Plasmon-induced strong interaction between chiral molecules and orbital angular momentum of light

Whether or not chiral interaction exists between the optical orbital angular momentum (OAM) and a chiral molecule remains unanswered. So far, such an interaction has not been observed experimentally. Here we present a T-matrix method to study the interaction between optical OAM and the chiral molecule in a cluster of nanoparticles. We find that strong interaction between the chiral molecule and OAM can be induced by the excitation of plasmon resonances. An experimental scheme to observe such an interaction has been proposed. Furthermore, we have found that the signal of the OAM dichroism can be either positive or negative, depending on the spatial positions of nanocomposites in the cross-sections of OAM beams. The cancellation between positive and negative signals in the spatial average can explain why the interaction has not been observed in former experiments.

Scientific RepoRts | 5:18003 | DOI: 10.1038/srep18003 To address this issue, we study the interaction between the OAM beams and the nanocomposite comprising chiral molecule and metallic NPs. A method to study such an interaction has been developed by using the T-matrix. Based on such a method, we study the interaction between OAM beams and chiral molecules located in the vicinity of NPs. We find that the strong interaction between the chiral molecule and OAM can be induced by the excitation of plasmon resonances of NPs. Experimentally, such an interaction could be observed if we define the OAM dichroism (OD) to be the difference of absorption rates between two focused linear polarized OAM beams with opposite topological indices. We should stress here that a nonzero spatial-average OD signal exists only for the nanocomposites located in a certain spatial region of OAM beams. In addition, we also find that the OD can be further enhanced when the orientations of nanocomposites are fixed.

Results and Discussions
We consider here a semi-classical hybrid system consisting of a molecules and a cluster of n NPs, which is excited by an OAM beam with B inc l represent electronic and magnetic fields of the OAM beam (their expressions in aplanatic system are given in supplementary materials). The intensity of the incident wave is considered to be weak enough that the mechanical interaction between the light and the hybrid system can be neglected. The molecules used here are assumed to be point-like two-level systems, no vibrational structure of transitions is considered. According to the previous investigations, the master equation for quantum states of the molecule can be written as 19,23,25 is the Hamiltonian of the molecule and  H 0 describes the internal electronic structure of the molecule, ρ is the density matrix and ρ σ =

E B E B e Re
T T tot tot i t represents the EM field acting on the molecule. Where ι e , ι m , ι r and ι  p are the electric charge, mass, position and momentum operators for the ι th charged particle in the molecule system. The electric quadrupole interaction term V quad bears the same order of magnitude with the magnetic dipole term − ⋅  m B T , whose expression is written as Q being the element of the electric quadrupole operator. In this work we set the electric dipole of the molecule to be aligned with the rotational symmetry axis of the nano system. Thus, this term makes no contribution to the dichroism and can be omitted here. ρ Γ ( ) ij is the relaxation term which describes the damping with ρ ρ γ ρ Γ ( ) = −Γ ( ) = − 11 22 22 22 , ρ γ ρ Γ ( ) = 12 21 12 and ρ γ ρ Γ ( ) = 21 21 21 . By solving this equation within rotating-wave approximation, and letting ρ σ = ω − e i t 21 21 , we can obtain energy absorption rate of the system: where Q mol represents the absorption rate of the molecule in the system, which is expressed as where ω 0 is the frequency of molecular transition, γ 21 is the relaxation term, µ µ =

