Nonreciprocal acoustic propagation and leaky-wave radiation in a waveguide with flow

Reciprocity in sound propagation is the principle for which the same transmission signal is expected when source and receiver switch places. In a multi-port system, this implies that the scattering matrix of a reciprocal system is inherently symmetric,^1,2 i.e., the acoustic signal transmitted from one port to a second one is identical to the signal transmitted from the second to the first when all other ports are impedance matched. Recent interest in acoustic and elastodynamic devices that break reciprocity has been driven by their applicability in several practical situations, including in sonar and ultrasound technology, or for full-duplex acoustic communications, in which asymmetric transmission or routing of signals to different portions of a transmit-receive module are desirable.^3–7 In a multi-port system, nonreciprocity provides protection from arbitrary backscatter from the loads, which can protect sensitive equipment and allows components to be added in a modular fashion. In antenna theory, reciprocity requires that an emitter's reception pattern is always identical to its transmission pattern for all angles. By breaking this reciprocity, an antenna can transmit more than it receives which shields it from echoes. Electromagnetic radiators with this broken reciprocity have been studied and fabricated, with applications including thermal management.^8,9 Reciprocity breaking is also integral to the study of topological acoustics, where it gives rise to unidirectional topological modes that can propagate in a crystal lattice immune to defects and disorder.^10–15 

It is known that a steady fluid flow is a simple avenue to break reciprocity for sound,^6,16 and recent work has shown how adding a slow material flow to a resonant device can make it, under a suitable design, highly nonreciprocal.¹⁷ In the following, we describe the operation, design, and performance of innovative acoustic isolators not based on resonant cavities. The first isolator design, an acoustic Mach-Zehnder interferometer, consists of a single mode duct split into two paths and later recombined. By exploiting the directional interference between these paths, it is possible to realize an efficient isolator. A more compact version of this $p$ isolator is then explored, at the expense of using bimodal input and output waveguides. We then also use bimodal ducts with a nonuniform velocity profile and a grating to induce a phononic transition. To conclude, we consider how simple duct flow may also realize non-reciprocal leaky-wave radiation for sound.

Reciprocity in acoustics is inherently rooted in the linearity and time-reversal symmetry of the wave equation, which in the time harmonic regime using the $e−iωt$ convention reads¹⁸ 

where $p$ is the acoustic pressure, $ω$ the angular frequency, $c$ the small-signal speed of sound, $i$ the imaginary unit, and $v→$ the time-invariant velocity field. Consider two sourceless time-harmonic fields, $p1$ and $p2$⁠, each satisfying this equation. Multiplying the wave equation for $p1$ by $p2$ and subtracting the result from the wave equation for $p2$ multiplied by $p1$ gives

In the case of no flow, the right-hand side of this equation vanishes, leaving a reciprocity relationship between fields 1 and 2. However, as the equation demonstrates, a reciprocity relationship is not guaranteed when a steady flow is present.

Reciprocity is a result of the time-reversibility of the fluid dynamics. The propagation of acoustic waves is underpinned by time-symmetric equations of classical mechanics and should exhibit reciprocity when all the relevant field quantities are transformed appropriately in time. The velocity vector field, $v→$⁠, representing the underlying motion of the fluid particles in steady flow, flips sign under the time reversal operation, $T$⁠. When this odd vector is not flipped between fields 1 and 2, the conditions for $T$ are not fully satisfied, and nonreciprocity is possible. A challenge of this way of breaking reciprocity is that the required flow velocity should be large compared to the group velocity of sound for the effects to be large.

It is also possible to break reciprocity using nonlinear techniques,^19–23 but nonreciprocity then becomes a function of the input intensity and does not allow for simultaneous excitation from multiple ports. The magnetic field, an odd vector field, can also be used to break reciprocity, as in magnetoelastic coupling,^3,4 but these phenomena tend to be very weak in natural materials. At first glance, it would seem that reciprocity would also not hold in the presence of loss when, under $T$⁠, material absorption is not transformed into gain. But reciprocity is described in terms of ratio of field quantities, such as the ratio of the transmitted vs incident powers. In this case, if the loss is maintained as loss in the reciprocal condition, the wave is attenuated when propagating in either direction and these ratios remain unchanged,²⁴ though there is the possibility of still breaking reciprocity if the loss is direction dependent.²⁵ In the following, we explore how to engineer acoustic devices to take advantage of small medium flows to realize large nonreciprocal effects.

