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Optical Materials Express

Optical Materials Express

  • Editor: David J. Hagan
  • Vol. 1, Iss. 4 — Aug. 1, 2011
  • pp: 670–677
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Influence of bandgap and polarization on photo-ionization: guidelines for ultrafast laser inscription [Invited]

Douglas J. Little, Martin Ams, and Michael J. Withford  »View Author Affiliations


Optical Materials Express, Vol. 1, Issue 4, pp. 670-677 (2011)
http://dx.doi.org/10.1364/OME.1.000670


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Abstract

An ultrafast laser was used to fabricate waveguides in Yb:QX phosphate glass and BK7 borosilicate glass using linearly polarized and circularly polarized beams. Circularly polarized pulses were found to induce a higher refractive index change in Yb:QX phosphate glass, while in BK7 borosilicate glass circularly and linearly polarized pulses were found to induce the same refractive index change. An explanation for these contrasting results is proposed based on the fundamental polarization-dependence of photo-ionization. This explanation reconciles observations made in this study and also in a previous study in fused silica glass.

© 2011 OSA

1. Introduction

Ultrafast lasers have generated substantial interest as an alternative platform for fabricating miniaturized photonic components due to the ability to induce permanent, highly localized refractive index changes inside bulk glass [1

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

]. While the underlying mechanism of this phenomenon is reasonably well understood, there is still debate over the effect of ultrafast-laser polarization. A number of groups have reported the formation of nano-gratings in ultrafast-laser-modified regions in fused silica, where the alignment of the nano-gratings have been shown to be dependent on the polarization of the ultrafast laser beam [2

2. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867–1869 (2005). [CrossRef] [PubMed]

4

4. Y. Bellouard, E. Barthel, A. A. Said, M. Dugan, and P. Bado, “Scanning thermal microscopy and Raman analysis of bulk fused silica exposed to low-energy femtosecond laser pulses,” Opt. Express 16(24), 19520–19534 (2008). [CrossRef] [PubMed]

]. Dynamics of plasma formation and the interference of plasmonic waves and the incoming ultrafast-laser pulse have been cited as explanations for this observed behavior.

Under conditions typical of writing waveguides (astigmatic beam-shaping, ~0.6 NA focusing objective), it has been observed that waveguides written with circularly polarized ultrafast beams can exhibit substantially different refractive changes than waveguides written with linearly polarized ultrafast beams under otherwise identical writing conditions [5

5. M. Ams, G. D. Marshall, and M. J. Withford, “Study of the influence of femtosecond laser polarisation on direct writing of waveguides,” Opt. Express 14(26), 13158–13163 (2006). [CrossRef] [PubMed]

,6

6. D. J. Little, M. Ams, P. Dekker, G. D. Marshall, J. M. Dawes, and M. J. Withford, “Femtosecond laser modification of fused silica: the effect of writing polarization on Si-O ring structure,” Opt. Express 16(24), 20029–20037 (2008). [CrossRef] [PubMed]

]. To date, a satisfactory explanation for this behavior has not yet been proposed. Manipulating the polarization of the ultrafast writing beam has become an increasingly common technique to alter or enhance the way in which ultrafast lasers modify different materials [7

7. S. Gross, M. J. Withford, and A. Fuerbach, “Direct femtosecond laser written waveguides in bulk Ti3+ sapphire,” Proc. SPIE 7589, 75890U (2010). [CrossRef]

9

9. R. S. Taylor, E. Simova, and C. Hnatovsky, “Creation of chiral structures inside fused silica glass,” Opt. Lett. 33(12), 1312–1314 (2008). [CrossRef] [PubMed]

]. Explanations into the underlying cause of this behavior will allow researchers to systematically optimize the polarization for any given application over a range of materials and writing systems.

2. Theory

Ultrafast-laser induced modification of glass is the end result of a cascade of physical processes initiated by nonlinear optical absorption and the generation of free carriers (photoionization) [10

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

]. Higher pulse energies almost always result in a larger refractive index change (whether positive or negative). Larger refractive index changes are therefore indicative of more energy being absorbed by the glass undergoing modification.

