  We theoretically study the quality and Purcell enhancement factors in
epsilon-near-zero (ENZ) optical microcavities with ideally lossless or slightly
lossy structures. In the case of ENZ rectangular cavities with perfect electric
conductor (PEC) walls, our results supported by numerical simulations, show
that both the quality and Purcell factors grow with increasing the length of
cavity, having respective functional dependencies of $n^{-3}$ and $n^{-2}$ on
the effective index of refraction, $n$. Further, we study extremely low-loss
all-dielectric Bragg reflection waveguide cavities, near their cutoff
frequencies. In such ENZ structures, the respective quality and Purcell factors
scale as \((n\kappa)^{-1}\) and $n/\kappa$, in terms of real ($n$) and
imaginary ($\kappa$) parts of the cavity effective index of refraction. We
believe such structures exhibiting remarkably high quality ($\sim10^6$) and
Purcell factors ($\sim10^3$), can drastically enhance various linear and
nonlinear optical processes, and thus find use in applications ranging from
laser development to optical sensing and metrology.
