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Abstract
The modification at the interface between glass plates induced by ultrafast laser is important for the glass welding strength, therefore the relationship between the modification size and processing parameters should be identified. The experimental method has its limitation in understanding the nature of the modification. In this study, a numerical model for the temperature distribution determining the modification size induced by ultrafast laser scanning is developed, in which a three-dimensional steady model for the beam propagation with a transient ionization model is established to estimate the free electron density by the single laser pulse, and then a heat accumulation model for multiple laser pulses is employed to describe energy transportation within the irradiated bulk. The experiment for the internal modifications in single-piece fused silica samples irradiated by a picosecond laser with different pulse energies and scanning velocities is performed to validate the present model.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Glass welding by ultrafast laser pulses has attracted a lot of attentions since the melting zone can be precisely controlled at the interface between glass plates and crack free can be also achieved for most commercially available glass materials. During glass welding process, ultrafast laser energy is nonlinearly absorbed in the interior of glass by multiphoton ionization followed by avalanche ionization, which enables internal melting in bulk glass without any damage on the surface. One of the most important factors for the glass welding strength is the size of the internal melting zone (i.e. modification zone) because the modification at the interface between the two pieces of glasses to be welded is to form the weld joint, in which the length of the modification region limits the range of laser defocusing distance and the width determines the area of the weld joint. Therefore, it is necessary to understand the relationship between the modification size and the ultrafast laser processing parameters.
The modification induced by ultrafast laser has been experimentally investigated. Schaffer et al. [1] found the cone structure in glass induced by ultrafast laser and this modification was influenced by the focus position [2–4], scanning velocity [5,6], pulse energy and repetition rate [7–9]. However, it is difficult to track the modification evolution using the experimental methods although the high speed camera can be used. And also it is hard to measure the free electron density as well as the temperature [10]. Therefore, numerical simulation is used to understand the spatiotemporal dynamics of the modification induced by ultrafast laser in glass.
Several models have been developed for single pulse energy absorption inside transparent matter. Maxwell’s equations [11–13] combined with the rate equation for free electron generation and a simplified geometrical model of laser beam focusing [14] were used to investigate the evolution of free electron density in the modified zone. Finite difference time domain was employed for the full set of Maxwell’s equations coupled with Drude model to describe the generated plasma [15]. The transient solution for Maxwell’s equations with the requirements of small time step and mesh size needs many computational resources. Therefore, the nonlinear Schrödinger equation (NLSE) was proposed to describe laser beam propagation [16]. The NLSE is an asymptotic parabolic approximation of Maxwell’s equations that was developed for describing unidirectional propagation of slowly varying envelopes of laser light [17], which can reduce the computational resource. The NLSE coupled with plasma equations [18–23] was to describe laser beam propagation and free electron density in the transparent material. These models are much more suitable for single pulse irradiation according to the computational time and memory.
The interaction between multiple ultrafast laser pulses and glass also has been investigated through some simplifications. Thermal diffusion was only considered in multiple pulses processing of glass [24–27]. The beam propagation and free electron generation were included in a simulation of ultrashort laser drilling of glass, in which heat accumulation calculation between pulses was considered [28]. However, two model parameters require experimental determination. Miyamoto et al. [29] put forward a numerical model to predict the shape of the internal modification in glass by multiple pulses through simplifying the distribution of the absorbed single pulse energy based on the experimental modified shape. Therefore, the single heat source highly depends on the experiments. In their subsequent work [30], temperature distribution was simulated at high pulse repetition rates based on rate equation model coupled with thermal conduction model. However, beam propagation is not included, and laser intensity distribution in glass is not clear. Sun et al. [31,32] established a numerical model for predicting the internal modification, in which a line heat source was considered using one dimensional plasma dynamics and nonlinear energy deposition, and then a thermal conduction model was utilized to calculate the temperature distribution. However, the one-dimensional beam propagation model cannot take the effects of the plasma on beam propagation in space into account.
