The usual square law still applies. Of course if you are inside the beam cone, it look way brighter than an object of 10^35 W should look at the same distance, but if you move twice as far away, the brightness will still drop to one quarter. And inversely if you are NOT in the cone it will look dimmer than an object of this intrinsic brightness should look. But the gain you get from bundeling the radiation into a cone instead of emitting isotropically is fixed and the scaling with distance is thus unaffected. Look up "effective isotropic radiated power" (EIRP) if you want to know more.
I think many people's first foray into understanding this is grokking the difference between dBi and dBd, and then understanding that "gain" in antennas is another way of looking at directivity.
The inverse-square law always applies at distances like this. But the GP's remark probably refers to the difference in intensity between slight changes in angle. I.e., if you are ten times as far away, but just a hair closer to the beam center, you might have more flesh ripped from your bones anyway.
Another way to say this is that inverse-square applies exactly on any exactly radial path. But, close to the beam, even an unnoticeably oblique path may yield radically different results. In real life, you rarely get exactness, so in such a circumstance you should be prepared to find intensity varying along a very different curve.
> Achieving a truly collimated beam where the divergence is 0 is not possible, but achieving an approximately collimated beam by either minimizing the divergence or maximizing the distance between the point of observation and the nearest beam waist is possible.
The diffraction limit is a well supported theoretical limit that is caused by the wave nature of light. There is ways to get around it in the near field (up to a few ten wavelength away from the source). But in the far field you can not get a perfectly divergence-free finite-width beam. [1] Nor can you focus a beam to a beam waist smaller than about the wavelength.
In practice the situation is even worse and you can't even get the performance that a perfect Gaussian beam would allow. We often express the performance of real beams by a thing called M^2 or beam quality parameter [2]. In some sense is measures how much wider the beam is than it needs to be, and this number is never less than one.
I don't think that's true. The flux cross section f(2r)/f(r) = 4 is derived specifically for an isotropic radiation source. You would need to re-derive it for whichever beam shape you want to model.
EIRP is describing that the total flux through a sphere at a given radius is constant regardless of the beam shape, but we are only interested in a specific cross section and are not accounting for the rest of it.
And yes, the inverse square law is strictly speaking only true for point sources. But it is an extremely good approximation once you are much further away than the size of the source (a distance ten times the diameter of the source is usually totally sufficient).
I had to re-derive it for a perfect cone to be sure. You are correct; at a distance where the size of the cross section of the beam is significantly larger than the source, deviation from inverse square becomes negligible.
