1. Introduction

Metallic nanostructures with helical shapes feature a broad spectrum of interesting radiation and polarization characteristics that can largely be controlled by varying their geometrical parameters [1]. In the field of plasmonics [2, 3], there have been many fascinating studies that focused on the dichroic properties of metal nano-helices excited using light with circular polarization [4–9]. Since the helix is a chiral shape, meaning that it cannot be superimposed to its mirror image, plasmonic nano-helices have chiral properties that make light-matter interaction sensitive to the handedness of circularly polarized radiation. In particular, differential scattering and absorption of left- and right-circularly polarized light, which is known as circular dichroism (CD), have been demonstrated using plasmonic nano-helices in the visible and infrared spectral range [10, 11]. However, most of the structures mentioned in these studies have sub-wavelength dimensions and their properties have been investigated only over a limited range of geometrical parameters with respect to the optical wavelength. In addition, the plasmonic behavior of nano-helices is often captured by considering their optical transmittance or scattering/extinction cross-sections, which fail to reveal the rich angular scattering and directivity properties that are unique to the nano-helices. As a result, it is very interesting to systematically explore the directional scattering of plasmonic nano-helices in the diffractive regime, where their geometrical features are comparable to the wavelength of light.

In this paper, using the rigorous Surface Integral Equation (SIE) method [12–15], we systematically study the far-field radiation characteristics of diffractively coupled gold (Au) nano-helices in the optical regime. By doing so, we demonstrate that the engineering of their geometrical parameters provides novel opportunities to achieve highly directional scattering along multiple directions with controlled polarization states in the visible regime. Our results establish general design rules that can be utilized to engineer novel directional nano-antennas of great interest for the development of sensors and filters with unprecedented beam forming and polarization capabilities.

We show in Fig. 1 the geometry of a representative nano-helical structure, which can be parameterized by the radius R of the helix, the helical pitch P, and the radius r of the cylindrical Au wire. The helix has a number of helical turns N, which together with R, P, and r define the set of the basic geometrical parameters that will be studied in the paper. As we will systematically discuss in this work, Au nano-helices feature vastly different radiation patterns and polarization characteristics of scattered plane waves depending on the values of their geometrical parameters with respect to the wavelength of light.

 

Fig. 1 Definition of geometrical parameters of a helix. r is the radius of the cylindrical wire that forms the helix, R is the radius of the helix, and P is the pitch (separation between consecutive helical turns) of the helix. In this case, the number of helical turns shown is N = 4. The orientation of the nano-helix is such that its helical axis is parallel to the x-axis.

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We focus our study on the excitation with linearly polarized plane waves that propagate along the helical axis. In this case, classical antenna theory [1, 16, 17], which is valid in the limit of perfectly metallic (i.e., no losses) helical wires with infinitesimally small r, provides general guidelines on the choice of the geometrical parameters that give rise to different radiation patterns. In particular, when the geometrical dimensions of the nano-helix become comparable to the wavelength of light, we expect the far-field radiation patterns to exhibit highly directional modes, known as axial modes, with directional and polarization characteristics controlled by the helical geometry [1]. However, due to the dispersive nature of the Au material in the visible regime, we have found in our study that the axial modes supported by nano-helices at optical frequencies are qualitatively different from the classical antenna case [1, 16, 17] and no longer have a well-defined circular polarization. Nevertheless, we have identified a class of thin-wire (r = 10nm) nano-helices that support perfectly circularly polarized backward radiation lobes when their pitch is equal to one wavelength. Moreover, when Au wires with larger r = 100nm are used, we discovered that highly directional modes can be formed in the optical regime resembling the radiation patterns of the traditional (i.e., radio frequency) axial modes over a wide range of R and P values. Therefore, in this paper we use the term “quasi-axial modes” to designate this class of highly directional (beaming) modes that are created in the optical regime, with various degrees of elliptical polarization, beyond the standard thin-wire approximation [16, 17].

