Processing apparatus for producing a cut surface in a transparent material (7, 8), comprising - a laser device, which is configured to effect separation within the transparent material (7, 8) by way of optical radiation and which has - an optical unit that focuses the optical radiation along an optical axis into a focus located in the material and has in the material (7, 8) an image field in which the focus is located and which has an image field size, - a device for adjusting a position of the focus transversely to the optical axis and along the optical axis, - a control device that is connected to the laser device and controls the laser device such that the device thereof for adjusting the position of the focus displaces the position of the focus of the optical radiation in the material (7, 8) along a trajectory, wherein the control device controls the laser device in a manner such that the cut surface at least in sections extends substantially parallel with respect to the optical axis or the cut surface at least in sections extends slightly at an angle to the optical axis, wherein the cut surface transversely to the optical axis follows a curve, and the control device during the movement along the curve specifies by a repeatedly oscillating adjustment the position of the focus between an upper axial focus position and a lower axial focus position, wherein - the device for adjusting the position of the focus transversely to the optical axis effects a displacement of the image field transversely to the optical axis and is configured for adjusting the position of the focus along the optical axis in two stages, with one stage for the slow adjustment with a large travel and one stage for the quick adjustment with a short travel, and - the control device controls the laser device such that the cut surface has, transversely to the optical axis, a maximum extent that is greater than the image field size and specifies for the device for adjusting the position of the focus transversely to the optical axis a movement of the position of the focus along the curve. - a laser device, which is configured to effect separation within the transparent material (7, 8) by way of optical radiation and which has - an optical unit that focuses the optical radiation along an optical axis into a focus located in the material and has in the material (7, 8) an image field in which the focus is located and which has an image field size, - a device for adjusting a position of the focus transversely to the optical axis and along the optical axis, - a control device that is connected to the laser device and controls the laser device such that the device thereof for adjusting the position of the focus displaces the position of the focus of the optical radiation in the material (7, 8) along a trajectory, wherein the control device controls the laser device in a manner such that the cut surface at least in sections extends substantially parallel with respect to the optical axis or the cut surface at least in sections extends slightly at an angle to the optical axis, wherein the cut surface transversely to the optical axis follows a curve, and the control device during the movement along the curve specifies by a repeatedly oscillating adjustment the position of the focus between an upper axial focus position and a lower axial focus position, wherein - the device for adjusting the position of the focus transversely to the optical axis effects a displacement of the image field transversely to the optical axis and is configured for adjusting the position of the focus along the optical axis in two stages, with one stage for the slow adjustment with a large travel and one stage for the quick adjustment with a short travel, and - the control device controls the laser device such that the cut surface has, transversely to the optical axis, a maximum extent that is greater than the image field size and specifies for the device for adjusting the position of the focus transversely to the optical axis a movement of the position of the focus along the curve.