Theory of “frozen waves”: modeling the shape of stationary wave fields

In this work, starting by suitable superpositions of equal-frequency Bessel beams, we develop a theoretical and experimental methodology to obtain localized stationary wave fields (with high transverse localization) whose longitudinal intensity pattern can approximately assume any desired shape within a chosen interval 0 < or = z < or = L of the propagation axis z. Their intensity envelope remains static, i.e., with velocity v = 0, so we have named "frozen waves" (FWs) these new solutions to the wave equations (and, in particular, to the Maxwell equation). Inside the envelope of a FW, only the carrier wave propagates. The longitudinal shape, within the interval 0 < or = z < or = L, can be chosen in such a way that no nonnegligible field exists outside the predetermined region (consisting, e.g., in one or more high-intensity peaks). Our solutions are notable also for the different and interesting applications they can have--especially in electromagnetism and acoustics--such as optical tweezers, atom guides, optical or acoustic bistouries, and various important medical apparatuses.