Vol.:(0123456789)1 3 https://doi.org/10.1007/s00170-023-11215-5 ORIGINAL ARTICLE Adjustment of the geometries of the cutting front and the kerf by means of beam shaping to maximize the speed of laser cutting Jannik Lind1,2  · Christian Hagenlocher1  · Niklas Weckenmann2 · David Blazquez‑Sanchez2  · Rudolf Weber1  · Thomas Graf1 Received: 20 December 2022 / Accepted: 3 March 2023 © The Author(s) 2023 Abstract The shape of the laser beam used for fusion cutting significantly influences the geometry of both the cutting front and the cut- ting kerf. The angle of the cutting front in turn impacts the local absorptivity, while the width of the kerf defines the amount of material, which has to be molten. The kerf’s geometry therefore determines the maximum possible cutting speed at which a successful cut is feasible with a given available laser power. The absorptivity, the width of the kerf, and the maximum possible cutting speed can be estimated from a simple model considering the conservation of energy and rough geometrical approximations. In order to verify the prediction of the model, the geometry of the cutting front and kerf resulting from dif- ferent processing conditions was observed by means of online high-speed X-ray diagnostics. The geometry of the interaction zone was recorded with a framerate of 1000 Hz during fusion cutting of 10-mm-thick samples of stainless steel. Comparing the results obtained with different shapes of the laser beam, it was found that the absorptivity is increased when the beam’s longitudinal cross-section (parallel to the feed) is enlarged. Reducing the width of the beam in the transversal direction nor- mal to the feed reduces the cross-sectional area of the cutting kerf. The findings show a good agreement with the geometric model which enabled the prediction of the absorptivity and the cross-sectional area of the cutting kerf and hence allows to reliably estimate the maximum cutting speed for different shapes of the laser beam, laser power, and sheet thicknesses. Keywords Laser beam cutting · Beam shaping · Online high-speed X-ray imaging · Cutting speed 1 Introduction The average laser power of solid-state lasers has increased continuously over the past few years. In laser cutting this allows for an increase of the maximum thickness of the cut workpieces [1. , 2. ]. This development is however associ- ated with an increased complexity in the choice of process parameters and laser beam properties in order to optimize the cut quality, and to increase the maximum cutting speed [3. –8. ]. Increasing the maximum cutting speed reduces the total time required to cut a workpiece and therefore offers the potential to increase the productivity of a laser cutting machine. The maximum cutting speed is reached when the absorbed power is not sufficient to melt the volume of the material per unit of time which is required to generate the cutting kerf. The amount of absorbed power is determined by the absorptivity. The required amount of molten material is determined by the cross-sectional area of the cutting kerf and the cutting speed [9. , 10. ]. The cross-sectional area of the cutting kerf is determined by the caustic of the laser beam [1. , 11. , 12. ]. It was observed that the cross-sectional area of the kerf is enlarged and the maximum cutting speed reduced, when the diameter of the beam is increased [12. ]. Additionally, the diameter of the beam also influences the absorptivity at the cutting front [13. ]. According to the geometric model of the cutting front introduced by Mahrle et al. [13. ], the global angle of incidence � on the cutting front decreases with increasing beam diameter. According to the Fresnel equations [14. ] for an unpolarized beam, the decrease of the angle of incidence from 90° to approximately 80° leads to an increased absorptivity, resulting in a higher maximum cutting speed [9. , 10. ]. Increasing the diameter of the beam however also enlarges the cross-sectional area * Jannik Lind jannik.lind@ifsw.uni-stuttgart.de 1 Institut für Strahlwerkzeuge (IFSW), University of Stuttgart, Pfaffenwaldring 43, 70569 Stuttgart, Germany 2 Precitec GmbH & Co. KG, Draisstraße 1, 76571 Gaggenau, Germany / Published online: 13 March 2023 The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 http://crossmark.crossref.org/dialog/?doi=10.1007/s00170-023-11215-5&domain=pdf http://orcid.org/0000-0002-2196-5616 http://orcid.org/0000-0003-2929-9723 http://orcid.org/0000-0002-3504-229X http://orcid.org/0000-0001-8779-2343 http://orcid.org/0000-0002-8466-073X 1 3 of the kerf, which has a contrary impact on the maximum cutting speed. A suitable analytical and experimentally veri- fied model for the resulting cross-sectional area of the cut- ting kerf and the absorptivity at the cutting front is therefore required in order to be able to reliably predict the maximum cutting speed for different shapes of the applied laser beam and a given available laser power. The experimental verifica- tion of the angles of incidence � predicted by the geometri- cal model in [13. ] is difficult to realize with conventional in situ diagnostics. In [13. ] and [15. ] the predicted angles were therefore compared with the ones extracted from lon- gitudinal sections taken from “frozen” cuts. A “frozen” cut is obtained by abruptly interrupting the cutting process by simultaneously switching off the laser beam and the cutting gas. While [13. ] reported a good agreement of thus deter- mined angles with the predicted global angle of incidence, large deviations were reported in [15. ]. The different find- ings could result from the deviations of the average angle measured from a “frozen” cut from the actual average angle of the front during the cutting process. In