Nonlinear helical dichroism in chiral and achiral molecules

Chiral interactions are prevalent in nature, driving a variety of biochemical processes. Discerning the two non-superimposable mirror images of a chiral molecule, known as enantiomers, requires interaction with a chiral reagent with known handedness. Circularly polarized light beams are often used as a chiral reagent. Here we demonstrate efficient chiral sensitivity with linearly polarized helical light beams carrying an orbital angular momentum of ±lħ, in which the handedness is defined by the twisted wavefront structure tracing a left- or right-handed corkscrew pattern as it propagates in space. By probing the nonlinear optical response, we show that helicity-dependent nonlinear absorption occurs even in achiral molecules and can be controlled. We model this effect by considering induced multipole moments in light–matter interactions. Design and control of light–matter interactions with helical light may open new opportunities in chiroptical spectroscopy, light-driven molecular machines, optical switching and in situ ultrafast probing of chiral systems and magnetic materials. Nonlinear absorption of helical light beams offers a new chiroptical detection scheme for both chiral and achiral molecules in liquid phase.

Our understanding of light-matter interactions is mainly based on the propagation of homogeneously polarized light and the dominance of the dipole-active transitions between different quantum states of matter. Higher-order multipole effects are often ignored. The strength of dipole transitions is governed by the frequency, intensity and polarization of the incident light. The optical phase, represented by the wavefront of the light beam, plays a minimal role in such transitions. Within the dipole approximation, chiral systems are often studied using circularly polarized light (CPL), in which the dynamical rotation of the electric-field vector around the propagation direction yields an effective spin angular momentum (SAM) ±ħ, and therefore the handedness.
Two correlated optical techniques are widely used to probe chiral systems based on the propagation of polarized light through a chiral sample: circular dichroism (CD) and optical rotation (OR). In CD, left-CPL and right-CPL are absorbed differently in the two enantiomers, leading to differential absorption 1,2 . In OR, on the other hand, left-CPL and right-CPL travel with unequal velocities in the two enantiomers, leading to polarization rotation 3,4 . The chiral sensitivity of CD is poor, on the order of 0.01-1% (ref. 5 ), because it involves coupling of electric and magnetic dipole transitions. CD can be enhanced by employing super chiral light 6 , plasmonic structures 7,8 and strong field techniques using elliptically polarized light 9,10 . A distinct gas-phase chiroptical technique that has emerged in recent years is photo-electron circular dichroism (PECD). In PECD, photo-ionization of a chiral molecule results in an asymmetric photo-electron angular distribution for the two enantiomers 11 . PECD is due to pure electric dipole transitions, so its efficiency is one to two orders of magnitude larger than that of CD 5,12 .
Light can also carry orbital angular momentum (OAM) of ±lħ associated with dynamical rotation of the wavefront structure [13][14][15] . The handedness of such helical light beams is defined by the twisting of the wavefront undergoing l intertwined rotations in one wavelength. This additional degree of freedom influences the characteristics of lightmatter interactions. The angular distribution 16 , time delay 17 and dynamics of photo-electrons 18,19 can be modified during the photo-ionization of atoms and molecules. Also, the OAM of light can be transferred to matter either externally by exerting a torque (as in optical tweezers 20 and Bose-Einstein condensates 21,22 ) or internally by rotating the electron distribution and resulting in modified selection rules for transitions that depend on the topological charge l (refs. 23,24 ). Article https://doi.org/10.1038/s41566-022-01100-0 on the laser polarization and are typically higher for linear than circular polarization [35][36][37] . Figure 2a,b shows HD(Type II) in fenchone as a function of peak laser fluence for l = ±1 and l = ±2, respectively. The results for limonene (C 10 H 16 ) are shown in Supplementary Section 1. Also shown in Fig. 2a is the differential absorption of two orthogonal linearly polarized Gaussian beams (dashed lines). The chiral signal in fenchone increases with laser fluence, reaching a maximum of ~4-6%, and also increases with the helicity (l value) of the beam (Fig. 2a,b). For l = 2, the onset of nonlinear absorption occurs at higher pulse energies. In experiments with different l values, the pulse energies were always varied from below the threshold for the onset of nonlinear absorption to approximately twice the threshold value, as shown in Fig. 1b. This ensured that the nonlinear response regime we investigated remained the same (~10 13 W cm −2 ), even though the peak fluence of OAM beams depends on the l value 38 (Supplementary Section 8). Chiral discrimination with linearly polarized helical light beams demonstrates the prominent role of the optical phase associated with the OAM of light.
