Evaluation of exact boundary mappings for one-dimensional semi-infinite periodic arrays

Periodic arrays are structures consisting of geometrically identical subdomains, usually called periodic cells. In this paper, by taking the Helmholtz equation as a model, we consider the definition and evaluation of the exact boundary mappings for general one-dimensional semi-infinite periodic arrays for any real wavenumber. The well-posedness of the Helmholtz equation is established via the limiting absorption principle. An algorithm based on the doubling procedure and extrapolation technique is proposed to derive the exact Sommerfeld-to-Sommerfeld boundary mapping. The advantages of this algorithm are the robustness and simplicity of implementation. But it also suffers from the high computational cost and the resonance wave numbers. To overcome these shortcomings, we propose another algorithm based on a conjecture about the asymptotic behaviour of limiting absorption principle solutions. The price we have to pay is the resolution of two generalized eigenvalue problems, but still the overall computational cost is significantly reduced. Numerical evidences show that this algorithm presents theoretically the same results as the first algorithm. Moreover, some quantitative comparisons between these two algorithms are given

Waveguide photonic limiters based on topologically protected resonant modes

We propose a concept of chiral photonic limiters utilising topologically protected localised midgap defect states in a photonic waveguide. The chiral symmetry alleviates the effects of structural imperfections and guaranties a high level of resonant transmission for low intensity radiation. At high intensity, the light-induced absorption can suppress the localised modes, along with the resonant transmission. In this case the entire photonic structure becomes highly reflective within a broad frequency range, thus increasing dramatically the damage threshold of the limiter. Here we demonstrate experimentally the principle of operation of such photonic structures using a waveguide consisting of coupled dielectric microwave resonators.Comment: 6 pages, 4 figure

Fast numerical methods for waves in periodic media

Periodic media problems widely exist in many modern application areas like semiconductor nanostructures (e.g.\ quantum dots and nanocrystals), semi-conductor superlattices, photonic crystals (PC) structures, meta materials or Bragg gratings of surface plasmon polariton (SPP) waveguides, etc. Often these application problems are modeled by partial differential equations with periodic coefficients and/or periodic geometries. In order to numerically solve these periodic structure problems efficiently one usually confines the spatial domain to a bounded computational domain (i.e.\ in a neighborhood of the region of physical interest). Hereby, the usual strategy is to introduce so-called \emph{artificial boundaries} and impose suitable boundary conditions. For wave-like equations, the ideal boundary conditions should not only lead to well-posed problems, but also mimic the perfect absorption of waves traveling out of the computational domain through the artificial boundaries. In the first part of this chapter we present a novel analytical impedance expression for general second order ODE problems with periodic coefficients. This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary conditions is then used for computing the bound states of the Schr\"odinger operator with periodic potentials at infinity. Other potential applications are associated with the exact artificial boundary conditions for some time-dependent problems with periodic structures. As an example, a two-dimensional hyperbolic equation modeling the TM polarization of the electromagnetic field with a periodic dielectric permittivity is considered. In the second part of this chapter we present a new numerical technique for solving periodic structure problems. This novel approach possesses several advantages. First, it allows for a fast evaluation of the Sommerfeld-to-Sommerfeld operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. In the sequel we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schr\"odinger equation and the Helmholtz equation with damping. Finally, in the third part we consider periodic arrays that are structures consisting of geometrically identical subdomains, usually called periodic cells. We use the Helmholtz equation as a model equation and consider the definition and evaluation of the exact boundary mappings for general semi-infinite arrays that are periodic in one direction for any real wavenumber. The well-posedness of the Helmholtz equation is established via the \emph{limiting absorption principle} (LABP). An algorithm based on the doubling procedure of the second part of this chapter and an extrapolation method is proposed to construct the exact Sommerfeld-to-Sommerfeld boundary mapping. This new algorithm benefits from its robustness and the simplicity of implementation. But it also suffers from the high computational cost and the resonance wave numbers. To overcome these shortcomings, we propose another algorithm based on a conjecture about the asymptotic behaviour of limiting absorption principle solutions. The price we have to pay is the resolution of some generalized eigenvalue problem, but still the overall computational cost is significantly reduced. Numerical evidences show that this algorithm presents theoretically the same results as the first algorithm. Moreover, some quantitative comparisons between these two algorithms are given

Fast numerical methods for waves in periodic media

Periodic media problems widely exist in many modern application areas like semiconductor nanostructures (e.g. quantum dots and nanocrystals), semi-conductor superlattices, photonic crystals (PC) structures, meta materials or Bragg gratings of surface plasmon polariton (SPP) waveguides, etc. Often these application problems are modeled by partial differential equations with periodic coefficients and/or periodic geometries. In order to numerically solve these periodic structure problems efficiently one usually confines the spatial domain to a bounded computational domain (i.e. in a neighborhood of the region of physical interest). Hereby, the usual strategy is to introduce so-called artificial boundaries and impose suitable boundary conditions. For wave-like equations, the ideal boundary conditions should not only lead to w ell-posed problems, but also mimic the perfect absorption of waves traveling out of the computational domain through the artificial boundaries ..

Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides

We study the propagation of time-harmonic acoustic or transverse magnetic (TM) polarized electromagnetic waves in a periodic waveguide lying in the semi-strip $(0,\infty)\times(0,L)$. It is shown that there exists a Riesz basis of the space of solutions to the time-harmonic wave equation such that the translation operator shifting a function by one periodicity length to the left is represented by an infinite Jordan matrix which contains at most a finite number of Jordan blocks of size $> 1$. Moreover, the Dirichlet-, Neumann- and mixed traces of this Riesz basis on the left boundary also form a Riesz basis. Both the cases of frequencies in a band gap and frequencies in the spectrum and a variety of boundary conditions on the top and bottom are considered

Windowed Green Function Method for Nonuniform Open-Waveguide Problems

This contribution presents a novel Windowed Green Function (WGF) method for the solution of problems of wave propagation, scattering and radiation for structures which include open (dielectric) waveguides, waveguide junctions, as well as launching and/or termination sites and other nonuniformities. Based on use of a "slow-rise" smooth-windowing technique in conjunction with free-space Green functions and associated integral representations, the proposed approach produces numerical solutions with errors that decrease faster than any negative power of the window size. The proposed methodology bypasses some of the most significant challenges associated with waveguide simulation. In particular the WGF approach handles spatially-infinite dielectric waveguide structures without recourse to absorbing boundary conditions, it facilitates proper treatment of complex geometries, and it seamlessly incorporates the open-waveguide character and associated radiation conditions inherent in the problem under consideration. The overall WGF approach is demonstrated in this paper by means of a variety of numerical results for two-dimensional open-waveguide termination, launching and junction problems.Comment: 16 Page

Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.Comment: to appear in SIAM Journal on Mathematical Analysi

The limiting absorption principle and a radiation condition for the scattering by a periodic layer

Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. In this paper, we consider a medium that is defined in the upper two-dimensional half-space by a penetrable and periodic contrast. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. Our method of proof seems to be new: By the Floquet-Bloch transform we first reduce the scattering problem to a finite-dimensional one that is set in the linear space spanned by all surface waves. In this space, we then compute explicitly which modes propagate along the periodic structure to the left or to the right. This finally yields a representation for our limiting absorption solution which leads to a proper extension of the well known upward propagating radiation condition. Finally, we prove uniqueness of a solution under this radiation condition