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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 15, Iss. 9 — Apr. 30, 2007
  • pp: 5674–5686
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Observation of pressure wave generated by focusing a femtosecond laser pulse inside a glass

M. Sakakura, M. Terazima, Y. Shimotsuma, K. Miura, and K. Hirao  »View Author Affiliations


Optics Express, Vol. 15, Issue 9, pp. 5674-5686 (2007)
http://dx.doi.org/10.1364/OE.15.005674


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Abstract

The pressure (or stress) wave generated by focusing a femtosecond laser pulse inside a glass has been considered one of the important factors in determining structures created in the laser focal region. In this paper, a method of the transient lens (TrL) analysis was proposed to characterize the pressure wave. Experimentally, the TrL signal exhibited damping oscillation within 2 ns. Simulations of the TrL signal showed that the shape of the oscillating signal depended on the width and amplitude of the pressure wave. Comparing the observed TrL signal with the simulated one, we estimated these properties of the pressure wave generated after femtosecond laser focusing inside a soda-lime glass.

© 2007 Optical Society of America

1. Introduction

Femtosecond lasers have been used as tools for microfabrication in most transparent materials [1–10

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21, 1729–1731 (1996). [CrossRef] [PubMed]

]. For example, optical waveguides can be produced inside glasses by translating the focal point of the femtosecond laser [1

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21, 1729–1731 (1996). [CrossRef] [PubMed]

, 2

2. K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71, 3329–3331 (1997). [CrossRef]

, 6–10

6. C. B. Schaffer, N. Nishimura, E. N. Glezer, A. M.-T. Kim, and E. Mazur, “Dynamics of femtosecond laser-induced breakdown in water from femtoseconds to microseconds,” Opt. Express 10, 196–203 (2002). [PubMed]

], and micro-voids created by tightly focused femtosecond laser inside a glass may be used as an optical binary memory [3

3. E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, “Threedimensional optical storage inside transparent materials,” Opt. Lett. 21, 2023–2025 (1996). [CrossRef] [PubMed]

, 4

4. C. B. Schaffer, A. Brodeur, J. F. Garcia, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett. 26, 93–95 (2001). [CrossRef]

]. Recently, the femtosecond laser microfabrication technique has been applied to cell surgery, and micro-disruption of biological tissues [11–13

11. K. König, I. Riemann, and W. Fritzsche, “Nanodissection of human chromosomes with near-infrared femtosecond laser pulses,” Opt. Lett. 26, 819–821 (2001). [CrossRef]

].

In the femtosecond laser microfabrication, nonlinear excitation is involved in the fabrication process, because of the strong electromagnetic field of the short pulse [1–13

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21, 1729–1731 (1996). [CrossRef] [PubMed]

]. This nonlinear excitation results in the electronic excitation in a small region at the focusing point. Furthermore, due to the short pulse, the elastic stress can be confined in a very small volume, and the subsequent relaxation of the stress results in the pressure wave generation [4

4. C. B. Schaffer, A. Brodeur, J. F. Garcia, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett. 26, 93–95 (2001). [CrossRef]

, 13–15

13. Vogel A., J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81, 1015–1047 (2005). [CrossRef]

]. A necessary condition for the confinement of large elastic stress [13

13. Vogel A., J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81, 1015–1047 (2005). [CrossRef]

, 14

14. M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005). [CrossRef]

, 16

16. G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103, 487–518 (2003). [CrossRef] [PubMed]

] is that the stress generation time (τstress) should be much faster than the acoustic transit time, which is defined by a/vac, where a and vac are the smallest dimension of the excited volume and the sound velocity in the excited material, respectively. For example, if one excites a volume of radius a=1 μm of material having vac=5 km/s, which is a typical value for glasses, the stress generation time must be much shorter than 200 ps. Therefore, pulse duration shorter than several ps is essential for generating the strong pressure waves.

Several researchers have suggested that strong pressure/stress waves or shock waves are important driving forces in creating a high refractive index structure or void structure [4

4. C. B. Schaffer, A. Brodeur, J. F. Garcia, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett. 26, 93–95 (2001). [CrossRef]

, 9

9. J. W. Chan, T. R. Huster, S. H. Risbud, and D. M. Krol, “Modification of the fused silica glass network associated with waveguide fabrication using femtosecond laser pulses,” Appl. Phys. A 76, 367–372 (2003). [CrossRef]

, 14

14. M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005). [CrossRef]

, 15

15. M. Sakakura and M. Terazima. “Oscillation of the refractive index at the focal region of a femtosecond laser pulse inside a glass,” Opt. Lett. 29, 1548–1550 (2004). [CrossRef] [PubMed]

], because the creation of such structures must be accompanied by a large density distribution change. On the other hand, it has sometimes been observed that the strong pressure waves could induce collateral damage in laser machining [17

17. A. Vogel, S. Busch, and U. Parlitz, “Shock wave emission and cavitation bubble generation by picosecond and nanosecond optical breakdown in water,” J. Acoust. Soc. Am. 100, 148–165 (1996). [CrossRef]

]. To understand and control the effects of the pressure wave on femtosecond laser machining, it is necessary to elucidate the generation and propagation processes of the pressure waves.

However, for determining the spatial profile of the refractive index from the TrL Image, a rather complicated calculation is required. Moreover, in order to trace the temporal profile, many TrL images at various delay times between the pump and probe pulses have to be analyzed. This procedure may not be convenient for routine studies of the pressure waves under various excitation conditions, and it is therefore desirable to develop an alternative method for characterizing the pressure wave more conveniently and easily.

In the present report, we demonstrated a simple way to determine the phase change and width of the pressure wave by the TrL signal. Previously, it was reported that the TrL signal after focusing a femtosecond pulsed laser showed a damping oscillation with a frequency of about 1 GHz (TrL oscillation) [Fig. 1(b)] [14

14. M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005). [CrossRef]

]. The origin of the oscillation was attributed to a propagation of a pressure wave. This observation and study lead us to consider that the feature of the pressure wave may be obtained by analyzing the TrL oscillation. We investigated the relation between the TrL oscillation and a pressure wave in detail by TrL signal simulations and found that the amplitude and the width of the pressure wave can be determined from the damping rate and the amplitude of the TrL oscillation.

