Abstract

Part of the chain in petawatt laser systems may involve extreme focusing conditions for which nonparaxial and vectorial effects have high impact on the propagation of radiation. We investigate the possibility of using propagation equations to simulate numerically the focal spot under these conditions. We derive a unidirectional propagation equation for the Hertz vector, describing linear and nonlinear propagation under situations where nonparaxial diffraction and vectorial effects become significant. By comparing our simulations to the results of vector diffraction integrals in the case of linear tight-focusing by a parabolic mirror, we establish a practical criterion for the critical f -number below which initializing a propagation equation with a parabolic input phase becomes inaccurate. We propose a method to find suitable input conditions for propagation equations beyond this limit. Extreme focusing conditions are shown to be modeled accurately by means of numerical simulations of the unidirectional Hertz-vector propagation equation initialized with suitable input conditions.

© 2015 Optical Society of America

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References

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  1. M. Kolesik and J. V. Moloney, “Modeling and simulation techniques in extreme nonlinear optics of gaseous and condensed media,” Rep. Prog. Phys. 77016401 (2014).
    [Crossref]
  2. A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de 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).
  3. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
    [Crossref]
  4. J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
    [Crossref] [PubMed]
  5. P. St and J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24, 4729 (2006).
    [Crossref]
  6. J. Liu, J. Dai, S. L. Chin, and X.-C. Zhang, “Broadband terahertz wave remote sensing using coherent manipulation of fluorescence from asymmetrically ionized gases,” Nat. Photonics 4, 627–631 (2010).
    [Crossref]
  7. M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2014).
    [Crossref]
  8. F. Buccheri and X.-C. Zhang, “Terahertz emission from laser-induced microplasma in ambient air,” Optica 2, 366–369 (2015).
    [Crossref]
  9. P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3, 381–387 (2007).
    [Crossref]
  10. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
    [Crossref]
  11. M. Apostol and M. Ganciu, “Polaritonic pulse and coherent X- and gamma rays from Compton (Thomson) backscattering,” J. Appl. Phys. 109, 013307 (2011).
    [Crossref]
  12. L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
    [Crossref] [PubMed]
  13. A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.-Usp. 54, 9 (2011).
    [Crossref]
  14. T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526, 157–172 (2014).
    [Crossref]
  15. T. M. Jeong, S. Weber, B. Le Garrec, D. Margarone, T. Mocek, and G. Korn, “Spatio-temporal modification of femtosecond focal spot under tight focusing condition,” Opt. Express 23, 11641 (2015).
    [Crossref] [PubMed]
  16. J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
    [Crossref]
  17. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A. 17, 2081–2089 (2000).
    [Crossref]
  18. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A. 17, 2090–2095 (2000).
    [Crossref]
  19. G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D 157, 112(2001).
    [Crossref]
  20. C. L. Arnold, A. Heisterkamp, W. Ertmer, and H. Lubatschowski, “Computational model for nonlinear plasma formation in high NA micromachining of transparent materials and biological cells,” Opt. Express 15, 10303 (2007)
    [Crossref] [PubMed]
  21. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
    [Crossref]
  22. E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Polarisation response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680 (2012).
    [Crossref]
  23. M. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, “Metastable electronic states and nonlinear response for high-intensity optical pulses,” Optica 1, 323–331 (2014).
    [Crossref]
  24. K.-Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15, 4577–4584 (2007).
    [Crossref] [PubMed]
  25. P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E. 69 (6), 066415 (2004).
    [Crossref]
  26. A. V. Balakin, A. V. Borodin, I. A. Kotelnikov, and A. P. Shkurinov, “Terahertz emission from a femtosecond laser focus in a two-color scheme,” J. Opt. Soc. Am. B. 27, 16–26 (2010).
    [Crossref]
  27. Y. R. Shen, The Principles of Nonlinear Optics (J. Wiley, 1984).
  28. G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E. 89, 013306 (2014)
    [Crossref]
  29. A. V. Borodin, N. A. Panov, O. G. Kosareva, V. A. Andreeva, M. N. Esaulkov, V. A. Makarov, A. P. Shkurinov, S. L. Chin, and X.-C. Zhang, “Transformation of terahertz spectra emitted from dual-frequency femtosecond pulse interaction in gases,” Opt. Lett. 38, 1906–1908 (2013).
    [Crossref] [PubMed]
  30. N. A. Panov, V. A. Makarov, V. Y. Fedorov, and O. G. Kosareva, “Filamentation of arbitrary polarized femtosecond laser pulses in case of high-order Kerr effect,” Opt. Lett. 38, 537–539 (2013).
    [Crossref] [PubMed]
  31. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E. 70(3), 036604 (2004).
    [Crossref]
  32. C. S. Milsted and C. D. Cantrell, “Vector effects in self focusing,” Phys. Rev. A.,  53(5), 3536–3542 (1996).
    [Crossref] [PubMed]

