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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 12 — Jun. 17, 2013
  • pp: 14291–14302
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Crack-free conditions in welding of glass by ultrashort laser pulse

Isamu Miyamoto, Kristian Cvecek, and Michael Schmidt  »View Author Affiliations


Optics Express, Vol. 21, Issue 12, pp. 14291-14302 (2013)
http://dx.doi.org/10.1364/OE.21.014291


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Abstract

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 OSA

1. Introduction

Internal modification of transparent material is one of the most interesting applications of ultrashort laser pulses (USLP), which includes waveguide formation [1

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21(21), 1729–1731 (1996). [CrossRef] [PubMed]

], 3-D memory [2

2. M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, “Three-dimensional optical data storage in vitreous silica,” Jpn. J. Appl. Phys. 37(Part 2, No. 12B), L1527–L1530 (1998). [CrossRef]

], selective etching [3

3. C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 47–61 (2006). [CrossRef]

], fusion welding [4

4. T. Tamaki, W. Watanabe, J. Nishii, and K. Itoh, “Welding of transparent materials using femtosecond laser pulses,” Jpn. J. Appl. Phys. 44(22), L687–L689 (2005). [CrossRef]

6

6. S. Richter, S. Döring, A. Tünnermann, and S. Nolte, “Bonding of glass with femtosecond laser pulses at high repetition rates,” Appl. Phys., A Mater. Sci. Process. 103(2), 257–261 (2011). [CrossRef]

] 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 [7

7. I. Miyamoto, A. Horn, and J. Gottmann, “Local melting of glass material and its application to direct fusion welding by ps-laser pulses,” J. Laser MicroNanoeng. 2(1), 7–14 (2007). [CrossRef]

] and Foturan glass [8

8. I. Miyamoto, K. Cvecek, Y. Okamoto, M. Schmidt, and H. Helvajian, “Characteristics of laser absorption and welding in FOTURAN glass by ultrashort laser pulses,” Opt. Express 19(23), 22961–22973 (2011). [CrossRef] [PubMed]

,9

9. S. Wu, D. Wu, J. Xu, Y. Hanada, R. Suganuma, H. Wang, T. Makimura, K. Sugioka, and K. Midorikawa, “Characterization and mechanism of glass microwelding by double-pulse ultrafast laser irradiation,” Opt. Express 20(27), 28893–28905 (2012). [CrossRef] [PubMed]

] 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 [10

10. M. Levesque, B. Labrnche, R. Forest, E. Savard, S. Deshaies, and A. Cournoyer, “Welding of glass pieces,” Physics Procedia 5, 139–144 (2010). [CrossRef]

].

A variety of papers have been published on USLP welding of glass, including evaluation of nonlinear absorptivity [9

9. S. Wu, D. Wu, J. Xu, Y. Hanada, R. Suganuma, H. Wang, T. Makimura, K. Sugioka, and K. Midorikawa, “Characterization and mechanism of glass microwelding by double-pulse ultrafast laser irradiation,” Opt. Express 20(27), 28893–28905 (2012). [CrossRef] [PubMed]

,11

11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

], preparation of joint interface [12

12. K. Cvecek, I. Miyamoto, J. Strauss, M. Wolf, T. Frick, and M. Schmidt, “Sample preparation method for glass welding by ultrashort laser pulses yields higher seam strength,” Appl. Opt. 50(13), 1941–1944 (2011). [CrossRef] [PubMed]

], evaluation of mechanical strength of weld joint [4

4. T. Tamaki, W. Watanabe, J. Nishii, and K. Itoh, “Welding of transparent materials using femtosecond laser pulses,” Jpn. J. Appl. Phys. 44(22), L687–L689 (2005). [CrossRef]

,6

6. S. Richter, S. Döring, A. Tünnermann, and S. Nolte, “Bonding of glass with femtosecond laser pulses at high repetition rates,” Appl. Phys., A Mater. Sci. Process. 103(2), 257–261 (2011). [CrossRef]

,8

8. I. Miyamoto, K. Cvecek, Y. Okamoto, M. Schmidt, and H. Helvajian, “Characteristics of laser absorption and welding in FOTURAN glass by ultrashort laser pulses,” Opt. Express 19(23), 22961–22973 (2011). [CrossRef] [PubMed]

