Dyadic analysis of a cylindrical wire consisting of a cover with fully-populated surface conductivity tensor

Shiva Hayati Raad; Zahra Atlasbaf

doi:10.1364/OE.27.021214

1. Introduction

Graphene, a 2D carbon material arranged in a honeycomb lattice, has been the subject of various scientific research due to its extraordinary properties [1]. It has been used in the design of various planar and non-planar structures, including but not limited to waveguides [2], invisible cloaks [3], oscillators [4], circulators, phase shifters, and isolators [5]. Specifically considering the graphene wrapped wires, a cylindrical waveguide with electrically or magnetically biased graphene cover supports highly confined low loss surface waves [6]. Also, graphene-coated single and double layer micro-tubes behave as refractive index sensors with high sensitivities and figure of merits (FOM) [7]. Moreover, graphene meta-surface can be used for the electromagnetic cloaking of the cylindrical objects by adjusting the surface reactance of the nano-patches [8]. Consequently, localized surface plasmons of graphene-coated cylindrical wires can be exploited for guided and radiative reconfigurable devices and applications [9–11].

Analysis of graphene-based planar structures is commonly conducted by incorporating appropriate surface conductivities in the boundary conditions of Maxwell’s equations. The surface conductivity of graphene takes various isotropic and anisotropic forms under different conditions. For instance, it can be approximated as a diagonal tensor for a densely packed graphene strip. Moreover, it is a tensor with equal diagonal and opposite off-diagonal components for graphene sheet under magnetic bias [12,13]. In general, closed-form formulas are not available for the surface conductivity of patterned sheets and the main challenge in this regard is the efficient modeling of the graphene surface conductivity. It is worth noting that patterned electrically and magnetically biased graphene sheets have promising light-matter interactions and they are of interest in many applications [14,15].

Two procedures were commonly used to analyze similar patterned structures with elements made of perfect electric conductors (PEC). 1) Using approximate formulas based on Floquet expansion and 2) implementing the periodic method of moments (PMoM). The first method is suitable for some common shapes in the sub-wavelength limit while the second approach can be generalized for any desired geometrical shape and any desired periodicity [16,17]. Similarly, for the sub-wavelength square graphene-based structures, closed-form formulas are extracted to calculate their surface conductivity under electric and magnetic biases [18,19]. For the arbitrarily shaped graphene elements, surface conductivity extraction based on full-wave simulation or measured data is suggested for both electric and magnetic biases [20,21].

Trying to model the conformal meta-surfaces based on Floquet expansion and PMoM, it is observed that the unit cell analysis cannot be conducted in the conformal structures due to lose of the periodicity [22]. Moreover, MoM implementation of cylindrical structures is mathematically heavy [23]. Therefore, the available formulas for the surface impedance of the periodic planar sheets are successfully used for the fast and accurate analysis of their cylindrical counterparts in the deep sub-wavelength regime [24]. Based on the above discussions, the aim of this paper is to numerically obtain the tensorial effective surface conductivity of the arbitrarily shaped graphene patches under magnetic bias. Afterward, the same structure is mounted on a cylinder and it is investigated by means of the associated dyadic Green’s function (DGF).

The paper is organized as follows. In Section 2, DGF of a monolayer cylinder with full-tensor surface conductivity boundary condition is formulated using the scattering superposition method. To validate the extracted coefficients, guided and radiative applications of these structures under electric bias are considered in Section 3. Later, densely packed graphene strips and magnetically biased graphene-based square patterns are analyzed by assigning appropriate formulas to the elements of the surface conductivity tensor. Finally, the details of extracting the effective tensor surface conductivity for the arbitrarily shaped graphene patterns are provided.

2. Dyadic Green’s function formulation

The structure under consideration is a monolayer cylinder with radius a, complex permittivity ε₂, and complex permeability μ₂. The constitutive parameters of the environment medium are respectively ε₁ and μ₁ and it is considered that the core cylinder is covered with an anisotropic 2D shell having full-tensor surface conductivity ${\bar{\bar{σ}}}_{ϕ z}$ defined as [25]:

{\bar{\bar{σ}}}_{ϕ z} = [\begin{matrix} σ_{ϕ ϕ} & σ_{ϕ z} \\ σ_{z ϕ} & σ_{z z} \end{matrix}] .

