Arbitrarily shaped thermal cloaks with non-uniform profiles in homogeneous media configurations

Guoqiang Xu; Haochun Zhang; Kexin Wang; Yan Jin; Yan Li

doi:10.1364/OE.26.025265

1. Introduction

Invisibility cloaks, which significantly provide internal invisibility without external field perturbations, have been promoted with the proposition of transformation optics (TO) [1]. The essence of TO based cloaking designs is to manipulate field distributions by mapping coordinate transformations onto the spatial distributions of conductive components, under the premise of form invariance for governing equations. By combining permittivities and permeabilities with coordinate transformations, the first design of invisible cloak [1, 2] has been demonstrated in electromagnetic field. Such cloaking design has aroused considerable interests rapidly, which indeed motivated the related researches and experiments in the fields of optics [3], electrics [4], acoustics [5], surface waves [6], etc. In addition, an optical illusion [7] only with positive components of anisotropic and inhomogeneous media has been proposed, which provided a novel way to achieve expected performances without complementary media. Furthermore, some diffusion strategies [8,9] of invisibility have also been demonstrated. Since the form-invariance of thermal conduction equation has been validated [10], such manipulative function was first introduced into thermal field to design conventional thermal cloaks. As a guidance, transformation thermodynamics [11] is one of the methods to design a class of conventional thermal cloaks, including circular [11–15] and arbitrarily shaped cloaks [10, 16], through directly mapping the coordinate transformations onto conductive components. However, singular and limited parameters occurred in the transformed processes, due to the extreme maps between initial point and tailored region. To make up the above imperfections in conventional thermal cloak, another technique, named scattering-cancellation [17–20], has been employed to theoretically and experimentally design 2D [17, 18] and 3D [19, 20] bi-layer thermal cloaks. Nevertheless, scattering-cancellation completely dependents on the regular geometries based on well-established effective medium theory, which restricts its applications in achieving arbitrarily structural cloaks. According to the findings of heat flux bending in bi-layer media configurations [21], a new class of unconventional thermal cloaks, such as ground cloak [22] and arbitrarily polygonal cloak [23], have been demonstrated through linear maps. It’s noted that the employment of linear maps is applicable to the unidirectional devices [24, 25], in which the transformed spaces can be described by lines directly. For the omni-directional devices, the transformed domains are required to be created by a point. That is, the conventional TO method [1] is still the effective way for omni-directional designs. Such novel attempts provide new avenues for the traditional cloaking studies of non-continuous shape cloaks. However, two non-negligible issues remain challenges both in currently conventional and unconventional cloaking researches. The first is imperfect and complicated processes for combining the conductive components and arbitrary structures. The second is same/quasi conformal profiles of the internal and external regions. The currently conventional thermal cloak techniques cannot simultaneously satisfy the demands of non-singularity and irregular structures, due to the imperfect point-region maps and geometric-dependent thermal neutrals. Hence, approximate transformations are not avoided. Besides, the unconventional thermal cloaks are realized by linear maps, which remain complicated processes for determining characteristic lines and employing local coordinate systems. Furthermore, there’s one common in the transformations for designing currently conventional/unconventional thermal cloaks. That is, the same/quasi-conformal internal and external regions (with same profiles) are the precondition, whether local rotations are employed in related regions. In other words, neither the conventional nor unconventional designs have achieved “complete” arbitrary thermal cloaks (equipped with different external and internal regions) with non-singularity and homogeneous parameters.

In this paper, the “complete” arbitrary thermal cloaks are first designed to realize the arbitrary cloaking behaviors in non-uniformly structural devices. In addition, rotatory linear map [26] is employed to determine the characteristic lines and implement the spatial transformations. Four non-uniform profile cloaking schemes, constructed with homogeneous media configurations, have been significantly demonstrated with no (few) external thermal perturbations and internal temperature gradients. The findings of this paper open up an avenue to further design “real meaning” arbitrary thermal cloak in varied regional profiles with homogeneous parameters. Both of the conventional and unconventional thermal cloaks can be also achieved with such novel methodology, as the same/quasi-conformal internal and external regions can be considered as the special cases of the “complete” arbitrary cloaking strategy.

2. Transformation process

As illustrated in Figs. 1(a) and 1(b), the current techniques can be only used to achieve the conventional and unconventional cloaks shaped in same/quasi conformal profiles. Such designs can be considered as the special cases of the “complete” arbitrary cloaks as shown in Fig. 1(c). That is, the current transformations are the basics for realizing the proposed non-uniform profile cloaks.

Fig. 1 Contrastive transformations of current techniques and proposed cloak. Among the subgraphs, the grey, orange, and write domains respectively denote the functional, thermal invisible and rotated initial regions. (a) Conventional shape cloak; (b) Unconventional shape cloak with non-continuous profiles. Such devices are achieved by two steps [23]: first, internal rotation of the initial region (from purple dot profile to write polygon); second, expansion of the rotated initial region (from write domain to orange region); (c) “complete” arbitrary cloak.

