Numerical simulation of multi-pulsed femtosecond laser ablation: effect of a moving laser focus

Yiwei Dong; Yiwei Dong; Zongpu Wu; Yancheng You; Chunping Yin; Wenhui Qu; Xiaoji Li

doi:10.1364/OME.9.004194

1. Introduction

Femtosecond lasers have been widely employed in the field of precision machining, owing to their extremely short pulse width and high peak power. Theoretically, these lasers can provide high-precision cold processing without a heat effect. However, the laser energy and process parameters used during actual processing still have a significant impact on the processing accuracy and efficiency.

To improve the processing accuracy and efficiency of femtosecond lasers, many experiments and numerical simulations have focused on the interactions between femtosecond lasers and materials. In addition, mathematical models for describing femtosecond laser ablation have been proposed. Ding et al. [1] simulated the processes of femtosecond and picosecond laser pulse ablation of copper using an improved one-dimensional fluid dynamics code, and they explored the variation in the ablation depth with the pulse width and the influence of the pulse repetition rate on the ablation. Nedialkov et al. [2] simulated the processing of holes with a high depth-to-diameter ratio on iron with 100 fs laser pulses, using a molecular dynamics model. The results indicated that the morphology of a hole was related to the interaction between the ablated particles and the wall of the hole. In other words, the ablated particles collided with the wall at a high speed under high energy-density pulses, leading to secondary ablation. According to the semiclassical two-temperature model, Chen et al. [3] studied heat conduction in copper films irradiated with a femtosecond laser pulse train. They found that if the pulse interval was relatively short (approximately 1 ps), then the number of pulses should be increased to increase the lattice temperature, whereas the number of pulses should be reduced to increase the lattice temperature for long pulse intervals (approximately 100 ps). Although these three models can be used to simulate the interactions between ultrafast laser pulses and materials, fluid dynamics and molecular dynamics models are complex, and require significant computational resources [4]. Therefore, the two-temperature model is a better choice when it is only necessary to explore parameters such as the temperature and ablation depth during interactions between femtosecond laser pulses and materials. Jiang et al. [5, 6] studied the process of machining materials with multiple femtosecond laser pulses and femto-nanosecond combined laser pulses based on a two-dimensional two-temperature model, and presented a series of results. In their proposed numerical model, the variation of the waist radius of the beam along its propagation direction was not considered in the laser source term. Therefore, the model is suitable only for simulating hole machining on metal films, whereas its accuracy in simulating the processing of deep holes in materials with thicknesses greater than the Rayleigh length of the laser is insufficient. Therefore, Zhang et al. [7] proposed a two-temperature model with an improved laser source term for processing deep holes with femtosecond laser pulses. This model provides an enhanced simulation accuracy, as it considers the variation in the beam waist radius with processing depth.

Currently, in the actual processing of deep holes, helical drilling is an effective approach for processing holes with a high aspect ratio. During helical drilling, the laser focus moves downward as material is removed to correct the defocusing distance generated during processing. However, there is no description of this process in current models; thus, shortcomings remain when modeling deep-hole processing with multi-pulse femtosecond laser ablation.

In this study, we propose an enhanced two-temperature model to describe the downward motion of the laser focus during femtosecond laser helical drilling. Copper was used as the target material in this study owing to its relatively simple thermal properties. To reduce the computational scale of the numerical model, the influence of the helical trajectory on the ablation process was neglected. We only simulated the process during which material was ablated by the femtosecond laser focus moving downward, to explore the influences of the pulse interval and the downward velocity of the laser focus on the electron-lattice temperature and the ablation depth in the material.

2. Computation model

2.1 Adapted two-temperature model and laser source model

Regarding the interactions between an ultrafast laser pulse and metallic materials, Anisimov et al. [8] proposed a two-temperature model to describe the energy distribution in the internal electronic system and lattice system of a material. According to this model, the lattice thermal conductivity is approximately two orders of magnitude smaller than the electronic thermal conductivity, thus ignored the heat conduction term in the lattice [9]. Nonetheless, according to Rethfeld et al. [10] and Xiong et al. [11], during a multi-pulse laser ablation process, heat conduction in the lattice must not be overlooked. Therefore, in this study we adopt a two-temperature model that considers lattice heat conduction, which is expressed as follows:

(1)$${C_e}\frac{{\partial {T_e}}}{{\partial t}} = \nabla \cdot [{k_e}\nabla {T_e}] - G({T_e} - {T_l}) + Q$$

