Three-Dimensional Spatiotemporal Pulse-Train Solitons

Experimental realization of three-dimensional spatiotemporal solitons, which were proposed several decades ago, is still considered a “grand challenge” in nonlinear science. Here, we present experimental observation of 3D optical spatiotemporal pulse-train solitons. A spatially bright temporally dark pulse-train beam is trapped in a bulk medium that supports two types of nonlinearities: slowly responding saturable self-focusing that collectively self-trap the beam in the transverse directions and fast self-phase modulation that self-localizes each dark notch temporally (longitudinally). This work opens the possibility for experimental investigations of various soliton phenomena, including soliton interaction in 3D, formation of multimode spatiotemporal solitons, and envisioning new entities like partially coherent spatiotemporal solitons.

Popular Summary

Localized waves of all kinds have a natural tendency to diverge as they evolve, like ripples in a pond. Remarkably, there are localized waves that maintain their shape during propagation because of a robust balance between linear and nonlinear effects. These special wave packets are called solitons, and they exhibit properties that are normally associated with particles, such as collision, attraction, repletion, fission, and fusion, all of which depend on the solitons’ dimensionality. Researchers have explored solitons that are self-trapped in one or two dimensions (in space, time, and spacetime) in many systems. Experimental demonstration of three-dimensional spacetime (spatiotemporal) solitons, on the other hand, is still considered a grand challenge in nonlinear science. In optics, such solitons are also called “light bullets.” We present the first experimental observation of these three-dimensional spatiotemporal solitons.

More specifically, we observe optical pulse-train spatiotemporal solitons, or a train of light bullets. We trapped a pulse-train beam in a homogeneous medium that supports two types of nonlinearities: a slowly responding, saturable, self-focusing mechanism that self-traps the beam transversely (spatially) and a fast self-phase modulation that self-localizes the pulses longitudinally (temporally).

Our work opens the possibility for experimental investigations of various soliton phenomena, including soliton interaction in three dimensions, formation of multimode spatiotemporal solitons, and envisioning new entities like partially coherent spatiotemporal solitons.

Figure 1

Experimental setup. A Ti:sapphire laser pulse-train beam ($\lambda =0.8\mu \mathrm{m}$) with 1-KHz repetition rate and 30-fs pulse duration is divided into reference and signal beams. The signal beam propagates through a pulse shaper designed to form a dark pulse (100-fs FWHM) centered at a relatively wide background pulse (1-ps FWHM). The dark pulse-train beam is then focused to the input face of a photorefractive SBN crystal ($5\times 5\times 15\mathrm{mm}$ and $6\times 6\times 6\mathrm{mm}$). The spatial intensity distributions at the input and output faces of the crystal are recorded by an imaging system and CCD. The temporal pulse profile is characterized through spectral interferometry by interfering the reference and signal beams in a spectrometer.

Figure 2

Three-dimensional spatiotemporal pulse-train soliton in a long crystal ($L=15\mathrm{mm}$). (a) Spatial intensity distribution at the input plane of the crystal ($\mathrm{FWHM}\approx 15\mu \mathrm{m}$). (b) The intensity distribution at the output plane of the crystal under approximately linear propagation condition (the screening self-focusing is turned off by applying zero external voltage and the Kerr nonlinearity is small because the intensity of the beam is low—$10\mu \mathrm{W}$ averaged power of the incoming beam). The beam broadens to $\mathrm{FWHM}\approx 278\mu \mathrm{m}$. Output plane intensity distributions when the screening self-focusing is turned on ($V=500\mathrm{V}$) at (c) $10\mu \mathrm{W}$ and (d) $70\mu \mathrm{W}$ averaged powers, demonstrating the transverse spatial self-trapping ($\mathrm{FWHM}\approx 29\mu \mathrm{m}$ and $\mathrm{FWHM}\approx 19\mu \mathrm{m}$, respectively). (e) Reconstructed intensity pulse profiles of (bottom up) input pulse (black), diffracted beam with $P_{\mathrm{avg}}=10\mu \mathrm{W}$ (red), and spatially trapped beam with $P_{\mathrm{avg}}=10$, 20, 25, 30, 40, 45, 50, 60, and $70\mu \mathrm{W}$. (f) The duration of the dark notch as a function of average power demonstrating the temporal confinement. The horizontal dashed line corresponds to the duration at the input plane.

Figure 3

Three-dimensional spatiotemporal pulse-train soliton in a short crystal ($L=6\mathrm{mm}$). (a) Spatial intensity distribution at the input plane of the crystal ($\mathrm{FWHM}\approx 15\mu \mathrm{m}$). (b) The intensity distribution at the output plane of the crystal under approximately linear propagation condition. The beam broadens to $\mathrm{FWHM}\approx 136\mu \mathrm{m}$. Output plane intensity distributions when the screening self-focusing is turned on ($V=1800\mathrm{V}$) at (c) $10\mu \mathrm{W}$ and (d) $60\mu \mathrm{W}$ averaged powers, demonstrating the transverse spatial self-trapping ($\mathrm{FWHM}\approx 26\mu \mathrm{m}$ and $\mathrm{FWHM}\approx 13\mu \mathrm{m}$, respectively). (e) Reconstructed intensity pulse profiles of (bottom up) input pulse (black), diffracted beam with $P_{\mathrm{avg}}=10\mu \mathrm{W}$ (red), and spatially trapped beam with $P_{\mathrm{avg}}=10$, 20, 30, 40, 50, and $60\mu \mathrm{W}$. (f) The duration of the dark notch as a function of average power demonstrating the temporal confinement. The horizontal dashed line corresponds to the duration at the input plane.