Abstract:
A spatially localized initial condition for an energy-conserving wave equation with periodic coefficients disperses (spatially spreads) and decays in amplitude as time advances. This dispersion is associated with the continuous spectrum of the underlying differential operator and the absence of discrete eigenvalues. The introduction of spatially localized perturbations in a periodic medium leads to defect modes, states in which energy remains trapped and spatially localized. In this paper we study weak, localized perturbations of one-dimensional periodic Schr\"odinger operators. Such perturbations give rise to such defect modes, and are associated with the emergence of discrete eigenvalues from the continuous spectrum. Since these isolated eigenvalues are located near a spectral band edge, there is strong scale-separation between the medium period and the localization length of the defect mode. Bound states therefore have a multi-scale structure: a "carrier Bloch wave" times a "wave envelope", which is governed by a homogenized Schr\"odinger operator with associated effective mass, depending on the spectral band edge which is the site of the bifurcation. Our analysis is based on a reformulation of the eigenvalue problem in Bloch quasi-momentum space, using the Gelfand-Bloch transform and a Lyapunov-Schmidt reduction to a closed equation for the near-band-edge frequency components of the bound state. A rescaling of the latter equation yields the homogenized effective equation for the wave envelope, and approximations to bifurcating eigenvalues and eigenfunctions.

Abstract:
Let $Q(x)$ denote a periodic function on the real line. The Schr\"odinger operator, $H_Q=-\partial_x^2+Q(x)$, has $L^2(\mathbb{R})-$ spectrum equal to the union of closed real intervals separated by open spectral gaps. In this article we study the bifurcation of discrete eigenvalues (point spectrum) into the spectral gaps for the operator $H_{Q+q_\epsilon}$, where $q_\epsilon$ is spatially localized and highly oscillatory in the sense that its Fourier transform, $\widehat{q}_\epsilon$ is concentrated at high frequencies. Our assumptions imply that $q_\epsilon$ may be pointwise large but $q_\epsilon$ is small in an average sense. For the special case where $q_\epsilon(x)=q(x,x/\epsilon)$ with $q(x,y)$ smooth, real-valued, localized in $x$, and periodic or almost periodic in $y$, the bifurcating eigenvalues are at a distance of order $\epsilon^4$ from the lower edge of the spectral gap. We obtain the leading order asymptotics of the bifurcating eigenvalues and eigenfunctions. Underlying this bifurcation is an effective Hamiltonian associated with the lower edge of the $(b_*)^{\rm th}$ spectral band: $H^\epsilon_{\rm eff}=-\partial_x A_{b_*,\rm eff}\partial_x - \epsilon^2 B_{b_*,\rm eff} \times \delta(x)$ where $\delta(x)$ is the Dirac distribution, and effective-medium parameters $A_{b_*,\rm eff},B_{b_*,\rm eff}>0$ are explicit and independent of $\epsilon$. The potentials we consider are a natural model for wave propagation in a medium with localized, high-contrast and rapid fluctuations in material parameters about a background periodic medium.

Abstract:
In this paper the characteristic matrix method is used to study the propagation of electromagnetic waves through one-dimensional lossy photonic crystals composed of negative and positive refractive index material layers with symmetric and asymmetric geometric structures with a defect layer at the center of the structure. First, the positive index material defect layer is considered, and the effects of the polarization and the angle of incidence on the defect mode in the transmission spectra of the both structures are investigated. The results show that the number of the defect modes in the transmission spectra depends on the geometry (symmetric or asymmetric) of the structure. In addition, it is shown that the defect mode frequency increases as the angle of incidence increases. This property is independent of the geometry of the structure. Then, for normal incidence, the negative index material defect layer is considered, and the properties of defect modes for both structures are investigated. The results can lead to designing new types of transmission narrow filters.

