where we have expressed the material polarization density, , in terms of the induced current density, (A/). Spatial dispersion is now introduced via the relationship between the induced current and the electric field [29],
where the dyadic tensor represents the response of the medium and is a volume of integration, centered at . We next restrict our attention to the case of weak spatial dispersion, where . We can then simplify (2.25) by using the three-dimensional Taylor expansion of truncated to the second order [29],
where the subscripts and run over , , and , which decomposes the second term in three terms and the third term in nine terms. Substituting (2.26) into (2.25) leads, after somewhat involved manipulations and simplifications [29], to
where is the Kronecker delta. The relation (2.28) corresponds to general second-order weak spatial dispersion constitutive relations. The presence of the spatial derivatives makes it practically cumbersome, and we shall therefore transform it into a simpler form [148]. For simplicity, we restrict our attention to the isotropic version of (2.28), which may be written as [29]
where is the permittivity, and , , and are complex scalar constants related to the parameters in (2.28). In order to simplify (2.29), we shall first demonstrate that Maxwell equations are invariant under the transformation [148]
rel="nofollow" href="#ulink_84bcecdf-bd6c-54f2-b886-dfb1c70ac029">2.24a)