# Analytical Model for the Depth Progress during Laser Micromachining of V-Shaped Grooves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Model for the Prediction of the Depth and Width of Laser Micromachined Grooves

_{C}and y − y

_{C}are the distances from the centre of the laser beam located at (x

_{C}, y

_{C}), w

_{0}is the beam radius and ϕ

_{0}denotes the peak fluence, which is given by

_{P}is the pulse energy. At normal incidence, material removal by ablation on the surface occurs when the locally absorbed fluence $A\cdot \varphi (y)$, where A is the material-specific absorptivity at the wavelength of the incident radiation, exceeds the value

_{E}denotes the effective penetration depth of the absorbed energy density and h

_{V}denotes the volume-specific enthalpy required for heating and complete vaporization of the material. The effective penetration depth l

_{E}is dominated either by the optical penetration depth or by the electron heat diffusion length, depending on the peak fluence of the incident radiation [22]. Additionally, l

_{E}and thus ϕ

_{th}decrease with increasing number of pulses applied to the surface [23] due to the so-called incubation effect [24]. The incubation effect saturates after about 100 pulses, whereupon the effective penetration depth and the ablation threshold are not significantly decreased further by additional pulses [23]. For the sake of simplicity, the energy penetration depth l

_{E}and thus the ablation threshold ϕ

_{th}are assumed to be constant over the entire process for the presented analytical model. The error caused by this simplification during the first 100 pulses is negligible as typically, more than several thousands of pulses are applied to each location for the production of laser machined grooves.

_{G}resulting from material removal corresponds to two times the ablation radius r

_{abl}and is calculated by [15]

_{x}is defined by

_{x}is the scanning speed and f

_{rep}is the pulse repetition rate. A constant groove width d

_{G}is typically achieved with a spatial pulse overlap ranging from 30% to 95% [2].

_{G,n}after n ∈ 1,2,…N scans can be calculated by

_{G,n − 1}denotes the groove depth after n − 1 scans and z

_{S,n}denotes the depth ablated by the nth scan (cf. Figure 1). The overall absorptance η

_{A}resulting from multiple reflections inside the V-shaped groove may be calculated assuming specular reflections of a ray, which is incident in z-direction and is found to be [25]

_{R}denotes the number of reflections of the ray until it leaves the groove again. This number of reflections depends on the aspect ratio of the V-shaped groove and is given by [25]

_{A,n}(x

_{j}) absorbed from one single pulse of a Gaussian laser beam during the nth scan along the groove at the location x in a stripe with the width dx inside the groove between the edges at y = ±d

_{G}/2 amounts to

_{C}was set to zero for the beam, which is centred on the groove.

_{A}(d

_{G}, z

_{G,n − 1}) only defines the amount of energy dE

_{A,n,j}(x) absorbed in the groove but does not specify the transversal distribution of the fluence in the y–z-plane (cf. Figure 1). As shown by raytracing simulations of V-shaped capillaries in [21], the effect of multiple reflections causes an elevated absorbed fluence near the tip of the groove. As a simple approximation for the transversal distribution of the absorbed fluence in the groove, it is assumed in the following that the absorbed fluence linearly increases with the depth along the sidewalls of the V-shaped groove in the y–z-plane. In analogy to the model presented for percussion drilling [15] and assuming that multiple reflections only occur normal to the axis of the groove, the distribution of the absorbed fluence at a given location x along the groove is assumed to start with ϕ

_{th}at the edge of the groove (at y = ±d

_{G}/2) and end with ϕ

_{tip,n,j}(x) at the tip of the groove (y = y

_{C}= 0). With this assumption, the energy dE

_{A,n,j}(x) absorbed at the location x from a single pulse j in a stripe of width dx amounts to