In the present numerical study, we solved the Maxwell’s equations using a Surface Integral Equations (SIE) [12–15] approach. This method is computationally advantageous compared to differential methods since only a discretization of the scatter‘s surface is required and the radiation condition at infinity is automatically satisfied [12]. In particular, we have used the Poggio-Miller-Chu-Harrington-Wu-Tsai formulation (PMCHWT) [12], employing the Rao-Wilton-Glisson (RWG) basis functions [18], and calculating the weakly singular integrals using the techniques described in [19]. We recently demonstrated that the PMCHWT formulation is among the most accurate SIE formulations for the modeling of the scattering by resonant plasmonic structures in both the far and near zone [14]. Specifically, we validated our FORTRAN implementation of this formulation against the analytical Mie solution for the scattering of a plane wave by a penetrable gold nano-sphere; evaluating its accuracy in both the near and the far field region as a function of the surface mesh density. Furthermore, in Ref [20]. we compared the PMCHWT and Mie solution for the problem of scattering by an Aluminum nano-sphere excited by a dipolar source located in the near-zone of the particle. We also successfully used this formulation in Ref [21]. for the modeling of bulk and surface second harmonic (SH) scattering from plasmonic structures and we validated its results against the analytical SH-Mie solution. Moreover, we have rigorously tested our code also in the presence of non-spherical plasmonic particles such as ellipsoids [14], cubes [14] and triangular nano-prisms [14], dolmens [22] and coated particles [23], comparing our solution to other numerical methods such as the null field method [24] and the discrete dipole approximation [25] and a modal expansion method [26]. We implemented our investigation of Au nano-helices using 9600 number of unknowns for each geometry (i.e. 3200 triangles). In the worse-case scenario, the L/λ ratio is 0.047, which corresponds to the case in Fig. 1(g) (R = 600nm, P = 1000nm, r = 100nm, N = 4, excited at λ = 600nm).

Without loss of generality, we focus our study on left-handed helical structures since all the results can be easily extended to right-handed cases as well. In particular, the radiation patterns of right-handed helices are the mirror-symmetric images of the left-handed cases along the helical axis, with polarization switched from left-handed to right-handed.

2. Scattering and Radiation Modes of Au Nano-helices

In this section we start by discussing the scattering properties of nano-helices with the helical axis oriented along the x-direction (see Fig. 1). We excite the nano-helices with a plane wave traveling along the positive x-direction, and linearly polarized in the positive z-direction. We start our analysis by considering an incident wavelength λ_exc = 600nm, and extend it across the visible spectrum by additionally considering the cases of 400nm and 500nm excitation. By focusing on these representative wavelengths we provide a general picture of how diffractively-coupled Au nano-helices interact with light in the visible spectrum.

It is well-known from classical antenna theory that helical antennas with sizes smaller than the wavelength produce isotropic radiation modes, called normal scattering modes, while larger helices with sizes comparable to the wavelength support highly directional modes called axial or beam modes [16, 17]. The formation of axial modes can be qualitatively understood based on a simple model (i.e., the array model) of a helical antenna. According to this picture, helical antennas are reduced to linear arrays of single-turn helical elements spaced by P. Axial modes then correspond to the end-fire radiation modes of such equivalent linear arrays. The conditions for the formation of normal and axial modes in radio frequency (RF) helical antennas have been studied in detail by Kraus [16], who summarized his results by constructing a diagram, known as the Kraus’ diagram, which captures the effect of different geometrical parameters on the radiation properties of helical antennas. This diagram provides a sort of “modal phase space’ that displays different radiation diagrams against the helical pitch P_λ and the circumference of the helical cross-section πD_λ (D=2R), all scaled by the excitation wavelength λ.

