Directional Generation of Graphene Plasmons by Near Field Interference

2.4 MB·

by  Lei Wang

Directional Generation of Graphene Plasmons by
Near Field Interference
Lei Wang,1,2,∗ Wei Cai,2,3,4,† Xinzheng Zhang,2,3,4 Jingjun Xu,2,3,4,‡ and Yongsong Luo1
1College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China
2The Key Laboratory of Weak-Light Nonlinear Photonics, Ministry of Education,
School of Physics and TEDA Applied Physics Institute, Nankai University, Tianjin 300457, China
3Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, China
4Synergetic Innovation Center of Chemical Science and Engineering, Tianjin 300071, China
(Dated: January 5, 2016)
6
The highly unidirectional excitation of graphene plasmons (GPs) through near-field interference
1
0 oforthogonallypolarizeddipolesisinvestigated. ThepreferredexcitationdirectionofGPsbyasin-
2 gle circularly polarized dipole can be simply understood with the angular momentum conservation
law. Moreover, the propagation direction of GPs can be switched not only by changing the phase
n
differencebetweendipoles,butalsobyplacingthez-polarizeddipoletoitsimageposition,whereas
a
thehandednessofthebackgroundfieldremainsthesame. TheunidirectionalexcitationofGPscan
J
be extended directly into arc graphene surface as well. Furthermore, our proposal on directional
4
generationofGPscanberealizedinasemiconductornanowire/graphenesystem,whereasemicon-

ductor nanowire can mimic the circularly polarized dipole when illuminated by two orthogonally
]
l polarized plane waves.
l
a
h PACSnumbers: 73.20.Mf,81.05.ue,78.67.Wj,78.20.Bh
-
s
e
I. INTRODUCTION reported methods, the use of near field interference of a
m
circularlypolarizedwaveiscornerstoneforactiveswitch-
t. Graphene plasmons (GPs), the intrinsic collective os- ing, along with very high extinction ratio between differ-
a cillationsofelectronscoupledwithelectromagneticwaves ent directions. However, the current experimental effort
m
in doped graphene, have attracted enormous interests to mimic a two-dimensional rotating dipole by oblique
d- for their unique properties, including such as inher- incidence is controversial due to the inevitable magnetic
n ently highly controllable, long-lived and extremely elec- induction currents[18]. It is well-known that two orthog-
o tromagnetic field confinement and enhancement in mid- onally oriented dipoles can be induced by two orthogo-
c infrared and terahertz spectral regimes[1–3]. Since theo- nally polarized incident plane waves. Aware of that a
[ retically proposed by Jablan et al in 2009[4], GPs have circularlypolarizeddipolecanbeefficientlymimickedby
1 been widely studied for electro-optical modulation[5, 6], a nanowire illuminated by two orthogonal plane waves
v quantum plasmonics[7], light harvesting[8], transforma- duetotheextremelylocalizationofGPs,itisnaturalfor
7 tion optics[9] and infrared biosensors[10] at nanometer us to consider realizing unidirectional generation of GPs
3 scale. Due to large wavevector mismatch between GPs by using a combined circularly polarized dipole in un-
4 and free light field, propagating GPs are usually excited structuredgraphene. Furthermore,thecombineddipoles
0
by deep sub-wavelength point-like sources[11]. In these can be separated in space compared to a single circu-
0
. cases, the sources such as emitters, absorbers and scat- larly polarized dipole, which provides us another degree
1 terers serve as dipoles perpendicular or parallel to the of freedom to control the excitation of GPs.
0
propagation plane of GPs[12]. However, the propaga-
6 Inthisstudy,twoorthogonallyorienteddipolesareem-
1 tion direction of excited GPs is usually isotropic in the ployedtoefficientlyexcitesymmetricandanti-symmetric
: graphene plane as a result of the symmetry of structures charge ordering modes in flat and arc graphene planes.
v
and excitation configurations. And the unidirectional
i As long as the constructive and destructive interferences
X launching of GPs is still unsolved problem although it
of near-fields take place in different propagation direc-
r is important in ultracompact plasmonic devices at the tions, the unidirectional launching of GPs occurs. Due
a chip scale. On the other hand, to achieve highly direc-
to the inherent phase difference between symmetric and
tionallaunchingofsurfacewavesinmetal,attemptshave
anti-symmetricevanescentmodesinducedbyx-polarized
beenmadetobreakthesymmetrybyintroducingoblique
andz-polarizeddipoles,theextraphasedifferenceofπ/2
incidence[13, 14], double slits[15] and circularly polar-
, e.g., circularly polarized dipoles, should be introduced.
