Continuous Solitons in a Lattice Nonlinearity

We study theoretically and experimentally the propagation of optical solitons in a lattice nonlinearity, a periodic pattern that both affects and is strongly affected by the wave. Observations are carried out using spatial photorefractive solitons in a volume microstructured crystal with a built-in oscillating low-frequency dielectric constant. The pattern causes an oscillating electro-optic response that induces a periodic optical nonlinearity. On-axis results in potassium-lithium-tantalate-niobate indicate the appearance of effective continuous saturated-Kerr solitons, where all spatial traces of the lattice vanish, independently of the ratio between beam width and lattice constant. Decoupling the lattice nonlinearity allows the detection of discrete delocalized and localized light distributions, demonstrating that the continuous solitons form out of the combined compensation of diffraction and of the underlying periodic volume pattern.

Figure 1

Nonlinear propagation in periodic systems. (a) Trapping in photonic lattices: the periodic pattern $\delta n_{\text{latt}}$ affects the spatial propagation but is not affected by the wave. (b) Trapping in lattice nonlinearity: optical field and lattice are mutually coupled and $\delta n_{\text{latt}}$ depends on the waveform.

Figure 2

Light propagation in the lattice nonlinearity embedded in a microstructured KLTN. (a) Sketch of the physical geometry and (b)–(c) optical lattice detection: (b) transmission microscopy image at $\vartheta ={\vartheta }_B$ and (c) its intensity Fourier transform revealing the grating period $\mathrm{\Lambda }=5.5\pm 0.3\mu \mathrm{m}$ (expanded view in the inset). (d) Observed beam dynamics from the starting delocalized discrete pattern to the continuous soliton. (d1) Input and (d2) output beam when the lattice nonlinearity is deactivated. (d3) Output discrete spatial distribution as soon as the lattice nonlinearity is enabled at $V=400\mathrm{V}$ and (d4) continuous soliton at the steady state. (e) Intensity Fourier transform of (d2) (red line), (d3) (cyan line), and (d4) (magenta line).

Figure 3

Soliton propagation: experimental and numerical results. Observed (a) input, (b) Diffracted, and (c) self-trapped output at applied bias field for (top) 1D beams, with FWHM of $7\mu \mathrm{m}$ (tight-binding regime), (middle) $22\mu \mathrm{m}$ (weak-binding regime), and (bottom) 2D beams. In all the cases, continuous localization emerges as confirmed numerically by (d) propagation and (f) associated spatial index of refraction modulation (blue line) compared with just the contribution of the photorefractive response in biased condition (orange line). (e) Experimental relation between normalized intensity and external field for 1D solitons with linear fit (dashed line).

Figure 4

Zero-field discrete dynamics after the soliton formation. (a)–(d) Measured time evolution: (a)–(b) discrete delocalized pattern, (c) localization with discrete features and (d) relaxation to the equilibrium. (e)–(f) Numerical results at $t=0$: (e) discrete propagation and (f) nonlinearity supporting the periodically modulated output light distribution compared with the previous soliton nonlinearity (orange line). (g) Comparison of the spectral properties of (a) (magenta line) and (b) (cyan line) with those of the numerical output in (e) (red line).