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Most implementations of the ground plane invisibility cloak are based on the isotropic design through the quasi-conformal transformation. However recent theoretical analysis predicts the unavoidable lateral shift of the scattering fields associated with these cloaks making them detectable. In this paper, we propose an alternative method to design the ground plane invisibility clock with electromagnetic beam modulation blocks through simple coordinate transformation discussed in our previous work. The ground plane cloak obtained with the rigorous transformation optics possesses moderate anisotropic distributions of material parameters, but results in no lateral shift of the scattering fields. To realize the design, a possible scheme is suggested by discretizing the ground plane cloak to several homogeneous sub-blocks. These sub-blocks can be realized with multilayer isotropic dielectrics with alignment angles that are determined by the effective medium theory. Thus the non-magnetic ground plane invisibility cloak can be constructed by several multilayered normal dielectrics aligned in different angles. The performance of the proposed cloak and its practical implementation is validated by full-wave electromagnetic simulations with both near field distributions and far field scattering patterns under different EM wave incident angles. The proposed cloak is composed of normal dielectric multilayers, thus can leads to easy experimental demonstration of non-magnetic ground plane cloak in the frequency range from microwave to optical.
©2010 Optical Society of America
1. Introduction
Recently, electromagnetic invisibility cloak [1,2] has intrigued intensive attention on both theoretical analysis and practical implementation [3]. The unprecedented ability of concealing objects could be attributed to coordinate transformation or transformation optics (TO) [4,5], which ensure the electromagnetic (EM) waves to be bent round a certain hidden region, leaving the object inside the region without interacting with the EM waves. Much effort has been focused on this area ever since [6–14]. The cloak structure derived from TO is independent of the geometry or size of the hidden object. However, the material parameters for an ideal invisibility cloak would be singular at the inner surface of the cloak, which is impossible for practical implementation. Although simplified cloak has been used to avoid the problem [3], the cloak itself has introduced unexpected scattering, which degrades the cloaking performance [6,8,11,12]. Besides the invisibility cloak, the TO method has also been successfully used to design many other interesting microwave and optical devices [15–22].
In addition to the cloaking of any object in the free space, a more practical idea for invisibility has been proposed to conceal the object on an infinite half-space ground plane under a so-called carpet or ground plane cloak [23–31]. The idea is then verified by experiments in both microwave and optical frequency range [24–26]. Recently, the design of carpet cloak has also been successfully extended to arbitrary shape and three dimensional cases [31]. Since the half-space ground could be approximately regarded as a perfect electrical conductor (PEC), the design target of the ground plane cloak is to minimize the irregular scattering from the reflective object on the ground, leaving the total reflection approaching to that of the PEC ground, as a specular reflection. To achieve this goal, TO method is employed by mapping the hidden region under a ground plane cloak structure to a two dimensional (2D) sheet, which is inherently invisible when it is sit on the flat PEC ground plane. Generally, most ground plane cloaks are based on quasi-conformal transformation, which allows minimal anisotropy and facilitates fabrication [23]. However, the material properties retrieved from this procedure are highly spatially variant, which restricts the maximum size of each mesh when discretizating the cloak in the implementation. Another drawback which has been recently predicted is that the isotropy structure derived from the quasi-conformal transformation will lead to lateral shift of the scattering fields, making the inside objects visible to near field detection [30].
In this paper, we propose an alternative attempt to construct the ground plane invisibility cloak by a pair of EM beam modulation blocks discussed in [18]. By keeping the in plane refraction index unchanged, we can further reduce the perfect cloak to its non-magnetic form for optical applications. We find that the proposed cloak is simpler to understand, less spatially variant, allowing larger sub meshes than that derived from the quasi-conformal transformation. Moreover, it in principle produces no near field lateral shift of scattering fields.
In the following, we first briefly review the design of EM beam modulation blocks through TO, and then apply such blocks to construct the ground plane invisibility cloak on an infinite PEC ground plane. To realize the design, a possible scheme is suggested by discretizing the ground plane cloak to several homogeneous sub-blocks. These sub-blocks (spatial invariant, but anisotropic) can be composed of multilayer isotropic dielectrics with alignment angles that are determined by the effective medium theory (EMT) [8,13,28,29]. Thus an optical ground plane invisibility cloak can be constructed by several multilayered normal dielectrics aligned in different angles. We provide the validation of the cloak performance by full-wave electromagnetic numerical simulations with both near field distributions and far field scattering patterns under different EM wave incident angles. We emphasis that the proposed coordinate transformation could overcome the problem of EM beam lateral shift that discussed in [30] and by applying the EMT the invisibility carpet could be easily realized through multilayered normal dielectrics in the frequency range from microwave to optical.
