Scattering of light by light does not occur in the solutions of Maxwell's equations (since they are linear and EM waves obey superposition), but it is a prediction of QED (the most significant Feynman diagrams have a closed loop of four electron propagators).
Has scattering of light by light in vacuum ever been observed, for photons of any energy? If not, how close are we to such an experiment?
This was demonstrated by "Experiment 144" at SLAC in 1997. Here is a list of publications from that project, for instance "Positron Production in Multiphoton Light-by-Light Scattering", whose abstract reads:
A signal of 106±14 positrons above background has been observed in collisions of a low-emittance 46.6 GeV electron beam with terawatt pulses from a Nd:glass laser at 527 nm wavelength in an experiment at the Final Focus Test Beam at SLAC. The positrons are interpreted as arising from a two-step process in which laser photons are backscattered to GeV energies by the electron beam followed by a collision between the high-energy photon and several laser photons to produce an electron-positron pair. These results are the first laboratory evidence for inelastic light-by-light scattering involving only real photons.
I guess you were asking about elastic photon-photon interaction. If that is the case, I remember reading some proposals (and argumentation of feasibility) of experiments using existing laser facilities in:
E. Lundström et al. Using High-Power Lasers for Detection of Elastic Photon-Photon Scattering. Phys. Rev. Lett. 96 no. 8, 083602 (2006). arXiv:hep-ph/0510076
Further references:
J. Lundin et al. Analysis of four-wave mixing of high-power lasers for the detection of elastic photon-photon scattering. Phys. Rev. A 74 no.4, 043821 (2006). arxiv:hep-ph/0606136.
Stephen D. H. Hsu and Brian M. Murray. Thermal gravity, black holes and cosmological entropy. Phys. Rev. D 73 no.4, 044017. arxiv:hep-th/0512033.
I have no idea if the experiments were really made, but those references are quite cool :)
I've a vague memory that light-on-light scattering puts an upper limit on the energy of photons that will propagate over cosmic distance scales. So one probe is very high energy gamma astronomy.
Vacuum magnetic birefringence basically involves the same loop diagram as light-light elastic scattering except that two of the four photons come from a magnet. Detecting this effect is the aim of the PVLAS experiment in Ferrara, Italy. See arXiv:1406.6518 and references within. The experiment is running at the moment but the sensitivity is not good enough yet to detect this fundamental process. We're working on it.
Evidence (4.4 standard deviations above background expectation) for light by light scattering ( $\gamma \gamma \rightarrow \gamma \gamma$ ) has also been seen recently at the Large Hadron Collider in collisions of lead atoms: https://arxiv.org/abs/1702.01625
