Detecting acoustic chirality with matched metamaterial vortex wave antennas

Acoustic vortex waves have recently emerged as a promising avenue for signal modulation in acoustic communications, and a variety of recent reports are approaching vortex wave generation from different, unique directions.^1–4 As illustrated in Fig. 1, vortex waves form a helical phase front, characterized by the wave chirality, or phase rotation per unit propagation length.⁵ These waves can take on a large (theoretically infinite) range of chirality values. As propagating harmonic excitations that carry both angular and linear momentum,⁶ they have been shown to be interesting in a variety of contexts, from enhanced imaging⁷ to particle manipulation,⁸ as well as in communications.^1,2,9,10 Because integer changes in wave chirality, or mode number, form an orthogonal basis set, this provides the potential to increasing information transfer rates with acoustic waves by modulating the angular mode number, accessing a spatial basis set for communication at low acoustic frequencies.

Many approaches for generating acoustic vortex waves have been presented in the literature. These typically fall into three categories: direct generation with source arrays,^1,8,11 coherent scattering from wavelength scale surfaces,^9,12,13 or sub-wavelength metamaterial approaches.^14–23 Previous demonstrations using acoustic vortex waves to increase data channel capacity have mainly relied on arrays of sources and receivers.^1,24–27 Shi et al. demonstrated acoustic vortex wave communication by using arrays of up to 64 active transducers to send and receive vortex wave signals. Vortex wave communication using active arrays involves a transducer array, coded with a digital processor to generate interference patterns for each vortex mode, including modulation of both phase and amplitude of each transducer independently. Once the vortex signal reaches the receiving array, the amplitudes and phases of each receive transducer are processed to measure the vortex charge. While that method is effective and accurate, it requires complex electronics and signal processing to code and de-code the vortex wave chirality and mode number of each vortex wave. The high electronics investment is also costly, both in terms of financial investment and assembly time of electronics components.

Recent work using acoustic metasurfaces has shown a promising avenue for building vortex wave systems while limiting the total number of needed sources.^3,16,25 These metamaterial approaches are good solutions to the electronic complexity seen in other types of vortex wave generators, but no authors have yet demonstrated both emission and detection using a metamaterial antenna capable of generating multiple modes with a single aperture. A design by Guo et al.²⁸ demonstrated vortex wave detection using a series of metasurfaces for demultiplexing using metamaterials. Alternatively, a geometrically simple design by Pilipchuk et al.²⁹ generated vortex waves by utilizing cylindrical resonances and breaking axial symmetry with high efficiency but did not allow for multiple vortex modes in a single device.

In this work, we demonstrate chirality transfer between two metamaterial vortex wave antennas, generating and sensing transmitted acoustic wave chirality with a small set of four omnidirectional microphones. Our metamaterial vortex antenna is a sub-wavelength geometry based on a pair of acoustic leaky wave surfaces³⁰ decorating a ring resonator waveguide. This analog emitter radiates and detects both positive and negative integer mode chirality. Through computational simulation and experimental verification, using three-dimensional (3D) printed waveguides, we show that the vortex mode chirality can be transferred between two opposing acoustic vortex antennas in a “pitch–catch” configuration. This opens potential use of the designed metamaterial vortex wave apertures as chiral data channel sources and receivers. In Sec. II, we describe the design of the coupled antennas, including a review of the leaky wave and ring resonator concepts that form the basis for the design. Section III describes finite element analysis (FEA) of the performance of the antennas both as individual units and as a coupled pair. Experimentally measured performance of the chirality detection mode is presented in Sec. IV. Finally, Sec. V summarizes the results and discusses opportunities and hurdles of implementation of this system in acoustic communications.

