1. Introduction

In the past few years, transformation optics has aroused a lot of scientific interest for its unprecedented flexibility in manipulating electromagnetic waves propagation. The first interesting application of this concept is the realization of invisibility cloaks [1–3], in which electromagnetic waves are guided around a certain region of space rendering the interior of the region invisible for external detector. Exploiting the coordinate transformation and the invariance of Maxwell's equations, one can determine the permittivity and permeability of the transformation medium. Various approaches like ray tracing [4], analytical method [5] and numerical methods (both finite-element method [6] and the finite-difference time-domain method [7]) have been used to simulate the electromagnetic properties of cloaks. However, this kind of cloak requires singular values on the inner boundary, and as a consequence it only works in a narrow frequency. Another kind of cloak called carpet cloak [8], which can conceal an object on a plane ground by mimicking the specular reflection of a mirror ground, is more practical for real application. Since it can be constructed with non-resonant components, carpet cloak has been experimentally realized both in microwave and optical frequency recently [9–11].

After the concept has been applied on invisible cloaks, further efforts have been taken to utilize transformation optics to design various kinds of interesting devices, such as wave concentrator [12], rotator [13], beam shifter [14], bends [15], focusing lenses [16,17], illusion devices [18,19], planar antenna [20], and waveguide connectors [21,22]. In principle, the constructive transformation medium is inhomogeneous and anisotropic, which is usually realized by artificially structured metamaterials, such as split ring resonator [3,9,23], and some natural materials, like birefringent calcite [10,11]. However, the anisotropy of natural material is weak and cannot be engineered as desired. A different easy way to implement the transformation medium is employing thin alternating layers of normal isotropic materials based on the effective medium theory. For instances, the concentric layered structures of alternating homogeneous isotropic dielectrics have been constructed to realize cylindrical cloaks [24,25] and spherical cloaks [26,27], while the oblique layered structures have been utilized to demonstrate carpet cloaks [28–30], beam shifters [31].

In free space, a planar reflector covered with transformed medium is presented for imitating the dihedral corner scattering response [32], and corner and wedge structures coated by special transformation medium can induce an electromagnetic scattering similar to that of a planar metallic sheet [33]. Recently, the utilization of periodically layered structure is extended into the domain of transformation acoustics, and a bending layered structure has been cut into particular shapes for acoustic cloaking and cavity, or trough illusion [34]. Following this study, in this paper, we address the design of transformation medium to fill up a cavity under a metal plane which is capable of controlling the scattering by mimicking the specular reflection of flat metallic mirror. Such reflection of the cavity might be very useful in electromagnetic wave anti-detection by rendering a reduction of the overall visibility of the cavity. The inverse problem is also presented that a cavity scattering could be imitated by a transformation coating laid on the metallic plane surface. Since the electromagnetic field scattered by cavity is much irregular, such a transformation medium coating for the metallic plane can enhance the electromagnetic wave scattering, leading many desirable applications such as target identification and illusion. In most cases, the designed transformation medium requires continuously inhomogeneous and anisotropic material parameters and sometimes results in extreme or singular material properties, making the realization quite difficult.

In the following, we consider the case of simplified polygonal cavity, whose boundary is made up of planar metal walls. By employing a general way of transformation optics, we find that the cavity filling medium of ideal parameters can be divided into several regions and each region only requires homogeneous but anisotropic parameters. Based on the effective medium theory, the comprising region of homogeneous anisotropy can be realized by oblique alternating layers of isotropic dielectrics. We provide the validation of the mimicking performance of both our ideas by full-wave finite-element numerical simulation with both near field distributions and average power outflow patterns at an observation semicircle under a TM Gaussian beam illumination.

2. Theory

As known, different coordinate transformations could be chosen to design the desirable transformation medium by such as space squeezing [1,8], rotating [13], and expanding [22]. In this work, considering a general scenario, a bi-dimensional cavity (identical along z-axis), which has arbitrary inner perfect electrical conductor (PEC) boundary, is filled up with our interest physical medium (grid region) in real space$\left(x\text{'}y\text{'}z\text{'}\right)$as illustrated in Fig. 1(a) . In order to induce the desirable specular reflection, we let the virtual space$\left(x\text{'}y\text{'}z\text{'}\right)$be an infinite PEC plane as show in Fig. 1(b). A simple transformation between the two space [the dash lined region (ADC) in Fig. 1(b) and the grid region(A’D’C’B’) in Fig. 1(a)] is given by

 

Fig. 1 Geometry of problems. (a) A cavity filled up with transformation medium in real space; (b) A PEC plane in virtual space; (c) An cavity illusion coating laid on a PEC plane in real space; (d) A empty cavity under a PEC plane in virtual space.

