1 Introduction

Percussion drilling is one of the most effortless laser applications to implement, as only a focusing lens is required to focus the laser beam on the workpiece. Micro-drilling with ultrashort laser pulses is, however, a highly dynamic process in which a multitude of mechanisms interact at high temperatures and pressures on very small temporal and spatial scales [1, 2]. At present, it is still a demanding task to separate and understand the many effects that occur during the drilling process. With the use of additional process strategies such as helical drilling [3,4,5,6], the use of process gas [3] and focus adjustment [7] as well as the availability of higher pulse energies of up to several millijoules [8, 9], not only the number of process parameters increases, but also the number of effects that occur. Thus, it is still challenging to reliably predict the drilling depth, borehole geometry, and drilling duration, especially with analytical models. An important step in this direction was to understand and describe the heat accumulation that occurs during machining with ultrashort laser pulses [10,11,12,13] and its consequences on the drilling process [14, 15]. When heat accumulation effects and particle-ignited plasmas can be avoided, it is possible to estimate the final depth of percussion-drilled microholes with an analytical model [16, 17] and the temporal evolution of the drilling depth can be predicted [18]. In the latter case, the use of optical coherence tomography (OCT) proved to be a promising diagnostic method to measure the drilling progress online.

For many high-tech applications, such as drilling of injection nozzles [19] or drilling of spinnerets for fibre production [4, 20], the quality of the microholes is of crucial importance. In particular, the quality of the edge and the shape accuracy of the microhole are relevant criteria. The quality-reducing effects that can occur during laser drilling are manifold and the impact of these effects include shape deviations, striations, burrs, side channels, and the formation of multiple holes. Although the studies published in [16] and [18] deal with the production of high-quality microholes, the classification of high-quality microholes primarily only refers to the exclusion of thermal damage and excessive melting due to heat accumulation. The quality of the microhole’s edge on the side where the laser beam exits and its dependence on processing parameters was not investigated in these studies and is now addressed in the present work. From previously published work, it is already known that side channels form at the far end of the hole when the fluence reaching the tip is close to the ablation threshold [21]. Such side channels appear as multiple holes on the rear side of the workpiece, but can be avoided by increasing the incident laser fluence [17]. Both studies indicate that the fluence has a significant influence on the quality of the microhole’s exit. The fluence reaching the far end of the hole changes with the evolving geometry of the hole being drilled [18]. As known from experiments, the same happens with the quality of the holes. A simplified analytical model is therefore proposed in the following to calculate the fluence reaching the microhole’s exit. By comparison to experimental results, we show that the quality of the holes significantly deteriorates as soon as the fluence reaching the far end of the hole drops below a value of 2.8 times the ablation threshold of the material.

In the following, the experimental setup is presented in Sect. 2 together with the method applied to classify and determine the edge quality of the holes. The experimental investigations on the edge quality of the hole’s exit are presented in Sect. 3, followed by the derivation of the analytical model for the calculation of the fluence at the hole’s exit in Sect. 4. Finally, by correlating the experimental results to the theoretical calculations, the conditions under which high-quality exits can be produced are derived in Sect. 5.

2 Methods

2.1 Experimental setup

The experiments were performed using an ultrafast titan sapphire laser system (Spitfire ACE, Spectra Physics) operating at a wavelength \(\lambda\) of 800 nm. The laser system provides ultrashort laser pulses with an adjustable pulse duration \(\tau\) from 35 fs to 5 ps and with pulse energies \({E}_{\text{P}}\) of up to 7 mJ at a repetition rate \({f}_{\text{R}}\) of 1 kHz. The pulse duration was set to 1 ps (FWHM, Gauss-fit). The circularly polarized Gaussian beam with a beam diameter of 3.7 mm (\(1/{e}^{2}\) diameter) was focused by a telecentric \(f\)-theta lens with a focal length of \(100\) mm to a focal diameter of \({d}_{\text{f}}=36~{\upmu}\text{m}\) (\(1/{e}^{2}\) diameter). The beam quality factor was measured to be M2 = 1.3. The focus was positioned 5 mm below the surface of the 0.5 mm thick workpiece made of cold-rolled stainless steel (St 1.4301/AISI 304). The beam diameter on the workpiece’s surface thus was \({d}_{\text{s}}=188~\upmu{\text{m}}\). The diameters were measured using a beam profiling camera. A galvanometer scanner was used to position the laser beam on the workpiece. Both the pulse energy (and with it the fluence) incident on the workpiece and the number of pulses applied to a single drilled hole were varied in the experiments. To account for statistical variability, 10 holes were drilled with each set of parameters.

