Tunable invisibility cloaking by using graphene-coated nanowires

0.34 MB·

by  Mahin Naserpour

Tunable invisibility cloaking by using
graphene-coated nanowires
Mahin Naserpour∗ and Carlos J. Zapata-Rodr´ıguez†
Department of Optics and Optometry and Vision Science,
University of Valencia,
Dr. Moliner 50, 46100 Burjassot, Spain
Slobodan M. Vukovi´c‡ and Milivoj R. Beli´c
Texas A&M University at Qatar, P.O. Box 23874, Doha, Qatar
7 (Dated: January 11, 2017)
1
We investigate, both theoretically and numerically, a graphene-coated nano-cylinder illuminated
0
by a plane electromagnetic wave in THz range of frequencies. We have derived an analytical for-
2
mula that enables fast evaluation of the spectral window with a substantial reduction in scattering

n efficiency for sufficiently thin cylinder. This effect leads to tunable resonant invisibility that can
a be achieved via modification of graphene chemical potential monitored by the gate voltage. A
J multi-frequencycloaking mechanism based on dimercoated nanowires is also discussed.

0
1 PACSnumbers:
Keywords: Nanowires,Plasmonics,THzOptics

]
s
c
I. INTRODUCTION ablesthecancellationofscatteringeffects. Drasticallyre-
i
t duced overall visibility of the scattering object that was
p
o conducted via graphene monolayer wrapping the cylin-
Recent progress in design and manufacturing of new
. der, for the first time, was extended to the far-infrared
s photonic materials has enabled novel models and mul-
c range of frequencies by Chen and Alu [10]. In that case,
tifunctional devices to achieve unprecedented control of
i the local polarizability of a nanowire with moderately
s light. Inparticular,byusingvarioustechniquesbasedon
y thick diameter and the graphene coating sheet with the
transformation optics, the possibilities have appeared to
h opposite signs can be mutually cancelled under suitable
electromagneticallyisolate a space regionfor certain fre-
p layout. Different schemes have been proposed with plas-
[ quenciesorrangeoffrequencies. Itmeansthatanobject
monic compounds, using a dipole moment of opposite
1 locatedinsidesuchaspaceregionwillpracticallystopin- phase to attain a scattering cancellation [11–13]. If the
teracting with illuminating light [1–3]. In fact, a remote
v incidentradiationisp-polarized,however,thespectralre-
observerwillnotbeabletodetectthepresenceoftheob-
7 sponse of the scattering crosssection seems to be totally
jects thatareshieldedandprotectedbyopticalcloaking.
8 dissimilar [14]. Then, a set formedby a peak, associated
4 Amajordrawbackinpracticalapplicationsofthesetech-
with the scattering via a narrow Mie mode, and a valley
2 niquesisduetothepropertiesofmaterialsneeded,either
offers the prospect to obtain invisibility with unprece-
0 artificialor natural,thatexhibit extreme permittivity or
dented control over the wire resonance spectral location
1. permeability. However,coatings can be designedsuch as
by changing the chemical potential.
0 metal-dielectric multilayer or metasurfaces to drastically
In the present paper we extend the previous concept
7 reduce the scattered signalfrom subwavelengthparticles
byemployingap-polarizedincidentplanewavethatillu-
1 and make the assembly nearly undetectable [4–7]. Of
: minatesagraphenecoateddielectricnanowireinorderto
v course,thelightinteractswiththenanoparticles,butthe
get near the fundamental localized surface plasmon po-
i destructive interference ofdifferent elements that consti-
X lariton. Subsequently we optimize the scattering cancel-
tute the complex nanostructure practically cancels out
lationofthenanocavity. Theproposedtheoreticalmodel
r the scattered radiation.
a is critically discussedwitha conclusionthat a character-
Graphene represents a groundbreaking and exciting istic lineshape appears comparable to the shape of Fano
material that combines suitable characteristics for the resonance [15]. The check of validity and estimation of
use in optoelectronic devices in far-infrared and THz re- our results is performed by using the full-wave Lorenz-
gion of frequencies due to the intrinsic ability to mold Mie method [16].