E B E E B B
inc l s inc l s , with E s and B s being the EM field of scattered wave by the NPs in the absence of the molecule. G is the function which describes the broadening of the resonance peak of the signal due to the interactions between the molecules and NPs, and its expression can be founded in ref. 25. The induced dipole of the molecule is given by 21 12 . Q NP in Eq. (2) is the absorption rate of NPs in the system. According to the law of energy conservation, it equals to the energy flows into them. Thus we have are the amplitudes of the total complex EM field in the space, and Ω ξ denotes a surface circumscribing around the ξth NP. If we define OAM dichroism (OD) of the system as the difference of absorption rate between two focused linear polarized OAM beams with opposite topological indices +l and −l, at the same time, they are space inversion of each other, it can be expressed as represents the contribution of the molecule in the molecule-NP nanocomposites, whose value is influenced by the NPs. should not be vanished. Also by analyzing the properties of + ± a v l and + ± b v l under space inversion, the orientation averaged OD of the system is given by (see methods section for detail) From the above equations, the orientation averaged OD of the system can be obtained. Figure 1(b,c) show the calculated orientation averaged OD as a function of wavelength for a chiral molecule being placed in the vicinity of a single gold sphere ( Fig. 1(a)) for the OAM incident beam with = ± l 1 and = ± l 2, respectively. Here the OAM beams used are x directional polarized Laguerre-Gaussian (LG) lights with a numerical aperture of ε α = = . . The focusing position of the beam locates at the origin of the coordinate, as shown in Fig. 1(a). The radius of the gold sphere is taken as 15 nm, the distance between the chiral molecular and the sphere is taken as 2 nm. The parameters of the molecular dipole are taken according to refs. 1 and 23: µ = e r For the dielectric functions of Au, the Johnson's data were adopted 37 , the permittivity of water is taken to be ε = . . The green line and red line in Fig. 1 correspond to OD mol and OD NP , respectively, the total OD is described by the blue line. The extinction cross section of the gold sphere is plotted as the orange line, and the plasmon resonance peak appears at the wavelength of 520 nm. Comparing the present OD spectra with that of pure chiral molecular (without the NP, insets of Fig. 1(b,c)), we find that the OD mol signal around λ = nm 300 is improved about 2 times. Moreover, a new OD band appears in the plasmon resonance spectral region. These results reveal not only a plasmon-enhanced chiral molecule-OAM interaction at the molecule resonance frequency, but also an OD effect at the plasmonic resonant frequency of metallic NP. Note that spherical NP itself is achiral, such an induced plasmonic OD correlated with the enhanced interaction between the chiral molecular and the OAM has not been reported in previous studies.
For a single sphere, the enhancement of OD signal by the NP is small and weak. When the chiral molecule is put in the hotspot of the dimer, as shown in Fig. 1(d), giant enhancement effect can be observed. The cases when the molecule is put in the gap of an Au dimer with an inter particle distance of 1nm is presented in Fig. 1(e,f) for the OAM beam with = ± l 1 and = ± l 2, respectively. Comparing it with the case of the single sphere, the total OD is significantly enhanced near the wavelength of coupled plasmon resonance. Specifically, the plasmonic OD at the hotspot can be 50 times larger than that of the single sphere when = ± l 1, while the enhancement of OD mol can also reach 30 times. The phenomena are similar for the OAM beam with different topological indices such as = ± l 2 illustrated in Fig. 1(f). Considering that the EM field of the OAM beam is nonuniform in space, we investigate the OD signal at different position of the OAM beams. Figure 2(a,b) correspond to the case of single sphere with = ± l 1 and = ± l 2, respectively, Fig. 2(c,d) to the case of dimer. The olive/red/black blue line in Fig. 2 represents the orientation averaged OD when the molecule is put at x = 50 nm/100 nm/150 nm in the focal plane (origin of the coordinate is set to be the focal center of the OAM beams). The corresponding OD for the mirror reflected samples are also plotted as green, pink and sky blue lines, as expected these signals are of same values but opposite signs. With the position of the nanocomposite is shifted away from the beam center, the OD signal decreases. In order to disclose such a phenomenon we rewrite Eq. (10) under the dipole approximation, which means the cut-off of v is set to be 1. If we consider the nanocomposite system being located not too close to the beam center, the electric and magnetic fields illuminated on it can be viewed as a constant vector. Using Eqs. (8) and (11), and expand the incident wave according to Eqs. (42)(43), the OD can be rewritten as (blue dark line), respectively. The corresponding signals for the mirror reflected system are presented as green ). (a,b) correspond to the system of a metal nanoparticle and chiral molecule with = ± l 1 and = ± l 2, respectively. (c,d) to the system of a NP dimer and a chiral molecule with = ± l 1 and = ± l 2, respectively. (e,f) describe ⋅ / as a function of position for the incident OAM beam with = l 1 and = l 2 at the wavelength of 300 nm, respectively.
Scientific RepoRts | 5:18003 | DOI: 10.1038/srep18003 where C is a real constant. It is worth to note when the molecule is put inside the gap of the dimer, Eq. (12)  are solved beyond the dipole approximation (see methods section for detail). Our calculations show that the first term in Eq. (12) plays a leading role in the OD signals, thus the signal is proportional to ⋅ l . This variable has been firstly introduced by Tang and Cohen to characterize the chirality of light 38 , since they are proportional to the difference of absorption rates between two oppositely handed molecules. Later it was Cameron et al. 39 who indicated that this term is in fact proportional to the helicity density which describes the 'screw action' of the EM field 40 . The calculated results for ⋅ / as a function of position are plotted in Fig. 2(e,f) for the OAM beam with = l 1 and = l 2 at the wavelength of 300 nm, respectively. It can be found the value of ⋅ / + (+ ) + is close to that of the circular polarized wave, of which value decreases with the increase of the relative distance between nanocomposites and the center of the beam, which directly leads to the decrease of the OD signal. This phenomenon has also been reported in ref. 41, where a large helicity density has been observed at the center of the Bessel beam with the OAM.
We would like to point out that the above OD signals originate from the interaction of the OAM and the chiral molecule, which are not because of the spin angular momentum, because we have used the linear polarized OAM light as the incident wave with no spin angular moment even when it is strongly focused by lens 42,43 .
The previous theoretical investigations have shown that only the spin angular momentum leads to a differential absorption, there is no interaction of the OAM of the beam with the chirality of the molecule 14,15,44,45 . This is because the signals calculated in these works are the average of contributions from all the molecules in spaces. In fact, our calculated results have also shown that the OD signals do not exist in the case of the whole spatial average either, which will be discussed in the following section.
OD for nanocomposites with whole spatial average. Because of heterogeneity of the OAM beam in space, the spatial average for the OAM dichroism is needed in some cases, for example, to compare the theoretical results with the experimental measurements as described in refs. 18,19. Here we consider a spatial averaged OD performed in the focal plane normal to the propagation direction of the OAM beam, which can be expressed as  The involved integration in Eq. (14) can be calculated analytically by using the integrations given in (supplementary materials). If we use s and ′ s to stand for a and b, the involved integration in Eq. (14) can be expressed as , it can be x y ). Thus from Eq. (14), we can see that ( ) ≡ Ω z OD 0 S 0 for any linear polarized OAM beam. In order to analyze the physical origin of the phenomenon, we plotted in Fig. 3(a) the ρ ( , ) Ω z OD 0 0 distribution in the focal plane at the wavelength of 300 nm for the molecule-NP system described in Fig. 1(a). The X-polarized OAM beam with = . NA 0 55 and = + l 1 is used. The corresponding intensity distribution of the incident electric field is plotted in Fig. 3(b) for comparison. One can see that the OD signal of this molecule-NP system change sign when it is moved away from the beam center. The cancellation between positive and negative signals in the spatial average explains why ( ) Ω z OD S 0 is zero. Here it is worth to note that even though the electric field intensity in the central region is weak, the OD signal in this region is still large due to the higher helicity density near the beam center.
In fact, from the above theory, the same conclusion can be drawn for the single molecule without NPs. This may explain why the influence of the OAM on the CD of chiral molecules has not been observed in the previous experiments 17,18 . Our theory has demonstrated that the OD signal from the chiral molecule can not be observed in the whole spatial average even with the aid of plasmonic NPs. It is worth to note that ( ) ≡ Ω OD z 0 S 0 is also true for systems of pure NPs since Eq. (10) is valid in the absence of the molecule. Thus such an OD signal can not be observed for the structural chirality 25,46-53 . Although the OD signal does not exist due to the whole spatial average, we can still observe the signal at some conditions. In the following, we will discuss such phenomena.

