Abstract

High-order harmonic generation (HHG) in gases is known to benefit from using mid-infrared driving laser fields, since, due to a favorable wavelength scaling of the electron ponderomotive energy, higher-energy photon production becomes feasible with longer-wavelength drivers. On the other hand, recent studies have revealed a number of physical effects whose importance for HHG increases with increasing laser wavelength. These effects, as a rule, result not only in a general decrease of the harmonic yield but also in a reshaping of the emission spectrum. Therefore, detailed study of the dependence of HHG yield on the laser wavelength has become an important issue for producing intense extremely short extreme ultraviolet (XUV) and x-ray pulses using HHG driven by long-wavelength laser fields. Here we address this issue by calculating the HHG spectra for laser wavelengths ranging from 2 to 20 µm. This study has been carried out in a frame of strong-field approximation modified properly to take into account the effect of the magnetic field of a laser pulse on the dynamics of the field-ionized electron and the atomic bound-state depletion. We show that different regions of the HHG spectrum behave differently with the laser wavelength and discuss the origins of this behavior. In particular, we show that in a weak ionization regime, the dipole-approximation scaling law for the harmonic yield, which is calculated as the integral over the spectral interval of fixed width and relative position with respect to the cutoff energy, obeys the power law, where the absolute value of the exponent is an integer equal to $\mu = {7}$ for the cutoff and $\mu = {8}$ for the plateau harmonics. Above a certain critical wavelength, due to the nondipole effects, the efficiency of HHG decreases more strongly than according to a power law, and this decrease is different for different regions of the spectrum. The analytical formulas are derived that match well the calculated wavelength scalings.

© 2019 Optical Society of America

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References

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2019 (2)

E. Migal, A. Pushkin, B. Bravy, V. Gordienko, N. Minaev, A. Sirotkin, and F. Potemkin, “3.5  mJ 150  fs Fe:ZnSe hybrid mid-IR femtosecond laser at 4.4  µm for driving extreme nonlinear optics,” Opt. Lett. 44, 2550–2553 (2019).
[Crossref]

D. J. Wilson, A. M. Summers, S. Zigo, B. Davis, S.-J. Robatjazi, J. A. Powell, D. Rolles, A. Rudenko, and C. A. Trallero-Herrero, “An intense, few-cycle source in the long-wave infrared,” Sci. Rep. 9, 6002 (2019).
[Crossref]

2018 (4)

2017 (1)

2016 (3)

2015 (4)

A. S. Emelina, M. Yu. Emelin, and M. Yu. Ryabikin, “Multi-keV ultrahigh-order harmonics produced in gases with subrelativistically intense mid-IR laser pulses,” J. Opt. Soc. Am. B 32, 2478–2487 (2015).
[Crossref]

M. V. Frolov, N. L. Manakov, W.-H. Xiong, L.-Y. Peng, J. Burgdörfer, and A. F. Starace, “Scaling laws for high-order-harmonic generation with midinfrared laser pulses,” Phys. Rev. A 92, 023409 (2015).
[Crossref]

B. Wolter, M. G. Pullen, M. Baudisch, M. Sclafani, M. Hemmer, A. Senftleben, C. D. Schröter, J. Ullrich, R. Moshammer, and J. Biegert, “Strong-field physics with mid-IR fields,” Phys. Rev. X 5, 021034 (2015).
[Crossref]

A. V. Mitrofanov, A. A. Voronin, D. A. Sidorov-Biryukov, A. Pugžlys, E. A. Stepanov, G. Andriukaitis, T. Flöry, S. Ališauskas, A. B. Fedotov, A. Baltuška, and A. M. Zheltikov, “Mid-infrared laser filaments in the atmosphere,” Sci. Rep. 5, 8368 (2015).
[Crossref]

2014 (3)

K.-H. Hong, C.-J. Lai, J. P. Siqueira, P. Krogen, J. Moses, C.-L. Chang, G. J. Stein, L. E. Zapata, and F. X. Kärtner, “Multi-mJ, kHz, 2.1  µm optical parametric chirped-pulse amplifier and high-flux soft x-ray high-harmonic generation,” Opt. Lett. 39, 3145–3148 (2014).
[Crossref]

