1. Introduction

Since the experimental verification [1], the electromagnetic cloaks based on the metamaterial technologies have attracted great attention as a perfect stealth technology that is effective against both monostatic and bistatic radars. As Maxwell’s equations are invariant under a coordinate transformation, the cloaks designed by the transformation optics (TO) technique allow an electromagnetic wave to propagate through the transformed medium to match that propagating outside the medium. However, a major drawback of this method is that anisotropic constitutive parameters are inevitable in order to realize the transformed medium, which is the major limitation on the practicality of the method. Optical conformal mapping (CM) may be an alternative that does not require anisotropic constitutive parameters [2]. However, the cloaks based on it have limited practicality since they have relatively narrow operating bandwidths [3,4]. To overcome this limitation, mantle cloaks based on two-dimensional metamaterials have demonstrated wideband cloaking properties by successfully canceling the scattered fields [5–8].

Another popular cloaking technique is the ground-plane cloak, which transforms an arbitrarily-shaped object on the ground plane to a flat sheet, is another successful implementation of the cloaking technique [9]. This technique is based on the quasi-conformal mapping (QCM). Since the method retains orthogonality of the grids in the space after transformation, the electromagnetic fields in the physical space are almost identical as those in the virtual space without using anisotropic constitutive parameters. The isotropic parameters of it allow the invisibility to be achieved over a broad bandwidth [9–11]. Furthermore, a two-dimensional (2D) free-space cloak structure can be realized simply by mirroring the ground-plane cloak [12,13]. However, such a free-space cloak has the following disadvantages: the cloak is unidirectional since the cloak transforms a diamond shaped object into a flat sheet. Also, although the anisotropy of the refractive indices of the transformed medium remains low, it may not be negligible for the free-space cloak. Finally, the relatively large size of cloak structure may not be attractive for practical applications.

In this study, a design method for the free-space cloak is proposed to minimize not only the size of the cloak, but also the scattering cross section (SCS) over a broad bandwidth as well as for a wide range of incident angles. First, an approximated model for a free-space cloak is proposed which takes into account not only the shrinkage of the medium along the vertical direction but also its expansion along the lateral direction. Using this model, it is demonstrated that the performance deterioration due to neglecting the non-unity anisotropy factor can be compensated by the proposed method of scaling the refractive index profile of the cloak medium, which controls its gradient. Therefore, the method can be applied to a reduced-size free-space cloak of which the performance deterioration is much significant due to the size reduction. The proposed method achieves up to 71% reduction in SCS with as much as 78.3% size reduction compared to the original cloak, making it one of the most practical cloaks demonstrated thus far. Finally, the improved invisibility not only over a broad bandwidth but also in a wide range of incident angles is verified by comparing the SCS of the proposed scaled cloak with that of an unscaled cloak via full-wave simulated and experimental results, both of which are fabricated using a three-dimensional (3D) printer.

2. Approximated model of isotropic free-space cloak

Figures 1(a) and 1(b) show the virtual space, which is the 2D free space with a flat plate, and the physical space of the free-space cloak, respectively. The latter is designed by mirroring the ground-plane cloak based on the quasi-conformal mapping (QCM) technique. The width and height of the cloak as well as of the diamond object inside the cloak are w and h as well as w_obj and h_obj, respectively. The physical space is generated by minimizing the modified Liao functional and applying the slipping boundary condition at all boundaries [9,14]. The effect of the non-unity anisotropy factor of the ground-plane cloak can be modelled by an intermediate space, whose vertical axis is compressed by the bottom boundary that is shifted upward [15]. However, this intermediate space may be insufficient for a precise estimation of the performance of the free-space cloak constructed by mirroring the ground plane structure as shown in Fig. 1(b). When the slipping boundary condition is applied to the boundary of the diamond object that is longer than the width of this object, the expansion of the grids in the horizontal direction is inevitable. This has a strong effect on the refractive indices of the cloak medium, and therefore must be taken into account.

 

Fig. 1 Free-space cloak designed using quasi-conformal mapping (QCM). (a) Virtual and (b) physical spaces. (c) Proposed approximated model. (d) Flowchart to obtain approximated model.

