*2.1.2 Non-retarded dispersion relation*

The same procedure can be repeated for the TM<sup>z</sup> illumination. The normalized

*n*¼�∞

j j *an* <sup>2</sup> (12)

NSCS <sup>¼</sup> <sup>X</sup><sup>∞</sup>

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 limita-

As in any resonant problem, additional information can be obtained by studying

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

*<sup>n</sup>*ð Þ *x*<sup>1</sup> *bn* ¼ 0

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

*<sup>n</sup>*ð Þþ *x*<sup>1</sup> *σ ωiR*1*j*

*ωε*0*ε*<sup>1</sup>

*bnJ* 0 *<sup>n</sup>*ð Þ *x*<sup>1</sup>

*<sup>n</sup>*ð Þ *x*<sup>1</sup> *hn*ð Þ¼ *x*<sup>2</sup> 0 (14)

(13)

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

scattering cross-section (NSCS) reads as:

*Nanoplasmonics*

tion can be obviated [25].

*2.1.1 Eigenmode problem and complex frequencies*

the electromagnetic field expansion.

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

*H*ð Þ<sup>1</sup>

*Dn* ¼ *hn*ð Þ� *x*<sup>2</sup> *j*

*<sup>n</sup>* ð Þ *<sup>x</sup>*<sup>2</sup> *an* � *<sup>k</sup>*<sup>1</sup>

*ε*1 *J* 0

*<sup>n</sup>* ð Þ *<sup>x</sup>*<sup>2</sup> *an* � *Jn*ð Þ *<sup>x</sup>*<sup>1</sup> *bn* <sup>¼</sup> *<sup>σ</sup>k*<sup>1</sup>

determinant must be equal to zero, a condition which can be written as:

LSP mode [26]:

**40**

When the size of the cylinder is small compared to the eigenmode wavelength, i.e., *<sup>λ</sup><sup>n</sup>* <sup>¼</sup> <sup>2</sup>*π<sup>c</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 *j <sup>n</sup>*ð Þ *<sup>x</sup>* <sup>≈</sup> *<sup>n</sup> <sup>x</sup>*<sup>2</sup> and *hn*ð Þ *<sup>x</sup>* <sup>≈</sup> � *<sup>n</sup> <sup>x</sup>*<sup>2</sup> [27]. Thus, the dispersion relation (14) adopts the form:

$$
tau\_0 \frac{\varepsilon\_1 + \varepsilon\_2}{\sigma} = \frac{n}{R\_1} \tag{15}$$

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

$$k\_{sp} = i\alpha\varepsilon\_0 \frac{\varepsilon\_1 + \varepsilon\_2}{\sigma},\tag{16}$$

it follows that the dispersion relation (14) for LSPs in dielectric cylinders wrapped with a graphene sheet can be written as:

$$k\_{sp} 2\pi R\_1 = 2\pi n \tag{17}$$

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 surface plasmon corresponding to the flat graphene sheet.

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 written as:

$$
\varepsilon\_1 + \varepsilon\_2 = \frac{\varepsilon^2 \mu\_c n}{\hbar^2 \pi \varepsilon\_0 R\_1 o(\omega + i \chi\_c)} \tag{18}
$$

which can be analytically solved for the plasmon eigenfrequencies,

$$
\rho\_n = \sqrt{\frac{\alpha\_0^2 n^2}{\varepsilon\_1 + \varepsilon\_2} - \left(\frac{\chi\_c}{2}\right)^2} - i \frac{\chi\_c}{2} \approx \frac{n \nu\_0}{\sqrt{\varepsilon\_1 + \varepsilon\_2}} - i \frac{\chi\_c}{2}, \tag{19}
$$

where *ω*<sup>2</sup> <sup>0</sup> <sup>¼</sup> *<sup>e</sup>*2*μ<sup>c</sup> πε*0ℏ2*R*<sup>1</sup> is the effective plasma frequency of the graphene coating. It is worth noting that the real part of *ω<sup>n</sup>* is proportional to ffiffiffiffi *μc* p , and as a consequence, 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 emitter [28, 29].

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


*Dn*,*q*ð Þ¼ *x*

*<sup>q</sup>* ¼ ffiffiffiffi *εq*

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

represented by the following matrices. We have:

design some novel optoelectronic devices.

