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Evolution of free-electron density in internal modification of glass by fs-laser pulses at high pulse repetition rates is simulated based on rate equation model, which is coupled with thermal conduction model in order to incorporate the effect of thermal ionization. Model shows that highly absorbing small plasma generated near the geometrical focus moves toward the laser source periodically to cover the region, which is much larger than focus volume. The simulated results agree qualitatively with dynamic motion of plasma produced in internal modification of borosilicate glass by fs-laser pulses at 1 MHz through the observation using high-speed video camera. The paper also reveals the physical mechanism of the internal modification of glass when heat accumulation is significant.
© 2016 Optical Society of America
1. Introduction
Internal modification of transparent dielectrics using ultrashort laser pulses (USLPs) has attracted much attention especially at high pulse repetition rates in a variety of applications including micro-fusion welding [1–3], selective etching [4] and waveguide formation [5–7], due to the advantages providing defect-free modification with large modified region at high throughput. Local element distribution can be also changed to produce highly efficient wave guiding structures with very large index contrast at high pulse repetition rates [8,9]. However, the laser-matter interaction at high repetition rates is not well understood, because the interaction is much more complicate than that of single laser pulse due to temperature rise cause by heat accumulation effect.
Effect of heat accumulation on internal modification was first reported by Schaffer et al [10] using simple thermal conduction model, showing that the modified region at high pulse repetition rates becomes much larger than single pulse. However, discrepancy between simulated and experimental modified dimensions is found to increase as the pulse numbers increases, since the effect of the temperature rise on the laser absorptivity is not incorporated in the model. Miyamoto et al [11,12] reported that the laser absorptivity increases as the pulse repetition rate increases by thermal conduction model, indicating the laser absorptivity is strongly affected by the temperature rise. They also showed that the laser absorption region can be extended toward the laser source where the laser intensity is much lower than the threshold of photoionization, suggesting that thermally ionized electrons play an important role in the ionization process at high pulse repetition rates. However, no papers have been published to simulate the ionization process by successive USLPs at high pulse repetition rates, while a lot of papers [13–18] have been reported to simulate the evolution of free-electrons in internal modification of transparent dielectrics by USLPs based on rate equation model.
In experimental approach, a new finding has been reported through the observation of the plasma behavior using high-speed video camera [19,20], showing that small plasma with a size of focus volume moves upward periodically to cover the region much larger than focus volume. While plasma motion is speculated to be gaseous volume driven by buoyancy in [20], the observed plasma motion seems to be too fast to attribute to buoyancy.
In the present paper, ionization process by successive fs-laser pulses at high pulse repetition rates is simulated based on rate equation model coupled with thermal conduction model to incorporate the effects of thermal ionization in the model. Simulation model is compared with the experimental observation of plasma using high-speed video camera.
2. Observation of internal modification process at high pulse repetition rate
Pump-probe technique [21,22] is useful to observe the time-resolved plasma evolution in single or repeated laser irradiation at low repetition rates of USLPs. However, this technique cannot be used for the repeated irradiation of USLPs at high pulse repetition rates, because the laser-matter interaction process can be changed pulse-by-pulse due to heat accumulation effects. It is suggested that the ionization is affected by the temperature rise through thermally ionized electrons, which seed the avalanche ionization in the following laser pulses [11,12]. Therefore we used high-speed video camera to record long-term process in internal modification at high pulse repetition rate, of which preliminary results are published in [19]. Here pictures taken by high-speed video camera are outlined.
Laser pulses with duration of τp = 550 fs (IMRA, wavelength λ = 1045 nm) is focused into a side-polished borosilicate glass sample (D263, Schott) with a thickness of 1 mm using NA0.65 lens at a depth of approximately 250 µm from the surface as shown in Fig. 1(a). Photographs are taken from the side (in y-direction) using high-speed video camera (Photron Fastcam) at a frame rate of 50 kHz with an exposure time of 5.4 µs without lighting. When the fs-laser pulses are irradiated at a pulse repetition rate of 1 MHz, for instance, pictures are taken at each 20 pulses with overlapped images of 5 ~6 pulses, which is too slow to observe the process pulse-by-pulse. However, as will be shown later, the frame rate is fast enough to observe the dynamic behavior of the plasma in longer time scale, which has never been observed in the past.