2015 (2)

2014 (5)

M. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, “Metastable electronic states and nonlinear response for high-intensity optical pulses,” Optica 1, 323–331 (2014).
[Crossref]

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E. 89, 013306 (2014)
[Crossref]

M. Kolesik and J. V. Moloney, “Modeling and simulation techniques in extreme nonlinear optics of gaseous and condensed media,” Rep. Prog. Phys. 77016401 (2014).
[Crossref]

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526, 157–172 (2014).
[Crossref]

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2014).
[Crossref]

2013 (2)

2012 (1)

E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Polarisation response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680 (2012).
[Crossref]

2011 (3)

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.-Usp. 54, 9 (2011).
[Crossref]

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de 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).

M. Apostol and M. Ganciu, “Polaritonic pulse and coherent X- and gamma rays from Compton (Thomson) backscattering,” J. Appl. Phys. 109, 013307 (2011).
[Crossref]

2010 (2)

J. Liu, J. Dai, S. L. Chin, and X.-C. Zhang, “Broadband terahertz wave remote sensing using coherent manipulation of fluorescence from asymmetrically ionized gases,” Nat. Photonics 4, 627–631 (2010).
[Crossref]

A. V. Balakin, A. V. Borodin, I. A. Kotelnikov, and A. P. Shkurinov, “Terahertz emission from a femtosecond laser focus in a two-color scheme,” J. Opt. Soc. Am. B. 27, 16–26 (2010).
[Crossref]

2007 (3)

2006 (2)

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[Crossref]

P. St and J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24, 4729 (2006).
[Crossref]

2004 (3)

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E. 70(3), 036604 (2004).
[Crossref]

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E. 69 (6), 066415 (2004).
[Crossref]

2003 (1)

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

2002 (1)

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
[Crossref] [PubMed]

2001 (1)

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D 157, 112(2001).
[Crossref]

2000 (2)

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A. 17, 2081–2089 (2000).
[Crossref]

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A. 17, 2090–2095 (2000).
[Crossref]

1996 (1)

C. S. Milsted and C. D. Cantrell, “Vector effects in self focusing,” Phys. Rev. A.,  53(5), 3536–3542 (1996).
[Crossref] [PubMed]

1939 (1)

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Andre, Y.-B.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Andreeva, V. A.

Apostol, M.

M. Apostol and M. Ganciu, “Polaritonic pulse and coherent X- and gamma rays from Compton (Thomson) backscattering,” J. Appl. Phys. 109, 013307 (2011).
[Crossref]

Arnold, C. L.

Balakin, A. V.

A. V. Balakin, A. V. Borodin, I. A. Kotelnikov, and A. P. Shkurinov, “Terahertz emission from a femtosecond laser focus in a two-color scheme,” J. Opt. Soc. Am. B. 27, 16–26 (2010).
[Crossref]

Biegert, J.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
[Crossref]

Borodin, A. V.

Bourayou, R.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Brambilla, E.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de 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).

Brown, J. M.

Buccheri, F.

Bulanov, S. V.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[Crossref]

Cantrell, C. D.

C. S. Milsted and C. D. Cantrell, “Vector effects in self focusing,” Phys. Rev. A.,  53(5), 3536–3542 (1996).
[Crossref] [PubMed]

Chin, S. L.

A. V. Borodin, N. A. Panov, O. G. Kosareva, V. A. Andreeva, M. N. Esaulkov, V. A. Makarov, A. P. Shkurinov, S. L. Chin, and X.-C. Zhang, “Transformation of terahertz spectra emitted from dual-frequency femtosecond pulse interaction in gases,” Opt. Lett. 38, 1906–1908 (2013).
[Crossref] [PubMed]

J. Liu, J. Dai, S. L. Chin, and X.-C. Zhang, “Broadband terahertz wave remote sensing using coherent manipulation of fluorescence from asymmetrically ionized gases,” Nat. Photonics 4, 627–631 (2010).
[Crossref]

Chu, L.

