Nonreciprocal Elastic Wave Beaming in Dynamic Phased Arrays

Beam forming using phased arrays forms the basis of several sonar communication and biomedical imaging techniques. However, to date, such arrays remain constrained by wave reciprocity in addition to being confined to their operational frequency; two limitations that have severely hindered imaginative advancements in this domain. In the context of sound propagation, nonreciprocity typically refers to unidirectional elastic and surface acoustic wave devices. However, a breakage of reciprocity in phased arrays manifests itself in reception and transmission patterns, which can be independently tuned, which has thus far been elusive. This work reports on a class of nonreciprocal phased arrays, which operate independently and simultaneously within different directions and frequency channels, thus breaking transmission-reception symmetry and offering enhanced capabilities in guided wave engineering. The system comprises an array of transceiving piezoelectric wafer discs bonded to an elastic medium and incorporates a prescribed dynamic modulation on top of a static phase gradient, which enables concurrent phononic transitions in energy and momentum spaces that mitigate the constraints imposed by Lorentz reciprocity. Following the theory and predictive analysis, the entire array and its associated capabilities are demonstrated experimentally.

Figure 1

STP PWD phased-array surface mounted on a thin aluminum plate. (a) Coordinate system and array geometry. (b) Schematic diagram of the different array components including piezoelectric discs, STP phase shifters, power divider (PD), and voltage generator. Different harmonics are generated and propagate in different directions.

Figure 2

STP PWD phased array in transmission and reception modes. (a),(b) Out-of-plane displacement field at $t=0.005$ s for a modulation depth of $\delta =1.5$ and $\delta =2.4$, respectively. The insets show the normalized FFT amplitudes of sensor locations indicated by stars as predicted by both theory and simulations. (c),(d) Directional breakdown and distribution of the FFT amplitudes for up-converted, fundamental and down-converted harmonics when $\delta =1.5$ and $\delta =2.4$, respectively. Theoretical principal transmission directions are plotted as dashed arrows for comparison. (e) Plane wave with a frequency of $\tilde{\omega }$ incident upon the array from an arbitrary angle $\tilde{\theta }$. (f) From left to right: normalized FFT amplitudes (both theory and simulations) of received signal at output channel when the incoming plane wave is incident from ${\tilde{\theta }}_{A,+1}$, ${\tilde{\theta }}_A$ and ${\tilde{\theta }}_A,-1$, respectively. (g) FFT magnitude of received signal at the output channel as a function of the incident angle for different harmonics. Theoretical principal reception directions are plotted as dashed lines for comparison. Here, $\omega /2\pi =\tilde{\omega }/2\pi =3\mathrm{kHz}$, ${\omega }^{(+1)}/2\pi ={\tilde{\omega }}^{(+1)}/2\pi =3.05\mathrm{kHz}$, ${\omega }^{(-1)}/2\pi ={\tilde{\omega }}^{(-1)}/2\pi =2.95\mathrm{kHz}$.

Figure 3

Experimental setup. (a) Illustration of the experimental setup showing the aluminum plate under study, mounting setup, laser vibrometer system (SLDV), controller circuit, and the perfectly matched layer (PML). The STP PWD phased array is highlighted in the closeup using blue lines. The arrays used for generating incident waves in reception are highlighted in dashed red boxes. (b) Schematic diagram of the controller circuit. Solid lines represent the circuit for the transmission experiment, dashed lines represent the circuit for the reception experiment, and the thicker lines represent the connections that remain constant for both experiments. (I) and (II) refer to the arrays generating the different incident waves shown in (a) for the reception experiment. The individual piezoelectric connections in the schematic diagram and their respective phase signals are color coded for clarity.

Figure 4

Experimental performance of the dynamic array. (a),(b) Out-of-plane velocity wave field for $\delta =1.5$ and $\delta =2.4$ at $t=0.001$ s. In (a) and (b), the insets show time- and frequency-domain measurements at sensor locations indicated by the red, blue, and green markers, which are placed at a radial distance of $450\mathrm{mm}$ from the center of the plate at principal transmission directions. (c),(d) Out-of-plane velocity wave fields of the generated incident waves approaching the array from principal reception directions. In (c) and (d), the top-left insets show the postprocessed individually modulated signals for each of the channels in light pink and the collected voltage signal in magenta. The top-right insets show the unmodulated signal from the first (navy-blue) and last (maroon) elements of the phased array. The bottom insets show the normalized FFT amplitude of the collected voltage signal at the output port. All parameters are the same as those used in Fig. 2.

Figure 5

Nonreciprocal wave beaming in the STP PWD phased array. (a) Transmission channels as a result of an input voltage signal with frequency of $\omega$. (b) Reception process and normalized FFT amplitudes (as predicted from both theory and simulations) of received signal at the output port corresponding to different transmission channels. (c) Characteristics of the received signal in different stages of the reception process. (d) Effect of temporal modulation speed on the radiation and reception patterns of the STP PWD phased array. The parameter used are as follows: $\omega /2\pi =\tilde{\omega }/2\pi =3\mathrm{kHz}$, ${\omega }^{(+1)}/2\pi ={\tilde{\omega }}^{(+1)}/2\pi =3.05\mathrm{kHz}$, ${\omega }^{(-1)}/2\pi ={\tilde{\omega }}^{(-1)}/2\pi =2.95\mathrm{kHz}$, ${\kappa }_c=5.4113\pi$, ${\kappa }_m=15.4821\pi$, and $\delta =1.5$.