Bifunctional arbitrarily-shaped cloak for thermal and electric manipulations

Lin Zhang; Yan Shi

doi:10.1364/OME.8.002600

1. Introduction

The past decade has witnessed a rapid development of metamaterials [1–3]. Among various achievements of the metamaterials, cloak that renders targets invisible to observers has attracted tremendous attentions due to its magical physics and potential military applications. Theoretical designs and experimental verifications of various cloaks have been realized, including electromagnetic cloaks [4–10], acoustic cloaks [11–14], conduction cloaks [15], matter wave cloaks [16], elastic wave cloaks [17], thermal cloaks [18–22], etc. But these cloaks only manipulate the fields in sole physical phenomenon. Later, so-called multifunctional cloaks to conceal objects in multiple physical fields have been developed. In [23], a bifunctional cloaking device was designed in the frame of coordinate transformation (CT). By implementing form invariances of conduction and heat equations, electric current and heat flux are manipulated, simultaneously. Although the CT-based cloaks demonstrate perfect invisibility performance for the objects with arbitrary shapes, the resultant constitute parameters of the cloaks are highly anisotropic and inhomogeneous, which poses huge challenges for the physical realization of the multifunctional cloaks.

In order to alleviate these difficulties, scattering cancellation (SC) method as an alternative to the CT technique has been used in the design of cloaks. The SC usually use a coating with homogeneous and isotropic parameters to suppress the overall electromagnetic scattering from objects by controlling the dominated orders of Mie scattering coefficient. With the SC technique, electromagnetic invisibility [24–27] and illusion cloaks [28–32], thermal invisibility cloaks [33], bifunctional cloaks for electric and thermal conductions [34–38] have been developed. Due to the use of the Mie series expansion, the conventional SC approach is limited to canonical shapes such as infinite cylinder and sphere.

In this work, the SC technique is combined with the discrete dipole approximation (DDA) approach [39–42] to design bifunctional cloaks with electric and thermal characteristics for the arbitrarily-shaped objects. The contributions of the proposed design are two folds: one is to design the arbitrarily-shaped cloaks for manipulation of the electric and thermal fields, which distinguishes our method from the previously reported SC designs; the other is that the homogeneous and isotropic constitute parameters of the designed coatings make the bifunctional cloaks easy to realize compared with the CT-based designs. The remainder of this paper is organized as follows. The integral equations for the static thermal and electric problems are first formulated, and the arbitrarily-shaped objects in the multiphysical fields are modelled by the DDA method. Then, relationship between the temperature/electric potential and the thermal/electric matrix obtained by the DDA is established according to the eigenvalue analysis. Next an analytical formulation to determine the parameters of the coating is derived for the electric-thermal invisibility or mimicry of the object, followed by an optimization procedure for the optimal invisibility and illusion performance of the designed cloaks. Finally, numerical examples are given to demonstrate good performance of the proposed design approach.

2. DDA method for the thermal-electric problem

Assume an arbitrary volume V’ in a background region with thermal conductivity $κ_{0}$ and permittivity $ε_{0}$ . A heat source with a volume density ${\dot{q}}_{v}$ and a charge source with a volume density $ρ$ exist within V’ simultaneously. According to static thermal and electric theories, the governing equations for the thermal and electric phenomena can be expressed as

\nabla^{2} T = - \frac{{\dot{q}}_{v}}{κ_{0}}

\nabla^{2} ϕ = - \frac{ρ}{ε_{0}}

where

T

and

ϕ

are temperature and electric potential, respectively. The general solution of above Poisson equation can be uniformly expressed as

f = \frac{1}{4 π υ} \int_{V^{'}} \frac{ζ ({\overset{⇀}{r}}^{'})}{| \overset{⇀}{r} - {\overset{⇀}{r}}^{'} |} d v^{'}

where

f = {\begin{cases} T for thermal \\ ϕ for electric \end{cases}, ζ = {\begin{cases} {\dot{q}}_{v} for thermal \\ ρ for electric \end{cases}, υ = {\begin{cases} κ_{0} for thermal \\ ε_{0} for electric \end{cases}

It can be seen from Eqs. (1) and (2) that there is a dual relationship between the static thermal and electric problems. Hence in this paper the static thermal field is solved, and the static electric one is obtained by the duality.

In order to model the static thermal problem, consider an arbitrarily-shaped object V with the thermal conductivity $κ$ illuminated by an initially uniform density of heat flux ${\overset{⇀}{q}}^{i}$ , as shown in Fig. 1. The density of heat flux $\overset{⇀}{q}$ in the whole space can be split into two parts: one is the incident flux ${\overset{⇀}{q}}^{i}$ , which is associated with the primary source in the absence of the object and the other is the scattered flux ${\overset{⇀}{q}}^{s}$ , which is associated with the equivalent induced source density. The superposition of the incident and scattered fluxes yields the original flux $\overset{⇀}{q}$ in the presence of the scatter.

