# High-order harmonic generation

Matter responds highy noninlearly to an intense laser field, generating high-harmonics, i.e, radiation whose frequency is a multiple of that of the driving field. These harmonics exhibit a region with comparable intensities, the so-called "plateau", followed by a sharp decrease in the yield, the so-called "cutoff".

It is well-known that high-order harmonic generation is the result of a three-step physical mechanism in which an electron reaches the continuum by tunneling of multiphoton ionization, propagates in the field and recombines with a bound state of its parent ion, generating harmonics. Below you will find a brief summary of our work on the topic

## Multielectron effects in molecular HHG

with Brad Augstein (PhD student/PDRA)

High-harmonic spectrum from N2 taking into account multiple orbitals. From Phys. Rev. A 81 (4), 043409 (2010)

In this project, the Strong-Field Approximation has been modified in order to include multiple molecular orbitals and/or more than one active electron, for a diatomic molecule. This has been performed employing simplified multi-orbital models [1] and Moller-Plesset perturbation theory around the highest-occupied molecular orbital [2]. We found that, as long as the multielectron effects are incorporated statically, they do not influence the spectra considerably. Furthermore, for an initial coherent superposition, different orbitals can be mapped to different regions in the spectra. Finally, we have investigated how nodal planes can be mapped into nodal surfaces in HHG spectra, if isoelectronic pairs consisting of a homonuclear and a heteronuclear molecule are taken [3]. We were one of the first groups worldwide to go beyond the single active electron, single active orbital approximation within the SFA. For a brief review of how our work fits into a broader context see [4].

[1] C. Figueira de Morisson Faria and B.B. Augstein, Phys. Rev. A 81 (4), 043409 (2010)

[2] B.B. Augstein and C. Figueira de Morisson Faria, J. Mod. Opt. 58(13), 1173-1187 (2011)

[3] B.B. Augstein and C. Figueira de Morisson Faria, J. Phys. B 44, 055601 (2011)

[4] B.B. Augstein and C. Figueira de Morisson Faria, Modern Physics Letters B

26(2), 1130002 (brief review) (2012)

## Orthogonally polarized fields

with Toni Das (MSc/PhD student and PDRA) and Brad Augstein (PDRA)

The main objective of this project was to steer electron dynamics using

orthogonally polarized fields and retrieve this information from high-order harmonic spectra. A striking feature encountered by us was an orbit-dependent dynamic shift that could be related to an electron’s angle of return and was superposed to purely structural interference effects [1]. These studies were first performed for single molecules and subsequently taking account macroscopic effects such as phase matching. This latter work was performed in collaboration with Prof Jon Marangos’s group at the Imperial College London, in which we found realistic conditions for which the above-mentioned shift could be observed in an experimental setting [2]. It was numerically challenging and required the use of High-Performance Computing. We have also used the field ellipticity as a tool for mapping molecular nodal planes and revealing further artefacts the SFA, as it allowed access to an inaccessible parameter range [3]. This work strengthened our links with experimental groups and broadened our research towards HHG propagation.

[1] T. Das, B. B. Augstein and C. Figueira de Morisson Faria, Phys. Rev. A 88, 023404 (2013)

[2] T. Das, B. B. Augstein, C. Figueira de Morisson Faria, L. E. Chipperfield, D. J. Hoffmann, J. P. Marangos, Phys. Rev. A 92, 023406 (2015)

[3] T. Das and C. Figueira de Morisson Faria, Phys. Rev. A 94,023406 (2016

Propagated high-harmonic spectra from a diatomic molecule in an rthogonally polarized field, highlighting how the contributions of a specific orbit can be extracted, with the electron's angle of return. From T. Das et al, Phys. Rev. A 92, 023406 (2015)

## Inhomogeneous media

with Carlos Zagoya (PDRA), Heloise Chomet (MSci student), Matt Bonner (MSci student), Emma Slade (summer student)

The goal of this work was to understand whether nanostructures are viable HHG sources. A key question was to understand the success of simplified theoretical models, which found that inhomogeneous fields extend the maximal high-order harmonic energy and increase the harmonic intensity in orders of

magnitude. Such studies, however, are mainly descriptive and thus fairly vague about the physical reasons behind this. Using a phase-space analysis, we have shown that the models used in the literature can be described by Mathieu’s equation. We have also identified the different time scales involved and provided analytical expressions that are formally similar to those for an

ensemble of ions in a Paul trap. This work provides a unique intake on plasmonic HHG enhancements, bringing together Dynamical Systems Theory and Strong-Field Physics.

C. Zagoya, M. Bonner, H. Chomet, E. Slade, C. Figueira de Morisson Faria, Phys. Rev. A 93, 053419 (2016)

Time-frequency maps and classical returning times (superimposed dots) as functions of the harmonic order and the field cycles, for several inhomogeneity parameters

## Control of High-order Harmonic Generation

Early career work

I have also made significant contributions to the control of HHG, in earlier phases of my career, in collaboration with scientists of several institutions, such as the MBI-Berlin, the MPIPKS-Dresden, and the MPQ, Munich. This line of research includes several studies of HHG with bi- and polychromatic driving fields [1-4], and additional binding potentials [5]. Using the time-dependent Schroedinger equation (TDSE) and time-frequency analysis, we have shown that it is possible to suppress and enhance groups of harmonics, or extend the maximal harmonic frequency without loss of intensity [5]. We have also investigated resonant enhancements in HHG [6]. Further studies include an analysis of quantum interference effects and control of HHG using attosecond pulses [7] and investigating quantum wires [8] and quantum wells [9] as possible HHG sources. The latter work lies at the borderline between attosecond and condensed-matter physics, and shows that confinement plays a key role in extending the HHG plateau. This has been confirmed and extended by our recent work on plasmonically enhanced HHG.

[1] C. Figueira de Morisson Faria, M. Dörr, W. Becker and W. Sandner, Phys. Rev. A 60(2), 1377-1384 (1999)

[2] C. Figueira de Morisson Faria, W. Becker, M. Dörr and W. Sandner, Laser Phys. 9(1), 388-394 (1999)

[3] C. Figueira de Morisson Faria, D. B. Milošević and G.G. Paulus, Phys. Rev. A 61(6), 063415 (2000)

[4] C. Figueira de Morisson Faria and M.L. Du, Phys. Rev. A 64(2), 023415 (2001)

[5] C. Figueira de Morisson Faria and J.M. Rost, Phys. Rev. A 62(5), 051402(R) (2000)

[6] C. Figueira de Morisson Faria, R. Kopold, W. Becker and J.M. Rost, Phys. Rev. A 65(2), 023404 (2002)

[7] C. Figueira de Morisson Faria, P. Salières, P. Villain and M. Lewenstein, , Phys. Rev. A 74 (5), 053416 (2006)

[8] O.A. Castro-Alvaredo, A. Fring and C. Figueira de Morisson Faria, Phys. Rev. B 67(12), 125405 (2003)

[9] C. Figueira de Morisson Faria and I. Rotter, Phys. Rev. A 66(1), 013402 (2002); Laser Phys. 13, 985-994 (2003).