Keep pulling the thread on inverse square law of light.
The perceived brightness of a searchlight beam viewed from the side decreases with the fourth power of the distance (1/d^4), combining the inverse square law for light traveling from the source and the inverse square law for light reflecting back to the observer.
The intensity of light from a source follows the inverse square law, where doubling the distance reduces the brightness to one-fourth, and tripling the distance reduces it to one-ninth.
Searchlight beams appear to terminate abruptly in the sky because their perceived brightness drops off so rapidly that the light quickly falls below the detection threshold of the human eye's retina.