Scattering suppression and wideband tunability of a flexible mantle cloak for finite-length conducting rods

In this section we analyze the scattering from an infinite conducting cylinder of radius $a$ surrounded by a mantle cloak of thickness ${{a}_{c}}-a$. The geometry and coordinate system under analysis are shown in figure 1, which shows an arbitrary impinging plane wave incident with angle $\left( \phi ,\theta \right)$ measured from the cylinder axis ${\bf \hat{z}}$ in vacuum $\left( {{\varepsilon }_{0}},{{\mu }_{0}} \right)$ under an ${{e}^{j\omega t}}$ time convention. A 2D cross-section of the geometry is shown in figure 1(b). The conductive cylinder is assumed to be covered by a dielectric shell coated with an ultrathin patterned conductive layer, and its finite-length realization, analyzed in the following section, is shown in figure 2.

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Since the thickness of the conducting cylinder and the metasurface are much larger than the skin depth at microwave frequencies, they may be well described as perfect electric conductors (PEC). Dielectric or conductive losses can be taken into account considering the loss tangent (${\rm tan} \delta$) and surface resistance [29], respectively. The dielectric layer provides insulation between the center conductor and the patterned conductive surface, and the choice of layer thickness and permittivity (${{\varepsilon }_{c}}$) is influential on the overall cloaking effect, bandwidth, loss, and realizability.

Since the patterned metal thickness and granularity are smaller than the wavelength of operation, we model the coating as a uniform impedance surface [6, 30, 31] across which the tangential electric field is continuous, i.e., ${\bf \hat{n}}\times {\bf E}(a_{c}^{+})={\bf \hat{n}}\times {\bf E}(a_{c}^{-})$ and satisfies

$$\begin{eqnarray}&&{\bf \hat{n}}\times {{{\bf E}}^{+}}\left( {{k}_{0}}{{a}_{c}} \right)={{\overline{{{\bf \bar{Z}}}}}_{{\bf s}}}\cdot {\bf \hat{n}}\times \left[ {\bf H}({{k}_{0}}a_{c}^{+})-{\bf H}({{k}_{c}}a_{c}^{-}) \right]={{\overline{{{\bf \bar{Z}}}}}_{{\bf s}}}\cdot {{{\bf J}}_{{\bf s}}}.\end{eqnarray} \tag{ 1 }$$

Equation (1) provides the relationship between tangential electric and magnetic fields on the anisotropic shunt surface impedance ${{\overline{{{\bf \bar{Z}}}}}_{{\bf s}}}$ surrounding the cylinder. This anisotropic impedance tensor can effectively model realistic deviations from an ideal isotropic cover, including the insurgence of polarization coupling. Even for normally incident TM plane waves, some cross-polarization coupling is expected, due to the granularity and possible asymmetries in the cover design; however, this effect is most visible at oblique incidence angles, for which it occurs even for ideally symmetric objects [26, 27]. The level of cross-polarization coupling depends on the complexity and number of subwavelength elements for a chosen surface, and it may be minimized by using simpler covers [22]. Using a square patch array, for instance, we may safely assume minimal polarization coupling and other non-ideal effects at normal incidence, which allows treating the surface impedance as a scalar ${{Z}_{s}}$.

Under this assumption, the total scattering width (SW) of the cylinder may be determined by the contribution from the most relevant scattering orders of the expansion:

$$\begin{eqnarray}&&{{\sigma }_{2D}}=\frac{2{{\lambda }_{0}}}{\pi } \sum \limits_{n=0}^{{{N}_{{\rm max} }}} {{\left| c_{n}^{TM} \right|}^{2}}\left( 2-{{\delta }_{n0}} \right),\end{eqnarray} \tag{ 2 }$$

where ${{\delta }_{n0}}$ is the Kronecker delta and ${{N}_{{\rm max} }}=5$ is used in the following, which is sufficient for the size of the considered rod. Here, $c_{n}^{{\rm TM}}$ are the electric scattering multipoles due to a normally incident TM plane waves, derived from Mie theory and written in the compact form $c_{n}^{{\rm TM}}=-U_{n}^{{\rm TM}}/\left( U_{n}^{{\rm TM}}-jV_{n}^{{\rm TM}} \right)$, where the determinants $U_{n}^{{\rm TM}}$ and $V_{n}^{{\rm TM}}$ can be obtained using an analysis analogous to the one described in [6, 13, 21]. Similar considerations apply for TE scattering contributions; however, here the TE scattering harmonics are negligible, since we consider conducting cylindrical targets of moderate electrical cross-section ($2a<{{\lambda }_{0}}/2$, where ${{\lambda }_{0}}$ is the free-space wavelength of operation), and are anyway not excited for normal TM incidence. Under these conditions, the omnidirectional scattering mode $n=0$ is the most significant term. Therefore, by properly tailoring a single cover, we may be able to significantly reduce the overall SW, leaving only higher-order residual and more directive scattering modes.

