Efficient manipulation of graphene absorption by a simple dielectric cylinder

Ting-Hui Xiao; Lin Gan; Zhi-Yuan Li

doi:10.1364/OE.23.018975

1. Introduction

In recent years, graphene, a flat monolayer of carbon atoms tightly packed into a two-dimensional honeycomb lattice, has aroused great interests for its superior electronic and optical properties [1]. It has been widely used in optoelectronics [2–5 ] and all-optical devices [6, 7 ]. Optical absorption of graphene, which is closely connected to the photon-carrier generation in optoelectronic devices and thermo-optic effect [8] in all-optical devices, plays a key role and evokes great concerns for designing these devices. The absorption properties of various graphene based structures such as graphene disks [9] and ribbons [10], graphene coated subwavelength dielectric gratings [11], gold ribbons [12] or photonic crystal [13], and graphene combined plasmonic nanoantenna [14] or Fabry-Perot cavity [15] have been studied. Researchers are dedicated to find simple and reliable ways to control and tune the absorption of graphene as they wish.

Construction of simple controllable geometric structure to tune optical property is a heuristic method [16]. Many researchers have focused on utilizing various nano/microstructures to manipulate the absorption and scattering properties of concerned materials. For examples, maneuvering the surface plasmon resonance of metal nanostructures including sphere, cube, rod, tetrahedron and octahedron through shape-controlled synthesis has been experimentally realized, which can tune the extinction, absorption and scattering spectra of these metal material [17–19 ]. It is expected that integrating graphene with a shape-controlled geometric nano/microstructure and making use of the geometric configuration of these hybrid structures can be a promising method to manipulate and modulate the graphene absorption.

In this work, we propose a simple while efficient approach to manipulate the absorption of graphene by coating graphene on the curved outer surface of a micrometer-size dielectric cylinder. The absorption properties of the hybrid graphene-cylinder are theoretically explored by using an analytical method. We engineer the absorption properties of graphene by taking into account several factors, including the geometric and physical configuration of the cylinder, the material property of graphene, and polarizations of the incident wave. The totally different absorption properties of two polarized waves (TE and TM) are found and the underlying physical reasons are analyzed. Following our analysis, we have found the optimized parameters of the model system that allow for realizing enhancement and tunability of graphene absorption around the telecommunication wavelength of 1.55 μm. Whispering gallery mode (WGM), which has been widely used to enhance the interaction between light and material [20, 21 ], is produced in our hybrid structure and utilized to assist the tunability of graphene absorption. Moreover, we demonstrate the continuous tunability of graphene absorption and its dependence on the cylinder radius, refractive index, and chemical potential. In addition, for further acquiring the physical insight, field distributions of our hybrid model have also been calculated and analyzed, from which the physical origin for the enhancement and tunability of graphene absorption in this hybrid microstructure are identified.

2. Theoretical model and formalisms

In this paper, we consider a two-dimensional (2D) hybrid system consisting of a thin-layer graphene coating the outer surface of a dielectric cylinder extending along the $z$ axis. Figure 1 illustrates the dielectric cylinder of radius $a$ and refractive index $m$ coated by a graphene layer. Here, the infinitesimally thin graphene layer at $r = a$ is characterized by a surface conductivity $σ$ determined by Kubo formula [22]. The isotropic surface conductivity $σ$ includes the intra-band and inter-band terms and highly depends on the working frequency and chemical potential. For the sake of simplicity, we suppose the host medium is air and the electromagnetic waves propagate in the $x y$ plane. Throughout this paper, we assume the cylinder is made from a homogeneous, isotropic and non-dispersive dielectric medium. The harmonic electric field $E$ and magnetic field $H$ with a time dependence $e^{- i ω t}$ satisfy the Maxwell equations as

\nabla \times \vec{H} = - i ω ε \vec{E},

\nabla \times \vec{E} = i ω μ \vec{H},

where

ω

,

ε

and

μ

are angular frequency, permittivity and permeability. With respect to the

z

direction, the electromagnetic fields are decomposed into transverse electric (TE) and transverse magnetic (TM) waves.

Fig. 1 Schematic of light incident on a graphene coated dielectric cylinder embedded in the air.

