Abstract
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.
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Introduction
Many biomolecules are chiral in nature. Chirality plays a pivotal role in biochemistry and the evolution of life itself1,2. When chiral molecules interact with circularly polarized photons, they exhibit optical activity, such as the circular dichroism (CD) effect describing the difference in molecular absorption of right- and left-handed circularly polarized photons. The related CD spectroscopic technique has been one of the central methods for optically probing the molecular chirality3,4. In addition to polarized light, which is intrinsically linked to the spin angular momentum of photons, photons can carry orbital angular momentum (OAM), which is associated with the helicity in spatial phase distribution5,6,7,8,9,10. Over the past decades, many researchers have studied the interactions between OAM photons and chiral molecules11,12,13,14,15,16,17,18. Although many of them have predicted an exchange can occur between the OAM of the photon and the center of mass motion of the molecule, whether OAM photons can interact with molecular chirality remains an open question.
Vortex beams with various OAMs have been experimentally realized in the optical domain5,6. The possibility of encoding large amounts of information in vortex beams due to the absence of an upper limit has raised the prospects of their applicability in information processing tasks7,8. The OAM beams have been widely used in manipulating and trapping microscopic particles9,10. The problem is whether vortex beams with OAMs can be used to probe molecular chirality? Does this additional degree of freedom play a role in CD? If the interaction between the OAM and the chiral molecule was found, it is potentially useful for a broad range of research areas and applications. Thus, the electromagnetic (EM) interactions of OAM photons with atoms and molecules have received considerable attention11,12,13,14,15,16,17,18. Some theoretical investigations predicted that the interaction of OAM light with atoms and molecules should be observable within the electric dipole approximation11,12,13, but the others obtained a contradictory result14,15,16. The latter outcome has been supported by recent experimental studies17,18, which verified that the influence of the OAM on CD of chiral molecules can not be detectable.
On the other hand, recent investigations have demonstrated that light-molecule interactions can be enhanced via resonant excitations of surface plasmon resonance (SPR) of metallic nanostructures. For CD probe of molecular chirality, it has been reported a large difference in absorption of right- and left-handed circularly polarized photons in the SPRs19,20,21,22,23,24,25,26,27. In addition, the SPRs have been also used to improve nanoparticle(NP)-assisted biosensing28,29,30 and Raman scattering31,32,33. In this work, we attempt to explore the role of SPRs in the promotion of the interactions between the OAMs and the chiral molecules.
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 
NPs, which is excited by an OAM beam with 
and 
, 
and 
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 as19,23,25
where 
is the Hamiltonian of the molecule and 
describes the internal electronic structure of the molecule, 
is the density matrix and 
is the corresponding matrix element. Here 
is the light-matter interaction operator, 
and 
are electric and magnetic operators and 
represents the EM field acting on the molecule. Where 
, 
, 
and 
are the electric charge, mass, position and momentum operators for the 
charged particle in the molecule system. The electric quadrupole interaction term 
bears the same order of magnitude with the magnetic dipole term 
, whose expression is written as 
, with 
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. 
is the relaxation term which describes the damping with 
, 
and 
. By solving this equation within rotating-wave approximation and letting 
, we can obtain energy absorption rate of the system:
where 
represents the absorption rate of the molecule in the system, which is expressed as
where 
is the frequency of molecular transition, 
is the relaxation term, 
and 
are the matrix elements of electric and magnetic dipole momentum operators. 
, with 
and 
being the EM field of scattered wave by the NPs in the absence of the molecule. 
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 
. 
