## Analysis of the effects of spherical aberration on ultrafast laser-induced refractive index variation in glass

Optics Express, Vol. 15, Issue 19, pp. 12395-12408 (2007)

http://dx.doi.org/10.1364/OE.15.012395

Acrobat PDF (1844 KB)

### Abstract

We propose a comprehensive analysis of the effects that spherical aberration may have on the process of ultrafast laser photowriting in bulk transparent materials and discuss the consequences for the generated refractive index changes. Practical aspects for a longitudinal photowriting configuration are emphasized. Laser-induced index variation in BK7 optical glass and fused silica (a-SiO2) affected by spherical aberration are characterized experimentally using phase-contrast optical microscopy. Experimental data are matched by analytical equations describing light propagation through dielectric interfaces. Corrective solutions are proposed with a particular focus on the spatial resolution achievable and on the conditions to obtain homogeneously photo-induced waveguides in a longitudinal writing configuration.

© 2007 Optical Society of America

## 1. Introduction

2. S. Valette, R. Le Harzic, N. Huot, E. Audouard, and R. Fortunier, “2-D calculations of the thermal effects due to femtosecond laser-metal interaction,” Appl. Surf. Sci. **247**, 238–242 (2005). [CrossRef]

3. 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]

7. S. Juodkasis, S. Matsuo, H. Misawa, V. Mizeikis, A. Marcinkevicius, H. B. Sun, Y. Tokuda, M. Takahashi, T. Yoko, and J. Nishii, “Application of femtosecond laser pulses for microfabrication of transparent media,” Appl. Surf. Sci. **197**, 705–709 (2002). [CrossRef]

8. K. Minoshima, A.M. Kowalevicz, I. Hartl, E.P. Ippen, and J.G. Fujimoto, “Photonic device fabrication in glass by use of nonlinear materials processing with a femtosecond laser oscillator,” Opt Lett. **26**, 1516–1518 (2001). [CrossRef]

10. J. P. McDonald, V. R. Mistry, K. E. Ray, and S. M. Yalisove, “Femtosecond pulsed laser direct write production of nano- and microfluidic channels,” Appl. Phys. Lett. **88**, 183113–183115 (2006). [CrossRef]

11. N. Takeshima, Y. Narita, S. Tanaka, Y. Kuroiwa, and K. Hirao, “Fabrication of high-efficiency diffraction gratings in glass,” Opt. Lett. **30**, 352–354 (2005). [CrossRef] [PubMed]

12. H. Zhang, S. M. Eaton, J. Li, A. H. Nejadmalayeri, and P. R. Herman, “Type II high-strength Bragg grating waveguides photowritten with ultrashort laser pulses,” Opt. Express **15**, 4182–4191 (2007). [CrossRef] [PubMed]

16. Q. Sun, H. Jiang, Y. Liu, Y. Zhou, H. Yang, and Q. Gong, “Effect of spherical aberrations on the propagation of a tightly focused femtosecond laser pulse inside fused silica,” Pure Appl. Opt. **7**, 655–659 (2005). [CrossRef]

16. Q. Sun, H. Jiang, Y. Liu, Y. Zhou, H. Yang, and Q. Gong, “Effect of spherical aberrations on the propagation of a tightly focused femtosecond laser pulse inside fused silica,” Pure Appl. Opt. **7**, 655–659 (2005). [CrossRef]

15. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. **98**, 013517 1–5 (2005). [CrossRef]

17. P. Török, P. Vagra, and G. Németh, “Analytical solution of the diffraction integrals and interpretation of wavefront distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. **A 12**, 2660–2671 (1995). [CrossRef]

18. J. S. H. Wiersma, T. D. Visser, and P. Török, “Annular focusing through a dielectric interface: scanning and confining the intensity,” Pure Appl. Opt. **7**, 1237–1248 (1998). [CrossRef]

19. M. J. Booth and T. Wilson, “Refractive-index-mismatch induced aberrations in single-photon and two-phton microscopy and the used of aberration correction,” J. Biomed. Opt. **6**, 266–272 (2001). [CrossRef] [PubMed]

