## Abstract

The modified rigorous coupled-wave analysis technique is developed to describe the optical characteristics of the plasmonic structures with the grating-gated delta-thin conductive channel in the far- and near-field zones of electromagnetic waves. The technique was applied for analysis of the resonant properties of AlGaN/GaN heterostructures combined with a deeply subwavelength metallic grating, which facilitates the excitation of the two-dimensional plasmons in the terahertz (THz) frequency range. The convergence of the calculations at the frequencies near the plasmon resonances is discussed. The impact of the grating’s parameters, including filling factor and thickness of the grating, on resonant absorption of the structure was investigated in detail. The spatial distributions of the electromagnetic field in a near-field zone were used for the evaluation of total absorption of the plasmonic structures separating contributions of the grating-gated two-dimensional electron gas and the grating coupler.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Nowadays, structures with spatially periodical lateral structurization/metasurfaces are a focus of studies as the key elements of the many opto- and optoelectronics devices with broad application areas, including spectroscopy, imaging, holography, optical lithography, biochemical sensing, and military application (for more information, see recent review in Ref. [1]). For example, diffractive gratings are widely utilized in different spectral ranges from microwaves to deep ultraviolet as dispersive optical components [2], antenna elements [3], polarizers [4], etc.

Recently, great attention has been paid to exploitation of the subwavelength metasurfaced structures in terahertz (THz) and far-infrared spectral ranges. Particularly, the semiconductor structures with surface-relief gratings [5,6], quantum well (QW) heterostructures [7,8] or graphene-based structures [9] incorporated with metallic gratings are widely discussed as potential efficient emitters and detectors of the THz radiation [10]. In such structures, the gratings play a role of the coupler, facilitating the resonant interaction between incident electromagnetic ($\textit{em}$) waves and charge density waves such as surface plasmon-polaritons or two-dimensional (2D) plasmons. Moreover, detailed investigations of the resonant properties of grating-based plasmonic structures can serve as additional tool for basic characterization of the 2D electron gas (2DEG) in QWs [11,12], graphene [13–17], and other novel 2D layered materials [18].

Mentioned research is faced with the problem of the rigorous electrodynamic simulations of optical characteristics of grating-based structures containing very thin or even atomically thin conductive layers. In the simulations, these layers should be treated as delta-thin with (2D) parameters such as the sheet conductance of the electron channel. For such structures, the technique of integral equations (IEs) is often applied for the solutions of the Maxwell’s equations. This technique uses Green function formalism and is based on a reduction of the Maxwell’s system of equations to the linear IEs. The latter can be solved, for example, using Galerkin schemes with guaranteed convergence. The IE technique has been exploited for the different problems including investigations of 2D plasmon instabilities under the grating [8,19,20], detection of THz radiation [21–23], and interaction of THz radiation with conductive-strip gratings [24,25]. In spite of apparent advantages with respect to fast convergence, this method is typically formulated for modeling structures with simple geometries, and grating is treated as delta-thin.

Another powerful method for solution of grating-related electrodynamic problems is known as rigorous coupled-wave analysis (RCWA) [26–28]. This method belongs to the matrix-type methods and operates with systems of algebraic equations, which are formulated for coefficients of the Fourier expansion of the actual components of the $\textit{em}$ field. The RCWA can be applied for arbitrary complexity of the grating-based structures with any geometry of gratings. Typically, the realization of the RCWA relates to the structures where each layer is described by the bulk parameters. The RCWA method for structures with delta-thin conductive channels requires essential modifications, which were recently discussed in Refs. [29,30] for plasmonic structures with graphene. The aim of this paper is to present such modifications in more detail, focusing on the study of convergence of the proposed method on the example of calculations of optical characteristics of a metallic grating-based resonant plasmonic structure with a QW in THz frequency range. The effect of the grating depth on the plasmon resonance, analysis of the near-field pattern, and comparison to other methods are discussed. A total absorption of the plasmonic structures separating contributions of the grating-gated 2DEG and metallic grating couplers is also simulated using spatial distributions of the $\textit{em}$ field in the near-field region.

Mathematical formalism of the modified RCWA method is presented in Section 2. RCWA will be formulated for the planar diffraction problem (plane of incidence is perpendicular to grating strips) including the cases of TM and TE polarizations. The investigations of the convergence of the proposed method and comparison with the IE technique will be illustrated in Section 3 on the example of the calculations of transmission, reflection, and absorption spectra of a QW plasmonic structure with a deeply subwavelength metallic grating. The effect of the finite thickness of the metallic grating will be studied in detail. In Section 4, we will perform the analysis of the near-field patterns under conditions of the plasmon resonances. The main results will be summarized in Section 5.

## 2. MATHEMATICAL FORMALISM

Let us assume that multilayered structure with the grating is illuminated by plane $\textit{em}$ wave of TM polarization with frequency, $\omega$, and angle of incidence, $\theta$ (see Fig. 1). The grating of period ${a_g}$ is formed by the infinitely long in $y$-direction rectangular bars with width, ${w_g}$, and height, ${h_g}$. The structure consists of $N$ layers with thicknesses ${d_j} = {z_j} - {z_{j - 1}}$, $j = 1\ldots N$. Each $j$-layer, including the grating region, is described by the own dielectric permittivity, ${\epsilon _{\omega ,j}}(x)$. The delta-thin conductive channel is placed between $j - 1$ and $j$ layers and is described by high-frequency conduction current, ${\vec J^{2D}}(x,t)\delta (z - {z_j})$. The whole structure is set between two non-absorbing half-spaces with ${\epsilon _0}$ (at $z \lt 0$) and ${\epsilon _{N + 1}}$ (at $z \gt D$, where $D$ is the total thickness of the structure).

