Different photon will not be able to interact with each other due to superposition (not quite correct because of non-linear phenomenon but close enough), and in particular there are no interferences. (if there are interactions due to non-linear phenomenon, they could create particles with mass)
When you see an interference pattern, each photon is interfering with *itself*. In fact, the same photon going through different slits give a wavefunction for each slit and these wavefunctions are always orthogonal.
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