## Cloak/anti-cloak interactions

Optics Express, Vol. 17, Issue 5, pp. 3101-3114 (2009)

http://dx.doi.org/10.1364/OE.17.003101

Acrobat PDF (544 KB)

### Abstract

Coordinate-transformation cloaking is based on the design of a metamaterial shell made of an anisotropic, spatially inhomogeneous “transformation medium” that allows rerouting the impinging wave around a given region of space. In its original version, it is generally believed that, in the ideal limit, the radiation *cannot* penetrate the cloaking shell (from outside to inside, and viceversa). However, it was recently shown by Chen *et al*. that electromagnetic fields may actually penetrate the cloaked region, provided that this region contains *double-negative* transformation media which, via proper design, may be in principle used to (partially or totally) “undo” the cloaking transformation, thereby acting as an “anti-cloak.” In this paper, we further elaborate this concept, by considering a more general scenario of cloak/anti-cloak interactions. Our full-wave analytical study provides new insightful results and explores the effects of departure from ideality, suggesting also some novel scenarios for potential applications.

© 2009 Optical Society of America

## 1. Introduction and background

1. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E **72**, 016623 (2005). [CrossRef]

2. G. W. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. London A **462**, 3027–3059 (2006). [CrossRef]

3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

4. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9794. [CrossRef] [PubMed]

5. U. Leonhardt, “Optical conformal mapping,” Science **312**, 1777–1780 (2006). [CrossRef] [PubMed]

6. U. Leonhardt and T. G. Philbin, General relativity in electrical engineering, New J. Phys. **8**, 247 (2006), http://www.iop.org/EJ/article/1367-2630/8/10/247/njp6_10_247.html. [CrossRef]

7. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**, 977–980 (2006). [CrossRef] [PubMed]

8. A. Alù and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. A **10**, 093002 (2008). [CrossRef]

*bending*the ray trajectories (which describe the high-frequency power flux) around the object to be cloaked. The design may be performed by first deriving the desired field distribution in a fictitious

*curved-coordinate*space containing a “hole,” and subsequently exploiting the formal invariance of Maxwell's equations under coordinate transformations to translate such distribution into a conventionally flat, Cartesian space, filled by a suitably anisotropic and spatially inhomogeneous transformation medium [3–6

3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

*exact*full-wave analytic studies in the spherical [9

9. H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. **99**, 063903 (2007). [CrossRef] [PubMed]

10. B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B **76**, 121101 (2007). [CrossRef]

*perfect cloaking*, i.e., zero external scattering and zero transmission into the cloaked region, at

*any*given frequency. Thus, for an “ideal” cloaking (implying a lossless, non-dispersive, anisotropic, spatially inhomogeneous metamaterial, with extreme values of the relative permittivities and permeabilities ranging from zero to infinity) of an isotropic object, the field

*cannot*penetrate from outside to inside, and vice-versa [11

11. B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. **100**, 063904 (2008). [CrossRef] [PubMed]

12. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. **99**, 113903 (2007). [CrossRef] [PubMed]

16. H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B **76**, 241104 (2007). [CrossRef]

17. H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express **16**, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603. [PubMed]

*et al*. introduced a new twist in this concept, showing that the above picture may not be longer valid if the cloaked region is allowed to contain another anisotropic, spatially inhomogeneous medium suitably designed to act as an “anti-cloak.” In particular, with reference to a two-dimensional (2-D) cylindrical scenario, they showed that it is possible, in principle, to design a transformation-medium shell (based on a linearly-decreasing radial coordinate transformation) that, when laid directly between the cloak and the object, is capable of “undoing” the cloaking transformation, restoring (partially or totally) the original scattering response. In terms of practical feasibility, the same limitations mentioned above for the cloak clearly hold for the anti-cloak as well, with a further complication given by the

*double negative*(DNG) [18] character of the anti-cloak transformation-medium, dictated by the monotonic negative slope of the corresponding coordinate-transformation [17

17. H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express **16**, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603. [PubMed]

*double-positive*(DPS), or

*single-negative*(SNG) transformation media, relaxing the complexity of the anti-cloak design and making its practical realization arguably more feasible. Moreover, we show that the effect may be preserved even in the presence of a vacuum shell separating cloak and anti-cloak, suggesting a counterintuitive “field-tunneling” mechanism. Finally, we explore the effects of the presence of penetrable objects inside the vacuum shell, as well as of the departure from ideal conditions, i.e., the presence of losses, and deviations from the ideal design parameters. These results provide new insights in the anti-cloaking mechanism, and may ensure further degrees of freedom in the design of an anti-cloak. Moreover, besides the potential application (as a cloaking cancellation) suggested in [17

