Spectral Properties of Thin-Film Photonic Crystals

Abstract

We study spectral properties of three-dimensional photonic crystals formed by a periodic array of air cubes separated by a thin film of optically dense dielectric material. The thickness δ of the dielectric component is assumed to be small, whereas its permittivity ε is large. The spectrum of the photonic crystal is studied as δ → 0 while εδ = η −1 is kept constant. Under the additional condition of a vanishing normal component of the magnetic field on the surface of the optically dense dielectric film, we carry out a thorough analytical and numerical analysis of the spectrum of the photonic crystal. In particular, for small η we obtain the linear and quadratic approximations of the spectral bands and, given δ, we estimate the minimum value of ε for which at least one spectral gap opens up.

Dispersion for the Schrödinger equation on networks

“…Function ωR ω f (x) can be analytically continued in a region containing the imaginary axis.Proof. The proof is an immediate consequence of decomposition(14) and of Proposition 3.2.…”

“…Differential operators on metric graphs arise in a variety of applications. We mention some of them: carbon nano-structures [26], photonic crystals [14], high-temperature granular superconductors [1], quantum waveguides [8], free-electron theory of conjugated molecules in chemistry, quantum chaos, etc. For more details we refer the reader to review papers [23], [24], [17] and [13].…”

Slow light in photonic crystals

“…To diagonalize T (ν) we use the approach outlined in [46] and used in [47]. Namely, there exists the following representation for the diagonal form X of T (ν) ζ = e −S(ν) W e S(ν) = W 0 + νζ 1 + ν 2 ζ 2 + · · · , S (ν) = νS 1 + ν 2 S 2 + · · · ,…”

“…The two forward modes contribute additively to the energy flux S T , but the contribution of the fast mode remains regular in the vicinity of ω a , while the contribution of the slow mode shows the same singular behavior as that described by Eqs. (47) and (49). Fig.…”

“…Particular constructions of the perturbation theory we discuss here follow from [46] and [47]. Suppose that W (ν) = W 0 + νW 1 + ν 2 W 2 + · · · (573) and T 0 is a diagonal matrix with distinct elements, i.e.…”