Designs for electromagnetic cloaking a three-dimensional arbitrary shaped star-domain

Alessandro Veltri

doi:10.1364/OE.17.020494

1. Introduction

The development of transformation techniques to design the distribution of the dielectric permittivity and magnetic permeability of an hollow shell to make it able to exclude every electromagnetic field from its cavity [1, 2] attracted much attention in last years: cloak based on geometric optics [3], high order transformation to get rid of the index matching problem [4, 5], different methods to obtain a broadband cloak [6, 7] have been considered. Also an extended review paper on the subject has recently appeared [8]. Experimental works have showed the applicability of these theoretical results to reduce visibility in the visible range of frequency [9, 10] and have added new interest on the argument, emphasizing the feasibility of such devices. Furthermore, the possibility to design cloaks of arbitrary shapes has been analyzed in 2d and [12, 13, 14, 15, 16, 17] and in 3d [18]. To achieve this outcome, some papers proposed analytical or numerical ways to modify the transformation [15, 16, 13] or to derive material parameter from the solution of Laplace’s equation [12], while others used segmentation design to approach the problem [17] or by using general coordinate transformations [18]. Anyway a straightforward general method to design the cloak of a 3d generic objects remains an unsolved problem.

Fig. 1. Cross section of the cloak for an arbitrary shaped object. The cloak is the hollow shell defined by its internal (a) and external (b) surfaces.

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Designing a proper cloak for a generic 3d-object means to consider a region of space around it and calculate the proper distribution for the dielectric and the magnetic tensors of this region. The transformation technique [1, 2] requires (i) to choose a suitable external surface of the cloak and (ii) to find a spatial transformation able to compress all the space in the volume inside the external surface of the cloak (b in Fig. 1) into the shell contained between the internal and the external surfaces (a and b in Fig. 1). One of the main problems in dealing with cloaking a generic 3d-object is then to find a mathematical expression for a and b as well as a way to calculate their spatial derivatives. In this paper we will set up a procedure to calculate these quantities starting from a set of geometric points more or less uniformly distributed on the surface of the object to be cloaked. These points could be mesh nodes or 3d body sensors, in a such a general way that, in principle it will be possible to use this algorithm as a subroutine of any kind of 3d-design software or also to realize a dynamic system able to real-time design the cloak of a time-dependent shape. In this work the external surface b will be defined through a transformation, self similar with respect to a point suitably chosen inside the object, applied to the object surface a. This procedure allows to obtain all the required information starting only from the localization of a set of points on the surface of the object to be cloaked.

2. Reconstructing the inner surface

Let us suppose, as indicated in the previous section, that the surface of the object, which corresponds also to the internal surface of its cloak, is identified through a set A of N points uniformly distributed on it: A={P ₁≡(x ⁽¹⁾ ₁,x ⁽¹⁾ ₂,x ⁽¹⁾ ₃),P ₂≡(x ⁽²⁾ ₁,x ⁽²⁾ ₂,x ⁽²⁾ ₃),…,P_N≡(x ^(N) ₁,x ^(N) ₂, x ^(N) ₃)} as in Fig. 2(a). In principle, this A-set is referred to a generic system of coordinates (SOC). We can then calculate the barycenter B≡(x ^(B) ₁, x ^(B) ₂, x ^(B) ₃) of the A-set of points and make it the origin of a new SOC, with respect to which which we can recalculate the coordinates of points belonging the A-set. As we shall see in the following, we need this change of coordinates because, in order to perform the proper transformation, the more natural way is to realize that in a SOC whose origin is contained inside the object we want to cloak.

Fig. 2. (a) a set of geometric points uniformly distributed on the object to cloak; (b) the discrete polar function a_i(φ_i,θ_i) corresponding to the set of points sketched in (a); (c) the continuous function a(φ,θ) obtained via interpolation from the discrete function a_i(φ_i,θ_i) presented in (b); (d) The object is smoothly reconstructed by reverting the points of function a(φ,θ) of sketch (c) in the Cartesian space.

