Attosecond Pulse Generation
High-harmonic generation in a gas: ponderomotive energy, cutoff photon energy I_p + 3.17 U_p, last odd harmonic, barrier-suppression intensity and the Fourier-limited attosecond pulse of the selected band.
Ponderomotive energy U_p
11.95 eV
barrier-suppression intensity 2.47e14 W/cm²
Cutoff photon energy
53.65 eV
I_p + 3.17 U_p; λ = 23.11 nm, last odd harmonic 33; Keldysh γ = 0.812
Fourier-limited pulse
54.24 as
Gaussian, ΔE = 33.65 eV from the band edge to the cutoff
Driver cycles in the FWHM
1.87
bursts every T/2 = 1.334 fs; harmonics 2ħω = 3.100 eV apart
In the three-step model an electron tunnels out near a field peak, is driven by the laser field and recombines, emitting up to I_p + 3.17 U_p, with U_p = e²I/(2ε₀cmω²) ∝ Iλ². A multi-cycle driver repeats this every half cycle, giving a train of attosecond bursts and odd harmonics 2ħω apart; an isolated pulse needs a driver of about one cycle or a gating scheme. The Fourier limit treats the selected band, from its low edge (a metal filter, say) to the cutoff, as a Gaussian FWHM. The intrinsic attochirp makes real pulses longer, and above the barrier-suppression intensity ionization depletes the gas before the peak. Since Γ grows as I^(3/2)λ⁴, long-wavelength drivers meet the magnetic-drift limit first (Γ = 1 for helium near 7 × 10¹⁵ W/cm² at 1.8 µm, 1.3 × 10¹⁴ W/cm² at 8 µm).
Cutoff energy against driver wavelength
- Cutoff, argon at 2.00e14 W/cm²
- Set wavelength
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Three-step model cutoff I_p + 3.17 U_p, U_p = e²I/(2ε₀cmω²), ionization energies from NIST. The Fourier limit treats the band from its low edge to the cutoff as a Gaussian FWHM; attochirp, phase matching and ionization depletion are not modelled. Barrier suppression from the classical over-the-barrier field; few-cycle drivers reach somewhat beyond it (Brabec & Krausz §VI–VII). The page flags the magnetic (Lorentz) drift when Klaiber's parameter √(2I_p)a₀³c/(16ω) exceeds 1, where recombination falls.
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- Brabec T., Krausz F. (2000). Intense few-cycle laser fields: frontiers of nonlinear optics. Rev. Mod. Phys. 72, 545–591, §VI–VII. doi:10.1103/RevModPhys.72.545
- Klaiber M., Hatsagortsyan K. Z., Müller C., Keitel C. H. (2008). Coherent hard x rays from attosecond pulse train-assisted harmonic generation. Opt. Lett. 33, 411–413. Drift criterion √(2I_p)ξ³c/(16ω) > 1. doi:10.1364/OL.33.000411
- Galloway B. R. et al. (2016). Lorentz drift compensation in high harmonic generation in the soft and hard X-ray regions of the spectrum. Opt. Express 24, 21818. doi:10.1364/OE.24.021818