Christophe Caloz - Electromagnetic Metasurfaces

( 2.12a)

( 2.12b)

Substituting ( 2.12a) into ( 2.12b) yields

( 2.13)

Following a similar procedure with a time-asymmetric function, i.e. , corresponding to a purely imaginary , we obtain

( 2.14)

Equations ( 2.13) and ( 2.14) are the sought Kramers–Kronig relations. They indicate that variations in the real part of the medium parameters are necessarily associated with specific variations in their imaginary parts. For a quantity such as the refractive index, for instance, this implies that transforming an electromagnetic wave (e.g. negatively refracting it) from the real part of the index (e.g. making it negative) necessarily results in a variation of the imaginary part of the index, which corresponds to inserting loss, and vice versa.

The Lorentz oscillator is a simple classical atomic model 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 is the electric charge and is the local field. 4We assume here that the magnetic force is negligible compared to the electric force, which is the case for nonrelativistic velocities, and that the nuclei, which are much heavier than the electrons, are not moving. The restoring force between the nuclei and the electrons can be expressed similarly to the force of a mass attached to a spring, i.e.

( 2.16)

where is the mass of the electron cloud, is a constant analogous to the stiffness of the spring, and is the displacement from equilibrium of the electron cloud. Finally, to model dissipation, we introduce the frictional force

(2.17)

where is the displacement velocity of the electron cloud and is a constant representing the friction coefficient. Applying Newton's second law with these three forces, we obtain

(2.18)

Rearranging the terms and noting that , we get

( 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 [140], where is the electron density. This transforms ( 2.19) into

( 2.20)

For electrically small particles, we may use the Clausius–Mosotti expression, which relates the local field to the total electric, and to the polarization density as [140], which reduces ( 2.20) into

(2.21)

where is the resonant frequency. In the harmonic regime, this equation becomes

(2.22)

where is the time-domain Fourier transform of and that of . Finally substituting , eliminating , and solving for yields the dispersive susceptibility