Abstract

Whereas the Kerr nonlinearity is well understood in the perturbative limit of nonlinear optics, there is considerable discussion about its functional form and magnitude at extreme intensities, at which point matter starts to ionize. Here, we introduce a concept to answer this question and theoretically analyze its feasibility. We demonstrate that seeded Kerr instability amplification provides clear signatures from which functional form and magnitude of the Kerr nonlinearity can be extracted in the non-perturbative limit of nonlinear optics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref]
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2018 (1)

2017 (1)

2016 (1)

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light–matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref] [PubMed]

2015 (5)

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonic generation from gases and solids,” Nature 522, 462–464 (2015).
[Crossref] [PubMed]

M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. W. Koch, M. Kira, and R. Huber, “Real time observation of interfering crystal electrons in high-harmonic generation,” Nature 523, 572–575 (2015).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All–optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher–order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114(9), 093901 (2015).
[Crossref] [PubMed]

S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, “Measurement of the nonlinear refractive index of air constituents at mid–infrared wavelengths,” Opt. Lett. 40(24), 5794–5797 (2015).
[Crossref] [PubMed]

2014 (2)

N. Jhajj, E. W. Rosenthal, R. Birnbaum, J. K. Wahlstrand, and H. M. Milchberg, “Demonstration of long–lived high–power optical waveguides in air,” Phys. Rev. X 4(1), 011027 (2014).

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

2013 (3)

A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Optical–field–induced current in dielectrics,” Nature 493, 70–74 (2013).
[Crossref]

M. Schultze, E. M. Bothschafter, A. Sommer, S. Holzner, W. Schweinberger, M. Fiess, M. Hofstetter, R. Kienberger, V. Apalkov, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Controlling dielectrics with the electric field of light,” Nature 493, 75–78 (2013).
[Crossref]

Y. Liu, Y. Brelet, G. Point, A. Houard, and A. Mysyrowicz, “Self–seeded lasing in ionized air pumped by 800 nm femtosecond laser pulses,” Opt. Express 21(19), 22791–22798 (2013).
[Crossref] [PubMed]

2012 (4)

J. Schwarz, P. Rambo, M. Kimmel, and B. Atherton, “Measurement of nonlinear refractive index and ionization rates in air using a wavefront sensor,” Opt. Express,  20(8), 8791–8803 (2012).
[Crossref] [PubMed]

M. Petrarca, Y. Petit, S. Henin, R. Delagrange, P. Bejot, and J. Kasparian, “Higher-order Kerr improve quantitative modeling of laser filamentation,” Opt. Lett. 37(20), 4347–4349 (2012).
[Crossref] [PubMed]

T. Vockerodt, D. S. Steingrube, E. Schulz, M. Kretschmar, U. Morgner, and M. Kovacev, “Low– and high–order harmonic generation inside an air filament,” Appl. Phys. B 106(3), 529–532 (2012).
[Crossref]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron–hole recollisions,” Nature 483, 580–583 (2012).
[Crossref] [PubMed]

2011 (6)

H. J. Wörner, J. B. Bertrand, B. Fabre, J. Higuet, H. Ruf, A. Dubrouil, S. Patchkovskii, M. Spanner, Y. Mairesse, V. Blanchet, E. Mevel, E. Constant, P. B. Corkum, and D. M. Villeneuve, “Conical intersection dynamics in NO2 probed by homodyne high-harmonic spectroscopy,” Science 334(6053), 208–212 (2011).
[Crossref] [PubMed]

S. Ghimire, A. D. Dichiara, E. Sistrunk, P. Agostini, and L. F. Dimauro, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

M. Krüger, M. Schenk, and P. Hommelhoff, “Attosecond control of electrons emitted from a nanoscale metal tip,” Nature 475, 78–81 (2011).
[Crossref] [PubMed]

V. Loriot, P. Bejot, W. Ettoumi, Y. Petit, J. Kasparian, S. Henin, E. Hertz, B. Lavorel, O. Faucher, and J. Wolf, “On negative higher–order Kerr effect and filamentation,” Laser Phys. 21(7), 1319–1328 (2011).
[Crossref]

P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011).
[Crossref] [PubMed]

P. Bejot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, “General approach to spatiotemporal modulational instability processes,” Phys. Rev. A 83(1), 013830 (2011).
[Crossref]

2010 (3)

P. Bejot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. Wolf, “Higher–order Kerr terms allow ionization–free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010).
[Crossref]

T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics 4, 188–193 (2010).

