[Rebel_Ace]

"...Over a period of time, the asteroid is deflected from its original course..."

"...Only if the thing is not rotating..."

Actually, yes, even if the thing is rotating, although that makes the energy requirements of the laser go up somewhat, but not much.

The laser heats a spot where you want the "jet" to occur, and with enough laser power, that happens in just a few seconds. As the asteroid rotates beneath the laser beam, other areas of the rock are vaporized, but the direction of the impulse remains the same. It would just create a linear scar on the asteroid, instead of a simple "hole" of ejected material. The key is that the craft delivering the laser is at "station keeping".

You can even use the laser to first stop the asteroid's tumbling, if desired.
Here is a link to a video testing a laser on a small rotating rock, to illustrate the principle: Small Scale Test of Laser Ablation

[Elsie]

Ablation.

THAT’s the word I was thinking of.

[Elsie]

The laser heats a spot where you want the "jet" to occur...

And the ideal spot would be 90° away from the direction it is heading towards us.

The problem; as I see it; is that the laser would be quite near Earth and the jet would be mainly on the lead surface.

Slowing it down, but not deflecting it away.

[Elsie]

Let’s do a lab experiment.

Hang a rock on a pedulum in a vacuum.

Pulse that sucker with the largest cutting laser we have now.

Measure the deflection.

Scale up.