Modeling of Subwavelength Gratings: Near-Field Behavior

Abstract

Subwavelength gratings have received considerable attention in the fields of photonics, optoelectronics, and image sensing. This paper presents simple analytical expressions for the near-field intensity distribution of radiation scattered by these gratings. Our proposed methodology employs a 2D point dipole model and a specialized version of perturbation theory. By validating our models via numerical techniques including boundary and finite element methods, we demonstrate their effectiveness, even for narrow slits.

1. Introduction

Subwavelength gratings (SWGs), which have periods shorter than the wavelength of the radiation they interact with, have receiving growing attention, with a range of applications in diverse fields. Their versatility has been demonstrated in improving the focusing of light beams [1], contributing to advances in silicon photonics [2], enabling the development of optoelectronic devices exploiting surface plasmons [3], serving as mirrors in vertical cavity surface-emitting lasers [4], and playing a crucial role in image sensing [5,6]. These structures offer a cost-effective and compact solution, with potential in manipulating the polarization properties of emitted radiation [7]. They can also enhance the performance of InGaAs infrared polarization imaging [8]. The incorporation of double SWGs has demonstrated promise in creating ultranarrow resonances in the optical response of systems, leading to gain factors of up to ${10}^5$ [9]. In hybrid multiplexing technology, subwavelength gratings (SWGs) play a crucial role in enhancing optical communication capacity by combining multiple wavelengths, modes, and dual polarizations [10]. Moreover, the near-field region’s structure formation can be achieved through laser ablation, facilitated via subwavelength scattering from both dielectric and metallic nano-objects [11]. Broadband achromatic full-Stokes imaging relying on phase metasurface polarimetry is achieved, with crosstalk avoided [12,13].

The theory of electrodynamics diffraction is based on Maxwell’s equations and the decomposition of their solutions into eigenfunctions. Rigorous coupled-wave analysis (RCWA), which was mainly developed for configurations with high symmetry, is part of this theory. Relief profiles with square-wave, triangular, or saw-tooth shapes are some examples of such configurations [14]. Additionally, other noteworthy examples include periodic arrays of parallel infinite circular cylinders [15,16,17]. When analyzing relief profiles and ribbons, plane waves are utilized as eigenfunctions, while cylindrical arrays rely on Bessel or Hankel functions as the foundation.

Various numerical methods are available for analyzing this problem, including the finite-difference time-domain (FDTD) [18], discrete dipole approximation (DDA) [19,20], boundary element method (BEM) [21], and finite element method (FEM) [22]. However, these methods have inherent limitations and potentially introduce numerical errors, making it difficult to compare approaches. Comparing methods with each other and with theoretical calculations is crucial. Approximation formulas can also be used to analyze the dependence of scattered radiation intensity on various task parameters, thus simplifying the selection of optimal solutions. Optimal parameter values can then be refined through numerical calculations. Therefore, the derivation of new simple analytical formulas remains a highly relevant task in conjunction with the development of numerical methods.

Since the period of an SWG is much smaller than the wavelength of radiation, it is common to simplify the grating as an effective continuous layer of a medium with an averaged dielectric permittivity. This approach provides precise results for reflection, transmission, and extinction, particularly when assuming that the effective medium is anisotropic [23]. However, the basic averaging method fails to capture complex interference phenomena such as the resonances occurring at the commencement of fresh diffraction modes (Rayleigh frequencies) [24,25] and near-field effects, for example, subdiffraction light focusing [26]. This study aimed to acquire analytical expressions for SWGs with cylindrical components. We used the 2D point dipole approximation, which is analogous to Markel’s 3D model describing the interaction between dielectric spheres [27]. Another innovative approach in this field is perturbation theory, applicable when the permittivity is close to unity with $|\epsilon -1|\ll 1$. The theory differs from the quantum-mechanical Born series due to specific boundary conditions in electrodynamics. A similar approach was previously utilized to study the scattering problem for two parallel cylinders when exposed to light with arbitrary polarization [28]. Below, we present a theory for p waves that excite plasmon resonances in metallic media. This method is also applicable to s waves and can be used for oblique incidence.

