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I was wondering, since a dyson fan blows out air in the ring, and the surrounding air gets sucked through with it, greatly amplifying the amount of air being moved, is such a mechanic also possible for photons?

Where a ring or light emitting photos create such a drag on surrounding photons or similar materials that can be affected by it so that the total quantity of photons(ignoring the spectrum those photons are moving in) moving in the same direction is greater than the emitted photons?

Or are photons too small to be affected by these mechanics?

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It's not a question of photons being too small, but a question of photons not interacting with each other in most conditions. You can cross two laser beams and they will be completely unaffected most of the time. Classically, this is because electromagnetic fields obey the principle of superposition. Modernly, this can be explained by the fact that photons are an electromagnetic phenomena but carry no charge and therefore don't attract or repulse each other.

For completeness sake, there are two situations in which photons can interact with each other: 1 - In some materials, when two photons of the right energy hit the atoms present in the material, some new effect occurs that would not occur if only one photon was there. You can learn more about this my googling non-linear optics. 2 - At very high energies (gamma rays), photons can create virtual particles and these particles can interact with each other to influence the trajectories of two nearby photons. Even then, I don't think you could get a directional effect as in the Dyson fan from any of these effects.

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