Optimal illusion and invisibility of multilayered anisotropic cylinders and spheres

Lin Zhang; Yan Shi; Chang-Hong Liang

doi:10.1364/OE.24.023333

1. Introduction

The subject of electromagnetic invisibility cloak has attracted great attention from both physics and engineering societies. The method of transformation optics (TO) independently developed by Pendry [1] and Leonhardt [2] is arguably the most popular for design of various invisibility cloaks. In the TO method, electromagnetic wave is controlled to go around the object, and then come back to the original trajectory. With theoretical verification [3, 4] and experimental realization [5–8], invisibility cloaks have experienced a tremendous development in the past few years, such as complementary cloak [9–11], carpet cloak [12] and anti-cloak [13, 14]. However, currently available technologies to convert the transformation-based cloaks into practical applications are still facing numerous bottlenecks. Particular challenges are laid on extreme electromagnetic parameters and narrow operating band. In an attempt to relieve the limitations of physical realization on the cloaks, many other means for manipulation of electromagnetic wave have been developed including conformal transformation [15, 16], surface wave cloak [17–19], and coding metamaterial [20], etc.

Alternatively, scattering cancellation method [21] has been proposed to achieve electromagnetic transparency using a shell with appropriately designed permittivity and permeability. Due to the ease of implementation, various invisibility [22–28], illusion [29], inhomogeneous [30], and bifunctional [31] cloaks based on the scattering cancellation method have been developed. On the other hand, with rapid advances in material technology, anisotropic materials allow for greater design flexibility and novel application in microwave engineering. Electromagnetic analysis of the anisotropic materials has been a subject of great interest [32, 33], and the scattering cancellation-based cloaks for anisotropic objects have been developed [34, 35]. However, to our best knowledge, no reports have been given to analyze invisibility and illusion effects of inhomogeneous anisotropic cylindrical and spherical objects. Moreover, how to design a cloak with the optimal invisibility and illusion performances according to the scattering cancellation method has not yet been studied.

The principal purpose of this paper is to achieve both the optimal invisibility and illusion for inhomogeneous anisotropic cylindrical and spherical objects with electrically small size. The inhomogeneous anisotropic objects are modeled as many thin layers of piecewise homogeneous layers. Following Mie series theory, Bessel and Hankel functions of the fractional orders are employed to expand the electromagnetic fields in each homogeneous anisotropic layer, and the corresponding scattering coefficients are determined according to the boundary conditions. With the resultant scattering coefficients, explicit expressions for geometric and electromagnetic parameters of the coating are derived to realize invisibility and illusion of the inhomogeneous anisotropic objects. By appropriately adjusting anisotropy parameters of the coating, the optimal illusion and invisibility effects can be achieved. Some cylindrical and spherical examples from the perspective of near and far fields are given to validate the proposed methodology.

2. Theoretical analysis

2.1 Illusion and Invisibility of inhomogeneous anisotropic cylinders and spheres

Consider an inhomogeneous object composed of n piecewise anisotropic layers illuminated by a plane wave. The time harmonic dependence of $e^{j ω t}$ is assumed. Without loss of generality, TM polarized plane wave is considered in this paper. TE polarized case can be obtained by interchanging the role of permittivity and permeability according to duality relationship. Wrapping the inhomogeneous object with a slab of homogeneous material whose radius, permittivity and permeability tensors are $r_{n + 1}$ , ${\bar{\bar{ε}}}_{n + 1}$ , and ${\bar{\bar{μ}}}_{n + 1}$ , respectively, to achieve invisibility and illusion performances.

2.2.1 Illusion and Invisibility of inhomogeneous anisotropic cylinder

At first, we analyze scattering behavior of a two-layer inhomogeneous cylinder (n = 2) concretely. In ith layer whose radius is $r_{i}$ , permittivity and permeability tensors in cylindrical coordinate are expressed as

{\bar{\bar{ε}}}_{i} = [\begin{matrix} ε_{i r} \\ ε_{i θ} \\ ε_{0} \end{matrix}], {\bar{\bar{μ}}}_{i} = [\begin{matrix} μ_{0} \\ μ_{0} \\ μ_{i z} \end{matrix}] .

A homogeneous material slab covers the inhomogeneous cylinder to achieve the illusion and invisibility. According to wave transformation relationship, a TM-polarized plane wave can be written in terms of a linear superposition of cylindrical waves [36]

H_{z}^{i n c} = \sum_{m = - \infty}^{\infty} j^{- m} J_{m} (k_{0} r) e^{j m θ},

in which

J_{m}

is mth-order Bessel function of the first kind. The total fields in each layer of the coated cylinder can be expressed in terms of non-integer Bessel and Hankel functions according to Mie series expansion method as follows:

{\begin{cases} H_{z} = \sum_{m = - \infty}^{\infty} j^{- m} A_{m} J_{1 m} (k_{1} r) e^{j m θ} r < r_{1} \\ H_{z} = \sum_{m = - \infty}^{\infty} j^{- m} [B_{m} J_{2 m} (k_{2} r) + C_{m} Y_{2 m} (k_{2} r)] e^{j m θ} r_{1} < r < r_{2} \\ H_{z} = \sum_{m = - \infty}^{\infty} j^{- m} [F_{m} J_{3 m} (k_{3} r) + G_{m} Y_{3 m} (k_{3} r)] e^{j m θ} r_{2} < r < r_{3} \\ H_{z} = \sum_{m = - \infty}^{\infty} j^{- m} [J_{m} (k_{0} r) + D_{m} H_{m}^{(1)} (k_{0} r)] e^{j m θ} r > r_{3} \end{cases},

where

J_{i m}

,

Y_{i m}

, and

H_{m}^{(1)}

represent imth-order Bessel functions of the first and second kinds, mth-order Hankel function of the first kind, respectively, and A_m, B_m, C_m, F_m, G_m, and D_m are corresponding expansion coefficients. Here

k_{i} = ω \sqrt{μ_{i z} ε_{i θ}}

and

i m = m \sqrt{A R_{i}}

(i = 0, 1, 2, 3)

, in which

A R_{i} = ε_{i θ} / ε_{i r}

denotes the electric anisotropy ratio. For an arbitrarily anisotropic material with

ε_{i r} \neq ε_{i θ}

, im is a non-integer, while im becomes an integer for an isotropic material. Hence the resultant scattering field can be written as

H_{z}^{s c a} = \sum_{m = - \infty}^{\infty} j^{- m} D_{m} H_{m}^{(1)} (k_{0} r) e^{j m θ} .

