aa r X i v : . [ phy s i c s . op ti c s ] A ug The Anti-Cloak
Huanyang Chen ∗ , Xudong Luo, and Hongru Ma Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China
C.T. Chan
Department of Physics, The Hong Kong University of Science and Technology,Clear Water Bay, Hong Kong, China
Abstract
A kind of transformation media, which we shall call the “anti-cloak”, is proposed to partiallydefeat the cloaking effect of the invisibility cloak. An object with an outer shell of “anti-cloak”is visible to the outside if it is coated with the invisible cloak. Fourier-Bessel analysis confirmsthis finding by showing that external electromagnetic wave can penetrate into the interior of theinvisibility cloak with the help of the anti-cloak.
PACS numbers: r = b − r b − a ( r ′ − b ) + b, the requiredparameters for a partial cylindrical cloak (transverse electric (TE) mode is considered here)are obtained as follow, µ r = r ′ − a r ′ , µ θ = r ′ r ′ − a , ε z = ( b − r b − a ) r ′ − a r ′ , (1)where a = a − r b − r b. This partial cloak can reduce the total scattering cross section of a perfectelectrical conductor (PEC) cylinder from its radius r ′ = a to an equivalent PEC cylinderwhose radius is r = r . In the limit as r goes to zero, the partial cloak becomes perfect [1].Now let us add another coordinate transformation inside the cloak ( c < r ′ < a ) asdepicted in Fig. 2(a), r = d − r c − a ( r ′ − c ) + d. The corresponding material parameters are then, µ r = r ′ − a r ′ , µ θ = r ′ r ′ − a , ε z = ( d − r c − a ) r ′ − a r ′ , (2)where a = ad − cr d − r . We note that these values are negative. We call this kind of transfor-mation media the ”anti-cloak” as we shall see that they cancel partially the effect of aninvisibility cloak. 2
IG. 1: (Color online) The schematic figure to illustrate the cloaking effect and anti-cloaking effect.
In the same spirit of the partial cloak, when a PEC cylinder with a radius r ′ = c is coatedwith the anti-cloak in direct contact with the partial cloak, the total scattering cross sectionwill be changed into that of an equivalent PEC cylinder whose radius is r = d . We notethat there are no PEC boundary between the cloak and the anti-cloak (at r ′ = a ), they arein direct contact.Doing the same Fourier-Bessel analysis in [11], we can obtain the electric fields in eachregion, 3 a) r’bacOr dr -1.0-0.50.00.5 µ r r’(b) bac -50050100150200 µ θ r’ (c)a bc ε z r’(d)c a b FIG. 2: (Color online) (a) The coordinate transformation of the cloak ( a < r ′ < b ) and anti-cloak( c < r ′ < a ). (b) µ r for cloak and anti-cloak. (c) µ θ for cloak and anti-cloak. (d) ε z for cloak andanti-cloak. The red solid lines in b-d denote the parameters of cloak, while the black dashed linesin b-d denote the parameters of anti-cloak. ( b ≤ r ) : E z = P l α inl J l ( k r ) exp( ilθ )+ α scl H l ( k r ) exp( ilθ ) , ( a ≤ r < b ) : E z = P l α l J l ( k ( r − a )) exp( ilθ )+ α l H l ( k ( r − a )) exp( ilθ ) , ( c ≤ r < a ) : E z = P l α l J l ( k ( r − a )) exp( ilθ )+ α l H l ( k ( r − a )) exp( ilθ ) . (3)where J l \ H l are the l -order Bessel \ Hankel function of the 1st kind, k is the wave vector of4he light in vacuum, k = b − r b − a k , k = d − r c − a k , α inl and α scl are the incident and scatteringcoefficients outside the cloak, α il ( i = 1 , , ,
4) are the expansion coefficients for the field inthe cloak and anti-cloak. The primes are dropped for aesthetic reasons from here. From thecontinuous boundary conditions (at r = b and r = a ) and the PEC boundary ( E z = 0 at r = c ), we can obtain that, α l = α l = α inl ,α l = α l = α scl ,α scl = − J l ( k d ) H l ( k d ) α inl . (4)This result confirms that the PEC cylinder with its radius r = c coated with the anti-cloakand cloak is equivalent to a PEC cylinder with its radius r = d in the view of outside world.To demonstrate the properties of the anti-cloak, we set a = 0 . m , b = 0 . m , c = 0 . m , d = 0 . m , r = 0 . m . We plot the parameters of the cloak and anti-cloak at differentradial positions in Fig. 2(b)-(d). All the parameters of anti-cloak are negative because of thenegative slope of the coordinate transformation. A plane wave is incident from left to rightwith the frequency 2 GHz . In Fig. 3(a), we plot the scattering pattern of a PEC cylinderwith a radius r . The tiny PEC cylinder causes little scattering for the incoming plane wavewhich can be treated as almost invisible. In Fig. 3(b), we plot the scattering pattern of aPEC cylinder with a radius a coated by a partial cloak. The outer radius of the cloak is b .We see that the partial cloak reduces substantially the scattering of the PEC cylinder withits radius a when we compare Fig. 3(a) and Fig. 3(b). When r is made as small as we like,the scattering becomes vanishing small. In Fig. 3(c), we plot the scattering pattern of aPEC cylinder with a radius c coated by an anti-cloak and a partial cloak [20]. The anti-cloakis located in c < r < a , the cloak locates in a < r < b . Without the anti-cloak, the wavebasically goes around the