1. Introduction

Diffraction gratings, periodic structures that diffract an incident electromagnetic wave into several different directions, have long been investigated for their numerous applications in spectroscopy, imaging, and external cavity lasers [1–6]. Recently, the concept of metagratings evolved from the conventional diffraction gratings, allowing wave manipulation with very high efficiency [7]. The prefix ‘meta’ implies that the grating is composed of a single or several subwavelength metamaterials, the diffraction properties of which can be engineered. The fundamental physical principle behind their operation is related to the Floquet-Bloch (FB) theorem, implying that when a plane wave is scattered by a periodic structure, a discrete set of diffracted waves can be generated, some of which are propagating. The undesired propagating waves can be eliminated by properly designing a metagrating, so all the incident power is transferred to a number of desired directions. These structures can realize different applications, such as an anomalous reflector, lens, and beam splitter [8–14].

More recently, graphene-based metagratings were introduced for instantaneous control over the diffraction pattern at THz regime [15,16]. Graphene, a two-dimensional material made of carbon atoms and arranged in a honeycomb lattice, has attracted huge research interest [17,18]. High electrical conductivity, gate-variable optical conductivity, optical transparency, controllable plasmonic properties, and high-speed operation are some of the unique electromagnetic properties of graphene, making it a promising material for a variety of applications such as perfect absorption, transformation optics, antennas, wavefront shaping, and analog computing [19–28]. In [15,16], the graphene strips were suggested for realizing reconfigurable metagratings wherein multiple functionalities were enabled such as anomalous reflection and retroreflection. Their characteristics can be tuned by electrostatically biasing the graphene strips.

Analysis and design of previously proposed graphene-based metagratings heavily relies on time-consuming full-wave simulations and sophisticated design procedures. Therefore, an analytical method for designing graphene-based metagratings is quite in demand. In [29–32], accurate analytical methods were presented to analyze a periodic arrays of graphene ribbons and disks based on the integral equations governing the surface current density on graphene. However, these methods are obtained under a quasi-static approximation; thus, the period of the structure must be smaller than the operating wavelength. Consequently, these analytical methods cannot be used to design graphene-based metagratings. Very recently, an analytical method was presented for the diffraction analysis of graphene ribbon arrays and the closed-form and analytical expressions were derived for the reflection coefficients of zeroth and higher diffracted orders [33]. However, in [33], the graphene ribbons are considered suspended in free space, which is impractical for designing a metagrating.

In this paper, we extend the work in [33] to enable a rigorous analysis of a PEC-backed array of graphene ribbons. We first derive an integral equation that describes the surface currents induced on the ribbons. This equation is solved by expanding the current density in terms of eigenfunctions related to the problem of a single graphene ribbon. Next, the electromagnetic field is expressed as a Rayleigh expansion and electromagnetic boundary conditions are applied to determine the reflection coefficients of diffracted orders. The accuracy of the proposed method will be demonstrated through numerical examples. Finally, a tunable beam splitter and a tunable retroreflector are designed at THz frequencies. The designed beam splitter can split the incident wave to different angles with good efficiency by electrostatically biasing the graphene ribbons. It will be shown that the proposed retroreflector is able to reflect waves back to the incident direction with high efficiency for different angles of incidence. The retroreflector designed in this paper has a $90\%$ power efficiency, which is better than previously reported graphene-based retroreflectors [15,16].

2. Current distribution on a PEC–backed array of graphene ribbons

Consider a periodic array of graphene ribbons which is placed at the distance of $h$ from a PEC plate as shown in Fig. 1. The width of the graphene ribbons and periodicity of the array along the $x$-axis are $w$ and $D$, respectively. The structure is infinite along the $y$-axis. A transverse-magnetic (TM) polarized plane wave is incident on the ribbon array with an angle of incidence $\theta _i$. Graphene is modeled as a surface conductivity

In the following, a time dependence of the form $\exp (j{\omega }t)$ is implicitly assumed.

 

Fig. 1. A periodic array of graphene ribbons at a distance $h$ from a PEC back plate. The ribbons are illuminated by an obliquely incident TM polarized wave. The angle of incidence is $\theta _i$.