previous work [16. ] we used high-speed X-ray diagnostics to observe the geometry of the cutting front during the cutting process and showed that the angle of incidence is subject to significant fluctuations. The measured angles of a “frozen” cut therefore can only represent a specific snap-shot and do not provide evidence of the prevailing average situation. To experimen- tally validate the theoretical model from [13. ], the in-pro- cess measurement of the global angle of incidence on the cutting front is still pending. The present paper therefore presents a space- and time- resolved experimental X-ray analysis of the influence of different shapes of the laser beam on the geometry of the cutting front and the kerf in order to validate the geometric model of the cut front introduced by Mahrle et al. [13. ]. An analytical equation to predict the cross-sectional area of the cutting kerf is presented and experimentally validated. The analytical equations describing the global angle of incidence on the front and the cross-sectional area of the kerf were combined with the energy balance to predict the maximum cutting speeds for different shapes of the laser beam, laser power, and sheet thicknesses. The overall relation between the shape of the laser beam, the absorptivity, and the cross- sectional area of the kerf was determined experimentally and by analytical equations. An optimization strategy to increase the maximum cutting was derived from the results. The theoretical findings are in good agreement with experi- mental results. 2 Theory A simple analytical approach to estimate the maximum cut- ting speed for laser beam fusion cutting with a given laser power P is to consider the energy balance [9. , 10. ], where the sum of the process power PP required to melt the mate- rial and the power PL which is lost by heat conduction, yields the required overall absorbed power of the process, where P is the incident laser power and ηA is the overall absorptance of the radiation at the cutting front in the kerf. Approximating the cutting front by an inclined plane and neglecting multiple reflections in the kerf, the absorptance �A equals the absorptivity A = A(�, �) , which depends on the applied wavelength � , the angle of incidence � , the processed material and its temperature and is essen- tially determined by the Fresnel equations [14. ]. Without evaporation of the material, the process power is given by where F is the cross-sectional area of the cutting kerf, � the density of the cut material, cp its specific heat capacity, ΔTP the difference between the temperature of the expelled melt and the ambient temperature, and hS the latent heat of fusion. The power which is lost by heat conduction depends on the thermal properties of the material and the cutting speed and may be expressed by the cutting depth s of the workpiece and the depth-specific power loss PL,s [3. ]. The depth-specific loss PL,s can be derived by solving the heat conduction equation for a constant power distribution along the beam axis at the cutting front, as reported by Petring [3. ]. With a given maxi- mum available laser power P and the above assumptions, the maximum cutting speed is obtained by inserting Eq. (2) and Eq. (3) in Eq. (1) with �A = A and solving for v. Figure 1 a) sketches the geometrical conditions in the kerf with the average global inclination of the cutting front, as indicated by the blue dashed line. The caustic of the beam is sketched by the red lines, where df is the diameter of the beam waist. According to the assumption, the length of the cutting front equals the extent of the beam’s cross-section in the direction of the feed at maximum cutting speed. With the definition of the inclination angle of the cutting front as shown in the figure, the average inclination angle �@vmax (1)Pabs = P ∙ �A = PP + PL (2)PP = F ∙ v ∙ � ⋅ ( cpΔTP + hS ) , (3)PL = PL,s ⋅ s, (4)vmax = P ∙ A − PL,s ⋅ s F ∙ � ⋅ ( cpΔTP + hS ) 1528 The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 also equals the average angle of incidence of the beam on the cutting front at maximum cutting speed. Figure 1 b) illus- trates the conditions with a beam oscillation in the direction of the feed, which was shown to reduce the global angle of incidence �@vmax [13. , 15. ]. According to Mahrle et al. [13. ], a beam oscillation with the amplitude of ±a results in an global angle of incidence where s is the thickness of the sheet and is the beam diameter at the distance z from the top surface of the sheet. The beam waist with its diameter df is located at z = zf . The Rayleigh-length is zR. The averaged angle of incidence on the cutting front can be used to calculate the absorptivity A = A(�) using the Fresnel equations [14. ]. Assuming an unpolarized beam with a wavelength of 1030nm , Fig. 2 shows the absorptivity as a function of the angle of incidence on a surface of liquid iron with the complex refractive index nc = 3.6 − 5.0i [18. , 18. ]. (5)�@vmax = arctan ( s d(z=0) 2 + d(z=s) 2 + 2a ) , (6)d(z) = df ∙ √√√√ 1 + (z − zf ) 2 z2 R For angles of incidence � < 86◦ , the absorptivity ranges between 31 and 41% and decreases significantly for angles exceeding 86◦ . With the geometric assumption that the length of the cutting front equals the extent of the beam’s cross-section in the direction of the feed at maximum cutting speed, it follows from Eq. (5) that an increased extent of the beam’s cross-section is associated with a reduced angle of incidence � and hence an increased Fig. 1 The global angle �@vmax describes the average inclination of the cutting front and equals the averaged angle of incidence of the beam on the cutting front at maximum cutting speed; a) constant feed, b) beam oscil- lation in feed