Fig. 1 (a) Experimental setup and coordinates. Laser beam (τp = 550 fs, f = 1 MHz, Q = 2.5 µJ) is focused into moving glass sample (v = 20 mm/s) at a depth of zf = 250 µm. (b) Cross-section of internal modification (y-z plane). (c) Side-view of the modified glass sample (x-z plane). The starting position is indicated by an arrow, and horizontal distance of 20 µm corresponds to 1,000 laser pulses. (d) Snap shot of laser-induced plasma taken in y-direction. GF is shown by a dotted line.
Figures 1(b) and (c) show a cross-section and a side view of the laser-irradiated glass sample obtained at a pulse repetition rate f = 1 MHz, pulse energy Q = 2.5 µJ and translation speed v = 20 mm/s. The cross-section exhibits a dual-structure [11] consisting of a teardrop-shaped inner structure and an elliptical outer structure. The contour of the inner structure is visible because the diffusion coefficient of the network modifiers are generally much larger than that of the network formers, so that the network modifiers are enriched around the laser-irradiated region to produce round or ring-shaped structure around the laser beam axis [12,23]. Thermal conduction model [2,11,12] shows that the laser energy is absorbed in the inner structure, whose contour corresponds to the temperature T ≈3,600°C in borosilicate glass D263, suggesting that the difference of the diffusion coefficient between the network formers and the network modifiers becomes significant at T > 3,600 °C. The model also shows that the glass is melted in the outer structure where working point (1,050 °C in D263) with a viscosity 104 dPas is reached. The dual-structure is also clearly observed in the side view [Fig. 1(c)]. The vertical size of the inner structure changes transiently for some distance at the beginning. Figure 1(d) shows a snap shot of the laser-induced plasma, indicating that the plasma with the size much smaller than the inner structure appears always within the inner structure.
Figure 2 shows a series of pictures taken by the high-speed video camera. The pictures show striking fact that the small plasmas with nearly the size of focus volume appear near the geometrical focus (GF), and move within the inner structure of the size as large as ≈100 µm. This means that the inner structure (laser absorption region) is not filled by the plasma as is reported [17] but partially filled by the small plasma which moves upward periodically. The thermal conduction model with continuous energy deposition (time-averaged model) predicts that the laser energy is absorbed in the inner structure [11]. Since the plasma size is much smaller than the inner structure, the local plasma temperature is expected to be much higher than the temperature simulated by the thermal conduction model assuming the inner structure is filled by the plasma [11,17].
Fig. 2 High-speed pictures taken at 50,000 frames per second under conditions of τp = 550 fs, f = 1MHz, Q = 2.5 µJ, zf = 250 µm, v = 20 mm/s in D263. Figures indicate relative frame number. #2 ~#31 correspond to 6th cycle in Fig. 3 where steady condition is nearly reached.
It is noted that the plasma is generated at locations slightly above GF (#2). Then it moves upward (#3~#24) to disappear at the highest position (#26~#31). Tandem plasmas move upward, and the moving speed is slowed down gradually as the height increases. Before reaching the highest position, the plasmas decrease their brightness and the size, and disappear eventually. It is interesting to note that a new plasma is produced near the GF when the plasma still exists at the highest position, as seen in #2~#7. This suggests that the upper plasma is partially transparent.
Figure 3 shows the variation of the height of the tandem plasma measured at the top and bottom with respect to the GF from the first laser pulse. It is noted that the frequency of the periodic plasma evolution is approximately one to two orders smaller than the repetition rate of the fs-laser pulses. The figure also shows steady height of the inner and outer structure are reached after ≈3,000 pulses in f = 1 MHz condition, in accordance with the variation of vertical size of the inner structure shown in Fig. 1(c).
Fig. 3 Vertical positions of the bright region at the top and bottom with respect to the GF are shown by closed and open circles, respectively, and are plotted from the first laser pulse (f = 1 MHz, Q = 2.5 µJ and v = 20 mm/s).
