1. Introduction

Nanoantennas that excel in confining optical energy in deep-subwavelength space have attracted significant research attention owing to their potential applications such as high-speed optical communication [1–6], quantum computing with single phonon sources [7–9] and detectors [10–12], nonlinear plasmonics [13–15], and single-molecule spectroscopy [16–18]. Recent advances in nanoantenna technology offer tremendous opportunities to control their functionality while maintaining extremely confined optical energy [19–21]. In particular, various nanoantenna configurations have been proposed to enable the control of the far-field radiation characteristics. For example, the Yagi–Uda [22–27], bull’s eye [28,29], and bimetallic nanoantennas [30] have demonstrated the ability to control the beam direction of the antenna from highly localized optical energy to the free space or vice versa. Here, the electromagnetic waves generated from a single feed nanoantenna are interfered by the electromagnetic waves from the surrounding antenna elements. These systems inevitably require the use of multiple antenna elements in an array form to adjust the phase interference and steer the beam direction. However, the method using multiple antenna elements not only consumes considerable space but also lowers the high-density optical energy by distributing energy to the surrounding antenna array. It is also challenging to ensure that all the array elements are precisely sized and positioned for accurate phase interference. The ability to steer the beam direction of a single nanoantenna would be a major breakthrough in the development of the next-generation nanoantenna-based photonic devices. For example, integrating a single quantum emitter into the engineered nanoantenna will enable a fast, bright and even directional quantum source in future quantum integrated circuits. It can also be applied not only to detect or focus light incident in a specific direction on nanometer space, but also to increase the efficiency of a single-molecule spectroscopy. One possible approach to steer the beam direction of a single nanoantenna is to utilize the higher order eigenmodes. This is because nanoantennas have several bright (even) and dark (odd) modes with relatively wide spectral bandwidths sufficient to overlap multiple eigenmodes [31,32].

In this paper, we propose an approach to steer the beam direction of a single nanoantenna by adjusting two antenna modes with opposite phase symmetry. Using a simple dipole antenna approximation, we theoretically verified that the beam of a single nanoantenna can be steered by exciting a mixture of even- and odd-symmetric modes. In particular, by setting the phase difference between the even- and odd-symmetric modes at π/2, the beam steering conditions are satisfied, and the steering angle is controlled by adjusting the amplitude ratio between the two modes. To implement our theory in a practical device, we introduced the asymmetric trapezoidal nano-slot antenna, with different side air-gaps of 10 and 50 nm. This structure simultaneously excites the dipole and quadrupole modes under planewave pumping conditions and has a beam steering angle of 35° or more, in proximity to the resonance of the quadrupole mode. In this case, the steering angle is adjusted according to the degree of asymmetry of the trapezoidal slot structure.

2. Theorical background

First, we investigated the theoretical possibility that the beam direction of a single nanoantenna could be steered by a combination of different eigenmodes of the antenna. To simplify the development of the theory, we approximated the antenna modes with dipole oscillations and calculated the far-field radiation patterns based on this model.

As shown in Fig. 1(a), we modeled the even-symmetric 1^st mode of the nanoantenna with a single dipole oscillation. Here, the dipole moment is expressed by ${{\textbf d}_{\textbf e}} = \alpha {E_e}\hat{{\textbf z}}$, where E_e is the amplitude of the near electric field of the antenna mode and $\alpha $ is the coefficient of polarizability. So, the electric field in the far-field regime (${\textrm{E}_\textrm{e}}^\textrm{f}$) is calculated by ${\textrm{E}_\textrm{e}}^\textrm{f} = C{E_e}sin \theta ({1/r} )exp({ikr} )\; \hat{{\textbf x}}$ (see Appendix A), where $\theta $ is the polar angle, r is the distance between the dipole and the measuring point, and C is the constant coefficient. Accordingly, the time-averaged radiation power per unit solid angle ($\Gamma = dP/d\Omega \propto {|{{{\textbf E}_{\textbf e}}^{\textbf f}} |^2}$) is proportional to ${|{\textrm{sin} \theta } |^2}$, as shown in Fig. 1(a).

