**1. Introduction**

Cylindrically layered structures have various exotic applications. For instance, a metal-core dielectric-shell nano-wire has been proposed for the cloaking applications in the visible spectrum. The functionality of this structure is based on the induction of antiparallel currents in the core and shell regions, and the design procedure is the so-called scattering cancelation technique [1]. Experimental realization of a hybrid gold/silicon nanowire photodetector proves the practicality of these structures [2]. As an alternative approach for achieving an invisible cloak, cylindrically wrapped impedance surfaces are designed by a periodic arrangement of metallic patches, and the approach is denominated as mantle cloaking [3].

Conversely, cylindrically layered structures can be designed in a way that they exhibit a scattering cross-section far exceeding the single-channel limit. This phenomenon is known as super-scattering and has various applications in sensing, energy harvesting, bio-imaging, communication, and optical devices [4, 5]. Moreover, a cylindrical stack of alternating metals and dielectrics behaves as an anisotropic cavity and exhibits a dramatic drop of the scattering cross-section in the transition from hyperbolic to elliptic dispersion regimes [6, 7]. The Mie-Lorenz theory is a powerful, an exact, and a simple approach for designing and analyzing the aforementioned structures.

Since graphene material is atomically thin, in order to consider its impact on the electromagnetic response of a given structure, boundary conditions at the interface can be simply altered. To this end, graphene surface currents that are proportional to its surface conductivity should be accounted for ensuring the discontinuity of tangential magnetic fields. In the infrared range and below, we can describe the graphene layer with a complex-valued surface conductivity *σ* which may be modeled using the Kubo formulas [18, 19]. The intraband and interband contributions of graphene surface conductivity under local random phase approximations

*kBT*

ln 2 cosh *<sup>μ</sup><sup>c</sup>*

2*kBT*

<sup>ℏ</sup>*<sup>ω</sup>* � <sup>2</sup>*μ<sup>c</sup>* ð Þ<sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup>*kBT* <sup>2</sup>

<sup>2</sup>*<sup>π</sup>* ln <sup>ℏ</sup>*<sup>ω</sup>* <sup>þ</sup> <sup>2</sup>*μ<sup>c</sup>* ð Þ<sup>2</sup>

� � � � (1)

#

(2)

ð Þ *ω* þ *i*Γ

� *i*

The parameters ℏ, *e*, *μc*, Γ, and *T* are reduced Plank's constant, electron charge, chemical potential, charge carriers scattering rate, and temperature, respectively. The above equations are valid for the positive valued chemical potentials. Moreover, graphene-based structures can be analyzed by assigning a small thickness of *Δ*≈ 0*:*35*nm* to the graphene interface and later approaching it to the zero. In this method, by defining the volumetric conductivity as *σ<sup>g</sup>*,*<sup>V</sup>* ¼ *σg=Δ* and using it in

*(a) and (b) the real and imaginary parts of graphene surface conductivity [20] and (c) and (d) the real and*

*<sup>σ</sup>*intrað Þ¼ *<sup>ω</sup>* <sup>2</sup>*ie*<sup>2</sup>

� � �

*π*ℏ<sup>2</sup>

ℏ*ω* � 2*μ<sup>c</sup>* 2*kBT*

*Scattering from Multilayered Graphene-Based Cylindrical and Spherical Particles*

read as [18]:

**Figure 2.**

**37**

*imaginary parts of graphene equivalent permittivity [21].*

*<sup>σ</sup>*inter <sup>≈</sup> *<sup>e</sup>*<sup>2</sup>

4ℏ

1 2 þ 1 *<sup>π</sup>* arctan

*DOI: http://dx.doi.org/10.5772/intechopen.91427*

Multilayered spherical structures have also attracted lots of interests in the field of optical devices. A dielectric sphere made of a high index material supports electric and magnetic dipole resonances which results in peaks in the extinction cross-section [8]. Moreover, by covering the dielectric sphere with a plasmonic metal shell, an invisible cloak is realizable, which is useful for sensors and optical memories [9]. By stacking multiple metal-dielectric shells, an anisotropic medium for scattering shaping can be achieved [10].

