Breaking acoustic reciprocity is essential to create robust one-way propagation where sound waves or elastic vibrations are permitted to travel in only one direction. This unidirectional response forms the basis for devices such as acoustic isolators and circulators, and it also unlocks new functionalities for complex systems such as acoustic topological insulators. After reviewing the principles of acoustic reciprocity, we look at techniques to achieve large reciprocity breaking, including nonlinearities, moving media, spatiotemporal modulation, and nonlinear bianisotropy. We then discuss the recent surge of progress in nonreciprocal surface acoustic wave devices and topological acoustic systems, areas which we predict will continue to flourish in the coming years. We anticipate that these and other applications of nonreciprocity will continue to enhance acoustic technology and form the basis for new acoustic devices. Reciprocity is a fundamental principle in wave manipulation, and techniques for breaking its symmetry will continue to be discovered, refined, optimized, and applied to several acoustic domains as the understanding of the underlying principles and new technologies mature.
The study of acoustics explores the excitation, propagation, and measurement of acoustic waves in natural and engineered environments. Among these, the study of acoustic propagation has flourished in the past 20 years, in large part due to the emergence of acoustic metamaterials,1–3 which have greatly extended our ability to manipulate sound waves. Early studies focused on the exploration of reciprocal functionalities. In this scenario, acoustic metamaterials, no matter the complexity of their structure and no matter how counterintuitive their routing of acoustic waves may be, are subject to the rule of reciprocity. That is, if we interchange the location of the sound source and the receiver, the received signal will always be the same, regardless of the microstructure of the material. This general rule is satisfied in both time and frequency domains.
The concept of reciprocity is easily confused with time-reversal symmetry, as they are both symmetries between two different wave fields with strong relations with each other. Time-reversal symmetry can be understood by visualizing a video of an acoustic wave played backward. When this backward-scrolling video is an allowed physical process (that is, it obeys the relevant laws of physics), time-reversal symmetry is intact. This global reversing of the wavefield in a region of space is the central focus of the field of time-reversed acoustics.4 For example, a time-reversed acoustic experiment may seek to create a converging spherical wave, the time-reversed version of a point source, where an array of sources is used to create the converging wave with a focus at the location where the original source would reside. Studies in acoustic reciprocity differ in that they are typically concerned with the symmetric relation between a source and a receiver. That is, instead of examining the relationship between a forward and a fully-reversed wave field, reciprocity establishes a relationship between field amplitudes and phases at locations or ports in wave fields that may appear globally quite different from each other. There is also the difference that strict time-reversal means that all involved physical processes must be reversed, whereas reciprocity is typically concerned with reversing propagation features. This difference is apparent in the presence of material loss, which is the most common example in which reciprocity and time-reversal symmetry diverge. When an acoustic wave travels through a lossy medium, the wave energy is dissipated through countless microscopic collisions of the particles in the medium. When time is reversed (i.e., the movie is played backward), the inverted collisions perfectly come together to create a gain for the reversed wave. The movie looks unrealistic because it shows an unlikely local spontaneous decrease in entropy, but the Newtonian equations describing the collisions have not been violated. In reciprocity, sources and receivers are swapped, but the material loss can still be present, implying that waves are equally attenuated in either direction, as it would occur in a realistic experimental setup. In a reciprocal system, transmission in opposite directions is still identical, even though the lossy channel obviously does not obey time-reversal symmetry as the signals decay in both directions. It is for this reason that, compared with time-reversal symmetry, reciprocity is often considered a more general phenomenon in wave engineering.5
The reciprocity principle was originally introduced by Helmholtz in 1860.6 It concerns the definition of acoustic reciprocity where, by interchanging the sound source and the receiver, both the amplitude and phase of the received signal remain the same. This general notion of reciprocity can be illustrated by considering the very familiar example of human communications. We assume that if we can hear someone talking to us, they should be able to hear us if we talk back to them with the same volume. We do not expect the shape of the room or any nearby sound absorbing materials to change this property, which is an intuitive understanding that reciprocity holds even in the presence of spatial inhomogeneities and linear dissipation.