 

Fig. 1. Schematic of approximate dipole oscillations and radiation patterns of (a) the even-symmetric 1^st mode, (b) the odd-symmetric 2^nd mode, and (c) the mixed mode of a single nanoantenna

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In a similar way, we approximated the odd-symmetric 2^nd mode of the nanoantenna as two dipole oscillations a distance L apart and with a phase difference of $\pi $. In this case, the electric field in the far-field regime (${{\textbf E}_{\textbf o}}^{\textbf f}$) is calculated by ${{\textbf E}_{\textbf o}}^{\textbf f} ={-} iC({kL/2} ){E_o}sin\theta cos\theta ({1/r} )\textrm{exp}({ikr} )\; \hat{{\textbf x}}$ (see Appendix B), where E_o is the amplitude of the near electric field of the odd-symmetric antenna mode and k is the wavenumber. It follows that the time-averaged radiation power per unit solid angle of the 2^nd mode is proportional to $\textrm{|}sin\theta cos\theta {\textrm{|}^\textrm{2}}$, as shown in Fig. 1(b).

When these two antenna modes are mixed together in a single nanoantenna, the radiation power per unit solid angle follows the relation $\Gamma \propto {|{{a_e}sin\theta - i{a_o}sin\theta cos\theta } |^2}$ (see Appendix C), where a_e and a_o are the complex coefficients that represent how much the even- and odd-symmetric modes are mixed in a single antenna. Thus, one can change the far-field radiation patterns by adjusting a_e and a_o, and under certain conditions, the radiation beam of the antenna will be steered.

We calculated the far-field radiation patterns based on Fig. 1(c), as a function of the mixture of a_e and a_o in polar coordinates, as shown in Figs. 2(a) and 2(b). When a single dipole oscillation (a_e : ia_o = 1 : 0) or a double-dipole oscillation (a_e : ia_o = 0 : 1) is present in the antenna, the far-field radiation exhibits a dipole or quadrupole radiation pattern, respectively. However, when the odd-symmetric quadrupole mode is added to the even-symmetric dipole mode (a_e = 1, ia_o ≠ 0) or vice versa (a_e ≠ 0, ia_o = 1), the antenna beam begins to be steered in a specific direction. As shown in Figs. 2(a) and 2(c), as ia_o is varied from 0 to ±1 at a_e = 1, the far-field radiation of the dipole mode is steered with an angle $\delta\left(=\pi / 2-\theta_{\max }\right)$ from 0° to ∓30°, where ${\theta _{max}}$ is the polar angle at the maximum radiation power. In this condition, the far-field radiation shows the unidirectional beam with a single value of $\delta $ in the range of −30° ∼ 30°.

 

Fig. 2. Normalized radiation power per unit solid angle (dP/dΩ) in polar coordinate, when (a) ia_o increases from 0 to 1 at a_e = 1 and (b) when a_e increases from 0 to 1 at ia_o = 1. Calculated steering angle (δ) of the radiation beam when (c) ia_o is between −1 and 1 at a_e = 1 and (d) when a_e is between −1 and 1 at ia_o = 1. (e) Energy ratio between + and – direction beams when a_e is between −1 and 1 at ia_o = 1.

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On the other hand, when the dipole mode is added to the quadrupole mode (or a_e is varied from 0 to ±1 at ia_o = 1), the bidirectional far-field pattern of the original quadrupole mode is transformed into the unidirectional pattern in a specific range of a_e, as shown in Figs. 2(d) and 2(e). For example, when a_e approaches from 0 to 1 at ia_o = 1, one of the two directional beams is shifted from -45° to -30°, while the other beam moves from 45° to 90°. Here, the energy of the former beam occupies a high proportion of the total energy during a_e approaches 1, while the energy of the latter beam weakens and eventually disappears, as shown in Fig. 2(e). For the condition of a_e : ia_o = 0.5 : 1, one beam is directed to −36°, and the other to 57°. Here, the radiation power of the beam at −36° accounts for 96% of the total radiation power. If this condition (a_e : ia_o = 0.5 : 1) is set as the boundary of the unidirectional beam, the maximum steerable angle of a unidirectional beam by the mixture of two eigenmodes of a single nanoantenna is ±36 °, as shown in Fig. 2(d).