In a rigid walled waveguide with transversely uniform flow, the phase velocity of a sound wave in the direction of the fluid flow differs from the sound speed according to $cph=c+v→⋅k̂$⁠, where $k̂$ is the wavevector direction. For the waveguide's plane wave mode, this means that it travels faster or slower by an amount $v=|v→|$ depending on whether it is propagating with or against the direction of flow. Higher-order modes, described as the superposition of waves reflecting off the sides of the duct at an angle $θ$⁠, have their phase speed along the waveguide axis increased or decreased by $v cos θ$⁠.

The dispersion relation in a waveguide with uniform flow along the x axis and no z-component of the wavenumber is, from Eq. (1)

For the plane wave mode, with $ky=0$⁠, the group velocity, $∂ω/∂k$⁠, is increased or decreased by $v$⁠. On a band diagram, this modification appears as a change of the slope of the plane wave line, which extends to zero frequency (Fig. 1). For higher-order modes the behavior is simplified when the Mach number, $v/c$⁠, is small. In this regime, the wavenumber of all modes is increased or decreased by $vω/c2$⁠. The flow creates an asymmetry in the dispersion diagram about the frequency axis. This asymmetry can be quantified by the nonreciprocal phase shift, $Δβ=|βp|−|βm|$⁠, the difference in the real part of the wavenumber along the waveguide axis, $kx$⁠, between a forward and a backward propagating wave. At low Mach numbers the phase difference is approximately equivalent in the forward and backward directions and it is linear with the fluid velocity, $Δβ=2δβ=2vω/c2$⁠, with $δβ$ being the wavenumber shift due to fluid flow. It is this nonzero nonreciprocal phase shift that can be exploited to create a sound isolator.

An isolator is a two-port device that allows large transmission in one direction while blocking backscattered waves.²⁶ A wave entering the device through an input port, say port 1, is transmitted out the exit port, port 2, while a wave coming from port 2 is blocked from exiting either port 1 or 2. Such a device clearly breaks reciprocity and linearity and passivity require that it includes loss, meaning its scattering matrix is inherently non-symmetric and non-unitary. Reciprocity must be broken because, if left intact, large transmission through the device from port 1 to port 2 necessarily implies large transmission through the device from port 2 to port 1. In addition, there must be a mechanism for loss because without loss the scattering matrix being unitary allows only non-reciprocity in the phase but not in the amplitude. The loss can come from cumulative effects, such as thermoviscous attenuation, or by including additional ports that collect the energy incident from port 2. A three-port device that breaks reciprocity, called a circulator, was recently demonstrated in acoustics using moderate flow in a circular resonant cavity.¹⁷ The small $δβ$ induced in the cavity was largely amplified by its resonance, implying a trade-off between bandwidth and flow velocity. A circulator can function as an isolator when one of the ports is terminated with a matched impedance, such as a wave absorbing anechoic wedge. The unused port provides the loss required by the isolator.

A Mach-Zehnder interferometer is a device that splits a monochromatic beam of light in two and recombines the beams after they travel down separated paths.^27,28 The amount of interference, constructive or destructive, at the point where the beams are recombined indicates the phase difference accumulated by the beams on their individual paths. To work as an isolator, an interferometer additionally needs the two essential features of an isolator: broken reciprocity and loss. Reciprocity is broken by embedding nonreciprocal phase shifters in the two arms of the Mach-Zehnder interferometer. These are the areas where the phase of the beam advances more in one direction than in the reverse direction. In electromagnetic Mach-Zehnder isolators, nonreciprocal phase shifters are implemented using magnetic materials^29,30 or suitably modulated material properties.^31,32 In our proposed acoustic Mach-Zehnder interferometer (Fig. 2), the nonreciprocal phase shifters are areas of flow. In these waveguide sections, the phase of a propagating sound wave advances more when travelling in the direction of flow than when travelling against it. The loss comes from extra ports included above and below the input and output ports. Sound incident from port 2 exits the system through these ports terminated with impedances matched to the waveguide impedance.