ized incident waves[16, 17], etc. Among the numerous
Moreover, whenthecircularlypolarizeddipoleisdecom-
posed into two linear polarized ones and separated on
different sides of the graphene, the behaviors of induced
∗ [email protected] charge distribution and handedness of the background
† [email protected] field are opposite. In further, the circularly polarized
‡ [email protected] dipole can be efficiently mimicked by a semiconductor
2
nanowireilluminatedbytwoorthogonallypolarizedplane satisfied, so the induced magnetic field has a relation of
waves in experiments, and the problem about magnetic H0(k ,z)∝ pz∓i. Onecanconcludethatthecomplete
y spp px
induction currents induced by oblique incidence can be interferencestakeplaceaslongasp andp haveequally
z x
solved in our considered system. We believe our find- modulus with phase difference of ±π/2. The contribu-
ings should be found applications in compact plasmonic tion from the graphene can be included by introducing
circuits in mid-infrared and terahertz regimes. reflected and transmitted fields, which can be calculated
via simply multiplying the individual angular spectrums
withcorrespondingFresnelcoefficientsrp andtp, respec-
II. THEORETICAL BACKGROUND tively. When the dimensionless conductivity α = 2πσ/c
in units of the fine-structure constant α (cid:39) 1/137 was
0
adopted,theFresnelcoefficientscanbewrittenas[11,12]:
ThephenomenaofunidirectionalexcitationofGPscan
be understood by considering a dipole placed at a sub- 2k(cid:48)
wavelength distance close to a free standing graphene rp(k )=1− z (1)
sheet. Fig. 1(a) illustrates the scheme employed in x (cid:15)skz+kz(cid:48) +2αkzkk0z(cid:48)
our design. A two-dimensional (2D) dipole with mo-
2(cid:15) k
smheenettu.mApC2Dart=esia[npx,cpozo]rdiisnaptleacseydsteambovise cahogseranphweinthe tp(kx)= (cid:15)skz+kz(cid:48)s+z2αkzkk0z(cid:48) (2)
the graphene sheet laying in z = 0 and the position
where (cid:15) is dielectric constant of substrate, k =
of the dipole is (0, zdip). Without loss of generality, (cid:112)k2−ks2 and k(cid:48) = (cid:112)(cid:15) k2−k2 are magnitudezs of
the result can be extended to three-dimensional (3D) 0 x z s 0 x
the longitudinal wavenumbers. For the reflected and
treatment directly[16]. The vector potential A induced
transmitted fields, one can obtain the angular spec-
by the dipole without graphene can be expressed as
A(r) = −iωµ G(r(cid:48),r)p, where G(r,r(cid:48)) = iH(1)(k |r− tra Hyref(kx,z) = rp(kx)iω4πpx[ppxz kkxz + 1]eikz(z+zdip) and
r(cid:48)|) is the 2D0 Green’s function in free s4pac0e[19,020], Hytr(kx,z) = tp(kx)iω4πpx[ppxz kkxz + 1]e−ikz(z−zdip), respec-
tively. The angular spectra of total magnetic fields
and k = ω/c is wavenumber in vacuum. The angular
0
in the spaces on top of and bottom of graphene are
spectrum decomposition of the vector potential can be
wkri=tte(cid:112)nka2s−Ak(r2)is=thωe4µπ0lopn(cid:82)g−∞it∞udeeikzw|zka−zvzednipu|meibkxexrd.kTx,huwshtehree cHaylc(ukxla,tze)d=asHHytry((kkxx,,zz)), r=espHecy0t(ikvxel,yz.)F+roHmyretf(hkex,czu)stoamnd-
z 0 x aryboundaryconditionandchargeconservationlaw,i.e.,
magnetic field can be deduced as H0(x,z) = 1 (∇ ×
A)y = 4iωπ (cid:82)−∞∞[pzkkxz ∓px]eikz|z−zdip|eyikxxdkx. Tµh0e an- ndu×ce(dHc2h−argHe1d)e=nsKity=ρisσndEi(cid:107)natnhde∇grsa·pKhen=eilωayρesr, tchaenibne-
