## Abstract

Inspired by a general theorem on non-radiating sources demonstrated by Devaney and Wolf, a unified theory for invisible and cloaking structures is here proposed. By solving Devaney-Wolf theorem in the quasi-static limit, a weak solution is obtained, demonstrating the existence of Anapole modes, Mantle Cloaking and Plasmonic Cloaking. Beyond the quasi-static regime, a strong solution of Devaney-Wolf theorem can be formulated, predicting general non-scattering devices based on directional invisibility, Transformation Optics, neutral inclusions and refractive index continuity. Both weak and strong solutions are analytically demonstrated to depend on the concept of contrast, mathematically defined as a normalized difference between constitutive parameters (or wave-impedance property) of a material and its surrounding background.

© 2016 Optical Society of America

## 1. Introduction: optical invisibility and Devaney-Wolf theorem

One of the first papers on experimental invisibility was presented in 1902 by Wood [1], starting with this sentence: “A transparent body, no matter what its shape, disappears when immersed in a medium with the same refractive index and dispersion”. As a matter of fact, one of the first theories to inspire (and fabricate) invisibility systems in optics was simply this: the continuity of the refractive index property. Invisibility conjectures have been initiated even earlier than 1902 [2], but robust theories started developing after Veselago postulated negative constitutive parameters in materials [3]: first investigations have shown how these new exotic properties have been exploited to induce harmonic cancellation effects for scattered waves, giving rise to the so called Plasmonic Cloaking technique [4]. With the mixed use of extreme and low material properties, the class of invisible systems was enriched by new architecture schemes, able to exclude fields from their interior and simulatenously switch off any radiation just outside their domain of definition: it was the begininning of Transformation Optics [5, 6]. By choosing the proper constitutive parameters, anisotropic material functions became responsible of bending waves around the hidden region without perturbing the sourroundings: for example, permittivity and permeability for optical cloaking [5, 6] and thermal conductivity and specific heat capacity for thermal cloaking [7]. The recent trend on non-scattering devices has moved from the design of volumetric inclusions forming *metamaterials* to patterned impedance loading creating *metasurfaces*. In particular, Scattering Cancellation theory has been enriched by Mantle Cloaking [8], presented in the literature as a complete different approach from the Plasmonic theory [4]: both based on field expansion in harmonics, the cancellation mechanism is now related to a proper surface impedance *Z _{s}* rather than less-than-one or negative nature of constitutive parameters

*ε*or

*µ*.

In this theoretical framework, we want to demonstrate a possible unification of all individual efforts between ancient and quite recent invisibility and cloaking theories in electromagnetics and suggesting a general theory for non-scattering structures based only on constitutive parameters and impedance models. It will be shown how existing recent results in the literature of cloaking can be predicted without solving any field decomposition or wave equation but just knowning the main constitutive parameters involved and wave impedance as defined in the scattering phenomenon.

As a starting point towards a general theory for invisibility and cloaking structures, we get inspired by Theorem III on non-radiating sources prooved in 1973 by Devaney and Wolf [9] and, throughout this work, it will be referred as Devaney-Wolf theorem. A necessary and sufficient condition for a current distribution
$\overrightarrow{j}({r}^{\prime})$, localized in *r*′ ∈ Ω, to be nonradiating is “the vanishing of certain Fourier components *J _{ω}* of the transverse part of the current density; namely those components for which
$\left|\overrightarrow{k}\right|=\omega /c$” [9]. This statement corresponds to

In the following sections, two kind of solutions will be obtained by inspection of Devaney-Wolf theorem and with analytical passages based also on the proper definition of the unit polarization vector $\widehat{p}$. Each solution will be expanded into a set of ancient invisible systems and into a set of recent cloaking theories, here generalized for arbitrary shape geometries in arbitrary homogeneous backgrounds.

