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.

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Sensing an incoming signal is typically associated with absorbing a portion of its energy, inherently perturbing the measurement and creating reflections and shadows. Here, in contrast, we demonstrate a non-invasive, shadow-free, invisible sensor for airborne sound waves at audible frequencies, which fully absorbs the impinging signal, without at the same time perturbing its own measurement or creating a shadow. This unique sensing device is based on the unusual scattering properties of a parity-time (PT) symmetric metamaterial device formed by a pair of electro-acoustic resonators loaded with suitably tailored non-Foster electrical circuits, constituting the acoustic equivalent of a coherent perfect absorber coupled to a coherent laser. Beyond the specific application to non-invasive sensing, our work broadly demonstrates the unique relevance of PT-symmetric metamaterials for acoustics, loss compensation and extraordinary wave manipulation.

Graded metasurfaces exploit the local momentum imparted by an impedance gradient toMetasurfaces are thin structured arrays that have attracted significant attention for the level of control of electromagnetic waves that they enable [1]- [6]. Phase-gradient metasurfaces, in particular, have recently been explored to tailor the electromagnetic wavefront in unprecedented ways to realize low-profile lenses, holograms, beam steerers and other ultrathin optical devices.

We aim at providing a global perspective on electromagnetic nonreciprocity and clarifying confusions that arose in recent developments of the field. We provide a general definition of nonreciprocity and classify nonreciprocal systems according to their linear time-invariant (LTI), linear time-variant (LTV), or nonlinear natures. The theory of nonreciprocal systems is established on the foundation formed by the concepts of time reversal, time-reversal symmetry, time-reversal symmetry breaking, and related Onsager-Casimir relations. Special attention is given to LTI systems, the most-common nonreciprocal systems, for which a generalized form of the Lorentz reciprocity theorem is derived. The delicate issue of loss in nonreciprocal systems is demystified and the so-called thermodynamics paradox is resolved from energyconservation considerations. An overview of the fundamental characteristics and applications of LTI, LTV, and nonlinear nonreciprocal systems is given with the help of pedagogical examples. Finally, asymmetric structures with fallacious nonreciprocal appearances are debunked.

Non-reciprocal components, which are essential to many modern communication systems, are almost exclusively based on magneto-optical materials, severely limiting their applicability. A practical and inexpensive route to magneticfree non-reciprocity could revolutionize radio-frequency and nanophotonic communication networks. Angular-momentum biasing was recently proposed as a means of realizing isolation for sound waves travelling in a rotating medium 1 , and envisaged as a path towards compact, linear integrated non-reciprocal electromagnetic components 2,3. Inspired by this concept, here we demonstrate a subwavelength, linear radio-frequency non-reciprocal circulator free from magnetic materials and bias. The scheme is based on the parametric modulation of three identical, strongly and symmetrically coupled resonators. Their resonant frequencies are modulated by external signals with the same amplitude and a relative phase di erence of 120 • , imparting an e ective electronic angular momentum to the system. We observe giant non-reciprocity, with up to six orders of magnitude di erence in transmission for opposite directions. Furthermore, the device topology is tunable in real time, and can be directly embedded in a conventional integrated circuit.

Metasurfaces characterized by a transverse gradient of local impedance have recently opened exciting directions for light manipulation at the nanoscale. Here we add a temporal gradient to the picture, showing that spatio-temporal variations over a surface may largely extend the degree of light manipulation in metasurfaces, and break several of their constraints associated to symmetries. As an example, we synthesize a non-reciprocal classical analogue to electromagnetic induced transparency, opening a narrow window of one-way transmission in an otherwise opaque surface. These properties pave the way to magneticfree, planarized non-reciprocal ultrathin surfaces for free-space isolation.

Optical analog signal processing has been gaining significant attention as a way to overcome speed and energy limitations of digital techniques. Metasurfaces offer a promising avenue towards this goal, due to their efficient manipulation of optical signals over deeply subwavelength volumes. To date, metasurfaces have been proposed to transform signals in the spatial domain, e.g., for beam steering, focusing or holography, for which angulardependent responses, or nonlocality, are unwanted features that must be avoided or mitigated. Here, we show that the metasurface nonlocality can be engineered to enable signal manipulation in the momentum domain over an ultrathin platform. We explore nonlocal metasurfaces performing basic mathematical operations, paving the way towards fast and power efficient ultrathin devices for edge detection and optical image processing.