The spatial distribution of the laser energy absorbed by nonlinear absorption process in bulk glass w(z) is determined and thermal cycles due to the successive ultrashort laser pulse (USLP) is simulated using w(z) based on the transient thermal conduction model. The thermal stress produced in internal melting of bulk glass by USLP is qualitatively analyzed based on a simple thermal stress model, and crack-free conditions are studied in glass having large coefficient of thermal expansion. In heating process, cracks are prevented when the laser pulse impinges into glass with temperatures higher than the softening temperature of glass. In cooling process, shrinkage stress is suppressed to prevent cracks, because the embedded molten pool produced by nonlinear absorption process behaves like an elastic body under the compressive stress field unlike the case of CW-laser welding where the molten pool having a free surface produced by linear absorption process is plastically deformed under the compressive stress field.
©2013 Optical Society of America
Internal modification of transparent material is one of the most interesting applications of ultrashort laser pulses (USLP), which includes waveguide formation , 3-D memory , selective etching , fusion welding [4–6] and so on. Among them fusion welding of glass has been attracting much attention, because crack-free welding of glass even with a large coefficient of thermal expansion (CTE) such as borosilicate glass  and Foturan glass [8,9] is available without pre- and post-heating, and welding is accomplished by selective melting only near the interface of glass plates. This is quite a contrast to existing CW-laser welding of glass where crack-free welding is possible only with glass having small CTE like fused silica, and melting of whole thickness of the upper glass plate is required in overlap welding of glass plates .
A variety of papers have been published on USLP welding of glass, including evaluation of nonlinear absorptivity [9,11], preparation of joint interface , evaluation of mechanical strength of weld joint [4,6,8,9,12,13], joining dissimilar glass materials  and so on. In spite that prevention of cracks is one of the most important tasks in welding of brittle material like glass, it is quite strange that no papers have been published to account for the mechanism of crack-free welding of glass having large CTE using USLP. While it is reported cracks are also produced depending on the laser parameters in USLP welding of glass , no systematic study on the cracking conditions has been reported so far. USLP welding of glass is characterized by steep temperature gradients due to extremely short duration of laser pulses and molten pool embedded in bulk glass caused by nonlinear absorption process. Thus large differences from conventional CW-laser welding procedures of glass are expected where mild temperature gradient is produced along with molten pool having a free surface.
In the present study, spatial distribution of laser energy absorbed in bulk glass by nonlinear absorption process in USLP welding of glass is analyzed and the thermal cycle due to successive USLP is simulated based on the thermal conduction model, assuming Gaussian beam propagation. Stress field in heating and cooling processes is qualitatively analyzed based on a simple thermal stress model to clarify crack-free conditions, showing that no cracks are produced with an embedded molten pool at high pulse repetition rates. It is also demonstrated that cracks are developed in cooling process even using USLP when the molten pool has a free surface.
2. Crack-free conditions in internal melting of glass by USLP
USLP with pulse duration of 10ps (wavelength λ = 1064nm) was tightly focused into bulk of borosilicate glass (D263, Schott) with a thickness of 1mm using a lens of NA 0.55. Experiments were conducted at different pulse repetition rates f (50kHz~1MHz) and pulse energies Q0 (<11µJ) at a translation speed of v = 20mm/s transversely to the laser axis.
Figure 1 shows the cross-sections observed by a transmission microscope at pulse energy of Q0 = 1.63µJ at different pulse repetition rates. At f≧200kHz, the internal melting without cracks is observed. This is a big contrast to CW-laser welding with CO2 laser  where cracks are produced in the molten region by the shrinkage stress, when the weld sample is cooled down to room temperature [15,16], and thus crack-free local melting is available only in glass having small CTE like fused silica [10,14,17]. This suggests that the stress field in USLP welding is different from that in CW-laser welding.
The cross-section of crack-free internal melting exhibits a dual-structure consisting of an elliptical outer structure and a teardrop-shaped inner structure as seen in Figs. 1(c)–1(e). The outer structure corresponds to the region where the forming temperature Tform (1,051˚C) with a viscosity η = 104dPas is reached. The outer region is concerned as the molten region from the fact that two glass plates coalesce in the outer region . The thermal conduction model shows that the inner structure coincides with the laser-absorbed region, and the bottom tip of the inner structure coincides with the geometrical focus of the laser beam assuming self-focusing is negligible . At lower pulse repetition rates of f = 50kHz and 100kHz, cracks are produced and no dual-structure is produced as seen in Figs. 1(a) and 1(b).
