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    One of the most significant factors in understanding the behaviour of laser beams during the propagation process through plasma is how the spot size of the beam changes with both laser and plasma parameters. Self-focusing and defocusing of the laser beam, therefore, play an important role in this study. This chapter is aimed at presenting a brief investigation into these common phenomena in laser-plasma interaction. In addition to a short overview on the types of self-focusing of laser beams through plasma, a detailed study on the relativistic self-focusing of high-intense laser beam in quantum plasma as a particular nonlinear medium is performed. In this case, the essential equations to model this phenomenon are derived; furthermore, a range of values of laser-plasma parameters which would satisfy these equations is considered.
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    A hypothesis of the beam auto-focusing in the relativistic vacuum diode with a needle anode is proposed. A physical model of the effect is elaborated. As is shown, auto-focusing of a high current beam can be caused by ions emitted from the anode. The initial (pre-focusing) stage of focusing is provided with ionization of the gas molecules absorbed in the anode surface. Then (at a quasi-steady stage), the focusing is driven by the metal ions being evaporated from the anode heated by the beam. Simulation of the process show that the duration of the initial pre-focusing stage would be within 0.1-0.2 nsec.
    Relativistic electron beam
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