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Electrically Tunable Optical Nonlinearities in Graphene-Covered SiN WaveguidesCharacterized by Four-Wave Mixing

Koen Alexander,∗ Bart Kuyken, and Dries Van ThourhoutPhotonics Research Group, INTEC, Ghent University-IMEC, Ghent B-9000, Belgium and

Center for Nano-and Biophotonics (NB-Photonics), Ghent University, Ghent B-9000, Belgium

N. A. Savostianova, S. A. MikhailovInstitute of Physics, University of Augsburg, D-86135 Augsburg, Germany

(Dated: June 2, 2017)

We present a degenerate four-wave mixing experiment on a silicon nitride (SiN) waveguide coveredwith gated graphene. We observe strong dependencies on signal-pump detuning and Fermi energy,i.e. the optical nonlinearity is demonstrated to be electrically tunable. In the vicinity of theinterband absorption edge (2|EF | ≈ ~ω) a peak value of the waveguide nonlinear parameter of ≈

6400 m−1W−1, corresponding to a graphene nonlinear sheet conductivity |σ(3)s | ≈ 4.3 · 10−19 A

m2V−3 is measured.

In recent years, there has been increasing interest inthe nonlinear optical properties of graphene. Both theo-retical predictions [1–6] and experimental studies [7–11]have indicated that graphene has a very high third-order

sheet conductivity σ(3)s , which leads to a strong nonlinear

optical response. Despite the consensus that nonlinear-ities in graphene are strong, very different values of thecorresponding material parameters have been reported(see, e.g., discussions in Refs. [3, 11]). Reasons for thiscan be found in the fact that in different experiments dif-ferent nonlinear effects (harmonics generation, four-wavemixing, Kerr effect) are probed at different wavelengths,in samples with different carrier densities and in differ-ent dielectric environments. Moreover, in experimentalstudies pulses with vastly different durations and opti-cal bandwidths have been used. All these factors cansignificantly influence the final result. Hence a detailedquantitative study of the nonlinear response of graphene,at different frequencies and in samples with different elec-tron densities, is imperative.

Another important research direction is the search forinterband resonances in the nonlinear response functionof graphene. It has been theoretically predicted [3–6]that the nonlinear parameters of graphene should haveresonances at frequencies corresponding to the interbandabsorption edge, e.g. at ~ω = 2|EF |/3 for third harmonicgeneration or at ~ω = 2|EF | for self-phase modulation(SPM) (SPM in graphene with fixed EF on a waveguidehas been measured e.g. in Ref. [12]), etc., where ω is theincident photon frequency and EF the Fermi energy. Fig.1a shows the band diagram of graphene, along with the

photon energy. SPM, which scales as Im[σ(3)s ], has been

predicted to peak at 2|EF | ≈ ~ω, after which it decreasessharply for further increasing |EF | (see for example Fig.3 in Ref. [13]). The majority of experiments have beenperformed in weakly doped graphene at high frequencies(near-IR, visible), where ~ω ≫ 2|EF |. An experimentalobservation of the Fermi energy related resonances would

not only be interesting for fundamental science but alsofor nonlinear devices since it would provide a way to con-trol the nonlinear optical response of practical systems.

In this Letter, we characterize the Fermi energy de-pendence of the third order nonlinear effects in graphene,tuning the graphene from intrinsic (2|EF | ≪ ~ω) to be-yond the interband absorption edge (2|EF | > ~ω). Wedo this by means of four-wave mixing (FWM) in an in-tegrated silicon nitride (SiN) waveguide, covered with amonolayer of graphene. Because of the coupling betweenthe evanescent tail of the highly confined waveguide modeand the graphene over a relatively long length, significantlight-matter interaction can be achieved. Studies of non-linear effects in graphene-covered silicon waveguides andresonators have been published previously [12, 14–16].However, an intrinsic disadvantage of using a silicon plat-form for the characterization of graphene nonlinearitiesis that silicon has a relatively strong nonlinear responseitself. The real part of the nonlinear parameter of a typ-ical Si waveguide is about γSi ≈ 300 m−1W−1 [17], asopposed to about γSiN ≈ 1.4 m−1W−1 [17] for a SiNwaveguide, which is negligible compared to the nonlinearparameters of the graphene-covered waveguide measuredin this work (γPL is the nonlinear phase shift acquiredover length L at power P , see Supplemental Materialor Ref. [18]). Using SiN, we can thus avoid any am-biguity about the origin of the strong nonlinear effects.Furthermore, we have achieved electrical tuning of EF

(gating) by using a polymer electrolyte [19]. We haveperformed measurements for a varying signal-pump de-tuning and for a broad range of charge carrier densitiesand demonstrate, for the first time to our knowledge, asignificant increase of the nonlinear response of graphenein the vicinity the interband absorption edge ~ω ≈ 2|EF |.Moreover, we demonstrate a good qualitative agreementwith theoretical calculations.

