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 the maximum energy the ionized electron can gain in the electric field of the laser. The cut-off energy is given by,
where Up is the ponderomotive energy from the laser field and Ip is the ionization potential.
This derivation of the cut-off energy is derived from a semi-classical calculation. The electron is initially treated quantum mechanically as it tunnel ionizes from the parent atom, but then its subsequent dynamics are treated classically. The electron is assumed to be born into the vacuum with zero initial velocity, and to be subsequently accelerated by the laser beam's electric field.
Half an optical cycle after ionization, the electron will reverse direction as the electric field changes, and will accelerate back towards the parent nucleus. Upon returning to the parent nucleus it can then emit bremsstrahlung-like radiation during a recombination process with the atom as it returns to its ground state. This description has become known as the recollisional model of high harmonic generation .
Some interesting limits on the HHG process which are explained by this model show that HHG will only occur if the driving laser field is linearly polarised. Ellipticity on the laser beam causes the returning electron to miss the parent nucleus. Quantum mechanically, the overlap of the returning electron wavepacket with the nuclear wavepacket is reduced. This has been observed experimentally, where the intensity of harmonics decreases rapidly with increasing ellipticity. Another effect which limits the intensity of the driving laser is the Lorentz force. At intensities above 1016 Wcm−2 the magnetic component of the laser pulse, which is ignored in weak field optics, can become strong enough to deflect the returning electron. This will cause it to 'miss' the parent nucleus and hence prevent HHG.
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