Field emission displays (FEDs) work in principle in a similar way with conventional cathode ray tubes (CRTs), namely emitted electrons excite phosphors which in turn emit visible light. However, FEDs obtain the electrons through field emission from a distributed network of sharp emitters, and there is no special deflection system for them. The size of the light spot associated with an island of emitters (a pixel) depends on many factors, such as display geometrical dimensions and operating conditions. In this article, a special self-focusing configuration for FEDs is analyzed using a combined analytical and numerical approach. The proposed configuration involves a distributed network of electron emitting areas which are concave in shape and not plane as usual, one for each phosphor pixel. The cathode concave areas are covered with a porous silicon layer, having sharp fibrils with several nanometers radius of curvature. The concave shape has a built-in electron self-focusing feature but, as a by-product, decreases the electric field compared to the plane cathode situation. However, the field multiplication approximation applies in this case, as concerns the electric field enhancement due to both the concave cathode and fibrils. It is shown that high enough electric fields are obtained, allowing electron field emission to occur. Analytical equations are provided for the electric field on the cathode concave surface. These equations are then used in a numerical model, taking into account the electric field enhancement associated with the fibrils, their mutual influence included. This approach allows the computation of the field emitted current as function of model parameters.