It is essential to understand the interaction between ultrafast pulses and glass, through which the temperature distribution induced by multiple pulses and thereby the dimension of the modification can be evaluated. Therefore, in this study, a three-dimensional steady model for laser beam propagation with a transient ionization model is established to estimate the free electron density by the single laser pulse. Subsequently, a heat accumulation model for multiple laser pulses is employed to describe energy transportation within the irradiated bulk. The present model for the temperature distribution is used to predict the modification size induced by laser scanning. The experimental modifications with different pulse energies and scanning velocities are investigated to validate the present model.
2. Mathematical model
Due to the Gaussian laser beam input, the light intensity within focal volume is much higher than the rest part of the bulk, where the plasma is generated and the light intensity is greatly attenuated, leaving an asymmetric ionized zone along the beam propagating direction. Hereafter, laser energy deposited within focal volume relaxes to the vicinity and the macroscopic heat transfer occurs. This multiple physics process involves the propagation of laser beam, the ionization of dielectrics and heat transfer, which are of femtosecond, picosecond and nanosecond timescale, respectively. To avoid the tremendous amount of computation induced by a transient electromagnetic model for laser beam propagation in which the grid size and time step are limited by the wavelength and frequency of electromagnetic wave, a steady model for the beam propagation with a transient ionization model is established to calculate single laser pulse energy deposited in bulk. Subsequently, a heat accumulation model for multiple laser pulses is employed to describe energy transportation within irradiated bulk. From all above, a comprehensive understanding of the modification within transparent dielectric bulk by multiple ultrafast laser pulses is attained.
2.1 Beam propagation model
The transient model of the beam propagation is based on Maxwell’s equations or NLSE. Maxwell’s equations are the basic equations used to describe the electromagnetic field. The complete set of Maxwell’s equations is accurate but requires many computer resources both in time and memory. The NLSE is an asymptotic parabolic approximation of Maxwell’s equations that is developed for describing unidirectional propagation of slowly varying envelopes of laser light. In this study, considering the establishment of electric field is instantaneous compared with the duration of laser pulse, a three-dimensional steady model with the maximum electric intensity and less computation resource can be used, which is based on paraxial Helmholtz equation assuming that electric intensity has the form of time periodic oscillation to eliminate the time oscillation term, and is expressed as Eq. (1).
2.2 Excitation model
The generation of free electron by multiphoton ionization and avalanche ionization is taken into consideration in the excitation model and the free electron relaxation is also considered. In glass, the electrons in the valence band can be excited as free electrons by strong electric fields via multiphoton ionization. Then free electrons are heated to even higher energy levels by the inverse bremsstrahlung absorption and exchange energy with other electrons by collisions to initialize the avalanche ionization. The free electrons also lose energy due to the relaxation and return to the valence band, as shown in Fig. 1.
The excitation model can give the relationship between electric field and free electron density through the rate equation coupled with light intensity. The rate equation for free electrons is given by Eq. (2) [33], in which the finite valence band electrons and the relaxation of conduction band electrons are considered.
Where $f = \sqrt \varepsilon$. Relative permittivity $\varepsilon $ is given by Eq. (6) [34].
where n is refractive index of glass, $\; \omega $ is laser angular frequency, $\tau $ is damping time, ${\omega _p} = e\sqrt {\frac{{{n_e}}}{{{m_e}{\varepsilon _0}}}} $ is plasma frequency, e is the electron charge, ${m_e}$ is the electron mass, ${\varepsilon _0}$ is the vacuum dielectric permittivity. It is obvious that the dielectric permittivity $\varepsilon $ is related to the free electron density ${n_e}$ and thus varies in the ionization process.Laser energy density F is proportional to the square of electric intensity E, and can be expressed as Eq. (7).
where c is the velocity of light. This means that the internal electric intensity with the input of laser intensity can be obtained based on the beam propagation model, then the distribution of the free electron density is achieved through the excitation model.It is noted that complex relative permittivity ${\varepsilon _{\textrm{avg}}}$ is time-domain averaged in the beam propagation model due to the steady analysis, and expressed as Eq. (8).