,9

9. S. Wu, D. Wu, J. Xu, Y. Hanada, R. Suganuma, H. Wang, T. Makimura, K. Sugioka, and K. Midorikawa, “Characterization and mechanism of glass microwelding by double-pulse ultrafast laser irradiation,” Opt. Express 20(27), 28893–28905 (2012). [CrossRef] [PubMed]

,12

12. K. Cvecek, I. Miyamoto, J. Strauss, M. Wolf, T. Frick, and M. Schmidt, “Sample preparation method for glass welding by ultrashort laser pulses yields higher seam strength,” Appl. Opt. 50(13), 1941–1944 (2011). [CrossRef] [PubMed]

,13

13. W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89, 021106 (2006).

], joining dissimilar glass materials [13

13. W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89, 021106 (2006).

] 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 [11

11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

], 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
Fig. 1 Cross-sections of internally melted D263 at 20mm/s at Q0 = 1.63µJ. Experimental values of nonlinear absorptivity Aex are (a) 45%, (b) 52%, (c) 66% (d) 76% and (e) 81%.
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 [14

14. Y. Arata, H. Maruo, I. Miyamoto, and S. Tackuchi, “Dynamic behavior of laser welding and cutting,” Proc Symp. Electron and ion beam Science and technologies, 7th Int. Conf. 111–128 (1976)

] where cracks are produced in the molten region by the shrinkage stress, when the weld sample is cooled down to room temperature [15

15. M. Watanabe and K. Satoh, Welding Mechanics and its Applications (Asakura, 1965), Chap. 8.

,16

16. T. Terasaki, “Welding distortion and residual stress,” J. Jpn. Welding Soc. 78, 139–146 (2009).

], and thus crack-free local melting is available only in glass having small CTE like fused silica [10

10. M. Levesque, B. Labrnche, R. Forest, E. Savard, S. Deshaies, and A. Cournoyer, “Welding of glass pieces,” Physics Procedia 5, 139–144 (2010). [CrossRef]

,14

14. Y. Arata, H. Maruo, I. Miyamoto, and S. Tackuchi, “Dynamic behavior of laser welding and cutting,” Proc Symp. Electron and ion beam Science and technologies, 7th Int. Conf. 111–128 (1976)

,17

17. D. O. MacCallum, G. A. Knorovsky, and S. T. Reed, “CO2 laser welding fused silica,” Proc. 24th Int. Cong. on Application of Lasers and Electro-Optics (ICALEO) 687–695 (2005)

]. 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 [11

11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

]. 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 [11

11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

]. 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
Fig. 2 Crack-free and cracking conditions at v = 20mm/s in D263. Crack-free internal melting is available at average laser power higher than 0.25W.
. 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.

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 [18

18. C. Hnatovsky, R. S. Taylor, P. P. Rajeev, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Pulse duration dependence of femtosecond-laser-fabricated nanogratings in fused silica,” Appl. Phys. Lett. 87(1), 014104 (2005). [CrossRef]

], 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

6. S. Richter, S. Döring, A. Tünnermann, and S. Nolte, “Bonding of glass with femtosecond laser pulses at high repetition rates,” Appl. Phys., A Mater. Sci. Process. 103(2), 257–261 (2011). [CrossRef]

,8

8. I. Miyamoto, K. Cvecek, Y. Okamoto, M. Schmidt, and H. Helvajian, “Characteristics of laser absorption and welding in FOTURAN glass by ultrashort laser pulses,” Opt. Express 19(23), 22961–22973 (2011). [CrossRef] [PubMed]

], 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 [19

19. Y. Liu, M. Shimizu, B. Zhu, Y. Dai, B. Qian, J. Qiu, Y. Shimotsuma, K. Miura, and K. Hirao, “Micromodification of element distribution in glass using femtosecond laser irradiation,” Opt. Lett. 34(2), 136–138 (2009). [CrossRef] [PubMed]

] 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 [8

8. I. Miyamoto, K. Cvecek, Y. Okamoto, M. Schmidt, and H. Helvajian, “Characteristics of laser absorption and welding in FOTURAN glass by ultrashort laser pulses,” Opt. Express 19(23), 22961–22973 (2011). [CrossRef] [PubMed]