Degenerate four-wave mixing (FWM) is a third ordernonlinear optical process in which two pump photons atfrequency ωp are converted into two photons at different

2

(b)(a) (d)

(e)

SiN

500 nm1600 nm

330 nm80 nm

VGS

VGS

(c)

ħω

EF

ωp

ωp

ωs

ωi

SiOx

SiOx

Ti/Au

SiOx

Graphene

SiN

Polymer Electrolyte

Pump Laser

Signal Laser

EDFA Filter

50/50

GrapheneWaveguides FBG

OSA

FIG. 1. a) Graphene band diagram. b) Degenerate FWM energy diagram. c) SEM image of the cross-section of a SiNwaveguide. d) Sketch of the gating scheme. e) Setup used for the FWM experiments.

frequencies, typically denoted as the signal ωs and idlerωi. Energy conservation dictates that ωs + ωi = 2ωp,which is schematically shown in Fig. 1b. For a smallsignal-pump detuning, ωp ≈ ωs ≈ ωi, the theory pre-dicts a vanishing FWM response beyond the interbandabsorption edge, 2|EF | > ~ωp, in analogy with SPM de-scribed before. However, as opposed to SPM, the theoryas it is published in Refs. [4, 5] does not predict a sharppeak at 2|EF | ≈ ~ωp. This is because the FWM responsescales as the absolute value of the third order conductiv-ity, |σ(3)

s |, in which Re[σ(3)s ] typically dominates Im[σ

(3)s ].

To investigate this experimentally, a pump at a fixedwavelength and a signal with variable wavelength are in-jected into the graphene-covered SiN waveguide. Underthe current experimental conditions, one can prove thatthe conversion efficiency η, defined as the ratio betweenthe idler power to the signal power at the output, isquadratically dependent on the nonlinear parameter γof the waveguide (see Supplemental Material),

η ≡ Pi(L)

Ps(L)≈ |γ(ωi;ωp, ωp,−ωs)|2

(ωi

ωp

)2

Pp(0)2L2

eff,

(1)

where L is the length of the graphene-covered waveguide

section, Leff ≈ 1−e−αL

α the effective length of the nonlin-ear process and α the linear waveguide loss. The effectof the phase mismatch is neglected since Lβ2∆ω2 ≪ 1in the presented experiment (L = 100 µm, ∆ω < 1013

rad/s and β2 of a SiN waveguide is on the order of 10−25

s2/m [17]; here β2 ≡ ∂2β∂ω2 , with β(ω) the propagation

constant of the optical mode). The nonlinear parameterγ of the waveguide is, to a good approximation, propor-

tional to the nonlinear conductivity σ(3)s of graphene (see

Supplemental Material),

γ(ωi;ωp, ωp,−ωs) ≈

i3σ

(3)s, xxxx(ωi;ωp, ωp,−ωs)

16P2p

∫

G

|e(ωp)‖ × ez|4dℓ ,(2)

where e(ωp)‖ is the electric field component tangentialto the graphene sheet at the pump frequency, ez is theunit vector along the propagation direction and Pp isthe power normalization constant of the optical mode.A set of straight waveguides was patterned in a 330

nm thick LPCVD SiN layer on top of a 3 µm burriedoxide layer on a silicon handle wafer. The sample wasthen covered with LPCVD oxide and planarized usinga combination of chemical mechanical polishing, reactiveion etching and wet etching. Subsequently, a CVD-growngraphene layer was transferred to the samples by Graphe-nea [20] and patterned using photolithography and oxy-gen plasma etching so that different waveguides were cov-ered with different lengths of graphene. Metallic contacts(Ti/Au; ≈ 5 nm/300 nm) were applied at both sides ofeach waveguide, with a spacing of 12 µm. Fig. 1c showsa SEM image of the waveguide cross-section, note that ≈80 nm of oxide is left on the waveguide. Finally the struc-tures were covered with a polymer electrolyte consistingof LiClO4 and polyethylene oxide (PEO) in a weight ra-tio of 0.1:1. Fig. 1d shows a sketch of the cross-section(not to scale). The gate voltage VGS can be used to gatethe graphene layer [19]. The dependence of EF on VGS

can be approximated by the following formula [19, 21]:

VGS − VD = sgn(EF )eE2

F

~2v2FπCEDL+

EF

e, (3)

where e is the electron charge, vF ≈ 106 m/s the Fermivelocity, CEDL the electric double layer capacitance andVD the Dirac voltage. Based on measurements of the

3

optical loss and the graphene sheet resistance versus thegate voltage we estimated CEDL ≈ 1.8 · 10−2 F m−2 andVD ≈ 0.64 V, see Supplemental Material.

The setup used for the FWM experiment is shown inFig. 1e. A pump laser (Syntune S7500, λp = 1550.18nm) is amplified using an Erbium-doped fiber amplifier(EDFA), a tunable band-pass filter suppresses the Am-plified Spontaneous Emission (ASE) of the EDFA. Thesignal is provided by a Santec Tunable Laser TSL-510.Pump and signal are coupled into the waveguide througha grating coupler. At the output a fiber Bragg grating(FBG) filters out the strong pump light and the signaland idler are visualized on an Anritsu MS9740A opticalspectrum analyser (OSA).

−1 −0.5 0 0.5 10

2

4

6 (a)

VGS (V)

Loss(·10−2dB/µm)

1,546 1,550 1,554−100

−80

−60

−40

−20

0

η

λs λi(b)

λ (nm)

Pow

er(dBm)

−10

0

10

−1

0

1

−70

−60

−50 (c)

λs − λp (nm)VGS (V)

η(dB)

FIG. 2. a) Waveguide loss as a function of gating voltage.b) Examples of the optical spectra (VGS = −0.5 V). Thepump peak (1550.18 nm) is filtered out by the FBG. Thesignal peaks can be seen on the left and the correspondingidler peaks on the right. Graphene section length: 100 µm.c) Conversion efficiency η as a function of VGS and detuningλs − λp.

