F.P.H. van Beckum

Numerical studies of 2D photonic crystals: Waveguides, coupling between waveguides and filters

In photonic crystals, light propagation is forbidden in a certain wavelength range, the bandgap. In a two-dimensional crystal composed of parallel high-refractive index rods in a low-index background a line defect can be formed by removing a row of these rods, which can act as a waveguide for frequencies in the bandgap of the crystal. In order to get more insight into the main features of such waveguides we have studied a number of properties, using simulation tools based on the finite difference time domain method and a finite element Helmholtz solver. We show conceptually simple methods for determining the bandgap of the crystal as well as the dispersion of a waveguide for wavelengths in this bandgap. For practical applications, it is also important to know how much light can be coupled into the waveguide. Therefore, the coupling of light from a dielectric slab waveguide into the photonic crystal waveguide has been examined, showing that a coupling efficiency of up to 83% can be obtained between a silicon oxide slab and a waveguide in a crystal of silicon rods. Finally, calculations on an ultra-compact filter based on reflectively terminated side-branches of waveguides (similar to tuned stubs in microwave engineering) are shown and discussed.

New true fourth-order accurate scalar beam propagation methods for both TE and TM polarization

New 2D scalar beam propagation methods for both TE and TM polarization are presented. Both second- and fourth-order accurate schemes, in the lateral stepsize, are shown. The methods use uniform discretization and can handle arbitrary positions of interfaces between materials with different refractive indices. Either Transparent Boundary Conditions or Perfectly Matched Layers are used at the boundary of the computational window.

Numerical solution of the transverse resistivity of superconducting cables under AC conditions

The authors develop a numerical method for calculating the transverse resistivity of superconducting cables. A superconducting cable consists of a twisted bundle of strands with a nonconducting inner region. If such a cable is placed in an external magnetic field, the induced currents will not merely flow in the axial direction, but also around the center, in the plane of the cross section. It is shown that the transverse transport current, which is induced by external fields acting on the cable, can saturate most of the filaments of the superconducting layer. This results in a smaller maximal value of a longitudinal transport current and small coupling losses. >

Calculations on the current density and the voltage-current relation under a.c. conditions of filaments

Technical applications of multifilamentary wires indicate that filaments are used in complex magnetic fields (a combination of non-parallel a.c./d.c. transverse and rotating fields) carrying an a.c./d.c. transport current of various frequency. Furthermore, due to technical manufacturing processes the filaments are heavily distorted. Therefore, a numerical model is developed to compute the current density of a filament of arbitrary shape in any external transverse field carrying an a.c./d.c. transport current. The great flexibility of the model is shown in several examples.

Coupling losses in superconducting, torus-shaped wires due to applied magnetic field changes

The stationary electric field, current pattern and coupling losses in a multfilamentary, superconducting, twisted, torus-shaped wire are calculated for a torus placed in a homogeneous magnetic field increasing in time at a constant rate and parallel to the torus plane. The radius of the wire is considered to be small compared to the mean radius of the torus. An important parameter for the problem is the ratio between the twist length of the superconducting filaments and the mean radius of the torus. In the configuration considered this parameter is small. The coupling losses are approximately inversely proportional to the square of this ratio. Furthermore, for the wire to have unsaturated parts, the analysis shows that the rate of change of the magnetic field must decrease when this ratio increases.

Calculations on 2-dimensional waveguides in photonic crystals

Using 2 different numerical methods (FDTD and a novel pandirectional planar Helmholtz solver), we analyzed the waveguide behaviour of a channel in a 2-dimensional photonic crystal. We calculated the bandgap of the crystal and the dispersion of a waveguide that is formed by a missing row of rods in the crystal. The coupling to this waveguide of both a Gaussian beam and a dielectric slab waveguide could be modeled and coupling efficiencies calculated.

NUMERICAL SOLUTIONS OF THE CURRENT DISTRIBUTION IN SUPERCONDUCTING RECTANGULAR CABLES

Cables are used in several technical applications such as accelarators, SMES and fusion projects. In all of these applications we find that a cable is placed in a combination of a transverse and a longitudinal applied field. Furthermore, we can define the problem periodical along the axis of the cable, because of the solinoidal or toroidal shape of the magnet. We will derive a simple numerical method for calculating the current distribution in a rectangular cable, where a cable is approximated by a small layer of wires with a nonconducting inner region, where the internal structure of the wire is neglected. A general outline for the unsaturated stationairy problem is given, but the described method can also be applied for instationary situations or cases were saturated regions may be expected.

Nonlinear Galerkin Method for Hamiltonian Systems

In its simplest form, the Galerkin method is the truncation of a differential equation by projection on a set of (spatial) base functions, in the present case: truncation to a certain number n of Fourier modes. But rather than neglecting all other modes completely, the nonlinear modification consists in taking some of the effects of the higher modes into account in the calculation of the first n modes. Specifically, if in the dynamic equations for the higher modes the time-derivative is set equal to zero, the equations simplify to quasi-stationary relations from which the higher modes can be solved, as function of the lower modes.

The Exact Solution of the Electromagnetic Field Configuration in Multifilamentary Wire in a Time-Dependent Field

In some macroscopic applications of superconductivity, multifilamen-tary wires are subjected to large time-dependent magnetic fields. To estimate the losses and the performance under such conditions, it is essential to have a thorough understanding of the electromagnetic field inside such a wire.