the surface current with low-loss rates [8, 9]. High con-
ductivity within atomically thin layer of graphene, and
significanttuningabilityviatheappliedbiasvoltage,en-
II. FORMULATION OF THE PROBLEM.
ANALYTICAL CONSIDERATIONS
Let us consider a cylindrical dielectric nanowire of ra-
∗AlsoatDepartmentofPhysics,ShirazUniversity,Iran
†Electronicaddress: [email protected] dius R and relative permittivity ǫd over-coated by an
‡AlsoatIHTM,UniversityofBelgrade,Serbia atomicallythingrapheneshell,andorientedalongz axis,
2
as illustrated in Fig. 1. The nanowire is assumed to be
placed in a lossless environment medium of the relative
permittivityǫandilluminatedbyaplaneelectromagnetic
wave that propagates along the x axis with the electric
field vector E in the xy plane. The graphene conduc-
in
tivity σ within the coating shell is described in the local
randomphaseapproximationbyusingKuboformula[17]
as a sum of intraband σ and interband σ contri-
intra inter
bution, where
FIG. 1: Illustration of the atomically thin graphene layer of
conductivityσcoatinganano-sizedscatteringcylinderofper-
2ie2k T µ
mittivityǫd,wheretheenvironmentmediumhasapermittiv- σ = B ln 2cosh , (1)
ityǫ. HereweconsideranincidentTEz-polarizedplanewave. intra π~2(ω+iΓ) 2k T
(cid:20) (cid:18) B (cid:19)(cid:21)
and
e2 1 1 ~ω 2µ i (~ω+2µ)2
σ = + arctan − ln . (2)
inter 4~"2 π (cid:18) 2kBT (cid:19)− 2π (~ω−2µ)2+(2kBT)2#
Here, e is the charge of an electron, ~ is the reduced may be set as [16]
−
Planck’s constant, k is Boltzmann’s constant, T is the
B
temperature,Γisthe chargecarriersscatteringrate,µ is +∞
thechemicalpotentialand~ωisthephotonenergy. Also, Hsca =zˆH0 aninHn(1)(kr)exp(inφ), (5)
the time harmonic dependence of the field is adopted to n=−∞
X
be of the form exp( iωt).
− and
Toestimateanalyticallythescatteringefficiencyofthe
coated nanowire, we followed the Lorenz-Mie scattering E = E +∞ a in rˆnHn(1)(kr) +φˆiH(1)′(kr)
method given in detail for instance in Refs. [10, 14, 18– sca − 0 n " kr n #
20]. Asmentionedabove,weassumethatthenanotubeis n=X−∞
illuminated by a TEz-polarized plane wave propagating exp(inφ),(6)
×
alongthexaxis,asillustratedinFig.1. Theelectromag-
netic field of the incident plane wave may be set as where H(1)() = J ()+iY () is the Hankel function of
n n n
· · ·
the first kind and order n, and the coefficients a must
n
be determined. The total magnetic field in the environ-
+∞ ment medium is simply H =H +H . Finally, the
Hin =zˆH0exp(ikx)=zˆH0 inJn(kr)exp(inφ), electromagnetic field in thteotdielecitnric corsecaof the coated
n=−∞ cylinder (r <R) is expressed as
X
(3)
and Ein =i Hin/ωǫǫ0, which reads as +∞
∇×
H =zˆH b inJ (k r)exp(inφ), (7)
d 0 n n d
n=−∞
+∞ X
J (kr)
Ein =−E0 in rˆn nkr +φˆiJn′ (kr) exp(inφ), and
n=−∞ (cid:20) (cid:21)
X
(4) k H +∞ J (k r)
where r and φ are the radial and azimuthal cylindri- Ed =−ωdǫ ǫ0 bnin rˆn nk rd +φˆiJn′ (kdr)
calcoordinates,respectively,H isaconstantamplitude, d 0 n=−∞ (cid:20) d (cid:21)
0 X
E = H k/ωǫǫ , J () is the Bessel function of the first exp(inφ), (8)
0 0 0 n
· ×
kind and order n, k = ω/c is the wavenumber in the
0
vacuum, and k = k0√ǫ. Here the prime appearing in where the wavenumber kd =k0√ǫd.
