An Efficient Method for Light Beaming from Subwavelength Slits Surrounded by Surface Gratings

Abstract

This paper investigates the directional beaming of metallic subwavelength slits surrounded by dielectric gratings. The design of the structure for light beaming was formulated as an optimization problem for the far-field angular transmission. A vertical mode expansion method was developed to solve the diffraction problem, which was then integrated into a genetic algorithm and an active set method to obtain the optimal result. By using the proposed method for a two-slit structure, we demonstrate that both the single- and dual-beaming effects can be efficiently achieved. Moreover, the beaming directions can be flexibly adjusted and precisely controlled.

1. Introduction

Since the discovery of extraordinary optical transmission through subwavelength hole arrays perforated in metal films [1], optical field diffraction by small apertures has been intensively studied from both the theoretical and application aspects. One of the interesting results is the light beaming from a subwavelength aperture with metal grating structures discovered by Lezec et al. [2]. Springing from the demonstration of this beaming effect, many efforts have been made to realize and optimize directional beaming. The most-popular models are the grating-based beaming structures, such as a single slit surrounded by metallic grooves or dielectric gratings [3,4,5,6,7,8,9], subwavelength slit arrays [10,11,12], and annular aperture array structures [13,14,15,16], to name a few. These methods generally rely on the excitation of surface plasmon polaritons (SPPs) on the corrugated metal surface and reradiated in a specific direction. In recent years, there have also been interesting methods for manipulating beams that involve the use of metasurfaces and photonic crystals, etc. [17,18,19,20,21,22].

The physical mechanism for light beaming from grating-based beaming structures has been thoroughly investigated [3,23,24,25,26,27]. These studies provide valuable information on choosing some key parameters for light beaming structures. However, the effects of many geometrical parameters are interlinked, and efficient numerical methods are still required to optimize the performance of light beaming structures in practical applications. This would facilitate the efficient and precise manipulation of light beams. General computational methods, such as the finite-difference time domain (FDTD) method and the finite-element method (FEM), have been used to study this problem [9,20]. However, they are time consuming, since a small grid or mesh size must be taken, which gives rise to a large indefinite linear system and is expensive to solve.

In this work, we developed a vertical mode expansion method (VMEM) for the light beaming structure, as shown in Figure 1. The structure can be separated into a number of regions, where each region is piecewise uniform along the vertical direction. In each uniform region, using a proper reference solution, the wave field can be expanded by eigenmodes of a one-dimensional differential operator. The expansion coefficients are solved from a linear system obtained by matching the wave field at the interfaces between neighboring regions. Since only the vertical direction is discretized, the method is extremely fast and efficient, which provides a powerful computational engine for the optimization of light beaming structures.

The aim of this paper was to demonstrate that beaming of light can be achieved by optimizing the far-field angular transmission over geometrical parameters that define the optical structures, and the beaming directions can be precisely controlled. In this work, the problem was formulated as a bound constrained optimization problem. The objective function, depending on practical purposes, was formulated as a functional of the far-field angular transmission. The problem was solved by the integration of the VMEM and optimization algorithms. More precisely, the objective function was evaluated using the VMEM, and the genetic algorithm and active set method were combined to obtain the optimal result.

The rest of this paper is organized as follows. In Section 2, we formulate the diffraction problem. In Section 3, we develop a vertical mode expansion method. In Section 4, the far-field angular transmission is defined and evaluated analytically by the VMEM. In Section 5, a two-slit beaming structure is considered, and both the single- and dual-beaming effects are realized to illustrate the effectiveness of the method we developed. The paper is concluded with some general remarks in Section 6.

2. Problem Formulation

We considered the diffraction of light beaming structures with metallic subwavelength slits surrounded by gratings in the output surface. The schematic approach of the structure with a single slit is shown in Figure 1.