Beam splitting in our proposed acoustic Mach-Zehnder interferometer is accomplished using a multimode interference coupler, a type of beam splitter found in optics.³³ It operates by decomposing the incident energy into many transverse modes. Output ports are placed on the other side of the splitter at locations where the modal interference pattern has high field strength so that ideally all of the input energy is equally split into the two output paths. In our design, a sound wave enters the system in the dominant mode (plane-wave like) from port 1 at a frequency below the cutoff of any higher-order modes. It enters the beam splitter where, because the input to the cavity is along the centerline, the interference pattern of the modes is transversely symmetric. By optimizing the length of the cavity and placing the two cavity outputs equidistant from the centerline, the sound is split equally into the upper and lower paths. The phase at the entrance to these two paths is also equal. The two separated waves traverse their individual paths before being recombined with another identical beam splitter on the output (right) side of the device. If the waves from the two paths have the same phase when they reach the second beam splitter, they excite the symmetric mode of the cavity and the wave predominantly gets out through the center waveguide. On the other hand, if they have a phase difference of 180 degrees, the antisymmetric mode is excited and there is no output power at the center waveguide, since this mode has a zero at the location of this waveguide. For any other phase difference between the two waveguides, the output power is between these two extreme cases. The intensity of the wave exiting the output port is then dependent on the difference between the phase length of the upper path versus the lower path.

In order to break reciprocity, it is also necessary to break the vertical symmetry of the device. Reciprocal phase shifting areas serve this purpose. Their necessity can be seen by considering a horizontal and vertical reflection of the geometry without them. The transformation from these two reflections returns the same device geometry even though the input and output ports are now flipped. The result is that transmission from port 1 to port 2 is the same as from port 2 to port 1. It is possible to break the symmetry between the two paths, and thus break reciprocity, by putting flow in just one of the two paths, negating the need for the reciprocal phase shifting segments. However, this then requires a higher rate of flow. An alternative way is to add reciprocal phase shifters to one of the paths, which is what we do here (upper path in Fig. 2). The reciprocal phase shifting segments in this case are simply extra duct length. The physical length of the upper path is then slightly longer than that of the lower path.

The device is modelled as a slab with acoustically hard walls. A wave entering the device from port 1 (input port, on left) ideally passes through the device and exits port 2 (output port, on right). However, a wave incident from port 2 ideally neither exits port 1 nor is reflected back out port 2. It instead exits the system through one of extra ports included above and below the input and output ports. These extra ports are needed for it to function as an isolator since the acoustic medium and waveguide walls are assumed lossless. In the upper path, the wave travels in the same direction as the material flow and in the lower path against the flow, producing a nonreciprocal phase response at the other end. The length is chosen to achieve a nonreciprocal phase difference of $−π/2$ between the two paths for propagation in the forward direction. The included reciprocal phase shifters make the upper path slightly longer than the lower path, yielding a $π/2$ reciprocal phase difference between them. In the forward direction, these two phase differences cancel out, and the waves constructively interfere at the beam splitter on the right side, giving large transmission to port 2. When a plane wave instead enters the device from the right (port 2), the reciprocal phase difference is still $π/2$⁠, but the nonreciprocal phase difference is now also $π/2$⁠. The beams arrive at the left beam splitter out of phase, and therefore destructively interfere. There is an interference null where port 1 branches off the beam splitter and interference maxima at the loss ports, where most of the energy exits the system. The interference can also be explained in terms of asymmetric wave speeds. In the forward direction, the wave in the upper path travels faster than in the lower path because it is in the same direction as the flow. However, the upper path wave also has to travel a longer distance because of the extra duct sections (the reciprocal phase shifters). The longer distance and faster speed cancel, and the upper path wave arrives in phase with the lower path wave for constructive interference at the point of recombination. In the reverse direction, the upper path wave has to both travel further and at a speed reduced by the flow. It arrives out of phase with the lower path wave and destructively interferes.

Isolation is given by $20 log |S21/S12|$⁠, where the scattering parameter $Sij$ is the ratio of the field exiting the system through port i to the field entering the system from port j. Using the finite element software COMSOL Multiphysics, we see that the peak isolation is 69 dB with a modest flow velocity of 3.2 m/s at an operating frequency of 1497 Hz. The nonreciprocal phase difference of $−π/2$ requires that $lΔβ=π/2$⁠, where $l$ is the length of the flow region. Using the relation between $Δβ$ and $v$ for the plane wave mode predicts a needed flow velocity of 4.8 m/s. This prediction is based on the isolator being an idealized Mach-Zehnder. The duct corners and imperfect multimode beam splitters cause it to deviate from this ideal model, as they add reactive loads to the system, reducing the required flow velocity. However, there is still a similar tradeoff between required flow rate and length of the flow region. The total footprint of the device is roughly 4 × 18 $λ$ at the operating frequency, with the smallest dimension being the height of the ducts.

