## Abstract

We experimentally determined the three-dimensional temperature distribution and modification mechanism in a soda-lime-silicate glass under irradiation of ultrafast laser pulses at high repetition rates by analyzing the relationship between the morphology of the modification and ambient temperature. In contrast to previous studies, we consider the temperature dependence of thermophysical properties and the nonlinear effect on the absorbed energy distribution along the beam propagation axis in carrying out analyses. The optical absorptivity evaluated with the temperature distribution is approximately 80% and at most 3.5% smaller than that evaluated by the transmission loss measurement. The temperature distribution and the strain distribution indicate that visco-elastic deformation and material flow play important roles in the laser-induced modification inside a glass.

© 2012 OSA

## 1. Introduction

When an ultrafast laser pulse is tightly focused inside transparent materials, the light energy is absorbed by the electrons around the focal volume through multiphoton and inverse bremsstrahlung absorption processes. The energy of the photoexcited electrons is transferred to the lattice, and the temperature around the focal volume is elevated [1]. In particular, at high repetition rates (>100 kHz), the thermal energy from each irradiation accumulates around the focal spot because the irradiation rate is faster than that in thermal diffusion, and the characteristic shape modification called “heat modification” occurs by thermal energy [2]. Heat modification has been applied to the fabrication of low-loss optical waveguides [3], laser welding [4], and space-selective control of the material properties of a glass [5–7]. For these applications, the three-dimensional temperature distribution during laser irradiation is important because it can affect the main factors of modification such as stress, material flow, crystallization, and glass transition [8]. Several researchers have investigated the temperature distribution and mechanism of heat modification during high-repetition-rate irradiation [2–4, 9–12]. However, in their analyses, the temperature dependence of thermophysical properties and the nonlinear effect on the absorbed energy distribution along the beam propagation axis have not been considered [13]. In this study, we take these factors into account to determine the three-dimensional temperature distribution and discuss the modification mechanism.

## 2. Experimental method

We used amplified femtosecond (fs) laser pulses (250 kHz, 70 fs, 800 nm) of a mode-locked Ti-sapphire laser oscillator (Coherent; Mira and RegA). The fs laser pulses were attenuated by a neutral density filter and focused inside a soda-lime-silicate glass plate (Schott B 270 Superwite) [14] with a 20× objective lens (NA = 0.40; Nikon LU Plan ELWD 20). The glass plate was placed on a temperature-controllable stage (Yonekura; MS-TPS), in which the ambient temperature (*T _{a}*) could be controlled by infrared radiation from a halogen lamp. The ambient temperature was measured with a thermocouple, which was calibrated by observing the fusions of several metals. Under controlled ambient temperature, the sample was irradiated with fs laser pulses at different pulse energies (1.0, 1.5, and 2.0 µJ) for 1 s.

## 3. Experimental results

Figures 1(a)–(c)
show the optical microscope images of the modified regions after irradiation of 2.0-µJ pulses. Two boundaries are observed in the images [2–4,9,10,15]. In this letter, the region inside the inner boundary is referred as the “inner-modified region” and that between the inner and outer boundaries is referred as the “outer-modified region.” The volume of the outer boundary is larger at higher ambient temperatures. As shown in Figs. 1(d)–(f), the ambient temperature-dependent volume of the outer-modified region can be understood by assuming that the modification occurs above the characteristic temperature threshold (*T _{out}*). We quantify the size of the outer-modified region by employing

*R*and

_{r}*R*, which are defined in Fig. 1(a). Figure 1(g) shows the ambient temperature dependence of

_{z}*R*and

_{r}*R*.

_{z}## 4. Analysis

We simulated the thermal energy distribution during laser irradiation to obtain the temperature distribution. The starting equation is Fourier’s law:

where**J(**

*t*

**,r)**is the flux of thermal energy,

*λ*(T) is the thermal conductivity,

*T*

**(**

*t*

**,r)**is the temperature, ∇ is nabla,

*t*is the time after photoexcitation by the first pulse, and

**r**is the position vector in Cartesian coordinates. From the definition of specific heat,where

*q*

**(**

*t*

**,r)**is the thermal energy per unit volume,

*ρ*is the density (weight per unit volume),

*C*(

_{p}*T*) is the specific heat under constant pressure, and

*T*is the ambient temperature. By the equation of continuity, Eqs. (1) and (2), and the heat source by laser irradiation, we obtain the following equation:

_{a}*D = λ*(

*T*(

*t,*

**r**))

*/*(

*ρC*(

_{p}*T*(

*t,*

**r**)), which corresponds to the thermal diffusivity. In the first step of the analysis, we assumed the heat source for the single pulse to be expressed as follows [10]:

*n*is the pulse number,

*Δt*is the pulse interval (= 4 µs),

_{L}*q*is the maximum absorbed energy,

_{0}*r =*(

*x*)

^{2}+ y^{2}*is the radial distance,*

^{1/2}*z*is the position along the beam propagation axis, and

*w*and

_{r}*l*are the width of absorbed energy in the radial direction (= 1.1 µm, which is determined by the diffraction limit) and that in the beam propagation direction, respectively [Fig. 2(a) ]. We used the room-temperature value (4.6 × 10

_{z}^{−7}m

^{2}/s) [14] of

*D*for all temperature ranges because there are experimental reports indicating that the change of

*D*is small from 300 K to 1650 K in soda-lime-silicate glasses [16,17]. By using the initial condition of Eq. (4), we can solve Eq. (3) analytically and obtain the time evolution of the energy distribution for one pulse:

*t’*is the time after the photoexcitation of a certain pulse. After irradiation with the

*N*pulse, the thermal energy distribution is expressed by

_{th}*n*is an integer. The thermal energy distribution can be transformed to a temperature distribution by Eq. (2). We used the temperature dependence of the specific heat shown in Fig. 2(b), which is obtained by shifting the temperature axis in the reported data on soda-lime-silicate glass to match T

_{g}[18]. We did not consider the temperature dependence of the density and used the room-temperature value (2.55 × 10

^{3}kg/m

^{3}) [14], because the change of the density is small from 1050 K to 1650 K in a soda-lime-silicate glass [19]. Therefore, the only temperature-dependent material parameter considered in this study is the specific heat. Given our assumption that the temperature at the outer boundary reaches the characteristic temperature

*T*during laser irradiation, the following relation is obtained at the outer boundary

_{out}**r**=

**r**

*:*

_{boundary}where *t _{ex}* is the exposure time of the 250-kHz laser pulses ( = 1 s). When

**r**= (

*R*, 0, 0) {or

_{r}**r**= (0, 0,

*R*)} is substituted into Eq. (7),

_{z}*R*(or

_{r}*R*) as a function of

_{z}*T*can be obtained. By fitting the relationship between

_{a}*R*(or

_{r}*R*) and

_{z}*T*by the Eq. (7),

_{a}*q*,

_{0}*l*, and

_{z}*T*, can be determined.

_{out}## 5. Discussion

We summarize the best-fitting parameters in Table 1
and show the energy distribution curves in Figs. 2(c) and (d). The open circles in Figs. 2(c) and (d) are the thermal energies after 1-s laser irradiation at the positions of **r** = (*R _{r}*, 0, 0) {or

**r**= (0, 0,

*R*)}, which were calculated by the right-hand side of Eq. (7) with

_{z}*T*and determined

_{a}*T*.

_{out}The thermal energy (*Q _{a}*) due to single photoexcitation can be obtained by integrating Eq. (7) over all space:

The optical absorptivities calculated by dividing *Q _{a}* by the pulse energy are shown in Fig. 3(a)
. As the reference, we also show the absorptivity determined by the transmission loss measurement [shown in the inset of Fig. 3(a)]. The optical absorptivity determined by

*Q*corresponds to the lower limit of absorptivity because the absorbed light energy should not include photoluminescence, thermal radiation, or stress energies. In contrast, transmission loss measurement cannot exclude these contributions. Figure 3(a) shows that the absorptivity calculated with

_{a}*Q*is at most 3.5% smaller than that evaluated by the transmission loss measurement. This difference indicates that the contribution of photoluminescence, thermal radiation, and stress energies should be less than 3.5% of the incident light energy.

_{a}The determined characteristic temperature thresholds (*T _{out}*) are shown in Fig. 3(b). The gray band indicates the temperature range where the estimated percentage of the stress relaxation after 1-s laser irradiation changed from 1% to 99%. This means that stress relaxation is almost complete when the temperature reaches above the gray band during the exposure time ( = 1 s). The stress relaxation is estimated by the Vogt-Kelvin model, which is the basic model for the visco-elastic deformation (see ref. 9 for details). The

*T*values for three pulse energies are located in the gray band. This indicates that, inside the outer boundary, visco-elastic stress relaxation had occurred during the exposure time. In contrast, outside the outer boundary, visco-elastic relaxation had not started during the exposure time. Therefore, the material outside of the boundary served as a wall for the materials inside the boundary. As the result, a clear boundary appeared.

