1. Introduction

Since the introduction of a perfect electromagnetic (EM) cloak based on transformation optics [1], this field has been developing rapidly. Besides cloaks, many other novel devices have also been designed using transformation optics [2–4]. However, perfect cloaks require singular material parameters. To overcome this difficulty in realization, much effort towards simplified models [5–7], experiments [8–11] and other invisibility mechanisms [13–15] has been made. Among these progresses, carpet cloaking [16] is a good compromise between functionality and feasibility. It utilizes a nonsingular mapping to transform a perfect electrical conductor (PEC) ground from flat to convex, making some space beneath for concealment. External observers above the PEC ground cannot perceive the existence of the bump. Singularity in material parameters is avoided by using the nonsingular transformation. Furthermore, the application of (quasi-) conformal mapping justifies the one-parameter characterization of the cloak. Different from perfect cloaks, only half-space cloaking above a PEC ground was achieved. In fact, half-space cloaking above a dielectric half-space was studied even earlier [17], where two matching strips are buried under a semi-cylindrical cloaking cover to produce the same reflection as that from the bare half-space. Unlike our earlier work [17], the present design of cloaking on a dielectric half-space uses only nonsingular isotropic material and does not need any structure underground. Very recently, special interest is greatly aroused by the experimental realizations of carpet cloaks at both microwave and optical frequencies [9–12].

The original theory for carpet cloaking proposed by Li and Pendry is only for invisibility on a PEC ground, and all subsequent works still stick to the same assumption. In this paper, invisibility on a PEC ground is extended to an arbitrary dielectric ground, so that the reflection from the carpet cloak is the same as that from a bare dielectric ground. In practical applications, a dielectric (especially with loss) ground is also much more common and significant than a PEC ground. An analytical conformal mapping is introduced for the first time for carpet cloak design. As one of the advantages, design and implementation processes are greatly simplified by using explicit expressions. Application of analytical functions also facilitates powerful tools of complex analysis for carpet cloak design. In comparison, in Li and Pendry’s and all other carpet cloak designs numerical recipes were adopted to introduce quasi-conformal mappings in a predefined region, where numerical optimization was performed to minimize the material anisotropy. The remaining small anisotropy is then removed by replacing with an isotropic material. This will cause a noticeable lateral shift of the reflected beam position, as pointed out in a recent study [18]. Such a lateral shift of the reflected beam position will not be an issue in our design because our analytical conformal mapping can generate an exactly isotropic material. In the configuration (see Fig. 1 ), the concealment volume is created by transforming the dielectric interface from flat to convex with a nonsingular mapping. An additional absorbing layer is introduced, instead of just a PEC boundary, to produce correct reflection coefficient and prevent EM wave from penetrating into the concealment volume. The design is elaborated in Section 2, followed by the numerical verification and discussions in Section 3 and conclusions in Section 4.

 

Fig. 1 Schematic diagram for carpet cloaking on a dielectric half-space: Air and the bare dielectric half-space in the virtual space (a) are transformed to the carpet cloak in the physical space (b).

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2. Design of the carpet cloak on a dielectric half-space

To design an invisibility cloak with transformation optics, we always start from a virtual space. Here we consider a virtual space with a dielectric half-space as shown in Fig. 1(a). The dielectric half-space is characterized by a complex permittivity of ε_g = ε_r – ε_i * i (harmonic time dependence exp(iωt) is assumed and suppressed throughout the paper). For half-space cloaking, only EM waves above the half-space concern us. Refracted field is effectively attenuated in the lossy half-space within a finite layer indicated by the dashed line in Fig. 1(a). This layer, together with air space above the half-space, constitutes the region to be transformed, under which nearly no refracted EM field exists. After the transformation, the layer becomes curved, forming some space for concealment in the physical space. In practice, the cloak should have a finite volume as enclosed by the blue line in Fig. 1(b). Therefore, outside this volume the transformation must be very close (if not equal) to identity so that the material parameters can be approximated by air and the reflected field remains nearly the same as that from the bare dielectric half-space. The configuration is assumed to be two-dimensional (2D) and transverse electrically (E_z-) polarized for simplicity.

Conformal mapping is preferred in transformation optics designs, since the resulted material can be described by only one parameter for a specified polarization, and consequently much easier to realize. Real and imaginary parts of analytical functions of a complex variable are naturally used to introduce conformal mappings [19,20]. Here we use a simple analytical function of z' = z – 1 / z, where z = x + y * i and z' = x' + y' * i represent 2D coordinate variables in the virtual and physical spaces, respectively. This function fulfills all the required properties mentioned above. z' approaches to z when z goes to infinity. The layer under the interface bounded by y = 1.3 and y = 1.4 (arbitrary unit is used throughout the paper except where specified) is transformed to the curved absorbing layer shown in Fig. 1(b). Expressing explicitly the real and imaginary parts of the function in terms of the spatial coordinates, we obtain the following transformation

Up till now, the dielectric half-space is assumed to be lossy, and thus the refracted EM wave is sufficiently attenuated inside the absorbing layer. On the other hand, if the dielectric half-space is lossless or has very low loss, the absorbing layer cannot be designed by using the previous procedure directly. However, the carpet cloak can still be introduced in a similar way. In this case, fictitious loss is added to the dielectric half-space in the virtual space to attenuate the refracted EM wave. Additional reflection may be caused by the fictitious loss adversely. In order to keep the additional reflection to a negligible level, we use a gradient loss of smooth function (e.g., quadratic) with the loss increasing gradually from 0 at the interface of the dielectric half-space. Now the refracted EM wave can be effectively attenuated within a finite layer as before. Then the absorbing layer of the carpet cloak is constructed by applying the transformation in the same way as before.