21. M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. **88**, 031109–031111 (2006). [CrossRef]

22. M. A. A. Neil, R. Juskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, “Active aberration correction for the writing of three-dimensional optical memory device,” Appl. Opt. **41**, 1374–1379 (2002). [CrossRef] [PubMed]

3. 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]

23. Z. Wu, H. Jiang, H. Yang, and Q. Gong, “The refocusing behaviour of a focused femtosecond laser pulse in fused silica,” Pure Appl. Opt. **5**, 102–107 (2003). [CrossRef]

## 2. Aberrated optical path difference

19. M. J. Booth and T. Wilson, “Refractive-index-mismatch induced aberrations in single-photon and two-phton microscopy and the used of aberration correction,” J. Biomed. Opt. **6**, 266–272 (2001). [CrossRef] [PubMed]

*A*. Due to refraction at the dielectric interface, the real focal point is shifted and its position depends on the opening angle

*α*: for low values of

^{’}*α*

^{’}, rays converge to the paraxial focus

*A*. For high values of α’, rays converge to the marginal focus

^{’}_{P}*A*. The longitudinal spherical aberration (LSA)

^{’}_{M}*l*, i.e. the algebric distance between the paraxial and the marginal focus, is then easily expressed as:

*A*in a matched medium,

*n*and

_{1}*n*are the refractive indices of the media before and after the interface respectively.

_{2}_{aberr}may be evaluated using the wave aberration in the object and image space respectively Δ

_{1}and Δ

_{2}. Indeed, according to Nijboer formulas in the case of a rotationally invariant problem [24], the wave aberration Δ

_{2}, as defined as the algebric distance along the optical ray in the image space between a reference sphere centered in the paraxial focus

*A*and the actual aberrated wavefront, is given by:

^{’}_{P}*n*.Δι

_{i}_{t}keeps constant along an optical ray [24], and getting rid of constant terms that do not affect the point spread function (PSF), one can deduce Δ

_{aberr}as follows:

_{aberr}is negative for

*n*<

_{1}*n*, as it should be the case in an overcorrected optical system. Equation (3) cannot be directly compared to expressions obtained in [19

_{2}19. M. J. Booth and T. Wilson, “Refractive-index-mismatch induced aberrations in single-photon and two-phton microscopy and the used of aberration correction,” J. Biomed. Opt. **6**, 266–272 (2001). [CrossRef] [PubMed]

20. M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched-media,” J. Microsc. **192**, 90–98 (1998). [CrossRef]

_{aberr}is conducted in the object space, and the reference sphere is centered in

*A*. By applying Gouy theorem from object to image spaces, and operating a change in the center of the reference sphere from

*A*to

*A*, the two expressions become identical, except the sign.

^{’}_{P}_{2}deduced from Eq. (2) may be compared to the Seidel expansion of spherical aberration, widely used in optical system design. Indeed, the lowest-order spherical aberration term in Seidel expansion is expressed as Δ

_{2}=-

*a α*

^{′}^{4}/4, where

*a*is the Seidel spherical aberration coefficient. A development of Eq. (2) to fourth order in α’ leads to:

*a*direct calculation of a according to Seidel theory, only valid for NA<0.2 [25].

*R*) with no azimuthal dependence [25]:

^{0}_{n}(ρ*R*(

^{0}_{n}*ρ*) may be expressed in terms of tabulated Legendre polynomials of the first kind [17

17. P. Török, P. Vagra, and G. Németh, “Analytical solution of the diffraction integrals and interpretation of wavefront distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. **A 12**, 2660–2671 (1995). [CrossRef]

*Z*(

^{0}_{n}*ρ*) and

*A*respectively the Zernike polynomial and coefficient defined in [19

^{’}_{n0}**6**, 266–272 (2001). [CrossRef] [PubMed]

20. M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched-media,” J. Microsc. **192**, 90–98 (1998). [CrossRef]

20. M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched-media,” J. Microsc. **192**, 90–98 (1998). [CrossRef]

*A*is NA-dependent and may be expressed analytically:

_{n0}*B*given by Eq.(13) of Ref. [20

_{n}(γ)**192**, 90–98 (1998). [CrossRef]

## 3. Characterization of the focal volume

*l*(LSA).