In the Cartesian coordinate system, the structure is invariant in y and parallel to the $xy$ plane. The lower boundary is placed on or slightly above $z=0$, and we assumed the half-plane ${\mathbb{R}}_{-}^2=\{(x,z)\in {\mathbb{R}}^2:z<0\}$ is the free space. The metal slab was assumed to be thick enough that it is impenetrable. The structure is multi-layered, and it can be separated into a number of regions $S_l,l=1,2,\cdots ,N$; in each region, the material is uniform in the x-direction. Denoting $x_1=-\infty$ and $x_{N+1}=+\infty$, then $S_l$ can be represented as

$$S_l=\{(x,z):x_l<x<x_{l+1},z\in \mathbb{R}\},l=1,2,\cdots ,N.$$

(1)

In each region $S_l$, the materials vary only in the vertical direction; let $\epsilon$ and $\mu$ be the relative permittivity and the relative permeability, respectively, then

$$\epsilon ={\epsilon }^{\left(l\right)}\left(z\right),\mu ={\mu }^{\left(l\right)}\left(z\right),\forall (x,z)\in S_l.$$

(2)

Above the structure, we specified a TM polarized incident plane wave $H_y^{\left(i\right)}=e^{i(\alpha x-\beta z)}$, where $\alpha =k_{0}sin{\theta }_{in}$ and $\beta =k_{0}cos{\theta }_{in}$, ${\theta }_{in}\in (-\pi /2,\pi /2)$ is the angle of incidence with respect to the negative z-axis, and $k_0$ is the wavenumber of the free space. For TM polarization ($H_y,E_x$ and $E_z$ are the nonzero components of the electromagnetic fields), the total field $H_y$ satisfies the following Helmholtz equation:

$$\frac{\partial }{\partial x}\left(\frac{1}{\epsilon }\frac{\partial H_y}{\partial x}\right)+\frac{\partial }{\partial z}\left(\frac{1}{\epsilon }\frac{\partial H_y}{\partial z}\right)+k_0^2\mu H_y=0.$$

(3)

To address the diffraction problem, we developed a new vertical mode expansion method, which provides a powerful tool for computing and optimizing plasmonic structures.

3. Vertical Mode Expansion Method

In this section, we developed a vertical mode expansion method for the problem formulated above. The light beaming structure is multi-layered and separated into a number of regions $S_l$, $l=1,2,\cdots ,N$. The basic idea of the vertical mode expansion method is to represent the total field in each region as the superposition of a reference field $H_y^{\left(l\right)}$ and a sub-field $H_y^{\left(s\right)}=H_y-H_y^{\left(l\right)}$, i.e.,

$$H_y=H_y^{\left(l\right)}+H_y^{\left(s\right)}.$$

(4)

The reference field $H_y^{\left(l\right)}$ was chosen such that it is a solution excited by the incident wave on a hypothetical infinite 1D structure with the same profile as $S_l$. Thus, $H_y^{\left(l\right)}$ also satisfies the Helmholtz equation and can be written as

$$H_y^{\left(l\right)}=e^{i\alpha x}{{\tilde{H}}_y}^{\left(l\right)}\left(z\right),$$

(5)

where ${{\tilde{H}}_y}^{\left(l\right)}\left(z\right)$ is a one-dimensional function of z and can be evaluated analytically; the details can be found in [28]. With the properly chosen reference field, the sub-field $H_y^{\left(s\right)}$ is shown to be outgoing as $z\to \pm \infty$ and satisfies the homogeneous Helmholtz equation in $S_l$:

$$\frac{\partial }{\partial x}\left(\frac{1}{{\epsilon }^{\left(l\right)}}\frac{\partial H_y^{\left(s\right)}}{\partial x}\right)+\frac{\partial }{\partial z}\left(\frac{1}{{\epsilon }^{\left(l\right)}}\frac{\partial H_y^{\left(s\right)}}{\partial z}\right)+k_0^2{\mu }^{\left(l\right)}{H}_y^{\left(s\right)}=0.$$

