# Theoretical Study: High Harmonic Generation by Light Transients  
[PERSON]  
1 Department fur Physik, Ludwig-Maximilians-Universitat Munchen, Am Coulombwall 1, D-85748 Garching, Germany; [EMAIL_ADDRESS] (M.W. ); [PERSON].H. )1  
Max-Planck Institut fur Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany1  
Footnote 1: email: {}  
Received: 11 April 2018; Accepted: 2 May 2018; Published: 5 May 2018  
[PERSON]  
1 Department fur Physik, Ludwig-Maximilians-Universitat Munchen, Am Coulombwall 1, D-85748 Garching, Germany; [EMAIL_ADDRESS] (M.W. ); [PERSON].H. )1  
[PERSON]  
1 Department fur Physik, Ludwig-Maximilians-Universitat Munchen, Am Coulombwall 1, D-85748 Garching, Germany; [EMAIL_ADDRESS] (M.W. ); [PERSON].H. )1  
[PERSON]  
1 Department fur Physik, Ludwig-Maximilians-Universitat Munchen, Am Coulombwall 1, D-85748 Garching, Germany; [EMAIL_ADDRESS] (M.W. ); [PERSON].H. )1  
Footnote 11:  
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In high harmonic generation (HHG), an increase in the period of the oscillating laser field leads to the increase in the electron's excursion time and therefore recollision energy and HHG cutoff. The longer wavelength of the driving field results in a higher excursion time \(\tau_{\mathrm{cr}}\propto\lambda_{0}\) of the freed electrons. However, quantum diffusion scales with \(\lambda^{-5}\) to \(\lambda^{-7}\), which dramatically reduces the recombination probability with the parent ion (\(\propto\tau^{-3}\)) and thus the yield of the harmonic radiation at keV photon energies [4]. On the other hand, the HHG cutoff scales linearly with the laser's peak intensity. However, increasing the peak intensity of pulses containing more than one optical cycle leads to undesired pre-ionization of the gas atoms, causing depletion of their ground state, reducing the phase-matching during pulse propagation and ultimately leading to an exponential drop in harmonic yield [5; 6]. High peak intensity also causes spatiotemporal distortions within the driving pulse. Defocusing and self-phase modulation reduces the peak intensity as the pulse propagates through the gas, thereby reducing the yield of high energy photons [7; 8]. In addition, the HHG yield depends on the coherent build-up of radiation emitted from many atoms, which is optimal when both the driving pulse and the generated light travel in phase through the medium. This condition is achieved when the positive dispersion of the neutral gas is balanced by the negative dispersion of the free electrons. However, this is only possible below a critical ionization rate, therefore favouring long wavelength driving fields, as they require a lower intensity to generate a specific cutoff energy. Generation of excessive free electrons in the medium at high peak intensity results in a phase mismatch [9]. The quantum phase of the electron wavefunction returning to the core can be optimized by shaping the laser field to enhance a particular harmonic feature or the intra-atomic phase matching. Therefore, the yield and efficiency of the HHG process can be increased by synthesis of two or three color sinusoidal pulses [10; 11; 12; 13] due to the enhanced tunnel ionization [14; 15]. However, this approach suffers from pre-ionization and ground state depletion, due to the multiple field cycles in the interfering pulses and their incomplete destructive field interference over tens of femtoseconds out of the temporal window of interest for HHG. Non-sinusoidal waveforms can be generated by field synthesis of few-cycle pulses at different central frequencies to confine the field's energy and therefore ionization within a single cycle electric field. To overcome these barriers and to push HHG cutoff energies beyond the keV levels, all the above-mentioned parameters must be carefully tailored and optimized. Despite preliminary studies on the waveform synthesis of few-cycle pulses [6; 13; 16], there has not been systematic investigation on optimized drivers for generating isolated attosecond pulses in the X-ray regime.