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Due to the concept of transformation optics [1], the transformation of the coordinate can be mapped into the change of material properties in the real space. Both the relative permittivity and permeability tensors of the transformation medium can be expressed as

For the inverse problem, we imitate an arbitrary-shaped cavity scattering [Fig. 1(a)] by designing a transformation medium coating which is laid on the metallic plane in real space$\left(x\text{'}y\text{'}z\text{'}\right)$as depicted in Fig. 1(c). For this case, the virtual space$\left(x\text{'}y\text{'}z\text{'}\right)$should include an empty cavity under the PEC plane surface as illustrated in Fig. 1(d). The transformation coordinate could be obtained by mapping the virtual cavity inner boundary $y_1\left(x\right)$(ABC) into real plane surface boundary$y=0$ (A’C’), leaving the physical outer boundary $y_2\left(x\right)$ invariant in transformation. The transformation functions between the two space [the dash lined region (ADCB) in Fig. 1(d) and the grid region (A’D’C’) in Fig. 1(c)] are expressed as

Similarly, we can get the Jacobian transformation matrix and relative permittivity and permeability tensor as same forms with Eqs. (2) and (3), respectively, but element parameters $K$and $P$are given as

Furthermore, the symmetrical permittivity and permeability tensor derived in Eq. (3) can be transformed into a diagonal matrix, which will be useful in the following demonstration of transformation medium realized by alternating layered structures. Next, both ideas proposed above will be verified by full-wave finite-element numerical simulation.

3. Design and performance of transformation medium

In this study, in order to validate ideas presented above, we suppose a semielliptical cavity under a standard metallic plane. It should be pointed out that the cavity boundary is also PEC material for absolute reflection of TM Gaussian incident wave (with the magnetic field along $z$ axis), which is also utilized in all the following numerical simulations. As a representative example, we let the center of the semiellipse be the coordinate origin (0,0) and the major axis of the ellipse be 2.0 and minor axis be 1.0, and the cavity boundary function is given as $y\text{'}=y_1\left(x\right)=-0.5\sqrt{1-x^2}$. Based on the imbedded transformation optics [21], for simplicity, we take the outer boundary of the transformed space as$y\text{'}=y_2\left(x\right)=-y_1\left(x\right)$, indicating that the transformed space takes a full elliptical shape. Inserting the boundary functions into Eqs. (2) and (3), the constitutive permittivity and permeability tensor of the transformed space could be obtained.

Figure 2(a) demonstrates the transverse magnetic field distribution of the semielliptical cavity filled up with transformation medium in a full ellipse shape. Here the incoming wave is properly selected to be a TM Gaussian beam of 2.0 GHz and its incident angle is set to be 45°, supposing the zero-angle along positive $x$ axis in $x-y$ plane. Similar to the carpet cloak, the cavity filled up with the designed transformation medium can reflect the incoming TM wave in the specular direction, mimicking that the incoming wave impinges onto a flat reflective surface (mirror) as depicted in Fig. 2(b). In Fig. 2(d), we plot the near magnetic field distribution of the empty cavity structure. As indicated, the incoming Gaussian wave could be confined around the left focus and then generate a broad and cured wavefronts. This scattering phenomenon could find interesting applications in target identification, enhancing radar-cross-section (RCS) scenarios. In order to imitate the diffuse scattering of this cavity, we can derive the constitutive properties of a transformation medium coating lain on a metallic plane. We let the same incident Gaussian beam illuminate the designed illusion coating, and the near field magnetic field is plotted in Fig. 2(c), from which it can be seen that the field outside the transformed space has a very excellent agreement with the case of empty cavity [Fig. 2(d)].

 

Fig. 2 Transverse magnetic field distribution of (a) a semielliptical cavity filled up with ideal transformation medium, (b) a PEC plane, (c) a cavity illusion coating laid on a PEC plane, (d) empty semiellipse, for an incident angle of 45°.

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In addition, the performance of both cases is still verified by simulating total average scattering power outflow at an observation semicircle with its radius of 1.95m and origin at (0,0) [see dark lined semicircle in Fig. 2(b)]. In Fig. 3 , it is obvious that the total average power outflow patterns show tiny difference between cases of physical medium and the PEC plane or bare cavity.