(6)

We then performed a separation of variables for $H_y^{\left(s\right)}$ such that

$$H_y^{\left(s\right)}={\varphi }^{\left(l\right)}\left(z\right)V^{\left(l\right)}\left(x\right),$$

(7)

then the function ${\varphi }^{\left(l\right)}$ and $V^{\left(l\right)}$ satisfy

$$\begin{array}{cc}\hfill {\epsilon }^{\left(l\right)}\frac{d}{dz}\left(\frac{1}{{\epsilon }^{\left(l\right)}}\frac{d{\varphi }^{\left(l\right)}}{dz}\right)+k_0^2{\epsilon }^{\left(l\right)}{\mu }^{\left(l\right)}{\varphi }^{\left(l\right)}& ={\left[{\eta }^{\left(l\right)}\right]}^2{\varphi }^{\left(l\right)},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \frac{d^2{V}^{\left(l\right)}}{d{x}^2}+{\left[{\eta }^{\left(l\right)}\right]}^2{V}^{\left(l\right)}& =0.\hfill \end{array}$$

(9)

Since the vertical axis z is unbounded, the PML technique can be applied to truncate z to a finite interval. A PML corresponds to a complex coordinate stretching, which replaces z by a complex $\widehat{z}$ [29]. In practice, we only need to replace $dz$ in (8) by $s\left(z\right)dz$ for some complex function $s\left(z\right)$. With the truncation of z by PMLs, the eigenvalue problem (8) can be solved by a pseudospectral method as in [30,31], and a discrete sequence of eigenpairs $\{{\eta }_j^{\left(l\right)},{\varphi }_j{\left(z\right)}^{\left(l\right)}\}$ for $j=1,2,3,\cdots$ can be obtained. Based on those vertical modes, the total field in region $S_l,l=1$, $2,\cdots ,N$ can be represented as

$$\begin{array}{cc}\hfill H_y& =H_y^{\left(l\right)}+\sum _{j=1}^{\infty }{\varphi }_j^{\left(l\right)}{V}_j^{\left(l\right)}.\hfill \end{array}$$

(10)

The other two components $E_x$ and $E_z$ can then be obtained from Maxwell’s equations, and we have

$$\begin{array}{cc}\hfill E_x& =E_x^{\left(l\right)}+\frac{1}{i{k}_0{\epsilon }^{\left(l\right)}}\sum _{j=1}^{\infty }\frac{d{\varphi }_j^{\left(l\right)}}{\partial z}{V}_j^{\left(l\right)},\hfill \\ \hfill E_z& =E_z^{\left(l\right)}-\frac{1}{i{k}_0{\epsilon }^{\left(l\right)}}\sum _{j=1}^{\infty }{\varphi }_j^{\left(l\right)}\frac{d{V}_j^{\left(l\right)}}{dx},\hfill \end{array}$$

(11)

where $E_x^{\left(l\right)}=e^{i\alpha x}{\tilde{E}}_z\left(z\right)$ and $E_z^{\left(l\right)}=e^{i\alpha x}{\tilde{E}}_z\left(z\right)$ are components of the reference solution. The general solution of $V_j^{\left(l\right)}$ in (9) is simply

$$V_j^{\left(l\right)}=a_j^l{e}^{i{\eta }_j^{\left(l\right)}(x-x_l)}+b_j^l{e}^{-i{\eta }_j^{\left(l\right)}(x-x_{l+1})},l=1,\cdots ,N,$$

(12)

where $a_j^l$ and $b_j^l$ are the coefficients of the left-going and right-going eigenmodes in region $S_l$. Noticing that, in region $S^{\left(1\right)}$ and $S^{\left(N\right)}$, the fields are outgoing as $x\to \pm \infty$, thus $a_j^1=b_j^N=0$, $j=1,2,3,\cdots$. A linear system:

$$A\mathit{u}=\mathit{c}$$

(13)

can be established by matching the continuity condition of $H_y$ and $E_z$ on the vertical boundaries between the interfaces $x_l$, $l=2,3,\cdots ,N$. The unknown $\mathit{u}$ is a vector for the coefficients $a_j^l$ and $b_j^l$. In the fully discretized version, the variable z is discretized by $N_z$ points $z_n$ for $1\le n\le N_z$. We obtain $N_z$ numerical modes, then the index j ranges from 1 to $N_z$, and A is a $2(N-1)N_z\times 2(N-1)N_z$ matrix. The right-hand side of (13) is related to the difference between the reference solutions in $S_l$ and $S_{l-1}$. Notice that the reference solutions for regions $S_l$ and $S_{l-1}$ are different. Therefore, the right-hand side of (13) is nonzero in general. After the linear system is solved, the field components in all regions can be easily evaluated.

4. Far-Field Angular Transmission

For light beaming, it is important to calculate the far-field angular transmission, and we derived it in this section. As depicted in Figure 1, the total transmitted power T can be represented as the limit of the integral of Poynting vector $\mathit{S}\left(\mathit{r}\right)$ over the half circle ${\mathcal{B}}_r$:

$$T=\underset{r\to \infty }{lim}{\int }_{{\mathcal{B}}_r}\mathit{S}·\widehat{\mathit{r}}ds={\int }_{-\pi /2}^{\pi /2}\underset{r\to \infty }{lim}r\mathit{S}·\widehat{\mathit{r}}d\theta ,$$

(14)

where $r=\left|\mathit{r}\right|$ is the radius of the circle and $\widehat{\mathit{r}}=\mathit{r}/r$ is the unit vector from the origin to the point $\mathit{r}$. The far-field angular transmission can, thus, be defined as $P\left(\theta \right)={lim}_{r\to \infty }r\mathit{S}·\widehat{\mathit{r}}$; since the far fields are transverse to $\widehat{r}$, the wave energy propagates only in the radial direction, and we have

$$P\left(\theta \right)=\frac{1}{2}\underset{r\to \infty }{lim}r{\left|H_y\left(\mathit{r}\right)\right|}^2.$$

(15)

For any point $\mathit{r}\in {\mathbb{R}}_{-}^2$, the magnetic field $H_y\left(\mathit{r}\right)$ is related to the electric field $E_x$ on $z=0$ by

$$H_y\left(\mathit{r}\right)=\frac{k_0}{2}{\int }_{-\infty }^{\infty }{H}_0^{\left(1\right)}(k_0|\mathit{r}-x\widehat{\mathit{x}}\left|\right)E_x(x,0)dx,$$

(16)

where $H_0^{\left(1\right)}$ is the zeroth-order Hankel function of the first kind and $\widehat{\mathit{x}}$ is the unit vector of the x-axis [32]. The far-field can then be simplified by using the asymptotic expression for the Hankel function, and we have

$$H_y\left(\mathit{r}\right)\approx \sqrt{\frac{k_0}{2\pi r}}{e}^{-i\frac{\pi }{4}}{\int }_{-\infty }^{\infty }{E}_x(x,0)e^{-i{k}_{0}sin\theta x}dx.$$

(17)

By substituting (17) into (15), the far-field angular transmission can, thus, be evaluated as

$$P\left(\theta \right)=\frac{k_0}{4\pi }{\left|{\int }_{-\infty }^{\infty }{E}_x(x,0)e^{-i{k}_{0}sin\theta x}dx\right|}^2=\frac{k_0}{4\pi }{\left|J\left(\theta \right)\right|}^2,$$

