Christophe Caloz - Electromagnetic Metasurfaces

It is not possible to further transform ( 2.37a) to eliminate the double spatial derivative associated with . However, it turns out that is most often negligible compared to and , as discussed in [29, 148], so that this term may generally be ignored. Solving then ( 2.37) for ultimately leads to the spatial-dispersion relations

(2.38a)

(2.38b)

where , , and , which are the isotropic (or biisotropic) counterparts of the bianisotropic constitutive relations ( 2.4).

Note that when , then and . This means that the permittivity and permeability have fundamentally different dispersive nature, a fact that has important consequences for metamaterials, as we will see throughout the book.

Reciprocity or nonreciprocity is a fundamental physical property of all media, structures, devices, and systems. “A nonreciprocal (reciprocal) system is defined as a system that exhibits different (same) received–transmitted field ratios when its source(s) and detector(s) are exchanged” [28].

A linear-time-invariant (LTI) system can be made nonreciprocal only via the application of an external bias field that is odd under time reversal, such as a magnetic field or a current. The most common example of a nonreciprocal device is the Faraday isolator, whose nonreciprocity is obtained by biasing a ferrite with an external static magnetic field [143].

This section derives the Lorentz reciprocity theorem for a bianisotropic LTI medium, which provides the general conditions for reciprocity in terms of susceptibility tensors. These relations naturally apply to LTI metasurfaces as well.

Let us consider a volume enclosed by a surface and containing the impressed electric and magnetic sources and , respectively. In the time-harmonic regime, these sources are related to the fields , , , and via Maxwell equations,

( 2.39a)

( 2.39b)

Similarly, the phase-conjugated impressed sources, 5 and , are related to the phase-conjugated fields , , , and as

( 2.40a)

( 2.40b)

Subtracting 2.39apre-multiplied by from 2.40bpre-multiplied by , and subtracting 2.40apre-multiplied by from 2.39bpre-multiplied by yields

(2.41a)

(2.41b)