## D-Scan measurement of ablation threshold incubation effects for ultrashort laser pulses |

Optics Express, Vol. 20, Issue 4, pp. 4114-4123 (2012)

http://dx.doi.org/10.1364/OE.20.004114

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### Abstract

We report the validation of the Diagonal Scan (D-Scan) technique to determine the incubation parameter for ultrashort laser pulses ablation. A theory to calculate the laser pulses superposition and a procedure for quantifying incubation effects are described, and the results obtained for BK7 samples in the 100 fs regime are compared to the ones given by the traditional method, showing a good agreement.

© 2012 OSA

## 1. Introduction

1. N. Bloembergen, “Laser-induced electric breakdown in solids,” IEEE J. Quantum Electron. **10**(3), 375–386 (1974). [CrossRef]

2. D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in Si0_{2} with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. **64**(23), 3071–3073 (1994). [CrossRef]

4. R. Stoian, A. Rosenfeld, D. Ashkenasi, I. V. Hertel, N. M. Bulgakova, and E. E. B. Campbell, “Surface charging and impulsive ion ejection during ultrashort pulsed laser ablation,” Phys. Rev. Lett. **88**(9), 097603 (2002). [CrossRef] [PubMed]

5. M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. **85**(9), 6803–6810 (1999). [CrossRef]

6. W. Kautek, J. Kruger, M. Lenzner, S. Sartania, C. Spielmann, and F. Krausz, “Laser ablation of dielectrics with pulse durations between 20 fs and 3 ps,” Appl. Phys. Lett. **69**(21), 3146–3148 (1996). [CrossRef]

8. M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. **80**(18), 4076–4079 (1998). [CrossRef]

2. D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in Si0_{2} with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. **64**(23), 3071–3073 (1994). [CrossRef]

9. A. P. Joglekar, H. Liu, G. J. Spooner, E. Meyhofer, G. Mourou, and A. J. Hunt, “A study of the deterministic character of optical damage by femtosecond laser pulses and applications to nanomachining,” Appl. Phys. B **77**, 25–30 (2003). [CrossRef]

10. S. Nolte, C. Momma, H. Jacobs, A. Tunnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B **14**(10), 2716–2722 (1997). [CrossRef]

11. E. G. Gamaly, A. V. Rode, B. Luther-Davies, and V. T. Tikhonchuk, “Ablation of solids by femtosecond lasers: ablation mechanism and ablation thresholds for metals and dielectrics,” Phys. Plasmas **9**(3), 949–957 (2002). [CrossRef]

*F*.

_{th}12. F. Costache, S. Eckert, and J. Reif, “Near-damage threshold femtosecond laser irradiation of dielectric surfaces: desorbed ion kinetics and defect dynamics,” Appl. Phys., A Mater. Sci. Process. **92**(4), 897–902 (2008). [CrossRef]

13. M. Mero, B. Clapp, J. C. Jasapara, W. Rudolph, D. Ristau, K. Starke, J. Kruger, S. Martin, and W. Kautek, “On the damage behavior of dielectric films when illuminated with multiple femtosecond laser pulses,” Opt. Eng. **44**(5), 051107 (2005). [CrossRef]

*F*values. The defects can be intrinsic to the material, or can be externally originated, such as laser created color centers [14

_{th}14. L. C. Courrol, R. E. Samad, L. Gomez, I. M. Ranieri, S. L. Baldochi, A. Zanardi de Freitas, and N. D. Vieira, “Color center production by femtosecond pulse laser irradiation in LiF crystals,” Opt. Express **12**(2), 288–293 (2004). [CrossRef] [PubMed]

*F*for subsequent pulses. These cumulative phenomena fall under the classification of incubation effects [15

_{th}15. D. Ashkenasi, M. Lorenz, R. Stoian, and A. Rosenfeld, “Surface damage threshold and structuring of dielectrics using femtosecond laser pulses: the role of incubation,” Appl. Surf. Sci. **150**(1-4), 101–106 (1999). [CrossRef]

17. Y. Jee, M. F. Becker, and R. M. Walser, “Laser-induced damage on single-crystal metal surfaces,” J. Opt. Soc. Am. B **5**(3), 648–659 (1988). [CrossRef]

*S*is [17

17. Y. Jee, M. F. Becker, and R. M. Walser, “Laser-induced damage on single-crystal metal surfaces,” J. Opt. Soc. Am. B **5**(3), 648–659 (1988). [CrossRef]