As Fig. 1 illustrates, graphene-based structures with different shapes can be successfully modeled by the proposed interface. Specifically, we are focusing on the conformal densely packed graphene strips and magnetically biased graphene patches with arbitrary shapes under magnetic bias. To this end, appropriate values should be assigned to the elements of the surface conductivity tensors. Under locally flat consideration of the conformal structures, we will use the closed-form approximate formulas based on effective medium theory (EMT) for the former [13], while for the latter, in general, we propose effective conductivity extraction based on numerical simulation in section 3. For the ease of analysis, sub-wavelength square patches with gap thickness of g and periodicity D are considered as the special case of arbitrarily shaped patches. Here, we are not repeating the available homogenized surface conductivity formulas of the literature for the brevity. Throughout the paper, the optical parameters of the graphene are considered to be controlled by relaxation time τ (ps), chemical potential μ_c (eV), and magnetic bias field B₀ (T). Densely packed graphene strips are characterized with the geometrical parameters W and L respectively representing the width and periodicity of the elements. Time-harmonic fields with $e^{i ω t}$ time dependency are considered throughout the paper and the rotation axis of the cylinder is assumed along the z-direction.

Fig. 1 (a) 2D view of a monolayer cylinder covered with a sheet with full-tensor surface conductivity ${\bar{\bar{σ}}}_{ϕ z}$ , including densely packed graphene strips and cylindrically wrapped sub-wavelength periodic square patches under magnetic biases and (b) its 3D view. As shown in the figure, in each case, proper values should be assigned to the elements of ${\bar{\bar{σ}}}_{ϕ z}$ . The rotation axis of the cylinder is set along the z-axis.

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Extracting DGF for this configuration based on the scattering superposition method is the aim of this section. Without loss of any generality, the localized point source is considered in region 1 and the outer and inner regions are denominated as region 1 and region 2, respectively. DGF for the structure with a source lying in region 2, can be deducted by the symmetry properties of DGF [26]. In scattering superposition technique, Green’s function is considered to result from a source radiating in an infinite homogeneous medium and the same source radiating in the presence of the obstacle. These two contributions are denominated as the free-space and scattering Green’s functions, respectively.

To formulate the problem, Green’s functions in each region of the solution domain should be expanded in terms of cylindrical vector wave functions $\bar{M}$ and $\bar{N}$ . They are defined based on the scalar wave function and represent the electric fields of TE (E_z = 0) and TM (H_z = 0) waves, respectively. It should be recalled that different references define TE and TM waves in different manners. We have translated all the definitions throughout the paper to the one that we have introduced here.

The scalar wave function in the cylindrical coordinate with the components $(r, ϕ, z)$ is defined as: $ψ_{μ} = Z_{n} (μ r) e^{i n ϕ} e^{i h z}$ , where Z_n is the Bessel function of the first kind in region 2 and the Hankel function of the first kind in region 1, both with the orders of n. Moreover, parameters $μ$ , n, and h are wave numbers in radial, azimuthal and longitudinal directions. They are depended via the wave number k_j of the region (j = 1,2) as $μ^{2} + h^{2} = k_{j}^{2}$ . Using the above $ψ_{μ}$ , vector wave functions can be readily obtained as:

{\bar{M}}_{μ} (h) = e^{i n ϕ} e^{i h z} [i n \frac{Z_{n} (μ r)}{r} \hat{r} - \frac{\partial Z_{n} (μ r)}{\partial r} \hat{ϕ}]

{\bar{N}}_{μ} (h) = \frac{1}{k_{j}} e^{i n ϕ} e^{i h z} [i h \frac{\partial Z_{n} (μ r)}{\partial r} \hat{r} - \frac{h n}{r} Z_{n} (μ r) \hat{ϕ} - μ^{2} Z_{n} (μ r) \hat{z}] .