Fig. 1. (a). The temporal evolution of the phase distribution of the probe beam after the femtosecond laser irradiation inside a glass. It was obtained by the TrL method in our previous study. (b) The TrL signal after the femtosecond laser irradiation inside a glass.

2. Principle

2.1 Principle of TrL method

When isotropic material is irradiated with a pump beam which has a spatially Gaussian form, the profile of the material response may have an axial symmetry. In the TrL technique, the material response is detected as a spatial change of another probe beam [14

14. M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005). [CrossRef]

, 15

15. M. Sakakura and M. Terazima. “Oscillation of the refractive index at the focal region of a femtosecond laser pulse inside a glass,” Opt. Lett. 29, 1548–1550 (2004). [CrossRef] [PubMed]

, 18–23

18. M. Terazima and N. Hirota, “Rise profile of the thermal lens signal: Contribution of the temperature lens and population lens,” J. Chem. Phys. 100, 2481–2486 (1994). [CrossRef]

]. A lot of dynamics can be observed using the TrL measurement, because the photo-induced refractive index change originates from various chemical or physical effects such as temperature change [18

18. M. Terazima and N. Hirota, “Rise profile of the thermal lens signal: Contribution of the temperature lens and population lens,” J. Chem. Phys. 100, 2481–2486 (1994). [CrossRef]

, 19

19. M. Terazima, “Temperature lens and temperature grating in aqueous solution,” Chem.Phys. 189, 793–804 (1994). [CrossRef]

], density change [14

14. M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005). [CrossRef]

, 15

15. M. Sakakura and M. Terazima. “Oscillation of the refractive index at the focal region of a femtosecond laser pulse inside a glass,” Opt. Lett. 29, 1548–1550 (2004). [CrossRef] [PubMed]

, 18

18. M. Terazima and N. Hirota, “Rise profile of the thermal lens signal: Contribution of the temperature lens and population lens,” J. Chem. Phys. 100, 2481–2486 (1994). [CrossRef]

], the creation of photoexcited states [20, 21], intermediate species or photoreaction products [18

18. M. Terazima and N. Hirota, “Rise profile of the thermal lens signal: Contribution of the temperature lens and population lens,” J. Chem. Phys. 100, 2481–2486 (1994). [CrossRef]

], and nonlinear optical effects [22

22. M. Terazima, “Ultrafast transient Kerr lens in solution detected by the dual beam ‘thermal lens’ method,” Opt. Lett. 20, 25–27 (1995). [CrossRef] [PubMed]

, 24

24. R. Boyd, Nonlinear Optics 2nd Edition (Academic Press, 2003), Chap. 4.

].

When a femtosecond laser pulse is tightly focused inside a glass [Fig. 2(a)], the nonlinear photoexcitation may be induced by the strong electromagnetic field in the laser focal region. Subsequently, plasma creation and annihilation [13

13. Vogel A., J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81, 1015–1047 (2005). [CrossRef]

], and rapid temperature rise caused by the energy transfer from the hot electrons to the lattice [13

13. Vogel A., J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81, 1015–1047 (2005). [CrossRef]

, 16

16. G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103, 487–518 (2003). [CrossRef] [PubMed]

] occur, and the rapid temperature rise induces confined stress generation. These dynamics will change the refractive index distribution (Δn(t,x,y,z)) around the photoexcited region. When a probe beam passes through the photoexcited region, the wave front of the beam is deformed by the refractive index distribution (Fig. 2(b) middle). The deformation of the wave front after passing through the photoexcited region corresponds to the spatial phase modulation, which is expressed by Δϕ(x,y). After the spatial phase modulation and the subsequent propagation in the free space, the spatial intensity distribution of the probe beam is altered due to the diffraction (Fig. 2(b) left to right). From this intensity distribution change, we can obtain the refractive index profile in the photoexcited material.

In the conventional TrL or thermal lens methods [18

18. M. Terazima and N. Hirota, “Rise profile of the thermal lens signal: Contribution of the temperature lens and population lens,” J. Chem. Phys. 100, 2481–2486 (1994). [CrossRef]

,20

20. M. Terazima and N. Hirota, “Population lens in thermal lens spectroscopy,” J. Phys. Chem. 96, 7147–7150 (1992). [CrossRef]

], the central intensity of the probe beam is detected in order to measure the refractive index change, because the probe beam is simply focused or expanded by a lens-like refractive index distribution. The TrL signal, which is the light intensity at the center of the probe beam plotted against the delay time with respect to the excitation laser pulse, reflects the temporal evolution of the refractive index change.

Fig. 2. Schematic illustration of the Transient Lens Method. (a) Red dotted lines indicate a pump beam, and blue solid lines indicate a probe beam. (b) Illustrates of how the probe beam profile is deformed by Δn(x,y,z) in the photoexcited region. Probe beam profile is Gaussian prior to entering photoexcited region (left). While passing through the photoexcited region, the wave front is deformed by Δn(x,y,z) (middle). Probe beam profile deformed by the diffraction is detected on the detection plane (right)

2.2 Calculation of TrL signal

As drastic changes in temperature and density are expected in the femtosecond laser induced structural change inside a glass, the refractive index distribution created around the photoexcited volume may not be of the same Gaussian shape as the pumping light. In such a case, we need to simulate the TrL signal using a model based on the Fresnel diffraction theory [14

14. M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005). [CrossRef]

, 19

19. M. Terazima, “Temperature lens and temperature grating in aqueous solution,” Chem.Phys. 189, 793–804 (1994). [CrossRef]

]. Let us consider the probe beam propagation along the z axis from left to right through the photoexcited region as shown in Fig. 2(a). If the electric field of the probe beam before entering the photoexcited region is written as E 0(x, y), the electric field at the exit plane of the photoexcited region [E0(x, y)] is given by

