Exact Dynamic Localization in Curved AlGaAs Optical
Rajiv Iyer1, Jun Wan2, Marc M. Dignam2, C. Martijn de Sterke3, J. Stewart Aitchison1
1 Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, Ontario, Canada, M5S 3G4.
2 Department of Physics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6.
3 Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS), University of Sydney, New South Wales 2006, Australia.
Abstract: We present the first experimental observations of exact dynamic localization of an
optical beam in periodically-curved strongly-coupled waveguide arrays. Spatial and spectral
measurements of two and four period devices agree well with theory.
©2007 Optical Society of America
OCIS codes: (130.3120) Integrated optics devices; (130.2790) Guided waves
The evolution of an electron wavepacket in a one-dimensional periodic spatial potential under the influence of a
time-dependent, ac electric field has attracted much recent attention [1-3]. Under special conditions, the interplay
between the potential and the ac electric field causes the wavepacket to exhibit dynamic localization (DL), the
periodic return of the wave packet to its original localized state, a phenomenon similar to Bloch oscillations in dc
fields. Because of the difficulties observing DL in electronic systems, we consider an alternative, but mathematically
equivalent, system: the Curved Coupled Optical Waveguide (CCOW) array [4-6], where the time-dependent ac field
is mimicked in the optical domain by a distance-dependent periodic curvature profile in the waveguides.
DL has been previously observed to arise for continuously varying ac fields in systems whose bandstructure
could be well described by the nearest-neighbor tight-binding (NNTB) approximation [1, 2, 3, 6]. This approximate
dynamic localization (ADL) was observed in a sinusoidally curved CCOW device by Longhi . In contrast, exact
dynamic localization (EDL) can only occur in general structures (e.g., non-NNTB) if the ac field is discontinuous
. Here we present an experimental demonstration of EDL in non-NNTB CCOWs with discontinuous curvature
profiles — the first demonstration of EDL in any system. We find that EDL persists for at least four periods, and
report excellent agreement of our measurements with theory for both DL evolution and wavelength dependence.
Device Design and Simulation
A schematic of our CCOW system is shown in Fig. 1. The single mode, polarization independent waveguides were
designed to have width, dw = 4.0 μm. The CCOW array was designed with period, d = 6.7 μm at the resonant
wavelength, λR = 1550 nm. We used the simplest discontinuous ac curvature profile, the square-wave, with a
distance dependent radius of curvature, R(v), which varied between +R0 to –R0 over one full period, Λ. For our
design, R0 = 35.238 mm, and Λ = 5 mm.
The beam evolution over two periods through the CCOW structure was simulated at λR = 1550 nm using the
Schrödinger equation in the one-band approximation (Fig. 2). At v=0, an initially localized beam begins to spread
(with a slight asymmetry in the u-dimension) to an oscillation amplitude of ΨDL ≈ 4d (i.e., the maximum beam
divergence is ±4 waveguides), and relocalizes at v=Λ, and v=2Λ, remaining localized over a length, ΔvDL ≈ 300 μm.
Fig. 1. Schematic of CCOW structure Fig. 2. Simulation results of beam propagation. (Log scale)
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The chips were fabricated on an AlGaAs wafer using photolithography and dry etching . A polarized CW beam
was coupled into each chip using a microscope objective; the output beam was captured using a traditional
waveguide imaging system. Only TM-launch results are presented below.
We tracked the beam propagation through our CCOW device to obtain a spatial map around the 2-period EDL
plane at 1550 nm (Fig. 3(a)). The measured performance agrees very well with the theoretical predictions from the
one-band Schrödinger model (using experimentally measured parameters), shown in Fig. 3(b). After 2 periods of
oscillation, we see the following expected behavior: (1) DL was clearly observed, (2) the beam remains localized for
ΔvDL ≈ 300 μm, (3) the oscillation amplitude, ΨDL ≈ 4d, and (4) the beam propagation was slightly asymmetric in
The wavelength dependence of our 4-period EDL device was conducted between 1480 and 1600 nm, shown in
Fig. 4(a). From the measurements, we see that after 4 periods, the beam has fully relocalized at a wavelength of
1555 nm, i.e., the actual resonant wavelength of the fabricated device. For comparison, the simulation result from
the one-band Schrödinger model (using experimentally measured parameters) is shown in Fig. 4(b). DL occurs only
close to the resonant wavelength; hence, the beam is seen to spread to ±3d at 1480 nm, and to ±2d at 1600 nm.
Because of this wavelength dependence, this device has been proposed for optical filtering applications .
Fig. 3(a). Measured beam propagation.
Fig. 3(b). Simulation results of beam
propagation. (Logarithmic scale)
Fig. 4. Wavelength dependency.
(a) Measured, and (b) Simulation results
(Both on a logarithmic scale)
This is the first time in any system, to the authors’ knowledge, that exact dynamic localization has been
experimentally observed, or that the evolution of DL of any kind has been experimentally mapped. We
demonstrated EDL over four periods in the optical domain, using closely coupled CCOW arrays, driven by an
equivalent ac square-wave field; we find excellent agreement with theory and with the design targets. We have
shown that the CCOW array serves as an ideal alternative system in which to test quantum electronic phenomena.
This device has applications in broadband optical filtering.
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