According to the Equivalence Principle, any distribution of electromagnetic fields can be generated within a closed region by suitable electric and magnetic sources lying on its bounding surface. In electromagnetic theory, these sources have been used as fictitious analytical tools. However, this has changed in recent years with the development of “Huygens's metasurfaces,” which are subwavelength thin sheets of artificial active and/or passive elements. With proper configuration, these elements can physically approximate the aforementioned sources, enabling precise manipulation of electromagnetic waves in extraordinary manners. Given their potential, this work aims to provide an overview of Huygens's metasurfaces with a focus on cloaking applications. After an exposition of general operation and design principles, both active and passive Huygens's cloaks are described.

Considered a foundation of classical electromagnetics, the “Equivalence Principle,” a generalization of the Huygens's principle, postulates that any electromagnetic field distribution within a region can be represented by equivalent electric and magnetic sources.1 Conventionally, these sources are considered fictitious tools, which help describe wave behavior. However, this has changed with recent advances in artificial materials and the development of so-called Huygens's metasurfaces (HMSs).2,3 Composed of subwavelength-scaled, composite elements (“meta-atoms”) arranged into thin sheets, HMSs can either actively or passively recreate the aforementioned sources. Along with providing a physical analog to theory, the ability to recreate these sources on demand enables HMSs to arbitrarily synthesize and control electromagnetic waves.

Here, we aim to present an overview of passive and active HMSs in the context of a cloaking paradigm. This is fitting as some of the earliest reported HMS concepts have focused on cloaking.4 General theory and design principles for HMSs are first formulated. These are then used to develop active and passive electromagnetic cloaks capable of concealing various electromagnetic scatterers. Finally, the effectiveness of these cloaks is verified through simulations and experiments.

For simplicity, we will restrict the problem to two-dimensions (/z=0). The HMS, denoted by the dashed line in Fig. 1, divides the problem into two regions. These are labeled V1 and V2 and contain electromagnetic fields E1,H1and{E2,H2}, respectively. Due to its deep subwavelength thickness, the HMS can be theoretically modeled as an infinitesimally thin sheet of spatially varying surface electric currents (Js) and/or surface magnetic currents (Ms). These so-called “Huygens's sources” give rise to discontinuities in the tangential electromagnetic fields according to2 

Js=n̂×(Ht,2Ht,1)n̂×ΔHt,Ms=(Et,2Et,1)×n̂ΔEt×n̂,
(1)

where the subscript t denotes the tangential field components.

FIG. 1.

Schematic of the 2D, two-region problem.

FIG. 1.

Schematic of the 2D, two-region problem.

Close modal

The currents Js and Ms can also be related to the average tangential electromagnetic fields across the HMS according to the Bianisotropic Sheet Transition Conditions (BSTCs),5 

[JsMs]=[Y¯¯seK¯¯emK¯¯meZ¯¯sm]·[Et,avHt,av],Et,avEt,1+Et,22,Ht,avHt,1+Ht,22,
(2)

where Y¯¯se and Z¯¯sm are 2 × 2 tensors defining the effective surface electric admittance and surface magnetic impedance, while K¯¯em and K¯¯me characterize the bianisotropic coupling.

The surface currents may interact with the electromagnetic fields to emit and/or dissipate power. In that case, the HMS can be classified as active and/or lossy. Conversely, passive and lossless HMSs are reactive and, thus, neither consume nor generate real power. Such surfaces must satisfy5 

Re{Y¯¯se}=Re{Z¯¯sm}=Im{K¯¯em}=Im{K¯¯me}=0.
(3)

Furthermore, a surface consisting of only reciprocal components must satisfy5 

Y¯¯seT=Y¯¯se,Z¯¯smT=Z¯¯sm,K¯¯emT=K¯¯me.
(4)

For practical reasons, (4) is assumed throughout this review.

