Is a 1mW red laser dangerous?

A 1mW red laser would mean a directly-injected laser diode that emits its radiation only in a very narrow band; no infrared would be generated whatsoever.

So, unless you shoot it *DIRECTLY* into your eye(s) for more than 250mS (¼ second), there should be no danger whatsoever.
 
Like The LED Museum said, a red laser would not output infrared radiation since it is not frequency-doubled, it is just a single wavelength diode. As long as you don't shine it in your eyes it is very safe, and even if you did by accident at that low power level I don't think it would cause any damage (but you should still be very careful!).

An actual infrared laser (Or frequency doubled laser without an IR filter which would put out a lot of IR) would be very dangerous because you can't see the infrared (so you won't see it and look away if it reflects in your eyes) and IR directly reflects off of most surfaces the way normal light does off of a mirror.
 
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