Nonlinear graphene plasmonics: amplitude equation

0.21 MB·English

by  A. V. Gorbach

Nonlinear graphene plasmonics: amplitude equation
A.V. Gorbach
Centre for Photonics and Photonic Materials, Department of Physics,
University of Bath, Bath BA2 7AY, United Kingdom
Using perturbation expansion of Maxwell equations, the amplitude equation is derived for non-
linear TM and TE surface plasmon waves supported by graphene. The equation describes inter-
play between in-plane beam diffraction and nonlinerity due to light intensity induced corrections
to graphene conductivity and susceptibility of dielectrics. For strongly localized TM plasmons,
graphene is found to bring the superior contribution to the overall nonlinearity. In contrast, non-
linear response of the substrate and cladding dielectrics can become dominant for weakly localized
3
TE plasmons.
1
0
PACSnumbers: 42.65.Wi;78.67.Wj;73.25.+i;78.68.+m
2

n
x
a I. INTRODUCTION x
J

7 Cladding dielectric
2 Applications of graphene in photonics and optoelec-
|
tronics arebeing actively discussedinrecentyears[1, 2]. H
| Graphene
] In particular, graphene plasmonics is considered as a ,
s |
E
c promissing alternative to conventional plasmonics with
|
i noble metals [3]. Recently, hybrid metal-graphene plas-
t
p monic structures have been proposed as the propitious
o platform for novel optical devices [4].
.
s Graphene supports two types of surface plasmons:
c Substrate dielectric
i transverse magnetic (TM) and transverse electric (TE)
s
modes [5, 6]. TM graphene plasmon is in many
y
ways analogous to the surface plasmon excited at a
h y z
p metal/dielectricinterface[7],althoughspecificfeaturesof
[ collective electron excitation in the purely 2D graphene
FIG. 1: (Color online) Schematic illustration of surface plas-
lead to qualitative differences in the spectra of plasmons
1 mon propagating along graphene sheet. Fields are exponen-
inthesetwosystems[6]. Comparedtoitsmetalanalogue,
v tionally localized in x (across the interface), as shown in the
4 TM plasmon supported by graphene offers susbstantial left panel.
0 enhancement of the field localization, accompained by
4 theconsiderabledecreaseofthepropagationloss–allbe-
6 ing crucialfor potential applications of surface plasmons
.
1 in miniature photonic components. The existence of TE
servation of four-wave mixing with graphene flakes [14]
0 plasmonisdirectlyrelatedtothelinear(Dirac)spectrum
andarangeofnonlineareffectsinagraphene-coatedpho-
3 of electrons in graphene [5], there is no analogue of such
1 surface wave in conventional plasmonics. TE plasmon is tonic crystal nano-cavity [15]. Altogether these findings
: put forward the great potential of graphene for building
v only weakly localized at the surface, however it is char-
functional nano-photonic devices.
i acterized by considerably low propagationlosses even at
X
room temperatures. Spectral characteristics of TE and
In this work we consider nonlinear surface waves sup-
r TMgrapheneplasmonsaredefinedbythechargedensity,
a portedbygrapheneinthesimpleplanargeometryshown
which can be controlled chemically [8] or electrically [9].
in Fig. 1. Allowing for light intensity corrections to the
This tunability represents another important advantage
surfacecurrentandtothesusceptibilityofdielectricssur-
of graphene plasmons over metal plasmons.
roundinggraphene,aswellasintroducingdiffractiondue
Opticalpropertiesofdopedgrapheneareencapsulated to a finite beam width in the unbound (y) direction, we
in the induced surface current . So far, graphene plas- develop asymptotic expansion of Maxwell equations and
K
monshavebeenstudiedundertheassumptionofalinear boundaryconditionstoobtainanamplitudeequationfor
relation between the current and the field amplitude quasi-TM and quasi-TE surface waves. The asymptotic
K
: =σ [19]. This is true only atlowlightintensities, expansionprocedureissimilartothatrecentlydeveloped
E K E
while generally the dependence ( ) is predicted to be for semiconductor and metal nano-waveguides [16, 17].
