Cloaking design for arbitrarily shape objects based on characteristic mode method

Yan Shi; Lin Zhang

doi:10.1364/OE.25.032263

1. Introduction

Interest in metamaterials has grown in recent years and various electromagnetic applications based on their anomalous properties have been developed [1–3]. One of the most exciting applications is realization of electromagnetic invisibility cloak [4–28]. Especially, transformation optics (TO) [8, 9] as a breakthrough technique to manipulate electromagnetic waves offers an unprecedented flexibility in design of the cloak. To-based cloaking strategies theoretically and experimentally demonstrated at various frequencies rely on the intuitive idea of bending incoming waves around an object smoothly without any scatterings [10–17]. Material properties based on the TO design are often strongly inhomogeneous, anisotropic and negative. These complexities of the material parameters limit the practical realization of these cloaks.

In recent years, scattering cancellation has been shown to be a simple way to design cloaking devices with moderately size objects [18–28]. By warping a shell with the designed permittivity and permeability on a target, the target becomes invisible [18–24] or disguised as another object [25–27]. The scattering cancellation-based cloak design is implemented by controlling the zeroth and first orders of Mie scattering coefficients. However, analytical expressions for the Mie scattering coefficients are only limited to canonical shapes, for instance an infinite cylinder and a sphere [18–27]. Extension of the scattering cancellation approach to the cloaking design for the arbitrarily-shaped objects has been reported [28]. The facing critical issue is how to determine the relationship between scattering section of the object and its equivalent polarizability tensor. A feasible implementation way to use numerical methods, for example method of moment (MoM) [29]. However, owing to time consuming, it becomes cumbersome to implement the numerical methods in the scattering cancellation-based cloaking design for the arbitrarily-shaped objects.

In this work, a characteristic mode method (CMM) [30–32] is developed to design invisibility and illusion cloaks for arbitrarily-shaped objects. Discrete dipole approximation (DDA) [33–38] as a flexible and powerful technique is used to solve electromagnetic wave scattering from object with arbitrary geometry and composition. With the CMM, a generalized eigenvalue equation for the matrix in the DDA method is formulated and physical mechanisms of the scattering problem are shed some light on according to the resultant generalized eigenvalues, which distinguishes the proposed method from all reported scattering cancellation approaches. The analytical formulations are derived for illusion and invisibility cloak designs. To our best knowledge, it is first time to theoretically design the cloak based on the CMM. The formulations of our theory are given in Section II. In Section III, we present the numerical simulations to achieve the optimal invisibility and illusion cloak designs for the arbitrarily-shaped objects. Conclusions are given in Section IV.

2. Cloaking of arbitrarily shaped objects

2.1 Discrete-dipole approximation (DDA)

The DDA method initially proposed by Purcell and Pennypacker [33] and further developed by Draine and associates [34–36] is used to solve interaction of electromagnetic wave and object with arbitrary shape in free space. Consider an incident plane wave with ${\overset{⇀}{E}}^{i n c} (\overset{⇀}{r}) = {\hat{u}}^{i n c} e^{i {\overset{⇀}{k}}^{i n c} \cdot \overset{⇀}{r}}$ scattering from a dielectric object with relative permittivity of $ε_{r}$ , where ${\overset{⇀}{k}}^{i n c} = k {\hat{k}}^{i n c}$ and ${\hat{u}}^{i n c}$ are incident vector and unit polarization vector, respectively. Here $e^{- i ω t}$ time dependence is assumed.

The DDA method starts from a volume integral equation (VIE) governing electric field inside a dielectric scatterer [39], i.e.,

\overset{⇀}{E} (\overset{⇀}{r}) = {\overset{⇀}{E}}^{i n c} (\overset{⇀}{r}) + \frac{i}{4 π ω ε_{0}} \int_{V} \bar{G} (\overset{⇀}{r}, {\overset{⇀}{r}}^{'}) \cdot \overset{⇀}{J} ({\overset{⇀}{r}}^{'}) d v^{'},

where

\overset{⇀}{J} (\overset{⇀}{r}) = - i ω [ε_{r} (\overset{⇀}{r}) - 1] ε_{0} \overset{⇀}{E} (\overset{⇀}{r})

is equivalent volume current density and free-space dyadic Green’s function is defined as

\bar{G} (\overset{⇀}{r}, {\overset{⇀}{r}}^{'}) = [k^{2} \bar{I} + \nabla \nabla] g (\overset{⇀}{r}, {\overset{⇀}{r}}^{'}) = g (\overset{⇀}{r}, {\overset{⇀}{r}}^{'}) [k^{2} (\bar{I} - \frac{\hat{R} \hat{R}}{R^{2}}) - \frac{1 - i k R}{R^{2}} (\bar{I} - 3 \frac{\hat{R} \hat{R}}{R^{2}})],

g (\overset{⇀}{r}, {\overset{⇀}{r}}^{'}) = \frac{e^{i k | \overset{⇀}{r} - {\overset{⇀}{r}}^{'} |}}{| \overset{⇀}{r} - {\overset{⇀}{r}}^{'} |}, \hat{R} = \frac{\overset{⇀}{R}}{R} = \frac{\overset{⇀}{r} - {\overset{⇀}{r}}^{'}}{| \overset{⇀}{r} - {\overset{⇀}{r}}^{'} |} .

