Impact of the non-measured infrared spectral range of the imaginary refractive index on the derivation of the real refractive index using the Kramers-Kronig transform

A quantitative analysis to asses the influence of a non-measured imaginary index spectrum on the extracted real refractive index is presented. The investigation was done on the Mid-IR spectral range, where the ''measured'' imaginary spectrum is defined between 800 and 4500 cm−1. The influence of bands of various locations and shapes in the non-measured IR spectral region (0-800 cm−1) on the n values obtained by the computational procedure of the Kramers-Kronig transform was investigated. Additional analysis was conducted to estimate the relevance of different assumptions that are commonly made with regard to the non-measured range (e.g. linear extrapolation or the effect of uncertainty in the precise band location). The results show that the contribution of an unmeasured band at any wavenumber ν˜0 is well described by a simple function of the band location and ν˜0, regardless of the band shape. Furthermore, the error caused by incorrect band location can also be described by a simple function of the band location, the band location error and ν˜0. The simple functions can be used to estimate the impact that ignoring or misplacing a band will have on the extracted n spectrum, without performing the whole KK integration. These relationships were validated on two data sets of optical constants of crystalline ammonium sulfate and water in the Mid-IR range.