Why don't photons collide with each other when traveling towards each other?
Photons in free space act almost exclusively as waves. Therefore, when they cross paths they merely set up an interference pattern for the very brief time of their interaction. No energy is exchanged and the quantum state of each photon is unchanged after they pass each other.
This interference pattern is akin to ripples on water that approach each other, form an interference pattern of peaks and troughs and then continue on their way.
If matter is present where the photons cross, non-linear effects caused by accelerated electric charges may allow the photons to interact. This interaction could be considered a collision of sorts, resulting in exchange of energies with many possible outcomes.
One such outcome is called frequency doubling, where two photons are combined to form one photon at twice the frequency.
Scott Wilber, President, ComScire - Quantum World Corporation
'The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.'