Adopting image theorem for rigorous analysis of a perfect electric conductor&#x2013;backed array of graphene ribbons

Mahdi Rahmanzadeh; Ali Abdolali; Amin Khavasi; Hamid Rajabalipanah

doi:10.1364/JOSAB.35.001836

Journal of the Optical Society of America B
Vol. 35,
Issue 8,
pp. 1836-1844
(2018)
•https://doi.org/10.1364/JOSAB.35.001836

Adopting image theorem for rigorous analysis of a perfect electric conductor–backed array of graphene ribbons

Mahdi Rahmanzadeh, Ali Abdolali, Amin Khavasi, and Hamid Rajabalipanah

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Mahdi Rahmanzadeh, Ali Abdolali, Amin Khavasi, and Hamid Rajabalipanah, "Adopting image theorem for rigorous analysis of a perfect electric conductor–backed array of graphene ribbons," J. Opt. Soc. Am. B 35, 1836-1844 (2018)

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Abstract

Analytical and numerical study of graphene ribbons has become a prime focus of recent research due to their potential applications in tunable absorption, wavefront manipulation, polarization conversion, and so on. In this paper, an accurate analysis of a perfect electric conductor (PEC)–backed array of graphene ribbons (PAGR) is presented based on the well-known electromagnetic (EM) image theorem, where the induced currents are theoretically derived under a transverse-magnetic-polarized incident wave. For the first time, the proposed analysis rigorously incorporates the EM coupling effects between the PEC back plate and the subwavelength array of graphene ribbons. It is proved that the strong interaction between the PEC back plate and graphene ribbons drastically affects the results, especially in ultra-thin PAGR structures, whereas it was neglected in the previous works. As a proof of principle, an ultra-thin graphene-assisted absorber ( $= 0.05 λ_{0}$ ) exhibiting tunable absorption at the terahertz regime is theoretically designed to verify the proposed analytical scheme. Unlike the previous studies, this paper reveals a more general, valid, and reliable analysis of PEC-backed graphene ribbons and can be simply extended to 2D geometries of PEC-backed graphene metasurfaces.

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Fig. 1. Periodic array of graphene ribbons with the width of

w

and periodicity of

D

placed (a) between two half-spaces and (b) at a distance of

h

from a PEC back plate (PAGR structure).

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Fig. 2. Adopting the image theorem for calculation of the surface currents on graphene ribbons in the configuration of Fig. 1.

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Fig. 3. Proposed equivalent circuit model for the PAGR structure of Fig. 1(b); the periodic array of graphene ribbons is modeled as parallel R-L-C branches, each representing one of the resonance modes.

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Fig. 4. (a) Reflectivity and (b) reflection phase of a periodic array of graphene ribbons with a width of

w = 35 μm

and periodicity of

D = 70 μm

at the distance of

h = 3 μm

above the PEC back plate; the graphene parameters are assumed as

τ = 1 ps

and

E_{f} = 1 eV

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Fig. 5. (a) Reflectivity and (b) reflection phase of a periodic array of graphene ribbons with a width of

w = 9 μm

and periodicity of

D = 10 μm

at the distance of

h = 1 μm

above the PEC back plate; the graphene parameters are assumed as

τ = 0.8 ps

and

E_{f} = 0.7 eV

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Fig. 6. (a) Schematic view of the PAGR configuration. (b) The computed normalized eigencurrents and (c) the simulated electric field distribution of the graphene ribbons in the PAGR configuration at the vicinity of the first three resonances.

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Fig. 7. (a) Optimum conductance value G of graphene ribbons for satisfaction of the resonance condition of Eq. (16a) in terms of

w / D

and

E_{f}

; (b) the optimum capacitance value C of graphene ribbons for satisfaction of the resonance condition of Eq. (16b) in terms of periodicity.

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Fig. 8. (a) Absorption spectra of a periodic array of graphene ribbons with a width of

w = 21.3 μm

and periodicity of

D = 41 μm

at the distance of

h = 5 μm

above the graphene parameters are assumed as

τ = 0.6 ps

and

E_{f} = 0.85 eV

; (b) the absorption spectra of the same PAGR structure for different Fermi energy levels.

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Fig. 9. Resonance frequency of absorption for different (a) Fermi energy levels, (b)

h / D

ratios, and (c) filling factors (

w / D

) of graphene ribbons when a periodic array of graphene ribbons with a width of

w = 21.3 μm

and periodicity of

D = 41 μm

is located at the distance of

h = 5 μm

above the PEC plate.

