Mode-dependent Far-field Radiation of Circularly Polarized Light by a Single Plasmonic Nanohelix

Abstract

Chiral plasmonic nanoantennas exhibit unique chiroptical properties in both the near-field and far-field, offering exciting opportunities for applications in chiroptical sensing, circularly polarized luminescence, and quantum communication. Recent advances in chiral plasmonics have facilitated the creation of nanoparticles with intrinsic chirality at the single-particle level. However, the far-field directionality of these chiral nanoantennas is not well understood. In this work, A numerical study is conducted on a single plasmonic nanoantenna that undergoes a size-preserving compression, transitioning from a straight nanorod to left- or right-handed helices with varying pitches. These nanostructures showcase distinct mode-dependent chiroptical far-field properties, encompassing positive or negative dissymmetry factors, 3D power radiation patterns, and angular distribution of circular polarization, under both plane-wave and point-source excitations. The findings provide new insights into angle-resolved spectroscopies of intrinsically chiral plasmonic nanoparticles and the applications of chiral plasmonic nanoantennas for controlling circularly polarized light emission in nanophotonics.

1 Introduction

Chiral plasmonic nanostructures, characterized by their asymmetric geometries and handedness,^[1] exhibit distinctive responses to left- and right-handed circularly polarized light, known as chiroptical activity that manifests diversely in different chiral spectroscopies.^{[2, 3]} These nanostructures play a crucial role in light-matter interactions by serving as chiral nanoantennas, facilitating the transmission of chiral light in both the near- and far-field. In the near-field, they can generate an intense superchiral field,^[4] which is essential for enantioselective recognition and detection.^[5] In the far-field, they enable the modulation, conversion, or detection of circular polarization and orbital angular momentum in light beams.^{[6, 7]} These abilities are particularly crucial for exploring novel optical effects and enabling advanced applications, such as chiral molecule sensing, chiral photocatalysis, circularly polarized light sources, quantum communication, and a variety of chiroptical nanophotonic devices.^[8-10]

Recent advancements in wet-chemistry synthesis and nanofabrication techniques have facilitated the production of individual plasmonic nanoparticles with intrinsic chirality and chiroptical responses in the visible frequencies.^{[1, 11, 12]} The far-field properties of these nanoparticles have been effectively characterized at the single-particle level using various techniques, such as photothermal circular dichroism (CD) microscopy,^{[13, 14]} dark-field scattering microscopy,^{[15, 16]} confocal transmission spectroscopy,^[17] and angle-resolved cathodoluminescence polarimetry.^[18] However, little attention has been given to the far-field directionality of a single chiral nanoantenna, despite its crucial role in light manipulation, as extensively demonstrated in previous studies on achiral plasmonic nanoantennas.^{[19, 20]} To the best of our knowledge, only a few works have mentioned the back focal plane imaging of single chiral plasmonic nanoparticles, which partially reveals their far-field angular power distribution.^[21-23] However, these studies do not shed light on how the far-field energy is distributed among different angles of space in relation to its polarization state, or how these behaviors can be influenced by different plasmon modes, which are vital for controlling chiral light. Therefore, it is necessary to further improve our understanding of the far-field directionality of different plasmon modes, considering the profound applications based on not only angular power distribution but also angular polarization distribution.

Herein, as a case study, we numerically explore the transformation of a single plasmonic nanoantenna from a straight nanorod to left- or right-handed nanohelices with different pitches, which have not been reported in previous works (Scheme 1). The nanohelices exhibit a single-loop structure. It is noteworthy that the dimensions of the nanorod or nanohelix, including its length and diameter, remain constant during the height compression, ensuring size consistency. The choice of silver as the example material is based on our previous findings that reveal the unique far-field directionality of individual silver nanorods.^{[24, 25]} A phenomenon known as the color routing effect, i.e., light with different colors can be routed into different directions, arises from the distinct far-field directionality of standing-wave-like multipole plasmon modes with odd and even symmetries, evident in both white-light scattering and plasmon-exciton coupling interactions.^{[24, 26, 27]} During the size-preserving compression process, one can expect that the helices can also support standing-wave-like multipole plasmon modes of different orders, and that CD can arise from the helical charge oscillations.^[28] Our research focuses on analyzing the changes in these multipole plasmon modes during the transition from achiral to chiral morphology and the resultant alterations in far-field directionality. We introduce a description of the far-field pattern that combines angular power distribution and the angular polarization distribution associated with the Stokes parameter S₃, revealing the routing effect of circularly polarized light. While silver is the primary focus of this study, other plasmonic metals are expected to exhibit similar properties under specific dimensional configurations. Our simulation outcomes can offer a simple yet straightforward understanding of how morphological variations impact the near- and far-field properties of a single plasmonic nanohelix, aiding in the assessment of their potential applications in nanoantenna design for circularly polarized luminescence.

