## Abstract

Chiral plasmonic nanostructures will be of increasing importance for future applications in the field of nano optics and metamaterials. Their sensitivity to incident circularly polarized light in combination with the ability of extreme electromagnetic field localization renders them ideal candidates for chiral sensing and for all-optical information processing. Here, the resonant modes of single plasmonic helices are investigated. We find that a single plasmonic helix can be efficiently excited with circularly polarized light of both equal and opposite handedness relative to that of the helix. An analytic model provides resonance conditions matching the results of full-field modeling. The underlying geometric considerations explain the mechanism of excitation and deliver quantitative design rules for plasmonic helices being resonant in a desired wavelength range. Based on the developed analytical design tool, single silver helices were fabricated and optically characterized. They show the expected strong chiroptical response to both handednesses in the targeted visible range. With a value of 0.45, the experimentally realized dissymmetry factor is the largest obtained for single plasmonic helices in the visible range up to now.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. INTRODUCTION

Chirality or handedness is a geometric property of all objects that cannot be superimposed onto their mirror image. The paradigm of a chiral object is represented by the helix. In nature, helices find their manifestations, e.g., in the geometry of amino acids, DNA, or climbing plants. In optics, the electric field vector of left (right) circularly polarized light follows a left-handed (right-handed) helix upon propagation for a fixed point in time [1]. Chiral light can be used as a probe for chiral matter [2–4]. However, the interaction of circularly polarized light with naturally occurring chiral objects is inherently weak. To significantly enhance light–matter interaction in the visible range, metal nanostructures can be employed [5–7]. These can exhibit collective oscillations of their free electron gas under visible light incidence, so-called plasmon polaritons, which in turn lead to extreme electromagnetic (EM) field concentrations. Accordingly, plasmonic helices constitute the nearly perfect choice in terms of maximized chiroptical interaction [8].

Apart from their strong chiroptical response, helical plasmonic antennas show a complex resonance behavior, which was not fully understood until now. For the radio frequency regime, an empirical description of the reception and emission behavior of helical antennas is known [9,10]. In the GHz range, a fully analytical treatment was developed for small chiral scatterers [11]. Towards the visible range, the mismatch between free space and plasmon wavelength on a metallic wire leads to an additional degree of freedom. Furthermore, losses start to play a crucial role. Commonly, the theoretical treatment of the helix response makes use of full-field modeling as it is well-suited for providing spectra to be compared with the experiment. Although modeling allows for directly displaying all fields, local charges, and currents, it is hard to deduce physical mechanisms from these. Earlier studies on plasmonic helices attributed observed spectral features to a superposition of the respective split-ring eigenmodes of a one-turn helix with Bragg resonances when moving to more than one turn [12]. This explains the large asymmetry in the transmission of gold helical arrays, which strongly scatter and absorb incident light matching the handedness of the helix geometry and are transparent to light of opposite handedness. Numerical investigations on single silver helices proposed an effective dipole length onto the wire to describe the resonant modes [13]. However, the scaling law found therein could not be verified in another study on arrays of silver and nickel helices. Instead, in the lower frequency range (visible and near-IR), a linear scaling with the total wire length was found [14]. Recently, a single loop gold helix was investigated in detail concerning the excitation of the fundamental resonance [15]. A multipole analysis revealed that the one-loop helix can be described as a point-like chiral molecule with nearly parallel electric and magnetic dipole moments on resonance [2]. All these studies concentrate on the opto-chiral response for equal handedness of the helix and incident field [16].

Another important observation is the possible zero crossing of the dissymmetry factor. The dissymmetry factor $g$ normalizes the difference in extinction between left-circularly and right-circularly polarized (LCP and RCP) light to the total extinction (cf. Supplement 1, Section SI). For the case of molecules, it is known that light of opposite handedness compared to the molecule handedness may also lead to efficient excitation [2]. Indeed, such a spectral response was experimentally observed before for the case of chiral “plasmonic molecules” of various geometries [17–24]. This zero crossing can be qualitatively described in terms of coupled oscillating dipoles within the Born–Kuhn model [18,25]. The Born–Kuhn model was developed to describe the optical activity of molecules. By using the dipole coupling strength and several further parameters for fitting, it provides an intuitive description of the excitation mechanism in dipole-dominated chiral systems and reproduces the zero crossing. Similarly, ensembles of plasmonic helices from different metals showed a distinct asymmetry in transmission (${T}_{\mathrm{RCP}}-{T}_{\mathrm{LCP}}$) with zero crossing [26–28]. To describe more complex chiral systems, the Born–Kuhn model was extended [29], and generalized dipole-based oscillator models were developed [30,31]. Furthermore, equivalent circuits were employed [32]. While all of these models provide design rules for chiral systems consisting of coupled achiral elements, none of them can be directly transferred to the case of plasmonic helices for which multiple reflections of plasmon polariton modes and retardation effects cause a wealth of higher order resonances and a rich spectral behavior. Hence, the full description of all types of resonant helix modes remains an open issue.

