Metaclusters for the Full Control of Mechanical Waves

We present a method for the control of waves based on inverse multiple scattering theory. Conceived as a generalization of the concept of metagrating, we call metaclusters to a finite set of scatterers whose position and properties are obtained by inverse design once we have defined their response to some external incident field. The particular focus is on designing passive metaclusters that do not require an external source of energy. The method is applied to the propagation of flexural waves in thin plates, and to the design of far-field patterns, although its generalization to acoustic or electromagnetic waves is straightforward. Numerical examples are presented to the design of uni- and bidirectional “anomalous scatterers,” which will bend the scattering energy along a specific direction, “odd pole” scatterers, whose radiation pattern presents an odd number of poles, and to the generation of vortical patterns. Finally, some considerations about the optimal design of these metaclusters are discussed.

Figure 1

Examples of the target patterns. Solid lines are normalized amplitudes, and dashed lines are normalized phases of the pattern functions (a finite number of 81 modes in Eq. (13) is assumed).

Figure 2

Examples of the optimal far-field patterns for square $3\times 3$ and $4\times 4$ arrays (lattice spacing $a$) and circular arrangements with $8$ and $10$ particles (radius $a$) for $ka=1$ based on Eqs. (20) and (21). Incidence angle $\theta =\pi$ (the $-x$ direction). For the vortex, Fig. 2, circular shapes are amplitude profiles while the spirals in the center are phases of the scattering pattern. All patterns are normalized.

Figure 3

The optimal pattern functions for the passive $2\times 2$ metacluster at two frequencies bounding a bandwidth of passive designs, $ka\in \{1.9,2.8\}$ (a), and the corresponding displacement field generated at $ka=1.9$ (b). Admittances of the cluster are give in Table 1. The efficiencies of the energy conversions are ${\eta }_{ka=1.9}=0.60$ and ${\eta }_{ka=2.8}=0.35$.

Figure 4

The optimal pattern function for the passive $2\times 2$ metacluster (a), and the corresponding displacement field generated at $ka=3.1$ (b). Admittances of the cluster are given in Table 2. The energy efficiency for this setup is $\eta =1$.

Figure 5

The optimal pattern function for the passive $2\times 2$ metacluster (a), and the corresponding displacement field generated at $ka=5.4$ (b). Admittances of the cluster are give in Table 3. The energy efficiency parameter is $\eta =0.17$.

Figure 6

The optimal pattern function for the passive circular metacluster of five particles (a), and the corresponding displacement field generated at $ka=5.3$ (b). Admittances of the cluster are given in Table 4. The energy efficiency is $\eta =0.84$ for this cluster configuration.

Figure 7

The optimal pattern function for the passive $2\times 2$ metacluster: amplitude (a) and phase (b); and the corresponding displacement field generated at $ka=5.5$: displacement magnitude (c) and phase (d). Admittances of the cluster are given in Table 5. The energy efficiency is $\eta =0.14$.