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Sound Isolation and Giant Linear Nonreciprocity in a Compact Acoustic Circulator

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Science  31 Jan 2014:
Vol. 343, Issue 6170, pp. 516-519
DOI: 10.1126/science.1246957

Acoustically Isolated

The control of sound transmission is desirable in a number of circumstances from noise suppression to imaging technologies. Fleury et al. (p. 516; see the cover; see the Perspective by Cummer) studied a subwavelength acoustic meta-atom consisting of a resonant ring cavity biased by an internally circulating fluid. The direction of rotational flow of the fluid (air) changed the resonant properties of the ring cavity, allowing the propagation of sound waves within the cavity to be controlled. With several ports connected to the cavity, sound could be directed to a certain port while isolating transmission in another.

Abstract

Acoustic isolation and nonreciprocal sound transmission are highly desirable in many practical scenarios. They may be realized with nonlinear or magneto-acoustic effects, but only at the price of high power levels and impractically large volumes. In contrast, nonreciprocal electromagnetic propagation is commonly achieved based on the Zeeman effect, or modal splitting in ferromagnetic atoms induced by a magnetic bias. Here, we introduce the acoustic analog of this phenomenon in a subwavelength meta-atom consisting of a resonant ring cavity biased by a circulating fluid. The resulting angular momentum bias splits the ring’s azimuthal resonant modes, producing giant acoustic nonreciprocity in a compact device. We applied this concept to build a linear, magnetic-free circulator for airborne sound waves, observing up to 40-decibel nonreciprocal isolation at audible frequencies.

Reciprocity refers to the symmetric wave transmission between two points in space. It is a basic property observed in many wave phenomena because it is directly associated to the symmetry of physical laws under time reversal (1). Reciprocal transmission is not always desirable—for example, when one wants to isolate or protect a region of space allowing wave transmission in one direction yet blocking it in the opposite one. For electromagnetic waves, several approaches are available to break reciprocity and achieve isolation by using both linear (210) and nonlinear (1114) techniques, the most common being based on magnetic biasing. In contrast, nonreciprocal isolation of acoustic waves has so far been based on nonlinear mechanisms (1517), which introduce inherent signal distortions and limitations on the amplitude of operation. Recently, proposals to achieve unidirectional sound propagation in linear components have been discussed (1824), but they rely on devices that are strictly symmetric to time-reversal, being therefore completely reciprocal (25, 26) and not attaining the highly desirable functionality of a nonreciprocal linear acoustic isolator. However, nonreciprocal acoustic wave propagation can occur in linear media, by using magneto-acoustic effects (27), or in moving fluids (28). These possibilities are typically associated with weak effects that are only observable over large volumes and often require bulky and impractical biasing equipment.

Inspired by the way magnetic bias produces electromagnetic nonreciprocity in magneto-optical media based on the electronic Zeeman effect, we introduce here an acoustic meta-atom able to realize a nonreciprocal, linear, compact acoustic isolator. In magneto-optical media, such as ferromagnetic materials or atomic vapors, an applied static magnetic field Embedded Image splits the energy levels corresponding to countercirculating electronic orbitals (Fig. 1A), inducing different refractive properties for right- and left-handed circularly polarized waves that depend nonreciprocally on the direction of propagation. An analogous phenomenon can be obtained for acoustic waves in our macroscopic meta-atom (Fig. 1B). Imparting a circular motion to the fluid filling a subwavelength acoustic resonant ring cavity splits the degenerate counter-propagating azimuthal resonant modes and, for a proper velocity and cavity design, induces giant nonreciprocity via modal interference. In this scenario, the angular momentum vector imparted by the circular flow takes the role of the static magnetic bias, and the proposed meta-atom experiences the acoustic analog of Zeeman splitting and therefore displays a nonreciprocal response.

Fig. 1 The acoustic Zeeman effect.

