An apparatus for laser machining a workpiece includes a beam shaping device for forming a focal zone from an input laser beam incident on the beam shaping device, and a telescope device for imaging the focal zone into a material of the workpiece. The beam shaping device is configured to impose a phase on a beam cross section of the input laser beam in such a way that the focal zone extends along a longitudinal centre axis which is curved at least in certain portions. The telescope device is assigned a beam splitting device for splitting an output laser beam output coupled from the beam shaping device into a plurality of polarized partial beams, each of which has one of at least two different polarization states. The focal zone has an asymmetrical cross section in a plane oriented perpendicular to the longitudinal centre axis.