E0xy=E0xyexp{iΔϕxy}
(1)

where Δϕ (x, y) is the phase distribution change due to the refractive index change. It should be noted that this phase change is an integrated phase change along the z axis for describing the refractive index distribution in the photoexcited volume. In this paper, we define a phase increase Δϕ>0 as delay of the phase, i.e, positive refractive index change. After the phase distribution change due to passage through the photoexcited region, the probe beam propagates a distance of z0 and reaches the detection plane. During the propagation from the photoexcited region to the detection plane, the intensity distribution of the probe beam is spatially modulated by a diffraction effect. The diffraction effect was calculated using the Fresnel diffraction integral [14

14. M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005). [CrossRef]

, 19

19. M. Terazima, “Temperature lens and temperature grating in aqueous solution,” Chem.Phys. 189, 793–804 (1994). [CrossRef]

, 25

25. K. Iizuka, Engineering Optics (Springer-Verlag, Berlin; Tokyo, 1985), Chap. 2.

] as follows. The electric field ESIG(x’, y’) and intensity distribution ISIG(x’, y’) of the probe beam are written as

ESIGxy=1z0E0xyexp(x2+y2λz0)
(2)
ISIGxy=A1λz0E0xyexp(x2+y2λz0)2
(3)

where λ is wavelength of the probe beam, (x’, y’) is the coordinate on the detection plane, and A is a constant. As the phase distribution change and intensity distribution of the probe beam are symmetrical around the z axis, the intensity distribution on the detection plane can be written as [25

25. K. Iizuka, Engineering Optics (Springer-Verlag, Berlin; Tokyo, 1985), Chap. 2.

]

ISIG(r)=A2πλz00E0(r)exp{i(Δϕ(r)+πr2λz0)}J0(2πλz0rr)rdr2
(4)

where r = (x 2 + y 2)1/2 and r’ = (x2 + y2)1/2 represent the distance from the z axis on the TrL plane and the detection plane, respectively, and the function J 0(x) is the 0th order Bessel function.

As the TrL signal intensity (ITrL) is defined by the central intensity of the probe beam [14

14. M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005). [CrossRef]

,19

19. M. Terazima, “Temperature lens and temperature grating in aqueous solution,” Chem.Phys. 189, 793–804 (1994). [CrossRef]

], we calculated it by

ITrL=SISIG(r)dSSIREF(r)dS
(5)

where IREF(r’) is the intensity distribution of the probe beam without a pump beam, S is the region inside a small circle whose center is located at the symmetric center of the probe beam. The Eqs. (4) and (5) and a time-dependent phase distribution function Δϕ(t,r) are used to simulate the TrL signal.

2.3 Expression of Δϕ(t,r)

After a femtosecond laser pulse is focused inside a glass, high-density plasmas are created only in the laser focal volume due to nonlinear absorption. As the plasma energy is transformed to the lattice energy in several picoseconds, the temperature in the laser focal volume is elevated immediately after photoexcitation. We assumed that the temperature distribution in the laser focal region [Fig. 3(a)] is expressed by

Txyz=T0+ΔTexp[r2(wth2)2](lth2zlth2)
(6)

where T 0 is the ambient temperature, ΔT is the maximum temperature increase, wth is the width of the temperature distribution, and lth is the length of the temperature distribution in the direction of the beam direction. The phase of the probe light changes due to the temperature dependence of the refractive index with a constant density. We expressed this phase change by Δϕ0 *exp[-{r/(wth/2)}2]. Here, we ignored the time dependence of this phase change, because the thermal diffusion took place in much slower time scale (>3 ns) [26

26. S. M. Eaton, H. Zhang, P. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Y. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate,” Opt. Express 13, 4708–4716 (2005). [CrossRef] [PubMed]

].

Fig. 3. (a). The heated region (colored by red) which appears in the laser focal region. The broken red lines represent excitation laser. (b). The density distributions Δρ(t,r) at t=50 ps, 100 ps, 260 ps, 800 ps and 1500 ps obtained by the calculation of thermoelastic wave equation. (c). Temperature distribution used for calculation of Δρ(t,r) shown in (b). In this case, wth=2.0 μm (d). The phase distribution of the pressure wave at 260 ps in case of wth=2.0 μm and definition of specific phase shift Δϕp and specific width wp of the pressure wave. In this case, Δϕp=1.0 and wp=1.26 μm.

The temperature change induces the thermoelastic stress, and the subsequent relaxation of the stress results in the density distribution change in the laser focal region. To obtain the phase change due to the density change ΔϕPW(t,r), we calculated the density distribution change Δρ(t,r) after the laser irradiation by solving a thermoelastic equation numerically [16

16. G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103, 487–518 (2003). [CrossRef] [PubMed]

,27

27. I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von. Rosenberg, and M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissure,” Proc. Natl. Acad. Sci. USA 92, 1960–1964 (1995). [CrossRef] [PubMed]

,28

28. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, Oxford, 1986), Chap. 3.

]. In the elastic calculation, the elastic constants of the soda-lime glass (Young modulus=72 GPa, Poisson ratio=0.29, density=2.54 g/cm3) were used. An example of the calculated density distribution change is shown in Fig. 3(b), which was calculated by the temperature distribution shown in Fig. 3(c). First, the density in the central region decreases due to the thermal expansion, and simultaneously the density at the periphery of the heated region increases. The density increase reaches maximum after 260 ps (in the case of wth=2.0 μm), and the region of high density propagates in the outward direction at the sound velocity vs~6.0 km/s. This propagating wave is the pressure wave. After calculating Δρ(t,r), the phase change due to the pressure wave was obtained approximately by

ΔϕPWtr=π(n021)Δρtrn0ρ0λl
(7)

where n 0 and ρ 0 are, respectively, the refractive index and the density of the glass before the laser irradiation, and l is the length of the pressure wave in the beam propagation direction. This equation was obtained according to the Newton-Drude model [29

29. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1984), Chap. II.

]. Although the length of the pressure wave cannot be determined in the TrL method, we can assume that it is as long as the Rayleigh range of the excitation beam.