shielded region, but if the anti-cloak is in contact with the cloak,EM wave from outside can go into the anti-cloak to interact with the object inside. Thescattering of the cloak is enlarged again to that of an equivalent PEC cylinder whose radiusis d . We plot the scattering pattern of the equivalent PEC cylinder in Fig. 3(d). When c = d , the anti-cloak together with the partial cloak becomes invisible, that means one candirectly see the PEC cylinders with radius r = c , and the anti-cloak cancels out the effect5 FIG. 3: (Color online) The electric filed distribution for, (a) a tiny PEC cylinder with a radius r (outlined by the write point); (b) a PEC cylinder with a radius a wearing a partial cloak (theouter radius is b , the inner and outer boundaries of the cloak are outlined by the write solid lines);(c) a PEC cylinder with a radius c wearing an anti-cloak and the same partial cloak as in (b)(the inner and outer boundaries of the cloak are outlined by the write solid lines while the innerboundary of the anti-cloak at r = c is outlined by the blue solid line); (d) a PEC cylinder with aradius d (outlined by the write solid line). of the partial cloak completely. For aesthetic reasons, if the electric field is larger than themaximum value in color bar in Fig. 3(c), we have replaced this overvalued field with themaximum value when plotting Fig. 3(c). If the electric field is smaller than the minimumvalue, we have replaced this overvalued field with the minimum value when plotting Fig.3(c).Due to the continuous coordinate transformation at r = a , the impedances are matched6 θ E z π/2 π 3π/2 2π FIG. 4: (Color online) The electric field distribution at r = a − . r (red dashed line), r = a (solidblue line) and r = a + 0 . r (green dotted line). at this touching boundary of the cloak and anti-cloak. The electric field is very large atthis touching boundary. To show this property, we plot the electric field for different anglesat fix radii near r = a in Fig. 4. Three fixed radii are chosen, r = a − . r , r = a and r = a + 0 . r . We find that the electric field near r = a is very large.Analytically, one can obtain the electric field at r = a as follow, E z = P l α inl J l ( k r ) exp( ilθ ) + α scl H l ( k r ) exp( ilθ )= P l α inl exp( ilθ )[ J l ( k r ) − J l ( k d ) H l ( k d ) H l ( k r )] . (5)The term H l ( k r ) becomes very large when r is small, that is why we obtain large electricfield above.Since we can make r as small as we like, we reach the conclusion that an almost perfectcloak can be defeated by an anti-cloak. In other words, the transformation media cloak is nota panacea as there exists some objects that it cannot hide. In the limit that r is exactly zero,the situation requires further mathematical analysis due to the singularity properties of theanti-cloak and cloak ( H l ( k r ) diverges when r goes to zero). From a physical standpoint,we may argue as follows. Near the inner boundary of the invisibility cloak, µ r goes to zero7nd µ θ goes to infinity and they are positive, while near the outer boundary of the anti cloak, µ r goes to zero and µ θ goes to infinity from the negative side. The positive singular valueshave to come from an in-phase resonance while the negative infinity comes from out-of-phaseresonance. If we put them in contact, the system response is canceled out, and the cloakingeffect is weaken or even destroyed. The surface mode resonance at r = a is excited andcontributes to the large electric field. In addition, if the losses are considered, the electricfield will become finite for r is exactly zero. The cylindrical anti-cloak concept could beextended to three dimensions.In conclusion, we find that the invisible cloak cannot hide the enclosed domain if theinside domain has a shell of anti-cloak. The properties are demonstrated by using theFourier-Bessel analysis and finite-element full wave simulations. The anti-cloak region is ananisotropic negative refractive shell that is impedance matched to the cloak outside, whichhas a positive refractive index. It is known that [21] a negative refractive index medium“cancels” the space of a positive index medium that has the same impedance. So, a heuristicway of understanding the operation of an anti-cloak is that it annihilates the functionalityof the interior part of the invisibility cloak, and effectively shifts the enclosed PEC regionoutwards to make contact with the outer part of the cloaking shell that is not “canceled”.This leads to a finite cross section. Acknowledgments
This work was supported by the National Natural Science Foundation of China un-der grand No.10334020 and in part by the National Minister of Education Program forChangjiang Scholars and Innovative Research Team in University, and Hong Kong CentralAllocation Fund HKUST3/06C. ∗ Correspondence should be addressed to: [email protected] [1] J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science
2] U. Leonhardt, “Optical conformal mapping,” Science
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