Generally, differential absorption defined by HD(Type I) = A (+l, s) − A(−l, s) is not expected in isotropic achiral molecules with CPL. This is also true even with helical light 29 . We did not observe any differential absorption of linearly polarized helical light with symmetric OAM beams where the null intensity region is at the centre of the beam. However, when the singularity in the OAM beam is displaced from the centre, achiral molecules with different point group symmetries exhibit dichroism (Fig. 3). The singularity within the focal region of a focused OAM beam can be shifted either by (1) detuning the q-plate, leading to a superposition of the incident Gaussian beam with the converted OAM beam 39 , or (2) physically shifting the centre of the q-plate with respect to the incident beam. We used the latter technique, because the former gives rise to a Gaussian component in the transmitted beam. The resultant asymmetric OAM beam introduces an asymmetry in the intensity profile in the focal region, as shown in Fig. 3a. The orientation of the laser polarization is perpendicular to the displacement of the singularity. In the experiment, alignment of the singularity at the centre of the OAM beam can only be defined before the objective, which translates to an uncertainty ~±100 nm at the focus (see Methods for calibration details). Figure 3b presents the differential absorption of linearly polarized helical light (l = ±1) in achiral molecules, HD(Type I), with no inversion symmetry-acetone (C 3 H 6 O), methanol (CH 3 OH) and air-for different positions of the singularity in the focused OAM beam. The differential absorption is averaged over a fluence range from the onset of nonlinear absorption (which depends on the ionization potential) in the normalized transmission curve to ~2 J cm −2 . In acetone (blue curve) and methanol (red curve), differential absorption exhibits a sinusoidal behaviour, with a change in sign as the singularity traverses the zero position from −1.8 μm to 1.8 μm. The small variation in the zero-crossing point for the two molecules (~100 nm) is due to uncertainty in the position of the singularity. Helicity-dependent absorption does not exist in air (empty cuvette in the experiment) and is independent of the position of the singularity (black curve). This represents the background noise introduced by our experimental set-up (Supplementary Section 5). Helicity-dependent absorption in benzene (C 6 H 6 ), a molecule with inversion symmetry, is shown in Fig. 3c (black curve). The observed behaviour is similar to that of non-centrosymmetric molecules (Fig. 3b), suggesting that the HD signal does not appear to depend on molecular symmetry.
To disentangle the roles of phase and field gradients in an asymmetric OAM beam, we studied the differential absorption of a circularly polarized, non-OAM, annular beam that mimics the profile of a Laguerre-Gaussian (LG) beam with zero OAM value (Supplementary Section 6). Dichroism in R(−)-fenchone (green curve), S(+)-fenchone (red curve) and acetone (blue curve) is independent of the position of the singularity and is less than 1%, within the standard error. Absence of differential absorption confirms that helicity-dependent absorption in achiral molecules is a phase effect and not only due to field gradients.
The possibility of no upper bound on the l value, while SAM can assume only two defined values, has generated substantial interest in using helical light beams carrying OAM as a chiral reagent. Early studies on chirality with twisted photons were not promising. Most experiments were performed in the linear absorption regime 25,26 , whereas theory focused solely on the coupling of the electric and magnetic dipole transition moments responsible for conventional CD 27 . Helical dichroism (HD), analogous to CD, was demonstrated by exploiting local field effects in molecules adsorbed on nanoparticle aggregates 28 , chiral metasurfaces 29 and non-chiral nanostructures 30 . Coupling of the OAM of light with a material's chirality without any intermediary was recently shown to require higher-order transition moments to be engaged along with SAM to observe any chiroptical effects 31,32 .