H. J. Wörner, J. B. Bertrand, D. V. Kartashov, P. B. Corkum, and D. M. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
[Crossref] [PubMed]

2009 (4)

2008 (2)

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser gas interactions,” Nature Photon. 2, 605–609 (2008).
[Crossref]

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

2007 (4)

P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3, 381–387 (2007).
[Crossref]

L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[Crossref]

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

C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and V. T. Tikhonchuk, “Conical forward THz emission from femtosecond–laser–beam filamentation in air,” Phys. Rev. Lett. 98(23), 235002 (2007).
[Crossref]

2006 (1)

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312(5772), 424–427 (2006).
[Crossref] [PubMed]

2004 (2)

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867–871 (2004).
[Crossref] [PubMed]

A. Vincotte and L. Berge, “χ(5) susceptibility stabilizes the propagation of ultrashort laser pulses in air,” Phys. Rev. A 70(6), 061802 (2004).
[Crossref]

2002 (1)

2000 (1)

A. Couairon and L. Berge, “Modeling the filamentation of ultra–short pulses in ionizing media,” Phys. of Plasmas,  7(1), 193–209 (2000).
[Crossref]

1998 (1)

1992 (1)

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46(7), 4202–4208 (1992).
[Crossref] [PubMed]

1990 (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quant. Electr. 26(4), 760–769 (1990).
[Crossref]

1989 (1)

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(11), 471–476 (1966).

Afonso, C. N.

Agostini, P.

S. Ghimire, A. D. Dichiara, E. Sistrunk, P. Agostini, and L. F. Dimauro, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2012).

Agrawal, G. P.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46(7), 4202–4208 (1992).
[Crossref] [PubMed]

Almasi, G.

L. Palfalvi, B. Toth, G. Almasi, J. Fulop, and J. Hebling, “A general Z–scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[Crossref]

Apalkov, V.

A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Optical–field–induced current in dielectrics,” Nature 493, 70–74 (2013).
[Crossref]

M. Schultze, E. M. Bothschafter, A. Sommer, S. Holzner, W. Schweinberger, M. Fiess, M. Hofstetter, R. Kienberger, V. Apalkov, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Controlling dielectrics with the electric field of light,” Nature 493, 75–78 (2013).
[Crossref]

Atherton, B.

Baker, S.

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312(5772), 424–427 (2006).
[Crossref] [PubMed]

Bart, G.

Barth, J. V.

A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Optical–field–induced current in dielectrics,” Nature 493, 70–74 (2013).
[Crossref]

Bejot, P.

J. Doussot, G. Karras, F. Billard, P. Bejot, and O. Faucher, “Resonantly enhanced filamentation in gases,” Optica 4(7), 764–769 (2017).
[Crossref]

M. Petrarca, Y. Petit, S. Henin, R. Delagrange, P. Bejot, and J. Kasparian, “Higher-order Kerr improve quantitative modeling of laser filamentation,” Opt. Lett. 37(20), 4347–4349 (2012).
[Crossref] [PubMed]

P. Bejot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, “General approach to spatiotemporal modulational instability processes,” Phys. Rev. A 83(1), 013830 (2011).
[Crossref]

V. Loriot, P. Bejot, W. Ettoumi, Y. Petit, J. Kasparian, S. Henin, E. Hertz, B. Lavorel, O. Faucher, and J. Wolf, “On negative higher–order Kerr effect and filamentation,” Laser Phys. 21(7), 1319–1328 (2011).
[Crossref]

P. Bejot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. Wolf, “Higher–order Kerr terms allow ionization–free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010).
[Crossref]

P. Maioli, R. Salame, N. Lascoux, E. Salmon, P. Bejot, J. Kasparian, and J. P. Wolf, “Ultraviolet–visible conical emission by multiple laser filaments,” Opt. Express 17(6), 4726–4731 (2009).
[Crossref] [PubMed]

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Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quant. Electr. 26(4), 760–769 (1990).
[Crossref]

Sherwin, M. S.

