1. Introduction

The possibility of manipulating the scattering cross-section of an object has drawn increasing attention in recent years [1–11]. Critical to such unique and large-scale applications as invisibility cloaking [12, 13], the ability to tailor scattering characteristics is of equal importance in the case of nanoscale elements, with many potential applications, including nanoantenna devices, integrated optical circuits, sensors with nanoparticles, photovoltaic systems, or perfect absorbers. Control of the scattering response has been actively discussed in the context of optical antennas, demonstrating color switching operation with bi-metallic structures or several coupled plasmonic nanodisks [2, 14], forward unidirectional light generation with a variety of antenna designs [5–8, 11, 15–18], or the possibility of angle-sweeping operation [19, 20]. Of special interest here are those elements capable of exhibiting both electric polarization and magnetization in response to an applied field. The interference effects in the radiation emanating from induced electric and magnetic dipole oscillators superimposed in space bring an additional degree of freedom in tailoring the scattering cross-section; in particular, they allow for the predictable unidirectional scattering of an incident field if a comparable strength of magnetic and electric response can be achieved [21]. The latter, in turn, can lead to other interesting possibilities, such as manipulating the optical force by the interference effects in radiation scattered from the dipoles [22,23], tailoring coupling between single-photon emitters and receivers formed by magneto-electric nanoparticles [24], as well as numerous unique processes arising with nonlinear interactions [25].

Few natural materials, however, exhibit intrinsic magnetic dipole transitions [26], and in those that do, the response is significantly weaker than the one originating from their electric dipole counterparts. As an alternative, an effective resonant magnetic response can be achieved with nonmagnetic materials by properly structuring the natural media [27] and leveraging the geometrical shape-based effective magnetic dipoles of plasmonic or high-index dielectric inclusions [28]. While the idea of a unidirectional scattering owing to magneto-electric interference has been discussed in detail theoretically [21, 23, 24, 29–33], obtaining comparable levels of electric and magnetic dipole-type responses from a physical nanoelement within the same frequency range requires a careful design of its geometry. Unidirectional scattering based on magneto-electric interference utilizing a particular nanoelement geometry has been demonstrated or predicted with spherical-based particle designs and their arrangements [1, 34–37]. In particular, a pronounced forward unidirectional scattering has been predicted utilizing high-permittivity and nano-shell particles designs [1, 34], while both forward and backward unidirectional operations have been demonstrated experimentally for silicon nanoparticles [37], which have been also previously predicted for silicon and germanium spherical-shape geometries [35, 36]. Such availability of backward unidirectional scattering exploiting the resonant nature of magnetic response [21, 32, 33], and thus an opportunity of color-switching between forward and backward directions, would definitely be advantageous in all of the above applications. On the other hand, the fact that a variety of nanoelements possess an effective magnetic resonant mode within some frequency range can make magneto-electric interference a method of choice in achieving such a switching operation if a proper design procedure is developed. While analytical solutions are available for spherical-based shapes, analysis and prediction of such a response from other, especially non axisymmetric, geometries would however require the use of other methods, such as the discrete dipole approximation [38, 39] or full-scale numerical simulations.

Here, we perform a full-scale numerical analysis to show the possibility of achieving such a switching operation by utilizing an off-resonant electric dipole response of plasmonic dimer nanoelements, matched in strength with their geometric shape-leveraged magnetic dipole resonance. We consider a geometry of plasmonic cut-wire pair dimers–nanoelements consisting of two metal strips separated by a thin layer of a dielectric. The geometry is known for exhibiting a magnetic-type resonance in the visible range [40–42] and has been widely explored as a building block in negative index media designs [43–53]. The latter designs do not require, however, matching the strength of an electric and a magnetic-type response. Additionally, in negative-index medium designs, the two types of resonance do not necessarily originate from the same element, with the electric-type response often coming from coupling between the elements in a periodic arrangement or from additional metallic wires introduced in the geometry [43–48, 50–52]. In the present work, no goal of obtaining negative values of the effective permeability and/or permittivity from periodically arranged dimers is pursued. Rather, we focus on the possibility of obtaining resonant magnetic and off-resonant electric responses of equal strength from an isolated dimer element, considering several dimer orientations to the incident field. Owing to the non-resonant nature of the electric response, the relative signs of magnetic and electric polarizabilities of the nanoelement are expected to change over the frequency span of the magnetic resonance, while their absolute values can stay relatively close with a proper design. Since the directionality of scattering is defined by the relative phases of the induced magnetic and electric dipoles, which in turn can be traced to the sign of the real part of the product of electric and magnetic polarizabilities of the element [21,23], the scattering direction is expected to change over the frequency span of the resonance. For example, given a positive value of the non-resonant electric polarizability, the change from forward to backward direction is observed when moving the excitation from frequencies below to those above the resonance. Note that, while the magnetic polarizability of the element changes sign around the resonance, no negative values of the effective permeability or permittivity are necessarily achieved in the case of a periodic arrangement of the elements designed in the above manner. The unidirectional scattering properties of a periodic arrangement will, however, remain similar to that of a single element, or can be improved with a proper design.

