Typically free-electron lasers (FELs) have employed electromagnets arranged in linearly periodic arrays to provide the magnetic field necessary to impart a transverse oscillatory acceleration to an electron beam thus causing it to radiate. With the advent of the high energy product permanent magnet materials, it has become practical in some cases to replace electromagnets with permanent magnet arrays. Difficulties with both procedures were obtaining desired magnetic field strengths at short wave lengths, utilization of only a small part of the beam energy, and impracticability of obtaining the desired bore cross sections at high field strengths and short periods. These problems can be ameliorated by permanent magnet arrays that offer unusually high fields and modest structural masses. In such structures, the electron beams follow circular rather than linear paths, the magnetic periodicity is azimuthal, the fields are axial and unidirectional, and the “wiggle” is in the radial direction. Because of its circular path, the e-beam traverses the array many times so that a much larger portion of the electron kinetic energy can be converted to radiation than in the usual linear arrays which the electron beam traverses only once. The circular beam arrays are based on both magic spheres and magic toroid structures in which the same field produces the wiggle and maintains the circular path. An example of the calculated results is that of a sphere in which the ratio of the outer to inner diameters is 3, and the extremes of the periodic field values are 3.4 and 3.7 T causing synchrotron radiation of 2.5×1013 Hz and wiggler radiation of 2.0×1014 Hz.