Thermal conduction model is presented, by which nonlinear absorptivity of ultrashort laser pulses in internal modification of bulk glass is simulated. The simulated nonlinear absorptivity agrees with experimental values with maximum uncertainty of ±3% in a wide range of laser parameters at 10ps pulse duration in borosilicate glass. The nonlinear absorptivity increases with increasing energy and repetition rate of the laser pulse, reaching as high as 90%. The increase in the average absorbed laser power is accompanied by the extension of the laser-absorption region toward the laser source. Transient thermal conduction model for three-dimensional heat source shows that laser energy is absorbed by avalanche ionization seeded by thermally excited free-electrons at locations apart from the focus at pulse repetition rates higher than 100kHz.
©2011 Optical Society of America
Internal modification of transparent material by ultrashort laser pulses has been drawing much attention due to attractive applications such as formation of optical waveguide  and splitter , and fusion welding [3,4]. In the internal modification process, the laser energy is absorbed by nonlinear process with multiphoton ionization followed by avalanche ionization , and the absorbed laser energy in the free electrons is transferred to the lattice to provide the temperature field in bulk glass. Despite the internal modification process is produced by the nonlinear absorption, study on the nonlinear absorptivity is quite limited.
Schaffer et al. reported the dimensions of the internal modification at high pulse repetition rates can be much larger than those of single pulse due to heat accumulation effect . Since then the internal modification with heat accumulation has attracted interest due to its new possibility in internal modification process [7,8], and several authors reported thermal conduction models for internal modification in bulk glass at high pulse repetition rates [9,10]. In their models, however, the nonlinear absorptivity was simply assumed, and the distribution of the absorbed laser energy was assumed to be of spherical symmetry despite the modified structure extends asymmetrically along the optical axis [4,8–10]. The laser energy absorbed in the laser-induced plasma has been also simulated based on the rate equation for free electrons at single pulse irradiation in distilled water [11–13]. The model, however, cannot be applied to multi-pulse irradiation at high repetition rates.
Recently, an experimental procedure to measure the nonlinear absorptivity of ultrashort laser pulses in bulk glass was developed , assuming the reflection and the scattering by the laser-induced plasma are negligible based on the measurement by Nahen et al. , and the nonlinear absorptivity was shown to increase significantly with increasing pulse repetition rate at durations of 10ps  and 400fs . It was also reported that the region of the laser energy absorption extends toward the laser source at high pulse repetition rates [4,15], suggesting the absorbed laser energy distributes asymmetrically along the laser axis. The evaluation of the nonlinear absorptivity, however, still remains challenging, since the accuracy of the evaluated nonlinear absorptivity has not been shown, and no thermal conduction model to cope with asymmetrical distribution is available.
In the present paper, a thermal conduction model is presented to simulate the nonlinear absorptivity of ultrashort laser pulses in bulk glass at different energies and the repetition rates of the laser pulse. The accuracy of the simulation is evaluated by comparing with the experimental values. The model shows the increase in the laser absorption needs the extension of the laser absorption region along the laser axis.
2. Experimental evaluation of nonlinear absorptivity
2.1 Cross-section of modified region
Ultrashort pulse laser system from Lumera Laser (Super Rapid; λ = 1064nm, M2 = 1.1) with a pulse duration of 10ps was used for internal modification of borosilicate glass sample (Schott D263) with a thickness of 1.1mm. The laser beam was focused into the bulk glass by an objective lens for microscope with a numerical aperture (NA) 0.55 at a depth of 260µm from the sample surface. The glass sample was translated transversely to the laser beam at a constant speed of 20mm/s. Amongst many parameters influencing the nonlinear absorptivity, we focus here on the energy Q0 and the repetition rate f of the laser pulse to find out their effects on the dimensions of modified region and thereby the nonlinear absorptivity. Pulse repetition rate was varied between 50kHz and 1MHz.
Figure 1 shows the cross-sections obtained at different energies and repetition rates of the laser pulse. Vertical microcracks were observed only at limited conditions of low pulse energies at 50kHz and 100kHz. The modified region consists of a teardrop-shaped inner structure and an elliptical outer structure. The contour of the outer structure was very clearly observed, although it became less clear at average laser power below 300mW. The contour of the inner structure was, however, not as clear as that of the outer structure.