Fig. 1 Schematic diagram for static thermal problem.

Download Full Size | PDF

Considering that there are no applied heat sources in the object, the density of heat flux $\overset{⇀}{q}$ in the V satisfies the following relationship:

\nabla \cdot \overset{⇀}{q} = 0

in which

\overset{⇀}{q} = - κ \nabla T

Substituting Eq. (6) into Eq. (5), the resulting equation can be rewritten as

\nabla \cdot (- κ_{0} \nabla T) = \nabla \cdot [- (κ_{0} + κ) \nabla T]

According to Eq. (7), we can know that

\nabla \cdot [- (κ_{0} + κ) \nabla T]

represents the equivalent induced heat source density and generates the scattered flux

{\overset{⇀}{q}}^{s}

. Introducing intensity of thermal polarization

\overset{⇀}{Q}

as

\overset{⇀}{Q} = - (κ_{0} + κ) \nabla T

the temperature T^s associated with

{\overset{⇀}{q}}^{s}

can be solved according to Eq. (3) as

T^{s} = \frac{1}{4 π κ_{0}} \int_{V} \frac{(\overset{⇀}{r} - {\overset{⇀}{r}}^{'}) \cdot \overset{⇀}{Q} ({\overset{⇀}{r}}^{'})}{{| \overset{⇀}{r} - {\overset{⇀}{r}}^{'} |}^{3}} d v^{'}

Therefore, the temperature T associated with

\overset{⇀}{q}

can be obtained as

T = T^{s} + T_{0} = T_{0} + \frac{1}{4 π κ_{0}} \int_{V} \frac{(\overset{⇀}{r} - {\overset{⇀}{r}}^{'}) \cdot \overset{⇀}{Q} ({\overset{⇀}{r}}^{'})}{{| \overset{⇀}{r} - {\overset{⇀}{r}}^{'} |}^{3}} d v^{'}

in which the temperature T₀ is associated with

{\overset{⇀}{q}}^{i}

. Equation (10) is denoted as the static temperature field integral equation.

In order to solve Eq. (10), the discrete-dipole approximation (DDA) method is used. The object is discretized into N cubes V_j (j = 1,…,N), each of which has a side length of d and is represented by a point dipole. In this scenario, Eq. (10) can be expressed as

\begin{array}{l} T ({\overset{⇀}{r}}_{j}) - \frac{1}{4 π κ_{0}} \int_{V_{j}} \frac{(\overset{⇀}{r} - {\overset{⇀}{r}}^{'}) \cdot \overset{⇀}{Q} ({\overset{⇀}{r}}^{'})}{{| \overset{⇀}{r} - {\overset{⇀}{r}}^{'} |}^{3}} d v^{'} = T_{0} ({\overset{⇀}{r}}_{j}) + \frac{1}{4 π κ_{0}} \sum_{\begin{array}{l} l = 1 \\ l \neq j \end{array}}^{N} \frac{({\overset{⇀}{r}}_{j} - {\overset{⇀}{r}}_{l}) \cdot \overset{⇀}{Q} ({\overset{⇀}{r}}_{l})}{{| {\overset{⇀}{r}}_{j} - {\overset{⇀}{r}}_{l} |}^{3}} Δ V_{l} \\ = T_{0} ({\overset{⇀}{r}}_{j}) + \frac{1}{4 π κ_{0}} \sum_{\begin{array}{l} l = 1 \\ l \neq j \end{array}}^{N} \frac{| {\overset{⇀}{r}}_{j} - {\overset{⇀}{r}}_{l} | \cos θ_{j l}}{{| {\overset{⇀}{r}}_{j} - {\overset{⇀}{r}}_{l} |}^{3}} Q ({\overset{⇀}{r}}_{l}) \cdot Δ V_{l} \end{array}

where

θ_{j l}

is the angle between

{\overset{⇀}{r}}_{j} - {\overset{⇀}{r}}^{'}_{l}

and

\overset{⇀}{Q} ({\overset{⇀}{r}}^{'}_{l})

, and

Δ V_{l} = d^{3}

is the volume of the lth cube. Defining the equivalent dipole moment

p_{l}^{H} = Q ({\overset{⇀}{r}}_{l}) \cdot Δ V_{l} \cos θ_{j l} | {\overset{⇀}{r}}_{j} - {\overset{⇀}{r}}_{l} |

