**3.1 Multilayered graphene-based spherical structures**

In this section, the most general graphene-based structure with *N* dielectric layers, as shown in **Figure 10**, is considered, and plane wave scattering is analyzed through extracting recurrence relations for modified Mie-Lorenz coefficients. It should be noted that since, in the TMM method, multiple matrix inversions are

#### **Figure 9.**

*Simultaneous super-scattering and super-cloaking using the structure of* **Figure 6***. NSCS for (a) super-cloaking and (b) super-scattering regimes and corresponding magnetic field distributions, respectively, in (c) and (d) [33].*

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

**Figure 10.**

**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.

In this section, multilayered graphene-coated particles with spherical morphol-

ogy are investigated, and corresponding modified Mie-Lorenz coefficients are extracted by expanding the incident, scattered, and transmitted electromagnetic fields in terms of spherical harmonics. It is clear that by increasing the number of graphene layers, further degrees of freedom for manipulating the optical response can be achieved. For the simplicity of the performance optimization, an equivalent RLC circuit is proposed in the quasistatic regime for the sub-wavelength plasmons,

In this section, the most general graphene-based structure with *N* dielectric layers, as shown in **Figure 10**, is considered, and plane wave scattering is analyzed through extracting recurrence relations for modified Mie-Lorenz coefficients. It should be noted that since, in the TMM method, multiple matrix inversions are

*Simultaneous super-scattering and super-cloaking using the structure of* **Figure 6***. NSCS for (a) super-cloaking and (b) super-scattering regimes and corresponding magnetic field distributions,*

**3. Graphene-coated spherical structures**

*Nanoplasmonics*

and various practical examples are presented.

**Figure 9.**

**46**

*respectively, in (c) and (d) [33].*

**3.1 Multilayered graphene-based spherical structures**

*Spherical graphene-dielectric stack (a) 2D and (b) 3D views [34]. Please note that the numbering of the layers is started from the outermost layer in order to preserve the consistency with the reference paper [35].*

necessary, unlike the cylindrically layered structures of the previous section, the spherical geometries are analyzed through recurrence relations. Also, scattering from a single graphene-coated sphere has been formulated elsewhere [16], and it can be simply attained as the special case of our formulation.

The scattering analysis is very similar to that of the single-shell sphere [16], unless the Kronecker delta function is used in the expansions in order to find the electromagnetic fields of any desired layer with terse expansions. Therefore [34]:

$$E\_i = E\_0 \sum\_{n} \sum\_{m} i^n \frac{2n+1}{n(n+1)} \left\{ \mathbf{M}\_{mn}^{(1)} - i \mathbf{N}\_{mn}^{(1)} \right\} \tag{25}$$

$$\begin{split} \mathbf{E}\_{\text{scat}}^{p} &= \mathbf{E}\_{0} \sum\_{n} \sum\_{m} i^{n} \frac{2n+1}{n(n+1)} \times \\ &\left\{ \left( \mathbf{1} - \delta\_{p}^{N} \right) \mathbf{B}\_{\text{H}}^{p} \mathbf{M}\_{mn}^{(1)} - i \left( \mathbf{1} - \delta\_{p}^{N} \right) \mathbf{B}\_{V}^{p} \mathbf{N}\_{mn}^{(1)} + \left( \mathbf{1} - \delta\_{p}^{1} \right) \mathbf{D}\_{\text{H}}^{p} \mathbf{M}\_{mn} - i \left( \mathbf{1} - \delta\_{p}^{1} \right) \mathbf{D}\_{V}^{p} \mathbf{N}\_{mn} \right\} . \end{split} \tag{26}$$

By considering *zn* as either *j <sup>n</sup>* or *<sup>h</sup>*ð Þ<sup>1</sup> *<sup>n</sup>* , which stand for the spherical Bessel and Hankel functions of the first kind with order *n*, respectively, and *Pm <sup>n</sup>* as the associated Legendre function of order (*n*, *m*), the vector wave functions are defined as follows:

$$\mathbf{M}\_{mn}\left(k\_p\right) = z\_n\left(k\_p r\right)e^{im\phi} \left[\frac{im}{\sin\theta}P\_n^m(\cos\theta)\hat{\theta} - \frac{dP\_n^m(\cos\theta)}{d\theta}\hat{\phi}\right] \tag{27}$$

$$\mathbf{N}\_{mn}\left(k\_p\right) = \frac{n(n+1)}{k\_p r}z\_n(k\_p r)P\_n^m(\cos\theta)e^{im\phi}\hat{r} + \tag{28}$$

$$\frac{1}{k\_p r}\frac{d\left[r z\_n(k\_p r)\right]}{dr}\left[\frac{dP\_n^m(\cos\theta)}{d\theta}\hat{\theta} + \frac{im}{\sin\theta}P\_n^m(\cos\theta)\hat{\phi}\right]e^{im\phi} $$

where super-indices (1) in the vector wave functions show that the Hankel functions are used in the field expansions. The boundary conditions at the interface of adjacent layers read as:

$$
\hat{\mathbf{r}} \times \mathbf{E}^{(p)} = \hat{\mathbf{r}} \times \mathbf{E}^{(p+1)} \tag{29}
$$

$$\mathbf{e}\frac{\mathbf{1}}{\operatorname{ico}\mu\_{p+1}}\hat{\mathbf{r}} \times \nabla \times \mathbf{E}^{(p+1)} - \frac{\mathbf{1}}{\operatorname{ico}\mu\_{p}}\hat{\mathbf{r}} \times \nabla \times \mathbf{E}^{(p)} = \sigma\_{[p+1]p}\hat{\mathbf{r}} \times \left(\hat{\mathbf{r}} \times \mathbf{E}^{(p)}\right) \tag{30}$$

Therefore, the linear system of equations resulting from the above conditions is:

*T<sup>H</sup> Pp* ¼

*TV Fp* ¼

*TV Pp* ¼

coefficients via:

**Figure 12**.

**49**

*kp*þ1*μp∂ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>ψ</sup>pp*

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

*kp*þ1*μ<sup>p</sup> <sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ξpp*

*kp*þ1*μ<sup>p</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ψpp*

where *<sup>g</sup>* <sup>¼</sup> *<sup>i</sup>ωσ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup>μpμp*þ1. By using *BN*

*Qext* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>*

the scattering shaping of various spherical geometries.

method, it is efficient in terms of memory as well.