From the above discussions, it can be deduced that tailoring the Mie-Lorenz resonances in the curved particles results in developing novel optical devices. In this chapter, we are going to extend the realization of various optical applications based on the excitations of localized surface plasmons (LSP) in graphene-wrapped cylindrical and spherical particles. To this end, initially we introduce a brief discussion of modeling graphene material based on corresponding surface conductivity or dielectric model. Later, we extract the modified Mie-Lorenz coefficients for some curved structures with graphene interfaces. The importance of developed formulas has been proven by providing various design examples. It is worth noting that graphenewrapped particles with a different number of layers have been proposed previously as refractive index sensors, waveguides, super-scatterers, invisible cloaks, and absorbers [11–15]. Our formulation provides a unified approach for the plane wave and eigenmode analysis of graphene-based optical devices.

Graphene is a 2D carbon material in a honeycomb lattice that exhibits extraordinary electrical and mechanical properties. In order to solve Maxwell's equations in the presence of graphene, two approaches are applied by various authors, and we will review them in the following paragraphs. It should be noted that although we are discussing the graphene planar model, we will use the same formulas for the curved geometries when the number of carbon atoms exceeds 10<sup>4</sup> , letting us neglect the effect of defects. Therefore, the radii of all cylinders and spheres are considered to be greater than 5 nm [16]. Moreover, bending the graphene does not have any considerable impact on the properties of its surface plasmons, except for a small downshift of the frequency. **Figure 1** shows the propagation of the graphene surface plasmons on the S-shaped and G-shaped curves [17].

*Scattering from Multilayered Graphene-Based Cylindrical and Spherical Particles DOI: http://dx.doi.org/10.5772/intechopen.91427*

Since graphene material is atomically thin, in order to consider its impact on the electromagnetic response of a given structure, boundary conditions at the interface can be simply altered. To this end, graphene surface currents that are proportional to its surface conductivity should be accounted for ensuring the discontinuity of tangential magnetic fields. In the infrared range and below, we can describe the graphene layer with a complex-valued surface conductivity *σ* which may be modeled using the Kubo formulas [18, 19]. The intraband and interband contributions of graphene surface conductivity under local random phase approximations read as [18]:

$$\sigma\_{\rm intra}(\boldsymbol{\alpha}) = \frac{2ie^2k\_BT}{\pi\hbar^2(\boldsymbol{\alpha} + i\Gamma)} \ln\left[2\cosh\left(\frac{\mu\_c}{2k\_BT}\right)\right] \tag{1}$$

$$\sigma\_{\text{inter}} \approx \frac{e^2}{4\hbar} \left[ \frac{1}{2} + \frac{1}{\pi} \arctan\left(\frac{\hbar\nu - 2\mu\_c}{2k\_B T}\right) - \frac{i}{2\pi} \ln \frac{\left(\hbar\nu + 2\mu\_c\right)^2}{\left(\hbar\nu - 2\mu\_c\right)^2 + \left(2k\_B T\right)^2} \right] \tag{2}$$

The parameters ℏ, *e*, *μc*, Γ, and *T* are reduced Plank's constant, electron charge, chemical potential, charge carriers scattering rate, and temperature, respectively. The above equations are valid for the positive valued chemical potentials. Moreover, graphene-based structures can be analyzed by assigning a small thickness of *Δ*≈ 0*:*35*nm* to the graphene interface and later approaching it to the zero. In this method, by defining the volumetric conductivity as *σ<sup>g</sup>*,*<sup>V</sup>* ¼ *σg=Δ* and using it in

#### **Figure 2.**

*(a) and (b) the real and imaginary parts of graphene surface conductivity [20] and (c) and (d) the real and imaginary parts of graphene equivalent permittivity [21].*

Conversely, cylindrically layered structures can be designed in a way that they exhibit a scattering cross-section far exceeding the single-channel limit. This phenomenon is known as super-scattering and has various applications in sensing, energy harvesting, bio-imaging, communication, and optical devices [4, 5]. Moreover, a cylindrical stack of alternating metals and dielectrics behaves as an anisotropic cavity and exhibits a dramatic drop of the scattering cross-section in the transition from hyperbolic to elliptic dispersion regimes [6, 7]. The Mie-Lorenz theory is a powerful, an exact, and a simple approach for designing and analyzing

Multilayered spherical structures have also attracted lots of interests in the field

of optical devices. A dielectric sphere made of a high index material supports electric and magnetic dipole resonances which results in peaks in the extinction cross-section [8]. Moreover, by covering the dielectric sphere with a plasmonic metal shell, an invisible cloak is realizable, which is useful for sensors and optical memories [9]. By stacking multiple metal-dielectric shells, an anisotropic medium