J′ (α) denotes derivative with respect to the variable α. The Lorenz-Mie scattering coefficients a and d , are
n n n
InEq.(3)weusedtheJacobi-Angerexpansionofaplane determinedbymeansoftheproperboundaryconditions.
wave in a series of cylindrical waves. The scattered elec- Due to the existence of graphene surface conductivity,
tromagnetic field in the environment medium, r > R, the boundary conditions at r = R are given by [21] in
3
order to ensure the continuity of tangential components Here, the coefficients with different index n can be
of the electromagnetic field at the graphene shell: treated separately due to the linear independence of the
multipolar components of the wave field. Finally, we ob-
φˆ Ed = φˆ (Esca+Ein), (9) tain the analytical expressions for the scattering coeffi-
· ·
zˆ H = zˆ (H +H )+φˆ E σ, (10) cients [14]:
d sca in d
· · ·
Since the electric field is assumed to be in the xy plane,
the φ-component of the effective conductivity has to be
taken into account, only. These two equations can be
written as
kdb J′ (k R) = k J′ (kR)+a H(1)′(kR) , (11)
ǫ n n d ǫ n n n
d
h i
b J (k R) = J (kR)+a H(1)(kR)
n n d n n n
ik
σ d b J′ (k R). (12)
− ωǫ ǫ n n d
d 0
k J′(k R) ǫJ (kR) ikσ J′(kR) kǫ J (k R)J′(kR)
d n d n − ωǫ0 n − d n d n
a = , (13)
n kH(1)′(kR) ǫhJ (k R)+ikdσJ′(k Ri) k ǫH(1)(kR)J′(k R)
n d n d ωǫ0 n d − d n n d
h ′ i
kǫ H(1) (kR)J (kR) H(1)(kR)J′(kR)
d n n − n n
b = . (14)
n kH(1)′(kR) ǫhJ (k R)+ikdσJ′(k R) k ǫH(1)(kiR)J′(k R)
n d n d ωǫ0 n d − d n n d
h i
tion is achieved when the scattering coefficients a si-
n
multaneously tend to zero. In contrast, resonances can
be attributed to the pole(s) of scattering coefficient(s).
Now, we would like to compare the scattering efficiency
of: (i) a bare dielectric cylinder, (ii) a graphene nan-
otube, and (iii) a uniform-graphene coated dielectric
nanocylinder. We assume the dielectric core made of
SiO with the relative permittivity ǫ = 3.9 and radius
2 d
R = 0.5 µm; the graphene monolayer with the chemical
potential µ = 0.5 eV; ambient temperature T = 300 K
and carrier scattering rate Γ = 0.1 meV. The last two
FIG. 2: Scattering efficiency, Qsca, of a bare cylinder of ra- parameters are kept fixed in all numerical calculations
dius R = 0.5 µm and permittivity ǫd = 3.9 immersed in air, that follow. The scattering efficiency spectrum Qsca(λ),
a graphene nanotube of chemical potential µ = 0.5 eV sur- whereλisthewavelengthoftheTEz-polarizedilluminat-
roundedbyair,andthegraphene-coatednanocylinder. Inall ing radiation, for each of the three objects is presented
calculations we set T =300 K and Γ=0.1 meV.
in Fig. 2. In the range of long wavelengths, the curve
pattern is formed by a peak resonance at λ = 36.4 µm
III. SCATTERING EFFICIENCY andaminimumofscatteringefficiencyatawavelengthof
28.1 µm. The resonance wavelength and the invisibility
wavelengthcanbefoundinthevicinitiesoftheresonance
Thescatteringcoefficientsa ,givenbyEq.(13),enable
n peak for the free-standing graphene nanotube, found at
the fast estimation of the scattering efficiency [16]:
λ=23.4µm. Withoutthepresenceofthegraphenecoat-
+∞ ing, the dielectric cylinder cannot exhibit any significant
2
Q = a 2. (15) featureofthescatteringefficiencyinsuchspectralrange.
sca n
kR | | One might presume that the set of peak and valley pat-
n=−∞
X
terned in the scattering efficiency follows the character-
Typically, the terms with low order n effectively con- isticlineshapeoftheFanoresonance,wherethe emission
tribute to the summation when the size of the scat- of electromagnetic wavesby the coated nanowire creates
tering object is small enough. The invisibility condi-
4
On the other side, in order to obtain an analytical ex-
pressionforthe fastevaluationofthe invisibility window
we use the following approximate expression for σ
intra
that dominates the contribution to graphene conductiv-
ityformoderateandlowfrequencies(~ω <µ),andlarge
doping (µ k T):
B
≫
ie2µ
σ = ( σ), (16)
intra π~2(ω+iΓ) ≈
In fact, this expression represents generalization of the
well known Drude model for application to graphene.