Scheme 1

Schematic illustration of the single-particle structure in the simulation model, depicting a size-preserving compression (length: 200 nm; diameter: 20 nm) from a straight nanorod to left- or right-handed helices with varying pitches. The axial pitch is varied from the central model to the two sides, set at 200, 180, 100, and 30 nm, respectively. The helix radius is determined by the axial pitch of each nanohelix (Table S1, Supporting Information).

2 Results and Discussion

We began with the spectral analysis of the plasmonic nanoantennas to identify the potential plasmon modes. The simulation model was set up as an achiral nanorod, helix 1-R, helix 2-R, or helix 3-R (Scheme 1), employing circularly polarized plane-waves for excitation, where RCP and LCP denote right- and left-handed circular polarization, respectively. Given previous works showing that multipole plasmon modes with even symmetry require non-normal incident light for excitation,^[24] we investigated five different incidence angles to account for the asymmetric nature of the helices (Figure 1A). The wavevectors k₁₀₀, k₀₁₀, and k₀₀₁ are oriented along x-, y-, and z-axis of the coordinate system. $${{{{\bf k}}}_{10\bar{1}}}$$ lies in the xz-plane at 45° with respect to the z-axis, whereas $${{{{\bf k}}}_{01\bar{1}}}$$ is in the yz-plane at 45° with respect to the z-axis. Extinction cross-section (σ_ext), as the sum of scattering and absorption cross-sections, was calculated for each model in the visible and near-infrared region. Differential extinction cross-section (Δσ_ext = σ_ext,LCP − σ_ext,RCP) was calculated by the difference of σ_ext under the excitation of RCP and LCP plane-waves, corresponding to the CD spectrum observed in experiments.

Figure 1

Spectral analysis of Ag nanorod and representative nanohelices. A) Schematic illustration of the incidence angles with respect to the nanostructure. B) Extinction cross-section (σ_ext) spectra of the Ag nanorod excited by RCP and LCP plane-waves with different incidence angles. C) Differential extinction cross-section (Δσ_ext = σ_ext,LCP − σ_ext,RCP) spectra of the Ag nanorod. D) Extinction cross-section spectra of the right-handed Ag nanohelices (pitch from left to right: 180, 100, and 30 nm) excited by RCP and LCP plane-waves with different incidence angles. Insets show the simulation models. E) Differential extinction cross-section spectra of the right-handed Ag nanohelices. N is defined as the longitudinal Fabry–Pérot order of plasmon modes in the nanorods.

In the case of the achiral Ag nanorod (Figures 1B and S1, Supporting Information), RCP and LCP plane-waves at different angles can selectively excite a few longitudinal plasmon modes, including the longitudinal dipole mode (N = 1), quadrupole mode (N = 2), and octupole mode (N = 3), with the transverse dipole plasmon mode on the shorter wavelength side. As it has identical responses to both RCP and LCP light, the differential extinction is completely zero (Figure 1C), which aligns with our intuitive expectations. Furthermore, these results exclude the presence of extrinsic chirality originating from the relative angle of the incident light and the nanorod orientation.^[29]

