There was an alignment issue to be checked:
The pump and the probe laser inside a thick material are subject to the Snell's law.
Since the probe has a non-zero incidence angle, the crossing point changes position inside the material according to how much material the beams have traveled in before the crossing point.
If the beams are not perfectly horizontal and well aligned the car be an asymmetry on the absorption signal if the beams imping on one surface or on the other one.

To check this, I measured a scan at the center of the sample from one side, and I flipped the sample to do the same measurement from the other surface.

Result: the two plots overlap quite well.

The arrow in the first plot indicates where the beams come from