Nonlinear localized modes in Glauber-Fock photonic lattices

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by  Alejandro J. Martínez

Nonlinear localized modes in Glauber-Fock photonic lattices
Alejandro J. Mart´ınez1,2, Uta Naether1,2, Alexander Szameit3, and Rodrigo A. Vicencio1,2
1Departamento de F´ısica and MSI-Nucleus on Advanced Optics, Facultad de Ciencias, Universidad de Chile, Santiago, Chile
2Center for Optics and Photonics (CEFOP), Casilla 4016, Conceptio´n, Chile
3Institute of Applied Physics, Friedrich-Schiller-Universita¨t Jena, Max-Wien-Platz 1, 07743 Jena, Germany
CompiledJanuary19,2012
We study a nonlinear Glauber-Fock lattice and the conditions for the excitation of localized structures. We
investigate the particular linear properties of these lattices, including linear localized modes. We investigate
numericallynonlinearmodescenteredineachsiteofthelattice.Wefoundastrongdisagreementofthegeneral
tendency between the stationary and the dynamical excitation thresholds, and we give a new definition based
2 onbothconsiderations.
1 (cid:13)c 2012 OpticalSocietyofAmerica
0 OCIS codes: 190.0190,190.5530,190.6135,230.4320
2

n
a Photonic crystals and waveguide arrays are one of the characterizethesolutions,weusetheintegralpower(P)
J
most fruitful areas of research in optics and nonlinear and the participation ratio (R) defined as

