Patent Publication Number: US-2021170530-A1

Title: System for asymmetric optical beam shaping

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of and claims priority under 35 U.S.C. § 120 to U.S. application Ser. No. 15/599,720, filed on May 19, 2017, which is a continuation of PCT Application No. PCT/EP2015/077172, filed on Nov. 19, 2015, which claims priority to German Application No. 10 2014 116 957.3, filed on Nov. 19, 2014 and German Application No. 10 2014 116 958.1, filed on Nov. 19, 2014. The entire contents of these priority applications are incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present invention relates to diffractive optical elements that are used in optical systems for beam shaping a laser beam and in particular for beam shaping a laser beam for processing materials that are essentially transparent for the laser beam. Moreover, the invention relates to a system and a method for laser material processing. 
     BACKGROUND 
     There are many possibilities for using absorption of light for processing a work-piece, in particular by introducing localized modifications into the work-piece. The so-called volume absorption, i.e., an absorption that is not limited to the surface, opens the possibility to process brittle-hard materials that are essentially transparent for the laser beam. Generally, volume absorption benefits from a kind of nonlinear absorption, at which an interaction with the material takes place only at a material dependent (threshold) intensity. 
     SUMMARY 
     Herein, a nonlinear absorption is understood as an intensity dependent absorption of light, that is not primarily based on the direct absorption of the light. Instead it is based on an increase of the absorption during interaction with the incident light, often a temporally limited laser pulse. Thereby, electrons can absorb that much energy by inverse bremsstrahlung that further electrons are set free by impacts, so that the rate of generating electrons overcomes that rate of recombination. Under specific conditions, those initial electrons, which are required for the avalanche-like absorption, may already be present from the start or may be generated by an existing rest-absorption by linear absorption. For example, for ns-laser pulses, an initial ionization may result in an increase in temperature that causes an increase of the number of free electrons and therefore of the following absorption. Under other conditions, such initial electrons may be generated by multi-photon ionization or tunnel ionization as examples of well-known nonlinear absorption mechanisms. For ultrashort laser pulses with, for example, sub-ns-pulse durations such an avalanche-like generation of electrons can be utilized. 
     A volume absorption may be used for materials, which are essentially transparent for the laser beam (herein in short referred to as transparent materials), for forming a modification of the material in an elongated focus zone. Such modifications may allow separating, drilling, or structuring of the material. For separating, for example, rows of modifications may be generated that cause a breaking within or along the modifications. Moreover, it is known to generate modifications for separating, drilling, and structuring that allow a selective etching of the modified areas (SLE: selective laser etching). 
     The generation of an elongated focus zone can be affected with the help of apodized Bessel beams (herein also referred to as quasi-Bessel beam). Such beam profiles may be formed, for example, with an axicon or a spatial light modulator (SLM: spatial light modulator) and an incident light beam having a Gaussian beam profile. A subsequent imaging into a transparent work-piece results in the intensities required for volume absorption. Quasi-Bessel beams—like Bessel beams—usually have a ring-shaped intensity distribution in the far field of the beam profile existing within the work-piece. Calculating phase distributions for beam shaping quasi-Bessel beams, e.g., with an SLM is disclosed in Leach et al., “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express 14, 5581-5587 (2006), the entire contents of which are incorporated by reference. 
     Moreover, systems are known for forming a line of intensity enhancements, e.g., with the help of multifocal lenses. Thereby, a phase modification of the laser beam to be focused is performed in the far field, i.e., during focusing, whereby the phase modification results in the formation of longitudinally displaced focus zones. 
     An aspect of the present disclosure has the objective to improve processing quality of laser material processing with elongated focus zones as they can be generated, for example, with a diffractive optical beam shaping element. In particular, the objective is, for laser processing applications, to provide in beam propagation direction elongated, slender beam profiles with a high aspect ratio for processing transparent materials such that a separating or cutting process can be performed with an increased precision of the cutting edge. 
     At least one of the objectives is solved by the methods, the optical systems, and the laser processing machines of the independent claims. Further developments are given in the dependent claims. 
     In an aspect, modifications are generated for material processing a material with a pulsed laser beam, wherein the material is in particular to a large extent transparent for the laser beam and the modifications are formed asymmetric transverse to a propagation direction of the laser beam. Thereby, the laser beam is shaped for forming an elongated focus zone in the material, wherein the focus zone is configured such that it includes at least one intensity maximum, which is transverse flattened in a flattening direction, or a transverse and/or axial sequence of asymmetric intensity maxima, which are flattened in a flattening direction. After the positioning of the focus zone in the material, a modification is generated, and the material and the focus zone are moved relative to each other in or across to the flattening direction or in or across to the sequence direction. 
     In embodiments, a diffractive optical beam shaping element is used for imposing a phase distribution on the laser beam, which is intended for laser processing a material, wherein the beam shaping element includes an (areally configured) phase mask, which is configured for imposing one or more beam shaping phase distributions on the laser beam, which is incident on the phase mask. A virtual or real optical image is attributed to at least one beam shaping phase distribution, wherein the optical image is imagable in an elongated focus zone for forming a modification in the material ( 9 ) to be processed. 
     In a further aspect, a method for guiding a crack during laser processing for processing a material, which is to a large extent transparent for a pulsed laser beam includes the following step for separating or preparing the separation of the material generating modifications in the material by focusing the laser beam in an asymmetric focus zone of the pulsed laser beam, wherein successive modifications are displaced with respect to each other along a relative movement direction between laser beam propagation direction and material, and each of the modifications includes a preferred direction in the crack formation, and the relative movement direction and the preferred direction are adapted to each other. 
     In a further aspect, an optical system for beam shaping of a laser beam for processing a material, which is in particular for the laser beam to a large extent transparent, by modifying the material in a focus zone, which is elongated in a propagation direction and in a further direction transverse to the propagation direction, includes an optical element, which is configured to shape the focus zone in propagation direction, wherein the optical element, which is configured to shape the focus zone in propagation direction, additionally shapes the focus zone in a further direction transverse to the propagation direction, or wherein a further optical element, which is configured to shape the focus zone in a further direction transverse to the propagation direction, is positioned downstream of the optical element configured to shape the focus zone in propagation direction. 
     In a further aspect, an optical system for beam shaping of a laser beam for processing a material, which is in particular for the laser beam to a large extent transparent, by modifying the material in a focus zone, which is elongated in a propagation direction, includes, a diffractive optical beam shaping element for imposing one or more phase distributions on the laser beam that are in particular essentially rotationally symmetric around the beam axis. The optical system further includes a near field optics, which is arranged downstream of the diffractive optical beam shaping element at a beam shaping distance and is configured to focus the laser beam into the focus zone. At least one imposed phase distribution is such that a virtual optical image of an elongated focus zone is attributed to the laser beam, the virtual optical image being located upstream of the diffractive optical beam shaping element, or a real optical image of an elongated focus zone is attributed to the laser beam, the real optical image being located after the diffractive optical beam shaping element. The beam shaping distance corresponds to a propagation length of the laser beam, within which the plurality of beam shaping phase distributions transforms the transverse input intensity profile into a transverse output intensity profile in the region of the near field optics, and in particular the transverse output intensity profile has, in comparison with the input intensity profile, at least one local maximum lying outside of a beam axis. 
     Herein, concepts are disclosed that allow to at least partly improve aspects of the prior art. In particular, additional features and their functionalisms result from the following description of embodiments on the basis of the drawings. The drawings show: 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  is a schematic illustration of an optical system for beam shaping of a laser beam; 
         FIG. 2  is a schematic illustration of a laser processing device with an optical system according to  FIG. 1  for material processing; 
         FIG. 3  is a schematic illustration of an optical system for explaining the optical functioning; 
         FIG. 4  is an example of a longitudinal intensity distribution in an elongated focus zone after imaging a virtual optical image; 
         FIG. 5  is a ZR-section of the longitudinal intensity distribution shown in  FIG. 4 ; 
         FIG. 6  is an exemplary experimental study on the modification of a transparent material in an elongated focus zone according to  FIGS. 4 and 5 ; 
         FIG. 7  is a schematic illustration for explaining the generation and imaging of a real intensity enhancement, 
         FIG. 8  is an example of a longitudinal intensity distribution in an elongated focus zone after imaging a real intensity enhancement according to  FIG. 7 ; 
         FIG. 9 ,  FIG. 10  and  FIG. 11  are schematic illustrations of examples for optical systems based on transmitting or reflective axicons; 
         FIG. 12  is a schematic illustration of an example of an optical system based on a spatial light modulator; 
         FIG. 13  is schematic illustration of an example of an optical system based on a transmitting diffractive optical element; 
         FIG. 14  is a schematic illustration of an example of a phase distribution in a diffractive optical element in an optical system according to  FIG. 13 ; 
         FIG. 15  is an exemplary intensity cross-section of an output intensity profile in an optical system according to  FIG. 13 ; 
         FIG. 16  is an XY-view of the output intensity profile of the intensity cross-section shown in  FIG. 15 ; 
         FIG. 17  is a schematic illustration of an example of an optical system with filtering non-phase-modulated beam portions; 
         FIG. 18  is a schematic illustration of an example of an optical system based on a diffractive optical element with a linear phase contribution for separating a phase-modulated beam portion; 
         FIG. 19  is a schematic illustration of an example of an optical system with a scan device; 
         FIG. 20  is a schematic illustration for explaining the imaging system of an optical system; 
         FIG. 21A ,  FIG. 21B  and  FIG. 21C  are schematic illustrations for explaining potential causes for crack formation; 
         FIG. 22A ,  FIG. 22B ,  FIG. 22C ,  FIG. 23A ,  FIG. 23B ,  FIG. 23C  to  FIG. 23D  are schematic illustrations for asymmetric beam shaping by introducing a beam aperture; 
         FIG. 24A  and  FIG. 24B  are XY- and ZY-cuts for illustrating an asymmetric array of intensity maxima; 
         FIG. 25A  and  FIG. 25B  are schematic illustrations of examples of a phase distribution in a diffractive optical element for asymmetric beam shaping; 
         FIG. 26  is a phase distribution segmented azimuthal; 
         FIG. 27  is an exemplary intensity cross-section for phase imposing according to  FIG. 26 ; 
         FIG. 28  is an XY-view of the output intensity profile of the intensity cross-section shown in  FIG. 27 ; 
         FIG. 29  is a ZX-cut of an elongated focus zone for a phase imposing according to  FIG. 26 ; 
         FIG. 30  is a ZY-cut of an elongated focus zone for a phase imposing according to  FIG. 26 ; 
         FIG. 31  is a phase distribution for generating two transverse displaced inverse quasi-Bessel beam profiles; 
         FIG. 32  is an amplitude distribution for a cut along the beam axis Z when propagating from the beam shaping element to the near field optics for a phase imposing ac-cording to  FIG. 31 ; 
         FIG. 33  is an XY-view of the output intensity profile for a phase imposing according to  FIG. 31 ; 
         FIG. 34A  and  FIG. 34B  are ZX- and XY-cuts of an elongated focus zone for a phase imposing according to  FIG. 31 ; 
         FIG. 35A, 35B  and  FIG. 35C  are beam profiles at z=10 mm, z=100 mm, z=150 mm for a phase imposing according to  FIG. 31 ; and 
         FIG. 36A, 36B  and  FIG. 36C  are transverse intensity distributions in X direction at z=10 mm, z=100 mm, z=150 mm for a phase imposing according to  FIG. 31 . 
     
    
    
     DETAILED DESCRIPTION 
     Aspects described herein are based partly on the realization that, due to the high intensities needed for laser processing, intensities may be present already during the preparation of the laser beam that result in damage of optical elements. In view thereof, it was further realized that the generation of an elongated focus zone within the work-piece may be based on the imaging of a virtual beam profile. By this concept of imaging a virtual beam profile, regions with intensity peaks can be reduced or even avoided in the optical system. It was further realized that a phase distribution attributed to the virtual beam profile may be imposed onto the laser beam that causes the desired change of the intensity distribution in the far field. In particular, it was realized that by a far field distribution, which originates from such a virtual beam profile, for example, inverse-Bessel beam-like or inverse quasi-Airy beam-like intensity distributions, specifically designed intensity distributions, and in particular superpositions of the same in the focus zone can be created. For such intensity distributions, a lateral energy entry into the focus zone can take place, which in particular enables the processing of transparent materials. It was further realized that, in comparison to systems for imaging a real intensity enhancement, the concept of the imaging of a virtual beam profile may lead to shorter configurations of such optical systems. 
