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
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
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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.
#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 10715
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
#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 10716
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.
#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 10717
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.
#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 10718
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.
#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 10719
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
#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 10720
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.
#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 10721
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
#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 10722
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
#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 10723
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
#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 10724
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.
#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 10725
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
#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 10726
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).
#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 10727
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