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Corkum, P. B.

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

Corti, T.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de 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).

Couairon, A.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de 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).

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
[Crossref] [PubMed]

Dai, J.

J. Liu, J. Dai, S. L. Chin, and X.-C. Zhang, “Broadband terahertz wave remote sensing using coherent manipulation of fluorescence from asymmetrically ionized gases,” Nat. Photonics 4, 627–631 (2010).
[Crossref]

Ertmer, W.

Esaulkov, M. N.

Fedorov, V. Y.

Fibich, G.

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D 157, 112(2001).
[Crossref]

Franco, M.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
[Crossref] [PubMed]

Frey, S.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Ganciu, M.

M. Apostol and M. Ganciu, “Polaritonic pulse and coherent X- and gamma rays from Compton (Thomson) backscattering,” J. Appl. Phys. 109, 013307 (2011).
[Crossref]

Glownia, J. H.

Gonoskov, A. A.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.-Usp. 54, 9 (2011).
[Crossref]

Gryaznov, G. A.

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E. 89, 013306 (2014)
[Crossref]

Hafizi, B.

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E. 69 (6), 066415 (2004).
[Crossref]

Hauri, C. P.

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2014).
[Crossref]

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

Heinrich, A.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

Heisterkamp, A.

Helbing, F. W.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

Ilan, B.

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D 157, 112(2001).
[Crossref]

Jakobsen, P.

Jeong, T. M.

Kapetanakos, C. A.

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E. 69 (6), 066415 (2004).
[Crossref]

Kasparian, J.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Keller, U.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

Khazanov, E. A.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.-Usp. 54, 9 (2011).
[Crossref]

Kim, K.-Y.

Kolesik, M.

M. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, “Metastable electronic states and nonlinear response for high-intensity optical pulses,” Optica 1, 323–331 (2014).
[Crossref]

M. Kolesik and J. V. Moloney, “Modeling and simulation techniques in extreme nonlinear optics of gaseous and condensed media,” Rep. Prog. Phys. 77016401 (2014).
[Crossref]

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de 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).

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E. 70(3), 036604 (2004).
[Crossref]

Korn, G.

Kornelis, W.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

Korzhimanov, A. V.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.-Usp. 54, 9 (2011).
[Crossref]

Kosareva, O. G.

Kotelnikov, I. A.

A. V. Balakin, A. V. Borodin, I. A. Kotelnikov, and A. P. Shkurinov, “Terahertz emission from a femtosecond laser focus in a two-color scheme,” J. Opt. Soc. Am. B. 27, 16–26 (2010).
[Crossref]

Krausz, F.

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

Lamouroux, B.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
[Crossref] [PubMed]

Le Garrec, B.

Lee, J.

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526, 157–172 (2014).
[Crossref]

Liu, J.

J. Liu, J. Dai, S. L. Chin, and X.-C. Zhang, “Broadband terahertz wave remote sensing using coherent manipulation of fluorescence from asymmetrically ionized gases,” Nat. Photonics 4, 627–631 (2010).
[Crossref]

Lubatschowski, H.

Majus, D.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de 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).

Makarov, V. A.

Margarone, D.

Mejean, G.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Milsted, C. S.

C. S. Milsted and C. D. Cantrell, “Vector effects in self focusing,” Phys. Rev. A.,  53(5), 3536–3542 (1996).
[Crossref] [PubMed]

Mocek, T.

Moloney, J. V.

M. Kolesik and J. V. Moloney, “Modeling and simulation techniques in extreme nonlinear optics of gaseous and condensed media,” Rep. Prog. Phys. 77016401 (2014).
[Crossref]

M. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, “Metastable electronic states and nonlinear response for high-intensity optical pulses,” Optica 1, 323–331 (2014).
[Crossref]

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E. 70(3), 036604 (2004).
[Crossref]

Mourou, G. A.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[Crossref]

Mysyrowicz, A.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
[Crossref] [PubMed]

Panov, N. A.

Peñano, J. R.

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E. 69 (6), 066415 (2004).
[Crossref]

Perezhogin, I. A.

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E. 89, 013306 (2014)
[Crossref]

Popov, A. M.

E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Polarisation response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680 (2012).
[Crossref]

Potravkin, N. N.