     An elongated focus zone relates herein to a three-dimensional intensity distribution defined by the optical system that determines the spatial extent of the interaction and thereby the modification within the to be processed material. The elongated focus zone determines thereby an elongated region in which a fluence/intensity is present within the to be processed material, which is beyond the threshold fluence/intensity being relevant for the processing/modification. Usually, one refers to elongated focus zones if the three-dimensional intensity distribution with respect to a target threshold intensity is characterized by an aspect ratio (extent in propagation direction in relation to the lateral extent) of at least 10:1, for example 20:1 and more, or 30:1 and more. Such an elongated focus zone can result in a modification of the material with a similar aspect ratio. In some embodiments, focus zones can be formed that are, for example, also in propagation direction parallel with respect to each other, wherein each of the focus zones has a respective aspect ratio. In general, for such aspect ratios, a maximal change of the lateral extent of the (effective) intensity distribution over the focus zone can be in the range of 50% and less, for example 20% and less, for example in the range of 10% and less. 
     Thereby, the energy within an elongated focus zone can be laterally supplied essentially over the complete length of the created modification. As a consequence, a modification of the material in the initial region of the modification does not have or hardly has any shielding effects on the part of the laser beam that causes a modification of the material downstream of the beam, i.e., for example, in the end region of the modification zone. In that sense, a Gaussian beam cannot generate a comparable elongated focus because the energy supply is performed essentially longitudinally and not laterally. 
     The transparency of a material, which is essentially transparent for a laser beam, relates herein to the linear absorption. For light below the threshold fluence/intensity, material, which is essentially transparent for a laser beam, may absorb, for example, along a length up to the back end of the modification, e.g., less than 20% or even less than 10% of the incident light. 
     Aspects described herein further are partly based on the realization that by a desired beam shaping, for example, with a diffractive optical element (DOE), the density of free electrons, which is created in the material by nonlinear absorption, may be tailored. Along the thereby created modifications, a crack formation may be specifically guided, which then results in the separation of the work-piece. 
     Aspects described herein further are based partly on the realization that, for a DOE, multiple phase distributions can be provided in the phase distribution of a phase mask, for example, in respective segments. Thereby, in particular the advantages of the concept of a virtual optical image, for example, an inverse quasi-Bessel beam (e.g., inverse quasi-Bessel like beam) shape, can be used at the superposition of the imaging of multiple such virtual images (in longitudinal or lateral direction), wherein also the interaction (e.g., interference) and spatial constellation of multiple imaging may have effects onto the formation of the common focus zone. In addition, it was recognized that in this manner asymmetric “common” focus zones can be created. For example, for material processing, asymmetric “common” focus zones create a preference for a specific movement direction or a specific separation direction. Moreover, it was recognized that, during the laser processing, such preferred directions may be adopted to desired processing trajectories by orienting/turning the DOE within an optical system. For digital phase masks (SLMs etc.), a direct controlling of the phase distribution may further be performed to adapt the preferred direction. 
     Aspects described herein further are based in part on the realization that, by the use of a DOE, additional phase distributions may be imposed onto the beam, which, for example, may simplify the setup of the underlying optical systems and/or the isolation of a usable beam portion. 
     In other words, disadvantages of the prior art may in some embodiments at least partly be overcome by an optic concept, in which the beam profile, which is positioned in the region of the work-piece and which is elongated in propagation direction, is affected by an imaging of a created virtual beam profile. In some embodiments, the optic concept further allows a filtering possibility for undesired beam portions, for example, in a region of the Fourier-plane of the beam profile and a separation of the beam shaping from the focusing. 
     The systems and methods resulting from these realizations can inter alia enable separating of transparent, brittle-hard materials with high velocity and with good quality of the cutting edge. Moreover, such systems and methods may further enable separating without a taper angle as it is created in ablating methods. In particular, when separating based on non-ablating modifications, there may be no or only a small removal, with the consequence that the material has only a few particles on the surface after the processing. 
     In the following, the underlying optical concept will be generally explained with reference to  FIGS. 1 to 8 . Then, exemplary embodiments of optical systems will be explained, which, on the one side, implement the optical system by conventional optics such as lenses and mirrors (see  FIGS. 9 to 11 ) and, on the other side, by diffractive optical elements (see  FIGS. 12 to 16 ). In connection with  FIGS. 17 to 20 , the combinability of the optical system with components and aspects for filtering and scanning as well as general aspects of the beam development within the optical system are explained. 
     In connection with  FIGS. 21A to 21C , potential causes are explained for formation of a preferred direction for a crack formation. 
     In the remaining figures, various concepts are proposed that relate to the interference of inverse quasi-Bessel beams in the formation of transverse asymmetries due to azimuthal segmentation ( FIGS. 26 to 30 ) and transverse displaced phase distributions ( FIGS. 31 to 36C ). 
       FIG. 1  shows a schematic illustration of an optical system  1  for beam shaping a laser beam  3  with the aim to create a focus zone  7 , which is elongated in a propagation direction  5 , within a to be processed material  9 . Generally, laser beam  3  is determined by beam parameters such as wavelength, spectral width, temporal pulse shape, formation of pulse groups, beam diameter, transverse input intensity profile, transverse input phase profile, input divergence, and/or polarization. According to  FIG. 1 , laser beam  3  is supplied to optical system  1  for beam shaping, i.e., for transforming one or more of the beam parameters. Usually, for laser material processing, laser beam  3  will be a collimated Gaussian beam with a transverse Gaussian intensity profile, which is generated by a laser beam source  11 , for example an ultrashort pulse high-intensity laser system. The transformation can be performed, for example, into an inverse Bessel or inverse Airy beam shape. 
     In the laser processing machine  21  shown in  FIG. 2 , optical system  1  may, for example, be used for material processing. Laser processing machine  21  includes a support system  23  and a work-piece positioning unit  25 . Support system  23  spans over work-piece positioning unity  25  and carries laser system  11 , which is integrated in  FIG. 2 , for example, in an upper crossbeam  23 A of support system  23 . In addition, optical system  1  is mounted at crossbeam  23 A to be displaceable in X direction, so that both components are arranged close to each other. In alternative embodiments, laser system  11  may be provided, for example, as a separate external unit, laser beam  3  of which is guided to optical system  1  by optical fibers or as a free propagating beam. 
     Work-piece positioning unit  25  carries a work-piece that extends in the X-Y-plane. The work-piece is the to be processed material  9 , for example, a glass plate or a plate in ceramic or crystalline embodiment such as sapphire or silicon, that is essentially transparent for the laser wave-length used. Work-piece positioning unit  25  allows displacing the work-piece in Y direction relative to support system  23 , so that, in combination with the displaceability of optical system  1 , a processing area is provided, which extends within the X-Y-plane. 
     According to  FIG. 2 , in addition, a relocatability is provided of e.g., optical system  1  or cross-beam  23 A in Z direction, such that the distance to the work-piece can be set. For a cut running in Z direction, the laser beam is usually also directed in the Z direction (i.e., normal) onto the work-piece. However, additional processing axes may be provided as exemplarily illustrated in  FIG. 2  by the boom arrangement  27  and the additional rotational axes  29 . Accordingly, boom arrangement  27  is optional in the embodiment of  FIG. 2 . In addition, redundant add-on axes may be provided for higher dynamics, as, for example, not the work-piece or the optical system, but more compact and respectively designed components are accelerated. 
     Laser processing machine  21  further includes a control unit not explicitly shown in  FIG. 1 , which is, for example, integrated within support system  23  and which in particular includes an interface for inputting operation parameters by a user. In general, the control unit includes elements for controlling electrical, mechanical, or optical components of laser processing machine  21 , for example, by controlling respective operation parameters such as pump laser power, cooling power, direction and velocity of the laser machine and/or the work-piece positioning unit, electrical parameters for setting an optical element (for example, of an SLM) and the spatial orientation of an optical element (for example, for rotation of the same). 
     Additional arrangements for laser processing machines with various degrees of freedom are disclosed, for example, in EP 1 688 807 A1, the entire contents of which are incorporated by reference. In general, for smaller work-pieces often only the work-piece is moved, and for larger work-pieces only the laser beam or—as in  FIG. 2 —the work-piece and the laser beam are moved. Moreover, two or more optical systems and, thus, focus zones may be supplied by a single laser system  11 . 
     The modifications within the material, which are generated by the laser processing machine, may be used, for example, for drilling, separating by induced tensions, welding, creating a modification of the refraction behavior, or for selective laser etching. Accordingly, it is important to control the geometry as well as the type of modification in a suitable manner. Besides parameters such as laser wavelength, temporal pulse shape, number of pulses, energy and temporal distance of the pulses within a pulse group creating an individual modification, as well as pulse energy or pulse group energy, the beam shape plays a decisive role. 
     In particular, an elongated volume modification allows processing of a, in beam propagation direction long extending, volume region within a single processing step. In particular, at one position in feed direction, the processing can take place over a large extent in only a single modification processing step. By the use of the optical systems described herein, beam shapes, and methods, one can achieve, on the one side, better work results (in comparison to single modifications that are positioned next to each other at one position in feed direction in succeeding modification processing steps) and, on the other side, one can reduce the processing time and the requirements for the system technology. Then, for single modifications, multiple working steps are needed that increase the time needed and that require a more involved ensuring of relative positions of the single modifications. 
     In addition, an elongated focus zone can be helpful when processing uneven materials, because essentially identical laser processing conditions are given along the elongated focus zone such that, in those embodiments, a respective readjusting in propagation direction may not be necessary or only be necessary starting at a larger deviation of the position of the to be processed material than the lengths of the elongated focus area (in consideration of the required processing/intrusion depth). 
     In general, it applies to the processing of transparent materials by elongated volume absorption that, as soon as absorption takes place, that absorption itself or the resulting changes in the material properties can influence the propagation of the laser beam. Therefore, it is advantageous, if beam portions, which should cause a modification deeper within the work-piece, i.e., in beam propagation direction downward, essentially propagate not through regions of considerable absorption. 
     In other words, it is favorable to lead those beam portions, which contribute to the modification further downward, under an angle to the interaction zone. An example for this is the quasi-Bessel beam, for which a ring-shaped far-field distribution is given, the ring width of which is typically small in comparison to the radius. Thereby, the beam portions of the interaction zone are led in essentially with that angle in rotational symmetry. The same applies for the inverse quasi-Bessel beam described herein or for modifications or extensions of the same such as the homogenized or modulated inverse quasi-Bessel beam. Another example is the inverse accelerated “quasi-Airy beam-like” beam, for which the beam portions are led into the modification under an offset angle, where this is done clearly tangential and—not as for the pure quasi-Bessel beam rotationally symmetric—to the curved modification zone, e.g., as for a curved inverse quasi-Bessel beam. 
     Moreover, it is desired to considerably pass the threshold for the nonlinear absorption only within the desired volume region and to choose the geometry of that volume area such that it is suitable for the desired application, but that also the propagation to further downstream positioned volume regions is not significantly disturbed. For example, it may be advantageous to keep secondary maxima of an apodized Bessel beam profile below a threshold intensity needed for nonlinear absorption. 
     In view of modifications being subsequent in the feed direction, the geometry of the modified volume may further be selected such that, for a row of multiple modifications in the feed direction, an earlier induced modification has only an insignificant influence on the formation of the following modifications. 
     As already mentioned, for fast processing, the generation of a single modification can be performed with only a single laser pulse/a single laser pulse group, so that a position on a work-piece is approached only once in this case. 
     Ultrashort pulse lasers can make intensities (power densities) available that allow causing a sufficiently strong material modification in respective long interaction zones. The geometric extent of the modification is thereby set with the help of beam shaping such that a long extending, high density of free electrons is created by nonlinear absorption in the material. The supply of energy in deeper regions is performed laterally, so that the shielding effect by an upstream interaction of the plasma can be avoided in comparison to a Gaussian focusing. For example, an electron density, which extends smoothly in longitudinal direction, or an electron density, which is modulated spatially with a high frequency, can be generated. 
     At the respective intensities, within regions with a sufficiently high density of free electrons, an explosive expansion of the material may be caused, whereby the thereby resulting shock-wave can create nanoscopic holes (nano-voids). Additional examples for modifications (modification zones) are changes in the refractive index, compressed and/or tensile stress induced regions, micro-crystallites, and local changes in stoichiometry. 
     As explained in the beginning, by the accumulation of such modification zones in feed direction, a course of a crack can be set. During processing, the work-piece is accordingly separated along a respective modified contour. The crack formation can then occur directly thereafter or can be induced by another process. For example, for the separation of non-pre-strained materials ultrasound ramps, or temperature ramps may be used in order to cause a later separation along the modified contour. A single modification usually does not lead to crack formation. 
     With the help of a tailored beam shape, various tension distributions within the material and between the modified regions can be created in order to adapt the separation process to a given material. In the process, strong spatial and temporal gradients can favor the formation of a micro- or nano-explosion. 
     The modification geometry is thereby primarily determined by the beam shaping (and not by the nonlinear propagation as, for example, the filamentation). The generation of spatial gradients can be achieved by the optical systems described herein, while the generation of the temporal gradients can be achieved by pulse trains or pulse shaping. 
