Space-Quasiperiodic and Time-Chaotic Parametric Patterns in a Magnonic Quasicrystal Active Ring Resonator

We demonstrate the dissipative spatiotemporal patterns that are generated by an active ring resonator based on a magnonic quasicrystal (MQC) with Fibonacci-type structure. The dissipative patterns are formed through the magnetostatic surface spin-wave (MSSW) parametric decay into exchange spin waves (SWs), temporal dispersion of the ring resonator and amplification. In the spatial domain, the MSSW and SW parametric patterns measured with the help of Brillouin light spectroscopy have the quasiperiodic spatial localization in crests and grooves of the MQC, respectively. In contrast to the optical quasiperiodic parametric spatial conservative solitons, the amplitude profiles of the magnonic quasiperiodic parametric spatial dissipative patterns correspond to the profiles of the crests and grooves of the MQC. In the time domain, the packets of the chaotic parametric pulses, which are analogs of temporal solitons, are generated at each point of the spatial patterns. The chaotic nature of such pulses is confirmed by an estimate of a highest Lyapunov exponent from the experimental time series. The pulse packets are formed through the time-filtering technique using the external microwave (MW) pulses to control a ring gain. It is shown that at a certain duty factor of the external MW pulses, the parametric MSSW pulses have the amplitude and phase profiles corresponding to the profiles of a conservative bright envelope soliton.

Figure 1

Scheme of a MQC active ring resonator.

Figure 2

Characteristics of the dissipative solitons generated in the single-mode regime under the external MW pulses: (a) the microwave power spectrum (solid line) of the dissipative solitons, (b) the time series of the envelope of the MW pulses (upper panel), the dissipative solitons (middle panel) and their phase profiles (lower panel), (c) the projections of a phase portrait on the parameter plane [$A(t),A(t+{\tau }_d)$], where ${\tau }_d=2{\tau }_r$, and (d) the iteration diagram of the envelope phase. In (a), the MQC transmission is marked with a dashed line.

Figure 3

Characteristics of the dissipative solitons generated in the multimode regime at three values of the duty factor $q$ of the external MW pulses: (a-i), (b-i), (c-i) 15; (a-ii), (b-ii), (c-ii) 8.25 and (a-iii), (b-iii), (c-iii) 3.125. In (a-i), (a-ii), and (a-iii) the microwave power spectra of the dissipative solitons, in (b-i), (b-ii), and (b-iii) the time series of the envelope of the MW pulses (upper panel), the dissipative solitons (middle panel), and their phase (lower panel), in (c-i), (c-i), and (c-iii) the projections of a phase portrait on the parameter plane $[A(t),A(t+{\tau }_d)]$.

Figure 4

Dependences of the highest Lyapunov exponent $\lambda$ on the duty factor $q$ calculated based on the experimental time series by two methods: (a) Wolf method and (b) Rosenstein method. Circles correspond to the single-mode and squares correspond to the multimode generation regime. For both methods, the calculations are performed for the embedding dimension $D_{\mathrm{emb}}=9$.

Figure 5

The spatial (a-i), (b-i) and temporal (a-ii), (a-iii), (b-ii), (b-iii) distributions of the square of the magnetization amplitude measured (a) at the MSSW frequency $f_{\mathrm{osc1}}=3241.3$ MHz and (b) at the SW frequency $f_{\mathrm{osc1}}/2$. In both cases, the temporal distributions are obtained for grooves (a-ii), (b-ii) and crests (a-iii), (b-iii). The symbols $x_1-x_6$ correspond to six values of the spatial coordinate: $x_1=0.2$ mm, $x_2=0.4$ mm, $x_3=0.65$ mm, $x_4=0.75$ mm, $x_5=1.9$ mm, $x_6=2.15$ mm. Even symbols correspond to the grooves, and odd symbols correspond to the crests.