Loss-compensated non-reciprocal scattering based on synchronization

Abstract

Breaking the reciprocity of wave propagation is a problem of fundamental interest, and a much-sought functionality in practical applications, both in photonics and phononics. Although it has been achieved using resonant linear scattering from cavities with broken time-reversal symmetry, such realizations have remained inescapably plagued by inherent passivity constraints, which make absorption losses unavoidable, leading to stringent limitations in transmitted power. In this work, we solve this problem by converting the cavity resonance into a limit cycle, exploiting the uncharted interplay between non-linearity, gain, and non-reciprocity. Remarkably, strong enough incident waves can synchronize with these self-sustained oscillations and use their energy for amplification. We theoretically and experimentally demonstrate that this mechanism can simultaneously enhance non-reciprocity and compensate absorption. Real-world acoustic scattering experiments allow us to observe non-reciprocal transmission of audible sound in a synchronization-based three-port circulator with full immunity against losses.

Fig. 1. Concept of synchronization-based loss-compensated non-reciprocal scattering in a circulator.

a, b State of the art. Inducing a Zeeman-type bias in a resonator cavity (a) leads to non-reciprocal transmission of incident waves (b). Due to the passivity of the resonance, non-reciprocal transmission will be lossy, even if the isolation of the blocked direction is perfect. c A limit cycle in a round cavity with multiple ports continuously radiates a signal through the ports. d A harmonic wave incident on the cavity can synchronize with and gain energy from the limit cycle’s emissions. In the presence of a bias, this enables loss-compensated non-reciprocal transmission. e In the absence of external forcing, the acoustic field is either spinning right or left. f Under external forcing, the spinning acoustic field becomes a standing wave. This figure uses one of the color maps introduced in ref. .

Fig. 2. Acoustic circulator based on a spinning cavity limit cycle.

a Experimental setup. A swirling flow with fixed axial and azimuthal components is imposed in the blue pipe which passes through a cylindrical cavity, leading to a limit cycle (whistling) at around 800 Hz. In contrast with classic wind instruments whose sound radiation originates from self-oscillating standing modes, our special whistle involves spinning modes. For the scattering experiments, three 70 cm long square waveguides are connected to the upstream wall of the round cavity at equispaced angular positions (120° apart). These waveguides are equipped with sound-absorbing foam at their left termination in order to provide quasi-anechoic conditions. For the scattering experiments, incident waves are sent to the cavity using a compression driver (source) mounted on one of the waveguides. Microphones in the waveguides are used to reconstruct the incident and outgoing waves. b Acoustic mode of the circulator cavity, also showing isosurfaces of the azimuthal component of the coherent vorticity fluctuations ${\tilde{\mathrm{\Omega }}}_{\theta }$ (see the “Methods” section). This inset uses one of the color maps introduced in ref. . c Zoomed-in view of this instantaneous snapshot of the vorticity fluctuations, which was obtained from stereoscopic particle image velocimetry.

Fig. 3. Experimental validation of a circulator with fully compensated transmission losses.

Amplitude transmission coefficients for increasing incident wave amplitudes $\mathrm{\Delta }\tilde{s}$ and Δs (expressed relative the intervals $\tilde{s}\in \left[0.04,2.1\right]$ and s ∈ [10, 120] Pa, respectively; see the “Comparison of theoretical and experimental results” section). a–c Theoretical results obtained from the non-linear coupled-mode theory described in the “Non-linear wave-mode coupling” section. d–f Experimental measurements performed on the setup described in the “Experimental realization” section. In both cases, a harmonic incident wave is imposed in port “1''. The blue and black curves refer to the transmitted wave amplitudes at ports “2” and “3'', respectively, relative to the incident wave amplitude and frequency.

Fig. 4. Sketch of the experimental setup used in this work (not for scale).