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Let us now calculate for each point of the A-set (P_n≡(x ⁽ⁿ⁾ ₁,x ⁽ⁿ⁾ ₂,x ⁽ⁿ⁾ ₃) its spherical coordinates P_n≡(φ ⁽ⁿ⁾,θ ⁽ⁿ⁾, r ⁽ⁿ⁾)). It is then possible to define the discrete function a_n(φ ⁽ⁿ⁾,θ ⁽ⁿ⁾)=r ⁽ⁿ⁾. The function a_n corresponding to the set of points presented in Fig 2(a) is plotted in Fig. 2(b) as an example. The continuous function a(φ,θ) (Fig 2(c)) can obviously be calculated for each value of θ and φ through interpolation of the discrete function a_n(φ ⁽ⁿ⁾,θ ⁽ⁿ⁾). Actually we used a linear Delaunay interpolation algorithm provided by the Open Source computational geometry package CGAL [19]. The equation r=a(φ,θ) allows to fully reconstruct the inner surface of the cloak (an example corresponding to the set sketched in Fig 2(a) is presented in Fig 2(d)).

3. Transformation

In order to calculate the value of dielectric tensor ε_ij(x ₁,x ₂,x ₃) inside the cloak, it is necessary to define a coordinate transformation designed to compress all the space inside an outer cloak surface, suitably chosen, defined by r=b(φ,θ) into the space in between the inner cloak surface and this outer cloak surface. As the origin of the SOC is now located in the barycenter of the A-set of points, a useful way to chose the outer surface is through a simple expansion of the inner surface b(φ,θ)=Sa(φ,θ), with S>1.

If the domain we want to cloak is or can be reduced to a 3d-star domain (which means any set A in Euclidean space R ³ if there exists a point P ₀ such that for all points P_i in A the line segment from P ₀ to P_i is in A), we can define an independent transformation for each couple of values of φ and θ, by compressing the interval 0≤r≤b(φ,θ) into the interval a(φ,θ)≤r≤b(φ,θ)

The simpler transformation of this type is a linear one:

r' = αr + a (φ, θ)

with α=S-1/S.

In order to calculate the matrix Λⁱ_j=∂x′_i/∂x_j, corresponding to the transformation between the two coordinate systems, it will be more useful to recast Eq. (1) to its Cartesian form, by introducing a vector function a(φ,θ), which is nothing but the vector from the barycenter to a point of the inner surface, identified by the couple of value φ,θ.

\begin{array}{c} {x'}_{1} = {αx}_{1} + a_{1} (φ, θ) \\ {x'}_{2} = {αx}_{2} + a_{2} (φ, θ) \\ {x'}_{3} = {αx}_{3} + a_{3} (φ, θ) \end{array}

where obviously φ=φ(x ₁,x ₂,x ₃) and θ=θ (x ₁,x ₂,x ₃) through the usual transformation from Cartesian to spherical coordinates.

From Eq. (2) the following transformation matrix can be derived:

Λ_{j}^{i} = \frac{{\partial x'}_{i}}{{\partial x}_{j}} = (\begin{array}{c} α + \frac{{\partial a}_{1}}{{\partial x}_{1}} & \frac{{\partial a}_{1}}{{\partial x}_{2}} & \frac{{\partial a}_{1}}{\partial z} \\ \frac{{\partial a}_{2}}{{\partial x}_{1}} & α + \frac{{\partial a}_{2}}{{\partial x}_{2}} & \frac{{\partial a}_{2}}{\partial z} \\ \frac{{\partial a}_{3}}{{\partial x}_{1}} & \frac{{\partial a}_{3}}{{\partial x}_{2}} & α + \frac{{\partial a}_{3}}{\partial z} \end{array})

and by applying this result to the Cartesian metrics g^kl we can calculate the metrics of this new deformed space:

g^{' ij} = Λ_{k}^{i} g^{kl} Λ_{l}^{j} = Λ_{l}^{i} δ^{kl} Λ_{l}^{j} = Λ_{l}^{i} Λ_{j}^{j} = Λ^{2}

which acts on light in a way equivalent to a dielectric tensor calculated as:

{ε'}^{ij} = {∣ \det ({Λ'}_{j}^{i}) ∣}^{- 1} {g'}^{ij} ε_{0}

where ε ₀ is the dielectric permittivity in free space. Magnetic permeability can be derived by applying the same transformation to free space so ε ^′ij and µ ^′ij are in fact the same symmetric tensor. Keeping in mind that, for reasons of suitability from now we will refer only to ε ^′ij.

