Narrow EM wave beam - how it propagates

Consider a narrow EM wave beam propagating in the Z direction, whose intensity is maximal at the YZ plane and fades quickly as we move away from the YZ plane. Let X be the direction of the E vector.
Consider a small (w.r.t wavelength) cube, whose sides are parallel to the X,Y and Z axes, respectively. Let the cube be placed some distance away from the YZ plane. It seems that the flux of E through the closer face parallel to the YZ axis is bigger than the flux of E through the further face parallel to the YZ plane, in contradiction to the Gauss' law. What is the explanation?
(Even when we take it into account that the beam is divergent, we can still place the cube so that two of its facets are perpendicular to E. It still seems to contradict the Gauss' law).

It would seem to me that since there is no source charge in the cube, Div E = 0. This qualitatively means that what flux lines comes in goes out, which in this case seems to be satisfied.

perhaps I am not understanding your question, but this is my perspective.

have fun

Also, as an analogy, if I place this same cube next to a huge amount of charge, but still with no charge within it, again you have huge flux on one side of the cube and less on the other, but again since there is no charge within, Div E = 0 and you can imagine that all flux lines that enter will leave the cube. Seems like a similar scenario.

There's no contradiction. Any propagating field other than a plane-wave, in particular fields which are confined in space or fields with space-dependent amplitude like the one you described (if I understand correctly), can be described as a superposition of plane-waves propagating in different directions via a fourier transform. Exactly one of them is propagating in the Z direction, others are propagating in various angles to the Z direction. So you have power flux not only through the forward and backward facing sides of the cube, but also from the top, bottom and sides. If you count all of them, Gauss' Law should be just fine. Read more about diffraction for more detailed descriptions.

TheArcane, clearly you do not understand what Fourier transform does.
The beam I described cannot be represented as a superposition of plane waves.

@htg: I followed your recent threads on waveguide feed and resonators. You should really get a book on waveguide basics, and study that first.