Tunable optical forces between nanophotonicwaveguidesJoris Roels1*, Iwijn De Vlaminck2, Liesbet Lagae2, Bjorn Maes1, Dries Van Thourhout1 and Roel Baets1

The confinement of light in components with nanoscale cross-sections in nanophotonic circuits significantly enhances the mag-nitude of the optical forces experienced by these components1,2.Here we demonstrate optical gradient forces between two nano-photonic waveguides, and show that the sign of the force can betuned from attractive to repulsive by controlling the relativephase of the optical fields injected into the waveguides. Theoptical gradient force could have applications in opticallytunable microphotonic devices and nanomechanical systems.

Optical forces can be divided into two major categories: radiationpressure and transverse gradient forces. Radiation pressure effectshave been studied extensively using high-finesse optical resonators3–5.These effects can be understood as momentum exchange betweenphotons and matter, so the force acts along the light propagation direc-tion. Radiation pressure effects also provide a route towards opticalcooling of a micromechanical resonator to its quantum ground state6.The gradient force acts transversely to the propagation direction of thelight and enables, for example, optical tweezing in free-space optics7.However, when harnessed on an optical chip, this force also enables dis-placement of integrated nanophotonic structures. Mechanical displace-ment through attractive forces between a suspended silicon waveguideand silica substrate have recently been demonstrated experimentally8.

In this Letter we demonstrate transverse optical forces between twoclosely spaced nanophotonic wires (wire width, w¼ 445 nm; wirethickness, t¼ 220 nm; gap, g¼ 220 nm) in high-index-contrastsilicon-on-insulator (Fig. 1). Although each of the nanophotonicwires supports just one transverse electric (TE) mode (with an effectiveindex neff¼ 2.23 for this mode and the electric field parallel to the sub-strate), two guided supermodes are present when the waveguides areplaced sufficiently close to each other. The propagating TE mode withthe highest (lowest) index is symmetric (anti-symmetric) in the trans-verse electric field E with respect to the symmetry plane in betweenthe waveguides. These modes will be referred to as the symmetric andantisymmetric modes throughout this Letter (with effective indicesnþ and n2, respectively).

Optical forces arise through an adiabatic change in the eigen-mode energy of the propagating modes. Each separate modalforce contribution is given by9

Fopt ¼ �PLc

ng

v

dvdgjk ð1Þ

where Fopt is the total time-averaged force acting on each waveguide, Pand v are the optical power and the eigenfrequency of the consideredmode, L is the length of the (free-standing) waveguide, c is the speedof light in vacuum, g is the gap between the waveguides, and ng and kare the group index and the wavevector of the considered mode,respectively. Negative values correspond to attractive forces. Fromequation (1) it can be understood that the force of the symmetric(antisymmetric) mode will correspond to an attractive (repulsive)

force, because its eigenfrequency decreases (increases) with smallergap size. The above equation yields 20.23 pNmm21 mW21 (attrac-tive symmetric mode) and 0.1 pN mm21 mW21 (repulsive antisym-metric mode) for the normalized optical forces of the currentdevice. Consequently, controlling the relative excitation of these twomodes enables force tuning from attractive to repulsive.

Integrating the parallel underetched waveguides as a coupler into aMach–Zehnder interferometer (MZI) with unequal arm lengths(DL� 113 mm) allowed the relative input phase between the inputwaveguides to be changed by adjusting the wavelength. As a result,the excitation of the supermodes can be controlled. An optimizedshallow etched multimode interferometer (MMI) 3 dB splitter10 wasused as a second coupler, ensuring that both input waveguides ofthe waveguide coupler received equal optical powers. Intuitively, itis clear that in-phase fields favour the symmetric mode, whereas aphase shift of 1808 will favour the antisymmetric mode. For otherphase shifts a superposition of the two modes can be expected. Byintegrating the Maxwell stress tensor over the waveguide edges theforce can be calculated directly11 using the eigenmode expansion soft-ware tool CAMFR12. A beating of the optical force is expected that isrelated to the beat period (�120 mm) of the supermodes. Because thelength of the underetched waveguide coupler (25 mm) is smaller thanthis beat period, one should be careful using the average optical force.Figure 2 shows the calculated force as a function of propagation

ΔL

L

Waveguidecoupler

MMIcoupler

PumpProbe

20 μm

Figure 1 | The Mach–Zehnder interferometer. a, Schematic showing that

one arm of the interferometer is longer (DL� 113 mm) than the other. The

device has two couplers: a multimode interference (MMI) 3 dB splitter, and

a waveguide coupler that consists of two underetched parallel waveguides of

length L� 25 mm separated by a gap of 220 nm. b, Scanning electron

micrograph showing where the pump and probe beams enter the device.

1Department of Information Technology (INTEC), Ghent University – IMEC, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium, 2IMEC, Kapeldreef 75, B-3001Leuven, Belgium. *e-mail: joris.roels@intec.ugent.be

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length for different relative excitations. In the most extreme case(equal excitation of symmetric and antisymmetric modes, black linein Fig. 2) the beating force amplitude is calculated to be �25% ofthe average optical force for our device.

