Standing-wave solutions in twisted multicore fibers

In the present work, we consider the existence and spectral stability of standing-wave solutions to a model for light propagation in a twisted multicore fiber with no gain or loss of energy. Numerical parameter continuation experiments demonstrate the existence of standing-wave solutions for sufficiently small values of the coupling parameter. Furthermore, standing waves exhibiting optical Aharonov-Bohm suppression, where there is a single waveguide which remains unexcited, exist when the twist parameter $\varphi$ and the number of waveguides $N$ is related by $\varphi =\pi /N$. Spectral computations and numerical evolution simulations suggest that standing-wave solutions where the energy is concentrated in a single site are neutrally stable. Solutions with asymmetric coupling and multipulse solutions are also briefly explored.

Figure 1

Schematic of twisted, multicore fiber consisting of $N$ waveguides arranged in a ring.

Figure 2

Schematic of symmetry relationship between nodes for $N=6$ and $N=7$. For nodes connected with arrows, the amplitudes $a_k$ are the same and the phases ${\theta }_k$ are opposite.

Figure 3

Standing-wave solution for $N=6$, $\omega =1{\text{mm}}^{-1}$, $k=0.25{\text{mm}}^{-1}$, and $\varphi =0.25$. Left panel shows amplitudes $a_n$ and phases ${\varphi }_n$ for solution at each node. Right panel is intensity of solution $|c_n|$ versus $z$ for nodes 1–4, which is constant in $z$. Evolution in $z$ computed using the fourth-order Runge-Kutta method.

Figure 4

Standing-wave solution for $N=7$, $\omega =1{\text{mm}}^{-1}$, $k=0.25{\text{mm}}^{-1}$, and $\varphi =0.25$. Left panel shows amplitudes $a_n$ and phases ${\varphi }_n$ for solution at each node. Right panel is intensity of solution $|c_n|$ versus $z$ for nodes 1–4, which is constant in $z$. Evolution in $z$ computed using the fourth-order Runge-Kutta method.

Figure 5

Amplitude of node 4 (minimum intensity) versus $\varphi$ for standing-wave solution with $N=6$ (top), and amplitude of nodes 4 and 5 (minimum intensity) versus $\varphi$ for standing-wave solution with $N=7$ (bottom). $\omega =1{\text{mm}}^{-1}$, coupling constants $k$ are ${\text{mm}}^{-1}$.

Figure 6

Left panel shows $k_0$ versus $N$ for dark node opposite peak intensity node for $N$ even, $\omega =1{\text{mm}}^{-1}$. Right panel shows $k_0$ vs $\omega$ together with least-squares linear regression line for $N=50$.

Figure 7

${\ell }^2$ norm of solution vs $k$ for $N=50$ with dark node opposite peak intensity node. $\omega =1{\text{mm}}^{-1}$.

Figure 8

Standing-wave solution for $N=6$ and $\varphi =\pi /6$. Left panel shows amplitudes $a_n$ and phases ${\varphi }_n$ for solution at each node. Right panel is intensity of solution $|c_n|$ versus $z$ for nodes 1–4, which is constant in $z$. Evolution in $z$ computed using the fourth-order Runge-Kutta method. Node 1 has maximum amplitude, and node 4 is a dark node. $\omega =1{\text{mm}}^{-1}$; $k=0.25{\text{mm}}^{-1}$.

Figure 9

Standing-wave solution for $N=7$ and $\varphi =\pi /7$. Left panel shows amplitudes $a_n$ and phases ${\varphi }_n$ for solution at each node. Right panel is intensity of solution $|c_n|$ versus $z$ for nodes 1–4, which is constant in $z$. Evolution in $z$ computed using the fourth-order Runge-Kutta method. Nodes 4 and 5 have equal and maximum amplitude, and node 1 is a dark node. $\omega =1{\text{mm}}^{-1}$, $k=0.25{\text{mm}}^{-1}$.

Figure 10

Spectrum of linearization of (2) about solution for even $N$ with a single dark node opposite a single bright node. $N=6$ (left panel) and $N=50$ (right panel). $k=0.25{\text{mm}}^{-1}$, $\omega =1{\text{mm}}^{-1}$, $\varphi =\pi /N$.

Figure 11

Amplitude $|c_n|$ for the first four nodes versus $z$ for solution with $N=6$, $\varphi =\pi /6$ (left panel) and $N=7$, $\varphi =\pi /7$ (right panel). Evolution performed using a fourth-order Runge-Kutta scheme, $k=0.25{\text{mm}}^{-1}$.

Figure 12

Amplitude $|c_n|$ for the first four nodes versus $z$ for solution with $N=10$ and $\varphi =\pi /10$. Initial condition is the solution to (8) with $k=0.45{\text{mm}}^{-1}$. Evolution performed with $k=0.35{\text{mm}}^{-1}$ (left) and $k=0.55{\text{mm}}^{-1}$ (right) using a fourth-order Runge-Kutta scheme.

Figure 13

Asymmetric standing-wave solution to (14) for $N=6$, $k_1=0.4{\text{mm}}^{-1}$, and $k_n=0.25{\text{mm}}^{-1}$ for all other $n$. Left panel shows amplitudes $a_n$ and phases ${\varphi }_n$ for solution at each node. Right panel is intensity of solution $|c_n|$ versus $z$ for nodes 1–4, which is constant in $z$. Evolution in $z$ computed using the fourth-order Runge-Kutta method. $\varphi =0.25$, $\omega =1{\text{mm}}^{-1}$.

Figure 14

Spectrum of linearization of (14) about asymmetric standing-wave solution from Fig. 13 (left panel). Amplitude $|c_n|$ for first four nodes versus $z$ for evolution of perturbation of this solution using the fourth-order Runge-Kutta scheme (right panel).

Figure 15

Amplitudes $a_n$ for double pulse solutions with two bright nodes in opposite positions of the ring. $N=8$, $\varphi =\pi /4$ (left panel) and $N=12$, $\varphi =\pi /6$ (right panel). $\omega =1{\text{mm}}^{-1}$, $k=0.25{\text{mm}}^{-1}$.

Figure 16

Spectrum of linearization of (2) about symmetric double pulse solution with $N=12$ and $\varphi =\pi /6$ (left panel). Amplitude $|c_n|$ for the first four nodes versus $z$ for evolution of the perturbation of this solution using the fourth-order Runge-Kutta scheme (right panel). $\omega =1{\text{mm}}^{-1}$, $k=0.25{\text{mm}}^{-1}$.