Abstract

It is well known that an intense pulse sent through a medium with an intensity-dependent index will acquire a stationary shape (spatial soliton) while focusing and collapsing into a point. Many nonlinear phenomena, such as THz generation, require the simultaneous propagation and collapse at two wavelengths. It is shown here that the Kerr effect introduces a coupling between pulses sent co-propagating into a transparent medium with a nonlinear index. Because of this coupling, the pulses at the different wavelengths reshape towards a stationary pair that evolves towards a common focus in time and space. The effect of normal and anomalous dispersion on two-color pulse collapse is investigated numerically. The model to be considered here is an extension of ($ 2 + 1 $)-dimensional nonlinear Schrodinger equations (NLSEs) by inclusion of dispersion and for a beam consisting of two frequencies. As such, our study centers on a system of coupled ($ 3 + 1 $)-D NLSEs describing the co-propagation of two pulses in the non-resonant regime under self-focusing. We should emphasize that this model gives new insight on the initial dynamics of the two-color filament. While inclusion of the small normal dispersion tends to a temporal split of the beam, anomalous dispersion facilitates collapse. In considering initial “short” (fs) and “long” (ps–ns) temporal pulses, our results present different scenarios of the initial evolution that include the role dispersion may have.

© 2019 Optical Society of America

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References

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  1. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [Crossref]
  2. A. Sukhinin, A. B. Aceves, J.-C. Diels, and L. Arissian, “Collapse events of two-color optical beams,” Phys. Rev. A 95, 031801 (2017).
    [Crossref]
  3. A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
    [Crossref]
  4. K. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
    [Crossref]
  5. G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240 (1999).
    [Crossref]
  6. A. Sukhinin and A. B. Aceves, “Optical UV filament and vortex dynamics,” J. Opt. 15, 044010 (2013).
    [Crossref]
  7. G. Fibich and G. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379–1381 (1997).
    [Crossref]
  8. J. E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. 17, 583–585 (1992).
    [Crossref]
  9. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990).
    [Crossref]
  10. B. Alonso, Í. J. Sola, J. S. Román, Ó. Varela, and L. Roso, “Spatiotemporal evolution of light during propagation in filamentation regime,” J. Opt. Soc. Am. B 28, 1807–1816 (2011).
    [Crossref]
  11. I. Gražulevičiūtė, G. Tamošauskas, V. Jukna, A. Couairon, D. Faccio, and A. Dubietis, “Self-reconstructing spatiotemporal light bullets,” Opt. Express 22, 30613–30622 (2014).
    [Crossref]
  12. C. Jing, Z. Wang, and Y. Cheng, “Characteristics and applications of spatiotemporally focused femtosecond laser pulses,” Appl. Sci. 6, 428 (2016).
    [Crossref]
  13. N. Aközbek, A. Becker, M. Scalora, S. Chin, and C. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
    [Crossref]
  14. A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B 48, 094021 (2015).
    [Crossref]
  15. I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
    [Crossref]
  16. M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
    [Crossref]
  17. A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
    [Crossref]
  18. R. I. Grynko, D. L. Weerawarne, and B. Shim, “Effects of higher-order nonlinear processes on harmonic-generation phase matching,” Phys. Rev. A 96, 013816 (2017).
    [Crossref]
  19. N. Jhajj, E. Rosenthal, R. Birnbaum, J. Wahlstrand, and H. Milchberg, “Demonstration of long-lived high-power optical waveguides in air,” Phys. Rev. X 4, 011027 (2014).
    [Crossref]

2017 (2)

R. I. Grynko, D. L. Weerawarne, and B. Shim, “Effects of higher-order nonlinear processes on harmonic-generation phase matching,” Phys. Rev. A 96, 013816 (2017).
[Crossref]

A. Sukhinin, A. B. Aceves, J.-C. Diels, and L. Arissian, “Collapse events of two-color optical beams,” Phys. Rev. A 95, 031801 (2017).
[Crossref]

2016 (1)

C. Jing, Z. Wang, and Y. Cheng, “Characteristics and applications of spatiotemporally focused femtosecond laser pulses,” Appl. Sci. 6, 428 (2016).
[Crossref]

2015 (1)

A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B 48, 094021 (2015).
[Crossref]

2014 (2)

N. Jhajj, E. Rosenthal, R. Birnbaum, J. Wahlstrand, and H. Milchberg, “Demonstration of long-lived high-power optical waveguides in air,” Phys. Rev. X 4, 011027 (2014).
[Crossref]

I. Gražulevičiūtė, G. Tamošauskas, V. Jukna, A. Couairon, D. Faccio, and A. Dubietis, “Self-reconstructing spatiotemporal light bullets,” Opt. Express 22, 30613–30622 (2014).
[Crossref]

2013 (1)