Acoustic reciprocity may also involve reciprocal couplings of the acoustic field with other phenomena, for example, electromagnetic waves. Many acoustic transducers used for sound generation and reception are based on the coupling of acoustic and electromagnetic phenomena in piezoelectrical materials. This coupling obeys reciprocity, hence, if the electrical and acoustic signals are interchanged, the piezoelectric and electric-piezo switching efficiencies are identical. With this property, an acoustic transducer can be utilized as both a sound source and an acoustic sensor.
The photoacoustic (or optoacoustic) effect, based on which ultrasound waves are generated by the thermal expansion of a material as it absorbs light, is another situation in which acoustic propagation involves other phenomena. Interestingly, reciprocity does not hold for this effect. Photoacoustic waves are generated by light absorption in a material sample, causing thermal expansion, and then resulting in ultrasound propagation.7 The reversed coupling effect, i.e., generating light by sound waves, has been rarely reported due to thermal phenomena being involved in the coupling. Actually, the reciprocity principle can be broken in several other situations. Aside from nonreciprocal coupling with other phenomena, any kind of acoustic nonlinearity or bias through moving media will result in nonreciprocity. In fact, the acoustic wave equation itself is nonlinear, indicating that acoustic propagation can become nonreciprocal as the intensities grow in the presence of geometrical asymmetries.8–11 Only in the small signal approximation can the wave equation be considered linear and admit a reciprocity principle.
Biased moving media, such as air currents between two speakers, also breaks the symmetrical transmission of reciprocity as sound propagation in the upstream and downstream directions is different. So why do we adopt acoustic reciprocity in our daily life? Why, when conversing with others, do we generally not notice any asymmetry? Because in most scenarios, the acoustic pressures involved are indeed very small and because the speeds of any nearby breeze are often tiny compared with the speed of sound. The resulting small amounts of nonreciprocity are too weak to be observed or matter. The aim of nonreciprocal acoustic wave engineering, then, is to strengthen these effects by developing techniques to enlarge nonreciprocity for small pressure amplitudes and slow-moving media. While any amount of nonlinearity or external modulation breaks the exact reciprocity symmetry, in practice systems must be carefully designed to achieve an effect large enough to be useful. Two useful metrics in quantifying nonreciprocity are isolation, commonly defined as the ratio between forward and backward transmission, and insertion loss, which is how much signal power is lost in the forward transmission due to the presence of the nonreciprocal element. We will first review statements of reciprocity in acoustics, then discuss techniques for large reciprocity breaking, highlighting some of the promising applications for nonreciprocity and discussing future possibilities for this vibrant field of research.
A. Example reciprocity statements in acoustics
A reciprocity relation is any symmetrical relation between field quantities when sources and receivers are interchanged. For example, suppose there is a monopole source distribution mi and a dipole source distribution di in the acoustic field (pi, vi), where i can be 1 or 2, indicating acoustic fields generated by source distributions 1 and 2, respectively. In the frequency domain, the sound equations can be written as
After some mathematical manipulation, we can arrive at the reciprocity relation
Suppose there is a monopole source located at and another monopole source located at . Substituting these into Eq. (3), we get
If we assume , Eq. (4) indicates that the acoustic pressure at produced by a monopole source at will be identical to the acoustic pressure at produced by a monopole source at , as sketched in Fig. 1(a). If there is instead a dipole source located at and another dipole source at , from Eq. (3), we get
This reciprocity relation is illustrated in Fig. 1(b). If we suppose and both of them are in the x-direction, we could infer , indicating that the x-component of the particle velocity at produced by an x-directed dipole source at will be identical to the x-component of the velocity at produced by an x-directed dipole source at . Other forms of reciprocity follow similar derivations, for example, a statement of reciprocity in an elastic string, where the induced displacements due to interchanged loads are equal [Fig. 1(c)], and in a general scattering formalism, where the reciprocity statement implies that the resulting scattering matrix is symmetric [Fig. 1(d)].