Relevant to the discussion in the previous paragraphs, we would like to emphasize that, while there is no fundamental limitation on the strength of unidirectionality in the case of forward scattering, there is a limitation preventing an ideal backward unidirectional scattering situation [32, 33]. In particular, according to the optical theorem [28], the extinction cross section (sum of absorption and scattering cross sections) is a function of the amplitude of light scattered precisely in the forward direction. Consequently, from power conservation considerations, approaching the limit of zero scattering in the forward direction (a condition required for an ideal backward unidirectional scattering case) necessarily assumes also approaching a zero overall scattering. This limitation, however, does not prevent the possibility of minimizing the forward-scattered field by canceling the in-phase contribution to the scattering in the forward direction (the latter contribution does not participate in power extraction from the incident wave), while maintaining a small but non-zero amplitude of the wave scattered forward in quadrature with the incident field [32]. In this case, for a considerably sub-wavelength particle, the amplitude of light scattered in all other directions can be made significantly larger than that in the forward direction, with no violation to the optical theorem. We are interested in the latter situation when considering a backscattering response in the present study. We will return to this discussion when presenting the simulated scattering results.

The paper is organized as follows. In the next section, we consider a dimer element oriented along the propagation direction of the incident field. We associate dimer resonances corresponding to the odd and even number of nodes of the magnetic field within the spacer layer with the predominantly magnetic or electric type response. We then identify the frequency ranges around the magnetic resonance, which allow for a unidirectional forward and backward scattering response, and discuss the limitations of such a geometry in parametric optimization. Next, we consider the dimer orientation orthogonal to the propagation direction in Section 3. Although even-order resonances, associated with the predominant electric-type response, are not excited with such an orientation, we demonstrate that a unidirectional forward and backward scattering is still achievable by exploiting other types of electric responses. Magnetic and electric resonances are relatively decoupled with this orientation, thus allowing for greater flexibility in parameters optimization. We then briefly consider periodic arrangements of dimer elements, showing the possibility of further improvement to unidirectionality. We also discuss the specific case of a perforated metal film, achieved by gradually bringing closer together dimer elements that are oriented along the propagation direction and spaced periodically within a single layer in the transverse direction. We discuss the differences between the forward-scattering response owing to an extraordinary optical transmission (EOT) achieved with such a film, and a unidirectional scattering from an isolated dimer element or from its dilute periodic arrangement.

2. Coupled wires parallel to the propagation direction

The geometry of a dimer oriented along the k-vector of the incident field is shown in Fig. 1(a). In this configuration, the main resonances of the nanoelement can be qualitatively understood as a standing wave pattern of the metal-dielectric-metal (MDM) nano-waveguide formed within the dielectric spacer and truncated on the sides by the dimer facets. We choose the initial spacer thickness t = 7 nm in order to provide a relatively high effective index for the formed nano-waveguide; this allows for a sub-wavelength size of the element while keeping the loss relatively low and ensuring that the propagation length of the fundamental MDM waveguide mode exceeds the dimer length [54]. The spectral dependence of the peak enhancement of the norm of the magnetic field within the dimer volume for the parameters indicated in Fig. 1(a), and field distribution of the eigenmodes within the dielectric spacer corresponding to the first two main resonances at 214 THz and 388 THz, are shown in Figs. 1(b) and 1(c). Here and further in the study, frequency-domain simulations employing the finite elements method incorporated within COMSOL Multiphysics commercial package (http://www.comsol.com) are used for the analysis. Air is assumed for the surrounding medium throughout the study.

 

Fig. 1 (a) The geometry of a dimer parallel to field propagation direction. The used parameters are L = w = 120 nm, t = 15 nm, d = 7 nm, the spacer refractive index n = 1.58. Silver dielectric function follows the data in [55]. (b) Spectral dependence of field enhancement within the dimer. Color insets: magnetic field distribution within the xz cross-section; Arrows: the direction of the total electric field within the same cross-section. (c) Top: Magnetic field distribution of the eigenmodes within the spacer layer for the first two resonances. Bottom: E_x component of the scattered field for the same resonances, showing the in-phase and out-of-phase field oscillations at the two sides of the dimer; xz cross-section shown; the dimer is located in the center of each color image, indicated by the arrow.

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As expected, the consecutive resonances are characterized by the increasing number of nodes of the magnetic field within the dimer length, roughly corresponding to the mλ/2 optical path length within the nano-waveguide along the propagation direction (m = 1, 2 for the two peaks mentioned above). We note though that this picture only gives a qualitative understanding, as the exact resonances spectral positions strongly depend on other parameters; in particular they shift to noticeably lower frequencies when the thickness t of the strips is reduced to comparable or below the skin depth values.