Applying continuous conditions of tangential components of the total fields at each interface, the scattering coefficient $D_{m}^{}$ can be derived as

D_{m}^{} = \frac{\begin{matrix} J_{1 m} (k_{1} r_{1}) & - J_{2 m} (k_{2} r_{1}) & - Y_{2 m} (k_{2} r_{1}) & 0 & 0 & 0 \\ \frac{k_{1}}{ω ε_{1 θ}} {J^{'}}_{1 m} (k_{2} r_{1}) & - \frac{k_{2}}{ω ε_{2 θ}} {J^{'}}_{2 m} (k_{2} r_{1}) & - \frac{k_{2}}{ω ε_{2 θ}} {Y^{'}}_{2 m} (k_{2} r_{1}) & 0 & 0 & 0 \\ 0 & J_{2 m} (k_{2} r_{2}) & N_{2 m} (k_{2} r_{2}) & - J_{3 m} (k_{3} r_{2}) & - Y_{3 m} (k_{3} r_{2}) & 0 \\ 0 & \frac{k_{2}}{ω ε_{2 θ}} {J^{'}}_{3 m} (k_{2} r_{2}) & \frac{k_{2}}{ω ε_{2 θ}} {N^{'}}_{3 m} (k_{2} r_{2}) & - \frac{k_{3}}{ω ε_{3 θ}} {J^{'}}_{3 m} (k_{3} r_{2}) & - \frac{k_{3}}{ω ε_{3 θ}} {Y^{'}}_{3 m} (k_{3} r_{2}) & 0 \\ 0 & 0 & 0 & J_{3 m} (k_{3} r_{3}) & Y_{3 m} (k_{3} r_{3}) & J_{m} (k_{0} r_{3}) \\ 0 & 0 & 0 & \frac{k_{3}}{ω ε_{3 θ}} {J^{'}}_{m} (k_{3} r_{3}) & \frac{k_{3}}{ω ε_{3 θ}} {N^{'}}_{m} (k_{3} r_{3}) & \frac{k_{0}}{ω ε_{0}} {J^{'}}_{m} (k_{0} r_{3}) \end{matrix}}{\begin{matrix} J_{1 m} (k_{1} r_{1}) & - J_{2 m} (k_{2} r_{1}) & - Y_{2 m} (k_{2} r_{1}) & 0 & 0 & 0 \\ \frac{k_{1}}{ω ε_{1 θ}} {J^{'}}_{1 m} (k_{2} r_{1}) & - \frac{k_{2}}{ω ε_{2 θ}} {J^{'}}_{2 m} (k_{2} r_{1}) & - \frac{k_{2}}{ω ε_{2 θ}} {Y^{'}}_{2 m} (k_{2} r_{1}) & 0 & 0 & 0 \\ 0 & J_{2 m} (k_{2} r_{2}) & Y_{2 m} (k_{2} r_{2}) & - J_{3 m} (k_{3} r_{2}) & - Y_{3 m} (k_{3} r_{2}) & 0 \\ 0 & \frac{k_{2}}{ω ε_{2 θ}} {J^{'}}_{m} (k_{2} r_{2}) & \frac{k_{2}}{ω ε_{2 θ}} {Y^{'}}_{2 m} (k_{2} r_{2}) & - \frac{k_{3}}{ω ε_{3 θ}} {J^{'}}_{3 m} (k_{3} r_{2}) & - \frac{k_{3}}{ω ε_{3 θ}} {Y^{'}}_{3 m} (k_{3} r_{2}) & 0 \\ 0 & 0 & 0 & J_{3 m} (k_{3} r_{3}) & N_{3 m} (k_{3} r_{3}) & - H_{m}^{(1)} (k_{0} r_{3}) \\ 0 & 0 & 0 & \frac{k_{3}}{ω ε_{3 θ}} {J^{'}}_{3 m} (k_{3} r_{3}) & \frac{k_{3}}{ω ε_{3 θ}} {Y^{'}}_{3 m} (k_{3} r_{3}) & - \frac{k_{0}}{ω ε_{0}} H_{m}^{(1)}^{'} (k_{0} r_{3}) \end{matrix}},

in which the prime denotes the derivative with respect to the argument. With the scattering coefficient

D_{m}^{}

, radar scattering section (RCS)

σ

and scattering cross section (SCS)

C_{s c a}

can be calculated, respectively, as

σ = \lim_{r \to \infty} [2 π r \frac{{| H_{z}^{s c a} |}^{2}}{{| H_{z}^{i n c} |}^{2}}], C_{s c a} = \frac{4}{k_{0}} \sum_{m = - \infty}^{\infty} {| D_{m}^{} |}^{2} .

In order to determine the effect of the number of the scattering coefficients $D_{m}^{}$ on the scattering section, we consider a nonmagnetic homogeneous anisotropic cylinder with a radius of a. Figure 1 illustrates variation of the SCS with the argument $k_{0} a$ for different number of the scattering coefficients $D_{m}^{}$ . For comparison, the SCS of a homogeneous isotropic cylinder is also plotted in Fig. 1. It can be seen that when $k_{0} a$ is small, the SCS solved by the scattering coefficients with m = 1 is coincident with that calculated by the scattering coefficients with m = 5. Hence the SCS is dominated by m = 0 and m = 1 terms of the scattering coefficients in the long wavelength limit.

Fig. 1 SCS of a cylinder versus the argument $k_{0} r$ .

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To achieve the illusion, the scattering coefficients of the coated anisotropic cylinder are set as same as that of an isotropic cylinder with a radius of $r_{e}$ , permittivity of $ε_{e}$ , and permeability of $μ_{e}$ . According to standard Mie series expansion method, scattering coefficient $S_{m}^{}$ of the isotropic illusion cylinder can be got as

S_{m}^{} = \frac{\begin{matrix} J_{m} (k_{e} r_{e}) & J_{m} (k_{0} r_{e}) \\ \frac{k_{e}}{ω ε_{e}} {J^{'}}_{m} (k_{e} r_{e}) & \frac{k_{0}}{ω ε_{0}} {J^{'}}_{m} (k_{0} r_{e}) \end{matrix}}{\begin{matrix} J_{m} (k_{e} r_{e}) & - H_{m}^{(1)} (k_{0} r_{e}) \\ \frac{k_{e}}{ω ε_{e}} {J^{'}}_{m} (k_{e} r_{e}) & - \frac{k_{0}}{ω ε_{0}} H_{m}^{(1)}^{'} (k_{0} r_{e}) \end{matrix}} .