gular spectrum of the magnetic field can be written as obtained from the difference of magnetic fields at each
Hy0(kx,z)= iω4πpx[ppxz kkxz ∓1]eikz|z−zdip|. side of the graphene
δ(z)∂ σE (x,0)
ρind(x)= x
s iω ∂x
δ(z) ∂
= (H (x,0−)−H (x,0+)). (3)
iω ∂x y y
From now on, the prefactor δ(z) will be omitted for con-
venience, thus the angular spectrum of ρind can be writ-
s
ten as
ik k
ρisnd(kx)=[tp−(1+rp)]4πx[pzkx +px]eikzzdip. (4)
z
Noting that rp and tp depend on the modulus of
k only, naturally, one can divide the contributions
z x
y of a circularly polarized dipole into two parts, i.e.,
d ρisnd(kx) = ρpz(kx) + ρpx(kx). The former induced by
z-polarized dipole (abbreviated as p for convenience)
z
x
satisfies ρpz(kx) ∝ pzkkxz2, while the latter induced by x-
FIG.1. Schematicofdirectionalexcitationofgrapheneplas- polarizeddipolesatisfiesρpx(kx)∝pxkx. Thefundamen-
monsbyadipolesource. Thedipolesourcecanbemimicked tal mechanism for directional generation is the charge
by a semiconductor nanowire. densityinducedbyp hasanevenparitybothinangular
z
spectrum and real space[21], whereas the opposite hold
Nextweturntothecasewherethedipoleislaidontop true for a px dipole. The superposition of ρpz and ρpx
of graphene. Due to the extremely large wavenumber of (nottheH withoppositeparities)leadstotheconstruc-
(cid:112) y
GPs, i.e., k =k (cid:29)k , and k = k2−k2 ≈ik is tive and destructive interferences in different directions.
x spp 0 z 0 x spp
3
When the p or p dipole is moved to its image po- (corresponding to λ≈10 µm). The dipole is situated at
z x
sition (0, -z ), noting that H˜ (k ,0−)−H˜ (k ,0+) = adistanceofd=0.01λontopofthegrapheneplaneand
dip y x y x
H (k ,0)(1 + rp − tp) = H (k ,0+) − H (k ,0−) and has a momentum of [1, pz]p , where the unit length mo-
0 x y x y x px x
the minus sign should be adopted before px in the ex- mentump˙x =−iωpx =1A·misadoptedforconvenience,
pressions of Hy thanks to z−(−zdip)>0, therefore the and the ratio pz is discussed later. Chemical potential
charge density can be written as of the doped gprxaphene is set as µ=0.4 eV, and ambient
temperature is set as T=300 K, the in-plane conductiv-
ip k2
ρ˜pz(kx)=(1+rp−tp)4πz kxeikzzdip =−ρpz(kx), ity of the graphene is computed within the local-random
z phase approximation (RPA)[22, 23] with an intrinsic re-
ρ˜px(kx)=(1+rp−tp)i(−4πpx)kxeikzzdip =ρpx(kx),(5) lµax=at3i0o0n0tcimm2e/Vτs)=, w1h2i0chfsis(aintdyipciactailnpgartahme emteorbdileitryiveodf
from experiments[10, 24]. Commercial software COM-
where the tildes means the quantity induced by dipoles
SOL Multiphysics based on FEM method is adopted to
located at its image position. Therefore, one can obtain
the relation of ρ˜pz(x) = −ρpz(x) and ρ˜px(x) = ρpx(x). solve the Maxwell equations. From the dispersion re-
lations of GPs, the wavenumber of GPs for this free-
Thisresultmeansthatmovingthep dipoletoitsimage √
z standing graphene is k = 1−α2 k ≈ 21.78 k , in-
position will switch the preferred propagation direction spp 0 0
dicatingtheplasmonwavelengthλ is459nm,andthe
of GPs, while moving the p dipole will not change the spp
x
preferred direction. This behavior is quite counterintu- extinction parameter is a∗ ≡ kkxz|kx=Re{kspp} = 0.9989i.
itive. Because the magnetic field induced by the dipole Therefor pz =±0.9989i will lead to completely destruc-
satisfiedH˜pz =Hpz andH˜px =−Hpx,whichmeansthat tiveinterfepxrenceoftheexcitedGPsinacertaindirection.
