Analysis and particle-in-cell (PIC) simulations of fast particles produced by a short laser pulse with duration of 40 fs and intensity ≥ 10¹⁸ W/cm² interacting with a foil target are performed. Initially, the plasma density distribution of the foil target has a smooth gradient with the scale-length of plasma density varying across it. The absorbed laser energy is transferred to fast electrons, which interact with the foil and are partially ejected from the foil surface. These electrons produce an electric field that causes an ion beam to be emitted from the foil. We analyze the different mechanisms of ion acceleration in the foil plasma and the influence of density gradient and other laser and plasma parameters on ion acceleration. The angular distributions of the ejected electrons and ions are calculated. The optimum laser-plasma parameters needed to achieve the most highly focused ion beam are analyzed.