  Optical cavities are an enabling technology of modern quantum science: from
their essential role in the operation of lasers, to applications as fly-wheels
in atomic clocks and interaction-enhancing components in quantum optics
experiments, developing a quantitative understanding of the mode-shapes and
energies of optical cavities has been crucial for the growth of the field.
Nonetheless, the standard treatment using paraxial, quadratic optics fails to
capture the influence of optical aberrations present in modern cavities with
high finesse, small waist, and/or many degenerate modes. In this work, we
compute the mode spectrum of optical resonators, allowing for both non-paraxial
beam propagation and beyond-quadratic mirrors and lenses. Generalizing prior
works [1-5], we develop a complete theory of resonator aberrations, including
intracavity lenses, non-planar geometries, and arbitrary mirror forms.
Harnessing these tools, we reconcile the near-absence of aberration in Ref. [6]
with the strongly evident aberrations in the seemingly similar cavity of Ref.
[7]. We further validate our approach by comparison to a family of non-planar
lens cavities realized in the lab, finding good quantitative agreement. This
work opens new prospects for cavities with smaller waists and more degenerate
modes.
