Table Of ContentEfficient second harmonic generation in a metamaterial with two resonant
modes coupled through two varactor diodes
ToshihiroNakanishi,1,a) YasuhiroTamayama,1 andMasaoKitano1
Department of Electronic Science and Engineering, Kyoto University, Kyoto 615-8510,
Japan
(Dated:26January2012)
We present an effective method to generate second harmonic (SH) waves using nonlinear metamaterial composed
of coupled split ring resonators (CSRRs) with varactor (variable capacitance) diodes. The CSRR structure has two
resonant modes: a symmetric mode that resonates at the fundamental frequency and an anti-symmetric mode that
2
1 resonates at the SH frequency. Resonantfundamentalwaves in the symmetric mode generate resonantSH waves in
0 theanti-symmetricmode. ThedoubleresonancecontributestoeffectiveSHradiation. Intheexperiment,weobserve
2 19.6dB enhancementin the SH radiation in comparisonwith the nonlinearmetamaterial that resonates only for the
fundamentalwaves.
n
a
J PACSnumbers: 78.67.Pt,42.65.Ky,41.20.-q
5 Keywords:metamaterials,nonlinearoptics,secondharmonicgeneration
2
] Metamaterials, which are composed of artificial sub- giontoestimatetheSHGefficiencyoftheCSRRs,comparing
s
wavelength structures, exhibit extraordinaryelectromagnetic thesinglyresonantmetamaterial.
c
i properties. Numerous studies have focused on the linear
t The varactor loaded split ring resonator (SRR) shown in
p response characteristics of metamaterials. Recent studies
Fig. 1(a) is a typical example of singly resonant nonlinear
o have also reported the development of nonlinear media and
metamaterials. It can be modeled as a series resonant cir-
. the control of the nonlinear properties of the metamaterials
s cuitcomposedofaninductor,varactor,andresistor,asshown
c through the introductionof nonlinear elements into the con-
si stituentsofmetamaterial. Highnonlinearitycanbe achieved in Fig. 1(b). The electromotive force induced by the exter-
nal magnetic field B is represented by the voltage source
y in resonant-type metamaterials such as split ring resonators
h (SRRs) byplacingnonlinearelementsatlocationswherethe V(t)=V0cosωtinthecircuitmodel.Ifweconsiderthevar-
p actorasalinearcapacitor,thecircuitisasimpleharmonicos-
electric (or magnetic) field is concentrated due to the res-
[ cillatordrivenbytheexternalforceandthevoltageacrossthe
onance effect.1 Nonlinear metamaterials have been studied
1 for property tuning,2–4 frequency mixing,5 imaging beyond capacitor,vC,reachesamaximumattheresonantangularfre-
quencyω =1/√LC.However,thevaractorhasnonlinearity
v diffractionlimit,6 anddevelopingbistable media.3,7,8 Several 0
6 kinds of nonlinear metamaterials for generating second har- inthecapacitanceanditsvoltagevC withrespecttothecharge
9 monic(SH)waveshavebeenreported. Mostofthemetama- qcanberepresentedasvC = q/C(q) = q/C +αq2. Anan-
1 harmonictermαq2contributestothegenerationofsecond-or
terials are designedsuchthat theyresonatewith the incident
5 higher-orderharmonicwaves. Usingtheperturbationmethod
1. wavesorthefundamentalwaves.2,9–13Thesearecalledsingly under a weak nonlinearity condition,16 the amplitude of the
resonant metamaterials. If the structure has resonant modes
0
notonlyforthefundamentalfrequencybutalsofortheSHfre-
2
quency,SHradiationcouldbeenhanced14,15. Inthispaper,we
1
v: pdrooupbolysereasomneatnhtocdontodiitmiopnl,eomreandtomuebtlaymreastoerniaanlstmsaetitsafmyiantgeritahle, L C(q) b
i E
X usingcoupledsplitringresonators(CSRRs)withtwovaractor
r diodes, whichgenerateSH wavesdueto nonlinearity. There V +- I a g w
a are two resonant modes in the CSRR structure: one for the c
fundamentalwaves;theotherfortheSHwaves. Theyarecou- B R
pledowingtothenonlinearityofthediodes. TheSHoscilla- (a) (b) (c)
tionexcitedinthevaractorsdirectlyexcitestheSHresonance L C(q 1 ) C(q2 ) L
modethroughthenonlinearity-assistedcoupling.Secondhar-
monic generation through the CSRRs is more efficient than 2Va
that through the previously proposed metamaterial with two Vs C' Vs
resonant modes, where SH oscillation indirectly excites the
resonantmodedesigned for the SH waves throughmagnetic R (d) R (e)
coupling.15Wedemonstrateexperimentsinthemicrowavere-
FIG. 1. (a) Varactor-loaded split ring resonator and (b) its circuit
model. (c)Coupledsplitringresonatorand(d)itscircuitmodel. (e)
a)Electronicmail:[email protected] Splitringresonatorasthereferencemetamaterial.
