## Scattering and cloaking of binary hyper-particles in metamaterials |

Optics Express, Vol. 18, Issue 19, pp. 19626-19644 (2010)

http://dx.doi.org/10.1364/OE.18.019626

Acrobat PDF (2033 KB)

### Abstract

We derive the *d*-dimensional scattering cross section for homogeneous and composite hyper-particles inside a metamaterial. The polarizability of the hyper-particles is expressed in multi-dimensional form and is used in order to examine various scattering characteristics. We introduce scattering bounds that display interesting results when *d* → ∞ and in particular consider the special limit of hyper-particle cloaking in some detail. We demonstrate cloaking via resonance for homogeneous particles and show that composite hyper-particles can be used in order to obtain electromagnetic cloaking with either negative or all positive constitutive parameters respectively. Our approach not only considers cloaking of particles of integer dimension but also particles with non-integer dimension such as fractals. Theoretical results are compared to full-wave numerical simulations for two interacting hyper-particles in a medium.

© 2010 Optical Society of America

## 1. Introduction

1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of *ε* and *μ*,” Sov. Phys. Usp. **10**, 509 (1968). [CrossRef]

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*ε*and negative permeability

*μ*simultaneously. Materials that have a negative refractive index are not found in nature so that experimental cloaking of an object has required the use of artificial inclusions such as wires and split-ring resonators [19

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

20. 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 **312**, 1780–1782 (2006). [PubMed]

21. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E **74**, 036621 (2006). [CrossRef]

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28. A. Alù and N. Engheta, “Robustness in design and background variations in metamaterial/plasmonic cloaking,” Radio Sci. **43**, RS4S01 (2008). [CrossRef]

29. V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express **18**(10), 9780 (2010). [CrossRef] [PubMed]

10. A. Sihvola, “Peculiarities in the dielectric response of negative-permittivity scatters,” Prog. Electromagn. Res. **66**, 191–198 (2006). [CrossRef]

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

31. M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E **75**, 036603 (2007). [CrossRef]

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

33. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**, 224–227 (2007). [CrossRef]

34. W. Li, J. Guan, Z. Sun, W. Wang, and Q. Zhang, “A near perfect-invisibility cloak constructed with homogeneous materials,” Opt. Express **17**, 23410–23416 (2009). [CrossRef]

*d*-dimensional theory, we study the behavior of composite systems for various known limits, eg,

*d*= 3 spherical particles in a medium. Using our generalized result for

*d*-dimensional scattering we focus on the interesting limit of cloaking of particles and achieve this via resonance of homogeneous particles, and then by using composite layered particles with negative and positive constitutive parameters respectively. Beyond examining canonical dimensional scattering, our approach via multi-dimensional representation is crucial in the understanding of how particle interactions change the effective scattering response of metamaterial structures. Lower dimensional interactions inside a medium are much easier analyzed if we transform such problems to interacting particles in higher dimension. All this also has experimental relevance since for metamaterial systems of infinite degrees of freedom, and after integrating these out, these can sometimes take the form of higher dimensional fluctuations (interactions). As a classic example we consider fractals where the proper description of a line for instance is in fact through fractional dimensions. To clarify this a little further, we can investigate the situation of spherical particles embedded in a medium, then by integrating out the degrees of freedom inside the medium, you may have spheres that behave like hyper-spheres to a first approximation. This is because the interactions between the medium and the spherical particles can be mimicked by interactions in hyper-space. Thus we can imagine a

*d*= 1 space with a medium between particles. The effective interactions that the particles induce on the medium can look like higher dimensional interactions. One other reason why higher dimensional interactions are easier to study is because dipole-dipole interactions induced by the electromagnetic fields vanish more rapidly than those for particles of smaller dimension-we show this in Section 2. This hints at the possibility of designing metamaterials with particles of dimension higher than three which would drastically reduce scattering effects, even if the particle dimension is only slightly higher that three (fractal), for example

*d*= 3.1 which is easier in terms of design and manufacture compared to higher integer dimensional particles. Most importantly, a