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron–hole recollisions,” Nature 483, 580–583 (2012).
[Crossref] [PubMed]

Shim, B.

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher–order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114(9), 093901 (2015).
[Crossref] [PubMed]

Shirvanyan, V.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light–matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref] [PubMed]

Sistrunk, E.

S. Ghimire, A. D. Dichiara, E. Sistrunk, P. Agostini, and L. F. Dimauro, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

Skupin, S.

L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[Crossref]

Smith, R. A.

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312(5772), 424–427 (2006).
[Crossref] [PubMed]

Solis, J.

Sommer, A.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light–matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref] [PubMed]

M. Schultze, E. M. Bothschafter, A. Sommer, S. Holzner, W. Schweinberger, M. Fiess, M. Hofstetter, R. Kienberger, V. Apalkov, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Controlling dielectrics with the electric field of light,” Nature 493, 75–78 (2013).
[Crossref]

Spanner, M.

H. J. Wörner, J. B. Bertrand, B. Fabre, J. Higuet, H. Ruf, A. Dubrouil, S. Patchkovskii, M. Spanner, Y. Mairesse, V. Blanchet, E. Mevel, E. Constant, P. B. Corkum, and D. M. Villeneuve, “Conical intersection dynamics in NO2 probed by homodyne high-harmonic spectroscopy,” Science 334(6053), 208–212 (2011).
[Crossref] [PubMed]

Steingrube, D. S.

T. Vockerodt, D. S. Steingrube, E. Schulz, M. Kretschmar, U. Morgner, and M. Kovacev, “Low– and high–order harmonic generation inside an air filament,” Appl. Phys. B 106(3), 529–532 (2012).
[Crossref]

Stockman, M. I.

A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Optical–field–induced current in dielectrics,” Nature 493, 70–74 (2013).
[Crossref]

M. Schultze, E. M. Bothschafter, A. Sommer, S. Holzner, W. Schweinberger, M. Fiess, M. Hofstetter, R. Kienberger, V. Apalkov, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Controlling dielectrics with the electric field of light,” Nature 493, 75–78 (2013).
[Crossref]

Stolen, R. H.

Suarez-Garcia, A.

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(11), 471–476 (1966).

Taylor, A. J.

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser gas interactions,” Nature Photon. 2, 605–609 (2008).
[Crossref]

Teng, H.

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312(5772), 424–427 (2006).
[Crossref] [PubMed]

Thire, N.

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All–optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonic generation from gases and solids,” Nature 522, 462–464 (2015).
[Crossref] [PubMed]

Tikhonchuk, V. T.

C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and V. T. Tikhonchuk, “Conical forward THz emission from femtosecond–laser–beam filamentation in air,” Phys. Rev. Lett. 98(23), 235002 (2007).
[Crossref]

Tisch, J. W. G.

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312(5772), 424–427 (2006).
[Crossref] [PubMed]

Toth, B.

L. Palfalvi, B. Toth, G. Almasi, J. Fulop, and J. Hebling, “A general Z–scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[Crossref]

Vampa, G.

M. Nesrallah, G. Vampa, G. Bart, P. B. Corkum, C. R. McDonald, and T. Brabec, “Theory of Kerr instability amplification,” Optica 5(3), 271–278 (2018).
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G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All–optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonic generation from gases and solids,” Nature 522, 462–464 (2015).
[Crossref] [PubMed]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quant. Electr. 26(4), 760–769 (1990).
[Crossref]

Vieillard, T.

P. Bejot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. Wolf, “Higher–order Kerr terms allow ionization–free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010).
[Crossref]

Villeneuve, D. M.

H. J. Wörner, J. B. Bertrand, B. Fabre, J. Higuet, H. Ruf, A. Dubrouil, S. Patchkovskii, M. Spanner, Y. Mairesse, V. Blanchet, E. Mevel, E. Constant, P. B. Corkum, and D. M. Villeneuve, “Conical intersection dynamics in NO2 probed by homodyne high-harmonic spectroscopy,” Science 334(6053), 208–212 (2011).
[Crossref] [PubMed]

H. J. Wörner, J. B. Bertrand, D. V. Kartashov, P. B. Corkum, and D. M. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
[Crossref] [PubMed]

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867–871 (2004).
[Crossref] [PubMed]

Vincotte, A.