From the above discussions, it can be deduced that tailoring the Mie-Lorenz resonances in the curved particles results in developing novel optical devices. In this chapter, we are going to extend the realization of various optical applications based on the excitations of localized surface plasmons (LSP) in graphene-wrapped cylindrical and spherical particles. To this end, initially we introduce a brief discussion of modeling graphene material based on corresponding surface conductivity or dielectric model. Later, we extract the modified Mie-Lorenz coefficients for some curved structures with graphene interfaces. The importance of developed formulas has been proven by providing various design examples. It is worth noting that graphenewrapped particles with a different number of layers have been proposed previously as refractive index sensors, waveguides, super-scatterers, invisible cloaks, and absorbers [11–15]. Our formulation provides a unified approach for the plane wave and eigen-

Graphene is a 2D carbon material in a honeycomb lattice that exhibits extraordinary electrical and mechanical properties. In order to solve Maxwell's equations in the presence of graphene, two approaches are applied by various authors, and we will review them in the following paragraphs. It should be noted that although we are discussing the graphene planar model, we will use the same formulas for the

the effect of defects. Therefore, the radii of all cylinders and spheres are considered to be greater than 5 nm [16]. Moreover, bending the graphene does not have any considerable impact on the properties of its surface plasmons, except for a small downshift of the frequency. **Figure 1** shows the propagation of the graphene surface

*Propagation of graphene surface plasmons on curved structures: (a) S-shaped and (b) G-shaped [17].*

, letting us neglect

the aforementioned structures.

*Nanoplasmonics*

for scattering shaping can be achieved [10].

mode analysis of graphene-based optical devices.

plasmons on the S-shaped and G-shaped curves [17].

**Figure 1.**

**36**

curved geometries when the number of carbon atoms exceeds 10<sup>4</sup>

Maxwell's curl equations, the equivalent complex permittivity of the layer can be obtained as [20]:

$$
\varepsilon\_{\mathbf{g},eq} = \left(-\frac{\sigma\_{\mathbf{g},i}}{\alpha \Delta} + \varepsilon\_0\right) + i \left(\frac{\sigma\_{\mathbf{g},r}}{\alpha \Delta}\right) \tag{3}
$$

In order to obtain the modified Mie-Lorenz coefficients, the incident, scattered, and internal electromagnetic fields are expanded in terms of cylindrical coordinates special functions which are, respectively, the Bessel functions and exponentials in the radial and azimuthal directions. In order to exploit a terse mathematical

*<sup>k</sup><sup>ρ</sup>* ^*<sup>ρ</sup>* � *<sup>Z</sup>*<sup>0</sup>

The complete explanation of the above vector wave functions and their self and mutual orthogonally relations can be found in the classic electromagnetic books [22]. In the above equation, *Zn* is the solution of the Bessel differential equation, and *n* is its order. It is clear that in the environment, the radial field contains the Hankel function of the first kind in order to account for the radiation condition at infinity, while in the medium region, the Bessel function is utilized to satisfy the finiteness

In the graphene-based cylindrical structures, the plasmonic state is achieved via illuminating a TE<sup>z</sup> wave to the structure. Therefore, for the normal illumination, the incident, scattered, and dielectric electromagnetic fields are shown with the

> X∞ *n*¼�∞

X∞ *n*¼�∞ *Ani*

*Bn i*

where *H*<sup>0</sup> and *E*<sup>0</sup> are the magnitudes of the incident electric and magnetic fields, respectively, and they are related via the intrinsic impedance of the free space. The wavenumber in the region *l* is denoted by *kl*. The coefficients½ � *An*, *Bn* are, respectively, ½ � 1, 1 , ½ � *an*, *bn* , and ½ � *cn*, *dn* for the incident, scattered, and core regions. Moreover, *an* and *bn* are the well-known Mie-Lorenz coefficients, which are called the modified Mie-Lorenz coefficients for the scattering analysis of graphene-based structures.

The boundary conditions at the graphene interface at *ρ* ¼ *R*<sup>1</sup> are the continuity of the tangential electric fields along with the discontinuity of tangential magnetic

By applying the boundary conditions in the expanded fields, the linear system of equations for extracting the unknowns can be readily obtained. The solution of the

h i

*ωε*<sup>0</sup> *J* 0 *<sup>n</sup>*ð Þ *k*2*R*<sup>1</sup> h i � *<sup>k</sup>*2*ε*<sup>1</sup> *Jn*ð Þ *<sup>k</sup>*1*R*<sup>1</sup> *<sup>J</sup>*

