Remote whispering metamaterial for non-radiative transceiving of ultra-weak sound

Transceiving ultra-weak sound typically relies on signal pre-amplification at the transmitting end via active electro-acoustic devices, which inherently perturbs the environment in the form of noise that inevitably leads to information leakage. Here we demonstrate a passive remote-whispering metamaterial (RWM) enabling weak airborne sound at audible frequencies to reach unprecedented signal enhancement without altering the detected ambient soundscape, which is based on the extraordinary scattering properties of a metamaterial formed by a pair of self-resonating subwavelength Mie meta-cavities, constituting the acoustic analogy of Förster resonance energy transfer. We demonstrate efficient non-radiative sound transfer over distances hundreds times longer than the radius of the meta-cavities, which enables the RWM to recover weak sound signals completely overwhelmed by strong noise with enhanced signal-to-noise ratio from −3 dB below the detection limit of 0 dB in free space to 17.7 dB.

Note that owing to the monopolar resonance nature, the pressure value at the receiving position | | 0 should be ∑ 0 , which equals to the 0th-order (6) The results at different distance are shown in Fig. S1, which clearly confirm the efficient enhancement in both near field in close proximity much shorter than a wavelength and far field much longer than a wavelength. (bottom x-axis) and operating wavelength λ (top x-axis).

Note 2. RWM at higher-order monopolar modes
In addition to the basic monopole acoustic mode, the system composed of high-refractiveindex particles are capable of exciting higher order monopole modes. Figure S2(a) shows the theoretical and simulated enhancement spectra / when placing two particles (  60) at the launch and receiving ends. The spectrum exhibits the monopole resonance, the second monopole resonance and the third monopole resonance at 518 Hz, 1205 Hz and 1901 Hz, respectively. It is worth noting that the enhancement value is highest at the monopole resonance mode and decreases with the increasement of the order. The corresponding pressure field distributions of these three modes at the receiving end are shown as Fig. S2(b)-S2(d). In each order of monopole resonance mode, the sound energy is highly localized within the highrefractive-index particle and the first monopole resonance mode exhibits the strongest intensity of the sound field. Thus, we focus on the first-order monopole resonance in the manuscript due to its strongest capability to enhance the detected pressure value while manipulate sound waves with larger wavelength. second-order, and third-order monopolar mode.

Note 3. Influence on radiation impedance of the speaker
The radiation capability of a sound source is normally characterized in terms of the real part of the acoustic radiation impedance Z, i.e., the radiation resistances Re(Z). For a given source like a speaker, the acoustic radiation resistances calculated from the complex ratio of sound pressure on the surface of a source to the corresponding normal velocity of the source with and without the metacavity are shown in Fig. S3. The results clearly exhibit a significant enhancement at the resonance frequency in the presence of the metacavity in comparison to the bare source. The changes in the radiation impedance thus indicate that more energy can be emitted to far field.

Figure S3
: Sound source's radiation resistance Re(Z) with and without the metacavity.

Note 4. Experimental setup
For the experimental setup and measurement, we employed a balanced armature speaker (Knowles, Model ID DWFK-31785-000, 5◊2.7◊3.9 mm 3 ) as the monopole source, with the speaker mouth embedded and sealed into a hole (6 mm radius) at the center of the element S (see Fig. S4). The miniature dimensions of this speaker and its efficient response at required  (c) Top view of the experimental setup. The speaker mouth and the condenser microphone are embedded and sealed into the holes at the center of the meta-cavity S and D, respectively.

