## Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses |

Optics Express, Vol. 19, Issue 11, pp. 10714-10727 (2011)

http://dx.doi.org/10.1364/OE.19.010714

Acrobat PDF (1363 KB)

### Abstract

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 10*ps* 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 100*kHz*.

© 2011 OSA

## 1. Introduction

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. D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, “Infrared photosensitivity in silica glasses exposed to femtosecond laser pulses,” Opt. Lett. **24**(18), 1311–1313 (1999). [CrossRef]

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

4. 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 Micro/Nanoengineering **2**, 7–14 (2007). [CrossRef]

5. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter **53**(4), 1749–1761 (1996). [CrossRef] [PubMed]

4. 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 Micro/Nanoengineering **2**, 7–14 (2007). [CrossRef]

*et al.*[14

14. K. Nahen and A. Vogel, “Plasma formation in water by picosecond and nanosecond Nd:YAG laser pulses – part II: transmission, scattering, and reflection,” J. Sel. Top. Quant. Electron. **2**, 861–871 (1996). [CrossRef]

*ps*[4

4. 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 Micro/Nanoengineering **2**, 7–14 (2007). [CrossRef]

*fs*[15

15. 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/Nanoengineering **2**, 57**-**63 (2007). [CrossRef]

**2**, 7–14 (2007). [CrossRef]

15. 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/Nanoengineering **2**, 57**-**63 (2007). [CrossRef]

## 2. Experimental evaluation of nonlinear absorptivity

### 2.1 Cross-section of modified region

*nm*,

*M*= 1.1) with a pulse duration of 10

^{2}*ps*was used for internal modification of borosilicate glass sample (Schott D263) with a thickness of 1.1

*mm*. 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 260

*µm*from the sample surface. The glass sample was translated transversely to the laser beam at a constant speed of 20

*mm/s*. Amongst many parameters influencing the nonlinear absorptivity, we focus here on the energy

*Q*and the repetition rate

_{0}*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 50

*kHz*and 1

*MHz*.

*kHz*and 100

*kHz*. 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 300

*mW*. The contour of the inner structure was, however, not as clear as that of the outer structure.

*µJ*independently of the pulse repetition rate. The bottom edge of the internal structure did not change its vertical position at pulse repetition rate

*f*≥300

*kHz*as shown by a horizontal line in Fig. 1. At pulse repetition rates

*f*≤200

*kHz*, however, the bottom edge of the inner structure extended below the geometrical focus at higher pulse energies; at

*f*= 50

*kHz*the inner structure penetrated through the contour of the outer structure.

### 2.2 Characteristic temperature of the modified structure

*T*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;

_{m}*T*= 1,225

_{m}*°C*in Schott AF45 [9

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

*T*= 560

_{m}*°C*in Schott B270 [10

10. 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**, 231112 (2008). [CrossRef]

*et al.*[16]. Assuming the glass coalesces at the forming temperature with the viscosity of 10

^{4}

*dPas*[17], the characteristic temperature of the outer structure of D263 is evaluated to be

*T*= 1,051

_{m}*°C*. The forming temperature of AF45 1,225

*°C*[18] agrees with the characteristic temperature speculated in Ref [9

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

*°C*[19], and the characteristic temperature 560

*°C*in B270 speculated from the relaxation time of the stress based on viscoelastic model [10

10. 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**, 231112 (2008). [CrossRef]

*T*= 1,051

_{m}*°C*as “melting temperature” and the outer region as “molten region” hereafter.

### 2.3 Experimental evaluation of nonlinear absorptivity

*ps*and 8

*ns*[14

14. K. Nahen and A. Vogel, “Plasma formation in water by picosecond and nanosecond Nd:YAG laser pulses – part II: transmission, scattering, and reflection,” J. Sel. Top. Quant. Electron. **2**, 861–871 (1996). [CrossRef]

*ps*pulse duration, the nonlinear absorptivity

*A*is given by [4

_{Ex}**2**, 7–14 (2007). [CrossRef]

*Q*is incident laser pulse energy,

_{0}*Q*transmitted laser pulse energy through the glass sample and

_{t}*R*Fresnel reflectivity. The validity of Eq. (1) is examined by comparing with the nonlinear absorptivity simulated in 3.2.

*Q*using a setup shown in the inset of Fig. 3 . The threshold pulse energy for the nonlinear absorption was approximately 0.4

_{t}*µJ*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.

*S*of the molten region corresponding to the outer structure plotted as a function of pulse energy

*Q*at different pulse repetition rates, assuming the modified region is elliptical. The cross-sectional area

_{0}*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

*W*( =

_{ab}*A*). Interestingly,

_{Ex}fQ_{0}*S*falls within a narrow band along a single curve, suggesting

*S*is given as a function of

*W*. It will be shown that

_{ab}*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

*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

*T*in a steady state is given by [20]where

_{0}*s*,

^{2}= x^{2}+ y^{2}+ (z-z’)^{2}*K*is thermal conductivity and

*α*is thermal diffusivity given by

*K/cρ*(

*c*= specific heat and

*ρ*= density). It is assumed the thermal properties of the material are independent of temperature, for simplicity. Despite the actual laser beam has finite spot size with pulsed energy delivery, the simple line heat source model with constant heat delivery was used to calculate the temperature at the contour of the molten zone, because the temperature rise at locations apart from the heat source is spatially and temporally averaged. The distribution

*w(z)*is determined by fitting the isotherm of the maximum cycle temperature in (

*y,z*) attained at

*x*where

*dT/dx*= 0 given byto the contour of the experimental modification structure. Then the nonlinear absorptivity

*A*is given byand

_{Cal}*l*is also determined. The advantage of our procedure is that the absolute laser energy absorbed in the bulk glass can be simulated even in the irradiation of multiple laser pulses at high repetition rates. In the following sections, the nonlinear absorptivity

*A*and the length of the absorbed region

_{Cal}*l*are simulated.

### 3.2. Nonlinear absorptivity

*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 bywhere

*a, b*and

*m*are positive constants. From Eq. (4),

*A*is written in a formIn the simulation, the thermal constants,

_{Cal}*ρ*= 2.51

*g/cm*and

^{3}*c*= 0.82

*J/gK*(mean value of 20~100

*°C*) [17] were used. Since the thermal conductivity of D263 is not available, we adopted

*K*= 0.0096

*W/cmK*(room temperature) of Corning 0211 [21], which is equivalent to Schott D263.

*T*= 1,051

_{m}*°C*simulated at 500

*kHz*for different

*m*using selected values of

*a*and

*b*, assuming

*T*= 25

_{0}*°C*. The isothermal line fits well to the contour of the experimental molten region, and

*l*= 50

*µm*and

*A*= 0.819 are obtained with minor effect of

_{Cal}*m*. The simulated nonlinear absorptivity 0.819 agrees well with the experimental value of

*A*= 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

_{Ex}*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

*A*and

_{Cal}*l*.

*kHz*, 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 [22

22. S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. **87**(21), 213902 (2001). [CrossRef] [PubMed]

*T*= 1,051°C and the experimental melt contour using ‘equivalent’ distribution of

_{m}*w(z)*given by Eq. (5) and resultant value

*A*= 85.6% is in good agreement with the experimental value of

_{Cal}*A*= 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

_{Ex}*T*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.

_{m}*A*simulated at different values of

_{Cal}*Q*and

_{0}*f*is plotted with closed circles in Fig. 3, showing excellent agreement with

*A*. Excellent agreement is obtained even at 50

_{Ex}*kHz*and 100

*kHz*, 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

*A*is plotted in Fig. 7 . It is rather surprising that despite the thermal constants at room temperature were used for simulation,

_{Cal}/A_{Ex}*A*ranges in a very narrow region of 1.02±0.03 excepting for limited condition of

_{Cal}/A_{Ex}*W*<300

_{ab}*mW*. The prime reason of the larger data scattering at smaller

*W*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

_{ab}*W*. Detailed calculation indicated that the bias of 0.02 can be removed to result in

_{ab}*A*= 1±0.03 by adopting slightly smaller thermal conductivity of

_{Cal}/A_{Ex}*K*= 0.0093

*W/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.

*A*and

_{Ex}*A*are derived independently based on different physical basis, and that the high accuracy of

_{Cal}*A*= 1±0.03 is attained under the conditions with a wide variety of

_{Cal}/A_{Ex}*Q*and

_{0}*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 10

*ps*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 [14

14. K. Nahen and A. Vogel, “Plasma formation in water by picosecond and nanosecond Nd:YAG laser pulses – part II: transmission, scattering, and reflection,” J. Sel. Top. Quant. Electron. **2**, 861–871 (1996). [CrossRef]

### 3.3 Length of laser absorption region

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

*z*is distance from the focus,

*λ*wavelength of the laser beam,

*M*beam quality factor, NA numerical aperture of focusing optics and

^{2}*n*refractive index of the bulk glass, the laser intensity

_{g}*I(z)*at

*z*is given byThen

*z*corresponding to the intensity

*I*is written in a form ofwhere

*τ*is the pulse duration of the laser beam. Relationship between

*z*and

*Q*for different values of

_{0}*I*is plotted for

*τ*= 10

*ps*with solid lines in Fig. 8 . In this calculation, we determined the radius of the beam waist

*ω*using the experimental value of the damage threshold of

_{0}*Q*= 0.4

_{th}*µJ*(Fig. 3) by

*ω*=

_{0}*I*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 260

_{th}*µm*. Assuming that the difference of the damage threshold between fused silica and D263 is small [24],

*ω*is evaluated to be 2.26

_{0}*µm*using the damage threshold of fused silica

*I*= 5x10

_{th}*for the super-polished surface [5*

^{11}W/cm^{2}5. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter **53**(4), 1749–1761 (1996). [CrossRef] [PubMed]

25. D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. **64**, 3071–3073 (1994). [CrossRef]

*ps.*

*l*is plotted

*vs. Q*at different pulse repetition rates

_{0}*f*with closed circles in Fig. 8. It is seen that

*l*increases with increasing pulse energy

*Q*and the rate of increase is obviously larger at higher pulse repetition rates. The experimental length

_{0}*l*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

_{Ex}*l*. It is interesting to note that at

*f*= 50

*kHz*, the laser intensity at the top edge of the inner structure agrees approximately with the breakdown threshold for multiphoton ionization 5x10

*[5*

^{11}W/cm^{2}5. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter **53**(4), 1749–1761 (1996). [CrossRef] [PubMed]

25. D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. **64**, 3071–3073 (1994). [CrossRef]

*kHz*and no multiphoton ionization occurs outside of this region, assuming the effects of the heat accumulation can be neglected at 50

*kHz*. The validity of the assumption will be verified latter.

*I(l)*decreases with increasing pulse repetition rate. At

*Q*≈4

_{0}*µJ*, for instance,

*I(l)*decreases with increasing pulse repetition rate, and reaches down to ≈10

^{10}

*W/cm*at 1

^{2}*MHz*. This value is approximately fifty times as low as that of 50

*kHz*, indicating no multiphoton ionization occurs there. This means that the laser energy is absorbed by avalanche ionization in the region of

*I(l)*<

*I<I*but without seed electrons by multiphoton ionization at pulse repetition rates

_{th}*f*≥100

*kHz*. Then a question arises: what is the source of the seed electrons for the avalanche ionization? That will be discussed in the next section.