For example, that model is particularly accurate for the
imaginary part of graphene conductivity for λ>10 µm,
if all the other relevant parameters stay the same as in
FIG. 3: Scattering efficiency spectra, Qsca, for a uniform- Fig. 2 & 3. In addition, the requirement ω Γ (or
graphene coated cylinder illuminated by a TEz-polarized λ 12.4 mm) gives the upper limit on the wav≫elength.
plane wave, when the radius R varies, maintaining fixed Th≪erefore, for the parameters used in the present paper,
ǫd = 3.9 and µ = 0.5 eV. We included analytical equations we may conclude that the expression:
(19) and (20) indicating the resonance condition and the in-
visibility condition, respectively. e2µ
Im(σ) , (17)
≈ π~2ω
theinterferencebetweenthenonresonantscatteringfrom can be used with the high accuracy. On the other hand,
the dielectric core and scattering by narrow Mie modes analytical approximations for the resonance wavelength
in the graphene nanotube [22]. The graphene-coated andtheinvisibilitywavelengthcanbededucedbymeans
nanowire exhibits a scattering efficiency dominated by of the electrostatics approximation [16, 27]. By compar-
thegraphenenanotube(surroundedbyair)atlongwave- ing the squaredmodulus of the scattering coefficients a
n
lengths, where the graphene coating isolates the central fornanosizedgraphene-coatedcylinders,onecanobserve
dielectric cylinder. On the other hand, the bare (un- that the first order dominates over the rest of orders in
coated)nanowiredominatesthescatteringpropertiesfor the spectralrangeofinterest. For a 2 and R=0.5 µm,
1
| |
shorterwavelengths,duetoitsstrongerresponsefarfrom we observe a maximum at 36.4 µm and a minimum at
theRayleighscatteringregime. NotethatinFig.2,addi- 28.1 µm, two features which are replicated in the scat-
tionalminimaofthescatteringefficiencycanbefoundin tering efficiency. In the latter, in fact anadditionalpeak
the visible and near-IR due to the contribution of high- isalsoobservedat25.7µmcorrespondingtoaresonance
order Fano resonances, as described in Refs. [23, 24]. peak of a 2. Further peaks and minima are difficult
2
| |
Itis necessaryto pointoutthatevaluationofthe elec- to be registered. For that reason, the scattering orders
tromagnetic field components, given by (5–8), as well n = 1 can be taken into account, only. Within the
±
as the scattering efficiency (15), can be alternatively electrostaticsapproximation,we further use aTaylorex-
achieved by using a matrix method to derive the scat- pansioninthelimitR 0,retainingonlythetermofthe
→
tering coefficients an and bn [25, 26]. In this respect, lowestorder,suchasJ1(x)≈x/2andH1(1)(x)≈−2i/πx.
the finite width of the graphene layer (tg) has to be in- Then,
troduced, that leads to the corresponding graphene per-
mittivity ǫg = 1+iσ/ǫ0ωtg. In particular, tg = 0.5 nm a = iπ (kR)2 ǫd−ǫ+iσ/ǫ0ωR. (18)
has been employed, where the calculated scattering effi- 1 4 ǫ +ǫ+iσ/ǫ ωR
d 0
ciency for multilayerednanotubes (not shown here) is in
excellent agreement with the results depicted in Fig. 2. Inotherwords,thecomplexscatteringobjectundercon-
sideration behaves like a solid cylinder of dielectric con-
Consequently, the presence of a peak resonance for a
stant ǫ + iσ/ǫ ωR, that assumes a bulk conductivity
graphene nanotube induces the Fano-like resonance in d 0
σ/R.
the complex graphene-dielectric nanowire. Therefore,
When ω Γ, it follows from Eq. (18) that the reso-
tuningthe resonancepeakofthe graphenenanotubewill
≫
nance appears at the frequency:
enable to shift the invisibility wavelength on demand.