Crack-free and cracking conditions are shown with circles and triangles at different pulse repetition rates and pulse energies, respectively in Fig. 2. The squares show the intermediate conditions where the images are not clear enough to recognize whether or not cracks are produced from the microscope images. At given pulse repetition rates, cracks are produced at pulse energies below some critical values. The crack-free conditions are above a solid line corresponding to the average laser power of W≈0.25W, suggesting average laser power higher than some critical laser power is needed to suppress cracks. The dual-structure is always found in crack-free region.Fig. 1 in accordance with our previous report [8,11].
3. Thermal conduction analysis
3.1 Transient thermal conduction model
When USLP is tightly focused into bulk glass, one might expect that highly excited free electrons produce microscopic structural change and chemical decomposition before reaching equilibrium between the free electrons and the lattice to provide temperature field. While microscopic structural changes such as nanogratings , for instance, can be produced by a single USLP to reduce the mechanical strength due to its characteristic narrow slot structures, they are annealed out in molten region at high pulse repetition rates.
The chemical decomposition can also result in non-uniform element distribution and even pores in the molten region. Although some pores are found in internally melted fused silica by USLP [6,8], no pores are produced in borosilicate glass as seen in Fig. 1. It is also noted that no evaporation loss occurs in the embedded molten pool in contrast to the case of conventional welding where selective element loss occurs due to th temperature dependent vapor pressure of the consisting elements of weld material because the molten pool has free surface. It is also reported that non-uniform element distribution  is produced in internal melting by USLP at high pulse repetition rates in glass containing a large amount of network modifiers due to the temperature dependent diffusivity of elements at high temperatures. However, a three-point-bending test of glass sample with internally melted region indicates no decrease in the bending strength in comparison with the base material , suggesting the effects of non-uniform element distribution on macroscopic mechanical strength are negligible at high pulse repetition rates.
Thus in the present study we assume that microscopic effects that can be produced before reaching equilibrium between the free electrons and the lattice are negligible, and only macroscopic thermal stress is analyzed qualitatively based on a thermal stress model where the temperature field due to the laser energy absorbed by nonlinear process is simulated by a classic thermal conduction model.
Transient temperature distribution in glass due to successive USLP is simulated by the thermal conduction model to study the crack-formation mechanism in heating process. The model is briefed here, since it is detailed in . Assuming Gaussian beam propagation in glass sample as shown in Fig. 3(a), the radius of the laser spot ω(z) is given by Fig. 3(b), the laser energy absorbed at (r,z) per unit volume per pulse q(r,z) is given by
3.2 Simulation of isothermal line
In order to simulate the transient temperature TN(x,y,z;t) by Eq. (3), w(z) has to be determined. While distribution of the absorbed laser energy in the plasma has been simulated by several authors based on rate equations for free electrons in single pulse irradiation [21–23], the model cannot be applied to the case of multi-pulse irradiation at high repetition rates. We introduce a moving line heat source model, where the absorbed laser energy w(z) is concentrated to an infinitesimally thin line along the laser axis with continuous heat delivery. Then the steady temperature distribution T(x,y,z) is given by Eq. (4) to the experimental modified pattern with the dual-structure seen in Fig. 1. While the laser beam has actually a finite spot size ω(z) and pulsed energy delivery, such a simple model can be used to simulate the isotherm, because the temperature apart from the heat source is spatially and temporally averaged. In order to minimize the number of the parameters to be determined by fitting, w(z) is assumed to be a simple function written in a form 
By fitting the simulated isotherm of the forming temperature to the experimental outer structure, Wab is determined precisely within uncertainty of ± 3% . However, the distribution of w(z) cannot be determined precisely by the fitting, since the melt isotherm is distant from the heat source. Thus the distribution of w(z) is determined by fitting the simulated isotherm Tin to the experimental inner structure where Tin is the characteristic temperature of the inner structure, since the inner isotherm is sensitively affected by the distribution of w(z) due to the shorter distance to the heat source.
Figure 4 shown w(z) providing best fitting of the isothermal lines. Figures 5(c)-5(d) show thus simulated isotherms at f = 200~500kHz, and the simulated isotherm of Tform = 1,051˚C plotted in the blue line agrees well with the contour of the experimental outer structure, and the simulated values of Acal agrees with Aex within error of ± 3% in accordance with . The simulated isotherms of Tin plotted by red lines are also seen to agree with the experimental inner structures in terms of shape and size. However, simulated characteristic temperatures Tin tend to increase as f increases in a range of 3,000˚C~3,800˚C within condition tested unlike the case of the outer structure where the characteristic temperature is deterministically given by 1,051 ˚C.
At lower repetition rates where no dual-structure is observed, the isotherm Tform was simulated using Aex shown in Fig. 1. At f = 100kHz, the simulated isotherm of Tform agrees with the experimental contour except at locations near the focus as seen in Fig. 1(b). At f = 50kHz, the simulated isotherm of Tform occupies only the narrow region near the laser beam axis, and the experimental contour (green line) is as low as approximately 500˚C, as seen in Fig. 5(a).