In this Article we introduce a conceptually new form of chiroptical detection technique based on nonlinear absorption of linearly polarized helical light beams in the liquid phase. We first demonstrate enantioselectivity in chiral molecules, directly without any intermediary, by probing the differential absorption between the two enantiomers for a specific helical light-we call this HD(Type II). Second, we show the differential absorption of left-and right-helical light in isotropic achiral and chiral molecules-HD(Type I)-which can be precisely controlled by displacing the singularity present in an OAM beam. Third, we reveal that HD(Type I) is scalable by changing the OAM value and that it can be further controlled by varying the laser polarization. This feature does not exist in polarization-based chiroptical techniques, such as CD. Finally, we show that HD(Type I, II) is a phase effect and does not necessarily require CPL. To understand HD, we model the light-matter interaction by considering multipole expansion. We find that HD arises from coupling of the electric dipole and electric quadrupole terms, and that it can be tuned by changing the laser polarization in addition to the OAM value.
To probe the dichroism, we measured the absorption of loosely focused femtosecond Gaussian and helical light pulses propagating through a liquid sample contained in a cuvette, as shown in Fig. 1a. A q-plate converted incident Gaussian light to an optical vortex beam (OVB) carrying an OAM value that is two times the topological charge, q (refs. 33,34 ). Further experimental details are provided in the Methods. Normalized transmission of left-and right-circularly polarized (s = ±1) helical light (l = ±1) is shown in Fig. 1b as a function of the peak laser fluence in S(+)-fenchone, C 10 H 16 O. For each successive laser pulse, the energy was varied using a combination of half-wave plate (HWP) and polarizer, and a fresh sample region was irradiated by translating the cuvette. Each curve in the figure is an average of three independent measurements, and the colour band represents the statistical standard error. At low peak laser fluences, absorption is negligible and all curves overlap. The laser fluence is not sufficient to induce multiphoton transitions. At the onset of multiphoton absorption (~1.4 J cm −2 ), the transmission starts to decrease monotonically. For a given helicity, transmission is nearly identical for left-and right-CPL, except at fluences greater than ~2.2 J cm −2 .
The normalized chiral signal in fenchone, defined as HD (Type II: ±l; ±s) = 2 D(±l; ±s)−L(±l; ±s) D(±l; ±s)+L(±l; ±s) (see Methods for notation), is shown in Fig. 1c (solid lines). For a Gaussian beam (l = 0), the chiral signal is less than 1% (dashed lines). Overlap of the signal with the error bands represents the sensitivity of our experiment. The chiral signal is enhanced significantly when the helical phase is introduced to the incident beam by switching from a Gaussian beam to an OAM beam. For l = ±1, HD(Type II) is ~5%. However, SAM associated with light polarization appears to play a minimal role. For a given helicity, the chiral signals for left-and right-CPL nearly overlap with each other within experimental error, similar to the Gaussian beam.
To isolate the role of SAM and OAM, we performed experiments with linearly polarized (s = 0) helical (±l) and Gaussian beams. It is worth noting that multiphoton absorption cross-sections depend Article https://doi.org/10.1038/s41566-022-01100-0 Figure 4a presents the HD(Type I) signal in limonene enantiomers for different positions of the singularity in an OAM beam. The spatial variation of HD(Type I) exhibits the same behaviour as achiral molecules (Fig. 3b,c). The key difference is that with asymmetric OAM beam, the helicity-dependent absorption in the two enantiomers is different, resulting in non-zero chiral signal given by HD(Type II). Figure 4b,c show that the HD(Type II) signal in limonene as a function of peak laser fluence is weakly dependent on the position of the singularity     To understand the chiral light-matter interaction, we modelled the rate of excitation (Γ) of the molecule using time-averaged induced multipoles (Methods). The coupling terms involving the electric-magnetic dipole (E1M1) and electric dipole-quadrupole (E1E2) moments (equation (8)) are pseudoscalars, so they change sign under improper rotation, contributing to HD(Type I, II) (equations (10) and (11)). HD(Type I) was modelled by evaluating equations (9) and (10), assuming full alignment, using the experimental parameters (equations (12)- (19); ω = 2.35 × 10 15 Hz, ω 0 = 2 μm) and approximating the response tensors G′ and A′′ with scalars (or pseudoscalar) 40 . The E1M1 term vanishes because we take the difference between left and right-helical beams for the same polarization. Hence, the E1E2 term is the major contributor to the HD(Type I) for asymmetric OAM beams. Figure 5a depicts the modelled E1E2 term (described in terms of differential optical helicity Δϒ) for different polarizations, described by ellipticity ε. The magnitude of differential absorption is maximal for circular polarization (ε = 1), decreases with ellipticity and goes to zero for perfect linear polarization (ε = 0). This is in agreement with the experimentally observed sinusoidal behaviour of HD(Type I) for the chiral molecule shown in Fig. 5b and the achiral molecules shown in Fig. 3b,c. The minimum ε value achieved experimentally was 0.05. In the experiments, the displacement of singularity is restricted by the focal spot, whereas the modelling of the E1E2 term does not include such boundary effects. As a result, at the extreme positions of the singularity in the experiment, the intensity profile mimics a symmetric Gaussian, leading to zero gradient force (Supplementary Section 9) and subsequently no differential absorption. However, in the modelling, the intensity profile does not converge to a symmetric Gaussian.
