Fig. 1. Linear band structure and diffraction properties of a 1D waveguide array. (a) Transmission spectrum consisting of bands of allowed propagation constants separated by forbidden gaps. (b) Modes in convex regions experience normal diffraction. (c) Modes in concave regions experience anomalous diffraction.

Fig. 2. 1D discrete or lattice solitons from the first band (a) 1D transmission spectrum showing the nonlinear propagation constants for the fundamental LS (base of the BZ zone) and spatial gap soliton (edge of BZ zone). (b) The fundamental LS has an in-phase structure that requires a self-focusing nonlinearity (positive defect). (c) The first-band gap soliton has a staggered phase structure that requires a defocusing nonlinearity (negative defect).

Fig. 3. Band structure and optical induction of a square waveguide array. (a,b) Band structure of a 2D square lattice, with high-symmetry points labeled in (b). (c) Interference of four plane waves to optically induce a 2D square array of 2D waveguides. (d) Photograph of a typical output face, showing a lattice with an 11µm period.

Fig. 4. Principles of grating-mediated waveguiding. (a) Type-I guide, with a bell-shaped y-profile. (b) Type-II guide, with a trough-shaped y-profile. (c) Index grating with three different amplitudes, corresponding to three different y-planes. (d) The dispersion/diffraction curves near the edge of the first Brillouin zone for gratings with index profile n(x,y)=n ₀[1+εA(y)cos(πx/D)]. (e,f) Typical index amplitudes A(y) of (e) Type-I gratings and (f) Type-II gratings. (g) The effective waveguide structure in y: Type-I beams need a cos(πx/D) dependence and Type-II beams need a sin(πx/D) dependence to experience grating-mediated waveguiding. Taken from [60].

Fig. 5. Experimental scheme and resutls of a Type-II (trough-shaped) grating-mediated waveguide. (a) Schematic of optical induction of waveguide. (b) Photograph of input beam. (c) Waveguiding when position and phase of (b) match with grating structure. (d–f) Diffraction in unguided conditions: (d) Free-space diffraction of (b) in absence of waveguide. (e) Diffraction when the input is not Bragg-matched with the grating. (f) Diffraction when the input (b) is Bragg-matched but has the “wrong” phase relative to the grating. In (b–f), the intensity in each figure is normalized to its own peak intensity. Taken from [60].

Fig. 6. Experimental observation of discrete diffraction and solitons in 1D optically-induced waveguide arrays. (a–c) Propagation for an on-axis input probe. Discrete diffraction (a) in the linear regime. Soliton intensity (b) and relative phase (c) in the nonlinear regime (+1.6kV/cm). The phase information (c) is obtained by interfering the output signal (b) with a plane wave. (d–f) Propagation at the Bragg angle, corresponding to the edge of the first Brillouin zone. Discrete diffraction (d) in the linear regime. Soliton intensity (e) and relative phase (f) in the nonlinear regime (-2.0kV/cm). Note that the central peak experiences destructive interference, while the surrounding lobes experience constructive interference. Note also the need for defocusing nonlinearity (negative voltage) to create the 1D gap soliton (e). Taken from [14].

Fig. 7. Experimental observation of discrete diffraction and solitons in 2D optically-induced waveguide arrays. (a–c) Propagation for an on-axis input probe. Discrete diffraction (a) in the linear regime. Soliton intensity (b) and relative phase (c) in the nonlinear regime (+1.6kV/cm). The phase information (c) is obtained by interfering the output signal (b) with a plane wave. (d–f) Propagation at the Bragg angle, corresponding to the corner of the first Brillouin zone (M-point in Fig. 4c). Discrete diffraction (d) in the linear regime. Soliton intensity (e) and relative phase (f) in the nonlinear regime (-1.6kV/cm). Note that the central peak experiences destructive interference, while the surrounding lobes experience constructive interference. Note also the need for defocusing nonlinearity (negative voltage) to create the 2D gap soliton (e). Taken from [26].

Fig. 8. Experimental observation of discrete vortex solitons in 2D optically-induced waveguide arrays. (a) Discrete diffraction and (b) spiral phase structure (formed by interfering the output (a) with a plane wave) in the linear regime. (b) Soliton intensity and (c) relative phase in the nonlinear regime (+1.6kV/cm) for a vortex with its singularity centered on-site. (d) Soliton intensity and (e) relative phase in the nonlinear regime (+1.6kV/cm) for a vortex with its singularity centered in-between sites. The vortex solitons keep their relative phase, despite the fact that the conservation of angular momentum (topological charge) is not guaranteed in a lattice. Taken from [56].

Fig. 9. Theoretical characterization of random-phase lattice soliton (RPLS). (a) Modes are taken statistically from regions of the transmission spectrum with the same band curvature. (b,c) Power spectrum in the (b) Fouier basis and (c) Floquet-Bloch basis. Note the sharp cut-offs in (c), implying that the F-B representation is more appropriate for lattice modes. (d,e) Statistical properties of RPLS. The blue lines indicate the centers of each waveguide in the 1D array. (d) Mutual coherence/correlation function centered on the central waveguide. (e) Soliton profile in black and correlation length in red. (f) Modal structure as given by waveguide theory. See text for its relation to the correlation behavior in (e). Taken from [82].