*ωε*<sup>0</sup> *J* 0 *<sup>n</sup>*ð Þ *k*1*R*<sup>1</sup> h i � *<sup>k</sup>*1*ε*2*H*ð Þ<sup>1</sup>

ð Þ *<sup>k</sup>*2*R*<sup>1</sup> *Jn*ð Þ� *<sup>k</sup>*2*R*<sup>1</sup> *<sup>H</sup>*ð Þ<sup>1</sup>

*ωε*<sup>0</sup> *J* 0 *<sup>n</sup>*ð Þ *k*1*R*<sup>1</sup> h i � *<sup>k</sup>*1*ε*2*H*ð Þ<sup>1</sup>

*N<sup>n</sup>* ¼ *kZn*ð Þ *kρ e*

� �*<sup>e</sup>*

*<sup>n</sup>*ð Þ *<sup>k</sup><sup>ρ</sup> <sup>ϕ</sup>*^

*in<sup>ϕ</sup>* (4)

*in<sup>ϕ</sup>*^*z* (5)

*<sup>n</sup>Nn*ð Þ *kl<sup>ρ</sup>* (6)

*<sup>n</sup>Mn*ð Þ *kl<sup>ρ</sup>* (7)

0 *<sup>n</sup>*ð Þ *k*2*R*<sup>1</sup>

> 0 *<sup>n</sup>*ð Þ *k*1*R*<sup>1</sup>

0 *<sup>n</sup>*ð Þ *k*1*R*<sup>1</sup> (10)

(11)

*<sup>n</sup>* ð Þ *k*2*R*<sup>1</sup> *J*

*<sup>n</sup>* ð Þ *k*2*R*<sup>1</sup> *J*

*<sup>ϕ</sup>*^*:E<sup>d</sup>* <sup>¼</sup> *<sup>ϕ</sup>*^*:*ð Þ *<sup>E</sup>sca* <sup>þ</sup> *<sup>E</sup>in* (8)

^*z:H<sup>d</sup>* <sup>¼</sup> ^*z:*ð Þþ *<sup>H</sup>sca* <sup>þ</sup> *<sup>H</sup>in <sup>ϕ</sup>*^*:Ed<sup>σ</sup>* (9)

*<sup>n</sup>* ð Þ *k*2*R*<sup>1</sup> *J*

0 *<sup>n</sup>*ð Þ *k*2*R*<sup>1</sup>

notation, the vector wave functions are introduced as [22]:

superscripts *l* ¼ *in*, *sca*, *d*, respectively, and they read as [23]:

*<sup>H</sup><sup>l</sup>* <sup>¼</sup> *<sup>H</sup>*<sup>0</sup> *kl*

*<sup>E</sup><sup>l</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>0</sup> *kl*

extracted equations for the scattering coefficients leads to:

*<sup>n</sup>*ð Þ *<sup>k</sup>*1*R*<sup>1</sup> *<sup>ε</sup>*<sup>2</sup> *Jn*ð Þ� *<sup>k</sup>*2*R*<sup>1</sup> *<sup>i</sup> <sup>k</sup>*2*<sup>σ</sup>*

*k*2*ε*<sup>1</sup> *H*ð Þ<sup>1</sup> *n* 0

ð Þ *<sup>k</sup>*2*R*<sup>1</sup> *<sup>ε</sup>*<sup>1</sup> *Jn*ð Þþ *<sup>k</sup>*1*R*<sup>1</sup> *<sup>i</sup> <sup>k</sup>*1*<sup>σ</sup>*

ð Þ *<sup>k</sup>*2*R*<sup>1</sup> *<sup>ε</sup>*<sup>1</sup> *Jn*ð Þþ *<sup>k</sup>*1*R*<sup>1</sup> *<sup>i</sup> <sup>k</sup>*1*<sup>σ</sup>*

condition at the origin of the structure.

*DOI: http://dx.doi.org/10.5772/intechopen.91427*

fields. Therefore:

*an* ¼

*bn* ¼

**39**

*k*<sup>1</sup> *J* 0

*k*2*H*ð Þ<sup>1</sup> *n* 0

*k*2*H*ð Þ<sup>1</sup> *n* 0 *<sup>M</sup><sup>n</sup>* <sup>¼</sup> *k in Zn*ð Þ *<sup>k</sup><sup>ρ</sup>*

*Scattering from Multilayered Graphene-Based Cylindrical and Spherical Particles*

where subscripts *r* and *i* represent the real and imaginary parts of the surface conductivity, respectively. Both models will be used in the following sections.