For asymmetric LG beams, the modelled E1E2 term predicts a higher magnitude of differential absorption for higher values of l, as shown in Fig. 5c. Figure 5d shows experimental HD(Type I) for asymmetric LG beams in limonene for two different OAM values of l = ±1, ±3. The magnitude of HD increases with the l value, in agreement with the modelled E1E2 term, where the limonene response tensor is approximated as a scalar quantity. The E1E2 term is non-zero even for asymmetric non-OAM annular beams. However, because there is no l dependence, HD(Type I) does not exist, as shown in Fig. 3c for the non-OAM, annular beam obtained through Fresnel diffraction.
The contribution of the E1E2 term to the HD signal also depends on the laser fluence. This vanishes at low laser intensity where the degree of molecular alignment is negligible. The extent of molecular alignment with an ~100-fs laser pulse depends on (1) the laser intensity [Υ + − Υ − ] dxdy is the differential optical helicity representing the E1E2 coupling term (see main text for details). The error bars represent the propagation error of the chiral signal for three independent measurements (sample size).
Our technique affords tunability of OAM-dependent differential absorption in both chiral and achiral systems by (1) changing the ellipticity of the laser polarization (Fig. 5a,b), (2) varying the l value (Fig. 5c,d), (3) superimposing Gaussian and OAM beams to manipulate the field distributions and (4) shifting the singularity. The technique is quite general and can be extended to chiral and achiral transparent solids, plasmonic metasurfaces and gas-phase molecules. The OAM of light offers an additional degree of freedom to control light-matter interactions. This opens new opportunities in next-generation chiroptical spectroscopy, asymmetric catalysis and light-driven molecular machines, where ultrashort light pulses can trigger interconversion of enantiomers for dynamic control of chirality.
A key advantage of using helical light is that the HD signal scales linearly with the l value, as predicted by our qualitative model. In Fig. 5, a linear increase in differential absorption was observed by changing from l = 1 to l = 3. Our model predicts a linear behaviour until l ≈ 15, therefore, an increase in differential absorption by a factor of 15 relative to l = 1. However, at very high l values, saturation behaviour can be expected, as observed recently in chiral nanostructures 29 , due to the increase in the size of the singularity and the finite beam waist determined by the focusing geometry. The enhanced differential absorption with asymmetric OAM beams at higher l values can be utilized in molecular optical switching for digital processing, in which light transmission can be controlled with helicity.
Another unique feature of the scaling behaviour of the HD signal is that it leads to higher chiral sensitivity. In contrast, the sensitivity of current chiroptical techniques that rely on light polarization is predetermined by the nature of the light-matter interaction, which is not scalable. In fenchone, the chiral signal increased from ~4% to ~6% when l was varied from 1 to 2 (Fig. 2). However, the chiral signal in limonene did not show such dependence. Although the l dependence of the chiral signal needs further investigation, the sensitivity of our technique is comparable to PECD and two-photon absorption CD 46 .
The demonstrated differential absorption in liquids suggests a degree of control over the ionization of gas-phase molecules when irradiated with an intense helical light beam. In addition to polarization and magnetic field, the OAM of light represents another parameter to control the continuum trajectory of ionized electrons. The OAM of light, together with a transverse magnetic field and longitudinal electric field, generate chiral continuum electrons scalable with the l value. In contrast to single-photon ionization, chiral electron trajectories generated by tunnel ionization have attosecond resolution and can be used to probe chiral dynamics with attosecond precision.