*d*-dimensional theory may prove very important in the understanding of broadband effective response in metamaterial structures. One approach that might be used in order to obtain broadband behavior seems to be via the utilization of fractal particle systems, since fractals are well known for their broadband characteristics. Our approach is used to predict electromagnetic cloaking for fractal particles which are subsequently verified by full-wave numerical simulations. These results seem promising and indicate that broadband cloaking for example may be achievable through metamaterial structures consisting of fractal particles. At this point it is worth emphasizing that by the use of the word ‘cloaking’ throughout the paper we mean a substantial reduction in the scattering of the fields in comparison to normal scattering effects. This seems to be appropriate because we do not believe that it is practically possible to achieve 100% invisibility using current technology, although this may change in the future. For this reason, in order to test our theoretical predictions for cloaking in particular, via full-wave numerical analysis, we consider binary particles interacting with the electromagnetic field since any perturbation of the field by one particle will be greatly enhanced by that of the other and vice-versa. It is common in the literature to consider cloaking of single particles inside a medium but the field perturbations by these single particles might be so small that it is not possible to ascertain whether this effect is attributed to cloaking or whether such scattering reductions are due to the particle size being small with corresponding small scattering area/volume in comparison to the wavelength. Our binary calculations show that indeed cloaking is occurring since, as expected, for two interacting particles in close vicinity to each other, normal scattering conditions cause large perturbations to the incident field. The paper is organized as follows: in Section 2 we start by investigating what the general scattering cross section is for hyper-particles and consider some interesting limits. In Section 3 we look closer at the polarization characteristics of hyper-particles and for the cloaking limit in particular, examine three ways of achieving this. One way is via resonance of homogeneous particles and another is via negative constitutive parameters while the third approach is via all positive constitutive parameters. We present full-wave numerical solutions for comparison. In Section 4 we examine cloaking for non-integer dimensional particles such as fractals and give theoretical and numerical solutions for binary interacting Koch fractal-particles. We conclude the paper in Section 5.

## 2. Scattering cross section of hyper-particles

*λ*

_{0}is the wavelength of the incident electromagnetic field. In what follows we will normalize the general

*d*-dimensional scattering cross section by the coefficient of ∣·∣

^{2}in Eq. (1), since the

*d*= 3 case can be used as the reference dimension without loss of generality. The field induces a total dipole moment

**p**

_{tot}such that

**p**

_{tot}=

*γ*

_{eff}**E**

_{0}which upon substitution into Eq. (1) gives (a similar approach can be utilized for the magnetic dipole moment),

*d*-dimensional dipole moment induced by the field is defined in terms of the polarizability of the hyper-particles. In general, for two interacting hyper-particles in close proximity, there is a mutual polarization between them which is strongly dependent on the distance that separates the two hyper-particles. However we find that because the hyper-particle size is smaller than the wavelength, the leading term for the polarization of each of the hyper-particles is sufficiently accurate and independent of their separations, a hypothesis we validate later in the paper when we compare theory with full wave numerical simulations. Hence the leading

*d*-dimensional dipole moment is determined to be:

*γ*of a hyper-particle consisting of double layered hyper-volumes, viz. Fig. 1, is given by

*x*) as the gamma-function. Note that the constitutive parameters consist of

*γ*

_{1}=

*ε*

_{2}/

*ε*

_{0}and

*γ*

_{2}=

*ε*

_{1}/

*ε*

_{2}if we are dealing with relative permittivities and

*γ*

_{1}=

*μ*

_{2}/

*μ*

_{0}and

*γ*

_{2}=

*μ*

_{1}/

*μ*

_{2}if we are dealing with relative permeabilities in

*γ*corresponding to electric or magnetic particles respectively. The parameter

*β*=

*a*

^{d}_{1}/

*a*

^{d}_{2}is the ratio of the hyper-radii, obtained via consideration of the ratio of the inner and outer hyper-volumes of the particles respectively-refer to Fig. 1. From Eq. (2) and (3) we obtain the total scattering cross section for

*d*-dimensional particles embedded in a medium as being,

*d*-dimensional particle embedded inside a medium of permittivity

*ε*

_{0}and permeability

*μ*

_{0}. For hyper-particles that are homogeneous with radius, ie, only one hyper-volume, we have

*a*

_{1}= 0 and so we see that

*β*= 0 in Eq. (4). In this limit Eq. (5) reduces to a form corresponding to homogeneous hyper-particles:

*γ*

_{1}=

*ε*

_{2}/

*ε*

_{0}and

*γ*

_{2}=

*ε*

_{1}/

*ε*

_{2}imply that if the hyper-particle is superconducting, so that its permittivity tends to infinity compared to the surrounding medium, we have