3. Device design

To implement our theory in a practical device, we studied beam steering based on the air-slot nanoantennas. The air-slot nanoantennas, like other types of nanoantennas, have multiple resonant eigenmodes over a broad spectral range. The nanoantenna with a rectangular air-slot on a 100 nm thick gold film, shown in Fig. 3(a), has three x-polarized resonance modes in the wavelength range of 400–2000nm. Here, the slot length (L) and gap size (g) are 400 nm and 50 nm, respectively. The 1^st and 3^rd modes have electric fields with even symmetry with respect to the xy-plane; the 2^nd mode has an electric field with odd symmetry. In the 1^st mode (even mode), the electric field has a half wavelength sinusoidal distribution along the longitudinal z-direction of the air-slot, described by $\mathbf{E}(z)= \; E_{e} \sin (\pi z / L) \hat{\mathbf{x}}$, as shown in Fig. 3(b). In the 2^nd mode (odd mode), the electric field has a full wavelength sinusoidal distribution along the z-direction, described by $\mathbf{E}(z)=E_{o} \sin (2 \pi z / L) \hat{\mathbf{x}}$. From the surface equivalence principle, the even-symmetric 1^st mode and the odd-symmetric 2^nd mode are approximated as a single magnetic dipole oscillation ($\mathbf{m}_{\mathrm{e}}=\alpha E_{e} \hat{\mathbf{z}}$) and two magnetic dipole oscillations ($\mathbf{m}_{\mathrm{o}}=\pm\left(\alpha E_{o} / 2\right) \hat{\mathbf{z}}$) with a phase difference of π, respectively [33–36] (see Appendix D). Thus, the 1^st and 2^nd modes show the dipole and quadrupole far-field radiation patterns, respectively, as shown in Figs. 3(c) and 3(d). Here, the simulations were conducted numerically by using a commercially available finite-difference time-domain (FDTD) software package (Lumerical Solutions, Inc.). The complex permittivity of gold was taken from Johnson and Christy’s experimental study [37] and fitted with the Drude model to conduct the simulation. The far-field patterns were calculated through the post processing of near-to-far field transformations using the near-field data obtained from the FDTD simulation [38].

 

Fig. 3. (a) Schematic of rectangular nano-slot antenna. (b) |E|² profiles of the 1^st and 2^nd resonant modes along the xz-plane. Arrows represent the directions of the electric fields. (c) |E|² profiles in logarithm scale along the yz-plane. (d) 3D far-field intensity patterns in spherical coordinates. (e) Spectrum of normalized energy stored in the antenna under x-polarized planewave pumping conditions.

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When a x-polarized planewave is incident to the rectangular-slot nanoantenna, the 2^nd mode with odd phase symmetry is not excited because of zero coupling with the normal-incident planewave ${\textbf k} = {k_y}\hat{{\textbf y}}$. Figure 3(e) shows the energy spectrum stored inside the nanoantenna under normal planewave pumping conditions, where all the energy values are normalized by the maximum energy value of the 1^st mode. As shown in this figure, the 2^nd mode is not excited. In order to steer the beam direction of a single nanoantenna, we must excite the odd-symmetric 2^nd mode under these pumping conditions.

One way to excite the odd-symmetric 2^nd mode is to break the symmetry of the slot geometry. As shown in Fig. 4(a), we reformed the symmetric rectangular slot with the asymmetric trapezoidal slot. Here, some amount of coupling of the 2^nd mode is expected due to the distorted near-and far-field profiles by the asymmetric slot shape. We fixed the gap size (g₂) on one side of the trapezoid at 50 nm and deformed the gap size (g₁) on the other side from 40 to 5 nm. We can see in Fig. 4(b) that as g₁ decreases, the symmetry of the near-field profiles of both 1^st and 2^nd modes is broken, and when g₁ < 10 nm, the near electric-field is severely biased into the smallest gap in both 1^st and 2^nd modes. The radiation patterns of the 1^st and 2^nd modes are also slightly distorted, as shown in Fig. 4(c).

 

Fig. 4. (a) Schematic of trapezoidal nano-slot antenna. (b) |E|² profiles of the 1^st and 2^nd resonant modes, along the xz-plane with different g₁. Arrows indicate the directions of the electric fields. (c) |E|² profiles in logarithm scale, along the yz-plane. (d) Spectrum of energy stored in the antenna under x-polarized planewave pumping conditions. Here, the energy spectra are normalized by the maximum energy of the 1^st mode for the antenna with g₁ = 50 nm.