Fig. 2 summarizes the experimental results obtainedfor a set of 1600 nm wide waveguides. By measuring thetransmission of a set of waveguides with varying graphenelengths, as well as the transmission as a function of VGS ,the propagation loss as a function of VGS was extrapo-lated, Fig. 2a. Note that the absorption drops sharplyfor negative voltages, indicating that the Fermi level ofthe graphene gets tuned beyond the interband absorp-tion edge. In Fig. 2b, some of the measured spectra forthe FWM experiment are plotted (VGS = −0.5 V), thepump laser light is filtered out by the FBG. The con-version efficiency η can be read as the ratio between theidler (λi) and signal (λs) peaks (we correct for variations

in the transmission of the grating couplers with chang-ing wavelength). The measured conversion efficienciesare plotted in Fig. 2c, for a range of different voltagesand signal wavelengths (λp = 1550.18 nm), with an es-timated on-chip pump power of Pp(0) = 10.5 dBm anda graphene length L = 100 µm. The FWM conversionefficiency is highly dependent on both detuning λs − λp

and the applied voltage.

Using Eqs. (1) and (2), the magnitude of the nonlin-ear parameter γ(ωi;ωp, ωp,−ωs) and of the third order

conductivity σ(3)s (ωi;ωp, ωp,−ωs) can be calculated. For

the calculation of Leff the loss measurement in Fig. 2awas used. The integral and power normalization con-stant Pp in Eq. (2) are calculated using a COMSOLMultiphysicsr-model of the cross-section in Fig. 1c.Figs. 3a and 3b show the results of these conversions.

The measured values for |γ| (|σ(3)s |) have a sharp reso-

nance as a function of detuning and a broad asymmetric

resonance as a function of EF . |γ| (|σ(3)s |) is about 2800

m−1W−1 (2·10−19 Am2/V3) at small |EF | (for minimumdetuning) and about 6400 m−1W−1(4.3 ·10−19 Am2/V3)at its absolute peak.

We can compare these experimental results with aslightly modified version of the theory published in Refs.[4, 5]. In these papers analytical expressions for the third

order conductivity σ(3)s, αβγδ(ω1+ω2+ω3;ω1, ω2, ω3, EF ,Γ)

were derived at T = 0, where the relaxation rate Γ wasassumed to be energy independent. However, under theseassumptions the theory does not predict the measured in-

crease of σ(3)s with increasing |EF | (see the solid line on

Fig. S5 in the Supplemental Material). To get a bettercorrespondence between theory and experiment, we canhowever assume that Γ(E) is a function of the electronenergy. Both theoretical (e.g. Ref. [22]) and experimen-tal (e.g. Ref. [23]) studies indicate that the relaxationrate Γ(E) ∝ |E|−α is a power-law function of energy E,at |E| & E0, with α being determined by the scatter-ing mechanism. According to theory, α = 1 for impurityscattering and E0 is related to the density of impurities[22]. Experimental data confirmed the power-law depen-dence of Γ(E) but showed a slightly smaller value of α,0.5 . α . 1 (see the inset in Fig. 2 in Ref. [23]). To beable to use a formula for Γ(E) at all energies includingthe limit E → 0 we adopt the model

Γ(E) =Γ0

(1 + E2/E20)

α/2, (4)

which has a correct asymptote Γ(E) ∝ |E|−α at largeenergies |E| ≫ E0 and gives a constant relaxation rateat E → 0; the quantities Γ0, E0 and α in Eq. (4) aretreated as fitting parameters. In addition, to take intoaccount the effects of nonzero temperatures, we can usethe formula (the frequency arguments are omitted for

4

−1 −0.5 0 0.5 10

2

4

6 (a)

VGS (V)

|γ|(·103/m/W

)

-2.88 nm-1.18 nm-0.38 nm0.52 nm1.42 nm3.12 nm

-0.4 -0.3 -0.2 0

1

2

3

4

EF (eV)

|σ(3

)s

|(·10−19Am

2/V

3)

−10 −5 0 5 100

2

4

6 (b)

λs − λp(nm)

|γ|(·103/m/W

)

0.2 V-0.2 V-0.5 V-0.7 V-0.9 V-1.2 V

6 4 2 0 -2 -4 -6

1

2

3

4

ωs − ωp(·1012rad/s)

|σ(3

)s

|(·10−19Am

2/V

3)

−0.5 −0.4 −0.3 −0.2 −0.1 00

20

40

60(c)

EF (eV)

|σ(3

)s

|(·10−19Am

2/V

3) -5 nm

-3 nm0 nm3 nm5 nm

−10 −5 0 5 100

20

40

60(d)

λs − λp(nm)

|σ(3

)s

|(·10−19Am

2/V

3) 0 eV

-0.20 eV-0.33 eV-0.40 eV-0.42 eV-0.45 eV

6 4 2 0 -2 -4 -6ωs − ωp(·10

12rad/s)

FIG. 3. (a) and (b): measured nonlinear parameter |γ(ωi;ωp, ωp,−ωs)|/graphene nonlinear conductivity |σ(3)s (ωi;ωp, ωp,−ωs)|.

As a function of gate voltage/Fermi energy for different values of detuning λs − λp (a). As a function of detuning for different

values of the gate voltage (b). (c) and (d): calculated values of |σ(3)s (ωi;ωp, ωp,−ωs)|, as a function of Fermi energy for different

wavelength detunings (c), and as function of detuning for a range of Fermi energies (d).

clarity) [4]:

σ(3)s, αβγδ(EF ,Γ0, E0, α, T ) =

1

4T

∫ +∞

−∞

σ(3)s, αβγδ(E

′F ,Γ0, E0, α, T = 0)

cosh2(

EF−E′

F

2T

) dE′F .