It has been observed previously that ultrafast-laser written waveguides exhibit a different refractive index change depending on whether the polarization of the ultrafast laser pulses was linear or circular under otherwise identical writing conditions [5

5. M. Ams, G. D. Marshall, and M. J. Withford, “Study of the influence of femtosecond laser polarisation on direct writing of waveguides,” Opt. Express 14(26), 13158–13163 (2006). [CrossRef] [PubMed]

,6

6. D. J. Little, M. Ams, P. Dekker, G. D. Marshall, J. M. Dawes, and M. J. Withford, “Femtosecond laser modification of fused silica: the effect of writing polarization on Si-O ring structure,” Opt. Express 16(24), 20029–20037 (2008). [CrossRef] [PubMed]

]. This behavior implies that the energy absorbed is polarization-dependent. Here, we hypothesize that the polarization dependence of photoionization cross-sections is the origin of this behavior. To test this hypothesis, the polarization-dependency on the refractive index change is measured for several glasses and checked for consistency with photoionization theory.

It has been established theoretically and experimentally that the relative photoionization cross-sections for linearly and circularly polarized beams (denoted σl and σc respectively) is most strongly dependent on the multiphoton order, N, which is defined as the number of photons an electron is required to absorb and cross the energy band gap of the glass to become a free carrier [11

11. H. R. Reiss, “Polarization effects in high-order multiphoton ionization,” Phys. Rev. Lett. 29(17), 1129–1131 (1972). [CrossRef]

,12

12. V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarizations,” Phys. Rev. Lett. 97(23), 237403 (2006). [CrossRef]

]. Reiss identified three behavioral regimes; First, if N = 1, 2 then σc / σl is approximately 1. Second, if N = 3, 4 then σc / σl > 1 and third, if N ≥ 5, then σc / σl < 1. In the context of ultrafast-laser modification, these behavioral regimes can be identified by comparing the refractive index change induced by linearly and circularly polarized pulses. In the first regime (N = 1, 2) the refractive index change induced by each polarization should be equal, while for the second and third regimes (N = 3, 4 and N ≥ 5) the polarization that yields the largest refractive index change should be circular and linear respectively.

2.1 Correcting for the Strong Laser Field

A photoionization event can be described in terms of two contributing ionization mechanisms, multiphoton ionization and tunneling ionization. Multiphoton ionization is the direct absorption of N photons by an electron, resulting in a free carrier being generated and is the dominant mechanism at low irradiances. Tunneling ionization is the tunneling of an electron through the potential barrier that is formed between the valence and conduction bands of the glass when distorted by a strong electric field and is the dominant mechanism at high irradiances. The relative dominance of multiphoton and tunneling ionization is quantified using the Keldysh parameter
γ=Eg2Up,
(1)
where Eg is the energy band gap of the glass and Up is the ponderomotive energy, given by
Up=e28π2c3ε0mnIλ2,
(2)
where I is the irradiance, λ is the wavelength of the optical field, m* is the reduced mass of the electron-hole pair and n is the refractive index of the glass. Multiphoton ionization is said to dominate when γ » 1. The peak irradiances used in this study vary between 42 TW/cm2 and 98 TW/cm2, corresponding to γ = 0.6-1.2, indicating a significant contribution from both multiphoton and tunneling ionization.

3. Experiment

Refractive index profiles of ultrafast-laser written waveguides were measured using a refractive near-field profilometer (Rinck Electronik) operating at 637 nm. The purpose of measuring the refractive index profile was threefold; to measure the peak refractive index contrast of the waveguide cross-section, to verify that no significant differences in waveguide morphology existed between waveguides written with different polarizations under otherwise identical conditions and to verify that nonlinear distortion of the ultrafast beam was not present. Note that peak refractive index changes quoted in this paper refer to positive refractive index changes unless stated otherwise.

