Formulation

As can be seen from Fig. 1, a cross-section of an infinitely long conducting circular cylinder coated with an eccentric anisotropic shell is shown. Three rectangular coordinate systems and one cylindrical coordinate system are defined, the z-axis which is common to these coordinate systems is not plotted. The global coordinate system (x, y) and the local coordinate system (x _c, y _c) are defined. The radii of cylinders are denoted by a and b, their axes are fixed at origins O _c and O, respectively. A Gaussian beam source which is located at (x ₁ = −r ₀, y ₁ = 0) is incident on the circular cylinder, making an angle θ ₀ clockwise with respect to the negative x-axis.

Fig. 1.

Geometry of the problem.

Transmitted wave

In the expressions for the electromagnetic fields, the time dependence exp (jωt) is omitted throughout. Consider a homogeneous anisotropic medium characterized by the following permittivity and permeability tensors in the X _cO _cY _c coordinate frame:

(1)$$\bar{\bar{\varepsilon }} = \left[{\matrix{ {\varepsilon_{xx}} \hfill & {\varepsilon_{xy}} \hfill & 0 \hfill \cr {\varepsilon_{yx}} \hfill & {\varepsilon_{yy}} \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & {\varepsilon_{zz}} \hfill \cr } } \right]\comma \;\bar{\bar{\mu }} = \left[{\matrix{ {\mu_{xx}} \hfill & {\mu_{xy}} \hfill & 0 \hfill \cr {\mu_{yx}} \hfill & {\mu_{yy}} \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & {\mu_{zz}} \hfill \cr } } \right].$$

The magnetic field in the annular region (part II) can be expressed in the local coordinates (x _c, y _c) as follows [Reference Ren and Wu12]

(2)$$H_z\lpar \rho _c\comma \;\theta _c\rpar = \sum\limits_{n ={-}\infty }^\infty {\,j^{{-}n}\sum\limits_{m ={-}\infty }^\infty {\lsqb c_m^{\lpar 1\rpar } F_{nm}^{\lpar 1\rpar } \lpar \rho _c\comma \;\theta _c\rpar + c_m^{\lpar 2\rpar } F_{nm}^{\lpar 2\rpar } \lpar \rho _c\comma \;\theta _c\rpar \rsqb } } \comma \;$$

where

(3)$$F_{nm}^{\lpar 1\rpar } \lpar \rho _c\comma \;\theta _c\rpar = \int_0^{2\pi } {J_n\lpar k\lpar \xi \rpar \rho _c\rpar e^{\,j\lsqb n\theta _c + \lpar m-n\rpar \xi \rsqb }d\xi } \comma \;$$

(4)$$F_{nm}^{\lpar 2\rpar } \lpar \rho _c\comma \;\theta _c\rpar = \int_0^{2\pi } {Y_n\lpar k\lpar \xi \rpar \rho _c\rpar e^{\,j\lsqb n\theta _c + \lpar m-n\rpar \xi \rsqb }d\xi } \comma \;$$

where $c_m^{\lpar 1\rpar }$ and $c_m^{\lpar 2\rpar }$ are unknown coefficients, J _n[k(ξ)ρ _c] and Y _n[k(ξ)ρ _c] are the Bessel functions of the first and second kind, respectively.

The tangential component of the electric field in the annular region (part II) can be expressed as

(5)$$j\omega \gamma E_\theta ^c ={-}\left[{\varepsilon_{\rho_c\rho_c}\lpar \theta_c\rpar \displaystyle{{\partial H_z^c } \over {\partial \rho_c}} + \varepsilon_{\theta_c\rho_c}\lpar \theta_c\rpar \displaystyle{1 \over {\rho_c}}\displaystyle{{\partial H_z^c } \over {\partial \theta_c}}} \right]\comma \;$$

where

(6)$$\eqalign{\varepsilon _{\rho _c\rho _c}\lpar \theta _c\rpar = & \;\varepsilon _ +\,{+} \, \varepsilon _-\cos 2\theta _c + \sigma _ + \sin 2\theta _c\comma \;\cr \varepsilon _{\theta _c\rho _c}\lpar \theta _c\rpar = & -\!\!\sigma _- + \sigma _ + \cos 2\theta _c-\varepsilon _-\sin 2\theta _c.} $$