*F*

_{th}_{,1}and

*F*

_{th}_{,}

*are the ablation threshold fluences for a single pulse and for the superposition of*

_{N}*N*pulses, respectively. Equation (1) arises from a simple probabilistic model [17

17. Y. Jee, M. F. Becker, and R. M. Walser, “Laser-induced damage on single-crystal metal surfaces,” J. Opt. Soc. Am. B **5**(3), 648–659 (1988). [CrossRef]

**5**(3), 648–659 (1988). [CrossRef]

20. H. W. Choi, D. F. Farson, J. Bovatsek, A. Arai, and D. Ashkenasi, “Direct-write patterning of indium-tin-oxide film by high pulse repetition frequency femtosecond laser ablation,” Appl. Opt. **46**(23), 5792–5799 (2007). [CrossRef] [PubMed]

*S*<1, incubation effects are present; if

*S*= 1, the ablation threshold does not depend on the pulses superposition, and for

*S*>1, the material becomes more resistant to ablation as the pulses etch it (laser conditioning) [21

21. J. Bonse, J. M. Wrobel, J. Krüger, and W. Kautek, “Ultrashort-pulse laser ablation of indium phosphide in air,” Appl. Phys., A Mater. Sci. Process. **72**(1), 89–94 (2001). [CrossRef]

12. F. Costache, S. Eckert, and J. Reif, “Near-damage threshold femtosecond laser irradiation of dielectric surfaces: desorbed ion kinetics and defect dynamics,” Appl. Phys., A Mater. Sci. Process. **92**(4), 897–902 (2008). [CrossRef]

14. L. C. Courrol, R. E. Samad, L. Gomez, I. M. Ranieri, S. L. Baldochi, A. Zanardi de Freitas, and N. D. Vieira, “Color center production by femtosecond pulse laser irradiation in LiF crystals,” Opt. Express **12**(2), 288–293 (2004). [CrossRef] [PubMed]

12. F. Costache, S. Eckert, and J. Reif, “Near-damage threshold femtosecond laser irradiation of dielectric surfaces: desorbed ion kinetics and defect dynamics,” Appl. Phys., A Mater. Sci. Process. **92**(4), 897–902 (2008). [CrossRef]

15. D. Ashkenasi, M. Lorenz, R. Stoian, and A. Rosenfeld, “Surface damage threshold and structuring of dielectrics using femtosecond laser pulses: the role of incubation,” Appl. Surf. Sci. **150**(1-4), 101–106 (1999). [CrossRef]

*F*value for many pulses, the established procedure consists in applying the “zero damage method” [22

_{th}22. J. M. Liu, “Simple technique for measurements of pulsed Gaussian-beam spot sizes,” Opt. Lett. **7**(5), 196–198 (1982). [CrossRef] [PubMed]

_{00}Gaussian beam, in different spots, and an exponential fit is performed to the data of the squared ablated diameter as a function of the pulse fluence. For this, the beam propagation must be known to estimate the beam size at the sample surface, and many measurements of the damage diameter must be done, originating large uncertainties for fluences close to the threshold, at which the damage size is close to zero and is difficult to be determined.

23. R. E. Samad and N. D. Vieira Jr., “Geometrical method for determining the surface damage threshold for femtosecond laser pulses,” Laser Phys. **16**(2), 336–339 (2006). [CrossRef]

24. R. E. Samad, S. L. Baldochi, and N. D. Vieira Jr., “Diagonal scan measurement of Cr:LiSAF 20 ps ablation threshold,” Appl. Opt. **47**(7), 920–924 (2008). [CrossRef] [PubMed]

## 2. Theory

23. R. E. Samad and N. D. Vieira Jr., “Geometrical method for determining the surface damage threshold for femtosecond laser pulses,” Laser Phys. **16**(2), 336–339 (2006). [CrossRef]

24. R. E. Samad, S. L. Baldochi, and N. D. Vieira Jr., “Diagonal scan measurement of Cr:LiSAF 20 ps ablation threshold,” Appl. Opt. **47**(7), 920–924 (2008). [CrossRef] [PubMed]

23. R. E. Samad and N. D. Vieira Jr., “Geometrical method for determining the surface damage threshold for femtosecond laser pulses,” Laser Phys. **16**(2), 336–339 (2006). [CrossRef]

*E*

_{0}is the pulse energy and ρ

_{max}is half of the ablation profile maximum transversal dimension. This maximum is located at the positions ±χ relative to the beamwaist, with χ being given by [23

**16**(2), 336–339 (2006). [CrossRef]

*w*

_{0}and

*z*

_{0}are the laser beam beamwaist and confocal parameter [25

25. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**(10), 1550–1567 (1966). [CrossRef] [PubMed]

*F*.

