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We study the roles of thermal ionization and electronic damage in the internal modification of bulk borosilicate glass by high repetition rate picosecond laser pulses. Laser-induced plasma generation, nonlinear energy deposition and steady temperature distribution are numerically analyzed. The simulated modified regions show good agreement with the experimental results, thereby revealing the roles of thermal damage and electronic damage in the internal modification. While the elliptical outer structure is recognized as the molten region, we found that the teardrop-shaped inner structure is the damaged zone caused by high-density free-electrons. In the formation of the inner structure, cascade ionization is seeded by thermal ionization instead of multi-photon ionization and dramatically increases the free-electron density to the damage threshold. The contour of the inner structure is found to be corresponding to a characteristic isotherm of around 3000 ~4000 °C.
© 2013 Optical Society of America
1. Introduction
The internal modification of transparent dielectrics by high repetition rate ultra-short lasers becomes highly attractive in various applications of micromachining [1] such as waveguide writing [2–7] and micro-welding [8–12]. When ultra-short laser pulses are focused inside bulk transparent dielectrics, laser pulse energy is nonlinearly absorbed through generating and heating free-electrons and then transferred to the lattice to heat the material. When the time interval between successive pulses is less than the thermal diffusion time, the temperature in the focal volume rises up to several thousand degrees [13–15] because of heat accumulation. The re-solidification of the molten material leads to a refractive index change and micro-joining. The internal modifications in the bulk transparent dielectrics by high repetition rate ultra-short laser pulses have been beautifully demonstrated in experiments [2–12]. The cross-section of the internal-modified region consists of an elliptical outer structure and a teardrop-shaped inner structure [6–9]. The outer modified region is recognized as the molten zone, but the formation mechanism of the inner structure is not clearly understood yet. In this paper, we discuss this mechanism by considering the role of thermal ionization. Thermal ionization generates high-density free-electrons in the temperature-accumulated volume, and when subsequent pulses interact with this volume, the free-electron density overcomes the damage threshold via cascade ionization. Therefore the inner structure is considered to be the damaged zone induced by high-density free-electrons.
In detail, we present a numerical model to study the internal modification in bulk borosilicate glass irradiated by high repetition rate picosecond laser pulses. Simulations are performed to study the plasma formation, nonlinear energy deposition, temperature distribution and modifications in bulk borosilicate glass. We get the modified regions with the temperature criterion and the critical free-electron density for laser damage, respectively. They are consistent and both agree well with the experimental results. The formation mechanisms of the inner structure are discussed based on thermal ionization and electronic damage.
2. Numerical model
In the numerical model of internal modification, nonlinear energy deposition of laser pulses is determined by beam propagation and a plasma model. Using the deposited energy as a heat source, we apply a thermal conduction model to derive the temperature distribution. Internal modification regions are estimated based on free-electron density and temperature distribution, respectively.
2.1 Nonlinear energy deposition
The schematic diagram of the setup for internal modification of bulk borosilicate glass with high repetition rate laser pulses is shown in Fig. 1. The laser system delivers 1064 nm, 10 ps laser pulses with a repetition rate of 500 kHz. Laser beam is focused at zf = 150 μm in the sample by an objective lens (NA = 0.55). The sample is borosilicate glass (Schott D263) with a thickness d of 200 μm. The beam quality factor M2 = 1.1. The scan speed v is 20 mm/s. The beam waist w0 in the material is 3.3 μm. The pulse energy Ep varies from 0.1 μJ to 2.46 μJ. For the borosilicate glass sample, the refractive index n0 is 1.52 [16] and the band-gap Eg is 3.7 eV [8].
Fig. 1 Schematic diagram of the setup for the internal modification of borosilicate glass sample with ultra-short laser pulses. Laser pulses propagate along the z axis and are focused inside the sample with an objective lens. The laser source scans along the x axis at a speed v. Cross-sections of the modification regions discussed in this study are in the y-z plane.