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Figure 4(d) shows the energy spectrum stored inside the antenna as g₁ decreases from 50 to 5 nm, under normal planewave pumping conditions. All the energy values are normalized by the maximum energy of the 1^st mode for the antenna with g₁ = 50 nm. We can clearly see that as g₁ decreases (or as the degree of slot asymmetry increases), the odd-symmetric 2^nd mode is highly excited. For the antenna with g₁ = 40 nm, the normalized maximum energies for the 1^st and 2^nd mode increase to 1.08 and 0.03, respectively. When g₁ decreases to 10 nm, the normalized maximum energies for 1^st and 2^nd modes further increase to 1.28 and 0.24, respectively. The quality factor of the 1^st mode is calculated to be 5 for the antenna with g₁ = 10 nm. This means that about 4.3% of the maximum energy of the 1^st mode (λ_1st = 1439 nm) exists near the resonance of the 2^nd mode (λ_2nd = 766 nm) in the spectrum, as shown in Fig. 5(b). This fact opens the possibility that the trapezoidal nano-slot antenna will have conditions (a_e ≠ 0, a_o ≠ 0) to steer the beam direction.

 

Fig. 5. (a) Far-field radiation patterns are projected from spherical coordinates to the xz-plane. (b) Normalized energy spectra of decomposed 1^st, 2^nd and 3^rd modes in a logarithm scale when g₁ = 10 nm and g₂ = 50 nm. (c) Far-field radiation patterns at wavelengths near the resonance of 2^nd antenna mode.

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4. Beam steering of trapezoidal-slot nanoantenna

Through computational simulations, we confirmed that the trapezoidal-slot nanoantenna can steer the normal-incident planewave beam. To display the far-field patterns, we projected the angle of the far-field radiation from spherical coordinates onto the xz-plane, as shown in Fig. 5(a). To see the spectral responses at first, we fixed g₁ and g₂ at 10 nm and 50 nm, respectively. As estimated in Fig. 4(d), the 2^nd mode is excited under normal planewave pumping, and the energy tail of the 1^st mode extends near the resonance of the 2^nd mode with a low quality-factor, as shown in Fig. 5(b). So, it is expected that the beam can be steered near the 2^nd resonance mode (a_o ≠ 0) because the energy of the 1^st mode remains (a_e ≠ 0) near the 2^nd mode and the phase difference of π/2 (ia_o = real) can be found near the resonance of the 2^nd mode [39]. As shown in Fig. 5(c), we examined the radiation patterns of the nanoantenna for wavelengths near the 2^nd mode from 700 to 780 nm. At λ = 760 nm, the radiation beam of the nanoantenna shows the maximum steering of the planewave beam with an angle (δ) of −35.5°, which is almost identical to the theoretical maximum steering angle of 36.0° discussed in Fig. 2. Here, a very weak second radiation beam is observed at δ = 44.5°. At λ = 725 and 700 nm, δ are calculated to be −22.7° and −8.2°, respectively, and at λ = 780 nm, the beam shows the bidirectional radiation pattern.

Finally, we studied the effect of the degree of slot asymmetry on the beam steering. Figure 6 shows the far-field radiation patterns at wavelengths with the maximum steering angle, when g₁ decreases from 40 to 10 nm and g₂ is fixed at 50 nm. It is clear that the stronger the degree of asymmetry of the trapezoidal shape, the greater the beam steering angle. For g₁ = 40, 30, and 20 nm, the maximum δ are calculated to be −0.9°, −13.6°, and −30.0°, respectively. At g₁ = 10 nm, the radiation beam shows the steering angle of −35.5° close to the theoretically predicted maximum steering angle of 36°. The beam divergences, defined as the full-widths at half-maximum (FWHM) of the beam angle, are calculated to be 46°, 69°, 56° and 39° for the antennas with g₁ = 40, 30, 20, and 10 nm, respectively. Here, the wide beam divergences are due to the small modal sizes of the 1^st (even) and 2^nd (odd) modes of the nanoantenna. Through the studies using the higher-order eigenmodes of the nanoantenna, we can reduce the width of the beam divergence. For example, the beam divergences of the 3^rd and 5^th modes show 27° and 20°, respectively.

 

Fig. 6. (a) Projected far-field radiation patterns, (b) normalized radiation power per unit solid angle in polar coordinate, and (c) |E|² profiles in logarithm scale along yz-plane near the resonance of the 2^nd antenna mode at wavelengths with the maximum steering angle, when g₁ decreases from 40 to 10 nm and g₂ is fixed at 50 nm.

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5. Conclusion

We have proposed a novel method for beam steering of a single nanoantenna by exciting multiple antenna modes with opposite phase symmetries. Our theoretical and numerical studies confirmed that the combination of even- and odd-symmetric modes, with π/2 phase difference, allows for effective beam steering of a single nanoantenna. Here, the steering angle is controlled by adjusting the degree of mixing of the even- and odd-symmetric modes. We believe that our beam steering approach will pave the way for beam engineering of a single nanoantenna.