*2.2.1 Application in mantle cloaking*

impedance equals *z*�<sup>1</sup>

*Jn kgRq*þ<sup>1</sup>

**Figure 5.**

**43**

*field distribution [32].*

� � � *tgkg <sup>J</sup>*

coefficients can be calculated as:

0 *<sup>n</sup> kgRq*þ<sup>1</sup> *z*�<sup>1</sup> *<sup>q</sup> J* 0

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

The argument of the above special functions is *<sup>x</sup>*<sup>1</sup> <sup>¼</sup> *kqx*, and the TE<sup>z</sup> wave

*an* <sup>¼</sup> *Tn*,21

let us consider each graphene interface as a thin dielectric with the equivalent complex permittivity defined in Eq. (3) and utilize the TMM formulation in the limiting case of a small radius at the graphene interface with the wave number of *kg*,

i.e., *Rq*þ<sup>1</sup> � *Rq* ¼ *tg* . At each boundary, using the Taylor expansion as *Jn kgRq*

*<sup>g</sup>* <sup>¼</sup> <sup>1</sup> �*iση*<sup>0</sup>

*<sup>g</sup>* <sup>¼</sup> 1 0 *iση*<sup>0</sup> 1

where the free-space impedance *η*<sup>0</sup> equals 377 ohms. Once *Tn* matrix is generated, the modified Mie-Lorenz coefficients and thus scattering cross-section are readily attainable. In the following subsections, the above equations will be used to

Widely tunable scattering cancelation is feasible by using patterned graphenebased patch meta-surface around the dielectric cylinder as shown in **Figure 5**. The surface impedance of the graphene patches can be simply and accurately calculated by closed-form formulas, to be inserted in the modified Mie-Lorenz theory [32].

*(a) Electromagnetic cloaking of a dielectric cylinder using graphene meta-surface and (b) corresponding electric*

*TTE*

*TTM*

*Tn*,21 þ *iTn*,22

In order to incorporate the graphene surface conductivity in the above formulas,

� � in the *Tn* matrix, the graphene interface can be

0 1 � � (23)

� � (24)

*Jn*ð Þ *x*<sup>1</sup> *Yn*ð Þ *x*<sup>1</sup>

" #

*<sup>q</sup> Y*<sup>0</sup> *<sup>n</sup>*ð Þ *x*<sup>1</sup>

p . After generating *Tn* matrix for the structure, *an*

(21)

(22)

� � <sup>¼</sup>

*<sup>n</sup>*ð Þ *<sup>x</sup>*<sup>1</sup> *<sup>z</sup>*�<sup>1</sup>

**Table 1.**

*Resonance frequencies ω<sup>n</sup> for the first four eigenmodes (*1≤*n*≤4*), R*<sup>1</sup> ¼ 30 *nm, μ<sup>c</sup>* ¼ 0*:*5 *eV, γ<sup>c</sup>* ¼ 0*:*1 *meV, ε*<sup>1</sup> ¼ 2*:*13*, μ*<sup>1</sup> ¼ 1*, ε*<sup>2</sup> ¼ 1*, and μ*<sup>2</sup> ¼ 1 *[26].*

**Figure 4.**

*Multilayered cylindrical structure consisting of alternating graphene-dielectric stacks under plane wave illumination. The 2D graphene shells are represented volumetrically for the sake of illustration [31].*

immersed in the vacuum. The graphene parameters are *μ<sup>c</sup>* ¼ 0*:*5 eV, *γ<sup>c</sup>* ¼ 0*:*1 meV, and *T* ¼ 300° *K*. **Table 1** shows the first four eigenfrequencies calculated by solving the full retarded (FR) dispersion Eq. (14) (second column) and by using the analytical approximation (AA) given by Eq. (19) (third column). Since the radius of the wire is small compared with the eigenmode wavelengths, good agreement is obtained between the complex FR and AA *ω<sup>n</sup>* values, even when the AA assigns <sup>I</sup>*<sup>m</sup> <sup>ω</sup><sup>n</sup>* ¼ �*γc=*2<sup>≈</sup> <sup>0</sup>*:*<sup>25</sup> � <sup>10</sup>�<sup>3</sup>*μm*�<sup>1</sup> to all multipolar plasmon modes.