A. Vincotte and L. Berge, “χ(5) susceptibility stabilizes the propagation of ultrashort laser pulses in air,” Phys. Rev. A 70(6), 061802 (2004).
[Crossref]

Vockerodt, T.

T. Vockerodt, D. S. Steingrube, E. Schulz, M. Kretschmar, U. Morgner, and M. Kovacev, “Low– and high–order harmonic generation inside an air filament,” Appl. Phys. B 106(3), 529–532 (2012).
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S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, “Measurement of the nonlinear refractive index of air constituents at mid–infrared wavelengths,” Opt. Lett. 40(24), 5794–5797 (2015).
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N. Jhajj, E. W. Rosenthal, R. Birnbaum, J. K. Wahlstrand, and H. M. Milchberg, “Demonstration of long–lived high–power optical waveguides in air,” Phys. Rev. X 4(1), 011027 (2014).

Weerawarne, D. L.

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher–order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114(9), 093901 (2015).
[Crossref] [PubMed]

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M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quant. Electr. 26(4), 760–769 (1990).
[Crossref]

Wolf, J.

V. Loriot, P. Bejot, W. Ettoumi, Y. Petit, J. Kasparian, S. Henin, E. Hertz, B. Lavorel, O. Faucher, and J. Wolf, “On negative higher–order Kerr effect and filamentation,” Laser Phys. 21(7), 1319–1328 (2011).
[Crossref]

P. Bejot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. Wolf, “Higher–order Kerr terms allow ionization–free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010).
[Crossref]

Wolf, J. P.

P. Maioli, R. Salame, N. Lascoux, E. Salmon, P. Bejot, J. Kasparian, and J. P. Wolf, “Ultraviolet–visible conical emission by multiple laser filaments,” Opt. Express 17(6), 4726–4731 (2009).
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L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[Crossref]

Wörner, H. J.

H. J. Wörner, J. B. Bertrand, B. Fabre, J. Higuet, H. Ruf, A. Dubrouil, S. Patchkovskii, M. Spanner, Y. Mairesse, V. Blanchet, E. Mevel, E. Constant, P. B. Corkum, and D. M. Villeneuve, “Conical intersection dynamics in NO2 probed by homodyne high-harmonic spectroscopy,” Science 334(6053), 208–212 (2011).
[Crossref] [PubMed]

H. J. Wörner, J. B. Bertrand, D. V. Kartashov, P. B. Corkum, and D. M. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
[Crossref] [PubMed]

Wright, E. M.

P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011).
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A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light–matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref] [PubMed]

Yakovlev, V. S.

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light–matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref] [PubMed]

M. Schultze, E. M. Bothschafter, A. Sommer, S. Holzner, W. Schweinberger, M. Fiess, M. Hofstetter, R. Kienberger, V. Apalkov, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Controlling dielectrics with the electric field of light,” Nature 493, 75–78 (2013).
[Crossref]

A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Optical–field–induced current in dielectrics,” Nature 493, 70–74 (2013).
[Crossref]

Zahedpour, S.

Zaks, B.

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron–hole recollisions,” Nature 483, 580–583 (2012).
[Crossref] [PubMed]

Zeidler, D.

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867–871 (2004).
[Crossref] [PubMed]

Appl. Phys. B (2)

T. Vockerodt, D. S. Steingrube, E. Schulz, M. Kretschmar, U. Morgner, and M. Kovacev, “Low– and high–order harmonic generation inside an air filament,” Appl. Phys. B 106(3), 529–532 (2012).
[Crossref]

L. Palfalvi, B. Toth, G. Almasi, J. Fulop, and J. Hebling, “A general Z–scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[Crossref]

IEEE J. Quant. Electr. (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quant. Electr. 26(4), 760–769 (1990).
[Crossref]

J. Opt. Soc. Am. B (2)

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(11), 471–476 (1966).

