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Abstract
The efficient manipulation of light-matter interactions in subwavelength all-dielectric nanostructures offers a unique opportunity for the design of novel low-loss visible- and telecom-range nanoantennas for light routing applications. Several studies have achieved longitudinal and transverse light scattering with a proper amplitude and phase balance among the multipole moments excited in dielectric nanoantennas. However, they only involve the interaction between electric dipole, magnetic dipole, and up to the electric quadrupole. Here, we extend and demonstrate a unidirectional transverse light scattering in a V-shaped silicon nanoantenna that involves the balance up to the magnetic quadrupole moment. Based on the long-wavelength approximation and exact multipole decomposition analysis, we find the interference conditions needed for near-unity unidirectional transverse light scattering along with near-zero scattering in the opposite direction. These interference conditions involve relative amplitude and phases of the electromagnetic dipoles and quadrupoles supported by the silicon nanoantenna. The conditions can be applied for the development of either polarization- or wavelength- dependent light routing on a V-shaped silicon and plasmonic nanoantennas.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Dielectric and metallic nanostructures provide photonic and plasmonic platforms to explore light-matter interactions at the nanoscale and to develop applications in many domains such as nanoscopic position sensing [1,2], optical routing and coupling devices [3–6]. Triggered by M. Kerker et al.’s seminal work [7], several reports have focused not only on the longitudinal light scattering (i.e., first Kerker’s condition) but also on the transverse light scattering (i.e., generalized Kerker’s condition) [8,9]. The longitudinal light scattering, with unity forward scattering and zero backscattering, is obtained with the balance between the amplitudes and phases of the in-plane electric and magnetic multipole moments of the same order. In contrast, the transverse light scattering requires the balance of that among out-of-plane dipole and high-order multipole modes. Plasmonic resonance modes in metallic nanoparticles were first explored and exploited for directional nanoantenna design. Unidirectional transverse light scattering has been demonstrated in a V-shaped and triangular plasmonic nanoantenna, where the transverse scattering is obtained with the interference between two multipoles [6,10–12] and in a U-shaped plasmonic nanoantenna with the interactions between three plasmonic resonances [13]. More complex nanostructures such as bimetallic plasmonic nanoantennas [3,14], plasmonic trimer [15], and a large-area of plasmonic nanoparticles [4] have shown unidirectional transverse light scattering. Although plasmonic nanoantennas provide directional scattering, their subwavelength mode confinement in all three dimensions leads to a high absorption loss and low transverse light scattering. In contrast, nanoantennas made of dielectric materials provide low-loss absorption and strong magnetic resonant modes and therefore, offer the ability to design efficient unidirectional transverse light scattering, even if the mode volume is not confined to sub-wavelength effective volume. Among the available dielectric materials for photonic platform, silicon has been the material of choice mainly because of the high-contrast index in the visible range (i.e., Δn ∼ 2.8) [16]. Such high-contrast index allows silicon-based antennas to support both strong electric and magnetic dipolar as well as high-order multipoles that, upon proper inter-coupling, scatters the incident light into the far field with a desirable direction [9,17,18]. Bidirectional transverse light scattering has been achieved in dielectric sphere nanoantennas at the expense of structured electromagnetic incident waves such as radially, or azimuthally polarized light [19]. Experimental reports have shown that a single dielectric nanoantenna can reach bidirectional scattering into diametrically opposite directions [1]. The scattering direction is controlled by the incident wavelength and is governed by the amplitude and phase balance between the electric dipole, magnetic dipole and electric quadrupole [20]. In addition, theoretical bidirectional light scattering was demonstrated with a silicon stair-like nanoantenna. The interference here was done with the interaction between an electric dipole, two magnetic dipoles and one electric quadrupole [21]. A pair of asymmetric silicon nanocuboids has been also proposed to obtain bidirectional light scattering [22]. Moreover, the isotropic transverse light scattering was suggested and experimentally demonstrated in the microwave regime [9]. However, few of them provide the condition for unidirectional transverse light scattering with the use of longitudinal electric dipole, transverse magnetic dipole, electric quadrupole and up to the magnetic quadrupole. The main reason is that the longitudinal electric dipole and magnetic quadrupole, necessary for unidirectional transverse light scattering, is not efficiently excited on nanostructures presenting two or more mirror symmetric planes such as spheres and cubes. Compared to bidirectional transverse light scattering, unidirectional transverse light scattering is well suitable for light coupling, routing, and high-accuracy position sensing applications. Indeed, unidirectional transverse light scattering has shown to be an extremely precise tool for position sensing, with angstrom-scale resolution [1,2,23].