_{out}Having determined the absorbed energy of one pulse, we evaluated the detailed temperature distribution with the absorbed energy fixed. Although we assumed an elliptically symmetric heat distribution to obtain the optical absorptivity and characteristic temperature, the actual heat distribution should be asymmetric in the beam propagation direction because of plasma dispersion and the Kerr effect [20]. In order to obtain a more plausible temperature distribution, we changed the shape of heat distribution only in the z direction to reproduce the entire shape of the outer boundary with the absorbed energy for one pulse fixed. We show the optical microscope image and the three-dimensional temperature distributions determined for 2.0 µJ in Figs. 4(a)–(d) .

Figures 4(b), (c), and (d) show the temperature distributions at 0, 1, and 4 µs, respectively, after 1-s irradiation at 2.0 µJ of pulse energy, with the contour lines of the characteristic temperatures of the outer and inner boundaries drawn in the figures. The sharp edge near the beam propagation axis in the contour lines of *T _{out}* at 0 µs disappears after 1 µs owing to thermal diffusion, and the difference between 1 and 4 µs is small. Such cycles were repeated every 4 µs until the exposure of laser pulses was stopped. In contrast, such a sharp edge is not observed in the optical microscope image. This difference implies that the modification is slow enough that the sharp edge of the temperature distribution cannot affect the modification

Miyamoto et al. [4] proposed that in Schott D263 glass, the characteristic temperature of the outer boundary (*T _{out}*) is 1324 K and that of the inner boundary (

*T*) is 3873 K. The estimated

_{in}*T*in our analysis is much lower than their value, although the glass transition temperature (T

_{out}_{g}= 830 K) and the softening temperature (T

_{s}= 1009 K) of D263 [14] are close to those (T

_{g}= 806 K and T

_{s}= 997 K) of B270 [14]. The difference in

*T*could be because of an invalid assumption in their analysis. They assumed

_{out}*T*to be the glass-forming temperature (at a viscosity of 4.0 dPas) because welding between two glass plates occurs inside the outer boundary. However, there is no evidence that welding does not occur below this temperature. In addition, the difference between

_{out}*T*and

_{out}*T*in our analysis is smaller than their value. This is because we consider the temperature dependence of the specific heat. In our results,

_{in}*T*is 1220 K, which is close to the glass-forming temperature (= 1306 K, viscosity of 4.0 dPas) of B270. To investigate the mechanism of inner boundary formation, we heat-treated the glass sample at T

_{in}_{g}(= 806 K) for 1 h after laser irradiation. In Figs. 4(e) and (f), we show the residual strain distribution before and after the heat treatment, measured by using a polarization microscope with a liquid crystal compensator [21]. Before the heat treatment, the direction of the slow axis in Fig. 4(d) indicates that the strain was directed in the radial direction in the outer-modified region. This implies that the outer-modified region is compressed owing to thermal expansion in the central high-temperature region during laser exposure. After the heat treatment, as is shown in Fig. 4(f), the strain in the inner-modified region still remained, whereas that in the outer modified region disappeared. Given that the strain cannot be removed by heat treatment at

*T*, formation of the inner-modified region is likely to include the flow of glass elements [15]. In addition, the changes cannot be reversed to the original state unless the temperature of the heat-treatment is so high that the elements in the glass can move freely. Blondin et al. suggested that the change in the refractive index distribution near the focal spot under high-repetition fs laser irradiation is likely related to the flow of glass elements [11]. Our experimental result supports their suggestion.

_{g}## 6. Conclusion

In conclusion, we determined three-dimensional temperature distributions and considered the modification mechanism in a glass exposed to laser irradiation. The optical absorptivity evaluated with the temperature distribution is approximately 80% and at most 3.5% smaller than that evaluated by the transmission loss measurement. Based on the temperature distribution and the strain distribution, we conclude that the formation of the outer-modified region is due to visco-elastic deformation, and that of the inner-modified region is due to the flow of glass elements.

## Acknowledgments

The authors thank Dr. Nishi and Mr. Iwahara at Kyoto University for fruitful discussions. This work was supported by a Grant-in-Aid for JSPS Fellows.

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