**2. Electromagnetic field**

By an inhomogeneous medium, we mean a medium in which the conductivity **σ r** !� �, dielectric **<sup>ε</sup> <sup>r</sup>** !� � and magnetic **<sup>μ</sup> <sup>r</sup>** !� � permittivity, and the current density **J** ! **r** !� � have an arbitrary, but differentiable, in the ordinary and in the generalized sense, dependence on coordinates points of the medium. Below we will not point

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out the explicit dependence of these and other quantities on time and coordinates, where this will not lead to misunderstanding. In an isotropic inhomogeneous medium, ε, μ, σ are scalar functions of coordinates. To derive the wave equation of the electromagnetic field in such a medium, we use the following well known fundamental and material Maxwell equations in a continuous isotropic and stationary medium:

$$\begin{aligned} \mathbf{1.} \ & \mathbf{V} \times \mathbf{\overrightarrow{H}} = \mathbf{\overrightarrow{J}} + \frac{\partial \mathbf{\overrightarrow{D}}}{\partial \mathbf{t}}; \quad \mathbf{2.} \ & \mathbf{V} \times \mathbf{\overrightarrow{E}} = - + \frac{\partial \mathbf{\overrightarrow{B}}}{\partial \mathbf{t}}; \quad \mathbf{3.} \ \mathbf{\overrightarrow{J}} = \mathbf{\sigma} \mathbf{\overrightarrow{E}};\\ \mathbf{4.} \ & \mathbf{7.} \ & \mathbf{\overrightarrow{J}} = - - \frac{\partial}{\partial \mathbf{t}} (\rho\_{\mathbf{f}} + \rho\_{\mathbf{ext}}); \quad \mathbf{5.} \ \mathbf{\overrightarrow{J}} = \mathbf{\overrightarrow{J}}\_{\mathbf{f}} + \mathbf{\overrightarrow{J}}\_{\mathbf{ext}}; \quad \mathbf{6.} \ \mathbf{\overrightarrow{D}} = \mathbf{e} \mathbf{\overrightarrow{E}};\\ \mathbf{7.} \ & \mathbf{\overrightarrow{B}} = \mathbf{u} \mathbf{\overrightarrow{H}}; \quad \mathbf{8.} \ \mathbf{e} = \mathbf{e}\_{0} \mathbf{e}\_{\mathbf{r}}; \quad \mathbf{9.} \ \mu = \mu\_{0} \mu\_{\mathbf{r}}, \end{aligned} \tag{1}$$

where *ρ <sup>f</sup>* is the density of free charges of the medium, and *ρext* is the density of external charges introduced into the medium, **ε<sup>r</sup>** and **μ<sup>r</sup>** are the relative dielectric and magnetic permeability of the medium, and **J** ! *ext* is the current density created by free and external charges. The wave equation for the electric vector in an inhomogeneous medium can be obtained, as for a homogeneous medium, excluding the vector of magnetic field strength from the system of Maxwell's equations. For this, we use the well-known vector analysis formulas [26] and take the rotor from the 2nd equation in system (Eq. (1)):

$$
\nabla \times \nabla \times \vec{\mathbf{E}} = \nabla \cdot \left(\nabla \cdot \overrightarrow{\mathbf{E}}\right) - \Delta \cdot \overrightarrow{\mathbf{E}} = -\frac{\partial}{\partial t} \nabla \times \overrightarrow{\mathbf{B}} \tag{2}
$$

In this case, the source of electromagnetic field is the electric current density **J** ! , so the divergence of the vector ∇ � **E** ! , we need to associate with a current density in the medium. For this, we use (Eq. (5)) of the Maxwell system of equations (Eq. (1)) and obtain ∇ � E ! ¼ 1 <sup>σ</sup> ∇ � J ! � E ! ð Þ ∇ � σ h i � � . Using the vector analysis formulas, we find the following expression:

$$\begin{split} \nabla \left( \nabla \cdot \overrightarrow{\mathbf{E}} \right) &= \overrightarrow{\mathbf{E}} \frac{\left( \nabla \cdot \boldsymbol{\sigma} \right)^{2}}{\sigma^{2}} - \left[ (\nabla \cdot \ln \boldsymbol{\sigma}) \nabla \right] \overrightarrow{\mathbf{E}} - \left( \overrightarrow{\mathbf{E}} \cdot \nabla \right) (\nabla \cdot \ln \boldsymbol{\sigma}) - (\nabla \cdot \ln \boldsymbol{\sigma}) \times \nabla \times \overrightarrow{\mathbf{E}} + \cdots \\ &+ \left( \nabla \cdot \overrightarrow{\mathbf{J}} \right) \left( \nabla \cdot \frac{\mathbf{1}}{\sigma} \right) + \frac{1}{\sigma} \nabla \left( \nabla \cdot \overrightarrow{\mathbf{J}} \right) \end{split} \tag{3}$$