(A) The classical electronic Zeeman effect involves an atom biased by an external magnetic field Embedded Image. The magnetic bias lifts the degeneracy of electronic eigenstates (here, Embedded Image orbitals, with energies Embedded Image) with a splitting proportional to Embedded Image. (B) The proposed acoustic Zeeman effect involves an acoustic meta-atom (here, a ring cavity, with degenerate counterpropagating modes) internally biased by a circulating fluid with angular momentum Embedded Image. The modes split in a similar fashion (30). (C) Frequency splitting as a function of the bias velocity. (D) A circulator as a three-port generalization of an isolator, its schematic model (top) and its implementation by using a Zeeman acoustic meta-atom coupled via small holes to three acoustic waveguides (bottom). Sound propagates from port 1 to 3, 3 to 2, and 2 to 1.

It is possible to qualitatively explain this phenomenon by considering the effective wavelength change for sound propagating in a moving medium. We assume that the azimuthally symmetric cavity is filled with a fluid on which we apply a circular motion with velocity Embedded Image along the azimuthal direction Embedded Image (Fig. 1B). In absence of rotation (Embedded Image), the ring resonates when its average circumference approximately equals an integer number Embedded Image of wavelengths, supporting degenerate counterpropagating eigenmodes with azimuthal dependence Embedded Image and frequencies Embedded Image, where Embedded Image is the speed of sound and Embedded Image is the mean radius. For Embedded Image, the sound effectively circulates with different velocities Embedded Image and Embedded Image in the moving fluid, depending on whether it travels with or against the flow. As a consequence, the resonant frequencies of the Embedded Image modes split according toEmbedded Image (1)The splitting is linear with respect to the biasing flow velocity, in perfect analogy to the electronic Zeeman effect (29)—an analogy that becomes even more apparent when the phenomenon is more rigorously studied with the quantum-mechanical approach developed in the supplementary materials (30). If the circulation is right-handed (RH), the RH mode Embedded Image shifts to a higher frequency, whereas the left-handed (LH) mode Embedded Image shifts down by the same amount. In order to validate this model, we numerically solve the eigenvalue problem for the dominant mode Embedded Image, for which the ring diameter is smaller than the wavelength, for a range of biasing fluid velocity. The cavity, whose geometry is detailed in (30), is designed to resonate around 800 Hz, and the corresponding eigenvalues (Fig. 1C) are found in perfect agreement with Eq. 1.

To demonstrate the possibility of getting large nonreciprocity through the proposed concept, we consider a three-port generalization of the isolator, also known in the microwave community as a circulator (Fig. 1D). Such a device allows acoustic power incident in port 1 to only be transmitted to port 3. From port 3, however, power only flows to port 2 and, likewise, from 2 to 1. The scattering matrix Embedded Image of the acoustic circulator is nonsymmetricalEmbedded Image (2)which is a symptom of its inherently nonreciprocal nature. An isolator is a subsystem of a circulator and can be readily obtained by impedance matching one of the circulator ports.

We realized such a device by coupling the proposed acoustic Zeeman meta-atom to three acoustic waveguides symmetrically placed at 120° intervals via small holes (Fig. 1D). As a result of mode splitting, an acoustic wave incident at port 1 unevenly couples to both RH and LH modes, in general with different amplitudes Embedded Imageand Embedded Image. Their interference supports different outputs at ports 2 and 3 despite the geometrical symmetry of the cavity. Using temporal coupled-mode theory, the power transmission coefficients at ports 2 and 3 are (2)Embedded Image (3)Embedded Image (4)where Embedded Image are the decay rates associated with RH and LH modes, with Embedded Image owing to symmetry. It follows that we can obtain Embedded Image and Embedded Image at frequency Embedded Image if the cavity modes are split so that Embedded Image. As seen in Eq. 1, this condition is directly satisfied for Embedded Image at the optimal bias velocity Embedded Image. After considering that Embedded Image and Embedded Image—where Embedded Image and Embedded Image are the sound velocity and the cavity quality factor, respectively—we obtain the optimal fluid velocity Embedded Image. Therefore, by choosing a sufficiently high Q-factor the proposed acoustic meta-atom can realize an ideal linear circulator (Eq. 2) within a subwavelength volume using arbitrarily low bias velocities.