Thus, the illusion condition can be obtained as

D_{0} = S_{0}

and

D_{1} = S_{1}

. Considering the approximations

J_{0} (x) ≅ 1

,

{J^{'}}_{0} (x) ≅ - x / 2

,

H_{0}^{(1)} = 1 + j 2 \ln (x / 2) / π

, and

H_{0}^{(1)}^{'} = - x / 2 + j 2 / (π x)

in the long wavelength limit, we can derive the illusion conditions as

{(\frac{r_{3}}{r_{1}})}^{2} = \frac{(μ_{2 z} - μ_{1 z}) + (μ_{3 z} - μ_{2 z}) {(\frac{r_{2}}{r_{1}})}^{2} + (μ_{e} - μ_{0}) {(\frac{r_{e}}{r_{1}})}^{2}}{μ_{3 z} - μ_{0}},

{(\frac{r_{3}}{r_{2}})}^{2 t_{3}} = \frac{ε_{3 θ} - t_{3} ε_{e}^{(2)}}{ε_{3 θ} + t_{3} ε_{e}^{(2)}} \frac{r_{3}^{2} (ε_{3 θ} + ε_{0} t_{3}) (ε_{e} + ε_{0}) - r_{e}^{2} (ε_{3 θ} - ε_{0} t_{3}) (ε_{e} - ε_{0})}{r_{3}^{2} (ε_{3 θ} - ε_{0} t_{3}) (ε_{e} + ε_{0}) - r_{e}^{2} (ε_{3 θ} + ε_{0} t_{3}) (ε_{e} - ε_{0})},

where

ε_{e}^{(2)} = \frac{\frac{ε_{2 θ} + ε_{e}^{(1)} t_{2}}{ε_{2 θ} - ε_{e}^{(1)} t_{2}} - {(\frac{r_{1}}{r_{2}})}^{2 t_{2}}}{\frac{ε_{2 θ} + ε_{e}^{(1)} t_{2}}{ε_{2 θ} - ε_{e}^{(1)} t_{2}} + {(\frac{r_{1}}{r_{2}})}^{2 t_{2}}} \frac{ε_{2 θ}}{t_{2}}, ε_{e}^{(1)} = {\begin{cases} \frac{ε_{1 θ}}{t_{1}} t h e c o r e i s d i e l e c t r i c \\ 0 t h e c o r e i s c o n d u c t o r \end{cases},

t_{i} = \sqrt{A R_{i}} (i = 1, 2, 3) .

Note that

ε_{e}^{(l)}

(l = 1, 2)

in Eq. (10) represents the equivalent permittivity of the region from the lth layer to the innermost layer.

We extend the above procedure to an inhomogeneous anisotropic cylinder composed of n piecewise homogeneous layers, the corresponding illusion conditions can be got as follows:

{(\frac{r_{n + 1}}{r_{1}})}^{2} = \frac{(μ_{2 z} - μ_{1 z}) + (μ_{3 z} - μ_{2 z}) {(\frac{r_{2}}{r_{1}})}^{2} + (μ_{4 z} - μ_{3 z}) {(\frac{r_{3}}{r_{1}})}^{2} + \dots + (μ_{(n + 1) z} - μ_{n z}) {(\frac{r_{n - 1}}{r_{1}})}^{2} + (μ_{e} - μ_{0}) {(\frac{r_{e}}{r_{1}})}^{2}}{μ_{(n + 1) z} - μ_{0}},

{(\frac{r_{n + 1}}{r_{n}})}^{2 t_{n + 1}} = \frac{ε_{(n + 1) θ} - t_{n + 1} ε_{e}^{(n)}}{ε_{(n + 1) θ} + t_{n + 1} ε_{e}^{(n)}} \frac{r_{n + 1}^{2} (ε_{(n + 1) θ} + ε_{0} t_{n + 1}) (ε_{e} + ε_{0}) - r_{e}^{2} (ε_{(n + 1) θ} - ε_{0} t_{n + 1}) (ε_{e} - ε_{0})}{r_{n + 1}^{2} (ε_{(n + 1) θ} - ε_{0} t_{n + 1}) (ε_{e} + ε_{0}) - r_{e}^{2} (ε_{(n + 1) θ} + ε_{0} t_{n + 1}) (ε_{e} - ε_{0})},

in which

ε_{e}^{(n)} = \frac{\frac{ε_{n θ} + ε_{e}^{(n - 1)} t_{n}}{ε_{n θ} - ε_{e}^{(n - 1)} t_{n}} - {(\frac{r_{n - 1}}{r_{n}})}^{2 t_{n}}}{\frac{ε_{n θ} + ε_{e}^{(n - 1)} t_{n}}{ε_{n θ} - ε_{e}^{(n - 1)} t_{n}} + {(\frac{r_{n - 1}}{r_{n}})}^{2 t_{n}}} \frac{ε_{n θ}}{t_{n}}, ε_{e}^{(1)} = {\begin{cases} \frac{ε_{1 θ}}{t_{1}} t h e c o r e i s d i e l e c t r i c \\ 0 t h e c o r e i s c o n d u c t o r \end{cases} .

It is worthwhile pointing out that invisibility conditions can be easily derived according to the illusion conditions only by replacing

ε_{e}

,

μ_{e}

, and

r_{e}

in Eqs. (12) and (13) by

ε_{0}

,

μ_{0}

, and

r_{n + 1}

.

2.2.2 Illusion and Invisibility of inhomogeneous sphere

Similarly, we starts with illusion and invisibility of a two-layer inhomogeneous anisotropic sphere coated by a layer of anisotropic dielectric. The nonmagnetic constitutive relation of each layer in spherical coordinate can be expressed as

{\bar{\bar{ε}}}_{i} = [\begin{matrix} ε_{i r} \\ ε_{i θ} \\ ε_{i θ} \end{matrix}] .

