Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries

Chao Li; Kan Yao; Fang Li

doi:10.1364/OE.16.019366

1. Introduction

Recently, transformation optics opens an exciting gateway to design optical and electromagnetic (EM) ‘invisibility cloak devices’ and becomes a topic of great interest in applied optics and electromagnetics society [1-23]. The basic principle is to squeeze a volume in a virtual space into a shell with complex medium parameters in the physical space, which excluding EM waves in the concealment volume [1-4]. Experimentally, an invisible cloak with simplified material parameters has been implemented at microwave regime [2] and a nonmagnetic cloak is designed at optical wavelengths [5]. The possibility of the ‘invisibility’ was lately studied intensively with analytical and numerical methods [6-14]. For example, the sensitivity of the field to the tiny perturbations of the cloak’s parameters was studied in Ref. 6 and 7, and the method to extend the bandwidth of the cloaks was studied in Ref. 8. Inspired by the idea of the invisible cloak, some interesting applications, such as the concentrators [15] and field rotators [16], have also been proposed. Up to now, most of the studies are focused on the spherical and cylindrical cloaks [1-14] with rotational symmetry. Latterly, cloaks possessing geometries with reduced symmetries, such as elliptic cylinders [17], eccentric elliptic cylinders [18] and square cloaks [15] were proposed. Most recently, design of cloaks with arbitrary shapes was investigated by the authors [19] and other research groups [20-22]. However, for all the structures mentioned above, the inner and outer boundaries of the cloaks are geometrically conformal. In practice, it is sometimes desirable to make the outer boundary of a cloak not conformal to its inner boundary. For example, Kwon and Werner have proposed a 2D elliptic cloak with a uniform thickness, which is more convenient to conceal objects with long and thin shapes [23]. However, their transformation procedure is only suitable for design of cloaks with their individual shape. Although the medium parameters of invisibility cloaks with arbitrary non-conformal boundaries can be obtained numerically as proposed by Hao et.al.[24], their method is still inconvenient for the analysis and design of cloaks since the medium parameters should be derived point by point with complex matrix operations.

Toward the practical and flexible realizations of EM cloaks, this Paper presents a general transformation method to design 2D cloaks with non-conformal inner and outer boundaries. The general and explicit expressions for the complex medium parameters are derived, which is the fundamental of a cloak design procedure. A 2D cloak device with irregular and non-conformal boundaries is designed. Full-wave simulation results are provided for verification. The invisibility of the cloak to external incident waves is also quantitatively evaluated based on the Huygens’ Principle, in which the scatter width is calculated from the simulated near field. The generalization in this Paper highly improves the flexibilities for cloak design.

2. Medium transformation for 2D EM cloaks with non-conformal boundaries

In the original space, a point can be described by (x, y, z) in Cartesian coordinate or (r, θ, z) in Cylindrical coordinate with relationship

x = r \cos θ, y = r \sin θ .

To compress the cylindrical volume defined by r≤R ₂(θ) in the original space into an annular volume defined by R₁(θ’)≤r’≤R ₂(θ’) in the transformed space, the coordinate transformation can be defined as

r^{'} = R_{1} (θ) + \frac{R_{2} (θ) - R_{1} (θ)}{R_{2} (θ)} r, θ^{'} = θ, z^{'} = z .

Outside this domain the identity transformation is adopted. Then,

x^{'} = r^{'} \cos θ^{'}, y^{'} = r^{'} \sin θ^{'}

is applied to complete the whole transformation between the transformed Cartesian coordinate (x’, y’, z’) and the original Cartesian coordinate (x, y, z), which can finally be derived as

x^{'} = \frac{x}{\sqrt{x^{2} + y^{2}}} R_{1} (\tan^{- 1} \frac{y}{x}) + x \frac{R_{2} (\tan^{- 1} \frac{y}{x}) - R_{1} (\tan^{- 1} \frac{y}{x})}{R_{2} (\tan^{- 1} \frac{y}{x})},

y^{'} = \frac{y}{\sqrt{x^{2} + y^{2}}} R_{1} (\tan^{- 1} \frac{y}{x}) + y \frac{R_{2} (\tan^{- 1} \frac{y}{x}) - R_{1} (\tan^{- 1} \frac{y}{x})}{R_{2} (\tan^{- 1} \frac{y}{x})},

z^{'} = z .

