St udy of  i nt ri nsi c l ocal i zed vi brat i onal  modesi n mi cromechani cal  osci l l at or arrays
著者Sat o Masayuki ,  Hubbard B. E. ,  Engl i sh L. Q. ,Si evers A. J . ,  I l i c B. ,  Czapl ewski  D. A. ,Crai ghead H. G.
j ournal  orpubl i cat i on t i t l e
vol ume1382
page range702- 715
year2003- 06- 01
URLht t p: //hdl . handl e. net /2297/3822
Study of intrinsic localized vibrational modes in micromechanicaloscillator arrays
M. Sato, B. E. Hubbard, L. Q. English, and A. J. SieversLaboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501
B. IlicCornell Nanofabrication Facility and Department of Applied and Engineering Physics, Cornell University,Ithaca, New York 14853-5403
D. A. Czaplewski and H. G. CraigheadDepartment of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853-3501
�Received 12 September 2002; accepted 21 November 2002; published 22 May 2003�
Intrinsic localized modes �ILMs� have been observed in micromechanical cantilever arrays, andtheir creation, locking, interaction, and relaxation dynamics in the presence of a driver have beenstudied. The micromechanical array is fabricated in a 300 nm thick silicon–nitride ﬁlm on a siliconsubstrate, and consists of up to 248 cantilevers of two alternating lengths. To observe the ILMs inthis experimental system a line-shaped laser beam is focused on the 1D cantilever array, and thereﬂected beam is captured with a fast charge coupled device camera. The array is driven near itshighest frequency mode with a piezoelectric transducer. Numerical simulations of the nonlinearKlein–Gordon lattice have been carried out to assist with the detailed interpretation of theexperimental results. These include pinning and locking of the ILMs when the driver is on,collisions between ILMs, low frequency excitation modes of the locked ILMs and their relaxationbehavior after the driver is turned off. © 2003 American Institute of Physics.�DOI: 10.1063/1.1540771�
An advance of the theory of nonlinear excitations in dis-crete lattices was the discovery that some localized vibra-tions in perfectly periodic but nonintegrable latticescould be stabilized by lattice discreteness. The modula-tional instability of extended large amplitude vibrationalmodes has been proposed as a mechanism for the realiza-tion of dynamical localization on the scale of the latticeconstant. Although theoretically a variety of methods toexcite the instability of a homogeneous vibrational modehave been proposed, these ideas have yet to be testedexperimentally. Since the observation of nanoscale local-ized vibrational modes still cannot be achieved there isdeﬁnite advantage to examining a macroscopic array,which is small enough so that the entire time dependenceof the instability dynamics occurs in a practical measure-ment interval. This has been accomplished by using mi-cromechanical silicon technology to fabricate up to 248identical cantilevers with a 40 micron lattice constant.Optical techniques have been used to track the motion ofindividual cantilevers in the presence of an inertialdriver. In addition to experimentally characterizing themodulational instability and identifying the best methodfor producing intrinsic localized modes a new discovery isthe locking of the local mode amplitude with thedriver frequency. Numerical simulations have been usedto better understand the nature of this synchronizationeffect.