As mentioned previously, the next step of the formulation is expanding the components of Green’s function in terms of the vector wave functions. Based on the Ohm–Rayleigh method, free-space Green’s function ${\bar{\bar{G}}}_{e 0}$ at the field point of $\bar{R}$ , resulting from a source lying outside the cylinder at the location of ${\bar{R}}^{'}$ , can be obtained as [26]:

\begin{array}{l} {\bar{\bar{G}}}_{e 0} (\bar{R}, {\bar{R}}^{'}) = - \frac{1}{k^{2}} \hat{r} \hat{r} δ (\bar{R} - \bar{R^{'}}) \\ + \frac{i}{8 π} \int_{- \infty}^{\infty} d h \sum_{n} \begin{array}{l}  \end{array} \frac{1}{η^{2}} \times [{\bar{M}}_{η}^{} (h) {\bar{M}}^{'}_{η}^{(1)} (- h) + {\bar{N}}_{η}^{} (h) {\bar{N}}^{'}_{η}^{(1)} (- h)] r < r^{'}, \end{array}

where superscript (1) denotes that Hankel functions of the first kind are used in the expansion of the vector wave functions. Moreover, the prime represents the wave functions corresponding to the coordinate of the source defined as

(r^{'}, ϕ^{'}, z^{'})

.

δ

is the Kronecker delta function and

δ_{0}

is the Kronecker delta function.To expand the scattering Green’s function, one should note that in general the electromagnetic waves are coupled in the cylindrical structures. They may decouple for some special cases that result in the diagonal surface conductivity tensors, e.g, normally incident plane waves. In our specific problem, due to the full tensor anisotropic nature of the cover, TE and TM waves are always coupled. Therefore, scattering Green’s functions in the environment medium (region 1) resulting from dipole residing in the same region can be written as:

\begin{array}{l} {\bar{\bar{G}}}_{e s}^{(11)} (\bar{R}, {\bar{R}}^{'}) = \frac{i}{8 π} \int_{- \infty}^{\infty} d h \sum_{n} \frac{1}{η^{2}} \times \\ {[A_{η} {\bar{M}}_{η}^{(1)} (h) + B_{η} {\bar{N}}_{η}^{(1)} (h)] {\bar{M}}^{'}_{η}^{(1)} (- h) + [C_{η} {\bar{N}}_{η}^{(1)} (h) + D_{η} {\bar{M}}_{η}^{(1)} (h)] {\bar{N}}^{'}_{η}^{(1)} (- h)}, \end{array}

where

A_{η}

,

B_{η}

,

C_{η}

, and

D_{η}

are the unknown coefficients of the DGF expansion. In the formulas, the superscript in

{\bar{\bar{G}}}_{e s}^{(f s)}

is used to indicate the position of the field (f) and source (s) points. Similarly, the scattering Green’s function in the core medium (region 2) resulting from the assumed dipole in region 1 can be expanded as:

\begin{array}{l} {\bar{\bar{G}}}_{e s}^{(21)} (\bar{R}, {\bar{R}}^{'}) = \frac{i}{8 π} \int_{- \infty}^{\infty} d h \sum_{n} \frac{1}{η^{2}} \times \\ {[a_{ξ} {\bar{M}}_{ξ} (h) + b_{ξ} {\bar{N}}_{ξ} (h)] {\bar{M}}^{'}_{η}^{(1)} (- h) + [c_{ξ} {\bar{N}}_{ξ} (h) + d_{ξ} {\bar{M}}_{ξ} (h)] {\bar{N}}^{'}_{η}^{(1)} (- h)}, \end{array}

where

a_{ξ}

,

b_{ξ}

,

c_{ξ}

, and

d_{ξ}

are expansion coefficients in the region 2. The unknowns will be found by applying appropriate boundary conditions. For the continuity of tangential electric Green's function one may write [26]:

\hat{r} \times [{\bar{\bar{G}}}_{e}^{(11)} (\bar{R}, {\bar{R}}^{'}) - {\bar{\bar{G}}}_{e}^{(21)} (\bar{R}, {\bar{R}}^{'})] = 0.