As for the three helices with different pitches, all of them display a series of plasmon modes with selective responses to the RCP and LCP light (Figure 1D), resulting in the presence of positive or negative peaks in the differential extinction cross-section spectra (Figure 1E). Several noteworthy observations can be highlighted: First, the peak wavelengths exhibit apparent similarities across the four structures, suggesting a common origin or nature of these modes. Second, for the same structure, the N = 1 and N = 3 peaks of differential extinction may exhibit positive or negative values depending on the incidence angle, as previously observed from the chiral assemblies of plasmonic nanoparticles.^[30] In contrast, the N = 2 and N = 4 peaks remain consistent in their sign, probably due to the mode symmetry. This observation helps interpret the relation between the CD spectra observed from individual chiral nanoparticles and the corresponding ensemble samples.^[31] The orientation-dependent CD highlights the significance of considering orientational averaging to accurately represent the nanoparticle's chirality in an ensemble. Taking the helix 2-R model as an example, we calculated the averaged differential extinction cross-section spectrum considering 18 different incidence angles in 3D space (Figure S2, Supporting Information). Third, all the plasmon modes supported by nanohelices show comparable intensities, unlike the huge intensity difference among the plasmon modes supported by the achiral nanorod, indicating their remarkable potential to influence light manipulation and light-matter interactions. Moreover, the correspondence of these peak wavelengths with those in the extinction spectrum, along with their well-separated nature in the spectra, suggests that these CD signals are likely a result of the inherent chirality associated with these multipole plasmon modes. When charges oscillate along the surface of nanohelix, they naturally generate magnetic dipoles and multipole moments. The CD signal can be attributed to the interaction of these electric and magnetic multipoles.

To further understand the near-field properties of these plasmon modes, we simulated the charge distribution and volume electric field distribution of N = 1, 2, 3, and 4 modes in the nanorod and nanohelices, under the excitation of RCP and LCP plane-waves (Figures 2 and S3, Supporting Information). A wavevector along $${{{{\bf k}}}_{10\bar{1}}}$$ was chosen as it allows for the excitation of all plasmon modes. Across different structures, standing wave-like charge oscillations appear from one end to the other, labeled as N = 1, 2, 3, and 4 from the fundamental mode to the higher-order multipole modes, with resonance wavelengths consistent with the peaks in Figure 1. In achiral nanorods, the multipole plasmon modes (N = 2, 3, 4, …) are typically considered equivalent to multiple aligned electric dipoles oscillating head-to-head, resulting in a total dipole momentum approaching zero.^[32] Consequently, these modes excited by RCP and LCP plane-waves exhibit significantly lower intensity compared to the dipole mode (N = 1), as shown in Figure 1B. Qualitatively, for the even-order plasmon modes in the straight nanorods, electric dipole pairs (N = 2) or two electric dipole pairs (N = 4) with opposite orientations located on the two sides of the central plane of the nanorods symmetrically, leading to zero total dipole moments and a complete field cancellation. In nanohelices, however, this cancellation effect is less pronounced due to the asymmetric curvature of the nanohelix. The total dipole moments of multipole plasmon modes do not tend toward zero. This explains the observation in Figure 1D that the peak intensities of the multipole modes are comparable to those of the dipole modes in the nanohelices. Moreover, these helical charge oscillations are the physical origin of the chiroptical responses in Figure 1E.

Figure 2

Charge distributions and volume electric field distributions of N = 1, 2, 3, and 4 modes in the four structures, under RCP plane-wave excitation. For each mode, the left column illustrates the charge distribution on the Ag surface, while the right column displays the volume electric field distribution within the Ag structure. The small arrows denote the current distribution.

When examining the far-field directionality of plasmonic nanoantennas, a common approach is to calculate the angular distribution of energy radiation.^{[19, 20]} Figure 3 shows the 3D far-field patterns of scattered light intensity at the resonant wavelengths of modes N = 1, 2, 3, and 4, from the four structures in response to the obliquely incident RCP and LCP plane-waves. For each specific mode of a given structure, the 3D far-field patterns exhibit little difference between the RCP and LCP excitations. This is because the near-field charge oscillations and electric field profiles induced by RCP and LCP plane-waves are actually quite similar (Figures 2 and S3, Supporting Information), differing only in their intensities. Consequently, this similarity predisposes the angular distributions of energy radiations. Nonetheless, a comparison of the far-field patterns between the nanorod and nanohelices reveals intriguing similarities and distinctions. The asymmetric nature of helices breaks the axial symmetry, leading to 3D far-field patterns that deviate from the axially symmetric patterns observed in achiral nanorods. Despite this, all N = 1 modes display a characteristic doughnut shape in the far-field, reflecting the shared feature of dipole plasmon mode. However, significant discrepancies emerge in the 3D far-field patterns of multiple plasmon modes, implying the complexity of far-field directionality after the introduction of chirality.