The object is discretized into N small cubic elements, each of which is approximately represented by a point dipole. The size length of each cubic element is $d$ and the corresponding volume is $V_{j} = d^{3}$ ( $j = 1, \dots, N$ ). Application of the discretization into the VIE leads to

\overset{⇀}{E} (\overset{⇀}{r}) - \frac{i}{4 π ε_{0} ω} \int_{V_{j}} \bar{G} (\overset{⇀}{r}, {\overset{⇀}{r}}^{'}) \cdot \overset{⇀}{J} ({\overset{⇀}{r}}^{'}) d v^{'} = {\overset{⇀}{E}}^{e x c} (\overset{⇀}{r}),

in which

{\overset{⇀}{E}}^{e x c} (\overset{⇀}{r}) = {\overset{⇀}{E}}^{i n c} (\overset{⇀}{r}) + \frac{i}{4 π ε_{0} ω} \sum_{\begin{array}{l} k = 1 \\ k \neq j \end{array}}^{N} \int_{V_{k}} \bar{G} (\overset{⇀}{r}, {\overset{⇀}{r}}^{'}) \cdot \overset{⇀}{J} ({\overset{⇀}{r}}^{'}) d v^{'} .

Equation (5) represents the field exciting the jth element, which consists of the incident field and the contribution from all other elements. Invoking the long wave approximation, we can re-express Eq. (5) as

{\overset{⇀}{E}}_{j}^{e x c} = {\overset{⇀}{E}}_{j}^{i n c} + \frac{i}{4 π ω ε_{0}} \sum_{\begin{array}{l} k = 1 \\ k \neq j \end{array}}^{N} \bar{G} ({\overset{⇀}{r}}_{j}, {\overset{⇀}{r}}_{k}) \cdot {\overset{⇀}{J}}_{k} Δ V_{k} .

The volume current density ${\overset{⇀}{J}}_{k}$ in the kth element can be related to a dipole moment

{\overset{⇀}{p}}_{k} = \frac{i}{4 π ω ε_{0}} V_{k} {\overset{⇀}{J}}_{k} .

Inserting Eq. (7) into Eq. (6), we can obtain

{\overset{⇀}{E}}_{j}^{e x c} = {\overset{⇀}{E}}_{j}^{i n c} + \sum_{\begin{array}{l} k = 1 \\ k \neq j \end{array}}^{N} \bar{G} ({\overset{⇀}{r}}_{j}, {\overset{⇀}{r}}_{k}) \cdot {\overset{⇀}{p}}_{k} .

Furthermore, the dipole moment in the

V_{j}

induced by

{\overset{⇀}{E}}_{j}^{e x c}

can be obtained as

{\overset{⇀}{p}}_{j} = α_{j} {\overset{⇀}{E}}_{j}^{e x c},

where

α_{j}

is defined as dipole polarizability. There are some formulations for calculation of the dipole polarizability [38]. One of the most popular forms is based on lattice dispersion relation (LDR), i.e.,

α_{j} = \frac{α_{j}^{C M}}{1 + (α_{j}^{C M} / d^{3}) [b_{1} + ε_{j} b_{2} + ε_{j} b_{3} S] {(k d)}^{2} - (2 / 3) i {(k d)}^{3}},

in which

b_{1} = - 1.8915316

,

b_{2} = 0.1648469

,

b_{3} = - 1.7700004

and the Clausius–Mossotti polarizability

α_{j}^{C M}

is

α_{j}^{C M} = \frac{3 d^{3}}{4 π} (\frac{ε_{j} - 1}{ε_{j} + 2}),

and

S = {(u_{x}^{i n c} k_{x}^{i n c})}^{2} + {(u_{y}^{i n c} k_{y}^{i n c})}^{2} + {(u_{z}^{i n c} k_{z}^{i n c})}^{2} .

Here

k_{t}^{i n c}

and

u_{t}^{i n c}

(t = x, y, z)

are components of the incident vector

{\overset{⇀}{k}}^{i n c}

and the polarization vector

{\overset{⇀}{u}}^{i n c}

of the incident field. Substituting Eq. (9) into Eq. (8), we can obtain a system of 3N complex linear equations:

\bar{A} \cdot \bar{p} = {\bar{E}}^{i n c},

where

A_{m n} = {\begin{cases} - \frac{e^{i k R_{m^{'} (m) n^{'} (n)}}}{R_{_{m^{'} (m) n^{'} (n)}}} [k^{2} ({\bar{I}}_{u^{'} (m) v^{'} (n)} - \frac{{\hat{R}}_{u^{'} (m)} {\hat{R}}_{v^{'} (n)}}{R_{_{m^{'} (m) n^{'} (n)}}^{2}}) - \frac{1 - i k R_{_{m^{'} (m) n^{'} (n)}}}{R_{_{m^{'} (m) n^{'} (n)}}^{2}} ({\bar{I}}_{u^{'} (m) v^{'} (n)} - 3 \frac{{\hat{R}}_{u^{'} (m)} {\hat{R}}_{v^{'} (n)}}{R_{_{m^{'} (m) n^{'} (n)}}^{2}})] m \neq n \\ α_{m^{'} (m)}^{- 1} m = n \end{cases} .