K E
highlynonlinear[10,11]. Theparticularlystrongnonlin- Further, we analyze the relative contribution from di-
ear repsonse of graphene has been confirmed in several electrics and graphene to the overall effective nonlinear-
experiments,including directmeasurementswith optical ity of the system for the two types of plasmons, and the
Kerr gate [12] and z-scan [13] techniques, as well as ob- impact of geometry on the nonlinearity enhancement.
2
II. SETUP AND ASYMPTOTIC EXPANSION highlydopedgraphene, µ kT andh¯ω <2µ,µisthe
| |≫ | |
OF MAXWELL EQUATIONS chemicalpotential[5]. Furthermore,weassumethatnon-
linearcorrectionstothedielectricsusceptibility χ E~ 2
3
∼ | |
We considerthe planargeometry,inwhichsinglelayer and graphene conductivity σ E~ 2 are of the same or-
3
graphene is sandwiched in-between two dielectrics. We der of smallness O(s). We l∼et th|e m| ode amplitude ψ to
choose x axis to be perpendicular to the interfaces, z is vary slowly with propagation distane, ∂ ψ βψ, and
z
thedirectionofpropagation,andy istheunbounddirec- consider weak diffraction, ∂ ψ = 0. Using≪asymptotic
y
tioninwhichlightcandiffract,seeFig.1. Formonochro- expansionofMaxwellEqs.(1) an6 d boundary conditions,
matic fields, ~ = 1E~e−iωt+c.c., ~ = 1H~e−iωt+c.c., in Eqs. (4)-(6), below we derive propagation equation for
E 2 H 2
eachdielectricdomainwesolvestationaryMaxwellequa- the mode amplitude ψ.
tions: Note, with the account of diffraction, the separation
into TM and TE modes can no longer be performed, in-
D~
~ ~ E~ = . (1) stead one deals with quasi-TM and quasi-TE modes.
∇×∇× ǫ
0
Here spatial coordinates are normalized to the inverse
III. QUASI-TM SURFACE PLASMON
wave number k = 2π/λ = ω/c. For homogeneous
isotropic dielectrics, dispacement vector takes the form:
We seek a guided mode solution in the form:
D~ = ǫ ǫE~ +N~ , (2)
0
h i E = A (ψ,x)+B (ψ,x)+O(s5/2) eiβz , (10)
x x x
N~ = 21χ3(cid:18)|E~|2E~ + 12E~2E~∗(cid:19) . (3) Ey = hC(ψ,x)+O(s2) eiβz , i (11)
(cid:2) (cid:3)
E = A (ψ,x)+B (ψ,x)+O(s5/2) eiβz , (12)
Adapting complex amplitude notation to the surface z z z
current ~ = 1K~e−iωt + c.c., K~ = [0,K ,K ]T, the h i
boundaryKcondi2tions can be written as: y z where ψ =ψ(z,y) is a slowly varying function: ∂zψ s,
∂ ψ s1/2, A s1/2, C s, B s3/2. The cho∼sen
y x,z
∼ ∼ ∼ ∼
∆[E ]=0, ∆[E ]=0, (4) orders of smallness are justified below by solving con-
y z
∆[H ]=icǫ ∆[∂ E ∂ E ]=K , (5) sistently boundary value problems, that emerge in dif-
y 0 z x x z z
− − ferent orders of s. Following substitution into Maxwell
∆[H ]=icǫ ∆[∂ E ∂ E ]= K , (6)
− z 0 x y− y x − y equations, in the order O(s1/2) we obtain the following
boundary value problem:
where operators ∆ and Θ are defined as:
∆[f(x)]= lim(f(x δ) f(x+δ)) , (7) LˆTMA~ = 0, (13)
δ→0 − −
Θ[f(x)]= 1 lim(f(x δ)+f(x+δ)) . (8) ∆[Az]=0, ∆[iβAx−∂xAz]=α(1I)Θ[Az], (14)
2δ→0 −
