Evaluation of nonlinear absorptivity in internal modification of ......Evaluation of nonlinear...

Evaluation of nonlinear absorptivity in internal

modification of bulk glass by ultrashort laser

pulses

Isamu Miyamoto,1,2,*

Kristian Cvecek,3 and Michael Schmidt

2,3,4

1Osaka University, 1-12, Yama-Oka, Suita, Osaka 565-0871, Japan 2 Erlangen Graduate School of Advanced Optical Technolies (SAOT), Paul-Gordon Str. 6, 91052 Erlangen, Germany

3Bayerisches Laserzentrum, Konrad-Zuse Str. 2-6, 91052 Erlangen, Germany 4University Erlangen-Nuremberg, Paul-Gordon Str. 6, 91052 Erlangen, Germany

*[email protected]

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 Optical Society of America

OCIS codes: (140.3390) Laser materials processing; (140.3440) Laser induced breakdown;

(140.7090) Ultrafast lasers; (160.2750) Glass and other amorphous materials; (190.3390) Laser

materials processing; (190.4180) Multiphoton processes.

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). 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).

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).

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). 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).

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).

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).

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).

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).

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).

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).

#142102 - $15.00 USD Received 1 Feb 2011; revised 20 Apr 2011; accepted 4 May 2011; published 17 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10714

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).

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). 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).

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).

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).

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). 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).

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

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] and splitter

[2], 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 [5], 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]. 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 [4], assuming the reflection and the scattering by the

laser-induced plasma are negligible based on the measurement by Nahen et al. [14], and the

nonlinear absorptivity was shown to increase significantly with increasing pulse repetition rate

at durations of 10ps [4] and 400fs [15]. 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 f300kHz as shown by a horizontal line

in Fig. 1. At pulse repetition rates f200kHz, 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] and Tm =

560°C in Schott B270 [10].

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. [16]. Assuming the glass coalesces at the forming

temperature with the viscosity of 104dPas [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]. 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] is


obviously underestimated. In D263, we refer Tm = 1,051°C as “melting temperature” and the

outer region as “molten region” hereafter.

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).

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]. 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]

2

0

11 ,

1

t

Ex

QA

Q R

(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. 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.


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.

Fig. 3. Nonlinear absorptivity vs. pulse energy at different pulse repetition rates. Solid lines

show experimental values, and closed circles simulated values.

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.


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.

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]

0

0

1 ( ')( , , ) exp ' ,

4 2

lw z v

T x y z x s dz TK s

(2)

where s2 = 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 by

3

0

( ')exp 1 ' 0

2 2

lw z v x v x

x s dzs rr

(3)

to the contour of the experimental modification structure. Then the nonlinear absorptivity ACal

is given by

0 0

1( ) ,

l

CalA w z dzfQ

(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

( ) ,mw z az b (5)

where a, b and m are positive constants. From Eq. (4), ACal is written in a form

1

0

1

1

m

Cal

aA l bl

fQ m

(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 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 [22]. 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.

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/cm1.5, 19500W/cm2 and 1.95*104W/cm3, respectively (Q0 = 1.59µJ, v = 20mm/s, T0 =

25°C).

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).

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].

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.


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

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]

22 2

0 02

0

( ) 1 ; ,g

M z Mz

NAn

(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

0

2

2( ) .

( )

QI z

z (8)

Then z corresponding to the intensity I is written in a form of

20

0

2,

gn Qz

NA I

(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. 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 = )/(2 thth IQ , 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], ω0 is evaluated to be 2.26µm

using the damage threshold of fused silica Ith = 5x1011

W/cm2 for the super-polished surface [5]

and the bulk [25] 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 5x1011

W/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.

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.

It is seen that the laser intensity at the top edge of the inner structure I(l) decreases with

increasing pulse repetition rate. At Q04µJ, 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 f100kHz. 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 Q0 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 [20]. In the present paper, the thermal conduction equation is

derived in the similar way for simulating more realistic energy distribution.

Fig. 9. Length of laser absorbed region l and average absorbed laser power per unit length Wab/l

vs. averaged absorbed laser power of Wab ( = ACalfQ0) at different energies Q0 and repetition

rates of laser pulse f.

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 by

21 1 2 2

1

3/2 110

' , ', ' ' ( ') ( ')1( , , ; ) exp ' ' '

8 4

N

i

q x v t if y z x x v t if y y z zT x y z t dx dy dz

c t ift if

(11)

As the first step we assume that the radial intensity distribution of the absorbed laser beam has

the Gaussian distribution with radius of 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 of

2

2 2

2 ( ) 2( , ) exp ; 0 ,

( ) ( )

w z rq r z z l

z f z

(12)

where r2 = x

2 + y

2. Substituting Eq. (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 of


21 2

21

2 1 2 110 0

21 1 ( ') ( ')

( , , ; ) exp '.4( ') 8 ( ') 8

lN x v t if yw z z z

T x y z t dzc f tz t if z t ift if

(13)

Figure 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;

NS500 pulses at 50kHz and NS3,000 pulses at 300kHz.

Fig. 10. Transient temperature on the laser beam axis at z = 2.5µm and z = l/2 at 20mm/s at pulse repetition rates of (a) 50kHz (Q0 = 3.72µJ, l = 22µm) and (b) 300kHz (Q0 = 3.9µJ, l =

72µm).

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 ΔT1,370°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.

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).

The free electron density excited to the conduction band is estimated, assuming that

molecules are thermalized and have a Maxwell-Boltzmann distribution [26] 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] 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/cm

3, 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].

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

f500kHz, 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]. 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

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 f100kHz.

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).


Evaluation of nonlinear absorptivity in internal modification of ......Evaluation of nonlinear...

Documents

Transcript of Evaluation of nonlinear absorptivity in internal modification of ......Evaluation of nonlinear...