High-harmonic generation (HHG)

High-harmonic generation (HHG) is the emission of high odd multiples of a laser's frequency by atoms or molecules exposed to a field comparable to the binding field of their outer electrons. A near-infrared ultrafast laser pulse of tens of femtoseconds, focused into a gas jet or gas-filled cell to $10^{14}$–$10^{15}$ W/cm², produces a comb of harmonics spaced by twice the laser photon energy. Their intensity falls over the first few orders, stays roughly constant across a broad plateau, then drops sharply at a cutoff. With an 800 nm Ti:sapphire laser in argon the cutoff lies near 44 eV (28 nm); neon and helium, with higher ionization potentials, tolerate higher intensities and reach 100–200 eV and beyond. HHG is the standard table-top source of coherent extreme-ultraviolet light and of attosecond pulses. Unlike perturbative processes such as third-harmonic generation, its efficiency does not fall steeply with harmonic order across the plateau.

The three-step model

Corkum's semiclassical model (1993), developed in parallel by Kulander, Schafer and Krause, describes each half-cycle of the drive field in three steps:

1. the strong field bends the atomic potential and an electron tunnels out near a field maximum;
2. the free electron is accelerated away, then driven back as the field reverses;
3. a fraction of returning electrons recombine with the parent ion and emit their kinetic energy plus the ionization potential $I_p$ as a single photon.

Classical trajectories give a maximum return kinetic energy of 3.17 times the ponderomotive energy $U_p$, the mean quiver energy of a free electron in the field. The cutoff photon energy is therefore

$$E_\text{max} = I_p + 3.17\,U_p$$ $$U_p\,[\mathrm{eV}] = 9.33 \times 10^{-14}\, I\,\lambda^2$$

with $I$ in W/cm² and $\lambda$ in µm. The quantum treatment of Lewenstein and coworkers (1994) keeps this form with a slightly larger coefficient on $I_p$.

For argon ($I_p$ = 15.76 eV) at 800 nm and $1.5 \times 10^{14}$ W/cm², $U_p$ = 8.96 eV and $E_\text{max}$ = 44.2 eV. The drive photon energy is 1.550 eV, so the cutoff falls at order 28.5, and the highest odd harmonic below it is the 27th, at 41.8 eV and 29.6 nm. Observed cutoffs are often a little lower, because ionization depletes the medium before the pulse peak.

Odd harmonics and attosecond pulses

The emission repeats every half-cycle with the sign of the field reversed. In a symmetric medium this half-cycle periodicity with alternating sign produces only odd harmonics, spaced by $2\hbar\omega$ = 3.10 eV at 800 nm. In the time domain the plateau harmonics form an attosecond pulse train with one burst per half-cycle, every 1.33 fs at 800 nm. Isolating a single burst requires a drive pulse only a few optical cycles long with a stabilized carrier-envelope phase, or a gating technique that confines emission to one half-cycle. Isolated pulses of 100–300 as are now common, and several groups have measured pulses below 100 as, including 43 as in 2017. Pierre Agostini, Ferenc Krausz and Anne L'Huillier received the 2023 Nobel Prize in Physics for experimental methods that generate attosecond pulses of light, work that began with the observation of the harmonic plateau in rare gases in 1987–1988.

Phase matching and efficiency

The single-atom response is weak, and useful flux requires the harmonic fields from many atoms to add in phase. The wavevector mismatch has contributions from neutral-gas dispersion, free-electron plasma dispersion, the Gouy phase of the focus and the intensity-dependent dipole phase. Phase matching is possible only while the ionization fraction stays below a critical value of a few percent, which limits the usable intensity. Loosely focused gas jets and cells and gas-filled hollow waveguides are the main geometries. Conversion efficiencies per harmonic are typically $10^{-6}$–$10^{-5}$ in argon, lower in neon and helium.

Because $U_p$ scales as $\lambda^2$, longer drive wavelengths extend the cutoff. At 1.8 µm and the same intensity, $U_p$ grows by a factor of 5.06 to 45.3 eV and the argon cutoff estimate rises to about 160 eV. Mid-infrared drivers built on optical parametric amplification and chirped pulse amplification have pushed HHG into the water window (about 280–530 eV) and beyond, at the cost of a single-atom yield that falls steeply with wavelength.

Measurement and applications

The harmonics copropagate with the intense drive beam, which is removed with thin metal filters (aluminium transmits roughly 20–70 eV) or grazing-incidence reflections. A grazing-incidence grating spectrometer with a microchannel plate or an XUV-sensitive CCD resolves the individual orders, and attosecond pulse durations are measured by photoelectron methods such as streaking and RABBITT. HHG sources serve attosecond spectroscopy of electron dynamics, angle-resolved photoemission, coherent diffractive imaging, and inspection of extreme-ultraviolet lithography masks.

Common questions

Why are only odd harmonics produced?

A gas of atoms is centrosymmetric, so the response to a field $E$ is the negative of the response to $-E$. A response that repeats every half-cycle with alternating sign contains only odd multiples of the drive frequency. Adding a weak second-harmonic field breaks this symmetry and produces even harmonics as well.

What limits the cutoff energy?

Raising the intensity raises $U_p$ but also ionizes the gas, destroying phase matching and depleting the atoms before the pulse peak. Gases with high $I_p$, short drive pulses and longer drive wavelengths are the main routes to higher cutoffs.

References: P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994 (1993); M. Lewenstein et al., "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994); F. Krausz and M. Ivanov, "Attosecond physics," Rev. Mod. Phys. 81, 163 (2009); T. Gaumnitz et al., "Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver," Opt. Express 25, 27506 (2017).