Note 5. Describing the physical model with a rigorous acoustic scattering theory
The meta-cavity labyrinth structure can be simplified into a previously demonstrated physical equivalent with an inner core of background medium, an outer layer (yellow region) of equivalent medium and a virtual layer (gray region), as shown in Fig. S5(a) [1]. The radius of the outer layer is with η the filling ratio of the narrow channels. The virtual layer is introduced to leave the total radius of the element unchanged, which has the identical fields at the two interfaces. The transfer system can be divided into 5 different regions using the simplified model: the inner and outer layer of the emitting (receiving) region and the free space. Setting , , , the pressure field obeys the 2D Helmholtz equation ∇ 0 , which can be decomposed into the incident (outgoing) and scattering cylindrical waves represented by Bessel(Jm) and Hankel(Hm) functions According to the above equation, we can obtain the detected pressure as: | |/| | . The ehancement of the measured pressure value is: Figure S5  Additionally, a careful analysis of the two efficiencies ⁄ and ⁄ as to how they relate to the overall performance ⁄ is presented. We compute the enhancement factor ⁄ for the enclosed speaker with the bare microphone and equivalently ⁄ for the opposite configuration, finally to be able to compare them to the system where both are enclosed with the respective enhancement factor ⁄ . In doing so, we varied the separation /λ between microphone and loudspeaker and defined the factor ⁄ to be able to assess whether the process is a product of emission and reception.
In the near-field regime, Fig. S6 clearly displays how the enhancement when both elements are enclosed by the RWM exceeds the product of the individual processes since , which stems from the strong monopole-monopole interaction occurring between two such resonators that is giving rise to the pronounced sound field enhancement including pressure oscillations. On the other hand, in the far-field the system can be broken down into two cascaded processes of emission and receiving the signal, and the total transfer efficiency is approximately the product of two efficiencies, . Figure S6: Contrast factor ⁄ between the enhancement when both elements are enclosed by the RWM and the product of the individual processes.

Note 6. Comparison with the classic acoustic Helmholtz resonators
To shed more light on the superiority of the Mie meta-cavities over classic acoustic elements, In principle, the metacavity and Helmholtz resonator are two distinct structures with localized resonances. Contrary to the inertial resonance nature of the Helmholtz resonator that creates a localized monopole resonance in its neck-cavity, which acts as mass-spring oscillator, the metacavity achieves a strong monopolar response via intrinsic resonances originating from the Mie scattering of the high-index geometry (nr > 1). Further, the fluid particles at the Helmholtz resonator's neck yields free-space sound radiation similar to an open-ended pipe in contrast to the metacavity, whose coherent symmetrical radial oscillation stems from a monopolar Mie resonance when the effective wavelength equals the resonator diameter. Those fundamentally different resonance-origins when comparing the two devices, explain the distinct performances when it comes to the acoustic pressure enhancement. Note 7. Sound pressure field emitted from a weak sound source in free space Figure S8 gives the sound pressure fields emitted from a weak sound source in the condition without and with the RWM system. The volume flow rate per unit length out from the source is /157. The detected pressure is only 16.1 dB for the free space case [see Fig. S8(a)], which is too weak to detect even in an extremely quiet environment, let alone in ordinary noisy background with low SNR. For comparison, the RWM system can enhance the detected sound pressure emitted from the same weak sound source up to 60 dB [see Fig. S8(b)]. Note that the enhancement in the receiving location is much larger than that in exterior area. As a result, we can achieve efficient signal amplification for the specified location while keeping the other areas relatively quiet, no matter using the same weak sound source. Figure S8: Comparison to a weak sound source with the volume flow rate / . (a) Sound pressure field emitted from the weak sound source directly, in free space without RWM system. The detected pressure at the receiving point is 16.1 dB. (b) Sound pressure field of the same weak sound source using RWM system. The detected pressure reaches up to 60 dB.