A blower and shop air from the laboratory supply the wind channel of the experimental setup. The respective air flows are monitored using mass flow meters. At the swirler, air is injected tangentially to the wall of the circular cross-section wind channel, and it mixes with the main air from the blower. This configuration enables full control of the swirl intensity of the jet flowing through the cavity. This deep axisymmetric cavity is the central element of the three-port circulator, which is equipped with three waveguides. Their angular positions are 120° apart at the same radius. One of the waveguides is equipped with a compression driver to acoustically force one of the three circulator’s ports. Microphones are placed along the waveguides and on the cavity for acoustic measurements. The wind channel has anechoic, catenoidal terminations which are filled with acoustic foam. Similarly, the ends of the waveguides attached to the cavity are also filled with anechoic foam to achieve low acoustic reflection.

Fig. 5. Some features of the theoretical model presented in this work.

a Typical damping non-linearity of the cavity modes considered in this work. In the absence of an incident wave, if the linear gain of the “−” mode, β₋, exceeds the linear damping γ, and if the non-linear component of the cavity gain decreases for increasing amplitude ∣a₋∣, a stable limit cycle with amplitude ∣a₀∣ occurs, which corresponds to an equilibrium between gain and loss mechanisms. In this example, the mode a₊ is linearly stable because β₊ < γ. b Sketch of the effective resistance associated with the non-linear cavity modes considered in this work. At large amplitudes ∣a_±∣, the system exhibits a finite resistive term tending to γ, i.e., its response is essentially equivalent to that of a linearly stable resonance.

Fig. 6. Bifurcation diagram of the circulator cavity.

Shown is the whistling intensity, defined as the squared acoustic RMS amplitude ${\overline{p}}^2$, measured inside the cavity for different values of the bulk flow speed $\overline{U}=U+V$ and the swirl number Σ = V/U (see Fig. 4 for a definition of U and V). The operating point at $\overline{U}=23.1$ m/s and Σ = 0.49 is marked by a red dot. The smaller insets show the variation of ${\overline{p}}^2$ (gray curves) along the cyan and green curves, respectively. The black lines in the smaller insets are linear regression lines to extrapolate the bifurcation points. This figure uses one of the color maps introduced in ref. .

Fig. 7. Synchronization dynamics of the non-linear cavity mode.

a Experimental Arnold tongue. Incident waves at fixed amplitudes s and frequencies f were sent to the obstacle. By analyzing the relative peak height of the power spectrum density (PSD) of the forced mode, synchronized states can be identified. b A synchronized mode has an isolated peak at the forcing frequency. A non-synchronized mode has one peak at the self-sustained limit cycle frequency and another at the forcing frequency, as well as less dominant harmonics. c Pressure dynamics of a synchronized mode: quasi-sinusoidal oscillation at quasi-constant amplitude. d Non-synchronized states show beating oscillations due to interference of the self-sustained mode and the external forcing. The small insets in (c) and (d) show a couple of cycles of the same time traces.

Fig. 8. Modeled variation of the eigenfrequencies f_±, the linear gains β_±, and the decay rate γ with respect to the swirl number Σ.

The bias introduced by the swirl governs the Zeeman splitting of the eigenfrequencies that are degenerate at Σ = 0. The values at the operating point are marked by red dots. The “+” and “−” branches correspond to the modes spinning with and against the swirling flow, respectively.

Fig. 9. Measured transmission coefficients in the no-flow case.

The data shown was measured for different values of the incident wave amplitude s ∈ [10, 120] Pa. The black dotted curves correspond to T_1→3, the solid blue curves correspond to T_1→2 and the red lines correspond to 100% transmission.

Fig. 10. Sensitivity of the measured amplitude transmission coefficients at Δs = 0.82.

The data shown in this figure were measured over different values of the axial (U) and azimuthal (V) flow components. The black dotted curves correspond to T_1→3, the solid blue curves correspond to T_1→2 and the red lines correspond to 100% transmission. The operating point is marked with a thick red box around the respective inset.