Closed Form Kramers–Kronig Relations to Extract the Refractive Index of Metamaterials

Closed form expressions of the Kramers-Kronig and of the multiply subtractive Kramers-Kronig relations are derived to predict the real part of the refractive index from the imaginary part, which is given in discrete frequency points. The accuracy and the convergence rate of the closed form expressions are investigated by calculating the refractive index of a hypothetical double negative metamaterial with predefined parameters. Then, a two-step procedure is presented, which utilizes the subtractive Kramers-Kronig relation to uniquely retrieve the effective refractive index of metamaterials even when the Kramers-Kronig relation fails. The developed procedure is demonstrated by extracting the effective parameters of multilayer fishnet metamaterials.