Investigation of methods to set up the normal vector field for the differential method

Abstract

In Fourier modal methods like the RCWA and the Differential Method the Li-rules for products in truncated Fourier space have to be obeyed in order to achieve good convergence of the results with respect to the mode number. The Lirules have to be applied differently for parts of the field that are tangential and orthogonal to material boundaries. This is achieved in the Differential Method by including a field of vectors in the calculation that are normal to the material boundaries. The same can be done laterally in each layer of an RCWA calculation of a 2-D periodic structure. It turns out that discontinuities in the normal vector field can disturb the computation especially when metallic materials are dominant in the structure which would make the usefulness of the normal vector method questionable. So it is of great importance to investigate how normal vector fields can be established with as few discontinuities as possible. We present various methods for the 2-D RCWA and the 1-D and 2-D Differential Method and compare the respective convergence behaviors. Especially we emphasize methods that are automatic and require as few user input as possible.

Local normal vector field formulation for periodic scattering problems formulated in the spectral domain

“…This computational overhead should be as low as possible, to allow for fast analysis and reconstruction algorithms. One quite general approach to generate normalvector fields has been put forward in [7,8], based on scattereddata interpolation.…”

“…A disadvantage of the existing normal-vector-field formulation is that the normal-vector field is required on the entire computational domain [1,3,7]. As a consequence, one cannot operate on isolated domains without taking care of connecting interfaces.…”

A Spectral Volume Integral Method Using Geometrically Conforming Normal-Vector Fields

“…Nevertheless, the staircase approximation in the aperiodic direction, which is inherent to RCWA, can still introduce scattering that is not present in the smooth structure that is approximated [6]. A perturbative technique based on tilted normal-vector fields has been proposed [8] that can mitigate the scattering from the staircase edges. Although the normal-vector field is not strictly geometrically conformal, it induces a faster convergence.…”

“…To study the effects of geometrically conforming normal-vector fields, we consider a case of 1D-periodic cylindrical rods, as it is considered one of the most difficult cases to compute [14] and an independent reference solution is available. However, the algorithm can handle gratings with arbitrary height profiles, similar to [8,14]. The 1D-periodic case enables us to study the behavior of the algorithm for a very large number of unknowns per direction with acceptable computation time and memory demand.…”

Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures

“…For non-rectangular shapes a zigzag approximation of the profile had to be used. This inconvenience has been removed by considering separately the tangential and normal components of the field at the interface [7][8][9][10]. Another important improvement was the introduction of the technique of adaptive spatial resolution [11].…”