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# Multipolar terahertz absorption spectroscopy ignited by graphene plasmons

## Abstract

Terahertz absorption spectroscopy plays a key role in physical, chemical and biological systems as a powerful tool to identify molecular species through their rotational spectrum fingerprint. Owing to the sub-nanometer scale of molecules, radiation-matter coupling is typically dominated by dipolar interaction. Here we show that multipolar rotational spectroscopy of molecules in proximity of localized graphene structures can be accessed through the extraordinary enhancement of their multipolar transitions provided by terahertz plasmons. In particular, specializing our calculations to homonuclear diatomic molecules, we demonstrate that a micron-sized graphene ring with a nano-hole at the core combines a strong near-field enhancement and an inherently pronounced field localization enabling the enhancement of the dipole-forbidden terahertz absorption cross-section of $${{\rm{H}}}_{2}^{+}$$ by 8 orders of magnitude. Our results shed light on the strong potential offered by nano-structured graphene as a robust and electrically tunable platform for multipolar terahertz absorption spectroscopy at the nanoscale.

## Introduction

The terahertz (THz) domain of the electromagnetic radiation spectrum hosts a plethora of complex mechanisms entailing radiation–matter interaction in physical1,2,3,4, chemical5,6,7,8,9, and biological10,11,12 systems. THz radiation plays a key role in noninvasive medical diagnostics because of the low photon energy ($$\simeq$$meV) that does not lead to ionization, thus not posing at risk living organisms and enabling imaging of the tooth structure, skin cancer, and tumors13,14. Furthermore, structural vibrations and rotations of molecules involve absorption resonances in the THz spectrum playing the role of peculiar barcodes enabling chemical identification of substances in security screening, food, and pharmaceutical inspection. In this respect, THz absorption spectroscopy entails the chemical resolution of dangerous and illicit substances concealed within dry packaging like pure and plastic explosives15, MDMA (ecstasy)16, methamphetamine hydrochloride17, and ephedrine18.

Generally, because the typical size of molecules is much smaller than the radiation wavelength, their effective interaction with radiation is rather weak and occurs within the so-called dipole approximation, which assumes molecules as point-like scatterers so that the spatial variation of the radiation field can be disregarded within the volume of the molecule itself. In the context of molecular rotational spectroscopy, the dipole-like radiation–molecule interaction implies that molecules should possess a permanent dipole moment and absorption among rotational levels occurs only for well-known selection rules $$\Delta J=\pm \! 1$$ and $$\Delta M=0,\pm \! 1$$, where $$J,M$$ are the norm and azimuthal rotational quantum numbers of the molecule19. Such selection rules limit the amount of information extractable from molecular absorption spectra. Furthermore, symmetric molecules that do not possess a permanent dipole moment do not absorb, and can not be probed by THz radiation.

Nanophotonics offers a platform to increase radiation–molecule interaction through the tight confinement and large field enhancement produced by plasmons, i.e., the collective excitations of conduction electrons in metals, thus enabling surface-enhanced infrared absorption (SEIRA) and surface-enhanced Raman scattering (SERS)20,21. Both SERS and SEIRA enable chemical resolution with a sensitivity that can reach the single-molecule detection limit22,23,24, and have been exploited for several applications like pregnancy tests based on metal colloids25 and cancer screening26. The lower photon energy in surface-enhanced THz absorption spectroscopy (SETAS) further enables the noninvasive chemical resolution of polymers and proteins27. Doped graphene offers an appealing alternative to noble metals because it sustains electrically tunable surface plasmons at infrared and THz frequencies28,29,30,31,32,33, thus enabling label-free SERS and SEIRA34,35,36,37,38,39.

Here, we explore the potential of localized graphene plasmons for multipolar SETAS of dipole-inactive molecules. In particular, we demonstrate that the strong-field enhancement and the deep-subwavelength features of THz plasmons in a graphene micro-ring with a nano-hole at the core generate sharp peaks at the inner and outer edges such that the multipolar absorption cross-section of nearby molecules increases enormously. Specializing our calculations to the simplest dipole-inactive homonuclear diatomic molecule, i.e., $${{\rm{H}}}_{2}^{+}$$, we observe that its plasmon-enhanced average quadrupolar absorption cross-section increases up to eight orders of magnitude with respect to homogeneous plane-wave excitation. Our calculations based on frequency-domain modeling of graphene plasmons in the random phase approximation (RPA) and Power–Zienau–Woolley radiation–molecule interaction further illustrate the detailed angular momentum exchange between the molecule and the impinging radiation, unveiling that plasmon-enhanced radiation spin–orbit interaction40 opens transition channels for all molecular orientations. Our results constitute a solid theoretical basis to support graphene as an ideal platform for electrically tunable multipolar SETAS at the nanoscale.

