## Morphology of femtosecond laser modification of bulk dielectrics |

Optics Express, Vol. 19, Issue 1, pp. 271-282 (2011)

http://dx.doi.org/10.1364/OE.19.000271

Acrobat PDF (993 KB)

### Abstract

Using 3D Finite-Difference-Time-Domain simulations, we study the morphology of the laser-created damage inside fused silica. Among the competing effects limiting the intensity in the dielectric, we find the most important is the pulse defocusing by the plasma lens, partially balanced by the Kerr effect. Less important are collisional energy dissipation and laser depletion by multi-photon absorption. We also found that the profile of generated plasma is asymmetrical in the transverse cross-section, with the plasma extended along the direction perpendicular to the laser polarization.

© 2011 Optical Society of America

## 1. Introduction

1. D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, “Infrared photosensitivity in silica glasses exposed to femtosecond laser pulses,” Opt. Lett. **24**, 1311–1313 (1999). [CrossRef]

2. V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically produced arrays of planar nanostructures inside fused silica,” Phys. Rev. Lett. **96**, 057404 (2006). [CrossRef] [PubMed]

3. E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. **71**, 882–884 (1997). [CrossRef]

4. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics **2**, 219–225 (2008). [CrossRef]

5. G. D. Valle, R. Osellame, and P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt. A: Pure Appl. Opt. **11**, 013001 (2009). [CrossRef]

13. L. Hallo, A. Bourgeade, V. T. Tikhonchuk, C. Mezel, and J. Breil, “Model and numerical simulations of the propagation and absorption of a short laser pulse in a transparent dielectric material: blast-wave launch and cavity formation,” Phys. Rev. B **76**, 024101 (2007). [CrossRef]

11. I. M. Burakov, N. M. Bulgakova, R. Stoian, A. Mermillod-Blondin, E. Audouard, A. Rosenfeld, A. Husakou, and I. V. Hertel, “Spatial distribution of refractive index variations induced in bulk fused silica by single ultrashort and short laser pulses,” J. App. Phys. **101**, 043506 (2007). [CrossRef]

8. D. M. Rayner, A. Naumov, and P. B. Corkum, “Ultrashort pulse non-linear optical absorption in transparent media,” Opt. Express **13**, 3208–3217 (2005). [CrossRef] [PubMed]

14. D. Grojo, M. Gertsvolf, H. Jean-Ruel, S. Lei, L. Ramunno, D. M. Rayner, and P. B. Corkum, “Self-controlled formation of microlenses by optical breakdown inside wide-band-gap materials,” App. Phys. Lett. **93**, 243118 (2008). [CrossRef]

*i.e.*, it does not possess limitations of the uni-directional nonlinear models presented in the majority of the previous work. In this study we will concentrate on a relatively mild focusing with a numerical aperture (NA) equal to 0.65, as in Ref. [14

14. D. Grojo, M. Gertsvolf, H. Jean-Ruel, S. Lei, L. Ramunno, D. M. Rayner, and P. B. Corkum, “Self-controlled formation of microlenses by optical breakdown inside wide-band-gap materials,” App. Phys. Lett. **93**, 243118 (2008). [CrossRef]

## 2. Model

### 2.1. Basic equations

*E⃗*and

*B⃗*are the components of electromagnetic field,

*D⃗*electric field displacement vector,

*H⃗*the magnetic field auxiliary vector,

*t*is the time, and

*c*is the speed of light. We extend the Yee discretization algorithm [16] by introducing the consitutive relations, and the current density In (2),

*χ*is the linear dispersionless susceptibility of the material and

_{l}*χ*is the Kerr susceptibility. The electromagnetic response of generated plasma is represented by the current density

_{k}*J⃗*whereas laser depletion due to multi-photon absorption is accounted for by the quantity

_{p}*J⃗*.