direction with an amplitude of ±a . The waist of the beam is located at the distance zf from the top surface of the cut sheet. In the right column the cross-sectional area F of the cutting kerf is shown by the orange area Fig. 2 Absorptivity as a function of the angle of incidence assuming liquid iron and unpolarized radiation with a wavelength of 1030 nm as calculated using Fresnel’s equations 1529The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 absorptivity A as long as � exceeds approximately 80°. With an increased absorptivity A , the maximum cutting speed vmax is increased according to Eq. (4). Considering the geometrical conditions of the beam’s cross-section in the plane normal to the feed seen on the right of Fig. 1, we make the basic assumption that the cross- sectional area of the cutting kerf corresponds to the area covered by the caustic of the beam in the y–z plane from the top to the bottom surface of the sample. An increased extent of the beam’s cross-section in the y–z plane therefore results in an increased cross-sectional area of the cutting kerf F , which reduces the maximum cutting speed vmax according to Eq. (4). It follows from Eq. (5) and Eq. (7) that the diameter of the beam d(z) and the oscillation amplitude a significantly influence both the cross-sectional area of the kerf F and the absorptivity A which results from the changed inclina- tion angle of the front. According to Eq. (4), both quantities influence the maximum cutting speed that can be achieved with a given laser power P. (7)F = z=s ∫ z=0 ⎛ ⎜⎜⎝ df ∙ ���� 1 + (z − zf ) 2 z2 R ⎞ ⎟⎟⎠ dz Fig. 3 Sketch of the experimental setup Fig. 4 a) Images of the inten- sity distribution at different z-positions of the beam shapes “top hat,” “annular,” and “line scan” and b) the time-averaged intensity distributions in the focal plane. The beam shapes “top hat” and “line scan” were measured with a Primes HighPower MicroSpotMonitor. The beam shape “annular” was measured with a Primes Focus Monitor 1530 The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 3 Experimental setup Fusion cutting of stainless steel 1.4301 was investigated using a Trudisk8001 laser (by Trumpf) with a wavelength of 1.03 µm in combination with different cutting heads from Precitec in order to verify the prediction of the model given by Eqs. (4), (5), and (7). A sketch of the experimental setup is shown in Fig. 3. The origin of the Cartesian coordinate system was set at the intersection point of the center line of the nozzle and the surface of the sample. The feed was achieved by moving the sample in x-direction. The X-ray imaging system consists of a tube emitting X-rays (green) that transirradiate the sample and of an imaging system (purple), which is composed of a scintil- lator, an image intensifier, and a high-speed camera, as described in [18. ]. The high-speed camera was used to cap- ture images of the process at a frame rate of 1000 fps with a spatial resolution of 37 pixels/mm. The acceleration volt- age of the X-ray tube was set to 140 kV at a tube power of 90 W. The width of the samples in y-direction was chosen to be 6 mm to allow for a suitable X-ray imaging. The X-ray videos were post-processed with a flat-field correction and Kalman filtering in order to enhance the image contrast and to reduce the noise [18. ]. Figure 4 a) shows the intensity distributions of the three applied laser beams at different z-positions and Fig. 4 b) Fig. 5 a) Tveraged X-ray image. b) Isometric view of the 3D reconstruction of the kerf and the cutting front. c) Front view of the 3D reconstruc- tion of the cutting front; “top hat”: P = 6kW,v = 1.3m∕min, p = 9bar, zf = 5.5mm, ΔzNoz = 0.5mm, dNoz = 5.0mm, s = 10mm 1531The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 shows the respective intensity distributions in the focal plane. The waist diameter df , the Rayleigh-length zr , and the beam propagation factor M2 were calculated using the 2nd order moments. The beam shape which we refer to as “top hat” in the following was obtained by connecting a ProCutter2.0 cutting head to a beam-delivery fiber with a core diameter of 100 μm and by focusing the beam to a waist diameter of approximately df ≈ 166�m , a Rayleigh- length of about zr ≈ 1.6mm , and a beam propagation factor of M2 ≈ 13 as measured with a Primes HighPower MicroS- potMonitor. Due to the fiber-optic beam delivery, the beam exhibited a top-hat-shaped intensity distribution at the waist. To obtain the beam shape referred to as “annular,” a ProCutterEdgeTec cutting head was connected to a beam- delivery fiber with a core diameter of 100 μm and the beam was focused to a waist with an outer diameter of approxi- mately df ≈ 900�m with an annular-shaped intensity distri- bution, a Rayleigh-length of about zr ≈ 8.3mm , and a beam propagation factor of M2 ≈ 74 as measured with a Primes Focus Monitor. To generate what is referred to as “line scan” in the following, a LightCutter cutting head was connected to a beam delivery fiber with a core diameter of 100 µm and the beam was focused to a waist diameter of approximately df ≈ 208�m , a Rayleigh-length of about zr ≈ 2.5mm , and a beam propagation factor of M2 ≈ 13 as measured with a Primes HighPower MicroSpotMonitor. The cutting head was connected to a 2D-galvanometer scanner, which oscillates the beam in x-direction with a frequency of 800 Hz and an amplitude of a ≈ 100�m , as indicated by the black arrow in Fig. 4. The influence that the shape of the laser beam has on the geometries of the cutting front and the kerf was investigated by studying cuts with a length of 40 mm. The laser power P, the distance ΔzNoz between the nozzle and the sample’s surface, the diameter dNoz of the nozzle, the cutting speed