Switchable invisibility of dielectric resonators

1.1 MB·

by  Mikhail V. Rybin

Switchable invisibilityofdielectricresonators
Mikhail V. Rybin1,2,∗ Kirill B. Samusev1,2, Polina V. Kapitanova3,
Dmitry S. Filonov3, Pavel A. Belov3, Yuri S. Kivshar3,4, and Mikhail F. Limonov1,2
1 Ioffe Institute, St. Petersburg 194021, Russia
2DepartmentofDielectricandSemiconductorPhotonics, ITMOUniversity,St.Petersburg197101, Russia
3DepartmentofNanophotonics andMetamaterials, ITMOUniversity, St.Petersburg197101, Russiaand
4Nonlinear Physics Center, Research School of Physics and Engineering,
Australian National University, Canberra ACT 2601, Australia
(Dated:February1,2017)
7 Thestudyofinvisibilityofaninfinitedielectricrodwithhighrefractiveindexisbasedonthetwo-dimensional
1 Miescatteringproblem,anditsuggestsstrongsuppressionofscatteringduetotheFanointerferencebetween
0 spectrallybroadnonresonant wavesandnarrowMie-resonantmodes. However, whenthedielectricrodhasa
2 finiteextension,itbecomesaresonatorsupportingtheFabry-Perotmodeswhichintroduceadditionalscattering
andeventually destroy theinvisibility. Herewereveal that forshorter rods withmodest values of theaspect
n
ratior/L(whererandLaretheradiusandlengthoftherod,respectively),thelowestspectralwindowofthe
a
J scatteringsuppressionrecoverscompletely,sothatevenafinite-sizeresonatormaybecomeinvisible.Wepresent
a direct experimental verification of the concept of switchable invisibility at microwaves using a cylindrical
1
finite-sizeresonatorwithhighrefractiveindex.
3

]
s I. INTRODUCTION where q = cot∆ is the Fano asymmetry parameter, Ω =
c
i (ω−ω0)/(Γ/2), whereω0 andΓcorrespondtothe position
pt Invisibility, mimicry, and cloakingare perpetualproblems andthewidthofthenarrowband,∆(ω)=ϕA(ω)−ϕB(ω)is
o in both wildlife and inorganic nature. A new fresh impe- thephasedifferencebetweenthenarrowresonantandcontin-
. tus to the progress in this field has been given by the con- uumstatescorrespondingly.Themainpointoftheapproachis
s
c ceptofmetamaterials1,allowingtodesignacloakingstructure basedonthecharacteristiclineshapeoftheFanointensitythat
i thatsurroundsanobjectmakingitinvisibletoanoutsideob- vanishesataspecificfrequencyωzero =ω0−qΓ/2.Itmeans
s
y server. Different approachesto achieve invisibility cloaking thatthescatteringbyanobjectcompletelyvanishes.Notethat
h have been proposed including multilayered shells that may ωzero = ω0 atq = 0,ωzero < ω0 atq > 0andωzero > ω0 at
p make even plasmonic particle invisible2 being also tuned by q <0.
[
graphene3ornonlinearlayersimbeddedintomulti-shellstruc- The second idea is to employ the well-known Mie reso-
1 tures4,ormagneto-opticalshellactivatedbyanexternalmag- nances. The case of an infinite cylinder, referred to as the
v netic field5. It was recognized that the invisibility cloaking Lorenz-Mie theory, correspondsto a 2D problem of scatter-
2 becomepossibleasaproductofthepowerfulconceptsofthe ingintheplanenormaltothesymmetryaxisz,anditinvolves
2 so-called transformation optics6–9, and it was studied for a cylindricalfunctionsintheinfiniteseriesoftheanalyticalMie
0 varietyofapplicationsincludingradardetection,sensorsand solution29. Asaresultoftheinterferenceofthenonresonant
9
detectors10–12.
0
. Unfortunately,mostofthesuggestedmethodsforinvisibil-
1 2r
ity cloaking require the creation of specially designed shells
0
or media with perfect parameters making difficult a practi-
7
1 calrealizationofmanyinvisibilityconcepts1,13. Recentlywe z
: suggested a novelapproachfor the realization of tunable in-
v
visibilitycloakingandadirectswitchingfromvisibilitytoin-
i
X visibilityregimesandback14. Thisapproachisbasedontwo
x
L ≈ r
r basicideas.
