Bastiaan P. de Hon

On the Convergence of the Eigencurrent Expansion Method Applied to Linear Embedding via Green's Operators (LEGO)

The scattering from a large complex structure comprised of many objects may be efficiently tackled by embedding each object within a bounded domain (brick) which is described through a scattering operator. Upon electromagnetically combining the scattering operators we arrive at an equation which involves the total inverse scattering operator S-1 of the structure: We call this procedure linear embedding via Greens operators (LEGO). To solve the relevant equation we then employ the eigencurrent expansion method (EEM)-essentially the method of moments with a set of basis and test functions that are approximations to the eigenfunctions of S-1 (termed eigencurrents). We have investigated the convergence of the EEM applied to LEGO in cases when all the bricks are identical. Our findings lead us to formulate a simple and practical criterion for controlling the error of the computed solution a priori.

Bending loss in optical fibers - a full-wave approach

The radiated power emanating from a bent single-mode fiber is computed for various radii of curvature by a full-vectorial analysis. The only approximation is the truncation of a spectral series, the accuracy of which can be controlled. Hence, the complex propagation coefficient of the fundamental mode approaches the exact value and consequently, the bending loss does as well. Two widely accepted bending-loss formulas, based on asymptotic approximations to scalar-field theory, are compared with our full-vectorial results. Both have a limited region of validity. For simplicity, the comparison is performed on a step-index fiber with a cladding of infinite extent. However, the full-wave method is capable of dealing with arbitrary index profiles.

Recursive evaluation of space-time lattice Green's functions

Up to a multiplicative constant, the lattice Greens function (LGF) as defined in condensed matter physics and lattice statistical mechanics is equivalent to the Z-domain counterpart of the finite-difference time-domain Greens function (GF) on a lattice. Expansion of a well-known integral representation for the LGF on a ν-dimensional hyper-cubic lattice in powers of Z−1 and application of the Chu–Vandermonde identity results in ν − 1 nested finite-sum representations for discrete space-time GFs. Due to severe numerical cancellations, these nested finite sums are of little practical use. For ν = 2, the finite sum may be evaluated in closed form in terms of a generalized hypergeometric function. For special lattice points, that representation simplifies considerably, while on the other hand the finite-difference stencil may be used to derive single-lattice-point second-order recurrence schemes for generating 2D discrete space-time GF time sequences on the fly. For arbitrary symbolic lattice points, Zeilbergers algorithm produces a third-order recurrence operator with polynomial coefficients of the sixth degree. The corresponding recurrence scheme constitutes the most efficient numerical method for the majority of lattice points, in spite of the fact that for explicit numeric lattice points the associated third-order recurrence operator is not the minimum recurrence operator. As regards the asymptotic bounds for the possible solutions to the recurrence scheme, Perrons theorem precludes factorial or exponential growth. Along horizontal lattices directions, rapid initial growth does occur, but poses no problems in augmented dynamic-range fixed precision arithmetic. By analysing long-distance wave propagation along a horizontal lattice direction, we have concluded that the chirp-up oscillations of the discrete space-time GF are the root cause of grid dispersion anisotropy. With each factor of ten increase in the lattice distance, one would have to roughly double the pulse width of the source signature to keep pulse distortion at bay. The GF time sequences can also be used for an efficient computation of discrete space-frequency LGFs, especially if one employs Aitkens δ2 process for the acceleration of the convergence of the consecutive partial sums.

A priori error estimate and control in the eigencurrent expansion method applied to linear embedding via Green's operators (LEGO)

Linear embedding via Greens operators (LEGO) [1, 2] is a domain decomposition method in which the electromagnetic scattering by an aggregate of N<inf>D</inf> bodies (immersed in a homogeneous background medium) is tackled by enclosing each object within an arbitrarily-shaped bounded domain D<inf>k</inf> (brick), k = 1, …, N<inf>D</inf> (e.g., see Fig. 1). The bricks are characterized electromagnetically by means of scattering operators S<inf>kk</inf>, which are subsequently combined to form the total inverse scattering operator S⁻¹ of the structure [1]. Finally, we use the eigencurrent expansion method (EEM) [1,3] to solve the relevant equation involving S⁻¹, viz.

Numerical modeling method for the dispersion characteristics of single-mode and multimode weakly-guiding optical fibers with arbitrary radial refractive index profiles

Accurate, reliable and fast numerical modeling methods are required to design the optimum radial refractive index profile for single and multimode fibers to give specific dispersion characteristics prior to or even obviating costly experimental work. Such profiles include graded index and multiple concentric cladding layers. In this paper, a new numerical method is introduced which enables the derivatives of the propagation coefficient to be calculated analytically up to the third order of a single mode or multimode weakly guiding optical fiber with an arbitrary radial refractive index profile. These quantities are required to determine the group delay, τg, chromatic dispersion, D, and dispersion slope of the fiber. The expansion of the modal fields in terms of Laguerre-Gauss polynomials in the Galerkin method offers certain benefits. In particular, due to simplicity of the basis functions it is possible to carry out further analytical work on the results such as repeated differentiation of the matrix equation resulting from the Galerkin method to define up to the third-order derivatives of the propagation coefficients with respect to wavelength. This avoids approximation errors inherent in numerical differentiation, giving better accuracy and, at the same time, significantly reduces the computation time. A computer program was developed to demonstrate the proposed method for single and multimode fibers with radially arbitrary refractive index profiles. The paper provides simulation results to validate the approach.

Scattering from a random distribution of numerous bodies with linear embedding via Green's operators

We discuss the application of linear embedding via Greens operators (LEGO) to the solution of the scattering of electromagnetic waves from random distributions of different objects. The latter are enclosed in simple-shaped bricks described via scattering operators that have to be computed only once for a given frequency. Therefore, the study of many distributions made of the very same objects but located in different positions can be efficiently carried out by re-using the scattering operators. Besides, the equation of LEGO is solved via the Moment Methods combined with Arnoldi basis functions — which allows the corresponding algebraic system to be effectively compressed. We investigate the properties of LEGO through a few numerical examples.