The light was coupled in and out of the chip by means of agrating coupler and a fibre with a 108 tilted angle. The gratingcoupler ensured that the excited waveguide modes were TE. Withan adiabatic taper the light was confined to a nanophotonic wireand launched into the 3 dB splitter.

The structures were fabricated using a deep ultraviolet lithographyprocess in high-index-contrast silicon-on-insulator. More details onthis fibre-to-chip coupling mechanism13 and the deep ultraviolet(CMOS compatible) lithography fabrication process can be foundelsewhere14. Finally, the underetch to release the waveguides wasperformed with buffered HF.

We made use of the mechanical resonance of the free-standingbeams (Qmech� 6,000) and excited the in-plane fundamental mech-anical vibration modes of the beams. For this purpose the device wasplaced in vacuum conditions (1� 1024 mbar) to avoid air dampingof the vibration. Clamping losses were the main dominant mechan-ical loss mechanism, because no special precautions were taken toavoid them. Owing to imperfections in the fabrication process,the two waveguides had a different mechanical resonance frequency(5.773 MHz and 5.497 MHz; see Fig. 3a).

The optomechanical transducer mechanism needed to transformthe induced vibration into a measurable optical output powerchange was provided by the device itself. A varying gap distance gbetween the waveguides changed the coupling, which translated intoa different distribution of the power in the two output arms of thewaveguide coupler15. Because the total device in principle is no morethan a MZI with delay length DL and one underetched waveguidecoupler, the transmission spectrum T(l) and transduction coefficientdT=dg of this device can be easily obtained in semi-analytical form:

T lð Þ ¼ 12

1þ sin2pDLneff

l

� �sin

2pL nþ � n�� �

l

� �� �ð2Þ

dTdg¼ pL

l

dðnþ � n�Þdg

sin2pDLneff

l

� �cos

2pL nþ � n�� �

l

� �ð3Þ

The measured transmission spectrum is shown in Fig. 3b. The fittingcurve (solid black line) allows neff� 2.33 and nþ2 n2� 0.014

to be extracted. Both differ by less than 5% from the theoreticallyexpected values (calculated with CAMFR). The transduction co-efficient is also shown in Fig. 3b and describes the relative outputpower change for a change in waveguide separation. It was measuredby recording the peak power spectral density (PSD) of the thermalresponse (of the 5.773 MHz beam) for each wavelength.These experimental data were calibrated with the estimated dis-placement noise 1.3 pm Hz21/2 (see also equation (4)) to arrive atthe curve plotted in Fig. 3b. For the transduction coefficient fit neffand nþ2 n2 were kept constant using the above determinedvalues. The extracted value for d(nþ2 n2)/dg� 0.00012 refractiveindex unit (RIU) nm21 again differs by less than 5% from the theor-etically expected value. Not surprisingly, the highest transduction isfound for wavelengths that correspond with peaks and valleys in the

Probe laser

Pump laser +electro-optical

modulator

Detector +ESA

Filter

Figure 4 | The experimental setup. The pump and probe lasers are

simultaneously injected into the device in different directions, with the pump

beam first passing through an electro-optical modulator. On leaving the

device the probe beam is first filtered and then analysed with an optical

detector and an electrical spectrum analyser (ESA).

0 20 40 60 80

0.15

0.10

0.05

0.00

−0.05

−0.10

−0.25

−0.15Forc

e (p

N m

W−1

μm

−1)

−0.20

100Propagation distance (μm)

Figure 2 | Force (calculated) versus distance for different relative mode

excitations. Forces above the dotted line are repulsive and forces below the

dotted line are attractive. Five different relative mode excitations are shown

(A, antisymmetric; S, symmetric): 100% A (pink); 95% A/5% S (red); 50%

A/50% S (black); 5% A/95% S (green); and 100% S (blue).

5.85.75.6Frequency (MHz)

Wavelength (nm)1,5541,5521,5501,5481,546

5.5

1e+0

Tran

smis

sion

(mW

)

1e−1

1e−2

1e−3

1e−4

1e−5

1e+0

Transduction coefficient (nm

−1)

1e−1

1e−2

1e−3

1e−4

1e−5

500a

b

400

300

200

PSD

(pW

Hz−1

)

100

0

Figure 3 | Calibration, transmission and transduction. a, Power spectral

density (PSD) of the thermal force used to calibrate the transduction

coefficient and optical force. b, Transmission spectrum (blue circles) and

transduction coefficient (red circles). The transmission spectrum is for a

probe laser with an input power of 20 mW (total insertion loss¼ 16 dB,

mainly due to coupling losses). The transduction coefficient was determined

by measuring how the peak PSD of the thermal force (a) varied with the

probe wavelength. The black solid lines are fits to equations (2) and (3).

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transmission spectrum. We chose to set our probe wavelengthto 1,548 nm.