The dominant magnetic nature of the lowest order resonance [40, 42] can be intuitively traced to the direction of the x-component of the electric field at the opposite facets of the dimer (sides A and B in the Fig. 1(b) inset) – the major field component that couples to the far-field. The direction is opposite in the case of odd-order resonances; thus, it can be expected that, in response to the excitation, the near field within the dimer will couple to the oscillating out-of phase propagating waves of the (scattered) field radiated from the two sides of the dimer. Such electric field pattern is a characteristic response produced by a magnetic dipole, and is seen pronounced in Fig. 1(c) for the 214 THz resonance. The opposite situation holds for even order resonances (388 THz in Fig. 1), where the x-component of a local field is co-directional at the opposite facets, and hence is expected to lead to the dominant effective electric-type response, with in-phase oscillating waves scattered from the opposite dimer sides; the expected pattern is indeed seen in Fig. 1(c) for the second resonance. As such, if a comparable relative strength of the two types of the resonance can be achieved for some spectral position in between the two resonances where both types of the response are pronounced, a magneto-electric interference might be expected at this spectral position. The latter leads to an interferometric cancelation of the scattered radiation on one side of the dimer and an interferometric doubling on the other side, effectively creating a unidirectional scattering pattern.

The strength of the magnetic response depends on the area encompassed by the circulating displacement field pattern. As such, assuming the length L of the nano-element is fixed by the choice of the resonant wavelength, the thickness t of the metal parts is chosen in order to ensure a sufficient volume to fit most of the magnetic resonant field penetrating the metal. On the other hand, as mentioned before, for thicknesses below the skin depth value, increasing size t shifts the resonances to higher frequencies, thus increasing the fractional size of the dimer compared to the wavelength. The 15 nm height used in the example gives some trade-off between keeping the dimer size within the λ/10 range while ensuring the noticeable strength of the resonance. While the scattering strength can additionally be regulated by the dimer size along the y direction, at normal incidence, this dimension does not significantly affect the directionality of the scattering response into the xz plane–the plane of interest in the case of a unidirectional scattering along the propagation direction, which we assume to be along z. In the presented examples, we fix this dimension to that along z (L = w). Note that, assuming the magnetic and the electric types of the response are produced by the two consecutive resonances as discussed above, the choice of the parameters for one resonance type for the most part fixes the strength of the other one. In that sense horizontal dimer orientation does not allow for much flexibility in designing the relative strength of magnetic and electric responses, as discussed in more detail later.

To verify the possibility of forward/backward unidirectional scattering with the above configuration, we perform the effective parameters retrieval for a single layer of dimers arranged periodically in the direction transverse to the propagation, as shown in Fig. 2(a). We employ a subwavelength period to avoid grating effects, while ensuring the period is large enough (500 nm) to avoid a significant coupling between the neighboring elements. With such a dilute arrangement, the effective electric and magnetic polarizabilities averaged over the volume of one unit cell can be approximated by the effective susceptibilities of the layer of the elements [56] which, in turn, are obtained by a transfer matrix retrieval [57]. The resulting parameters are shown in Fig. 2(b). As seen from the figure, the electric and magnetic susceptibilities do have close absolute values in the circled regions below and above the 214 THz resonant point, with the opposite signs of the effective susceptibilities occurring above the resonance. The directionality of the response arising from magneto-electric interference is related to the sign of the real part of the product of the effective electric (α^e) and magnetic (α^m) polarizabilities of the dimer elements as R ≡ Re(α^e α^m*) [23], with the positive sign leading to forward scattering. (In the above expression, the asterisk denotes complex conjugate.) A unidirectional scattering is therefore expected within the circled regions, with the direction switching from forward to backward once passing the resonance as the excitation moves towards higher frequencies. Note that, while there is also a region above the 388 THz resonance where electric and magnetic susceptibilities come close, the size of the dimer gets closer to that of the wavelength in this frequency range and higher-order multipoles acquire a more pronounced relative response. Because of the additional complexity of the system, we do not discuss it further in this study.

 

Fig. 2 Effective susceptibilities retrieved for a dilute layer of dimer elements. (a) The geometry used in the retrieval. (b) Retrieved effective electric (χ^e) and magnetic (χ^m) susceptibilities.

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The scattered field in the xz plane produced by an isolated dimer element when illuminated by a plane wave normally incident on the dimer and polarized along x-axis is shown in Fig. 3. As expected, the unidirectional response is observed at the circled spectral regions, with the scattering direction changing from forward to backward around the resonance point at 214 THz. The spectral dependence of the ratio of forward to backward scattered field intensity (D ≡ 10log₁₀ |E_far (θ = 0°)|² / |E_far (θ = 180°)|², with E_far(θ) being the amplitude of the far field scattered in the θ direction and θ = 0° indicating forward scattering) is given in Fig. 3(c), showing up to about ±12 dB unidirectionality strength.