where A~ = [A ,A ]T, α =σ /(cǫ ), and operator Lˆ
x z 1 1 0 TM
Takingintoaccoundfirstordernonlinearcorrectionsto is defined as:
the relationship betwen surface current and electric field
ahmarpmliotnuidces gK~e(nE~e)ra[t1i1o]na,nwdenoebgtlaeicnt:ing the effect of higher LˆTM =(cid:20) βi2β−∂xǫ −∂iβx2x∂x−ǫ (cid:21) . (15)
K =Θ σ E + σ3 E~ 2E + 1E~2E∗ , (9) We choose the solution in the form A~ = I1/2ψ(z,y)~e,
y,z (cid:20) 1 y,z 2 (cid:18)| | y,z 2 y,z(cid:19)(cid:21) where~e=[e ,e ]T is the linear surface plasmon mode:
x z
If the nonlinear response is neglected altogether (σ3 = x<0 : e =eqsx , e = −iβeqsx (16)
0, χ = 0), and no losses due to electron-phonon scat- z x
3 q
s
tering or defects are considered at zero temperature
iβ
(Re(σ ) = 0), the above system admits solutions in x>0 : e =e−qcx , e = e−qcx (17)
1 z x
q
the form of surface plasmons propagating in z direc- c
tion: E,H eiβz. Field amplitudes in such solutions q = β2 ǫ , (18)
are exponen∼tially localized at the interface x = 0 and s,c q − s,c
constant along the unbound direction y. For the case of
ǫ and ǫ correspond to dielectric layers at x < 0 (sub-
s c
positive/negativeimaginarypartofconductivity σ only
1 strate) and x > 0 (cladding), respectively. Propagation
TM/TE surface plasmon exists [5].
constant β is defined through the dispersion relation [6]:
Below we consider a weakly dissipative case, so that
dσu1m=mσy1(Rsm) +alliσp1(aIr)aamnedteσr.1(RT)/hσis1(Ia)s∼sumsp≪tio1n,iwshvearleidsfoisr aa β2ǫs ǫ + β2ǫc ǫ =α(1I) (19)
s c
− −
p p
3
The normalization factor I is chosen in a way that Finally, computing integrals in the r.h.s. of Eq. (29),
ψ 2 is the power density (measured in watts per meter) we obtain the amplitude equation:
| |
carried in the z direction [16]:
∂ψ 1 ∂2ψ
i + +iΛψ+Υψ 2ψ =0, (31)
2βk ∂(z/k) 2βk∂(y/k)2 | |
I = , (20)
ǫ cQ wherethenonlinearparameterΥcombinescontributions
0
+∞ β2 ǫ ǫ of graphene and dielectrics:
Q = ǫe 2dx= s + c (21)
Z−∞ | x| 2 (cid:18)qs3 qc3(cid:19) Υ = g(γG+γD), (32)
iα k2 1
Collecting terms of the order O(s) we obtain: γG = 2ǫ cβ32P2Θ(cid:20)|~e|2|ez|2+ 2~e2(e∗z)2(cid:21) , (33)
0
∆[C(]β=20−,ǫ)C−∆∂[∂x2xCC=I−1I/21/∂2∂ψyeψ](i=βezα+(I∂)xΘe[xC)],.((2232)) γD = 2ǫ0ckβ22P2 Z−+∞∞χ3(cid:18)|~e|4+ 12|~e2|2(cid:19)dx, (34)
x − y x − 1 +∞ 2β2 ǫ 2β2 ǫ
P = ~e2dx= − s + − c , (35)
From ~ E~ = 0 in the order O(s1/2) it follows that Z−∞ | | 2qs3 2qc3
∇ ·
iβez+∂xex =0,andthereforeC solvesthehomogeneous the surface-inducednonlinearityenhancementfactor g is
equation. It is non-zero due to simultaneous diffraction [16]:
(∂ ψ = 0) and discontinuity of e component at the in-
y 6 x g = (1+η)−2 , (36)
terface, see Eq. (23). Substituting C = I1/2∂ ψe , it is
y y
easyto see thatey satisfiesthe samehomogeneousequa- η = −i∆[e∗e ]= 1 1 + 1 , (37)
tion as e . Comparing boundary conditions for e and βP z x −P (cid:18)q q (cid:19)
z y s c
e , we obtain e =( i/β)e .