v, the pressure p of the nitrogen processing gas, and the waist position zf were adapted in order to achieve stable cutting conditions. Figure 5 a) shows a time-averaged X-ray image of the cut- ting process in a s = 10 mm thick sheet of stainless steel. The gray-scale values in the image represent the local transmit- tance of the X-ray radiation through the sample. In order to avoid overexposure of the camera above and below the sam- ple, lead apertures were positioned at the upper and lower edge of the sample. As a result, only the central 8.5 mm of a 10-mm-thick sample can be observed in z-direction. The top and bottom 0.75 mm of the sample are not visible. A clear contrast between the solid sample material (dark, high absorption of X-rays) and the cutting kerf (bright, low absorption of X-rays) is visible in the X-ray image of Fig. 5 a). The contour of the center line of the cutting front along the cut- ting depth is highlighted by the white dotted line. Connecting the front of the kerf at the top surface with the one at the bottom surface of the sample (red dashed line) delivers the experimen- tally determined angle �exp which corresponds to the average angle of incidence of the radiation on the cutting front. The gray-scale value of each pixel contains information about the thickness of the irradiated material and therefore of the width of the cutting kerf. Figure 5 b) and c) show the 3D geometry of the cutting kerf that was reconstructed from this information. The reconstruction method is based on the Lambert–Beer-Law and assumes a mirror symmetrical geom- etry of the kerf to the x–z plane, as described in [18. ]. The cross-sectional area of the front view in the y–z plane is the experimentally determined cross-sectional area Fexp∗ of the kerf, as highlighted by the black dotted line in Fig. 5 c). The upper and lower areas Flead,t and Flead,b (gray areas in Fig. 5 c) of the kerf front could not be reconstructed from the X-ray images due to the lead apertures mentioned above. Instead, they were assumed to have the same width as the kerf that is visible next to the edge to the apertures and extend to the very top and bottom of the cut sample. The sum then corresponds to the experimentally determined cross- sectional area with which the cutting front moves through the sheet and is highlighted by the green dotted line in Fig. 5 c). 4 Adjusting the cross‑sectional area of the cutting kerf by means of beam shaping The basic assumption in the prediction of the cross-sec- tional area of the cutting kerf is that it corresponds to the cross-sectional area covered by the caustic of the beam in the y–z plane from the top to the bottom surface of the sample, as described by Eq. (7). In the following we compare the calculated area F of the beam caustic with the measured cross-sectional area Fexp of the cutting kerf for different shapes of the laser beam and different positions of the beam waist to validate the assumption. 4.1 Kerf shaping by adjusting the shape of the laser beam Figure 6 shows the caustics of the beam shapes “top hat,” “annular,” and “line scan” in the y–z plane, which were cal- culated with Eq. (6). The cross-sectional area F for the beam shapes is marked by the red, yellow, and green area. The results confirm that the width of the “annular” beam in the y–z plane is larger than the ones obtained with the “top hat” beam and the “line scan”. Figure 7 shows the experimentally determined cross-sectional areas Fexp (dots (8)Fexp = Fexp∗ + Flead,t + Flead,b 1532 The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 and crosses) of the reconstructed cutting kerfs as a func- tion of the cutting speed. The cross-sectional areas F of the caustics, as calculated by Eq. (7), are represented by the horizontal lines. The scatter band (shaded area) of the cal- culated cross-sectional area F represents a deviation of the measured beam propagation factor M2 by ±15% . The data points of the experimentally determined cross-sectional are Fexp,top hat of the “top hat” beam show the average value from three analyzed cutting processes and the lengths of the error bars indicates the range between the minimum and maximum measured values. The data points of the experi- mentally determined cross-sectional area Fexp,annular of the “annular” beam give the value of one single analyzed cut- ting process. The data points with the cross represent the cut limit vmax . The samples were not cut successfully when the cutting speed was increased further, which is referred to a loss of cut. The cross-sectional area Fannular of the “annular” beam in the y–z plane and the experimentally determined cross-sec- tional area Fexp,annular are larger than the one of the “top hat” beam. The results show that the experimentally determined cross-sectional areas Fexp slightly decrease with increasing cutting speed and increase with an increased extent of the cross-sectional area of the beam caustic. The calculated cross-sectional areas F of the “top hat” beam are reasonably consistent with the measured ones especially at higher cut- ting speeds. In the case of the “annular” beam, the deviation between the measured area Fexp and the predicted area F at maximum cutting speed is 0.8mm2 , which is a difference of 10%. Hence, the reasonable agreement between the meas- ured cross-sectional areas Fexp of the cutting kerf and the area F covered by the caustic of the beams in the y–z plane given by Eq. (7) confirms the validity of the assumptions. An analysis of the cross-sectional area for the beam shape “line scan” was omitted because its cross-sectional area of the caustic in the y–z plane is not influenced by the beam oscillation in the direction of the feed. Fig. 6 The caustics of the beams “top