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I read this question, and I thought the following thing:

As photons can collide with each other, can we use photons to control the direction of another photon? As far as I know photons have a acceleration, so if we can throw a photon extremely close to the collision point, this photon will have a small acceleration and will just hit the another photon and change the direction of both..

for example:

  1. photon 1 is thrown
  2. when photon 1 is very close to the collision point photon 2 is thrown
  3. Both photons collide to each other and goes to different direction (like pool balls)
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    $\begingroup$ "like pool balls" is the issue. To deal with photons you need to deal with quantum theory and this doesn't work out like classical mechanics with pool balls. $\endgroup$ – StephenG Feb 1 '17 at 18:26
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Photons do not collide with each other. Or at least only very very rare. In this rare process the direction of the scattered photon has an angular dependency. That means you cant really control the scattered photon to go into a specific direction. Can we use photons to control the direction of another photon? No, not with your two photon set up.

Of course you could build some set up where a light beam activates a 'machine' that bends another light beam into a specific direction..

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Photon-photon collisions do happen. As Mr. Puh said, it's very uncommon. In technical terms, the scattering cross-section is very low. High-energy photons have a higher cross-section than low-energy photons, so they are much more likely to collide. The probability of a collision goes up with frequency to the 6th power, but unfortunately I can't find a quick comparison of how likely the collision is compared to particles like electrons or protons.

Trying to deflect photons via collision runs into Heisenberg's uncertainty principle. With pool balls, we can know where they are and how fast they're moving to an incredibly high degree of accuracy. We can predict the outgoing momentum very well. With light, we'll need to narrow down the width of our target area to that cross-section that I mentioned before. (I really wish I could find a number for that.) This means the light's momentum will be uncertain, and small changes in the momentum could result in a different direction for the other photon.

One could come up with a probability distribution that describes where the photons are most likely to go afterward, but it would be much less precise than if you did this with larger objects like pool balls or even two atomic nuclei.

One other small item: You say that photons have an acceleration. This is typically not true when they're passing through empty space. They only change speed when they're moving from one medium to another (like from air to water). I'm not sure why you're bringing in acceleration here; it doesn't seem to have much to do with the question.

I hope that helps!

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Photon photon scattering belongs to the framework of quantum mechanics, as photons are elementary particles of the standard model.

This means that there is a probability distribution of scattering and the calculations depend on quantum electrodynamics , with the evaluation of Feynman diagrams.

twophot

This diagram for frequencies of light lower than twice the mass of an electron calculated will give the probability. The four electromagnetic vertices mean that the probability is small, order of (1/137)^4. So a photon passes a photon most probably undisturbed.

For energies where pair production can take place, an e+e- pair for example, there are only two electromagnetic vertices, and then the probability gets higher, and gamma gamma scattering has been proposed to form a collider.

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