Adjustment of the geometries of the cutting front and the kerf by means of beam shaping to maximize the speed of laser cutting

The shape of the laser beam used for fusion cutting significantly influences the geometry of both the cutting front and the cutting 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 different 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 normal 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.


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 associated 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 of the kerf, which has a contrary impact on the maximum cutting speed. A suitable analytical and experimentally verified model for the resulting cross-sectional area of the cutting 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 verification of the angles of incidence predicted by the geometrical 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 longitudinal 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 determined angles with the predicted global angle of incidence, large deviations were reported in [15. ]. The different findings 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 experimentally validate the theoretical model from [13. ], the in-process measurement of the global angle of incidence on the cutting front is still pending.
The present paper therefore presents a space-and timeresolved 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 crosssectional 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 experimental results.

Theory
A simple analytical approach to estimate the maximum cutting 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 P P required to melt the material and the power P L which is lost by heat conduction, yields the required overall absorbed power of the process, where P is the incident laser power and η A 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 essentially 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, c p its specific heat capacity, ΔT P the difference between the temperature of the expelled melt and the ambient temperature, and h S 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 P L,s [3. ]. The depth-specific loss P L,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 maximum 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 d f 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 also equals the average angle of incidence of the beam on the cutting front at maximum cutting speed. Figure 1 b) illustrates 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 d f is located at z = z f . The Rayleigh-length is z R . 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 n c = 3.6 − 5.0i [18. , 18. ].
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 oscillation in feed direction with an amplitude of ±a . The waist of the beam is located at the distance z f 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 With an increased absorptivity A , the maximum cutting speed v max 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 crosssectional 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 v max 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 inclination 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.

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 scintillator, an image intensifier, and a high-speed camera, as described in [18. ]. The high-speed camera was used to capture images of the process at a frame rate of 1000 fps with a spatial resolution of 37 pixels/mm. The acceleration voltage 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. ].
shows the respective intensity distributions in the focal plane. The waist diameter d f , the Rayleigh-length z r , and the beam propagation factor M 2 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 d f ≈ 166 m , a Rayleighlength of about z r ≈ 1.6mm , and a beam propagation factor of M 2 ≈ 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 beamdelivery fiber with a core diameter of 100 μm and the beam was focused to a waist with an outer diameter of approximately d f ≈ 900 m with an annular-shaped intensity distribution, a Rayleigh-length of about z r ≈ 8.3mm , and a beam propagation factor of M 2 ≈ 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 d f ≈ 208 m , a Rayleigh-length of about z r ≈ 2.5mm , and a beam propagation factor of M 2 ≈ 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 Δz Noz between the nozzle and the sample's surface, the diameter d Noz of the nozzle, the cutting speed v, the pressure p of the nitrogen processing gas, and the waist position z f were adapted in order to achieve stable cutting conditions. Figure 5 a) shows a time-averaged X-ray image of the cutting process in a s = 10 mm thick sheet of stainless steel. The gray-scale values in the image represent the local transmittance of the X-ray radiation through the sample. In order to avoid overexposure of the camera above and below the sample, 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 cutting 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 experimentally 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 geometry 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 F exp * of the kerf, as highlighted by the black dotted line in Fig. 5 c). The upper and lower areas F lead,t and F lead,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 crosssectional area with which the cutting front moves through the sheet and is highlighted by the green dotted line in Fig. 5 c).

Adjusting the cross-sectional area of the cutting kerf by means of beam shaping
The basic assumption in the prediction of the cross-sectional 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 F exp of the cutting kerf for different shapes of the laser beam and different positions of the beam waist to validate the assumption. Figure 6 shows the caustics of the beam shapes "top hat," "annular," and "line scan" in the y-z plane, which were calculated with Eq. (6). The cross-sectional area F for the beam shapes is marked by the red, yellow, and green area.

Kerf shaping by adjusting the shape of the laser beam
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 F exp (dots and crosses) of the reconstructed cutting kerfs as a function 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 calculated cross-sectional area F represents a deviation of the measured beam propagation factor M 2 by ±15% . The data points of the experimentally determined cross-sectional are F exp,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 experimentally determined cross-sectional area F exp,annular of the "annular" beam give the value of one single analyzed cutting process. The data points with the cross represent the cut limit v max . 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 F annular of the "annular" beam in the y-z plane and the experimentally determined cross-sectional area F exp,annular are larger than the one of the "top hat" beam. The results show that the experimentally determined cross-sectional areas F exp 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 cutting speeds. In the case of the "annular" beam, the deviation between the measured area F exp and the predicted area F at maximum cutting speed is 0.8mm 2 , which is a difference of 10%. Hence, the reasonable agreement between the measured cross-sectional areas F exp 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": d f ≈ 166 m , z r ≈ 1.6mm , M 2 ≈ 13 , z f = 5.5mm ; "annular": d f ≈ 900 m , z r ≈ 8.3mm , M 2 ≈ 74 , z f = 1.0mm ; "line scan": d f ≈ 208 m , z r ≈ 2.5mm , M 2 ≈ 13 , z f = 6.0mm Fig. 7 Experimentally determined cross-sectional areas F exp 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, z f = 5.5mm, Δz Noz = 0.5mm, d Noz = 5.0mm, s = 10mm ; "annular":P = 8kW, p = 12bar, z f = 1.0mm, Δz Noz = 1.0mm, d Noz = 2.5mm, s = 10mm Fig. 8 Experimentally determined cross-sectional area F exp of the cutting kerf (dots) and the calculated cross-sectional area F of the caustic of the "top hat" beam (solid line) in the y-z plane as a function of the waist position z f ; "top hat": P = 6kW,v = 1.3m∕min, p = 9bar, Δz Noz = 0.5mm, d Noz = 5.0mm,s = 10mm Figure 8 compares the experimentally determined cross-sectional areas F exp 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 z f of the waist. The scatter band (shaded area) of the calculated cross-sectional area F represents a deviation of the measured beam propagation factor M 2 by ±15% . The cutting speed was v = 1.3m∕min. Figure 8 shows that the experimental results agree reasonably well with the calculated cross-sectional areas when varying the waist position. The experimentally determined areas F exp 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 v max as assumed by the model. In order to ensure a successful cut in the full range of the different waist positions, the cutting speed was reduced to v = 0.46 • v max , 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 position shown in Fig. 8 prove that the calculation of the crosssectional area of the caustic with Eq. (7) can be used to estimate the cross-sectional area of the cutting kerf for different shapes of the laser beam and waist positions at the maximum cutting speed.