When an intense picosecond laser pulse is propagating inside the bulk borosilicate glass, nonlinear propagation effects may occur, such as self-focusing (Kerr effect), self-phase modulation and plasma defocusing. The most important factor governing the spatial dynamics of laser beam is self-focusing. For a Gaussian beam, self-focusing overcomes diffraction and leads to collapse only if the input peak power exceeds a critical threshold [17]:
where n2 is the nonlinear Kerr coefficient. For our borosilicate glass sample, n2 = 3.45 × 10−20 m2/W (BK7 glass) [18], Pcrit = 3.2 MW for the above parameters, and the corresponding critical pulse energy Ecrit is 32 μJ. In this study, the applied pulse energy (< 2.46 μJ) is far less than Ecrit and filament-like modification regions are not observed in the experimental results [8], so we neglect the nonlinear propagation effects and take the transport equation to describe the beam propagation. Optical aberrations induced by the air-glass interface will change the geometric focus position, but have a minimal effect on the focal volume as the focal depth of the laser beam is quite small. In the simulations, the optical aberrations are not taken into account.Laser pulse propagation in the bulk borosilicate glass can be described by a transport equation
where I(z, t) is laser intensity, zR and zf are the Rayleigh length and focus position of the laser beam, respectively, and α(z,t) is the total absorption coefficient given bywhere ηphoto is the photo-ionization coefficient, k is the photon number for MPI and equals to the integer part of (Eg/ħω + 1), ηcasc is the cascade ionization coefficient, ρ(z, t) is laser-induced free-electron density, and Eav = 2.25 Eg is the average energy of free-electrons, which is the sum of the kinetic energy and the band-gap Eg [19]. Therefore nonlinear absorption dynamics of laser energy and optical absorptivity of the dielectrics can be evaluated numerically with the absorption coefficient α(z,t).Laser-induced damage or modification in transparent dielectrics occurs from ablation for pulse duration tp ≤ 10 ps and from conventional melting, boiling, and fracture for tp ≥ 50 ps [20]. In this study, the pulse duration used in the internal modification of bulk borosilicate glass is 10 ps. Therefore the internal modification mechanism is based on the ablative regime dominated by collisional and multiphoton ionization and plasma formation, rather than the long-pulse, thermally dominated regime. In transparent dielectrics, strong electric fields excite the electrons in the valence band to be free electrons via photo-ionization and cascade ionization. The temporal evolution of free-electron density ρ is described by a rate equation [19–22] like
where the terms on the right side denote the rates for photo-ionization, cascade ionization and recombination, respectively, and ηrec is the recombination coefficient. The laser intensity used in this study is smaller than 2 × 1012 W/cm2 and the Keldysh parameter [22] is larger than 2.5. Thus multi-photon ionization (MPI) rather than tunneling ionization dominates in the photo-ionization process. The MPI coefficient ηphoto is taken from the Kennedy’s approximation of the Keldysh model [22]. The expression for the cascade ionization coefficient ηcasc is given in Ref [22]. Two-body recombination mechanism is used in the plasma model in Eq. (4) and the recombination rate ηrec of the sample glass is taken as 2 × 10−9 cm3/s. When the free-electron density ρ is decaying from 1020 cm−3 to 5 × 1018 cm−3, the laser-induced electron plasma lifetime 1/(ηrecρ) is in the range of from 5 ps to 100 ps, which is consistent with the typical lifetime of a free-electron gas in borosilicate glasses [23], where no self-trapping occurs.The top surface of the bulk borosilicate glass sample is located at z = 0 and the sample thickness is d. The laser pulse shape is a Gaussian profile I(0, t) = I0 exp[-4 ln2 (t - n0zf/c - 2tp)2/ tp2], where c is the light speed in vacuum, tp is the pulse duration and I0 is the peak intensity. As the interaction volume is far away from the sample surfaces, the Fresnel reflections on the surfaces are negligible as well as the scattering of laser light in the interaction volume.
2.2 Heat source and thermal conduction
In the focal volume of ultra-short pulses, the laser energy is absorbed via MPI and the inverse bremsstrahlung process as described by Eqs. (2) and (4). The deposited energy is firstly carried by free-electrons and then transferred to the lattice through electron-phonon collision, resulting in thermal diffusion and heat modification.