(7)$$\eqalign{\varepsilon _ \pm{=} & \displaystyle{1 \over 2}\lpar \varepsilon _{xx} \pm \varepsilon _{yy}\rpar \comma \;\cr \sigma _ \pm{=} & \displaystyle{1 \over 2}\lpar \varepsilon _{xy} \pm \varepsilon _{yx}\rpar .} $$

Incident and scattered waves

Consider the case where a beam source (e.g. a horn antenna) is located at x ₁ = −r ₀ as illustrated in Fig. 1. The z-component of the magnetic field of the incident Gaussian beam source is expressed in the global coordinate system (x, y) as follows [Reference Kozaki11]

(8)$$H_z^{inc} \lpar x_1 ={-}r_0\comma \;y_1\rpar = e^{-\beta ^2y_1^2 }\comma \;$$

where

(9)$$\beta ^2 = a_0^2 + jb_0^2 \comma \;$$

where 1/|β| corresponds to the beamwidth of the incident wave. The parameters a ₀, b ₀ are determined from the distributions of amplitude and phase.

The incidence beam can be expressed as following [Reference Kozaki11]

(10)$$H_z^{inc} \lpar x_1\comma \;y_1\rpar = \displaystyle{1 \over {2\sqrt \pi \beta }}\int_{-\infty }^\infty {e^{{(-\alpha ^{2/4}\beta ^{2})}{-j(x_1 + r_0)\sqrt {k_0^2 -\alpha ^2} -j\alpha y_1}} d\alpha } .$$

For polar coordinates (ρ, θ), equation (10) can be expressed as:

(11)$$H_z^{inc} \lpar \rho \comma \;\theta \rpar = \sum\limits_{n ={-}\infty }^\infty {\,j^{{-}n}e^{\,jn\theta }J_n\lpar k_0\rho \rpar A_n^h } \comma \;$$

where

(12)$$A_n^h \cong A_{n1}^h = \displaystyle{{e^{{-}jk_0r_0}} \over {\sqrt {1-jZ_0} }}e^{-{\lpar \lpar n/k_0\rpar \beta \rpar }^2/\lpar 1-jZ_0\rpar }\comma \;$$

where Z ₀ = (2β ²r ₀)/k. This expression is valid for |(βλ)²| < 0.3 and b/λ < 3.0. For (b/λ) ≅ (3 ~ 5), an improved representation must be used:

(13)$$A_n^h \cong A_{n2}^h = A_{n1}^h \left[{1-2{\left({\displaystyle{\beta \over {k_0\sqrt {1-jZ_0} }}} \right)}^4n^2 + \displaystyle{4 \over 3}{\left({\displaystyle{\beta \over {k_0\sqrt {1-jZ_0} }}} \right)}^6n^4 + \ldots } \right].$$

Meanwhile, the scattered magnetic field in the free space region (part III) is expressed as

(14)$$H_z^s \lpar \rho \comma \;\theta \rpar = \sum\limits_{n ={-}\infty }^\infty {B_nj^{{-}n}H_n^{\lpar 2\rpar } \lpar k\rho \rpar e^{\,jn\theta }} \comma \;$$

where B _n are unknown coefficients.

The tangential component of the electric field can be written in the global coordinate system (x, y) as follows:

(15)$$E_\theta ^{inc} \lpar \rho \comma \;\theta \rpar = j\sqrt {\displaystyle{{\mu _0} \over {\varepsilon _0}}} \sum\limits_{n ={-}\infty }^\infty {A_nj^{{-}n}{{J}^{\prime}}_n\lpar k\rho \rpar e^{\,jn\lpar \theta + \theta _0\rpar }} \comma \;\quad \lpar \rho \ge b\rpar \comma \;$$

(16)$$E_\theta ^s \lpar \rho \comma \;\theta \rpar = j\sqrt {\displaystyle{{\mu _0} \over {\varepsilon _0}}} \sum\limits_{n ={-}\infty }^\infty {B_nj^{{-}n}H{_n^{\lpar 2\rpar } }^{\prime}\lpar k\rho \rpar e^{\,jn\theta }} \comma \;\quad \lpar \rho \ge b\rpar .$$

Boundary conditions

1. (1) Boundary Conditions on the Core Surface:

Assume the surface of the conducting circular cylinder is represented by ρ _c = a, the electric field vanishes on the surface of the conducting circular cylinder, the boundary condition can be written as

(17)$$E_\theta ^c \lpar \rho _c\comma \;\theta _c\rpar = 0\comma \;\quad \lpar \rho _c = a\rpar .$$

i.e.