Download Full Size | PDF

Also, the fact that the free-space cloaks developed from the ground-plane cloaks, even when they are properly designed with a low enough anisotropy factor [12,13], may not be explained with the intermediate space. In this work, the approximated model in Fig. 1(c) is proposed for an accurate analysis of this. Because the cloak based on the QCM technique controls the wave paths before and after the waves are reflected by the hidden object, the waves scattered from the vertex of the diamond object at x = ±w_obj/2 must be considered. Hence, the diamond object in the cloak is modelled as a hexagonal object which has a reduced height $h_{obj}^{\prime }<h_{obj}$. Because the transformation is dominant along the vertical axis, its width w_obj is maintained. Owing to the non-zero height $h_{obj}^{\prime }$, the vertical axis of the approximated model shrinks to δ_yh, where δ_y < 1 [15]. Most importantly, in this work, the width of the medium in the approximated model is wider than that of the physical space. This is achieved by setting the width to δ_xw, where w is the width of the physical space and δ_x > 1. By taking into account the width expansion of each grid in the physical space, the proposed method provides a more accurate model of the cloak.

The approximated model which consists of the hexagonal object surrounded by a homogeneous medium can be obtained through the recursive method shown in Fig. 1(d). Firstly, for a given h, the height of the hexagonal object $h_{obj}^{\prime }$ in Fig. 1(c) is predetermined as a non-zero initial value to calculate the shrink ratio along the y axis, ${\delta }_y=\left(h-h_{obj}^{\prime }\right)/h$. Secondly, the expansion ratio along the x axis is calculated by δ_x = δ_y × α. The anisotropy factor α is calculated by $\alpha ={\delta }_x^{\prime }/{\delta }_y^{\prime }$, where ${\delta }_x^{\prime }$ and ${\delta }_y^{\prime }$ are the expansion and shrink ratios of the width and height of the rectangular grids in Fig. 1(b) compared to those in Fig. 1(a), respectively. Although the anisotropy factor is inhomogeneous, the inhomogeneity is usually neglected. Thirdly, the constitutive parameters of the medium surrounding the hexagonal object are calculated using

To validate the accuracy of the proposed approximated model, the cloak demonstrated in Fig. 1(b) is designed for h, w, h_obj, and w_obj of 120, 104, 18, and 42 mm, which are 4λ₀, 3.47λ₀, 0.6λ₀, and 1.4λ₀ at 10 GHz, respectively, where λ₀ is the free space wavelength. In this case, the anisotropy factor α is 1.044, which comparable to previous cloak designs [9,11]. The final design parameters of the approximated model, which are obtained via the recursive method in Fig. 1(d), are δ_x = 1.0014, δ_y = 0.9592, n = 1.0204, and $h_{obj}^{\prime }=4.9\text{mm}$. Figure 2(a) shows the calculated isotropic refractive index profile of the cloak. Figures 2(b) and 2(c) show the simulated 2D scattered electric fields E_s for the incident plane wave of |E_inc| =1 V/m, without and with the cloak, respectively. Figure 2(d) shows the E_s of the approximated model. The full-wave simulated results are calculated using the commercial software COMSOL Multiphysics.

 

Fig. 2 Free-space cloak based on QCM. (a) Calculated refractive index profile. Simulated scattered fields of (b) diamond object (PEC), (c) cloak, and (d) approximated model. (e) Far-field scattered power densities of (b), (c), and (d).

Download Full Size | PDF

To better assess the validity of the proposed model in Fig. 2(d), the patterns of P_s,far are compared in Fig. 2(e). The excellent match between the power densities of the cloak and the approximated model especially in the forward direction validates the proposed model. By integrating P_s,far over 0 ≤ ϕ ≤ 2π, the total scattered power density can be calculated. From this, the normalized scattering cross section (SCS) can be defined as the ratio of the total scattered power density with and without cloak. The normalized SCSs are 0.888 and 0.898 for the cloak and the approximated model, respectively. The error between the two is 1.1%, showing an excellent match. A normalized SCS smaller than one is an evidence that the cloak effectively decreases the total scattered power density. However, as shown in Fig. 2(e), the forward scattering increases with the cloak than without it. This is largely due to neglecting the non-unity anisotropy factor α.

