Opportunities for wideband wavelength conversion in ... · IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. , NO. , 2015 1 Opportunities for wideband wavelength conversion

Vrije Universiteit Brussel

Opportunities for wideband wavelength conversion in foundry-compatible siliconwaveguides covered with grapheneVermeulen, Nathalie; Cheng, Jinluo; Sipe, J.; Thienpont, Hugo

Published in:IEEE J. Sel. Top. Quantum Electron.

DOI:10.1109/JSTQE.2015.2490550

Publication date:2016

Document Version:Final published version

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Citation for published version (APA):Vermeulen, N., Cheng, J., Sipe, J., & Thienpont, H. (2016). Opportunities for wideband wavelength conversionin foundry-compatible silicon waveguides covered with graphene. IEEE J. Sel. Top. Quantum Electron., 22(2), 1-13. [8100113]. https://doi.org/10.1109/JSTQE.2015.2490550

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https://doi.org/10.1109/JSTQE.2015.2490550

IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. , NO. , 2015 1

Opportunities for wideband wavelength conversionin foundry-compatible silicon waveguides

covered with grapheneNathalie Vermeulen,Member, IEEE,JinLuo Cheng, J. E. Sipe, and Hugo Thienpont,Member, IEEE

Abstract—We numerically examine the opportunities for wide-band wavelength conversion through four-wave mixing (FWM)in a foundry-compatible 220 nm-thick silicon-on-insulator (SOI)waveguide covered with the highly nonlinear two-dimensionalmaterial of graphene. As a case study, we consider a foundry-compatible SOI waveguide shaped as a double spiral and coveredwith two separate graphene sheets, which are covered in turnby two solid polymer electrolyte gates. When combining sub-watt level pump powers with a short waveguide length of only afew hundreds of microns, ‘perfectly phase-matched’ conversionwith significant efficiencies close to−20 dB can be obtainedover a more than 40 THz-wide signal band adjacent to thepump frequency. Because of the tunability of the grapheneproperties using the electrolyte top gates, it is also possibleto obtain ‘quasi-phase matched’ FWM conversion through aperiodic sign reversal of the graphene third-order nonlinearityalong the waveguide. Conversion efficiencies exceeding−30 dB

can be achieved over a3.4 THz-wide signal band that is situatedas much as58 THz away from the pump frequency. Finally, thetunability of the graphene also allows for switching the converterfrom the perfectly phase-matched to the quasi-phase-matchedoperation mode.

Index Terms—Integrated optics, nonlinear wave propagation,phase matching, silicon on insulator technology, thin film devices,waveguide components

I. I NTRODUCTION

NONLINEAR optical wavelength conversion, a researcharea which emerged together with the invention of the

laser in 1960, continues to be widely investigated today witha broad range of application possibilities in spectroscopy,metrology, optical telecommunications, and a host of otherfields [1]. Over the past decade much attention has beenpaid to the development of waveguide-based nonlinear opticalwavelength converters that can be embedded in photonicintegrated circuits [2]-[17]. The process most often employedin these devices is the third-order nonlinear interaction of Kerrfour-wave mixing (FWM), which entails the conversion of asignal input frequencyνs to an idler output frequencyνi in thepresence of a pump at frequencyνp, with νp − νs = νi − νp.Various waveguide converters have already been demonstrated,

N. Vermeulen ([email protected]) and H. Thienpont are withtheBrussels Photonics Team (B-PHOT), Department of Applied Physics andPhotonics (IR-TONA), Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel,Belgium. J. E. Sipe is with the Department of Physics and the Institute forOptical Sciences, University of Toronto, 60 St. George St.,Toronto ON,M5S 1A7, Canada. J. L. Cheng is with both Vrije Universiteit Brussel andUniversity of Toronto.

Manuscript received June 7, 2015; revised September 12, 2015; acceptedOctober 7, 2015.

several of which have been implemented in the silicon-on-insulator (SOI) material platform allowing for strong light con-finement and high-precision manufacturing using the matureCMOS technology [1].

When targeting efficient FWM conversion over a wide rangeof signal wavelengths, wideband phase matching is required.This is quite challenging in SOI waveguides because ofsilicon’s strong material dispersion [4]. One approach to tacklethis problem is to engineer the cross-sectional dimensions, andhence the dispersion, of SOI waveguides to establish (almost)perfect phase matching (PPM) over a wide continuous signalfrequency band adjacent to the pump frequency. Turner and co-workers succeeded in designing an SOI waveguide exhibitingdispersion-engineered PPM over a broad pump-to-signal band-width of 52 THz in the near-infrared telecom domain [11].However, the waveguide had to be made as long as1.5 cmto attain significant conversion efficiencies around−20 dBat a pump input power of110 mW . Such waveguide lengthsresult in large device footprints of minimally10000µm2. Therequired length could be somewhat reduced to sub-cm-valueswhen suppressing the free carrier absorption in the waveguideby means of carrier-extracting p-i-n diodes. Nevertheless, theparticular SOI waveguide developed by Turneret al. [11]exhibits another drawback: Its highly customized cross-sectiondoes not comply with the fabrication constraints of multi-project-wafer-oriented silicon photonics foundries, which em-ploy a standard waveguide thickness of220 nm [18]. Sincethe use of such foundries can pave the way to the large-volumefabrication of integrated photonic components at low cost [1],adhering to these foundry standards is a natural strategy toexploit the full potential of silicon photonic devices.

Moving from the customized SOI waveguide as discussedabove [11] to foundry-compatible220 nm-thick SOI waveg-uides does not have a strong influence on the required propa-gation length for efficient conversion, but it inevitably reducesthe achievable PPM bandwidth [8]. However, it is possible toextend the wavelength range of phase-matched FWM outsidethe PPM bandwidth using the concept of quasi-phase-matching(QPM). When employing QPM for a given set of pump, signaland idler wavelengths, the phase mismatch for the consideredset of wavelengths is periodically compensated by varyingthe waveguide’s optical properties along the propagation path.Different schemes for QPM of FWM processes in siliconwaveguides were recently introduced, including QPM basedon an adiabatic variation of the dispersion along an SOIwaveguide [15] - [17], and ‘automatic’ QPM where the

This is the author’s version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available athttp://dx.doi.org/10.1109/JSTQE.2015.2490550

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anisotropy of silicon’s Kerr tensor is exploited to induce aspatially varying Kerr nonlinearity for Transverse Electric (TE)modes propagating along a uniform SOI ring or spiral [12]-[13]. These QPM schemes can be implemented while alsoallowing PPM conversion around the pump frequency [13] -[16], albeit with a small yet distinguishable penalty in PPMconversion efficiency. This penalty is induced by the spatialwaveguide variations required for QPM, yielding less effi-cient PPM conversion than possible with uniform waveguideproperties optimized for PPM all along the propagation path.More importantly, the efficiency of these QPM conversionschemes is less than optimal in that none of them achievethe ‘ideal’, most effective scenario where the third-orderFWMnonlinearity periodically reverses sign along the waveguide, inanalogy with periodically poled second-order nonlinear media[19]. As such, the existing QPM schemes for FWM processesin silicon waveguides are in fact not optimal when targetingwideband yet efficient wavelength conversion in foundry-compatible220 nm-thick SOI waveguides with preferably ashort propagation length.

In this paper, we numerically examine the opportunitiesfor wideband wavelength conversion in a foundry-compatible220 nm-thick SOI waveguide combined with the highly non-linear two-dimensional material ofgraphene. When limitingthe waveguide length to a few hundreds of microns to enablea small device footprint, we find that the deposition of agraphene top layer on the SOI waveguide leads to a signif-icant conversion efficiency improvement in the PPM regime.Furthermore, the graphene deposition enables a new schemefor establishing QPM. Indeed, because of the tunability of thegraphene properties through control of its chemical potentialµ, it is possible to periodically reverse the sign of the effectiveFWM nonlinearity experienced by the TE waveguide modesalong their propagation path, enabling for the first time QPMof FWM processes employing the ‘ideal’ and most effectivescenario. The tunability of the top layer also allows forreconfiguration between the PPM and QPM conversion modes,so that both can be realized within the same waveguide withouta penalty in PPM conversion efficiency. As a case study, wenumerically simulate the wavelength conversion performanceof a spiral-shaped graphene-covered foundry-compatible SOIwaveguide in both the PPM and QPM regimes, and compareit with the performance attainable in the bare SOI waveguide.

This paper is organized as follows: In Section II, we addressthe basic concepts of PPM and QPM wavelength conversionin SOI waveguides covered with graphene. In Section III,we discuss the design of a spiral-shaped graphene-coveredfoundry-compatible SOI waveguide allowing both PPM andQPM conversion, and evaluate its performance in the PPMregime (Section III.A) and the QPM regime (Section III.B).Finally, we conclude in Section IV.