We performed full-wave numerical simulations in order to investigate the behavior of the proposed device, assuming an acoustic wave incident from port 1. The magnitude of the pressure transmission coefficient at ports 2 and 3 in the absence of biasing motion is shown in Fig. 2A. In this case, the transmission coefficients at the two output ports are identical, which is due to symmetry. The unbiased cavity simply forms a power divider, which at resonance sends 4/9 of the power to each of the output ports, and the remaining 1/9 is reflected. The altered transmission spectrum when the device is appropriately biased with optimal fluid velocity Embedded Image is shown in Fig. 2B. As predicted by our model, the transmission to port 2 is dramatically reduced at the operating frequency Embedded Image, whereas transmission to port 3 reaches unity, indicating that all energy is now directed there. On the contrary, when the excitation is incident on port 3 the acoustic wave flows to port 2 instead of port 1. The effect of varying the biasing fluid velocity on the transmission from port 1 to ports 2 and 3 is shown in Fig. 2C. When Embedded Image (unbiased device), the amplitude transmission coefficients are equal to 2/3. As the velocity increases, the transmission to port 2, Embedded Image, gradually goes down to zero, whereas the transmission to port 3, Embedded Image, increases to reach unity for the specific value of bias velocity used in Fig. 2B. This value coincides with the optimal bias velocity Embedded Image calculated above and provides the correct amount of acoustic Zeeman splitting to obtain an ideal acoustic circulator. Beyond this value, Embedded Image increases again, whereas Embedded Image decreases. The proposed effect is quite robust to fluctuations in the mean velocity, and our simulations predict a large degree of isolation Embedded Image over a moderately broad range of velocities around the optimal value.

Fig. 2 Full-wave simulations for excitation at port 1.

(A) Case of no bias. Transmission at ports 2 and 3 is identical. (B) Case of optimal biasing velocity for maximum nonreciprocity, with zero transmission to port 2 and total transmission to port 3 at Embedded Image. (C) Effect of varying the biasing fluid velocity on the nonreciprocal transmission properties of the device at Embedded Image. (D) Acoustic pressure field distribution inside the unbiased device. (E) Same as (D), but when the circulator is optimally biased. Black arrows represent the average acoustic power flow.

To gain further insights into the response of the proposed device, we show the acoustic pressure field distribution for unbiased operation (Fig. 2D) and for optimal biasing velocity (Fig. 2E). In the first case, the resonant modes are degenerate and evenly excited, resulting in a field distribution inside the cavity that is totally symmetric with respect to the axis of port 1. Ports 2 and 3, which are symmetrically placed, are therefore evenly excited, and the response is fully reciprocal. The average power flow, represented by the black arrows, is split evenly between the two output ports. When the device is biased, on the contrary, the mode splitting is perfectly balanced to produce an asymmetric field distribution with respect to port 1 and create, by destructive interference between the two modes, a null of pressure field at port 2 and conversely, through constructive interference, a maximum at port 3. In this scenario, the power flow is routed exclusively toward the output port on the left of the input (the direction opposite to the velocity bias), independent of the feeding port.