     Generally, a scaling of the intensity distribution of a beam shape can be achieved by the imaging ratio of the system, in particular by the focal length and the numerical aperture of the near field optics of the imaging system. Additional possibilities for scaling result from the use of an additional lens as well as the shifting of the beam shaping element and/or the far field optics (see the description in connection with  FIGS. 17 and 22 ). Thus, the lateral and longitudinal extent of the beam profile within the work-piece can be influenced. In addition, spatial filters and apertures may be used within the beam path for beam shaping, in order to prepare the beam. 
     Exemplary laser beam parameters for, for example, ultrashort pulse laser systems and parameters of the optical system and the elongated focal zone, which can be applied within the range of this disclosure, are: 
     pulse energy Ep: 1 μJ to 10 mJ (e.g., 20 μJ to 1000 μJ), 
     energy of a pulse group Eg: 1 μJ to 10 mJ 
     ranges of wavelength: IR, VIS, UV (e.g., 2 μm&gt;λ&gt;200 nm; e.g., 1550 nm, 1064 nm, 1030 nm, 515 nm, 343 nm) 
     pulse duration (FWHM): 10 fs to 50 ns (e.g., 200 fs to 20 ns) 
     interaction duration (depending on the feed velocity): smaller 100 ns (e.g., 5 ps-15 ns) 
     duty cycle (interaction duration to repetition time of the laser pulse/the pulse group): less than or equal to 5%, e.g., less than or equal to 1% 
     raw beam diameter D (1/e2) when entering the optical system: e.g., in the range from 1 mm to 25 mm 
     focal lengths of the near field optics: 3 mm to 100 mm (e.g., 10 mm to 20 mm) 
     numerical aperture NA of the near field optics: 0.15≤NA≤0.5 
     length of beam profile within the material: larger 20 μm 
     maximal lateral extent of the beam profile within the material, where applicable in the short direction: smaller 20λ 
     aspect ratio: larger 20 
     modulation in propagation direction: larger 10 periods over the focus zone 
     feed dv between two neighboring modifications e.g., for separating applications: 
     100 nm&lt;dv&lt;10*lateral extent in feed direction feed during interaction duration: e.g., smaller 5% of the lateral extent in feed direction 
     Thus, the pulse duration of the laser pulse and the interaction duration relate to a temporal range, within which, for example, a group of laser pulses interacts with the material for the formation of a single modification at a location. Thereby, the interaction duration is short regarding the present feed velocity, so that all laser pulses of a group contribute to a modification at one position. 
     If the work-piece is thinner than the focus zone is long, the focus zone is positioned partially outside of the work-piece, so that modifications may be caused that are shorter than the focus zone. Such a situation may be advantageously used to make the processing process robust also with respect to varying the distance between the optics and the work-piece. In some embodiments, a modification may be advantageous that does not reach through the complete work-piece. In particular, the length of the focus zone and/or its position within the work-piece may be adapted. In general it is noted, that, due to different thresholds for the nonlinear absorption, a focus zone with assumed identical intensity may cause differently large modifications in differing materials. 
     The aspect ratio relates to the geometry of the beam profile (the focus zone) within the to be processed material as well as the geometry of the modification created with a beam profile. For asymmetric or in lateral direction modulated (for example, non-rotationally symmetric or ring-shaped) beam profiles, the aspect ratio is given by the ratio of the length of the modification with respect to a maximum lateral extent in the shortest direction that is present within that range of length. If the beam profile thereby includes a modulation in lateral direction, for example, for ring-shaped beam profiles, then the aspect ratio relates to the width of a maximum, for a ring-shaped beam profile, for example, to the strength of the ring. When a plurality of modification volumes, which are displaced in lateral direction, are formed, the aspect ratio relates to the lateral extent of a single modification. For a beam profile modulated in propagation direction (e.g., due to interferences), the aspect ratio relates to the higher ranking total length. 
     Assuming a distance d between the beam shaping element and the focusing lens (near field optics), which is in particular larger than the focal length fN of the near field optics, and an NA of the near field optics with respect to air &gt;0.15, the used angular spectrum α of the beam shaping element can be in the range tan(α)&lt;f*NA/d&lt;NA/2 and preferably tan(α)&gt;f*NA/(d*   4   ). 
     The previously mentioned ranges for parameters may allow the processing of a material thickness up to, for example, 5 mm and more (typically 100 μm to 1.1 mm) with roughness of the cutting-edge Ra, for example, smaller than 1 μm. 
     Optical system  1  may further include a beam processing unit  13  for adapting beam parameters such as beam diameter, input intensity profile, input divergence, and/or polarization of laser beam  3 . For example, the laser beam of a pulsed laser system is coupled into optical system  1  with, for example, a beam diameter of 5 mm, pulse duration of 6 ps at wavelengths around 1030 nm and is led to processing unit  13 . 
       FIG. 3  shows the schematic setup of optical system  1  for explaining the functionality. Optical system  1  is based on a beam shaping element  31  and an imaging system  33 . Beam shaping element  31  is adapted to receive laser beam  3 . Accordingly, it is adapted to a transverse input intensity profile  41  of laser beam  3 . In addition, beam shaping element  31  is adapted to impose onto laser beam  3  a beam shaping phase distribution  43  (schematically indicated by dashes in  FIG. 1 ) over transverse input intensity profile  41 . Imposed phase distribution  43  is such that a virtual optical image  53  (essentially) of elongated focus zone  7  is attributed to laser beam  53 , virtual optical image  53  being located in front of beam shaping element  31 . Beam shaping element  31  creates in this manner a virtual beam profile that is located upstream of beam shaping element  31 , but does not correspond to the real path of the beam being at that position. 
     Imaging system  33  is construed such that the virtual beam profile is imaged into the area of the laser processing machine, in which the work-piece is positioned during the processing. In  FIG. 3 , imaging system  33  includes for that purpose an, in beam direction, first focusing element, which is referred to herein as far field optics  33 A, and an, in direction of the beam, second focusing element, which is referred to herein as near field optics  33 B. 
     Far field optics  33 A is provided in the area of phase imposing and is illustrated in  FIG. 3  exemplarily downwards of beam shaping element  31  by a lens shape. As will be explained in the following, far field optics  33 A may also be arranged shortly before beam shaping element  31 , composed of components before and after the beam shaping element, and/or completely or partially integrated in the same. 
     After the imposing of the phase within beam shaping element  31 , laser beam  3  propagates in accordance with imaging system  33  over a beam shaping distance Dp to near field optics  33 B. Beam shaping distance Dp corresponds thereby to a propagation length of the laser beam  3 , within which imposed phase distribution  43  transforms the transverse input intensity profile  41  into a transverse output intensity profile  51  at near field optics  33 B. Herein, output intensity profile  51  includes those transverse intensity profiles in the optical system that are determined by the phase imposing. This is usually completed at the latest in the area of the focal length before the near field optics or within the area of the near field optics. 
     For implementing the concept of a virtual beam profile, there are the following considerations for the propagation length (from beam shaping element  31  to near field optics  33 B), which laser beam  3  has to propagate within the optical system. In general, the optical system forms an imaging system  33  with a far field focusing action and a near field focusing action. The latter is determined by near field optics  33 B and thereby by near field focal length fN. The first is determined by a far field focusing action and a respective far field focal length fF. Far field focal length fF can be realized by the separate far field optics  33 A and/or can be integrated into the beam shaping element. See in this respect also  FIG. 20 . Imaging system  33  has an imaging ratio of X to 1, whereby X for a demagnification of the virtual image usually is larger than 1. For example, imaging ratios are implemented that are larger than or equal to 1:1 such as larger than or equal to 5:1, 10:1, 20:1, or 40:1. In other words, with this definition of the imaging, the factor X resembles the magnification of the lateral size of the focus zone into the virtual profile. The angle is respectively demagnified. Attention should be paid to the fact that the imaging ratio goes quadratic into the length of the profile. Accordingly, the longitudinal length of a virtual image becomes smaller, for example, for an imaging ratio of 10:1 by a factor of 100 and for an imaging ratio of 20:1 by a factor of 400. 
     At an imaging ratio of 1:1, there is fN=fF, an overlapping alignment of the focal planes is assumed. In general, there is fF=X fN. If the far field optics  33 A is integrated into the beam shaping element, it is positioned, e.g., at a distance fN+fF from the near field optics, i.e., typically in the range of the sum of the focal lengths of both optical elements. For a 1:1 or a de-magnifying imaging system, the propagation length corresponds therefore at least to twice the focal length of the near field optics. 
     Separating far field optics  33 A and beam shaping element  31  and assuming, that the virtual optical image should not overlap (in particular not within the intensity region being relevant for the focus zone) with the beam shaping element, the beam shaping element is arranged at at least at a distance of ½ downward of the longitudinal center of virtual beam profile  53 . Here, the length I is the longitudinal extent of virtual beam profile  53  with respect to the relevant intensity area. The longitudinal center of virtual beam profile  53  is located e.g., at the entrance side focal plane of far field optics  33 A, which is located at a distance fN+fF from near field optics  33 B. In this case, the propagation length is d=fN+2fF−I/2=(1+2X)fN−I/2, therefore smaller than fN+2fF=(1+2X)fN, or, in other words, smaller than the distance between the optical elements plus 
     For the distance fN+2fF=(1+2X)fN, also for increasing beam enlargements a respectively increasing length I of virtual beam profile  53  can be imaged, whereby—as explained later—a defined end of the profile can be maintained. 
     In general it is mentioned that, due to raw beam divergences and convergences as well as for deviating adjustment of the imaging system, deviations from the above considerations may occur. In contrast to a comparable image of a real intensity enhancement, i.e., images with comparable imaging ratios, the beam shaping element is located closer (see the respective discussion on  FIGS. 7 and 8 ). A common distance therefore lies in a range (1+2X)fN≥d≥2fN. 
     Due to the imposed phase, transverse output intensity profile  51  includes, in comparison to input intensity profile  41 , at least one local maximum  49  located outside of a beam axis  45 . Local maximum  49  being located outside beam axis  45  results in a lateral energy entry into focus zone  7 . Depending on beam shaping element  31 , local maximum  49  of transverse output intensity profile  51  can be made rotationally symmetric with respect to beam axis  45 —as indicated in  FIG. 3  in the cut view—or it can be formed in an azimuthal angular range (see, e.g.,  FIGS. 29 and 30 ). Usually, the beam axis is defined by the center of the beam of the lateral beam profile. The optical system can usually be related to an optical axis, which usually runs through a symmetry point of the beam shaping element (e.g., through the center of the DOE or the tip of the reflective hollow cone axicon). For rotationally symmetric beams and a respective exact alignment, the beam axis may coincide with the optical axis of the optical system at least in sections. 
     The local maximum can be considered a generic feature of output intensity profile  51 , where in particular for inverse quasi-Bessel beam shapes, a typical substructure with a steep and slowly falling flank can be formed. That substructure can invert itself due to the focusing action of the beam forming element and/or the far field optics in the range of an associated far field focal plane. In particular, the output intensity profile can show within the range of that far field focal plane the local maximum particularly “sharp” or, for example, for inverse quasi-Bessel beam shapes, the local maximum can form itself quite fast after the beam forming element. However, the aspects of the substructure may vary due to the various possibilities in the phase imposing. 
     The concept of a virtual beam profile can, on the one side, reduce the constructional length of optical system  1  and, on the other side, it can avoid the formation of an elongated beam profile with significant intensity enhancement within optical system  1 . Imaging system  33  is configured such that, within optical system  1 , the far field of the virtual beam profile is formed and that the focusing in the near field optics  33 B can be done using a common focusing component such as a lens, a mirror, a microscopic objective, or a combination thereof. In that case, “common” is understood herein in the sense of that the characteristic beam shape is essentially imposed by beam shaping element  31  and not by near field optics  33 B. 
     In  FIG. 3 , a path of the beam is indicated for illustration that corresponds to a beam herein referred to as an inverse quasi-Bessel beam. For that purpose, the path of the beam is illustrated downstream of beam shaping element  31  with solid lines. Upstream of beam shaping element  31 , instead of incident collimated beam  3 , the virtual beam profile is sketched in analogy to a real quasi-Bessel beam by dashed lines. 
