Discrete self-trapping vs. defocusing in nonlinear waveguide arrays

Abstract

We predict a sharp crossover from self-defocusing to discrete localization in photonic lattices with defocusing nonlinearity for increasing index contrast. We demonstrate experimentally discrete self-trapping of light by single site excitation in LiNbO₃ waveguide arrays.

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Discrete Self-Trapping vs. Defocusing in Nonlinear Waveguide Arrays
Christian R. Rosberg1, Michal Matuszewski1,2, Dragomir N. Neshev1, Andrey A. Sukhorukov1, Arnan
Mitchell4, Marek Trippenbach2,3, Michael W. Austin4, Wieslaw Krolikowski1, and Yuri S. Kivshar1
1Nonlinear Physics Centre and Laser Physics Centre,
Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS),
Research School of Physical Sciences and Engineering,
The Australian National University, Canberra ACT 0200, Australia
2Institute for Theoretical Physics, Warsaw University, Poland
3Soltan Institute for Nuclear Studies, Warsaw, Poland
4School of Electrical and Computer Systems Engineering, RMIT University, Melbourne, Australia
We predict a sharp crossover from self-defocusing to discrete localization in
photonic lattices with defocusing nonlinearity for increasing index contrast.
We demonstrate experimentally discrete self-trapping of light by single site
excitation in LiNbO3waveguide arrays.
c ? 2005 Optical Society of America
OCIS codes: 190.4420, 190.5940
Nonlinear periodic photonic structures offer unique possibilities to engineer and control optical wave diffraction and
nonlinear self-action of light1. In this paper, we study the self-action of a narrow Gaussian beam propagating in a
nonlinear defocusing medium with a periodic refractive index modulation in the transverse direction. We demonstrate
that a sharp transition from self-defocusing to discrete self-trapping occurs when the depth of the index modulation is
increased, and we observe experimentally the formation of a self-trapped state from single-site excitation in a nonlinear
defocusing waveguide array.
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Fig. 1. (a,b) Bandgap structure for a refractive index contrast of 1.1×10−4and 2.8×10−4, respectively. (c) Minimum
gap soliton width vs. refractive index contrast. Insets show the soliton profiles corresponding to (a,b). (d) Beam
self-trapping efficiency vs. refractive index contrast, calculated as the power fraction remaining in the 20 central
waveguides at the output for optimized input power (solid) compared with low power linear diffraction (dashed).
Nonlinear propagation of light in periodic photonic structures is commonly described by a discretizing tight-binding
approximation, idan/dz + βan+ C(an−1+ an+1) + γ|an|2an= 0, where z is the propagation coordinate, an(z) is the
n−th mode amplitude, β is the propagation constant, C is the coupling coefficient, and the last term accounts for
nonlinear mode detuning. In the framework of this discrete model, there is a full equivalence between the two types
of nonlinearities since the above equation remains unchanged after the transformation: an→ (−1)na∗
Provided only a single waveguide is excited, the only difference is in the phase structure of the nonlinear localized
solution. Indeed, a self-focusing nonlinearity (γ > 0) supports light self-localization in the form of unstaggered discrete
solitons1,2, whereas a defocusing nonlinearity (γ < 0) can lead to self-trapping in the form of staggered solitons when
the amplitude of the modes in neighboring waveguides is out of phase3.
The discrete model is generally valid when the contrast of the refractive index modulation is sufficiently large.
However, in the case of defocusing nonlinearity, the beam will experience nonlinearly enhanced defocusing instead of
self-trapping if the lattice modulation is absent or very weak. In order to study beam propagation beyond the limits
of the discrete model we numerically solve a normalized continuous nonlinear Schr¨ odinger equation for the slowly
varying field envelope E(x,z), i∂E/∂z + D∂2E/∂x2+ F(|E|2)E + ρ ∆n(x) E = 0, where D = zsλ/(4πn0x2
diffraction coefficient, ρ = 2πzs/λ, x is normalized to xs, and z is normalized to zs. To match our experiments,
the linear refractive index change ∆n is taken as ∆n(x) = ξ?
nand γ → −γ.
s) is the
nexp[−(x − n d)2/w2], where the waveguide width
w = 12µm, the waveguide spacing d = 19µm, and ξ defines the modulation depth. The refractive index contrast
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Citation: Rosberg, C, Matuszewski, M, Neshev, D, Sukhorukov, A, Mitchell, A, Trippenbach, M, Austin, M, Krolikowski, W and
Kivshar, Y 2006, 'Discrete self-trapping vs. defocusing in nonlinear waveguide arrays', in Proceedings of the Conference on
Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS 2006), California, USA, 21-26 May 2006.

