Rapid Publications: The European Optical Society
Rapid Publications: The European Optical Society
Rapid Publications: The European Optical Society
Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) www.jeos.org
T HScattering
E E U R O P E A N
cancellation by metamaterial cylindrical
O P T I CAL SO CI ET Y
multilayers
R A P ID P U B LIC A T IO N S
Simone Tricarico
Filiberto Bilotti
bilotti@uniroma3.it
Department of Applied Electronics, University ”Roma Tre”, Via della Vasca Navale,
84 – 00146 Rome, Italy
Department of Applied Electronics, University ”Roma Tre”, Via della Vasca Navale,
84 – 00146 Rome, Italy
Lucio Vegni Department of Applied Electronics, University ”Roma Tre”, Via della Vasca Navale,
84 – 00146 Rome, Italy
In this paper, we present the theoretical analysis and the design of cylindrical multilayered electromagnetic cloaks based on the scattering
cancellation technique. We propose at first the analysis and the design of bi-layered cylindrical shells, made of homogenous and isotropic
metamaterials, in order to effectively reduce the scattered field from a dielectric cylindrical object. The single shell and the bi-layered
shell cases are compared in terms of scattering reduction and loss effects. The comparison shows that the bi-layered configuration
exhibits superior performances. The scattering cancellation approach, is, then, extended to the case of generic multilayered cylindrical
shells, considering again homogeneous and isotropic metamaterials. The employment of the proposed technique to the case of cloaking
devices working at multiple frequencies is also envisaged and discussed. Finally, some practical layouts of cylindrical electromagnetic cloaks
working at optical frequencies are also proposed. In these configurations, the homogenous and isotropic metamaterials are replaced by their
actual counterparts, obtained using alternating stacked plasmonic and non-plasmonic layers. The theoretical formulation and the design
approaches presented throughout the paper are validated through proper full-wave numerical simulations. [DOI: 10.2971/jeos.2009.09021]
1 INTRODUCTION
Recently, the possibility to synthesize properly engineered objects by exploiting the unusual properties of their local neg-
cloaks of invisibility by employing metamaterials and plas- ative polarizability. This approach has the advantage of using
monic media has determined a growing interest in the study homogeneous covers, reducing the complexity of the design,
of transparency and scattering-free phenomena. Several and is not based on intrinsically resonant phenomena, provid-
groups worldwide have proposed alternative solutions [1]– ing good performances in terms of bandwidth and response
[10] based on really different mechanisms, such as anomalous to geometric variations [8, 9] and losses. Since the covers must
localized resonances [1] or coordinate transformations [2]. be inherently dispersive to obtain the desired combination of
constitutive parameters, the cloaking effect is practically lim-
In particular, the approach relying on conformal mapping ited by the dispersion of the involved media.
techniques is quite elegant and has led to some successful
results, both at optical and microwave frequencies [3]–[5]. At optical frequencies, noble metals exhibit already the re-
The main advantage of such cloaking devices is that they are quired negative polarizability to enable the scattering can-
mostly object independent [6]. This is because the approach cellation [10]. However, such materials exhibit a strong plas-
results eventually in a rerouting of the electromagnetic ra- monic behavior and high losses at the visible frequencies. A
diation around the obstacle, effectively reducing the scatter- negative polarizability may be obtained also using materi-
ing outside the cloaked object. Some issues related to this als with close-to-zero values of the permittivity. Such materi-
approach concern the operative bandwidth, the losses, and als, called Epsilon-Near-Zero (ENZ) metamaterials, show su-
the complex design of the covers, based on inhomogeneous perior performances in terms of operational bandwidth and
and anisotropic materials. Such drawbacks limit the valuable losses. However, the problem is that nature does not offer
cloaking effect to a narrow range of frequencies and to a given ENZ materials at visible frequencies and, thus, they have to
polarization. be obtained artificially. One possibility has been presented
in [11], where the authors have shown how the employment of
An interesting alternative solution relies on the scattering can- alternating stacked plasmonic and non-plasmonic layers can
cellation technique proposed in [7]. The scattering cross sec- be effectively used to synthesize the required permittivity pro-
tion of a given object can be, in fact, drastically reduced by em- file and values at optical frequencies.