     Similar to a common quasi-Bessel beam, also the inverse quasi-Bessel beam has a ring structure in the focal plane of far field optics  33 A. However, divergent beam areas  55 A,  55 B indicated in the schematic cut view, which impinge on far field optics  33 A, do not result from a “real” quasi-Bessel beam profile, but they result directly from the interaction of beam shaping element  31  with incident laser beam  3 . Due to the direct interaction, beam areas  55 A,  55 B are shaped in their lateral intensity distribution by transverse beam profile  41  of laser beam  3 . Accordingly, for a Gaussian input beam, the intensity decreases in the radial direction principally in beam areas  55 A,  55 B from the inside to the outside. Due to the divergence of beam areas  55 A,  55 B, typically an area of low (in the ideal case no) intensity is formed accordingly on the beam axis for the phase-modulated beam portions. In that case, the divergence of a beam portion, accordingly also a divergent beam portion, relates herein to a beam portion that moves away from the beam axis. However, in that area, a beam portion of a phase unmodulated beam and/or also an additional, phase-modulated beam portion may be superimposed. With respect to the development of the beam within the optical system during the shaping of an inverse Bessel beam, it is referred to the description of  FIGS. 33 and 34 . This intensity behavior is schematically indicated in transverse intensity courses (e.g., transverse intensity beam profile segments)  57 A and  57 B. It is noted that the intensity courses along the propagation length can change due to imposed phase distribution  43 . At least, however, within the initial area (i.e., beam areas  55 A,  55 B laying close to beam shaping element  31 ) and due to the beam shaping element  31  acting in general as a pure phase mask, the incident intensity profile of laser beam  3  dominates the divergent phase-modulated beam portions. 
     For a clear explanation of an inverse quasi-Bessel beam, further intensity courses  57 A′ and  57 B′ are schematically indicated in  FIG. 3 . Here, it is assumed that beam shaping element  31  influences only the phase and not the amplitude. One recognizes that the focusing by far field optics  33 A (or the respective far field action of beam shaping element  31 ) reverses the intensity course at the exit of optical system  1  such that, during the formation of elongated focus zone  7  on beam axis  45 , at first low intensities are superposed, which originate from the decreasing flanks of the incident Gaussian beam profile. Thereafter, the higher intensities super-pose, which originate from the central area of the incident Gaussian beam profile. In this context it is noted that not only the intensity on the beam shaping element, but also the contributing area has to be acknowledged. For rotationally symmetry, the distance enters accordingly quadratic. As in particular illustrated in connection with  FIG. 4 , the longitudinal intensity profile ends exactly in that area, in which the beam portions from the center of the input profile cross. In the center, although the highest intensity is present, the area goes to zero. Moreover, it is noted that, after the focus zone, a reversed intensity course is present again, which corresponds to intensity courses  57 A,  57 B after the beam shaping element (assuming no interaction with a material). 
     Due to imaging with imaging system  33 , there are incident virtual intensity courses  57 A″ and  57 B″, which are accordingly schematically indicated with respect to the virtual beam shaping in  FIG. 3 . Those correspond in principle to intensity courses  57 A′ and  57 B′. 
     Those intensity courses, which are inverted in comparison to a quasi-Bessel beam, cause a specific longitudinal intensity course for the inverse quasi-Bessel beam for focus zone  7  as well as in the virtual beam profile, i.e., optical image  53 , because here the superposition of beam portions  55 A,  55 B is done virtually. For the respective discussion of the intensity course for a conventional quasi-Bessel beam, it is referred to  FIGS. 7 and 8  and the respective description. 
       FIG. 4  exemplarily illustrates a longitudinal intensity distribution  61  within elongated focus zone  7  as it can be calculated for the imaging of virtual optical image  53  of an inverse quasi-Bessel beam shape. Depicted is a normed intensity I in Z direction. It is noted that a propagation direction according to a normal incidence (in Z direction) onto material  9  is not required and, as illustrated in connection with  FIG. 2 , can take place alternatively under an angle with respect to the Z direction. 
     One recognizes in  FIG. 4  an initially slow intensity increase  61 A over several  100  micrometer (initial superposition of low (outer) intensities) up to an intensity maximum, followed by a strong intensity decrease  61 B (superposition of the high (central) intensities). For an inverse Bessel beam shape, the result is therefore a hard border of the longitudinal intensity distribution in propagation direction (the Z direction in  FIG. 4 ). As one can in particular recognize in view of intensity courses  57 A′ and  57 B′ shown in  FIG. 3 , the hard border is based on the fact that the end of longitudinal intensity distribution  61  relies on the contributions of the beam center of the incident laser beam having admittedly a lot of intensity, but on a strongly reduced (going to zero) area. In other words, the end relies on the imaging of a virtual beam profile in which in the center for the inverse quasi-Bessel beam a hole is created. The strong gradient at the intensity decrease at the end relies on the high intensity in the center of the input profile, limited, however, by the disappearing area. For an ideal imaging system, the longitudinal extent of intensity distribution  61  is defined by the position of the virtual profile and the imaging scale. If in addition the work-piece includes a large refractive index, the beam profile is accordingly lengthened. 
     In this context it is added that the hard border has the consequence in laser processing machines that the, in propagation direction, front end of a modification is essentially stationary in propagation direction also if the incident transverse beam profile is increased. The modification changes its extent only in the back part, i.e., it can lengthen in direction to the near field optics, if the input beam diameter of the laser beam enlarges. A once set position of the hard border with respect to the work-piece support or the work-piece itself can thereby avoid high intensities downstream of the modification. In contrast thereto, an enlargement of the input beam diameter, when imaging a real intensity enhancement, causes an elongation of the modification in propagation direction, i.e., for example into a work-piece support, which can result in damages of the same. 
       FIG. 5  shows an exemplary X-Y-cut  63  of the intensity within focus zone  7  for the longitudinal intensity distribution  61  shown in  FIG. 4 . One recognizes the elongated formation of focus zone  7  over several hundred micrometer at a transverse extent of some few micrometer. Together with the threshold behavior of the nonlinear absorption, such a beam profile can cause a clearly defined elongated modification within the work-piece. The elongated shape of focus zone  7  includes, for example, an aspect ratio, i.e., the ratio of the length of the focus zone to a maximal extent in the lateral shortest direction being present within that length—the latter for non-rotationally symmetric profiles, in the range from 10:1 to 1000:1, e.g., 20:1 or more, for example 50:1 to 400:1. 
     If one frees oneself from the beam shape—shown in  FIG. 4 —of an inverse quasi-Bessel beam, which is not modified in propagation direction with respect to amplitude, beam shaping element  31  can additionally create an amplitude redistribution in the far field, which e.g., can be used for an intensity modification in propagation direction. 
     However, the thereby created intensity distribution in front of focus zone  7  can no longer be presented in a very clear manner. Nevertheless, often initial stages of inversions will show up in the beginning region or in the end region of the longitudinal intensity profile, for example a slow increase and a steep decrease. However, a (phase caused) amplitude redistribution by the phase description of beam shaping element  31  may just exactly be set to an inverted intensity distribution, in order to cause, for example, a form of a longitudinal flat top intensity profile. 
     Additionally, the following feature for distinguishing from a “real” beam shape may be maintained: For the case of a real Gaussian input beam, there exists, e.g., for a real Axicon, a plane between near field optics and focus zone at which the demagnified 
     Gaussian transverse beam profile of the input beam is present and can accordingly be made visible. A respective imaging exists for the virtual optical image. However, in this case, the image plane, in which the demagnified Gaussian transverse beam profile is present, lies behind the focus zone. The transverse beam profile can accordingly be made visible. This applies generally to phase masks for the herein disclosed inverse beam shapes, if those are illuminated with a Gaussian beam profile. Specifically, the demagnified Gaussian transverse beam profile is positioned in the image plane of the beam shaping element and therefore usually directly downstream of the focus zone. Due to the already performed divergence, it is therefore significantly larger than the transverse beam profile of the inverse quasi-Bessel beam in the focus zone. Also, it is much lower in the intensity. 
     One can recognize the position of the imaged Gaussian transverse beam profile of the input beam by a fast turn up of the structure of the beam profile, i.e., a strong change over a small lateral area. For example, the transverse intensity profile of the inverse quasi-Bessel beam is present in the focus zone. When passing through the image plane of the beam shaping element, then “quasi” immediately the dark spot in the center is formed. For an inverse quasi-Bessel beam, this is different at the beginning of the focus zone. There, due to the increased superposition of the border areas of the Gaussian beam profile, a slow transition is made from a dark center to the transverse intensity profile of the inverse quasi-Bessel beam, which is filled in the center. In other words, in longitudinal direction, the intensity increases over a larger area then it decreases at the end. At the end, that transition is accordingly clearly sharply limited. It is added that, when imaging a real Bessel intensity enhancement, the behavior at the end and the behavior at the beginning are interchanged, i.e., at the end of the Bessel beam profile, the dark spot forms more slowly. 
     As previously explained, the concept of using a virtual beam profile therefore has an effect inter alia on the phase imposing to be applied and the resulting intensity courses in focus zone  7 . 
       FIG. 6  illustrates modification zones  65  that were created in the context of an experimental study for examining the formation of modifications in a transparent material. Each modification zone  65  goes back to the interaction with a group of laser pulses, for example two  6  ps pulses at a temporal separation of about 14 ns. The shape of the modification zones correspond to the shape of elongated focus zone  7  as assumed in accordance with  FIGS. 4 and 5 . The maximal length is limited by the geometry of elongated focus zone  7  at a required intensity/fluence. 
     The upper four images illustrate the threshold behavior for pulse group energies Eg from about 20 μJ to 40 μJ. The lower four images illustrate the shaping of the elongated modification zones  65  at pulse group energies Eg from about 30 μJ to 200 μJ. With increasing total energy Eg, the modification zone lengthens in the direction of the beam entrance (near field optics), because the threshold intensity for the nonlinear absorption is reached within a longer area of focus zone  7 . The end of the modification in beam propagation direction is in its position essentially stationary, and even in particular without secondary correction of the distance of a near field optics ( 33 B) to the to be processed work-piece. At lower energies, an initial walk in beam direction of the back end may occur due to the existing gradient in longitudinal direction, in particular if the modification threshold lies at small intensities within the beam profile. However, the walk decreases at medium and high energies, because the generation of the inverse quasi-Bessel beam profile includes in propagation direction an implicit maximal back end. 
     A similar behavior in the change of the longitudinal extent of the modification is also created for a radially increasing beam diameter of incident laser beam  3 . Also in that case, the modification zone is lengthening in direction of the beam entrance (near field optics), because the intensity areas of incident laser beam  3 , which are added in a radial direction at the outside, guide energy into the longitudinal intensity area in the area of slow intensity increase  61 A (i.e., intensity increase with slow gradient). The maximum of the intensity distribution will accordingly be shifted in direction of the beam entrance. The end of the modification in beam propagation direction is in contrast in its position essentially stationary, because that position is supplied with energy by the center of the beam of incident laser beam  3 . In addition it is noted that this behavior can be observed also for modified inverse quasi-Bessel beam shapes. For example, for a flat top beam shape as discussed in connection with  FIGS. 23 to 26 , the position of the end of the modification would essentially not change for a change in the beam diameter. For such a changed incident intensity profile, the beam shaping element may further eventually no longer result in an optimized flat top structure so that this may result in modulations in the intensity and eventually a variation of the beginning. 
       FIG. 7  serves as an illustration of a beam guidance at which a real intensity enhancement  71  is generated by a beam shaping optics  73  such as an axicon. This corresponds to the known formation of a quasi-Bessel beam. Intensity enhancement  71  is then imaged by a telescope  75  into work-piece  9  by forming a focus zone  77 . As shown in  FIG. 7 , in such a setup, there is the danger that the real intensity enhancement  71  damages a far field optics  79  of telescope system  75 , in particular if a small setup length is to be realized. The herein disclosed optical system (see, e.g.,  FIG. 3 ), which implements the concept of a virtual image, bypasses that risk of a damage to the beam guiding optics. 
       FIG. 8  illustrates for completeness a longitudinal intensity distribution  81  in Z direction that results from the setup of  FIG. 7 . After an ab initio steep increase  81 A, an intensity maximum is reached, beginning at which the intensity decreases again. At lower intensities, a slowly vanishing drop  81 B (vanishing drop of small gradient) begins. One sees the general reversal of the longitudinal intensity distributions  61  and  68  of  FIGS. 4 and 8 , at which the “hard border” at the end is replaced by a “hard beginning”. 
     For such a quasi-Bessel beam, the passing through an axicon with a laser beam having an incident Gaussian beam profile  83  will result in superposed beam portions  85 A,  85 B, the intensity weights of which result in real longitudinal intensity distribution  81  (at first superposition of the intensities of the central area of Gaussian beam profile  83 , then superposition of lower (outer) intensities of Gaussian beam profile  83 ). For explaining, again schematic intensity courses  87 A and  87 B are indicated downstream of far field optics  79 , and intensity courses  87 A′ and  87 B′ are indicated upstream of focus zone  77 . 