_{th}*F*, the pulses superposition that etches ρ

_{th,N}_{max}at the χ locus must be known. We hypothesize here that this superposition

*N*at χ is given by the sum of the intensities generated at this spot by all the pulses that hit the sample during a D-Scan, normalized by the intensity generated by the pulse centered at χ. Then, to determine

*N*, we initially calculate the intensity generated at the position (

*q*,χ) by a single pulse impinging on the sample at the position (0,

*y*), as indicated in Fig. 2 . In this Fig., the

*y*axis lies on the sample surface, and the

*z*axis, which is the laser beam longitudinal axis, is transversal to the sample surface. The distance

*q*from the

*y*axis is used for generality.

*y*) has a spot size

*w*(

*z*), as shown in Fig. 2, and an intensity distribution given by [23

**16**(2), 336–339 (2006). [CrossRef]

*P*

_{0}is the pulse power,

*r*is the radial distance and

*w*(

*z*) is given by the usual Gaussian propagation law [25

25. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**(10), 1550–1567 (1966). [CrossRef] [PubMed]

*q*,χ) is then:

*t*

_{0}=0 a pulse hits the sample at

*y*= χ, generating the intensity

*I*(

*q*, χ,

*z*). At this position, the beam generates the profile maximum, ρ

*, and by definition [23*

_{max}**16**(2), 336–339 (2006). [CrossRef]

*z*= χ in the longitudinal axis; consequently, its intensity at (

*q*,χ) is

*I*(

*q*, χ, χ). The next pulse will hit the sample after a time 1/

*f*, where

*f*is the pulses repetition rate; after this interval, the sample has been displaced by

*v*/

_{y}*f*and

*v*/

_{z}*f*in the

*y*and

*z*directions, where

*v*and

_{y}*v*are the displacement speeds in these directions, respectively [24

_{z}24. R. E. Samad, S. L. Baldochi, and N. D. Vieira Jr., “Diagonal scan measurement of Cr:LiSAF 20 ps ablation threshold,” Appl. Opt. **47**(7), 920–924 (2008). [CrossRef] [PubMed]

*q*,χ) by the

*n*pulse to hit the sample is

^{th}*q*,χ),

*I*, is the summation of the contribution of all pulses that hit the sample:

_{Tot}*w*(χ+

*nv*/

_{z}*f*) does not change significantly around χ, and can be considered to be constant and equal to

*w*(χ), which is the spot size at

*t*=

*t*

_{0}. This approximation considers that the transversal (

*v*) and longitudinal (

_{y}*v*) sample displacement speeds and the focusing lens should be chosen to produce an elongated ablation profile. Also, the approximation takes into account that the pulses, which have a spatial Gaussian intensity profile, contribute negligibly to the intensity at (

_{z}*q*,χ) when distant from χ by a few spot sizes. Along with these considerations and the substitution of Eq. (3) in Eq. (5), which leads to

*N*, Eq. (9) has now to be normalized by the intensity

*Ι*

_{0}of a single pulse centered at χ. Substituting

*n*= 0 in Eq. (7) provides

*I*

_{0}, which is the term on the left of the summation symbol in Eq. (9), and consequently, the ratio

*N*=

*I*/

_{Tot}*I*

_{0}is simply:

_{3}[26, 27

27. Wolfram Research Inc, “Jacobi theta function ϑ_{3}” (1998–2011), retrieved 2011, http://functions.wolfram.com/EllipticFunctions/EllipticTheta3/06/01/03/.

*N*.

*v*/(

_{y}*f*ρ

_{max}) is always positive, the second argument of the Jacobi theta function on Eq. (11) can only assume values between 0 and 1, limiting the range of

*N*from 1 to infinity [26, 27

27. Wolfram Research Inc, “Jacobi theta function ϑ_{3}” (1998–2011), retrieved 2011, http://functions.wolfram.com/EllipticFunctions/EllipticTheta3/06/01/03/.