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In the case of TE wave, there are only three field components $E_{z}$ , $H_{r}$ and $H_{ϕ}$ in the cylindrical polar coordinate system. So Eq. (1) can be written as

\frac{1}{r} \frac{\partial}{\partial r} (r H_{ϕ}) - \frac{1}{r} \frac{\partial H_{r}}{\partial ϕ} = - i ω ε E_{z},

and Eq. (2) can be written as

\begin{array}{l} \frac{1}{r} \frac{\partial E_{z}}{\partial ϕ} = i ω μ H_{r}, \\ - \frac{\partial E_{z}}{\partial r} = i ω μ H_{ϕ} . \end{array}

Substituting Eq. (4) into Eq. (3), we can obtain

\frac{\partial^{2} E_{z}}{\partial r^{2}} + \frac{1}{r} \frac{\partial E_{z}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} E_{z}}{\partial ϕ^{2}} + k^{2} E_{z} = 0,

where

k

equals to

ω \sqrt{ε μ}

. As Eq. (5) is a typical Helmholtz equation, the solutions are of the form

E_{z} = \sum_{n = - \infty}^{\infty} B e s s e l (n, k r) \cdot e^{i n ϕ},

where

B e s s e l (n, k r)

is a kind of Bessel function of integral order

n

determined by specific conditions. For the TE incident wave, the incident electric field

E_{i z}

can be written as

E_{i z} = E_{0} e^{- i k_{0} r \cos ϕ},

where

E_{0}

is the amplitude of incident electric field and

k_{0}

is the wave number in the air. We can expand Eq. (7) in the cylindrical polar coordinate system as

E_{i z} = \sum_{n = - \infty}^{\infty} A_{n} J_{n} (k_{0} r) \cdot e^{i n ϕ},

where

A_{n}

is the expansion coefficient, which equals to

E_{0} {(- 1)}^{n}

, and

J_{n}

is the Bessel function of the first kind of integral order

n

. Therefore, the magnetic fields

H_{r}

and

H_{ϕ}

can be acquired by substituting Eq. (8) into Eq. (4):

\begin{array}{l} H_{i r} = \frac{1}{i ω μ r} \frac{\partial E_{i z}}{\partial ϕ} = \frac{n}{ω μ r} \sum_{n = - \infty}^{\infty} A_{n} J_{n} (k_{0} r) \cdot e^{i n ϕ}, \\ H_{i ϕ} = - \frac{1}{i ω μ} \frac{\partial E_{i z}}{\partial r} = - \frac{k_{0}}{i ω μ} \sum_{n = - \infty}^{\infty} A_{n} J_{n}^{'} (k_{0} r) \cdot e^{i n ϕ} . \end{array}

Similar to the incident electric field, the internal field, which includes the electric field and magnetic field in the cylinder, can be expressed as

\begin{array}{l} E_{c z} = \sum_{n = - \infty}^{\infty} C_{n} J_{n} (m k_{0} r) \cdot e^{i n ϕ}, \\ H_{c r} = \frac{1}{i ω μ r} \frac{\partial E_{c z}}{\partial ϕ} = \frac{n}{ω μ r} \sum_{n = - \infty}^{\infty} C_{n} J_{n} (m k_{0} r) \cdot e^{i n ϕ}, \\ H_{c ϕ} = - \frac{1}{i ω μ} \frac{\partial E_{c z}}{\partial r} = - \frac{m k_{0}}{i ω μ} \sum_{n = - \infty}^{\infty} C_{n} J_{n}^{'} (m k_{0} r) \cdot e^{i n ϕ}, \end{array}

where

C_{n}

is the expansion coefficient of internal field. As the fields must be finite at the origin,

J_{n}

is the appropriate Bessel function for the field in cylinder. Additionally, the scattered field can be written as

\begin{array}{l} E_{s z} = - \sum_{n = - \infty}^{\infty} B_{n} H_{n} (k_{0} r) \cdot e^{i n ϕ}, \\ H_{s r} = \frac{1}{i ω μ r} \frac{\partial E_{s z}}{\partial ϕ} = - \frac{n}{ω μ r} \sum_{n = - \infty}^{\infty} B_{n} H_{n} (k_{0} r) \cdot e^{i n ϕ}, \\ H_{s ϕ} = - \frac{1}{i ω μ} \frac{\partial E_{s z}}{\partial r} = \frac{k_{0}}{i ω μ} \sum_{n = - \infty}^{\infty} B_{n} H_{n}^{'} (k_{0} r) \cdot e^{i n ϕ}, \end{array}