**Figure 2(a)** and **(b)** shows the real and imaginary parts of graphene surface conductivity at the temperature of *T* = 300°K. The real part of the conductivity accounts for the losses, while the positive valued imaginary parts represent the plasmonic properties [20]. Moreover, the real and imaginary parts of the graphene equivalent bulk permittivity are shown in **Figure 2(c)** and **(d)**. The negative valued real relative permittivity represents the plasmonic excitation, and the imaginary part of the permittivity represents the losses [21]. It should be noted that all of the formulas of this chapter are adapted with exp ð Þ �*iωt* time-harmonic dependency.

## **2. Graphene-coated cylindrical tubes**

In this section, the modified Mie-Lorenz coefficients of a single-layered graphene-coated cylindrical tube will be extracted. The formulation is expanded into the multilayered graphene-based tubes through exploiting the TMM method, and later, various applications of the analyzed structures, including emission and radiation properties, complex frequencies, super-scattering, and super-cloaking, will be explained.

#### **2.1 Scattering from graphene-coated wires**

Let us consider a graphene-wrapped infinitely long cylindrical tube. The structure is shown in **Figure 3(a)**, and it is considered that a TE<sup>z</sup> -polarized plane wave illuminates the cylinder. In general, TE and TM waves are coupled in the cylindrical geometries. For the normally incident plane waves, they become decoupled, and they can be treated separately. For simplicity, we consider the normal incidence of plane waves where the wave vector *k* is perpendicular to the cylinder axis.

#### **Figure 3.**

*(a) A single-layered graphene-coated cylinder under TE<sup>z</sup> plane wave illumination and (b) corresponding scattering efficiency for ε<sup>1</sup> = 3.9 and μ<sup>c</sup> = 0.5 eV. The normalization factor in this figure is the diameter of the cylinder [23].*

*Scattering from Multilayered Graphene-Based Cylindrical and Spherical Particles DOI: http://dx.doi.org/10.5772/intechopen.91427*

In order to obtain the modified Mie-Lorenz coefficients, the incident, scattered, and internal electromagnetic fields are expanded in terms of cylindrical coordinates special functions which are, respectively, the Bessel functions and exponentials in the radial and azimuthal directions. In order to exploit a terse mathematical notation, the vector wave functions are introduced as [22]:

$$\mathbf{M}\_n = k \left( \dot{m} \frac{Z\_n(k\rho)}{k\rho} \hat{\rho} - Z\_n'(k\rho)\hat{\phi} \right) e^{in\phi} \tag{4}$$

$$\mathbf{N}\_n = kZ\_n(k\rho)e^{in\phi}\hat{\mathbf{z}}\tag{5}$$

The complete explanation of the above vector wave functions and their self and mutual orthogonally relations can be found in the classic electromagnetic books [22]. In the above equation, *Zn* is the solution of the Bessel differential equation, and *n* is its order. It is clear that in the environment, the radial field contains the Hankel function of the first kind in order to account for the radiation condition at infinity, while in the medium region, the Bessel function is utilized to satisfy the finiteness condition at the origin of the structure.

In the graphene-based cylindrical structures, the plasmonic state is achieved via illuminating a TE<sup>z</sup> wave to the structure. Therefore, for the normal illumination, the incident, scattered, and dielectric electromagnetic fields are shown with the superscripts *l* ¼ *in*, *sca*, *d*, respectively, and they read as [23]:

$$\mathbf{H}\_{l} = \frac{H\_{0}}{k\_{l}} \sum\_{n=-\infty}^{\infty} A\_{n} i^{n} \mathbf{N}\_{n}(k\_{l}\rho) \tag{6}$$

$$E\_l = \frac{E\_0}{k\_l} \sum\_{n = -\infty}^{\infty} B\_n i^n \mathbf{M}\_n(k\_l \rho) \tag{7}$$

where *H*<sup>0</sup> and *E*<sup>0</sup> are the magnitudes of the incident electric and magnetic fields, respectively, and they are related via the intrinsic impedance of the free space. The wavenumber in the region *l* is denoted by *kl*. The coefficients½ � *An*, *Bn* are, respectively, ½ � 1, 1 , ½ � *an*, *bn* , and ½ � *cn*, *dn* for the incident, scattered, and core regions. Moreover, *an* and *bn* are the well-known Mie-Lorenz coefficients, which are called the modified Mie-Lorenz coefficients for the scattering analysis of graphene-based structures.