Assuming that all components of the $\textit{em}$ field oscillate in time as $\exp (- i\omega t)$, the Maxwell’s equations written for amplitudes take the form

Equation (4) can be rewritten in terms of the vector ${\textbf{H}_{y,j}}$,

Thus, the second boundary condition in Eq. (8) can be rewritten in the form

Equation (6) composes the system of ordinary second-order differential equations with constant coefficients. The solution of this system can be expressed in terms of the eigenvalues, ${\lambda _{m,j}}$, and eigen vectors, ${\vec w_{m,j}}$, of the matrix $\hat{\boldsymbol A}$,

Matching magnetic and electric fields on $j $-interface according to the boundary conditions in Eqs. (8) and (9), we come to the following recurrence relationship between constants of integration, $C_{\nu ,j}^ \pm$ and $C_{\nu ,j + 1}^ \pm$:

Equation (13) allows us to couple amplitudes of the reflected wave (in region $z \lt {z_0}$) and transmitted wave (in the region $z \gt {z_N}$), and consequently to calculate transmission and reflection coefficients for different diffraction orders. Indeed, components of the Fourier vectors of $y $-magnetic and $x $-electric fields in the region $z \lt 0$ ($j = 0$) are

Using Eqs. (14) and (15), boundary conditions in Eqs. (8) and (9), and Eqs. (13), we found that Fourier vectors $\textbf{R}$ and $\textbf{T}$ are coupled through following matrix equations:

It should be noted that usage of this system of equations, written in the present form, applying to the situation of deep surface grating and optically dense materials, can face the problem of the computational instability. This instability is associated with the procedure of the numerical inversion of the second matrix in the right-hand side of Eq. (16) when exponentially small terms, $\exp (- {k_0}{\bar \lambda _{\nu ,j}}{d_j})$ (standing in the ${\hat \Lambda _j}$), become smaller than machine precision. To avoid this obstacle, authors in Ref. [28] proposed to use the following decomposition of the badly inverted matrix:

Components of the vectors $\textbf{T}$ and $\textbf{R}$ provide the transmission, ${T_m}$, reflection, ${R_m}$, and coefficients for any $m$th diffraction order as well as total absorption, $L$. Particularly,

This method can be extended for the case of TE polarization of the incident radiation. Now, actual components of the $\textit{em}$ waves are ${E_y}$, ${H_x}$, and ${H_z}$. Matrix Eq. (6) is formulated for the Fourier vectors ${\textbf{E}_{y,j}}$, where

The Fourier vectors of the magnetic field components are given as follows:The proposed modified RCWA method is applied for investigation of particular plasmonic structures with a grating-gated 2DEG channel. Such structures possess the resonant properties in the THz frequency range (wavelength of order of 100 µm) for TM-polarized incident radiation due to excitations of plasmons in conductive channel of 2DEG [7,11,19,39]. Below, we will study spectral characteristics of the AlGaN/GaN-based plasmonic structure with a deeply subwavelength (micron period) metallic grating, calculating transmission (${T_0}$), reflection (${R_0}$) and absorption (${L_0}$) coefficients, their convergence versus number of the Fourier harmonics, and the near-field mapping. Also, we will pay attention to the dependence of the plasmon resonances versus geometry of the grating.

## 3. FAR-FIELD CHARACTERISTICS AND THEIR CONVERGENCE

Here and below, we will study the case of a TM-polarized incident $\textit{em}$ wave with incidence angle $\theta = 0$. The structure under test is formed by $N = 3$ media, embedded into the air, including the rectangular metallic grating with ${\epsilon _{\omega ,1}}(x) = {\epsilon _{\omega ,M}}\Theta ({w_g} - x) + {\epsilon _0}\Theta (x - {w_g})$ (where $\Theta (x)$ stands the Heaviside step function, $x \in [0,{a_g}]$, ${\epsilon _{\omega ,M}} = 1 + 4\pi i{\sigma _M}/\omega$, and ${\sigma _M} = 4 \times {10^{17}}\,{{\rm s}^{- 1}}$ that corresponds to the gold), and the AlGaN barrier and GaN buffer layers with constant dielectric permittivities ${\epsilon _2} = 9.2$ and ${\epsilon _3} = 8.9$, respectively. The 2D conductive channel is formed in the plane $z = {z_2}$. The matrix ${[\Gamma _j^{2D}]_{m,m^\prime}} = 4\pi /c \times \sigma _{\omega ,m}^{2D}{\delta _{m,m^\prime}}{\delta _{j,2}}$, where for description of the high-frequency properties of 2DEG we used Drude–Lorentz model $\sigma _{\omega ,m}^{2D} = {e^2}{n_{2D}}{\tau _{2D}}/{m^*}(1 - i\omega {\tau _{2D}})$ with electron effective mass, ${m^*} = 0.22 \times {m_e}$, concentration of 2DEG, ${n_{2D}} = 6 \times {10^{12}}\,{{\rm cm}^{- 2}}$, and effective scattering time, ${\tau _{2D}} = 0.5 \;{\rm ps}$. Other geometrical parameters of the structure are listed in the caption for Fig. 2. The selected parameters are close to the parameters of the experimental structures recently studied in Ref. [11].