One-dimensional numerical simulations of plasma dynamics and nonlinear energy deposition along the z axis are performed at the center of laser beam spot, i.e. y = 0 μm. It is assumed that all the energy of the free-electrons is transferred to the lattice and contributes to thermal effects. That is, the energy gained by the free-electrons from the laser pulse serves as the heat source for thermal diffusion. Thus by integrating the nonlinear absorption coefficient in Eq. (3), the volume density distribution of the deposited energy of an individual laser pulse along the z axis can be expressed by
The deposited energy distribution on the x-y plane is approximately proportional to the intensity distribution I(z,t) [7, 8, 13, 21] because of cascade ionization and heat accumulation. Since the beam size in the focal volume is much smaller than the heat affected zone, the line density of the deposited energy can be treated as a line heat source, which is calculated by integrating Qv(z) on the transverse plane aswhere w(z) = w0[1 + (z-zf)2/zR2]1/2 is the beam size distribution along the z axis.When a pulsed laser at a repetition rate of frep scans from x = -∞ at a constant speed v along the x axis in bulk borosilicate glass, where w0/v >> 1/frep, the deposited energy can be treated as a continuous heat delivery. At the same time, the assumption about the continuous heat delivery is valid only when the deposited energy of laser pulses inside bulk dielectrics merely serves as a heat source, which spreads energy in the surrounding volume by thermal diffusion. In the internal modification of some glass systems with ultrashort pulsed lasers, such as fused silica, the formation of self-organized periodic patterns consisting of single or multiple bubbles buried in bulk materials has been reported as a consequence of cumulative energy deposits [24–26]. Bubble formation and decomposition of the glass matrix into a gas phase consume a portion of the deposited energy of laser pulses and modify the local heat transfer conditions. Therefore the deposited energy cannot be simply assumed as a continuous heat delivery. In the experimental results studied in this paper, however, bubbles or voids were not observed on the cross-section of the internal modified zone and the smooth waveguide-like modified channel rather than the periodic patterns was resulted along the scanning direction of laser pulses. Thus the assumption about the continuous heat delivery is suitable in this study. The material characteristics, e.g. temperature distributions, at the writing/welding front keep in a constant state for successive pulses [15] and the resulting modification channel along the x axis is homogenously distributed with a constant diameter [6, 7, 27]. That means, the interactions of the focal volume with successive pulses are approximately the same and produce the identical deposited energy distribution Ql(z) by each pulse. Thus the continuous heat delivery is given by the line power density, i.e. frep Ql(z), and the steady temperature distribution in the three-dimensional space is calculated by [8]
where T0 = 25 °C is the initial temperature of the dielectrics, r = [x2 + y2 + (z-z′)2]1/2, κ = λc/(nM·cp) is the thermal diffusivity, λc = 0.96 W/(m·K) is the thermal conductivity, nM = 2.51 × 103 kg/m3 is the mass density, and cp = 0.82 × 103 J/(kg·K) is the specific heat capacity of the dielectrics.Using the line density distribution Ql(z) of the deposited energy of an individual pulse, one can evaluate the optical absorptivity. When the incident pulse energy is Ep, the absorptivity A is defined as the ratio of the absorbed energy to the pulse energy, i.e.
2.3 Modification criteria
At laser wavelength of 1064 nm, the critical free-electron density for laser breakdown [19] is ρcrit = ω2meε0/e2 = 1.0 × 1021 cm−3, where ω is the laser angular frequency, me is the electron mass, ε0 is the vacuum dielectric permittivity and e is the electron charge. The damage threshold usually corresponds to a conduction electron density for which the energy density of these electrons equal to the lattice binding energy [17]. In glasses, the critical free-electron density for laser damage is ρd ≈1020 cm−3 [21, 22, 28], above which the material is modified because of the energy released from high-density free-electrons, so called electronic damage or electronic modification in this paper. The shape and size of the electronic-damage tracks correspond to the zone where the electron density created by photo-ionization and cascade ionization is close to ρd. The term “electronic damage or modification” here denotes an ablation mechanism based on the free-electron density rather than on the temperature. We distinguish it from thermal damage/ablation, but do not specify exactly what this damage means on the chemistry level. ρd is used as the criterion for the electronic modification here.