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Let us first consider the case where the graphene ribbons are suspended in free space (Fig. 2). Due to incidence of the TM-polarized wave and the uniformity of the structure along the $y$-axis, surface currents will be induced on the ribbons along the $x$-axis. We denote these currents by $J_{x}^{1}$. Based on Floquet theorem, current density on each ribbon (designated by an index $l$) can be related to the current density $J_{x}^{1}(x)$ on a reference ribbon ($l=0$) by

 

Fig. 2. Periodic array of graphene ribbons suspended in free space.

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The electric field at each point on a ribbon is the sum of the external (incident) electric field and the electric field generated in free space by the induced surface currents on all the ribbons. Using (3), the integro-differential equation governing the surface currents induced on the reference ribbon can thus be written as [33]

In [33], the integral Eq. (5) was analytically solved by expanding the current density in terms of the eigenfunctions arising in the problem of scattering by a single graphene ribbon. However, (5) cannot describe the induced surface current on the ribbons in presence of a PEC back plate, as it does not take account of currents induced on the latter. In [32], it was shown that the effect of the PEC is strong and cannot be neglected. In order to take account of the PEC, we adopt the image theory by replacing the PEC plate with a virtual current distribution $-J_x(x)$ at $z=2h$ (Fig. 3). At the same time, the external electric field must be replaced by the incident field plus the field reflected by the PEC plate in the absence of graphene ribbons,

 

Fig. 3. Adopting the image theorem for calculating the surface currents on a PEC-backed array of graphene ribbon.

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The surface current density of a PEC-backed array of graphene ribbons will then satisfy the integral equation

Here $G_{x}^{2}(x-x')$ accounts for the electric field generated by the PEC plate (image currents) and is given by

To solve (8), consider the following eigenvalue problem

The functions $\psi _{n}^{0}(x)$ and eigenvalues $q_{n}^{0}$ were presented in [30]. The first three eigenfunctions can be obtained as listed in Table 1. The higher-order eigenfunctions ($n > 3$) can be approximately determined by $\sqrt {2/w} \cos (\frac {n\pi }{w}x)$ and $\sqrt {2/w} \sin (\frac {n\pi }{w}x)$ for odd and even orders, respectively.

Table 1. The first three eigenfunctions for the problem of a single graphene ribbon.

View Table

Using first order perturbation theory, we next approximate the eigenvalue in the original eigenvalue problem as [33]

To calculate $q_n$, we use the following relationships for ${G_{x}^{1}}(x - x')$ and ${G_{x}^{2}}(x - x')$ (see Appendix)

Upon replacing ${G_{x}^{1}}(x - x')$ and ${G_{x}^{2}}(x - x')$ by (15) in (14), we obtain

Unlike eigenvalues, we neglect the effect of perturbation on the eigenfunctions, and take $\psi _{n}(x)$ to be the same as $\psi _{n}^{0}(x)$ in the unperturbed problem describing the surface currents induced on a single graphene ribbon. Note that for the method to be valid, the width of the ribbons must be much smaller than the wavelength ($k_0 w <<1$) while the period of the structure is arbitrary.

We may next write down the solution of (8) in terms of the eigenfunctions as follow

3. Reflection and transmission coefficients of diffraction orders

The distribution of surface current on the PEC-backed graphene ribbons was derived in the previous section. In this section, we use the result obtained to calculate the reflection coefficient of diffraction orders. To that end, we express the total electromagnetic field in all the regions as a Rayleigh expansion. An incident field of unit amplitude with the wave vector $\mathbf {k}=k_x \hat {\mathbf {x}}+k_z\hat {\mathbf {z}}$ will be reflected by the grating/PEC system into various diffraction orders with the amplitude $R_m$ (Fig. 4). The electromagnetic field in the region $z<0$ may be written as

In the region $0<z<h$ the electromagnetic field is expanded as

 

Fig. 4. Diffraction of an obliquely incident TM-plane wave by a PEC-backed array of graphene ribbons. The reflection coefficient for the $m$-th order diffracted wave is denoted by $R_{m}$

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In these equations, $k_{z,m}$ and $k_{x,m}$ are the wave-vector components in $z$ and $x$ directions, respectively, and are given by (18). $k_{z,m}$ is either negative real (propagating wave) or positive imaginary (evanescent wave).

