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Electromagnetic scattering from radially inhomogeneous cylindrical structures using the state space method

Noushin Vaseghi and Mohammad Sadegh Abrishamian

Open Access Open Access

Abstract

In this paper, the state space method is developed for analysis of the scattering of radially inhomogeneous cylindrical structures. The developed technique is based on obtaining and using the state transition matrix of cylindrical layers. The most valuable and interesting property of the presented method is its systematic approach, allowing one to easily apply this method to any types of cylindrical inhomogeneous media and simply implement it in a programming language supporting matrix manipulations. The validity of the presented method as well as the correctness of the formulation is achieved by providing some numerical examples and comparing the obtained results with the results of previously published works.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Waves and fields in inhomogeneous media is a recognized subject of major importance when it comes to electromagnetic research. The exact solution of the wave equation in such media is only applicable to a few particular problems; thus, scattering from inhomogeneous media has been intensively investigated and several approaches have been presented such as Richmond method [1], solving Riccati equation [2], full-wave analysis [3–6], finite difference method [7], Taylor and Fourier series expansions [8,9], method of moments [10], cascading thin linear layers method [11] and so on.

In addition, several methods have been introduced for the analysis of scattering from inhomogeneous cylindrical Structures. The problem of electromagnetic scattering by a two-dimensional cylinder has only exact solutions for a limited class of geometries [12–14]. When the cylindrical structure is coated with an inhomogeneous material, the problem should be solved numerically. Over the years, numerous methods have been presented such as surface integral equations [15–17], surface-volume integral equations [18–20], hybrid finite element method [21] and so on.

As an advantageous, the surface integral equation is only valid for homogeneous and piecewise homogeneous materials. In most of published works, the efficiency of proposed methods is not clearly manifested for handling inhomogeneous materials.

In this contribution, the objective of the present work is to characterize electromagnetic scattering from radially inhomogeneous cylindrical structure using the state space method. Here, the proposed method consists of using the state space method by exploiting the state transition matrix for determining scattering cross section from radially inhomogeneous cylindrical structures. Generally, application of common procedures of determining the RCS of radially inhomogeneous cylindrical structures is not simply possible. The state transition matrix method has been well described in scattering problems, including isotropic, anisotropic, and bi-anisotropic planar layered media, over the years [22–24]. This study presents the details of its application in the formulation for scattering problems including inhomogeneous cylindrical structures.

The organization of this paper is as follows. In Section 2, the basic concepts of state space method will be presented. Section 3 deals with the formulation of state transition matrix method for inhomogeneous cylindrical structures. The proposed method will be verified along with example computations in Section 4. Finally, summary and conclusions are made in Section 5.

2. Basic concepts of the state of space approach

Consider a radially inhomogeneous cylindrical shell as shown in Fig. 1 characterized by a set of constitutive relations.

$$\begin{array}{l} {\bar{D} = \varepsilon ({\rho ,\omega } )\bar{E}}\\ {\bar{B} = \mu ({\rho ,\omega } )\bar{H}} \end{array}$$
$$\left\{ {\begin{array}{l} {\nabla \times \bar{H} = j\omega \varepsilon ({\rho ,\omega } )\bar{E}\; \; \; }\\ {\nabla \times \bar{E} ={-} j\omega \mu ({\rho ,\omega } )\bar{H}} \end{array}} \right.$$
where ${\mathrm{\varepsilon}}({\mathrm{\rho},\mathrm{\omega}} )$ and $\mathrm{\mu}({\mathrm{\rho},\mathrm{\omega}} )$ are permittivity and permeability respectively.

 figure: Fig. 1.

Fig. 1. Geometry of the scattering problem, the radially inhomogeneous cylindrical coating is assumed to be homogeneous in the $\varphi$ and Z directions

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The cylindrical structure is infinite extent along the z-direction. So, when the oblique wave incidences on the structure, we have ${\raise0.7ex\hbox{$\partial $} \!\mathord{\left/ {\vphantom {\partial {\partial z}}} \right.}\!\lower0.7ex\hbox{${\partial z}$}} = j{k_z}$, where ${k_z} = {k_0}\cos {\theta _i}$, and in order to satisfy the periodicity condition along φ, where ${\raise0.7ex\hbox{$\partial $} \!\mathord{\left/ {\vphantom {\partial {\partial \varphi }}} \right.}\!\lower0.7ex\hbox{${\partial \varphi }$}} = jn$. By eliminating ρ-components of electric and magnetic fields from curl Maxwell’s equations we can write:

$$\frac{\partial }{\partial \rho}\left[ {\begin{array}{c} {{{\bar{E}}_T}}\\ {{{\bar{H}}_T}} \end{array}} \right] = \Gamma \left[ {\begin{array}{c} {{{\bar{E}}_T}}\\ {{{\bar{H}}_T}} \end{array}} \right]$$
where ${\bar{E}_T} = ({{E_z},{E_\varphi }} )$ and ${\bar{H}_T} = ({{H_z},{H_\varphi }} )$ are transverse components of electric and magnetic fields, respectively, and $\Gamma $-matrix is given by:
$${\Gamma ^n} = \left[ {\begin{array}{@{}cccc@{}} 0&0&{\frac{{jn{\beta_z}}}{{\rho \omega \varepsilon ({\rho ,\omega } )}}}&{ + j\omega \mu ({\rho ,\omega } )- \frac{{j\beta_z^2}}{{\omega \varepsilon ({\rho ,\omega } )}}}\\ 0&{\frac{{ - 1}}{\rho }}&{\frac{{j{n^2}}}{{{\rho^2}\omega \varepsilon ({\rho ,\omega } )}} - j\omega \mu ({\rho ,\omega } )}&{\frac{{ - jn{\beta_z}}}{{\rho \omega \varepsilon ({\rho ,\omega } )}}}\\ {\frac{{ - jn{\beta_z}}}{{\rho \omega \mu ({\rho ,\omega } )}}}&{\frac{{j\beta_z^2}}{{\omega \mu ({\rho ,\omega } )}} - j\omega \varepsilon }&0&0\\ { + j\omega \varepsilon ({\rho ,\omega } )- \frac{{j{n^2}}}{{{\rho^2}\omega \mu ({\rho ,\omega } )}}}&{\frac{{jn{k_z}}}{{\rho \omega \mu ({\rho ,\omega } )}}}&0&{\frac{{ - 1}}{\rho }} \end{array}} \right]$$
where ω is the angular frequency. Notice that above equations are valid for any integer n.

3. Theory and formulation

3.1. State transition matrix of an inhomogeneous cylindrical shell

Defining a 4×4 state transition matrix-$\phi$ with 2×2 sub-matrices $({{\phi_1},{\phi_2},{\phi_3},{\phi_4}} )$ as it relates to the transverse components of electric and magnetic fields at the two boundaries of the inhomogeneous cylindrical shell, we can write

$${\left[ {\begin{array}{c} {{{\bar{E}}_T}}\\ {{{\bar{H}}_T}} \end{array}} \right]_{\rho = b}} = \phi {\left. {\left[ {\begin{array}{c} {{{\bar{E}}_T}}\\ {{{\bar{H}}_T}} \end{array}} \right]} \right|_{\rho = a}} = \left[ {\begin{array}{cc} {{\phi_1}}&{{\phi_2}}\\ {{\phi_3}}&{{\phi_4}} \end{array}} \right]{\left. {\left[ {\begin{array}{c} {{{\bar{E}}_T}}\\ {{{\bar{H}}_T}} \end{array}} \right]} \right|_{\rho = a}}$$

The computation of state transition matrix of an inhomogeneous cylindrical layer is more complicated than that of a cylindrical homogeneous layer. In fact, if the layer is homogeneous, similar to state-space analysis in linear time-invariant systems, the STM is exp(-Γ(b-a)). For the computation of such a matrix, many methods have been proposed including expansion of Φ in a power series, Cayley–Hamilton theorem and etc [25]. For inhomogeneous layers, Γ-matrix is dependent on ρ and so the state transition matrix cannot be computed using exp(-Γ(b-a)). The most straight forward method is subdividing the inhomogeneous slab into N homogeneous electrically thin layers, as shown in Fig. 2. Then, the state transition matrix of inhomogeneous cylindrical layer can be written as

$$\phi = {\phi ^{({layer1} )}}{\phi ^{({layer2} )}} \ldots {\phi ^{({layerN} )}}$$

 figure: Fig. 2.

Fig. 2. Modeling of an inhomogeneous cylindrical layer as a stratified structure of N homogeneous layers

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Notice that using more homogeneous layers can yield a more accurate state transition matrix for inhomogeneous layer. Once the state transition matrix of inhomogeneous cylindrical layer is determined, reflection and transmission coefficients can be subsequently identified.