Finally, the chiral motion of free electrons also results in a magnetic field along the laser propagation axis that can be controlled via the optical OAM. The magnetic field will be dominant around the singular part of the beam where the electric field is weak. For OAM single-cycle pulses, the magnetic field can be switched on a timescale of ~1 fs. This opens a potentially new way to in situ, ultrafast probing of magnetic materials.

Differential absorption measurements
A Ti:sapphire laser amplifier system, operating in an external trigger mode and producing 45-fs, 800-nm pulses with a maximum pulse energy of 2.5 mJ, was used in the transmission measurements. An aspheric objective lens (numerical aperture (NA) of 0.3) was used to focus the femtosecond pulses into a cuvette (10-mm thick) containing liquid samples of chiral or achiral molecules. A second aspheric objective with the same or higher NA (0.5) collected and collimated the transmitted light onto a photodiode (PD2), positioned immediately after the objective. For every laser shot, the transmitted light signal on PD2 was normalized with the incoming light signal on PD1, reflected off a glass plate positioned in the beam path at an angle of ~20° to avoid Brewster's angle. The signals generated by PD1 and PD2 were stretched by an electronic pulse stretcher, discretized, and recorded by a data acquisition card. A combination of an HWP and a polarizer was used to vary the pulse energy (power control in Fig. 1a). The incident pulse energies were measured before the objective. During the measurement, for every laser shot, the pulse energy was increased by ~3 nJ and the sample was translated by 5 μm to avoid microbubbles. Multiple transmission curves similar to those in Fig. 1a were obtained for each sample, to be averaged and smoothed. The difference in the normalized transmission of left-and right-helical light (l = ±1) is proportional to HD(Type I) (that is, differential absorption). To ensure a shortest pulse in the interaction region, a negative chirp was introduced and optimized by measuring the second harmonic generation in a barium borate crystal placed at the location of the cuvette. A single-shot autocorrelator then continuously monitored the pulse duration. The pulse duration at the interaction region is ~100 fs. Transmission measurements were always performed in an empty cuvette before each experiment to determine and minimize background errors resulting from any discrepancies between the PDs (Supplementary Section 5). In addition, the measured single-shot beam profile, pulse spectrum and OAM value remained unchanged after transmission through the samples (Supplementary Sections 2-4).

Sign dependence of HD in chiral molecules
We used conventional labelling of a chiral molecule by the sign of the direction of rotation of polarized light, dextrorotatory (d; +) and levorotatory (l; −). Chiral molecules are also labelled as (S)-and (R)-based on their absolute chemical configuration, representing the left-and right-handed isomers. This labelling refers to the spatial orientation of groups at the chiral centre and not to the optical rotation of polarized light. As a result, it is possible for an isomer to be S(+), S(−), R(+) and R(−). The sign dependence of the HD signals arises from optical rotation by an enantiomer. HD(Type II) signals in fenchone (Fig. 2) and limonene (Fig. 4b,c) are of opposite signs because they rotate the plane of linear polarization in opposite directions. R(+)-limonene and S(+)-fenchone rotate the plane of polarization clockwise, and S(−)-limonene and R(−)-fenchone rotate counterclockwise.

Generation of OAM beams
Light beams carrying orbital and/or spin angular momentum were generated and controlled by an OAM/SAM unit (Fig. 1a) consisting of a combination of a HWP and quarter-wave plates (QWPs), a linear polarizer (LP) and a birefringent liquid-crystal-based phase plate called a q-plate 33 . When an incident Gaussian beam propagates through the q-plate with a topological charge q, it acquires an OAM defined by l = ±2q, with a phase singularity and hence a null intensity region at the centre of the beam-an optical vortex. The wavefront structure of such beams undergoes l intertwined rotations in one wavelength, and the direction of rotation is determined by the sign of the input polarization. The conversion efficiencies of the q-plates were 91 ± 2% for l = 1, 3 and ~80 ± 2% for l = 2. Circularly (linearly) polarized Gaussian beams were produced by a QWP. The ellipticity of CPL at the sample was 97 ± 2%. Circularly (linearly) polarized OAM light s = ±1, l = ±1 (s = 0, l = ±1) was generated using a combination of QWP, q-plate, QWP, LP and QWP. The ellipticity of circularly (linearly) polarized OAM light reaching the sample was 95 ± 2% (5 ± 2%).