Note 8. System robustness
System robustness is highly desirable for the practical realization of relevant functional devices. We have studied the performance of the RWM system in the condition of large obstacle or random scatterers. The results show good robustness against various variations and objects. and S9(c). Without the RWM device, the sound intensity is extremely low at the receiving ends as the obstacle blocks the transmission of most sound waves. On contrary, for the result with RWM system, the emission efficiency is improved by the meta-cavity at the launch end and the sound energy is strongly localized into the receiving area by the meta-cavity at the receiving end.
We also examine the robustness against the scattering layer and the results are shown in Fig The maximum enhancement reaches 293, which is even higher than the free space case. The sound pressure field distributions shown in Figs. S9(e) and S9(f) intuitively confirm the distinct sound pressure fields especially at the receiving end.
For comparison, Figure S10 shows the enhancement spectrum / of the system consisting of Helmholtz resonators in the condition with and without the large size solid obstacle. The obstacle has a significant negative influence on the performance of the system as the maximum amplification reduces by 66%. Thus, the proposed Mie meta-cavities also exhibit the superiority of better robustness compared with classic acoustic Helmholtz resonators.  Enhancement spectrum / in the condition with and without a large size solid obstacle.

Note 9. Multi-target remote whispering
As shown in Fig. S11(a), receiving regions Ⅰ, Ⅱ, and Ⅲ and the point source in the center are covered by proposed Mie resonance meta-cavities. The volume flow rate per unit length out from the source is /157 and the distances are 1.3 m, 0.5 m and 3 m for illustration. Figure S11(b) gives the sound pressure field radiated from the weak sound source using the multi-target remote whispering. The detected sound pressure level in the center of regions Ⅰ, Ⅱ, and Ⅲ are 60.6 dB, 63.2 dB and 57.7 dB, which is significantly higher than the average value of 38.3 dB in surrounding environment. Thus, we can amplify signals in several designated locations while keeping the other space quiet in real time using the proposed scheme.

Note 10. Overcoming the conventional acoustic detection limit
The volume of the noise speaker in Fig 5(a) was adjusted to control the signal-to-nose ratio (SNR) of the detected signal. Figure S12 gives the comparison of the SNR of the signal measured in free space and in the RWM system. The blue solid line is the fitted curve with a slope of about 116, indicating more than 116 times enhancement (that is, about 20.7 dB) of SNR achieved in the RWM system under different background noise conditions, compared with that obtained in free space. This will enable the RWM system to detect weak acoustic signals below the detection limit of a conventional acoustic sensor (that is, SNR<1 in free space).
Figure S12: SNR measured in the RWM system compared with that obtained in free space.
The input signal from the speaker was a series of Gaussian pulses with a center frequency of 563 Hz and a band width of 75 Hz. In the gray highlighted zone, the simulated free-space SNR is < 1, indicating that the input signal is below the detection limit of conventional acoustic detection system.

Note 11. Anti-interference remote whispering in an extreme case
As shown in Fig. S13(a), we also investigate the performance of the proposed RWM system in an extreme case when the strong external interference source is placed between the sound source and the receiving location. The distance between the noise and the receiving location is set as 0.325m, which is only a quarter of the distance between the sound source and the location. Figures S13(b) and S13(c) show the time domain and frequency domain pulse signal measurements in free space (top) and in by using the RWM (bottom). Compared with the results in Fig. 5(b) and 5(c), the influence of the noise is stronger in free space while the detected signal in RWM system does not suffer from any obvious distortions.