B. Nonreciprocity research in electromagnetics
Research on reciprocity breaking in electromagnetics began earlier than in acoustics in part because magneto-optical materials, which can strongly break reciprocity through the application of a dc magnetic bias, are relatively common in nature. Hence, the earliest nonreciprocal wave devices were mostly based on ferrites and found application in both microwave engineering and optics. The principle of magneto-optical nonreciprocity is to utilize the biasing static magnetic field to influence the electron cyclotron orbiting motion inside the material, which nonreciprocally affects electromagnetic wave propagation through the material. This is the working principle of the earliest electromagnetic waveguide circulators and isolators.12,13 However, nonreciprocal devices based on magneto-optical media are usually bulky and large because the underlying phenomena are rather weak, which prevent on-chip integration.
Later, several magnetic-free approaches were proposed to break reciprocity. Inspired by the biased magnetic field to break reciprocity, an effective approach was to use dynamic modulation to create an effective magnetic field.14 In this technique, although there is no real magnetic field biasing the structure, a gauge field is formed that mimics the magnetic field and can manipulate light nonreciprocally.15 Spatiotemporal modulation on the basis of resonant systems was then proposed by mimicking angular momentum as a way of replacing the magnetic bias, which removes the degeneracy between opposite resonant states, to realize giant nonreciprocity with modest requirements on the modulation frequency and amplitude. This approach was first applied by modulating effective material parameters,16 and it was later introduced into circuit designs17 and finally realized in on-chip integration.18 Other approaches to break electromagnetic reciprocity are based on nonlinear materials combined with geometrical asymmetries.19–21
II. BREAKING RECIPROCITY IN ACOUSTICS
Similar techniques have found success in breaking acoustic reciprocity. In general, reciprocity theorems cease to apply in the presence of effects that violate the underlying assumptions of linearity, passivity, and time-invariance, though their presence does not guarantee broken reciprocity. For example, as sketched in Fig. 2(a) for an acoustic wave, reciprocity may be broken as the wave passes through a nonlinear medium, but only if there is an accompanying spatial asymmetry. Aside from nonlinearity, reciprocity can also be violated by the presence of any quantity that is odd under time reversal, meaning that their sign changes with a change in sign of the time coordinate (the position of the particle is time even—the length quantity is unchanged by the time reversal; the velocity of a particle is time odd—the time reversal switches its direction). For this reason, reciprocity does not hold when an acoustic wave is traveling through moving media, such as an ocean current. However, reciprocity is restored if the time-odd quantities are reversed along with the source and receiver positions [Fig. 2(b)]. Magnetic fields, which arise due to moving charged particles, break reciprocity unless the direction of the charged particles (and hence the direction of the magnetic bias) is also reversed [Fig. 2(c)]. And reciprocity only holds for nonreciprocal media, where the homogenized material properties arise from hidden sources of motion such as gyroscopic spinning as in Fig. 2(d), when the directions of the hidden motions are reversed. As examples of techniques to break reciprocity, we now discuss nonlinearity, moving media, spatiotemporal modulation, and bianisotropy in greater detail.
In principle, the acoustic wave equation is nonlinear, and strong nonlinear features may be observed as the intensities grow. Nonlinearity in acoustics can introduce frequency harmonics, seen in the time response as waveform distortions, and an amplitude-dependent response, which results from effective material properties that vary with the amplitude. The systematic study of nonreciprocal acoustics started by looking at nonlinearities. In 2009, by combining a phononic crystal with nonlinear materials, the so-called acoustic diode was proposed to realize sound energy isolation.8 The idea is to place a phononic crystal adjacent to a nonlinear material so that the phononic crystal is able to block sound waves at the fundamental frequency while allowing transmission of the second harmonic. If the acoustic wave encounters the phononic crystal first, it is totally blocked. But, when incident from the opposite direction, the wave passes instead through the nonlinear material first where some of the acoustic energy is converted into the second harmonic and is then free to pass through the phononic crystal. Due to the effective filter function of the phononic crystal, the transmission asymmetry for this system can be very high. However, since there is no phase-matching in the system, energy conversion from the fundamental frequency to the harmonic is very limited, resulting in a comparatively low working efficiency. This proposal was implemented experimentally by utilizing the high nonlinearity of ultrasound contrast agent microbubbles.9 To overcome the low efficiency in energy conversion, an active approach was then proposed.10 Here, Helmholtz resonators were used for frequency filtering in place of a phononic crystal and a loudspeaker was used in place of a nonlinear media. In 2011, a different approach based on bifurcation-based acoustic switching was proposed for sound isolation.11 Similar designs have included modulating geometric nonlinearity22 and utilizing nonlinear interfaces.23 These nonlinear approaches to nonreciprocity do not require frequency shifting and energy conversion, and hence are able to operate at the fundamental frequency. However, due to the involved nonlinearities, waveform distortion is unavoidable.24,25
B. Moving media
Nonreciprocal devices based on nonlinearity suffer from several drawbacks. First, the working properties are generally amplitude dependent. Second, waveform distortion is always involved due to media nonlinearity. Third, to obtain large levels of transmission asymmetry, the operation amplitudes need to be relatively high. And, fourth, there is possible degradation in performance when both forward and backward waves are present at the same time, a problem known in photonics as dynamic reciprocity,26 with a trade-off between forward transmission and nonreciprocal intensity range.19 These drawbacks have encouraged research into alternative routes to acoustic nonreciprocity.
