1. Introduction

In laser cutting, one problem is to cut through a piece of material with a certain thickness. How to set the optimum laser and scanning parameters to get through-cuts at the fastest speed is very interesting. For thick materials, taper of sidewalls needs to be considered, while for thin materials (thickness much smaller than the Rayleigh length), taper is usually not a concern, and the main goal is to achieve through-cuts. Here, “thin” means that one is concerned with whether or not a piece of material with a certain thickness can be cut through, without the consideration of wall taper. In this paper, we are interested in laser cutting of thin materials and investigate how to maximize the cutting speed with a given laser power.

Many models have been proposed to explain the mechanisms of ultrafast laser ablation [1–5]. Laser scanning parameters have been optimized to get higher ablation efficiency [6–9]. However, most of the optimization methods are based on maximizing the ablation rate, which is often defined as the ablated volume within a certain amount of time. In laser cutting of thin materials, however, ablation volume is less of a concern than ablation depth. Instead of finding the parameters to get the largest ablation volume, the parameters to get the largest scanning speed that leads to through-cut perforation are more important in these cases. Furthermore, in many industrial applications, the exact properties of materials under processing are unknown or difficult to obtain. A model with as few material properties as possible is preferred for such scenarios. In this paper, we have developed a laser ablation model to maximize the laser scanning speed to cut through thin materials and we also show an example of how to apply the model to cutting through a material without knowing the material properties in advance. This work is important for the practical use of ultrafast lasers in applications such as processing thin films, non-woven fabrics, and display panels.

2. Model derivation

Our model is based on the work of Neuenschwander et al. [10] in which an analytical model was developed to maximize ablation volume in laser drilling when the laser beam is stationary. Even though the model can produce results that are consistent with the experiments of laser cutting, a model developed specifically for dynamical laser cutting is needed to fully understand and optimize laser cutting operations. We start with the logarithmical relationship between ablation depth ${z_{abl}}$ and laser fluence $\phi $ [3,11]

We assume the laser spot has a Gaussian profile with spot radius ${w_0}$ (measured at $1/{e^2}$ level), so the fluence distribution at radial position $r$ on the material surface can be expressed as

Due to the existence of threshold ${\phi _{th}}$, ablation by a single pulse is limited to a circular region with radius R. Therefore, ablation depth ${z_{abl}}$ is a piecewise function of radial position r.

In deriving Eq. (4), we have used Eqs. (1) and (2). R can be written as

Similarly, ${\phi _{th}}$ can be expressed as

Equations (4) to (6) have been verified in Neuenschwander’s work [12].

Now we consider a moving laser spot. As illustrated in Fig. 1(a), the first laser pulse irradiates the surface and is centered at point O. Without loss of generality, we choose a “point of interest” A that lies on the circle corresponding to the threshold condition of this pulse, so ablation depth at point A from the first pulse can be considered as zero. When the spot moves along the x-direction, as shown in Fig. 1(b), point A receives successive pulses that gradually increase the ablation depth pulse by pulse. Here we assume that ablation depth from a single pulse at a given point is determined by the “local fluence” at that particular point. Specifically, we can determine the fluence of the n-th pulse at point A as

 

Fig. 1. Schematic representation of the multi-pulse ablation on a point of interest. The material ablation is high in the pulse center and low in the edge. A circular region with radius R is defined with the edge fluence has zero ablation depth. When the laser is scanning, the point of interest experiences multi-pulse ablation with each pulse causing a different ablation depth due to different local fluences.

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Notice that the first term in Eq. (8) is zero, which can be confirmed by substituting Eq. (6) into Eq. (1). Thus, the total ablation depth ${z_{abl}}_T$ caused by all pulses can be calculated as

Using the identities of integer summation,

Equation (9) can be written as

Substituting Eq. (9) into Eq. (13), we obtain

Equation (14) can be used to derive the “critical condition” when the total ablation depth ${z_{ab{l_T}\;}}$ equals material thickness d.

Equation (15) is a quadratic equation with respect to v and by solving the equation, the critical speed at which, the laser beam just cuts through the material can be expressed as in Eq. (16).

Substituting R using Eq. (5) yields

Equation (17) gives an explicit relationship between ${v_c}$ and other related parameters, and it can be used in multiple ways to optimize laser cutting performance. For example, one can use this equation to derive or compute numerically the maximum critical speed ${v_{cmax}}$ at an optimized focal spot radius ${w_0}_{opt}$, as it will be discussed later.

