# Electromagnetic Scattering Analysis of SHDB Objects Using Surface Integral Equation Method

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## Abstract

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## 1. Introduction

## 2. General Linear Boundary Condition

## 3. Surface Integral Equation Method for SHDB

## 4. Numerical Results

#### 4.1. Special Cases of DB and SH Surfaces

#### 4.2. SHDB Disk, Oblique Incidence Angle

## 5. Perspectives into Realization of the SHDB Boundary

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A Soft-and-Hard/DB (SHDB) disk with a diameter of $3\lambda $ and a thickness of $0.04\lambda $. The green lines indicate the direction of the vector ${\mathbf{a}}_{t}$ on the surface of the disk. Vectors ${\mathbf{u}}^{i}$ and ${\mathit{E}}^{i}$ show the directions of the incident wave and the incident electric field.

**Figure 2.**Bistatic radar cross sections (RCS) for the SHDB disk shown in Figure 1 at the frequency of $300\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{MHz}$ with the incident direction ${\mathbf{u}}^{i}=-\mathbf{z}$. (

**a**) ${T}_{d}=1$, ${T}_{s}=0$ (DB surface). (

**b**) ${T}_{d}=0$, ${T}_{s}=1$ (SH surface).

**Figure 3.**Bistatic RCS for perfect electric conductor (PEC) targes at the frequency of $300\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{MHz}$. (

**a**) Bistatic RCS in the E plane for a PEC sphere with a radius of $1.5\lambda $. (

**b**) Bistatic RCS in the E plane for a PEC disk. The size of the PEC disk and the incident field are the same as in Figure 2b.

**Figure 4.**Surface current distributions of the SHDB disk shown in Figure 1 at the frequency of $300\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{MHz}$ for an incident wave with ${\mathbf{u}}^{i}=-\mathbf{z}$. (

**a**) Real $\left\{{\eta}_{0}\mathit{J}\right\}$ for ${T}_{d}=1$, ${T}_{s}=0$ (DB surface). (

**b**) Real $\left\{\mathit{M}\right\}$ for ${T}_{d}=1$, ${T}_{s}=0$ (DB surface). (

**c**) Real $\left\{{\eta}_{0}\mathit{J}\right\}$ for ${T}_{d}=0$, ${T}_{s}=1$ (SH surface). (

**d**) Real $\left\{\mathit{M}\right\}$ for ${T}_{d}=0$, ${T}_{s}=1$ (SH surface).

**Figure 5.**Components of the reflection dyadic of an infinite SHDB plane with ${T}_{d}={T}_{s}=1$ for a plane wave incident at ${\theta}^{\mathrm{i}}={45}^{\circ}$ as functions of $\beta $.

**Figure 6.**Bistatic RCS of the disk in Figure 1 with ${T}_{s}={T}_{d}=1$ illuminated by a plane wave with ${\theta}^{\mathrm{i}}={45}^{\circ}$ at the frequency of $300\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{MHz}$. The results are shown in the x-z plane. (

**a**) $\beta ={0}^{\circ}$. (

**b**) $\beta ={38.7}^{\circ}$. (

**c**) $\beta ={90}^{\circ}$. (

**d**) Bistatic RCS at $\theta =-{45}^{\circ}$ as a function of $\beta $.

**Figure 7.**Levels of materialization of a given boundary: the medium whose surface imitates the boundary condition is described by a set of electromagnetic constitutive material parameters. On the other hand, the ultimate physical material itself can be modeled as a collection of polarizable particles whose electric and magnetic dipole moments effectively homogenize into these constitutive dyadics.

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**MDPI and ACS Style**

Kong, B.; Ylä-Oijala, P.; Sihvola, A.
Electromagnetic Scattering Analysis of SHDB Objects Using Surface Integral Equation Method. *Photonics* **2020**, *7*, 134.
https://doi.org/10.3390/photonics7040134

**AMA Style**

Kong B, Ylä-Oijala P, Sihvola A.
Electromagnetic Scattering Analysis of SHDB Objects Using Surface Integral Equation Method. *Photonics*. 2020; 7(4):134.
https://doi.org/10.3390/photonics7040134

**Chicago/Turabian Style**

Kong, Beibei, Pasi Ylä-Oijala, and Ari Sihvola.
2020. "Electromagnetic Scattering Analysis of SHDB Objects Using Surface Integral Equation Method" *Photonics* 7, no. 4: 134.
https://doi.org/10.3390/photonics7040134