Abstract

Nonlinear optical methods have become ubiquitous in many scientific areas, from fundamental studies of time-resolved electron dynamics to microscopy and spectroscopy applications. They are, however, often limited to a certain range of parameters such as pulse energy and average power. Restrictions arise from, for example, the required field intensity as well as from parasitic nonlinear effects and saturation mechanisms. Here, we identify a fundamental principle of nonlinear light–matter interaction in gases and show that paraxial nonlinear wave equations are scale-invariant if spatial dimensions, gas density, and laser pulse energy are scaled appropriately. As an example, we apply this principle to high-order harmonic generation and provide a general method for increasing peak and average power of attosecond sources. In addition, we experimentally demonstrate the implications for the compression of short laser pulses. Our scaling principle extends well beyond those examples and includes many nonlinear processes with applications in different areas of science.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (2)

2014 (6)

F. Böhle, M. Kretschmar, A. Jullien, M. Kovacs, M. Miranda, R. Romero, H. Crespo, U. Morgner, P. Simon, R. Lopez-Martens, and T. Nagy, “Compression of CEP-stable multi-mJ laser pulses down to 4  fs in long hollow fibers,” Laser Phys. Lett. 11, 095401 (2014).
[Crossref]

J. Rothhardt, M. Krebs, S. Hädrich, S. Demmler, J. Limpert, and A. Tünnermann, “Absorption-limited and phase-matched high harmonic generation in the tight focusing regime,” New J. Phys. 16, 033022 (2014).
[Crossref]

J. Limpert, A. Klenke, M. Kienel, S. Breitkopf, T. Eidam, S. Hädrich, C. Jauregui, and A. Tuennermann, “Performance scaling of ultrafast laser systems by coherent addition of femtosecond pulses,” IEEE J. Sel. Top. Quantum Electron. 20, 5 (2014).
[Crossref]

H. Fattahi, H. G. Barros, M. Gorjan, T. Nubbemeyer, B. Alsaif, C. Y. Teisset, M. Schultze, S. Prinz, M. Haefner, U. M. A. A. L. Vámos, A. Schwarz, O. Pronin, J. Brons, X. T. Geng, G. Arisholm, M. Ciappina, V. S. Yakovlev, D.-E. Kim, A. M. Azzeer, K. N. D. Sutter, Z. Major, T. Metzger, and F. Krausz, “Third-generation femtosecond technology,” Optica 1, 45–63 (2014).
[Crossref]

S. Leone, C. McCurdy, J. Burgdörfer, L. Cederbaum, Z. Chang, N. Dudovich, J. Feist, C. Greene, M. Ivanov, R. Kienberger, U. Keller, M. Kling, Z.-H. Loh, T. Pfeifer, A. Pfeiffer, R. Santra, K. Schafer, A. Stolow, U. Thumm, and M. Vrakking, “What will it take to observe processes in ‘real time’?” Nat. Photonics 8, 162–166 (2014).
[Crossref]

K. Cassou, S. Daboussi, O. Hort, O. Guilbaud, D. Descamps, S. Petit, E. Mével, E. Constant, and S. Kazamias, “Enhanced high harmonic generation driven by high-intensity laser in argon gas-filled hollow core waveguide,” Opt. Lett. 39, 3770–3773 (2014).
[Crossref]

2013 (1)

P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. 84, 073103 (2013).
[Crossref]

2012 (4)

C.-T. Chiang, A. Blättermann, M. Huth, J. Kirschner, and W. Widdra, “High-order harmonic generation at 4  MHz as a light source for time-of-flight photoemission spectroscopy,” Appl. Phys. Lett. 101, 071116 (2012).
[Crossref]

C. M. Heyl, J. Güdde, A. L’Huillier, and U. Höfer, “High-order harmonic generation with μJ laser pulses at high repetition rates,” J. Phys. B 45, 074020 (2012).
[Crossref]

M. Miranda, T. Fordell, C. Arnold, A. L’Huillier, and H. Crespo, “Simultaneous compression and characterization of ultrashort laser pulses using chirped mirrors and glass wedges,” Opt. Express 20, 688–697 (2012).
[Crossref]

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hermández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. Murnane, and H. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[Crossref]

2011 (1)

P. Tzallas, E. Skantzakis, L. Nikolopoulos, G. Tsakiris, and D. Charalambidis, “Extreme-ultraviolet pump-probe studies of one-femtosecond-scale electron dynamics,” Nat. Phys. 7, 781–784 (2011).
[Crossref]

