Scattering from a Sphere Under the Impedance Boundary Condition

Abstract:

The scattering from a sphere is analyzed by a Method of Moments, solving the Electric Field Integral Equation with the Impedance Boundary Condition approximation. Results are compared with the Mie theory.

Electric Field Integral Equation in case of the Impedance Boundary Condition

For an homogeneous dielectric or conducting object, the scattering problem can be solved by the surface integral equation approach, involving surface equivalent electric and magnetic currents. Such equation enforce the boundary condition at the surface of the object of the tangential component of the electric and magnetic field [2], [1]. The application of the impedance boundary condition (IBC) results in halving the number of unknown, reducing the size of the matrix of the Method of Moments and consequently reducing the computation time and the dinamic memory required [3].

The impedance boundary condition states that the surface component of the electric $\vec E$ and magnetic $\vec H$ field are related by

$$\vec E\vert _S \times \hat n = \zeta_c \left[\hat n \times (\vec H \times \hat n )\vert _S \right]$$ (1)

where $\vec n$ is the outer normal of the object surface $S$ and the $\zeta_c$ is the impedance of the object. Some discussion about the validity of the impedance boundary condition can be found in [4]. For homogeneous isotropic scatterers the limits of validity of the IBC are:

$$\vert N\vert \gg 1$$ (2)

$$\vert\Im(N)\vert k_0 a_{min} \gg 1$$ (3)

being $N$ the refractive index of the object, $k_0$ the wavenumber for the outside medium and $a_{min}$ the smallest radius of curvature of the surface of the object.

When the IBC can be applied, the equivalent surface electrical $J_s$ and magnetical $J_{ms}$ currents are related by

$$\hfill \vec J_{ms} = \zeta_c \vec J_s \times \hat n.$$ (4)

The Electric Field Integral Equation (EFIE) can be written starting from eq. (1) and writing the tangential component of the magnetic field by the equivalent electric current

$$\hfill (\vec E_i + \vec E_s) \times \hat n\vert _S = \zeta_c \vec J_s \times \hat n\vert _S.$$ (5)

where $\vec E_i$ and $\vec E_s$ are the incident and the scattered electric field, respectively. Using the mixed potential to express the scattered electric field and the eq. 4, we obtain

$$\begin{matrix}\hfill \hat n \times \vec E_i \vert _S = & \zeta_c ... ...t_{S} \vec J_{ms}(\vec r') g(\vec r, \vec r') dS' \right] . \hfill \end{matrix}$$ (6)

with $\vec r \in S$ , $\omega=2 \pi f$ , $f$ is the frequency, $\mu$ and $\epsilon$ are the magnetic and the dielectric constant, respectively, and $g(\vec r,\vec r')$ is the free spece Green's function. After the extraction of the singular point contribution ( $\vec r = \vec r'$) of the last term, we can write the EFIE as

$$\begin{matrix}\hfill \hat n \times \vec E_i \vert _S = & \frac{1}... ...vec J_{ms}(\vec r') \times \nabla' g(\vec r, \vec r') dS' , \hfill \end{matrix}$$ (7)

where $\int^{pv}$ indicates that the integral has to be evaluated in its principal value, and $\vec J_{ms}$ is related to the unknown equivalent electric current $\vec J_s$ by eq. 4. The eq. 7 is the EFIE that we solve with the Method of Moments.

Once the electric equivalent current has been evaluated, the scattered field in the far region can be evaluated as [3]

$$\vec E_s(r,\theta,\phi) = \frac{e^{-jkr}}{r} \left[ F_1(\theta,\phi) \hat \theta + F_2(\theta,\phi) \hat \phi \right] ,$$ (8)

where

$$F_1(\theta,\phi) = -\frac{j\omega \mu_0}{4 \pi} \int_S \left[ \vec J_s(\vec r') \c... ... n(\vec r') \cdot \hat \phi(\theta,\phi) \right] e^{jk\hat r \cdot \vec r'} dS'$$ (9)

$$F_2(\theta,\phi) = -\frac{j\omega \mu_0}{4 \pi} \int_S \left[ \vec J_s(\vec r') \c... ...vec r') \cdot \hat \theta(\theta,\phi) \right] e^{jk\hat r \cdot \vec r'} dS' .$$ (10)

Results

In the following we include some results, in terms of the bistatic scattering cross section, defined as

$$\sigma(\theta,\phi) = \lim_{r \rightarrow \infty} 4 \pi r^2 \left\vert \frac{\vec E_s(r,\theta,\phi)}{E_0} \right\vert^2 ,$$ (11)

normalized to the cross section ($\pi a^2$ ) of a sphere of radius $a$ , with $E_0$ the intensity of the incident plane wave. The results obtained by MoM with the IBC formulation are compared to those obtained by the Mie series [3], for a sphere with electrical radius $ka=2$, and different values of the normalized impedance of the medium $Z=\zeta_c / \zeta_0$.

In order to apply the MoM, the Rao-Wilton-Glisson (RWG) basis function are used [5], with a sphere discretized in triangles with an approximate dimension of $\lambda/20$ , for a mesh with 1078 nodes and 2152 triangles.

Bibliography

1

J. J. H. Wang, Generalized Moment Methods in Electromagnetics, John Wiley and Sons, 1991.

2

K. Umashankar, A. Taflove, and S. M. Rao, ``Electromagnetic scattering by arrbitrary shaped three-dimensional homogeneous lossy dielectric objects,'' IEEE Trans. on Antennas and Propagat., vol. 34, 1986, pp. 758-766.

3

A. Sebak, L. Shafai, ``Scattering from arbitrarily-shaped objects with impedance boundary condition,'' IEE Proceedings, vol. 136, 1989, pp. 371-376.

4

D. S. Wang, ``Limits and Validity of the Impedance Boundary Condition on Penetrable Surfaces,'' IEEE Trans. on Antennas and Propagat., vol. 35, 1987, pp.453-457.

5

S. M. Rao, D. R. Wilton, A. W. Glisson, ``Electromagnetic Scattering by Surfaces of Arbitrary Shape, IEEE Trans. on Antennas and Propagat., vol. 30, 1982, pp. 409-418.