z y − z andtheeffectivelinearabsorptionparameterisgivenby:
IntheorderO(s3/2)weobtainthe followingboundary
value problem: α(R)k
Λ=g1/2 1 Θ[e 2]. (38)
z
2βP | |
Lˆ B~ =I1/2J~, (24)
TM In the above derivations we used the auxiliraly relation
∆[Bz]=0, (25) g1/2Q = β2P, which can be obtained by using iβez =
∆[iβBx+∂zψI1/2ex−∂xBz]=−iα(1R)I1/2ψΘ[ez] −∂Exxepxraenssdiotnakfionrggbryapphaerntseinntoenglrinaelainr Ecoqe.ffi(3c5ie)n[t16in]. Eq.
+α1(I)Θ[Bz]− 2iα3I3/2|ψ|2ψΘ |~e|2ez+ 12~e2e∗z ,(26) (33) canbe replacedby the integralsimilarto the one in
(cid:2) (cid:3) Eq. (34), following introduction of an effective graphene
where α3 =σ3/(cǫ0) and nonlinear susceptibility:
Jx = ∂zψ(2iβex−∂xez)+∂y2yψ(ex−∂xey) χ3(gr) =iα3δ(x)= iσǫ(c3)δ(x), (39)
+ψ 2ψ n , (27) 0
x
| | where δ(x) is the Dirac delta function. Note however
J = ∂ ψ∂ e +∂2 ψ(e iβe )
z − z x x yy z − y the different structure of the term under the integral,
+ψ 2ψ nz , (28) which is due to the surface nature of nonlinear response
| |
in graphene, cf. Eqs. (9) and (3). In the limit of high
nx,z =INx,z(ex,0,ez). localization, β ǫs,c, for the guided mode one obtains
≫
Next, we project Eq. (24) onto the linear mode ~e: simple relation e = ie , and therefore:
x z
±
1 1
+∞ +∞ ~e2 e 2+ ~e2(e∗)2 ~e4 , ~e2 0. (40)
~e∗ Lˆ B~ dx=I1/2 ~e∗ J~ dx, (29) | | | z| 2 z ≈ 2| | ≈
TM
Z−∞ (cid:16) · (cid:17) Z−∞ (cid:16) · (cid:17) Apparently,inthislimit,theeffectivenonlinearresponse
+∞ 0 +∞ ofgrapheneistwiceweakerthanthatofainfinitesimally
take −∞ = −∞+ 0 in the l.h.s., apply integration thin Kerr medium with the susceptibility χ(gr).