2
SHoscillationisobtainedas top plate
(brass)
αV2 250mm
q˜(2ω)= 0 , (1) m
ω3Z(2ω)Z(ω)2 x m
2
where Z(ω) = R i[ωL 1/(ωC)] is the impedance of z 12 150mm
− − y
the circuit. When the circuit resonates at ω = ω , Z(ω)
0
| | ground plate
takes a minimum value and q˜(2ω) is maximized. As a re- 25mm
| | (aluminum)
sult,theenhancedSHsignalisradiatedfromthesinglyreso-
nantmetamaterial.Here,wecanexpectthat q˜(2ω) isfurther
| |
enhancedbyalsoreducingtheimpedanceat2ω,i.e. Z(2ω).
| | FIG.2.SchematicofstriplineTEM-cellwaveguide.
This implies that more efficient SHG can be achieved when
the metamaterialresonatesfor both the fundamentaland SH
frequencies.
Itshouldbenotedthatifthediodesarearrangedinthesame
ForfurtherenhancementoftheSHradiation,weintroduce
direction on the outer ring, the fundamental current and the
the CSRR shown in Fig. 1(c). Two ring structures share a
induced SH currentflow in the symmetric mode, which res-
gapwitha capacitanceC′ atthecenterofthestructure. Fig-
onatesonlyforthefundamentalwave. FromEq.(4),theam-
ure 1(d) representsthe circuit modelof the CSRR. It should
plitudeoftheSHoscillationisobtainedas
benotedthattherearetwo diodesoppositelydirectedonthe
outerring. We denotethe chargesof the varactorsasq1 and αV2
qth2e. eInquteartimonssooffthmeovtaiorinasbaleres,eqxsp=resqs1ed+aqs2 andqa = q1−q2, q˜a(2ω)= 2ω3Za(2ωs)Zs(ω)2. (5)
ItiseasilydeducedfromEq.(2)thattherearenoSHwavesin
d2q dq q
L s+R s+ s +αq q =2V cosωt, (2) thesymmetricmode. ThisisbecausetheinducedSHelectro-
dt2 dt C s a s motivevoltagesinthetwodiodescanceleachotherduetothe
Ld2qa+Rdqa+ qa + αq2+ αq2 =2V cosωt, (3) anti-symmetricarrangementsofthediodesforthesymmetric
dt2 dt C 2 s 2 a a mode. TheSHoscillationexcitedonlyintheanti-symmetric
a
modeformsanelectricdipolewithmagnitudep˜= q˜ (2ω)g¯,
where 1/C = 1/C + 2/C′. The excitation voltage V is a
a s whereg¯isaneffectivedipolelengthandisdeterminedbythe
induced by the magnetic flux through the rings and voltage
current or charge distributions. The electric dipole oscilla-
V is induced by the electric field at the central gap of the
a tion contributes to efficient SH radiation. Both Z (ω) and
structure. Equations(2)and(3)representthatthesymmetric | s |
Z (2ω) inEq.(5)aresmall,becauseofthedoublyresonant
current (dashed line) and anti-symmetric current (solid line) | a |
condition,ω = ω = ω /2. Consequently,the amplitudeof
formtworesonantmodeswithresonantangularfrequenciesof s a
thesecondharmonicoscillationcanbesignificantlyenhanced
ω = 1/√LC andω = 1/√LC ,respectively;thesemodes
s a a incomparisontothesinglyresonantmetamaterial,wherethe
are coupled through the nonlinearity α. We set the gap ca-
SH currenthas to flow through the high impedance Z(2ω)
pacitanceC′ = 2C/3, so thatthe doublyresonantcondition | |
inEq.(1).