Figure 1
Fig. 1 (Color online) Refractive index profiles of waveguides written in a) Kigre Yb:QX with circularly polarized pulses, b) Kigre Yb:QX with linearly polarized pulses, c) Schott BK7 with circularly polarized pulses, d) Schott BK7 with linearly polarized pulses. All waveguides in this image were written with a pulse energy of 700 nJ. The ultrafast writing beam propagated from left to right.
below shows the measured refractive index profiles of waveguides written in Kigre Yb:QX phosphate glass and Schott BK7 borosilicate glass with circularly and linearly polarized ultrafast pulses possessing a pulse energy of 700 nJ. The circular symmetry of the modified regions verified that spatial distortion of the ultrafast writing beam was negligible. While there is a slight asymmetry in the waveguide shape along the x-direction, this is caused by depletion of the pulses as they pass through focus due to photo-ionization, resulting in a greater amount of energy being absorbed before focus than after. For the waveguides depicted in Fig. 1, the measured 1/e2 diameters were 7.7 ± 0.2 μm and 12.6 ± 0.2 μm in X and Y respectively for waveguide a) (Kigre Yb:QX; circularly polarized pulses), 7.5 ± 0.2 μm and 13.0 ± 0.2 μm in X and Y for waveguide b) (Kigre Yb:QX; linearly polarized pulses), 5.0 ± 0.2 μm and 7.5 ± 0.2 μm for waveguide c) (Schott BK7; circularly polarized pulses) and 5.0 ± 0.2 μm and 7.9 ± 0.2 μm for waveguide d) (Schott BK7; linearly polarized pulses). Similar agreement was found for waveguides written at other pulse energies. This verified that no significant differences in morphology existed between waveguides written with different polarizations under otherwise identical writing conditions.

4. Results

The peak refractive index change of waveguides written in Schott BK7 borosilicate glass as a function of the peak irradiance of the writing beam is shown in Fig. 2
Fig. 2 Peak refractive index change induced in BK7 by circularly and linearly polarized ultrafast pulses as a function of the peak irradiance.
. The refractive index change induced by linearly polarized pulses was very slightly greater than the refractive index induced by circularly polarized pulses, in most cases the difference was less than the experimental uncertainty.

The peak refractive index change of waveguides written in Kigre Yb:QX phosphate glass is shown in Fig. 3
Fig. 3 Peak refractive index change induced in Yb:QX by circularly and linearly polarized ultrafast pulses as a function of the peak irradiance. The data points at 42 TW/cm2 overlap with one another.
. In contrast to the results in BK7 glass, circularly polarized pulses clearly induce a higher refractive index change for all the tested pulse energies except at 300 nJ, where the difference was negligible.

5. Discussion

Experimental observations can be explained through calculation of the modified multiphoton order and linking it back to one of the three regimes of polarization-dependent behavior identified by Reiss. By calculating the band gap for each glass and using Eq. (3). N’ was calculated as a function of irradiance. A summary of these calculations is shown below in Fig. 4
Fig. 4 N’ (solid lines) and (2Eg – Up)/2Ephoton (dashed lines) as a function of irradiance for Schott Lithosil fused silica, Kigre Yb:QX phosphate glass and Schott BK7 borosilicate glass.
.

For Schott BK7 borosilicate glass, the energy band gap was calculated from the transmission spectrum of the glass to be 4.28 eV using the method outlined by Mott and Davis [14

14. N. F. Mott and E. F. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, 1979).

]. Assuming a reduced mass close to that of fused silica (m* = 0.64me [15

15. A. Couairon, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses,” Phys. Rev. B 71(12), 125435 (2005). [CrossRef]

]) for irradiances of 42-98 TW/cm2, N’ is calculated to be 2. This corresponds to the regime where σc / σl ≈1 and the refractive index change induced by linearly and circularly polarized ultrafast pulses is roughly equal, which agrees with what is observed when writing waveguides in BK7.

For Kigre Yb:QX phosphate glass, the energy band gap was calculated to be 5.31 eV from the cut-off wavelength measured by Yoneda et al. [16

16. H. Yoneda, K. Yamaguchi, and K. Ueda, “Dispersion of optical refractive index of Yb3+ doped laser glass and their fitting to a Lorentzian model,” Jpn. J. Appl. Phys. 38(Part 2, No. 6A/B), L639–L641 (1999). [CrossRef]

]. Assuming the same value of m* as before, for irradiances from 42 to 69 TW/cm2, N’ is calculated to be 3, whereas for irradiances of 69-98 TW/cm2, N’ is calculated to be 2. As σc / σl > 1 when N’ = 3, 4, it is expected that circularly polarized pulses will induce the highest refractive index change up to a peak irradiance of around 69 TW/cm2. For irradiances above this, it is expected that the refractive index change induced by linearly and circularly polarized pulses is roughly equal. It is observed however that circularly polarized pulses induce a higher refractive index change all the way up to 98 TW/cm2. This difference between prediction and observation can be resolved by considering the presence of optical losses due to photo-ionization and inverse Bremsstrahlung absorption which would act to reduce the peak irradiance of the focused ultrafast pulse, potentially reducing it below 69 TW/cm2, even for the highest pulse energy used.