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2.1 Theoretical derivation of transformation process

As demonstrated in Fig. 1, arbitrarily polygonal thermal cloaks with same internal and external profiles can be obtained by adopting local rotations (step 1) and expansions (step 2) in the transformed domains [23, 26]. Considering the geometrical relations of polygonal profiles, the general vertexes in initial and transformational domains for the transformation processes can be written as follows:

\begin{array}{l} A_{n} (r_{2} \sin \frac{(2 n - 1) π}{N}, r_{2} \cos \frac{(2 n - 1) π}{N}); B_{n} (r_{2} \sin \frac{(2 n + 1) π}{N}, r_{2} \cos \frac{(2 n + 1) π}{N}); \\ A_{_{n}}^{'} (r_{1} \sin \frac{2 n π}{N}, r_{1} \cos \frac{2 n π}{N}); A_{n}^{''} (r_{0} \sin \frac{2 n π}{N}, r_{0} \cos \frac{2 n π}{N}); \\ E_{n}^{'} (r_{1} \sin \frac{2 (n - 1) π}{N}, r_{1} \cos \frac{2 (n - 1) π}{N}); E_{n}^{''} (r_{0} \sin \frac{2 (n - 1) π}{N}, r_{0} \cos \frac{2 (n - 1) π}{N}) . \end{array}

The above general vertexes are obtained by considering the rotary effects only in the internal polygons of a certain angle π/N around O (0, 0), where N is the side quantity (N≥3), n denotes the series number for functional regions ranging from 0 to N-1; r₂ denote the radii of external polygons both in the original and transformational domains; and r₁ and r₀ respectively represent the initial and transformed radii of internal polygons. Here, singular values are significantly avoided in the transformation process, once r₁ is larger than 0. That is, a finite-size polygonal hole with an initial radius r₁ is employed in the original domain to operate entire transformations through line-line maps [23, 26]. Hence, it’s important to select the initial interior conductivity κ, as it is the basis of the transformation process and the expected cloaking performance would be influenced by its variation. For simplicity and generality, the initial conductivities of the interior and background domain are same in this paper.

The rotated conductivities (step 1) are equal to the initial ones, i.e., κ' = R(θ)∙κ∙R(θ)', due to the unimodular rotation matrices R(θ). Following the previous designs [23, 26], the functional regions I with two vertices on the external polygons, and the functional regions II with two vertices on the internal polygons can be created by mapping the triangular elements of A_nB_nA_n' and A_nA_n'E_n' onto A_nB_nA_n” and A_nA_n”E_n”, respectively. Thus, the transformation equations for achieving arbitrarily polygonal thermal cloaks with same internal and external profiles can be expressed as:

{\begin{matrix} x_{I/II}^{''} = a_{1, I/II} x_{I/II}^{'} + b_{1, I/II} y_{I/II}^{'} (x_{I/II}^{'}) + c_{1,I/II} \\ y_{I/II}^{''} (x_{I/II}^{''}) = d_{1, I/II} x_{I/II}^{'} + e_{1, I/II} y_{I/II}^{'} (x_{I/II}^{'}) + f_{1,I/II} \\ z_{I/II}^{''} = z_{I/II}^{'} \end{matrix} .

where, the subscripts I/II represent the type of the functional regions; (x', y', z') and (x”, y”, z”) denote the coordinates for the initial domains with certain rotations and the transformed spaces; a₁, b₁, c₁, and d₁, e₁, f₁ are the scaled components along the x' and y' directions. By employing varied side quantity N and series number n under certain radial setups, the conductive components (κ”) for each functional regions (step 2) with same external and internal profiles can be obtained.

To further realize the such cloaking behaviors in arbitrary domains with non-uniform configurations of internal and external side-quantity, additional transformations are required to map the internal/external profiles derived by Eq. (2) onto the objective regions with side quantity N_in /N_ext. It is noted that there are two cases for the further transformations. Case 1: more external sides and less internal sides, that is, the external profile is unchanged (N) and the internal polygon changes accordingly (N_in). Case 2: more internal sides and less external sides, i.e., the internal polygon keeps uniform (N) and the external profile varies (N_ext). The transformed process for Case 1 is shown in Fig. 2.

Fig. 2 Transformations for the Case 1. (a) Unconventional transformation for arbitrarily polygonal thermal cloak; (b) Further transformation for Case 1. Such transformation is realized through mapping the internal black dot polygon (original region) onto the gold line polygon (objective polygon), i.e., the external profile keeps unchanged. Concomitantly, the functional regions are changed. (c) Enlarged view for the proposed maps. Note that, the original vertexes can be mapped onto either the objective vertexes or the adjacent sides.

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As seen, the vertexes (A_n”, B_n”, E_n”) of originally internal polygons will be mapped onto the sides of the objective profiles, due to the decreasing interior angle with few side quantities. Hence, such transformation can be operated once the objective points (A_n”', B_n”, E_n,in”') are determined through the following internal side expression on the basis of the general vertexes [24].

\begin{array}{r} y = \tan \frac{(2 n_{t, i n} - 1) \cdot 2 π}{N_{in}} \cdot \tan \frac{2 π}{N_{in}} x + r_{0} \cos \frac{2 n_{t, i n} π}{N_{in}} \\ - r_{0} \cdot \tan \frac{2 π}{N_{in}} \cdot \tan \frac{(2 n_{t, i n} - 1) \cdot 2 π}{N_{in}} \cdot \sin \frac{2 n_{t, i n} π}{N_{in}} . \end{array}

