Attosecond Theory Theoretical work undertaken in the project **Contents** **Introduction** Understanding the fundamental physics underlying the process of high harmonic generation (HHG) is an essential part of developing and utilising the attosecond light source. To this end we have carried out detailed theoretical investigations to provide support and guidance to the experimental project. The theory of HHG can be split into two main areas. Firstly, the response of single atoms to intense laser fields. Secondly, how the high frequency light emitted by isolated atoms coherently sum to form the net emission from a gas. **Single atom** **Fig. 1:** Time resolved harmonic emission (from a wavelet transform ) calculated using the TDSE model. The overlaid black curves are the quantum orbits given by the quantum orbit model, corresponding to electron trajectories which have spend under one cycle in the continuum. The electric field of the laser pulse is shown below and the HHG spectrum is plotted on the left. As can be seen the features present in the TDSE model are reproduced by the quantum orbit model. | Several methods are utilised to calculate the response of a single atom to the intense laser pulse, however, two models dominate. The first proceeds by numerically solving the time-dependent Schrodinger equation (TDSE) in one-dimension. This includes the full interaction of the electron with the atomic potential and laser field. The second model employs the strong field approximation (SFA) and then decomposes the resulting integral into a sum of complex electron trajectories, or quantum orbits. The quantum orbit model enables the direct elucidation of the temporal structure of HHG. The approach of applying the SFA is valid for intense laser fields; for 800nm radiation this implies that the intensity should exceed 100TWcm^{-2}. The process of HHG in this strong field limit can be seen as a three-stage process: (1) tunnel ionisation of the most weakly bound electron near the local peak of the electric field, (2) acceleration of the electron in the time-varying laser field and (3) recombination of the continuum electron into the bound electronic state, about two-thirds of a cycle later, accompanied by the emission of HHG radiation. The time structure of HHG emerges naturally within this picture, since the period of emission is inherently brief (typically a few hundred attoseconds). The formulation of this theory in terms of electron trajectories provides a powerful insight into the quantum physics. It allows for individual trajectories to be studied in isolation, enabling their particular contribution to the total spectrum to be found and hence building a deeper understanding of how features in the spectrum are formed. **Fig. 2:** Experimental and theoretical positions of half-cycle cut-offs vs. CEP. The theoretical HCO positions are given by the black curves. The vertical coloured bands are spatially integrated experimental HHG spectra for different values of CEP from the lock point, with the data smoothed in the spectral direction to remove the modulation from individual harmonics. | **Propagation** Although in a dilute gas each atom responds to the laser field in ignorance of its neighbours, HHG is inherently a multi-atom process. Due to the coherence of the laser field only atomic emissions that are well phase-matched will survive propagation to be detected in an experiment. The large variations in the intensity and phase of the laser field throughout the interaction region makes the process highly selective, filtering out much of the spectrum seen in the single-atom calculations. It also means that the geometry of the laser focus, in particular the location of the gas jet with repect to the laser focus, is of critical importance. Investigations into how individual electron trajectories contributed to the final propagated spectrum led to the discovery that there were significant features in the final spectrum that were produced by single pairs of electron trajectories. The contributuions from these pairs of trajectories are dependent on the form of the laser field during their time in the continuum. For the pair's cut-off energy this corresponds to just over a half-cycle of the laser field and are therefore called half-cycle cut-offs (HCOs); as opposed to the cut-off for the entire spectrum. It is these HCOs that are visible in the final spectrum and their energies provide half-cycle snapshots of the electric-field. Working alongside the experimental team this was developed into a highly accurate technique for reconstructing few-cycle electric-fields. |