*γ*= 1. On the other hand if the permittivity of the medium tends to infinity compared to the permittivity of the hyper-particle we have the so called holes limit:

*γ*of composite double layered or homogeneous hyper-particles is bound by:

*σ*and superconducting limit

_{H}*σ*:

_{S}*d*→ ∞) the holes limit in Eq. (9) implies that the hyper-particles lose their polarizations, ie,

*γ*= 0. In this instance the particles are cloaked since the impinging electromagnetic field is not perturbed by their presence. At the same time we also see that the general scattering cross section Eq. (5) also tends to zero because for any arbitrary constitutive relative parameters in Eq. (4),

*γ*→ 0 as

*d*→ ∞. Hence as the dimension of the particles increases, the scattering cross section reduces considerably. This is shown in Fig. 2 for the canonical geometrical particles

*d*= 1 line particle,

*d*= 2 circular disk (or cylindrical cross section),

*d*= 3 sphere and the hyper-particle

*d*= 4 for the typical constitutive parameters given. Furthermore, the results in Fig. 2 show that higher dimensional particles have reduced scattering cross section as expected but also exhibit a somewhat broad range of reduced

*σ*compared to lower dimensional particles. In all cases we observe narrow band resonances where the scattering cross section becomes singular, ie, where the polarization of the particles approaches zero very rapidly. Even though we can reduce

*σ*considerably in such regions for any given value

*ε*

_{2}for the outer hyper-volume as shown in Fig. 2 for example, the scattering cross section is still far from being zero and there will always be field interaction with the hyper-particles and hence overall there will be scattering. In the next section we will theoretically predict the conditions that give rise to particle cloaking and independently test these results using full-wave numerical simulations.

## 3. Multi-dimensional cloaking

*σ*and

_{H}*σ*including the zero polarization case

_{S}*γ*= 0. In what follows we will derive a general result which represents these important limits and which will allow us to consider more closely the condition for particle cloaking. We will show that the latter can be achieved by three different approaches; (i) by hyper-particle resonance, (ii) by using negative constitutive parameters and (iii) via the more interesting approach of obtaining cloaking via all positive constitutive parameters. This latter approach does not require the use of negative permittivity or permeability particles or their practical equivalents such as wires and split-ring resonators. Instead, cloaking is achieved via hyper-particles consisting of homogeneous composite hyper-layers where each layer has only positive values for the permittivity or permeability respectively. The ease of fabrication of such particles for experiments and technological use is obvious compared to traditional approaches such as field manipulation methods, eg, optical transformations or coordinate transforms and so on, that for any practical realization, always require left-handed behavior. We begin by firstly looking at creating standing waves (resonances) inside homogeneous hyper-particles and by doing so show that it is possible to resonate the field inside them so that the scattering is reduced substantially. We demonstrate this for the important

*d*= 3 limit consisting of two particles (spheres) embedded in a medium such as air, while other dimensional particles can be treated in a similar fashion. From Fig. 1 we find that for homogeneous particles

*a*

_{1}= 0 so let

*a*

_{2}=

*a*then we have:

*n*

_{2}is the refractive index and

*λ*

_{0},

*λ*

_{2}are the wavelengths. These and other parameters are shown in Fig. 1. For non-magnetic spheres we take

*μ*

_{2}= 1 so that now we have,

*λ*

_{2}inside them matches the radius

*a*(

*D*= 2

*a*is the diameter). This means that we must find an integer number

*m*= 1,2,3,…, such that

*λ*

_{0}= 15 cm corresponding to a frequency

*f*

_{0}= 2 GHz, particle radius of

*a*= 5 cm and permittivity

*ε*

_{2}= 9. From Eq. (15) we predict that

*m*= 2 gives the desired cloaking effect as can be shown in Fig. 3. The full wave numerical simulation, Fig. 3(b), shows that there are indeed

*m*= 2 wavelengths inside each of the spherical particles that give

*λ*

_{2}=

*a*from Eq. (15) and which cloak the particles compared to the top simulation, Fig. 3(a), depicting normal scattering. Note that a similar approach can be used for non-homogeneous double-layered spherical particles too. Given that resonance effects can be used for scattering reduction or enhancement we note that the Fröhlich condition which is very important in the study of surface resonances of particles can be obtained from Ω

_{F}(

*d*):

