We present experimental observations of exact dynamic localization of an optical beam in a periodically curved AlGaAs waveguide array. The dynamic localization of the beam is “exact” in that it is observed even when the photonic band of the array is not well described in the nearest-neighbor tight-binding approximation. We present the spatial evolution of the beam around the two-period plane in the structure, explicitly demonstrating the delocalization and subsequent relocalization of the beam. We also demonstrate the strong wavelength dependence of the beam relocalization for a four period structure.
©2007 Optical Society of America
The evolution of an electron wavepacket in a one-dimensional periodic spatial potential under the influence of a time-dependent, periodic electric field has attracted a great deal of attention over the last 20 years [1, 2, 3, 4, 5, 6, 7]. Much of the interest arises from the subtle interplay between the periodic potential and the electric field, which, under some conditions, can counteract the tendency of the wavepacket to spread. For cases when the wavepacket returns to its original localized state, dynamic localization (DL) occurs, a phenomenon that is particularly interesting, for neither the periodic potential nor the electric field on their own inhibit the wavepacket’s delocalization.
Dynamic localization is closely related to Bloch oscillations (BOs) [8, 9], which occur when the applied field, Fdc, is constant in time, rather than periodic. In a spatially periodic potential, a localized wavepacket in such a field initially spreads; however, the spreading is subsequently arrested and reversed, so that the wavepacket returns to its original state. This motion is periodic in time, with a Bloch period that scales as F -1 dc. This type of electron localization was proposed more than 50 years ago by James , and was first experimentally verified in optically-excited semiconductor superlattices in 1992 .
The behavior of a wavepacket undergoing DL is similar to BOs: for an ac field with an appropriately chosen shape and amplitude , the wave packet returns to its initial state, with a period given by the period of the ac field. DL was first theoretically shown to arise for sinu-soidally varying electric fields [1, 2]. However, it has been shown that DL can only occur with sinusoidal fields (or any continuous periodic ac fields) in systems in which adjacent spatial wells are weakly coupled, so that the bandstructure is well described by the NNTB (nearest-neighbor tight-binding) approximation . In this approximate dynamic localization (ADL), the maximum beam divergence, or oscillation amplitude is small, and the beam does not re-localize in general (e.g., non-NNTB) structures. In contrast, it has been theoretically shown that exact dynamic localization (EDL) can occur in these general structures only if the electric field is discontinuous , allowing for rigorous relocalizations, with much larger oscillation amplitudes — a phenomena that has not yet been experimentally demonstrated.
The experimental observation of EDL in electronic systems is very challenging. Large amplitude, discontinuous ac fields (on the order of 10 kV/cm) in the THz frequency range (such that the period is shorter than electron decoherence times of ∼ 1 ps) are extremely difficult to generate. Electronic systems have the additional problem that electron-electron and electron-phonon interactions significantly alter the dynamics of the wavepacket evolution. As well, indirect measurement techniques must be used (e.g., four-wave mixing [11, 12] or terahertz emission [5, 7, 13]), since direct imaging of the electron wavepacket is not possible.
To overcome the difficulties of observing DL in electronic systems, two alternative systems have been considered: cooled atomic arrays , and optical waveguide arrays [15, 16, 17], the second of which is investigated in this paper. The spatial propagation of light in such arrays is directly analogous to the temporal evolution of an electron wavepacket in a superlattice. The effect of the applied electric field on the superlattice is obtained in the waveguide structure by inducing a transverse refractive index gradient across the array (attained, for example, by a temperature , or structural gradient ). In the work presented in this paper, the index gradient was created using Curved Coupled Optical Waveguide (CCOW) arrays [20, 21], in which the periodically varying waveguide curvature profile serves as the optical analog of the ac electric field. Of note, CCOW structures do not suffer the drawbacks of electronic systems: the 1–10 mm period necessary for the waveguide curvature, and the curvature discontinuities are easily patterned using standard photolithographic techniques; the electron-electron and electron-phonon interactions have no direct linear optical analog; and the CCOW system permits the spatial monitoring of the evolution of the optical beam.
Longhi et al.recently presented the first experimental observations of DL in a sinusoidally-shaped CCOW array — a demonstration of ADL, because of its continuously varying curvature profile . In contrast, in this paper, we present the first experimental demonstration of EDL in CCOWs with discontinuous waveguide curvature profiles — this being the first demonstration of EDL in any system. We report excellent correspondence of our experimental results with theory for both DL evolution and wavelength dependence; notably, we find that the EDL persists for at least four periods.