2. Beam modulation blocks designed from transformation optics
In principle, countless coordinate transformations could be chosen to design the ground plane cloak by space squeezing. Among these approaches, a quasi-conformal transformation has been widely used [23–27], for the best advantage of reducing anisotropy of the cloak. On the other side of the issue, the material parameters retrieved from the quasi-conformal transformation are of high spatial variation. It requires careful calculation and metamaterial design to ensure the correct material parameter distribution in the ground plane cloak. Besides, the near-field lateral shift of the scattered field is unavoidable. This is due to the disaccord of free space and transformed space when introducing isotropy, in which, the phase propagation is preserved while the energy propagation will be distorted, rendering reflected beam with lateral deviation from the ideal specular path and thus the cloak will become detectable [30].
Here we propose an alternative scheme for the transformation, which was previously used for beam modulation [18] to overcome the shortcomings of quasi-conformal transformation. For simplicity, we restrict the problem to a 2D case. In the proposed transformation, only the y coordinate has been transformed in a uniform linear proportion within the cloak area, similar with the so-called “transfinite transformation” discussed in [23]. As discussed in [23], the “transfinite transformation” leads to rather moderate material distributions, compared with quasi-conformal transformation.
Consider a rectangular region (actually the geometry of the region is not limited to rectangular, i.e., the method could be used for other shapes) on top of an infinite ground plane (PEC plane) as shown in Fig. 1(a) . The interested region indicated with light grey in the original space [Fig. 1(a)] is modulated to a pair of compressor (left) and expander (right) in the real space [Fig. 1(b)], leaving a small triangular region (a bump on the ground plane) with no coordinate meshes, as illustrated in Fig. 1. Under this coordinate transformation, the original space which represents any isotropic normal dielectric has been changed into two parts, one with twisted coordinate meshes (in dark grey) which is equivalent to the rectangular light grey region in the original space, representing the ground plane cloak, and the other one with no coordinate meshes, representing the region to be concealed. The originally uniform EM wave propagation will be squeezed into the upper modulated blocks, keeping the lower triangular region unaffected by the EM waves. The coordinate transformation relationship is given as the following,
in which, η represents the modulation coefficient, J indicates the Jacobian matrix, with the element of , and , represents the original or the transformed material permittivity and permeability tensor, respectively. Equation (1-1) and (1-2) represent transformation relationships for the compressor and expander respectively. The perfect ground plane cloak could be constructed using these parameters.
Fig. 1 (a) The original space (light gray) in host medium, and (b) the transformed space (dark gray) with a pair of EM beam compressor and expander and the void triangular space for “invisible” objects.
For transverse magnetic (TM) incident waves (with magnetic field along the x 3 coordinate), it is also possible to remove the requirement of magnetic response of the material, which is difficult for natural materials, especially in optical frequency range. Therefore, we could reduce the material parameters for a perfect cloak to the non-magnetic form by keeping the following product unchanged similar to the procedure used in [6],
where is the permittivity tensor element of the anisotropic dielectric in the x 1 – x 2 plane), and represents the in-plane anisotropic permittivity tensor elements of the non-magnetic cloak.3. Design and performance of the ground plane cloak
Suppose the whole beam modulation block structure is embedded in a host medium extending to infinity, which is represented by absorbing boundary (AB) in the numerical calculation. Any reflective bump on the ground plane with the height of or larger than the wavelength scale will produce obvious scattering of the illuminating EM waves, denoted as diffuse reflection. To make the bump invisible to detectors, the reflection pattern of the bump should be specular, mimicking that of a flat PEC plane. Utilizing the coordinate transformation described in Fig. 1, we are able to design a perfect ground plane cloak with the proposed beam modulation blocks. By covering the perfect conducting bump with the beam modulation blocks, any object concealed under the bump will not interact with the illuminating EM waves, leaving an ideal specular EM reflection, which imitates the case for a flat PEC ground plane.