2.2 Analysis of the drilled holes

The edges of the exits of the microholes were quantitatively evaluated by automated analysis of microscope images. The microscope images used had a resolution of 0.3 µm/px. For the automated analysis, a supervised learning approach was used to reliably determine the contour shape of the hole’s opening [22, 23]. The image analysis was performed with the software Mathematica 12.1. Figure 1 shows the microscope image of a microhole’s exit (a) and the opening of the microhole identified by supervised learning (b). An ellipse was fitted to the determined contour of the hole’s exit with the least squares method (see Fig. 1c). Figure 1d shows the radial deviation of the hole’s edge from the fitted ellipse. Based on these data, the amplitude of striations \(\xi\) and the perimeter ratio \(\sigma\) can be determined, which are introduced in the following.

Fig. 1
figure 1

Microscope image of a microhole’s exit (a). Shape of the hole’s exit as extracted by image processing (b). Ellipse (red) fitted to the determined contour of the hole’s exit (c). The plot in d shows the radial deviation of the hole’s edge from the fitted ellipse and the striations amplitude \(\xi\). The geometrical properties of the hole’s exit are summarized in (e)

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The striations are quantified based on the deviation of the hole’s contour from the fitted ellipse. The distance between the 5% and 95% quantile of this deviation is used as a measure of the amplitude of the striations \(\xi\) (see Fig. 1d). The perimeter ratio \(\sigma\) is defined as the ratio between the length of the hole´s contour and the circumference of the fitted ellipse and thus is a measure of the frequentness of the striations. These two quantities allow to assess different edge qualities of microholes, as shown by the exemplary images in Fig. 2 used to define for four different quality types: ideal, round exit (a), exit with one larger striation (b), exit with many small striations (c), and exit with many larger striations (d). Only exits with both, a small \(\xi\) and a small \(\sigma\), indicate that it is an exit with good edge quality.

Fig. 2
figure 2

Exemplary images (500  × 500 pixels) and geometrical properties for different types of edge quality: ideal, round exit (a), exit with one larger striation (b), exit with many small striations (c), and exit with many larger striations (d)

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3 Evolution of the edge quality during the drilling process

The evolution of the edge quality of the holes’ exits during the drilling process is shown in Fig. 3 for microholes which were percussion drilled with a peak fluence of 8.6 J/cm2 in a Gaussian beam. Figure 3 shows the striations amplitude \(\xi\) (a) and the perimeter ratio \(\sigma\) (c) as well as selected microscope images (b) of the exits of the microholes as a function of the total number of applied pulses. With the parameters used, the exits of the microholes after the breakthrough exhibit a high quality with an even, almost circular contour. Only if more than 10,000 pulses are applied, the edge quality deteriorates significantly and striations become clearly apparent on the wall and consequently shape the contour of the exit. Based on the parameters \(\xi\) and \(\sigma\), the exits of the microholes can be assigned to two quality classes using a k-means clustering algorithm [24]. The range in which the applied number of pulses leads to exits with low values of \(\xi\) and \(\sigma\), hence producing holes with high-quality exits, is marked in green. With the parameters used, the breakthrough was reached after about 5000 pulses. The width of the range leading to high-quality exits is \({\Delta N}_{\text{q}}=(5000 \pm 500)\) pulses after breakthrough.