ω =2ω issatisfied.
a s WefabricatedtheCSRRillustratedinFig.1(c)with35µm-
We willsolve q andq in the presenceofa smallnonlin-
s a thick copper film on a polyphenylene ether (PPE) substrate
earity α, using a perturbativeapproach. We assume that the
withathicknessof0.8mm. Thedimensionsarea=14mm,
solutionforα=0isgivenbyqi(0) =q˜i(0)(ω)e−iωt+c.c.(i= b = 24mm, c = 4mm, g = 0.5mm, and w = 1mm.
s,a). From Eqs. (2) and (3), the amplitudesof q(0) and q(0) For comparison, we also prepared a reference metamaterial,
s a
oscillating at ω are written as q˜(0)(ω) = Vi (i = s,a), shown in Fig. 1(e), which has the same structure, exceptfor
i ωZi(ω) the absence of the central structure and the direction of the
where Z (ω) = R i[ωL 1/(ωC)] and Z (ω) = R
s a
− − − diodes. The loop current in this structure also resonates at
i[ωL 1/(ωC )] aretheimpedancesforthesymmetricand
a
− thesamefundamentalfrequencyasthatoftheCSRR,andthe
anti-symmetricmodes,respectively. Whentheexcitationan-
SHelectromotivevoltagesgeneratedinthetwovaractorscon-
gular frequency ω is tuned close to ω , q˜(0) becomes large
s | s | tribute to the loop current, which does not resonate. There-
duetoresonance.Ontheotherhand,theanti-symmetricmode
fore, the ratio of SHG efficiency between CSRRs and SRRs
ishardlyexcited;q˜a(0) 0. canbeinterpretedastheenhancementfactorowingtotheres-
∼
Thefirst-ordersolutionofqawithrespecttoαsatisfies onanceeffectfortheSHwaves.
Before SHG demonstration, we conducted transmission
Ld2qa(1)+Rdqa(1)+ qa(1) + α q(0) 2 =0. (4) measurements to identify the resonant modes and resonant
dt2 dt C 2{ s } frequencyofthemetamaterials. We usedawaveguidecalled
a
a striplineTEM-cell, as shownin Fig. 2.16,17 Thewaveguide
It is found that the resonant oscillation qs(0) in the symmet- supportsaTEMmodepropagatinginthez direction. Theta-
ricmodeinducestheSH currentintheanti-symmetricmode peredstructuresensurethatthewaveimpedanceismaintained
throughthe last term α q(0) 2. The nonlinearityof the var- at50Ωalongthe waveguideandthe tipsof the topplate are
2{ s }
actor results in a coupling between the two resonant modes. connectedtotheinputandoutputports,whichhaveSMAcon-
3
nectors.WeplacedthreeSRRsorCSRRsatintervalsof5cm (a) Signal Generator Spectrum Analyzer
inside the waveguide, as shown in Fig. 2. In the transmis-
LPF -10dB HPF
sionspectra,weobservedtworesonantdipsat1.36GHzand
2.72GHzfortheCSRRsandadipat1.36GHzfortheSRRs. TEM-Cell
<1400 MHz >2275MHz
Thus, it is found that the common dips at 1.36GHz corre- isolator
spond to resonancein the symmetric (or loop-current)mode (b)m-50
B CSRR
and the higherresonantmode at 2.72GHzfor the CSRRs is d-60
the anti-symmetric mode. This CSRR structure satisfies the r / -70 SRR
e
doublyresonantcondition,2ω =ω . w-80
s a o
Figure 3(a) shows the experimental setup to measure the G p-90
SH power generated in the metamaterials. A low pass filter H-100
S 2.4 2.5 2.6 2.7 2.8 2.9 3.0
(LPF)wasusedtosuppressresidualharmonicsfromthesig-
SH frequency / GHz
nal generator (Agilent N5183A). The output signal contain- (c)m -20
ing the SH wave passes through a high pass filter (HPF) to B CSRR
d -40
rejectthefundamentalwaveandissenttothespectrumana- r / SRR
lyzer(ADVANTESTU3751). Whilesweepingthefrequency we -60
of the input signal, the power density at the SH frequency, po -80 19.6dB
or SHG power, was observed. Figure 3(b) shows the SHG G
H-100
power obtained for the CSRR (circles) and SRR (triangles) S -9 -6 -3 0 3 6 9
metamaterialsforaninputpowerof 3dBm. Thepeaksare input power / dBm
−
foundaround2.7GHzforbothcases, andthepeakvaluefor
the CSRRs is higher than that of the SRRs by about 20dB. FIG. 3. (a) Experimental setup for SHG measurements. (b) SHG
ThisisclearevidencethatSHGintheCSRRmetamaterialis powerfor−3dBminput.(c)SHGpeakpowerasafunctionofinput
power.