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 cutting 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 calculated 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). Following the abovementioned model introduced by Mahrle et al. [13. ], the caustic resulting from the "line scan" is approximated 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": 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, was reached for at a cutting speed of v max = 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 approximately 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 represents a deviation of the measured beam propagation factor M 2 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 acceptable in view of the simplified geometrical approach adopted from [13. ].
In order to predict the maximum cutting speed v max 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 theoretically predicted angle of incidence A( @vmax ) at maximum cutting speed (horizontal lines). The scatter band (shaded area) of the calculated absorptivity A( @vmax ) represents a deviation of the measured beam propagation factor M 2 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 changing 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 maximum 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, z f = 5.5mm, Δz Noz = 0.5mm, d Noz = 5.0mm, s = 10mm ; "annular":P = 8kW, p = 12bar, z f = 1.0mm, Δz Noz = 1.0mm, d Noz = 2.5mm, s = 10mm ; "line scan": P = 6kW,p = 15bar, z f = 6.0mm, Δz Noz = 0.7mm, d Noz = 5.0mm ,a = 0.1mm, s = 10mm Fig. 12 Absorptivity resulting at the experimentally determined angles of incidence 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, z f = 5.5mm, Δz Noz = 0.5mm, d Noz = 5.0mm, s = 10mm ; "annular":P = 8kW, p = 12bar, z f = 1.0mm, Δz Noz = 1.0mm, d Noz = 2.5mm, s = 10mm ; "line scan": P = 6kW ,p = 15bar, z f = 6.0mm, Δz Noz = 0.7mm, d Noz = 5.0mm,a = 0.1mm, s = 10mm 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).

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 P L according 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 conduction equation however indicates that the power loss P L 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 stainless steel used in the experiments, the value of the depthspecific power loss was found to be P L,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 investigated 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 absorptivity A from Eq. (4) allows the calculation of the maximum cutting speed v max . Figure 13 compares the experimentally determined maximum cutting speed v max,exp (crosses) with the predicted maximum cutting speed v max (solid lines) as a function of the laser power P for different thicknesses s of the cut sheets, when cutting with a conventional top-hat-shaped intensity distribution. The scatter band (shaded area) of the calculated predicted maximum cutting speed v max represents a deviation of the measured beam propagation factor M 2 by ±15% . The process parameters are given in Table 2.
As expected, the results show that the maximum cutting speed increases with increasing laser power P and decreases with increasing thickness s of the sheet. Increasing the power from 3kW to 8kW increases the maximum cutting speed v max,exp from 1.0m∕min to 4.1m∕min , when cutting a s = 10mm thick sheet. The predicted cutting speeds v max 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%.  13 Experimentally determined maximum cutting speed v max,exp (crosses) and calculated maximum cutting speed v max (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  The good agreement between the measured and calculated results proves that the calculated cross-sectional area F , angle of incidence @vmax and hence the absorptivity A( @vmax ) can be used to predict the maximum cutting speed v max according to Eq. (4) for different laser power and sheet thicknesses.

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 approximately 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 v max,exp (crosses) and predicted maximum cutting speed v max (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 v max represents a deviation of the measured beam propagation factor M 2 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 cutting 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 asymmetrical "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 crosssectional area F and thus provides highest cutting speeds.

Conclusion
In summary we presented 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. ]. 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 crosssection 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 direction 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 equations 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 and calculated maximum cutting speed v max (solid lines) as a function of the laser power P for different beam shapes; "top hat": P = 6kW,p = 9bar, z f = 5.5mm, Δz Noz = 0.5mm, d Noz = 5.0mm, s = 10mm ; "annular":P = 8kW, p = 12bar, z f = 1.0mm, Δz Noz = 1.0mm, d Noz = 2.5mm, s = 10mm ; "line scan": P = 6kW,p = 15bar, z f = 6.0mm, Δz Noz = 0.7mm, d Noz = 5.0mm ,a = 0.1mm, s = 10mm 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.

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.
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