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E. 89, 013306 (2014)
[Crossref]

Prade, B.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
[Crossref] [PubMed]

Ramírez-Góngora, O. de J.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de 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).

Rodriguez, G.

Rodriguez, M.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Russell, J.

Salmon, E.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Sauerbrey, R.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Sergeev, A. M.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.-Usp. 54, 9 (2011).
[Crossref]

Shalaby, M.

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2014).
[Crossref]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (J. Wiley, 1984).

Shkurinov, A. P.

Sprangle, P.

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E. 69 (6), 066415 (2004).
[Crossref]

St, P.

Stratton, J.

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Sudrie, L.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
[Crossref] [PubMed]

Tajima, T.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[Crossref]

Taylor, A. J.

Teleki, A.

Tikhonova, O. V.

E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Polarisation response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680 (2012).
[Crossref]

Török, P.

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A. 17, 2090–2095 (2000).
[Crossref]

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A. 17, 2081–2089 (2000).
[Crossref]

Tzortzakis, S.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
[Crossref] [PubMed]

Varga, P.

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A. 17, 2090–2095 (2000).
[Crossref]

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A. 17, 2081–2089 (2000).
[Crossref]

Volkova, E. A.

E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Polarisation response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680 (2012).
[Crossref]

Weber, S.

Wille, H.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
[Crossref]

Wolf, J.-P.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Woste, L.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Wright, E. M.

Yu, J.

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Zhang, X.-C.

Ann. Phys. (Berlin) (1)

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526, 157–172 (2014).
[Crossref]

Appl. Phys. B. (1)

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B. 79, 673 (2004).
[Crossref]

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

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de 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).

J. Appl. Phys. (1)

M. Apostol and M. Ganciu, “Polaritonic pulse and coherent X- and gamma rays from Compton (Thomson) backscattering,” J. Appl. Phys. 109, 013307 (2011).
[Crossref]

J. Lightwave Technol. (1)

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

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A. 17, 2081–2089 (2000).
[Crossref]

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A. 17, 2090–2095 (2000).
[Crossref]

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

A. V. Balakin, A. V. Borodin, I. A. Kotelnikov, and A. P. Shkurinov, “Terahertz emission from a femtosecond laser focus in a two-color scheme,” J. Opt. Soc. Am. B. 27, 16–26 (2010).
[Crossref]

Nat. Commun. (1)

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2014).
[Crossref]

Nat. Photonics (1)

J. Liu, J. Dai, S. L. Chin, and X.-C. Zhang, “Broadband terahertz wave remote sensing using coherent manipulation of fluorescence from asymmetrically ionized gases,” Nat. Photonics 4, 627–631 (2010).
[Crossref]

Nat. Phys. (1)

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

Opt. Express (3)

Opt. Lett. (2)

Optica (2)

Phys. Rev. (1)

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Phys. Rev. A. (1)

C. S. Milsted and C. D. Cantrell, “Vector effects in self focusing,” Phys. Rev. A.,  53(5), 3536–3542 (1996).
[Crossref] [PubMed]

Phys. Rev. E. (3)

P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E. 69 (6), 066415 (2004).
[Crossref]

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E. 89, 013306 (2014)
[Crossref]

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E. 70(3), 036604 (2004).
[Crossref]

Phys. Rev. Lett. (1)

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89, 186601 (2002).
[Crossref] [PubMed]

Phys.-Usp. (1)

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.-Usp. 54, 9 (2011).
[Crossref]

Physica D (1)

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D 157, 112(2001).
[Crossref]

Quantum Electron. (1)

E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Polarisation response of a gas medium in the field of a high-intensity ultrashort laser pulse: high order Kerr nonlinearities or plasma electron component?” Quantum Electron. 42, 680 (2012).
[Crossref]

Rep. Prog. Phys. (1)

M. Kolesik and J. V. Moloney, “Modeling and simulation techniques in extreme nonlinear optics of gaseous and condensed media,” Rep. Prog. Phys. 77016401 (2014).
[Crossref]

Rev. Mod. Phys. (1)

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[Crossref]

Science (1)

J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Woste, “White light filaments for atmospheric analysis,” Science 301, 61 (2003).
[Crossref] [PubMed]

Other (2)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
[Crossref]

Y. R. Shen, The Principles of Nonlinear Optics (J. Wiley, 1984).

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Figures (7)