We assumed that the geometrical focus is located at the bottom edge of the inner structure at pulse energy near the threshold of the internal modification, which was approximately 0.4µJ independently of the pulse repetition rate. The bottom edge of the internal structure did not change its vertical position at pulse repetition rate f≥300kHz as shown by a horizontal line in Fig. 1. At pulse repetition rates f≤200kHz, however, the bottom edge of the inner structure extended below the geometrical focus at higher pulse energies; at f = 50kHz the inner structure penetrated through the contour of the outer structure.
2.2 Characteristic temperature of the modified structure
In the present study, the temperature distribution in bulk glass is simulated to determine the isothermal line by the thermal conduction model, and the nonlinear absorptivity is evaluated by fitting the simulated isothermal line to the contour of the experimental modification structures. Then it is essential to evaluate the characteristic temperature of the modified structure. Although the characteristic temperature Tm of the outer structure in borosilicate glass has been speculated in two papers, large difference is found in these values despite the both materials have similar thermal properties; Tm = 1,225°C in Schott AF45  and Tm = 560°C in Schott B270 .
We focused laser beam at the interface of overlapped glass plates of D263 moving transversely to the laser beam, and observed the cross-section of the sample by an optical microscope. As shown in Fig. 2 , the original interface of the glass plates disappears within the outer structure, indicating two glass plates are actually melted and coalesced within the outer region. This is in accordance with the observation of overlap welding in borosilicate glass plates of Corning 0211 by Bovatsek et al. . Assuming the glass coalesces at the forming temperature with the viscosity of 104 dPas , the characteristic temperature of the outer structure of D263 is evaluated to be Tm = 1,051°C. The forming temperature of AF45 1,225°C  agrees with the characteristic temperature speculated in Ref . On the other hand, the forming temperature of B270 is 1,033°C , and the characteristic temperature 560°C in B270 speculated from the relaxation time of the stress based on viscoelastic model  is obviously underestimated. In D263, we refer Tm = 1,051°C as “melting temperature” and the outer region as “molten region” hereafter.
2.3 Experimental evaluation of nonlinear absorptivity
It is reported that the reflection and the scattering of the laser energy by the laser-induced plasma in distilled water are negligible at pulse duration of 30ps and 8ns . Assuming that the reflection and the scattering of the laser energy by the laser-induced plasma in bulk glass are also negligible at 10ps pulse duration, the nonlinear absorptivity AEx is given by Eq. (1) is examined by comparing with the nonlinear absorptivity simulated in 3.2.
The nonlinear absorptivity in the bulk glass was determined by measuring the transmitted laser energy Qt using a setup shown in the inset of Fig. 3 . The threshold pulse energy for the nonlinear absorption was approximately 0.4µJ in accordance with the threshold energy assuming nonlinear absorption is associated with material modification that can be seen with the microscope. The measured nonlinear absorptivity shown by solid lines increases with increasing pulse energy reaching as high as approximately 90%. The rate of increase is larger at higher repetition rates, indicating the nonlinear absorptivity increases with increasing pulse repetition rate.
Figure 4(a) shows the cross-sectional area S of the molten region corresponding to the outer structure plotted as a function of pulse energy Q0 at different pulse repetition rates, assuming the modified region is elliptical. The cross-sectional area S increases with increasing pulse energy, and the slope becomes steeper as the pulse repetition rates increases. In Fig. 4(b), S is re-plotted against the average absorbed laser power Wab ( = AExfQ0). Interestingly, S falls within a narrow band along a single curve, suggesting S is given as a function of Wab. It will be shown that S can be simulated based on the thermal conduction model in 3.2.