2010 (3)

2009 (3)

O. Varela, A. Zaïr, J. Román, B. Alonso, I. Sola, C. Prieto, and L. Roso, “Above-millijoule super-continuum generation using polarisation dependent filamentation in atoms and molecules,” Opt. Express 17, 3630–3639 (2009).
[Crossref]

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

E. Esarey, C. Schroeder, and W. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81, 1229–1285 (2009).
[Crossref]

2008 (1)

2007 (1)

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

2006 (2)

A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis, F. W. Helbing, U. Keller, and A. Mysyrowicz, “Self-compression of ultra-short laser pulses down to one optical cycle by filamentation,” J. Mod. Opt. 53, 75–85 (2006).
[Crossref]

S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnurer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: experiments versus numerical simulations,” Phys. Rev. E 74, 056604 (2006).
[Crossref]

2005 (3)

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub-10-fs, 5-mj-optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett. 86, 11 (2005).
[Crossref]

A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30, 2657–2659 (2005).
[Crossref]

V. Strelkov, V. Platonenko, and A. Becker, “High-harmonic generation in a dense medium,” Phys. Rev. A 71, 053808 (2005).
[Crossref]

2004 (1)

C. Hauri, W. Kornelis, F. 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, 673–677 (2004).
[Crossref]

2003 (1)

P. Russell, “Applied physics: photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref]

2002 (3)

E. Takahashi, Y. Nabekawa, T. Otsuka, M. Obara, and K. Midorikawa, “Generation of highly coherent submicrojoule soft x rays by high-order harmonics,” Phys. Rev. A 66, 021802 (2002).
[Crossref]

E. Takahashi, Y. Nabekawa, and K. Midorikawa, “Generation of 10-μJ coherent extreme-ultraviolet light by use of high-order harmonics,” Opt. Lett. 27, 1920–1922 (2002).
[Crossref]

J.-F. Hergott, M. Kovacev, H. Merdji, C. Hubert, Y. Mairesse, E. Jean, P. Breger, P. Agostini, B. Carré, and P. Salières, “Extreme-ultraviolet high-order harmonic pulses in the microjoule range,” Phys. Rev. A 66, 021801 (2002).
[Crossref]

2001 (1)

A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[Crossref]

2000 (2)

H. Conrads and M. Schmidt, “Plasma generation and plasma sources,” Plasma Sources Sci. Technol. 9, 441–454 (2000).
[Crossref]

G. Fibich and A. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000).
[Crossref]

1999 (4)

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

N. Bloembergen, “From nanosecond to femtosecond science,” Rev. Mod. Phys. 71, S283–S287 (1999).
[Crossref]

M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: Extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[Crossref]

E. Constant, D. Garzella, P. Breger, E. Mével, C. Dorrer, C. L. Blanc, F. Salin, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: model and experiment,” Phys. Rev. Lett. 82, 1668–1671 (1999).
[Crossref]

1997 (1)

1995 (1)

1994 (2)

P. Corkum, N. Burnett, and M. Ivanov, “Subfemtosecond pulses,” Opt. Lett. 19, 1870–1872 (1994).
[Crossref]

M. Lewenstein, P. Balcou, M. Ivanov, A. L’Huillier, and P. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref]

1991 (2)

A. L’Huillier, K. J. Schafer, and K. C. Kulander, “Theoretical aspects of intense field harmonic-generation,” J. Phys. B 24, 3315–3341 (1991).
[Crossref]

G. Mainfray and G. Manus, “Multiphoton ionization of atoms,” Rep. Prog. Phys. 54, 1333–1372 (1991).
[Crossref]

1976 (1)

J. Reintjes, R. Eckardt, C. She, N. Karangelen, R. Elton, and R. Andrews, “Generation of coherent radiation at 53.2  nm by fifth-harmonic conversion,” Phys. Rev. Lett. 37, 1540–1543 (1976).
[Crossref]

1975 (1)

J. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

1961 (1)

P. Franken, A. Hill, C. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–120 (1961).
[Crossref]

Agostini, P.

J.-F. Hergott, M. Kovacev, H. Merdji, C. Hubert, Y. Mairesse, E. Jean, P. Breger, P. Agostini, B. Carré, and P. Salières, “Extreme-ultraviolet high-order harmonic pulses in the microjoule range,” Phys. Rev. A 66, 021801 (2002).
[Crossref]

E. Constant, D. Garzella, P. Breger, E. Mével, C. Dorrer, C. L. Blanc, F. Salin, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: model and experiment,” Phys. Rev. Lett. 82, 1668–1671 (1999).
[Crossref]

Akturk, S.