As an example, we set the parameters η = 3/4, a = 4λ, b = 4λ (λ denotes the working wavelength) and the host medium with ε = 2.2, μ = 1, the material parameters for the cloak covering the bump could be directly retrieved from Eq. (2) following the TO procedure described in [4,5]. For verification of the cloak performance, we use finite element method (FEM) to analyze the EM behavior predicted by the EM theory and TO method. A TM Gauss beam (with magnetic field perpendicular to the x 1 – x 2 plane) with a waist of 3λ impinges along the direction with an azimuth angle φ = 135°. Figure 2 shows the near field transverse magnetic field distributions for four different cases: the flat PEC ground plane [Fig. 2(a)], the reflective PEC bump on the ground plane [Fig. 2(b)], the bump covered with perfect ground plane cloak [Fig. 2(c)], and the bump covered with non-magnetic cloak [Fig. 2(d)]. As indicated in the results, the field scattered by the reflective PEC bump [Fig. 2(b)] is quite irregular, while in the cases where the bump is covered with either perfect or non-magnetic ground plane cloak [Fig. 2(c) or 2(d)], the magnetic scattering field is confined highly in the specular direction, mimicking that in the case of a flat reflective PEC ground plane [Fig. 2(a)].
Fig. 2 Simulation results from finite element method for (a) the flat ground plane, (b) the bare bump, (c) the bump covered with perfect ground plane cloak, and (d) the bump with non-magnetic cloak. The red lines denote the power flow paths (from left to right). PEC and AB indicate perfect electrical conductor and absorbing boundary, respectively.
We conclude that the perfect ground plane cloak derived from TO is equivalent to the case of a PEC ground plane, which is the direct consequence of the invariance of Maxwell’s equations in different coordinates. The non-magnetic ground plane cloak for the TM wave illumination is also efficient. This is due to the reason of keeping the in-plane refraction index unchanged that maintains the wave trajectory as that in the case of the perfect cloak. The red lines in Fig. 2 denote the power flow trajectories. We find the reflected beam is symmetry with the incident beam in Fig. 2(c) or 2(d). This result confirms obviously that the proposed method does not generate any lateral shift of the scattered fields, which is different from the case of using quasi-conformal transformation as discussed in [30].
Through the near-to-far-field extrapolation algorithm, we calculate the normalized far field pattern of the scattering fields in Fig. 3 . We compare the four cases corresponding to the four plots in Fig. 2, which show that either the flat ground plane or the bump with either the perfect ground plane cloak or the non-magnetic cloak reflects the incident wave in the specular direction, while the bare bump scatters the incident wave irregularly and results in a power gap between two side lobes at around 15° and 75°. These patterns give convincing evidence for the theoretical analysis and verify the near field calculations. We have also checked the cloaking performance under different incident azimuth angle which shows the similar result with a specular reflection.
4. Practical implementation of the ground plane cloak
The above analysis indicates that the proposed simple transformation is valid for designing ground plane cloak. Based on this coordinate transformation, the designed cloak could shield objects underneath a triangular PEC bump without producing any lateral shift of the scattered fields. For TM incident waves, the cloak with the reduced non-magnetic parameters only requires spatially variant anisotropic in-plane permittivity elements . The spatially variant permittivity distribution could be realized with graded index metamaterials as suggested in [24]. However, to design the graded index metamaterial structure is a complicate work, especially when the number of the mesh elements is very large. It is noted that the transformation relationship in Eqs. (1-1) and (1-2) leads to a rather moderate distribution of material parameters, which allows coarse discretization to several sub-divisions. The sub-divided block could then be represented by the material parameters at its geometrical center. Therefore through discretization the 2D anisotropic medium of each sub-divided block becomes spatial invariant. The graded index metamaterials used in [24] are essentially resulted from much finer sub-division, demanded by the larger variant range of the material parameter distribution based on the quasi-conformal transformation, while this proposed transformation, although simple, only requires a moderate distribution range of the material parameters.
Through coarse discretization, the spatial invariant but anisotropic sub-divided blocks could finally be realized by aligned alternate dielectric multilayered structures based on EMT [28,29]. In the practical implementation, there are four parameters required to be defined: the thickness ratio r of the alternating dielectrics, the permittivity of the two dielectric εa and εb, and the alignment angle θ. They are connected in the following equations. The anisotropic in-plane permittivity for each sub-divided block is determined by
in which, represents the non-magnetic permittivity at the geometrical center of the sub-division retrieved from the TO, θ is the alignment angle of the multilayered dielectric structure with respect to the x′ 1 axis, and are the in-plane eigen-permittivitiy, while could be arbitrary for TM waves. Assuming , we can then determine the eigen-permittivitiy from the alternating dielectric multilayers composed of two natural dielectrics with permittivity εa and εb. We haveThus, the four values, the thickness ratio r, permittivity εa and εb, and the alignment angle θ can be determined through Eqs. (3)–(5).