*3.1.1 Quasistatic approximation and RLC model*

*kp*þ1*μ<sup>p</sup> <sup>∂</sup>ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* � *<sup>∂</sup>ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* � �

*kp*þ1*μ<sup>p</sup> <sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* � *<sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* � �

*kp*þ1*μ<sup>p</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* � *<sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* � �

*<sup>H</sup>*,*<sup>V</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>1</sup>

ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>B</sup>*<sup>1</sup>

where symbol ℜ represents the real part of the summation. In order to verify the extracted coefficients, the extinction efficiencies of three graphene-coated structures is provided in **Figure 11**. In the graphical representation of the structures, the dashed lines illustrate graphene interfaces, while the solid line shows a PEC core. The optical and geometrical parameters are *R*<sup>1</sup> = 200 nm, *R*<sup>2</sup> = 100 nm, *R*<sup>3</sup> = 50 nm, *μ*<sup>c</sup> = 0.3 eV,*T* = 300° K, and *τ* = 0.02 ps. The analytical results are compared with the

*<sup>n</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>ψ</sup>pp*

*<sup>n</sup> <sup>∂</sup>ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>∂</sup>ξpp*

*<sup>n</sup> <sup>∂</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>∂</sup>ψpp*

*<sup>V</sup>* <sup>þ</sup> *<sup>B</sup>*<sup>1</sup> *H* *<sup>n</sup> ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> <sup>∂</sup>ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> <sup>∂</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>H</sup>*,*<sup>V</sup>* ¼ 0, the recurrence relations

� � (42)

(39)

(40)

(41)

(43)

*<sup>n</sup>* � *kpμp*þ<sup>1</sup>*∂ψpp*

*<sup>n</sup>* � *kpμp*þ1*ξpp*

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

*<sup>n</sup>* � *kpμp*þ1*ψpp*

can be started, and the field expansion coefficients in any desired layer can be obtained. The extinction efficiency is related to the external modified Mie-Lorenz

> *<sup>k</sup>*<sup>2</sup> <sup>ℜ</sup>X<sup>∞</sup> *n*¼1

numerical results of CST 2017 commercial software, and good agreement is

achieved. Moreover, the analytical formulation provides a fast and accurate tool for

In order to realize the priority of the closed-form analytical formulation with respect to the numerical analysis, the simulation times of both methods are included in **Table 2**. Considerable time reduction using the exact solution is evident. Moreover, since 3D meshing and perfectly matched layers are not required in this

Based on the results of Section 3.1, the modified Mie-Lorenz coefficients of the graphene-based spherical particles form infinite summations in terms of spherical Bessel and Hankel functions. In general, graphene plasmons are excited in the subwavelength regime, and only the leading order term of the summation is sufficient for achieving the results with acceptable precision. In this regime, the polynomial expansion of the special functions can also be truncated in the first few terms [22]. Later, the extracted modified Mie-Lorenz coefficients can be rewritten in the form of the polynomials. To further simplify the real-time monitoring and performance

optimization of the graphene-coated nanoparticles, an equivalent RLC circuit can be proposed by representing the rational functions in the continued fraction form as [36]:

The equivalent circuit corresponding to the above representation is shown in

1 *<sup>Z</sup>*<sup>1</sup> <sup>þ</sup> <sup>1</sup> *<sup>Z</sup>*2<sup>þ</sup> <sup>1</sup> *<sup>Z</sup>*3<sup>þ</sup> …

*YTE=TM* ¼ *Y*<sup>0</sup>

*ξpp <sup>n</sup> Bp H ∂ξpp <sup>n</sup> Bp V* " # þ *ψpp <sup>n</sup> D<sup>p</sup> <sup>H</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � � *∂ψpp <sup>n</sup> Dp <sup>V</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � � 2 6 4 3 7 <sup>5</sup> <sup>¼</sup> *<sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>B</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *H <sup>∂</sup>ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>B</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *V* 2 4 3 5 þ *ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> D*ð Þ *<sup>p</sup>*þ<sup>1</sup> *H ∂ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> D*ð Þ *<sup>p</sup>*þ<sup>1</sup> *V* 2 4 3 5 (31) *kp*þ<sup>1</sup> *μ<sup>p</sup>*þ<sup>1</sup> *<sup>∂</sup>ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>B</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *H <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>B</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *V* 2 4 3 5 þ *∂ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> D*ð Þ *<sup>p</sup>*þ<sup>1</sup> *H ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> D*ð Þ *<sup>p</sup>*þ<sup>1</sup> *V* 2 4 3 5 0 @ 1 <sup>A</sup> � *kp μp ∂ξpp <sup>n</sup> <sup>B</sup><sup>p</sup> H ξpp <sup>n</sup> Bp V* " # þ *∂ψpp <sup>n</sup> D<sup>p</sup> <sup>H</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � � *ψpp <sup>n</sup> D<sup>p</sup> <sup>V</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � � 2 6 4 3 7 5 0 B@ 1 CA <sup>¼</sup> �*iωσ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> ξpp <sup>n</sup> <sup>B</sup><sup>p</sup> H ∂ξpp <sup>n</sup> <sup>B</sup><sup>p</sup> V* " # þ *ψpp <sup>n</sup> D<sup>p</sup> <sup>H</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � � *∂ψpp <sup>n</sup> D<sup>p</sup> <sup>V</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � � 2 6 4 3 7 5 0 B@ 1 CA (32)

where *ψpq <sup>n</sup>* ¼ *j <sup>n</sup> kpRq* � � , *ξpq <sup>n</sup>* <sup>¼</sup> *<sup>h</sup>*ð Þ<sup>1</sup> *<sup>n</sup> kpRq* � �, *∂ψpq <sup>n</sup>* <sup>¼</sup> <sup>1</sup> *<sup>ρ</sup>* d *ρj <sup>n</sup>*ð Þ *<sup>ρ</sup>* � �� � *ρ*¼*kpRq* , and *∂ ξpq <sup>n</sup>* ¼ 1 *<sup>ρ</sup>* <sup>d</sup> *<sup>ρ</sup>h*ð Þ<sup>1</sup> *<sup>n</sup>* ð Þ *ρ* h i� � � *ρ*¼*kpRq* (d is defined as a symbol for the derivative with respect to the radial component). By rearranging the above equations, the coefficients of the layer (*p* + 1) can be written in terms of the coefficients of the layer *p* as:

$$
\begin{bmatrix} B\_{H,V}^{(p+1)} \\ D\_{H,V}^{(p+1)} \end{bmatrix} = \begin{bmatrix} \mathbf{1} & R\_{fp}^{H,V} \\ T\_{fp}^{H,V} & T\_{fp}^{H,V} \\ R\_{fp}^{H,V} & \mathbf{1} \\ T\_{Pp}^{H,V} & T\_{Pp}^{H,V} \end{bmatrix} \begin{bmatrix} B\_{H,V}^p \\ D\_{H,V}^p + \delta\_p^1 \end{bmatrix} \tag{33}
$$

where the sub/superscripts *H* and *V* represent the TE and TM waves, respectively. The directions of propagation of these waves are realized thought the subscripts *F* (outgoing waves) and *P* (incoming waves). The effective reflection coefficients are extracted as:

$$R\_{\rm Fp}^{H} = \frac{k\_{p+1}\mu\_p \partial \nu\_n^{(p+1)p} \varphi\_n^{pp} - k\_p \mu\_{p+1} \partial \nu\_n^{pp} \, \nu\_n^{(p+1)p} + \mathcal{g} \, \nu\_n^{pp} \, \nu\_n^{(p+1)p}}{k\_{p+1}\mu\_p \partial \nu\_n^{(p+1)p} \, \xi\_n^{pp} - k\_p \mu\_{p+1} \partial \xi\_n^{pp} \, \nu\_n^{(p+1)p} + \mathcal{g} \, \xi\_n^{pp} \, \nu\_n^{(p+1)p}} \tag{34}$$

$$R\_{Pp}^{H} = \frac{k\_{p+1}\mu\_p \partial \xi\_n^{(p+1)p} \xi\_n^{pp} - k\_p \mu\_{p+1} \partial \xi\_n^{pp} \xi\_n^{(p+1)p} + \mathbf{g}\_n \xi\_n^{pp} \xi\_n^{(p+1)p}}{k\_{p+1}\mu\_p \partial \xi\_n^{(p+1)p} \varphi\_n^{pp} - k\_p \mu\_{p+1} \partial \varphi\_n^{pp} \xi\_n^{(p+1)p} + \mathbf{g}\_n \varphi\_n^{pp} \xi\_n^{(p+1)p}} \tag{35}$$

$$R\_{fp}^V = \frac{k\_{p+1}\mu\_p \nu\_n^{(p+1)p} \partial \nu\_n^{pp} - k\_p \mu\_{p+1} \nu\_n^{pp} \partial \nu\_n^{(p+1)p} + \mathbf{g} \, \partial \nu\_n^{pp} \partial \nu\_n^{(p+1)p}}{k\_{p+1}\mu\_p \nu\_n^{(p+1)p} \, \partial \xi\_n^{pp} - k\_p \mu\_{p+1} \xi\_n^{pp} \partial \nu\_n^{(p+1)p} + \mathbf{g} \, \partial \xi\_n^{pp} \partial \nu\_n^{(p+1)p}} \tag{36}$$

$$\mathcal{R}\_{p\_p}^V = \frac{k\_{p+1}\mu\_p \,\xi\_n^{(p+1)p} \,\partial\xi\_n^{pp} - k\_p \mu\_{p+1} \xi\_n^{pp} \,\partial\xi\_n^{(p+1)p} + \mathbf{g}\,\partial\xi\_n^{pp} \,\partial\xi\_n^{(p+1)p}}{k\_{p+1}\mu\_p \,\xi\_n^{(p+1)p} \,\partial\eta\_n^{pp} - k\_p \mu\_{p+1} \eta\_n^{pp} \,\partial\xi\_n^{(p+1)p} + \mathbf{g}\,\partial\eta\_n^{pp} \,\partial\xi\_n^{(p+1)p}} \tag{37}$$

Moreover, it can be readily shown that the transmission coefficients read as:

$$T\_{Fp}^{H} = \frac{k\_{p+1}\mu\_p \left(\partial\!\!\!\!\!\!/\eta\_n^{(p+1)p}\,\xi\_n^{(p+1)p} - \!\!\!\!\!\!\!\/)^{(p+1)p}\_n \partial\!\!\!\!\!\/)^{(p+1)p}\_n}{k\_{p+1}\mu\_q \partial\!\!\!\/\eta\_n^{(p+1)p}\,\xi\_n^{pp} - k\_p\mu\_{p+1}\partial\!\!\!\!\!\/)^{(p+1)p}\_n \partial\!\!\!\/^{(p+1)p}\_n} + \text{g.t.}\tag{38}$$

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

$$T\_{p\_p}^H = \frac{k\_{p+1}\mu\_p \left(\partial \xi\_n^{(p+1)p} \,\mu\_n^{(p+1)p} - \partial \mu\_n^{(p+1)p} \xi\_n^{(p+1)p}\right)}{k\_{p+1}\mu\_p \partial \xi\_n^{(p+1)p} \,\mu\_n^{pp} - k\_p \mu\_{p+1} \partial \mu\_n^{pp} \xi\_n^{(p+1)p} + \mathbf{g} \,\mu\_n^{pp} \xi\_n^{(p+1)p}} \tag{39}$$

$$T\_{Fp}^V = \frac{k\_{p+1}\mu\_p \left(\nu\_n^{(p+1)p} \partial \xi\_n^{(p+1)p} - \xi\_n^{(p+1)p} \partial \nu\_n^{(p+1)p}\right)}{k\_{p+1}\mu\_p \nu\_n^{(p+1)p} \partial \xi\_n^{pp} - k\_p \mu\_{p+1} \xi\_n^{pp} \partial \nu\_n^{(p+1)p} + \mathbf{g} \ \partial \xi\_n^{pp} \partial \nu\_n^{(p+1)p}} \tag{40}$$

$$T\_{Pp}^V = \frac{k\_{p+1}\mu\_p \left(\xi\_n^{(p+1)p} \partial \mathsf{w}\_n^{(p+1)p} - \mathsf{w}\_n^{(p+1)p} \partial \xi\_n^{(p+1)p}\right)}{k\_{p+1}\mu\_p \xi\_n^{(p+1)p} \partial \mathsf{w}\_n^{pp} - k\_p \mu\_{p+1} \mathsf{w}\_n^{pp} \partial \mathsf{f}\_n^{(p+1)p} + \mathsf{g} \ \partial \mathsf{w}\_n^{pp} \partial \mathsf{f}\_n^{(p+1)p}} \tag{41}$$