a The first idea is to archive the scattering cancellation us- L
y
ing the propertiesof the characteristic lineshape of the Fano
resonance15,16. WecanexpectaninteractionoftheFano-type
H
ibfroaasdpespctercatlrluymna(rfreoawturbealnedssvbiratcukagllryouanndy)otrhirgoiungihntaenraicnttsewrfeitrh- k TE-wave
ence effect constructively or destructively. It is well known E
that the Fano resonance has been observed across many dif-
ferent branchesof physics17–20 including plasmonic21–23 and L >> r H || z
dielectricstructures24–28. Iftherearenopartsofbackground
avoiding the above interaction, the Fano relation takes the
FIG. 1: Schematic of the cylindrical dielectric resonator. TE-
form14 polarizedplanewaveincidentstothecylindrical resonator. Aspect
(q+Ω)2 r-to-Lratioisvariedfromr ≈L(disk-shapedresonator)tor ≪L
2
I(ω)= sin [∆(ω)], (1) (rod-shapedresonator).
1+Ω2
2
0
ε=60 r/L << 1 TE010 TE r/L = 0.33
110
PEC
TM
111
-10 TE01 TE11 TE21
TE
02
-20
(a) (b)
-30
0
B) r/L = 0.4 TM111 r/L = 0.5
d
y ( TM111
c
n -10
e
ci
fi
f
e
g
n -20
ri
e
att
Sc (c) (d)
-30
0
r/L = 0.66 TE TE r/L = 1.0
010 110
TE
020
-10 TM111 TE210
-20
(e) (f)
-30
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Size parameter χ
FIG.2: SpectraldependenciesofthetotalscatteringefficiencyoftheinfinitehomogeneousdielectriccylinderQ0 (a)andfinitecylindrical
resonator Qr/L for different ratio r/L (b-f) - solid curves. Spectral dependencies of the infinite Q∞PEC (a) and finite QrP/ELC (b-f) perfect
conductingmetalcavities–dashedcurves.TE-polarization,ε1 =60,ε2 =1.Thesizeparameterχ=rω/c=2πr/λ.
andresonantscattering,thespectraofthesquaredLorenz-Mie tivity of heated water in a plastic tube14. In contrast to this
scatteringcoefficients|a |2 demonstrateasymmetricprofiles earlier study on the scattering and invisibility of an infinite
n
witheithersharpincreaseordropattheresonancefrequencies cylinder, here we study the a finite-lengthcylindricaldielec-
ofthecylindereigenmodes. Thescatteringcoefficients|a |2 tric resonator (Fig. 1) and demonstrate novel features of the
n
canbepresentedintheformofinfinitecascadesofresonances invisibilityphenomena.
eachofwhichisdescribedbytheconventionalFanoformula
(1) as was shown for the first time numerically14,29 and then
analyticallyforthecasesofinfinitecylinder30andsphere31.
II. SCATTERINGFROMDIELECTRICRESONATORS
Employingthesetwobasicideas,onecandemonstratethat
the resonant Mie scattering from an infinite dielectric cylin- As the starting point, we use the previous results for the
dervanishesatfrequenciesofωzeroandthecylinderbecomes elastic scattering by a homogeneousinfinite dielectric cylin-
invisibleatanyangleofobservation. Importantly,forahigh- der of the radius r with dielectric permittivity ε1 embedded
indexdielectriccylinderatypicalasymmetricFanoprofilehas inatransparenthomogeneousmediumwithdielectricpermit-
a local maximum and local minimum located close to each tivityofε214. Fortheelectromagneticwaveincidentperpen-
other. Such a proximity allows to realize switchable invisi- dicular to the infinite cylinder axis z, the two transverse po-
bility of a macroscopic object transforming from the visible larizations appear decoupled. As a result, the modes can be
toinvisibleregimesandback,withoutusinganycoatinglay- classified as either TE modes (E , E , H ) with the elec-
x y z
ers. The idea was confirmed experimentally at microwaves tric field confined to the (x, y) plane and the magnetic field
by employing high temperature-dependentdielectric permit- polarized along the cylinder axis z or TM modes (H , H ,
x y
3
E )forwhichthemagneticfieldisorientedperpendicularto 0
z (a) TE
the cylinder axis z. Here, we consider the TE-polarization, TE01 TE11 TE21 02
at which the Mie resonances excited in the infinite cylinder
are denoted as TEnk, (n is integer and k is positive integer) -10
wherenisthemultipoleorder,andkistheresonancenumber.