In this paper, the experimental results using USLP with pulse duration of 10ps are discussed. It is worthwhile to compare the duration of USLP in terms of the contribution of multiphoton ionization and avalanche ionization in the laser-absorption process. While breakdown process is dominated by multiphoton ionization, it is noted that 100~1000 times more free electrons are produced by avalanche ionization at the end of the laser pulse when single infrared laser pulse with a duration as short as 100fs is focused in water . This suggests that the contribution of avalanche ionization is always dominant in internal melting process using commercially available USLP in fs~ps regimes at high pulse repetition rates, although the multiphoton ionization is essential to start avalanche ionization even at high pulse repetition rates.
3.3 Transient temperature distribution
Substituting w(z) shown in Fig. 4, the transient temperature distribution TN(0,0,z;t) is simulated. Figure 6 shows the temperature variation TN(0,0,z;t) simulated on the laser axis (r = 0) at z = 1µm and z = 12µm. In this simulation, it is assumed that q(r,z) is independent of N, for simplicity. At time t = 0, the temperature instantaneously rises by ΔT(z), which is given byFigure 7 shows TSB(z) and TSP(z) plotted vs. z where TSB(z) and TSP(z) are the temperature at (0,0,z) at time just before and after the impingement of the laser pulse at N→∞, respectively, and are given by
While the number of the laser pulse before reaching steady value TSB(z) increases as the pulse repetition rate increases, the time needed for TNB(z) to reach steady value TSB(z) is unchanged, approximately 5ms, which corresponds to the moving distance of x = 100µm at v = 20mm/s. Interestingly TSB(z) increases as z increases until the maximum is reached near the middle despite ΔT(z) is largest at z = 0. This is because the cooling rate between laser pulses increases as z = 0 is approached due to decrease in spot size ω(z). It is also noted that ΔT(0) decreases as f increases, because the laser energy reaching the focus decreases due to increased laser absorption in the longer and hotter plasma column as f increases.
4. Mechanism of crack-free welding of glass using USLP
4.1 Stress model in laser welding of glass
The thermal stress in glass in laser welding can be qualitatively analyzed by a simple model consisting of bar A and bar B, which correspond to welding region and surrounding region, respectively, and are connected to the rigid body [15,16] for CW-laser and USLP as shown in Figs. 8 and 9, respectively. Then cracks are developed, when the tensile stress exceeds the strength of the glass if the temperature of the glass is not high enough to provide ductility. In heating process where the laser energy is being deposited in bar A, compressive and tensile stresses are produced in bars A and B, respectively, since the thermal expansion in bar A is constrained by bar B (a).
In heating process in CW-laser welding, cracks in bar B are normally suppressed, since bar B gains ductility by the temperature rise due to the thermal conduction from bar A. When bar A is melted, the compressive stress in bar A is released due to the plastic deformation of the molten region as shown in Fig. 8(b). It should be noted that the plastic deformation occurs because the molten region in bar A contains a free surface in CW-laser welding (b). When bar A is cooled down to room temperature, tensile and compressive stresses are produced in bars A and B, respectively, since the thermal shrinkage of bar A is constrained by bar B (c). Cracks are produced in bar A if the tensile stress in bar A exceeds the strength of the material due to brittleness of glass at room temperature. Therefore crack-free welding is available only in glass having small CTE like fused silica.
The thermal stress developed in USLP welding of glass is illustrated in Fig. 9 where two features are found that are different from CW-laser welding. First, the tensile stress is produced in bar B before the temperature rise due to the thermal conduction from bar A to bar B occurs (a). Thus cracks are developed, unless the temperature in bar B just before the impingement of the laser pulse is high enough to provide ductility. Second, the nonlinear absorption process produces the molten pool embedded in the bulk glass, which behaves like an elastic body (b). This is a big contrast to the case of CW-laser welding where the molten pool has a free surface and hence is plastically deformed by the compressive stress. Since the elastic strain is reversible, no shrinkage stress is left in bar A when it is cooled down to room temperature (c), resulting in crack-free welding even with glass having large CTE unless no cracks are produced in the heating process.
In the following sections, crack-free conditions in heating and cooling processes in USLP welding of glass at N→∞ are qualitatively analyzed based on the thermal conduction model.
4.2 Crack-free condition in heating process ˚C
Since thermal stress and ductility of glass are dependent on ΔT and TSB, respectively, the tendency toward cracking can be qualitatively estimated by the relationship between ΔT and TSB as shown in Fig. 10 that is the re-plot of Fig. 6. As a general trend in Fig. 10, the peak of ΔT decreases and the minimum of TSB increases as the pulse repetition rate increases, suggesting that tendency toward cracking is reduced at higher pulse repletion rates. It is also seen that ΔT increases and TSB decreases as the focus is approached, suggesting that cracks tend to be produced near the focus.