(2)$${C_l}\frac{{\partial {T_l}}}{{\partial t}} = \nabla \cdot [{k_l}\nabla {T_l}] + G({T_e} - {T_l})$$

where $C$ is the heat capacity, $T$ is the temperature, $k$ is the conductivity, G is the electron-lattice coupling factor, Q is the source term. The subscripts e and l denote electron and lattice, respectively. As ${C_e}$, ${C_l}$, ${k_e}$, ${k_l}$ and G are all temperature dependent, based on the model proposed by Chen et al. [12,13] and Xiong et al. [14], we used the following equation to describe the previously given parameters:

(3)$${C_e}({T_e}) = \left\{ \begin{array}{l} {C_{e0}}{T_e}\quad \quad \quad \quad \quad{T_e} \le \frac{{{T_F}}}{{{\pi^2}}}\\ \frac{{2{C_{e0}}{T_e}}}{3} + \frac{{C_e^{\prime}({T_e})}}{3}\quad \,\,\frac{{{T_F}}}{{{\pi^2}}} < {T_e} \le \frac{{3{T_F}}}{{{\pi^2}}}\\ N{k_B} + \frac{{C_e^{\prime}({T_e})}}{3}\quad \,\,\;\frac{{3{T_F}}}{{{\pi^2}}} < {T_e} \le {T_F}\\ \frac{{3N{k_B}}}{2}\quad \quad \quad \quad \quad{T_e} > {T_F} \end{array} \right.$$

where ${C_{e0}}$ is the coefficient for electron heat capacity, ${T_F}$ is the Fermi temperature, ${k_B}$ is the Boltzmann constant, N and $C_e^{\prime}$ can be calculated by:

(4)$$N = \frac{{{N_A}}}{V}$$

(5)$$C_e^{\prime}({T_e}) = \frac{{{C_{e0}}{T_F}}}{{{\pi ^2}}} + \frac{{{{3N{k_B}} \mathord{\left/ {\vphantom {{3N{k_B}} 2}} \right.} 2} - {{{C_{e0}}{T_F}} \mathord{\left/ {\vphantom {{{C_{e0}}{T_F}} {{\pi^2}}}} \right.} {{\pi ^2}}}}}{{{T_F} - {{{T_F}} \mathord{\left/ {\vphantom {{{T_F}} {{\pi^2}}}} \right.} {{\pi ^2}}}}}({T_e} - {{{T_F}} \mathord{\left/ {\vphantom {{{T_F}} {{\pi^2})}}} \right.} {{\pi ^2})}}$$

The electron heat conductivity coefficient can be expressed as follows:

(6)$${k_e}({T_e},{T_l}) = \chi {\upsilon _e}\frac{{{{(\upsilon _e^2 + 0.16)}^{1.25}}(\upsilon _e^2 + 0.44)}}{{{{(\upsilon _e^2 + 0.092)}^{0.5}}(\upsilon _e^2 + \eta {\upsilon _l})}}$$

where ${\upsilon _e}\textrm{ = }{{{T_e}} \mathord{\left/ {\vphantom {{{T_e}} {{T_F}}}} \right.} {{T_F}}}$ and ${\upsilon _l}\textrm{ = }{{{T_l}} \mathord{\left/ {\vphantom {{{T_l}} {{T_F}}}} \right.} {{T_F}}}$, $\chi $ and $\eta $ are material-dependent constants. For copper, $\chi = 377\,\textrm{W/(m} \cdot \textrm{K)}$ and $\eta = 0.139$ [3].

The electron-lattice coupling coefficient can be expressed as follows:

(7)$$G({T_e},{T_l}) = {G_{RT}}\left[ {\frac{{{A_e}}}{{{B_l}}}({T_e} + {T_l}) + 1} \right]$$

where ${G_{RT}}$ is the electron–lattice coupling factor at room temperature, ${A_e}$ and ${B_l}$ are material constants, respectively.

The lattice heat capacity is given by

(8)$${C_l}({T_l}) = \left\{ \begin{array}{ll} 313.7 + 0.324{T_l} - 2.687 \times {10^{ - 4}} \times T_l^2 + 1.257 \times {10^{ - 7}} T_l^3 & {T_l} \le {T_m}\\ 510.1 & {T_l} > {T_m} \end{array} \right.$$

where ${T_m}$ is the melting temperature and the lattice heat conductivity coefficient is

(9)$${k_l}({T_e},{T_l}) = {{{k_e}({T_e},{T_l})} \mathord{\left/ {\vphantom {{{k_e}({T_e},{T_l})} {99}}} \right.} {99}}$$

Regarding multi-pulse femtosecond laser helical drilling and the downward motion of the laser focus, the interaction between the laser beam and the material as the focus moves into the material can be simplified as shown in Fig. 1, because the effects of the helical trajectory on ablation are not considered.

Fig. 1. Interaction between the laser beam and material.

