To investigate the energy dissipation process after focusing a femtosecond laser pulse inside a zinc borosilicate glass, the time-dependent lens effect in the laser focal region was observed by a transient lens (TrL) method. We found that the TrL signal after 100 ns can be explained clearly by thermal diffusion. By fitting the observed signal, we obtained the phase change due to temperature increase, the initial diameter of the heated volume and the thermal diffusivity. On the basis of the results, the temperature increase and the cooling rate were estimated to be about 1800 K and 1.7×108 Ks-1, respectively. We have also observed the signal change on a 100 ns scale, which can not be explained by the thermal diffusion model. This change was attributed to the relaxation of the heated material.
© 2007 Optical Society of America
Optical waveguides can be fabricated inside glasses by focusing femtosecond (fs) laser pulses [1–7]. One of the important factors in the production of the optical waveguide is the repetition rate of the laser irradiation, because the laser irradiation with high repetition rate (>200 kHz) causes heat accumulation in the laser focal volume [2–7]. For example, it has been reported that the heat accumulation effects result in a waveguide having an effective region larger than the photoexcited volume and create smooth structures without any cracks. Although several researchers have investigated the heat accumulation effects in the production of the waveguides [7–9], the thermal diffusion dynamics after photoexcitation have never been elucidated by the time-resolved technique. By the observation of the thermal diffusion, the initial heated volume and the thermal diffusion rate can be determined. The volume of the heat accumulation during the fs laser machining is also determined by the initial heated volume and the thermal diffusion rate (i.e. cooling rate); therefore, it is important to elucidate the heating process and the thermal diffusion by a time-resolved method.
The transient grating [10–12] and transient lens (TrL) [13–17] methods have been utilized to observe thermalization and thermal diffusion dynamics in various laser-induced reactions so far. In the TrL method, the refractive index lens in the sample is observed by detecting a spatial deformation of a probe beam passing through the photoexcited volume, which is called a ‘lens effect’. As a temperature change in the material causes a refractive index change (Δn), the temperature distribution change due to the thermal diffusion can be detected by the TrL method. In this article, we report a time-resolved observation of the thermalization and the thermal diffusion process after a single shot irradiation of a fs laser pulse inside a glass by the TrL method.
2. Experimental methods
Experimental setup for the TrL method is shown schematically in Fig. 1(a). Femtosecond laser pulses from (Femtolite C-10-SP; IMRA) were amplified by a regenerative amplifier (IFRIT; Cyber Laser, 780 nm, 220 fs) with a repetition rate of 100 Hz. A portion of this laser light was detected by a photodiode and the frequency of this signal was electrically reduced to 3 Hz by a frequency divider. By using a mechanical shutter triggered by this 3 Hz-signal, laser pulses was selected from the 100 Hz-pulse train with a repetition rate of 3 Hz. The opening time of the shutter was synchronized to the pulse train by a delay generator (DG535; Stanford Research Systems). The selection of one pulse from the pulse train was confirmed by monitoring the selected pulse with a photodiode and oscilloscope. The repetition rate of 3 Hz was slow enough to avoid multiple excitations at one spot of the material by translating the sample The fs laser pulse was focused inside a glass plate (Corning 0211; zinc borosilicate glass) placed on a XYZ stage by a 20X objective lens with a numerical aperture (NA) of 0.4 (Sigma Koki; OBL-20), and a material in the laser focal region was photoexcited. To avoid multiple excitations in the same region, the glass sample was moved vertically to the laser beam axis about 2 ms after the photoexcitation. After the photoexcitation, the time-dependent refractive index distribution, i.e. a TrL, appears around the photoexcited region. To probe the TrL, a CW He-Ne laser beam (λprobe=633 nm) was passed through the TrL region and the central region of the beam was detected by a photomultiplier (HAMAMATSU; R955). The time-dependent light intensity (ISig(t)) detected by the photomultiplier was recorded by a digital oscilloscope (Textronix; TDS 784D).
The TrL signal ITrL(t) was defined by
where Iprobe(t) is the intensity of the probe beam at the beam center without the photoexcitation. The TrL signal reflects the lens shape created in the sample. For example, as a convex refractive index lens focuses the probe beam at far field in the upper of Fig. 1(b), the central intensity of the probe beam is larger than that without photoexcitation, i.e. ITrL>1.
In order to observe the TrL signal, the focus position of the probe beam should be different from the TrL region. The distance between the focal point of the probe beam and the TrL region is called a ‘focal mismatch’, and is denoted as d (Fig. 1(b)). Here, we defined that d is positive when the probe beam focuses prior to the TrL. The focal mismatch is important for interpreting the origin of the TrL signal, because it also affects the signal decay rate [14–17]. For example, the probe beam is focused at far field by a transient convex-type refractive index lens at d>0, while it is expanded at far field by the same type lens at d<0 as shown in Fig. 1(b). In this experiment, d was changed by using a beam expander.
3. Results and Discussion
3.1 TrL signals and simulation by the thermal diffusion model
Figure 2(a) shows the TrL signals from the fs laser focused area inside a Corning 0211 glass with various excitation pulse energies. These signals were measured at d=+0.07 mm. The signals rose within 1 µs and decayed in about 6 µs. The rising component and the maximum signal intensity became larger as the excitation energy increased. Also the time when the signal reaches maximum depended on the excitation energy; it is slower at stronger excitation energy.