(5)

Figs. 3c and 3d show thus obtained theoretical depen-dencies of the absolute value of the third order conduc-tivity |σ(3)

s, xxxx(ωi;ωp, ωp,−ωs)| on the Fermi energy andthe detuning λs − λp. The parameters ~Γ0 = 2.5 meV,E0 = 250 meV and α = 0.8 have been chosen so thatgood qualitative agreement was obtained with the exper-imental plots shown in Figs. 3a and 3b. One can see thatthe theory indeed describes the most important featuresof the FWM response: a narrow resonance as a functionof λs − λp and a broad strongly asymmetric shape as afunction of EF ; the inflection point at EF ≈ −0.4 eV cor-responds to ~ωp ≈ 2|EF |. Quantitatively, the theory pre-dicts about one order of magnitude larger response thanwas experimentally observed (this contrasts to a numberof previous publications, where the measured nonlinearresponse was claimed to be larger than the theoreticallycalculated one, see discussions in Refs. [3, 4, 13]). This

discrepancy should be subject to further investigation.Experimental errors could have an influence, such as anoverestimated pump power Pp(0) or effective length Leff,errors on the exact dimensions of the waveguide cross-section, etc. Inhomogeneities in the doping level of thegraphene could also have an influence, effectively creat-ing inhomogeneous broadening of the measured responsein the |EF |-direction and diminishing the height of thepeak. Finally, the difference could partly be due to im-perfections in the graphene which the theory might failto fully take into account.

In conclusion, we have performed a degenerate four-wave mixing experiment on a graphene-covered SiNwaveguide. A polymer electrolyte enabled us to gate thegraphene over a relatively large window. The experimentshows that the nonlinear conductivity of graphene has asharp resonance as a function of signal-pump detuning,also a broad asymmetric resonance shape in the vicinityof the absorption edge 2|EF | = ~ωp is observed. Qualita-tive agreement was obtained between these experimentaldata and an adapted version of previously published the-ory [4, 5], in which we introduced an energy-dependentrelaxation rate Γ(E). From an application perspective, it

5

is important to note that the measured nonlinear param-eter of the waveguide |γ| is tunable by applying a gatevoltage, and that optimizing this voltage the parametersurpasses ≈ 2000 m−1W−1 over the full measured band-width of 20 nm, with peak values over ≈ 6000 m−1W−1.This is more than 3 orders of magnitude larger than thenonlinear parameter of a standard SiN waveguide. Themost obvious trade-off is the strongly increased linear ab-sorption (more than 2 orders of magnitude, though thisabsorption is also tunable with voltage). The question ofwhether a graphene-covered integrated waveguide plat-form could serve for actual applications requires furtherresearch, such as experiments that quantify not only themagnitude, but also the phase of γ, experiments to testhow the material behaves over larger bandwidths andat higher optical powers, optimizations of the waveguidecross-section, etc. The quantitative gap between theoret-ical and experimental results is another issue that needsfurther investigation.

We thank Prof. Daniel Neumaier and Dr. MuhammadMohsin for useful discussions and for providing the poly-mer electrolyte. We also thank Dr. Owen Marshall forgiving useful advice. The work has received funding fromthe European Unions Horizon 2020 research and innova-tion programme GrapheneCore1 under Grant AgreementNo. 696656. K. A. is funded by FWO Flanders.

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[10] Lili Miao, Yaqin Jiang, Shunbin Lu, Bingxin Shi, Chu-jun Zhao, Han Zhang, and Shuangchun Wen. Broad-band ultrafast nonlinear optical response of few-layersgraphene: toward the mid-infrared regime. PhotonicsResearch, 3(5):214–219, Oct 2015.

[11] Evdokia Dremetsika, Bruno Dlubak, Simon-PierreGorza, Charles Ciret, Marie-Blandine Martin, StephanHofmann, Pierre Seneor, Daniel Dolfi, Serge Massar,Philippe Emplit, and Pascal Kockaert. Measuring thenonlinear refractive index of graphene using the opticalkerr effect method. Optics Letters, 41(14):3281–3284, Jul2016.

[12] Nathalie Vermeulen, David Castello-Lurbe, JinLuoCheng, Iwona Pasternak, Aleksandra Krajewska, Tymo-teusz Ciuk, Wlodek Strupinski, Hugo Thienpont, andJurgen Van Erps. Negative kerr nonlinearity of grapheneas seen via chirped-pulse-pumped self-phase modulation.Physical Review Applied, 6(4):044006, 2016.

[13] Nathalie Vermeulen, JinLuo Cheng, John E. Sipe, andHugo Thienpont. Opportunities for wideband wavelengthconversion in foundry-compatible silicon waveguides cov-ered with graphene. IEEE Journal of Selected Topics inQuantum Electronics, 22(2):347–359, 2016.

[14] Tingyi Gu, Nick Petrone, James F McMillan, Arendvan der Zande, Mingbin Yu, Guo-Qiang Lo, Dim-LeeKwong, James Hone, and Chee Wei Wong. Regenerativeoscillation and four-wave mixing in graphene optoelec-tronics. Nature Photonics, 6(8):554–559, 2012.