*l*. By developing Eq. (1) at lowest order in NA,

*l*is simplified into

*l*, with

_{Seidel}=a NA^{2}/n^{2}_{2}*a*given by Eq. (4), in agreement with Seidel theory valid for low NA. Another expression for the LSA may be deduced from scalar diffraction theory limited to the on-axis intensity, following the procedure detailed in [19

**6**, 266–272 (2001). [CrossRef] [PubMed]

*u*and

_{1}*u*characterizing the longitudinal extension of spherical aberration are defined such that:

_{2}*I*the on-axis intensity, 0<

*ε*<1 and

*u*and

_{1}*u*may be determined analytically by relating the normalized radius at which the ray passes through the pupil plane to the normalized on-axis position

_{2}*u*at which it crosses the focal area.

*l*, given by [25]:

_{confocal}*l*with NA is added in Fig. 2. For high enough NA (depending on depth), spherical aberration governs.

_{confocal}17. P. Török, P. Vagra, and G. Németh, “Analytical solution of the diffraction integrals and interpretation of wavefront distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. **A 12**, 2660–2671 (1995). [CrossRef]

18. J. S. H. Wiersma, T. D. Visser, and P. Török, “Annular focusing through a dielectric interface: scanning and confining the intensity,” Pure Appl. Opt. **7**, 1237–1248 (1998). [CrossRef]

**A 12**, 2660–2671 (1995). [CrossRef]

18. J. S. H. Wiersma, T. D. Visser, and P. Török, “Annular focusing through a dielectric interface: scanning and confining the intensity,” Pure Appl. Opt. **7**, 1237–1248 (1998). [CrossRef]

*A*given by Eq. (6). By developing Eq. (10) at lowest order in NA,

_{2,0}*dz*is simply expressed as the half of the LSA, in agreement with Seidel theory. Evolution of

*dz*with depth for different NA is plotted in Fig. 3 for fused silica. As expected, for NA>0.4, the difference between half Seidel LSA and Eq. (10) overpasses 11%.

*σ*satisfies

_{Δ}*σ*

_{Δ}≪

*λ/(2π)*, the Strehl ratio R

_{s}may be expressed analytically as :

*A*the Zernike coefficient defined by Eq. (6). In Eq. (11), the involved OPD is Δ

_{2j,0}_{aberr}where the focus term has been corrected. Indeed, this enables to have a straightforward access to the OPD in the plane of the best focus.

## 4. Experiments

23. Z. Wu, H. Jiang, H. Yang, and Q. Gong, “The refocusing behaviour of a focused femtosecond laser pulse in fused silica,” Pure Appl. Opt. **5**, 102–107 (2003). [CrossRef]

_{2}and BK7 by single and multiple pulses of 170 fs at different depths are given in Fig. 4 for a certain range of input energies. Illustrative for the elongation effect of aberrations, the affected regions extend on several tens of µm depending on the focusing depth, and several gray level regions are visible. A black region corresponds to a positive refractive index change, while the white regions are equally attributed to a decrease of the refractive index or to the appearance of a light scattering center. It was shown before that white regions can be associated with a maximum rate of energy deposition and highest temperatures achieved in the material [26

26. I. M. Burakov, N. M. Bulgakova, R. Stoian, A. Mermillod-Blondin, E. Audouard, R. 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 1–7 (2007). [CrossRef]

_{2}are plotted in Figs. 5 and 6.

27. L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. **191**, 333–339 (2001). [CrossRef]

## 5. Correction of spherical aberration

*R*above 0.8 as defined by the Rayleigh criterion [25], for various corrections of spherical aberration, and for various depths. In Eq. (5), the aberrated OPD

_{s}*Δ*is expressed as a function of Zernike polynomials. Supposing that the first N Zernike polynomials are corrected, the remaining OPD after correction is

_{aberr}*Δ*:

_{remain}*A*given by Eq. (6).

