Focus wave modes: a dilemma with causality

It is demonstrated that the equivalent aperture distribution for a well-collimated FWM (focus wave mode) 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 various source-free synthesis options, and it is verified here directly, within the context of aperture radiation. The 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