C. L. Arnold, B. Zhou, S. Akturk, S. Chen, A. Couairon, and A. Mysyrowicz, “Pulse compression with planar hollow waveguides: A pathway towards relativistic intensity with table–top lasers,” New J. Phys. 12, 073015 (2010).
[Crossref]

Ališauskas, S.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hermández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. Murnane, and H. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[Crossref]

Alonso, B.

Alsaif, B.

Andrews, R.

J. Reintjes, R. Eckardt, C. She, N. Karangelen, R. Elton, and R. Andrews, “Generation of coherent radiation at 53.2  nm by fifth-harmonic conversion,” Phys. Rev. Lett. 37, 1540–1543 (1976).
[Crossref]

Andriukaitis, G.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hermández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. Murnane, and H. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[Crossref]

Arisholm, G.

Arnold, C.

Arnold, C. L.

C. L. Arnold, B. Zhou, S. Akturk, S. Chen, A. Couairon, and A. Mysyrowicz, “Pulse compression with planar hollow waveguides: A pathway towards relativistic intensity with table–top lasers,” New J. Phys. 12, 073015 (2010).
[Crossref]

Arpin, P.

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A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis, F. W. Helbing, U. Keller, and A. Mysyrowicz, “Self-compression of ultra-short laser pulses down to one optical cycle by filamentation,” J. Mod. Opt. 53, 75–85 (2006).
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C. M. Heyl, J. Güdde, A. L’Huillier, and U. Höfer, “High-order harmonic generation with μJ laser pulses at high repetition rates,” J. Phys. B 45, 074020 (2012).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hermández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. Murnane, and H. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
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J. Limpert, A. Klenke, M. Kienel, S. Breitkopf, T. Eidam, S. Hädrich, C. Jauregui, and A. Tuennermann, “Performance scaling of ultrafast laser systems by coherent addition of femtosecond pulses,” IEEE J. Sel. Top. Quantum Electron. 20, 5 (2014).
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F. Böhle, M. Kretschmar, A. Jullien, M. Kovacs, M. Miranda, R. Romero, H. Crespo, U. Morgner, P. Simon, R. Lopez-Martens, and T. Nagy, “Compression of CEP-stable multi-mJ laser pulses down to 4  fs in long hollow fibers,” Laser Phys. Lett. 11, 095401 (2014).
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J. Rothhardt, M. Krebs, S. Hädrich, S. Demmler, J. Limpert, and A. Tünnermann, “Absorption-limited and phase-matched high harmonic generation in the tight focusing regime,” New J. Phys. 16, 033022 (2014).
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P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. 84, 073103 (2013).
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J.-F. Hergott, M. Kovacev, H. Merdji, C. Hubert, Y. Mairesse, E. Jean, P. Breger, P. Agostini, B. Carré, and P. Salières, “Extreme-ultraviolet high-order harmonic pulses in the microjoule range,” Phys. Rev. A 66, 021801 (2002).
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E. Constant, D. Garzella, P. Breger, E. Mével, C. Dorrer, C. L. Blanc, F. Salin, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: model and experiment,” Phys. Rev. Lett. 82, 1668–1671 (1999).
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S. Leone, C. McCurdy, J. Burgdörfer, L. Cederbaum, Z. Chang, N. Dudovich, J. Feist, C. Greene, M. Ivanov, R. Kienberger, U. Keller, M. Kling, Z.-H. Loh, T. Pfeifer, A. Pfeiffer, R. Santra, K. Schafer, A. Stolow, U. Thumm, and M. Vrakking, “What will it take to observe processes in ‘real time’?” Nat. Photonics 8, 162–166 (2014).
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S. Leone, C. McCurdy, J. Burgdörfer, L. Cederbaum, Z. Chang, N. Dudovich, J. Feist, C. Greene, M. Ivanov, R. Kienberger, U. Keller, M. Kling, Z.-H. Loh, T. Pfeifer, A. Pfeiffer, R. Santra, K. Schafer, A. Stolow, U. Thumm, and M. Vrakking, “What will it take to observe processes in ‘real time’?” Nat. Photonics 8, 162–166 (2014).
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A. L’Huillier, K. J. Schafer, and K. C. Kulander, “Theoretical aspects of intense field harmonic-generation,” J. Phys. B 24, 3315–3341 (1991).
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H. Conrads and M. Schmidt, “Plasma generation and plasma sources,” Plasma Sources Sci. Technol. 9, 441–454 (2000).
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S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnurer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: experiments versus numerical simulations,” Phys. Rev. E 74, 056604 (2006).
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M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: Extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
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Schrauth, S.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hermández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. Murnane, and H. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
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M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: Extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
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S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnurer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: experiments versus numerical simulations,” Phys. Rev. E 74, 056604 (2006).
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S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnurer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: experiments versus numerical simulations,” Phys. Rev. E 74, 056604 (2006).
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Nat. Photonics (1)

