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It is well-known that laser light is directional because of coherence from stimulated emission. This means that the light is emitted ONLY in the direction of propagation, and not in any type of sideways direction (unlike a standard light bulb). Therefore we should not 'see' the laser light until it actually hits its target, and is partially reflected back to our own observation point.

So why is it that despite our own observation point say, being at right angles to the direction of propagation of the laser beam, we can still see the laser beam when no light is travelling to us before it reaches its target? And in particular, would we still be able to observe the laser beam in a vacuum before it reaches its target, where we are positioned orthogonal to the beam?

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If you're in a normal environment (classroom, standard lab, etc.), there are particles of dust in the air that can scatter part of the beam in any direction. If I understand your question correctly, that's what you see when "positioned orthogonal to the beam." In a vacuum, as you suggested, that would not happen because there would be nothing against which it could scatter.

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