Appendix A: Radiation pattern of a single dipole oscillation

In the radiation zone ($kr \gg 1$), the electric-field (${{\textbf E}_{\textbf e}}^{\textbf f}$) of a single dipole, located at the origin, is given by,

(1)$${{\textbf E}_{\textbf e}}^{\textbf f} = \frac{{ - {Z_0}}}{{4\pi }}{k^2}({\hat{{\textbf r}} \times {\textbf d}} )\frac{{{e^{ikr}}}}{r} = \frac{{ - {Z_0}}}{{4\pi }}{k^2}\frac{{\alpha {E_e}{e^{ikr}}}}{r}({sin\theta sin\phi \hat{{\textbf x}} - sin\theta cos\phi \hat{{\textbf y}}} )$$

where $\hat{{\textbf r}}$ is the normal vector directed into the observation point, $\hat{{\textbf r}} = \; sin\theta cos\phi \hat{{\textbf x}} + sin\theta sin\phi \hat{{\textbf y}} + cos\theta \hat{{\textbf z}}$, and ${\textbf d}$ is the dipole moment, ${\textbf d} = \alpha {E_e}\hat{{\boldsymbol z}}$. Here we define a coefficient, ${C_o} ={-} ({\alpha {Z_0}{k^2}} )/4\pi $ to simplify the expression.

(2)$${{\textbf E}_{\textbf e}}^{\textbf f} = {C_o}{E_e}\frac{{{e^{ikr}}}}{r}({sin\theta sin\phi \hat{{\textbf x}} - sin\theta cos\phi \hat{{\textbf y}}} )$$

Then, the time-averaged radiating power per unit solid angle on the yz-plane ($\phi = {\pi }/2$) is

(3)$$\Gamma = \frac{{dP}}{{d\Omega }} = \frac{1}{2}\textrm{Re}[{{r^2}\hat{{\textbf r}} \cdot ({{{\textbf E}_{\textbf e}}^{\textbf f} \times {{\textbf H}_{\textbf e}}{{^{\textbf f}}^{\ast }}} )} ]\propto \textrm{Re}[{{{\textbf E}_{\textbf e}}^{\textbf f} \cdot {{\textbf E}_{\textbf e}}{{^{\textbf f}}^{\ast }}} ]\propto si{n^2}\theta $$

Appendix B: Radiation pattern of two dipole oscillations

Suppose there are two oscillating dipoles, one of which is located at $({L/2} )\; \hat{{\textbf z}}$ with the dipole moment ${{\textbf d}_1} = ({\alpha {E_o}/2} )\hat{{\textbf z}}$, and the other is at ${\; } - ({L/2} )\; \hat{{\textbf z}}$ with the dipole moment ${{\textbf d}_2} ={-} ({\alpha {E_o}/2} )\hat{{\textbf z}}$. Then, the radiated electric field (${{\textbf E}_{\textbf o}}^{\textbf f}$) is given by

(4)$$\begin{aligned} \mathbf{E}_{\mathbf{o}}^{\mathbf{f}} =\mathbf{E}_{\mathbf{1}}^{\mathbf{f}}+\mathbf{E}_{\mathbf{2}}^{\mathbf{f}}=\frac{-Z_{0}}{4 \pi} k^{2}\left(\hat{\mathbf{r}} \times \mathbf{d}_{\mathbf{1}}\right) \frac{e^{i k | \mathbf{r}-L / 2 \hat{\mathbf{z}}} |}{r} \\ +\frac{-Z_{0}}{4 \pi} k^{2}\left(\hat{\mathbf{r}} \times \mathbf{d}_{2}\right) \frac{e^{i k\left|\mathbf{r}+L / 2^{\hat{\mathbf{z}}}\right|}}{r} \end{aligned}$$

In the radiation zone, ${\textbf r} \gg L/2$, we use a Taylor series,

(5)$$\left|{{\textbf r} \mp \frac{L}{2}\hat{{\textbf z}}} \right|= \sqrt {{r^2}sin{\theta ^2}cos{\phi ^2} + {r^2}sin{\theta ^2}sin{\phi ^2} + {{\left( {rcos\theta \mp \frac{L}{2}} \right)}^2}} \approx r \mp \frac{1}{2}Lcos\theta $$