The expression for the rotor of the magnetic field induction vector has the form

$$\nabla \times \overrightarrow{\mathbf{B}} = \nabla \times \left(\mu \overrightarrow{\mathbf{H}}\right) = \mu \left(\nabla \times \overrightarrow{\mathbf{H}}\right) + \left(\nabla \cdot \mu\right) \times \overrightarrow{\mathbf{H}} \tag{4}$$

Using equations 1 and 2 of the system of Maxwell equations and expressions (Eq. (2)) and (Eq. (3)), we find that � *<sup>∂</sup> <sup>∂</sup><sup>t</sup>* ∇ � **B** � �! ¼ �*μσ <sup>∂</sup> <sup>∂</sup><sup>t</sup>* **E** ! � *<sup>μ</sup> <sup>∂</sup> ∂t* **J** ! � *εμ <sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>t*<sup>2</sup> **E** ! þ ð Þ� ∇ � ln *μ* ∇ � **E** � �! , and the desired equation for an electric vector with a field source, in which the charge flux from the volume occupied by the current is not equal to zero, has the following form:

*Green's Function Method for Electromagnetic and Acoustic Fields in Arbitrarily… DOI: http://dx.doi.org/10.5772/intechopen.94852*

$$\begin{split} \epsilon\mu \frac{\partial^2}{\partial t^2} \overrightarrow{\mathbf{E}} + \mu\sigma \frac{\partial}{\partial t} \overrightarrow{\mathbf{E}} - \Delta \overrightarrow{\mathbf{E}} - (\nabla \cdot \ln \mu \sigma) \times \left(\nabla \times \overrightarrow{\mathbf{E}}\right) + \overrightarrow{\mathbf{E}} \left(\frac{\nabla \cdot \sigma}{\sigma}\right)^2 - \left[ (\nabla \cdot \ln \sigma) \nabla \right] \overrightarrow{\mathbf{E}} - \left(\overrightarrow{\mathbf{E}} \cdot \nabla\right) (\nabla \cdot \ln \sigma) = \\ \epsilon = -\mu \frac{\partial}{\partial t} \overrightarrow{\mathbf{J}} - \left(\nabla \cdot \overrightarrow{\mathbf{J}}\right) \left(\nabla \cdot \frac{\mathbf{1}}{\sigma}\right) - \frac{1}{\sigma} \nabla \left(\nabla \cdot \overrightarrow{\mathbf{J}}\right) \end{split} \tag{5}$$

If there is no injection of external charges into the medium, the value ∇ � **J** ! is equal to zero and ∇ **J** ! ¼ ∇ **J** ! *ext* ¼ � *<sup>∂</sup> <sup>∂</sup><sup>t</sup> ρext* otherwise. When deriving (Eq. (5)), no conditions on the field frequency were used. Therefore, the equation is valid up to frequencies that correspond to wavelengths λ larger than the sizes of atoms or molecules. The smallness of the ratio of the first and second terms of equation (Eq. (5)) corresponds to the condition of quasi-stationarity of the electromagnetic field. For a monochromatic field with the angular frequency *ω*, the modulus of their ratio is equal *<sup>ε</sup> <sup>σ</sup> ω* and small under conditions of high conductivity, low dielectric constant, or low the angular frequency. In this case, the propagation of the field in the medium will have a predominantly diffusion character and will be described by the following equation:

$$\begin{split} \mu \sigma \frac{\partial}{\partial t} \overrightarrow{\mathbf{E}} - \Delta \overrightarrow{\mathbf{E}} - (\nabla \cdot \ln \mu \sigma) \times \left( \nabla \times \overrightarrow{\mathbf{E}} \right) + \mathbf{E} \left( \frac{\nabla \cdot \sigma}{\sigma} \right)^2 - \left[ (\nabla \cdot \ln \sigma) \nabla \right] \overrightarrow{\mathbf{E}} - \left( \overrightarrow{\mathbf{E}} \cdot \nabla \right) (\nabla \cdot \ln \sigma) = \\ \sigma = -\mu \frac{\partial}{\partial t} \overrightarrow{\mathbf{J}} - \left( \nabla \cdot \overrightarrow{\mathbf{J}} \right) \left( \nabla \cdot \frac{\mathbf{1}}{\sigma} \right) - \frac{\mathbf{1}}{\sigma} \nabla \left( \nabla \cdot \overrightarrow{\mathbf{J}} \right) \end{split} \tag{6}$$

