In this paper, some generalizations of electromagnetic scattering problems by elementary shapes are presented. In particular, the aim of the paper is to provide solutions to the scattering problem by multiple objects with simple shapes, either in concentric configuration or arbitrarily distributed in the space. The vector harmonics, representing the fields, and their properties are applied in order to solve five different problems: the electromagnetic scattering by an infinitely long circular stratified cylinder, by a multilayered sphere, by an ensemble of parallel cylinders, by an ensemble of multi-spheres, and ultimately by a sphere embedded in a circular cylinder. Numerical results in particularly important configurations are shown.
1. INTRODUCTION
In the first part of this tutorial [1], canonical scattering problems of a plane wave by a circular cylinder and a sphere were presented. The vector harmonics, also known as vector wave functions, are applied [2] for this purpose. These harmonics simplify the representation of the fields for a better readability and, more importantly, make the imposition easy of the boundary conditions. In this second part of the tutorial, the objective is to show how the use of the vector harmonics leads to the solution to scattering problems in the cases in which multiple targets are present.
The solution to scattering problems and its potential extension for scenarios related to multiple targets was established in the early stages of scattering literature for its relevant scientific interest. In fact, the case of electromagnetic radiation interacting with a single object with a canonical shape is rare—more often the radiation interacts with several objects. Sometimes, it is also possible to idealize all the objects with the same canonical shape; in the opposite case, several different shapes must be considered. While the scattering by several objects with the same shape was widely analyzed in the literature [3–13], the case of scatterers with different shapes was less studied [14–18]. The reason can be found in the difficulties faced in obtaining an analytical solution, even when simple shapes are considered. In the literature, solutions to the scattering by objects with different shapes were tackled, both numerically and analytically. In the last case, only solutions in the form of an integral equation were found with an exception for the simplest case of a sphere embedded in a cylinder [19–21]. In the last years, the scattering radiation by an inhomogeneous electromagnetic wave was also introduced, to take into account the loss of media where the scatterers were immersed, since in nature most of the materials contain water [22,23].
The list of physical scenarios where several scatterers could be considered is endless: in the biomedical case, for example, a single human body or multiple human bodies can be represented as a combination of cylinders and spheres. Then biological tissues, such as veins and capillaries, can be represented as multiple cylinders and red globules in a vein as small spheres inside a cylinder and so on [24–27]. Applications in civil engineering are other good examples, such as the several pipelines and metal filaments present for an electromagnetic radar applied to inspect a street or a wall, just to name a few [28–31]. Moreover, several applications of electromagnetic sensing can be found in the remote inspection of heritage conservation on statues with multiple layers or on ancient buildings with several columns [32,33]. Finally, in the field of optics and spectroscopy, the electromagnetic scattering approach was used by many researchers for their studies [34–37].
Scattering problems by multiple objects can be classified into two different classes: the case of concentric objects and the case of non-concentric objects. In several applications, it is possible to model scatterers as concentric objects of specific shapes, e.g., a pipeline can be seen as a cylinder with multiple layers, the cover and the inner part, just to cite the main ones. While the scattering by concentric objects with a simple shape is relatively simple to solve when the scattering by a single object is known, scattering by non-concentric objects arbitrarily displaced in the space is a hard task to find. When the electromagnetic fields are expressed in the vector harmonics formalism, each vector harmonic must be expressed in a referenced frame with the origin on the object’s center to apply the boundary condition on each scatterer. For this purpose, special transformations of the vector harmonics must be introduced; such transformations are known as addition theorems, and they will be presented in the body of the paper. The solution to a scattering problem by multiple objects with different shapes is even more difficult. In this case, the field can be decomposed into single scattered fields for each object, and thus, each field is represented by vector harmonics of different kinds, e.g., cylindrical or spherical. Moreover, these harmonics must be converted into one another to impose the boundary conditions. To obtain this result, we introduce special expansions to represent cylindrical vector harmonics as a function of spherical ones and vice versa.
The structure of the paper is as follows: in Section 2, the scattering by concentric objects is introduced, in particular, in Section 2.A, the scattering by a multilayered cylinder is analyzed, while in Section 2.B, the scattering by a multilayered sphere is considered; in Section 3, the case of scattering by objects arbitrarily displaced in space is illustrated, in particular, in Section 3.A, the scattering by parallel circular cylinders is considered, while in Section 3.B, the scattering by several spheres arbitrarily displaced in space is presented. In Section 4, the scattering by a sphere embedded into a circular cylinder is described. In each section, numerical results are also shown. Finally, in Section 5, the conclusions are drawn.
2. MULTILAYER SCATTERER
This section deals with the scattering problems for stratified objects. In particular, two cases are presented: scattering by a multilayer cylinder and scattering by a multilayer sphere. In both cases, the vector harmonics formalism is applied, and the scattering coefficients are determined by the transfer-matrix method [38]. Moreover, in both cases, the incident field is supposed to be an elliptically polarized plane wave with an arbitrary direction of propagation. Throughout this paper, a time dependence ${e^{- i\omega t}}$ is assumed and always omitted, $\omega$ being the angular frequency of the incident field. Moreover, the scatterers will always be considered in free space, filled with a medium with relative electric permittivity ${\varepsilon _e}$, relative magnetic permeability ${\mu _e}$, and conductivity ${\sigma _e}$.
A. Multiconcentric Cylinders
The scenario in Fig. 1 is taken into consideration. A multilayer cylinder with $N$ layers placed in free space is illustrated. Each layer is characterized by the radius of its external surface ${a_j}$, its relative electric permittivity ${\varepsilon _j}$, its relative magnetic permeability ${\mu _j}$, and its electric conductivity ${\sigma _j}$, with $j = 1, \ldots,N$. We name ${k_j}$ the propagation constant of the $j$-th layer. The most external layer of the cylinder is assumed to be the first, and the subscript “$i$” is used for the external electromagnetic parameters.
The incident field is an elliptically polarized plane wave, forming an angle ${\vartheta _i}$ with the $z$ axis, and an angle ${\varphi _i}$ with the $x$ axis. Without loss of generality, we suppose the $z$ component of the propagation vector to be positive. An elliptically polarized plane wave can be expanded in cylindrical vector harmonics [1,8,39,40]:
(1)$${{\bf E}_i}({k_i}{\bf r}) = \sum\limits_{m = - \infty}^{+ \infty} \left[{{a_m}{\bf M}_m^{(1)}({k_i}{\bf r}) + {b_m}{\bf N}_m^{(1)}({k_i}{\bf r})} \right],$$
where (2)$${a_m} = \frac{{{E_{{hi}}}}}{{{k_{{i\rho}}}}}{i^{m + 1}}{e^{- im{\varphi _i}}}\quad {b_m} = - \frac{{{E_{{vi}}}}}{{{k_{{i\rho}}}}}{i^m}{e^{- im{\varphi _i}}},$$
(3)$${k_{{iz}}} = {k_i}\cos {\vartheta _i}\quad {k_{{i\rho}}} = {k_i}\sin {\vartheta _i},$$
and with (4)$${{\bf M}_m}({\bf r}) = {{\bf m}_m}({k_{{i\rho}}}\rho){e^{{im\varphi}}}{e^{i{k_{{iz}}}z}},$$
(5)$${{\bf N}_m}({\bf r}) = {{\bf n}_m}({k_{{i\rho}}}\rho){e^{{im\varphi}}}{e^{i{k_{{iz}}}z}},$$
where the radial-dependent vectors can be written as follows: (6)$${{\bf m}_m}({k_{{i\rho}}}\rho) = im\frac{{{Z_m}({k_{{i\rho}}}\rho)}}{\rho}{\boldsymbol \rho} - {k_{{i\rho}}}\frac{{\partial {Z_m}({k_{{i\rho}}}\rho)}}{{\partial \rho}}{\boldsymbol \varphi},$$
(7)$$\begin{split}{{\bf n}_m}({k_{{i\rho}}}\rho) &= i\frac{{{k_{{iz}}}{k_{{i\rho}}}}}{k}\frac{{\partial {Z_m}({k_{{i\rho}}}\rho)}}{{\partial \rho}}{\boldsymbol \rho} - \frac{{m{k_{{iz}}}}}{k_i}\frac{{{Z_m}({k_{{i\rho}}}\rho)}}{\rho}{\boldsymbol \varphi} \\&\quad+ \frac{{k_{{i\rho}}^2}}{k_i}{Z_m}({k_{{i\rho}}}\rho){\boldsymbol z}.\end{split}$$
The function ${Z_m}({k_\rho}\rho)$ represents the generic Bessel function of the first, second, third, or fourth kind, i.e., ${Z_m}({k_\rho}\rho) = \{{J_m}({k_\rho}\rho),{Y_m}({k_\rho}\rho),H_m^{(1)}({k_\rho}\rho),H_m^{(2)}({k_\rho}\rho)\}$, in particular, the two last functions are also known as Hankel functions of the first and second kinds, respectively [41]. Furthermore, the use of the apices (1), (2), (3), and (4) was to represent the vector harmonics (${{\bf M}_m}$ and ${{\bf N}_m}$) containing the Bessel functions ${J_m}$, ${Y_m}$, $H_m^{(1)}$, and $H_m^{(2)}$, respectively. For more details, see the first part of the tutorial [1]. For the sake of brevity, it is convenient to write the radial-dependent vectors in a more compact way as follows: (8)$${{\bf m}_m}({k_{{i\rho}}}\rho) = {m_\rho}({k_{{i\rho}}}\rho){\boldsymbol \rho} + {m_\varphi}({k_{{i\rho}}}\rho){\boldsymbol \varphi},$$
(9)$${{\bf n}_m}({k_{{i\rho}}}\rho) = {n_\rho}({k_{{i\rho}}}\rho){\boldsymbol \rho} + {n_\varphi}({k_{{i\rho}}}\rho){\boldsymbol \varphi} + {n_z}({k_{{i\rho}}}\rho){\boldsymbol z}.$$