3. Simulation of nonlinear absorptivity
3.1 Thermal conduction model for evaluating nonlinear absorptivity
When a line heat source with continuous heat delivery of w(z) appears at x = y = 0 in a region of 0<z<l in an infinite solid moves at a constant speed of v along x-axis, the temperature at (x,y,z) with an initial temperature T0 in a steady state is given by 
3.2. Nonlinear absorptivity
The function w(z) is expected to be not a simple form, since the density of the laser absorption increases as the focus is approached, and on the other hand the diameter of the laser absorption region increases with stepping away from the focus. We examined a variety of functions for w(z), and satisfactory results were obtained only when w(z) is a monotonically increasing function. Here as the first step we assumed w(z) is a simple function of z with least parameters given byEq. (4), ACal is written in a form17] were used. Since the thermal conductivity of D263 is not available, we adopted K = 0.0096W/cmK (room temperature) of Corning 0211 , which is equivalent to Schott D263.
Figure 5 shows the examples of the isothermal line of Tm = 1,051°C simulated at 500kHz for different m using selected values of a and b, assuming T0 = 25°C. The isothermal line fits well to the contour of the experimental molten region, and l = 50µm and ACal = 0.819 are obtained with minor effect of m. The simulated nonlinear absorptivity 0.819 agrees well with the experimental value of AEx = 0.81 (see in Fig. 3(a)). On the other hand, the isothermal line of 3,600°C, which provides the closest isotherm to the inner structure, is sensitively affected by m, and best fitting was found with m = 1, showing that the inner and the outer structures are produced by the common thermal origin. The value m = 1 is used hereafter for evaluating ACal and l.
At 50kHz, on the other hand, larger discrepancy is observed between the simulated isothermal line of 3,600°C and the experimental inner structure as is shown in Fig. 6 ; the experimental inner structure extends further below z = 0 presumably due to self focusing and defocusing by electron cloud . Nevertheless satisfactory agreement is found between the simulated isothermal line of Tm = 1,051°C and the experimental melt contour using ‘equivalent’ distribution of w(z) given by Eq. (5) and resultant value ACal = 85.6% is in good agreement with the experimental value of AEx = 85%. The characteristic temperature evaluated in the inner structure ranged in a rather wide region of 3,600±300°C with accompanying deformed shape, suggesting only the thermal effects cannot explain the shape of the inner structure. Thus the isothermal line of Tm for the outer structure was adopted for evaluating the nonlinear absorptivity hereafter, since the outer structure is more clearly observed and provides more reproducible nonlinear absorptivity.
The nonlinear absorptivity ACal simulated at different values of Q0 and f is plotted with closed circles in Fig. 3, showing excellent agreement with AEx. Excellent agreement is obtained even at 50kHz and 100kHz, in spite that large discrepancy is found between the isothermal line of 3,600°C and the inner structure. For analyzing the accuracy of the evaluated values, the ratio of ACal/AEx is plotted in Fig. 7 . It is rather surprising that despite the thermal constants at room temperature were used for simulation, ACal/AEx ranges in a very narrow region of 1.02±0.03 excepting for limited condition of Wab<300mW. The prime reason of the larger data scattering at smaller Wab is caused by the fact that the contour of the outer structure becomes increasingly unclear as the absorbed laser power decreases. Secondary reason is the outer structure deviates significantly from ellipse at smaller Wab. Detailed calculation indicated that the bias of 0.02 can be removed to result in ACal/AEx = 1±0.03 by adopting slightly smaller thermal conductivity of K = 0.0093W/cmK. The simulated cross-sectional area S is also plotted in Fig. 4(b) by a solid line, and is in excellent agreement with the experimental values.
It should be emphasized that AEx and ACal are derived independently based on different physical basis, and that the high accuracy of ACal/AEx = 1±0.03 is attained under the conditions with a wide variety of Q0 and f. This suggests that both the experimental measurement and the simulation model have the uncertainty less than at least ±0.03, since the error is accumulated in the calculation of the ratio. This in turn indicates the assumption that the reflection and the scattering of the laser energy by the laser-induced plasma in bulk glass at 10ps pulse duration are negligible is justified. This is considered to be because even though the free electron density exceeds the critical value in bulk glass to become highly reflective, the absorbing plasma with lower electron density existing in the surrounding area can absorb the reflected laser energy .
