  Much of our physical intuition about black hole quasinormal modes in general
relativity comes from the eikonal/geometric optics approximation. According to
the well-established eikonal model, the fundamental quasinormal mode represents
wavepackets orbiting in the vicinity of the black hole's geodesic photon ring,
slowly peeling off towards the event horizon and infinity. Besides its strength
as a "visualisation" tool, the eikonal approximation also provides a simple
quantitative method for calculating the mode frequency, in close agreement with
rigorous numerical results. In this paper we move away from Einstein's theory
and its garden-variety black holes and go on to consider spherically symmetric
black holes in modified theories of gravity through the lens of the eikonal
approximation. The quasinormal modes of such black holes are typically
described by a set of coupled wave equations for the various field degrees of
freedom. Considering a general, theory-agnostic, system of two equations for
two perturbed fields, we derive eikonal formulae for the complex fundamental
quasinormal mode frequency. In addition we show that the eikonal modes can be
related to the extremum of an effective potential and its associated "photon
ring". As an application of our results we consider a specific example of a
modified theory of gravity with known black hole quasinormal modes and find
that these are well approximated by the eikonal formulae.
