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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 19626–19644
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Scattering and cloaking of binary hyper-particles in metamaterials

A. Alexopoulos and K. S. B. Yau  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 19626-19644 (2010)
http://dx.doi.org/10.1364/OE.18.019626


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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

Since the initial ideas of Veselago [1

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

], there has been enormous interest in the understanding and development of metamaterial structures. The underlying physics requires that both the permittivity and permeability of such materials be simultaneously negative. This in turn implies a negative refractive index. It is of no surprise that these results have sparked research into the scattering effects of metamaterials with particles embedded in them in both two and three dimensions. In the case of two-dimensional systems in particular, the results are useful for the construction of metasurfaces which have the advantage of being less lossy compared to bulk metamaterial structures while they have application in such areas as smart surface design, cavity resonators, waveguide structures and radar absorbing surface structures to name only a few. In the last years in particular, there has been research done in order to understand the effective behavior of media with artificial particles embedded in them, with a view to obtaining some desired electromagnetic response [2–18

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]

]. One of the most interesting areas of research has been in particle cloaking. This phenomenon requires that the field be ‘wrapped’ around an object without any, or very little perturbation. To bend waves around an object in the practical sense requires materials that possess negative permittivity 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

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]

]. It is possible to theoretically predict how a material medium must be constructed in order to accommodate such bending of the fields around objects and this approach has numerous naming conventions in the literature such as field transformation method, coordinate transforms, optical transformations and transformation media to name a few [21–24

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]

]. A similar approach to these is via the use of conformal mapping but the method is restricted to two-dimensions [25–27

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

]. There is also cloaking of particles achieved by making use of their plasmonic response or via the use of plasmonic covers [6–9

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

] and [28

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

, 29

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]

]. This is an interesting approach as it also ties in with the manipulation of particle polarization [10

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

] and [30

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]

]. Other advances have been made using these concepts to obtain metamaterial structures for cloaking [31

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

], strong reduction of electromagnetic scattering using non-magnetic cloaks [32

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]

], even for optical frequencies [33

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

] and cloaking using homogeneous materials [34

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]

]. In what will follow, we shall derive the scattering characteristics of particles embedded in a medium in multi-dimensional form which also requires that these particles be expressed as hyper-particles. From our 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

σ=8π33λ04γeff2
(1)

where 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 = γeff E 0 which upon substitution into Eq. (1) gives (a similar approach can be utilized for the magnetic dipole moment),

σ=8π33λ04ptotE02
(2)

The total 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:

Fig. 1. A hyper-particle is shown in two dimensions (d = 2) with all constitutive parameters defined. Higher or lower dimensional particles have a similar configuration depending on their dimension. The definitions here hold for integer dimensions while for fractal dimensions we consider a different approach, refer to Section 4.
ptot=E0dπd2a2dΓ(d2+1)γ
(3)

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

γ=γ11γ1+d1[1+β(γ21)[(d1)γ1+1](γ11)(γ2+d1)][1+β(d1)(γ11)(γ21)(γ1+d1)(γ2+d1)]1
(4)

Here we take Γ(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 = ad 1/ad 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,

σ=8d2πd+3a22d3λ04Γ2(d2+1)γ2
(5)

Equation (5) is the total scattering cross section of a (composite) double layer 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:

σ=8d2πd+3a22d3λ04Γ2(d2+1)(γ11γ1+d1)2
(6)
Fig. 2. Scattering cross section for d = 1,2,3 and 4 non-homogeneous composite layered particles for the parameters chosen. The scattering cross section for particles with dimension d = 1 is given by the lightest curve while particles with increasing dimension d = 2,3 and 4 are represented by the darker curves respectively. Hence a particle with dimension d = 4 (darkest curve) has the smallest scattering cross section compared to other dimensions. In all cases, ε 2 = 1 means that σ approaches the holes limit (σH) while at ε 2 = 2 it means that σ tends towards the superconducting limit (σS). As the polarizability of the particles decreases, the scattering cross section becomes singular as shown by the sharp dips (σ → 0).

