Final Report Summary - ATTOSECOND OPTICS (Harnessing attosecond nonlinear optics for controlling and enhancing high harmonic generation and producing useful coherent x-rays on a tabletop) Project context and objectivesHigh harmonic generation (HHG) is a nonlinear process that is used for production of coherent light at extreme ultraviolet and soft x-ray spectral regions. In HHG, high-intensity visible/Infrared laser light is upconverted into extreme ultraviolet and x-ray radiation. This HHG light is emitted in a directed, narrow divergence beam that can be fully coherent,  and can have a pulse duration as short as hundred attosecond (1 as = 10^ -18 seconds). [2-5] This table-top x-ray light source has found broad range of applications during the past few years. [1-7] The energy range of the photons emitted in HHG is very broad and ends up with a cut-off energy that scales linearly with the applied laser intensity. [8,9] The conversion efficiency of HHG is unfortunately relatively low; [10-5] and smaller, primarily due to collective de-phasing effects (low phase matching condition). Moreover, the conversion efficiency to shorter wavelengths decreases rapidly due to de-phasing. [6,8] The dominant source for de-phasing of the high harmonics emitted from many atoms is the associated photo-ionisation of the nonlinear medium. The large and dynamically varying index of refraction of the free-electron plasma speeds up the phase velocity of the driving laser with respect to the generated harmonics. This light-induced dispersion prevents the constructive interference of the x-ray waves that were emitted from all the atoms in the gas. A main research objective of this International Reintegration Grant (IRG) was developing schemes for correcting the de-phasing effect and therefore enhances the conversion efficiency of HHG. Other objectives focused on developing schemes for controlling the spectral, temporal, spatial and polarisation properties of the produced HHG beam.During the IRG grant period, we developed two approaches for correcting the de-phasing problem and therefore enhance the HHG efficiency. We also developed schemes for controlling the spatio-spectral structure of attosecond pulses that are produced through HHG as well as the polarisation states of HHG beams. Finally, we have been developing a compact HHG-based emitter of coherent extreme ultraviolet radiation that is driven by a laser with very long pulses (quasi-continuous-wave).- In Ref. 9, we showed that a sawtooth phase-modulation is the optimal profile for grating assisted phase matching (GAPM) a technique for quasi phase matching of nonlinear processes, including high harmonic generation. Perfect (sharp) sawtooth modulation fully corrects the phase-mismatch, exhibiting conversion equal to conventional phase matching, while smoothened, approximate sawtooth structures are more efficient than sinusoidal or square GAPM modulations that were previously studied. As an example, we demonstrate numerically optically-induced sawtooth GAPM for high harmonic generation. Sawtooth GAPM is the most efficient method for increasing the conversion efficiency of high harmonic generation through quasi-phase-matching, with an ultimate efficiency that closely matches the ideal phase-matching case. Experiments for demonstrating sawtooth GAPM is currently on-going in our Lab. Preliminary results are encouraging.- In Ref. 10, we demonstrated experimentally pressure-tuning phase matching of high harmonic generation that is produced in planar hollow waveguide. We also showed that that the regime of pressure-tuning phase matching in high harmonic generation is extended to shorter wavelengths when the process is driven by focusing light (positive intensity gradient) within a planar waveguide.- In Ref. 11, we and our collaborators showed, theoretically and experimentally, that increasing the wavelength of the driving laser increases the cut-off frequency for pressure tuning phase matching of HHG and therefore also increases the HHG flux at the x-ray spectral region.- In Ref 12, we proposed a scheme for producing attosecond pulses with sophisticated spatio-spectral waveforms. The profile of a seed attosecond pulse is modified and its central frequency is up-converted through interaction with an infrared pump pulse. The transverse profile of the infrared beam and a spatiotemporal shift between the seed and infrared pulses are used for manipulating the spatio-spectral waveform of the generated pulse beam. We present several examples of sophisticated isolated attosecond pulse beam generation, including spatio-spectral Airy beam that exhibits prismatic self-bending effect and a beam undergoing auto-focusing to a sub-micron spot without the need of a focusing lens or nonlinearity.- In Refs. 13, we proposed a method for obtaining high-harmonic radiation with desirable elliptical polarisation. Atoms are shined by a combination of a strong linearly-polarised laser fields and an additional weak field, which is elliptically-polarised in a plane perpendicular to the polarisation direction of the strong field. The strong driver ionises and recollides electrons with their parent ion, while the weak field perturbatively drives the electrons away from "head-on" collision. Upon recombination, new elliptically-polarised harmonics with same ellipticity as the weak driver are emitted at efficiency that is proportional to the intensity of the weak beam and independent of its elilipticity.- In Refs. 14 and 15, we propose a device that emits ultra-narrow bandwidth high-order harmonics of a continuous-wave driving laser. The device consists of nano antennas that are coupled to a whispering gallery micro-resonator. We develop this device with our collaborators.