In laser-plasma interactions,a magnetic field with a strength up to 100 T magnitudes can be generated as a result of different physical mechanisms.In recent years,many achievements have been made in theoretical and experimental research on self-generated magnetic fields,including proton radiography diagnostics of magnetic fields.

In Ref.[12],time-resolved monoenergetic proton radiography was used to measure the magnetic field generated in nanosecond laser foil interactions.The 2D hydrodynamic program LASNEX was used to simulate magnetic field generation and evolution,and the hybrid particle-in-cell program LSP was used to calculate the image formed by the proton beam deflection due to the applied magnetic field.

The circular ring structure and overall deformation structure are reproduced almost exactly by LASNEX+LSP simulations.However,considering the mean radius of the ring and the deflection of each beamlet,there remain differences between the experimental results and the LASNEX+LSP results at 0.6 ns,0.9 ns,and 1.2 ns.

In this study,we attempt to determine the magnetic field distribution that matches the experimental results using the Monte Carlo particle transport program.

FLUKA is a versatile Monte Carlo particle transport program that can perform calculations about particle transport and interactions with matter.As it can reasonably consider the interaction between protons and matter and couple a magnetic field with a certain spatial distribution into the program,it can be used in the calculations of the proton radiography of the self-generated magnetic field,which is the focus of this study.

Based on the understanding of the generation mechanisms of the magnetic field,the features of the magnetic field distributions obtained via LASNEX simulations,and the displacement of each proton beamlet obtained from the experimental results in Ref.[12],we assume that the magnetic fields have the form B=B0exp(?|r?Rb|lb)×d2Rb2, where B₀is the peak intensity,R_bcorresponds to the radius of the bubble surface,l_bis the scale length of the magnetic field,r is the distance from the field point to the laser focal point,and d is the vertical distance from the field point to the z-axis.Moreover,simulations are performed using different parameters.The one-dimensional displacement of the proton beamlets is compared with the experimental results.Good consistency is achieved with the proton radiography images in Ref.[12].

The spatial distributions and time evolution of the magnetic field deduced from FLUKA and those calculated from LASNEX are compared and analyzed.First,we find that the peak intensity of the magnetic field increases during the laser pulse,reaching 38 T at 0.9 ns,and decreases when the laser is switched off(Fig.4),which is consistent with the LASNEX results.Second,we roughly estimate the speed of plasma expansion when the laser is on according to the evolution of the radius of magnetic field(Fig.6).Third,we find that the range of the magnetic field inferred from the FLUKA calculations(Fig.7 and Table 1)is larger than that calculated from the LASNEX simulations.One main reason for this is the influence of magnetic field dissipation.Another reason is the existence of an under-dense plasma region with a small temperature gradient before the critical surface.

The distribution of the magnetic field in long-pulse laser-plasma interactions is achieved in this study.We achieve good consistency with the proton radiography images obtained in Ref.[12]using the FLUKA simulations.The evolution of the peak intensity and bubble radius are examined.The range of the magnetic field is compared with that calculated using the hydrodynamic program,LASNEX.The field range is found to be larger than that obtained from the LASNEX simulations,and possible reasons for this are analyzed.FLUKA can be an efficient tool for obtaining a detailed and reasonable interpretation of experimental data,which is vital in proton radiography studies.