Quantitative analysis of circular dichroism at higher-order resonance of extrinsic plasmonic chiral nanostructures using multipole decomposition combined with the optical theorem

Plasmonic chirality, which has garnered significant attention in recent years due to its ability to generate strong near-field enhancement and giant circular dichroism (CD). Currently, various theories have been proposed to explain plasmonic extrinsic chirality, however, a comprehensively quantitative explanation for the high-order optical response of extrinsic metamolecule has yet to be established. Herein, we present a concise and quantitative explanation of the giant high-order extrinsic CD of a plasmonic nanocrescent, which origins from multipole decomposition in combination with the optical theorem. Our findings indicate that the high-order resonance modes exhibit giant CD comparable to dipolar modes and can be conveniently applied to the chiral recognition of metamolecules. Furthermore, the nonradiative electric quadrupole resonance exhibits enormous electric field enhancement near metamolecule, which has great application potential in the fields of molecular recognition and sensing in the visible region.


Introduction
Chirality refers to the property of an object that cannot be superimposed onto its mirror image through translation and rotation.This unique characteristic has captured significant attention due to its prevalent presence in various biological and chemical systems.The analysis of chirality is enormously important in biology, chemistry, physics, and medicine [1].However, the optical response of biomolecules is quite weak because of the slight difference in absorption cross-sections between the left and right circularly polarized light (CPL).In contrast to chiral molecules in nature, artificial metamolecules have greater optical activity, which has a wide range of potential applications.More importantly, metamolecules enable a large chiral near-field enhancement for detecting chiral molecules [2][3][4][5][6][7][8].Considerable efforts have been focused on studying and designing nanostructures that support chirality due to their unique properties and promising applications [9,10].Various chiral structures have been designed and fabricated, such as nanocrescents [11][12][13][14], split ring resonators [15][16][17], heterogeneous spheres [18], and the like [19][20][21][22][23][24][25][26][27].In particular, extrinsic chiral metamolecules with symmetrical structures have received increasing attention due to their great optical activity under oblique incidence.Such symmetric extrinsic chiral metamolecules may provide avenues for designing a variety of chiral devices, including reflective optical elements [25,28], polarization-resolved ultracompact optical devices [29,30] and ultrasensitive biomolecule sensors [2,31,32].
Therefore, understanding and controlling the optical activity of chiral metal nanostructures are crucial issues at the frontiers of nanophotonic and chiral molecule sensing.Many methods have been put forward to explain the extrinsic chirality.From the molecular point of view, Plum et al explained extrinsic chirality phenomenologically on basis of the interaction of collinear components of an electric dipole (ED) and magnetic dipole (MD) [23,24].Hu et al quantitatively interpret the extrinsic chirality at the dipole resonance of two heterogeneous ellipsoids employing discrete dipole approximation [15,33].However, several experiments and simulations show that in practice, when the high-order resonances of symmetric metamolecules are excited, the huge extrinsic chiral response also exists and plays a dominant role [12,13,15,16,34].Based on the T matrix, the polarizability retrieval is applied to explore the high-order multipole moment and its influence on the extrinsic chirality, requiring multiple calculations of different incident fields to retrieve the T matrix [12,35,36].Although the chiral response of high-order resonance has been experimentally measured in the above-mentioned works, the chiral response of this part still needs to be analyzed more intuitively and concisely to be better utilized in practice.
In this paper, we analyze the well-known nanocrescent structure by using the multipole decomposition combined with optical theorem [37][38][39][40][41], and explore the chiral response of nanocrescent under CPL for different incident angles.Compared with the traditional ED-MD interaction, the high-order resonances of extrinsic metamolecule are interpreted, which has also successfully explained the intrinsic chirality in our previous work [42].With this theory, we give a quantitative explanation to the origin of high-order chiral response.Moreover, these high-order modes, possessing higher energies than the dipole modes and appearing generally in the visible light region, can be used in the sensing of chiral molecules and other fields.Therefore, the theory proposed here can not only deeply explore the contribution of high-order multipole moments to the chiral response, but also open up a route for the design of novel chiral metamolecule devices.