Now, we apply the electromagnetic boundary conditions at $z=0$,

By substituting (20) and (22) in (23), we have after some straightforward mathematical manipulations,

Multiplying both sides of (24) by $e^{jk_{x,m} x}$ and taking the integral over one period, the reflection coefficients are readily obtained as

Although we have so far assumed the graphene ribbons to be deposited on a foam substrate with a permittivity of $\varepsilon _0$, our proposed method can be readily generalized to the case where ribbons are mounted on an arbitrary substrate. The free-space periodic Green’s function ($G_{x}(x - x')$) must then be replaced with the periodic Green’s function of the multilayered configuration. In the Appendix, we present and discuss this generalized case.

4. Results, discussions and applications

4.1 Numerical results

In this subsection, we verify the accuracy of the proposed analytical method through some numerical examples. As the first example, we consider an array of graphene ribbons with $w=8\mu m$ and $D=50\mu m$. The distance from the PEC plate is $h=2 \mu m$. The charges relaxation time and Fermi energy are assumed to be $\tau = 2 ps$, and $E_F=1.5$eV, respectively. The second example describes a PEC-backed array of graphene ribbons under oblique TM incidence. The parameters of the structure are $D=75\mu m$, $w=10\mu m$, $h=10\mu m$, $E_F=0.5$eV, and $\tau =0.75 ps$. The diffraction efficiency of zeroth (specular) diffracted and higher diffracted orders for different angles of incidence are depicted in Fig. 5 and Fig. 6. For the sake of comparison, we have also included results obtained using the subwavelength approximation [32] as well as the finite integration technique (FIT). The latter is utilized as a reliable benchmark to verify the results and is provided by the commercial program, CST Microwave Studio software 2019. Graphene is modeled as an impedance surface with zero thickness in CST, with a surface impedance given by $1/\ \sigma _s$. In addition, periodic boundary conditions were applied in both $x$ and $y$ directions, while the perfectly matched-layer (PML) and PEC boundary conditions were applied in the $z$ direction.

 

Fig. 5. Diffraction efficiency of (a) zeroth-order mode and (b) higher order modes of an array of graphene ribbons with $w= 8 \mu m$ and $D= 50 \mu m$ at a distance $h=2 \mu m$ from the PEC sheet. A normally incident TM plane wave illuminates the graphene ribbons. The graphene parameters are $\tau = 2 ps$ and $E_F=1.5$eV.

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Fig. 6. Diffraction efficiency of a PEC-backed periodic array of graphene ribbons illuminated by an TM polarized wave. The graphene ribbons parameters are $D=75\mu m$, $w=10\mu m$, $h=10\mu m$, $E_F=0.5$eV, and $\tau =0.75 ps$. (a) and (b) are the diffraction efficiencies of zeroth-order mode for different angles of incidence. (c) Diffraction efficiencies of higher orders at normal incidence. (d) $DE_{-1}$ for incident angles $\theta _i=30^\circ$ and $\theta _i=60^\circ$.

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Excellent agreement is observed between the FIT results and our analytical method demonstrating the accuracy of the proposed method. By contrast, the method of [32] can not predict the reflection spectra of graphene ribbons as it uses a quasi-electrostatic approximation which is clearly not valid here. Hence, it cannot be used to analyze and design metagratings for which a non-zero diffraction order is always propagating.

The resonance features observed in Fig. 5 and Fig. 6 are due to the formation of plasmonic standing waves in individual ribbons and can be explained by the obtained analytical expressions. Equation (19b) shows these resonances to occur when, for a particular eigenvalue, $q_n\approx 1/\sigma _s$, and lead to large changes in the the reflection coefficients. It is found from (16) that the change in the magnitude of $q_n$ is small when the angle of incidence is varied. As a result, we expect that the resonance frequency to change slightly with $\theta _i$, as seen in Fig. 6.