Nonreciprocal phenomena are commonly encountered in underwater acoustics and aerodynamics, as in the case of large ocean currents or high-speed airflow. Giant nonreciprocity typically requires very high moving speeds and/or very long distances to accumulate the nonreciprocal effect. However, in 2014, by introducing biased angular momentum into a circular resonator with sharp resonance, giant acoustic isolation was achieved with a slow airflow in a subwavelength scale metamaterial.27 The introduced angular momentum split the resonance mode for clockwise and anti-clockwise waves into two different frequencies, mimicking the Zeeman effect in quantum mechanics. Though in this setup, the frequency splitting is proportional to the biased flow speed, because the resonator supported a very sharp resonance, even a tiny frequency difference could result in large sound isolation. This design was then soon applied in proposals for acoustic Chern insulators,28,29 with an experimental realization recently reported.30 Nonreciprocal acoustic isolators based on the Mach–Zenhder interferometer have also been reported where, by coupling two acoustic waveguides with different moving media (in reality, typically one waveguide is with static media while the other is with moving flow), one can realize constructive interference in one direction and destructive interference in the opposite direction.31,32
C. Spatiotemporal modulation
While biased moving media is a good solution for sound isolation, the working bandwidth is typically narrow and the robustness to dissipation is quite weak. In addition, the moving flow will typically cause undesired additional sound, for example, noise from fans that are used to create a steady flow. Moreover, the moving media approach is practically limited to fluids, which prevent its applicability in elastic wave propagation. Spatiotemporal modulation, which can mimic the effect of moving media, was subsequently proposed. By dynamically changing the effective bulk modulus of three connected resonant cavities, acoustic isolation levels of over 40 dB with insertion losses as low as 0.3 dB were seen in simulations of a noise-free, integrable, frequency scalable subwavelength device.33 In this study, the volume of the cavities was alternately modulated in the clockwise direction, which introduces an effective angular momentum to break the reciprocity among the ports. In another example, by modulating two tightly coupled resonators inside an acoustic waveguide, an isolation factor greater than 25 dB has also been reported.34 This approach has additionally been utilized in nonreciprocal sound transmission between two acoustic resonators.35 Specifically, by introducing an initial spatial phase bias to the space-time modulated coupled resonators, acoustic energy could be always confined in one of the two connected resonators. All above-mentioned spatiotemporal modulations involve changes in the volumes of resonators, which is a modulation of the effective bulk modulus. To instead achieve nonreciprocity through a time-varying effective density, a scheme has been proposed to modulate elastic membranes.36 Using piezoelectric transducers to modulate solid media, similar nonreciprocal transmission has also been reported for elastic waves.37,38
D. Nonreciprocal bianisotropy
In regular acoustic media, momentum change is a function of acoustic velocity and volume change is a function of acoustic pressure. Acoustic bianisotropy, also called Willis coupling, considers the coupling between acoustic pressure and velocity, resulting in momentum and volume change as functions of both acoustic pressure and velocity.39 In Willis media, the mass conservation equation and the momentum equation can be written as40–44
Here, and are the effective bulk modulus and density, while and are the Willis coefficients, quantifying the coupling between acoustic pressure and velocity. The dispersion relation of the above two equations is
The first term with plus and minus signs indicates the reciprocal portion of the wavenumber, while the second term indicates the nonreciprocal portion of the wavenumber. When and but , the nonreciprocal portion of the wavenumber is zero, indicating that the presence of Willis coupling does not necessarily imply that reciprocity is broken. However, when , Willis coupling can result in large nonreciprocity. Based on active Willis materials, broadband sound isolation has been reported in both airborne acoustics45,46 and mechanical waves.47 On the passive side, by modulating both elastic moduli and mass density in space and time in a wave-like fashion, nonreciprocal wave propagation has also been reported.48–50 There is some debate about whether this method of introducing effective moving materials should be considered a passive approach, because it requires extra energy to operate the system. However, this type of modulation only converts the energy in the moving parts while leaving the acoustic energy conserved and therefore it is useful to distinguish it from modulation schemes which explicitly input acoustic energy. A nonreciprocal acoustic lens based on passive modulation of Willis coupling that only images an object from one side has also been proposed.51