(18)

where we denote the integral as $J\left(\theta \right)={\int }_{-\infty }^{\infty }{E}_x(x,0)e^{-i{k}_{0}sin\theta x}dx$. For slit–groove structures, i.e., slits surrounded by metallic grooves, the evaluation of $J\left(\theta \right)$ can be simplified if a PEC or a surface impedance boundary condition is applied, since by placing the bottom surface of the structure on $z=0$, the infinite integral interval can be reduced to the finite opening areas of the slits and grooves [3,33]. However, for a real metal structure or a structure with a grating protruding through the metal surface, the integral has to be evaluated on the infinite interval. In such cases, if the domain is truncated or discretized in the x-direction, the evaluation will be impossible or inefficient. One attractive advantage of the VMEM for light beaming is that $J\left(\theta \right)$ can be calculated analytically.

In the VMEM, the fields on $z=0$ are given by the expansions (10) and (11), and the integral can be written as the summation over each region; we obtain

$$\begin{array}{c}\hfill J\left(\theta \right)=\sum _{l=1}^N\left\{\widetilde{E_x}\left(0\right)I_1^l+\frac{1}{i{k}_0{\epsilon }_0}\sum _{j=1}^{\infty }\frac{d{\varphi }_j^{\left(l\right)}}{dz}\left(0\right)I_{2,j}^l\right\},\end{array}$$

(19)

where

$$\begin{array}{cc}\hfill I_1^l& ={\int }_{x_l}^{x_{l+1}}{e}^{i\alpha x}{e}^{-i{k}_{0}sin\theta x}dx,\hfill \\ \hfill I_{2,j}^l& ={\int }_{x_l}^{x_{l+1}}{V}_j^{\left(l\right)}{e}^{-i{k}_{0}sin\theta x}dx.\hfill \end{array}$$

(20)

By substituting the general solution of $V_j^{\left(l\right)}$ (12) into (20), the integrals above can be evaluated analytically. More precisely, in region $S_l,l=2,3,\cdots ,N-1$, we have

$$\begin{array}{cc}\hfill I_1^l& =\frac{e^{i(\alpha -k_{0}sin\theta )x_{l+1}}-e^{i(\alpha -k_{0}sin\theta )x_l}}{i(\alpha -k_{0}sin\theta )},\hfill \\ \hfill I_{2,j}^l& =a_j^l\frac{e^{i({\eta }_j^{\left(l\right)}-k_{0}sin\theta ){\delta }_l}-1}{i({\eta }_j^{\left(l\right)}-k_{0}sin\theta )}{e}^{-i{k}_{0}sin\theta x_l}\hfill \\ \hfill & +b_j^l\frac{e^{i({\eta }_j^{\left(l\right)}+k_{0}sin\theta ){\delta }_l}-1}{i({\eta }_j^{\left(l\right)}+k_{0}sin\theta )}{e}^{-i{k}_{0}sin\theta x_{l+1}},\hfill \end{array}$$

(21)

where ${\delta }_l=x_{l+1}-x_l$ is the width of region $S_l$. In the first and last regions, the widths of the regions are infinite, and the integrals should be considered separately. Since the structure is impenetrable in $S_1$ and $S_N$, the reference fields vanish at $z=0$ in these two regions, and we can simply let $I_1^l=0$, for $l\in \{1,N\}$. In addition, the fields in these two regions are outgoing as $x\to \pm \infty$; thus, $a_j^1=0$ and $b_j^N=0$, and this yields

$$\begin{array}{cc}\hfill I_{2,j}^1& =b_j^1\frac{i{e}^{-i{k}_{0}sin\theta x_2}}{{\eta }_j^{\left(1\right)}+k_{0}sin\theta },\hfill \\ \hfill I_{2,j}^N& =a_j^N\frac{i{e}^{-i{k}_{0}sin\theta x_N}}{{\eta }_j^{\left(N\right)}-k_{0}sin\theta }.\hfill \end{array}$$

(22)

After the linear system (13) is solved, $J\left(\theta \right)$ can then be evaluated. Consequently, the far-field angular transmission $P\left(\theta \right)$ is obtained from (18), and the total transmitted power is

$$T=\frac{k_0}{4\pi }{\int }_{-\pi /2}^{\pi /2}{\left|J\left(\theta \right)\right|}^{2}d\theta .$$

(23)

Light beaming can be achieved by manipulating the far-field angular transmission, and we demonstrate this in the next section.