In Fig. 2 we have calculated the Kraus’ diagram corresponding to Au nano-antennas of helical shape with realistic dispersion data [27] at 600nm and for different geometrical parameters. As we will show, wavelength scaling rules inspired by RF antenna theory are generally valid to approximate the scattering behavior of thin-wire (r = 10nm) nano-helices in the visible spectral range. On the other hand, the results (λ_exc = 600nm) summarized in the diagram include cases for both thin-wire (r = 10nm) Au nano-helices (Figs. 2(a)-2(d), circles) and thick-wire (r = 100nm) nano-helices (Figs. 2(e)-2(h), crosses). Our data clearly demonstrate the wide range of radiation patterns that can be achieved using Au nano-helices with different geometrical parameters. Moreover, we note that in the case of thin-wire nano-helices, classical RF antenna theory [16, 17] qualitatively predicts the respective regions of the parameter space for normal modes and axial mode formation (Fig. 2). In particular, we report in Fig. 2(a) a calculated radiation diagram corresponding to a typical normal mode region (labeled (a) in the Kraus’ diagram), while in Fig. 2(c) we show the radiation diagram of a structure in a shaded region of the Kraus’ diagram where axial modes are predicted to exist by antenna theory. We notice that the radiation diagram in Fig. 2(c) consistently features a predominant radiation lobe in the forward direction. In addition, the radiation diagrams shown in Fig. 2(b) and 2(d), which correspond to helical structures with parameters outside the Kraus’ region of axial mode formation, do not feature directional radiation modes, in agreement with the prediction of classical antenna theory. Besides, the axial mode shown in Fig. 2(c) can simply be designed by following the well-known rules [16, 17]:

 

Fig. 2 Representative radiation patterns of Au nano-helices and their corresponding positions in Kraus’ diagram for RF helical antenna radiation patterns [7, 8]. Plane wave excitation at λexc = 600nm is used. The region under the black dashed curve, where the length of one helical turn (L, where L2 = (πD)2 + P2) is smaller than half of the wavelength, corresponds to radiation normal modes in Kraus’s diagram [7, 8]. The region shaded blue corresponds to the region of axial modes [8]. Outside this region, higher order modes exist, with more complex radiation patterns. The Au nano-helices have N = 4, and: a) R = 20nm, P = 30nm, r = 10nm; b) R = 300nm, P = 100nm, r = 10nm; c) R = 100nm, P = 150nm, r = 10nm; d) R = 300nm, P = 600nm, r = 10nm; e) R = 200nm, P = 300nm, r = 100nm; f) R = 200nm, P = 600nm, r = 100nm; g) R = 600nm, P = 1000nm, r = 100nm; h) R = 200nm, P = 1000nm, r = 100nm.

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In Fig. 2 we additionally summarize results obtained on a group of helices with thicker Au wire radius of r = 100nm and have found that such helical structures produce radiation patterns with very good directionality (Figs. 2(e)-2(h)). We refer to these high-directivity modes as “quasi-axial modes” since they are obtained in the optical regime using thick Au wires with realistic dispersion properties. Interestingly such modes, which are qualitatively very similar to the axial modes of RF antenna theory, cannot be obtained using thin wires in the optical regime. We also stress that quasi-axial modes provides very good directionality even outside the axial mode region of Kraus’ diagram (see Fig. 2) and that they can be excited in a large range of either R (Figs. 2(h)-2(g)) or P (Figs. 2(e)-2(f)) values. Finally we remark that this group of nano-helices do not support normal scattering modes since, due to their much thicker wire radius, their overall size is comparable to the wavelength of light [17, 28].

This preliminary discussion clearly demonstrates that classical antenna theory, even when augmented by wavelength scaling arguments [29], can only serve as a qualitative guideline for the design of optical nano-antennas with complex shapes and strongly dispersive materials. As a result, we believe that there is presently a compelling need to embark in a systematic numerical study of the scaling properties of thin-wire and thick-wire Au helical nano-antennas with geometrical parameters comparable to the optical wavelength. In the next sections we will start this study by examining how variations in geometrical parameters of Au nano-helices affect their radiation properties.