Here

m^{'} (m)

and

n^{'} (n)

are the indexes of the cubic elements corresponding to the mth row and the nth column of the matrix, respectively, and

u^{'} (m)

is the Cartesian components in the

m^{'} th

cubic element. Note that

\bar{A}

in Eq. (13) is a symmetry matrix. By solving Eq. (13), we can obtain the polarization of each element. Furthermore, the scattered electric field with the scattering vector

{\overset{⇀}{k}}^{s c a} = k {\hat{k}}^{s c a}

and the radar cross section (RCS) can be calculated as, respectively

{\overset{⇀}{E}}^{s c a} (\overset{⇀}{r}) = \frac{e^{i k r}}{r} k^{2} \sum_{m = 1}^{N} e^{- i k {\overset{⇀}{r}}_{m} \cdot {\hat{k}}^{s c a}} {\overset{⇀}{p}}_{m},

σ = \lim_{r \to \infty} 4 π r^{2} {| \frac{{\overset{⇀}{E}}^{s c a}}{{\overset{⇀}{E}}^{i n c}} |}^{2} = 4 π k^{4} {| \sum_{m = 1}^{N} e^{- i k {\overset{⇀}{r}}_{m} \cdot {\hat{k}}^{s c a}} {\overset{⇀}{p}}_{m} |}^{2} .

2.2 Characteristic mode method (CMM)

The CMM originally defined by Garbacz [30] and later reformulated by Harrington and Mautz [31,32] determines mode currents and modal fields for the objects with arbitrary shapes. The CMM offers deep insights into the analysis of scattering and radiation problems. According to $\bar{A}$ in Eq. (13), its real and imaginary parts can be defined as

\bar{R} = \frac{1}{2} (\bar{A} + {\bar{A}}^{*}), \bar{X} = \frac{1}{2 i} (\bar{A} - {\bar{A}}^{*}) .

Here asterisk denotes complex conjugate. Because

\bar{A}

is symmetry,

\bar{R}

and

\bar{X}

are real symmetry. From Eq. (17), we can know

\bar{A} = \bar{R} + i \bar{X}

.

Case 1 $\bar{X} \neq 0$

For the matrix $\bar{A}$ with nonzero imaginary part, we consider the following generalized eigenvalue equation

\bar{A} \cdot {\bar{q}}_{n} = (1 + i λ_{n}) \bar{R} \cdot {\bar{q}}_{n},

in which

λ_{n}

are eigenvalues and

{\bar{q}}_{n}

are corresponding eigenfunctions. Further, Eq. (18) can be reduced to

\bar{X} \cdot {\bar{q}}_{n} = λ_{n} \bar{R} \cdot {\bar{q}}_{n} .

Due to the real symmetry property of $\bar{R}$ and $\bar{X}$ , the eigenvalues $λ_{n}$ and the eigenfunctions ${\bar{q}}_{n}$ are real. Moreover, the eigenfunctions ${\bar{q}}_{n}$ satisfy the orthogonality relationship [31,32], i.e.,

{\bar{q}}_{m}^{T} \cdot \bar{Z} \cdot {\bar{q}}_{n} = 0 (\bar{Z} = \bar{A}, \bar{R}, \bar{X}),

where

m \neq n

. When the eigenfunctions

{\bar{q}}_{n}

are normalized according to

{\bar{q}}_{n}^{T} \cdot \bar{R} \cdot {\bar{q}}_{n} = 1,

we can obtain

{\begin{cases} {\bar{q}}_{m}^{T} \cdot \bar{R} \cdot {\bar{q}}_{n} = δ_{m n} \\ {\bar{q}}_{m}^{T} \cdot \bar{X} \cdot {\bar{q}}_{n} = λ_{n} δ_{m n} \\ {\bar{q}}_{m}^{T} \cdot \bar{A} \cdot {\bar{q}}_{n} = (1 + i λ_{n}) δ_{m n} \end{cases},

where

δ_{m n}

is the Kronecker delta. It can be seen from (22) that the choice of

{\bar{q}}_{n}

as the basis functions leads to diagonal matrix representations of

\bar{A}

,

\bar{R}

and

\bar{X}

. Therefore, we shall call

{\bar{q}}_{n}

the characteristic dipole moments. Then the dipole moment

\overset{⇀}{p}

of the object is expanded in terms of

{\bar{q}}_{n}

, i.e.,

\bar{p} = \sum_{l} α_{l} {\bar{q}}_{l} .

Substituting Eq. (23) into Eq. (13) and considering Eq. (22), we have

\bar{p} = \sum_{l} \frac{V_{l}^{i n c}}{1 + i λ_{l}} {\bar{q}}_{l},

in which

V_{l}^{i n c} = {\bar{q}}_{l}^{T} \cdot {\bar{E}}^{i n c} .

Substituting Eq. (24) into Eq. (16), the RCS is rewritten as

σ = 4 π k^{4} {| \sum_{l} \frac{V_{l}^{i n c} V_{l}^{s c a}}{1 + i λ_{l}} |}^{2},

where

V_{l}^{s c a}

can be obtained by replacing

{\hat{k}}^{i n c}

in Eq. (25) by

{\hat{k}}^{s c a}

.

