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In this paper we develop homogenization theory for spatiotemporally modulated wire medium. We first solve for the modal waves that are supported by this composite medium, we show peculiar properties such as extraordinary waves that propagate at frequencies below the cut-off frequency of the corresponding stationary medium. We explain how these unique solutions give rise to an extreme Fresnel drag that exists already with weak and slow spatiotemporal modulation. Next, we turn to derive the effective material permittivity that corresponds to each of the first few supported modes, and write the average fields and Poyntings vector. Nonlocality, nonreciprocity, and anisotropy due to the spatiotemporal modulation direction, are three inherent properties of this medium, and are clearly seen in the effective material parameters. As a figure of merit, we also derive the effective permittivity of a plasma medium with spatiotemporally modulated plasma frequency. This comparison is interesting since the plasma medium can be considered as the effective medium that is obtained by a stationary wire medium. We validate that homogenization and spatiotemporal variation are not necessarily interchangeable operations. And indeed, in certain parameter regimes the homogenization should be performed directly on spatiotemporally modulated composite medium, rather than first homogenize the stationary medium and then phenomenologically introduce the effect of the space-time modulation.
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