The understanding and the control of high-power laser propagation into under-dense plasma is important to achieve inertial confinement fusion. During this process, the interaction of the laser with the plasma filling the hohlraum can lead to significant losses of laser energy which prevent ignition. Self-focusing or filamentation of the laser light is one of these phenomena which are desired to be mitigated since they also affect the uniformity of the laser illumination on the hohlraum wall.In order to improve our understanding of the laser-plasma interaction phenomena at play, we describe an experimental and numerical study involving an intense laser pulse between 1014 W.cm-2 and 1016 W.cm-2 , and which interacts with millimetric and under-dense plasma (having density of few % of the critical density). This work presents two experiments fielding a series of diagnostics aimed at well characterizing the laser propagation (Hisac camera) together with heat deposition in plasmas using Thomson scattering. Experimental results will be presented and discussed in the light of detailed simulations performed with the 3D laser propagation code Hera. In order to take into account the temperature gradients within the plasma during the laser propagation, Hera (laser propagation code) and FCI2 (radiation-hydrodynamic code) have been coupled. Besides, proton radiography has been used in order to access to electric fields. The measurements led to the implementation of a new and promising numerical tool using the Hera and Diane codes (Diane is a Monte Carlo particle tracing code). 3D proton radiography modelling opens new possibilities for users of this temporally and spatially resolved diagnostic.
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