The key-role of the Poynting vector in justifying the linear sampling method

A new and very general justification of the performances of the linear sampling method is presented. The constraint expressed by the far-field equation, together with energy conservation along the flow strips (or tubes in the 3D case) of the (time-averaged) Poynting vector associated to the field whose far-field pattern is the difference between the two sides of the far-field equation, force any approximate solution of the latter to behave as indicator function for the support of the unknown scatterer, provided that this approximation is good enough and the flow lines have a regular behaviour from a neighbourhood of the sampling point z up to the far-field region.