Note 12. High transfer efficiency for sound signal & information
The proposed RWM device can exhibit valuable high efficiency from the viewpoint of signal & information transfer and detection. Concerning such matter of efficiency, please note that the SNR obtained by RWM could be significantly higher than that obtained by a bare receiver of the same surface area.
To corroborate this claim, we have investigated the SNR by taking the receiver area into account. As shown in the schematic diagrams in Figs. S14(a) and S14(b), we calculated the SNR of the sound signals received by a standalone small receiver with a diameter of 0.64 cm (the same size as the 1/4-inch microphone used in experiments) and a standalone fictitious large receiver with a diameter of 8 cm (the same size as the Mie resonator), respectively, all in the absence of the metamaterial. Here, a series of Gaussian modulated sinusoidal pulses (blue) is generated from the source speaker to mimic a whispered weak sound signal, which is mixed with broadband white noise (yellow) radiated from a strong external interference source. The middle and bottom row in Fig. S14 show the corresponding calculated time domain and frequency domain signals. It can be seen that the size of the receiver has no obvious effect on the quality of the receiver signal, in spite of the obvious amplitude differences that however carry no additional information. Increasing the size of the receiver from 0.64 cm to 8 cm only achieves a minor SNR improvement of about 0.035 dB, which is negligible in practice. For comparison, the unique enhancement effect can be achieved by the small receiver enclosed by our RWM system subject to the same source signal and noise, as shown in Fig. S14(c). Note that the RWM brings an SNR improvement of about 20.7 dB, which signifies a remarkable increase. This SNR improvement originates from the enhanced directionality and frequency selectiveness induced by the monopole-monopole interaction of two Mie-resonators, which is highly efficient and NOT supported by a receiver of arbitrary surface area.
Additionally, to further elaborate on the high efficiency of SNR enhancement by our RWM compared to the bare receiver, Fig. S15 presents the distinction between the SNR of the signal obtained by our RWM system with a small receiver and the SNR obtained in free space by a large receiver. The blue solid line is the fitted curve with a slope of about 116, which indicates a 116 times enhancement (about 20.7 dB) of the RWM SNR under different background noise conditions, compared with the SNR of the same small but bare receiver. This enhancement enables the RWM system to detect extremely weak acoustic signals far below the detection limit of a conventional acoustic receiver (that is, SNR < 1 in free space), which exhibit high efficiency in signal & information transfer and detection. On the contrary, the purple solid line has a slope of only 1.008, indicating almost the same SNR obtained by both the large area receiver and small area receiver without using the metamaterial.
In conclusion, we have clearly demonstrated a unique and substantial improvement of the SNR which is not achievable by an arbitrary regular receiver, confirming that the practical prospects for remote whispering sound transfer are extremely efficient and realistic beyond its physics related beauty.  , where = atmospheric pressure in unit of kPa and = ambient temperature in unit of ℃.
Thus, the characteristic impedance of air, , decreases with an increment of .
To elaborate on the achievable reconfigurability of RWM, Fig. S16 shows the adjustable range of the operation frequency, which can be tuned from 534 Hz to 574 Hz when changes from 10 ℃ to 30 ℃. Note that even a wider adjustable range can be obtained by further changing the value of and/or .

Note 14. RWM system for underwater sound
Full-wave numerical simulations are conducted to characterize the acoustic response of a RWM system for underwater sound. As shown in Fig. S17(a), the external surface of each metacavity is covered with a rubber coating layer to isolate the air medium inside the structure from exterior water background (in order to avoid the serious viscous effect and thin-wall vibration when the meta-cavity is filled with water instead, while significantly lower the working frequency so as to make the meta-cavity in deep subwavelength scale). The simulation is performed by the acoustic-solid interaction mode in COMSOL to characterize the intensive interaction between fluid and solid. For illustration, the transfer distance is set to 100 m, while the inner and outer radii of the rubber layer are only 4 cm and 4.5 cm, respectively. The Young's modulus and Poisson's ratio of the rubber layer are set as 0.6 MPa and 0.49, respectively. Figure   S17(b) presents the simulated transfer enhancement PSD/P0 of the proposed underwater RWM system. The spectra exhibit a maximum enhancement peak of 12.97 at the extremely low frequency of 26.5 Hz, and the corresponding sound pressure field is shown in Fig. S17(c). The enhancement of the detected signals and the reduction of the ambient sound leakage can also be observed.

Note 15. RWM system for electromagnetic signal
Moreover, our presented approach can in fact be extended beyond the sole use for acoustic wave transfer. Under the same principle, simulations of the RWM but for electromagnetic (EM) signals are shown in Fig. S18, which depicts how the received EM signal can be substantially enhanced with co-existing dipolar or quadrupolar modes in RWM system (blue solid curve) than ordinary detectable signals in free space (red dashed curve).