Figure 3

3D far-field patterns of scattered light intensity of N = 1,2,3, and 4 modes based on the four structures, under RCP and LCP plane-wave excitation, respectively. The orientation of each far-field pattern aligns with that of the corresponding structure. For each mode, the left column displays the far-field pattern under RCP excitation, while the right column shows the far-field pattern under LCP excitation.

We further examined the near-field charge distribution and far-field directionality of these nanoantennas using an electric dipole as the point-source for excitation, relevant to the plasmon-enhanced emission process of nanoemitters coupled with nanoantennas.^{[33, 34]} Our previous studies have covered different cases of achiral nanorods,^{[26, 27]} where the main conclusion is that the radiation of electric dipoles can be enhanced by individual nanorods through the Purcell effect, with modulation of their far-field directionality by dipole or multipole plasmon modes. In the context of nanohelices, for the sake of simplicity, we focused on the helix 2-R model with the pitch of 100 nm as an example to explore the impact of different electric dipole configurations. Initially, we investigated the influence of distance by placing a z-polarized electric dipole near one end of the helix 2-R, varying the distance from 2 to 18 nm (Figure S4, Supporting Information). Our findings reveal that the energy emitted from the coupled system decreases with increasing distance, aligning well with the Purcell effect. Moreover, the shapes of the 3D far-field patterns are predominantly influenced by the coupled plasmon modes (N = 1, 2, 3, and 4) and remain largely consistent with those observed under plane-wave excitation. We then explored the impact of dipole orientation and position, demonstrating the excitation of multipole plasmon modes across all configurations, resulting in highly similar 3D far-field patterns (Figures S5,S6, Supporting Information). In addition, we examined the coupled system involving right-handed nanohelices coupled with circularly polarized dipoles rotating in the xy- or yz-plane (Figure S7, Supporting Information). Consistent far-field patterns with the same mode-dependency were identified, reaffirming the crucial role of the plasmonic nanohelix in far-field modulation. These results indicate that, within the coupled system comprising an electric dipole and a nanohelix, the predominant factor influencing far-field radiation directionality is the specific plasmon mode rather than the configuration of the electric dipole. It is worth noting that the shapes of these far-field patterns are consistent with those under plane-wave excitation (Figure 3), suggesting that these are inherent far-field characteristics of the plasmonic nanohelix.

Another key aspect of far-field directionality analysis is the 3D angular distribution of circular polarization in the radiated light. For this purpose, we introduced Stokes parameters S₀ and S₃ and thereby defined the degree of circular polarization (DCP) to elucidate the polarization characteristics:^[35]

$$$$\begin{equation}{{S}_0} = {{I}_\parallel } + {{I}_ \bot } = {{\left| {{{E}_\parallel }} \right|}^2} + {{\left| {{{E}_ \bot }} \right|}^2}\ \end{equation}$$$$(1)

$$$$\begin{equation}{{S}_3} = {{I}_{{\mathrm{LCP}}}}\ - \ {{I}_{{\mathrm{RCP}}}} = {\mathrm{\ i}}\left( {{{E}_\parallel }E_ \bot ^* - {{E}_ \bot }E_\parallel ^*} \right)\end{equation}$$$$(2)

$$$$\begin{equation}{\mathrm{DCP\ }} = {{S}_3}/{{S}_0} = {\mathrm{\ i}}\left( {{{E}_\parallel }E_ \bot ^{\mathrm{*}} - {{E}_ \bot }E_\parallel ^{\mathrm{*}}} \right)/\left( {{{{\left| {{{E}_\parallel }} \right|}}^2} + {{{\left| {{{E}_ \bot }} \right|}}^2}} \right)\end{equation}$$$$(3)

