Laser interferometry is used by gravitational wave detectors like LIGO. The general idea behind it is relatively simple.
Gravitational waves stretch and squeeze space. This means that some distances become longer and some shorter. A laser interferometer measures the resulting distance by splitting a laser beam into two, sending each of the two beams along different directions in space (each along one of the two arms of the detector), and then recombining the beams. If the beams travelled exactly the same distance, then a detector detecting the recombined beam will see a beam of the same brightness as the original beam, or nothing at all, depending on how the detector is set up. That's because the two waves of the beams will either be in exact harmony, bumping each other up to the original brightness, or exactly cancel each other out. If the travel distance of the two beams are not the same, because a gravitational wave has stretched one arm of the detector and squeezed the other, then the beams won't line up in a constructive or destructive way, but instead be slightly offset from each other. The resulting interference of the beams can be measured.
All this is best explained visually. Here is a very nice explanation due to the NSF, LIGO, Caltech and MIT.
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This article is part of our Stuff happens: The physics of events project, run in collaboration with FQXi. Click here to see more articles and videos about gravitational waves.