OD for nanocomposites with fixed orientations in a defined region.
We consider some NPs locate in some defined regions, as shown in Fig. 4(a). The samples are concentrated in some circular areas. In these areas, the samples have fixed axis relative to the incident waves. In a real experiment, this corresponds to the situation where the molecules or the composite systems are fixed on the substrate. Here the direction of the molecule's electric dipole is set to be aligning with the incident OAM wave and the symmetry axis of the composite system. If the incident wave is oblique, the OD signal appears even when the structure is achiral. This effect also occurs for circular polarized waves known as extrinsic dichroism caused by mutual orientations of incident waves and samples 54-56 . In the following, we consider the cases where the OD signals are gathered from circular areas around the beam center under the vertical incidence (see Fig. 4(b)). Specifically the OD signal is rewritten as  18) also gives the result for the whole spatial average, it is ∆ ≡ 0 for all the cases (single molecule or molecule-NP systems), which is the same with the orientation averaged case discussed above. However, if the spatial average is calculated for some defined regions, the situation becomes different. The olive, red and dark blue lines in Fig. 4 are the calculated results for the orientation fixed OD that are averaged in some spatial regions as shown in Fig. 4(b), which correspond to the region ρ = − nm 0 50 , − nm 50 100 and − nm 100 150 , respectively. The corresponding OD for the samples with opposite chirality are shown by green,   Figure 4(c,d) show the calculated results for the nanocomposite consisting of an Au nanoparticle and a chiral molecule under the OAM incident beam with = ± l 1 and = ± l 2, respectively, while (e,f) are for the system with Au dimer and a chiral molecule. The parameters of nanocomposites are taken identical with those in Fig. 1. The corresponding OD signals for the single molecule without NPs are plotted in the insets of Fig. 4(c,d). In contrast to the case under the whole spatial average, we find that strong plasmon-induced OD signals appear for the region spatial average. Generally they decrease with the increase of the relative distances between nanocomposites and the center of the beam, that is, the maximum appears in the region ρ = − nm 0 50 . However, it is different for the case in Fig. 4(f ) for the dimer system with = ± l 2, the maximum of OD signals appear in the region ρ = − nm 50 100 . This is because the relative value of the longitude part of the electric field ρ φ ρ φ ( , )/ ( , ) at various wavelengths has different behaviors for the OAM incident beam with = ± l 1 and = ± l 2.
In Fig. 4(g,h), we plot the value of ∫ inc z l inc l 0 2 at the focal plane as a function of the wavelength and the relative distances between the center of the x-polarized OAM beam and nanocomposites for the case with = l 1 and = l 2, respectively. The relative values of the longitude electric field in the region ρ = − nm 0 50 is obviously larger than those in other regions for the case with = ± l 1. This is in contrast to the case with = ± l 2, where the maximum of the longitude electric field appears in the region ρ = − nm 50 100 . The magnitude of the longitude electric field determines the field intensity in the hotspots of the dimer. In panels A and B of Fig. 4, we plot the intensity distribution of the electric field in the dimer under the OAM incident beam with = l 2 at the wavelength of 605 nm. Panel A corresponds to the case where the dimer is put at = x nm 100 , while Panel B is for the same system situated at the center of the beam. Since the OAM beam with = l 2 has a vanishing E z at the center of the beam 57 , no hotspot is generated. This means that the electric field is not greatly amplified in the position of the molecule.
The calculated results shown in Fig. 4 are only for the case under the spatial average in some defined regions when the orientations of nanocomposites are fixed. In fact, if we consider random orientations of nanocomposites and do orientation average for such a case again, the OD signals still exist. Figure 5 displays the calculated results. The results in Fig. 5(a-d) correspond to those in Fig. 4(c-f), respectively. Comparing them, we find that the OD signals decrease when the orientation averages are performed, however, plasmon-induced OD signals are still large.