3.2. Reflection and transmission matrices

By introducing the reflection and transmission matrices of [R] and [T] as

$${ {E_T^r} |_{\rho = b}} = [R ]{ {E_T^i} |_{\rho = b}} = \left[ {\begin{array}{cc} {{R_{zz}}}&{{R_{z\varphi }}}\\ {{R_{\varphi z}}}&{{R_{\varphi \varphi }}} \end{array}} \right]{ {E_T^i} |_{\rho = b}}$$
$${ {E_T^t} |_{\rho = a}} = [T ]{ {E_T^i} |_{\rho = b}} = \left[ {\begin{array}{cc} {{T_{zz}}}&{{T_{z\varphi }}}\\ {{T_{\varphi z}}}&{{T_{\varphi \varphi }}} \end{array}} \right]{ {E_T^i} |_{\rho = b}}$$
where superscripts i, r and t denote the incident, reflected and transmitted field, respectively. As proved in Appendix, we can write
$$\begin{aligned} & [R] = {({Z_r^{ - 1} - Z_i^{ - 1}} )^{ - 1}}[{\phi {}_3 + Z_t^{ - 1}\phi {}_4 - Z_i^{ - 1}\phi {}_1 - Z_i^{ - 1}{\phi_2}\,Z_t^{ - 1}} ]\\ &\qquad\qquad {[{\phi {}_3 + Z_t^{ - 1}\phi {}_4 - Z_r^{ - 1}\phi {}_1 - Z_r^{ - 1}{\phi_2}\,Z_t^{ - 1}} ]^{ - 1}}({Z_i^{ - 1} - Z_r^{ - 1}} )\end{aligned}$$
$$[T ]= {({\phi {}_3 + Z_t^{ - 1}\phi {}_4 - Z_r^{ - 1}\phi {}_1 - Z_r^{ - 1}{\phi_2}\,Z_t^{ - 1}} )^{ - 1}}({Z_i^{ - 1} - Z_r^{ - 1}} )$$
and wave impedance matrices are defined as
$$\left\{ {\begin{array}{l} {E_T^i(b )= {Z_1}H_T^i(b )}\\ {E_T^r(b )= {Z_2}H_T^r(b )}\\ {E_T^t(a )= {Z_3}H_T^t(a )} \end{array}} \right.$$
where in
$${Z_1} = {\left. {\left[ {\begin{array}{cc} 0&{j\omega \mu \frac{{{J_n}({k\rho } )}}{{\frac{\partial }{{\partial \rho }}[{{J_n}({k\rho } )} ]}}}\\ {\frac{{ - 1}}{{j\omega \varepsilon }}\frac{{\frac{\partial }{{\partial \rho }}[{{J_n}({k\rho } )} ]}}{{{J_n}({k\rho } )}}}&0 \end{array}} \right]} \right|_{\rho = b}}$$
$${Z_2} = {\left. {\left[ {\begin{array}{cc} 0&{j\omega \mu \frac{{H_n^2({k\rho } )}}{{\frac{\partial }{{\partial \rho }}[{H_n^2({k\rho } )} ]}}}\\ {\frac{{ - 1}}{{j\omega \varepsilon }}\frac{{\frac{\partial }{{\partial \rho }}[{H_n^2({k\rho } )} ]}}{{H_n^2({k\rho } )}}}&0 \end{array}} \right]} \right|_{\rho = b}}$$
$${Z_3} = {\left. {\left[ {\begin{array}{cc} 0&{j\omega \mu \frac{{{J_n}({k\rho } )}}{{\frac{\partial }{{\partial \rho }}[{{J_n}({k\rho } )} ]}}}\\ {\frac{{ - 1}}{{j\omega \varepsilon }}\frac{{\frac{\partial }{{\partial \rho }}[{{J_n}({k\rho } )} ]}}{{{J_n}({k\rho } )}}}&0 \end{array}} \right]} \right|_{\rho = a}}$$

3.3. Scattering cross section of cylindrical structures

At the end of this section, whereas it is often desired to know the electromagnetic waves scattering at large distances, we would like to get this by using the reflection matrix [R]. Considering presented discussions in [26], 2D scattering cross section for TE and TM cases can be written as:

$$\sigma _{2D}^{TM} = \frac{{4c}}{\omega }{\left|{\mathop \sum \limits_{n ={-} \infty }^\infty {R_{zz}}(n )\frac{{{J_n}\left( {\frac{\omega }{c}b} \right)}}{{H_n^{(2 )}\left( {\frac{\omega }{c}b} \right)}}{e^{jn\varphi }}} \right|^2}$$
$$\sigma _{2D}^{TE} = \frac{{4c}}{\omega }{\left|{\mathop \sum \limits_{n ={-} \infty }^\infty {R_{\varphi \varphi }}(n )\frac{{\mathop {{J_n}}\limits^{\prime} \left( {\frac{\omega }{c}b} \right)}}{{H_n^{{{(2 )}^{\prime}}}\left( {\frac{\omega }{c}b} \right)}}{e^{jn\varphi }}} \right|^2}$$
where ${\textrm{R}_{\textrm{zz}}}(\textrm{n} )$ and ${\textrm{R}_{{\varphi \varphi }}}(\textrm{n} )$ are the computed reflection coefficients assuming n-th harmonic.

4. Numerical examples and results

In this section, three examples are provided to illustrate the accuracy of the proposed method in the computation of scattering cross section of radially inhomogeneous cylindrical structures. Notice that due to computational goals and limitations, we should truncate summations in (15) and Eq. (16) at a positive integer M.