hat,” “annular,” and “line scan” in the y–z plane (solid lines). “top hat”: df ≈ 166�m , zr ≈ 1.6mm , M2 ≈ 13 , zf = 5.5mm ; “annular”: df ≈ 900�m , zr ≈ 8.3mm , M2 ≈ 74 , zf = 1.0mm ; “line scan”: df ≈ 208�m , zr ≈ 2.5mm , M2 ≈ 13 , zf = 6.0mm Fig. 7 Experimentally determined cross-sectional areas Fexp of the cutting kerf as a function of the cutting speed (dots and crosses) compared to the areas covered by the caustic of the beam in the y–z plane from the bottom to the top surface of the sample (solid lines); “top hat”: P = 6kW,p = 9bar, zf = 5.5mm, ΔzNoz = 0.5mm, dNoz = 5.0mm, s = 10mm ; “annular”:P = 8kW, p = 12bar, zf = 1.0mm, ΔzNoz = 1.0mm, dNoz = 2.5mm, s = 10mm Fig. 8 Experimentally determined cross-sectional area Fexp of the cut- ting kerf (dots) and the calculated cross-sectional area F of the caus- tic of the “top hat” beam (solid line) in the y–z plane as a function of the waist position zf ; “top hat”: P = 6kW,v = 1.3m∕min, p = 9bar, ΔzNoz = 0.5mm, dNoz = 5.0mm,s = 10mm 1533The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 4.2 Kerf shaping by adjusting the waist position Figure 8 compares the experimentally determined cross-sec- tional areas Fexp of the cutting kerf (dots) with the calculated cross-sectional areas F of the caustic of the “top hat” beam in the y–z plane (solid line) as a function of the position zf of the waist. The scatter band (shaded area) of the calculated cross-sectional area F represents a deviation of the measured beam propagation factor M2 by ±15% . The cutting speed was v = 1.3m∕min. Figure 8 shows that the experimental results agree rea- sonably well with the calculated cross-sectional areas when varying the waist position. The experimentally determined areas Fexp are systematically larger than the ones calculated by Eq. (7), with a maximum deviation of less than 25%. This can be attributed to the fact that these experiments were not performed at the maximum possible cutting speed vmax as assumed by the model. In order to ensure a successful cut in the full range of the different waist positions, the cut- ting speed was reduced to v = 0.46 ∙ vmax , which leads to a widening of the cutting kerf. The good agreement between the measured and calculated results at maximum cutting speed shown in Fig. 7 and the reasonable agreement of the measured and calculated results with varying waist posi- tion shown in Fig. 8 prove that the calculation of the cross- sectional area of the caustic with Eq. (7) can be used to estimate the cross-sectional area of the cutting kerf for dif- ferent shapes of the laser beam and waist positions at the maximum cutting speed. 5 Adjusting the angle of incidence by means of beam shaping The basic assumption for the prediction of the average angle of incidence on the cutting front is that the length of the cut- ting front equals the extent of the beam’s cross-section in the direction of the feed at maximum cutting speed. According to the model presented in Sect. 2, the different shapes of the beam allow to influence the average angle of incidence of the radiation on the cutting front. In the following we compare the calculated angle of incidence �@vmax with the average inclination angle of the contours of the cutting fronts for the beam shape “annular.” Furthermore, we compare the calcu- lated angle of incidence �@vmax with the measured angle of incidence �exp of the cutting front for different cutting speeds and shapes of the laser beam to validate the assumption. In the case of the “top hat” and the “annular” beam, the caustic in the x–z plane may be calculated using Eq. (6). Fol- lowing the abovementioned model introduced by Mahrle et al. [13. ], the caustic resulting from the “line scan” is approxi- mated by adding the amplitude 2a to the diameter d of the oscillating beam. Figure 9 shows the thus determined caustics in the x–z plane. The linear interpolation of the cutting front with the global angle of incidence � given by Eq. (5) obtained for the “annular” beam is shown by the yellow dashed line. Figure 10 compares the shape of the center line in the x–z plane at y = 0 on the cutting fronts for different cutting speeds, which were determined from X-ray images in case of cutting with the “annular” beam with the corresponding beam caustic. At a cutting speed of v = 2.5m∕min , the 10-mm-thick sample could not be cut through completely. The cut limit Fig. 9 Caustic of the beam shapes “top hat,” “annular,” and “line scan” in the x–z plane (solid lines); “top hat”: df ≈ 166�m , zr ≈ 1.6mm , M2 ≈ 13 , zf = 5.5mm ; “annular”: df ≈ 900�m , zr ≈ 8.3mm , M2 = 74 , zf = 1.0mm ; “line scan”: df ≈ 208�m , zr ≈ 2.5mm , M2 ≈ 13 , zf = 6.0mm a = 0.1mm Fig. 10 Contours of the center line in the x–z plane of the cutting fronts for different cutting speeds; “annular”: P = 8kW, p = 12bar, zf = 1.0mm, ΔzNoz = 1.0mm, dNoz = 2.5mm, s = 10mm , df ≈ 900�m , zr ≈ 8.3mm , M2 ≈ 74 1534 The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 was reached for at a cutting speed of vmax = 2.0m∕min . The linear interpolation between the front of the kerf at the top surface with the one at the bottom surface of the sample at maximum cutting speed (yellow dashed line) delivers the experimentally determined global angle of incidence �exp on the cutting front. This linear interpolation is congruent with the linear interpolation calculated from the caustic as described by Eq. (5) and shown in Fig. 9, which results in a good agreement of the calculated and experimentally determined global angle of incidence � . The results prove that when the cutting process is at maximum cutting speed, the length of the cutting front in x-direction equals approxi- mately the extent of the beam’s cross-section. Figure 11 compares the experimentally