Exact solutions and dynamics of the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity

This paper investigate the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity by using the method of dynamical systems. The functions $q(x,t)=\phi(\xi)\exp(i(-kx+\omega t))$ are solutions of the equation (1.1) that governs the propagation of Raman solitons through optical metamaterials, where $\xi=x-vt$ and $\phi(\xi)$ in the solutions satisfy a singular planar dynamical system (1.5) which has two singular straight lines. By using the bifurcation theory method of dynamical systems to the equation of $\phi(\xi)$, bifurcations of phase portraits for this dynamical system are obtained under 28 different parameter conditions. Based on those phase portraits, 62 exact solutions of system (1.5) including periodic solutions, heteroclinic and homoclinic solutions, periodic peakons and peakons as well as compacton solutions are derived.     

The interaction of electron beams and electromagnetic fields in relativistic Cerenkov generators

A method of numerical analysis of linear and nonlinear nonstationary processes in relativistic Cerenkov generators based on periodic superdimensional waveguides is proposed. The main idea lies in considering a nonregular waveguide as a sequence of wave transformers and using a cross-section method. Configurations of eigenwave fields of periodic waveguides with a high-current relativistic electron beam are considered. The processes of generation development in a section of a relativistic multiwave generator are studied. We show that the system frequency is proved to be determined by the longitudinal resonances of surface waves and internal feedback.     

Gap opening and split band edges in waveguides coupled by a periodic system of small windows

Using an example of two coupled waveguides, we construct a periodic second-order differential operator acting in a Euclidean domain and having spectral gaps whose edges are attained strictly inside the Brillouin zone. The waveguides are modeled by the Laplacian in two infinite strips of different width that have a common interior boundary. On this common boundary, we impose the Neumann boundary condition, but cut out a periodic system of small windows, while on the remaining exterior boundary we impose the Dirichlet boundary condition. It is shown that, by varying the widths of the strips and the distance between the windows, one can control the location of the extrema of the band functions as well as the number of the open gaps. We calculate the leading terms in the asymptotics for the gap lengths and the location of the extrema.     

Second kind integral equation formulation for the mode calculation of optical waveguides

We present a second kind integral equation (SKIE) formulation for calculating the electromagnetic modes of optical waveguides, where the unknowns are only on material interfaces. The resulting numerical algorithm can handle optical waveguides with a large number of inclusions of arbitrary irregular cross section. It is capable of finding the bound, leaky, and complex modes for optical fibers and waveguides including photonic crystal fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation is well conditioned even in the case of nonsmooth geometries. Our method is highly accurate and thus can be used to calculate the propagation loss of the electromagnetic modes accurately, which provides the photonics industry a reliable tool for the design of more compact and efficient photonic devices. We illustrate and validate the performance of our method through extensive numerical studies and by comparison with semi-analytical results and previously published results.     

Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equations

For a formally self-adjoint elliptic system of partial differential equations with periodic coefficients in the space ℝ ⁿ , we show that, by conferring contrast properties to the coefficients of differential operators, one can open a gap in the essential spectrum of the system. We suggest a method based on the derivation of an asymptotically sharp generalized Korn inequality and the use of the maximin principle; this method applies to perforated media as well as to periodic layered and quasi-cylindrical waveguides.     

Bifurcations and Exact Solutions of the Raman Soliton Model in Nanoscale Optical Waveguides with Metamaterials

In this paper, we study Raman soliton model in nanoscale optical waveguides with metamaterials, having polynomial law non-linearity. By using the bifurcation theory method of dynamical systems to the equations of $\phi(\xi)$, under 24 different parameter conditions, we obtain bifurcations of phase portraits and different traveling wave solutions including periodic solutions, homoclinic and heteroclinic solutions for planar dynamical system of the Raman soliton model. Under different parameter conditions, 24 exact explicit parametric representations of the traveling wave solutions are derived. The dynamic behavior of these traveling wave solutions are meaningful and helpful for us to understand the physical structures of the model.     

Resolving Operators for a Three-Dimensional Periodic Waveguide

We construct the solving operators of a three-dimensional periodic dielectric waveguide with absorption. The results obtained can be applied to the problem describing a dropping wave onto the interface of two periodic dielectric waveguides with absorption. Bibliography: 6 titles.     

Opening gaps in the spectrum of the water-wave problem in a periodic channel

It is shown that the essential spectrum of a problem in the theory of linear water waves in a periodic channel can contain any prescribed number of gaps. One of such waveguides consists of a periodic family of identical ponds of unit size connected by narrow shallow channels. The effect of gap opening is achieved by decreasing a geometric parameter describing the size of these channels.     

Darboux transformation and soliton solutions for the coupled cubic-quintic nonlinear Schrödinger equations in nonlinear optics

In this paper, by virtue of the Darboux transformation (DT) and symbolic computation, the quintic generalization of the coupled cubic nonlinear Schrödinger equations from twin-core nonlinear optical fibers and waveguides are studied, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations is obtained and the corresponding DT is constructed. Moreover, one-, two- and three-soliton solutions are presented in the forms of modulus. Features of solitons are graphically discussed: (1) head-on and overtaking elastic collisions of the two solitons; (2) periodic attraction and repulsion of the bounded states of two solitons; (3) energy-exchanging collisions of the three solitons.     