_{G,n − 1}at the location x by the jth pulse during the nth scan is given by

_{tip,n,j}(x) with a spatial offset of δ

_{x}each absorbed at the tip of the groove around an arbitrary point x

_{0}. For the sake of clarity, the figure is divided into two parts showing the pulses with j ≤ 0 in Figure 2a) and the pulses with j ≥ 0 in Figure 2b). The pulses are numbered in such a way that the beam axis coincides with x

_{0}at the moment when the 0th pulse hits the workpiece (x

_{C,0}= x

_{0}). Considering this diagram, it becomes evident that from the perspective of a point (x = x

_{0}, y = 0) located at x

_{0}somewhere along the centre line of the groove, the individual pulses of a scan can only contribute to the ablation of the groove at this point x

_{0}as long as the fluence absorbed at the tip ϕ

_{tip,n,j}(x

_{0}) > ϕ

_{th}exceeds the ablation threshold ϕ

_{th}. The fluence that is absorbed at the tip of the groove from each of the pulses j of one scan (with j = …, −3, −2, −1, 0, 1, 2, 3, …) at the location x = x

_{0}is given by the intersection of the fluence distribution ϕ

_{tip,n,j}(x) with the ordinate at x = x

_{0}, as indicated by the colored small arrows in Figure 2.

_{0}, y = 0), as only their fluences ϕ

_{tip,n,j}(x

_{0}) exceed the ablation threshold ϕ

_{th}, whose value is indicated by the black dotted line. The intersection ${\varphi}_{\mathrm{tip},n,j}\left(x={j}_{\mathrm{abl},n}\cdot {\delta}_{x}\right)={\varphi}_{\mathrm{th}}$ of the fluence distribution ϕ

_{tip,n,j}(x) with the ablation threshold ϕ

_{th}determines the maximum number of pulses j

_{abl,n}contributing to ablation in this direction. Using Equation (12) and solving for j

_{abl,n}yields

_{S,n}as seen by the spot x = x

_{0}located on the centre line of the groove corresponds to the accumulated depth ablated by the pulses $j=-\lfloor {j}_{\mathrm{abl},n}\rfloor $ to $j=\lfloor {j}_{\mathrm{abl},n}\rfloor $ and can be calculated by

_{P,n,j}(x) denotes the depth ablated by the pulse j during the nth scan. According to the logarithmic ablation law [23,26], the depth increment ablated by a single pulse is given by

_{A}(d

_{G}, z

_{G,n − 1}) as given by Equations (8) and (9) is assumed during one scan over the groove. This induces a negligible error since η

_{A}changes very slowly with an increasing number n of scans as long as z

_{S,n}<< d

_{G}, which is typically the case in micromachining processes with a reasonable pulse overlap Ω

_{x}in the range of 30–95%.

_{G,n}can be recursively calculated as a function of the number n of scans. A useful way to proceed is by starting with the calculation of the constant parameters that are not affected by the recursive calculation, such as the spatial offset δ

_{x}between the impact locations of two consecutive pulses using Equation (6). Furthermore, the peak fluence ϕ

_{0}and ablation threshold ϕ

_{th}can be calculated with Equations (2) and (3), respectively, in order to determine the width of the groove d

_{G}using Equation (4). With the first scan (n = 1) at the beginning of the recursive calculation, a very small value should be chosen for the initial groove depth, e.g., z

_{G,0}= 1 nm (z

_{G,0}≠ 0), so as not to divide by 0 in the subsequent calculation of the absorptance η

_{A}(d

_{G}, z

_{G,0}) in Equations (8) and (9). Then, the maximum number of pulses j

_{abl,1}contributing to ablation in each direction is calculated using Equation (13), followed by the calculation of the fluence ϕ

_{tip,1,j}(x) deposited at the tip of the groove with Equation (12) for each contributing pulse j during this first scan. The depth increment z

_{P,1,j}(x) ablated by each pulse j is calculated using Equation (15) and accumulated according to Equation (14). Then, the accumulated depth of the first scan z

_{S,1}is added to the initial groove depth z

_{G,0}as given by Equation (7). The calculation of the absorptance η

_{A}(d

_{G}, z

_{G,1}) of the groove with increased depth z

_{G,1}starts the second loop of the recursive calculation. This procedure must be repeated n times to receive the groove depth z

_{G,n}after micromachining with n scans.