Laser Phys. (1)

V. Loriot, P. Bejot, W. Ettoumi, Y. Petit, J. Kasparian, S. Henin, E. Hertz, B. Lavorel, O. Faucher, and J. Wolf, “On negative higher–order Kerr effect and filamentation,” Laser Phys. 21(7), 1319–1328 (2011).
[Crossref]

Nat. Photonics (3)

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

M. Farsaril and B. N. Chichkov, “Materials processing: two–photon fabrication,” Nat. Photonics 3, 450–452 (2009).
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T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics 4, 188–193 (2010).

Nat. Phys. (2)

P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3, 381–387 (2007).
[Crossref]

S. Ghimire, A. D. Dichiara, E. Sistrunk, P. Agostini, and L. F. Dimauro, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

Nature (9)

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron–hole recollisions,” Nature 483, 580–583 (2012).
[Crossref] [PubMed]

M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. W. Koch, M. Kira, and R. Huber, “Real time observation of interfering crystal electrons in high-harmonic generation,” Nature 523, 572–575 (2015).
[Crossref] [PubMed]

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867–871 (2004).
[Crossref] [PubMed]

H. J. Wörner, J. B. Bertrand, D. V. Kartashov, P. B. Corkum, and D. M. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
[Crossref] [PubMed]

A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit, T. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W. Schweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kienberger, K. Yabana, N. Karpowicz, M. Schultze, and F. Krausz, “Attosecond nonlinear polarization and light–matter energy transfer in solids,” Nature 534, 86–90 (2016).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonic generation from gases and solids,” Nature 522, 462–464 (2015).
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M. Krüger, M. Schenk, and P. Hommelhoff, “Attosecond control of electrons emitted from a nanoscale metal tip,” Nature 475, 78–81 (2011).
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A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Mühlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Optical–field–induced current in dielectrics,” Nature 493, 70–74 (2013).
[Crossref]

M. Schultze, E. M. Bothschafter, A. Sommer, S. Holzner, W. Schweinberger, M. Fiess, M. Hofstetter, R. Kienberger, V. Apalkov, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Controlling dielectrics with the electric field of light,” Nature 493, 75–78 (2013).
[Crossref]

Nature Photon. (1)

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser gas interactions,” Nature Photon. 2, 605–609 (2008).
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Opt. Express (4)

Opt. Lett. (3)

Optica (2)

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A. Couairon and L. Berge, “Modeling the filamentation of ultra–short pulses in ionizing media,” Phys. of Plasmas,  7(1), 193–209 (2000).
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Phys. Rep. (1)

A. Couairon and A. Mysyrowicz,” Femtosecond filamentation in transparent media,” Phys. Rep. 441(47), 47–189 (2007).
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Phys. Rev. A (3)

A. Vincotte and L. Berge, “χ(5) susceptibility stabilizes the propagation of ultrashort laser pulses in air,” Phys. Rev. A 70(6), 061802 (2004).
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Phys. Rev. Lett. (6)

C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and V. T. Tikhonchuk, “Conical forward THz emission from femtosecond–laser–beam filamentation in air,” Phys. Rev. Lett. 98(23), 235002 (2007).
[Crossref]

P. Bejot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J. Wolf, “Higher–order Kerr terms allow ionization–free filamentation in gases,” Phys. Rev. Lett. 104(10), 103903 (2010).
[Crossref]

P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011).
[Crossref] [PubMed]

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher–order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114(9), 093901 (2015).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All–optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Phys. Rev. X (1)

N. Jhajj, E. W. Rosenthal, R. Birnbaum, J. K. Wahlstrand, and H. M. Milchberg, “Demonstration of long–lived high–power optical waveguides in air,” Phys. Rev. X 4(1), 011027 (2014).

Rep. Prog. Phys. (1)

L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[Crossref]

Science (2)

H. J. Wörner, J. B. Bertrand, B. Fabre, J. Higuet, H. Ruf, A. Dubrouil, S. Patchkovskii, M. Spanner, Y. Mairesse, V. Blanchet, E. Mevel, E. Constant, P. B. Corkum, and D. M. Villeneuve, “Conical intersection dynamics in NO2 probed by homodyne high-harmonic spectroscopy,” Science 334(6053), 208–212 (2011).
[Crossref] [PubMed]

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312(5772), 424–427 (2006).
[Crossref] [PubMed]

Other (2)

R. W. Boyd, Nonlinear Optics, 3rd. ed. (Academic Press, 2003).

G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2012).