4.1. Example 1: Cylindrical homogeneous dielectric shell in free space

As the first example, consider a cylindrical homogeneous dielectric shell with inner radius $a = 0.25\lambda$ and outer radius $b = 0.3\lambda$ with characteristics ${\varepsilon _r} = 4$ and ${\mu _r} = 1$. The normalized TE and TM scattering cross sections of this structure assuming $M = 20$ is computed by the proposed method and compared with presented result in [28]. As shown in Fig. 3 and Fig. 4, there is an excellent agreement between the results.

 figure: Fig. 3.

Fig. 3. Normalized scattering cross section of homogeneous dielectric cylindrical shell for (a) $T{M_z}$ and (b) $T{E_z}$ polarizations [27].

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 figure: Fig. 4.

Fig. 4. Normalized scattering cross section of the conducting cylinder coated with an inhomogeneous dielectric layer by steady state method and exact method presented in 26.

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4.2. Example 2: Conducting cylinder with an inhomogeneous dielectric coating

In this example, we consider a conducting cylinder coated with an inhomogeneous dielectric layer. Assume a conducting cylinder with radius a=2cm with an inhomogeneous dielectric coating of ${\varepsilon _r}(\rho )= 11 - 5({\rho - a} )/({b - a} )$ and ${\mu _r} = 1$ with thickness 0.33 cm at frequency of 9 GHz. The normalized scattering cross section of this structure is computed by the proposed method and plotted versus the angle of observation from 0° to 180° for TM polarization. An excellent agreement can be seen between the obtained result by the proposed method and the presented one in [28]. The time consumed for this examples presented in this paper is less than 10 seconds using a computer with Intel Core I5 CPU and MATLAB program.

5. Conclusion

A technique has developed for calculating the scattered fields of a dielectric cylinder or cylindrical shell of arbitrary cross section shape. The proposed method can be summarized as determining Γ-matrix of an inhomogeneous layer, determining state transition matrix-Φ, computation of reflection and transmission matrices and finally computation of scattering cross section.

All of these steps have been presented in detail in this paper. Finally, several examples of validation of scattering cross section of inhomogeneous cylindrical structures have been considered in the last section. Since the presented approach is very systematic, it can be simply implemented in programming languages supporting matrix manipulations such as MATLAB. The consumed time for this method is much less than that for numerical methods such as finite difference. In future, the proposed method is expected to be extended and used for reconstructing constitutive parameters of more complex structures such as anisotropic or bianisotropic cylindrical and spherical inhomogeneous structures [29,30].

Appendix

Appendix A. State space matrix $\Gamma$

For getting state space matrix-$\Gamma $, we expand first and second curl Maxwell’s equations

$$\left\{ {\begin{array}{l} {\frac{1}{\rho }\left( {\frac{{\partial {H_z}}}{{\partial \varphi }} - \frac{{\partial ({\rho {H_\varphi }} )}}{{\partial z}}} \right) = j\omega \varepsilon {E_\rho }\; \; \; \; }\\ {\frac{{\partial {H_\rho }}}{{\partial z}} - \frac{{\partial {H_z}}}{{\partial \rho }} = j\omega \varepsilon {E_\varphi }\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }\\ {\frac{1}{\rho }\left( {\frac{{\partial ({\rho {H_\varphi }} )}}{{\partial \rho }} - \frac{{\partial ({{H_\rho }} )}}{{\partial \varphi }}} \right) = j\omega \varepsilon {E_z}} \end{array}} \right.$$
$$\left\{ {\begin{array}{l} {\frac{1}{\rho }\left( {\frac{{\partial {E_z}}}{{\partial \varphi }} - \frac{{\partial ({\rho {E_\varphi }} )}}{{\partial z}}} \right) ={-} j\omega \mu {H_\rho }}\\ {\frac{{\partial {E_\rho }}}{{\partial z}} - \frac{{\partial {E_z}}}{{\partial \rho }} ={-} j\omega \mu {H_\varphi }\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }\\ {\frac{1}{\rho }\left( {\frac{{\partial ({\rho {E_\varphi }} )}}{{\partial \rho }} - \frac{{\partial {E_\rho }}}{{\partial \varphi }}} \right) ={-} j\omega \mu {H_z}} \end{array}} \right.$$
By eliminating $\rho$–components of electric and magnetic fields from above equations we have
$$\frac{\partial }{{\partial \rho }}\left[ {\begin{array}{l} {{E_z}}\\ {{E_\varphi }}\\ {{H_z}}\\ {{H_\varphi }} \end{array}} \right] = \Gamma \left[ {\begin{array}{l} {{E_z}}\\ {{E_\varphi }}\\ {{H_z}}\\ {{H_\varphi }} \end{array}} \right]$$
$$\left\{ \begin{array}{l} \frac{{\partial {E_z}}}{{\partial \rho }} ={+} j\omega \mu {H_\varphi }\\ \frac{{\partial {E_\varphi }}}{{\partial \rho }} ={-} j\omega \mu {H_z} - \frac{1}{\rho }{E_\varphi } + \frac{1}{\rho }\frac{{\partial {E_\rho }}}{{\partial \varphi }} ={-} \frac{1}{\rho }{E_\varphi } + \left[ { - j\omega \mu + \frac{{j{n^2}}}{{\omega \varepsilon {\rho^2}}}} \right]{H_z} \end{array} \right.$$
$$\left\{ \begin{array}{l} \frac{{\partial {H_z}}}{{\partial \rho }} ={-} j\omega \varepsilon {E_\varphi }\\ \frac{{\partial {H_\varphi }}}{{\partial \rho }} ={-} j\omega \varepsilon {E_z} - \frac{1}{\rho }{H_\varphi } + \frac{1}{\rho }\frac{{\partial {H_\rho }}}{{\partial \varphi }} ={-} \frac{1}{\rho }{H_\varphi } + \left[ {j\omega \varepsilon - \frac{{j{n^2}}}{{\omega \mu {\rho^2}}}} \right]{E_z} \end{array} \right.$$
which is given in Eq. (4).