The deposited energy mediated by the free-electrons results in a temperature rise in the interaction volume. Due to the slow scan and high repetition rate of the laser source, the spatial overlap 1-v/(2w0frep) of adjacent pulses is 99.4% and there are more than one hundred laser pulses hitting the same spot with a size of 2w0. The temperature T increases with pulse number because of heat accumulation and the material is modified because of visco-elastic deformation and glass element flow [29]. Firstly, we follow the assumptions of the heat modification criteria proposed by Miyamoto et al. [8] that the isothermal lines at Tm = 1051 °C and Tb = 3600 ± 300 °C are the boundaries of two different heat modification regions. Tm is referred as the melting temperature and Tb is a fitting temperature to reproduce the contour of the inner structure in the internal modification. Therefore the heat modification criteria in this paper are that the area where the temperature T > Tb is the inner structure while the zone where Tm < T < Tb is the outer molten region. Secondly, the choice of Tb is explained based on the numerical and experimental results and meanwhile, the roles of electronic damage and thermal damage in the internal modification of bulk borosilicate glass are compared.
3. Results and discussion
3.1 Plasma dynamics and energy deposition
Using the numerical model, we analyze the internal interaction process in bulk borosilicate glass irradiated by picosecond laser pulses with a pulse energy Ep = 2.46 μJ, including nonlinear absorption of photons, plasma formation and energy deposition. Figure 2(a) shows laser-induced free-electron density distributions along the z axis at different time instants. The laser pulse is a Gaussian profile and converges before arriving at the focus, so the leading-edge of the laser pulse produces high-density free-electrons at the focus firstly. For the subsequent portion of the laser pulse, the plasma formation front moves towards the laser source. Therefore the laser intensity after t = 15 ps is consumed before arriving at the focus as shown in Fig. 2(b).
Fig. 2 Distributions of (a) free-electron density ρ, (b) laser intensity I, (c) maximum free-electron density ρM during the ionization process and (d) volume density Qv and line density Ql of the deposited energy along the z axis. Pulse energy Ep = 2.46 μJ. Graph (a) has the same legend as that shown in (b). The red arrow indicates the laser incident direction. The focus of laser beam is located at z = 150 μm indicated by a thin gray line.
In order to study the electronic damage, the maximum free-electron density ρM during the ionization process along the z axis is obtained and shown in Fig. 2(c). The electronic modification forms in the area where ρM > ρd. So the length of the electronically damaged region is calculated to be 68 μm, which will be compared in Section 3.4 to the inner-structure length of the modification region observed in the experiment and predicted by the temperature Tb.
Using Eqs. (5) and (6), we calculated the volume density Qv and the line density Ql of the deposited energy along the z axis, which are shown in Fig. 2(d). Due to the plasma expansion towards the laser source, the peak of the volume density along the z axis is in front of the focus. The line density distribution is an approximately triangular profile because of the focus and its peak is located at z = 100 μm, far from the geometric focus. By integrating Ql(z) from z = 0 to d, the absorbed energy is calculated to be 1.92 μJ. Thus with Eq. (8), the absorptivity A is evaluated to be 78%.
3.2 Temperature distribution
The simulated line density Ql of the deposited energy along the z axis in bulk borosilicate glass irradiated by laser pulses with various pulse energies are shown in Fig. 3(a). At the threshold case (Ep = 0.41 μJ), the energy distribution is located near the focus. When Ep > 0.41 μJ, due to the larger peak intensity, the deposited energy distribution expands towards the laser source with the similar profile and the absorbed energy increases. Using Eq. (8) and the data in Fig. 3(a), the optical absorptivity A of the dielectrics is calculated as a function of pulse energy Ep as shown in Fig. 3(b). The absorptivity A nonlinearly increases with Ep and finally gets saturated, which agrees very well with the experimental results got by the transmission loss measurement in Ref [8]. Therefore the present model can be used to predict and evaluate the nonlinear absorptivity before the experimental implementation.