17. H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express **16**, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603. [PubMed]

## 2. Problem geometry and formulation

*R*

_{2}, with dielectric permittivity

*ε*

_{1}and magnetic permeability

*μ*

_{1}, immersed in vacuum in the auxiliary space (

*x*′,

*y*′,

*z*′). In the associated cylindrical (

*r*′,

*ϕ*′,

*z*′) reference system, this configuration may be parameterized by the

*ε*

_{0}and

*μ*

_{0}denoting the vacuum dielectric permittivity and magnetic permeability, respectively. We map the above configuration onto the actual physical space (

*x*,

*y*,

*z*) via the piecewise linear radial (in the associated (

*r*,

*ϕ*,

*z*) cylindrical reference system) coordinate transformation (see Fig. 1(b)):

*R*

_{2}<

*r*<

*R*

_{3}, an arbitrary coordinate mapping

*f*(

*r*) < 0 can be assumed, in light of the theory developed in what follows, and therefore it has not been specified in (2) and in Fig. 1(b). When the (negligibly small) parameters Δ

_{2}and Δ

_{3}are zero, it is readily recognized that the outermost layer

*R*

_{3}<

*r*<

*R*

_{4}corresponds to a standard invisibility cloak [3

3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

*R*

_{1}<

*r*<

*R*

_{2}corresponds to the anti-cloak introduced in [17

**16**, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603. [PubMed]

*r*<

*R*

_{4}into the concentric annular layer

*R*

_{3}<

*r*<

*R*

_{4}, thereby creating a “hole” of radius

*R*

_{3}wherein an object may be concealed. The anti-cloak transformation may be interpreted as a somehow

*reverse*operation, where an “anti-hole” of radius

*R*

_{2}>

*R*

_{1}is created around a cylinder of radius

*R*

_{1}. From a topological viewpoint, their combination results in a four-layer cylindrical configuration of radii

*R*=1,…,4 (see Fig. 1(c)), where the transformed regions

_{ν}*R*

_{3}<

*r*<

*R*

_{4}(cloak) and

*R*

_{1}<

*r*<

*R*

_{2}(anti-cloak) are characterized by

*curved*coordinates, while the regions

*r*<

*R*

_{1}and

*r*>

*R*

_{4}maintain the flat, Cartesian metrics. Conversely, the layer

*R*

_{2}<

*r*<

*R*

_{3}does not admit any physical image (i.e.,

*r*′ > 0) in the auxiliary (

*x*′,

*y*′,

*z*′) space, thereby constituting a

*inaccessible*to the EM fields. It is instructive to observe in Fig. 1(c) how the curved coordinates in the cloak and anti-cloak regions bend in a somehow complementary fashion around the cloaked layer, while matching the flat, Cartesian coordinates in the regions

*r*<

*R*

_{1}and

*r*>

*R*

_{4}.

*globally flat*space by filling up the transformed regions with anisotropic, spatially inhomogeneous transformation media, whose permittivity and permeability tensors may be readily derived from the Jacobian matrix of the transformation (2) (see [4

4. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9794. [CrossRef] [PubMed]

15. W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. **91**, 111105 (2007). [CrossRef]

*R*

_{3}<

*r*<

*R*

_{4}(cloak) and

*R*

_{1}<

*r*<

*R*

_{2}(anti-cloak) are filled by the transformation media arising from (3), while the regions

*r*<

*R*

_{1}and

*r*>

*R*

_{4}remain filled by isotropic, homogeneous media (with constitutive parameters

*ε*

_{1},

*μ*

_{1}and

*ε*

_{0},

*μ*

_{0}, respectively), and the layer

*R*

_{2}<

*r*<

*R*

_{3}is filled by vacuum. A few considerations are in order here:

- Note that, as in [10,12
10. B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B

**76**, 121101 (2007). [CrossRef]], we introduced in the transformation (2) two small dimensional parameters Δ12. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations,” Phys. Rev. Lett.