#### **2.2 Multilayered graphene-based cylindrical structures**

In this section, multilayered cylindrical tubes with multiple graphene interfaces are of interest. In order to ease the derivation of the unknown expansion coefficients, matrix-based TMM formulation is generalized to the tubes with several graphene interfaces. Initially, consider a layered cylinder constructed by the staked ordinary materials under TE<sup>z</sup> plane wave illumination, as shown in **Figure 4**. The total magnetic field at the environment can be expressed as the superposition of incident and scattered waves as in Section 2.1. The unknown expansion coefficients of the scattered wave can be determined by means of the *Tn* matrix defined as [30]:

$$T\_n = \left[D\_{n,C}(R\_1)\right]^{-1} \cdot \left\{ \prod\_{q=1}^{N} D\_{n,q}(R\_q) \cdot \left[D\_{n,q}(R\_{q+1})\right]^{-1} \right\} .cdot D\_{n,N+1}(R\_{q+1}) \tag{20}$$

where *C* represents the core layer. In the above equation, the dynamical matrix *Dn*,*<sup>q</sup>* of each region is constructed based on its constitutive and geometrical parameters distinguished through the subscript *q* (*q* = 1, 2, … , *N*). We have:

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

$$D\_{n,q}(\mathbf{x}) = \begin{bmatrix} J\_n(\mathbf{x}\_1) & Y\_n(\mathbf{x}\_1) \\ z\_q^{-1} J\_n'(\mathbf{x}\_1) & z\_q^{-1} Y\_n'(\mathbf{x}\_1) \end{bmatrix} \tag{21}$$

The argument of the above special functions is *<sup>x</sup>*<sup>1</sup> <sup>¼</sup> *kqx*, and the TE<sup>z</sup> wave impedance equals *z*�<sup>1</sup> *<sup>q</sup>* ¼ ffiffiffiffi *εq* p . After generating *Tn* matrix for the structure, *an* coefficients can be calculated as:

$$a\_n = \frac{T\_{n,21}}{T\_{n,21} + iT\_{n,22}} \tag{22}$$

In order to incorporate the graphene surface conductivity in the above formulas, let us consider each graphene interface as a thin dielectric with the equivalent complex permittivity defined in Eq. (3) and utilize the TMM formulation in the limiting case of a small radius at the graphene interface with the wave number of *kg*, i.e., *Rq*þ<sup>1</sup> � *Rq* ¼ *tg* . At each boundary, using the Taylor expansion as *Jn kgRq* � � <sup>¼</sup> *Jn kgRq*þ<sup>1</sup> � � � *tgkg <sup>J</sup>* 0 *<sup>n</sup> kgRq*þ<sup>1</sup> � � in the *Tn* matrix, the graphene interface can be represented by the following matrices. We have:

$$T\_{\mathcal{g}}^{TE} = \begin{bmatrix} \mathbf{1} & -i\sigma\eta\_0 \\ \mathbf{0} & \mathbf{1} \end{bmatrix} \tag{23}$$

$$T\_{\rm g}^{\rm TM} = \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ i\sigma\eta\_0 & \mathbf{1} \end{bmatrix} \tag{24}$$

where the free-space impedance *η*<sup>0</sup> equals 377 ohms. Once *Tn* matrix is generated, the modified Mie-Lorenz coefficients and thus scattering cross-section are readily attainable. In the following subsections, the above equations will be used to design some novel optoelectronic devices.

#### *2.2.1 Application in mantle cloaking*

Widely tunable scattering cancelation is feasible by using patterned graphenebased patch meta-surface around the dielectric cylinder as shown in **Figure 5**. The surface impedance of the graphene patches can be simply and accurately calculated by closed-form formulas, to be inserted in the modified Mie-Lorenz theory [32].