Appendix B. Impedance matrices ${Z_{1,2,3}}$

With considering both $T{E_Z}$ and $T{M_Z}$ polarizations, we can determine ${Z_1}$, ${Z_2}$ and ${Z_3}$. For example if we suppose $T{M_Z}$ polarization we will have

$$\left\{ {\begin{array}{l} {E_z^i = \mathop \sum \limits_{n ={-} \infty }^\infty {j^{ - n}}{J_n}({k\rho } ){e^{jn\varphi }}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }\\ {H_\varphi^i = \frac{1}{{j\omega \mu }}\frac{{\partial {E_z}}}{{\partial \rho }} = \frac{1}{{j\omega \mu }}\mathop \sum \limits_{n ={-} \infty }^\infty {j^{ - n}}\frac{\partial }{{\partial \rho }}[{{J_n}({k\rho } )} ]{e^{jn\varphi }}} \end{array}} \right.$$
$${\left[ {\begin{array}{c} {E_z^i}\\ {E_\varphi^i} \end{array}} \right]_{\rho = b}} = \left[ {\begin{array}{cc} 0&A\\ B&0 \end{array}} \right]{\left. {\left[ {\begin{array}{c} {H_z^i}\\ {H_\varphi^i} \end{array}} \right]} \right|_{\rho = b}}$$
and we can get A from this by solving systems of equations for A. So, we have
$${\left. {A\,.\,\frac{1}{{j\omega \mu }}\,\frac{\partial }{{\partial \rho }}[{{J_n}(k\rho )\,} ]} \right|_b} = {J_n}(kb)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,A = \frac{{j\omega \mu .{J_n}(kb)}}{{{{\left. {\,\frac{\partial }{{\partial \rho }}[{{J_n}(k\rho )\,} ]} \right|}_b}}}$$
and for $T{E_Z}$ polarization
$$\left\{ {\begin{array}{l} {\textrm{H}_\textrm{z}^\textrm{i} = \mathop \sum \limits_{\textrm{n} ={-} \infty }^\infty {\textrm{j}^{ - \textrm{n}}}{\textrm{J}_\textrm{n}}({\textrm{k}\mathrm{\rho}} ){\textrm{e}^{\textrm{jn}{\mathrm{\varphi}}}}}\\ {\textrm{E}_{\mathrm{\varphi}}^\textrm{i} = \frac{{ - 1}}{{\textrm{j}\mathrm{\omega}{\mathrm{\varepsilon}}}}\frac{{\partial {\textrm{H}_\textrm{z}}}}{{\partial \mathrm{\rho}}} = \frac{{ - 1}}{{\textrm{j}\mathrm{\omega}{\mathrm{\varepsilon}}}}\mathop \sum \limits_{\textrm{n} ={-} \infty }^\infty {\textrm{j}^{ - \textrm{n}}}\frac{\partial }{{\partial \mathrm{\rho}}}[{{\textrm{J}_\textrm{n}}({\textrm{k}\mathrm{\rho}} )} ]{\textrm{e}^{\textrm{jn}{\mathrm{\varphi}}}}} \end{array}} \right.$$
and we can get B from this by solving systems of equations for B. So, we have
$$\begin{aligned}&B\mathop \sum \limits_{n ={-} \infty }^\infty {j^{ - n}}{J_n}({k\rho } ){e^{jn\varphi }} = \frac{{ - 1}}{{j\omega \varepsilon }}\mathop \sum \limits_{n ={-} \infty }^\infty {j^{ - n}}\frac{\partial }{{\partial \rho }}[{{J_n}({k\rho } )} ]{e^{jn\varphi }}\\ &\Rightarrow B = \frac{{ - 1}}{{j\omega \varepsilon }}\frac{{\frac{\partial }{{\partial \rho }}{{[{{J_n}({k\rho } )} ]}_b}}}{{{J_n}({kb} )}} \end{aligned}$$
In a similar manner we can determine ${Z_2}$ and ${Z_3}$ as given in Eqs. (13) and (14).