### 3.4 Transient thermal conduction model for three-dimensional heat source

**53**(4), 1749–1761 (1996). [CrossRef] [PubMed]

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

**2**, 861–871 (1996). [CrossRef]

26. P. K. Kennedy, “A first-order model for computation of laser-induced breakdown thresholds in ocular and aqueous media: part I – theory,” IEEE J. Quantum Electron. **31**, 2241–2249 (1995). [CrossRef]

**2**, 7–14 (2007). [CrossRef]

15. 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/Nanoengineering **2**, 57**-**63 (2007). [CrossRef]

*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 by

*ω(z)*given by Eq. (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 ofwhere

*r*. Substituting Eq. (12) into Eq. (11), the transient temperature rise

^{2}= x^{2}+ y^{2}*T(x,y,z;t)*at

*(x,y,z)*at time

*t*after the generation of

*N-*th pulse is derived in a form of

*z*= 2.5

*µm*and

*z = l/2*at

*f*= 50

*kHz*and

*f*= 300

*kHz*. The simulation was made at nearly the same pulse energy,

*Q*= 3.72

_{0}*µJ*for 50

*kHz*and

*Q*= 3.9

_{0}*µJ*for 300

*kHz*, 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 11

*th*pulse, and thereafter only the base temperature,

*T*(

_{B}*0,0,z;t*) with

_{f}*t*, 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

_{f}= 1/f*T*(

_{BS}*0,0,z;t*) between two pulse repetition rates. The number of laser pulse

_{f}*N*for

_{S}*T*to reach steady temperature

_{B}(0,0,z;t_{f})*T*increases in proportion to the pulse repetition rate;

_{BS}(0,0,z;t_{f})*N*≈500 pulses at 50

_{S}*kHz*and

*N*≈3,000 pulses at 300

_{S}*kHz*.

*ω(z)*. At 300

*kHz*, for instance, the steady temperature of

*T*(

_{BS}*0,0,z;t*)

_{f}*=*3,180°C is reached at

*z*= 2.5

*µm*as the result of the instantaneous temperature rise per pulse of

*ΔT*= 1,350°C. Higher value of

*T*(

_{BS}*0,0,z;t*)

_{f}*≈*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 50

*kHz*; higher

*T*(

_{BS}*0,0,z;t*)

_{f}*≈*1,200 °C is reached with smaller value of

*ΔT*≈1,370°C at

*z = l*/2. The fact that steady temperatures of

*T*(

_{BS}*0,0,z;t*) at 50

_{f}*kHz*for both

*z*= 2.5

*µm*and

*z = l/*2 are much lower than those at 300

*kHz*is caused not only by longer cooling time but by the fact that the laser beam size in laser absorption region at 50

*kHz*is much smaller than that of 300

*kHz*.

*T*(

_{BS}*0,0,z;t*) at

_{f}*z*= 2.5

*µm*and

*z*=

*l*/2 were simulated at nearly constant pulse energy of

*Q*= 3.9±0.18

_{0}*µJ*corresponding to the experimental conditions along dotted line in Fig. 8 (only 1,000

*kHz*:

*Q*= 3.1

_{0}*µJ*). Figure 11(a) shows the steady values of

*T*(

_{BS}*0,0,z;t*) at

_{f}*z*= 2.5

*µm*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 50

*kHz*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,

*T*(

_{BS}*0,0,z;t*) increases rapidly, reaching up to approximately 4,000

_{f}*°C*.

26. P. K. Kennedy, “A first-order model for computation of laser-induced breakdown thresholds in ocular and aqueous media: part I – theory,” IEEE J. Quantum Electron. **31**, 2241–2249 (1995). [CrossRef]

*T*(

_{BS}*0,0,z;t*). In this calculation, the band gap energy of D263, Eg = 3.7

_{f}*eV*, determined by a Tauc plot of optical transmission spectroscopy data [27] was used. Figure 11(b) shows the free electron density calculated at the temperatures of

*T*(

_{BS}*0,0,z;t*) at

_{f}*z*= 2.5

*µm*and

*z = l*/2 plotted vs. pulse repetition rate

*f*. At 50

*kHz*, the thermally excited free electron density at

*z = l*/2 is ≈5x10

^{6}/

*cm*, 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 50

^{3}*kHz*. This result is supported by the fact that the laser intensity at the upper edge of the inner structure at 50