where

B_{n}

is the expansion coefficient of scattered field and

H_{n}

is the Hankel function of integral order

n

. Applying the boundary conditions at

r = a

\begin{array}{l} {\hat{e}}_{r} \times ({\vec{E}}_{i z} + {\vec{E}}_{s z} - {\vec{E}}_{c z}) = 0, \\ {\hat{e}}_{r} \times ({\hat{H}}_{i ϕ} + {\hat{H}}_{s ϕ} - {\hat{H}}_{c ϕ}) = σ {\vec{E}}_{c z}, \end{array}

where

σ

is the conductivity of graphene that can be calculated by using the Kubo formula, we can obtain all the expansion coefficients. Thus,

R_{t e n}

, which is the ratio of

B_{n}

to

A_{n}

, can be calculated as

R_{t e n} = \frac{B_{n}}{A_{n}} = \frac{J_{n} (m k_{0} a) J_{n}^{'} (k_{0} a) - m J_{n} (k_{0} a) J_{n}^{'} (m k_{0} a) - i ω μ σ J_{n} (m k_{0} a) J_{n} (k_{0} a)}{J_{n} (m k_{0} a) H_{n}^{'} (k_{0} a) - m H_{n} (k_{0} a) J_{n}^{'} (m k_{0} a) - i ω μ σ J_{n} (m k_{0} a) H_{n} (k_{0} a)} .

In the case of TM incident wave, the three existing field components are $H_{z}$ , $E_{r}$ and $E_{ϕ}$ . According to Eq. (1) and Eq. (2), we can obtain the similar equations to Eqs. (3) and (4) for the TM incident wave as

\frac{1}{r} \frac{\partial}{\partial r} (r E_{ϕ}) - \frac{1}{r} \frac{\partial E_{r}}{\partial ϕ} = i ω μ H_{z},

\begin{array}{l} \frac{1}{r} \frac{\partial H_{z}}{\partial ϕ} = - i ω ε E_{r}, \\ - \frac{\partial H_{z}}{\partial r} = - i ω ε E_{ϕ} . \end{array}

The similar Helmholtz equation for

H_{z}

is

\frac{\partial^{2} H_{z}}{\partial r^{2}} + \frac{1}{r} \frac{\partial H_{z}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} H_{z}}{\partial ϕ^{2}} + k^{2} H_{z} = 0,

which can be acquired by substituting Eq. (15) into Eq. (14). For the TM incident wave, the incident magnetic field can be expressed as

H_{i z} = H_{0} e^{- i k_{0} r \cos ϕ} .

It can also be expanded in a series of Bessel function of first kind as

H_{i z} = \sum_{n = - \infty}^{\infty} D_{n} J_{n} (k_{0} r) \cdot e^{i n ϕ},

where

D_{n}

is the expansion coefficient, which equals to

H_{0} {(- 1)}^{n}

. And the corresponding incident electric field components are

\begin{array}{l} E_{i r} = - \frac{1}{i ω ε_{0} r} \frac{\partial H_{i z}}{\partial ϕ} = - \frac{n}{ω ε_{0} r} \sum_{n = - \infty}^{\infty} D_{n} J_{n} (k_{0} r) \cdot e^{i n ϕ}, \\ E_{i ϕ} = \frac{1}{i ω ε_{0}} \frac{\partial H_{i z}}{\partial r} = \frac{k_{0}}{i ω ε_{0}} \sum_{n = - \infty}^{\infty} D_{n} J_{n}^{'} (k_{0} r) \cdot e^{i n ϕ} . \end{array}

