Electromagnetic Metasurfaces. Christophe Caloz

Electromagnetic Metasurfaces

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 bold r depends on its excitation at position bold r prime [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.

Graphs depict the experimental dispersion curves of the permittivity of silver, gold, and aluminum. (a) Real part. (b) Imaginary part.

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) StartLayout 1st Row 1st Column bold upper D 2nd Column equals epsilon 0 bold upper E plus bold upper P equals epsilon 0 bold upper E plus StartFraction 1 Over j omega EndFraction bold upper J Subscript ind Baseline comma EndLayout

(2.24b) StartLayout 1st Row 1st Column bold upper H 2nd Column equals mu 0 Superscript negative 1 Baseline bold upper B comma EndLayout

where we have expressed the material polarization density, bold upper P , in terms of the induced current density, bold upper J Subscript ind (A/ normal m squared ). Spatial dispersion is now introduced via the relationship between the induced current and the electric field [29],

(2.25) bold upper J Subscript ind Baseline left-parenthesis bold r right-parenthesis equals integral Underscript upper V Endscripts ModifyingAbove Above ModifyingAbove upper K With bar With bar left-parenthesis bold r comma bold r Superscript prime Baseline right-parenthesis dot bold upper E left-parenthesis bold r Superscript prime Baseline right-parenthesis normal d upper V comma

where the dyadic tensor upper K overbar overbar represents the response of the medium and upper V is a volume of integration, centered at bold r . We next restrict our attention to the case of weak spatial dispersion, where upper V much-less-than lamda . We can then simplify (2.25) by using the three-dimensional Taylor expansion of bold upper E left-parenthesis bold r prime right-parenthesis truncated to the second order [29],

(2.26)

where the subscripts alpha and beta 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 a Subscript ellipsis are related to upper K overbar overbar . Inserting then (2.27) into 2.24a yields

(2.28)

where delta Subscript i j is the Kronecker delta. The relation (Скачать книгу