], 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 [11

11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

]. Assuming Gaussian beam propagation in glass sample as shown in Fig. 3(a)
Fig. 3 (a) Instantaneous Gaussian heat source with radius of ω(z) and intensity of q(r,z) at repetition rate of f. (b) Line heat source of continuous heat delivery with average intensity distribution w(z).
, the radius of the laser spot ω(z) is given by [20

20. A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993). [CrossRef]

]
ω(z)=ω01+(M2λzπω02ng)2;ω0=M2λπNA,
(1)
where z is distance from the focus along the optical axis, λ wavelength of the laser beam, ω0 radius of laser spot at z = 0, M2 beam quality factor, NA numerical aperture of focusing optics and ng refractive index of the bulk glass. Assuming w(z) is time-averaged laser power absorbed per unit length in the laser-induced plasma in a region of 0<z<l as shown in Fig. 3(b), the laser energy absorbed at (r,z) per unit volume per pulse q(r,z) is given by
q(r,z)=2w(z)πω2(z)fexp{2r2ω2(z)};0zl,
(2)
where r = (x2 + y2)0.5. Assuming that q(r,z) appears instantaneously at a repetition rate of f in an infinite body, which moves transversely to the laser beam at a constant speed of v, the temperature (x,y,z) at time t after the impingement of Nth pulse is given by
TN(x,y,z;t)=1πcρfi=0N11πα(tif1)0lw(z')ω2(z')+8α(tif1)×exp[2{(x+v(tif1))2+y2}ω2(z')+8α(tif1)(zz')24αt]dz'.
(3)
where α is thermal diffusivity, c specific heat and ρ density. In this model, thermal properties of glass are assumed to be independent of temperature, for simplicity.

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

21. Y. R. Shen, The Principles of Nonlinear Optics (John Wiley and Sons, 1984), Chap. 27.

23

23. C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells,” Opt. Express 15(16), 10303–10317 (2007). [CrossRef] [PubMed]

], 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 [11

11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

]
T(x,y,z)=14πK0lw(z')sexp{v2α(x+s)}dz'+T0,
(4)
where s2 = x2 + y2 + (z-z’)2 and K is thermal conductivity. In the present study, w(z) is determined by fitting the simulated isotherm of 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 [11

11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

]
w(z)=azm+b;0<z<l,
(5)
where a, b and m are positive constant. Assuming Acal is nonlinear absorptivity, the average absorbed laser power Wab is given by
Wab=AcalW=0lw(z)dz,
(6)
where W0 is average incident laser power.

By fitting the simulated isotherm of the forming temperature to the experimental outer structure, Wab is determined precisely within uncertainty of ± 3% [11

11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

]. 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
Fig. 4 Simulated w(z) at different pulse repetition rates corresponding to Fig. 5.
shown w(z) providing best fitting of the isothermal lines. Figures 5(c)
Fig. 5 Simulated isothermal lines of Tout = 1051°C (blue line). (a) The outer structure (500°C) is shown by a green line (Aex = 45%), (b) Tin = 3000°C (Aex = 52%,), (c) Tin = 3000°C (Acal = 64.1%), (d) Tin = 3300°C (Acal = 74.2%), (e) Tin = 3800°C (Acal = 79.9%). (D263, m = 2 and M2 = 3.5).
-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 [11

11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

]. 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 [22

22. J. Noack and A. Vogel, “Laser-induced plasma formation in water at nanosecond to femtosecond time scales: Calculation of thresholds, absorption coefficients and energy density,” IEEE J. Quantum Electron. 35(8), 1156–1167 (1999). [CrossRef]

]. 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
Fig. 6 Temperature variations on the laser axis TN(0,0,z;t) at z = 1µm and 12µm are plotted until N = 12, and thereafter only TNB(0,0,z;τ) is plotted. Pulse repetition rates f (kHz): (a) 50, (b) 200 and (c) 500 (v = 20mm/s, Q0 = 1.63µJ, M2 = 3.6)
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 by
ΔT(z)=TN(0,0,z;0)TN1(0,0,z;τ)=q(0,z)cρ;(τ=1/f),
(7)
and decreases down to the base temperature TNB = TN(0,0,z;τ) due to thermal diffusion between laser pulses. In the figure, the temperature variation within each pulse is plotted until 12th pulse, and thereafter only the base temperature TNB = TN(0,0,z;τ) is plotted for better viewability. The base temperature TNB(z) increases as the number of the laser pulse increases due to the heat accumulation between laser pulses. Figure 7
Fig. 7 Distribution of TSB(z) and TSP(z) plotted by thick and thin lines, respectively, at f = 50kHz, 200kHz and 500kHz (N→∞).
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

