Optimising point source irradiation of a capsule for maximum uniformity

Inertial Confinement Fusion involves the implosion of a spherical capsule containing thermonuclear fuel. The implosion is driven by irradiating the outside of the capsule by X-rays or by optical laser irradiation, where in each case the highest uniformity of irradiation is sought. In this paper we consider the theoretical problem of irradiation of a capsule by point sources of X-rays, and seek configurations which maximise uniformity. By studying the root-mean-square deviation in terms of different order harmonic modes, we rationalise the dependence of uniformity on distance $d$ of the point sources from the centre of a capsule. After investigating simple configurations based on the Platonic solids, we use a global optimisation algorithm (basin-hopping) to seek better arrangements. The optimum configurations are found to depend strongly on $d$; at certain values which minimise nonuniformity, these involve grouping of sources on the vertices of octahedra or icosahedra, which we explain using a modal decomposition. The effect of uncertainties in both position and intensity is studied, and lastly we investigate the illumination of a capsule whose radius is changing with time.