Case 2 $\bar{X} \approx 0$

If $\bar{X}$ is approximately negligible compared with $\bar{R}$ , the eigenvalue Eq. (18) becomes meaningless. In this scenario, the eigenvalue equation should be reconsidered as

\bar{A} \cdot {\bar{q}}_{n} = \bar{R} \cdot {\bar{q}}_{n} = λ_{n} {\bar{q}}_{n} .

With the normalization and orthogonality of eigenfunctions ${\bar{q}}_{n}$ , we have

{\bar{q}}_{m}^{T} \cdot \bar{A} \cdot {\bar{q}}_{n} = λ_{n} δ_{m n} .

Following the above derivation, we can obtain

\bar{p} = \sum_{l} \frac{V_{l}^{i n c}}{λ_{l}} {\bar{q}}_{l},

Finally, the RCS can be expressed as

σ = 4 π k^{4} {| \sum_{l} \frac{V_{l}^{i n c} V_{l}^{s c a}}{λ_{l}} |}^{2} .

It is worthwhile pointing out that according to Eqs. (26) and (30) the eigenvalue $λ_{n}$ ranging from $- \infty$ to $+ \infty$ is very important for scattering phenomenon. When $| λ_{n} | = 0$ , the corresponding eigenfunction ${\bar{q}}_{n}$ is an externally resonant mode which is most efficient in scattering energy. On the contrary, the eigenfunction ${\bar{q}}_{n}$ with $| λ_{n} | = \infty$ corresponds to the trivial mode, which does not lead to scattering problem. When $λ_{n} > 0$ ( $λ_{n} < 0$ ), ${\bar{q}}_{n}$ is an inductive (capacitive) mode which stores predominantly magnetic (electric) energy. Figure 1 shows the eigenvalue distributions and the RCSs of two spheres with the same radius and different permittivity at 700MHz. The radius of the sphere is $0.15 λ$ , in which $λ$ is wavelength in free space. Relative permittivities of two spheres are 5 and 1.02, respectively. We can observe that the eigenvalues of the sphere with the relative permittivity of 5 distribute at zero nearby, while the eigenvalues of the nearly air-filled sphere tend to be infinite. Hence, the RCS of the former is far larger than that of the latter.

Fig. 1 Two spheres with the same radius and the different permittivities. (a) Eigenvalue distribution. (b) RCS.

Download Full Size | PDF

2.3 Illusion and invisibility conditions

According to Eqs. (26) and (30), we can observe that the RCS of the object is tightly related to its eigenvalues. Assume a dielectric object with an arbitrary shape and relative permittivity $ε_{1}$ . In order to make the object have an illusion image as another object with relative permittivity $ε_{e}$ , the object is wrapped by a dielectric shell with relative permittivity $ε_{2}$ so that the shape of the resultant coated object is same as that of the illusion object, as shown in Fig. 2.

Fig. 2 Diagram of the illusion effect.

Download Full Size | PDF

We use the DDA method to solve the plane wave scattering from the coated object and the illusion object, respectively. The corresponding linear equation systems of the coated and the illusion objects are

{\bar{A}}^{c o a t e d} {\bar{p}}^{c o a t e d} = {\bar{E}}^{i n c},

{\bar{A}}^{I l l u s i o n} {\bar{p}}^{I l l u s i o n} = {\bar{E}}^{i n c} .

Considering that the coated and illusion objects have the same shapes, the dimensions of the matrices in Eqs. (31) and (32) are same when the same cubic elements are used to discretize these two objects. According to the CMM, we have

{({\bar{R}}^{c o a t e d})}^{- 1} {\bar{X}}^{c o a t e d} {\bar{q}}^{c o a t e d} = λ^{c o a t e d} {\bar{q}}^{c o a t e d},

{({\bar{R}}^{I l l u s i o n})}^{- 1} {\bar{X}}^{I l l u s i o n} {\bar{q}}^{I l l u s i o n} = λ^{I l l u s i o n} {\bar{q}}^{I l l u s i o n} .

If the coated object has the same eigenvalues and eigenfunctions as those of the illusion object, the coated object behaves same as the illusion object. However, this rigorous condition is almost impossible to achieve. An approximate to the rigorous condition is to keep summation of all eigenvalues same for the coated and illusion objects, i.e., the same traces of the matrices. Specifically, we have

t r [{\bar{X}}^{I l l u s i o n} {({\bar{R}}^{I l l u s i o n})}^{- 1}] = t r [{\bar{X}}^{c o a t e d} {({\bar{R}}^{c o a t e d})}^{- 1}] .

