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.
© 1997 Optical Society of America
Rafael Piestun, Yoav Y. Schechner, and Joseph Shamir, "Self-imaging with finite energy," Opt. Lett. 22, 200-202 (1997)