OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 24301–24306
« Show journal navigation

Light filaments with higher-order Kerr effect

Haitao Wang, Chengyu Fan, Pengfei Zhang, Chunhong Qiao, Jinghui Zhang, and Huimin Ma  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 24301-24306 (2010)
http://dx.doi.org/10.1364/OE.18.024301


View Full Text Article

Acrobat PDF (952 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The influence mechanism of higher-order Kerr effect on the propagation of laser beam is investigated by a modified model, which indicates that a collapsing wave will transform into a universe blowup profile. The analysis of higher-order terms of the nonlinear refractive index shows that the filamentation process can be induced by Kerr self-focusing without the occurrence of the ionization effect. The determining role of the combination of self-focusing and spontaneous defocusing and the energy reservoir in formation of lengthy filament is confirmed visually.

© 2010 OSA

1. Introduction

In the view of practical applications, much attention has been focused on the long range propagation of filamentation in air for higher power laser beams in recent years [7

7. W. Liu, J.-F. Gravel, F. Th’eberge, A. Becker, and S. L. Chin, “Background reservoir its crucial role for longdistance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80(7), 857–860 (2005). [CrossRef]

, 8

8. E. P. Silaeva and V. P. Kandidov, “Propagation of a High-Power Femtosecond Pulse Filament through a Layer of Aerosol,” Atmos. Oceanic Opt. 22(1), 26–34 (2009). [CrossRef]

]. The usual mechanism responsible for the long range propagation of these filaments is identified as the dynamic balance between self-focusing due to the nonlinear refractive index of the air and self-defocusing due to the axial plasma generated by multiphoton ionization (MPI) of the air molecules and the diffraction of the laser beam, when the input power of the beam is higher than the critical power of self-focusing. Theoretically, several models based on self-channeling a moving focus [9

9. A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997). [CrossRef] [PubMed]

] or dynamic spatial replenishment [10

10. M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83(15), 2938–2941 (1999). [CrossRef]

] addressed a dynamic equilibrium to comprehend the process of the filamentation. However, Méchain et al. [11

11. G. M’ejean, A. Couairon, Y.-B. Andr’e, 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]

] experimentally observed a new propagation regime with noionizated channels extending over the distance of 2 kilometers in air. They measured the maximum intensity that does not exceed a few1012 W/cm2 in the light channels. Subsequently, Ruiz et al. [12

12. C. Ruiz, J. San Román, C. Méndez, V. Díaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95(5), 053905 (2005). [CrossRef] [PubMed]

] proposed the spontaneous generation of a filament without ionization in air due to soliton propagation for laser power lower than the collapse threshold. Based on numerical calculation, the observation of nonlinear saturation compressing the collapse of the self-focusing was reported in Ref [13

13. G. Fibich, N. Gavish, and X. P. Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211(3-4), 193–220 (2005). [CrossRef]

].

2. Numerical simulation model

We consider a linearly polarized incident electric field at λ0 = 800 nm, with cylindrical symmetry around the propagation axis z. The scalar envelope is assumed to be slowly varying in time and along z axis, it evolves according to the nonlinear propagation Schrödinger equation will be modified as follows:
Ez=i2k0(2x2+2y2)Eik''22t2E+ik0n2n0|E|2E+ik0n0(j=25n2*j|E|2*j)Eβ(K)2|E|(2K2)E
(1)
Where z is the longitudinal distance propagation, while 2=2/x2+2/y2 accounts for the diffraction. This is a multispecies code and there, in the case of air, it sortsN2,O2andAr, with the variable indices coefficients n2*j as described in the Ref [14

14. 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 (2009) (Erratum in Opt. Express 18, 3011 ) (2010). [CrossRef] [PubMed]

]. In Eq. (1), k0=ω0/c=2πn0/λ0, and ω0=2πc/λ0 are the wave number and the angular frequency of the carrier wave, respectively. The critical power for self-focusing is defined by pcr=3.77λ02/8πn0n2 and in air it takes the value pcr2.54GW for the laser foundational wavelength λ0 = 800 nm, n0is the linear refractive index. The second-order temporal derivation refers to normal group-velocity dispersion (GVD) with coefficient k"=k/ω|ω0=0.2fs2/cm. The higher-order Kerr terms at the right hand fourth term are considered here, because the incident laser intensity may increase to saturation levels for which χ(2*j+1) susceptibilities becomes efficient, i.e., the χ(5) susceptibility contributes to n4 is negative that acts as a defocusing effect. It seems that it can stabilize the propagation of laser beam with no occurrence collapse and lead to decrease both the maximal plasma density and the maximal on-axis intensity of the beam. The last term describes the power dissipation assured by multiphoton absorption (MPA) with coefficient β(K)=3.1×1098cm2K3/WK1, K = 8 is the number of photons needed to extract electrons from neutral molecules with the lowest ionization potential, i.e., dioxygen molecules, in air, the ionization of oxygen molecules with gap potential Ui=12.1eV.

