Spatiotemporal Binary Acoustic Metasurfaces

Over recent years, digital acoustic metasurfaces have been developed rapidly as a highly active research area owing to their unique and flexible manipulation of acoustic wavefronts. Nevertheless, nearly all recent attention in the acoustic community has been concentrated on space-encoded architectures, leaving the room free for benefiting from the unique features of spatiotemporally modulated metasurfaces. By entering the world of time, here, we propose a space-time-coding acoustic digital metasurface with exotic ability to dynamically transfer the energy of the carrier acoustic signal to a series of harmonic components, with equivalent magnitudes and phases that can be precisely and independently engineered. The transmission phases of the contributing elements are dynamically controlled in time by exploiting an elaborately designed electromechanical controller system. The study is supported by theoretical, numerical, and experimental verifications which are in excellent agreement. The results demonstrate that by distributing the coding sequences in both space and time dimensions, diverse scattering functionalities can be elaborately acquired for one or multiple harmonic frequencies in a programmable way. This paradigm of acoustic metasurfaces, without resorting to high-cost nonlinear components, opens up unprecedented potential for efficient harmonic control used in adaptive beamforming and acoustic imaging systems.

Figure 1

Conceptual illustration of the nonlinear acoustic wave manipulation for real-time scattering pattern control in both space and frequency domains based on a space-time-coding digital metasurface comprising electromechanically programmable acoustic meta-atoms. Once the cavity size of the acoustic meta-atom is altered by means of the electromechanical programming system, the transmission phase can be dynamically switched between different states while maintaining high energy transmission. The electromechanical system is composed of a magnet and a coil which is dynamically fed by predetermined AC time-coding sequences.

Figure 2

(a) Front view of the designed acoustic meta-atom, where the incident sound propagates along the positive $z$-direction. The influence of the cavity height variation on the measured and simulated (b) transmission amplitude and (c) phase responses of the electromechanically programmable meta-atom at $f_c=3\mathrm{kHz}$.

Figure 3

The harmonic phases/intensities distribution of the time-domain digital acoustic metasurface at 3 kHz with the pulse duration $\tau =10\mathrm{ms}$ under different time-varying coding sequences with (a) $({\varphi }_1,{\varphi }_2)=(0,{45}^{\circ })$, (b) $({\varphi }_1,{\varphi }_2)=(0,{90}^{\circ })$, (c) $({\varphi }_1,{\varphi }_2)=(0,{135}^{\circ })$, (d) $({\varphi }_1,{\varphi }_2)=(0,{180}^{\circ })$, and (d) $({\varphi }_1,{\varphi }_2,{\varphi }_3,{\varphi }_4)=(0,{90}^{\circ },{180}^{\circ },{270}^{\circ })$.

Figure 4

The equivalent phase coverages of the time-domain digital acoustic metasurface at different harmonic frequencies. The transmission phase of the acoustic elements is periodically switched between (a) $({\varphi }_1,{\varphi }_2)=(0,{180}^{\circ })$ and (b) arbitrary (${\varphi }_1$, ${\varphi }_2$) ($L>2$).

Figure 5

The calculated results of 2D polar patterns for different 1-bit and 2-bit equivalent space codes of (a) 00000000, (b) 01230123, (c) 13322012, (d) 10010011, (e) 11000110, and (f) 03331021 at different harmonics (a) $-3\mathrm{rd}$-, (b) $+1\mathrm{st}$-, (c) $-1\mathrm{st}$-, (d) $+3\mathrm{rd}$-, (e) $-1\mathrm{st}$-, and (f) $+3\mathrm{rd}$-order harmonic frequencies. The blue, red, and green lines in the transmission patterns of the 1-bit equivalent space codes stand for the time-domain control signals with $({\varphi }_1,{\varphi }_2)=(0,{45}^{\circ }),(0,{90}^{\circ })$, and $(0,{180}^{\circ })$, respectively. The length of time sequence code is $L=2$.

