  We present our results for simultaneous measurement of the refractive indices
of gallium arsenide (GaAs) and aluminum gallium arsenide
(Al$_\mathrm{x}$Ga$_\mathrm{1-x}$As) from $2.0$ to $7.1\,\mathrm{\mu m}$
($5000$ to $1400\,\mathrm{cm^{-1}}$). We obtain these values from a
monocrystalline superlattice Bragg mirror of excellent purity (background
doping $\leq 1 \times 10^{-14}\,\mathrm{cm^{-3}}$), grown via molecular beam
epitaxy. To recover the refractive indices over such a broad wavelength range,
we fit a dispersion model for each material. In a novel combination of
well-established methods, we measure both a photometrically accurate
transmittance spectrum of the Bragg mirror via Fourier-transform infrared
spectrometry and the individual physical layer thicknesses of the structure via
scanning electron microscopy. To infer the uncertainty of the refractive index
values, we estimate relevant measurement uncertainties and propagate them via a
Monte-Carlo method. This highly-adaptable approach conclusively yields
propagated relative uncertainties on the order of $10^{-4}$ over the measured
spectral range for both GaAs and Al$_{0.929}$Ga$_{0.071}$As. The fitted model
can also approximate the refractive index for MBE-grown
Al$_\mathrm{x}$Ga$_\mathrm{1-x}$As for $0\leq x \leq 1$. Both these updated
values and the measurement approach will be essential in the design,
fabrication, and characterization of next-generation active and passive optical
devices in a spectral region that is of high interest in many fields, e.g.,
laser design and cavity-enhanced spectroscopy in the mid-infrared spectral
region.