In Eq. (3), N_in is the side quantity of transformed internal objective; n_t,in denote the series number in such transformed system. The next consideration is to determine the locations of transformed vertexes. Obviously, the transformed component in the x direction can be expressed as: $r_{0}^{'''} \cos θ_{1}^{'''}$ , where, $r_{0}^{'''}$ is the distance between the transformed vertexes and origin, $θ_{1}^{'''}$ is the bending of $r_{0}^{'''}$ to the principle axis (y). Considering the related geometrical relations and Eq. (3), the coordinate components for transformed vertexes can be written as follows:

x^{'''} = r_{0}^{'''} \sin θ^{'''} = \frac{r_{0} \cos \frac{(2 n_{t, i n} - 1) π}{N_{in}} \sin θ_{1}^{'''}}{\cos (θ_{1}^{'''} - \frac{(2 n_{t, i n} - 1) π}{N_{in}})}, θ_{1}^{'''} \in [\frac{2 (n_{t, i n} - 1) π}{N_{in}}, \frac{2 n_{t, i n} π}{N_{in}}],

y^{'''} = r_{0} \tan \frac{(2 n_{t, i n} - 1) \cdot 2 π}{N_{in}} \cdot \tan \frac{2 π}{N_{in}} \cdot (\frac{x^{'''}}{r_{0}} - \sin \frac{2 n_{t, i n} π}{N_{in}}) + r_{0} \cos \frac{2 n_{t, i n} π}{N_{in}} .

Thus, the transformed regions I and II for Case 1 can be designed through respectively mapping A_nB_nA_n” and A_nA_n”E_n” onto A_nB_nA_n,in”' and A_nA_n,in”'E_n,in”'. The scaled components for transformed regions I and II in Case 1 can be obtained as follows using the similar relations to Eq. (2):

(\begin{matrix} a_{2 (1), I} \\ b_{2 (1), I} \\ c_{2 (1), I} \end{matrix}) = {(\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} & r_{2} \cos \frac{(2 n - 1) π}{N} & 1 \\ r_{2} \sin \frac{(2 n + 1) π}{N} & r_{2} \cos \frac{(2 n + 1) π}{N} & 1 \\ r_{0} \sin \frac{2 n π}{N} & r_{0} \cos \frac{2 n π}{N} & 1 \end{matrix})}^{- 1} \cdot (\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} \\ r_{2} \sin \frac{(2 n + 1) π}{N} \\ x_{E'''}^{'''} \end{matrix}) .

(\begin{matrix} d_{2 (1), I} \\ e_{2 (1), I} \\ f_{2 (1), I} \end{matrix}) = {(\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} & r_{2} \cos \frac{(2 n - 1) π}{N} & 1 \\ r_{2} \sin \frac{(2 n + 1) π}{N} & r_{2} \cos \frac{(2 n + 1) π}{N} & 1 \\ r_{0} \sin \frac{2 n π}{N} & r_{0} \cos \frac{2 n π}{N} & 1 \end{matrix})}^{- 1} \cdot (\begin{matrix} r_{2} \cos \frac{(2 n - 1) π}{N} \\ r_{2} \cos \frac{(2 n + 1) π}{N} \\ y_{E'''}^{'''} \end{matrix}) .

(\begin{matrix} a_{2 (1), II} \\ b_{2 (1), II} \\ c_{2 (1), II} \end{matrix}) = {(\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} & r_{2} \cos \frac{(2 n - 1) π}{N} & 1 \\ r_{0} \sin \frac{2 n π}{N} & r_{0} \cos \frac{2 n π}{N} & 1 \\ r_{0} \sin \frac{2 (n - 1) π}{N} & r_{0} \cos \frac{2 (n - 1) π}{N} & 1 \end{matrix})}^{- 1} \cdot (\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} \\ x_{A'''}^{'''} \\ x_{E'''}^{'''} \end{matrix}) .

(\begin{matrix} d_{2 (1), II} \\ e_{2 (1), II} \\ f_{2 (1), II} \end{matrix}) = {(\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} & r_{2} \cos \frac{(2 n - 1) π}{N} & 1 \\ r_{0} \sin \frac{2 n π}{N} & r_{0} \cos \frac{2 n π}{N} & 1 \\ r_{0} \sin \frac{2 (n - 1) π}{N} & r_{0} \cos \frac{2 (n - 1) π}{N} & 1 \end{matrix})}^{- 1} \cdot (\begin{matrix} r_{2} \cos \frac{(2 n - 1) π}{N} \\ y_{A'''}^{'''} \\ y_{E'''}^{'''} \end{matrix}) .

In Eqs. (6)–(9), a₂₍₁₎, b₂₍₁₎, c₂₍₁₎, and d₂₍₁₎, e₂₍₁₎, f₂₍₁₎ are the scaled components along the x” and y” directions for Case 1; $x_{A^{'''}}^{'''}$ , $x_{E^{'''}}^{'''}$ , $y_{A^{'''}}^{'''}$ , and $y_{E^{'''}}^{'''}$ are the coordinate components of general vertexes.

The transformation process for Case 2 is shown in Fig. 3. Similar to Case 1, the vertexes of originally external polygon is mapped onto the adjacent side of the transformed profile with fewer boundaries. Hence, the transformed external side expression can be expressed as:

Fig. 3 Transformations for the Case 2. (a) Unconventional transformation for arbitrarily polygonal thermal cloak; (b) Further transformation for Case 2. Such transformation is realized through mapping the external black dot polygon (original region) onto the gold line polygon (objective polygon), i.e., the internal profile keeps unchanged. Concomitantly, the functional regions are changed. (c) Enlarged view for the maps. Note that, the original vertexes can be mapped onto either the objective vertexes or the adjacent sides.

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\begin{array}{r} y = - \tan \frac{(2 n_{t,ext} - 1) \cdot 2 π}{N_{ext}} \cdot \tan \frac{2 π}{N_{ext}} x + r_{2} \cos \frac{(2 n_{t,ext} - 1) π}{N_{ext}} \\ + r_{2} \cdot \tan \frac{2 π}{N_{ext}} \cdot \tan \frac{(2 n_{t,ext} - 1) \cdot 2 π}{N_{ext}} \cdot \sin \frac{(2 n_{t,ext} - 1) π}{N_{ext}} . \end{array}

In Eq. (10), N_ext is the side quantity of transformed external objective; n_t,ext denote the series number in such transformed system. The distance between the origin and objective vertexes can be determined as r₂”' with Eq. (10). Considering the geometrical relations inside the transformed regions, the directional components of objective vertex can be expressed as:

x^{'''} = r_{2}^{'''} \sin θ^{'''} = \frac{r_{2} \cos \frac{(2 n_{t,ext} - 1) π}{N_{ext}} \sin θ_{2}^{'''}}{\cos (θ_{2}^{'''} - \frac{(2 n_{t,ext} - 1) π}{N_{ext}})}, θ_{2}^{'''} \in [\frac{(2 n_{t,ext} - 1) π}{N_{ext}}, \frac{(2 n_{t,ext} + 1) π}{N_{ext}}],

y^{'''} = r_{2} \tan \frac{2 π}{N_{ext}} \cdot \tan \frac{(2 n_{t,ext} - 1) \cdot 2 π}{N_{ext}} \cdot (\sin \frac{(2 n_{t,ext} - 1) π}{N_{ext}} - \frac{x^{'''}}{r_{2}}) + r_{2} \cos \frac{(2 n_{t,ext} - 1) π}{N_{ext}} .

where,

θ_{2}^{'''}

denotes the bending between r₂”' and principle axis (y). Hence, the transformed regions I and II for Case 2 can be obtained through respectively mapping A_nB_nA_n” and A_nA_n”E_n”onto F_n,ext”'G_n,ext”'A_n” and F_n,ext”'A_n”E_n”.

The scaled components for the transformed regions I and II in Case 2 can be calculated as follows:

(\begin{matrix} a_{2 (2), I} \\ b_{2 (2), I} \\ c_{2 (2), I} \end{matrix}) = {(\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} & r_{2} \cos \frac{(2 n - 1) π}{N} & 1 \\ r_{2} \sin \frac{(2 n + 1) π}{N} & r_{2} \cos \frac{(2 n + 1) π}{N} & 1 \\ r_{0} \sin \frac{2 n π}{N} & r_{0} \cos \frac{2 n π}{N} & 1 \end{matrix})}^{- 1} \cdot (\begin{matrix} x_{F'''}^{'''} \\ x_{G'''}^{'''} \\ r_{0} \sin \frac{2 n π}{N} \end{matrix}) .

(\begin{matrix} d_{2 (2), I} \\ e_{2 (2), I} \\ f_{2 (2), I} \end{matrix}) = {(\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} & r_{2} \cos \frac{(2 n - 1) π}{N} & 1 \\ r_{2} \sin \frac{(2 n + 1) π}{N} & r_{2} \cos \frac{(2 n + 1) π}{N} & 1 \\ r_{0} \sin \frac{2 n π}{N} & r_{0} \cos \frac{2 n π}{N} & 1 \end{matrix})}^{- 1} \cdot (\begin{matrix} y_{F'''}^{'''} \\ y_{G'''}^{'''} \\ r_{0} \cos \frac{2 n π}{N} \end{matrix}) .

(\begin{matrix} a_{2 (2), II} \\ b_{2 (2), II} \\ c_{2 (2), II} \end{matrix}) = {(\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} & r_{2} \cos \frac{(2 n - 1) π}{N} & 1 \\ r_{0} \sin \frac{2 n π}{N} & r_{0} \cos \frac{2 n π}{N} & 1 \\ r_{0} \sin \frac{2 (n - 1) π}{N} & r_{0} \cos \frac{2 (n - 1) π}{N} & 1 \end{matrix})}^{- 1} \cdot (\begin{matrix} x_{A'''}^{'''} \\ r_{0} \sin \frac{2 n π}{N} \\ r_{0} \sin \frac{2 (n - 1) π}{N} \end{matrix}) .

(\begin{matrix} d_{2 (2), II} \\ e_{2 (2), II} \\ f_{2 (2), II} \end{matrix}) = {(\begin{matrix} r_{2} \sin \frac{(2 n - 1) π}{N} & r_{2} \cos \frac{(2 n - 1) π}{N} & 1 \\ r_{0} \sin \frac{2 n π}{N} & r_{0} \cos \frac{2 n π}{N} & 1 \\ r_{0} \sin \frac{2 (n - 1) π}{N} & r_{0} \cos \frac{2 (n - 1) π}{N} & 1 \end{matrix})}^{- 1} \cdot (\begin{matrix} y_{F'''}^{'''} \\ r_{0} \cos \frac{2 n π}{N} \\ r_{0} \cos \frac{2 (n - 1) π}{N} \end{matrix}) .