(18)$$\sum\limits_{m ={-}\infty }^\infty {\lsqb c_m^{\lpar 1\rpar } E_{nm}^{\lpar 11\rpar } \lpar a\rpar + c_m^{\lpar 2\rpar } E_{nm}^{\lpar 12\rpar } \lpar a\rpar \rsqb } = 0\comma \;$$

where

(19)$$\eqalign{E_{nm}^{\lpar 11\rpar } \lpar a\rpar = & -j\sqrt {\displaystyle{{\varepsilon _0} \over {\mu _0}}} \int_0^{2\pi } {\displaystyle{{k\lpar \xi \rpar } \over {\omega \gamma }}e^{\,j\lpar m-n\rpar \xi }} \cr & \quad \times \left[{\,j\varepsilon_{\rho_c\rho_c}\lpar \xi \rpar {{J}^{\prime}}_n\lpar k\lpar \xi \rpar a\rpar -\displaystyle{{n\varepsilon_{\theta_c\rho_c}\lpar \xi + {\pi / 2}\rpar } \over {k\lpar \xi \rpar a}}J_n\lpar k\lpar \xi \rpar a\rpar } \right]d\xi \comma \;} $$

(20)$$\eqalign{E_{nm}^{\lpar 12\rpar } \lpar a\rpar = & -j\sqrt {\displaystyle{{\varepsilon _0} \over {\mu _0}}} \int_0^{2\pi } {\displaystyle{{k\lpar \xi \rpar } \over {\omega \gamma }}e^{\,j\lpar m-n\rpar \xi }} \cr & \quad \times \left[{\,j\varepsilon_{\rho_c\rho_c}\lpar \xi \rpar {{Y}^{\prime}}_n\lpar k\lpar \xi \rpar a\rpar -\displaystyle{{n\varepsilon_{\theta_c\rho_c}\lpar \xi + {\pi / 2}\rpar } \over {k\lpar \xi \rpar a}}Y_n\lpar k\lpar \xi \rpar a\rpar } \right]d\xi .} $$

1. (1) Boundary Conditions on the External Surface: The tangential components of the electric and magnetic fields are continuous on the surface of anisotropic-free space, the boundary conditions can be expressed as

   (21)$$H_z^c = H_z^{inc} + H_z^s \comma \;\quad \lpar \rho = b\rpar \comma \;$$
   (22)$$E_\theta ^c = E_\theta ^{inc} + E_\theta ^s \comma \;\quad \lpar \rho = b\rpar .$$

This requires that the fields be expressed in terms of the X _cO _cY _c coordinate frame be translated to the XOY coordinate frame using the addition theorem for cylindrical functions, i.e. [Reference Stratton13]:

(23)$$J_n\lpar k\lpar \xi \rpar \rho _c\rpar e^{\,jn\theta _c} = \sum\limits_{m ={-}\infty }^\infty {J_m\lpar k\lpar \xi \rpar \rho \rpar J_{m-n}\lpar k\lpar \xi \rpar \rho _0\rpar e^{\,j\lsqb m\theta -\lpar m-n\rpar \varphi _0\rsqb }} \quad \lpar \rho \gt \rho _0\rpar \comma \;$$

(24)$$Y_n\lpar k\lpar \xi \rpar \rho _c\rpar e^{\,jn\theta _c} = \sum\limits_{m ={-}\infty }^\infty {Y_m\lpar k\lpar \xi \rpar \rho \rpar J_{m-n}\lpar k\lpar \xi \rpar \rho _0\rpar e^{\,j\lsqb m\theta -\lpar m-n\rpar \varphi _0\rsqb }} \quad \lpar \rho \gt \rho _0\rpar .$$