*γ*

_{1},

*γ*

_{2}and

*β*for a particle of dimension

*d*, any unknown constitutive parameter can be determined which obeys the Fröhlich condition. More precisely, Eq. (16) determines the maximum scattering that can be obtained for any given constitutive parameters in opposition to the case where there is scattering reduction or cloaking. We now consider obtaining the cloaking condition Ω

_{C}for

*d*-dimensional particles using negative constitutive parameters for the hyper-particles. This is achieved by,

*γ*

_{1}and

*γ*

_{2}for non-magnetic particles we have

*μ*

_{1},

*μ*

_{2}and

*μ*

_{0}. Equation (19) determines the value of

*ε*

_{1}of the inner layer of the hyper-particles that is needed in order to make them ‘invisible’. We consider once again

*d*= 3 binary particles and for the parameters shown in Fig. 3 predict the permittivity

*ε*

_{1}required for the inner volume so that the spheres become invisible to the field using Eq. (19). We find that the value obtained for the inner layer permittivity must be negative,

*ε*

_{1}< 0 as expected, and the cloaking effect predicted by the theory is confirmed in Fig. 3(c) using full wave analysis. Aside from the approach above, we can alternatively solve the equation for

*d*which indicates which hyper-particle is best suited for

*σ*= 0 given arbitrary values in the parameters

*γ*

_{1},

*γ*

_{2}and

*β*. Solving Eq. (17) for the fixed parameters shown in Fig. 4, different values of

*γ*

_{2}are chosen so that

*d*= 5 particles are needed if cloaking is to be achieved. Equation (17) requires numerical solution for integer

*d*. In some instances obtaining exact integer values for

*d*may require changing the parameters to accommodate. For non-integer dimensions the solutions pertain to fractal particles, see Section 4. We next consider a general expression for the hyper-particle polarizability which allows us to also study the three important limits discussed so far, namely, the holes, superconducting and cloaking limits:

*f*(

_{mn}*d*) is an arbitrary

*m*×

*n*polarization matrix for

*m,n*= 0,1,… with elements given by:

*a*

_{1}be less than the outer hyper-volume

*a*

_{2}or

*a*

_{1}≤

*a*

_{2}or alternatively

*a*

_{1}/

*a*

_{2}∈ [0,1]. Hence by the use of Eq. (21) we obtain the following interesting limits:

*f*

_{01}(

*d*) ≡

*f*

_{10}(

*d*) = 0 (

*cloaking*)

*f*(

_{mn}*d*) =

*α*(

*t*) (

*n*,

*m*≠ 0,1)

*f*(

_{mn}*d*) =

*α*(

*t*) represents the case where the polarizability of the hyper-particles is manipulated on the basis of an arbitrary function

*α*(

*t*) that could vary in time or in terms of some other parameter. In other words, suppose that the scattering profile of the hyper-particles is manipulated to achieve a desired effect, then strictly speaking, this requires that the polarizability of the hyper-particles is changed accordingly by a function

*α*(

*t*) and so for

*m*=

*n*= 2 say, we have

*f*

_{22}(

*d*) =

*α*(

*t*), where

*α*(

*t*) can be chosen so that scattering enhancement or reduction is obtained respectivley. While

*α*(

*t*) can be arbitrary and not just dependent on time, it must however be bound by the limits −1/(

*d*− 1) ≤

*α*(

*t*) ≤ 1 corresponding to the holes and superconducting limits respectively. The simplest variation with time is a linear profile such as

*α*(

*t*) =

*ct*, where

*c*is a constant. Notice that for

*t*= 0, this coincides with the cloaking limit.Before examining the cloaking limit again, it is worth mentioning that in order to obtain superconducting particles we require that the constitutive parameters of the particles be infinite in value, ie, for example

*ε*

_{2}→ ∞ as determined in the previous sections. However it can be shown that via the use of Eq. (20), it is possible to choose practical (non-infinite) values for these parameters which induce superconducting behavior for the particles at those values. This is done by adjusting the radii of the hyper-particles for any given constitutive parameters so that for

*Re*(

*β*) we obtain the superconducting limit. Furthermore, observe that from Eqs. (20), (21) and (22), as

*d*→ ∞ for the holes limit, the particles behave the same way as the cloaking limit since,

*ε*

_{2}for the outer hyper-volumes that make

*d*= 1,2,3,4,5 particles invisible to the field. Using the fixed parameters shown in Fig. 5, we chose the value