*kHz*agrees with the threshold intensity

*I*for multiphoton ionization (Fig. 8). As the pulse repetition rate

_{th}*f*increases, the free electron density

*n*at

_{eB}(0,0,z;t_{f})*z = l*/2 increases, and reaches nearly constant value of ≈10

^{18}/

*cm*, which is high enough to seed the avalanche ionization [12

^{3}12. J. Noack and A. Vogel, “Laser-induced plasma formation in water at nanosecond to femtosecond time scales: calculation of thresholds, absorption coefficient and energy density,” IEEE J. Quantum Electron. **35**, 1156–1167 (1999). [CrossRef]

*z =*2.5

*µm*) also increases with increasing pulse repetition rate reaching up to ≈10

^{18}/

*cm*, nearly equal to that of

^{3}*l*/2 at pulse repetition rates

*f*≥500

*kHz*, 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.

*fs*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 [12

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

## 4. Summary

*kHz*.

## Acknowledgments

## References and links

1. | K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. |

2. | D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, “Infrared photosensitivity in silica glasses exposed to femtosecond laser pulses,” Opt. Lett. |

3. | T. Tamaki, W. Watanabe, J. Nishii, and K. Itoh, “Welding of transparent materials using femtosecond laser pulses,” Jpn. J. Appl. Phys. |

4. | 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 Micro/Nanoengineering |

5. | B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter |

6. | C. B. Schaffer, J. F. Garcia, and E. Mazur, “Bulk heating of transparent materials using high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. |

7. | R. Osellame, N. Chiodo, V. Maselli, A. Yin, M. Zavelani-Rossi, G. Cerullo, P. Laporta, L. Aiello, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Optical properties of waveguides written by a 26 MHz stretched cavity Ti:sapphire femtosecond oscillator,” Opt. Express |

8. | S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. |

9. | S. M. Eaton, H. Zhang, P. R. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate,” Opt. Express |

10. | M. Sakakura, M. Shimizu, Y. Shimotsuma, K. Miura, and K. Hirao, “Temperature distribution and modification mechanism inside glass with heat accumulation during 250 |

11. | Y. R. Shen, |

12. | J. Noack and A. Vogel, “Laser-induced plasma formation in water at nanosecond to femtosecond time scales: calculation of thresholds, absorption coefficient and energy density,” IEEE J. Quantum Electron. |

13. | 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 |

14. | K. Nahen and A. Vogel, “Plasma formation in water by picosecond and nanosecond Nd:YAG laser pulses – part II: transmission, scattering, and reflection,” J. Sel. Top. Quant. Electron. |

15. | 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/Nanoengineering |

16. | J. Bovatsek, A. Araia, and C. B. Schaffer, “Three-dimensional micromachining inside transparent materials using femtosecond laser pulses: new applications,” Proceedings of CLEO/Europe - EQEC2005 (2005). |

17. | http://www.schott.com/special_applications/english/download/d263te.pdf. |

18. | http://www.schott.com/special_applications/english/download/af45e.pdf. |

19. | http://psec.uchicago.edu/glass/Schott%20B270%20Properties%20%20Knight%20Optical.pdf. |

20. | H. S. Carslaw and J. C. Jaeger, |

21. | |

22. | S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. |

23. | A. E. Siegman and S. W. Townsent, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. |

24. | I. Miyamoto and T. Hermann, “Characteristics of internal melting of glass for fusion welding using ps laser pulses with average power up to 8W,” Proc. 8th Int. Symp. On Laser Precision Microfabrication- LPM2007 (2007). |

25. | D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. |

26. | P. K. Kennedy, “A first-order model for computation of laser-induced breakdown thresholds in ocular and aqueous media: part I – theory,” IEEE J. Quantum Electron. |

27. | K. Morigaki, |

**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 Systems

**History**

Original Manuscript: February 1, 2011

Revised Manuscript: April 20, 2011

Manuscript Accepted: May 4, 2011

Published: May 17, 2011

**Citation**

Isamu Miyamoto, Kristian Cvecek, and Michael Schmidt, "Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses," Opt. Express **19**, 10714-10727 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10714

