different choice for α$_{1}$ , α$_{2}$ and α$_{3}$ , which broadens the spectral range where 2Γ( λ ) remains smaller than ∼ 1, is α$_{1}$ = α$_{2}$ = α$_{3}$ = λ$_{0}$/ (3 an$_{s}$ ). In this case, Γ $^{′}$( λ$_{0}$ ) and Γ $^{′′}$( λ$_{0}$ ) do not vanish, however two additional zeros of Γ are introduced at λ ≃ 516 nm and λ ≃ 618 nm, as shown by the dotted curve in Fig.1(c). The correspond- ing behavior of polychromatic BOs, for a beam covering a spectral extent Δ λ ≃ 130 nm, is shown in Fig.4. Different algorithms, based on optimization methods (e.g., trying to minimize ∫ dλ | Γ( λ ) | ), might be investigated to flatten the function Γ( λ ) over a prescribed spectral range. As a general rule, the increase of the number N of phase gra- dients makes it the flattening procedure more flexible, and thus the polychromatic imaging of higher quality or applicable to a broader spectral range (for instance by increasing the spectral flattening order M ). In conclusion, broadband self-imaging based on poly- chromatic BOs in circularly-curved waveguide arrays has been proposed by insertion of lumped phase slips, which combat the dispersion of BO period for the various spec- tral beam components. Author E-mail address: longhi@fisi.polimi.it References 1. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisen- berg, and Y. Silberberg, Phys. Rev. Lett. 83 , 4756 (1999). 2. T. Pertsch, P. Dannberg, W. Elflein, A. Br¨ auer, and F. Lederer, Phys. Rev. Lett. 83 , 4752 (1999). 3. D.N. Christodoulides, F. Lederer, and Y. Silberberg, Na- ture 424 , 817 (2003). 4. R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C.J. Oton, and L. Pavesi, Phys. Rev. Lett. 91 , 263902 (2003). 5. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96 , 243901 (2006). 6. R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, Opt. Express 15 , 3212 (2007). 7. G. Lenz, I. Talanina, and M. 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Louradour, A. 3 can check that Eq.(7) is satisfied up to the order M = 2 by simply assuming α$_{1}$ = α$_{3}$ = λ$_{0}$/ (2 an$_{s}$ ) and α$_{2}$ = 0. This corresponds to the introduction, at the arc lengths s$_{1}$ = s$_{B}$ ( λ$_{0}$ ) and s$_{3}$ = 3 s$_{B}$ ( λ$_{0}$ ), a lumped π phase shift uniformly across the array. The corresponding behavior of Γ( λ ) is shown by the dashed curve in Fig.1(c). Poly- chromatic BOs and (approximate) self-imaging over the entire spectral range (540,620) nm is demonstrated in Fig.3(a), where the intensity evolution of the polychro- matic beam of Fig.2(a) is now computed with insertion of the phase gradients at s$_{1}$ = s$_{B}$ ( λ$_{0}$ ) = 2 cm and at s$_{3}$ = 3 s$_{B}$ ( λ$_{0}$ ) = 6 cm. In the numerical simulations, the lumped π phase slips are simply obtained by tilting the waveguides (tilting angle θ = θ$_{B}$ = 25 mrad) as in [11]. The spectrally-resolved output intensity profiles of the polychromatic beam are also shown in Fig.3(b). Note that, with this choice of α$_{1}$ , α$_{2}$ and α$_{3}$ , the function Γ( λ ) remains reasonably small in the range λ ∼ (540 , 620) nm, however it rapidly increases outside this interval. A Fig. 4. (Color online) Same as Fig.3, but with the in- sertion of three lumped waveguide tilts at s$_{1}$ = 2 cm, s$_{2}$ = 4 cm and s$_{3}$ = 6 cm (tilt angle θ = 2 θ$_{B}$/ 3 ≃ 16.7 mrad). -125 0 GLYPH<c104> GLYPH<c109> ( m) GLYPH<c104> GLYPH<c109> ( m) -125 125 0 s (cm) 8 0 (a) (b) 125 GLYPH<c108> =500GLYPH<.notdef>nm GLYPH<c108> =580GLYPH<.notdef>nm GLYPH<c108> =600GLYPH<.notdef>nm GLYPH<c108> =630GLYPH<.notdef>nm GLYPH<c108> =530GLYPH<.notdef>nm Fig. 3. (Color online) Polychromatic BOs as obtained by insertion of two lumped waveguide tilts at s$_{1}$ = 2 cm and s$_{3}$ = 6 cm (tilt angle θ = θ$_{B}$ = 25 mrad). In (b) the output intensity profiles at the output plane for the different spectral components are depicted. -125 0 GLYPH<c104> GLYPH<c109> ( m) GLYPH<c104> GLYPH<c109> ( m) -125 125 0 s (cm) 8 0 (a) (b) 125 GLYPH<c108> =540GLYPH<.notdef>nm GLYPH<c108> =580GLYPH<.notdef>nm GLYPH<c108> =600GLYPH<.notdef>nm GLYPH<c108> =620GLYPH<.notdef>nm GLYPH<c108> =560GLYPH<.notdef>nm