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Recent Developments in the Theory of Microwave Metamaterials

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The original work by Veselago on the electrodynamics of materials with negative constitutive parameters suggested that they had interesting properties, such as backward wave propagation. His paper has 7000 citations and led to the field of metamaterials, where three dimensional arrays of electrical and optical circuit elements are used to create artificial materials which exhibit phenomena such as negative refraction. However two fundamental errors occur repeatedly in papers on metamaterials. The first assumption is that a bulk material can possess negative constitutive parameters. We will demonstrate that this violates causality and such materials cannot exist in reality as they would be unstable. The second assumption is that metamaterials with circuit elements which are small with respect to the wavelength support single mode propagation. We will show that the necessary and sufficient condition for causal single mode propagation is that the network inductance matrix is the inverse of the capacitance matrix, as is the case for the homogeneous N-wire transmission line. This condition is impossible to realise in real metamaterials, due to the complexity of the inductance matrix. Thus all metamaterials must support multi-valued propagation constants which are the eigenvalues of the network matrix. Examples of multi-mode propagation producing negative refraction in finite-thickness microwave and optical metamaterials will be presented. Finally we will show that a correct understanding of multi-mode propagation can greatly simplify the analysis and design of 2-D networks such as waffle-iron filters.

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