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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 22 — Nov. 4, 2013
  • pp: 26921–26928
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Modulation of laser induced-cracks inside a LiF single crystal by fs laser irradiation at multiple points

Masaaki Sakakura, Yuki Ishiguro, Naoaki Fukuda, Yasuhiko Shimotsuma, and Kiyotaka Miura  »View Author Affiliations


Optics Express, Vol. 21, Issue 22, pp. 26921-26928 (2013)
http://dx.doi.org/10.1364/OE.21.026921


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Abstract

Crack formations inside a LiF single crystal after femtosecond laser irradiation at multiple points were investigated. In the case of sequential laser irradiation at three points, the propagations of some cracks were prevented by the dislocation bands generated by the previous laser irradiation. On the other hand, in the case of simultaneous laser irradiation at three points with a spatial light modulator, cracks in all the <100> directions from the photoexcited regions were generated clearly, but the length of one crack depended on the distribution of laser irradiation positions. The simulation of elastic dynamics after fs laser irradiation at three points elucidated that the interference of laser induced stress waves depended on the distributions of the irradiation positions. We found that the constructive interference of stress waves at a crack tip should have prevented the crack from propagating further and the tensile stress by destructive interference of stress waves along a crack should have facilitated the propagation of the crack.

© 2013 Optical Society of America

1. Introduction

In laser processing, control of the structural changes after the photoexcitation is an important issue to improve the accuracy and efficiency of the processing [1

1. G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103(2), 487–518 (2003). [CrossRef] [PubMed]

, 2

2. A. Vogel, J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005). [CrossRef]

]. In particular, inside single crystals, the structural change depends on the cracks and line defects (dislocations), which appear in the specific directions from the photoexcited region [3

3. B. Lawn, Fracture of Brittle Solids (Cambridge University, 1993).

, 4

4. W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics (John Wiley, 1976), Chaps. 4 and 14.

] and modify the mechanical property around the photoexcitation. Therefore, to control the structural change in laser processing, we have to understand how cracks and dislocations appear after the photoexcitation.

2. Method

2.1. Setup for fs laser irradiation

Fig. 2 (a) Schematic illustration of optical setup of a parallel laser processing. L1, L2 are lenses of the focal lengths of 500 mm and 150 mm, respectively. MP is a metal plate, which blocked a 0th order diffraction and unnecessary light spots. M is a mirror. (b) The pattern of three light spots for photoexcitation inside a LiF single crystal. The red points are the photoexcited region.
Figure 2(a) shows a schematic of a simultaneous laser irradiation system. The laser source was a mode-locked Ti:Sapphire laser and regenerative amplifier (Mira-Legend, Coherent Inc. a pulse duration of 120 fs, a central wavelength of 800 nm, and repetition rate of 1 kHz). The laser pulses were modulated after reflection on an SLM (LCOS-SLM, X10468-02, Hamamatsu Photonics K.K.). The phase modulated laser pulses passed through a telescope with two convex lenses (L1 and L2) and was focused inside a sample with an objective lens (50 × , NA = 0.8; Nikon LU Plan). The sample was a LiF single crystal, the (001) surface of which had been cleaved and optically polished. The laser pulses were incident normal to the (001) plane of the crystal and focused at a depth of about 150 μm below the surface. Because we confirmed that the pulse duration change after the objective lens was less than 20 fs, we did not pre-compensate the chirp of the laser pulse.

The distributions of the light spots in the focal plane were controlled by a computer generated hologram (CGH) displayed on the SLM. The light spots were formed at the focus both in the telescope and at the focus of the objective lens. In the telescope, the 0 order diffraction light and unnecessary light spots were blocked by a thin steel plate, and three light spots were formed inside a sample. The distributions of the light spots for photoexcitation are shown in Fig. 2(b). Three light spots are located at the corner of an isosceles triangle whose base line is parallel to the <100> of a LiF crystal. r denotes the length between two spots and θ denotes the vertex angle. In this study, r was the same (r = 25 μm) in all shots. Only a single laser pulse was focused at each spot. The pulse energy was about 3 μJ at each spot.

