Analysis and particle-in-cell (PIC) simulations of fast particles produced by a short laser pulse of duration 40 fs and intensity larger than 10¹⁹ W/cm² interacting with a foil target are performed. Fast particles generated by laser-plasma interactions can be used in many applications, from manufacturing to medicine and even for the initiation of tabletop nuclear reactions. Fast ion generation by the interaction of an ultra-short high intensity laser pulse with a plasma has been demonstrated in recent theoretical [1] and experimental [2] papers, with maximum ion energies of up to 0.5 GeV having been observed. Different methods of fast ion generation have been proposed for both gas [3] and solid [4] targets. It has been shown that the energy of a laser pulse can be efficiently converted into fast ion energy using foil targets. Simulations [5] have shown that the mechanisms for generating ion acceleration are the ambipolar field and the Coulomb explosion. It has also been shown that fast electrons ejected from the foil by the laser field create a strong ambipolar field, which is the main source of acceleration of ions ejected from the back of the foil. Thus, a collimated ion beam can be produced by focusing an intense laser onto the surface of a solid film. Fast ions accelerate normally to the foil surface because this is the direction of the ambipolar field [6]. It is clear that these ions could be focused by a curve of some sort in the foil surface. This effect has been demonstrated numerically for plasmas with sharp density gradients and experimentally [7], but there is not analytical development of this process. Most experimental high power lasers produce a pre-pulse, which generates a plasma layer with a smooth density gradient on the surface of the foil. We attempt to develop an analytical model to analyze the mechanisms of ion acceleration in plasma layers with smooth density gradients and optimize gradient scale. Based on analytical model tested 2D PIC simulations, we obtain space-angular ion beam distribution for curved laser foil targets.

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