The interaction between the incident wave and the cylinder can be taken into account by considering a scattered wave in free space and an internal wave in the first layer of the cylinder. The scattered wave can be written as a series of cylindrical vector harmonics:
(10)$${{\bf E}_s}\!({k_i}{\bf r}) = \sum\limits_{m = - \infty}^{+ \infty} \left[{{e_m}{\bf M}_m^{(3)}({k_i}{\bf r}) + {f_m}{\bf N}_m^{(3)}({k_i}{\bf r})} \right].$$
Concerning the internal waves, in each layer, incoming waves toward the center and outgoing waves from the center of the cylinder will be present. Due to the linearity of the problem, again, the superposition of these waves can be written as a series of cylindrical vector harmonics. If the $j$-th cylinder is considered, the field inside the layer can be formulated as follows: (11)$$\begin{split}{{\bf E}_j}({k_j}{\bf r}) &= \sum\limits_{m = - \infty}^{+ \infty} [r_m^j{\bf M}_m^{(1)}({k_j}{\bf r}) + s_m^j{\bf N}_m^{(1)}({k_j}{\bf r})]\\[-3pt]&\quad + \sum\limits_{m = - \infty}^{+ \infty} [u_m^j{\bf M}_m^{(2)}({k_j}{\bf r}) + v_m^j{\bf N}_m^{(2)}({k_j}{\bf r})],\end{split}$$
with $j = 1, \ldots,N - 1$. Moreover, the $j$-th field was expressed as a superposition of the Bessel spherical functions of first and second kinds ${J_n}$ and ${Y_n}$, respectively, because they are stationary waves and the point $\rho = 0$ is external to the considered domain. For this reason, the last internal field must be represented as (12)$${{\bf E}_N}({k_N}{\bf r}) = \sum\limits_{m = - \infty}^{+ \infty} \left[{r_m^N{\bf M}_m^{(1)}({k_N}{\bf r}) + s_m^N{\bf N}_m^{(1)}({k_N}{\bf r})} \right].$$
Before going forward, observe once again that the choice made for the vector harmonics is important: in Eq. (10), the radial dependence of the harmonics is of a Hankel function type; this is the result of the external wave going towards free space with no reflections so that the wave is analogous to a progressive wave.On the other hand, in Eqs. (11) and (12), the radial dependence is of a Bessel function type of the first and second kinds, because in the layers, both the direct and reflected waves propagate, and the result of their superposition is analogous to a stationary wave. These analogies are the same that arise between the Bessel and Hankel functions and the trigonometric and imaginary exponential functions.At this point, all the electric fields that contribute to the problem as a function of vector harmonics are represented, and the boundary conditions on the surfaces of the $N$ layers can be imposed. Initially, we consider the $j$-th layer, and on this surface ($\rho = {a_j}$), we impose the continuity of the tangential components of the fields:
(13)$$\left({{{\bf E}_{i_j}} + {{\bf E}_{s_j}} - {{\bf E}_{{i_{j + 1}}}} - {{\bf E}_{{s_{j + 1}}}}} \right) \times {{\boldsymbol \rho}_{\!i}} = 0\quad \text{for}\;\rho = {a_{\!j}},$$
(14)$$\left({{{\bf H}_{i_j}} + {{\bf H}_{s_j}} - {{\bf H}_{{i_{j + 1}}}} - {{\bf H}_{{s_{j + 1}}}}} \right) \times {{\boldsymbol \rho}_{\!i}} = 0\quad \text{for}\;\rho = {a_{\!j}}.$$
Due to the proportionality between the electric field and the curl of the magnetic field, both equations can be expressed as a function of the electric fields: (15)$$\left({{{\bf E}_{i_j}} + {{\bf E}_{s_j}} - {{\bf E}_{{i_{j + 1}}}} - {{\bf E}_{{s_{j + 1}}}}} \right) \times {{\boldsymbol \rho}_{\!i}} = 0\quad \text{for}\;\rho = {a_{\!j}},$$
(16)$$\!\!\!\big[{\nabla \times ({{\bf E}_{i_j}} + {{\bf E}_{s_j}} - {{\bf E}_{{i_{j + 1}}}} - {{\bf E}_{{s_{j + 1}}}})} \big] {{\boldsymbol \rho}_{\!i}} = 0\quad \text{for}\;\rho = {a_{\!j}}.\!$$
Considering the orthogonal properties of the vector harmonics (see [1,42]) and Eqs. (8) and (9), we can obtain (17)$${{\bf m}_m}({k_\rho}\rho) \times {\boldsymbol \rho} = {m_{\varphi m}}({k_\rho}\rho){\boldsymbol z},$$
(18)$${{\bf n}_m}({k_\rho}\rho) \times {\boldsymbol \rho} = {n_{\varphi m}}({k_\rho}\rho){\boldsymbol z} - {n_{{zm}}}({k_\rho}\rho){\boldsymbol \varphi},$$
(19)$$\left[{\nabla \times {{\bf m}_m}({k_\rho}\rho)} \right] \times {\boldsymbol \rho} = k{n_{\varphi m}}({k_\rho}\rho){\boldsymbol z} - k{n_{{zm}}}({k_\rho}\rho){\boldsymbol \varphi},$$
(20)$$\left[{\nabla \times {{\bf n}_m}({k_\rho}\rho)} \right] \times {\boldsymbol \rho} = - k{m_{\varphi m}}({k_\rho}\rho){\boldsymbol z}.$$
For more details on the orthogonal properties of the vector harmonics, see the first part of the tutorial [1]. Using the cited properties and Eq. (11) in Eqs. (15) and (16), the terms of the boundary conditions on the cylindrical surface between the $j$-th and $j + 1$-th layers can be represented as follows: (21)$$\left\{{\begin{split}& s_m^{j + 1}n_{\varphi m}^{(1)}({k_{j + 1}}{\rho _j}) + v_m^{j + 1}n_{\varphi m}^{(2)}({k_{j + 1}}{\rho _j}) + r_m^{j + 1}m_{\varphi m}^{(1)}({k_{j + 1}}{\rho _j}) + u_m^{j + 1}m_{\varphi m}^{(2)}({k_{j + 1}}{\rho _j})\\&\quad = s_m^jn_{\varphi m}^{(1)}({k_j}{\rho _j}) + v_m^jn_{\varphi m}^{(2)}({k_j}{\rho _j}) + r_m^jm_{\varphi m}^{(1)}({k_j}{\rho _j}) + u_m^jm_{\varphi m}^{(2)}({k_j}{\rho _j})\\&{k_{j + 1}}[- s_m^{j + 1}m_{\varphi m}^{(1)}({k_{j + 1}}{\rho _j}) - v_m^{j + 1}m_{\varphi m}^{(2)}({k_{j + 1}}{\rho _j}) + r_m^{j + 1}n_{\varphi m}^{(1)}({k_{j + 1}}{\rho _j}) + u_m^{j + 1}n_{\varphi m}^{(2)}({k_{j + 1}}{\rho _j})] \\&\quad = {k_j}[- s_m^jm_{\varphi m}^{(1)}({k_j}{\rho _j}) - v_m^jm_{\varphi m}^{(2)}({k_j}{\rho _j}) + r_m^jn_{\varphi m}^{(1)}({k_j}{\rho _j}) + u_m^jn_{\varphi m}^{(2)}({k_j}{\rho _j})]\\& s_m^{j + 1}n_{{zm}}^{(1)}({k_{j + 1}}{\rho _j}) + v_m^{j + 1}n_{{zm}}^{(2)}({k_{j + 1}}{\rho _j}) = s_m^jn_{{zm}}^{(1)}({k_j}{\rho _j}) + v_m^jn_{{zm}}^{(2)}({k_j}{\rho _j})\\&{k_{j + 1}}[r_m^{j + 1}n_{{zm}}^{(1)}({k_{j + 1}}{\rho _j}) + u_m^{j + 1}n_{{zm}}^{(2)}({k_{j + 1}}{\rho _j})] = {k_j}[r_m^jn_{{zm}}^{(1)}({k_j}{\rho _j}) + u_m^jn_{{zm}}^{(2)}({k_j}{\rho _j})].\end{split}} \right.$$
Now, the linear system in the matrix formalism can be introduced:(22)$$\begin{split}&\left[{\begin{array}{cccc}{n_{{zm}}^{(1)}({k_{j + 1}}{\rho _j})}&{n_{{zm}}^{(2)}({k_{j + 1}}{\rho _j})}&0&0\\{n_{\varphi m}^{(1)}({k_{j + 1}}{\rho _j})}&{n_{\varphi m}^{(2)}({k_{j + 1}}{\rho _j})}&{m_{\varphi m}^{(1)}({k_{j + 1}}{\rho _j})}&{m_{\varphi m}^{(2)}({k_{j + 1}}{\rho _j})}\\{- {\zeta _j}m_{\varphi m}^{(1)}({k_{j + 1}}{\rho _j})}&{- {\zeta _j}m_{\varphi m}^{(2)}({k_{j + 1}}{\rho _j})}&{{\zeta _j}n_{\varphi m}^{(1)}({k_{j + 1}}{\rho _j})}&{{\zeta _j}n_{\varphi m}^{(2)}({k_{j + 1}}{\rho _j})}\\0&0&{{\zeta _j}n_{{zm}}^{(1)}({k_{j + 1}}{\rho _j})}&{{\zeta _j}n_{{zm}}^{(2)}({k_{j + 1}}{\rho _j})}\end{array}} \right]\left({\begin{array}{c}{s_m^{j + 1}}\\{v_m^{j + 1}}\\{r_m^{j + 1}}\\{u_m^{j + 1}}\end{array}} \right) \\&\quad = \left[{\begin{array}{cccc}{n_{{zm}}^{(1)}({k_j}{\rho _j})}&{n_{{zm}}^{(2)}({k_j}{\rho _j})}&0&0\\{n_{\varphi m}^{(1)}({k_j}{\rho _j})}&{n_{\varphi m}^{(2)}({k_j}{\rho _j})}&{m_{\varphi m}^{(1)}({k_j}{\rho _j})}&{m_{\varphi m}^{(2)}({k_j}{\rho _j})}\\{- m_{\varphi m}^{(1)}({k_j}{\rho _j})}&{- m_{\varphi m}^{(2)}({k_j}{\rho _j})}&{n_{\varphi m}^{(1)}({k_j}{\rho _j})}&{n_{\varphi m}^{(2)}({k_j}{\rho _j})}\\0&0&{n_{{zm}}^{(1)}({k_j}{\rho _j})}&{n_{{zm}}^{(2)}({k_j}{\rho _j})}\end{array}} \right]\left({\begin{array}{c}{s_m^j}\\{v_m^j}\\{r_m^j}\\{u_m^j}\end{array}} \right),\end{split}$$
having indicated with ${\zeta _j} = {k_{j + 1}}/{k_j}$. Imposing $[A]$ for the first matrix and $[B]$ for the matrix on the second side of the previous equation, the following is obtained: (23)$$\left({\begin{array}{c}{s_m^{j + 1}}\\{v_m^{j + 1}}\\{r_m^{j + 1}}\\{u_m^{j + 1}}\end{array}} \right) = \left[{\begin{array}{c}{{A^{- 1}}B}\end{array}} \right]\left({\begin{array}{c}{s_m^j}\\{v_m^j}\\{r_m^j}\\{u_m^j}\end{array}} \right) = \left[{\begin{array}{c}M\end{array}} \right]\left({\begin{array}{c}{s_m^j}\\{v_m^j}\\{r_m^j}\\{u_m^j}\end{array}} \right).$$
At this stage, the transmission from the more external layers towards the more internal ones can be analyzed. Considering the following transmitted field as (24)$$\left({\begin{array}{c}{s_m^N}\\0\\{r_m^N}\\0\end{array}} \right) = \prod\limits_{i = N - 1}^1 [{M_i}]\left({\begin{array}{c}{e_m}\\{a_m}\\{f_m}\\{b_m}\end{array}} \right),$$
where $v_m^N$, $u_m^N = 0$ because in the last layer $\rho = 0$, the linear system becomes (25)$$\left\{\begin{split}{M_{11}}{e_m} + {M_{12}}{a_m} + {M_{13}}{f_m} + {M_{14}}{b_m} &= s_{{mn}}^N\\{M_{21}}{e_m} + {M_{22}}{a_m} + {M_{23}}{f_m} + {M_{24}}{b_m} &= 0\\{M_{31}}{e_m} + {M_{32}}{a_m} + {M_{33}}{f_m} + {M_{34}}{b_m} &= r_{{mn}}^N\\{M_{41}}{e_m} + {M_{42}}{a_m} + {M_{43}}{f_m} + {M_{44}}{b_m} &= 0.\end{split} \right.$$
Solving the above linear system, the unknown scattering coefficients ${e_m}$ and ${f_m}$ are obtained. These coefficients, used in Eq. (10), can lead to the scattered electric field.In Fig. 2(a), the amplitude of the scattered electric field as a function of frequency in the visible range is represented; the incident wave is a linearly polarized plane wave in the $ y $ direction, propagating along the $ z $ direction, with a magnitude 1 V/m; the scattered field is evaluated at the point $(2{a_1},0,0)$, with ${a_1} = 200\; \text{nm}$. A two-layer cylinder, with internal radius ${a_2} = {a_1}/2$, and with electromagnetic parameters ${\mu _e} = {\mu _1} = {\mu _2} = 1$, ${\sigma _e} = {\sigma _1} = {\sigma _2} = 0\;\text{S}/\text{m}$, ${\varepsilon _e} = {\varepsilon _2} = 1$, and ${\varepsilon _1} = 3$ (solid line), and a one-layer cylinder, i.e., without the core layer (dashed line), are considered. In particular, it is shown how the presence of the core in the cylinder can vary the scattered electric field. Figure 2(b) shows the trend of the electric field when the dielectric permittivity of the shell changes from two to six (all other parameters remain unchanged). As noticed, there is a decrease in the maximum value of the diffused electric field, due to the higher value of the dielectric constant of the shell.
Figure 3 shows the scattered electric field distribution calculated as $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ at an incident wavelength of 600 nm and ${\varepsilon _1} = 3$; all other parameters remain unchanged compared to the previous case.