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At the beginning of this process, the laser focus was on the initial surface of the material ($z = 0$), and the center of the spot was at $\textrm{r} = 0$. Material was removed as the process progressed, forming a new material surface, and the laser focus moved downward over time. Therefore, the laser beam propagation equation can be expressed as follows:

(10)$$\omega (z,t) = {\omega _0}\sqrt {1 + {{\left( {\frac{{z - vt}}{{{z_R}}}} \right)}^2}} $$

In this equation, ${\omega _0}$ is the focal radius, ${z_R}$ is the Rayleigh length, v is the speed of focus downward.

When the laser pulse irradiated a new surface, the pulse propagation was blocked, owing to the opacity of the material. The energy distributed inside the material was affected by the optical penetration depth $\delta $ of the material and the electronic ballistic length ${\delta _b}$. Thus the material only absorbed a portion of the energy. Because the material was removed, no interaction occurred between the laser beam and material on the new surface, implying that it is not necessary to consider the laser energy distributed outside the material. Therefore, the distribution of the laser energy in the material can be expressed as follows:

(11)$$S(r,z,t) = \frac{{1 - R({T_e},{T_l})}}{{(\delta + {\delta _b})}} \cdot \frac{{2{E_p}}}{{\pi \cdot {\omega ^2}(z,t)}} \cdot \exp \left( { - \frac{{z - {z_s}}}{{\delta + {\delta_b}}} - \frac{{2{r^2}}}{{{\omega^2}(z,t)}}} \right)\quad \quad (z \ge {z_s})$$

where ${E_p}$ is the single pulse energy, ${z_s}$ is the interfacial location, R is the reflectivity of material to laser, which has a significant influence on the laser energy distribution on the material. The dielectric critical point model without Lorentzian terms is used to describe the temperature-dependent reflectivity [15]:

(12)$$\varepsilon = {\varepsilon _\infty } - \frac{{\xi _p^2}}{{\xi (\xi + i\gamma )}} = {\varepsilon _\infty } - \frac{{\xi _p^2}}{{{\xi ^2} + {\gamma ^2}}} + i\frac{{\xi _p^2 \cdot \gamma }}{{\xi ({\xi ^2} + {\gamma ^2})}} = {\alpha _1} + i{\alpha _2}$$

where $\xi _p^{}$ is the plasma frequency, $\xi $ is the laser frequency, ${\varepsilon _\infty } = 9.428$, $\gamma = \frac{1}{{{\tau _e}}}$, and the relaxation time of electrons ${\tau _e}$ can be expressed as follows [1,16]:

(13)$${\tau _e} = \frac{1}{{{A_e}T_e^2 + 1.41{\upsilon _{ep}}}}$$

where ${\upsilon _{ep}}$ is the electron-phonon collision rate, which is defined as [4,16]:

(14)$$\begin{array}{l} {\upsilon _{ep}} = \frac{{{\Xi ^2}}}{{8\pi {\varepsilon _F}{k_F}{\rho _0}{v_s}}} \cdot \frac{{{m_{opt}}}}{{{m_e}}} \cdot \left\{ {\int_0^{{q_b}} {\frac{{{e^{{\phi_l}}} - {e^{{\phi_e}}}}}{{({e^{{\phi_l}}} - 1)({e^{{\phi_e}}} - 1)}}{q^4}dq + \varsigma \int_0^{{q_b}} {\frac{{{e^{{\phi_l}}} - {e^{{\phi_e}}}}}{{({e^{{\phi_l}}} - 1)({e^{{\phi_e}}} - 1)}}{q^3}dq} } } \right.\\ \quad \quad \left. { + {q_b}\frac{{{e^{{\varphi_l}}} + {e^{{\varphi_e}}}}}{{({e^{{\varphi_l}}} - 1)({e^{{\varphi_e}}} - 1)}} \times \frac{{{{(2{k_F})}^4} - q_b^4}}{4} - 4\varsigma q_b^2k_F^2\frac{{{e^{{\varphi_l}}} - {e^{{\varphi_e}}}}}{{({e^{{\varphi_l}}} - 1)({e^{{\varphi_e}}} - 1)}}} \right\} \end{array}$$

where ${\varepsilon _F}$ is the Fermi energy, ${k_F}$ is the Fermi radius, ${\rho _0}$ is the density of material at room temperature, ${v_s}$ is the longitudinal sound velocity in solid, q is the phonon wave vector, $\Xi = 3.99\textrm{eV}$, ${q_b} = 8.97 \times {10^9}{\textrm{m}^{\textrm{ - 1}}}$, $\hbar = {h \mathord{\left/ {\vphantom {h {2\pi }}} \right.} {2\pi }}$, ${\phi _l} = \frac{{\hbar q{v_s}}}{{{k_B}{T_l}}}$, ${\phi _e} = \frac{{\hbar q{v_s}}}{{{k_B}{T_e}}}$, ${\varphi _l} = \frac{{\hbar {q_b}{v_s}}}{{{k_B}{T_l}}}$, ${\varphi _e} = \frac{{\hbar {q_b}{v_s}}}{{{k_B}{T_e}}}$, ${m_{opt}} = 1.39{m_e}$, and $\varsigma = 2{m_{opt}} \cdot {v_s}/\hbar $. It is worth noting that the last two terms in the bracket of Eq. (14) are important at room temperature (approximately 300K) and above, while the first two terms only dominant at low temperatures, which implies that the first two terms can be discounted during the calculation.

Then, the refractivity and the extinction coefficient can be determined as ${n_r} = \sqrt {\frac{{{\alpha _1} + \sqrt {\alpha _1^2 + \alpha _2^2} }}{2}} $ and ${n_i} = \sqrt {\frac{{ - {\alpha _1} + \sqrt {\alpha _1^2 + \alpha _2^2} }}{2}} $, respectively.

According to the Fresnel equation, the reflectivity is given by

(15)$$R({T_e},{T_l}) = \frac{{{{({n_r} - 1)}^2} + {n_i}}}{{{{({n_r} + 1)}^2} + {n_i}}}$$

Regarding the processing time, the temporal profile of the output laser pulse follows a Gaussian distribution. Thus, the distribution of the laser energy of the pulse over time can be expressed as follows:

(16)$$T(t) = \left\{ \begin{array}{l} \frac{1}{{{t_p}}} \cdot \sqrt {\frac{{4\ln 2}}{\pi }} \cdot \exp \left[ { - 4\ln 2{{\left( {\frac{{t - 2{t_p}}}{{{t_p}}}} \right)}^2}} \right]\quad \quad \quad \, (0 \le t \le 4{t_p})\\ 0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (4{t_p} < t < \frac{1}{f}) \end{array} \right.$$

Equation (16) shows that the first pulse was output at $t = 0$, reached its peak when $t = 2{t_p}$, and ended at $t = 4{t_p}$, ${t_p}$ is the pulse width. In addition, $T(t)$ was set to zero following the end of the first pulse and before the start of the next pulse, to ensure that no further laser energy was input into the material.

We obtained the laser source term to simulate drilling considering the downward motion of the laser focus from the previous equations for the spatial and temporal distributions of laser energy:

(17)$$Q(r,z,t) = S(r,z,t) \cdot T(t)$$

2.2 Phase transformation model

In the proposed laser source model, determining the new material surface ${z_s}$ is key to determine whether the model can accurately describe ablation via the downward motion of multiple focused femtosecond laser pulses. Ultrafast laser ablation involves many complicated physical mechanisms, such as Coulomb explosions, mechanical fractures, and phase explosions. Existing numerical simulation methods are divided into the fixed mesh method and deformed mesh method. The fixed mesh method has been used extensively to numerically simulate femtosecond laser ablation, owing to its simple mathematical structure and low computational cost. However, this method can only be used to determine the position of the phase transformation interface. If the grid does not deform, then the change in the material structure during actual processing cannot be determined. Therefore, this method is unsuitable for simulating the multi-pulse femtosecond laser drilling process as the laser focus moves downward. The deformed mesh method can transform the calculation unit based on the material removal mechanism during simulation. Thus, the position of the phase transformation interface needs to be determined based on the real-time position of ${z_s}$. Therefore, we adopted the critical-point separation model proposed by Wu et al. [17]. According to this model, under the action of an ultrafast laser pulse with a high energy density (>1 $J/c{m^2}$), we first heated the workpiece to the highest temperature that would not cause a significant change in the density of the material, and then the density decreased according to a relationship close to $T \propto {\rho ^{{2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}$. Owing to thermodynamic instability, the irradiated material was transformed into a bubble-drop transition layer as the expansion trajectory entered the unstable region near the critical point. The region above was ablated and separated from the parent material, and the region below resolidified on the parent material. Thus, we considered the initial depth of regions in the material forming the transition layer as the ablation depth. The phase-separation temperature of the material is related to the critical temperature, and can be determined based on the following equation:

(18)$${T_{sep}} = {T_c} \cdot {(\frac{{{\rho _0}}}{{{\rho _c}}})^{{2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}}$$

Hence, to obtain the position of ${z_s}$, only the temperature field in the material must be determined by solving the two-temperature equation. The position of the region whose maximum temperature is equal to the phase-separation temperature can subsequently be determined. During the calculation, to force the material interface to move to the critical-point separation temperature line, we used a dynamic mesh technique proposed by Dold [18], where the speed of sound in the material ${v_s}$ is used to describe the deformation of the boundary surface, to track the position of the processed surface.

