Models for supercontinuum generation beyond the slowly-varying-envelope approximation

We show numerically that both the modified Korteweg–de Vries and the sine-Gordon models are conducive to the generation of supercontinua with spectral bandwidths of several octaves, when an intense short pulse is launched as the initial condition. These models beyond the slowly-varying-envelope approximation could play an important role in modeling supercontinuum generation in gas-filled hollow waveguides.

Figure 1

Evolution of the spectral widths of the harmonics with respect to the harmonic order. The blue dotted line is the spectrum of a pulse evolving according to the mKdV equation at $z=2$, using the initial parameters $\Delta t=100,\nu =0.3$, and $A=0.7$. The red solid lines are Gaussian fits of the peaks, assuming that the spectral width of the $n\text{th}$ harmonic is proportional to $\sqrt{n}$, in accordance with analytical results. The green dotted lines are Gaussian spectral profiles having all the same width as the fundamental, for comparison (logarithmic scale).

Figure 2

Temporal evolution of an initial pulse with $\Delta t=100,\nu =0.3$, and $A=0.7$, according to the mKdV model. The input pulse splits into a set of FCP solitons (the soliton speed is compensated for clarity).

Figure 3

Evolution of an initial pulse with $\Delta t=100,\nu =0.3$, and $A=0.7$, according to the mKdV model (same as Fig. 2): (a) initial pulse $z=0$, (b) set of emerging FCP solitons $z=999$, and (c) zoom on some FCP solitons in (b) (the soliton speed is compensated for clarity).

Figure 4

Evolution of the spectrum (on a logarithmic scale) of the initial pulse with $\Delta t=100,\nu =0.3$, and $A=0.7$, shown in Fig. 2. A very broad spectrum is reached quickly.

Figure 5

First stage of the evolution according to the mKdV model of the spectrum of the initial pulse with $\Delta t=100,\nu =0.3$, and $A=0.7$, shown in Fig. 2 (blue solid lines), compared to the effect of self-phase modulation (red dotted lines), for $z=0$, 2, 4, 10, and 20.

Figure 6

Evolution of the spectrum (on a logarithmic scale) of an initial pulse with $\Delta t=100,\nu =0.3$, and $A=0.17$. The spectral broadening evolves periodically, showing some recurrence phenomena.

Figure 7

Evolution of the superbreather generated by an initial pulse with $\Delta t=100,\nu =0.3$, and $A=0.17$.

Figure 8

Evolution of the profile of an initial pulse with $\Delta t=100,\nu =0.3$, and $A=1$, according to the sG model: (a) input $\left(z=0\right)$, (b) formation of the first FCP soliton $\left(z=9.2\right)$, (c) the second FCP soliton is just formed $\left(z=11.16\right)$, and (d) end of our computation $\left(z=20\right)$.

Figure 9

Evolution of an initial pulse with $\Delta t=100,\nu =0.3$, and $A=2.5$, according to the sG model.

Figure 10

Evolution of the spectrum of an initial pulse with $\Delta t=100,\nu =0.3$, and $A=1$, according to the sG model (on a logarithmic scale). From bottom to top, $z=0$, 4, 8, 9.2 (arising of the first soliton), 11.16 (the second soliton is just formed), and 20.

Figure 11

Evolution of the spectrum of an initial pulse with $\Delta t=80,\nu =0.375$, and $A=5$, according to the sG model (on a logarithmic scale). Notice the high number of harmonics and the oscillations of their amplitudes.

Figure 12

First stage of the evolution of the spectrum of the initial pulse with $\Delta t=100,\nu =0.3$, and $A=1$, according to the sG model (blue solid lines), compared to the effect of self-phase modulation (red dotted lines), for $z=0$, 2.4, 4.8, 7.2, and 9.6.

Figure 13

Evolution of the spectrum of an initial pulse with $\Delta t=100,\nu =0.3$, and $A=2.5$ (on a logarithmic scale, in dB), against wavelength $\lambda$, according to three models: the mKdV one, with $c_1=0,c_2=0.0784$, and $c_3=1$ (red dashed line); the sG one, with $c_1=12.5$ and $c_2=c_3=0$ (blue dash-dotted line); and the mKdV-sG one, with $c_1=6.25,c_2=0.0392$, and $c_3=0.5$ (light blue solid line). The propagation distances are $z=0$, 6, 12, and 16.