A.-T. Le, H. Wei, C. Jin, V. N. Tuoc, T. Morishita, and C. D. Lin, “Universality of returning electron wave packet in high-order harmonic generation with midinfrared laser pulses,” Phys. Rev. Lett. 113, 033001 (2014).
[Crossref]

A. S. Emelina, M. Y. Emelin, and M. Y. Ryabikin, “On the possibility of the generation of high harmonics with photon energies greater than 10  keV upon interaction of intense mid-IR radiation with neutral gases,” Quantum Electron. 44, 470–477 (2014).
[Crossref]

2013 (1)

C. Hernández-Garsía, J. A. Pérez-Hernández, T. Popmintchev, M. M. Murnane, H. C. Kapteyn, A. Jaron-Becker, A. Becker, and L. Plaja, “Zeptosecond high harmonic keV x-ray waveforms driven by midinfrared laser pulses,” Phys. Rev. Lett. 111, 003002 (2013).
[Crossref]

2012 (6)

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

C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature 483, 194–197 (2012).
[Crossref]

A. D. DiChiara, E. Sistrunk, C. I. Blaga, U. B. Szafruga, P. Agostini, and L. F. DiMauro, “Inelastic scattering of broadband electron wave packets driven by an intense midinfrared laser field,” Phys. Rev. Lett. 108, 003002 (2012).
[Crossref]

C. Vozzi, M. Negro, and S. Stagira, “Strong-field phenomena driven by mid-infrared ultrafast sources,” J. Mod. Opt. 59, 1283–1302 (2012).
[Crossref]

Y. Deng, A. Schwarz, H. Fattahi, M. Ueffing, X. Gu, M. Ossiander, T. Metzger, V. Pervak, H. Ishizuki, T. Taira, T. Kobayashi, G. Marcus, F. Krausz, R. Kienberger, and N. Karpowicz, “Carrier-envelope-phase-stable, 1.2  mJ, 1.5 cycle laser pulses at 2.1  µm,” Opt. Lett. 37, 4973–4975 (2012).
[Crossref]

M. V. Frolov, N. L. Manakov, A. M. Popov, O. V. Tikhonova, E. A. Volkova, A. A. Silaev, N. V. Vvedenskii, and A. F. Starace, “Analytic theory of high-order-harmonic generation by an intense few-cycle laser pulse,” Phys. Rev. A 85, 033416 (2012).
[Crossref]

2011 (2)

M. V. Frolov, N. L. Manakov, A. A. Silaev, N. V. Vvedenskii, and A. F. Starace, “High-order harmonic generation by atoms in a few-cycle laser pulse: carrier-envelope phase and many-electron effects,” Phys. Rev. A 83, 021405 (2011).
[Crossref]

G. Andriukaitis, T. Balciunas, S. Ališauskas, A. Pugžlys, A. Baltuška, T. Popmintchev, M. C. Chen, M. M. Murnane, and H. C. Kapteyn, “90  GW peak power few-cycle mid-infrared pulses from an optical parametric amplifier,” Opt. Lett. 36, 2755–2757 (2011).
[Crossref]

2009 (2)

M. V. Frolov, N. L. Manakov, T. S. Sarantseva, and A. F. Starace, “Analytic formulae for high harmonic generation,” J. Phys. B 42, 035601 (2009).
[Crossref]

M. V. Frolov, N. L. Manakov, T. S. Sarantseva, M. Yu. Emelin, M. Yu. Ryabikin, and A. F. Starace, “Analytic description of the high-energy plateau in harmonic generation by atoms: Can the harmonic power increase with increasing laser wavelengths?” Phys. Rev. Lett. 102, 243901 (2009).
[Crossref]

2008 (2)

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4, 386–389 (2008).
[Crossref]

M. V. Frolov, N. L. Manakov, and A. F. Starace, “Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. 100, 173001 (2008).
[Crossref]