Generation of annular beams and displacement of singularity
An annular light beam with no OAM (l = 0) was generated by a circular aperture and exploiting the Fresnel diffraction of the incident Gaussian beam to produce an Airy pattern beam with a null region in the centre of the beam (Supplementary Section 6). A Galilean beam expander and QWP were used to magnify the beam by a factor of four and produce circular polarization. The singularity/null intensity region in the OAM (non-OAM beam) beam was displaced by translating the q-plate (circular aperture), mounted on a x,y-stage, with a step size of 250 ± 40 μm. When focused by the objective, this translated to a displacement step size of 300 ± 20 nm with respect to the centre of the beam. The calibration was achieved by measuring the total translation required to displace the singularity to the periphery of the defocused beam and comparing it to the measured spot size of 2 ± 0.2 μm obtained by knife-edge measurements.

Multipole expansion of the light-matter interaction
We consider a monochromatic electromagnetic field incident on a molecule. The resultant time-harmonic charge and current distributions are described by a multipole expansion. The single-photon rate of excitation of the molecule was expressed in terms of time-averaged induced multipoles.
The induced electric-dipole μ , electric quadrupole θ and magnetic dipole m, are expressed in multipole expansion terms as 47,48 where the Greek alphabet represents the Cartesian indices. α αβ is the electric dipole polarizability, χ αβ is the magnetic susceptibility, Ã αβγ is the electric dipole-quadrupole polarizability, where Ã αβγ =ã γαβ , and G αβ is the electric-magnetic dipole polarizability, where G αβ = −g αβ . The field gradient ∇ βẼ γ is defined by Ẽ βγ (refs. [47][48][49] ). The complex electric and magnetic fields are defined as Ẽ (t) =Ẽ 0 e −iωt and B (t) =B 0 e −iωt , where Ẽ 0 and B 0 are arbitrary complex vectors and ω is the frequency of the incident light, The terms with tilde are complex quantities

The dynamic multipole interaction Hamiltonian 47,48 is given by
The rate of change of the Hamiltonian gives the rate of energy absorption by the induced multipoles 47,48 , that is, energy absorbed from the EM fields, expressed in terms of time average: where μ ,ṁ,θ, E α = (B α e −iωt +B * α e iωt ) are real quantities. By considering the component of dipole moment along the electric field direction, we replace β with α (ref. 47 ) to get Magnetic dipole-dipole excitation (M1-M1) is given by

Nature Photonics
Article https://doi.org/10.1038/s41566-022-01100-0 Electric-magnetic dipole excitation (E1-M1) results in the interaction of the electric field with the electric dipole moment induced by the magnetic field and vice versa. The expression is given by To describe the bulk response, the tensor quantity G ′ αβ must be averaged over all degrees of molecular orientation. For partial orientation of molecules, due to the laser induced dipole force (see below), we consider anisotropic averaging ⟨G ′ αβ ⟩ ρ → ρG ′ αα , where ρ is an orientation-dependent weighting factor influenced by the degree of molecular alignment 50,51 .
