Spectral analysis of focus wave modes

Abstract

The focus wave mode (FWM), which is a time-dependent beam field that propagates at the speed of light without dispersion and retains its shape in space, is an interesting wave object with possible implications for synthesizing focused fields under transient conditions. To explore this potential, it is necessary to understand fully the properties of this wave field. It is already known that the FWM is a homogeneous solution of the wave equation, which is related in a special way to the field of a source moving on a complex trajectory parallel to the real axis of propagation. This suggests that there may be a connection between the FWM and the conventional free-space Green’s function. It is shown here that the FWM is related, in fact, to a source-free combination of causal and anticausal free-space Green’s functions and that one can formulate a bilateral transform pair relating these solutions. This new representation is then analyzed by using the spectral theory of transients to establish the properties of the FWM in terms of a distribution of transient plane waves. The spectral decomposition in the spatial wave-number domain reveals that the FWM is synthesized by both forward- and backward-propagating plane waves that are restricted to the visible spectrum. Asymptotic considerations show that the dominant mechanism is constructive interference of the backward-propagating waves. Taken together, the Green’s-function and spectral approaches grant further insight into the physical and spectral properties of the FWM. The conclusions cast doubt on the possibility of embedding the FWM within a causal excitation scheme.