determined global angle of incidence �exp on the cutting front (dots and crosses) as a function of the cutting speed with the global angles of incidence �@vmax (horizontal lines) at the maximum cutting speed as calculated with Eq. (5). The scatter band (shaded area) of the calculated global angle of incidence �@vmax rep- resents a deviation of the measured beam propagation factor M2 by ±15% . The data points obtained with the “top hat” beam represent the average value of three analyzed cutting processes and the lengths of the error bars correspond to the range between the minimum and maximum measured values. The data points obtained in the case of the “annular” beam and the “line scan” represent the value of one single analyzed cutting process. It can be seen that the experimentally determined global angle of incidence �exp of the laser beam on the cutting front decreases as expected with increasing cutting speed. The minimum angle of incidence furthermore decreases with increasing width of the beam’s cross-sectional area in the x–z plane. The angle of incidence �@vmax calculated for the maximum possible cutting speed is viably consistent with the measured ones (crosses) in the case of the “top hat” and “annular” beams. In the case of the “line scan” the two angles deviate by 1.2◦ , which is still considered to be accept- able in view of the simplified geometrical approach adopted from [13. ]. In order to predict the maximum cutting speed vmax using Eq. (4), the absorptivity A was calculated with the Fresnel equations, as shown in Fig. 2. Figure 12 shows the calculated absorptivity resulting from the experimentally determined angle of incidence A(�exp) (dots and crosses) as a function of the cutting speed and the one corresponding to the theoreti- cally predicted angle of incidence A(�@vmax) at maximum cut- ting speed (horizontal lines). The scatter band (shaded area) of the calculated absorptivity A(�@vmax) represents a deviation of the measured beam propagation factor M2 by ±15%. As expected, the absorptivity A(�exp) increases with increasing cutting speed. In addition, the absorptivity is shown to increase when the extent of the cross-sectional area of the beam in the x–z-plane beam is enlarged by chang- ing from the “top hat” beam to the “line scan” and to the “annular beam” (cf. Figure 9). Using the experimentally determined angle of incidence �exp , the absorptivity at maxi- mum cutting speed is found to reach a value of 38.1% for the Fig. 11 Experimentally determined global angle of incidence �exp on the cutting front (dots and crosses) as a function of the cutting speed and global angle of incidence �@vmax at the maximum cutting speed (horizontal lines) as given by Eq. (5); “top hat”: P = 6kW,p = 9bar, zf = 5.5mm, ΔzNoz = 0.5mm, dNoz = 5.0mm, s = 10mm ; “annular”:P = 8kW, p = 12bar, zf = 1.0mm, ΔzNoz = 1.0mm, dNoz = 2.5mm, s = 10mm ; “line scan”: P = 6kW,p = 15bar, zf = 6.0mm, ΔzNoz = 0.7mm, dNoz = 5.0mm ,a = 0.1mm, s = 10mm Fig. 12 Absorptivity resulting at the experimentally determined angles of inci- dence A(�exp) (dots and crosses) as a function of the cutting speed and the one obtained for the theoretical angle of incidence A(�@vmax) at maximum cutting speed (horizontal lines) as given by Eq. (5); “top hat”: P = 6kW,p = 9bar, zf = 5.5mm, ΔzNoz = 0.5mm, dNoz = 5.0mm, s = 10mm ; “annular”:P = 8kW, p = 12bar, zf = 1.0mm, ΔzNoz = 1.0mm, dNoz = 2.5mm, s = 10mm ; “line scan”: P = 6kW ,p = 15bar, zf = 6.0mm, ΔzNoz = 0.7mm, dNoz = 5.0mm,a = 0.1mm, s = 10mm 1535The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 annular beam whereas it amounts to only 26.7% in the case of the “top hat” beam. The maximum absorptivity achieved with the “line scan” is 34.4%. The absorptivity of the global angle of incidence A(�@vmax) calculated for the maximum cutting speed (horizontal lines) is viably consistent with the ones obtained with the measured angles (crosses). 6 Determination of the maximum cutting speed In order to predict the attainable maximum cutting speeds in laser beam cutting with different shapes of the laser beam according to Eq. (4), multiple reflections are not considered and the amount of power loss by heat conduction PL accord- ing to Eq. (3) was estimated by solving the heat conduction equation, as reported by Petring [3. ]. The power which is lost by heat conduction in general depends on the cutting speed and the width of the kerf. The solution of the heat conduc- tion equation however indicates that the power loss PL can be assumed to be independent of the cutting speed and of the width of the kerf for the parameter ranges used in the present study. Note that this assumption is not valid for very low cutting speeds [3. , 10. ]. For a given cutting speed of v = 2.0m∕min , a kerf width of |y| = 0.5mm and the stain- less steel used in the experiments, the value of the depth- specific power loss was found to be PL,s = 30 W mm . For the sake of simplicity this value was used to calculate the maximum cutting speed according to Eq. (4) for all of the three investi- gated shapes of the laser beam. The temperature of the melt is assumed to be equal to the melting temperature and local overheating of the melt above the melting temperature is not considered. The material properties are shown in Table 1. Inserting the material properties of stainless steel as listed in Table 1, Eqs. (3), (5), and (7) and the absorptiv- ity A from Eq. (4) allows the calculation of the maximum cutting speed vmax . Figure 13 compares the experimentally determined maximum cutting speed vmax,exp (crosses) with the predicted maximum cutting speed vmax (solid lines) as a function of the laser power P for different thick- nesses s of the cut sheets, when cutting with a