## Results

### Terahertz graphene plasmons

Because multipolar absorption spectroscopy relies on field localization, it is essential to excite dipole-inactive molecules with deep-subwavelength THz fields. Surface plasmons in metal-based41 and graphene-based42 waveguides and localized structures are crucial in this context because they enable strong-field localization well below the wavelength. The appeal of graphene stems from its linear energy dispersion $${\bf{\cal{E}}}={v}_{{\rm{F}}}{k}_{{\rm{e}}}$$ for low-energy electrons with wave vector $${{\bf{k}}}_{{\rm{e}}}$$, where $${v}_{{\rm{F}}}\simeq c/300$$ is the Fermi velocity and $$c$$ is the speed of light in vacuum. Thanks to such linear dispersion, electrical doping through an external bias voltage enables a substantial shift of the Fermi energy up to $$\simeq 1\ {\rm{eV}}$$43. In turn, while metal-based structures possess fixed resonances lacking post-fabrication tunability, graphene overcomes this limit through electrical gating. Furthermore, graphene plasmons have an effective wavelength $${\lambda }_{{\rm{eff}}}$$ that can reach $${\lambda }_{{\rm{eff}}}\simeq \!\lambda /100$$ (where $$\lambda$$ is the vacuum wavelength) along with reduced absorption and thus are superior to metals in terms of field localization and resonance quality factor44,45,46. Our theoretical description of the graphene response to Terahertz waves is based on the local random phase approximation (RPA) (see Supplementary Note 1), with an experimentally realistic electron mobility set to $$\mu =1\ {{\rm{m}}}^{2}\,{{\rm{V}}}^{-1}\,{{\rm{s}}}^{-1}$$ throughout all the calculations reported here (see below).

In order to illustrate multipolar absorption spectroscopy, we consider a graphene micro-ring with external radius $$b$$ and an inner hole at the core with radius $$a$$ (see Fig. 1a). Due to rotational invariance, radiation scattering by the micro-ring is accompanied by conservation of the component of angular momentum along the micro-ring axis, and hence it is convenient to consider an impinging field with a well-defined value of such component. Such an impinging field also allows to illustrate the detailed angular momentum exchanges between THz radiation and the molecule, and hence we assume that the graphene ring is illuminated by a left-handed circularly polarized monochromatic plane wave at normal incidence with electric field $${{{{\bf{E}}}}}_{{\rm{in}}}={\rm{Re}}[{E}_{0}{{\rm{e}}}^{i{k}_{0}z-i\omega t}{\hat{{\bf{e}}}}_{{\rm{L}}}]$$, where $${E}_{0}$$ is the complex field amplitude, $${k}_{0}=2\pi /\lambda$$, $$\omega ={k}_{0}c$$, and $${\hat{{\bf{e}}}}_{{\rm{L}}}$$ is the complex unit vector of left circular polarization. We hereafter set $$\lambda =55.92\ \upmu {\rm{m}}$$ as the fixed vacuum wavelength, so that the photon energy $$\hslash \omega$$ is resonant with the $$J=0\to J=2$$ rotational transition of $${{\rm{H}}}_{2}^{+}$$ (see below). Owing to the graphene discontinuity and the deep-subwavelength polarization40, the spin–orbit interaction (SOI) of radiation provides a scattered field $${{{\bf{E}}}}_{{\rm{scat}}}\,({\bf{r}},t)={\rm{Re}}[{\bf{E}}({\bf{r}}){{\rm{e}}}^{-i\omega t}]$$ with amplitude