_{tip,n,j}(x) at the tip of the groove decreases with increasing groove depth due to the increasing length of the sidewalls $\sqrt{{d}_{\mathrm{G}}^{2}+4\cdot {z}_{\mathrm{G},n-1}^{2}}$. This reduction is partially compensated by an increased absorptance η

_{A}(d

_{G}, z

_{G,n−1}) due to the increasing number N

_{R}of reflections within the groove (cf. Equations (8) and (9)). The maximum attainable groove depth z

_{G,∞}obtained after n → ∞ scans is reached when the fluence ϕ

_{tip,n,j}(x = x

_{C}) at the tip of the V-shaped groove converges to the value of the ablation threshold ϕ

_{th}. The maximum groove depth z

_{G,∞}can therefore be found with Equation (12) by setting ϕ

_{tip,n,j}(x = x

_{C}) = ϕ

_{th}, and solving for z

_{G,∞}, which yields

_{∞}= η

_{A}(d

_{G}, z

_{G,∞−1}) denotes the absorptance of a groove micromachined with ∞−1 scans. As the absorptance, in turn, depends on the groove depth z

_{G,n−1}(cf. Equations (8) and (9)), the maximum groove depth z

_{G,∞}cannot be calculated directly but has to be found by a recursive calculation using Equation (7). Assuming a high aspect ratio z

_{G,∞}/d

_{G}of the final groove, the absorptance can, however, be approximated to be η

_{∞}≈ 1, and the maximum achievable groove depth z

_{G,∞}obtained with a given parameter set can directly be estimated using Equation (16) by setting η

_{∞}= 1. Equation (16) also shows that the maximum achievable groove depth does neither depend on the repetition rate f

_{rep}nor on the scanning parameters such as the scanning speed v

_{x}and that—for a given beam radius w

_{0}and with the material-specific value of ϕ

_{th}—it can only by increased by increasing the pulse energy E

_{P}.

_{P}, the repetition rate f

_{rep}, the radius w

_{0}of the laser beam, the scanning speed v

_{x}, and the number of scans n, as well as the three material parameters, absorptivity A, energy penetration depth l

_{E}, and the enthalpy h

_{V}for heating and complete vaporization of the material.

## 3. Experimental Verification of the Analytical Model

_{0}= 55 ± 5 µm. The focus position was always set on the surface of the samples. Grooves with a length of 10–35 mm were micromachined in the Ti-samples with different pulse energies E

_{P}, repetition rates f

_{rep}, scanning speeds v

_{x}and number of scans n, as summarized in Table 1. The spatial offset δ

_{x}of the impact locations of two consecutive pulses and the corresponding pulse overlap Ω

_{x}were calculated according to Equation (6) and Equation (5), respectively.

_{G}= ${138}_{-5}^{+8}$ μm independent of the number of scans, which is also in good agreement with the results shown in [2]. The V-shape clearly dominates the shape of the shown grooves for n ≥ 800. Deviations from the V-shape can be seen for n = 300 and n = 600 due to rough structures at the bottom of the grooves. Bending of the tip of the groove occurred for n = 10,000, which was also observed in [8] for micromachining of deep grooves in a Ni-alloy and drilling of deep microholes in CVD diamond [27]. The cause for the bending of the tip has not been conclusively clarified yet, but a polarization-dependent behaviour was found in [27].