ploying plasmonic and metamaterial cloaks, which can sup-
press the multi-polar scattering of relatively electrically large In this paper, we extend the scattering cancellation approach
Received December 01, 2008; published May 07, 2009 ISSN 1990-2573
Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) S. Tricarico et. al.
to the case of multi-layered cylindrical cloaks, which effec- and order n. The total field E outside the covered cylinder is
tively reduce the scattering cross-section of cylindrical objects. given by the sum of Ei and scattered electric field Es , which
A theoretical analysis of the structure is firstly presented, lead- can be consistently written as:
ing to the design formulas for multi-layered cylindrical cloaks.
The actual implementation of the cylindrical shells work- ∞ h i
∑ j−n δn
(2)
E = Ei + Es = ẑ Jn (k0 r ) + cn Hn (k0 r ) cos nφ,
ing at optical frequencies is obtained using the approach al- n =0
ready proposed in [11]. Finally, some results obtained through
proper full-wave simulations are presented, in order to vali- (2)
where r ≥ r3 , Hn is the Hankel function of the second kind
date the proposed theoretical approach and to show its appli-
of integer order n, and cn the unknown scattering coefficients.
cability also to the case of multi-frequency cloaking operation.
The fields inside each cylindrical region can be factorized in
terms of radial and azimuthal functions. The electric field in
2 THEORY AND DESIGN OF A the cylindrical object can be, thus, expressed as:
BI-LAYERED CYLINDRICAL CLOAK
∞
Let’s consider a time harmonic (the dependence e jωt is as- E(1) = ẑ Ez,1 = ẑ ∑ j−n δn `n Jn (k1 r) cos nφ, 0 ≤ r ≤ r1
sumed in the paper) monochromatic plane wave with unit n =0
amplitude, which normally impinges on an infinitely long cir-
cular cylinder of radius r1 , with permittivity ε 1 and perme- where k1 is the wave number in the medium given
√ √
ability µ1 . We firstly assume that the cylinder is covered by by k1 = ω µ1 ε 1 = k0 µr1 ε r1 , being (ε r1 , µr1 ) the relative
two concentric homogeneous shells, with radii r2 and r3 , and permittivity and permeability of the material, respectively.
constitutive parameters (ε 2 , µ2 ) and (ε 3 , µ3 ), respectively (Fig-
ure 1). The entire structure is surrounded by vacuum (ε 0 , µ0 ). In the two cylindrical shells, the solution can be expressed as:
We study the case of a TMz polarized impinging wave, while
(2)
the results for the TEz polarization may be obtained by dual- E(2) = ẑ Ez
ity. ∞
= ẑ ∑ j−n δn [an Jn (k2 r) + bn Yn (k2 r)] cos nφ, r1 ≤ r ≤ r2
n =0
(3)
E(3) = ẑ Ez
∞
= ẑ ∑ j−n δn [dn Jn (k3 r) + f n Yn (k3 r)] cos nφ, r2 ≤ r ≤ r3
n =0
(2)
√ √ √ √
where k2 = ω µ2 ε 2 = k0 µr2 ε r2 , k3 = ω µ3 ε 3 = k0 µr3 ε r3 ,
and Yn is the Bessel function of the second kind and order n.
an , bn , dn , f n are unknown coefficients to be determined.
∞
k2
∑ j−n δn
(2)
an Jn0 (k2 r ) + bn Yn0 (k2 r ) cos nφ,
Hφ =
jωµ2 n =0
r1 ≤ r ≤ r2
∞
k3
∑ j−n δn
(3)
dn Jn0 (k3 r ) + f n Yn0 (k3 r ) cos nφ,
Hφ =
jωµ2 n =0
FIG. 1 TM polarized plane wave impinging on an infinitely long circular cylinder, cov- r2 ≤ r ≤ r3
ered with a bi-layered homogeneous shell.
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Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) S. Tricarico et. al.