Also, due to the presence of anisotropic surface currents at the interface, discontinuity of tangential magnetic Green’s function requires that:

\hat{r} \times (\frac{\nabla \times {\bar{\bar{G}}}_{e}^{(11)}}{μ_{1}} - \frac{\nabla \times {\bar{\bar{G}}}_{e}^{(21)}}{μ_{2}}) = i ω {\bar{\bar{σ}}}_{ϕ z} . {\bar{\bar{G}}}_{e}^{(21)},

where full-tensor surface conductivity of the interface was introduced previously. By imposing the boundary conditions in Eqs. (7) and (8) and using the orthogonality of the vector wave harmonics, linear systems of equations for determining the unknowns can be obtained as:

[\begin{matrix} - \frac{\partial H_{n}^{(1)} (η a)}{\partial a} & - \frac{1}{k_{1}} \frac{h n}{a} H_{n}^{(1)} (η a) & \frac{\partial J_{n} (ξ a)}{\partial a} & \frac{1}{k_{2}} \frac{h n}{a} J_{n} (ξ a) \\ 0 & - \frac{1}{k_{1}} η^{2} H_{n}^{(1)} (η a) & 0 & \frac{1}{k_{2}} ξ^{2} J_{n} (ξ a) \\ - \frac{1}{μ_{1}} \frac{h n}{a} H_{n}^{(1)} (η a) & - \frac{k_{1}}{μ_{1}} \frac{\partial H_{n}^{(1)} (η a)}{\partial a} & \frac{1}{μ_{2}} \frac{h n}{a} J_{n} (ξ a) + σ_{1} & \frac{k_{2}}{μ_{2}} \frac{\partial J_{n} (ξ a)}{\partial a} + σ_{2} \\ \frac{1}{μ_{1}} η^{2} H_{n}^{(1)} (η a) & 0 & - \frac{1}{μ_{2}} ξ^{2} J_{n} (ξ a) + σ_{3} & σ_{4} \end{matrix}] \times [\begin{matrix} A_{η} \\ B_{η} \\ a_{ξ} \\ b_{ξ} \end{matrix}] = [\begin{matrix} \frac{\partial J_{n} (η a)}{\partial a} \\ 0 \\ \frac{1}{μ_{1}} \frac{h n}{a} J_{n} (η a) \\ - \frac{1}{μ_{1}} η^{2} J_{n} (η a) \end{matrix}]

[\begin{matrix} - \frac{1}{k_{1}} \frac{h n}{a} H_{n}^{(1)} (η a) & - \frac{\partial H_{n}^{(1)} (η a)}{\partial a} & \frac{1}{k_{2}} \frac{h n}{a} J_{n} (ξ a) & \frac{\partial J_{n} (ξ a)}{\partial a} \\ - \frac{1}{k_{1}} η^{2} H_{n}^{(1)} (η a) & 0 & \frac{1}{k_{2}} ξ^{2} J_{n} (ξ a) & 0 \\ - \frac{k_{1}}{μ_{1}} \frac{\partial H_{n}^{(1)} (η a)}{\partial a} & - \frac{1}{μ_{1}} \frac{h n}{a} H_{n}^{(1)} (η a) & \frac{k_{2}}{μ_{2}} \frac{\partial J_{n} (ξ a)}{\partial a} + {σ^{'}}_{1} & \frac{1}{μ_{2}} \frac{h n}{a} J_{n} (ξ a) + {σ^{'}}_{2} \\ 0 & \frac{1}{μ_{1}} η^{2} H_{n}^{(1)} (η a) & {σ^{'}}_{3} & - \frac{1}{μ_{2}} ξ^{2} J_{n} (ξ a) + {σ^{'}}_{4} \end{matrix}] \times [\begin{matrix} C_{η} \\ D_{η} \\ c_{ξ} \\ d_{ξ} \end{matrix}] = [\begin{matrix} \frac{1}{k_{1}} \frac{h n}{a} J_{n} (η a) \\ \frac{1}{k_{1}} η^{2} J_{n} (η a) \\ \frac{k_{1}}{μ_{1}} \frac{\partial J_{n} (η a)}{\partial a} \\ 0 \end{matrix}],

where:

σ_{1} = i ω σ_{z ϕ} \frac{\partial J_{n} (ξ a)}{\partial a}, σ_{2} = i ω \frac{σ_{z z}}{k_{2}} ξ^{2} J_{n} (ξ a) + i ω σ_{z ϕ} \frac{h n}{k_{2} a} J_{n} (ξ a)

σ_{3} = i ω σ_{ϕ ϕ} \frac{\partial J_{n} (ξ a)}{\partial a}, σ_{4} = i ω σ_{ϕ ϕ} \frac{1}{k_{2}} \frac{h n}{a} J_{n} (ξ a) + i ω σ_{ϕ z} \frac{1}{k_{2}} ξ^{2} J_{n} (ξ a)

{σ^{'}}_{1} = i ω σ_{z z} \frac{1}{k_{2}} ξ^{2} J_{n} (ξ a) + i ω σ_{z ϕ} \frac{h n}{k_{2} a} J_{n} (ξ a), {σ^{'}}_{2} = i ω σ_{z ϕ} \frac{\partial J_{n} (ξ a)}{\partial a}

{σ^{'}}_{3} = i ω σ_{ϕ ϕ} \frac{1}{k_{2}} \frac{h n}{a} J_{n} (ξ a) + i ω σ_{ϕ z} \frac{1}{k_{2}} ξ^{2} J_{n} (ξ a), {σ^{'}}_{4} = i ω σ_{ϕ ϕ} \frac{\partial J_{n} (ξ a)}{a} .

Apparently, the general forms of the equations are the same as a single layer dielectric cylinder except for some corrections proportional to the elements of the surface conductivity tensor [26]. Moreover, it is interesting to note that trying to simulate the cylindrically wrapped graphene with tensor surface conductivity using commercial software packages various problems emerge. Some of them include 1) due to the infinite extent of the structure, simulation using 3D simulators (e.g., CST, HFSS) requires 3D simulation with periodic boundary condition in the infinite direction. This leads to huge memory and time requirement 2) Magnetically biased graphene is not present in most of the simulators, therefore the related formulas/data should be inserted manually 3) Full tensor nature of the conductivity along with curvature of the structure make the simulation more difficult for applying the boundary conditions and meshing the structure, respectively. Using our formulation, analysis of these structures can be conducted easily.

3. Results and discussion

To verify the validity of the extracted formulas, two cases are considered: 1) calculating the propagation constant of graphene covered cylindrical waveguides and 2) discussing the plane wave scattering by graphene-based cylindrical wires. To this end, guided modes are treated as the poles of the Sommerfeld integrals appeared in Eqs. (4)-(6). Moreover, the radiated modes are attained by approximating the aforementioned integrals using the saddle point method. Also, the electrically biased shells are considered by setting the diagonal terms of the surface conductivity tensor equal to the graphene isotropic surface conductivity and the off-diagonal terms equal to zero.

3.1 Guided mode applications

To obtain the propagation and attenuation constants of the localized surface plasmons for graphene-coated cylindrical waveguides (GCWs), the characteristic equation should be solved. To this end, the pole singularities of the expansion coefficients should be obtained by setting the determinant of the coefficient matrices to zero. In the following, the characteristic equations for the dominant n = 0 modes of the electrically and magnetically biased graphene-based waveguides will be obtained. It is worth noting that we have provided this section merely for proving the consistency of our formulas with those obtained by the other methods. For the electrically biased graphene cover, we observed that the determinants of both systems of equations Eqs. (9)-(10) are identical and they can be written as the multiplication of two decoupled terms, one for TE and the other for TM modes. The fully-retarded characteristic equation of TE and TM modes respectively are:

\frac{ω ε_{2}}{ξ} \frac{J_{1} (ξ a)}{J_{0} (ξ a)} - \frac{ω ε_{1}}{η} \frac{H_{1}^{1} (η a)}{H_{0}^{(1)} (η a)} + i σ_{d} = 0

\frac{ξ}{ω μ_{2}} \frac{J_{0} (ξ a)}{J_{1} (ξ a)} - \frac{η}{ω μ_{1}} \frac{H_{0}^{(1)} (η a)}{H_{1}^{1} (η a)} + i σ_{d} = 0,

where the derivatives of the special functions are simplified using

\partial Z_{0} (η a) / \partial a = - η Z_{1} (η a)

and

\partial Z_{0} (ξ a) / \partial a = - ξ Z_{1} (ξ a)

. The former equation is the same as the one obtained in [6]. For the magnetically biased graphene sheet, the elements of the surface conductivity tensor can be considered as:

σ_{ϕ ϕ} = σ_{z z} = σ_{d}, σ_{ϕ z} = - σ_{z ϕ} = σ_{0}

, where σ_d and σ₀ can be represented in Drude form for most practical applications [12]. Nullifying the determinant of the coefficient matrices results:

\begin{array}{l} [\frac{ω ε_{2}}{ξ} \frac{H_{1} (ξ a)}{H_{0} (ξ a)} - \frac{ω ε_{1}}{η} \frac{J_{1} (η a)}{J_{0} (η a)} - j σ_{d}] \\ \times [\frac{η}{ω μ_{1}} \frac{J_{0} (η a)}{J_{1} (η a)} - \frac{ξ}{ω μ_{2}} \frac{H_{0} (ξ a)}{H_{1} (ξ a)} - j σ_{d}] = - σ_{0}^{2} . \end{array}

The above equation corresponds to the hybrid TE and TM modes and it is the same as that of [6]. It is apparent that higher-order guided modes can be treated readily by assigning the desired mode number to the variable n. The resulted equations are complex functions with complex roots and numerical solutions of them with purely numerical algorithms are complicated in general [13]. An alternative is to connect the positions of maxima and widths of resonances in scattering cross section spectra to the complex frequencies of the cylindrical cavities [27]. The discussions of the subsequent section will be fruit-full for this purpose.

Importantly, the effect of spatial dispersions due to residing graphene on top of high-dielectric-constant materials had been considered for the planar structures, previously. A full-tensor surface conductivity with the elements proportional to the propagation constant of plasmons is essential for investigating this effect [17,28]. By our fully-populated tensor boundary conditions, the impact of spatial dispersions on the localized surface plasmons of the wire can be analyzed, readily. Moreover, translating the high dielectric constant core with a semi-infinite half-space with the same material based on the Bohr model [29] suggests that the spatial dispersion effects in the practical applications of the cylindrical geometries should be considered. Eventually, the plane wave scattering from graphene-based conformal structures under the tensor surface conductivity assumption will be considered in the next section.

3.2 Calculating scattering properties

In this section, plane wave scattering from graphene-based conformal structures will be obtained. For the derivation of the Green’s function in section 2, we have considered a localized point source outside of the cylinder which resembles a plane wave at the far distance. In this region, corresponding Sommerfeld integrals can be approximated by employing the saddle point technique. It can be readily shown that under this condition: $h = k_{1} \cos θ$ , where $θ$ represents the incident angle of the plane wave [26]. Therefore, the total scattering cross section efficiency (TSCS) can be calculated as [30]

T S C S = \frac{1}{k_{1} a} \sum_{n = - \infty}^{\infty} [{| A_{η} |}^{2} + {| B_{η} |}^{2}]

for TE waves. The scattering efficiency of the TM polarized waves can be calculated from the above formula by replacing the coefficients

A_{η}

and

B_{η}

with

C_{η}

and

D_{η}

, respectively. The latter coefficient of each polarization denotes the effect of cross coupling of TE and TM waves. In the above formula, the normalization factor is the total scattering cross section of an infinite PEC cylinder which equals to 4a [7].