y y y y
moving a p dipole to its image position will change the
x
incident dipole fields, while they keep unchanged when
5
moving pz dipoles. Combination of these two facts leads 4 (a) (b)
3
toanamazingresultthattheincidentandinducedfields
havedifferentpreferredpropagationdirections. Remark- 21 kx=-kspp kx=kspp
ably, the finally preferred direction of GPs is determined 0
z -1
by the induced charge pattern rather than the incident
-2
fielTdh.ere are a lot of parameters to quantitative describe --43 0|Hy(t0)| 0(×.1506 A/m1)
-5
the asymmetrical excitation. Among them, the angu-
lar spectrum ratios satisfy R [F] ≡ |F(k )/F(−k )| =
k x x
|−ppppxzxzkkkkxzxz++11|,(F ∈{Hy,Ez,Ex,ρ})whichdependon ppxz kkxz (c) 3 (d)
only. Moreover, the spatial dependent near field ratios 2
defined as R [F] ≡ |F(x)/F(−x)|, (F ∈ {H ,ρ,P })
x y x
are also very important for GPs which can be obtained Rx [ ]
via full-field simulations and verified by near field ex- RRxx [[ ] ]
periments directly. In engineering, another important
parameter to quantify the asymmetrical transmission is
extinction ratio, which is defined as the logarithm of en-
FIG. 2. Directional excitation of GPs by a circularly po-
ergy flux ratio in opposite directions η = 10log(P /P ).
r l larized dipole. (a) Magnetic field distributions for GPs ex-
The right and left energy flux along the graphene can cited by a 2D circularly polarized dipole p = [1,a∗]p ,
2D 0
be obtained by integrating the relative Poynting vector where −iωp = 1A · m. (b) Angular momentum spectra
0
along z direction far from the dipole source of initial (H0), reflective (Href) and total (H ) magnetic
y y y
field magnitude of the polarized dipole (solid lines) and an
P (x→∞)=(cid:90) ∞ 1Re{E H∗}dz. (6) ideal circularly polarized (dashed lines). The dotted lines in-
x 2 y z dicate the wavenumbers of GPs. (c) The simulated (solid
−∞
line) and analytically calculated (marked by red circle) spa-
tialdependentchargedensityingraphene. (d)Spatialdepen-
dent near field ratio R [Href] = |Href(x)/Href(−x)|(colored
III. NUMERICAL SIMULATIONS x y y y
inblack),R [ρ]=|ρ(x)/ρ(−x)|(coloredinred)andR [P ]=
x x x
|P (x)/P (−x)|(colored in blue).
x x
We present several scenarios in which the proper
choices of dipoles close to graphene sheet provide possi-
bilities for directional excitation of GPs. First of all, we The simulated distribution of H field |Re{H }| is de-
y y
considerthebasicmodeldescribedinSecII,andcompare picted in Fig. 2(a). Clearly, the plasmon mode is uni-
the simulated results to the analytical results calculated directionally excited by the circularly polarized dipole,
from angular spectra. In the simulation, the frequency which has a much larger amplitude along +x than −x.
of electromagnetic field emitted by the dipole is 30 THz Besides, one can find that the background field is an-
4
ticlockwise rotational due to the individual rotational 5
dipole source. Thus the angular momentum density of 4
3 200 nm
the background field is L = (cid:15)0r×(E×B) = c12r×S, 2
which is along −y direction, where S is the Poynting 1
0
vector[25, 26]. Moreover, the angular momentum direc- z -1
tion of preferred exited GPs is consistent with the an- -2
gular momentum direction of background fields due to --43 (a) 0|Hy(t0)| 0(×.1506 A/m1) (b)
theconservationofangularmomentum. Whenthephase
-5
difference between p and p changes from π/2 to −π/2
x z
or placing the rotational polarized dipole on bottom of
thegrapheneplane,thedirectionsofangularmomentum
as well as the propagating GPs inverse. Therefor one
can determine the preferred excitation directions simply
by the direction of angular momentum. On the other
hand, the directional excitation of GPs in real space can
be understood by the asymmetry of the angular spec- (c) (d)
tra in different directions. The angular spectra of ini-
tial, reflected and total magnetic fields are shown in Fig.