4.2 Design of a tunable beam splitter

To illustrate the capability of our method, we design a THz beam splitter based on the concept of metagratings using analytical expressions derived in the previous section. We assume that the PEC-backed array of graphene ribbons is illuminated by a normally incident TM plane wave. For simplification of the design procedure, we assume that 0 (specular) and $(\pm 1)$ orders are the only propagating modes. According to (18), the periodicity of the structure must be chosen such that $\lambda _0<D<2 \lambda _0$. Due to the symmetry of the structure, we expect diffraction efficiency of $(-1)$ and $(+1)$ to be equal. By eliminating specular reflection, the incident power is split at the angle $\theta _{\pm 1}$, as shown in Fig. 7(a). $\theta _{\pm 1}$ can be written in terms of the wavelength ($\lambda _0$) and the periodicity of graphene (D) as

 

Fig. 7. (a) Schematic representation of the designed tunable beam splitter. By eliminating the specular reflection, power splitting can be achieved. (b) $DE_{\pm 1}$ of the designed beam splitter for different Fermi energy levels. The geometrical and material parameters of graphene ribbons are $D=39.2\mu m$, $w=3.6\mu m$ and $h=8.5\mu m$, $\tau =1ps$. (c) Distribution of the real part of the electric field phasor for $E_F=0.9$eV at 9.31 THz. (d) Distribution of the real part of the electric field phasor for $E_F=1.3$eV at 11.246 THz. The diffraction efficiencies and the E-field distributions are obtained by CST Microwave Studio 2019.

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We next design a beam splitter for the operating frequency of 10 THz and a splitting angle of $\theta =50^\circ$. Based on (28), periodicity must be chosen as $D=39.2 \mu m$. Graphene ribbons may be considered to be deposited on a foam substrate for the proposed beam splitter with a permittivity of $\varepsilon =\varepsilon _0$ at the THz spectrum [34]. A relaxation time of $\tau =1ps$ is assumed for the ribbons. To extract the other parameters ($E_F$, $w$, and $h$), we utilize the genetic algorithm (GA) to minimize the specular reflection. Using the proposed method, we define the cost function $DE_0^2+1/(DE_1^2+DE_{-1}^2)$ at 10 THz. The other parameters of the structure are extracted as $w=3.6 \mu m$, $h=8.5\mu m$, and $E_F=1$eV. The diffraction efficiencies of the optimized structure are plotted in Fig. 7(b). It can be seen in Fig. 7(b) that the incident power is mostly reflected as the ($\pm 1$) FB mode at 10 THz. The results demonstrate an efficiency of around $80\%$, which is better than previous graphene-based splitters [35–37]. The relative bandwidth of the designed beam splitter is 5$\%$ (in this paper, bandwidth is defined as the frequency range in which the power efficiency is above $75\%$), which is comparable with other graphene-based beam splitters. [16,35] We also investigate the effect of electrostatic bias on the performance of the beam splitter in Fig. 7(b). As can be observed, changing $E_F$ in the range of $0.8-1.3$eV gives the possibility of tuning the splitting frequency between 8.78 and 11.246 THz. As mentioned before, according to 28, the angle of $\theta _{\pm 1}$ is proportional to the period of the structure and frequency. The period of the structure is fixed; hence the angle of $\theta _{\pm 1}$ can be scanned by varying the frequency. Based on 28, at 8.78 THz and 11.246 THz, the power is split into angles $55^\circ$, and $42^\circ$, respectively. Therefore, the proposed structure may split the incident wave to different angles by changing $E_F$. The electric field distributions are presented in Fig. 7 for different frequencies.

4.3 Design of a tunable retroreflector

Retroreflectors are defined as electromagnetic structures that can reflect an incident wave into the same direction as the latter. Recently, retroreflectors made of metagratings have attracted much attention due to their high power efficiency and much less complex fabrication process in comparison with other approaches such as metasurfaces [15,16,38]. In this subsection, we use the proposed method to design a tunable retroreflector.

Consider a PEC-backed array of graphene ribbons under oblique TM incidence. Usually, in a metagrating, all the power carried by the incident plane wave should be transferred to the $(-1)$ order. In the retroreflector the incident and the diffracted wave angles are identical ($\theta _{-1}=-\theta _i$, the so-called auto-collimation blazing) as shown in Fig. 8(a). Thus, to achieve auto-collimation, the period of the structure must be set as

 

Fig. 8. (a) Schematic representation of the proposed retroreflector (b) Diffraction efficiencies of the designed metagrating for $\theta _i=30$. Structure parameters are $D=60\mu m$, $w=13.7\mu m$, $h=17.5\mu m$, $E_F=1.15$eV, and $\tau =1ps$. (c) The real part of electric field distribution for $E_F=1.15$eV at 5THz. CST Microwave Studio 2019 was used to extract the E-field distribution. (d) Diffraction efficiencies of the same structure with Fermi energy levels of $1.3$eV and $0.95$eV, illuminated by obliquely incident TM polarized plane waves with incident angles $\theta _i=25^\circ$ and $\theta _i=35^\circ$, respectively. In (b) and (d), the frequency at which auto-collimation can be achieved is denoted by a dashed line.