3. Scaling of refractive index profile

In the previous section, the proposed approximated model identified two major factors that contribute to deterioration of the free-space cloaking performance: the non-unity refractive index and the non-zero thickness of the transformed hidden region which can be modeled with a hexagonal object. In this section, a method is proposed that compensates the two. Furthermore, a substantially reduction in the cloak size can be achieved since the associated performance deterioration can be compensated by the same method.

To investigate the effect of the homogeneous medium with a non-unity refractive index in the approximated model, its refractive index is scaled by a factor β that is the inverse of n, i.e. β = 1/n. This is equivalent to removing the homogeneous medium which was generated by neglecting non-unity anisotropy factor α, leaving only the hexagonal object in the free space. This reduces the normalized SCS from 0.888 to 0.776. This removes the effect of non-unity refractive index, but that of the hexagonal object with a non-zero thickness still remains.

Figure 3 shows the far-field scattered power densities P_s,far in the forward direction and the normalized SCSs of the approximated model, calculated for various β. Results suggest that there is an optimal β which minimizes the forward scattered field or its normalized SCS. In Fig. 3 it is seen that the optimal scaling factor that minimizes the forward scattering of the approximated model is β_opt = 0.987. Thus, by scaling the refractive index profile of the cloak medium with β_opt = 0.987, the forward scattered power density is reduced by 75% as shown in Fig. 4(a). Also, the normalized SCS of the cloak is reduced to 0.711. This is because scaling adjusts the gradient of the refractive index profile such that diverging of the electromagnetic waves is minimized as they scattered by the hexagonal object. Therefore, both the effect of the non-unity refractive index, but also the scattering by the hexagonal object can be controlled by the scaling factor β, and can be optimized.

 

Fig. 3 Far-field scattered power densities in forward direction and normalized SCSs of approximated model vs β.

Download Full Size | PDF

 

Fig. 4 Results of free-space cloak based on QCM and its approximated model after scaling refractive indices by scaling factor β = 0.987. (a) Far-field scattered power densities. (b) Effect of optimal scaling on normalized SCS of free-space cloak over broad frequency range.

Download Full Size | PDF

Figure 4(b) compares the normalized SCSs in the 1–30 GHz range before and after scaling by β_opt. Electromagnetic cloaking is achieved from 1 GHz up to 10 GHz for both of the cloaks. However, the normalized SCS of the unscaled cloak increases rapidly beyond 11 GHz, while that of the scaled cloak remains below one up to 26 GHz, showing an unprecedented cloaking bandwidth property. This shows that simply by optimizing the refractive index of the medium, the cloaking performance can be improved substantially over a very wide bandwidth, without requiring complex structures [16,17].

Although the electromagnetic cloaking performance can be enhanced over a broad bandwidth by optimally scaling the refractive index profile of the cloak structure, it remains to be theoretical. This is because for practically all cases, the scaled cloak medium requires a wide range of refractive indices, including those below one. This inherently limits the bandwidth since their realization requires resonant-type metamaterials [1,18]. An option is to approximate the parts with refractive indices below one to a free space, i.e. n = 1 [12].

Figure 5 shows the refractive index profiles of the reduced-size free-space cloaks and the ray tracing simulated results at 10 GHz. The refractive index profile of Fig. 5(a) is taken from Fig. 2(a) in the ranges of −21 mm< x <21 mm and −20.3 mm< y <20.3 mm, where those below one are approximated to one. The effect of this approximation can be compensated with optimizing the height of the cloak. Here, the optimum height is determined as 40.6 mm which reduces the normalized SCS to 0.408. This is even lower than the normalized SCS of 0.711 of the cloak with the refractive index scaled optimally. This indicates that the effects of the two factors that determine both the path and phase of the waves propagating through the cloak, the anisotropy factor α and the gradient of the refractive index, can be controlled effectively to enhance the cloaking performance by truncating the cloak structure.

 

Fig. 5 Reduced-size cloaks. (a) Refractive index profile and ray simulation results for h = 40.6 mm. (b) Optimally-scaled refractive index profile and ray simulation results for h = 26 mm. (c) Far-field scattered power densities.

Download Full Size | PDF

Despite the substantial reduction in the SCS, the ray tracing results still shows the electromagnetic waves diverge out as they pass through the cloak. This suggests that the cloaking performance still remains to be improved. This can be compensated by scaling the refractive index with an optimum factor. The cloak can be miniaturized further in this process, since the waves travel along a path in the forward direction rather than along a path that spreads out. In fact, the optimal height of the cloak is h = 26 mm with β_opt = 1.259, which minimizes P_s,far to −57.2 dB/m² in the forward direction with a normalized SCS of 0.258.