In Eqs. (13)–(16), a₂₍₂₎, b₂₍₂₎, c₂₍₂₎, and d₂₍₂₎, e₂₍₂₎, f₂₍₂₎ are the scaled components along the x” and y” directions for Case 2; $x_{F^{'''}}^{'''}$ , $x_{G^{'''}}^{'''}$ , $y_{F^{'''}}^{'''}$ , and $y_{G^{'''}}^{'''}$ are the coordinate components of the general vertexes. The above derivations present the proposed transformations for Cases 1 and 2. Note that, varied kinds of such transformations for each functional region can be achieved by employing different $r_{0}^{'''}$ / $r_{2}^{'''}$ and $θ_{1}^{'''}$ / $θ_{2}^{'''}$ once the transformed internal/external polygon is determined. For simplifying the entire processes, the adjacent vertexes or the central points on the transformed profiles are selected as the objective points (A_n,in”', E_n,in”' or F_n,ext”', G_n,ext”') for operating the transformations in this paper. Taking Eq. (2) into Eqs. (6)–(9) and Eqs. (13)–(16) respectively and independently, the conductive components for realizing the proposed cloaking of Cases 1 and 2 can be obtained through κ”' = J· κ'·J'/det(J) = [a₃, b₃, d₃, e₃]· κ' . Here, κ”' is the transformed conductivities, a₃, b₃, d₃, and e₃ are directional elements of the entire transformations, and J is the related Jacobi matrix, which can be written as:

J = [\begin{matrix} a_{2 (1 / 2), I/II} & b_{2 (1 / 2), I/II} \\ d_{2 (1 / 2), I/II} & e_{2 (1 / 2), I/II} \end{matrix}] \cdot [\begin{matrix} a_{1, I/II} & b_{1, I/II} \\ d_{1, I/II} & e_{1, I/II} \end{matrix}] .

In general, the above transformations provide a novel avenue to generally achieve “complete” arbitrary shaped thermal cloaks. Owing to the premise of form invariance for governing equations [1], the demonstrated spatial transformations can be also employed in some other fields, such as electromagnetics [1, 2] and optics [4], to design a class of unidirectional invisibilities. Moreover, various transformed effects (expansion, compression, and rotation) can be independently or simultaneously achieved through manipulating $r_{0}^{'''}$ / $r_{2}^{'''}$ and $θ_{1}^{'''}$ / $θ_{2}^{'''}$ in the above derivations. That is, some representative unidirectional TO devices, including arbitrary shaped concentrator [26], lens [27], illusions [28], and camouflage [29], can be also designed in varied physical fields through implementing the above transformed derivations with appropriate adjustments.

2.2 Obtainment of the objective structure

The conductive components for each functional region can be theoretically obtained with the above derivations. The following considerations is to design such non-uniform profile cloaking devices with homogeneous bulk media. For general and typical validations, two schemes with hexagon-triangle (N = 6, N_in = 3) and pentagon-square (N = 5, N_in = 4) structures are designed for Case 1; another two schemes with square-pentagon (N_ext = 4, N = 5) and triangle-hexagon (N_ext = 3, N = 6) profiles are proposed for Case 2. For all of the schemes, the radii of transformed external and internal polygons are 0.1 m and 0.02 m, i.e., r₂ = 0.1 m and r₀ = 0.02 m. The initial internal radii r₁ are selected as 0.001 m, in order to avoid the singularities and limited conductivities during the transformations. As [27] pointed out, the diagonal conductivity components can be calculated with the directional elements of κ”' with the fact of b₃ = d₃ and $β^{'''} = \frac{π}{2} - \frac{1}{2} arctan (\frac{2 d_{3}}{a_{3} - e_{3}})$ based on effective medium theory. Here 𝛽”' denotes the bending between the initial system and principle axis system. Thus, the normalized diagonal conductivity components for each functional region of Cases 1 and 2 can be obtained and respectively presented in Tables 1 and 2, where, κ_∥ and κ_⊥ denote the parallel and perpendicular components to the y axis. Due to the adjacent point-point transformations, some functional regions in Case 1 and the triangle-hexagon scheme of Case 2 are merged.

Table 1. Normalized diagonal conductivity components (κ”' /κ') for Case 1

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Table 2. Normalized diagonal conductivity components (κ”' /κ') for Case 2

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To demonstrate such cloaking behaviors, the background medium is selected as Nickel steel (40% Ni, κ = 10 W∙m⁻¹∙K⁻¹) with dimensions of 400 mm × 400 mm. Following effective medium theory [21, 26], the diagonal conductivities can be achieved by alternatively configuring negative and positive conductivities. For achieving such homogeneous cloaking behaviors, 5 positive and 5 negative medium layers with uniform width are employed in each functional region. The appropriate media can be determined based on Eqs. (16) and (17) in [26]. Note that, the fractions of positive and negative media approach 55% and 45% due to the abovementioned layered configurations. Hence, the appropriate media with related conductivities for each functional region of Cases 1 and 2 can be selected and shown in Tables 3 and 4.

