Amit Hochman

Comparison of different methods for rigorous modeling of photonic crystal fibers

We present a summary of the simulation exercise carried out within the EC Cost Action P11 on the rigorous modeling of photonic crystal fiber (PCF) with an elliptically deformed core and noncircular air holes with a high fill factor. The aim of the exercise is to calculate using different numerical methods and to compare several fiber characteristics, such as the spectral dependence of the phase and the group effective indices, the birefringence, the group velocity dispersion and the confinement losses. The simulations are performed using four rigorous approaches: the finite element method (FEM), the source model technique (SMT), the plane wave method (PWM), and the localized function method (LFM). Furthermore, we consider a simplified equivalent fiber method (EFM), in which the real structure of the holey fiber is replaced by an equivalent step index waveguide composed of an elliptical glass core surrounded by air cladding. All these methods are shown to converge well and to provide highly consistent estimations of the PCF characteristics. Qualitative arguments based on the general properties of the wave equation are applied to explain the physical mechanisms one can utilize to tailor the propagation characteristics of nonlinear PCFs.

Efficient and spurious-free integral-equation-based optical waveguide mode solver

Modal analysis of waveguides and resonators by integra-lequation formulations can be hindered by the existence of spurious solutions. In this paper, spurious solutions are shown to be eliminated by introduction of a Rayleigh-quotient based matrix singularity measure. Once the spurious solutions are eliminated, the true modes may be determined efficiently and reliably, even in the presence of degeneracy, by an adaptive search algorithm. Analysis examples that demonstrate the efficacy of the method include an elliptical dielectric waveguide, two unequal touching dielectric cylinders, a plasmonic waveguide, and a realistic micro-structured optical fiber. A freely downloadable version of an optical waveguide mode solver based on this article is available.

Analysis of strictly bound modes in photonic crystal fibers by use of a source-model technique

We describe a source-model technique for the analysis of the strictly bound modes propagating in photonic crystal fibers that have a finite photonic bandgap crystal cladding and are surrounded by an air jacket. In this model the field is simulated by a superposition of fields of fictitious electric and magnetic current filaments, suitably placed near the media interfaces of the fiber. A simple point-matching procedure is subsequently used to enforce the continuity conditions across the interfaces, leading to a homogeneous matrix equation. Nontrivial solutions to this equation yield the mode field patterns and propagation constants. As an example, we analyze a hollow-core photonic crystal fiber. Symmetry characteristics of the modes are discussed and exploited to reduce the computational burden.

A stabilized discrete empirical interpolation method for model reduction of electrical, thermal, and microelectromechanical systems

We present a few modifications that stabilize nonlinear reduced order models generated by discrete empirical interpolation methods. We combine a different approach to linearization with a multipoint stabilization technique. The examples used to demonstrate our methods effectiveness are a nonlinear transmission line, a micromachined switch, and a nonlinear thermal model for an RF amplifier.

Rigorous modal analysis of metallic nanowire chains

Nanowire chains (NCs) are analyzed by use of a rigorous, full-wave, Source-Model Technique (SMT). The technique employs a proper periodic Greens function which converges regardless of whether the structure is lossless or lossy. By use of this Greens function, it is possible to determine the complex propagation constants of the NC modes directly and accurately, as solutions of a dispersion equation. To demonstrate the method, dispersion curves and mode profiles for a few NCs are calculated.

A Numerical Methodology for Efficient Evaluation of 2D Sommerfeld Integrals in the Dielectric Half-Space Problem

The analysis of 2D scattering in the presence of a dielectric half-space by integral-equation formulations involves repeated evaluation of Sommerfeld integrals. Deformation of the contour to the steepest-descent path results in a well-behaved integrand, that can be readily integrated. A well-known drawback of this method is that an analytical expression for the path is available only for evaluation of the reflected fields, but not for the evaluation of the transmitted fields. A simple scheme for numerical determination of the steepest-descent path, valid for both cases, is presented. The computational cost of the numerical determination is comparable to that of evaluating the analytical expression for the steepest-descent path for reflected fields. When necessary, contributions from branch-cut integrals and a second saddle point are taken into account. Certain ranges of the input parameters, which result in integrands that vary rapidly in the neighborhood of the saddle point, require special treatment. Alternative paths and specialized Gaussian quadrature rules for these cases are also proposed. An implementation of the proposed numerically determined steepest-descent path (ND-SDP) method is freely available for download.

Reduced-Order Models for Electromagnetic Scattering Problems

We consider model-order reduction of systems occurring in electromagnetic scattering problems, where the inputs are current distributions operating in the presence of a scatterer, and the outputs are their corresponding scattered fields. Using the singular-value decomposition (SVD), we formally derive minimal-order models for such systems. We then use a discrete empirical interpolation method (DEIM) to render the minimal-order models more suitable to numerical computation. These models consist of a set of elementary sources and a set of observation points, both interior to the scatterer, and located automatically by the DEIM. A single matrix then maps the values of any incident field at the observation points to the amplitudes of the sources needed to approximate the corresponding scattered field. Similar to a Greens function, these models can be used to quickly analyze the interaction of the scatterer with other nearby scatterers or antennas.

Calculation of confinement losses in photonic crystal fibers by use of a source-model technique

We extend our previous work on photonic-crystal fibers (PCFs) using the source-model technique to include leaky modes of fibers having a finite-sized photonic bandgap crystal (PBC) cladding. We concentrate on a hollow-core PCF and calculate the confinement losses by means of two different methods. The first method is more general but also more computationally expensive; we use sources that have a complex propagation constant and seek a transverse resonance in the complex plane. The second method, applicable only to modes with small confinement losses, uses sources with a real propagation constant to approximate leaky modes that have a propagation constant that is close to the real axis. We then apply Poyntings theorem to calculate the attenuation constant in a manner akin to the perturbation methods used to calculate the losses in finite-conductivity metal waveguides. This first approximation can be improved through iterative application of the algorithm, i.e., by use of sources with the attenuation constant found in the first approximation. The two methods are shown to be in good agreement with each other and with previously published results for solid-core PCFs. Numerical results show that, for the hollow-core PCF analyzed, many layers of PBC cladding are needed to attain confinement losses that are acceptable for telecommunications.

Modal dynamics in hollow-core photonic-crystal fibers with elliptical veins

Modal characteristics of hollow-core photonic-crystal fibers with elliptical veins are studied by use of a recently proposed numerical method. The dynamic behavior of bandgap guided modes, as the wavelength and aspect ratio are varied, is shown to include zero-crossings of the birefringence, polarization dependent radiation losses, and deformation of the fundamental mode.

On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems

A computational scheme for solving 2D Laplace boundary-value problems using rational functions as the basis functions is described. The scheme belongs to the class of desingularized methods, for which the location of singularities and testing points is a major issue that is addressed by the proposed scheme, in the context he 2D Laplace equation. Well-established rational-function fitting techniques are used to set the poles, while residues are determined by enforcing the boundary conditions in the least-squares sense at the nodes of rational Gauss-Chebyshev quadrature rules. Numerical results show that errors approaching the machine epsilon can be obtained for sharp and almost sharp corners, nearly-touching boundaries, and almost-singular boundary data. We show various examples of these cases in which the method yields compact solutions, requiring fewer basis functions than the Nystrom method, for the same accuracy. A scheme for solving fairly large-scale problems is also presented.