Abstract
The recent development of subwavelength photonic and phononic crystals shows the possibility of controlling wave propagation at deep subwavelength scales. Subwavelength bandgap phononic crystals are typically created using a periodic arrangement of subwavelength resonators, in our case small gas bubbles in a liquid. In this work, a waveguide is created by modifying the sizes of the bubbles along a line in a dilute two-dimensional bubbly crystal, thereby creating a line defect. Our aim is to prove that the line defect indeed acts as a waveguide; waves of certain frequencies will be localized to, and guided along, the line defect. The key result is an original formula for the frequencies of the defect modes. Moreover, these frequencies are numerically computed using the multipole method, which numerically illustrates our main results.
Highlights
Line defects in bandgap photonic or phononic bandgap crystals are of interest due to their possible applications in low-loss waveguides
We consider a line defect in a phononic bandgap crystal comprised of gas bubbles in a liquid
Our results are more general than previous weak localization results since we explicitly show how the defect band depends on the perturbation size
Summary
Line defects in bandgap photonic or phononic bandgap crystals are of interest due to their possible applications in low-loss waveguides. The main mathematical problem of interest is to show that the spectrum of the defect operator has a non-zero overlap with the original bandgap. For previous works regarding line defects in bandgap crystals we refer to [11, 23, 24, 13, 14, 15, 16, 18]. We consider a line defect in a phononic bandgap crystal comprised of gas bubbles in a liquid. It is possible to create subwavelength bandgap crystals, which operate at wavelengths much larger than the unit cell size of the microstructured material. Mathematical properties of bubbly phononic bandgap materials have been studied in, for example, [1, 3, 6, 7, 8, 12], and subwavelength phononic bandgap materials have been experimentally realised in [25, 26, 27]
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