where *<sup>g</sup>* <sup>¼</sup> *<sup>i</sup>ωσ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup>μpμp*þ1. By using *BN <sup>H</sup>*,*<sup>V</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>1</sup> *<sup>H</sup>*,*<sup>V</sup>* ¼ 0, the recurrence relations can be started, and the field expansion coefficients in any desired layer can be obtained. The extinction efficiency is related to the external modified Mie-Lorenz coefficients via:

$$Q\_{\rm ext} = \frac{2\pi}{k^2} \Re \sum\_{n=1}^{\infty} (2n+1) \left( B\_V^1 + B\_H^1 \right) \tag{42}$$

where symbol ℜ represents the real part of the summation. In order to verify the extracted coefficients, the extinction efficiencies of three graphene-coated structures is provided in **Figure 11**. In the graphical representation of the structures, the dashed lines illustrate graphene interfaces, while the solid line shows a PEC core. The optical and geometrical parameters are *R*<sup>1</sup> = 200 nm, *R*<sup>2</sup> = 100 nm, *R*<sup>3</sup> = 50 nm, *μ*<sup>c</sup> = 0.3 eV,*T* = 300° K, and *τ* = 0.02 ps. The analytical results are compared with the numerical results of CST 2017 commercial software, and good agreement is achieved. Moreover, the analytical formulation provides a fast and accurate tool for the scattering shaping of various spherical geometries.

In order to realize the priority of the closed-form analytical formulation with respect to the numerical analysis, the simulation times of both methods are included in **Table 2**. Considerable time reduction using the exact solution is evident. Moreover, since 3D meshing and perfectly matched layers are not required in this method, it is efficient in terms of memory as well.

#### *3.1.1 Quasistatic approximation and RLC model*

Based on the results of Section 3.1, the modified Mie-Lorenz coefficients of the graphene-based spherical particles form infinite summations in terms of spherical Bessel and Hankel functions. In general, graphene plasmons are excited in the subwavelength regime, and only the leading order term of the summation is sufficient for achieving the results with acceptable precision. In this regime, the polynomial expansion of the special functions can also be truncated in the first few terms [22]. Later, the extracted modified Mie-Lorenz coefficients can be rewritten in the form of the polynomials. To further simplify the real-time monitoring and performance optimization of the graphene-coated nanoparticles, an equivalent RLC circuit can be proposed by representing the rational functions in the continued fraction form as [36]:

$$Y\_{\rm TE/TM} = Y\_0 \frac{1}{Z\_1 + \frac{1}{Z\_2 + \frac{1}{Z\_3 + \dotsb}}} \tag{43}$$

The equivalent circuit corresponding to the above representation is shown in **Figure 12**.

Therefore, the linear system of equations resulting from the above conditions is:

<sup>5</sup> <sup>¼</sup> *<sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>B</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup>

2 4

> 3 5

> > 3 7 5

1 CA

� �, *∂ψpq*

radial component). By rearranging the above equations, the coefficients of the layer

*R<sup>H</sup>*,*<sup>V</sup> Fp T<sup>H</sup>*,*<sup>V</sup> Fp*

2 4 *Bp H*,*V* 3

*<sup>n</sup> ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> <sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> ∂ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> <sup>∂</sup>ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> <sup>∂</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> <sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*<sup>n</sup> <sup>∂</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>*

*Dp <sup>H</sup>*,*<sup>V</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p*

*<sup>n</sup> <sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>ψ</sup>pp*

*<sup>n</sup> <sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>ξ</sup>pp*

*<sup>n</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>ξ</sup>pp*

*<sup>n</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>ψ</sup>pp*

*<sup>n</sup> <sup>∂</sup>ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>∂</sup>ψpp*

*<sup>n</sup> <sup>∂</sup>ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>∂</sup>ξpp*

*<sup>n</sup> <sup>∂</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup>∂ξpp*

*<sup>n</sup> <sup>∂</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup>∂ψpp*

*<sup>n</sup> <sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* <sup>þ</sup> *<sup>g</sup> <sup>ξ</sup>pp*

1 *T<sup>H</sup>*,*<sup>V</sup> Pp*

where the sub/superscripts *H* and *V* represent the TE and TM waves, respectively. The directions of propagation of these waves are realized thought the subscripts *F* (outgoing waves) and *P* (incoming waves). The effective reflection

1 <sup>A</sup> � *kp μp*

*H <sup>∂</sup>ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>B</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *V*

> 0 B@

*<sup>n</sup>* <sup>¼</sup> <sup>1</sup> *<sup>ρ</sup>* d *ρj <sup>n</sup>*ð Þ *<sup>ρ</sup>* � �� � *ρ*¼*kpRq*

(d is defined as a symbol for the derivative with respect to the

3 5 þ

*∂ξpp <sup>n</sup> <sup>B</sup><sup>p</sup> H*

" #

*ξpp <sup>n</sup> Bp V* 2 4

þ

2 6 4

*ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> D*ð Þ *<sup>p</sup>*þ<sup>1</sup> *H ∂ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> D*ð Þ *<sup>p</sup>*þ<sup>1</sup> *V*

> *∂ψpp <sup>n</sup> D<sup>p</sup>*

*ψpp <sup>n</sup> D<sup>p</sup>* 3

*<sup>H</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � �

*<sup>V</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � �

, and *∂ ξpq*

5 (33)

5 (31)

3 7 5

(32)

*<sup>n</sup>* ¼

(34)

(35)

(36)

(37)

(38)

1 CA <sup>¼</sup>

*ξpp <sup>n</sup> Bp H*

" #

*Nanoplasmonics*

2 4 þ

*<sup>∂</sup>ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>B</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *H <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>B</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *V*

> *ξpp <sup>n</sup> <sup>B</sup><sup>p</sup> H ∂ξpp <sup>n</sup> <sup>B</sup><sup>p</sup> V*

*<sup>n</sup>* ¼ *j*

� � *ρ*¼*kpRq*

coefficients are extracted as:

*RH*

*RH*

*RV*

*RV*

*T<sup>H</sup> Fp* ¼

**48**

0 B@

" #

2 6 4

*ψpp <sup>n</sup> D<sup>p</sup>*

*∂ψpp <sup>n</sup> Dp*

> 3 5 þ

> > þ

*<sup>n</sup> kpRq* � � , *ξpq*

2 4

2 6 4

*<sup>H</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � � 3 7

*∂ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> D*ð Þ *<sup>p</sup>*þ<sup>1</sup> *H ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> D*ð Þ *<sup>p</sup>*þ<sup>1</sup> *V*