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Tables (4)

Table 1. First Three Eigenfunctions and Eigenvalues for the Problem of a Single Graphene Ribbon

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Table 2. First Three Eigenvalues for the Problem of a PAGR Structure ( $H / D = 0.01$ )

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Table 3. First Three Eigenvalues for the Problem of a PAGR Structure ( $H / D = 0.05$ )

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Table 4. First Three Eigenvalues for the Problem of a PAGR Structure ( $H / D = 0.1$ )

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Equations (21)

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σ_{G} (ω) = σ_{G}^{intra} (ω) + σ_{G}^{inter} (ω),

σ_{G}^{intra} (ω) = \frac{2 k_{B} e^{2} T}{π ℏ^{2}} \ln [2 \cosh (\frac{E_{f}}{2 k_{B} T})] \frac{i}{ω + i τ^{- 1}},

σ_{G}^{inter} (ω) = \frac{e^{2}}{4 ℏ} [H (\frac{ω}{2}) + \frac{4 i ω}{π} \int_{0}^{\infty} \frac{H (ζ) - H (ω / 2)}{ω^{2} - 4 ξ^{2}} d ξ],

H (ζ) = \sinh (\frac{ℏ ζ}{k_{B} T}) / [\cosh (\frac{E_{f}}{k_{B} T}) + \cosh (\frac{ℏ ζ}{k_{B} T})] .

τ = E_{f} μ / v_{f}^{2} e,

J_{x, i} (x) / σ_{G} = E_{x, i}^{ext} (x) + \frac{1}{j ω ϵ_{0}} \frac{d}{d x} \int_{d_{i}}^{d_{i} + w} G_{0} (x - x^{'}) \frac{d J_{x, i} (x^{'})}{d x^{'}} d x^{'} + \frac{1}{j ω ϵ_{0}} \frac{d}{d x} \sum_{l \neq i} \int_{d_{l}}^{d_{l} + w} G_{0} (x - x^{'}) \frac{d J_{x, l} (x^{'})}{d x^{'}} d x^{'} .

G_{0} (x - x^{'}) \approx - \frac{1}{2 π} \ln (k_{0} | x - x^{'} |),

J_{x} (x) / σ_{G} = E_{x, i}^{ext} (x) - \frac{1}{2 π j ω ϵ_{0}} \frac{d}{d x} \int_{- w / 2}^{w / 2} \ln (k_{0} | x - x^{'} |) \frac{d J_{x} (x^{'})}{d x^{'}} d x^{'} - \frac{1}{2 π j ω ϵ_{0}} \frac{d}{d x} \sum_{l \neq 0} \int_{- w / 2}^{w / 2} \ln (k_{0} | x - x^{'} + l D |) \frac{d J_{x} (x^{'})}{d x^{'}} d x^{'} + \frac{1}{2 π j ω ϵ_{0}} \frac{d}{d x} \sum_{l = - \infty}^{\infty} \int_{- w / 2}^{w / 2} \ln (k_{0} | \sqrt{{(x - x^{'} + l D)}^{2} + {(2 h)}^{2}} |) \frac{d J_{x} (x^{'})}{d x^{'}} d x^{'} .

- k_{p} J_{x} (x) + \frac{1}{π} P \int_{- w / 2}^{w / 2} \frac{1}{x - x^{'}} \frac{\partial J_{x} (x^{'})}{\partial x^{'}} d x^{'} + \frac{1}{π} \sum_{l \neq 0} \int_{- w / 2}^{w / 2} \frac{1}{x - x^{'} + l D} \frac{\partial J_{x} (x^{'})}{\partial x^{'}} d x^{'} - \frac{1}{π} \sum_{l = - \infty}^{\infty} \int_{- w / 2}^{w / 2} \frac{x - x^{'} + l D}{{(x - x^{'} + l D)}^{2} + {(2 h)}^{2}} \frac{\partial J_{x} (x^{'})}{\partial x^{'}} d x^{'} = 2 j ϵ_{0} E_{x, i}^{ext} (x),

\frac{1}{π} P \int_{- w / 2}^{w / 2} \frac{1}{x - x^{'}} \frac{\partial ψ_{m} (x^{'})}{\partial x^{'}} d x^{'} + \frac{1}{π} \sum_{l \neq 0} \int_{- w_{0} / 2}^{w_{0} / 2} \frac{1}{x - x^{'} + l D} \frac{\partial ψ_{m} (x^{'})}{\partial x^{'}} d x^{'} - \frac{1}{π} \sum_{l = - \infty}^{\infty} \int_{- w / 2}^{w / 2} \frac{x - x^{'} + l D}{{(x - x^{'} + l D)}^{2} + {(2 h)}^{2}} \frac{\partial ψ_{m} (x^{'})}{\partial x^{'}} d x^{'} = q_{m} ψ_{m} (x) .

J_{x, i} (x) = \sum_{m = 1}^{\infty} A_{m} ψ_{m} (x),

A_{m} = \frac{2 j ω ϵ_{0}}{q_{m} - k_{p} (ω)} \int_{- w / 2}^{w / 2} E_{x}^{ext} (x) ψ_{m} (x) d x .