II. BASIC CONCEPTS OFPPM AND QPM WAVELENGTH

CONVERSION IN GRAPHENE-COVEREDSOI WAVEGUIDES

The spatial evolution of the pump, signal and idler fieldsat frequenciesνp/s/i or wavelengthsλp/s/i in a graphene-covered SOI waveguide converter is governed by nonlinear

propagation equations that describe Kerr-nonlinear interac-tions, including not only the FWM conversion process butalso Kerr-induced phase modulation. We exclude Raman-nonlinear interactions since we can assume that the FWMtransitions are detuned far away from the Raman resonances ofgraphene [20]. Furthermore, we focus on (quasi-) continuous-wave operation in the strong-pump approximation [21], andassume all fields to be TE polarized. Thus, the polarization ofthe fields is within the plane of the graphene top layer; havingout-of-plane field components would not be beneficial for theFWM efficiency, since the Kerr tensor elements of grapheneare considered negligible for the polarization direction per-pendicular to the graphene sheet [22]. Finally, although agraphene-covered SOI waveguide is a heterogeneous medium,we will describe the fields in the same way as those in ahomogeneous waveguide and introduceeffectivewaveguideparameters (see Appendix) to take into account the medium’shybrid structure. As such, we can use the following setof generic equations for describing the steady-state spatialevolution of the slowly varying pump, signal, and idler fieldamplitudesAp(ζ), As(ζ), Ai(ζ) [12], [21], [23]:

∂Ap

∂ζ= iγS|Ap|2 Ap − αpAp

−(

ωr

ωp

)2(σFCA

2− i

ωp

cσFCI

)

NAp (1)

∂As

∂ζ= iγC1|Ap|2As + iγM1A

2pA

∗i e

−i∆kζ − αsAs

−(

ωr

ωs

)2(σFCA

2− i

ωs

cσFCI

)

NAs (2)

∂Ai

∂ζ= iγC2|Ap|2Ai + iγM2A

2pA

∗s e

−i∆kζ − αiAi

−(

ωr

ωi

)2(σFCA

2− i

ωi

cσFCI

)

NAi (3)

where ζ is the spatial coordinate along the waveguide, andwhere Ap/s/i(ζ) are normalized such that|Ap/s/i|2 corre-sponds to power. The FWM terms are those that contain boththe linear phase mismatch∆k = −2kp + ks + ki (with kp/s/ibeing the pump, signal and idler wave numbers), and theeffective coefficientsγM1/M2 = γM1,Si/M2,Si + γM1,g/M2,g,comprising contributions from the SOI waveguide and thegraphene sheet. In Eq. (1) the effective coefficientγS containsγS,Si accounting for self-phase modulation and two-photonabsorption at the pump wavelength in the SOI waveguide, andalso comprisesγS,g capturing the corresponding phenomenain the graphene layer. In Eqs. (2)-(3) the effective coefficientsγC1/C2 consist ofγC1,Si/C2,Si covering cross-phase modula-tion and cross two-photon absorption in the SOI waveguidefor pump/signal and pump/idler photons, respectively, andalsocompriseγC1,g/C2,g representing the equivalent effects in thegraphene. The factorsαp/s/i = αp,Si/s,Si/i,Si + αp,g/s,g/i,g

account for the linear losses in the SOI waveguide and thegraphene sheet. The formulas for all these effective parameterscan be found in Appendix. We remark that they all are function




of the spatial coordinateζ, as is required for QPM operationand will be discussed in detail in the next section. The lastterm in Eqs. (1)-(3) represents free-carrier effects withσFCA

andσFCI coefficients quantifying the efficiency of free-carrierabsorption and free-carrier index change, respectively [9], andwith ωp/s/i = 2πνp/s/i and ωr = 2πc/(1550 nm) where cindicates the speed of light. The factorN in this term is theeffective free-carrier density in the waveguide:

N =τeff

2hνpA′

(

Im(2γS,Si) + rD Im(2γS,g))

|Ap|4

+τeffrDhνpA′

×(

2αp,g|Ap|2 + 2αs,g|As|2νpνs

+ 2αi,g|Ai|2νpνi

)

(4)

with h Planck’s constant. The first term at the right-hand sideof Eq. (4) represents the free-carrier generation induced bytwo-photon absorption in the SOI waveguide and the corre-sponding absorption contribution in the graphene top layer, andthe second term indicates the free-carrier generation inducedby one-photon absorption in the graphene layer only. Thegraphene contribution to these two terms has been expressedina rather phenomenological way since not all photons absorbedin the graphene sheet give rise to the creation of free carriersand instead can contribute to, amongst others, intra-bandtransitions. The factorτeff in Eq. (4) indicates the effectivefree-carrier lifetime. Because of the short free-carrier lifetimein graphene, only the graphene-generated free carriers thatdiffuse into the silicon waveguide will effectively contributeto the free-carrier effects, as shown earlier [24]. As such,in our numerical simulations we are allowed to employ thevalues forτeff , σFCA and σFCI of the bare SOI waveguide,and incorporate the contribution from the graphene-generatedfree carriers by including a graphene-to-SOI diffusion ratio rDin Eq. (4). This ratio quantifies the fraction of the free carriersgenerated in the graphene that diffuses into the SOI waveguide.Finally, the factorA′ in Eq. (4) represents the waveguide cross-sectional area over which the free carriers are distributed.

Turning now to the FWM terms in Eqs. (1)-(3), both thelinear phase mismatch∆k = −2kp+ks+ki and the nonlinearphase mismatch contribution, function ofRe(γS) and thepump powerPp, need to be added up to obtain the full phasemismatch∆ktotal. Using a Taylor series expansion for thelinear contribution,∆ktotal can be expressed as

∆ktotal = β2∆ω2ps + (1/12)β4∆ω4

ps + 2Re(γS)Pp (5)

with βi representing the i-th order dispersion at the pumpwavelength and with∆ωps = 2π|νs−νp|. Taking into accountthat the second-order dispersion term generally dominatesoverthe fourth-order dispersion term, one can engineerβ2 to obtainPPM (i.e.∆ktotal ≃ 0) within a certain operation bandwidth inthe vicinity of the pump frequencyνp. As the evolution of theidler power |Ai|2 with position ζ depends on the phase mis-

match along the expression1 ∂ |Ai|2/∂ζ ∝ cos(−∆ktotal ζ),the idler experiences a continuous power growth in the PPMregime along the waveguide, and no phase-mismatch-inducedtransition from idler growth to idler depletion can occur.

Outside the PPM operation bandwidth, the phase mismatch∆ktotal can deviate significantly from zero due to the fourth-order dispersion term in Eq. (5), inducing a change in thefields’ phase relation so that the termcos(−∆ktotal ζ) de-termining the idler power evolution will periodically evolvealong the waveguide fromcos(0) = 1 to cos(π) = −1 andback. As a result, there will be a reversal of the conversionprocess with idler photons being annihilated, deteriorating thenet idler growth. The latter can be overcome by employingQPM, so that for a discrete set of signal and idler frequenciesvery far away from the pump frequency one can also attain anefficient idler growth. Generally speaking, QPM aims at pe-riodically compensating the phase-mismatch-induced changein the fields’ phase relation while avoiding the detrimentalreversal of the conversion process, so that an overall efficientgrowth of the idler power is achieved [12]. Ideally, in thesections where the sign ofcos(−∆ktotal ζ) has changed, oneshould also have a sign reversal in the Kerr nonlinearity, sothatboth sign changes annihilate each other and the idler continuesto grow.

As mentioned, the different effective coefficients in Eqs.(1)-(3) take into account the contributions from both the SOIwaveguide and the graphene top layer. The impact of thelatter on both the loss parameters and the nonlinear parameterscan be very strong. Experimental investigations with isolatedgraphene at photon energies above the one-photon absorptiononset (i.e.hν > 2 |µ|) have revealed extremely high valuesfor the nonlinearity of the two-dimensional material [26] -[28]. Our recent calculations indicate that, when the chemicalpotential µ is tuned such that the photon energy is justbelow the onset of one-photon absorption, the nonlinearitiesalso become very strong due to the presence of a resonancepeak [22], [29]. This is not surprising, since conventionalsemiconductors also exhibit strong nonlinearity just below thethreshold for single photon absorption [30]. This implies thatthe strong nonlinearity enhancement reported in [27] upongraphene deposition on a silicon waveguide can be combinedwith low linear absorption losses, favoring efficient PPMconversion. In addition, when moving from the one-photonto the two-photon absorption threshold a sign change occursin the graphene nonlinearities, which is also in line with thebehavior of the nonlinearities of direct-bandgap semiconduc-tors [30]. For fixed photon energies this sign change can becontrolled by changingµ. Hence, by spatially varying thechemical potentialµ of a graphene layer on top of an SOIwaveguide using, e.g., locally deposited electrolyte gates, itshould be possible to establish QPM conversion along the‘ideal’ scheme where the nonlinearity periodically reversessign along the propagation path. What is more, because ofthe voltage-tunability of graphene, one is also able to switchfrom the PPM conversion mode with a uniform, maximum

1As shown earlier [16], [25], the idler evolution is not merelydeterminedby ∆k as one might expect from Eq. (3), but by∆ktotal in which also thephase modulation effects are incorporated.




��l p

��l s

PPM inputP

��l

��l p

��l i

PPM outputP

��l s

��l

Distancealong spiral

��Im( )s(3)

PPM operation

UPPM

UPPM

UQPM1

UQPM2

��l p

��l s

QPM inputP

��l

��l p

��l s

QPM outputP

��l i

��l

Distancealong spiral

QPM operation

��Im( )s(3)

(a)

(b)

Fig. 1. Basic concept of the graphene-covered spiral-shaped SOI waveguideconverter with (a) PPM operation and (b) QPM operation. The upper andlower spiral halves are covered with two separate graphene sheets. Thesegraphene sheets are covered in turn with separate solid polymer electrolytegates, indicated as the green-colored, dark-shaded and red-colored, light-shaded areas. On top of each gate and on its underlying graphene sheetelectrical contacts are placed, across which a voltage is applied to tune thegraphene properties. In the case of PPM operation, the same voltage is appliedto both gates so that the optical fields propagating in the spiral experiencea constant graphene nonlinearity along the spiral path (seeright-hand sideof figure (a)). In the case of QPM operation, different voltages are appliedto the two gates so that the propagating fields experience a spatially varyinggraphene nonlinearity along the spiral path (see right-hand side of figure (b)).

nonlinearity all along the waveguide, to the QPM conversionmode with a periodic sign reversal of the nonlinearity alongthe propagation path.

