The Kerr effect is present in many different transparent media, but not in vacuum. You're probably right that sufficient power density would give rise to self-focusing by gravitational lensing, but it seems like the necessary power density would be very high — the gravitational acceleration needs to be a substantial fraction of c very frequently, like 0.001c over every interval where diffraction gives you a divergence of one milliradian. A narrower beam moves the escaping light you need to capture closer to the center of mass of the beam, but also increases that divergence.

It isn't obvious to me how to calculate this. Can anyone help?

You likely couldn't get gravitational lensing, you would probably get pair production first, which would screw with your energy density.

Yes, I'd think the Schwinger limit[0] would be reached first, which is currently the dream of most of us in this field. The Irradiance for the Schwinger limit is 10^29 W/cm^2.

By contrast, the planned ELI[1] experiment will have pulse with a intensity of 10^24 W/cm^2. The current state of the art lasers of high peak intensity is ~10^21 W/cm^2 (current record I believe is 10^22 W/cm^2). There are proposals to getting to the Schwinger limit but they are still in the hypothetical stage.

[0] https://en.wikipedia.org/wiki/Schwinger_limit

[1] https://en.wikipedia.org/wiki/Extreme_Light_Infrastructure

> what power density would it take to get self-focusing in a vacuum by gravitational lensing?

Not this is a question I'd look forward to be answered by Randall Munroe in his http://what-if.xkcd.com/

Yes. Send it!

Sent, I would love to see Randall Munroe's take on it.

You likely couldn't get self-focusing in a vacuum by gravitational lensing. Pair production would likely dominate the process first and kill your energy density.