$${\bf{E}}({\bf{r}}) = \, {E}_{{\rm{L}}}\left({\bf{r}}\right){\hat{{\bf{e}}}}_{{\rm{L}}}+{E}_{{\rm{R}}}\left({\bf{r}}\right){\hat{{\bf{e}}}}_{{\rm{R}}}+{E}_{{\rm{z}}}\left({\bf{r}}\right){\hat{{\bf{e}}}}_{{\rm{z}}}\\ = \, {\tilde{E}}_{{\rm{L}}}\left({r}_{\perp },z\right){\hat{{\bf{e}}}}_{{\rm{L}}}+{\tilde{E}}_{{\rm{R}}}\left({r}_{\perp },z\right){{\rm{e}}}^{2i\varphi }{\hat{{\bf{e}}}}_{{\rm{R}}}+{\tilde{E}}_{{\rm{z}}}\left({r}_{\perp },z\right){{\rm{e}}}^{i\varphi }{\hat{{\bf{e}}}}_{{\rm{z}}},$$
(1)

where $${\hat{{\bf{e}}}}_{{\rm{R}}}$$ is the complex unit vector of right circular polarization, $${\hat{{\bf{e}}}}_{{\rm{z}}}$$ is the unit vector over the $$z$$ direction, $${{\bf{r}}}_{\perp }$$ is the in-plane position vector with modulus $${r}_{\perp }$$ and azimuthal angle $$\varphi$$, and the components’ amplitudes $${\tilde{E}}_{{\rm{L}}}$$, $${\tilde{E}}_{{\rm{R}}}$$, and $${\tilde{E}}_{{\rm{z}}}$$ are functions only of $${r}_{\perp }$$ and $$z$$. We emphasize that the peculiar topological charges of the $${\rm{R}}$$ and $${\rm{z}}$$ components arise from radiation SOI under the conservation of the total angular momentum (graphene ring $$+$$ radiation) along $${\hat{{\bf{e}}}}_{{\rm{z}}}$$ owing to cylindrical symmetry (see the Methods and Supplementary Note 1). Thus, the graphene ring acts as a nanogenerator of THz vortices in the different field components and, as we show below, such vortices play a crucial role in the enhancement of the molecule absorption cross-section over all orientations.

Depending on the inner and outer radii, the graphene ring sustains plasmon resonances at peculiar Fermi energies $${E}_{{\rm{F}}}$$ where the absorption cross-section is maximum (see Fig. 1b). In order to achieve a strong plasmon resonance at an experimentally feasible Fermi energy, we hereafter set $$a=15\ {\rm{nm}}$$ and $$b=4\ \upmu {\rm{m}}$$ for which the resonance occurs at $${E}_{{\rm{F}}}=0.63\ {\rm{eV}}$$. Due to above discussed angular momentum conservation, the induced surface charge density over the graphene micro-ring has topological charge equal to the impinging angular momentum, i.e., $${\sigma }_{{\rm{sur}}}\left({{\bf{r}}}_{\perp }\right)={\tilde{\sigma }}_{{\rm{sur}}}\left({r}_{\perp }\right){{\rm{e}}}^{i\varphi }$$. It is worth noting that, at the peculiar Fermi energy where electron oscillations are resonant, the radial part of the induced surface charge density $${\tilde{\sigma }}_{{\rm{sur}}}$$, as reported in Fig. 1c, is particularly large close to the inner ring edge (where $${\tilde{\sigma }}_{{\rm{sur}}}\simeq 9000\ {\epsilon }_{0}{E}_{0}$$) entailing a very strong and rapidly varying scattered field inside the ring nano-hole (see Supplementary Note 2 and Supplementary Fig. 1).

Owing to the resonant excitation, the scattered field is much larger than the impinging field $$|{\bf{E}}| \, > > \, {E}_{0}$$, see Fig. 2a–c, where we plot the field profile of every component at resonance, at different distances from the micro-ring. Note that, thanks to the excitation of graphene plasmons in the ring, the THz wave gets squeezed at the nanoscale and it also displays spatial features whose scale is comparable with the molecular scale, as shown in the three insets. The peculiar topological charges of the field components $${\rm{R}}$$ and $${\rm{z}}$$ are responsible for their vortex-like structures, which are observed as an azimuthal angle periodicity of imaginary (or real) part of their field amplitudes (see Figs. 2d–f). Note that the inner edge of the graphene ring produces a local field intensity enhancement of the order $$|{\bf{E}}/{E}_{0}|\simeq 1{0}^{6}$$, while at the outer edge $$|{\bf{E}}/{E}_{0}|\simeq 1{0}^{3}$$, which is consistent with previously reported results47.

