Acoustically Induced Transparency Resonance and Flat Band in 1D Quasi-Periodic Structure According to an Arithmetic Sequence

In this paper, we study a perfect quasi-periodic system, consisting of a periodicity of segments of length d
1
and N' closed resonators of lengths d
2j
, in which the resonators lengths depend on each other according to an arithmetic sequence. Firstly, we take a single resonator between two semi-infinite guides and we indicate the existence of the AIT resonance and the resonant modes move towards the lower frequencies with the variation of a resonator length d
21
. Secondly, we show the appearance of new discrete modes located between two transmission zeros. These discrete modes are located around specific frequencies with higher transmission rates and quality factors. If we start increasing the number of resonators, we get more AIT resonances, and their resonance frequencies depend on the interaction between the modes of each resonator. The creation of resonators of different lengths in a perfect comb-like quasi-periodic system gives rise to very thin flat bands located inside the band gaps. If we consider that we have (j) resonators of different lengths, in this case there will be (j-1) flat bands, which gives a general rule of an arithmetic sequence such as
${{\mathrm{d}}_{2\mathrm{j}}} = {{\mathrm{d}}_{21}} + ({\mathrm{j} - 1}) \times \text{step}$
. On the other hand, we study the effect of introducing five defects at the resonators located in the middle of the quasi-periodic structure, these defects have different lengths
${{\mathrm{d}}_{02\mathrm{j}}}[{\mathrm{j} = 1 - 5}]$
, the creation of geometrical defects give rise to defect modes in the band gaps. These results are suitable for the application of multichannel filters.