Model
Schematic of the investigated nanocrescent is shown in figures 1(a) and (b).We consider a gold nanocrescent located in the x-y plane, with the incident light propagating in the y-z plane.The outer radius of the structure was set to R 1 , the inner radius was set as R 2 , and the width and height of the nanocrescent were set as D 1 and H, respectively.All numerical simulations were done by using finite element commercial software (COMSOL Multiphysics 5.5).The nanocrescent was put in a medium with a refractive index n = 1.The permittivity of gold nanocrescent was adopted according to Johnson and Christy's works [43].The incident background field was set to E 0 e −i(k0 sin θy+k0 cos θz) , where the amplitude of the electric field E 0 = 1V/m, and the incidence elevation angle θ denotes the angle between the wave vector k and the z axis.Perfect matched layer was applied to minimize scattering from the outer boundary.
The circular dichroism (CD) was calculated as the difference in extinction cross-section under left circularly polarized light (LCP) and right circularly polarized light (RCP).When the nanocrescent is irradiated with incident light with θ = 45 • , as shown in figure 1(b), an enormous CD can be obtained at the dipole resonance (1390 nm).Remarkably, the strong CD induced by the high-order resonance (680 nm) is comparable to the case of the dipole one (figures 1(c) and (d)).We also calculate the imaginary part of the mixed electric and magnetic polarizability G ′ ′ of the nanocrescent under CPL illumination, which is derived from the chiral molecules [15,44,45].As shown in figure 1(d), the ED-MD interaction well explains the absorption difference of metamolecule in the dipole resonance (1390 nm), but it does not allow for the high-order region (wavelength below 680 nm), and the difference in the scattering of metamolecules has a greater impact on CD (CD = Ext LCP − Ext RCP ) than the difference in absorption.The theory of ED-MD interaction derived from molecules does not consider the scattering effect, so it cannot directly explain the CD of metamolecules.

Multipole decomposition of the scattering and extinction spectrum
Here, we use the multipole decomposition combined with the optical theorem to explore the origin of the chiroptical response of metamolecule, the induced polarization P of the metamolecule is expanded up to third order with Taylor expansion [37,38,42], which can be extracted from the finite element method: Integrating these terms over the volume of the structure, we get the exact ED moment (p), MD moment (m), electric quadrupole (EQ) moment (Q e ), and magnetic quadrupole (MQ) moment (Q m ): ) Substituting the multipole moment into the scattering formula, we can get the contribution of each moment to the scattering cross-section, According to the optical theorem, the extinction cross-section is determined by the expression: Substituting the multipole moments into the extinction cross-section formula, the multipolar components of the extinction cross-section can be obtained: From formulas ( 6) and ( 8), we can get the contribution of multipole moments to scattering and extinction cross-section.It is of great help to analyze the chirality of metamolecule, which has considerable scattering and absorption effects.
Figure 2 shows the scattering and extinction spectra of the multipole moments of the nanocrescent under different CPL light with θ = 45 • .We find that the ED plays a dominant role in the scattering and extinction throughout the wavelength range (figures 2(a) and (b)).Although the contribution of MD and EQ to scattering cross-section is weak, figure 2(c) shows that their influence on ∆S is not negligible.The EQ has a small contribution to ∆S at 850 nm, but its contribution to ∆S at 680 nm is much greater than that of the MD.Figures 3(d) and (e) show the contributions of the multipole moments to the extinction cross-section.The contribution of the EQ to the total extinction cross section at 850 nm and 680 nm wavelengths under RCP excitation is also much greater than that of the MD.Moreover, EQ is one of the main sources of CD at 850 nm and 680 nm, and its contribution far exceeds the MD component and can even be comparable to the  ED component.Due to the antisymmetric charge distribution of EQ, the EQ radiates weakly into the far field.Therefore, the EQ component is weak in ∆S, but makes a significant contribution to CD via the absorption difference ∆A.
To deeply explore the influence of high-order multipole moments on the chiral response, the chirality of metamolecule under different incident angles is calculated.We find that contributions of multipole moments are quite different under various incident angles (θ).Figures 3(a  moment, and EQ moment to the CD at the resonance wavelength (680 nm, 850 nm, and 1390 nm) under different incident angles.When the oblique incident angle is 45 • , the CD response is the largest and decreases as the angle deviates from 45 • .It can be seen that the ED moment dominates in CD at all incident angles, the MD moment has a larger contribution to CD than the EQ moment at the dipole resonance (1390 nm), and the EQ moment has a larger contribution than the MD at the high-order resonance.It is worth noting that the contribution of the EQ moment to CD at the 850 nm resonance is even equivalent to that of the ED moment.This non-radiative resonance has great application potential in chiral recognition, enhanced Raman.It also proves that the interaction of the electric and MD moments is not applicable to explain the origin of chirality in the high-order resonance region.
From equations ( 2)-( 4), we can get the multipole moment components of the metamolecule at resonance wavelength.As shown in figure 4, the chiral response at the dipole resonance wavelength of 1390 nm is mainly derived from the p x and m z components.At all resonance wavelengths, the response of the z-component m z of the MD moment under different CPL has a major influence on the chirality.Moreover, the EQ moment components Q e xy and Q e yx at the dipole resonance and EQ resonance exhibit large intensity differences under different CPL excitations, which is the main source of the contribution of the EQ moment to CD.The EQ moments Q e xx and Q e yy also contribute greatly to the chiral response at 680 nm.The multipole moment analysis calculated here has a very important guiding significance to find the origin of chirality and design novel metasurfaces with giant CD.

