Edge modes in active systems of subwavelength resonators

Abstract

Wave scattering structures with amplification and dissipation can be modelled by non-Hermitian systems, opening new ways to control waves at small length scales. In this work, we study the phenomenon of topologically protected edge states in acoustic systems with gain and loss. We demonstrate that localized edge modes appear in a periodic structure of subwavelength resonators with a defect in the gain/loss distribution, and explicitly compute the corresponding frequency and decay length. Similarly to the Hermitian case, these edge modes can be attributed to the winding of the eigenmodes. In the non-Hermitian case, the topological invariants fail to be quantized, but can nevertheless predict the existence of localized edge modes.

Functional analytic methods for discrete approximations of subwavelength resonator systems

“…In order for localized modes to exist, there must be an eigenvalue µ = µ α j (b 0 ) of C edge which is constant in α for some b = b 0 . We can then compute b as b = b ± , where Based on these ideas, we can prove the following result [18]. Theorem 4.27.…”

“…Most importantly, we allow the wave speeds to be complex, giving a non-Hermitian capacitance formulation similarly as in Section 4.3. Details of this analysis are found in [18].…”

Wave interaction with subwavelength resonators

“…Moreover, a structure which exhibits asymptotic unidirectional reflectionless transmission at certain frequencies is designed. In [15], the phenomenon of topologically protected edge states in systems of subwavelength resonators with gain and loss is studied. It is demonstrated that localized edge modes appear in a periodic structure of subwavelength resonators with a defect in the gain/loss distribution, and the corresponding frequencies and decay lengths are explicitly computed.…”

Unidirectional edge modes in time-modulated metamaterials

“…This approach allows to establish for the first time a complete mathematical and numerical framework for metamaterials: (i) a discrete approximation for computing subwavelength resonances in both finite and periodic systems of subwavelength resonators has been introduced [2,3,11]; (ii) the existence of Dirac singularities was proved [8,15]; (iii) the role that Dirac points play in the origin of robust (topologically protected) edge states has been explored [6], and (iv) it has been shown that the introduction of time-modulations into systems of subwavelength resonators can shift the Dirac singularities to the origin of the Brillouin zone [13]. This approach has been also extended to the analysis of exceptional points, non-hermitian systems of subwavelength resonators and their applications in sensing at subwavelength scales [4,5,12].…”