Here is a detailed presentation on the topic:
https://www.ucolick.org/~max/289/Lectures%202016/Lecture%2010%20Laser%20Guide%20Stars/Lecture10.2016.v2.pdf

One of the things that made the use of Laser guide stars so challenging was the lack of lasers that could reach the exact wavelength needed- in the early 2000's there simply was no easy way to generate 10's of Watts of laser power at 589nm.

The first observatory systems had to use dye lasers - dye lasers are very ugly beasts.

A dye molecule was synthesized that - when 'pumped' by intense laser light at 532nm - would fluoresce over a broad range of wavelengths around 589nm. The dye in our case was dissolved into ~50 gallons of pure ethanol.

This flammable mixture went to 3 laser subsystems: Master oscillator, Pre-amplifier, and Amplifier.

The master oscillator generates a very low power of light locked to the sodium transition at 589nm (this is the seed light that gets amplified)- the dye is happy to lase at any of many wavelengths around this wavelength, so there is a complex setup of etalons, sodium vapor cell, and a feedback loop to ensure it only lased at the exact wavelength we needed. The dye was excited by a Neodymium YAG laser, that was frequency doubled by a non-linear crystal to output ~30 Watts of green 532nm light.

The seed light was transported to the laser room bolted to the side of the telescope where the pre-amplifier and amplifier were housed.

The pre-amplifier is a glass cell, the dye is flowing through this glass cell at a high rate. The pulses of seed light from the master oscillator pass through this cell of flowing dye. Additionally,intense pulses of green light from more Neodymium YAG lasers- precisely timed to overlap the pulse of seed light- were focused into the same volume of dye that the seed is passing through. The green light excites the dye, and the seed light stimulates the dye molecules to emit at the same wavelength of 589nm.

The pre-amp boosted the power of the 589nm light to ~100mW - this light is then sent to the final amplifier where the same process was repeated to boost it to ~10 Watts.

this light was then transmitted out to create the laser guide star.

Technology has advanced greatly since then, and the dye laser is no longer used. Its now much easier to achieve lasers at 589nm... The first Dye laser based sodium guide star was fielded at the Lick observatory, and the second was at Keck- both were built by a top-notch team of physicists and engineers at Lawrence Livermore National Lab.

Now, you can buy one that uses far friendlier and efficient technology from commercial vendors (IF you have the money)