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For more simplicity in the design process, we need to ensure that all higher-order modes other than $(-1)$ are evanescent. Hence according to (18) and (29), we restrict the angle of incidence to $19.48^\circ < \theta _i<90^\circ$. To design a retroreflector for $\theta _i=30^\circ$ at 5 THz, period of the ribbon grating is chosen as $D=60 \mu m$ according to (29). As in the previous section, we use a foam substrate and assume the relaxation time of graphene to be $1ps$. Again, a GA is run (using the proposed analytical method) to maximize the first negative-order power. The cost function used is $DE_0^2+1/DE^2_{-1}$ at the desired frequency. The optimal values of other structure parameters are obtained as $w=13.7\mu m$, $h=17.5 \mu m$, and $E_F=1.15$eV . The diffraction efficiencies of the designed metagrating are plotted in Fig. 8(b). An excellent agreement is observed between the numerical results and those predicted by our analytical approach. As can be seen from Fig. 8(b), the incident power in transferred to the $(-1)$ order in the 4.4- 6 THz range with very good efficiency. According to (29), auto-collimation for $\theta _i=30$ occurs at 5THz in which it can be observed that the incident wave couples to the (-1) mode. The designed structure has an efficiency of about $90\%$, which is a remarkable achievement compared with previously reported graphene-based metagratings($80 \%$) [15,16] and metasurface-based retroreflectors ($80 \%$) [39,40]. The relative bandwidth of the proposed structures is $30\%$, which is better than the previously reported metagrating designs [9,11,15,16,41,42]. The electric field distribution plotted at 5THz demonstrates that the incident wave is reflected in the direction of incidence (Fig. 8(c)). If we change the value of $E_F$ to $1.3$eV and $0.95$eV, auto-collimation can be achieved at the angle of $25^\circ$ and $35^\circ$, respectively. The diffraction efficiencies and auto-collimation frequency of these cases are also depicted in Fig. 8(d). The results show that auto-collimation can occur for different incident angles when the Fermi level energy changes.

5. Conclusion

In this paper, based on the integral equations governing the surface current density induced on the PEC-backed graphene ribbons, an analytical method was proposed to calculate the reflection coefficients of diffracted orders. The proposed method was validated through numerical examples, showing excellent agreement with full-wave simulation results. Unlike previous works, our method is valid even for wavelengths smaller than the period of the structure. Using the proposed analytical method, a tunable beam splitter, and a tunable retroreflector were designed at THz frequencies with high power efficiency. We showed that the operating angle and frequency of the designed structures can be controlled by changing the electrostatic biasing of graphene. The proposed analytical approach paves the way for efficient design of tunable metagratings with a wide range of applications in the context of wavefront shaping.

6. Appendix

Proof of Eq. (15)

Here we aim to prove (15). The Hankel function $\frac {1}{{4j}} H_0^{(2)}\left ( {{k_0}\left | {\textbf{r} - \textbf {{r'}}} \right |} \right )$ , with $\textbf{r} = \left ( {x,y} \right )$ , satisfies

(30)$$\left( {\partial _x^2 + \partial _y^2 + k_0^2} \right)\frac{1}{{4j}}H_0^{(2)}\left( {{k_0}\left| {\textbf{r} - \textbf{{r'}}} \right|} \right) ={-} \delta \left( {\textbf{r} - \textbf{{r'}}} \right)$$

and may be represented as [43]

(31)$$\frac{1}{{4j}}H_0^{(2)}\left( {{k_0}\left| {\textbf{r} - \textbf{{r'}}} \right|} \right) = \frac{1}{{2j}}\int\limits_{ - \infty }^\infty {\frac{{{e^{ - j{q_x}\left( {x - x'} \right) - j\left| {y - y'} \right|\sqrt {k_0^2 - q_x^2} }}}}{{\sqrt {k_0^2 - q_x^2} }}} \frac{{d{q_x}}}{{2\pi }}$$