> *<sup>H</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � �

*<sup>V</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � �

*<sup>n</sup> kpRq*

1 *T<sup>H</sup>*,*<sup>V</sup> Fp*

*R<sup>H</sup>*,*<sup>V</sup> Pp T<sup>H</sup>*,*<sup>V</sup> Pp*

*<sup>n</sup>* � *kpμ<sup>p</sup>*þ<sup>1</sup>*∂ψpp*

*<sup>n</sup>* � *kpμ<sup>p</sup>*þ<sup>1</sup>*∂ξpp*

*<sup>n</sup>* � *kpμ<sup>p</sup>*þ<sup>1</sup>*∂ξpp*

*<sup>n</sup>* � *kpμ<sup>p</sup>*þ<sup>1</sup>*∂ψpp*

*<sup>n</sup>* � *kpμ<sup>p</sup>*þ<sup>1</sup>*ψpp*

*<sup>n</sup>* � *kpμ<sup>p</sup>*þ<sup>1</sup>*ξpp*

*<sup>n</sup>* � *kpμ<sup>p</sup>*þ<sup>1</sup>*ξpp*

*<sup>n</sup>* � *kpμ<sup>p</sup>*þ<sup>1</sup>*ψpp*

Moreover, it can be readily shown that the transmission coefficients read as:

*kp*þ<sup>1</sup>*μ<sup>p</sup> <sup>∂</sup>ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* � *<sup>ψ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ξ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup>* � �

*<sup>n</sup>* � *kpμ<sup>p</sup>*þ<sup>1</sup>*∂ξpp*

*<sup>n</sup>* <sup>¼</sup> *<sup>h</sup>*ð Þ<sup>1</sup>

(*p* + 1) can be written in terms of the coefficients of the layer *p* as:

*<sup>V</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> *p* � �

> *ψpp <sup>n</sup> D<sup>p</sup>*

*∂ψpp <sup>n</sup> D<sup>p</sup>*

*B*ð Þ *<sup>p</sup>*þ<sup>1</sup> *H*,*V D*ð Þ *<sup>p</sup>*þ<sup>1</sup> *H*,*V*

3 5 ¼

2 4

*Fp* <sup>¼</sup> *kp*þ<sup>1</sup>*μp∂ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>ψ</sup>pp*

*Pp* <sup>¼</sup> *kp*þ<sup>1</sup>*μp<sup>∂</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>ξ</sup>pp*

*Fp* <sup>¼</sup> *kp*þ<sup>1</sup>*μpψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ψpp*

*Pp* <sup>¼</sup> *kp*þ<sup>1</sup>*μ<sup>p</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ξpp*

*kp*þ<sup>1</sup>*μp∂ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>ξ</sup>pp*

*kp*þ<sup>1</sup>*μp<sup>∂</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>ψ</sup>pp*

*kp*þ<sup>1</sup>*μpψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ξpp*

*kp*þ<sup>1</sup>*μ<sup>p</sup> <sup>ξ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>∂</sup>ψpp*

*kp*þ<sup>1</sup>*μq∂ψ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup> <sup>n</sup> <sup>ξ</sup>pp*

*∂ξpp <sup>n</sup> Bp V*

> 0 @

�*iωσ*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup>*

where *ψpq*

*<sup>n</sup>* ð Þ *ρ* h i�

*kp*þ<sup>1</sup> *μ<sup>p</sup>*þ<sup>1</sup>

1 *<sup>ρ</sup>* <sup>d</sup> *<sup>ρ</sup>h*ð Þ<sup>1</sup>

The elements of the equivalent circuit for the TM coefficients read as:

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

<sup>2</sup> <sup>¼</sup> �*σ*0�<sup>1</sup>

*The proposed equivalent circuit for the scattering analysis of electrically small graphene-coated spheres [36].*

In order to illustrate the application of Mie analysis for the graphene-wrapped structures, let us consider vertical and horizontal dipoles in the proximity of a graphene-coated sphere, as shown in **Figure 13**. Although in the Mie analysis, the excitation is considered to be a plane wave, by using the scattering coefficients, the total decay rates can be calculated for the dipole emitters, and it can be proven that the localized surface plasmons of the graphene-wrapped spheres can enhance the total decay rate, which is connected to the Purcell factor [16, 37]. The amount of electric field enhancement for the radial-oriented and tangential oscillating dipoles

*bn*ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>P</sup>*<sup>1</sup>

*<sup>n</sup>* ð Þþ *xd ibnP*<sup>1</sup>

*<sup>n</sup>*ð Þ <sup>0</sup> *<sup>h</sup>*ð Þ<sup>1</sup>

*<sup>n</sup>*ð Þ 0

*<sup>n</sup>* ð Þ *xd xd*

> *xdh*ð Þ<sup>1</sup> *<sup>n</sup>* ð Þ *xd* h i<sup>0</sup>

> > *xd*

9 >=

>;

<sup>1</sup> *<sup>σ</sup>*0�<sup>2</sup> <sup>2</sup> *<sup>σ</sup>*0<sup>2</sup> 3 *<sup>x</sup>*<sup>2</sup> , *<sup>Z</sup>*<sup>0</sup>

<sup>3</sup> <sup>¼</sup> *<sup>x</sup>σ*0<sup>2</sup>

1*σ*0<sup>4</sup> 2*σ*0�<sup>4</sup>

<sup>3</sup> (45)

(46)

(47)

*Y*0

**Figure 12.**

*E E*0

**51**

<sup>¼</sup> <sup>1</sup> <sup>þ</sup>X<sup>∞</sup>

*n*¼1 *i*

*3.1.3 Application in super-scattering*

<sup>0</sup> <sup>¼</sup> *<sup>x</sup>*<sup>3</sup> , *<sup>Z</sup>*<sup>0</sup>

*3.1.2 Application in emission*

<sup>1</sup> ¼ �*σ*<sup>0</sup>

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

with the distance of *xd*, respectively, read as:

*E E*0

*<sup>n</sup>* ð Þ 2*n* þ 1 *n n*ð Þ þ 1

<sup>¼</sup> <sup>1</sup> <sup>þ</sup>X<sup>∞</sup>

8 ><

>:

*n*¼1 *i n*þ1

*anP*<sup>0</sup>

*<sup>n</sup>*ð Þ <sup>0</sup> *<sup>h</sup>*ð Þ<sup>1</sup>

**Figure 13(b)** shows the local field enhancement for the average orientation of the dipole emitter in the vicinity of the sphere with *R*<sup>1</sup> = 20 nm, coated by a graphene material with the chemical potential of *μ<sup>c</sup>* = 0.1 eV. As the figure shows, an enhanced electric field in the order of �104 is obtained for the dipole distance of 1 nm with averaged orientation, and it decreases as the dipole moves away from the sphere.

