Energetic proton acceleration from concave targets, the front of which were irradiated with 40 fs laser pulses with an intensity of 1020 W cm2, has been studied as a function of the depth of the concave shape. Three kinds of targets, a triangular concave target, a circular concave target and a parabolic concave target are considered. When the depth of the concave shape was varied, the peak proton energy showed a maximum. The underlying mechanism for the existence of a maximum peak proton energy is presented by tracing the proton trajectory. It is concluded that a parabolic concave target is the best, among the targets considered, for accelerating a proton beam, since a proton beam from a parabolic concave target goes through the strongest electric field.