### Plasmon-assisted multipolar absorption spectroscopy

In general, multipolar absorption is inherently small because it ensues from the electromagnetic field spatial variation within the molecule19. However, over recent years it has been shown that such “forbidden” transitions can be accessed by the deep-subwavelength scattered field produced by graphene nanostructures47,48,49,50,51,52,53,54,55. Note in Figs. 2a–c that the sharp peak at the inner ring radius produces spatial features of the THz scattered wave that are of the order of $$\simeq \! 1\ \mathring{\rm{A}}$$ (see Supplementary Note 2, where we report analytical approximations of the sharp peak in terms of elliptic integrals). Thus, multipolar absorption is expected to be particularly efficient for molecules nearby the inner ring radius. Such a peculiar feature of graphene thus opens otherwise precluded possibilities for multipolar terahertz absorption sensing. Indeed, in the far-infrared and terahertz regions of the electromagnetic spectrum molecules typically possess purely-rotational transitions, while purely vibrational or mixed roto-vibrational transitions typically happen in the mid-infrared19. In order to illustrate plasmon-assisted multipolar absorption spectroscopy, we consider the simplest homonuclear diatomic molecule (dipole inactive), i.e., ionized molecular hydrogen $${\rm{H}}_{2}^{+}$$, which possesses rotational energy levels $${E}_{J}=BJ\,(J+1)$$, where $$B=29.8\ {{\rm{cm}}}^{-1}$$ (see the Methods and Supplementary Note 2). We investigate the molecule–radiation interaction by resorting to the Power–Zienau–Woolley scheme56, whose interaction Hamiltonian is $${H}_{{\rm{I}}}=-\int {\rm{d}}^{3}{\bf{r}}{\bf{P}}({\bf{r}})\cdot {\bf{E}}({\bf{r}},t)$$ where $${\bf{E}}({\bf{r}},t)$$ is the total electric field around the graphene disk and $${\bf{P}}({\bf{r}})$$ is the molecule polarization density. By solving perturbatively the density matrix equations of the molecule with such an interaction Hamiltonian in principle enables the calculation the molecular absorption cross-section $${\sigma }_{{\rm{mol}}}$$ accounting for all multipolar transitions (see the Methods section and Supplementary Note 2).

## Discussion

For quadrupole transitions, averaging over all possible molecular orientations accounted by the azimuthal quantum number $$M$$, we obtain the results in Fig. 3, where in (a) we depict the dependence of the plasmon-enhanced molecule absorption cross-section $${\sigma }_{{\rm{mol}}}$$ over the in-plane radius $${r}_{\perp }$$ and altitude $$z$$, while in (b) we illustrate the altitude dependence of $$\langle {\sigma }_{{\rm{mol}}}\rangle$$ averaged over the disk $$z={z}_{0}$$, $$0\le {r}_{\perp }\le b$$. Both the absorption cross-section and its average on the disk are rescaled to the absorption cross-section of molecules in vacuum $${\sigma }_{{\rm{vac}}}$$. Note in Fig. 3a that graphene enhances the quadrupolar transition cross-section by $$16$$ orders of magnitude locally for molecules nearby the inner graphene edge. The enhancement factor reduces for radii $$a \, < \, r_{\perp} \, < \, b$$, but the minimum is $$1{0}^{4}$$. At the outer radius $${r}_{\perp }=b$$, the enhacement factor rises again sharply owing to the high field gradient provided by the graphene edge discontinuity and reaches $$1{0}^{9}$$. The overall performance of the plasmon-enhanced spectroscopic technique is evaluated through the averaged enhancement factor $$\langle {\sigma }_{{\rm{mol}}}\rangle /{\sigma }_{{\rm{vac}}}$$, which ranges from $$1{0}^{6}$$ to $$1{0}^{9}$$ for molecules at altitudes between $$1\ {\rm{nm}}$$ and $$50\ {\rm{nm}}$$ (see Fig. 3b). We emphasize that such an enhancement factor is averaged over all molecular orientations and nevertheless it remains large due to the electromagnetic SOI enabling an efficient vortex generation over the right circular polarization and longitudinal components (see Fig. 2; Supplementary Note 2).

