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Phase mismatch and group velocity mismatch are two dispersion effects which limit the effectiveness of optical frequency conversion processes. Here we propose a condition for a high harmonic generation in which a phase mismatch and a group-velocity mismatch work together through constructive inter sub-cycle interference to provide an efficient, phase-matched optical frequency conversion.
This condition, which we call an anomalous phase matching (APM) can be valuable to the emerging extreme non-linear optics regime of a long wavelength drive field and a high gas pressure. The APM condition is fulfilled once both phase mismatch and group velocity mismatch (GVM) are present in the optical system, which is the most general case to consider. The effect of a phase mismatch is that after a propagation of one coherence length the emission start to destruct coherently, leading to a decrease in the up-conversion efficiency. The effect of the GVM is that after a particular harmonic order is being generated by the fundamental pulse, it starts to drift away from its creation time in the reference frame moving with the pump pulse, due to different group velocities between the pump and that harmonic. If a harmonic emission arrives, due to GVM, to an emission with the opposite polarity after traversing a coherence length the interference of both emissions would be constructive. Actually, this condition ensures constructive interference over the whole interaction length allowing for an increased HHG flux.
We derive a mathematical condition for an APM and show numerically that APM can be achieved and wavelength-tuned in several different setups: by varying the peak intensity of the pump laser in a pre-ionized gaseous medium, by varying the wave-guide radius in a hollow wave guide filled with a neutral gas and by varying the pressure of a neutral gas in a gas cell at moderate pressures and long fundamental beam wavelengths.
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