To numerically investigate the attainable conversion effi-ciency in these different regimes, we consider as a case studya foundry-compatible220 nm-thick SOI waveguide convertershaped as a double spiral and covered with graphene. Asshown in Figs. 1(a)-(b), the upper and lower spiral halvesare covered with two separate graphene sheets with a smallinterspacing. These sheets are covered in turn with solidpolymer electrolyte gates [31], indicated as the green-colored,dark-shaded area (‘gate 1’) and red-colored, light-shadedarea(‘gate 2’). These two electrolyte gates are also separated bya small spacing. On top of each gate and on its underlyinggraphene sheet electrical contacts are placed, across which avoltage is applied to tune the graphene properties. By applyingeither the same or different bias voltages to the two gates, theoptical fields propagating in the spiral waveguide experienceeither a constant high graphene nonlinearity along the spiralpath as desired for PPM operation, or a graphene nonlinearitywith periodically changing sign along the spiral as required forQPM operation. Neither PPM nor QPM operation will be af-fected by the spacing between the two graphene sheets as longas it is kept below a few micron, which is very well feasibleusing, e.g., photolithographic graphene patterning [32].Theparticular converter design sketched in Fig. 1 offers severalpractical advantages: While the use of a spiral waveguideenables a small device footprint, the large-area gate patterningon top is less prone to fabrication errors than when depositingindividual gates on each spiral section separately. As well,the solid polymer electrolyte gates made of, e.g.,LiClO4

and Poly(ethylene) Oxide (PEO) allow tuning the chemicalpotential of graphene to very high values using only lowvoltages (e.g.,|µ| > 0.8 eV using a voltage of only3 V [31]),and can nowadays be patterned with submicron accuracy [33].Another important asset of the spiral design is that in the QPMregime the periodicity of the positive nonlinear and negativenonlinear waveguide sections is chirped. As is the case forbroadband second-harmonic generation in chirped periodicallypoled nonlinear media [19], we find here that this converterdesign allows QPM operation not just for one discrete signalwavelength far away from the pump wavelength, but for acontinuous band of signal wavelengths, hence enabling trulywideband conversion.

In what follows we will determine the design parametersof the targeted graphene-covered SOI waveguide converter.Tofind the optimal values for the graphene chemical potential|µ|,we will use graphs generated with our earlier presented theory[22], [29] for both the linear and nonlinear conductivitiesofgraphene. This way we can determine at which|µ|-valueswe have both low linear absorption and strong nonlineareffects. Our theoretical curves for the linear conductivity arein line with experimental data and with the widely used Kubo-formalism (see, e.g., Supplementary Information of [27]),so that we can directly implement these theoretical data inour numerical simulations for the converter. In contrast, thetheoretical curves for the nonlinear conductivity are system-atically lower than what has been experimentally observed




[22], [29], so we take them only as a qualitative guide. Weuse them only to estimate at which values of|µ| with lowlinear absorption we can expect nonlinearities as strong asthose observed experimentally at|µ|-values where the linearabsorption was high. We then take these values of|µ| withsmall linear absorption as our working point, and implementthe experimentally determined values of the nonlinearity inour numerical simulations.

III. D ESIGN AND MODELING RESULTS FOR THE

SPIRAL-SHAPED FOUNDRY-COMPATIBLE SOI WAVEGUIDE

CONVERTER COVERED WITH GRAPHENE

A. PPM conversion

We first investigate the optimal graphene properties for PPMconversion. The two gates in Fig. 1 should be subjected tothe same bias voltageUPPM, or, in other words, all graphenesections should be tuned to the same chemical potentialµPPM. This chemical potential should be chosen such thatthe corresponding graphene nonlinearity is high and the linearabsorption loss is low. As PPM conversion occurs betweenpump, signal and idler wavelengths that lie relatively closeto each other, the nonlinearityσ(3)(−ωs, ωp, ωp) for FWMbetween the pump, signal and idler can be approximatedby the nonlinearity for self-phase modulation at the pumpwavelength. When assuming a telecom pump wavelengthλp

of 1625 nm (this corresponds to a photon energy of0.763 eV )and a conservative carrier scattering rate of33 meV forboth inter- and intra-band scattering in graphene, our earlierpresented theory [22], [29] predicts that the graphene linearconductivityσ(1)(ωp) and its third-order conductivity for self-phase modulationIm(σ(3)(−ωp, ωp, ωp)) will vary with thechemical potential as shown in Figs. 2 and 3. We point out thatthe relation between these conductivities and their correspond-ing bulk susceptibilities is given byχ = σ/(−iωǫ0dg) withǫ0 the dielectric permittivity anddg = 0.3 nm the effectivethickness of graphene. Fig. 2 for the linear conductivity showsthat, at|µ| > 0.763 eV

2 for which the considered pump photonenergy is below the one-photon absorption threshold, the one-photon absorption loss quantified byRe(σ(1)) decreases fromthe universal conductivityσ0 to values far belowσ0. In thisregion one no longer has absorption associated with interbandtransitions in the graphene, but the intraband transitionscanstill induce losses. As such,Re(σ(1)) will not entirely vanish,as has also been verified experimentally [34], [35]. Regardinggraphene’s nonlinear conductivity, to our knowledge onlyexperimental data for|µ| ≤ 0.26 eV are available. Forexample, Gu and co-workers [27] experimentally quantifiedthe nonlinearity of a graphene sample with|µ| = 0.26 eV fornarrow-band FWM in the near-infrared telecom domain, andthey obtained a Kerr nonlinear indexn2 = 10−13m2/W anda two-photon absorption coefficientβ = 3000× 10−11m/W .Following the approach in [36] and taking into account thatat |µ| = 0.26 eV the linear conductivityσ(1) ≈ σ0 yields acomplex refractive index, we find that the corresponding non-linear susceptibility is given byχ(3)(−ωp, ωp, ωp) = (3.39 +i3.46) × 10−15m2/V 2. This yields a nonlinear conductivityof σ(3)(−ωp, ωp, ωp) = (1.84 − i1.80)σ0 × 10−16m2/V 2.

ImRe

2

1

0

1.20.80.40

~ω = 0.763 eV

σ(1);xx(ω)/σ0

|µ| (eV)

Fig. 2. Graphene’s linear conductivity at room temperature for the photonenergy of0.763 eV versus chemical potential [22], [29]. The inter- and intra-band scattering rates are taken to be33 meV .

400

300

200

100

0

1.20.80.40

~ω = 0.763 eV

Im[σ

(3);xxxx(−

ω,ω

,ω)]/σ

0(×

10−19

m2V−2)

|µ| (eV)

Fig. 3. Graphene’s nonlinear conductivity for self-phase modulation at thephoton energy of0.763 eV versus chemical potential, as calculated usingour earlier presented theory [22], [29]. In the calculations, room temperatureis assumed, and the inter- and intra-band scattering rates are taken to be33 meV . Only the qualitative trends of this curve should be considered;the quantitative nonlinearity values used in the numerical simulations for theconverter differ from the values indicated in this graph (see text).

Although experimental data are available only at relativelysmall chemical potentials, the qualitative trends of the curve inFig. 3 show that at|µ| = 0.46 eV the self-phase modulationnonlinearity is expected to be as large as at|µ| = 0.26 eV ,allowing us to adopt theσ(3)-value of Gu and co-workers[27], while the single-photon loss in Fig. 2 is much lowerthan at|µ| = 0.26 eV (Re(σ(1)) = 0.155σ0). Hence, tuningboth gates 1 and 2 to a bias voltage at which the underlyinggraphene sheet acquires|µPPM| = 0.46 eV is an appropriateworking point for the targeted PPM conversion. A detailedoverview of the graphene conductivities is provided in TableI. This table also specifies the contributions of the graphenelayer to the effective parameters for the graphene-coveredSOIconverter with PPM operation (see formulas in Appendix).

We now choose an appropriate cross-sectional SOI waveg-uide geometry with which we can fulfill the PPM condition∆ktotal ≃ 0. We do not want our choice to depend on thepump power, since we target efficient conversion at different




TABLE IGRAPHENE CONDUCTIVITIES(ADOPTED FROM THEORY AND

EXPERIMENTS AS EXPLAINED IN THE TEXT) AND GRAPHENE’ S

CONTRIBUTIONS TO THE EFFECTIVE PARAMETERS OF THE CONVERTERIN

THE PPM REGIME

PPM – gate 1 PPM – gate 2Re(σ(1)(ωp))/σ0 (-) 0.155 0.155Re(σ(1)(ωs))/σ0 (-) 0.155 0.155Re(σ(1)(ωi))/σ0 (-) 0.155 0.155σ(3)(−ωp, ωp, ωp)/σ0

(×10−16 m2/V2) 1.84 - i 1.80 1.84 - i 1.80σ(3)(−ωs, ωp, ωp)/σ0

(×10−16 m2/V2) - i 1.80 - i 1.80αp,g (×103 m−1) 1.27 1.27αs,g (×103 m−1) 1.27 1.27αi,g (×103 m−1) 1.27 1.27γS,g (×102 m−1W−1) 8.90 + i 9.09 8.90 + i 9.09γC1,g (×102 m−1W−1) 17.80 + i 18.18 17.80 + i 18.18γC2,g (×102 m−1W−1) 17.80 + i 18.18 17.80 + i 18.18γM1,g (×102 m−1W−1) 8.90 8.90γM2,g (×102 m−1W−1) 8.90 8.90

pump powers. An appropriate way to approach∆ktotal ≃ 0regardless the pump power used is by choosing a waveguidegeometry for whichβ2 vanishes atλp = 1625 nm, makingthe linear phase mismatch zero in the vicinity of the pumpwavelength. We consider a220 nm-thick strip SOI waveguideon which an oxide cladding is deposited that afterwards isetched down to the silicon top surface. As such, a planarizedgeometry is created [37] (see Fig. 4) which facilitates the trans-fer of a graphene layer on top using, e.g., the electrochemicaldelamination technique [38]. We note that, since widebandFWM is targeted, the waveguide input and output shouldbe provided with broadband edge couplers [39] rather thangrating couplers. The graphene layer, patterned [32] to createtwo separate sheets on top of the two spiral halves, is coveredby solid polymer PEO electrolyte [33] to enable gating. Forreasons that will be explained in the next section, we assumethe waveguide spiral is constructed with an average radius of23 µm, which for an Archimedean spiral outline as shown inFig. 1 yields a total propagation distance of350 µm and adevice footprint as small as2200 µm2. For a220 nm-thickSOI waveguide with an average bending radius of23 µm,numerical mode simulations with the graphene cover layerand PEO cladding included2 show that a waveguide width of670 nm yields a close-to-zeroβ2 or dispersion parameterDfor the TE mode atλp = 1625 nm, as shown in Fig. 4.