Near-field enhancement and superchiral field sensing
Plasmonic chiral structures are widely used in the detection and sorting of chiral molecules due to great chiral near-field enhancement.The absorption of chiral molecules [44] where G ′ ′ is the imaginary part of the isotropic mixed ED-MD polarizability, satisfying • B ± , as the superchiral field, can be used for measurements of chiral molecules.Here, we calculated the chiral near-field enhancement around the nanocrescent at the resonance peaks under different CPL excitation, and we find that the nanocrescent particles have strong chiral near-field enhancement under oblique incidence.The enhancement factor is defined as g C = C/ |C CPL |, where C CPL = ϵ 0 /2cωE 2 0 is the optical chirality for CPL with electric field amplitude E 0 .In chiral molecules measurements, the molecules are distributed randomly over the whole probe volume, where the signal of the molecular chirality may increase at one location and decrease at another location.Therefore, the volume-average optical chirality ⟨ Ĉ⟩ = 1 V ´V g C dV is applied in two spheres with a radius 100 nm around the tips of the nanocrescent.
From figure 5(a), we achieve that the metamolecule can induce strong near-field enhancement and superchiral field enhancement.Especially, the electric field enhancement g E = |E| /E 0 around the tips of the nanocrescent is up to 56, which is promising for chiral molecule sensing.At the EQ resonance wavelength 680 nm, the electric field enhancement factor g E reaches up to 47.8, which is higher than that at the dipole resonance, and the electric field enhancement factor g E is greatly different under the excitation of different CPL.This difference can be exploited to selectively enhance the electric field by adjusting the circular polarization of the incident light.Moreover, the EQ resonance with higher energy in the visible region exhibits high localization.Figure 5(b) shows that the superchiral field enhancement at the dipole resonance can reach −34.5, and the enhancement factor at the EQ resonance is much smaller than that at the dipole resonance.We attribute this to the fact that the superchiral field is derived from the ED-MD interaction.In addition, we believe that the chiral enhancement effect of the large electric field at the EQ resonance still needs to be further explored, and the EQ resonance may have a better enhancement effect on large chiral molecules.In figures 5(c) and (d), we observe that the chiral spectrum of the V 2 region is zero near 1250 nm under LCP excitation, and can reach 1.5 under RCP excitation, which can selectively enhance the chiral response by switching the circular polarization state.