3. Analysis of laser-matter interaction based on numerical simulation
3.1 Simulation model
The evolution of electron density ρ(z,t) in conduction band under the influence of ultrashort laser pulses is given by the following rate equation [15]
where ηphoto is photoionization coefficient, ηcasc is cascade ionization coefficient, ηrec is recombination coefficient, K is number of photon for photoionization, and I(z,t) is laser intensity propagating in glass. Omitting the collision losses, cascade ionization coefficient ηcasc is given by [15,17]where τ is the time between collisions, c is the vacuum speed of light, e is electron charge, ε0 is vacuum permittivity, n0 is refractive index of the glass at frequency ωL, and m is electron mass. In Eq. (2), the diffusion loss of free-electrons from the focus volume is neglected, since the diffusion distance is much smaller than focus volume size in fs-laser pulse duration. When the laser beam having pulse energy Q and Gaussian pulse shape is focused into bulk glass, I(z,t) is given by [24,25]where Eg is band gap energy of the glass, τp is the duration of laser pulse, zf is the distance between the GF and the glass surface [Fig. 1(a)], and α(z,t) is the absorption coefficient of the laser beam. We assumed that the radius of laser beam spot ω(z) is given by [24]where zR is Rayleigh length and ω0 is laser spot radius at the beam waist. The laser power absorbed by free-electron per unit volume α(z,t)I(z,t) given byis transferred to the lattice to elevate the temperature of glass. At the laser pulse energy Q = 2.5 µJ used in this study, the peak laser power reaches approximately 4.5 MW, at which the effect of self-focusing cannot be neglected [30]. However, no appreciable filament is observed in Fig. 1, suggesting self-focusing effect on the internal modification process is limited. Thus in the simulation, the effect of self-focusing was neglected as the first step for simplicity. Assuming that the bulk glass moving at a constant speed of v along x axis is irradiated by USLPs successively, the temperature rise of the glass from room temperature at time t at location (x,y,z) with the coordinate fixed to the laser beam axis shown in Fig. 1(a) is given by [11]where cg is heat capacity, ρg is density, k is thermal diffusivity, f is pulse repetition rate and q(z’) is absorbed laser energy per unit length given byIt is assumed that radiation loss and dependence of thermal properties on temperature are negligible, and that laser energy with radius of ω(z) is absorbed instantaneously since the pulse duration is much shorter than thermal diffusion time. We also assume that the thermally ionized electron density ρtherm(T) in the conduction band is given by [18]which contributes as the seed electrons for avalanche ionization where ρbound is bounded electron density in glass. Thus Eq. (1) is numerically solved with the initial condition3.2 Laser-matter interaction in single laser pulse irradiation
Free-electron density and laser intensity propagating in glass are numerically simulated, assuming that multi-photon ionization (MPI) dominates in the photoionization process, since Keldysh parameter γ [26] is significantly larger than 1 within the simulation condition in this study as will be described below. We used Kennedy’s approximation [16] of the Keldysh model [26] for ηphoto. In cascade ionization, the electron collision time of 8 fs was used, while various values of τ were employed in different reports in a range of τ = 1 ~23.3 fs [15,27–29]. For the recombination rate, ηrec = 2 × 10−15 m3/s [17,30] was used. The band gap energy of D263, Eg = 3.7 eV, determined by a Tauc plot of optical transmission spectroscopy data [11,31] was used. Simulation is carried out using τp = 550 fs, Q = 2.5 µJ, f = 1 MHz, zf = 250 µm, NA = 0.65, n0 = 1.52 and v = 20 mm/s in accordance with the experimental conditions, and the simulated results are shown along z-axis at x = y = 0 throughout the paper [see Fig. 1(a)]. A rather larger value of zR = 15 µm is used taking into consideration that the spherical aberration is significant when laser beam is focused deep into optically denser material using high NA lens [32], which provides spot radius of ω0 = 3.3 µm at the waist.
Figure 4 shows the time-dependent free-electron density ρ(z,t) due to the single irradiation of the laser pulse simulated at different z. ρ(z,t) still keeps high value after it reaches a maximum value ρmax near the peak of I(z,t), while electron loss by recombination becomes appreciable at ρ > 1026 /m3. Figure 5 shows the time-dependent laser intensity passing through the plasma I(z,t) in comparison with I0(z,t) without plasma absorption [α(z,t) = 0] at different values of z. By comparing these two values, the threshold laser intensity of nonlinear absorption Ith is estimated, assuming Ith is the value where the difference between two values is appreciable. It is seen Ith ≈2 × 1016 W/m2, which is close to the experimental damage threshold of fused silica at τp = 550 fs reported in [14]. It is noted that the effective pulse width is narrowed as the distance passing through the plasma with higher free-electron density increases.