Section 2 introduces the point dipole approximation (PDA), which allows for the derivation of simplified analytical formulas. In Section 3, the perturbation series, based on the Fourier series, is explained for near-field distribution. Numerical modeling using BEM and FEM is discussed in Section 4, which is focused on determining the near-field intensity card for electric and magnetic fields, as well as comparing it to analytical formulas. Finally, Section 5 offers a summary of the findings.

2. Point Dipole Approximation

2.1. Polarizability

We study an endless series of cylindrical components with a radius of a and a periodicity of L. An electromagnetic wave approaches from beneath in the positive y direction, and our coordinate system is arranged such that the z axis aligns with the parallel cylinders’ axes, while x is perpendicular to them. We analyze a fundamental unit cell, as shown in Figure 1a, with periodic boundary conditions at $x=-L/2$ and $x=L/2$. The cylindrical elements, with a dielectric permittivity of $\epsilon$, are represented using circles. The surrounding medium is free space with a permittivity of $\epsilon =1$. The distinctive features of the p-wave scattering of this grating under normal incidence illumination are mainly apparent in the nearby surroundings of the cylinders, i.e., the near field. In this regard, the study of near-field behavior is the primary focus of the investigation of subwavelength scattering within the lattice.

Discrete dipole approximation (DDA) is a widely used numerical method for calculating light scattering from particles of various shapes. There is no biased language or filler words, and formal register is maintained by avoiding contractions, informal expressions, and unnecessary jargon. The method requires dividing the particle into small domains and replacing each domain with a dipole. The resulting system of equations for the coupled dipoles is then solved to determine the dipole moments and the distribution of the scattered field. Technical terms are explained when first used, and the language is clear, objective, and neutral throughout the text. The structure is well organized and follows a logical progression. The use of common academic sections and formatting features adheres to style guides, and quotes are clearly marked.

In this section, we simplify the process by replacing each cylinder with a single point dipole located at its center, as shown in Figure 1b. This dipole is induced by both the incident wave and the infinite number of remaining dipoles. Although this approach is expected to be less accurate than DDA, it enables the derivation of an explicit approximate analytical expression for the distribution of scattered field intensity. We use the point dipole approximation (PDA) method for reference.

The alignment of cylinders has an effective lattice polarizability coefficient. Under normal incidence, the phase of the incident field is identical for all cylinders. Consequently, the two-dimensional dipoles that are excited in them have identical phases. When assessing the near-field consequences from these dipoles in the subwavelength regime, our computations may be managed within the electrostatic limit, overlooking any changes in the individual dipole phase at the observation point.

The electric field vector of the incident wave lies along the x-axis

$${\mathbf{E}}_x=E_0{\mathbf{e}}_x,$$

(1)

where ${\mathbf{e}}_x$ is the unit orth. The total field $\mathbf{E}$ acting on each cylinder is the sum of the external field ${\mathbf{E}}_0$ and the field of all neighboring dipoles ${\mathbf{E}}^{\prime }$:

$$\mathbf{E}={\mathbf{E}}_0+{\mathbf{E}}^{\prime }.$$

(2)

The dipole moment per unit length $\mathbf{d}$ is proportional to the total field $\mathbf{E}$: $\mathbf{d}=\alpha \mathbf{E},$ where $\alpha$ is the polarizability of a dielectric cylinder in a homogeneous external field [29]:

$$\alpha =\frac{a^2}{2}\frac{\epsilon -1}{\epsilon +1}.$$

(3)

A two-dimensional dipole with number n directed along the x axis creates a field

$${\mathbf{E}}_n^{\prime }=\frac{2\mathbf{d}}{{\left(nL\right)}^2},n\ne 0$$

(4)

in the neighborhood of the “central” cylinder with number 0. In the subwavelength (electrostatic) limit, the total field from all other cylinders at the center of cylinder $n=0$ ($x=0$) is given by the sum

$${\mathbf{E}}^{\prime }=2\sum _{n=1}^{\infty }{\mathbf{E}}_n^{\prime }.$$

(5)

In the 2-dimensional case, the summation of the reciprocals of the squares of the natural numbers is known [30]:

$$\sum _{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}.$$

(6)

Hence, the field of dipoles is

$${\mathbf{E}}^{\prime }=\frac{2{\pi }^2}{3L^2}\alpha ({\mathbf{E}}_0+{\mathbf{E}}^{\prime }).$$

