##### 4.2.1. Theoretical Analysis

In several practical applications, especially in the radar community, 2D geometries and elongated objects may be of particular interest for cloaking applications. Similar to the spherical geometry and analogous to Equations (7)–(10), suitably chosen geometrical and electromagnetic properties of plasmonic covers may realize significant scattering suppression in the case of an infinite cylinder.74 Here, we consider a general model for cylindrical objects, which includes the effects of arbitrary incidence angle, and which may be applied to cloaking of 3D elongated objects.45 In this case, we consider an oblique plane wave impinging on a coated infinite cylinder at an angle *α* (Figure 3b). The radii of core and shell are *a* and *a*_{c}, respectively, and the electric and magnetic properties are given in each region with the same parameters used in the previous section. The scattering coefficients can be written similar to Equation (4),74 with determinants

- (11)

where *J*_{n}(·) and *Y*_{n}(·) are cylindrical Bessel and Neumann functions of order *n*,78 respectively. The characteristic impedance and wavenumber *k*_{l} are defined for each region *l*. Oblique incidence arises in the expression of the transverse wavenumber , where β = *k*_{0}cos (α). Similar to the spherical case, the scattering coefficients *c*^{TM}_{n} are written in terms of Equation (11), and *c*^{TE}_{n}, associated to TE-polarized cylindrical harmonics, can be obtained by electromagnetic duality. Also in this case, cloaking may be achieved by enforcing *U*^{TM}_{n} = 0 and/or *U*^{TE}_{n} = 0 and, as a quantitative measure of the cloak's ability to significantly reduce the scattering of a cylindrical object, we consider the total scattering width (SW)79

- ((12))

In the long-wavelength limit *k*_{0}*a*, *k*_{c}a_{c} ≪ 1, the scattering coefficients in Equation (11) are uncoupled and weakly dependent on the angle of incidence. Therefore, akin to the conditions in Equations (7) and (10),74 scattering cancellation of the dominant scattering multipoles for magnetodielectric cylinders can be written as

- ((13))

Cloaking of thin perfect electric conducting (PEC) or conducting cylinders is more challenging than conducting spheres. The reason is that the dominant scattering coefficient *c*^{TM}_{0} cannot be cancelled in the quasistatic limit due to the induced axial conduction currents. Nevertheless, significant reduction of all other scattering terms can be achieved by properly choosing the plasmonic cover constitutive parameters and thickness to cancel the subsequent multipolar terms. Under the quasistatic limit, these conditions are given by74

- ((14))

Notice that Equation (14) requires the use of a magnetic permittivity cover to cancel the most significant scattering terms, other than *c*^{TM}_{0}, in the quasistatic limit. However, it is important to note here that the cancellation of the dominant *c*^{TM}_{0} scattering term may be possible for a thin PEC cylinder, but the cover composition and thickness solution space must be studied using the full dynamic equations in Equation (11) rather than under the quasistatic limit.

Cloaking of cylindrical or other anisotropically-shaped objects presents other challenges: inherent polarization coupling arises for oblique illumination and when considering finite length and low conductivity, additional axial resonances usually arise, which may affect the overall cloaking performance.74, 82, 83 However, for thin cylindrical objects with large aspect ratios, the scattering coefficients in Equation (11) are virtually uncoupled and are only weakly affected by oblique excitation.74

We have previously shown74 that, by using a single-layered plasmonic cloak, significant scattering reduction may be obtained for long dielectric cylinders with cross-sections up to a half wavelength, and the scattering suppression is still quite promising up to one wavelength. Properly designed plasmonic cloaks provide significant scattering suppression even for large obliquely incident angles and arbitrary polarization. It is noted that, while the relative reduction of scattering may be less at near-grazing angles than for normal incidence, this is not necessarily relevant, since the strongest scattering occurs at normal incidence for parallel polarization.

##### 4.2.2. Metamaterial Design and Experimental Realization

A number of experimental cloaking verifications have been recently reported for cylindrical targets. Yet these approaches have been usually limited to 2D environments or to reducing the scattering only over a limited angular spectral range, such as in the case of bumps on a mirror or a ground plane (as in the carpet cloaking technique).37–44, 46–53 Photothermal deflection, utilizing the *mirage effect*, has also been shown as an alternative venue to suppress the scattering for specific positions of the observer.55 We recently reported the first 3D demonstration of metamaterial cloaking in free-space, applicable to arbitrary excitation angle and observer's position, by applying the plasmonic cloaking approach.45 A metamaterial design at microwave frequencies has confirmed that a single-layer plasmonic shell can drastically reduce the total SCS of a finite-length, moderate-size, dielectric cylinder by suppressing the dominant *c*^{TM}_{0} multipole.45 The realized 3D plasmonic cloak was particularly attractive due to its ease of fabrication, relative invariance to near-grazing incidence, and minor deviation in performance compared to an infinite cylinder.74 Additionally, since the scattering cancellation mechanism is inherently nonresonant, it has proved quite tolerant to fabrication imperfections and Ohmic losses.

