Negative refraction and spatial echo in optical waveguide arrays

0.18 MB·English

by  Ramaz Khomeriki

Negative refraction and spatial echo in optical waveguide arrays
Ramaz Khomeriki1,2,∗ and Lasha Tkeshelashvili2,3
1Max-Planck Institute for the Physics of Complex Systems,
No¨thnitzer Str. 38, 01187 Dresden, Germany
2Physics Department, Tbilisi State University,
Chavchavadze 3, 0128 Tbilisi, Georgia
3Institute of Physics, Tbilisi State University,
Tamarashvili 12, 0169 Tbilisi, Georgia
∗Corresponding author: [email protected]
3
1
0 CompiledJanuary17,2013
2 ThespecialsymmetrypropertiesofthediscretenonlinearSchro¨dingerequationallowacompleterevivalofthe
initial wavefunction. That is employed in the context of stationary propagation of light in a waveguide array.
n
Asaninvertingsystem wepropose ashortarrayofalmostisolatedwaveguides whichcause arelativeπ phase
a
shiftintheneighboringwaveguides.Bymeansofnumericalsimulationsofthemodelequationswedemonstrate
J
anovelmechanismforthenegative refractionofspatialsolitons. (cid:13)c 2013 OpticalSocietyofAmerica
6 OCIS codes: 190.4370, 190.6135.
1

]
s The optical properties of nano-structured metallic sys- uides, and leads to the negative refraction of solitons in
c
tems, often called metamaterials, can be tailored to al- the nonlinear optical lattices.
i
t
p most any need [1]. The left-handed materials with neg- Ourconsiderationisbasedonthefactthatinquantum
o ative refractive index represent a striking example of mechanics the Schr¨odinger equation is invariant under
. suchkind of artificialstructures [2]. It is importantthat the simultaneous transformations t t and ψ ψ∗.
s
→ − →
c the negative refraction is predicted to exist for nonlin- Therefore, the conjugation of the system wave function
i
s earmetamaterialsaswell[3].Apparently,the conceptof atthearbitrarytimemomentleadstothereversalofthe
y negativerefractionimpliesgreatpotentialforthecontrol system dynamics. Alternatively, it is possible to change
h
and manipulation of optical solitons [1,4]. It should be the sign of the whole Hamiltonian. For example, this
p
noted, however, that according to the Kramers-Kronig can be done in the context of cold atoms in optical lat-
[
relations, the left-handed materials are inherently ab- tices by simultaneously changing the signs of hopping