\frac{1}{π} P \int_{- w / 2}^{w / 2} \frac{1}{x - x^{'}} \frac{\partial ψ_{m} (x^{'})}{\partial x^{'}} d x^{'} = k_{m} ψ_{m} (x),

{\begin{cases} \int_{- w / 2}^{w / 2} ψ_{m} (x) ψ_{n} (x) d x = δ_{m n} \\ ψ_{m} (0) = ψ_{m} (w) = 0 \end{cases},

q_{m} = k_{m} - \frac{1}{π} \sum_{- \infty (l \neq 0)}^{\infty} \int_{- w / 2}^{w / 2} \int_{- w / 2}^{w / 2} \ln | x - x^{'} + l D | \frac{d ψ (x_{m})}{d x} \frac{d ψ (x_{m}^{'})}{d x^{'}} d x d x^{'} + \frac{1}{π} \sum_{- \infty}^{\infty} \int_{- w / 2}^{w / 2} \int_{- w / 2}^{w / 2} \ln | \sqrt{{(x - x^{'} + l D)}^{2} + {(2 h)}^{2}} | \frac{d ψ (x_{m})}{d x} \frac{d ψ (x_{m}^{'})}{d x^{'}} d x d x^{'} .

R_{m} = \frac{D}{S_{m}^{2}} \frac{π ℏ^{2}}{e^{2} E_{f} τ}, L_{m} = \frac{D}{S_{m}^{2}} \frac{π ℏ^{2}}{e^{2} E_{f}}, C_{m} = \frac{S_{m}^{2}}{D} \frac{2 ϵ_{0}}{q_{m}},

[\begin{matrix} A & B \\ C & D \end{matrix}] = \prod_{m} (\begin{matrix} 1 & 0 \\ Y_{g_{m}} & 1 \end{matrix}) (\begin{matrix} \cosh γ_{g} h & \sinh γ_{g} h / Y_{c} \\ Y_{c} \sinh γ_{g} h & \cosh γ_{g} h \end{matrix}),

R = (B - η_{0} D) / (B + η_{0} D) .

Y_{i n}^{tot} = Y_{i n}^{line} + Y_{g} = j B + \frac{1}{R + j X} = \frac{R}{R^{2} + X^{2}} + j (B - \frac{X}{R^{2} + X^{2}}),

R_{opt} = \frac{η_{0}}{1 + {(B η_{0})}^{2}},

X_{opt} = ω L - \frac{1}{ω C} = \frac{η_{0}^{2} B}{1 + {(B η_{0})}^{2}} .

Eigenvalue	Eigenfunction
$0.737 π / w$	$ψ_{1} = \sqrt{w} [1.2 \sin (\arccos 2 x / w) - 1.06 \sin (3 \arccos 2 x / w)]$
$1.753 π / w$	$ψ_{2} = \sqrt{w} [1.254 \sin (2 \arccos 2 x / w) - 0.302 \sin (4 \arccos 2 x / w)]$
$2.748 π / w$	$ψ_{2} = \sqrt{w} [0.308 \sin (\arccos (2 x / w)) + 1.19 \sin (3 \arccos (2 x / w)) - 0.484 \sin (5 \arccos (2 x / w))]$

$w / D$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$q_{1} π / w$	0.365	0.232	0.173	0.139	0.116	0.1	0.088	0.079	0.071
$q_{2} π / w$	1.22	0.826	0.624	0.502	0.421	0.363	0.319	0.286	0.26
$q_{3} π / w$	2.277	1.661	1.289	1.051	0.887	0.768	0.676	0.604	0.545

$w / D$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$q_{1} π / w$	0.668	0.562	0.476	0.412	0.363	0.323	0.290	0.261	0.228
$q_{2} π / w$	1.732	1.625	1.481	1.342	1.221	1.117	1.031	0.961	0.918
$q_{3} π / w$	2.744	2.695	2.58	2.431	2.276	2.128	1.991	1.861	1.723

$w / D$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$q_{1} π / w$	0.716	0.667	0.611	0.558	0.509	0.465	0.423	0.379	0.323
$q_{2} π / w$	1.751	1.732	1.688	1.626	1.556	1.487	1.424	1.377	1.366
$q_{3} π / w$	2.747	2.743	2.729	2.695	2.643	2.576	2.498	2.404	2.271

Eigenvalue	Eigenfunction
$0.737 π / w$	$ψ_{1} = \sqrt{w} [1.2 \sin (\arccos 2 x / w) - 1.06 \sin (3 \arccos 2 x / w)]$
$1.753 π / w$	$ψ_{2} = \sqrt{w} [1.254 \sin (2 \arccos 2 x / w) - 0.302 \sin (4 \arccos 2 x / w)]$
$2.748 π / w$	$ψ_{2} = \sqrt{w} [0.308 \sin (\arccos (2 x / w)) + 1.19 \sin (3 \arccos (2 x / w)) - 0.484 \sin (5 \arccos (2 x / w))]$

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