Electrons See the Guiding Light

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 combining the beams would be NI. If all of them were in phase, with their wave trains aligned, however, the peak intensity would be N²I. But, alas, that dividend of coherent superposition is unavailable to the well-meaning hobbyist who just purchased a wheelbarrow full of $5 laser pointers.

Suppose you bypass the problem by simply buying a laser that, when tightly focused, gives an electric field of 10⁷ V/m. Then the second problem can be expressed this way: “Hey genius, where’s my meter?” The heckler is pointing out that the focused laser beam is, at most, tens of microns wide—far less than 1 m. So the field is better expressed as the ratio 200 V/(20 µm), and your laser purchase would provide (at most) only 200 V of accelerating potential across the focal spot. Actually, it wouldn’t even remotely do that, courtesy of the third problem.

That problem is that laser light in free space is a high-frequency transverse electromagnetic wave, with fields orthogonal to the beam direction: An electron would do a high-frequency sideways shimmy, with negligible net acceleration and energy gain.