The spectra of the far-field characteristics such as transmission, reflection, and absorption coefficients calculated for three depths of the metallic grating are illustrated in Fig. 2. As seen, all spectra possess a strong resonance at the frequency of 1.7 THz and much more weaker resonances at frequencies of 3.5 THz and 4.75 THz. The emergence of these resonances relates to the grating-assisted interaction of the incident $\textit{em}$ wave with the plasmons in the channel of the 2DEG. At the resonance, plasmon excitation of 2DEG with wave vectors determined by the grating period can effectively absorb energy of the incident $\textit{em}$ wave. Physics of 2D plasmons and their resonant interaction with $\textit{em}$ radiation are well-described in Refs. [7,19,39–41].

The considered plasmonic structure can provide considerable absorption of the THz radiation. For example, at 1.7 THz, the absorption coefficient ${L_0} \sim 40\%$. The RCWA calculations show that absorption coefficient of the plasmonic structure is almost independent of the depth of the metallic grating [see Fig. 2(c)]. Moreover, we show that results of the IE method [19,21,23], developed for the same plasmonic structure but with delta-thin grating (see black dashed lines), and the present RCWA for the grating with ${h_g} = 0.05\;{\unicode{x00B5}{\rm m}}$ almost coincide. An increase of the grating depth only leads to the additional dispersion of transmission and reflection coefficients in the higher frequency range. This dispersion is also observed in modeling structure without 2DEG (see dashed–dotted lines). Very weak dependence of the absorption of the plasmonic structure versus grating depth (even for deep grating with ${h_g} = 5\;{\unicode{x00B5}{\rm m}}$) indicates that subwavelength highly conductive grating plays the role of almost-lossless waveguide for incident TM-polarized wave of THz frequencies. For example, absorption of the deep grating in the structure without 2DEG does not exceed 2% in the considered spectral range. The calculations of the partial losses associated with grating and 2DEG gas in the grating-2DEG plasmonic structure can be performed using the pattern of the near-field, and one will be done below in the Section 4.

The convergence of the RCWA method versus number of Fourier harmonics, M, is illustrated in Figs. 2(d) and 2(e) on example of ${T_0}$ coefficient. The proposed method provides fast convergence, and results with reasonable accuracy can be already obtained using $M \approx 30 - 50$ for all considered cases in Figs. 2(d) and 2(e). In order to quantify the convergence of the RCWA method, we introduce relative error defined as follows: ${\delta _M} = |T_0^{(M)} - T_0^{(200)}|/T_0^{(200)}$ [see insets in Figs. 2(d) and 2(e)]. For example, for shallow grating (black circles), ${\delta _{30}} = 0.2\%$ for resonant frequency 1.7 THz, and ${\delta _{30}} = 0.01\%$ for the frequency 3.5 THz. Convergence becomes worse for the deep gratings: at resonant frequency 1.7 THz, ${\delta _{30}} = 0.5\%$ (for ${h_g} = 1\;{\unicode{x00B5}{\rm m}}$) and ${\delta _{30}} = 1.3\%$ (for ${h_g} = 5\;{\unicode{x00B5}{\rm m}}$); at non-resonant frequency 3.5 THz, ${\delta _{30}} = 0.1\%$ (for ${h_g} = 1\;{\unicode{x00B5}{\rm m}}$) and ${\delta _{30}} = 0.2\%$ (for ${h_g} = 5\;{\unicode{x00B5}{\rm m}}$). Thus, estimations show that convergence of the RCWA method exhibits dependence on grating depth and frequency of the incident radiation. The cases of the deep gratings and resonant frequencies of the plasmon excitation require account of the larger numbers of Fourier harmonics.

In the case of the plasmonic structure with narrow-slit grating, ${w_g} = 0.85\;{\unicode{x00B5}{\rm m}}$ and ${a_g} = 1\;{\unicode{x00B5}{\rm m}}$, our calculations predict much more pronounced features in the optical characteristics including intensity of the plasmon resonances versus grating depth (see Fig. 3). Moreover, narrow-slit grating provides more efficient coupling between incident radiation and plasmon excitations that leads to an emergence of well-pronounced multiple plasmon resonances, which are redshifted in comparison to the previous case. The redshift of the resonant frequency is the result of a larger contribution of the gated region of 2DEG where phase velocity of the plasmons is smaller than in the ungated region of 2DEG [42].

The first plasmon resonance occurs at frequency of 1.38 THz; at this, absorption of the THz waves reaches a value of ${\sim}50\%$. However, in this spectral range, the effect of the grating thickness is still a weak. Starting from the frequencies larger than 2 THz, spectral characteristics are essentially modified by grating thickness. As seen from Fig. 3(c), deep grating suppresses plasmonic mechanisms of the absorption of THz radiation. The absorption coefficient ${L_0}$ at resonant frequency of 3.78 THz is decreased from 28% for shallow grating (${h_g} = 0.05\;{\unicode{x00B5}{\rm m}}$) to 15% for the deepest grating (${h_g} = 5\;{\unicode{x00B5}{\rm m}}$). Apparently, this effect relates to an essential increase of the reflectivity of the plasmonic structures with thicker gratings as shown in Fig. 3(b). It means that for the deeper gratings, a smaller portion of the $\textit{em}$ energy is concentrated in 2DEG as it will be further illustrated in Section 4.