B. Multiconcentric Spheres
The geometry of the problem is depicted in Fig. 4. An elliptically polarized plane wave impinges on a multilayer sphere. Both the external medium and the media of the layers are dissipative. Similar to the previous case, the sphere is characterized by an external radius ${a_1}$ and by an undefined number $N$ of layers, each defined by a radius ${a_j}$, with $j = 1, \ldots,N$. The most general form of an elliptically polarized plane traveling wave as a function of the spherical unit vectors is
(26)$${{\bf E}_i}({\bf r}) = \left({E_i^H{{\boldsymbol \vartheta}_0} + E_i^E{{\boldsymbol \varphi}_0}} \right){e^{i{{\bf k}_i} \cdot {\bf r}}},$$
with (27)$${{\bf k}_i} = {k_i}\left({\sin {\vartheta _i}\cos {\varphi _i} {{\bf x}_i} + \sin {\vartheta _i}\sin {\varphi _i} {{\bf y}_i} + \cos {\vartheta _i} {{\bf z}_i}} \right),$$
(28)$${{\boldsymbol \vartheta}_0} = \cos {\vartheta _i}\cos {\varphi _i} {{\bf x}_i} + \cos {\vartheta _i}\sin {\varphi _i} {{\bf y}_i} - \sin {\vartheta _i} {{\bf z}_i},$$
(29)$${{\boldsymbol \varphi}_0} = {-}\! \sin {\varphi _i} {{\bf x}_i} + \cos {\varphi _i} {{\bf y}_i},$$
where ${{\bf x}_i}$, ${{\bf y}_i}$, ${{\bf z}_i}$ are the Cartesian unit vectors, while $E_i^E$ and $E_i^H$ are the parallel ($ E $) and perpendicular ($ H $) polarizations of the incident electric field, respectively. The wave vector of the radiation forms an angle ${\vartheta _i}$ with the $z$ axis, and its projection on the interface forms an angle ${\varphi _i}$ with the $x$ axis. In the case of the external medium being dissipative, the angle ${\vartheta _i}$ would be complex. Let ${\varepsilon _i}$, ${\mu _i}$, ${k_i}$ be the relative permittivity, relative permeability, and the wavenumber of the environment, respectively, while ${\varepsilon _j}$, ${\mu _j}$, ${k_j}$ are the relative permittivity, relative permeability, and the wavenumber of each spherical layer ($j = 1, \ldots,N$), respectively. The elliptically polarized plane wave can be expanded in spherical vector harmonics, and it will assume the following form [1,8,39,40]: (30)$$\begin{split}{{\bf E}_i}({\bf r}) &= \left({E_i^H{{\boldsymbol \vartheta}_0} + E_i^E{{\boldsymbol \varphi}_{0i}}} \right){e^{i{k_i}\left({\sin {\vartheta _i}{{\bf x}_i} + \cos {\vartheta _i}{{\bf z}_i}} \right)}}\\ &= \sum\limits_{n = 1}^\infty \sum\limits_{m = - n}^n \left[{{a_{{mn}}}{\bf M}_{{mn}}^{(1)}({k_i}{\bf r}) + {b_{{mn}}}{\bf N}_{{mn}}^{(1)}({k_i}{\bf r})} \right],\end{split}$$
with (31)$${\bf M}_{{mn}}^{(1)}({k_i}{\bf r}) = {j_n}({k_i}r){{\bf m}_{{mn}}}(\vartheta ,\varphi),$$
(32)$${\bf N}_{{mn}}^{(1)}({k_i}{\bf r}) = \frac{{{j_n}({k_i}r)}}{{{k_i}r}}{{\bf p}_{{mn}}}(\vartheta ,\varphi) + \frac{1}{{{k_i}r}}\frac{{d[r{j_n}({k_i}r)]}}{{dr}}{{\bf n}_{{mn}}}(\vartheta ,\varphi),$$
where the apex (1) points out that the radial dependence is represented by the spherical Bessel functions of the first kind ${j_n}$. The tesseral vectorial harmonics [1] (also known as vector spherical harmonics) can be expressed as follows: (33)$${{\bf m}_{{mn}}}(\vartheta ,\varphi) = {e^{{im\varphi}}}\left[{i\pi _n^m(\cos \vartheta){{\boldsymbol \vartheta}_i} - \tau _n^m(\cos \vartheta){{\boldsymbol \varphi}_i}} \right],$$
(34)$${{\bf n}_{{mn}}}(\vartheta ,\varphi) = {e^{{im\varphi}}}\left[{\tau _n^m(\cos \vartheta){{\boldsymbol \vartheta}_i} + i\pi _n^m(\cos \vartheta){{\boldsymbol \varphi}_i}} \right],$$
(35)$${{\bf p}_{{mn}}}(\vartheta ,\varphi) = {e^{{im\varphi}}}n(n + 1)P_n^m(\cos \vartheta){{\bf r}_i},$$
where $P_n^m(\cos \vartheta)$ are the associated Legendre functions, and the azimuthal dependence is expressed by the following functions, sometimes called tesseral scalar harmonics: (36)$$\pi _n^m(\cos \vartheta) = \frac{m}{{\sin \vartheta}}P_n^m(\cos \vartheta),$$
(37)$$\tau _n^m(\cos \vartheta) = \frac{{dP_n^m(\cos \vartheta)}}{{d\vartheta}}.$$
Applying the procedure explained in [8,17], the following expressions for the expansion coefficients are reached: (38)$${a_{{mn}}} = {i^n}\frac{{2n + 1}}{{n(n + 1)}}\frac{{(n - m)!}}{{(n + m)!}}\big({E_i^H {{\boldsymbol \vartheta}_0} + E_i^E {{\boldsymbol \varphi}_0}} \big) \cdot {{\bf m}_{{mn}}}({\vartheta _i},{\varphi _i}),$$
(39)$${b_{{mn}}} = {i^{n - 1}}\frac{{2n + 1}}{{n(n + 1)}}\frac{{(n - m)!}}{{(n + m)!}}\big({E_{\!i}^H {{\boldsymbol \vartheta}_0} + E_{\!i}^E {{\boldsymbol \varphi}_0}} \big) \cdot {{\bf n}_{{mn}}}({\vartheta _i},{\varphi _i}),$$
with (40)$$\begin{split}{{\bf m}_{{mn}}}({\vartheta _i},{\varphi _i}) = {e^{im{\varphi _i}}}\left[{i\frac{m}{{\sin {\vartheta _i}}}P_{\!n}^m(\cos {\vartheta _i}){{\boldsymbol \vartheta}_i} - \frac{{\textit{dP}_{\!n}^m(\cos {\vartheta _i})}}{{d\vartheta}}{{\boldsymbol \varphi}_i}} \right]\!,\\\end{split}$$
(41)$$\begin{split}{{\bf n}_{{mn}}}({\vartheta _i},{\varphi _i}) = {e^{im{\varphi _i}}}\!\left[\!{\frac{{\textit{dP}_n^m(\cos {\vartheta _i})}}{{d\vartheta}}{{\boldsymbol \vartheta}_i} + i\frac{m}{{\sin {\vartheta _i}}}P_n^m(\cos {\vartheta _i}){{\boldsymbol \varphi}_i}}\! \right]\!.\\\end{split}$$
Now, the scattered field by the stratified sphere will be expressed. In this case, the expansion in vector spherical functions has to take into account the outgoing nature of the phenomena; to do this, the spherical Bessel function of the third kind is considered [42]: (42)$${{\bf E}_s}({k_i}{\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^n \left[{{e_{{mn}}}{\bf M}_{{mn}}^{(3)}({k_i}{\bf r}) + {f_{{mn}}}{\bf N}_{{mn}}^{(3)}({k_i}{\bf r})} \right],$$
where the coefficients ${e_{{mn}}}$ and ${f_{{mn}}}$ are the unknowns of our problem. Due to the linearity of the model, the field inside each layer can always be written as a superposition of a spherical wave incoming toward the center of the spheres and one outgoing the center (see Fig. 4): (43)$$\begin{split}\!\!\!\!{{\bf E}_j}({k_j}{\bf r})&= {{\bf E}_{i_j}}({k_j}{\bf r}) + {{\bf E}_{s_j}}({k_j}{\bf r}) \\[-2pt]& = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^n [r_{{mn}}^j{\bf M}_{{mn}}^{(1)}({k_j}{\bf r}) + s_{{mn}}^j{\bf N}_{{mn}}^{(1)}({k_j}{\bf r})] \\[-2pt]&\quad {+}\sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^n [u_{{mn}}^j{\bf M}_{{mn}}^{(2)}({k_j}{\bf r}) + v_{{mn}}^j{\bf N}_{{mn}}^{(2)}({k_j}{\bf r})],\!\end{split}$$
with $j = 1, \ldots,N$. It can be noted that the $j$-th field is expressed as a superposition of the Bessel spherical functions of first and second kinds ${j_n}$ and ${y_n}$, respectively, because the point $r = 0$ is external to the considered domain. For this reason, the last electric field must be represented as (44)$$\!\!\!{{\bf E}_N}({k_N}{\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^n \big[{r_{{mn}}^N{\bf M}_{{mn}}^{(1)}({k_N}{\bf r}) + s_{{mn}}^N{\bf N}_{{mn}}^{(1)}({k_N}{\bf r})} \big].\!$$
At this point, all the electric fields that contribute to the problem formulation have been introduced. Thus, the boundary conditions on the $N$ layers can be imposed. Initially, the continuity of the tangential part of the fields is imposed on the external surface of the $j$-th layer. Due to the proportionality between the curl of the electric field and the magnetic field, both equations can be formulated as a function of only the electric fields: (45)$$\left({{{\bf E}_{i_j}} + {{\bf E}_{s_j}} - {{\bf E}_{{i_{j + 1}}}} - {{\bf E}_{{s_{j + 1}}}}} \right) \times {{\bf r}_i} = 0\quad \text{for}\; r = {a_j},$$
(46)$$\!\!\!\big[{\nabla \times ({{\bf E}_{i_j}} + {{\bf E}_{s_j}} - {{\bf E}_{{i_{j + 1}}}} - {{\bf E}_{{s_{j + 1}}}})} \big] \times {{\bf r}_i} = 0\quad \text{for}\; r = {a_j}.\!$$
Generally, using Eqs. (31)–(35), the fields can be written in the two following forms: (47)$$\begin{split}{{\bf E}_i}({\bf r}) &= \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^n \left\{a_{{mn}}^j{{\bf m}_{{mn}}}(\vartheta ,\varphi){j_n}({k_j}r)\right. \\[-2pt]& \quad \left. {+}b_{{mn}}^j\left[{{\bf n}_{{mn}}}(\vartheta ,\varphi){j^\prime _n}({k_j}r) + {{\bf p}_{{mn}}}(\vartheta ,\varphi)\frac{{{j_n}({k_j}r)}}{{{k_j}r}}\right]\right\} ,\end{split}$$
(48)$${{\bf E}_i}({\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^n \left[a_{{mn}}^j{\bf M}_{{mn}}^{(1)}({k_j}{\bf r}) + b_{{mn}}^j{\bf N}_{{mn}}^{(1)}({k_j}{\bf r})\right],$$
namely, using the tesseral function formalism or the vector spherical harmonics formalism, respectively. Setting ${z^\prime _n}(kr) = \frac{1}{{kr}}\frac{{d[r{z_n}(kr)]}}{{dr}}$, where ${z_n} = {j_n},{y_n},h_n^{(1)},h_n^{(2)}$, the following properties of the spherical harmonics can be employed [42]: (49)$${{\bf m}_{{mn}}}(\vartheta ,\varphi) \times {{\bf r}_i} = - {{\bf n}_{{mn}}}(\vartheta ,\varphi),$$
(50)$${{\bf n}_{{mn}}}(\vartheta ,\varphi) \times {{\bf r}_i} = {{\bf m}_{{mn}}}(\vartheta ,\varphi),$$
(51)$$\nabla \times {{\bf M}_{{mn}}}(kr,\vartheta ,\varphi) = k{{\bf N}_{{mn}}}(kr,\vartheta ,\varphi) = k{{\bf m}_{{mn}}}(\vartheta ,\varphi){z^\prime _n}(kr),$$