I. INTRODUCTION
The concept of nonlinear energy localization in periodiclattices characterizes a new class of dynamical excitations,namely, intrinsic localized modes �ILMs�.1 In addition to the-oretical and numerical studies involving nonlinear crystaldynamics,2–7 applications to other topics have appeared suchas magnetic systems,8–17 electron–phonon systems,18,19 reac-tion dynamics,20 molecular biophysics,21,22 and lattice-assisted energy/charge transfer in polarizable matter.23 Someof these efforts devoted to examining the nonlinear dynamicsof nanoscale lattices have made contact with other possibleapplications for ILMs such as in friction24 and crackpropagation.25 Still other larger scale applications deal withILMs in Josephson arrays,26,27 E&M ILMs in opticalswitches,28andinnonlinearphotoniccrystalwaveguides.29,30 The largest scale application has to do with lo-calized multibunch modes in accelerators.31 Thus from con-densed matter physics to arrays used in high technology, onesees a new class of problems emerging, which share a com-mon denominator.At the smallest scale, details of ILM quantization18,32,33
are still to be explored. For a classical nonlinear oscillatorarray, there are a number of characteristic ILM properties,probed theoretically, such as their interaction with an acdriver,14,34–36 their propagation5,37–40 and amplitude depen-dent mobility4,6,40–42 in a discrete lattice potential,43,44 aswell as their interactions with impurities,45–50 that still need
CHAOSVOLUME 13, NUMBER 2JUNE 2003
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to be examined experimentally. Note that strongly excitedILMs42 can be trapped anywhere in the lattice, so they alsocould approach impurity mode behavior. Thus the explora-tion of the amplitude-dependent properties of ILMs is one ofthe important experimental issues at the present time.Although some experimental studies have been reportedfor large scale mechanical systems,51,52 for somewhatsmaller Josephson-junction arrays,26,27,53,54 and for nanoscalelattices,15–17,55 none of these studies have examined thedriven amplitude dependent trapping phenomena. The lowresonant frequency of the mechanical systems makes thetime scale too long to perform the necessary experiments.ILMs can be seeded in Josephson-junction arrays and theyare stable so that trapped ILMs can be measured; however,the high oscillation frequency, 1 GHz to 10 THz,56 makes itdifﬁcult to observe their dynamical motion. Finally, observ-ing the dynamics in nanoscale lattices has not yet been ac-complished.Recentlymicro-electro-mechanicalsystem�MEMS� silicon technology has matured sufﬁciently so thatnow it is relatively straight forward to make many identicalelements.57–59 While previous experimental studies of indi-vidual micromechanical oscillators have focused on issues ofnonlinearity and speciﬁc applications,60–63 in this paper wedescribe our experimental investigation of ILM creation, re-laxation, locking, and interaction in 1D coupled oscillatorarrays of the cantilever design.In the next section the experimental and numerical-simulation procedures are described. An optical arrangementwith a 1D charge coupled device �CCD� is used to displaythe motion of localized modes. Numerical simulations havebeen made using coupled nonlinear Klein–Gordon equationsto represent the oscillator array. In Sec. III, the experimentalresults are presented and compared with the simulations. Thebreakup of the uniform mode excitation, the development oflocalized modes, and stationary localized modes locked tothe driver frequency have been observed in this time depen-dent investigation. Simulations play an important role as theyare used to interpret and understand these experimental ob-servations. The conclusions are presented in Sec. IV.
II. EXPERIMENTAL DETAILS
A. Physical
1. Fabrication and linear measurements tocharacterize cantilever coupling
To fabricate SiNx cantilever arrays on a silicon substrate,the starting ﬁlm is a low stress silicon nitride layer, siliconrich to alleviate tensile stresses. After coating with a photo-resist mask it is exposed and then etched via a CF4 plasma ina reactive ion chamber. Next, the silicon substrate is undercutusing an anisotropic KOH etch, thus releasing the SiNx can-tilevers. A top view of the resulting structure is shown in Fig.1�a�. The relative scale of the cantilevers and overhang canbe seen. The overhang region provides the coupling betweenthe cantilevers. A 3D rendition of one unit cell of the result-ing array is shown in Fig. 1�b�. The physical characteristicsof the four oscillator arrays studied here are given in Table I.To examine the coupling properties of the overhang be-tween the cantilevers, some initial tests have been carried out