The ray simulation results in Fig. 5(b) show that the proposed method of scaling maintains more rays to travel in the forward direction as they propagate through the cloak. Although there is a limitation that the rays reflected near the vertex on the right-hand side of the diamond object are not controlled in the forward direction due to the refractive indices approximated to one, the significant reduction in the scattered fields especially in the forward direction can be seen in Fig. 5(c). This is also confirmed by the reduction in the normalized SCS from 0.408 to 0.258.

However, the cloaking performance cannot be compensated when the size reduced excessively. This is because then the original function of controlling the wave paths is lost completely, and cannot be recovered. Also, when the cloak is designed originally with an excessively low height, the anisotropy factor α increases too high such that the original function of the coordinate transformation significantly deteriorated when α is neglected. This results in a normalized SCS close to one, which is difficult to reduce with the proposed method of scaling. Designing the cloak with a sufficiently large h retains reasonably low α to maintain the original function of the coordinate transformation, while the performance can be enhanced by optimally scaling the refractive index.

Table 1 summarizes the forward far-field scattered power densities and the normalized SCSs of the cloaks at 10 GHz before and after size reduction, with and without optimally scaling the refractive index. The normalized SCS of the cloak originally designed with h = 120 mm can be reduced further from 0.888 to 0.258 with the optimization of the height to 26 mm followed by the scaling of the refractive index profile with the optimum scaling factor β_opt = 1.259. This is a 71% reduction in the normalized SCS over the unscaled original cloak, with a 78.3% reduction in the size.

Table 1. Comparison of scattered power densities in forward direction and normalized SCSs at f =10 GHz.

View Table

4. Experimental verification

For experimental verification, the cloak is fabricated using drilled-hole type unit cells. To this end, the refractive index profile of the cloak is discretized using square unit cells, whose refractive indices are sampled at their central locations. The length of the unit cell is 2.45 mm, which is shorter than λ_min/10, where λ_min is the shortest wavelength in X band. After then, the thickness and the radius of drilled-hole type unit cells are designed using full-wave simulations and an algorithm that extracts the permittivity and permeability of unit cells [19] to provide the refractive index that matches that of designed value. The cloak structure is fabricated using a 3D printer of which the printing material has a relative permittivity of 2.95 with a dielectric loss tangent of 0.025 at 10 GHz.

In the previous section, it is seen that the optimal height of the scaled cloak is h=26 mm. However, the demonstrated cloak in this section is designed with h = 34 mm. This is the minimum h that can be fabricated with refractive indices lower than 1.7, which is the highest that can be realized using the 3D printer and the printing material. For h = 34 mm, the optimal scaling factor is β_opt = 1.055, which minimizes the far-field scattered power density P_s,far in the forward direction at 10 GHz. For comparison, a cloak with h = 39.2 mm is also fabricated, which is the optimal height that minimizes P_s,far in the forward direction without scaling the refractive indices. This is somewhat lower than h = 40.6 mm in the previous section, which is due to the non-zero loss tangent of the printing material. The fabricated unscaled and proposed cloaks are shown in Figs. 6(a) and 6(b), respectively. Then the scattered fields of the fabricated cloaks are measured using an electromagnetic field scanner, shown in Fig. 6(c).

 

Fig. 6 Fabricated reduced-size free-space cloaks and measurement system. (a) Unscaled (h =39.2 mm) cloak, (b) proposed (h =34 mm) cloak, and (c) electromagnetic field scanner.

Download Full Size | PDF

Figure 7 shows the measured results at 10 GHz. For comparison, simulated results are also shown. Figures 7(a)–7(c) show the simulated scattered fields of the diamond object and with the unscaled and the proposed cloaks for the incident plane wave of |E_inc| =1 V/m, respectively. Figures 7(e)–7(g) show the measured scattered fields of each case. The scattered fields can be plotted by subtracting the fields measured before and after placing an object in the field scanner. All the measured scattered fields are normalized so that the range of the field intensities matches those of the simulated versions. The measured scattered field Fig. 7(g) is in a great match with the simulated result in Fig. 7(c). More importantly, comparing Fig. 7(g) with Figs. 7(e) and 7(f) reveals that the measured scattered field is reduced considerably, verifying the effectiveness of the proposed cloak.