The previous relations can be written as a system in the form: where the positions r2 = α r1 and r3 = β r1 , being β > α > 1,
have been used.
an Jn (k2 r1 ) + bn Yn (k2 r1 ) = `n Jn (k1 r1 )
dn Jn (k3 r2 ) + f n Yn (k3 r2 ) = an Jn (k2 r2 ) + bn Yn (k2 r2 ) In order to minimize the scattering cross section of the cylin-
(2)
Jn (k0 r3 ) + cn Hn (k0 r3 ) = dn Jn (k3 r3 ) + f n Yn (k3 r3 ) drical object, thus, this first order scattering coefficient c0 does
go to zero. This happens when the following condition holds:
k2 k
an Jn0 (k2 r1 ) + bn Yn0 (k2 r1 ) = 1 `n Jn0 (k1 r1 )
ωµ2 ωµ1
ε r1 + α2 − 1 ε r2 + β2 (ε r3 − 1) − α2 ε r3 = 0 (12)
k3 0
dn Jn (k3 r2 ) + f n Yn0 (k3 r2 )
ωµ3
This means that the ratio α between the radius of the first shell
k
= 2 an Jn0 (k2 r2 ) + bn Yn0 (k2 r2 ) and the radius of the object should match the relation:
ωµ2
s
(2)
Jn0 (k0 r3 ) + cn Hn 0 (k0 r3 ) ε r2 − ε r1 − β2 (ε r3 − 1)
α0 = (13)
k3 0 ε r2 − ε r3
dn Jn (k3 r3 ) + f n Yn0 (k3 r3 )
=
ωµ3
while for the higher order terms n = 1, 2, . . . it can be proven
(4)
that the relation becomes:
or, in matrix form as S · wT = v, where v =
{0, 0, 0, 0, Jn (k0 r3 ) , Jn0 (k0 r3 )} and S is the matrix of the αn =
system coefficients defined by Eq. (5). " #1
(µr2 + µr3 ) β2n(µr3 − 1)(µr1 − µr2 )+(µr3 + 1)(µr1 + µr2 ) + Mn 2n
gation), then the cn vanish, being |Sc | zero (two columns are
linearly dependent), while for ε r3 = µr3 = 1 we reduce to the
If the object has no magnetic properties, that is
single cover case [13].
µr1 = µr2 = µr3 = 1, then the zero scattering condition
for the generic order n, may be obtained when:
We can also express the scattering coefficients cn in a more
convenient form, which will be useful in the next section, s
2n+2 ε r2 − ε r1 − β2n+2 (ε r3 − 1)
when we will extend this approach to multiple layers. αn = , n = 0, 1, 2 . . . (16)
ε r2 − ε r3
(2)
Remembering that Hn = Jn − jYn and using the multi-linear Since the higher order scattering coefficients vanish at a higher
properties of the determinants, we have: rate (as shown in Figure 2), in the quasi static limit the condi-
Bn tion Eq. (13) is sufficient to guarantee a sensible reduction of
cn = − (7)
Bn + j An the scattering cross section of the cylindrical object.
where Bn and An are given by Eqs. (8) and (9) respectively.
Using this form, which is similar to [13], it results that the ze-
roes of cn occur when Bn = 0, being the denominator a finite
quantity.
( k 0 r1 )2 h i
c0 ≈ jπ ε r1 + α2 − 1 ε r2 + β2 (ε r3 − 1) − α2 ε r3 FIG. 2 Relative amplitude of the scattering coefficients of order n > 0 for a TM po-
4 larized plane wave impinging on an infinitely long dielectric circular cylinder with
+ o ( k 0 r1 )4 (11) ε r1 = 4, ε r2 = 2, ε r3 = 3 and α = 1.5, β = 2.
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Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) S. Tricarico et. al.