A second disagreement between prediction and observation at 42 TW/cm2, where the refractive index induced by circularly and linearly polarized pulses is equal, can be attributed to the presence of defect states in the band structure of Yb:QX; energy states that lie between the valence and conduction bands, which potentially reduces the size of the electronic band gap, reducing N’ to 2 or lower [16

16. H. Yoneda, K. Yamaguchi, and K. Ueda, “Dispersion of optical refractive index of Yb3+ doped laser glass and their fitting to a Lorentzian model,” Jpn. J. Appl. Phys. 38(Part 2, No. 6A/B), L639–L641 (1999). [CrossRef]

]. As defect states are sparsely populated, they are quickly depleted and so this effect diminishes at higher irradiances.

In light of this reasonably strong agreement between prediction and observation, it is worth re-examining past experimental work done in Schott Lithosil fused silica by the authors [6

6. D. J. Little, M. Ams, P. Dekker, G. D. Marshall, J. M. Dawes, and M. J. Withford, “Femtosecond laser modification of fused silica: the effect of writing polarization on Si-O ring structure,” Opt. Express 16(24), 20029–20037 (2008). [CrossRef] [PubMed]

]. It was observed in fused silica that for peak irradiances less than 42 TW/cm2 a higher refractive index change was obtained using linearly polarized pulses, whereas for peak irradiances greater than 42 TW/cm2 a higher refractive index change was observed using circularly polarized pulses. The behavior indicates that there is a switch in regime from σc / σl < 1 (N’ ≥ 5) to σc / σl > 1 (N’ = 3, 4). For fused silica, with a band gap of 7.76 eV calculated from the transmission spectrum, the irradiance at which N’ switches from 5 to 4 is about 48 TW/cm2, which is in close agreement to the observed value of 42 TW/cm2.

It should be noted that there is substantial amount of uncertainty associated with these calculations, mainly regarding to the value of the reduced mass of the electron-hole pair in the various glasses. Other factors such as optical loss and the non-parabolic band structure of the glasses will also influence these figures. Despite this, strong qualitative agreement is observed between experiment and the predicted behavior.

There are some important practical implications that arise from this explanation for the observed dependence of ultrafast-laser induced refractive index change on the polarization of the ultrafast pulses. Firstly, for large band gap media such as fused silica and crystalline materials the polarization of the ultrafast pulses is expected to strongly influence the resultant modification (at a wavelength of 800 nm) since N is large and thus falls into a regime where the modifications induced by linearly and circularly polarized pulses is very different. In contrast, for small band gap media such as chalcogenide, the polarization of the ultrafast pulses is not expected to influence the resultant modification since N is small.

For ultrafast lasers with shorter wavelengths, the corresponding N for these materials will be lower, which will tend to shift the process into regimes where circularly polarized pulses yield greater modification (N = 3, 4) or where both polarizations yield the same polarization (N = 1, 2). Note too that the ponderomotive energy depends on λ 2 so the polarization-dependence is much slower to switch to different regimes with increasing irradiance. For ultrafast lasers with longer wavelengths, the corresponding N will be larger which will tend to shift the process into regimes where linearly polarized pulses yield greater modification. This however, is offset by the fact that the ponderomotive energy increases more rapidly with irradiance, which will tend to push the process into the N = 3, 4 or N = 1, 2 regimes. A recent paper by Eaton et al. found when writing waveguides in fused silica with a 532 nm wavelength laser, circularly polarized pulses yielded waveguides with a smaller mode-field diameter than those written with linearly polarized pulses [17

17. S. M. Eaton, M. L. Ng, R. Osellame, and P. R. Herman, “High refractive index contrast in fused silica waveguides by tightly focused, high-repetition rate femtosecond laser,” J. Non-Cryst. Solids 357(11-13), 2387–2391 (2011). [CrossRef]

]. This is indicative of a higher refractive index change being induced by circularly polarized pulses, which is consistent with prediction, as N’ = 3, 4 up to an irradiance of 218 TW/cm2 for 532 nm in fused silica. It is anticipated that future studies comparing waveguides written with circularly and linearly polarized ultrafast pulses at wavelengths other than 800 nm will continue to test the predictions outlined herein.