We note that the only component of the angular momentum conserved is the projection over the $$z$$-axis $${J}_{{\rm{z}}}$$. The system radiation$$+$$molecule is invariant under rotations around the $$z$$-axis only for molecules positioned over the $$z$$-axis. Thus, angular momentum conservation implies that for $${r}_{\perp }=0$$ the only transitions $$J\to J+2$$, $$M\to M+1$$ are allowed, which we obtain analytically (see Supplementary Note 2). Conversely, off-axis molecules are driven by a THz field that is not rotationally invariant over their axis, and thus all transitions $$J\to J+2$$, $$M\to 0,\,M\pm 1,\,M\pm 2$$ are ignited.

A relevant practical aspect of the proposed sensing technique lies in the electrical control of the graphene Fermi energy, which allows the fine tuning of the desired plasmon resonance to match the molecule rotational transition. This requires a gating device such that a gate at the bottom is combined with a contact for the graphene ring. For instance, electrical connection may be attained by a thin transparent insulating layer, an approach that has already been exploited for the active control of graphene disk plasmons57.

Our calculations demonstrate that enhanced multipolar THz absorption spectroscopy can be attained through graphene plasmons, which enable to identify the chemical species of dipole-inactive adsorbed molecules. The enhancement factor of the molecular absortpion cross-section locally reaches values as high as $$1{0}^{16}$$, while in average lies between $$1{0}^{7}$$ and $$1{0}^{9}$$, depending on the altitude of the molecule with respect to the graphene area. We envisage that our findings are relevant in a number of other sensing applications in the THz and far-infrared, as in particular Raman spectroscopy. Furthermore, the strong SOI of radiation mediated by graphene acts as a nanogenerator of electromagnetic vortices that plays a key role for chiral sensing of molecule enantiomers. Graphene enhanced multipolar terahertz absorption spectroscopy can potentially open disruptive applications in pharmaceutical sciences.

## Methods

### Scattering of radiation with well-defined angular momentum by a graphene micro-ring

In the presence of the impinging plane wave $${{\bf{E}}}^{\left({\rm{i}}\right)}\left({\bf{r}}\right)={{\bf{E}}}_{0}{\rm{e}}^{i{k}_{0}z}$$ ($${k}_{0}=\omega /c$$), the graphene micro-ring at the plane $$z=0$$ scatters the field $${\bf{E}}\left({\bf{r}}\right)$$ satisfying, in the long-wavelength limit, the equation

$${\bf{E}}\left({\bf{r}}\right)={{\bf{E}}}_{0}{\rm{e}}^{i{k}_{0}z}+\frac{i\sigma }{4\pi {\varepsilon }_{0}\omega }\nabla \int {\rm{d}}^{2}{{\bf{r}}}{\prime}_{\perp }\frac{{\nabla }{\prime}_{\perp } \cdot \left[f\left({r}{\prime}_{\perp }\right){{\bf{E}}}_{\perp }\left({{\bf{r}}}{\prime}_{\perp } \right)\right]}{\sqrt{{|{{\bf{r}}}_{\perp }-{{\bf{r}}}{\prime}_{\perp } |}^{2}+{z}^{2}}}$$
(2)

where $$\sigma$$ is the graphene surface conductivity at frequency $$\omega$$, $${{\bf{r}}}_{\perp }=x{\hat{{\bf{e}}}}_{{\rm{x}}}+y{\hat{{\bf{e}}}}_{{\rm{y}}}$$, $${{\bf{E}}}_{\perp }\left({{\bf{r}}}_{\perp }\right)={E}_{{\rm{x}}}\left({{\bf{r}}}_{\perp }\right){\hat{{\bf{e}}}}_{{\rm{x}}}+{E}_{{\rm{y}}}\left({{\bf{r}}}_{\perp }\right){\hat{{\bf{e}}}}_{{\rm{y}}}$$ is the transverse component of the electric field on the ring, $${\nabla }_{\perp }=\frac{\partial }{\partial x}{\hat{{\bf{e}}}}_{{\rm{x}}}+\frac{\partial }{\partial y}{\hat{{\bf{e}}}}_{{\rm{y}}}$$, and $$f\left({r}_{\perp }\right)$$ is the ring geometrical filling factor which is equal to zero if $${r}_{\perp} < \, a$$ or $${r}_{\perp } > \, b$$ and it is equal to one if $$a+\delta \, < \, r_{\perp} \, < \, b - \delta$$, where $$\delta$$ is a suitable edge layer thickness58. The electrostatic approximation allows one to introduce the potential $$V({{\bf{r}}}_{\perp })$$ for the electric field produced by the ring on itself according to $${{\bf{E}}}_{\perp }\left({{\bf{r}}}_{\perp }\right)={{\bf{E}}}_{0}-{\nabla }_{\perp }V\left({{\bf{r}}}_{\perp }\right)$$ and Eq. (2) turns into a scalar integral equation for $$V({{\bf{r}}}_{\perp })$$ (see Supplementary Note 2). Due to the ring rotational invariance, the $$z$$- component of the radiation angular momentum is conserved upon scattering, so that the choice of an impinging plane wave with well-defined angular momentum entails a dimensionality reduction of the scattering problem. Specifically, we have chosen a left hand circularly polarized impinging plane wave with $${{\bf{E}}}_{0}={E}_{0}{\hat{{\bf{e}}}}_{{\rm{L}}}$$, which has angular momentum equal to one and, as shown in the Supplementary Note 1, this implies that the electrostatic potential has a definite topological charge equal to the radiation angular momentum, i.e.,

