Energy Coupled Mode Theory for Electromagnetic Resonators

There is recent interest in the inter/intra-element interactions of metamaterial (MM) unit cells. To calculate the effects of these interactions, which can be substantial, an ab-initio general coupled mode equation, in the form of an eigenvalue problem, is derived. The solution of the master equation gives accurate estimates of the coupled frequencies and fields in terms of an arbitrary number of uncoupled modes, having equal or different frequency values. By doing so, the problem size is limited to the number of modes rather than the usually large discretized spatial and temporal domains obtained by rigorous full-wave solvers. Therefore, the method can be considered as an approximate numerical recipe, which determines the behavior of a complex system once its simpler ingredients are known. Besides quantitative analysis, the coupled mode equation proposes a pictorial view of the split rings’ hybridization. It can be regarded as the electromagnetic analog of molecular orbital theory. Although being an approximate method, the solution of the eigenvalue problem for different configurations gives valued information and insight about the coupling of MMs unit cells. Unlike rigorous numerical techniques, closed-form expressions can be derived using the coupled mode equation. For instance, it is shown that the behavior of split rings as a function of the relative position and orientation can be systematically explained. This is done by singling out the effect of each relevant parameter such as the coupling coefficient and coupling induced frequency shift coefficients.