**99**, 113903 (2007). [CrossRef] [PubMed]_{2}and Δ_{3}, which parameterize the departure of the cloak and anti-cloak from the ideal case (Δ_{2}, Δ_{3}→ 0). These are necessary in the design of the anti-cloak, as it will become clear in the following. - In our configuration, unlike that in [17
**16**, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603. [PubMed]*R*_{2}<*r*<*R*_{3}), which, as observed above, constitutes the cloaked region in the limit Δ_{2},Δ_{3}→ 0. In this same limit, the external cloak is perfectly matched with vacuum at the interface*r*=*R*_{4}, and the anti-cloak is perfectly matched with the inner medium at the interface*r*=*R*_{1}; this yields*zero scattering*outside the region*r*>*R*_{4}. Moreover, as a consequence of the*non-monotonic*behavior of the transformation (2), the interfaces bounding the cloaked layer*r*=*R*_{2}and*r*=*R*_{3}are imaged in the same point*r*′ = 0 in the auxiliary space (see Fig. 1(b)). This suggests the possibility of an intriguing and somehow counterintuitive mechanism of field transfer between these interfaces, via the “tunneling” through the cloaked layer. - In view of the negative slope of the transformation (2) in the anti-cloak layer
*R*_{1}<*r*<*R*_{2}, the corresponding constitutive parameters (3) are*opposite in sign*to those of the inner cylinder (*ε*_{1},*μ*_{1}). This suggests*four*possible configurations of interest, involving the possible combinations of DPS and DNG, or alternatively epsilon-negative (ENG) and mu-negative (MNG), media.

## 3. Full-wave analytical solution

*iωt*)), TM-polarized plane wave, with unit-amplitude

*z*-directed magnetic field, impinging from the positive

*x*-direction on the four-layer cloak/anti-cloak configuration in Fig. 1(d). It is expedient to represent the incident magnetic field in the associated (

*r*,

*φ*,

*z*) cylindrical coordinate system in terms of a Fourier-Bessel series:

*λ*

_{0}being the corresponding wavelength), and

*J*denotes the

_{n}*n*th-order Bessel function of the first kind [19, Chap. 9]. Following [10

**76**, 121101 (2007). [CrossRef]

*x*′,

*y*′,

*z*′) space (plane-wave scattering by a homogeneous, isotropic circular cylinder, cf. Fig. 1(a)). Introducing, for notational convenience the “dummy” parameters

*R*

_{0}= 0 and

*R*

_{5}=∞, the Fourier-Bessel expansion of the above field can be compactly written as:

*g*(

*r*) =

*k*

_{0}

*r*in the cloaked layer

*R*

_{2}<

*r*<

*R*

_{3}, and

*a*

^{(ν)}

_{n}and

*b*

^{(ν)}

_{n}denote the unknown expansion coefficients (to be computed by enforcing the boundary and tangential-field-continuity conditions),

*Y*denotes an

_{n}*n*th-order Bessel function of the second kind [19, Chap. 9

9. H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. **99**, 063903 (2007). [CrossRef] [PubMed]

*δ*is the Kronecker delta (accounting for the presence of the incident field (4) in the vacuum region

_{pq}*r*>

*R*

_{4}). Note that the field-finiteness condition at

*r*= 0 and the radiation-at-infinity condition imply that

*b*

^{(1)}

_{n}= 0 and

*b*

^{(5)}

_{n}=

*ia*

^{(5)}

_{n}, respectively. The corresponding electric field components can be readily derived from (5) via the relevant Maxwell’s curl equation:

*r*=

*R*

_{1}and

*r*=

*R*

_{4}, we readily derive

*r*=

*R*

_{2}and

*r*=

*R*

_{3}require particular care since, in the limit of ideal cloak (Δ

_{3}→ 0) and anti-cloak (Δ

_{2}→ 0), the transformation in (2) vanishes, thereby causing the Bessel functions of second kind in the expansion (5) to exhibit a singular behavior. In this ideal scenario, the anti-cloak design would not be possible, since the inner layer of the cloak is perfectly impenetrable. However, in the limit for which Δ

_{2}and Δ

_{3}tend both to zero (but are not exactly zero), it is still possible to tailor the design of a suitable anti-cloak, as discussed in the following.