Fig. 3 (a) Simulated line density Ql of the deposited energy along the z axis in bulk borosilicate glass irradiated by laser pulses with different pulse energies. (b) Absorptivity A as a function of pulse energy Ep given by our numerical model and the experiment in Ref [8]. In (a), the red arrow indicates the laser incident direction and the focus of laser beam is located at z = 150 μm indicated by a thin gray line.
Using the deposited heat source Ql in Fig. 3(a) and the heat conduction model in Eq. (7), we obtain the cross-sections of the steady temperature distribution at different pulse energies as shown in Fig. 4(a). The maximum temperature on the cross section increases with pulse energy Ep and exceeds 6000 °C when Ep >1.21 μJ. This is consistent with the Raman temperature measurements [14] and the temperature estimation based on the transient refractive index change measurement [15]. The high temperature region (e.g. T > 1000 °C) extends along the y direction because of more absorbed energy and along the z direction because of the heat source expansion towards the laser source.
Fig. 4 Cross-sections of (a) simulated temperature distribution and (b) simulated modification regions defined by two isothermal lines (Tm = 1051 °C and Tb = 3600 °C) and (c) internal modification in the reported experiment in Ref [8]. at different pulse energies. The red arrow indicates the laser incident direction. The focus of laser beam is indicated by horizontal lines.
Based on the temperature distribution in Fig. 4(a), two different heat modification regions are defined by the two isothermal lines (Tm and Tb), as shown in Fig. 4(b). The outer and inner contours are both smooth and regular because of heat diffusion. The simulated cross-sections of the internal modifications in bulk borosilicate glass are in great agreements with the experimental results [8] in Fig. 4(c) in a wide range of pulse energy. It is widely considered that the outer modified region is the molten zone [8, 9]. The different morphology of the inner structure, however, indicates another modification mechanism, i.e. electronic damage, which will be discussed in the following section.
3.3 Electronic damage and thermal damage
In order to explore the formation mechanism of the inner structure, we compare the electronically damaged region defined by the threshold electron density ρd and the modified region defined by the temperature criterion Tb. The distributions of the maximum free-electron density ρM during the ionization process along the z axis in bulk borosilicate glass irradiated by laser pulses with various pulse energies are obtained as shown in Fig. 5(a). At the threshold case (Ep = 0.41 μJ), ρM ≈ρd, thus the electronic modification starts to occur. When increasing the pulse energy, the peak of the maximum free-electron density exceeds ρd and the laser-induced plasma expands towards the laser source because of the larger peak intensity. Therefore the length of the electronically damaged region, where ρM > ρd, is increasing with the pulse energy, whereas the lower end is always located at the focus as observed in the experiment (see Fig. 4(c)).
Fig. 5 (a) Distributions of maximum free-electron density ρM along the z axis in bulk borosilicate glass irradiated by laser pulses with different pulse energies. (b) Size comparison of the inner structure in the modification region along the z axis defined by the free-electron density ρd and by the temperature Tb as well as the experimental measurement [8] at various pulse energies Ep. In (a), the red arrow indicates the laser incident direction and the focus of laser beam is located at z = 150 μm indicated by a thin gray line.
Figure 5(a) shows that the maximum free-electron density in the internal modification of bulk borosilicate glass saturates around 3 × 1020 cm−3 when the pulse energy Ep increasing from 0.41 μJ to 2.46 μJ. The saturating peak density is well below the critical free-electron density ρcrit for laser breakdown [28]. Therefore the transient refractive index change because of the presence of free-electron density [30] is minimal and thereby the in-process reflection and diffraction induced by the plasma can be neglected in the internal modification.
In order to determine the damage mechanism of the inner structure in the internal modification region, in Fig. 5(b) we compare the sizes (along the z axis) of the electronic damage track defined by the damage threshold density ρd and the inner modification region defined by the temperature Tb as well as the inner structure measured in the experiments [8]. The length estimations of the inner structure based on the electronic and heat modifications are both in great agreement with the experimental results. Therefore the electronic damage and the heat modification induced by temperatures larger than Tb seem to be consistent. In order to understand the physical picture behind, we consider thermal ionization in the following section and give an explanation based on thermal ionization.