The chosen cross-sectional waveguide geometry yields across-sectional areaA′ = 1.5 × 10−13 m2, an effective free-carrier lifetimeτeff around0.5 ns [11], and a linear SOI prop-agation lossα(p/s/i),Si = 34.5 m−1 (the latter corresponds toa power loss of3 dB/cm). For the free-carrier absorption ef-ficiency and index change efficiency of the SOI waveguide weuseσFCA = 1.45× 10−21 m2 andσFCI = −5.3× 10−27 m3

2For the mode simulations we used the commercial mode solver LumericalMODE. For silicon we adopted the refractive index data outlined in [40]for wavelengths above1.2µm. For SiO2 we implemented the Sellmeiercoefficients of [41]. The refractive index of the PEO electrolyte covering thegraphene was assumed to be 1.45 [42]. The graphene layer was implementedas a thin, three-dimensional layer with a refractive index determined by itslinear conductivity.

x (mm)

-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 -0.2 -0.4 -0.6 -0.8 -1.0

0.9

0.5

0.1

-0.3

-0.7

y (mm)SiO2 SiO2

PEOgraphene

Si

Fig. 4. (Top) TE mode profile2 and (bottom) dispersion parameterD around1625 nm for a 670 nm-wide, 220 nm-thick strip SOI waveguide with abend radius of23 µm and with oxide at the sides and with a graphene sheetfeaturing|µ| = 0.46 eV covered by a solid polymer PEO electrolyte on topof the waveguide.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−70

−60

−50

−40

−30

−20

−10

Pump input power (W)

Con

vers

ion

effic

ienc

y (d

B)

graphene−covered SOIgraphene−covered SOIwith zero linear absorptionSOISOI when combiningPPM with QPM

Fig. 5. Signal-to-idler conversion efficiency in the PPM regime versuspump input power for: the graphene-covered converter (solidblack line); thegraphene-covered converter in the absence of linear absorption (solid greyline); the bare SOI waveguide converter (dashed black line); the bare SOIwaveguide converter in case PPM and QPM operation are combined (dottedgrey line). For all simulations the signal input powerPs,in is taken to be250 µW .

[9]. Regarding the graphene-to-SOI diffusion rate we canderive from earlier reported experiments [24] that in the caseof weakly doped graphene (i.e., with|µ| ≃ 0) deposited onan SOI waveguiderD is very small and below0.01. Since wehere work with strongly doped graphene, the Shottky barrierbetween the graphene sheet and the intrinsic silicon underneath[43] will exceed that of [24] so thatrD will most likely be evensmaller than0.01. Nevertheless, in our simulations we assumerD = 0.01 to avoid underestimating the graphene-inducedfree-carrier absorption and refractive index change. Finally, forthe nonlinear parameters of the silicon waveguide we take anonlinear refractive indexn2 = 6× 10−18 m2/W and a two-photon absorption coefficientβ = 0.5 × 10−11 m/W [44],yielding χ(3) = 2.56× 10−19 + i2.76× 10−20m2/V 2.

To determine the attainable conversion efficiencies in thePPM regime for the considered graphene-covered SOI waveg-




43 THz

��l p

Fig. 6. Signal-to-idler conversion efficiency of the graphene-covered SOIwaveguide converter in the PPM regime versus signal wavelength for a pumpinput power of500 mW and a signal input power of250 µW . The−3 dBPPM bandwidth is found to be43 THz.

uide converter and for its bare counterpart, we numericallysolve Eqs. (1)-(4) forλs,i ≈ λp, and for pump input powersPp,in ranging between10 mW and 900 mW , 3 to calculatethe generated idler output powerPi,out. For all simulationsthe signal input powerPs,in is taken to be250 µW , and theconversion efficiency is determined by calculatingPi,out/Ps,in

(see Fig. 5). To quantify the PPM bandwidth for the graphene-covered converter, we also solve Eqs. (1)-(4) for varyingλs

and λi in the vicinity of λp, and with the dispersion ofthe graphene sheet taken into account. The correspondingconversion efficiencies are displayed in Fig. 6. The softwareused for these calculations is the numerical solver packageofMATLAB for ordinary differential equations.

When comparing the solid and dashed black curves in Fig.5, we find that the graphene-covered SOI waveguide exhibitsa significantly higher conversion efficiency than the bare SOIwaveguide. Especially at low pump powers of a few tens ofmilliwatts, strong performance improvements up to8 dB canbe achieved. In case PPM and QPM operation4 are combinedin the same device, the dashed black curve for the bare SOIwaveguide converter will slightly shift down to the level ofthe dotted grey curve because of the performance penaltydiscussed in the Introduction, whereas the solid black curvefor the graphene-covered converter will remain unchanged.The conversion efficiency of the graphene-covered converterapproaches−20 dB for sub-watt level pump powers, showingthat considerable efficiencies are feasible even for propaga-tion distances as short as350 µm and device footprints assmall as 2200 µm2. At pump powers approaching1 W ,the efficiency curve starts to flatten as a result of nonlinearabsorption. We point out that, for the dispersion-engineered

3Although these pump powers are substantial, no absorption-induced satura-tion effects are expected in the graphene cover layer since the power residingin the graphene is less than0.1% of the total power and will give rise tolimited absorption below the single-photon absorption threshold. For muchhigher pump powers (i.e. when using pulsed excitation), saturation couldoccur. In that case the linear graphene absorption decreases, but also thenonlinearity is expected to go down [45], and a separate study will be neededto assess the resulting conversion efficiency.

4For the bare SOI converter we here consider the combination ofPPMoperation with ‘automatic’ QPM operation based on a spatially varying Kerrnonlinearity along the spiral, but a similar result is obtained in case the PPMoperation is combined with QPM based on a spatial variation ofthe dispersionalong the spiral.

SOI waveguide converter considered earlier [11], a−20 dBconversion efficiency was achieved at a pump input power ofonly 110 mW , but the propagation length was more than 40times longer and the footprint more than 4 times larger thanfor the graphene-covered foundry-compatible SOI waveguideconverter considered here. The solid grey curve in Fig. 5shows that, when totally neglecting the linear absorption lossof the graphene on top as is done in some studies (see, e.g.,[46]), considerably higher performances exceeding−20 dBare predicted. However, even at our working point below thethreshold for single-photon absorption induced by interbandtransitions, the graphene will still cause non-negligiblelossassociated with intraband transitions, as discussed earlier onand verified experimentally. The solid black curve in Fig. 5properly accounts for the influence of this loss.

As shown in Fig. 6, our simulation results for the graphene-covered converter indicate a large PPM bandwidth. The−3dB-bandwidth is found to be43 THz, which is not aslarge as the52 THz bandwidth obtained for the extensivelydispersion-engineered SOI waveguide considered earlier [11]but still represents a very significant value for a foundry-compatible waveguide with limited dispersion engineeringpossibilities. The difference between the two values is lessthan 20 %. Hence, although graphene exhibits, like silicon,a strong material dispersion as shown by the imaginary partof the linear conductivity in Fig. 2, the graphene-covered SOIwaveguide converter allows for broadband PPM operation.

B. QPM conversion

For signal and idler wavelengths situated outside the PPMbandwidth, QPM conversion can be employed instead. Asa case study, we consider QPM conversion betweenλs =2370 nm (i.e. a signal photon energy of0.523 eV ) andλi = 1236.4 nm (i.e. an idler photon energy of1.003 eV ),while pumping atλp = 1625 nm. For the converter of Fig. 1 tooperate in the QPM regime, gates 1 and 2 should induce differ-ent chemical potentials in the underlying graphene sections sothat these acquire FWM nonlinearities of opposite sign. Likeinthe previous section, the linear absorption lossRe(σ(1)) shouldbe kept low. As shown in Fig. 7, the chemical potentials wherethis requirement is met at all three involved photon energiesranges from approximately0.6 eV to 1 eV . To establishefficient QPM conversion, we thus need to determine twochemical potentials within this range that yield strong FWMnonlinearities with opposite sign. Our earlier presented theory[22], [29] predicts that graphene’s third-order conductivityIm(σ(3)(−ωs, ωp, ωp)) for FWM at the considered photonenergies will vary with the chemical potential as shown inFig. 8. The qualitative trends of this curve indicate0.6 eV and0.77 eV as two interesting chemical potential values for QPMconversion. To our knowledge no experimental data are avail-able for wideband FWM in graphene at such high chemicalpotentials, but Hendry and co-workers carried out widebandFWM experiments in graphene with|µ| ≈ 0 eV [26]. Thenonlinearities reported by Hendryet al. for their widebandFWM experiment and by Guet al. [27] for their narrow-bandFWM experiment are very similar. Therefore, like we did in




TABLE IIGRAPHENE CONDUCTIVITIES(ADOPTED FROM THEORY AND

EXPERIMENTS AS EXPLAINED IN THE TEXT) AND GRAPHENE’ S

CONTRIBUTIONS TO THE EFFECTIVE PARAMETERS OF THE CONVERTERIN

THE QPM REGIME

QPM – gate 1 QPM – gate 2Re(σ(1)(ωp))/σ0 (-) 0.074 0.074Re(σ(1)(ωs))/σ0 (-) 0.133 0.114Re(σ(1)(ωi))/σ0 (-) 0.056 0.111σ(3)(−ωp, ωp, ωp)/σ0

(×10−16 m2/V2) 1.84 - i 1.80 1.84 - i 1.80σ(3)(−ωs, ωp, ωp)/σ0

(×10−16 m2/V2) - i 1.80 i 1.80αp,g (×103 m−1) 0.61 0.61αs,g (×103 m−1) 1.19 1.02αi,g (×103 m−1) 0.35 0.70γS,g (×102 m−1W−1) 8.80 + i 9.00 8.80 + i 9.00γC1,g (×102 m−1W−1) 15.05 + i 15.38 15.05 + i 15.38γC2,g (×102 m−1W−1) 14.90 + i 15.22 14.90 + i 15.22γM1,g (×102 m−1W−1) 3.89 -3.89γM2,g (×102 m−1W−1) 7.46 -7.46

