## Off-resonance and non-resonant dispersion of Kerr nonlinearity for symmetric molecules [Invited] |

Optics Express, Vol. 19, Issue 23, pp. 22486-22495 (2011)

http://dx.doi.org/10.1364/OE.19.022486

Acrobat PDF (779 KB)

### Abstract

The exact formula is derived from the “sum over states” (SOS) quantum mechanical model for the frequency dispersion of the nonlinear refractive index coefficient *n*_{2} for centrosymmetric molecules in the off-resonance and non-resonant regimes. This expression is characterized by interference between terms from two-photon transitions from the ground state to the even-symmetry excited states and one-photon transitions between the ground state and odd-symmetry excited states. When contributions from the two-photon terms exceed those from the one-photon terms, the non-resonant intensity-dependent refractive index *n*_{2}>0, and vice versa. Examples of the frequency dispersion for the three-level SOS model are given. Comparison is made with other existing theories.

© 2011 OSA

## 1. Introduction

*n*

_{2}(when the photon energies are all much smaller than the energy to the first excited state), the Kerr nonlinear refractive index coefficient due to transitions between the electronic states of atoms and molecules [1

1. M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. **25**(16), 1183–1185 (2000). [CrossRef] [PubMed]

4. C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, “A simplified three-level model for describing the molecular third-order nonlinear-optical susceptibility,” Int. J. Quantum Chem. **43**(1), 27–36 (1992). [CrossRef]

5. D. Lu, G. Chen, J. W. Perry, and W. A. Goddard III, “Valence-bond charge-transfer model for nonlinear optical properties of charge-transfer organic molecules,” J. Am. Chem. Soc. **116**(23), 10679–10685 (1994). [CrossRef]

12. J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mod. Phys. **37**(1), 1–18 (1965). [CrossRef]

5. D. Lu, G. Chen, J. W. Perry, and W. A. Goddard III, “Valence-bond charge-transfer model for nonlinear optical properties of charge-transfer organic molecules,” J. Am. Chem. Soc. **116**(23), 10679–10685 (1994). [CrossRef]

7. M. G. Kuzyk and C. W. Dirk, “Effects of centrosymmetry on the nonresonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A **41**(9), 5098–5109 (1990). [CrossRef] [PubMed]

13. B. J. Orr and J. F. Ward, “Perturbation theory of the non-linear optical polarization of an isolated system,” Mol. Phys. **20**(3), 513–526 (1971). [CrossRef]

15. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express **17**(16), 13429–13434 (2010). [CrossRef]

16. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express **18**(3), 3011–3012 (2010). [CrossRef]

*n*

_{2}is obtained essentially from the linear susceptibility and contains a phenomenological nonlinear “force” constant [17

17. W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulae,” Opt. Express **18**(7), 6613–6620 (2010). [CrossRef] [PubMed]

*n*

_{2}nor can the magnitude of

*n*

_{2}be calculated from measurable parameters. There is also a model due to Brée et. al which was applied to atomic argon [18

18. C. Brée, A. Demircan, and G. Steinmeyer, “Saturation of the all-optical Kerr effect,” Phys. Rev. Lett. **106**(18), 183902 (2011). [CrossRef] [PubMed]

*n*

_{2}and is labeled here as the “two photon resonance model”. Neither of these approaches describe completely the third order nonlinearity of symmetric molecules, nor does the two level SOS model, since such molecules have zero permanent dipole moment.

*n*

_{2}[12

12. J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mod. Phys. **37**(1), 1–18 (1965). [CrossRef]

13. B. J. Orr and J. F. Ward, “Perturbation theory of the non-linear optical polarization of an isolated system,” Mol. Phys. **20**(3), 513–526 (1971). [CrossRef]

*m*and

*n*to be non-zero requires a change in the symmetry between the wave functions of the two states, i.e. one state has to have even spatial symmetry (gerade) and the other odd spatial symmetry (ungerade). In atoms and centrosymmetric molecules, the ground state wavefunction is of even symmetry. A three-level model with parameters diagrammed in Fig. 1 has been explored previously based on the general SOS formalism of Orr and Ward [13

13. B. J. Orr and J. F. Ward, “Perturbation theory of the non-linear optical polarization of an isolated system,” Mol. Phys. **20**(3), 513–526 (1971). [CrossRef]