A. Sukhinin and A. B. Aceves, “Optical UV filament and vortex dynamics,” J. Opt. 15, 044010 (2013).
[Crossref]

2011 (2)

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

B. Alonso, Í. J. Sola, J. S. Román, Ó. Varela, and L. Roso, “Spatiotemporal evolution of light during propagation in filamentation regime,” J. Opt. Soc. Am. B 28, 1807–1816 (2011).
[Crossref]

2010 (1)

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

2004 (1)

M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[Crossref]

2003 (2)

N. Aközbek, A. Becker, M. Scalora, S. Chin, and C. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
[Crossref]

K. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[Crossref]

2000 (1)

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
[Crossref]

1999 (1)

G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240 (1999).
[Crossref]

1997 (1)

1992 (1)

1990 (1)

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Aceves, A.

A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B 48, 094021 (2015).
[Crossref]

Aceves, A. B.

A. Sukhinin, A. B. Aceves, J.-C. Diels, and L. Arissian, “Collapse events of two-color optical beams,” Phys. Rev. A 95, 031801 (2017).
[Crossref]

A. Sukhinin and A. B. Aceves, “Optical UV filament and vortex dynamics,” J. Opt. 15, 044010 (2013).
[Crossref]

Aközbek, N.

N. Aközbek, A. Becker, M. Scalora, S. Chin, and C. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
[Crossref]

Alonso, B.

Arissian, L.

A. Sukhinin, A. B. Aceves, J.-C. Diels, and L. Arissian, “Collapse events of two-color optical beams,” Phys. Rev. A 95, 031801 (2017).
[Crossref]

A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B 48, 094021 (2015).
[Crossref]

Babushkin, I.

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

Becker, A.

N. Aközbek, A. Becker, M. Scalora, S. Chin, and C. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
[Crossref]

Berge, L.

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

Birnbaum, R.

N. Jhajj, E. Rosenthal, R. Birnbaum, J. Wahlstrand, and H. Milchberg, “Demonstration of long-lived high-power optical waveguides in air,” Phys. Rev. X 4, 011027 (2014).
[Crossref]

Bowden, C.

N. Aközbek, A. Becker, M. Scalora, S. Chin, and C. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
[Crossref]

Brambilla, E.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Cheng, Y.

C. Jing, Z. Wang, and Y. Cheng, “Characteristics and applications of spatiotemporally focused femtosecond laser pulses,” Appl. Sci. 6, 428 (2016).
[Crossref]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Chin, S.

N. Aközbek, A. Becker, M. Scalora, S. Chin, and C. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
[Crossref]

Corti, T.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Couairon, A.

I. Gražulevičiūtė, G. Tamošauskas, V. Jukna, A. Couairon, D. Faccio, and A. Dubietis, “Self-reconstructing spatiotemporal light bullets,” Opt. Express 22, 30613–30622 (2014).
[Crossref]

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Diels, J.-C.

A. Sukhinin, A. B. Aceves, J.-C. Diels, and L. Arissian, “Collapse events of two-color optical beams,” Phys. Rev. A 95, 031801 (2017).
[Crossref]

A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B 48, 094021 (2015).
[Crossref]

Dubietis, A.

Elsaesser, T.

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

Faccio, D.

Fibich, G.

K. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[Crossref]

G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240 (1999).
[Crossref]

G. Fibich and G. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379–1381 (1997).
[Crossref]

Gaeta, A. L.

K. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[Crossref]

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
[Crossref]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Gražuleviciute, I.

Grynko, R. I.

R. I. Grynko, D. L. Weerawarne, and B. Shim, “Effects of higher-order nonlinear processes on harmonic-generation phase matching,” Phys. Rev. A 96, 013816 (2017).
[Crossref]

Herrmann, J.

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

Jhajj, N.

N. Jhajj, E. Rosenthal, R. Birnbaum, J. Wahlstrand, and H. Milchberg, “Demonstration of long-lived high-power optical waveguides in air,” Phys. Rev. X 4, 011027 (2014).
[Crossref]

Jing, C.

C. Jing, Z. Wang, and Y. Cheng, “Characteristics and applications of spatiotemporally focused femtosecond laser pulses,” Appl. Sci. 6, 428 (2016).
[Crossref]

Jukna, V.

Koehler, C.

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

Kolesik, M.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[Crossref]

Kuehn, W.

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

Majus, D.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Milchberg, H.

N. Jhajj, E. Rosenthal, R. Birnbaum, J. Wahlstrand, and H. Milchberg, “Demonstration of long-lived high-power optical waveguides in air,” Phys. Rev. X 4, 011027 (2014).
[Crossref]

Moll, K.

K. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[Crossref]

Moloney, J.

M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[Crossref]

Papanicolaou, G.