When the object is of moderately size, the diagonal elements of the matrices

{\bar{A}}^{c o a t e d}

and

{\bar{A}}^{I l l u s i o n}

in Eqs. (31) and (32) are dominant compared with the non-diagonal elements. Hence these non-diagonal elements are reasonably omitted. In this scenario, the matrices in Eq. (35) can be written as

{\bar{X}}^{I l l u s i o n} = {\bar{X}}^{c o a t e d} = d i a g [\begin{matrix} - \frac{2}{3} k^{3} & \dots & - \frac{2}{3} k^{3} \end{matrix}],

{\bar{R}}^{I l l u s i o n} = d i a g [\begin{matrix} β_{e} & \dots & β_{e} \end{matrix}],

{\bar{R}}^{c o a t e d} = d i a g [\begin{matrix} β_{2} & \dots & β_{2} & β_{1} & \dots & β_{1} & β_{2} & \dots & β_{2} \end{matrix}],

where

β_{s} = \frac{4 π}{3 d^{3}} \frac{ε_{s} + 2}{ε_{s} - 1} + \frac{k^{2}}{d} (b_{1} + b_{2} ε_{s})

(s = 1, 2, e)

. Assume that the total numbers of the dipole moments for both the coated and illusion objects are N, and the numbers of the dipole moments for the original object and the coating shell are N₁ and N₂, respectively. Therefore, we have N = N₁ + N₂. Inserting Eqs. (36)-(38) into Eq. (35), we can obtain

\frac{N_{1}}{β_{1}} + \frac{N_{2}}{β_{2}} = \frac{N}{β_{e}} .

Solving Eq. (39), the relative permittivity of the coating shell can be obtained as

\begin{array}{l} ε_{2} = \frac{3 β_{2} d^{3} - 3 k^{2} (b_{1} - b_{2}) d^{2} - 4 π}{6 b_{2} k^{2} d^{2}} - \frac{1}{6 b_{2} k^{2} d^{2}} {9 d^{6} [(b_{1} + b_{2}) k^{2} d^{2} - 18 (b_{1} + b_{2}) k^{2} β_{2} d^{5} +, \\ {+ π k^{2} d^{2} (24 b_{1} - 120 b_{2}) + 9 β_{2}^{2}] - 24 π β_{2} d^{3} + 16 π^{2}}}^{\frac{1}{2}} \end{array}

in which

β_{2} = \frac{N_{2} β_{1} β_{e}}{(N β_{1} - N_{1} β_{e})} .

In order to make the original object invisible, we only require to replace the relative permittivity $ε_{e}$ of the illusion object by one. In this case, we have $β_{e} = \infty$ and the elements in the ${\bar{X}}^{I l l u s i o n}$ are far smaller than those in the ${\bar{R}}^{I l l u s i o n}$ . Hence we can obtain $t r [{\bar{R}}^{c o a t e d}] = \infty$ , which means no scattering phenomena.

Note that the relative permittivity of the coating determined by (40) does not rigorously satisfy the condition that the coated object has the same eigenvalues as those of the illusion object. Therefore, the relative permittivity in (40) may be considered as an initial value. By optimizing the relative permittivity, we search for the optimal invisibility and illusion effects of the arbitrarily shaped object. Here we define the following evaluation functions for the optimization procedure [27]:for illusion

σ_{1} = \frac{1}{4 π} \int_{0}^{π} \int_{0}^{2 π} | σ_{c o a t e d} (θ, ϕ) - σ_{i l l u s i o n} (θ, ϕ) | \sin θ d θ d ϕ .

and for invisibility

σ_{2} = \frac{1}{4 π} \int_{0}^{π} \int_{0}^{2 π} | σ_{c o a t e d} (θ, ϕ) | \sin θ d θ d ϕ .

By suitably choose the relative permittivity of the coating shell, we can minimize

σ_{1}

and

σ_{2}

to achieve the optimal invisibility and illusion effects of the arbitrarily-shaped objects.

3. Simulations and discussion

To validate the above theoretical design, some numerical examples are given in this section. A finite element based numerical simulation is used to study illusion and invisibility cloak performance. An incident plane wave is considered in the simulation and the computational domain is terminated by the scattering boundary conditions.

As the first example, consider a slightly lossy cubic object with a side length of 0.05 m and relative permittivity of $ε_{1} = 3.8 + i 0.5$ . In order to make the cubic object have the same illusion image as a sphere with a radius of 0.075 m and relative permittivity of $ε_{e} = 5$ at the frequency of 600 MHz, a coating with relative permittivity of $ε_{2} = 5.142$ is designed according to Eq. (40). Figure 3 shows RCS comparisons in XOY and XOZ planes between the coated and illusion objects, in good agreement. Further, Fig. 4 demonstrates near-field distributions of the coated and illusion objects. Two nearly identical field distributions are observed. Backscattering comparison between the coated and illusion objects over the frequencies is plotted in Fig. 5. It can be seen that backscattering cross sections of the coated and illusion objects are almost same in a wide frequency band ranging from 100MHz to 1GHz.

Fig. 3 RCS comparison between the coated object and the illusion object. (a) XOY plane. (b) XOZ plane.

Download Full Size | PDF

Fig. 4 Comparison of near field distribution. (a) Coated object. (b) Illusion object.

Download Full Size | PDF

Fig. 5 Backscattering comparison between the coated and illusion objects over frequencies.