2(b). Onecanfindtheangularspectrahaveconstructive
and destructive interferences at +k and −k , respec- FIG. 3. Directional excitation of GPs by two separated
x x
orthogonal polarized dipoles. (a) Magnetic field distribu-
tively. To understand this effect, the Fresnel coefficient
rp is considered within plasmon pole approximation ( tions for GPs excited by two orthogonal polarized dipoles
p = [1,0]p and p = [0,a∗]p located at (0, 0.01λ) and
k (cid:39) ±k ). The coefficient rp ∝ 1 has peaks x 0 z 0
x spp k2−k2 (0,-0.01λ),respectively. Theinsertfigureshowstheenlarged
x spp
when k = ±Re{k }. Specifically, when pz equals to excitation region. (b) Angular momentum spectra of initial
a∗, Hy0(xkspp,0+) ∝sppppxz kkxz|kx=Re{kspp} +1 =px2, this is a gmraagpnheetnicepfilealdnem,tahgeniitnuitdieal(aHny0d)ianndduccehdaqrgueandteintiseistyh(aρv)eionppthoe-
peak due to constructive interference and the intensity
site preferred direction for excitation of GPs. (c) The simu-
is twice larger than the magnitude of magnetic field in-
lated (solid line) and analytically calculated (marked by red
duced by px or pz individually. In the same manner, circle)spatialdependentchargedensityingraphene. (d)The
the valley exists due to constructive interference occurs spatial dependent extinction ratio for a single circularly po-
at |H0(−k ,0+)| = 0. As a result, the angular spec- larized dipole (case I, colored in black), and two separated
y spp
tra of initial, reflected and total fields have peaks at orthogonal polarized dipoles (case II, colored in red).
+k as well as valleys at −k for pz = a∗. As to
ansipdpeal rotational polarized dispppole, e.pgx. pz equals to
px
1inig, atnodaH∗y0≈(ks1pip.,0H+o)w∝evkkexzr,|kxth=eRree{kissppa}+re1miasrkaabboluetd2iffoewr-- results are shown in Fig. (3). One can find that the
rotational direction of background field in Fig. 3(a) is
ence near k = −k , where it is a peak instead of val-
x spp the same as that in Fig. 2(a), however, the propaga-
ley for the ideal circularly polarized dipole. This comes
tiondirectionofexcitedGPsinverses. Thisresultmeans
fromthat|rp|ismaximaat−k and|H0(−k ,0+)|∝
spp y spp that one cannot distinguish these two cases from the far
|−i/a∗+1| is a slowly varying quantity. In further, the
fields excepted for the preferred propagation direction of
spatial distribution of charge density is a vital physical
excited GPs. Considering that the H is discontinuous
quantity to describe the collective oscillations, such as y
across the graphene plane, the Href changes from up-
plasmons. InFig. 2(c),thesimulatedchargedensitydis- y
side to downside of graphene when the dipole is placed
tribution in the graphene plane is compared to the ana-
to its image position. Meanwhile, H0 and the induced
lyticalresultfromEq. (4). Theyareinperfectagreement y
charge ρ remain unique. Similar to the results in Fig.
and have apparently constructive and destructive inter-
2(b), the angular spectra of H0 and ρ are also shown in
ferences in x > 0 and x < 0, respectively. The spatial y
Fig. 3(b) to demonstrate the mechanism of directional
dependent near field ratios of exited GPs are shown in
excitationinthisscenario. Remarkably,thepreferreddi-
Fig. 2(d). We can see that the near field ratios of re-
rections of background field and induced charge are op-
flectedmagnetic, chargedensityandenergyfluxareover
posite,whicharealong+xand−x,respectively. Thisre-
100 for x<3λ .
spp sultisrathercounterintuitiveandmeansthattheangular
A circularly polarized dipole can be decomposed into momentum is ’non-conservation’ at first sight. Actually,
twoorthogonalpolarizeddipoles, thusitisinterestingto onecanfindthispuzzlingresultcomesfromthemagnetic
see what will happen if these two dipoles are placed on field discontinuity at upper and lower sides of graphene
the different sides of graphene. Specifically, two orthog- (r (cid:54)= t − 1), and the angular momentum is conserva-
onal polarized dipoles with momenta p = [1,0]p and tion. In our proposed scheme, the angular momentum is
x 0
p = [0,a∗]p are considered, where they are located at zero at the original point with both positive and nega-
z 0
(0, 0.01λ) and (0, -0.01λ), respectively. The simulated tive signs in z =0 plane simultaneously. Thus the use of
5
theconservationofangularmomentumcannotdetermine to the extra minus sign, the preferrer direction of ρ and
the preferred direction of GPs directly. To demonstrate H0 is always opposite in this condition. The spatial de-
y
the physical factor to determine the preferred directions, pendent induced charge density is plotted in Fig. 3(c).