To design a general HMS for arbitrary wave manipulation, the desired fields on either side of the device are first stipulated. Next, an appropriate design in the form of implicit electric and magnetic currents or explicit surface parameters can be found by solving (1) or (2), respectively.

To construct an electromagnetic cloak, we first assume that the HMS completely encloses the target while isolating it from all external illuminations (known a priori). Consequently, we have

Et,2=Eti,Ht,2=Hti,Et,1=Ett,Ht,1=Htt,
(5)

where the superscripts i and t denote incident and transmitted fields, respectively. The absence of reflections renders the target undetectable from all external observers.

There are two approaches to synthesize the field discontinuities described by (5). These are examined subsequently, starting with the active approach, followed by the passive alternative. For simplicity, we assume, henceforth, that the electromagnetic fields are strictly TMz-polarized (E=ẑE). Cloaks that generate nonradiative external reflections in the form of cross-polarized surface waves6 are noted but not discussed further in this paper.

According to (1) and (5), the HMS cloak needs to support the surface currents,

Js=n̂×(HtiHtt),Ms=(EtiEtt)×n̂.
(6)

In an active cloaking approach, these spatially varying continuous current sheets are discretized into n electric (ρe=Jslh) and magnetic (ρm=Mshl) dipole moments.7 Here, h represents the target height, whenever applicable.4 Each moment can then be physically realized by radiating sources,7 such as conventional antennas with inter-elemental spacing l, serving as the constitutive meta-atoms. Although other antennas may be used, the most direct means of realizing ρe and ρm are through current driven dipoles (Ie=ρe/h) and loops (Im=ρm/(jωμA)) of area A [Fig. 2(a)].

FIG. 2.

Possible physical meta-atom designs for (a) active and (b) passive HMS cloaks.

FIG. 2.

Possible physical meta-atom designs for (a) active and (b) passive HMS cloaks.

Close modal

The impressed sources of an active HMS cloak grant us full control over the surface current densities regardless of the illumination. Therefore, with a priori knowledge of Eti and Hti, we are at liberty to choose the fields in volume 1. The most natural choice is to set Ett=Htt=0 such that the target enclosed by the cloak, no matter its size, shape, and material parameters, will not interact with any electromagnetic fields. Hence, it will not cause scattering.

The inherent reconfigurability of their impressed sources allows active HMS cloaks to be dynamically tuned in response to different illuminations. However, this flexibility comes at the cost of constant power requirement and the need for complex control circuits.

For applications involving fixed illumination, such as electromagnetic interference reduction,8 it is possible to passively implement the HMS cloak without any active or lossy components, leading to easier fabrication and integration processes.

Substituting (5) and (6) into (2), we obtain an underdetermined system of equations that can be solved to assess the required surface parameters. For the TMz problem considered herein, we can fix several superfluous degrees of freedom by assuming bi-isotropy. However, this is still not sufficient to guarantee passivity and losslessness. To do that, we can impose the additional constraint of local (pointwise) power conservation on the tangential fields,9 

Re{Eti×(Hti)*}=Re{Ett×(Htt)*}.
(7)

With the known incident fields, we can solve (7) for the required transmitted fields to conserve local power. This now ensures that the surface parameters are passive and lossless.

The required transmitted fields, as determined by (7), can be interpreted as auxiliary fields, which balance the power flow at every point on the metasurface.9 These fields are permitted to penetrate into the target if it consists of a dielectric material. As such, interaction between the target and the auxiliary field is confined within region 1 and remains unobservable from region 2 [Fig. 3(a)]. In contrast, targets made from perfect electric conductors (PECs) do not permit internal fields and, thus, demand special treatment. One possible approach is to introduce a dielectric coating between the cloak and the target. As depicted in Fig. 3(b), this additional region admits nonzero transmitted fields {Et,Ht}, which facilitates the satisfaction of (7) on the HMS, subject to the constraint of vanishing Et along the PEC.10 

FIG. 3.