After the laser irradiation, the morphology of the structural change was observed with a transmission optical microscope, and the birefringence around the structural change was analyzed by a polarization microscope with a liquid crystal compensator (CRI, Inc. LC-Polscope).

2.2. Simulation of stress waves

3. Results

3.1 Cracks by sequential irradiations

First, the fs laser irradiations at three points were carried out point by point from #1 to #3 (sequential irradiations). Transmission optical microscope images of cracks by sequential irradiations are shown in Fig. 3(a).
Fig. 3 (a) Transmission optical microscope images and (b) birefringence images of fs laser-induced cracks by sequential laser irradiations at two (left) or three (right) points inside a LiF single crystal. The sequence of the laser irradiation was #1, #2 and #3. (Media 1) In (b), the color and brightness indicate the azimuth and retardance of the birefringence, respectively. In the left images in (b), white dotted circles indicate the positions of the third photoexcited regions.
Four cracks were generated in the <100> directions from the first and second photoexcited regions (the left images of Fig. 3(a)). However, the crack from the third photoexcited region was different completely and depended on the vertex angle (θ). Atθ = 75°, the downward crack from the third photoexcited region was not generated. On the other hand, only the downward crack from the third photoexcited region was clear at θ = 105°. At the intermediate angle (θ = 90°), the appearance of all the cracks from the third photoexcited region were not as clear as other cracks. In addition, some cracks from the first and second photoexcited regions, for example, the upward cracks at θ = 75°, became thinner after the third photoexcitation.

The inhibition of crack generation from the third photoexcited region should be due to the mechanical property change of the crystal by the first and second laser irradiations. For example, dislocation bands generated from the photoexcited region increase the mechanical properties around the dislocation bands, because disordered lattice in dislocation bands prevents cracks from propagating [19

19. T. H. Courtney, Mechanical Behavior of Materials (McGraw-Hill, 1990) pp. 102–106 and Chap. 5.

, 20

20. N. F. Mott, “Dislocations, work-hardening and creep,” Nature 175(4452), 365–367 (1955). [CrossRef]

]. To examine the effect of the dislocation bands, birefringence image of the irradiation region were obtained by a polarization microscope. The birefringence images corresponding to Fig. 3(a) were shown in Fig. 3(b). Birefringence is generated in the <100> and <110> directions from the first and second photoexcited regions. The birefringence in the <100> directions is due to cracks, because the interface of the crack changes the polarization of the reflected light [21

21. G. R. Fowles, Introduction to Modern Optics (Dover, 1975), Chap. 2.

]. On the other hand, the birefringence in the <110> directions is due to dislocation bands, because the stress generated around dislocations induces birefringence [19

19. T. H. Courtney, Mechanical Behavior of Materials (McGraw-Hill, 1990) pp. 102–106 and Chap. 5.

, 22

22. M. Ohring, Engineering Materials Science (Academic, 1995) Chap. 7.

]. The dislocation bands from the first and second photoexcited regions cross near the third photoexcited region (white dotted circles in Fig. 3(b)). Because dislocations increase the mechanical strength of a crystal [19

19. T. H. Courtney, Mechanical Behavior of Materials (McGraw-Hill, 1990) pp. 102–106 and Chap. 5.

, 20

20. N. F. Mott, “Dislocations, work-hardening and creep,” Nature 175(4452), 365–367 (1955). [CrossRef]

], the mechanical strength increase due to the dislocation bands could affect the crack formation inside a LiF crystal. Atθ = 75°, the third photoexcited region was at the upper side of the crossing region. Therefore, the downward region of the third photoexcited region had been mechanically strong so that any crack was not generated in the downward direction. In the case of θ = 90°, four cracks from the third photoexcited region were clearly observed in the birefringence image. At this angle, the third photoexcited region was just at the crossed region of the dislocation bands, so four cracks were generated. At θ = 105°, the third photoexcited region was at the lower side of the crossing region and surrounded by four dislocation bands. Because the cracks except that in the downward direction needed to pass through the dislocation bands, these cracks were shorter than other cracks.