## 2. ANALYTICAL MODEL

Figure 1 depicts the geometrical setting of this work. Circularly polarized light is incident on a plasmonic helix. The plane wave propagates along the helix axis. To understand the chiroptical response of such a helical nanostructure, we split the problem into two parts. First, we study the existence of plasmonic modes, and, in a second step, we investigate how these modes may be excited. To do so, let us start with a straight metallic wire. Upon excitation by an external EM field of a certain frequency, an excitation of a standing wave may occur. Optically, the associated charge density oscillation can be understood as a plasmonic Fabry–Perot mode [33] [cf. Fig. 2(a)]. Standing wave patterns evolve when the wavelength meets the following condition:

The effective wavelength ${\lambda}_{\mathrm{eff}}$ of the plasmonic mode is reduced in comparison to the exciting free-space wavelength $\lambda $ and follows a nearly linear scaling behavior [34],

In the visible range, the propagation constant $\gamma (\lambda )$ is inversely proportional to the incident wavelength $\lambda $. The correction term involves the wire radius ${r}_{\mathrm{wire}}$ and the order of the mode $n$.Together, equations Eqs. (1) and (2) allow us to calculate the excitation wavelengths for Fabry–Perot resonances of different orders $n$ on linear plasmonic antennas. These modes can be regarded as a discrete set of eigenmodes for the helical geometry as long as the interaction between the helix turns is negligible. In the visible range, this is safely ensured for pitch heights larger than 200 nm due to the rapid decay of the plasmonic near-field outside the helix. Since the wire diameter is also accounted for, the assumption is valid for a wide range of geometries as further detailed in Supplement 1, Sections SIII and SV.

Now let us tackle the geometry dependent efficiency of excitation of the eigenmodes. A mode is efficiently excited if the spatial distribution of the incident electric field vectors resembles that of the local mode pattern on the helix surface [35,36]. For the case of a single-turn helix, this is true when the handedness of the light matches the handedness of the structure. As shown in Fig. 2(b), the incident RCP light follows the arc geometry of the right-handed helix, and, thus, drives the free electron gas along a curved path. The resulting mode is dipolar ($n=1$) [15]. But also for LCP fields, the field vectors may match the local mode pattern for higher field energy and smaller incident wavelengths, respectively. The corresponding mode pattern of the $n=3$ mode can be viewed as lineup of three local dipoles along the helix turn.

This intuitive reasoning can be formalized by analyzing the extinction [37],

Therefore, the overlap integral describes the mode matching between incident and resonator fields [36]. According to the identified set of eigenmodes, the current density ${\mathit{j}}_{\mathrm{wire},n}$ is given by mapping a sine function with effective wavelength ${\lambda}_{\mathrm{eff}}$ onto a helical path. When parameterizing the amplitude of the current along the $z$ direction, the integral Eq. (3) reduces to a one-dimensional (1D) problem (cf. Supplement 1, Section IV). We, therefore, term our analytical model 1D mode matching.

The corresponding solution for incident circularly polarized light reads as

The obtained formula fully quantifies the excitation efficiencies for helix resonances of all orders excited by any handedness of the incident light. The excitation efficiency of a respective Fabry–Perot eigenmode is determined by its fitting onto the wire length and its match with the exciting external field. Generally spoken, all modes with approximately 1 dipole per turn are efficiently excited by light with the handedness of the helix [14]. All modes with approximately 3 dipoles per turn can be excited by light with the opposite handedness compared to the helix. Beyond the quantitative description, the geometrical considerations of the 1D mode matching provide a straightforward design route for plasmonic helices.