_{MPA}*J⃗*is calculated from the fluid equations for the electron component of the generated plasma: where

_{p}*n*and

*u⃗*are electron particle density and fluid velocity, respectively, ∇

*p*is the pressure gradient, and the generation rate of free electrons via six-photon absorption is given by [7

7. J. R. Peñano, P. Sprangle, B. Hafizi, W. Manheimer, and A. Zigler, “Transmission of intense femtosecond laser pulses into dielectrics,” Phys. Rev. E **72**, 036412 (2005). [CrossRef]

*I*is laser intensity,

*σ*= 2 × 10

^{13}cm

^{−3}psec

^{−1}(cm

^{2}/TW)

^{6}is the six-photon absorption cross-section [7

7. J. R. Peñano, P. Sprangle, B. Hafizi, W. Manheimer, and A. Zigler, “Transmission of intense femtosecond laser pulses into dielectrics,” Phys. Rev. E **72**, 036412 (2005). [CrossRef]

*n*is the saturation particle density. The quantity

_{s}*n*can be estimated as the electron density when every molecule of silica is singly-ionized:

_{s}*n*=

_{s}*ρ*/

_{m}N_{A}*M*, where

_{μ}*ρ*is the mass density of fused silica,

_{m}*M*its molar mass and

_{μ}*N*Avogadro number. Substitution of the appropriate constants into this expression results in

_{A}*n*≈ 10

_{s}*n*, where

_{cr}*n*is the critical particle density for electrons,

_{cr}*i.e.*, the density for which the electron plasma frequency equals the laser frequency;

*n*= 1.75 × 10

_{cr}^{21}cm

^{−3}for the wavelength considered. In the actual calculations, we approximate the laser intensity by the square of instantaneous electric field: Eq. (5) assumes that the ionization occurs on a time scale slower than a wave period. The approximation given by Eq. (6) introduces an inaccuracy of order

*is the electron collision frequency. Neglecting collisions with neutrals, this frequency is [18] where*

_{e}*n*is the electron particle density, ln Λ ≈ 10, and

_{e}*T*electron temperature. This temperature is approximately equal to the energy of electron quiver motion,

_{e}*E*is the electric field in the laser pulse,

*ω*

_{0}is laser frequency and

*e*,

*m*are electron charge and mass, respectively. The laser intensity in the dielectric is limited by the ionization threshold, equal to (1 ÷ 1.5) × 10

^{13}W/cm

^{2}[14

14. D. Grojo, M. Gertsvolf, H. Jean-Ruel, S. Lei, L. Ramunno, D. M. Rayner, and P. B. Corkum, “Self-controlled formation of microlenses by optical breakdown inside wide-band-gap materials,” App. Phys. Lett. **93**, 243118 (2008). [CrossRef]

*v*≈ 3 × 10

_{q}^{−3}

*c*, where

*c*is the speed of light in vacuum. This velocity corresponds to an electron kinetic energy in eV range. Substituting this value into Eq. (8), one can obtain a pessimistic estimation for the damping factor A more complete consideration with a temperature-dependent damping factor will be developed and published elsewhere.

*p*= 0. We also assume that the plasma fluid velocity

*u⃗*in the collision-dominated regime is small and thus (

*u⃗*∇)

*u⃗*≈ 0. Finally, we assume that the dominant mechanism causing changes to the particle density of electrons is ionization rather than the hydrodynamic advance of the plasma,

*i.e.*, ∇(

*nu⃗*) ≪

*∂n*/

_{MPI}*∂t*. Under these approximations, Eq. (4) simplifies to The last equation in (11) can be also identified with equation of motion for free particles in the Drude model for conductive media [19]. One may calculate the current of free plasma electrons

*J⃗*is obtained by setting the laser energy density depletion rate equal to the product (

_{MPA}*J⃗*·

_{MPA}*E⃗*) [7

7. J. R. Peñano, P. Sprangle, B. Hafizi, W. Manheimer, and A. Zigler, “Transmission of intense femtosecond laser pulses into dielectrics,” Phys. Rev. E **72**, 036412 (2005). [CrossRef]

*J⃗*is parallel to

_{MPA}*E⃗*one obtains where

*W*is the ionization energy, equal to 9 eV for the case of fused silica.