, Eq. (11) is rewritten as

T ({\overset{⇀}{r}}_{j}) - \frac{1}{4 π κ_{0}} \int_{V_{j}} \frac{(\overset{⇀}{r} - {\overset{⇀}{r}}^{'}) \cdot \overset{⇀}{Q} ({\overset{⇀}{r}}^{'})}{{| \overset{⇀}{r} - {\overset{⇀}{r}}^{'} |}^{3}} d v^{'} = T_{0} ({\overset{⇀}{r}}_{j}) + \sum_{\begin{array}{l} l = 1 \\ l \neq j \end{array}}^{N} G_{T} ({\overset{⇀}{r}}_{j}, {\overset{⇀}{r}}_{l}) p_{l}^{H}

in which

G_{T} ({\overset{⇀}{r}}_{j}, {\overset{⇀}{r}}_{l}) = \frac{1}{4 π κ_{0}} \frac{1}{{| {\overset{⇀}{r}}_{j} - {\overset{⇀}{r}}_{l} |}^{3}}

and the superscript “H” represents the static heat problem.

It can be seen from Eq. (12) that its left-hand side represents the temperature exciting the jth cube, and thus the equivalent dipole moment in the V_j can be obtained as

p_{j}^{H} = α_{j}^{H} {T ({\overset{⇀}{r}}_{j}) - \frac{1}{4 π κ_{0}} \int_{V_{j}} \frac{(\overset{⇀}{r} - {\overset{⇀}{r}}^{'}) \cdot \overset{⇀}{Q} ({\overset{⇀}{r}}^{'})}{{| \overset{⇀}{r} - {\overset{⇀}{r}}^{'} |}^{3}} d v^{'}}

Here

α_{j}^{H}

is defined as thermal dipole polarizability. Combining Eqs. (12) and (14), a matrix equation can be obtained as

{\bar{A}}^{H} \cdot {\bar{p}}^{H} = {\bar{T}}_{0}

in which

{\bar{T}}_{0}

represents the temperature vector associated with the incident heat flux, and

A_{i j}^{H} = {\begin{cases} \frac{1}{4 π κ_{0} {| {\overset{⇀}{r}}_{i} - {\overset{⇀}{r}}_{j} |}^{3}} i \neq j \\ 1 / α_{j}^{T} i = j \end{cases}

By the duality, a similar matrix equation for the static electric problem can be obtained as

{\bar{A}}^{E} \cdot {\bar{p}}^{E} = {\bar{ϕ}}_{0}

in which

A_{i j}^{E} = {\begin{cases} \frac{1}{4 π ε_{0} {| {\overset{⇀}{r}}_{i} - {\overset{⇀}{r}}_{j} |}^{3}} i \neq j \\ 1 / α_{j}^{E} i = j \end{cases}

Here

{\bar{ϕ}}_{0}

is an electric potential vector associated with the incident static electric field,

α_{j}^{E}

is defined as electric dipole polarizability, and the superscript “E” represents the static electric problem.

The widely used Clausis-Mossotti (CM) polarizability holds for the static electric and thermal problems [39]. Hence the thermal and electric dipole polarizabilities can be uniformly written as

α_{l}^{S} = \frac{3 d^{3}}{4 π} \frac{m_{l} - m_{0}}{m_{l} + 2 m_{0}} S = H, E

in which

m = {\begin{cases} κ S = H \\ ε S = E \end{cases}

Once Eqs. (15) and (17) are solved for the equivalent dipole moment, respectively, the temperature T and the electric potential

ϕ

in far region can be calculated as

U^{S} = \sum_{j = 1}^{N} \frac{1}{4 π m_{0} r^{3}} p_{j}^{S} (S = H, E)

in which

U^{S} = {\begin{cases} T S = H \\ ϕ S = E \end{cases}

3. Thermal-electric cloaking for arbitrarily-shaped objects

3.1 Eigenvalue analysis of the thermal-electric problem

According to above discussion, we can know that a matrix equation can be formulated to model thermal/electric problem of the arbitrarily-shaped object. It can be seen from Eqs. (16) and (18) that the corresponding matrix is real and symmetry. Therefore, we consider the following eigenvalue problem

{\bar{A}}^{S} \cdot {\bar{q}}_{k}^{S} = λ_{k}^{S} \cdot {\bar{q}}_{k}^{S} (S = H, E and k = 1, \dots, N)