It should be noted that application of the RCWA methods for accurate calculations of far-field spectral characteristics of the plasmonic structure with narrower-slit grating requires a larger number of Fourier harmonics [see Figs. 3(d) and 3(e)]. Now, the relative errors ${\delta _{30}}$ for resonant frequency 1.38 THz are equal to 0.6%, 2.2%, and 2.5% for ${h_g} = 0.05\; \unicode{x00B5}{\rm m}$, 1 µm, and 5 µm, respectively. Relative errors less than 1% for deep gratings are achievable at $M \gt 60$. Similarly to the previous case, the convergence of the RCWA method is improved at higher frequencies. So, at frequency of 3.78 THz, accuracy of computation with relative errors ${\delta _M} \lt 1\%$ is achieved at $M \gt 30$. All spectra shown in Figs. 2 and 3 are obtained at $M = 100$.

## 4. NEAR-FIELD STUDY

Together with calculations of optical characteristics relating to the far-field, the RCWA method allows us to study geometry of the near-field. Especially, we will pay attention to the spatial distributions of the ${E_x}(x,z)$ and ${E_z}(x,z)$ components of the $\textit{em}$ fields. Absolute values of these components determine the local absorption of the $\textit{em}$ wave and can be used for extraction of partial losses in metallic grating and 2DEG. Having RCWA data on Fourier vectors ${\textbf{T}_1}$ [taking from the solutions of master system Eq. (16)], we can find vectors of the integration constants $\vec C_j^ \pm$ in each $j$ layer of the structure,

The spatial distribution of $|{H_y}(x,z)|$ for the particular case of the plasmonic structure with shallow grating is shown in Fig. 4(a). This component is tangential to the grating sides and exhibits smooth behavior with partial penetration into the grating bar. Calculations give that skin depth, $\delta = c/\sqrt {2\pi {\sigma _M}\omega}$, of the gold at frequency of 1.7 THz is equal to 0.057 µm, which is comparable with height of the grating bar. The cold zone of ${H_y}$ component occupies the middle region of the bar near the bottom face. In the plane of 2DEG, $|{H_y}(x,z)|$ is a discontinuous quantity according boundary conditions in Eq. (8).

The electric components ${E_x}$ [Fig. 4(b)] and ${E_z}$ [Fig. 4(c)] show more interesting behavior with highly non-uniform distributions. The ${E_z}(x,z)$ can be directly obtained from the second relationship in Eq. (7) (using already found Fourier vectors, ${\textbf{H}_{y,j}}(z)$) and Eq. (3). For the correct reconstruction of the ${E_x}$ component, we follow the method discussed in Refs. [43,44]. In the region of the grating, $z \in [0,{h_g}]$, the $x$ component of the electric field is the normal to the grating bar’s sides, and one has a discontinuity. It is more effective to reconstruct a continuous quantity, the component of the displacement field, ${D_x}(x,z)$, which can be easily calculated from the derivative of the ${H_y}$ component with respect to the $z$ coordinate [see the first equation within Eq. (2)]. Then ${E_x}(x,y) = {D_x}(x,z)/{\epsilon _1}(x,z)$, where dielectric permittivity ${\epsilon _1}(x,z)$ is the known discontinuous function. Such method allows us to partially avoid an emergence of the unphysical spurious oscillations, known as Gibb’s phenomenon. Nevertheless, reconstruction of the near-field patterns requires account of much more Fourier harmonics than for calculations of the far-field characteristics. This circumstance was discussed in Ref. [44].

As seen, both ${E_x}$ and ${E_z}$ components demonstrate the field concentration effect. The energy of the $\textit{em}$ field is mainly concentrated near the ridges (hot zone I) of the metallic bars and in the region between grating bars and 2DEG (hot zone II). In the hot zones (I) and (II), both electric components are essentially enhanced. In the hot zone (II), the ${E_z}$ component predominantly dominates. The specific formation of the cold zone for the ${E_z}$ component at the vertical axis $x = {w_g}/2$ and $x = ({a_g} + {w_g})/2$ reflects the quadruple-related symmetry of the near-field (for details, see Ref. [39]). In the hot zones, amplitudes of the electric components can be in several tens of times larger than the amplitude of the incident wave. In spite of the magnetic component, the penetration of the electric components inside the metallic bar is strongly suppressed, which is the result of the edge effects. As seen, the ${E_x}$ component mainly penetrates to the grating’s bars from the upper and back faces, and ${E_z}$ from the side faces as the tangential ones for corresponding faces.

A similar geometry of the near-fields is realized for the case of the deep grating (see Fig. 5). The mappings of the ${E_x}$ and ${E_z}$ components show that the incident wave passes through the subwavelength grating in the form of TEM mode, i.e., in the grating slit, the wave has predominantly polarization along the $x$ direction with almost constant amplitude.

Additionally, we used COMSOL Multiphysics [45] to validate independently the obtained results by the finite element method. The Wave Optics module [45] is used to solve Maxwell’s equations for the system, which is shown in Fig. 1. The 2DEG was introduced as the surface current density at the interface of AlGaN and GaN. The uniform quadratic mesh with 5 nm size is used to resolve near-field components (see Fig. 6). COMSOL’s results of the electric component distributions for the case of the shallow grating with ${h_g} = 0.05\;{\unicode{x00B5}{\rm m}}$ are shown in Fig. 6. Excellent agreement between the modified RCWA and finite element methods is demonstrated in both far- and near-field studies.

