# Angle-resolved RABBIT: theory and numerics

## Abstract

Angle-resolved (AR) RABBIT measurements offer a high information content measurement scheme, due to the presence of multiple, interfering, ionization channels combined with a phase-sensitive observable in the form of angle and time-resolved photoelectron interferograms. In order to explore the characteristics and potentials of AR-RABBIT, a perturbative 2-photon model is developed; based on this model, example AR-RABBIT results are computed for model and real systems, for a range of RABBIT schemes. These results indicate some of the phenomena to be expected in AR-RABBIT measurements, and suggest various applications of the technique in photoionization metrology.

Article history

• Update 23/05/17 - new section added discussing approximations in more explicit detail.

## Introduction

The RABBIT methodology - “reconstruction of attosecond harmonic beating by interference of two-photon transitions” (Muller 2002) - essentially defines a scheme in which XUV pulses are combined with an IR field, and the two fields are applied to a target gas. The gas is ionized, and the photoelectrons detected. In the typical case, the IR field is at the same fundamental frequency $$\omega$$ as the field used to drive harmonic generation, and the XUV field generated is an atto-second pulse train with harmonic components $$n\omega$$, with odd-$$n$$ only. In this case, if the intensity of the IR field is low to moderate, the resultant photoelectron spectrum will be comprised of discrete bands corresponding to direct 1-photon XUV ionization, and sidebands corresponding to 2-photon XUV+IR transitions (Muller 2002). (The energetics of this situation are illustrated in fig. \ref{fig:pathways}.) Temporally, if the XUV pulses are short relative to the IR field cycle, the sidebands will also show significant time-dependence, since they will be sensitive to the optical phase difference between the XUV and IR fields, with an oscillatory frequency of $$2\omega$$. In this case, a measurement which is angle-integrated, or made at a single detection geometry, can be viewed as a means to characterising the properties of the XUV pulses (spectral content and optical phase), provided that the ionizing system is simple or otherwise well-characterised (Muller 2002); RABBIT can therefore be utilised as a pulse metrology technique (Muller 2002, Krausz 2009), and this is the typical usage.

Conversley, RABBIT can also be regarded as a photoelectron metrology technique, since it is sensitive to the magnitudes and phases of the various photoionization pathways accessed. In contrast to most traditional (energy-resolved) photoelectron spectroscopy techniques, RABBIT has the distinction of interfering pathways resulting from different 1-photon transition energies: it is thus sensitive to the energy-dependence of the photoionization dynamics, as well as to the partial-wave components within each pathway. An angle-resolved (AR) RABBIT measurement is particularly powerful in this regard, since the partial-wave phases are encoded in the angular part of the photoelectron interferogram. Although this is a potentially powerful technique, the underlying photoionization dynamics may be extremely complicated, hence quantitative analysis of experimental results is challenging.