## 2. The weak solution: Anapole modes, Mantle Cloaking and Plasmonic Cloaking

Here, we first consider into the Devaney-Wolf theorem the quasi-static limit: for *ω* → 0, the exponential function in Eq. (1) approaches unity and the equation becomes

*weak solution*of Devaney-Wolf theorem. In electromagnetic theory, the electric source $\overrightarrow{j}({r}^{\prime})$ can be written in terms of Volume and Surface Equivalence principles [10], where $\overrightarrow{j}({r}^{\prime})$ is explicited as a function of electric (or magnetic) field configurations supported by certain combinations of volumetric (or surface) materials, i.e.,

*ε*(

_{r}*r*′). Dielectrics are able to sustain (polarization) volumetric sources ${\overrightarrow{j}}_{v}$ in a Σ region, due to their electric susceptibility function

*χ*(

_{ε}*r*′)

*≡ ε*(

_{r}*r*′) −1 (n.b., dimensionless), with an electromagnetic response directly proportional to their internal total electric field ${\overrightarrow{E}}_{t}$. On the other hand, surface electric sources are sustained by scatterers made up of metallic materials, with surface sources ${\overrightarrow{j}}_{s}$ computed as the jump of total magnetic fields ${\overrightarrow{H}}_{t}$ above (

^{+}) and below (

^{−}) the defined surface contour Γ, with its proper normal $\widehat{n}$. The insertion of Eq. (3) and Eq. (4) into the weak solution can give rise to proper non-radiating conditions depending on the dielectric and/or metallic materials used to build the overall non-scattering system.

As a first example, we consider the case of a single homogeneous dielectric particle, thus able to sustain volumetric sources: for this reason, Eq. (3) can be inserted into Eq. (2) and the condition becomes

*p*-component of the total electric field. Due to the fact that the electric susceptibility is taken out from the integral sign, due to its homogeneous value in the overall domain Σ, the zero condition on scattered fields is mainly addressed to the topology of ${\tilde{E}}_{p}$ within the dielectric particle.

Depending on the 2D (e.g., cylindrical) or 3D (e.g., spherical) structure under investigation, two types of configurations can be imagined for cancellation effects in quasi-static regime, directly relating the volumetric source
${\overrightarrow{j}}_{v}({r}^{\prime})$ with the average total electric field component
${\tilde{E}}_{p}$. As reported in Fig. 1, for an incident wave traveling energy along the radial direction (i.e.,
$\widehat{\rho}$ for cylindrical or
$\widehat{r}$ for spherical coordinate systems), a cancellation can take place for a certain *p*-component of the field (in a scalar sense) with positive and negative orientation (e.g., assuming
$\widehat{p}\equiv \widehat{z}$ in 2D systems), as in Fig. 1 (left), whereas cancellation effects can be achieved (in a vector sense) with closed-loop configurations (e.g., assuming
$\widehat{p}\equiv \widehat{\theta}$ in 3D systems), as in Fig. 1 (right).

Phase/antiphase contributions in the source term
${\overrightarrow{j}}_{v}({r}^{\prime})$ can be implemented with piecewise homogeneous dielectric/plasmonic strucutures [4], whereas pure dielectric nanoparticle can sustain a loop-induced mode: named as *anapole*, it was experimentally observed in a recent paper [11] with corresponding electric field lines that are closed. This is consistent with the weak solution, due to the fact that, in quasi-static conditions, the simple averaging of volumetric sources within the nanoparticle is sufficient to *qualitatively* describe the non-scattering contribution in the far-field as addressed by Devaney-Wolf theorem. We are currently investigating the formation of the anapole mode beyond quasi-static regime.

Once fixed the geometry (domain Σ or contour Γ) and object to be hidden (dielectric or metal), a cloaking system can be build with composite (e.g., metal-dielectric) or non-composite materials (e.g., dielectric-dielectric), just solving the weak solution for these two different design architectures. From now on, we will consider 2D cylindrical structures with reference system $(\widehat{\rho},\widehat{\varphi},\widehat{z})$ as in Fig. 1 (left), even if this methodology can also be applied to 3D systems.

As a second example, we investigate a composite cloaking system made up of an arbitrary shape dielectric object with homogeneous permittivity *ε*_{1}, located in the area *A*_{1}, to be cloaked with a metallic patterned surface of arbitrary contour *C*. The weak solution can be splitted in two contributions (dielectric object plus patterned metallic cloak regions), giving

*A*

_{1}or contour

*C*): thus, a condition for the ratio of the average tangential fields follows, component by component. For incident fields with TM

*polarization, with $\widehat{z}$ defined along the cylinder’s axis, the polarization unit vector becomes $\widehat{p}\equiv \widehat{z}$ and the normal $\widehat{n}\equiv \widehat{\rho}$, thus the result reads*

_{z}*A*

_{1}(=

*πa*

^{2}) and contour

*C*(= 2

*πb*) give rise to the same equation as found in [12] for object radius

*a*and cloak diamater 2

*b*. However, as noticed in [13], the dispersion of ${Z}_{s}^{TM}$ in Eq. (7) does not obey to Foster’s theorem [14] for cloaking natural dielectrics having ${\chi}_{{\epsilon}_{1}}>0$, whereas artificial dielectrics possessing ${\chi}_{{\epsilon}_{1}}<0$ can be hidden by a reactive (capacitive) network of Foster type. Retracing the mathematical passages up to now, it is clear how the explicit

*ω*-dependence in the surface impedance comes out from the volumetric source in Eq. (3).