2.3 Heat accumulation model
Heat accumulation model includes the heat source induced by single laser pulse and the heat accumulation induced by multiple laser pulses. The heat source is formed instantaneously since the energy transferred from free electrons to the lattice is much faster than thermal diffusion. The single heat source Q is given by Eq. (9) [35].
where Ui is band gap energy. It is assumed that all the energy of free electrons including kinetic energy and band gap energy is converted into heat.As shown in Fig. 2, based on Green's function for the scanning strategy with the repetition rate f and the scanning speed v along z axis, the temperature solution for the location (x0, y0, z0) at time t after the Nth pulse is given by Eq. (10).
3. Numerical model
According to the mathematical models involved in the interaction between ultrafast laser and glass, the numerical model can be also divided into three parts, i.e. beam propagation model, excitation model and heat accumulation model, which are decoupled from each other because of different time scales. The calculation flowchart can be seen in Fig. 3.
The excitation model is defined on a single point of glass, including three equations: the ionization rate equation (Eq. (2)) to obtain the number of free electrons, light intensity distribution equation (Eq. (3)) to describe the development of light intensity, optical model (Eqs. (4)–(6)) to solve optical parameters which are coupled with free electron density in the duration of laser pulse. The excitation model is a transient model, where physical characteristics can be solved in every time step, therefore the maximum free electron density used for the heat accumulation model and the averaged optical parameters used for the beam propagation model can be obtained at a given light intensity. The parameter values that a=$4\; c{\textrm{m}^2}/\textrm{J}$[33], ${\delta _N} = 6 \times {10^8}\; \textrm{c}{\textrm{m}^9}\textrm{p}{\textrm{s}^{ - 1}}\textrm{T}{\textrm{W}^{ - 6}}$[33], N=6 [33], nv=1028 m−3 [36], ${\tau _1}$=150 fs [37] in Eq. (2), tp=8 ps in Eq. (3), n=1.45, $\tau $=1 fs [38], e=1.602×10−19 C, me=9.109×10−31 kg, ${\varepsilon _0}$=8.8542×10−12 F/m in Eq. (6) are used, respectively. The excitation model is solved in time step by Runge-Kutta method.
Through the excitation model, free electron density in glass after a single laser pulse can be obtained and the energy of free electrons (Eq. (9)) is input into the heat accumulation model as the single heat source. The single heat source is superimposed during the scanning process to obtain the temperature at any position and time via Green's function (Eq. (10)). Due to the small modification volume, the glass is considered big enough and thus the heat conduction is performed in an infinite space. The parameter values that Ui=1.44×10−18 J [33] in Eq. (9), Cl=1.6×106 J/m3K, α=1.3 W/Km, T0=293.15 K in Eq. (10) are used, respectively. The heat accumulation model is analytically solved.
Beam propagation model is a three-dimensional steady model, which can give the distribution of maximum electric intensity in glass with Eq. (1). It is assumed that every point in laser irradiated volume experiences the same process to generate free electrons described as the excitation model. The output of electric intensity by the beam propagation model is regarded as the input of the excitation model. Beam propagation model is solved by finite element method through COMSOL Multiphysic. The geometry of the beam propagation model includes three layers with the air layers on the upper and lower sides, and the glass layer in the middle, as shown in Fig. 4, in which the x axis is defined as the symmetric axis. The air layers are used for the laser reflection at the top and bottom surfaces of the glass. Based on the symmetry of the governing equation and laser input, a quarter cylindrical domain can be used in the beam propagation model. The thickness of air layer and glass layer is 0.3 mm and 2 mm respectively, and the radius of the quarter cylindrical domain is 80 µm. The laser focal plane is 0.3 mm above the bottom surface of the glass.
For the beam propagation model, the input boundary #1 is defined by Eq. (11).