2007 (3)

J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[Crossref]

C. Vozzi, F. Calegari, E. Benedetti, S. Gasilov, G. Sansone, G. Cerullo, M. Nisoli, S. De Silvestri, and S. Stagira, “Millijoule-level phase-stabilized few-optical-cycle infrared parametric source,” Opt. Lett. 32, 2957–2959 (2007).
[Crossref]

M. Uiberacker, Th. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond real-time observation of electron tunnelling in atoms,” Nature 446, 627–632 (2007).
[Crossref]

2006 (1)

V. V. Strelkov, A. F. Sterjantov, N. Y. Shubin, and V. T. Platonenko, “XUV generation with several-cycle laser pulse in barrier-suppression regime,” J. Phys. B 39, 577–589 (2006).
[Crossref]

2005 (2)

X. M. Tong and C. D. Lin, “Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B 38, 2593–2600 (2005).
[Crossref]

M. Yu. Emelin, M. Yu. Ryabikin, A. M. Sergeev, M. D. Chernobrovtseva, T. Pfeifer, D. Walter, and G. Gerber, “High-efficiency generation of attosecond pulses during atomic ionization from excited electronic states,” Europhys. Lett. 69, 913–919 (2005).
[Crossref]

2003 (1)

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611–615 (2003).
[Crossref]

2002 (1)

C. C. Chirilă, N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Nondipole effects in photon emission by laser-driven ions,” Phys. Rev. A 66, 063411 (2002).
[Crossref]

2001 (2)

N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Photon emission by ions interacting with short intense laser pulses: beyond the dipole approximation,” J. Phys. B 34, L55–L61 (2001).
[Crossref]

B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A 65, 011804 (2001).
[Crossref]

2000 (2)

V. D. Taranukhin, “Relativistic high-order harmonic generation,” Laser Phys. 10, 330–336 (2000).
[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. Envelopes of the spectra of harmonics generated from different sources with wavelengths from 2 to 16 µm (see legend). Calculations were carried out without taking into account the electron magnetic drift and the atomic bound-state depletion.
Fig. 2.
Fig. 2. Envelopes of the spectra of harmonics generated from the sources with different wavelengths: (a) 8 µm; (b) 14 µm; (c) 20 µm. Calculations were carried out taking into account the atomic bound-state depletion, both in the dipole approximation (i.e., without taking into account the electron magnetic drift) and beyond it (see legend).
Fig. 3.
Fig. 3. HHG yield versus $\lambda $ for different positions of the integration window (see legend). Both the electron magnetic drift and the atomic bound-state depletion are taken into account. Dashed lines are the approximations by the ${\lambda ^{ - \mu }}$ scaling law.
Fig. 4.
Fig. 4. Same as in Fig. 3 but calculated in the dipole approximation.
Fig. 5.
Fig. 5. Same as in Fig. 4 but calculated by neglecting the atomic bound-state depletion.
Fig. 6.
Fig. 6. Same as in Fig. 4 but with the correction factor Eq. (22) taken into account.
Fig. 7.
Fig. 7. Relation between parameters $a$ and β, which, according to Eqs. (25),(28), gives the $ {p_ \bot } $ value as a function of the relative position of the harmonic frequency within the plateau, determined by parameter $a$ , for the electron short and long trajectories contributing predominantly to the harmonic photon emission.
Fig. 8.
Fig. 8. Normalized magnitude of the electric field at the instants of launching the electron short and long trajectories as a function of the relative position of the harmonic frequency within the plateau.
Fig. 9.
Fig. 9. Same as in Fig. 3 but with correction factors Eqs. (22) and (24) taken into account.