3. Numerical simulations and discussions

However, the spatial solitonlike beam of self-channeling effect is not steady due to modulational instabilities in the transverse cross-section of the beam, which will exhibit a complex spatial feature. Figure 1 illustrates that the propagation dynamical processes will lead to the intensity spikes, and the spatial intensity profiles and their variation at different positions along the propagation axis are shown later for further details. At lower power, no ring appears during the self-focusing stage. The higher-order Kerr terms that occurs beyond the nonlinear focus does not even dig any hole in the transverse distribution. In contrast, at slightly higher power, inhomogeneities of the beam are amplified during the self-focusing stage and clearly engender a ring at z>9.8m [see Fig. 1(b)] as a result of modulational instability, and the higher-order Kerr effect is not significant until z~10.8m, as shown in Fig. 1(c). The tendency is even more enhanced at z~11.2m, where a deeper ring structure appears during the second self-focusing stage [Fig. 1(d)] and finally coalesces into a single filament, as can be seen in Figs. 1(e) and 1(f). Typically, the filament possessing a central core from tens to several hundreds of microns can propagate over tens of meters in air, much longer than the Rayleigh length [17

17. M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev. Lett. 103(4), 043901 (2009). [CrossRef] [PubMed]

].

At the low level of intensity, the energy losses due to MPI and dissipation caused by plasma absorption do not play dominant roles in the filamentation. When the central single spike is formed, the converged intensity increases quickly to the maximum. Then the higher-order terms of the Kerr effect will become to play a dominant role, and produce an additional nonlinear perturbation into the phase of the beam. Later on, modulational instability of the transverse cross-section gives rise to the gradient of the intensity flowing perpendicular to the propagation axis. A small portion of the energy of neighboring filaments will leak out and form a ring surrounding the main core that is shown in Fig. 2
Fig. 2 (Color online) The intensity profile of the beam at (a) z = 12.7 m and (b) z = 14.1m, showing the presence of a narrow core surrounded by a weaker ring. It is shown that the robustness of the filament comes from the transverse low-intensity pedestal that controls the formation of the central hot spot.
for two different places. Subsequently the central intensity will undergo a decrease to form a hollow pipe and the higher-order effect also becomes weaker than the Kerr quadratic and the energy begin to flows from the peripheral region towards both the center filament and the near-filament region before the occurrence of another peak. Once more, the higher-order effect due to the increasing energy in the central region corresponding to the energy extraction from the external regions will cause defocusing. As a result, the combination of this focusing and spontaneous defocusing can maintain the long range propagation of the laser beam.

It should be noticed that the slow multiple refocusing of low-intensity background can continually replenish the energy loss in the core and supports the filamentation process over long distance. When the holes are dipped in the transverse beam section, they always appear near the nonlinear focus, as the intensity increase abruptly as shown in Figs. 3(a)
Fig. 3 (Color online) (a) The energy fluence distribution in the filament along the propagation axis z; (b) On-axis peak intensity vs propagation distance, and(c) Iso-intensity (iso = 0.5) distribution of the filament.
and 3(b). Figure 3(c) shows the three-dimensional plot of the surface corresponding to equal values of the intensity along the propagating axis. They only result from the defocusing effect of the higher-order terms in the Eq. (1).

4. Conclusion

Acknowledgments

This work was supported by the National High Technology Development Program of China (Grant Nnos.A825021 and A825011), and Computational Center of Hefei Institutes of Physical Science, Chinese Academy of Sciences (Grant No.0330405002-7).

References and links

1.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995). [CrossRef] [PubMed]

2.

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

3.

A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008). [CrossRef] [PubMed]

4.

A. Couairon, S. C. Himadri, and B. G. Mette, “From single-cycle self-compressed filaments to isolated attosecond pulses in noble gases,” Phys. Rev. A 77(5), 053814 (2008). [CrossRef]

5.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003). [CrossRef] [PubMed]

6.

J.-F. Daigle, G. Méjean, W. Liu, F. Théberge, H. L. Xu, Y. Kamali, J. Bernardt, A. Azarm, A. Sun, P. Mathieu, G. Roy, J. R. Simard, and S. L. Chin, “Long range trace detection in aqueous aerosol using remotefilament-induced breakdown spectroscopy,” Appl. Phys. B 87(4), 749–754 (2007). [CrossRef]

7.

W. Liu, J.-F. Gravel, F. Th’eberge, A. Becker, and S. L. Chin, “Background reservoir its crucial role for longdistance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80(7), 857–860 (2005). [CrossRef]

8.