2.3 Establishment of the computation model

According to the two-temperature and phase transformation models, we established a computational model to describe the downward motion of the laser focus during multi-pulse femtosecond laser ablation, and used the finite element method to solve this. Because a multi-pulse femtosecond laser ablation process that only considers only the downward motion of a laser focus is axially symmetric, we employed a two-dimensional axisymmetric model in the simulation to reduce the computational cost. The computational model and boundary conditions are illustrated in Fig. 2.

Fig. 2. Simulation model and boundary conditions.

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During the calculation, the laser focus radius was 10 µm in the simulation. Thus, we set the maximum mesh size to be less than 0.06 µm in the irradiated region (0 < r < 15 µm), to ensure that the laser energy could be correctly resolved. Because the pulse time of the femtosecond laser is extremely short, whereas the pulse period is relatively long, the time scale of the solution is large, and the computational cost is high. Therefore, we described multiple laser pulses using pulse trains, where each pulse train consisted of three pulses, as shown in Fig. 3. We only explored the influence of the time interval $\triangle d$ between three pulses in one pulse train on the electron–lattice temperature and ablation depth.

Fig. 3. Pulse distribution.

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We selected a copper workpiece as the target material for laser ablation as it has relatively complete thermophysical parameters. The relevant material parameters and laser parameters used in the simulation are shown in Table 1. We solved this model using the backward difference method. To ensure that the energy could be resolved correctly in terms of the irradiation time, we set the initial step size to 0.1 fs and kept the solution step size constant at 10 fs during irradiation, while the step size was kept constant at 200 fs between pulses. Meanwhile, to decrease the calculation time we used an adaptive step size for the rest of the simulation.

Table 1. Parameters in the computation model

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3. Results and analysis

3.1 Verification of simulation model

To verify the rationality of the computational model proposed in Section 2, we simulated a single-pulse laser ablation process by varying the power density of the incident laser, and compared the ablation depth results with those from existing research [4]. The laser energy density ranged from 2 to 10 ${J \mathord{\left/ {\vphantom {J {c{m^\textrm{2}}}}} \right.} {c{m^\textrm{2}}}}$, yielding a variation in the ablation depth with pulse energy density, as shown in Fig. 4.

Fig. 4. Laser ablation depth as a function of the pulse fluence.

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As shown in Fig. 4, Dasallas et al. [4] proposed a numerical model exploring the influence of the laser fluence on the ablation depth in copper with s and p polarization at vertical angles of incidence. From the obtained laser fluence-dependent ablation depth curve, at a lower laser fluence (< 4 ${J \mathord{\left/ {\vphantom {J {c{m^\textrm{2}}}}} \right.} {c{m^\textrm{2}}}}$), the ablation depth is sensitive to changes in the fluence, which implies that a small increase in the laser fluence can lead to a significant increase in the ablation depth. Subsequently, the ablation depth becomes less sensitive to changes in the laser fluence, and the ablation depth tends to increase at a smooth rate.

To further verify the proposed computational model, we used the laser parameters obtained from Wang et al. [20,21] to perform an additional calculation. The comparison between the simulated and published results is shown in Fig. 5. As can be seen on the figure, the ablation depths calculated by the proposed axisymmetric model are found to be in good agreement, including the trend and values.

Fig. 5. Comparison of ablation depth simulated with the axisymmetric model and published results

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The laser fluence-dependent ablation depth curve calculated using the proposed model exhibits the same tendency, which indicates that the proposed model is reasonable for describing the influence of the laser fluence on the ablation depth. However, it is worth noting that there remains a gap between the ablation depth calculated with the proposed model and the results obtained from the existing literatures, which may relate to differences between the methods used in the models to determine the depth. In our proposed model, a critical-point phase separation model is used to describe the phase transition, and the dynamic interface is used to move the material interface to the isophase separation temperature line with a constant sound velocity ${v_s}$ to determine the deformation of the boundary surface. In the model proposed by Wang et al. [21], a grid would be removed when its calculated temperature reached to 90% of the thermodynamic equilibrium critical temperature (7696 K), which can be regarded as the material removal. Furthermore, compared with the method proposed by Dasallas et al. [4], which did not consider the phase transition, only the temperature field during the ablation process can be determined. Moreover, Dasallas et al. regarded the ablation depth as the depth at which the temperature reaches the material vaporization temperature (9100 K), which leads to the difference in the calculated ablation depths.

Based on the above analysis, one can conclude that our proposed model can be used to describe the influence of parameters such as laser fluence on the electron temperature, lattice temperature, and ablation depth in the material during femtosecond laser processing. Therefore, it is acceptable to use the proposed model to further investigate the effects of parameters such as the pulse interval time and downward velocity of the focus on the electron temperature, lattice temperature, and ablation depth.