TSB(z)=limNTN(0,0,z;τ),TSP(z)=limNTN(0,0,z;0).
(8)

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

15. M. Watanabe and K. Satoh, Welding Mechanics and its Applications (Asakura, 1965), Chap. 8.

,16

16. T. Terasaki, “Welding distortion and residual stress,” J. Jpn. Welding Soc. 78, 139–146 (2009).

] for CW-laser and USLP as shown in Figs. 8
Fig. 8 Thermal stress produced in CW-laser welding of glass where molten pool contains free surface. Tensile stress is produced in bar A when bar A is cooled down to room temperature ( + and represent tensile and compressive stress, respectively).
and 9
Fig. 9 Thermal stress in USLP welding of glass where molten pool contains no free surface. While tensile stress is developed in heating period in bar B, no tensile is produced in Bar A, when bar A is cooled down to room temperature.
, 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
Fig. 10 Relationship between ΔT and TSB at Q0 = 1.63µJ at different pulse repetition rates in D263. The region highlighted by green color corresponds to the crack-free condition in heating process.
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 [24

24. I. Miyamoto, A. Horn, J. Gottmann, D. Wortmann, I. Mingareev, F. Yoshino, M. Schmidt, Y. Okamoto, Y. Uno, and T. Hermann, “Novel fusion welding technology of glass using ultrashort pulse lasers,” Proc. 27th International Congress on Applications of Lasers and Electro-Optics (ICALEO) 112–121 (2010). [CrossRef]

], 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,
TSB>Tsoft
(9)
has to be fulfilled.

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) [25

25. C. B. Schaffer, A. O. Jamison, and E. Mazur, “Morphology of femtosecond laser-induced structural changes in bulk transparent materials,” Appl. Phys. Lett. 84(9), 1441–1443 (2004). [CrossRef]

], AF45 (αcte = 4.5 x10−6/˚C: Schott) [26

26. S. M. Eaton, H. Zhang, P. R. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Y. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate,” Opt. Express 13(12), 4708–4716 (2005). [CrossRef] [PubMed]

,27

27. S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51(16-18), 2533–2542 (2004). [CrossRef]

], B270 (αcte = 9.4 x10−6/˚C: Schott) [28

28. M. Sakakura, M. Shimizu, Y. Shimotsuma, K. Miura, and K. Hirao, “Temperature distribution and modification mechanism inside glass with heat accumulation during 250 kHz irradiation of femtosecond laser pulses,” Appl. Phys. Lett. 93(23), 231112 (2008). [CrossRef]

], Foturan (αcte = 8.6 x10−6/˚C: Schott) [8

8. I. Miyamoto, K. Cvecek, Y. Okamoto, M. Schmidt, and H. Helvajian, “Characteristics of laser absorption and welding in FOTURAN glass by ultrashort laser pulses,” Opt. Express 19(23), 22961–22973 (2011). [CrossRef] [PubMed]

]. 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)
Fig. 11 (a) Experimental set up (f = 1MHz, Q0 = 1.6µJ). (b) Laser-irradiated region showing in region i internal melting (no cracks), (ii) cracks with melt region appeared at the rear surface and iii) no melting.
. 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).

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.

Acknowledgments

This work was partially supported by Erlangen Graduate School in Advanced Optical Technologies (SAOT).

References and links

1.

K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21(21), 1729–1731 (1996). [CrossRef] [PubMed]

2.

M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, “Three-dimensional optical data storage in vitreous silica,” Jpn. J. Appl. Phys. 37(Part 2, No. 12B), L1527–L1530 (1998). [CrossRef]

3.

C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys., A Mater. Sci. Process. 84(1-2), 47–61 (2006). [CrossRef]

4.

