my cents :shhh:
The expander is used to expand a beam of the finished laser (like a DPSS green laser with a ~1mm beam). A simple design is a combination of a negative (divergent) lens and a positive (focusing) lens.
The collimation is mostly done for laser diodes with a very big divergent. Mostly a single aspehric lens is sufficient to get a good beam. Though collimated beam sizes are not large (<15 mm) as good optics for large NA & diameter are very expensive.
Yes. Some tricks should be done to correct a laser diode astigmatism.
A philosophical question:
What do you call a collimated laser beam? :na:
(The answer should give an insight about a relationship between a focal length, a beam diameter, and the farthest focusing distance within Gaussian optics.)