 

Fig. 7 Simulated and measured results of unscaled and proposed reduced-size free-space cloaks at 10 GHz. Simulated scattered fields of (a) diamond object (PEC), (b) unscaled cloak, and (c) proposed cloak. (d) Normalized far-field scattered power densities. Measured scattered fields of (e) diamond object (aluminum), (f) fabricated unscaled cloak, and (g) proposed cloak. (h) Normalized far-field scattered power densities. (i) Simulated and measured normalized SCSs in the X band. (j) Normalized SCSs vs incident angles at 10 GHz.

Download Full Size | PDF

Figures 7(d) and 7(h) show the normalized far-field scattered power densities P_s,far calculated from the simulated and measured results, respectively. The normalized P_s,far for measured results are calculated using the surface equivalence theorem [20]. The scattered magnetic fields H_s along the x and the y axes are estimated using the measured E_s fields along the z axis via the discretized Maxwell’s equations. Then, the far-field scattered power density P_s,far can be calculated using the equivalent electric and magnetic surface current densities on a rectangular boundary −200 mm< x <200 mm and −200 mm< y <200 mm. For comparison, the calculated P_s,far is normalized to the peak value of the diamond object. Results show that the normalized SCS of the proposed cloak from the measured results is 0.401, which is 15% lower than that of the unscaled version, 0.472. The 15% decrease of normalized SCS from the experimental results matches greatly with the 15.5% decrease from 0.414 to 0.35 in the simulated results.

Figure 7(i) compares the simulated and measured normalized SCSs in the X band, demonstrating enhanced invisibility over a broad bandwidth. The simulated 3dB cloaking bandwidth of the proposed cloak is 41.9% from 8.3 GHz to 12.7 GHz, which can be enhanced even further, at the cost of increased SCS at the design frequency. The measured 3 dB cloaking bandwidth is relatively narrower than the simulated bandwidth. Major factors that contribute to the discrepancy include discretization of the cloak structure [21], the small air gap between the sample and the top plate of the measurement system that cause additional scattering [18], and reduce the average refractive index of the cloak material [22], tolerance of the 3D printer [23] in dimensional accuracy and the electromagnetic properties of the printed material depending on the printing conditions, limited bandwidth of the measurement system including the probe patterns, and approximation and simplification due to the minimum printable thickness of 0.5 mm for the 3D printer used in this work. Nevertheless, an enhancement in the invisibility over a relatively broad bandwidth can still be verified. Finally, Figure 7(j) shows that the proposed cloak provides improved invisibility over a wide range of incident angles at 10 GHz. Results reveal that the SCS remains at substantially low levels for incident angles up to ϕ = 20°, and stays mostly under one even at an incident angle as high as ϕ = 90°. This suggests that the unidirectional cloak property can be overcome with the proposed method, which is a major limitation of the cloaks based on QCM [12,13].

5. Conclusions

A design method has been proposed to improve the invisibility of an isotropic free-space electromagnetic cloak based on the quasi-conformal mapping. By optimally scaling the refractive index profile of the cloak, its performance can be enhanced. Consequently, the normalized scattering cross section (SCS) can be reduced dramatically over a broad bandwidth. This is because the optimal scaling factor can compensate for the deterioration of the cloaking performance due to neglecting the non-unity anisotropy factor. At the same time, a substantial reduction in size is also possible. Based on the proposed method, the height of the cloak can be reduced by 78.3%, while the SCS can be reduced by 71% at 10 GHz. Together with the extremely simple design process, these properties make the proposed cloak one of the most practical cloaks reported thus far. Finally, the proposed method is verified experimentally by fabricating using a 3D printer. Compared to the unscaled cloak, the enhanced cloaking performance of the proposed cloak is verified not only over a relatively broad bandwidth but also in a wide range of incident angles. An inherent limitation of the proposed cloak is that its effectiveness is limited to transverse electric (TE) waves. This is a common problem for this type of cloaks that leaves 1D objects after transformation [24,25]. The improvement of polarization independence remains as a future work. The application of the proposed method to other optical devices such as lenses [26] and antennas [27] remains as potential areas of study.