Asymptotics of bound states and bands for waveguides coupled through small windows

A system of two waveguides coupled laterally through small windows is considered. The asymptotics (in the width of windows) of ground state close to the threshold is obtained for the case of finite number of apertures. The cases of periodic system of coupling windows is studied. The asymptotics of the band edges is obtained. The technique is matching of asymptotic expansions of the solutions.     

On the explicit parametric description of waves in periodic media

A method for the parameterization of the one-dimensional wave equation is proposed that makes it possible to find its solution by quadratures under an arbitrary dependence of the refraction index on the current wave phase. The form of the solution found is used to investigate the structure of the wave function for a periodic refraction index. Explicit expressions for the fundamental system of solutions and for the Floquet index are obtained. Examples of applying the proposed method to the optimal synthesis of multilayer interference mirrors and Bragg waveguides are discussed.     

Asymptotic behavior of spectral gaps in a regularly perturbed periodic waveguide

Since the spectrum of a periodic waveguide is the union of a countable family of closed bounded segments (spectral bands), it can contain opened spectral gaps, i.e., intervals in the real positive semi-axis that are free of the spectrum but have both endpoints in it. A cylindrical waveguide has an intact spectrum that is a closed ray. We consider a small periodic perturbation of the waveguide wall, and, by means of an asymptotic analysis of the eigenvalues in the model problem on the periodicity cell, we show how a spectral gap opens when the cylindrical waveguide converts into a periodic one. Indeed, a cylindrical waveguide can be interpreted as a periodic one with an arbitrary period, but all its spectral bands touch each other. A periodic perturbation of the waveguide wall provides the splitting of the band edges. This effect is known in the physical literature for waveguides of different shapes, and, in this paper, we provide a rigorous mathematical proof of the effect. Several variants of the edge splitting (alone and coupled, simple and multiple knots) are examined. Explicit formulas are obtained for a plane waveguide.     

Absolute Continuity in Periodic Waveguides

We study second order elliptic operators with periodic coefficientsin two-dimensional simply connected periodic waveguides withthe Dirichlet or Neumann boundary conditions. It is proved thatunder some mild smoothness restrictions on the coefficients,such operators have purely absolutely continuous spectra. Theproof follows a method suggested previously by A. Morame totackle periodic operators with variable coefficients in dimension2. 2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.     

Resonances for multistratified acoustic waveguides

The equation of acoustic oscillations in multistratified waveguides is considered. It is assumed that the properties of the medium do not depend on the longitudinal coordinate in a neighbourhood of infinity and may be different at different ends of the waveguides. It is proved that the truncated resolvent of the corresponding operator admits an analytical continuation through the continuous spectrum. The singularities (poles, branching points) of the truncated resolvent on the continuous spectrum are investigated. The large time asymptotic behavior of the compulsory oscillations due to periodic forces is obtained.     

Quasiconformal mappings and periodic spectral problems in dimension two

We study spectral properties of second-order elliptic operators with periodic coefficients in dimension two. These operators act in periodic simply-connected waveguides, with either Dirichlet, or Neumann, or the third boundary condition. The main result is the absolute continuity of the spectra of such operators. The cornerstone of the proof is an isothermal change of variables, reducing the metric to a flat one and the waveguide to a straight strip. The main technical tool is the quasiconformal variant of the Riemann mapping theorem. This work is supported by The Royal Society.     

Spectrum created by line defects in periodic structures

We study a Helmholtz‐type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three‐dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.     

Gap detection in the spectrum of an elastic periodic waveguide with a free surface

A three-dimensional periodic elastic waveguide is constructed whose continuous spectrum (the frequencies that admit propagating waves) contains a gap, i.e., an interval that has its ends in the continuous spectrum but contains at most a discrete spectrum. The waveguide consists of an infinite chain of massive bodies connected by short thin links, and its surface is assumed to be free. The method for detecting a gap also applies to plane problems, including scalar ones. Periodic elastic waveguides with different shapes or contrasting properties are indicated in which a gap can also be detected.     

Non-spectral Field Representations in Dielectric Fibre Guides

Open waveguiding structures, such as acoustic, piezoelectric,gyrotropic and dielectric waveguides, are used for a varietyof applications. An example of such open waveguides is the dielectricfibre guide used in the construction of optical communicationsystems. An understanding of the modal properties of this guideis therefore of fundamental interest to the understanding ofthe modal properties of similar open structures. We analysein detail the transverse electric (TE) and transverse magnetic(TM) polarized electromagnetic fields within a circularly cylindricaldielectric waveguide with arbitrary non-vanishing analytic radialpermittivity variation. It is shown that the field can be representedas an infinite sum of functions, including the finite numberof spectral (surface wave) and an infinity of non-spectral (complex)eigenfunctions. Additional functions are also included. Boththe field polarization and the behaviour of the permittivityat the waveguide boundary are found to affect the form and thevalidity of the representation.     

Diffraction problem for periodic waveguides

The unique solvability of the diffraction problem on a junction of three periodic waveguides and continuous dependence of the solution on the data are established. Bibliography: 7 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 37, 2008, pp. 23–36.     

Trapped modes for periodic structures in waveguides

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. Copyright © 2004 John Wiley & Sons, Ltd.