we turn to see the dependence of directional generation One can find that the charge oscillates only in the -x di-
on phase difference ∆φ = φpz −φpx = arg{pz}. Sim- rection which is in good agreement with the analytical
px
ilar to the superposition of polarizations, these two op- result. The comparison of spatial dependent extinction
posite sense of rotations lead to a classification of vi- ratio of a single circularly polarized dipole (named af-
bration ellipses according to their handedness, which is ter case I) and two orthogonal polarized dipoles placed
decided by the phase difference of two vibration vectors. at both sides of the graphene (named after case II) are
If the phase difference satisfies ∆φ=mπ,m=0,1,2···, shown in Fig. 3(d). One can see the extinction ratio is
the superpositions are linear polarized dipoles, and their over20forx<4λspp,andtheratioincaseIIislessthan
angular momenta is zero due to r//S, thus the excited thatincaseI,thisoriginatesfromtheoppositepreferred
GPs should be isotropic without any other asymmetry directions of the initial and induced fields in case II. The
to fulfill the conservation of angular momentum. When differenceonextinctionratiobetweenthesetwocasescan
the phase difference satisfies ∆φ=π/2±2mπ, the near be ignored when x(cid:38)5λspp.
fields rotates in the anticlockwise sense, it is said to be
left-handed. If extra π phase is introduce to the ∆φ,
3
(a) (b)
the handedness and preferred direction of excitation will
2
change. In the proposed system, there are three impor-
tant factors to determine the handedness and preferred 01 r=1.4 mm
direction. The first one is the initial phase difference z-1
∆φ which is from the dipoles themselves, e.g., if the ini-
tial arg{ppxz} changes from π/2 to π/2 ± π, the hand- --32 0|Hy(t0)| (×1106 A/m2)
edness, i.e., rotational direction of background field and -3 -2 -1 0 1 2 3
preferred direction will change. The second factor is the
dipole position relative to the graphene. From the rela- 3 (c) (d)
tion Hy =Hypz +Hypx =i/µ[kxAz+sgn(zdip−z)kzAx], 2
one can known that moving p dipole to its image posi- 1
x
tionwillintroducedaminussignduetothesignfunction, 0
z
which is equivalent to introduce extra π phase difference -1
when talking about the handedness and the initial elec- -2 |Hy(t0)| (×106 A/m)
tromagneticfield,whilethereisnoextraphasedifference -3 0 1 2
whenmovingthep dipolestotheirimageposition. The -3 -2 -1 0 1 2 3
z
totalextraphasedifferencefromaforementionedtwofac-
tors will determine the handedness and the preferred di-
FIG. 4. Directional excitation of GPs on curved free-
rection of initial field. However, they are insufficient to
standing graphene. The distribution of magnetic field
determine the preferred direction of exited GPs. Not- |Re{H }| for GPs excited by the configuration of case I(a)
z
ing that the scattering field of upper and lower sides of and two separated orthogonal polarized dipoles (c). Sim-
graphenesatisfied−r =(t−1)forfreestandinggraphene, ulated and analytically calculated spatial dependent charge
which introduced a minus sign compared to the continu- density for the case I(b) and case II(d). The phase in (c) is
ous boundary condition r = t−1. This is the last vital set as π/4 in order to show the two individual sources.
factor to determine the preferred direction of induced
field. Actually, the aforementioned counterintuitive re- Except for the scenario of directional excitation of
sult is originated from the minus sign, which can not be GPs in a flat graphene plane, the directional genera-
treatedasextraphasedifferenceasbeforebecauseitonly tion of GPs on a curved free-standing graphene sheet
acts on induced field and do not affect the handedness is also investigated. The curvature breaks the mir-
and the distribution of initial magnetic field. In a word, ror symmetry relative to graphene plane, then the in-
therearethreefactorsforpx toaffectthepreferreddirec- duced radiative loss will affect the excitation and prop-
tionofexcitedGPs,whileonlytwofactorsforpz toaffect agation efficiency of GPs. The critical curvature radius
the preferred direction of excited GPs in our considered which permits confined wave exist can be calculated by
