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Localization of Classical Waves I: Acoustic Waves.
 Commun. Math. Phys
, 1996
"... We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existe ..."
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Cited by 37 (0 self)
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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the selfadjoint operators A = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almost the wh...
BandGap Structure Of Spectra Of Periodic Dielectric And Acoustic Media. I. Scalar Model
 I. Scalar model, SIAM J. Appl. Math
, 1996
"... . We investigate the bandgap structure of the spectrum of secondorder partial differential operators associated with the propagation of waves in a periodic twocomponent medium. The medium is characterized by a realvalued positiondependent periodic function "(x) that is the dielectric constant f ..."
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Cited by 25 (5 self)
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. We investigate the bandgap structure of the spectrum of secondorder partial differential operators associated with the propagation of waves in a periodic twocomponent medium. The medium is characterized by a realvalued positiondependent periodic function "(x) that is the dielectric constant for electromagnetic waves and mass density for acoustic waves. The imbedded component consists of a periodic lattice of cubes where "(x) = 1. The value of "(x) on the background is assumed to be greater than 1. We give the complete proof of existence of gaps in the spectra of the corresponding operators provided some simple conditions imposed on the parameters of the medium. Key words: propagation of electromagnetic and acoustic waves, bandgap structure of the spectrum, periodic dielectrics, periodic acoustic media. AMS subject classification. 35B27, 73D25, 78A45. 1. INTRODUCTION. One of the main observations in the quantum theory of solids is that the energy spectrum of an electron in a ...
Localization of Classical Waves II: Electromagnetic Waves.
 Commun. Math. Phys
, 1997
"... We consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r \Theta 1 " 0 (x) r \Theta has a gap in the spectrum, were r \Thet ..."
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Cited by 20 (0 self)
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We consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r \Theta 1 " 0 (x) r \Theta has a gap in the spectrum, were r \Theta \Psi = r\Theta\Psi. We prove the existence of localized waves, i.e., finite energy solutions of Maxwell's equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times. Localization of electromagnetic waves is a consequence of Anderson localization for the selfadjoint operators M = r \Theta 1 "(x) r \Theta . We prove that, in the random medium described by "(x), the random operator M exhibits Anderson localization inside the gap in the spectrum of M 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almo...
Spectral Properties of Classical Waves in High Contrast Periodic Media
 SIAM J. Appl. Math
, 1998
"... We introduce and investigate the band gap sturcture of the frequency spectrum for classical electromagnetic and acoustic waves in a high contrast twocomponent periodic medium. The asymptotics with respect to the high contrast is considered. The limit medium is described in terms of appropriate sefl ..."
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Cited by 15 (5 self)
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We introduce and investigate the band gap sturcture of the frequency spectrum for classical electromagnetic and acoustic waves in a high contrast twocomponent periodic medium. The asymptotics with respect to the high contrast is considered. The limit medium is described in terms of appropriate sefladjoint operators and the convergence to the limit is proven. These limit operators give an idea of the spectral structure and suggest new numerical approaches as well. The results are obtained in arbitrary dimension and for rather general geometry of the medium. In particular, 2D photonic band gap structures and their acoustic analogs are covered. Key words: propagation of electromagnetic and acoustic waves, band gap structure of the spectrum, periodic dielectrics, periodic acoustic media, high contrast. AMS subject classiifcation: 35B27, 73D25, 78A45. To appear in SIAM Journal on Applied Mathematics. y This work was supported by the U.S. Air Force Grant F496209410172DEF z This ...
Localization of Electromagnetic and Acoustic Waves in Random Media. Lattice Models
 Lattice Model, J. Stat. Phys
, 1994
"... We consider lattice versions of Maxwell's equations and of the equation that governs the propagation of acoustic waves in a random medium. The vector nature of electromagnetic waves is fully taken into account. The medium is assumed to be a small perturbation of a periodic one. We prove rigorously t ..."
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Cited by 13 (4 self)
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We consider lattice versions of Maxwell's equations and of the equation that governs the propagation of acoustic waves in a random medium. The vector nature of electromagnetic waves is fully taken into account. The medium is assumed to be a small perturbation of a periodic one. We prove rigorously that localized eigenstates arise in a vicinity of the edges of the gaps in the spectrum. A key ingredient is a new Wegnertype estimate for a class of lattice operators with offdiagonal disorder. Key words: localization, random media, electromagnetic waves, acoustic waves, lattice model. The work is supported by U. S. Air Force grant AFOSR910243 y The work is partially supported by NSF grant DMS 9208029 1 Introduction Decades after P. W. Anderson [1] described the remarkable phenomenon of the localization in space of electron wave functions in disordered solids, physicists have begun to ask whether other waves, say electromagnetic or acoustic, can be localized if the propagating m...