Download Full Size | PDF

In the following, a mantle cloaking for a target with a large loss is designed. We consider a cubic object with a side length of 0.2λ₀ and relative permittivity of $ε_{1} = 4.96 + i 5.55$ . To achieve the lossy object invisible, the object is wrapped by a material shell with a thickness of 0.1λ₀. According to Eq. (40), the relative permittivity of the coating is obtained as $ε_{2} = 0.61$ . The object is illuminated by a TE polarized plane wave with a wavelength of $λ_{0} = 30.2 μ m$ in the direction of $θ = 90^{o}$ and $ϕ = 0^{o}$ , and Fig. 6 demonstrates the comparison of the near-field distribution in XOZ plane between the lossy object without and with the cloaking. With the designed cloaking, the object becomes invisible. Figure 7 depicts the RCSs of the lossy object with and without the coating in XOY and XOZ planes, respectively. We can observe that a RCS reduction of 7dB is obtained by using the designed coating. According to Figs. 3-7, we can know that the cloak designed by the proposed method is independent of the dissipation of the target.

Fig. 6 Comparison of near field distribution. (a) Sole object. (b) Coated object.

Download Full Size | PDF

Fig. 7 RCS comparison between the coated object and the sole object. (a) XOY plane. (b) XOZ plane.

Download Full Size | PDF

Next, the illusion cloak design for a target composed of an anisotropic material is performed. Consider a cylinder with a radius of 0.1λ₀. The anisotropic relative permittivity of the cylinder is given as follows:

{\bar{ε}}_{1} = [\begin{matrix} 6.25 \\ 4 \\ 1 \end{matrix}] .

A TE polarized plane wave with a frequency of 3GHz is incident on the cylinder. In order to make the cylindrical object disguised as a cylinder with a radius of 0.2λ₀ and the relative permittivity of

ε_{e} = 6.5

, a coating with the relative permittivity of

ε_{2} = 7

is designed according to Eq. (40). Figures 8 and 9 show comparisons of the near-field distribution in the XOY plane and the RCS between the anisotropic cylinder with and without the covering, respectively. Good illusion performance can be observed and thus the proposed method is suitable to the cloaking design of the anisotropic objects.

Fig. 8 Comparison of near field distribution. (a) Coated object. (b) Illusion object.

Download Full Size | PDF

Fig. 9 RCS comparison between the coated object and the sole object.

Download Full Size | PDF

Further, an invisible cloaking design for a target with a dispersive material is implemented. Here consider a cubic object with a side length of 0.6 m. The object is made of the Lorentz medium with the following relative permittivity

ε_{1} = ε_{\infty} + \frac{(ε_{s} - ε_{\infty}) ω_{p}^{2}}{ω_{p}^{2} + j 2 ω δ_{p} - ω^{2}}

in which

ε_{\infty} = 3

,

ε_{s} = 4.5

,

δ_{p} = 10^{8} s^{- 1}

, and

ω_{p} = 4 \times 10^{8} s^{- 1}

. The real and imaginary parts of the relative permittivity of the cube are given in Fig. 10 (a). In order to achieve the invisibility of the dispersive object, a mantle coating is designed according to the relative permittivity of the object at 200 MHz. The relative permittivity of the coating is determined as

ε_{2} = 0.39

from Eq. (40). Figures 10(b) and 11 show comparisons of the backscattering cross section and the near-field distribution between the sole object and the coated object, respectively. At 200 MHz, a RCS reduction of 20 dB can be obtained, and thus good invisibility is achieved. At the frequencies below 200 MHz, good RCS reductions are still observed because there is a slight change in the relative permittivity of the dispersive object compared with that at 200 MHz. However, with the increase of the frequency, the relative permittivity of the dispersive object greatly varies. Hence the invisibility effect caused by the coating based on the relative permittivity of the object at 200 MHz becomes worsen.

Fig. 10 Invisibility of the dispersive object. (a) Relative permittivity of the object. (b) Backscattering comparison between the coated and illusion objects over frequencies.

Download Full Size | PDF

Fig. 11 Comparison of near field distribution at 200 MHz. (a) Sole object. (b) Coated object.

Download Full Size | PDF

In the fifth example, consider a goblet-shaped object with relative permittivity of $ε_{1} = 2.5$ , as shown in Fig. 14. A TE polarized plane wave with the frequency of 500 MHz is incident on the object from the direction of $θ = 90^{o}$ and $ϕ = 0^{o}$ . For the purpose of suppressing its RCS, the object is wrapped by a material shell. According to Eq. (40), we have $ε_{2} = 0.76$ . The relative permittivity of the coating is reasonably adjusted to obtain the minimal evaluation function $σ_{2}$ . As shown in Fig. 12, $σ_{2}$ reaches the minimum value with $ε_{2} = 0.45$ . Figure 13 shows the RCS reduction of the goblet-shaped object coated by a designed shell. It can be seen that the RCS reductions of 5.8 dB at $ϕ = 0^{o}$ in XOY plane and at $θ = 90^{o}$ in XOZ plane are obtained when the unoptimized coating is used. With the use of the optimized coating, the RCS reductions of 14dB at $ϕ = 0^{o}$ in XOY plane and at $θ = 90^{o}$ in XOZ plane are achieved. A comparison of the near-field distribution between the sole object and the coated object with the optimized coating is shown in Fig. 14. It can be seen that the object becomes invisible when the designed coating is used. Further, the RCS comparisons between the sole object and the coated object with the optimized coating for the obliquely incident plane waves with TE and TM polarizations are given in Figs. 15 and 16, respectively. It can be seen that whatever the polarization of the incident wave is, the RCS of the object at $θ = 0^{o}$ is reduced over 10dB.