Then, the electric field is given by

(6)$${{\textbf E}_{\textbf o}}^{\textbf f} = \frac{{ - {Z_0}}}{{4\pi }}{k^2}\frac{{\alpha ({{E_o}/2} ){e^{ikr}}}}{r}\left[ {\left( { - 2isin\theta sin\phi sin\left( {\frac{{kLcos\theta }}{2}} \right)} \right)\hat{{\textbf x}} + \left( {2isin\theta cos\phi sin\left( {\frac{{kLcos\theta }}{2}} \right)} \right)\hat{{\textbf y}}} \right]$$

When the antenna’s dimensions are too small in comparison to the wavelength, we can approximate $kL \ll 1$. In this case, the electric field is

(7)$${{\textbf E}_{\textbf o}}^{\textbf f} = {C_o}\frac{{{E_o}}}{2}\frac{{{e^{ikr}}}}{r}[{({ - ikLsin\theta sin\phi cos\theta } )\hat{{\textbf x}} + ({ikLsin\theta cos\phi cos\theta } )\hat{{\textbf y}}} ]$$

Therefore, the time-averaged radiating power per unit solid angle (yz-plane, $\phi = {\pi }/2$) is

(8)$$\Gamma = \frac{{dP}}{{d\Omega }} \propto \textrm{Re}[{{{\textbf E}_{\textbf o}}^{\textbf f} \cdot {{\textbf E}_{\textbf o}}{{^{\textbf f}}^{\ast }}} ]\propto si{n^2}\theta co{s^2}\theta $$

Appendix C: Beam steering by mixing two resonant modes

In this method, we determined the radiation pattern of the even and odd modes as follows:

(9)$${{\textbf E}_{\textbf e}}^{\textbf f} = {C_o}{E_e}\frac{{{e^{ikr}}}}{r}({sin\theta sin\phi \hat{{\textbf x}} - sin\theta cos\phi \hat{{\textbf y}}} )$$

(10)$${{\textbf E}_{\textbf o}}^{\textbf f} = {C_o}\frac{{{E_o}}}{2}\frac{{{e^{ikr}}}}{r}[{({ - ikLsin\theta sin\phi cos\theta } )\hat{{\textbf x}} + ({ikLsin\theta cos\phi cos\theta } )\hat{{\textbf y}}} ]$$

When the two modes are excited together with a relative phase shift ($\Delta $), the total electric field is given by:

(11)$$ \begin{aligned}{{\textbf E}_{\textbf T}}^{\textbf f} &= \left( { - i\frac{{kL}}{2}{E_o}sin\theta sin\phi cos\theta {e^{i\Delta }} + {E_e}sin\theta sin\phi } \right)\frac{{{C_o}{e^{ikr}}}}{r}\hat{{\textbf x}} \\ &+ \left( {i\frac{{kL}}{2}{E_o}sin\theta cos\phi cos\theta {e^{i\Delta }} - {E_e}sin\theta cos\phi } \right)\frac{{{C_o}{e^{ikr}}}}{r}\hat{{\textbf y}}\end{aligned}$$

We then define the complex coefficients as ${a_o} = ({kL/2} ){E_o}{e^{i\Delta }}$ and ${a_e} = {E_o}$. Then, the time-averaged radiating power per unit solid angle in the yz-plane ($\phi = \pi /2$) is

(12)$$\Gamma = \frac{{dP}}{{d\Omega }} \propto Re[{{{\textbf E}_{\textbf T}}^{\textbf f} \cdot {{\textbf E}_{\textbf T}}{{^{\textbf f}}^\ast }} ]\propto {C_o}^2{\left|{{E_e}sin\theta - i\frac{{kL}}{2}{E_o}sin\theta cos\theta {e^{i\Delta }}} \right|^2} = {|{{a_e}sin\theta - i{a_o}sin\theta cos\theta } |^2}$$

When there is a phase shift of π/2 between the even and odd modes, ${a_e}$ and $i{a_o}$ become real values. The steering angle can be determined by finding the angle at which the radiating power is maximum,

(13)$$\begin{aligned} \frac{{d\Gamma }}{{d\theta }} &= 2sin\theta cos\theta {({{a_e} - i{a_0}cos\theta } )^2} + 2i{a_0}si{n^3}\theta ({{a_e} - i{a_0}cos\theta } )\\ &= 2sin\theta ({{a_e} - i{a_0}cos\theta } )({2i{a_0}co{s^2}\theta - {a_e}cos\theta - i{a_0}} ) \\ &= 0 \end{aligned}$$

