An official website of the United States government
The paper deals with forward and inverse homogenization of Maxwell's equations with a geometry on a microscopic scale given by a quasiperiodic distribution of piece-wise constant components defined by the use of a mapping R : ℝ n → ℝ m , m > n, and a periodic unit cell in ℝ m . Inverse homogenization makes use of a Stieltjes analytic representation for the effective complex permittivity, which depends upon R, unlike for the periodic case.
more »
« less
Multiple scale method applied to homogenization of irrational metamaterials
We adapt the multiple scale method introduced over 40 years ago for the homogenization of periodic structures [1], to the quasiperiodic (cut-and-projection) setting. We make use of partial differential operators (gradient, divergence and curl) acting on periodic functions of m variables in a higher-dimensional space that are projected onto operators acting on quasiperiodic functions in the n-dimensional physical space (m>n). We replace heterogeneous quasiperiodic structures, coined irrational metamaterials in [2], by homogeneous media with anisotropic permittivity and permeability tensors, obtained from the solution of annex problems of electrostatic type in a periodic cell in higher dimensional space. This approach is valid when the wavelength is much larger than the period of the higher dimensional elementary cell.
more »
« less
- Award ID(s):
- 1715680
- PAR ID:
- 10283755
- Date Published:
- Journal Name:
- 2020 Fourteenth International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials)
- Page Range / eLocation ID:
- 162 to 164
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We consider a discrete non-linear Schrödinger equation on Z and show that, after adding a small potential localized in the time-frequency space, one can construct a three-parametric family of non-decaying spacetime quasiperiodic solutions to this equation. The proof is based on the Craig–Wayne–Bourgain method combined with recent techniques of dealing with Anderson localization for two-dimensional quasiperiodic operators with degenerate frequencies.more » « less
-
Abstract We obtain a perturbative proof of localization for quasiperiodic operators on$$\ell ^2(\mathbb Z^d)$$ with one-dimensional phase space and monotone sampling functions, in the regime of small hopping. The proof is based on an iterative scheme which can be considered as a local (in the energy and the phase) and convergent version of KAM-type diagonalization, whose result is a covariant family of uniformly localized eigenvalues and eigenvectors. We also prove that the spectra of such operators contain infinitely many gaps.more » « less
-
Analytic representation formulas and power series are developed describing the band structure inside non-magnetic periodic photonic three-dimensional crystals made from high dielectric contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasiperiodic source-free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. A convergent power series for the Bloch wave spectrum is recovered from the representation formulas. Explicit conditions on the contrast are found that provide lower bounds on the convergence radius. These conditions are sufficient for the separation of spectral branches of the dispersion relation for any fixed quasi-momentum.more » « less
-
Analytic representation formulas and power series are developed describing the band structure inside non-magnetic periodic photonic three-dimensional crystals made from high dielectric contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasiperiodic source-free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. A convergent power series for the Bloch wave spectrum is recovered from the representation formulas. Explicit conditions on the contrast are found that provide lower bounds on the convergence radius. These conditions are sufficient for the separation of spectral branches of the dispersion relation for any fixed quasi-momentum.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/Metamaterials49557.2020.9284990