While developed in two dimensions for conceptual simplicity, this design can naturally be extended to three dimensions, as the plane wave mode has no out-of-plane wavevector component. However, the fields inside the cavities are not propagating plane waves, so the locations of the adjoining ducts need to be chosen appropriately to get the desired energy splitting. Membranes could be used to contain flow in the desired regions. With no tension applied, such membranes have a response dictated by the bending stiffness and can present small impedance under proper conditions.^34,35 Air flow could enter the device through regions that present a high impedance to the passing acoustic waves. Helmholtz resonators, quarter wave resonators, or subwavelength metamaterials could be used to this end.³⁶ The flow in the device is shown as unidirectional, but could be part of a flow system that loops back on in a region external to the device.

An improved isolator design that reduces the required footprint can be obtained by considering two transverse modes at the input and output waveguides (Fig. 3). An incoming plane wave from the input port is split into two by a hard wall, modelled as perfectly thin, that runs down the middle of the waveguide. The upper beam encounters a side cavity, which reciprocally shifts its phase relative to the lower beam. The other necessary component of our isolator, a nonreciprocal phase shift, again comes from flow regions. As the separating wall ends, the beams are recombined in phase, and a plane wave is transmitted through the output port. If instead the system is excited from the right, an incoming plane wave is split into two paths which are out of phase at the point of recombination because of the nonreciprocal phase shift. Since the waveguide supports an odd second-order mode, instead of being reflected back the two signals efficiently couple to the odd mode, which then propagates away from the device through the input port.

Using this device in combination with mode filters at the entrance and exit to absorb the odd mode completes the isolating behavior. From the left, an input plane wave passes through the device with high transmission, but if the same plane wave enters the device from the right, it is converted to the odd mode and absorbed by the filter on the left, implying large nonreciprocity and isolation. The device peak isolation is 29 dB, with a lower insertion loss than the first design. The required flow velocity is roughly 4 m/s with a device footprint of 2 × 15 $λ$ at the optimal frequency.

Spatiotemporal modulation in wave-carrying media can induce transitions to other modes of the system with a resultant change in frequency and wavenumber.³⁷ This mode conversion is known in optics as a photonic transition, in analogy to quantum electronic transitions in solid state physics. For example, an electromagnetic wave travelling in a silicon waveguide can be transitioned from one mode of propagation, say mode 1, to another, say mode 2. The transition takes place as the wave travels through a region where the material property is modulated with a travelling wave perturbation of the form $cos (Δωt−Δkz)$ where $t$ is time, the z-direction is along the waveguide axis, and $Δω$ and $Δk$ are the frequency and wavenumber difference between modes 1 and 2. The amplitudes of modes 1 and 2 oscillate in the modulated region whose length is chosen so that only mode 2 is present at its end. Full oscillations between modes 1 to 2 are possible, and the small modulation lends the problem to an analytical treatment using perturbation theory. When a wave is instead incident as mode 1 from the opposite side, very little mode conversion takes place and it remains in mode 1 form. This happens because the incident wave is now travelling in the direction opposite to that of the material property perturbation, so the mode matching condition is no longer satisfied. The total effect is large nonreciprocal mode conversion. When the system is combined with appropriate filters, the phenomenon is used to create highly-nonreciprocal linear isolators for light.^38,39

In acoustics, fluid flow modifies the phase speed for passing waves and can be used to induce transitions between modes. However, a travelling wave perturbation of the velocity field is difficult to achieve as the background fluid flow is not governed by a wave equation. If instead steady flow is used, nonreciprocal mode conversion is still possible, but extra features are needed to make the effect large and the analysis is not as straightforward. Time-invariant flow implies that mode conversion can only occur between modes of the same frequency. The system in this case is governed by the convected wave equation [Eq. (3)], where there is no mechanism for frequency conversion.

