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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13429–13434
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Measurement of high order Kerr refractive index of major air components

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 13429-13434 (2009)
http://dx.doi.org/10.1364/OE.17.013429


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Abstract

We measure the instantaneous electronic nonlinear refractive index of N2, O2, and Ar at room temperature for a 90 fs and 800 nm laser pulse. Measurements are calibrated by post-pulse molecular alignment through a polarization technique. At low intensity, quadratic coefficients n2 are determined. At higher intensities, a strong negative contribution with a higher nonlinearity appears, which leads to an overall negative nonlinear Kerr refractive index in air above 26 TW/cm2.

© 2009 Optical Society of America

1. Introduction

The aim of the present work is the measurement of the nonlinear refractive index of the main air components (N2, O2 and Ar) at filamentation laser intensities. The optical technique used for that purpose is insensitive to the plasma [5

5. P. Sprangle, E. Esarey, and B. Hafizi, “Propagation and stability of intense laser pulses in partially stripped plasmas,” Phys. Rev. E 565894–5907 (1997).

] and allows quantitative measurement from the analysis of the time-dependent laser-induced birefringence. We show that the nonlinear Kerr index exhibits a large variation and becomes negative above few tens of TW/cm2. Drastic effects on pulse propagation are awaited from this strong nonlinear behaviour.

2. Polarization technique : principle and alignment contribution

The method exploits the occurrence of the post pulse molecular alignment. It is now well established that the interaction of molecules with strong ultra-short laser pulse induces periodic transient molecular alignment under field-free condition (i.e. after the pulse extinction) [6

6. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

]. At the occurrence of post-pulse alignment, the sample becomes significantly birefringent. Additionally, the instantaneous Kerr effect during the pulse contributes to an extra birefringence. The principle of the technique is to compare the birefringence contribution resulting from both effects. The well known alignment signal is thus used to calibrate the instantaneous Kerr effect. The experimental setup makes use of the strong field polarization technique [7

7. V. Renard, M. Renard, S. Guérin, Y. T. Pashayan, B. Lavorel, O. Faucher, and H. R. Jauslin, “Postpulse molecular alignment measured by a weak field polarization technique,” Phys. Rev. Lett. 90(15), 153601 (2003). [CrossRef] [PubMed]

] implemented for measuring the degree of post-pulse molecular alignment. This pump-probe technique consists in measuring the birefringence of a gas sample that interacts with a “pump” pulse through the depolarization of a time-delayed weak “probe” pulse.

In the present experiment, both pulses are derived from a Ti:Sapphire chirped pulse amplified system working at 1 KHz (pulse duration of 90 fs at 800 nm). The two laser beams are focused with the same lens of focal length f=20 cm and overlapped at a small angle (4°) in a gas cell. The energy of the vertically polarized pump pulse is controlled by means of a half-wave plate and a polarizer. The probe pulse is initially polarized at 45° with respect to the pump. The amount of depolarized light passing through a crossed analyzer, placed after the cell, is measured with a photomultiplier. The pump-probe delay is scanned by means of a motorized delay line. The expression of the homodyne signal S homo(t) can be written as [8]

Shomo(t)Ipr(t)(Δn(t))2=Ipr(t)(n(t)n(t))2
(1)

where Δn(t) is the difference of refractive index along the polarization axis n (t) and perpendicular to it n (t), and I pr(t) is the intensity envelop of the probe pulse. Due to the quadratic response of the homodyne detection, the sign of the birefringence is lost, whereas the sensitivity is enhanced. In order to provide the sign of Δn(t), heterodyne detection can be implemented by inducing an additional birefringence 𝓟 by means of a phase plate inserted between the two crossed polarizers. The pure heterodyne signal S hetero(t) is then obtained by the subtraction of two experimental signals recorded with opposite signs of 𝓟[7

7. V. Renard, M. Renard, S. Guérin, Y. T. Pashayan, B. Lavorel, O. Faucher, and H. R. Jauslin, “Postpulse molecular alignment measured by a weak field polarization technique,” Phys. Rev. Lett. 90(15), 153601 (2003). [CrossRef] [PubMed]

, 8

8. V. Loriot, P. Tzallas, E. P. Benis, E. Hertz, B. Lavorel, D. Charalambidis, and O. Faucher, “Laser-induced field-free alignment of the OCS molecule,” J. Phys. B 40(12), 2503–2510 (2007). [CrossRef]

].