Taking account of these two contributions, we used the following formula to express the phase distribution change:

Δϕtr=ΔϕPWtr+Δϕ0exp{r(wth2)}2
(8)

where the first term is the phase shift due to the pressure wave and the second term represents the time-independent phase shift due to the thermal expansion.

For characterizing Δϕpw(t,r), we used the width of the pressure wave (wp), and the maximum phase shift (ΔϕP) at the time when the density increase due to the pressure wave becomes maximum. For example, when wth, is 2.0 μm, the pressure wave is largest at 260 ps. Therefore, in the case of wth,=2.0 μm, Δϕp corresponds to the maximum phase shift at 260 ps. The definitions of Δϕp and the width of the pressure wave (wp) are shown in Fig. 3(d). By the elastic calculation, we found that wp is proportional to wth, (wp=0.63* wth) for all wth. We simulated the TrL signals by Eq. (8) with various wth, and Δϕp, and compared the simulated TrL signals with the experimental results.

3. Experiments

Figure 4(a) is a schematic illustration of an experimental setup of the TrL method. We used a laser pulse at a central wavelength (λp) of 780 nm and a pulse duration of about 220 fs from a Ti: sapphire laser based regenerative amplifier (IFRIT; Cyber laser Inc.). This laser pulse was split by a 1:1 beam splitter, and one was used for excitation and the other was used to generate a probe beam. The excitation laser pulse was focused inside a soda-lime silicate glass plate (S-1225; MATSUNAMI) by a 20X objective lens (NA=0.40), and the material in the laser focal region was photoexcited. The estimated spot size and Rayleigh range of the excitation laser at the laser focus were about 1.2 μm and 9 μm, respectively.

The other pulse was frequency doubled by being passed through a BBO crystal to produce a probe beam of λ=390 nm. The probe beam was optically delayed against the excitation pulse and focused inside a sample. The energy of the probe beam was attenuated sufficiently weak (<1 nJ/pulse) so as not to photoexcite the sample. The distance of the focal point of the probe beam from the photoexcited region, which is an important parameter to obtain a strong lens signal [30

30. K. Uchiyama, A. Hibara, H. Kimura, T. Sawada, and T. Kitamori, “Thermal Lens Microscope,” Jpn. J. Appl. Phys. 39, 5316 (2000). [CrossRef]

], was adjusted to d=-0.06 mm by changing the distance between two lenses (L1 and L2). The position of d is shown in Fig. 2(a). The probe beam after passing through the glass plate was collimated by a lens (L3;f0=30 mm) and the intensity distribution at the position of L3 was imaged on a CCD camera. A blue filter and a prism are used for separating the excitation pulse completely from the probe beam. The images of the probe beam detected by the CCD camera were transferred to a computer, and the TrL signal was obtained by plotting the central intensities of the images [broken red lines depicted in Figs. 4(b) and 4(c)] against the delay time of the probe beam. The spatial width of the probe beam in front of the objective lens was about 4 mm, and the diameter of the central region for calculating TrL signal intensity was about 0.4 mm. The TrL signal intensity was normalized by the probe light intensity without the excitation pulse [Eq. (5)]. To detect the dynamics induced by only one pulse, the sample glass plate was translated in the vertical direction with respect to the beam axis at ~15 mm/s during data acquisition. The repetition rate of the pulses was 100 Hz, so that the photoexcited volumes at each shot are 150 μm apart from each other. This distance is sufficiently far apart to avoid multi-excitation at the same volume. As the extinction ratio of the source laser is less than 1/100, the observation is not affected by the pre- and pos-pulses.

Fig. 4. (a). Schematic of an experimental setup for the TrL method. L1, L2 and L4: lenses of f0= 150 mm; L3: lens of f1=30 mm; HS: harmonic separator for Ti: sapphire laser; BF: blue filter for attenuating the pump pulse. Arrow from CCD camera means that images detected by a CCD camera are transferred to a computer. (b) and (c) Probe beam profiles at the position of L3, which were imaged on a CCD camera without and with a pump pulse, respectively. TrL signal intensity is light intensity inside a red dotted circle.

4. Results

4.1 TrL signal due to pressure wave propagation

Figure 5(a) shows the TrL signals simulated at various wth with fixed Δϕp =0.5. Oscillation is clearly observed even after 2 ns in the TrL signal at wth=0.8 μm, while those at larger wth decays until 2 ns. This tendency suggests that the TrL oscillation decays faster as wth increases. Moreover, the oscillation amplitude also depended on wth. From Fig. 5(a), we can see that the oscillation amplitude at wth=1.6 and 3.2 μm are larger than that at wth=0.8 μm.

Next, the TrL signals were simulated at various Δϕp with fixed wth =2.0 μm. The simulated TrL signals are shown in Fig. 5(b). We found that the oscillation amplitude becomes larger Δϕp increases, while the decay rate of the oscillation is not so sensitive to Δϕp.

Fig. 5. (a). TrL signals simulated by Eq. (6) with various width wth and Δϕp=0.50, and (b) ones with various amplitude Δϕp and wth=2.0 μm. (c) The definition of oscillation amplitudes Aosci1 and Aosci2. The solid lines are baselines of the TrL signals.
Fig. 6. Plots of the oscillation decay ratio and Aosci1 against wth (a) and Δϕp (b). The blue closed circles and the red open circles are Rdecay and Aosci1, respectively.

To clarify the shape of the oscillation, we defined the oscillation amplitudes at the first and second peaks (Aosci1 and Aosci2, respectively) and an oscillation decay ratio (Rdecay). The definition of Aosci1 and Aosci2 are shown in Fig. 5(c). The oscillation decay ratio is defined by RdecayAosci2/Aosci1, which is an indicator of the damping rate, i.e. smaller Rdecay means faster decay of the oscillation.

Rdecay and Aosci1 are plotted against wth in Fig. 6(a). It shows clearly that Rdecay becomes smaller as wth becomes larger. The oscillation amplitude Aosci1 becomes larger as wth increases until wth~3.2 μm. The relationship between Rdecay and wth indicates that wth can be estimated from experimentally determined Rdecay Once wth is determined from Rdecay, the width of the pressure wave, wp, can be calculated from the empirical relation of wp=0.63* wth.

