The invention is directed to an apparatus and a method for generating separating fissures in a substrate through action of a laser beam having an elliptical beam cross section with a beam spot length in a movement direction along a separating fissure to be generated in the substrate. The apparatus and method are particularly suitable for separating fissures which start or end on the substrate surface. A suitable diaphragm prevents radiation from impinging on the substrate before the start and after the end of the fissure.