− Jn (k1 r1 ) Jn (k2 r1 ) Yn (k2 r1 ) 0 0 0
k1 0 k2 0 k2 0
− ωµ J ( k 1 r1 ) ωµ2 Jn ( k 2 r1 ) ωµ2 Yn ( k 2 r1 ) 0 0 0
1 n
0 Jn (k2 r2 ) Yn (k2 r2 ) − Jn (k3 r2 ) −Yn (k3 r2 ) 0
S= (5)
k2 0 k2 0 k3 0 k3
0 ωµ2 Jn ( k 2 r2 ) ωµ2 Yn ( k 2 r2 ) − ωµ J ( k 3 r2 )
3 n
− ωµ Y 0 ( k 3 r2 )
3 n
0
(2)
0 0 0 Jn (k3 r3 ) Yn (k3 r3 ) − Hn (k0 r3 )
k3 0 k3 0 (2)
0 0 0 ωµ3 Jn ( k 3 r3 ) ωµ3 Yn ( k 3 r3 ) − Hn 0 (k0 r3 )
Jn (k1 r1 ) Jn (k2 r1 ) Yn (k2 r1 ) 0 0 0
k1 0 k2 0 k2 0
ωµ1 Jn ( k 1 r1 ) ωµ2 Jn ( k 2 r1 ) ωµ2 Yn ( k 2 r1 ) 0 0 0
Jn (k2 r2 ) Yn (k2 r2 ) − Jn (k3 r2 ) −Yn (k3 r2 )
0 0
k2 0 k2 0 k3 0 k3
Y 0 ( k 3 r2 )
0 ωµ2 Jn ( k 2 r2 ) ωµ2 Yn ( k 2 r2 ) − ωµ J ( k 3 r2 )
3 n
− ωµ 3 n
0
0 0 0 Jn (k3 r3 ) Yn (k3 r3 ) Jn (k0 r3 )
k3 0 k3 0 Jn0 (k0 r3 )
0 0 0 ωµ3 Jn ( k 3 r3 ) ωµ3 Yn ( k 3 r3 )
cn = − (6)
|S|
Jn (k1 r1 ) Jn (k2 r1 ) Yn (k2 r1 ) 0 0 0
k1 0 k2 0 k2 0
ωµ1 Jn ( k 1 r1 ) ωµ2 Jn ( k 2 r1 ) ωµ2 Yn ( k 2 r1 ) 0 0 0
Jn (k2 r2 ) Yn (k2 r2 ) − Jn (k3 r2 ) −Yn (k3 r2 )
0 0
Bn = k2 0 k2 0 k3 0 k3 (8)
Y 0 ( k 3 r2 )
0 ωµ2 Jn ( k 2 r2 ) ωµ2 Yn ( k 2 r2 ) − ωµ J ( k 3 r2 )
3 n
− ωµ 3 n
0
0 0 0 Jn (k3 r3 ) Yn (k3 r3 ) Jn (k0 r3 )
k3 0 k3 0 Jn0 (k0 r3 )
0 0 0 ωµ3 Jn ( k 3 r3 ) ωµ3 Yn ( k 3 r3 )
Jn (k1 r1 ) Jn (k2 r1 ) Yn (k2 r1 ) 0 0 0
k1 0 k2 0 k2 0
ωµ1 Jn ( k 1 r1 ) ωµ2 Jn ( k 2 r1 ) ωµ2 Yn ( k 2 r1 ) 0 0 0
Jn (k2 r2 ) Yn (k2 r2 ) − Jn (k3 r2 ) −Yn (k3 r2 )
0 0
An = k2 0 k2 0 k3 0 k3 (9)
Y 0 ( k 3 r2 )
0 ωµ2 Jn ( k 2 r2 ) ωµ2 Yn ( k 2 r2 ) − ωµ J ( k 3 r2 )
3 n
− ωµ 3 n
0
0 0 0 Jn (k3 r3 ) Yn (k3 r3 ) −Yn (k0 r3 )
k3 0 k3 0 −Yn0 (k0 r3 )
0 0 0 ωµ3 Jn ( k 3 r3 ) ωµ3 Yn ( k 3 r3 )
β2 α2 ε r1 − ε r2
ε r1 − ε r2
2
+ 2 = 1, where = γ2 , = η2
γ η 1 − ε r3 ε r3 − ε r2
(17)
Only a given set of values of the electric parameters satisfies
this relation. The equation, in fact, admits solutions only if the
curve represented by Eq. (16) lies within the region delimited
by the condition β > α > 1, depicted in Figure 3. Eq. (17)
is a quadratic form that, according to the relative permittivity
values ε r1 , ε r2 , ε r3 , may represent in the plane (α, β) either a
circle, or an ellipse, or an hyperbola.