This paper is structured as follows: a brief review of the theory of DL in CCOWs, and details of the device parameters are presented in Section 2. The experimental results, including the wavelength dependence of the beam at the 4-period EDL plane, and spatial field evolution of the beam around the 2-period EDL plane are presented in Section 3. We discuss our results and conclude in Section 4.
2. Dynamic localization in curved coupled optical waveguides
2.1. Theory of dynamic localization in CCOWs
The CCOW system is a periodic array of curved waveguides. The structure is shown schematically in Fig. 1. The width of the individual waveguides, dw, and the transverse period of the array, d, are kept constant, while the array as a whole is curved, with a radius of curvature, R(v), that depends on the propagation distance, v, along the center of the central waveguide.
We have shown previously  that in the slowly-varying envelope approximation, the propagation equation for the beam in a CCOW array maps directly onto the Schrödinger equation for an electron of mass, m*, in a periodic potential, U(u), in the presence of a uniform ac electric field, F(t), via
where u is the spatial coordinate perpendicular to v in the plane of the structure, t is time, c is the speed of light in a vacuum, h is Planck’s constant, λ is the free space wavelength, e is the magnitude of the charge on an electron, n(u) is the refractive index profile along u (obtained after reducing the 3D system to 2D using the effective index method ), nˉ is the effective index of the fundamental mode in a single waveguide, and δn 2(u) = n 2(u)-nˉ2. The mapping of t onto v, and F(t) onto R(v), allows us to design the CCOW analog of the electronic system in which DL should arise. We note that because the effective ac electric field depends explicitly on λ (Eq. (1)) and DL only occurs for particular field amplitudes, DL in the CCOW system only occurs at resonant wavelengths, λR. This idea has been exploited in a previous paper to propose that such structures could be used as optical filters .
It is useful to characterize the properties of a CCOW system by first considering an array with straight waveguides. Due to the periodicity of the array, the modes form photonic bands .
The propagation constant, β, of the modes in the lowest band can be expressed as
where βp are the Fourier expansion coefficients of the propagation constant, and ku is the Bloch wavevector of the mode (-π/d < ku ≤ π/d) . The bandwidth of this lowest band is defined to be B = max[β(ku)]- min[β(ku)]. Larger bandwidths correspond to increased coupling between the waveguides.
It has been shown that DL of any kind can only occur in the one-band approximation , (i.e., where only the lowest band is considered), and when coupling to radiation modes (i.e., bend loss) is small; the coupling to radiation modes in CCOWs is analogous to Zener tunneling to higher bands in an electronic system. As mentioned in Section 1, ADL occurs in CCOW structures with continuous curvature profiles only if the photonic band is well described in the NNTB approximation, i.e., βp ≃ 0 for |p| ≥ 2 (see Eq. (2)) . Longhi, et al.  have experimentally demonstrated ADL in a CCOW structure with continuously varying, sinusoidally-shaped waveguides, indicating that their structure could be well described using the NNTB approximation. Experimental investigations on multiband effects in CCOWs under various excitation geometries has been conducted in a related work .
In general CCOW structures (e.g., non-NNTB structures), rigorous DL (i.e., EDL) is only possible if the radius of curvature profile, R(v), satisfies a number of conditions. Two important (but not sufficient) conditions are that R(v) must be discontinuous at every sign change, and R(v) must satisfy
Of the many different curvature profiles that can produce EDL , the simplest is the square-wave field, with curvature period, Λ, and radius of curvature profile,
where m is an integer. The required radius of curvature amplitude, Ro, using Eq. (3), is found to be
In the structures that are investigated in this paper, we have chosen Q = 1, in order to maximize Ro and thereby minimize the coupling to radiation modes.
The EDL in this Q=1 square-wave case is easily understood by considering the beam evolution over one full curvature period: the beam undergoes one BO in the first half of the curvature period (where R(v) = +Ro), and undergoes another BO in the second half of the curvature period (where R(v) = -Ro).
The measure of the delocalization between relocalization points is defined as the oscillation amplitude, ΨDL, (in units of transverse periods, d), which is proportional to the photonic bandwidth B, multiplied by the curvature period, Λ . Therefore, in order to observe relatively large, unambiguous oscillation amplitudes (i.e., ΨDL ≥ 3d), systems with strong coupling are required (which, by definition cannot be described by the NNTB approximation). For this reason, we have chosen to demonstrate rigorous EDL in a non-NNTB system using the discontinuous square-wave curvature profile described in Eq. (4). Details of the design are presented in the next subsection.
