Electromagnetic Metasurfaces. Christophe Caloz

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]

(2.29)

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]

(2.30a)

      (2.30b)

where is an arbitrary vector. To prove this invariance, consider Maxwell–Ampère equation

(2.31)

Substituting (2.30) into this relation yields

      (2.32)

      which clearly reduces to

(2.33)

hence proving the equivalence of (2.33) and (2.31), and therefore demonstrating the invariance of Maxwell equations under the transformation (2.30) for any [148]. We now substitute (2.29) into (2.30) along with , which yields

(2.34a)

      (2.34b)

Further substituting Maxwell–Faraday equation, , into (2.34) yields

      (2.35a)

      (2.35b)

      Finally substituting these relations with into (2.30) leads to the relations

      (2.36a)

      (2.36b)

      which take the compact form

(2.37a)

      (2.37b)

where

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