One of the cases that satisfies the above equation is

(14)$$2i{a_0}co{s^2}\theta - {a_e}cos\theta - i{a_0} = 0\; \Leftrightarrow cos\theta = \; \frac{{{a_e}}}{{4i{a_o}}}\left( {1 - \sqrt {1 + 8\frac{{{{({i{a_o}} )}^2}}}{{a_e^2}}} } \right)$$

Of the two cases, the latter case represents steering behavior. Here, we define the steering angle as $\delta = \pi /2 - \theta $

(15)$$\delta \textrm{ = }\frac{\pi }{2}\textrm{ - }{\cos ^{ - 1}}\left( {\frac{{{a_e}}}{{4i{a_o}}}\left( {1 - \sqrt {1 + 8\frac{{{{({i{a_o}} )}^2}}}{{a_e^2}}} } \right)} \right)$$

If us assume $\delta $ is small, the following expression is obtained

(16)$$|\delta |\approx \frac{{{a_e}}}{{4i{a_o}}}\left( {4\frac{{{{({i{a_o}} )}^2}}}{{a_e^2}}} \right) = \frac{{i{a_o}}}{{{a_e}}}$$

This represents the fact that the steering angle increases as the odd mode is excited more.

Appendix D: Dipole oscillation modeling of slot-antenna modes

According to the surface equivalence principle, the fields outside an imaginary infinite 2D-plane are obtained by placing electric- and magnetic-current densities over the surface. For a small aperture in an infinitely extended perfect electric conductor, the aperture can be replaced solely by the magnetic surface current density (${{\textbf M}_{\textbf s}}$). Here, the magnetic surface current density can be determined with the knowledge of the tangential electric field (${{\textbf E}_{\textbf t}}$) on the surface of the aperture, ${{\textbf M}_{\textbf s}} ={-} 2\hat{{\textbf n}} \times {{\textbf E}_{\textbf t}}$, where $\hat{{\textbf n}}$ is the normal vector directed into the region of interest. Then, in the presence of the surface magnetic-current source, the electric field is given by

(17)$${\textbf E} ={-} \nabla \times \smallint \frac{{{{\textbf M}_{\textbf s}}{e^{ik|{{\textbf r} - {\textbf r^{\prime}}} |}}}}{{4\pi |{{\textbf r} - {\textbf r^{\prime}}} |}}\textrm{d}{A^{\prime}}$$

where the unprimed and the primed coordinates represent the observation point and the source point, respectively. $|{{\textbf r} - {\textbf r^{\prime}}} |$ is the distance from the source to the observation point. In the radiation zone $|{\textbf r} |\gg |{{\textbf r^{\prime}}} |$, we may approximate the above equation using Taylor series

(18)$$\frac{{{e^{ik|{{\textbf r} - {\textbf r^{\prime}}} |}}}}{{|{{\textbf r} - {\textbf r^{\prime}}} |}} \approx \frac{{{e^{ikr}}{e^{ - ik\hat{{\textbf r}}\Delta {\textbf r^{\prime}}}}}}{r} \approx \frac{{{e^{ikr}}}}{r}({1 - ik\hat{{\textbf r}}\cdot {\textbf r^{\prime}} + \cdots } )$$

Additionally, by replacing $\nabla $ with $i{\textbf k}$ in the radiation zone, the electric field becomes

(19)$$\begin{aligned} {\textbf E} &={-} \nabla \times \smallint \frac{{{{\textbf M}_{\textbf s}}{e^{ikr}}}}{{4\pi r}}({1 - ik\hat{{\textbf r}}\cdot {\textbf r^{\prime}} + \cdots } )\textrm{d}{A^{\prime}} \\ &={-} i{\textbf k} \times \frac{{{e^{ikr}}}}{{4\pi r}}\smallint {{\textbf M}_{\textbf s}}({1 - ik\hat{{\textbf r}}\cdot {\textbf r^{\prime}} + \cdots } )\textrm{d}{A^{\prime}} \end{aligned}$$

In the far-zone, we assume that the leading term in the integration is dominant, $k\hat{{\textbf r}}\cdot {\textbf r^{\prime}} \ll 1$

(20)$${\textbf E} = ik\hat{{\textbf r}} \times \frac{{{e^{ikr}}}}{{2\pi r}}\smallint ({{\textbf M}_{\textbf s}} = \hat{{\textbf n}} \times {{\textbf E}_{\textbf t}})\textrm{d}{A^{\prime}}$$

Comparing Eq. (20) with the radiation pattern for magnetic dipole radiation,

(21)$${\textbf E} = \frac{{ - {Z_0}}}{{4\pi }}{k^2}({\hat{{\textbf r}} \times {\textbf m}} )\frac{{{e^{ikr}}}}{r}$$

This led us to the equation for the effective magnetic dipole moment, which is expressed in terms of integrals of the tangential electric field in the aperture.