In Fig. 6(b), Aosci1 and Rdecay are plotted against Δϕp. Clearly, Aosci1 becomes larger as Δϕp increases, while Rdecay is not so sensitive to ΔϕP. This result suggests that the amplitude of the pressure wave can be determined roughly from Aosci1 in the observed TrL signal, only if wth is determined.

4.2 Experimental results and comparison with simulation

Figure 7 shows the experimentally observed TrL signals at d=-0.06 mm with an excitation energy of 0.1 ~ 0.5 μJ/pulse. All TrL signals showed oscillations with damping, which indicated that a pressure wave was generated at all excitation energies. As the excitation energy per pulse (Iex) increased, the amplitude of the oscillation increased and the oscillation damped faster. To clearly show the excitation energy dependence of the TrL oscillation, Aosci1 and Rdecay were plotted against Iex in Figs. 8(a) and 8(b) respectively. The negative slope of Rdecay vs Iex means that wth became larger as excitation energy increased, because the simulation showed that Rdecay becomes smaller with increasing wth. On the other hand, it is difficult to conclude that larger Aosci1 means larger Δϕp, because Aosci1 is sensitive to both wth and Δϕp. In other words, Δϕp can be determined only when wth is fixed. Therefore, first, we determined wth from Rdecay, and next, Δϕp was determined from Aosci1 using this wth. The determined wth and Δϕp were plotted against Iex in Figs. 9(a) and 9(b). The result shows that the width of the pressure wave became larger with increasing the excitation energy. On the other hand, the amplitude of the pressure wave increased at Iex less than 0.3 μJ/pulse, but it was saturated at Iex>0.3 μJ/pulse. This saturation may be attributed to the nonlinear temperature dependence of the thermal expansion coefficient and the heat capacity of the glass, because the stronger excitation induces a very high temperature increase. To understand the origin of this saturation clearly, it is necessary to study the TrL signals for various glasses with different thermal properties in future.

Fig. 7. TrL signals measured with various excitation laser energies. (a) 0.10~0.30 μJ/pulse (b) 0.40 and 0.50 μJ/pulse. Each signal is offset for clarity. The broken line is the baseline (ITrL=1.0) for each TrL signal.
Figs. 8. (a). and (b). Plots of the oscillation decay ratio and the oscillation amplitude against the excitation pulse energy, respectively.
Figs. 9. (a). and (b). The width wth and the phase amplitudes Δϕp of the pressure wave are plotted against excitation pulse energy Iex, respectively.

5. Discussion

In this study, we showed that the properties of the propagating pressure wave induced by the femtosecond laser irradiation can be extracted from the oscillation of the TrL signal. Previously, Cho et al. also observed the probe beam propagating through the photoexcited region [31

31. S. -H. Cho, H. Kumagai, and K. Midorikawa, “In situ observation of dynamics of plasma formation and refractive index modification in silica glasses excited by a femtosecond laser,” Opt. Commun. 207, 243–253 (2002). [CrossRef]

]. However, they did not observe the oscillating intensity change of the probe beam. The most important difference between our study and Cho’s is the detection method of the probe beam. In our study, the central intensity of the probe beam was monitored, so that the diffraction due to the induced refractive index change was detected. On the other hand, in Cho’s study, the intensity of whole probe beam was monitored. Under this condition, light intensity change due to diffraction cannot be detected. Since the probe beam is diffracted by the refractive index change due to the pressure wave, the oscillation can be observed only by our method.

The pressure or stress waves from femtosecond laser focused regions inside transparent materials such as water [5

5. E. N. Glezer, C. B. Schaffer, N. Nishimura, and E. Mazur, “Minimally disruptive laser-induced breakdown in ter,” Opt. Lett. 22, 1817 (1997). [CrossRef]

, 6

6. C. B. Schaffer, N. Nishimura, E. N. Glezer, A. M.-T. Kim, and E. Mazur, “Dynamics of femtosecond laser-induced breakdown in water from femtoseconds to microseconds,” Opt. Express 10, 196–203 (2002). [PubMed]

, 13

13. Vogel A., J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81, 1015–1047 (2005). [CrossRef]

] and polymer [32

32. A. A. Babin, A. M. Kiselev, D. I. Kulagin, K. I. Pravdenko, and A. N. Stepanov, “Shock-Wave Generation upon Axicon Focusing of Femtosecond Laser Radiation in Transparent Dielectrics,” JETP Letters 80, 298–302 (2004). [CrossRef]

] have been observed by several researchers. For example, Glezer et al. observed the pressure wave generation in the femtosecond laser focal region inside water by a time-resolved direct imaging [5

5. E. N. Glezer, C. B. Schaffer, N. Nishimura, and E. Mazur, “Minimally disruptive laser-induced breakdown in ter,” Opt. Lett. 22, 1817 (1997). [CrossRef]

, 6

6. C. B. Schaffer, N. Nishimura, E. N. Glezer, A. M.-T. Kim, and E. Mazur, “Dynamics of femtosecond laser-induced breakdown in water from femtoseconds to microseconds,” Opt. Express 10, 196–203 (2002). [PubMed]

], in which the image of the pressure wave was focused directly onto a camera. All other researchers used the direct imaging technique, because the direct imaging can visualize the pressure wave. However, the direct imaging is not sensitive to the phase change compared with the TrL method, because of the following reason. In the TrL method, a small phase change in the probe region can be detected, because the diffracted probe light propagates a long distance after the phase modulation by the pressure wave (Fig. 2). The propagation of the long distance enhances the diffraction of the probe light. However, on the other hand, the direct imaging is not so sensitive to the phase change, because the diffracted probe light very close to the place of the phase change is imaged on the detector by a lens system. This propagation length is so short that small phase modulation cannot induce detectable light intensity modulation by diffraction effect. Therefore, it is necessary to use a rather strong laser pulse, often stronger than 1 μJ/pulse, to detect a pressure wave by the direct imaging. In the present study, the pressure wave generation could be observed as the oscillating signal even at a laser energy as weak as 0.1 μJ/pulse (Fig. 7). This high sensitivity should be very valuable for the investigation of the fs-laser induced structural change inside a glass, because the waveguide formations inside glasses by fs-laser are usually conducted with excitation pulse energies of 0.1~1 μJ/pulse.

