  Near-unstable cavities have been proposed as an enabling technology for
future gravitational wave detectors, as their compact structure and large beam
spots can reduce the coating thermal noise of the interferometer. We present a
tabletop experiment investigating the behaviour of an optical cavity as it is
parametrically pushed to geometrical instability. We report on the observed
degeneracies of the cavity's eigenmodes as the cavity becomes unstable and the
resonance conditions become hyper-sensitive to mirror surface imperfections. A
simple model of the cavity and precise measurements of the resonant frequencies
allow us to characterize the stability of the cavity and give an estimate of
the mirror astigmatism. The significance of these results for gravitational
wave detectors is discussed, and avenues for further research are suggested.