This procedure can be carried out by simply modifying
the radius of the graphene nanotube [27]. Scattering ef- e2µ
ω = , (19)
ficiency for a graphene-coated dielectric nanowire as a res sπ~2ǫ0R(ǫd+ǫ)
function of radius R is presented in Fig. 3. It is evi-
dent that peak-valley lineshape is preserved, but shifted Assuming that ǫ and ǫ are positive, a resonance peak
d
towards the longer wavelengths when the radius of the will occur since Im(σ) > 0 in the spectral range of in-
cylinder R increases. terest. Note that the peak resonance is associated to a
5
localized surface plasmon in the graphene-coated cylin-
der, and its dispersion relation can be interpreted such
that this bound mode cyclically propagates along the
cylinder perimeter exactly one effective wavelength cor-
responding to the flat graphene sheet [27]. More in-
teresting in our study, a drastic reduction of the scat-
tering efficiency will be experienced under the constrain
ǫ ǫ Im(σ)/ǫ ωR = 0. The frequency for invisibility
d 0
− −
will surge at
e2µ
ω = , (20)
inv sπ~2ǫ0R(ǫd ǫ)
−
Asaconsequence,theinvisibilitywindowdependsonthe
parameter µ/R∆ǫ, where ∆ǫ stands for the difference of
dielectric constants between the core and the environ-
ment media. In order to meet either the invisibility or
the scattering resonanceconditions, for givenmonochro-
matic field, the nanowire radius R has to be adjusted.
This is illustrated in Fig. 3, where the analytical expres-
sions(19)and(20)havebeenincludedforresonanceand FIG.4: Modulusofthenormalizedtotalelectricfield,|E/E0|,
for a graphene-coated cylinder of radius R = 0.5 µm, keep-
invisibility, respectively. Excellent agreement is evident, z
ing ǫd =3.9 and µ=0.5 eV, illuminated by a TE -polarized
especially for low radius nanowires. Finally, we would
plane wave of wavelength (a) λ=10.0 µm, (b) λ=28.1 µm
liketoemphasisethatthescatteringefficiencycanbeac-
coinciding with theinvisibility window, (c) λ=36.4 µm sat-
curately calculated as Qsca = 4a1 2/kR for sufficiently isfying the resonance condition, and (d) λ=50.0 µm.
| |
thin graphene-coated nanowires, that is when the dipo-
lartermgivenwithinthe electrostaticsapproximationin
Eq. (18) dominates. Therefore, the spectral dependence a iπ(kR)2/4. In contrast, for longer wavelengths
1
of the scattering efficiency can be written as grap≈heneconductivitycanbe neglectedandthe complex
nanowirebehaveslikeanuncoatedcylinder. Inthatcase,
Q ω3 ω2−ωi2nv 2+(ωΓ)2. (21) a1 iπ(kR)2(ǫd ǫ)/4(ǫd+ǫ). The invisibility window
sca ∝ (ω2 ω2 )2+(ωΓ)2 is m≈anifested at t−he wavelength λ = 28.1 µm, where a
(cid:0) − res(cid:1)
drasticreductionofthescatteringwavefieldisapparent.
Interestingly, when the regions of invisibility and reso-
Here the first-order scattering coefficient can straight-
nance start to overlap, i.e. when ω ω ω in
inv − res ≪ forwardly approximated by a1 πωinvΓ∆ǫR2/8c2,
thespectralwindowofinterest, Q approachesthe well ≈ −
sca and therefore the scattering efficiency increases with the
known Fano resonance lineshape [15]:
cylinder radius as Q R3 at ω . Note that the
sca inv
∝
(FΓ/2+ω ω )2 electric field is conserved inside and outside the coated
Q − res . (22) nanowire. On the contrary, the resonant condition will
sca ∝ (ω ωres)2+(Γ/2)2 be satisfied nearby at λ = 36.4 µm, where an enhanced
−
scattered field is emitted by the coated cylinder.
Here, the so called Fano parameter F =
(ω ω )/(Γ/2). The validity of such an ap-
res inv
−
proximation is limited to the case ǫ ǫ, as discussed
d
≫ IV. TUNING INVISIBILITY
below. In the case of anisotropic plasmonic nanotubes
such behaviour has been highlighted in Ref. [28].