In the following we demonstrate the implementation with a simple example. We divide the whole ground plane cloak proposed in section 3 into eight sub-divisions as shown schematically in Fig. 4(a) . Due to the symmetry, the cloak is only composed of four different blocks, and each block can be realized with a certain aligned alternating dielectric multilayer structure. The material parameters and the alignment angles are summarized in Table 1 . We assume the thickness ratio r = 1 for simplicity. The thickness of each layer is set to λ/20 as to abide the essential condition of EMT. The moderate permittivity of the dielectric layers in the cloak can be easily realized with normal dielectrics as indicated in Table 1. The cloak performance is verified again through FEM-based simulations as shown in Fig. 4(b) and 4(c). We compare both the case where the cloak is composed with eight block of homogeneous anisotropic medium [Fig. 4(b)] and the case where the eight blocks in the cloak are realized by aligned multilayered dielectrics. These two implementations both inherit the performance of an ideal ground plane cloak, which confirms the design procedure. Note that slight reflection might occur at the incidence plane due to impedance mis-matching when the spatial variant cloak is sub-divided coarsely, which could be minimized by finer sub-division. We also find very slight lateral shift between the power flow path and the specular ray trace. We attribute this slight excursion to the coarse sub-division, for the reason that the lateral deviation is only about λ/10, much less than the characteristic lateral shift in the isotropic cloak derived from quasi-conformal mapping, which is in the order of the bump height (one wavelength) [30].
Fig. 4 Schematic view of the cloak realized with sub-blocks of dielectric multilayer (a), and the calculated magnetic field distributions from finite element method for (b) sub-divided cloak; (c) multilayer cloak.
Table 1. Permittivity and alignment angels for the sub-blocks of the compressor in the cloak
The far field pattern for the normalized scattering fields is shown in Fig. 5 . We find that both the sub-divided anisotropic non-magnetic cloak and the multilayer dielectric cloak could shield the PEC bump from outside detection. We could see that either the near field (Fig. 4) or the far field result (Fig. 5) shows little difference between the anisotropic non-magnetic sub-divided cloak and the multilayer dielectric cloak, indicating that the approach of using alternating layered dielectrics to realize the sub-divided blocks is a reasonable and feasible solution.
We have also verified the cloaking performance of the multilayer cloak for incident EM waves with different incident angles. One example for the incidence at the azimuth angle φ = 120° is shown in Fig. 6 with both the near field distribution and far field scattering pattern. These results are very similar to those in Fig. 5 with the incident azimuth angle of 135°, except the reflected lobe direction.
Fig. 6 (a) Near field distribution of magnetic fields, and (b) the far field scattering pattern for case with incident azimuth angle φ = 120°.
The proposed multilayer cloak is composed of several kinds of normal dielectrics with moderate permittivity as shown in Table 1, which do not rely on any EM resonant structures such as split ring resonator [3] or metallic cut wire structure [6] used in other cloak implementations. The working bandwidth of the proposed cloak is only limited by the dispersion of the dielectrics, which could be quite wide at microwave or terahertz wavebands. It may also lead to wideband optical ground plane cloak by designing the dielectrics with engineered weak dispersive dielectric thin films.
5. Conclusion
In conclusion, we propose a ground plane cloak constructed with the EM beam modulation blocks through simple coordinate transformation. We find that this coordinate transformation is easy to understand, results in moderate permittivity variation that allows large sub-division meshes, and most importantly, leads to no lateral shift of the scattered fields. Material parameters for both the perfect ground plane cloak, and the non-magnetic cloak for optical applications, have been obtained from the TO procedure, and the cloak performance have been verified with near-field distribution and the far field scattering patterns. For practical implementation, we demonstrate that the proposed ground plane cloak can be discretized to several sub-division blocks of homogeneous anisotropic dielectrics, and can be then realized with aligned alternate layered structure of normal isotropic dielectrics. The realization procedure is provided with clear analytical expressions for the material and geometric parameters and verified by numerical simulations. Both near field distributions and far field scattering patterns prove the validation of the implementation with good cloak performance. The proposal only requires multilayer of normal dielectrics, thus may lead to easy experimental demonstrations of non-magnetic ground plane cloak in either the microwave or the optical frequency range.
Acknowledgements
This work is supported by the National Basic Research Program of China (Grant No. 2004CB719800) and the National Nature Science Foundation of China (Grant Nos. 6090320, 60990322, 60671002, and 60801001).
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