2.2. CCOW design
The EDL structures were designed using an AlGaAs wafer [19, 27]. The out-of-plane modal confinement was achieved by a 1.5 μm Al18Ga82As core layer, surrounded by Al24Ga76As cladding layers (1.5 μm above, and 4.0 μm below), grown on a GaAs substrate (Fig. 1). The in-plane confinement was achieved by defining waveguides with an etch depth of h = 1.35 μm and width of dw = 4.0 μm. The waveguides were designed to be single-mode and essentially polarization independent at 1550 nm, with an effective refractive index calculated to be nˉ = 3.261. The array was designed with a transverse period, d = 6.7 μm, with the band’s first five Fourier coefficients (see Eq. (2)) being β 1 = -9.70 cm-1, β 2 = 1.06 cm-1, β 3 = -0.190 cm-1, β 4 = 0.04 cm-1, and β 5 = -0.01 cm-1. With this design, the photonic bandwidth of the lowest band was computed to be B = 39.6 cm-1.
Our structure was designed to satisfy a number of criteria: that (1) the photonic bandwidth, B, was wide enough to create a relatively a large oscillation amplitude, (2) the higher Fourier coefficients of the photonic band (i.e., βp for |p| ≥ 2) were large enough to invalidate the NNTB approximation, such that DL would not occur with continuous curvature waveguide profiles, and (3) the radius of curvature was large enough such that radiation losses (i.e., the coupling to higher-band radiation modes) were relatively small .
To demonstrate EDL, the square-wave curvature profile was used (see Eqs. (4) and (5)), with a DL period, Λ = 5 mm, and an amplitude, Ro = 35.238 mm, at the resonant wavelength, λR = 1550 nm. The spatial evolution of the beam over four periods through the CCOW structure was calculated using (a) a 2D Beam Propagation Method (BPM) simulation (with no assumptions), and (b) Eqs. (1) to map the beam evolution onto the Schrodinger equation and solving the dynamics using an expansion in the basis of the Wannier functions of the lowest band [6,16]. This second method explicitly omits any coupling between bands and is henceforth referred to as the one-band Schrödinger model. The results are plotted in Fig. 2 on a logarithmic scale in the (u, v) coordinate system. The relocalized beam profile at v/Λ = 4 is shown adjacent to both spatial maps. From these simulations, we note the following: (1) DL in our system occurs every v = mΛ, (2) the DL persists for a span of ΔvDL ≃ 300 μm about each localization plane, (3) the oscillation amplitude, ΨDL ≃ 4d, and (4) the beam propagation is slightly asymmetric (in the u dimension). The radiation modes seen in Fig. 2(a) introduce less than 5% loss per DL period, and create the ripples seen between relocalizations due to interference with the light still confined in the array. These radiation modes (and ripples) do not appear in the Schrödinger simulation results (Fig. 2(b)), because the higher photonic bands are excluded in the one-band model. Since the one-band model provides an excellent description of our system, it was used to produce the theoretical results in the remainder of the paper.
Because the second Fourier coefficient, β 2, in our structure’s band dispersion is not negligible (compared to β 1), this system cannot be described in the NNTB approximation. Thus, a continuously varying waveguide curvature profile (corresponding to a continuous ac field in an electronic system) should not yield DL (i.e., ADL should not be possible). To demonstrate this, we calculated the spatial evolution of the light in the same non-NNTB CCOW structure, but with a sinusoidal curvature profile, R(v) = 1/(RAsin(2πv/Λ)). It has been shown that for such a curvature profile, ADL can only occur for an NNTB structure when the amplitude, RA, satisfies: J 0(nˉdΛ/λRA)) = 0, where J 0(x) is the zeroth order Bessel function of the first kind [1, 2, 17, 28]. In Fig. 3, we plot the beam evolution through this CCOW array using the smallest root of the Bessel function. The beam has relocalized reasonably well at v/Λ = 1; however, in this non-NNTB structure, DL clearly has broken down, the beam evolution has lost its periodicity, and at v/Λ = 4, the beam has spread to a width greater than 5d (plotted in the beam profile to the right of the spatial map).