The characteristic of the TrL method is that the amplitude (phase shift, Δϕp) and width (wp) of the pressure wave can be obtained easily by the comparison of the signal shape with the simulation. These values can be used to estimate the energy of the laser induced pressure wave. According to the elastic mechanics, the work qcompress, required for a compression of Δρ per unit volume is given by [16

16. G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103, 487–518 (2003). [CrossRef] [PubMed]

]

qcompress=92B(Δρρ0)2
(9)

where B is the bulk modulus and ρ0 is the density under the ambient pressure. If the phase change of the probe light, i.e., refractive index change, is proportional to the density change, the energy of the pressure wave Eq. (9) can be written as

qcompress(r)=92B(n0λπ(n021)lΔϕPWtr)2
(10)

Using Eq. (10) and normalized phase function of the pressure wave ΔϕPw(t,r/wth)/ΔϕP, one may find that the total energy of the pressure wave, QP (=qV ; V is the compressed volume) is given by:

QP=92B(n0λπ(n021)l)2lΔϕp2r0(ΔϕPW(t,rwth)Δϕp)22πrdr
(11)

where the lower limit of the integral (r0) should be selected not to contain the compression region in the heated material. We chose r0=1.5wth, because this value is sufficiently far from the heated region and only the contribution of the pressure wave will be calculated. The integration is conducted at the time when the pressure wave is sufficiently separated from the heated region, i.e. tVp>3wth, for example, wth=1.0 μm gives t>500 ps. The square of the normalized phase function of the pressure wave at tvp=4wth is shown in Fig. 10(a). When the integral in Eq. (11) was calculated at various t, we found that this integral is almost equal to 5.1wth 2. Therefore,

Qp=92×5.1wth2Δϕp2B(n0λπ(n021)l)2l
(12)

which means that Qp is proportional to ΔϕP2 wth2 . Figure 10(b) shows a plot of ΔϕP2 wth2 against the excitation laser energy on a logarithmic scale. For example, when the excitation energy is Eex=0.2 μJ/pulse, the energy of the pressure wave is calculated to be Qp~0.51×10-9 J for soda lime glass (B~58 GPa and n 0=1.5) at l=20 μm. Therefore, the energy conversion efficiency from the excitation pulse energy to the pressure wave is QP/Eex=0.26 %. This efficiency is the same order as that of shock wave in water by femtosecond laser irradiation, which was calculated by Vogel et al [13

13. Vogel A., J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81, 1015–1047 (2005). [CrossRef]

, 17

17. A. Vogel, S. Busch, and U. Parlitz, “Shock wave emission and cavitation bubble generation by picosecond and nanosecond optical breakdown in water,” J. Acoust. Soc. Am. 100, 148–165 (1996). [CrossRef]

].

Fig. 10. (a). Red line is plot of the square of normalized ΔϕpW at t * vp=4* wth (red line), broken blue line is the lower limit of the integral for calculating the compression energy of the pressure wave QP, the area filled by green mean the region of the pressure wave, and the heated region is filled by yellow. (b) Δϕp2 wth2 is plotted against Iex.

Finally, we propose one of the interesting applications of the TrL method for the microfabrication. We may be able to control the amplitude of the pressure wave and heated volume by changing excitation condition and monitoring them by the TrL method during laser machining. For example, it is important to minimize the heated volume for producing microstructures inside a glass. In this study, we showed that the width of the heated volume wth is smaller at larger Rdecay of the TrL signal. Therefore, we may be able to minimize the heated volume by controlling various excitation conditions to obtain the largest Rdecay This idea will realize the real time optimization of the excitation condition for the femtosecond laser microfabrication inside a glass. The TrL method will be a powerful tool not only for elucidating the mechanism of the laser induced damage but also for developing a novel technique of bulk machining by the femtosecond laser.

6. Conclusions

Acknowledgments

The authors would like to thank Koichiro Tanaka, Koji Fujita, Shingo Kanehira and Masayuki Nishi from Kyoto University and Jianrong Qiu from Zhejiang University for helpful discussions. This work was supported by the computational materials science unit in Kyoto University and the Ministry of Education, Science, Sports and Culture (MEXT), Grant-in-Aid for Young Scientists (B), 2006, No. 18760548.

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, 1729–1731 (1996). [CrossRef] [PubMed]

2.

K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. 71, 3329–3331 (1997). [CrossRef]

3.

E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, “Threedimensional optical storage inside transparent materials,” Opt. Lett. 21, 2023–2025 (1996). [CrossRef] [PubMed]

4.

C. B. Schaffer, A. Brodeur, J. F. Garcia, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,” Opt. Lett. 26, 93–95 (2001). [CrossRef]

5.

E. N. Glezer, C. B. Schaffer, N. Nishimura, and E. Mazur, “Minimally disruptive laser-induced breakdown in ter,” Opt. Lett. 22, 1817 (1997). [CrossRef]

6.

C. B. Schaffer, N. Nishimura, E. N. Glezer, A. M.-T. Kim, and E. Mazur, “Dynamics of femtosecond laser-induced breakdown in water from femtoseconds to microseconds,” Opt. Express 10, 196–203 (2002). [PubMed]

7.

A. M. Streltsov and N. F. Borrelli, “Study of femtosecond-laser-written waveguides in glasses,” J. Opt. Soc. Am. B 19, 2496–2504 (2002). [CrossRef]

8.

C. B. Schaffer, A. Brodeur, and E. Mazur, “Laser-induced breakdown and damage in bulk transparent materials induced by tightly focused femtosecond laser pulses,” Meas. Sci. Technol. 12, 1784–1794 (2001). [CrossRef]

9.

J. W. Chan, T. R. Huster, S. H. Risbud, and D. M. Krol, “Modification of the fused silica glass network associated with waveguide fabrication using femtosecond laser pulses,” Appl. Phys. A 76, 367–372 (2003). [CrossRef]

10.