The spatial distributions of the $|{E_x}|(x,z)$ and $|{E_z}|(x,z)$ can be used for calculation of the partial losses in the grating, ${L_{\textit{gr}}}$ and 2DEG, ${L_{2DEG}}$,

For the case in Fig. 4, we obtained that ${L_{\textit{gr}}} = 0.184\%$, which consists of $0.0741\%$ contribution of the ${E_x}$ component and 0.11% contribution of the ${E_z}$-component. Losses in 2DEG are considerably larger, ${L_{2DEG}} = 37.92\%$. Total losses from the near-field patterns are ${L_0} = {L_{2DEG}} + {L_{\textit{gr}}} = 38.11\%$. Calculations of the ${L_0}$ from far-field characteristics give the almost same value 38.21%. For the case of deep grating (see Fig. 5), we obtained the increase of absorption in the grating bars, ${L_{\textit{gr}}} = 0.56\%$ (with 0.072% and 0.49% contributions for ${E_x}$ and ${E_z}$ components, respectively), with almost same value of absorption in 2DEG ${L_{2DEG}} = 37.97\%$. The total losses ${L_0} = 38.53\%$ that almost coincide with the number 38.58% were obtained from far-field characteristics.

Calculations of the partial losses at the frequency of the first-order plasmon resonance indicate that the incident $\textit{em}$ wave is mainly absorbed by 2DEG and this absorption weakly depends on thickness of grating bars. This fact is illustrated by the spatial distribution of the amplitude of the $x $-component of the electric field, $|{E_x}(x,{z_2})|$, in the plane of 2DEG, calculated at the frequencies of the first [Fig. 7(a)] and second [Fig. 7(b)] plasmon resonances at three values of the grating depth. As seen, all three (grey, red, green) curves for lower frequency ($\omega /2\pi = 1.7\,{\rm THz}$) almost coincide, and all of them exhibit non-uniform, oscillating-like behavior in the gated region with almost flat distribution in the ungated region. The obtained distribution denotes that a larger part of $\textit{em}$ energy is absorbed in the gated region, i.e., under the metallic strip.

For the higher frequency ($\omega /2\pi = 3.5\,{\rm THz}$), spatial distribution of the $|{E_x}(x,{z_2})|$ quantity [Fig. 7(b)] acquires a more complicated form with several spatial oscillations in the gated region. At this, the effect of the grating thickness becomes visible, i.e., the deep gratings start to screen the interaction of the $\textit{em}$ wave with 2DEG. The emergence of the several spatial oscillations in the distribution of $|{E_x}(x,{z_2})|$ leads to suppression of absorptivity of the plasmonic structures at higher order plasmon resonances. Also, according the Drude model, at higher frequencies, the response of electron gas on the $\textit{em}$ wave becomes weaker, which leads to a decrease of the prefactor standing in Eq. (30). This prefactor, $4\pi {\rm Re}[\sigma _\omega ^{2D}]/c$, for two considered frequencies $\omega /2\pi = 1.7$ and $\omega /2\pi = 3.5\,{\rm THz}$ is equal to 0.049% and 0.012%, respectively. Using the obtained distributions in Fig. 7(b), we found that for ${h_g} = 0.05, 1, 5\;{\unicode{x00B5}{\rm m}}$, ${L_{2DEG}} = 4.42, 4.3$ and 3.8%, and the corresponding values of total losses are calculated from far-field characteristics, ${L_0} = 4.8, 5.3, 6.3\%$. Note that, for the structure with the deepest grating, the absorptions in 2DEG and the grating bars become comparable.

The plasmonic structure with narrow-slit grating provides more efficient coupling between 2DEG and $\textit{em}$ radiation. The distributions of $|{E_x}(x,{z_2})|$ calculated for the structure with ${w_g} = 0.85\;{\unicode{x00B5}{\rm m}}$ at two resonant frequencies $\omega /2\pi = 1.38\,{\rm THz}$ (first-order plasmon resonance) and $\omega /2\pi = 3.78\,{\rm THz}$ (third-order plasmon resonance) are shown in Fig. 8. As seen, the geometry of the distributions obtained for the frequency of first-order plasmon resonance [Fig. 8(a)] is similar to the previous case depicted in Fig. 7(a). However, the wider gated region of 2DEG integrally provides larger contribution to the absorption of the $\textit{em}$ wave by 2DEG. The corresponding values of ${L_{2DEG}}$ are following: $47.5\%$ (for ${h_g} = 0.05\;{\unicode{x00B5}{\rm m}}$), $46.6\%$ (for ${h_g} = 1\;{\unicode{x00B5}{\rm m}}$), and $41.2\%$ (for ${h_g} = 5\;{\unicode{x00B5}{\rm m}}$). At this, ${L_0} = 48.2, 47.6, 43.9\%$, respectively. The distributions in Fig. 8(b) obtained at the frequency of third-order plasmon resonance demonstrate multiple spatial oscillations. The number of such oscillations is proportional to the order of plasmon resonances. Also, we see that the deepest grating with ${h_g} = 5\;{\unicode{x00B5}{\rm m}}$ essentially suppresses the plasmon absorption of the $\textit{em}$ wave. The corresponding values of ${L_{2DEG}}$ are the following: 27.2% (for ${h_g} = 0.05\;{\unicode{x00B5}{\rm m}}$), 24.1% (for ${h_g} = 1\;{\unicode{x00B5}{\rm m}}$), and 11.8% (for ${h_g} = 5\;{\unicode{x00B5}{\rm m}}$). At this, ${L_0} = 28.7\%$, 26.1%, and 15.1%, respectively.

## 5. SUMMARY

We have developed a computationally stable RCWA method for the solution of Maxwell’s equations in the case of multilayered plasmonic structures with a delta-thin grating-gated conductive channel. The method was formulated for a planar diffraction problem for both TM and TE polarization of an incident wave. The method was implemented for investigation of far- and near-field characteristics of the particular plasmonic structures based on AlGaN/GaN heterostructure with a deeply subwavelength metallic grating coupler.