While other continuous curvature profiles (besides sinusoidal) may lead to improved ADL performance, true DL cannot be observed without curvature discontinuities, as previously mentioned . Hence, in the structures presented in this paper, square-wave curvature profiles were employed to demonstrate EDL performance unambiguously.
3. Experiment and results
Fabrication of the optical chips were conducted using hard-contact photolithography in a Karl Suss MA6 mask aligner with chrome-on-quartz mask plates. Ultra-violet/temperature cycle curing of the Shipley SPR 511 positive photoresist prior to etching was performed in order to preserve the near-rectangular profile necessary for vertical waveguide sidewalls. The chips were etched using an Oxford Plasmalab System100 ICP/RIE dry etcher. Once etched, the chips’ front- and end-facets were cleaved using using standard diamond scribing techniques. The etch depth was measured to be 1.33 ± 0.005 μm, or 1.5% shallower than the target of h =1.35 μm; the post-etch waveguide width was found to hit the target of dw = 4.0 ± 0.05 μm; and the waveguide separation was measured to hit the target of d = 6.7 μm ± 0.025 μm. Layouts of the photomask designs (which are described in detail in the next two subsections) are presented in Fig. 4. In these figures, the waveguide curvature of the arrays (highlighted by the exaggerated curvature of their measurement lines) are barely visible, because the lateral amplitude of the curved waveguide path (in the u dimension) is only 22 μm; the curved propagation distance, v, is therefore effectively equivalent to the linear distance, z, to within 0.02% (see Fig. 1).
The experimental setup consisted of a collimated CW beam (from a JDS SWS 15101 tunable laser source) focused into the front-facet of each chip using a Newport FL40B coupling lens. A half-waveplate + polarizer combination was used to select either TM or TE polarized light for the launch. The beam from the chip’s end-facet was captured using a traditional waveguide imaging system, using a Hamamatsu C2741 IR camera and a New Focus 5723-BH aspheric lens for optimal image quality.
An independent single-mode waveguide coupled to the central waveguide of each 51-waveguide CCOW array guaranteed a localized input mode for every experiment (see Fig. 4(a)). A full polarization characterization of the experiments revealed that the CCOW design was essentially polarization independent, as expected (Section 2.2). Therefore, only experimental and simulation results for the TM launch are presented below.
3.1. Wavelength dependence
As described in Section 2.2, dynamic localization in our CCOW device should occur at λR = 1550 nm. We wished to examine the wavelength dependence of our CCOW device at precisely the forth EDL plane (i.e., at v/Λ = 4). Since achieving a cleaving tolerance less than ±0.5 mm is difficult using standard cleaving procedures, we implemented a novel technique which allows the chip to be cleaved at an arbitrary z-plane, while still being able to image the desired EDL plane to within an accuracy of a few microns.
A sufficiently long and wide (6 and 3 mm, respectively) output-slab waveguide (OSlab) was photolithographically defined, originating at the termination of the forth EDL plane, as shown in the photomask layout in Fig. 4(b). Note that the OSlab must be adequately wide to avoid reflections from its side walls, and consequently multimode interference patterns.
Upon cleaving the chip through the OSlab at an arbitrary z position, the forth EDL plane can be brought into transverse focus (in the u dimension) by literally imaging through the OSlab; whatever beam profile appears at the output of the CCOW array is allowed to propagate without lateral confinement through the OSlab, as if in a block of transparent material (with an index equal to the OSlab’s effective-index, nˉslab). We define the resulting post-cleaved length of the OSlab to be Zlen. The imaging system, initially focussed on the end-facet (i.e., at the output of the OSlab) must be brought closer to the chip by a travel distance of Zfocus = Zlen/nˉslab to bring the output of the CCOW array into lateral focus. Zlen must be kept as small as possible to minimize the effect of spherical aberration on the beam. In our experiment, Zlen was measured to be approximately 1 mm, resulting in a Zfocus ≃ 0.300 mm. This imaging scheme allows for the observation of the EDL plane to a tolerance of approximately 1 μm (achieved by photolithography), and limited only by the z-travel resolution of the imaging system.