 

Fig. 3 (a) Total average power outflow patterns of the semielliptical cavity filled up with ideal transformation medium and perfect metallic plane; (b) Total average power outflow patterns of a cavity illusion coating laid on a PEC plane and empty semielliptical cavity.

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In most cases, the designed transformation medium based on the optical transformation requires continuously inhomogeneous and anisotropic material parameters and sometimes results in extreme or singular material properties, which is impossible for practical realization even with artificially structured metamaterials. For the above instances, a set of constitutive parameters of the two models above can be derived from Eq. (3), and the spatial variable parameters are simulated and shown in Fig. 4 . Both ends of the major axis of the ellipse or semiellipse require infinite values.

 

Fig. 4 Variable constitutive parameters. (a) (b) Permittivity elements of the cavity filling medium; (c) (d) Permittivity elements of the cavity illusion coating.

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4. Practical implementation of the transformation medium

The above analysis indicates that the designed transformation medium derived from the embedded transformation optics is valid for our proposed ideas. Now we intend to simplify the cavity structure, which has constant constitutive parameters, thus leading an ease of practical fabrication. We let the constitutive parameters $K$and $P$be constant and the necessary condition is that both inner and outer boundary functions should be proportioned and linear functions. For the sake of simplicity, we still make inner and outer boundaries symmetrical along $x$ axis. Now we can suppose that the cavity is composed of planar walls, then the cross section of the transformed space takes a symmetrical polygonal shape, which can be divided into several regions and each region only requires spatial invariant but anisotropic material parameters. As pointed in last paragraph in Section 2, the constitutive tensor is symmetrical, which can be obtained from a diagonal tensor through rotating the optical axis by angle$\theta$.

In subwavelength scale, the layered structure of alternating homogeneous isotropic materials could be effectively treated as single anisotropic medium with effective dielectric permittivity of $\overline{\overline{\epsilon }}=\left[\begin{array}{ccc}{\epsilon }_{\parallel }& {\epsilon }_{\perp }& {\epsilon }_{\perp }\end{array}\right]$, where ${\epsilon }_{\parallel }=\left({\epsilon }_1+\kappa {\epsilon }_2\right)/\left(1+\kappa \right)$, ${\epsilon }_{\perp }=\left(1+\kappa \right){\epsilon }_1{\epsilon }_2/\left(\kappa {\epsilon }_1+{\epsilon }_2\right)$ and $\kappa =d_2/d_1$ denotes the ratio of the two layer widths [35]. In fact, the element value of ${\epsilon }_{33}=\epsilon {\text{'}}_{33}$could be arbitrary for TM wave. So in order to map the desired constitutive tensor into the anisotropy of layered structure, we let $\epsilon {\text{'}}_{11}={\epsilon }_{\parallel }=2{\epsilon }_1{\epsilon }_2/\left({\epsilon }_1+{\epsilon }_2\right)$and$\epsilon {\text{'}}_{22}=\epsilon {\text{'}}_{33}={\epsilon }_{\perp }=\left({\epsilon }_1+{\epsilon }_2\right)/2$, where we have made the thickness of the layers kept identical ($\kappa =1.0$) and sufficiently thin compared with the wavelength. For a TM Gaussian wave incidence, the parameters are simplified as ${\epsilon }_1$,${\epsilon }_2$ and${\mu }_{33}$, which can be determined as

In the following, we demonstrate the implementation of transformation medium with a tri-sided polygonal cavity example. The designed alternating layered system is consisting of three regions denoted with Roman numbers as depicted in Fig. 5(a) . The essential parameters including cavity inner boundary functions, permittivity of the two layers ${\epsilon }_1$ and${\epsilon }_2$, alignment angles and ratios of layer thickness to wavelength for all regions are summarized in Table 1 .

 

Fig. 5 (a) Schematic view of cavity filling medium realized by isotropic layers; Magnetic field distribution of a tri-sided polygonal cavity filled up with (b) ideal transformation medium and (c) isotropic layers for an incident angle of 45°; Magnetic field distribution of a tri-sided polygonal cavity filled up with (d) ideal transformation medium and (e) isotropic layers for an incident angle of 18°.

