Pure-quartic solitons

Abstract

Temporal optical solitons have been the subject of intense research due to their intriguing physics and applications in ultrafast optics and supercontinuum generation. Conventional bright optical solitons result from the interaction of anomalous group-velocity dispersion and self-phase modulation. Here we experimentally demonstrate a class of bright soliton arising purely from the interaction of negative fourth-order dispersion and self-phase modulation, which can occur even for normal group-velocity dispersion. We provide experimental and numerical evidence of shape-preserving propagation and flat temporal phase for the fundamental pure-quartic soliton and periodically modulated propagation for the higher-order pure-quartic solitons. We derive the approximate shape of the fundamental pure-quartic soliton and discover that is surprisingly Gaussian, exhibiting excellent agreement with our experimental observations. Our discovery, enabled by precise dispersion engineering, could find applications in communications, frequency combs and ultrafast lasers.

Figure 1. Concept of pure-quartic solitons and their experimental demonstration.

(a) Schematics of pure-quartic solitons: (Left) Fourth-order dispersion (FOD) gives rise to temporal pulse broadening (blue output pulse versus black input pulse in time) without affecting the spectrum; (Centre) self-phase modulation (SPM) generates spectral broadening (red output pulse versus black input pulse in frequency) without affecting the temporal pulse shape; (Right) the interplay of FOD and SPM can give rise to pure-quartic solitons which remain nearly unperturbed (green output pulses versus black input pulses in both frequency and time); (b) Frequency-resolved electrical gating set-up: mode locked laser (MLL), photonic crystal waveguide (PhC-wg), tunable delay, ultrafast photodiode (PD), Mach–Zehnder modulator (MZM), and optical spectrum analyser (OSA); (c) Scanning electron microscope image of the sample; (d) Measured dispersion of the silicon photonic crystal waveguide used in our experiments: group index (n_g), second-order dispersion parameter (β₂) and fourth-order dispersion parameter (β₄).

Figure 2. Experimental and modelling results.

(a) Frequency and (b) time domain results for different input powers. The dashed red lines represent the intensity measurements, the blue solid lines represent the intensity simulations, the black dashed line represents the measured phase, and the solid black line represents the simulated phase. The green solid line at 0.7 W represents the normalized input intensity. The yellow box encompasses the fundamental pure-quartic soliton, showing nearly unperturbed propagation and flat temporal phase. The turquoise box includes two cases of higher-order pure-quartic solitons, showing temporal compression and nonlinear spectral broadening. The higher-order pure-quartic solitons observed here are greatly perturbed by the presence of free carriers.

Figure 3. Simulations of the propagation of a fundamental and a higher-order pure-quartic soliton along five quartic dispersion lengths, L_FOD.

(a) Fundamental (P₀=0.7 W) pure-quartic soliton with only self-phase modulation and quartic dispersion present, and (c) in the more realistic scenario for our silicon waveguide with two-photon absorption and free carriers; (b) and (d) are similar but for a higher power level (P₀=4.5 W) where a higher-order pure-quartic soliton results.

Figure 4. Approximate solutions to a fundamental pure-quartic soliton and phase diagram.

Comparison between the variational and local approximate solutions for the fundamental pure-quartic soliton and the numerical output after propagating over thirty quartic dispersion lengths for (a) a Gaussian input, (b) a hyperbolic secant input, and (c) a super Gaussian input of order 4. In the Gaussian case the measured output pulse at 0.7 W is shown in the background (dot-dash cyan curve). (d) Phase shift induced by the fourth-order dispersion (dashed purple) and the self-phase modulation (solid turquoise) independently and its combined phase shift (black).

Figure 5. The cancellation of nonlinear and dispersive phase components on the fundamental pure-quartic soliton.

(a) FOD-induced (red) and SPM-induced (blue) frequency chirps after a propagation of L_FOD/10 for the Gaussian pure-quartic soliton of equation (3); (b) similar, but for the sech² type solutions obtained taking ref. with β₂=0. The dashed black curves in the background of (a) and (b) represent the input pulse intensity: the Gaussian solution of equation 3 and the sech² solution in ref. respectively.