with small arrays composed of nine identical cantilevers.These are samples A and B in Table I. A piezoelectric trans-ducer �PZT� with variable frequency is used to drive themonocantilever array to obtain the frequency dependent re-sponse of individual cantilevers.The optical apparatus used to measure the dependence ofthe spectral bandwidth on the overhang is shown in Fig. 2.The sample is attached to the PZT and situated in an evacu-ated chamber maintained below 1 mTorr at ambient tempera-ture. A cantilever is chosen, and a laser beam focused on it,as can be seen in the ﬁgure. A variable frequency ac voltagedrives the PZT which uniformly shakes the entire sample sothat the cantilever array experiences a common acceleration.The resulting deﬂection of the laser beam from the oscillat-ing cantilever is measured by a position sensitive photodiodedetector �PSD�, with output voltage proportional to the opti-cal spot position. For a linear response of the array the PZTac voltage is typically set to 0.05 V; for examining the non-linear shift of a resonance frequency, and hence the anhar-monicity of the system, the driving voltage is �0.1 V. ForILM creation it is �10 V.Figures 3�a� and 3�b� show the signal versus driving fre-quency for test samples A and B, respectively. These twosamples, which are identical except for different overhangwidths, produce different spectral bandwidths as expected.Figures 3�a� and 3�b� clearly show that the bandwidth in-creases with the length of the overhang. Ignoring for themoment any normal modes of the overhang itself, the canti-levers should produce nine degrees of vibrational freedom.The nine peaks observed are the resonant frequencies ofthese nine normal modes. The solid and dotted curves denotethe spectra taken of the center and edge cantilever, respec-tively, so that all modes are counted.With the coupling between oscillators characterized, thedesign of the large oscillator arrays can now be considered. It
FIG. 1. Characteristic dimensions of a di-element type silicon nitride canti-lever array. �a� Top view photograph. White region to the right is the 300 nmthick SiNx array and overhang, while the dark gray region on the left is theﬁlm supported by the substrate. The parameters for this sample �D� aregiven in Table I. �b� 3D sketch showing one unit cell.
703Chaos, Vol. 13, No. 2, 2003Study of ILMs in MEMS cantilever arrays
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is known that cantilevers of this design have a hardnonlinearity.61,62 The procedure of choice for the productionof ILMs is to drive a uniform mode to large amplitude so thatit becomes unstable and breaks up into localized excitations.To achieve the large amplitude uniform mode instability foran array with hard anharmonicity, the highest frequency uni-form mode of the array needs to be driven.64
For a mono-type cantilever 1D lattice this would be thezone-boundary excitation. �Although our arrays have ﬁxedboundary conditions they are sufﬁciently large that terminol-ogy consistent with periodic boundary conditions will beused throughout the paper.� To excite the zone boundarymode shown in Fig. 4�a� would require a special driver, onethat could produce � out-of-phase amplitude on neighboringcantilevers. In order to use the PZT driver, which acceleratesthe entire lattice uniformly, two different-length cantileversper unit cell have been constructed in arrays C and D, asdisplayed in Fig. 1. With this di-element array, the dispersioncurve is folded over �see Fig. 4�b�� so that the highest fre-quency vibrational mode is now at the zone center. Thisoptic-like mode can be excited with the PZT driven at theappropriate frequency. Any ILMs should appear near the topof the upper band as shown in the ﬁgure.
2. Linear measurements of large di-element cantileverarrays
The frequency dependent linear response of individualcantilevers for a large di-element array �sample D� is shownin Fig. 5. The laser beam, shown in Fig. 2, is now focused ona cantilever, near the center of the sample. The drive fre-quency is incremented in 100 Hz steps in the lower fre-quency region and 50 Hz steps in the higher frequency re-gion. For each frequency point, a measurement takes aboutthree seconds. Optic-like normal modes can be activated bythe uniform driver, and can be seen in this ﬁgure. As ex-pected, there are two pass bands, with the upper band�143.2–147.0 kHz� much narrower than the lower one�60.8–127.7 kHz�. Due to the coupling of the driver to thenormal modes the frequency dependence of the response de-creases with increasing frequency in the lower band, andincreases with increasing frequency in the upper band. Thereis at least one other band at around 235–240 kHz, whichmay be related to the vibration of the overhang. However,
FIG. 3. Linear spectra for mono-element arrays showing the dependence ofthe bandwidth on the overhang. �a� Sample A with 28.5 �m overhang, and�b� sample B with 35.6 �m overhang. Samples A and B contain nine canti-levers; see Table I for more details. Solid and dotted curves correspond tothe spectrum for the center and for the edge cantilever, respectively.
TABLE I. Summary of the characteristics of the micromechanical samples.
