Colloquium: Looking at a soliton through the prism of optical supercontinuum

A traditional view on solitons in optical fibers as robust particlelike structures suited for information transmission has been significantly altered and broadened over the past decade when solitons have been found to play the major role in generation of octave broad supercontinuum spectra in photonic crystal and other types of optical fibers. This remarkable spectral broadening is achieved through complex processes of dispersive radiation being scattered from, emitted, and transformed by solitons. Thus solitons have emerged as the major players in nonlinear frequency conversion in optical fibers. Unexpected analogies of these processes have been found with dynamics of ultracold atoms and ocean waves. This Colloquium focuses on recent understanding and new insights into physics of soliton-radiation interaction and supercontinuum generation.

Figure 1

(Color online) Photograph of supercontinuum generated by an $800\mathrm{nm}$ femtosecond pulse in a photonic-crystal fiber and projected onto a screen (http://www.bath.ac.uk/physics/groups/cppm/).

Figure 2

(Color online) (a) Group index, that is, speed of light divided by the group velocity $n_g=c{\partial }_{\delta }k$ (dashed line) and GVD ${\beta }_2={\partial }_{\delta }^{2}k$ (full line) typical for photonic-crystal fibers used in supercontinuum experiments. The zero GVD point is at $\simeq 790\mathrm{nm}$ $\left(380\mathrm{THz}\right)$ and the normal GVD range is for wavelengths $<790\mathrm{nm}$. (b) XFROG spectrogram showing simultaneous frequency and time-domain pictures of the supercontinuum from Fig. 3 at $z=1.5\mathrm{m}$. Pump wavelength is $850\mathrm{nm}$ $\left(350\mathrm{THz}\right)$.

Figure 3

(Color online) Numerical simulation of supercontinuum generation in a photonic-crystal fiber pumped with $200\mathrm{fs}$ pulses at $850\mathrm{nm}$ and having $6\mathrm{kW}$ peak power. The fiber dispersion as in Fig. 2a. One can see three stages in the supercontinuum expansion. Symmetric spectral broadening due to SPM happens over first $5\mathrm{cm}$. Antisymmetric spectral broadening due to soliton fission accompanied by the emission of the resonant Cherenkov radiation and soliton-radiation interaction both develop between $\sim 5$ and $\sim 15\mathrm{cm}$. After that the dynamics of the short-wavelength edge is determined by the radiation trapping.

Figure 4

(Color online) Soliton evolution in (a) $(z,t)$ and (b) $(z,\delta )$ planes in the presence of the Raman effect. $k\left(\delta \right)=\frac{1}{2}{\beta }_2{\delta }^2$. Note that the dispersive waves in (a) are not reflected by the soliton. (c) Phase-matching diagrams between the soliton and dispersive waves [see Eq. (10)]. Straight lines are the soliton wave numbers $k_s$ for two different frequencies ${\delta }_s$ [see Eq. (7)]. Point 1 indicates the initial soliton frequency ${\delta }_s$ and point 2 is ${\delta }_s$ at some distance down the fiber. Parabola shows the dispersion of linear waves $k\left(\delta \right)$. All units are dimensionless.

Figure 5

(Color online) The same as Fig. 4, but with ${\beta }_3>0$. $k\left(\delta \right)=\frac{1}{2}{\beta }_2{\delta }^2+\frac{1}{6}{\beta }_3{\delta }^3$. Intersections of the straight lines with $k\left(\delta \right)$ [see (c)] correspond to the Cherenkov resonances.

Figure 6

(Color online) The same as Fig. 4 but with ${\beta }_3<0$. $k\left(\delta \right)=\frac{1}{2}{\beta }_2{\delta }^2+\frac{1}{6}{\beta }_3{\delta }^3$. Intersections of the straight lines with $k\left(\delta \right)$ [see (c)] correspond to the Cherenkov resonances.

Figure 7

(Color online) A weak Gaussian pulse in normal GVD range initially placed behind the soliton experiences multiple reflections from the latter. After each collision some light gets through the soliton and some is reflected back. The reflected light is localized close to the soliton and almost indistinguishable from it in the $(z,t)$ plot (a). However, the reflected light is clearly visible in the spectral $(z,\delta )$ plot (b) because its frequency is blueshifted after each collision.

Figure 8

(Color online) Experimentally measured XFROG spectrogram showing emission of the strong long-wavelength radiation and its reflection from the second soliton in a fiber with ${\beta }_3<0$ (32). The strong resonant radiation (wave 1) is emitted by the soliton 1. The weaker soliton 2 reflects a part of this wave backward and simultaneously transforms frequency of the reflected radiation (wave 2). Dotted line marks the zero GVD wavelength.

Figure 9

(Color online) Scattering of radiation from a soliton. (a) and (c) Wave number matching diagrams and (b) and (d) XFROG spectrograms for the fiber pumped with the soliton and cw (Raman effect is disregarded). (a) and (b) ${\beta }_3>0$ and (c) and (d) ${\beta }_3<0$. The black curved lines in (a) and (c) are $k={\beta }_2{\delta }^2∕2+{\beta }_3{\delta }^3∕6$. The upper and lower horizontal lines in (a) and (c) correspond to the FWM resonances given by Eqs. (14, 15), respectively. The middle horizontal lines in (a) and (c) give Cherenkov resonances [Eq. (10)].

Figure 10

(Color online) Experimentally measured XFROG spectrograms showing the radiation-soliton interaction. (a) The resonance given by Eq. (14) (34) and (b) given by Eq. (15) (33). (a) The case of the forward reflected wave as in Fig. 9d. Both (a) and (b) measurements have been taken for ${\beta }_3<0$.

Figure 11

(Color online) Spectral evolution along the fiber length. (a) Experiment and (c) modeling. (b) XFROG diagram computed for the propagation distance $z=2\mathrm{m}$. (d) Group index as a function of wavelength. The decrease in height of the shaded triangles shows the decreasing frequency difference between the incident and reflected waves (41).

Figure 12

(Color online) Trapping potential and quasibound radiation modes. (a) Full line shows the potential $V$. The dashed line shows the potential with the contribution from the inertial force disregarded, which removes the minimum and makes the radiation trapping impossible. Level lines indicate ground state, second, and tenth modes. (b) The ground state mode of the potential $V$ and (c) The second and tenth modes.

Figure 13

(Color online) Spectral evolution of the soliton-radiation bound states with (a) $n=1$ and (b) $n=3$ given by Eqs. (17, 18).

Figure 14

(Color online) (a) Spectral and (b) time-domain evolutions of the soliton and trapped radiation modeled using the realistic frequency dependence of GVD [cf. (a) with Fig. 13]. (b) is plotted in the reference frame accelerating with $g_0$, which explains the curvature of the soliton trajectory opposite to the one in Fig. 5a.

Figure 15

(Color online) Bouncing of wave packets in linear potentials. (a) Series of images of the condensate bouncing off a light sheet (19). Time interval between the images is $2\mathrm{ms}$ along the horizontal axis. (b) Numerical modeling of the dispersive radiation bouncing off the decelerating soliton. $\xi$ is the dimensionless time measured in the reference frame moving together with the soliton and $z$ is the distance along the fiber. (c) Frequency $\delta$ conversion of the bouncing radiation (42).