Localized Classical Waves Created by Defects
 J. Stat. Phys
, 1997
"... We study acoustic and electromagnetic waves in a periodic medium (or any other background medium with a spectral gap) disturbed by a single defect, i.e., a local disturbance analogous to a well potential in solid state physics. We show that defects do not change the essential spectrum of the associa ..."
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Cited by 11 (2 self)
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We study acoustic and electromagnetic waves in a periodic medium (or any other background medium with a spectral gap) disturbed by a single defect, i.e., a local disturbance analogous to a well potential in solid state physics. We show that defects do not change the essential spectrum of the associated nonnegative operators and can only create isolated eigenvalues of finite multiplicity in a gap of the periodic medium, with the eigenmodes decaying exponentially. We give a constructive and simple description of defects in acoustic and dielectric media, including a simple condition on the parameters of the medium and of the defect, which ensures the rise of a localized eigenmode with the corresponding eigenvalue in a specified subinterval of the given gap of the periodic medium. KEY WORDS: Electromagnetic waves, acoustic waves, localization, photonic localization, periodic medium, spectral gap, photonic crystals, photonic band gaps, defects. 1 Introduction Localization of classical wav...
Midgap Defect Modes In Dielectric And Acoustic Media
 SIAM J. Appl. Math
, 1998
"... . We consider three dimensional lossless periodic dielectric (photonic crystals) and acoustic media having a gap in the spectrum. If such a periodic medium is perturbed by a strong enough defect, defect eigenmodes arise, localized exponentially around the defect, with the corresponding eigenvalues i ..."
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Cited by 8 (1 self)
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. We consider three dimensional lossless periodic dielectric (photonic crystals) and acoustic media having a gap in the spectrum. If such a periodic medium is perturbed by a strong enough defect, defect eigenmodes arise, localized exponentially around the defect, with the corresponding eigenvalues in the gap. We use a modified BirmanSchwinger method to derive equations for these eigenmodes and corresponding eigenvalues in the gap, in terms of the spectral attributes of an auxiliary HilbertSchmidt operator. We prove that in three dimensions, under some natural conditions on the periodic background, the number of eigenvalues generated in a gap of the periodic operator is finite, and give an estimate on the number of these midgap eigenvalues. In particular, we show that if the defect is weak there are no midgap eigenvalues. Key words. photonic crystal, photonic bandgap, periodic acoustic medium, periodic dielectric medium, midgap states, defect modes, localization of light AMS subject...
Photonic Pseudogaps for Periodic Dielectric Structures
 J. Stat. Phys, 74, Issue
, 1994
"... We consider the problems of existence and structure of gaps (pseudogaps) in the spectra associated with Maxwell equations and equations that governs the propagation of acoustic waves in periodic two component media. The dielectric constant " is assumed to be real and positive, and the value of " = " ..."
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Cited by 2 (2 self)
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We consider the problems of existence and structure of gaps (pseudogaps) in the spectra associated with Maxwell equations and equations that governs the propagation of acoustic waves in periodic two component media. The dielectric constant " is assumed to be real and positive, and the value of " = " b on the background is supposed to be essentially larger than the value of " = " a on the embedded component. We prove the existence of pseudogaps in the spectra of the relevant operators. In particular, we give an accurate treatment of the term "pseudogap". We also show that if the contrast " b =" a approaches infinity then the bands of the spectrum shrink to a discrete set which can be identified with the set of eigenvalues of a Neumann type boundary value problem and, thus, can be effectively calculated. Key words: waves, periodic dielectrics, periodic acoustic media, pseudogaps in the spectrum. Introduction The idea of finding and designing periodic and disordered dielectric materials...
Enhancement of &xchi;² Cascading Processes in 1D Photonic Band Gap Structures
"... We theoretically analyze the nonlinear phase shifts induced by cascaded (2) : (2) processes in1D photonic band gap (PBG) structures. We find that the enhancement of the density of modes near the band edge, coupled with a suitable choice of relative phase mismatch, leads to a new remarkable eff ..."
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We theoretically analyze the nonlinear phase shifts induced by cascaded (2) : (2) processes in1D photonic band gap (PBG) structures. We find that the enhancement of the density of modes near the band edge, coupled with a suitable choice of relative phase mismatch, leads to a new remarkable effect: the relative phase shift of the fundamental field upon transmission can be of order over a distance of 7 microns, with input intensities on the order of only 10 MW/cm 2 . 2 The past two decades have witnessed an intense investigation of electromagnetic wave propagation phenomena at optical frequencies in periodic structures. These structures are usually referred to as photonic band gap (PBG) crystals [1], and many practical applications of this new technology have been suggested at least in one dimension, for nonlinear [2,3] and linear systems [4,5]. These devices [25] are based on the physics of the photonic band edge. In this letter we point out yet another effect that can oc...