When *<sup>ε</sup> <sup>σ</sup> ω*> > 1 the field propagation in the media is of the wave-type mainly. At the present time, there are no methods for finding exact solutions of equations of the type (Eq. (5)) and (Eq. (6)). Solutions satisfying a given accuracy can be obtained in two ways. The first is to use numerical methods. The second, which we will follow, consists in passing from the differential equation (5) to the integral equation using the tensor Green's function of the Helmholtz equation for the Fourier - the spectrum of the vector of the electric field strength. The solution to an integral equation can be written in the form of a sequence of approximate solutions, in which each subsequent term is more accurate. It is important that such a procedure for finding a solution is applicable for arbitrary differentiable, both in the usual and in the generalized sense, dependences of σ, ε, and μ on coordinates. For this, we express the vector of the electric field **E** ! **r** !, *t* � � and the current density **<sup>J</sup>** ! **r** !, *t* � � through their Fourier spectra **E** ! **r** !, *ω* � � and **<sup>J</sup>** ! **r** !,*ω* � �

$$\overrightarrow{\mathbf{E}}\left(\overrightarrow{\mathbf{r}},t\right) = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \overrightarrow{\mathbf{E}}\left(\overrightarrow{\mathbf{r}},\omega\right) e^{i\alpha t} d\alpha,\\ \overrightarrow{\mathbf{J}}\left(\mathbf{r},t\right) = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \overrightarrow{\mathbf{J}}\left(\overrightarrow{\mathbf{r}},\omega\right) e^{i\alpha t} d\alpha. \tag{7}$$

For high frequencies, when the dependence of ε and μ from the field frequency cannot be neglected, but spatial dispersion and nonlinear effects can be neglected, **J** ! **r** !, *ω* � � <sup>¼</sup> *σ ω*, *<sup>r</sup>* ! � �**<sup>E</sup>** ! *ω*, *r* ! � �, *<sup>D</sup>* ! *ω*, *r* ! � � <sup>¼</sup> *ε ω*, *<sup>r</sup>* ! � �**<sup>E</sup>** ! *ω*, *r* ! � � <sup>и</sup> *<sup>B</sup>* ! *ω*, *r* ! � � <sup>¼</sup> *μ ω*, *r* ! � �*<sup>H</sup>* ! *ω*, *r* ! � �. Spatial dispersion plays a minor role in comparison with temporal dispersion and is significant in media with the mean free path of the charge or its

$$
\sigma \left( \overrightarrow{\mathbf{r}} \right) = \sigma\_\varepsilon + \sigma\_1 \left( \overrightarrow{\mathbf{r}} \right), \mu \left( \overrightarrow{\mathbf{r}} \right) = \mu\_\varepsilon + \mu\_1 \left( \overrightarrow{\mathbf{r}} \right), \varepsilon \left( \overrightarrow{\mathbf{r}} \right) = \varepsilon\_\varepsilon + \varepsilon\_1 \left( \overrightarrow{\mathbf{r}} \right) \tag{8}
$$

$$\left(\alpha^2 \mathbf{e}\left(\stackrel{\rightarrow}{\mathbf{r}}\right) \mu\left(\stackrel{\rightarrow}{\mathbf{r}}\right) - i\alpha \mu\left(\stackrel{\rightarrow}{\mathbf{r}}\right) \sigma\left(\stackrel{\rightarrow}{\mathbf{r}}\right) - \left(\frac{\nabla \cdot \sigma\left(\stackrel{\rightarrow}{\mathbf{r}}\right)}{\sigma\left(\stackrel{\rightarrow}{\mathbf{r}}\right)}\right)^2\right) = \mathbf{k}^2 \left(\stackrel{\rightarrow}{\mathbf{r}}, \alpha\right),$$