_{n0}*A*is also NA-dependent, one obtains the maximum NA that still satisfies

_{n0}*Rs*>0.8. We checked that this result is in good agreement with the direct resolution of Rayleigh criterion, specifying that the maximum peak-valley OPD affected by spherical aberration that ensures

*Rs*>

*0.8*is λ/4 [25].

*ϕ*=1.22

*λ*/(2

*NA*) FWHM for the maximum tolerable NA. The spatial optical resolution is plotted in Fig. (7) for various values of N. As refractive indices of BK7 and fused silica are close, the two curves nearly overlap. Moreover, it appears that N=6, i.e. only 3 Zernike polynomials, is largely enough to correct nearly all spherical aberration for depth up to several millimeters. It is to be noted that Fig. (7) does not describe the minimum size of modified area in the material but the minimum size of the optical spot. Physical phenomena inducing the index variation are not described here.

*R*is simply replaced by

_{s}*R*=

^{′}_{S}*R*with

^{k}_{S}*k*being the number of near-infrared photons necessary to bridge the bang gap of the material. The results, plotted in Fig. (8), confirm that, for both glasses, homogeneous ~cm-long waveguides may be longitudinally photo-induced with a NA=0.45 objective as long as spherical aberration is corrected up to N=6.

30. N. Sanner, N. Huot, E. Audouard, C. Larat, and J. P. Huignard, “Direct ultrafast microstructuring of materials using programmable beam shaping,” Opt. Laser Eng. **45**, 737–741 (2007). [CrossRef]

## 6. Conclusion

## Acknowledgments

## References and links

1. | B. N. Chichkov, C. Momma, S. Nolte, F. von Albenstein, and A. Tünnermann, “Femtosecond, picosecond and nanosecond laser ablation of solids,” Appl. Phys. |

2. | S. Valette, R. Le Harzic, N. Huot, E. Audouard, and R. Fortunier, “2-D calculations of the thermal effects due to femtosecond laser-metal interaction,” Appl. Surf. Sci. |

3. | K Miura, J. Qiu, H. Inouye, T. Mitsuyu, and K. Hirao, “Photowritten optical waveguides in various glasses with ultrashort pulse laser,” Appl. Phys. Lett. |

4. | 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. |

5. | 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. |

6. | A. Mermillod-Blondin, I. M. Burakov, R. Stoian, A. Rosenfeld, E. Audouard, N. M. Bulgakova, and I. V. Hertel, “Direct observation of femtosecond laser induced modifications in the bulk of fused silica by phase contrast microscopy,” J. Laser Micro/Nanoeng. |

7. | S. Juodkasis, S. Matsuo, H. Misawa, V. Mizeikis, A. Marcinkevicius, H. B. Sun, Y. Tokuda, M. Takahashi, T. Yoko, and J. Nishii, “Application of femtosecond laser pulses for microfabrication of transparent media,” Appl. Surf. Sci. |

8. | K. Minoshima, A.M. Kowalevicz, I. Hartl, E.P. Ippen, and J.G. Fujimoto, “Photonic device fabrication in glass by use of nonlinear materials processing with a femtosecond laser oscillator,” Opt Lett. |

9. | M. H. Hong, B. Luk’Yanchuk, S. M. Huang, T. S. Ong, L. H. Van, and T. C. Chong, “Femtosecond laser application for high capacity optical data storage,” Appl. Phys. |

10. | J. P. McDonald, V. R. Mistry, K. E. Ray, and S. M. Yalisove, “Femtosecond pulsed laser direct write production of nano- and microfluidic channels,” Appl. Phys. Lett. |

11. | N. Takeshima, Y. Narita, S. Tanaka, Y. Kuroiwa, and K. Hirao, “Fabrication of high-efficiency diffraction gratings in glass,” Opt. Lett. |

12. | H. Zhang, S. M. Eaton, J. Li, A. H. Nejadmalayeri, and P. R. Herman, “Type II high-strength Bragg grating waveguides photowritten with ultrashort laser pulses,” Opt. Express |

13. | A. Marcinkevicius, V. Mizeikis, S. Juodkasis, S. Matsuo, and H. Misawa, “Effects of refractive index-mismatch on laser microfabrication in silica glass,” Appl. Phys. |

