Solitary-wave solutions for few-cycle optical pulses

Propagation of short optical pulses in a one-dimensional nonlinear medium is considered without the use of the slow envelope and unidirectional propagation approximations. The existence of uniformly moving solitary solutions is predicted for a Sellmeier-type dispersion function in the anomalous dispersion domain. A four-parametric family of such solutions is found that contains the classical envelope soliton in the limit of large pulse durations. In the opposite limit we get another family member, which, in contrast to the envelope soliton, strongly depends on the nonlinearity model and represents the shortest and the most intense pulse that can propagate in a stationary manner.

Figure 1

(Color online) Real part of $ϵ\left(\omega \right)$ for a fluoride glass (solid line) and a fit (5) (dotted line). We have chosen ${\omega }_0=1\text{PHz}$ and obtained $\overline{ϵ}=1.552$ and ${\mu }^2=0.0121$ for the best fit between two inflection frequencies (thick points). For comparison, the eighth-order Taylor expansion of $ϵ\left(\omega \right)$ with the central frequency ${\omega }_0$ (close to the ZDF of 0.98 PHz) is also shown (dashed line).

Figure 2

(Color online) Effective potential $U\left(b\right)$ from Eq. (31) for $\Omega =1$ and three values of the parameter $s$: $8s^2=\left(\text{A}\right)0.97$, (B) 1, and (C) 1.03.

Figure 3

(Color online) Normalized electric field ${\widehat{u}}_x={\phantom{|}{u}_x\left(\xi \right)/\sqrt{{\lambda }^2-1}|}_{z=0}$ and pulse envelope ${\phantom{|}b\left(\xi \right)|}_{z=0}$ computed from Eq. (31) for $\Omega =1$ and different values of the parameter $s$.