2.1 Effect of R and P on the radiation properties of Au nano-helices

Unlike the broadband quasi-axial modes produced by thick-wire nano-helices, we discovered that the radiation characteristics of thin-wire nano-helices are more sensitive to the variation of their geometrical parameters. In this section we will investigate the effects of varying R and P in thin-wire helices (r = 10nm) with a fixed number of turns (N = 4). The additional effects of varying N and r will be separately addressed in the following two sections of the paper.

Figure 3 shows the calculated radiation diagrams of nine representative structures with different values of R and P, excited at 400nm (blue), 500nm (green), and 600nm (red). In each row of the figure, the value of R is kept constant while P increases from 100nm to 700nm. Similarly, along each column R varies from 100nm to 700nm, while P is kept constant. Our results show that, at each fixed incident wavelength, varying P (with a fixed R) dramatically affects the directional properties of the backward lobes. On the other hand, varying R at a fixed P has more influence on the characteristics of the forward lobes.

 

Fig. 3 The normalized 2D radiation patterns of Au nano-helices with fixed r = 10nm, N = 4, and a) R = P = 100nm; b) R = 100nm, P = 400nm; c) R = 100nm, P = 700nm; d) R = 400nm, P = 100nm; e) R = P = 400nm; f) R = 400nm, P = 700nm; g) R = 700nm, P = 100nm; h) R = 700nm, P = 400nm; i) R = P = 700nm. All the structures are excited with linearly polarized plane waves along the helical axis. The normalized radiation patterns are plotted for the equatorial plane with $\theta ={90}^{\circ }$. The three colors indicate different excitation wavelengths: 400nm (blue), 500nm (green), and 600nm (red).

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Based on our numerical study we also notice that variations of P on the length scale of the optical wavelength increase significantly the total number of side lobes, both in the forward and backward scattering direction. This is due to the fact that, as P increases, the helices better approximate the well-known behavior of linear wire antennas [16, 17]. In particular, in the case of Fig. 3(c), the small value of R makes the structure more closely resembling a linear wire antenna compared to the cases shown in Fig. 3(f) and 3(i), where the values of R are larger. As a result, in Fig. 3(c), the side lobes (the ones that are almost perpendicular to the helical axis) correspond to conical scattering lobes formed because of the increased linear-wire character of the helical antennas [30].

On the other hand, it is known that helical structures additionally possess a “loop antenna” contribution, which becomes more prominent for larger values of R. Larger R effectively provides better directivities and more forward-directed scattered radiation. In this case, the excited portions of the helical “loop” (parallel to the incident electric field) behave as two effective dipolar sources excited in phase, which explains the increase in the number of forward lobes in proportion to R [17].

Moreover, the radiation side lobes for longer wavelengths occur at larger angles demonstrating the relevance of radiative coupling (i.e., diffraction) in helical nano-antennas. This fact can be directly appreciated in Fig. 3(d), where the ratio $\sin \theta /{\lambda }_{exc}$remains constant, demonstrating the consistency with the Bragg’s diffraction law. In the next section we will discuss the role of the number of helical turns N on the far-field radiation properties. In addition, backward lobes at 180° appear in the radiation diagrams in the second column of Fig. 3 (Figs. 3(b), 3(e), and 3(h)), when excited at 400nm (blue curves). These backward lobes are produced when the pitch of the helix matches the wavelength of the incident light along the helical axis. As we will discuss further in the next section, such phase matching condition is essential to achieve circularly polarized scattered radiation.

