Lieb lattice has been proved to host various extraordinary properties due to its unique Dirac-flat band structure. However, the realization of tunable Lieb lattices with controllable configurations still remains a significant challenge. We demonstrate the first realization of a robust and tailorable plasma Lieb lattice in dielectric barrier discharges by use of uniquely designed mesh-water electrodes. Fast reconfiguration between square lattice, Lieb lattice, and various Lieb superlattices has been achieved in a wide range of discharge parameters even in ambient air. An active control on the symmetry, size, and fine structures of plasma elements in Lieb lattices is realized. Three distinct discharge stages in plasma Lieb lattice are proposed on the basis of fast camera diagnostics. The Dirac-flat band structure of plasma Lieb lattice is demonstrated. An experimental verification of the photonic band gap for Lieb lattice is provided. Moreover, the Gierer-Meinhardt reaction diffusion model with spatial modulations is established to simulate the formation of different Lieb lattices. Experimental observations and numerical simulations are in good agreement. The results provide an important step forward in the ongoing effort to realize tunable Lieb Lattices, which may find promising applications in the manipulation of microwaves.
Diffusion plays a crucial role in the forming and evolving of Turing patterns. Generally, the diffusion processes in complex systems do not comply to the complete random walk theory, which means that the diffusion is abnormal rather than normal, such as super-diffusion, sub-diffusion and anisotropic diffusion. However, most of previous studies focused on the pattern formation mechanism under the normal diffusion. In this paper, a two-component reaction-diffusion model with anisotropic diffusion is used to study the effect of anisotropic diffusion on Turing patterns in heterogeneous environments. Three different types of anisotropic diffusions are utilized. It is shown that the system gives rise to stripe patterns when the degree of anisotropic diffusion is high. The directions of stripes are determined by the degree of the diffusion coefficient deviating from the bifurcation point. In a low degree of anisotropic diffusion, the pattern type is the same as the counterpart in a low degree of the isotropic diffusion. When the diffusion coefficient grows linearly in the space, different types of patterns compete with each other and survive in different regions under the influence of spatial heterogeneity. When the diffusion coefficient is modulated by a one-dimensional periodic function, both type and wavelength of the pattern are determined by the modulated wavelength and the intrinsic wavelength. The system can exhibit alternating two-scale mixed patterns of different types when the modulated wavelength is larger than the intrinsic wavelength. Note that each of the diffusion coefficients of some special anisotropic media is a tensor, which can be expressed as a matrix in two-dimensional cases. We also study the influence of off-diagonal diffusion coefficient <inline-formula><tex-math id="M1">\begin{document}$ D $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M1.png"/></alternatives></inline-formula> on Turing pattern. It is found that the Turing pattern induced by off-diagonal diffusion coefficient always selects the oblique stripe pattern. The off-diagonal diffusion coefficient <inline-formula><tex-math id="M2">\begin{document}$ D $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M2.png"/></alternatives></inline-formula> not only affects the pattern selection mechanism, but also expands the parameter range of Turing space. The critical diffusion coefficient <inline-formula><tex-math id="M3">\begin{document}$ {D_{\text{c}}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M3.png"/></alternatives></inline-formula> increases linearly with the diagonal diffusion coefficient <inline-formula><tex-math id="M4">\begin{document}$ {D_u} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M4.png"/></alternatives></inline-formula>increasing. The intrinsic wavelength of the oblique stripe pattern decreases as the off-diagonal diffusion coefficient <inline-formula><tex-math id="M5">\begin{document}$ D $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M5.png"/></alternatives></inline-formula> increases. It is interesting to note that the critical wavelength corresponding to the critical diffusion coefficient <inline-formula><tex-math id="M6">\begin{document}$ {D_{\text{c}}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M6.png"/></alternatives></inline-formula> is independent of the diagonal diffusion coefficient <inline-formula><tex-math id="M7">\begin{document}$ {D_u} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M7.png"/></alternatives></inline-formula>. These results not only provide a new insight into the formation mechanism of Turing patterns, but also increase the range and complexity of possible patterns.
We present the experimental realization of tunable honeycomb superlattice plasma photonic crystals (PPCs) in dielectric barrier discharge by utilizing mesh-liquid electrodes. Fast reconfiguration among the simple honeycomb lattice, honeycomb superlattice, and honeycomb-snowflake superlattice is achieved. A dynamic control on the sizes of center scattering elements in the honeycomb superlattice has been realized. A phenomenological activator-inhibitor reaction diffusion model is established to demonstrate the formation and transition of the honeycomb superlattice. The simulations reproduce well the experimental observations. The photonic band diagrams of different honeycomb PPCs are studied by using finite element method. The addition of large center elements in the honeycomb superlattice yields remarkable omnidirectional band gaps that are about 2.5 times larger than the simple honeycomb lattice case. We propose an effective scheme to fabricate spatiotemporally controllable honeycomb lattices that enable great improvement in band gap size and dynamic control of microwave radiations for wide applications.
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