Christophe Caloz - Electromagnetic Metasurfaces

Figure 2.1 Classification of (bianisotropic) metamaterials in terms of their space–time variance and dispersion. The I- and D-notations refer to the inverse and direct space/time domains.

Temporal dispersion is the property according to which the response of a medium, at a given instant of time, depends on the excitation at previous times. This can be mathematically expressed by the time-domain counterparts 3of ( 2.3), as

(2.5a)

(2.5b)

where only the time dependence of the fields is explicitly mentioned and the spatial dependence is omitted for conciseness. These expressions indicate that the material responses, and , are the temporal convolution of the susceptibilities and excitation functions. They may be alternatively written as

( 2.6a)

( 2.6b)

Since the convolution in the time domain corresponds to the multiplication in the frequency domain, the time-domain Fourier transform of ( 2.6) corresponds to ( 2.3).

We have mentioned that matter is temporally dispersive and shown that the corresponding medium parameters at a given instant of time depend on the excitation at previous times. Combining these facts with the concept of causality allows one to relate the real and imaginary parts of the material parameters to each other. The corresponding relations are referred to as the Kramers–Kronig relations. Given their crucial importance and given their unclear presentation in the literature, we shall now precisely derive them.

The fundamental tenet of causality is that an effect cannot precede its cause, so that, in particular, matter cannot respond before being excited. Therefore, assuming that a medium starts to be excited at , its response at must necessarily be zero. We have thus [59]

( 2.7)

where represents the variation of any of the susceptibility components in ( 2.3) from the steady-state regime, is the Heaviside step function, and is an arbitrary function that satisfies for . Transforming ( 2.7) to the frequency domain, we get

( 2.8)

where

( 2.9)

with representing the Fourier transformation operation, being the Dirac delta distribution, and PV denoting the principal value of its argument. Substituting ( 2.9) into ( 2.8) leads to the explicit convolution relation

( 2.10)

Noting that is a complex quantity, we can rewrite ( 2.10) in the more explicit form

( 2.11)

where and are the real and imaginary parts of , respectively. Since may take arbitrary values for , we chose such that it is time-symmetric (or even), i.e. , which leads to a purely real function . We can then separate ( 2.11) into its real and imaginary parts as