Table 3. Selected media and related conductivities for Case 1

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Table 4. Selected media and related conductivities for Case 2

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The media of internal structures are also selected as Nickel steel (40% Ni). The physical models of each scheme are shown in Fig. 4. Because of the transformations, the positive media are not completely uniform in each independent part of same functional region. Considering the different demands of negative conductivities in related regions, synthetic epoxy by mixing hexagonal boron nitride (h-BN) particle can be adopted to match the varied requirements [28]. Besides, pure epoxy can be employed in regions II directly. Full-wave simulations operated by with COMSOL Multiphysics 5.2 to demonstrate the proposed schemes. In the general setups, the left and right boundaries are respectively set as constant temperature sides with T_L = 373 K and T_R = 293 K, while insulative boundaries are employed in the top and bottom sides. Moreover, the ambient temperatures are set as 293 K.

Fig. 4 Geometrical model for the proposed “complete” arbitrary cloak. (a) Hexagon-triangle scheme; (b) Pentagon-square scheme; (c) Square-pentagon scheme; (d) Triangle-hexagon scheme.

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3. Numerical demonstration and discussion

3.1 Demonstration of the arbitrarily shaped cloaking performances

According to the proposed transformations for Cases 1 and 2, the proposed schemes are examined and the temperature distributions are demonstrated in Fig. 5.

Fig. 5 Temperature distributions of the proposed schemes. Among the above subgraphs, (a) and (b) present the temperature distributions of the hexagon-triangle and pentagon-square schemes for Case 1; (c) and (d) demonstrate the thermal profiles of the square-pentagon and triangle-hexagon schemes for Case 2. The white lines represent the isotherms.

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As illustrated in Figs. 5(a) and 5(b), the isothermal lines distributed smoothly both in the backgrounds of the hexagon-triangle and pentagon-square schemes with little perturbation at the structural deformations (sharp corners). Such little perturbation was caused by the locally anisotropies and would be obviously reduced by increasing the internal anisotropies with more media layer configurations [18, 26]. The heat flux was extremely restored by the alternative layer configurations inside functional regions, which force the ladder-shaped distributions to be occurred on the isothermal lines. With the transformed effects of functional regions, the external thermal energies were simultaneously compressed onto the sharp corners of the internal profiles by regions I, and expanded to the internal polygon sides by regions II. Thus, the profiles of isothermal distributions were gradually changed from the external polygons to the internal structures, as well as the same temperatures could be guaranteed on the internal polygon boundaries. It’s noted that asymmetric thermal distributions were observed in the functional regions of hexagon-triangle scheme due to the asymmetrically transformed combinations of functional regions II [Fig. 2(a)]. However, such the asymmetric transformations led no (few) effects on the cloaking behaviors, once the employed conductivities followed the transformational demands. Owing to the combined effects of the functional regions, uniform temperature distributions, i.e., no temperature gradients, were observed inside the internal polygons. Hence, significant cloaking behaviors were obtained in the hexagon-triangle and pentagon-square schemes of Case 1. The temperature distributions for the square-pentagon and triangle-hexagon schemes of Case 2 were demonstrated in Figs. 5(c) and 5(d). Similar to those of Case 1, smooth thermal profiles were observed in the backgrounds, as well as uniform temperature distributions were obtained in the internal regions. Restored and symmetrical thermal distributions were observed in the functional regions, due to the symmetrical transformations. In general, “complete” arbitrary cloaking behaviors have been significantly demonstrated in arbitrary domains with non-uniformly internal and external side configurations, as varied schemes of Cases 1 and 2 are validated in Fig. 4. Hence, the proposed methodology is effective to design such cloaking devices.

3.2 Discussion on the proposed cloaking performances

In other to further investigate the proposed cloaking performances, the symmetry lines (y = 0) are employed to describe the cloaking performances along the parallel distributions quantitatively. In addition, the temperatures on the line of y = 0 of the bare plate made of pure Nickel steel are also demonstrated to make fair contrasts. As illustrated in Fig. 6, the contrast temperatures of the bare plate distributed averagely along the parallel distributions, due to the homogeneous thermal conductivity. The temperature gradient of the central point (0, 0) was 200 K/m, which can be indicated from Fig. 6. Compared with the contrasts, all of the proposed schemes performed significant cloaking behaviors in the internal polygons, as the temperatures were uniform, and the temperature gradient of the central point approached to 0 K/m.

Fig. 6 Temperatures on the lines of y = 0 m. (a) and (b) present the measured temperatures of the hexagon-triangle and pentagon-square schemes for Case 1; (c) and (d) present the measured temperatures of the square-pentagon and triangle-hexagon schemes for Case 2.

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The symmetry line of the hexagon-triangle scheme shown in Fig. 6(a) is selected to describe the cloaking behaviors in detail. As seen, the temperatures of each point on the background approached to the corresponding ones of the contrast bare plate. Such approximate effects guaranteed the same/similar background temperature distributions, which were contributed to achieve the illusive background thermal profiles as seen in Fig. 5(a). That is, the non-uniform hexagon-triangle scheme based on the transformations can perform simultaneous internal cloaking and external illusion. Moreover, the temperatures inside the functional regions perturbed extremely, once the heat fluxes crossed the functional regions. The effective functions forced the thermal profiles to perform ladder shapes, owing to the manipulations of the bi-layer media configurations inside the functional regions. Such strong manipulations made the temperatures approach to be same inside the internal polygons, as well as bended the heat flux smoothly on the surroundings of the external profiles. Hence, the cloaking behaviors were achieved in the internal polygons. In analogy, same thermal performances on the symmetry lines (y = 0) were also observed in the rest schemes as shown in Figs. 6(b)–6(d). It obviously indicates that good cloaking abilities can be obtained in arbitrarily non-uniform profiles with the proposed transformed strategy, as no temperature gradients and perturbations were observed inside the internal domains, and perfect overlaps of the background thermal profiles were illustrated in the external regions. Here, some differences on the ranges of the functional regions were observed among the proposed schemes, due to the actual geometrical structures of the functional regions. Such differences have no effects on the cloaking performances due to the proposed designs of the conductive maps. Generally, the findings shown in Fig. 6 provides a quantitative description of such cloaking performances, and further validate the feasibility of the proposed strategy for designing non-uniform profile cloaking devices, as no temperature gradients at the central points of the demonstrated schemes were obtained, and similar/same thermal profiles were obviously achieved in the external backgrounds.