In this work, we investigate the unidirectional transverse light scattering in a V-shaped silicon nanoantenna upon illumination with a linearly polarized plane-wave. We use the long-wavelength approximation to find the conditions required for near-unity unidirectional transverse light scattering, the exact multipole decomposition and full-wave simulation based on the finite element method to confirm results. We found that the balance between the amplitude and phase of the longitudinal component of the electric dipole and that of the transverse component of the magnetic dipole, electric and magnetic quadrupoles are necessary for near-unity unidirectional transverse light scattering. This near-unity unidirectional scattering comes along with near-zero scattering of light in the opposite direction. We present proper far-field radiation patterns, directivities and scattering cross sections spectra as optical response of the silicon nanoantenna.
2. Methods and materials
The silicon nanoantenna consists of a V-shaped prism (Fig. 1(a)). For the silicon material dispersion, we used the data from Palik’s handbook [16]. The silicon antenna is embedded in a homogenous host medium with permittivity ε (i.e., in air) and illuminated with a x-polarized plane wave propagating along the z direction. Under this configuration, we take advantage of the even mirror symmetry normal to the y-axis to analyze the problem. The mirror symmetry is broken in a plane normal to the x-axis.
Fig. 1. Unidirectional transverse light scattering. (a) Schematic of the V-shaped silicon nanonatenna. An incident wave, polarized along the x-axis propagates along the z-axis. The direction of the scattered light is in the -x-axis direction. (b) A schematic vector diagram of the multipoles for the unidirectional transverse light scattering.
As previously said, the incident light excites the electric and magnetic resonance modes and, upon interactions, scatters the light into the far field with a desirable direction. The excited modes can be decomposed into the electric dipole (px, 0, pz), magnetic dipole (0, my,0), electric quadrupole (Qxz), and magnetic quadrupole (Myz and Mxy).
2.1 Long-wavelength approximation
In the long-wavelength approximation, the total radiation far-field is given by [24,25]
Fig. 2. Radiation patterns of the individual multipoles and interference. Far field radiation patterns of (a) the z-component of electric dipole moment (pz); (b) the transverse magnetic dipole moment (my); (c) the electric quadrupole moment (Qxz), and (d) the magnetic quadrupole moment (Mxy). The unidirectional transverse scattering in right (θ = π/2 rad) or left (θ = 3π/2 rad) directions resulting from a phase difference of 0 or π between (e) the electric and magnetic dipoles and (f) the electric and magnetic quadrupoles. The arrows on the xz-plane scattering patterns indicate the phase of the radiated field.
The arrows denote the phase symmetry of the far-field radiation patterns. A phase symmetry analysis provides a concise way to show the interference between two or more multipole moments [8]. The pz and the Mxy show even symmetry, while the my and the Qxz show an odd one. The interference between one multipole with even symmetry with another of odd one originates unidirectional transverse light scattering. This condition can be expressed as ${\textrm{p}_\textrm{z}}\mathrm{\ \pm }\frac{{{\textrm{m}_\textrm{y}}}}{\textrm{c}} = 0$, where the unidirectional right (θ = π/2 rad) or left (θ = 3π/2 rad) transverse light scattering arises from an in- or out-of- phase between the pz and my multipolar moments, respectively (Fig. 2(e)). For the quadrupoles, the condition is expressed as $\frac{{\textrm{ik}}}{\textrm{6}}{\textrm{Q}_{\textrm{xz}}} \mp \frac{{\textrm{ik}}}{{\textrm{2c}}}{\textrm{M}_{\textrm{xy}}} =\,0$, (Fig. 2 (f)).
