3 show that the external force can restore a real-valued energy spectrum. The lattice consists of an array of equally-spaced single-mode waveguides with alternating gain ( V$_{I}$ > 0) and loss ( V$_{I}$ < 0) channels, i.e. V$_{R}$ ( x ) = ∑ $_{l}$h ( x − la − a/ 2) and V$_{I}$ ( x ) = ∑ $_{l}$( − 1) $^{l}$h ( x − la − a/ 2), where h ( x ) = h ( − x ) defines the (real) index profile of the waveguide channel (see Fig.3(a)). Note that, for α = 0, the lattice period is 2 a . A typical band diagram is shown in Fig.3(b). In the tight-binding approximation and for α ≪ 1, light transport in the lattice is described by coupled-mode equations for the modal amplitudes a$_{l}$ of light trapped in the waveguides [3, 16, 18, 21] at propagation distances corresponding to Bragg reflec- tion from the lattice, a rather abrupt amplification of the scattered beam is observed [see the inset of Fig.2(b)]. This dynamical regime, which can be referred to as uni- directional BO, has no counterpart in ordinary lattices and is related to violation of Friedel’s law of Bragg scat- tering by complex potentials [7, 20], i.e. breakdown of Bragg diffraction invariance by crystal inversion. Ana- lytical insights into unidirectional BO can be gained by writing Eq.(1) in momentum space and assuming that the lattice is excited by a broad wave packet at normal incidence, i.e. ψ ( x, 0) = ∫ dkg ( k ) exp( − ikx ), where the plane-wave spectrum g ( k ) is narrow at around k = 0 with a spectral width much smaller than the Bragg wave number k$_{B}$ = 2 π/a . Let us indicate by ψ$_{0}$ ( x, z ) the prop- agated wave packet that one would observe in absence of the lattice, i.e. the solution to Eq.(1) with V = 0. The center of mass 〈 x 〉 ( z ) of ψ$_{0}$ ( x, z ) follows the parabolic path 〈 x 〉 ( z ) = − ( F/ 2 n$_{s}$ ) z 2 due to the external force F , whereas the mean value of momentum 〈 k 〉 grows lin- early, 〈 k 〉 ( z ) = − F z . In presence of the complex lattice, Bragg diffraction sets in when 〈 κ 〉 gets close to − k$_{B}$/ 2, but not when 〈 κ 〉 ∼ k$_{B}$/ 2; this nonreciprocal behavior of Bragg scattering by complex potentials is discussed e.g. in [7, 20]. In presence of the lattice, one can show that the wave packet is expressed by a superposition of retarded wave packets ψ$_{0}$ ( x, z − nz$_{B}$ ) with amplitudes B$_{n}$ ( z ), namely In Eq.(5), B$_{0}$ ( z ) = 1, whereas for n ≥ 1 the amplitudes B$_{n}$ can be iteratively calculated from the relation B$_{n}$ ( z ) = − i V$_{0}$ λ ∫ z dξ exp[ − iϕ$_{n}$ ( ξ )] B$_{n}$$_{−}$$_{1}$ ( ξ ) , (6) FIG. 3: (color online) BO in a complex binary lattice ( a = 10 µ m). (a) Profiles of V R and V I , and (b) corresponding band diagram for α = 0 . 01 (real and imaginary parts of energy E ). (c) Resonance curves in the (Δ /f, g/f ) plane corresponding to a real-valued WS ladder spectrum. (d) Breathing BO mode (single waveguide input excitation) for F = 1 . 