Besides the thermal behaviors along the parallel directions on the symmetry lines, the external perturbations on the surroundings are also the evaluation index for TO cloaking behaviors. Hence, the temperatures on the lines of x = 0.1 m are selected both in the proposed schemes and the contrast bare plate. The related temperatures of each scheme are presented in Fig. 7. It indicates that temperature deviations occurred at the structural deformations in each proposed scheme, which were caused by the rapid changes in the internal anisotropies at the geometrical deformations [22, 23, 26] and the approximate conductivities of the employed media. Note that, the asymmetric deformations were observed in the hexagon-triangle scheme due to the abovementioned asymmetric combinations and transformations. Though the observed deviations would perturb the external thermal distributions, such effects were quite small compared with the entire thermal fields.

Fig. 7 Temperatures on the lines of x = 0.1 m. (a) and (b) present the measured temperatures of the hexagon-triangle and pentagon-square schemes for Case 1; (c) and (d) present the measured temperatures of the square-pentagon and triangle-hexagon schemes for Case 2. The black dash lines denote the temperature band gaps between the observed highest and lowest temperatures.

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As seen from Fig. 7, the strongest perturbation, i.e., the band gap between the highest and lowest observed temperatures, were obtained in the hexagon-triangle schemes. The values of the temperature band gap was 0.664 K, which is far smaller than the measured temperatures on the line of x = 0.1 m. Hence, the external thermal field can be considered as uniform distributions. Indeed, the temperature deformations would not occur in the ideal schemes, i.e., continuous functional regions with satisfied conductive components. However, such ideal conditions cannot be achieved in the practical deigns, due to the natural restrictions of the homogeneous media. As the previous researches indicate [30], TO based performances are dominated only by anisotropy inside the functional regions. Hence, it can be predicted that such thermal deformations can be significantly reduced by increasing the internal anisotropies through increasing the layer configurations [23, 26]. In general, the external temperature deformations were far small than the observed temperatures, which contributed to maintain approximate thermal profiles as that of the contrast plate. Thus, the findings further verify the effectiveness of such transformations on designing non-uniform profile cloaks. In addition, the deformations can be significantly suppressed through increasing the layer configurations. However, such operations would enhance the processing difficulty in practical applications. Hence, the proposed schemes with 10 layers in each functional regions seem to be a compromise choice, as the observed good cloaking performances in each scheme.

4. Summary

In summary, a general strategy for achieving “complete” arbitrary cloaking behaviors in non-uniform profile devices with homogenous media has been proposed based on the rotatory linear maps [26]. Such transformations first provides a significant function to combine the external and internal domains, which are shaped in different profiles. Four kinds of non-uniform cloaking schemes, respectively considered as 2 cases, have been designed and numerically presented with homogenous bulk media based on the transformational results. Demonstrations reveal that all of the proposed non-uniformly polygonal cloaking devices performed effective thermal cloaking performances, as no temperature gradients have been observed in the internal regions, and smooth thermal fields with no (few) perturbation have been obtained in the external background. Investigations on the measured lines further illustrate the significances of the proposed cloaking performances and the transformed strategy. The findings obtained in this work can be used to achieve cloaking performances in “real meaning” arbitrary domains, which are no longer restricted by local rotations and regional profiles. Furthermore, such methodology can be further employed to design other thermal TO devices, including lens [27], illusions [28], camouflage [29], and concentrator [30, 31]. Meanwhile, it also presents great potentials in other physical fields, such as optics [32], electromagnetics [33], and surface waves [34], to achieve TO devices without the restrictions of regional structures.

5. Funding

National Natural Science Foundation of China (Grant Nos. 51776050 and 51536001).

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Hexagon-triangle scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
κ_∥ = 0.441 κ_⊥ = 2.261	κ_∥ = 0.475 κ_⊥ = 2.102	κ_∥ = 0.469 κ_⊥ = 2.134	κ_∥ = 0.441 κ_⊥ = 2.261	κ_∥ = 0.475 κ_⊥ = 2.102	κ_∥ = 0.469 κ_⊥ = 2.134
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
Merged	κ_∥ = 0.0489 κ_⊥ = 21.901	Merged	κ_∥ = 0.0489 κ_⊥ = 21.901	Merged	κ_∥ = 0.0489 κ_⊥ = 21.901
Pentagon-square scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
κ_∥ = 0.497 κ_⊥ = 1.999	κ_∥ = 0.495 κ_⊥ = 2.021	κ_∥ = 0.473 κ_⊥ = 2.122	κ_∥ = 0.473 κ_⊥ = 2.122	κ_∥ = 0.495 κ_⊥ = 2.021	/
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
κ_∥ = 0.0423 κ_⊥ = 22.023	κ_∥ = 0.0423 κ_⊥ = 22.023	κ_∥ = 0.0477 κ_⊥ = 22.437	Merged	κ_∥ = 0.0477 κ_⊥ = 22.437	/