Schematics of passive HMS cloaks for (a) penetrable dielectric targets and (b) impenetrable PEC targets.

FIG. 3.

Schematics of passive HMS cloaks for (a) penetrable dielectric targets and (b) impenetrable PEC targets.

Close modal

Regardless of the type of target, the theoretically derived surface parameters can be realized using a modified version of the passive meta-atoms previously used to construct planar HMSs.11 This original planar design can also be generalized to build curved surfaces.10 As depicted by the illustrative example in Fig. 2(b), the meta-atom consists of an asymmetric stack of three curved electrically polarizable scatterers. In this particular design, the outer layers behave like capacitors to z-directed electric fields, whereas the middle layer can be inductive or capacitive depending on its geometry. Notably, since each layer contains just a conductive pattern, the overall HMS cloak does not require any active or lossy elements.

Finally, we note that the proposed approach is applicable to targets of all electrical sizes, distinguishing it from the well-known passive mantle cloaks intended to operate within the quasi-static regime.12 

Having established the basic theory, we now present experimental measurements and simulation results, which demonstrate the validity of both cloaking approaches.

Figure 4 shows the 2D layout and corresponding experimental apparatus of an active HMS cloak. Here, a metallic five-sided polygonal target (measuring: L = 272.73 mm, W = 218.18 mm, h = 38 mm, and ϕ=45°) is illuminated at θ=30° incidence by a monopole positioned ρ=600 mm from the target's center. Next, an h = 38 mm wide parallel-plate waveguide is formed by sandwiching the target between two 1626 × 1094 mm aluminum plates. This enforces 2D conditions and suppresses any non-TEM modes as plate spacing is less than half a wavelength at the operational frequency of 1.2 GHz. Finally, radiation boundaries are synthesized by lining the waveguide's perimeter with absorbing foams.

FIG. 4.

Active HMS cloak: layout and experiment.

FIG. 4.

Active HMS cloak: layout and experiment.

Close modal

The incident electric field radiated by the monopole source is a cylindrical wave described by

Eti=ẑE0H0(2)(k|ρ(xtar,ytar)ρ|),
(8)

where H0(2) is the zeroth-order Hankel function of the second kind, k the free-space wavenumber, ρ(xtar,ytar) an observation point on the target's surface, and ρ the source location.14 Since the target is conductive, all electric surface currents are shorted out (Js=0), meaning that only magnetic sources are required to realize the active HMS. Following the formulation previously derived, the magnetic surface current required to cloak the target is15,16

Ms=n̂×ẑE0H0(2)(k|ρ(xtar,ytar)ρ|).
(9)

To construct the cloak array, 15 evenly spaced monopoles are mounted 14 mm from the target's surface.13 The close proximity of these monopoles to the target results in their electric currents being mirrored across the conductive plane, creating a virtual current loop.17,18 This element design also reduces the cloak's profile and improves robustness.

We plot the measured total electric field inside the waveguide, with and without the HMS cloak, in Figs. 5(a) and 5(b), respectively. When the incident wave impinges on the uncloaked object [Fig. 5(a)], the resultant scattering manifests as distortion near the illuminated lower right surface of the target. Additionally, a shadow region forms on the opposite side of the illuminated area due to the target occluding the impinging wave. Conversely, enabling the cloak cancels out scattering and restores the original incident field. This can be seen in Fig. 5(b) where frontal/side distortion is reduced and the shadow region has largely disappeared. To provide a quantitative performance metric, Fig. 5(c) shows the measured bistatic radar cross section (RCS) of the target, referenced to the frame depicted in Fig. 4. It can be seen that scattering suppression is achieved over the entire azimuth with an average reduction of 8.6 dB.

FIG. 5.