Fig. 4 Time-dependent free-electron density ρ(z,t) at different values of z by single laser pulse. Maximum value of ρ(z,t) at z is designated by ρmax (τp = 550 fs, Q = 2.5 µJ, zR = 15 µm, zf = 250 µm).
Fig. 5 Waveform of laser intensity simulated without and with plasma, I0(z,t) and I(z,t), respectively, at z = (a) 202 µm, (b) 211 µm, (c) 224 µm and (d) 239 µm.
The maximum values of the laser intensity propagating in glass at z with and without plasma, which are designated by Imax and I0max, respectively, are plotted vs. z in the single laser pulse irradiation in Fig. 6. In the figure, ρmax and the maximum value of the corresponding absorption coefficient αmax are also plotted. It is seen ρmax reaches appreciable value at z ≈200 µm (A’) at the threshold laser intensity of Imax ≈Ith (2 × 1016 W/m2). The highest free-electron density of ρmax ≈7 × 1026/m3 is reached at a location approximately 30 µm upstream of the GF. It is noted that the highest value of the laser intensity Imax is reduced down to as low as approximately 1/6 of I0max, because the laser energy is consumed upstream for producing free-electrons. Thus Keldysh parameter is large enough to adopt MPI approximation at this laser intensity.
Fig. 6 Free-electron density ρmax, laser intensity (Imax: in plasma, I0max: without absorption) plotted vs. z by single laser pulse irradiation. Laser beam propagates from left to right.
3.3 Temperature variation in multi-pulse laser irradiation
Figure 7 illustrates the temperature variation simulated at z = 240 µm, for instance, when five laser pulses impinge at a pulse repetition rate of f = 1 MHz. It is seen that temperature rises stepwise at the moment of the laser pulse impingement and is then cooled down by the thermal diffusion between the laser pulses. In the figure, the temperatures just before and after the impingement of the 4th laser pulse, for instance, are represented by TJBP and TJAP, respectively; the 4th laser pulse interacts with the material having temperature TJBP given by Eq. (6). Free-electron density is simulated by numerically solving Eq. (1) with initial condition, ρ(z,0) = ρtherm(z), which is determined by Eq. (7).
Fig. 7 Temperature variation at x = 0 and z = 240 µm, when five laser pulses (Q = 2.5 µJ) are irradiated. Temperature just before and after 4th laser pulse are represented by TJBP and TJAP, respectively.
Figure 8 shows the distributions of the temperature TJBP along z-axis at different numbers of laser pulse n simulated based on simple heat accumulation model where the effect of thermal ionization is neglected [10]. The temperature rises monotonically with little change in its peak location, since the laser pulse having the same energy distribution is absorbed in the bulk glass repeatedly. A dotted line shows 3,600 °C, which corresponds to the temperature at the contour of the laser absorption region (inner structure) predicted by thermal conduction model with continuous heat delivery [11,12]. In the figure, the locations of TJBP = 3,600 °C at 65th pulse, for instance, are shown by A0 and B0. The region where TJBP reaches 3,600 °C becomes continuously wider as the pulse number increases.
Fig. 8 Distribution of TJBP along z-axis at different pulse numbers simulated with simple heat accumulation model. Dotted line shows T = 3,600 °C, which is the temperature at the contour of the inner structure (laser absorption region). Locations where TJBP reaches 3,600 °C at 65th pulse, for instance, are shown by A0 and B0.
Figure 9 shows the distributions of TJBP along z-axis at different n simulated when the effect of thermal ionization is incorporated in the model. While the temperature distribution is nearly the same as Fig. 8 up to n ≈20 providing the peak temperature at the fixed position of z ≈222 µm, the temperature curve changes its shape dramatically in n > 20 showing dynamic variation of the peak value, the peak position and the extent of high temperature region periodically. The temperature distributions are plotted in every cycle separately to make the periodicity clearer. In 20 < n < 41, the temperature distribution provides two peaks showing a pair of plasma moving together toward the laser source with narrowing the high temperature region. The front peak reaches the highest value at n ≈36, and then decreases with slowdown of the forward motion. At the end of the decreasing phase of the front peak (36 < n < 41), the front peak is absorbed by the growing rear peak. As the cycle number increases, the plasma moves further toward the laser source with increasing the peak temperature as seen in Figs. 9(b)-9(d). As the result, wider region of TJBP > 3,600 °C is covered by the motion of the plasma, although each temperature distribution is narrower than Fig. 8.