To obtain a simplified form of Eq. (17), we assume $N \gg 1$. Therefore, we can make approximations $\frac{{N({N + 1} )}}{2} \approx \frac{{{N^2}}}{2}$ and $\frac{{N({N + 1} )({2N + 1} )}}{6} \approx \frac{{{N^3}}}{3}$. With these two approximations, Eq. (13) can be simplified to

Substitute Eq. (10) and Eq. (5) into Eq. (18), we obtain

Equation (19) can be written in a different form by substituting Eq. (5) into Eq. (19).

Again, we can set the condition that the total ablation depth ${z_{abl}}_T$ is equal to the thickness d, then

Substitute Eq. (3) into Eq. (21), we arrive at a simplied form of the critical speed ${v_c}$.

Equation (22) can be used to obtain the maximum ${v_c}$ with respect to the focal spot radius ${w_0}$ by computing the derivative

Correspondingly, at ${v_c} = {v_{cmax}}$, the peak fluence is ${e^3} \approx 20$ times of the threshold fluence.

The factor ${e^3}$ is a consequence of maximizing the ablation depth compared to the factor of ${e^2}$ by maximizing the ablation volume given Gaussian profile. Equations (24) and (25) suggest that for a cutting operation with all other parameters fixed, there exists an “optimal focal spot radius” ${w_0}_{opt}$ with which the highest cutting speed is obtained to achieve through-cuts for material thickness d. This conclusion can be understood intuively using Fig. 1. If ${w_0}$ is large, there will be a large number of pulses irraidating at point A. However, the peak fluence will be low, which leads to a low cutting speed because the local fluence for many pulses is below threshold. On the other hand, if ${w_0}$ is small, pulse-to-pulse overlap will be small, which results in a low cutting speed in order to obtain continuous through-cuts. Therefore, an optimal focal spot radius excists.

The key findings of this work are Eq. (17) and Eq. (22), which tell us that with all other parameters fixed, critical cutting speed ${v_c}$ to achieve through-cuts for material thickness d can be increased by finding an optimal focal spot radius ${w_0}_{opt}$.

3. Experimental study

Experiments have been performed to verify the theory. The experimental setup is schematically illustrated in Fig. 2. A femtosecond laser (Light Conversion, Pharos) was used that has a maximum average power of 6 W at 6 kHz repetition frequency. The pulse duration is measured as 193 fs and the center wavelength is 1030 nm. A motorized linear stage (Newport, ILS100CC) was used to allow precise linear motion control of the sample, which is a thin nonwoven sheet with a thickness of 170 µm provided by Kimberly-Clark. Since the material properties are unknown, and the only required material properties are the threshold fluence ${\phi _{th}}$ and the energy penetration depth $\mathrm{\delta }$ in the model, an experiment was conducted to obtain the relationship between the single-pulse ablation depth and peak fluences from which, a fit was performed to get the ${\phi _{th}}$ and $\mathrm{\delta }$.

 

Fig. 2. Schematic diagram of the experimental setup.

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In the experiment, the laser was focused by a lens with focal length 1000 mm to achieve a minimum focal spot radius of 184.6 µm ($1/{e^2}$ intensity level). Laser fluence was controlled by changing the average output power of the laser. Different numbers of pulses were delivered to hit different locations on the sample until the threshold number of pulses for each fluence that resulted in a complete through-drilling was obtained. The power and corresponding threshold pulse numbers are listed in Table 1.

Table 1. Average laser power and number of pulses that drill through the sample.

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The total thickness divided by the number of pulses is the ablation depth per pulse at that fluence [1]. Here an assumption was made that the ablation depth of each pulse in multi-pulse ablation is not influenced by previous pulses [12]. The fitting results of the energy penetration depth $\delta $ and the threshold fluence ${\phi _{th}}$ can be seen in Fig. 3. In all the experiments, no burning is observed on both the front and back side of the sample.

 

Fig. 3. The fitting of the energy penetration depth $\delta $ and the threshold fluence ${\phi _{th}}$ from the ablation depth per pulse at different peak fluences as described in Eq. (1). The fitting results are ${\phi _{th}} = 0.217 \pm 0.075\;J/c{m^2},\;\delta = 17.42 \pm 4.77\;\mu m.$

