  Optical cavities are widely used to enhance the interaction between atoms and
light. Typical designs using a geometrically symmetric structure in the
near-concentric regime face a tradeoff between mechanical stability and high
single-atom cooperativity. To overcome this limitation, we design and implement
a geometrically asymmetric standing-wave cavity. This structure, with mirrors
of very different radii of curvature, allows strong atom-light coupling while
exhibiting good stability against misalignment. We observe effective
cooperativities ranging from $\eta_{\rm eff}=10$ to $\eta_{\rm eff}=0.2$ by
shifting the location of the atoms in the cavity mode. By loading $^{171}$Yb
atoms directly from a mirror magneto-optical trap into a one-dimensional
optical lattice along the cavity mode, we produce atomic ensembles with
collective cooperativities up to $N\eta=2\times 10^4$. This system opens a way
to preparing spin squeezing for an optical lattice clock and to accessing a
range of nonclassical collective states.