3.2 Effect of pulse interval

To determine the effect of the pulse interval on multi-pulse femtosecond laser ablation with a fixed focus, we varied the pulse interval time ($\triangle d = 1,\;5,\;10,20,50,100,200,300\;\textrm{ps}$) to simulate the ablation process. The pulse interval time-dependent maximum electron temperature, pulse interval time-dependent maximum lattice temperature, and pulse interval time-dependent ablation depth are shown Figs. 6, 7, and 8, respectively.

Fig. 6. Maximum electron temperature at different pulse intervals. Pulse width is 250fs, laser wave length is 800nm, single pulse energy is 10µJ (laser fluence is 6.37 $J/c{m^2}$)

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Fig. 7. Maximum lattice temperature at different pulse intervals. Pulse width is 250fs, laser wave length is 800nm, single pulse energy is 10µJ (laser fluence is 6.37 $J/c{m^2}$)

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Fig. 8. Ablation depths at different pulse intervals. Pulse width is 250fs, laser wave length is 800nm, single pulse energy is 10µJ (laser fluence is 6.37 $J/c{m^2}$)

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Figure 6 shows that three pulses can cause the maximum electron temperature to rise stepwise when the pulse interval is small ($\triangle d = 1,\;10\textrm{ps}$). The maximum electron temperature and temperature increase that can be obtained after the three pulses are higher when the pulse interval time is smaller. Figure 7 also shows the same trend in the maximum lattice temperature, and the lattice temperature is high. The reason is that, for the short pulse intervals, since the second and third pulses arrive, the material temperature keeps at a high level due to the heat accumulation in the ablated zone, therefore, the maximum lattice temperature can reach to a relatively high level. Thus, using the two-temperature model to solve the lattice temperature, the instantaneous maximum temperature would be much higher than the phase-separation temperature of the material, and it decreases as the ablation continues. This effect was also mentioned and discussed in a previous published paper [17].

Although the second and third pulses can also increase the electron and lattice temperatures as the pulse interval increases ($\triangle d = 20,\;200\textrm{ps}$), the temperature increase decreased when the pulse interval time increases, and the electron and lattice temperature changes caused by the third pulse are insignificant.

From the time-dependent ablation depth curve shown in Fig. 8, one can observe that the maximum ablation depth increases rapidly when the interval time is less than 20 ps within the selected pulse interval range. When the interval time ranges from 20 ps to 200 ps, the sensitivity of the maximum ablation depth to the pulse interval is considerably reduced, at which point the increased pulse interval time has an insignificant effect on increasing the ablation depth. When the pulse interval time is greater than 200 ps, the increased pulse interval time cannot affect a further increase in the maximum ablation depth. Furthermore, the maximum ablation depth begins to decrease as the pulse interval time increases. Therefore, one can conclude that a smaller pulse interval leads to faster ablation, although the ablation ends earlier and the maximum ablation depth is highly sensitive to the pulse interval when the pulse interval is less than 20 ps.

The short pulse intervals ($\triangle d < 20\textrm{ps}$) can generate higher maximum electron and lattice temperatures with a smaller ablation depth arises because heat cannot be transferred from electrons and the lattice to the non-laser-affected zone as the pulse interval time decreases. Therefore, when a subsequent pulse arrives, the material temperature in the laser ablation zone remains very high, and the heat accumulation effect is obvious. Thus, the maximum electron and lattice temperatures are much higher. Shortly after the maximum lattice temperature exceeds the phase separation temperature, the electron and lattice temperatures reach equilibrium and rapidly decrease, resulting in a short ablation duration, which affects the ablation depth in the material.

Figure 7 shows that the maximum lattice temperature is always higher than the phase separation temperature ${T_{sep}}$ between the first and last pulses when the pulse interval time is less than 20 ps, i.e., material ablation is still in progress, and there is no stagnation between pulses. At this point, the ablation time is determined by the pulse interval. Therefore, when the pulse interval is less than 20 ps, the maximum ablation depth is highly sensitive to variations in the pulse interval time.

As the pulse interval increases ($20\textrm{ps} < \triangle d < 200\textrm{ps}$), the electron and lattice temperature increases caused by subsequent pulses decrease, and the maximum lattice temperature is higher than the phase separation temperature for a longer period of time, increasing the ablation duration. Therefore, the ablation depth is greater than the ablation depth with a short pulse interval. However, as the pulse interval time increases further ($\triangle d > 200\textrm{ps}$), the electron and lattice temperatures have already reached equilibrium before the arrival of the next pulse and are lower than the phase separation temperature. Thus, ablation stagnates and the heat accumulation effect is weakened. Hence, the lattice temperature increases and exceeds the phase separation temperature again after subsequent pulses, but the duration for which the maximum lattice temperatures are above the phase separation temperature is smaller than the pulse interval time (200 ps). Thus, the maximum ablation depth decreases. One can infer that the heat accumulation effect will be further weakened if the pulse interval increases further, and the influence of the current pulse on the subsequent pulse will be reduced. Therefore, the pulse train technique for the ablation process will lose its advantage.