#### **Figure 5.**

*(a) Electromagnetic cloaking of a dielectric cylinder using graphene meta-surface and (b) corresponding electric field distribution [32].*

immersed in the vacuum. The graphene parameters are *μ<sup>c</sup>* ¼ 0*:*5 eV, *γ<sup>c</sup>* ¼ 0*:*1 meV, and *T* ¼ 300° *K*. **Table 1** shows the first four eigenfrequencies calculated by solving the full retarded (FR) dispersion Eq. (14) (second column) and by using the analytical approximation (AA) given by Eq. (19) (third column). Since the radius of the wire is small compared with the eigenmode wavelengths, good agreement is obtained between the complex FR and AA *ω<sup>n</sup>* values, even when the AA assigns

*Multilayered cylindrical structure consisting of alternating graphene-dielectric stacks under plane wave illumination. The 2D graphene shells are represented volumetrically for the sake of illustration [31].*

*<sup>n</sup>* FR *<sup>μ</sup>m*�**<sup>1</sup>** ð Þ AA *<sup>μ</sup>m*�**<sup>1</sup>** ð Þ

 <sup>0</sup>*:*<sup>8868268</sup> � *<sup>i</sup>*0*:*<sup>4105015</sup> � <sup>10</sup>�<sup>3</sup> <sup>0</sup>*:*<sup>8873162</sup> � *<sup>i</sup>*0*:*<sup>2532113</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>254676</sup> � *<sup>i</sup>*0*:*<sup>2532181</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>254855</sup> � *<sup>i</sup>*0*:*<sup>2532113</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>536735</sup> � *<sup>i</sup>*0*:*<sup>2531644</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>536877</sup> � *<sup>i</sup>*0*:*<sup>2532113</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>774508</sup> � *<sup>i</sup>*0*:*<sup>2531757</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>774632</sup> � *<sup>i</sup>*0*:*<sup>2532113</sup> � <sup>10</sup>�<sup>3</sup>

*Resonance frequencies ω<sup>n</sup> for the first four eigenmodes (*1≤*n*≤4*), R*<sup>1</sup> ¼ 30 *nm, μ<sup>c</sup>* ¼ 0*:*5 *eV, γ<sup>c</sup>* ¼ 0*:*1 *meV,*

In this section, multilayered cylindrical tubes with multiple graphene interfaces are of interest. In order to ease the derivation of the unknown expansion coefficients, matrix-based TMM formulation is generalized to the tubes with several graphene interfaces. Initially, consider a layered cylinder constructed by the staked ordinary materials under TE<sup>z</sup> plane wave illumination, as shown in **Figure 4**. The total magnetic field at the environment can be expressed as the superposition

> � �*: Dn*,*<sup>q</sup> Rq*þ<sup>1</sup> � � � � �<sup>1</sup>

where *C* represents the core layer. In the above equation, the dynamical matrix *Dn*,*<sup>q</sup>* of each region is constructed based on its constitutive and geometrical param-

*:Dn*,*N*þ<sup>1</sup> *Rq*þ<sup>1</sup>

� � (20)

( )

<sup>I</sup>*<sup>m</sup> <sup>ω</sup><sup>n</sup>* ¼ �*γc=*2<sup>≈</sup> <sup>0</sup>*:*<sup>25</sup> � <sup>10</sup>�<sup>3</sup>*μm*�<sup>1</sup> to all multipolar plasmon modes.

of incident and scattered waves as in Section 2.1. The unknown expansion coefficients of the scattered wave can be determined by means of the *Tn*

*Dn*,*<sup>q</sup> Rq*

eters distinguished through the subscript *q* (*q* = 1, 2, … , *N*). We have:

**2.2 Multilayered graphene-based cylindrical structures**

*:* Y *N*

*q*¼1

matrix defined as [30]:

**42**

**Table 1.**

*Nanoplasmonics*

**Figure 4.**

*ε*<sup>1</sup> ¼ 2*:*13*, μ*<sup>1</sup> ¼ 1*, ε*<sup>2</sup> ¼ 1*, and μ*<sup>2</sup> ¼ 1 *[26].*

*Tn* <sup>¼</sup> ½ � *Dn*,*<sup>C</sup>*ð Þ *<sup>R</sup>*<sup>1</sup> �<sup>1</sup>