 

Fig. 3 (a) Orientation of the induced electric and magnetic dipoles assumed in the theory derivation. (b) Top: Near field scattered into the xz plane by a single isolated dimer element with the geometry parameters used in Figs. 1 and 2. The excitations frequencies are taken below (209 THz), at (214 THz), and above (219 THz) the resonance. Bottom: Far field intensity scattered into the xz plane for the same excitation positions. Blue, Red, and Green curves respectively correspond, to the |E_x|², |E_z|² scattered far field components, and total scattered far field (E_y-component is negligible in the xz scattering plane). (c) Ratio D of forward to backward scattered field intensity for a dimer with the same geometry.

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The dilute, subwavelength arrangement employed for the retrieval in Fig. 2 largely reduces the impact of a collective response on the resonance spectral position, and, as a result, the above retrieval procedure can be useful in estimating the feasibility of a unidirectional scattering for a given geometry of a nanoelement. Also, while applied to the case of nano-strip dimers, it can equally be used for nanoelements of other shapes, as long as a geometric shape-based magnetic-dipole-type response is exhibited by the element. The prediction based on the effective parameters is still, however, related to a layer of periodically arranged elements, with the retrieved effective polarizabilities being averaged over the volume of each unit cell, while we are aiming at the analysis of the response produced by a single isolated element.

As seen in Fig. 3(b), at the resonance (214 THz), both E_x and E_z components of the electric scattered far field show clearly pronounced features of dipole-level contributions. This suggests that an insight into the conditions leading to a unidirectional scattering from an isolated dimer can be achieved by modeling the nanoelement as a linear combination of oscillating electric and magnetic dipole sources induced in a dimer by the incident field. Relating the radiation pattern scattered from such a set of dipoles to a numerically determined scattered far field allows for a multipole analysis of the scattered far field assuming for the latter an expansion up to the dipole terms. While solving an inverse scattering problem for a quantitative retrieval of the components of the effective polarizability of a single dimer would be beyond the scope of this work, below we explain the considerations elucidating the conditions that would lead to a unidirectional response.

Assuming an x-polarized incident field and performing the analysis of the angular dependence of the radiation scattered in the three orthogonal (xy, yz, and xz) planes, we identify the components of the far-field scattering response to be originating from the x-oriented electric and y-oriented magnetic dipoles induced within the dimer. The presence of scattering from the dipoles induced in other orientations, although allowed in the analysis (to be presented in more detail elsewhere), is found to be negligible in the far-field pattern (two or more orders of magnitude weaker), with either the horizontal or orthogonal orientation of the dimer to the incident field. The resulting expression for, e.g., the E_θ far field component of the electric field scattered in the xz (φ = 0) plane is then as follows (see Appendix for derivation), with angles θ and φ in standard spherical coordinates:

The expression in Eq. (1) converges to that of a field scattered by an isotropic small magneto-electric particle excited by the x-polarized incident plane wave [28], with the induced dipole moments related to the polarizability tensors of the dimer as $\mathbf{p}={\epsilon }_0\overline{\overline{{\alpha }^e}}{\mathbf{E}}_0$ and $\mathbf{m}=\overline{\overline{{\alpha }^m}}{\mathbf{H}}_0$. In the latter expressions, E₀ = (E_x0, 0, 0)^T and H₀ = (0, E_x0/η, 0)^T are the vector amplitudes of the incident field, while p = (p_x, 0, 0)^T and m = (0, m_y, 0)^T denote vectors formed by the components of the induced electric and magnetic dipole moments, respectively.

In the case of an equal strength of x-oriented electric and y-oriented magnetic ( $\frac{1}{{\epsilon }_0}{p}_x=\eta m_y\equiv \kappa$) dipole responses, forward radiation (θ = 0) is doubled ( $E_{\theta }=2\kappa \frac{1}{4\pi }{\left(\frac{\omega }{c}\right)}^2\frac{e^{ikr}}{r}$), while back-ward scattering (θ = π) is suppressed (E_θ = 0) with the same sign of the induced magnetic and electric dipoles (in-phase dipole oscillations). Suppression in the forward direction is expected for the out-of-phase dipole oscillations (opposite dipole signs), in agreement with the previous qualitative discussion based on the relative phase of the scattered electric and magnetic fields.