*ε*

_{2}= 5 randomly and considered

*d*= 3 binary particles in order to perform full wave simulations and examine whether there is any cloaking effect. In fact as Fig. 6 shows, we have reduced the scattering effects shown on the left in Fig. 6(a) and 6(c) to ‘zero’ as can be seen from the results on the right in Fig. 6(b) and 6(d). For the field plots on the left of Fig. 6 we see that the bottom field plot [Fig. 6(c)] shows scattering in the

*x*–

*z*plane for parameters

*f*

_{0}= 2 GHz,

*ε*

_{2}= 7.0,

*a*

_{2}=

*a*= 5.0 cm. The top plot [Fig. 6(a)] is the scattering sampled in the

*x*–

*z*and

*x*–

*y*planes respectively for an incident polarization in the

*x*–

*z*plane. For the field plots on the right of Fig. 6, the bottom field plot [Fig. 6(d)] shows cloaking of the spherical particles for parameters

*f*

_{0}= 2 GHz,

*ε*

_{2}= 5.0,

*ε*

_{1}= 2.0,

*a*

_{2}= 5 cm and

*a*

_{1}= 4.64159 cm. The bottom plot [Fig. 6(d)] shows the cloaking in the

*x*–

*z*plane while the top plot [Fig. 6(b)] also shows cloaking in the

*x*–

*y*plane thus indicating that the cloaking effect is 3D in nature. The centers of the two particles are separated by

*R*= 20 cm. We also perform full-wave numerical simulations for scattering and cloaking of

*d*= 2 particles (disks or cylindrical cross sections) shown in Fig. 7. The top field plots, Fig. 7(a) and 7(b), correspond to the case where the particles are surrounded by air so that

*ε*

_{0}= 1 or

*μ*

_{0}= 1 for magnetic particles, and the cloaking effect shown in Fig. 7(b) is via negative constitutive parameters for the inner layer. The bottom field plots [Figs. 7(c) and 7(d)] are for particles surrounded by a medium with

*ε*

_{0}= 2.5 or

*μ*

_{0}= 2.5, while the cloaking effect in Fig. 7(d) is achieved via all positive constitutive parameters. The parameters used for the homogeneous scattering [left plots Figs. 7(a) and 7(c)] in the simulations are:

*ε*

_{2}= 5.0 and

*a*

_{2}=

*a*= 5 cm. The parameters for the composite cloaked particles on the right are: [Fig. 7(b):

*a*

_{1}= 2.5 cm,

*a*

_{2}= 5.0 cm,

*ε*

_{1}= −2/7 and

*ε*

_{2}= 2.0] and [Fig. 7(d):

*a*

_{1}= 4.4096 cm,

*a*

_{2}= 5.0 cm,

*ε*

_{1}= 2.0 and

*ε*

_{2}= 5.0]. The separation of the two particle centers is once again

*R*= 20 cm. For

*d*= 1 particles, by their very nature, ie, by having a very small cross section to begin with, do not show any discernable differences in the field plots when comparing simulations between normal scattering and cloaking. For this reason we have instead considered the scattering cross section and as Fig. 8 shows, the scattering cross section (bistatic RCS) extracted from the full-wave numerical simulations confirms our theoretical predictions for

*d*= 1 binary particles (line particles). In particular, Fig. 8 shows the backward and forward scattering for homogeneous

*d*= 1 particles (blue curve) in air which have low

*σ*anyway, but also cloaking of such particles in air (red curve) and in a surrounding medium with

*ε*

_{0}= 2.5 (green curve). For the

*d*= 1 particles in air (red curve) we have used

*a*

_{1}=

*l*

_{1}= 0.025 m,

*a*

_{2}=

*l*

_{2}= 0.05 m,

*w*= 0.001 m (

*w*is the width of the

*d*= 1 particles),

*ε*

_{2}= 5.0,

*ε*

_{1}= 0.5 and

*f*

_{0}= 2 GHz. For the case where the particles are surrounded by a medium with

*ε*

_{0}= 2.5 the parameters are the same except that

*l*

_{1}= 0.0333 m and

*ε*

_{1}= 2.0. Notice that in all cases, all-positive constitutive parameters were used. Next we consider the situation where the outer hyper-layer of the particles might possess a frequency response

