  We describe a qualitatively new regime of cavity quantum electrodynamics, the
super strong coupling regime. This regime is characterized by atom-field
coupling strengths of the order of the free spectral range of the cavity,
resulting in a significant change in the spatial mode functions of the light
field. It can be reached in practice for cold atoms trapped in an optical
dipole potential inside the resonator. We present a nonperturbative scheme that
allows us to calculate the frequencies and linewidths of the modified field
modes, thereby providing a good starting point for a quantization of the
theory.