As seen from Fig. 3(c), while unidirectionality is strongly pronounced, the suppression of the scattering on one of the sides is not perfect in either the forward or backward case, which might be expected due to a non-ideal matching of the strength of magnetic and electric responses seen in Fig. 2. The directionality can be further improved by regulating the strength of the magnetic response relative to that of the electric one, e.g., by increasing the spacer thickness d. An example of such an optimization is shown in Fig. 4. For the same dimer parameters, a strong unidirectionality of about +20 and −18 dB for the forward and backward scattering directions, respectively, is reached with the spacer thickness of about 15 nm. Further increase of spacer thickness reduces the magnetic resonance strength leading, eventually to a split of the effective MDM nano-waveguide mode into two separate plasmon modes propagating along each strip [58]. A simultaneous reduction of the directionality strength follows. The reduction is however rather gradual, allowing for some flexibility in the parameters choice. A faster reduction in the backward scattering strength can be understood by the increased mismatch in the imaginary parts of the electric and magnetic polarizabilities. The signs of these imaginary parts remain the same (positive) for lossy elements, preventing a complete matching of the two types of the polarizabilities (with the opposite signs) required in the backward scattering case, in agreement with the discussion of the general limitations brought by the optical theorem [32,33], as it was outlined in the Introduction.

 

Fig. 4 Improving directionality by regulating the strength of a magnetic resonance. (a) Effective parameters of a layer of horizontally oriented dimers, for several thicknesses of the spacer layer. (b) Spectral dependence of the ratio D of forward to backward scattering from a single isolated dimer with the same parameters.

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The particular optimum value of the spacer thickness depends on the employed thickness t of the metal strips, as well as on the operational frequency range, materials choice, and other parameters, and can be regulated by the design. Interestingly however, although a layer of elements was used in the retrieval in Figs. 2 and 4(a), versus a scattering from a single isolated dimer shown in Figs. 3 and 4(b), the spectral positions of extremes in the directionality ratio in Fig. 4(b) match almost precisely the spectral positions where the effective electric and magnetic susceptibilities values match, either with the same or with the opposite signs, for all spacer values in Fig. 4; once again suggesting the above procedure provides a good quantitative prediction of the unidirectional scattering response.

The present configuration, however, with both types of the responses linked to Fabry-Perot resonances of the same cavity, does not leave much room for engineering of the relative resonance strength: desirable modifications of one of the resonances (e.g., changing strip separation d performed above, or changing the dimer length L) also changes the properties of the other one. A slight increase in the effective electric susceptibility values, simultaneously with the increase in strength of the magnetic response, can be seen, for example, in Fig. 4a for different spacer values. While the overlap of the electric and magnetic susceptibilities was noticeably improved in the above example, we find that the use of more lossy materials such as gold, or the use of higher-index spacers (which increases both the real and the imaginary parts of the effective refractive index for the MDM waveguide mode) usually prevents a significant improvement of the electric and magnetic parameters overlap with this orientation of a dimer, leaving the directionality strength in the range of ±10 dB. Increasing height t of the metal parts, on the other hand, in addition to shifting both resonant frequencies, adds a non-resonant part to the electric-type response, further enlarging the separation between electric and magnetic susceptibilities. Additionally, the use of horizontal orientation presents difficulties for the experimental implementation. We show in the next section that one can overcome the above limitations while preserving the forward/backward switching capability with orthogonal orientation of dimers to the incident field and assuming some additional geometric modifications.

3. Coupled wires orthogonal to propagation direction

The spectrum of the resonant response of an orthogonally oriented dimer having otherwise the same geometry as in Figs. 1 and 3 is shown in Fig. 5. The even-order (in the number of nodes of magnetic field) resonances are not excited with such a configuration: e.g., the 388 THz resonance is missing in the produced response in Fig. 5(b). As a result, no significant one-side suppression of the scattering radiation is produced with the same (orthogonally oriented) geometry parameters, as seen in Fig. 5(c), which shows only up to a 6 dB unidirectionality ratio strength. A different electric-type response, the one corresponding to a resonant frequency of a single metal stripe is, however, pronounced with this orientation, as seen in the inset in Fig. 5(b) for the resonance around 515 THz. This resonance is defined by the length of the metal stripe and does not depend on the separation distance between the two stripes. Hence, it is relatively decoupled from the Fabry-Perot-type resonances of the effective MDM waveguide mode, which, in addition to the the stripe length, have a strong dependence on the spacer thickness. The configuration thus allows for a rather independent adjustment of the strength of the two types of the response. For example, regulating the strength of a magnetic response by varying the spacer thickness d and stripes thickness t (the latter affecting the area of the effective magnetic dipole loop current) leaves now the position and the strength of the electric resonance defined by the length L unaffected.

 

Fig. 5 (a) Dimer orientation with respect to the incident field. (b) Solid curve: spectral response of a dimer with the same geometry as in Figs. 1 and 3 but oriented orthogonally to propagation direction, as shown in Fig. 5(a). The dashed curve shows the response from Fig. 1(b) (horizontal orientation). (c) Scattering directionality ratio D for this geometry.