**76**, 121101 (2007). [CrossRef]

_{2},Δ

_{3}→ 0) the small argument approximations of the Bessel functions of first [19, Eq. (9.1.7)] and second kind [19, Eqs. (9.1.8) and (9.1.9)], we obtain for the zeroth-order coefficients:

*n*≠ 0) th-order coefficients:

*J*

_{1}(

*k*

_{0}

*R*

_{2})

*Y*

_{1}(

*k*

_{0}

*R*

_{3})-

*J*

_{1}(

*k*

_{0}

*R*

_{3})

*Y*

_{1}(

*k*

_{0}

*R*

_{2}) and

*all*the zeroth-order coefficients in (8) and (9) do not depend on Δ

_{2}, and, as already observed for the standard cloak case (see, e.g., [10

**76**, 121101 (2007). [CrossRef]

**99**, 113903 (2007). [CrossRef] [PubMed]

*logarithmically*to zero in the limit of an ideal cloak (Δ

_{3}→ 0). Also the (

*n*≠ 0) th-order coefficients

*a*

^{(3)}

_{n},

*b*

^{(3)}

_{n},

*b*

^{(4)}

_{n},

*a*

^{(5)}

_{n}, and

*b*

^{(5)}

_{n}in (11) and (12) do not depend on Δ

_{2}, and they tend

*exponentially*to zero in the limit Δ

_{3}→ 0. Conversely, the remaining (

*n*≠ 0) th-order coefficients

*a*

^{(1)}

_{n}and

*a*

^{(2)}

_{n}in (10) depend (exponentially) on the

*ratio*Δ

_{3}/Δ

_{2}. This suggests that letting Δ

_{2},Δ

_{3}→ 0, while keeping their ratio

*finite*, it is possible to combine the cloak and anti-cloak functions, suppressing (via the cloak) the field in the vacuum layer

*R*

_{2}<

*r*<

*R*

_{3}(as well as the one scattered in the vacuum region

*r*>

*R*

_{4}) and having it “restored” (via the anti-cloak) in the cylinder

*r*<

*R*

_{1}. This justifies the previous assumption that a field “tunneling” is responsible for transmitting the field from the cloak inner interface to the anti-cloak outer interface, consistent with (2). It is worth pointing out that the field restored in the cylinder

*r*<

*R*

_{1}is a distorted version of the incident field (4), since the expansion coefficients (cf. (8) and (10)) are clearly different; in particular, the zeroth-order terms are vanishingly small.

## 4. Representative results

*ε*

_{1}=

*ε*

_{0},

*μ*

_{1}=

*μ*

_{0}) and, consequently (in view of (2) and (3)), a DNG anti-cloak, with

*R*

_{1}= 0.4

*λ*,

*R*

_{2}= 0.75

*λ*

_{0},

*R*

_{3}= 1.7

*λ*

_{0}and

*R*

_{4}= 2.5

*λ*

_{0}. We assume very slight losses (tanδ=10

^{-4}) and, for the small parameters, we choose Δ

_{2}=

*R*

_{2}/200 and Δ

_{3}=

*R*

_{3}/200, which yields constitutive parameters of the transformation media ranging (in absolute values) between nearly zero and 200.

*r*>

*R*

_{4}), experiences a response essentially equivalent to that of a standard transformation-based cloak, with very low scattering and very mild distortion of the original planar wavefronts. On the other hand, an internal observer, placed inside the region

*r*<

*R*

_{1}, would experience a substantially different situation, as compared with a standard cloak. In fact, the field is very weak, almost zero, in the layer

*R*

_{2}<

*r*<

*R*

_{3}, which can thus be thought as effectively cloaked, but it is restored (via the anti-cloak) in the inner cylinder

*r*<

*R*

_{1}. The power flow may be better understood in the time-domain animation in Fig. 2(a) (Media 1) and the magnified detail (with the normalized Poynting vector map superimposed) in Fig. 2(b). Once again, outside the cloak, the picture is essentially equivalent to the standard cloak. Inside the cloak, the anti-cloak and the inner cylinder form a “resonating cavity” which, via the vanishingly small coupling through the cloaked layer is able to restore a modal field. In particular, it is interesting to observe the power circulation in the “cavity,” with a forward flow in the inner cylinder, directed parallel to the impinging plane wave, which circulates back in the anti-cloak region.