Full-wave simulations are relied on to optimize the setup (Fig. 4), with guidelines taken from the perturbation analysis of photonic transitions. For one, optimal coupling between modes come from maximizing their overlap integral in the presence of the flow, $∫vP0P1$⁠, integrated over the waveguide cross section in the flow region. In our case, $P0$ is the profile of the plane wave mode—a constant. $P1$ is the profile of the first higher-order mode, given by the cosine function. To maximize the integral with these modal profiles, the velocity function $v$ should be anti-symmetric over the waveguide cross-section. Limiting the flow profile to a single speed and only pointed in the forward direction, the integral is maximized when the flow covers half the waveguide height [Fig. 4(a)].

Nonreciprocal mode conversion is now possible, but a large velocity is still needed for good performance. This requirement can be relaxed by introducing a grating to provide a z-directed modulation, analogous to the $Δk$ modulation in a photonic transition. The $Δk$ difference between the two waveguide modes predicts a notch spacing of 5.7 $λ$⁠, close to the optimal result, 5.9 $λ$⁠, obtained from simulations. Unlike the case for photonic transitions, there is still considerable mode oscillation that occurs when a wave passes through the system in the backwards direction. Modal oscillations occur in both directions, but with different periodicities. The length of the flow region is chosen to enhance the nonreciprocal response. Ideally, a plane wave mode incident from the left oscillates until it exits the system out the right as a plane wave mode. However, a plane wave mode incident from the right oscillates in that same distance to the higher order mode, exiting the system on the left.

In the geometry presented here, whether from the right or left, an incident plane wave exits the system on the other side as a linear combination of the two modes, but the weight of each mode is asymmetric in the forward and backward directions [Fig. 4(b)]. In practice, mode filters on either side would remove the second mode. The isolation is then given as the ratio between how much of the plane wave mode makes it through the device when entering from the left versus the right. The phononic transition waveguide provides 15 dB of isolation at a flow velocity of 12.4 m/s when its length is ∼100 $λ$ (17 grating notches).

We use broken reciprocity from flow to create a nonreciprocal leaky-wave antenna (Fig. 5). Flow fills the full waveguide height. The waveguide is coupled to free space by shunts and is terminated on the right with a matched absorber. The free space background is modelled as a semi-infinite medium with a hard boundary condition in the plane on top of the antenna. As expected from reciprocity, in the absence of flow reception and transmission patterns are equivalent [Fig. 5(b), left]. However, when flow is added, a large asymmetry between the beampatterns arises. The behavior can be understood by looking at the dispersion lines for the lower order (plane wave like) mode in Fig. 1(b), with the blue lines representing the waveguide modes in the flow region and the free space sound cone lying on the lower black line (equivalent to the plane wave dispersion). In transmission, the wave travels in the direction of flow, which increases its phase velocity, reducing the wavenumber so the blue line on the right side is within the sound cone. The wavenumber is shifted outside the sound cone in the reverse direction. The transmit beampattern is effectively rotated counterclockwise by including flow, while the receive pattern is rotated clockwise. The angle of rotation can be estimated from the simple waveguide with flow by $cos−1((1+v/c)−1)$⁠, which for a 12.4 m/s flow is 15 degrees. In the main emission direction, the receive sensitivity is over 10 dB smaller. The effect relies on operation near endfire, where the rotation of the receive beampattern pushes its main lobe outside of the sound cone. The amount of nonreciprocity seen at the angle of peak emission is dictated by the length of the antenna, which affects the angular width of the lobes, and the flow velocity, which determines how much the lobes in the transmit and receive patterns are rotated away from each other.

Steady fluid flow is a powerful way to break time-reversal symmetry in acoustic systems, as it is a property that naturally remains constant regardless of the direction of incoming waves. We have shown that low Mach number flow through straight channels is sufficient to achieve a significant amount of isolation in waveguiding structures. Acoustic Mach-Zehnder interferometry, where incoming sound is split into two paths and later recombined, allows for isolating devices with footprints of a few wavelengths. Alternatively, a waveguide that induces phononic transitions also provides isolation because the transitions are directionally dependent. Frequency-preserving transitions can be enabled by a nonuniform transverse profile combined with a grating. We showed how a directional antenna that violates the typical symmetry between its direction of transmission and reception can be formed by coupling a waveguide with flow to free space. These one-way structures provide a valuable tool in the creation of complex acoustic networks where the ability to isolate one subsection can greatly facilitate system design.

This work was partially funded by the Office of Naval Research, Air Force Office of Scientific Research and the National Science Foundation.