6. Conclusion

The dependence of ultrafast-laser induced refractive index on the polarization of the ultrafast pulses was explained using the regimes of polarization-dependency of photoionization identified by Reiss. The contrasting behavior in three different glasses; Schott BK7 borosilicate glass, where the refractive index change induced by circularly and linear pulses was equal; Kigre Yb:QX phosphate glass, where the highest refractive index change was induced by circularly polarized pulses and Schott Lithosil fused silica, where the highest refractive index change induced was induced by linearly polarized pulses below 42 TW/cm2 and circularly polarized pulses above 42 TW/cm2, was successfully reconciled using this explanation, with good agreement between observation and prediction in all three glasses. This insight allows researchers to take a more systematic approach when optimizing the polarization of the ultrafast laser for their chosen applications.

Acknowledgments

This work was carried out with the assistance of the Australian Research Council under the ARC Centers of Excellence Program.

References and links

1.

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

2.

C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867–1869 (2005). [CrossRef] [PubMed]

3.

Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91(24), 247405 (2003). [CrossRef] [PubMed]

4.

Y. Bellouard, E. Barthel, A. A. Said, M. Dugan, and P. Bado, “Scanning thermal microscopy and Raman analysis of bulk fused silica exposed to low-energy femtosecond laser pulses,” Opt. Express 16(24), 19520–19534 (2008). [CrossRef] [PubMed]

5.

M. Ams, G. D. Marshall, and M. J. Withford, “Study of the influence of femtosecond laser polarisation on direct writing of waveguides,” Opt. Express 14(26), 13158–13163 (2006). [CrossRef] [PubMed]

6.

D. J. Little, M. Ams, P. Dekker, G. D. Marshall, J. M. Dawes, and M. J. Withford, “Femtosecond laser modification of fused silica: the effect of writing polarization on Si-O ring structure,” Opt. Express 16(24), 20029–20037 (2008). [CrossRef] [PubMed]

7.

S. Gross, M. J. Withford, and A. Fuerbach, “Direct femtosecond laser written waveguides in bulk Ti3+ sapphire,” Proc. SPIE 7589, 75890U (2010). [CrossRef]

8.

A. H. Nejadmalayeri and P. R. Herman, “Ultrafast laser waveguide writing: Lithium niobate and the role of circular polarization and picosecond pulse width,” Opt. Lett. 31(20), 2987–2989 (2006). [CrossRef] [PubMed]

9.

R. S. Taylor, E. Simova, and C. Hnatovsky, “Creation of chiral structures inside fused silica glass,” Opt. Lett. 33(12), 1312–1314 (2008). [CrossRef] [PubMed]

10.

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

11.

H. R. Reiss, “Polarization effects in high-order multiphoton ionization,” Phys. Rev. Lett. 29(17), 1129–1131 (1972). [CrossRef]

12.

V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarizations,” Phys. Rev. Lett. 97(23), 237403 (2006). [CrossRef]

13.

M. Ams, G. D. Marshall, D. J. Spence, and M. J. Withford, “Slit beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses,” Opt. Express 13(15), 5676–5681 (2005). [CrossRef] [PubMed]

14.

N. F. Mott and E. F. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, 1979).

15.

A. Couairon, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses,” Phys. Rev. B 71(12), 125435 (2005). [CrossRef]

16.

H. Yoneda, K. Yamaguchi, and K. Ueda, “Dispersion of optical refractive index of Yb3+ doped laser glass and their fitting to a Lorentzian model,” Jpn. J. Appl. Phys. 38(Part 2, No. 6A/B), L639–L641 (1999). [CrossRef]

17.