4. Calculation of the Transformation matrix

The transformation described in the previous section is radial with respect to the origin; in spherical coordinates this mean φ(x_k)=φ(x′_k) and θ(x_k)=θ (x′_k). Using these relations, we can calculate

{x'}_{i} = (α + \frac{a}{r}) x_{i}

and its inverse function

x_{i} = \frac{1}{α} (1 - \frac{a}{r'}) {x'}_{i}

and it is possible to demonstrate as well

\frac{{x'}_{i}}{r'} = \frac{x_{i}}{r}

Using the relations (6) and (8), the transformation matrix presented in Eq. (3) can now be expressed as

Λ_{j}^{i} = {αδ}_{j}^{i} + \frac{a}{r} (δ_{j}^{i} - \frac{x_{i} x_{j}}{r^{2}}) + (\frac{\partial a}{\partial φ} \frac{\partial ϕ}{{\partial x}_{j}} + \frac{\partial ϕ}{\partial θ} \frac{\partial θ}{{\partial x}_{j}}) \frac{x_{i}}{r}

where ∂a/∂θ and ∂a/∂φ can be numerically evaluated from the interpolated function a(φ,θ), while ∂θ/∂x_j and ∂φ/∂x_j can be analytically calculated in a standard way from the spherical transformations.

Fig. 3. Matrix of images representing a color map of the dielectric tensor ε^ij in cut at y=0. Some elements shows negative values and ε ¹¹ is very high near the sharper borders.

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By substituting Eq. (9) in Eq. (5) we are able to calculate the dielectric tensor everywhere in the cloak. It is worth noting that the values obtained using these formulas must be attributed to the points of the cloak identified by Eq. (6), or in other words, that to obtain the value of the dielectric tensor in a particular point x′_k of the cloak we have first to calculate using Eq. (7) the corresponding values of x_k and then to evaluate Eq. (9) in correspondence of these values.

In Fig. 3 we present the matrix of images corresponding to a cut in the xz-plane of ε^ij in y=0 realized in the case of object presented in Fig. 2 covered with a cloak of size S=1.5: it is noticeable that the values of some elements are negative and that the value of ε ¹¹ can become very high near to the sharper borders.

5. Ray tracing

In order to verify the validity of the proposed algorithm we have performed a geometric ray tracing simulations in the calculated material. By assuming plane wave solutions with slowly varying coefficients in Maxwell equations and utilizing ε^ij=µ^ij, we can obtain in a standard way [2] the following dispersion relation:

\frac{c^{4}}{ω^{4}} \frac{1}{\det (ε)} {[k_{i} ε_{ij} k_{j} - \frac{ω^{2}}{c^{2}} \det (ε)]}^{2} = 0

here ω is the frequency and k the wave-vector of the considered wave. From Eq. (10) it is easy to calculate

ω = c^{2} \frac{k_{i} ε_{ij} k_{j}}{\det (ε)} .

We can now write the usual geometrical optics ray tracing equations:

\frac{{dk}_{e}}{dt} = - \frac{\partial ω}{{\partial x}_{e}} = - \frac{c^{2}}{2 ω} k_{i} \frac{\partial}{{\partial x}_{e}} (\frac{ε_{ij}}{\det (ε)}) k_{j}

\frac{{dx}_{e}}{dt} = \frac{\partial ω}{{\partial k}_{e}} = \frac{c^{2}}{ω} \frac{ε_{ei} k_{i}}{\det (ε)} (i, j, e = 1, \dots, 3)

here we used the dielectric tensor symmetry (ε_ie=ε_ei).

We choose ω ^-1 for time normalization and c/ω for length normalization we get k_i→k̂_i=k_ic/ω, t→τ=tω ^-1 and x→ξr=ω/c. Thus Eq. (12) can be written as:

\frac{d {\hat{k}}_{r}}{d τ} = - \frac{1}{2} {\hat{k}}_{i} {\hat{k}}_{j} \frac{\partial}{{\partial ξ}_{r}} (\frac{ε_{ij}}{\det (ε)})

\frac{{d ξ}_{r}}{dτ} = \frac{ε_{ri} {\hat{k}}_{i}}{\det (ε)}

Although some works [4, 5] exist describing high-order transformations which allow to realize index matching between the cloak and the free space, the determination of the dielectric tensor presented in this work is linear and generate a discontinuity on the external surface of the cloak: this means that the refraction has to be considered. When reaching the external surface of the cloak, the wave-vector k̂⁽¹⁾ will be refracted into k̂⁽²⁾, which we determine using the following properties:

1. k̂⁽¹⁾ _⊥=k̂⁽²⁾ _⊥ (in respect to the surface)

2. k̂⁽²⁾ must satisfy Eq. (11)

here n is the unit vector normal to the cloak external surface. Requirement 1 means

