The electric field of a tightly focused milliwatt laser pointer can reach an impressive 50 000 V/m, a gradient that could, in principle, accelerate the electrons in a dental x-ray tube. With lots of cheap laser pointers and some assembly time at your kitchen table, can you build a 10 million V/m compact electron accelerator? Sadly, it wouldn’t work, and you’d needlessly annoy the neighbors.

Three main problems undermine the laser-accelerator project. First, although the light wave in a laser beam is a coherent electromagnetic field oscillation, with peaks and valleys aligned in lockstep, the train of peaks and valleys from the second laser is randomly phase shifted in time with respect to the first. So the fields from multiple lasers would interfere, as peaks from one laser cancel valleys from another.

If one has N such randomly phased laser beams, each with intensity I, the peak intensity from...

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