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Supplemental Material

THEORY OF FOUR-WAVE MIXING IN GRAPHENE-COVERED SIN WAVEGUIDES

In this section the expressions for the nonlinear parameters and the linear loss of a waveguide

covered with graphene are derived. In this derivation, the nonlinearities and linear losses will be

treated as perturbations to the ideal lossless, linear waveguide. In addition, an expression for the

degenerate four-wave mixing conversion efficiency η is derived.

The complex amplitude of an unperturbed waveguide mode at frequency ωj can be written as:

E0(ωj , r) =A0e(ωj, r⊥)√

Pj

eiβjz , (S1)

H0(ωj , r) =A0h(ωj , r⊥)√

Pj

eiβjz . (S2)

Where e(ωj , r⊥) and h(ωj , r⊥) are the vectorial electric and magnetic mode profiles, in what follows,

we will often omit the arguments r and r⊥ for brevity. A0 is the complex amplitude of the mode. βj

is the mode propagation constant and Pj is the power normalization constant, defined so that the

total power of the mode equals |A0|2:x

A∞

1

2ℜ|A0|2

e(ωj)√Pj

× h∗(ωj)√Pj

· ezdA ≡ |A0|2

⇒ Pj =1

4

x

A∞

e(ωj)× h∗(ωj) + e∗(ωj)× h(ωj)) · ezdA .

(S3)

A∞ is the plane perpendicular to the waveguide propagation direction. ez is the unit vector in the

propagation direction z. By definition, these modes obey the Maxwell curl equations,

∇× E0(ωj) =iωjµ0H0(ωj) , (S4)

∇×H0(ωj) =− iωjǫ0n2E0(ωj) , (S5)

n(r⊥) is the index of the unperturbed waveguide cross-section. One can include the effect of pertur-

bations, such as linear losses and nonlinearities, by introducing complex slowly varying amplitudes

Aj(z). The perturbed waveguide modes are then written as:

E(ωj, r) =Aj(z)e(ωj , r⊥)√

Pj

eiβjz , (S6)

H(ωj, r) =Aj(z)h(ωj , r⊥)√

Pj

eiβjz . (S7)

2

In practice, we will consider the total field to be a superposition of a number of monochromatic

waves:

E(r, t) =∑

j

ℜAj(z)e(ωj , r⊥)√

Pj

e−i(ωjt−βjz) , (S8)

H(r, t) =∑

j

ℜAj(z)h(ωj , r⊥)√

Pj

e−i(ωjt−βjz) . (S9)

These perturbed modes should also obey the Maxwell curl equations, where the influence of the

graphene sheet can be incorporated as a current density, J(ωj),

∇× E(ωj) =iωjµ0H(ωj) , (S10)

∇×H(ωj) =− iωjǫ0n2E(ωj) + J(ωj) . (S11)

This current density can be written as the sum of a linear and a nonlinear contribution,

J(ωj) =JL(ωj) + JNL(ωj)

=σ(1)(ωj)E(ωj) +1

4

∑

ωj=ωk+ωl+ωm

σ(3)(ωj;ωk, ωl, ωm)...E(ωk)E(ωl)E(ωm) ,

(S12)

σ(1) and σ(3) are the first and third order conductivity tensors. To derive the coupled-wave equations,

we can start from the conjugated form of the Lorentz reciprocity theorem [S1]:

x

A∞

∇ · F =∂

∂z

x

A∞

F · ezdA . (S13)

A∞ is the surface perpendicular to the propagation direction. The F-field can be constructed from

the perturbed and unperturbed waveguide mode fields as F ≡ E∗0(ωj) × H(ωj) + E(ωj) × H∗

0(ωj).

Substituting this in Eq. (S13) yields:

x

A∞

(∇×E∗0(ωj)) ·H(ωj)− E∗

0(ωj) · (∇×H(ωj))

+(∇×E(ωj)) ·H∗0(ωj)− E(ωj) · (∇×H∗

0(ωj))dA

=∂

∂z

x

A∞

A∗0Aj(z)

Pj

e(ωj)× h∗(ωj) + e∗(ωj)× h(ωj)) · ezdA .

(S14)

The left hand side of Eq. (S14) can be simplified by substituting Eqs. (S4)-(S5) and (S10)-(S11).

The right hand side can be simplified by using the normalization condition (Eq. (S3)). Eventually

this gives

∂

∂zAj = − e−iβjz

4√

Pj

x

A∞

e∗(ωj) · J(ωj)dA , (S15)

substituting Eqs. (S6) and (S7) in Eq. (S12), and subsequently in Eq. (S15), one gets a general

3

coupled-wave equation for the set of slowly varying amplitudes:

∂

∂zAj =− Aj

4Pj

x

A∞

e∗(ωj) · σ(1)(ωj)e(ωj)dA

−∑

ωj=ωk

+ωl+ωm

AkAlAmei(βk+βl+βm−βj)z

16√

PjPkPlPm

x

A∞

e∗(ωj) · σ(3)(ωj;ωk, ωl, ωm)...e(ωk)e(ωl)e(ωm)dA ,

(S16)

here the summation goes over all possible combinations of 3 frequencies that add up to ωj, including

the negative frequencies. Moreover, since the time-dependent electrical fields are real-valued, one

can make use of the equality e(−ω) = e∗(ω).