{\hat{k}}_{⊥ i}^{(2)} = {\hat{k}}_{⊥ i}^{(1)} = (δ_{ij} - n_{i} n_{j}) k_{j}^{(1)}

while requirement 2 means

{\hat{k}}_{i}^{(2)} ε_{ij} {\hat{k}}_{j}^{(2)} - \det (ε) = 0

using k̂⁽²⁾ _i=k̂⁽²⁾ _‖ n_i+k̂⁽²⁾ _⊥ and taking into account Eq. (16) we get

{\hat{k}}_{i}^{(2)} = [{\hat{k}}_{∥}^{(2)} - ({\hat{k}}^{(1)} \cdot n)] n_{i} + {\hat{k}}_{i}^{(1)}

by substituting (18) in (17) we finally obtain the following second order equation

[k̂⁽²⁾ _‖i n_i+(δ_ir-n_in_r)k̂⁽¹⁾ _r]ε_ij[k̂⁽²⁾ _‖j n_j+(δ_js-n_jn_s)k̂⁽¹⁾ _s]-det(ε)=0

whose solution is ${\hat{k}}_{∥}^{(2)} = \frac{αγ - β \pm \sqrt{β^{2} - αδ - \det (ε)}}{α}$ with α=n_iε_ijn_j, β=n_iε_ij k̂⁽¹⁾ _j, γ=k̂⁽¹⁾·n and δ=k̂⁽¹⁾ _i ε_ijk̂⁽¹⁾ _j. Substituting k̂⁽²⁾ _‖ in Eq. (18) it is now possible to calculate k̂⁽²⁾.

Fig. 4. Ray tracing simulation for the object described in Fig. 2 cloaked using S=1.5 and put on a reflecting surface (ground). The object to cloak is represented in gray-scale, the external surface of the cloak is sketched as a green grid while the light beams are plotted in red: (a) top-right view of the ray tracing in presence of the cloaked object (b) top-right view of the ray tracing on a empty surface (c) side view of the ray tracing in presence of the cloaked object (d) side view of the ray tracing on a empty surface. The light beams plotted in (a) and (c) are perfectly superimposed to the ones plotted in (b) and (d) in the section of space outside the cloak, this means the object will be invisible for any external detector.

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In Fig. 4 we present the result of a ray tracing simulation relative to the object presented in Fig. 2 covered with a cloak of size S=1.5 as the one represented in Fig. 3. A reflecting plane (let say the ground) is considered in this example just to take into account a more general situation in which the object, the source and the detector can be in different positions inside a realistic environment. The light beams bend inside the cloak [Fig. 4(a, c)] in such way that it is impossible for an external detector to distinguish this situation from the one in which there is nothing on the ground [Fig. 4(b, d)].

6. Conclusions

In this paper we have presented a straightforward method to design the electromagnetic cloak of a 3d-generic shaped object spatially defined by a set of points distributed over it. The object is then reconstructed via interpolation and the derivatives of the surface required for the calculation of the transformation matrix are calculated numerically. This approach can be used on any three-dimensional object realized with commercial or free 3d-design software by acquiring the mesh nodes position as the set of point needed by our method. The use of higher order transformation in order to avoid refraction at the external surface can be easily introduced in our algorithm and will be the object of a future work, this will allow to get rid also scattering on the surfaces which is not taken into account in a ray tracing scheme. Our approach does not require any particular geometrical symmetry. It is true that the transformation we used is radial with respect to the origin, so that the algorithm proposed in this paper will be limited to star domain objects, but in our opinion this does not represent a serious limitation because, on the one hand it is always possible to include more complex domains in a wider star domain, and on the other hand it is possible to couple our algorithm with a more complex strategy to eventually perform non-radial transformation. Once solved this limit, since there is no strict-requirements on the spatial points distribution, in principle it will be possible to use our algorithm to dynamically recalculate the dielectric tensor needed to cloak an object defined by a set of moving points such as the body sensors of a cloaking suit.

Acknowledgments

The author acknowledges Dr. Dalice Piñero Cruz and Dr. Mario Crea for their help and support, and Dr. Ashod Aradian for the useful scientific discussions. This work about invisibility is at the end dedicated to the “Precari Invisibili della Ricerca - UNICAL” (invisible unpermanents of research), whom are still fighting for their visibility and to survive as scientists in the Italian University: for the passion, the anger and the rain we shared in these dark days.

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Designs for electromagnetic cloaking a three-dimensional arbitrary shaped star-domain

Abstract

1. Introduction

2. Reconstructing the inner surface

3. Transformation

4. Calculation of the Transformation matrix

5. Ray tracing

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (18)

Optics Express