Oscillatory and Localized Perturbations of Periodic Structures and the Bifurcation of Defect Modes

V. Duchêne; I. Vukićević; M. I. Weinstein

doi:10.1137/140980302

Oscillatory and Localized Perturbations of Periodic Structures and the Bifurcation of Defect Modes

SIAM Journal on Mathematical Analysis ◽

10.1137/140980302 ◽

2015 ◽

Vol 47 (5) ◽

pp. 3832-3883 ◽

Cited By ~ 2

Author(s):

V. Duchêne ◽

I. Vukićević ◽

M. I. Weinstein

Keyword(s):

Periodic Structures ◽

Defect Modes

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Asymptotic estimates for localized electromagnetic modes in doubly periodic structures with defects

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2006.1800 ◽

2007 ◽

Vol 463 (2080) ◽

pp. 1045-1067 ◽

Cited By ~ 10

Author(s):

A.B Movchan ◽

N.V Movchan ◽

S Guenneau ◽

R.C McPhedran

Keyword(s):

Periodic Structures ◽

Numerical Models ◽

Electric Polarization ◽

Silica Matrix ◽

Effective Diameter ◽

Localized Modes ◽

Helmholtz Equations ◽

Defect Modes ◽

Electric And Magnetic Fields ◽

Thin Bridge

The paper presents analytical and numerical models describing localized electromagnetic defect modes in a doubly periodic structure involving closely located inclusions of elliptical and circular shapes. Two types of localized modes are considered: (i) an axi-symmetric mode for the case of transverse electric polarization with an array of metallic inclusions; (ii) a dipole type localized mode that occurs in problems of waveguide modes confined in a defect region of an array of cylindrical fibres, and propagating perpendicular to the plane of the array. A thin bridge asymptotic analysis is used for case (i) to establish double-sided bounds for the frequencies of localized modes in macro-cells with thin bridges. For the case (ii), the electric and magnetic fields independently satisfy Helmholtz equations, but are coupled through the boundary conditions. We show that the model problem associated with localized vibration modes is the Dirichlet problem for the Helmholtz operator. We characterize defect modes by introducing a parameter called the ‘effective diameter’. We show that for circular inclusions in silica matrix, the effective diameter is accurately represented by a linear function of the inclusion radius.

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Homogenized description of defect modes in periodic structures with localized defects

Communications in Mathematical Sciences ◽

10.4310/cms.2015.v13.n3.a9 ◽

2015 ◽

Vol 13 (3) ◽

pp. 777-823 ◽

Cited By ~ 8

Author(s):

Vincent Duchêne ◽

Iva Vukićević ◽

Michael I. Weinstein

Keyword(s):

Periodic Structures ◽

Defect Modes ◽

Localized Defects

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Waves in lattices with imperfect junctions and localized defect modes

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.0255 ◽

2008 ◽

Vol 464 (2096) ◽

pp. 2037-2054 ◽

Cited By ~ 5

Author(s):

V.V Zalipaev ◽

A.B Movchan ◽

I.S Jones

Keyword(s):

Vibration Mode ◽

Periodic Structures ◽

Lattice Models ◽

Theory Of Elasticity ◽

Defect Modes ◽

Thin Walled Structures ◽

Discrete Lattices ◽

Thin Walled ◽

Type Boundary ◽

Type Boundary Condition

A correspondence between continuum periodic structures and discrete lattices is well known in the theory of elasticity. Frequently, lattice models are the result of the discretization of continuous mechanical problems. In this paper, we discuss the discretization of two-dimensional square thin-walled structures. We consider the case when thin-walled bridges have defects in the vicinity of junctions. At these points, the displacement satisfies an effective Robin-type boundary condition. We study a defect vibration mode localized in the neighbourhood of the damaged junction. We analyse dispersion diagrams that show the existence of standing waves in a structure with periodically distributed defects.

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