Fig. 3
figure 3

Striations amplitude \(\xi\) (a), microscope images (b), b and perimeter ratio \(\sigma\) (c) of the microhole exits which were produced in 0.5 mm thick samples of stainless steel (St 1.4301/AISI 304) with different number of pulses with a peak fluence of 8.6 J/cm2 in Gaussian beam \((\lambda =800~\text{nm,} \tau =1~\text{ps,} {f}_{\text{R}}=1~\text{kHz,} {E}_{\text{P}}=1.2~\text{mJ).}\) The green section highlights the range \({\Delta N}_{\text{q}}\) after breakthrough in which high-quality exits are formed

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As seen from Fig. 4, this abrupt change of the quality can be observed also when drilling with other pulse energies. Thereby, it was found that the range in which exits of high quality are formed decreases with decreasing peak fluence to \((4000 \pm 500)\) pulses after breakthrough at 6.5 J/cm2 and to \((2000 \pm 500)\) pulses after breakthrough at 4.3 J/cm2, respectively. For microholes drilled at a further reduced fluence of 2.6 J/cm2, no high-quality microholes were formed, hence \({\Delta N}_{\text{q}}\) = 0 (data not shown).

Fig. 4
figure 4

Striations amplitude \(\xi\) (a, d), microscope images (b, e), and perimeter ratio \(\sigma\) (c, f) of the exits of microholes which were produced in 0.5 mm thick samples of stainless steel (St 1.4301/AISI 304) with a peak fluence of 6.5 J/cm2 (left) and 4.3 J/cm2 (right) in Gaussian beam \((\lambda =800~\text{nm,} \tau =1~\text{ps,} {f}_{\text{R}}=1~\text{kHz,} {E}_{\text{P}}=0.9~\text{mJ (left) and }{E}_{\text{P}}=0.6~\text{mJ (right))}\). The green section highlights the range of pulse numbers \({\Delta N}_{\text{q}}\) after breakthrough in which high-quality exits are formed

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The results show that high-quality exits are obtained only during the phase in which the diameter of the exits undergoes a significant widening rate. The development of the mid-range diameter of the exits and entrances of the drilled microholes is shown in Fig. 5. As expected from the model [18] presented in Sect. 4, the diameter of the entrances remains almost constant with increasing number of pulses. The evaluation of the quality of the entrances reveals that the entrances are always characterized by a high degree of striations throughout the entire drilling process (data not shown). The striations amplitude \(\xi\) and perimeter ratio \(\sigma\) for the entrances of the microholes generally lie in the range of \(\xi =(21\pm 5)~\upmu\text{m}\) and \(\sigma =1.48\pm 0.1\) for all three fluences and are thus well above the values of the exits. The extent and location of the striations within the drilling channel can be seen in the two insets of Fig. 5, each showing the cross section of a microhole produced with a peak fluence of 8.6 J/cm2. The microhole shown on the left was drilled with a total of 8000 pulses and thus lies within the quality range. In this case, striations are only present in the upper two thirds of the drilling channel. The microhole shown on the right was drilled with a total of 11,000 pulses and thus lies outside the quality range. Here, the striations already extend over the entire drilling channel and thus shape the microhole exit.

Fig. 5
figure 5

Development of the mid-range diameter of the exits and the entrances of the microholes which were produced in 0.5 mm thick samples of stainless steel (St 1.4301/AISI 304) with a peak fluence of 8.6 J/cm2 (violet), 6.5 J/cm2 (orange) and 4.3 J/cm2 (blue) in Gaussian beam \((\lambda =800~\text{nm,} \tau =1~\text{ps,} {f}_{\text{R}}=1~\text{kHz)}\). The insets show the cross section of microholes which were produced with a with a peak fluence of 8.6 J/cm2 and 8000 pulses (left) and 11,000 pulses (right)

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4 Analytical model of the fluence at the exit of percussion-drilled microholes

In the previous section, it was shown that the edge quality of the microhole’s exit changes significantly during the drilling process. In particular, it has been observed that high-quality exits are formed immediately after the breakthrough and that the edge quality deteriorated only after an increased number of pulses by an intensified formation of striations. The experimental study also reveals that the range of the number of applied pulses after breakthrough, in which high-quality exits were maintained, decreases with decreasing peak fluence of the incident Gaussian beam. This may suggest the hypothesis that the abrupt deterioration of the edge quality at the upper end of this range could be caused when the fluence reaching the far exit of the holes drops below a certain limit. To investigate whether the fluence at the exit can explain a change of the edge quality, the following section presents an analytical estimation of the fluence reaching the exit of the microholes. Assuming that a minimum fluence at the microhole’s exit is required to obtain edges of high quality, conditions can be derived under which high-quality microhole exits can be produced as later verified in Sect. 5.