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An example of such an adjustment leading to an increase of the unidirectionality ratio to a large value of above 25 dB for forward scattering while also improving backward scattering strength to around −17 dB, is shown in Fig. 6. A structure with an increased strip thickness (t = 30 nm) has been used in the study. As seen from Fig. 6a, the change in spacer thickness d from 7 nm to 11 nm leads to an increase in strength of the effective magnetic resonance and leaves the effective electric susceptibility practically unaffected; therefore, this gradually increases the overlap of the strength of the two types of the response. The retrieval was performed for a layer of dimer elements similar to what was done in the Fig. 2 calculation. The corresponding ratio of the scattering strength produced by an isolated dimer element with the same parameters is shown in Fig. 6(b), demonstrating the relevant gradual increase of the unidirectionality of the response, which reaches a compelling value of 25 dB for the forward direction. The unidirectionality remains strong with some further increase of the spacer thickness (e.g., the +25 dB/−15 dB values persist with a 15 nm separation), and is eventually reduced due to the reduction of the magnetic resonance strength. As in the case with horizontal orientation, a noticeably faster growth of the forward-scattering directionality compared to the backward one observed in Fig. 6(b) can be understood by an incomplete matching of the two types of the polarizabilities (with the opposite signs) required in the backward scattering case, due to the fact that the signs of the imaginary parts of both electric and magnetic polarizabilities remain positive for lossy elements. As also seen from the figure, the extremes of the directionality ratio again match almost precisely the spectral position where the electric and magnetic susceptibilities values match. An example of the scattered far field distribution for a structure with d = 11 nm near the peaks of the unidirectionality ratio is shown in Fig. 6(c). We observed similar strong ratios with even reduced strip lengths of about 90 nm, as well as with other parameters variations, exploiting a better geometry flexibility for matching the strengths of the resonances with this orientation.

 

Fig. 6 (a) Effective parameters retrieved for a layer of (sparsely spaced) orthogonally oriented dimer elements with t = 30 nm, L = w = 120 nm, and d = 7 nm, 9 nm, and 11 nm; (b) Ratio D of forward to backward scattered field intensity produced by a single isolated orthogonally oriented dimer with the same geometries as in (a). (c) Top: Scattered far field intensity projected on the surface of a sphere for an isolated dimer with d = 11nm, for the spectral positions below (292 THz) and above (307 THz) the resonance. Bottom: Angular distribution of the scattered into the xz plane electric far field for the same spectral points. Color indicates field components similarly as in Fig. 3(b), bottom.

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The presented response, while allowing for a good suppression of scattering in either the forward or backward directions, encompasses, however, some range of scattering angles in addition to these directions. The directionality, as well as the strength of the response, can be significantly improved with periodic dimer arrangements, as we discuss next. Additionally, in the case of horizontally oriented dimers arranged periodically within a single layer transverse to the propagation direction (see Fig. 2(a)), the geometry highly resembles that of a perforated metal film. This geometry converts precisely to the periodically perforated film case once the spacing between the dimers is reduced to match the size d of the slit. Such a film has been regularly associated with the extra-ordinary transmission (EOT) effect [59,60], which similarly assumes a suppression of backward-scattering radiation. It would be of interest, therefore, to analyze the relation between the above unidirectional forward scattering produced by a single isolated dimer and the EOT achieved with a perforated film. We consider these questions in the next section.

4. Periodic dimer arrangements

Suppression of radiation in the direction transverse to propagation can easily be achieved by utilizing a periodic dimer arrangement, with dimers spaced with a subwavelength period within a single layer oriented transversely to the propagation direction. In the case of an orthogonal (as in Fig. 5(a)) orientation of a dimer, however, such periodic arrangement regularly leads to a coupling between neighboring dimers within the layer, noticeably modifying the total response. Moving dimers further apart would, on the other hand, lead to a pronunciation of Woods anomalies [61, 62], which again would undesirably modify the original strength of the electric and magnetic responses. Coupling between the elements is significantly less pronounced with horizontal dimer orientation which, however, has limitations on an independent adjustment of the strength of the two types of responses, as discussed in Section 1. The considerable (120 nm) subwavelength size achieved for the nanoelements in the previous section is thus of most importance in relation to a periodic arrangement, allowing for a subwavelength periodicity in the transverse direction with no significant coupling between the elements within the layer even in the case of an orthogonal dimer orientation. A noticeable improvement of directionality and of the strength of the response with such a periodic arrangement is shown in Fig. 7. It is of interest that a considerable improvement is achieved with just a small number (four) of elements, keeping the total size of the structure significantly on the nanoscale.

 

Fig. 7 Norm of the far scattered field with the increasing number of elements arranged periodically in the direction transverse to propagation, as shown in the scheme on the left. (a) Normalized by the response of 4 elements. (b) Normalized by the peak value in each case, showing improved directionality with the same periodic arrangements.

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We note that, as seen in Figs. 1 and 5, the presented structures provide more than an order of magnitude enhancement of the incident field amplitude (two orders enhancement in intensity). Such enhanced interaction generally leads to increased absolute values of scattering cross sections, compared to those from non-resonant geometries, or structures exhibiting a weaker resonance. The drawback of this is that the absorption is also enhanced, i.e. the enhancement occurs for the total extinction cross section (scattering + absorption). Including realistic materials losses in the analysis, as has been done in the present study, is therefore of major importance.