The scattering cross section of either double layered hyper-particles Eq. (5) or homogeneous hyper-particles Eq. (6) must both have upper and lower bounds of validity. To examine these bounds we consider the quasi-static limit, ie, at the frequency range where according to the Landau-Lifshitz theory the permeability of a particle is extremely weak and in fact ceases to have any physical meaning. The relative parameters 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

γ1=ε2ε0
γ2=ε1ε2=1
(7)

and hence Eq. (4) becomes 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:

γ1=ε2ε00
γ=1d1
(8)

Hence the polarization γ of composite double layered or homogeneous hyper-particles is bound by:

1d1γ1
(9)

As a consequence of Eq. (9), the scattering cross section is also bound and lies in the interval between the holes limit σH and superconducting limit σS:

σHσσS
(10)

where we have

σH=8d2πd+3a22d3λ04(d1)2Γ2(d2+1)
(11)

and

σS=8d2πd+3a22d3λ04Γ2(d2+1)
(12)

3. Multi-dimensional cloaking

In the last section we discussed three important limits for the scattering of hyper-particles, namely the holes, superconducting and zero polarization or scattering limits respectively. Evidently all scattering profiles are bound between σH and σS including the zero polarization case 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:

n2=ε2μ2=(λ0λ2)
(13)

where 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,

λ2=λ0ε2
(14)

In order to create a standing wave inside the spheres we require that the wavelength λ 2 inside them matches the radius a (D = 2a is the diameter). This means that we must find an integer number m = 1,2,3,…, such that

2a=mλ2m=2aε2λ0
(15)

Consider an incident electromagnetic field with wavelength 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 ΩF(d):

ΩF(d)(γ1+d1)(γ2+d1)+(d1)(γ21)(γ11)(a1a2)d=0
(16)

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

ΩC(d)=Ω(d)(γ11)(γ2+d1)(γ21)[(d1)γ1+1]+(a1a2)d=0
(17)

Solving Eq. (17) for γ 2 (or γ 1) we obtain

γ2=[1(d1)(γ11)β(γ1(d1)+1)][1+(γ11)β(γ1(d1)+1)]
(18)
Fig. 3. Field plots for d = 3 binary interacting particles (spheres) using full wave numerical analysis. The top plot (a) with ε 2 = 5.0 shows scattering by two homogeneous spherical particles. Particles cloaked via resonance (b) with ε 2 = 9 and via negative constitutive parameters for the inner core volume (c) with a 1 = 2.5 cm, ε 2 = 2.0, ε 1 = −1.69231 and β = (0.5)3. In all cases the spherical particles are surrounded by air so that ε 0 = 1 or μ 0 = 1. Common parameters are: a 2 = a = 5 cm and f 0 = 2 GHz.

and upon substitution of γ 1 and γ 2 for non-magnetic particles we have

ε1=ε2[ε2(d1)+ε0(d1)(a2a1)d(ε2ε0)ε2(d1)+ε0+(a2a1)d(ε2ε0)]
(19)

Fig. 4. Equation (17) is solved for fixed parameters as shown but varying γ 1 in order to determine which integer-dimensional particles, for the parameters given, result in σ = 0. Curve (1) (orange) indicates that for γ 1 = 0.3 the intersection with the horizontal axis at zero gives a value d > 4 which is rejected-(this would involve fractal particles). The value γ 1 = 1.1 represented by (3) (blue) indicates a value of d ≈ 6, again rejected because it is a non-integer dimension. Finally when γ 1 = 1.72 we see that the curve (2) (red) intersects the horizontal axis at d = 5. Hence a d = 5 particle is best suited in this case to cancel all scattering for the parameters given.
(a1a2)d=(γ2+d1)[(γ11)fmn(d)(γ1+d1)](γ21)[(d1)γ1+1]fmn(d)(d1)(γ11)(γ21)
(20)

where fmn(d) is an arbitrary m × n polarization matrix for m,n = 0,1,… with elements given by:

fmn(d)=(1)m(d1)mδmn
(21)

We note that in order for Eq. (20) to make physical sense, we require that the radius of the inner hyper-volume a 1 be less than the outer hyper-volume a 2 or a 1a 2 or alternatively a 1/a 2 ∈ [0,1]. Hence by the use of Eq. (21) we obtain the following interesting limits:

f00(d)=1(superconducting)
(22)
Fig. 5. The optimum ratio for the hyper-radii is calculated as a function of the permittivity (or permeability) of the outer hyper-volume that makes cloaking for d = 1,2,3,4 particles possible for all positive constitutive parameters as considered here. Notice that for any arbitrary value for ε 2, cloaking can be achieved by more than one choice of hyper-particle, provided that the ratio a 1/a 2 is adjusted accordingly. This gives us enormous insight as to the cloaking behavior of higher dimensional particles and vice-versa.

f11(d)=1d1(holes)

f 01(d) ≡ f 10(d) = 0 (cloaking)

fmn(d) = α(t) (n,m ≠ 0,1)