Where ${\boldsymbol E}(x,y,z)$ is the electric intensity, E0 is the magnitude of electric field at the center of beam waist, r0 is the radius of beam waist, $r(x) = {r_0}\sqrt {1 + \frac{{{x^2}}}{{x_R^2}}}$ is the radius of beam along x axis, ${x_R} = \pi r_0^2/\lambda$ is the Rayleigh length, $R(x) = x + \frac{{x_R^2}}{x}$ is the curvature radius of wave front, $\eta (x) = \arctan \left( {\frac{x}{{{x_R}}}} \right)$ is the Gouy phase-shift. The relationship between the magnitude of electric field E0 and laser intensity I0 at the center of laser beam can be expressed as Eq. (12).
The boundaries #2 and #3 are defined as perfect electric and magnetic boundary respectively. Because the boundary #4 (i.e. the cylindrical surface) is far away from the beam, the electromagnetic intensity is very small and thus it maintains the perfect electric boundary. The boundary #5 is defined as the matched boundary to make laser pulse pass through without reflection. All the boundary conditions for the beam propagation model are listed in Table 1.
Table 1. Boundary conditions for the beam propagation model
4. Experimental setup
A picosecond laser system (TruMicro 5050) with the pulse duration of 8 ps, the wavelength of 1030 nm, the maximum repetition rate and pulse energy of 400 kHz and 125 µJ was employed to perform the internal modification experiments. The movement of laser beam was realized through a two-axis galvanometer system coupled with a focus lens. The focal length of the lens is 100 mm and the beam size is about 20 µm at the focal plane. Commercial fused silica with the thickness of 2 mm was used. A schematic diagram of the experimental setup is illustrated in Fig. 5.
The laser processing parameters used in the experiment are listed in Table 2. After the ultrafast laser treatment, the samples were then sectioned with the saw machine and ground with sand trays of particle size from 800 #, 1200 #, 2000 # and 3000 #. Finally, the glass was carefully polished with 10 µm, 5 µm and 1 µm diamond paste. The modification zone was observed from the cross section view by a transmission optical microscope (Olympus BX51M).
Table 2. Laser processing parameters in the experiment
5. Results and discussions
Figure 6 shows the numerical results of the temperature distribution on the cross section of the glass with laser pulse energy of 50 µJ, pulse repetition rate of 400 kHz and scanning speed of 20 mm/s. The softening temperature of 1938K of fused silica is considered as the isotherm to identify the profile of the internal modification region. The isothermal line of the softening temperature shows a teardrop-shaped structure. The head of the teardrop is close to the laser incident point, and the tail is far away from the laser incident point. This shape formation can be explained as follows.
Because of the Gaussian distribution in time of laser pulse where the light intensity increases in the first half of the pulse duration and then decreases in the second half, the glass at the laser focal plane is firstly modified with the energy increase, and then the plasma is generated, which prevents laser from continuing downward propagation. In the numerical model, the imaginary part of the refractive index represents the loss of light, which indicates the shielding effect of plasma to light. The larger the light intensity is, the more negative the imaginary part of the refractive index is and then the greater the loss of light is. Therefore, the area below the laser focus cannot be modified and the tail of the teardrop is formed at the laser focal plane. As the laser energy continues to increase in the first half of the pulse duration, the modified morphology can only grow up upstream above the laser focus where the laser beam diameter is bigger, leading that the modified region is wider compared to that at the focal plane, resulting in the formation of a conical structure. In the second half of the pulse duration, because of the reduction of laser energy, the width of the modified region is narrowed to form the head of the teardrop. Therefore, the profile of the modified region is finally determined.
Figure 7(a) shows the experimental images of the cross section of the glass with different pulse energies at the scanning velocity of 50 mm/s. With the increase of the pulse energy, the length and width of the teardrop morphology increase correspondingly.