3.4 Heat accumulation and thermal ionization
In the internal interaction volume of bulk borosilicate glass with high repetition rate ultra-shot laser pulses, heat accumulation results in a high temperature up to several thousand degrees [13–15]. Therefore thermal ionization becomes crucial to produce high-density free-electrons [8], which serves as the initial free-electrons for the plasma formation induced by subsequent pulses. Thus cascade ionization can be seeded by thermally-ionized free-electrons without the contributions of MPI. Thermal ionization makes the high-temperature region semi-transparent or even opaque.
Assuming the molecules of borosilicate glass are thermalized, the free-electron density in the conduction band can be expressed by [31]
where ħ is the reduced Planck constant, Ec is the boundary of the conduction band, kB is Boltzmann constant. The free-electron density ρti induced by thermal ionization as a function of the transient sample temperature T is shown in Fig. 6(a). When the temperature T reaches 3600 °C, the thermal-ionized electron density is around 5 × 1018 cm−3.Fig. 6 (a) Free-electron density ρti induced by thermal ionization in borosilicate glass as a function of the sample temperature T. (b) The line densities Ql of the deposited energy along the z axis at different temperatures of the bulk borosilicate glass. In (a), the red spot on the curve indicates the thermal-ionized free-electron density at the temperature of 3600°C. In (b), the red arrow indicates the laser incident direction and the focus of laser beam is indicated by a thin gray line.
In order to study the effect of the accumulated temperature on the energy deposition of the subsequent laser pulse, we make a simple assumption that laser pulses with pulse energy of 2.46 μJ interact with the bulk borosilicate glass sample with an initial homogenous temperature distribution ranging from 25 °C to 4000 °C. The temperature is transient so phase transition does not occur in borosilicate glass. That is, the bulk borosilicate glass has a homogenous thermally-ionized free-electron density distribution and thereby is not completely transparent. The line densities Ql of the deposited energy along the z axis at different temperatures are compared in Fig. 6(b). When the sample temperature is larger than 3000 °C, the pulse energy is almost linearly absorbed and the bulk borosilicate glass becomes opaque.
Therefore we can conclude that in the volume where T > 3000 °C, pulse energy is strongly absorbed. Thermally-ionized free-electrons serve as the seed electrons for cascade ionization and the free-electron density rapidly rises beyond the threshold value ρd for the electronic damage. The contour of the electronic damage is consistent with the isotherm of the characteristic temperature Tb, which is in the range of 3000°C ~4000 °C [32]. This is the reason that the length of the inner structure is accurately estimated by taking the characteristic temperature Tb = 3600°C in Section 3.3.
Based on the above numerical analysis, the inner structure of the internal modification is found to be the electronically damaged zone and consistent with the volume where T > Tb. Since the electronic damage occurs in a shorter time scale than thermal modification, the electronically damaged region (the darker inner structure) could be further modulated because of phase transition (T > Tm) and glass element flow [29] and thereby covered by the heat affected zone (the outer molten region) as we have found in the surface ablation of borosilicate glass [22]. The thermally damaged zone and electronically damaged region can be clearly distinguished, as shown in Fig. 7.
Fig. 7 The thermal damage and electronic damage on the cross section of the internal modification [8].
4. Conclusion
In summary, we have numerically investigated the internal modification in bulk borosilicate glass irradiated by picosecond laser pulses at a repetition rate of 500 kHz. Free-electron dynamics, nonlinear energy deposition and thermal diffusion are analyzed. The maximum free-electron density in the interaction volume was shown to saturate around 3 × 1020 cm−3 when increasing pulse energy, and it is well below the breakdown threshold density (1021 cm−3 at 1064 nm). The simulated cross-sections of heat modification regions, as well as the nonlinear absorptivity, agree quite well with the experimental results at various pulse energies. When the elliptical outer structure is known as a molten zone, we found that the teardrop shaped inner structure is the electronically damaged zone as well as the energy-absorption region. Heat accumulation and thermal ionization play a major role in the induction of the electronic damage and the formation of the inner structure. In the high-temperature volume induced by heat accumulation, cascade ionization seeded by thermal ionization results in the electronic damage. The characteristic temperature for the electronic damage is found to be in the range of 3000 ~4000 °C.
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