$$V\left({{\bf{r}}}_{\perp }\right)=\tilde{V}\left({r}_{\perp }\right){\rm{e}}^{i\varphi }.$$
(3)

Accordingly, the electric field generated by such potential has the general form as in Eq. (1) and the radial part $$\tilde{V}\left({r}_{\perp }\right)$$ satisfies a one-dimensional integral equation on the interval $$a < r_{\perp} < b$$, which we have numerically solved resorting to representation theory (see Supplementary Note 1). Once the potential is evaluated on the ring, Eq. (2) yields the scattered electric field (in the near-field) which formally coincides with the field produced by the surface charge density $${\sigma }_{{\rm{sur}}}\left({{\bf{r}}}_{\perp }\right)=-\frac{i\sigma }{\omega }{\nabla }_{\perp }\cdot \left[f\left({r}_{\perp }\right){{\bf{E}}}_{\perp }\left({{\bf{r}}}_{\perp },0\right)\right]$$ which, due to conservation of angular momentum, has a well-defined topological charge in turn equal to the radiation angular momentum, i.e.,

$${\sigma }_{{\rm{sur}}}\left({{\bf{r}}}_{\perp }\right)={\tilde{\sigma }}_{{\rm{sur}}}\left({r}_{\perp }\right){\rm{e}}^{i\varphi }.$$
(4)

### Interaction of the scattered field with hydrogen molecular ions

We have investigated the molecule–radiation interaction by means of the Power–Zienau–Woolley56 scheme according to which the interaction hamiltonian is

$${H}_{{\rm{I}}}=-\int {\rm{d}}^{3}{\bf{r}}\ {\bf{P}}\left({\bf{r}}\right)\cdot {\bf{E}}\left({\bf{r}},t\right),$$
(5)

where $${\bf{E}}\left({\bf{r}},t\right)={\rm{Re}}\left[{\bf{E}}\left({\bf{r}}\right){\rm{e}}^{-i\omega t}\right]$$ is the real electric field and $${\bf{P}}\left({\bf{r}}\right)$$ is the polarization density which, for a system of point charges (of charge $${q}_{\alpha }$$ and located at $${{\bf{r}}}_{\alpha }$$) localized around the point $$\bar{{\bf{R}}}$$, is

$${\bf{P}}\left({\bf{r}}\right) = \sum _{\alpha } q_{\alpha} \left({\bf{r}}_{\alpha } - \bar{\bf{R}}\right) \int_0^1 {\rm{d}}u\ \delta \left[{\bf{r}}-\bar{\bf{R}} - u\left({\bf{r}}_{\alpha }-\bar{\bf{R}}\right)\right].$$
(6)

