Rotating solitons supported by a spiral waveguide

We investigate numerically light propagation in a single spiral waveguide formed in a nonlinear dielectric medium for a low spatial frequency of the waveguide rotation. We present a general procedure for finding solitonic solutions in spiral waveguiding structures, as well as the variational approach to calculate soliton parameters analytically. Solitons supported by the spiral waveguide perform robust stable rotational oscillatory motion, with the period predicted by their static characteristics, without any signatures of wave radiation or soliton decay over many rotation periods and diffraction lengths.

Figure 1

Sketch of the single spiral waveguide.

Figure 2

Fundamental soliton intensity profile for $\mu =2L_D^{-1}$. Parameters: $\mathrm{\Omega }=-0.2\mathrm{rad}/L_D$, $P_w=0.044$, $R=3$, $\mathrm{\Gamma }=30$.

Figure 3

Fundamental soliton intensity profiles (top) and the corresponding potential profiles (bottom) at $y=0$. Parameters are as in Fig. 2, except $P_w=0.05$ ($I_{w0}=0.1$, $W_w=0.4$).

Figure 4

Fundamental soliton power, width, and peak intensity as functions of the propagation constant. Black dots represent the stable rotary solitonic solutions, red (gray) dots the unstable solutions. Parameters are as in Fig. 3.

Figure 5

Peak intensity as a function of the propagation distance for several different values of the propagation constant. Parameters are as in Fig. 3.

Figure 6

Typical trajectory of a stable rotating soliton [red (gray) line] supported by the spiral waveguide (black line): The fundamental soliton oscillates regularly around the waveguide during propagation. (a) Three-dimensional (3D) view for the propagation distance $z=100L_D=3.183\mathrm{\Lambda }$. (b) 2D view for $z=\mathrm{\Lambda }$. Parameters are as in Fig. 3, $\mu =15L_D^{-1}$.

Figure 7

(a) The initial soliton center position $x_c$ as a function of the propagation constant $\mu$. Black dots represent the stable rotary solitonic solutions, red (gray) dots the unstable solutions. Parameters are as in Fig. 3. (b) The quantity $(R-x_c)/\left({\mathrm{\Omega }}^2{x}_c\right)$ is practically independent of the frequency of rotation $\mathrm{\Omega }$. Parameters are as in Fig. 3, except for $P_w=0.1$ and $P=0.5$.

Figure 8

The period of small oscillations $T$ as a function of the propagation constant. For the stable (black dots) and unstable [red (gray) dots] rotating solitons period is given by Eq. (13); blue (gray) circles represent results obtained in numerical simulations. Parameters are as in Fig. 3.

Figure 9

Untypical trajectory of a rotating soliton [red (gray) line] supported by the spiral waveguide (black line): The fundamental soliton is launched counterclockwise and rotates in this direction during propagation, while the waveguide spirals in the opposite direction. (a) 3D view for the propagation distance $z=400L_D$. (b) 2D view for $z=294L_D$ (one period). The spiral waveguide channel has a hyper-Gaussian shape, with $P_w=0.044$, $\mu =26L_D^{-1}$, and $P=98.9$; other parameters are as in Fig. 3.

Figure 10

Fundamental soliton power and width as functions of the peak intensity. Black dots represent stable rotary solitonic solutions obtained numerically; black solid line represents results of the variational approach. Parameters are as in Fig. 3.