3.2.1 Electrically biased continuous graphene shell

Figure 2 shows TSCS of TE and TM waves for an infinite length cylinder with a = 50 μm, ε₂ = 2.4, τ = 1ps, and μ_c = 0.25 eV. The results are compared with those of the frequency domain solver of CST commercial software. It is observed that both results are in good agreement with each other. The above-mentioned parameters are used from [7], in which TE wave scattering is investigated and its refractive index sensing capability is discussed. For this polarization, the localized surface plasmons (LSP) and whispering-gallery (WG) modes are excited in the graphene coating and dielectric cylinder, respectively. Moreover, under TM illumination the localized surface plasmons of the wire are not excited since the electric field in the graphene cover induces currents along the infinite cylinder axis [27]. But, it was shown that at the higher frequencies up to visible light through telecommunication frequencies and under the condition that the core radius is larger than a critical value, TM surface plasmons can be excited [31]. This clarifies the importance of analyzing scattering by a TM polarized plane wave which is commonly neglected. Importantly, thanks to the dyadic analysis, TE and TM waves scattering are treated by a unified formulation. On the other hand, the duality principle is not readily applicable to this problem because of the graphene surface currents. This means that without dyadic formulation these two cases should be treated separately.

Fig. 2 TSCS of an electrically biased graphene-coated infinite cylinder with a = 50 μm, ε₂ = 2.4, τ = 1ps, and μ_c = 0.25 eV for (a) TE and (b) TM waves.

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In principle, TE and TM modes of the cylindrical structures are coupled for the oblique incident plane waves. Studying the effect of incident angle on the scattering performance of a plasmonic coating around a dielectric cylinder is important for the design of photonic devices and it was previously investigated for a graphene cover under electric bias [32]. Figure 3 shows the effect of coupling coefficient $B_{η}$ on TSCS of the structures in Fig. 2(a) for various incident angles. The effect of mode coupling on the scattering cross section gets larger by increasing the deviation of the incident angle from the normal to cylinder axis (θ = 90°). Also, it is confirmed that for the normally incident plane wave, the coupling coefficient equals to zero. Using our formulas, the examination can be extended to the graphene shells with anisotropic surface conductivities.

Fig. 3 The effect of coupling coefficient $B_{η}$ on TSCS of the structure of Fig. 2(a) for various incident angles under TE illumination.

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3.2.2 Densely packed cylindrically conformal graphene strips

So far, we have provided some examples in order to show the validity of the extracted coefficients appeared in DGF expansion. In order to investigate the applications of the cover with full-tensor surface conductivity, we are focusing on two special cases in the following sub-sections. Initially, let us consider magnetically biased densely packed graphene strips with the periodicity L = 420 nm, width W = 400 nm, chemical potential μ_c = 0.5 eV, and relaxation time τ = 1ps (Fig. 1(a)). The strips are wrapped around a hollow cylinder with the radius of a = 50 μm. The planar counterpart of the proposed structure with the similar geometrical parameters behaves as a two-dimensional hyperbolic metamaterial [13]. Here, we are using the reconfigurable hyperbolic nature of the graphene strips by wrapping them around an electrically large cylindrical core. This approach can be considered as another way of achieving cylindrical hyperbolic metamaterials obtained by alternative metal-dielectric shells at the sub-wavelength-scale [33]. Moreover, Recently, it has been shown that highly confined surface waves present in the hyperbolic meta-surfaces, wrapped around the tubes, enhance light interactions with localized emitters or molecules [34].

Figure 4 shows TSCS of the proposed structure for various superconductive magnets with the magnetic biases B₀ ranging from 20 to 40 T. By increasing the external magnetic bias field, the position of the scattering peaks experiences a blue-shift. This feature is not achievable with metal-dielectric structures. The range of variation of the electrostatic, magnetic field is chosen based on [5]. Moreover, reflectance, transmittance, and absorbance analysis of a free-standing graphene grating were studied prior to this research and it reveals the excitation of the surface plasmon resonances for TE states and Rayleigh anomalies for TM states [35] and the research can be continued by translating these effects to the cylindrical geometries.