In the case of degenerate four-wave mixing, there are 3 monochromatic waves involved, the pump,

signal and idler, at equally spaced frequency intervals (∆ω = ωs − ωp = ωp − ωi). For this specific

case, the coupled wave equations (Eq. (S16)) can be simplified to:

∂Ap

∂z= iγ(ωp;ωp, ωp,−ωp)|Ap|2Ap + 2γ(ωp;ωp, ωs,−ωs)|As|2Ap

+ 2γ(ωp;ωp, ωi,−ωi)|Ai|2Ap + 2γ(ωp;ωs, ωi,−ωp)AsAiA∗pe

−i∆βz − α(ωp)

2Ap ,

(S17)

∂As

∂z= i

ωs

ωpγ(ωs;ωs, ωs,−ωs)|As|2As + 2γ(ωs;ωs, ωp,−ωp)|Ap|2As

+ 2γ(ωs;ωs, ωi,−ωi)|Ai|2As + γ(ωs;ωp, ωp,−ωi)ApApA∗i e

i∆βz − α(ωs)

2As ,

(S18)

∂Ai

∂z= i

ωi

ωpγ(ωi;ωi, ωi,−ωi)|Ai|2Ai + 2γ(ωi;ωi, ωp,−ωp)|Ap|2Ai

+ 2γ(ωi;ωi, ωs,−ωs)|As|2Ai + γ(ωi;ωp, ωp,−ωs)ApApA∗se

i∆βz − α(ωi)

2Ai ,

(S19)

where Ap(z), As(z) and Ai(z) are the complex amplitudes of respectively the pump, signal and idler,

normalized so that |Ap,s,i|2 equals the total power in the respective mode. ∆β = 2β(ωp)−β(ωs)−β(ωi)

is the phase mismatch term. α(ω) represents the linear loss and γ(ωp + ωq + ωr;ωp, ωq, ωr) is the

nonlinear parameter of the waveguide. Comparing with Eq. (S16), these parameters become:

α(ωj) =1

2Pj

x

A∞

e∗(ωj) · σ(1)(ωj)e(ωj)dA , (S20)

γ(ωj = ωp + ωq + ωr;ωp, ωq, ωr) =

i3

N(p,q,r)

∑

k,l,m

1

16√

PjPkPlPm

x

A∞

e∗(ωj) · σ(3)(ωj;ωk, ωl, ωm)...e(ωk)e(ωl)e(ωm)dA .

(S21)

In Eq. (S21), the summation parameters (k, l,m) take all different permutations of the set (p, q, r).

N(p,q,r) is the number of permutations of the set (p, q, r). In the specific case of graphene, σ(1) and

σ(3) are only present in a very thin layer. The effects can be very well described using first and third

order sheet conductivities, σ(1)s and σ

(3)s . The surface integrals then become line integrals over the

4

graphene,

α(ωj) =1

2Pj

∫

G

e∗(ωj) · σ(1)s (ωj)e(ωj)dℓ , (S22)

γ(ωj = ωp + ωq + ωr;ωp, ωq, ωr) =

i3

N(p,q,r)

∑

k,l,m

1

16√

PjPkPlPm

∫

G

e∗(ωj) · σ(3)s (ωj ;ωk, ωl, ωm)

...e(ωk)e(ωl)e(ωm)dℓ .(S23)

In the specific case of the four-wave mixing experiment described in this work, the coupled-wave

equations in Eqs. (S17)-(S19) can be strongly simplified. Firstly, the pump carries a much higher

power than the signal, moreover the idler will be orders of magnitude weaker (|Ap| > |As| ≫ |Ai|).Secondly, in Ref. [S2] it was demonstrated for a similar waveguide platform that nonlinear absorption

only starts affecting the overall power transmission significantly for power levels above 1 W. In the

experiments presented here the on-chip power levels were kept on the order of 10 mW or lower. Hence

self-phase/amplitude modulation is expected to be much weaker than linear absorption (|γ||Ap|2 ≪|α(ω)

2|). Thirdly, the phase mismatch is negligible (Lβ2∆ω2 ≪ 1, L = 100 µm, ∆ω < 1013 rad/s and

β2 ≡ ∂2β∂ω2 of a SiN waveguide is on the order of 10−25 s2/m [S17]). All these assumptions lead to

heavily simplified coupled-wave equations:

∂Ap

∂z≈− α(ωp)

2Ap , (S24)

∂As

∂z≈− α(ωs)

2As , (S25)

∂Ai

∂z≈i

ωi

ωp

γ(ωi;ωp, ωp,−ωs)ApApA∗s −

α(ωi)

2Ai . (S26)

Under these conditions the conversion efficiency η, defined as the ratio of the idler power to the

signal power, has a quadratic dependence on the nonlinear parameter γ [S4]:

η ≡ Pi(L)

Ps(L)=

|Ai(L)|2|As(L)|2

≈ |γ(ωi;ωp, ωp,−ωs)|2(ωi

ωp

)2

Pp(0)2L2

effeα(ωs)−α(ωi)L

≈ |γ(ωi;ωp, ωp,−ωs)|2(ωi

ωp

)2

Pp(0)2L2

eff ,

(S27)

where the effective interaction length is defined as:

Leff ≡ 1− e−α(ωp)+α(ωs)/2−α(ωi)/2L

α(ωp) + α(ωs)/2− α(ωi)/2≈ 1− e−αL

α. (S28)