Appendix C. Reflection and transmission matrices R and T

After getting impedance matrices, we can determine reflection and transmission matrices to calculate scattering from the cylindrical structure. Considering Eq. (5) and Eq. (11), we can write

$$E_T^i(b )+ E_T^r(b )= \phi {}_1E_T^t(a )+ {\phi _2}\,Z_3^{ - 1}\,E_T^t(a )$$
$$H_T^i(b )+ H_T^r(b )= \phi {}_3E_T^t(a )+ {\phi _4}\,Z_3^{ - 1}\,E_T^t(a )$$
By multiplying Eq. (27) by $\textrm{Z}_2^{ - 1}$, we obtain
$$Z_2^{ - 1}\,E_T^i(b )+ Z_2^{ - 1}E_T^r(b )= Z_2^{ - 1}\phi {}_1E_T^t(a )+ Z_2^{ - 1}{\phi _2}\,Z_3^{ - 1}\,E_T^t(a )$$
Subtracting Eq. (29) from Eq. (28), yields the following equation
$$({Z_1^{ - 1} - Z_2^{ - 1}} )\,E_T^i(b )= ({\phi {}_3 + Z_3^{ - 1}\phi {}_4 - Z_2^{ - 1}\phi {}_1 - Z_2^{ - 1}{\phi_2}\,Z_3^{ - 1}} )\,E_T^t(a )$$
Similarly, multiplying Eq. (27) by $\textrm{Z}_1^{ - 1}$, we get
$$Z_1^{ - 1}E_T^i(b )+ Z_1^{ - 1}E_T^r(b )= [{Z_1^{ - 1}\phi {}_1 + Z_1^{ - 1}{\phi_2}\,Z_3^{ - 1}} ]E_T^t(a )$$
and then subtracting Eq. (31) from Eq. (28) yields
$$({Z_2^{ - 1} - Z_1^{ - 1}} )E_T^r(b )= [{\phi {}_3 + Z_3^{ - 1}\phi {}_4 - Z_1^{ - 1}\phi {}_1 - Z_1^{ - 1}{\phi_2}\,Z_3^{ - 1}} ]E_T^t(a )$$
Solving Eq. (30) with respect to $\textrm{E}_\textrm{T}^{\textrm{t}}(\textrm{a} )$ and substituting into Eq. (32) and then multiplying by inverse $({\textrm{Z}_2^{ - 1} - \textrm{Z}_1^{ - 1}} )$, we finally get
$${({Z_2^{ - 1} - Z_1^{ - 1}} )^{ - 1}}({Z_2^{ - 1} - Z_1^{ - 1}} )E_T^r(b )= {({Z_2^{ - 1} - Z_1^{ - 1}} )^{ - 1}}[{\phi {}_3 + Z_3^{ - 1}\phi {}_4 - Z_1^{ - 1}\phi {}_1 - Z_1^{ - 1}{\phi_2}\,Z_3^{ - 1}} ]E_T^t(a )$$
So we can write reflection matrix as given in Eq. (9). As a similar manner, subtracting Eq. (29) from Eq. (28) yields the transmission matrix as follows
$$\begin{aligned} & E_T^r(b )= {({Z_2^{ - 1} - Z_1^{ - 1}} )^{ - 1}}[{\phi {}_3 + Z_3^{ - 1}\phi {}_4 - Z_1^{ - 1}\phi {}_1 - Z_1^{ - 1}{\phi_2}\,Z_3^{ - 1}} ]\\ & \qquad \qquad{[{\phi {}_3 + Z_3^{ - 1}\phi {}_4 - Z_2^{ - 1}\phi {}_1 - Z_2^{ - 1}{\phi_2}\,Z_3^{ - 1}} ]^{ - 1}}({Z_1^{ - 1} - Z_2^{ - 1}} )E_T^i(b )\end{aligned}$$
So we can write transmission matrix as given in Eq. (10).

Disclosures

The authors declare no conflicts of interest.