Consider, next, the summation

(32)$$\sum_{n ={-} \infty }^\infty {\frac{1}{{4j}}{e^{j{k_x}nD}}H_0^{(2)}\left[ {{k_0}\sqrt {{{\left( {x + nD - x'} \right)}^2} + {{\left( {y - y'} \right)}^2}} } \right]}$$

Which may be rewritten as

(33)$$\begin{aligned} &\sum\limits_{n ={-} \infty }^\infty {{e^{j{k_x}nD}}} \frac{1}{{2j}}\int\limits_{ - \infty }^\infty {\frac{{{e^{ - j{q_x}\left( {x + nD - x'} \right) - j\left| {y - y'} \right|\sqrt {k_0^2 - q_x^2} }}}}{{\sqrt {k_0^2 - q_x^2} }}} \frac{{d{q_x}}}{{2\pi }}\\ &= \frac{1}{{2j}}\int\limits_{ - \infty }^\infty {\sum\limits_{n ={-} \infty }^\infty {{e^{j{k_x}nD}}\frac{{{e^{ - j{q_x}\left( {x + nD - x'} \right) - j\left| {y - y'} \right|\sqrt {k_0^2 - q_x^2} }}}}{{\sqrt {k_0^2 - q_x^2} }}} } \frac{{d{q_x}}}{{2\pi }}\\ &= \frac{1}{{2j}}\int\limits_{ - \infty }^\infty {\left[ {\sum\limits_{n ={-} \infty }^\infty {{e^{j\left( {{k_x} - {q_x}} \right)nD}}} } \right]} \frac{{{e^{ - j{q_x}\left( {x - x'} \right) - j\left| {y - y'} \right|\sqrt {k_0^2 - q_x^2} }}}}{{\sqrt {k_0^2 - q_x^2} }}\frac{{d{q_x}}}{{2\pi }} \end{aligned}$$

We may next use the relationship

(34)$$\sum_{n ={-} \infty }^\infty {{e^{j\left( {{k_x} - {q_x}} \right)nD}}} = \frac{{2\pi }}{D}\sum_{n ={-} \infty }^\infty {\delta \left( {{k_x} - {q_x} + \frac{{2\pi n}}{D}} \right)}$$

Which yields

(35)$$\begin{aligned}&\sum\limits_{n ={-} \infty }^\infty {\frac{1}{{4j}}{e^{j{k_x}nD}}H_0^{(2)}\left[ {{k_0}\sqrt {{{\left( {x + nD - x'} \right)}^2} + {{\left( {y - y'} \right)}^2}} } \right]}\\ &= \frac{1}{{2j}}\int\limits_{ - \infty }^\infty {\left[ {\frac{{2\pi }}{D}\sum\limits_{n ={-} \infty }^\infty {\delta \left( {{k_x} - {q_x} + \frac{{2\pi n}}{D}} \right)} } \right]} \frac{{{e^{ - j{q_x}\left( {x - x'} \right) - j\left| {y - y'} \right|\sqrt {k_0^2 - q_x^2} }}}}{{\sqrt {k_0^2 - q_x^2} }}\frac{{d{q_x}}}{{2\pi }}\\ &= \frac{1}{{2jD}}\sum\limits_{n ={-} \infty }^\infty {\frac{{{e^{ - j\left( {{k_x} + \frac{{2\pi n}}{D}} \right)\left( {x - x'} \right) - j\left| {y - y'} \right|\sqrt {k_0^2 - {{\left( {{k_x} + \frac{{2\pi n}}{D}} \right)}^2}} }}}}{{\sqrt {k_0^2 - {{\left( {{k_x} + \frac{{2\pi n}}{D}} \right)}^2}} }}} \end{aligned}$$

As a result

(36)$$\begin{aligned}&\sum\limits_{n ={-} \infty }^\infty {\frac{1}{{4j}}{e^{j{k_x}nD}}H_0^{(2)}\left[ {{k_0}\sqrt {{{\left( {x + nD - x'} \right)}^2} + {{\left( {2h} \right)}^2}} } \right]}\\ &= \frac{1}{{2jD}}\sum\limits_{n ={-} \infty }^\infty {\frac{{{e^{ - j\left( {{k_x} + \frac{{2\pi n}}{D}} \right)\left( {x - x'} \right) - j\left| {2h} \right|\sqrt {k_0^2 - {{\left( {{k_x} + \frac{{2\pi n}}{D}} \right)}^2}} }}}}{{\sqrt {k_0^2 - {{\left( {{k_x} + \frac{{2\pi n}}{D}} \right)}^2}} }}} \end{aligned}$$