Figure 2(a) demonstrates the total Mie scattering efficiency
Q0 = χ2 P∞n=−∞|an|2 (χ = rω/c = 2πr/λ)fortheinfinite B)-20 r/L << 1
cylinderinthelow-frequencypartofthespectrumatε1 =60 d 0
where the resonant modes TE01, TE11, TE21, and TE02 are y ( (b) TE010 TE110
observed29. For a reference scattering intensity we choose nc-10
standard scattering efficiency Q0PEC for the infinite cylinder cie
mdoawdeoffroinmvipsiebrifleitcytcwointhduQct0in<gmQe0tal(PisEcCl,eεar1ly→se∞en).nAearwtihne- effi-20
PEC g TM111
TE11Mieresonanceatχ≈0.5[Fig.2(a)]. n r/L = 0.6
Next we continue with numericalcalculationsof the scat- ri-30
e 0
teringfromafinitedielectriccylinderQr/L,namely,afinite- att (c) TE010 TE110
extentdielectricresonatorwiththeradiusrandlengthL.The Sc-10 experiment TE210
resonatorwith the dielectricpermittivityε1 embeddedin the theory
transparent and homogeneoussurroundingmedium with the PEC
dielectricpermittivityofε2 = 1. Thespectrawerecalculated -20
byusingtheCSTMicrowaveStudiosoftware. Wecalculated
r/L = 1.0
thescatteringefficiencyQr/Lastheradarcrosssectionofthe -30
0.2 0.4 0.6 0.8
cylindricalresonatordividedbytheprojectedcrosssectionof
the resonator S = 2rL. Figures 2(b-f) show the total Mie Size parameter χ
scatteringefficiencyfromfinitecylindricalresonatorQr/Lfor
differentaspect ratiosr/L together with the scattering spec- FIG.3: Low-frequencyregionofthecalculatedscatteringefficiency
traofinfiniteperfectconductingmetalcavitiesQr/L . Using Qr/L(blackshort-dashedcurve),scatteringefficiencyoftheperfect
PEC
the standardnomenclature32, the resonatormodesare identi- metallic conductor PEC QrP/ELC (green long-dashed curve), and the
fiedtobeTM andTE wherenisintegerrelatedtothe experimentally measured scattering efficiency of a cylindrical tube
rotational behnakvpior of thenkmpode, k > 1 identifies resonance filledwithwater(redcurve): (a)r/L = 0(r/L = 0.03inexperi-
numberforthemodeswithfixedsymmetryn,andpenumer- ment),(b)r/L= 0.6,and(c)r/L= 1. TE-polarization,ε1 = 60,
ε2=1.
atesmoderelativetothecylinderaxisdirection. Forarather
long cylindricalresonator (L > r), the low-frequencyspec-
trum Qr/L demonstratesseveralstrongadditional‘resonator
modes’incomparisonwiththespectrumoftheinfinitecylin- without any coating layers, we use a plastic cylinder filled
derQ0 (Fig.2). Resonatormodeshaveanobviousoriginbe- with water that is characterized by dielectric permittivity of
ing related directly to the appearance of the end faces of a ε ≈ 80 at roomtemperatureand ε ≈ 50 at 90◦C in the mi-
finitecylinder. crowave frequencyrange33. The microwave scattering spec-
New(Fabry-Perotorresonator)modesintroduceadditional tra Qr/L of the cylindrical resonator depending on the r/L
scatteringthatsuppresspartlyorcompletelyspectralwindows parameter and water temperature were measured in an ane-
ofthe invisibility,speciallyforthe lowestFano-windowcor- choicchamber. Arectangularhornantenna(TRIM0.75GHz
responding to the TE11 Mie resonance. With increasing of to 18 GHz; DR) connected to a transmitting port of a vec-
r/L, all resonances that have a genetic link with the Mie tornetworkanalyzerAgilentE8362Cisusedtoapproximate
modesoftheinfinitecylinderdemonstrateonlyaslighthigh- a plane-wave excitation. The cylindrical rod of water with
frequencyshift. Alternatively,exactlywhatonemightexpect, radiusr = 20mmandvariableheightisplacedintothe far-
the resonatormodesdemonstratea strongshiftfrom the low fieldregionofthehornantennaonthedistanceof2.5m. The
tohigherfrequencieswithdecreasingofthenormalizedlength similarhornantennaisemployedasareceiverandplacedon
L/r, as clearly illustrated by the behaviorof the TM111 res- 2.5mdistancefromthecylindricalrod. FortheTEpolariza-
onator mode in Fig. 2. As a result, for a resonator with pa- tion, the electromagnetic wave incident perpendicular to the
rameters r/L ≥ 1 the lowest TE110 window of invisibility resonator axis z and the magnetic field is oriented along the
at χ ≈ 0.6 completely recovers (marked by red circles in axis z. Using the opticaltheorem34 we calculate the scatter-
Figs.2aand2f). ing efficiency from the imaginary part of the measured for-
ward scattering amplitude. The dielectric permittivity of the
plastictubeismuchsmallerthanthatforwaterinmicrowave
III. EXPERIMENTALAPPROACH frequencyrange and the tube has no effect on the scattering
efficiency.