For simulating more precise temperature distribution, one can use complex numerical analysis using temperature dependent thermal properties if they are known. Temperature dependent thermal properties, however, are actually not available, and we believe the simple analytical equation using the effective value of the thermal constants can provide the temperature distribution, which is precise enough to evaluate the nonlinear absorptivity.
3.3 Length of laser absorption region
Assuming that self-focusing and defocusing by free electrons are negligible, and that the radius of the laser beam propagating in the bulk glass is given by Fig. 8 . In this calculation, we determined the radius of the beam waist ω0 using the experimental value of the damage threshold of Qth = 0.4µJ (Fig. 3) by ω0 = , where Ith is the laser intensity of the damage threshold, since the spherical aberration is not negligible in our experimental condition using the high NA lens for focusing laser beam at a depth of 260µm. Assuming that the difference of the damage threshold between fused silica and D263 is small , ω0 is evaluated to be 2.26µm using the damage threshold of fused silica Ith = 5x1011W/cm2 for the super-polished surface  and the bulk  at 10ps.
The simulated length of the absorbed region l is plotted vs. Q0 at different pulse repetition rates f with closed circles in Fig. 8. It is seen that l increases with increasing pulse energy Q0 and the rate of increase is obviously larger at higher pulse repetition rates. The experimental length lEx measured from the geometrical focus to the top edge of the inner structure is also plotted in the figure by open circles, exhibiting excellent agreement with l. It is interesting to note that at f = 50kHz, the laser intensity at the top edge of the inner structure agrees approximately with the breakdown threshold for multiphoton ionization 5x1011W/cm2 [5,25]. This suggests that the breakdown occurs due to the contribution of the free electrons provided by multiphoton ionization in the inner structure of 50kHz and no multiphoton ionization occurs outside of this region, assuming the effects of the heat accumulation can be neglected at 50kHz. The validity of the assumption will be verified latter.
It is seen that the laser intensity at the top edge of the inner structure I(l) decreases with increasing pulse repetition rate. At Q0≈4µJ, for instance, I(l) decreases with increasing pulse repetition rate, and reaches down to ≈1010 W/cm2 at 1MHz. This value is approximately fifty times as low as that of 50kHz, indicating no multiphoton ionization occurs there. This means that the laser energy is absorbed by avalanche ionization in the region of I(l)<I<Ith but without seed electrons by multiphoton ionization at pulse repetition rates f≥100kHz. Then a question arises: what is the source of the seed electrons for the avalanche ionization? That will be discussed in the next section.
The value l in Fig. 8 is re-plotted against the experimental average absorbed power Wab in Fig. 9 . Interestingly again, the data falls on a very narrow band along a single line like the case of the cross-sectional area seen in Fig. 4(b), indicating Wab is closely related with l. Some scattering of the evaluated values is found at lower values of Wab at 50kHz and 100kHz, because the contour of the outer structure is not clear enough to evaluate the value l exactly. The averaged value of the absorbed laser power per unit length Wab/l is also plotted in Fig. 9, showing monotonic increase with increasing Wab. This indicates that increase in the absorbed laser energy needs the increase in the length of the absorbed region, when Q 0 or f increases.
3.4 Transient thermal conduction model for three-dimensional heat source
Most probable source of the free electrons other than multiphoton ionization for seeding avalanche ionization is the thermally excited free electrons in conduction band, since higher temperature is attained due to heat accumulation at higher pulse repetition rates. Although the influence of impurities on the seed electrons has been reported in many papers [5,12,14,26], little has been discussed on the density of the thermally excited free electrons in bulk glass without impurity at high pulse repetition rates. For evaluating the density of the thermally excited free electrons, it is essential to simulate the temperature distribution in the glass sample at the moment of the laser pulse impingement. For this purpose, we have developed the thermal conduction model with taking into consideration the spatial and temporal distribution of the absorbed laser energy. In the previous papers [4,15], the transient temperature distribution in the laser-irradiated glass sample was simulated by the thermal conduction model assuming that a rectangular solid heat source with uniform intensity is successively deposited in an infinite solid moving at a constant speed. In this model, the thermal conduction equation was derived by spatial and temporal integration of the solution of instantaneous point heat source . In the present paper, the thermal conduction equation is derived in the similar way for simulating more realistic energy distribution.