Reciprocity is a powerful concept in the analysis, design, and operation of acoustic systems. Knowing that a system obeys reciprocity implies that there are underlying symmetries that can in many cases simplify the analysis of complex systems. For instance, a complex measurement of an acoustic quantity can in many instances be substituted by a simpler one when reciprocity holds. Consider what a driver hears when operating a vehicle. How much of the total noise produced by the vehicle is due to the airborne sound generated by the tires? Answering this question directly could mean placing microphones in the ears of an artificial head in the driver's seat and then deactivating or disconnecting all other sources of sound, such as any structure-borne vibrations. This is no small feat. But the same information can be obtained through a reciprocal measurement. In this case, sound sources are instead placed in the ears of the artificial head and the resulting pressure is then measured at points on the tire surface.52 For many years, reciprocity has served as a useful tool in physical acoustics, including for calibrating microphones,53 for deriving analytical scattering solutions in the presence of a flaw,54 and for inverse parameter estimation in seismic experiments.55
While the absence of reciprocity might make the analysis more difficult, it enables novel wave guiding phenomena, such as in the examples so far discussed. Nonreciprocity finds application in all typical acoustic wave domains: air, water, biological tissues, and solids. We discuss here, in particular, two areas of application that are poised to make lasting contributions in the acoustics community. The first is surface acoustic wave (SAW) devices, which are a promising platform for nonreciprocity because of their widespread industrial and research use. The second application area is topological acoustics, where nonreciprocity is tied to the time-reversal symmetry breaking that is essential to many topological effects.
A. Nonreciprocal SAWs
Waves confined to an interface between two materials are called surface waves and are characterized by an exponential decay of the wave amplitude into the materials. The lower symmetry and tighter energy confinement of surface waves relative to bulk waves have made them a promising avenue for a large nonreciprocal response. In these waves, the matter at the interface undergoes harmonic motion as displacement from equilibrium is counterbalanced by a restoring force. In the case of ocean waves, the displaced water at the water/air interface is restored by the gravitational force. In solid media, the dominate restoring force is the elastic resistance to deformation. Surface waves on solids are called Rayleigh waves, after Lord Rayleigh who described their existence in an 1885 paper.56 In these waves, the solid particles undergo elliptical motion where the size of the ellipses shrinks with depth. In an isotropic solid, these ellipses lie fully in the plane made by the direction of travel and the surface normal. And, in a semi-infinite elastic block, these waves are non-dispersive.
Rayleigh waves are of technological significance as they are the primary type of wave used by SAW devices. In these devices, metal electrodes launch and receive Rayleigh waves on a piezoelectric substrate. Because the Rayleigh wave speed in a typical substrate is a few thousand meters per second, the wavelengths in an SAW signal are many orders of magnitude smaller than they would be if the signal was in the form of an electromagnetic wave. The ability of SAW devices to process signals in a small volume makes them well suited as components of integrated circuits and they are consequently widely used in modern electronic systems (a typical cell phone contains dozens of SAW filters).57,58 SAW devices are also widely used in microfluidics, such as in the manipulation of biological matter, and are also being increasingly utilized in basic quantum research, for example, as a way to controllably transmit single electrons.57 To date, these devices have been largely constrained by reciprocity. But the recent effort has begun to show how this restriction might be efficiently overcome, opening even more possibilities for these microelectromechanical devices. For example, incorporating nonreciprocity could allow a single SAW device to serve dual roles as a filter and an isolator. Nonreciprocity can also help minimize the degrading effect of reflections from the substrate boundaries.