In previous work, we presented a method of generating acoustic vortex waves based on a metamaterial aperture and a single sound source.³⁰ The approach involved wrapping a one-dimensional (1D) leaky wave antenna (LWA) about a central axis to form a circular, radiating surface, as depicted in Figs. 1(a) and 1(b). A LWA is a type of waveguide antenna that loses energy to its surroundings, with designed directional dispersion.^31,32 For guided acoustic waves in air, this leakage can be accomplished by adding sub-wavelength holes (ports) to the top of a rigid, hollow, guide channel. In a linear LWA [Fig. 1(a)], the propagation direction of the resulting radiating wave, $θ ( f )$⁠, is given by the dispersive ratio of the interior guided wavevector, $β x ( f )$⁠, to the free space propagation wavevector, k₀: $sin θ = β x ( f ) / k 0 ,$ where $k 0 = 2 π f / c air$⁠, with f being the excitation frequency, and $c air = 340 m / s$ is the speed of sound in air.

Our curved LWA approach, illustrated in Fig. 1(b), maps the linear leaky wave output onto a rotational propagation, forming a vortex wave with helical planes of consent phase whose pitch angle is given by $θ ( f )$⁠. The harmonic pressure fields of these waves have the approximate general form⁹  $p ( r , ϕ , z ) ∝ J L ( k 0 r ) e i L ϕ + k z z$⁠. This pressure field is propagating in $z ̂$⁠, radially modulated with an Lth order Bessel function, $J L ( · )$⁠, with the additional azimuthal phase term $e i L ϕ$⁠. The phase field of the radiated pressure for the ideal vortex wave antenna (VWA) is shown in Fig. 1(c) and can be compared to the phase field for a Bessel function source in Fig. 1(d). The Bessel function shows a perfect null at the source center. In comparison with the VWA, phase field does not have a perfect null point.

The design implements a discrete port cross section, as opposed to a single leaky slit, with dimensions (⁠ $X ′ · W ′$⁠), as shown in Fig. 2(c). Discrete ports decrease the VWA radiation efficiency,³⁰ which was a design choice made to ensure that significant acoustic energy would remain in the waveguide and fully circulate in the waveguide before radiating.

Using Eq. (2), it is possible to determine the relationship between frequency and mode number for the designed geometry, and that relationship is plotted in Fig. 3. Figure 3 also shows the relationship between the dispersion and radiated pitch angle, $θ ( f )$⁠. Since these relationships are controlled by the exact port and waveguide geometry used, changes to those geometries will affect the propagation direction. Frequencies of integer mode numbers are read directly from the results of the FEA simulations.

Additionally, using the same Eq. (2) approach, it is possible to predict the mode numbers for different geometry variations, such as vortex radii, as shown in Fig. 4. Understanding of the ring radius to frequency relationship is an important design factor if considering the multiplexing approach of nesting rings of various sizes. The vertical line at 4 kHz indicates that several simultaneous modes are possible at a single excitation frequency by varying only the radius.

FEA (COMSOL Multiphysics) is used to calculate the radiated acoustic pressure field, inside and outside the VWAs, predicting the chirality detection emitted from one VWA received by the other VWA. FEA is also used to calculate the radiated pressure efficiency. For simulation, all boundaries are modeled as sound hard (rigid) except the top of the VWA ports [colored yellow in Fig. 2(c)]. The gap, g, between the transmit and receive VWAs was varied from 10 to 400 mm. In this figure, only the interior air channels of the waveguide are shown. An input plane wave field is created at the waveguide input port (labeled “In,” colored red), and a plane wave radiation boundary condition allows non-reflecting termination of the wave at the output port (labeled “Out,” colored green). The air volume outside the simulation region (not shown) is surrounded by an absorbing, perfectly matched layer domain, simulating an open system outside the waveguide.

The presented antenna arrangement enables the simultaneous transfer of both positive and negative chirality to microphones B_o and A_o, respectively, from A_i and B_i at a single acoustic frequency. In the A_i source configuration, the $B o − A o$ (blue line) pressure amplitude difference corresponds to the transfer of a CW rotating spiral wave transmitting from the transmit to the receive VWA at microphone B_o.

To show the effect of beam divergence, we include a study of the VWA geometries coupled to a sound hard waveguide. The length of the waveguide, l, is varied in a range scaled ten times larger than in the free space experiment. In Fig. 5, we plot the received amplitude asymmetry for distances of l = 100, 200, and 400 mm. The waveguide effect is to constrain the divergence of the vortex wave energy without altering the transmitted phase. As can be seen, in this case, the transmitted asymmetry is unaffected by the distance between the two VWAs.