H. Zhang, S. M. Eaton, and P. R. Herman, “Low-loss Type II waveguide writing in fused silica with single picosecond laser pulses,” Opt. Express 14, 4826–4834 (2006). [CrossRef] [PubMed]

11.

K. König, I. Riemann, and W. Fritzsche, “Nanodissection of human chromosomes with near-infrared femtosecond laser pulses,” Opt. Lett. 26, 819–821 (2001). [CrossRef]

12.

P. S. Tsai, B. Friedman, A. I. Ifarraguerri, B. D. Thompson, V. L. Ram, C. B. Schaffer, Q. Xiong, R. Y. Tsien, J. A. Squier, and D. Kleinfeld, “All-Optical Histology Using Ultrashort Laser Pulses,” Neuron 39, 27–41 (2003). [CrossRef] [PubMed]

13.

Vogel A., J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81, 1015–1047 (2005). [CrossRef]

14.

M. Sakakura and M. Terazima, “Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass,” Phys. Rev. B 71, 024113 (2005). [CrossRef]

15.

M. Sakakura and M. Terazima. “Oscillation of the refractive index at the focal region of a femtosecond laser pulse inside a glass,” Opt. Lett. 29, 1548–1550 (2004). [CrossRef] [PubMed]

16.

G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103, 487–518 (2003). [CrossRef] [PubMed]

17.

A. Vogel, S. Busch, and U. Parlitz, “Shock wave emission and cavitation bubble generation by picosecond and nanosecond optical breakdown in water,” J. Acoust. Soc. Am. 100, 148–165 (1996). [CrossRef]

18.

M. Terazima and N. Hirota, “Rise profile of the thermal lens signal: Contribution of the temperature lens and population lens,” J. Chem. Phys. 100, 2481–2486 (1994). [CrossRef]

19.

M. Terazima, “Temperature lens and temperature grating in aqueous solution,” Chem.Phys. 189, 793–804 (1994). [CrossRef]

20.

M. Terazima and N. Hirota, “Population lens in thermal lens spectroscopy,” J. Phys. Chem. 96, 7147–7150 (1992). [CrossRef]

21.

M. Terazima, “Transient lens spectroscopy in a fast time scale; Photoexcitation of Rhodamine 6G and Methyl Red solution,” Chem. Phys. Lett. 230, 87–92 (1994). [CrossRef]

22.

M. Terazima, “Ultrafast transient Kerr lens in solution detected by the dual beam ‘thermal lens’ method,” Opt. Lett. 20, 25–27 (1995). [CrossRef] [PubMed]

23.

J. F. Power, “Pulsed mode thermal lens effect detection in the near field via thermally induced probe beam spatial phase modulation: a theory,” Appl. Opt. 29, 52–63 (1990). [CrossRef] [PubMed]

24.

R. Boyd, Nonlinear Optics 2nd Edition (Academic Press, 2003), Chap. 4.

25.

K. Iizuka, Engineering Optics (Springer-Verlag, Berlin; Tokyo, 1985), Chap. 2.

26.

S. M. Eaton, H. Zhang, P. Herman, F. Yoshino, L. Shah, J. Bovatsek, and A. Y. Arai, “Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate,” Opt. Express 13, 4708–4716 (2005). [CrossRef] [PubMed]

27.

I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von. Rosenberg, and M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissure,” Proc. Natl. Acad. Sci. USA 92, 1960–1964 (1995). [CrossRef] [PubMed]

28.

L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, Oxford, 1986), Chap. 3.

29.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1984), Chap. II.

30.

K. Uchiyama, A. Hibara, H. Kimura, T. Sawada, and T. Kitamori, “Thermal Lens Microscope,” Jpn. J. Appl. Phys. 39, 5316 (2000). [CrossRef]

31.

S. -H. Cho, H. Kumagai, and K. Midorikawa, “In situ observation of dynamics of plasma formation and refractive index modification in silica glasses excited by a femtosecond laser,” Opt. Commun. 207, 243–253 (2002). [CrossRef]

32.

A. A. Babin, A. M. Kiselev, D. I. Kulagin, K. I. Pravdenko, and A. N. Stepanov, “Shock-Wave Generation upon Axicon Focusing of Femtosecond Laser Radiation in Transparent Dielectrics,” JETP Letters 80, 298–302 (2004). [CrossRef]

33.

J. B. Lonzaga, S. M. Avenesyan, S. C. Langford, and J. T. Dickinson, “Color center formation in soda-lime glass with femtosecond laser pulses,” J. Appl. Phys. 94, 4332–4340 (2003). [CrossRef]

OCIS Codes
(140.3390) Lasers and laser optics : Laser materials processing
(320.7120) Ultrafast optics : Ultrafast phenomena
(350.5340) Other areas of optics : Photothermal effects

ToC Category:
Ultrafast Optics

History
Original Manuscript: March 13, 2007
Revised Manuscript: April 16, 2007
Manuscript Accepted: April 17, 2007
Published: April 25, 2007

Citation
M. Sakakura, M. Terazima, Y. Shimotsuma, K. Miura, and K. Hirao, "Observation of pressure wave generated by focusing a femtosecond laser pulse inside a glass," Opt. Express 15, 5674-5686 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5674