Due to the symmetry of

{\bar{A}}^{S}

, the eigenvalues

λ_{k}^{S}

are real and the eigenfunctions

{\bar{q}}_{k}^{S}

satisfy the orthogonality. With the following normalization

{({\bar{q}}_{k}^{S})}^{T} \cdot {\bar{A}}_{k}^{S} \cdot {\bar{q}}_{k}^{S} = 1

the equivalent dipole moments in the thermal/electric problem can be expressed in terms of the eigenfunctions as

{\bar{p}}^{S} = \sum_{k = 1}^{N} \frac{{({\bar{q}}_{k}^{S})}^{T} \cdot {\bar{U}}_{0}^{S}}{λ_{k}^{S}} {\bar{q}}_{k}^{S}

in which superscript T denotes the transpose operator and

{\bar{U}}_{0}^{S} = {\begin{cases} {\bar{T}}_{0} S = H \\ {\bar{ϕ}}_{0} S = E \end{cases}

Inserting Eq. (25) into Eq. (21), the temperature T and the electric potential ϕ in far region can be calculated as

U^{S} = \frac{1}{4 π m_{0} r^{3} λ_{k}^{S}} W \cdot \sum_{k = 1}^{N} [{({\bar{q}}_{k}^{S})}^{T} \cdot {\bar{U}}_{0}^{S}] {\bar{q}}_{k}^{S}

where

W = [\begin{matrix} 1 & 1 & \dots & 1 \end{matrix}]

. It can be seen from Eq. (27) that the temperature T and the electric potential ϕ are tightly related to the eigenvalues of the corresponding system matrix

{\bar{A}}^{S}

. Therefore, we can manipulate the temperature T and the electric potential ϕ according to the respective eigenvalues.

3.2 Design of thermal-electric cloak

In order to design the thermal-electric cloak, a scattering cancellation technique based on the eigenvalues is developed. Consider an arbitrarily-shaped object with thermal conductivity $κ_{1}$ and permittivity $ε_{1}$ . A shell with thermal conductivity $κ_{2}$ and permittivity $ε_{2}$ is used to wrap the object for generating the illusion or invisibility effects in the static thermal and electric fields, simultaneously. Assume that the coated object behaves like another object with the same geometry but different thermal conductivity $κ_{e}$ and permittivity $ε_{e}$ . Following the above DDA procedure to model the static thermal and electric problems, four matrix equations for the coated object and the illusion object can be obtained, respectively, as

{\bar{A}}_{Q}^{S} \cdot {\bar{p}}_{Q}^{S} = {\bar{U}}_{0}^{S} (S = H, E; Q = c o a t e d, i l l u s i o n)

Note that the same geometries of the coated and illusion objects lead to the same dimensions of four matrices in Eq. (28). According to Eq. (27), either the temperature T or the electric potential ϕ depends on the eigenvalues of the corresponding matrix in Eq. (28). Hence the coated object has the same thermal illusion image as the illusion object when the eigenvalues of the two thermal matrices are identical. Similar conclusion on the electric illusion image is valid for the coated and illusion objects. In general, however, it is very difficult to guarantee this rigorous condition for objects with arbitrary shapes and constitute parameters. A good approximation to this rigorous condition is that two thermal/electric inverse matrices have same traces, i.e.,

Trace {{[{\bar{A}}_{c o a t e d}^{S}]}^{- 1}} = Trace {{[{\bar{A}}_{i l l u s i o n}^{S}]}^{- 1}}

On the other hand, we can know from Eqs. (16) and (18) that the diagonal elements in both the thermal and electric matrices are far larger than the non-diagonal elements. Hence it is reasonable to omit the non-diagonal elements in both the thermal and electric matrices. In this scenario, the trace of the matrix in Eq. (29) become the summation of the diagonal elements. Hence we have

\sum_{j = 1}^{N} α_{j, c o a t e d}^{S} = \sum_{j = 1}^{N} α_{j, i l l u s i o n}^{S}

If the original object, the coating, and the illusion object consist of homogeneous materials, Eq. (30) can be reduced to

N_{1} α_{o r i g n a l}^{S} + N_{2} α_{c o a t i n g}^{S} = N α_{i l l u s i o n}^{S}

in which N₁, N₂, and N are the numbers of the equivalent dipole moments for the original object, the coating and the illusion object, respectively. Therefore, we have N₁ + N₂ = N. Substituting Eqs. (19) and (20) into Eq. (31), the illusion condition for the thermal-electric problem can be derived as