(52)$$\nabla \times {{\bf N}_{{mn}}}(kr,\vartheta ,\varphi) = k{{\bf M}_{{mn}}}(kr,\vartheta ,\varphi) = - k{{\bf n}_{{mn}}}(\vartheta ,\varphi){z_n}(kr).$$
In this way, the terms of the boundary conditions for the $j$-th and $j + 1$-th layer can be represented as follows: (53)$$\left\{\begin{array}{l}r_{{mn}}^j{j_n}({k_j}{r_j}) + u_{{mn}}^j{y_n}({k_j}{r_j}) = r_{{mn}}^{j + 1}{j_n}({k_{j + 1}}{r_j}) + u_{{mn}}^{j + 1}{y_n}({k_{j + 1}}{r_j})\\{k_j}[r_{{mn}}^j{j^\prime_n}({k_j}{r_j}) + u_{{mn}}^j{y^\prime_n}({k_j}{r_j})] = {k_{j + 1}}[r_{{mn}}^{j + 1}{j^\prime_n}({k_{j + 1}}{r_j}) + u_{{mn}}^{j + 1}{y^\prime_n}({k_{j + 1}}{r_j})]\\s_{{mn}}^j{j^\prime_n}({k_j}{r_j}) + v_{{mn}}^j{y^\prime_n}({k_j}{r_j}) = s_{{mn}}^{j + 1}{j^\prime_n}({k_{j + 1}}{r_j}) + v_{{mn}}^{j + 1}{y^\prime_n}({k_{j + 1}}{r_j})\\{k_j}[s_{{mn}}^j{j_n}({k_j}{r_j}) + v_{{mn}}^j{y_n}({k_j}{r_j})] = {k_{j + 1}}[s_{{mn}}^{j + 1}{j_n}({k_{j + 1}}{r_j}) + v_{{mn}}^{j + 1}{y_n}({k_{j + 1}}{r_j})].\end{array} \right.$$
Next, the focus is to express the fields of the $j + 1$-th layer as a function of the fields present in the $j$-th layer. The above conditions are independent two by two; therefore, the conditions in the matrix form can be written as [22,43–46] (54)$$\begin{split}\left({\begin{array}{c}{r_{{mn}}^{j + 1}}\\{u_{{mn}}^{j + 1}}\end{array}} \right) &= \frac{1}{{{A_{j + 1}}}}\left[{\begin{array}{cc}{{y^\prime_n}({k_{j + 1}}{r_j})}&{- \frac{{{y_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}\\{- {j^\prime_n}({k_{j + 1}}{r_j})}&{\frac{{{j_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}\end{array}} \right]\\&\quad\times\left[{\begin{array}{cc}{{j_n}({k_j}{r_j})}&{{y_n}({k_j}{r_j})}\\{{j^\prime_n}({k_j}{r_j})}&{{y^\prime_n}({k_j}{r_j})}\end{array}} \right]\left({\begin{array}{c}{r_{{mn}}^j}\\{u_{{mn}}^j}\end{array}} \right),\end{split}$$
(55)$$\begin{split}\left({\begin{array}{c}{s_{{mn}}^{j + 1}}\\{v_{{mn}}^{j + 1}}\end{array}} \right) &= - \frac{1}{{{A_{j + 1}}}}\left[{\begin{array}{cc}{{y_n}({k_{j + 1}}{r_j})}&{- \frac{{{y^\prime_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}\\{- {j_n}({k_{j + 1}}{r_j})}&{\frac{{{j^\prime_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}\end{array}} \right]\\&\quad\times\left[{\begin{array}{cc}{{j^\prime_n}({k_j}{r_j})}&{{y^\prime_n}({k_j}{r_j})}\\{{j_n}({k_j}{r_j})}&{{y_n}({k_j}{r_j})}\end{array}} \right]\left({\begin{array}{c}{s_{{mn}}^j}\\{v_{{mn}}^j}\end{array}} \right),\end{split}$$
having put ${\zeta _j} = \frac{{{k_{j + 1}}}}{k_j}$, and with (56)$$\!\!\!{A_{j + 1}} = {j_n}({k_{j + 1}}{r_j}){y^\prime _n}({k_{j + 1}}{r_j}) - {y_n}({k_{j + 1}}{r_j}){j^\prime _n}({k_{j + 1}}{r_j}),\!$$
using the recurrence relations of the spherical Bessel functions [41]: (57)$${z^\prime _n}(z) = {z_{n - 1}}(z) - \frac{{n + 1}}{z}{f_n}(z),$$
(58)$${z^\prime _n}(z) = {z_{n + 1}}(z) - \frac{n}{z}{f_n}(z).$$
Replacing the above equations in Eq. (56), the following is obtained: (59)$${A_{j + 1}} = \frac{1}{{{{({k_{j + 1}}{r_j})}^2}}}.$$
Performing some algebra, we obtain (60)$$\left({\begin{array}{c}{r_{{mn}}^{j + 1}}\\{u_{{mn}}^{j + 1}}\end{array}} \right) = [{M_j}]\left({\begin{array}{c}{r_{{mn}}^j}\\{u_{{mn}}^j}\end{array}} \right),$$
(61)$$\left({\begin{array}{c}{s_{{mn}}^{j + 1}}\\{v_{{mn}}^{j + 1}}\end{array}} \right) = [{N_j}]\left({\begin{array}{c}{s_{{mn}}^j}\\{v_{{mn}}^j}\end{array}} \right),$$
with (62)$$\left[{M_j} \right] = \frac{1}{{{A_{j + 1}}}}\left[{\begin{array}{cc}{{j_n}({k_j}{r_j}){y^\prime_n}({k_{j + 1}}{r_j}) - \frac{{{j^\prime_n}({k_j}{r_j}){y_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}&\quad {{y_n}({k_j}{r_j}){y^\prime_n}({k_{j + 1}}{r_j}) - \frac{{{y^\prime_n}({k_j}{r_j}){y_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}\\{- {j_n}({k_j}{r_j}){j^\prime_n}({k_{j + 1}}{r_j}) + \frac{{{j^\prime_n}({k_j}{r_j}){j_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}&\quad {- {y_n}({k_j}{r_j}){j^\prime_n}({k_{j + 1}}{r_j}) + \frac{{{y^\prime_n}({k_j}{r_j}){j_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}\end{array}} \right],$$
(63)$$\left[{N_j} \right] = \frac{1}{{{A_{j + 1}}}}\left[{\begin{array}{cc}{- {j^\prime_n}({k_j}{r_j}){y_n}({k_{j + 1}}{r_j}) + \frac{{{j_n}({k_j}{r_j}){y^\prime_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}&\quad {- {y^\prime_n}({k_j}{r_j}){y_n}({k_{j + 1}}{r_j}) + \frac{{{y_n}({k_j}{r_j}){y^\prime_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}\\{{j^\prime_n}({k_j}{r_j}){j_n}({k_{j + 1}}{r_j}) - \frac{{{j_n}({k_j}{r_j}){j^\prime_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}&\quad {{y^\prime_n}({k_j}{r_j}){j_n}({k_{j + 1}}{r_j}) - \frac{{{y_n}({k_j}{r_j}){j^\prime_n}({k_{j + 1}}{r_j})}}{{{\zeta _j}}}}\end{array}} \right],$$
considering $u_{{mn}}^N$, $v_{{mn}}^N = 0$ because in the last layer, $r = 0$. Now, analyzing the transmission through all the layers is quite trivial; in fact, the transmitted field can be written as (64)$$\left({\begin{array}{c}{r_{{mn}}^N}\\0\end{array}} \right) = \prod\limits_{i = N - 1}^1 [{M_i}]\left({\begin{array}{c}{a_{{mn}}^1}\\{e_{{mn}}^1}\end{array}} \right),$$
(65)$$\left({\begin{array}{c}{s_{{mn}}^N}\\0\end{array}} \right) = \prod\limits_{i = N - 1}^1 [{N_i}]\left({\begin{array}{c}{b_{{mn}}^1}\\{f_{{mn}}^1}\end{array}} \right).$$
These equations are known as transfer matrix equations [38]. Moreover, in the matrices $[{M_1}]$ and $[{N_1}]$, the spherical Bessel function ${y_n}({k_j}{r_j})$ should be replaced with $h_n^{(1)}({k_j}{r_j})$, because on the first spherical interface, the Bessel spherical function of the third kind traveling in the external environment also plays a role. In fact, on the first layer, unlike the others, the scattered field assumes the form expressed by Eq. (42), instead of (43). Expressing it as a system of equations, (66)$$\left\{\begin{split}{M_{11}}{a_{{mn}}} + {M_{12}}{e_{{mn}}} &= r_{{mn}}^N\\{M_{21}}{a_{{mn}}} + {M_{22}}{e_{{mn}}} &= 0,\end{split} \right.$$
(67)$$\left\{\begin{split}{N_{11}}{b_{{mn}}} + {N_{12}}{f_{{mn}}} &= s_{{mn}}^N\\{N_{21}}{b_{{mn}}} + {N_{22}}{f_{{mn}}} &= 0,\end{split} \right.$$
the unknowns of the problem are found: (68)$$\left\{\begin{split}{e_{{mn}}} &= - {a_{{mn}}}\frac{{{M_{21}}}}{{{M_{22}}}}\\{f_{{mn}}} &= - {b_{{mn}}}\frac{{{N_{21}}}}{{{N_{22}}}},\end{split} \right.$$
(69)$$\left\{\begin{split}r_{{mn}}^N &= {a_{{mn}}}\frac{{\det[M]}}{{{M_{22}}}}\\s_{{mn}}^N &= {b_{{mn}}}\frac{{\det[N]}}{{{N_{22}}}},\end{split} \right.$$
with (70)$$\det[{M_j}] = \frac{A_j}{{{\zeta _j}{A_{j + 1}}}},$$
(71)$$\det[{N_j}] = \frac{{{\zeta _j}{A_j}}}{{{A_{j + 1}}}}.$$
The coefficients of the series are directly proportional to the coefficients of the incident field represented as a superposition of vector spherical harmonics (also known as vector spherical wave functions or vector partial waves), as in the case of a classic sphere.In Fig. 5(a), the magnitude of the scattered electric field is expressed as a function of frequency in the visible range. The considered scenario is an incident wave linearly polarized along the $ x $ axis and propagating along the $ z $ axis, with a magnitude of 1 V/m. A two-layer sphere with external radius ${a_1} = 200\;\text{nm} $, internal radius ${a_2} = {a_1}/2$, and electromagnetic parameters ${\mu _e} = {\mu _1} = {\mu _2} = 1$, ${\sigma _e} = {\sigma _1} = {\sigma _2} = 0\;\text{S}/\text{m}$, and ${\varepsilon _e} = {\varepsilon _2} = 1$, ${\varepsilon _1} = 3$ (solid line) and a one-layer sphere, i.e., without the core layer (dashed line), are considered. The scattered field is evaluated at the point $(2{a_1},0,0)$. In particular, it is shown how the presence of the core in the sphere can vary the scattered electric field. Figure 5(b) shows the trend of the electric field when the dielectric permittivity of the shell changes from two to six (all other parameters are the same). As noticed, there is a decrease in the maximum value of the diffused electric field due to the higher value of the dielectric constant of the shell. In addition, the resonance peaks due to the increase in the dielectric constant are highlighted.
Figure 6 shows the scattered electric field distribution calculated as $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ at an incident wavelength of 600 nm, and ${\varepsilon _1} = 3$; all other parameters remain unchanged compared to the previous case.
3. MULTIPLE SCATTERERS
In this section, the scattering by multiple objects is illustrated. There are several methods to face the scattering by multiple objects. Here, the so-called $ T $-matrix approach is considered [9,47], applying the Foldy–Lax multiple scattering equations (FLMSEs) [48,49], in order to impose the continuity of the tangential components of the electromagnetic fields on the surface of each scatterer. As we will see, this approach was chosen because it is suitable to incorporate the vector harmonics formalism providing an elegant solution to the problem.