(7)

From Equation (7), we obtain for the field of other dipoles and the induced dipole moment:

$$\begin{array}{c}\hfill {\mathbf{E}}^{\prime }=\frac{2{\pi }^2\alpha }{3L^2-2{\pi }^2\alpha }{\mathbf{E}}_0,\end{array}$$

(8)

$$\begin{array}{c}\hfill \mathbf{d}=\alpha ({\mathbf{E}}_0+{\mathbf{E}}^{\prime })=\frac{3L^2\alpha }{3L^2-2{\pi }^2\alpha }{\mathbf{E}}_0.\end{array}$$

(9)

Using Formula (3), we have

$$\mathbf{d}={\alpha }_g{\mathbf{E}}_0.$$

(10)

In this context, we can refer to the proportionality factor

$${\alpha }_g=\frac{3L^2\alpha }{3L^2-2{\pi }^2\alpha }=\frac{3a^2{L}^2(\epsilon -1)}{6L^2(\epsilon +1)-2a^2{\pi }^2(\epsilon -1)}$$

(11)

as the “grating polarizability”; resonant field enhancement occurs in areas where the denominator (11) is small within the slits. This phenomenon is known as lattice dipole plasmon resonance. Specifically, this peak can be seen in metallic cylinder lattices in the optical frequency range with a real dielectric constant value ($\mathrm{Re}\epsilon$) that is less than zero.

2.2. Near Field

The near-field approximation for the nth dipole at the observation point with Cartesian coordinates $x,y$ is described by the expression

$${\mathbf{E}}_n=\frac{4{\mathbf{r}}_n({\mathbf{r}}_n·\mathbf{d})}{r_n^4}-\frac{2\mathbf{d}}{r_n^2},$$

(12)

where the dot inside parentheses denotes the scalar product, ${\mathbf{r}}_n={\mathbf{e}}_{x}x+{\mathbf{e}}_{y}y$; vectors ${\mathbf{e}}_{x,y}$ are unit orthants of the coordinate axes, $x_n=x-nL$, $r_n=\sqrt{x_n^2+y^2}$.

According to (12), the total field $E_x$ from all cylinders of the lattice is described by the infinite sum:

$$E_x=\sum _{n=-\infty }^{\infty }({\mathbf{e}}_x·{\mathbf{E}}_n)=d\sum _{n=-\infty }^{\infty }\frac{-y^2+{(x-Ln)}^2}{{[y^2+{(x-Ln)}^2]}^2}.$$

(13)

For the y component of the cylinder field $E_y$ at the observation point, respectively, we obtain:

$$E_y=\sum _{n=-\infty }^{\infty }({\mathbf{e}}_y·{\mathbf{E}}_n)=d\sum _{n=-\infty }^{\infty }\frac{4y(x-Ln)}{{[y^2+{(x-Ln)}^2]}^2}.$$

(14)

Note that both sums provide the periodic function of x.

Summing up the series, we obtain:

$$\begin{array}{c}\hfill E_x=\frac{{\pi }^2}{L^2}d\left[csc\left(\pi \right(x-iy)/L)+csc\left(\pi \right(x+iy)/L)\right],\end{array}$$

(15)

$$\begin{array}{c}\hfill E_y=\frac{i{\pi }^2}{L^2}d\left[csc\left(\pi \right(x-iy)/L)-csc\left(\pi \right(x+iy)/L)\right].\end{array}$$

(16)

The total electric field intensity of the scattered radiation is given by a simple formula

$$I_s=|E_x{|}^2+{|E_y|}^2=\frac{16{\pi }^4{d}^2}{L^4{\left(cos\frac{2\pi x}{L}-cosh\frac{2\pi y}{L}\right)}^2}.$$

(17)

According to Equation (17), the scattered radiation displays periodic variations in intensity along the x axis, whereas it symmetrically and rapidly diminishes along the y axis. Figure 2 illustrates the intensity’s distribution of the scattered field in PDA near a cylinder, demonstrating the existence of saddle points in the gap’s central region. Accordingly, the saddle points occur at the positions of $x=(n+1/2)L$ and $y=0$.

3. Perturbation Theory

3.1. Fourier Series

The PDA model offers an adequate depiction of the scattered electric field in the outer region beyond the cylinders. However, the model predicts a singularity within each cylinder, resulting in an infinite field. Additionally, the model does not consider the scattered magnetic field, as it sets the rotor (curl) of the magnetic field to zero. Nevertheless, in reality, the scattered magnetic field is not zero. To obtain a more precise representation of the electric field, we suggest considering the following equation:

$$\nabla \times \nabla \times \mathbf{E}=i{k}_0\nabla \times \mathbf{H}=\epsilon \left(\mathbf{r}\right)k_0^2\mathbf{E}.$$

(18)

Derived from Maxwell’s equations, this paragraph describes the electric field $\mathbf{E}$ in a medium with magnetic vacuum permeability of 1 and space-variable dielectric permittivity $ϵ\left(\mathbf{r}\right)$.

The scattering of p waves is often explained using a single Helmholtz equation for the magnetic field. However, solving a periodic problem for a magnetic field with the Fourier method can prove challenging due to the discontinuity of its derivative at the boundary caused by imposed boundary conditions. To effectively solve the issue at hand, the following pair of equations for a two-dimensional electric field are utilized. This pair guarantees the electric field’s and its derivative continuity, making for easier and more manageable solutions.

For equidistant parallel cylinders, function $\epsilon \left(\mathbf{r}\right)$ is expressed through the use of theta functions:

$$\epsilon \left(\mathbf{r}\right)=1+(\epsilon -1)\sum _{m=-\infty }^{\infty }\theta \left(a-\sqrt{{(x-Lm)}^2+y^2}\right).$$

(19)

For p waves, $E_z=0$ and components $E_x(x,y)$ and $E_y(x,y)$ remain in Equation (18). Represent the components, the periodic functions of x, as Fourier series:

$$E_x=\sum _{m=-\infty }^{\infty }{U}_m\left(y\right)e^{2\pi mx/L},E_y=\sum _{m=-\infty }^{\infty }{V}_m\left(y\right)e^{2\pi mx/L}.$$

(20)

At normal incidence of the external field, this representation is natural. Substituting the sums, multiplying by $exp(-2\pi ipx/L)$, and integrating over an elementary cell, as shown in Figure 1a, from $-L/2$ to $L/2$, we obtain the following equations for the amplitudes:

$$\begin{array}{c}\hfill U_p^{″}-\frac{2\pi ip}{L}{V}_p^{\prime }+k_0^2{U}_p\\ \hfill =-(\epsilon -1)\theta (a-|y\left|\right)\sum _{m=-\infty }^{\infty }\frac{U_m\left(y\right)}{\pi (m-p)}sin\frac{2\pi (m-p)\sqrt{a^2-y^2}}{L},\\ \hfill -\frac{4{\pi }^2{p}^2}{L^2}{V}_p-\frac{2\pi ip}{L}{U}_p^{\prime }+k_0^2{V}_p\\ \hfill =-(\epsilon -1)\theta (a-|y\left|\right)\sum _{m=-\infty }^{\infty }\frac{V_m\left(y\right)}{\pi (m-p)}sin\frac{2\pi (m-p)\sqrt{a^2-y^2}}{L}.\end{array}$$

(21)

Express the equations for first-order amplitudes more clearly by beginning with Equation (21). The equations may be written as follows: $U_p=U_p^{\left(0\right)}+U_p^{\left(1\right)}+\dots$ and $V_p=V_p^{\left(0\right)}+V_p^{\left(1\right)}+\dots$. We consider the right side in Equation (21) as small and write the amplitudes as perturbation series. For the unperturbed, zero-order solution, $U_{p\ne 0}^{\left(0\right)}$ and $V_{p\ne 0}^{\left(0\right)}$ are both equal to zero; otherwise, an exponential increase along y would occur. $V_0^{\left(0\right)}$ is zero due to perpendicular incidence. We take $U_0^{\left(0\right)}=e^{i{k}_{0}y}$ for clarity, which gives the electric field at zero order the specific physical meaning of the external field falling from bottom to top onto the grating. The equations for the first-order amplitudes, $U_p^{\left(1\right)}$ and $V_p^{\left(1\right)}$, are:

$$\begin{array}{c}\hfill U_p^{\left(1\right)″}-\frac{2\pi ip}{L}{V}_p^{\left(1\right)\prime }+k_0^2{U}_p^{\left(1\right)}=-(\epsilon -1)\theta (a-|y\left|\right)\frac{e^{i{k}_{0}y}}{\pi p}sin\frac{2\pi p\sqrt{a^2-y^2}}{L},\\ \hfill q_p^2{V}_p^{\left(1\right)}+\frac{2\pi ip}{L}{U}_p^{\left(1\right)\prime }=0,q_p^2=\frac{4{\pi }^2{p}^2}{L^2}-k_0^2.\end{array}$$

(22)

3.2. Solutions

At $p\ne 0$, we find a solution decreasing at infinity:

$$\begin{array}{c}\hfill U_p^{\left(1\right)}= (\epsilon -1)q_p^2{\int }_{-a}^a{G}_p(y,\tilde{y})\frac{e^{i{k}_0\tilde{y}}}{\pi p}sin\frac{2\pi p\sqrt{a^2-{\tilde{y}}^2}}{L}d\tilde{y},\\ \hfill V_p^{\left(1\right)}=-\frac{2\pi ip}{L{q}_p^2}{U}_p^{\left(1\right)\prime },G_p(y,\tilde{y})=-\frac{e^{-q_p|y-\tilde{y}|}}{2q_p}.\end{array}$$

(23)

At $p=0$, there is no decreasing; then, a different approach must be used. First, we take any Green’s function and describe the general solution

$$\begin{array}{c}\hfill U_0^{\left(1\right)}=C_0^{\left(1\right)}{e}^{i{k}_{0}y}+D_0^{\left(1\right)}{e}^{-i{k}_{0}y}\\ \hfill -(\epsilon -1)k_0^2{\int }_{-a}^a\left[\theta (y-\tilde{y})\frac{e^{i{k}_0(y-\tilde{y})}-e^{-i{k}_0(y-\tilde{y})}}{2i{k}_0}\right]\frac{2\sqrt{a^2-{\tilde{y}}^2}}{L}{e}^{i{k}_0\tilde{y}}d\tilde{y}.\end{array}$$

(24)

The convenient Green’s function is the one that is zero when $y<\tilde{y}$.

The coefficients $C_0^{\left(1\right)}$ and $D_0^{\left(1\right)}$ are calculated from the scattering boundary conditions:

$$U_0(y\to -\infty )=e^{i{k}_{0}y}+A_0{e}^{-i{k}_{0}y},V_0(y\to +\infty )=B_0{e}^{i{k}_{0}y}.$$

(25)

Substituting $U_0=U_0^{\left(0\right)}+U_0^{\left(1\right)}$ into condition (25), we have

$$C_0^{\left(1\right)}=0,D_0^{\left(1\right)}=A_0^{\left(1\right)}=\frac{i(\epsilon -1)k_0}{2L}\pi a^2\left[J_0\left(2k_{0}a\right)+J_2\left(2k_{0}a\right)\right],$$

(26)

where $J_0,J_2$ are the Bessel functions of the zeroth and second order, respectively [30]. Within the first order $B_0^{\left(1\right)}=i(\epsilon -1)k_0\pi a^2/2L$, i.e., practically coinciding with $A_0^{\left(1\right)}$. At last, the magnetic field can be found as a rotor of the electric field vector:

$$H_z(x,y)=\frac{1}{i{k}_0}\left[{\left(E_y\right)}_x-{\left(E_x\right)}_y\right]=-\frac{U_0^{\left(1\right)\prime }\left(y\right)}{i{k}_0}-i{k}_0\sum _{p\ne 0}\frac{U_p^{\left(1\right)\prime }\left(y\right)}{q_p^2}{e}^{2\pi px/L}.$$

(27)

Estimating using Equation (26), we can determine the small perturbation parameter by comparing the first and zeroth orders, given as:

$$|\epsilon -1|\left(k_{0}a\right)\frac{a}{L}\ll 1.$$

(28)

This observation suggests that the method remains valid for SWGs, even when $|\epsilon -1|\sim 1$, while $k_{0}a\ll 1$. The expansion is particularly effective for sparse gratings when $a\ll L$.

