Self-similar propagation and asymptotic optical waves in nonlinear waveguides

The properties of self-similar optical waves propagating in a tapered cubic-quintic nonlinear waveguide are investigated. Using a lens-type transformation we obtain the exact analytical self-similar solutions which describe the propagation of bright-shaped solitons, dark-shaped solitons, kink-shaped solitons, and antikink-shaped solitons. The stability of the solutions is examined by numerical simulations such that stable bright solitons are found. Beyond the exact analytical solutions, asymptotic optical waves are also found by employing a direct ansatz. These waves possess linear chirps and can propagate self-similarly. The possibility of controlling the shape of output asymptotic optical waves is demonstrated. The analytical results are confirmed by numerical simulations. Finally, we investigate the generation and propagation properties of self-similar optical waves in a quintic nonlinear medium.

Figure 1

Profiles of the tapering function, width, CQ nonlinearities, gain, and chirp function for $n=G=1$.

Figure 2

Compressions of the bright and dark self-similar solitons in the sech${}^2$-profile CQ nonlinear waveguide. (a) Intensity profile of the bright soliton given by Eqs. (4) and (8) for $N_b=0.5$ and (b) intensity profile of the dark soliton given by Eqs. (4) and (9) for $N_d=2$. The other parameters are ${\zeta }_0=0,\tau =G=1,v=3$.

Figure 3

Compressions of the kink and antikink self-similar solitons in the sech${}^2$-profile CQ nonlinear waveguide. Intensity profiles of (a) the kink soliton for $a_k=-1$ and (b) the antikink soliton for $a_k=1$ given by Eqs. (4) and (10). The other parameters are ${\zeta }_0=0,G=v={\omega }_k=1$.

Figure 4

(a) The stable evolution of the bright self-similar soliton given by Eqs. (4) and (8), with the same parameters as in Fig. 2 except for $v=0$. Panels (b) and (c) show the unstable evolutions of the dark and kink self-similar solitons given by Eqs. (4), (9), and (10), respectively, with the same parameters as in Figs. 2 and 3 except for $v=0$. (d) Evolution of a stable secant pulse in the CQ nonlinear waveguide within the framework of Eq. (3) when $\sigma \gamma =\tau \delta =1,G=0$.

Figure 5

(a) The self-similar evolution of the compact pulse in the CQ nonlinear waveguide, starting with the Gaussian input pulse $U(0,X)=exp(-X^2/2)/{\pi }^{1/4}$. From top to bottom, the propagation distance is $Z=6,5.6$, and 5, respectively. (b) The pulse's width and amplitude as functions of $Z$. (c) The phase (phase offset is ignored) of the pulse at propagation distance $Z=6$. Here, and in the following figures, the solid lines and circles represent results of the direct numerical simulations of Eq. (3) and the analytical predictions, respectively. The dotted lines in (a) show the pulse intensity at $Z=6$ when the quintic nonlinearity is absent. The parameters are $\gamma =G=K=1,F=G^2/60$, and $c_0=0.05$ such that $\mu =0.8069$.

Figure 6

The same as in Fig. 5 except that the input pulse is $U(0,X)=\mathrm{sech}\left(X\right)/2^{1/2}$, i.e., the hyperbolic secant input pulse.

Figure 7

(a) Compressions of the bright self-similar soliton given by Eqs. (4) and (20) in the sech${}^2$-profile nonlinear media. (b) The stable evolution of the bright self-similar soliton with initial zero velocity. The parameters are ${\zeta }_0=0,-\tau =G=1,v=3$.

Figure 8

(a) The evolution of the compact self-similar solution in the nonlinear media, starting with the Gaussian input pulse $U(0,X)=exp(-X^2/2)/{\pi }^{1/4}$. From top to bottom, the propagation distance is $Z=4,3.8,3.6$, and $3.4$, respectively. (b) The solution's width and amplitude as functions of $Z$. (c) The phase (phase offset is ignored) of the pulse at propagation distance $Z=4$. The parameters are $G=-K=1,F=G^2/60$, and $c_1=0.5$ such that $\mu =\sqrt{2}/\pi$.