Fig. 6. Scattering and cloaking of d = 3 spherical particles embedded in a medium with permittivity (or permeability) of ε 0 = 2.5 (μ 0 = 2.5). Field plots on the left (a) and (c) show normal scattering of homogeneous particles while the field plots on the right (b) and (d) show cloaking via layered particles and using all positive constitutive parameters.
limd(1d1)=f01(d)=f10(d)0
(23)

We now return to the limit for cloaking of the hyper-particles given in Eq. (22). In this limit, Eq. (20) becomes:

(a1a2)d=(γ11)(γ2+d1)(γ21)[(d1)γ1+1]
(24)

Fig. 7. Scattering and cloaking of d = 2 particles (disks or cylinders). The top field plots (a) and (b) correspond to the case where the particles are surrounded by air so that ε 0 = 1 or μ 0 =1 and the cloaking effect shown top right (b) is via negative constitutive parameters for the inner layer. The bottom field plots (c) and (d) are for the particles surrounded by a medium with ε 0 = 2.5 or μ 0 = 2.5. Cloaking in plot (d) is achieved via all positive constitutive parameters.
Fig. 8. The scattering cross section (RCS) of d = 1 homogeneous and composite layered line segments, ie, d = 1 particles. The full-wave numerical simulations shown here confirm our theoretical predictions.
Fig. 9. The optimum ratio for the radii is shown for a frequency dependent outer layer based on the Drude model that makes d = 1,2,3,4 particles invisible to the field. As the particle dimension increases d → ∞, we find that the curves approach unity, ie, a 1a 2. Notice that the vertical axis implies that a realistic solution exists iff a 1a 2 so a 1/a 2 ∈ [0,1].
ε2(ω)=1ωp2ω2+αωi
(25)

for a metallic outer layer where ωp is the plasma frequency and 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]

]:

μ2(ω)=1ω2ω2ωm2+αωi
(26)

where ωm is the magnetic resonance and 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,

a1a2=(ε0ε2(ω))(ε1+(d1)ε2(ω))(ε1ε2(ω))(ε0+(d1)ε2(ω))d
(27)

Fig. 10. Numerical simulations showing cloaking effect for d = 3 (spherical) particles. In plots (a) and (b) the parameters are the same as those of Fig. 6(d) and we show here that the condition for cloaking is independent of the particle separations, even when the gap between them is zero as in (b) (touching). In (c) and (d) we have scaled the size of the particles by a factor of 1/10 and changed the frequency to f 0 = 20 GHz to verify that the cloaking effect is also independent of such things as incident frequency, particle size or number of particles.
Fig. 11. For an n = 1 iterated Koch fractal with dimension d = 1.13093, normal scattering is shown bottom left for a homogeneous binary pair and the cloaked version on the right for a composite layered Koch snowflake fractal pair predicted by our d-dimensional theory.

4. Cloaking of particles with fractal dimension

In this section we consider the very interesting case where the hyper-particles have fractal dimension. In order to examine whether our theory is consistent with full wave numerical simulations we consider a simple fractal particle that is known as the Koch snowflake [35

35. J. Gleick, Chaos: Making a New Science, (New York: Penguin Books, 1988).

, 36

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).

]. Our approach is to firstly derive the dimension of such a particle which should exist in the range 1 < 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:

dn=log(Sn)log(Nn)
Sn=4Sn1n2andS1=12
Nn=3Nn1n2andN1=9
(28)

where 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 → ∞:

An=3s24(1+3k=1n4k19k)
(29)

Taking the ratio of pseudo-areas for two initiators with side lengths s 1 and s 2 respectively we have:

s1s2=(γ11)(γ2+d1)(γ21)[(d1)γ1+1]
(30)

Hence the ratio of the lengths of two fractal initiators 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 = 0.837409s 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

We have derived a general multi-dimensional representation for the scattering of hyper-particles. We have concluded that fractal dimensional particles as well as higher integer dimensional particles have greatly reduced scattering cross section compared to particles with familiar canonical geometries. We have shown that cloaking is achievable if particles lose their polarization which could be accomplished via the fabrication of particles with increased dimension for example. Electromagnetic cloaking is shown for homogeneous particles via resonance as well as for composite particles with negative and all positive constitutive parameters. In the case of composite hyper-particles, we have studied cloaking for the situation where the particles have a frequency response according to the Drude model. The theoretical derivations have been validated via full-wave numerical simulations. The results show that it is possible to theoretically predict the conditions for electromagnetic cloaking of multi-dimensional particles. This knowledge means that we can then make use of 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.

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A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15(12), 7578–7590 (2007). [CrossRef] [PubMed]

9.

A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008). [CrossRef] [PubMed]

10.

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

11.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (New York: Wiley, 1983).

12.

