Acoustic Kerr nonlinearity of wave propagation in a planar nanoelectromechanical waveguide
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Acoustic Kerr nonlinearity of wave propagation in a planarnanoelectromechanical waveguide
M. Kurosu,
1, 2, ∗ D. Hatanaka, ∗ and H. Yamaguchi
1, 2 NTT Basic Research Laboratories, NTT Corporation,Atsugi-shi, Kanagawa 243-0198, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan
Abstract
Nonlinearity is the key to introducing novel concepts in various technologies utilizing travelingwaves. In contrast to the field of optics, where highly functional devices have been developedusing optical Kerr nonlinearity , such a nonlinear effect in acoustic devices has yet to be fullyexploited. Here, we show that most fundamental nonlinear phenomena of self-phase modula-tion (SPM), cross-phase modulation (XPM) and four-wave mixing (FWM) caused by the acousticKerr effect are quantitatively characterized using a newly developed platform consisting of a pla-nar nanoelectromechanical waveguide (NEMW). Combining the cutting-edge technology of a highcrystalline quality NEMW with a piezoelectric interdigital transducer (IDT), we efficiently excitean intense and long-lived traveling wave sufficiently to induce and characterize acoustic nonlinear-ity. The observed nonlinear phenomena are precisely described by the model using the nonlinearSchr¨odinger (NLS) equation, so that this architecture enables the nonlinear dynamics to be per-fectly tailored. The flexible and integratable platform extends the ability to manipulate acousticwave propagation on a chip, thus offering the potential to develop highly functional devices andstudy novel nonlinear acoustics. . There have been a number of studieswith the aim of utilizing acoustic waves for signal processing application such as microwave-to-optical converters and quantum computing . This is because, compared with electro-magnetic waves, acoustic devices have the distinct advantages of a short wavelength andsmall energy loss in on-chip applications. The key to improving the ability of the on-chipacoustic manipulation is a nonlinear effect, which allows various types of advanced controlto be realized including short pulse generation, frequency conversion and amplification, asalready demonstrated in nonlinear optics .However, compared with work in the optics field, there have been few studies of nonlineartraveling acoustic wave due to the lack of a suitable platform to realize low-loss wave-guidingand a transducer capable of exciting an intense nonlinear wave. Although some pioneeringstudies show that nonlinear wave propagation has been observed in solid crystals , astrong laser pulse is needed to induce nonlinearity and the experiments must be conductedat low temperature. Such conditions are unsuitable for practical use and monolithic on-chipintegration. In contrast, nonlinear stress-strain relation induced by geometric nonlinearity,has been intensively investigated in nanomechanical resonators. A number of intriguingresults have been reported including mode coupling, a phononic frequency comb, frequencystabilization, and chaos . Inspired by the development of nanomechanical technology, anovel acoustic platform for a NEMW has recently been realized that is constructed froma suspended semiconductor membrane array. This hosts its excellent properties, such asengineerable dispersion, low propagation loss, design flexibility and semiconductor-basedintegration. Thus, this architecture has enabled the demonstration of electrical phononmanipulation, energy focusing by dispersion, and the active manipulation of phononic bandstructures on a chip .By combining a newly designed 33-mm long acoustic waveguide structure with an IDTthat enables the efficient excitation of strong acoustic vibration with a moderate inputamplitude, we demonstrate the fundamental nonlinear phenomena of SPM, XPM and FWMinduced by the acoustic Kerr effect and confirm that they can be controlled by adjusting theinput excitation amplitude and propagation distance. While optical Kerr effects originate inrefractive index variation due to the electric field of light, acoustic Kerr effects are caused bygeometric nonlinearity . These nonlinear propagation dynamics can be observed because of2he low-loss, single-mode and long transmission channel with a high quality GaAs/AlGaAssingle crystal heterostructure. This platform provides proof of the capacity to manipulatenonlinear acoustic wave propagation on a chip and will pave the way to the development ofnonlinear acoustic devices.A vibrating membrane is fabricated by sacrificially etching an Al . Ga . As layer as shownin Fig. 1c, and this is 33-mm long when folded to