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derivative with respect to the arc length. Here we con- sider a waveguide array with constant curvature κ = 1 /R , into which a sequence of lumped phase gradients are superimposed at the arc lengths s$_{1}$ , s$_{2}$ ,...., i.e. we as- sume in Eq.(2) κ ( s ) = 1 /R + α$_{1}$δ ( s − s$_{1}$ )+ α$_{2}$δ ( s − s$_{2}$ )+ ... , where the parameters α$_{1}$ , α$_{2}$ ,... define the phase slips ϕ$_{1}$ , ϕ$_{2}$ ,... introduced uniformly across the array at the arc lengths s$_{1}$ , s$_{2}$ ,.... The relation between the phase slip ϕ$_{l}$ and the parameter α$_{l}$ is simply obtained after integration of Eq.(2) in the infinitesimal interval ( s = s − l , s = s + l ), yielding ϕ$_{l}$ = 2 πn$_{s}$aα$_{l}$/λ . In practice, a lumped phase gradient can be introduced (i) by a sudden tilt of waveg- uide axis [11] by a small angle θ ; in this case α$_{l}$ = θ and ϕ$_{l}$ = πθ/θ$_{B}$ , where θ$_{B}$ = λ/ (2 an$_{s}$ ) is the Bragg angle; (ii) by local waveguide segmentation, channel narrowing or index change modulation [15]. To investigate the self- imaging property of the arrayed structure, it is enough to consider the impulse response of the array correspond- ing to excitation of the waveguide n = 0. The solution to Eq.(2) with the initial condition c$_{n}$ (0) = δ$_{n,}$$_{0}$ reads [12] 

where J$_{n}$ is the Bessel function of order n , Γ( λ ) = Δ( λ ) | q ( s, λ ) | and 

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The condition for self-imaging after a propagation length s , i.e. | c$_{n}$ ( s ) | 2 = δ$_{n,}$$_{0}$ , is thus q ( s, λ ) = 0. For the BO problem in absence of phase slips, κ ( s ) = 1 /R and | q ( s, λ ) | = s | sin( πs/s$_{B}$ ) | / ( πs/s$_{B}$ ), where 

is the BO period for the spectral field component of wavelength λ . For a monochromatic beam with carrier wavelength λ$_{0}$ , periodic self-imaging is thus attained at propagation lengths s which are integer multiples of the BO cycle s$_{B}$ ( λ$_{0}$ ). Unfortunately, for spectrally-broad beams different wavelength components λ show a shifted BO cycle s$_{B}$ ( λ ), making self-imaging and polychromatic BOs to rapidly washing out for a relatively broad spec- trum of the incoming beam. As an example, Fig.2(a) shows the breakdown of self-imaging and smearing out of a breathing BO mode for polychromatic light in a L = 8- cm-long circularly-curved array as obtained by numeri- cal simulations of Eq.(1) for an index profile of the array shown in Fig.1(b) and for a radius R = 40 cm. The inten- sity distribution of Fig.2(a) is obtained by the incoher- ent superposition of five monochromatic fields of equal power (at λ =540, 560, 580, 600, and 620 nm), which excite simultaneously the waveguide n = 0. For com- parison, Fig.2(b) shows the evolution of beam intensity for the carrier spectral component solely ( λ = λ$_{0}$ = 580 nm), which undergoes periodic self imaging with spatial period s$_{B}$ ( λ$_{0}$ ) = λ$_{0}$R/ ( n$_{s}$a ) = 2 cm. Note that the full propagation length L cm comprises exactly N = 4 BO 

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cycles at the carrier wavelength. The washing out of self imaging at the output plane for a spectrally broad beam is related to the rapid deviation from zero of Γ( λ ) as λ varies by a a few percents around the reference wave- length λ$_{0}$ , as shown by the solid curve in Fig.1(c). The allowed spectral band Δ λ for (approximate) self-imaging can be roughly estimated from the condition 2 | Γ( λ ) | < 1 [see Eq.(3)], and turns out to be Δ λ ∼ 20 nm for the solid curve of Fig.1(c). The introduction of appropriate phase slips across the array may be exploited to flatten the function Γ( λ ) at around λ = λ$_{0}$ . To this aim, let us consider ( N − 1) lumped phase gradients at s$_{1}$ = s$_{B}$ ( λ$_{0}$ ), s$_{2}$ = 2 s$_{B}$ ( λ$_{0}$ ),..., s$_{N}$$_{−}$$_{1}$ = ( N − 1) s$_{B}$ ( λ$_{0}$ ), i.e. let us assume κ ( s ) = 1 /R + ∑ N$_{−}$ 1 k =0 α$_{k}$δ ( s − s$_{k}$ ) with s$_{0}$ = 0 and α$_{0}$ = 0 for definiteness. From Eq.(4) with s = L , the expression of q can be calculated in closed form and reads 

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where we have set F ( ξ ) = ∑ N$_{−}$ 1 k =0 exp( − 2 iξρ$_{k}$ ) and ρ$_{k}$ = k + ( an$_{s}$/λ$_{0}$ )( α$_{0}$ + α$_{1}$ + ...α$_{k}$ ). Appropriate choice of the values α$_{k}$ may flatten the shape of Γ( λ ) at around its zero λ = λ$_{0}$ . A first approach consists of considering the Taylor expansion of Γ( λ ) at around λ$_{0}$ and imposing Γ = ( ∂ Γ /∂λ ) = ... = ( ∂ $^{M}$Γ /∂λ $^{M}$) = 0 at λ = λ$_{0}$ up to a certain order M ≥ 1, which defines the spectral flatness order. This requires F ( π ) = F $^{′}$( π ) = ... = F ( M$_{−}$ $^{1)}$( π ) = 0, which yields 

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for l = 0 , 1 , .., M − 1. As an example, let us consider the case N = 4, previously discussed in Fig.2. Then one 