Effect of dielectric substrate

Here, we generalize our proposed method to the case where the graphene ribbons are deposited on a PEC-backed substrate with a relative permittivity of $\varepsilon _{r2}$ and thickness of $h$. We assume the oblique TM polarized wave to be incident on the graphene ribbons from a homogeneous isotropic media with a relative permittivity of $\varepsilon _{r1}$ as shown in Fig. 9. To include the effect of the substrate, we can rewrite the integral equation (8) as [44]

(37)$$\frac{1}{\sigma }{J_x}(x) = E_x^{ext} + \frac{1}{{j\omega }}\frac{d}{{dx}}\int\limits_{ - w/2}^{w/2} {{G_{x1,p}}(x - x')} \,\frac{{d\,{J_x}(x')}}{{dx'}}dx' - j\omega \int\limits_{ - w/2}^{w/2} {{G_{x2,p}}(x - x')\,} {J_x}(x')dx'$$

where

(38)$$E_x^{ext} = {e^{ - j{k_{x,0}}x}}\frac{{2{\xi _{1,0}}{\xi _{2,0}}(1 - {e^{ - 2j{k_{z2,0}}h}})}}{{{\xi _{1,0}}(1 + {e^{ - 2j{k_{z2,0}}h}}) + {\xi _{2,0}}(1 - {e^{ - 2j{k_{z2,0}}h}})}}$$

 

Fig. 9. An oblique TM polarized wave with the angle of $\theta _i$ is incident on a PEC-backed array of graphene ribbons. The ribbons are on a substrate with relative permittivity $\varepsilon _{r2}$.

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Here $E_x^{ext} (x)$ is the incident field plus the field reflected by the PEC-backed substrate in the absence of graphene ribbons and [44]

(39)$$\begin{aligned} G_{x1,p}(x - x') &= \frac{1}{D}\sum\limits_{p ={-} \infty }^{ + \infty } {\{ \frac{1}{{2j{\varepsilon _0}{k_{z1,p}}}}\frac{{\left( {\Gamma _0^L - 1} \right)\left( {1 - {e^{2j{k_{z2,p}}h}}} \right)}}{{\Gamma _0^L + {e^{2j{k_{z2,p}}h}}}} + }\\ &\frac{{{\mu _0}{\omega ^2}}}{{2jk_{x,p}^2{k_{z1,p}}}}\left( {\frac{{1 + \Gamma _0^L{e^{2j{k_{z2,p}}h}}}}{{\Gamma _0^L + {e^{2j{k_{z2,p}}h}}}} - \frac{{ - 1 + \Gamma _0^T{e^{2j{k_{z2,p}}h}}}}{{\Gamma _0^T - {e^{2j{k_{z2,p}}h}}}}} \right)\} {e^{ - j{k_{x,p}}(x - x')}} \end{aligned}$$

(40)$${G_{x2,p}}(x - x') = \frac{1}{D}\sum_{p ={-} \infty }^{ + \infty } {\frac{{{\mu _0}}}{{2j{k_{z1,p}}}}\frac{{\left( {\Gamma _0^T + 1} \right)\left( {1 - {e^{2j{k_{z2,p}}h}}} \right)}}{{\Gamma _0^T - {e^{2j{k_{z2,p}}h}}}}{e^{ - j{k_{x,p}}(x - x')}}}$$

where

(41a)$${k_{x,p}} = {k_0}\sqrt {{\varepsilon _{r1}}} \,\sin \,({\theta _i})\, + \frac{{2\,\pi \,p}}{D}$$

(41b)$${k_{zi,p}} ={-} j{k_0}\sqrt {{{(\sqrt {{\varepsilon _{r1}}} \,\sin \,({\theta _i}) + \frac{{p{\lambda _0}}}{D})}^2} - {\varepsilon _{ri}}} \,\,\,\,\,\,\,\,i = 1,2$$

(42)$$\Gamma _0^T = \frac{{{k_{z1,p}} - {k_{z2,p}}}}{{{k_{z1,p}} + {k_{z2,p}}}}\,\,\,\,~~~~~~,~~~~~~~\,\,\,\,\Gamma _0^L = \frac{{{k_{z1,p}}/{\varepsilon _{r1}} - {k_{z2,p}}/{\varepsilon _{r2}}}}{{{k_{z1,p}}/{\varepsilon _{r1}} + {k_{z2,p}}/{\varepsilon _{r2}}}}$$