S±(t)Ipr(t)(Δn(t)±)2
(2)
Shetero(t)=S+(t)S(t)Ipr(t)(Δn(t))
(3)

For linear molecules, the degree of alignment can be characterized by the expectation value 〈cos2 θ〉, with θ the angle between the molecular axis and the field [6

6. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

]. The temporal dependence of alignment can be accurately calculated for linear molecules. The birefringence induced by the alignment is given by

Δnrot(t)=3ρ4n0ε0Δα{cos2θ(t)13},
(4)

with Δα the polarizability anisotropy (Δα(N2)=4.6 a.u. [9

9. Maroulis, “Accurate electric multipole moment, static polarizability and hyperpolarizability derivatives for N2,” J, Chem. Phys. 118, 2673–2687 (2003). [CrossRef]

], Δα(O2)=7.25 a.u. [10

10. E. Hertz, B. Lavorel, O. Faucher, and R. Chaux, “Femtosecond polarization spectroscopy in molecular gas mixtures: Macroscopic interference and concentration measurements,” J. Chem. Phys. 113(16), 6629–6633 (2000). [CrossRef]

]), ρ the gas density, n 0 the linear refractive index of the gas, and e0 the dielectric constant of the vacuum. The alignment signal obtained by the polarisation technique in homodyne (resp. heterodyn) detection is calculated by substituting Δn by Δn rot in Eq. (1) (resp. Eq. (3)). Knowing the degree of alignment, it is thus possible to calibrate the magnitude of the other birefringence signal contributions.

3. Electronic Kerr terms

3.1. Low field: Measurement of the nonlinear index n2

The instantaneous Kerr effect produces a variation of the refractive index along the pump polarization axis defined by n kerr‖=n 2 I. Far from any resonances, using the tensor properties of the susceptibility χ (3), the perpendicular component can be written nkerr=13n2I [11

11. R. W. Boyd, Nonlinear opticsthird ed.Academic Press (2007).

]. The resulting birefringence is the subtraction of the refractive index between the parallel and the perpendicular axis Δnkerr=23n2I. The total birefringence created by the pump beam is the sum of the rotational and the instantaneous Kerr contribution. The low field homodyne detection signal S homo(t) can be then written as

Shomo(t)Ipr(t)(3ρΔα4n0ε0(cos2θ13)+23n2I)2.
(5)

At low intensity, the term n 2 I is sufficient to describe the Kerr effect. An experimental result obtained with nitrogen in these conditions is shown in Fig. 1. The first birefringence peak centered around the zero delay [Fig. 1(a)] is composed from the electronic ultrafast response and the inertial rotational response which is slightly delayed towards positive times (around 120 fs). The corresponding variation of the refractive index is indicated on Fig. 1(b). The full signal is fitted with expression (5) using a magnitude factor and n 2 as free parameters. The post-pulse molecular alignment determines the global magnitude factor while the value of n 2 is determined through the shape and the relative amplitude of the first peak. For the experiment performed in argon, the signal only depends on the electronic Kerr effect. It has been compared to the N2 post-pulse signal recorded separately but in the same experimental conditions as in [12

12. V. Loriot, E. Hertz, B. Lavorel, and O. Faucher, “Field-free molecular alignment for measuring ionization probability,” J. Phys. B 41(1), 015604 (2008). [CrossRef]

]. The electronic Kerr contribution was then calibrated through the alignment of nitrogen. Compared to molecules, the uncertainty is slightly reduced, because the instantaneous and the rotational contributions are very well separated in two different signals. A statistic number of this experiment has been realized in order to considerate the additional uncertainty due to the gas change. The measured values are presented in the table 1.

Table 1. Measured n2 coefficients at 1 bar for nitrogen, oxygen, argon, and air in unit of 10-7cm2/TW

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Fig. 1. (a) Recorded homodyne birefringence signal [dots] versus pump-probe delay in 1 bar of N2 at room temperature. The mean intensity of the pump is estimated around 500 GW/cm2. The simulated signal [full line] has been adjusted to the experimental data [dots]. From this adjustment, the corresponding variation of the refractive index Δn (b) due to the total nonlinear birefringence [full line] composed by the instantaneous Kerr [dots] and rotational [dash] components is deduced.