The final expressions in Eqs. (S27) and (S28) are only valid when the approximation α(ωp) ≈ α(ωs) ≈α(ωi) ≡ α holds, i.e. when the frequencies detuning is small (∆ω ≪ ωp). For the specific experiments

described in this work, the expressions for α(ω) and γ(ωi;ωp, ωp,−ωs) can be further simplified. It is

assumed that a flat sheet of graphene lies in the xz plane (this can easily be generalized to arbitrary

graphene shapes). Firstly, the linear conductivity has only two nonzero elements, which are equal:

5

σ(1)xx = σ

(1)zz . Now we can treat the linear conductivity as a scalar parameter and calculate the linear

loss as:

α(ωj) =σ(1)s,xx(ωj)

2Pj

∫

G

|e(ωj)‖|2dℓ , (S29)

where e(ωj)‖ is the electric field component tangential to the graphene sheet. Using symmetry

considerations one can prove that the third order conductivity tensor of graphene has only the

following nonzero elements [S5]:

σ(3)s, xxxx = σ(3)

s, zzzz , (S30)

σ(3)s, xxzz = σ(3)

s, zzxx , (S31)

σ(3)s, xzxz = σ(3)

s, zxzx , (S32)

σ(3)s, xzzx = σ(3)

s, zxxz , (S33)

σ(3)s, xxxx = σ(3)

s, xxzz + σ(3)s, xzxz + σ(3)

s, xzzx . (S34)

Moreover, simulations show that the modes in the SiN waveguides used in this work are quasi-

transversal, meaning that ex ≫ ez. This implies that the term containing σ(3)s, xxxx in Eq. (S23) is

about two orders of magnitude larger than any of all other terms, the expression for the nonlinear

parameter can be simplified to:

γ(ωi;ωp, ωp,−ωs) ≈ i3σ

(3)s, xxxx(ωi;ωp, ωp,−ωs)

16Pp

√PiPs

∫

G

e∗(ωi)xe(ωp)xe(ωp)xe∗(ωs)xdℓ (S35)

≈ i3σ

(3)s, xxxx(ωi;ωp, ωp,−ωs)

16P2p

∫

G

|e(ωp)x|4dℓ . (S36)

To arrive to the second expression, we have used the assumption that e(ωp) ≈ e(ωs) ≈ e(ωi), which

is the case when one considers the same spatial modes and small detunings (∆ω ≪ ωp). We can

further generalize this expression to arbitrary graphene shapes:

γ(ωi;ωp, ωp,−ωs) ≈ i3σ

(3)s, xxxx(ωi;ωp, ωp,−ωs)

16P2p

∫

G

|e(ωp)‖ × ez|4dℓ . (S37)

In this Letter, the electric field profile e was calculated for the cross-section of the waveguide using

COMSOL Multiphysicsr. The above formula was then used to convert between the waveguide

nonlinear parameter γ and the material parameter σ(3)s .

GRAPHENE GATING USING POLYMER ELECTROLYTE

In this work, a polymer electrolyte (LiClO4 and polyethylene oxide in a weight ratio of 0.1:1) is

used to gate the graphene, i.e. to electrostatically change the carrier density or Fermi energy of

the graphene. Fig. S1a shows a sketch of the sample cross-section. Each waveguide is covered

by patterned graphene, which is contacted at both sides, making a simple resistance measurement

possible by applying a voltage VDS. The whole sample is covered with the polymer electrolyte, by

6

Graphene

Polymer Electrolyte

VGSVDS

Ti/Au

SiOxSiN

100 μm

FIG. S1. a) Sketch of the sample cross-section and gating scheme (not to scale). b) Optical microscope image of aset of waveguides. The SiN waveguides can be seen clearly, as well as the grating couplers. Every other waveguideis covered with a section of graphene. Graphene is not visible on the optical microscope image but the extent of thegraphene (under the contacts) is shown by the dashed lines. The graphene is contacted at both sides. On top of thisstructure the polymer electrolyte is spin-coated (not in this image).

applying a voltage VGS to a ‘gate’ contact (in principle any isolated contact in the vicinity of the

waveguide, in the Letter the contact on the adjacent waveguide is used) the carrier density in the

graphene can be tuned. The dependence of the Fermi energy EF on the gate voltage VGS can be

approximated by the following formula [S6, S7]:

VGS − VD = sgn(EF )eE2

F

~2v2FπCEDL+

EF

e, (S38)

with e the electron charge, vF ≈ 106 m/s the Fermi velocity and CEDL the electric double layer

capacitance. VD is the Dirac voltage, the voltage at which the graphene becomes intrinsic and at

which the conductance reaches a minimum. Note that Eq. (S38) is derived at temperature T=0 K,

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2

1,000

1,200

1,400

1,600

VGS (V)

Resistance

(Ω)

FIG. S2. The measured electrical resistance over the graphene as function of the gate voltage VGS .

7

−1.4−1.2 −1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 10

2

4

6·10−2

VGS (V)

Waveguideloss

(dB/µ

m)

measurement

fit ∝ ℜσ(1)s

FIG. S3. The measured optical loss and corresponding fit, proportional to the real part of the linear conductivity of

graphene σ(1)s .

but the difference with room temperature is negligibly small (see Fig. S4). To obtain an estimate

of VD, an electrical resistance measurement of the gated graphene is used. Fig. S2 shows the total

resistance between the source and drain electrode as a function of the VGS voltage. Based on this

measurement we estimate VD ≈ 0.64 V. To estimate the capacitance CEDL, a measurement of the

optical loss through the waveguide, as a function of VGS can be used. Fig. S3 shows the measured

loss and corresponding fit, which is proportional to the real part of the linear conductivity σ(1)s ,

which was calculated using the Kubo formula [S8–S10]. The obtained value for CEDL is 1.8 · 10−2 F

m−2 (other used parameters for this fit were ~Γ = 10 meV and T = 293 K). The resulting relation

between Fermi energy and gate voltage is shown in Fig. S4.