     Before various exemplary configurations of optical systems, which implement the concept of virtual intensity enhancement, will be explained in the following, it is referred again to  FIG. 3 , and in particular transverse output intensity profile  51  at near field optics  33 B. In the area of output intensity profile  51 , one can influence, for example, with an aperture  50  the portions that contribute to the focus zone. Depending on the orientation and width of the aperture opening and in general the shape of the aperture, a modification of the longitudinal intensity distribution  61  can be caused in elongated focus zone  7  and can be even adapted. For a rotationally symmetric—as shown in  FIGS. 3 and 7 —output intensity profile  51  or for a respective symmetric phase distribution in  FIG. 14 , a rotation of the aperture around the beam axis can cause a rotation of the orientation of the modification in focus zone  7 . 
     Such a mechanical control of an aperture  50  can e.g., be performed alternatively by a rotation of a previously addressed and in connection with the following figures further explained permanently written DOE or can be implemented electronically by time-dependent setting of a programmable DOE. Accordingly, the laser processing machines, which are intended for a respective laser cutting and in particular guiding, include rotation mechanics and/or electric control devices. As addressed above, here the output intensity or the optical element of an inverse as well as a regular quasi-Bessel beam can be modified or rotated. 
     This relates in particular to the systems explained in the following that include beam shaping elements in the transmission and reflection, wherein the imposing of the phase distribution is performed in particularly refractive, reflective, or diffractive. It is referred to the preceding description with respect to the already described components such as laser system  11 . 
     In view of the distances of beam shaping optics  73  from the near field optics, the following values can apply similar to the considerations for the virtual image. For a real beam profile, one would typically position the center of the to be imaged real beam profile of length I in the entrance-side focal length of the far field optics. A typical distance would then be at least 
     fN+2fF+I/2=(1+2X)fN+I/2, thus larger than fN+2fF, in other words, larger than the dis-tance between the optical elements plus fF. 
       FIG. 9  shows a refractive beam shaping with the help of a hollow cone axicon  131 A. This creates a virtual inverse quasi-Bessel beam profile  153 A upward of hollow cone axicon  131 A. The same is indicated in  FIG. 9  by dashed lines, a real intensity enhancement is not present in that area. In addition, in the embodiment of  FIG. 9 , the far field optics is configured in beam propagation direction downstream of hollow cone axicon  131 A as plano-convex lens  133 A. Near field optics  33 B causes the focusing of the laser beam into focus zone  7 , so that the virtual inverse quasi-Bessel beam profile  153 A is related to the laser beam as virtual optical image of focus zone  7 . 
       FIG. 10  shows an embodiment with a hollow cone axicon lens system  131 B that is used as a refractive beam shaping element. Here, the far field optics is integrated in the beam shaping element as convex lens surface  133 B, which is positioned at the entrance side of the hollow cone axicon. This setup creates similarly a virtual inverse quasi-Bessel beam profile  153 B. 
       FIG. 11  illustrates an embodiment with a reflective beam shaping element, in particular a reflective axicon mirror system  131 C. A highly reflective surface of the beam shaping element is shaped such that the beam shaping feature of a reflective axicon is combined with the far field forming component of a focusing hollow mirror. Accordingly, axicon mirror system  131 C has the function of beam shaping as well as of the far field optics. A virtual inverse quasi-Bessel beam profile  153 C is indicated at the backside of axicon mirror system  131 C, thus in an area that is not passed by laser beam  3 . 
     As is further shown in  FIG. 11 , after beam adaptation unit  13 , laser beam  3  of laser system  11  is coupled into optical system  1  by a deflection mirror  140 . Deflection mirror  140  is, for example, arranged on the optical axis between axicon mirror system  131 C and near field optics  33 B and guides the beam to beam shaping element  131 C. In some embodiments, the deflection mirror may, for example, be centrally drilled through to guide as less as possible light onto the central area of beam shaping element  131 C, which potentially has optical errors. In addition to those aspects of filtering described in the following in connection with  FIGS. 17 and 18 , it is already noted at this stage that deflection mirror  140  at the same time blocks an undesired central beam portion such that the same is not focused by near field optics  33 B. 
       FIGS. 12 and 13  show embodiments of the optical system with digitalized beam shaping elements. Here, the digitalization can relate to the use of discrete values for the phase shift and/or the lateral structure (for example, pixel structure). The use of spatial light modulators (SLMs) is one of many different possibilities to realize the beam shaping with programmable or also permanently written diffractive optical elements (DOE). 
     In addition to the simple generation of one or more virtual beam profiles, e.g., according to the phase imposing of one or more hollow cone axicons, diffractive optical elements allow the desired modification, for example, for homogenizing of the longitudinal intensity distribution. 
     For this, deviations in the phase can exemplarily be used in the range equal to or smaller than 50%, e.g., equal to or smaller than 20% or equal to or smaller than 10% with respect to, for example, the hollow cone axicon phase (and thereby of an inverse quasi-Bessel beam). In general, SLMs allow very fine phase changes at a lateral rough resolution, in contrast to, for example, lithographically generated, permanently written DOEs. Permanently written DOEs include e.g., plano-parallel steps, the thickness of which determine the phase. So, the lithographic manufacturing allows a large lateral resolution. Binary steps can result in real and virtual beam profiles. Only a number of more than two phase steps can result in a differentiation in the sense of a preferred direction for the virtual beam profile. For example, four or eight or more phase steps allow an efficient beam shaping with respect to the virtual beam profile. However, the discretization can cause secondary orders that can, for example, be filtered out. 
     Herein, the structural element of a diffractive optical beam shaping element, which causes the phase imposing and is configured in an areal shape, be it an adjustable SLM or a permanently written DOE, is referred to as a phase mask. Depending on the type of configuration of the DOE, it may be used in transmission or in reflection to impose a phase distribution on a laser beam. 
     In  FIG. 12 , a spatial light modulator  31 A is used in reflection for phase imposing. For example, spatial light modulator  31 A is based on a “liquid crystal on silicon” (LCOS) that enables a phase shift that is programmable for the individual pixels. Spatial light modulators can further be based on micro-systems (MEMS), micro-opto-electro-mechanical systems (MOEMS), or micro-mirror-matrix systems. In SLMs, the pixels can, for example, be controlled electronically to cause a specific phase imposing over the transverse input intensity profile. The electronic controllability allows, for example, the online-setting of phases and, thus, the adaptation of focus zone  7 , e.g., in dependence of the to be processed material or in reaction of fluctuations of the laser. In the configuration of  FIG. 12 , the function of a diffractive axicon for the generation of a virtual inverse quasi-Bessel beam profile can be combined, for example, with the far field forming action of a far field optics by the phase shifting of the spatial light modulator  31 A. Alternatively, a permanently written reflective DOE can be used as beam shaping element  31 A. 
       FIG. 13  is a schematic view of an optical system based on a DOE  31 B, for which the phase imposing is permanently written in DOE  31 B. DOE  31 B is used in that case in transmission. As in  FIG. 12 , the phase shift, which, for example, results in a virtual quasi-Bessel beam profile, as well as the focusing property of far field optics are combined within the DOE  31 B. 
     The optical systems of  FIGS. 9 to 13  can result in output intensity profiles that correspond to inverse quasi-Bessel beam profiles and that are attributed to virtual optical images. The optical system of  FIG. 7  as well as analog setups for  FIGS. 9 to 13 , for example, with a regular in transmission used axicon can result in output intensity profiles of (regular) quasi-Bessel beam shapes. For the utilization of cutting procedures based on local effects, inverse quasi-Bessel beam profiles and regular quasi-Bessel beam profiles can similarly be used, wherein of course the herein disclosed differences need to be considered. 
       FIG. 14  illustrates an example of a phase distribution  243  as it can be provided e.g., in DOE  31 B. Phase distribution  243  is rotationally symmetric. One recognizes ring-shaped phase distributions, the frequency of which is modulated in radial direction. The rings point to the generation of a rotationally symmetric virtual quasi-Bessel beam profile. The frequency modulation points to the integration of the phase component of the far field optics in the phase distribution for beam shaping. In  FIG. 14 , the phases are indicated in the range of ±π. In alternative configurations, discrete phase distributions such as binary phase distributions or multi-step (for example, 4 or more levels in the range of the phase shift from 0 to 2π) phase distributions can be implemented in DOE phase masks. 
       FIGS. 15 and 16  illustrate exemplarily an output intensity profile  251  within the intensity cross-section ( FIG. 15 ) and in the  2 D-top view ( FIG. 16 ). One recognizes an intensity maximum  249  extending in a ring shape around beam axis  45 . There is hardly any intensity in the beam center. 
     In some embodiments, the transition into the inverse quasi-Bessel beam will not be complete such that accordingly a non-phase-modulated remaining beam, for example with a Gaussian beam shape, is superposed to the ring-shaped intensity profile.  FIG. 15  indicates schematically such a non-phase-modulated beam portion  252  by a dash-dotted line. 
     Maximum  249  of that intensity distribution in  FIG. 15  is an example of a local intensity maximum, with which an original input intensity profile (e.g., a Gaussian beam profile) was modified in the area of the transverse output intensity profile. The rotational symmetry of the ring structure is due to the rotational symmetry of the inverse quasi-Bessel beam profile. In alternative embodiments, the local intensity maximum is limited to an azimuthal angular range. In addition, a superposition of azimuthal limited and/or ring-shaped local maxima may be given. 
     When using a refractive hollow cone axicon (see  FIGS. 9 and 10 ) for the generation of an inverse quasi-Bessel beam-shaped output intensity profile, undesired beam portions may be created under undesired angles for a non-perfect tip of the axicon. Also for diffractive beam shaping elements, non-desired beam portions may appear. For example, a non-phase-modulated beam portion, which cannot be neglected, or additional orders of diffraction in the far field of the laser beam can be present. 
     The herein disclosed optical systems simplify, by using the far field components, the insertion and the shape selection of filters to filter out such disturbing beam portions. In particular, these undesired beam portions can be separated from the desired beam portions (beam for use) in a simple manner in the area of the Fourier plane. 
     Referring to the non-phase-modulated beam portion  252  of  FIG. 15 ,  FIG. 17  shows an exemplary optical system that is based on the optical system shown in  FIG. 3 . However, additionally a filtering of non-phase-modulated portions is performed in the area of the Fourier plane of imaging system  33 . Exemplarily, a spatial filter unit  220  is indicated upward of near field optics  33 B in  FIG. 17 . 
     Filter unit  220  includes a central area around beam axis  45  that blocks, for example, the Gaussian intensity distribution—indicated in  FIG. 15 —of the non-phase-modulated beam portion  252 . Filter unit  220  can additionally include sections, which are located radially further outside, for blocking higher orders of diffraction by the DOE or the SLM. 
     In general, filter unit  220  is provided for the suppression of non-phase-modulated base modes and higher diffraction orders as well as of scattered radiation of the various herein disclosed refractive, reflective, or diffractive beam shaping elements. For rotationally symmetric output intensity profiles, usually also the filter unit is made rotationally symmetric. In some embodiments, only some portions of filter unit  220  or no filtering at all is provided. 
     Diffractive beam shaping elements allow a further approach for suppressing the non-phase-modulated beam portions. For this, an additional phase contribution is imposed to deflect the phase-modulated beam portion. 
       FIG. 18  shows, for example, an optical system in which the diffractive optical element  31  is additionally provided with a linear phase contribution. The linear phase contribution results in a deflection  230  of phase-modulated beam  203 A. Non-phase-modulated beam portion  203 B is not deflected and impinges, for example, on a filter unit  222 . 
       FIG. 19  shows a further embodiment of an optical system that utilizes the use of the far field component additionally for the implementation of a scan approach. In general, a scan system allows shifting focus zone  7  within a certain range. In general, it is possible by the separation of the beam shape from the near field focusing to provide favorable telecentric scan approaches, in particular for the volume absorption. In some embodiments, in addition the location as well as the angle can be set. Accordingly, such scanner systems can allow writing fine contours into a work-piece. 
     In the configuration of  FIG. 19 , a scanner mirror  310  is positioned at the image side focal plane of a near field optics  333 B. Scanner mirror  310  deflects the laser beam in the range of the output intensity distribution onto near field optics  333 B positioned at the side. The deflection in the Fourier plane results in that the propagation direction in the work-piece is preserved despite the displacement in location. The scanning region itself is determined by the size of near field optics  333 B. 
     If scanner mirror  310  is not correctly positioned in the focal plane of near field optics  333 B or if it can be moved with respect thereto, then an orientation of the elongated focus zone, in particular an angular deviation from the Z direction in  FIG. 2 , can be set. 
     With the help of a configuration in accordance with the optical system shown in  FIG. 13 ,  FIG. 20  explains exemplarily the underlying imaging features. The optical system includes a beam shaping element  31  that operates also as a far field optics and is therefore characterized by a focal length fF. In addition, the optical system includes near field optics  33 B that is characterized by focal length fN. In  FIG. 20 , the focal planes of the far field optics and the near field optics coincide. Accordingly, in  FIG. 20  only one focal plane  340  is indicated by a dashed line. In that configuration of overlapping focal planes, the imaging system images for incidence of a plane wave front generally a virtual beam shape  253  onto elongated focus zone  7 , for example, an inverse quasi-Bessel beam profile, inverse modulated or homogenized quasi-Bessel beam profiles as examples for inverse quasi-Bessel/Airy beam shapes (e.g., inverse quasi-Bessel like beam shapes, inverse quasi-Airy like beam shapes). 