Assuming the instantaneous heat q(x’,y’,z’) appears at a repetition rate of f in an infinite solid moving at a constant speed of v along x-axis, the temperature at (x,y,z) at time t after the incidence of N-th pulse is given byEq. (7), for simplicity. Assuming w(z) given by Eq. (5) is redistributed to provide instantaneous heat source with a repetition rate of f, q(r,z) is written in a form ofEq. (12) into Eq. (11), the transient temperature rise T(x,y,z;t) at (x,y,z) at time t after the generation of N-th pulse is derived in a form ofFigure 10 shows the temperature change simulated on the laser beam axis at z = 2.5µm and z = l/2 at f = 50kHz and f = 300kHz. The simulation was made at nearly the same pulse energy, Q0 = 3.72µJ for 50kHz and Q0 = 3.9µJ for 300kHz, which correspond to the experimental conditions. The horizontal axis indicates the number of laser pulse in logarithmic scale. The temperature change within each pulse is plotted until 11th pulse, and thereafter only the base temperature, TB(0,0,z;tf) with tf = 1/f, just before the impingement of the laser pulse is plotted. Large amplitudes of the temperature change are observed indicating the instantaneous temperature rise due to each pulse is cooled down to the base temperature just before the impingement of the next laser pulse. The increment of the base temperature per pulse, however, is also observed, which is caused by the heat accumulation. The increment of the base temperature per pulse increases with increasing pulse repetition rate due to shorter cooling time between pulses. This results in large difference in the steady temperature of TBS(0,0,z;tf) between two pulse repetition rates. The number of laser pulse NS for TB(0,0,z;tf) to reach steady temperature TBS(0,0,z;tf) increases in proportion to the pulse repetition rate; NS≈500 pulses at 50kHz and NS≈3,000 pulses at 300kHz.
Heat accumulation is also affected by the radius of the laser beam ω(z). At 300kHz, for instance, the steady temperature of TBS(0,0,z;tf) = 3,180°C is reached at z = 2.5µm as the result of the instantaneous temperature rise per pulse of ΔT = 1,350°C. Higher value of TBS(0,0,z;tf) ≈3,800°C is reached despite of much smaller value of ΔT = 75°C at z = l/2. This is because larger spot size at z = l/2 provides smaller temperature gradient, resulting in slower cooling rate. Similar situation is also observed at 50kHz; higher TBS(0,0,z;tf)≈1,200 °C is reached with smaller value of ΔT≈1,370°C at z = l/2. The fact that steady temperatures of TBS(0,0,z;tf) at 50kHz for both z = 2.5µm and z = l/2 are much lower than those at 300kHz is caused not only by longer cooling time but by the fact that the laser beam size in laser absorption region at 50kHz is much smaller than that of 300kHz.
The steady values TBS(0,0,z;tf) at z = 2.5µm and z = l/2 were simulated at nearly constant pulse energy of Q0 = 3.9±0.18µJ corresponding to the experimental conditions along dotted line in Fig. 8 (only 1,000kHz: Q0 = 3.1µJ). Figure 11(a) shows the steady values of TBS(0,0,z;tf) at z = 2.5µm and z = l/2 plotted vs. pulse repetition rate. Both curves show basically similar behavior except that the temperature increases faster at z = l/2 with increasing pulse repetition rate. It is noted that the temperature at 50kHz is as low as 800~1,000°C, indicating that heat accumulation effect is very small and thus the laser pulses impinges always in the bulk glass with low temperatures. As the pulse repetition rate increases, TBS(0,0,z;tf) increases rapidly, reaching up to approximately 4,000°C.