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To verify the predictions of Eqs. (17) and (22), cutting experiments were performed at different situations to determine the maximum threshold cutting speed ${v_{cmax}}$ and the corresponding optimal focal spot radius ${w_0}_{opt}$. The lens was replaced by another lens of 35 mm focal length to achieve wide tunability in focal spot radius (discussed later). The power delivered to the sample was kept at 1.2 W. This is to simulate a common scenario in practice where the laser is running at a certain maximum power to achieve the highest processing throughput. All other parameters were kept the same. Initially, the sample was placed on the focal plane of the lens. Different scanning speeds were applied to find the critical speed for a through-cut, which is determined by visual examination or by a microscope. Here the cutting strategy is to do a single pass cut at each parameter setting. Other cutting strategies such as multiple cutting passes at high speeds are not discussed here. The threshold cutting speed was recorded with this spot radius. Then, the sample was shifted by the stage toward the laser incoming direction so that the spot radius was increased on the sample. The same cutting test was repeated to find the threshold cutting speed for a different focal spot radius. Multiple spot radii and threshold speeds were recorded and displayed with the model predictions in Fig. 4.

 

Fig. 4. The real cut-through speeds measured by experiments and predicted by Eq. (17) and Eq. (22) using the fitted energy penetration depth and threshold fluence (a) without fluence correction α and (b) with fluence correction $\alpha $ . The pulse overlap (reading from the right y axis) shows the validity of reducing Eq. (17) to Eq. (22) at the assumption of $N \gg 1$.

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It is found that the predictions and the experiments show a similar trend, i.e, there is a maximum in the threshold cutting speed ${v_c}$. The prediction of Eq. (22) is slightly lower than that of Eq. (17), which is due to the approximation of high pulse overlap ratio made in the derivation (discussed in the next section). However, the experimental results are found to be lower than the predictions. The difference is attributed to effects that are not accounted for in our model. One of these effects is plasma shielding and particle shielding [13–15], which reduces the laser energy delivered on the sample due to obstructions from plasma plume, particles, etc. Another possible reason is that not all laser energy delivered to the sample could be absorbed. Absorptance could also be modified by pulses [16]. Therefore, our model is corrected by multiplying the laser fluence with a coefficient $\alpha $.

It is found that $\alpha = 2.3$ gives a good comparison with the experimental result, as seen in Fig. 4(b).

4. Discussion

From the modeling and experiments, we can find that there is an optimized spot radius that can lead to the maximum scanning speed that cuts through the thin materials. The complete cut-through speed model expressed in Eq. (17) is reduced to Eq. (22) when the pulse overlap calculated in Eq. (26) is beyond ∼96% in this case, which is consistent with the assumption that the pulse number that has ablation contribution to a point of interest is much larger than 1 ($N \gg 1$).

In cases where the pulse overlap is not that high, the complete model of speed in Eq. (17) has to be used and a numerical solution can be obtained to get the cut-through speed at various spot radii for thin material perforation.

When the simplified model is applied as expressed in Eq. (22), the optimum fluence is now ${e^3}$ times the threshold fluence, which is different from the optimum fluence ${e^2}$ times the threshold fluence in the optimum ablation volume derivation [12].

A comparison of parameters (maximum speed, optimum spot radius and pulse overlap) among the two models and experiment has been summarized in Table 2. The difference in optimum spot radius is more obvious which can be explained as the incomplete use of laser energy in cutting due to limited absorption and shielding effects.

Table 2. A comparison of laser/scanning parameters in models and experiment.

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Usually, there is an incubation effect in multi-pulse ablation [18,19], which describes the fact that the threshold fluence for one pulse is different from multiple pulses. In our experiment used to fit the energy penetration depth $\delta $ and the threshold fluence ${\phi _{th}}$, the incubation effect is not considered. This could lead to the deviation between the modelling from experimental results discussed above. This could explain why the scaling factor is needed. The model developed is for single-pass processes. Multiple passes require the sample to be stationary, which may not be possible for certain applications. One can revise Eq. (1) to that ${z_{abl}}$ is a function of both fluence and number of passes. This requires a new derivation of the model which is beyond the scope of current work. Given the fact that many effects such as the incubation effect discussed above is excluded from the model, the scaling factor is therefore a phenomenological factor that relates to specific laser-matter interaction processes and need to be obtained on a case-by-case basis.

5. Conclusion

In this paper, we present an ablation model that gives the optimized cut-through scanning speeds at various spot radii in thin material perforation. This model focuses on maximizing the ablation depth rather than ablation volume, which is closer to the case of thin material perforation. When the pulse overlap is high, the formulation can be simplified, where optimum spot radius and optimum peak fluence have explicit expressions, which are compared to experimental results. The optimum laser parameters in this new model are different from those obtained at maximizing the ablation volume and can be practical for industrial applications.