that describes the interaction of a time-harmonic field with matter. It was initially developed to describe the resonant behavior of an electron cloud. This model has been widely used to describe the temporal-frequency dispersive nature of materials and their related frequency-dependent refractive index [140]. It turns out to be also particularly useful in describing the responses of metamaterials, as they are generally made of resonant scattering particles.
Let us consider that an electron cloud is subjected to the Lorentz electric force,
(2.15)
where
(2.16)
where
(2.17)
where
(2.18)
Rearranging the terms and noting that
(2.19)
This equation may be written in a more convenient form by noting that the displacement from equilibrium is related to the electric polarization density as
(2.20)
For electrically small particles, we may use the Clausius–Mosotti expression, which relates the local field to the total electric,
(2.21)
where