To date, the most studied technique for SAW nonreciprocity is the magnetoelastic effect. The amount of nonreciprocity achieved by the coupling of magnetic and elastic fields is dependent on the relative orientations of the applied magnetic field, the direction of wave propagation, and any special directions dictated by various symmetries of the system such as the crystal structure and material shape.59 The first studies on surface wave magnetic nonreciprocity were conducted decades ago, but only modest effects were demonstrated. In one early study, nonreciprocity was observed in a single-crystal aluminum block where the applied magnetic field caused the conduction electrons to undergo cyclotron orbits, influencing the elastic wave propagation nonreciprocally due to the presence of the symmetry-breaking surface. The nonreciprocity was small, and the setup needed to be kept at very low temperature (4 K).60
Through a recent surge of interest, devices are now being introduced that can obtain large nonreciprocal effects while at room temperature. In a remarkable demonstration, a magnetic SAW device operating at 1435 MHz was recently reported to achieve 48.4 dB of isolation, which is substantially higher than that of commercial Faraday isolators (∼20 dB) in this frequency range.61 The setup, illustrated in Fig. 3(a), consists of SAW waves generated on a piezoelectric LiNbO3 substrate and then sent through a ferromagnetic (FeGaB) bilayer stack where resonant magnetoelastic coupling between SAW and magnetic spin waves induces large dissipation. This magnetically induced interaction is nonreciprocal and, as plotted in Fig. 3(b), depends on the relative angles of the anisotropy direction of the ferromagnetic layer, applied magnetic field (the highest isolation was seen at ∼10 Oe), and direction of SAW propagation. In another study, a similar ferromagnetic bilayer structure was found to achieve high amounts of nonreciprocity over a wide 6 GHz frequency range in simulations.62 Nonreciprocity has also been studied in devices that use Dzyaloshinskii–Moriya interactions in a ferromagnetic/metal bilayer63,64 and in platforms with magnetic nanowires.65,66 Another recent approach was to achieve nonreciprocity through magneto-rotation, as illustrated in Fig. 3(c).67 SAW devices with magnetic layers have also shown promise as magnetic field sensors68 and nonreciprocal spin wave devices.69
While not yet demonstrated experimentally, spatiotemporal modulation is another feasible route to SAW nonreciprocity. A pair of papers, published simultaneously in the same journal, analyzed one possibility of this where an elastic surface is loaded with an array of masses with modulated spring constants, as shown in Fig. 3(d).70,71 Connecting SAW devices to modulating circuit components, such as switches or varactors, is another direction being pursued.72,73
B. Topological acoustics
Nonreciprocal acoustics has also found application in the dynamic field of topological acoustics, which is the study of acoustic systems that can be characterized by quantities (topological invariants) that remain unchanged even when the system is distorted through continuous transformations. Both topological acoustics and its counterpart in electromagnetics, topological photonics, arose from pioneering work on the integer quantum Hall effect in condensed matter physics. In this effect, seen in a collection of electrons constrained to move in a plane and subject to a magnetic field which breaks the time-reversal symmetry, there are electron states that are confined to the outer surface, largely unaffected by the shape of the surface. Analogs to this effect have been shown in acoustics where time-reversal symmetry is broken through the use of a moving fluid or time modulation. If the acoustic system is then excited by a source in a suitable frequency range, the vibrations counterintuitively travel exclusively around the perimeter of the system, though the middle remains open. Figures 4(a) and 4(b) show a realization of this effect in elastic wave propagation on a polyester plate with attached time-modulated piezoelectric disks.74 The spatiotemporal modulation breaks time-reversal symmetry and opens topological bandgaps in the system that support topologically protected modes propagating along the edges of the lattice. Topological effects also arise in passive systems that obey reciprocity, such as the acoustic counterparts of the spin Hall and valley Hall effects. These phenomena are interesting as they exhibit robust propagation effects similar to the integer Hall effect without the need to break time-reversal symmetry, although this means that they are often more susceptible to breakdown due to perturbations that break the needed spatial symmetries in the system.