Similar to the TE wave, the fields inside cylinder can be expressed as

\begin{array}{l} H_{c z} = m \sum_{n = - \infty}^{\infty} G_{n} J_{n} (m k_{0} r) \cdot e^{i n ϕ}, \\ E_{c r} = - \frac{1}{i ω ε_{m} r} \frac{\partial H_{c z}}{\partial ϕ} = - \frac{n m}{ω ε_{m} r} \sum_{n = - \infty}^{\infty} G_{n} J_{n} (m k_{0} r) \cdot e^{i n ϕ}, \\ E_{c ϕ} = \frac{1}{i ω ε_{m}} \frac{\partial H_{c z}}{\partial r} = \frac{k_{0}}{i ω ε_{0}} \sum_{n = - \infty}^{\infty} G_{n} J_{n}^{'} (m k_{0} r) \cdot e^{i n ϕ}, \end{array}

where

G_{n}

is the expansion coefficient of internal field. The scattered field is

\begin{array}{l} H_{s z} = \sum_{n = - \infty}^{\infty} F_{n} H_{n} (k_{0} r) \cdot e^{i n ϕ}, \\ E_{s r} = - \frac{1}{i ω ε_{0} r} \frac{\partial H_{s z}}{\partial ϕ} = - \frac{n}{ω ε_{0} r} \sum_{n = - \infty}^{\infty} F_{n} H_{n} (k_{0} r) \cdot e^{i n ϕ}, \\ E_{s ϕ} = \frac{1}{i ω ε_{0}} \frac{\partial H_{s z}}{\partial r} = \frac{k_{0}}{i ω ε_{0}} \sum_{n = - \infty}^{\infty} F_{n} H_{n}^{'} (k_{0} r) \cdot e^{i n ϕ}, \end{array}

where

F_{n}

is the expansion coefficient of internal field. If we apply the boundary conditions at

r = a

\begin{array}{l} {\hat{e}}_{r} \times ({\vec{E}}_{i ϕ} + {\vec{E}}_{s ϕ} - {\vec{E}}_{c ϕ}) = 0, \\ {\hat{e}}_{r} \times ({\hat{H}}_{i z} + {\hat{H}}_{s z} - {\hat{H}}_{c z}) = σ {\vec{E}}_{c ϕ}, \end{array}

we can obtain all the coefficients. The ratio

R_{t m n}

of

F_{n}

to

D_{n}

can be calculated as

R_{t m n} = \frac{F_{n}}{D_{n}} = \frac{J_{n}^{'} (m k_{0} a) J_{n} (k_{0} a) - m J_{n}^{'} (k_{0} a) J_{n} (m k_{0} a) - \frac{i σ k_{0}}{ω ε_{0}} J_{n}^{'} (m k_{0} a) J_{n}^{'} (k_{0} a)}{J_{n}^{'} (m k_{0} a) H_{n} (k_{0} a) - m H_{n}^{'} (k_{0} a) J_{n} (m k_{0} a) - \frac{i σ k_{0}}{ω ε_{0}} J_{n}^{'} (m k_{0} a) H_{n}^{'} (k_{0} a)} .

After we obtain the coefficients of the scattered field, the scattering and extinction cross sections per unit length of the graphene coated cylinder can be calculated as follows [23]:

\begin{array}{l} C_{s c a} = \frac{4}{k_{0}} [{| R_{0} |}^{2} + 2 \sum_{n = 1}^{\infty} ({| R_{n} |}^{2})], \\ C_{e x t} = \frac{4}{k_{0}} Re {R_{0} + 2 \sum_{n = 1}^{\infty} R_{n}}, \end{array}

where

R_{n} = R_{t e n}

for the TE wave while

R_{n} = R_{t m n}

for the TM wave. The absorption cross section per unit length can be acquired by the equation

C_{a b s} = C_{e x t} - C_{s c a} .

3. Results and discussions

Now we can investigate the absorption properties of graphene manipulated by the dielectric cylinder based on the analytical method. In the first step the refractive index m of the dielectric cylinder is fixed to 1.45 and the chemical potential of graphene is set to be 0.3 eV. In the case of TM incident wave, the wavelength of incident wave is set at 1.55 μm. Firstly, we explore the dependence of the absorption property on the radius of the cylinder. We find our hybrid structure will exert a strong modulation on the optical absorption of graphene when the radius of the cylinder is at the scale of several micrometers. Figure 2(a) displays the calculated absorption cross section as a function of the radius of the graphene coated cylinder. It can be seen from the figure that several absorption cross section peaks appear. The distance between adjacent peaks is a constant, which equals to 0.175 μm. Additionally, the values of peaks firstly enhance to the maximum then decrease. To unveil the physical origin of strong modulation on the optical absorption of graphene, the magnetic field $(H_{z})$ distributions of different cylinders whose radii are 4.140, 4.325 and 4.510 μm, corresponding to the three adjacent absorption peaks in Fig. 2(a) are respectively calculated and displayed in Figs. 2(b)-2(d). It shows that when the radius is set at an absorption peak in Fig. 2(a), a resonance of electromagnetic fields recirculating in phase along the internal surface of the cylinder is produced. The three cylinders in Figs. 2(b)-2(d) respectively support 20, 21 and 22 recirculating phase periods.