Electric dipole-quadrupole excitation (E1-E2) arises from two contributions: (1) the interaction of the electric field with the electric dipole moment induced by the field gradient and (2) the interaction of the field gradient with the quadrupole moment induced by the electric field. The expression is given by We consider anisotropic averaging ⟨A ′′ where ρ is an orientation-dependent weighting factor influenced by the degree of molecular alignment. For complete alignment (ρ = 1), considering the principal molecular axis is oriented along the propagation axis (γ), the tensor quantity can be expressed as ⟨A ′′ 47 ). Because the dominant contribution to the dipole moment is along the electric field direction (α), we limit to A ′′ ααγ . This tensor quantity vanishes for randomly oriented molecules, ρ = 0 (isotropic averaging). The higher multipole transitions such as E1M2, E2M1 and E2E2 are ignored because the molecular response tensor for these transitions is very small. The total rate of excitation is a sum of all four absorption rates for an asymmetric LG beam: The δ is the asymmetrical parameter, δ = 0 for a symmetric LG beam and δ ≠ 0 for an asymmetric LG beam. The sign in Γ ± represents the sign of the OAM. The above equation is generalized to multiphoton absorption (Supplementary Section 10). E1 and M1 are the electric and magnetic dipoles, respectively, and E2 is the electric quadrupole. The coupling terms E1M1 and E1E2 are pseudoscalars and change sign under improper rotation. For a specific chiral system, the sign change of these quantities upon reflection would lead to non-zero differential absorption, defined as HD (Type I, II) dxdy (9) and integrated over the beam cross-section. HD(Type I) is defined as the difference between the absorption of left-and right-handed helical light for the same molecule (Figs. 3 and 4a): HD(Type II) is defined as the difference in absorption between the two enantiomers for a specific helicity and polarization (Figs. 2 and 4b,c): where R and S represent the two enantiomers. The parameter (3) and (4) is called optical chirality, a local measure of the degree of circular polarization, and we define Re [(Ẽ ± δ ) * ⋅ ∇Ẽ ± δ ] as optical helicity. Both are time-even psuedoscalars. Optical chirality is often used to describe the phenomenon of CD in molecules 6,47,52 and metasurfaces 29,53,54 . Optical helicity, ϒ, is a quantity describing the handedness of helical light. It contains the gradient of the electric field, giving rise to a linear l dependence. The above equations, assuming full molecular alignment, show that the chiral light-matter interaction depends on both molecular transitions (contained in G′ and A′′) and optical chirality C, and optical helicity, ϒ. The contribution of the E1E2 term was modelled in Fig. 5 by considering the optical helicity term ϒ (dominant component defined by approximating A′′ as a scalar).
The following qualitative behaviour emerges by evaluating equations (9) and (10) for HD(Type I) and equations (9) and (11) for HD (Type II): • HD(Type I) is a beam-dominated property and does not exist for symmetric LG beams. For asymmetric LG beams, the E1M1 term vanishes (because we take the difference between the left-and right-helical beams for the same polarization) and the E1E2 coupling term is non-zero. For a circularly polarized Gaussian beam, the E1M1 term is non-zero and gives rise to conventional CD. • HD(Type II) is a material-dominated property where both E1M1 and E1E2 coupling terms contribute to differential absorption for asymmetric LG beams. For symmetric LG beams, the E1E2 term averages out to zero, but the E1M1 term is non-zero and can contribute to HD. E1M1 and E1E2 are of similar magnitudes 47,55 .
For asymmetric LG beams with the same polarization but different helicities, the E1E2 term is responsible for the change in the sign of the HD curves. Because the E1M1 term changes sign only with polarization, its finite magnitude leads to an offset in the HD curves (Fig. 4b,c).
The E1E2 coupling term is often ignored because it vanishes for a random orientation of molecules due to rotational isotropic averaging. However, if the molecules are preferentially aligned, the quadrupolar interactions containing A′′ cannot be neglected 56 . This is the case for asymmetric LG beams, in which the spatial inhomogeneity in the intensity profile at the interaction region gives rise to an optical dipole force defined by F = α 1 2 ∇E 2 + α d dt (E × B), where the first term defines the gradient force and the second term the scattering force, which can be neglected when the Poynting vector remains constant over an optical cycle. A net non-zero force is exerted on the induced dipoles directed towards the extrema of the radiation field (Fig. 3a). The resultant torque preferentially aligns the molecular axis parallel to the polarization plane 57 , leading to a non-zero averaged E1E2 contribution to HD. The E1E2 contribution to HD vanishes for symmetric LG beams because the gradient force is zero. The E1M1 term is non-zero even for randomly oriented molecules.

Asymmetric LG beams
We introduced an asymmetry parameter, δ, into the symmetric LG beam to obtain an asymmetric LG beam. We also consider the longitudinal component of the field (E z ) as a correction to the paraxial regime 58,59 to take into account its finite l (OAM) contribution. The importance of the longitudinal component in the light-matter interaction has been recently demonstrated in differentiating nanoparticle aggregates using helical light 28 . When light is focused tightly using a higher-NA objective, the contribution of the longitudinal component becomes significant.

Data availability
The minimum dataset necessary to interpret the results can be obtained from the corresponding authors upon reasonable request.

Code availability
The simulation data were obtained by evaluating the equations using standard technical software. The code is available upon reasonable request to the corresponding authors.