According to the form-invariant transformation theorem of Maxwell’s Equations, the associated permittivity and permeability tensors of the transformation media become [3]

ε_{i^{'} j^{'}} = {∣ \det (Λ_{i}^{i^{'}}) ∣}^{- 1} Λ_{i}^{i^{'}} Λ_{j}^{j^{'}} ε_{ij}, μ_{i^{'} j^{'}} = {∣ \det (Λ_{i}^{i^{'}}) ∣}^{- 1} Λ_{i}^{i^{'}} Λ_{j}^{j^{'}} μ_{ij},

where

$Λ_{i}^{i^{'}} = \frac{\partial q_{i^{'}}}{\partial q_{i}}$

is the Jacobian matrix between the transformed and the initial coordinates. Substitute Eq.(4) into (5), the resultant permittivity and permeability tensors become

ε_{x^{'} x^{'}} = \frac{{{[r^{'} - R_{1} (θ^{'})]}^{2} + U^{2}} \cos^{2} θ^{'} - 2 U r^{'} \sin θ^{'} \cos θ^{'} + r^{'^{2}} \sin^{2} θ^{'}}{r^{'} [r^{'} - R_{1} (θ^{'})]},

ε_{x^{'} y^{'}} = ε_{y^{'} x^{'}} = \frac{{U^{2} - R_{1} (θ^{'}) [2 r^{'} - R_{1} (θ^{'})]} \sin θ^{'} \cos θ^{'} + U r^{'} (\cos^{2} θ^{'} - \sin^{2} θ^{'})}{r^{'} [r^{'} - R_{1} (θ^{'})]},

ε_{y^{'} y^{'}} = \frac{{{[r^{'} - R_{1} (θ^{'})]}^{2} + U^{2}} \sin^{2} θ^{'} + 2 U r^{'} \sin θ^{'} \cos θ^{'} + r^{'^{2}} \cos^{2} θ^{'}}{r^{'} [r^{'} - R_{1} (θ^{'})]},

ε_{z^{'} z^{'}} = \frac{r^{'} - R_{1} (θ^{'})}{r^{'}} {[\frac{R_{2} (θ^{'})}{R_{2} (θ^{'}) - R_{1} (θ^{'})}]}^{2} .

where

U = \frac{[r^{'} - R_{1} (θ^{'})] R_{1} (θ^{'}) \frac{d R_{2} (θ^{'})}{d θ^{'}} - [r^{'} - R_{2} (θ^{'})] \frac{d R_{1} (θ^{'})}{d θ^{'}} R_{2} (θ^{'})}{R_{2} (θ^{'}) [R_{2} (θ^{'}) - R_{1} (θ^{'})]} .

The permeability tensor µ⇉′ is equal to ε⇉′. Here

$\frac{d R_{1} (θ^{'})}{d θ^{'}} and \frac{d R_{2} (θ^{'})}{d θ^{'}}$

represent the first order derivative of R₁(θ’) and R₂(θ’) over θ’. Eqs. (6)-(7) gives the general and explicit expressions of the medium parameters for 2D cloaks with arbitrarily non-conformal boundaries. For the special case R₁(θ’)=τR₂(θ’), where τ<1 represents the linear compression ratio between the inner and outer boundaries, the tensors in Eq. (6) can be simplified as

ε_{x^{'} x^{'}} = \frac{{[r^{'} - τ R_{2} (θ^{'})]}^{2} \cos^{2} θ^{'} + τ^{2} {[\frac{d R_{2} (θ^{'})}{d θ^{'}}]}^{2} \cos^{2} θ^{'} - 2 τ r^{'} \frac{d R_{2} (θ^{'})}{d θ^{'}} \sin θ^{'} \cos θ^{'} + r^{'^{2}} \sin^{2} θ^{'}}{r^{'} [r^{'} - τ R_{2} (θ^{'})]}

ε_{x^{'} y^{'}} = ε_{y^{'} x^{'}}^{'} = \frac{- τ R_{2} (θ^{'}) [2 r^{'} - τ R_{2} (θ^{'})] \sin θ^{'} \cos θ^{'} + τ^{2} \sin θ^{'} \cos θ^{'} {[\frac{d R_{2} (θ^{'})}{d θ^{'}}]}^{2} + τ r^{'} \frac{d R_{2} (θ^{'})}{d θ^{'}} (\cos^{2} θ^{'} - \sin^{2} θ^{'})}{r^{'} [r^{'} - τ R_{2} (θ^{'})]}

ε_{y^{'} y^{'}} = \frac{{[r^{'} - τ R_{2} (θ^{'})]}^{2} \sin^{2} θ^{'} + τ^{2} {[\frac{d R_{2} (θ^{'})}{d θ^{'}}]}^{2} \sin^{2} θ^{'} + 2 τ r^{'} \frac{d R_{2} (θ^{'})}{d θ^{'}} \sin θ^{'} \cos θ^{'} + r^{'^{2}} \cos^{2} θ^{'}}{r^{'} [r^{'} - τ R_{2} (θ^{'})]}