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Figures (4)
Fig. 1.
Fig. 1. Geometry of the scattering problem, the radially inhomogeneous cylindrical coating is assumed to be homogeneous in the $\varphi$ and Z directions

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Fig. 2.
Fig. 2. Modeling of an inhomogeneous cylindrical layer as a stratified structure of N homogeneous layers

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Fig. 3.
Fig. 3. Normalized scattering cross section of homogeneous dielectric cylindrical shell for (a) $T{M_z}$ and (b) $T{E_z}$ polarizations [27].

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Fig. 4.
Fig. 4. Normalized scattering cross section of the conducting cylinder coated with an inhomogeneous dielectric layer by steady state method and exact method presented in 26.

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Equations (34)

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D ¯ = ε ( ρ , ω ) E ¯ B ¯ = μ ( ρ , ω ) H ¯
{ ∇ × H ¯ = j ω ε ( ρ , ω ) E ¯ ∇ × E ¯ = − j ω μ ( ρ , ω ) H ¯
∂ ∂ ρ [ E ¯ T H ¯ T ] = Γ [ E ¯ T H ¯ T ]
Γ n = [ 0 0 j n β z ρ ω ε ( ρ , ω ) + j ω μ ( ρ , ω ) − j β z 2 ω ε ( ρ , ω ) 0 − 1 ρ j n 2 ρ 2 ω ε ( ρ , ω ) − j ω μ ( ρ , ω ) − j n β z ρ ω ε ( ρ , ω ) − j n β z ρ ω μ ( ρ , ω ) j β z 2 ω μ ( ρ , ω ) − j ω ε 0 0 + j ω ε ( ρ , ω ) − j n 2 ρ 2 ω μ ( ρ , ω ) j n k z ρ ω μ ( ρ , ω ) 0 − 1 ρ ]
[ E ¯ T H ¯ T ] ρ = b = ϕ [ E ¯ T H ¯ T ] | ρ = a = [ ϕ 1 ϕ 2 ϕ 3 ϕ 4 ] [ E ¯ T H ¯ T ] | ρ = a
ϕ = ϕ ( l a y e r 1 ) ϕ ( l a y e r 2 ) … ϕ ( l a y e r N )
E T r | ρ = b = [ R ] E T i | ρ = b = [ R z z R z φ R φ z R φ φ ] E T i | ρ = b
E T t | ρ = a = [ T ] E T i | ρ = b = [ T z z T z φ T φ z T φ φ ] E T i | ρ = b
[ R ] = ( Z r − 1 − Z i − 1 ) − 1 [ ϕ 3 + Z t − 1 ϕ 4 − Z i − 1 ϕ 1 − Z i − 1 ϕ 2 Z t − 1 ] [ ϕ 3 + Z t − 1 ϕ 4 − Z r − 1 ϕ 1 − Z r − 1 ϕ 2 Z t − 1 ] − 1 ( Z i − 1 − Z r − 1 )
[ T ] = ( ϕ 3 + Z t − 1 ϕ 4 − Z r − 1 ϕ 1 − Z r − 1 ϕ 2 Z t − 1 ) − 1 ( Z i − 1 − Z r − 1 )
{ E T i ( b ) = Z 1 H T i ( b ) E T r ( b ) = Z 2 H T r ( b ) E T t ( a ) = Z 3 H T t ( a )
Z 1 = [ 0 j ω μ J n ( k ρ ) ∂ ∂ ρ [ J n ( k ρ ) ] − 1 j ω ε ∂ ∂ ρ [ J n ( k ρ ) ] J n ( k ρ ) 0 ] | ρ = b
Z 2 = [ 0 j ω μ H n 2 ( k ρ ) ∂ ∂ ρ [ H n 2 ( k ρ ) ] − 1 j ω ε ∂ ∂ ρ [ H n 2 ( k ρ ) ] H n 2 ( k ρ ) 0 ] | ρ = b
Z 3 = [ 0 j ω μ J n ( k ρ ) ∂ ∂ ρ [ J n ( k ρ ) ] − 1 j ω ε ∂ ∂ ρ [ J n ( k ρ ) ] J n ( k ρ ) 0 ] | ρ = a