These theoretical and full-wave simulation results are confirmed by our experimental investigations. Photographs of the fabricated device are presented in Fig. 3, A and B. The device consists of a ring cavity that resonates in the audible range at Embedded Image and contains three low-noise central processing unit (CPU) cooling fans, placed at 120° intervals so as to generate the desired circulating air flow. Details of the setup are provided in (30). The ring cavity, with the top cover removed to show the fan positions, is presented in Fig. 3A, and the closed cavity connected to the three acoustic waveguides is presented in Fig. 3B. By varying the input current at the fans, we are able to control the air velocity in the cavity and the corresponding biasing angular momentum. Shown in Fig. 3, C and D, are the measured transmission spectrum at ports 2 and 3 normalized to the no-bias case (Fig. 3C). As expected, when the fans are not operating (Fig. 3C) transmission at ports 2 and 3 is equal, and the system is fully reciprocal. On the other hand, for an input current to the fans Embedded Image (Fig. 3D), nonreciprocity is clearly observed, which is in excellent agreement with our theory. The effect of varying the fan input current, directly correlated with the fan speed, on the measured transmission coefficients is shown in Fig. 3E. Our measurements corroborate the theoretical predictions of Fig. 2C, showing the evolution from a fully reciprocal system for Embedded Image to a close-to-ideal acoustic circulator for Embedded Image. To quantify the performance of the realized device, the measured isolation Embedded Image is shown as a function of the input current (Fig. 3F). Around the optimal bias value, this device produces very large values of isolation, up to 40 dB. During the experiments, we have tested the device response at all ports, confirming its nonreciprocal operation and circulation of the input acoustic signal. Because the bias is electrically controlled, our design provides a large degree of tunability, with the possibility of switching from reciprocal to nonreciprocal operation and reversing the handedness of the circulator by simply changing the polarity of the input current. In our experiments, narrowband signals with carrier frequency Embedded Image were transmitted through the realized device, and acoustic nonreciprocity, isolation, and circulation could be directly verified by ear.

Fig. 3 Experimental results.

(A) Photograph of the fabricated cavity (without the top cover). The ring cavity is biased by using three low-noise CPU fans connected to a current source. (B) Photograph of the fabricated device. The biased cavity is closed and connected to three acoustic waveguides. Sound is incident from port 1, and transmission to ports 2 and 3 is measured. (C and D) Measured pressure transmission spectrum normalized to the unbiased case when (C) the fans are not powered and (D) the fan velocity is adjusted to produce optimal nonreciprocal behavior. (E) Measured transmission spectrum at Embedded Image normalized to the peak transmission in the unbiased case as a function of the input current. (F) Measured isolation in decibels as a function of input current at the fans.

A similar effect may be realized in liquids by using cylindrical cavities biased with magnetic stirrers. The fluid motion is a convenient way to impart the desired angular momentum to the resonant cavity, but other solutions, such as involving spatially modulated micromechanical resonators, may be considered in order to realize this effect in a fully integrated design and at higher frequencies. Our realized prototype is moderately subwavelength, but even smaller dimensions may be obtained by loading the resonant cavity. The proposed effect is largely tunable and scalable from audible to ultrasonic frequencies and provides a compact solution to realize acoustic switching, energy rectification, isolation, and circulation. Targeting higher frequencies may result in impractically small cavity sizes; however, it is possible to work with higher-order modes to circumvent this issue. We envision that this concept may open new directions in acoustics research, including advances in noise control, transducer technologies, energy harvesting systems, acoustic imaging, and sensing.

Supplementary Materials

www.sciencemag.org/content/343/6170/516/suppl/DC1

Materials and Methods

Fig. S1

References (3135)

References and Notes

  1. Materials and methods are available as supplemenatry materials on Science Online.
  2. Acknowledgments: This work has been supported by the Defense Threat Reduction Agency Young Investigator Program (YIP) award HDTRA1-12 1-0022 and the Air Force Office of Scientific Research YIP award FA9550-11-1-0009. A provisional U.S. patent has been filed with title “Non-reciprocal acoustic devices based on angular momentum bias” (61/868,178). R. F., D.L.S., and A.A. developed the concept presented in this paper. R.F. and D.L.S. carried out the analytical and numerical modeling and built the device. R.F. and C.F.S. conducted the measurements. M.R.H. contributed to the design and realization of the experimental set-up. A.A. supervised the entire project. All authors discussed the results and commented on the article.
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