Conclusions
In summary, we have developed a T-matrix method to study the interaction between optical OAM and the chiral molecule in a cluster of nanoparticles. Our results have revealed that the strong interaction between the chiral molecule and OAM can be induced by the excitation of plasmon resonances. Such an interaction leads to the OAM dichroism effect, which depends on the geometrical configuration of the molecule-NP system, such as the orientations and spatial positions of nanocomposites, in the illumination of OAM beam. It is important to note that the sign of the OAM dichroism signal can be either positive or negative, depending on the spatial positions of the nanocomposite in the cross-section of OAM beams. In this point, experimental observation of a nonzero OAM dichroism signal from the molecule-NP system is challenging, since it requires spatial arrangement of nanocomposites in a very limited space region of the incident OAM beam. This leads to the OAM wave improper for the detection of the chirality of samples which are randomly distributed. However, it does not prevent the OAM beam from becoming as an alternative probe of the chirality of an individual nano structure. Since the OAM adds another dimension for the judgment of the chirality of nano structure in addition to the spin angular momentum, we believe it may play an important role in the realization and improvement of chirality detection at the nanoscale 55,58 . Our theoretical study reveals here the possibility to observe the interactions between chiral molecules and OAM beams, which paves the way to explore a novel plasmon-based spectroscopic technique for optically detecting the molecular chirality.

Methods
T-matrix formula for the absorption rate. In this section, we provide the T-matrix formula for the absorption rate. The incident wave can be expanded as a series of VSFs in a coordinate system of which origin is set to the position of the molecule: where a v and b v are the expansion coefficients of the incident wave propagating along the z direction, ′ R vv is the 2 × 2 rotation block matrix, which is related to two sets of VSFs by the following relation where ε (µ) and ε 0 (µ 0 ) are the relative and absolute permittivity (permeability) in the space and vacuum, respectively. ξ r is the coordinate of the ξ th sphere, ξ ( ) ′ T vv ij are elements of the coupled T-matrix ( ξ ( ) T ) of the ξth NP, and they are related to the general single particle T-Matrix T p by the following equations:  (19)(20)(21)(22)(23), the absorption rate of the molecule can be expressed as 21 22 are the expansion coefficients of the scattered wave from the ξ th NP irritated by a dipole with a momentum of µ 21 , the calculated method for them can be found in the supplement materials of ref. 25. Similarly,  If the space inversions are performed for both the NP-molecule system and the incident wave, the absorption rate of the system 〈 〉 /  Q mol NP should be unchanged because of parity conservation 61 . Since the electric field possesses an odd parity while the magnetic field has an even parity, from Eqs. (42) and (43)