To validate the coefficients of Table 1, the simulation of the overall signal has been successfully compared to a set of data recorded in 1 bar of air. Our values are consistent with the ones measured by Nibbering et al. [2

2. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz “Determination of the inertial contribution to the nonlinear refractive index of air N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–660 (1997). J-F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependance of n2 in air,” Opt. Commun. 135, 310 (1997). [CrossRef]

] for N2, but differ for O2 and Ar. However, they are limited to a narrower interval, as in ref. [13] although with lower predicted values (i.e. 1.5-1.7·10−7 cm2/TW).

3.2. Strong field: Measurement of the higher nonlinear terms

At stronger field, the expansion of the electronic part is not sufficient to describe the whole intensity dependency of the refractive index. Pure heterodyne detection is employed here in order to obtain the sign of the birefringence Δn(t). The intensity dependence for argon and nitrogen is shown in Fig. 2. To prevent any spatio-temporal distortion of the pump pulse during its propagation in the gas cell, the pressure was reduced to 100 mbar. In Fig. 2(a1) the super-position of the orientational and electronic responses results in a global positive signal around the zero delay. In contrast, at higher intensity, the signal drops rapidly and becomes negative, as evidenced by the comparison between Fig. 2(a2) and (a3) that only differ by ≈20% in intensity. The baseline observed for positive delay in Fig. 2(a) is due to permanent molecular alignment [6

6. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

, 7

7. V. Renard, M. Renard, S. Guérin, Y. T. Pashayan, B. Lavorel, O. Faucher, and H. R. Jauslin, “Postpulse molecular alignment measured by a weak field polarization technique,” Phys. Rev. Lett. 90(15), 153601 (2003). [CrossRef] [PubMed]

] which depends on the intensity. It should be noted that simulation performed with the appropriate intensity reproduces satisfactorily the post-pulse contribution indicating a minor influence of propagation effects under the present experimental conditions.

An expansion of the Kerr development is required to justify the experimental observation around the zero delay. The refractive index variation along the polarisation axis is developed as n kerr‖=n 2 I+n 4 I 2+⋯+n 10 I 5. In order to interpret the measurements, the birefringence resulting from each term in the series is considered. The relationship between parallel and perpendicular is known for n 2 and n 4 [11

11. R. W. Boyd, Nonlinear opticsthird ed.Academic Press (2007).

, 14

14. J. Arabat and J. Etchepare, “Nonresonant fifth-order nonlinearities induced by ultrashort intense pulses,” J. Opt. Soc. Am. B 10(12), 2377–2382 (1993). [CrossRef]

] from the symmetry properties of the tensor χ (3) and χ (5). This can be generalized to the higher order terms Δn (2×j)=2j/(2j+1)n (2×j) Ij with j∊ℕ*. For high intensity, the pure heterodyne signal Shetero becomes

Shetero(t)Ipr(t)(3ρΔα4n0ε0(cos2θ13)Δnrot(t)+23n2I+45n4I2+67n6I3+89n8I4+1011n10I5Δnkerr(t)).
(6)

Table 2. Measured coefficients of the nonlinear refractive index expansion of nitrogen, oxygen, argon, and air with Iinv the intensity leading to nKerr‖=0. The uncertainty corresponds to two standard deviations of the fitted values over a set of experimental records.

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bound excited electrons. The high-order terms of the Kerr effect have to be taken into account when modelling the light propagation in air. We recall that the filamentation process in gas is usually defined by the balancing between Kerr focusing and ionization defocusing. The present work suggests that the Kerr effect contributes to the defocusing mechanism. For some gases, the filamentation can be governed by the instantaneous effect, while the ionization plays a minor role [15

15. V. Loriot, P. Bájot, E. Hertz, O. Faucher, B. Lavorel, S. Henin, J. Kasparian, and J.-P. Wolf, “Higher-order Kerr terms allowing ionization-free filamentation in air,” Phys. Rev. Lett. (Submitted to).

, 16

16. G. Méchain, A. Couairon, Y.-B. André, C. d’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79(3), 379–382 (2004). [CrossRef]

]. In general, it will strongly depends on the gas and experimental conditions.