The wavelength dependence of the 4-period EDL CCOW structure using the output slab technique was conducted between 1480 and 1600 nm. The output images taken over the wavelength range were combined and expanded using cubic interpolation to provide the data plot shown in Fig. 5(a). Notably, the actual resonant wavelength is seen to be 1555 nm or +0.32% higher than the target λR = 1550 nm. Upon performing a back calculation of Eq. (5), the actual waveguide effective-index was found to be 3.271, or +0.32% higher than the target nˉ = 3.261. This effective-index deviation is within the cumulative tolerances of its defining parameters: the aluminum percentage of the cladding and core layers, the thickness of the cladding and core layers, the etch depth, and the waveguide width.
The simulation results obtained from the one-band Schrödinger model (using the experimentally obtained values for d, dw, nˉ, and h) are presented in Fig. 5(b). There is excellent correspondence of the performance of our device with theory, particularly in the spreading of the beam to ±3d at 1480 nm, and to ±2d at 1600 nm. Both the measurement and simulation results at 1555 nm are shown in Fig. 5(c) revealing excellent dynamic localization of the beam in our device. A video of the wavelength dependence is provided in Fig. 6, highlighting the correspondence of the beam divergence between the measured and simulation results.
3.2. Spatial map
Direct imaging of the spatial evolution of the beam through a CCOW structure has been demonstrated in Er:Yb doped phosphate glass ; however this approach was not well suited for the AlGaAs waveguides used for our CCOW devices. As such, a traditional end-facet imaging system was employed to capture the light emitted from the experiments, as described earlier.
To capture the progression of the beam through our structure, a staggered multi-experiment technique was employed , where a set of 21 identical EDL CCOW experiments were placed side-by-side on the same chip (Fig. 4(c)). Each CCOW array was 3Λ in length, and was staggered by an offset of Δz = 250 μm with respect to its preceding neighbor. Upon performing the end-facet cleave roughly through the 2-period plane of the middle (i.e. the 11th) experiment, at v/Λ ≃ 2, we were left with 21 cutbacks of an identical experiment at incrementally varying lengths, covering a total span of 5 mm, i.e., one full EDL period. The capture of the relocalized mode was assured, since the stagger, Δz, was less than the predicted DL propagation length ΔvDL ≃ 300 μm (see Section 2.2).
The experiment was conducted at λ = 1550 nm, and an example of the output from one of the 21 experiments is shown in Fig. 7(a). Horizontal slices from each of the output images (illustrated by the dashed white line in Fig. 7(a)) were stacked to produce a single 21-pixel tall image, which was then subsequently expanded using cubic interpolation. The resulting plot is shown in Fig. 7(b); the horizontal line represents the data slice from Fig. 7(a).
To validate the performance of our CCOW design, the simulation results from the one-band Schrödinger model over one full period around the 2Λ EDL plane, are shown in Fig. 7(c) (using the experimentally obtained parameters, as discussed in Subsection 3.1). Apart from the presence of some low-intensity scattered and radiated light, the experimental results agree very well with the theoretical predictions: (1) DL is clearly observed, (2) the measured DL propagation span is ΔvDL ≃ 300 nm, (3) the oscillation amplitude, ΨDL ≃ 4d, and (4) the beam propagation is seen to be slightly asymmetric (in the u dimension).
4. Discussion and conclusion
This is the first time in any system, to the authors knowledge, that (1) the evolution of DL has been spatially mapped, and (2) exact dynamic localization has been experimentally demonstrated. In this report, EDL was demonstrated over four periods in the optical domain, using non-NNTB CCOW arrays, driven by an equivalent ac square-wave field. Novel observation techniques (staggered experiments and the output slab technique) were employed to obtain very clear device dynamics demonstrating the following: the initially localized beam diverged and subsequently relocalized unambiguously at the 2- and 4-period planes of the device; the beam had an oscillation amplitude of 4 waveguides; and the device had a strong wavelength dependence measured over the range from 1480 to 1600 nm. The devices showed excellent correspondence with theory (as predicted by both the one-band Schrödinger model as well as 2D-BPM), and in very close agreement with the design targets.
Work is currently underway to investigate more complex EDL ac driving functions in CCOW arrays, as well as other novel structures.
The authors would like to thank the Emerging Communications Technology Institute at the University of Toronto, Toronto, Canada, as well as the Institute for Microstructural Sciences at the National Research Council, Ottawa, Canada, for providing the photolithographyand etching facilities, respectively, required to fabricate the optical microchips. This work was produced with the assistance of the Natural Sciences and Engineering Research Council of Canada and the Australian Research Council under the ARC Centres of Excellence program.
