Fourier finite element modeling of light emission in waveguides: 2.5-dimensional FEM approach

Computationally fast EM field propagation through axi-symmetric media using cylindrical harmonic decomposition

We describe and provide a systematic procedure for computationally fast propagation of arbitrary vector electromagnetic (EM) fields through an axially symmetric medium. A cylindrical harmonic field propagator is chosen for this purpose and in most cases, this is the best and the obvious choice. Firstly, we describe the cylindrical harmonic decomposition technique in terms of both scalar and vector basis for a given input excitation field. Then we formulate a generalized discrete Fourier-Hankel transform to achieve efficient vector basis decomposition. We allow a slower, pre-computation step, that finds a representation of the axi-symmetric medium as a transfer matrix in a discrete, cylindrical-harmonic basis. We find this matrix from a series of axi-symmetric (2D) finite element simulations (also known as the 2.5D technique). This transfer matrix approach significantly reduces the computational load when the transverse size or range exceeds about 30 wavelengths. This matrix is independent of the input excitation field for a given space-bandwidth product and hence makes it reusable for different excitation fields. We numerically validate the above approaches for different axi-symmetric EM scattering media which include a hemispherical gradient-index Maxwell's fish-eye lens, a transformation optics designed spherical invisibility cloak, a thin aspheric lens, and a cylindrical perfect lens.