In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field which does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillating charges and currents). Even when there in fact is an electromagnetic wave produced (e.g. by a transmitting antenna) one can still identify as an evanescent field the component of the electric or magnetic field that cannot be attributed to the propagating wave observed at a distance of many wavelengths (such as the far field of a transmitting antenna).
The hallmark of an evanescent field is that there is no net energy flow in that region. Since the net flow of electromagnetic energy is given by the average Poynting vector, that means that the Poynting vector in these regions, as averaged over a complete oscillation cycle, is zero.
In many cases one cannot simply say that a field is or is not evanescent. For instance, in the above illustration energy is indeed transmitted in the horizontal direction. The field strength drops off exponentially away from the surface, leaving it concentrated in a region very close to the interface, for which reason this is referred to as a surface wave. However there is no propagation of energy away from (or toward) the surface (in the z direction), so that one could properly describe the field as being "evanescent in the z direction." This is one illustration of the inexactness of the term. In most cases where they exist, evanescent fields are simply thought of and referred to as electric or magnetic fields, without the evanescent property (zero average Poynting vector in one or all directions) ever being pointed out. The term is especially applied to differentiate a field or solution from cases where one normally expects a propagating wave