In the transmission optical microscope images, some cracks from the second and third photoexcited region became thinner after the laser irradiation at the third region (The right images of Fig. 3(a)). For example, θ = 75° in Fig. 3(a), the upward cracks from the photoexcited regions on the bottom became thinner after the third photoexcitation. Similar phenomena have been observed in a CaF2 single crystal by Qian et al [7

7. B. Qian, J. Song, G. Dong, L. Su, B. Zhu, X. Liu, S. Sun, Q. Zhang, and J. Qiu, “Formation and partial recovery of optically induced local dislocations inside CaF2 single crystal,” Opt. Express 17(10), 8552–8557 (2009). [CrossRef] [PubMed]

]. They found that some fs laser induced cracks disappeared after laser irradiation at the neighboring area. The thinning of some cracks may be due to the propagation of stress waves, because the stress wave can compress cracks transiently.

3.2 Cracks by simultaneous irradiation

Next, we investigated the crack formation by simultaneous fs laser irradiation at three points inside a LiF single crystal. Figure 4(a) shows transmission optical microscope images of the laser induced cracks. At all the spots’ distributions, four cracks were generated from every photoexcited points, but the length of some cracks depended on θ. At θ = 90°, the lengths of the cracks in all directions were almost same. On the other hand, at θ = 75° and 105°, the lengths of the downward cracks from the vertex were clearly different than other cracks (indicated by red arrows); at θ = 75°, the crack was shorter than the other cracks and looks as if it had been shut by compression, while at 105°, the crack was about 1.8 times longer than other cracks. The length of the crack downward from the vertex was plotted against θ in Fig. 4(b), which shows that the crack was the longest at θ = 105° and the shortest at θ = 75°.
Fig. 4 (a) Transmission optical microscope images of fs laser-induced cracks by simultaneous laser irradiation at three points inside a LiF single crystal. (Media 2) (b) The length of the downward crack from the vertex plotted against θ. (c) Pulse energy dependence of the crack lengths at θ = 105°. The red circles are the length of the downward crack from the vertex, and the blue ones are those of the other cracks.

In addition, the elongation of the crack at θ = 105° depended on the laser pulse energy. Figure 4(c) shows how the length of the crack at θ = 105° depended on the laser pulse energy. At lower pulse energy (<2.5 μJ/pulse·spot), the lengths of cracks in all the directions were almost the same, but the crack downward from the vertex became longer than other cracks as the pulse energy increased. This pulse energy dependence suggests that the interaction between three photoexcited regions becomes strong enough to affect the crack length as the pulse energy increases.

3.3 Crack length and interference of stress waves

The dependence of crack length on the focal spots’ distribution (θ) suggests that something from three photoexcited points had influenced each other after the photoexcitation. One possibility is the interference of stress waves [8

8. M. Sakakura, T. Tochio, M. Eida, Y. Shimotsuma, S. Kanehira, M. Nishi, K. Miura, and K. Hirao, “Observation of laser-induced stress waves and mechanism of structural changes inside rock-salt crystals,” Opt. Express 19(18), 17780 (2011). [CrossRef] [PubMed]

], because different stress distributions could be formed by different distributions of photoexcited points. To confirm this hypothesis, we simulated the transient density distributions after the photoexcitation at three points of θ = 75° and 105° and r = 25 μm. After the photoexcitation at three points, three stress waves were generated at the same time and propagated to interfere with each other.
Fig. 5 Simulated transient density distributions after simultaneous photoexcitation at three points inside a LiF single crystal. The regions in which the crack became shorter or longer in the experiment are magnified in the right.
Figure 5 shows the simulated transient density distributions at the times when stress waves interfere with each other. The simulated density distributions show that stress waves interfere constructively or destructively along the cracks differently at different θ. At θ = 75°, the stress waves from the two photoexcited points on the bottom collides at 2400 ps, and the interference of these stress wave compresses the tip of the crack from the photoexcited point at the vertex (a white arrow in Fig. 5(a)). This compressed region propagates along the crack as if the compression shut this crack until 3000 ps. This interference of stress waves suggests that the crack was prevented from propagating by the compressive stress between 2400 ps and 3000 ps. On the other hand, at θ = 105°, the stress waves from the two photoexcited regions on the bottom collides at 3000 ps. Because the collision time of stress waves at θ = 105° was about 1.5 times later than at θ = 75°, the compression due to the interference of stress waves is about 1.5 times smaller than that at θ = 75° (Fig. 5(b)). The compressive stress along the crack disappears in 200 ps (at 3200 ps), and subsequently the tensile stress appears along the crack at 3500 ps. It is possible that this tensile stress could facilitate the crack propagation.