Fig. 2 (a) Calculated absorption cross section as a function of the radius of graphene coated cylinder at the wavelength of 1.55 μm at the incidence of TM wave. (b)-(d) Calculated magnetic field $(H_{z})$ distributions of different cylinders with radius of (b) 4.140 μm, (c) 4.325 μm and (d) 4.510 μm, corresponding to the absorption peaks in (a).

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These electromagnetic modes produced in the hybrid structure are the so-called WGMs. This means that the dielectric cylinder works as a microcavity. Therefore, the tunability of the graphene absorption can be readily explained. The reason why several peaks of absorption are in equal distance with each other along the axis of radius of cylinder as shown in Fig. 2(a) is that the perimeter of the cylinder has to be suitable to form a WGM. So the equal distance $Δ r$ between adjacent peaks in Fig. 2(a) should basically satisfy the equation $2 π Δ r = λ / n$ , where n is an integer. Therefore, the interval of adjacent peaks is a constant, which is consistent with the calculated value 0.175 μm. The magnetic field distributions of different cylinders supporting continuous phase periods (20, 21, 22) in Figs. 2(b)-2(d) that correspond to the adjacent peaks in Fig. 2(a) is a proof. Moreover, only when the radius of the cylinder is at the scale of several micrometers can the WGM be excited and the modulation depth of absorption by WGM is the highest. Additionally, the coupling efficiency from the incident wave to the WGM that affects the value of absorption peak is also determined by the radius of the cylinder. With the further increase of the radius, the coupling efficiency will decrease and the modulation will become weaker. The reason is that a larger fraction of energy will pass through the cylinder rather than transfer to the WGM.

In addition, according to our calculation, when the chemical potential of graphene is at 0.3 eV, the absorbance of a single layer graphene in the air is about 2.1% at the wavelength of 1.55 μm. As the effective cross section per unit length of incident wave for the cylinder is the diameter of the cylinder, the absorbance in our case can be calculated by $C_{a b s} / 2 r$ , where $r$ is the radius of the cylinder. For example, when the radius is at 4.88 μm (peak position in Fig. 2(a)) for TM incident wave at the wavelength of 1.55 μm, the absorbance is 7.8%. The enhancement is about four times. Thus, for TM incident wave, the dielectric cylinder can realize enhancement of graphene absorption by utilizing WGM.

In order to explore the tunability of graphene absorption by utilizing the hybrid structure, we modify the cylinder radius, refractive index and chemical potential of graphene coating in a small range. Considering the dispersion of graphene, we calculate the absorption cross section of the hybrid system with different radii of cylinder at the wavelength of incident wave ranging from 1.4 to 1.6 μm. The radius varies from 4.10 to 4.19 μm by a step of 0.01 μm. As is illustrated in Fig. 3(a) , the absorption peak gradually shifts to the longer wavelength with the increase of radius. The peak shift versus the radius change is about 5 nm per 10 nm. Figure 3(b) shows that the magnitude of the absorption peak linearly enhances with the increase of radius. When the radius of the cylinder is set at 4.14 μm and the refractive index of the dielectric cylinder varies from 1.41 to 1.50 with a step of 0.01, it can be observed in Fig. 3(c) that the peak position not only shifts to the longer wavelength but also becomes narrower. The magnitude of the absorption peak also enhances when the refractive index of cylinder increases, which is presented in Fig. 3(d). However, when the chemical potential of the graphene coating layer varies from 0.30 to 0.39 eV, the position of absorption peak keeps unchanged while the values of absorption peaks decrease with the increase of the chemical potential, as demonstrated in Figs. 3(e) and 3(f).

Fig. 3 Calculated absorption cross sections and peak values at different radius [(a) and (b)], refractive index [(c) and (d)] and chemical potential [(e) and (f)] of the graphene coated cylinder at the incidence of TM wave.