\frac{ε_{2} / ε_{0}}{κ_{2} / κ_{0}} = \frac{2 N η_{e}^{E} - 2 N_{1} η_{1}^{E} + N_{2}}{2 N η_{e}^{H} - 2 N_{1} η_{1}^{H} + N_{2}} \frac{- N η_{e}^{H} + N_{1} η_{1}^{H} + N_{2}}{- N η_{e}^{E} + N_{1} η_{1}^{E} + N_{2}}

where

η_{1}^{E} = \frac{ε_{1} - ε_{0}}{ε_{1} + 2 ε_{0}}, η_{e}^{E} = \frac{ε_{e} - ε_{0}}{ε_{e} + 2 ε_{0}}, η_{1}^{H} = \frac{κ_{1} - κ_{0}}{κ_{1} + 2 κ_{0}}, η_{e}^{H} = \frac{κ_{e} - κ_{0}}{κ_{e} + 2 κ_{0}}

Particularly, the equation holds for invisible case when the thermal conductivity

κ_{e}

and the permittivity

ε_{e}

of the illusion object are replaced by the thermal conductivity

κ_{0}

and the permittivity

ε_{0}

of the background region, i.e.,

κ_{e} = κ_{0}

and

ε_{e} = ε_{0}

. Therefore, the invisibility condition for the thermal-electric problem can be obtained as

\frac{ε_{2} / ε_{0}}{κ_{2} / κ_{0}} = \frac{C_{1} ε_{1} + 2 ε_{0}}{C_{2} ε_{1} + ε_{0}} \frac{C_{2} κ_{1} + κ_{0}}{C_{1} κ_{1} + 2 κ_{0}}

in which

C_{1} = \frac{(η - 2)}{(η + 1)} C_{2} = \frac{(η + 1)}{(2 η - 1)}, η = \frac{N_{2}}{N_{1}}

3.3 Optimization of illusion and invisibility effects

The material parameters of the coating determined from Eqs. (32) and (34) can achieve the illusion and invisibility of the objects with arbitrary shapes and constitute parameters. Considering that some approximations are made in deriving Eqs. (32) and (34), some optimizations can be implemented for better illusion and invisibility performance. To evaluate the performance of the designed cloak, a deformation parameter is defined as

De = | U_{c o a t e d}^{S} - U_{I l l u s i o n}^{S} |

where

U_{c o a t e d}^{S}

and

U_{I l l u s i o n}^{S}

are the temperature or electric potential of the coated and illusion objects, respectively. For the invisibility, the illusion object is same as the background, and thus

U_{I l l u s i o n}^{S}

becomes the temperature/electric potential associated with incident thermal/electric field. In this case, an optimized function can be defined as follows:

δ = \frac{1}{L} \int_{L} De d l

in which l is a reference path outside the coated object, and L is length of the path. To obtain optimal performance, the material parameters of the coating are first solved from Eq. (32) or Eq. (34) as the initial parameters. By minimizing Eq. (37), we search for the material parameters of the cloak to achieve the optimal illusion and invisibility performance.

4. Numerical examples

In this section, some numerical examples are given to demonstrate the performance of the proposed cloak designs with the thermal-electric characteristic. A commercial software COMSOL MULTIPHYSICS is used to implement the full-wave simulation. In the simulation, the target objects are located at the origin of the coordinate. The constant electric potential and the temperature are chosen at two boundary surfaces parallel to the yoz plane, and therefore the static heat flux and electric flux along + x direction are applied. At the left and right boundary surfaces, the electric potentials are 6 V and 1 V, respectively, and the temperature are, respectively, 383 K and 293 K. The thermal insulation and dielectric shielding boundary conditions are simultaneously implemented at other boundary surfaces. It is worthwhile pointing out that the thermal radiation effect is reasonably ignored in all numerical examples due to the small sizes of the objects.

As the first example, consider a three-dimensional cylinder object with an asteroid profile, whose governing equation is $x^{2 / 3} + y^{2 / 3} = a^{2 / 3}$ with a = 2 cm. The height of the object is 2 cm. The thermal and electric parameters of the object are $κ_{1} = 0.02 κ_{0}$ and $ε_{1} = 0.3 ε_{0}$ , and the background with $κ_{0} = 65 W / m K$ and $ε_{0} = 1 / 36 π \times 10^{- 9} F / m$ is used. In order to achieve the thermal-electric invisibility of the object, a coating is designed to warp the object so that the coated object becomes a spherical object with a radius of 2.1 cm. The parameters of the coating are determined as $κ_{2} = 2 κ_{0}$ and $ε_{2} = 1.56 ε_{0}$ according to Eq. (34). A reference line at x = 2.15 cm and z = 0 cm is used, where the temperature and potential are sampled, as shown in Fig. 2. The designed parameters as the initial values are optimized by using Eq. (37). The optimized parameters of the coating are $κ_{2} = 2.1 κ_{0}$ and $ε_{2} = 1.5 ε_{0}$ . Figure 2 shows the temperature and potential distributions with and without the optimized coating in xoy plane. It can be seen that there is a set of the isopotential and isothermal lines outside the designed cloak which are straight and parallel to each other. It means that the object inside the shell becomes invisible. Figure 3 demonstrates the variation of optimization function δ with the parameters of the coating. It can be seen that the optimized parameters of the coating correspond to the minimal values of the optimization function. Furthermore, Fig. 4 shows comparison of the thermal and electric deformation parameters on the reference line between the target object, the coated object with initial design, and the coated object with the optimized design. It can be observed that with the coating, deformations of the thermal and electric fields are greatly reduced. With the optimized parameters, the minimized deformations can be achieved for the thermal and electric fields simultaneously.