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### References

- 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]
- D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, “Infrared photosensitivity in silica glasses exposed to femtosecond laser pulses,” Opt. Lett. 24(18), 1311–1313 (1999). [CrossRef]
- T. Tamaki, W. Watanabe, J. Nishii, and K. Itoh, “Welding of transparent materials using femtosecond laser pulses,” Jpn. J. Appl. Phys. 44, L687–L689 (2005). [CrossRef]
- 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 Micro/Nanoengineering 2, 7–14 (2007). [CrossRef]
- B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B Condens. Matter 53(4), 1749–1761 (1996). [CrossRef] [PubMed]
- C. B. Schaffer, J. F. Garcia, and E. Mazur, “Bulk heating of transparent materials using high-repetition-rate femtosecond laser,” Appl. Phys., A Mater. Sci. Process. 76, 351–354 (2003). [CrossRef]
- R. Osellame, N. Chiodo, V. Maselli, A. Yin, M. Zavelani-Rossi, G. Cerullo, P. Laporta, L. Aiello, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Optical properties of waveguides written by a 26 MHz stretched cavity Ti:sapphire femtosecond oscillator,” Opt. Express 13(2), 612–620 (2005). [CrossRef] [PubMed]
- S. Nolte, M. Will, J. Burghoff, and A. Tünnermann, “Ultrafast laser processing: new options for three-dimensional photonic structures,” J. Mod. Opt. 51, 2533–2542 (2004). [CrossRef]
- S. M. Eaton, H. Zhang, P. R. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate,” Opt. Express 13(12), 4708–4716 (2005). [CrossRef] [PubMed]
- M. Sakakura, M. Shimizu, Y. Shimotsuma, K. Miura, and K. Hirao, “Temperature distribution and modification mechanism inside glass with heat accumulation during 250kHz irradiation of femtosecond laser pulses,” Appl. Phys. Lett. 93, 231112 (2008). [CrossRef]
- Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
- J. Noack and A. Vogel, “Laser-induced plasma formation in water at nanosecond to femtosecond time scales: calculation of thresholds, absorption coefficient and energy density,” IEEE J. Quantum Electron. 35, 1156–1167 (1999). [CrossRef]
- 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]
- K. Nahen and A. Vogel, “Plasma formation in water by picosecond and nanosecond Nd:YAG laser pulses – part II: transmission, scattering, and reflection,” J. Sel. Top. Quant. Electron. 2, 861–871 (1996). [CrossRef]
- 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/Nanoengineering 2, 57-63 (2007). [CrossRef]
- J. Bovatsek, A. Araia, and C. B. Schaffer, “Three-dimensional micromachining inside transparent materials using femtosecond laser pulses: new applications,” Proceedings of CLEO/Europe - EQEC2005 (2005).
- http://www.schott.com/special_applications/english/download/d263te.pdf .
- http://www.schott.com/special_applications/english/download/af45e.pdf .
- http://psec.uchicago.edu/glass/Schott%20B270%20Properties%20%20Knight%20Optical.pdf .
- H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 353 (Oxford at the Clarendon Press, 1959).
- http://www.coresix.com/images/0211.pdf .
- S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87(21), 213902 (2001). [CrossRef] [PubMed]
- A. E. Siegman and S. W. Townsent, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993). [CrossRef]
- I. Miyamoto and T. Hermann, “Characteristics of internal melting of glass for fusion welding using ps laser pulses with average power up to 8W,” Proc. 8th Int. Symp. On Laser Precision Microfabrication- LPM2007 (2007).
- D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. 64, 3071–3073 (1994). [CrossRef]
- P. K. Kennedy, “A first-order model for computation of laser-induced breakdown thresholds in ocular and aqueous media: part I – theory,” IEEE J. Quantum Electron. 31, 2241–2249 (1995). [CrossRef]
- K. Morigaki, Physics of Amorphous Semiconductors (Imperial College Press, 1999).

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