Todemonstrateexperimentallytherealizationoftheinvis- Weobserveexperimentallyastrongsuppressionofscatter-
ibility of the cylindricalresonator with high refractive index ingatfrequenciescorrespondingtotheFanodipsinthefunc-
4
r/L << 1 r/L = 0.6 r/L = 1.0
(a) (b) (c) (d) (e) (f)
0
1
x
m
o
o
Z
(g) (h) (i) (j) (k) (l)
FIG.4: Numericalresults. MagneticfieldmapfortheH componentaroundcylinder(a)-(f)andinsidecylinder(g)-(l). Shownare: strong
z
Miescattering(a),(g)forr/L = 0,χ = 0.48, (c),(i)forr/L = 0.6, χ = 0.52, (e,k)forr/L = 1,χ = 0.57,andtheFanoinvisibility
regime(b),(h)forr/L=0,χ=0.50,(d),(j)forr/L=0.6,χ=0.54,(f,l)forr/L=1,χ=0.59.TE-polarization,ε1=60,ε2=1.
tionQr/L. Figure3showsaninvisibilityeffect,namelysup-
suppressed
pressionofscatteringlowerthanPEClevelQr/L duetothe 1.0 scattering
PEC
TE110-Fanoresonanceatχ ≈ 0.5forr/L = 0(r/L = 0.03 L
/
in experiment) [Fig. 4(a)], at χ ≈ 0.55 for r/L = 0.6 r
o 0.8
[Fig. 4(b)], and at χ ≈ 0.6 for r/L = 1 [Fig. 4(c)]. There- i
t
a
fore the incident TE-polarized electromagnetic wave passes r
the cylindrical resonator without scattering making it invisi- ct
e0.6
blefromanyangleofobservation.Forr/L=0.6,theTM111 p
s
resonatormodeatχ≈0.58partlysuppressspectralwindows A
of invisibility. Nevertheless, with future decreasing of r/L,
0.4
the TM111 resonator mode shifts to higher frequencies and
the lowestTE110 Fano-windowofinvisibilitycompletelyre- 0.4 0.5 0.6 0.7 0.8
covers. Size parameter χ
To demonstrate dramatic difference in the scattering be-
tween the maximum and minimum of the Fano Mie line-
FIG.5: Invisibilitymap for a homogeneous dielectricresonator in
shape, wecalculatednumericallya structureofthemagnetic
thelow-frequency spectralregion. Shaded areasdefinetheregions
fieldaroundandinsidethecylindricalresonatordependingon of strongscattering. Open domains mapsthe regionof suppressed
r/L. Figure 4 shows that for the TE110 Mie resonance in scatteringQr/LincomparisonwiththePECscatteringQr/L. TE-
PEC
the regime of Fano invisibility we observe practically com- polarization,ε1 =60,ε2=1.
plete suppression of the scattering, see panels (b), (d), and
(f). For those normalized frequencies the resonant H field
z
near the cylinder’s boundaryinside the rod mimics the inci-
tionalcloakinglayers.
dentfielddistribution,Fig.4(h),(j),(l).Asaresult,anincident
Toanalyzetheinvisibilitydynamics,wecalculatedthedif-
TE-polarizedelectromagneticwavepassesthecylindricalres-
onatorwithoutscatteringthatistheresonatorbecomesinvis- ferenceQr/L−QrP/ELCasafunctionoftheresonatordielectric
iblefromanyangleofobservationwithoutadditionalcoating permittivity ε1. We found that the scattering efficiencies of
layers. Atthesametimeinsidetheresonatoronecanobserve dielectriccylindricalresonatorandPEC coincidewhen ε1 is
verystrongMieresonancewhichneverthelessdonotdisturb about10.Atε1 >10theinvisibilitydynamicsappearsforthe
theincidencewavebecauseoftheFanocondition. TE110 Mie resonance. The invisibility map demonstrated in
Fig.5showstheregionsofinvisibility[Qr/L < Qr/L ]with
PEC
respecttothecontrollableparametersχandr/Lforε1 =60.
