Reducing physical appearance of electromagnetic sources

Paul-Henri Tichit; Shah Nawaz Burokur; André de Lustrac

doi:10.1364/OE.21.005053

1. Introduction

In the telecommunications domain, there are actually growing interests in the miniaturization of devices, particularly for antennas in transport and aeronautical fields. In most cases it is the physics itself that limits the possibility of size reduction. The transformation optics (or coordinate transformation) concept [1–5], an innovative approach to design new class of electromagnetic devices, can prove its usefulness for miniaturization since it allows making a link between space, time and material. The main idea of coordinate transformation is to make an equivalence between Maxwell equations described in an initial coordinate system and these same equations described in another arbitrary transformed one. The result is a direct link between the permittivity and permeability of the material and the metric tensor of the transformed space containing the desired electromagnetic properties [6–8]. This method was first used by U. Leonhardt [1] and J. B. Pendry [2] to design an electromagnetic invisibility cloak in 2006 [9]. Since then, the invisibility cloak has been a subject of intensive studies [10] and later, other systems resulting from coordinate transformation have emerged. Thus, concentrators [11], rotators [12], lenses [13–16], artificial wormholes [17], waveguide bends and transitions [18–23], electromagnetic cavities [24,25], illusion systems [26,27] and antennas [28–35] have emerged. In most cases, the generated materials are inhomogeneous and anisotropic since the created virtual spaces make use of arbitrary coordinates. Devices generated by transformation optics can then be fabricated through the use of metamaterials, which are subwavelength engineered artificial structures that derive their properties from their structural geometry.

In this paper, transformation optics concept is applied to transform the signature of a radiating source. We show that a linear space compression followed by a space expansion, make the radiation pattern of a small aperture antenna appear like that of a large one. The material parameters generated from the transformation are discussed and the results are validated by numerical simulations performed using sources of different shapes and lengths. We further show that the proposed transformation can also be applied to an array of miniaturized electromagnetic radiators.

2. Transformation formulations

To achieve the transformation of a small aperture source into a much larger one, we discretize the space around the latter radiating element into two different zones; a first zone which will make our source appear bigger than its real physical size and a second zone which ensures the impedance matching with the surrounding radiation environment. The operating principle is shown by the schematic in Fig. 1(a) . In a space point of view, the technique consists in compressing a circular region of space of radius R₁/q₁ (with q₁ < 1), delimited by the red circle in Fig. 1(a)) in a region of radius R₁. In the studied transformation, our space is described by polar coordinates and the angular part of these coordinates remains unchanged. The second part of the transformation consists in an impedance matching with the surrounding space through an annular expansion zone defined between circular regions with radius R₁ and R₂, as illustrated in Fig. 1(b). This space expansion can be performed using three different transformations: a positive exponential transformation, a negative exponential transformation, and a linear one. We denote below and in the rest of the paper the two different zones by the index i, where i = 1 corresponds to the first zone and i = 2 to the second zone. The final virtual space describing our device is represented in Fig. 1(c). Figure 1(d) summarizes the different transformations considered. To secure the impedance and metric matching, continuity of our transformations is assured at the boundary of the first region (point A in Fig. 1(d)) and at the outer boundary of the device (point B in Fig. 1(d)).

Fig. 1 Representation of the proposed coordinate transformation: (a) initial and (b) virtual space. (c) The region 1 of the material enlarges the aperture of the source whereas region 2 allows matching the impedance with that of the surrounding environment. (d) Operating principle of the transformation, which consists in firstly a compression of the central space (0 < r’ < R₁) and secondly, an expansion to match the space metric (R₁ < r’ < R₂). Continuity of the transformations is assured at the boundary of the compressed region (point A) and at the outer boundary of the device (point B).

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Mathematically, the transformation in the different regions can be written as:

{\begin{matrix} r' = f_{i} (r, θ) \\ θ' = θ \\ z' = z \end{matrix}

The Jacobian matrix of the transformation is given in the cylindrical coordinate system as:

\bar{\bar{J_{c y l}}} = (\begin{matrix} \frac{\partial r'}{\partial r} & \frac{\partial r'}{\partial θ} & \frac{\partial r'}{\partial z} \\ \frac{\partial θ'}{\partial r} & \frac{\partial θ'}{\partial θ} & \frac{\partial θ'}{\partial z} \\ \frac{\partial z'}{\partial r} & \frac{\partial z'}{\partial θ} & \frac{\partial z'}{\partial z} \end{matrix}) = (\begin{matrix} f_{i, r} & f_{i, θ} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})

where f_i,r and f_i,θ represent the respective derivatives of f_i with respect to r and θ. To calculate permittivity and permeability tensors directly from the coordinate transformation in the cylindrical and orthogonal coordinates, we need to express the metric tensor in the initial and virtual spaces. The final Jacobian matrix needed for the permeability and permittivity tensors of our material is then given as:

\bar{\bar{J_{i}}} = (\begin{matrix} f_{i, r} & \frac{f_{i, θ}}{r} & 0 \\ 0 & \frac{r'}{r} & 0 \\ 0 & 0 & 1 \end{matrix})

The coefficient of our material can be written as $\bar{\bar{ψ_{i}}} = \frac{J_{i} J_{i}^{T}}{\det (J_{i})}$ in the cylindrical coordinates. The material parameters obtained using the transformation in Eq. (1) are:

{\begin{matrix} {(ψ_{r r})}_{i} = \frac{r f_{i, r}}{r'} + \frac{f_{i, θ}^{2}}{r r' f_{i, r}} \\ {(ψ_{r θ})}_{i} = \frac{f_{i, θ}}{r f_{i, r}} \\ {(ψ_{θ θ})}_{i} = \frac{r' f_{i, r}}{r} \\ {(ψ_{z z})}_{i} = \frac{r}{r' f_{i, r}} \end{matrix}

These parameters are relatively simple for the transformation in the first zone since it leads to constant values. But the permittivity and permeability components have to be expressed in the Cartesian coordinate system so as to have a perfect equivalence in Maxwell’s equations and also to physically design our device. Using matrix relations between cylindrical and Cartesian coordinates, we have:

\begin{array}{l} \bar{\bar{ε}} = (\begin{matrix} ψ_{x x} & ψ_{x y} & 0 \\ ψ_{y x} & ψ_{y y} & 0 \\ 0 & 0 & ψ_{z z} \end{matrix}) ε_{0} \bar{\bar{μ}} = (\begin{matrix} ψ_{x x} & ψ_{x y} & 0 \\ ψ_{y x} & ψ_{y y} & 0 \\ 0 & 0 & ψ_{z z} \end{matrix}) μ_{0} \\ with {\begin{matrix} ψ_{x x} = ψ_{r r} \cos^{2} (θ) + ψ_{θ θ} \sin^{2} (θ) - ψ_{r θ} \sin (2 θ) \\ ψ_{x y} = ψ_{y x} = (ψ_{r r} - ψ_{θ θ}) \sin (θ) \cos (θ) + ψ_{r θ} \cos (2 θ) \\ ψ_{y y} = ψ_{r r} \sin^{2} (θ) + ψ_{θ θ} \cos^{2} (θ) + ψ_{r θ} \sin (2 θ) \end{matrix} \end{array}

The angular part in the coordinate transformation described above allows obtaining more general and adjustable parameters for a possible physical realization of the device. However, in the present study, we consider f_i,θ = 0 to simplify the calculations. To apply our proposed coordinate transformation, we consider a radial compression of the space in region 1. This leads to a material with high permittivity and permeability tensors. For the transformation, we choose $r' = f_{1} (r) = q_{1} r$ with q₁ being a coefficient lower than 1. The physical meaning of the factor q₁ is the compression factor applied in the central region. This factor has a transition value which can be defined as $q_{0} = \frac{R_{1}}{R_{2}}$ where the material of the matching zone (region 2 in Fig. 1(c)) switch from a right-handed (positive refractive index) to a left-handed (negative refractive index) material. Indeed when q₁ < q₀ the material presents a negative index and the final apparent size of the source can be larger than 2R₂. Now if this embedded source has a small aperture, much smaller than the wavelength, then after transformation this antenna will behave like one with a large aperture, typically q₁ times larger and potentially much greater than the wavelength. A small aperture antenna is well known to radiate isotropically. The same antenna embedded in the material defined by Eq. (5) will present a directive radiation and therefore electrically appear as if its size is larger than the working wavelength. Moreover, we can obtain the radiation of a conventional array of antennas using much smaller dimensions for the latter array embedded in zone 1. To assure a good impedance matching for the radiated fields, a matching zone (region 2) is added around region 1. To design this zone, we consider three different possible transformations that match the space from R₁ to R₂. The first studied transformation for this matching region is a linear one that takes the form $r' = f_{2} (r) = \frac{1}{α} [r + R_{2} (α - 1)]$ whereas the two other transformations have logarithmic forms that can be expressed as $r' = f_{2} (r) = \frac{1}{q_{2}} \ln (\frac{r - d}{p})$ where $d = R_{2} - p e^{q_{2} R_{2}}$ and $p = \frac{R_{2} - q_{1} R_{1}}{e^{q_{2} R_{2}} - e^{q_{1} R_{1}}}$ are constant values. In these two cases, the inverse transformation defining r from r’ has an exponential form defined by: $F_{2} (r') = d + p e^{q_{2} r}$ . This exponential transformation can be characterized by the factor q₂ that indicates the shape of the progressive metric matching to vacuum ( $\bar{\bar{g}} = \bar{\bar{I}}$ ), as illustrated in Fig. 1(d). A small value of q₁ indicates a high compression of the space in the first region. To compensate this high compression, the transformation in the second region gives negative electromagnetic parameters due to the relative positions of points A and B (Fig. 1(d)), as presented in Fig. 2 . In such case, the wave propagates with a backward phase in this region. Figure 2 shows the variations of the different components of the permittivity and permeability tensors for the matching region 2. For the linear transformation, the minimum and maximum of the material parameters depend on the geometrical properties of the problem and thus they depend only on α and γ which are given by:

α = \frac{R_{2} - \frac{R_{1}}{q_{1}}}{R_{2} - R_{1}} a n d γ = 1 + \frac{R_{2} (1 - α)}{R_{1} α}

where q₁ is defined on ]0, 1]. α is therefore defined on ]-∞,1] and vanishes at q₁ = q₀. Thus, γ is a function of α and is larger than 1 for q₁ > q₀ and is negative for q₁ < q₀. In this last case such a medium is a left-handed material.

Fig. 2 (a) Representation of the transformations in regions 1 and 2. The blue and red traces correspond respectively to q₁ = 1/40 and q₁ = 1/16 and the continuous and dashed traces correspond respectively to a linear and exponential transformations. (b)–(c) Variation of the components in Cartesian coordinates of the matching region 2. The permittivity and permeability are respectively plotted for the linear transformation with q₁ = 1/40, R₁ = 2 mm (continuous blue traces case) and for the exponential transformation with q₁ = 1/16, q₂ = 15 and R₁ = 5 mm (continuous and dashed red traces case).

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We can note that the trends of the permittivity and permeability values in the Cartesian coordinates are quite similar for both linear and exponential transformation. The values depend only on R₁, R₂, q₁ and q₂ for the linear transformation. For the case of the exponential transformation in region 2, the parameters considered are q₁ = 1/16, q₂ = 15, R₁ = 5 mm and R₂ = 45 mm and as it can be observed, the calculated components ψ_xx, ψ_yy and ψ_zz are always negative.

3. Numerical validation

In order to validate the proposed concept, we use the commercial software Comsol MULTIPHYSICS to perform numerical simulations of the different transformation cases presented above. All the simulations are run in the microwave domain at 10 GHz. The validation of our design is performed in a two-dimensional configuration in a transverse electric mode (TEz) (E parallel to the z-axis). Different current sources perpendicular to the xy plane are used as radiating elements in order to show that our transformation can be applied to any type of source embedded in the region 1. Continuity and matched conditions are applied respectively to the boundary of zone 1 and zone 2.

To verify our design, we fix R₁ = 2 mm and R₂ = 45 mm. The results obtained from linear transformations both in region 1 and 2, as defined by the continuous blue trace in Fig. 2(a), are presented in Fig. 3 . In Fig. 3(a), the electric field distribution of a current source radiating in free space is plotted. The source is supposed to have a width d = 80 mm (2.7λ at 10 GHz). For such a large size, the radiation is equivalent to that of an array of several elements and therefore, the radiated field is directive. Figure 3(b) shows a similar source but with a much smaller size d = 2 mm (λ/15 at 10 GHz) embedded in the metamaterial shell having a compression factor q₁ = 1/40. In this scenario, a radiation pattern similar to the large aperture source is observed, demonstrating that small aperture antennas inserted in the proposed material shell present the same electromagnetic behavior as much larger aperture antennas in free space. However, this same miniature source will radiate in an isotropic manner in free space (Fig. 3(c)). The same observations can be made when replacing the linear current source by a crossed-type one, as illustrated in Figs. 3(d)-3(f).

Fig. 3 Electric field distribution at 10 GHz of a linear source: (a) with dimension d = 80 mm radiating in free space, (b) with dimension d = 2 mm embedded in a metamaterial shell, and (c) with dimension d = 2 mm radiating in free space. Electric field distribution at 10 GHz of a crossed-type source: (d) with dimension d = 80 mm radiating in free space, (e) with dimension d = 2 mm embedded in a metamaterial shell, and (f) with dimension d = 2 mm radiating in free space. The metamaterial shell is defined by a double linear transformation where R₁ = 2 mm, R₂ = 45 mm and q₁ = 1/40.