ε_{z^{'} z^{'}} = {(\frac{1}{1 - τ})}^{2} \frac{r^{'} - τ R_{2} (θ^{'})}{r^{'}}

Equation (8) is the medium parameters of 2D cloaks with conformal boundaries, as developed in Ref.19. For a cloak with uniform thickness T, we can set R₂(θ’)=R₁(θ’)+T in Eq. (6) to obtain the medium parameters. In fact, R₁(θ’) and R₂(θ’) can be chosen as arbitrary continuous functions with period 2π to represent closed contours with arbitrary shapes. They can be generally expressed by a Fourier series as

\sum_{n = 0}^{\infty} A_{n} \cos (n θ^{'}) + \sum_{n = 1}^{\infty} B_{n} \sin (n θ^{'})

If

$\frac{d R_{1} (θ^{'})}{d θ^{'}}$

and

$\frac{d R_{2} (θ^{'})}{d θ^{'}}$

are both continuous, the medium parameters will be continuously varying in the cloak region. If

$\frac{d R_{1} (θ^{'})}{d θ^{'}}$

or

$\frac{d R_{2} (θ^{'})}{d θ^{'}}$

is discontinuous in certain θ′_d, which means the cloaks have sharp corners, the medium parameters will be discontinuous at the corresponding positions. Fortunately, it has been verified by the square cloak in Ref.15 that, such discontinuity does not break any fundamental cloaking properties. The above discussion means the generalization in this Paper can be specialized to all of the formerly designed cloaks.

Fig. 1. Generalized coordinate transformation. (a) The original space. (b) The transformed space. The region with r≤R₂(θ) (shaded) in (a) is transformed to the region with R₁(θ’)≤r’≤R₂(θ’) (shaded) in (b).

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3. Electromagntic properties of a 2D cloak with non-conformal boundaries

To show the flexibility of the approach to design 2D irregular cloaks with non-conformal inner and outer boundaries,

R_{1} (θ^{'}) = [12 + 2 \cos (θ^{'}) + \sin (2 θ^{'}) - 2 \sin (3 θ^{'})] / 30,

R_{2} (θ^{'}) = [10 + \sin (θ^{'}) - \sin (2 θ^{'}) + 2 \cos (5 θ^{'})] / 12,

are chosen as an example. In this section, the interactions of the electromagnetic waves with the irregular cloak are studied. The exciting TM plane wave has an electric filed polarized in z’ direction with unit amplitude E⇀_in′=ẑ′exp(-jk₀ x′), and incident upon the cloak along the +x’ direction. The frequency of the time harmonic incident wave is set to be 1 GHz. Theoretically, the inner region of the cloak is shielded from external fields and consequently an object of any shape and material placed in it has no impact on the external fields. And the field distribution in the transformed space can be analytically mapped from the field distribution in the original space based on the transformation theory.

E_{i^{'}} = \frac{\partial q_{i}}{\partial q_{i^{'}}} E_{i}

Figure 2 shows the theoretical electric field distribution by field transformation. The inner and outer boundaries of the cloak are shown as the black contours. The green lines indicate the electromagnetic power-flow lines. It’s seen that the electromagnetic waves are naturally guided around the inner region by the cloak with non-conformal inner and outer boundaries.

Fig. 2. Electric field distribution in the vicinity of the cloak obtained based on analytical field transformation. The green lines show the flow of EM power.

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Fig. 3. Full-wave simulation results for electric field distribution in the vicinity of the PEC cylinder without cloak.

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To further verify the field transformation and the medium parameters, the cloak performance is also investigated numerically based on finite-element method (FEM). In simulation, we fill the inner region of the cloak with perfect electric conductor (PEC) and see whether it can be “seen” from outside. Perfect matched layers (PMLs) are applied to terminate the computational domain in ±x’ and ±y’ directions to avoid unwanted reflections. In Fig. 3, the irregular PEC cylinder is directly exposed to the incoming wave, where strong scattering can be observed. Figure 4 shows the electric field distribution near the cloaked structure. It’s seen that, the wave is smoothly bent around the cloaked area and the phase fronts are perfectly restored when the wave exits the cloak. The field distribution agrees well with the theoretical results in Fig. 2.

Fig. 4. Full-wave simulation results for electric field distribution in the vicinity of the cloaked PEC cylinder with non-conformal boundaries (excited by a TM plane wave).

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Fig. 5. Full-wave simulation results for electric field distribution in the vicinity of the cloaked PEC cylinder with irregular shape (excited by a TM cylindrical wave).