realize a small device footprint as shownin Fig. 1a. This waveguide hosts a continuous transmission band between 2.4 and 7.4MHz except in the 7.4-7.8 MHz bandgap regime, which is caused by Bragg reflection ofa vibration from a periodically arrayed air holes along the waveguide as shown in Fig.1b. This phononic crystal structure modulates the group velocity dispersion (GVD) of thedevice as shown in Fig. 1e, indicating that the dispersive effect can be tuned by engineeringthe periodic structure or selecting the operating frequency. Considering our goal, which isto confirm the ability to control nonlinear wave propagation, the dispersion effect on thepropagation dynamics should be minimized. In our work, we chose an IDT electrode pitchof 20 µ m to correspond with the operating frequency of a low GVD regime. Therefore,the IDT can excite large flexural vibrations around 5.38 MHz at which the GVD coefficient k is ∼ − × − s m − estimated with a finite element method (FEM) simulation asshown in Fig. 1d and 1e. The resultant waves can be measured at various distances inthe waveguide by adjusting the laser spot position of an optical interferometer. All theexperiments described here were performed at room temperature and in a moderate vacuum( ∼
10 Pa).The mechanical motion of the suspended vibrating plate is governed by the Euler-Bernoulli equation . From this, we derive a wave equation, namely the NLS equation,which can be used to predict the nonlinear dynamics of acoustic wave propagation in awaveguide . Here, the envelope of the vibrating pulse centered around wavenumber k andthe angular frequency ω was assumed to vary slowly in the temporal and spatial domains.Considering the amplitude A ( x, t ) where the x is the propagation distance and t is time, theNLS equation is given by, ∂A∂x = − α A − ik ∂ A∂T + iξ | A | A, (1)where α , k = ∂ k∂ω and ξ denote the linear loss, GVD coefficient and an effective nonlinearparameter, respectively. T = t − x/v g is the time in a moving frame where v g is group3elocity. In this device, α is set at 0.29 dB mm − , which is determined by a time-of-flight measurement described in detail elsewhere . In this new structure, the value wasgreatly improved compared with that in our previous report because of the process anddevice design optimization. It is worth noting that the third term on the right hand side ofequation (1) contains | A | representing the squared amplitude. This governs the acoustic Keffnonlinearity and cannot be negligible when the intensity of the traveling wave is significantlylarge and induces third-order nonlinearity in the system. As a result, the phase of the waveis modulated during propagation, which is known as SPM, especially in the field of nonlinearoptics . When the GVD effect can be ignored, the maximum phase shift θ max caused bySPM is written as , θ max ( x ) = ξ | A ( x, | x eff (2)where x eff is the effective distance, which is defined by x eff = − exp( − αx ) α . As seen fromequation (2), the SPM-induced phase shift is proportional to the instantaneous squaredamplitude of the pulse, and thus its temporal response is followed by the pulse envelopeas shown in Fig. 2a. The sign of the nonlinear parameter ξ determines the polarity ofthe SPM. Additionally, the phase is accumulated while propagating in the waveguide. Ourhigh crystalline quality NEMW with a transmission channel long enough to allow us tocharacterize the phase accumulation is a suitable platform on which to observe this nonlineardynamics.The SPM process is at the heart of nonlinear phenomena and thus, it is of prime impor-tance to investigate the effect. To that end, an intense pulsed acoustic wave is efficientlyexcited using the IDT electrode located at one end of the waveguide, where a Gaussian-shaped pulse A (0 , T ) = A exp (cid:16) − T T (cid:17) is used as the input. The time evolution of theamplitude and phase of the pulse is measured at distances of 0, 10, and 19.5 mm from theIDT as shown in Fig. 2b-2d and 2e-2g, respectively. The amplitude of a pulse envelope ofwidth T = 40 µ s increases as the excitation voltage is increased to 1.2 V rms at x = 0 mm(Fig. 2b), whereas the temporal response of the phase is nearly flat and is invariant evenwhen the excitation voltage is changed (Fig. 2e). However, as the wave propagates, thephase is being modulated through the SPM process, and it finally reaches -50 degrees at x = 19 . rms excitation voltage (Fig. 2g). This negative phase shift due tothe sign of ξ becomes apparent at larger excitation amplitudes. We note that the negative4hase shift indicates the leading phase because we used the conventions of exp( − i ωt ) as afundamental wave . Furthermore, to validate these results, we simulate the propagationdistance dependence of the phase shift at various voltages using equation (1) as shown inFig. 2h (see also Methods). The experimentally observed variation in the phase can bewell reproduced by a simulation that involves applying the split-step Fourier method to theNLS equation (see Supplementary Information for details), where the larger SPM occursat a longer distance and a larger amplitude. From this comparison, the effective nonlinearparameter of this NEMW is obtained as ξ = −