     Though the focal planes do not need to overlap always. For example, the imaging system can be adapted to a given beam divergence, but laser beam  3  may be incident with another divergence. In those cases, still a virtual optical image being positioned in front of the beam shaping element is attributed to elongated focus zone  7 , but it does not need to be a perfect imaging. A similar situation may be given for an intended misalignment of the imaging system, for example, in connection with a scanner device. 
       FIG. 20  illustrates in addition the terms “far field optics” and “near field optics”. The far field optics creates the far field of virtual beam path  253  in the range of far field focal length fF. As previously already explained, the far field optics can be distributed in its function, for example, be made of one or more components, which are arranged before and/or after the beam shaping element and displaced with respect to the same, and/or be integrated into the beam shaping element. The near field optics focuses the beam with the smaller focal length fN in the direction of the work-piece and thereby forms the focus zone. Thus, the far field of virtual beam profile  53  with respect to the far field optics, as well as the far field of focus zone  7  with respect to near field optics  33 B is positioned in the area of focal plane  340 . 
     Also for non-perfect imaging (e.g., non-overlapping focus planes of far field optics and near field optics), essentially an acceptable intensity distribution in the focus zone can be given, because the intensity profile, which impinges onto the near field optics, changes only a little. 
     For example, in the case of an inverse quasi-Bessel beam shape, the first focusing by the far field optics within the optical system causes an adaptation of the ring size on the near field optics. In that manner, the far field optics has a focusing action onto the ring diameter, which, as indicated in the figures, decreases up to some type of intermediate focus. 
       FIG. 21  illustrates the beam path in an optical system for the case that a convergent laser beam  3 ′ impinges on beam shaping element  31 . Phase-modulated portion  303 A of the laser beam is focused onto elongated focus zone  7 . Due to the convergence of incident laser beam  3 ′ (and eventually due to a separate focusing far field optics or an integration into the phase distribution of beam shaping element  31 ), the non-phase-modulated portion  303 B (dash dotted line) decreases further during the propagation length Dp and impinges on a central area of near field optics  33 B. Thereby, a focus  350  for non-phase-modulated beam portion  303 B is formed that is closer to near field lens  33 B than it is elongated focus zone  7 . The non-phase-modulated portion is strongly divergent after focus  350 , so that those intensities are no longer reached within the work-piece with respect to the non-phase-modulated beam portion  303 B that result in nonlinear absorption. In such a configuration, one can do without filtering non-phase-modulated beam portions  303 B. 
     Nevertheless, a spatially localized filter unit can be provided in the area of focus  350  (or even between far field optics and near field optics, if the beam is strongly focused) such that non-phase-modulated beam portion  303 B is kept out of the interaction zone and the work-piece. 
     Regarding further embodiments, it is in particular referred to the priority application DE 2014 116 958.1 of the same applicant filed on 19 Nov. 2014. The content of that application is incorporated herein in its entirety. For example, one or more additional lenses may be arranged upstream of the beam shaping element for the adaptation of the beam divergence. 
     Diffractive optical elements allow a digitalized and e.g., pixel based phase adaptation over the input intensity profile. Starting from the intensity distribution of an inverse quasi-Bessel beam shape, a longitudinal flat top intensity profile can, for example, be generated in focus zone  7 . For that purpose, the phase distribution within the beam shaping element can be influenced such that intensity contributions in the output intensity profile are taken out of the area, which forms the intensity maximum and the tails of the Bessel beam, and are radially redistributed by a phase change such that, for the later focusing by near field optics  33 B, the increasing area  61 A and the decreasing area  61 B are magnified or far extending tails are avoided to the most part (e.g., by pushing power from the tails into the homogenized area). 
     Modification zones are formed essentially always over the same range of extent in Z direction within work-piece  9 . This is based on the essentially constant intensity having only a short increase and drop off. With increasing energy, however, not only the strength but also the lateral extent of the modification zones increases. 
     Moreover, supplemental phase imposing can generally be done in the area of the image side focal plane of near field optics  33 B such as lateral and/or longitudinal multi-spot phase imposing. In particular, the formation of an accelerated Airy beam shape is possible. 
     In some embodiments, an optical system is configured, for example, such that a real intensity enhancement in accordance with  FIG. 7  as well as a virtual intensity enhancement in accordance with  FIG. 3  is created. Thereby, the longitudinal extent of modification zones can be widened. 
     An inverted quasi-Bessel beam can be generated with the herein disclosed refractive, reflective, and diffractive optical systems, for example, with the hollow cone axicon systems and the DOE systems. A DOE system can be based, for example, on the phase distribution of a phase mask shown in  FIG. 14 , in which a focusing phase contribution is provided in addition to the phase required for the inverse quasi-Bessel beam. 
     Usually, a laser beam having a rotationally symmetric Gaussian beam profile is irradiated onto the beam shaping element. A Gaussian beam profile includes a transverse amplitude course that runs through the beam center in a Gaussian manner. 
     Due to the pure phase mask, a Gaussian beam profile and a Gaussian amplitude course are present directly after the beam shaping element still similar to the Gaussian beam. Then, a sharply limited hole forms immediately, however, caused by the imposed phase, which yields the divergence that is continuously growing. At the same time, a ring area with higher amplitude is formed. 
     Ring area is sharply limited towards the inside which can be seen at a step shape in the radial amplitude/intensity distribution. A flank of the circumferential step faces towards that beam axis/towards the beam center. With increasing z values, the opposing sections of flank get separated, i.e., the central sharply limited hole grows fast in diameter (D 1 &lt;D 2 ). 
     In the radial amplitude/intensity distribution, ring area drops towards the outside with increasing z values faster and faster. In the far field, i.e., for example in the overlapping focal planes of the imposed focusing far field action and the near field optics, a sharp ring is formed within beam profile, that thereafter diverges. Thereby, now a sharp edge is performed at the outer side, i.e., the step of the inner flank now faces towards the outside. 
     This general behavior of the beam profile and the amplitude courses enable a test of an optical system with a Gaussian input beam, for which at first a hole forms with a steep flank facing the inside and thereby results in a local maximum outside of the beam axis in the far field. An imaging of the beam profile from the inner area as well as in the area of the focus zone can identify the respective beam profile. The use of the optical system is thereby not necessarily limited to Gaussian beams. In addition, it is to note that the figures are a result of calculations for the ideal case. For example, if a non-ideal DOE is used, the addressed non-phase-modulated portion for higher orders or a portion of a real quasi-Bessel beam (such as for a bi-nary mask) can be on the beam axis and can fill the “hole” with intensity. 
     An inverse quasi-Bessel beam can therefore include a step with a steep flank in the amplitude course and accordingly in the intensity distribution. The same can in particular face to the inside in the area close to the beam shaping element, for example, in the area up to half of the far field, and in particular in the area of a focus length of the far field optics downstream of the beam shaping element. For a “simple” inverse quasi-Bessel beam without base at the beam axis, the amplitude/intensity increases in the range of the step from almost zero to the maximum of the phase-modulated beam portion. Thereby, the formation of the step (within the phase-modulated beam portion) is also given for an exemplary incident beam having essentially a constant radial intensity (radial flat top) across the beam shaping element, because the step concerns essentially the beam center. 
     The beam characteristic described before upstream of the far field focal plane is thereafter radially inverted up to the focus zone. After that focus zone, it inverts radially another time such that again a step shape can be formed at that position—without interaction with a material to be processed. The beam profile can, for example, be analyzed by taking the beam at a respective position, be it within the optical system after the beam shaping element or before or after the focus zone. In particular, for setups that allow a blocking of a central disturbing beam, one can analyze the intensity distribution of the phase-modulated beam portion before or after the focus area. 
     In this context, it is further referred to the German patent application DE 10 2014 116 957.3 entitled “Optisches Strahlformungselement” filed by the same applicant on Nov. 19, 2014 that in particular discusses optical systems for beam shaping. The content of that application is herein incorporated in its completeness. As is explained therein generally, inter alia inverse quasi-Bessel beams can be used for laser material processing. 
     The herein disclosed concepts allow influencing the focus zone in longitudinal direction and lateral direction. In particular, the use of a DOE as an example for an areally configured phase mask enables a simultaneous imposing of multiple phase distributions on laser beam  3 . For generating an inverse quasi-Bessel/Airy beam, a virtual optical image is attributed to at least one of the phase distributions, wherein the virtual optical image can be imaged into an elongated focus area for forming a modification in the material to be processed by the laser beam. In the presence of two such phase distributions, that result in at least partially overlapping focus zones or focus zones, which at least influence each other, one can shape the geometry of the modification(s) of the material to be processed—generated by a laser pulse or a group of laser pulses. 
     In general, such phase distributions can form one or more ring structures, a ring segment structure limited to an angular range (see e.g.,  FIG. 28 ), and/or one or more local maxima (see e.g.,  FIG. 33 ) in a transverse far field intensity distribution (exit intensity distribution). 
     Several such phase distributions can be imposed in various manners. An association of segments on the phase mask is most obvious (see e.g.,  FIG. 26 ). These segments can be separate areal regions, wherein the separate regions can e.g., join radially and/or azimuthal with respect to each other and can transition into each other, for example, abrupt or weighted in the border region. Moreover, the segments can be at least partially encapsulated into each other (see e.g.,  FIG. 31 ). Finally, the phase increase, which is created by an (areal) portion of the areally configured phase mask, can be a combination of phase contributions that are respectively attributed to such a phase distribution. Besides the DOE configurations described in the following, for example, also respectively combined hollow cone axicons or reflective axicons (with respect to an inverse Bessel beam) and axicons in general (with respect to regular Bessel beams) can reproduce an areal segmentation. One should understand the following examples for explaining potential concepts accordingly, wherein the concepts can also be realized with other approaches for phase imposing that were addressed before and are herein disclosed. In general, several optical elements can be combined within a DOE, by determining e.g., the transmission function of all elements (e.g., hollow cone axicon(s) and lens(es); adding the individual phase functions (exp(−1i(phi1+phi2+ . . . )))). In addition or alternatively, some type of superposition of individual transmission functions can be done. For the determination of the phase distributions, it was initially referred to the publication of Leach et al. 
     For segments, which are rotationally symmetric, also the intensity distribution is rotationally symmetric and each interference maximum corresponds to a volume area, in which the intensity/fluence can be above a threshold intensity/fluence. 
     The aspects described in the following are based on that initially addressed realization that the density of free electrons, which can develop within the material by nonlinear absorption, can be tailored by a targeted beam shaping, for example, with a diffractive optical element (DOE). Along the resulting modifications, a crack formation can be specifically guided that then causes the separation of the work-piece. Moreover it was recognized that regular and inverse Bessel beam types are suitable in particular for using an aperture for the targeted beam shaping because they include a ring-shaped far field for introducing the aperture. 
     The crack formation can occur or be initiated after introducing all modifications, which delimit a contour, by applying an in particular thermal or mechanical induced tension. Depending on the choice of parameters of the laser and depending on the material, the crack formation can already take place during the writing of the modifications. When initiating the crack formation during the creation of modifications, the crack is preferably guided to follow the latest generated modification, i.e., the crack forms against the feed direction. 
     For example, it was in particular recognized that, for the buildup of the preferred direction for separating transparent materials, an intensity distribution, which deviates from the rotational symmetry, can make a crack creation at the same time or in a successive step along the set preferred direction. In other words, the intensity distribution is no longer rotationally symmetric in that XY-plane, i.e., in the plane orthogonal to the beam propagation direction, instead it is e.g., ellipse-like. An intensity maximum in the focus zone is surrounded by a respective area of increasing high intensity, wherein herein both together is understood, in particular in view of the behavior for crack formation, generally as intensity maximum of a focus zone, and in this context the intensity maximum should be understood not to be reduced to the maximum value itself. 
     In general, a focus zone is assumed that has an elongated extent in propagation direction of the laser beam as well as an in particular asymmetric extent in the plane transverse to the propagation direction. A similarly shaped modification is created, respectively. If the focus zone includes in its extent in array of several focus points with local intensity maxima within the envelope of the focus zone, a respective array of modifications with sub-focus zones is formed. Also the intensity distributions of such focus points can have an in particular asymmetric extent in the plane transverse to the propagation direction. If the envelope shape of the focus zone is flattened in the plane transverse to the propagation direction, focus points can be shaped by interference effects, wherein the focus points are also flattened in orthogonal direction with respect to the flattening of the envelope (see  FIGS. 20B and 23B ). 