$$\stackrel{\rightarrow}{\mathbf{f}}\mathbf{\bar{f}}\_{\text{ext}}\left(\alpha, \stackrel{\rightarrow}{\mathbf{r}}\right) = i\alpha\mu\left(\stackrel{\rightarrow}{\mathbf{r}}\right) \overline{\mathbf{J}}\left(\stackrel{\rightarrow}{\mathbf{r}}, \alpha\right) + \left(\nabla \overline{\mathbf{J}}\left(\stackrel{\rightarrow}{\mathbf{r}}, \alpha\right)\right) \left(\frac{\nabla \sigma\_1\left(\stackrel{\rightarrow}{\mathbf{r}}\right)}{\sigma^2\left(\stackrel{\rightarrow}{\mathbf{r}}\right)}\right) + \frac{1}{\sigma\left(\stackrel{\rightarrow}{\mathbf{r}}\right)} \nabla \left(\stackrel{\rightarrow}{\mathbf{r}}\right) \overline{\mathbf{J}\left(\stackrel{\rightarrow}{\mathbf{r}}, \alpha\right)}\right), \tag{9}$$

$$\begin{split} \stackrel{\cdot}{\vec{\mathbf{f}}} \left( \mathbf{o}, \stackrel{\cdot}{\mathbf{r}} \right) &= -\left( \frac{\operatorname{v}\_{\sf{\sf{e}}} \left( \vec{\mathbf{r}} \right)}{\mu \left( \vec{\mathbf{r}} \right)} + \frac{\operatorname{v}\_{\sf{\sf{e}}} \left( \vec{\mathbf{r}} \right)}{\sigma \left( \vec{\mathbf{r}} \right)} \right) \times \left( \nabla \times \stackrel{\cdot}{\mathbf{E}} \left( \stackrel{\cdot}{\mathbf{r}}, \mathbf{o} \right) \right) - \left( \frac{\operatorname{v}\_{\sf{\sf{e}}} \left( \vec{\mathbf{r}} \right)}{\sigma \left( \vec{\mathbf{r}} \right)} \nabla \right) \stackrel{\cdot}{\mathbf{E}} \left( \stackrel{\cdot}{\mathbf{r}}, \mathbf{o} \right) - \\ \stackrel{\cdot}{\mathbf{E}} \left( \stackrel{\cdot}{\mathbf{E}} \left( \stackrel{\cdot}{\mathbf{r}}, \mathbf{o} \right) \cdot \nabla \right) \left( \frac{\operatorname{v}\_{\sf{\sf{e}}} \left( \stackrel{\cdot}{\mathbf{r}} \right)}{\sigma \left( \vec{\mathbf{r}} \right)} \right) . \text{ Substituting expressions (Eq. (8)) and (Eq. (9)) into} \\ \stackrel{\cdot}{\mathbf{e}} \text{equation (Eq. (5)), we arrive at the following equation:} \end{split}$$

$$
\Delta \overrightarrow{\mathbf{E}} \left( \overrightarrow{\mathbf{r}}, \boldsymbol{\omega} \right) + k^2 \left( \overrightarrow{\mathbf{r}}, \boldsymbol{\omega} \right) \overrightarrow{\mathbf{E}} \left( \overrightarrow{\mathbf{r}}, \boldsymbol{\omega} \right) = \overrightarrow{\mathbf{f}} \left( \overrightarrow{\mathbf{r}}, \boldsymbol{\omega} \right) + \overrightarrow{\mathbf{f}}\_{\text{ext}} \left( \overrightarrow{\mathbf{r}}, \boldsymbol{\omega} \right) \tag{10}
$$