*ω*so that

*ε*

_{2}(

*ω*). Using the Drude model we can express the response as:

*ω*is the plasma frequency and

_{p}*α*is the damping constant. Alternatively if the particles are magnetic then we can obtain a frequency response

*μ*

_{2}(

*ω*) [16

16. J. Pendry, A. J. Holden, D. D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. **47**, 2075–2084 (1999). [CrossRef]

*ω*is the magnetic resonance and

_{m}*α*in this case is the magnetic damping constant. From Eq. (25) we can determine the optimum values for the radii

*a*

_{1}and

*a*

_{2}of the hyper-particles that for any given frequency band

*ω*make the particles transparent via the expression,

*μ*

_{0},

*μ*

_{1}and

*μ*

_{2}. Results for the Drude model are shown in Fig. 9. In all the numerical simulations so far we have used parameters that are the same. This was done deliberately in order to examine whether for a given set of identical parameters, the cloaking of hyper-particles is consistent. In other words if we had used one set of parameters for

*d*= 3 particles and then another set of parameters for

*d*= 2 particles for example, it might have appeared as though the parameters were ‘fudged’ in order to accommodate the cloaking effect for a particular particle type. Using the same fixed parameters to obtain cloaking for different hyper-particles confirms the validity of our approach. On the other hand it is interesting to ascertain whether our method is applicable in the more general sense. For example in Section 2 we stated that the dipole-dipole interactions between hyper-particles is negligible and that only the leading dipole-moment of each hyper-particle is sufficient. The latter has been the central idea in formulating our theory of hyper-particle cloaking. In order to test this hypothesis, we have performed simulations for

*d*= 3 spherical particles (for brevity) and the results are shown in Fig. 10. Using the same parameters as those in Fig. 6(d) we show in Fig. 10(a) and 10(b) that the cloaking of the spherical particles is independent of their separations, even when they are touching. The gap between the particles in Fig. 10(a) is

*r*= 5 cm while in Fig. 10(b) it is

*r*= 0 cm. Compare these results for the particle separation in Fig. 6(d) at

*r*= 10 cm. In all cases we confirm that cloaking is independent of particle separation. Furthermore, we show in Fig. 10(c) and Fig. 10(d) that the cloaking of particles is independent of the constitutive parameters

*ε*

_{0},

*ε*

_{1}and

*ε*

_{2}(or

*μ*

_{0},

*μ*

_{1}and

*μ*

_{2}) even if we scale down the size of the particles and change the frequency of the incident field. Thus for the same constitutive parameters as for the results in Fig. 6(d), 10(a) and 10(b), we consider changing the size of the particles to

*a*

_{2}= 0.5 cm and calculate the inner volume radius to be

*a*

_{1}= 0.464159 cm. At the same time the gap separation of the particles in Fig. 10(d) is

*r*= 2.0 cm. For a frequency of

*f*

_{0}= 20 GHz (

*λ*

_{0}= 1.5 cm) the results show that the approach considered in the paper for hyper-particle cloaking is also scalable and applicable to smaller hyper-particle sizes as well as different frequencies. Figure 10(c) also shows that cloaking is independent of the number of particles present thus also confirming that inter-hyper-particle interactions are negligible. In the next Section we consider cloaking of hyper-particles with non-integer dimension.

## 4. Cloaking of particles with fractal dimension

*d*< 2. The second issue is to deal with the fact that for fractal particles we can no longer define radii

*a*

_{1}and

*a*

_{2}since these parameters are non-invariant given that the boundaries (surfaces) of such particles change with each iteration. Furthermore, the particle dimension is dependent on the number of iterations

*n*:

*S*is the number of bounding segments which make up the fractal after iteration

*n*and

*N*is the number of times the initiator is segmented to produce the fractal of order

*n*. As

*n*→ ∞ we find that the dimension corresponds to the well known value for the Koch curve or snowflake, ie,

*d*

_{∞}= 1.2618…, see Fig. 11(c). In order to perform full wave simulations, we consider only the

*n*=1 iteration for the particles [Fig. 11(b)] so that from Eq. (28) we have

*d*

_{1}=

*d*=1.13093. Clearly this value lies in the interval 1 <

*d*< 2 and is therefore a fractal. Since it is physically unfeasible to assign radii