G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240 (1999).
[Crossref]

G. Fibich and G. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379–1381 (1997).
[Crossref]

Ramírez-Góngora, O. D. J.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Reimann, K.

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

Román, J. S.

Rosenthal, E.

N. Jhajj, E. Rosenthal, R. Birnbaum, J. Wahlstrand, and H. Milchberg, “Demonstration of long-lived high-power optical waveguides in air,” Phys. Rev. X 4, 011027 (2014).
[Crossref]

Roso, L.

Rothenberg, J. E.

Scalora, M.

N. Aközbek, A. Becker, M. Scalora, S. Chin, and C. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
[Crossref]

Shim, B.

R. I. Grynko, D. L. Weerawarne, and B. Shim, “Effects of higher-order nonlinear processes on harmonic-generation phase matching,” Phys. Rev. A 96, 013816 (2017).
[Crossref]

Silberberg, Y.

Skupin, S.

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

Sola, Í. J.

Sukhinin, A.

A. Sukhinin, A. B. Aceves, J.-C. Diels, and L. Arissian, “Collapse events of two-color optical beams,” Phys. Rev. A 95, 031801 (2017).
[Crossref]

A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B 48, 094021 (2015).
[Crossref]

A. Sukhinin and A. B. Aceves, “Optical UV filament and vortex dynamics,” J. Opt. 15, 044010 (2013).
[Crossref]

Tamošauskas, G.

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

Varela, Ó.

Wahlstrand, J.

N. Jhajj, E. Rosenthal, R. Birnbaum, J. Wahlstrand, and H. Milchberg, “Demonstration of long-lived high-power optical waveguides in air,” Phys. Rev. X 4, 011027 (2014).
[Crossref]

Wang, Z.

C. Jing, Z. Wang, and Y. Cheng, “Characteristics and applications of spatiotemporally focused femtosecond laser pulses,” Appl. Sci. 6, 428 (2016).
[Crossref]

Weerawarne, D. L.

R. I. Grynko, D. L. Weerawarne, and B. Shim, “Effects of higher-order nonlinear processes on harmonic-generation phase matching,” Phys. Rev. A 96, 013816 (2017).
[Crossref]

Woerner, M.

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

Appl. Phys. B (1)

N. Aközbek, A. Becker, M. Scalora, S. Chin, and C. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
[Crossref]

Appl. Sci. (1)

C. Jing, Z. Wang, and Y. Cheng, “Characteristics and applications of spatiotemporally focused femtosecond laser pulses,” Appl. Sci. 6, 428 (2016).
[Crossref]

Eur. Phys. J. Spec. Top. (1)

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. D. J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

J. Opt. (1)

A. Sukhinin and A. B. Aceves, “Optical UV filament and vortex dynamics,” J. Opt. 15, 044010 (2013).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B 48, 094021 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (2)

A. Sukhinin, A. B. Aceves, J.-C. Diels, and L. Arissian, “Collapse events of two-color optical beams,” Phys. Rev. A 95, 031801 (2017).
[Crossref]

R. I. Grynko, D. L. Weerawarne, and B. Shim, “Effects of higher-order nonlinear processes on harmonic-generation phase matching,” Phys. Rev. A 96, 013816 (2017).
[Crossref]

Phys. Rev. E (1)

M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[Crossref]

Phys. Rev. Lett. (4)

I. Babushkin, W. Kuehn, C. Koehler, S. Skupin, L. Berge, K. Reimann, M. Woerner, J. Herrmann, and T. Elsaesser, “Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases,” Phys. Rev. Lett. 105, 053903 (2010).
[Crossref]

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[Crossref]

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
[Crossref]

K. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[Crossref]

Phys. Rev. X (1)

N. Jhajj, E. Rosenthal, R. Birnbaum, J. Wahlstrand, and H. Milchberg, “Demonstration of long-lived high-power optical waveguides in air,” Phys. Rev. X 4, 011027 (2014).
[Crossref]

SIAM J. Appl. Math. (1)