A. Multicylinders
In this section, we consider the scattering by several parallel cylinders in free space, as shown in Fig. 7. An arbitrary number $L$ of dielectric cylinders, with relative permittivities ${\varepsilon _j}$, with $j = 1, \ldots,N$, infinite length, and radii ${r_j}$ in free space filled by a medium, in general, dissipative, with relative permittivity ${\varepsilon _i}$, relative permeability ${\mu _i}$, and electric conductivity ${\sigma _i}$, is considered. As usual, an elliptically polarized plane wave is considered as an incident field. The field on the surface of the $q$-th cylinder needs to be taken into consideration to apply FLMSEs; this field is called the exiting field. It is the superposition of the incident field and all the scattered fields by the cylinders:
(72)$${\bf E}_{\text{ex}}^q = {{\bf E}_i} + \sum\limits_{p = 1\atop p \ne q}^L {{\bf E}_s^p} .$$
The incident field as a function of vector cylindrical harmonics centered on the $q$-th cylinder is [1] (73)$$\begin{split}{{\bf E}_i}({k_i}{{\boldsymbol \rho}_q}) &= \left[{{E_{v0}}{\boldsymbol v} + {E_{h0}}{\boldsymbol h}} \right]{e^{i{{\bf k}_i} \cdot {{\boldsymbol \rho}_q}}}{e^{i{{\bf k}_i} \cdot ({\boldsymbol \rho} - {{\boldsymbol \rho}_q})}} \\& = \sum\limits_{m = - \infty}^{+ \infty} \big[{{{\tilde a}_m}{\bf M}_m^{(1)}({k_i},{\boldsymbol \rho} - {{\boldsymbol \rho}_q}) + {{\tilde b}_m}{\bf N}_m^{(1)}({k_i},{\boldsymbol \rho} - {{\boldsymbol \rho}_q})} \big],\end{split}$$
with (74)$${\tilde a_m} = {a_m}{e^{i{{\bf k}_i} \cdot {{\boldsymbol \rho}_q}}},$$
(75)$${\tilde b_m} = {b_m}{e^{i{{\bf k}_i} \cdot {{\boldsymbol \rho}_q}}}$$
the incident translated coefficients on the $q$-sphere. The exiting field of the $q$-th cylinder is (76)$$\begin{split}{\bf E}_{\text{ex}}^q({k_i}{{\boldsymbol \rho}_q}) &= \sum\limits_{m = - \infty}^{+ \infty} \big[w_m^q{\bf M}_m^{(1)}({k_i},{\boldsymbol \rho} - {{\boldsymbol \rho}_q}) \\[-2pt]&\quad+ v_m^q{\bf N}_m^{(1)}({k_i},{\boldsymbol \rho} - {{\boldsymbol \rho}_q}) \big],\end{split}$$
while the scattered electric field from $p \ne q$-th cylinder is (77)$$\begin{split}{\bf E}_s^p({k_i}{{\boldsymbol \rho}_p}) &= \sum\limits_{m^\prime = - \infty}^{+ \infty} \big[T_{{m^\prime}}^Mw_{{m^\prime}}^p{\bf M}_m^{(3)}({k_i},{\boldsymbol \rho} - {{\boldsymbol \rho}_p}) \\&\quad+ T_{{m^\prime}}^Nv_{{m^\prime}}^p{\bf N}_m^{(3)}({k_i},{\boldsymbol \rho} - {{\boldsymbol \rho}_p}) \big],\end{split}$$
having indicated with $T_{{m^\prime}}^M$ and $T_{{m^\prime}}^N$ the scattering coefficients for the dielectric cylinder case, i.e., the $ T $-matrix coefficients [48,49]: (78)$$T_m^M = - \frac{{{J^\prime_m}({k_{{i\rho}}}a)}}{{H^{\prime(1)}_m({k_{{i\rho}}}a)}} ,$$
(79)$$T_m^N = - \frac{{{J_m}({k_{{i\rho}}}a)}}{{H_m^{(1)}({k_{{i\rho}}}a)}}.$$
Applying the addition theorem on the vectorial cylindrical harmonics (VCH) function, we obtain (80)$$M_{{m^\prime}}^{(3)}(k,{\boldsymbol \rho} - {{\boldsymbol \rho}_p}) = \sum\limits_{m = - \infty}^{+ \infty} {A_{{mm^\prime}}}M_m^{(1)}(k,{\boldsymbol \rho} - {{\boldsymbol \rho}_q}),$$
(81)$$N_{{m^\prime}}^{(3)}(k,{\boldsymbol \rho} - {{\boldsymbol \rho}_p}) = \sum\limits_{m = - \infty}^{+ \infty} {A_{{mm^\prime}}}N_m^{(1)}(k,{\boldsymbol \rho} - {{\boldsymbol \rho}_q}),$$
(82)$$M_{{m^\prime}}^{(1)}(k,{\boldsymbol \rho} - {{\boldsymbol \rho}_p}) = \sum\limits_{m = - \infty}^{+ \infty} {B_{{mm^\prime}}}M_m^{(1)}(k,{\boldsymbol \rho} - {{\boldsymbol \rho}_q}),$$
(83)$$N_{{m^\prime}}^{(1)}(k,{\boldsymbol \rho} - {{\boldsymbol \rho}_p}) = \sum\limits_{m = - \infty}^{+ \infty} {B_{{mm^\prime}}}N_m^{(1)}(k,{\boldsymbol \rho} - {{\boldsymbol \rho}_q}),$$
with (84)$${A_{{mm^\prime}}} = H_{m - m^\prime}^{(1)}(k|{{\boldsymbol \rho}_p} - {{\boldsymbol \rho}_q}|){e^{- i(m - m^\prime){\varphi _{{pq}}}}},$$
(85)$${B_{{mm^\prime}}} = J_{m - m^\prime}^{(1)}(k|{{\boldsymbol \rho}_p} - {{\boldsymbol \rho}_q}|){e^{- i(m - m^\prime){\varphi _{{pq}}}}}.$$
Adding all fields inside the FLMSEs and using the orthogonality properties of the VCHs, the following linear system is achieved: (86)$$w_m^q = {\tilde a_m} + \sum\limits_{m^\prime = - \infty}^{+ \infty} \sum\limits_{p = 1\atop p \ne q} {{A_{{mm^\prime}}}T_{{m^\prime}}^Mw_{{m^\prime}}^p} ,$$
(87)$$v_m^q = {\tilde b_m} + \sum\limits_{m^\prime = - \infty}^{+ \infty} \sum\limits_{p = 1\atop p \ne q} {{A_{{mm^\prime}}}T_{{m^\prime}}^Nv_{{m^\prime}}^p} .$$
At this point, the linear system can be solved and the coefficients $w_m^q$ and $v_m^q$ determined. Since the scattered field by the $q$-th cylinder is writable as a superposition of VCHs, as (88)$$\!\!\!{\bf E}_s^q = \sum\limits_{m = - \infty}^{+ \infty} \big[{e_m^q{\bf M}_m^{(3)}({k_i},{\boldsymbol \rho} - {{\boldsymbol \rho}_q}) + f_m^q{\bf N}_m^{(3)}({k_i},{\boldsymbol \rho} - {{\boldsymbol \rho}_q})} \big],\!$$
the coefficients of the $q$-th cylinder can be expressed as follows: (89)$$e_m^q = T_m^Mw_m^q,$$
(90)$$f_m^q = T_m^Nv_m^q.$$
Hence, the total scattered field can be reached: (91)$${{\bf E}_s} = \sum\limits_{q = 1}^L {\bf E}_s^q.$$
As an example, we consider the scattering by eight equidistant parallel cylinders with their centers on a circumference with a radius of 200 nm [see Fig. 8(a), small figure at the top left]. All the cylinders’ centers are located on the plane containing the wave vector and the electric vector of the incident field. In Fig. 8(b), the magnitude of the scattered electric field as a function of the frequency in the visible range for five different values of the cylinders’ dielectric constants is represented. An incident plane wave, linearly polarized along the direction of the cylinders’ axes, propagating along the $z$ direction, and with a magnitude of 1 V/m, is considered. The field is computed at the point $(2a,0,0)$, where $a = 50\;\text{nm} $ is the radius of each cylinder. The following fictitious electromagnetic parameters are considered: ${\mu _e} = {\mu _1} = {\mu _2} = 1$, ${\sigma _e} = {\sigma _1} = {\sigma _2} = 0\;\text{S}/\text{m}$, and ${\varepsilon _i} = 1\,e\,{\varepsilon _1} = [2,3,4,5,6]$. As can be seen in Fig. 8(b), an increase in the dielectric permittivity leads to a highlight of the resonance peaks and a displacement of the same towards shorter wavelengths.
Figure 9 shows that the scattered electric field distribution calculated is $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ at an incident wavelength of 600 nm, and ${\varepsilon _1} = 3$; all other parameters have remain unchanged compared to the previous case.
B. Multispheres
In this section, the scattering by multiple spheres, with arbitrary radii and electromagnetic properties, in free space is analyzed (see Fig. 10). Here, the use of the FLMSE is adopted again, taking into account the $q$-th sphere:
(92)$${\bf E}_{\text{ex}}^q = {{\bf E}_i} + \sum\limits_{p = 1\atop p \ne q}^L {{\bf E}_s^p} .$$
The same steps as in the previous section are followed, except that the spherical vector harmonics are considered: (93)$${{\bf E}_i}({k_i}{\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^{+ n} \left[{{a_{{mn}}}{\bf M}_{{mn}}^{(1)}({k_i}{\bf r}) + {b_{{mn}}}{\bf N}_{{mn}}^{(1)}({k_I}{\bf r})} \right],$$
with (94)$${a_{{mn}}} = {i^n}\frac{{2n + 1}}{{n(n + 1)}}\frac{{(n - m)!}}{{(n + m)!}}\left({E_i^H {{\boldsymbol \vartheta}_0} + E_i^E {{\boldsymbol \varphi}_0}} \right) \cdot {{\bf m}_{{mn}}}({\vartheta _i},{\varphi _i}),$$
(95)$${b_{{mn}}} = {i^{n - 1}}\frac{{2n + 1}}{{n(n + 1)}}\frac{{(n - m)!}}{{(n + m)!}}\big({E_i^H {{\boldsymbol \vartheta}_0} + E_i^E {{\boldsymbol \varphi}_0}} \big) \cdot {{\bf n}_{{mn}}}({\vartheta _i},{\varphi _i}).$$
The exiting field of the $q$-th cylinder is (96)$$\begin{split}{\bf E}_{\text{ex}}^q({k_I}{{\bf r}_q}) &= \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^{+ n} \big[w_{{mn}}^q{\bf M}_{{mn}}^{(1)}({k_i},{\bf r} - {{\bf r}_q})\\&\quad + v_{{mn}}^q{\bf N}_{{mn}}^{(1)}({k_I},{\bf r} - {{\bf r}_q}) \big],\end{split}$$
while the scattered electric field for the $p \ne q$-th cylinder is (97)$$\begin{split}{\bf E}_s^p({k_i}{{\bf r}_p}) &= \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^{+ n} \big[T_{{m^\prime}}^Mw_{{m^\prime}}^p{\bf M}_{{mn}}^{(3)}({k_i},{\bf r} - {{\bf r}_p}) \\[-2pt]&\quad+ T_{{m^\prime}}^Nv_{{m^\prime}}^p{\bf N}_{{mn}}^{(3)}({k_i},{\bf r} - {{\bf r}_p}) \big],\end{split}$$
$T_{{m^\prime}}^M$ and $T_{{m^\prime}}^N$ indicating the scattering coefficients for the single sphere, i.e., the $ T $-matrix coefficients are (98)$$T_{{mn}}^M = - {a_{{mn}}}\frac{{{{\dot j}_n}({k_i}a){j_n}({k_1}a) - \chi {j_n}({k_i}a){{\dot j}_n}({k_1}a)}}{{\dot h_n^{(1)}({k_i}a){j_n}({k_1}a) - \chi h_n^{(1)}({k_i}a){{\dot j}_n}({k_1}a)}},$$
(99)$$T_{{mn}}^N = - {b_{{mn}}}\frac{{{j_n}({k_i}a){{\dot j}_n}({k_1}a) - \chi {{\dot j}_n}({k_i}a){j_n}({k_1}a)}}{{h_n^{(1)}({k_i}a){{\dot j}_n}({k_1}a) - \chi \dot h_n^{(1)}({k_i}a){j_n}({k_1}a)}},$$
with ${a_{{mn}}}$, ${b_{{mn}}}$ defined in Eqs. (94) and (95) and with $\chi$ refractive index contrast, i.e., it is the ratio between the refractive indices of the scatterer and the environment. Moreover, these coefficients can be easily generalized to include magnetic permeability [50]. The above $ T $-matrix coefficients are represented in the “symmetrized” Mie coefficients form as written by Mie in 1908 [14,51]. To applythe boundary conditions on the surface of each sphere, we consider the addition theorem for the sferical vectorial harmonic functions (SVHFs) [8,12,13,52]: (100)$$\begin{array}{c}{{\bf M}_{{mn}}^{(1)}(k{\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^{+ n} \left[{A_{\mu \nu mn}^{11}(k{{\bf r}_i}){\bf M}_{\mu \nu}^{(1)}(k{\bf r^\prime}) + B_{\mu \nu mn}^{11}(k{{\bf r}_i}){\bf N}_{\mu \nu}^{(1)}(k{\bf r^\prime})} \right]}\\{{\bf N}_{{mn}}^{(1)}(k{\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^{+ n} \left[{A_{\mu \nu mn}^{11}(k{{\bf r}_i}){\bf N}_{\mu \nu}^{(1)}(k{\bf r^\prime}) + B_{\mu \nu mn}^{11}(k{{\bf r}_i}){\bf M}_{\mu \nu}^{(1)}(k{\bf r^\prime})} \right]}\\[8pt]\end{array}\quad \text{for}\;{{\bf r}_i} \gt {\bf r^\prime},$$