With the wave transformation relationship, the incident wave with x-polarization can be written in terms of a linear superposition of spherical waves as [36]

E_{θ}^{i n c} = E_{0} \frac{\cos φ}{k_{0} r} \sum_{n = 1}^{\infty} j^{- n} (2 n + 1) {\hat{J}}_{n} (k_{0} r) P_{n}^{} (\cos θ),

where

{\hat{F}}_{l} (x) = \sqrt{π x / 2} F_{l} (x)

and

P_{n} (c o s θ)

is nth-order Legendre polynomial. The total fields in each layer can be expressed by spherical wave functions as follows:

{\begin{cases} E_{θ} = E_{0} \frac{\cos φ}{ω} \sum_{m = 1}^{\infty} j^{- m} \frac{2 m + 1}{m (m + 1)} A_{m} {\hat{J}}_{1 m} (k_{1} r) P_{m}^{1} (\cos θ) r < r_{1} \\ E_{θ} = E_{0} \frac{\cos φ}{ω} \sum_{m = 1}^{\infty} j^{- m} \frac{2 m + 1}{m (m + 1)} [B_{m} {\hat{J}}_{2 m} (k_{2} r) + C_{m} {\hat{Y}}_{2 m} (k_{2} r)] P_{m}^{1} (\cos θ) r_{1} < r < r_{2} \\ E_{θ} = E_{0} \frac{\cos φ}{ω} \sum_{m = 1}^{\infty} j^{- m} \frac{2 m + 1}{m (m + 1)} [F_{m} {\hat{J}}_{3 m} (k_{3} r) + G_{m} {\hat{Y}}_{3 m} (k_{3} r)] P_{m}^{1} (\cos θ) r_{2} < r < r_{3} \\ E_{θ} = E_{0} \frac{\cos φ}{ω} \sum_{m = 1}^{\infty} j^{- m} \frac{2 m + 1}{m (m + 1)} [{\hat{J}}_{m} (k_{0} r) + D_{m} {\hat{H}}_{m}^{(1)} (k_{0} r)] P_{m}^{1} (\cos θ) r > r_{3} \end{cases},

in which

P_{n}^{1} (c o s θ) = d P_{n} (c o s θ) / d θ

,

k_{i} = ω \sqrt{μ_{0} ε_{i θ}}

, and

i m = \sqrt{2 m (m + 1) A R_{i} + 0.25} - 0.5

(i = 1, 2, 3)

. Therefore, the corresponding scattering field is obtained as

E_{θ} = E_{0} \frac{\cos φ}{ω} \sum_{m = 1}^{\infty} j^{- m} \frac{2 m + 1}{m (m + 1)} D_{m} {\hat{H}}_{m}^{(1)} (k_{0} r) P_{m}^{1} (\cos θ)

The scattering coefficients $D_{m}^{}$ of the coated anisotropic sphere are determined by matching the boundary conditions which require continuity of the tangential components of the total fields on the interface, e.g.,

D_{m}^{} = \frac{\begin{matrix} {\hat{J}}_{1 m} (k_{1} r_{1}) & - {\hat{J}}_{2 m} (k_{2} r_{1}) & - {\hat{Y}}_{2 m} (k_{2} r_{1}) & 0 & 0 & 0 \\ \frac{k_{1}}{ω ε_{1 θ}} {\hat{J^{'}}}_{1 m} (k_{1} r_{1}) & - \frac{k_{2}}{ω ε_{2 θ}} {\hat{J^{'}}}_{2 m} (k_{2} r_{1}) & - \frac{k_{2}}{ω ε_{2 θ}} {\hat{Y^{'}}}_{2 m} (k_{2} r_{1}) & 0 & 0 & 0 \\ 0 & {\hat{J}}_{2 m} (k_{2} r_{2}) & {\hat{Y}}_{2 m} (k_{2} r_{2}) & - {\hat{J}}_{3 m} (k_{3} r_{2}) & - {\hat{Y}}_{3 m} (k_{3} r_{2}) & 0 \\ 0 & \frac{k_{2}}{ω ε_{2 θ}} {\hat{J^{'}}}_{2 m} (k_{2} r_{2}) & \frac{k_{2}}{ω ε_{2 θ}} {\hat{Y^{'}}}_{2 m} (k_{2} r_{2}) & - \frac{k_{3}}{ω ε_{3 θ}} {\hat{J^{'}}}_{3 m} (k_{3} r_{2}) & - \frac{k_{3}}{ω ε_{3 θ}} {\hat{Y^{'}}}_{3 m} (k_{3} r_{2}) & 0 \\ 0 & 0 & 0 & {\hat{J}}_{3 m} (k_{3} r_{3}) & {\hat{Y}}_{3 m} (k_{3} r_{3}) & {\hat{J}}_{m} (k_{0} r_{3}) \\ 0 & 0 & 0 & \frac{k_{3}}{ω ε_{3 θ}} {\hat{J^{'}}}_{3 m} (k_{3} r_{3}) & \frac{k_{3}}{ω ε_{3 θ}} {\hat{Y^{'}}}_{3 m} (k_{3} r_{3}) & \frac{k_{0}}{ω ε_{0}} {\hat{J^{'}}}_{m} (k_{0} r_{3}) \end{matrix}}{\begin{matrix} {\hat{J}}_{1 m} (k_{1} r_{1}) & - {\hat{J}}_{2 m} (k_{2} r_{1}) & - {\hat{Y}}_{2 m} (k_{2} r_{1}) & 0 & 0 & 0 \\ \frac{k_{1}}{ω ε_{1 θ}} {\hat{J^{'}}}_{1 m} (k_{1} r_{1}) & - \frac{k_{2}}{ω ε_{2 θ}} {\hat{J^{'}}}_{2 m} (k_{2} r_{1}) & - \frac{k_{2}}{ω ε_{2 θ}} {\hat{Y^{'}}}_{2 m} (k_{2} r_{1}) & 0 & 0 & 0 \\ 0 & {\hat{J}}_{2 m} (k_{2} r_{2}) & {\hat{Y}}_{2 m} (k_{2} r_{2}) & - {\hat{J}}_{3 m} (k_{3} r_{2}) & - {\hat{Y}}_{3 m} (k_{3} r_{2}) & 0 \\ 0 & \frac{k_{2}}{ω ε_{2 θ}} {\hat{J^{'}}}_{2 m} (k_{2} r_{2}) & \frac{k_{2}}{ω ε_{2 θ}} {\hat{Y^{'}}}_{2 m} (k_{2} r_{2}) & - \frac{k_{3}}{ω ε_{3 θ}} {\hat{J^{'}}}_{3 m} (k_{3} r_{2}) & - \frac{k_{3}}{ω ε_{3 θ}} {\hat{Y^{'}}}_{3 m} (k_{3} r_{2}) & 0 \\ 0 & 0 & 0 & {\hat{J}}_{3 m} (k_{3} r_{3}) & {\hat{Y}}_{3 m} (k_{3} r_{3}) & - {\hat{H}}_{m}^{(1)} (k_{0} r_{3}) \\ 0 & 0 & 0 & \frac{k_{3}}{ω ε_{3 θ}} {\hat{J^{'}}}_{3 m} (k_{3} r_{3}) & \frac{k_{3}}{ω ε_{3 θ}} {\hat{Y^{'}}}_{3 m} (k_{3} r_{3}) & - \frac{k_{0}}{ω ε_{0}} {\hat{H}}_{m}^{(1)}^{'} (k_{0} r_{3}) \end{matrix}} .