Fig. 2. Pure heterodyne signal in a 100 mbar of N2 and argon at room temperature for low, medium, and high intensity. (a 1a 3) N2 at 22 TW/cm2, 42 TW/cm2, and 49 TW/cm2, respectively. (b 1b 3) Ar at 18 TW/cm2, 24 TW/cm2, and 30 TW/cm2), respectively. A sign reversal of the Kerr component at zero delay is observed.
Fig. 3. Nonlinear refractive index variation of air constituents versus intensity at room temperature and 1 atm. (a) N2, (b) O2, (c) Ar, and (d) air.

4. Conclusion

New experimental determination of the nonlinear Kerr index of refraction of N2, O2, and Ar constituents have been performed. The time resolved birefringence method allows to measure the Kerr coefficients calibrated with the postpulse molecular alignment without the detrimental plasma contribution. At high intensity, the saturation of the electronic Kerr effect is observed, followed by a sign inversion above an intensity of few tens of TW/cm2 for the three gaseous components. This work reports, to our knowledge, the first experimental evidence of the sign inversion of Kerr terms at high intensity. The present result is expected to play a dominant role in the self guiding of ultrashort laser pulses. In particular, the usual description of a filament with the plasma as main defocusing contribution becomes questionable in consideration of the high-order Kerr terms revealed in the present work. An illustration of their influence reported in [15

15. V. Loriot, P. Bájot, E. Hertz, O. Faucher, B. Lavorel, S. Henin, J. Kasparian, and J.-P. Wolf, “Higher-order Kerr terms allowing ionization-free filamentation in air,” Phys. Rev. Lett. (Submitted to).

] demonstrates the possibility of plasma-free filamentation.

Acknowledgments

This work was supported by the Conseil Régional de Bourgogne, the ANR COMOC, and the FASTQUAST ITN Program of the 7th FP.

References and links

1.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007). [CrossRef]

2.

E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz “Determination of the inertial contribution to the nonlinear refractive index of air N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–660 (1997). J-F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependance of n2 in air,” Opt. Commun. 135, 310 (1997). [CrossRef]

3.

A. Couairon, “Dynamics of femtosecond filamentation from saturation of self-focusing laser pulses,” Phys. Rev. A 68(1), 015801 (2003). [CrossRef]

4.

R. Nuter and L. Bergé, “Pulse chirping and ionization of O2 molecules for the filamentation of femtosecond laser pulses in air,” J. Opt. Soc. Am. B 23, 874–884 (2006). [CrossRef]

5.

P. Sprangle, E. Esarey, and B. Hafizi, “Propagation and stability of intense laser pulses in partially stripped plasmas,” Phys. Rev. E 565894–5907 (1997).

6.

H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

7.

V. Renard, M. Renard, S. Guérin, Y. T. Pashayan, B. Lavorel, O. Faucher, and H. R. Jauslin, “Postpulse molecular alignment measured by a weak field polarization technique,” Phys. Rev. Lett. 90(15), 153601 (2003). [CrossRef] [PubMed]

8.

V. Loriot, P. Tzallas, E. P. Benis, E. Hertz, B. Lavorel, D. Charalambidis, and O. Faucher, “Laser-induced field-free alignment of the OCS molecule,” J. Phys. B 40(12), 2503–2510 (2007). [CrossRef]

9.

Maroulis, “Accurate electric multipole moment, static polarizability and hyperpolarizability derivatives for N2,” J, Chem. Phys. 118, 2673–2687 (2003). [CrossRef]

10.

E. Hertz, B. Lavorel, O. Faucher, and R. Chaux, “Femtosecond polarization spectroscopy in molecular gas mixtures: Macroscopic interference and concentration measurements,” J. Chem. Phys. 113(16), 6629–6633 (2000). [CrossRef]

11.

R. W. Boyd, Nonlinear opticsthird ed.Academic Press (2007).

12.

V. Loriot, E. Hertz, B. Lavorel, and O. Faucher, “Field-free molecular alignment for measuring ionization probability,” J. Phys. B 41(1), 015604 (2008). [CrossRef]

13.

P. Neogrády, M. Medvĕd, I. Černušàk, and M. Urban, “Benchmark calculations of some molecular properties of O2, CN and other selected small radicals using the ROHF-CCSD(T) method,” Mol. Phys. 100, 541 (2002). [CrossRef]

14.