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Since the proposed cloak has no symmetry in any directions, it’s necessary to study its interaction with EM waves from different orientations. An effective way is to investigate its property under the illumination of a cylindrical wave, which can be decomposed to different planar wave components. In the simulation, a line source is set at the position x’=-0.8m, y’=- 0.8m to generate the cylindrical wave. The results are given in Fig. 5. It’s seen that the cylindrical wave is perfectly guided around the cloaked object, and the circular phase fronts are remained outside the cloak without any obvious scattering.

To quantitatively evaluate the cloaking performance, the scatter width σ (the 2D equivalent of a radar cross section) is calculated based on the Huygens’ Principle. To determine σ, the scattered electric field in far field region is calculated by the integration of the simulated near field [19],

σ = \frac{k_{0} {∣ {\hat{r}}_{0} \times ∲_{C} [(\hat{n} \times {\vec{E}}_{c}) - η_{0} {\hat{r}}_{0} \times (\hat{n} \times {\vec{H}}_{c})] \exp (i k {\vec{r}}^{'} \cdot {\hat{r}}_{0}) d l ∣}^{2}}{{4 ∣ {\vec{E}}^{i} ∣}^{2}}

where E⃗_c and H⃗_c is the EM fields on the integration contour C, r̂₀ is the unit vector of the scattering direction, r⃗′ is the position vector on the contour C, and η₀ is the free space wave impedance. In Fig. 6, the scatter widths of the PEC cylinder with and without cloaks are both calculated for comparison. Considering the non-symmetry of the structure, the scatter widths for TM plane waves from four different incident directions are studied. Table1 lists some parameters to describe and compare the scattering properties, including the averaged scatter widths σ_avg, the maximum scatter widths σ_max, and the ratios between the cases with and without cloaks. It’s seen that the cloak greatly reduces the scatter width in different scattering angles. The total scatter power (equivalent to the averaged scatter width) of the irregular PEC cylinder is reduced more than 30 times and the maximum scatter width is reduced more than 160 times. No doubt the ratios could be pushed even larger with finer meshes in simulation. A more interesting phenomenon is that the scattering power of the cloaked structure is almost isotropy over all the angles, which is very different from conventional scattering from objects with irregular shapes.

Fig. 6. The scatter width of the cloaked and uncloaked PEC cylinder for different incident directions. The dotted lines are for PEC cylinder without cloak. The solid lines are for PEC cylinder with cloak. The blue, red, black, and green lines are for the incident direction (angle)+x(θ i=0°), -x(θ i=180°), +y(θ _i=90°), -y(θ _i=270°), respectively.

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Table 1. Some parameters to describe and compare the scatter properties

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4. Conclusion

A general transformation procedure for designing 2D cloaks with non-conformal inner and outer boundaries is demonstrated. The explicit expressions for the complex medium parameters are developed, which can be readily specialized to all of the previously designed 2D cloaks. A peculiar cloak device with irregular and non-conformal boundaries is designed as an example. Full-wave simulations combined with Huygens’ Principle are applied to verify the invisibility of the cloak to external incident waves. All the theoretical and numerical results verify the effectiveness of the proposed method. Although we confine ourselves to 2D cases in this paper, the method can be readily extended to design three dimensional (3D) cloaks with non-conformal boundaries. The generalization in this Paper highly improves the flexibilities for cloak design.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (60501018), the National Basic Research Program of China under Grant (2004CB719800), and the Knowledge Innovation Program of Chinese Academy of Sciences.

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Incident angle (direction)		θ_i=0° (+x)	θ_i=180° (-x)	θ_i=90° (+y)	θ_i=270° (-y)
Averaged Scatter width (m)	no cloak (σ_avg1)	1.8977	1.9073	1.8925	1.8972
Averaged Scatter width (m)	with cloak (σavg2)	0.0577	0.0576	0.0574	0.0574
σ_avg1/σ_avg2		32.89	33.11	32.97	33.05
Maximum Scatter width (m)	no cloak (σ_max1)	13.490	13.213	12.735	13.9316
Maximum Scatter width (m)	with cloak (σ_max2)	0.0724	0.0746	0.0752	0.0757
σ_max1/σ_max2		186.33	177.12	169.34	184.04

Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries

Abstract

1. Introduction

2. Medium transformation for 2D EM cloaks with non-conformal boundaries

3. Electromagntic properties of a 2D cloak with non-conformal boundaries

4. Conclusion

Acknowledgements

References and links

Cited By

Figures (6)

Tables (1)

Equations (21)

Optics Express