Assuming that the cross-section of bar A is much larger than that of bar B, the tensile stress σ = εE is produced in bar B when thermal strain ε = ΔTαcte in bar A is constrained by bar B where αcte is CTE and E is Young’s modulus. Then the tensile stress σ reaches σtens = 150MPa, which corresponds to the tensile strength of the material , at the critical temperature rise of ΔTcr≈285°C where E = 7.3x104MPa and αcte = 7.2x10−6/°C. It is seen that ΔT easily exceeds ΔTcr in USLP welding of glass as is seen in Fig. 10. Therefore in order to avoid cracks in heating process,
Αt f = 50kHz, ΔT reaches as high as 1,300°C, which is much higher than ΔTcr, while TSB is as low as 320~440°C, which is significantly lower than Tsoft = 736°C of D263, suggesting that cracks cannot be avoided in the whole laser-absorption region. Actually cracks are found above and below the laser absorption region as seen in Fig. 1(a). At f = 100kHz, while the peak of ΔT is also significantly higher than ΔTcr, TSB is higher than Tsoft except near the focus, suggesting the crack-free condition Eq. (9) is not fulfilled near the focus. In accordance with the simulated curve of ΔT-TSB, the small crack is found near the focus in Fig. 1(b).
At pulse repetition rates f≧200kHz, TSB is higher than not only Tsoft but even Tform in the whole laser-irradiated region, indicating the crack-free condition Eq. (9) is more safely fulfilled. The dual-structure visualizes the fulfillment of the safer crack-free condition TSB>Tform, since the teardrop-shaped inner structure corresponding to the laser absorption region is surrounded by the elliptical molten region in the dual-structure. Several authors have reported that dual-structure provides no cracks in different glasses with large CTE including 0211 (αcte = 7.4 x10−6/˚C: Corning) , AF45 (αcte = 4.5 x10−6/˚C: Schott) [26,27], B270 (αcte = 9.4 x10−6/˚C: Schott) , Foturan (αcte = 8.6 x10−6/˚C: Schott) . These results support our stress model of crack-free internal melting of glass in heating process.
4.3 Crack-free condition in cooling process
In order to clearly demonstrate that cracks are avoided in the cooling process when the molten pool is embedded, two experiments were conducted at pulse repetition rate of 1MHz where no cracks are produced in heating process.
In the first experiment, USLP was focused into the tilted glass plate near the rear surface, and the sample was moved horizontally at 20mm/s as shown in Fig. 11(a). Figure 11(b) shows the appearance of the molten region near the bottom surface. No cracks are found when the molten pool is embedded in bulk glass (region i), while cracks are produced when the molten region is exposed to the bottom surface (region ii).
In the second experiment, a sample pair having varying clearance d between glass plates was prepared where two glass plates with optical contact was cramped mechanically on one side and a metallic foil with a thickness of 30µm was inserted from the other side, and was welded by USLP. Figure 12 shows the top view of the weld bead illuminated by the light of 530nm. While the crack-free welding is produced at clearances smaller than some critical value of dc≈500nm, minute cracks were developed at clearances d>dc. It should be noted that the value of dc is of rough estimation, since the clearance d changes due to the shrinkage stress in cooling process. This suggests that at d>dc the viscosity resistance between the glass plates is reduced so that plastic deformation of the molten pool cannot be neglected while the glass is at temperatures high enough to have small viscosity. Small droplets were found to spread out of the laser-irradiated region at larger clearances suggesting the internal pressure of the laser-heated glass is large. This experimental result also indicates that the optical contact is strongly recommended in overlap welding of glass plates.
4. Summary and conclusions
Crack-free conditions in USLP welding of glass in heating and cooling processes are qualitatively analyzed based on the simple thermal stress model. It is found that USLP welding produces the stress field different from that of CW-laser welding, resulting in crack-free welding of glass even with large CTE because of the nonlinear absorption of the laser energy, and cracks in heating process because of the short laser pulse.
In heating process, transient temperature distributions in glass due to the successive USLP at N→∞ (N = number of the laser pulse) are simulated assuming Gaussian beam propagation. Our simulation revealed that cracks are prevented when the laser pulse impinges into glass with temperature higher than softening temperature of glass. Cracks are more safely prevented when dual-structure is produced, which consists of elliptical outer structure (molten pool) and teardrop-shaped inner structure (laser absorption region).
In cooling process cracks are suppressed even in glass having large CTE when a molten pool is embedded in bulk glass, since the embedded molten pool behaves like elastic body so that shrinkage stress of the molten region is suppressed when it is cooled down to room temperature.
This work was partially supported by Erlangen Graduate School in Advanced Optical Technologies (SAOT).
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