14. | D. Liu, Y. Li, R. An, Y. Dou, H. Yang, and Q. Gong, “Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing,” Appl. Phys. |

15. | C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations,” J. Appl. Phys. |

16. | Q. Sun, H. Jiang, Y. Liu, Y. Zhou, H. Yang, and Q. Gong, “Effect of spherical aberrations on the propagation of a tightly focused femtosecond laser pulse inside fused silica,” Pure Appl. Opt. |

17. | P. Török, P. Vagra, and G. Németh, “Analytical solution of the diffraction integrals and interpretation of wavefront distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. |

18. | J. S. H. Wiersma, T. D. Visser, and P. Török, “Annular focusing through a dielectric interface: scanning and confining the intensity,” Pure Appl. Opt. |

19. | M. J. Booth and T. Wilson, “Refractive-index-mismatch induced aberrations in single-photon and two-phton microscopy and the used of aberration correction,” J. Biomed. Opt. |

20. | M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive-index-mismatched-media,” J. Microsc. |

21. | M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. |

22. | M. A. A. Neil, R. Juskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, “Active aberration correction for the writing of three-dimensional optical memory device,” Appl. Opt. |

23. | Z. Wu, H. Jiang, H. Yang, and Q. Gong, “The refocusing behaviour of a focused femtosecond laser pulse in fused silica,” Pure Appl. Opt. |

24. | A. Maréchal, |

25. | M. Born and E. Wolf, |

26. | I. M. Burakov, N. M. Bulgakova, R. Stoian, A. Mermillod-Blondin, E. Audouard, R. 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. |

27. | L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Study of damage in fused silica induced by ultra-short IR laser pulses,” Opt. Commun. |

28. | N. Sanner, N. Huot, E. Audouard, C. Larat, P. Laporte, and J. P. Huignard, “100 kHz diffraction-limited femtosecond laser machining,” Appl. Phys. |

29. | N. Sanner, N. Huot, E. Audouard, C. Larat, B. Loiseau, and J. P. Huignard, “Programmable spatial beam shaping of a 100 kHz amplified femtosecond laser,” Opt. Lett. |

30. | N. Sanner, N. Huot, E. Audouard, C. Larat, and J. P. Huignard, “Direct ultrafast microstructuring of materials using programmable beam shaping,” Opt. Laser Eng. |

**OCIS Codes**

(140.3390) Lasers and laser optics : Laser materials processing

(140.7090) Lasers and laser optics : Ultrafast lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: April 26, 2007

Revised Manuscript: June 11, 2007

Manuscript Accepted: June 15, 2007

Published: September 14, 2007

**Citation**

N. Huot, R. Stoian, A. Mermillod-Blondin, C. Mauclair, and E. Audouard, "Analysis of the effects of spherical aberration on ultrafast laser-induced refractive index variation in glass," Opt. Express **15**, 12395-12408 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12395