_{g}and the first odd symmetry excited state 1B

_{u}and between that excited state and the dominant even-symmetry excited state mA

_{g}respectively [4

4. C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, “A simplified three-level model for describing the molecular third-order nonlinear-optical susceptibility,” Int. J. Quantum Chem. **43**(1), 27–36 (1992). [CrossRef]

11. D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photon. **2**(1), 60–200 (2010). [CrossRef]

*n*

_{2}due to one and two photon transitions are proportional to

11. D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photon. **2**(1), 60–200 (2010). [CrossRef]

*n*

_{2}exceed those due to one photon transitions, and vice-versa. This model has been successfully applied to the explanation of the nonlinearity, including its sign in the non-resonant regime, for linear organic molecules such as squaraine dyes, CS

_{2}, and conjugated polymers [19

19. K. S. Mathis, M. G. Kuzyk, C. W. Dirk, A. Tan, S. Martinez, and G. Gampos, “Mechanisms of the nonlinear optical properties of squaraine dyes in poly(methyl methacrylate) polymer,” J. Opt. Soc. Am. B **15**(2), 871–883 (1998). [CrossRef]

*n*

_{2}in terms of electric dipole transition moments and locations of the excited states which for simple atoms and molecules can be calculated from first principles. It will be shown that the relative importance of the contributions of the one- and two-photon transitions still determines the sign of the non-resonant nonlinearity.

## 2. Sum over states for symmetric molecules

**20**(3), 513–526 (1971). [CrossRef]

*g*) electron to all of the excited states (subscript

*m*). First order perturbation theory is used to calculate the probability for transitions into the excited states

*m*in terms of the transition dipole moments defined in Eq. (1). This yields the induced polarization in each state

*m*by each field component which then gives the linear atomic/molecular susceptibility.

*n*) from the ground state and all of the previously excited states

*m*(due to the first interaction). This leads to the second order atomic/molecular susceptibility

*i*with the nonlinear polarization induced. Each of

*v*,

*m*and

*n*are each over all of the excited states (with the exclusion of the ground state). The frequency terms are

*m’*th state to decay to the

*n*’th state.

_{g}→6B

_{u}→5A

_{g}→8B

_{u}→1A

_{g}which involve two different odd symmetry (one photon) states which are not allowed in the simple three level model. As a result there are more possible terms for two photon transitions than one photon transitions.

## 3. Linear symmetric molecules

*z*-polarized incident fields and, since the interest here is in

*n*

_{2}, this restricts the subscripts of the macroscopic third order susceptibility

*z,z,z,z*. The further detailed discussion addresses linear molecules since the non-resonant

*n*

_{2}has been measured recently for air and its primary constituents, namely the linear molecules O

_{2}and N

_{2}and its dispersion calculated via the “extended Miller formulas” [15

15. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express **17**(16), 13429–13434 (2010). [CrossRef]

17. W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulae,” Opt. Express **18**(7), 6613–6620 (2010). [CrossRef] [PubMed]

15. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express **17**(16), 13429–13434 (2010). [CrossRef]

16. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express **18**(3), 3011–3012 (2010). [CrossRef]

24. J. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependence of *n*_{2} in air,” Opt. Commun. **135**(4-6), 310–314 (1997). [CrossRef]

*z*-axis, the net contribution is only 1/5th

*n*

_{2}(defined by

*I*is the intensity) is given bywhich includes all three possible permutations of the input -

*ω*that are required to describe the instantaneous interaction which produces

*n*

_{2.}The refractive index

*n*in Eq. (4) is the average over all possible orientations and is proportional to

*N*is the density of molecules and

*ε*(

_{r}*ω*) is the relative dielectric constant at the frequency

*ω*. It is important to realize that

*all*of the terms in Eqs. (3) and (4) contribute to an intensity-dependent refractive index and absorption. However, only the case in which

*ω*=

*n*

_{2}in the non-resonant or off resonance limits.

*n*

_{2}, which is a formidable task, especially near and on the one- and two-photon resonances. However, it has proven possible to obtain closed form solutions for

*n*

_{2}in the off-resonance regime which corresponds simply to neglecting the imaginary parts of the denominators of all of the terms [11

11. D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photon. **2**(1), 60–200 (2010). [CrossRef]

*n*

_{2}due to both two- and one-photon transitions is given below.