Im0.523 eV: Re

Im0.763 eV: Re

Im1.003 eV: Re

~ω

3

2

1

0

0.2

0.15

0.1

0.05

01.20.80.40

Im[σ

(1);xx(ω)]/σ

0

Re[σ(1);xx(ω)]/σ

0

|µ| (eV)

Fig. 7. Graphene’s linear conductivity at room temperature for the photonenergies of0.523 eV , 0.763 eV and 1.003 eV versus chemical potential[22], [29]. The inter- and intra-band scattering rates are taken to be33 meV .The values for the real and imaginary parts are shown on the left and rightaxes, respectively, and for the real part only that part of the curve is shownthat is below0.2σ0.

the previous section for narrow-band FWM, we can assumethat also for broadband FWM

∣

∣Im(σ(3)(−ωs, ωp, ωp))∣

∣ =1.80σ0×10−16m2/V 2 for a small chemical potential. Takinginto account that the trends in Fig. 8 indicate that the nonlin-earities at|µ| = 0.6 eV and 0.77 eV are at least equally aslarge in absolute values as those at a small chemical potential,we adoptIm(σ(3)(−ωs, ωp, ωp)) = 1.80σ0 × 10−16m2/V 2

and−1.80σ0 × 10−16m2/V 2 at |µ| = 0.6 eV and0.77 eV ,respectively. Tuning the bias voltages of gates 1 and 2 such thatthe underlying graphene sections acquire

∣

∣µQPM1

∣

∣ = 0.77 eV

and∣

∣µQPM2

∣

∣ = 0.6 eV , respectively, thus is an appropriateworking point for the targeted QPM conversion. A detailedoverview of the corresponding graphene conductivities andof graphene’s contributions to the effective parameters oftheconverter with QPM operation can be found in Table II.

Turning now to the properties of the SOI spiral waveguide,the earlier mentioned choice of an average bending radiusvalue of 23µm relies on the requirements for QPM. In theQPM regime the sections where the sign ofcos(−∆ktotal ζ)

40

20

0

−20

1.20.80.40

~ωs = 0.523 eV

~ωp = 0.763 eV

Im[σ

(3);xxxx(−

ωs,ω

p,ω

p)]/σ

0(×

10−19

m2V−2)

|µ| (eV)

Fig. 8. Graphene’s nonlinear conductivity for four-wave mixing with a pumpphoton energy of0.763 eV and a signal photon energy of0.523 eV versuschemical potential, as calculated using our earlier presented theory [22], [29].In the calculations, room temperature is assumed, and the inter- and intra-band scattering rates are taken to be33 meV . Only the qualitative trends ofthis curve should be considered; the quantitative nonlinearity values used inthe numerical simulations for the converter differ from the values indicatedin this graph (see text).

has changed should correspond to the sections where theFWM nonlinearity has a reversed sign as well. For a converterconfiguration as in Fig. 1 the sign reversal of the nonlinearityoccurs every time the spiral angle varies overπ. Following thesame reasoning as in our earlier work [12], [14] and takinginto account that the nonlinear phase mismatch is small ascompared to the linear part, the condition for having FWM inthe QPM regime is given by

Ravg =1

|∆k| (6)

with Ravg representing the average radius of the spiral-shapedwaveguide. For the pump, signal and idler wavelengths underconsideration, the linear phase mismatch∆k in the graphene-covered waveguide5 equals4.4× 104 m−1, for which Eq. (6)yieldsRavg = 23µm.

Using the graphene parameter values specified above, weobtain for the effective parameters defined in Appendix thevalues outlined in Table II. For the remaining simulationparameters we employ the same settings as previously.

To calculate the attainable conversion efficiencies in theQPM regime for the considered graphene-covered SOI waveg-uide converter and for its bare counterpart, we solve Eqs.(1)-(4) for signal and idler wavelengths of2370 nm and1236.4 nm, and for pump input powersPp,in ranging between10 mW and900 mW , and we plot the corresponding conver-sion efficiencies in Fig. 9. To quantify the QPM bandwidth ofthe graphene-covered converter, we also solve Eqs. (1)-(4)forvaryingλs andλi in the vicinity of 2370 nm and1236.4 nm,respectively. The resulting conversion efficiencies are shownin Fig. 10. To illustrate what happens inside the spiral, Fig.

5This linear phase mismatch has been calculated using numericalmodesimulations with a graphene chemical potential of0.69 eV , which is theaverage of the chemical potentials of the graphene sheets under gates 1 and2.




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−70

−60

−50

−40

−30

−20

Pump input power (W)

Con

vers

ion

effic

ienc

y (d

B)

graphene−covered SOIgraphene−covered SOIwith zero linear absorptionSOI (best−case scenario)

Fig. 9. Signal-to-idler conversion efficiency in the QPM regime versuspump input power for: the graphene-covered converter (solidblack line); thegraphene-covered converter in the absence of linear absorption (solid greyline); the bare SOI waveguide converter along the best-casescenario of havinga 12 dB enhancement of the conversion efficiency obtained without QPM(dashed black line). For all simulations the signal input power Ps,in is takento be250 µW .

��l p

58 THz

3.4 THz

Fig. 10. Signal-to-idler conversion efficiency of the graphene-covered SOIwaveguide converter in the QPM regime for a pump input power of500 mWand a signal input power of250 µW as a function of signal wavelengthcentered around2370 nm. The latter corresponds to a pump-signal frequencyspacing of58 THz. The−3 dB QPM bandwidth is found to be3.4 THz.

11 shows how the idler power evolves within the graphene-covered spiral for a pump input powerPp,in of 500 mW .For all simulation results the signal input powerPs,in is againtaken to be250 µW . In principle, the efficiency curve in Fig.9 for the bare SOI waveguide converter in the QPM regimedepends on which of the existing QPM schemes for SOIwaveguides is being used. However, all schemes seem to havein common that an efficiency enhancement up to12 dB can beproduced as compared to a waveguide without QPM (see, e.g.,[13], [15] where QPM is obtained through a Kerr-nonlinearityvariation and a dispersion variation, respectively). This12 dBenhancement in fact represents the best-case scenario and canonly be obtained at specific pump powers [13]. Nevertheless,to ensure that the performance benchmark we set for evalu-ating the graphene-covered converter is sufficiently high,wedetermine the efficiency curve for the bare SOI waveguideconverter in Fig. 9 by applying for all pump powers a12 dBenhancement of the conversion efficiency obtained withoutQPM.

Although we have deliberately plotted the best-case-scenario conversion efficiency for the bare SOI converter inFig. 9, we find that the graphene-covered SOI waveguidewith the new QPM scheme introduced here yields significantlyhigher conversion efficiencies. At low pump powers, perfor-

0 0.5 1 1.5 2 2.5 3 3.5x 10

−4

0

0.5

1

1.5

2

2.5

3x 10−7

Position within the spiral (m)

Idle

r po

wer

(W

)

Fig. 11. Spatial evolution of the idler power within the spiral converter in theQPM regime for a pump input power of500 mW and a signal input powerof 250 µW .

mance improvements up to8 dB can be achieved. The attain-able conversion efficiencies of the graphene-covered converterin the QPM regime are lower than in the PPM regime, butstill can exceed−30 dB for sub-watt level pump powers anda propagation distance of only350 µm. The lower conversionefficiencies in the QPM regime as compared to the PPMregime are a direct consequence of the basic concept of QPM,where the non-zero phase mismatch, although periodicallycompensated, inevitably results in varying growth rates forthe idler along the spiral (see Fig. 11). The main strengthof the QPM concept as compared to PPM operation is thatit can be applied to signal wavelengths far outside the PPMbandwidth. In this particular case, the pump-signal frequencydifference equals58 THz which significantly goes beyond the43 THz bandwidth in the PPM regime (see Section III.A),and even beyond the52 THz record PPM bandwidth of theextensively dispersion-engineered SOI waveguide consideredearlier [11]. We remark that, when totally neglecting the lineargraphene absorption in the graphene-covered converter, QPMperformances close to−20 dB are predicted as shown by thesolid grey curve in Fig. 9.

Fig. 10 illustrates that thanks to the chirped nature of theQPM periodicity in the spiral-shaped graphene-covered SOIwaveguide, efficient QPM operation is not just obtained atone discrete signal wavelength, but over a continuous bandaround the design signal wavelengthλs = 2370 nm. Its−3dB-bandwidth equals3.4 THz. As such, we find thatthe considered graphene-covered foundry-compatible spiral-shaped SOI waveguide converter not only allows for broad-band PPM conversion but also for QPM conversion overconsiderable bandwidths centered at signal wavelengths veryfar away from the pump wavelength.

IV. CONCLUSION AND OUTLOOK

The use of graphene as a waveguide cover layer opensup different routes towards wideband FWM conversion infoundry-compatible SOI structures. For the spiral-shapedgraphene-covered foundry-compatible SOI waveguide studiedhere with a length of a few hundreds of microns, we foundPPM conversion efficiencies that approach−20 dB and thatare up to8 dB higher than those of the bare SOI waveguide.At the same time, the PPM signal bandwidth is less than20% smaller than the record PPM bandwidth achievable in




extensively dispersion-engineered SOI waveguides that arenot foundry-compatible. The latter can also yield−20 dBconversion efficiencies with pump powers that are up to 10times lower than those considered here, but in that casewaveguide lengths more than 40 times longer and footprintsmore than 4 times larger than those used here are required. Wealso found that the new graphene-enabled QPM scheme wherethe ‘ideal’ scenario of periodically reversing the nonlinearitysign is implemented yields conversion efficiencies that areup to 8 dB higher than those attainable with the existingQPM schemes for bare SOI waveguides in the best-casescenario. QPM conversion efficiencies exceeding−30 dBhave been shown for a3.4 THz-wide band centered at a signalwavelength that is even outside the record PPM bandwidthof SOI waveguides incompatible with foundry fabricationconstraints. The possibility of switching between the PPMand QPM conversion modes is an additional asset that furtherleverages the practical usability of graphene-covered foundry-compatible SOI waveguide converters.