16 nm − m − . This value is sufficiently largeto observe novel nonlinear phenomena. For example, acoustic soliton compression is alsopossible with realistic parameters as shown in Supplementary Information.Although SPM works only with a single acoustic wave, acoustic Kerr nonlinearity canalso be exploited to tune and manipulate the propagation dynamics through the inter-action between different acoustic waves. An incident wave can interact with a strongly-excited wave through this third-order nonlinear effect, thus resulting in the phase of theincident wave being modulated. This nonlinear interaction can generate a new acousticwave, whose phenomenon is known as FWM. These effects have been intensively studiedin the field of nonlinear optics leading to the development of a range of light manipulationtechniques including wavelength conversion, amplification, supercontinuum generation andsolitons . Figure 3a-3f show the spectral response of a continuous wave (signal) andan intense pulsed wave (pump) excited from the IDT at f s = 5.264 MHz and f p = 5.383MHz at various positions in the waveguide as shown in Fig. 1a. As described above, thestrong pump wave activates the nonlinearity and this induces a temporal phase shift inthe signal wave, which is called XPM. Hence, the temporally varying phase modifies theinstantaneous frequency of the pulse, resulting in the bottom of the signal spectrum beingbroadened. On the other hand, under appropriate conditions that satisfy energy and mo-mentum conservation, the FWM process also permits the generation of a new wave, calledan idler. The output spectra reveal the idler wave at f i = 5.502 MHz, which satisfies theenergy conservation requirement 2 f p = f s + f i . The momentum conservation requirement2 k p = k s + k i is largely met due to the small GVD and the frequency separation between thewaves involved, namely k p ∼ k s ∼ k i where k p , k s and k i are the wavenumber of the pump,signal and idler, respectively. The spectral responses resulting from the XPM and FWMprocesses can be numerically calculated using equation (1) with the parameters obtained5rom the SPM experiment, as shown by the solid blue lines in Fig. 3a-3f. The numericallycalculated and experimental spectra are good agreement, where the spectral broadening inthe signal via XPM and the idler generation via FWM are well reproduced.To further elucidate the nonlinear acoustic process, the propagation dynamics of thesignal, pump, idler and broadened signal due to XPM are investigated by measuring theamplitude of each wave at distances of 0 to 31 mm as shown in Fig. 4. At the beginningof the propagation, the idler peak amplitude (green) increases greatly and it reaches itsmaximum value of ∼
20 pm at x = 7 ∼
10 mm. However, the pump (red) and signal(blue) are attenuated due to the loss, which suppresses the efficiency of the FWM process.Additionally, the idler experiences energy dissipation during propagation. Therefore, idlergeneration is overwhelmed by the loss, resulting in the output amplitude being reduced.The dynamics of nonlinear wave propagation is quantitatively captured by a numericalcalculation based on the NLS equation (solid lines). The results provide proof that thetransmission properties of the signal, pump and idler waves originate from the nonlineareffects of the SPM, XPM and FWM processes. To the best of our knowledge, ours is thefirst report of the experimental demonstration and perfect modeling of SPM, XPM andFWM using a solid acoustic waveguide.In conclusion, we demonstrated SPM, XPM and FWM resulting from the acoustic Kerreffect in a planar NEMW and confirmed the effectiveness of using the NLS equation toinvestigate the acoustic wave dynamics. Combined with the phononic crystal property,which enables the GVD to be engineered, the novel acoustic platform offers us the abilityto generate and fully manipulate a range of nonlinear phenomena including solitons andsupercontinuum generation. This will enhance the availability of acoustic waves andphonons for use in the fields of acoustic signal processing and opto-/electro-mechanics aswell as investigations into fundamental nonlinear physics.
MethodsMeasurement setup.