4.1 Fluence at the microhole’s exit

The fluence at the exit of the microhole can be derived from the analytical drilling model published in [18], which predicts the temporal evolution of the drilling depth of blind holes. The model assumes a conical shape of the evolving microhole, where \({z}_{\text{tip}}\) is its depth and \({r}_{\text{entr}}\) is the radius of the hole’s entrance (see Fig. 6a, modified for the following discussion of through-holes). Assuming that the laser beam incident on the top of the workpiece has a Gaussian fluence distribution \(\Phi (\mathrm{r})={\phi }_{0}\bullet {\text{exp}}\left(-2 {r}^{2}/{w}^{2}\right)\) the radius of the entrance of the drilled hole is given by [25, 26]

$$r_{{{\text{entr}}}} = w\sqrt {\frac{1}{2}\ln \left( {\frac{{A \phi_{{0}} }}{{\phi_{{\text{abs,th}}} }}} \right)} ,\quad {\Phi }_{0} \ge {\Phi }_{{\text{abs,th}}} /A,$$
(1)
Fig. 6
figure 6

a Conical geometry of the considered microholes. At the edge of the microhole’s entrance the fluence is assumed to correspond to the ablation threshold and at the tip of the cone the fluence is \({\Phi }_{\eta,{\text{tip}}}\) following the model of [18] as shown by (b)

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where A is the material-specific absorptivity on the surface of the workpiece and

$$\phi_{{0}} = 2E_{{\text{P }}} /\left( {\pi w^{2} } \right)$$
(2)

is the peak fluence on the workpiece in the centre of the Gaussian beam and \(w\) is the beam radius on the surface of the workpiece [27].

The distribution of the pulse energy \({E}_{\text{P}}\) of the incident laser pulses on the inner walls of the microhole is influenced by multiple reflections. It was shown in [18] that a good agreement with experimental results is obtained with the assumption that the absorbed fluence increases linearly along the depth of the microhole starting from a value given by the ablation threshold fluence \({\Phi }_{\text{abs,th}}\) at the edge of the entrance to the value [18]

$${\Phi }_{{\eta ,{\text{tip}}}} = \frac{{3 \eta \left( {1 - \frac{{\phi_{{\text{abs,th}}} }}{{A\,\phi_{{0}} }}} \right)E_{{\text{P }}} }}{{A_{{{\text{cone}}}} }} - 2{\Phi }_{{\text{abs,th}}} ,\quad A_{{{\text{cone}}}} > 0,$$
(3)

at the tip of the blind microhole, where \(\eta\) is the integral absorptance of the hole,

$$A_{{{\text{cone}}}} = \pi r_{{{\text{entr}}}} \left( {r_{{{\text{entr}}}}^{2} + z_{{{\text{tip}}}}^{2} } \right)^{1/2}$$
(4)

is the area of the lateral surface of the conically shaped hole (see Fig. 6b) and, as an amendment to the expression given in [18], we here take into account that only the fraction

$$\frac{1}{{E_{\text{P}} }}\mathop \int \limits_{0}^{2\pi } \mathop \int \limits_{0}^{{r_{{{\text{entr}}}} }} {\Phi }\left( {\text{r}} \right)r~ {\text{d}}r{\text{d}}\varphi = \left( {1 - \frac{{\phi_{{\text{abs,th}}} }}{{A\,\phi_{{0}} }}} \right)$$
(5)

of the pulse energy is incident inside the entrance opening of the hole.