The boundary conditions at the graphene interface at *ρ* ¼ *R*<sup>1</sup> are the continuity of the tangential electric fields along with the discontinuity of tangential magnetic fields. Therefore:

$$
\hat{\boldsymbol{\phi}}.\mathbf{E}\_d = \hat{\boldsymbol{\phi}}.(\mathbf{E}\_{\text{sca}} + \mathbf{E}\_{\text{in}})\tag{8}
$$

$$
\hat{z}.\mathbf{H}\_d = \hat{z}.(\mathbf{H}\_{\kappa a} + \mathbf{H}\_{\dot{m}}) + \hat{\phi}.\mathbf{E}\_d\sigma\tag{9}
$$

By applying the boundary conditions in the expanded fields, the linear system of equations for extracting the unknowns can be readily obtained. The solution of the extracted equations for the scattering coefficients leads to:

$$\sigma\_{n} = \frac{k\_{1}f\_{n}'(k\_{1}R\_{1})\left[\varepsilon\_{2}I\_{n}(k\_{2}R\_{1}) - i\frac{k\_{2}\sigma}{\alpha c\_{0}}f\_{n}'(k\_{2}R\_{1})\right] - k\_{2}\varepsilon\_{1}I\_{n}(k\_{1}R\_{1})f\_{n}'(k\_{2}R\_{1})}{k\_{2}H\_{n}^{(1)'}(k\_{2}R\_{1})\left[\varepsilon\_{1}I\_{n}(k\_{1}R\_{1}) + i\frac{k\_{1}\sigma}{\alpha c\_{0}}f\_{n}'(k\_{1}R\_{1})\right] - k\_{1}\varepsilon\_{2}H\_{n}^{(1)}(k\_{2}R\_{1})f\_{n}'(k\_{1}R\_{1})} \tag{10}$$

$$b\_{n} = \frac{k\_{2}e\_{1}\left[H\_{n}^{(1)'}(k\_{2}R\_{1})I\_{n}(k\_{2}R\_{1}) - H\_{n}^{(1)}(k\_{2}R\_{1})I\_{n}'(k\_{2}R\_{1})\right]}{k\_{2}H\_{n}^{(1)'}(k\_{2}R\_{1})\left[e\_{1}I\_{n}(k\_{1}R\_{1}) + i\frac{k\_{1}\sigma}{av\_{0}}f\_{n}'(k\_{1}R\_{1})\right] - k\_{1}e\_{2}H\_{n}^{(1)}(k\_{2}R\_{1})f\_{n}'(k\_{1}R\_{1})} \tag{11}$$

Maxwell's curl equations, the equivalent complex permittivity of the layer can be

*ωΔ* <sup>þ</sup> *<sup>ε</sup>*<sup>0</sup> 

where subscripts *r* and *i* represent the real and imaginary parts of the surface conductivity, respectively. Both models will be used in the following sections. **Figure 2(a)** and **(b)** shows the real and imaginary parts of graphene surface conductivity at the temperature of *T* = 300°K. The real part of the conductivity accounts for the losses, while the positive valued imaginary parts represent the plasmonic properties [20]. Moreover, the real and imaginary parts of the graphene equivalent bulk permittivity are shown in **Figure 2(c)** and **(d)**. The negative valued real relative permittivity represents the plasmonic excitation, and the imaginary part of the permittivity represents the losses [21]. It should be noted that all of the formulas of this chapter are adapted with exp ð Þ �*iωt* time-harmonic dependency.

In this section, the modified Mie-Lorenz coefficients of a single-layered graphene-coated cylindrical tube will be extracted. The formulation is expanded into the multilayered graphene-based tubes through exploiting the TMM method, and later, various applications of the analyzed structures, including emission and radiation properties, complex frequencies, super-scattering, and super-cloaking,

Let us consider a graphene-wrapped infinitely long cylindrical tube. The struc-

illuminates the cylinder. In general, TE and TM waves are coupled in the cylindrical geometries. For the normally incident plane waves, they become decoupled, and they can be treated separately. For simplicity, we consider the normal incidence of

plane waves where the wave vector *k* is perpendicular to the cylinder axis.

*(a) A single-layered graphene-coated cylinder under TE<sup>z</sup> plane wave illumination and (b) corresponding scattering efficiency for ε<sup>1</sup> = 3.9 and μ<sup>c</sup> = 0.5 eV. The normalization factor in this figure is the diameter of the*

<sup>þ</sup> *<sup>i</sup> <sup>σ</sup>g*,*<sup>r</sup> ωΔ* 

(3)


*<sup>ε</sup>g*,*eq* ¼ � *<sup>σ</sup>g*,*<sup>i</sup>*

**2. Graphene-coated cylindrical tubes**

**2.1 Scattering from graphene-coated wires**

ture is shown in **Figure 3(a)**, and it is considered that a TE<sup>z</sup>

obtained as [20]:

*Nanoplasmonics*

will be explained.