8
physics. During the last years these ideal experimen-
1 tal and theoretical scenarios enabled the prediction and (cid:88)N P2
P ≡ |E |2 and R≡ . (2)
] demonstration of many fundamental properties of the n (cid:80)N |E |4
s general area of discrete nonlinear systems [1,2]. Studies n=1 n=1 n
c
i of periodic and non-periodic systems allowed to formu-
t
p late the necessary conditions for the excitation of non-
o linear localized structures (i.e., discrete solitons) in dif-
s. ferent dimensions and geometries. For example, surface
c solitons have been an intense area of research [3] where
i
s stationary localized solutions exist only above a power
y
threshold.
h
The development of new experimental techniques has
p
[ allowedavarietyofinterestingconfigurations.Inpartic-
ular, we mention here the laser direct-writing approach
1
usingfocusedpulsedfemtosecondlaserradiation[4].The
v
possibilities range from ordered to disordered systems
1
3 in 1D and 2D settings. For example, main properties
8 of chirped arrays have been theoretically [5] and exper- Fig. 1. Intensity distribution (|Eν|2) of linear modes.
3 imentally [6] analyzed. Very recently, in the context of n
.
1 quantumopticsinwaveguidearrays,Glauber-Fock(GF) Linear modes of model (1) are given by Eν(z) =
√ √ n
0 photoniclattices[7,8]havebeenstudied.Authorsfound, n!2−n/2H (λ / 2)exp(iλ z), with H being an Her-
n ν ν n
2
forexample,theappearanceofanewfamilyofquantum mitepolynomial[7].Theeigenvaluespectrum{λ }does
1 ν
correlations which are absent in uniform arrays. not form a compact band, but rather a set of well sep-
:
v In this letter, we study the excitation of nonlinear lo- arated discrete frequency levels. The system is invariant
i
X calized states in a GF lattice. This system corresponds under the staggered-unstaggered transformation:
toasingle-modeopticalwaveguidearraywhosecoupling
r
a between sites increases as a square root. In the coupled- En[−ν+(N+1)/2] =(−1)nEn[ν+(N+1)/2] ∀ν ∈{1,(N−1)/2},
mode approximation, the evolution of the field ampli-
with λ = −λ . Therefore, only
tudeisdescribedbyadiscretenonlinearSchr¨odinger-like [ν+(N+1)/2] [−ν+(N+1)/2]
half of the modes are shown in Fig.1 for N =59, follow-
equation [1,2,7,8]:
ing Ref. [8]. The spatial distribution of the linear modes
−idEn =V E +V E +γ|E |2E , (1) isveryinhomogeneous.Formodeswithfrequenciesclose
dz n+1 n+1 n n−1 n n to zero (ν ∼30), light is extended across the array with
whereE denotestheamplitudeofthemodeinthen-th awelldefinedmainlobecenteredatdifferentpositionsof
n
waveguide for an array of N sites. z corresponds to the the lattice [see Fig.1]. For modes with frequencies closer
propagation coordinate along the waveguide, γ to the to the band edges (ν ∼ 1 or 59), profiles are well local-
√
nonlinear parameter and V = nV to an increasing ized.[ForanhomogenousDNLSlattice,alllinearmodes
n 1
coupling coefficient (GF lattice). Without loss of gener- are spatially extended (R ∼ 2N/3) without any main
ality, we define V = 1 and γ = 1 (focusing case). To lobe or peak [1,2]].
1
1
As a consequence of the linear properties (mode pro-
files and frequency gaps), nonlinear modes will exist in
different frequency domains. For example, modes ν = 1
and 59 are the most localized ones (R≈9), both having
a peak located around n∼55. These modes have no ex-
tended tail and they are a result of the ramping in the
coupling coefficient (for chirped systems, there is always
atleastonelinearlocalizedmodeclosetooneofthebor-
ders of the lattice [5,6]). Therefore, we expect to find a
nonlinearsolutionbifurcatingwithzeropowerthreshold
close to this region. On the other hand, the center mode
(λ = 0, R ≈ 18) has a main peak located at n = 1.
30
This mode possesses a non-negligible tail which will for-
bid the excitation of a zero-threshold fundamental non-
linear solution (one-peaked bright solitons, i.e. an odd
mode [2]). In addition, modes ν =28, 29, 31 and 32 also
possess a main peak located at n=1. Therefore, we ex-
pectthatdifferentlinearmodes-atdifferentfrequencies
-contributeintheformationoftheprofile/tailstructure
while the solution decreases its power and interact with
the linear spectrum. For example, modes ν =26, 27, 33
and 34 possess a main peak located at n=2. Therefore,
the respective nonlinear mode will not have an unique
frequency origin, and it will probably experience some
power threshold. By observing Fig. 1, we expect that
the power and frequency thresholds increase as the po-
sition of the linear mode peak increases with n (up to
the region where we expect a zero threshold). Then, the
power threshold should increase again because there is
no further linear mode peaked at n>56.
We look for stationary solutions of Eq.(1) of the form 20 c)
E (z)=A exp(iλz), where A are real numbers and λ
n n n Pst10
corresponds to the solution frequency. We compute one-
peak localized stationary solutions by using a Newton- 0
Raphson iterative method starting at λ (cid:29) λ ≈ 14. 1 10 20 30 40 50 59
59
Oncewegetanumericalsolutionwemakeasweepinλto Site,n
constructthewholefamily.Foreachsolution,weperform
astandardstabilityanalysis[3]and,asageneralconven- Fig.2.(a)and(b)P versuspropagationconstant(λ)for
tion,weuseblack(gray)linesforstable(unstable)solu- solutions peaked at site n = 1 and n = 5, respectively
tions. Fig.2(a) shows a Power versus Frequency diagram (vertical dashed lines correspond to different λν). Insets
for solutions with a main peak located at n = 1. De- show profiles {An} for parameters indicated in each fig-
pending on the accuracy of our numerical continuation, ure. (c) Pst versus the peak position n.
we detect some “resonances” of the nonlinear solutions
withsomelinearbandmodes(verticaldashedlinesindi-
cate different λν). If we numerically continue these solu- In contrast, the solution peaked at n = 56 exhibits a
tions-byincreasingthefrequency-weseehowthetails power that drops to almost zero, because it bifurcates
increase with an structure similar to the linear modes from the linear localized mode ν = 59. We character-
at those regions [see divergent curves in Figs.2(a)-(b)]. izeallthestationarysolutionsbydefiningthestationary
Solutions peaked close to n=1 present similar features; power threshold (P ) as the last set {λ,P} before the
st
see, for example, Fig.2(b). Insets in Figs.2(a)-(b) show solution becomes unstable [horizontal lines in Figs.2(a)-
some profiles for these families. We see how high fre- (b)].Fig.2(c)showsP versusthesitewherethesolution
st
quency (power) solutions are well localized and how low hasitsmainpeak.Weseeanapproximatelysquare-root-
frequency solutions interact with the linear modes (see dependenceofthepowerincrementfromn=1uptothe
the tails of these profiles). If we excite solutions peaked region n ∼ 56, where this threshold abruptly drops to
at larger n, we find a phenomenology typical for surface zero. For the further solutions, the stationary threshold
modes [3]; i.e., the power and frequency decrease up to increases again up to a maximum.
a point where ∂P/∂λ < 0, solutions become unstable In a next step we want to determine the dynami-
and are not allowed to exist below a power threshold. cal power threshold for the excitation of one-site solu-
2
tions centered in different sites of the lattice. We in- trivial issue due to the long tail of the solutions. From
tegrate numerically model (1) with a one-site input Fig.3 we see how the tendency of our definition agrees
√
condition E (0) = P δ , where P is the input perfectly with the dynamical result. Nevertheless, the
n 0 n,n0 0
power and n the input position. For the set {n ,P } pseudo-stationarypredictionwillalwaysbesmallerthan
0 0 0
we compute the space-averaged fraction of power: T ≡ the dynamical one (for example, bulk DNLS solutions
1/(P z )(cid:82)zmax|E (z)|2dz, remaining at the initial have a stationary threshold equal to zero and a dynam-
0 max 0 n0
waveguideafterafixedpropagationdistancez .Fig.3 ical one around 4).
max
Additionally, we have studied systems with different
functional forms of V (exponential, quadratic, linear
n
and logarithmic). All these systems possess a similar
bandstructureinthesenseofnon-compactnessandspa-
tial distribution of linear modes. The numerical compu-
tationofT showssimilarfeaturesthantheonessketched
in Fig.3, except the global shape of the increment which
isproportionaltotheshapeoftheparticularV function.
n
In order to experimentally prove the threshold behavior
andobservenonlinearlocalizedmodesinsuchstructures,
the range of the coupling function V is constrained to
n
the length of the sample. A large value of V implies a
n
strongcouplingandafastertransportfromoneguideto
another and, therefore, requires a very long crystal for
observing such phenomenology.
Inconclusion,westudiedthelinearpropertiesandthe
stationary and dynamical excitation of nonlinear local-
ized modes in GF lattices. Contrary to homogenous sys-
tems, we found that close to the lattice-edges solutions
possess smaller stationary thresholds. We introduced a
new quantity, the P , which showed its utility when
ps
comparing stationary and dynamical properties.
Fig.3.T asafunctionofP andn .Continuous,dashed TheauthorsacknowledgesupportfromCONICYTfel-
0 0
and dotted white lines correspond to T = 0.8, P and lowship, FONDECYT Grant 1110142, Programa ICM
st
P , respectively. Inset: zoom of region n∈{45,59}. P10-030-F,ProgramadeFinanciamientoBasaldeCON-
ps
ICYT(FB0824/2008)andtheGermanMinistryofEdu-
shows our results where T = 1 (lighter color) means cationandResearch(CenterforInnovationCompetence
that100%ofthelightisconfinedintheinputwaveguide program, grant 03Z1HN31).
whereas T = 0 (darker color) means that all the light
has escaped from the input position. In this figure, we References
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3