Ultrafast chirality: the road to efficient chiral measurements

Abstract

Today we are witnessing the electric-dipole revolution in chiral measurements. Here we reflect on its lessons and outcomes, such as the perspective on chiral measurements using the complementary principles of "chiral reagent" and "chiral observer", the hierarchy of scalar, vectorial and tensorial enantio-sensitive observables, the new properties of the chiro-optical response in the ultrafast and non-linear domains, and the geometrical magnetism associated with the chiral response in photoionization. The electric-dipole revolution is a landmark event. It has opened routes to extremely efficient enantio-discrimination with a family of new methods. These methods are governed by the same principles but work in vastly different regimes - from microwaves to optical light; they address all molecular degrees of freedom - electronic, vibrational and rotational, and use flexible detection schemes, i.e. detecting photons or electrons, making them applicable to different chiral phases, from gases to liquids to amorphous solids. The electric-dipole revolution has also enabled enantio-sensitive manipulation of chiral molecules with light. This manipulation includes exciting and controlling ultrafast helical currents in vibronic states of chiral molecules, enantio-sensitive control of populations in electronic, vibronic and rotational molecular states, and opens the way to efficient enantio-separation and enantio-sensitive trapping of chiral molecules. The word "perspective" has two meanings: an "outlook" and a "point of view". In this perspective article, we have tried to cover both meanings.

Fig. 1. Two challenges for ultrafast chiral spectroscopy: spatial and temporal scales of electron dynamics in molecules. (Left) The electric field vector of circularly polarized light makes a helix in space. The pitch of the optical helix (light wavelength) is orders of magnitude larger than the size of small chiral molecules, chiral moieties, or local chiral centers such as asymmetric carbons. This usually leads to extremely weak enantio-sensitive signals in such systems. (Right) The optical cycle is orders of magnitude longer than the characteristic time scale of electron motion in molecules (10–100 as).

Fig. 2. Two roads to efficient enantio-discrimination: chiral reagent (approaches that substitute the inefficient (non-local) light helix in space by an efficient (local) light helix in time) and chiral observer (approaches that do not rely on the chiral properties of light).

Fig. 3. Left- and right-chewing (chiral) cows convert in-plane rotation of the jaw, characterized by the pseudo-vector L⃑, into out-of-plane food motion, characterized by the vector J⃑.

Fig. 4. (a) Circularly polarized light (CPL) defines a (chiral) helix in space and thus it can lead to enantio-sensitive scalar observables. (b) In the electric–dipole approximation, where the helical structure of CPL is neglected, the in-plane rotation of the electric-field vector defines a pseudo-vector, which can couple to molecular pseudoscalars and lead to enantio-sensitive vectorial observables. (c) The Lissajous figure of a two-colour field with orthogonally polarized ω and 2ω frequency components does not define an oriented plane, and therefore it does not provide a pseudo-vector. It can lead to enantio-sensitive tensorial observables.

Fig. 5. Two electric dipole revolutions in chiral measurements. Left box: Traditional chiroptical methods rely on the spatial helix of light. Right box: The first electric dipole revolution introduces efficient methods which can detect enantio-sensitive vectorial and tensorial observables without relying on the spatial helix of light. These methods rely on the principle of a chiral observer – a chiral set-up formed by combinations of achiral fields and detectors. The second electric dipole revolution introduces an efficient chiral reagent – it substitutes the inefficient (non-local) light helix in space by an efficient (local) chiral light trajectory in time.

Fig. 6. Photoelectron circular dichroism (left) vs. photoexitation circular dichroism (right). In PECD circularly polarized light induces a chiral current in the continuum, while in PXCD, it induces a chiral current in the bound states.

Fig. 7. Chiral electronic currents in molecules: trajectories traced by the tip of the induced dipole in randomly oriented chiral molecules. (a) Helical attosecond electron current in PXCD for a model chiral molecule. Time is in units of the period set by the energy difference between the excited states. (b) Time-dependent polarization driven in randomly oriented propylene oxide by of an elliptically polarized field with intensity 5 × 10¹³ W cm⁻², wavelength 1770 nm and 5% of ellipticity (see ref. for details of the calculations). The polarization of the driving field is depicted by the dashed red ellipse. The ultrafast polarization response is 3D, chiral, and enantio-sensitive: the in-plane (achiral) polarization components are identical in left- and right-handed molecules, whereas the (chiral) out-of plane component is out of phase in opposite enantiomers.