The magnetic field in the annular region (part II in Fig. 1) can be expressed in the global coordinates (x, y) as follows

(25)$$H_z^c \lpar \rho \comma \;\theta \rpar = \sum\limits_{n ={-}\infty }^\infty {\,j^{{-}n}\sum\limits_{m ={-}\infty }^\infty {\lsqb c_m^{\lpar 1\rpar } F_{nm}^{\lpar 1\rpar } \lpar \rho \comma \;\theta \rpar + c_m^{\lpar 2\rpar } F_{nm}^{\lpar 2\rpar } \lpar \rho \comma \;\theta \rpar \rsqb } } \comma \;$$

where

(26)$$F_{nm}^{\lpar 1\rpar } \lpar \rho \comma \;\theta \rpar = \sum\limits_{i ={-}\infty }^\infty {e^{\,j\lsqb i\theta -\lpar i-n\rpar \varphi _0\rsqb }\int_0^{2\pi } {J_i\lpar k\lpar \xi \rpar \rho \rpar J_{i-n}\lpar k\lpar \xi \rpar \rho _0\rpar e^{\,j\lpar m-n\rpar \xi }d\xi } } \comma \;$$

(27)$$F_{nm}^{\lpar 2\rpar } \lpar \rho \comma \;\theta \rpar = \sum\limits_{i ={-}\infty }^\infty {e^{\,j\lsqb i\theta -\lpar i-n\rpar \varphi _0\rsqb }\int_0^{2\pi } {Y_i\lpar k\lpar \xi \rpar \rho \rpar J_{i-n}\lpar k\lpar \xi \rpar \rho _0\rpar e^{\,j\lpar m-n\rpar \xi }d\xi } } .$$

Applying the boundary conditions (21) and (22), two equations can be obtained as

(28)$$\eqalign{& \sum\limits_{n ={-}\infty }^\infty {\,j^{l-n}e^{{-}j\lpar l-n\rpar \varphi _0}\sum\limits_{m ={-}\infty }^\infty {\lsqb c_m^{\lpar 1\rpar } F_{nm}^{\lpar 21\rpar } \lpar b\rpar + c_m^{\lpar 2\rpar } F_{nm}^{\lpar 22\rpar } \lpar b\rpar \rsqb } } \cr & = A_lJ_l\lpar kb\rpar e^{\,jl\theta _0} + B_lH_l^{\lpar 2\rpar } \lpar kb\rpar \comma \;} $$

(29)$$\eqalign{& \sum\limits_{n ={-}\infty }^\infty {\,j^{l-n}e^{{-}j\lpar l-n\rpar \varphi _0}\sum\limits_{m ={-}\infty }^\infty {\lsqb {c_m^{\lpar 1\rpar }} {E_{nm}^{\lpar 21\rpar }} \lpar b\rpar + c_m^{\lpar 2\rpar } E_{nm}^{\lpar 22\rpar } \lpar b\rpar \rsqb } } \cr & = A_l{{J}^{\prime}}_l\lpar kb\rpar e^{{\,jl\theta_{0}}} + B_l{H}{\prime}_l^{\lpar 2\rpar } \lpar kb\rpar \comma \;} $$

where

(30)$$F_{nm}^{\lpar 21\rpar } \lpar b\rpar = \int_0^{2\pi } {J_l\lpar k\lpar \xi \rpar b\rpar J_{l-n}\lpar k\lpar \xi \rpar \rho _0\rpar e^{\,j\lpar m-n\rpar \xi }d\xi } \comma \;$$

(31)$$F_{nm}^{\lpar 22\rpar } \lpar b\rpar = \int_0^{2\pi } {Y_l\lpar k\lpar \xi \rpar b\rpar J_{l-n}\lpar k\lpar \xi \rpar \rho _0\rpar e^{\,j\lpar m-n\rpar \xi }d\xi } \comma \;$$