Fig. 12 Variation of the evaluation function with the relative permittivity of the coating.

Download Full Size | PDF

Fig. 13 RCS comparison between the coated object and the sole object. (a) XOY plane. (b) XOZ plane.

Download Full Size | PDF

Fig. 14 Comparison of near-field distribution. (a) Sole object. (b) Coated object.

Download Full Size | PDF

Fig. 15 RCS comparison between the coated object and the sole object for TE polarized, obliquely incident wave. (a) YOZ plane. (b) XOZ plane.

Download Full Size | PDF

Fig. 16 RCS comparison between the coated object and the sole object for TM polarized, obliquely incident wave. (a) YOZ plane. (b) XOZ plane.

Download Full Size | PDF

Finally, consider a Chinese map-shaped object with an area of 0.412 m² and a height of 0.2 m and the relative permittivity of 6.5, as shown in Fig. 19. A cover is designed to make the object disguised as a similar object with an area of 1.1 m² and the relative permittivity of 5 at the frequency of 300 MHz. According to Eq. (40), the relative permittivity of the coating is obtained, i.e., $ε_{2} = 4.03$ . Further we determine the optimal coating parameter of $ε_{2} = 4.38$ for the minimal evaluation function $σ_{1}$ , as shown in Fig. 17. Figures 18 and 19 demonstrate comparisons of the RCS and the near-field distribution between the coated object with the optimal coating and the illusion object, respectively. Good illusion performance is observed in this example.

Fig. 17 Variation of the evaluation function with the relative permittivity of the coating.

Download Full Size | PDF

Fig. 18 RCS comparison between the coated object and the illusion object. (a) XOY plane. (b) XOZ plane.

Download Full Size | PDF

Fig. 19 Comparison of near field distribution. (a) Coated object. (b) Illusion object.

Download Full Size | PDF

According to above numerical examples, the invisibility and illusion mantle cloaks designed by the proposed method consists of a homogeneous and isotropic material with the relative permittivity larger than zero, whatever the target is composed of isotropic or anisotropic, lossless or lossy, dispersive or nondispersive material. Therefore, the proposed mantle cloaking is more easily achieved. A possible implementation for the cloak is to use the metasurface. Some metasurface elements, such as capacitively coupled short dipole and spiral resonator elements [18], square and triangle elements [23], and Jerusalem cross element [24], have been designed to achieve the invisibility and the illusion images of the objects with canonical shapes including sphere and cylinder.

4. Conclusions

We have successfully demonstrated invisibility and illusion cloak designs for the arbitrarily shaped objects based on the CMM. We have theoretically investigated the relationship between the scattering field and generalized eigenvalues of the matrix in the DDA method. With the proposed design method, the cloaks have arbitrarily shapes and provide ideal invisibility and illusion performance in microwave and optical frequencies, independent of the incident wave and the target.

Funding

National Natural Science Foundation of China (No. 61771359); Technology Innovation Research Project of the CETC; Fundamental Research Funds for the Central Universities (No. SPSZ031410).

References and links

1. F. Capolino, Applications of Metamaterials (CPC, 2009).

2. T. J. Cui, D. R. Smith, and R. P. Liu, Metamaterials Theory, Design and Applications (Springer, 2010).

3. D. H. Werner and D. H. Kwon, Transformation Electromagnetics and Metamaterials Fundamental Principles and Applications (Springer, 2014).

4. W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Tuning plasmonic cloaks with an external magnetic field,” Phys. Rev. Lett. 111(21), 215504 (2013). [CrossRef] [PubMed]

5. W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Molding the flow of light with a magnetic field: plasmonic cloaking and directional scattering,” J. Opt. Soc. Am. A 31(9), 1969–1976 (2014). [CrossRef] [PubMed]

6. M. V. Rybin, D. S. Filonov, P. A. Belov, Y. S. Kivshar, and M. F. Limonov, “Switching from visibility to invisibility via Fano resonances: theory and experiment,” Sci. Rep. 5(1), 8774 (2015). [CrossRef] [PubMed]

7. R. Peng, Z. Xiao, Q. Zhao, F. Zhang, Y. Meng, B. Li, J. Zhou, Y. Fan, P. Zhang, N. Shen, T. Koschny, and C. M. Soukoulis, “Temperature-controlled chameleonlike cloak,” Phys. Rev. X 7(1), 011033 (2017). [CrossRef]

8. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

10. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006). [CrossRef] [PubMed]

11. W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett. 93(19), 194102 (2008). [CrossRef]

12. Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]

13. F. F. Huo, L. Li, T. Li, Y. M. Zhang, and C. H. Liang, “External invisibility cloak for multi-objects with arbitrary geometries,” IEEE Antennas Wirel. Propag. Lett. 13, 273–276 (2014). [CrossRef]

14. Y. Shi, W. Tang, and C. H. Liang, “A minimized invisibility complementary cloak with a composite shape,” IEEE Antennas Wirel. Propag. Lett. 13, 1880 (2015).