where E_⊥ and E_∥ represent the polar and azimuthal components of the far-field electric field in the spherical coordinate, I_∥ and I_⊥ are the corresponding intensity, I_RCP and I_LCP are the intensity of RCP and LCP radiation. The angular distribution of DCP is a crucial metric to assess the far-field directionality of RCP and LCP light, i.e., DCP = +1 indicates pure LCP, and DCP = −1 indicates pure RCP. Its combination with the angular distribution of far-field power can generate a new 3D far-field pattern with the shape representing the far-field power while the red and blue colors denote the positive and negative DCP (Figure S8, Supporting Information). For example, we simulated single electric dipoles with linear or circular polarization in free space and recorded their electric field in the near- and far-field regions (Figure 4A,B, Supporting Information). The linear dipole exhibits a characteristic doughnut-shaped far-field pattern with zero DCP, whereas the circularly polarized dipoles display capsule-shaped far-field power patterns, with opposing DCP in the upper and lower hemispheres. The accuracy of the DCP calculation has been checked by the analytical solution of the circularly polarized electric dipole:^{[35, 36]}

$$$$\begin{equation}{\mathrm{DCP}}\left( {\varphi ,\theta } \right) = \pm \ \frac{{{{{\left( {\cos \varphi + 1} \right)}}^2} - {{{\left( {\cos \varphi - 1} \right)}}^2}}}{{{{{\left( {\cos \varphi + 1} \right)}}^2} + {{{\left( {\cos \varphi - 1} \right)}}^2}}} = \pm \frac{{2\cos \varphi }}{{{{{\cos }}^2}\varphi + 1}}\end{equation}$$$$(4)

where φ is the polar angle relative to the z-axis, θ is the azimuthal angle. The notation “±” signifies that the positive sign corresponds to the left-handed circularly polarized dipole around the positive z-axis, and the negative sign corresponds to the right-handed one.

Figure 4

Radiation from an electric dipole modulated by the Ag nanohelix. A) Spatial distribution of time-domain electric field generated by in-plane single electric dipoles with linear or circular polarization. The white arrow represents the polarization of each dipole. B) 3D far-field patterns of the single electric dipoles, with color indicating DCP and shape indicating far-field power. LCP is represented by red color while RCP is represented by blue color. C–F) 3D far-field patterns of an electric dipole coupled with a plasmonic nanohelix. The linear polarization of the dipole is along z-axis, whereas the circularly polarized dipoles are rotating in the xz-plane. The nanohelix is either right- or left-handed, as illustrated in the left column. The shape of the 3D far-field patterns indicates the intensity of far-field power, with the red and blue colors denoting the positive and negative DCP.

We employed the same methodology to simulate the coupled system of an electric dipole and a nanohelix, focusing on positioning the dipole near one end of the nanohelix while varying the electric dipole's polarization between linear and circular, and the nanohelix's handedness between right- and left-handed (Figure 4C–F). In all these scenarios, the far-field patterns exhibit strong dependence on the plasmon modes and show significant differences from those of free-standing single dipoles, showcasing the effectiveness of modulating far-field emission behaviors in these systems. The mode-dependent far-field directionality is evident, both in far-field power and DCP. Comparing Figure 4C,D, the angular distributions of far-field power and DCP remain consistent for each mode, regardless of whether a linear dipole or a circularly polarized dipole is used. In comparing Figure 4D–F, the circularly polarized dipole is coupled to right- and left-handed nanohelices, respectively, resulting in mirror-symmetric far-field patterns with opposite DCP. These results show that the far-field patterns are mainly determined by the structure and chirality of nanohelix. In particular, the N = 3 mode shows the capacity to route RCP and LCP light selectively toward the ends or sides of the nanohelix. This is an interesting example of the simultaneous modulation of RCP and LCP light by a single plasmonic nanoparticle. However, considering these circularly polarized lights with different radiation angles in single-particle CD spectroscopy measurements may lead to an averaging effect, ultimately weakening the chiroptical response.