Similarly, the phase symmetry between pz and Qxz also provides unidirectional transverse scattering, ${\textrm{p}_\textrm{z}} \mp \frac{{\textrm{ik}}}{\textrm{6}}{\textrm{Q}_{\textrm{xz}}}=\,0$, where the scattering radiation patterns are the same as in Fig. 2(f). Moreover, unidirectional transverse light scattering condition is also obtained as $\frac{{{\textrm{m}_\textrm{y}}}}{\textrm{c}}\mathrm{\ \pm }\frac{{\textrm{ik}}}{{\textrm{2c}}}{\textrm{M}_{\textrm{xy}}}=\,0$ (scattering patterns are the same as in Fig. 2(e)). Finally, we notice that the phase symmetry of my and Qxz multipole modes are odd and the radiated far electric fields are opposite, equivalently, the my multipole scatters to the right and Qxz to left transverse directions. As a result, the relative in- and out-of- phase conditions provide zero (in both directions) and bidirectional (with zero forward and backward scattering) transverse light scattering, respectively.
Based on the previous analysis, we obtain the condition for unidirectional transverse light scattering, that it is, a near-unity directional transverse scattering to the left side (i.e., θ = 3π/2 rad) while a zero directional transverse light scattering to the right side (i.e., θ = π/2 rad). The scattering cross sections are thus given by:
Fig. 3. Influence of the x-component of the electric dipole on the radiation pattern. Radiation patterns result from (a) the absence (my+pz+Mxy+Qxz) and (b) the addition of the x-component of the electric dipole (px+my+pz+Mxy+Qxz). In (a), the phase difference is π rad between my and pz, zero between Mxy and Qxz, and π rad between my and Qxz. In (b), the same phase relationship as (a) holds but with an additional phase difference of π rad between my and px.
It is worth noting that, although we assumed equal amplitudes for each multipole mode, our conclusion still holds for different relative magnitude configurations. The generalized unidirectional transverse scattering theory can also be applied to other unidirectional side scattering conditions [10,20,21]. In addition, even if the transverse electric dipole component px is excited, we show that the condition of the extra component px in Eq. (3) has negligible effect on the scattering direction even the π rad phase difference between px and my. Figure 3(b) shows the calculated scattering radiation pattern that results from the addition of px component.
We observe that when the magnitude of the px component is weak compared to that of my (i.e., c|px/my|= 0.3), the direction of scattering pattern is in the θ = 3π/2 rad but the scattering intensity decreases. Furthermore, when the magnitude of px equals that of my, the light scattering direction is not exactly in the θ = 3π/2 rad but has an angle of deviation of about θ = 0.05π rad (blue line pattern in Fig. 3(b)).
2.2 Beyond the long-wavelength approximation
The long-wavelength approximation, developed above, is valid for scatter dimensions with negligible size compared to the wavelength of the incident light. To confirm our theory, we obtain and present the unidirectional transverse light scattering obtained with the use of the exact expressions for the multipole decomposition in the silicon V-shaped nanoantenna in the visible range ranging from 560 nm to 800 nm. The exact expressions for the multipole moments are valid for any wavelength and size dimensions [26].
In addition, we perform full-wave numerical calculations based on the finite element method implemented by a commercially available software by COMSOL Multiphysics. The tetrahedral meshes are generated by using the built-in algorithm of the software. The V-shaped nanoantenna is enclosed by a sphere whose diameter is 2 times the wavelength of the incident light. Outside the sphere is a perfect matched layer with the thickness of 0.5 times the wavelength of incident light. The incident light direction and polarization is as shown in Fig. 4(a)
Fig. 4. Unidirectional transverse light scattered by a V-shaped silicon nanoantenna. Transverse scattering light patterns in (a) 3D and (b) 2D are demonstrated at the wavelength of λ = 640 nm. In (b), the red and blue solid lines are the radiation pattern in the xz- and xy-planes, respectively. (c) Wavelength dependence of the radiation angle. (d) Wavelength dependence of the radiation directivities. The dash lines in (c) and (d) denote the maximum directivity (L/R ratio) at the wavelength of λ = 640 nm.
