We present an inversion method called pBasex aimed at reconstructing the original Newton sphere of expanding charged particles from its two-dimensional projection by fitting a set of basis functions with a known inverse Abel integral. The basis functions have been adapted to the polar symmetry of the photoionization process to optimize the energy and angular resolution while minimizing the CPU time and the response to the cartesian noise that could be given by the detection system. The method presented here only applies to systems with a unique axis of symmetry although it can be adapted to overcome this restriction. It has been tested on both simulated and experimental noisy images and compared to the Fourier-Hankel algorithm and the original Cartesian basis set used by [Dribinski et al.Rev. Sci. Instrum. 73, 2634 (2002)], and appears to give a better performance where odd Legendre polynomials are involved, while in the images where only even terms are present the method has been shown to be faster and simpler without compromising its accuracy.

Two-dimensional charged particle image inversion using a polar basis function expansion

Angle-resolved photoelectron spectroscopy has been performed for more than 70 years in various guises, but recently its potential to help solve in detail problems in the photoionization dynamics and intramolecular dynamics of gas-phase molecules has been recognized. One key development has been the design of experiments in appropriate geometries to extract information that pertains to the molecular frame, another has been the development of imaging spectrometers, and a third is the use of ultrafast lasers to cause photoionization. In this review, which is aimed at experimentalists, simple expressions for photoelectron angular distributions (PADs) in various experimental geometries are given and their applications explained.

Photoelectron angular distributions.

Shine a light: a circular dichroism effect in the ±10 % regime on randomly oriented chiral molecules in the gas phase is demonstrated. The signal is derived from images of photoelectron angular distributions produced by resonance-enhanced multiphoton ionization and allows the enantiomers to be distinguished. To date, this effect could only be generated with a synchrotron source. The new tabletop laser-based approach will make this approach far more accessible.

Circular dichroism in the photoelectron angular distributions of camphor and fenchone from multiphoton ionization with femtosecond laser pulses.

In this Perspective we discuss photoelectron circular dichroism (PECD), a relatively novel technique that can detect chiral molecules with high sensitivity. PECD has an enantiomeric sensitivity of typically 1–10%, which is two to three orders of magnitude larger than that of the widely employed technique of circular dichroism (CD). In PECD a chiral molecule is photoionized with circular polarized light, and the photoelectron angular scattering distribution is detected using particle imaging techniques. We present the general physical principles of photoelectron circular dichroism and we address both single- and multiphoton excitation. PECD has been measured with synchrotron radiation in single-photon ionization as well as, very recently, with femtosecond laser radiation in multiphoton ionization. We discuss the experimental implementation of PECD, focusing on velocity map coincidence imaging where the momentum distribution of both the electron and the coincident ion is measured. The coincident detection of the mass and momentum of the ion adds very powerful mass-correlated information to the PECD measurement of the chiral molecule. We illustrate the capabilities and the potential of PECD with various experimental examples and introduce computational methods that are able to model quantitatively experimental PECD results. We conclude with an outlook on novel developments and (analytical) implementations of PECD that may further broaden the application of PECD for the sensitive detection of chirality in molecules.

Detecting chirality in molecules by imaging photoelectron circular dichroism

Here, we provide a detailed account of novel experiments employing electron-ion coincidence imaging to discriminate chiral molecules. The full three-dimensional angular scattering distribution of electrons is measured after photoexcitation with either left or right circular polarized light. The experiment is performed using a simplified photoelectron-photoion coincidence imaging setup employing only a single particle imaging detector. Results are reported applying this technique to enantiomers of the chiral molecule camphor after three-photon ionization by circularly polarized femtosecond laser pulses at 400 nm and 380 nm. The electron-ion coincidence imaging provides the photoelectron spectrum of mass-selected ions that are observed in the time-of-flight mass spectra. The coincident photoelectron spectra of the parent camphor ion and the various fragment ions are the same, so it can be concluded that fragmentation of camphor happens after ionization. We discuss the forward-backward asymmetry in the photoelectron angular distribution which is expressed in Legendre polynomials with moments up to order six. Furthermore, we present a method, similar to one-photon electron circular dichroism, to quantify the strength of the chiral electron asymmetry in a single parameter. The circular dichroism in the photoelectron angular distribution of camphor is measured to be 8% at 400 nm. The electron circular dichroism using femtosecond multiphoton excitation is of opposite sign and about 60% larger than the electron dichroism observed before in near-threshold one-photon ionization with synchrotron excitation. We interpret our multiphoton ionization as being resonant at the two-photon level with the 3s and 3p Rydberg states of camphor. Theoretical calculations are presented that model the photoelectron angular distribution from a prealigned camphor molecule using density functional theory and continuum multiple scattering X alpha photoelectron scattering calculations. Qualitative agreement is observed between the experimental results and the theoretical calculations of the Legendre moments representing the angular distribution for the two enantiomers. The electron-ion coincidence technique using multiphoton ionization opens new directions in table-top analytical mass-spectrometric applications of mixtures of chiral molecules.

Imaging photoelectron circular dichroism of chiral molecules by femtosecond multiphoton coincidence detection

It is shown that the angular distribution of photoelectrons ejected on absorption of left or right circularly polarized light by a mixture of randomly oriented dextrorotatory and levorotatory optical isomers behaves as $A\ifmmode\pm\else\textpm\fi{}Bcos\ensuremath{\theta}+C{cos}^{2}\ensuremath{\theta}$, where $\ensuremath{\theta}$ is the angle between the photoelectron momentum and photon direction of incidence and $\ifmmode\pm\else\textpm\fi{} Bcos\ensuremath{\theta}$ is introduced by the presence of unequal numbers of each isomer. All coefficients of ${cos}^{n}\ensuremath{\theta}$ are of order $\ensuremath{\alpha}$, and analysis shows that $B$ can be as large as $A$ and $C$.

Theory of the angular distribution of photoelectrons ejected from optically active molecules and molecular negative ions

Formulas are presented for the angular distribution of photoelectrons produced on absorption of circularly polarized photons by randomly oriented, optically active (chiral) molecules or molecular negative ions. Results are presented to order ${\ensuremath{\alpha}}^{3}$. The results for nonchiral molecules can be recovered by setting the appropriate coefficients equal to zero. These findings suggest measurement of the angular distribution difference for absorption of left and right circularly polarized photons, whose leading term behaves as $2{B}_{e}cos\ensuremath{\theta}$, where ${B}_{e}$ is a coefficient appropriate for the electric dipole ($E1$) process and $\ensuremath{\theta}$ is the angle between the direction of the emerging beam of photoelectrons and the direction of incidence of the photon beam. The integrated cross-section difference behaves as $8\ensuremath{\pi}({A}_{m}+\frac{{C}_{m}}{3})$, where these coefficients are appropriate for the electric-dipole, magnetic-dipole ($E1M1$) interference; this term is responsible for the circular dichroism of the molecule in the region of continuous absorption.

Burke Ritchie

Papers

Theory of the angular distribution of photoelectrons ejected from optically active molecules and molecular negative ions

Theory of the angular distribution for ejection of photoelectrons from optically active molecules and molecular negative ions. II