1
sorptive at the relevant frequencies [5]. Although the and interaction constants [11]. The formal analogy of
v
dielectric nano-structures cannot be described by effec- thequantum-mechanicalsystemswithlightpropagation
0
7 tive medium parameters, it is still possible to apply the inwaveguidearrays[12]allowsoneto expectthe similar
7 generalized Snell’s refraction law to such systems (e.g. effects in the context of optics. In particular, the com-
3 see [6]). In this sense, the negative refraction can be re- plete image reconstructionin a system of coupled linear
.
1 alizedinperiodicdielectricsystemsaswell[1].Thephys- waveguides was demonstrated theoretically [13] and re-
0 icalmechanismsthatgoverntheelectromagneticfieldsin alized experimentally [14].
3 dielectric nanostructures are different from those in the In order to explore the above mentioned analogy, let
1
metallic systems, and therefore, the constraintbased on usstartwiththeDiscreteNonlinearSchr¨odinger(DNLS)
:
v the Kramers-Kronigrelations does not apply. equation:
Xi Recently,in[7]thenegativerefractionofopticalbeams
atinterfacesbetweenbulkdielectricsandtiltedphotonic
ar lattices was studied both theoretically and experimen- i∂ψm =J(ψm+1+ψm−1)+θ(z)ψm 2ψm, (1)
∂z | |
tally. Moreover, it was shown that the discussed effect
persists in the presence of nonlinear self-focusing. The
whereψ standsforanon-sitefunctionofitsargument,
negative refraction was predicted for linear waves at in- m
andθ(z)is astepwise functiondefined asθ(z <0)= 1
terfaces between two photonic crystal waveguide arrays
−
and θ(z > 0) = 1. It is easy to see that this equa-
as well [8]. Later, in [9] it was demonstrated that the
tionisinvariantunderthesimultaneoustransformations
solitary wave refraction can be effectively controlled in
ψ ( 1)mψ and z z. Furthermore, let us sup-
nonlinear optical lattices. Here, we show that the nega- m m
→ − → −
posethatEq.(1)isvalidintheinterval z >ℓandallow
tive refraction and spatial echo for solitons can be real-
| |
the system to evaluate from the initial state ψ ( L) to
ized in nonlinear periodic dielectrics. In particular, the m
−
ψ ( ℓ). Now, if the system can somehow be brought
solitonpropagationinanarrayofcoupledopticalwaveg- m
−
to the state ψ (ℓ)=( 1)mψ ( ℓ), the initial state re-
uides[4,10]ischosenasamodelsystem.Wedemonstrate m m
− −
emerges at z = L with ψ (L) = ( 1)mψ ( L). The
thattheproperlydesignedscattererinthesystemcauses m m
− −
transition from ψ ( ℓ) to ψ (ℓ) = ( 1)mψ ( ℓ) can
thedesiredrelativephaseshiftintheneighboringwaveg- m m m
− − −
be achieved by evaluating the system according to the
1
following equation:
∂ψ 1 π
i m = ( 1)m+ ψ , (2)
m
∂z (cid:20) − 2(cid:21)2ℓ
intheinterval z <ℓ.Itshouldbenotedthattheneigh-
| |
boring sites gain π phase difference in this interval.
Therefore, for our purpose, the waveguide array de-
sign must be governed by (1) in the interval z > ℓ,
| |
while for z < ℓ the approximate model equation (2)
| |
must be valid. The discrete equations (1) and (2) repre-
sentgoodapproximationsforthecoldatomdynamicsin
deep harmonic potential or light propagation in optical
waveguidearrays.Amoregeneraltheoreticalframework
is given by:
∂Ψ ∂2Ψ
i = +δnΨ+χ(z)Ψ2Ψ, (3)
∂z ∂x2 | |
Fig. 1. Light intensity distributions for a) and b) repre-
for z > ℓ. Here, δn = W cos(x), and χ(z) = χ( z) sent the results of numerical simulations with different
| | − −
stands for the Kerr-type nonlinearity constant. More- boundary conditions of (3) for L < z < ℓ and of (4)
| |
over, for z <ℓ: for z < ℓ. Here, L = 750 and ℓ = 50. The values of
| | | |
nonlinear coefficients are shown in the graphs. Dashed
∂Ψ ∂2Ψ
vertical lines indicate the boundaries of the scatterer.
i = +δnΨ, (4)
∂z ∂x2 c) schematically displays the refractive index profiles in
′ the corresponding arrays (see the text for the values of
where δn = W cos(x)+(π/4ℓ)cos(x/2). In these equa-
modulations).
tions the variables x and z are scaled in units of 1/K
and k/K2, respectively. K represents the inverse spac-
ing of harmonic transverse modulations of the refrac-
′
tive index, and k is the carrier wavenumber given by In the case of Eq. (4) the lattice of depth W is mod-
k = n0ω/c. Here, ω is the laser beam frequency, and ulated with the amplitude π/4ℓ. As far as this modula-
n0 is the averaged refractive index. Note that, δn tion is assumed to be small, similar to Eq. (5) we can
k2[n(x) n0]/n0K2 definesthescaledrelativevariatio≡n write ϕ (x) exp √W′(x 2πm)2/√8 . Neverthe-
ofthelin−earrefractiveindexinthecorrespondingregions m ∼ n− − o
less, the harmonic modulation affects the corresponding
of the waveguide array.
eigenenergies of the modes and leads to a staggered en-
Let us first examine Eq. (3). Note that, as far as the
ergy structure. Using again the expansion (6) for large
case of deep lattices is concerned, the wavefunction can ′
values of W we obtain the negligible overlap between
be assumed to be localized aroundthe maximums of re-
neighboringmodes J 0.Therefore,we canequate the
fractive index profile, i.e. at the points x = 2πm (m is →
hopping terms to zero which finally gives the discrete
integer). Moreover, the fundamental mode for the mth
equation (2).
potential well can be approximated by:
We performed the numerical simulations based on
Eqs. (3) and (4) with the initial conditions at z = L
ϕm(x)∼expn−√W(x−2πm)2/√8o. (5) (L ℓ). Initially, the launched spatial soliton pro−pa-
≫
gates along the trajectory z = vx. Here, the parame-
Inthiscasethetight-bindingapproximationapplies[15–
ter v determines the direction of the optical beam. Due
18],andthe full wavefunction Ψ(x,z)canbe writtenas
to the effective index mismatch, the scattering process
follows:
takes place at the interfaces of the waveguide array re-
Ψ(x,z)= ψ (z)ϕ (x). (6)
m m
gion ℓ < z < ℓ. That causes the change of the beam
Xm −
propagation trajectory to z = vx, and eventually, re-
Furthermore, assuming that the mode overlap of the −
sults in the negative refraction of the soliton. Moreover,
neighboringsitesissmall,(3)reducestotheDNLSequa-
thearbitraryinitialprofileofthewavepacketatz = L
tion (1) where the coupling constantJ can be easily de- −
is expected to be reproduced at z =L.
termined from the following expression [15]:
In the numerical simulations (see Fig. 1) we choose
J = δn(x)ϕmϕm+1 ∂ϕm∂ϕm+1 dx. (7) tLiv=e i7n5d0exanmdoℓd=ula5t0i.oInnitshδenre=gicoons(ℓx<),|azn|d<tLhethneonrleifnreaacr-
Z (cid:20) − ∂x ∂x (cid:21)
coefficient is χ = 0.002 (for L < z < ℓ we have
| | − −
Herethenormalizationofeigenmodesϕ in(5)ischosen χ < 0, while χ > 0 for ℓ < z < L). Moreover, in the
m
such that in DNLS equation (see Eq. (1)) the nonlinear range z <ℓ refractive index is modulated according to
| |
coefficient value becomes exactly one. δn=3cos(x)+(π/4ℓ)cos(x/2), and the nonlinear coef-
2
5053) is also acknowledged.
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This work is supported by joint grant No 09/08 from
RNSF (Georgia)andCNRS (France);The funding from
Science and Technology Center in Ukraine (Grant No
3