When we say, for example, an antenna has spec of 22 dBi;
Does it mean that its power is 22 dB stronger for any point at fixed direction than if it were to emit spherically?
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For an antenna, a gain of 22 dBi means a gain of 22 dB with respect to a theoretical isotropic antenna.
If an isotrpoic antenna radiates 1 watt, it would radiate that power uniformly over the 4π steradians around it, so at a power density of about 80 mW per steradian.
For your antenna with 22 dBi gain, 22 dB is a power ratio of 158, the emitted power density in the peak direction would be 158 * 80m = 12.5 W/steradian for the same total 1 watt radiated.
You don't get something for nothing, and the beam width at this gain would be very small.
As you move further away from the antenna, the gain would stay at 22 dBi, and the power per steradian would stay the same. As the distance increases, the actual area covered by one steradian increases, and the power per area would fall following the standard inverse square law relation.
For reference, there's no physical isotropic radiator. A simple dipole has a gain of 3 dBi, because it radiates its power preferrentially around its equator, and no power towards either pole.