2.2 Effect of N on the radiation properties of Au nano-helices

As we introduced in Section 2.1, intuitive insights on the characteristics of the radiation patterns of helical nano-antennas can be obtained by regarding them as linear arrays of radiative elements composed by single helical turns periodically distributed with pitch spacing P. Within the validity limits of the simple array model of helical antennas, the effect of N on the radiation diagrams can be qualitatively understood by multiplying the radiation diagram corresponding to a single helical turn (i.e., the element pattern) with a linear array factor of period P [1, 16, 17]. We recall that the array factor (for the electric field) that contributes to the far-field radiation pattern can be expressed as [17]:

While it is more complicated to determine the phase ψ in the case of helical nano-antennas excited by light and operating outside the axial modes, we can show the relevance of the array model using assumed values of ψ. For our particular choice of geometrical parameters, R = 300nm, P = λ_exc = 600nm, r = 10nm, we can calculate the array factors for each case, assuming that the forward lobe at 0° is produced because the field of each helical element are in phase (since P = λ_exc). This is the same mechanism for producing axial mode in RF helical antenna (ψ = −2π, φ = 0° and p = 1) [17]. The results of our numerical simulations, shown in Fig. 4, support the validity of the array model for diffractively-coupled helical nano-antennas in the optical regime. Figure 4 shows the rigorously calculated radiation patterns (blue lines) of representative helical nano-antenna structures of increasing number of helical turns N using the SIE method. The SIE results are compared with the approximated ones (red lines) obtained by multiplying the radiation pattern of the single-turn helix (Fig. 4(a)) with the array factors associated with each values of N, calculated using Eq. (5) and assuming ψ = −2π. The results in Fig. 4 show a remarkable agreement between the SIE simulations and the efficient estimations based on pattern multiplication [16, 17].

 

Fig. 4 The normalized 2D radiation patterns (blue solid curves) of R = 300nm, P = 600nm, r = 10nm, Au nano-helices, with a) N = 1, b) N = 2, c) N = 3, d) N = 4, e) N = 5, f) N = 6. All the structures are excited by plane wave along helical axis at λexc = 600nm. The radiation patterns are plotted for the equatorial plane with $\theta ={90}^{\circ }$. The red dashed curves are the theoretical predictions by antenna theory using pattern multiplication in each case.

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We notice that in Fig. 4 we chose P = λ_exc = 600nm, which makes the elements of the arrays (i.e., the helical turns) in phase with each other and determines the direction of the forward lobe (e.g. Figure 4(d)) to be at φ = 0°, in agreement with the numerical results (red curves in Fig. 4). On the other hand, the backward lobe (φ = 180°) has a very interesting origin. To better understand this point, we remind that the incident plane wave can be decomposed into to circularly polarized bases of opposite handedness and that only radiation with a matching polarization handedness with respect to the one of the helical structure can be efficiently transmitted [4, 5] Therefore, since the radiation handedness is interchanged upon reflection, only the right-handed component can be reflected by the left handed helix creating the backward lobe by constructive interference. Therefore, as observed in our simulations for helices with P = λ, the forward and backward scattering components of the radiation diagram originate from opposite polarization handedness. We conclude this section by noticing that our numerical data in Fig. 4 clearly show that a relatively small number of turns (N = 4) is already sufficient to capture the general characteristics of radiation patterns of Au helical nano-antennas. From a manufacturing standpoint, it is very relevant that, the general radiation properties of dispersive nano-helical antennas already develop in structures with as little as two helical turns.

2.3 Effect of r on the radiation properties of Au nano-helices

The previous discussion restricted the value of the Au wire radius to r = 10nm, which ensured a qualitative agreement with the simple design rules of antenna theory [16, 17, 28]. However, we will now discuss the case of helical nano-antennas of increasing r, and show that the wire radius plays a significant role in the far-field radiative properties of Au nano-helices at optical wavelengths, giving rise to new directional modes.

In the present analysis we restrict the range for the value of r to be between 10 nm and 100 nm, since such values are either studied by other groups or can be presently achieved using different nano-fabrication techniques [4–9, 31]. We found that increasing r directs progressively more energy into the forward scattering direction irrespective of the values of R and P (within the investigated diffractive range). Figure 5 shows calculated radiation patterns of representative structures of increasing r while keeping all other geometrical parameters constant.

