Topological analog signal processing

Abstract

Analog signal processors have attracted a tremendous amount of attention recently, as they potentially offer much faster operation and lower power consumption than their digital versions. Yet, they are not preferable for large scale applications due to the considerable observational errors caused by their excessive sensitivity to environmental and structural variations. Here, we demonstrate both theoretically and experimentally the unique relevance of topological insulators for alleviating the unreliability of analog signal processors. In particular, we achieve an important signal processing task, namely resolution of linear differential equations, in an analog system that is protected by topology against large levels of disorder and geometrical perturbations. We believe that our strategy opens up large perspectives for a new generation of robust all-optical analog signal processors, which can now not only perform ultrafast, high-throughput, and power efficient signal processing tasks, but also compete with their digital counterparts in terms of reliability and flexibility.

Fig. 1

Robust topological analog signal processing. We consider the possibility to process time-domain wave signals by engineering their transfer function as they propagate through an engineered solver. a A first-order differential equation solver is constructed from resonant tunneling through a crystal defect. The output signal is the solution of the differential equation associated with the transfer function of the system. b In the presence of geometrical defects, like slight position shifts, the signal processing functionality achieved with the trivial equation solver of panel (a) is completely destroyed. c To make the signal processing robust, we propose instead to build the target transfer function of the system from resonant tunneling through a topological edge mode. d Markedly different from the trivial equation solver of panels (a) and (b), the output of the topological solver is left totally unaffected by the disorder

Fig. 2

Numerical demonstration of the topological differential equation. We compare a topological (a) and a trivial (b) acoustic signal processors designed such that the envelope of their output $\tilde{f}\left(t\right)$ is the solution of the differential equation f′(t) + αf(t) = βg(t), where g(t) is envelope of the input signal $\tilde{g}\left(t\right)$ modulated with carrier frequency f₀. a An arbitrarily chosen signal envelope g(t) is applied to the input of the topological equation solver. The transfer function of the system H(f) (green line), which reproduces exactly the mathematical target defined by the equation (dashed line), is not affected by the presence of disorder (bottom signal path). As a result, the envelope of the output signal f(t) matches exactly the solution even in the presence of disorder. b Conversely, in a topologically trivial processor, the presence of disorder-induced localized states creates spurious peaks and shifts the transfer function of the system, which makes it deviate from the targeted transfer function (dashed line). The parameters of the linear differential equation are chosen to be α = β = 2π, and the position disorder strength is 18% of the lattice period in both cases

Fig. 3

Effect of various defect types on the topological equation solver. a A topological interface made from tight-binding SSH chains (top), consisting of resonators with resonance frequency ω₀ coupled to each other via detuned hopping amplitudes K and J > K, supports an edge mode protected by chirality. The transmission spectrum of the chain (bottom) shows a mid-gap resonance, which corresponds to the topological edge mode. b Some disorder is added to the hopping amplitudes of the system (top), which preserve chiral symmetry. The bottom panel demonstrates the robustness of the transmission peak (averaged over 20 realizations of disorder) as the disorder strength (DS) is increased. c Same as panel (b) except that the disorder is applied to the on-site potentials of the chain, hereby breaking chiral symmetry. The transmission peak is sensitive to arbitrarily weak disorder. d−f Same as (a−c) but for the proposed acoustic equation solver. The resonance line-shape of the edge mode is robust to the position movement (normalized to the lattice constant) of the rods inside the waveguide (panel e), which does not break the symmetry $M_{\mathrm{cell}}^2=1$ (see Methods). In contrast, detuning the radii of the obstacles breaks this property, and causes degradation in the performance of the equation solver (panel f)

Fig. 4

Experimental demonstration of the topological equation solver. The acoustic waveguide is a square transparent tube and the scatterers are made from black Nylon rods. As in our numerical investigations of Fig. 2, we compare the trivial signal processors in terms of robustness to position defects. A frequency domain measurement allows us to extract the transfer function H(f) (green line), which we compare to the ideal target (dashed line) and simulation (gray). An additional measurement then performed in time domain by directly sending an input signal with envelope g(t) into the system, and recording the coda with envelope f(t) at the output. a The topological equation solver is indeed found to be immune to the shits in rods position. b Very differently, the trivial equation solver is severely affected. The parameters of the linear differential equation are chosen to be α = 2.7π, β = 10π/3, and the position disorder has the same strength in both cases

Fig. 5

Robust resolution of a second-order differential equation. a The second-order transfer function associated with the resolution of the equation f″(t) + 6πf^'(t) + 8π²f(t) = 4π²g(t) can be achieved by proper subtraction of two first-order transfer functions. b Implementation of the scheme in panel (a) with two topological first-order differentiators. The signal subtraction is realized with a rat-race coupler (circular component connecting the two systems). The bottom panels represent full-wave numerical simulations of the complete 3D structure in the case of a Gaussian pulse input, demonstrating that the targeted signal processing task is indeed performed by the system. c Experimental realization of the second-order differential equation solver. The measured output signal envelope (f(t), purple lines) is found to be in perfect agreement with both the numerical simulation (gray) and with the exact solution of the corresponding second-order differential equation (dashed line)

Fig. 6

Topology of the bands. We define the topology of the bands as the number of times the contours $\mathcal{C}$ crosses the axis of the cone defined in Eq. 20. a For the trivial lattice, the contour $\mathcal{C}$ does not cross the axis of the cone, corresponding to a zero topological invariant. b When the system goes through phase transition, the contour $\mathcal{C}$ touches the tip of the cone. The topological invariant cannot be defined in this case. c Same as panels (a) and (b) but for the topological lattice. The contour $\mathcal{C}$ crosses the axis of the cone one time in this case, which corresponds to a nontrivial topology