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Any field in a half-space can be described by a continuous spectrum of cylindrical waves. If this spectrum comprises substantially only waves whose propagation constant is very close to the plane wave propagation constant, the field can be resolved into a set of elementary wave beams which are characterized by Laguerre polynomials. They satisfy orthogonality relations like the wave modes in a waveguide. The elementary beams or "beam modes" can be reiterated and guided by reconstituting the cross-sectional phase distribution at certain intervals. Reiterative beams are utilized in the beam waveguide. The finite size of the phase resetting devices effects a modification of the reiterative beam modes and causes diffraction losses. These losses decrease very rapidly with increasing diameter of the phasing devices.