Fleury, Sounas and Alù demonstrated (Nature Communications, 2015) an invisible sensor for airborne sound: a parity-time (PT) symmetric pair of loudspeakers loaded by tailored non-Foster electrical circuits, which fully absorbs and senses the impinging wave while remaining reflection-free and shadow-free, as the acoustic analog of a coherent-perfect-absorber–laser pair. The demonstrated device operates only in a narrow band around 250 Hz, and the authors envisioned, without demonstrating, broadband operation via more complex non-Foster load dispersion engineering and 2D/3D arrays of such gain–loss cells for loss-immune negative refraction, focusing, and subwavelength imaging. No experimental broadband gain–loss PT invisible acoustic sensor and no experimental 2D/3D loss-immune negative-refracting PT acoustic array have been demonstrated to date. The problem of extending the PT-symmetric invisible sensor concept to broadband operation and array-scale loss-immune wave control therefore remains open.

Background

In wave physics, a system is said to possess parity-time (PT) symmetry when its governing operator is invariant under simultaneous mirror reflection (parity) and time reversal; in a scattering medium this is realized by balancing gain and loss at mirror-symmetric positions, which makes the effective Hamiltonian non-Hermitian. Non-Hermitian PT-symmetric systems can exhibit exceptional points — degeneracies at which eigenvalues and eigenvectors coalesce — where unusual scattering occurs. One such effect is unidirectional invisibility (also called an anisotropic transmission resonance): zero reflection and unit-magnitude transmission for waves incident from one side, but full reflection from the other, while the wave is transmitted with a characteristic phase advance. A related object is the coherent perfect absorber (CPA), the time reverse of a laser: an interferometric scatterer that fully absorbs incident waves, and its combination with a gain element forms a CPA–laser pair. A non-Foster electrical circuit is an active circuit that synthesizes a negative capacitance or negative inductance; because its reactance dispersion violates the Foster reactance theorem obeyed by passive one-ports, it can relax the bandwidth–size trade-off that otherwise constrains subwavelength resonant transducers. Negative refraction means that the wave vector component transmitted across an interface bends to the opposite side of the normal, enabling planar focusing and subwavelength imaging by a flat slab rather than a curved lens. In 2015, Fleury, Sounas and Alù built a PT-symmetric acoustic cell consisting of two electro-acoustic transducers (loudspeakers) in a subwavelength-separated waveguide, one loaded by a resistive circuit (loss, the sensor) and one by a non-Foster circuit providing gain. At the design frequency of 250 Hz the scattering matrix reaches an exceptional point, so that from the sensing side the device is unidirectionally invisible: the passive loudspeaker absorbs twice the incident power (the incident wave plus a synchronized replica emitted by the active loudspeaker, the acoustic analog of a CPA coupled to a laser) while creating no reflection and no shadow. Stability was ensured by design: the poles of the acoustic admittance were placed with positive imaginary part, and Routh–Hurwitz criteria (a classical algebraic test for stability of a polynomial's root locations) were applied to the load impedance. The theory underlying the envisioned extensions had been developed by the same group, which showed that PT-symmetric metasurfaces can in principle support loss-immune and broadband wave manipulation, negative refraction, and planar focusing, and that the concept carries over to electromagnetic waves and light. What the experiment realized, however, was a single cell at a single frequency, and the bandwidth of the unidirectional invisibility was narrow, limited by the dispersion of the electrical loads; the paper explicitly notes that increasing the bandwidth requires engineering that dispersion with more complex (non-Foster) circuit designs, and envisions 2D or 3D arrays of such inclusions for negatively refracting or focusing acoustic waves, with applications in loss-compensated sound focusing, loss-immune phase compensation, and non-invasive subwavelength acoustic imaging and sensing.

Problem Statement

The demonstrated parity-time-symmetric invisible acoustic sensor consists of a single gain–loss cell of two non-Foster-loaded electro-acoustic transducers that is unidirectionally invisible and fully sensing only in a narrow band around its 250 Hz design frequency. The open problem, as envisioned in the source work, is to generalize this cell into a practical PT-symmetric invisible-sensing and loss-immune wave-control platform, with one common objective: experimental gain–loss PT-symmetric acoustic devices whose exceptional-point invisibility (zero reflection, unit transmission, full absorption of the sensed signal, no shadow) and loss compensation hold over a frequency band spanning at least one octave about the design frequency (a factor of two in frequency — for the 250 Hz design, at least about 177–354 Hz), and at array scale. Concretely, this comprises three targets stated in the source: (1) broadband operation — design, stabilize and experimentally validate electrical-load dispersions (more complex non-Foster circuit designs, with stability margins robust under realistic component tolerances and drift) that keep the unidirectional-invisibility scattering condition of the PT cell satisfied over the same one-octave frequency band about the design frequency rather than at a single frequency; (2) array scaling — realize and experimentally characterize two- or three-dimensional arrays of such gain–loss PT inclusions, including their collective behavior and inter-cell coupling, that negatively refract, focus, or form subwavelength images of acoustic waves in a loss-immune manner, as the array-level extension of the loss-immune wave manipulation that PT-symmetric metasurface theory predicts for such gain–loss inclusions; and (3) platform transfer — tailored stable realizations of the same gain–loss invisible-sensing concept for other frequency ranges and transducer technologies, and for electromagnetic waves and light, where stable gain elements must be engineered. The research objective covers both the theory and design of such broadband, array-scale, stable PT-symmetric devices and their experimental demonstration with quantified bandwidth, scattering and imaging performance. Targets (1) and (2) are posed jointly, and target (3) is a further generalization. The one-octave broadband condition applies to the arrays of target (2) as well, so they must be shown over the octave band rather than only at a single design frequency.

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