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Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) S. Tricarico et. al.
FIG. 4 Region of the (α,β) plane in which the zero scattering condition admits solutions FIG. 6 Regions of the (α,β) plane in which the zero scattering condition admits solu-
when γ2 = η 2 = ζ 2 > 2. tions when γ > 1, η > 1, (case γ > η).
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Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) S. Tricarico et. al.
η 2 involving the electrical parameters of the object and of the In this case, we have an upper limit for the values of β, and
shells, in fact, both quantities cannot be negative at the same the design equations for the two shells are summarized as:
time. Moreover the electric parameters, which determine the
sign of either γ2 or η 2 , inherently impose some limitations on ε r1 > 1 > ε r2 > ε r3 or ε r1 < 1 < ε r2 < ε r3 (29a)
the ranges of allowable values for such quantities. Let’s con- β2 α2
sider, for instance, the case η 2 < 0, which is verified once ei- − 2 =1 (29b)
γ2 |η |
ther ε r1 < ε r2 < ε r3 or ε r1 > ε r2 > ε r3 . In this case, Eq. (17)
ε r1 − ε r3 ε − ε r2
becomes: < β2 < r1 (29c)
β2 α2 1 − ε r3 1 − ε r2
2
− 2 = 1. (24)
γ |η |
Only if γ2 < η 2 1 + η 2 Eq. (24) does not admit solutions,
Since we are interested only in a portion of the (α, β) plane, since the entire curve does not intersect the region of interest.
implying 2
2 2
η To summarize, we may state that, given an object made of a
β min > 1 ⇒ γ > , (25)
1 + |η 2 | regular dielectric (ε r1 > 1), the choice of the effective parame-
ters for the cover shells such that ε r3 < ε r2 < 1, always leads
in order to have solutions it should be at least γ2 η 2 > 1.
to a suitable solution. These anomalous values of the relative
Only in this case, in fact, we may have one intersection β2min >
permittivity can be achieved, thus, employing proper meta-
1 with the line α = 1:
materials with the required either negative or near-zero per-
γ2 ε r1 − ε r3 mittivity values.
β2min =
2
2
1 + η = >1 (26)
|η | 1 − ε r3
Before concluding this sections, it is worth noticing that if in
(i.e. either ε r1 > 1 > εp
r3 or ε r1 < 1 < ε r3 ), being the asymptote Eq. (17) it results that either γ2 → ∞ or η 2 → ∞, the struc-
of the hyperbola β = γ2 / |η 2 |α within the region β > α > 1 ture collapses into the one with one cover shell only and, thus,
(see Figure 7). the solutions turns into those ones already presented in [13].
γ2 → ∞ , in fact, implies that ε r3 = 1, which means that the
outer shell is removed. η 2 → ∞ implies, instead, that ε r3 = ε r2 ,
which means that the two shells merge into one.
In this particular case, the design formulas for the two shells
can be summarized as:
γ2 η 2
ε − ε r2 FIG. 8 TM-polarized plane wave impinging on an infinitely long circular cylinder, cov-
2
β max = 2 = r1 (28) ered with a m-layered inhomogeneous shell.
| η | − γ2 1 − ε r2
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Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) S. Tricarico et. al.
This approach is conveniently summarized in [14], where the is a valid approximation even for relatively electrically larger
following recursive formula is derived: object. This approach may be, indeed, also extended to a more
general case. It is, in fact, always possible to consider a finite
Am+1,n Umn Wmn Am,n set of scattering coefficients from Eq. (10), and impose numer-
= (30)
Bm+1,n Vmn Xmn Bm,n ically that they shall vanish at the desired design frequency.
