Abstract: In the case of a method for producing a temperature-controlling hollow structure in a substrate, first of all a substrate consisting of a substrate material is provided. The substrate is surveyed in order to ascertain where inclusions are in the substrate. Then, a temperature-controlling hollow structure is worked into the substrate by focusing a processing light beam with a beam axis aligned along a standard direction successively onto processing locations at which the temperature-controlling hollow structure is to be produced. As a result, the substrate material is modified or removed at the processing locations. If an inclusion is on the beam axis aligned along the standard direction, the direction of the beam axis relative to the mirror substrate is changed such that the beam axis does not intersect the inclusion.

Abstract: In a method for incorporating temperature-regulating hollow structures into a substrate, in particular into a substrate for an optical element, such as a mirror for an EUV projection exposure apparatus, there are the following steps: (A) providing a substrate; (B) progressively focusing a processing light beam on ablation locations at which temperature-regulating hollow structures are intended to arise; (C) a scanning process is carried out in which the processing light beam is guided with a focus in such a way that an ablation focus is moved along a scanning trajectory; (D) the scanning trajectory comprises a plurality of scanning patterns; (E) the scanning positions are spaced apart from one another in a longitudinal direction of the temperature-regulating hollow structure to be produced; and (F) the scanning trajectory additionally comprises pattern jump paths.

Abstract: A hollow structure (28) is produced in a workpiece (25) to form a substrate for a mirror through material-removing processing with pulsed laser radiation (35). The pulsed laser radiation is focused into a focal region (39), forming a removal front (46) for the areal removal of material of the workpiece (25) by moving the focal region (39) along a movement pattern (41), and producing the hollow structure (28) by moving the removal front (46) within the workpiece (25). The removal front is not aligned perpendicular to an incoming radiation direction (Z) of the pulsed laser radiation (35) at the radiation entrance side (27) of the workpiece at least intermittently during the production. The hollow structure is produced in the form of a channel through which a fluid is able to flow.