Consider now an infinite conductive rod with cross-section $2a={{\lambda }_{0}}/4$ at 3 GHz covered by a lossless dielectric shell (${{\varepsilon }_{c}}=4{{\varepsilon }_{0}}$) of thickness ${{a}_{c}}-a=1\;{\rm mm}$. Figure 3 shows the surface reactance (black line) required to cancel the dominant scattering order, calculated using equation (2). A suitably designed metasurface can synthesize the required surface impedance at the frequency of interest ${{f}_{0}}=3\;{\rm GHz}$, but it can meet the required reactance only at a single frequency, given the negative dispersion of the required impedance that would violate Foster's reactance theorem. This is consistent with the discussion in [16], which shows the necessity of non-Foster (active) metasurfaces to be able to track the required surface reactance dispersion over a broad bandwidth. In the figure we also show an ideal dispersionless cover that meets the required reactance at the figure of interest (red dotted line) and the one of a realistic patch array (blue dashed line) for an infinite (2D) cylinder, which synthesizes the required surface reactance at the design frequency, but that has a positive dispersion. The corresponding scattering widths for these three scenarios are shown in figure 4. Specifically, the figure shows the SW of the bare cylinder (solid black line) and of the same object after applying the ideal isotropic cover with dispersionless surface reactance $X_{S}^{{\rm Mie}}=-25\;\Omega$, tailored to meet the condition $\left| c_{n=0}^{{\rm TM}} \right|=0$. Scattering resonances are visible at higher frequencies, when $V_{n}^{{\rm TM}}=0$. These resonances are somewhat intensified since losses are not considered in this figure, but they are consistent with the general theorem derived in [32] that requires the overall scattering of a cloaked object, when integrated over all frequencies, to be necessarily larger than the uncovered object. This implies that, while we are able to suppress the scattering at the target frequency, we are necessarily forced to distribute it at other frequencies. We also show in figure 4 the response of an ideal non-Foster surface $\left( X_{S}^{{\rm NIC}} \right)$ that tracks the required surface reactance and significantly reduces the scattering across our band of interest. Such an active surface was first envisioned for dielectric rods in [16] and may be synthesized by loading the metasurface with negative impedance converter elements (NIC).

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Any of the proposed metasurfaces produce a clear scattering dip at 3 GHz in figure 4, at which the omnidirectional scattering term is nullified by the capacitive surface. The required reactance may be realized with different metasurface geometries, such as Jerusalem crosses, cross dipoles, horizontal strips, or patch arrays [22]. A passive patch array is chosen here for its simplicity, bandwidth performance, and design flexibility, with the possibility of using asymmetric patches for dual-polarization operation [23] and the possibility of easily loading it with circuit elements to potentially realize a non-Foster response. As a starting point for the design, the required capacitive value $X_{S}^{{\rm Mie}}=-25\;\Omega$ is synthesized based on an analytical model available in the planar case, with patch periodicity $D$ separated by gaps $g$, and dielectric backing ${{\varepsilon }_{c}}$ [22, 30, 31],

$$\begin{eqnarray}&&X_{S}^{{\rm patches}}=-\frac{\pi }{\omega {{\varepsilon }_{0}}({{\varepsilon }_{c}}+1)D{\rm ln} \left[ {\rm csc} \left( \frac{\pi g}{2D} \right) \right]}.\end{eqnarray} \tag{ 3 }$$

It is important to notice in (3) that there is no angle dependence for this averaged description of the dielectric-backed patch array for TM polarized wavefronts, since this model is based on a simplified dominant Floquet analysis [22, 30, 31]. When applied to the cylinder, the curvature and presence of a close-by ground plane is expected to introduce parasitic effects, modifying the surface response.

The blue dashed curves in figures 3 and 4 show the calculated dispersion of the corresponding surface impedance and scattering width for this metasurface. Not surprisingly, the scattering reduction in this realistic scenario is narrower in bandwidth than in the dispersionless scenario (red dotted curve), due to the fact that its reactance diverges more quickly from the required value for frequencies different than ${{f}_{0}}$. However, the possibility of actively loading such a surface is expected to dramatically extend the bandwidth of such passive covers, as shown in the non-Foster case. We discuss circuit loaded metasurfaces in the closing section of this paper.

