Electromagnetic Metasurfaces. Christophe Caloz

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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

(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.

      

      2.2.2 Lorentz Oscillator Model

      The Lorentz oscillator is a simple classical atomic

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