InFig.4weshowthetotalelectricfieldinsideacoated A possibility of tuning the invisibility window, as well
cylinder, E , and the field in the enviroment medium, as the window of the resonance peak, by modifying the
d
E + E , normalized to the electric field of the in- chemical potential of the graphene coating that can be
in sca
cident plane wave, E , at different wavelengths. Out monitoredbygatevoltageseemstobeveryimportantfor
0
of the spectral band where the invisibility and reso- applications. The spectrum of Q for a varying chem-
sca
nance exist, as illustrated for (a) λ = 10 µm, and (d) ical potential µ is shown in Fig. 5(a), assuming a fixed
λ = 50 µm, a moderately intense signal that corre- core of radius R = 0.5 µm and permittivity ǫ = 3.9.
d
sponds to a dipolar scattered field is evident. In this We observe that the peak-to-valley sinuosity governing
case,Rayleighscatteringisgoverning,whichischaracter- the scattering spectrum is blueshifted for an increasing
ized by a dominant coefficient a λ−2. For ω ω value ofµ. As longasthe chemicalpotentialisenlarged,
1 res
∝ ≪
the graphene layer acts as a perfect electric conductor thegrapheneconductivityalsogrowsproducingashield-
(PEC), and the graphene-coated dielectric cylinder be- ing effect in the nanowire, which can be compensated
haves exactly like a bare PEC nanowire. In that case, by shifting the working wavelength to the near infrared.
6
FIG.6: FEA-basednumericalevaluationofthescatteringeffi-
ciencyofaclusteroftwographene-coatednanowiresofradius
R=0.5 µm, core permittivity ǫd =3.9, and graphene chem-
ical potential µ = 0.5 eV, immersed in air. The nanowire
cluster is oriented transversally (red solid line) and along
(bluesolidline)thepropagationdirectionoftheincidentlight.
For comparison, we includethe spectrum of Qsca for a single
nanoparticle in dashed green line.
range of interest, dispersion of that semiconductor ǫ (ω)
d
can be very well described by the Drude model with the
plasmon wavelength located at λ =5.41 µm [29]. Min-
p
imum of the scattering efficiency and the peak of the
resonance both shift to longer wavelengths with an in-
creaseof the core permittivity. For the core permittivity
ǫ thatisonlyslightlyhigherthanǫ,theinvisibilitywin-
d
dow significantly widens at the rate dω /dǫ , which is
inv d
notably higher than dω /dǫ , for the resonance peak
FIG.5: ContourplotofQscaforgraphenecoatedcylindersil- res d
z window. For instance, the scattering minimum spans
luminatedbyaTE -polarizedplanewave,when(a)thechem-
ical potential µ is varied, keeping ǫd = 3.9 and R = 0.5 µm, from 11.89 µm to 16.62 µm in the interval where ǫd
and(b)thepermittivityǫdofthedielectriccorevariesandthe changesfrom1.5to 2.0,whereasthe variationofthe res-
environment medium has a dielectric constant ǫ = 1, main- onant peak is produced from a wavelength of 26.09 µm
taining fixed R=0.5 µm and µ=0.5 eV. We also represent to 28.53 µm. Moreover, when ǫ < ǫ, the conducting
d
| |
the curve of resonance given by Eq. (19) in dot-dashed line, graphene sheet can excite a resonantpeak but the invis-
andtheinvisibilityconditionofEq.(20)drawnindashedline, ibility window vanishes. On the other hand, the max-
whentheexternalmediumhasapermittivityǫ=1(inblack)
imum and the minimum of the curve virtually coincide
and, for comparison, when ǫ=4 (in yellow).
for an epsilon-near-zero surrounding material. We point
outthatanonneglectingvalueofthe imaginarypartofǫ
imposes a correctionofEqs.(19) and(20), thus prevent-
For example, the invisibility spectral band is 42.25 µm ingthe peak-valleycollapse. Inversely,highvaluesofthe
wide when changing the chemical potential from 0.1 eV environment permittivity enable to move the resonant
for λinv = 62.25 µm to 1.0 eV for λinv = 20.00 µm. No- peak further from the invisibility window, as depicted in
tice, however, that the spectral width of the invisibility Fig. 5(b) for ǫ=4.
