# Simulating Photons and Plasmons in a Three-dimensional Lattice Page: 4 of 13

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as left-handed materials support electromagnetic waves with a very counter-

intuitive behavior: their group and phase velocities oppose each other. Left-

handed materials can be used for developing, for example, "perfect" lenses [2]

capable of sub-wavelength resolution.

Key to understanding the bulk properties of the infinitely extended composite

materials is the accurate determination of the band structure of the electro-

magnetic waves by solving Maxwell's equations. In the frequency domain, the

wave equation becomes

Vx (--'.V x E) = c E-E (1)

where E is the electric field and p (E) the permeability (permittivity) of the

medium. Both e and p are spatially varying tensors. The perfect conductor

boundary condition E x n = 0, where n is the normal, is applied at the

conductor boundaries. Because of the system periodicity, and according to

Floquet's theorem, the electric field can be expressed as E exp ik-x, where

E(x+L) = E(x) is a periodic function, L is any one of the periodicity vectors,

and k is inside the Brillouin zone. Therefore, it is sufficient to solve Eq. (1)

inside the unit cell, which we assume to be a parallelepiped with the dimensions

Lx Lx XL2. The equation for E can be obtained from (1) by replacing V by

V + ik:

W2

a

(V + ik) x [ i-' (V + ik) x E] = -c E-. (2)

C2

Here, k-L represents, up to a term 27r(m+n+1) (m, n and 1 are integers), the

phase shift experienced by the wave across the unit cell. For m = n = l = 0,

k can be thought of as the wave vector of an incident wave. From now on we

will focus on E and drop the-for notational simplicity.

Note that we have chosen here to write the equation for E. The equation for

the magnetic field H is similar to (2) except for e and p playing opposite roles.

Hence, the pairs (E, E) and (pt, H*) can be regarded as dual of each other (the

complex * and Hermitian t conjugates are required to derive the Poynting

flux involving E only from an expression solely based on H and vice-versa).

This duality extends to the jump conditions across permittivity discontinuities

([[n-E-E]] = 0) and permeability discontinuities ([[n-p-H]] = 0) but not, how-

ever, to the boundary conditions at the surface of perfect conductors. While

the tangential E (Et) must be set to zero there, it is the normal component of

H (Hn) that is required to vanish at the conductor so that the mode structure

of magnetic and electric fields will in general to be fundamentally different. It

turns out that solving for H is from a coding viewpoint straightforward be-

cause Hn = 0 are the default (homogeneous natural) boundary conditions of2

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Pletzer, A. & Shvets, G. Simulating Photons and Plasmons in a Three-dimensional Lattice, report, September 3, 2002; Princeton, New Jersey. (digital.library.unt.edu/ark:/67531/metadc734157/m1/4/: accessed October 15, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.