 

Fig. 5 The normalized radiation patterns of Au nano-helices with fixed R = 300nm, P = 600nm, N = 4, a) r = 10nm, b) r = 50nm, c) r = 100nm. The structures are all excited by plane wave along helical axis at λexc = 600nm. The normalized radiation patterns are plotted for the equatorial plane with $\theta ={90}^{\circ }$.

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The general trend shown in Fig. 5 was observed in all investigated structures characterized by different values of N, R, and P. Clearly this scaling effect with respect to the wire radius is purely diffractive in nature and it is more easily observed in the optical regime (with or without material dispersion) compared to the RF regime [16, 17]. This is also the reason why we refer to the radiation patterns such as in Fig. 5(c) and Figs. 2(e)-2(h) as “quasi-axial modes”. In these cases, the radiation energy is mostly concentrated in the forward scattering direction, and the polarization is not necessarily circular. Consistently, we note that this phenomenon is not unique to dispersive metallic structures. In fact, we have also found that dielectric nano-helices (n = 3.5) show enhanced forward scattering radiation as their wire radius is increased. This effect results purely from diffraction in structures with r comparable to λ and with a sufficiently large volume fraction of scattering material, in analogy with the prevalently forward scattering nature of diffractive spherical particles described in the Mie theory [32]. In the next sections we will show that it is also possible to design Au helical nano-antennas that exhibit both directional radiation and ideal circular polarization, paving the way to novel potential applications to active nanostructures.

3. Polarization Control of Au Nano-helices

In this section we will study the effect of the geometrical parameters on the polarization states of nano-helices. As discussed in Section 2, quasi-axial modes with pronounced forward lobes are obtained for a large range of geometrical parameters, R and P, when the radius of the wire r is large (r = 100nm). However, we have found that the general polarization state of quasi-axial modes is elliptical, with different degrees of eccentricity. Similarly, the forward lobes of other cases of Au helical nano-antennas with thinner wires are all found to be elliptical to various degrees. As we mentioned previously, these behavior results from the losses associated to the dispersion of Au in the visible regime. To overcome this limitation, we consider thin-wire helical nano-antennas, and identify the special conditions that allow us to obtain perfect circular polarization scattering.

We focus now on the radiation patterns that are supported by structures with small r and a pitch equal to one wavelength, excited using linearly polarized plane waves directed along the helical axis. In this particular case, backward lobes with perfectly circularly polarized radiation can be obtained when P = λ, similar to the cases already shown in Figs. 4(b) and 4(f), and Figs. 5(a) and 5(b). Moreover, these structures feature radiation patterns with left-handed circularly polarized radiation in the backward lobe, while showing elliptically polarized forward lobes. This interesting phenomenon is analogous to the selective reflection of light by chiral liquid crystals, which occurs when the pitch of the chiral liquid crystal is equal to the wavelength of normally incident light [33, 34]. In Fig. 6(a) we show the calculated AR values for both the back and forward scattered lobes of this interesting class of nano-helices as a function of N for two different wire thicknesses of Au nano-helices. In particular, the AR values of backward lobes with r = 10nm (red) overlap with the corresponding cases with r = 50nm (pink), and are identically equal to one. We also plot in the figure the theoretical AR values calculated according to RF antenna theory for the axial-mode (black dotted line). These results demonstrate that, for thin-wire Au nano-helices with pitch equal to one wavelength, the AR of the backward lobe can be well-below the RF limit of axial modes supporting perfectly circular polarization states. As shown in Fig. 6(a), even with a thicker wire of r = 50nm, we still have a reflection of circularly polarized light in the backward direction. In particular, the N = 4 case now corresponds to the radiation pattern in Fig. 5(b). Therefore, significant reflection of circularly polarized light can be achieved even with a thicker Au wire diameter (2r = 100nm), which can easily be achieved using current fabrication techniques [4–9, 31].