15. W. X. Jiang and T. J. Cui, “Radar illusion via metamaterials,” Phys. Rev. E 83(2), 026601 (2011). [CrossRef] [PubMed]

16. W. X. Jiang, C. W. Qiu, T. C. Han, S. Zhang, and T. J. Cui, “Creation of ghost illusion using wave dynamics in metamaterials,” Adv. Funct. Mater. 23(32), 4028–4034 (2013). [CrossRef]

17. Y. Shi, W. Tang, L. Li, and C. H. Liang, “Three-dimensional complementary invisibility cloak with arbitrary shapes,” IEEE Antennas Wirel. Propag. Lett. 14, 1550–1553 (2015). [CrossRef]

18. Z. H. Jiang and D. H. Werner, “Quasi‐three‐dimensional angle‐tolerant electromagnetic illusion using ultrathin metasurface coatings,” Adv. Funct. Mater. 24(48), 7728–7736 (2014). [CrossRef]

19. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72(1), 016623 (2005). [CrossRef] [PubMed]

20. A. Alú and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double-positive metamaterial layers,” J. Appl. Phys. 97(9), 094310 (2005). [CrossRef]

21. A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100(11), 113901 (2008). [CrossRef] [PubMed]

22. Y. Ni, L. Gao, and C. W. Qiu, “Achieving invisibility of homogeneous cylindrically anisotropic cylinders,” Plasmonics 5(3), 251–258 (2010). [CrossRef]

23. P. Y. Chen and A. Alú, “Mantle cloaking using thin patterned metasurfaces,” Phys. Rev. B 84(20), 205110 (2011). [CrossRef]

24. N. Xiang, Q. Cheng, H. B. Chen, J. Zhao, W. X. Jiang, H. F. Ma, and T. J. Cui, “Bifunctional metasurface for electromagnetic cloaking and illusion,” Appl. Phys. Express 8(9), 092601 (2015). [CrossRef]

25. L. Zhang, Y. Shi, and C. H. Liang, “Achieving illusion and invisibility of inhomogeneous cylinders and spheres,” J. Opt. 18(8), 085101 (2016). [CrossRef]

26. F. Yang, Z. L. Mei, W. X. Jiang, and T. J. Cui, “Electromagnetic illusion with isotropic and homogeneous materials through scattering manipulation,” J. Opt. 17(10), 105610 (2015). [CrossRef]

27. L. Zhang, Y. Shi, and C. H. Liang, “Optimal illusion and invisibility of multilayered anisotropic cylinders and spheres,” Opt. Express 24(20), 23333–23352 (2016). [CrossRef] [PubMed]

28. S. Tricarico, F. Bilotti, A. Alù, and L. Vegni, “Plasmonic cloaking for irregular objects with anisotropic scattering properties,” Phys. Rev. E 81(2), 026602 (2010). [CrossRef] [PubMed]

29. R. F. Harrington, Field Computation by Moment Methods (IEEE, 1993).

30. R. J. Garbacz and R. H. Turpin, “A generalized expansion for radiated and scattered fields,” IEEE Trans. Antenn. Propag. 19(3), 348–358 (1971). [CrossRef]

31. R. F. Harrington and J. R. Mautz, “Theory of characteristic modes for conducting bodies,” IEEE Trans. Antenn. Propag. 19(5), 622–628 (1971). [CrossRef]

32. R. F. Harrington, J. R. Mautz, and J. Y. Chang, “The theory of characteristic modes for dielectric and magnetic bodies,” IEEE Trans. Antenn. Propag. 20(2), 194–198 (1972). [CrossRef]

33. E. M. Purcell and C. R. Pennypacker, “Scattering and adsorption of light by nonspherical dielectric grains,” J. Astrophys. 186, 705 (1973). [CrossRef]

34. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” J. Astrophys. 333, 848 (1988). [CrossRef]

35. B. T. Draine and J. J. Goodman, “Beyond clausius-mossotti—wave-propagation on a polarizable point lattice and the discrete dipole approximation,” J. Astrophys. 405, 685 (1993). [CrossRef]

36. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491 (1994). [CrossRef]

37. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Ra. 79, 775–824 (2003). [CrossRef]

38. M. A. Yukin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Ra. 106(1), 558–589 (2007). [CrossRef]

39. R. Mittra, Computer Techniques for Electromagnetics (Permagon, 1973).

Cloaking design for arbitrarily shape objects based on characteristic mode method

Abstract

1. Introduction

2. Cloaking of arbitrarily shaped objects

2.1 Discrete-dipole approximation (DDA)

2.2 Characteristic mode method (CMM)

Case 1 $\bar{X} \neq 0$

Case 2 $\bar{X} \approx 0$

2.3 Illusion and invisibility conditions

3. Simulations and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (19)

Equations (45)

Optics Express

Abstract

1. Introduction

2. Cloaking of arbitrarily shaped objects

2.1 Discrete-dipole approximation (DDA)

2.2 Characteristic mode method (CMM)

Case 1X¯≠0

Case 2 X¯≈0

2.3 Illusion and invisibility conditions

3. Simulations and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (19)

Equations (45)

Optics Express

Case 1 $\bar{X} \neq 0$

Case 2 $\bar{X} \approx 0$