T. Tamaki, W. Watanabe, J. Nishii, and K. Itoh, “Welding of transparent materials using femtosecond laser pulses,” Jpn. J. Appl. Phys. 44(22), L687–L689 (2005). [CrossRef]

5.

I. Miyamoto, A. Horn, J. Gottmann, D. Wortmann, and F. Yoshino, “Fusion welding of glass using femtosecond laser pulses with high-repetition rates,” J. Laser Micro Nanoeng. 2(1), 57–63 (2007). [CrossRef]

6.

S. Richter, S. Döring, A. Tünnermann, and S. Nolte, “Bonding of glass with femtosecond laser pulses at high repetition rates,” Appl. Phys., A Mater. Sci. Process. 103(2), 257–261 (2011). [CrossRef]

7.

I. Miyamoto, A. Horn, and J. Gottmann, “Local melting of glass material and its application to direct fusion welding by ps-laser pulses,” J. Laser MicroNanoeng. 2(1), 7–14 (2007). [CrossRef]

8.

I. Miyamoto, K. Cvecek, Y. Okamoto, M. Schmidt, and H. Helvajian, “Characteristics of laser absorption and welding in FOTURAN glass by ultrashort laser pulses,” Opt. Express 19(23), 22961–22973 (2011). [CrossRef] [PubMed]

9.

S. Wu, D. Wu, J. Xu, Y. Hanada, R. Suganuma, H. Wang, T. Makimura, K. Sugioka, and K. Midorikawa, “Characterization and mechanism of glass microwelding by double-pulse ultrafast laser irradiation,” Opt. Express 20(27), 28893–28905 (2012). [CrossRef] [PubMed]

10.

M. Levesque, B. Labrnche, R. Forest, E. Savard, S. Deshaies, and A. Cournoyer, “Welding of glass pieces,” Physics Procedia 5, 139–144 (2010). [CrossRef]

11.

I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express 19(11), 10714–10727 (2011). [CrossRef] [PubMed]

12.

K. Cvecek, I. Miyamoto, J. Strauss, M. Wolf, T. Frick, and M. Schmidt, “Sample preparation method for glass welding by ultrashort laser pulses yields higher seam strength,” Appl. Opt. 50(13), 1941–1944 (2011). [CrossRef] [PubMed]

13.

W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett. 89, 021106 (2006).

14.

Y. Arata, H. Maruo, I. Miyamoto, and S. Tackuchi, “Dynamic behavior of laser welding and cutting,” Proc Symp. Electron and ion beam Science and technologies, 7th Int. Conf. 111–128 (1976)

15.

M. Watanabe and K. Satoh, Welding Mechanics and its Applications (Asakura, 1965), Chap. 8.

16.

T. Terasaki, “Welding distortion and residual stress,” J. Jpn. Welding Soc. 78, 139–146 (2009).

17.

D. O. MacCallum, G. A. Knorovsky, and S. T. Reed, “CO2 laser welding fused silica,” Proc. 24th Int. Cong. on Application of Lasers and Electro-Optics (ICALEO) 687–695 (2005)

18.

C. Hnatovsky, R. S. Taylor, P. P. Rajeev, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Pulse duration dependence of femtosecond-laser-fabricated nanogratings in fused silica,” Appl. Phys. Lett. 87(1), 014104 (2005). [CrossRef]

19.

Y. Liu, M. Shimizu, B. Zhu, Y. Dai, B. Qian, J. Qiu, Y. Shimotsuma, K. Miura, and K. Hirao, “Micromodification of element distribution in glass using femtosecond laser irradiation,” Opt. Lett. 34(2), 136–138 (2009). [CrossRef] [PubMed]

20.

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993). [CrossRef]

21.

Y. R. Shen, The Principles of Nonlinear Optics (John Wiley and Sons, 1984), Chap. 27.

22.

J. Noack and A. Vogel, “Laser-induced plasma formation in water at nanosecond to femtosecond time scales: Calculation of thresholds, absorption coefficients and energy density,” IEEE J. Quantum Electron. 35(8), 1156–1167 (1999). [CrossRef]

23.

C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells,” Opt. Express 15(16), 10303–10317 (2007). [CrossRef] [PubMed]

24.