by paRrts andRuse boRundary conditions in Eqs. (14) and 3
(26) to obtain:
IV. QUASI-TE SURFACE PLASMON
+∞
~e∗ Lˆ B~ dx=
TM
Z−∞ (cid:16) · (cid:17) For the case of quasi-TE mode we use the ansatz:
∗ ∗ ∗ ∗
∆[iβ(e B +e B ) e ∂ B +B ∂ e ]
z x x z − z x z z x z Ex = Cx(ψ,x)+O(s2) eiβz , (41)
=−∂zψI1/2∆[exe∗z]−iα(1R)ψI1/2Θ[|ez|2] E = (cid:2)A(ψ,x)+B(ψ,x)(cid:3)+O(s5/2) eiβz , (42)
y
iα ψ 2ψI3/2Θ ~e2 e 2+ 1~e2(e∗)2 .(30) h i
−2 3| | (cid:20)| | | z| 2 z (cid:21) Ez = Cz(ψ,x)+O(s2) eiβz , (43)
(cid:2) (cid:3)
4
where ∂ ψ s, ∂ ψ s1/2, A s1/2, C s, B Performingprojectioninther.h.s. ofEq.(55),andusing
z y x,z
∼ ∼ ∼ ∼ ∼
s3/2. Followingsubstitution intoMaxwell’sequations,in boundaryconditionsinEqs.(45)and(57),weobtainthe
theorderO(s1/2)weobtainthefollowingboundaryvalue amplitude equation (31) with the following coefficients:
problem:
Υ = γ +γ , (59)
TE G,TE D,TE
LˆTEA = 0, (44) i3α k2
γ = 3 Θ e 4 , (60)
∆[A]=0, ∆[∂xA]=−α(1I)Θ[A], (45) G,TE 2ǫ0cβ2P2 (cid:2)| y| (cid:3)
Lˆ = β2 ǫ ∂2 . (46) 3k2 +∞
We choose theTsoElution in−the−forxmx A=I1/2ψ(z,y)e , γD,TE = 2ǫ0cβ2P2 Z−∞ χ3|ey|4dx, (61)
y
where e is the surface plasmon mode: α(R)k
y Λ = 1 Θ[e 2]. (62)
TE y
2βP | |
x<0 : e =eqsx , (47)
y
x>0 : e =e−qcx , (48) Forquasi-TEmodetheenhancementfactorgisabsent.
y
Also,duetothelinearmodebeingscalar,inthiscasethe
qs,c are defined in Eq.(18), andthe normalizationfactor nonlinear response of graphene is completely analogoues
I ensures ψ 2 gives the power density carried in the z to that of a infinitesimally thin Kerr medium with the
| |
direction: susceptibility χ(gr) in Eq. (39).
3
4k
I = , (49)
βǫ cP
0
V. ANALYSIS AND DISCUSSION
+∞ 1 1
P = e 2dx= + . (50)
y
Z−∞ | | 2qs 2qc Conductivity of graphene consists of intra- and inter-
Dispersion relation for the TE plasmon is given by: band contributions, σ1 = σintra +σinter. For the case
of a highly doped graphene, µ kT, µ is the chemical
| |≫
β2 ǫ + β2 ǫ = α(I) . (51) potential, intra- and inter-band terms are given by the
p − s p − c − 1 semi-classical formalism [5]:
In the order O(s) we obtain:
ie2 1
Lˆ C~ = I1/2∂ ψ[∂ e ,iβe ]T , (52) σintra(Ω) = , (63)
TM y x y y π¯h · Ω+iν
− intra
∆[C ]=0, ∆[iβC ∂ C ]=α(I)Θ[C ], (53) ie2 2 Ω iν
z x− x z 1 z σinter(Ω) = ln −| |− inter , (64)
4π¯h 2+ Ω +iν
where C~ = [C ,C ]T, operator Lˆ is defined in | | inter
x z TM
Eq. (15). Substituting in the above equations C~ = where Ω=h¯ω/µ, excitation below interband absorption
I1/2∂ ψ~e and eliminating e , we obtain: threshould is assumed: Ω < 2, coefficients ν = ¯h/(µτ)
y x | |
take into account losses due to electron scatterings at
Lˆ e =iβǫ−1Lˆ e =0. (54) finite temperatures, below we take τ = 100fs and
TE z TE y intra
τ = 1ps [6, 15]. For the doping level of µ = 0.1eV
inter
Inotherwords,e solvesthesamehomogeneousequation
z we obtain ν 0.066, ν 0.007, and the inter-
intra inter
as ey. Comparing boundary conditions for ey, Eq. (45), band absorption ≈threshould is at≈ω = 2µ/¯h 3 1014
th
ande ,Eq.(53),weobtaine =(i/β)e ande 0. Itis ≈ ·
z z y x rad/s (λ 6.3µm). The corresponding dimensionless
≡ th
easy to check that this choice also satisfies the condition ≈
conductivity α is plotted in Fig. 2. Imaginary part of
1
~ E~ =0 in the order O(s). α changes its sign at Ω 1.67, linear TM plasmons
∇I·n the order O(s3/2) the following boundary value ex1ist for Ω < Ω (i.e. w0he≈n α(I) > 0), while TE – for
problem is obtained: 0 1
Ω > Ω (α(I) < 0) [5]. For the chosen doping level, Ω
0 1 0
Lˆ B~ =I1/2 i2β∂ ψe ∂2 ψ(iβe +∂ e ) ,(55) corresponds to λ0 7.5µm.