_{ion}13. L. Hallo, A. Bourgeade, V. T. Tikhonchuk, C. Mezel, and J. Breil, “Model and numerical simulations of the propagation and absorption of a short laser pulse in a transparent dielectric material: blast-wave launch and cavity formation,” Phys. Rev. B **76**, 024101 (2007). [CrossRef]

### 2.2. Model benchmark

*i*the imaginary unit. At the entrance of the plasma, the wave envelope is assumed to be Gaussian with the width Δ

*T*: After propagation a distance

*x*, the envelope becomes where ℱ

^{−1}is the inverse Fourier transformation and

*k*(

*ω*) is evaluated from Eq. (14).

*J*≡ 0,

_{p}*χ*= 0, and propagate a plane electromagnetic wave through a 40-micron-thick slab of fused silica. Figure 1a shows the absolute value of Poynting vector of the pulse just before it enters the slab (at

_{k}*t*= 0) and right after it exits the slab (

*t*= 350 fs). The transmitted pulse is cut at intensity ∼ 10

^{13}W/cm

^{2}. The dependence of the peak transmitted intensity on peak incident pulse intensity is given in Fig. 1b. It is seen that as soon as the incident laser exceeds the threshold intensity ∼ 10

^{13}W/cm

^{2}, the pulses become depleted via ionization, and the transmitted intensity is capped by the threshold value. Due to the reflections from the vacuum-dielectric and dielectric-vacuum interfaces, the actual intensity threshold inside the dielectric should be slightly larger. The found value is consistent with the experimental measurements [8

8. D. M. Rayner, A. Naumov, and P. B. Corkum, “Ultrashort pulse non-linear optical absorption in transparent media,” Opt. Express **13**, 3208–3217 (2005). [CrossRef] [PubMed]

*+ ℰ*

_{em}*is conserved throughout the simulation with*

_{abs}*J*≡ 0, where ℰ

_{p}*is the total electromagnetic energy in the domain and ℰ*

_{em}*the absorbed energy.*

_{abs}### 2.3. Laser source excitation

**?**]. This method generally requires an exact knowledge of the electromagnetic field being solution to the Maxwell equations at the boundary. To evaluate this field we use the model of the focused laser pulse, previously employed in [21

21. K. I. Popov, V. Yu. Bychenkov, W. Rozmus, R. D. Sydora, and S. S. Bulanov, “Vacuum electron acceleration by tightly focused laser pulses with nanoscale targets,” Phys. Plasmas **16**, 053106 (2009). [CrossRef]

22. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. **56**, 99–107 (1939). [CrossRef]

*A*is the mirror surface,

*k*the length of the wave vector,

*n⃗*the inner normal to the integration surface and with

*u*being the distance between the observer and the point at the surface:

*u*= |

*r⃗*–

_{p}*r⃗*|. The integrals in (17) are evaluated numerically. When integration is performed over an unclosed surface, as the surface of the mirror is, they approach a solution to Maxwell equations with the given boundary conditions asymptotically, as

_{s}*R*≫

_{m}*λ*with NA kept constant, where

*R*is the mirror radius and

_{m}*λ*the laser wavelength.

*R*, with

_{g}*R*= 0.5

_{g}*R*. The actual mirror radius used in the simulations was equal to 1 mm. The resulting focused pulse had full width at intensity half-maximum (in silica, in the absence of nonlinear effects) equal to 1

_{m}*μ*m in the transverse direction and 7.5

*μ*m in the longitudinal direction.

### 3. Morphology of the generated plasma

*λ*= 800 nm, peak laser intensity at the focus (in the absence of nonlinear effects and dispersion)

*I*= 10

^{14}W/cm

^{2}and the laser pulse length (defined as the full width at the half-maximum within a Gaussian envelope) equal to 50 fs. The corresponding laser pulse energy is 4 × 10

^{−7}J. In all the examples shown below we assume the medium possesses a linear refraction index

*n*

_{0}≈ 1.45. The Kerr susceptibility was assumed to be

*χ*

_{3}= 1.9 × 10

^{−15}esu.