(32)$$\eqalign{E_{nm}^{\lpar 21\rpar } \lpar b\rpar = & -j\sqrt {\displaystyle{{\varepsilon _0} \over {\mu _0}}} \int_0^{2\pi } {\displaystyle{{k\lpar \xi \rpar } \over {\omega \gamma }}J_{l-n}\lpar k\lpar \xi \rpar \rho _0\rpar e^{\,j\lpar m-n\rpar \xi }} \cr & \quad \times \left[{\,j\varepsilon_{\rho \rho }\lpar \xi \rpar {J}^{\prime}_l\lpar k\lpar \xi \rpar b\rpar -\displaystyle{{l\varepsilon_{\theta \rho }\lpar \xi + {\pi / 2}\rpar } \over {k\lpar \xi \rpar b}}J_l\lpar k\lpar \xi \rpar b\rpar } \right]d\xi \comma \;} $$

(33)$$\eqalign{E_{nm}^{\lpar 22\rpar } \lpar b\rpar = & -j\sqrt {\displaystyle{{\varepsilon _0} \over {\mu _0}}} \int_0^{2\pi } {\displaystyle{{k\lpar \xi \rpar } \over {\omega \gamma }}J_{l-n}\lpar k\lpar \xi \rpar \rho _0\rpar e^{\,j\lpar m-n\rpar \xi }} \cr & \quad \times \left[{\,j\varepsilon_{\rho \rho }\lpar \xi \rpar {Y}^{\prime}_l\lpar k\lpar \xi \rpar b\rpar -\displaystyle{{l\varepsilon_{\theta \rho }\lpar \xi + {\pi / 2}\rpar } \over {k\lpar \xi \rpar b}}Y_l\lpar k\lpar \xi \rpar b\rpar } \right]d\xi .} $$

In order to obtain numerical results, the infinite series need to be truncated under the prerequisite of achieving the solution convergence. Truncating these equations for −N ≤ n ≤ N and −N ≤ m ≤ N, they can be formed in vectors and matrices, the unknown coefficients can be obtained from these equations finally, and the electromagnetic field can be calculated while the radar cross-section per unit length (RCS) can be written as:

(34)$$\displaystyle{\sigma \over \lambda }\lpar \theta \comma \;\theta ^{inc}\rpar = \displaystyle{2 \over \pi }\left\vert {\sum\limits_{n ={-}\infty }^{ + \infty } {B_ne^{\,jn\theta }} } \right\vert ^2.$$

Numerical results

In this section, the variations of RCS versus the eccentricity, the distance between the laser and the scatterer are depicted; the total field distribution around the scatterer is listed and discussed. In order to check the validity and accuracy of the proposed method and associated code, the angular distributions of RCS are shown in Figs 2 and 3. A conducting circular cylinder covered with an eccentric anisotropic shell approaches a concentrically circular geometry in both cases. For the plane wave incidence case [Reference Ren and Wu12, Reference Beker, Umashankar and Taflove14] in Fig. 2, both a ₀ and b ₀ are set to be zero, while r ₀/λ = 3.0, ka = 1, kb = 2, ρ ₀ = 0, φ₀ = 0°, θ ₀ = 0°, N = 12. The elements in the permittivity tensor and the permeability tensor are: ɛ_xx = 1.5ɛ₀, ɛ_yy = 2.5ɛ₀, ɛ_xy = −ɛ_yx = 3ɛ₀, μ _zz = 1.5μ ₀. For the Gaussian beam incidence case in Fig. 3, the parameters are: ɛ_xx = 4ɛ₀, ɛ_yy = ɛ₀, ɛ_xy = ɛ_yx = 0, μ _zz = 2μ ₀, ka = π, kb = 2π, ρ ₀ = 0, φ₀ = 0°, θ ₀ = 0°, r ₀/λ = 30.0, a ₀λ = 0.2, b ₀λ = 0.4, N = 15.The results come into excellent agreement with those from [Reference Mao, Zhang, Gao and Wu6] for the concentric geometry case.

Fig. 2.

H-polarization, bistatic radar cross-sections RCS dB, ɛ_xx = 1.5ɛ₀, ɛ_yy = 2.5ɛ₀, ɛ_xy = −ɛ_yx = 3ɛ₀, μ _zz = 1.5μ ₀, (r ₀/λ) = 3.0, ka = 1, kb = 2, ρ ₀ = 0, φ₀ = 0°, θ ₀ = 0°.

Fig. 3.

