If I have a green laser (500 nm wavelength) with initial beam diameter 3 mm, with proper lenses I can easily focus it to 1 mm or 0.5 mm for small distances like 10 cm.
My question is can I focus it from 3mm to 1 mm on long distances like 1 km or 100 m? From common sense, I feel that it is not possible from diffraction limits point of view, but I could not find exact explanation and equation how to calculate it. There are equations how to calculate maximum possible resolution for microscope/telescope, but not the minimum size of focal spot depending on the focal length/wave length of the lens.
For example, in the picture below if X is a few centimetres it looks possible, but if X is 100 meters or 1 km is it possible or not theoretically and practically?
Let's consider a simple case when I have a laser beam, which usually has Gaussian energy distribution, but for simplicity let's assume I cut sides of this Gaussian beam with the diaphragm and it almost flat (not really Gaussian). Anyway, 80% of energy is in the middle of Gaussian beam and if I cut the sides I will lose only 20%.
Or I can ask how to calculate what is the minimal Gaussian beam waist can be achieved by lens depending on focal length of the lens, beam wavelength and initial beam waist size. If I read this article, I could not find the answer as well.
For lasers there is the only way to calculate it properly is Ray Transfer matrices for Gaussian beams as written here. It depends not only on initial beam size, but on initial beam curvature (how fast it diverges in space).
I asked one professor specialized in lasers and the answer was: In reality, focusing the laser beam to small size can be achieved for several meters with some optics (lenses etc), but long distance is practically not possible.