Elliptic-type soliton combs in optical ring microresonators

Soliton crystals are periodic patterns of multispot optical fields formed from either time or space entanglements of equally separated identical high-intensity pulses. These specific nonlinear optical structures have gained interest in recent years with the advent and progress in nonlinear optical fibers and fiber lasers, photonic crystals, wave-guided wave systems, and most recently optical ring microresonator devices. In this work an extensive analysis of characteristic features of soliton crystals is carried out, with an emphasis on their one-to-one correspondence with elliptic solitons. With this purpose in mind, we examine their formation, their stability, and their dynamics in ring-shaped nonlinear optical media within the framework of the Lugiato-Lefever equation. The stability analysis deals with internal modes of the system via a $2\times 2$-matrix Lamé-type eigenvalue problem, the spectrum of which is shown to possess a rich set of bound states consisting of stable zero-fequency modes and unstable decaying as well as growing modes. Turning towards the dynamics of elliptic solitons in ring-shaped fiber resonators with Kerr nonlinearity, we first propose a collective-coordinate approach, based on a Lagrangian formalism suitable for elliptic-soliton solutions to the nonlinear Schrödinger equation with an arbitrary perturbation. Next we derive time evolutions of elliptic-soliton parameters in the specific context of ring-shaped optical fiber resonators, where the optical field evolution is thought to be governed by the Lugiato-Lefever equation. By solving numerically the collective-coordinate equations an analysis of the amplitude, the position, the phase of internal oscillations, the phase velocity, the energy, and phase portraits of the amplitude is carried out and reveals a complex dynamics of the elliptic soliton in ring-shaped optical microresonators. Direct numerical simulations of the Lugiato-Lefever equation are also carried out seeking for stationary-wave solutions, and the numerical results are in very good agreement with the collective-coordinate approach.

Figure 1

Elliptic-soliton solution to the NLSE: ${\beta }_2=-0.5, \beta =0.5$ and $\gamma =0.9, \kappa =0.97$.

Figure 2

Real parts of the internal modes $v(\phi$) (left) and $u(\phi$) (right) given in (25) and (26). Top: $\kappa =0.8$, bottom: $\kappa =97$.

Figure 3

Plots of the internal-mode eigenvalues $\nu$ (right) and $\omega$ (left) given by (27), as a function of the Jacobi elliptic modulus $\kappa$. In the two plots the shaded region (blue region in color) denotes the stability domain.

Figure 4

Time evolutions of the elliptic-soliton parameters $\eta \left(t\right), q\left(t\right), \varphi \left(t\right)$, and $\delta \left(t\right)$ for ${\alpha }_1=0.001$ and ${\alpha }_2=0.5$.

Figure 5

Time evolutions of the elliptic-soliton amplitude $\eta \left(t\right)$ for ${\alpha }_2=0.01$. From (a) to (d): ${\alpha }_1=0.001, {\alpha }_1=0.005{\alpha }_1=0.01, {\alpha }_1=1$.

Figure 6

Time evolutions of the elliptic-soliton amplitude $\eta \left(t\right)$, for ${\alpha }_1=0.001$. From (a) to (d): ${\alpha }_2=0.005, {\alpha }_2=0.5, {\alpha }_2=1, {\alpha }_2=1$.

Figure 7

Phase portraits of the amplitude for ${\alpha }_1=0.001$. From top to bottom and from left to right: ${\alpha }_2=0.05, {\alpha }_2=0.5, {\alpha }_2=0.7, {\alpha }_2=1$.

Figure 8

Phase portraits of the amplitude for ${\alpha }_2=0.01$ and ${\alpha }_1=0.005$ (left), ${\alpha }_1=0.01$ (right).

Figure 9

Time variation of the elliptic-soliton energy. From graphs (a) to (f) the linear loss parameter is constant (i.e., ${\alpha }_1=0.001$), while the pump detuning frequency is varied as: (a) ${\alpha }_2=0.001$, (b) ${\alpha }_2=0.01$, (c) ${\alpha }_2=0.1$, (d) ${\alpha }_2=0.5$.

Figure 10

Soliton-crystal profiles from numerical simulations for $F_0={\alpha }_1={\alpha }_2=0$.

Figure 11

Soliton-crystal profiles from numerical simulations for $F_0=0.1, \beta \tau =0.5$, and ${\alpha }_1=0.1$. ${\alpha }_2$ is varied as 0.05 (left graph), 0.1 (middle graph), and 0.5 (right graph).

Figure 12

Oscillating amplitude, width, and repetition rate of pulses in the soliton-crystal structure, as suggested by direct numerical simulations of Eq. (46) at different times $\tau$. $F_0=0.1, {\alpha }_1=0.1, {\alpha }_2=0.5$ and from left to right: $\beta \tau =0.5$, 10, 15, and 20.