$$\begin{split} \mathbf{k}^2 \left( \overrightarrow{\mathbf{r}} \right) &= \left( \mathbf{o}^2 \mathbf{e}\_c \mu\_\mathbf{c} - \mathbf{i} \mathbf{o} \mu\_\mathbf{c} \sigma\_\mathbf{c} \right) + \mathbf{o}^2 \left( \mathbf{e}\_c \mu\_\mathbf{1} \left( \overrightarrow{\mathbf{r}} \right) + \mathbf{e}\_1 \left( \overrightarrow{\mathbf{r}} \right) \mu\_\mathbf{c} + \mathbf{e}\_1 \left( \overrightarrow{\mathbf{r}} \right) \mu \left( \overrightarrow{\mathbf{r}} \right) \right) \\ &- \mathbf{i} \mathbf{o} \left( \sigma\_\mathbf{c} \mu\_\mathbf{1} \left( \overrightarrow{\mathbf{r}} \right) + \sigma\_\mathbf{1} \left( \overrightarrow{\mathbf{r}} \right) \mu\_\mathbf{c} + \sigma\_\mathbf{1} \left( \overrightarrow{\mathbf{r}} \right) \mu \left( \overrightarrow{\mathbf{r}} \right) \right) \cdot \left( \frac{\nabla \cdot \sigma\_\mathbf{1} \left( \overrightarrow{\mathbf{r}} \right)}{\sigma \left( \overrightarrow{\mathbf{r}} \right)} \right)^2 = \mathbf{k}\_\mathbf{c}^2 + \mathbf{k}\_\mathbf{1}^2(\mathbf{r}) \end{split} \tag{11}$$

*Green's Function Method for Electromagnetic and Acoustic Fields in Arbitrarily… DOI: http://dx.doi.org/10.5772/intechopen.94852*

Thus equation (Eq. (10)) can be written as:

$$
\Delta \overrightarrow{\mathbf{E}} \left( \overrightarrow{\mathbf{r}}, \boldsymbol{\alpha} \right) + k\_c^2 \overrightarrow{\mathbf{E}} \left( \overrightarrow{\mathbf{r}}, \boldsymbol{\alpha} \right) = \overrightarrow{\mathbf{f}}\_l \left( \boldsymbol{\alpha}, \overrightarrow{\mathbf{r}} \right) + \overrightarrow{\mathbf{f}}\_{\text{ext}} \left( \boldsymbol{\alpha}, \overrightarrow{\mathbf{r}} \right), \tag{12}
$$

where f! <sup>1</sup> ω, r ! � � <sup>¼</sup> <sup>f</sup> ! ω, r ! � � � k2 <sup>1</sup> r !, ω � �<sup>E</sup> ! r !, ω � �.

Formally, we can consider Eq. (12) as an inhomogeneous Helmholtz equation and using the Green's function for it, we can rewrite Eq. (12) in the form of an integral equation. For vector fields, the Green's function [7–10] is a tensor of the second rank. In an orthogonal coordinate system, Eq. (5) decomposes into a system of three scalar equations for the projections of the field E (r, ω) on the coordinate axis. This simplifies the form of the Green's tensor and it has only diagonal elements that are not equal to zero. It can be represented as the vector

*G* ! **r** ! � **<sup>r</sup>** ! 1 � � <sup>¼</sup> <sup>P</sup><sup>3</sup> *<sup>i</sup>*¼<sup>1</sup>*<sup>n</sup>* ! *iGi* **r** ! � **<sup>r</sup>** ! 1 � �, where *Gi* **<sup>r</sup>** ! � **<sup>r</sup>** ! 1 � � is the components of which are the Green's functions of the one-dimensional Helmholtz equation and **n***<sup>i</sup>* are the unit vectors of the coordinate axes. Let the area Ω in which we describe the field be large enough so that on its borders the field and its derivatives can be equated to zero. Using the Green tensor, we can rewrite Eq. (5) for the electric vector at the point *r* ! ∈ Ω of the in the form of the following integral equation

$$\overrightarrow{\mathbf{E}}\left(\overrightarrow{\mathbf{r}},\omega\right) = \sum\_{i=1}^{3} \mathbf{n}\_{i} \left[ \mathbf{G}\_{i}\left(\overrightarrow{\mathbf{r}} - \overrightarrow{\mathbf{r}}\_{1}\right) \left[ \mathbf{f}\_{\text{ext}}^{i}\left(\boldsymbol{\omega},\overrightarrow{\mathbf{r}}\_{1}\right) + \mathbf{f}\_{1}^{i}\left(\boldsymbol{\omega},\overrightarrow{\mathbf{r}}\_{1}\right) \right] d\overrightarrow{\mathbf{r}}\_{1} \tag{13}$$

where **r** !, **r** ! 1 � �<sup>∈</sup> <sup>Ω</sup>, **<sup>f</sup>** *i ext*ð Þ *ω*, **r**<sup>1</sup> and **f** *i* <sup>1</sup>ð Þ *ω*, **r**<sup>1</sup> are the projections of vectors **f** ! *ext ω*, **r** ! � � and **<sup>f</sup>** ! <sup>1</sup> *ω*, **r** ! � � on to the coordinate axes. Integration is performed over the volume occupied by inhomogeneities, which are secondary sources of the field. In practice, the volume should be chosen such that secondary and higher order sources make a noticeable contribution to the field. Due to the rapid decrease in the amplitude of the Green's function and, especially with a strong absorption of the electromagnetic field by the medium, the region of integration can be about 1/α where α is the absorption coefficient of the field.