IV. INVISIBILITYMAP Theregionsoftheinvisibilityareshadedinwhite.FromFig.5
itisclearlyseenthattheinvisibilityinlow-frequencyspectral
Thecalculationsshowthatbyadjustingthefrequencyand regionsfollowstrongFanoresonanceforTE110Miemode.
dielectricpermittivity,ahomogeneoushigh-indexcylindrical AscanbeseenfromFigs.2and5, foraratherlongcylin-
resonatorcanbemadeessentiallyinvisiblewithoutanyaddi- drical resonator (L > r), the resonator mode TM111 intro-
5
Frequency (GHz) imentally, we take advantage of the strong temperature de-
0.8 1.0 1.2 1.4 1.6 1.8 pendence of the water permittivity ε1 in the microwave fre-
0 quency range33. It changes from ε1 = 80 at 20◦C to ε1 =
ε1=72 50 at 90◦C. Such unique dependence of ε1 leads to a con-
siderabletemperatureshiftofthemaxima(strongscattering)
and minima(possible invisible regime)of the Fano-Mie res-
-20 -ε113= d7B1 onancemodes,asseen inFig.6 forTE010 andTE110 bands.
When high-frequency wing of TE110 Fano lineshape moves
throughafixedfrequencyweobserveastrongchangeofthe
B) (b) -ε216= d6B9 scattering intensity from high to very small values. There-
d fore,theFanoresonancemakespossibletoswitchthescatter-
(
y -40 ingofacylindricalresonatorfiledwithwaterfromthevisible
nc -ε319= d6B8 regime of strong Mie scattering (varepsilon1 = 64, water
cie atT = 80◦C) to theinvisibilityregime(varepsilon1 = 72,
effi ε1=67 wateratT =35◦C)atthesamefrequencyof1.22GHz.
g -60 -52 dB
n
i
er ε1=66 VI. CONCLUDINGREMARKS
tt -65 dB
a
c
S -80 We have demonstrated that different Mie resonances of a
(a) ε1=65 homogeneousdielectric resonator with high refractive index
-78 dB
areresponsibleforsharpFanoresonancesintheTE-polarized
TE010 TE110 scattered spectra. The unique property of the characteristic
ε1=64 Fanolineshapeisthatthetransmissionintensityvanishesata
-100 -91 dB
specialfrequency. Thismeansthatthescatteringcompletely
PEC vanishes at this frequency, and an object becomes invisible
atanyangleofobservation. TheeffectoftheFanoinvisibil-
0.4 0.5 0.6 0.7 0.8
ity has been suggestedearlier for an infinite dielectric cylin-
Size parameter χ
derinlow-frequencyspectralregion14. Here,wehavestudied
thescatteringfromafinite-lengthdielectricresonatorandre-
FIG.6:Experimentalresults.(a)Measuredtemperaturedependence vealedthatthespectraofaratherlongresonator(L>r)con-
ofthetotalscatteringefficiencyQr/Lofafinite-length(r/L=0.6) tainadditionalTM andTE (p 6= 0)modesthatsuper-
cylindrical resonator filled with water (TE polarization, ε2 = 1). imposein the low-nfrkepquencyrnegkpionand introduceadditional
Curvesareshifted verticallyby thevalues marked on theplot. In-
scattering suppressing or completely removing the spectral
serts(a,b)showthecalculatedmagneticfieldsatthefrequency1.22
windowsoftheinvisibility. However,whentheratior/Lbe-
GHzintheregimesoftheFanoinvisibilityandstrongMiescattering,
comeslarger,theresonatormodesshifttohigherfrequencies
respectively.
and the invisibility window completely recover at r/L ≥ 1.
Asaresult,afinite-lengthdielectricresonatorbecomesinvis-
duces additional scattering within TE110 window of invisi- ible from any angle of observationat certain frequenciesfor
bility. The effect regains with increasing of r/L when the the TE-polarized waves. We have verified these theoretical
resonator modes demonstrate a strong shift from the low to predictionsinmicrowaveexperiments.
higher frequencies. For structural parameters r/L ≥ 1 the
lowestwindowofinvisibilitycompletelyrecovers.
Acknowledgments
V. DEMONSTRATIONOFSWITCHABLEINVISIBILITY We acknowledge fruitful discussions with A.A. Kaplyan-
skii. This work was supported by the Russian Foundation
Figure2showsthatforthehigh-indexdielectricresonator for Basic Research (No. 16-02-00461), and the Australian
an asymmetricFano profilehasa maximumand a minimum Research Council. The experimentalstudies have been sup-
located close to each other. This proximity can be used for portedby the grantof Russian Science Foundation(No. 15-
thedemonstrationoftheswitchingbetweenvisibilityandin- 19-30023). P.K.acknowledgesascholarshipofthePresident
visibility. To prove the concept of Fano invisibility exper- oftheRussianFederation.
∗ Electronicaddress:[email protected] (2015).
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