Here we apply the previous theoretical results to the problem of cloaking a 3D conducting rod of length $L=23.1\;{\rm cm}$. The finite length introduces obvious differences compared to the infinite case, mostly associated with end effects and longitudinal standing-wave resonances of the rod. The surface impedance $X_{S}^{{\rm Mie}}=-25\;\Omega$, calculated by Mie theory (2), to suppress the scattering from an infinite cylinder (cf figure 2) is therefore not expected to be the same impedance required in the case of a finite-length rod. While Mie theory can provide good physical insights into the type of cover needed to suppress the scattering from dielectric and conductive rods of finite length, design refinements and optimization are required to minimize the total RCS of finite length rods, using the analytical results of the previous section as starting point.

In figure 5, we show the results of our optimization around the design formula (3), which leads to a patch array with dimensions $D=24.3\;{\rm mm}$, $g=1.43\;{\rm mm}$, ${{a}_{c}}-a={\rm 1}\;{\rm mm}$, backed by a flexible dielectric spacer with ${{\varepsilon }_{c}}=4$, capable of suppressing the scattering of our finite-length cylinder. In the figure we plot the total integrated RCS of the bare and covered object, showing large scattering suppression, over 14 dB (96%) at ${{f}_{0}}=2.9\;{\rm GHz}$. Figure 6 shows the calculated effective impedance across the band of interest for the optimized planar patch array, and compare the results extracted from full-wave simulations to the closed-form analytical model in (3). The effective surface impedance of the planar array is extracted using a simple transmission line approach, where the infinite planar array is represented by a shunt impedance (in the inset of figure 6) [31]. The comparison between (3) and the extracted impedance is excellent, especially at lower frequencies, due to the large patch density.

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Figure 6 essentially shows an estimate of the required surface impedance for the finite length rod, which is seen to be quite different from the one of the infinite case at the design frequency. The required additional surface capacitance is associated with the additional scattering produced by the rod truncation and the associated current distribution induced on the object. We have indeed verified that the finite-length bare cylinder scatters comparatively more than the infinite geometry. In comparison, our previous work on dielectric rods [13] showed that the infinite cylinder scenario provided a description closer to the finite-length case. It appears that the conductive cylinder provides additional challenges in this 3D case, which are addressed by fine tuning the analytical design with proper optimization.

To provide further insights into the scattering suppression mechanism of the optimized cover, figure 7 shows the calculated three-dimensional scattering patterns with and without cloak for a finite-length aluminum rod of length $L=23.1\;{\rm cm}$ at the design frequency 2.9 GHz. As evident in the figure, the bare rod (panel a) scattering is dominated by the omnidirectional TM harmonic ($n=0$); however, other scattering modes are clearly present due to its moderate cross-section and finite length. When covered by the optimized lossless mantle cloak (b), a strong suppression of this scattering term leaves a much reduced residual scattering, formed by a superposition of higher-order scattering harmonics. Panels (c–d) consider the presence of losses in the dielectric spacers, using realistic material models: low-loss ceramics or flexible dielectrics [33, 34] (c), and flexible Neoprene (d). In all cases we included the realistic finite conductivity of copper and aluminum. By integrating the calculated patterns we obtain a total 3D RCS respectively of: (a) $\sigma _{3D}^{{\rm bare}}$ = −17.2 dBm², (b) $\sigma _{3D}^{{\rm lossless}}$ = −31.3 dBm². In the lossy examples, the scattering reduction is slightly worsened to (d) $\sigma _{3D}^{{\rm high}-{\rm loss}}$ = −24.9 dBm², with larger contributions in the shadow region. This is fully consistent with the forward scattering theorem and power conservation, as the presence of absorption inherently requires an increase in forward scattering. These effects are further illustrated in figure 8, which considers the most relevant H-plane cut (cf figure 1 $\theta =90{}^\circ$). For low-loss covers, the scattering suppression all around the cloak is excellent, even in the forward direction, i.e., the cloak allows restoring the impinging field distribution even in the back of the obstacle. In the case of larger dielectric loss in the cloak spacer, the shadow is increased due to absorption, while the mono-static cross-section ($\phi =0{}^\circ$) decreases. Nevertheless, even with this lossy dielectric spacer we achieve an overall scattering suppression of nearly 8 dB, further showing the mantle cloak's resiliency to loss. As noticed in [35], this is due to the inherent robustness of scattering cancellation techniques, which are inherently non-resonant in their operation.