This can be done by truncating the scattering series at a suffi-
where:
ciently high order and minimizing it, as also suggested in [7].
Umn = µm k m+1 Jn (k m rm ) Yn0 (k m+1 rm ) In this case, the scattering cancellation condition can be easily
obtained with a multi-layered structure, employing materials
− µm+1 k m Jn0 (k m rm ) Yn (k m+1 rm )
with suitable electric parameters, retrieved by a numerical op-
Vmn = µm+1 k m Jn (k m+1 rm ) Jn0 (k m rm ) timization procedure.
− µm k m+1 Jn0 (k m+1 rm ) Jn (k m rm )
Wmn = µm k m+1 Yn (k m rm ) Yn0 (k m+1 rm )
− µm+1 k m Yn0 (k m rm ) Yn (k m+1 rm )
Xmn = µm+1 k m Yn0 (k m rm ) Jn (k m+1 rm )
− µm k m+1 Yn (k m rm ) Jn0 (k m+1 rm )
(31)
√ √
and k m = ω µm ε m = k0 µrm ε rm . The amplitude of the scat-
tering coefficients is related to Eq. (30) by the equivalent of
Eq. (7), as:
B M+1,n
cn = − (32)
B M+1,n + jA M+1,n
The first entries in the recursion formula may be assumed as:
2k0 µ0
Um0 ≈ + o (k0 rm )
πk0 rm
1
Vm0 ≈ k20 µ0 rm (ε rm+1 − ε rm ) + o (k0 rm )3
2
2k µ ε rm
Wm0 ≈ 2 0 0 log + o (k0 rm )
π k0 rm ε rm1 FIG. 9 Amplitude of the exact recursion matrix coefficients Um0 , Xm0 and of the
2k µ
Xm0 ≈ 0 0 + o (k0 rm ) approximated ones Ûm0 , X̂m0 for two successive layers with dielectric contrast
πk0 rm ε rm+1 − ε rm = 1.
(34)
Substituting Eqs. (34) in Eq. (30) one can get a straight itera- 4 NUMERICAL RESULTS
tive rule to easy design multi-layered cylindrical covers. It is
straightforward to verify that in the case of a bi-layered cover In order to validate the theoretical analysis for the multi-
the recursion procedure here outlined returns exactly Eq. (15) layered cylindrical cloaking structures presented so far, we
obtained previously. returns exactly obtained previously. Dur- performed a set of numerical full-wave simulations through
ing the recursion procedure, one may neglect the terms which a commercial software based on the Finite Integration Tech-
are infinitesimal of higher order, but obviously increasing the nique. In the simulations we have taken into account some
numbers of layers it is more difficult to satisfy the quasi static real life aspects, such as material losses, and compared them
approximation, because the global thickness of the structure to the ideal results obtained in the previous sections.
progressively get electrically larger. Nevertheless, following
the results presented in Figure 8, Eqs. (34) may be considered Since we are interested in a configuration working at optical
as accurate in a certain range of electrical thicknesses of the frequencies, we firstly considered a dielectric cylindrical ob-
shells. In fact, even if the proposed closed formulas have been ject of finite length L = 500 nm and radius r1 = 30 nm made
obtained in the case of electrically small objects, the dipolar of silica (SiO2 ) and illuminated by a monochromatic plane
term used to describe the scattering behavior of the structure wave at the frequency of 600 THz. Since the far field scattering
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Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) S. Tricarico et. al.
The field lines, which are deformed by the scattering from the
object in the case of the bare cylinder, are restored to the ones
of a plane wave passing through the structure in the case of
the cylindrical object covered by the cloak. This result is high-
lighted by the phase distributions shown in Figure 14.
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Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) S. Tricarico et. al.
FIG. 13 Magnetic field distribution at the cloak frequency f = 600 THz. An example of the actual material implementation, related to
the design of Figure 15, is reported in Figure 16. The cylindri-
cal object made of silica is surrounded by two material shells.
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Journal of the European Optical Society - Rapid Publications 4, 09021 (2009) S. Tricarico et. al.
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