## 3. DESIGN TOOL

Based on this, we aim to design a plasmonic helix showing both types of resonances, sensitive to matching and opposite handedness, and switching its response in the visible range. We chose silver as the plasmonic metal described by a Drude permittivity fitted to tabulated data [39]. The wire radius is around 30 nm, and the excitation wavelength is in the middle of the desired range, e.g., around 800 nm. According to Eq. (2), the corresponding effective wavelength of a linear antenna is 550 nm. This value will serve as single-turn wire length. The wire can be coiled up to a helix with a radius 60 nm and a single pitch of height 310 nm. We choose the helix to be right-handed and investigate the obtained resonant behavior using the 1D mode-matching model. Figure 3 shows the excitation efficiencies for the fundamental modes $n=1\u20133$ dependent on the incident circular polarization state. As the 1D model provides for discrete values of the excitation efficiencies, we plot the corresponding efficiencies by using a Lorentzian “sample function.” The sample function reflects the match of the respective effective wavelength with the wire length of the one-turn helix. The insets of Fig. 3 show the results of numerical modeling for the identified resonance conditions of the $n=1$ and $n=3$ eigenmodes.

Fig. 3(a) displays the extinction power for RCP light incident on the right-handed helix, i.e., matching handedness. The fundamental dipole mode ($n=1$) is most efficiently excited for RCP light at a wavelength around 1450 nm. This corresponds to an effective wavelength of 1100 nm, doubling the wire length. As can be seen from the inset of Fig. 3(a), the incident electric field (red arrows) follows the helical geometry and matches the local charge distribution. In contrast, the LCP light counteracts the current flow and suppresses the excitation of the $n=1$ mode. The same reasoning applies for the $n=3$ mode, as shown in Fig. 3(b). Here, the LCP excitation drives the charges along the observed mode pattern. Hence, LCP light efficiently excites this mode, while RCP excitation is suppressed. The external field vectors match with the respective dipole patterns of the numerically modeled surface charge distributions, thereby proving the expected excitation behavior.

The magenta curves in Fig. 3 correspond to the excitation efficiency of the $n=2$ mode, which deserves special attention. Here, the wire length coincides with the effective wavelength on the wire. In case of the one-turn helix, this mode can be excited for either handedness of the incident light. The external field drives the charges such that the induced overall dipole moment resembles that of the fundamental resonance of a planar split-ring excited with polarization perpendicular to the gap [15,40]. However, for pitch numbers $m>1$, the dipole moment of one turn may be compensated by the ones of neighboring turns. Hence, light of both handednesses excites this resonance with equal efficiency. As a consequence, this point defines the observed zero crossing of the dissymmetry, and we therefore name it the toggle point.

The identified toggle point can now serve as the starting point for the design of plasmonic helical antennas being resonant in a specific wavelength range for a desired handedness of the incident electric field. One starts by selecting the excitation wavelength where the zero crossing should occur. Next, the corresponding effective wavelength for the plasmonic wire made of metal a, with thickness $b$, can be calculated. Finally, this wavelength serves as the single-turn wire length of the envisaged plasmonic helix to be fabricated. Adding further turns to the helix does not change the geometrical excitation conditions. Instead it leads to an increasing number of higher order Fabry–Perot eigenmodes, which may be excited according to the reasoning above.

It has to be noted that this principle behavior is not dependent on the choice of the metal. Here, we chose silver, since it exhibits the smallest losses of all coinage metals. By switching to, e.g., gold, only the effective wavelength scaling onto the wire has to be adapted to the new material parameters (i.e., the permittivity) to define the wire length. Another point is that the geometric overlap conditions are not dependent on the radius of the helix. Hence, the radius can serve as an additional degree of freedom and be adjusted such that the near-field interaction between neighboring helix turns is negligible while the extinction power is maximized.

Based on the identified geometrical parameters, Fig. 4 compares the analytical model to full-field simulations for a helix with four turns ($m=4$). The dashed blue lines in Fig. 4(a) display analytically calculated Fabry–Perot modes of a straight silver cylinder with a radius of 32 nm, a length of 2016 nm, and round end caps. They agree well with the respective modes retrieved from full-field modeling (not displayed in this graph). The dots in Fig. 4(a) display the efficiency of energy transfer to the respective Fabry–Perot eigenmodes dependent on the handedness of the incident light according to equation Eq. (4). Black dots stand for LCP excitation, and gray ones for RPC excitation. The mode matching has been performed for the exact resonance positions for sinusoidal helix currents. The quantitative agreement only requires one additional parameter. The helix radius was corrected according to $r+\delta r$ to account for the near-surface character of plasmonic currents for higher mode orders; $\delta r$ was chosen to be 10 nm (see Supplement 1, Section SIV). The numerically calculated extinction efficiencies ${Q}_{\mathrm{ext}}$ of a right-handed helix with four turns are plotted as black (LCP) and gray (RCP) curves [Fig. 4(a)]. The chiroptical response is characterized by efficient excitation of resonant modes around 1000 nm for RCP light, a transition region with no excitations around 800 nm, and the LCP excitation range around 600 nm. Both ranges of strong dissymmetry in Fig. 4(a) contain plasmonic Fabry–Perot modes of different orders, as can be seen in Fig. 4(b), showing surface charge distributions for two selected modes. At $\lambda =1070\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, RCP light excites a mode of order $n=5$, while for LCP excitation at $\lambda =530\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, the excited mode has the order $n=13$. The 1D design tool provides the envelope of geometrically excitable modes and determines the excitation efficiencies of the underlying Fabry–Perot eigenmodes (see Supplement 1, Section SV).