We consider next a periodic arrangement of dimers parallel to the k-vector of the incident field orientation. As discussed in the previous section, such a configuration is strongly related to the perforated metal film geometry when the structural period is significantly sub-wavelength. The latter geometry is regularly associated with the EOT effect which, in turn, implies a suppression of scattering on the reflected side. We present below some analyses of the relation between the unidirectional scattering produced by isolated horizontally oriented dimer elements and the EOT effect observed in a perforated film case.

In Fig. 8, we consider the change in the spectral response of a layer of horizontally oriented dimers while reducing the spacing between the elements. The geometry is indicated in Fig. 8(a). To reduce computational time, here we consider dimers having the xz cross-section close to that in Section 1 but infinitely extended along the y direction, allowing the use of 2D geometry in numerical simulations. With the normal incidence, the directionality of scattering into the xz plane is not significantly affected while switching between the 2D and 3D dimer configurations in the above way. We assume air (n = 1) for the material of the spacer, to avoid double-periodicity when reducing the spacing between the nanoelements; moreover, we slightly modify the cross-section parameters (L = 275 nm, d = 10 nm) compared to that in Section 1 to optimize the scattering unidirectionality for the air-filled isolated dimer element. Also, gold [55] is used for metal in this example. The achieved maximum forward to backward scattering ratio into the xz plane for an isolated (infinitely extended in the ”y” direction) dimer with such parameters is about +9.5 dB and −7 dB for the positions below (190 THz) and above (210 THz) the magnetic resonance, respectively. We start from a dilute arrangement with the elements separated by a = 565 nm. We then bring dimers closer together until the spacing between the neighboring elements is reduced to the size of the slit between the strips (a = d = 10 nm curve). The latter case corresponds precisely to a perforated gold film forming a one-dimensional metallic grating with the structural period p equal to p = a + t = d + t.

 

Fig. 8 Analysis of EOT versus the unidirectional scattering produced by horizontally oriented dimer elements. (a) The geometry under consideration. The spacing between the neighboring elements is decreased until the dimers form a single-periodic perforated film. (b) Spectral response while decreasing the separation between the elements. (c) Transmission curve with the largest and the smallest separations.

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The corresponding change in the transmission properties of the layer as the dimer spacing is reduced is shown in Figs. 8(b) and 8(c). As seen from Fig. 8(b), reducing the periodicity leads to a gradual appearance of new resonances around 267 THz and 475 THz, the spectral positions in between the original Fabry-Perot resonances of an isolated dimer element (200 THz and 364 THz). The latter resonances, on the contrary, get gradually extinguished, reflecting a gradual change in the eigenmode structure for the modified geometry. Note that precisely new resonances appear, rather than the old resonances getting shifted to new spectral positions.

As seen in Fig. 8(c), the newly appeared resonances correspond to the enhanced transmission of the formed layer. We note here that while the term resonance is used to indicate field enhancement peaks in Fig. 8(b), it has in fact a different meaning in the cases of isolated dimers (e.g., a = 500 nm) and the perforated film (a = 10 nm). While corresponding to a regular absorptive resonance leading to transmission minimums in the case of isolated dimers, as seen in Fig. 8(c) for the a = 500 nm curve, an enhanced transmission is observed at the “resonance” locations (267 THz and 475 THz) in the case of a perforated film (a = 10 nm curve in Fig. 8(c)). The qualitative difference between the two types of “resonances” can also be noted from the retrieved effective refractive index curves for the above two arrangements, as shown in Fig. 9(a): a resonance in the effective refractive index is observed for the layer with a dilute dimer arrangement (a = 500 nm, 140 nm) at the spectral location of the first resonance (200THz) of an isolated dimer. This resonance is gradually extinguished as the dimers are brought closer, leading to a non-resonant spectral dependence of the effective index once the dimers form a perforated film (a = 10 nm). Note also that high transmission values seen in Fig. 8(c) for the a = 500 nm curve in the region of EOT resonances (in between the dimer resonances at 200 THz and 364THz) are merely due to a dilute arrangement of dimer elements with such a large period.

 

Fig. 9 (a) Effective refractive index of a layer formed by dimers spaced periodically in the transverse direction (see Fig. 8(a)), for several spacing values. (b) Evolution of the unidirectionality ratio D while reducing the spacing a between the dimers in a layer.

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As seen from Fig. 9(a), the non-resonant effective refractive index of the effective layer formed by the perforated film is n_eff = 2.04 and n_eff = 2.28 at the EOT frequencies of 267 THz and 475 THz, respectively. With the employed length of the gold strips of L = 275 nm, the corresponding optical thicknesses of the layer are 561 nm and 627 nm, thus matching almost precisely the half and the full wavelength of these newly emerging resonances (561 nm and 631 nm for the 267 THz and 475 THz resonances, respectively). We found a similar relation with all considered geometries of horizontally-oriented dimers arranged periodically in the transverse to propagation direction. The EOT effect observed in Fig. 8(c) at these spectral locations can thus be understood as a response of an effective half-wave layer formed by the perforated film.