G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240 (1999).
[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. (a) Two-color spatial soliton [ $ {{\cal E}_1} $ (solid red), $ {{\cal E}_2} $ (dash blue)] with powers $ {P_{1cr}} \approx 0.085,{P_{2cr}} \approx 1.773 $ , $ {P_{cr}} = {P_{1cr}} + {P_{2cr}} \approx 1.858 $ ; (b), (c) Gaussian fits $ {a_i}\exp ( - {r^2}/w_i^2) $ (dotted black) with parameters $ {a_1} = 0.337,{w_1} = 1.592,{a_2} = 2.106,{w_2} = 1.295 $ , and powers $ {P_{1gaus}} \approx 0.07,{P_{2gaus}} \approx 1.86 $ , $ {P_{gaus}} \approx 1.93 $ .
Fig. 2.
Fig. 2. Simulation parameters: $ {\epsilon _1} = 0,{\epsilon _2} = 0.02 $ , $ {{A_1} = 1.9},$ $ {{A_1} = 1.9},$ ${A_2} = 1.1,\tau = 1.5 $ . (a)  $ z = 0 $ , (b)  $ z = 0.554 $ , (c)  $ z = 0.652 $ , and (d)  $ z = 0.676 $ .
Fig. 3.
Fig. 3. Simulation parameters: $ {\epsilon _1} = 0,{\epsilon _2} = - 0.02 $ , $ {{A_1} = 1.9},$ ${A_2} = 1.1,\tau = 1.5 $ . (a)  $ z = 0 $ , (b)  $ z = 0.772 $ , (c)  $ z = 0.923 $ , and (d)  $ z = 0.978 $ .
Fig. 4.
Fig. 4. Simulation parameters: $ {\epsilon _1} = 0.02,{\epsilon _2} = 0 $ , $ {A_1} = 1.9,$ ${A_2} = 1.1,\tau = 1.5 $ . (a)  $ z = 0 $ , (b)  $ z = 0.504 $ , (c)  $ z = 0.736 $ , and (d)  $ z = 0.764 $ .
Fig. 5.
Fig. 5. Simulation parameters: $ {\epsilon _1} = - 0.02,{\epsilon _2} = 0 $ , $ {A_1} = 1.9,$ ${A_2} = 1.1,\tau = 1.5 $ . (a)  $ z = 0 $ , (b)  $ z = 0.604 $ , (c)  $ z = 0.774 $ , and (d)  $ z = 0.776 $ .
Fig. 6.
Fig. 6. Simulation parameters: $ {\epsilon _1} = 0.02,{\epsilon _2} = 0.02 $ , $ {A_1} = 1.9,$ ${A_2} = 1.1,\tau = 1.5 $ . (a)  $ z = 0 $ , (b)  $ z = 0.403 $ , (c)  $ z = 0.543 $ , and (d)  $ z = 0.655 $ .
Fig. 7.
Fig. 7. Simulation parameters: $ {\epsilon _1} = - 0.02,{\epsilon _2} = - 0.02 $ , $ {A_1} = 1.9,$ ${A_2} = 1.1,\tau = 1.5 $ . (a)  $ z = 0 $ , (b)  $ z = 0.904 $ , (c)  $ z = 0.942 $ , and (d)  $ z = 0.956 $ .
Fig. 8.
Fig. 8. Simulation parameters: $ {\epsilon _1} = - 0.02,{\epsilon _2} = 0.02 $ , $ {A_1} = 1.9,$ ${A_2} = 1.1,\tau = 1.5 $ . (a)  $ z = 0 $ , (b)  $ z = 0.504 $ , (c)  $ z = 0.665 $ , and (d)  $ z = 0.683 $ .
Fig. 9.
Fig. 9. Simulation parameters: $ {\epsilon _1} = 0.02,{\epsilon _2} = - 0.02 $ , $ {A_1} = 1.9,$ ${A_2} = 1.1,\tau = 1.5 $ . (a)  $ z = 0 $ , (b)  $ z = 0.844 $ , (c)  $ z = 0.919 $ , and (d)  $ z = 0.969 $ .

Equations (9)

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i E 1 z + 1 2 k 1 Δ E 1 + k 1 2 2 E 1 t 2 + n 2 k 1 n 0 ( | E 1 | 2 + 2 | E 2 | 2 ) E 1 = 0 , i E 2 z + 1 2 k 2 Δ E 2 + k 2 2 2 E 1 t 2 + n 2 k 2 n 0 ( 2 | E 1 | 2 + | E 2 | 2 ) E 2 = 0 ,
i γ E 1 z + Δ E 1 + ε 1 2 E 1 t 2 + 1 γ 2 ( | E 1 | 2 + 2 | E 2 | 2 ) E 1 = 0 ,
i E 2 z + Δ E 2 + ε 2 2 E 1 t 2 + ( 2 | E 1 | 2 + | E 2 | 2 ) E 2 = 0 ,
E 1 ( z = 0 ) = A 1 E 1 ,
E 2 ( z = 0 ) = A 2 E 2 e t 2 / τ 2 ,
P 1 = 0.334 , P 2 = 2.139 , P = P 1 + P 2 = 2.472 ,
P 1 = 3.93 P 1 c r ,   P 2 = 1.2 P 2 c r ,   P 1 / P 2 = 0.156 ,   P = 1.33 P c r .
E 1 ( z = 0 ) = A 1 E 1 e t 2 / τ 2 , E 2 ( z = 0 ) = A 2 E 2 ,
E 1 ( z = 0 ) = A 1 E 1 e t 2 / τ 2 , E 2 ( z = 0 ) = A 2 E 2 e t 2 / τ 2 .

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