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Active remote cloaking and optical transformations from a boundary element

perspective

H. H. Zheng, J. J. Xiao, Y. Lai, and C. T. ChanDepartment of Physics and William Mong Institute of NanoScience and Technology,

The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

(Dated: January 3, 2011)

We show that the scheme of exterior cloaking using active sources for the two dimensionalHelmholtz equation [Phys. Rev. Lett. 103, 073901 (2009)] can be extended to create arbitraryillusion effects. Using a boundary element approach, we show that active sources can create a silentdomain which can conceal any objects inside and at the same time can make the objects look likeany other object. The sources can be placed in continuous curves that do not encircle the objectsto be cloaked.

PACS numbers: 41.20.Jb, 42.79.-e

The classical wave scattering cross section of an ob-ject can be significantly larger than or smaller thanthe geometric cross section.1,2 Recipes to achieve in-visibility (zero cross section) are particularly intrigu-ing,3,4,5,6,7,8,9,10 enabled by the concept of transforma-tion media and artificial metamaterials.6,11,12,13 Invisi-bility devices based on transformation media typicallyencircle the object to be cloaked. It was then proposedthat remote cloaking can be achieved,14 and the conceptcan be extended to create illusions (e.g. change an appleoptically to an banana).15 These recipes are based on pas-sive transformation media, and usually have bandwidthlimitations. Active cloaking, using active sources, havealso been proposed.16 It was recently shown that thosedevices can be remote devices,17 meaning that they donot need to encircle the object. In this paper, we willshow that we can have a clear understanding of active re-mote cloaking and a straightforward recipe to determinethe sources if we consider the problem using a boundaryelement approach. More interestingly, we show that itis clear from the boundary element formulation that wecan create illusions (e.g. change an apple optically to anbanana) without bandwidth limitations.

In the quasistatic limit, it is possible to have cloak-ing effects on polarizable line dipoles that lie outside thecloaking shell.18,19 Most recently, external (or remote)cloaking working at a finite frequency for finite-sized ob-jects was proposed.14 These schemes are based on pas-sive metamaterials. Recently, Miller showed that activesource can be used to make a region of space invisible ortransparent to waves.16 Vasquez et al.17 have analyticallyconstructed active sources to achieve exterior cloaking fortwo dimensional (2D) Laplace equation and numericallydemonstrated that multipole point sources have the sameeffect for the 2D Helmholtz equation in a broadband fash-ion.17,20

In this work, we demonstrate that such active cloakingeffects can be formulated using a 2D boundary integralequation.21 In particular, we show numerically that spe-cific values of fields and their normal gradients on somecurves (labeled as Γa in Fig. 1) will force a certain domainΩc ⊂ R2 to be “silent” with respect to incoming waves

and thus sources that create those fields will render anyobject inside invisible. Recently, it was recognized thatillusion effects can be created with properly designed pas-sive devices that does not encompass the object.15 Here,we show that active sources can go beyond remote cloak-ing and can be used to create illusion effects so that oneobject looks like another object. In other words, we areproposing an active wave-generating device, which whenplaced next to arbitrary objects, can make them appearlike another object. Invisibility is a special case in whichthe object looks like air. The active sources serve to gen-erate outgoing fields that mimic those scattered from an-other object Vi under the same incoming wave, renderingan arbitrary object placed inside the silent domain Ωc toappear like the object Vi for any observers outside somespecific domain Γb that encloses the whole system. Aschematic figure for the configuration is shown in Fig. 1.The predesigned object Vi , which is the illusion we wantto create, can be changed on demand as long as we cancompute the scattered fields of this object due to theincoming probing waves.

The active sources are placed on the red lines whichdo not need to enclose the object to be cloaked. Theseactive sources generate fields that cancels the incomingwaves to create a “quiet” zone Ωc inside which the totalfield is zero so that an arbitrary object (say an apple)can be concealed, and at the same time outgoing waveswhich is exactly those of another object (say a banana)outside a boundary Γb and thereby creating an illusion.Observers outside Γb would see a banana, although it isan apple that is inside Ωc. In this approach, the devicedoes not need to be tailored to the object and illusion, incontrast to the passive ones,15 and there is no intrinsicbandwidth limitation. However, one need to know inadvance the probing wave, or one must set up sufficientlyfast-responsive sensors to capture the information of theprobing wave on the boundary of Ωc.