#### *2.2.2 Application in super-scattering*

Let us consider a triple shell graphene-based nanotube under plane wave illumination, as shown in **Figure 6(a)**. This structure is used to design a dual-band superscatterer in the infrared frequencies. To this end, modified Mie-Lorenz coefficients of various scattering channels should have coincided with the proper choice of geometrical and optical parameters. In order to construct the *Tn* matrix for this geometry, one needs to multiply nine 2 � 2 dynamical matrices, which is mathematically complex for analytical scattering manipulation. Therefore, the associated planar structure, shown in **Figure 6(b)**, is used to develop the dispersion engineering method as a quantitative design procedure of the super-scatter. The separations of the free-standing graphene layers are *d*<sup>1</sup> = *d*<sup>2</sup> = 45 nm in the planar structure, and the transmission line model is used to analyze it. Moreover, the chemical potential of lossless graphene material is *μ*<sup>c</sup> = 0.2 eV in all layers. The dispersion diagram of the planar structure is illustrated in **Figure 7(a)**, which predicts the presence of three plasmonic resonances in each scattering channel of the tube at around the frequencies that fulfill *βReff* ¼ *n*, where *Reff* is the mean of the radii of all layers and *β* is the propagation constant of the plasmons in the planar structure. This condition is known as Bohr's quantization formula [30], and its validity for our specific structure is proven by means of the previously developed formulas in **Figure 7(b)**. Eqs. (12), (22), and (23) are used to obtain this figure.

In order to design a dual-band super-scatterer, the plasmonic resonances of two scattering channels have coincided by fine-tuning the results of the Bohr's model. The optimized geometrical and constitutive parameters are *Rc* = 45.45 nm, *d*<sup>1</sup> = 45.05 nm, *d*<sup>2</sup> = 43.23 nm, *ε*<sup>1</sup> = 3.2, *ε*<sup>2</sup> = 2.1, *ε*<sup>3</sup> = 2.2, and *ε*<sup>4</sup> = 1. **Figure 8** shows the NSCS and magnetic field distribution for the dual operating bands of the structure. It is clear that NSCS exceeds the single-channel limit by the factor of 4, and in the corresponding magnetic field, there is a large shadow around the nanometer-sized cylinder at each operating frequency. Other designs are also feasible by altering optical and geometrical parameters. Furthermore, the far-field radiation pattern is a hybrid dipolequadrupole due to simultaneous excitation of the first two channels. It should be noted that an inherent characteristic of the super-scatterer design using plasmonic graphene material is extreme sensitivity to the parameters. Moreover, in the presence of losses, the scattering amplitudes do not reach the single-channel limit anymore, and this restricts the practical applicability of the concepts to low-frequency windows.

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

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

As another example, the dispersion diagram of **Figure 7(a)** along with Foster's theorem has been used to conclude that each scattering channel of the triple shell tube contains two zeros which are lying between the plasmonic resonances, predicted by the Bohr's model. Later, we have coincided the zeros and poles of the first two scattering channels in order to observe super-scattering and super-cloaking simultaneously [33]. The optimized material and geometrical parameters are *ε*<sup>c</sup> = 3.2, *ε*<sup>1</sup> = *ε*<sup>2</sup> = 2.1, *Rc* = 45.45 nm, *d*<sup>1</sup> = 46.25 nm, and *d*<sup>2</sup> = 46.049 nm. The NSCS curves corresponding to the super-cloaking and super-scattering regimes are illustrated in

*(a) and (b) The NSCS of dual-band super-scatterer respectively, in the first and second operating frequencies*

*and (c) and (d) corresponding magnetic field distributions [30].*

*2.2.3 Application in simultaneous super-scattering and super-cloaking*

**Figure 8.**

**45**

#### **Figure 6.**

#### **Figure 7.**

*(a) Dipole and quadruple Mie-Lorenz scattering coefficients for the tube of* **Figure 6** *and (b) dispersion diagram of the associated planar structure [30]. f1p, f2p, and f3p are the plasmonic resonances of the dipole mode predicted by the planar configuration. The prime denotes the same information for the quadruple mode. f1c, f2c, and f3c are the same information calculated by the exact modified Mie-Lorenz theory of the multilayered cylindrical structure.*

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

In order to design a dual-band super-scatterer, the plasmonic resonances of two scattering channels have coincided by fine-tuning the results of the Bohr's model. The optimized geometrical and constitutive parameters are *Rc* = 45.45 nm, *d*<sup>1</sup> = 45.05 nm, *d*<sup>2</sup> = 43.23 nm, *ε*<sup>1</sup> = 3.2, *ε*<sup>2</sup> = 2.1, *ε*<sup>3</sup> = 2.2, and *ε*<sup>4</sup> = 1. **Figure 8** shows the NSCS and magnetic field distribution for the dual operating bands of the structure. It is clear that NSCS exceeds the single-channel limit by the factor of 4, and in the corresponding magnetic field, there is a large shadow around the nanometer-sized cylinder at each operating frequency. Other designs are also feasible by altering optical and geometrical parameters. Furthermore, the far-field radiation pattern is a hybrid dipolequadrupole due to simultaneous excitation of the first two channels. It should be noted that an inherent characteristic of the super-scatterer design using plasmonic graphene material is extreme sensitivity to the parameters. Moreover, in the presence of losses, the scattering amplitudes do not reach the single-channel limit anymore, and this restricts the practical applicability of the concepts to low-frequency windows.