5. Optimization and Discussion

In this section, we considered the diffraction of a two-slit structure and demonstrate that light beaming can be achieved by manipulating the far-field angular transmission. Depending on the practical purposes, the objective functions can be formulated as functionals of the far-field angular transmission $P\left(\theta \right)$. Light beaming can then be achieved by optimizing the objective function over incident angles and geometrical parameters that define the plasmonic structures, and the beaming angles can be precisely controlled.

We considered a structure with two slits surrounded by a few surface dielectric gratings, as depicted in Figure 2. The thickness of the metal slab is 0.3 μm, and the widths of the slits are $d_1$ and $d_2$. We set $d_1+d_2=0.1$ μm to maintain a constant total slit width. The slits are surrounded by dielectric gratings in the output surface, with a height of $t=0.08$ μm. The structure has five, three, and five dielectric bars on the left, middle, and right part, respectively. The dielectric bars are periodically arranged in each part, with a fill factor of 0.5 and periods of $p_1,p_0$ and $p_2$. The offsets distances are denoted as $l_i(i=0,1,2,3)$ and are depicted in Figure 2. A TM polarized plane wave with wavelength $\lambda =0.532$ μmm is incident on the top surface of the structure. At this wavelength, the refractive index of the metal slab and the dielectric bar are $0.13+3.19i$ and 1.72, respectively. We adopted a Cartesian coordinate system with the bottom of the structure at $z=0$ and the origin at the center of the middle part of the structure. For this structure, our purpose was to show that both the single- and dual-directional beaming effects can be efficiently achieved, and the beaming angles can be precisely controlled.

We first considered the single-beaming effect; it is expected to achieve highly efficient directional beaming with any desired angle by optimizing the structure parameters and incident angle. We define the normalized far-field angular transmission as $P\left(\theta \right)/T$, which describes the far-field distribution, where T is the total transmission, as given in (23). For a given predefined beaming angle $\theta$, we denote the parameters to be optimized as $\mathit{x}=[p_0,p_1,p_2,l_0,l_1,l_2,l_3,d_1,{\theta }_{in}]$. The optimization problem can then be formulated as

$$\underset{\mathit{x}\in \mathsf{\Lambda }}{min}-P(\theta ,\mathit{x})/T,$$

(24)

where $\mathsf{\Lambda }$ is the feasible set of $\mathit{x}$. In this work, $\mathsf{\Lambda }$ represents the bound constraints of the variables, and we set the search ranges as follows: $p_i(i=0,1,2)\in (0,2\lambda ]$, $l_i(i=0,1,2,3)\in (0,2\lambda ]$, $d_1\in (0,0.1)$, and ${\theta }_{in}\in (-\pi /2,\pi /2)$. The optimization problem (24) was solved by the integration of the VMEM and efficient optimization algorithms. Specifically, the objective function was evaluated using the VMEM, and a genetic algorithm was employed to work out the optimal result to the first significant decimal place. This result was then used as an initial guess, and the final optimal result was obtained using the active set method [34].

For illustration purposes, we considered predefined beaming angles ranging from $0^{\circ }$ to ${30}^{\circ }$ in steps of ${10}^{\circ }$, using the proposed method, the final optimal results were obtained and are listed in

Table 1. The optimum structural parameters for single-beaming effect.