Conclusion
In this work, the high-order CD of extrinsic chiral metamolecule, excited in the visible region, is quantitatively demonstrated by multipole scattering combined with optical theorem.This combined approach is not only suitable for the exploration of dipole resonance but also can explain the high-order resonance that is not involved in the previous ED-MD interaction.It provides insights into the underlying physical processes governing the CD and allows for the characterization and optimization of these structures for various applications in areas such as sensing, spectroscopy, and optical manipulation.Meantime, the CD at the EQ resonance is comparable to that at the dipole resonance, indicating that the EQ resonance can also be conveniently employed to characterize the chirality of metamolecule.Furthermore, the measurement is relatively easy to implement since the EQ resonance occurs within the visible region.In addition, when the optical responses of metamolecule excited by CPL are investigated at different incident angles, a very large optical response can be obtained at the incident angle of about 45 • .The near-field enhancement of chirality, especially near the rounded corners, can reach up to 34.5.Finally, the volume-averaged chiral enhancements of two spherical area with radius R at the corner are calculated, and we observe that the dipolar resonance excited by the same CPL can exhibit distinctly different or even opposite chiral field enhancements in the V 1 and V 2 regions.And this diametrically opposite enhancement of chirality can potentially be in the detection of chiral molecules which also provides a reference for the design of new extrinsic chiral metasurfaces.

Figure 1 .
Figure 1.Structures investigated and their chiral spectral responses.(a), (b) Schematic diagrams of gold nanocrescent from different viewing angles.The nanocrescent structure in (a) has a height of 50 nm, outer radius R1 = 150 nm, inner radius R2 = 120 nm, center distance D2 = 60 nm, and the width of nanocrescent D1 = 90 nm.Furthermore, the tips of the nanocrescent are rounded with a radius of 5 nm.The incident direction of the excitation is indicated in (b).(c) Cross-section spectra of the nanocrescent under RCP and LCP excitations (θ = 45 • ).(d) Cross-section difference spectra of the nanocrescent (black, red, blue curves) and the imaginary part of the mixed electric and magnetic polarizability of the structure under CPL illumination (yellow square).

Figure 2 .
Figure 2. (a), (b) The multipole decomposition of scattering cross-section spectra at incident angle 45 • .The contribution of the ED dominates the entire scattering spectrum.To show the contribution of other multipole moments in panels (a) and (b), these contributions are multiplied by a factor of 5. (c) The scattering cross-section difference between right CPL and left CPL.(d) and (e) The multipole decomposition of extinction cross-section spectra.(c) and (f) depict the contribution of multipole moments to the scattering cross-section difference ∆S and CD, respectively, under different CPL.

Figure 3 .
Figure 3. (a), (b) Extinction spectra of metamolecule excited by CPL at different incident angles.(c) CD spectra excited by different incident angles.(d)-(f) The dependence of the contribution of different multipole moments to the CD spectra at the resonance wavelength of 680 nm, 850 nm, and 1390 nm as a function of the incident angles.
) and (b) show the extinction spectra of the metamolecule under the excitation of left and right CPL at different incident angles.The intensity of the dipole resonance at 1390 nm is closely related to the incident angle.When the oblique incident angle is 45 • , the CD spectrum is the strongest, accompanied with moderate resonance (figure3(c)).The high-order resonance has similar behaviors.Figures3(d)-(f) describes the respective contribution of ED moment, MD

Figure 4 .
Figure 4.The electric dipole, magnetic dipole components and the tensor components of the electric quadrupole moment of the metamolecule at resonance wavelength 680 nm, 850 nm, and 1390 nm.

Figure 5 .
Figure 5. (a), (b) The distributions of electric field enhancement and chiral near-field enhancement.(c) The schematic diagram of two spherical area V1 and V2 with radius R = 100 nm at the tips of metamolecule.(d) The volume-averaged chiral spectra of V1 and V2 under different CPL.The positive and negative signs represent the incidence of left and right CPL, respectively.