The first-order correction is shown in Figure 3, depicting 221 harmonics (with $\left|m\right|⩽110$) selected for their capacity to minimize the phase differences between adjacent cylinders ($\delta \phi \ll 1$). These plots exhibit the highest scattered electric field intensities near the interfaces of dielectric and free space situated on both sides, as presented in Figure 3a. Saddle points occur at the geometric centers of the slits, resembling those found in the PDA. The magnetic field distribution indicates the highest values at the top and bottom of the cylinders. Figure 3b displays a notable region with low field intensity at $\left|y\right|<0.25\mathsf{\mu }$m.

Perturbation theory was utilized to remove the cylinder’s center singularity, a characteristic feature of the PDA. First-order perturbation theory yields comprehensive distributions of electric and magnetic field intensities. The analytical expressions obtained with this methodology possess immense potential in optimizing resonant periodic structures.

4. Numerical Modeling

To determine the scattered field, we utilized the boundary element method (BEM) and leveraged Floquet’s theorem. Our approach involved computing an effective Green’s function, represented as a series. The theorem simplified the problem by focusing on the scattering by a single unit cell. Unlike the previous study [31], our current calculation does not consider the influence of a dielectric substrate, which simplifies the approach. The BEM calculation grid uses a division of the cylinder circumference into 360 panels. Figure 4a provides a visual representation of the near-field intensity distribution outside the cylinders. The field inside the cylinders is not calculated. Figure 4b displays the density plot of the results obtained from a sparse lattice using FEM in COMSOL Multiphysics^® Figure 4b displays the density plot of the results obtained from a sparse lattice using FEM in COMSOL Multiphysics^® [32]. The figure depicts a finite grating consisting of $N=20$ cylinders, with only the central part shown. Notably, COMSOL software (Version 6.2) was used to construct a grid of triangles with a side of 2 nm for FEM computation.

When comparing Figure 1a,b to the polarization direction antenna depicted in Figure 2, similar patterns can be observed. Both the boundary element method (BEM) and finite element method (FEM) calculations show saddle points at the midpoint between neighboring cylinders, while maxima are reached near the interface between the free space and the dielectric medium. Figure 3 presents the squared magnetic field computed using the BEM. The magnetic field does not require derivative calculations. Consequently, the computational domain covers the complete diagram, including internal areas. The magnetic field distribution peaks at the top and bottom points of the diagram, near the dielectric material.

When comparing the patterns in Figure 4 and Figure 5 with the representations provided from perturbation theory shown in Figure 3, it is apparent that there are notable similarities between the two sets of results. These similarities comprise the electric field having maxima at points on the right and left, the magnetic field having maxima at the top and bottom, the occurrence of saddle points for the electric field, and a minimum strip for the magnetic field appearing at small values of y.

We repeated the calculations for the same period L but with a narrow slit of $\delta =10$ nm. The finite element method (FEM) also yielded saddle points. The field enhancement factor increased by approximately six times in this case, as demonstrated in Figure 6.

5. Conclusions and Discussion

Subwavelength gratings have various uses in the fields of photonics and optoelectronics. Our study aimed to analyze a periodic system that comprises parallel cylinders. We utilized the 2D dipole model and Born series for this purpose. By employing perturbation theory, we established equations for the electric and magnetic components in the near vicinity. To demonstrate the accuracy of our analytical model, we validated it through numerical simulations. Saddle points appearred precisely at the geometric centers of the slits featured in the dipole model. The observation was later confirmed through numerical simulations. The dipole model quantifies the field within the slit, whereas perturbation theory provides an estimate of the intensity across the entire plane.

The dipole model and perturbation theory require a thorough examination of metallic (plasmonic) elements, scattering of s waves, and oblique incident scenarios. This study aimed to investigate the complexities that arise from inconsistent and smooth spatial distributions of $\epsilon \left(\mathbf{r}\right)$ within an element, various cross-sectional shapes of a cylinder, and the grating on a dielectric substrate. Furthermore, ongoing research includes constructing the Born series to determine the anisotropic dielectric permittivity of the medium and analyzing subsequent orders. These distinct phases of the study are projected to enhance understanding, facilitate the development of optimized designs, and significantly contribute to the advancement of cutting-edge optical technologies.