*n*

_{2}can be quite complicated, including multiple sign changes. In the non-resonant limit (

*ω*→0),

*n*

_{2}will still vanish because of centrosymmetry and the expression will again separate into positive and negative terms similar in form to the one given by Eq. (9). Thus, a positive value of

*n*

_{2}will always be associated with two-photon transitions.

## 4. Three-level model for linear symmetric molecules

*n*

_{2}is shown below in the limit that the decay times are small, i.e.

*n*

_{2}for a few scenarios in which both one- and two-photon transitions are important. Specifically

*n*

_{2}was calculated for the ratio

*n*

_{2}is negative in the frequency range between the one and two photon resonances. There is a pronounced dispersion resonance in the nonlinearity near the normalized frequency of the two photon absorption peak which in this case appears at

*n*

_{2}changes sign twice in the vicinity of this resonance. This occurs over a narrow range of

*n*

_{2}remains negative since the one photon transitions dominate. For values of

*ω*= 0 and the two photon transitions dominate.

## 5. Spherically symmetric molecules and atoms

## 6. Comparison with other models of *n*_{2}

25. J. Pérez Moreno and M. G. Kuzyk, “Fundamental limits of the dispersion of the two-photon absorption cross section,” J. Chem. Phys. **123**(19), 194101 (2005). [CrossRef] [PubMed]

26. J. H. Andrews, J. D. V. Khaydarov, K. D. Singer, D. L. Hull, and K. C. Chuang, “Characterization of excited states of centrosymmetric and noncentrosymmetric squaraines by third-harmonic spectral dispersion,” J. Opt. Soc. Am. **12**(12), 2360–2371 (1995). [CrossRef]

19. K. S. Mathis, M. G. Kuzyk, C. W. Dirk, A. Tan, S. Martinez, and G. Gampos, “Mechanisms of the nonlinear optical properties of squaraine dyes in poly(methyl methacrylate) polymer,” J. Opt. Soc. Am. B **15**(2), 871–883 (1998). [CrossRef]

## 7. Concluding remarks

*n*

_{2}in the off-resonance and non-resonant regimes. The net result is that the non-resonant sign of

*n*

_{2}can be used to determine whether one-photon or two-photon transitions dominate the nonlinear response of molecules. The frequency dispersion of

*n*

_{2}is complicated and none of the previous theories discussed here have captured completely the essential physics. Using a three-level model, it has been shown that the sign of

*n*

_{2}can change at least twice, depending on the details of the molecular properties.

**17**(16), 13429–13434 (2010). [CrossRef]

16. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express **18**(3), 3011–3012 (2010). [CrossRef]

## Acknowledgments

## References and links

1. | M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. |

2. | C. W. Dirk and M. G. Kuzyk, “Damping corrections and the calculation of optical nonlinearities in organic molecules,” Phys. Rev. B Condens. Matter |

3. | M. G. Kuzyk, “Compact sum-over-states expression without dipolar terms for calculating nonlinear susceptibilities,” Phys. Rev. A |

4. | C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, “A simplified three-level model for describing the molecular third-order nonlinear-optical susceptibility,” Int. J. Quantum Chem. |

5. | D. Lu, G. Chen, J. W. Perry, and W. A. Goddard III, “Valence-bond charge-transfer model for nonlinear optical properties of charge-transfer organic molecules,” J. Am. Chem. Soc. |

6. | Reviewed in J. M. Hales and J. W. Perry, “Organic and polymeric 3rd-order nonlinear optical materials and device applications,” in |

7. | M. G. Kuzyk and C. W. Dirk, “Effects of centrosymmetry on the nonresonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A |

8. | C. W. Dirk and M. G. Kuzyk, “Squarylium dye-doped polymer systems as quadratic electrooptic materials,” Chem. Mater. |

9. | M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, “Mechanisms of quadratic electrooptic modulation of dye-doped polymer systems,” J. Opt. Soc. Am. B |

10. | Y. Z. Yu, R. F. Shu, A. F. Garito, and C. H. Grossman, “Origin of negative |