Like section 2, $J_x (x)$ can be expanded in terms of the corresponding eigenfunctions as

(43)$${J_x}(x) = \sum_{n = 1} {{A_n}\psi _n^0(x)}$$

(44)$${A_n} = \frac{{{\sigma _s}}}{{1 - {q_n}{\sigma _s}}}\int\limits_{ - w/2}^{w/2} {\psi _n^0(x)E_x^{ext}(x)} \,dx$$

After some mathematical manipulation, $q_n$ is obtained as

(45)$${q_n} = \frac{{ - 1}}{{2jD\,\omega \,{\varepsilon _0}}}[\sum_{p ={-} \infty }^{ + \infty } {\sqrt {k_{x,p}^2 - k_0^2} \frac{{\left( {\Gamma _0^L - 1} \right)\left( {1 - {e^{2j{k_{z2,p}}h}}} \right)}}{{\Gamma _0^L + {e^{2j{k_{z2,p}}h}}}}} {\left| {\,{f_{pn}}} \right|^2}$$

To calculate the reflection coefficients of the structure, we expand the total field in the region $z<0$ as

(46a)$${H_{1y}} = {e^{ - j{k_{z1,0}}z}}{e^{ - j{k_{x,0}}x}} + \sum_m {{R_m}{e^{j{k_{z1,m}}z}}{e^{ - j{k_{x,m}}x}}}$$

(46b)$${E_{1x}} = {\xi _{1,0}}{e^{ - j{k_{z1,0}}z}}{e^{ - j{k_{x,0}}x}} - \sum_m {{\xi _{1,m}}{R_m}{e^{j{k_{z1,m}}z}}{e^{ - j{k_{x,m}}x}}}$$

For the substrate layer ($0<z<h$), the total field can be written as follows

(47a)$${H_{2y}} = \sum_m {{A_m}{e^{ - j{k_{z2,m}}z}}{e^{ - j{k_{x,m}}x}}} + \sum_m {{B_m}{e^{j{k_{z2,m}}z}}{e^{ - j{k_{x,m}}x}}}$$

(47b)$${E_{2x}} = \sum_m {{\xi _{2,m}}{A_m}{e^{ - j{k_{z2,m}}z}}{e^{ - j{k_{x,m}}x}}} - \sum_m {{\xi _{2,m}}{B_m}{e^{j{k_{z2,m}}z}}{e^{ - j{k_{x,m}}x}}}$$

where

(48)$${\xi _{i,m}} = {k_{zi,m}}/(\omega {\varepsilon _0 \varepsilon_{ri}}) ~~~~~~; \,\ i=1,2$$

After applying the boundary conditions, the reflection coefficients can be calculated. Equation (26) is still valid with the following correction in $C_m$

(49)$${C_m} = \frac{{{\xi _{1,m}}(1 + {e^{ - 2j{k_{z2,m}}h}})}}{{{\xi _{2,m}}(1 - {e^{ - 2j{k_{z2,m}}h}})}}$$

We next validate and verify the accuracy of the proposed method through a numerical example. Consider a PEC-backed array of graphene ribbons with the parameters $D=75\mu m$, $w=8\mu m$, $h=3\mu m$, $\varepsilon _{r1}=1$, $\varepsilon _{r2}=4$, $E_F=1.5$eV, and $\tau =2 ps$. The diffraction efficiencies of diffracted orders for different incident angles are plotted as function of frequency in Fig. 10. A full-wave simulation is performed to validate the proposed method, and an excellent agreement between the numerical results and those predicted by our analytical approach is observable.

 

Fig. 10. Comparing the results of proposed method with full wave simulations: (a) $DE_0$ (b) $DE_{\pm 1}$ for different incident angles. Graphene ribbon parameters are $D=75\mu m$, $w=8\mu m$, $h=3\mu m$, $\varepsilon _{r1}=1$, $\varepsilon _{r2}=4$, $E_F=1.5$eV, and $\tau =2 ps$. Full wave simulations are obtained by CST Microwave Studio.

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Funding

Sharif University of Technology.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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