The possibility of a super-scatterer design using graphene-coated spherical particles is illustrated in **Figure 14**. The design parameters are *ε*<sup>1</sup> = 1.44, *R*<sup>1</sup> = 0.24 μm, and *μ*<sup>c</sup> = 0.3 eV. The structure can be simply analyzed by the modified Mie-Lorenz coefficients. The general design concepts are similar to their cylindrical counterparts, namely, dispersion engineering using the associated planar structure, as shown in the inset of the figure. Due to the excitation of TM surface plasmons, the normalized extinction cross-section is five times greater than the bare dielectric

2*σ*0�<sup>1</sup> <sup>3</sup> , *Z*<sup>0</sup>

#### **Figure 11.**

*The extinction efficiencies of graphene-based particles with different number of layers: (a) two, (b) three, and (c) four [34].*


#### **Table 2.**

*Comparing the simulation time of CST and our codes [34].*

The continued fraction representation for the TM coefficients is:

$$b\_1 = \frac{\mathbf{x}^3}{-\sigma\_2'\sigma\_3'^{-1} + \frac{\mathbf{x}^2}{-\sigma\_1'^{-1}\sigma\_2'^{-2}\sigma\_3'^2 + \frac{\mathbf{x}^2}{\sigma\_1'^2\sigma\_2'^{-4}\sigma\_3'^{-4} + O'(x)}}}\tag{44}$$

where *σ*<sup>0</sup> <sup>0</sup> ¼ *σ*<sup>0</sup> 3 �1 *σ*0 2, *σ*<sup>0</sup> <sup>1</sup> <sup>¼</sup> *<sup>σ</sup>*0�<sup>1</sup> <sup>2</sup> *σ*<sup>0</sup> <sup>5</sup> � *<sup>σ</sup>*0�<sup>2</sup> <sup>2</sup> *σ*<sup>0</sup> 3*σ*0 4, *σ*<sup>0</sup> <sup>2</sup> <sup>¼</sup> *<sup>i</sup> <sup>m</sup>* <sup>3</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>*<sup>g</sup>* <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> ð Þ, *σ*0 <sup>3</sup> <sup>¼</sup> <sup>2</sup>*<sup>m</sup>* <sup>9</sup> �<sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>g</sup>* <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> ð Þ, *<sup>σ</sup>*<sup>0</sup> <sup>4</sup> <sup>¼</sup> *im* <sup>3</sup> �<sup>1</sup> <sup>þ</sup> *<sup>g</sup>* <sup>þ</sup> <sup>2</sup>*gm*<sup>2</sup> <sup>5</sup> <sup>þ</sup> <sup>9</sup>*m*<sup>2</sup> <sup>10</sup> , *<sup>σ</sup>*<sup>0</sup> <sup>5</sup> <sup>¼</sup> <sup>4</sup>*gm* <sup>45</sup> <sup>1</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> ð Þ. *Scattering from Multilayered Graphene-Based Cylindrical and Spherical Particles DOI: http://dx.doi.org/10.5772/intechopen.91427*

**Figure 12.**

*The proposed equivalent circuit for the scattering analysis of electrically small graphene-coated spheres [36].*

The elements of the equivalent circuit for the TM coefficients read as:

$$Z\_0' = \mathfrak{x}^3 \quad , \quad Z\_1' = -\sigma\_2' \sigma\_3'^{-1} \quad , \quad Z\_2' = \frac{-\sigma\_1'^{-1} \sigma\_2'^{-2} \sigma\_3'^2}{\mathfrak{x}^2} \quad , \quad Z\_3' = \mathfrak{x} \sigma\_1'^2 \sigma\_2'^4 \sigma\_3'^{-4} \tag{45}$$

#### *3.1.2 Application in emission*

In order to illustrate the application of Mie analysis for the graphene-wrapped structures, let us consider vertical and horizontal dipoles in the proximity of a graphene-coated sphere, as shown in **Figure 13**. Although in the Mie analysis, the excitation is considered to be a plane wave, by using the scattering coefficients, the total decay rates can be calculated for the dipole emitters, and it can be proven that the localized surface plasmons of the graphene-wrapped spheres can enhance the total decay rate, which is connected to the Purcell factor [16, 37]. The amount of electric field enhancement for the radial-oriented and tangential oscillating dipoles with the distance of *xd*, respectively, read as:

$$\frac{E}{E\_0} = \mathbf{1} + \sum\_{n=1}^{\infty} i^{n+1} b\_n (2n+1) P\_n^1(\mathbf{0}) \frac{h\_n^{(1)}(\mathbf{x}\_d)}{\mathbf{x}\_d} \tag{46}$$

$$\frac{E}{E\_0} = \mathbf{1} + \sum\_{n=1}^{\infty} i^n \frac{(2n+1)}{n(n+1)} \left\{ a\_n P\_n'(\mathbf{0}) h\_n^{(1)}(\mathbf{x}\_d) + i b\_n P\_n^1(\mathbf{0}) \frac{\left[ \mathbf{x}\_d h\_n^{(1)}(\mathbf{x}\_d) \right]'}{\mathbf{x}\_d} \right\} \tag{47}$$

**Figure 13(b)** shows the local field enhancement for the average orientation of the dipole emitter in the vicinity of the sphere with *R*<sup>1</sup> = 20 nm, coated by a graphene material with the chemical potential of *μ<sup>c</sup>* = 0.1 eV. As the figure shows, an enhanced electric field in the order of �104 is obtained for the dipole distance of 1 nm with averaged orientation, and it decreases as the dipole moves away from the sphere.