The transformation of the directionality ratio while reducing the spacing a between the elements is shown in Fig. 9(b). As seen, once the spacing is reduced to relatively small values of below 50 nm, the unidirectional response around the 200 THz resonance gradually disappears at the same rate as the reduction of field enhancement at this resonant frequency seen in Fig. 8(b). A new, strongly pronounced forward-scattering unidirectional response does, on the contrary, appear at the spectral locations corresponding to the EOT resonances in Figs. 8(b) and 8(c), which gradually move from 340 THz with a = 24 nm to about 285 THz with a = 14 nm and further down to 267 THz with a = 10 nm. Note that the latter forward-scattering resonances are equally present with larger values of the period (a > 24 nm), for which the magnetic resonance and the unidirectional forward/backward scattering response of a single dimer around 200 THz is still simultaneously pronounced; for a dilute dimer arrangement they are located, however, at much higher frequencies, beyond the range shown in Figs. 8 and 9. This forward-scattering peak then gradually moves toward lower frequency spectral positions once dimers are brought closer together forming an optically denser effective layer with a larger optical path length within the layer thickness, as seen in Fig. 9(b). For all spacings, the spectral position of this forward-scattering EOT peak is such that the optical thickness of the formed effective layer amounts to a half-wavelength at the spectral location of the peak. Note also that, as seen in Fig. 9(b), no backward scattering occurs next to the newly appeared EOT unidirectionality peak. This relationship of the unidirectional scattering at EOT peaks to the integer number of half-wavelengths within the optical path length is thus in qualitative contrast with the magneto-electric interference-based forward/backward unidirectional response of an isolated dimer, or of a dilute periodic arrangement of the dimer elements discussed in the previous two sections.

Note that, due to the fact that perforation with a deeply sub-wavelength period is considered in our study, the effective refractive index is not very high (n ≈ 2, as seen in Fig. 9(a)). No overall high reflection is therefore observed in the frequency regions away from the λ/2 transmission peaks, leading to all-positive values of the directionality ratio in these regions in Fig. 9(b). The situation might be different for structures with a comparable or larger than the wavelength perforation period. Such structures however have not been considered here.

5. Conclusions

In conclusion, we have numerically demonstrated the transition as excitation frequency increases between forward and backward unidirectional scattering in plasmonic structures consisting of two metal stripes separated by a thin layer of a dielectric. We attributed the response to the presence of a dipolar magneto-electric interference between the resonant magnetic and non-resonant electric-type response of an isolated dimer. We have shown that a simple retrieval procedure performed on a dilute layer of nanoelements arranged in order to minimize the collective response provides a simple method to predict the feasibility of a unidirectional, magneto-electric interference based, scattering from each nanoelement. Forward and backward unidirectional scattering can be obtained from such an isolated nanoelement oriented either along or orthogonally to the k-vector of the incident field. The latter orientation allows however for a greater flexibility in the parameters optimization due to a relative decoupling of magnetic and electric resonances contributing to the unidirectional response. We presented numerical examples of obtaining such a response from an isolated dimer with a subwavelength size of 120 nm, indicating the possibility of a strong directionality ratio of up to 26 dB for the forward scattering and of 17 dB for the backward scattering with an orthogonally oriented dimer. The pronounced unidirectionality of 20 dB in forward and 18 dB in backward directions has been also shown with a horizontal dimer orientation.

We next demonstrated that the unidirectional response can be further enhanced with periodic arrangements, with even just several elements arranged periodically in the direction transverse to propagation, yielding a significant improvement in the directionality.

In discussing periodic arrangements, we considered the specific case of a perforated gold film forming a one-dimensional metallic grating with a sub-wavelength structural period, obtained by gradually bringing closer together the periodically arranged dimers. In the latter case the dimers are assumed infinitely extended in the direction normal to the scattering plane of interest. We have shown that, while the EOT effect obtained with such a geometry equivalently leads to a forward unidirectional scattering response, the mechanism for the response is different than in the case of an isolated dimer or of its dilute periodic arrangement. In particular, Fabry-Perot transmission resonances through an effective layer with an optical thickness equal to an integer number of half-wavelengths at the EOT frequencies is responsible for the EOT achieved with the perforated film. Magneto-electric interference allowing for both forward and backward unidirectional response is, however, in place in the case of dimers.

The predicted unidirectional response can find numerous applications in nanoantennas devices, integrated optics circuits, sensors with nanoparticles, energy harvesting systems, or perfect absorbers, while the option of switching between forward and backward unidirectional scattering can create interesting possibilities for manipulating optical pressure forces.