J. Arabat and J. Etchepare, “Nonresonant fifth-order nonlinearities induced by ultrashort intense pulses,” J. Opt. Soc. Am. B 10(12), 2377–2382 (1993). [CrossRef]

15.

V. Loriot, P. Bájot, E. Hertz, O. Faucher, B. Lavorel, S. Henin, J. Kasparian, and J.-P. Wolf, “Higher-order Kerr terms allowing ionization-free filamentation in air,” Phys. Rev. Lett. (Submitted to).

16.

G. Méchain, A. Couairon, Y.-B. André, C. d’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79(3), 379–382 (2004). [CrossRef]

OCIS Codes
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(260.5950) Physical optics : Self-focusing
(320.2250) Ultrafast optics : Femtosecond phenomena
(350.5400) Other areas of optics : Plasmas

ToC Category:
Nonlinear Optics

History
Original Manuscript: April 29, 2009
Revised Manuscript: June 15, 2009
Manuscript Accepted: June 15, 2009
Published: July 21, 2009

Citation
Vincent Loriot, Edouard Hertz, Olivier Faucher, and Bruno Lavorel, "Measurement of high order Kerr refractive index of major air components," Opt. Express 17, 13429-13434 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13429


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References

  1. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007). [CrossRef]
  2. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14(3), 650–660 (1997). [CrossRef]
  3. A. Couairon, “Dynamics of femtosecond filamentation from saturation of self-focusing laser pulses,” Phys. Rev. A 68(1), 015801 (2003). [CrossRef]
  4. R. Nuter and L. Bergé, “Pulse chirping and ionization of O2 molecules for the filamentation of femtosecond laser pulses in air,” J. Opt. Soc. Am. B 23(5), 874–884 (2006). [CrossRef]
  5. P. Sprangle, E. Esarey, and B. Hafizi, “Propagation and stability of intense laser pulses in partially stripped plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(5), 5894–5907 (1997).
  6. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]
  7. V. Renard, M. Renard, S. Guérin, Y. T. Pashayan, B. Lavorel, O. Faucher, and H. R. Jauslin, “Postpulse molecular alignment measured by a weak field polarization technique,” Phys. Rev. Lett. 90(15), 153601 (2003). [CrossRef] [PubMed]
  8. V. Loriot, P. Tzallas, E. P. Benis, E. Hertz, B. Lavorel, D. Charalambidis, and O. Faucher, “Laser-induced field-free alignment of the OCS molecule,” J. Phys. B 40(12), 2503–2510 (2007). [CrossRef]
  9. G. Maroulis, “Accurate electric multipole moment, static polarizability and hyperpolarizability derivatives for N2,” J. Chem. Phys. 118(6), 2673–2687 (2003). [CrossRef]
  10. E. Hertz, B. Lavorel, O. Faucher, and R. Chaux, “Femtosecond polarization spectroscopy in molecular gas mixtures: Macroscopic interference and concentration measurements,” J. Chem. Phys. 113(16), 6629–6633 (2000). [CrossRef]
  11. R. W. Boyd, Nonlinear optics third ed. Academic Press (2007).
  12. V. Loriot, E. Hertz, B. Lavorel, and O. Faucher, “Field-free molecular alignment for measuring ionization probability,” J. Phys. B 41(1), 015604 (2008). [CrossRef]
  13. P. Neogrády, M. Medved, I. Cernušàk, and M. Urban, “Benchmark calculations of some molecular properties of O2, CN and other selected small radicals using the ROHF-CCSD(T) method,” Mol. Phys. 100, 541 (2002). [CrossRef]
  14. J. Arabat and J. Etchepare, “Nonresonant fifth-order nonlinearities induced by ultrashort intense pulses,” J. Opt. Soc. Am. B 10(12), 2377–2382 (1993). [CrossRef]
  15. V. Loriot, P. Béjot, E. Hertz, O. Faucher, B. Lavorel, S. Henin, J. Kasparian, and J.-P. Wolf, “Higher-order Kerr terms allowing ionization-free filamentation in air,” Phys. Rev. Lett. (Submitted to).
  16. G. Méchain, A. Couairon, Y.-B. André, C. d’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79(3), 379–382 (2004). [CrossRef]

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