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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 11 — May. 23, 2011
  • pp: 10714–10727
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Evaluation of nonlinear absorptivity in internal modification of bulk glass by ultrashort laser pulses

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


Optics Express, Vol. 19, Issue 11, pp. 10714-10727 (2011)
http://dx.doi.org/10.1364/OE.19.010714


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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 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 OSA

1. Introduction

Internal modification of transparent material by ultrashort laser pulses has been drawing much attention due to attractive applications such as formation of optical waveguide [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]

] and splitter [2

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]

], and fusion welding [3

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

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]

]. In the internal modification process, the laser energy is absorbed by nonlinear process with multiphoton ionization followed by avalanche ionization [5

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]

], 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 [6

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. 76, 351–354 (2003). [CrossRef]

]. Since then the internal modification with heat accumulation has attracted interest due to its new possibility in internal modification process [7

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 13(2), 612–620 (2005). [CrossRef] [PubMed]

,8

8. 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]

], and several authors reported thermal conduction models for internal modification in bulk glass at high pulse repetition rates [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]

,10

10. 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]

]. 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

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]

,8

8. 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]

10

10. 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]

]. 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

11. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

13

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 15(16), 10303–10317 (2007). [CrossRef] [PubMed]

]. 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 [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]

], assuming the reflection and the scattering by the laser-induced plasma are negligible based on the measurement by Nahen 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]

], and the nonlinear absorptivity was shown to increase significantly with increasing pulse repetition rate at durations of 10ps [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]

] and 400fs [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]

]. It was also reported that the region of the laser energy absorption extends toward the laser source at high pulse repetition rates [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]

,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]

], 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
Fig. 1 Cross-sections obtained at different pulse energies Q0 at pulse repetition rates f of (a) 50kHz, (b) 200kHz and (c) 500kHz at 20mm/s in D263. Geometrical focus is assumed to be at the bottom edge of internal modification at threshold pulse energy shown by a horizontal line. (NA0.55, τ = 10ps). (d) Coordinate and direction of laser beam (the sample is translated along x-axis, which is perpendicular to the paper plane).
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 [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]

] and Tm = 560°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 250kHz irradiation of femtosecond laser pulses,” Appl. Phys. Lett. 93, 231112 (2008). [CrossRef]

].

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
Fig. 2 Focused laser beam was irradiated at the interface of two glass plates of D263 (thickness: 1mm, v = 20mm/s. The glass plates are welded together within the outer structure of the modified region.
, 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. [16

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).

]. Assuming the glass coalesces at the forming temperature with the viscosity of 104 dPas [17], 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 [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]

]. On the other hand, the forming temperature of B270 is 1,033°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 250kHz irradiation of femtosecond laser pulses,” Appl. Phys. Lett. 93, 231112 (2008). [CrossRef]

] 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 [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]

]. 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 [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]

]
AEx=1QtQ01(1R)2,
(1)
where Q0 is incident laser pulse energy, Qt transmitted laser pulse energy through the glass sample and R Fresnel reflectivity. The validity of 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
Fig. 3 Nonlinear absorptivity vs. pulse energy at different pulse repetition rates. Solid lines show experimental values, and closed circles simulated values.
. 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)
Fig. 4 (a) Cross-sectional area S within isothermal line of Tm vs. laser pulse energy Q0 at different pulse repetition rates f. (b) S vs. AExfQ0. Closed circles are experimental values and solid line simulated.
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 [20

20. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 353 (Oxford at the Clarendon Press, 1959).

]
T(x,y,z)=14πK0lw(z')sexp{v2α(x+s)}dz'+T0,
(2)
where s2 = x2 + y2 + (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 by
0lw(z')sexp{v2α(x+s)}{xr3v2α(xr1)}dz'=0
(3)
to the contour of the experimental modification structure. Then the nonlinear absorptivity ACal is given by
ACal=1fQ00lw(z)dz,
(4)
and 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 ACal and the length of the absorbed region l are simulated.