The NEMW is excited by applying an alternating voltage orGaussian-shaped voltage from a function generator (NF WaveFactory 1974). The resultantflexural vibrations are measured with a laser Doppler interferometer (Melectro V1002). Thedemodulated electric signals are sent to a vector signal analyzer (HP89410A) in the spectralmeasurements as shown in Figs. 1d and 3. The signals are also sent to a lock-in amplifier6Zurich Instruments UHFLI) where in-phase ( X ) and out-of-phase ( Y ) components withrespect to the reference frequency ω are detected using a phase-sensitive detection method.Then, both signals are passed through low-pass filter with a bandwidth of 470 kHz and atime constant of 102.6 ns, and then measured with an oscilloscope (Agilent DSO6014A) forthe temporal measurements shown in Figs. 1e and 2b-2g. As a result, the amplitude andphase of the pulse can be estimated from √ X + Y and tan − ( Y /X ), respectively.
Data availability.
The data that support the findings of this study are available from thecorresponding author in response to a reasonable request. ∗ Electronic address: kurosu˙megumi˙[email protected]; [email protected] Marin-Palomo, P., et al.
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We acknowledge the stimulated discussion in the meeting of the Cooperative ResearchProject of the Research Institute of Electrical Communication, Tohoku University.This work is partly supported by a MEXT Grant-in-Aid for Scientic Research on Innova-tive Areas “Science of hybrid quantum systems” (Grant No. JP15H05869 and JP15K21727).
Author Contributions
M.K. performed the measurements, the data analysis and the simulation, and fabricatedthe device with the help of D.H. and H. Y. M.K., D.H. and H.Y. wrote the manuscript andH.Y. planned the project. 9 bc d e
PumpSignalFrequencyTime + Time + Frequency + Energy transfer from pump
FIG. 1: Nanoelectromechanical waveguide. a A schematic showing the device and the measurementsetup. Radio-frequency signals are sent from a function generator to the IDT electrode, where asingle pulsed wave, or a continuous wave and pulsed waves (signal and pump) are injected. Theleft and right insets show the input and output signal configurations in an FWM experiment (seeFigs. 3 and 4). b Microscope photograph of the IDT electrode and suspended membrane. Thewidth and pitch of the IDT are 5 and 20 µ m, respectively. The width and air-hole pitch of thewaveguide are 27.7 and 8 µ m, respectively. c A false-colored SEM image of a cross-section withIDT electrodes (yellow). The GaAs membrane with a thickness of 200 nm (pink) is suspended byetching an Al . Ga . As sacrificial layer (blue). The scale bar is 5 µ m. d The frequency responsesof the device excited by the IDT (blue) and a conventional piezotransduer (red) with 1.0 V rms ,where they are measured at distances x = 5 and 1 mm from each transducer, respectively. TheIDT hosts narrow excitation, whereas the activated vibration is larger than that from the usualtransducer of around 5.4 MHz. The inset shows an enlarged view from 5.0 MHz to 5.6 MHz. e GVD coefficient k as function of frequency. The experimental results (green dots) are estimatedfrom time-of-flight measurements and smoothed by Gaussian smoothing. The red line is calculatedwith the FEM simulation, where the internal stress between the GaAs and Al . Ga . As layers isincluded . bcd efgh FIG. 2: Self-phase modulation. a Phase change of a Gaussian-shaped pulse (red solid line) due toSPM. The positive (negative) nonlinear parameter ξ activates a positive (negative) phase shift asshown by the dashed (solid) blue line. The solid green line denotes the phase of an input pulse. b-g Temporal response of the amplitude ( b-d ) and phase ( e-g ) of an acoustic wave pulse measuredat a distance x = 0 mm ( b, e ), 10 mm ( c, f ), and 19.5 mm ( d, g ) when excited by the IDT withvoltages ranging from 0 V rms to 1.2 V rms . As the amplitude is small around the leading and trailingedges in the pulse, the phase fluctuates and is not determined. Therefore, it is removed from thedata. There is a slight slope in the time evolution of the phase on the baseline of the pulse. Thismight be frequency chirp caused by dispersion effect induced by the periodic electrode array in theIDT. h The propagation length dependence of the SPM induced the maximum phase differencewith respect to the phase of the pulse with 0.3 V rms . The solid lines show numerical simulationresults for ξ = −
16 nm − m − using the NLS equation. The colored bands indicate the numericalsimulation results for ξ = − ∼ −
17 nm − m − . mm 7 mm 12 mm17 mm 22 mm 27 mm a b cd e f FIG. 3: Four-wave mixing. a-f