For this reason, as a third example, there is the possibility to employ a non-composite cloaking system, using materials with the same constitutive property (e.g., dielectric-dielectric systems, magnetic-magnetic devices, etc.), in order to get rid of the *ω*-dependence during the splitting of the integral in the weak solution. For dielectric-dielectric systems (e.g., one object and one cloaking layer), the integral for volumetric sources is splitted as

*A*

_{1}and

*A*

_{2}are the domains in which object susceptibility ${\chi}_{{\epsilon}_{1}}$ and cloak susceptibility ${\chi}_{{\epsilon}_{2}}$ are defined, respectively. Due to the subwavelength condition, the electric field function ${\overrightarrow{E}}_{t}({r}^{\prime})$ is so slowly varying both in

*A*

_{1}and

*A*

_{2}that both integrals get simplified, due to the fact that fields within object and cloak domains are equal. As a consequence, the main responsible for the scattering cancellation is now uniquely related to susceptibility terms and the final condition for non-composite cloaking structures reduces to

*A*=

*A*

_{1}+

*A*

_{2}(e.g., ${\chi}_{{\epsilon}_{1}}$ in

*A*

_{1}and ${\chi}_{{\epsilon}_{2}}$ in

*A*

_{2}). In the case of dielectric-dielectric systems, the weak solution becomes Eq. (9) and we can recognize it as a generalization of Plasmonic Cloaking [4] for arbitrary shape scatterers: in order to achieve Eq. (9), a condition of the type ${\chi}_{{\epsilon}_{1}}{\chi}_{{\epsilon}_{2}}<0$ is needed. For electric (or magnetic) scattering case, the same conditions on permittivity (or permeability) as found in [4] are obtained particularizing the geometry for circular shape cylinders with

*A*

_{1}=

*πa*

^{2}and

*A*

_{2}=

*πb*

^{2}−

*A*

_{1}.

As reported in Fig. 2, it seems that the use of composite or non-composite cloaking systems is *interchangeable*: (I) a dielectric-dielectric cloaking systems is dispersionless but it needs opposite sign in the susceptibility term for cancellation effects; (II) a metal-dielectric structure defines on its touching boundary a surface impedance but its dispersion behaviour is of non-Foster type for the cloaking problem [13]. This is the reason why the effects of a plasmonic structure can be implemented without using negative materials but employing composite metal-dielectric structures with positive permittivity but limited frequency performances, as demonstrated recently [15].

Here, inspired by the literature on Inverse Scattering Problems [16], we propose to *extend the concept of susceptibility* towards other constitutive or impedance parameters in terms of a dimensionless parameter, named as the *contrast*, i.e.,

*P*is the property of the background (e.g., permittivity, permeability, wave impedance, wavenumber) and

_{b}*P*is the

_{m}*same*property in the material under investigation.

The power of such general theory resides on the fact that completely ignoring the scattered field expansion and without solving any wave equations, in quasi-static conditions it is possible to build a simple invisibility system just concentrating only on the involved material properties. For an arbitrary 2D (or 3D) invisible system localized in the area *A* (or volume *V*), the general condition for quasi-static cloaking depends only on the amount of contrast values (i.e., not only defined in terms of permittivity) weighted by their domain of definition, i.e.,

*χ*

_{1}in the area

*A*

_{1}(or volume

*V*

_{1}) has to be hidden by a cloak with contrast

*χ*

_{2}in the area

*A*

_{2}=

*A*−

*A*

_{1}(or in a volume

*V*

_{2}=

*V*−

*V*

_{1}).

