Lasers are bright, coherent light sources that propagate in a well-defined beam. Those properties make them useful for communications, interferometry, and even traffic speed measurement. But laser beams are also relatively small in diameter because the cost and complexity goes up as the mirrors and the active gain media inside get larger. But turning that 5-mm collimated beam into a 50-mm beam is easy. "Collimated" means "travelling without changing size," so a beam expander converts a small diameter beam into a large one, still staying collimated.
Determine the desired magnification. As an example, assume a 4-mm diameter beam needs to be expanded to a 40-mm diameter. That's a magnification of 10X.
Select a converging lens larger than the input beam. As far as focal length, a good place to start is about three or four times the input beam diameter.
If, for example, the input beam is 4mm in diameter, a 12.5-mm diameter lens is convenient. A convenient focal length would be 16mm.
Find the focal length of the collimating lens. The collimating lens focal length is the product of the converging beam focal length and the magnification.
In the example, the magnification is 10X and the converging lens focal length is 20mm. The collimating lens focal length is 10 * 16mm = 160mm.
Find the diameter of the collimating lens. The diameter of the output beam will be equal to the diameter of the input beam times the magnification. The collimating lens needs to be a little larger than the output beam.
In the example, the input beam is 4mm and the magnification is 10X which means (as expected) the output beam is 4mm * 10 = 40mm. A good choice for the collimating lens is therefore a 50-mm diameter, 160-mm focal length lens.
Place the collimating lens in the laser beam path, centring it on the laser beam. The other lens --- the converging lens --- will be closer to the laser, so leave the room.
Measure from the collimating lens back towards the laser. Place the converging lens closer to the laser by a distance equal to the sum of the two focal lengths. Center the converging lens in the laser beam.
In the example, the converging lens has a focal length of 16mm and the collimating lens focal length is 160mm, which means the two lenses will be separated by 176mm.
Adjust the position of the collimating lens to make the laser beam stay at the same diameter as it propagates along the optical path. This is a quick method to verify collimation.
With a converging lens, this is called a Keplerian beam expander. If the first lens is replaced by a diverging lens, the resulting beam expander will be more compact, without an intermediate laser focus point. That's a Galilean, or afocal, beam expander. Using off-the-shelf lenses, it's rare to find the perfect lens combination. Find lenses that are "close enough" and trim the beam down to size with an aperture. If building your own beam expander is too problematic, many companies offer off-the-shelf beam expanders in a range of magnifications for a range of wavelengths.
Watch for stray reflections. Particularly when inserting and aligning optical elements, laser reflections can fly through the lab. Lasers can easily damage retinas --- don't take chances with your sight.