It is noted in the experimental images that the black damage area appears at the upper center of the teardrop shape with higher pulse energy and is different from the change of refractive index of glass. In the center of the modified area, the temperature is the higher than that at surrounding areas due to the Gaussian distribution of laser intensity, then the huge temperature difference on the micron scale brings about extremely sharp temperature gradient which produces an expansion force that makes the surrounding glass dense and the central glass sparse, resulting in some small voids at the center of the modification. Due to the high temperature at the center, the glass melts with a certain fluidity. Therefore the small voids move upward driven by the buoyancy through the molten glass and gather into larger cavities at the upper part of the center. Finally, these cavities are fixed as the temperature decreases, forming the black irregular damage area.
Figure 7(b) shows the numerical results of the temperature distribution at the cross section of the glass with the same parameters in Fig. 7(a). The numerical and experimental pictures have the same scale. It can be seen that the numerical results agree with the experimental results. The length and width of the modification increase with the pulse energy. From the numerical results, the minimum modified shape with the length of 0.30 mm and the width of 0.012 mm is found with the pulse energy of 12.5 µJ, and the maximum one with the length of 1.20 mm and the width of 0.110 mm is produced by the pulse energy of 100 µJ.
The increase of the modification zone with the increase of the pulse energy can be attributed to two aspects. The first one is that the larger pulse energy input into the glass can generate a single heat source with more energy. Figure 8 shows the distribution of free electron density caused by a single pulse with the pulse energy 25 µJ and 100 µJ, respectively. Due to the symmetrical model, only half of the region is plotted. Figure 8(a) shows that the maximum free electron density of $2.38 \times {10^{27}}\; 1/{\textrm{m}^3}$ with the pulse energy of 25 µJ can be achieved at the center and $2.80 \times {10^{27}}\; 1/{\textrm{m}^3}$ is obtained with the pulse energy of 100 µJ, as shown in Fig. 8(b). The inhomogeneity of relative permittivity in the glass makes that the light intensity distribution is not concentrated as an ellipsoid, which results in that the distribution of free electron density in Fig. 8(b) is not an ellipsoid.
Moreover, the size of the heat source induced by higher pulse energy is larger, which causes the second reason that the effect of heat accumulation is more significant with higher pulse energy. Due to the same scanning velocity and repetition rate used for low and high pulse energies, the two neighbored pulses have the same interval in time and space. Therefore, the larger single pulse heat source can provide more heat accumulation, which also means that more pulses are required to achieve the temperature stability, as shown in Fig. 9. It shows the temperature change during laser scanning with different pulse energies of the reference point moving along with laser beam. In this study, the reference point is 0.1 mm above the lower surface of the glass based on the modification size obtained from the numerical model. Serrated waves can be found in the first half of the curves, because the temperature right at the current pulse is the highest and decreases as the distance from the current pulse increases until the next pulse comes. The second half of the curves is smooth because the reference point is selected before the pulse arrives, which can be convenient to observe when the temperature is stable. With the pulse energy of 100 µJ, the serrated fluctuation is more intense, the stable temperature reaches 18800 K after 1000 pulses. While the stable temperature reaches 3140 K after 500 pulses for the pulse energy of 12.5 µJ.
Based on these two aspects, the larger final profile of the modification can be achieved with the higher pulse energy. Also, it can be found that the head of the modification obtained from the numerical model is not round as the one in experiment at the high pulse energy, which can be explained as follows.
In the numerical model, thermal conduction is performed in an infinite space, while the upper and lower surfaces of the glass are convective with the air in experiment. Especially, the two ends of the modification are close to the surfaces of the glass at high pulse energy, which makes the heat easier to disperse to the upper and lower surfaces. Because the heat is concentrated in the upper part of the modified zone, it is easier for the heat to transfer to the upper surface according to Fourier law of heat conduction, thus widening the upper part of the modified region, illustrated as shown in Fig. 10. Moreover, the voids at the upper of the modification caused by thermal stress are not considered in this numerical model, and the voids also make the head of the modification round.
