We analyze the propagation of waves in unbounded photonic crystals. Waves are described by a Helmholtz equation with x-dependent coefficients, the scattering problem must be completed with a radiation condition at infinity. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new radiation condition to analyze the transmission problem where, at fixed frequency, a wave hits the interface between free space and a photonic crystal. We show that the vertical wave number of the incident wave is a conserved quantity. Together with the frequency condition for the transmitted wave, this condition leads (for appropriate photonic crystals) to the effect of negative refraction at the interface.
Accepté le :
DOI : 10.1051/m2an/2018026
Mots clés : Helmholtz equation, radiation, waveguide, Bloch analysis, outgoing wave condition, photonic crystal, transmission problem, negative refraction
@article{M2AN_2018__52_5_1913_0,
author = {Lamacz, A. and Schweizer, B.},
title = {Outgoing wave conditions in photonic crystals and transmission properties at interfaces},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1913--1945},
publisher = {EDP-Sciences},
volume = {52},
number = {5},
year = {2018},
doi = {10.1051/m2an/2018026},
mrnumber = {3885700},
zbl = {1414.35222},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018026/}
}
TY - JOUR AU - Lamacz, A. AU - Schweizer, B. TI - Outgoing wave conditions in photonic crystals and transmission properties at interfaces JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1913 EP - 1945 VL - 52 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018026/ DO - 10.1051/m2an/2018026 LA - en ID - M2AN_2018__52_5_1913_0 ER -
%0 Journal Article %A Lamacz, A. %A Schweizer, B. %T Outgoing wave conditions in photonic crystals and transmission properties at interfaces %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1913-1945 %V 52 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018026/ %R 10.1051/m2an/2018026 %G en %F M2AN_2018__52_5_1913_0
Lamacz, A.; Schweizer, B. Outgoing wave conditions in photonic crystals and transmission properties at interfaces. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1913-1945. doi : 10.1051/m2an/2018026. http://www.numdam.org/articles/10.1051/m2an/2018026/
[1] and , Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pure Appl. 77 (1998) 153–208. | DOI | MR | Zbl
[2] , and , Diffractive behavior of the wave equation in periodic media: weak convergence analysis. Ann. Mat. Pura Appl. 188 (2009) 561–589. | DOI | MR | Zbl
[3] , and , Diffractive geometric optics for Bloch wave packets. Arch. Ration. Mech. Anal. 202 (2011) 373–426. | DOI | MR | Zbl
[4] and , Guided waves in a photonic bandgap structure with a line defect. SIAM J. Appl. Math. 64 (2004) 2018–2033. | DOI | MR | Zbl
[5] , and , Analysis of the radiation properties of a planar antenna on a photonic crystal substrate. Math. Methods Appl. Sci. 24 (2001) 1021–1042. | DOI | MR | Zbl
[6] , , and , Diffraction by a defect in an open waveguide: a mathematical analysis based on a modal radiation condition. SIAM J. Appl. Math. 70 (2009) 677–693. | DOI | MR | Zbl
[7] and , Negative refraction in periodic and random photonic crystals. New J. Phys. 7 (2005).
[8] and , Homogenization of Maxwell’s equations in a split ring geometry. Multiscale Model. Simul. 8 (2010) 717–750. | DOI | MR | Zbl
[9] and , Tunable double negative band structure from non-magnetic coated rods. New J. Phys. 12 (2010) 083010. | DOI
[10] and , Inverse Acoustic and Electromagnetic Scattering Theory. Vol. 93 of Applied Mathematical Sciences, 2nd edn. Springer-Verlag, Berlin (1998). | MR | Zbl
[11] and , A Bloch wave numerical scheme for scattering problems in periodic wave-guides. SIAM J. Numer. Anal. 56 (2018) 1848–1870. | DOI | MR | Zbl
[12] and , Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability. Solid State Commun. 129 (2004) 643–647. | DOI
[13] and , Midgap defect modes in dielectric and acoustic media. SIAM J. Appl. Math. 58 (1998) 1748–1773. | DOI | MR | Zbl
[14] , A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides. SIAM J. Sci. Comput. 35 (2013) B438–B461. | DOI | MR | Zbl
[15] and , Wave propagation in locally perturbed periodic media (case with absorption): numerical aspects. J. Comput. Phys. 231 (2012) 1244–1271. | DOI | MR | Zbl
[16] and , Solutions of the time-harmonic wave equation in periodic waveguides: asymptotic behaviour and radiation condition. Arch. Ration. Mech. Anal. 219 (2016) 349–386. | DOI | MR | Zbl
[17] , and , Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides. BIT 55 (2015) 81–115. | DOI | MR | Zbl
[18] , Theorie der Luftschwingungen in Röhren mit offenen Enden. J. Reine Angew. Math. 57 (1860) 1–72. | MR | Zbl
[19] , The limiting absorption principle for a periodic semi-infinite waveguide. SIAM J. Appl. Math. 71 (2011) 791–810. | DOI | MR | Zbl
[20] and , Absence of bound states for waveguides in two-dimensional periodic structures. J. Math. Phys. 55 (2014) 033506. | DOI | MR | Zbl
[21] and , Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps. Opt. Express 16 (2008) 17383–17399. | DOI
[22] , , and , Photonic Crystals – Molding the Flow of Light. Princeton University Press (2008). | Zbl
[23] , An elementary introduction to the construction and the analysis of perfectly matched layers for time domain wave propagation. SeMA J. 57 (2012) 5–48. | DOI | MR | Zbl
[24] , and , Exact boundary conditions for periodic waveguides containing a local perturbation. Commun. Comput. Phys. 1 (2006). | Zbl
[25] and , Effective Maxwell equations in a geometry with flat rings of arbitrary shape. SIAM J. Math. Anal. 45 (2013) 1460–1494. | DOI | MR | Zbl
[26] and , A negative index meta-material for Maxwell’s equations. SIAM J. Math. Anal. 48 (2016) 4155–4174. | DOI | MR | Zbl
[27] , , and , All-angle negative refraction without negative effective index. Phys. Rev. B 65 (2002) 201104. | DOI
[28] , , and , Electromagnetic Bloch waves at the surface of a photonic crystal. Phys. Rev. B 44 (1991) 10961–10964. | DOI
[29] , and , Enhanced transmission and beaming of light via photonic crystal surface modes. Phys. Rev. B 69 (2004) 121402. | DOI
[30] , Negative refraction makes a perfect lens. Phys. Rev. Lett. 85 (2000). | DOI
[31] , New limiting absorption and limit amplitude principles for periodic operators. Z. Angew. Math. Phys. 66 (2015) 253–275. | DOI | MR | Zbl
[32] , Über das asymptotische Verhalten der Lösungen von Δu + λu = 0 in unendlichen Gebieten. Jber. Deutsch. Math. Verein. 53 (1943) 57–65. | MR | Zbl
[33] , Eighty years of Sommerfeld’s radiation condition. Historia Math. 19 (1992) 385–401. | DOI | MR | Zbl
[34] , The low-frequency spectrum of small Helmholtz resonators. Proc. A 471 (2015) 20140339. | MR | Zbl
[35] , Die Greensche Funktion der Schwingungsgleichung. Jbr. Deutsch. Math.-Verein. 21 (1912) 309–353. | JFM
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