As pointed out in the paper, the broadband character ofgraphene’s nonlinear optical behavior has already been exper-imentally investigated above the one-photon absorption onset(i.e. hν > 2 |µ| or |µ| < hν/2) through, amongst others,wideband FWM experiments in graphene with|µ| ≈ 0 eV[26]. Figs. 3 and 8 plotted for given photon energies show that,as long as|µ| ≪ hν/2, graphene’s nonlinear characteristicsare practically constant as a function of chemical potential.Equivalently, when fixing the chemical potential, graphene’snonlinearities only weakly depend on the photon energy aslong ashν ≫ 2 |µ|. As a result, graphene samples with|µ| ≈0 eV indeed exhibit very broadband nonlinear optical prop-erties from the microwave domain to the visible region [26],[47]. In this paper, however, we have considered graphene’soptical nonlinearities at a working pointbelowthe one-photonabsorption threshold. We have found that also below theone-photon absorption threshold strong nonlinearities can beobtained over a broad range of photon energies, providedthat the graphene chemical potential is tuned to a properlychosen value. Taking into account that with electrolyte topgates the chemical potential can be varied (in absolute value)from 0 eV to beyond1 eV , the operation bandwidth of thegraphene-covered waveguide converter presented here couldhypothetically also range from the microwave to the visibledomain. In practice, however, it will only cover the spectraldomains where the waveguide material is transparent, whichin the case of a silicon waveguide comprises the near- andmid-infrared.

Besides employing graphene, it is also interesting to con-sider using more recently introduced two-dimensional materi-als such as MoS2 [48], [49] and black phosphorus [50], [51]for wavelength conversion purposes. Graphene turns out tohave one of the highest Kerr nonlinear indices over the widestbandwidth as compared to other two-dimensional materials(see, e.g., [49] and [51]) because of the resonant enhancementover a broad range of photon energies in graphene’s linear,gapless band structure. Another special feature of grapheneis its electrically or chemically tunable chemical potentialwhich allows controlling both its nonlinear optical properties

and its linear absorption losses. These characteristics couldalso be controlled to some extent in the above mentionedtwo-dimensional materials with a bandgap, since it is pos-sible to modify their bandgap size [50]. Whereas the gappedtwo-dimensional materials are likely to be outperformed bygraphene when targeting wideband wavelength conversionbased on third-order nonlinear processes, they do offer theadvantage that they naturally exhibit a second-order opticalnonlinearity [52] whereas graphene does not. From a practicalpoint of view, materials like MoS2 and black phosphorus arestill quite challenging to prepare and process [50], but thiswill most likely improve over time as has been the case withgraphene. Therefore, for future nonlinear optical devicesonecould think of using multilayer hetero-structures involvingboth graphene and the emerging gapped two-dimensionalmaterials, as this could allow combining the best of bothworlds.

APPENDIX A

We assume a graphene-covered silicon waveguide runningin the ζ direction. The electric field associated with awaveguide mode is written as proportional to

eµ(x, y)eikµζ ,

whereµ = p, s, i indicates the mode profile appropriate forthe pump, signal or idler; we take the normal to the top of thewaveguide to bey. When determining these modal fields, weinclude in the Maxwell equations the linear optical propertiesof silicon (assumed lossless at the frequencies of interest),and treat the graphene as a thin layer with thicknessdg ontop of the waveguide. For the modal field simulations, it isallowed to takedg in a range around0.3 nm for whichthe mode simulation results converge. We include only theimaginary part of the linear graphene conductivityσ(1)(ω) indetermining the waveguide mode, converting it to an effectivedielectric tensor affecting the optical response of the thinlayer only in thex and ζ directions. We neglect any opticalresponse of the graphene in they direction. The real part ofthe linear graphene conductivity, which describes linear loss,and the nonlinear effects are included by allowing the modeamplitudes to be slowly varying, in the usual “coupled modeequation” approach. Our approach is similar in spirit to that ofLin et al. [9], but we use a strategy [53] that allows for bothmodal and material dispersion. The group velocities, includingmodal and material dispersion, of the modes are defined as

vµ =dωµ(k)

dk,

and thus include both contributions to the full dispersionrelation ωµ(k) of mode µ. This slightly modifies the nor-malization condition of the modes [54]; the strategy hasbeen used earlier for the description of quantum nonlinearoptical effects [55], and the inclusion of nonlinear effectsmirrors the treatment here. We use the functionsSi(x, y)(sg(x, y)) to indicate the location of the silicon (graphene),taking sSi(x, y) = 1 (sg(x, y) = 1) where the silicon (thethin layer modeling the graphene) is present andsSi(x, y) = 0(sg(x, y) = 0) where the silicon (the thin layer modeling




the graphene) is not present. We find the following formulasfor the effective parameters of the graphene-covered siliconwaveguide:




αp = αp,Si + αp,g

= αp,Si + (2vpdgIp)−1

Re(σ(1)(ωp))

∫

e∗p(x

′, y′) · e‖p(x′, y′)sg(x′, y′) dx′ dy′,

αs = αs,Si + αs,g

= αs,Si + (2vsdgIs)−1

Re(σ(1)(ωs))

∫

e∗s(x

′, y′) · e‖s(x′, y′)sg(x′, y′) dx′ dy′,

αi = αi,Si + αi,g

= αi,Si + (2vidgIi)−1

Re(σ(1)(ωi))

∫

e∗i (x

′, y′) · e‖i (x′, y′)sg(x′, y′) dx′ dy′,

γS = γS,Si + γS,g

=(

4v2pI2p

)−13ǫ0ωpχ

(3),ijklS

∫

(

eip(x′, y′)

)∗ejp(x

′, y′)(

ekp(x′, y′)

)∗elp(x

′, y′)sSi(x′, y′) dx′dy′

+(

4v2pdgI2p

)−13iσ

(3),ijklS

∫

(

eip(x′, y′)

)∗ejp(x

′, y′)(

ekp(x′, y′)

)∗elp(x

′, y′)sg(x′, y′) dx′ dy′,

γC1 = γC1,Si + γC1,g

= (2vpvsIpIs)−1

3ǫ0ωsχ(3),ijklC1

∫

(

eis(x′, y′)

)∗ejp(x

′, y′)(

ekp(x′, y′)

)∗els(x


+(2vpvsdgIpIs)−1

3iσ(3),ijklC1

∫

(

eis(x′, y′)

)∗ejp(x

′, y′)(

ekp(x′, y′)

)∗els(x

′, y′)sg(x′, y′) dx′ dy′,

γC2 = γC2,Si + γC2,g

= (2vpviIpIi)−1

3ǫ0ωiχ(3),ijklC2

∫

(

eii(x′, y′)

)∗ejp(x

′, y′)(

ekp(x′, y′)

)∗eli(x


+(2vpvidgIpIi)−1

3iσ(3),ijklC2

∫

(

eii(x′, y′)

)∗ejp(x

′, y′)(

ekp(x′, y′)

)∗eli(x

′, y′)sg(x′, y′) dx′ dy′,

γM1 = γM1,Si + γM1,g

=(

4vpIp√

vsviIsIi

)−1

3ǫ0ωsχ(3),ijklM1

∫

(

eis(x′, y′)

)∗(

eji (x′, y′)

)∗

ekp(x′, y′)elp(x


+(

4vpIp√

vsviIsIidg

)−1

3iσ(3),ijklM1

∫

(

eis(x′, y′)

)∗(

eji (x′, y′)

)∗


′, y′)sg(x′, y′) dx′ dy′

γM2 = γM2,Si + γM2,g

=(

4vpIp√

vsviIsIi

)−1

3ǫ0ωiχ(3),ijklM2

∫

(

eii(x′, y′)

)∗ (ejs(x

′, y′))∗



+(

4vpIp√

vsviIsIidg

)−1

3iσ(3),ijklM2

∫

(

eii(x′, y′)

)∗ (ejs(x

′, y′))∗

ekp(x′y′)elp(x

′, y′)sg(x′, y′) dx′ dy′

(7)




Here, the fieldse‖µ(x, y) indicate only thex andζ componentsof eµ(x, y). Furthermore,

Iµ ≡∫

dxdy ǫ0×

(vphase,x(x, y;ωµ)

vgroup,x(x, y;ωµ)n2x(x, y;ωµ)|exµ(x, y)|2

+vphase,y(x, y;ωµ)

vgroup,y(x, y;ωµ)n2y(x, y;ωµ)|eyµ(x, y)|2

+vphase,ζ(x, y;ωµ)

vgroup,ζ(x, y;ωµ)n2ζ(x, y;ωµ)|eζµ(x, y)|2)

where

vphase,(x,y,ζ)(x, y;ω) =c

nx,y,ζ(x, y;ω), (8)

and

vgroup,(x,y,ζ)(x, y;ω) =vphase,(x,y,ζ)(x, y;ω)

(

1 + ωnx,y,ζ(x,y;ω)

∂nx,y,ζ(x,y;ω)∂ω

)

(9)are respectively the local material phase and group velocities.For (x, y) at a point in the silicon we havenx(x, y;ω) =ny(x, y;ω) = nζ(x, y;ω) =

√ǫSi, whereǫSi is the (assumed

real) relative dielectric constant of silicon. For(x, y) at a pointin the thin layer modeling the graphene we havenx(x, y;ω) =

nζ(x, y;ω) =√

1− Im(

σ(1)(ω))

/ (ǫ0ωdg), where we onlytake the lossless part of the graphene response into accountindetermining the waveguide modes;ny(x, y;ω) = 1 since weneglect any optical response of graphene perpendicular to theplane in which it lies. Finally,nx(x, y;ω) = ny(x, y;ω) =nζ(x, y;ω) =

√ǫSiO2

(=√ǫPEO) in the SiO2 (PEO) sur-

rounding the waveguide, andnx(x, y;ω) = ny(x, y;ω) =nζ(x, y;ω) = 1 in air.