*μ*m × 32

*μ*m × 32

*μ*m simulation domain, with 45

^{3}grid points per

*μ*m

^{3}. These simulations were performed on a 300-cpu computer cluster, with a single simulation requiring 650 Gb of RAM and approximately 24 hours of runtime. For all the results shown below, the laser pulse is propagating from left to right, with the laser axis passing trough the center of the simulation domain.

*M*is the ion mass, is thus on the order

*c*∼ 10

_{s}^{3}cm/s. The sound velocity defines the scale of hydrodynamic expansion of plasma. In this way, if recombination and subsequent solidification, that are not accounted for by our model, happen on the ∼ns or a shorter time scale, the pattern of plasma, generated on the fs-scale, coincides with the shape of the laser-inflicted damage.

*x̂*direction. The profile is characterized by a narrowing shape in the direction of laser propagation (cf. [6], [14

**93**, 243118 (2008). [CrossRef]

*J⃗*= 0,

_{MPA}*i.e.*, with the effect of multi-photon absorption turned off, which is equivalent to assuming an infinite supply of photons. The resulting plasma profile is shown in Fig. 3b. It is seen that in the absence of multi-photon absorption, the maximum density of the generated plasma increases by ∼ 50%. However, the shape itself is essentially unchanged. Further, contrary to our expectations, no catastrophic ionization occurs. On the other hand, if all the nonlinear effects except for multi-photon absorption are switched off, the maximum density of plasma approaches saturation density

*n*= 10

_{s}*n*(Fig. 3c). The shape of the damaged region also changes considerably. We conclude that although multi-photon absorption is an important effect, it is not the main limiting mechanism of the laser intensity inside the plasma, at least in our considered parameter regime.

_{cr}*i.e.*, for collisionless plasma. It will be shown below that collisional energy dissipation is responsible for a large part of absorbed energy by the medium. However, as follows from Fig. 3a,d, this energy dissipation has a little effect on the maximum plasma density, though it does increase the total number of electrons.

*S⃗*|, and the right column the plasma density at

*t*= 300 fs (top),

*t*=320 fs (middle), and

*t*=340 fs, where

*t*is the time after the start of the simulation. As the laser focuses, its intensity grows to an above-threshold value already at the pulse leading edge, corresponding to |

*S⃗*|

*≈ (2 ÷3) × 10*

_{max}^{13}W/cm

^{2}(

*t*= 300 fs in Fig. 4). As a result, plasma is generated. As the plasma density grows, the laser field appears to be expelled from the damaged region, and thus the intensity profile forms a ring-like structure in the transverse cross-section (

*t*= 320, 340 fs). Due to this modification the laser intensity decreases to approximately the threshold value and as a result the ionization process stops. The ring-like intensity merges back to a spot behind the generated plasma (

*t*= 340 fs), but intensity there is still below the ionization threshold, and almost no new plasma is generated.

*, total ionization-absorbed energy ℰ*

_{em}*, instantaneous kinetic energy of all the free particles in the domain ℰ*

_{abs}*and their sum ℰ*

_{k}*+ ℰ*

_{em}*+ ℰ*

_{abs}*.*

_{k}*t*= 0, there is no electromagnetic field in the domain, the total energy is zero. At

*t*> 0 the pulse starts entering the domain from its left boundary. By

*t*≈ 260 fs, most of the pulse is inside the domain. At

*t*≈ 290 fs, the ionization process starts. Eventually the total energy starts to decrease, indicating conversion of the particle kinetic energy into heat. At the end of simulation (

*t*= 425 fs), the total electromagnetic energy in the domain has decreased approximately two times, with ∼ 1/3 of this change caused by the multi-photon absorption and ∼ 2/3 of the change by the thermal energy dissipation.

*t*= 320 and 340 fs. The rest of the defocused pulse does not contribute to additional plasma generation. A portion of the first half of the pulse that has created the plasma and does not experience defocusing continues propagating undisturbed in the forward direction. This results in an asymmetric plasma profile, elongated and sharpened in the forward direction, as shown in Figs. 3a and 4 at

*t*= 340 fs.

