12. D. Caruso, A. Troisi, Proc. Natl. Acad. Sci. U.S.A. 109,
13. J. Cabanillas-Gonzalez et al., Phys. Rev. B 75, 045207 (2007).
14. A. Devižis, A. Serbenta, K. Meerholz, D. Hertel,
V. Gulbinas, Phys. Rev. Lett. 103, 027404 (2009).
15. J. Cabanillas-Gonzalez et al., Phys. Rev. Lett. 96, 106601 (2006).
16. L. Sebastian, G. Weiser, H. Bässler, Chem. Phys. 61,
17. I. H. Campbell, T. W. Hagler, D. L. Smith, J. P. Ferraris,
Phys. Rev. Lett. 76, 1900–1903 (1996).
18. Y. Sun et al., Nat. Mater. 11, 44–48 (2011).
19. T. S. van der Poll, J. A. Love, T.-Q. Nguyen, G. C. Bazan,
Adv. Mater. 24, 3646–3649 (2012).
20. N. Blouin, A. Michaud, M. Leclerc, Adv. Mater. 19,
21. J. A. Love et al., Adv. Funct. Mater. 23, 5019–5026 (2013).
22. A. A. Bakulin, J. C. Hummelen, M. S. Pshenichnikov,
P. H. M. van Loosdrecht, Adv. Funct. Mater. 20,
23. C. Mayer et al., Adv. Funct. Mater. 19, 1173–1179 (2009).
24. L. G. Kaake, Y. Sun, G. C. Bazan, A. J. Heeger, Appl. Phys.
Lett. 102, 133302 (2013).
25. S. H. Park et al., Nat. Photonics 3, 297–302 (2009).
26. Z. He et al., Adv. Mater. 23, 4636–4643 (2011).
27. N. C. Miller et al., Nano Lett. 12, 1566–1570 (2012).
28. B. Collins et al., Adv. Energy Mater. 3, 65–74 (2013).
29. F. Liu, Y. Gu, J. W. Jung, W. H. Jo, T. P. Russell, J. Polym.
Sci. B Polym. Phys. 50, 1018–1044 (2012).
30. D. Amarasinghe Vithanage et al., Nat. Commun. 4, 2334
31. R. C. I. MacKenzie, J. M. Frost, J. Nelson, J. Chem. Phys.
132, 064904 (2010).
32. H. Tamura, M. Tsukada, Phys. Rev. B 85, 054301 (2012).
33. L. G. Kaake, D. Moses, A. J. Heeger, J. Phys. Chem. Lett.
4, 2264–2268 (2013).
34. A. C. Morteani et al., Adv. Mater. 15, 1708–1712 (2003).
Acknowledgments: We thank the Engineering and Physical
Sciences Research Council, and the Winton Programme
(Cambridge) for the Physics of Sustainability for funding.
S.G. acknowledges funding from the Fonds québécois de
recherche sur la nature et les technologies; A.R. thanks Corpus
Christi College for a Research Fellowship; A.K. thanks
National Research Foundation Singapore for a scholarship;
J.C. thanks the Royal Society for a Dorothy Hodgkin Fellowship;
and T.S.v.d.P. acknowledges funding from the Center for
Energy Efficient Materials, an Energy Frontier Research Center
funded by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences under Award DC0001009.
Figs. S1 to S14
20 September 2013; accepted 25 November 2013
Published online 12 December 2013;
Sound Isolation and Giant Linear
Nonreciprocity in a Compact
Romain Fleury,1 Dimitrios L. Sounas,1 Caleb F. Sieck,1,2 Michael R. Haberman,2,3 Andrea Alù1*
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 re-
versal (1). Reciprocal transmission is not always
desirable—for example, when one wants to iso-
late 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 reci-
procity and achieve isolation by using both linear
(2–10) and nonlinear (11–14) techniques, the most
common being based on magnetic biasing. In con-
trast, nonreciprocal isolation of acoustic waves
has so far been based on nonlinear mechanisms
(15–17), which introduce inherent signal distor-
tions and limitations on the amplitude of opera-
tion. Recently, proposals to achieve unidirectional
sound propagation in linear components have
been discussed (18–24), 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 re-
quire 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 ferro-
magnetic materials or atomic vapors, an applied
static magnetic field B →0 splits the energy levels
corresponding to countercirculating electronic or-
bitals (Fig. 1A), inducing different refractive pro-
perties for right- and left-handed circularly polarized
waves that depend nonreciprocally on the direc-
tion of propagation. An analogous phenomenon
can be obtained for acoustic waves in our macro-
scopic meta-atom (Fig. 1B). Imparting a circular
motion to the fluid filling a subwavelength acous-
tic 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 sce-
nario, the angular momentum vector imparted by
the circular flow takes the role of the static mag-
netic bias, and the proposed meta-atom experiences
the acoustic analog of Zeeman splitting and there-
fore displays a nonreciprocal response.
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 v along the azimuthal direction e →ϕ (Fig. 1B). In absence of rotation (v ¼ 0), the ring resonates when its average
circumference approximately equals an integer
number m of wavelengths, supporting degenerate
counterpropagating eigenmodes with azimuthal
dependence e Timϕ and frequencies wm ¼ mc=Rav,
where c is the speed of sound and Rav is the mean
radius. For v ≠ 0, the sound effectively circulates
with different velocities c þ v and c − v in the
moving fluid, depending on whether it travels with
or against the flow. As a consequence, the resonant
frequencies of the e Timϕ modes split according to
þ− m ¼ wm þ−mv Rav ð1Þ
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 mate-
rials (30). If the circulation is right-handed (RH),
the RH mode eimϕ shifts to a higher frequency,
whereas the left-handed (LH) mode e−imϕ shifts
down by the same amount. In order to validate this
model, we numerically solve the eigenvalue problem
for the dominant mode m ¼ 1, 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
1Department of Electrical and Computer Engineering, The
University of Texas at Austin, Austin, TX 78712, USA. 2Applied
Research Laboratories, The University of Texas at Austin, Austin,
TX 78758, USA. 3Department of Mechanical Engineering, The
University of Texas at Austin, Austin, TX 78712, USA.
*Corresponding author. E-mail: firstname.lastname@example.org