The calculations of the far-field characteristics including transmission, reflection, and absorption coefficients for zero diffraction order were performed in THz frequency range where the considered structure has multiple resonances related to the excitations of 2D plasmons in the conductive channel of AlGaN/GaN heterostructure. The dependence of these characteristics versus grating parameters and their convergence versus the number of the Fourier harmonics were analyzed.

We found that spectra of transmission and reflection coefficients in the lower frequency range, 0.2 THz, have weak dependence on grating depth. Results for both shallow (${h_g}/{a_g} = 0.05 \;{\unicode{x00B5}{\rm m}}/1\;{\unicode{x00B5}{\rm m}}$) and deep (${h_g}/{a_g} = 1$, $5\, {\unicode{x00B5}{\rm m}}/1\;{\unicode{x00B5}{\rm m}}$) gold grating are almost identical and coincide with the results of the IE method where grating is treated as delta-thin. In a higher frequency range, 3.5 THz, increase of the grating depth suppresses transmission with the increasing of the reflection coefficients. At the same time, the absorption spectrum remain less sensitive to the grating depth. We showed that dispersion of far-field characteristics on ${h_g}$ becomes more pronounced for a narrower-slit grating with ${w_g}/{a_g} = 0.85\;{\unicode{x00B5}{\rm m}}/1\;{\unicode{x00B5}{\rm m}}$ than for a wide-slit grating with ${w_g}/{a_g} = 0.5\;{\unicode{x00B5}{\rm m}}/1\;{\unicode{x00B5}{\rm m}}$. We showed that convergence of the calculations depends on geometrical parameters of the grating and frequencies. Better convergence is achieved for shallow and wide-slit grating with relative errors of ${\sim}0.05 {-} 0.2\%$ (in dependence on frequency) with 30 Fourier harmonics. For deep and narrow-slit grating, relative errors of ${\sim}0.1..1\%$ are achieved at $M \sim 50$.

Procedure of the calculations of the near-field characteristics was discussed in detail. Analysis of spatial distribution of the amplitudes of the $\textit{em}$ wave’s components in the near-field zone reveals the main physical peculiarities of the interaction of the plasmonic structure with incident radiation. It was shown that subwavelength metallic grating plays the role of a perfect waveguide for the incident wave, concentrator of the $\textit{em}$ energy, and polarization rotator. In the region of the grating slit, the $\textit{em}$ wave has predominantly lateral polarization with amplitude close to the amplitude of the incident wave. The hot zone is formed in the region between grating bars and 2DEG where the $\textit{em}$ wave has predominantly vertical polarization with amplitudes that can in 100 times exceed the amplitude of the incident wave.

The pattern of the near-field also was used for the calculations of the partial losses related to the grating and 2DEG. It was shown that at the frequencies of the plasmon resonances, the structure can efficiently absorb THz radiation with absorption coefficient values in the order of 20–50% (independent of the grating filling factor and order of plasmon resonance). We found that the contribution of 2DEG to the total losses is dominant at low-frequency plasmon resonances with weak dependence on the grating depth. At high-frequency plasmon resonances, the effect of the grating depth becomes essential. The deep gratings can effectively screen interaction of the $\textit{em}$ waves with plasmon oscillation in 2DEG that leads to a decrease of the total absorption of THz radiation.

Also, it should be noted that the proposed modified RCWA has several
advantages over conventional *volumetric*
RCWA. First, our realization of the RCWA method allows us to avoid
additional numerical manipulation with matrices that can reduce the
computational time. For the considered structure, we have 25% in terms of
computation time savings in comparison with conventional RCWA at the
volumetric treatment of the conductive layer. This value can be increased
in simulation of structures with a stack of 2D conductive layers. Second,
we operate with one parameter, 2D concentration, ${n_{2D}}$, instead of two independent parameters
of bulk concentration, ${n_{3D}}$, and thickness of the layer,
$d$. It can be convenient for the metrology
of the structures at the processing of the experimental data.

We suggest that the proposed modified RCWA algorithm can be effectively used for the modeling of the optical characteristics of various kinds of plasmonic structures with 2D conductive channels, including QW- or graphene-based structures, and the results of the paper provide deeper insight on physics of the interaction of THz radiation with grating-gated plasmonic structures.

## Funding

Lietuvos Mokslo Taryba (DOTSUT-184); European Regional Development Fund (No. 01.2.2-LMT-K-718-03-0096; Bundesministerium für Bildung und Forschung (VIP+ “Nanomagnetron”).

## Acknowledgment

The authors thank Prof. V. A. Kochelap (ISP NASU, Ukraine) and Dr. I. Kašalynas (FTMC, Lithuania) for fruitful discussions of the various aspects of this work. V. J. and V. V. K. acknowledge the support from the Research Council of Lithuania (Lietuvos mokslo taryba) through the “T-HP” Project (Grant No. DOTSUT-184) funded by the European Regional Development Fund according to the supported activity “Research Projects Implemented by World-class Researcher Groups” under the Measure No. 01.2.2-LMT-K-718-03-0096. S. M. K. would like to acknowledge the support of the Bundesministerium für Bildung und Forschung (VIP+ “Nanomagnetron”).

## Disclosures

The authors declare no conflicts of interest.