Figure S1b shows an optical microscope image of an actual set of contacted graphene-covered

waveguides. The SiN waveguides can be seen, as well as the grating couplers used to couple to the

optical fiber. To provide enough space for the contact needles, only every other waveguide is covered

−1.2 −1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 1.2

−0.4

−0.2

0

0.2

VGS (V)

EF(eV)

T = 0 KRoom temperature

FIG. S4. Estimated relation between gate voltage VGS and Fermi energy EF of the graphene covering the waveguideused in this work.

8

with a section of graphene. The graphene is not visible on the optical microscope image, therefore

its extent is shown by the dashed lines. The graphene is contacted at both sides. On top of this

structure the polymer electrolyte is spin-coated (not in this image).

EVALUATION OF σ(3)s,xxxx

−0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1−5 · 10−2 00

20

40

60

80

100

EF (eV)

|σ(3)

s|(·10

−19Am

2/V

3)

T = 0 K, Γ = Γ0

T = 0 K, Γ(E)

Room T , Γ(E)

FIG. S5. The absolute value of the third order conductivity |σ(3)s,xxxx(ωp;ωp, ωp,−ωp)| as a function of Fermi energy at

λs = λp = 1550.18 nm. Solid curve: T = 0, ~Γ = ~Γ0 = 2.5 meV; dashed (T = 0) and dotted (kT = 25 meV) curvesare plotted at ~Γ(E) determined by Eq. (4) of the main text with Γ0 = 2.5 meV, E0 = 250 meV and α = 0.8.

We compare our experimental results with the theory of Refs. [S11, S12]. Analytical expressions

for the third order conductivity σ(3)s,αβγδ(ωi;ωp, ωp,−ωs) have been derived in these papers at T = 0

and at the relaxation rate Γ = Γ0 independent of energy. The calculated |σ(3)s,xxxx(ωp;ωp, ωp,−ωp)|

is shown in Fig. S5 by the solid curve; one can see that the Fermi-energy dependence of the third

order conductivity at T = 0 and Γ = Γ0 has a step-like shape (compare with Fig. 9(b) from

Ref. [S12]) which does not look similar to the experimental curves in Fig. 3(a) of the main text.

However, as discussed in the main text, a more realistic model of the relaxation rate supposes an

energy dependence of Γ(E) given by Eq. (4) of the main Letter. Using this model for Γ(E) we plot

|σ(3)s,xxxx(ωp;ωp, ωp,−ωp)| by the dashed curve at T = 0 and by the dotted curve at room temperature

(kT = 25 meV); a good qualitative agreement of the dotted curve with the experimental data is now

evident. The parameters Γ0, α and E0 are chosen to quantitatively fit the characteristic features

of the curves |σ(3)(EF )| and |σ(3)(λs − λp)| (e.g. the linewidth, the maxima and minima) to the

experimental ones.

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Green, and Y. A. Vlasov. Engineering nonlinearities in nanoscale optical systems: physics and applications indispersion-engineered silicon nanophotonic wires. Advances in Optics and Photonics, 1(1):162–235, 2009.

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[S2] Koen Alexander, Yingtao Hu, Marianna Pantouvaki, Steven Brems, Inge Asselberghs, Simon-Pierre Gorza,Cedric Huyghebaert, Joris Van Campenhout, Bart Kuyken, and Dries Van Thourhout. Electrically control-lable saturable absorption in hybrid graphene-silicon waveguides. In Conference on Lasers and Electro-Optics(CLEO), pages 1–2, 2015.

[S3] David J. Moss, Roberto Morandotti, Alexander L. Gaeta, and Michal Lipson. New cmos-compatible platformsbased on silicon nitride and hydex for nonlinear optics. Nature Photonics, 7(8):597–607, 2013.

[S4] Utsav D. Dave, Bart Kuyken, Francois Leo, Simon-Pierre Gorza, Sylvain Combrie, Alfredo De Rossi, FabriceRaineri, and Gunther Roelkens. Nonlinear properties of dispersion engineered ingap photonic wire waveguidesin the telecommunication wavelength range. Optics Express, 23(4):4650–4657, 2015.

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[S6] Vrinda Thareja, Ju-Hyung Kang, Hongtao Yuan, Kaveh M. Milaninia, Harold Y. Hwang, Yi Cui, Pieter G.Kik, and Mark L. Brongersma. Electrically tunable coherent optical absorption in graphene with ion gel. NanoLetters, 15(3):1570–1576, 2015.

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[S8] L. A. Falkovsky and A. A. Varlamov. Space-time dispersion of graphene conductivity. The European PhysicalJournal B, 56(4):281–284, 2007.

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[S10] L. A. Falkovsky and S. S. Pershoguba. Optical far-infrared properties of a graphene monolayer and multilayer.Physical Review B, 76(15):153410, 2007.

[S11] J. L. Cheng, N. Vermeulen, and J. E. Sipe. Numerical study of the optical nonlinearity of doped and gappedgraphene: From weak to strong field excitation. Physical Review B, 92:235307, Dec 2015.

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