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
where 
are the amplitudes of the total complex EM field in the space and 
denotes a surface circumscribing around the 
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 
and 
, at the same time, they are space inversion of each other, it can be expressed as
where 
represents the contribution of the molecule in the molecule-NP nanocomposites, whose value is influenced by the NPs. 
is the molecule-induced OD for the nanoparticles, where 
and 
can be written as (see methods section for detail)
where 
is arranged as 
with 
and 
are set to be the vector spherical function (VSF) expansion coefficients of an OAM beam propagating along the 
direction (see supplementary material). 
is introduced here for notation simplification and the convention 
when 
and 
is used. Absorption of waves in any direction can be calculated by changing the Euler angles 
, 
and 
. 
is the rotation matrix which depends only on the direction of the incident wave which is given in refs 34, 35, 36, while 
is a matrix determined only by the property of the illuminated system, which has the form of
Based on Eqs. (4, we can obtain OD through numerical calculations. However, the results depend on geometrical configuration of the system. For example, the OD signals for the NP-molecule system being fixed in a certain region in space but possessing random orientations are different from those for the case being distributed uniformly in the whole space with random orientations. They are also different from the case, which the direction of the chiral system is fixed and the system is distributed uniformly in some certain regions. In the following, we will deal with each case separately.
OD for nanocomposites with fixed positions and random orientations
In practice, molecule-NP nanocomposites in liquid environment have random orientations, averaging over the solid angles of the directions of the samples need to be done. This is equivalent with averaging over the solid angles of the directions of the incident light as has been considered for the calculation of absorption and CD spectra19,23,25,26. Likewise, we did the average OD calculations for a system with random orientations but fixed position. The absorption for the OAM light by nanoparticles and the molecule should be rewritten as
with
where 
represents the averaging over the solid angle. For a chiral system, 
or 
should not be vanished. Also by analyzing the properties of 
and 
under space inversion, the orientation averaged OD of the system is given by (see methods section for detail)
with 
. 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 
and 
, respectively. Here the OAM beams used are x directional polarized Laguerre-Gaussian (LG) lights with a numerical aperture of 
and having a filling factor of 
. 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: 
and 
with a resonance wavelength of 300 nm. The direction of the electric dipole of the molecule is set to be align with the symmetry axis of the NP. In our calculations, we use 
Å, 
and 
. For the dielectric functions of Au, the Johnson’s data were adopted37, the permittivity of water is taken to be 
. If the OD signals are in units of 
, they have to be multiplied by 
where 
is Avogadro’s number19,23, I is related to the square of the incident electric field through 
.
(a) System of coordinates and schematics of a complex composed of a gold nanoparticle and chiral molecule. (b,c) Orientation averaged OD as a function of wavelength for a metal nanoparticle and chiral molecule under the OAM incident beam with 
and 
, respectively. The insets represent the calculated results for the single chiral molecule without NPs. (d) System of coordinates and schematics of a NP dimer and a chiral molecule. (e,f) Calculated OD as a function of wavelengths for the Au dimer and a chiral molecule under the OAM incident beam with 
and 
, respectively. Calculated extinctions for the corresponding systems are also shown. The radius of NPs are taken 15 nm and the separation between two NPs is 1 nm.
The green line and red line in Fig. 1 correspond to 
and 
, 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 
signal around 
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 
and 
, respectively. Comparing it with the case of the single sphere, the total 
is significantly enhanced near the wavelength of coupled plasmon resonance. Specifically, the plasmonic 
at the hotspot can be 50 times larger than that of the single sphere when 
, while the enhancement of 
can also reach 30 times. The phenomena are similar for the OAM beam with different topological indices such as 
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 
and 
, 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 
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
Orientation averaged OD as a function of wavelength under the OAM incident beam when the nanocomposites locate at 
(olive line), 
(red line) and 
(blue dark line), respectively. The corresponding signals for the mirror reflected system are presented as green (
), pink (
), blue (
). (a,b) correspond to the system of a metal nanoparticle and chiral molecule with 
and 
, respectively. (c,d) to the system of a NP dimer and a chiral molecule with 
and 
, respectively. (e,f) describe 
as a function of position for the incident OAM beam with 
and 
at the wavelength of 300 nm, respectively.
where 
is a real constant. It is worth to note when the molecule is put inside the gap of the dimer, Eq. (12) is still valid if 
and 
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 
. This variable has been firstly introduced by Tang and Cohen to characterize the chirality of light38, 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 field40. The calculated results for 
as a function of position are plotted in Fig. 2(e,f) for the OAM beam with 
and 
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 lens42,43.