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For further understanding of the tunability characteristics of the graphene absorption, we further calculate the magnetic field distributions of our model structure for TM wave. As is illustrated in Fig. 4 , for the radius of the cylinder of 4.14 μm, the wavelengths of the incident wave as 1.55 μm and 1.5 μm correspond to absorption peak and non-peak positions. When the incident wave is set at the wavelength of absorption peak, where the field distribution is shown in Fig. 4(a), the magnetic field is localized around the boundary of the cylinder where the graphene exists. It can be seen that the field propagates along the boundary of the cylinder, forming a WGM. However, when the incident wave is not at the wavelength of absorption peak, we cannot find the similar resonance from the field distribution as illustrated in Fig. 4(b).

Fig. 4 Calculated magnetic field $(H_{z})$ distributions at the incident wavelength of (a) 1.55 μm and (b) 1.5 μm (b) for TM wave. The radius of cylinder is 4.14 μm.

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Therefore, the tunabiliy of graphene absorption can be attributed to the excited WGM. The shift of absorption peak in spectrum and the enhancement of peak values illustrated in Fig. 3 can be explained. On one hand, when the radius of the cylinder is increased, the length of the microcavity is increased. So the position of absorption peak has to shift to the longer wavelength in order to satisfy the condition of forming the WGM of the same order. As the absorption peaks in Fig. 3(a) correspond to the same order of WGM, the field distributions for the cylinders of different radii are nearly equal. As the peak shifts just in a narrow spectrum, the dispersion of graphene nearly does not affect the absorption. When the radius increases, the linear increase of the perimeter of the cross section of cylinder corresponding to the interaction length between graphene and electromagnetic field leads to the nearly linear increase of the magnitude of the absorption peak as illustrated in Fig. 3(b). On the other hand, when the refractive index of the cylinder is increased, the effective length of the microcavity is increased. Thus the absorption peak also has a red shift for the same reason. Additionally, with the increase of the refractive index, the speed of the photon decreases in the microcavity, which leads to the increase of the photon lifetime. So the Q factor of the microcavity enhances and the absorption peak becomes narrower, which is consistent with the calculation results shown in Fig. 3(c), and there is no doubt that the absorption peak values will enhance, as shown in Fig. 3(d). The Q factor increases from 83 to 319 when the refractive index increases from 1.41 to 1.50. In contrast, when the incident wave is not at the wavelength of absorption peak, it will not form a WGM as shown in Fig. 4(b). It can be safely concluded from all the above analyses that the strong modulation on optical absorption for TM wave originates from the structure resonance but not the property of graphene material. Thus, the variation of chemical potential of graphene will only modify the value but not the position of absorption peak as presented in Figs. 4(e) and 4(f). The higher chemical potential, which means few vacant states for electrons in conduction band, will decrease the absorption.

At the same time, we consider the situation of TE wave incidence while maintain the other parameters unchanged. The wavelength of the incident wave is still set at 1.55 μm. When the radius of the cylinder varies, the value of the absorption cross section increases linearly, which is illustrated in Fig. 5 . Moreover, the absorption cross section of TE incident wave in the same radius range is three orders of magnitude smaller than that of TM incident wave. Thus, the dielectric cylinder will hinder the graphene absorption for TE incident wave. Similar to the TM incident wave, we have modified the radius and the refractive index of the cylinder and calculated the absorption spectra of graphene. The radius varies from 3.5 to 4.4 μm with a step 0.1 μm while the refractive index remains at 1.45 [Fig. 6(a) ]. The refractive index varies from 1.05 to 1.95 with a step 0.1 while the radius remains at 4.14 μm [Fig. 6(c)].

Fig. 5 Calculated absorption cross section as a function of the radius of the graphene coated cylinder at the wavelength of 1.55 μm for TE wave.

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Fig. 6 Calculated absorption cross sections and peak values at different radius [(a) and (b)], refractive index [(c) and (d)] and chemical potential [(e) and (f)] of the graphene coated cylinder for TE wave.

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It can be seen that the absorption peak does not shift in the spectrum when we change these parameters. The wavelength of the peak remains at 2.5 μm. However, the changes of absorption peak values with the variation of these two parameters are quite different. The absorption peak value linearly increases with the increase of the radius while maintain nearly unchanged with the increase of the refractive index, which are illustrated in Fig. 6(b) and 6(d) respectively. Differently, the absorption peak not only has a blue shift but also becomes narrower and the peak value decreases with the increase of chemical potential of graphene, as shown in Figs. 6(e) and 6(f).

