis of the sample). So the phase relation of the left bar indicates a dynamically stable profile while a larger
and right half of the beam can be switched to be ei- bar means a stronger oscillation around some minimum
ther in-phase or out-of-phase. With a 4f imaging sys- (stationary solution). For low level of power, we observe
tem, composedby lenses L1 and L2, the beam is imaged that both solutions are essentially stable. This coincides
onto a double-hole amplitude mask (AM). By using a with our analysis in the region of the first bifurcation
microscope objective, the beam is injected into a sat- point, where the appearance of the asymmetric solution
urable nonlinear coupler (SNC) fabricated by titanium weakly destabilizes the antisymmetric mode. Then, for
5
is the antisymmetric one, as shown in Fig. 6. The large
barforthelastasymmetric(red)pointindicatesthatthe
system has saturated and thadit the effective potential
has a shape like the thick line in Fig. 2. It is impor-
tant to mention that our experimental output profiles in
Fig.6-low-rowhaveastaggeredphase,whatisevidentby
observing the zero amplitude in between the two waveg-
uides. This gives us an extra support for relating the
observationofthe multi-stable regimewith theexistence
of an unstable IS, because the observed profiles are well
connected in phase space.
V. CONCLUSIONS
In conclusion, we have observed, for the first time, a
multi-stable regimeoffundamentalmodesin anonlinear
saturablecoupler. Acompletemapofnonlinearsolutions
hasbeenconstructed,includingtheirstabilityproperties,
andeffectivepotential. Numerically,wehavedetermined
the regions where the fundamental profiles are expected
to be stable, showing an excellent agreement with the
Figure6: (ColorOnline)Averagedcenterofmassatthesam- developed theory. We fabricated a nonlinear saturable
ple output vs input power for a LiNbO3 saturable coupler. coupler in LiNbO3 and observed stable propagation of
Red and bluesymbols, connected by full lines, correspond to fundamentalmodesforintermediatelevelofpower. This
the experimentally measured value of xout for an asymmet- constitutes the first experimental evidence, in any phys-
ric and antisymmetric input condition (symbols connected ical system, of the existence of intermediate solutions
by dashed lines), respectively. The lower row shows some
in discrete nonlinear lattices. Moreover, for larger pow-
(non-averaged)experimentaloutputprofilesfortheindicated
ers we observed the absence of the asymmetric solution
powerswithantisymmetric(ant)andasymmetric(asy)input
as a clear indication of the saturation of the nonlinear-
profiles. (The filled area denotes thesame that in Fig. 5).
ity. All these results support strongly recent theoretical
and numerical developments based on DNLS-like mod-
els in several contexts of physics and open, as a direct
powers in the region 300,1300 nW, a typical cubic-
∼ { } consequence, new opportunities of research on nonlin-
like picture is observed: the asymmetric solution is sta-
ear discrete systems. In addition, the understanding of
ble while the antisymmetric one is not. However, above
smallsystemsbecomesverycrucialwhenthinkingonthe
1300nWasaturablephenomenologyemerges. Thesat-
∼ implementation of photonic lattices for realistic applica-
urablenatureofthe nonlinearityallowsoscillationofthe
tions.
stability properties, with a richer dynamics when com-
paring with usual cubic systems. For this sample, the
region 1300,2300 nWcorrespondsto a regime where
the two∼fu{ndamental}nonlinear solutions of this problem Acknowledgments
becomesimultaneouslystable(asanexampleseethepro-
files for 1500nW). Therefore, we experimentally observe The authors thank M. Johansson for useful discus-
a multi-stable regime as an indirect evidence of the ex- sions. This work was supported in partby FONDECYT
istence of intermediate solutions in a saturable dimer. grant 1110142, CONICYT fellowships, Programa ICM
This observation gives a strong support to the theory P10-030-F,Programade FinanciamientoBasalde CON-
(and model) developed for this type of lattices. Above ICYT (FB0824/2008), and the Deutsche Forschungsge-
some given power ( 2300nW), the only stable solution meinschaft (Ki 482/14-1).
∼
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