Photonic lattices have provided in recent years a lab- oratory tool to visualize optical analogues of coherent phenomena generally encountered in solid-state physics, such as Bloch oscillations (BOs) [1–4] and dynamic lo- calization (DL) [5, 6] in lattices driven by external dc or ac fields. In waveguide arrays, a common engineer- ing approach to realize optical BOs and DL is to tailor the local curvature of the array axis [7, 8]. DL and BOs have been observed in bent waveguide arrays with either periodically-varying or constant curvature to mimic ac or dc fields [5, 6, 9, 10]. In optics, BOs and DL enable light diffraction management, with the possibility of periodic image reconstruction [8, 11]. Such a self-imaging effect, however, is generally limited to relatively narrow spec- tral beams. In case of DL, destruction of self-imaging for broadband beams stems from the resonant nature of the DL condition [12]. Here, periodic self-imaging is exactly fulfilled solely for a target wavelength, whereas other spectral components experience residual diffrac- tion. In [13], Garanovich and coworkers suggested spe- cial profiles of waveguide axis curvature that mix first and second resonances of DL, leading to approximate DL over an extremely broad spectral region. The exper- imental demonstration of broadband DL has been very recently reported by Szameit and collaborators [14]. In case of BOs, destruction of exact self-imaging with poly- chromatic beams has a different reason. Here periodic beam reconstruction is exactly attained for each spec- tral beam component, however the self-imaging period, given by the Bloch period s$_{B}$ , turns out to be propor- tional to the wavelength λ : therefore, for a beam spec- trally broadened by less than ∼ 10% around its carrier, field reconstruction is fully lost just after few BO cycles. In this Letter it is shown that polychromatic optical BOs can be approximately achieved over a broad spectral range by suitable insertion of lumped phase slips in the array, which combat the dispersion of BO period with wavelength. Let us consider light propagation in low- contrast-index one-dimensional waveguide arrays with a weakly curved axis in the ( x, z ) plane. In the curvilinear coordinate system ( s, η ) of Fig.1(a), where s is the arc length of the curved waveguide axis, the propagation for where Δ( λ ) is the coupling rate, at wavelength λ , be- tween adjacent waveguides, and the dot denotes the bulk refractive index, n ( η ) is the periodic index profile of the array with lattice period a , and κ ( s ) is the lo- cal curvature of waveguide axis. In the single-band and nearest-neighbor tight-binding approximations, Eq.(1) reduces to standard coupled mode equations for ampli- tudes c$_{n}$ ( s ; λ ) of modes trapped in the n -th waveguide of the array [8,13] Fig. 1. (Color online) (a) Schematic of a curved waveg- uide array. (b) Refractive index profile n ( η ) − n$_{s}$ of the array used in numerical simulations ( a = 8 µ m, n$_{s}$ = 1 . 45). (c) Behavior of Γ( λ ), entering in Eq.(3), versus wavelength for (i) the 8-cm-long circularly-curved array without lumped phase slips (solid curve); (ii) α$_{1}$ = α$_{3}$ = λ$_{0}$/ (2 an$_{s}$ ) and α$_{2}$ = 0 (dashed curve); (iii) for α$_{1}$ = α$_{2}$ = α$_{3}$ = λ$_{0}$/ (3 an$_{s}$ ) (dotted curve). (d) Behavior of waveguide coupling rate Δ versus wavelength. the spectral field component ψ ( η, s ; λ ) at wavelength λ is described by the Schr¨ odinger-like equation where ¯ h = λ/ (2 π ) is the reduced wavelength, n$_{s}$ is the i ¯ h ∂ψ ∂s = − ¯ h 2 2 n$_{s}$ ∂ $^{2}$ψ ∂η 2 + [ n$_{s}$ − n ( η )] ψ + n$_{s}$κ ( s ) ηψ (1) i ˙ c$_{n}$ = − Δ( λ )( c$_{n}$$_{+1}$ + c$_{n}$$_{−}$$_{1}$ ) + 2 πn$_{s}$κ ( s ) an λ c$_{n}$ (2) arXiv:1001.0988v1 [physics.optics] 6 Jan 2010 GLYPH<c104> s R = 1 GLYPH<c107> ( ) s x z (a) -20 -10 0 10 20 0 1 2 xGLYPH<.notdef>10 -3 GLYPH<c104> GLYPH<c109> ( m) n n s - a (b) GLYPH<c71>GLYPH<c40>GLYPH<c108> ) GLYPH<c108> GLYPH<c109> ( m) (c) GLYPH<c68> (mm ) -1 GLYPH<c108> GLYPH<c109> ( m) (d) 0.5 0.55 0.6 0.65 0 1 2 3 0.5 0.55 0.6 0.65 0.1 0.2 0.3 0.4 0.5 GLYPH<c108>GLYPH<c61>GLYPH<c108>$_{GLYPH<c48>}$ Polychromatic Optical Bloch Oscillations Stefano Longhi Dipartimento di Fisica and Istituto di Fotonica e Nanotecnologie del CNR, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy Compiled June 2, 2018 Bloch oscillations (BOs) of polychromatic beams in circularly-curved optical waveguide arrays are smeared out owing to the dependence of the BO spatial period on wavelength. Here it is shown that restoring of the self-imaging property of the array and approximate BOs over relatively broad spectral ranges can be achieved by insertion of suitable lumped phase slips uniformly applied across the array. c ○ 2018 Optical Society of America OCIS codes: 130.2790, 230.3120, 230.7370, 000.1600 1