I've been introduced to the concept of equivalent isotropically radiated power (EIRP) and so far I've used it to calculate RF power density at distance of 100m radiated by a directional antenna with a gain of 20dBi at 10W output power. The EIRP is 60dBm (1kW) so using the inverse square law 1kW/4pi100^2 got me ~7.9mW/m^2. I'm not sure I haven't made a mistake so please correct me if I'm wrong but under the assumption that I was in fact correct, a question arose.
If EIRP is supposed to represent a hypothetical isotropic radiator that would result in the same signal strength as produced by the directional antenna (at least in the area covered by the latter) then what happens if we were to measure power density close to the antenna? If we perform the same calculation as above but for 1m we end up with 1kW/4pi1^2 and that gives me ~79W/m^2. Now, ignoring near-field radiation and other circumstantial effects, why is the power density so high?
I feel like this is a good proof that I don't really understand gain but the argumentation that it's just the same power as from the isotropic source, just focused down to a beam doesn't cut it for me. I am assuming that if I input 1W into a 100% efficient isotropic antenna I get a total of 1W output power in all directions, now if I were to focus that to a point/small area I will get closer and closer to 1W but never above it. Where am I wrong?
Edited: I think it would be wise to clarify my misunderstanding now that I know the answer. The confusion in the above question stems from the fact that you can get a higher power density than input power. Of course, this doesn't violate any laws of physics as it's not the power that's higher but the area that gets smaller than m^2 so the density increases.