In order to apply this model to the discussion of through-holes, it is assumed that the hole within the workpiece with thickness \({z}_{\text{M}}\) evolves the same way as if the workpiece was thicker than \({z}_{\text{tip}}\) also when \({z}_{\text{tip}}>{z}_{\text{M}}\). This implies that multiple reflections on the walls of the hole do not constitute a significant transport of radiation from the lowest part (\(z\in [{z}_{\text{M}},{z}_{\text{tip}}]\)) back towards the upper part (\(z\in [0,{z}_{\text{M}}]\)) of the cone and that the radiation absorbed on the lower part approximately corresponds to the one that is transmitted through the opening at \(z={z}_{\text{M}}\) in the case of the through-hole. As seen from the good agreement with the experimental results discussed in Sect. 5, this appears to be a valid first-order assumption despite the strong simplification. The absorptance η used for our calculation is thus the one of the complete cone, which was originally introduced by Gouffé [28], with minor corrections by Hügel and Graf [27]

$$\eta = \frac{{1 + \left( {1 - A} \right) \left( {S_{\text{G}} - \frac{{{\Omega }_{\text{G}} }}{2\pi }} \right)}}{{A \left( {1 - S_{\text{G}} } \right) + S_{\text{G}} }},$$
(6)

where \({S}_{\text{G}}\) represents the ratio between the opening of the microhole and the complete surface area of the cone including the opening and \({\Omega }_{\text{G}}\) is the solid angle under which the aperture is seen from the tip of the hole. For a conical geometry, \({S}_{\text{G}}\) is given by

$$S_{\text{G}} = 1/\left( {1 + \sqrt {1 + \frac{{z_{{{\text{tip}}}}^{2} }}{{r_{{{\text{entr}}}}^{2} }}} } \right)$$
(7)

and \({\Omega }_{\text{G}}\) is given by

$${\Omega }_{\text{G}} = 4\pi \sin^{2} \left( {\frac{1}{2}\arctan \left( {\frac{{r_{{{\text{entr}}}} }}{{z_{{{\text{tip}}}} }}} \right)} \right).$$
(8)

The incremental increase of

$$z_{{{\text{tip}}}} \left( N \right) = z_{{{\text{tip}}}} \left( {N - 1} \right) + z_{{{\text{abl}}}} \left( N \right)$$
(9)

provoked by the Nth laser pulse is given by

$$z_{{\text{abl}}} \left( {N} \right) = l_{{{\text{ep}}}} \ln \left( {\frac{{{\Phi }_{{\eta ,{\text{tip}}}} \left( { N} \right)}}{{{\Phi }_{{\text{abs,th}}} }}} \right),\quad {\Phi }_{\eta ,{\text{tip}}} \ge {\Phi }_{{\text{abs,th}}},$$
(10)

as long as the fluence \({\Phi }_{{\eta},\text{tip}}\) absorbed at the tip exceeds the ablation threshold \({\Phi }_{\text{abs,th}}\), where \({l}_{\text{ep}}\) is the effective penetration depth and \(f(N)\) denotes the respective quantities as given by the geometry of the hole after \(N\) laser pulses with \({z}_{\text{tip}}\left(0\right)=0\) [18].

With the above assumptions and the linear increase of the fluence inside the hole as shown in Fig. 6b, the fluence of the Nth pulse at the depth \({z}_{\text{M}}\le {z}_{\text{tip}}(N)\) is found to be

$${\Phi }_{{\eta ,{\text{exit}}}} \left( N \right) = \frac{{z_{{\text{M}}} }}{{z_{{{\text{tip}}}} \left( N \right)}}\left( {\Phi_{{\eta ,{\text{tip}}}} \left( N \right) - {\Phi }_{{\text{abs,th}}} } \right) + {\Phi }_{{\text{abs,th}}}.$$
(11)

4.2 Fluence limit for high-quality microhole exits

In view of the hypothesis, that a minimum fluence \({\Phi }_{{\eta},{\text{q}}}\) is required to obtain exits of high quality we further need to calculate at what depth \({z}_{\text{q}}\) the fluence of the  Nth pulse equals \({\Phi }_{{\eta},{\text{q}}}\). As seen from Fig. 6b, this depth is given by

$${\text{z}}_{{\text{q}}} \left( N \right) = \frac{{{\Phi }_{{\eta ,{\text{q}}}} - {\Phi }_{{\text{abs,th}}} }}{{\Phi_{{\eta ,{\text{tip}}}} \left( N \right) - {\Phi }_{{\text{abs,th}}} }}z_{{{\text{tip}}}} \left( N \right), {\Phi }_{{\eta ,{\text{tip}}}} > {\Phi }_{{\text{abs,th}}}.$$
(12)