window, tuned by changing the graphene-shell chemical Next, we would like to analyze the scattering effi-
potential, depends on dielectric core radius R and per- ciency of the clusters of two invisible graphene-coated
mittivity, as well as ǫ. These parameters are responsible nanocylinders. In Fig. 6 we present the spectrum Q
sca
for cloaking activation at other frequencies. Therefore, that corresponds to dimmer nanowires, each of the ra-
the resultspresentedin Fig.5(a) aredevotedspecifically dius R = 0.5 µm and permittivity ǫ = 3.9 over-
d
to R=0.5 µm, and ǫd =3.9. coatedby auniform-grapheneshellwithchemicalpoten-
In Fig. 5(b) we present the scattering efficiency spec- tialµ=0.5eV. Thepairofnanowirescanbeorientedei-
trum Q for graphene coated dielectric nanowire as a theralongthepropagationdirectionoftheincidentplane
sca
functionofcorepermittivityǫ ,whenimmersedintheair wave, or perpendicularly. It is assumed that the axis of
d
(ǫ=1),whilekeepingfixedR=0.5µm,andµ=0.5eV. the cylinders is always in the z-direction. The scatter-
Negativevaluesofthe permittivity ǫ canbe realizedvia ingefficiencyisnumericallyevaluatedbyusingCOMSOL
d
semiconductor doping, e.g. n-InGaAs. In the spectral Multiphysics, which is a commercially available solver of
7
theMaxwell’sequationsbasedonthefiniteelementanal- niscencesasolidnanowirewithabulkconductivityequal
ysis (FEA). In the first case, a set of a peak resonance to the ratio of graphene surface conductivity and the
and a scattering minimum, found at λ =35.2 µm and wire radius. In that case, some analytical expressions
res
λ = 28.0 µm respectively, stands out in the scatter- (see equations(18)–(20)) havebeenderivedenabling the
inv
ing lineshape which is comparable to the spectral pat- characterizationof the spectral window for the invisibil-
ternforasinglegraphene-coatednanowire,exceptmaybe ity cloaking. We have examined the possibility of tuning
in an increased efficiency magnitude. For the case of the invisibility window by monitoring the applied gate
two directly adjacent coated nanowires illuminated by voltage to control the chemical potential of graphene.
light polarized along the interparticle y axis, two min- Moreover, we reveal that the spectral distribution of
ima areidentified atwavelengthsλ =28.0µm, coin- scattering efficiency, for the high index ratio between
inv 1
ciding with the previously analyzed invisibility window, the core and the surrounding medium, approaches the
and λ = 35.2 µm. In addition, two resonant peaks well-known Fano resonance line shape. We propose an
inv 2
are traced at λ = 33.6 µm and λ = 44.6 µm. adequate multi-frequency invisibility mechanism that is
res 1 res 2
The strong coupling between the graphene plasmons of based on a hybrid picture with clusters of several invis-
eachindividualcoatednanowireiswelldescribedbyahy- ible graphene-coated nano-cylinders. Our findings can
bridization picture relying on the analogy between plas- be straightforwardly applied to more complex graphene
monsandthewavefunctions ofsimplequantumsystems coating structures [32, 33]. Finally, we would like to em-
[30]. phasize that the proposed scheme seems to be tunable
Finally, it is worth noting that scattering cancellation enough to be implemented in ultra-thin reconfigurable
can be expected to appear at more different frequencies cloaking devices. Within the THz range of frequencies
when dealing with clusters formed by a higher number this can be applied to low-noise sensing and imaging.
of graphene-coated nanowires. Previously proposed al-
ternate multi-frequency cloaking mechanism is based on
several graphene layers implementation at different gate
charges [31].
V. CONCLUSIONS
We have demonstratedthat the scattering behaviorof
a sufficiently thin graphene-coated nano-cylinder remi-
[1] U.Leonhardt, Science 312, 1777 (2006). [15] B.Lukyanchuk,N.I.Zheludev,S.A.Maier,N.J.Halas,
[2] J.B. Pendry,D.Schurig,andD.R.Smith,Science312, P.Nordlander,H.Giessen,andC.T.Chong,Nat.Mater.
1780 (2006). 9, 707 (2010).