Fig. 2 Comparison of potential and temperature distributions between the original object and the coated object. (a) potential distribution of original object. (b) potential distribution of coated object with optimized coating. (c) temperature distribution of original object. (b) temperature distribution of coated object with optimized coating.

Download Full Size | PDF

Fig. 3 Variation of the optimization function with material parameters of the coating. (a) Electric optimization function. (b) Thermal optimization function.

Download Full Size | PDF

Fig. 4 Potential and temperature deformation parameters on the reference line. (a) Potential deformation. (b) Temperature deformation.

Download Full Size | PDF

In the second example, an illusion concept for three-dimensional object with an elliptical shape, whose profile is given by the equation ${(x / a)}^{2} + y^{2} + z^{2} = b$ with a = 1.5 cm and b = 1 cm, is presented. The thermal and electric parameters of the object and the host background are $κ_{1} = 2.5 κ_{0}$ , $ε_{1} = 6 ε_{0}$ , $κ_{0} = 44.5 W / m K$ , and $ε_{0} = 1 / 36 π \times 10^{- 9} F / m$ , respectively. A coating is designed to wrap the elliptically shaped object so that the coated object behaves like a cubic object with $κ_{e} = 4 κ_{0}$ and $ε_{e} = 3.5 ε_{0}$ . The coated object and the cubic object have same shape and those volumes are 20 cm³. According to Eq. (32), the constitute parameters of the coating can be determined as $κ_{2} = 5 κ_{0}$ and $ε_{2} = 2.86 ε_{0}$ . To obtain optimal performance, a reference line at x = 1.65 cm and z = 0 cm is chosen. The optimization function δ given by Eq. (37) can be minimized for $κ_{2} = 5 κ_{0}$ and $ε_{2} = 2.3 ε_{0}$ , as shown in Fig. 5. The comparisons of the temperature and potential distributions at the reference line between the coated and illusion objects are shown in Fig. 6. It can be seen that there are two nearly same thermal-electric distributions. And with the optimized materials of the coating, the better illusion performance can be achieved.

Fig. 5 Variation of the optimized function with material parameters of the coating. (a) Electric optimization function. (b) Thermal optimization function.

Download Full Size | PDF

Fig. 6 Comparison of temperature and potential distributions on the reference line between the illusion object, coated object with initial design, and coated object with optimized design. (a) Temperature distribution (b) Potential distribution.

Download Full Size | PDF

Next, consider a cubic object with a side length of 1.5 cm and the material parameters of $κ_{1} = 2 κ_{0}$ and $ε_{1} = 2.2 ε_{0}$ , as shown in Fig. 7. The material parameters of the background are $κ_{0} = 1 W / m K$ and $ε_{0} = 1 / 36 π \times 10^{- 9} F / m$ . To achieve the invisibility of the object, a cubic shell with the thickness of 0.45 cm is designed. The initially designed and optimized parameters of the coating are ( $κ_{2} = 0.5 κ_{0}$ , $ε_{2} = 0.45 ε_{0}$ ) and ( $κ_{2} = 0.5 κ_{0}$ , $ε_{2} = 0.48 ε_{0}$ ), respectively. The variations of the optimization functions with the parameters of the coating for the thermal and electric problems are given in Fig. 8. Hence a reference line at x = 0.98 cm and z = 0 cm is chosen for the optimization. In order to investigate the effect of the designed cloak on the incident field, the coated object is rotated around y axis with a rotation angle of θ. Figure 9 shows the temperature and potential deformations on the observation lines in xoy and xoz planes for the object with the optimized coating and without the coating, respectively. Here the observation lines for the cases of θ = 0°, 22.5°, and 45° are, respectively, chosen at the locations corresponding to maximum x coordinates of the coated object in xoy and xoz planes, i.e., x = 0.98 cm, 1.29 cm, and 1.4 cm. It can be seen that for different rotation angles, the temperature and potential deformations for the coated object in two cut planes are greatly reduced. Hence the designed cloak is independent of the directions of the static thermal and electric fields.

