**3. Graphene-coated spherical structures**

In this section, multilayered graphene-coated particles with spherical morphology 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, and various practical examples are presented.

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

*is started from the outermost layer in order to preserve the consistency with the reference paper [35].*

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

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

*Spherical graphene-dielectric stack (a) 2D and (b) 3D views [34]. Please note that the numbering of the layers*

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]:

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

*VN*ð Þ<sup>1</sup>

Legendre function of order (*n*, *m*), the vector wave functions are defined as follows:

*zn kpr* � �*Pm*

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

*<sup>n</sup>* ð Þ cos *θ <sup>d</sup><sup>θ</sup>* ^*<sup>θ</sup>* <sup>þ</sup>

*mn* <sup>þ</sup> <sup>1</sup> � *<sup>δ</sup>*<sup>1</sup>

*M*ð Þ<sup>1</sup>

*p* � �*D<sup>p</sup>*

*<sup>n</sup>* ð Þ cos *<sup>θ</sup>* ^*<sup>θ</sup>* � *dP<sup>m</sup>*

*<sup>n</sup>* ð Þ cos *θ e*

*im* sin *θ Pm*

� �*<sup>e</sup>*

*mn* � *<sup>i</sup>N*ð Þ<sup>1</sup>

*mn* n o (25)

*<sup>H</sup>Mmn*�*<sup>i</sup>* <sup>1</sup> � *<sup>δ</sup>*<sup>1</sup>

*<sup>n</sup>* , which stand for the spherical Bessel and

*<sup>n</sup>* ð Þ cos *θ <sup>d</sup><sup>θ</sup> <sup>ϕ</sup>*^

� � (27)

*im<sup>ϕ</sup>*^*r*<sup>þ</sup>

^*<sup>r</sup>* � *<sup>E</sup>*ð Þ *<sup>p</sup>* <sup>¼</sup> ^*<sup>r</sup>* � *<sup>E</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> (29)

^*<sup>r</sup>* � <sup>∇</sup> � *<sup>E</sup>*ð Þ *<sup>p</sup>* <sup>¼</sup> *<sup>σ</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> *<sup>p</sup>* ^*<sup>r</sup>* � ^*<sup>r</sup>* � *<sup>E</sup>*ð Þ *<sup>p</sup>* � � (30)

*<sup>n</sup>* ð Þ cos *<sup>θ</sup> <sup>ϕ</sup>*^

*imϕ*

*p* � �*D<sup>p</sup>*

*<sup>n</sup>* as the associated

*VNmn*<sup>o</sup>

(26)

(28)

can be simply attained as the special case of our formulation.

X *n*

�

� *<sup>i</sup>* <sup>1</sup> � *<sup>δ</sup><sup>N</sup> p* � �*Bp*

X *m i*

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

Hankel functions of the first kind with order *n*, respectively, and *Pm*

*im<sup>ϕ</sup> im* sin *θ Pm*

*dP<sup>m</sup>*

*iω μ<sup>p</sup>*

*E<sup>i</sup>* ¼ *E*<sup>0</sup>

*<sup>n</sup>* 2*n* þ 1 *n n*ð Þ þ 1

� � <sup>¼</sup> *zn kpr* � � *<sup>e</sup>*

� � <sup>¼</sup> *n n*ð Þ <sup>þ</sup> <sup>1</sup> *kpr*

*d rzn kpr* � � � � *dr*

^*<sup>r</sup>* � <sup>∇</sup> � *<sup>E</sup>*ð Þ *<sup>p</sup>*þ<sup>1</sup> � <sup>1</sup>

*Nmn kp*

1 *kpr*

*Ep scat* ¼ *E*<sup>0</sup>

**Figure 10.**

X *n*

*mn* <sup>n</sup>

*Mmn kp*

of adjacent layers read as:

1 *iω μ<sup>p</sup>*þ<sup>1</sup>

**47**

<sup>1</sup> � *<sup>δ</sup><sup>N</sup> p* � �*Bp*

X *m i*

*HM*ð Þ<sup>1</sup>

By considering *zn* as either *j*
