Figure 1

The silicon-based structures (both on a SiO${}_2$ substrate) studied in this Brief Report. (a) GVD of quasi-TE (dashed line) and -TM (solid line) modes of the Si-SiO${}_2$ slot waveguide. Inset below: TM-mode profile at ${\lambda }_0=2592$ nm. Inset above: geometry and materials of the structure. Parameters are $h=700$ nm, $w_1=220$ nm, $w_2=60$ nm, $w_3=600$ nm. (b) GVD of quasi-TE (dashed line) and -TM (solid line) modes of the Si-SiNc slot waveguide. Inset below: TM-mode profile at ${\lambda }_0=2511$ nm. Inset above: geometry and materials of the structure. Parameters are $h=700$ nm, $w_1=190$ nm, $w_2=80$ nm, $w_3=570$ nm. In both figures, the red dot indicates the wavelength ${\lambda }_0$ at which ${\beta }_3=0$, and the gray area is the region where the quartic approximation of the GVD, and therefore the use of Eq. (2), is valid.Reuse & Permissions

Figure 2

(a) Propagation of an input pulse ${\psi }_{\mathrm{in}}=N\text{sech}\left(\tau \right)$, with $N=10$ and ${\delta }_4=-0.01$, no shock term ($s=0$) and no 3OD (${\delta }_3=0$). Due to the nonintegrability of Eq. (2), which induces the presence of a solitonic friction, the pulse splits into several fundamental quartic solitons, approximated by Eqs. (3) and (4). (b) Same as in (a), but in the presence of a shock term ($s=0.03$) and 3OD (${\delta }_3=0.0017$). The shock term induces an asymmetric splitting of the solitons, which all decrease their velocity with respect to the case of (a). (c) Position of the parameters ($a$,$k$) of each soliton formed in (b) after a propagation distance $\xi =10$ (dots), on the predicted curve calculated by using Eqs. (3) and (4), indicated with a solid line. The dashed line indicates the relation between $a\left(q\right)$ and $k\left(q\right)$ that would exist for conventional Schrödinger solitons, for which $a\left(q\right)=k\left(q\right)$. The squares indicate the extracted parameters $a\left(q\right)$ and $k\left(q\right)$ obtained by fitting with a function $f=a\left(q\right)\mathrm{sech}\left[k\right(q\left)\tau \right]$. It is therefore evident that all the localized structures formed [which are numbered from 1 to 5 in (b)–(d)] are quartic solitons and not simple Schrödinger solitons. (d) XFROG spectrogram of (b) at $\xi =10$.Reuse & Permissions

Figure 3

Formation of a QS-induced continuum. The bottom part shows the GVD (dashed-dotted line) of the quasi-TM mode vs wavelength for the structure shown in Fig. 1a. The top part shows the input pulse spectrum (dotted line) and the output spectrum (solid red line) after a propagation of $\xi =10$. Input pulse duration is $t_0=40$ fs, input wavelength is ${\lambda }_0=2.6$ $\mu$m, at which ${\beta }_2\simeq -0.05$ ps${}^2$/m, ${\beta }_3\simeq 1.9\times {10}^{-5}$ ps${}^3$/m, and ${\beta }_4\simeq -2.5\times {10}^{-5}$ ps${}^4$/m. The nonlinear coefficient for the structure of Fig. 1a is calculated by using the vector model of Ref. [23], and is $\gamma \simeq 42$ m${}^{-1}$ W${}^{-1}$. The corresponding fundamental power is $P_0=0.74$ W. Input pulse shape is $N\text{sech}\left(t/t_0\right)$, with soliton order $N=5$.Reuse & Permissions