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 May 1, 2018 An ’optical ratchet’ for discretized light in photonic lattices, which enables to observe rectification of light refraction at any input beam conditions, is theoretically presented, and a possible experimental implementation based on periodically-curved zigzag waveguide arrays is proposed. c ○ 2018 Optical Society of America OCIS codes: 130.2790, 230.3120, 230.7370, 000.1600 Transport of discretized light in photonic lattices has attracted a great attention in recent years from both fundamental and applied viewpoints [1, 2]. Engineered photonic lattices offer the possibility of tailoring diffrac- tion [3], refraction [4], and of shaping polychromatic light beams [5–7]. They also provide a rich laboratory system to mimic the dynamics of quantum particles driven or disordered periodic lattices, with the observation of the optical analogues of Bloch oscillations [1, 8], Anderson localization [9, 10] and dynamic localization [11–13] to name a few. Recently, the transport of ac-driven quan- tum particles in periodic lattices has received a great interest also for the possibility to achieve Hamiltonian ratchet effects, i.e. a rectified transport in absence of a net bias force and dissipation. Different ratchet models based on optical lattices, which use ultracold atoms as ideal systems, have been proposed [14–19], and detailed studies have investigated the relation between rectifica- tion and broken space-time symmetries [15, 17], the ef- fects of atomic interactions [19], and the possibility to transport quantum information [16, 18]. In optics, the idea of realizing an ’optical ratchet’ has received few at- tention to date, and limited to solitary waves in non- linear systems [20–22]. In particular, in [20] a setup to observe ratchet effects for cavity solitons has been pro- posed using coupled waveguide optical resonators with an adiabatically-shaken holding pump beam, whereas in [21] control of soliton dragging in dynamic optical lat- tices has been investigated. In this Letter a simple ratchet scheme is proposed in a linear optical system corresponding to rectification of discretized light refraction in an array of periodically- curved zigzag waveguides, which may be realized with currently available femtosecond laser writing technology [23]. We consider two interleaved arrays of single-mode channel waveguides, a primary array A$_{n}$ and an auxil- iary array B$_{n}$, in the zigzag geometry shown in Fig.1(a), which is similar to the one recently realized in Ref. [23] to investigate second-order coupling effects. As opposed to Ref. [23], the propagation constant of modes in waveg- uides B$_{n}$ is assumed to be detuned by a relatively large amount σ from the propagation constant of modes in waveguides A$_{n}$, a condition that can be achieved in prac- In Eqs.(2) and (3), κ$_{aa}$ and κ$_{bb}$ are the coupling rates between adjacent waveguides in the primary and auxil- iary arrays, respectively, κ$_{ab}$ is the cross-coupling rate [see Fig.1(a)], Θ( z ) = πn$_{s}$ ˙ x 0 ( z ) d/λ , d is the spatial pe- riod of the arrays, and σ is the propagation constant mismatch between waveguides in the two arrays. For straight arrays and large mismatch | σ | ≫ κ$_{ab}$, κ$_{aa}$, κ$_{bb}$ , light dynamics in the two arrays is decou- pled and a broad beam, which excites the primary ar- ray A$_{n}$ at an incidence angle θ smaller than the Bragg 1 where ¯ h = λ/ (2 π ) is the reduced wavelength, n$_{s}$ is the refractive index of the substrate medium, n ( x, y ) is the refractive index profile of the (straight) arrayed struc- ture, ∇ 2 is the transverse Laplacian, and the dot de- notes the derivative with respect to z . As discussed e.g. in [11], light propagation in the optical system mimics the temporal dynamics of a quantum particle in a bind- ing potential n$_{s}$ − n ( x, y ) driven by an external (inertial) force − n$_{s}$ ¨ x 0 ( z ) with zero mean. In the tight-binding ap- proximation, light transfer among the waveguides may be described by a set of coupled mode equations for the amplitudes a$_{n}$ and b$_{n}$ of modes trapped in waveguides A$_{n}$ and B$_{n}$, respectively, which may be cast in the fol- lowing form (see, for instance, [24]) tice by e.g. lowering the writing velocity of the auxiliary array. The optical axis of the waveguides, which lies in the ( X, Z ) plane, is assumed to be periodically-curved along the paraxial propagation direction Z with a bend- ing profile x 0 ( Z ) of period 2 z 0 . In the waveguide ref- erence frame z = Z and x = X − x 0 ( Z ), where the arrays appear to be straight, under scalar and parax- ial assumptions light propagation at wavelength λ is de- scribed by the following Schr¨ odinger-like equation (see, for instance, [11]) i ¯ h ∂ψ ∂z = − ¯ h 2 2 n$_{s}$ ∇ $^{2}$ψ + [ n$_{s}$ − n ( x, y )] ψ + n$_{s}$ ¨ x 0 ( z ) xψ (1) i ˙ a$_{n}$ = − κ$_{aa}$ [ a$_{n}$ +1 exp( − 2 i Θ) + a$_{n}$$_{−}$ 1 exp(2 i Θ)] + (2) − κ$_{ab}$ [ b$_{n}$ exp( iσz i Θ) + b$_{n}$$_{−}$ 1 exp( iσz + i Θ)] , i ˙ b$_{n}$ = − κ$_{bb}$ [ b$_{n}$ +1 exp( − 2 i Θ) + b$_{n}$$_{−}$ 1 exp(2 i Θ)] + (3) − κ$_{ab}$ [ a$_{n}$ exp( − iσz + i Θ) + a$_{n}$ +1 exp( − iσz − i Θ)] , arXiv:1001.0992v1 [physics.optics] 6 Jan 2010 Rectification of light refraction in curved waveguide arrays