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Waves in slowly varying band-gap media
- Publication Year :
- 2016
-
Abstract
- This paper is concerned with the asymptotic description of high-frequency waves in locally periodic media. A key issue is that the Bloch-dispersion curves vary with the local microstructure, giving rise to hidden singularities associated with band-gap edges and branch crossings. We describe an asymptotic approach for overcoming this difficulty, and take a first step by studying in detail the simplest case of 1D Helmholtz waves. The method entails matching adiabatically propagating Bloch waves, captured by a multiple-scale Wentzel-Kramers-Brillouin (WKB) approximation, with complementary multiple-scale solutions spatially localised about dispersion singularities. Within the latter regions the Bloch wavenumber is nearly critical; this allows their homogenisation, following the method of high-frequency homogenisation (HFH), over a naturally arising scale intermediate between the periodicity (wavelength) and the macro-scale. Analogously to a classical turning-point analysis, we show that close to a spatial band-gap edge the solution is an Airy function, only that it is modulated on the short scale by a standing-wave Bloch eigenfunction. By carrying out the asymptotic matching between the WKB and HFH solutions we provide a detailed description of Bloch-wave reflection from a band gap. Finally, we implement the asymptotic theory for a layered medium and demonstrate excellent agreement with numerical computations.
- Subjects :
- DYNAMICS
REFLECTION
Mathematics, Applied
PHOTONIC CRYSTALS
FOS: Physical sciences
02 engineering and technology
01 natural sciences
Homogenization (chemistry)
WKB approximation
NEGATIVE REFRACTION
Mathematics - Analysis of PDEs
Negative refraction
Airy function
0102 Applied Mathematics
0103 physical sciences
LIGHT-PROPAGATION
FOS: Mathematics
010306 general physics
Mathematical Physics
Mathematics
periodic media
Science & Technology
Applied Mathematics
Mathematical analysis
Mathematical Physics (math-ph)
021001 nanoscience & nanotechnology
Wavelength
Geometric phase
Physical Sciences
Gravitational singularity
HIGH-FREQUENCY HOMOGENIZATION
BERRY PHASE
singular perturbations
OPTICS
0210 nano-technology
Bloch waves
Bloch wave
Optics (physics.optics)
Analysis of PDEs (math.AP)
Physics - Optics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....51b5c347f6a759a5d01b53296126c397