We interpreted the pulse energy dependence of the crack length at θ = 105° (Fig. 4(c)) based on the interference of stress waves. Figure 5(b) shows that the tensile stress along the crack appeared at 3500 ps. If the crack was in the process of propagation at this time, the crack length should have been about 14 μm. This length can be calculated by the velocity of the inner stress wave (~4 μm/ns), which induces the crack propagation [8

8. M. Sakakura, T. Tochio, M. Eida, Y. Shimotsuma, S. Kanehira, M. Nishi, K. Miura, and K. Hirao, “Observation of laser-induced stress waves and mechanism of structural changes inside rock-salt crystals,” Opt. Express 19(18), 17780 (2011). [CrossRef] [PubMed]

]. If it is assumed that only a propagating crack could be affected by the tensile stress at 3500 ps, the crack of shorter than 14 μm had not been affected by the tensile stress, because the crack propagation had already stopped by this tensile stress appeared. This is consistent to Fig. 4(c), because it suggests that the downward crack of 12 μm length from the vertex was not affected by the interference of stress waves. Since this interpretation is tentative, we need to investigate the dependence of the crack length on the distributions of photoexcited points to verify this interpretation.

The mechanisms of annihilation and facilitation of crack propagation by the interference of stress waves are illustrated schematically in Fig. 6.
Fig. 6 Proposed mechanism of modulation of crack propagation based on the simulation of the interference of stress waves. (a) θ = 75°, at which the compressive stress is formed at the crack tip and propagates along the crack. (b) θ = 105°, at which the compressive stress appears along the crack, but it changes to a tensile stress in 500 ps.
In short, at θ = 75°, interference of stress waves generates a compressive stress at the tip of the crack, which could prevent the crack from propagating further. After that, the compressive region propagates along the crack and the compression may shut the crack gradually (Fig. 6(a)). On the other hand, at θ = 105°, the compressive stress along the crack appears but disappears in a short time, and then, a tensile stress appears along the crack, which could facilitate crack propagation (Fig. 6(b)). Because stress waves can be generated also using picosecond (ps) laser [1

1. G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103(2), 487–518 (2003). [CrossRef] [PubMed]

, 2

2. A. Vogel, J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005). [CrossRef]

], similar phenomena can be observed in a ps laser processing. In addition, as stronger stress wave is generated by a ps laser pulse, stronger mechanical effects could be induced.

4. Conclusion

Acknowledgments

This research was supported by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Young Scientists (B), No. 22750187 and that for Scientific Research (A), No. 23246121.

References and links

1.

G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev. 103(2), 487–518 (2003). [CrossRef] [PubMed]

2.

A. Vogel, J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005). [CrossRef]

3.

B. Lawn, Fracture of Brittle Solids (Cambridge University, 1993).

4.

W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics (John Wiley, 1976), Chaps. 4 and 14.

5.

Z. Y. Wang, M. P. Harmer, and Y. T. Chou, “Laser-induced controlled cracking in ceramic crystals,” Mater. Lett. 7(5–6), 224–228 (1988). [CrossRef]

6.

S. Kanehira, K. Miura, K. Fujita, K. Hirao, J. Si, N. Shibata, and Y. Ikuhara, “Optically produced cross patterning based on local dislocations inside MgO single crystals,” Appl. Phys. Lett. 90(16), 163110 (2007). [CrossRef]

7.