The full setup for the optical force experiment is outlined inFig. 4. The probe and pump laser were inserted at different sidesof the chip. The pump laser light was modulated using an externalelectro-optical modulator. An optical pass filter (with central wave-length 1,548 nm) was inserted in front of the detector to make surethat only probe light reached the detector. This way, for each pumpwavelength, a radio-frequency vibration spectrum was recorded anda force for each wavelength could subsequently be extracted fromthese spectra. Calibration of the transduction coefficient and theoptical forces was possible by measuring the thermal response(Fig. 3b) of the beams and extracting the thermal force16.Consequently, the thermal displacement noise could also be deter-mined assuming the mechanical susceptibility of the beam (seeSupplementary Information) was known. This thermal force is(according to the fluctuation-dissipation theorem) given byequation (4) (where T is temperature, kB the Boltzmann constant,BW the measurement bandwidth, meff the oscillator effectivemass �2 pg)17:

Fbrown ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 kBTmeffvresBW

Qmech

sð4Þ

The measured normalized forces are shown in Fig. 5b. For each datapoint in this curve a full vibration radio-frequency spectrumwas taken (see Fig. 5a) and the magnitude of the force extractedfor each spectrum (more detailed information on this procedurecan be found in the Supplementary Information).

At first sight it is not possible to extract the direction of the force(attractive/repulsive) directly. One option is to check the phase shiftof the vibration output signals (compared to the reference signal ofthe signal generator) at 1,551.4 nm and 1,553.5 nm, because thesewavelengths correspond to the peak values for the force magnitude.We indeed found the signals to be in counter phase as expected,which proves that these different wavelengths generate a repulsiveand attractive force. The magnitude of the 1,551.4 nm signal,however, was approximately twice as large compared to the1,553.5 nm signal, which allows the former to be identified asattractive because this ratio was predicted by theory. In addition itis also possible to extract the sign of the force in a direct way bylooking at the interference of the signal with the thermo-opticalbackground signal (see Supplementary Information). The blacksolid line in Fig. 5b was fitted assuming a simple superposition ofthe spatially averaged attractive and repulsive eigenmode forces(Ftot ¼ Fþ cos2 pDLneff=lð Þ þ F� sin2 pDLneff=lð Þ), where Fþ andF2 are the forces for the symmetric and antisymmetric modesrespectively. Although we ignore the force beating (includingthis beating in the model would be difficult given the devicegeometry), agreement between theory and experiment is stillvery good. With an estimated modulated pump power of1.5 mW (18 mW modulated input power with 8 dB fibre-to-chipcoupling loss and approximately 3 dB insertion loss in theelectro-optical modulator) in the coupling region we also findthe normalized force values of 20.2 pN mm21 mW21 (attractive)and 0.1 pN mm21 mW21 (repulsive), both of which agreewith theory. The error bars in Fig. 5b are determined by the uncer-tainty regarding the exact optical power in the device (estimatedto be �20%).

Finally, note that thermomechanically induced vibrations wouldnot show repulsive or attractive behaviour and can clearly be neg-lected here, as was also expected from 3D FEM-COMSOL simu-lations, which predict the thermomechanical force to be at leastthree orders of magnitude smaller than the optomechanical force.Also, the thermo-optical background noise signal was estimated tobe 26 dB smaller compared to the optomechanical signal at themechanical resonance frequency (see Supplementary Information).

Received 22 May 2009; accepted 18 June 2009;published online 13 July 2009

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−8a

b

−10

−12

−14

−16

−18

−20

−22

−24

−26

−28

−30

0.2

0.1

0.0

−0.1

−0.2

−0.3

−0.4

5,765

1,549 1,550 1,551 1,552Wavelength (nm)

Opt

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forc

e (p

N μ

m−1

mW

−1)

1,553 1,554 1,555

5,770 5,775f (MHz)

P mod

/Pcw

(dB)

5,780

Figure 5 | Determining the optical force. a, Driven vibration spectra for

wavelengths of 1,551.4 nm (red circles) and 1,553.5 nm (blue) with

corresponding fits (black lines). b, The normalized optical force (extracted

from vibration spectra like those shown in a for different pump

wavelengths). Forces above the dotted line are repulsive and those below are

attractive. Error bars originate from the estimated error on the actual optical

power inside the device. The black line was fitted assuming a superposition

of the (spatially averaged) attractive and repulsive force.

LETTERS NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2009.186

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AcknowledgementsThe authors thank the Research Foundation—Flanders (FWO) for financial support. Wealso thank L. Haentjes for help with the construction of the vacuum chamber, M. Verbistfor taking the SEM image, W. Bogaerts for help with the design of the waveguides andS. Verstuyft for help with processing.

Author contributionsJ.R. and I.D.V. conceived and designed the experiments. J.R. performed the experiments,analysed the data and wrote the paper. J.R. and B.M. performed numerical simulations.L.L., D.V.T. and R.B. contributed materials and analysis tools. All authors discussed theresults and provided feedback and comments on the manuscript.

Additional informationSupplementary information accompanies this paper at www.nature.com/naturenanotechnology. Reprints and permission information is available online athttp://npg.nature.com/reprintsandpermissions/. Correspondence and requests for materialsshould be addressed to J.R.

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