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 by
w(z)=azm+b,
(5)
where a, b and m are positive constants. From Eq. (4), ACal is written in a form
ACal=1fQ0(am+1lm+1+bl)
(6)
In the simulation, the thermal constants, ρ = 2.51g/cm3 and 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 = 0.0096W/cmK (room temperature) of Corning 0211 [21], which is equivalent to Schott D263.

Figure 5
Fig. 5 Isothermal lines of 1,051°C and 3,600°C at f = 500kHz simulated at different values of m. In this calculation, b = 73W/cm (independently of m) and a for m = 0.5, 1.0 and 2.0 are 1032W/cm 1.5, 19500W/cm2 and 1.95*104 W/cm3, respectively (Q0 = 1.59µJ, v = 20mm/s, T0 = 25°C).
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
Fig. 6 Isothermal lines of 1,051°C and 3,600°C simulated at f = 50kHz (Q0 = 10.3µJ, v = 20mm/s, m = 1, T0 = 25°C).
; 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]

]. 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
Fig. 7 Ratio of ACal/AEx plotted vs. average absorbed laser power Wab (K = 0.0096W/cmK).
. 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 [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]

].

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 [23

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]

]
ω(z)=ω01+(M2λzπω02ng)2;ω0=M2λπNA,
(7)
where z is distance from the focus, λ wavelength of the laser beam, M2 beam quality factor, NA numerical aperture of focusing optics and ng refractive index of the bulk glass, the laser intensity I(z) at z is given by
I(z)=2Q0πτω2(z).
(8)
Then z corresponding to the intensity I is written in a form of
z=ngNA2Q0πτIω02,
(9)
where τ is the pulse duration of the laser beam. Relationship between z and Q0 for different values of I is plotted for τ = 10ps with solid lines in Fig. 8
Fig. 8 Simulated and experimental length of laser absorbed region vs. pulse energy at different pulse repetition rates. Relationship between Q0 and z for different values of I given by Eq. (9) is plotted by solid lines, where ω0 = 2.26µm is used.
. 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 = 2Qth/(πτIth), 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 [24

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).

], ω0 is evaluated to be 2.26µm using the damage threshold of fused silica Ith = 5x1011W/cm2 for the super-polished surface [5

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]

] and the bulk [25

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]

] 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

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

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]

]. 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.

3.4 Transient thermal conduction model for three-dimensional heat source

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)
Fig. 11 (a) Temperature TBS(0,0,z;tf) attained at steady condition at z = 2.5µm and z = l/2. (b) Thermally excited free electron density at TBS(0,0,z;tf) (Q0 = 3.9±0.18µJ, Eg = 3.7eV, v = 20mm/s).
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 [26

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]

] 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 [27

27. K. Morigaki, Physics of Amorphous Semiconductors (Imperial College Press, 1999).

] 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 [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]

].

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 [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]

]. 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.

4. Summary

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.

Acknowledgments

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).

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.

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]

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. 76, 351–354 (2003). [CrossRef]

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 13(2), 612–620 (2005). [CrossRef] [PubMed]

8.

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]

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]

10.

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]

11.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

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]

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 15(16), 10303–10317 (2007). [CrossRef] [PubMed]

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]

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]

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, Conduction of Heat in Solids, 353 (Oxford at the Clarendon Press, 1959).

21.

http://www.coresix.com/images/0211.pdf.

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]

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]

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. 64, 3071–3073 (1994). [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]

27.

K. Morigaki, Physics of Amorphous Semiconductors (Imperial College Press, 1999).

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

  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]
  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. 76, 351–354 (2003). [CrossRef]
  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 13(2), 612–620 (2005). [CrossRef] [PubMed]
  8. 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]
  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]
  10. 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]
  11. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
  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]
  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 15(16), 10303–10317 (2007). [CrossRef] [PubMed]
  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]
  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]
  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, Conduction of Heat in Solids, 353 (Oxford at the Clarendon Press, 1959).
  21. http://www.coresix.com/images/0211.pdf .
  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]
  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]
  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. 64, 3071–3073 (1994). [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]
  27. K. Morigaki, Physics of Amorphous Semiconductors (Imperial College Press, 1999).

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