B. Qian, J. Song, G. Dong, L. Su, B. Zhu, X. Liu, S. Sun, Q. Zhang, and J. Qiu, “Formation and partial recovery of optically induced local dislocations inside CaF2 single crystal,” Opt. Express 17(10), 8552–8557 (2009). [CrossRef] [PubMed]

8.

M. Sakakura, T. Tochio, M. Eida, Y. Shimotsuma, S. Kanehira, M. Nishi, K. Miura, and K. Hirao, “Observation of laser-induced stress waves and mechanism of structural changes inside rock-salt crystals,” Opt. Express 19(18), 17780 (2011). [CrossRef] [PubMed]

9.

T. Tochio, M. Sakakura, Y. Shimotsuma, M. Nishi, K. Hirao, and K. Miura, “Transient stress imaging after irradiation with a focused femtosecond laser pulse inside a single crystal,” Jpn. J. Appl. Phys. 51, 126602 (2012). [CrossRef]

10.

K. Yamamoto, E. Ohmura, N. Hasaka, and H. Morita, “Crack propagation in glass by laser irradiation along laser scribed line,” J. Manuf. Sci. Eng. 131(5), 051002 (2009). [CrossRef]

11.

Y. Hayasaki, M. Isaka, A. Takita, and S. Juodkazis, “Time-resolved interferometry of femtosecond-laser-induced processes under tight focusing and close-to-optical breakdown inside borosilicate glass,” Opt. Express 19(7), 5725–5734 (2011). [CrossRef] [PubMed]

12.

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(2), 024113 (2005). [CrossRef]

13.

S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-induced microexplosion confined in the bulk of a sapphire crystal: evidence of multimegabar pressures,” Phys. Rev. Lett. 96(16), 166101 (2006). [CrossRef] [PubMed]

14.

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87(3), 031101 (2005). [CrossRef]

15.

K. Matthews, The Crystran Handbook of Infra-Red and Ultra-Violet Optical Materials (Crystran Ltd. 2011) p. 48. Available: http://issuu.com/crystran/docs/handbook?e=2724951/2745150.

16.

L. S. Birks, Electron Probe Microanalysis (Chemical Analysis) (John Wiley, 1963), Chap. 6.

17.

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

18.

C. V. Briscoe and C. F. Squire, “Elastic constants of LiF from 4.2 K to 300 K by ultrasonic methods,” Phys. Rev. 106(6), 1175–1177 (1957). [CrossRef]

19.

T. H. Courtney, Mechanical Behavior of Materials (McGraw-Hill, 1990) pp. 102–106 and Chap. 5.

20.

N. F. Mott, “Dislocations, work-hardening and creep,” Nature 175(4452), 365–367 (1955). [CrossRef]

21.

G. R. Fowles, Introduction to Modern Optics (Dover, 1975), Chap. 2.

22.

M. Ohring, Engineering Materials Science (Academic, 1995) Chap. 7.

OCIS Codes
(140.3390) Lasers and laser optics : Laser materials processing
(140.7090) Lasers and laser optics : Ultrafast lasers
(160.3220) Materials : Ionic crystals
(320.5390) Ultrafast optics : Picosecond phenomena
(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors
(070.6120) Fourier optics and signal processing : Spatial light modulators

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: September 13, 2013
Revised Manuscript: October 18, 2013
Manuscript Accepted: October 18, 2013
Published: October 30, 2013

Virtual Issues
November 15, 2013 Spotlight on Optics

Citation
Masaaki Sakakura, Yuki Ishiguro, Naoaki Fukuda, Yasuhiko Shimotsuma, and Kiyotaka Miura, "Modulation of laser induced-cracks inside a LiF single crystal by fs laser irradiation at multiple points," Opt. Express 21, 26921-26928 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-26921