     With respect to the generated separating effects, the intensity areas are—based on the underlying laser parameters and that underlying material properties—decisive that lie with respect to a separating effect above a respective threshold. If this is the case for individual sub-focus zones, the same can influence the crack formation; if essentially only one zone in the shape of the envelope is above the threshold, this one will influence the crack formation. 
       FIGS. 21A and 21B  illustrate the asymmetric beam shape in the X-Y-plane exemplarily as an elongated ellipse  400 —be it a single modification or the modification due to a sub-focus zone. 
     A quasi-forced formation of a preferred direction of the crack formation can e.g., allow using larger distances between modifications, i.e., in total less modifications are required. The separating can itself be improved, because one can avoid a random preferred direction of cracking. In contrast, the “random” preferred direction of cracking can establish itself for modifications, which are in the X-Y-plane “point”-shaped or in space spherical or round pillar-shaped (rotationally symmetric), partly by material-specific or separating path-specific parameters such as e.g., a (pre-)tension in the material, a curvature of the material, and an implicitly given crystal structure. 
     The herein proposed crack and guiding can allow, in a controlled manner and in particular for fragile structures, an insertion of a crack close to a border or a previously generated contour. Moreover, the crack guiding can take place during the modifying without further processing steps, so that a time-saving can occur by less process steps. 
     When using asymmetric focus zones for generating modifications with a preferred direction, the asymmetry can be parallel to the feed direction. In  FIG. 21A , this would be a feed direction in X direction. This can result in the generation of a shockwave within the material, which may cause a crack formation along the modification (in  FIG. 21A  in X direction). I.e., depending on the material, an alignment of a, for example, in the X-Y-cut ellipse-like shape along the feed direction may cause a crack formation in direction of the length axis of the ellipse. This will be magnified by the respective sequence of modifications in feed direction. 
     The generation of shockwaves is well-known in the prior art. Thereby, around the modification, strong tensile stress is created, which acts orthogonal to the modification borders. (In the hereinafter explained  FIG. 21A , these were solid arrows  402 .) For elongated modifications, therefore the largest tension is created along the largest extent(s). For a modification array, the action of the individual modifications accumulates, so that in an asymmetric array the separating plane also runs in direction of the largest extent of the array. 
     With the help of a suitable beam shape and a suitable selection of laser parameters in dependence of the material parameters, the density of free electrons, which are generated within the material by nonlinear absorption, can be set such that the shockwave effect or the thermal stretching can be used for inserting a separating plane and/or a crack. Depending on the material, the degree of precision, the processing velocity, and further boundary conditions such as the distance between separating lines, the one or the other effect may be more advantageous. 
     Also if, for the creation of the modification, a melted volume with asymmetry in feed direction is created (in particular with low viscosity), for example, in the XY-cut in the shape of an ellipse, that alignment can cause a crack formation in direction of the asymmetry, for example, in direction of the length axis of the ellipse. 
     The asymmetry can further be present across to the feed direction, if asymmetric focus zones are used for creating modifications with a preferred direction. In  FIG. 21A , this would be a feed direction in Y direction. Also this orientation, which intuitively deviates from the thought of a “cutting knife”, can achieve results that for some materials and beam parameters increased the quality of the separating surface in comparison to the parallel alignment or rotationally symmetric intensity distributions. 
     It is assumed that by the thermal expansion of the modification (the material becomes in some cases heavy viscous) a stretching and thereby pressure tension along the solid arrows  402  in  FIGS. 21A and 21B  onto the surrounding material is created. The expansion can be e.g., transient or permanent. Due to the elongated shape, the pressure tensions induce a tensile tension along the dash dotted arrows  404 , i.e., in X direction and across to the elongated shape of the intensity distributions in the plane transverse to the beam direction. The effect of a plurality of modifications, in particular also of smaller extent, can add onto each other, so that—as indicated in  FIG. 21B —a crack  406  or a preferred location of formation for the crack forms between ellipses  400 . 
     It is assumed that, for most transparent materials, the use of the shockwave as well as the thermal expansion can be utilize for influencing the crack formation. Which effect is caused, depends on the setting of the laser parameters, i.e., wavelength, pulse duration, repetition rate, fundamental frequency, pulse energy, pulse group energy, etc. As, when creating a shockwave, a preferred direction is formed in the direction of the largest extent of the asymmetric focus zone, the focus zone is to be parallel to the feed direction, when using that effect. As, for the thermal expansion, the preferred direction is generated across to the largest extent, the focus zone is to be guided across to the feed direction, when using that effect. For an array of multiple foci, attention has to be paid that the stretched out separation plane is guided parallel to the feed direction. 
     For an elliptical individual modification, for example, also only a tension can be formed within the material (still without crack formation); the material is then only set under pre-tension. Only when applying an external tension, the desired crack forms according to the distribution of tension being present. So, several transverse elliptical modifications can create a crack between the same, which extends across to the orientation of elliptical base shape  400 . 
     In some cases, it was recognized that a crack formation does not yet occur for a single modification, instead cracks extent only for a sequence of several modifications, which were one after the other introduced in feed direction, from a preceedingly introduced to or up to within the subsequently introduced modification. Thereby, a crack can be avoided that runs ahead of the modification in feed direction and would be difficult to control in its orientation. In particular, if the tensile stress, which is required for the crack formation, is formed only in a transient manner, the simultaneous formation of several modifications being arranged in feed direction is advantageous as it is illustrated in  FIG. 21B  for a cut in that XY-plane in the case of ellipses. 
     If one includes the above considerations in the material processing of a material, which is in particular for the laser beam to a large extent transparent, asymmetric shaped modifications are formed transverse to the propagation direction of the laser beam. In that case, the laser beam can be appropriately focused for forming the elongated focus zone in the material, wherein the focus zone is configured such that it includes at least one intensity maximum, which is transverse flattened in a flattening direction. In  FIGS. 21A and 21B , the respective intensity area/intensity maximum is flattened in Y direction. Moreover, sequences can be created, transverse to the propagation direction (see  FIG. 21C  in comparison with  FIGS. 22C and 23C ) and/or axially with respect to that propagation direction (see  FIG. 21C  in comparison with  FIGS. 29 and 30 —wherein  FIG. 30  also shows a transverse sequence). Also those transverse and/or axial sequences of asymmetric intensity maxima can be flattened in sequence direction and thereby cause the effects for crack formation and crack guiding, which are herein described in particular in connection with  FIGS. 21A to 21C , if in particular the material and the focus zone are moved relative to each other, so that, for successive laser pulses, modifications are created, which are displaced transverse in feed direction, along which the crack formation should take place with e.g., respective precision. According to the laser parameter and the material features, the modifications can create (be it on the plane of the focus zone or on the plane of a sub-focus zone) the above explained preferred direction for the crack formation, be it along or across to the length direction of the asymmetric shape. 
       FIG. 21C  transfers the illustration of  FIG. 21B  to an array of three neighboring intensity distributions, wherein the intensity distributions are additionally modulated in Z direction. I.e., individual ellipsoids  410  are arranged on top of each other or next to each other, and e.g., are lined up in Z direction. Around each ellipsoid  400 ′, the tension distribution of a potential separating zone  406 ′ is indicated that can be created by the tensile stress along the (here solid) a-rows  404 , wherein separation zones  406 ′ can result together in the formation of a separating plane  410 . 
     In general, such tensions can also be present in Z direction between modifications that are also lined up in Z direction. Separating plane  410  results from adding up the tensile stresses  404 ′ that were created by ellipsoids  400 ′. 
     In the following, examples are disclosed for the generation of the asymmetry of the focus zone. On the one side, an aperture (see aperture  50  in  FIGS. 3 and 7 ) allows forming a slit, which can be set in thickness and orientation, in the area of the imaging system  33 , in particular focal plane  340 . Two positions for the aperture will be explained in connection with  FIGS. 22A to 23D . On the other side, a suitable asymmetry can be imposed by calculating the phase distribution of the DOEs. Exemplary phase distributions are shown in  FIGS. 24A and 24B  that can correspond to the positions of the aperture shown in  FIGS. 22A and 23A . 
       FIG. 22A  illustrates exemplary an output intensity profile  451  in the intensity cross-section after passing aperture  50 . Moreover, aperture opening  460  is exemplarily indicated in  FIG. 22A . One recognizes two intensity maxima  449  that extend in a ring segment shape around beam axis  45 . There is no intensity in the beam center.  FIG. 22B  illustrates an intensity distribution for a cut across the beam axis Z, and 
       FIG. 22C  shows an intensity distribution for a cut through beam axis Z within the focus area, as the intensity distribution could be the result of blocking out the border area in  FIG. 22A . 
       FIGS. 22B and 22C  show an elongated focus zone  477  with a sequence of about  5  strong intensity maxima  461  in propagation direction, wherein the intensity maxima with almost similar intensities can reach into the material over an area of several hundred p.m in Z direction. Interference maxima  461  are deformed in the cut across to the beam axis in a longitudinal manner, herein generally referred to as elliptical. The longitudinal axis of interference maxima  461  is lined up across to the direction of the sequence. For materials and laser parameters, which along with the considerations to  FIG. 21B , cause a crack formation across to the length axis of the ellipse and accordingly in  FIG. 22B  along with the Z direction, the material is separated accordingly along the sequence of interference maxima  461 . 
     For a smaller aperture opening  460 ′ shown in  FIG. 23A ,  FIGS. 23B and 23C  illustrate that the number of interference maxima  461  increases when closing the aperture, and that the elliptical deformation in X direction increases. Elongated focus zone  477  shows now more than 10 intensity maxima  461 . 
       FIG. 23D  is an example for material processing of alumosilicate glass (gorilla glass) with a double pulse of a pulse duration of about 7 ps at a wavelength of 1030 nm. The total energy of 125 μJ results from two single pulse energies of 62.5 μJ that are irradiated at the temporal distance of 200 μs. The processing is performed e.g., at a base frequency of 2 kHz. 
     Regular and inverse Bessel beam-types are specifically suitable for the use of apertures to generate elongated focus zones because they include a ring far field for inserting of the aperture. 
     Both ways for creating asymmetry can allow steerability in the process of material processing. For example, the geometry can be set via the thickness of the aperture. A small slid results in broader ellipses as well as in more ellipses in e.g., feed direction. 
     For example, the width of the aperture can be opened wider for radii to be cut in order to shorten a modification zone. In addition or alternatively, tracking a curve can be supported by alignment of the beam profile. For example, the complete optical system or the work-piece can be rotated. Furthermore, (assuming a rotationally symmetric input beam) the aperture and/or the DOE can be rotated around the beam axis, so that the preferred direction can be changed during the material processing. In addition, online adaptable phase distributions can be used. In particular, for an aperture both sides of the aperture can be changed synchronously or the sides of the aperture can be controlled individually. 
     In general, modifications can be created such that the created tension decreases at ends of the array, in order to avoid an accidental swerve of the crack (e.g., forward). Similarly, modifications can be created, so that the array creates “towards the back” further more large tension, so that the crack extends to the foregoing introduced modification. The result is kind of a guiding together with the crack or a guiding tracking the crack (following). 
     Exemplarily,  FIGS. 24A, 24B  show respective cut views of intensity distribution  477 ′ in the X-Y-plane (transverse to the beam) and in the Y-Z-plane (along the beam) of a linear array of focus zones extending into that material. One sees that intensities in the array increase against the feed direction V (here the feed is performed in the negative Y-direction). Accordingly, crack formations can extend towards previous modifications. In direction towards the not yet processed material, however, the tendency for crack formation is reduced. Such intensity distributions can be created by specific DOE-phase distributions and/or by superposing multiple beams. 
     Alternatively, the radiation field can be separated into two or more partial radiation fields, which are coupled into the material in a displaced manner, to achieve the desired asymmetry also when using rotationally symmetric radiation fields and partial radiation fields. 
       FIGS. 25A and 25B  show, alternatively to the use of an aperture respectively configured DOEs. One recognizes a phase distribution on a phase mask that has an azimuthal segmentation. Two segments are opposed to each other—each in triangular shape with a rectangular peak of the triangle positioned respectively in the center. The areas of the phase mask positioned between the segments form two opposing segments with a phase distribution that scatters the light into spatial regions that are not interesting. In general, an incident Gaussian beam is directed onto the beam shaping element such that the center of the beam shaping element coincides with the beam axis of the incident beam. 
     The angular range used in the DOE of  FIG. 25B  is reduced (analog to the aperture opening in  FIG. 23A ) such that also here an elongation of the elliptical shape and an increase in the number of intensity maxima in the focus zone are caused. 