It is well known that natural materials, such as noble metals and some semiconductors at infrared and visible frequencies show Drude-type dispersions due to atomic and electronic transitions at higher energy.79 To implement a realistic Drude-like metamaterial at lower frequencies,84 a cylindrical structure composed of *N* metallic implants embedded in a host dielectric ϵ_{diel}, as illustrated in Figure 3c, when excited with electric field parallel to the axis, provides the effective permittivity

- ((15))

As illustrated in Figure 3c,d, a properly designed cylindrical metamaterial cover can be homogenized as an equivalent bulk material of permittivity given by Equation (15). By selecting the number of metallic implants, inner and outer radii, and the host dielectric, we may be able to tailor the required effective permittivity over a large range of values. In the practical design, the number of metallic implants cannot be too small, since this may also reduce the effectiveness and isotropy of the metamaterial. Also, the metamaterial disc should be fairly thin compared to the operating wavelength, since unwanted resonances may appear across the host dielectric region *a*_{c} − *a* for large cover thicknesses, especially for a large dielectric constant of the filling material *ϵ*_{diel}.85 In addition, a small gap needs to be considered at each of the interfaces *r* = *a* and *r* = *a*_{c}:84

- ((16))

where *r* is the radial distance relative to the cylinder center. These recommended “virtual interfaces” in Equation (16) allow sufficient distance to neglect the effects of evanescent Floquet modes at the disc interface. These concepts were put forward and verified numerically74, 84 and later, demonstrated by the first experimental 2D plasmonic cloak40 and experimental 3D plasmonic cloak.45

Here, we provide a few numerical full-wave simulations, using CST Microwave Studio,86 with the aim of validating the effectiveness of this plasmonic cloaking technique for finite-length, moderate-size, dielectric cylinders in free space. We use the same cloak design as our recent experimental paper,45 but we provide here additional numerical and experimental results, providing further physical insights into its functionality. The metamaterial-assembled plasmonic cloak was designed to significantly reduce the scattering of a dielectric cylinder of length *L* = 1.8*λ*_{0} and diameter , where *λ*_{0} is the free space wavelength at the design frequency *f*_{0} = 3 GHz. The length of the dielectric object was chosen to be about two wavelengths, which may induce axial resonances especially in the case of oblique illumination and TM^{z} polarization.45 The dielectric cylinder has constitutive parameters (3ϵ_{0}, μ_{0}) and we consider realistic dielectric losses with tan *δ* = 0.002|_{3 GHz}. A plasmonic cylindrical cloak is designed, following the model of Figure 3, for TM^{z} impinging wave (with electric field polarized parallel to the cylinder axis) at normal incidence, which produces the largest total SCS. The cloak is tuned to cancel the dominate scattering multipole *c*^{TM}_{0}, which dominates the total SCS for these dimensions. To avoid the excitation of higher-order resonances across the shell thickness *a*_{c} − *a*, a conformal plasmonic cover of was implemented. The metal assumed for the metallic implants is copper, with finite conductivity σ_{Cu} = 5.8 × 10^{7}S/m.

In order to suppress the dominant dipolar scattering, the required effective permittivity of the plasmonic shell is ϵ_{c} = −13.6ϵ_{0}, in accordance with (7). To achieve this permittivity value, a conformal metamaterial cloak was implemented with *N* = 8 vertical metallic strips, which are inserted into a filling delectric of ϵ_{diel} = 16ϵ_{0} of thickness *a*_{c} − *a* = 0.0375*λ*_{0}. The number of vertical metallic strips was necessary to maintain the isotropy of plasmonic shell, while maintaining a relatively low filling dielectric. Additionally, the radial length of the metallic fins was designed according to Equation (16) to obviate the effects of evanescent Floquet modes at the two radial interfaces.

**Figure** **4** shows full-wave simulations comparing the scattering of the uncloaked (Figure 4a) and cloaked (Figure 4b) dielectric cylinders in the azimuthal plane at the design frequency, when excited by a plane wave propagating along the axis, with electric field polarized along the axis. The near-field field distribution of the cloaked object Figure 4b shows that the phase front of the impinging plane wave is almost completely restored even just outside the cloak, and higher-order scattering coefficients contribute very little to the overall residual scattering. The total SCS far-field pattern of the cloaked cylinder in Figure 4d shows a dramatic reduction compared to the uncloaked case Figure 4c**,** close to 12 dB at 3.13 GHz. The pattern of the uncloaked dielectric cylinder shows a large forward scattering (shadow), which is almost completely suppressed when covered by the plasmonic cloak. In Figure 4d, the residual scattering (zoomed in to see it in a different scale) shows minimal higher-order multipole contributions. It is important to point out that losses were considered in the simulations of Figure 4, and there is clearly no significant detrimental effect on the cloaking performance, which further demonstrates the robustness of this scattering-cancellation technique.

