Generalized dispersion Kerr solitons

We report a continuum of pulselike soliton solutions to the generalized nonlinear Schrödinger equation with both quadratic and quartic dispersion and a Kerr nonlinearity. We show that the well-known nonlinear Schrödinger solitons, which occur in the presence of only negative (anomalous) quadratic dispersion, and pure-quartic solitons, which occur in the presence of only negative quartic dispersion, are members of a large superfamily, encompassing both. The members of this family, none of which are unstable, have exponentially decaying tails, which can exhibit oscillations. We find analytic solutions for positive quadratic dispersion and negative quartic dispersion and investigate the soliton dynamics. We also find evidence that a combination of the quadratic and quartic dispersion, rather than exclusively quadratic or quartic dispersion, is likely to improve the performance of soliton lasers.

Figure 1

(a) ${\beta }_2$ versus ${\beta }_4$ parameter space for fixed $\mu =1.78$ ${\mathrm{mm}}^{-1}$. The horizontal axis (red) represents conventional NLS solitons while the vertical axis (blue) represents PQSs. The grey dashed curve represents the family of Karlsson-Höök solutions at this value of $\mu$. (b)–(e) Power (logarithmic scale) versus time for stationary solutions indicated in (a).

Figure 2

(a) $\mu$ versus ${\beta }_2$ parameter diagram of solutions to Eq. (8), for fixed ${\beta }_4=-1 {\mathrm{ps}}^4 {\mathrm{mm}}^{-1}$. (b)–(d) Configurations of roots $\lambda$ in the complex plane [see Eq. (10)], corresponding to each of the colored regions in (a). (e) Power (logarithmic scale) versus time for the stationary solution corresponding to (b). (f) Same as in (e), but corresponding to (c).

Figure 3

(a) Power versus time for highly oscillatory solution (blue solid curve) for ${\beta }_4=-1 {\mathrm{ps}}^4 {\mathrm{mm}}^{-1}, {\beta }_2=0.81 {\mathrm{ps}}^2 {\mathrm{mm}}^{-1}, \mu =1 {\mathrm{mm}}^{-1}$, and $\gamma =1 {\mathrm{W}}^{-1} {\mathrm{mm}}^{-1}$. The corresponding meta-envelope is given by the red dashed curve. (b) Spectral amplitude of the stationary solution of (a). (c) Two-dimensional Fourier transform of the stationary solution of (a), with the spectral amplitude from (b) shown by the color scale. The linear dispersion relation is represented by the blue dashed curve.

Figure 4

Dispersion relation (17) (solid curve) for ${\beta }_2>0$ and ${\beta }_4<0$. When the frequency spectrum of the field is concentrated around the two maxima, the dispersion relation can be approximated by the dashed curve.

Figure 5

Percentage error in the meta-envelope peak power compared with numerical solutions as a function of the expansion parameter $\epsilon$.

Figure 6

(a) Power versus time at various propagation distances for stationary solution of Fig. 3 multiplied with an amplitude multiplier $N=2$. (b) Power versus time for the input pulse in (a). (c) Pulse in (b) after propagation by five metasoliton periods $\pi T_0^2/\left(4\right|{\beta }_2\left|\right)$.

Figure 7

Time-bandwidth product versus GDKS shape parameter $\sigma$.

Figure 8

Energy of stationary solutions versus specified values of ${\beta }_2$ and ${\beta }_4$ (varying according to the straight part of the contour in Fig. 1) and pulse width. Red solid line represents the NLS solitons while the blue dashed line represents the PQSs.

Figure 9

(a) Linearized eigenspectrum for the GDKS with ${\beta }_2=0.7 {\mathrm{ps}}^2 {\mathrm{mm}}^{-1}, {\beta }_4=1 {\mathrm{ps}}^4 {\mathrm{mm}}^{-1}$, and $\mu =1 {\mathrm{mm}}^{-1}$. Red points represent high-spatial-frequency perturbations which are radiated as dispersive waves. The green star represents the zero eigenvalues corresponding to the translational and phase invariance of Eq. (4). Blue diamonds represent discrete internal modes. (b) Conjugate mode profiles $f$ (thick red solid curve) and $g$ (thin blue solid curve) corresponding to the larger internal mode eigenvalue of (a), with the soliton shown for comparison (yellow dashed curve). (c) Power versus time for the GDKS in (a) perturbed by the internal mode in (b) over five internal mode oscillation periods $T_{\text{int}}$.