Consequently, when the femtosecond laser pulse train was used to drill, there exists a reasonable pulse interval time, which allows the lattice temperature to be kept above the phase separation temperature for a relatively long time as the pulse moves downward, thereby achieving a larger ablation depth and a higher drilling efficiency.

3.3 Effect of downward focus velocity

One can conclude from Section 3.2 that the ablation depth can reach a higher value when the pulse interval is 200 ps. Meanwhile, the calculation of Section 3.1 shows that the ablation depth is approximately 98.28 nm when the energy of a single pulse is 10µJ (laser fluence: 6.37 $J/c{m^2}$). To verify that the proposed laser source term can be used to describe ablation as the laser focus moves downward, ablation was simulated at seven different downward focus velocities ($v = 0,\;250,\;500,\;750,1000,1250,1500\;{\textrm{m} \mathord{\left/ {\vphantom {\textrm{m} \textrm{s}}} \right.} \textrm{s}}$) with a pulse interval time of 200 ps and a pulse train in which single pulse energy is 10µJ (laser fluence: 6.37 $J/c{m^2}$). These simulations include cases in which positive and negative defocusing arises as the laser focus moves downward during the ablation process (here we define a focus position that is higher than the surface of the material as positive defocusing, and vice versa as negative defocusing). To better demonstrate the change of ablation depth with the focus down velocity, we set the ablation depth when the speed of focus down is $1500\;{\textrm{m} \mathord{\left/ {\vphantom {\textrm{m} \textrm{s}}} \right.} \textrm{s}}$ as the reference value (set to 0), then calculated the differences between this value and other ablation depths under different focus downward velocities. These differences were scaled up as relative ablation depth increments and represented as shown in Fig. 9.

Fig. 9. Relative ablation depth increment at different focus downward velocities.

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Figure 9 shows that the maximum ablation depth initially increases and subsequently decreases as the downward focus velocity increases. The maximum ablation depth corresponds to a focus velocity of 500 m/s. As discussed in a published work [22], there exists a critical value of the focus downward velocity, which can maximize the ablation depth. If the focus downward velocity is lower than the critical value, the ablation depth will be increased as the focus downward velocity increases, otherwise, the ablation depth will be decreased as velocity increases. These experimental results are consistent with the variation trend of the ablation depth shown in Fig. 9.

These simulation results show that the ablation depth generated by the first pulse is 98.28 nm, and the positive defocusing is 98.28nm since the focus remains fixed. After the second pulse, the ablation depth and positive defocusing continue to increase. The laser fluence irradiating the surface of the ablated material is weaker than that at the focus, causing the decrease of the ablation depth during subsequent pulses. When the laser focus moves down at a constant speed of 250 m/s, the focus is at $z = 50\textrm{nm}$, while the material surface is at ${z_s} = 98.28\textrm{nm}$ at the beginning of the second pulse. Thus the positive defocusing is 48.28 nm, increasing the ablation depth during the second pulse, owing to the higher fluence compared to a fixed focus. When the focus moves down at a constant speed of 500 m/s, the focus is at $z = 100\textrm{nm}$ at the beginning of the second pulse, and the negative defocusing is 1.72 nm, resulting in a higher laser fluence after the second pulse than in the cases where the focus remains fixed or moves downward at 250 m/s. Thus, the ablation depth can be further increased. When the focus moves downward at 750 m/s or faster, the second pulse has a negative defocusing of at least 51.72 nm, and the ablation depth decreases.

The small difference in the ablation depths at different downward velocities is a result of the limited computational resources for the simulation. The number of calculated pulses is only three, while the total solution time is only 1000 ps, and the defocus is only on the order of nm. Meanwhile, the Rayleigh length of the laser used in the simulation reaches 392.65 nm, and the change in the laser fluence caused by defocusing on the order of nm is small in comparison. Furthermore, the implemented simulation only examines three pulses, and the maximum ablation depth is only approximately 307.4 nm. Thus, the difference of these maximum ablation depth is small. In an actual industrial application, such as the laser helical processing of deep holes, tens of millions of pulses are generally used, and the accumulated difference can reach the order of µm. One can see that the velocity of the focus has a certain influence on the ablation efficiency during ablation processing, owing to defocusing during each pulse. Thus, the laser fluence changes during processing. Therefore, when using multiple femtosecond laser pulses for processing deep holes, the downward focus velocity can be determined based on the ablation depth of the material during a single pulse and the pulse interval time. This can be used to ensure that the negative defocus during the process is always less than 50 nm, or the laser focus can always be placed on the material surface during processing.

