At 60A, with pulses short enough, a very low rds mosfet and carefully regulated, low voltage this may be doable as long as the resistor is almost perfectly non-inductive. 60A pulses will store a significant energy in even a minor inductance and will then ret released as inductive spoke that may d up your laser and/mosfet. If the current needs to be adjustable, then this will not be a reasonable solution.
A constant current source that drives a mosfet or IGBT in linear region would almost certainly exceed soa of any mosfet or IGBT. The only solution I think would be to parallel numerous power BJTs, but those are quite obsolete for any normal use nowadays.
A switch mode source would have to be really, really fast, think MHz range to provide sharp enough rise/fall times, especially at such high current. Normal fets that can handle such current (few in parallel) will require a ton of gate charge and consequently a very high current have driver to switch fast enough. That is if you manage to handle the inductive kickback from literally any piece of conductor in the current path.
Maybe GaN would be a solution? I have no practical experience with it though.
Maybe you could also run some form of forward converter with transformer and primary switches operating in current mode? It would mean that you would have to somehow handle the negative part of the switching cycle so as to not f up your laser diode, but that way you could avoid 60A of current through a semiconductor in linear region.