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 molecule14,15,44,45. 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
where 
is the angular averaged OD signal of the system which is located at 
. Substituting Eq. (10) into Eq. (13), we find
The involved integration in Eq. (14) can be calculated analytically by using the integrations given in (supplementary materials). If we use 
and 
to stand for 
and 
, the involved integration in Eq. (14) can be expressed as
with
where 
and 
. The terms 
, 
, 
and 
are polarization dependent coefficients, their expressions are written as
The expressions for 
and the meaning for all the involved variables have been given in supplementary materials. Substitute Eq. (14) into Eq. (12), using the condition 
, it can be founded that 
for linear polarized beams (
). Thus from Eq. (14), we can see that 
for any linear polarized OAM beam.
In order to analyze the physical origin of the phenomenon, we plotted in Fig. 3(a) the 
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 
and 
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 
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.
(a) The OD distribution in the focal plane at the wavelength of 300 nm for the molecule-NP system described in Fig. 1a. The OAM beams have 
and 
. (b) The corresponding intensity distribution of the electric field at the wavelength of 300 nm.
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 experiments17,18. 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 
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 chirality25,46,47,48,49,50,51,52,53. 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 samples54,55,56. 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
(a) The distribution of nanocomposites with fixed orientations in a circular region. (b) Schematics of the OAM beam and the regions being considered. The green region marks = 0–50 nm, red region corresponds to ρ = 50–100 nm, blue region to ρ = 100–150 nm. (c,d) Orientation fixed OD averaged in some regions as a function of wavelengths for a metal nanoparticle and a chiral molecule under the OAM incident beam with 
and 
, respectively. The insets represent the corresponding results for the single chiral molecule without NPs. (e,f) Orientation fixed OD averaged in some regions as a function of wavelengths for the Au dimer and a chiral molecule under the OAM incident beam with 
and 
, respectively. The olive/red/dark blue lines correspond to the signals in 
= 0–50 nm/
= 50–100 nm/
= 100–150 nm regions. The corresponding signals for the mirror reflected system are presented as green/pink/blue lines. (g,h) Averaged relative values of the longitude electric field magnitude as a function of the relative distances between the center of the x-polarized OAM beam and nanocomposites for the case with 
and 
, respectively. Panel A: electric field distribution for a dimmer when the molecule is put at 
in the focal plane; Panel B: electric field distribution for a dimmer when the molecule is put at the center of the beam. The other parameters are taken identical with those in Fig. 1.
When setting 
and 
, Eq. (18) also gives the result for the whole spatial average, it is 
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 
, 
and 
, respectively. The corresponding OD for the samples with opposite chirality are shown by green, pink and sky blue lines. The results are calculated from Eq. (18) numerically and multiplied by a factor of 
for normalization. Here 
is equal to 
.
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 
and 
, 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 
. However, it is different for the case in Fig. 4(f) for the dimer system with 
, the maximum of OD signals appear in the region 
. This is because the relative value of the longitude part of the electric field 
at various wavelengths has different behaviors for the OAM incident beam with 
and 
.
In Fig. 4(g,h), we plot the value of 
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 
and 
, respectively. The relative values of the longitude electric field in the region 
is obviously larger than those in other regions for the case with 
. This is in contrast to the case with 
, where the maximum of the longitude electric field appears in the region 
. 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 
at the wavelength of 605 nm. Panel A corresponds to the case where the dimer is put at 
, while Panel B is for the same system situated at the center of the beam. Since the OAM beam with 
has a vanishing 
at the center of the beam57, 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.
(a,b) Orientation averaged OD in some regions as a function of wavelength for a metal nanoparticle and chiral molecule under the OAM incident beam with 
and 
, respectively. The insets represent the corresponding results for the single chiral molecule without NPs. (c,d) Orientation averaged OD in some regions as a function of wavelength for the Au dimer and a chiral molecule under the OAM incident beam with 
and 
, respectively. The olive/red/dark blue lines correspond to the signals in 
= 0–50 nm/
= 50–100 nm/
= 100–150 nm regions. The corresponding signals for the mirror reflected system are presented as green/pink/blue lines.