X. Zhou and G. Hu, “Design for electromagnetic wave transparency with metamaterials,” Phys. Rev. E 74, 026607 (2006). [CrossRef]

13.

J. S. McGuirk and P. J. Collins, “Controlling the transmitted field into a cylindrical cloak’s hidden region,” Opt. Express 16, 17560–17572 (2008). [CrossRef] [PubMed]

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 16, 9942–9950 (2008). [CrossRef] [PubMed]

15.

H. R. Stuart and R. W. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006). [CrossRef]

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]

17.

A. Alexopoulos, “Effective response and scattering cross section of spherical inclusions in a medium,” Phys. Lett. A 373(35), 3190–3196 (2009). [CrossRef]

18.

A. Alexopoulos, “Effective-medium theory of surfaces and metasurfaces containing two-dimensional binary inclusions,” Phys. Rev. E 81, 046607 (2010). [CrossRef]

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]

22.

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). [CrossRef] [PubMed]

23.

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirrage effect,” Opt. Lett. 32, 1069–1071 (2007). [CrossRef] [PubMed]

24.

A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43 (2008). [CrossRef]

25.

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

26.

U. Leonhardt, “Notes on conformal invisibility devices,” N. J. Phys. 8, 118 (2006). [CrossRef]

27.

U. Leonhardt, “General relativity in electrical engineering,” N. J. Phys. 8, 247 (2006). [CrossRef]

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]

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]

35.

J. Gleick, Chaos: Making a New Science, (New York: Penguin Books, 1988).

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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References

  1. V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
  2. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
  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 84, 4184 (2000).
  4. R. W. Ziolkowski, and A. D. Kipple, "Application of double negative materials to increase the power radiated by electrically small antennas," IEEE Trans. Antenn. Propag. 51, 2626-2640 (2003). [CrossRef]
  5. N. Engheta, and R. W. Ziolkowski, "A positive future for double-negative metamaterials," IEEE Trans. Microw. Theory Tech. 53, 1535-1556 (2005). [CrossRef]
  6. A. Alù, and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005). [CrossRef]
  7. A. Alù, and N. Engheta, "Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights," Opt. Express 15(6), 3318-3332 (2007). [CrossRef] [PubMed]
  8. A. Alù, and N. Engheta, "Cloaking and transparency for collections of particles with metamaterial and plasmonic covers," Opt. Express 15(12), 7578-7590 (2007). [CrossRef] [PubMed]
  9. A. Alù, and N. Engheta, "Multifrequency optical cloaking with layered plasmonic shells," Phys. Rev. Lett. 100, 113901 (2008). [CrossRef] [PubMed]
  10. A. Sihvola, "Peculiarities in the dielectric response of negative-permittivity scatters," Prog. Electromagn. Res. 66, 191-198 (2006). [CrossRef]
  11. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (New York, Wiley, 1983).
  12. X. Zhou and G. Hu, "Design for electromagnetic wave transparency with metamaterials," Phys. Rev. E 74, 026607 (2006). [CrossRef]
  13. J. S. McGuirk, and P. J. Collins, "Controlling the transmitted field into a cylindrical cloak’s hidden region," Opt. Express 16, 17560-17572 (2008). [CrossRef] [PubMed]
  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 16, 9942-9950 (2008). [CrossRef] [PubMed]
  15. H. R. Stuart, and R. W. Pidwerbetsky, "Electrically small antenna elements using negative permittivity resonators," IEEE Trans. Antenn. Propag. 54(6), 1644-1653 (2006). [CrossRef]
  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]
  17. A. Alexopoulos, "Effective response and scattering cross section of spherical inclusions in a medium," Phys. Lett. A 373(35), 3190-3196 (2009). [CrossRef]
  18. A. Alexopoulos, "Effective-medium theory of surfaces and metasurfaces containing two-dimensional binary inclusions," Phys. Rev. E 81, 046607 (2010). [CrossRef]
  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]
  22. 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). [CrossRef] [PubMed]
  23. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007). [CrossRef] [PubMed]
  24. A. V. Kildishev, and V. M. Shalaev, "Engineering space for light via transformation optics," Opt. Lett. 33, 43 (2008). [CrossRef]
  25. U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
  26. U. Leonhardt, "Notes on conformal invisibility devices," N. J. Phys. 8, 118 (2006). [CrossRef]
  27. U. Leonhardt, "General relativity in electrical engineering," N. J. Phys. 8, 247 (2006). [CrossRef]
  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]
  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]
  35. J. Gleick, Chaos: Making a New Science, (New York: Penguin Books, 1988).
  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).

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