The free electron density excited to the conduction band is estimated, assuming that molecules are thermalized and have a Maxwell-Boltzmann distribution  for temperature TBS(0,0,z;tf). In this calculation, the band gap energy of D263, Eg = 3.7eV, determined by a Tauc plot of optical transmission spectroscopy data  was used. Figure 11(b) shows the free electron density calculated at the temperatures of TBS(0,0,z;tf) at z = 2.5µm and z = l/2 plotted vs. pulse repetition rate f. At 50kHz, the thermally excited free electron density at z = l/2 is ≈5x106/cm3, which provides no thermally excited free electrons in the volume of the inner structure. This means that the multiphoton ionization is always needed for seeding avalanche ionization at 50kHz. This result is supported by the fact that the laser intensity at the upper edge of the inner structure at 50kHz agrees with the threshold intensity Ith for multiphoton ionization (Fig. 8). As the pulse repetition rate f increases, the free electron density neB(0,0,z;tf) at z = l/2 increases, and reaches nearly constant value of ≈1018/cm3, which is high enough to seed the avalanche ionization .
The free electron density near the focus (z = 2.5µm) also increases with increasing pulse repetition rate reaching up to ≈1018/cm3, nearly equal to that of l/2 at pulse repetition rates f≥500kHz, indicating that the laser energy can be absorbed by avalanche ionization seeded not only by multiphoton ionization but by thermally excited free electrons. This result also suggests that there exists a possibility that the avalanche ionization can occur even without the free electrons provided by multiphoton ionization, although no experimental evidence has been obtained at the moment. We have, however, some feeling that multiphoton ionization is still needed for stabilizing the absorption process. Further study is needed to clarify the absorption process at high pulse repetition rates from experimental and theoretical points of view.
Finally we should clarify what the term “nonlinear absorption” implies, since our results indicate that avalanche ionization at high pulse repetition rates governs the breakdown dynamics much more strongly than the case of single pulse, and at locations apart from the focus position the seed electrons for avalanche ionization are totally provided by thermal excitation to the conduction band. We also showed even a possibility that the laser-induced plasma can be sustained without multiphoton ionization. Nevertheless, however, the multiphoton ionization is still needed at least to provide the initial free electrons for avalanche ionization even at high pulse repetition rates. This is qualitatively similar to the case of single laser pulse with durations down to 100fs where breakdown process is dominated by the multiphoton ionization until approximately the maximum of the laser pulse, thereafter the avalanche ionization starts to govern the breakdown dynamics to produce more free electrons than multiphoton ionization . Thus we refer to the absorption that needs multiphoton ionization for providing initial seed electrons for avalanche ionization as nonlinear absorption. However, if it is proven that avalanche ionization can be sustained even without multiphoton ionization, we will have to reconsider the term of nonlinear absorption.
Thermal conduction model with moving line heat source with continuous heat delicery has been developed to simulate the dimensions of the molten region in internal modification of bulk glass by ultrashort laser pulses, and the nonlinear absorptivity and the length of the laser absorption region are evaluated by fitting the isothermal line of melting temperature to the contour of the molten region. The nonlinear absorptivity has been also experimentally determined by measuring the transmitted pulse energy through the glass sample. Excellent agreement is found between experimental and simulated nonlinear absorptivity with maximum uncertainty of ±3% when pulse energy and pulse repetition rate are widely changed at 10ps pulse duration in borosilicate glass of D263. Cross-sectional area of the molten region and the length of the laser-absorption region are closely related with the average absorbed laser power Wab. The nonlinear absorptivity increases with increasing energy and repetition rate of the laser pulse, and the increase in Wab is accompanied by the extension of the laser-absorption region toward the laser source.
The thermal conduction model for simulating transient temperature distribution has been developed, and the temperatures simulated at the moment just before the laser pulse impingement indicate that thermally excited free electrons to the conduction band due to heat accumulation cause the increase in the nonlinear absorptivity and the extension of the laser absorption region toward the laser source at high repetition rates f≥100kHz.
The authors wish to thank Dr. J. Gottmann and Dipl.-Phys. D. Esser, Lehrstuhl für Lasertechnik LLT, RWTH Aachen University, for their measurement of band gap energy of the glass sample. This work was partially supported by Erlangen Graduate School in Advanced Optical Technologies (SAOT).
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