and (101)$$\begin{array}{c}{{\bf M}_{{mn}}^{(3)}(k{\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^{+ n} \left[{A_{\mu \nu mn}^{31}(k{{\bf r}_i}){\bf M}_{\mu \nu}^{(1)}(k{\bf r^\prime}) + B_{\mu \nu mn}^{31}(k{{\bf r}_i}){\bf N}_{\mu \nu}^{(1)}(k{\bf r^\prime})} \right]}\\{{\bf N}_{{mn}}^{(3)}(k{\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^{+ n} \left[{A_{\mu \nu mn}^{31}(k{{\bf r}_i}){\bf N}_{\mu \nu}^{(1)}(k{\bf r^\prime}) + B_{\mu \nu mn}^{31}(k{{\bf r}_i}){\bf M}_{\mu \nu}^{(1)}(k{\bf r^\prime})} \right]}\end{array}\quad \text{for}\;{{\bf r}_i} \lt {\bf r^\prime}.$$
Adding the expressions obtained above in the FLMSEs, the following is achieved:
(102)$$\begin{split}&\sum\limits_{{mn}} w_{{mn}}^q{\bf M}_{{mn}}^{(1)}({k_i},{\bf r} - {{\bf r}_q}) + v_{{mn}}^q{\bf N}_{{mn}}^{(1)}({k_i},{\bf r} - {{\bf r}_q}) \\[-3pt]&\quad = \sum\limits_{\mu \nu} \sum\limits_{{mn}} {a_{\mu \nu}}\left[{A_{\mu \nu mn}^{11}({k_i},{\bf r} - {{\bf r}_q}){\bf M}_{\mu \nu}^{(1)}({k_i},{\bf r} - {{\bf r}_q}) + B_{\mu \nu mn}^{11}({k_i},{\bf r} - {{\bf r}_q}){\bf N}_{\mu \nu}^{(1)}({k_i},{\bf r} - {{\bf r}_q})} \right] \\[-3pt] &\qquad +{b_{\mu \nu}}\left[{B_{\mu \nu mn}^{11}({k_i},{\bf r} - {{\bf r}_q}){\bf M}_{\mu \nu}^{(1)}({k_i},{\bf r} - {{\bf r}_q}) + A_{\mu \nu mn}^{11}({k_i},{\bf r} - {{\bf r}_q}){\bf N}_{\mu \nu}^{(1)}({k_i},{\bf r} - {{\bf r}_q})} \right] \\[-3pt] &\qquad +\sum\limits_{p \ne q} \sum\limits_{\mu \nu} \sum\limits_{{mn}} w_{{mn}}^pT_\nu ^{{Mp}}\left[{A_{\mu \nu mn}^{31}({k_i},{\bf r} - {{\bf r}_q}){\bf M}_{\mu \nu}^{(1)}({k_i},{\bf r} - {{\bf r}_q}) + B_{\mu \nu mn}^{31}({k_i},{\bf r} - {{\bf r}_q}){\bf N}_{\mu \nu}^{(1)}({k_i},{\bf r} - {{\bf r}_q})} \right] \\[-3pt] &\qquad{+}v_{{mn}}^pT_\nu ^{{Mp}}\left[{B_{\mu \nu mn}^{31}({k_i},{\bf r} - {{\bf r}_q}){\bf M}_{\mu \nu}^{(1)}({k_i},{\bf r} - {{\bf r}_q}) + A_{\mu \nu mn}^{31}({k_i},{\bf r} - {{\bf r}_q}){\bf N}_{\mu \nu}^{(1)}({k_i},{\bf r} - {{\bf r}_q})} \right],\end{split}$$
and applying the orthogonality properties of the VSHFs with the use of some algebra, the following expressions for the unknown coefficient are obtained: (103)$$w_{\mu \nu}^q = \sum\limits_{\mu \nu} {\tilde a_{\mu \nu}} + \sum\limits_{\mu \nu}\! \sum\limits_{p = 1\atop p \ne q} {\big[w_{\mu \nu}^pT_\nu ^{{Mq}}A_{\mu \nu mn}^{(31)} + v_{\mu \nu}^pT_\nu ^{{Nq}}B_{\mu \nu mn}^{(31)}\big]} ,$$
(104)$$v_{\mu \nu}^q = \sum\limits_{\mu \nu} {\tilde b_{\mu \nu}} + \sum\limits_{\mu \nu} \sum\limits_{p = 1\atop p \ne q} {\big[w_{\mu \nu}^pT_\nu ^{{Mq}}B_{\mu \nu mn}^{(31)} + v_{\mu \nu}^pT_\nu ^{{Nq}}A_{\mu \nu mn}^{(31)}\big]} ,$$
where (105)$$\begin{split}{\tilde a_{\mu \nu}} &= {a_{\mu \nu}}A_{\nu\mu mn}^{(11)} + {b_{\mu \nu}}B_{\nu\mu mn}^{(11)},\\{\tilde b_{\mu \nu}} &= {a_{\mu \nu}}B_{\nu\mu mn}^{(11)} + {b_{\mu \nu}}A_{\nu\mu mn}^{(11)}.\end{split}$$
At this point, the scattered field by the whole set of spheres can be expressed as (106)$$\begin{split}{\bf E}_s^q(k{\bf r}) &= \sum\limits_{j = 1}^L \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^{+ n} [e_{{mn}}^j{\bf M}_{{mn}}^{(3)}(k,{\bf r} - {{\bf r}_j}) \\&\quad+ f_{{mn}}^j{\bf N}_{{mn}}^{(3)}(k,{\bf r} - {{\bf r}_j}) ],\end{split}$$
with (107)$$\left({\begin{array}{c}{a_{{mn}}^j}\\{b_{{mn}}^j}\end{array}} \right) = \left[{\begin{array}{cc}{{T^M}}&0\\0&{{T^N}}\end{array}} \right]\left({\begin{array}{c}{w_{{mn}}^j}\\{v_{{mn}}^j}\end{array}} \right).$$
As an example, we consider the scattering by eight equidistant spheres with the centers on a circumference with radius 200 nm [see Fig. 11(a), small figure at the top left]. All the spheres’ centers are located on the plane containing the wave vector and the electric vector of the incident field. In Fig. 11(b), the magnitude of the scattered electric field as a function of the frequency in the visible range for five different values of the spheres’ dielectric permittivities is represented. An incident plane wave, linearly polarized in the direction of the cylinders’ axes, propagating along the $z$ direction, and with a magnitude of 1 V/m, is considered. The field is computed at the point $(2a,0,0)$, where $a = 50\;\text{nm} $ is the radius of each sphere. The following fictitious electromagnetic parameters are considered: ${\mu _e} = {\mu _1} = {\mu _2} = 1$, ${\sigma _e} = {\sigma _1} = {\sigma _2} = 0\;\text{S}/\text{m}$, and ${\varepsilon _i} = 1$ and ${\varepsilon _1} = [2,3,4,5,6]$. As can be seen in Fig. 11(b), an increase in the dielectric permittivity leads to a highlight of the resonance peaks.
Figure 12 shows the scattered electric field distribution calculated as $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ at an incident wavelength of 600 nm, and ${\varepsilon _1} = 3$; all other parameters remain unchanged compared to the previous case.
4. SCATTERING BY A SPHERE EMBEDDED IN A CYLINDER
In this section, we face the interaction of an electromagnetic wave with two different geometries, simultaneously. The scattering by a sphere embedded in an infinite circular cylinder is considered, in particular, the center of the sphere is assumed to be on the cylinder’s axis. Primarily, the focus of this scenario is related to the interactions between two different waveforms: the cylindrical and the spherical harmonics.
To solve this problem, three reference frames need to be considered: a Cartesian $(x,y,z)$, a cylindrical $(\rho ,\varphi ,z)$, and a spherical frame $(r,\vartheta ,\varphi)$, all of them with the same origin. The relation between the unit vectors for each reference frame is
(108)$${\boldsymbol \rho} = \cos \varphi {\boldsymbol x} + \sin \varphi {\boldsymbol y},$$
(109)$${\boldsymbol \varphi} = {-} \sin \varphi {\boldsymbol x} + \cos \varphi {\boldsymbol y},$$
(110)$${\boldsymbol r} = \sin \vartheta \cos \varphi {\boldsymbol x} + \sin \vartheta \sin \varphi {\boldsymbol y} + \cos \vartheta {\boldsymbol z},$$
(111)$${\boldsymbol \vartheta} = \cos \vartheta \cos \varphi {\boldsymbol x} + \cos \vartheta \sin \varphi {\boldsymbol y} + \sin \vartheta {\boldsymbol z}.$$
The free space around the cylinder is filled with a material with relative permittivity ${\epsilon _i}$ and relative permeability ${\mu _i}$ (${k_i}$ wavenumber in the external environment). The circular cylinder, with the axis coincident with the $z$ axis and with radius ${a_c}$, is filled with a material with relative permittivity ${\epsilon _c}$ and relative permeability ${\mu _c}$ (${k_c}$ wavenumber in the cylinder), while the sphere, centered in the origin, with radius ${a_s} \lt {a_c}$, is filled with a material with relative permittivity ${\epsilon _s}$ and relative permeability ${\mu _s}$ (${k_s}$ wavenumber in the sphere). All the materials are assumed to be lossless. A plane wave incident on the cylinder with propagation vector ${k_i}$ forming an angle ${\vartheta _i}$ with the $z$ axis, and the relative projection on the $z$ plane that forms an angle ${\varphi _i}$ with the $x$ axis are considered (see Fig. 13). The propagation vector can be expressed as follows: (112)$$\!\!\!{{\bf k}_{\bf i}} = {k_i}{{\boldsymbol k}_i} = {k_i}(\sin \vartheta \cos \varphi {\boldsymbol x} + \sin \vartheta \sin \varphi {\boldsymbol y} + \cos \vartheta {\boldsymbol z}),\!$$
where ${{\boldsymbol k}_i}$ is a unit vector. The first interaction occurs with the cylinder, and the incident plane wave can then be decomposed in cylindrical harmonics: (113)$${{\bf E}_i}({\bf r}) = \sum\limits_{m = - \infty}^{+ \infty} \left[{{c_m}{\bf M}_m^{(1)}({\bf r}) + {d_m}{\bf N}_m^{(3)}({k_{{i\rho}}},{k_{{iz}}},{\bf r})} \right].$$
The scattered field and the internal field of the cylinder can be written as a superposition of cylindrical harmonics as well: (114)$${{\bf E}_{\text{sc}}}({\bf r}) = \sum\limits_{m = - \infty}^{+ \infty} \left[{{c_m}{\bf M}_m^{(3)}({k_e},{\bf r}) + {d_m}{\bf N}_m^{(3)}({k_e},{\bf r})} \right],$$
(115)$${{\bf E}_{\text{ic}}}({\bf r}) = \sum\limits_{m = - \infty}^{+ \infty} [{e_m}{\bf M}_m^{(1)}({k_c},{\bf r}) + {f_m}{\bf N}_m^{(1)}({k_c},{\bf r})].$$
As noted again, the radial dependence of the scattered field follows the Hankel function of the first type because it is traveling wave in free space, while the radial dependence of the internal field follows the Bessel function of the first kind because it is a stationary field.The internal field inside the cylinder interacts with the sphere. The interaction can be taken into account with two other fields: the scattered field by the sphere, ${{\bf E}_{\text{ss}}}$, and the internal field to the sphere, ${{\bf E}_{\text{is}}}$. Both fields can be expressed as a superposition of vector spherical harmonics with unknown coefficients:
(116)$$\!\!\!{{\bf E}_{\text{ss}}}({\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^n \big[{{g_{{mn}}}{\bf M}_{{mn}}^{(3)}({k_{{sf}}}{\bf r}) + {l_{{mn}}}{\bf N}_{{mn}}^{(3)}({k_{{sf}}}{\bf r})} \big],\!$$
(117)$$\!\!\!{{\bf E}_{\text{is}}}({\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^n \big[{{p_{{mn}}}{\bf M}_{{mn}}^{(1)}({k_{{sf}}}{\bf r}) + {q_{{mn}}}{\bf N}_{{mn}}^{(1)}({k_{{sf}}}{\bf r})} \big].\!$$
The radial dependence of the scattered field follows the spherical Hankel function of the first type because it is a traveling wave, while the radial dependence of the internal field follows the spherical Bessel function of the first kind because it is a stationary field. It is important to note that at this point, we should consider another interaction between ${E_{\text{ss}}}$ and the internal surface of the cylinder; such an interaction would generate another field inside the cylinder, superimposing to ${E_{\text{ic}}}$, and another field outside the cylinder, superimposing to ${E_{\text{sc}}}$. However, since ${E_{\text{ic}}}$ and ${E_{\text{sc}}}$ are unknown, these supplementary fields have already been considered in ${E_{\text{ic}}}$ and ${E_{\text{sc}}}$ themselves.To solve the scattering problem, a three-step process is considered: first the boundary conditions on the sphere’s surface are imposed, obtaining the scattering coefficients of the sphere as a function of the scattering coefficients of the cylinder, then the boundary conditions on the cylinder’s surface are imposed, obtaining a linear system where the only unknowns are the cylinder’s scattering coefficients; finally, the linear system is solved.