16

We will formulate and discuss the problem using theboundary element approach. We note that the surfaceintegral equation (SIE) of the 2D Helmholtz equation(∇2 + k2)φ(r) = 0, where k is the wavenumber, can be

2

“quite” zone

illusion control curve

Active sources

Arbitrary object cloaked

Γa

Γc

cΩ

Γb

=

illusion object Vi

FIG. 1: (Color online) Schematic figure for the illusion devicescontaining exterior active sources on the boundaries Γa (redlines) that would create a cloaked region Ωc which can containarbitrary scatter and render the total system response like anillusion scatter Vi outside the boundary Γb.

written as21

φ(r)∣

∣

∣

r∈Ω=

∮

∂Ω

ds[

g(s, r)∂nφ(s) − φ(s)∂ng(s, r)]

. (1)

Here n is the outward unit normal vector on the bound-ary and ∂n represents the normal gradient. The 2D SIEtells us that inside a homogeneous domain Ω, the scalarwave function φ(r) is completely determined by the fieldsand its normal derivatives on the boundary, connected bythe Green’s function g(r, r′). The counterpart of Eq. (1)for an open domain which might also involve an inci-dent field φinc(r), reads φ(r)|

r∈R2−Ω = φinc(r)+φbc(r),where

φbc(r) = −

∮

∂Ω

ds[

g(s, r)∂nφ(s) − φ(s)∂ng(s, r)]

. (2)

We note that Ω is a virtual domain, and the materialproperties are the same inside and outside the boundary,hence the fundamental Green’s function is the same inEq. (1) and Eq. (2). Inspired by Eq. (2), one can con-struct additional fields (φbc) in the domain, by appropri-ately choosing active sources to create φ(s) and ∂nφ(s)on the boundary. The issue is that whether it is possi-ble to use such kind of active sources to construct fieldswhich can cancel the incident field φinc inside the “quiet”zone Ωc, and simultaneously in the region outside Γb (i.e.,R2 − Ωb) mimics the scattering field, i.e.

φbc(r) =

−φinc(r), ∀ r ∈ Ωc ,

φsc(r), ∀ r ∈ R2 − Ωb ,

(3)

such that inside Ωc the total field is zero, while outsideΩb the total field is the superimposition of the incomingwaves and the scattered wave of a pre-designed illusionobject.

The solution can be numerically determined using theboundary element method (BEM),21 a method based onthe SIE. BEM approximates the surface integrals by dis-cretizing the surface ∂Ω into N surface elements ℓα on

which the functions φ(s) and ψ(s) = ∂nφ(s) are approxi-mated as constants that represent the value of functionsφ(s) and ψ(s) across the entire element ℓα, respectively.In other words, a local step function basis with regard toℓα is used to expand φ(s) and ψ(s) over the entire sur-face ∂Ω, with the expansion coefficients denoted as φα

and ψα, where α = 1, 2, · · ·N . From the perspective ofBEM, these expansion coefficients can be viewed as 2Nactive sources to be determined, and Eq. (3) representsthe conditions to determine these 2N unknowns. Gener-ally, one can set the restrictions of Eq. (3) at a numberof points inside the two domains listed and then solve alinear problem to obtain the solution.

There exists a uniqueness theorem22 which can help tosimplify the condition in Eq. (3). In general situations,fixing the values of φ on the boundary (Dirichlet bound-

ary condition) can already guarantee a unique solutionin the enclosed domain for the 2D Helmholtz equation.The constraints in Eq. (3) can be replaced by

φbc(r) =

−φinc(r), for r ∈ Γc ,

φsc(r), for r ∈ Γb .(4)

According to Eq. (4), one can choose Nc sample pointson Γc and Nb points on Γb and get a total of Nc + Nb

conditions to determine the 2N active sources φα andψα. For simplicity, let us set Nc +Nb = 2N , we can thenestablish the following linear equations

[

Hca Gca

Hba Gba

] [

Φa

Ψa

]

=

[

−Φincc

Φscb

]

, (5)

where Hba, Gba and Hca, Gca represent the interactingmatrices relating the “source points” on Γa to the fieldpoints on Γb and Γc, and have elements defined as

Hαβ =

∫

ℓβ

∂ng(s, rα)dl , (6)

Gαβ = −

∫

ℓβ

g(s, rα)dl . (7)

Here ℓβ ⊂ Γa, rα ∈ Γc ∪ Γb and Φincc on the right hand

side are the incoming probing wave fields φinc sampledat the inner quite zone boundary Γc ≡ ∂Ωc, Φ

scb are the

sampled scattering fields on the outer boundary Γb ≡∂Ωb that would have been scattered by the object Vi

under the illumination of the same probing wave φinc.Thus, the field outside Γb (see, for example, the dashed-line in Fig. 1) is almost the same as (there is numericalerror due to finite sampling) that of φinc scattered bythe object Vi. If we set Φ

scb = 0, we achieve the active

external invisible cloaking proposed by Vasquez et al.17

Φa and Ψa are bothN -dimensional vectors and representthe total field and field gradients on Γa, which requireactive sources to generate. We can see that BEM is quiteefficient in calculating the active sources. To determinethe sources explicitly from the fields, we can employ theinversion procedures suggested by Miller.16 To make theidea simple, we will not distinguish between the total

3

(c) (d)

(b)(a)

FIG. 2: (Color online) Two examples of active source external cloak. (a) and (b) show the total fields, (c) and (d) the“scattering” fields. The sources are arranged on the boundaries (the white dotted-lines) of three circles in the left panels andof two crescents for the case depicted in the right panels.

fields on Γa and the the active sources that generate thosefields thereafter.