3 Conclusion

In this work, we numerically investigate the transition of single Ag nanoantennas from achiral to chiral configurations, under plane-wave and point-source excitation, and analyze the resulting near- and far-field behaviors of the standing-wave-like plasmon modes. The far-field behaviors are described by far-field patterns that combine angular distributions of far-field power and DCP, showcasing the routing effect of circularly polarized light. Such routing effect is determined by the specific plasmon mode and structural chirality of the plasmonic nanoantenna, rather than the configuration and chirality of the electric dipole. We have demonstrated that emission from linearly polarized dipole can be modulated into circularly polarized far-field radiation by single plasmonic nanohelices, which is favorable for constructing circularly polarized light sources. The radiation from the coupled systems may exhibit multi-polarization characteristics along different directions. As a proof of principle, we have shown that the N = 3 mode has the ability to selectively guide RCP and LCP light toward the ends or sides of the nanohelix. Our simulation results show that the relative position of the point source and its polarization state do not affect DCP of emitted light, which holds inspirational significance for related experiments involving point sources, such as luminescent molecules. In experiments, the choice of optical path, the orientation of plasmonic nanoparticles, and the averaging of signals over various detection directions are among the critical factors influencing the measured single-particle CD spectra,^[13-18] which may differ significantly from those measured from ensemble samples. In addition, our research establishes a crucial link between the angular far-field distribution of the energy and the polarization states and investigates how these behaviors are affected by the multipolar plasmon modes in the nanohelices. These explorations are pivotal for the precise manipulation of chiral light. In summary, our findings not only highlight the great potential of chiral plasmonic nanoantennas in far-field directionality modulation, but also offer valuable insights for experiments involving angle-resolved spectroscopies and plasmon-exciton coupled systems.^{[33, 34]}

4 Experimental Section

Simulations

All the electromagnetic simulations were conducted using the COMSOL Multiphysics 2019b R6. Achiral nanorod was modeled as a cylinder with two hemispherical ends, while chiral nanohelices were modeled as the helical structure. The refractive index of the air was set at 1.00, and the dielectric function of silver was fitted from Johnson and Christy's data. For plane-wave excitation, a total-field scattered-field source was employed to simulate the near- and far-field properties. Circularly polarized plane-wave was simulated by combining two linearly polarized plane-waves with perpendicular polarization directions but parallel wavevectors, having equal amplitudes and a phase disparity of π/2. LCP/RCP was defined by the counterclockwise/clockwise electric field vectors from the point of view of the source. For point-source excitation, the linearly polarized electric dipole was simulated by an electric dipole, whereas the circularly polarized electric dipole was simulated by combining two linearly polarized dipoles with perpendicular polarization directions, equal amplitudes, and a phase disparity of π/2. In both cases, a perfectly matched layer was applied as the boundary to absorb the outgoing radiation. The surface charge density ρ_f was calculated by the continuity boundary condition of the electric displacement field at the interface, i.e., ρ_f = n·D, where n is the unit normal vector perpendicular to the metal surface and D is the electric displacement field. The distributions of the volume electric field and the surface electric field are described by $$| E | = \sqrt {E_x^2 + E_y^2 + E_z^2} $$. A near-to-far-field transformation was performed to extract the angular distribution of far-field radiation. The angular distribution of DCP was calculated by Equation 3.

Acknowledgements

Y.C. and Y.C. contributed equally to this work. X.Z. acknowledges financial support from the Natural Science Foundation of Guangdong Province (2023A1515110091), the Program for Guangdong Introducing Innovative and Entrepreneurial Teams (2019ZT08L101), and the Science and Technology Innovation Commission of Shenzhen (JSGGKQTD20221101115701006). Y.C. acknowledges financial support from the China Postdoctoral Science Foundation (2024M753075). B.X. acknowledges support from the National Key R&D Program of China (No. 2023YFA1407700), the National Natural Science Foundation of China (62475225, 12404187), Stable Support Program for Higher Education Institutions of Shenzhen（20220817185604001), Guangdong Basic and Applied Basic Research Foundation (2024A1515012031).

Conflict of Interest

The authors declare no conflict of interest.

Open Research

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.