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### References

- B. N. Chichkov, C. Momma, S. Nolte, F. von Albenstein and A. Tünnermann, "Femtosecond, picosecond and nanosecond laser ablation of solids," Appl. Phys. A 63, 109 (1996).
- S. Valette, R. Le Harzic, N. Huot, E. Audouard and R. Fortunier, "2-D calculations of the thermal effects due to femtosecond laser-metal interaction," Appl. Surf. Sci. 247, 238-242 (2005). [CrossRef]
- 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]
- 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]
- 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. 23, 93-95 (2001). [CrossRef]
- A. Mermillod-Blondin, I. M. Burakov, R. Stoian, A. Rosenfeld, E. Audouard, N. M. Bulgakova and I. V. Hertel, "Direct observation of femtosecond laser induced modifications in the bulk of fused silica by phase contrast microscopy," J. Laser Micro/Nanoeng. 1, 155-160 (2006). [CrossRef]
- S. Juodkasis, S. Matsuo, H. Misawa, V. Mizeikis, A. Marcinkevicius, H. B. Sun, Y. Tokuda, M. Takahashi, T. Yoko and J. Nishii, "Application of femtosecond laser pulses for microfabrication of transparent media," Appl. Surf. Sci. 197, 705-709 (2002). [CrossRef]
- K. Minoshima, A.M. Kowalevicz, I. Hartl, E.P. Ippen and J.G. Fujimoto, "Photonic device fabrication in glass by use of nonlinear materials processing with a femtosecond laser oscillator," Opt Lett. 26, 1516-1518 (2001). [CrossRef]
- M. H. Hong, B. Luk’Yanchuk, S. M. Huang, T. S. Ong, L. H. Van, andT. C. Chong, "Femtosecond laser application for high capacity optical data storage," Appl. Phys. A 79, 791-794 (2004).
- J. P. McDonald, V. R. Mistry, K. E. Ray, and S. M. Yalisove, "Femtosecond pulsed laser direct write production of nano- and microfluidic channels," Appl. Phys. Lett. 88, 183113-183115 (2006). [CrossRef]
- N. Takeshima, Y. Narita, S. Tanaka, Y. Kuroiwa and K. Hirao, "Fabrication of high-efficiency diffraction gratings in glass," Opt. Lett. 30, 352-354 (2005). [CrossRef] [PubMed]
- H. Zhang, S. M. Eaton, J. Li, A. H. Nejadmalayeri and P. R. Herman, "Type II high-strength Bragg grating waveguides photowritten with ultrashort laser pulses," Opt. Express 15, 4182-4191 (2007). [CrossRef] [PubMed]
- A. Marcinkevicius, V. Mizeikis, S. Juodkasis, S. Matsuo and H. Misawa, "Effects of refractive index-mismatch on laser microfabrication in silica glass," Appl. Phys. B 76, 257-260 (2003).
- D. Liu, Y. Li, R. An, Y. Dou, H. Yang and Q. Gong, "Influence of focusing depth on the microfabrication of waveguides inside silica glass by femtosecond laser direct writing," Appl. Phys. A 84, 257-260 (2006).
- C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner and P. B. Corkum, "High-resolution study of photoinduced modification in fused silica produced by a tightly focused femtosecond laser beam in the presence of aberrations," J. Appl. Phys. 98, 013517 1-5 (2005). [CrossRef]
- Q. Sun, H. Jiang, Y. Liu, Y. Zhou, H. Yang and Q. Gong, "Effect of spherical aberrations on the propagation of a tightly focused femtosecond laser pulse inside fused silica," Pure Appl. Opt. 7, 655-659 (2005). [CrossRef]
- P. Török, P. Vagra and G. Németh, "Analytical solution of the diffraction integrals and interpretation of wavefront distortion when light is focused through a planar interface between materials of mismatched refractive indices," J. Opt. Soc. Am. A 12, 2660-2671 (1995). [CrossRef]
- J. S. H. Wiersma, T. D. Visser and P. Török, "Annular focusing through a dielectric interface: scanning and confining the intensity," Pure Appl. Opt. 7, 1237-1248 (1998). [CrossRef]
- M. J. Booth and T. Wilson, "Refractive-index-mismatch induced aberrations in single-photon and two-phton microscopy and the used of aberration correction," J. Biomed. Opt. 6, 266-272 (2001). [CrossRef] [PubMed]
- M. J. Booth, M. A. A. Neil and T. Wilson, "Aberration correction for confocal imaging in refractive-index-mismatched-media," J. Microsc. 192, 90-98 (1998). [CrossRef]
- M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi and S. Miyata, "Predictive aberration correction for multilayer optical data storage," Appl. Phys. Lett. 88, 031109-031111 (2006). [CrossRef]
- M. A. A. Neil, R. Juskaitis, M. J. Booth, T. Wilson, T. Tanaka and S. Kawata, "Active aberration correction for the writing of three-dimensional optical memory device," Appl. Opt. 41, 1374-1379 (2002). [CrossRef] [PubMed]
- Z. Wu, H. Jiang, H. Yang and Q. Gong, "The refocusing behaviour of a focused femtosecond laser pulse in fused silica," Pure Appl. Opt. 5, 102-107 (2003). [CrossRef]
- A. Maréchal, Imagerie géométrique, aberrations, (Edition de la revue d’optique théorique et instrumentale, Paris 1952).
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