Page 2

is therefore ∆nmax= 0.442 ξ due to overlap of the individual index profiles. Other parameters are: λ = 0.532µm,
n0= 2.234, xs= 1µm, zs= 1mm, and F(I) = 1.5(1 + I)−1for photovoltaic defocusing nonlinearity.
The periodic modulation of the refractive index results in the formation of a bandgap structure [Fig. 1(a,b)]. In
the case of defocusing nonlinearity, staggered solitons can form at the bottom edge of the first band. The size of
the gap [see Fig. 1(a,b)] limits the minimum width of self-trapped beams as shown in Fig. 1(c). For a small index
contrast, the narrowest soliton spans several waveguides, but localizes at a single waveguide when the index contrast
is increased [see insets in Fig. 1(c)]. We study this transition by modeling the propagation of a narrow Gaussian input
beam matching the width of a single waveguide. In Fig. 1(d) we plot, as a function of the index contrast, the relative
power that remains in the central section of the array after propagation over a distance of many diffraction lengths
(1m). We compare linear diffraction (dashed line) to the effect of nonlinearity (solid line) when the power is optimized
to maximize the fraction which remains in the central section of the array. The nonlinear self-action results in beam
self-defocusing in the region where the two curves coincide in Fig. 1(d), but a bifurcation appears for increasing index
contrast, corresponding to the formation of initially broad gap solitons. As the minimum width of the soliton decreases
and approaches that of a single waveguide, self-trapping with almost 100% efficiency becomes possible, representing
a sharp crossover from self-defocusing to discrete self-trapping at a certain index contrast threshold.
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INTERFEROGRAM
SATURATED ZOOM
Fig. 2. (a) Output intensity distribution for linear propagation at low laser power (0.1 µW). (b) Corresponding
intensity profiles: Solid line – experiment; shading – numerical solution of continuous model; crosses – discrete
model lattice site amplitudes. (c) Simulated beam intensity along the sample for low input power. (d-f) Same as
(a-c) for nonlinear propagation at high laser power (0.1 mW). (g) Saturated image of the localized state in (d).
(h) Interferogram confirming the staggered phase structure of the output beam.
In previous experimental studies4,5of self-localization in periodic structures with defocusing nonlinearity, the input
beam was specially prepared to reflect the phase structure of the staggered soliton. However, observation of soliton
localization at a single waveguide was not reported. In our experiments, we study single-site discrete self-trapping
in an array of closely spaced single mode waveguides fabricated by Titanium indiffusion into a mono-crystal lithium
niobate wafer, producing an index contrast ∆nmax= 2.8 × 10−4. The LiNbO3sample exhibits a strong photovoltaic
effect which leads to a self-defocusing nonlinear response. An extraordinarily polarized beam from a cw Nd:YVO4
laser is tightly focused onto a single waveguide at the input of the array. At low laser power (∼ 0.1 µW) the probe
beam experiences typical discrete diffraction [Fig. 2(a)]. The corresponding output intensity profile in Fig. 2(b, solid
line) matches numerical calculations performed using both the discrete model (crosses) and the full continuous model
(shading). The simulated propagation inside the array is depicted in Fig. 2(c). When the laser power is increased
(∼ 0.1 mW) the defocusing photovoltaic nonlinearity leads to strong beam localization at a single waveguide [Fig. 2(d-
f)], in a similar way as in focusing waveguide arrays. To confirm that the nonlinear state is indeed localized inside the
Bragg reflection gap, and not in the total internal reflection gap, its staggered phase structure is verified by saturated
image detection of zero intensity lines [Fig. 2(g)] and by interferometric measurement with a reference wave [Fig. 2(h)].
In conclusion, we have studied the crossover between beam self-defocusing and discrete self-trapping in waveguide
arrays with defocusing nonlinearity. We have shown that an abrupt transition is observed in the beam dynamics with
the increase of the refractive index, and demonstrated experimentally the generation of strongly localized staggered
states in nonlinear defocusing waveguide arrays by single waveguide excitation.
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