3. Results and discussion
Figure 4(a) shows the schematics of the silicon nanoantenna with sizes H = 250 nm, L = 200 nm, D = 160 nm, and angle α = 150° and the calculated 3D scattering far-field radiation pattern that results upon an incident plane wave polarized along the x-axis at λ = 640 nm. The calculated radiation pattern with the use of the exact multipole decomposition agrees with that calculated with the finite element method.
A unidirectional transverse light scattering is clearly obtained in the –x-axis direction (θ = 3π/2 rad). For analysis, we plot the 2D radiation patterns in the xz- and xy- planes in Fig. 4(b) as red solid and blue dashed lines, respectively. From the plots, we obtain the angle of maximum radiation, and we calculate the ratios left-to-right (L/R ratio) and forward-to-backward (F/B ratio), which are known as directivity in antenna theory (Fig. 4(c)-Fig. 4(d)). For showing the clear directivity, we follow the method presented in Ref. [3]. The V-shaped nanostructure in our simulation is enclosed by a sphere, as shown in Fig. 1(a). The scattered power in left (-x) hemisphere is calculated as the left power. The scattered power in right (+x) hemisphere is calculated as the right power. The directivity left-to-right ratio is defined as 10*1og10(left/right). Similarly, the forward-to-backward ratio is the ratio by the scattered power in front (+z) hemisphere and back (-z) hemisphere. The unidirectional transverse light scattering is obtained in a broadband spectral range ranging from 600 nm to 670 nm, where the maximum scattering angle is close to θ = 3π/2 rad. The scattering directivity reaches a peak at λ = 640 nm with the maximum value of 6.4 dB. The F/B ratio is between -1 and +1 dB across a range between 600 nm to 670 nm.
In addition to the radiation pattern, we show the scattering cross sections of the electric dipole (p), magnetic dipole (m), electric quadrupole (Q), and magnetic quadrupole (M) calculated with the exact multipole expansion and the total scattering cross section calculated via direct full-wave numerical simulation (Fig. 5(a)). As expected, the main contributions to the transverse light scattering are pz, my, Qxz, and Mxy.
Fig. 5. Scattering cross section of the multipoles and the generalized transverse Kerker’s condition. (a) The total scattering cross section and the multipoles contribution and of the V-shaped Si nanoantenna. The black circle represents the SCSs summation of p, m, Q and M. The total value is obtained from the full-wave numerical simulation. (b) The real and imaginary parts of the coefficients A. The translucent yellow area shows the spectral range of unidirectional transverse light scattering. The dash line in (b) denotes the maximum directivity at λ = 640 nm.
We define $\textit{A = }\frac{{{\textit{m}_\textit{y}}}}{\textit{c}}\textit{ - }\frac{{\textit{ik}}}{\textit{6}}{\textit{Q}_{\textit{xz}}}\textit{ + (}{\textit{p}_\textit{z}}\textit{ + }\frac{{\textit{ik}}}{{\textit{2c}}}{\textit{M}_{\textit{xy}}}\textit{)}$, the real and imaginary parts of A are calculated and plotted in Fig. 5(b). The near-unity unidirectional transverse light scattering in the –x-axis arises when the transverse light scattering in the opposite direction + x-axis is suppressed (i.e., both Re{A}=0, and Im{A}=0). The near-zero A is obtained in a range between 600 nm and 670 nm (yellow area in Fig. 5(b)) with a maximum directivity at λ = 640 nm. The far-field radiation patterns also confirm this broadband unidirectional transverse scattering (Fig. 6(a)).
Fig. 6. Unidirectional transverse light scattering. (a) Far-field scattering patterns of the unidirectional transverse light scattering in the wavelength region from 600 nm to 670 nm. (b) Vector diagram depicts the complex multipolar coefficients at λ = 640 nm.
