Christophe Caloz - Electromagnetic Metasurfaces

( 2.27)

where the parameters are related to . Inserting then ( 2.27) into 2.24ayields

( 2.28)

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]

( 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 is a coupling parameter associated to chirality [26, 155] and where the parameter corresponds to artificial magnetism [122]. The constitutive relations ( 2.37) reveal, via ( 2.29), that chirality is related to spatial dispersion of the first order via in , while artificial magnetism is related to spatial dispersion of the second order via . Both chirality and artificial magnetism depend on the excitation frequency via . It may seem surprising that artificial magnetism is related to spatial dispersion but consider the following simple example, which is one of the easiest ways of creating an effective magnetic dipole. Consider, two small metal strips, one placed at a subwavelength distance to the other in the direction of wave propagation. If the conditions are met, a mode with an odd current distribution may be excited on the strips resulting in an effective magnetic dipole. In that case, the electric field on one of the strip slightly differs from the one on the other strip since they are separated by a distance , thus implying that the effective magnetic response of the strips spatially depends on the exciting electric field. If , then the difference of the electric field on the strips would vanish as would the effective magnetic dipole moment.