Electromagnetic analysis of arbitrarily shaped pinched carpets

We derive the expressions for the anisotropic heterogeneous tensors of permittivity and perme- ability associated with two-dimensional and three-dimensional carpets of an arbitrary shape. In the former case, we map a segment onto smooth curves whereas in the latter case we map a non convex region of the plane onto smooth surfaces. Importantly, these carpets display no singularity of the permeability and permeability tensor components, and this may lead to some broadband cloaking.

💡 Research Summary

The paper presents a comprehensive transformation‑optics framework for designing two‑dimensional (2‑D) and three‑dimensional (3‑D) “carpet” metamaterials of arbitrary shape that avoid the singular material parameters typical of conventional invisibility cloaks. Starting from the well‑known relation ε₀ = ε T⁻¹ and μ₀ = μ T⁻¹, where T = J Jᵀ/det(J) and J is the Jacobian of the coordinate map, the authors construct linear mappings that stretch a line segment (in 2‑D) or a planar domain (in 3‑D) onto smooth curves or surfaces while leaving the outer boundary unchanged.

In the 2‑D case the map (2)‑(4) sends the interval a < x < b on the x‑axis to a prescribed curve y₁(x) and keeps the upper curve y₂(x) fixed. The resulting inverse metric tensor T⁻¹ has only five non‑zero components, and its eigenvalues λ₁ = 1/α, λ₂, λ₃ are shown analytically to be strictly positive for any differentiable y₁, y₂ with α = (y₂−y₁)/y₁ > 0. Consequently the effective permittivity and permeability tensors are finite everywhere, eliminating the inner‑boundary singularity that plagues “blow‑up” cloaks. A concrete example using exponential‑sinusoidal profiles demonstrates that λ₂, λ₃ remain within a modest range (≈1.5–3.5), which is favorable for practical implementation.

The electromagnetic response for p‑polarized fields (H₃ component) is governed by Eq. (7). Finite‑element simulations (COMSOL) of a plane wave and a 45° Gaussian beam show that the carpet‑covered deformed mirror reproduces the scattering of a flat mirror, whereas the bare bump produces a markedly different field pattern. The authors discuss how the required anisotropic ε can be realized with dilute metallic wires and how the magnetic response (μ₃₃ = λ₁) can be achieved with split‑ring resonators, referencing earlier experimental demonstrations.

For the 3‑D carpet, the map (8)‑(10) lifts a planar domain D into a surface z₁(x,y) while preserving the top surface z₂(x,y). The Jacobian now contains ∂z/∂x₀ and ∂z/∂y₀, leading to a symmetric T⁻¹ with seven independent entries. Again, all eigenvalues are positive and bounded, as illustrated by iso‑contour plots (Fig. 4). The full vector Maxwell system is solved using curl‑conforming second‑order edge elements and perfectly matched layers to emulate an unbounded domain. Simulations of normal incidence and oblique (π/4) incidence of both plane waves and Gaussian beams confirm that the 3‑D carpet, even with a highly non‑convex bump, reflects the incident field almost identically to a flat ground plane, while the uncovered bump scatters strongly.

The paper concludes by emphasizing that the one‑to‑one geometric transform guarantees non‑singular material parameters, enabling broadband cloaking without the extreme parameter values required by traditional transformation‑optics cloaks. Practical realization could employ thin metallic wires for ε anisotropy and split‑ring resonators for μ, technologies already demonstrated at microwave and optical frequencies. Recent experimental progress on carpet cloaks near optical frequencies suggests that the proposed arbitrary‑shape designs are within reach. Overall, the work advances the state of the art by providing a general, mathematically rigorous, and numerically validated method for designing singularity‑free, broadband carpet cloaks of any shape.