The reason that leads to the linear increase of absorption cross section shown in Fig. 5 can be explained as follows. In the case of TE incident wave, the direction of electric field is parallel to the axis of cylinder. The current produced in the graphene layer, which is responsible for the absorption of graphene, is also parallel to the axis of cylinder. The perimeter of the cross section of cylinder will increase linearly with the increase of the radius of cylinder. Therefore, the current and the absorption cross section also linearly increase. This reason can also be used to explain the results in Figs. 6(b) and 6(d). The increase of the radius of cylinder that will linearly increase the perimeter of the cross section of cylinder also results in the linear increase of the absorption peak value. However, the variation of the refractive index of cylinder nearly does not affect the absorption when the radius of the cylinder is at the scale of several micrometers. When the refractive index varies from 1.05 to 1.95 at the wavelength of 2.5 μm, the absorption cross-section per unit length varies from 2.1654 nm to 2.1651 nm. The variation is only about 0.014%. Additionally, according to our calculation, with the increase of the radius of cylinder, the influence of the refractive index on the absorption will become greater. In this model, there are only three parameters, the conductivity of graphene, the refractive index and the radius of cylinder, that affect the position of absorption peak in spectrum. Moreover, the results and analyses above exclude the effect of refractive index and the radius of cylinder. So the conductivity of graphene becomes the only parameter that affects the position of absorption peak in the absorption spectrum. This suggests that the absorption characteristics of the graphene coated cylinder in the case of TE wave incidence is mainly attributed to the material property of graphene rather than the configuration of the dielectric cylinder. The results in Figs. 6(e) and 6(f) confirm such an assumption. The absorption peak position alters with the variation of material property of graphene. The blue shift and narrowing of absorption peak with the increase of chemical potential are attributed to the fact that more lower vacant states in conduction band of graphene are occupied by electrons and the effective vacant state distribution becomes higher in density and narrower in energy extension. Similar to the analysis of TM wave, the electric field distributions at the incident wavelengths of absorption peak (2.5 μm) and non-peak (1.5 μm) are illustrated in Fig. 7 . It can be seen from the figure that whether the wavelength of incident wave is at the absorption peak or not, there are no special electric field distributions caused by the hybrid structure.

Fig. 7 Calculated electric field $(E_{z})$ distributions at the incident wavelength of (a) 2.5 μm and (b) 1.5 μm for TE wave. The radius of cylinder is 4.14 μm.

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4. Conclusions

In summary, we have demonstrated efficient manipulation of graphene absorption by a simple micrometer-size dielectric cylinder theoretically by using an analytical method. We have systematically investigated the dependence of the absorption spectra of graphene on a series of geometric and physical parameters of the hybrid graphene-cylinder system, including the cylinder radius and refractive index, the chemical potential of graphene, and the incident wave polarization. These theoretical results clearly show a continuous tunability of the graphene absorption in a broad range of wavelength, and based on them, the optimized parameters of the hybrid structure for optimized absorption of graphene at the wavelength of 1.55 μm can be derived. Furthermore, the physical origin for these fruitful characteristics of the graphene absorption manipulation and modulation have been analyzed and identified. In particular, the WGM produced in the dielectric cylinder model has played an important role and can be used to explain in detail what structure modulation will affect the graphene absorption. The multiple absorption peaks produced simultaneously in a broad range of frequency spectrum by using this simple hybrid structure can find potential applications in broadband or multiple wavelengths graphene-based optoelectronic devices. Our results provide a simple, effective and experimentally realizable way to enhance and tune the interaction between light and graphene. In addition, with the development of graphene coating fabrication technology, graphene absorption manipulated by various dielectric optical microstructures and nanostructures other than the current simple dielectric cylinder with excellent optical property can be further explored and more fruitful functionalities can be expected. All these hybrid structures can be deeply exploited for many applications in optoelectronic and all-optical devices.

Acknowledgments

This work is supported by the 973 Program of China at No. 2013CB632704 and 2011CB922002, and the National Natural Science Foundation of China at No. 11204365.

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Efficient manipulation of graphene absorption by a simple dielectric cylinder

Abstract

1. Introduction

2. Theoretical model and formalisms

3. Results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (25)

Optics Express