*v*/(

_{y}*f*ρ

_{max}) ≈0 and the second argument of ϑ

_{3}approaches the unity; in this limit the Jacobi theta function for real arguments can be approximated by:

*x*

^{2}}≅1-

*x*

^{2}for

*x*≈0 in the result, leads to:providing a simple formula to determine the pulses superposition at the position χ from the experimental data. For high transversal translation speeds or small repetition rates,

*v*/(

_{y}*f*ρ

_{max}) tends to infinity, invalidating the approximation show in Eq. (12), and consequently, Eq. (13) cannot be used; in this case Eq. (11) must be used, providing

*N*values close to 1, as expected for the superposition of few pulses. Figure 3 shows a comparison between the superposition values calculated by the Jacobi theta Function (Eq. (11), red thick line) and the approximation for the superposition of many pulses (Eq. (13), black dashed line). It can be seen that the approximate solution, Eq. (13), departs from the analytical one, Eq. (11), only for

*v*/(

_{y}*f*ρ

*)>1, differing by less than 2.5% for values of this argument below 1.5. As a rule of thumb, Eq. (13) should be used to calculate*

_{max}*N*from the experimental parameters (

*v*,

_{y}*f*and ρ

_{max}) due to its simplicity; nevertheless, when this Eq. returns values smaller than 2, the Jacobi theta function, Eq. (11), must be used to provide the correct value of the superposition.

*n*is limited to the finite range from –

*m*to +

*m*. This represents the superposition of 2

*m*+ 1 pulses, and the results for

*m*= 1, 10, 100 and 1000 are shown in Fig. 3 as blue lines. It can be seen that the finite summation only diverges from the ϑ

_{3}function (summation from -∞ to + ∞, red line in the graph) when the superposition

*N*is larger than

*m*. This means that only 2

*m*+ 1≈2

*N*pulses have to be considered to obtain the superposition of

*N*pulses at (

*q*,χ); once the experimental conditions (beam focusing and displacement speeds) for generating an elongated profile are satisfied, these 2

*N*pulses are in the immediate vicinity of χ, and the more distant pulses do not contribute for the incubation effects at this position, as assumed by us when deriving Eq. (9). This validates our approximation.

*N*defined here by Eq. (11) is not the same as the one used in the traditional method, in which the

*N*pulses completely overlap. Nevertheless, we assume that these superpositions are equivalent once the numerical result of considering a finite number of terms on Eq. (10) demonstrates that only the pulses on the immediate vicinity of χ contribute to the D-Scan superposition, in a condition that is similar to the traditional superposition. Additionally, the present definition numerically allows fractionary superpositions, meaning that pulses whose spot size does not encompass the ρ

*position have a contribution that adds linearly to the incubation effects at this point.*

_{max}_{max}); the use of the experimental values of

*v*,

_{y}*f*and ρ

_{max}in expressions (2) and (11) or (13) provides the

*F*value for

_{th}*N*pulses superposition, quantifying the incubation effects for this experimental condition.

## 3. Results

*F*and

_{th}*S*in BK7 samples by the traditional and by the D-Scan methods, and the results were compared. All irradiations were done in air, and after etching the samples were cleaned with isopropyl alcohol in an ultrasonic cleaner to remove redeposited ablation debris. After cleaning, the samples were observed and photographed on an optical microscope, and the ablation dimensions were measured in the micrographs.

*v*[24

_{y}**47**(7), 920–924 (2008). [CrossRef] [PubMed]

*v*) and longitudinal (

_{y}*v*) displacements speeds, were chosen in order to produce elongated etched profiles as shown in Fig. 6 , satisfying the approximations that lead to Eq. (9). Also, the ratio

_{z}*v*/

_{y}*v*was kept constant between scans to produce profiles with the same length. Some combinations of

_{z}*f*and

*v*resulted in fractional superpositions. For each condition of energy and superposition, two scans were done and the etched profiles maximum transversal dimension were measured in the micrographs. Although the theory predicts that the etched profile must be symmetrical [23

_{y}**16**(2), 336–339 (2006). [CrossRef]

*was always measured on the left lobe, and the ablation threshold fluence values,*

_{max}*F*, were calculated using Eq. (2). As in the traditional method, the blur around each ablated region was considered to be the uncertainty of ρ

_{th}_{max}, and the final ρ

_{max}value is the average of the two measurements. Figure 6 shows two micrographs of the ablation profiles obtained for different superpositions, in which the measurements were taken. The software Mathematica was used to calculate all the superposition values (

*N*) using the Jacobi theta function, Eq. (11). The ablation threshold results obtained by the D-Scan method for each energy are shown in Fig. 5 as squares. In this Fig. is possible to see that the ablation threshold values obtained by the traditional and by the D-Scan methods show a good agreement, all dropping by factor close to 7 when going from a single pulse to the superposition of more than 1500 pulses. This graph demonstrates that the superposition defined in this work is equivalent to the traditional one in which the pulses overlap completely.