The example considered by Vasquez et al.17,20 for re-mote invisibility cloaking is achieved by simply settingΦ

scb = 0 in Eq. (5). The choice of plane wave exp(ikx)

as the incoming source is just for simplicity. The con-figurations of the cloaking devices are shown in Fig. 2for two kinds of source arrangements. Figures 2(a) and2(b) show the total fields which are generated by the “ac-tive sources” [solutions to Eq. (5)] placed on the whitedotted-lines and together with the incoming plane wave.The total fields φ(r) outside Γb (marked by the blackdashed-line), a circle of r = 20 units, are the same as theincident plane waves. At the same time, we achieve a“quiet” zone Ωc (bounded by Γc as marked by the greenlines) within a circle of r = 2 inside which the total fieldvanishes. Figures 2(c) and 2(d) show the correspond-ing “scattering fields” φsc(r) = φ(r) − φinc(r), i.e., totalfield minus the incident field, created by the active de-vices. Concomitantly, φsc(r) vanishes outside Γb and isthe reverse of φinc(r) inside the “quiet” zone Ωc. It isfurther shown (figure not presented here) that one singleconnected cloaking device can already achieve invisiblecloaking.

Next, we want to demonstrate an illusion effect such

that whatever objects placed inside the quite zone Ωc willappear like another object, chosen here to be a cylinderwith refractive index n = 1.5 and radius r = 10 units.Figures 3(a) and 3(b) show the scattered and total fieldof such a cylindrical scatterer (yellow circle) under the il-lumination of a plane wave exp(ikx). We first arbitrarilychoose the control boundary Γb (marked as the dashed-line) as a circle with radius r = 20 outside which theillusion shall be observed. This requires one to set Φ

scb

in Eq. (5) as the scattering fields due to a cylinder ofr = 10 [see Fig. 3(a)] on Γb. A total of N = 900 samplepoints are used here in the numerical calculation, andΦ

incc in Eq. (5) is the function exp(ikx) sampled over a

circle r = 2 (the boundary of the “quiet” zone Ωc markedby the green solid circle). By comparing the upper pan-els and the lower panels, it is clearly seen that the totalsystem responses to the incoming plane wave [Fig. 3(d)]in a way as the r = 10 cylinder does [Fig. 3(b)]. Wenote that we can put any non-radiating object inside Ωc

without affecting the total scattering pattern outside Γb

[see Fig. 3(c)]. We stress that we do not care what ob-ject is inside Ωc, since all the field inside Ωc is zero as isshown in Fig. 3(d), and any added passive object insidewould not affect the total fields in any other parts of thesystem. For example, we can put an apple inside Ωc, and

4

FIG. 3: (Color online) Optical illusion effect by active sources. Panels (a) and (c) show the “scattering” fields, (b) and (d)the total fields. The upper panels are for a dielectric cylinder with r = 10. The lower panels are for an illusion device, whichcomprises of sources on the boundaries of three circles, which gives the same scattering effect as the cylindrical scatter outsidethe dotted lines. Any object concealed inside the quite zone will appear as if it is a dielectric cylinder with r = 20 if an observeris sitting outside the dotted circle.

it will look like a dielectric cylinder to any observer out-side Γb. We can choose to change the sources so that theoutside observer see whatever we want them to see. Anypassive object inside Ωc does not to “talk” to the otherparts, rendering this illusion device workable for multipleand arbitrary objects. Figure 3(c) shows the scatteringfields without any object in Ωc. In such case the scat-tered fields are just the negative of the incoming planewave, as expected.

In conclusion, we used a boundary element methodto demonstrate both cloaking and illusion effects for the2D Helmholtz equation using active sources to generatefields on boundaries that do not encircle the object tobe cloaked. The scalar wave formalism applies both to

acoustic waves and EM waves in 2D. It works for arbi-trary objects and there is no intrinsic bandwidth limita-tion. Experimental demonstration for an arbitrary time-varying source would depend on the availability of sensorsthat can detect the fields quick enough on the boundariesand to respond fast enough.16

Acknowledgments

This work was supported by Hong Kong RGC GrantNo. 600209. Computation resources are supported byShun Hing Education and Charity Fund. We acknowl-edge discussions with Z. H. Hang and Jeffery Lee.

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