In subwavelength condition, non-composite structures of any arbitrary shape need to have positive/negative behaviour in the contrast parameters of object/cloak materials for cancellation effects (i.e., areas and volumes are always defined positive): however, negative contrasts do not necessary mean negative constitutive parameters as deducible from the definition in Eq. (10). In background scenarios with high material parameters (e.g., dielectric-filled waveguides with *ε _{b}* = 6

*ε*

_{0}[15]), it is straightforward to build negative contrasts with common materials (e.g., bakelite

*ε*= 5

*ε*

_{0}).

As Veselago has done in [3] only for the permittivity and permeability properties, there is the possibility to extend this concept to other parameters, also adding an active/passive nature to the considered material, as represented in Fig. 3.

## 3. The strong solution: directional invisibility, Transformation Optics, neutral inclusions and Wood’s invisibility

The main limitation of the weak solution is related to its frequency range of validity, dictated by the quasi-static limit. However, by inspection of the Devaney-Wolf theorem, another solution exists which is frequency-independent, i.e.,

and here it will be referred as the*strong solution*of Devaney-Wolf theorem. The reason behind such names for these solutions is now clear: Eq. (2) is the mathematical weak form of Eq. (12). The strong solution, which appear to be trivial, can have two different interpretations.

One aspect is related to the polarization unit vector $\widehat{p}$, which is implicitly dependent from the wavenumber unit vector $\widehat{k}$, due to the orthogonal condition

From this dependence, it can be imagined a device for which the strong solution is valid for a certain direction ${\widehat{k}}_{1}$, but it is non-zero for another impinging direction ${\widehat{k}}_{2}$, for example. Such exotic functionality or*directional invisibility*can be implemented with a particular design using metasurfaces as reported in [17] and it will be described in terms of strong solution in the next section.

The second aspect is related to reformulate the strong solution in terms of wave impedance with a zero contrast formalism. Considering as a reference case a complete background scenario defined everywhere with its proper wave impedance *Z _{b}*(

*ω,m*), for each angular frequency

*ω*and scattering harmonic index

*m*, a zero contrast impedance can be defined in a tensor formalism as

*free*propagation, with no equivalent sources $\overrightarrow{j}({r}^{\prime})$ induced anywhere, whereas other non-trivial solutions can exist looking at Eq. (14) and consider, for example, a medium with both dielectric and magnetic properties

*ε*and

_{b}*µ*, according to a transformation matrix ${\underset{\_}{\underset{\_}{T}}}_{H}$ and ${\underset{\_}{\underset{\_}{T}}}_{E}$, respectively. From the general contrast definition in Eq. (10), the zero difference between the new intrinsic impedance

_{b}*Z*′ and background intrinsic impedance ${Z}_{b}=\sqrt{{\mu}_{b}/{\epsilon}_{b}}$ directly solves Eq. (14) if the condition is and, as a consequence, the contrast impedance tensor achieves zeros in each considered direction.

As a matter of fact, the zero contrast impedance tensor can be interpreted as the passage from a complete background scenario to a cloaking system, where the intrinsic impedance remains an invariant of such transformation [18]: the strong solution for the wave impedance can be recognized to be Transfromation Optics [5, 6]. We are currently developing the contrast impedance tensor not for the intrinsic values but for the near-field impedance, as ratio of electric and magnetic fields polarized according to the unit vector
${\widehat{p}}_{E}$ and
${\widehat{p}}_{H}$, respectively. Only in the far-field limit at *r*^{*}

*Z*and it depends only on the constitutive parameters of the medium. However, in the near-field, the shape of the incoming wave (e.g., plane, cylindrical, spherical) has to be taken into account, thus even a complete background scenario has a different non-trivial wave impedance as a function of the incoming illumination [19].

_{FF}In addition to reflectionless solutions, if there is no difference between a scenario and the insertion of a novel scatterers (in terms of *ε*, *µ* or *Z*) in such background, a zero contrast occurs as reported in the center of the near-zero zone in Fig. 3: such point is a strong solution and it can be also interpreted as a *neutral inclusion* for this system [20].