*a*

_{1}and

*a*

_{2}to fractal particles we consider the length

*s*of one side of the triangular initiator [Fig. 11(a)]. The hyper-volume of such a fractal particle is now more like a pseudo-area

*A*, and is determined by

*n*→ ∞:

*s*

_{1}and

*s*

_{2}respectively we have:

*s*

_{1}and

*s*

_{2}can be used to obtain cloaking conditions for any given constitutive parameters in Eq. (30) analogous to integer dimensional hyper-particles with radii

*a*

_{1}and

*a*

_{2}. The bottom of Fig. 11 shows the field plots obtained by full wave numerical simulations for two interacting particles of fractal-dimension corresponding to

*n*=1 iterations, refer to Fig. 11(b). The field plot on the left is for normal scattering of the fractal particles while the field plot on the right confirms the theoretical predictions for their cloaking via the use of Eq. (30). Parameters used are:

*ε*

_{0}= 2.5,

*ε*

_{1}= 2.0,

*ε*

_{2}= 5.0,

*s*

_{1}= 0.837409

*s*

_{2}cm and

*s*

_{2}= 7.5 cm. We once again point out that the cloaking effect for the binary fractal particles is obtained via all positive constitutive parameters.

## 5. Conclusion

*d*= 1,2,3, … particles as fundamental building ‘blocks’ to construct larger particles that would also exhibit cloaking. Among other things, all this may be useful in many other areas such as the design of surface structures for radar cross section (RCS) manipulation or be utilized as components for passive devices.

## References and links

1. | V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

3. | D. R. Smith, W. J. Padilla, D. C. Vier, N. Nasser, and S. C. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. E |

4. | R. W. Ziolkowski and A. D. Kipple, “Application of double negative materials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag. |

5. | N. Engheta and R. W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. |

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

7. | A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express , |

8. | A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express |

9. | A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett. |

10. | A. Sihvola, “Peculiarities in the dielectric response of negative-permittivity scatters,” Prog. Electromagn. Res. |

11. | C. F. Bohren and D. R. Huffman, |

12. | X. Zhou and G. Hu, “Design for electromagnetic wave transparency with metamaterials,” Phys. Rev. E |

13. | J. S. McGuirk and P. J. Collins, “Controlling the transmitted field into a cylindrical cloak’s hidden region,” Opt. Express |

14. | S. Tomita, T. Yokoyama, H. Yanagi, B. Wood, J. B. Pendry, M. Fujii, and S. Hayashi, “Resonant photon tunneling via surface plasmon polaritons through one-dimensional metal-dielectric metamaterials,” Opt. Express |

15. | H. R. Stuart and R. W. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. |

16. | J. Pendry, A. J. Holden, D. D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. |

17. | A. Alexopoulos, “Effective response and scattering cross section of spherical inclusions in a medium,” Phys. Lett. A |

18. | A. Alexopoulos, “Effective-medium theory of surfaces and metasurfaces containing two-dimensional binary inclusions,” Phys. Rev. E |

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

20. | 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 |

21. | S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E |

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

23. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirrage effect,” Opt. Lett. |

24. | A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. |

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

26. | U. Leonhardt, “Notes on conformal invisibility devices,” N. J. Phys. |

27. | U. Leonhardt, “General relativity in electrical engineering,” N. J. Phys. |

28. | A. Alù and N. Engheta, “Robustness in design and background variations in metamaterial/plasmonic cloaking,” Radio Sci. |

29. | V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express |

30. | G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. Lond. A Math Phys. Sci. |

31. | M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E |

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

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

34. | W. Li, J. Guan, Z. Sun, W. Wang, and Q. Zhang, “A near perfect-invisibility cloak constructed with homogeneous materials,” Opt. Express |

35. | J. Gleick, |

36. | J. W. Harris and H. Stocker, “Koch’s Curve” and “Koch’s Snowflake,” 4.11.5-4.11.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 114–115, (1998). |

**OCIS Codes**

(160.3918) Materials : Metamaterials

(290.5839) Scattering : Scattering, invisibility

**ToC Category:**

Scattering

**History**

Original Manuscript: August 25, 2010

Revised Manuscript: August 25, 2010

Manuscript Accepted: August 26, 2010

Published: August 31, 2010

**Citation**

Aris Alexopoulos and Bobby Yau, "Scattering and cloaking of binary hyper-particles in metamaterials," Opt. Express **18**, 19626-19644 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19626

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