Harmonic generation

Perturbative Harmonic Generation

Perturbative Harmonic Generation is a process where by laser lightof frequency ω and photon energy ħω can be usedto generate new frequencies of light. The newly generatedfrequencies are integer multiples of the fundamentalnħω and this process was first discovered in 1961 byFranken et al [P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett.7, 118 (1961).] , with a ruby laser andcrystalline quartz as the non linear medium.

Harmonic generation in dielectric solids is well understood andextensively used in modern laser physics (see second harmonic generation). In 1967 New et al observed the first third harmonicgeneration in a gas [G. H. C. New and J. F. Ward, Phys. Rev. Lett. 19, 556 (1967).] . In monatomic gases it is onlypossible to produce odd numbered harmonics for reasons ofsymmetry. Harmonic generation in the perturbative (weak field)regime is characterised by rapidly decreasing efficiency withincreasing harmonic order and harmonics up to the 11th order havebeen observed under these conditions [J. Wildenauer, Journal of Applied Physics 62, 41 (1987).] . This behaviour can beunderstood by considering the electron having absorbed n photonsthen emitting a single high energy photon. The probability ofabsorbing n photons decreases as n increases, explaining the rapiddecrease in the initial harmonic intensities.

High Harmonic Generation (HHG)

The first High Harmonic Generation (HHG) was observed in1988 [M. Ferray et al., Journal of Physics B-Atomic Molecular and OpticalPhysics 21, L31 (1988).] , with surprising results. The highharmonic spectra were found to decrease in intensity at low ordersbut then form a plateau, whereby the intensity of the harmonicsremain approximately constant over many orders [X. F. Li, A. L'Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, PhysicalReview A 39, 5751 (1989).] .Plateau harmonics spanning hundreds of eV have been measured whichextend into the soft x-ray regime [J. Seres et al., Nature 433, 596 (2005).] . This plateau ends abruptly at a position calledthe High Harmonic Cut-off.

Properties of High Harmonics

High harmonics have a number of interesting properties. They are atunable table-top source of XUV/Soft X-rays, synchronised with thedriving laser and produced with the same repetition rate. Theharmonic cut-off varies linearly with increasing laser intensity upuntil the saturation intensity I_sat where harmonic generationstops [T. Brabec and F. Krausz, Reviews of Modern Physics 72, 545 (2000).] . The saturation intensity can beincreased by changing the atomic species to lighter noble gasesbut these have a lower conversion efficiency so there is a balanceto be found depending on the photon energies required.

High harmonic generation strongly depends on the driving laserfield and as a result the harmonics have similar temporal andspatial coherence properties [A. L'Huillier, K. J. Schafer, and K. C. Kulander, Journal of Physics B AtomicMolecular and Optical Physics 24, 3315 (1991).] . High harmonics are often generatedwith pulse durations shorter than that of the driving laser. Thisis due to phase matching and ionisation. Often harmonics are onlyproduced in a very small temporal window when the phase matchingcondition is met. Depletion ofthe generating media due to ionisation also means that harmonicgeneration is mainly confined to the leading edge of the drivingpulse [K. J. Schafer and K. C. Kulander, Physical Review Letters 78, 638 (1997).] .

High harmonics are emitted co-linearly with the driving laser andcan have a very tight angular confinement, sometimes with lessdivergence than that of the fundamental field and near Gaussianbeam profiles [J. W. G. Tisch et al., Physical Review A 49, R28 (1994).] .

Semi-classical approach to describe HHG

The maximum photon energy producible with high harmonic generation is given by the cut-off of the harmonic plateau. This can be calculated classically by examining themaximum energy the ionized electron can gain in the electric field of the laser. The cut-off energy is given by,

$E\_\{max\}=I\_p+3.17U\_p$

Where U_p is the Ponderomotive Energy from the laser field and I_p is the
ionization potential.

This derivation of the cut-off energy is derived from asemi-classical calculation whereby, the electron is treated quantummechanically while it tunnel ionises from the parent atom (see tunnel ionization)and its subsequent dynamicsare treated with classically. The electron is assumed to be born intothe continuum with zero initial velocity and is then acceleratedby the laser E-field. Half an optical cycle after ionisation the electron will reverse direction as theE-field changes and accelerate back towards the parent nucleus. Upon returning to the parent nucleus it can then emit bremsstrahlung-like radiation during a recombinationprocess with the atom as it returns back to its ground state.This description has become known as the recollisional model of high harmonic generation [P. B. Corkum, Physical Review Letters 71, 1994 (1993).] .

Some interesting limits on the HHG process which are explained by this model show that HHG will only occur ifthe driving laser field is linearly polarised. Ellipticity on the laser beam causes the returning electron to miss the parent nucleus. Quantummechanically, the overlap of the returning electron wavepacketwith the nuclear wavepacket is reduced. This has been observedexperimentally, where the intensity of harmonics decreases rapidlywith increasing ellipticity [P. Dietrich, N. H. Burnett, M. Ivanov, and P. B. Corkum, Physical ReviewA 50, R3585 (1994).] . Another effect which limits theintensity of the driving laser is the Lorentz force. Atintensities above 10¹⁶ Wcm^-2 the magnetic component ofthe laser pulse which is ignored in weak field optics, can becomestrong enough to deflect the returning electron. This will causeit to 'miss' the parent nucleus and hence prevent HHG.

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