References and links
1. D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged-particle moving under the influence of an electric-field,” Phys. Rev. B 34,3625–3633 (1986). [CrossRef]
3. X. G. Zhao, “Motion of Bloch electrons in time-dependent electric-fields with off-diagonal effects,” Phys. Lett. A 167,291–294 (1992). [CrossRef]
4. A. W. Ghosh, A. V. Kuznetsov, and J. W. Wilkins, “Reflection of THz radiation by a superlattice,” Phys. Rev. Lett. 79,3494–3497 (1997). [CrossRef]
5. W. X. Yan, S. Q. Bao, X. G. Zhao, and J. Q. Liang, “Dynamic localization versus photon-assisted transport in semiconductor superlattices driven by dc-ac fields,” Phys. Rev. B 61,7269–7272 (2000). [CrossRef]
7. A. Z. Zhang, L. J. Yang, and M. M. Dignam, “Influence of excitonic effects on dynamic localization in semiconductor superlattices in combined dc and ac electric fields,” Phys. Rev. B 67,205318-1–205318-7 (2003). [CrossRef]
8. F. Bloch, “Uber die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. A: Hadrons Nucl. 52,555–600 (1928).
9. C. Zener, “A theory of the electrical breakdown of solid dielectrics,” Proc. R. Soc. London, Ser. A , Containing Papers of a Mathematical and Physical Character145,523–529 (1934). [CrossRef]
10. H. M. James, “Electronic states in perturbed periodic systems,” Phys. Rev. 76,1611–1624 (1949). [CrossRef]
11. J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, “Optical investigation of Bloch oscillations in a semiconductor superlattice,” Phys. Rev. B 46,7252–7255 (1992). [CrossRef]
12. R.-B. Liu and B.-F. Zhu, “Degenerate four-wave-mixing signals from a dc- and ac-driven semiconductor superlattice,” Phys. Rev. B 59,5759–5769 (1999). [CrossRef]
13. C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, “Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice,” Phys. Rev. Lett. 70,3319–3322 (1993). [CrossRef] [PubMed]
14. K. W. Madison, M. C. Fischer, R. B. Diener, Q. Niu, and M. G. Raizen, “Dynamical Bloch band suppression in an optical lattice,” Phys. Rev. Lett. 81,5093–5096 (1998). [CrossRef]
15. G. Lenz, R. Parker, M. C. Wanke, and C. M. de Sterke, “Dynamic localization and ac Bloch oscillations in periodic optical waveguide arrays,” Opt. Commun. 218,87–92 (2003). [CrossRef]
16. J. Wan, M. Laforest, C. M. de Sterke, and M. M. Dignam, “Optical filters based on dynamic localization in curved coupled optical waveguides,” Opt. Commun. 247,353–365 (2005). [CrossRef]
17. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96,243901-1–243901-4 (2006). [CrossRef] [PubMed]
18. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83,4752–4755 (1999). [CrossRef]
19. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and K. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83,4756–4759 (1999). [CrossRef]
20. G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83,963–966 (1999). [CrossRef]
21. M. Heiblum and J. H. Harris, “Analysis of curved optical-waveguides by conformal transformation,” IEEE J. Quantum Electron. 11,75–83 (1975). [CrossRef]
23. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90,053902-1-053902-4 (2003). [CrossRef] [PubMed]
24. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13,1780–1796 (2005). [CrossRef] [PubMed]
25. K. Drese and M. Holthaus, “Anderson localization in an ac-driven two-band model,” J. Phys.: Condens. Matter 8,1193–1206 (1996). [CrossRef]
26. S. Longhi, M. Lobino, M. Marangoni, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Semiclassical motion of a multiband Bloch particle in a time-dependent field: Optical visualization,” Phys. Rev. B 74,155116-1–11 (2006). [CrossRef]
27. S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” J. Appl. Phys. 87,7825–7837 (2000). [CrossRef]
28. P. Domachuk, C. M. de Sterke, J. Wan, and M. M. Dignam, “Dynamic localization in continuous ac electric fields,” Phys. Rev. B 66,165313-1–165313-10 (2002). [CrossRef]
29. N. Chiodo, G. Della Valle, R. Osellame, S. Longhi, G. Cerullo, R. Ramponi, U. Laporta, and P. Morgner, “Imaging of Bloch oscillations in erbium-doped curved waveguide arrays,” Opt. Lett. 31,1651–1653 (2006). [CrossRef] [PubMed]