## REFERENCES AND NOTE

**1. **E. Popov, ed. *Gratings: Theory and Numeric
Applications*, 1st ed.
(Presses Universitaires de Provence,
2012).

**2. **W. Neumann, *Fundamentals of Dispersive Optical
Spectroscopy Systems* (SPIE,
2014).

**3. **H. F. Hammad, Y. M. M. Antar, A. P. Freundorfer, and M. Sayer, “A new dielectric grating
antenna at millimeter wave frequency,” IEEE
Trans. Antennas Propag. **52**,
36–44 (2004). [CrossRef]

**4. **S. Shena, Y. Yuana, Z. Ruana, and H. Tan, “Optimizing the design of an
embedded grating polarizer for infrared polarization light field
imaging,” Results Phys. **12**, 21–31
(2019). [CrossRef]

**5. **G. A. Melentev, V. A. Shalygin, L. E. Vorobjev, V. Y. Panevin, D. A. Firsov, L. Riuttanen, S. Suihkonen, V. V. Korotyeyev, Y. M. Lyaschuk, V. A. Kochelap, and V. N. Poroshin, “Interaction of surface
plasmon polaritons in heavily doped GaN microstructures with terahertz
radiation,” J. Appl. Phys. **119**, 093104
(2016). [CrossRef]

**6. **V. Janonis, S. Tumenas, P. Prystawko, J. Kacperski, and I. Kašalynas, “Investigation of *n*-type gallium nitride grating for applications
in coherent thermal sources,” Appl. Phys.
Lett. **116**, 112103
(2020). [CrossRef]

**7. **V. V. Popov, D. V. Fateev, O. V. Polischuk, and M. S. Shur, “Enhanced electromagnetic
coupling between terahertz radiation and plasmons in a grating-gate
transistor structure on membrane substrate,”
Opt. Express **18**,
16771–16776 (2010). [CrossRef]

**8. **V. V. Korotyeyev, V. A. Kochelap, S. Danylyuk, and L. Varani, “Spatial dispersion of the
high-frequency conductivity of two-dimensional electron gas subjected
to a high electric field: collisionless case,”
Appl. Phys. Lett. **113**,
041102 (2018). [CrossRef]

**9. **V. Ryzhii, T. Otsuji, and M. Shur, “Graphene based plasma-wave
devices for terahertz applications,” Appl.
Phys. Lett. **116**, 140501
(2020). [CrossRef]

**10. **T. Otsuji and M. Shur, “Terahertz plasmonics: good
results and great expectations,” IEEE Microw.
Mag. **15**(7),
43–50
(2014). [CrossRef]

**11. **D. Pashnev, T. Kaplas, V. Korotyeyev, V. Janonis, A. Urbanowicz, J. Jorudas, and I. Kašalynas, “Terahertz time-domain
spectroscopy of two-dimensional plasmons in AlGaN/GaN
heterostructures,” Appl. Phys. Lett. **117**, 051105
(2020). [CrossRef]

**12. **D. Pashnev, V. Korotyeyev, V. Janonis, J. Jorudas, T. Kaplas, A. Urbanowicz, and I. Kašalynas, “Experimental evidence of
temperature dependent effective mass in AlGaN/GaN heterostructures
observed via THz spectroscopy of 2D plasmons,”
Appl. Phys. Lett. **117**,
162101 (2020). [CrossRef]

**13. **B. Yan, J. Fang, S. Qin, Y. Liu, Y. Zhou, R. Li, and X.-A. Zhang, “Experimental study of plasmon
in a grating coupled graphene device with a resonant
cavity,” Appl. Phys. Lett. **107**, 191905
(2015). [CrossRef]

**14. **M. M. Jadidi, A. B. Sushkov, R. L. Myers-Ward, A. K. Boyd, K. M. Daniels, D. K. Gaskill, M. S. Fuhrer, H. D. Drew, and T. E. Murphy, “Tunable terahertz hybrid
metal–graphene plasmons,” Nano Lett. **15**, 7099–7104
(2015). [CrossRef]

**15. **B. Zhao and Z. M. Zhang, “Strong plasmonic coupling
between graphene ribbon array and metal gratings,”
ACS Photon. **2**,
1611–1618 (2015). [CrossRef]

**16. **H. Lu, J. Zhao, and M. Gu, “Nanowires-assisted excitation
and propagation of mid-infrared surface plasmon polaritons in
graphene,” J. Appl. Phys. **120**, 163106
(2016). [CrossRef]

**17. **S. M. Kukhtaruk, V. V. Korotyeyev, V. A. Kochelap, and L. Varani, “Interaction of THz radiation
with plasmonic grating structures based on graphene,”
in International Conference on Mathematical Methods in
Electromagnetic Theory, Lviv, Ukraine
(2016),
pp. 196–199.