$\mathit{\theta }$	${\mathit{p}}_0$	${\mathit{p}}_1$	${\mathit{p}}_2$	${\mathit{l}}_0$	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{d}}_1$	${\mathit{\theta }}_{\mathit{i}\mathit{n}}$
$0^{\circ }$	0.427	0.400	0.400	0.203	0.157	0.157	0.203	0.0500	$0^{\circ }$
${10}^{\circ }$	0.418	0.359	0.439	0.132	0.027	0.130	0.171	0.0346	$45.{54}^{\circ }$
${20}^{\circ }$	0.343	0.306	0.510	0.261	0.152	0.138	0.004	0.0588	$0.{34}^{\circ }$
${30}^{\circ }$	0.312	0.285	0.592	0.268	0.466	0.232	0.000	0.0678	$8.{97}^{\circ }$

. It was found that the periods of the gratings have some regularity. As the beaming angle $\theta$ increased, the period $p_2$ became larger, while $p_0$ and $p_1$ became smaller. Furthermore, for all the off-axis cases (i.e., $\theta \ne 0^{\circ }$), $p_1<p_2$. The periods of the gratings play a crucial role in determining the beaming angles.

In order to directly observe the beaming effect, the magnetic field intensity $|H_y{|}^2$ and far-field angular transmission for the optimal cases are shown in Figure 3 and Figure 4. Figure 3a corresponds to the on-axis beaming case (i.e., $\theta =0^{\circ }$), while Figure 3b–d correspond to the off-axis beaming cases with $\theta ={10}^{\circ },{20}^{\circ }$, and ${30}^{\circ }$, respectively. The far-field angular spectrum in shown in Figure 4; it can be seen that the angles correspond to the peak intensities coinciding with the predefined angles; this means that the beaming angles are precisely controlled.

We next considered the dual-beaming effect. For light beaming of a two-slit plasmonic structure, a natural question to consider is whether dual-beams can be realized and well-separated. While a method for rendering bundle beams from multiple subwavelength slits was proposed in [12], it relies on the destructive interference effects of beams, and the resulting beaming angles can only be very close to the on-axis direction and cannot be controlled. In this paper, we propose a method for achieving separated beams by optimizing the far-field angular transmission. Suppose the predefined two beaming angles are ${\theta }_1$ and ${\theta }_2$; if the far-field angular transmission has two separated peaks at these two angles and the intensities are very similar to each other, then dual-separated beams can be realized. To achieve this, we formulated the optimization problem as follows:

$$\underset{\mathit{x}\in \mathsf{\Lambda }}{min}-\frac{P({\theta }_1,\mathit{x})+P({\theta }_2,\mathit{x})-P(\frac{{\theta }_1+{\theta }_2}{2},\mathit{x})}{T}+\gamma \left|P({\theta }_2,\mathit{x})-P({\theta }_1,\mathit{x})\right|,$$

(25)

where the feasible set $\mathsf{\Lambda }$ and the parameter $\mathit{x}$ to be optimized are the same as the single-beaming cases, and $\gamma >0$ is a penalty parameter. The first term of the objective function aims to produce two separated peaks in the far-field angular transmission at ${\theta }_1$ and ${\theta }_2$, while the second term penalizes differences in the peak intensities to ensure that they are similar. The optimization problem (25) can be solved using the same method as in the single-beaming cases.

To illustrate the results, we considered both symmetric and non-symmetric dual-beaming effects (i.e., ${\theta }_1+{\theta }_2=0$ and ${\theta }_1+{\theta }_2\ne 0$, respectively). The optimal results are listed in

Table 2. The optimum structural parameters for dual-beaming effects.

