A method for generating separating fissures in a substrate through action of a laser beam with an elliptical beam cross section, comprising the steps of: - receiving and positioning a substrate in a workpiece receptacle; - generating a laser beam with an elliptical beam cross section to form an elliptical beam spot with a beam spot length and a beam spot width on the surface of the received substrate; characterised by the following steps and features: - directing the laser beam onto the received substrate, wherein the beam spot length is oriented in a direction of a separating fissure which is to be generated and which has a start and an end, guiding the laser beam along the separating fissure to be generated from the start of the fissure to the end of the fissure, wherein the laser beam is cut off at least periodically, namely decreasingly cut off from the start of the fissure and increasingly cut off toward the end of the fissure, by a diaphragm which is arranged in a diaphragm plane and which is guided at least periodically with the laser beam, so that the substrate is not acted upon by the laser beam before the start of the fissure or beyond the end of the fissure, and wherein the laser beam is directed onto the substrate in such a way that, irrespective of the diaphragm, the laser beam abuts with its beam spot length on the start of the fissure at the commencement of beam guidance in extension of the separating fissure and abuts with its beam spot length on the end of the fissure at the termination of beam guidance in extension of the separating fissure, wherein, as a result of the provided diaphragm, the laser beam is cut off completely in each instance by the diaphragm, whereby each point between the start of the fissure and the end of the fissure is acted upon by the laser beam with the same laser power that would also be applied if the laser beam moved beyond the start of the fissure or the end of the fissure without being cut off, wherein decreasing and increasing cutting off of the laser beam is carried out in that a rigid diaphragm (42.2) with an edge region (42.3) and with a rigid diaphragm aperture (43.2) is used as diaphragm, and the rigid diaphragm (42.2) whose rigid diaphragm aperture (43.2) abuts on the start of the separating fissure remains stationary relative to the guided laser beam at the commencement of beam guidance, while the laser beam is guided into the rigid diaphragm aperture (43.2) until the edge region (42.3) thereof no longer has a cutting off effect and is then guided along with the laser beam at a speed corresponding to the beam guidance until the rigid diaphragm aperture (43.2) encounters the end of the fissure and then comes to rest again relative to the guided laser beam, while the laser beam continues to be guided along the edge region (42.3) until it is completely cut off. - receiving and positioning a substrate in a workpiece receptacle; - generating a laser beam with an elliptical beam cross section to form an elliptical beam spot with a beam spot length and a beam spot width on the surface of the received substrate; - directing the laser beam onto the received substrate, wherein the beam spot length is oriented in a direction of a separating fissure which is to be generated and which has a start and an end, guiding the laser beam along the separating fissure to be generated from the start of the fissure to the end of the fissure, wherein the laser beam is cut off at least periodically, namely decreasingly cut off from the start of the fissure and increasingly cut off toward the end of the fissure, by a diaphragm which is arranged in a diaphragm plane and which is guided at least periodically with the laser beam, so that the substrate is not acted upon by the laser beam before the start of the fissure or beyond the end of the fissure, and wherein the laser beam is directed onto the substrate in such a way that, irrespective of the diaphragm, the laser beam abuts with its beam spot length on the start of the fissure at the commencement of beam guidance in extension of the separating fissure and abuts with its beam spot length on the end of the fissure at the termination of beam guidance in extension of the separating fissure, wherein, as a result of the provided diaphragm, the laser beam is cut off completely in each instance by the diaphragm, whereby each point between the start of the fissure and the end of the fissure is acted upon by the laser beam with the same laser power that would also be applied if the laser beam moved beyond the start of the fissure or the end of the fissure without being cut off,