*ε*

_{1}= -(

*ε*

_{0},

*μ*

_{1}= -

*μ*

_{0}) inner cylinder and a DPS anti-cloak (see also the time-domain animation in Fig. 3(a) (Media 2)), for which the same qualitative considerations about the power flow, mentioned above, hold. Figure 4 corresponds to the case of an ENG ((

*ε*

_{1}=-(

*ε*

_{0},

*μ*

_{1}=

*μ*

_{0}) inner cylinder and an MNG anti-cloak (see also the time-domain animation in Fig. 4(a) (Media 3)), while Fig. 5 pertains to the dual configuration featuring an MNG ((

*ε*

_{1}= (

*ε*

_{0},

*μ*

_{1}= -

*μ*

_{0}) inner cylinder and an ENG anti-cloak (see also the time-domain animation in Fig. 5(a) (Media 4)). In these last two scenarios, the power flow inside the “cavity” exhibits a less clean-cut behavior, with the presence of “loops” at the boundary between the cylinder and the anti-cloak, possibly due to localized resonances.

**16**, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603. [PubMed]

- They may relax some of the practical-feasibility limitations of the DNG anti-cloak introduced in [17
**16**, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603. [PubMed]*ε*_{1}=-(*ε*_{0},*μ*_{1}=-*μ*_{0}) are uniform and finite. - They suggest interesting scenarios for applications, where one may be able to cloak a penetrable object inside the vacuum shell and yet maintain the capability of “sensing” the outside field from the inside by creating, via the anti-cloak, an “invisible observation window.”

*Q*is readily recognized to be the total scattering cross-sectional width per unit length (normalized by the vacuum wavelength), which quantifies the visibility of the overall configuration to a far-field exterior observer, whereas the

_{e}*interior*parameter

*Q*, quantifies the capability of coupling the field inside the inner cylinder

_{i}*r*<

*R*

_{1}while maintaining a very weak intensity in the (ideally cloaked) layer

*R*

_{2}<

*r*<

*R*

_{3}. In the ideal case (Δ

_{2}, Δ

_{3}→ 0 and lossless materials), both parameters should vanish.

_{3}(for a fixed ratio Δ

_{2}/Δ

_{3}=

*R*

_{2}/

*R*

_{3}≈ 0.44), and for various values of the loss-tangent ranging from zero to 10

^{-2}. In particular, the blue dots denote the level of losses corresponding to the example in Fig. 2. For the lossless case, one observes the anticipated monotonic reduction of both parameters as Δ

_{3}→ 0. The scattering width

*Q*(see Fig. 6(a)) turns out to be only mildly dependent on the losses (with a significant departure from the lossless behavior observable for tanδ~10

_{e}^{-2}), and, as also observed in the standard cloak case, essentially decreases monotonically with the parameter Δ

_{3}. The effect of losses is much more evident in the interior parameter

*Q*(see Fig. 6(b)). In particular, increasing the losses, one observes the appearance (for small values of Δ

_{i}_{3}) and progressive enlargement of regions where

*Q*can be as high as ~1, thereby implying that the average values of the field intensity in the inner cylinder and in the cloaked layer are actually comparable. In such regions, while the exterior visibility of the configuration can still be relatively small (like in the standard cloak case), the peculiar coupling (via “tunneling”) effects observed in Figs. 2–5 are effectively destroyed. However, outside these ranges of parameters,

_{t}*Q*decreases with an

_{i}*oscillating*behavior, exhibiting minima that may be acceptably small (~10

^{-2}) even in the presence of moderate losses (tanδ ~10

^{-2}). This leads to the conclusion that, in the presence of losses, the “ideal” condition Δ

_{3}→ 0 is not necessarily “optimal,” and that a suitable tradeoff between the two parameters can be achieved for

*finite*values of Δ

_{3}. This is expected, since the ideal configuration Δ

_{2}, Δ

_{3}→ 0 would create a barrier for the EM fields at the inner boundary of the cloak, that necessitates to be relaxed when losses are present. The resonant nature of this cloak/anti-cloak interaction and of the field tunneling described above is clearly evident, which indeed may be significantly affected by losses and parameter deviations. Qualitatively similar trends, not shown here for sake of brevity, are observed for the other cloak/anti-cloak combinations (cf. Figs. 3–5).

*R*=

_{a}*λ*

_{0}and

*R*= 1.5

_{b}*λ*

_{0}(shown dashed), and permittivity

*ε*= 2

_{obj}*ε*

_{0}(tan (

*δ*= 10

^{-4}). The coaxial annular shell geometry was chosen in order to preserve the analytical tractability of the problem (via straightforward generalization of (5)). As one can see, also by comparison with Fig. 2, the dielectric annular layer is effectively cloaked. From a topological viewpoint, this may also be considered as the first example of

*selective*cloaking of a

*multiply-connected*penetrable object.