Therefore, the RCS and SCS are expressed in terms of the scattering coefficients

D_{m}^{}

respectively as

σ = \lim_{r \to \infty} [4 π r^{2} \frac{{| E_{}^{s c} |}^{2}}{{| E_{}^{i n c} |}^{2}}], C_{s c a} = \frac{2 π}{k_{0}^{2}} \sum_{m = 1}^{\infty} (2 m + 1) {| D_{m}^{} |}^{2} .

Figure 2 shows the SCS for the homogeneous anisotropic and isotropic spheres with the radius of a. It can be seen that for small argument $k_{0} a$ , i.e., in the quasistatic limit, only one term $D_{m}^{}$ dominates the SCS.

Fig. 2 SCS of a sphere versus the argument $k_{0} a$ .

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To make the coated anisotropic sphere behave like an isotropic sphere with a radius of $r_{e}$ , permittivity of $ε_{e}$ , and permeability of $μ_{e}$ , $D_{m}^{}$ must be equal to the corresponding scattering coefficients of the isotropic sphere given by

S_{m}^{} = \frac{\begin{matrix} {\hat{J}}_{m} (k_{e} r_{e}) & {\hat{J}}_{m} (k_{0} r_{e}) \\ \frac{k_{e}}{ω ε_{e}} {\hat{J^{'}}}_{m} (k_{e} r_{e}) & \frac{k_{0}}{ω ε_{0}} {\hat{J^{'}}}_{m} (k_{0} r_{e}) \end{matrix}}{\begin{matrix} {\hat{J}}_{m} (k_{e} r_{e}) & - {\hat{H}}_{m}^{(1)} (k_{0} r_{e}) \\ \frac{k_{e}}{ω ε_{e}} {\hat{J^{'}}}_{m} (k_{e} r_{e}) & - \frac{k_{0}}{ω ε_{0}} {\hat{H}}_{m}^{(1)}^{'} (k_{0} r_{e}) \end{matrix}} .

Therefore, the illusion conditions for the two-layer anisotropic sphere can be obtained by setting

D_{1} = S_{1}

. Further, with a generalized procedure, the illusion conditions for a n-layer anisotropic sphere can be get as

{(\frac{r_{n + 1}}{r_{n}})}^{2 t_{n + 1} + 1} = \frac{ε_{(n + 1) θ} - \frac{1}{2} (1 + t_{n + 1}) ε_{e}^{(n)}}{ε_{(n + 1) θ} + \frac{1}{2} t_{n + 1} ε_{e}^{(n)}} \frac{r_{n + 1}^{3} (2 ε_{0} + ε_{e}) (ε_{(n + 1) θ} + \frac{1}{2} t_{n + 1} ε_{0}) + r_{e}^{3} (ε_{0} - ε_{e}) (ε_{(n + 1) θ} - t_{n + 1} ε_{0})}{r_{n + 1}^{3} (2 ε_{0} + ε_{e}) [ε_{(n + 1) θ} - \frac{1}{2} (1 + t_{n + 1}) ε_{0}] + r_{e}^{3} (ε_{0} - ε_{e}) [ε_{(n + 1) θ} + (1 + t_{n + 1}) ε_{0}]},

in which

ε_{e}^{(n)} = \frac{2 {(\frac{r_{n}}{r_{n - 1}})}^{2 t_{n} + 1} - 2 \frac{ε_{n θ} - \frac{1}{2} (1 + t_{n}) ε_{e}^{(n - 1)}}{ε_{n θ} + \frac{1}{2} t_{n} ε_{e}^{(n - 1)}}}{(1 + t_{n}) {(\frac{r_{n}}{r_{n - 1}})}^{2 t_{n} + 1} + t_{n} \frac{ε_{n θ} - \frac{1}{2} (1 + t_{n}) ε_{e}^{(n - 1)}}{ε_{n θ} + \frac{1}{2} t_{n} ε_{e}^{(n - 1)}}} ε_{n θ}, ε_{e}^{(1)} = {\begin{cases} \frac{2 ε_{1 θ}}{1 + t_{1}} t h e c o r e i s d i e l e c t r i c \\ 0 t h e c o r e i s c o n d u c t o r \end{cases},

t_{i} = \sqrt{2 A R_{i} + 0.25} - 0.5 (i = 1, 2 \dots n + 1) .

The invisibility conditions of inhomogeneous anisotropic sphere can be obtained by substituting the equations

ε_{e} = ε_{0}

and

r_{n + 1} = r_{e}

into Eq. (22).

2.2 Optimization of illusion and invisibility effects

According to the above discussion, we can know that when a coating with the parameters determined by Eqs. (12), (13) and (22) is used, the invisibility and illusion phenomena of the inhomogeneous anisotropic cylindrical and spherical objects appear. Given the radius of the coating, its different anisotropic parameters generate different invisibility and illusion effects. Specifically, electromagnetic parameters of the coating in Eqs. (12), (13) and (22) depend on the electric anisotropy ratio $A R_{i}$ . Hence, we can find the optimal invisibility and illusion of the inhomogeneous anisotropic object by suitably adjusting $A R_{i}$ of the coating. In order to measure the optimal invisibility and illusion effects, we define the following evaluation functions:

σ_{1} = \frac{1}{4 π} \int_{0}^{π} \int_{0}^{2 π} | σ_{c} (θ, φ) - σ_{i} (θ, φ) | \sin θ d φ d θ for illusion,

σ_{2} = \frac{1}{4 π} \int_{0}^{π} \int_{0}^{2 π} | σ_{c} (θ, φ) | \sin θ d φ d θ for invisibility,

where

σ_{c} (θ, φ)

and

σ_{i} (θ, φ)

represent the RCS of the coated cylinder (sphere) and the corresponding illusion object, respectively. It is worthwhile noticing that if the scattering fields from the cylinder (sphere) are of the symmetry, two plane cuts through the symmetric axis, called the principal plane (E-plane or H-plane) are sufficient to represent the scattering fields. In this scenario, the evaluation functions around all the solid angle can be reduced to that in two principle planes. In the following numerical examples, the evaluation functions in two principle planes are used. According to Eq. (25), we can know that the best illusion effect can be obtained, when

σ_{1}

reaches the minimum value. Likewise, the best invisibility effect occurs with the minimum

σ_{2}

. Therefore,

A R_{i}

is searched for to minimize

σ_{1}

and

σ_{2}

for the optimal illusion and invisibility. In addition, note that the formulations for the invisibility and illusion of inhomogeneous cylinder and sphere, e.g., Eqs. (12), (13) and (22), are derived under the condition of the quasi-static approximation. Hence the electrical sizes of the cylindrical and spherical objects should be small. With the increase of the electrical sizes of the objects, more scattering coefficients are required to derive the invisibility and illusion condition.

