In this paper we explain the origin of the ring structure that sometimes forms a sharp border surrounding field-emission electron microscopy patterns from carbon nanotubes ͑CNTs͒ at high current. The rings turn out to be due to the self-focusing of thermal-field electrons emitted from the near-cap shank of a CNT that reaches high temperature through Joule heating. To prove this we have simulated the electrostatic fields, the dependence of electron emission on field, temperature, and position on the CNT, and the emitted electron trajectories for different CNT/electrode geometries. The rings are formed when the manifold of emitted electron trajectories folds over onto itself due to self-focusing by the back-support plane. Sufficient thermal electron currents can only be emitted because CNTs can maintain a continuous high-temperature self-heating state, a state for now unique to carbon nanotubes. We do not need to evoke space charge effects and a program based on Green's functions is to calculate the field emission current for all values of field and temperature. The original observation of this phenomenon goes back to at least the pulsed high current experiments on W emitters in the 1950s ͓W.