Fig. 7 The cubic cloak for the cubic object.

Download Full Size | PDF

Fig. 8 Variation of the optimized function with material parameters of the coating. (a) Electric optimization function. (b) Thermal optimization function.

Download Full Size | PDF

Fig. 9 Potential and temperature deformations on the observation line in xoy and xoz planes for different rotation angles. (a) Potential deformation in xoy plane. (b) Temperature deformation in xoy plane. (c) Potential deformation in xoz plane. (d) Temperature deformation in xoz plane.

Download Full Size | PDF

Finally, consider a spherical object with a radius of 0.2 cm which is made of high density polyethylene with $κ_{1} = 0.21 κ_{0}$ and $ε_{1} = 0.54 ε_{0}$ . The background is quartz glass with $κ_{0} = 1.46 W / m K$ and $ε_{0} = 3.7 / 36 π \times 10^{- 9} F / m$ . In order to make the sphere invisible, a spherical coating composed of granite is designed and the resultant radius of the coated sphere is 0.28 cm. The initially designed and optimized parameters of the coating are ( $κ_{2} = 1.78 κ_{0}$ , $ε_{2} = 1.35 ε_{0}$ ) and ( $κ_{2} = 1.6 κ_{0}$ , $ε_{2} = 1.35 ε_{0}$ ), respectively. Figures 10 (a) and (b) show the variations of the optimization functions with the parameters of the coating for the thermal and electric problems. Hence the reference line at x = 0.28 cm and z = 0 cm is chosen for the optimization. As shown in Figs. 10 (c) and (d), the potential and temperature deformations on the reference line for the object with the optimized coating and without the coating, respectively. It can be seen that with the designed coating, the sphere becomes invisible.

Fig. 10 The cloak design for the spherical object. (a) Variation of the electric optimized function with material parameters of the coating. (b) Variation of the thermal optimized function with material parameters of the coating. (c) Potential deformation on the reference line. (d) Temperature deformation on the reference line.

Download Full Size | PDF

5. Conclusion

In this paper, we proposed a method to design the bifunctional cloak with the thermal and electric characteristics for arbitrarily-shaped object. The discrete-dipole approximation method is used to solve the static thermal and electric field integral equations. With the eigenvalue analysis, physical mechanisms of the scattered thermal and electric fields are explored. Following the scattering cancellation procedure, the arbitrarily-shaped cloaks are designed and optimized. Good invisibility and mimicry characteristics in the multiphysical fields are validated by numerical results.

Funding

National Natural Science Foundation of China (No. 61771359); Natural Science Basic Research Plan in Shaanxi Province (2018JM6006); Technology Innovation Research Project of the CETC; Fundamental Research Funds for the Central Universities (No. JBF180202).

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. V. G. Leselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Phys. Uspekhi 10(4), 509–514 (1968). [CrossRef]

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

5. 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]

6. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8(10), 248 (2006). [CrossRef]

7. W. Yan, M. Yan, Z. Ruan, and M. Liu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. 10(4), 043040 (2008). [CrossRef]

8. 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]

9. Y. Shi, L. Zhang, W. Tang, L. Li, and C. H. Liang, “Design of a minimized complementary illusion cloak with arbitrary position,” Int. J. Antenn. Propag. 2015, 932495 (2015).

10. Y. Shi, W. Tang, and C. H. Liang, “A minimized invisibility complementary cloak with a composite shape,” IEEE Antennas Wirel. Propag. Lett. 13, 1800–1803 (2014). [CrossRef]

11. Y. Li, M. Wang, and W. Li, “Sound scattering of double concentric elastic spherical shell with multilayered medium cloak,” 2017 IEEE Underwater Technology (UT), Busan, 2017 (2017), pp. 1-6.