Sample labelABCD
TypeMono-elementMono-elementDi-elementDi-elementThickness �nm�300300300300Pitch ��m�1101106540Length ��m�505050/5550/55Width ��m�10101515Total number99152248Overhang ��m�28.535.67067f a �kHz�a180.43186.2136.1147.0f b �kHz�a133.0143.2f c �kHz�a121.2127.7f d �kHz�a171.43158.472.760.8
aResonant frequencies at the upper zone center, upper zone boundary, lower zone center, and lower zoneboundary, as shown in Fig. 7. For the mono-element type arrays, only two frequencies are speciﬁed whichcorrespond to the same modes in folded dispersion curve as if the samples were of the di-element type.
FIG. 2. Experimental setup for a one-cantilever linear response measure-ment. The cantilever array is in a vacuum chamber. A PZT is employed todrive the sample. A beam from a He–Ne laser is focused on one cantilever.A position-sensitive photodiode detector �PSD� which outputs voltage pro-portional to the laser beam position is used to pick up the deﬂection of thereﬂected laser beam. By scanning the frequency of the oscillator and mea-suring the ac voltage with the lock-in ampliﬁer, linear spectra are obtained.
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this region is beyond the frequency limit of the photosensitvedetector and the lock-in ampliﬁer, and it has not been mea-sured accurately.To examine the response of the system to large ampli-tude excitation and in particular to explore ILM dynamics, itis necessary to measure a large number of cantilevers simul-taneously. Figure 6 shows the experimental setup for mea-suring ILM dynamics versus time. In Fig. 6�a� the He–Nelaser beam is focused with a cylindrical lens into a line alongthe static array. The reﬂected beam is then imaged onto a 1DCCD camera. The PZT is driven with a voltage-controlledoscillator for variable or constant frequency operation asshown in the ﬁgure. The switch and the ramp generator aresynchronized with the camera by using a pulse generator.Images are captured by a computer, which is triggered by thepulse generator. The speed of the camera, about 18 kHz, isinsufﬁcient to monitor the sinusoidal motion of the cantile-vers. However, as the vibration of a cantilever grows, thereﬂected laser beam increasingly misses the 1D CCD cameraas illustrated in Fig. 6�b�, and the image of that particularcantilever becomes darker.The maximum lateral capturing speed of the camera canbe deﬁned as the array pitch divided by the camera period,i.e., 18�103 a/s, where a is the cantilever pitch. This issufﬁcient to observe the fastest traveling ILMs, which moveat speeds of up to 14�103 a/s. It should be emphasized thatILMs excited near the top of the band are far slower than thismaximum speed. Consequently, the capturing speed of thecamera is fast enough to observe both the lateral motion ofan ILM and its time development. This relatively straightfor-ward observational method permits a systematic monitoring
of the excitation pattern for large amplitude ILM creationand interaction with a driver.It has been shown theoretically that changing the fre-quency �chirping� of a large amplitude driver is an effectivemethod by which to excite a nonlinear system to large am-plitude so that the resulting instability produces ILMs.35,65
Since the resonant frequency of the nonlinear cantilevers in-
FIG. 4. Schematic dispersion curves. �a� Mono-element cantilever array: anILM can be expected to appear at the zone boundary frequency for hardnonlinearity. �b� Di-element cantilever array: the dispersion curve for themono-element array �thin dotted line� is folded back and a stop band ap-pears. For hard anharmonicity ILMs will be created at the zone center.
FIG. 5. Linear spectrum obtained for a large di-element cantilever array.The laser was focused on a cantilever near the middle of sample D thatcontains 248 cantilevers. Frequency is step incremented in 100 Hz stepsover the lower frequency region and in 50 Hz steps in the higher frequencyregion. The lower band begins at 60.8 kHz and ends at 127.7 kHz. Theupper band begins at 143.2 kHz and ends at 147.0 kHz.