conven- tional top-hat-shaped intensity distribution. The scatter band (shaded area) of the calculated predicted maximum cutting speed vmax represents a deviation of the measured beam propagation factor M2 by ±15% . The process param- eters are given in Table 2. As expected, the results show that the maximum cut- ting speed increases with increasing laser power P and decreases with increasing thickness s of the sheet. Increas- ing the power from 3kW to 8kW increases the maximum cutting speed vmax,exp from 1.0m∕min to 4.1m∕min , when cutting a s = 10mm thick sheet. The predicted cutting speeds vmax at the powers of 3kW and 8kW deviate by 0.1m∕min from the experimentally determined maximum cutting speed. For a sheet thickness of s = 30mm and a power of P = 12kW , the deviation is 0.02m∕min , which is a difference of 4%. Table 1 Material properties of stainless steel 1.4301 Density � in kg m3 7880 Specific heat capacity cp in J kg∙K 480 Process temperature TP in K 1683 Latent heat of fusion hS in kJ kg 290 Heat conductivity �th in W m⋅K 40 Fig. 13 Experimentally determined maximum cutting speed vmax,exp (crosses) and calculated maximum cutting speed vmax (solid lines) as a function of the laser power P for different sheet thicknesses s and a “top hat” beam shape. The process parameters are given in Table 2 Table 2 Process parameters for different sheet thicknesses stainless steel 1.4301 Sheet thickness s in mm 6 10 20 25 30 Waist diameter df in µm 166 166 166 200 200 Waist positio zf in mm 2.0 5.5 10.5 12.0 22.0 Gas pressure p in bar 18 9 20 18 25 Nozzle distance ΔzNoz in mm 1.0 0.5 0.5 0.3 0.2 Nozzle diameter dNoz in mm 2 5 5 5 5 1536 The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 The good agreement between the measured and cal- culated results proves that the calculated cross-sectional area F , angle of incidence �@vmax and hence the absorptiv- ity A(�@vmax) can be used to predict the maximum cutting speed vmax according to Eq. (4) for different laser power and sheet thicknesses. 7 Optimizing the maximum cutting speed with adjusting the shape of the beam Increasing the diameter of the beam decreases the angle of incidence which leads to an increased absorptivity for angles of incidence ranging between 90° and approxi- mately 80°. Increasing the diameter of the beam however also enlarges the cross-sectional area of the kerf, which has a contrary impact on the maximum cutting speed. While an increased absorptivity increases the maximum cutting speed according to Eq. (4), the cutting speed is reduced by an enlarged cross-sectional area of the kerf. Figure 14 shows the comparison of the experimentally determined maximum cutting speed vmax,exp (crosses) and predicted maximum cutting speed vmax (solid lines) as a function of the laser power P for different shapes of the laser beam. The scatter band (shaded area) of the predicted maximum cutting speed vmax represents a deviation of the measured beam propagation factor M2 by ±15%. The results show that at the same laser power the beam shape “line scan” allowed to attain the highest maximum cutting speed. As shown in Fig. 12 and Fig. 7, the beam shape “annular” leads to the highest absorptivity but also to a significantly enlarged cross-sectional area of the cut- ting kerf as compared to the results obtained with the “top hat” beam and the “line scan.” This results in a reduced maximum cutting speed, as predicted by Eq. (4). The asym- metrical “line scan” has the advantage of achieving a larger absorptivity than the “top hat” beam with approximately the same cross-sectional area of the kerf. The highest maximum cutting speed per unit power was therefore achieved with the “line scan.” The results show that the “line scan” combines the benefits of an increased absorptivity A and low cross- sectional area F and thus provides highest cutting speeds. 8 Conclusion In summary we presented a space- and time-resolved experi- mental X-ray analysis of the influence of different shapes of the laser beam on the geometry of the cutting front and the kerf in order to validate the geometric model of the cut front introduced by Mahrle et al. [13. ]. From the results, one can draw the following conclusions: • The length of the cutting front in the direction of the feed equals approximately the extent of the beam’s cross- section at maximum cutting speed. • The absorptivity increases when the extension of the beam’s cross-section in the direction of the feed is enlarged. • Reducing the width of the beam in the transversal direc- tion reduces the cross-sectional area of the cutting kerf. • A “line scan” in the direction of the feed yields a high absorptivity while keeping the cross-sectional area of the kerf small, which results in an increased cutting speed. These relationships were modelled by analytical equa- tions based on the geometry of the caustic of the laser beam and considering the energy balance. The good agreement between the measured and calculated results proves that the simple analytical model can be used to predict the maximum cutting speed for different shapes of the laser beam, laser power, and sheet thicknesses. The knowledge of these relationships allows for an optimized absorbed power and cross-sectional area of the kerf in laser fusion cutting Fig. 14 Experimentally determined maximum cutting speed vmax,exp (crosses) and calculated maximum cutting speed vmax (solid lines) as a function of the laser power P for different beam shapes; “top hat”: P = 6kW,p = 9bar, zf = 5.5mm, ΔzNoz = 0.5mm, dNoz = 5.0mm, s = 10mm ; “annular”:P = 8kW, p = 12bar, zf = 1.0mm, ΔzNoz = 1.0mm, dNoz = 2.5mm, s = 10mm ; “line scan”: P = 6kW,p = 15bar, zf = 6.0mm, ΔzNoz = 0.7mm, dNoz = 5.0mm ,a = 0.1mm, s = 10mm 