  Decades ago, the losses of glasses in the near infrared (near-IR) were
investigated in views of developments for optical telecommunications. Today,
properties in the mid-infrared (mid-IR) are of interest for molecular
spectroscopy applications. In particular, high-sensitivity spectroscopic
techniques based on high-finesse mid-IR cavities hold high promise for medical
applications. Due to exceptional purity and low losses, whispering gallery mode
microresonators based on polished alkaline earth metal fluoride crystals (i.e
the $\mathrm{XF_2}$ family, where X $=$ Ca, Mg, Ba, Sr,...) have attained
ultra-high quality (Q) factor resonances (Q$>$10$^{8}$) in the near-IR and
visible spectral ranges. Here we report for the first time ultra-high Q factors
in the mid-IR using crystalline microresonators. Using an uncoated chalcogenide
(ChG) tapered fiber, light from a continuous wave quantum cascade laser (QCL)
is efficiently coupled to several crystalline microresonators at 4.4 $\mu$m
wavelength. We measure the optical Q factor of fluoride crystals in the mid-IR
using cavity ringdown technique. We observe that $\mathrm{MgF_2}$
microresonators feature quality factors that are very close to the fundamental
absorption limit, as caused by the crystal's multiphonon absorption
(Q$\sim$10$^{7}$), in contrast to near-IR measurements far away from these
fundamental limits. Due to lower multiphonon absorption in $\mathrm{BaF_2}$ and
$\mathrm{SrF_2}$, we show that ultra-high quality factors of Q $\geqslant$ 1.4
$\times 10^{8}$ can be reached at 4.4 $\mu$m. This corresponds to an optical
finesse of $\mathcal{F}>$4$\cdot$ 10$^{4}$, the highest value achieved for any
type of mid-IR resonator to date, and a more than 10-fold improvement over the
state-of-the-art. Such compact ultra-high Q crystalline microresonators provide
a route for narrow linewidth frequency-stabilized QCL or mid-IR Kerr comb
generation.
