... the posted curve shows "manual" switching of an old prototype driver with three parallelized LM338 constant current regulators, so not the really used "pulse trains" for laser-engraving - this initial "slope" could be caused by contact resistance from the mechanical switch, so not typical for this type of driver -- the next days I want to rebuild the old LM338-driver with better signal feeding and digital switching, so can post some more images then ...

A more actual constant-current driver with OpAmps and MOSFET's gives this waveform when switched with 25 microsecond long pulses with a current of 12 Amps (measured 1.2 Volts across a resistor with 0,1 Ohms):

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With the last driver I've reduced the negative OFF-spike to some ten Milliamperes and trying to eliminate it totally ...