*F*

_{th}_{,1}, are in agreement for all measurements, being consistent with the ones reported in the literature [28

28. A. Ben-Yakar and R. L. Byer, “Femtosecond laser ablation properties of borosilicate glass,” J. Appl. Phys. **96**(9), 5316 (2004). [CrossRef]

29. N. Sanner, O. Utéza, B. Bussiere, G. Coustillier, A. Leray, T. Itina, and M. Sentis, “Measurement of femtosecond laser-induced damage and ablation thresholds in dielectrics,” Appl. Phys., A Mater. Sci. Process. **94**(4), 889–897 (2009). [CrossRef]

_{max}/ρ

_{max}) is small. Computing the average weighed by the squared variances (ponderate mean) for each parameter obtained by the D-Scan method gives

*F*

_{th}_{,1}= (3.29 ± 0.16) J/cm

^{2}and S = (0.76 ± 0.01). These values are in excellent agreement with the ones obtained by the traditional method, validating the D-Scan as a method to measure incubation effects. Also, they demonstrate that incubation effects play an important role in the ablation of BK7 by ultrashort pulses, making the material easier to be processed as the pulses superposition increases.

## 4. Conclusions

_{max}location, allowing the quantification of incubation effects. To that purpose, we defined a mathematical superposition for the D-Scan, and found an analytical expression for it. We applied the method to a common optical material, and demonstrated that the values obtained are in agreement with the ones given by the traditional method and by the literature, for more than three orders of magnitude. This convergence also confirmed our hypotheses that the superposition

*N*at a certain position is given by the sum of the intensity of many pulses displaced from this locus, and it is equivalent to the one of the traditional method. Moreover, the expression found is general and allows fractionary superposition of pulses, indicating that incubation effects are a linear sum of the contribution (intensity) of many pulses that are not required to be in the immediate vicinity of the ablated region.

**5**(3), 648–659 (1988). [CrossRef]

15. D. Ashkenasi, M. Lorenz, R. Stoian, and A. Rosenfeld, “Surface damage threshold and structuring of dielectrics using femtosecond laser pulses: the role of incubation,” Appl. Surf. Sci. **150**(1-4), 101–106 (1999). [CrossRef]

## Acknowledgments

## References and links

1. | N. Bloembergen, “Laser-induced electric breakdown in solids,” IEEE J. Quantum Electron. |

2. | D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in Si0 |

3. | J. Reif and F. Costache, “Femtosecond laser interaction with solid surfaces: explosive ablation and self-assembly of ordered nanostructures,” in |

4. | R. Stoian, A. Rosenfeld, D. Ashkenasi, I. V. Hertel, N. M. Bulgakova, and E. E. B. Campbell, “Surface charging and impulsive ion ejection during ultrashort pulsed laser ablation,” Phys. Rev. Lett. |

5. | M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. |

6. | W. Kautek, J. Kruger, M. Lenzner, S. Sartania, C. Spielmann, and F. Krausz, “Laser ablation of dielectrics with pulse durations between 20 fs and 3 ps,” Appl. Phys. Lett. |

7. | L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys, JETP-USSR |

8. | M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. |

9. | A. P. Joglekar, H. Liu, G. J. Spooner, E. Meyhofer, G. Mourou, and A. J. Hunt, “A study of the deterministic character of optical damage by femtosecond laser pulses and applications to nanomachining,” Appl. Phys. B |

10. | S. Nolte, C. Momma, H. Jacobs, A. Tunnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B |

11. | E. G. Gamaly, A. V. Rode, B. Luther-Davies, and V. T. Tikhonchuk, “Ablation of solids by femtosecond lasers: ablation mechanism and ablation thresholds for metals and dielectrics,” Phys. Plasmas |

12. | F. Costache, S. Eckert, and J. Reif, “Near-damage threshold femtosecond laser irradiation of dielectric surfaces: desorbed ion kinetics and defect dynamics,” Appl. Phys., A Mater. Sci. Process. |