618 m $^{$_{−}$1}$, cor- responding to a Bloch period z B ≃ 39 . 1 mm. ψ ( x, z ) ≃ + ∞ ∑ n =0 B$_{n}$ ( z ) ψ$_{0}$ ( x, z − nz$_{B}$ ) . (5) jacent waveguides, f = F a/λ , and g is the real-valued power gain/loss coefficient in the waveguides, which is proportional to α . For the potential of Fig.3(a) with α = 0 . 01, from the band diagram of Fig.3(b) (the two merged bands n = 1 and n = 2) one can estimate g ≃ 1 . 46 cm − 1 and Δ ≃ 14 cm − $^{1}$. For F = 0, the spectrum of Eq.(7) is purely continuous and composed by two merged minibands [the two lowest bands of Fig.3(b)], namely [16] Since complex-conjugate eigenvalues appear near the band edges for any value of g > 0, the lattice has zero threshold for P T symmetry-breaking, i.e. α$_{c}$ = 0. In presence of the external force F , the spectrum becomes E ( κ ) = ± λ √ (Δ / 2) $^{2}$cos$^{2}$( κa ) − ( g/ 2) 2 , | κ | ≤ π/ 2 a. (8) where we have set ϕ$_{n}$ ( z ) = ( λ $^{2}$k$_{B}$ / 2 n$_{s}$F )[( F z/λ − k$_{B}$n ) 2 + k$_{B}$ ( F z/λ − k$_{B}$n ) + k 2 $_{B}$/ 3] . In the limit z$_{B}$ ≫ ( n$_{s}$a/πF ) 1 / 2 and for F < 0, | B$_{n}$$_{≥}$$_{1}$ ( z ) | remains small be- cause of ϕ$_{n}$ ( z ) has no stationary points for z > 0. Physi- cally, this corresponds to the absence of Bragg scattering (the condition 〈 k 〉 ∼ − k$_{B}$/ 2 is never satisfied), and the scenario of Fig.2(a) is observed. Conversely, for F > 0 the phase ϕ$_{n}$ has a stationary point at z = z$_{B}$ ( n − 1 / 2), and one obtains | B$_{n}$ ( z ) | ∼ Γ $^{n}$H ( z − nz$_{B}$ + z$_{B}$/ 2), where Γ ≡ ( V$_{0}$/λ )( n$_{s}$a/F ) 1 / 2 and H ( ξ ) is the Heaviside step function [ H ( ξ ) = 0 for ξ < 0, H ( ξ ) = 1 for ξ > 0]. In this case, Eq.(5) clearly shows that a periodic regen- eration and amplification of the wave packet ψ$_{0}$ due to Bragg scattering occurs at z = z$_{B}$/ 2 , 3 z$_{B}$/ 2 , 5 z$_{B}$/ 2 , ... , explaining the behavior observed in Fig.2(b). For param- eter values used in the simulations, the predicted value of amplification factor Γ is Γ ≃ 2 . 186, with which the staircase behavior of beam power shown in the inset of Fig.3(b) is reproduced with excellent accuracy. As a second example, we consider a complex binary lattice [16] with broken P T symmetry at α$_{c}$ = 0, and where Δ / 4 is the real-valued coupling rate between ad- i da$_{l}$ dz = − Δ 4 ( a$_{l}$$_{+1}$ + a$_{l}$ ) + fla$_{l}$ + i ( − 1) $_{l}$g 2 a$_{l}$ (7) -1 0 1 xGLYPH<.notdef>10 -3 xGLYPH<.notdef>10 -6 GLYPH<c45>GLYPH<c112> GLYPH<c50>GLYPH<c107> a GLYPH<c45>GLYPH<c48>GLYPH<c46>GLYPH<c57>GLYPH<c112> GLYPH<c112> GLYPH<c48>GLYPH<c46>GLYPH<c57>GLYPH<c112> 8 0 -8 GLYPH<c45>GLYPH<c112> GLYPH<c50>GLYPH<c107> a GLYPH<c112> 0 n=2 n=1 n=4 n=3 n=3,4 Re{ ( )} E n GLYPH<c107> Im{ ( )} E n GLYPH<c107> (b) -200 0 0.5 1 1.5 2 2 4 6 8 10 GLYPH<c68>GLYPH<c47> f g f GLYPH<c47> (c) n=1,2 x ( m) GLYPH<c109> 200 z (cm) 10 0 (d) -10 0 10 -1 -0.5 0 0.5 1 xGLYPH<.notdef>10 -3 V x $_{I}$( ) V x $_{R}$( ) a x ( m) GLYPH<c109> (a) potential V A