soyfsgteramp.heWneh,eHnretfh(ez p=x 0a)ndalwpzaylsocdaetneoitnesththeesmamagenseitdiec rc ≈ Im{kz}(kk0spp−k0) ≈0.048λspp, thus a circular radii as
y r = 1.4 µm ≈ 3λ is chosen, which is a typical value
spp
field in the dipole side, thus the angular momentums of
in flexible transformation plasmonics[27]. A circularly
H0 and ρ have the same sign owing to the conservation
y polarized dipole with momentum as the same as in flat
of angular momentum. When they are in the different
grapheneisplacedabovethegraphenecircleatadistance
side, Href in z = 0 induced by the two dipoles denotes
y of100nm. TheconfigurationandsimulatedH fieldam-
y
different sides of graphene, and the preferred direction
plitude are shown in Fig. 4(a). Remarkably, the mode
of GPs should be decided by ρ rather than H0. Due
y propagates mainly along clockwise direction. which is in
6
coincidewiththeresultinflatgraphene(caseI).Wecan by two orthogonally polarized plane waves. Due to the
describe the induced charge density by waveinterference,thebackgroundstandingwavesatisfies
√ √
E ∝ 2cos(k x+π/4), E ∝ 2isin(k x+π/4) and
x x z x
ρ(l)≈ρf(x)+ρf(x∓2πr), l∈(−πr,πr] (7) p /p = E /E = itan(k x+π/4), thus the amplitude
z x z x x
condition of ideal circular dipole requires k D (cid:28) π/4,
x
the upper (lower) sign in Eq. 7 applies to l > 0 (l < 0), where D is the dimension of nanowire. In the configura-
where l = rθ is arc length away from the dipole. The tionofdirectionalexcitationofmetallicplasmons,k D ∼
√ x
spatial charge density ratio is shown in Fig. 4(b). One 1/ 2,theinduceddipolemomentistoononuniformtobe
canknowthatthedirectionalexcitationbehaviorofGPs treatedasidealcirculardipolesource. Thatistosay,this
in arc surface can be understood well by flat graphene methodisnotsuitablefordirectionalexcitationofmetal-
with the same parameter. When the circularly polar- lic plasmons. However, this is not a limitation any more
ized dipole is decomposed into two dipoles located both in excitation of GPs due to the deep sub wavelength of
√
above and below the graphene, the simulated Hy field thenanowiresizeininfraredspectrum, ie, kxD (cid:28)1/ 2.
amplitude shown in 4(c) and charge density distribution Inoursimulation,anIn Ga Asnanowirewithdiam-
0.53 0.47
shownin4(d)canbeunderstoodwellfromflatgraphene eter D of 100 nm (0.01λ) is used to mimic the rotational
in configuration of case II. These results show that di- polarizeddipole. Drudemodelwasadoptedtomodelthe
rectional propagation of GPs can be extended into arc dielectric constant of In Ga As. In this model, the
0.53 0.47
surfaces directly. ω2
dielectricfunctionisgivenby(cid:15)(ω)=(cid:15) − p ,where
∞ ω(ω+iγ)
(cid:112)
ω = nq2/m∗(cid:15) istheplasmafrequency,(cid:15) isthehigh
(a) 500 nm (b) frpequency dieleect0ric constant, and γ = q/µ∞m∗ is damp-
z 1 2 e e
x ingrate. Extractedfromthereferencein[28],theparame-
ters of In Ga As are (cid:15) =12.15, τ =γ−1 =0.1 ps,
0.53 0.47 ∞
and m∗ = 0.523m . Moreover, n = 6.3×1018/cm3 is
e
|Ez(t0)| (V/m) used to realize the resonance of the nanowire at 30 THz.
0 1 2
Theabsorptioncrosslengthnormalizedtogeometrycross
lengthfordifferentdiametersofthenanowireareshowed
in the insert figure of Fig. 5(a), the absorption peak lays
at 30 THz and is independent on the diameter of the
nanowirebecausetheelectrostaticapproximationissatis-
fied. Twotimeharmonicorthogonalincidentplanewaves
(c) (d) withamplitudeof1V/mandphasedifferenceofπ/2are
taken to illuminate the wire, the schematic and simu-
lated electric field distribution are depicted in Fig. 5(a),
wheretheincidentfieldhasbeensubtractedfromtheto-
FIG. 5. Directional excitation of GPs using In Ga As tal field. The expression of the incident fields adopted in
0.53 0.47
nanowires illuminated by two orthogonally polarized plane the simulation is expressed as
waves. (a) Schematics representation and electric field dis-
tributions |Ez| for excited GPs, the insert figure is normal- E = √1 (xˆ+zˆ)eikxx−ikzz−iωt
ized absorption cross section of the In0.53Ga0.47As nanowire 1 2
with radii of 20 nm, 50 nm and 100 nm. (b) The depen-
1
dence of the angular spectrum ratios Rk(kspp) on extinc- E2 = √ (xˆ−zˆ)e−ikxx−ikzz−iωt−iπ/2.