H-polarization, bistatic radar cross-sections RCS dB, ɛ_xx = 4ɛ₀, ɛ_yy = ɛ₀, ɛ_xy = ɛ_yx = 0, μ _zz = 2μ ₀, ka = π, kb = 2π, φ₀ = 0°, θ ₀ = 0°, (r ₀/λ) = 30.0, a ₀λ = 0.2, b ₀λ = 0.4.

Figure 3 also gives the effects of the different distance between the conducting and the anisotropic cylinder shell, i.e. the influence of the eccentricity on the values of RCS. This example consists of media with $\hat{x}$ and $\hat{y}$ principal axes. Compared with the concentric case, the graphics of non-concentric radar scattering are no longer symmetrically distributed with the scattering angles. The maximum values of RCS increase with the values of core eccentricity.

Figure 4 shows the effects of the distance between the laser source and the circular cylinder. The parameters are: ɛ_xx = 2ɛ₀, ɛ_yy = ɛ₀, ɛ_xy = −ɛ_yx = ɛ₀, μ _zz = 2μ ₀, ka = π, kb = 2π, φ₀ = 0°, θ ₀ = 0°, a ₀λ = 0.2, b ₀λ = 0.5, ρ ₀/a = 0.2, N = 15. The whole RCS decline with the increase of the source distance.

Fig. 4.

H-polarization, bistatic radar cross-sections RCS dB, ɛ_xx = 2ɛ₀, ɛ_yy = ɛ₀, ɛ_xy = −ɛ_yx = ɛ₀, μ _zz = 2μ ₀, ka = π, kb = 2π, φ₀ = 0°, θ ₀ = 0°, a ₀λ = 0.2, b ₀λ = 0.5, (ρ ₀/a) = 0.2.

Figures 5–8 show the absolute value of the total field near the scatterer, the arrow indicates the direction of the incident wave. Figures 5 and 6 represent two cases of magnetic field. It can be seen that there is a similar figure in concentric and non-concentric cases. The PEC acts as an impedance dielectric medium for magnetic field, it carries the Gaussian beam from the right to the anisotropic medium. Figures 7 and 8 represent two cases of electric field. The role of PEC in magnetic field and electric field is different. The tangential components of electric field vanish in PEC. The PEC acts like a mirror, blocking the transmission of electric field.

Fig. 5.

The absolute value of the total magnetic field |H _z| near the scatterer. ɛ_xx = 2ɛ₀, ɛ_yy = ɛ₀, ɛ_xy = ɛ_yx = 0, μ _zz = 2μ ₀, ka = π, kb = 2π, θ ₀ = 180°, (r ₀/λ) = 30.0, a ₀λ = 0, b ₀λ = 0, (ρ ₀/a) = 0, φ₀ = 0°.

Fig. 6.

The absolute value of the total magnetic field |H _z| near the scatterer. ɛ_xx = 2ɛ₀, ɛ_yy = ɛ₀, ɛ_xy = ɛ_yx = 0, μ _zz = 2μ ₀, ka = π, kb = 2π, θ ₀ = 180°, (r ₀/λ) = 30.0, a ₀λ = 0.3, b ₀λ = 0.4, (ρ ₀/a) = 0.2, φ₀ = 0°.

Fig. 7.

The absolute value of the total electric field $\vert {E_\theta } \vert$ near the scatterer. ɛ_xx = 2ɛ₀, ɛ_yy = ɛ₀, ɛ_xy = ɛ_yx = 0, μ _zz = 2μ ₀, ka = π, kb = 2π, θ ₀ = 180°, (r ₀/λ) = 3.0, a ₀λ = 0, b ₀λ = 0, (ρ ₀/a) = 0, φ₀ = 0°.

Fig. 8.

The absolute value of the total electric field $\vert {E_\theta } \vert$ near the scatterer. ɛ_xx = 2ɛ₀, ɛ_yy = ɛ₀, ɛ_xy = ɛ_yx = 0, μ _zz = 2μ ₀, ka = π, kb = 2π, θ ₀ = 180°, (r ₀/λ) = 3.0, a ₀λ = 0.3, b ₀λ = 0.4, (ρ ₀/a) = 0.2, φ₀ = 0°.