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Figures (4)

Fig. 1
Fig. 1 Schematic of the nonlinear index measurement. A strong, monochromatic pump beam (thick solid blue arrow), interacts non-collinearly with a weak monochromatic seed beam (thin dashed red arrow), inclined at an angle θo (angles exaggerated for clarity), both at the same frequency in a nonlinear medium. The seed beam is refracted at an angle θ, and then through nonlinear instability, it is amplified (thicker dashed red arrow). The pump beam remains approximately undistorted. The non-collinear angle is used to calculate the nonlinear index of refraction as shown in this work.
Fig. 2
Fig. 2 (a) Possible forms for f(nn) defined by Eq. (13) for s = 0 (solid blue line), s = 1/2 (green dashed line), s = 1 (dotted red line), and s = 2 (dashed dotted purple line) in air for pump wavelength λp = 0.8 μm, n = 1.0003 and n2 = 5 × 10−19 cm2/W, both taken from [17]. The Raman parameters for air are ν1 = 16 THz, ν2 = 13 THz and r = 1/2 [18]. Saturation intensity is estimated to be Ip = 17 TW/cm2, consistent with Loriot [29]. (b) tan θ̄ defined by Eq. (8) corresponding to each case in (a) for seed wavelength λs = 0.8 μm. (c) tan θ̄ defined by Eq. (8) corresponding to each case in (a) for seed wavelength λs = 1.6 μm. Linestyles are consistent with s values in (a)
Fig. 3
Fig. 3 Comparing θ̄(Ip) with and without the delayed Raman response. All parameters and linestyles used are exactly the same as in Fig. (2)(c), except in (a) r = 0 (no delayed Raman response). In (b), r = 1/2, as in Fig. (2)(c).
Fig. 4
Fig. 4 The maximum gain as a function of pump intensity, given by Eq. (5). All parameters and linestyles used are exactly the same as in Fig. (2); in (a) λs = 0.8 μm as in Fig. (2)(b). In (b), λs = 1.6 μm, as in Fig. (2)(c).

Equations (13)

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P ( 3 ) = 3 0 χ i ( 3 ) E ( x , t ) R ( t t ) | E ( x , t ) | 2 d t ,
P ( 3 ) 0 [ E p exp ( i ω p t i k p z ) + ε ( x , t ) ] [ f ( n n ) + Δ n n ( x , t ) f ( n n ) ] ,
Δ n n ( x , t ) = 3 χ i ( 3 ) E p ( 1 r ) ( ε ( x , t ) e i ω p t + i k p z + ε * ( x , t ) e i ω p t i k p z ) + 3 χ i ( 3 ) E p r h ( t t ) ( ε ( x , t ) e i ω p t + i k p z + ε * ( x , t ) e i ω p t i k p z ) d t ,
k ¯ = ( k p 2 D u 2 ) ( σ 2 1 ) ,
g ¯ = k n k n ( ) k p 2 σ 2 D u 2 ,
D u ( Ω ) = c 1 [ η ( ω p ) Ω + η g ( Ω ) Ω + η u ( Ω ) ω p ] ,
D g ( Ω ) = c 1 [ η g ( Ω ) Ω + η u ( Ω ) ω p ]
tan θ ¯ = k ¯ k p ( 1 + Δ u ) ,
tan 2 θ ¯ = ( 1 Δ u 1 + Δ u ) ( Δ i + Δ r ) n n f ( n n ) + Δ g n p 2 + f ( n n ) ,
n n f ( n n ) ( 1 + Δ u ) tan 2 θ ¯ ( n n ) ( 1 Δ u ) ( Δ i + Δ r ) [ n p 2 + f ( n n ) ] + Δ g = 0 .
f ( n n ) = n p 2 { exp [ 0 n n ( 1 + Δ u ) tan 2 θ ¯ ( n n ) ( 1 Δ u ) ( Δ i + Δ r ) d n n n n ] 1 } ,
f ( n n ) = n p 2 { exp [ 0 n n tan 2 θ ¯ ( n n ) d n n n n ] 1 } .
f ( n n ) = n n α n n 2 ( 1 + n n / n s ) s

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