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References

  1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, "Writing waveguides in glass with a femtosecond laser," Opt. Lett. 21, 1729-1731 (1996). [CrossRef] [PubMed]
  2. K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, "Photowritten optical waveguides in various glasses with ultrashort pulse laser," Appl. Phys. Lett. 71, 3329-3331 (1997). [CrossRef]
  3. E. N. Glezer, M. Milosavljevic, L. Huang, R. J. Finlay, T.-H. Her, J. P. Callan, and E. Mazur, "Threedimensional optical storage inside transparent materials," Opt. Lett. 21, 2023-2025 (1996). [CrossRef] [PubMed]
  4. C. B. Schaffer, A. Brodeur, J. F. Garcia, and E. Mazur, "Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy," Opt. Lett. 26, 93-95 (2001). [CrossRef]
  5. E. N. Glezer, C. B. Schaffer, N. Nishimura, and E. Mazur, "Minimally disruptive laser-induced breakdown in water," Opt. Lett. 22, 1817 (1997). [CrossRef]
  6. C. B. Schaffer, N. Nishimura, E. N. Glezer, A. M.-T. Kim, and E. Mazur, "Dynamics of femtosecond laser-induced breakdown in water from femtoseconds to microseconds," Opt. Express 10, 196-203 (2002). [PubMed]
  7. A. M. Streltsov and N. F. Borrelli, "Study of femtosecond-laser-written waveguides in glasses," J. Opt. Soc. Am. B 19, 2496-2504 (2002). [CrossRef]
  8. C. B. Schaffer, A. Brodeur, and E. Mazur, "Laser-induced breakdown and damage in bulk transparent materials induced by tightly focused femtosecond laser pulses," Meas. Sci. Technol. 12, 1784-1794 (2001). [CrossRef]
  9. J. W. Chan, T. R. Huster, S. H. Risbud and D. M. Krol, "Modification of the fused silica glass network associated with waveguide fabrication using femtosecond laser pulses," Appl. Phys. A 76, 367-372 (2003). [CrossRef]
  10. H. Zhang, S. M. Eaton, P. R. Herman, "Low-loss Type II waveguide writing in fused silica with single picosecond laser pulses," Opt. Express 14, 4826-4834 (2006). [CrossRef] [PubMed]
  11. K. König, I. Riemann, W. Fritzsche, "Nanodissection of human chromosomes with near-infrared femtosecond laser pulses," Opt. Lett. 26, 819-821 (2001). [CrossRef]
  12. P. S. Tsai, B. Friedman, A. I. Ifarraguerri, B. D. Thompson, V. L. Ram, C. B. Schaffer, Q. Xiong, R. Y. Tsien, J. A. Squier, and D. Kleinfeld, "All-Optical Histology Using Ultrashort Laser Pulses," Neuron 39, 27-41 (2003). [CrossRef] [PubMed]
  13. A. Vogel, J. Noack, G. Huttman, and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissues," Appl. Phys. B 81, 1015-1047 (2005). [CrossRef]
  14. M. Sakakura, and M. Terazima, "Initial temporal and spatial changes of the refractive index induced by focused femtosecond pulsed laser irradiation inside a glass," Phys. Rev. B 71, 024113 (2005). [CrossRef]
  15. M. Sakakura and M. Terazima. "Oscillation of the refractive index at the focal region of a femtosecond laser pulse inside a glass," Opt. Lett. 29, 1548-1550 (2004). [CrossRef] [PubMed]
  16. G. Paltauf and P. E. Dyer, "Photomechanical processes and effects in ablation," Chem. Rev. 103, 487-518 (2003). [CrossRef] [PubMed]
  17. A. Vogel, S. Busch, and U. Parlitz, "Shock wave emission and cavitation bubble generation by picosecond and nanosecond optical breakdown in water," J. Acoust. Soc. Am. 100, 148-165 (1996). [CrossRef]
  18. M. Terazima and N. Hirota, "Rise profile of the thermal lens signal: Contribution of the temperature lens and population lens," J. Chem. Phys. 100, 2481-2486 (1994). [CrossRef]
  19. M. Terazima, "Temperature lens and temperature grating in aqueous solution," Chem.Phys. 189, 793-804 (1994). [CrossRef]
  20. M. Terazima and N. Hirota, "Population lens in thermal lens spectroscopy," J. Phys. Chem. 96, 7147-7150 (1992). [CrossRef]
  21. M. Terazima, "Transient lens spectroscopy in a fast time scale; Photoexcitation of Rhodamine 6G and Methyl Red solution," Chem. Phys. Lett. 230, 87-92 (1994). [CrossRef]
  22. M. Terazima, "Ultrafast transient Kerr lens in solution detected by the dual beam 'thermal lens' method," Opt. Lett. 20, 25-27 (1995). [CrossRef] [PubMed]
  23. J. F. Power, "Pulsed mode thermal lens effect detection in the near field via thermally induced probe beam spatial phase modulation: a theory," Appl. Opt. 29, 52-63 (1990). [CrossRef] [PubMed]
  24. R. Boyd, Nonlinear Optics 2nd Edition (Academic Press, 2003), Chap. 4.
  25. K. Iizuka, Engineering Optics (Springer-Verlag, Berlin; Tokyo, 1985), Chap. 2.
  26. S. M. Eaton, H. Zhang, P. Herman, F. Yoshino, L. Shah, J. Bovatsek, A. Y. Arai, "Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate," Opt. Express 13, 4708-4716 (2005). [CrossRef] [PubMed]
  27. I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von. Rosenberg, and M. S. Feld, "The thermoelastic basis of short pulsed laser ablation of biological tissure," Proc. Natl. Acad. Sci. USA 92, 1960-1964 (1995). [CrossRef] [PubMed]
  28. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, Oxford, 1986), Chap. 3.
  29. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1984), Chap. II.
  30. K. Uchiyama, A. Hibara, H. Kimura, T. Sawada, and T. Kitamori, "Thermal Lens Microscope," Jpn. J. Appl. Phys. 39, 5316 (2000). [CrossRef]
  31. S. -H. Cho, H. Kumagai, and K. Midorikawa, "In situ observation of dynamics of plasma formation and refractive index modification in silica glasses excited by a femtosecond laser," Opt. Commun. 207, 243-253 (2002). [CrossRef]
  32. A. A. Babin, A. M. Kiselev, D. I. Kulagin, K. I. Pravdenko, and A. N. Stepanov, "Shock-Wave Generation upon Axicon Focusing of Femtosecond Laser Radiation in Transparent Dielectrics," JETP Letters 80, 298-302 (2004). [CrossRef]
  33. J. B. Lonzaga, S. M. Avenesyan, S. C. Langford, and J. T. Dickinson, "Color center formation in soda-lime glass with femtosecond laser pulses," J. Appl. Phys. 94, 4332-4340 (2003). [CrossRef]

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