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
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.
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