I. Miyamoto, A. Horn, J. Gottmann, D. Wortmann, I. Mingareev, F. Yoshino, M. Schmidt, Y. Okamoto, Y. Uno, and T. Hermann, “Novel fusion welding technology of glass using ultrashort pulse lasers,” Proc. 27th International Congress on Applications of Lasers and Electro-Optics (ICALEO) 112–121 (2010). [CrossRef]

25.

C. B. Schaffer, A. O. Jamison, and E. Mazur, “Morphology of femtosecond laser-induced structural changes in bulk transparent materials,” Appl. Phys. Lett. 84(9), 1441–1443 (2004). [CrossRef]

26.

S. M. Eaton, H. Zhang, P. R. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Y. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate,” Opt. Express 13(12), 4708–4716 (2005). [CrossRef] [PubMed]

27.

S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51(16-18), 2533–2542 (2004). [CrossRef]

28.

M. Sakakura, M. Shimizu, Y. Shimotsuma, K. Miura, and K. Hirao, “Temperature distribution and modification mechanism inside glass with heat accumulation during 250 kHz irradiation of femtosecond laser pulses,” Appl. Phys. Lett. 93(23), 231112 (2008). [CrossRef]

OCIS Codes
(140.3390) Lasers and laser optics : Laser materials processing
(140.3440) Lasers and laser optics : Laser-induced breakdown
(140.7090) Lasers and laser optics : Ultrafast lasers
(160.2750) Materials : Glass and other amorphous materials
(190.4180) Nonlinear optics : Multiphoton processes

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: January 22, 2013
Revised Manuscript: April 6, 2013
Manuscript Accepted: April 10, 2013
Published: June 7, 2013

Citation
Isamu Miyamoto, Kristian Cvecek, and Michael Schmidt, "Crack-free conditions in welding of glass by ultrashort laser pulse," Opt. Express 21, 14291-14302 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-14291