tion parameter |p /p |. The blue circle indicates the pa- 2
z x
rameter of the considered nanowire. The insert figure is the
It would be expected that the nanowire serves as a cir-
polarized circle of the nanowire with (solid line) and with-
cularly polarized dipole with p /p = i. From the fig-
out (broken line) graphene sheet. (c) The simulated and z x
ure, one can see that induced near field along +x with
theoretically calculated spatial dependent charge density in
a much larger amplitude than the one along −x, which
graphene. The thick line indicates the charge distribution
induced by ideal circular polarized dipole with dipole mo- is very similar to the case of a circularly polarized dipole
mentum as −iωpx = 76.72 pA · m, the thin line indicate with pz/px = i. The induced dipole of the nanowire in
the case of dipole momentum as −iωp = 61.08 pA·m and the diameter of 100 nm is −iωp = 76.72 pA·m and
x x
pz = −0.0624+0.9953i, respectively. The dot marked line pz = i, which means that the semiconductor nanowire
ipnxdicatesthesimulatedresultofsemiconductornanowire. (d) cpaxn serve as an ideal circularly polarized dipole as ex-
Theenergyfluxratioofidealdipole(ppxz =i,solidline),actual pected. Whenagraphenesheetisintroducedclosetothis
dipole(pz =−0.0624+0.9953i,brokenline)andsimulatedre- nanowire, the electric field reflected from the graphene
px
sult of semiconductor nanowire (thick line). willactonthenanowireaswell,andthischangesthepa-
rameters of the induced dipole to −iωp =61.08 pA·m
x
Next,weturntodiscusshowtorealizeourproposalin and pz =−0.0624+0.9953i, respectively. The vibration
px
real experiments. The dipole employed in the paper can ellipses of the induced dipole with and without graphene
be mimicked by a semiconductor nanowire illuminated are shown in the insert figure of Fig. 5(b). From the an-
7
gular momentum ratio shown in Fig. 5(b), the angular metric dipoles with orthogonally polarizations can direc-
momentumratiocanover1000forthenanowire. Thein- tionalgeneratepropagatingGPs. Theviewpointofangu-
duced charge density distribution is shown in Fig. 5(c), larmomentumconservationisveryefficienttodetermine
one can find that the charges oscillate only in the +x the preferred direction of exited GPs. When the dipoles
direction. The simulated charge distribution is in good are laid in different sides of graphene, the spatial charge
agreementforanalyticalcalculationwhenthedipolemo- densityratherthanthemagneticfieldshouldbeadopted
ment is set as the actual value of −iωp = 61.08 pA·m to analysis the excited GPs due to the extra minus sign
x
and pz =−0.0624+0.9953i,respectively. Thechargedis- from the discontinuous of magnetic field. In this condi-
px
tributionofunperturbedidealcircularpolarizeddipoleis tion,themagneticfieldofdipoleandinducedchargedis-
shown in thick line, one can see that the oscillation am- tributionhaveoppositepreferreddirectionsandtheprop-
plitude is a bit larger than actual situation. The energy ertiesofexcitedGPsshouldbedescribedbythebehavior
flux ratios are shown in the Fig. 5(d), the asymmetri- of induced charge. Moreover, the direction generation of
cal energy flux is very apparent, the unperturbed ideal GPs can be extended into arc surface directly. Further-
resultisgivenforcomparisonaswell. Theenergyfluxra- more, a semiconductor nanowire can be regarded as a
tio with extinction parameter of pz =−0.0624+0.9953i localized source to mimic the polarized dipoles, which
is similar to the ideal case excepptxfor small extra oscil- can be realized in real experiments.
lation and less magnitude due to the existence of real
part of the extinction parameter. The simulated result
ofnanowireissimilartotheanalyticalresultwithactual
extinction parameter, one can see that the energy flux ACKNOWLEDGMENTS
ratio mimicked by nanowire exceed 100 when the propa-
gation length is less than 2λspp, and the extinction ratio W Cai, X Zhang, and J Xu acknowledge support
exceed 10 in the whole calculation window. from the National Basic Research Program of China
(2013CB328702), Program for Changjiang Scholars and
Innovative Research Team in University (IRT0149),
IV. CONCLUSION the National Natural Science Foundation of China
(11374006) and the 111 Project (B07013). Y. Luo ac-
We demonstrated here that near field interference of knowledge support from the National Natural Science
a circularly polarized dipole and two mirror image sym- Foundation of China (61574122).
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