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There is recent interest in the inter/intra-element interactions of metamaterial 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 the coupled frequencies and fields in terms of the uncoupled modes. 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 full-wave solvers. Therefore, the method can be considered as a 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. The solution of the eigenvalue problem for different configurations gives valued information and insight about the coupling of metamaterials unit cells. 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 coupled induced frequency shift coefficients.

The theory and operation of various devices and systems, such as wireless power transfer via magnetic resonant coupling, magneto-inductive wave devices, magnetic resonance spectroscopy probes, and metamaterials can rely on coupled tuned resonators. The coupling strength is usually expressed in terms of the coupling coefficient κ, which can have electrical κE and/or magnetic κM components. In the current article, general expressions of κ are derived. The relation between the complex Poynting equation in its microscopic form and κ is made and discussed in detail. It is shown that κ can be expressed in terms of the interaction energy between the resonators' modes. It thus provides a general form that combines the magnetic and electric components of κ. The expressions make it possible to estimate the frequencies and fields of the coupled modes for arbitrarily oriented and spaced resonators. Thus, enabling the calculation of system specific parameters such as the transfer efficiency of wireless power transfer systems, resonator efficiency for electron spin resonance probes, and dispersion relations of magneto-inductive and stereo-metamaterials structures.

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