13. | M. Mero, B. Clapp, J. C. Jasapara, W. Rudolph, D. Ristau, K. Starke, J. Kruger, S. Martin, and W. Kautek, “On the damage behavior of dielectric films when illuminated with multiple femtosecond laser pulses,” Opt. Eng. |

14. | L. C. Courrol, R. E. Samad, L. Gomez, I. M. Ranieri, S. L. Baldochi, A. Zanardi de Freitas, and N. D. Vieira, “Color center production by femtosecond pulse laser irradiation in LiF crystals,” Opt. Express |

15. | D. Ashkenasi, M. Lorenz, R. Stoian, and A. Rosenfeld, “Surface damage threshold and structuring of dielectrics using femtosecond laser pulses: the role of incubation,” Appl. Surf. Sci. |

16. | S. Martin, A. Hertwig, M. Lenzner, J. Kruger, and W. Kautek, “Spot-size dependence of the ablation threshold in dielectrics for femtosecond laser pulses,” Appl. Phys., A Mater. Sci. Process. |

17. | Y. Jee, M. F. Becker, and R. M. Walser, “Laser-induced damage on single-crystal metal surfaces,” J. Opt. Soc. Am. B |

18. | Y. C. Lim, P. E. Boukany, D. F. Farson, and L. J. Lee, “Direct-write femtosecond laser ablation and DNA combing and imprinting for fabrication of a micro/nanofluidic device on an ethylene glycol dimethacrylate polymer,” J. Micromech. Microeng. |

19. | D. Gomez and I. Goenaga, “On the incubation effect on two thermoplastics when irradiated with ultrashort laser pulses: Broadening effects when machining microchannels,” Appl. Surf. Sci. |

20. | H. W. Choi, D. F. Farson, J. Bovatsek, A. Arai, and D. Ashkenasi, “Direct-write patterning of indium-tin-oxide film by high pulse repetition frequency femtosecond laser ablation,” Appl. Opt. |

21. | J. Bonse, J. M. Wrobel, J. Krüger, and W. Kautek, “Ultrashort-pulse laser ablation of indium phosphide in air,” Appl. Phys., A Mater. Sci. Process. |

22. | J. M. Liu, “Simple technique for measurements of pulsed Gaussian-beam spot sizes,” Opt. Lett. |

23. | R. E. Samad and N. D. Vieira Jr., “Geometrical method for determining the surface damage threshold for femtosecond laser pulses,” Laser Phys. |

24. | R. E. Samad, S. L. Baldochi, and N. D. Vieira Jr., “Diagonal scan measurement of Cr:LiSAF 20 ps ablation threshold,” Appl. Opt. |

25. | H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. |

26. | M. Abramowitz and I. A. Stegun, |

27. | Wolfram Research Inc, “Jacobi theta function ϑ |

28. | A. Ben-Yakar and R. L. Byer, “Femtosecond laser ablation properties of borosilicate glass,” J. Appl. Phys. |

29. | N. Sanner, O. Utéza, B. Bussiere, G. Coustillier, A. Leray, T. Itina, and M. Sentis, “Measurement of femtosecond laser-induced damage and ablation thresholds in dielectrics,” Appl. Phys., A Mater. Sci. Process. |

**OCIS Codes**

(160.2750) Materials : Glass and other amorphous materials

(160.4670) Materials : Optical materials

(220.4610) Optical design and fabrication : Optical fabrication

(320.2250) Ultrafast optics : Femtosecond phenomena

(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: November 22, 2011

Revised Manuscript: January 27, 2012

Manuscript Accepted: January 31, 2012

Published: February 3, 2012

**Virtual Issues**

Vol. 7, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Leandro Matiolli Machado, Ricardo Elgul Samad, Wagner de Rossi, and Nilson Dias Vieira Junior, "D-Scan measurement of ablation threshold incubation effects for ultrashort laser pulses," Opt. Express **20**, 4114-4123 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4114

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### References

- N. Bloembergen, “Laser-induced electric breakdown in solids,” IEEE J. Quantum Electron.10(3), 375–386 (1974). [CrossRef]
- D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in Si02 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett.64(23), 3071–3073 (1994). [CrossRef]
- J. Reif and F. Costache, “Femtosecond laser interaction with solid surfaces: explosive ablation and self-assembly of ordered nanostructures,” in Advances in Atomic Molecular, and Optical Physics, V. 53, G. Rempe and M. O. Scully, eds. (Elsevier Academic Press Inc, 2006), pp. 227–251.
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