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Table 1. Cavity Boundary Functions, Permittivity, Alignment Angles and Ratios of Layer Thickness to Wavelength for All Regions of Layered System for Cavity Filling Medium

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Figures 5(b) and 5(c) shows the near magnetic field distribution of the tri-sided polygonal cavity structure filled up with ideal anisotropic transformation medium and isotropic layers, respectively, for an incident angle of 45° from $x$ axis. We find that the cavity filled up with the designed multilayer medium of alternating homogenous dielectrics can confine the incoming Gaussian wave highly in the specular direction as expected, and it is difficult to observe the difference among cases of Figs. 5(b) and 5(c) and Fig. 2(b) in field distribution outside the outer boundary. Furthermore, we investigate the robust performance of designed cavity filling medium for a much lower incident angle of 18°. The magnetic field distribution could be compared between the cavity filled up with ideal anisotropic medium and isotropic layers shown in Figs. 5(d) and 5(e).

We also simulated the total average power outflow at an observation semicircle defined in Fig. 2(b). In Fig. 6(a) , we plot the average power outflow patterns of cases in Figs. 5 (b) and 5(c) and Fig. 2(b), which shows almost the same, for an incident angle of 45°. A similar plot is also simulated for the incident angle of 18° as shown in Fig. 6(b), from which a bit difference could be seen between cases of the isotropic layers and anisotropic parameters for angle from 120° to 160°. Figure 6 further confirms that the approach of the utilizing alternating layers to realize the anisotropic material is quite convincing with much less sensitive response to the angular variations of the excitation conditions.

 

Fig. 6 The total average power outflow patterns of different cases at the observation semicircle for an incident angle of (a) 45°, (b) 18°.

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Next, as the inverse problem, we design a layered coating laid on the metallic flat to mimic the empty tri-sided polygonal cavity designed above. In Figs. 7(a) and 7(b), we plot the magnetic field distribution of the empty cavity for incident angles of 45° and 18°, respectively. As indicated, the magnetic field scattered by the empty cavity boundary is quite irregular, which might be very useful in target identification and communication. Now we construct an illusion coating laid on the metallic plane realized by alternating layers system which is also consisting of three regions denoted with Roman numbers as shown in Fig. 8(a) . The parameters for each constitutive layered structure are summarized in Table 2 . It shows that cavity boundary functions have been taken the same, leading same values of permittivity for the alternating layers but different alignment angles compared with Table 1. Figure 8 shows magnetic field distribution of the illusion coating constructed with anisotropic parameters and isotropic layers for incident angles of 45° and 18°. It is obviously seen that outer magnetic field distribution plotted in Fig. 8 has little difference with the reference distribution of empty cavity shown in Fig. 7, indicating the illusion coating is valid for imitating electromagnetic scattering of the empty cavity.

 

Fig. 7 Magnetic field distribution of the empty tri-sided polygonal cavity for incident angle of (a) 45° and (b) 18°.

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Fig. 8 (a) Schematic view of cavity illusion coating realized by isotropic layers; Magnetic field distribution of the tri-sided polygonal cavity illusion coating with (b) ideal transformation medium and (c) isotropic layers, for an incident angle of 45°; Magnetic field distribution of the tri-sided polygonal cavity illusion coating with (d) ideal transformation medium and (e) isotropic layers, for an incident angle of 18°.

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Table 2. Cavity Boundary Functions, Permittivity, Alignment Angles and Ratios of Layer Thickness to Wavelength for All Regions of Layered System for Cavity Illusion Coating

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At last, we have also calculated the average power outflow patterns at the observation semicircle defined above for incident angles of 45° and 18°. As clearly shown in Fig. 9 , when the illusion coating laid on the metallic plane is realized by isotropic layers, the average power outflow pattern still has very small error compared with the cases of the empty cavity and the ideal transformation medium coating for both incident angles, behaving a robust illusion performance with excitation from different angles.

 

Fig. 9 The total average power outflow patterns of different cases at the observation semicircle for an incident angle of (a) 45°, (b) 18°.

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4. Conclusions

In conclusion, we have presented a general approach to design transformation medium to fill up an arbitrary-shaped cavity under a metallic plane for reduction of the electromagnetic scattering. The inverse problem is also investigated that the empty cavity scattering could be imitated by an illusion coating laid on the metallic plane. We have verified our both presented ideas by taking a semielliptical cavity as a representative example. According to the effective medium theory, we find alternating layers of normal isotropic dielectrics can realize the transformation medium as one desired in case that the cavity takes a polygonal shape. This scheme of constructing transformation medium simply by isotropic alternating layers behaves a good illusion performance of both PEC plane and empty cavity under the excitation for different angles.