12. M. D. Guild, A. J. Hicks, M. R. Haberman, A. Alù, and P. S. Wilson, “Acoustic scattering cancellation of irregular objects surrounded by spherical layers in the resonant regime,” J. Appl. Phys. 118(16), 016623 (2015). [CrossRef]

13. C. A. Rohde, T. P. Martin, M. D. Guild, C. N. Layman, C. J. Naify, M. Nicholas, A. L. Thangawng, D. C. Calvo, and G. J. Orris, “Experimental demonstration of underwater acoustic scattering cancellation,” Sci. Rep. 5(1), 13175 (2015). [CrossRef] [PubMed]

14. W. Akl and A. Baz, “Analysis and experimental demonstration of an active acoustic metamaterial cell,” J. Appl. Phys. 111(4), 044505 (2012). [CrossRef]

15. T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 685 (2008). [CrossRef]

16. S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008). [CrossRef] [PubMed]

17. M. Farhat, S. Guenneau, and S. Enoch, “Ultrabroadband elastic cloaking in thin plates,” Phys. Rev. Lett. 103(2), 024301 (2009). [CrossRef] [PubMed]

18. C. Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett. 92(25), 251907 (2008). [CrossRef]

19. R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: molding the flow of heat,” Phys. Rev. Lett. 110(19), 195901 (2013). [CrossRef] [PubMed]

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

21. K. P. Vemuri, F. M. Canbazoglu, and P. R. Bandaru, “Guiding conductive heat flux through thermal meatmaterials,” Appl. Phys. Lett. 105(19), 193904 (2014). [CrossRef]

22. Y. G. Ma, L. Lan, W. Jiang, F. Sun, and S. L. He, “A transient thermal cloak experimentally realized through a rescaled diffusion equation with anisotropic thermal diffusivity,” NPG Asia Mater. 26(5), e73 (2014).

23. J. Y. Li, Y. Gao, and J. P. Huang, “A bifunctional cloak using transformation media,” J. Appl. Phys. 108(7), 074504 (2010). [CrossRef]

24. A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express 15(6), 3318–3332 (2007). [CrossRef] [PubMed]

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

26. J. C. Soric, A. Monti, A. Toscano, F. Bilotti, and A. Alu, “Multiband and wideband bilayer mantle cloaks,” IEEE Trans. Antenn. Propag. 63(7), 3235–3240 (2015). [CrossRef]

27. P. Y. Chen and A. Alú, “Mantle cloaking using thin patterned metasurfaces,” Phys. Rev. B 84(20), 3825–3833 (2011). [CrossRef]

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

29. 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]

30. 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]

31. Y. Shi and L. Zhang, “Cloaking design for arbitrarily shape objects based on characteristic mode method,” Opt. Express 25(26), 32263–32279 (2017). [CrossRef]

32. 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]

33. M. Farhat, P. Y. Chen, H. Bagci, C. Amra, S. Guenneau, and A. Alù, “Corrigendum: Thermal invisibility based on scattering cancellation and mantle cloaking,” Sci. Rep. 6(1), 19321 (2016). [CrossRef] [PubMed]

34. R. Tarkhanyan and D. Niarchos, “Coexisting of thermal and electric cloaking effects in bi-layer nanoporous composite,” International Congress on Advanced Electromagnetic Materials in Microwave and Optics (Oxford, 2015), pp. 304–306. [CrossRef]

35. M. Raza, Y. Liu, and Y. Ma, “A multi-cloak bifunctional device,” J. Appl. Phys. 117(2), 4130 (2015). [CrossRef]

36. Y. Ma, Y. Liu, M. Raza, Y. Wang, and S. He, “Experimental demonstration of a multiphysics cloak: manipulating heat flux and electric current simultaneously,” Phys. Rev. Lett. 113(20), 205501 (2014). [CrossRef] [PubMed]

37. C. W. Lan, K. Bi, Z. H. Gao, B. Li, and J. Zhou, “Achieving bifunctional cloak via combination of passive and active schemes,” Appl. Phys. Lett. 109(20), 201903 (2016). [CrossRef]

38. T. Yang, X. Bai, D. Gao, L. Wu, B. Li, J. T. Thong, and C. W. Qiu, “Invisible sensor: Simultaneous sensing and camflaging in multiphysical fields,” Adv. Mater. 27(47), 7752–7758 (2015). [CrossRef] [PubMed]

39. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]

40. V. L. Y. Vincent, M. P. Mengüç, and T. A. Nieminen, “Discrete-dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Ra. 112(11), 1711–1725 (2011). [CrossRef]

41. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16(15), 1198–1200 (1991). [CrossRef] [PubMed]

42. E. Purcell and C. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186(2), 705–714 (1973). [CrossRef]

Bifunctional arbitrarily-shaped cloak for thermal and electric manipulations

Abstract

1. Introduction

2. DDA method for the thermal-electric problem

3. Thermal-electric cloaking for arbitrarily-shaped objects

3.1 Eigenvalue analysis of the thermal-electric problem

3.2 Design of thermal-electric cloak

3.3 Optimization of illusion and invisibility effects

4. Numerical examples

5. Conclusion

Funding

References and links

Cited By

Figures (10)

Equations (37)

Optical Materials Express