Abstract:
The Laplace operator is considered for waveguides perturbed by a periodic structure consisting of N congruent obstacles spanning the waveguide. Neumann boundary conditions are imposed on the periodic structure, and either Neumann or Dirichlet conditions on the guide walls. It is proven that there are at least N (resp. N-1) trapped modes in the Neumann case (resp. Dirichlet case) under fairly general hypotheses, including the special case where the obstacles consist of line segments placed parallel to the waveguide walls. This work should be viewed as an extension of "Periodic structures on waveguides" by Linton and McIvor.

Abstract:
We present an approach for the description of fluorescence from optically active material embedded in layered periodic structures. Based on an exact electromagnetic Green's tensor analysis, we determine the radiative properties of emitters such as the local photonic density of states, Lamb shifts, line widths etc. for a finite or infinite sequence of thin alternating plasmonic and dielectric layers. In the effective medium limit, these systems may exhibit hyperbolic dispersion relations so that the large wave-vector characteristics of all constituents and processes become relevant. These include the finite thickness of the layers, the nonlocal properties of the constituent metals, and local-field corrections associated with an emitter's dielectric environment. In particular, we show that the corresponding effects are non-additive and lead to considerable modifications of an emitter's luminescence properties.

Abstract:
We report dissipative surface solitons forming at the interface between a semi-infinite lattice and a homogeneous Kerr medium. The solitons exist due to balance between amplification in the near-surface lattice channel and two-photon absorption. The stable dissipative surface solitons exist in both focusing and defocusing media, when propagation constants of corresponding states fall into a total semi-infinite and or into one of total finite gaps of the spectrum (i.e. in a domain where propagation of linear waves is inhibited for the both media). In a general situation, the surface solitons form when amplification coefficient exceeds threshold value. When a soliton is formed in a total finite gap there exists also the upper limit for the linear gain.

Abstract:
We consider the discrete eigenvalues of the operator $H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x)$, where $V(\x)$ is periodic and $Q(\y)$ is localized on $\R^d,\ \ d\ge1$. For $\eps>0$ and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schr\"odinger operator, $H_0 = -\Delta_\x+V(\x)$, into spectral gaps. The nature of the bifurcation depends on the homogenized Schr\"odinger operator $L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y)$. Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of the bifurcation.

Abstract:
Transmission properties of one-dimensional lossy photonic crystals composed of negative and positive refractive index layers with one lossless defect layer at the center of the crystal are investigated by the characteristic matrix method. The results show that as the refractive index and thickness of the defect layer increase the frequency of the defect mode decreases. In addition, it is shown that the frequency of the defect mode is sensitive to the incidence angle, polarization and physical properties of the defect layer but it is insensitive to the small lattice loss factor. The height of the defect mode is very sensitive to the loss factor, incidence angle, polarization, refractive index and thickness of the defect layer. It was also shown that the height and the width of the defect mode are affected by the number of the lattice period and the loss factor. The results can lead to designing new types of narrow filter structures and other optical devices.

Abstract:
This paper stuides numerically the model equation in a one dimensional defective photonic lattice by modifying the potential function to a periodic function. It is found that defect modes (DMs) can be regarded as Bloch modes which are excited from the extended photonic band-gap structure at Bloch wave-numbers with kx = 0. The DMs for both positive and negative defects are considered in this method.

Abstract:
A theoretical study of optical properties of phase shift defects in one-dimensional asymmetrical photonic structures consisting of two rugate segments with different periodicities at both normal and oblique incidence is presented. Using the propagation matrix method we numerically calculated transmittance spectra, defect wavelengths, energy density distributions, and group velocities for TE and TM waves, respectively. Our study shows that by adjusting the periodicity of one rugate segment, the defect wavelengths can be shifted toward either a shorter wavelength or a longer wavelength. The differences of the energy density distributions of TE and TM waves at different angles of incidence are explained with the help of group velocity. Effects of the change of the period of one rugate segment on the peak energy densities of defect modes and minimum group velocities at different angles of incidence are also investigated.