_{V}= 47.1 J/mm³, which is required to heat and vaporize the material. The values are listed in Table 2. The absorptivity of titanium at normal incidence and at a wavelength of 1030 nm was set to A = 0.51 [30]. The effective penetration depth was used as a fit parameter. A good agreement between the calculated and the experimental results was found with l

_{E}= 30 nm. This value corresponds to an absorbed threshold fluence of ϕ

_{th}= 0.14 J/cm² (cf. Equation (3)). The fitted value of l

_{E}= 30 nm is consistent with experimentally determined values of the optical penetration depth of 26 nm for Ti6Al4V [31] and 30 nm for titanium [22].

_{G}= 126 µm and d

_{G}= 100 µm, respectively. The experimentally determined widths of ${138}_{-5}^{+8}$ µm (P1) and ${113}_{-5}^{+4}$ µm (P2) are slightly larger. The moderate deviations of less than 15% may be explained by the fact that no incubation effect is taken into account in the model.

_{G,}

_{n}as a function of the number n of scans was recursively calculated as described above. The calculations are compared to the experimental results in Figure 4. The groove depths as calculated by the model derived in the previous section and as measured from the cross sections for the different parameter combinations P1–P5 (cf. Table 1) are represented in different colors with dotted lines and data points, respectively. The value of the data points corresponds to the average values measured from up to five grooves micromachined with identical parameters. The error bars represent the deviation to the maximum and minimum measured groove depth of each parameter set.

_{G}/d

_{G}≈ 1.5, the measured groove depth increases almost linearly with the number of scans. The progress of the depth is found to slow down for aspect ratios beyond z

_{G}/d

_{G}> 1.5. At constant repetition rate f

_{rep}and scanning speed v

_{x}, higher depth progress and deeper grooves were achieved for higher pulse energies (cf. P1 and P2). For constant pulse energy E

_{P}and constant pulse overlap Ω

_{x}, the groove depth as a function of number of scans is similar (cf. P2 and P3). However, the net processing time is divided in half for P3 in comparison to P2 due to double the scanning speed v

_{x}at a twofold repetition rate f

_{rep}. At constant pulse energy E

_{P}and constant repetition rate f

_{rep}, higher depth progress is achieved with lower scanning speeds v

_{x}(cf. P3, P4 and P5). The relations observed in this work regarding the depth progress in micromachining of grooves in Ti6Al4V confirm the observations made for semiconductors in [2,9] and for a Ni-alloy in [8]: The groove depth increases with increasing number of scans, and at high pulse energies and low scanning speeds, a greater increase in depth was observed with each scan. The maximum groove depth of ${624}_{-38}^{+28}$ µm was achieved with the highest investigated pulse energy E

_{P}= 181 µJ and the highest number of scans n = 10,000 for this parameter combination (P1). The maximum measured groove depth for a constant pulse energy E

_{P}= 69 µJ and different scanning parameters (from P2 to P5) is in the range of ${306}_{-14}^{+13}$ µm.

_{P}= 69 µJ, but micromachined with different scanning parameters (from P2 to P5) converge to the same maximum groove depth, which agrees well with the theoretical prediction of the model of 326 µm calculated by Equation (16) for η

_{∞}= 1. Deviations from calculation and measurement might result from uncertainties regarding the material parameters used for the calculation, in particular the fitted value for the effective penetration depth l

_{E}, or from deviations of the assumed ideal V-shape, as shown before in Figure 3 for n = 10,000, with the bending of the tip. Complete vaporization is assumed in the proposed analytical model, whereas additional effects such as melting and spallation can cause a deviating process enthalpy and thus a different ablation rate [32].

_{G}and groove width d

_{G}, as calculated by the model derived in the previous section.

_{P}and f

_{rep}, the scanning speed v

_{x}and beam radius w

_{0}, and the three material parameters A, l

_{E}and h

_{V}, the model allows for the prediction of the groove dimensions as a function of the number of scans n and maximum achievable groove depth.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**V-shaped groove produced by a pulsed laser beam which is scanned along the x-axis. The Gaussian distribution of the fluence of the individual laser pulses is shown by the red curve. The width of the groove is denoted by d

_{G}= 2 r

_{abl}, which corresponds to two times the ablation radius r

_{abl}. The incrementally increased depth of the groove is denoted by z

_{G,n}, where n is the number of applied scans and z

_{S,n}= z

_{G,n}− z

_{G,n − 1}is the incremental increase in the depth produced by the nth scan along the groove. The cross section of the volume ablated during the nth scan is highlighted by the red hatched cross section.