It is worthwhile mentioning that the strong solution can extend the neutral inclusion concept from constitutive parameters to other wave characteristics. This is the case for the wavenumber: for example, a general wave phenomenon is propagating from material to material with a certain $\overrightarrow{k}$,

where*N*(

*r*′,

*ω*) is the refractive index function, as defined in terms of relative constituive parameters, with dispersive behaviour. A strong solution (or zero contrast) can be defined as the difference between the wavevector $\overrightarrow{k}$ in the material region and ${\overrightarrow{k}}_{b}$ in the homogeneous background, characterized by

*N*(

_{b}*ω*). Depending on the specific direction, the result in terms of wavenumber gets simplified in terms of refractive index, giving

## 4. Strong and weak solutions: surface and volumetric examples

In order to validate the strong and weak solutions, two examples are here reported: the non-scattering waveguide, proposed in [17], and a volumetric cloak for arbitrary dielectric shape scatterers in quasi-static regime.

The contrast impedance tensor formalism in Eq. (14) can be used to explain the behaviour of the patterned metasurface reported in [17], with non-scattering behaviour for a particular direction (in cartesian or curvilinear coordinate) and another functionality in the orthogonal one. As reported in Fig. 4, the simple design rule for such directional invisibility mechanism was having, on top of a grounded dielectric slab, rectangular patch loading as an elongated version of the squared ones all around, with one side that remains in common for both shapes [17]. For the wave traveling across the strong solution path, which is $\widehat{k}\equiv \widehat{y}$ in Cartesian system (left) or $\widehat{k}\equiv \widehat{\rho}$ in the polar case (right), the same impedance is encountered, thus zero contrast impedance, and no scattering is performed: numerical results and experimental measurement are reported in [17] (and reference therein).

For the weak solution case, we demonstrate that in subwavelength condition even an arbitrary domain can be cloaked with a dielectric-dielectric system, as reported in Fig. 5. Arbitrary cross section shapes have been drawn in CST Microwave Studio [21], collocating arbitrary points and using spline interpolation scheme for the cloak contour [22] and object perimeter [23]. Once defined the background (i.e., free space), object area *A*_{1}, cloak region *A*_{2} and object contrast
${\chi}_{{\epsilon}_{1}}=1.50$ (*ε*_{1} = 2.50*ε*_{0}) are fixed and the scenario is illuminated by an electromagnetic TM* _{z}* wave (
$\widehat{z}$ is the cylinder’s axis). The resulting cloak contrast is computed according to Eq. (9), giving
${\chi}_{{\epsilon}_{2}}=-0.91$ (

*ε*

_{2}= 0.09

*ε*

_{0}). Numerical results of the scattered field are shown in Fig. 5 for the uncloaked case (left) and for the covered structure (right). Even if slightly beyond subwavelength (axis are normalized to the working wavelength), the weak solution is still able to confine the radiation within the device itself with respect to the uncloaked case, both illuminated by a monochromatic incident wave with $\widehat{k}=\widehat{x}$.

## 5. Conclusions

With a compact and comprehensive unified methodology, a general theory for non-scattering structures has been derived as weak or strong solutions of the Devaney-Wolf theorem. The existence of Anapole modes, Mantle Cloaking and Plasmonic Cloaking can be explained by the weak solution, whereas, beyond the quasi-static limit, a strong solution of Devaney-Wolf theorem confirms general non-scattering devices based on directional invisibility, Transformation Optics, neutral inclusions and refractive index continuity.

If a contrast can be correctly defined in terms of constitutive or impedance property, new cloaking systems and exotic devices can be found also in other physical scenarios: improving cloaking effect by impressing a strong solution into the near-field impedance of an existing metasurface structure in conformal state [24] is currently under investigation.

## Acknowledgments

The authors would like to acknowledge all the invisible discussions, efforts and incitements behind this work.

## References and links

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**21. **Microwave Studio, Computer Simulation Technology, 2016.

**22. **The cloak external contour is plotted according to this points sequence: [x_{clk},y_{clk}] = [(0, 8);(6, 4);(6,−0);(6,−4);(0,−8);(−4,−6);(−6,−4); (−4,−2);(−6,−0);(−4, 2);(−6, 4);(−4, 6);(0, 8)].

**23. **The object contour is plotted according to this points sequence: [x_{obj},y_{obj}] = [(4,0);(2,2);(2,4);(0,6);(−2,4);(−4,4);(−2,0);(−4,−4);(−2,−4);(0,−6);(2, −4);(2, −2); (4,0)].

**24. **L. Matekovits and T. S. Bird, “Width-modulated microstrip-line based mantle cloaks− for thin single- and multiple cylinders,” IEEE Trans. Antennas Propag. , **62**(5), 2606–2615 (2014). [CrossRef]