significantlyenhancedbytheresonanceattheSHfrequency.
As shown in Fig. 3(b), we fit experimental data with theo-
retical curves(a solid line for the CSRR and dashed line for
the SRR), which can be derived from the fact that the SHG excites the resonant mode owing to the arrangement of the
powerisquadraticallyproportionaltoEq.(1)fortheSRRand nonlinearelements. However,it is actually difficultto attain
Eq. (5) for the CSRR. The fitting curves reproduce the ex- strongmagneticcouplingwithfiniteelementdimensionsand
perimental results well. Figure 3(c) shows the peak values alsodifficulttooptimizetheparametersofthemetamaterials,
of the SHG spectra. The circles and triangles correspond to considering the resonant frequency shifts induced by strong
the data for the CSRR and SRR, respectively. For weak in- coupling. TheCSRRmetamaterialattainsamuchhigheren-
put power, both SHG spectra exhibit quadratic dependence, hancement factor, just by designing the structures to satisfy
whichisrepresentedbythesolidlinefortheCSRRs andthe ωa =2ωs.
dashedlinefortheSRRs. Inthisregion,thesecondorderper- Inthispaper,weproposedamethodtoenhanceSHGwith
turbationisvalid,whereasforgreaterinputpower,third-order CSRR metamaterial. The arrangement of the nonlinear el-
nonlinearity should be taken into consideration. In fact, in ements induces coupling between the two resonant modes,
thehigherinputpowerregion,wheretheSHGspectraarebe- which satisfies the doubly resonant condition. Owing to the
lowthesolidordashedlines,theresonancelinewidthbroad- direct excitation in the resonant mode for SH waves, we at-
ens and a small resonant frequency shift is observed due to tainedsignificantenhancementofuptotwoordersofmagni-
the self-phase modulation. Fromthe abovereasons, it is ap- tude.ThepreciseandquantitativeevaluationofSHGefficien-
propriatetoestimatetheSHGenhancementoftheCSRR for ciesandoptimizationoftheCSRRstructureareleftasfuture
weak input power without third- or higher-order nonlineari- problems. We expect that metamaterials with nonlinearity-
ties. Thedifferencebetweenthesolidlineanddashedlineis assisted coupling could be employed for investigating other
19.6dB,whichcorrespondstotheenhancementfactorowing typesofnonlinearphenomena.
totheresonanceeffectfortheSHwave. The present research was supported by Grants-in-Aid for
Scientific Research Nos. 22109004 and 22560041, by the
Asforothertypesofdoublyresonantmetamaterials,mag-
GlobalCOEprogramofKyotoUniversity,andbyaresearch
netically coupled split ring resonators have been theoreti-
cally analyzed.14 Magnetically coupled cut-wire resonators grant from The Murata Science Foundation. One of the au-
thors (Y.T.) would like to acknowledgethe support of a Re-
has been studied and 6.6dB enhancementof SHG has been
achieved.15 Both of the metamaterials are composite meta- search Fellowship of the Japan Society for the Promotionof
ScienceforYoungScientists.
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