S. M. Eaton, M. L. Ng, R. Osellame, and P. R. Herman, “High refractive index contrast in fused silica waveguides by tightly focused, high-repetition rate femtosecond laser,” J. Non-Cryst. Solids 357(11-13), 2387–2391 (2011). [CrossRef]

OCIS Codes
(140.7090) Lasers and laser optics : Ultrafast lasers
(260.5210) Physical optics : Photoionization
(130.2755) Integrated optics : Glass waveguides

ToC Category:
Laser Materials Processing

History
Original Manuscript: June 22, 2011
Revised Manuscript: July 19, 2011
Manuscript Accepted: July 20, 2011
Published: July 21, 2011

Virtual Issues
Femtosecond Direct Laser Writing and Structuring of Materials (2011) Optical Materials Express

Citation
Douglas J. Little, Martin Ams, and Michael J. Withford, "Influence of bandgap and polarization on photo-ionization: guidelines for ultrafast laser inscription [Invited]," Opt. Mater. Express 1, 670-677 (2011)
http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-1-4-670


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References

  1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21(21), 1729–1731 (1996). [CrossRef] [PubMed]
  2. C. Hnatovsky, R. S. Taylor, E. Simova, V. R. Bhardwaj, D. M. Rayner, and P. B. Corkum, “Polarization-selective etching in femtosecond laser-assisted microfluidic channel fabrication in fused silica,” Opt. Lett. 30(14), 1867–1869 (2005). [CrossRef] [PubMed]
  3. Y. Shimotsuma, P. G. Kazansky, J. Qiu, and K. Hirao, “Self-organized nanogratings in glass irradiated by ultrashort light pulses,” Phys. Rev. Lett. 91(24), 247405 (2003). [CrossRef] [PubMed]
  4. Y. Bellouard, E. Barthel, A. A. Said, M. Dugan, and P. Bado, “Scanning thermal microscopy and Raman analysis of bulk fused silica exposed to low-energy femtosecond laser pulses,” Opt. Express 16(24), 19520–19534 (2008). [CrossRef] [PubMed]
  5. M. Ams, G. D. Marshall, and M. J. Withford, “Study of the influence of femtosecond laser polarisation on direct writing of waveguides,” Opt. Express 14(26), 13158–13163 (2006). [CrossRef] [PubMed]
  6. D. J. Little, M. Ams, P. Dekker, G. D. Marshall, J. M. Dawes, and M. J. Withford, “Femtosecond laser modification of fused silica: the effect of writing polarization on Si-O ring structure,” Opt. Express 16(24), 20029–20037 (2008). [CrossRef] [PubMed]
  7. S. Gross, M. J. Withford, and A. Fuerbach, “Direct femtosecond laser written waveguides in bulk Ti3+ sapphire,” Proc. SPIE 7589, 75890U (2010). [CrossRef]
  8. A. H. Nejadmalayeri and P. R. Herman, “Ultrafast laser waveguide writing: Lithium niobate and the role of circular polarization and picosecond pulse width,” Opt. Lett. 31(20), 2987–2989 (2006). [CrossRef] [PubMed]
  9. R. S. Taylor, E. Simova, and C. Hnatovsky, “Creation of chiral structures inside fused silica glass,” Opt. Lett. 33(12), 1312–1314 (2008). [CrossRef] [PubMed]
  10. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008). [CrossRef]
  11. H. R. Reiss, “Polarization effects in high-order multiphoton ionization,” Phys. Rev. Lett. 29(17), 1129–1131 (1972). [CrossRef]
  12. V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarizations,” Phys. Rev. Lett. 97(23), 237403 (2006). [CrossRef]
  13. M. Ams, G. D. Marshall, D. J. Spence, and M. J. Withford, “Slit beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses,” Opt. Express 13(15), 5676–5681 (2005). [CrossRef] [PubMed]
  14. N. F. Mott and E. F. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, 1979).
  15. A. Couairon, L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, “Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses,” Phys. Rev. B 71(12), 125435 (2005). [CrossRef]
  16. H. Yoneda, K. Yamaguchi, and K. Ueda, “Dispersion of optical refractive index of Yb3+ doped laser glass and their fitting to a Lorentzian model,” Jpn. J. Appl. Phys. 38(Part 2, No. 6A/B), L639–L641 (1999). [CrossRef]
  17. S. M. Eaton, M. L. Ng, R. Osellame, and P. R. Herman, “High refractive index contrast in fused silica waveguides by tightly focused, high-repetition rate femtosecond laser,” J. Non-Cryst. Solids 357(11-13), 2387–2391 (2011). [CrossRef]

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