To find the origin of the observed TrL signals, TrL signals were simulated by the thermal diffusion model . First, we obtained the time-dependent temperature distribution ΔT(t,x,y,z) by solving the thermal diffusion equation under the initial temperature distribution given by
where r=(x 2+y 2)1/2, wth is the diameter of the initial heated volume, ΔT0 is the maximum temperature increase after the excitation, and lz is the length of the heated volume along the beam axis. Here, we assumed that the initial temperature distribution ΔT(t=0,r) was created immediately after the photoexcitation. This assumption is reasonable, because our previous study showed that the temperature increased within several ps after the photoexcitation . The solution of the thermal diffusion equation under the initial condition given by Eq. (2) is
where Dth is the thermal diffusivity of the material. As this temperature change induces the refractive index change, the phase distribution of the probe beam is modulated. In our model, the phase shift was calculated by integrating the refractive index change due to ΔT(t,x,y,z) along the z axis. Therefore, the phase distribution change is expressed as
where Δϕth is the maximum phase change due to the temperature increase. We calculated the spatial distribution of the probe beam on the collimation lens by the following equation, which can be derived from the diffraction theory [14–16]
where A is a constant, E 0(t,r,d) is the electric field of the probe beam before entering the photoexcited region, r’ is the distance from the beam center at the collimation lens, z0 is the distance from the photoexcited region to the collimation lens, and the function J 0(x) is 0th order Bessel function. ISig(t) and IRef(t) in Eq. (1) were calculated by Eq.(5) at r’=0 with Δϕ(t,r) from Eq. (4) and Δϕ(t,r)=0, respectively .
If the refractive index change by the temperature increase dn/dT is a constant, Δϕth is given by
Strictly speaking, the value of dn/dT is temperature dependent. However, we consider that the assumption of the constant dn/dT does not cause significant error in the calculation, because dn/dTs of various alkali contained-silicate glasses were reported to be almost constant up to 2000 K . By substituting this phase distribution change to the equation based on the diffraction theory [14–16], the TrL signal can be simulated. There are three parameters for the simulation: Δϕth, wth, and Dth.
The TrL signals simulated at wth=2.0 µm and Dth=0.46 µm2µs-1, which is the value at the ambient temperature of Corning 0211,  with different Δϕth are shown in Fig. 2(b). The simulated TrL signals are similar to the experimentally observed ones after 100 ns. The simulated signal rose gradually in 1 µs and decayed in about 5 µs. Therefore, the slower rise and decay in the TrL signal should be caused by the thermal diffusion. Like the excitation energy dependence of the observed TrL signals, the maximum intensity became larger as Δϕth increased. This excitation energy dependence of the TrL signal means that the temperature change ΔT0, which is proportional to Δϕth, becomes larger as the excitation energy increases.
To confirm the above assignment of the origin of the TrL signal after 100 ns, we measured TrL signals at various d and compared them with simulated signals. Figure 3(a) shows the TrL signals with Iex=0.6 µJ/pulse at four different ds. The TrL signals depend on d strongly. At d>0, the time of the peak is delayed as d becomes larger, and the signal intensity at d=+0.07 mm is smaller than those at d=0.03 mm and 0.01 mm. At d<0, TrL signal is opposite to those at d>0. This d dependence means that this TrL signal comes from the refractive index lens, not from light absorption. The TrL signals simulated by Eqs. (4) and (5) at various ds are shown in Fig. 3(b). Clearly, the shapes after 100 ns and d-dependence of the simulated TrL signals look similar to those of the observed ones. Therefore, we can conclude that the observed TrL signal after 100 ns originates from the refractive index distribution change due to the thermal diffusion.
Qualitatively, the rise-decay profile of the simulated TrL signal can be explained as follows. Initially, the TrL signal is weak, because the photoexcited region is much smaller than the radius of the probe beam. Due to the thermal diffusion to the outward with time, the temperature elevated region becomes wider and comparable to that of the probed region. Hence, the TrL signal intensity increases with increasing delay time. Further thermal diffusion leads to further broadening of the temperature profile, which causes the decrease of the TrL signal.
3.2 Estimation of ΔT0, wth, and Dth
The comparison between the observed and simulated TrL signals shown above gives us an idea that we can determine the parameters in Eq. (4) by fitting the observed TrL signals. We found that the observed TrL signals with Iex=0.6 µJ/pulse can be fitted well by the TrL signals simulated with parameters Δϕth=5.8, wth=1.7 µm and Dth=0.75 µm2µs-1 at all ds. The fitted signals are shown in Fig. 4(a). The positive Δϕth suggests that the molar electric polarizability contribution to dn/dT  is larger than that of the thermal expansion. In case of this experiment, the thermal expansion could be suppressed, because the heated area is very small and confined in the solid material.