11. | D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photon. |

12. | J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mod. Phys. |

13. | B. J. Orr and J. F. Ward, “Perturbation theory of the non-linear optical polarization of an isolated system,” Mol. Phys. |

14. | Reviewed in S. Barlow and S. R. Marder, “Nonlinear optical properties of organic materials,” in |

15. | V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express |

16. | V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express |

17. | W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulae,” Opt. Express |

18. | C. Brée, A. Demircan, and G. Steinmeyer, “Saturation of the all-optical Kerr effect,” Phys. Rev. Lett. |

19. | K. S. Mathis, M. G. Kuzyk, C. W. Dirk, A. Tan, S. Martinez, and G. Gampos, “Mechanisms of the nonlinear optical properties of squaraine dyes in poly(methyl methacrylate) polymer,” J. Opt. Soc. Am. B |

20. | G. Stegeman and H. Hu, “Refractive nonlinearity of linear symmetric molecules and polymers revisited,” Photon. Lett. Poland |

21. | P. McWilliams, P. Hayden, and Z. Soos, “Theory of even-parity state and two-photon spectra of conjugated polymers,” Phys. Rev. B |

22. | G. I. Stegeman, “Nonlinear optics of conjugated polymers and linear molecules,” Nonlinear Opt., Quantum Opt. (to be published). |

23. | G. I. Stegeman and R. A. Stegeman, |

24. | J. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependence of |

25. | J. Pérez Moreno and M. G. Kuzyk, “Fundamental limits of the dispersion of the two-photon absorption cross section,” J. Chem. Phys. |

26. | J. H. Andrews, J. D. V. Khaydarov, K. D. Singer, D. L. Hull, and K. C. Chuang, “Characterization of excited states of centrosymmetric and noncentrosymmetric squaraines by third-harmonic spectral dispersion,” J. Opt. Soc. Am. |

27. | J. Kasparian, P. Béjot, and J.-P. Wolf, “Arbitrary-order nonlinear contribution to self-steepening,” Opt. Lett. |

28. | W. Ettoumi, P. Béjot, Y. Petit, V. Loriot, E. Hertz, O. Faucher, B. Lavorel, J. Kasparian, and J.-P. Wolf, “Spectral dependence of purely-Kerr-driven filamentation in air and argon,” Phys. Rev. A |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.3270) Nonlinear optics : Kerr effect

(190.5940) Nonlinear optics : Self-action effects

**ToC Category:**

Nonlinear Absorption and Dispersion

**History**

Original Manuscript: August 30, 2011

Revised Manuscript: September 27, 2011

Manuscript Accepted: October 5, 2011

Published: October 25, 2011

**Virtual Issues**

Nonlinear Optics (2011) *Optical Materials Express*

**Citation**

George Stegeman, Mark G. Kuzyk, Dimitris G. Papazoglou, and Stelios Tzortzakis, "Off-resonance and non-resonant dispersion of Kerr nonlinearity for symmetric molecules [Invited]," Opt. Express **19**, 22486-22495 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22486