4. Conclusions

The influences of the downward focus velocity and pulse interval time during helical drilling on thick plates using a femtosecond laser were investigated in this study. An adapted two-temperature model was proposed to describe the effect of a moving laser focus during drilling, as well as a numerical model for describing the ablation of holes with a high aspect ratio in metal with femtosecond laser pulses. A comparison of the single pulse ablation simulation results with results from the literatures verifies the validity of the proposed model. The effects of the pulse interval and focus velocity on the electron and lattice temperatures, as well as the ablation depth during processing, were studied. It was found that using a shorter pulse interval can increase the maximum electron and lattice temperatures, but it does not significantly increase the maximum ablation depth in the material. However, heat does not significantly accumulate when the pulse interval time is longer, and the electron and lattice temperatures do not increase during successive pulses. Moreover, the ablation depth saturates at a maximum value. The downward velocity of the focus can directly affect the defocusing during processing. Positive defocusing arises at a low downward focus velocity. Meanwhile, negative defocusing arises at high downward focus velocity, and the defocus will increase as processing continues, causing the processing efficiency to decrease. A higher processing efficiency can be achieved when a suitable focus velocity is adopted to maintain the negative defocus at less than 50 nm. Therefore, the model proposed in this work can provide reliable guidance for the reasonable selection of a pulse interval time and downward focus velocity. The lattice temperature, absorption, and processing efficiency can be increased during actual processing by using a reasonable pulse interval time and determining the appropriate downward focus velocity based on the ablation depth for a single pulse.

Funding

National Natural Science Foundation of China (51705440); Fundamental Research Funds for the Central Universities (XMU 20720180072); Chinese Aeronautical Establishment (20170368001); Shenzhen Fundamental Research Program (JCYJ20170818141303656); Natural Science Foundation of Fujian Province (2019J01044).

Acknowledgments

The authors would like to appreciate Ertai Wang, Saitao Zhang, Tao Liao, and Rui Zhao for their generosity, useful discussions and helpful assistance.

Disclosures

The authors declare no conflicts of interest.

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Parameters [unit]	Value
$A_{e}$ [ $K^{- 2} \cdot s^{- 1}$ ]	$1.75 \times 10^{7}$ [14]
$B_{l}$ [ $K^{- 1} \cdot s^{- 1}$ ]	$1.98 \times 10^{11}$ [18]
$C_{e 0}$ [ $kg / (m \cdot s^{2} \cdot K^{2})$ ]	70.361 [1]
$E_{p}$ [µJ]	10
f [kHz]	100
$G_{R T} [W / (m^{3} \cdot K)]$	$5.551 \times 10^{16}$ [19]
h[ $J \cdot s$ ]	$6.626 \times 10^{- 34}$
$k_{B}$ [J/K]	$1.38 \times 10^{- 23}$ [14]
$k_{F}$ [ $m^{- 1}$ ]	$1.358 \times 10^{10}$
$m_{e}$ [ $kg$ ]	$9.109 \times 10^{- 31}$
$N_{A}$ [ ${mol}^{- 1}$ ]	$6.02 \times 10^{23}$
$T_{c}$ [K]	8280 [1]
$T_{F}$ [K]	81600 [3]
$T_{m}$ [K]	1358 [18]
$t_{p}$ [fs]	250
V $[m^{3} / mol]$	$7.14 \times 10^{- 6}$
$v_{s}$ [m/s]	4760 [4]
$z_{R}$ [µm]	392.65
$ε_{F}$ [J]	$1.126 \times 10^{- 18}$
$λ$ [nm]	800
$ρ_{0}$ [ $kg / m^{3}$ ]	8900 [18]
$ρ_{c}$ [ $kg / m^{3}$ ]	1040 [18]
$ω_{0}$ [µm]	10
$δ$ [nm]	13 [4]
$δ_{b}$ [nm]	15 [4]
$ξ$ [Hz]	$3.75 \times 10^{14}$
$ξ_{p}$ [Hz]	$2.1327 \times 10^{15}$ [13]

Numerical simulation of multi-pulsed femtosecond laser ablation: effect of a moving laser focus

Abstract

1. Introduction

2. Computation model

2.1 Adapted two-temperature model and laser source model

2.2 Phase transformation model

2.3 Establishment of the computation model

3. Results and analysis

3.1 Verification of simulation model

3.2 Effect of pulse interval

3.3 Effect of downward focus velocity

4. Conclusions

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (9)

Tables (1)

Equations (18)

Optical Materials Express