# Vortex Creation without Stirring in Coupled Ring Resonators with Gain and Loss

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## Abstract

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## 1. Introduction

## 2. The Model

## 3. Stationary Solutions

## 4. Narrow Coupling Dynamics

## 5. Broad Coupling Dynamics

## 6. On Vortex Creation without Stirring

## 7. Experimental Proposal: Plasmonic and Metamaterial Effects in Arrays of Nanorings

## 8. Conclusions

## Supplementary Materials

- Animation 1: Symmetric.gif. Symmetric oscillations observed for ${J}_{0}=4$ (see Figure 4)
- Animation 2: Transition.gif. Transition from symmetric to asymmetric oscillations observed for ${J}_{0}=4.5$ (see Figure 6)
- Animation 3: Asymmetric.gif. Asymmetric oscillations observed for ${J}_{0}=5$ (see Figure 5)

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Absolute values of antisymmetric stationary states after propagation time $T=100$ in the coupled double-ring system (1) obtained for the initial conditions (4) with $\gamma =3$ and $\mathrm{\Gamma}=1$. Left panel: Antisymmetric states calculated for different coupling strengths and fixed coupling width ($w=1$). The black line represents the homogeneous state for the respective uncoupled system. Right panel: Antisymmetric states calculated for different coupling widths and fixed normalized coupling (${J}_{0}=1$).

**Figure 2.**Absolute value (left) and the phase (right) of stationary solutions observed in the case of narrow coupling $w=0.01$. The black curve represents antisymmetric solution ${\psi}_{1}\left(x\right)=-{\psi}_{2}\left(x\right)$ (${J}_{0}=1$), and blue and red curves show the absolute value of both channels in the asymmetric state (${J}_{0}=1.5$). The phase oscillating term is eliminated, so that ${\varphi}_{1}(x=\pm \pi )=0$. The relative phase of the second channel in the antisymmetric case is not shown, as ${\varphi}_{2,antisym}={\varphi}_{1,antisym}+\pi $. Note that ${\varphi}_{asym1}-{\varphi}_{asym2}$ equals zero at $x=\pm \pi $.

**Figure 3.**Top row: Contour plots of absolute values and phases of the propagated wavefunction ${\psi}_{1}$ in the stationary (asymptotic) regime for two different coupling strengths ${J}_{0}$ and the fixed width $w=1$. The phase oscillation term is eliminated, so that ${\varphi}_{1}(x=\pi )=0$. Center row: Norms in both channels (${N}_{1}$ and ${N}_{2}$) during propagation from initial perturbed symmetric homogeneous states, defined in Equation (4). Bottom row: Norms of both channels and average $({N}_{1}+{N}_{2})/2$ during one period of oscillations in the limit cycle regime. Note that we shifted the time in order to show directly the length of the period of oscillations. Panels on the left show symmetric oscillations for ${J}_{0}=4$, and panels on the right show asymmetric oscillations for ${J}_{0}=5$.

**Figure 4.**Snapshots of wavefunction propagation, representing the half period of symmetric oscillations with coupling strength ${J}_{0}=4$, $w=1$, $\gamma =3$ and $\mathrm{\Gamma}=1$. All frames are presented in pairs, where the top frames (a1-h1) show the absolute values of both wavefunctions (blue and red curves) and the rescaled coupling potential (green curve), while the bottom frames (a2-h2) show the relative phases in both rings. Phases are plotted so that point $x=\pi $ for the blue curve is fixed at zero, to eliminate the phase oscillation term.

**Figure 5.**Snapshots of wavefunction propagation, representing the full period of asymmetric oscillations with coupling strength ${J}_{0}=5$ in the broad coupling regime, $w=1$, $\gamma =3$ and $\mathrm{\Gamma}=1$. All frames are presented in pairs; the top frames (

**a1**–

**l1**) show absolute values of both wavefunctions (blue and red curves) and rescaled coupling potential (green curve), and bottom frames (

**a2**–

**l2**) show relative phases in both rings. The phase oscillation term was eliminated as in Figure 4.

**Figure 6.**Left panel: Frequency of the oscillations of the limit cycle solutions for three different values of $\gamma $. Symmetric oscillations (see Figure 4) are marked as continuous lines and asymmetric (see Figure 5) with dashed lines. The other parameters: $w=1$, $\mathrm{\Gamma}=1$. Right panel: The norm in both channels at ${J}_{0}=4.5$, where the blue line on the left panel has discontinuity. We observe that after initial propagation (not shown), the system develops into a symmetrically-oscillating state (shown in the time interval from $t=0$ to $t\approx 10$) and after several (typically three or four, depending on the initial perturbation) periods of oscillation, the system finally evolves into asymmetrical oscillations.

**Figure 7.**Snapshots of wavefunction propagation, representing relaxation from the point of vortex generation (Point b from Figure 5) after turning off the coupling, for $\gamma =3$ and $\mathrm{\Gamma}=1$. All frames are presented in pairs; the top frames (

**a1**–

**d1**) show absolute values of both wavefunctions (blue and red curves), and bottom frames (

**a2**–

**d2**) show the relative phases in both rings. The phase oscillation term was eliminated as in Figure 4.

**Figure 8.**(

**a**,

**b**) SEM images of the circular and c-shaped nanorings produced by shadow nanosphere lithography (SNL). (

**c**) Schematic of the assembly of the coupled nanoring arrays (left panel) and the resulting examples of possible offset-dependent ring alignment configurations.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ramaniuk, A.; Hung, N.V.; Giersig, M.; Kempa, K.; Konotop, V.V.; Trippenbach, M.
Vortex Creation without Stirring in Coupled Ring Resonators with Gain and Loss. *Symmetry* **2018**, *10*, 195.
https://doi.org/10.3390/sym10060195

**AMA Style**

Ramaniuk A, Hung NV, Giersig M, Kempa K, Konotop VV, Trippenbach M.
Vortex Creation without Stirring in Coupled Ring Resonators with Gain and Loss. *Symmetry*. 2018; 10(6):195.
https://doi.org/10.3390/sym10060195

**Chicago/Turabian Style**

Ramaniuk, Aleksandr, Nguyen Viet Hung, Michael Giersig, Krzysztof Kempa, Vladimir V. Konotop, and Marek Trippenbach.
2018. "Vortex Creation without Stirring in Coupled Ring Resonators with Gain and Loss" *Symmetry* 10, no. 6: 195.
https://doi.org/10.3390/sym10060195