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### References

- M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. 25(16), 1183–1185 (2000). [CrossRef] [PubMed]
- C. W. Dirk and M. G. Kuzyk, “Damping corrections and the calculation of optical nonlinearities in organic molecules,” Phys. Rev. B Condens. Matter 41(3), 1636–1639 (1990). [CrossRef] [PubMed]
- M. G. Kuzyk, “Compact sum-over-states expression without dipolar terms for calculating nonlinear susceptibilities,” Phys. Rev. A 72(5), 053819 (2005). [CrossRef]
- C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, “A simplified three-level model for describing the molecular third-order nonlinear-optical susceptibility,” Int. J. Quantum Chem. 43(1), 27–36 (1992). [CrossRef]
- D. Lu, G. Chen, J. W. Perry, and W. A. Goddard, “Valence-bond charge-transfer model for nonlinear optical properties of charge-transfer organic molecules,” J. Am. Chem. Soc. 116(23), 10679–10685 (1994). [CrossRef]
- Reviewed in J. M. Hales and J. W. Perry, “Organic and polymeric 3rd-order nonlinear optical materials and device applications,” in Introduction to Organic Electronic and Optoelectronic Materials and Devices, S.-S. Sun and L. Dalton, eds. (CRC, 2008), Chap. 17.
- M. G. Kuzyk and C. W. Dirk, “Effects of centrosymmetry on the nonresonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A 41(9), 5098–5109 (1990). [CrossRef] [PubMed]
- C. W. Dirk and M. G. Kuzyk, “Squarylium dye-doped polymer systems as quadratic electrooptic materials,” Chem. Mater. 2(1), 4–6 (1990). [CrossRef]
- M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, “Mechanisms of quadratic electrooptic modulation of dye-doped polymer systems,” J. Opt. Soc. Am. B 7(5), 842–858 (1990). [CrossRef]
- Y. Z. Yu, R. F. Shu, A. F. Garito, and C. H. Grossman, “Origin of negative χ3 in squaraines: experimental observation of two-photon states,” Opt. Lett. 19(11), 786–788 (1994). [CrossRef] [PubMed]
- D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photon. 2(1), 60–200 (2010). [CrossRef]
- J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mod. Phys. 37(1), 1–18 (1965). [CrossRef]
- B. J. Orr and J. F. Ward, “Perturbation theory of the non-linear optical polarization of an isolated system,” Mol. Phys. 20(3), 513–526 (1971). [CrossRef]
- Reviewed in S. Barlow and S. R. Marder, “Nonlinear optical properties of organic materials,” in Functional Organic Materials: Syntheses, Strategies and Applications, T. J. J. Muller and U. H. F. Bunz, eds. (Wiley, 2007), Chap. 11.
- V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17(16), 13429–13434 (2010). [CrossRef]
- V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express 18(3), 3011–3012 (2010). [CrossRef]
- W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulae,” Opt. Express 18(7), 6613–6620 (2010). [CrossRef] [PubMed]
- C. Brée, A. Demircan, and G. Steinmeyer, “Saturation of the all-optical Kerr effect,” Phys. Rev. Lett. 106(18), 183902 (2011). [CrossRef] [PubMed]
- K. S. Mathis, M. G. Kuzyk, C. W. Dirk, A. Tan, S. Martinez, and G. Gampos, “Mechanisms of the nonlinear optical properties of squaraine dyes in poly(methyl methacrylate) polymer,” J. Opt. Soc. Am. B 15(2), 871–883 (1998). [CrossRef]
- G. Stegeman and H. Hu, “Refractive nonlinearity of linear symmetric molecules and polymers revisited,” Photon. Lett. Poland 1(4), 148–150 (2009). [CrossRef]
- P. McWilliams, P. Hayden, and Z. Soos, “Theory of even-parity state and two-photon spectra of conjugated polymers,” Phys. Rev. B 43(12), 9777–9791 (1991). [CrossRef]
- G. I. Stegeman, “Nonlinear optics of conjugated polymers and linear molecules,” Nonlinear Opt., Quantum Opt. (to be published).
- G. I. Stegeman and R. A. Stegeman, Nonlinear Optics: Phenomena, Materials and Devices (J. Wiley, in press).
- J. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependence of n2 in air,” Opt. Commun. 135(4-6), 310–314 (1997). [CrossRef]
- J. Pérez Moreno and M. G. Kuzyk, “Fundamental limits of the dispersion of the two-photon absorption cross section,” J. Chem. Phys. 123(19), 194101 (2005). [CrossRef] [PubMed]
- J. H. Andrews, J. D. V. Khaydarov, K. D. Singer, D. L. Hull, and K. C. Chuang, “Characterization of excited states of centrosymmetric and noncentrosymmetric squaraines by third-harmonic spectral dispersion,” J. Opt. Soc. Am. 12(12), 2360–2371 (1995). [CrossRef]
- J. Kasparian, P. Béjot, and J.-P. Wolf, “Arbitrary-order nonlinear contribution to self-steepening,” Opt. Lett. 35(16), 2795–2797 (2010). [CrossRef] [PubMed]
- W. Ettoumi, P. Béjot, Y. Petit, V. Loriot, E. Hertz, O. Faucher, B. Lavorel, J. Kasparian, and J.-P. Wolf, “Spectral dependence of purely-Kerr-driven filamentation in air and argon,” Phys. Rev. A 82(3), 033826 (2010). [CrossRef]

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