TE z y− yy z x x The nonlinear co≈nductivity coefficient σ for graphene
(cid:2) ∆[B]=0, (cid:3) (56) 3
is given by [11]:
∆[∂ B ∂ C ]=iα(R)I1/2ψΘ[e ]
−xα(1I−)Θ[By ]x+i43α31I3/2|ψ|2ψΘ[|yey|2ey],(57) σ3(Ω)=−i332πe¯h2 (eVµF4Ω)23¯h2(1+iαT). (65)
ProjectingEq.(55)ontothemodee andfollowingessen-
y
Hereweintroducedcoefficientα toaccountfortwopho-
tiallythesamestepsasdescribedintheprevioussection, T
ton absorption in graphene, recent experiments suggest
in the l.h.s. we obtain:
α 0.1 [15]. Negative imaginary part of σ suggests
+∞ T ≈ 3
e∗Lˆ Bdx= Θ[e∗∂ B B∂ e∗]. (58) that the nonlinear response is of self-focusing type, cf.
Z−∞ y TE − y x − x y Eqs. (31), (33) and (60).
5
3000 0.5
0.04
a) intra
0.4
0.02
2000
(I)α1 0 full β 0.3/k
Λ
0.2
1000
−0.02
inter 0.1
−0.04
0 0
0.5 1 1.5 2 0.5 1 1.5
Ω Ω
FIG. 3: (Color online) Dispersion of TM plasmon in the
4
b) silicon-graphene-air geometry: propagation constant β and
loss parameter Λ/k as functions of Ω. Imaginary part of β
R) 3 as computed from the full dispersion relation with complex-
(α1 valued α1 is shown with squares.
3 2
0 full
1 inter stantlygrowsasΩincreasestowardsthethreshouldvalue
1 Ω . Propagationlossesarerelativelylow: Λ/(kβ)<10−3
0
intra
when 0.5 < Ω < 1.6, which is due to the low absorp-
0 tion rate in graphene below the interband absorption
0.5 1 1.5 2
Ω threshould, cf. Fig. 2(b).
Due to the high localization of TM plasmon, the
surface-induced enhancement factor g is large and is
FIG. 2: (Color online) Dimensionless conductivity of growing nearly exponentially with increasing Ω, see
graphene α1: imaginary (a) and real (b) parts. Dashed, dot- Fig. 4(a). This growth over-ballances the decay of
ted and solid curvescorrespond tointra-, inter- and full con- graphene nonlinear response σ Ω−3, cf. Eq. (65),
3
ductivity, respectively. The chemical potential is set to µ = ∼
and causes the considerable increase of the nonlinear co-
0.1eV, and relaxation times are τintra =100fs, τinter =1ps.
effficient Υ with frequency, see Fig. 4(b).