Fig. 9 Distribution of TJBP along z-axis at different pulse numbers simulated with incorporating the effect of thermal ionization in the model. The locations whereTJBP reaches 3,600 °C in each cycle is shown by Ath and Bth.
Figure 10(a) illustrates the peak position of the temperature TJBP plotted vs. number of laser pulse up to n = 100, which shows a periodic variation with increasing amplitude as the cycle number increases. The curve indicates that the moving speed of the peak position is slowed down as the plasma front moves upstream in qualitative agreement with the experimental observation shown in Fig. 3. This also agrees qualitatively with the recently published experimental observation of ps-microwelding of bulk glass using the high-speed camera [33].
Fig. 10 (a) Location of maximum temperature TJBP with thermal ionization. (b) The region where temperature exceeds 3,600 °C, without (black line) and with (dashed red line) thermal ionization. Location of maximum temperature is also plotted by blue line.
The temperature distribution is simulated up to n = 1,000, and the locations where TJBP = 3,600 °C is reached without and with the effects of thermal ionization are plotted by (A0, B0) and (Ath, Bth: outermost value in each cycle), respectively, in Fig. 10(b). While the distance between A0 and B0 expands to saturate to approximately 40 µm as n increases, it is considerably smaller than the experimental length of the inner structure. On the other hand, the distance between Ath and Bth reaches approximately 85 µm at n = 1,000, and is still increasing. The simulated value is expected to approach the experimental inner structure size of 100 µm around n ≈3,000.
Interestingly there exists some fluctuation of the peak location of TJBP with a range of z = 234 ± 3 µm on the downstream side as seen in Fig. 10(b), while little fluctuation exists in the upstream side. This suggests that some nondeterministic process is included in the downstream, despite that the ionization process near the focus is basically dependent on the seed electrons for avalanche ionization produced by MPI alone, which should be deterministic. This is because the laser intensity near the GF is affected by the laser absorption in the moving plasma upstream, providing longer term fluctuation of transmitted laser energy through the moving plasma. We suppose that this fluctuation corresponds to the uneven depth of the inner structure observed in Fig. 1(c), although the cycle of the observed fluctuation is much larger than that of the simulated value.
3.4 Mechanism of periodic plasma motion
In order to gain an insight into the periodic plasma motion discussed in Section 3.3, ρmax, αmax and Imax are plotted vs. z in Fig. 11 at laser pulse numbers of n = 52 ~66, which correspond to the third cycle in Fig. 9(d). The temperature TJBP and the corresponding thermally ionized electron density ρtherm determined by Eq. (7) are also plotted in the figure. Each curve shows complicate variation, since TJBP and Imax are affected by the laser energy consumption for ρmax and vice versa.
Fig. 11 Free-electron densities (ρtherm & ρmax), temperature TJBP, laser intensity (Imax: with plasma, I0max: without plasma) and absorption coefficient α max are plotted vs. z at every two pulses. Laser beam propagating from left to right is focused at GF zf = 250 µm.
The contribution of ρtherm is realized by comparing between ρmax attained with and without ρtherm at the same Imax. Based on a simplified model [14] assuming that MPI and avalanche ionization can be separated and that avalanche ionization occurs during the second half of the laser pulse after the creation of the initial electrons by MPI and thermal ionization with neglecting recombination electron loss, free-electron density ρmax can be roughly estimated by (ρMPI + ρtherm) × exp[ηcascImaxτp/2] where ρMPI is free-electron density produced by MPI alone. At the location z ≈200 µm (A: outside of the plasma) in Fig. 11(c), for instance, the threshold laser intensity Imax ≈2 × 1016 W/m2 with ρtherm ≈0 provides infinitesimal free-electron density ρmax. At z = 212 µm (B: inside of the plasma), on the other hand, the same value of Imax with ρtherm ≈8 × 1025 /m3 (C: TJBP = 5,000 °C) produces free-electron density as large as ρmax ≈1.9 × 1027 m−3, which provides the absorption coefficient as large as αmax ≈5 × 105 m−1. This leads to the conclusion that the plasma moves pulse-by-pulse toward the laser source, because the highly absorbing plasma having large αmax is produced by the existence of ρtherm so that the laser energy absorption is concentrated to very thin front layer of the plasma volume. The concentrated laser energy absorption provides local increase of TJBP and hence ρtherm at the plasma front face at the next pulse. It is noted that the avalanche ionization at z = 212 µm is seeded dominantly by thermally ionized electrons ρtherm (ρthrem » ρMPI), while the avalanche ionization at z = 200 µm is seeded by predominantly seeded by ρMPI.
