Adaptation: Numerical solution of the Kramers-Kronig transforms by trapezoidal summation as compared to a Fourier method

Abstract

Absorption and reflection spectra have successfully been used for substance identification, which is also applicable to the THz spectral range. For optical spectroscopy, the Kramers–Kronig transformation (KKT) is a powerful tool to determine the complex refractive index n˜ (with k, n absorption and refractive indices, respectively) from either an absorption or reflection measurement. By terahertz time-domain spectroscopy (THz-TDS) materials are probed with short pulses of radiation. The detection is sensitive to the sample on both the amplitude and the phase of the electrical field. Thus, THz-TDS spectroscopy can provide more information than conventional Fourier-transform spectroscopy, by which a power spectrum is measured. In the case of transmission measurements, for instance, formulae exist by which the frequency-dependent complex refractive index is directly calculated from the time-dependent electrical field (waveform) without the necessity of a KKT. In the case of reflection experiments, a comparable computation is possible for the frequency-dependent phase angle; here a KKT can help to recover essential parameters. We present a combination of KKT and TDS methods for the calculation of optical constants in the THz regime.