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In Figs. 4(a) -4(c), the linear transformation is followed by an exponential one with q₂ = 15 in region 2 and the compression factor in region 1 is decreased to q₁ = 1/40. The transformation is defined by the dashed blue trace in Fig. 2(a). The small size linear current source with d = λ/15 is embedded in the metamaterial shell defined by the proposed coordinate transformation. A directive emission is observed as in the previous case and as illustrated in the enlarged view of Fig. 4(b), we can clearly note the exponential form of the radiated field. We can also observe the perfect impedance matching between the regions 1 and 2 and between the region 2 and free space. This is clearly confirmed by the continuity of the electric field norm at the interface r = R₁ and by the absence of stationary waves in region 2 in Fig. 4(c).

Fig. 4 (a)-(c) Norm of the electric field of a line source with dimension d = 2 mm. The metamaterial shell is defined by a linear transformation with q₁ = 1/40 followed by an exponential one with q₂ = 15 where R₁ = 2 mm and R₂ = 45 mm. (d)-(f) Norm of the electric field of a crossed-type source with dimension d = 2 mm. The metamaterial shell is defined by a linear transformation with q₁ = 1/16 followed by an exponential one with q₂ = 15 where R₁ = 5 mm and R₂ = 45 mm.

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In Figs. 4(d)-4(f), the material in region 1 is defined by q₁ = 1/16 and in region 2, the material is defined by an exponential transformation with q₂ = 15. This transformation corresponds to the dashed red trace of Fig. 2(a) and the crossed-type source is embedded in the metamaterial shell. In this case also, a bidirectional directive beam can be observed even if the size of the source is very small compared to the working wavelength. In each case, the small aperture size of the radiating element has been transformed into a larger one: 40 times for the linear source and 16 times for the crossed-type source.

In the absence of the matching region, there is a high impedance mismatch at the boundary of the region 1 and all the energy emitted by the source is reflected at the boundary and confined in this latter region. This phenomenon is illustrated in Fig. 5 by the norm of the electric field. Stationary waves appear in the structure due to reflection at r = R₁.

Fig. 5 Norm of the electric field of a line source with dimension d = 2 mm. The metamaterial shell is defined by only a compression region presenting a linear transformation with q₁ = 1/40. No matching region is used in this case.

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The above results show that we are indeed able to hide the physical appearance of radiating sources by miniaturizing their physical dimensions without altering their radiation diagrams. Furthermore, we show that our transformation still holds for an array of small antennas. We have simulated an array of three sources of length L = 12.5 mm, spaced by a distance a = 5 mm and with a 30° phase shift between each element. These sources radiate in vacuum and as illustrated in Fig. 6(a) , we observe a radiated beam pointing in an off-normal direction due to the phase shift applied between the different elements of the array. When the dimensions of these antennas are reduced by a factor of 25 (q₁ = 1/25) the dimensions of the array become smaller compared to the wavelength and the radiated field becomes isotropic as shown in Fig. 6(b). By embedding the small sources in a material defined from the double linear transformation, we are able to recover the beam steering of the source array as shown in Fig. 6(c). This last example confirms the ability of our transformation to change the electromagnetic appearance of a group of radiators. Figures 6(d) and 6(e) show that in both cases, with and without transformed material, the impedance matching between the metamaterial shell and free space is perfect.

Fig. 6 3 sources with length L = 12.5 mm spaced by a distance a = 5 mm and with a 30° phase shift between each element radiate in free space (a, d) at 10 GHz (z-components of the electric field). When all the dimensions are reduced by a factor of 25 (L = 0.5 mm and a = 0.2 mm) the electric field distribution (z-component) in free space is represented in (b). Embedding the miniaturized sources in the metamaterial shell defined by a double linear transformation leads to a similar radiation pattern as the original sized sources as shown in (c) with perfect matching to free space (e).

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4. Conclusion

This work points out the use of transformation electromagnetics concept to design an artificial shell which allows hiding the physical appearance of electromagnetic sources by miniaturizing them. The latter concept makes use of two transformations; the first one to compress space and the second one to expand it. Numerical simulations have confirmed the operating principle of the transformations on sources of different geometries. We have shown that a very small source can emit a directive radiation comparable to an antenna with a large aperture. Furthermore, the concept has also been applied to an array of miniaturized radiating elements which is able to show an off-normal directive beam direction. The proposed idea constitutes an important step towards miniaturized devices in order to achieve performances that have been till now possible only with large physical devices. We can also imagine that such a device can be benefit in the design of headphones with a sound fidelity comparable to classical stereo speaker. High efficiency low profile field concentrators can also be imagined for energy harvesting.

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Reducing physical appearance of electromagnetic sources

Abstract

1. Introduction

2. Transformation formulations

3. Numerical validation

4. Conclusion

References and links

Cited By

Figures (6)

Equations (6)

Optics Express