An exciting recent development has been the experimental demonstration of topological pumping, where an edge state is adiabatically carried through the bulk to the opposite edge. Using a motorized crank to rotate a helical waveguide boundary as shown in Fig. 4(c), researchers transported an acoustic wave from one end to the other of a waveguide using topological pumping.75 The energy is transported from one end of the system to the other as the modal structure evolves from being localized at the right end to being localized at the left end [Fig. 4(d)]. As the pumping is topological, it is robust to perturbations. This is clearly a nonreciprocal result, and energy sent in from one direction is not equivalently carried back in the opposite direction. One limitation of topological pumping is that the transfer of the localized states must occur slowly enough for the adiabatic condition to apply. The speed of the topological energy transfer from one edge state to another can be increased by exploring more complex modulations of couplings, as examined recently with optimal control methods.76 Further progress can be made by tapping into the rich literature on shortcuts to adiabaticity.77
An alternative approach for edge-to-edge pumping is the use of two periodic, yet incommensurate, lattices connected to each other.78 This is demonstrated in Fig. 4(e) where lattices of hollow cylinders share a narrow middle spacer. By translating the top lattice, which is mounted on a sliding groove, a sound wave injected at the left end can be adiabatically transferred through the bulk to the right end. As the top lattice is continuously translated, multiple pumping cycles are completed and a receiver at the right end picks up a series of pulses, one for each cycle completion [Fig. 4(f)]. Unlike other realizations of topological pumping, using an incommensurate bilayer leads to bulk spectral gaps that do not vary with the pump parameter.
Topological acoustics is currently a highly active area of inquiry with new phenomena being demonstrated almost weekly. By no means exhaustive, a list of other nonreciprocal topological advances described in recent articles include a hybrid integer/valley Hall acoustic effect,79 particle pulling using topological chiral modes,80 feedback control as a versatile platform for topological effects,81 and observation of bulk-edge correspondence in a non-Hermitian active mechanical system.82
IV. DISCUSSION AND OUTLOOK
Nonreciprocity in acoustic wave propagation is useful because it dampens the effect of obstacles. Vibrations can travel to an intended destination in spite of perturbations that would otherwise impede their progress. For example, the inclusion of an isolator as part of an acoustic signal chain guarantees that no signal will be reflected back to the source, regardless of what comes after it in the signal chain. In addition, a device with a topologically protected edge mode ensures one-way acoustic propagation even through convoluted material interfaces. This simplifies the design of complete acoustic systems, because it introduces extra latitude in the design—many designs can have the same functionality.
What ingredients are required to violate reciprocity is well understood. But there is still substantial work to be done in developing the most effective nonreciprocity recipes for various applications. A feel for this large variation in potential applications can be obtained by considering three common scenarios of engineered acoustic propagation and the frequency ranges involved. First, airborne acoustics, where most devices deal with frequencies that fall in the range of human hearing (20–20 000 Hz), though higher frequencies are also routinely utilized in devices such as ultrasonic vehicle detectors. Second, biomedical ultrasound, where vibrations traveling through complex biological tissue fall in the 2–15 MHz range for diagnostic imaging and 0.25–2 MHz for high-intensity heating or ablating of tissue. Third, underwater sonar, where common working frequencies cover a range from about 1 kHz to 1 MHz. The large range of length scales and material properties at play in applied acoustics indicates that a diverse range of strategies for efficient nonreciprocal phenomena is needed. In this regard, it can be useful to consider which nonreciprocity techniques have or have not yet been applied in different acoustic wave platforms. Examples of these combinations are shown in Fig. 5. Ideas for future work can be found by considering combinations of techniques mentioned in the top row, representing different mechanisms to achieve nonreciprocity paired with the acoustic platforms mentioned in the lower row. The arrows indicate combinations that have been already at least partially explored, but several others remain to be explored.
