A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides

This works deals with one dimensional infinite perturbation - namely line defects - in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and the computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann (DtN) approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. On contrary to existing methods, this one is exact but there is a price to be paid : the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature

A Bloch wave numerical scheme for scattering problems in periodic wave-guides

We present a new numerical scheme to solve the Helmholtz equation in a wave-guide. We consider a medium that is bounded in the $x_2$-direction, unbounded in the $x_1$-direction and $\varepsilon$-periodic for large $|x_1|$, allowing different media on the left and on the right. We suggest a new numerical method that is based on a truncation of the domain and the use of Bloch wave ansatz functions in radiation boxes. We prove the existence and a stability estimate for the infinite dimensional version of the proposed problem. The scheme is tested on several interfaces of homogeneous and periodic media and it is used to investigate the effect of negative refraction at the interface of a photonic crystal with a positive effective refractive index.Comment: 25 pages, 10 figure

Theoretical and numerical studies of wave-packet propagation in tokamak plasmas

Theoretical and numerical studies of wave-packet propagation are presented to analyze the time varying 2D mode structures of electrostatic fluctuations in tokamak plasmas, using general flux coordinates. Instead of solving the 2D wave equations directly, the solution of the initial value problem is used to obtain the 2D mode structure, following the propagation of wave-packets generated by a source and reconstructing the time varying field. As application, the 2D WKB method is applied to investigate the shaping effects (elongation and triangularity) of tokamak geometry on the lower hybrid wave propagation and absorbtion. Meanwhile, the Mode Structure Decomposition (MSD) method is used to handle the boundary conditions and simplify the 2D problem to two nested 1D problems. The MSD method is related to that discussed earlier by Zonca and Chen [Phys. Fluids B 5, 3668 (1993)], and reduces to the well-known "ballooning formalism" [J. W. Connor, R. J. Hastie, and J. B. Taylor, Phys. Rev. Lett. 40, 396 (1978)], when spatial scale separation applies. This method is used to investigate the time varying 2D electrostatic ITG mode structure with a mixed WKB-full-wave technique. The time varying field pattern is reconstructed and the time asymptotic structure of the wave-packet propagation gives the 2D eigenmode and the corresponding eigenvalue. As a general approach to investigate 2D mode structures in tokamak plasmas, our method also applies for electromagnetic waves with general source/sink terms, either by an internal/external antenna or nonlinear wave interaction with zonal structures.Comment: 24 pages, 14 figure

Numerical methods for scattering problems in periodic waveguides

In this paper, we propose new numerical methods for scattering problems in periodic waveguides. Based on [20], the “physically meaningful” solution, which is obtained via the Limiting Absorption Principle (LAP) and is called an LAP solution, is written as an integral of quasi-periodic solutions on a contour. The definition of the contour depends both on the wavenumber and the periodic structure. The contour integral is then written as the combination of finite propagation modes and a contour integral on a small circle. Numerical methods are developed and based on the two representations. Compared with other numerical methods, we do not need the LAP process during numerical approximations, thus a standard error estimation is easily carried out. Based on this method, we also develop a numerical solver for halfguide problems. The method is based on the result that any LAP solution of a halfguide problem can be extended to the LAP solution of a fullguide problem. At the end of this paper, we also give some numerical results to show the efficiency of our numerical methods

High order methods to simulate scattering problems in locally perturbed periodic waveguides

In this paper, two high order numerical methods, the CCI method and the decomposition method, are propose to simulate wave propagation in locally perturbed periodic closed waveguides. As is well known the problem is not always uniquely solvable due to the existence of guided modes, the limiting absorption principle is a standard way to get the unique physical solution. Both methods are based on the Floquet-Bloch transform which transforms the original problem to an equivalent family of cell problems. The CCI method is based on a modification of integral contours of the inverse transform, and the decomposition method comes from an explicit definition of the radiation condition. Due to the local perturbation, the family of cell problems are coupled thus the whole system is actually defined in 3D. Based on different types of singularities, high order methods are developed for faster convergence rates. Finally we show the convergence results by both theoretical explanations and numerical examples

Dissipative solitons in pattern-forming nonlinear optical systems : cavity solitons and feedback solitons

Many dissipative optical systems support patterns. Dissipative solitons are generally found where a pattern coexists with a stable unpatterned state. We consider such phenomena in driven optical cavities containing a nonlinear medium (cavity solitons) and rather similar phenomena (feedback solitons) where a driven nonlinear optical medium is in front of a single feedback mirror. The history, theory, experimental status, and potential application of such solitons is reviewed

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

Hydro-micromechanical modeling of wave propagation in saturated granular media

Biot's theory predicts the wave velocities of a saturated poroelastic granular medium from the elastic properties, density and geometry of its dry solid matrix and the pore fluid, neglecting the interaction between constituent particles and local flow. However, when the frequencies become high and the wavelengths comparable with particle size, the details of the microstructure start to play an important role. Here, a novel hydro-micromechanical numerical model is proposed by coupling the lattice Boltzmann method (LBM) with the discrete element method (DEM. The model allows to investigate the details of the particle-fluid interaction during propagation of elastic waves While the DEM is tracking the translational and rotational motion of each solid particle, the LBM can resolve the pore-scale hydrodynamics. Solid and fluid phases are two-way coupled through momentum exchange. The coupling scheme is benchmarked with the terminal velocity of a single sphere settling in a fluid. To mimic a pressure wave entering a saturated granular medium, an oscillating pressure boundary condition on the fluid is implemented and benchmarked with one-dimensional wave equations. Using a face centered cubic structure, the effects of input waveforms and frequencies on the dispersion relations are investigated. Finally, the wave velocities at various effective confining pressures predicted by the numerical model are compared with with Biot's analytical solution, and a very good agreement is found. In addition to the pressure and shear waves, slow compressional waves are observed in the simulations, as predicted by Biot's theory.Comment: Manuscript submitted to International Journal for Numerical and Analytical Methods in Geomechanic