The absolute value of the radiated pressure field and phase, 10 mm above a single VWA for the integer mode frequencies indicated with arrows in Fig. 5, is shown in Figs. 6(a)–6(d). The pressure fields show the formation of a central acoustic null, indicative of a vortex/Bessel wave, and the phase plots show the clear, well-defined spiral phases for L = 4, 5, 6, and 7, with corresponding numbers of $L · 2 π$ phase shifts. The pressure field inside the waveguide is plotted on the bottom row of Fig. 6 showing uniform phase around the waveguide and between the VWA and coupling arc. The radiated efficiency of a single vortex antenna, determined from the FEA-predicted radiated pressure field at 0.1 mm above the antenna, is found to be –8.8 dB.

The absolute values of the radiated pressure field and phase, 10 mm above a single VWA, are shown in Figs. 7(a)–7(c) for the frequencies indicated with dots in Fig. 5(b). These frequencies are selected to show the nature of the radiated acoustic field, below the L = 6 ring mode resonance (5450 Hz), just above the vortex mode resonance (5710 Hz), and at the transmission valley between resonances (6000 Hz). For these cases, the plots show azimuthal nodes in the pressure field due to phase mismatch between the ring resonator modes and the circularity of the radiating fields.

The lack of well-defined chirality seen in the phase plots of Figs. 7(a) and 7(b) result in the equal pressure fields at both output microphones seen in Fig. 5. At 6000 Hz, we see the reversal of chirality in the ring resonator. This reversal of the mode sign is also reflected in an unbalanced output at the receiving microphones. In Fig. 8, we plot the FEA calculated acoustic energy flow direction using the instantaneous intensity, $I = p v$⁠, inside the transmit and receive VWA waveguide mid-planes. The off resonance frequencies of Fig. 8(a) show an opposing flow, standing wave excitation, inside the transmit VWA and nearly balanced energy flow at the receive ports, A_o and B_o. Generation of the L = 5 mode at 5570 Hz in Fig. 8(b) results in efficient CW rotation of the energy flow with the transmit VWA ring resonator and corresponding increased signal on the receiving B_o port.

Detuning from the ring resonator resonance condition of Eq. (2) to higher frequencies, such as in Fig. 8(c) at 6000 Hz, leads to a reversed phase matching condition inside the transmit VWA resonator. The detuning of the driving frequency above the ring resonance condition leads to phase reversal above resonance, typical of all driven harmonic oscillators.³⁴ Although the off resonance driving condition is not efficient, it swaps the direction of energy flow within the VWA ring resonator and produces the CCW vortex wave. This results in weakly enhanced transmission to the receiver A_o output port. For a more complete discussion of counter-propagating modes in ring resonators, the reader is referred to Manolatou et al.³⁵ 

Mode purity is an important metric when assessing how well an aperture produces vortex modes. In this case, we define mode purity as the degree to which, for a given desired mode value, the produced vortex has phase that corresponds to a pure OAM mode with phase term $e i L ϕ$⁠. By implementing the approach used by Jack et al.,³⁶ which uses a Fourier transform analysis to decompose individual OAM modes, we find mode purity values of 84.4%, 80.2%, 91.5%, and 85.9% for modes 4–7, respectively. The spectral analysis results from this method are shown in Fig. 9. Since this calculation is based on the finite element predicted waveforms, only positive mode values are shown since negative mode values would be identical due to symmetry of the simulation.

To experimentally test our design, the geometry is converted into 3D digital representation with CAD (Dassault Systemes Solidworks, Waltham, MA) software and physically realized with a PolyJet (J750, Stratasys, Rehovot, Israel) 3D printer as shown in Fig. 2(b). Experimental deviation from the FEA model is made in the form of coupling sections, used to connect 2 m long polycarbonate [25 mm inner diameter (ID)] tubes. These input waveguide tubes are used to move the source speakers (CE32A-8, Dayton Audio, Springboro, OH) physically far from the VWA to allow for time gating of the transmit and receive signals. Two source speakers are placed at the ends of the polycarbonate tubes, as the A_i and B_i inputs on the transmit (front) side of the setup shown in Fig. 2(b). Two receiver microphones (type 4187, Brüel and Kjaer, Naerum, Denmark) are placed 300 mm from the VWA ports on the receiver side VWA. The A_o microphone is highlighted in Fig. 2(b). Using this setup, a chirped waveform (6 ms duration, 2.5–10 kHz bandwidth) is launched from either the A_i or B_i speaker, and gated time series are collected simultaneously from the receiver microphones A_o and B_o. Figure 10 shows the measured vortex wave detection with comparison to the FEA-predicted results. Although the experimentally measured data are clearly less uniform, the major features are all found in both simulation and experiment.