Experimental results for active cloaking of a polygonal aluminum cylindrical target with 30° incidence: (a) cloak off, (b) cloak on, and (c) bistatic RCS. Simulated results for an active multi-target cloak with the polygonal target illuminated at 30° incidence: (d) cloak off, (e) cloak on, and (f) bistatic RCS. Simulation results for passive cloaking of a quasitriangular PEC cylinder: (g) cloak off, (h) cloak on, and (i) normal power density at the HMS. Parts (a)–(c) reproduced with permission from P. Ang and G. V. Eleftheriades, Sci. Rep. 10, 1–11 (2020).;13 Copyright 2020 Authors, licensed under a Creative Commons Attribution (CC BY) license.

FIG. 5.

Experimental results for active cloaking of a polygonal aluminum cylindrical target with 30° incidence: (a) cloak off, (b) cloak on, and (c) bistatic RCS. Simulated results for an active multi-target cloak with the polygonal target illuminated at 30° incidence: (d) cloak off, (e) cloak on, and (f) bistatic RCS. Simulation results for passive cloaking of a quasitriangular PEC cylinder: (g) cloak off, (h) cloak on, and (i) normal power density at the HMS. Parts (a)–(c) reproduced with permission from P. Ang and G. V. Eleftheriades, Sci. Rep. 10, 1–11 (2020).;13 Copyright 2020 Authors, licensed under a Creative Commons Attribution (CC BY) license.

Close modal

Another method of configuring this active cloak is to calculate the weights directly from incident field measurements. This enables operation in complex environments such as the simulated (in ANSYS HFSS) multi-target scenario in Fig. 5(d). Here, the same polygonal target is illuminated at θ=30° incidence by a monopole positioned ρ=600 mm from its center. However, a second metallic, circular target (r = 112 mm) is positioned 600 mm from the polygon's center and cloaked with an additional 15 monopole elements. A simulated empty waveguide is then used to obtain 30 incident field measurements at positions on the targets' surfaces corresponding to cloak element locations. Figure 5(e) confirms that weights calculated using this method are effective at hiding both targets with an average combined scattering reduction of 15.3 dB [Fig. 5(f)].

Next, in order to substantiate the developed passive cloaking concept, we present a local power-conserving bi-isotropic HMS cloak designed to conceal an electrically large quasi-triangular metallic cylinder from the fields radiated by an electric line source (with a frequency of 4.4 GHz). As hinted by a previous work, the cross section of the HMS cloak need not match that of the target.19 In fact, it is sometimes beneficial to have a differently shaped cloak, as its associated dielectric layer may provide enhanced structural rigidity. Nonetheless, it is noted that a minimal overall profile is achieved when the cross sections of the HMS and the target are parallel.

As a proof of concept, we present a cloak with a quasi-square cross section. To allow better visualization of the auxiliary fields, the permittivity of the dielectric layer is selected to be 1; its thickness is also exaggerated, resulting in an unnecessarily thick profile. The HMS is numerically implemented in COMSOL Multiphysics using field-dependent electric and magnetic surface current densities.19 

In Fig. 5(g), we show the simulated total electric field without the cloak. As expected, the metallic target generates a significant amount of reflections and casts a shadow. Figure 5(h) depicts the fields with the cloak present. Here, the wavefronts emanating from the line source remain unperturbed. Despite the highly intricate appearance of the transmitted auxiliary fields, an external observer will not perceive any scattering. To explicitly demonstrate the passivity and losslessness of the cloak, we plot the normal power density inside (S) and outside (S+) the HMS in Fig. 5(i), as a function of the azimuthal angle ϕ. A perfect match between the two power profiles means that the total power density StotS++S is identically zero. Hence, the surface neither generates nor absorbs power.

To conclude, we presented a general overview on the theory of Huygens's metasurfaces, along with their application to the design of active and passive electromagnetic cloaks. Next, practical realization schemes for both classes of HMSs were discussed. Experimental results for active cloaking and full-wave simulation results for passive cloaking confirm the effectiveness and robustness of HMSs for realizing electromagnetic invisibility.

This work was financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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