[strainer]

You get a lower exponent than the square with a beam. For a perfectly parallel beam the exponent is technically 0 (same intensity regardless of distance - like a laser pointer) For a focused beam intensity will actually increase with distance up to the focal point. Some energy can be absorbed by the medium (air/moisture/rain), but that is not covered by the inverse square law.

[abstrakraft]

Not with a planar or linear array. I'm skeptical, but prepared to learn something: what sort of array can generate a square beam?

Even lasers spread. This guy agrees with me: https://www.quora.com/Is-the-light-from-lasers-reduced-by-th...

[strainer]

The cross section of the beam can be any shape, the important thing is the signal source is an area not a point. The value of the inverse square (of distance from source) is dominates power density when the signal source is effectively a point - this becomes the situation for a laser or anything at long enough distance, but at distances where it is effectively a beam, the strength of power density does not dissipate by that value. The geometry of the beam concentrates the power density at its focal point, which can be far behind the source or in front of it. For an ideal beam focused to infinity (parallel rays) its power would never dissipate with distance - as all of the power goes in the same direction. We don't need to form an ideal beam to say its doesn't follow the inverse square rule, any focusing of rays breaks the rule (at scales where it is reasonable to treat it as a beam and not just rays emanating from a point)

I hope that helps picture the situation. My original comment that a beam can have a different exponent was incorrect except in some approximate sense. The inverse square value will still apply, but from the beams real or imaginary focal point (if it is a point) But it the case of a transmitter beaming a signal at another, that focal length can be far beyond the reciever, so the 'rule' can be completely confounded.

[jakobegger]

It depends. The inverse square law applies to spherical wavefronts. Due to diffraction, wave fronts always become spherical in the far field.

But in the near field, that's not the case.

Beamforming arrays have planar wavefronts close to the source, so the inverse square law does not apply. The wave fronts will become spherical again at a "large" distance from the emitter, where the meaning of "large" depends on wavelength and emitter size.

Focussed lasers also do not have spherical wave fronts in the near field. The distance at which the inverse square law starts to apply to lasers depends on beam width, coherence, and focus.