I hate to break your heart, but 2 Watts of Green tightly focused will give just the ability to burn the surface.
Thick wood is very, very, difficult to laser cut. Much deeper then 7-8 mm requires enormous power levels and massive amounts of assist gas.
180 Watts of Far Infrared Co2 is useful for the fast cutting of 1/4th Inch (6.25 mm) plywood, but the absolute practical limit for wood is about 10 mm in the under 500 watt class.
It is very, very, difficult to go deep in fibrous organics. The amount of smoke produced is enormous. Vaporizing the resulting carbon char is difficult. The gas cloud formed in the cut gets very adsorptive, too. The edges of the finished product are discolored too.
I've been there, done that...
Steve