Question about Nicoll-Dyson lasers

Perhaps somone can answer this for me.

While the beam from a Nicoll-Dyson wouldn't be directly visible (except to those unfortunates directly on line to the target, wouldn't it be visible through the effects on the interstellar medium? Even if it didn't pass though something like a nebula, wouldn't it heat up the interstellar gas along it's path?

Just wondering, because I was looking up FermiLab's Dyson Sphere search, and I was wondering if somebody had thought to look for traces of planet-killing lasers.