We estimated the maximum temperature increase, ΔT 0 from the determined phase change Δϕth and Eq. (6). Assuming lz=50 µm and using n 0=1.5 and dn/dT=3.4×10-6 K-1 of a borosilicate glass , we obtained ΔT 0~1790 K. This temperature increase is higher than the melting temperature of the glass. Therefore, we can speculate that the material in the laser focal region could melt after photoexcitation until heat dissipation begins. Previously, it was reported that a contribution of the thermal radiation to a thermal diffusion becomes significant at high temperature . Hence, it is rather surprising to find that the observed signal can be reproduced well by the simple thermal diffusion model even though the temperature change is more than 1000 K. This suggests that the energy emitted by a thermal radiation within the thermal diffusion time is much smaller than the thermal energy in the photoexcited region so that the thermal diffusion process is determined by phonon diffusion mainly.
The determined wth (=1.7 µm) is smaller than the diffraction limited beam diameter; (1.22λex)/NA~2.4 µm. This smaller diameter should be due to nonlinear excitation. The obtained thermal diffusivity (0.75 µm2µs-1) was larger than that of a borosilicate glass at room temperature (0.46 µm2µs-1). This difference should come from the temperature dependence of Dth.
By Eqs. (4) and (5), the obtained ΔT 0, wth and Dth, the temperature change at the center was calculated and shown in Fig. 4(b). The temperature decreases from ~1800 °C to ~50 °C within 10 µs. Hence, an averaged temperature cooling rate is calculated to be 1.7×108 Ks-1. This cooling rate is much faster than the fastest rate achieved by conventional glass production methods; for example, 105–106 Ks-1 by a roller quenching method . It is expected that the equilibrated structure at high temperature is fixed by the cooling process. The structure of high fictive temperature, which is defined by the temperature at which the structure is in equilibrium , is probably created at the laser focal region by this rapid cooling. This mechanism is consistent with the research of Chan et al., in which they observed Raman bands originating from the structure of a fictive temperature of 1500 °C at the fs laser focal region .
As described in the introduction, the cooling rate is important for determining the heat accumulation. According to Fig. 4(b), the temperature of the material cooled down to <100 °C in 10 µs. This suggests that the heat accumulation becomes critical when the repetition rate of the irradiation is higher than 100 kHz. This is consistent with Hermann’s study, which showed that the broadening of the heated area became apparent above 200 kHz .
3.3 Dynamics in 100 ns
Although the observed TrL signals can be explained well by the thermal diffusion model, the fitting is rather poor for the TrL signals before 100 ns. In Fig. 5, the TrL signals in 1 µs are shown. Clearly, the signal rises within 100 ns followed by the slower rise in about 200 ns, and the simulated signals do not reproduce the observed ones before 100ns. We found that the intensity of the faster rise becomes larger and the rise time becomes slower as the excitation energy increases, which is shown clearly in Fig. 5(b). Because the temperature elevation is higher at larger excitation energy and the temperature remains high within 100 ns according to Fig. 4(b), we speculated that the faster rise should be related to the material under high temperature. For a temporal interpretation, we attributed this rise to the dynamics in which the glass network in the heated region is relaxed to the structure in equilibrium at the temperature. This interpretation can explain the excitation energy dependence of the faster rise: as the temperature after irradiation becomes higher, the relaxation of the glass network needs longer time and the refractive index change due to the structural relaxation becomes larger. As a result, the rise time becomes slower and the intensity change in the rise becomes larger as the excitation energy increases.
The results in the study by Osellame et al. suggested the existence of the dynamics in 100 ns . In their study, it was found that an optical waveguide could not be formed in some glasses by the fs laser irradiation at 26 MHz (i.e. laser shot every 38 ns), although it can be formed at 250 kHz. Although the effect of the dynamics in 100 ns to the waveguide formation is not clear, it is possible that this dynamics is related to the structural change by the laser irradiation at several ten MHz.
As a summary of this study, we show the picture of the observed the dynamics in Fig. 6. The temperature increases immediately after laser irradiation. The temperature increase at the center of the irradiated material is about 1800 K at Iex=0.6 µJ/pulse. The material at the heated region melts and the glass network relaxes to equilibrium within 100 ns. The thermal diffusion occurs in several µs. This picture suggests that there are two important times: several tens ns and several µs. The slower time should determine whether the heat accumulation occurs or not under the laser irradiation with a repetition rate of several hundred kHz [2,4,7]. The faster rate could explain why the waveguide does not form under the laser irradiation at several ten MHz in some glasses such as a phosphate glass and a silica glass .
In conclusion, the dynamics accompanied by the thermalization in the fs laser focal region inside the glass was observed by the TrL method. The observed TrL signals can be explained by the thermal diffusion model, and the initial temperature distribution and the thermal diffusion rate were obtained. Therefore, the TrL method will be an important technique for elucidating the effect of heat as well as the mechanism of fs-laser induced structural change.
The authors thank Koichiro Tanaka, Koji Fujita, Shingo Kanehira and Masayuki Nishi from Kyoto University and Jianrong Qiu from Zhejiang University for discussions and supports. This research was supported by New Energy and Industrial Technology Development Organization and Science, Sports and Culture (MEXT), Grant-in-Aid for Young Scientists (B), 2006, No. 18760548.
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