1537The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 1 3 by adapting the beam shape. This allows to increase the productivity of a laser cutting machine by increasing the maximum cutting speed. The derived model was used to demonstrate a process optimization strategy in which the laser beam is oscillated in the direction of the feed. This optimization strategy was experimentally validated for a wide range of different laser powers. Author contribution All authors contributed to the study conception. Methodology, data curation, conceptualization, and writing—original draft preparation were performed by Jannik Lind. Supervision and writing—reviewing were performed by Christian Hagenlocher, Niklas Weckenmann, David Blazquez-Sanchez, Rudolf Weber, and Thomas Graf. All authors read and approved the final manuscript. Funding Open Access funding enabled and organized by Projekt DEAL. This research was supported by Precitec GmbH & Co. KG. The laser beam source TruDisk8001 (DFG object number:625617) was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – INST 41/990-1FUGG. Declarations Ethical approval There are no ethical issues to declare. Consent to participate All the authors participate in the research work of this paper. Consent for publication All authors agree to publish the research results in this paper. Competing interests The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. References 1. Wandera C, Kujanpää V (2011) Optimization of parameters for fibre laser cutting of a 10 mm stainless steel plate. Proc Inst Mech Eng B J Eng Manuf 225(5):641–649 2. Wandera C, Salminen A, Kujanpaa V (2009) Inert gas cutting of thick-section stainless steel and medium-section aluminum using a high power fiber laser. J Laser Appl 21(3):154–161 3. Petring D (1995) Anwendungsorientierte Modellierung des Laser- strahlschneidens zur rechnergestützten Prozeßoptimierung. Shaker, Aachen  4. Petring D, Abels P, Beyer E (1988) Absorption distribution on idealized cutting front geometries and its significance for laser beam cutting. Proceedings of SPIE - The International Society for Optical Engineering 5. Duan J, Man HC, Yue TM (2001) Modelling the laser fusion cut- ting process: I. Mathematical modelling of the cut kerf geom- etry for laser fusion cutting of thick metal. J Phys D Appl Phys 34(14):2127–2134 6. Mahrle A, Lütke M, Beyer E (2010) Fibre laser cutting: beam absorption characteristics and gas-free remote cutting. Proc Inst Mech Eng C J Mech Eng Sci 224(5):1007–1018 7. Tamsaout T, Amara EH, Bouabdallah A (2020) Numerical approach for hydrodynamic behavior in the kerf with a quasi- complete model of the laser cutting process. J Opt Soc Am A Opt Image Sci Vis 37(11):C86–C94 8. Pocorni J, Powell J, Frostevarg J, Kaplan AF (2018) The geometry of the cutting front created by fibre and CO2 lasers when profiling stainless steel under standard commercial conditions. Opt Laser Technol 103:318–326 9. Steen WM, Mazumder J (2010) Laser Material Processing. Springer, London, London 10. Hügel H, Graf T (2009) Laser in der Fertigung. Vieweg+Teubner, Wiesbaden 11. Vasileska E, Pacher M, Previtali B (2022) In-line monitor- ing of focus shift by kerf width detection with coaxial ther- mal imaging during laser cutting. Int J Adv Manuf Technol 118(7–8):2587–2600 12. Goppold C, Zenger K, Herwig P, Wetzig A, Mahrle A, Beyer E (2014) Experimental analysis for improvements of process effi- ciency and cut edge quality of fusion cutting with 1μm laser radia- tion. Phys Procedia 56:892–900 13. Mahrle A, Beyer E (2009) Theoretical aspects of fibre laser cut- ting. J Phys D: Appl Phys 42(17):175507 14. Demtröder W (2016) Experimentalphysik, Atome, Moleküle und Festkörper. (Band 3) 15. Scintilla LD, Tricarico L, Mahrle A, Wetzig A, Beyer E (2011) Experimental investigation on the cut front geometry in the inert gas laser fusion cutting with disk and CO2 lasers, Proc. of the ICALEO 40–49 16. Lind J, Hagenlocher C, Blazquez-Sanchez D, Hummel M, Olow- insky A, Weber R, Graf T (2022) Influence of the laser cutting front geometry on the striation formation analysed with high- speed synchrotron X-ray imaging. IOP Conference Series: Mate- rials Science and Engineering 17. Seibold G, Dausinger F, Hugel H (1999) Wavelength and tempera- ture dependence of laser radiation absorption of solid and liquid metals. Proceeding of the 7th NOLAMP Confernce, Finnland, 526–535  18. Dausinger F (1995) Strahlwerkzeug Lase: Energieeinkopplung und Prozesseffektivität. Stuttgart 19. Abt F, Boley M, Weber R, Graf T, Popko G, Nau S (2011) Novel X-ray system for in-situ diagnostics of laser based processes – first experimental results. Phys Procedia 2011:761–770 20. Kalman RE (1960) A new approach to linear filtering and predic- tion problems. J Basic Eng 82(1):35–45 21. Lind J, Fetzer F, Hagenlocher C, Blazquez-Sanchez D, Weber R, Graf T (2020) Transition from stable laser fusion cutting condi- tions to incomplete cutting analysed with high-speed X-ray imag- ing. J Manuf Process 60:470–480 Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 1538 The International Journal of Advanced Manufacturing Technology (2023) 126:1527–1538 http://creativecommons.org/licenses/by/4.0/ Adjustment of the geometries of the cutting front and the kerf by means of beam shaping to maximize the speed of laser cutting Abstract 1 Introduction 2 Theory 3 Experimental setup 4 Adjusting the cross-sectional area of the cutting kerf by means of beam shaping 4.1 Kerf shaping by adjusting the shape of the laser beam 4.2 Kerf shaping by adjusting the waist position 5 Adjusting the angle of incidence by means of beam shaping 6 Determination of the maximum cutting speed 7 Optimizing the maximum cutting speed with adjusting the shape of the beam 8 Conclusion References