Self-imaging with finite energy.

General solutions and conditions are presented for paraxial waves that image themselves with different scales through free propagation. These waves, represented as superpositions of Gauss-Laguerre modes, have finite energy and thus finite effective width. The self-imaging wave fields described by Montgomery [J. Opt. Soc. Am. 57, 772 (1967)], which possess a Fourier transform that is confined to a ring structure, are obtained as a specific limiting case of an infinite aperture.