We were able to verify the expressions above for theeffective linear loss parametersαp/s/i by comparing the valuesof αp/s/i with the linear loss values that the commercialLumerical MODE software readily calculates for graphene-covered waveguides. An excellent correspondence was ob-tained with relative differences of only a few percent.

Turning now to the expressions for the nonlinear parametersγS/C1/C2/M1/M2, we can assume in first order that all nonlin-ear susceptibilitiesχ(3) of silicon used in these formulas areequal. The nonlinear conductivities of graphene are given by

σ(3),ijklS = σ(3),ijkl(−ωp;ωp,−ωp, ωp),

σ(3),ijklC1 = σ(3),ijkl(−ωs;ωp,−ωp, ωs),

σ(3),ijklC2 = σ(3),ijkl(−ωi;ωp,−ωp, ωi),

σ(3),ijklM1 = σ(3),ijkl(−ωs;−ωi, ωp, ωp),

σ(3),ijklM2 = σ(3),ijkl(−ωi;−ωs, ωp, ωp).

For the numerical simulations in the paper we approximateσ(3)C1/C2 by σ

(3)S and σ

(3)M1 by ωs

ωiσ(3)M2 . This approximation,

which is generally used in the literature for silicon, allowsfor relating all nonlinear conductivities of graphene in Eq.(7) to experimentally determined values. We point out that,although to our knowledge the PEO electrolyte we consideris not known to exhibit particularly strong nonlinear opticaleffects, there exist other polymer electrolytes with relativelyhigh nonlinear optical coefficients of the order ofχ(3) =10−10 esu = 1.4 × 10−18m2/V 2 [56]. But even when usingsuch a highly nonlinear electrolyte cover layer, its contributionto the waveguide’s effective nonlinear parameters would bemore than 30 times smaller than that from the graphene sheet.Therefore, we can disregard the nonlinearity contributionfromthe electrolyte cover layer in the formulas for the effectivenonlinear parameters in Eq. (7). We do take the silicon waveg-uide into account in these formulas because, even thoughsilicon’sχ(3) is smaller than1.4×10−18m2/V 2, it contributessignificantly more to the effective nonlinear parameters thanthe electrolyte. This is due to the much higher field strengthin the silicon waveguide as compared to that in the electrolytecover layer. Finally, we remark that all effective coefficients inEq. (7) depend on the position along the waveguide becauseof the position dependence of the graphene chemical potentialrealized with the electrolyte gates on top.

ACKNOWLEDGMENT

This work has been supported by the ERC-FP7/2007-2013grant 336940, by the FWO-Vlaanderen project G.A002.13N,by the National Sciences and Engineering Research Councilof Canada, by IAP BELSPO (IAP P7-35), VUB-Methusalem,and VUB-OZR.

REFERENCES

[1] G. T. Reed (Ed.),Silicon photonics: the state of the art. West Sussex,UK: John Wiley, 2008, ch. 9.

[2] V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover,P. T.Ho, “Optical signal processing using nonlinear semiconductor microringresonators,”IEEE J. Sel. Top. Quantum Electron., vol. 8, no. 3, pp. 705-713, May-June 2002.

[3] D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phasematch-ing and nonlinear optical processes in silicon waveguides,” Opt. Exp., vol.12, no. 1, pp. 149-160, Jan. 2004.

[4] R. L. Espinola, J. I. Dadap, R. M. Osgood, Jr., S. J. McNab,andY. A. Vlasov, “C-band wavelength conversion in silicon photonic wirewaveguides,”Opt. Exp., vol. 13, no. 11, pp. 4341-4349, May 2005.

[5] Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadbandparametric generation and wavelength conversion in siliconwaveguides,”Opt. Exp., vol. 14, no. 11, pp. 4786-4799, May 2006.

[6] M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M.Lipson, andA. L. Gaeta, “Broad-band optical parametric gain on a siliconphotonicchip,” Nature, vol. 441, pp. 960-963, 2006.

[7] Y. -H. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen,“Demonstration of wavelength conversion at 40 Gb/s data ratein siliconwaveguides,”Opt. Exp., vol. 14, no. 24, pp. 11721-11726, Nov. 2006.

[8] M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta,“Broad-band continuous-wave parametric wavelength conversion in sili-con nanowaveguides,”Opt. Exp., vol. 15, no. 20, pp. 12949-12958, Oct.2007.

[9] Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomenain silicon waveguides: Modeling and applications,”Opt. Exp., vol. 15,no. 25, pp. 16604-16644, Nov. 2007.

[10] M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M.Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integratedwaveguide structures,”Nat. Photonics, vol. 2, pp. 737-740, 2008.




[11] A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M.Lipson, “Frequency conversion over two-thirds of an octavein siliconnanowaveguides,”Opt. Exp., vol. 18, no. 3, pp. 1904-1908, Feb. 2010.

[12] N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont,“Wavelength Conversion Based on Raman- and Non-Resonant Four-WaveMixing in Silicon Nanowire Rings Without Dispersion Engineering,”IEEE J. Sel. Top. Quantum Electron., vol. 17, no. 4, pp. 1078-1091,July-Aug. 2011.

[13] J. B. Driscoll, R. R. Grote, X. Liu, J. I. Dadap, N. C. Panoiu, and R.M. Osgood, “Directionally anisotropic Si nanowires: on-chip nonlineargrating devices in uniform waveguides,”Opt. Lett., vol. 36, no. 8, pp.1416-1418, April 2011.

[14] N. Vermeulen, J. E. Sipe, L. G. Helt, and H. Thienpont, “Opportunitiesfor wavelength conversion with on-chip diamond ring resonators,” LaserPhotonics Rev., vol. 6, no. 6, pp. 793-801, 2012.

[15] J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K.Bergman, and R. M. Osgood, Jr., “Width-modulation of Si photonicwires for quasi-phase-matching of four-wave-mixing: experimental andtheoretical demonstration,”Opt. Exp., vol. 20, no. 8, pp. 9227-9242, April2012.

[16] Y. Lefevre, N. Vermeulen, and H. Thienpont, “Quasi-Phase-Matchingof Four-Wave-Mixing-Based Wavelength Conversion by Phase-MismatchSwitching,” J. Lightwave Technol., vol. 31, no. 13, pp. 2113-2121, May2013.

[17] Y. Lefevre, N. Vermeulen, and H. Thienpont, “Simultaneous Quasi-Phase Matching of Two Arbitrary Four-Wave-Mixing Processes,” J.Lightwave Technol., vol. 33, no. 9, pp. 1726-1736, May 2015.

[18] See, e.g., www.epixfab.eu/technologies/imec-siphotonics; www.a-star.edu.sg/ime/RESEARCH/NANO-PHOTONICS-PROGRAMME

[19] D. S. Hum and M. M. Fejer, “Quasi-phase matching,”C. R. Physique,vol. 8, pp. 180-198, 2007.

[20] A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M.Lazzeri, F.Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, and A.K. Geim,“Raman Spectrum of Graphene and Graphene Layers,”Phys. Rev. Lett.,vol. 97, 187401, Nov. 2006.

[21] M. Krause, H. Renner and E. Brinkmeyer, “Analysis of Ramanlasingcharacteristics in silicon-on-insulator waveguides,”Opt. Exp., vol. 12, no.23, pp. 5703-5710, Nov. 2004.

[22] J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order opticalnonlinearity of graphene,”New J. Phys., vol. 16, 053014, 2014.

[23] N. Vermeulen, C. Debaes, A. A. Fotiadi, K. Panajotov, andH. Thien-pont, “Stokes-anti-stokes iterative resonator method for modeling Ramanlasers,”IEEE J. Quantum. Electron., vol. 42, no. 11, pp. 1144-1156, Nov.2006.

[24] Z. Cheng, H. K. Tsang, X. Wang, K. Xu, and J.-B. Xu, “In-PlaneOptical Absorption and Free Carrier Absorption in Graphene-on-SiliconWaveguides,” IEEE J. Sel. Top. Quantum Electron., vol. 20, no. 1,4400106, Jan.-Feb. 2014.

[25] G. P. Agrawal,Nonlinear fiber optics - Third edition. San Diego, USA:Academic Press, 2001, ch. 10.

[26] E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A.Mikhailov,“Coherent Nonlinear Optical Response of Graphene,”Phys. Rev. Lett.,vol. 105, 097401, Aug. 2010.

[27] T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu,G. Q. Lo,D. L. Kwong, J. Hone and C. W. Wong, “Regenerative oscillation andfour-wave mixing in graphene optoelectronics,”Nat. Phot., vol. 6, pp.554-559, Aug. 2012.

[28] H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, andP. Kockaert, “Z-scan measurement of the nonlinear refractiveindex ofgraphene,”Opt. Lett., vol. 37, no. 11, pp. 1856-1858, June 2012.

[29] J. L. Cheng, N. Vermeulen, and J. E. Sipe, “Third order nonlinearity ofgraphene: effects of phenomenological relaxation and finitetemperature,”arXiv:1503.07564, accepted for publication inPhys. Rev. B

[30] M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland,“Dispersion of Bound Electronic Nonlinear Refraction in Solids,” IEEEJ. Quantum Electron., vol. 27, no. 6, pp. 1296-1309, June 1991.

[31] A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S. K. Saha, U. V.Waghmare, K. S. Novoselov, H. R. Krishnamurthy, A. K. Geim, A. C.Ferrari, and A. K. Sood, “Monitoring dopants by Raman scattering in anelectrochemically top-gated graphene transistor,”Nat. Nanotechn., vol. 3,pp. 210-215, April 2008.

[32] Y. Ye, L. Gan, L. Dai, Y. Dai, X. Guo, H. Meng, B. Yu, Z. Shi,K.Shanga and G. Qin, “A simple and scalable graphene patterningmethodand its application in CdSe nanobelt/graphene Schottky junction solarcells,” Nanoscale, vol. 3, pp. 1477-1481, March 2011.

[33] D. J. Carrad, A. M. Burke, R. W. Lyttleton, H. J. Joyce, H.H. Tan,C. Jagadish, K. Storm, H. Linke, L. Samuelson, and A. P. Micolich,

“Electron-Beam Patterning of Polymer Electrolyte Films To Make Mul-tiple Nanoscale Gates for Nanowire Transistors,”Nano Lett., vol. 14, pp.94-100, 2014.

