Focus Wave Modes: A Dilemma with Causality

Focus wave modes (FWM) are source-free wave fields that behave like well-collimated beams moving through space at the speed of light. Because of this property the FWM have aroused interest as possible prototypes for launching focused electromagnetic pulses or, more generally, as basis functions for synthesizing such pulses. Previous investigations have established that the FWM can be generated either by an oscillating source and “antisource” moving together at the speed of light along a straight trajectory in a complex coordinate space, by a superposition of conventional causal and anticausal Green’s functions of the time-dependent equation if the source location and initiation time are allowed to be complex and in a moving coordinate frame, or by continuous superposition in the spatial wavenumber domain of forward and backward propagating transient plane waves. Each of these representations provides its own interpretation of the nature of the FWM, but all contain evidence of the simultaneous presence of forward and backward propagating wave fields. With the latter playing a dominant role in the well-collimated regime, this poses a fundamental difficulty when such collimated FWM basis functions are to be used for synthesis of causal excitations over an initial aperture plane. As demonstrated in the present paper, the equivalent aperture distribution for a well-collimated FWM radiates essentially backward; i.e., the important propagator in the forward propagating FWM is time-reversed (anticausal). This feature has already been noted for each constituent in the various source-free synthesis options listed above, and it is verified here directly, within the contest of aperture radiation. The unhappy conclusion is that despite their appealing physical properties, these source-free wave fields, when well collimated so that their focusing properties are taken advantages of, are inherently unsuited as elements for direct and efficient modeling of time-dependent causal phenomena.