## 4. EXPERIMENTAL REALIZATION

Finally, the designed helical plasmonic geometry is realized experimentally. Figure 5(a) shows scanning electron micrographs of the fabricated right-handed silver helices with $m=3\u20135$ turns, with a helix radius of ${r}_{h}=60\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ and a wire radius of ${r}_{w}=32\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, as determined from the SEM images. The employed setup for confocal white light transmission spectroscopy is shown in Fig. 5(b). Two Cassegrain objectives allow for dispersion-free extinction spectroscopy of single nanostructures in the wavelength range from 550 nm to 1050 nm. The helices were directly written onto transparent substrates using electron-beam-induced deposition (EBID), currently being the most flexible and precise three-dimensional (3D) nanostructuring technique [41–44]. Silver was subsequently deposited onto the EBID scaffold by means of glancing angle deposition [15,45]. The thickness of the resulting silver shell exceeds the skin depth, such that an optical description as solid silver helix is valid. Exactly 120 helices with different numbers of turns were spectroscopically examined for incident LCP and RCP broadband light in the range of $\lambda =500\u20131100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ (see Supplement 1, Section SII).

For the case of a four-pitch helix, Fig. 5(c) displays extinction maps for both incident polarizations at two selected wavelengths of $\lambda =600\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ and $\lambda =1000\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$. According to the design, a strong response of the right-handed helix to RCP light at 1000 nm occurs, while the response to LCP light is maximized at 600 nm. We quantify the strength of the chiroptical response by the dissymmetry factor $g$. Physically, $g$ describes the difference in the extinction for different handedness of incident circularly polarized light normalized by the total extinction,

The obtained dissymmetry factors of the investigated helices with $m=3$, 4, and 5 turns are plotted in Fig. 5(d). Solid lines display the average experimental dissymmetry factors ${g}_{1-T}$ of 30 single helix measurements for each number of turns. The dashed lines display ${g}_{\mathrm{Cext}}$ and are calculated based on full-field modeling with the geometrical parameters taken from SEMs. The modeling reproduces the observed spectral behavior well. The maximum value of the dissymmetry factor deduced from the simulations is close to the theoretical limit of 2. Experimentally, much smaller dissymmetry factors around 0.45 are observed. This discrepancy is mainly due to the chosen experimental setting in which light is focused to a spot size that exceeds the helix diameter (cf. Supplement 1, Section SII). Furthermore, scattering losses due to surface roughness and grain boundaries may cause an additional achiral loss background and thereby decrease the experimentally observed chiroptical response. The small spectral deviations can be attributed to geometrical imperfections of the real structures, which have a larger influence for the single helix measurement. Despite these limitations, the experimentally measured dissymmetry is the highest observed up to now for a single plasmonic nanohelix (see Supplement 1, Section SII).

## 5. CONCLUSION

Here, the interaction of single plasmonic helices with circularly polarized light was investigated, both theoretically and experimentally. Our findings lead to a complete understanding of all resonant features of plasmonic helices of any geometry and material. An analytical 1D mode-matching model was developed that quantitatively describes the excitation efficiencies of all types of resonant modes. Furthermore, the mode matching provides a straightforward design route for plasmonic helices being resonant in a desired wavelength range. We have employed the developed design tool to design helical plasmonic antennas with a strong chiroptical response for either handedness of incident light in the visible range. The obtained helical geometry was experimentally realized and showed an extraordinarily high dissymmetry for resonant modes responding to both handedness of incident circular polarization. In conclusion, an efficient route for the design and fabrication of helical nanoantennas is established. Thereby, potentially groundbreaking applications of plasmonic helices in all-optical information processing may be triggered, either based on the spin angular momentum of light [47] or based on their strong nonlinear response [48].