#### *3.1.3 Application in super-scattering*

The possibility of a super-scatterer design using graphene-coated spherical particles is illustrated in **Figure 14**. The design parameters are *ε*<sup>1</sup> = 1.44, *R*<sup>1</sup> = 0.24 μm, and *μ*<sup>c</sup> = 0.3 eV. The structure can be simply analyzed by the modified Mie-Lorenz coefficients. The general design concepts are similar to their cylindrical counterparts, namely, dispersion engineering using the associated planar structure, as shown in the inset of the figure. Due to the excitation of TM surface plasmons, the normalized extinction cross-section is five times greater than the bare dielectric

The continued fraction representation for the TM coefficients is:

<sup>5</sup> � *<sup>σ</sup>*0�<sup>2</sup> <sup>2</sup> *σ*<sup>0</sup> 3*σ*0 4, *σ*<sup>0</sup>

<sup>3</sup> �<sup>1</sup> <sup>þ</sup> *<sup>g</sup>* <sup>þ</sup> <sup>2</sup>*gm*<sup>2</sup>

<sup>3</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> �*σ*0�<sup>1</sup> <sup>1</sup> *σ*0�<sup>2</sup> <sup>2</sup> *σ*0<sup>2</sup>

**Figure 11(a)** 0.053214 s 32 h, 50 m, 18 s **Figure 11(b)** 0.045831 s 33 h, 45 m, 25 s **Figure 11(c)** 0.151555 s 33 h, 34 m, 55 s

*The extinction efficiencies of graphene-based particles with different number of layers: (a) two, (b) three, and*

<sup>3</sup><sup>þ</sup> *<sup>x</sup> σ*02 1*σ*0<sup>4</sup> <sup>2</sup> *<sup>σ</sup>*0�<sup>4</sup> <sup>3</sup> <sup>þ</sup>*O*0ð Þ *<sup>x</sup>*

<sup>2</sup> <sup>¼</sup> *<sup>i</sup> <sup>m</sup>*

, *σ*<sup>0</sup> <sup>5</sup> <sup>¼</sup> <sup>4</sup>*gm*

**Analytical CST**

<sup>5</sup> <sup>þ</sup> <sup>9</sup>*m*<sup>2</sup> 10

<sup>3</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>*<sup>g</sup>* <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> ð Þ,

<sup>45</sup> <sup>1</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> ð Þ.

(44)

*<sup>b</sup>*<sup>1</sup> <sup>¼</sup> *<sup>x</sup>*<sup>3</sup>

**Structure Simulation time**

�*σ*<sup>0</sup> 2*σ*0�<sup>1</sup>

<sup>1</sup> <sup>¼</sup> *<sup>σ</sup>*0�<sup>1</sup> <sup>2</sup> *σ*<sup>0</sup>

*Comparing the simulation time of CST and our codes [34].*

<sup>4</sup> <sup>¼</sup> *im*

where *σ*<sup>0</sup>

*σ*0 <sup>3</sup> <sup>¼</sup> <sup>2</sup>*<sup>m</sup>*

**50**

**Table 2.**

**Figure 11.**

*(c) four [34].*

*Nanoplasmonics*

<sup>0</sup> ¼ *σ*<sup>0</sup> 3 �1 *σ*0 2, *σ*<sup>0</sup>

<sup>9</sup> �<sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>g</sup>* <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> ð Þ, *<sup>σ</sup>*<sup>0</sup>

**Figure 13.**

*(a) Vertical and horizontal dipole emitters in the proximity of the graphene-coated sphere and (b) the local field enhancement for various dipole distances with averaged orientation [37].*

**Figure 14.**

*(a) Atomically thin super-scatterer and associated planar structure shown in the inset and (b) corresponding normalized scattering cross-sections by considering lossless and lossy graphene shells [38].*

proper design, a single graphene coating can eliminate the dipole resonace in a single reconfigurable frequency. The radius of the sphere is *R*<sup>1</sup> = 100 nm and its core permittivity is *ε*<sup>1</sup> = 3. It can be concluded that double graphene shells can suppress the scattering in the dual frequencies since each graphene shell with different geometrical and optical properties can support localized surface plasmon resonances in a specific frequency. By further increase of the graphene shells, other frequency bands can be generated. **Figure 16(b)** shows the cloaking performance of a spherical particle with multiple graphene shells. The radii of the spheres are 107.5, 131.5, and 140 nm, and the corresponding chemical potentials are 900, 500, and

*(a) Single and (b) multi-frequency cloaking using single/multiple graphene shells around a spherical*

700 meV, respectively. The permittivity of the dielectric filler is 2.1 [21].

As another example, a dielectric-metal core-shell spherical resonator (DMCSR) with the resonance frequency lying in the near-infrared spectrum is considered. In order to increase the optical absorption, the outer layer of the structure is covered with graphene. The localized surface plasmons of graphene are mainly excited in the far-infrared frequencies and in the near-infrared and visible range; it behaves like a dielectric. By hybridizing the graphene with a resonator, its optical absorption can be greatly enhanced. **Figure 17** shows the performance of the structure for

*3.1.6 Application in electromagnetic absorption*

*Wide-band cloaking using graphene disks with varying radii [39].*

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

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

various core radii [15].

**53**

**Figure 15.**

**Figure 16.**

*particle [21].*

sphere. Moreover, similar to the cylindrical super-scatterers, by considering a small amount of loss for the graphene coating by assigning Γ ¼ 0*:*11*meV*, the performance is considerably degraded [38].

### *3.1.4 Application in wide-band cloaking*

By pattering graphene-based disks with various radii around a dielectric sphere, it is feasible to design a wide-band electromagnetic cloak at infrared frequencies. The geometry of this structure is illustrated in **Figure 15**. In order to analyze the proposed cloak by the modified Mie-Lorenz theory, the polarizability of the disks can be inserted in the equivalent conductivity method. The extracted equivalent surface conductivity can be used to tune the surface reactance of the sphere for the purpose of cloaking [39].