The boundary conditions on the surfaces of the sphere ($r = {a_s}$) and of the cylinder ($\rho = {a_c}$) are the following:
(118)$${{\boldsymbol \rho}_i} \times \left({{{\bf E}_i} + {{\bf E}_{\text{sc}}} - {{\bf E}_{{cy}}}} \right) = 0\quad \text{per}\;\rho = {a_c},$$
(119)$${{\boldsymbol \rho}_i} \times \left[{\nabla \times \left({\frac{{{{\bf E}_i} + {{\bf E}_{\text{sc}}}}}{{{\mu _e}}} - \frac{{{{\bf E}_{\text{ic}}} + {{\bf E}_{\text{ss}}}}}{{{\mu _c}}}} \right)} \right] = 0\quad \text{per}\;\rho = {a_c},$$
(120)$${{\bf r}_i} \times \left({{{\bf E}_{\text{ic}}} + {{\bf E}_{\text{ss}}} - {{\bf E}_{\text{is}}}} \right) = 0\quad \text{per}\; r = {a_s},$$
(121)$${{\bf r}_i} \times \left[{\nabla \times \left({\frac{{{{\bf E}_{\text{ic}}} + {{\bf E}_{\text{ss}}}}}{{{\mu _c}}} - \frac{{{{\bf E}_{\text{is}}}}}{{{\mu _s}}}} \right)} \right] = 0\quad \text{per}\; r = {a_s}.$$
Looking at the boundary conditions in Eqs. (14)–(17), the fields expressed in cylindrical coordinates are computed on the sphere’s surface, while the fields expressed in spherical coordinates are computed on the cylinder’s surface. Therefore, in order to impose such conditions, the cylindrical harmonics need to be converted into spherical ones and vice versa. In particular, in Eqs. (14) and (15), ${{\bf E}_{\text{ss}}}$ is expressed as a superposition of cylindrical harmonics, while in Eqs. (16) and (17), ${{\bf E}_{\text{ic}}}$ is written as a superposition of spherical harmonics. The conversion formulas between vectorial cylindrical and spherical harmonics are well known in the literature. In particular, spherical harmonics expressed as an integral of cylindrical harmonics are [53] (122)$$\begin{split}{\bf M}_{{mn}}^{(p)}({\bf r}) &= \frac{{{i^{m - n - 1}}}}{{2k}}\int\limits_C \big[{\tau _{{mn}}}(\cos \alpha){\bf M}_m^{(p)}({\bf r}) \\&\quad+ {\pi _{{mn}}}(\cos \alpha){\bf N}_m^{(p)} \big]\text{d}\alpha ,\end{split}$$
(123)$$\begin{split}{\bf N}_{{mn}}^{(p)}({\bf r}) &= \frac{{{i^{m - n - 1}}}}{{2k}}\int\limits_C \big[{\pi _{{mn}}}(\cos \alpha){\bf M}_m^{(p)}({\bf r}) \\&\quad+ {\tau _{{mn}}}(\cos \alpha){\bf N}_m^{(p)} \big]\text{d}\alpha ,\end{split}$$
where $C$ is the integration path on $\zeta$ complex plane $(C = \zeta = i\infty \to \pi - i\infty)$.On the other hand, the cylindrical harmonics expressed as a series of spherical harmonics [53] are
(124)$${\bf M}_m^{(p)}({\bf r}) = \sum\limits_{n = m}^{+ \infty} {\gamma _{{mn}}}\big[{{\tau _{{mn}}}(\cos {\vartheta _i}){\bf M}_m^{(p)}({\bf r}) + i{\pi _{{mn}}}(\cos {\vartheta _i}){\bf N}_m^{(p)}} \big],$$
(125)$${\bf N}_m^{(p)}({\bf r}) = \sum\limits_{n = m}^{+ \infty} {\gamma _{{mn}}}\big[{i{\pi _{{mn}}}(\cos {\vartheta _i}){\bf M}_m^{(p)}({\bf r}) + {\tau _{{mn}}}(\cos {\vartheta _i}){\bf N}_m^{(p)}} \big],$$
where (126)$${\gamma _{{mn}}} = k{i^{n - m + 1}}\frac{{(2n + 1)(n - m)!}}{{n(n + 1)(n + m)!}}\sin {\vartheta _i}.$$
By using Eqs. (20) and (21), the field in Eq. (11) can be formulated again as follows: (127)$$\begin{split}{{\bf E}_{\text{ic}}}({\bf r}) &= \sum\limits_{m = 1}^{+ \infty} \sum\limits_{n = m}^{+ \infty} {\gamma _{{mn}}}\{{e_m}[{\tau _{{mn}}}(\cos {\vartheta _i}){\bf M}_m^{(1)}({\bf r}) + i{\pi _{{mn}}}(\cos {\vartheta _i}){\bf N}_m^{(1)}] \\ &\quad{+}{f_m}[i{\pi _{{mn}}}(\cos {\vartheta _i}){\bf M}_m^{(1)}({\bf r}) + {\tau _{{mn}}}(\cos {\vartheta _i}){\bf N}_m^{(1)}]\} ,\end{split}$$
where ${\vartheta _{\text{ic}}}$ denotes the angle that the wave vector inside the cylinder forms with the $z$ axis. The value of ${\vartheta _{\text{ic}}}$ is known, and in fact, ${k_{{cz}}} = {k_c}\cos {\vartheta _{\text{ic}}} = {k_{{iz}}} = {k_e}\cos {\vartheta _{\text{ic}}}$, because of the continuity of the $z$ component of the wave vector on the cylinder’s surface. Similarly, by using Eqs. (18) and (19), the field in Eq. (12) is as follows: (128)$$\begin{split}{{\bf E}_{\text{ss}}}({\bf r}) &= \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^n \frac{{{i^{m - n - 1}}}}{{2k}} \\ &\quad {g_{{mn}}}\int\limits_C \left[{{\tau _{{mn}}}(\cos \alpha){\bf M}_m^{(3)}({\bf r}) + {\pi _{{mn}}}(\cos \alpha){\bf N}_m^{(3)}} \right]\text{d}\alpha \\ &\quad + {l_{{mn}}}\int\limits_C \left[{{\pi _{{mn}}}(\cos \alpha){\bf M}_m^{(3)}({\bf r}) + {\tau _{{mn}}}(\cos \alpha){\bf N}_m^{(3)}} \right],\end{split}$$
where $\alpha$ is the angle, in the integral, between the wave vector of the elementary wave and the $z$ axis. At this stage, the boundary conditions are arranged, starting with the conditions on the sphere’s surface. Inserting Eq. (24) into Eq. (16) and cross multiplying by ${\boldsymbol r}$, the following conditions are reached: (129)$$\begin{split}&{-}{b_{{mn}}}\frac{{k_{e \rho}^2}}{k_e}{J_m}({k_{e \rho}}{a_c}) - {d_m}\frac{{k_{e \rho}^2}}{k_e}H_m^{(1)}({k_{e \rho}}{a_c}) + {f_m}\frac{{k_{c \rho}^2}}{k_c}{J_m}({k_{c \rho}}{a_c}) \\ &\quad {+}\sum\limits_{n = 1}^{+ \infty} \frac{{{i^{m - n - 1}}k_{c \rho}^2}}{{2k_c^2}}H_m^{(1)}({k_{c \rho}}{a_c})[{e_m}{\Theta _{{mn}}}({\vartheta _{\text{ic}}}) \\&\quad+ {f_{{mn}}}{\Phi _{{mn}}}({\vartheta _{\text{ic}}})] = 0,\end{split}$$
(130)$$\begin{split}&{-}{a_m}{J^\prime _m}({k_{e \rho}}{a_c}) - {b_m}\frac{{m{k_z}}}{{{k_e}{a_c}}}{J_m}({k_{e \rho}}{a_c}) - {c_m}H_m^{(1)^\prime}({k_{e \rho}}{a_c}) + \\ &\quad -{d_m}\frac{{m{k_z}}}{{{k_e}{a_c}}}H_m^{(1)}({k_{e \rho}}{a_c}) + {e_m}{J_m}({k_{c \rho}}{a_c}) + {f_m}\frac{{m{k_z}}}{{{k_c}{a_c}}}{J_m}({k_{c \rho}}{a_c}) \\ &\quad+ \sum\limits_{n = 1}^{+ \infty} \frac{{{i^{m - n - 1}}}}{{2{k_c}}}\{H_m^{(1)^\prime}({k_{c \rho}}{a_c})[{e_m}{\Psi _{{mn}}}({\vartheta _{\text{ic}}}) + i{f_{{mn}}}{\Theta^*_{{mn}}}({\vartheta _{\text{ic}}})] \\ &\quad +\frac{{m{k_z}}}{{{k_c}{a_c}}}H_m^{(1)}({k_{c \rho}}{a_c})[{e_m}{\Theta _{{mn}}}({\vartheta _{\text{ic}}}) + i{f_{{mn}}}{\Phi _{{mn}}}({\vartheta _{\text{ic}}})] = 0,\end{split}$$
(131)$$\begin{split}& - {a_{{mn}}}\frac{{k_{e \rho}^2}}{k_e}{J_m}({k_{e \rho}}{a_c}) - {c_m}\frac{{k_{e \rho}^2}}{k_e}H_m^{(1)}({k_{e \rho}}{a_c}) \\&\quad+ {\zeta _{12}}{e_m}\frac{{k_{c \rho}^2}}{k_c}{J_m}({k_{c \rho}}{a_c}) \\&\quad +{\zeta _{12}}\sum\limits_{n = 1}^{+ \infty} \frac{{{i^{m - n - 1}}k_{c \rho}^2}}{{2k_c^2}}H_m^{(1)}({k_{c \rho}}{a_c})[{e_m}{\Psi _{{mn}}}({\vartheta _{\text{ic}}}) \\&\quad+ i{f_{{mn}}}{\Theta^*_{{mn}}}({\vartheta _{\text{ic}}})] = 0,\end{split}$$
(132)$$\begin{split}&-{b_m}{J^\prime _m}({k_{e \rho}}{a_c}) - {a_m}\frac{{m{k_z}}}{{{k_e}{a_c}}}{J_m}({k_{e \rho}}{a_c}) - {d_m}H_m^{(1)^\prime}({k_{e \rho}}{a_c}) + \\ &\quad -{c_m}\frac{{m{k_z}}}{{{k_e}{a_c}}}H_m^{(1)}({k_{e \rho}}{a_c}) + {\zeta _{12}}{f_m}{J^\prime _m}({k_{c \rho}}{a_c}) \\&\quad+ {\zeta _{12}}{e_m}\frac{{m{k_z}}}{{{k_c}{a_c}}}{J_m}({k_{c \rho}}{a_c}) \\&\quad +{\zeta _{12}}\sum\limits_{n = 1}^{+ \infty} \frac{{{i^{m - n - 1}}}}{{2{k_c}}}\{H_m^{(1)^\prime}({k_{c \rho}}{a_c})[{e_m}{\Theta _{{mn}}}({\vartheta _{\text{ic}}}) \\&\quad+ {f_{{mn}}}{\Phi _{{mn}}}({\vartheta _{\text{ic}}})] \\ &\quad{+}\frac{{m{k_z}}}{{{k_c}{a_c}}}H_m^{(1)}({k_{c \rho}}{a_c})[{e_m}{\Psi _{{mn}}}({\vartheta _{\text{ic}}}) + i{f_{{mn}}}{\Theta^*_{{mn}}}({\vartheta _{\text{ic}}})] = 0,\end{split}$$
with (133)$${\Theta _{{mn}}}({\vartheta _{\text{ic}}}) = {\pi _{{mn}}}(\cos {\vartheta _{\text{ic}}}){\tau _{{mn}}}(\cos {\vartheta _{\text{ic}}})({G_{{mn}}} + i{L_{{mn}}}),$$
(134)$${\Theta^*_{{mn}}}({\vartheta _{\text{ic}}}) = {\pi _{{mn}}}(\cos {\vartheta _{\text{ic}}}){\tau _{{mn}}}(\cos {\vartheta _{\text{ic}}})({G_{{mn}}} - i{L_{{mn}}}),$$
(135)$${\Phi _{{mn}}}({\vartheta _{\text{ic}}}) = i{G_{{mn}}}\pi _{{mn}}^2(\cos {\vartheta _{\text{ic}}}) + {L_{{mn}}}\tau _{{mn}}^2(\cos {\vartheta _{\text{ic}}}),$$
(136)$${\Psi _{{mn}}}({\vartheta _{\text{ic}}}) = {G_{{mn}}}\tau _{{mn}}^2(\cos {\vartheta _{\text{ic}}}) + i{L_{{mn}}}\pi _{{mn}}^2(\cos {\vartheta _{\text{ic}}}),$$
and with (137)$${G_{{mn}}} = {\gamma _{{mn}}}\frac{{{\zeta _{{cs}}}{j^\prime_n}({k_s}{a_s}){j_n}({k_c}{a_s}) - {j_n}({k_s}{a_s}){j^\prime_n}({k_c}{a_s})}}{{{j_n}({k_s}{a_s})h_n^{(1)^\prime}({k_c}{a_s}) - {\zeta _{{cs}}}{j^\prime_n}({k_s}{a_s})h_n^{(1)}({k_c}{a_s})}},$$
(138)$${L_{{mn}}} = {\gamma _{{mn}}}\frac{{{\zeta _{{cs}}}{j_n}({k_s}{a_s}){j_n}({k_c}{a_s}) - {j^\prime_n}({k_s}{a_s}){j^\prime_n}({k_c}{a_s})}}{{{j^\prime_n}({k_s}{a_s})h_n^{(1)}({k_c}{a_s}) - {\zeta _{{cs}}}{j_n}({k_s}{a_s})h_n^{(1)^\prime}({k_c}{a_s})}},$$
(139)$${P_{{mn}}} = {\gamma _{{mn}}}\frac{{{j_n}({k_s}{a_s}))h_n^{(1)^\prime}({k_c}{a_s}) - {j^\prime_n}({k_s}{a_s}))h_n^{(1)}({k_c}{a_s})}}{{{j_n}({k_s}{a_s})h_n^{(1)^\prime}({k_c}{a_s}) - {\zeta _{{cs}}}{j^\prime_n}({k_s}{a_s})h_n^{(1)}({k_c}{a_s})}},$$
(140)$${Q_{{mn}}} = {\gamma _{{mn}}}\frac{{{j^\prime_n}({k_s}{a_s})h_n^{(1)}({k_c}{a_s}) - {j_n}({k_s}{a_s})h_n^{(1)^\prime}({k_c}{a_s})}}{{{j^\prime_n}({k_s}{a_s})h_n^{(1)}({k_c}{a_s}) - {\zeta _{{cs}}}{j_n}({k_s}{a_s})h_n^{(1)^\prime}({k_c}{a_s})}}.$$
Solving the linear system, the unknown coefficients and the scattered field are found: (141)$${{\bf E}_{\text{sc}}}({\bf r}) = \sum\limits_{n = 1}^{+ \infty} \sum\limits_{m = - n}^{+ n} \left[{{c_{{mn}}}{\bf M}_m^{(3)}({k_{{i\rho}}},{k_{{iz}}},{\bf r}) + {d_{{mn}}}{\bf N}_m^{(3)}({k_{{i\rho}}},{k_{{iz}}},{\bf r})} \right].$$
In Fig. 14, the magnitude of the scattered electric field as a function of frequency in the visible range is shown. The incident wave is a linearly polarized plane wave, along the cylinder axis, propagating along the $ x $ axis with a magnitude of 1 V/m. The scattered field is computed at the point $(2{a_c},0,0)$ with ${a_c} = 200\;\text{nm} $ being the external cylinder radius and ${a_s} = {a_c}/2$ the inner spherical radius. The fictitious electromagnetic properties are ${\mu _e} = {\mu _c} = {\mu _s} = 1$, ${\sigma _e} = {\sigma _c} = {\sigma _s} = 0\;\text{S}/\text{m}$, and ${\varepsilon _e} = {\varepsilon _s} = 1$, and ${\varepsilon _c} = [2,3,4,5,6]$. As can be seen in Fig. 14, an increase in the dielectric permittivity leads to a highlight of the resonance peaks and a displacement of the same towards the shorter wavelengths.