## APPENDIX A: METHODS

Glass cover slips were sputter-coated with a 50 nm thick amorphous layer of indium tin oxide (AJA International Inc.) to serve as transparent conductive substrates for direct electron beam writing. Helices were written in a FEI Strata DB 235 dual-beam focused-ion-beam microscope using the metal–organic precursor dimethyl-gold-acetylacetonate [${\mathrm{Me}}_{2}$ Au(acac)] by directly addressing the patterning board using stream files. Therewith, the electron beam was scanned in a circular path with a radius of 60 nm, a pitch of 600 pm, and a stepwise increasing dwell time ${t}_{d}=8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}+(n-1)\times 4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$, with the number of turns $n$. The applied step function compensates for the decrease of the vertical growth rate in the deposition with increasing structure height. The resulting helices have a constant pitch height of around 310 nm and arm thicknesses slightly below 20 nm and were used as a scaffold to be subsequently covered with the desired metal. Conformal coverage with silver was achieved by sputter coating (Plassys) under glancing angle incidence with an emission current of 110 mA and a deposition rate of 0.4 nm/s. The evaporated film thickness of 50 nm resulted in a silver layer of 21.5 nm onto the helices. This shell thickness exceeds the skin depths, making their plasmonic response in the visible range equivalent to that of a pure silver helix [43]. The resulting arm thickness of the helices is 64 nm.

Optical spectroscopy of single silver helices was carried out using a home-built confocal microscope. The collimated and IR-filtered emission from an incoherent halogen lamp covering the spectral range from 400 nm to 1000 nm (Dolan-Jenner Model DC-950) is focused onto the sample using an all-reflective Cassegrain objective (ARO) with a numerical aperture of 0.5 (5002-000, Beck Optronics Solutions Ltd.) resulting in a spot with a diameter of approximately 50 μm. LCP and RCP light in the complete visible spectral region was generated by a combination of a Glan–Taylor linear polarizer (PGL 15, B. Halle Nachf.) and a rotatable superachromatic quarter-wave plate (WPA4415, Union Optics). The light transmitted through the sample is collected by a second ARO (5006-000, Beck Optronics Solutions Ltd.) and imaged onto a pinhole with a diameter of 75 μm in the back focal plane of the ARO, limiting the observation area on the sample to approximately 1 μm. A 100 mm lens is afterwards used to focus the light onto the entrance slit of a spectrometer (SP2150, Princeton Instruments) in combination with a liquid-nitrogen-cooled CCD camera (Pixis eXcelonBR, Princeton Instruments). By raster-scanning the sample with the helices through the focus with a nominal step size of 250 nm perpendicular to the propagation direction of the beam using a 3D piezo scanner (P-611.3 Nanocube from Physik Instrumente), maps of the linear extinction of the silver helices are obtained.

Full-field modeling was carried out based on the finite element method (Comsol Multiphysics) and on the finite-difference time-domain technique (Lumerical Solutions) with the material response taken from reference [39]. The corresponding models were optimized concerning accuracy and convergence, and the calculated spectra agreed perfectly for both techniques (see Supplement 1, Section SV). The helices were modeled with round end caps, and geometrical parameters were taken from the SEM images. Analytical calculations were carried out using Mathematica and later transferred to Python. The linear wavelength scaling of Novotny [34] has been included in the corresponding code. The EM near-fields of different helix turns may interact and shift the excitation wavelength compared to the linear cylinder. Therefore, the near-field extension around the wire was calculated to define a minimum pitch height for the helix down to which the analytical treatment is justified (see Supplement 1, Section SIII). The developed analytical design tool allows for identifying an appropriate helix geometry for a desired resonant wavelength range and is available for download [49].

## Funding

Helmholtz Association (Postdoctoral Fellowship PD-140); Bundesministerium für Bildung und Forschung (BMBF) (NanoMatFutur); Deutsche Forschungsgemeinschaft (DFG) (SPP1839 Tailored Disorder); German-Israeli Foundation for Scientific Research and Development (GIF) (1256).

## Acknowledgment

The authors thank P. Wozniak and I. Fernandez-Corbaton for fruitful discussions. K. H. acknowledges funding from the Helmholtz association by the Postdoctoral Fellowship PD-140. M. S. acknowledges funding from the Federal Ministry of Education and Research within the NanoMatFutur program. Furthermore, C. L. thanks the Deutsche Forschungsgemeinschaft (SPP1839 Tailored Disorder) and the German-Israeli Foundation (GIF grant no. 1256) for financial support.

See Supplement 1 for supporting content.

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