Understanding the functionality of an array of invisibility cloaks

This paper describes the operation and the interaction of cloaking devices when they are periodically arranged. The main focus is on analyzing the dispersion relation of structures, which should mimic that of the vacuum in the ideal scenario. We distinguish between two cloaking mechanisms: cloaks designed within the framework of transformation optics and cloaks designed on the basis of the scattering cancellation technique. The difference between the two approaches is that the first operates independently of the frequency by assuming nondispersive materials, whereas the latter is designed to operate for a single frequency. Our numerical simulations demonstrate that arrays made of such invisible dielectric obstacles act like a homogeneous medium with permittivity and permeability equal to those of the surrounding medium, except for a countable set of eigenfrequencies associated with Mie resonances for the former type of (transformation-based) cloak. For the latter type of (plasmonic) cloak, the marginal scattering response indicates the effectiveness of cloaking arrays of individual particles. Our spectral (Floquet-Bloch) approach to cloaking might be useful to implement realistic applications such as biomedical sensing, noninvasive probing, sensing networks, or multiobjective camouflaging.

Figure 1

(a) Schematic of an array of cloaked dielectric cylinders in the two-dimensional (2D) geometry (for illustration we consider the example of plasmonic cloaks); this crystal is infinitely extended in all three directions. (b) First Brillouin zone $\Gamma XM$ where $\Gamma =(0,0)$, $X=(\pi /a,0)$, and $M=(\pi /a,\pi /a)$, with $a$ being the period of the array; dispersion diagrams will be computed along these directions. (c) Polarized plane wave incident on the structure composed of the cloaked cylinders.

Figure 2

Dispersion diagram for a doubly periodic array (pitch $a$) of cloaked electromagnetic obstacles (in dashed line) with a shell of inner and outer radii of $0.2a$ and $0.4a$, respectively, in the first Brillouin zone $\Gamma XM$ where $\Gamma =(0,0)$, $X=(\pi /a,0)$, and $M=(\pi /a,\pi /a)$ (schematic representing the unit cell of the array is drawn in the inset (a) of the figure). The triangles represent the dispersion in vacuum and are given for comparison as well as to test the efficiency of the cloak. In the insets (a) and (b), the first two eigenmodes of the structure are depicted. We show here a snapshot of the electric field in the unit cell at point $X$, where $X=(\pi /a,0$), and the streamlines that seem to be curved when passing through the cloaking device as predicted by the geometrical transformations. The inset (c) gives the fourth mode at point $M$, $M=(\pi /a,\pi /a)$, which is a localized one corresponding to the resonance of a cavity of radius $R_1$ [a Mie resonance called an almost-trapped eigenstate in the context of quantum cloaks by Greenleaf et al. (Ref. 30)]. It is important to note that we found a discrete set of such dispersionless modes at higher frequencies, all of which were associated with Mie resonances of the cavity (monopole, dipole, quadrupole, and so forth).

Figure 3

(a) Radial dependence of reduced electromagnetic parameters ${\mu }_r$ and ${\mu }_{\phi }$ obtained from the geometric transform in blue (light gray) and red (dotted line) and corresponding staircase approximation ${\mu }_r\left(i\right)$, $i=1,...,20$, for the layered cloak consisting of ten cells of alternating isotropic homogeneous layers (black solid line). The relative permittivity ${\varepsilon }_3$ is constant and equal to $0.5$. (b) The same as Fig. 2 but for a cloak made of the multilayered system described by Eq. (4). The dispersion of ideal cloak (dashed line), vacuum (dots) are compared to those of the cloak of Fig. 3b. (dotted-dashed line).

Figure 4

Scattered electric energy from the obstacle in vacuum (a) and when the cloak is turned on (b). The upper panels of (a) and (b) show the the electric field component $E_3$, while the lower ones represent the total energy. (c) Radar cross sections of the bare (solid line) and the cloaked cylinder (dashed line).

Figure 5

(a) Dispersion of a plasmonic cloak in blue (dashed) line compared to vacuum in solid line and the bare obstacle in dashed-dotted line. Simple case where the permittivity of the medium is taken to be constant, ${\varepsilon }_{\mathrm{ideal}}=-3.5454+\delta i$, where $\delta$ is a small imaginary part, to mimic loss. (b) Zoom on the direction $\Gamma$$X$, where the scattering is governed by the dipole term, and where we expect the structure to be invisible: the dispersion curves of the core-shell system almost coincide with those of vacuum, and the agreement is better in the zero-frequency limit (quasistatic case) within the homogenization regime as indicated by the dotted ellipses in (a).

Figure 6

Dispersion of a plasmonic cloak. (a) The permittivity of the shell follows the well-known Drude model (dashed line gives the dispersion of the ideal cloak, green (light gray) one stands for vacuum, while dashed-dotted line corresponds to the bare obstacle and black (dark gray) gives the dispersion of the Drude’s cloak). (b) Zoom on the region where $\text{Re}\left(\varepsilon \right)={\varepsilon }_{\mathrm{ideal}}=-3.5454$ and where the Drude shell is supposed to cloak perfectly the dielectric obstacle. This can be observed, since the two curves have an intersection point at $\omega /{\omega }_p\approx 0.47$ [as indicated by the arrow in (a)]. (c) Complete dispersion diagram showing the normalized frequency versus the modulus of the Bloch vector in the $X\Gamma$ direction for the bare obstacle (red), the vacuum (green), and the obstacle with the ideal shell (blue) and with the Drude shell (black).