  We performed high-resolution Fourier-transform infrared (FTIR) spectroscopy
of a polymethyl methacrylate (PMMA) sphere of unknown size in the Mie
scattering region. Apart from a slow, oscillatory structure (wiggles), which is
due to an interference effect, the measured FTIR extinction spectrum exhibits a
ripple structure, which is due to electromagnetic resonances. We fully
characterize the underlying electromagnetic mode structure of the spectrum by
assigning two mode numbers to each of the ripples in the measured spectrum. We
show that analyzing the ripple structure in the spectrum in the wavenumber
region from about $3000\,$cm$^{-1}$ to $8000\,$cm$^{-1}$ allows us to both
determine the unknown radius of the sphere and the PMMA index of refraction,
which shows a strong frequency dependence in this near-infrared spectral
region. While in this paper we focus on examining a PMMA sphere as an example,
our method of determining the refractive index and its dispersion from
synchrotron infrared extinction spectra is generally applicable for the
determination of the index of refraction of any transparent substance that can
be shaped into micron-sized spheres.
