The implosion of spherical capsules by incoherent X-rays emitted from the surrounding high-Z hot cavity walls seems to be a scheme relevant to inertial confinement fusion. The quality of the illumination for a spherical capsule enclosed within an axisymmetric radiating cavity is discussed for the two extreme cases of (optically) thick or thin, non-uniform, emitting cavity surface layers. When the specific intensity on the radiating surface is represented by an expansion in Legendre polynomials, it is found, for spherical radiating cavities, that the n term contribution to the power density upon the capsule can be factorized as the product of a function of the position on the capsule surface times a function f(n,x) of its radius x. Furthermore, it is found that values of x may exist for which f(n,x)=0, this being true both for thick and thin modelling for almost the same values of x. Simple criteria to obtain the desired degree of uniformity are suggested. The effects of the cavity shape on the illumination quality are also studied as a function of the cavity aspect ratio for spherical capsules enclosed within ellipsoidal cavities (oblate or prolate).