Figure 15 shows the scattered electric field distribution calculated as $\sqrt {E_{s_x}^2 + E_{s_y}^2 + E_{s_z}^2}$ at an incident wavelength of 600 nm, and ${\varepsilon _1} = 3$; all other parameters remain unchanged compared to the previous case.
5. CONCLUSION
This tutorial presents some advanced scattering problems. Applying the methods presented in [1], several scenarios with multiple targets have been taken into consideration. In particular, the solutions to the scattering problem by a stratified circular cylinder in free space, and to the scattering problem by a stratified sphere are introduced. Moreover, the problem of scattering by multiple objects arbitrarily displaced in free space has been also addressed, and in particular, the solution to the problem of parallel circular cylinders and spheres in free space has been obtained. Finally, the scattering problem by multiple objects with different shapes has been faced, and in particular, the case of scattering by a sphere embedded in an infinite circular cylinder has been studied. In all presented scenarios, we have considered the general case of dissipative materials. In the related solutions, several properties of the vector harmonics such as the addition theorems for Bessel and spherical Bessel functions, and the conversion formula to represent a cylindrical harmonic as a superposition of spherical harmonics and vice versa have been introduced.
All the proposed scattering problems have been solved by the method of vector harmonics, obtaining analytical solutions in terms of a series of special functions. The vector harmonics method allows emphasizing the similarity among several problems; more important than the solution to the particular problem is the way that this gives the instrument to the researcher to face new and more complicated scattering problems.
Disclosures
The authors declare no conflicts of interest.
REFERENCES
1. F. Frezza, F. Mangini, and N. Tedeschi, “Introduction to electromagnetic scattering: tutorial,” J. Opt. Soc. Am. A 35, 163–173 (2018). [CrossRef]
2. Z. Xue-Song, Vector Wave Functions in Electromagnetic Theory (Aracne, 1990).
3. J. H. Bruning and Y. T. Lo, “Electromagnetic scattering by two spheres,” Proc. IEEE 56, 119–120 (1968). [CrossRef]
4. O. I. Sindoni, F. Borghese, P. Denti, R. Saija, and G. Toscano, “Multiple electromagnetic scattering from a cluster of spheres. II. Symmetrization,” Aerosol Sci. Technol. 3, 237–243 (1984). [CrossRef]
5. S.-C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990). [CrossRef]
6. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef]
7. T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998). [CrossRef]
8. L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theory and Applications (Wiley, 2000).
9. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
10. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
11. A. V. Osipov and S. A. Tretyakov, Modern Electromagnetic Scattering Theory with Applications (Wiley, 2017).
12. S. Batool, A. Benedetti, F. Frezza, F. Mangini, and Y. L. Xu, “Effect of finite terms on the truncation error of addition theorems for spherical vector wave function,” PhotonIcs & Electromagnetics Research Symposium, Spring (PIERS, SPRING), Rome, Italy, 2019, pp. 2795–2801.
13. S. Batool, F. Frezza, F. Mangini, and Y. L. Xu, “Scattering from multiple PEC sphere using translation addition theorems for spherical vector wave function,” J. Quant. Spectrosc. Radiat. Transfer 248, 106905 (2020). [CrossRef]
14. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1940).
15. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
16. J. G. Van Bladel, Electromagnetic Fields, 2nd ed. (Wiley, 2007).
17. T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).
18. C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012).
19. F. Mangini and N. Tedeschi, “Scattering of an electromagnetic plane wave by a sphere embedded in a cylinder,” J. Opt. Soc. Am. A 34, 760–769 (2017). [CrossRef]
20. K. Sarabandi and P. F. Polatin, “Electromagnetic scattering from two adjacent objects,” IEEE Trans. Antennas Propag. 42, 510–517 (1994). [CrossRef]
21. H. He, N. Zeng, W. Li, T. Yun, R. Liao, Y. He, and H. Ma, “Two-dimensional backscattering Mueller matrix of sphere-cylinder scattering medium,” Opt. Lett. 35, 2323–2325 (2010). [CrossRef]
22. F. Frezza and F. Mangini, “Electromagnetic scattering of an inhomogeneous elliptically polarized plane wave by a multilayered sphere,” J. Electromagn. Waves Appl. 30, 492–504 (2016). [CrossRef]
23. F. Mangini, L. Dinia, and F. Frezza, “Electromagnetic scattering by a cylinder in a lossy medium of an inhomogeneous elliptically plane wave,” J. Telecommun. Inf. Technol. 4, 36–42 (2019). [CrossRef]
24. H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaf, and J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990). [CrossRef]
25. E. Figueiras, L. F. Requicha Ferreira, F. F. M. de Mul, and A. Humeau, “Monte Carlo methods to numerically simulate signals reflecting the microvascular perfusion,” in Numerical Simulations–Applications, Examples and Theory, L. Angermann, ed. (InTech, 2011), Chap. 7.
26. H. He, N. Zeng, R. Liao, T. Yun, W. Li, Y. He, and H. Ma, “Application of sphere-cylinder scattering model to skeletal muscle,” Opt. Express 18, 15104–15112 (2010). [CrossRef]
27. F. Frezza, F. Mangini, M. Muzi, and E. Stoja, “In silico validation procedure for cell volume fraction estimation through dielectric spectroscopy,” J. Biol. Phys. 41, 223–234 (2015). [CrossRef]
28. V. Ferrara, F. Troiani, F. Frezza, F. Mangini, L. Pajewski, P. Simeoni, and N. Tedeschi, “Design and realization of a cheap ground penetrating radar prototype @ 2.45 GHz,” in 10th European Conference on Antennas and Propagation (EuCAP) (2016), p. 4 [CrossRef] .
29. Y. J. Kim, L. Jofre, F. De Flaviis, and M. Q. Fen, “Microwave reflection tomographic array for damage detection of civil structures,” IEEE Trans. Antennas Propag. 51, 3022–3032 (2003). [CrossRef]
30. R. E. Grimm, E. Heggy, S. Clifford, C. Dinwiddie, R. McGinnis, and D. Farrell, “Absorption and scattering in ground-penetrating radar: analysis of the Bishop Tuff,” J. Geophys. Res. 111, E06S02 (2006). [CrossRef]
31. F. Mangini, P. P. Di Gregorio, M. Muzi, L. Pajewski, and F. Frezza, “Wire-grid modelling of metallic targets for ground penetrating radar applications,” in IMEKO International Conference on Metrology for Archaeology and Cultural Heritage (MetroArchaeo 2017) (2019), https://www.imeko.org/publications/tc4-Archaeo-2017/IMEKO-TC4-ARCHAEO-2017-022.pdf.
32. B. Bisceglia, R. De Leo, A. P. Pastore, S. von Gratowski, and V. Meriakri, “Innovative systems for cultural heritage conservation. Millimeter wave application for non-invasive monitoring and treatment of works of art,” J. Microwave Power Electromagn. Energy 45, 36–48 (2011). [CrossRef]
33. A. Macchia, F. Mangini, S. A. Ruffolo, M. Muzi, L. Rivaroli, M. Ricca, M. F. La Russa, and F. Frezza, “A novel model to detect the content of inorganic nanoparticles in coatings used for stone protection,” Prog. Org. Coat. 106, 177–185 (2017). [CrossRef]
34. V. I. Ovod, “Modeling of multiple scattering from an ensemble of spheres in a laser beam,” Part. Syst. Charact. 16, 106–112 (1999). [CrossRef]
35. Y. C. Shen, T. Lo, P. Taday, B. Cole, W. Tribe, and M. Kemp, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005). [CrossRef]
36. H. Zhong, A. Redo-Sanchez, and X. C. Zhang, “Identification and classification of chemicals using terahertz reflective spectroscopic focal-plane imaging system,” Opt. Express 14, 9130–9141 (2006). [CrossRef]
37. S. Kang, S. Jeong, W. Choi, H. Ko, T. D. Yang, J. H. Joo, J.-S. Lee, Y.-S. Lim, Q.-H. Park, and W. Choi, “Imaging deep within a scattering medium using collective accumulation of single-scatterer waves,” Nat. Photonics 9, 253–258 (2015). [CrossRef]
38. H. A. Kramers and G. H. Wannier, “Statistics of the two-dimensional ferromagnet. Part I,” Phys. Rev. 60, 252–262 (1941). [CrossRef]
39. A. J. Devaney and E. Wolf, “Multipole expansion and plane wave representation of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974). [CrossRef]
40. G. Cincotti, F. Gori, F. Frezza, F. Furnò, M. Santarsiero, and G. Schettini, “Plane-wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993). [CrossRef]
41. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover1972).
42. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
43. F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Electromagnetic interaction with two eccentric spheres,” J. Opt. Soc. Am. A 31, 783–789 (2014). [CrossRef]
44. F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Homogenization of a multilayer sphere as a radial uniaxial sphere: features and limits,” J. Electromagn. Waves Appl. 28, 916–931 (2014). [CrossRef]
45. F. Frezza, F. Mangini, and N. Tedeschi, “Electromagnetic scattering by two concentric spheres buried in a stratified material,” J. Opt. Soc. Am. A 32, 277–286 (2015). [CrossRef]
46. F. Mangini and F. Frezza, “Analysis of the electromagnetic reflection and transmission through a stratified lossy medium of an elliptically polarized plane wave,” Math. Mech. Complex Syst. 4, 153–167 (2016). [CrossRef]
47. G. Videen, “Light scattering from a particle on or near a perfectly conducting surface,” Opt. Commun. 115, 1–7 (1995). [CrossRef]
48. L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2000).
49. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics (Wiley, 2000).
50. D. Tzarouchis and A. Sihvola, “Light scattering by a dielectric sphere: perspectives on the Mie resonances,” Appl. Sci. 8, 22 (2018). [CrossRef]
51. G. Mie, “Beiträge zur Optik trüber medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908). [CrossRef]
52. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962). [CrossRef]
53. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier Science & Technology, 1991).
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