The steps for finding **E** ! **r** !,*ω* � � by the method of successive approximations can be as follows. We find the zeroth approximation **E** ! <sup>0</sup> **r** !,*ω* � � for the field, which is valid in a homogeneous medium with parameters *σc*, *μc*, *ε<sup>c</sup>*

$$\overrightarrow{\mathbf{E}}\_{0}(\overrightarrow{\mathbf{r}},\boldsymbol{\omega}) = \sum\_{i=1}^{3} \overrightarrow{\mathbf{n}}\_{i} \left[ G \left( \overrightarrow{\mathbf{r}} - \overrightarrow{\mathbf{r}}\_{1} \right) \left[ \left( \overrightarrow{\mathbf{n}}\_{i} \cdot \overrightarrow{\mathbf{f}}\_{\text{ext}} \left( \boldsymbol{\omega}, \overrightarrow{\mathbf{r}}\_{1} \right) \right) \right] d\overrightarrow{\mathbf{r}}\_{1} \right] \tag{14}$$

Integration is performed over the volume *Ω*<sup>1</sup> occupied by the external current (primary source of the field). This solution describes the primary field created by an external current. Using obtained by Eq. (14) expression **E** ! <sup>0</sup> **r** !,*ω* � � and expression (9), we find **f***<sup>i</sup>* <sup>1</sup> *ω*, **r** ! 1 � �. Using (Eq. (13)) and integrating, we obtain a more accurate first **E** ! <sup>1</sup> **r** !, *ω* � � approximation for **<sup>E</sup>** ! **r** !,*ω* � �, which takes into account the influence of medium inhomogeneities on the field. To find the second approximation, it is necessary to substitute **E** ! <sup>1</sup> **r** !,*ω* � � into **<sup>f</sup>***<sup>i</sup>* <sup>1</sup> *ω*, **r** ! 1 � � and using (Eq. (13)) to obtain the

second more accurate approximation. Similarly, more accurate solutions are obtained that take into account multiple field scattering by medium inhomogeneities. At these stages, the integration is performed over the volume occupied by inhomogeneities, which are secondary sources of the field. If the source of the field in an inhomogeneous medium is an external field with an electric vector **E** ! *ext* **r** !, *ω* � � it should be used as the vector **E** ! <sup>0</sup> **r** !, *ω* � �.

For determining the magnetic field component one uses Maxwell's equations and writes them in terms of magnetic and electric fields Fourier spectrums: ∇ �

$$\overrightarrow{\mathbf{E}}\left(\overrightarrow{\mathbf{r}},\omega\right) = -\omega \overrightarrow{B}\left(\overrightarrow{r},\omega\right). \text{ Substituting (Eq. (13)) in this equation one obtains } \overrightarrow{\mathbf{r}}$$

$$\overrightarrow{\mathbf{H}}\left(\overrightarrow{\mathbf{r}},\omega\right) = i\frac{\mathbf{1}}{\alpha\mu\left(\overrightarrow{\mathbf{r}}\right)}\left[\sum\_{i=1}^{3}\overrightarrow{\mathbf{n}}\_{i} \times \nabla G\left(\overrightarrow{\mathbf{r}} - \overrightarrow{\mathbf{r}}\_{1}\right)\right] \mathbf{f}\_{\text{ext}}^{i}\left(\boldsymbol{\omega},\overrightarrow{\mathbf{r}}\_{1}\right) + \mathbf{f}\_{1}^{i}\left(\boldsymbol{\omega},\overrightarrow{\mathbf{r}}\_{1}\right)\right]d\overrightarrow{\mathbf{r}}\_{1}.\tag{14a}$$

Using the Green's function, as the experience of its use shows (e.g. [7]) in such tasks, significantly reduces the requirements for computing resources and reduces the computation time. Note that the proposed procedure can be effective in simulating the optical properties of metamaterials, nanocomposites, and nanostructures.