3. Numerical simulations

To validate the above theoretical design, some numerical examples are given in this section. A finite element solver COMSOL MULTIPHYSICS is used to simulate illusion and invisibility of inhomogeneous anisotropic cylinders and spheres. A TM incident wave is considered in the simulation and the scattering boundary conditions are utilized to terminate the simulation region.

As the first example, consider a conducting sphere coated by a continuously inhomogeneous anisotropic nonmagnetic material. The relative permittivity tensor of the material is $ε_{r} = 2.734 \times 10^{5} r^{3} - 3.139 \times 10^{4} r^{2} + 742.5 r + 15$ and $ε_{θ} = ε_{φ} = 1.519 \times 10^{5} r^{3} - 1.924 \times 10^{4} r^{2} + 442.5 r + 14$ , as shown in Fig. 3(a). In order to model the inhomogeneous material, the material slab is divided into three layers of piecewise homogeneous anisotropic layers. Figure 3(b) shows the electromagnetic and geometric parameters of each layer. To realize the illusion of the inhomogeneous sphere, an anisotropic coating is designed so that the resultant RCS of the coated sphere is similar to that of an anisotropic sphere with $r_{e} = 4 λ_{0} / 15$ and $ε_{e} = 3.874 ε_{0}$ . According to Eq. (22), the parameters of the coating are determined as $r_{4} = 3.5 λ_{0} / 15$ , $ε_{4 r} = 14.6139 ε_{0} / 1.2$ , and $ε_{4 θ} = ε_{4 φ} = 14.6139 ε_{0}$ . Figures 4(a) and 4(b) show magnetic field distributions for the coated and illusion spheres at 900MHz in XOY plane, respectively. Two similar magnetic field distributions can be observed. Further, comparison of bistatic RCS between the coated and illusion spheres is given in Fig. 4(c). A very small difference of the RCS between them can be observed, which validates the proposed methodology.

Fig. 3 A continuously inhomogeneous sphere. (a) constitutive parameters. (b) approximation model with piecewise homogeneous layers.

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Fig. 4 Illusion for a continuously inhomogeneous sphere. (a) magnetic field distribution of the coated sphere in the XOY plane at 900MHz. (b) magnetic field distribution of the illusion sphere in the XOY plane at 900MHz. (c) bistatic RCSs of the coated and illusion spheres in XOY plane.

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In the second example, we consider an inhomogeneous anisotropic sphere composed of three nonmagnetic homogeneous layers. The parameters of the inhomogeneous sphere are as follows: $ε_{1 r} = 40 ε_{0} / 3$ , $ε_{1 θ} = ε_{1 φ} = 5 ε_{0}$ , $ε_{2 r} = 2 ε_{0} / 0.28$ , $ε_{2 θ} = ε_{2 φ} = 2 ε_{0}$ , $ε_{3 r} = 6 ε_{0} / 0.28$ , $ε_{3 θ} = ε_{3 φ} = 6 ε_{0}$ , $r_{1} = λ_{0} / 29$ , $r_{2} = 2 λ_{0} / 29$ , and $r_{3} = 3 λ_{0} / 29$ . In order to make the inhomogeneous sphere behave like a nonmagnetic isotropic sphere with $ε_{e} = 4.13 ε_{0}$ and $r_{e} = 4.5 λ_{0} / 29$ , parameters of the coating are determined according to Eq. (22) as follows: $ε_{4 r} = 9 ε_{0} / 0.055$ , $ε_{4 θ} = ε_{4 φ} = 9 ε_{0}$ , and $r_{4} = 4 λ_{0} / 29$ . According to designer’s intention, the outermost radius of the coated sphere can be arbitrarily chosen. In this example, the outermost radius of the coated sphere is $4 λ_{0} / 29$ , which is different from that of the illusion sphere $4.5 λ_{0} / 29$ . As shown in Figs. 5 (a) and 5(b), magnetic field distribution for the coated sphere is similar to that for the illusion sphere. Figures 5(c) and 5(d) show comparison of bistatic RCS between the coated and illusion spheres in XOY and XOZ planes, respectively. With the designed coating, a small difference of the RCS between them can be observed.

Fig. 5 Illusion for a two-layer inhomogeneous sphere. (a) magnetic field distribution of the coated sphere in the XOY plane at 500MHz. (b) magnetic field distribution of the illusion sphere in the XOY plane at 500MHz. (c) bistatic RCS of the coated and illusion spheres in XOY plane. (d) bistatic RCSs of the coated and illusion spheres in XOZ plane

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To demonstrate how to obtain the best illusion effect, consider a homogeneous sphere with a radius of $r_{1} = 2 λ_{0} / 18$ . Its permittivity tensor in spherical coordinate is $ε_{1 r} = 24 ε_{0} / 0.48$ and $ε_{1 θ} = ε_{1 φ} = 24 ε_{0}$ . A coating with a radius of $r_{2} = 4 λ_{0} / 18$ is designed such that the coated sphere has the most similar image to an isotropic sphere with $r_{e} = 5 λ_{0} / 18$ and $ε_{e} = 3.3 ε_{0}$ . Figure 6(a) illustrates the dependence of $σ_{1}$ on anisotropic ratio AR of the coating. From the Fig. 6(a) we can see that AR = 1 leads to the minimal $σ_{1}$ . Substituting AR = 1 into Eq. (22), the parameters of the coating can be determined as $ε_{2 r} = 16.225 ε_{0}$ , and $ε_{2 θ} = ε_{2 φ} = 16.225 ε_{0}$ . Figure 6(b) shows bistatic RCS in XOY plane for different AR of the coating. It can be seen that RCS for AR = 1 approximately coincides with that of the illusion sphere. Figure 7 shows comparison of magnetic field distribution between the coated sphere with the optimal AR and the illusion sphere. Very good illusion effect can be obtained.

Fig. 6 Optimal illusion effect of a homogeneous sphere. (a) evaluation function versus AR. (b) bistatic RCSs for different AR.

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Fig. 7 Illusion for a homogeneous anisotropic sphere. (a) magnetic field distribution of the coated sphere with the optimal AR in the XOY plane at 400MHz. (b) magnetic field distribution of the illusion sphere in the XOY plane at 400MHz.

