I'm asking. On another forum, somebody said a laser's power doesn't have anything to do with it's visibility.
Untrue - presuming you mean output power, of course.
I objected, using green lasers as an example, saying that lower powered ones can't be seen in day light, while higher powered ones can, thus proving power does affect visibility.
Keep in mind that many of the 'higher powered' ones sold to consumers these days do also happen to be the green lasers you're referring to, while the 'lower power' ones tend to be the classic red lasers. The human eye is more sensitive to green than it is to red and can resolve greater detail at green wavelengths (hence the Bayer pattern of digital cameras having double the green photosites). Thus, the color affects visibility as well.
Another person said lasers can only be seen, because their light is bouncing off dust and stuff in the atmosphere.
Correct. Light is just photons. For you to see those photons, they have to reach your eye. The only way for the photons from a laser to eventually reach your eye is either by having the laser pointed at your eyes directly (uhhm... don't. seriously, do not do that.) or for the photons to bounce off of other surfaces and then to your eye.
In 'air', that tends to be dust particles, soot particles from traffic, water vapor, etc.
That one I'm curious about. What if a laser was shot in a clean vacuum, with no dust for the light to bounce off of? Would we be able to see the laser, provided it had enough power and it was of the right wavelength? (lets just use the typical 532nm green laser for an example)
No, regardless of the strength of the laser. It could be that crazy multimegawatt laser they use in high energy physics experiments and (if were actually visible light they used) you still wouldn't see a beam.
*however*
At some point you are going to create lasers that are so incredibly powerful - absolutely ridiculously powerful - that physics as you and I know it go right out the window, and the lasers can conceivable create matter out of their own energy.. when that happens, there's once again matter for the beam to reflect off of, and you would see the beam. Technically, though, it would no longer be a perfect vacuum environment as the matter would be the contaminant.
This is one of many recent articles on that topic:
http://www.gizmodo.com.au/2010/08/lasers-so-powerful-they-destroy-themselves/
I'm not a physicist, though - I deal more with biology and mineralogy - so don't ask for details as to the above process
Are laser light waves similiar to radio waves in that the amplitude of the wave increases with more power, thus making it a wider/larger wave, and is that why the more powerful lasers have a thicker beam?
No. Radio waves are only related to light in terms of wavelengths. Light itself is an oddball particle/wave dual thingymabob (did I mention I'm not a physicist) that you can't apply other electromagnetic field properties to so easily.
Most simply put - the more powerful lasers have a -brighter- beam simply because they emit more photons per given timeframe. The 'thickness' of the beam is entirely a function of the collimator being used. Lasers are often regarded as having a constant thickness.. but shine a laserpointer far enough (kilometers, not meters) and you'll see that they do typically diverge a little bit. Similarly, an appropriate assembly can converge the laser to just a tiny pinprick. This has been used to create a 3D display, for example, exhibited at Siggraph by turning the air into a plasma at the focal point of the laser.
http://www.youtube.com/watch?v=He2QTpelAjE