 

Fig. 6 a) The axial ratio (AR) of the forward lobe (blue) and the backward lobe (red) by Au nano-helices with R = 300nm, P = 600nm, r = 10nm, and of the forward lobe (green) and the backward lobe (pink) by Au nano-helices with thicker wire of r = 50nm. The results are plotted against increasing N. The black dashed curve is the theoretical value of AR for axial mode helix, calculated as AR = (2N + 1)/(2N). b) The relative intensity of the backward lobe to the forward lobe as a function of wavelength. Fixed R = 300nm, r = 10nm and N = 4. The inset shows the wavelength and pitch at which the backward lobe with perfect circular polarization is obtained, in the case of Au nano-helices with fixed r = 10nm, R = 300nm, and N = 4.

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Figure 6(a) also shows that the AR of the forward lobe decreases with increasing number of turns, i.e., the polarization states in the forward direction are more circular. However, as mentioned, for helical nano-antennas with realistic Au dispersion data, the polarization states that can be achieved in the forward direction are always far from ideal circular polarization.

Furthermore, in Fig. 6(b) we show the bandwidth of the circularly-polarized backward lobe. We vary the wavelength of the incident light in the optical and near-infrared regime for Au nano-helices with three different pitches equal to the wavelength of blue (400nm), green (500nm) and red (600nm) light. The data clearly indicate that the circularly polarized backward lobe can only be obtained when the wavelength is close to one helical pitch, with a bandwidth (Full Width at Half Maximum, FWHM) that is roughly 50nm around the central wavelength. The calculated results agree well with the analogous case observed in chiral liquid crystals, where a single reflection bandgap is created when the wavelength equals one helical pitch [33]. The inset of Fig. 6(b) demonstrates that, for diffractively coupled Au nano-helices, the condition to obtain circularly polarized backward lobe ($P=\lambda$) can be extended from the visible to the near-infrared regime.

Finally, we notice that in the case of thin-wire nano-helices with $P=\lambda$ the polarization of the forward lobe has the same handedness as the nano-helix, in agreement with the mechanism already discussed in Section 2.2.

4. Conclusion

Based on the rigorous SIE method, we have conducted a systematic study of the radiative properties of diffractively-coupled Au nano-helices in the optical regime. When the wire radius is small, we have found a variety of highly directional radiation patterns depending on the ratio of P and R with respect to the incident wavelength. For this class of Au nano-helices, RF antenna theory provides a good guide to find the regions in the parameter space (Kraus’ diagram) for normal and axial modes. However, due to the dispersive nature of Au in the optical regime, axial modes no longer produce circularly polarized radiation. On the other hand, when$P=\lambda$, we have found radiation modes that support perfectly circularly polarized light. The radiated intensity in both forward and backward directions become comparable when the wire radius is thin. However, as the radius of the helical wire increase, radiation properties start to deviate from the RF antenna theory. Au nano-helices with R and P comparable to the incident wavelength are found to operate in the quasi-axial modes when their wire radius is large (r ~100nm). In quasi-axial modes, most of the radiated energy is directed forward along the helical axis, and the polarization of radiation in the forward direction are always elliptical with various degrees of eccentricity.

Based on these results, engineering nano-helical antennas in the visible regime with multidirectional operation can be realized with thin-wire (r = 10nm) nano-helices with R and P comparable to the wavelength. In this case, the radiation pattern is very sensitive to the values of geometrical parameters R and P, and the number of elliptically polarized forward lobes in the equatorial plane is proportional to R_λ. Besides, when P is equal to one wavelength, the backward lobe at 180° is perfectly circularly polarized with the same handedness as the nano-helix. These radiation properties can be achieved with as little as 2 to 6 number of helical turns. Also, we have found that we can achieve this behavior with a thicker wire of r up to around 50nm. Furthermore, we have found that the resonant condition $P=\lambda$for circularly polarized backward lobe is general and can be extended to near-IR regime.

These results are important as they provide novel opportunities for the design of nano-helical antennas that can be used to engineer sensors, filters and plasmonic components with unprecedented beam forming and polarization capabilities in the optical and near-infrared spectral range.