Aside from the study of techniques to break reciprocity, a field that will continue to be studied in its own right, nonreciprocal components will continue to find a place in demonstrations of novel acoustic phenomena. An example of this is seen in acoustic arrays, used widely for beamforming in sonar communications and biomedical imaging, where the breaking of reciprocity means that the reception and transmission can be independently tuned. Initial studies have numerically studied how this might be accomplished,31,83 but experimental demonstrations are still lacking. There is also work to be done with nonreciprocal acoustic metasurfaces. Breaking reciprocity for a surface breaks the constraint that the reflection and transmission coefficients are equivalent between a pair of incidence angles. That is, breaking reciprocity allows and , opening new degrees of freedom for transmissive and reflective surfaces.84
As highlighted above, two application areas with high potential for further investigation in this context are SAW devices and topological acoustics. While nonreciprocal SAW devices based on magnetoelastic coupling have received extensive recent attention, further work could develop nonreciprocal SAW devices based on nonlinearity, building on established nonlinear SAW theory.85,86 Even within the magnetoelastic paradigm, there is much left to explore in terms of heterostructure composition and theoretical descriptions of the complex spin/elastic wave interactions in these structures. It would also be instructive to consider the topological features of unidirectional SAW devices. The current wave of discoveries in topological acoustics will certainly continue forward. In addition, though topological effects can arise in reciprocal systems, the much larger degree of robustness given to these systems by nonreciprocity implies that methods to break reciprocity will continue to remain pertinent and important for this field. One promising direction is pursuing time modulation as a synthetic spatial dimension to realize higher-dimensional topological effects in lower spatial dimensionalities. Another possibility going forward is using nonreciprocity techniques to break time-reversal symmetry in order to create new forms of acoustic Weyl semimetals, which have been realized so far only through the breaking of inversion symmetries.87–89 Nonlinear topological phenomena, such as topological phase transitions triggered by the intensity of the input signal, have also been recently discussed,90,91 on the model of their electromagnetic counterparts,92,93 but much of the potential of nonlinear topological acoustics remains untapped.
The study of nonreciprocity also intersects with research in out-of-equilibrium systems, such as active matter, where the constituent particles of the wave-bearing materials have internal sources of energy. This includes the study of the nonreciprocal wave dynamics in everything from flocking birds to biological swimmers and colloidal spinners.94,95 Continued study of such nonreciprocal systems will lead to demonstrations of novel nonreciprocal physics and provide more insight into the complex emergent behavior frequently observed in these systems. Platforms allowing precise control over particle couplings, such as electromechanical devices attached to programmable microcontrollers,81,96 are a promising way to demonstrate new wave phenomena. In general, as the speeds involved in wave propagation in mechanical and acoustical systems are much smaller than in other wave physics domains and as the scales are often conveniently macroscopic, they are an attractive realm for demonstrating increasingly complex nonreciprocal wave phenomena.
In this article, we have discussed the fundamental principles of acoustic reciprocity and explained its relationship with time-reversal symmetry. While various other forms of reciprocity in fluid dynamics97 and elastodynamics98 exist, we have reviewed as examples reciprocity relations for simple sources in an acoustic medium. We have shown how, by breaking the underlying assumptions of reciprocity, there are a number of opportunities for breaking acoustic reciprocity and pointed out recent experimental demonstrations of these techniques. Research in acoustic nonreciprocity is now mature to the point that applications are possible as we have shown through our discussion of nonreciprocal SAW devices and topological acoustics. We have also highlighted the many other possibilities for further investigation.
Physical symmetries are transformations that leave a system unchanged and offer powerful frameworks for understanding a wide variety of phenomena. Reciprocity is a symmetry in that it is a transformation that interchanges sources and receivers while leaving certain field quantities unchanged. Much like other symmetries, it offers a simplifying view of a number of related phenomena. The study of this symmetry, how to break it, and how it can be harnessed to control acoustic fields is a rich endeavor that will continue to inspire new research for many years to come.
This work was supported by the National Science Foundation EFRI program and the Simons Foundation.
The data that support the findings of this study are available within the article.