### *3.1.5 Application in multi-frequency cloaking*

The other application that can be adapted to our proposed formulation of multilayered spherical structures is multi-frequency cloaking. As **Figure 16** shows, by

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

**Figure 15.** *Wide-band cloaking using graphene disks with varying radii [39].*

**Figure 16.**

sphere. Moreover, similar to the cylindrical super-scatterers, by considering a small amount of loss for the graphene coating by assigning Γ ¼ 0*:*11*meV*, the perfor-

*(a) Atomically thin super-scatterer and associated planar structure shown in the inset and (b) corresponding*

*normalized scattering cross-sections by considering lossless and lossy graphene shells [38].*

*(a) Vertical and horizontal dipole emitters in the proximity of the graphene-coated sphere and (b) the local*

*field enhancement for various dipole distances with averaged orientation [37].*

By pattering graphene-based disks with various radii around a dielectric sphere, it is

The other application that can be adapted to our proposed formulation of multilayered spherical structures is multi-frequency cloaking. As **Figure 16** shows, by

feasible to design a wide-band electromagnetic cloak at infrared frequencies. The geometry of this structure is illustrated in **Figure 15**. In order to analyze the proposed cloak by the modified Mie-Lorenz theory, the polarizability of the disks can be inserted in the equivalent conductivity method. The extracted equivalent surface conductivity can be used to tune the surface reactance of the sphere for the purpose of cloaking [39].

mance is considerably degraded [38].

**Figure 13.**

*Nanoplasmonics*

**Figure 14.**

**52**

*3.1.4 Application in wide-band cloaking*

*3.1.5 Application in multi-frequency cloaking*

*(a) Single and (b) multi-frequency cloaking using single/multiple graphene shells around a spherical particle [21].*

proper design, a single graphene coating can eliminate the dipole resonace in a single reconfigurable frequency. The radius of the sphere is *R*<sup>1</sup> = 100 nm and its core permittivity is *ε*<sup>1</sup> = 3. It can be concluded that double graphene shells can suppress the scattering in the dual frequencies since each graphene shell with different geometrical and optical properties can support localized surface plasmon resonances in a specific frequency. By further increase of the graphene shells, other frequency bands can be generated. **Figure 16(b)** shows the cloaking performance of a spherical particle with multiple graphene shells. The radii of the spheres are 107.5, 131.5, and 140 nm, and the corresponding chemical potentials are 900, 500, and 700 meV, respectively. The permittivity of the dielectric filler is 2.1 [21].

#### *3.1.6 Application in electromagnetic absorption*

As another example, a dielectric-metal core-shell spherical resonator (DMCSR) with the resonance frequency lying in the near-infrared spectrum is considered. In order to increase the optical absorption, the outer layer of the structure is covered with graphene. The localized surface plasmons of graphene are mainly excited in the far-infrared frequencies and in the near-infrared and visible range; it behaves like a dielectric. By hybridizing the graphene with a resonator, its optical absorption can be greatly enhanced. **Figure 17** shows the performance of the structure for various core radii [15].

**References**

[1] Kim K-H, No Y-S. Subwavelength core/ shell cylindrical nanostructures for novel plasmonic and metamaterial devices. Nano Convergence. 2017;**4**(1):1-13

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

anisotropy. Scientific Reports. 2016;**6**:

[12] Correas-Serrano D, Gomez-Diaz JS, Alù A, Melcón AÁ. Electrically and magnetically biased graphene-based cylindrical waveguides: Analysis and applications as reconfigurable antennas. IEEE Transactions on Terahertz Science and Technology. 2015;**5**(6):951-960

[13] Li R et al. Design of ultracompact graphene-based superscatterers. IEEE Journal of Selected Topics in Quantum Electronics. 2016;**23**(1):130-137

Cloaking of single and multiple elliptical cylinders and strips with confocal elliptical nanostructured graphene metasurface. Journal of Physics: Condensed Matter. 2015;**27**(18):185304

resonators. Scientific Reports. 2017;**7**(1):32

[16] Christensen T, Jauho A-P, Wubs M, Mortensen NA. Localized plasmons in graphene-coated nanospheres. Physical

[17] Xiao T-H, Gan L, Li Z-Y. Graphene surface plasmon polaritons transport on curved substrates. Photonics Research.

[18] Falkovsky LA. Optical properties of graphene and IV–VI semiconductors. Physics-Uspekhi. 2008;**51, 9**:887

[19] Hanson GW. Dyadic Green's functions and guided surface waves for

[14] Bernety HM, Yakovlev AB.

[15] Wan M et al. Strong tunable absorption enhancement in graphene using dielectric-metal core-shell

Review B. 2015;**91**(12):125414

2015;**3**(6):300-307

[11] Velichko EA. Evaluation of a graphene-covered dielectric microtube as a refractive-index sensor in the terahertz range. Journal of Optics. 2016;

34775

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

**18**(3):035008

[2] Fan P, Chettiar UK, Cao L, Afshinmanesh F, Engheta N,

Photonics. 2012;**6**(6):380

Physics. 2012;**112**(3):034907

light from subwavelength

[5] Qian C et al. Experimental

2010;**105**(1):013901

2016;**381**:234-239

2011;**19**(6):4815-4826

**55**

Brongersma ML. An invisible metal– semiconductor photodetector. Nature

[3] Padooru YR, Yakovlev AB, Chen P-Y, Alù A. Analytical modeling of conformal mantle cloaks for cylindrical objects using sub-wavelength printed and slotted arrays. Journal of Applied

[4] Ruan Z, Fan S. Superscattering of

nanostructures. Physical Review Letters.

observation of superscattering. Physical Review Letters. 2019;**122**(6):063901

[6] Naserpour M, Zapata-Rodríguez CJ. Tunable scattering cancellation of light using anisotropic cylindrical cavities. Plasmonics. 2017;**12**(3):675-683

[7] Díaz-Aviñó C, Naserpour M, Zapata-Rodríguez CJ. Conditions for achieving invisibility of hyperbolic multilayered nanotubes. Optics Communications.

[9] Monticone F, Argyropoulos C, Alù A. Layered plasmonic cloaks to tailor the optical scattering at the nanoscale. Scientific Reports. 2012;**2**:912

[8] Garcia-Etxarri A et al. Strong magnetic response of submicron silicon particles in the infrared. Optics Express.

[10] Liu W, Lei B, Shi J, Hu H. Unidirectional superscattering by multilayered cavities of effective radial

**Figure 17.**

*Strong tunable absorption using a graphene-coated spherical resonator with fixed dielectric core refractive index of n and silver shell thickness of t [15].*

The provided examples are just a few instances for scattering analysis of graphene-based structures. Based on the derived formulas, other novel optoelectronic devices based on graphene plasmons can be proposed. Moreover, since assemblies of polarizable particles fabricated by graphene exhibit interesting properties such as enhanced absorption, negative permittivity, giant near-field enhancement, and large enhancements in the emission and the radiation of the dipole emitters [40–43], the research can be extended to the multiple scattering theory.