After breakthrough, the fluence \({\Phi }_{{\eta}, {\text{exit}}}\left(N\right)\) at the exit of the hole exceeds the value of \({\Phi }_{{\eta},{\text{q}}}\) as long as \({{z}}_{\text{q}}\left(N\right)\le {z}_{\text{M}}\). According to the hypothesis, the quality of the holes therefore is expected to deteriorate as soon as \({{z}}_{\text{q}}\left(N\right)>{z}_{\text{M}}\). According to this model, \({z}_{\text{q}}\) is the depth to which the striations reach into the hole and marks the transition from a corrugated drilling channel to the one with a smooth surface as seen from the left inset of Fig. 5.

Let \({N}^{*}\) be the number of pulses at which the tip breaks through the material with the thickness \({{z}}_{\text{M}}\), hence

$$z_{{{\text{tip}}}} \left( {N^{*} } \right) = z_{\text{M}} ,$$
(13)

and \({N}^{**}\) the number of pulses at which \({\Phi }_{{\eta},{\text{exit}}}\) falls below \({\Phi }_{{\eta},{\text{q}}}\), hence

$${{z}}_{{\text{q}}} \left( {N^{**} } \right) = z_{{\text{M}}}.$$
(14)

The maximum number of pulses \({\Delta N}_{\text{q}}={N}^{**}-{N}^{*}\) that can be applied after breakthrough before the fluence \({\Phi }_{{\eta},{\text{exit}}}\) falls below \({\Phi }_{{\eta},{\text{q}}}\) is found by solving Eqs. (13) and (14) for \(N\) using the Eq. (1) through (12). Due to the implicit formulation of the equations, the solution needs to be performed iteratively. To this end, Eq. (9) may be interpreted as a differential equation

$$\frac{{{\text{d}} z_{{{\text{tip}}}} \left( N \right)}}{{{\text{d}} N}} = z_{{{\text{abl}}}} \left( {z_{{{\text{tip}}}} \left( N \right), N} \right),\quad z_{{{\text{tip}}}} \left( 0 \right) = 0,$$
(15)

in order to take advantage of corresponding numerical solvers.

5 Experimental verification

Figure 7 shows the experimentally determined values of \({\Delta N}_{\text{q}}\) (black dots) from Sect. 3 as a function of the peak fluence \({\Phi }_{0}\) (lower abscissa) or the pulse energy \({E}_{\text{P}}\) (upper abscissa) of the incident Gaussian beam together with the calculated values of \({\Delta N}_{\text{q}}\) with \({\Phi }_{{\eta},{\text{q}}}=(2.8\pm 0.2)\:{\Phi }_{\text{abs,th}}\). The material-specific and laser-specific parameters used for the calculation are listed in Table 1. According to [18], the ablation threshold \({\Phi}_{\text{th}}\) corresponds to the single-pulse ablation threshold of iron [29] and AISI 304 [30], so that \({\Phi}_{\text{abs,th}}\)= \(A \bullet {\Phi}_{\text{th}}=0.128~\text{J/cm}^{2}\). The effective penetration was set to a value of 35 nm [31]. The parameter \({\Phi }_{{\eta},{\text{q}}}\) was estimated by a least squares method. Figure 7 shows that the model with \({\Phi }_{{\eta},{\text{q}}}=2.8\:{\Phi}_{\text{abs,th}}\) (= \(0.36~\)J/cm2) describes the basic course of the experimental data. According to the model, the formation of striations can be explained by the fact that after \({\Delta N}_{\text{q}}\) pulses the fluence at the edge of the microhole’s exit falls below a limit value of \({\Phi }_{{\eta},{\text{q}}}=\) \(2.8~\) \({\Phi}_{\text{abs,th}}\).

