Conformal mapping was used to find a closed-form solution to Laplace's equation in the thin-film cathode field-emission triode, a structure commonly used in field-emission display applications. The equation was used to find the electric field on the surface of the cathode and this, together with the Fowler-Nordheim equation and published data on a carbon nanotube film, was used to find the emitted current density across the surface of the cathode. The equation of motion of the electron was integrated using a Runge-Kutta algorithm to find the electron trajectories. A range of anode voltages was investigated, and it was found that the beam divergence that occurred at low anode voltages could be turned into beam convergence by an increase in the anode voltage sufficient such that the vertical field in the device due to the anode was greater than that due to the gate. Furthermore, the modeling showed that when the anode voltage was this high, emission occurred preferentially from the center of the cathode, rather than from the edges, which occurred at lower anode voltages. This modeling provides a method of controlling beam divergence in field-emission-display pixels, without having to resort to additional focusing electrodes, hence, reducing fabrication complexity and system cost. (C) 2004 American Vacuum Society.