Fig. 4 TSCS of a densely packed cylindrically wrapped graphene strips with L = 420 nm, W = 400 nm, μ_c = 0.5 eV, and τ = 1ps for different magnetic biases.

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3.2.3 Sub-wavelength cylindrically conformal arbitrary shaped graphene patches

As another instance, TSCS of a hollow magnetically biased cylindrical structure with periodically arranged square patches is studied. The geometrical and optical parameters are D = 0.5 μm, g = 0.5 μm, μ_c = 0.5 eV, and τ = 1 ps. The structure is shown in Fig. 1(a). Figure 5 illustrates its performance of the structure for B₀ = 0 T and B₀ = 10 T. By changing the magnetic bias field, it is possible to switch between low and high scattering regimes at the same frequency. Similar switching capability between cloaking and scattering regimes are proposed using phase change material Ge₂Sb₂Te₅ (GST) [36]. Moreover, as depicted in Fig. 4, essentially by varying the electrostatic bias voltage, the target working frequency can be tuned. This idea can be possibly used for developing novel reconfigurable optical devices. Free-standing micro-patches with the aforementioned parameters behave as a frequency selective surface and we have used the corresponding closed-form approximate formulas to model the cylindrical graphene patterns [37].

Fig. 5 TSCS of a hollow magnetically biased cylindrical structure with periodically arranged square patches with D = 0.5 μm, g = 0.5 μm, B₀ = 1 T, μ_c = 0.5 eV, and τ = 0.5 ps wrapped around it.

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The above analysis can be generalized to the arbitrary shaped graphene-based unit cells by full-wave electromagnetic simulation. Considering a planar unit cell surrounded by two media with the refractive indices of n₁ and n₂, one may write [21]:

σ_{x, y} = \frac{n_{1} - n_{2} - S_{11}^{x, y} (n_{1} + n_{2})}{1 + S_{11}^{x, y}},

where

S_{11}^{x, y}

are the components of the S-matrix and their superscripts indicate the polarization of the incident electric field. The arbitrary shape of the elements provides various degrees of freedom for the manipulation of their light-matter interaction, including the possibility of tailoring the zero crossing point of the surface conductivity [15]. Finite element method under periodic boundary condition and Floquet excitation can be used for the analysis. Finally, the research can be continued by considering arbitrary sources in the presence of the proposed structure through convolution integral [26]. For instance, emission and radiation properties of a line dipole in the proximity of an isotropic graphene-coated wire are discussed in terms of spontaneous emission rate and the radiation efficiency [38] and the idea can be further continued for more general models of the graphene surface conductivity by means of our formulas.

4. Conclusion

Dyadic Green’s function for a monolayer cylinder with tensor surface conductivity boundary condition is formulated by the scattering superposition method. The obtained linear systems of equations for determining the expansion coefficients are similar to those of the dielectric cylinders except for some corrections proportional to diagonal and off-diagonal terms of graphene anisotropic surface conductivity. By setting the off-diagonal terms of the surface conductivity equal to zero, Green’s function formulas of an electrically biased graphene-based structure can be obtained, readily. The proposed formulas can model the anisotropic graphene patterns on curved surfaces, not easily attainable with commercial software packages.

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Dyadic analysis of a cylindrical wire consisting of a cover with fully-populated surface conductivity tensor

Abstract

1. Introduction

2. Dyadic Green’s function formulation

3. Results and discussion

3.1 Guided mode applications

3.2 Calculating scattering properties

3.2.1 Electrically biased continuous graphene shell

3.2.2 Densely packed cylindrically conformal graphene strips

3.2.3 Sub-wavelength cylindrically conformal arbitrary shaped graphene patches

4. Conclusion

References

Cited By

Figures (5)

Equations (19)

Optics Express