[3] W.Cai,U.K.Chettiar,A.V.Kildishev,andV.M.Sha- [16] C. F. Bohren and D. R. Huffman, Absorption and scat-
laev, Nat. Photonics 1, 224 (2007). tering of light by small particles (Wiley,1998).
[4] D.S.Filonov,A.P.Slobozhanyuk,P.A.Belov,andY.S. [17] A.Y.Nikitin,F.Guinea,F.Garcia-Vidal,andL.Martin-
Kivshar, Phys.StatusSolidi RRL6, 46 (2012). Moreno, Phys. Rev.B 84, 195446 (2011).
[5] P.-Y.Chen,J.Soric,andA.Alu`,Adv.Mater.24,OP281 [18] G. A.Shah,Mon. Not. R.Astron. Soc. 148, 93 (1970).
(2012). [19] H.E.BusseyandJ.H.Richmond,IEEETrans.Antennas
[6] A. Monti, A. Alu`, A. Toscano, and F. Bilotti, J. Appl. Propag. 23, 723 (1975).
Phys.117, 123103 (2015). [20] E. A.Velichko, J. Opt.18, 035008 (2016).
[7] J.C.Soric,A.Monti,A.Toscano,F.Bilotti,andA.Alu`, [21] Y. Gao, G. Ren, B. Zhu, H. Liu, Y. Lian, and S. Jian,
IEEE Trans. AntennasPropag. 63, 3235 (2015). Opt. express22, 24322 (2014).
[8] Y.Zhu,S.Murali,W.Cai,X.Li,J.W.Suk,J.R.Potts, [22] T. J. Arruda, A. S. Martinez, and F. A. Pinheiro, Phys.
and R.S. Ruoff,Adv.Mater. 22, 3906 (2010). Rev. A 92, 023835 (2015).
[9] F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, Nat. [23] A. Mirzaei, A. E. Miroshnichenko, I. V. Shadrivov, and
photonics 4, 611 (2010). Y. S. Kivshar,Sci. Rep.5, 9574 (2015).
[10] P.-Y.Chen and A.Alu`, ACS Nano 5, 5855 (2011). [24] M. V. Rybin, D. S. Filonov, P. A. Belov, Y. S. Kivshar,
[11] A.AluandN.Engheta,Phys.Rev.E72,016623(2005). and M. F. Limonov, Sci. Rep. 5, 8774 (2015).
[12] B. Edwards, A. Alu,M. G. Silveirinha, and N. Engheta, [25] C. D´ıaz-Avin˜´o, M. Naserpour, and C. J. Zapata-
Phys.Rev.Lett. 103, 153901 (2009). Rodr´ıguez, Opt.Express 24, 18184 (2016).
[13] S. Tricarico, F. Bilotti, and L. Vegni, J. Eur. Opt. Soc, [26] C. D´ıaz-Avin˜´o, M. Naserpour, and C. J. Zapata-
Rapid Publ. 4, 09021 (2009). Rodr´ıguez, Opt.Commun. 381, 234 (2016).
[14] M. Riso, M. Cuevas, and R. A. Depine, J. Opt. 17, [27] M. Cuevas, M. A. Riso, and R. A. Depine, J. Quant.
075001 (2015). Spectrosc. Radiat. Transf. 173, 26 (2016).
8
[28] C. D´ıaz-Avin˜´o, M. Naserpour, and lander, Chem. Rev.111, 3913 (2011).
C. J. Zapata-Rodr´ıguez, Plasmonics [31] M.Farhat,C.Rockstuhl,andH.Bag˘cı,Opt.Express21,
pp. 1–9 (2016), ISSN 1557-1963, URL 12592 (2013).
http://dx.doi.org/10.1007/s11468-016-0313-3. [32] P.-Y.Chen,J.Soric,Y.R.Padooru,H.M.Bernety,A.B.
[29] J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaft- Yakovlev,and A. Alu`, NewJ. Phys.15, 123029 (2013).
man, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, [33] M.DanaeifarandN.Granpayeh,J.Opt.Soc.Am.B33,
Nanophotonics 4, 44 (2015). 1764 (2016).
[30] N. J. Halas, S. Lal, W.-S. Chang, S. Link, and P. Nord-