On The Localization Of Photons In Some Dielectric Media

The propagation of guided waves in 3-D photonic crystals is studied. In this paper, we suppose that the spectrum of reference medium (a perfectly 3-D photonic crystal) has band gaps. If a line defect is introduced along one of the lattice vectors, we prove that spectrum can be created inside the gaps of the reference spectrum if the line defect is sufficiently strong. Furthermore, we also prove that the generalized eigenfunctions corresponding to the generalized eigenvalues created by line defect are exponentially decaying away from the line defect.Photonic crystals (PCs) are periodically structured dielectric media, which are designed to favor band gaps. PCW can be created if a line defect along one of the lattice vectors is introduced in a PC. If the frequency of the electromagnetic waves lies in the band gap in the spectrum of the reference medium, it is proposed that an electromagnetic wave propagation along the line defect can be localized in the transverse plane of the PCW. Such electromagnetic waves are called guided waves in physics.Consider a 3-D photonic crystals medium characterized by the electric permittivity (?)0(x) and magnetic permeabilityμ0 (x), where (?)0(x),μ0(x) are bounded measurable functions in R³. Without loss of generality, we assume that (?)0(x) andμ0(x) are periodic functions with period Y = R²/Z²(the torus), i.e.,We introduce a Maxwell operator M₀, on the weighted Hilbert space L²(R³;μ(x)dx).Floquet-Bloch theory tells us that the spectrum of the periodic operator M₀ is the union of a countable number of bands; More precisely, there exist continuous periodic functionsλj(k), j = 1,2,3 ... on R³ with period 2Ï€, such thatwhere Q = [-Ï€,Ï€)³ is called Brillioun zone in physical literature [12, 22]. Bands can overlap and fill all the real semi-axis, or can be separated by the gaps. Existence of band gaps for some periodic dielectric and acoustic media have been studied in [3, 4, 5, 8, 10]. In this paper, we assume that the spectrum of the operator M₀ has at least one gap. We denote G = (a,b) as a gap in the spectrum of M₀, where 0 < a < b <∞. Now, we introduce a line defect, denoted asΩl, along the x1 -direction:whereΩl = lΩis the support of the line defect in the transverse plane (x2,x3). We suppose thatΩis the measurable compact subset of R². Without loss of generality, we also assume that the origin is an inner point of the setΩ. In this paper, we consider that the perturbed medium is homogenous inside the line defectΩl. More precisely:where (?)1,μ1 are constants. We adapt M as the perturbed operator according to the perturbed medium which is defined analogously to M₀. For a vector v∈Cⁿ, n∈N, we set For simplicity of notations, we shall denote x = (x1,x2,x3)∈R³, the transverse components by x' = (x2,x3).Lemma 1. Suppose M is a non-negative self-adjoint operator on the Hilbert space L²(R³,μ(x)dx;C³), and M needn't to be banded, so we havedistwhere ||·||_μis the norm of the space L²(R³,μ(x)dx; C³).Theorem 2. Suppose G is a band gap in the spectrum of the operator M₀, and the interval I_(?),d satisfies (5). The interval I_(?),d contains at least one point in the spectrum of the perturbed operator M, if the following inequality is satisfiedwhere nk' is unit vector which parallels with the vector (?)x'φ.We need an auxiliary estimate on the resolvent in a finite volume. It is often called Combes-Thomas estimates in mathematical physics literature.Lemma 3.There exists a constant mz, which only depend on the distant from z to the edge of the band gap in the spectrum of M₀, and C, the following estimates are satisfiedwhere the norms on R(z) and (?)×R(z) in the left hand are the operator norms in L²(R³;C³).We will show that the generalized eigenfunction created by a line defect decays exponentially away from the defect strip. Theorem 4. Suppose that G is a band gap in the spectrum of the operator M, and z∈σ(M)∩G, u is a generalized eigenfunction of the M according to z, then we havewhere Xx(y) is the characteristic function of the cube {y ||yj-xj|≤1, j = 1,2,3} centered at x.∏1(z) and∏2(z) are two positive constants depending on z.