Transmission properties of space‐time modulated metamaterials

Abstract

We prove the possibility of achieving exponentially growing wave propagation in space‐time modulated media and give an asymptotic analysis of the quasifrequencies in terms of the amplitude of the time modulation at the degenerate points of the folded band structure. Our analysis provides the first proof of existence of k‐gaps in the band structures of space‐time modulated systems of subwavelength resonators.

Scattering from time-modulated subwavelength resonators

“…Let falsefalse{Difalsefalse}1≤i≤N be a disjoint collection of N resonators distributed inside U. We denote the region taken up by the resonators by D:=⋃i=1NDi.In the one-dimensional case considered in this paper, we will follow the notation introduced in [5]. In particular, we denote the resonators by Di:=false(xi−,xi+false), where false(xi±false)1≤i≤N are the 2N boundary points satisfying xi+<xi+1−, for any 1≤i≤N−1.…”

“…The reason we choose this specific definition of κifalse(tfalse) is such that its inverse has a finite Fourier expansion. Note that we do not assume ρ to be time-modulated due to the fact that time-modulating ρ does not affect the resonant quasi-frequencies at leading order in δ, see [5].…”

“…In this section, we derive two novel approximation formulae to the subwavelength resonant quasi-frequencies of (2.10); in other words, the values ω=ωi∈C such that there is a non-zero solution to (2.6) with uin=0. Firstly, we introduce a higher-order (in terms of the contrast parameter δ) capacitance matrix approximation for a general N in order to obtain the imaginary parts of the subwavelength resonant quasi-frequencies, which are not captured by the already known capacitance matrix approximation [5]. In the single-resonator case, this formula simplifies and admits explicit solutions.…”

“…Let Di=false(xi−,xi+false). Then for any smooth function f:Rfalse→R we define Inormal∂Difalse[ffalse] by [5] Inormal∂Difalse[ffalse]:=dfdx2.047em2.047em|−false(xi−false)−dfdx2.047em2.047em|+false(xi+false).Next, we define the capacitance matrix C=false(Cijfalse)i,j=1,…,N by Cij:=Inormal∂Difalse[Vjfalse]=dVjdx|−false(xi−false)−dVj…”

“…From [5], we start by claiming that for all i=1,…,N, vnscfalse(xfalse)1.2em1.2em|false[xi−,xi+false]=ci,n+Ofalse(δ1−γfalse).We know that vnscfalse(xfalse)1.2em1.2em|false(xi+,xi+1−false)=αni eikfalse(nfalse)x+βni e−ikfalse(nfalse)x.From [10], we know that the function Vi defined by (3.4) obeys Vifalse(xfalse)1.2em1.2em|false[xi+,xi+1−false]=−1ℓi(i+…”