### 6.9 Phase coherence of light

An interesting and less well known method of constraining non-systematic Lorentz violation is looking at
Airy rings (interference fringes) from distant astrophysical objects. In order for an interference pattern from
an astrophysical source to be observed, the photons reaching the detector must be in phase across the
detector surface. However, if the dispersion relation is fluctuating, then the phase velocity is
also changing. If the fluctuations are uncorrelated, then initially in-phase collections of photons will lose
phase coherence as they propagate. Uncorrelated fluctuations are reasonable, since for most of their
propagation time, photons that strike across a telescope mirror are separated by macroscopic distances.
Observation of Airy rings implies that photons are in phase and hence limits the fluctuations
in the dispersion relation [204, 245, 233]. The aggregate phase fluctuation is given by [233]
where is the Planck length, is the distance to the source, and is the wavelength of the
observed light. This technique was originally applied by [204], but the magnitude of the aggregate
phase shift was overestimated. PKS1413+135, a galaxy at a distance of 1.2 Gpc, shows Airy
rings at a wavelength of . Demanding that the overall phase shift is less than
yields constraints for and constraints of order for . Hence
this type of constraint is only able to minimally constrain Lorentz violating non-systematic
models. In principle the frequency of light used for the measurement can be increased, however, in
which case this type of constraint will improve. However, Coule [92] has argued that other
effects mask the loss of phase coherence from quantum gravity, making even this approach
uncertain.