An accurate modified Kramers–Kronig transformation from reflectance to phase shift on attenuated total reflection
Authors: Bertie John E. , Lan Zhida 
Published: July 2002

A new and simple procedure is presented for the calculation of the infrared real, n, and imaginary, k, refractive index spectra from s-polarized attenuated total reflection (ATR) spectra by a modified Kramers–Kronig transform of the reflectance to the phase shift on reflection. The procedure consists of two parts, first a new modified Kramers–Kronig (KK) transform, and second a new, wave number-dependent, correction to the phase shift. The procedure was tested with ATR spectra which were calculated from refractive index spectra that were synthesized under the classical damped harmonic oscillator model. The procedure is far more accurate than previous procedures for the real case of a wave number-dependent refractive index of the incident medium, and yields n and k values that are accurate to ≤0.1% provided that no errors are introduced by the omission of significant reflection bands. This new procedure can be used to obtain optical constants from any ATR experiment that yields the spectrum of Rs, the reflectance polarized perpendicular to the plane of incidence. In this laboratory Rs spectra are obtained from samples held in the Spectra-Tech CIRCLE cell in a Bruker IFS 113 V spectrometer. Accordingly the ATR spectra used to test the new procedure were calculated for the optical configuration of this system, which is m reflections at 45° incidence with equal intensities of s- and p-polarized light and retention of polarization between reflections. For the previously studied [J. S. Plaskett and P. N. Schatz, J. Chem. Phys. 38, 612 (1963); J. A. Bardwell and M. J. Dignan, ibid. 83, 5468 (1985)], but unreal, case of constant refractive index of the incident medium, n0, the new transform gave better results than either of two previously studied procedures. In this case the phase shift at each wave number was corrected by a constant which ensured that the correct phase shift was obtained at the highest wave number in the transform, 7800 or 8000 cm−1. In contrast to a previous study [J. Chem. Phys. 83, 5468 (1985)] it was found that the normal KK transform is inferior for this case to a previous modified KK transform [J. Chem. Phys. 38, 612 (1963)], and it is also inferior to the new modified KK transform. Further, the new transform has only the usual singularity of a KK transform, and this makes it numerically superior to the previous modified KK transform which has an additional singularity at 0 cm−1. For the real case, in which the refractive index of the incident medium changes with wave number, the new transform was used with a new simple wave number-dependent additive correction to the phase shift. This new correction is calculated with the actual value of n0 at each wave number. For molecular liquids such as methanol and benzene the new transform is markedly superior to the previous two transforms. It yields real and imaginary refractive index values that are accurate to better than 0.1% provided the reflection spectrum is known down to 2 cm−1. The latter condition is rarely fulfilled, and the effect of the omission of low wave number bands is illustrated. A method to reduce the impact of missing low-wave number parts of the reflectance spectrum is described, and its effectiveness is illustrated.