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In the following, the above method is used to design an invisibility cloak of a two-layer inhomogeneous sphere with a PEC core. The parameters of the inhomogeneous sphere are $ε_{2 r} = 35 ε_{0} / 1.68$ , $ε_{2 θ} = ε_{2 φ} = 35 ε_{0}$ , $r_{1} = λ_{0} / 14$ , and $r_{2} = 1.5 λ_{0} / 14$ . In order to dramatically reduce RCS of the inhomogeneous sphere, a shell is introduced to cover the sphere. According to Eq. (22), parameters of the shell can be determined as $ε_{3 r} = 0.34 ε_{0}$ , $ε_{3 θ} = ε_{3 φ} = 0.24 ε_{0}$ , and $r_{3} = 2 λ_{0} / 14$ . Figures 8(a) and 8(b) depict magnetic field distributions for the uncoated and coated spheres at 600MHz, respectively. A good invisibility of the coated sphere can be observed. Further, Figs. 9(a) and 9(b) show comparison of bistatic RCS between the uncoated and coated spheres in XOY and XOZ planes, respectively. In XOY plane, RCS reduction of 19dB at $θ = 0^{o}$ and 23dB at $θ = 180^{o}$ can be obtained, and in XOZ plane, RCS reduction larger than 19dB can be achieved. For comparison, the TE polarization case is given in Figs. 9(c) and 9(d). It can be seen from Figs. 9(c) and 9(d) that RCS reduction of 9dB at $θ = 180^{o}$ in XOY plane and 4dB at $θ = 0^{o}$ in XOZ plane can be obtained, respectively. According to Fig. 9, we can know that the invisibility performance for TE polarization is worse than that for TM polarization.

Fig. 8 Invisibility for a two-layer inhomogeneous sphere. (a) magnetic field distribution of the uncoated sphere in the XOY plane at 600MHz. (b) magnetic field distribution of the coated sphere in the XOY plane at 600MHz.

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Fig. 9 Comparison of bistatic RCS between the coated and uncoated spheres. (a) RCS in XOY plane for TM polarization. (b) RCS in XOZ plane for TM polarization. (c) RCS in XOY plane for TE polarization. (d) RCS in XOZ plane for TE polarization.

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In order to further decrease RCS of the inhomogeneous sphere, electric anisotropy ratio AR of the coating is optimized. Figures 10(a) and 10(b) show variation of evaluation function $σ_{2}$ with AR of the coating and bistatic RCS for different AR of the coating, respectively. It can be seen that the smallest RCS can be obtained when AR is equal to 3.01, which corresponds to the minimal $σ_{2}$ . Compared with the previous design with AR = 0.71, RCS with the optimized AR decreases 20dB at $θ = 0^{o}$ and 54dB at $θ = 180^{o}$ . Note that when AR of the coating increases to a certain extent, RCS of the coated sphere obviously increases. This is because with the increase of AR, electric size of the coated sphere increases accordingly such that the quasistatic limit condition cannot be satisfied.

Fig. 10 Optimal invisibility effect of an inhomogeneous sphere. (a) evaluation function versus AR. (b) bistatic RCSs for different AR.

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Next, consider a two-layer inhomogeneous cylinder whose permittivity and permeability tensors are $ε_{1 r} = 6.25 ε_{0}$ , $ε_{1 θ} = 4 ε_{0}$ , $ε_{2 r} = 0.777 ε_{0}$ , $ε_{2 θ} = 0.304 ε_{0}$ , $μ_{1 z} = 3 μ_{0}$ , and $μ_{2 z} = μ_{0} / 3$ . The radius of each layer is $r_{1} = λ_{0} / 138$ and $r_{2} = 2 λ_{0} / 138$ . Scattering effect of the inhomogeneous cylinder is manipulated by using a coating such that the resultant electromagnetic response is similar to that of an isotropic cylinder with a radius of $r_{e} = 4.5 λ_{0} / 138$ , permittivity of $ε_{e} = 6.28 ε_{0}$ , and permeability of $μ_{e} = 4 μ_{0}$ . According to Eqs. (12) and (13), parameters of the coating are obtained as $ε_{3 r} = 56.25 ε_{0}$ , $ε_{3 θ} = 36 ε_{0}$ , $μ_{3 z} = 4.0625 μ_{0}$ , and $r_{3} = 4 λ_{0} / 138$ . Figures 11(a) and 11(b) show magnetic field distributions for the coated and illusion cylinders, respectively. Scattering field depicted in Fig. 11(a) is in good agreement with that shown in Fig. 11(b). Further, in order to obtain the optimal illusion effect, Fig. 12(a) demonstrates variation of evaluation function $σ_{1}$ with $A R$ of the coating, when the radius of the coating is fixed. It can be seen that when AR = 0.64, $σ_{1}$ reaches the minimum value. Figure 12(b) shows bistatic RCSs for different AR. The RCS for AR = 0.64 coincides with that of the illusion cylinder, thus leading to a best illusion effect.

Fig. 11 Illusion for a two-layer inhomogeneous cylinder. (a) magnetic field distribution at 300MHz for the coated cylinder. (b) magnetic field distribution at 300MHz for the illusion cylinder.

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Fig. 12 Optimal illusion effect of an inhomogeneous cylinder. (a) evaluation function versus AR. (b) bistatic RCSs for different AR.

