Electromagnetic Metasurfaces. Christophe Caloz

Au 9.026 (137.36 nm) 0.0267 Al 14.75 (84.06 nm) 0.0818

2.3 Spatial Dispersion

In addition of being temporally dispersive, a medium may also be spatially dispersive. As temporal dispersion is a temporally nonlocal phenomenon, spatial dispersion is a spatially nonlocal phenomenon, whereby the response of the medium at the position depends on its excitation at position [9, 29, 52, 148]. This effect is important in metamaterials because it is fundamentally related to bianisotropy and artificial magnetism, as we shall now demonstrate.

Figure 2.3 Experimental dispersion curves of the permittivity of silver, gold, and aluminum [72, 102]. (a) Real part. (b) Imaginary part.

Note that an extensive treatment of the topic of spatial dispersion would be beyond the scope of this book. Here, we thus limit ourselves to a brief and simplified description of this phenomenon, while more advanced presentations may be found in [9, 29, 52, 148].

      In order to show how spatial dispersion brings about bianisotropy and artificial magnetism in the constitutive relations, consider a medium with the conventional constitutive relations

(2.24a)

      (2.24b)

      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],

(2.25)

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],

(2.26)

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

(2.27)

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

(2.28)

where is the Kronecker delta. The relation (