Fig. 8. Diagrams for various nonlinear enantio-sensitive or chiral-sensitive processes. Different colors mark different directions of light polarization (e.g. green – x, red – y, blue – z). Straight up and down arrows indicate photon absorption and photon emission, wavy arrows indicate induced polarization. (a) Diagrams for photo-excitation circular dichroism (PXCD), enantio-sensitive microwave spectroscopy (EMWS), and difference-frequency generation (DFG); (b) sum-frequency generation (SFG). For SFG, the field pseudovector is given by L⃑ = E⃑(ω₁) × E⃑(ω₂) and thus it vanishes when ω₁ = ω₂, i.e. second harmonic generation requires more than two photons. (c) Second harmonic generation (SHG) is indeed possible for higher-order processes because these are associated with field pseudovectors such as L⃑ = [E⃑(ω₁)·E⃑(ω₁)][E⃑*(ω₁) × E⃑(ω₁)], which are non-zero for elliptically polarized light and record its rotation direction.

Fig. 9. Chiral HHG driven using light with “forward” ellipticity. (a) Schematic representation of a non-collinear setup for creating a field with “forward” ellipticity; (b) a Gaussian beam acquires a strong longitudinal component upon tight focusing, also leading to “forward ellipticity”, which has opposite signs on opposite sides of the laser beam axis. (c) Achiral response corresponds to odd harmonics, chiral response corresponds to even harmonics and is out of phase in opposite enantiomers. (d) Spectral overlap of odd-order and even-order nonlinear-optical response in a few cycle pulse leads to opposite rotation of the polarization ellipse in opposite enantiomers.

Fig. 10. The concept of synthetic chiral light. (a and b) Schematic representation of two circularly polarized fields with opposite handedness in a given point in space: the electric-field (grey) and magnetic-field (purple) vectors are confined to the xy plane, orthogonal to the propagation direction of the wave (black). The two fields of opposite handedness cannot be superimposed by rotation, and therefore are chiral. However, in the electric–dipole approximation, which neglects the spatial structure of the wave and therefore its propagation direction and magnetic-field component, the two fields become identical, and thus achiral. (c and d) Synthetic chiral light can be created by combining a field that is elliptically polarized in the xy plane with frequency ω and a field that has twice the frequency and is linearly polarized along z. The resulting field is locally chiral because the tip of its electric-field vector draws a chiral Lissajous figure in time, at every point in space. Indeed, the Lissajous figures in c and d, corresponding to opposite two-colour phase delays, are mirror images which cannot be superimposed by rotation. Colour indicates positive (red) and negative (blue) values of E_z.

Fig. 11. Enantio-sensitivity in absorption. (a) Absorption occurs in a three-level system driven by three-color locally chiral light with frequencies ω₁, ω₂, and ω₃ polarized along x, y, and z, respectively. The lack of inversion symmetry in a chiral molecule allows for dipole couplings between all states. The second-order (two-photon induced) polarization at ω₃ = ω₁ + ω₂ is generated along z in randomly oriented chiral media. (b) Enantio-sensitive absorption for two-colour locally chiral light. Different colors mark different directions of light polarization (e.g. green – x, red – y, blue – z).

Fig. 12. Locally and globally chiral light. (a) Synthetic chiral light that is locally and globally chiral can be created with two non-collinear beams carrying cross polarized ω and 2ω colours. In the overlap region, the total ω field is elliptical in the xy plane, the 2ω field is z-polarized, generating the chiral Lissajous curve in the inset. (b) Even harmonic intensity emitted by randomly oriented left- and right-handed propylene oxide and the chiral response, see ref. for details. The field's chirality, and thus the enantio-sensitive response of the medium, is fully controlled by the ω, 2ω phase delays in the two beams.

Fig. 13. Polarization of charge versus polarization of chirality. (a) 1D arrangement of charged units that is: (i) neutral and unpolarized, and (ii) neutral and polarized. (b) 1D arrangement of chiral units that is: (i) achiral (racemic) and unpolarized, and (ii) achiral and polarized.

Fig. 14. Chirality polarized light can be created using the setup of Fig. 12a, but adjusting the ω, 2ω phase delays so that the field's handedness, characterized by its 5th-order chiral correlation function h⁽⁵⁾, is not maintained globally in space, in contrast to ref. . Here, it creates a periodic structure of dipoles of chirality. (a) h⁽⁵⁾, its phase (i.e. the field's handedness) is encoded in the colours. The arrows indicate the direction of polarization of chirality, which is imprinted in the nonlinear response of chiral matter. (b) 12th-harmonic emission from randomly oriented left- and right-handed fenchone, see ref. for λ = 1030 nm, F⁽¹⁾_ω = F⁽²⁾_ω = 0.015 a.u. and F_2ω = F_ω/10.

Fig. 15. An object displaying compound chirality in the form of two independent handedness: a helix made of a more tightly bound helix.