## 4. Laser polarization effect

23. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. **179**, 1–7 (2000). [CrossRef]

24. K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas **15**, 013108 (2008). [CrossRef]

*r*, having its axis along

*x̂*, located in space filled with another dielectric. Let the cylinder be composed of dielectric with permittivity

*ɛ*

_{2}, and the background dielectric have permittivity

*ɛ*

_{1}. Let also the field far from the cylinder be parallel to

*ŷ*and the origin be located at a point at the cylinder axis. Then it can be shown that The quantity

*E*

_{1}is the magnitude of local field at the cylinder boundary in YZ cross-section at

*y*= ±

*r*,

*z*= 0, and is perpendicular to the boundary.

*E*

_{2}is the local field magnitude at the boundary points

*y*= 0,

*z*= ±

*r*, and is parallel to the boundary. As can be seen from Eq. (20), if

*ɛ*

_{2}<

*ɛ*

_{1}, as is the case for a plasma in a dielectric background,

*E*

_{1}<

*E*

_{2}. Thus there is a field enhancement along direction perpendicular to the external field. This is what we observe in our simulations. Although the interaction problem is electromagnetic rather than electrostatic, this reasoning should be approximately valid for a cylinder radius smaller than the laser wavelength.

*i.e.*,

*ẑ*for a laser field polarized along

*ŷ*). This field enhancement produces more plasma in the

*ẑ*direction. This process results in the asymmetry of the generated plasma pattern shown in Fig. 7a.

## 5. Conclusion

## 6. Acknowledgments

## References and links

1. | D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, “Infrared photosensitivity in silica glasses exposed to femtosecond laser pulses,” Opt. Lett. |

2. | V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically produced arrays of planar nanostructures inside fused silica,” Phys. Rev. Lett. |

3. | E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. |

4. | R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics |

5. | G. D. Valle, R. Osellame, and P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt. A: Pure Appl. Opt. |

6. | L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. |

7. | J. R. Peñano, P. Sprangle, B. Hafizi, W. Manheimer, and A. Zigler, “Transmission of intense femtosecond laser pulses into dielectrics,” Phys. Rev. E |

8. | D. M. Rayner, A. Naumov, and P. B. Corkum, “Ultrashort pulse non-linear optical absorption in transparent media,” Opt. Express |

9. | A. Q. Wu, I. H. Chowdhury, and X. Xu, “Femtosecond laser absorption in fused silica: numerical and experimental investigation,” Phys. Rev. B |

10. | C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells,” Opt. Express |

11. | I. M. Burakov, N. M. Bulgakova, R. Stoian, A. Mermillod-Blondin, E. Audouard, A. Rosenfeld, A. Husakou, and I. V. Hertel, “Spatial distribution of refractive index variations induced in bulk fused silica by single ultrashort and short laser pulses,” J. App. Phys. |

12. | P. P. Rajeev, M. Gertsvolf, C. Hnatovsky, E. Simova, R. S. Taylor, P. B. Corkum, D. M. Rayner, and V. R. Bhardwaj, “Transient nanoplasmonics inside dielectrics,” J. Phys. B |

13. | L. Hallo, A. Bourgeade, V. T. Tikhonchuk, C. Mezel, and J. Breil, “Model and numerical simulations of the propagation and absorption of a short laser pulse in a transparent dielectric material: blast-wave launch and cavity formation,” Phys. Rev. B |

14. | D. Grojo, M. Gertsvolf, H. Jean-Ruel, S. Lei, L. Ramunno, D. M. Rayner, and P. B. Corkum, “Self-controlled formation of microlenses by optical breakdown inside wide-band-gap materials,” App. Phys. Lett. |

15. | A. Taflove and S. C. Hagness, |

16. | K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

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

18. | NRL Plasma Formulary (2002), p. 28. |

19. | C. A. Brau, |

20. | A. Taflove and S. C. Hagness, |

21. | K. I. Popov, V. Yu. Bychenkov, W. Rozmus, R. D. Sydora, and S. S. Bulanov, “Vacuum electron acceleration by tightly focused laser pulses with nanoscale targets,” Phys. Plasmas |

