Numerical computation of the complex dielectric permittivity using Hilbert transform and FFT techniques

A procedure is described to efficiently obtain the complex dielectric permittivity of semiconductor materials with numerical routines available in current software applications. This procedure calculates the real component of the dielectric permittivity from its imaginary component by making use of the Hilbert transform properties of the Kramers–Kronig relations. We show the reliability of this approach by reconstructing the real component from the experimentally known dispersion relations of GaAs and InAs. In addition, we use this procedure for the case when the imaginary component is obtained from analytical expressions as is the case for the Franz–Keldysh effect in the absorption of photons in GaAs. We show how the spectral photoreflection of structures where this effect is present can be easily modeled and simulated by this method.