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Our near-field measurements show strong scattering suppression and near-complete restoration of non-uniform incident wavefronts excited by a 10 dB gain horn antenna placed close to the target $\sim 1.5{{\lambda }_{0}}$. These measurements were conducted by raster scanning the transverse electric field throughout a plane at the central height of the cylindrical target. A highly repeatable (30 micron resolution) automated robotic arm, with a short electric field probe was programmed to take field measurements with a spatial sampling resolution of $0.06{{\lambda }_{0}}$, and over a scan area of $2.5{{\lambda }_{0}}\times 4.8{{\lambda }_{0}}$. For more details of our robotic arm scanner setup, see [10, 13, 36]. Figures 9 and 10 show the measurements of three different test scenarios (free space, bare, and covered) at 2.98 GHz and 2.84 GHz for normal ($\theta =90{}^\circ$) and oblique ($\theta =70{}^\circ$) incidence, respectively. Normal incidence is chosen as our original excitation geometry, which ensures maximum scattering signature for the bare cylinder. We also show results for $\theta =70{}^\circ$ incidence to show the robustness of the cloaking mechanism to variations in excitation. Our E-field probe samples the total field around the cloaked and uncloaked object, showing excellent restoration of the impinging fields and shadow suppression, even in very close proximity to the object. In the boxed regions of each panel the probe scans the field directly above the device, explaining why these measurements appear out of phase with respect to the rest of each image. Raw complex transverse electric field data are imaged with arbitrary units in figures 9 and 10 to highlight the differences of the fields for each of the test scenarios. To illustrate this cloaking effect, a free space measurement is shown in the left column, with the bare (center) and covered (right) cases. It is clear that the incident field is well restored all around the cloaked object, making the device much less detectable, even in its very near field. It is particularly impressive to realize that the cloak was designed for a far-field plane-wave at normal incidence, and yet it performs well in our near-field experiments, despite the change in excitation, the potential for near-field coupling between horn antenna and object under test, and the fabrication errors of our in-house realization.

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In order to further characterize the mantle cloak performance, we have also performed an extensive campaign of bistatic far-field measurements. Two ETS-Lindgren 3115 ultra-wideband microwave horns were placed equidistant (1.5 m) to the measurement setup [37]. For H-plane, bistatic measurements, one horn remained stationary, while the other was moved around the device under test in 20° steps, while keeping the distance constant. E-plane bistatic measurements were taken by keeping the horns stationary, but now moving the device under test about its axis in steps of 20°. By measuring the bistatic returns of our three tests (background, bare, and cloaked geometries, consistent with our near-field measurements), we are then able to estimate the level of RCS reduction, or scattering gain, by normalizing the cloaked RCS by that of the bare case. This normalization technique, along with conventional time-gating and vector background subtraction [13, 38], allowed us to compensate for the non-anechoic nature of our testing facility, as well as possible small calibration issues due to our test setup and environment.

Figure 11 shows the bistatic scattering reduction in the H-plane $\left( \theta =90{}^\circ \right)$ in the back half space, comparing our measurements with full-wave simulations. Over 10 dB scattering reduction is shown with a highly stable frequency of operation. Notice that some of the sharp features seen in the simulations appear smoother in our measurements. This can be explained by the presence of dielectric loss in our prototype, as well as other fabrication imperfections, which tend to smooth the resonant response in the object. In our simulations we considered a low-loss dielectric spacer, to highlight the performance that may be achieved with lower loss. However, previous experience taught that exact matching of loss between simulation and experiment does not lead to better agreement, due to the larger effects of dimensional imperfections in manual construction [10]. As seen in the figures, despite losses, a strong and stable scattering non-resonant cloaking effect is clearly visible, and it matches our full-wave simulations well.

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Next, we consider the scattering features in the forward H-plane direction (figure 12). There is a consistent 5.5 dB measured scattering suppression behind the cloaked conductive cylinder. This is quite impressive, also in light of the stable frequency response. We also consider oblique incidence angles for the back and forward scattering half-planes along the E-plane in figure 13 $\left( \phi =0{}^\circ \right)$ and figure 14 $\left( \phi =180{}^\circ \right)$. Strong and stable scattering reduction is still observed around the design frequency, confirming our far-field measurements and simulations, and the robustness to incidence angle of the cloaking phenomenon. The small detuning in the cloaking response for large incidence angles stems from the fact that the realistic anisotropic surface impedance has an inherent dependence on angle of incidence clearly not captured in (3). Yet, even in this worst-case scenario, the detuning is quite limited.

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