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References

  1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett.21(21), 1729–1731 (1996). [CrossRef] [PubMed]
  2. M. Watanabe, H. Sun, S. Juodkazis, T. Takahashi, S. Matsuo, Y. Suzuki, J. Nishii, and H. Misawa, “Three-dimensional optical data storage in vitreous silica,” Jpn. J. Appl. Phys.37(Part 2, No. 12B), L1527–L1530 (1998). [CrossRef]
  3. C. Hnatovsky, R. S. Taylor, E. Simova, P. P. Rajeev, D. M. Rayner, V. R. Bhardwaj, and P. B. Corkum, “Fabrication of microchannels in glass using focused femtosecond laser radiation and selective chemical etching,” Appl. Phys., A Mater. Sci. Process.84(1-2), 47–61 (2006). [CrossRef]
  4. T. Tamaki, W. Watanabe, J. Nishii, and K. Itoh, “Welding of transparent materials using femtosecond laser pulses,” Jpn. J. Appl. Phys.44(22), L687–L689 (2005). [CrossRef]
  5. I. Miyamoto, A. Horn, J. Gottmann, D. Wortmann, and F. Yoshino, “Fusion welding of glass using femtosecond laser pulses with high-repetition rates,” J. Laser Micro Nanoeng.2(1), 57–63 (2007). [CrossRef]
  6. S. Richter, S. Döring, A. Tünnermann, and S. Nolte, “Bonding of glass with femtosecond laser pulses at high repetition rates,” Appl. Phys., A Mater. Sci. Process.103(2), 257–261 (2011). [CrossRef]
  7. I. Miyamoto, A. Horn, and J. Gottmann, “Local melting of glass material and its application to direct fusion welding by ps-laser pulses,” J. Laser MicroNanoeng.2(1), 7–14 (2007). [CrossRef]
  8. I. Miyamoto, K. Cvecek, Y. Okamoto, M. Schmidt, and H. Helvajian, “Characteristics of laser absorption and welding in FOTURAN glass by ultrashort laser pulses,” Opt. Express19(23), 22961–22973 (2011). [CrossRef] [PubMed]
  9. S. Wu, D. Wu, J. Xu, Y. Hanada, R. Suganuma, H. Wang, T. Makimura, K. Sugioka, and K. Midorikawa, “Characterization and mechanism of glass microwelding by double-pulse ultrafast laser irradiation,” Opt. Express20(27), 28893–28905 (2012). [CrossRef] [PubMed]
  10. M. Levesque, B. Labrnche, R. Forest, E. Savard, S. Deshaies, and A. Cournoyer, “Welding of glass pieces,” Physics Procedia5, 139–144 (2010). [CrossRef]
  11. I. Miyamoto, K. Cvecek, and M. Schmidt, “Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses,” Opt. Express19(11), 10714–10727 (2011). [CrossRef] [PubMed]
  12. K. Cvecek, I. Miyamoto, J. Strauss, M. Wolf, T. Frick, and M. Schmidt, “Sample preparation method for glass welding by ultrashort laser pulses yields higher seam strength,” Appl. Opt.50(13), 1941–1944 (2011). [CrossRef] [PubMed]
  13. W. Watanabe, S. Onda, T. Tamaki, K. Itoh, and J. Nishii, “Space-selective laser joining of dissimilar transparent materials using femtosecond laser pulses,” Appl. Phys. Lett.89, 021106 (2006).
  14. Y. Arata, H. Maruo, I. Miyamoto, and S. Tackuchi, “Dynamic behavior of laser welding and cutting,” Proc Symp. Electron and ion beam Science and technologies, 7th Int. Conf. 111–128 (1976)
  15. M. Watanabe and K. Satoh, Welding Mechanics and its Applications (Asakura, 1965), Chap. 8.
  16. T. Terasaki, “Welding distortion and residual stress,” J. Jpn. Welding Soc.78, 139–146 (2009).
  17. D. O. MacCallum, G. A. Knorovsky, and S. T. Reed, “CO2 laser welding fused silica,” Proc. 24th Int. Cong. on Application of Lasers and Electro-Optics (ICALEO) 687–695 (2005)
  18. C. Hnatovsky, R. S. Taylor, P. P. Rajeev, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Pulse duration dependence of femtosecond-laser-fabricated nanogratings in fused silica,” Appl. Phys. Lett.87(1), 014104 (2005). [CrossRef]
  19. Y. Liu, M. Shimizu, B. Zhu, Y. Dai, B. Qian, J. Qiu, Y. Shimotsuma, K. Miura, and K. Hirao, “Micromodification of element distribution in glass using femtosecond laser irradiation,” Opt. Lett.34(2), 136–138 (2009). [CrossRef] [PubMed]
  20. A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron.29(4), 1212–1217 (1993). [CrossRef]
  21. Y. R. Shen, The Principles of Nonlinear Optics (John Wiley and Sons, 1984), Chap. 27.
  22. J. Noack and A. Vogel, “Laser-induced plasma formation in water at nanosecond to femtosecond time scales: Calculation of thresholds, absorption coefficients and energy density,” IEEE J. Quantum Electron.35(8), 1156–1167 (1999). [CrossRef]
  23. C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells,” Opt. Express15(16), 10303–10317 (2007). [CrossRef] [PubMed]
  24. I. Miyamoto, A. Horn, J. Gottmann, D. Wortmann, I. Mingareev, F. Yoshino, M. Schmidt, Y. Okamoto, Y. Uno, and T. Hermann, “Novel fusion welding technology of glass using ultrashort pulse lasers,” Proc. 27th International Congress on Applications of Lasers and Electro-Optics (ICALEO) 112–121 (2010). [CrossRef]
  25. C. B. Schaffer, A. O. Jamison, and E. Mazur, “Morphology of femtosecond laser-induced structural changes in bulk transparent materials,” Appl. Phys. Lett.84(9), 1441–1443 (2004). [CrossRef]
  26. S. M. Eaton, H. Zhang, P. R. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Y. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate,” Opt. Express13(12), 4708–4716 (2005). [CrossRef] [PubMed]
  27. S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt.51(16-18), 2533–2542 (2004). [CrossRef]
  28. M. Sakakura, M. Shimizu, Y. Shimotsuma, K. Miura, and K. Hirao, “Temperature distribution and modification mechanism inside glass with heat accumulation during 250 kHz irradiation of femtosecond laser pulses,” Appl. Phys. Lett.93(23), 231112 (2008). [CrossRef]

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