Figure 11(a) shows the experimental modifications with different scanning velocities at the pulse energy of 50 µJ. With the increase of the scanning velocity, the length and width of the teardrop morphology decrease correspondingly. Also it can be found that the black damage area appears at the upper center of the modified area at low scanning velocity. Similar to the case at high pulse energy, the head of teardrop-shaped area is round at lower velocity.
Figure 11(b) shows the numerical results of the temperature distribution on the cross section of glass with the same parameters in Fig. 11(a). With the increase of the scanning velocity, the width of the teardrop-shaped modification region decrease, which agrees with the experimental images. The width of the modified area is 0.044 mm with the scanning velocity of 100 mm/s, and becomes 0.166 mm as the scanning velocity decreases to 5 mm/s.
With the increase of the scanning velocity, the width of the modified region decreases, which is due to the smaller heat accumulation effect. When the repetition rate is fixed, the two adjacent pulses have the same time interval but a larger space interval for higher scanning speed, therefore the heat accumulation effect is smaller. Figure 12 shows the temperature change with different scanning velocities. The stable temperature reaches 28350 K after 10000 pulses for the case of 5 mm/s, while the stable temperature of 24360 K after 400 pulses is obtained for case of 100 mm/s. Therefore, the stable temperature as well as the corresponding pulse number is higher for low scanning velocity, leading larger width of the modification.
However, the length of the teardrop-shaped modification obtained from the numerical model is not consistent with the experimental results. The reason can be explained from the difference of the boundary condition between the numerical model and experiment, as shown in Fig. 13. In experiment, the convective boundary makes the heat dissipation faster and the thermal conduction to the upper and lower surfaces more significant. At low speed, due to the large heat accumulation effect, the upward and downward heat transfer brought by the convection leads that the teardrop-shaped modification becomes longer, as shown in Fig. 13(a). When the scanning velocity increases, the heat accumulation effect decreases, which causes the temperature decrease and thus the modification to shorten due to the convection at the top and bottom surfaces of the glass, as shown in Fig. 13(b).
Fig. 12. Temperature change during laser scanning with different scanning velocities (50 µJ, 400 kHz)
Figure 14 summarizes the length and width of the modified region obtained from the numerical model and experiment. The red circle presents the experimental results and the blue line stands for the numerical results. The two pictures on the left show the change of the length and width of the modified region with different pulse energies, and the two pictures on the right show the change of the length and width of the modified region with different scanning velocities. For the pulse energy, the length and the width in simulation are close to those in experiment, which implies the numerical results agree well with the experimental results. The length and width of the modified region is approximately linear with the pulse energy. Due to the small effect of the heat accumulation with the scanning speed of 50 mm/s, the length in numerical model is larger than that in experiment.
When the scanning speed is larger, the smaller heat accumulation can be obtained, thus resulting in the smaller length of the modification. However, this description does not fully agree with experimental length of the modification in Fig. 14 that the length increases then decreases with the increase of scanning speed, which can be explained as follows. The voids at the center of the modification at a relatively low velocity will affect the beam propagation, causing the shielding effect. Therefore, the modification length at scanning speed of 5 mm/s is smaller compared with that induced by scanning speed of 20 mm/s due to the stronger shielding effect resulted from the more voids at scanning speed of 5 mm/s.
According to the small difference of the boundary condition in the direction of the modification width between the numerical model and experiment, the numerical and experimental results of the modification width agree well, and the width is more useful for the welding strength control compared to the length of the modification.
6. Conclusions
Single-piece fused silica samples are irradiated by a picosecond laser with different pulse energies and scanning velocities. The modification zone on the cross section (xz-plane) is experimentally observed. A numerical model for the temperature distribution induced by ultrafast laser scanning is developed to investigate the size of the modification zone, in which a three-dimensional steady model for the beam propagation with a transient ionization model is established to estimate free electron density by single laser pulse, and then a heat accumulation model for multiple laser pulses is employed to describe energy transportation within irradiated bulk. The present model is able to predict the size of the teardrop shape, and the width of the modification agrees well with that obtained from experiments for a range of pulse energies and scanning velocities.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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