Energy Bands in Periodic Lattices—Green's Function Method

Ham, Frank S.; Segall, B.

doi:10.1103/physrev.124.1786

Cited by 355 publications

(114 citation statements)

References 16 publications

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“…These difficulties are overcome within the MS method, an approach based on the well-known in the electronic band-structure community Korringa-Kohn-Rostoker (KKR) theory [10,11,[26][27][28]. The success of this theory in both electronic and electromagnetic [29][30][31] band-structure calculations was a strong motivation for its application in the acoustic/elastic problem as well.…”

Section: Multiple Scattering Methods 21 Introductionmentioning

confidence: 99%

Classical vibrational modes in phononic lattices: theory and experiment

Sigalas¹,

Kushwaha²,

Economou³

et al. 2005

Zeitschrift Für Kristallographie - Crystalline Materials

209

120

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Abstract. We present a review, through selected illustrative examples, of the physics of classical vibrational modes in phononic lattices, which elaborates on the theory, the formalism, the methods, and mainly on the numerical and experimental results related to phononic crystals. Most of the topics addressed here, are written in a selfconsistent way and they can be read as independent individual parts.

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Section: Multiple Scattering Methods 21 Introductionmentioning

confidence: 99%

Classical vibrational modes in phononic lattices: theory and experiment

Sigalas¹,

Kushwaha²,

Economou³

et al. 2005

Zeitschrift Für Kristallographie - Crystalline Materials

209

120

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“…The scalar Green's function has been previously reported for energy bands calculations using Schrodinger's equation in a periodic potential [4]. Our objective is to deduct its dyadic form in electromagnetism for a lattice of point sources.…”

Section: Dyadic Green's Function For Periodic Structuresmentioning

confidence: 99%

Electromagnetic Scattering of Finite and Infinite 3D Lattices in Polarizable Backgrounds

Gallinet¹,

Martin

2009

AIP Conference Proceedings

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Abstract. A novel method is elaborated for the electromagnetic scattering from periodical arrays of scatterers embedded in a polarizable background. A dyadic periodic Green's function is introduced to calculate the scattered electric field in a lattice of dielectric or metallic objects. The method exhibits strong advantages: discretization and computation of the field are restricted to the volume of the scatterers in the unit cell, open and periodic boundary conditions for the electric field are included in the Green's tensor, and finally both near and far-fields physics are directly revealed, without any additional computational effort. Promising applications include the design of periodic structures such as frequency-selective surfaces, photonic crystals and metamaterials.Keywords: Electromagnetic Scattering, Dyadic Green Function, Periodic Structures PACS: 42.70. Qs,78.67.Bf,02.70.Pt Recent developments in the fabrication of periodic structures with tailored optical periodic structures call for specific modeUng tools. Most real structures are threedimensional but still very few techniques are able to simulate them without considerable computational efforts [1,2]. There is therefore a need to seek for a general method supporting full 3D vectorial calculations, enabling the investigation of both near and far-fields physics. Our approach is based on the volume integral equation for scattering in free space and at surfaces [3].We first develop the Bloch modes calculations in lattices using the Green's tensor formalism. The method exhibits strong advantages, especially about the restriction of the discretization and the minimal computational efforts required to reveal both near and far-field physics. Then two examples of calculations are reported to illustrate some physical effects that can be simulated: grating effects, propagation in multi-dimensional finite and infinite size lattices. Promising appUcations include the design of periodic structures such as frequency-selective surfaces, photonic crystals, and metamaterials. DYADIC GREEN'S FUNCTION FOR PERIODIC STRUCTURESThe scalar Green's function has been previously reported for energy bands calculations using Schrodinger's equation in a periodic potential [4]. Our objective is to deduct its dyadic form in electromagnetism for a lattice of point sources. Consider a scattering system described by a dielectric function e(r) embedded in an infinite homogeneous background medium EB. Nonmagnetic materials and harmonic fields with an exp(-/a)f) dependance are assumed. When this system is illuminated by an incident field E°(r) propagating in the background medium, the total electric field (incident plus scattered field) is a solution of the vectorial wave equation:where feg = ap-/c^ is the vacuum wave number The permittivity e(r) is assumed to have a spatial periodicity: e(r + l) = e(r), where 1 = wai +ma2 + pa3 {m,n,p integers) is a translation vector of the lattice and ai, a2, as are the translation primitive vectors. The volume of the unit cell is called Q.. If t...

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“…These singularities correspond to resonances of the unit cell of the lattice (i.e., eigenvalues of the negative Laplacian on a cell of the lattice with quasi-periodic boundary conditions; see Ham and Segall (1961) and Barnett and Greengard (2010)). In the context of photonic crystals, they can be thought of as the result of the dispersion relation intersecting a light line.…”

Section: The Helmholtz Green's Function For a Two-dimensional Latmentioning

confidence: 99%

“…Most notable is the use of Ewald summation (Ham and Segall (1961); Jordan et al (1986);and Moroz (2006)) which produces a doubly infinite sum that is exponentially convergent, though for the case of a rectangular lattice, a method which leads to an exponentially convergent integral representation has also been developed by Dienstfrey et al (2001); see also Linton (2010, Sec. 2.2.2).…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional, phase modulated lattice sums with application to the Helmholtz Green’s function

Linton

2015

Journal of Mathematical Physics

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A class of two-dimensional phase modulated lattice sums in which the denominator is an indefinite quadratic polynomial Q is expressed in terms of a single, exponentially convergent series of elementary functions. This expression provides an extremely efficient method for the computation of the quasi-periodic Green’s function for the Helmholtz equation that arises in a number of physical contexts when studying wave propagation through a doubly periodic medium. For a class of sums in which Q is positive definite, our new result can be used to generate representations in terms of θ-functions which are significant generalisations of known results.

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Energy Bands in Periodic Lattices—Green's Function Method

Cited by 355 publications

References 16 publications

Classical vibrational modes in phononic lattices: theory and experiment

Classical vibrational modes in phononic lattices: theory and experiment

Electromagnetic Scattering of Finite and Infinite 3D Lattices in Polarizable Backgrounds

Two-dimensional, phase modulated lattice sums with application to the Helmholtz Green’s function

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