     The phase distributions of  FIGS. 25A and 25B  are obviously not rotationally symmetric. A rotation of the respective DOE results in a respective rotation of the focus zone and accordingly in a change in the orientation of the preferred direction for the crack formation—similarly in analogy to the rotation of the linear aperture opening around the beam axis because the beam portions along the X direction are subject to the phase distribution of the “X”-segments  970 A and beam portions along the Y direction are subject essentially to the phase distribution of the “Y”-segments  970 B. 
     Often preferred directions for the crack formation form that do not coincide with the desired separating plane. This can be caused, for example, by anisotropic work-pieces, for example crystals, by deviations in the symmetry of the work-piece symmetry, or by an astigmatism due to the beam entrance surface, for example, for a beam incidence deviating from orthogonal incidence, or a curved beam entrance surface. Often, such “random” preferred directions disturb the processing. By the specifically set asymmetry of the beam profile, the “random” preferred direction can be oversteered advantageously and can be transferred into a desired preferred direction. 
     Summarizing, an orientation of the preferred directional/the beam profile relative to the feed can allow an improved separation in feed direction, in particular a faster, more efficient, more robust separation of high quality. 
     In general, for the herein disclosed crack-supported separation, which is based on asymmetry, the intensity distribution can be introduced only into the material or it can be introduced to extend up to one or both surfaces. 
     In addition, besides multi-spot configurations, also modulated intensity distributions can be used, wherein the modulation can be present in the beam direction and/or transverse to the beam direction. 
     In other words, in particular the use of an aperture allows adapting the crack length to the contour. For example, a short extension in feed direction can be set at sharp curves and edges, and a long extent in feed direction can be set for straight separating lines. 
       FIG. 26  shows a phase distribution  970  of a phase mask with azimuthal segmentation. A pair of “X” segments  970 A are opposed to each other—each in triangular shape with a rectangular peak of the triangle positioned respectively in the center (corresponding to the beam axis). The areas of the phase mask positioned between “X” segments  970 A form two opposing “Y” segments  970 B—also in triangular shape with a rectangular peak of the triangle positioned in the center. In general, an incident Gaussian beam is directed onto the beam shaping element such that the center of the beam shaping element coincides with the beam axis of the incident beam. 
     In the example of  FIG. 26 , the transition of the phase distributions between the individual segments takes place abrupt. The phase distribution of  FIG. 26  is obviously not rotationally symmetric because beam portions along the X direction are subject to the phase distribution of the “X”-segments  970 A and beam portions along the Y direction are essentially subject to the phase distribution of the “Y”-segments  970 B. 
       FIGS. 27 and 28  show an intensity distribution  971 A in X direction and a central portion  971 B of an XY-view on an intensity profile as they can form in the far field focal plane. As the phase mask of  FIG. 26  does not include a focusing phase portion, the same is used with a separate and therefore for both segments identical far field optics. In intensity distribution  971 A and in portion  971 B, one recognizes a two-part outer ring segment  972 A of an intensity enhancement being located radially outside. In portion  971 B, one further recognizes a two-part inner ring segment  972 B of an intensity enhancement being located radially inside. The latter includes essentially no contribution in X direction (y=0). Accordingly, it is also not viewable as an intensity enhancement in intensity distribution  971 A. Each part of ring segments  972 A and  972 B extends over 90°—according to the azimuthal segmentation. In consequence, the azimuthal segmented phase mask of  FIG. 26  results in an asymmetric intensity distribution in the far field. Moreover, a longitudinal interference structure can form due to differing angle portions. The asymmetry in the beam shape originates from the asymmetry in the segments. For identical angle portions in the segments and a phase shift of the segments of PI, e.g., an asymmetric beam shape without modulation can form, for which the distance of the thereby created inverse quasi-Bessel beams can be in the range of the beams themselves. The interference of the respective inverse quasi-Bessel beam shapes can accordingly result in an asymmetry/modulation in the transverse formation of the intensity distribution. 
     Exemplarily,  FIG. 29  shows a cut  973 A in ZX-plane through a common elongated focus zone  973  of an intensity distribution, which originates from an output intensity distribution according to  FIG. 28 .  FIG. 30  shows a respective cut in ZY plane  973 B of the intensity distribution. One recognizes sequences of essentially linearly arranged intensity maxima  975 . Intensity maxima reach significant intensities in a single row in  FIG. 29  and in three rows in  FIG. 30 . In  FIG. 30 , the maxima of the outer rows in Z direction are thereby displaced with respect to the inner row. If one selects now the Y direction as feed direction for the material processing, then a single laser pulse (or a group of laser pulses) forms a focus zone/modification zone that is elongated in feed direction, i.e., is asymmetric. Accordingly, the width of the focus zone/modification zone is reduced in the separating direction, i.e., in the YZ-plane. Accordingly, the result is an arrangement of three “cutting” rows of intensity maxima  975 . It is noted that herein grey scale illustrations of  FIGS. 29 and 30  are based on color illustrations, so that maximal values of the intensity/amplitude are illustrated dark. 
     In other words, the asymmetry, which was created by the segmentation of the phase mask, in combination with inverse quasi-Bessel beam shapes can be used for the formation of a geometric direction of preference during separation. Also in this configuration, the end region of the focus zone/modification zone can be essentially independent from the irradiated energy and the beam diameter of the incident beam. 
     A further example for an interaction space in material  9  having asymmetric geometry is explained in connection with  FIGS. 31 to 36C .  FIG. 31  shows a phase distribution  1043  of a phase mask that is based on a superposition of two phase distributions. Each of the phase distributions belongs to an inverse quasi-Bessel beam as it can be individually be generated, for example, with a hollow cone axicon. However, the centers of the phase distributions in X direction are displaced with respect to each other by Ax. Phase distribution  1043  includes further a superposition with a centrally arranged spherical phase, i.e., a focusing far field action is integrated in the phase mask being e.g., configured as a diffractive optical beam shaping element. 
       FIG. 32  shows an amplitude distribution for a cut along beam axis Z in the range from z=0 mm to z=250 mm as it can be the result of imposing phase distribution  1048 . One recognizes a dark central area  1010 A that widens in Z direction. Due to the only small laterally displaced phase distributions, diverse interference structures  1034  are formed in the bright intensity area  1010 B that adjoins radially to outside central area  1010 A. 
     The focusing far field action of phase distribution  1048  forms a ring in the respective focal plane, which is structured in its intensity. A respective output intensity profile  1051  is exemplary illustrated in  FIG. 33 . One recognizes local maxima, the extent of which is the largest in azimuthal direction in the X axis. The azimuthal extent decreases with increasing distance from the X axis and along the ring. 
       FIG. 34A  shows a ZX cut through the longitudinal intensity profile and  FIG. 34B  shows an XY cut through the longitudinal intensity profile within the interaction area, as it is the result of the focusing of output intensity profile  1051 . In particular, two elongated focus zones  1007 A and  1007 B are formed that are displaced in X direction and that extend in Z direction. Besides the main maximum, respective multiple secondary maxima are formed. The pulse energy or the pulse group energy can be set such that, in particular for nonlinear absorption, only the strongest maximum or the strongest of maxima of each focus zone results in a modification of the material. 
     If one scans the laser beam, which was formed in that manner, in Y direction over a material to be processed, a track of two modification zones at a distance is formed. Thereby, an intended tension distribution within the material can be created, which can e.g., start a separation preferably within an intermediate area  1050  between the elongated modification zones. For example, pressure tensions can build up in the modification zones, which result in the formation of tensile stress in the intermediate area that then supports the respective separating process. Here, the X direction would be again the separating direction and the Y direction would be the feed direction. 
     If one sets the laser parameters in dependence of the material such that a shockwave is created for the focus distribution as shown in  FIGS. 34A and 34B , then the intensity distribution can also be used for the X direction as a feed direction for inserting a preferred direction of cracking by creating a separation plane between the elongated focus zones  1007 A and  1007 B. The two focus zones  1007 A and  1007 B may be in addition advantageously be generated with a small distance there-between. 
     The development of intensities in the respective optical system downstream of the diffractive optical beam shaping element will again have—corresponding to an inverse quasi-Bessel beam shape—a step structure in the radial intensity distribution. Due to the lateral displacement of the beam portions for the two inverse quasi-Bessel beams, interference structures  1034 , however, form which can overlay with the step structure. 
     Despite the interference structures  1034 , one can recognize areas in beam profiles  1040 A to  1040 C for z=10 mm, z=100 mm, and z=150 mm, which are reproduced in  FIGS. 35A to 35C , that have higher intensities at the radial inner side.  FIGS. 36A to 36C  show respective intensity distributions  1042 A to  1042 C that extend radially in X direction. In particular, in  FIGS. 36B and 36C , one recognizes the formation of a steep flank  1047  that surrounds an inner area of lower intensity. Herein, the intensity radially decays to the outside with a slowly decreasing flank  1047 . However, the formation of the flanks is strongly dependent on the direction due to the interference, as it is shown, for example, in  FIGS. 35A to 35C . 
     The above explained examples are based on values of two phase distributions provided on the phase mask. However, more than two phase distributions can be provided. For example, more than two phase distributions can be provided in radial and azimuthal segments, or can be included in combinations of phase steps. 
     Further embodiments and/or further developments of the herein disclosed aspects are summarized in the following: 
     In general, the herein disclosed focusing elements such as the far field optics and the near field optics can be configured as, for example, lens, mirror, DOE, or a combination thereof. 
     Moreover, additional optical elements can be inserted into optical systems such as the herein disclosed embodiments. Inter alia intermediate images can be inserted in the imaging system, to be able to realize, for example, a filter function as well as a scan movement in the area of the image-side focal plane. Thereby, e.g., the image-side focal plane (e.g., image plane  340  in  FIG. 20 ) can itself be imaged by an additional optical system. Alternatively or additionally, such optical intermediate systems can allow, for example, realizing an enlarged working distance and/or a magnification of the working field for scanner application. 
     Regarding further developments of the diffractive optical beam shaping element, at least a plurality of beam shaping phase distributions  43  can be configured such that an incident laser beam  3  having a Gaussian intensity distribution is transferred into at least one divergent beam area  55 A,  55 B attributed to the virtual optical image  53 . Downstream of the diffractive optical beam shaping element  31 , the divergent beam area includes a transverse intensity distribution  57 A,  57 B, which decreases from the inside to the outside and is present in particular before a far field focal length (fF) attributed to a focusing action of the phase mask, and/or at least one of the plurality of beam shaping phase distributions  43  is configured such that an incident laser beam  3  is transferred into at least one divergent beam area  55 A,  55 B) attributed to the virtual optical image  53 . Downstream of the diffractive optical beam shaping element  31 , the divergent beam area including a transverse intensity distribution  57 A,  57 B that includes a section of a step-shaped intensity increase, which includes a steep flank ( 907 ) facing radially to the inside, and that is present in particular before a far field focal length (fF) attributed to a focusing action of the phase mask. 
     At least two segment specific phase distributions can be associated respectively with a segment-specific virtual optical image that can be imaged in a segment-specific focus zone, and the respective segment-specific focus zones are arranged with respect to each other such that they contribute together to the formation of a modification zone. 
     At least two segments may be composed of spatial structures that are at least partly encapsulated into each other, and/or segments of the plurality of segments join radially and/or azimuthal, wherein in particular a weighted transition between the respective neighboring phase distributions can be set in the transition area of neighboring segments of the plurality of segments. 
     The, in particular segment-specific, focus zones can be superposed with respect to each other and/or spatially complement each other, and/or at least two, in particular segment-specific, images of the virtual optical images are superposed while interfering, and/or at least two, in particular segment-specific, images of the virtual optical images can form a common elongated focus zone. 
     Moreover, to the beam shaping element, an imaging system can attribute an image plane downstream of the longitudinal center of the image of the virtual optical image, and a transverse beam profile of the laser beam can be present at the beam shaping element in the image plane. In particular, in the area of the image plane, a fast turn up in longitudinal direction from a lateral beam profile being present in the focus zone to a lateral beam profile with a dark center may be present, the latter in particular for an essentially lateral Gaussian beam profile of the laser beam and in particular with respect to beam portions of the incident laser beam, which create a divergent beam portion, which is attributed to the virtual optical image, and/or the optical system can be configured such that essentially only a central area of the incident laser beam contributes to a downstream positioned end of the focus zone attributed to the virtual image, so that a change of a beam diameter of the incident laser beam does not result in a significant longitudinal displacement of the downstream positioned end of the focus zone. 
     It is explicitly stated that all features disclosed in the description and/or the claims are intended to be disclosed separately and independently from each other for the purpose of original disclosure as well as for the purpose of restricting the claimed invention independently of the composition of the features in the embodiments and/or the claims. It is explicitly stated that all value ranges or indications of groups of entities disclose every possible intermediate value or intermediate entity for the purpose of original disclosure as well as for the purpose of restricting the claimed invention, in particular as limits of value ranges. 
     A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other embodiments are within the scope of the following claims.