$({\mathit{\theta }}_1,{\mathit{\theta }}_2)$	${\mathit{p}}_0$	${\mathit{p}}_1$	${\mathit{p}}_2$	${\mathit{l}}_0$	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{d}}_1$	${\mathit{\theta }}_{\mathit{i}\mathit{n}}$
$(-5^{\circ },5^{\circ })$	0.404	0.396	0.396	0.457	0.148	0.148	0.457	0.0501	$23.{78}^{\circ }$
$(-{10}^{\circ },{10}^{\circ })$	0.446	0.451	0.451	0.148	0.098	0.098	0.148	0.0498	$-10.{37}^{\circ }$
$(-{15}^{\circ },5^{\circ })$	0.326	0.477	0.411	0.457	0.125	0.157	0.212	0.0485	$26.{64}^{\circ }$
$({10}^{\circ },{20}^{\circ })$	0.102	0.332	0.469	0.209	0.018	0.224	0.917	0.0580	$-22.{43}^{\circ }$

. To directly observe the beaming effect, we present the magnetic field intensity $|H_y{|}^2$ and far-field angular transmission for the optimal cases in Figure 5 and Figure 6. Symmetric dual-beaming effects are shown in Figure 5a,b, with beaming angles of $(-5^{\circ },5^{\circ })$ and $(-{10}^{\circ },{10}^{\circ })$, respectively. Non-symmetric dual-beaming effects are shown in Figure 5c,d, with beaming angles of $(-{15}^{\circ },5^{\circ })$ and $({10}^{\circ },{20}^{\circ })$. It can be seen that the beaming angles can be flexibly adjusted to reside on the same or opposite side with respect to the on-axis direction. Figure 6 presents the far-field angular spectrum, in which two separated peaks are clearly displayed with almost identical intensity. Additionally, the positions of the peaks closely match the predefined angles, indicating precise control of the beaming angles.

Finally, we give some discussions on the physical mechanism of the directional beaming in the design of plasmonic structures. Firstly, the two slits were illuminated with TM polarized incident light from above, with the slit width much smaller than a wavelength. Consequently, only fundamental modes were considered to exist in the slits and the propagation constants were determined by the slit widths [35]. Secondly, the transmitted light through the slits diffracted into gratings, and under certain resonance conditions, surface plasmon polaritons (SPPs) were excited, with energy confined to the bottom of the dielectric bars. The SPPs were then radiated into free space, forming the directional beaming. In this study, it was evident that the incident angle and slit width were the parameters used to tune the optical phase retardation at the exits of the slits. The beaming angles are mainly determined by the periods of the gratings from the grating equation [35]. More explicitly, for a predefined beaming angle $\theta$, the periods of the gratings are related to $\theta$ approximately as follows:

$$p_1\approx \frac{2\pi }{k_{spp}+k_{0}sin\theta },p_2\approx \frac{2\pi }{k_{spp}-k_{0}sin\theta }.$$

(26)

Here, $k_{spp}$ denotes the wave number of the excited SPP. This explains the data in Table 1, which show that, as the beaming angle $\theta$ increases, $p_1$ decreases and $p_2$ increases, and $p_1<p_2$ for $\theta >0$. In fact, the SPP in the left part of the structure propagates left, with the direction of the radiated power determined by $p_1$ as in the first equation of (26). Similarly, the SPP in the right part of the structure propagates right, and the direction of the radiated power is determined by $p_2$. When the left-going and right-going SPPs are radiated in the same direction, a single-beaming effect was observed. For the dual-beaming cases, however, the SPPs are radiated into two different directions.

6. Conclusions

In this paper, we investigated the phenomenon of light beaming in plasmonic structures comprised of a metallic slab with subwavelength slits surrounded by gratings. The design of the structure for light beaming was formulated as a bound constrained optimization problem. To solve the diffraction problem efficiently, we developed a vertical mode expansion method, which served as a direct solver for the optimization problem. The optimal result was obtained through the integration of the VMEM and efficient optimization algorithms, namely the genetic algorithm and active set method. Numerical experiments demonstrated that light beaming can be achieved by optimizing the far-field angular transmission over the incident angle and the geometrical parameters that define the structure. For a two-slit structure, we showed that both the single- and dual-beaming effects can be realized, and the beaming directions can be flexibly adjusted and precisely controlled. These results have some potential applications in directional antennas, optical sensing, data storage, and so on.