The analysis of the quadrupolar absorption spectroscopy is based on the Taylor expansion of the electric field up to the first-order around the molecule center of the mass. As shown in the Supplementary Note 2, the probability amplitude $${M}_{{\rm{fi}}}$$ of the transition $$(n,{J}_{{\rm{i}}},{M}_{{\rm{i}}})\to (n,{J}_{{\rm{f}}},{M}_{{\rm{f}}})$$, where $$\left(n,J,M\right)$$ are the molecule vibrational, rotational, and magnetic quantum numbers, respectively, is

$${M}_{{\rm{fi}}}={I}_{{J}_{{\rm{i}}},{J}_{{\rm{f}}};n}\sum _{q=-2}^{2}\left\langle 2,{J}_{{\rm{i}}};q,{M}_{{\rm{i}}}|{J}_{{\rm{f}}},{M}_{{\rm{f}}}\right\rangle {V}^{q}$$
(7)

where $${I}_{{J}_{{\rm{i}}},{J}_{{\rm{f}}};n}$$ is independent on the field, $$\left\langle {j}_{1},{j}_{2};{m}_{1},{m}_{2}|{j}_{3},{m}_{3}\right\rangle$$ are the Clebsch–Gordan coefficients and $${V}^{q}$$ are the five spherical components of the field gradient tensor $${\partial }_{i}{E}_{j}$$ evaluated at the molecule center of the mass. Conservation of angular momentum upon scattering from the ring implies that they have defined topological charge

$${V}^{q}\left({r}_{\perp },\varphi,z\right)={\tilde{V}}^{q}\left({r}_{\perp },z\right){\rm{e}}^{i\left(1-q\right)\varphi }.$$
(8)

The absorption cross-section of the transition $${J}_{{\rm{i}}}\to {J}_{{\rm{f}}}={J}_{{\rm{i}}}+2$$ is (see Supplementary Note 2)

$${\sigma }_{{\mathrm{mol}}}\left({r}_{\perp },z\right)=g{|{I}_{{J}_{{\mathrm{i}}},{J}_{{\mathrm{i}}+2};n}|}^{2}\sum _{q=-2}^{2}{w}_{q}^{\left({J}_{{\mathrm{i}}}\right)}|{\tilde{V}}^{q}\left({r}_{\perp },z\right)|^{2}$$
(9)

where $$g$$ is a constant and $${w}_{q}^{\left({J}_{{\rm{i}}}\right)}=\sum _{{M}_{{\rm{i}}}=-{J}_{{\rm{i}}}}^{{J}_{{\rm{i}}}}{\left\langle 2,{J}_{{\rm{i}}};q,{M}_{{\rm{i}}}|{J}_{{\rm{i}}}+2,{M}_{{\rm{i}}}+q\right\rangle }^{2}$$. The absorption cross-section in the absence of the graphene ring, is

$${\sigma }_{{\rm{vac}}}=g{|{I}_{{J}_{{\rm{i}}},{J}_{{\rm{i}}+2};n}|}^{2}{w}_{1}^{\left({J}_{{\rm{i}}}\right)}{|{k}_{0}{E}_{0}|}^{2}.$$
(10)

so that the efficiency of the graphene ring of enhancing the rotational absorption cross-section of a molecule quantified by the ratio

$$\frac{\sigma_{\rm{mol}}\left(r_{\perp },z\right)}{\sigma_{\rm{vac}}} = \sum_{q=-2}^{2} \frac{w_q^{\left(J_{\rm{i}}\right)}}{w_1^{\left(J_{\rm{i}}\right)}}\left|\frac{{\tilde V}^q\left(r_{\perp },z\right)} {k_0 E_0}\right|^2.$$
(11)

## Data availability

The data and codes that support the findings of this study are available from the corresponding authors on request.

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## Acknowledgements

A.C. acknowledges the U.S. Army International Technology Center Atlantic for financial support (grant number W911NF-14-1-0315). C.C. acknowledges the H2020 QuantERA Quomplex project (grant number 731743). A.M. acknowledges support from the Rita Levi Montalcini Fellowship (grant number PGR15PCCQ5) funded by the Italian Ministry of Education, Universities and Research (MIUR). We acknowledge the PRIN 2017 Project no. 20177PSCKT.

## Author information

Authors

### Contributions

A.M. and A.C. conceived the idea and developed the approach. A.C. carried out the analytical analysis and performed the numerical simulations. A.M., A.C. and C.C. discussed the results and wrote the paper.

### Corresponding authors

Correspondence to Alessandro Ciattoni or Andrea Marini.

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Ciattoni, A., Conti, C. & Marini, A. Multipolar terahertz absorption spectroscopy ignited by graphene plasmons. Commun Phys 2, 111 (2019). https://doi.org/10.1038/s42005-019-0213-x

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• DOI: https://doi.org/10.1038/s42005-019-0213-x