Figure 10 is a plot of the received pressure amplitude difference between microphones A_o and B_o as a function of input frequency. The designed VWA preferentially transfers sound to the microphone located in the direction of the radiated vortex chirality, with good agreement between the finite element results and the data. Two experimental input configurations are shown, with each of the two sources activated as illustrated separately in Fig. 2(b).

Correspondingly, a CCW vortex wave propagates between the two VWAs in the B_i source configuration, preferentially delivering increased pressure amplitude to the A_o (red line) microphone at vortex wave resonance frequencies. We attribute deviations from the idealized transfer functions of the FEA model, with its clean symmetric line shapes, to experimental error, specifically, the impedance coupling conditions at the junction of the VWA and the polycarbonate waveguides, limits on 3D printing fidelity, and non-uniform microphone transfer functions.

The salient feature is the presence of distinct transmission difference peaks at 3970, 4760, 5570, and 6380 Hz as plotted in Fig. 6. Notably, these frequencies are slightly different from those predicted by Eq. (2), and this is due to the simplified assumptions made in Eq. (2), such as no end correction on the radiation from the antenna ports. The finite element simulation is used to calculate the detection efficiency of OAM waves using the two antennas. Although the efficiency is frequency-dependent, the average detection efficiency of the integer mode frequencies is roughly −20 dB.

In Fig. 11, we show the measured change in the amplitude difference spectra as the two VWA are moved apart from 10 to 40 mm. Although we note a significant reduction in the received amplitude, the chirality transfer is still evident. Indeed, the SNR of the reduced signal at g = 40 mm is still near 10 over the measured frequency range. This reduction in collected energy is attributable to divergence of the acoustic beam. The overall size of our antenna aperture, 58 mm, is comparable to the acoustic wavelengths (50–110 mm). This small aperture size leads to a fast divergence of the beam energy. Metamaterial-based vortex emitters with large apertures⁴ have recently been shown to emit vortex waves over long distances with high fidelity.

We have shown that we can improve the mode quality of an acoustic vortex antenna compared to our previous leaky wave metasurface previous design,³⁰ producing integer mode vortex waves. This was done in an analogous fashion to the generation of optical vortex waves in optical ring resonators, by adding an interior evanescent coupling port between an acoustic waveguide and a closed loop, ring resonator. The ring resonator was decorated on its top with a LWA such that high quality angular momentum modes with mode number $L = ± 4 , ± 5 , ± 6 ,$ and ±7 were radiated into the half-space above the VWA. Our computational and experimental results show successful generation of both positive and negative chirality integer vortex modes based on source direction and phase matching conditions of the driving and resonant field. The chiral mode information was transferred between identical, 3D printed “source” and “receiver” VWA over sub-wavelength, free space gaps.

Finally, although vortex waves are promising for applications in acoustic communications, there are practical limitations for using the current setup as-is for that application. The first limitation is the rigid boundary assumption in the design of the VWA, which is only applicable for an air-acoustic antenna and cannot be directly translated to work in water. It may be possible to implement other designs, such as those explored by Broadman et al.,³⁷ to achieve underwater vortex waves. The second key limitation of the design presented here is that the size of the antennas used causes very divergent beams, making them impractical for communication over any significant distance. Additionally, from a propagation standpoint, vortex waves are not stable in complex environments³⁸ or when interacting with boundaries,³⁹ although that may be a hurdle that is overcome by careful design and understanding of the environment.

This work was supported by the Office of Naval Research. The authors have no conflicts of interest to report. The data that support the findings of this study are available from the corresponding author upon reasonable request.