To further clarify the analysis, we plot the main complex multipolar coefficients in a vector diagram (Fig. 6(b)). The length of arrows indicates the amplitude of scattering cross sections and the orientations represent the phases.
The relative phase difference between my and Qxz is close to π rad, which results in transverse light scattering in the x-axis while the pz and Mxy components are close to π rad. Additionally, the amplitude ratio between px and my is 0.3 and the phase difference between px and my is π/2 rad, interference between px and my results no obvious directivity in z direction. This condition is like that presented in section 2 (Fig. 2(b)). We conclude that the origin of unidirectional transverse light scattering in the V-shaped silicon nanoantenna is the balance between the amplitude and phases of the out-of-plane electric dipole (pz), the in-plane magnetic dipole (my), the out-of-plane electric quadrupole (Qxz), and the in-plane magnetic quadrupole (Mxy). In addition, the near π phase difference between the pz + Mxy and my results in strong unidirectional transverse light scattering. To support our conclusion, we calculate the transverse light scattering properties of V-shaped silicon nanoantennas with angles opening of 180°, 120°, and 90°, keeping the same sizes H, L, and D (see suppl. Info. Section 2 and 3). With these angles, the magnetic quadrupole is weakly excited and therefore, the balance is not completely fulfilled decreasing directivity (see suppl. info). Nevertheless, the silicon nanoantenna with opening angle 120° and 90°, still provide transverse light scattering. Bidirectional transverse light scattering is obtained for the V-shaped silicon nanoantenna with opening angle α = 180° at 602 nm due to the additional mirror symmetry plane (i.e., the pz and Mxy are not excited).
As a benchmark case, we calculated the unidirectional transverse light scattering in a V-shaped gold nanoantenna (see suppl.info. Section 4). To allow comparison with the V-shaped silicon nanoantenna, we chose the size parameters L = 200 nm, W = 50 nm, H = 50 nm, and opening angle α = 120° to maximize unidirectional transverse light scattering. The length of the gold nanoantenna was fixed such that the resonance position overlaps with that of the silicon nanoantenna. In contrast to the dielectric counterpart, the incident light here is polarized along the y-axis. Under this configuration, the mirror symmetry, in the plane perpendicular to the polarization direction of incident light, is opposite to that in the silicon antennas. The unidirectional transverse scattering is obtained in the visible range with a largest L/R ratio of about -3.6 dB. However, the V-shaped gold nanostructure only supports the electric and magnetic dipoles, and the electric quadrupole. The magnetic quadrupole in the gold nanoantenna is weaker than that of the silicon counterpart. As a result of previous conditions, the transverse light scattering direction is in a opposite direction with respect to that of the silicon nanoantenna. Finally, we remark that the intensity of the light scattered by the silicon nanoantenna is two times higher than that of the gold nanoantenna.
4. Conclusion
We theoretically and numerically demonstrated the unidirectional transverse light scattering in a silicon V-shaped nanoantenna illuminated with a linearly polarized plane-wave. The nanoantennas do not require structured illumination such as radially or azimuthally polarized light, which experimentally simplifies the setup. The unidirectional transverse light scattering arises from the interferences of electric and magnetic dipoles as well as electric and magnetic quadrupoles, even if the relative phases and amplitudes multipole moments are not equal. The addition of the x-component of the electric dipole (px) does not reduce significantly the unidirectional transverse light scattering. Specifically, we demonstrate the suppression of the transverse light scattered in one transverse direction and enhanced in the opposite one even if the magnitude of the px is equal to that of the y-component of the magnetic dipole (i.e., c|px/my| = 1). The efficient excitation of the magnetic quadrupole in the V-shaped silicon nanoantenna provides novel possibilities to realize highly directive nanoantennas in transverse direction for nanoscale light routing (see suppl. Info. Section 5).
Funding
National Natural Science Foundation of China (51175436, 51575455).
Acknowledgments
Part of the computations were carried out with the resources of the French regional Grand-Est HPC Center ROMEO.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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