Fig. 1
Fig. 1 Focusing geometry with a parabolic mirror. (a) case of calculations using vector diffraction integrals, (b) propagation equations and (c) propagation equations with suitable input conditions. Black solid lines (a, c) and red solid lines (b, c) show the beam path simulated by using vector diffraction integrals and propagation equations, respectively. Dashed lines state the case f # = 2 and solid lines state the case f # = 0.5. Blue vertical lines locate the planes where initial conditions for the UPPE are set (b) and calculated using VDI (c).
Fig. 2
Fig. 2 Comparison of beam propagation through the focus (peak intensity vs propagation distance) with vector diffraction integrals (dotted curves) and with numerical simulation of the UPPE (solid curves). The input condition of Fig. 1(b) corresponds to a parabolic phase (gray curves). The results obtained with suitable input condition of Fig. 1(c) are shown by the solid orange curve.
Fig. 3
Fig. 3 Distribution of |Ex|2 at the best focus loci (z = 0 for the calculation with vector diffraction integrals and z = 25 µm for the simulation with UPPE) for the case of f # = 5. Values on the colorbar indicates I/I0.
Fig. 4
Fig. 4 Distance δz from the mirror to the plane z = −f (input condition) at the beam edge (red solid curve, analytical result) as a function of f -number. Confocal parameter b(top hat) for a cylindric beam (curve marked by black squares, result of a simulation with the vectorial diffraction integral). Confocal parameter b(Gauss) for the paraxial propagation of a Gaussian beam (black solid curve, analytical expression). The focal length is f = 1 cm for all curves.
Fig. 5
Fig. 5 Simulation results with the UPPE and input condition Eq. (12) for different focal lengths and f -numbers. For the same f -number, the suitability of the input condition improves with the decrease of the focal length. The vertical axis shows the dimensionless intensity. The intensity is normalized to the product of input intensity and squared focal length of the mirror, normalized to 1 cm. This is needed to ensure similar vertical scale for different focal lengths.
Fig. 6
Fig. 6 Intensity distributions in the waist for a x-polarized top-hat beam in case of f = 150 µm and f# = 2 > f#crit. (a, b) Nonparaxial diffraction without and with vectorial effects (simulation by UPPE and UHPE respectively, z = 4 µm). (c) Simulation with vector diffraction integral (z = 0).
Fig. 7
Fig. 7 Intensity distributions in the waist for a x-polarized top-hat beam in case of f = 150 µm and f# = 1 < f#crit. (a, b) Nonparaxial diffraction without and with vectorial effects (simulation by UPPE and UHPE respectively, z = 9 µm). (c) Simulation with vector diffraction integral (z = 0). (d) Nonparaxial diffraction with vectorial effects after translation of input conditions to the plane z = 30 µm.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

D = ρ B = 0 × E = B t × H = J + D t
E = i ω A ϕ , B = × A ϕ = , A = i ω μ 0 ε .
Δ ϕ + k 2 ( ω ) ϕ = 1 ε ( Φ ε ( ω ) l ) Δ A + k 2 ( ω ) A = i μ 0 ω ( Φ ε ( ω ) l ) ,
Δ + k 2 = 1 ε ( Φ ε l ) .
l z = Φ z ε .
E = ( ) + k 2 l , B = μ 0 ε × ( i ω ) .
( E x E y E z ) = ( k 2 k x 2 k x k y k x k y k 2 k y 2 k x k z k y k z ) ( x y ) ( i l k x i l k y Φ z / ε ) ,
( x y ) = 1 k 2 k z 2 ( k 2 k y 2 k x k y k x k y k 2 k x 2 ) [ ( E x E y ) + i l ( k x k y ) ] .
E z = i k z E + i ω 2 k z c 2 Φ 2 ε 0 .
E z = i k z E + i ω 2 k z c 2 Φ 2 ε 0 i ε k z k ( k Φ 2 ε 0 ) ,
z = i k z + i 2 k z ( Φ ε i l k ) ,
E x ( r , z = f ) = Θ ( r 0 r ) × exp ( i k 0 r 2 2 f ) ,
E x ( θ s , φ s ) = Θ ( θ s θ m ) ,
f # = 1 2 cot θ m = f 2 r 0 r 0 8 f .
f # c r i t = 1 4 ( 2 k 0 f 4 1 2 k 0 f 4 ) 1 4 2 k 0 f 4 ,
f # c r i t ( f , λ ) = 3.1 × f [ cm ] λ [ μ m ] 4 .

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