Square-pentagon scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
κ_∥ = 0.556 κ_⊥ = 1.767	κ_∥ = 0.581 κ_⊥ = 1.724	κ_∥ = 0.531 κ_⊥ = 1.986	κ_∥ = 0.531 κ_⊥ = 1.986	κ_∥ = 0.581 κ_⊥ = 1.724	/
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
κ_∥ = 0.0488 κ_⊥ = 21.986	κ_∥ = 0.0488 κ_⊥ = 21.986	κ_∥ = 0.0480 κ_⊥ = 21.478	κ_∥ = 0.0443 κ_⊥ = 22.567	κ_∥ = 0.0480 κ_⊥ = 21.478	/
Triangle-hexagon scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
κ_∥ = 0.388 κ_⊥ = 2.576	Merged	κ_∥ = 0.685 κ_⊥ = 1.459	Merged	κ_∥ = 0.685 κ_⊥ = 1.459	Merged
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
κ_∥ = 0.0462 κ_⊥ = 21.833	κ_∥ = 0.0462 κ_⊥ = 21.833	κ_∥ = 0.0461 κ_⊥ = 21.683	κ_∥ = 0.0454 κ_⊥ = 22.049	κ_∥ = 0.0454 κ_⊥ = 22.049	κ_∥ = 0.0461 κ_⊥ = 21.683

Hexagon-triangle scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
Synthetic epoxy (1.78) 4142H Steel (42.6)	Synthetic epoxy (2.13) Bronze C62500 (38.9)	Synthetic epoxy (1.78) 4142H Steel (42.6)	Synthetic epoxy (1.78) 4142H Steel (42.6)	Synthetic epoxy (2.13) Bronze C62500 (38.9)	Synthetic epoxy (1.78) 4142H Steel (42.6)
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
/	Epoxy (0.2) Copper (398)	/	Epoxy (0.2) Copper (398)	/	Epoxy (0.2) Copper (398)
Pentagon-square scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
Synthetic epoxy (2.15) Mold steel (38)	Synthetic epoxy (2.15) Mold steel (38)	Synthetic epoxy (2.04) Ni-Cu C70600 (40)	Synthetic epoxy (2.04) Ni-Cu C70600 (40)	Synthetic epoxy (2.15) Mold steel (38)	/
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	/	Epoxy (0.2) Copper (398)	/

Square-pentagon scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
Synthetic epoxy (3.4) 2515 steel (34.3)	Synthetic epoxy (3.4) 2515 steel (34.3)	Synthetic epoxy (3.345) Mold steel (38)	Synthetic epoxy (3.345) Mold steel (38)	Synthetic epoxy (3.4) 2515 steel (34.3)	/
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	/
Triangle-hexagon scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
Synthetic epoxy (1.78) Ni-Cu C78200 (48)	/	Synthetic epoxy (3.47) Ni-Cu C71500 (29)	/	Synthetic epoxy (3.47) Ni-Cu C71500 (29)	/
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)	Epoxy (0.2) Copper (398)

Hexagon-triangle scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
κ_∥ = 0.441 κ_⊥ = 2.261	κ_∥ = 0.475 κ_⊥ = 2.102	κ_∥ = 0.469 κ_⊥ = 2.134	κ_∥ = 0.441 κ_⊥ = 2.261	κ_∥ = 0.475 κ_⊥ = 2.102	κ_∥ = 0.469 κ_⊥ = 2.134
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
Merged	κ_∥ = 0.0489 κ_⊥ = 21.901	Merged	κ_∥ = 0.0489 κ_⊥ = 21.901	Merged	κ_∥ = 0.0489 κ_⊥ = 21.901
Pentagon-square scheme
I, n = 0	I, n = 1	I, n = 2	I, n = 3	I, n = 4	I, n = 5
κ_∥ = 0.497 κ_⊥ = 1.999	κ_∥ = 0.495 κ_⊥ = 2.021	κ_∥ = 0.473 κ_⊥ = 2.122	κ_∥ = 0.473 κ_⊥ = 2.122	κ_∥ = 0.495 κ_⊥ = 2.021	/
II, n = 0	II, n = 1	II, n = 2	II, n = 3	II, n = 4	II, n = 5
κ_∥ = 0.0423 κ_⊥ = 22.023	κ_∥ = 0.0423 κ_⊥ = 22.023	κ_∥ = 0.0477 κ_⊥ = 22.437	Merged	κ_∥ = 0.0477 κ_⊥ = 22.437	/

Arbitrarily shaped thermal cloaks with non-uniform profiles in homogeneous media configurations

Abstract

1. Introduction

2. Transformation process

2.1 Theoretical derivation of transformation process

2.2 Obtainment of the objective structure

3. Numerical demonstration and discussion

3.1 Demonstration of the arbitrarily shaped cloaking performances

3.2 Discussion on the proposed cloaking performances

4. Summary

5. Funding

References

Cited By

Figures (7)

Tables (4)

Equations (17)

Optics Express