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 nanoscale55,58. 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 
and 
are the expansion coefficients of the incident wave propagating along the 
direction, 
is the 2 × 2 rotation block matrix, which is related to two sets of VSFs by the following relation
where 
and 
are the spherical coordinates of the same evaluated point in the coordinate system 
and 
, respectively. The coordinate system 
is obtained from the 
through the Euler rotation 
. The EM field 
scattered by NPs in the absence of the molecule, are related to the incident wave by the T-Matrix and they can be expressed as
where 
(
) and 
(
) are the relative and absolute permittivity (permeability) in the space and vacuum, respectively. 
is the coordinate of the 
sphere, 
are elements of the coupled T-matrix (
) of the 
NP and they are related to the general single particle T-Matrix 
by the following equations:
with 
and the block-matrix components of 
are written as 
and 
(
). Here 
and 
are transformation matrices for the coordinate systems defined in Appendix B of ref. 34 and 
is the identity matrix. From Eqs (19, 20, 21, 22, 23), the absorption rate of the molecule can be expressed as
with
Here 
=
. The method to calculate the resonance broadening function 
has been described in ref. 25, of which value is independent with the incident wave. From Eqs. (1, 2, 3, 4 in the text, the absorption of NPs can be calculated. The total field around the 
th NP is given by the following equation
where 
is the total scattered field of the 
NP, which is expressed as
Here 
and 
represent the scattered fields of the 
th particle, which are caused by the external incident field and the irradiative molecule, respectively.
where 
and 
are the expansion coefficients of the scattered wave, which are given by
and
with 
. Here 
and 
are the expansion coefficients of the scattered wave from the 
NP irritated by a dipole with a momentum of 
, the calculated method for them can be found in the supplement materials of ref. 25. Similarly, 
denotes the total incidental wave on the 
NP, which 
and 
are composed of the excited wave and the scattered fields from other NPs.
where 
and 
are related to the scattering coefficients 
and 
, that is
Using Eqs. (26, 27), the absorption of NPs can be decomposed into the extinction part and the scattering part:
with 
and 
and
Using the orthogonality conditions of the VSFs, we have:
and
From Eqs. (26, 27, 28, 29, we have
with
Here when ‘type’ is set to be ‘ext’, 
and 
, while 
and 
for ‘sct’. From the above equations, Eq. (6) in the text can be obtained.
Symmetry analysis of 
In this section, we present symmetry analysis of 
. The expansion coefficients of any wave, 
and 
, can also be expressed as59,60
where 
is related to the associated Legendre function, which is given by
If the space inversions are performed for both the NP-molecule system and the incident wave, the absorption rate of the system 
should be unchanged because of parity conservation61. Since the electric field possesses an odd parity while the magnetic field has an even parity, from Eqs. (42) and (43), we have
where 
and 
are expansion coefficients of the space inversed incident wave, while 
is determined only by the property of the enantiomorphic system. According to Eq. (10) in the main text, we have
Form Eqs (45, 46), it can be seen that 
and 
are connected with 
and 
through
and
At the same time 
and 
should satisfy
and
if Eq. (46) will be satisfied for all kinds of incident waves.
According to the equations above, the dichroism of the system can be written as
For the special case of the OAM beams, Eqs. (10, 11) in the text can be readily gotten.
Additional Information
How to cite this article: Wu, T. et al. Plasmon-induced strong interaction between chiral molecules and orbital angular momentum of light. Sci. Rep. 5, 18003; doi: 10.1038/srep18003 (2015).
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11274042, 61421001 and 11174033).
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Numerical results and theoretical method are presented by T.W. the idea and physical analysis are given by X.Z. and R.W. All authors reviewed the manuscript.
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Wu, T., Wang, R. & Zhang, X. Plasmon-induced strong interaction between chiral molecules and orbital angular momentum of light. Sci Rep 5, 18003 (2015). https://doi.org/10.1038/srep18003
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DOI: https://doi.org/10.1038/srep18003
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