Remarkably, the relative contribution of silicon sub-
strateto the overalnonlinearityremainsnegligibly small
Below we consider surface plasmons in configurations
within the entire frequency window of existence of TM
with air (ǫ = 1) and silicon (ǫ = 12) as dielectrics. For
silicon we take χ =(4/3)cǫ ǫ n , n =4 10−18m2/W. plasmon, as illustrated in Fig. 4(b). Also, due to the
3 0 s 2 2
· largeβ, the diffractionterminEq.(31)canbe neglected
Two photon absorption in silicon is negligible for λ >
for typical beam widths L > 1µm. Indeed, for Ω = 1
2µm [18]. For simplicity we neglect dispersion of linear y
(λ 12.4µm) and the beam width of L = 10µm the
and nonlinear dielectric constants. y
diffr≈action length is L =L2βk 50mm, which is more
D y ≈
than six orders of magnitude larger that the apparent
plasmon wavelength λ = 2π/(βk) 10nm. Neglecting
A. Quasi-TM plasmon p
≈
nonlinear response of dielectrics and beam diffraction,
as well as disregarding linear and nonlinear absorption
First, we consider TM plasmon in the configuration
(R) (R)
(σ = σ = 0), one can find stationary solutions
with silicon substrate and air cladding. The correspond- 1 3
ing dispersion is plotted in Fig. 3. To validate our the- of Maxwell equations E~(x,y,z)= Iψ0~e(x,y;βNL)eiβNLz
ory, the propagation loss parameter Λ/k is compared with the nonlinear boundary condition in Eqs. (4-6) an-
against imaginary part of the propagation constant (see alytically. The corresponding dispersion relation reads:
opensquaresinFig.3). The latteris computedfromthe
full dispersion relation that takes into account complex- ǫs + ǫc =α(I)
valued α and obtained by replacing α(I) with iα in √βN2L−ǫs √βN2L−ǫc 1
Eq. (19).1 The results are found to be 1in perfec−t ag1ree- +α(3I)I ~e2 e 2+ 1~e2(e∗)2 ψ 2 . (66)
2 | | | z| 2 z | 0|
ment. (cid:2) (cid:3)
TM plasmon is characterized by a considerable light At the same time, substituting ψ(y,z) ψ eiβNLz into
0
≡
confinement in a wide range of frequencies: even at the theamplitudeequation(31)andassumingΛ=Im(Υ)=
frequency as low as Ω = 0.25 (ω = 7.5 1013rad/s, λ 0, we obtain β = β + (Υ/k)ψ 2. Solving the dis-
NL 0
· ≈ | |
25µm) the propagation constant is β 240, and it con- persion relation in Eq. (66) numerically, we found both
≈
6
0.015 3
1013 a)
5
. 0.01 2 k
1011 0) Λ/
ε
g − 6
2 0
9 β 0.005 1 1
10 (
107 0 0
0.5 1 1.5 1.7 1.8 1.9 2
Ω Ω
0.5 1 1.5
100
FIG. 6: (Color online) Dispersion of TE plasmon in the
] b) silicon-graphene-silicon geometry: Plasmon localization fac-
−1 80 1 |G torq=pβ2−ǫ,propagationlossΛ/k andimaginarypartof
W γ thepropagation constant (squares).
60 /
µ D
[ γ
ϒ) 40 0.56 | 0.1 200
( 0
e 1 ]
R 20 1 0.08
− 150
W

0 0.5 1 1.5 0 M 0.06 |G
Ω ) [ 0.04 100γ/D
ϒ γ
( |
FIG. 4: (Color online) Effective nonlinearity of quasi-TM e 0.02 50
R
plasmon: (a) surface enhancementfactor g;(b)nonlinearco-
efficient Υ and the relative dielectric nonlinearity.
0 0
1.7 1.8 1.9 2
Ω
FIG.7: (Coloronline)ThesameasFig.4(b)butforquasi-TE
300 plasmon in thesilicon-graphene-silicon geometry.