[34] H. Yan, F. Xia, W. Zhu, M. Freitag, C. Dimitrakopoulos, A.A. Bol,G. Tulevski, and P. Avouris, “Infrared Spectroscopy of Wafer-ScaleGraphene,”ACS Nano, vol. 5, no. 12, pp. 9854-9860, 2011.

[35] J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao,H. A.Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang,“Drude conductivity of Dirac fermions in graphene,”Phys. Rev. B, vol.83, 165113, 2011.

[36] R. del Coso and J. Solis, “Relation between nonlinear refractive indexand third-order susceptibility in absorbing media,”J. Opt. Soc. Am. B,vol. 21, no. 3, pp. 640-644, March 2004.

[37] X. Luo, Y. Cao, J. Song, X. Hu, Y. Cheng, C. Li, C. Liu, T.-Y.Liow, M. Yu, H. Wang, Q. J. Wang and P. G.-Q. Lo, “High-throughputmultiple dies-to-wafer bonding technology and III/V-on-Sihybrid lasersfor heterogeneous integration of optoelectronic integrated circuits,”Front.Mat., vol. 2, 28, April 2015.

[38] Y. Wang, Y. Zheng, X. Xu, E. Dubuisson, Q. Bao, J. Lu, and K. P. Loh,“Electrochemical Delamination of CVD Grown Graphene Film: Towardthe Recyclable Use of Copper Catalyst,”ACS Nano, vol. 5, no. 12, pp.9927-9933, 2011.

[39] V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaperfor compactmode conversion,”Opt. Lett., vol. 28, no. 15, pp. 1302-1304, 2003.

[40] H. H. Li, “Refractive index of silicon and germanium and its wavelengthand temperature derivatives,”J. Phys. Chem. Ref. Data, vol. 9, no. 3, pp.561-658, 1980.

[41] I. H. Malitson, “Interspecimen Comparison of the Refractive Index ofFused Silica,”J. Opt. Soc. Am., vol. 55, no. 10, pp. 1205-1208, 1965.

[42] J. D. Ingham and D. D. Lawson, “Refractive indexmolecular weightrelationships for poly(ethylene oxide),”J. Polymer Science Part A, vol.3, no. 7, pp. 2707-2710, July 1965.

[43] C.-C. Chen, M. Aykol, C.-C. Chang, A. F. J. Levi, and S. B.Cronin,“Graphene-Silicon Schottky Diodes,”Nano Lett., vol. 11, pp. 1863-1867,April 2011.

[44] H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts,A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photonabsorption and self-phase modulation in silicon waveguidesat 1.5 umwavelength,”App. Phys. Lett., vol. 80, pp. 416-418, 2002.

[45] Z. Zhang and P. L. Voss, “Full-band quantum-dynamical theory ofsaturation and four-wave mixing in graphene,”Opt. Lett., vol. 36, no.23, pp. 4569-4571, Dec. 2011.

[46] S. Shen, K. Wu, L. Sun, and C. Jiang, “Theoretical study of graphene-silicon nitride-silicon hybrid photonic crystal waveguides for four-wavemixing enhancement,”Appl. Opt., vol. 54, no. 12, pp. 3640-3644, April2015.

[47] Z. Zheng, C. Zhao, S. Lu, Y. Chen, Y. Li, H. Zhang, and S. Wen,“Microwave and optical saturable absorption in graphene,”Opt. Exp.,vol. 20, no. 21, pp. 23201-23214, Oct. 2012.

[48] H. Zhang, S. B. Lu, J. Zheng, J. Du, S. C. Wen, D. Y. Tang, and K. P.Loh, “Molybdenum disulfide (MoS2) as a broadband saturable absorberfor ultra-fast photonics,”Opt. Exp., vol. 22, no. 6, pp. 7249-7260, March2014.

[49] R. Wang, H.-C. Chien, J. Kumar, N. Kumar, H.-Y. Chiu, and H.Zhao,“Third-Harmonic Generation in Ultrathin Films of MoS2,” ACS Appl.Mater. Interf., vol. 6, pp. 314-318, 2014.

[50] Y. Chen, G. Jiang, S. Chen, Z. Guo, X. Yu, C. Zhao, H. Zhang, Q. Bao,S. Wen, D. Tang, and D. Fan, “Mechanically exfoliated black phosphorusas a new saturable absorber for both Q-switching and mode-locking laseroperation,”Opt. Exp., vol. 23, no. 10, pp. 12823-12833, May 2015.

[51] S. B. Lu, L. L. Miao, Z. N. Guo, X. Qi, C. J. Zhao, H. Zhang, S. C.Wen, D. Y. Tang, and D. Y. Fan, “Broadband nonlinear optical responsein multi-layer black phosphorus: an emerging infrared and mid-infraredoptical material,”Opt. Exp., vol. 23, no. 9, pp. 11183-11194, May 2015.

[52] L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. dePaula, “Observation of intense second harmonic generation from MoS2atomic crystals,”Phys. Rev. B, vol. 87, 201401, May 2013.

[53] The details of our strategy for the derivation in Appendix will bepublished in a separate paper.

[54] N. A. R. Bhat and J. E. Sipe, “Hamiltonian treatment of the electro-magnetic field in dispersive and absorptive structured media,” Phys. Rev.A, vol. 73, 063808, June 2006.

[55] Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous parametric down-conversion in waveguides: a backward Heisenberg picture approach,”Phys. Rev. A, vol. 77, no. 3, 033808, March 2008.

[56] J. A. Chilton, M. Goosey (Eds.),Special Polymers for Electronics andOptoelectronics. London, UK: Chapman and Hall, 1995, ch. 4.




Nathalie Vermeulen was born in Duffel, Belgium,in 1981. She received her M. Sc. Degree in electricalengineering with majors in photonics from the VrijeUniversiteit Brussel (VUB), Brussels, Belgium, in2004, and obtained the Ph.D. degree from VUB in2008. Since October 2013 she holds a Tenure Trackprofessor position in the Brussels Photonics Team(B-PHOT) at VUB.

She is (co-)author of 24 peer-reviewed jour-nal publications, 32 conference proceedings, and 2patents. Her research interests include modeling and

demonstrating novel concepts for nonlinear optics in photonic integratedcircuits, and developing mid-infrared solid-state lasers.

Prof. Vermeulen has been an invited speaker at 7 international conferences.In 2007 she was awarded the Newport Spectra-Physics Research ExcellenceAward and in 2010 she received the European Photonics21 Innovation Award.In 2013 she obtained a Starting Grant from the European Research Council(ERC), and she also became coordinator of a European Future and EmergingTechnologies (FET) project. Also in 2013, she was elected Member to theYoung Academy of Belgium. In 2014 she received the VUB I. VanderschuerenAward and in 2015 she was awarded the international LIGHT2015 YoungWomen in Photonics prize. She is a member of the International Societyfor Optical Engineers, Optical Society of America, and IEEE Laser andElectrooptics Society.

JinLuo Cheng graduated with majors in AppliedPhysics in 2002 and received his PhD in CondensedMatter Physics in 2007, both at University of Sci-ence and Technology of China. He currently is aPostDoc fellow in the Brussels Photonics Team (B-PHOT) at the Vrije Universiteit Brussel (VUB) andis also affiliated to the University of Toronto.

His current research interests include optical non-linearity calculations for different materials includ-ing graphene and diamond. He has coauthored 23peer-reviewed journal papers. Dr. Cheng has been

an invited speaker at 1 international conference.

John E. Sipereceived the Ph.D. degree from the De-partment of Physics, University of Toronto, Toronto,ON, in 1975.

In 1981, he joined the professorial staff at theUniversity of Toronto. He is a theorist, who has donemost of his work in close collaboration with exper-imentalists. His research interests include nonlinearand quantum optics, optical properties of semi-conductors, quantum interference effects in solids,foundations of quantum mechanics, and optical sens-ing.

Prof. Sipe has been an invited speaker at numerous conferences. He is aFellow of the Optical Society of America, the American Physical Society, andthe Royal Society of Canada.

Hugo Thienpont (M ‘99) was born in Ninove, Bel-gium, in 1961. He graduated as an electrotechnicalengineer in 1984 and received the Ph.D. degree inapplied sciences in 1990, both at the Vrije Univer-siteit Brussel (VUB), Brussels, Belgium. In 1994 hebecame Professor at the Faculty of Engineering. In2004 he was elected Chair of the Department of Ap-plied Physics and Photonics. Within this department,he is the research director of the Brussels PhotonicsTeam (B-PHOT). Currently he coordinates severalresearch and networking projects at the European

level and manages micro-photonics related industrial projects in collaborationwith companies like Barco, Melexis, Best, Tyco, and Umicore. He authoredmore than 700 SCI-stated journal papers and international conference pro-ceedings publications.

Prof. Thienpont received the International Commission for Optics PrizeICO in 1999 and the Ernst Abbe medal from Carl Zeiss. In 2003, hewasawarded the title of “IEEE Photonics Distinguished Lecturer.” In 2005 hereceived the SPIE Presidents Award for dedicated services to the EuropeanPhotonics Community, and in 2007 the international MOC Award for hiscontributions in the field of micro-optics and the prize “Prof. Roger van Geen”for his scientific achievements during his research career atVUB. In 2008 heobtains the prestigious “Methusalem” status for top-scientist from the Flemishgovernment for his research track-record in photonics. In 2011 he received theUniversity medal “Alma Mater Bene Merentibus” of the Warsaw University ofTechnology. In 2013 he was elected Member to the Royal Flemish Academyof Belgium for Science and Art and in 2013 he receives the price for sciencecommunication.

Hugo Thienpont is a member of the Board of Stakeholders of the Tech-nology Platform Photonics21, the International Society for Optical Engineers,and the IEEE Laser and Electrooptics Society. Since 2012 he is Vice-Rectorfor innovation and industrial policy of the Vrije Universiteit Brussel.



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