σ 2 D T M = 4 c ω | ∑ n = − ∞ ∞ ⁡ R z z ( n ) J n ( ω c b ) H n ( 2 ) ( ω c b ) e j n φ | 2
σ 2 D T E = 4 c ω | ∑ n = − ∞ ∞ ⁡ R φ φ ( n ) J n ′ ⁡ ( ω c b ) H n ( 2 ) ′ ( ω c b ) e j n φ | 2
{ 1 ρ ( ∂ H z ∂ φ − ∂ ( ρ H φ ) ∂ z ) = j ω ε E ρ ∂ H ρ ∂ z − ∂ H z ∂ ρ = j ω ε E φ 1 ρ ( ∂ ( ρ H φ ) ∂ ρ − ∂ ( H ρ ) ∂ φ ) = j ω ε E z
{ 1 ρ ( ∂ E z ∂ φ − ∂ ( ρ E φ ) ∂ z ) = − j ω μ H ρ ∂ E ρ ∂ z − ∂ E z ∂ ρ = − j ω μ H φ 1 ρ ( ∂ ( ρ E φ ) ∂ ρ − ∂ E ρ ∂ φ ) = − j ω μ H z
∂ ∂ ρ [ E z E φ H z H φ ] = Γ [ E z E φ H z H φ ]
{ ∂ E z ∂ ρ = + j ω μ H φ ∂ E φ ∂ ρ = − j ω μ H z − 1 ρ E φ + 1 ρ ∂ E ρ ∂ φ = − 1 ρ E φ + [ − j ω μ + j n 2 ω ε ρ 2 ] H z
{ ∂ H z ∂ ρ = − j ω ε E φ ∂ H φ ∂ ρ = − j ω ε E z − 1 ρ H φ + 1 ρ ∂ H ρ ∂ φ = − 1 ρ H φ + [ j ω ε − j n 2 ω μ ρ 2 ] E z
{ E z i = ∑ n = − ∞ ∞ ⁡ j − n J n ( k ρ ) e j n φ H φ i = 1 j ω μ ∂ E z ∂ ρ = 1 j ω μ ∑ n = − ∞ ∞ ⁡ j − n ∂ ∂ ρ [ J n ( k ρ ) ] e j n φ
[ E z i E φ i ] ρ = b = [ 0 A B 0 ] [ H z i H φ i ] | ρ = b
A . 1 j ω μ ∂ ∂ ρ [ J n ( k ρ ) ] | b = J n ( k b ) ⇒ A = j ω μ . J n ( k b ) ∂ ∂ ρ [ J n ( k ρ ) ] | b
{ H z i = ∑ n = − ∞ ∞ ⁡ j − n J n ( k ρ ) e jn φ E φ i = − 1 j ω ε ∂ H z ∂ ρ = − 1 j ω ε ∑ n = − ∞ ∞ ⁡ j − n ∂ ∂ ρ [ J n ( k ρ ) ] e jn φ
B ∑ n = − ∞ ∞ ⁡ j − n J n ( k ρ ) e j n φ = − 1 j ω ε ∑ n = − ∞ ∞ ⁡ j − n ∂ ∂ ρ [ J n ( k ρ ) ] e j n φ ⇒ B = − 1 j ω ε ∂ ∂ ρ [ J n ( k ρ ) ] b J n ( k b )
E T i ( b ) + E T r ( b ) = ϕ 1 E T t ( a ) + ϕ 2 Z 3 − 1 E T t ( a )
H T i ( b ) + H T r ( b ) = ϕ 3 E T t ( a ) + ϕ 4 Z 3 − 1 E T t ( a )
Z 2 − 1 E T i ( b ) + Z 2 − 1 E T r ( b ) = Z 2 − 1 ϕ 1 E T t ( a ) + Z 2 − 1 ϕ 2 Z 3 − 1 E T t ( a )
( Z 1 − 1 − Z 2 − 1 ) E T i ( b ) = ( ϕ 3 + Z 3 − 1 ϕ 4 − Z 2 − 1 ϕ 1 − Z 2 − 1 ϕ 2 Z 3 − 1 ) E T t ( a )
Z 1 − 1 E T i ( b ) + Z 1 − 1 E T r ( b ) = [ Z 1 − 1 ϕ 1 + Z 1 − 1 ϕ 2 Z 3 − 1 ] E T t ( a )
( Z 2 − 1 − Z 1 − 1 ) E T r ( b ) = [ ϕ 3 + Z 3 − 1 ϕ 4 − Z 1 − 1 ϕ 1 − Z 1 − 1 ϕ 2 Z 3 − 1 ] E T t ( a )
( Z 2 − 1 − Z 1 − 1 ) − 1 ( Z 2 − 1 − Z 1 − 1 ) E T r ( b ) = ( Z 2 − 1 − Z 1 − 1 ) − 1 [ ϕ 3 + Z 3 − 1 ϕ 4 − Z 1 − 1 ϕ 1 − Z 1 − 1 ϕ 2 Z 3 − 1 ] E T t ( a )
E T r ( b ) = ( Z 2 − 1 − Z 1 − 1 ) − 1 [ ϕ 3 + Z 3 − 1 ϕ 4 − Z 1 − 1 ϕ 1 − Z 1 − 1 ϕ 2 Z 3 − 1 ] [ ϕ 3 + Z 3 − 1 ϕ 4 − Z 2 − 1 ϕ 1 − Z 2 − 1 ϕ 2 Z 3 − 1 ] − 1 ( Z 1 − 1 − Z 2 − 1 ) E T i ( b )
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