**Figure 3.**

**38**

*cylinder [23].*

#### *Nanoplasmonics*

The same procedure can be repeated for the TM<sup>z</sup> illumination. The normalized scattering cross-section (NSCS) reads as:

$$\text{NSCS} = \sum\_{n=-\infty}^{\infty} |a\_n|^2 \tag{12}$$

where *j*

i.e., *<sup>λ</sup><sup>n</sup>* <sup>¼</sup> <sup>2</sup>*π<sup>c</sup>*

approximated by:

written as:

where *ω*<sup>2</sup>

emitter [28, 29].

**41**

*j <sup>n</sup>*ð Þ *<sup>x</sup>* <sup>≈</sup> *<sup>n</sup>*

the wire cylinder.

*<sup>n</sup>*ð Þ¼ *<sup>x</sup> <sup>J</sup>*<sup>0</sup>

*<sup>n</sup>*ð Þ *x xJn*ð Þ *<sup>x</sup>* , *hn*ð Þ¼ *<sup>x</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.91427*

*2.1.2 Non-retarded dispersion relation*

*<sup>x</sup>*<sup>2</sup> and *hn*ð Þ *<sup>x</sup>* <sup>≈</sup> � *<sup>n</sup>*

*H*ð Þ<sup>1</sup> <sup>0</sup> *<sup>n</sup>* ð Þ *x xH*ð Þ<sup>1</sup>

*Scattering from Multilayered Graphene-Based Cylindrical and Spherical Particles*

*iωε*<sup>0</sup>

Taking into account that in the non-retarded regime the propagation constant of the plasmon propagating along perfectly flat graphene sheet can be

*ksp* ¼ *iωε*<sup>0</sup>

wrapped with a graphene sheet can be written as:

*ω<sup>n</sup>* ¼

<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*2*μ<sup>c</sup> πε*0ℏ2*R*<sup>1</sup>

surface plasmon corresponding to the flat graphene sheet.

it follows that the dispersion relation (14) for LSPs in dielectric cylinders

where *n* is the LSP multipole order. The dispersion Eq. (17), known as Bohr condition, states that the *n*–th LSP mode of a graphene-coated cylinder accommodates along the cylinder perimeter exactly *n* oscillation periods of the propagating

For large doping (*μ<sup>c</sup>* ≫ *kBT*) and relatively low frequencies (ℏ*ω* ≪ *μc*), the intraband contribution to the surface conductivity (1), the Drude term, plays the leading role. In this case, the non-retarded dispersion equation Eq. (14) is

> *<sup>ε</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>*<sup>2</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>*μcn* ℏ2

which can be analytically solved for the plasmon eigenfrequencies,

� *<sup>γ</sup><sup>c</sup>* 2 � �<sup>2</sup> � *i γc*

the net effect of the chemical potential increment is to increase the spectral position of the resonance peaks when the structure is excited with a plane wave or a dipole

In the following example, we consider a graphene-coated wire with a core radius *R*<sup>1</sup> ¼ 30 nm, made of a non-magnetic dielectric material of permittivity *ε*<sup>1</sup> ¼ 2*:*13

<sup>2</sup> <sup>≈</sup> *<sup>n</sup>ω*<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ε*<sup>1</sup> þ *ε*<sup>2</sup> p � *i*

is the effective plasma frequency of the graphene coating. It is

*μc*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*ω*2 0*n*<sup>2</sup> *ε*<sup>1</sup> þ *ε*<sup>2</sup>

worth noting that the real part of *ω<sup>n</sup>* is proportional to ffiffiffiffi

s

LSPs, and it determines the complex frequencies in terms of all the parameters of

When the size of the cylinder is small compared to the eigenmode wavelength,

*ε*<sup>1</sup> þ *ε*<sup>2</sup> *<sup>σ</sup>* <sup>¼</sup> *<sup>n</sup> R*1

*ε*<sup>1</sup> þ *ε*<sup>2</sup>

*<sup>ω</sup><sup>n</sup>* > >*R*1, where *c* is the speed of light in free space, Eq. (13) can be approximated by using the quasistatic approximation as follows. Using the small argument asymptotic expansions for Bessel and Hankel functions, the functions