Figure 6

The harmonic phases/intensities distribution of the proposed time-domain digital acoustic metasurface at 3 kHz with the pulse duration $\tau =10\mathrm{ms}$ when the transmission phase of the acoustic elements is periodically switched between 0 and $\pi$ under the optimized time-varying coding sequences of (a) 00111100, (b) 10011011, (c) 10010110, (d) 11010101, and (e) 1111100110 for maximum possible coupling of the carrier power to (a) $\pm 1\mathrm{st}$-, (b) $\pm 2\mathrm{nd}$-, (c) $\pm 3\mathrm{rd}$-, (d) $\pm 4\mathrm{th}$-order harmonics, or (e) distributing it equally between these components.

Figure 7

Numerical performance of the proposed space-time-coding acoustic metasurface with different 1-bit equivalent space codes of (a) 00000000, (b) 00001111, (c) 00110011, (d) 00100100, and (e) 11010010 at different harmonics (a) $+2\mathrm{nd}$-, (b) $+1\mathrm{st}$-, (c) $-1\mathrm{st}$-, (d) $-3\mathrm{rd}$-, and (e) $-2\mathrm{nd}$-order harmonic frequencies. The time sequence length is $L=8$.

Figure 8

Numerical performance of the proposed space-time-coding acoustic metasurface with different 2-bit equivalent space codes of (a) 00112233 ($G_2$), (b) 3012230112300123 ($G_1\otimes G_{-4}$), (c) 01230123 ($G_1$), (d) 01302312 ($G_1\otimes G_2$), (e) 0123012312301230 ($G_1\otimes 0000000011111111$), and (f) 00113322 ($00001111\oplus 00110011$) at different harmonics (a) $-4\mathrm{th}$-, (b) $+2\mathrm{nd}$-, (c) $-1\mathrm{st}$-, (d) $+3\mathrm{rd}$-, (e) $+4\mathrm{th}$-, and (f) $-3\mathrm{rd}$-order harmonic frequencies. The time sequence length is $L=8$.

Figure 9

Numerical performance of the proposed space-time-coding acoustic metasurface with different 3-bit equivalent space codes of (a) 01234567, (b) 72107210, (c) 0167123023457456, (d) 761103325547, (e) 74451267, and (f) 63515172 (random) at different harmonics (a) $-2\mathrm{nd}$-, (b) $+4\mathrm{th}$-, (c) $-1\mathrm{st}$-, (d) $+1\mathrm{st}$-, (e) $+2\mathrm{nd}$-, and (f) $+3\mathrm{rd}$-order harmonic frequencies. The time sequence length is $L=8$.

Figure 10

Demonstrating the capability of the proposed space-time-coding acoustic metasurface in control of multiple harmonic frequencies at the same time by introducing suitable space-dependent time delays to the time-coding sequences. The equivalent space codes are 03614725, 01230123, 01234567, 00000000, 76543210, 32103210, and 52741630 (from left to right). (a) 3D and (2) 2D far-field patterns are plotted for different harmonic orders scanning the whole upper space. The monochromatic sound impinges on the metasurface from the bottom side. The time sequence length is $L=10$.

Figure 11

(a) Fabricated sample of the space-time-coding acoustic metasurface. (b) Schematic illustration of the experimental setup in which a Plexiglass plate is placed above the sample (not shown).

Figure 12

(a),(b) The measured and calculated harmonic conversion efficiencies when all coding elements share the time-coding sequence “$010101\cdots$” with $\mathrm{\Delta }\varphi ={90}^{\circ }$ and ${180}^{\circ }$, respectively. (c) Experimental demonstration of the harmonic-beam-steering capability of the designed acoustic metasurface by showing the harmonic far-field patterns when the coding elements are fed by the time-coding sequence “1111100110” ($\mathrm{\Delta }\varphi ={90}^{\circ }$) with $T_m/8$ ($T_m=40\mathrm{ms}$) consecutive time delays.