In the increasing phase of the peak value of ρmax [Figs. 11(b)-11(d)], the laser energy is absorbed in increasingly thinner layer, because high temperature region concentrates further to the narrower front region due to the increase in αmax. In this situation, no ionization occurs near the GF, because the laser energy is strongly absorbed by the plasma and the transmitted laser intensity does not reach the threshold of MPI (≈2 × 1016 W/m2).
As the plasma moves further toward the laser source, the forward plasma motion is slowed down by the decline of avalanche ionization to shift into decreasing phase of ρmax. The decline occurs because free-electron density produced by the avalanche ionization term of exp[ηcascImaxτp/2] is decreased due to the decrease of Imax as the plasma moves upstream. The decline of avalanche ionization results in the decrease of TJBP, hence ρtherm and αmax so that larger fraction of laser energy is transmitted through the plasma. When the transmitted laser intensity near the GF reaches threshold laser intensity Ith [at #61 between Fig. 11(e) and Fig. 11(f)], the plasma is produced near GF where the seed electrons are produced dominantly by MPI [Figs. 11(f)]. As shown in Fig. 10, the plasma motion extends further upstream as the cycle number increases due to heat accumulation in the laser absorption region until the absorbed laser energy and the heat diffusion are in final balance in a long time scale.
Our simulation model agrees qualitatively with experimentally observed periodic plasma motion by high-speed video camera, enabling understanding of the physical meaning of heat accumulation at high repetition rate. However, we find some discrepancies that the cycle of the simulated periodic plasma motion is significantly shorter than the experimental observation, and that simulated free-electron density seems to be overestimated. We do not know the reasons for the discrepancies exactly at the moment. One of the possible reasons for the discrepancies is considered to be because the numerical simulation was carried out using physical constants with uncertainty in a wide temperature range from room temperature to as high as ≈10,000 °C. Besides the model includes the rather complex interplay of processes including optical and thermal ionizations, laser beam propagation and heat diffusion. Actually our model is simplified due to limited capability of computer for simulating large number of laser pulses. While the simulation presented in this paper is limited to n = 1,000 laser pulses, this pulse number is still below the steady condition of 2,000 ~3,000 pulses. Further improvement of the model and exact knowledge of the physical properties at high temperatures are needed for qualitative agreement.
4. Summary
Spatial and temporal evolution of free-electron density in internal modification of glass by fs-laser pulses at high pulse repetition rates is simulated based on rate equation model and thermal conduction model to incorporate the effect of thermal ionization in the model.
Model shows that highly absorbing plasma with focus volume size generated near the geometrical focus moves toward the laser source periodically to cover the region with the size much larger than focus volume. It is shown that thermally ionized electrons play an important role in increasing free-electron density in plasma by seeding avalanche ionization to provide very large absorption coefficient, so that the laser energy is absorbed in thin front layer of the plasma volume. The small plasma appeared near the geometrical focus moves toward the laser source periodically to cover the laser-absorption region, which is much larger than focus volume.
The simulated results agree qualitatively with dynamic plasma motion observed using high-speed video camera in internal modification of borosilicate glass by fs-laser pulses at 1 MHz. The model also agrees with the experimentally observed heat accumulation effect where the laser absorptivity and hence the modified region increase with increasing pulse repetition rate at given laser pulse energies. However, the cycle of the simulated periodic plasma motion is significantly shorter than that of the experimental observation. Further study is needed for the quantitative agreement.
Acknowledgments
The experiment in this study was carried out using the fs-laser from IMRA with the technical support of Aisin Co. The authors wish to thank their support. This work was partially supported by Erlangen Graduate School in Advanced Optical Technologies (SAOT)
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