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In the following, a four-layer inhomogeneous cylinder is considered. In the cylinder, the innermost layer is a conductor, and electromagnetic parameters of the remaining layers are $ε_{2 r} = 40.5 ε_{0}$ , $ε_{2 θ} = 18 ε_{0}$ , $ε_{3 r} = 8 ε_{0} / 0.36$ , $ε_{3 θ} = 8 ε_{0}$ , $ε_{4 r} = 5 ε_{0} / 0.64$ , and $ε_{4 θ} = 5 ε_{0}$ . Dimensions of the four-layer cylinder are $r_{1} = λ_{0} / 30$ , $r_{2} = 1.5 λ_{0} / 30$ , $r_{3} = 2 λ_{0} / 30$ , and $r_{4} = 3 λ_{0} / 30$ . To greatly reduce RCS of the cylinder, a coating whose parameters are $ε_{5 r} = 0.33 ε_{0}$ , $ε_{5 θ} = 0.08 ε_{0}$ , and $r_{5} = 4 λ_{0} / 30$ is designed. Figures 13(a) and 13(b) illustrate magnetic field distributions for the uncoated and coated cylinders, respectively. Figure 13(c) shows bistatic RCSs of the uncoated and coated cylinders, and a significant reduction of 18dB at $θ = 0^{o}$ and 12.5dB at $θ = 180^{o}$ is observed. As shown in Fig. 13, the cylindrical object is well hidden by the use of the coating from the perspective of near and far fields. We further search for the optimal invisibility performance by adjusting the coating’s anisotropy ratio. Figure 14(a) shows variation of the evaluation function $σ_{2}$ with AR of the coating. We can see that the coating with AR = 7 results in the minimum $σ_{2}$ . The bistatic RCSs for different AR of the coating are compared in Fig. 14(b). The best invisibility effect with RCS reduction of 19dB at $θ = 0^{o}$ and 41dB at $θ = 180^{o}$ occurs when AR = 7. It is worthwhile pointing out that when AR of the coating becomes large enough, the resultant RCS increases due to the invalid long-wavelength limit. Further, the invisibility of the loss object is studied. Firstly, assume ε_θ of each layer lossy with lossless ε_r. Specifically, ε_θ of each layer is as follows: $ε_{2 θ} = (18 - 3 j) ε_{0}$ , $ε_{3 θ} = (8 - 2 j) ε_{0}$ , and $ε_{4 θ} = (5 - 3 j) ε_{0}$ . Figure 15 shows the magnetic field distribution and the bistatic RCS of the lossy cylinder with the above designed lossless coating. It can be seen that the cylindrical object is well hidden. In the following, consider lossy ε_r of each layer with lossless ε_θ. The corresponding ε_r of each layer is as follows: $ε_{2 r} = (40.5 - 0.5 j) ε_{0}$ , $ε_{3 r} = (8 / 0.36 - 0.5 j) ε_{0}$ , and $ε_{4 r} = (5 / 0.64 - 0.2 j) ε_{0}$ . As shown in Fig. 16, good invisibility of the loss cylinder can be realized with the use of the designed lossless coating. It is worthwhile pointing out that the electromagnetic parameters of the coating in Eqs. (12), (13) and (22) are determined with the assumption of inhomogeneous anisotropic lossless object. When the loss of the target is small, good invisibility and illusion are still achieved with the designed coating.

Fig. 13 Invisibility for a four-layer inhomogeneous cylinder. (a) magnetic field distribution at 800MHz for the uncoated target. (b) magnetic field distribution at 800MHz for the coated target. (c) bistatic RCSs of the uncoated and coated cylinders.

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Fig. 14 Optimal invisibility effect of an inhomogeneous cylinder. (a) evaluation function versus AR. (b) bistatic RCSs for different AR.

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Fig. 15 An inhomogeneous cylinder with lossy ε_θ. (a) magnetic field distribution at 800MHz for the uncoated target. (b) magnetic field distribution at 800MHz for the coated target. (c) bistatic RCSs of the uncoated and coated cylinders.

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Fig. 16 An inhomogeneous cylinder with lossy ε_r. (a) magnetic field distribution at 800MHz for the uncoated target. (b) magnetic field distribution at 800MHz for the coated target. (c) bistatic RCSs of the uncoated and coated cylinders.

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Finally, we consider a practical engineering implementation for illusion of a conducting cylinder with a radius of $λ_{0} / 10$ by using composite optical material, as shown in Fig. 17(a). In order to manipulate the scattering field of the PEC cylinder similar to a dielectric cylinder with a radius of $λ_{0} / 5$ and permittivity of $ε_{e} = 3.473 ε_{0}$ , a coating with a radius of $λ_{0} / 5$ and permittivity tensor of $ε_{2 r} = 2.878 ε_{0}$ and $ε_{2 θ} = 6.769 ε_{0}$ is used, according to Eqs. (12) and (13). A set of plasmonic spherical nanoparticles with a random distribution embedded in a dielectric host can be equivalent to a slab material with anisotropic dielectric [37, 38] in THz frequency band. According to the Maxwell Garnett effective medium theory, the permittivity tensor can be obtained as [37, 38]

ε_{r} = ε_{h} \frac{(1 + f) ε_{m} + (1 - f) ε_{h}}{(1 + f) ε_{h} + (1 - f) ε_{m}}, ε_{θ} = f ε_{m} + (1 - f) ε_{h},

where f is the filling factor of nanoparticles in the dielectric host,

ε_{h}

is permittivity of the dielectric host, and

ε_{m}

is permittivity of the spherical nanoparticle, which follows Drude model according to

ε_{m} = ε_{\infty} - \frac{ω_{p}^{2}}{ω^{2} - j ω γ},

in which

ε_{\infty}

is high frequency dielectric constant,

ω_{p}

is plasma frequency, and

γ

is damping term. In order to realize the electromagnetic parameters of the coating, the dielectric host is chosen as Poly Tetra Fluor Ethylene with the relative permittivity of 1.7689 [39]. The nanoparticles are made of silver with

ε_{\infty} = 5

,

ω_{p} = 13666 rad/s

, and

γ = 2733 rad/s

[40], and the filling factor of particles f is 0.5. Figure 17(b) shows RCS comparison between the conducting and the coated cylinders at 3THz, with a good agreement with each other. In addition, comparison of magnetic field distribution between them is given in Figs. 17(c) and 17(d). Two very similar magnetic distributions can be observed.

Fig. 17 Illusion of a conducting cylinder. (a) schematic figure of the structure. (b) bistatic RCS of the uncoated and coated cylinders. (c) magnetic field distribution at 3THz for the coated cylinder. (d) magnetic field distribution at 3THz for the illusion cylinder.

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4. Conclusion

This paper focuses on the design of illusion and invisibility cloaks for arbitrarily inhomogeneous anisotropic cylindrical and spherical objects based on Mie scattering theory. With the proposed formulations, parameters of the coating can be analytically solved to achieve illusion and invisibility of the object in the long-wavelength limit. Much efforts have been put on the analyses of the effects of the coating’s electric anisotropy ratio on the RCS. By optimizing the electric anisotropy ratio of the coating, the optimal illusion and invisibility can be achieved. A fairly good agreement between the simulation and the theory validates the proposed method.

Funding

The Program for the New Scientific and Technological Star of Shaanxi Province (No.2013KJXX-66, No. BD11015020008); Technology Innovation Research Project of the CETC; Fundamental Research Funds for the Central Universities (No. SPSZ031410).

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Optimal illusion and invisibility of multilayered anisotropic cylinders and spheres

Abstract

1. Introduction

2. Theoretical analysis

2.1 Illusion and Invisibility of inhomogeneous anisotropic cylinders and spheres

2.2.1 Illusion and Invisibility of inhomogeneous anisotropic cylinder

2.2.2 Illusion and Invisibility of inhomogeneous sphere

2.2 Optimization of illusion and invisibility effects

3. Numerical simulations

4. Conclusion

Funding

References and links

Cited By

Figures (17)

Equations (28)

Optics Express