Effective impedance modeling of metamaterial structures

Abstract

We present methods for retrieving the effective impedance of metamaterials from the Fresnel reflection coefficients at the interface between two semi-infinite media. The derivation involves the projection of rigorous modal expansions onto the dominant modes of the two semi-infinite media. It is shown that the effective impedance can also be written as a ratio of averaged field quantities. Thus a number of effective impedance formulas, previously obtained by field averaging techniques, can also be derived from the scattering-based formalism, by an appropriate choice of projection. Within the effective medium limit, it is observed that a simple semi-analytic modelling technique based on the effective impedance can be used to reliably compute the reflection coefficients of metamaterials over a wide range of incidence angles. We use this technique to model planar metamaterial waveguides or surface modes.

Evaluating resonances in PCSEL structures based on modal indices

“…This refractive index pattern η(x, z) may be produced by actual, direct material growth and/or by some post-growth fabrication process. For example, appropriate variation in the material composition of the vertical layers and/or thicknesses (y-axis) is often used to produce a variation in the vertical guided mode that may then be considered as creating an effective-index pattern η(x, z) in the x-z plane [18] corresponding to that in Fig. 1.…”

“…However, henceforth in this paper the study is restricted to a twodimensional (2D) analysis (in the x-z plane); i.e. essentially η(x, y, z) = η(x, z) and the excitation is such that any non-zero field component, F(x, y, z) = F(x, z) [18], i.e. ∂/∂y ≃ 0 is applicable; then, approximately, two independent categories of field solutions, one with polarisation, TE y ⇒ E y = 0 and the other with TM y ⇒ H y = 0 can be sustained [16,19].…”

“…Such limitations are removed if the CMT is used [16], but the mathematical complexities involved are somewhat daunting for the average user and requires very considerable effort and time to implement. The FDTD is a purely numerical method and may be used to solve for fields in almost any kind of structure [17]; typically, however, considerable computer resources are needed and significantly long computation times are required [18]. In view of the above, there is a noticeable need to develop another technique to solve for fields in 2D PCs -pertinent to PCSELswhich is relatively simple, easy to implement and requires modest resources and time while retaining a 'physical feel'.…”