**18. **T. Low, A. Chaves, J. D. Caldwell, A. Kumar, N. X. Fang, P. Avouris, T. F. Heinz, F. Guinea, L. Martin-Moreno, and F. Koppens, “Polaritons in layered
two-dimensional materials,” Nat.
Mater. **16**,
182–194 (2017). [CrossRef]

**19. **S. A. Mikhailov, “Plasma instability and
amplification of electromagnetic waves in low-dimensional electron
systems,” Phys. Rev. B **58**, 1517–1532
(1998). [CrossRef]

**20. **V. V. Korotyeyev and V. A. Kochelap, “Plasma wave oscillations in a
nonequilibrium two-dimensional electron gas: electric field induced
plasmon instability in the terahertz frequency range,”
Phys. Rev. B **101**,
235420 (2020). [CrossRef]

**21. **D. V. Fateev, V. V. Popov, and M. S. Shur, “Plasmon spectra
transformation in grating-gate transistor structure with spatially
modulated two-dimensional electron channel,”
Semiconductor **44**,
1406–1462 (2010). [CrossRef]

**22. **V. V. Popov, D. V. Fateev, T. Otsuji, Y. M. Meziani, D. Coquillat, and W. Knap, “Plasmonic terahertz detection
by a double-grating-gate field-effect transistor structure with an
asymmetric unit cell,” Appl. Phys.
Lett. **99**, 243504
(2011). [CrossRef]

**23. **Y. M. Lyaschuk and V. V. Korotyeyev, “Theory of detection of
terahertz radiation in hybrid plasmonic structures with drifting
electron gas,” Ukr. J. Phys. **62**, 889–902
(2017). [CrossRef]

**24. **O. V. Shapoval, J. S. Gomez-Diaz, J. Perruisseau-Carrier, J. R. Mosig, and A. I. Nosich, “Integral equation analysis of
plane wave scattering by coplanar graphene-strip gratings in the THz
range,” IEEE Trans. Terahertz Sci.
Technol. **3**,
666–674
(2013). [CrossRef]

**25. **O. V. Shapovala and A. I. Nosich, “Finite gratings of many thin
silver nano strips: optical resonances and role of
periodicity,” AIP Adv. **3**, 042120 (2013). [CrossRef]

**26. **M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave
analysis of planar-grating diffraction,” J.
Opt. Soc. Am. **71**,
811–818 (1981). [CrossRef]

**27. **M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and
efficient implementation of the rigorous coupled-wave analysis of
binary gratings,” J. Opt. Soc. Am. A **12**, 1068–1076
(1995). [CrossRef]

**28. **M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the
rigorous coupled-wave analysis for
surface-relief gratings: enhanced transmittance matrix
approach,” J. Opt. Soc. Am. A **12**, 1077–1086
(1995). [CrossRef]

**29. **S. Inampudi and H. Mosallaei, “Tunable wideband-directive
thermal emission from SiC surface using bundled graphene
sheets,” Phys. Rev. B **96**, 125407
(2017). [CrossRef]

**30. **S. Inampudi, V. Toutam, and S. Tadigadapa, “Robust visibility of graphene
monolayer on patterned plasmonic substrates,”
Nanotechnology **30**,
015202 (2019). [CrossRef]

**31. **L. Li, “Use of Fourier series in the
analysis of discontinuous periodic structures,”
J. Opt. Soc. Am. A **13**,
1870–1876 (1996). [CrossRef]

**32. **E. Popov, M. Nevière, and N. Bonod, “Factorization of products of
discontinuous functions applied to Fourier–Bessel
basis,” J. Opt. Soc. Am. A **21**, 46–52
(2004). [CrossRef]

**33. **E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Surface-integral equations
for electromagnetic scattering from impenetrable and penetrable
sheets,” IEEE Antennas Propag. Mag. **35**(6),
14–25 (1993). [CrossRef]

**34. **T. L. Zinenko and A. I. Nosich, “Plane wave scattering and
absorption by resistive-strip and dielectric-strip periodic
gratings,” IEEE Trans. Antennas
Propag. **46**,
1498–1505 (1998). [CrossRef]

**35. **V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Sum rules for the optical and
Hall conductivity in graphene,” Phys. Rev.
B **75**, 165407
(2007). [CrossRef]

**36. **L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared
properties of a graphene monolayer and multilayer,”
Phys. Rev. B **76**,
153410
(2007). [CrossRef]

**37. **M. V. Balaban, O. V. Shapoval, and A. I. Nosich, “THz wave scattering by a
graphene strip and a disk in the free space: integral equation
analysis and surface plasmon resonances,” J.
Opt. **15**, 114007
(2013). [CrossRef]

**38. **S. M. Kukhtaruk, V. A. Kochelap, V. N. Sokolov, and K. W. Kim, “Spatially dispersive
dynamical response of hot carriers in doped graphene,”
Physica E **79**,
26–37 (2016). [CrossRef]

**39. **Y. M. Lyaschuk and V. V. Korotyeyev, “Interaction of a terahertz
electromagnetic wave with the plasmonic system ‘grating–2D–gas.’
Analysis of features of the near field,” Ukr.
J. Phys. **59**,
495–504
(2014). [CrossRef]

**40. **A. V. Chaplik, “Absorption and emission of
electromagnetic waves by two-dimensional plasmons,”
Surf. Sci. Rep. **5**,
289–335 (1985). [CrossRef]

**41. **M. Dyakonov and M. Shur, “Shallow water analogy for a
ballistic field effect transistor: new mechanism of plasma wave
generation by dc current,” Phys. Rev.
Lett. **71**,
2465–2467 (1993). [CrossRef]

**42. **Parametrical studies of the plasmon resonance at different
configurations of the grating coupler are reported in Ref. [39] and in the recent
experimental paper Ref. [11].

**43. **K.-H. Brenner, “Aspects for calculating local
absorption with the rigorous coupled-wave method,”
Opt. Express **18**,
10369–10376 (2010). [CrossRef]

**44. **M. Weismann, D. F. G. Gallagher, and N. C. Panoiu, “Accurate near-field
calculation in the rigorous coupled-wave analysis
method,”J. Opt. **17**, 125612 (2015). [CrossRef]

**45. **COMSOL
AB, “COMSOL Multiphysics v. 5.5,” https://www.comsol.com.