3. Numerical verification and discussions

In order to validate and visualize the present design of carpet cloaking on a dielectric half-space, numerical simulations based on the finite element method are carried out. The configuration for simulations is the same as that shown in Fig. 1. Perfectly matched layers are used to terminate the computational domain. An obliquely incident E_z-polarized Gaussian beam can be used as an excitation. Free space wavelength is set to 0.12. Two different dielectric half-spaces are considered in the following simulations. In the first example, the dielectric half-space is described by ε_g / ε ₀ = 5 – 5i (a typical permittivity for soils with certain moisture at microwave frequencies [21]). The material parameters for the upper part and the absorbing layer of the carpet cloak are then implemented with Eq. (3). For the convenience of comparison, three different settings are simulated with the same excitation and snapshots of the electric field distributions are shown in Fig. 2 . In Fig. 2(a), a PEC bump is placed on the dielectric half-space, and the incident Gaussian beam is scattered violently over a large range of direction. Then the carpet cloak is applied to hide the PEC bump in Fig. 2(b), where the reflected field is restored to a Gaussian beam in an expected direction. The refracted wave is efficiently absorbed within the absorbing layer, and no field reaches the PEC bump. Figure 2(c) presents the reflection of the incident Gaussian beam from the bare dielectric half-space. By comparison, one sees the restored Gaussian beam in Fig. 2(b) is of the same direction and intensity as that in Fig. 2(c).

 

Fig. 2 Snapshots of the electric field distributions for a Gaussian beam impinging upon (a) a PEC bump on a dielectric half-space characterized by ε_g / ε ₀ = 5 – 5i, (b) the PEC bump covered by a carpet cloak, and (c) the bare dielectric half-space.

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In the above example, the relative permittivity ranges from 0.87 to 30. When a background material other than air [9–11] is considered, the range can be scaled up to above 1, making it realizable with conventional dielectrics or non-resonant metamaterials [9] at microwave frequencies. Then the broadband behavior of this carpet cloak can be expected when we use some low dispersion material at microwave frequencies. To illustrate this broadband behavior, we magnify the permittivity for the background material (and consequently the cloak and the half-space as well) by a factor of 1:0.87 so that the minimal value of the relative permittivity profile is 1. We also specifically choose meter for the length unit in the broadband simulation so that the operating frequency is in the order of GHz. Numerical simulations at three frequencies for this carpet cloak are performed and the results are shown in Fig. 3 , where one can see only two clear beams (the incident beam and the reflected beam in the expected direction) outside the carpet cloak in each sub-figure (no scattering in any other direction). From this figure (and the comparison with the corresponding reflection from the bare dielectric half-space) we can conclude that the present carpet cloak works well in a broad frequency band ranging at least from 0.5 GHz to 2 GHz.

 

Fig. 3 Snapshots of the electric field distributions for a Gaussian beam impinging upon a PEC bump covered by a carpet cloak on a dielectric half-space at (a) 0.5 GHz; (b) 1 GHz; and (c) 2 GHz.

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In the second simulation example, a lossless dielectric half-space with ε_g / ε ₀ = 2 is considered. As elucidated in the previous section, the absorbing layer of the carpet cloak is now designed by adding some fictitious gradient loss. The permittivity is modified to a complex one, ε_g [1 – 150 (1.4 – y)² i] within the layer of 1.3 ≤ y ≤ 1.4 (as before), which is then transformed to obtain the absorbing layer in the physical space. Our simulation shows that this strategy works well. The relative permittivity distribution of the carpet cloak is implemented by using Eq. (3), ranging from 0.87 to 12.00. Similarly, the range can be scaled up to above 1, making it realizable at optical frequencies with air diluted SiO₂ and Si structures [10,11]. All the other simulation settings are the same as those in the first simulation example. Simulation results are shown in Fig. 4 . The carpet cloak again restores the reflection of the incident Gaussian beam. Refracted EM wave is effectively attenuated in the absorbing layer. No additional reflection is observed. Comparison of Fig. 4(a) and Fig. 4(b) reveals that the reflected beams have the same amplitude and direction, rendering the PEC bump invisible to external observers.

 

Fig. 4 Snapshots of the electric field distributions for a Gaussian beam impinging upon (a) a carpet cloak applied to a PEC bump on a lossless dielectric half-space (ε_g / ε ₀ = 2), and (b) the bare dielectric half-space.

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Verified by the two simulation examples, the proposed carpet cloak can make objects on a dielectric half-space invisible. However, a close observation of Fig. 2 and Fig. 4 discloses a minor shift of the reflected Gaussian beam from the carpet cloak with respect to that from the bare dielectric half-space. This imperfection could be partially due to the effect of the truncation of the conformal mapping. A further reduction of this shift can be expected by optimally selecting analytical functions.

4. Conclusions

In summary, carpet cloaking on a dielectric half-space has been proposed in this paper. As an extension of earlier carpet cloaking on a PEC ground, the present design is applicable for more extensive situations. Similar to the conventional carpet cloak, the planar interface of the dielectric half-space in a virtual space is transformed to convex by a conformal mapping, forming a concealment volume. Here, a conformal mapping specified by an analytical complex function has been utilized for simplicity and analyticity. For both lossy and lossless dielectric half-spaces, an absorbing layer is introduced at the bottom of the carpet cloak to give correct reflection coefficient and to attenuate the refracted EM wave. The carpet cloak is characterized by only one nonsingular material parameter and thus broadband cloaking is feasible experimentally at both microwave and optical frequencies. Numerical simulations based on the finite element method have been performed to verify the effectiveness of the present carpet cloak. 2D EM simulations are implemented in this paper. However, it should be noted that the 2D transformation design also works for three-dimensional EM waves as long as the full material parameter set (both permittivity and permeability) is implemented.