**Figure 2.**Absorbed fluence distributions ϕ

_{tip,n,j}(x) along the x axis at y = 0 of the incident pulses (

**a**) from j = −3 to j = 0 and (

**b**) from j = 0 to j = 3. The coloured small arrows indicate the absorbed fluence at the location x

_{0}. The intersection of the fluence distributions ϕ

_{tip,n,j}(x) with the ablation threshold ϕ

_{th}determines the maximum number of pulses j

_{abl,n}contributing to ablation in this direction along the x axis.

**Figure 3.**Cross sections of grooves micromachined in Ti6Al4V with the parameter set P1. The depth and width of each groove are indicated by a yellow double arrow and a green double arrow, respectively.

**Figure 4.**Calculated groove depth (dotted lines, “Model”) and measured groove depth (data points, “Measured”) as a function of the number of scans for grooves micromachined in Ti6Al4V using the different parameter sets as given in Table 1.

**Figure 5.**Cross sections of grooves micromachined in Ti6Al4V with (

**a**) P1, n = 1500, (

**b**) P2, n = 10,000, (

**c**) P3, n = 8000, (

**d**) P4, n = 8000, and (

**e**) P5, n = 5000. An isosceles triangle with the dimensions of the calculated depth and width of the corresponding groove using the model presented in Section 2 is inserted for each parameter set in the respective colour.

P_{av} in W | E_{P} in µJ | ϕ_{0} in J/cm² | f_{rep} in kHz | v_{x} in m/s | δ_{x} in µm | Ω_{x} | Number of Scans n | |
---|---|---|---|---|---|---|---|---|

P1 | 9.05 | 181 | 3.81 | 50 | 1.2 | 24 | 78% | 300…10,000 |

P2 | 3.45 | 69 | 1.45 | 50 | 1.2 | 24 | 78% | 300…10,000 |

P3 | 6.90 | 69 | 1.45 | 100 | 2.4 | 24 | 78% | 600…20,000 |

P4 | 3.45 | 69 | 1.45 | 50 | 2.4 | 48 | 56% | 600…20,000 |

P5 | 3.45 | 69 | 1.45 | 50 | 0.6 | 12 | 89% | 150…5000 |

Material Parameter | Value |
---|---|

Density | $4506\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ [28] |

Heat capacity for solid titanium | $523\frac{\mathrm{J}}{\mathrm{kg}\cdot \mathrm{K}}$ [28] |

Melting temperature | 1668 °C [28] |

Latent heat of melting | $440\frac{\mathrm{kJ}}{\mathrm{kg}}$ [29] |

Vaporization temperature | 3287 °C [28] |

Latent heat of vaporization | $8305\frac{\mathrm{kJ}}{\mathrm{kg}}$ [29] |

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**MDPI and ACS Style**

Holder, D.; Weber, R.; Graf, T.
Analytical Model for the Depth Progress during Laser Micromachining of V-Shaped Grooves. *Micromachines* **2022**, *13*, 870.
https://doi.org/10.3390/mi13060870

**AMA Style**

Holder D, Weber R, Graf T.
Analytical Model for the Depth Progress during Laser Micromachining of V-Shaped Grooves. *Micromachines*. 2022; 13(6):870.
https://doi.org/10.3390/mi13060870

**Chicago/Turabian Style**

Holder, Daniel, Rudolf Weber, and Thomas Graf.
2022. "Analytical Model for the Depth Progress during Laser Micromachining of V-Shaped Grooves" *Micromachines* 13, no. 6: 870.
https://doi.org/10.3390/mi13060870