Equations (29)

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x ( t ) = i 0 t d τ ( π ε + i τ / 2 ) 3 / 3 2 2 d x ( p s t ( t , τ ) A ( t ) c ) × d x ( p s t ( t , τ ) A ( t τ ) c ) F ( t τ ) × exp [ i S ( p s t , x , t , τ ) i S m ( p s t , t , τ ) ] × exp [ 0 t Γ ( t ) 2 d t 0 t τ Γ ( t ) 2 d t ] + c . c .
d x ( p ) = i 2 7 / 7 2 2 π p x ( p 2 + 1 ) 3 ,
S ( p x , t , τ ) = t τ t [ 1 2 ( p x A ( t ) c ) 2 + I p ] d t ,
S m ( p , t , τ ) = t τ t 1 2 { p z [ p x A ( t ) c 2 A 2 ( t ) 2 c 3 ] } 2 d t ,
p s t , x ( t , τ ) = 1 τ t τ t A ( t ) c d t ,
p s t , z ( t , τ ) = p s t , x 2 c 1 2 c τ t τ t A 2 ( t ) c 2 d t ,
Γ ( t ) = 4 | F ( t ) | exp ( 2 3 | F ( t ) | ) exp ( 12 | F ( t ) | ) .
I ( E Ω ) I ( Ω ) = | 1 T 0 T x ¨ ( t ) exp ( i Ω t ) d t | 2 .
I a = 1 Δ E a ( 3.173 U p + I p ) a ( 3.173 U p + I p ) + Δ E I ( E Ω ) d E Ω ,
I ( E Ω ) = Ω I a t o m i c ( F ~ , ω ) W ( E ) σ a t o m i c ( r ) ( E ) ,
I a t o m i c ( F ~ , ω ) = 8 ω 2 π F ~ F 0 2 exp ( 2 3 F ~ ) ,
W ( E ) = p ( δ / δ I 0 I 0 ) 2 / 2 3 3 Ai 2 ( ξ ) τ 3 ,
ξ = ( δ / δ I 0 I 0 ) 1 / 1 3 3 ( E E max ) ;
σ a t o m i c ( r ) ( E ) = 32 π c 3 exp ( 4 p a r c t a n ( p ) ) p 2 ( p 2 + 1 ) 2 ( 1 exp ( 2 π p ) ) ,
3.173 U p I p Δ E / 2 ,
I ( E Ω ) λ 7 Ai 2 ( ξ ) ,
ξ = ( δ / δ I 0 I 0 ) 1 / 1 3 3 [ ( 1 a ) 3.173 U p + ( 1.32 a ) I p Δ E / Δ E 2 2 ] .
ξ ( δ / δ I 0 I 0 ) 1 / 1 3 3 ( 1 a ) 3.173 U p = 2.65 × 10 8 ( 1 a ) λ 2 ( a . u . ) = 9.47 ( 1 a ) λ 2 ( µ m ) .
Ai ( | ξ | ) sin ( 2 3 | ξ | 3 / 3 2 2 + π 4 ) π | ξ | 1 / 1 4 4 .
Ai ( | ξ | ) | ξ | 1 / 1 4 4
R d e p l = exp ( α Γ T ) ,
R d e p l = exp ( 1.6 Γ λ / c ) .
Γ p ( t ) = Γ 0 exp ( 2 3 | F ( t ) | ( 2 I p + p 2 ) 3 / 2 ) .
R N D ( a ) = exp { 2 3 | F ( a ) | [ ( 2 I p ) 3 / 2 ( 2 I p + p 2 ( a ) ) 3 / 2 ] } ,
p = 1 τ t τ t z ˙ ( t ) d t = F 0 2 c ω τ t τ t d t t τ t [ cos ( ω t ) cos φ ] sin ( ω t ) d t = β U p c ,
β = 1 2 ω τ [ sin ( 2 ω τ + 2 φ ) sin ( 2 φ ) ] cos ( 2 φ ) ,
t g φ = ω τ sin ( ω τ ) cos ( ω τ ) 1 ,
a = E + I p 3.173 U p + I p E 3.173 U p = 2 3.173 [ cos ( ω τ + φ ) cos φ ] 2 .
R N D ( 1 ) = exp { 2 3 × 0.951 F 0 × [ 1 ( 1 + ( 0.809 U p c ) 2 ) 3 2 ] } .

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