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I have a green laser. Texts usually say you can only see visible light when it is reflected from another material. But you can clearly see a laser ray all the way from the device to the surface pointed at, what is that ray then? Is it a reflection off of air? I can also see tiny sparkling green "dots" hovering around the central ray, sometimes they shine brighter than others. What are they? dust particles reflecting? On a surface pointed at I can also see many scattered "dots", if I put the laser through a transparent surface, the scattering and the density of the dots changes (if I hold either the laser or the surface in my hands they constantly change looking like a "static noise on a TV".

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The laser ray that you see in the air is, as you guessed, mostly diffusion and reflection by dust particles (the classical experiment is chalk dust on the path of the laser).

What you see on a surface (a wall say) is called speckle, and is caused by the slight imperfection of the wall. It only happens because you're using a laser. You're actually seeing an interference pattern. This only happens because the laser is extremely monochromatic (which explain why you don't see that with usual light). When you're putting a transparent medium on the path of the laser, you're modifying the phase of the wave, and changing the pattern accordingly, the resulting figure depends on details of the wall up to a scale of $0.1 \ \mu m$ (your wavelength), hence it is quite random and hard to predict.

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As well as lighting up dust particles along its path, the beam is also lighting up water droplets in the air, particularly when outdoors.

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