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J. Rothhardt, M. Krebs, S. Hädrich, S. Demmler, J. Limpert, and A. Tünnermann, “Absorption-limited and phase-matched high harmonic generation in the tight focusing regime,” New J. Phys. 16, 033022 (2014).
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M. Mlejnek, M. Kolesik, J. Moloney, and E. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83, 2938–2941 (1999).
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P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. 84, 073103 (2013).
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P. Russell, “Applied physics: photonic crystal fibers,” Science 299, 358–362 (2003).
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Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1.

Illustration of scale-invariant nonlinear optics: a laser pulse is focused (with focal length f ) into a gas medium with length L and density ρ . Nonlinear propagation effects lead to a modification of the spatiotemporal pulse profile. Identical spatiotemporal modifications can be expected if a more intense laser pulse is focused more weakly (with f η f to reach the same intensity) into a larger medium with length η 2 L and lower density ρ / η 2 . Note that the beam diameter before focusing is kept constant in this illustration.

Fig. 2.
Fig. 2.

Scaling filamentation and HHG. (a), (b) Simulated spatiotemporal intensity distributions (normalized individually) in a focused laser beam in Ar for three different positions along the propagation axis and two input parameter sets, scaled according to the presented scaling relations: (a)  τ = 20 fs , ε in = 2 mJ , p = 1.2 bar , W 0 = 40 μm ; (b)  τ = 20 fs , ε in = 128 mJ , p = 18.75 mbar , W 0 = 320 μm ). (e), (f) Simulated spatiotemporal intensity distributions for high-harmonic emission (above 31.5 eV) in Ar [same color scale as used for (a) and (b)] at three positions within the nonlinear medium: (e)  τ = 10 fs , ε in = 62.5 μJ , p = 256 mbar , W 0 = 10.6 μm , L = 2 mm ; (f)  τ = 10 fs , ε in = 16 mJ , p = 1 mbar , W 0 = 169.6 μm , L = 0.51 m . For both filamentation and HHG, the longitudinal position is specified with respect to the position of the geometrical focus; in (a) and (b) in units of the respective Rayleigh lengths, and in (e) and (f) in units of the length of the generation medium L . (c) Characteristic length, i.e., filament and gas cell length, respectively (blue, left axis) and gas pressure (red, right axis) as a function of η and ε in . η was arbitrarily set to unity for ε in = 1 mJ . (d) Integrated relative scaling error for the filament scaling presented in (a) and (b) for intensity (dots) and fluence (circles) (see Supplement 1).

Fig. 3.
Fig. 3.

Experimental filament scaling. (a), (b) Measured temporal intensity profiles as well as spectral power [(b) solid lines] and phase [(b) dashed lines] for six different parameter sets, shown in (c). For better visualization, the plotted datasets are vertically offset from each other. The measurement was performed by selecting the broadband radiation on the optical axis more than a focal length distance behind the filament. For reference, the input spectrum (gray shaded area) is shown in (b). The solid lines in (c) represent fits to the experimental data points, as defined by the presented scaling relations, indicating the expected scaling performance for input laser pulse energies within and beyond the measured parameter range. The gray data points in (c) visualize the extrapolated parameters shown in Table 2.

Tables (2)

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Table 1. Scaling Relations Derived in This Work a

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Table 2. Extrapolation of Typical Parameters for Filamentation and HHG a

Equations (3)

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[ z i 2 k ( ω , ρ ) Δ i k ( ω , ρ ) ] E ^ = i ω 2 2 k ( ω , ρ ) c 2 ε 0 P ^ NL .
[ z i c 2 ω Δ ] E ^ = 0 .
[ z i 2 k ( ω , ρ ) Δ i K ( ω , ρ ) ] E ^ = i ω 2 2 k ( ω , ρ ) c 2 ε 0 P ^ NL ( ρ ) ,

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