Next, we show in **Figure** **5** the frequency dispersion of three significant scattering metrics: total SCS, forward-, and back-scattering cross-sections of the cloaked object normalized to the uncloaked one.79 The normalized total SCS shows a relevant scattering reduction over a relatively broad bandwidth , defined as the fractional bandwidth over which the total scattering is reduced compared to the uncloaked object. Distinct SCS dips are observed for all three cases. For the back-scattering, a slight shift in dip position and narrower BWs are obtained. The main reason of this apparent bandwidth reduction is that the back-scattering is already quite low for an uncloaked dielectric object, as most of the scattering is in the forward direction (shadow). Very impressive scattering suppression is indeed found in the forward direction, ensuring much reduced shadow from the cylindrical object.

The metamaterial shell in Figure 4 was recently fabricated and measured in both the near-field and far-field regions to characterize its scattering profile in 3D.45 The host dielectric was assembled by separate sectoral pieces of Cuming Microwave C-Stock dielectric material,87 and the vertical metallic implants were made from precisely cutting copper tape strips with 66.0 μm thickness. The dielectric sectors are held together by Teflon end caps, which showed no significant change in the cloak operation. The “virtual interfaces” were optimized for *δ*(*a*) = 0.98 mm and *δ*(*a*_{c}) = 1.3 mm. We present here the measured scattering profiles in the near-field of the uncloaked and cloaked dielectric cylinders at the cloaking frequency, as shown in **Figure** **6**, when illuminated by a Gaussian beam produced by a horn antenna placed in the near-field of the objects. Each one of the targets is also compared to wave propagation in free space (i.e., the test environment without any object). Figure 6 shows the total near-field distribution for normal (*α* = 90°) and oblique incidence (*α* = 60°). In each near-field panel, the scan area in the *y-z* plane is a square azimuthal cut of 2.16*λ*_{0} × 2.16*λ*_{0} for normal incidence, and 3.10*λ*_{0} × 3.10*λ*_{0}, for oblique incidence, with equal step sizes of Δ*y* = Δ*z* = 0.0674*λ*_{0}, where *λ*_{0} = 3.11 GHz. At normal incidence, an electric-field probe was programmed to scan at a constant height of approximately 80% of the height of the object under test. The boxed regions in each panel show where the object under test was placed, and the probe was programmed to scan directly above that region in the indicated area to avoid collisions. In these areas, obviously, the fields appear out of phase compared to the rest of the image, as they refer to a different incidence plane. For oblique incidence, the measurement plane is approximately 50% of the object's height and, in the corresponding panels, the dashed line is below the scan plane and the solid line is above it. The data acquisition system, composed of an Agilent 5071C vector network analyzer (VNA), triggers a Fanuc LR Mate robot arm to make high precision and highly repeatable samples of the electric field in the scan area.

Strong scattering reduction is confirmed by comparing uncloaked and cloaked near-field images. Remarkably, it is seen that for both incident angles the cloak provides excellent cancellation of the scattered fields, as compared to the object alone. In the very near-field, the total fields as measured by the electric-field probe appear almost exactly as the “free space” background measurement in the third column. It should be also stressed that the illuminating Gaussian field pattern contains a more complex angular spectrum than that of the plane wave used in our full-wave simulations, ensuring that the cloak functionality is independent of the illumination form for TM polarization.

**Figure** **7** shows far-field bistatic RCS measurements along the elevation plane for the cloaked and uncloaked dielectric cylinders; here the scattering gain (in decibels) is specifically defined as the ratio between the RCS of covered and uncovered cylinder, which clearly quantifies the level of scattering suppression by the plasmonic cover. The inset of Figure 7 shows the cross-section of the real plasmonic cloak, as previously described and characterized.45 In our measurement, two broadband microwave horns were placed in the far-field of the target (*R* = 14*λ*_{0} = 140 cm), where the transmitting antenna was fixed at broadside (ϕ_{tx} = 0°, θ_{tx} = 90°), while the receiving antenna was positioned at several elevation angles (ϕ_{rx} = 0°, −90° ≤ θ_{rx} ≤ 30°) to characterize the back- and forward-scattering of the cloaked and uncloaked targets. At each of the viewing angles, relevant RCS suppression is observed around 3.13 GHz. It is remarkably evident here that one may obtain more than 10 dB of suppression of the forward scattering or “shadow” (ϕ_{rx} = 0°, θ_{rx} = −90°), which are, in general, the most difficult to suppress. Such a considerable forward scattering reduction further validates the diminishment of the total SCS, consistent with the optical theorem.79 Here, a slight frequency shift is also observed for increased angles of incidence. As expected, this shift is due to the TM-TE cross polarization coupling for dielectric cylinders.75, 82, 83 Based on these near- and far-field scattering measurements, in addition to evidence already previously presented,45 we have confirmed the robustness and effectiveness of the plasmonic cloak to arbitrary illumination.