### *2.2.3 Application in simultaneous super-scattering and super-cloaking*

As another example, the dispersion diagram of **Figure 7(a)** along with Foster's theorem has been used to conclude that each scattering channel of the triple shell tube contains two zeros which are lying between the plasmonic resonances, predicted by the Bohr's model. Later, we have coincided the zeros and poles of the first two scattering channels in order to observe super-scattering and super-cloaking simultaneously [33]. The optimized material and geometrical parameters are *ε*<sup>c</sup> = 3.2, *ε*<sup>1</sup> = *ε*<sup>2</sup> = 2.1, *Rc* = 45.45 nm, *d*<sup>1</sup> = 46.25 nm, and *d*<sup>2</sup> = 46.049 nm. The NSCS curves corresponding to the super-cloaking and super-scattering regimes are illustrated in

#### **Figure 8.**

*(a) and (b) The NSCS of dual-band super-scatterer respectively, in the first and second operating frequencies and (c) and (d) corresponding magnetic field distributions [30].*

*2.2.2 Application in super-scattering*

*Nanoplasmonics*

Eqs. (12), (22), and (23) are used to obtain this figure.

**Figure 6.**

**Figure 7.**

**44**

*cylindrical structure.*

*R1 is denoted with Rc in the text.*

Let us consider a triple shell graphene-based nanotube under plane wave illumination, as shown in **Figure 6(a)**. This structure is used to design a dual-band superscatterer in the infrared frequencies. To this end, modified Mie-Lorenz coefficients of various scattering channels should have coincided with the proper choice of geometrical and optical parameters. In order to construct the *Tn* matrix for this geometry, one needs to multiply nine 2 � 2 dynamical matrices, which is mathematically complex for analytical scattering manipulation. Therefore, the associated planar structure, shown in **Figure 6(b)**, is used to develop the dispersion engineering method as a quantitative design procedure of the super-scatter. The separations of the free-standing graphene layers are *d*<sup>1</sup> = *d*<sup>2</sup> = 45 nm in the planar structure, and the transmission line model is used to analyze it. Moreover, the chemical potential of lossless graphene material is *μ*<sup>c</sup> = 0.2 eV in all layers. The dispersion diagram of the planar structure is illustrated in **Figure 7(a)**, which predicts the presence of three plasmonic resonances in each scattering channel of the tube at around the frequencies that fulfill *βReff* ¼ *n*, where *Reff* is the mean of the radii of all layers and *β* is the propagation constant of the plasmons in the planar structure. This condition is known as Bohr's quantization formula [30], and its validity for our specific structure is proven by means of the previously developed formulas in **Figure 7(b)**.

*(a) Multilayered cylindrical nanotube with three graphene shells and (b) associated planar structure [30].*

*(a) Dipole and quadruple Mie-Lorenz scattering coefficients for the tube of* **Figure 6** *and (b) dispersion diagram of the associated planar structure [30]. f1p, f2p, and f3p are the plasmonic resonances of the dipole mode predicted by the planar configuration. The prime denotes the same information for the quadruple mode. f1c, f2c, and f3c are the same information calculated by the exact modified Mie-Lorenz theory of the multilayered* **Figure 9(a)** and **(b)**, as well as the expected phenomenon, is clearly observed. The corresponding magnetic field distributions, shown in **Figure 9(c)** and **(d)**, also manifest the reduced and enhanced scatterings in the corresponding operating bands, respectively. Similar to the dual-band super-scatterer of the previous section, the performance of this structure is very sensitive to the optical, material, and geometrical parameters. By further increasing the number of graphene shells, other plasmonic resonances and zeros can be achieved for the manipulation of the optical response.