_{3}→ 0 , even in the presence of a significantly lossy object. In particular, the scattering width

*Q*depends only mildly on the object parameters, whereas the interior parameter

_{e}*Q*, while remaining always rather small (<10

_{i}^{-4}), exhibits a stronger dependence. Finally, Fig. 9 shows the more realistic results pertaining to a slightly-lossy cloak/anti-cloak configuration (tan

*δ*= 10

^{-4}, as in Fig. 7). As for the case in the absence of the object (cf. Fig. 6), the scattering width

*Q*is not sensibly affected, whereas the effects are more dramatic for the interior parameter

_{e}*Q*which exhibit a decreasing oscillatory behavior, with local minima which may still be acceptably small. These minima tend to increase with increasing the object permittivity and losses, as one could intuitively expect by observing that the cloak/anti-cloak interactions should vanish in the limit of an impenetrable layer. Further results pertaining to progressively less penetrable objects (up to the perfectly-conducting case) confirm this intuitive picture, with values of

_{i}*Q*comparable with those in Figs. 8 and 9 (thereby indicating that the cloak mechanism is still effective for ideally impenetrable objects placed in the vacuum layer between cloak and anti-cloak) but

_{e}*Q*(thereby indicating the expected destruction of the anti-cloak tunneling mechanism in the case of impenetrable objects).

_{i}## 5. Conclusions and outlook

## References and links

1. | A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E |

2. | G. W. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. London A |

3. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

4. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express |

5. | U. Leonhardt, “Optical conformal mapping,” Science |

6. | U. Leonhardt and T. G. Philbin, General relativity in electrical engineering, New J. Phys. |

7. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science |

8. | A. Alù and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. A |

9. | H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. |

10. | B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B |

11. | B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. |

12. | Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. |

13. | M. Yan, Z. C. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. |

14. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photonics |

15. | W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. |

16. | H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B |

17. | H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express |

18. | R. W. Ziolkowski and N. Engheta, “Introduction, history and fundamental theories of double-negative (DNG) metamaterials,” in |

19. | M. Abramowitz and I. A. Stegun, |

20. | COMSOL MULTIPHYSICS - User's Guide (COMSOL AB, 2005). |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(260.2110) Physical optics : Electromagnetic optics

(230.3205) Optical devices : Invisibility cloaks

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 13, 2009

Revised Manuscript: February 9, 2009

Manuscript Accepted: February 13, 2009

Published: February 17, 2009

**Citation**

Giuseppe Castaldi, Ilaria Gallina, Vincenzo Galdi, Andrea Alù, and Nader Engheta, "Cloak/anti-cloak interactions," Opt. Express **17**, 3101-3114 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3101

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### References

- A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005). [CrossRef]
- G. W. Milton and N. A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London A 462, 3027-3059 (2006). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9794. [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006), http://www.iop.org/EJ/article/1367-2630/8/10/247/njp6_10_247.html. [CrossRef]
- D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006). [CrossRef] [PubMed]
- A. Alù and N. Engheta, "Plasmonic and metamaterial cloaking: physical mechanisms and potentials," J. Opt. A 10, 093002 (2008). [CrossRef]
- H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007). [CrossRef] [PubMed]
- B. Zhang, H. S. Chen, B. I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007). [CrossRef]
- B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, "Extraordinary surface voltage effect in the invisibility cloak with an active device inside," Phys. Rev. Lett. 100, 063904 (2008). [CrossRef] [PubMed]
- Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007). [CrossRef] [PubMed]
- M. Yan, Z. C. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233901 (2007). [CrossRef]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007). [CrossRef]
- W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007). [CrossRef]
- H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. R. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007). [CrossRef]
- H. Chen, X. Luo, H. Ma, and C. T. Chan, "The anti-cloak," Opt. Express 16, 14603 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14603. [PubMed]
- R. W. Ziolkowski and N. Engheta, "Introduction, history and fundamental theories of double-negative (DNG) metamaterials," in Metamaterials: Physics and Engineering Explorations, N. Engheta and R. W. Ziolkowski, eds., (Wiley-IEEE Press, 2006).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).
- COMSOL MULTIPHYSICS - User's Guide (COMSOL AB, 2005).

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