E. P. Silaeva and V. P. Kandidov, “Propagation of a High-Power Femtosecond Pulse Filament through a Layer of Aerosol,” Atmos. Oceanic Opt. 22(1), 26–34 (2009). [CrossRef]

9.

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997). [CrossRef] [PubMed]

10.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83(15), 2938–2941 (1999). [CrossRef]

11.

G. M’ejean, A. Couairon, Y.-B. Andr’e, 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]

12.

C. Ruiz, J. San Román, C. Méndez, V. Díaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95(5), 053905 (2005). [CrossRef] [PubMed]

13.

G. Fibich, N. Gavish, and X. P. Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211(3-4), 193–220 (2005). [CrossRef]

14.

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 (2009) (Erratum in Opt. Express 18, 3011 ) (2010). [CrossRef] [PubMed]

15.

T. D. Grow, A. A. Ishaaya, L. T. Vuong, A. L. Gaeta, N. Gavish, and G. Fibich, “Collapse dynamics of super-Gaussian Beams,” Opt. Express 14(12), 5468–5475 (2006). [CrossRef] [PubMed]

16.

V. P. Kandidov, O. G. Kosareva, I. S. Golubstov, W. Liu, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Self-transformation of a powerful femtosecond laser pulse into a white-light laser pulse in bulk optical media(or supercontinuum generation),” Appl. Phys. B 77(2-3), 149–165 (2003). [CrossRef]

17.

M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev. Lett. 103(4), 043901 (2009). [CrossRef] [PubMed]

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(260.5950) Physical optics : Self-focusing

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 10, 2010
Revised Manuscript: September 26, 2010
Manuscript Accepted: October 22, 2010
Published: November 5, 2010

Citation
Haitao Wang, Chengyu Fan, Pengfei Zhang, Chunhong Qiao, Jinghui Zhang, and Huimin Ma, "Light filaments with higher-order Kerr effect," Opt. Express 18, 24301-24306 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24301


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995). [CrossRef] [PubMed]
  2. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007). [CrossRef]
  3. A. Houard, Y. Liu, B. Prade, V. T. Tikhonchuk, and A. Mysyrowicz, “Strong enhancement of terahertz radiation from laser filaments in air by a static electric field,” Phys. Rev. Lett. 100(25), 255006 (2008). [CrossRef] [PubMed]
  4. A. Couairon, S. C. Himadri, and B. G. Mette, “From single-cycle self-compressed filaments to isolated attosecond pulses in noble gases,” Phys. Rev. A 77(5), 053814 (2008). [CrossRef]
  5. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003). [CrossRef] [PubMed]
  6. J.-F. Daigle, G. Méjean, W. Liu, F. Théberge, H. L. Xu, Y. Kamali, J. Bernardt, A. Azarm, A. Sun, P. Mathieu, G. Roy, J. R. Simard, and S. L. Chin, “Long range trace detection in aqueous aerosol using remotefilament-induced breakdown spectroscopy,” Appl. Phys. B 87(4), 749–754 (2007). [CrossRef]
  7. W. Liu, J.-F. Gravel, F. Th’eberge, A. Becker, and S. L. Chin, “Background reservoir its crucial role for longdistance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80(7), 857–860 (2005). [CrossRef]
  8. E. P. Silaeva and V. P. Kandidov, “Propagation of a High-Power Femtosecond Pulse Filament through a Layer of Aerosol,” Atmos. Oceanic Opt. 22(1), 26–34 (2009). [CrossRef]
  9. A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997). [CrossRef] [PubMed]
  10. M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83(15), 2938–2941 (1999). [CrossRef]
  11. G. M’ejean, A. Couairon, Y.-B. Andr’e, 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]
  12. C. Ruiz, J. San Román, C. Méndez, V. Díaz, L. Plaja, I. Arias, and L. Roso, “Observation of spontaneous self-channeling of light in air below the collapse threshold,” Phys. Rev. Lett. 95(5), 053905 (2005). [CrossRef] [PubMed]
  13. G. Fibich, N. Gavish, and X. P. Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211(3-4), 193–220 (2005). [CrossRef]
  14. 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 (2009) (Erratum in Opt. Express 18, 3011 ) (2010). [CrossRef] [PubMed]
  15. T. D. Grow, A. A. Ishaaya, L. T. Vuong, A. L. Gaeta, N. Gavish, and G. Fibich, “Collapse dynamics of super-Gaussian Beams,” Opt. Express 14(12), 5468–5475 (2006). [CrossRef] [PubMed]
  16. V. P. Kandidov, O. G. Kosareva, I. S. Golubstov, W. Liu, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Self-transformation of a powerful femtosecond laser pulse into a white-light laser pulse in bulk optical media(or supercontinuum generation),” Appl. Phys. B 77(2-3), 149–165 (2003). [CrossRef]
  17. M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev. Lett. 103(4), 043901 (2009). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited