A differential form of the Kramers-Kronig relation for determining a Lorentz-type of refractive index

The integral forms of the Kramers-Kronig (KK) relations that relate the real and imaginary parts, n´ and n", of a refractive index require the integral to be conducted over the full spectrum. In this paper, we derive a differential form of the KK relation that removes the need for knowing n" at all frequencies (ω) in order to calculate n´, which is more useful in measurements. Moreover, we show that, in a finite frequency range, our differential form of the KK relation can more accurately generate a resonant effective n´ for an artificial medium composed of a periodic structure than do the integral forms of the KK relations, making use of n" as obtained from a full-wave simulation.