(22)$${{\textbf m}_{{\textbf eff}}} = \frac{{ - 2i}}{{k{Z_0}}}\smallint (\hat{{\textbf n}} \times {{\textbf E}_{\textbf t}})\textrm{d}{A^{\prime}} = \frac{2}{{i{\mu _0}\omega }}\smallint (\hat{{\textbf n}} \times {{\textbf E}_{\textbf t}})\textrm{d}{A^{{\prime}}}$$

where ${Z_0}$ is the characteristic impedance and $\omega $ is the angular frequency of the light.

For the narrow-gap metal-insulator metal (MIM) trench, the electric field distribution follows the electric field polarized in the x-direction, due to the strongly enhanced field in that direction. The electric fields, polarized in the x-direction, of the even and odd modes are shown in Fig. 7. For the MIM rectangular trench, we approximated the electric field distribution (${{\textbf E}_{{\textbf even}}}$ and ${{\textbf E}_{{\textbf odd}}}$) of the resonant modes with sinusoidal functions, as follows:

(23)$${{\textbf E}_{{\textbf even}}} = {E_e}sin\frac{{\pi z}}{L}\hat{{\textbf x}}$$

(24)$${{\textbf E}_{{\textbf odd}}} = {E_o}sin\frac{{2\pi z}}{L}\hat{{\textbf x}}$$

where E_o and E_e represent the electric field amplitude at the antinode positions.

 

Fig. 7. Eigenmodes of the rectangular-slot antenna and corresponding effective dipole oscillations.

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As explained in the above section, the effective dipole moment can be obtained by integrating the tangential field in the rectangular aperture. Thus, the even mode can be modeled by a single dipole oscillation, and the corresponding dipole moment (${{\textbf m}_{\textbf e}}$) is given by

(25)$${{\textbf m}_{\textbf e}} = \frac{2}{{i{\mu _0}\omega }}\smallint (\hat{{\textbf n}} \times {{\textbf E}_{{\textbf even}}})\textrm{d}{A^{\prime}} = \frac{2}{{i{\mu _0}\omega }}\mathop \smallint \limits_0^w \mathop \smallint \limits_0^L {E_e}sin\frac{{\pi z}}{L}\textrm{d}z\textrm{d}x = \frac{{2w{E_e}}}{{i{\mu _0}\omega }}\left( {\frac{{2L}}{\pi }} \right)\hat{{\textbf z}} = \alpha {E_e}\hat{{\textbf z}}$$

Then, we model the odd mode using two dipole oscillations. To obtain each effective dipole moment (${{\textbf m}_{\textbf o}}$), we divide the area into two sections as follows:

(26)$${\textbf m}_{\textbf o}^1 = \frac{2}{{i{\mu _0}\omega }}\mathop \smallint \limits_0^w \mathop \smallint \limits_0^{L/2} {E_o}sin\frac{{2\pi z}}{L}\textrm{d}z\textrm{d}x = \frac{{2w{E_o}}}{{i{\mu _0}\omega }}\left( {\frac{L}{\pi }} \right)\hat{{\textbf z}} = \alpha \frac{{{E_o}}}{2}\hat{{\textbf z}}$$

(27)$${\textbf m}_{\textbf o}^2 = \frac{2}{{i{\mu _0}\omega }}\mathop \smallint \limits_0^w \mathop \smallint \limits_{L/2}^L {E_o}sin\frac{{2\pi z}}{L}\textrm{d}z\textrm{d}x ={-} \frac{{2w{E_o}}}{{i{\mu _0}\omega }}\left( {\frac{L}{\pi }} \right)\hat{{\textbf z}} ={-} \alpha \frac{{{E_o}}}{2}\hat{{\textbf z}}$$

where $\alpha $ is the effective polarizability.

Funding

National Research Foundation of Korea (2017R1D1A1B03036010, 2019M3E4A1078663, 2020R1A2C2010967); KU-KIST School (2E30620-20-051); Korea University (Korea University Grant); Korea Institute of Science and Technology (KU-KIST School Project).

Disclosures

The authors declare no conflicts of interest.

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