]
1
−
m 200
B. Quasi-TE plasmon
µ
[ Ω=1.25
k 100 Ω=1 As follows from the dispersion relation in Eq. (51),

β in order to excite TE plasmon in an assymetric geome-
∆
try with different substrate and cladding dielectrics one
0 has to make conductivity of graphene strong enough:
0 5 10 α(I) > ǫ ǫ . We would like to note however,
ψ 2 | | | s− c|
| | [W/m] in contraspt to the case of a conventional dielectric slab
0 waveguide,hereonlythe differencebetweenthe cladding
and substrate dielectric constants matters. Apparently,
FIG. 5: (Color online) Nonlinear index shift ∆β = βNL−β for the chosen doping level of graphene, TE plasmon
vs power density. Solid curves correspond to the numerical does not exist in the silicon-graphene-air configuration,
solution of the dispersion relation in Eq. (66), dashed lines - cf. Fig. 2(a). Instead, we consider the fully symmet-
totheresultgivenbytheamplitudeEq. (31): ∆βk=Υ|Ψ0|2. ric configuration with silicon in the cladding and sub-
strate: ǫ = ǫ = 12. The corresponding dispersion is
s c
plotted in Fig. 6. Due to the low values of α(I) , TE
| |
plasmon is only weakly localized. However, one benefits
frommuchsmallerpropagationlossesperplasmonperiod
results to be in perfect agreementat low power densities T =2π/(βk), compared to TM plasmons.
ψ 2, see Fig. 5. As the result of weak localization, typical values of
0
| |
7
the effective nonlinear coefficient Υ for TE plasmons Induced surfacecurrentingrapheneshows stronglynon-
are nearly twelve orders of magnitude below those for linear response to the applied electromagnetic field. We
TM plasmons, see Fig. 7. Remarkably, for TE plas- haveshownthatthisleadstotheeffectivefocussingKerr
mon nonlinear contribution from the dielectrics is domi- type nonlinearity. For TE plasmon this nonlinearity is
nant over graphene, they become comparable only when fully analogous to that of an infitesimally thin dielectric
the frequency Ω approaches the interband absorption layer. However,forTMplasmonthestructureofthecor-
threshould Ω =2. responding nonlinear coefficient is different and reflects
th
With the account of small propagation constants β, the unique surface-only response of graphene.
diffraction term and associated effects due to its inter- For typical doping levels of graphene of the order of
playwiththefocusingnonlinearitybecomeimportantfor 0.1eV, TM plasmons are strongly localized. This causes
TE plasmons with typical widths of the order of sev- the significant enhancement of nonlinearity, we predict
eral micrometres. Taking Ω = 1.9 (corresponding to thatconsiderablenonlinearphaseshiftscanbeobserveed
λ 6.5µm), for Ly = 5µm the diffraction length be- for powerdensities as lowas few micro-Wattsper metre.
com≈es LD = L2yβk ≈ 30µm, and the plasmon period is Remarkably, graphene contribution to the overall non-
2π/(βk) 0.5µm. However,due to the low nonlinearity, linearity is shown to be strongly dominant over that of
≈
one requires considerably high powers to observe basic dielectrics in this case. In contrast, TE plasmons are
effects such as self-focusing. For instance, to form a spa- only weakly localized, and the major part of the overall
tialsolitonofwidthLy thepeakpowerdensityshouldbe nonlinearity is due to dielectric substrate and cladding.
ψ0 2 =(ΥLD)−1 3 1010W/m, andtherefore the total Typical values of the nonlinear coefficient for TE plas-
| | ≈ ·
power must be of the order of several mega-Watts. monsarefoundtobeabout12ordersofmagnitudebelow
those for TM plasmons.
VI. SUMMARY
Acknowledgments
Using asymptoticexpansionofMaxwellequationsand
boundary conditions, we have derived amplitude equa-
tion for nonlinear TM and TE surface plasmon waves We acknowledge usefull discussions with Dmitry
inthedielectric-graphene-dielectricplanarconfiguration. Skryabin.
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