Fig. 7
figure 7

Number of pulses after breakthrough with high-quality results \({\Delta N}_{\text{q}}\) (black dots) as a function of the peak fluence \({\Phi }_{0}\) (lower abscissa) or the pulse energy \({E}_{\text{P}}\) (upper abscissa) of the incident Gaussian beam with the calculated values of \({\Delta N}_{\text{q}}\) with \({\Phi }_{{\eta},{\text{q}}}=(2.8\pm 0.2)\:{\Phi }_{\text{abs,th}}\). The absorbed ablation threshold \({\Phi }_{\text{abs},\text{th}}= A\bullet {\Phi }_{\text{th}}=0.128~\text{J}/\mathrm{cm}^{2}\) corresponds to the single-pulse ablation threshold of iron and AISI 304. The green area highlights the range \({\Delta N}_{\text{q}}\) of pulses after breakthrough in which high-quality exits are expected according to the model assumptions

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Figure 7 further reveals that the range, where high-quality exits are formed, can be increased by increasing the peak fluence of the incident Gaussian beam. However, it must be taken into account that when the peak fluence is increased to beyond 10 J/cm2, intensity-dependent effects such as plasma formation and air breakthrough [32, 33] or heat accumulation effects may occur [13, 34], which can lead to altered ablation and drilling progress.

Taking the criterion \({\Phi }_{{\eta},{\text{q}}}=2.8\:{\Phi}_{\text{abs,th}}\) as fluence limit to obtain high-quality exits of the microholes, the evolution of \({z}_{\text{q}}(N)\) can be determined according to Eq. (12). Figure 8 shows both \({z}_{\text{tip}}(N)\) (black solid line) and \({z}_{\text{q}}(N)\) (blue dashed line) as calculated for a peak fluence of 8.6 J/cm2 in a Gaussian beam with a diameter of 188 µm on the surface of the workpiece at \(z=0\). The horizontal extent of the green unicoloured area at a given depth \(z={z}_{\text{M}}\) corresponds to the range \({\Delta N}_{\text{q}}\), in which exits with high quality are expected according to the model assumptions when drilling through a workpiece with the thickness \({z}_{\text{M}}\). It reveals that \({\Delta N}_{\text{q}}\) initially increases and then decreases again with increasing depth, so that in this example high-quality exits are only possible up to a material thickness of around \(1.2~{\text{mm}}\). It should be mentioned that an increasing material thickness leads to an increased aspect ratio, which may influence the material expulsion. This effect is not accounted for in the proposed model.

Fig. 8
figure 8

Progression of \({z}_{\text{tip}}\) (black solid line) according to Eq. (15) and \({z}_{\text{q}}\) (blue dashed line) according to Eq. (12) for a quality limit fluence of \({\Phi }_{{\eta},{\text{q}}}=2.8\:{\Phi}_{\text{abs,th}}\) as a function of the number of pulses \(N\) for a peak fluence of 8.6 J/cm2 in Gaussian beam with a diameter of 188 µm at \(z=0\) \((\text{St }1.4301 /\text{ AISI }304, \lambda =800~\text{nm,} \tau =1~\text{ps,} {f}_{\text{R}}=1~\text{kHz,} {E}_{\text{P}}=1.2~\text{mJ)}\). The horizontal extent of the green unicoloured area at a given depth \(z={z}_{\text{M}}\) corresponds to the range \({\Delta N}_{\text{q}}\), in which exits with high quality are expected according to the model assumptions, in this figure exemplary shown for \({z}_{\text{M}}=0.5~{\text{mm}}\)

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6 Conclusion

By comparing results from an experimental study in stainless steel to calculations based on a model for the distribution of the absorbed fluence in conically shaped holes, it was shown that the edge quality of the exit of percussion-drilled microholes strongly depends on the fluence that is absorbed at this location. It was found that the quality of the hole’s exit quickly deteriorates with increasing number of pulses as soon as the fluence that is absorbed near the edge of the exit falls below a limit value of 2.8 times the ablation threshold. The number of pulses that can be applied after breakthrough until the quality starts to deteriorate can be increased by increasing the peak fluence of the applied Gaussian beam. The success of common process strategies for the production of high-quality microholes, such as helical drilling and dynamic focus adjustment, might therefore be explained by a significant increase of fluence that is absorbed near the microhole’s exit.

Table 1 Material-specific parameters for iron and laser-specific parameters used for the model calculation
Full size table