The laser cutter can work with wood of varying thickness up to half of an inch (or so I'm told. I've never seen it work with something quite that thick) depending on the wood. Slower cuts with higher power will often get you through darker/thicker wood but at some point it just won't make it through.
I've had mixed results with plywood. Voids and knots, which are usually only present in the middle plys if at all in nicer plywood (and thus impossible to detect until it's too late) are particularly bad for the laser cutter because the speed and power required to get through them is different from that required for the rest of the cuts..
Baltic birch plywood is generally used for such applications because of its dimensional stability and low (depending on grade) voids and knots. I've had pretty good luck with Baltic birch that I got from Woodcraft up to a quarter inch.
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