22. | J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. |

23. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. |

24. | K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas |

**OCIS Codes**

(140.3330) Lasers and laser optics : Laser damage

(260.3230) Physical optics : Ionization

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: November 2, 2010

Revised Manuscript: December 8, 2010

Manuscript Accepted: December 13, 2010

Published: December 22, 2010

**Citation**

K. I. Popov, C. McElcheran, K. Briggs, S. Mack, and Lora Ramunno, "Morphology of femtosecond laser modification of bulk dielectrics," Opt. Express **19**, 271-282 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-271

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### References

- D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, “Infrared photosensitivity in silica glasses exposed to femtosecond laser pulses,” Opt. Lett. 24, 1311–1313 (1999). [CrossRef]
- V. R. Bhardwaj, E. Simova, P. P. Rajeev, C. Hnatovsky, R. S. Taylor, D. M. Rayner, and P. B. Corkum, “Optically produced arrays of planar nanostructures inside fused silica,” Phys. Rev. Lett. 96, 057404 (2006). [CrossRef] [PubMed]
- E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. 71, 882–884 (1997). [CrossRef]
- R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219–225 (2008). [CrossRef]
- G. D. Valle, R. Osellame, and P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt. A, Pure Appl. Opt. 11, 013001 (2009). [CrossRef]
- L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
- J. R. Peñano, P. Sprangle, B. Hafizi, W. Manheimer, and A. Zigler, “Transmission of intense femtosecond laser pulses into dielectrics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 036412 (2005). [CrossRef]
- D. M. Rayner, A. Naumov, and P. B. Corkum, “Ultrashort pulse non-linear optical absorption in transparent media,” Opt. Express 13, 3208–3217 (2005). [CrossRef] [PubMed]
- A. Q. Wu, I. H. Chowdhury, and X. Xu, “Femtosecond laser absorption in fused silica: numerical and experimental investigation,” Phys. Rev. B 72, 085128 (2005). [CrossRef]
- C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells,” Opt. Express 15, 10303–10317 (2007). [CrossRef] [PubMed]
- I. M. Burakov, N. M. Bulgakova, R. Stoian, A. Mermillod-Blondin, E. Audouard, A. Rosenfeld, A. Husakou, and I. V. Hertel, “Spatial distribution of refractive index variations induced in bulk fused silica by single ultrashort and short laser pulses,” J. Appl. Phys. 101, 043506 (2007). [CrossRef]
- P. P. Rajeev, M. Gertsvolf, C. Hnatovsky, E. Simova, R. S. Taylor, P. B. Corkum, D. M. Rayner, and V. R. Bhardwaj, “Transient nanoplasmonics inside dielectrics,” J. Phys. B 40, S273–S282 (2007). [CrossRef]
- L. Hallo, A. Bourgeade, V. T. Tikhonchuk, C. Mezel, and J. Breil, “Model and numerical simulations of the propagation and absorption of a short laser pulse in a transparent dielectric material: blast-wave launch and cavity formation,” Phys. Rev. B 76, 024101 (2007). [CrossRef]
- D. Grojo, M. Gertsvolf, H. Jean-Ruel, S. Lei, L. Ramunno, D. M. Rayner, and P. B. Corkum, “Self-controlled formation of microlenses by optical breakdown inside wide-band-gap materials,” Appl. Phys. Lett. 93, 243118 (2008). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics, 3rd. ed. (Artech House, 2005), pp. 58–79.
- K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. AP-14, 302–307 (1966).
- L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).
- N. R. L. Plasma Formulary, (2002), p. 28.
- C. A. Brau, Modern Problems in Classical Electrodynamics (Oxford Univ. Press, 2004), pp. 342–347.
- A. Taflove and S. C. Hagness, Computational Electrodynamics, 3rd. ed. (Artech House, 2005), pp. 186–213.
- K. I. Popov, V. Yu. Bychenkov, W. Rozmus, R. D. Sydora, and S. S. Bulanov, “Vacuum electron acceleration by tightly focused laser pulses with nanoscale targets,” Phys. Plasmas 16, 053106 (2009). [CrossRef]
- J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939). [CrossRef]
- S. Quabis, R. Dorn, M. Eberler, O. Gl¨ockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]
- K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15, 013108 (2008). [CrossRef]

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