*<sup>n</sup>* ð Þ *<sup>x</sup>* . This condition is the dispersion relation of

*<sup>x</sup>*<sup>2</sup> [27]. Thus, the dispersion relation (14) adopts the form:

*<sup>σ</sup>* , (16)

*ksp*2*πR*<sup>1</sup> ¼ 2*πn* (17)

*πε*0*R*1*ω ω* <sup>þ</sup> *<sup>i</sup>γ<sup>c</sup>* ð Þ (18)

*γc*

<sup>2</sup> , (19)

p , and as a consequence,

(15)

where the normalization factor is the single-channel scattering limit of the cylindrical structures. In order to have some insight into the scattering performance of graphene-wrapped wires, the scattering efficiency for *ε*<sup>1</sup> = 3.9 and *μ<sup>c</sup>* = 0.5 eV is plotted in **Figure 3(b)** by varying the radius of the wire. As the figure illustrates, a peak valley line shape occurs in each wavelength. They correspond to invisibility and scattering states and will be further manipulated in the next sections to develop some novel devices. The excitation frequency of the plasmons is the complex poles of the extracted coefficients [24] which will be discussed in the next subsection. Interestingly, the scattering states of graphene-coated dielectric cores are polarization-dependent. By using a left-handed metamaterial as a core, this limitation can be obviated [25].

#### *2.1.1 Eigenmode problem and complex frequencies*

As in any resonant problem, additional information can be obtained by studying the solutions to the boundary value problem in the absence of external sources (eigenmode approach). Although, from a formal point of view, this approach has many similar aspects with those developed in previous sections, the eigenmode problem presents an additional difficulty related to the analytic continuation in the complex plane of certain physical quantities. Due to the fact that the electromagnetic energy is thus leaving the LSP (either by ohmic losses or by radiation towards environment medium), the LSP should be described by a complex frequency where the imaginary part takes into account the finite lifetime of such LSP. The eigenmode approach is not new in physics, but its appearance is associated to any resonance process (at an elementary level could be an RLC circuit), where the complex frequency is a pole of the analytical continuation to the complex plane of the response function of the system (e.g., the current on the circuit). Similarly, in the eigenmode approach presented here, the complex frequencies correspond to poles of the analytical continuation of the multipole terms (Mie-Lorenz coefficients) in the electromagnetic field expansion.

In order to derive complex frequencies of LSP modes in terms of the geometrical and constitutive parameters of the structure, we use an accurate electrodynamic formalism which closely follows the usual separation of variable approach developed in Section 2.1. We can obtain a set of two homogeneous equations for the *m*–th LSP mode [26]:

$$\begin{aligned} \frac{k\_2}{\varepsilon\_2} H\_n^{(1)'} (\varkappa\_2) a\_n - \frac{k\_1}{\varepsilon\_1} f\_n'(\varkappa\_1) b\_n &= 0\\ H\_n^{(1)} (\varkappa\_2) a\_n - f\_n(\varkappa\_1) b\_n &= \frac{\sigma k\_1}{\alpha \varkappa\_0 \varepsilon\_1} b\_n f\_n'(\varkappa\_1) \end{aligned} \tag{13}$$

where the prime denotes the first derivative with respect to the argument of the function and *xj* ¼ *kjR*<sup>1</sup> (*j* ¼ 1, 2). For this system to have a nontrivial solution, its determinant must be equal to zero, a condition which can be written as:

$$D\_n = h\_n(\mathbf{x}\_2) - j\_n(\mathbf{x}\_1) + \sigma \, o \, iR\_1 j\_n(\mathbf{x}\_1) h\_n(\mathbf{x}\_2) = \mathbf{0} \tag{14}$$

*Scattering from Multilayered Graphene-Based Cylindrical and Spherical Particles DOI: http://dx.doi.org/10.5772/intechopen.91427*

where *j <sup>n</sup>*ð Þ¼ *<sup>x</sup> <sup>J</sup>*<sup>0</sup> *<sup>n</sup>*ð Þ *x xJn*ð Þ *<sup>x</sup>* , *hn*ð Þ¼ *<sup>x</sup> H*ð Þ<sup>1</sup> <sup>0</sup> *<sup>n</sup>* ð Þ *x xH*ð Þ<sup>1</sup> *<sup>n</sup>* ð Þ *<sup>x</sup>* . This condition is the dispersion relation of LSPs, and it determines the complex frequencies in terms of all the parameters of the wire cylinder.