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References

  1. G. Paltauf and P. E. Dyer, “Photomechanical processes and effects in ablation,” Chem. Rev.103(2), 487–518 (2003). [CrossRef] [PubMed]
  2. A. Vogel, J. Noack, G. Huttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B81(8), 1015–1047 (2005). [CrossRef]
  3. B. Lawn, Fracture of Brittle Solids (Cambridge University, 1993).
  4. W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics (John Wiley, 1976), Chaps. 4 and 14.
  5. Z. Y. Wang, M. P. Harmer, and Y. T. Chou, “Laser-induced controlled cracking in ceramic crystals,” Mater. Lett.7(5–6), 224–228 (1988). [CrossRef]
  6. S. Kanehira, K. Miura, K. Fujita, K. Hirao, J. Si, N. Shibata, and Y. Ikuhara, “Optically produced cross patterning based on local dislocations inside MgO single crystals,” Appl. Phys. Lett.90(16), 163110 (2007). [CrossRef]
  7. B. Qian, J. Song, G. Dong, L. Su, B. Zhu, X. Liu, S. Sun, Q. Zhang, and J. Qiu, “Formation and partial recovery of optically induced local dislocations inside CaF2 single crystal,” Opt. Express17(10), 8552–8557 (2009). [CrossRef] [PubMed]
  8. M. Sakakura, T. Tochio, M. Eida, Y. Shimotsuma, S. Kanehira, M. Nishi, K. Miura, and K. Hirao, “Observation of laser-induced stress waves and mechanism of structural changes inside rock-salt crystals,” Opt. Express19(18), 17780 (2011). [CrossRef] [PubMed]
  9. T. Tochio, M. Sakakura, Y. Shimotsuma, M. Nishi, K. Hirao, and K. Miura, “Transient stress imaging after irradiation with a focused femtosecond laser pulse inside a single crystal,” Jpn. J. Appl. Phys.51, 126602 (2012). [CrossRef]
  10. K. Yamamoto, E. Ohmura, N. Hasaka, and H. Morita, “Crack propagation in glass by laser irradiation along laser scribed line,” J. Manuf. Sci. Eng.131(5), 051002 (2009). [CrossRef]
  11. Y. Hayasaki, M. Isaka, A. Takita, and S. Juodkazis, “Time-resolved interferometry of femtosecond-laser-induced processes under tight focusing and close-to-optical breakdown inside borosilicate glass,” Opt. Express19(7), 5725–5734 (2011). [CrossRef] [PubMed]
  12. 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. B71(2), 024113 (2005). [CrossRef]
  13. S. Juodkazis, K. Nishimura, S. Tanaka, H. Misawa, E. G. Gamaly, B. Luther-Davies, L. Hallo, P. Nicolai, and V. T. Tikhonchuk, “Laser-induced microexplosion confined in the bulk of a sapphire crystal: evidence of multimegabar pressures,” Phys. Rev. Lett.96(16), 166101 (2006). [CrossRef] [PubMed]
  14. Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett.87(3), 031101 (2005). [CrossRef]
  15. K. Matthews, The Crystran Handbook of Infra-Red and Ultra-Violet Optical Materials (Crystran Ltd. 2011) p. 48. Available: http://issuu.com/crystran/docs/handbook?e=2724951/2745150 .
  16. L. S. Birks, Electron Probe Microanalysis (Chemical Analysis) (John Wiley, 1963), Chap. 6.
  17. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, 1986).
  18. C. V. Briscoe and C. F. Squire, “Elastic constants of LiF from 4.2 K to 300 K by ultrasonic methods,” Phys. Rev.106(6), 1175–1177 (1957). [CrossRef]
  19. T. H. Courtney, Mechanical Behavior of Materials (McGraw-Hill, 1990) pp. 102–106 and Chap. 5.
  20. N. F. Mott, “Dislocations, work-hardening and creep,” Nature175(4452), 365–367 (1955). [CrossRef]
  21. G. R. Fowles, Introduction to Modern Optics (Dover, 1975), Chap. 2.
  22. M. Ohring, Engineering Materials Science (Academic, 1995) Chap. 7.

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