FIG. 6. �a� Experimental setup for the ILM measurements. A beam from aHe–Ne laser is focused along the array by using a cylindrical lens. Theelliptical spot from the cylindrical lens is positioned so that the short axis ofthe ellipse lies along the length of the cantilever, while the longer axisencompasses many cantilevers. A 1D CCD is used to detect the beam re-ﬂected from the cantilevers. A voltage controlled oscillator �vco�, switch�sw� and ampliﬁer are used to drive the PZT. The frequency of the vco iscontrolled by a ramp generator. A pulse generator, which controls the switch,and the ramp generator are synchronized to the camera. A scanner attachedto the mirror is driven by an oscillator �not shown in this ﬁgure�. Two-dimensional images can be captured by synchronized scanning of the mirrorand the camera. This feature is used to observe the amplitude pattern ofstationary ILMs. �b� Schematic showing the relation between cantilever am-plitude and detector response. At large vibration amplitudes the deﬂection ofthe laser beam from the cantilever changes and the image at the CCD sitebecomes darker. In this picture the elements of the 1D CCD camera arearranged horizontally.
705Chaos, Vol. 13, No. 2, 2003Study of ILMs in MEMS cantilever arrays
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creases with amplitude, it is necessary to increase the driverfrequency with time to track the highest resonant frequency.In addition, since the oscillators are necessarily damped, cwdriving is required for steady state observation of the result-ant ILMs.A scanner attached behind the mirror in front of the CCDas shown in Fig. 6�a� can be used to obtain a two-dimensional image of the cantilevers. This oscillator drivenscanner is synchronized with the camera. During the timedevelopment measurement of ILMs, which is the main focusof most experiments, this mirror is mechanically ﬁxed toobserve individual cantilever motion.
B. Numerical
The lowest frequency � of a simple cantilever with oneend ﬁxed is given by66
�2��3.52�2
L4 �EI�tw� ��3.52�2
12L3Em t3w�km .�1�
Here E is the Young’s modulus, the second moment-of-areaI�t3w/12, where t is the thickness of the beam, and w is itswidth �see Table I for these values�, L is the length of thebeam, and � is the density of the material.Our experimental system consists of an array of cantile-ver beams that are coupled together by the overhang regionbetween them. Since the two cantilevers in a unit cell havedifferent lengths, then, according to Eq. �1�, both their linearspring constants and masses will be different. As a ﬁrst ap-proximation, the dynamics of the overhang will be ignored,so that it acts only as a massless coupler between cantilevers.For small oscillations an individual cantilever in the array isassumed to obey Eq. �1� but with an effective spring andmass, in order to include some contribution of the overhang.For the large amplitude problem, each cantilever is then rep-resented by a mass and an onsite potential which has bothharmonic �quadratic� and anharmonic �hard quartic� terms.The coupling between cantilevers is assumed to be harmonicfor all cantilever amplitudes. Such a di-element lattice modelobeys the nonlinear Klein–Gordon equation,67 namely,
dt2 xai�ma�ddt xai�k2axai�k4axai3
�kI�2xai�xbi�xbi�1��ma�,�2�
dt2 xbi�mb�ddt xbi�k2bxbi�k4bxbi3
�kI�2xbi�xai�1�xai��mb�,�3�
where the subscripts a and b correspond to the differentlength cantilevers, xai and xbi are the displacements of thecantilever ends, ma and mb are their masses, � is a linear�energy� lifetime, k2a and k2b are the onsite harmonic springconstants, k4a and k4b are onsite quartic spring constants,and kI is the harmonic coupling constant. The PZT does notapply a force directly to the cantilevers, but causes accelera-tion and deceleration of the cantilever’s inertial frame withmagnitude �.
The masses are estimated from the density of siliconnitride and the size of the cantilevers. The lifetime � is ex-perimentally determined. In order to determine the springconstants k2a , k2b , and kI experimentally, four frequencieshave been measured: the maximum and minimum frequen-cies for the upper and for the lower bands. These points on adispersion curve are identiﬁed in Table I.By choosing the top three experimental frequencies forthe ﬁt to the small oscillation case, a good approximation tothe upper branch of the dispersion curve can be obtained.The lowest experimental frequency �lower branch, zone cen-ter� is not reproduced within our model. One way to obtaindifferent bandwidths is to introduce longer-range interac-tions, which would stem from the dynamical properties ofthe overhang. To reproduce the entire experimental disper-sion curve takes up to sixth nearest-neighbor interactions.Since ILMs are created at the top of the upper branch, theexact shape of the lower branch is not expected to play acrucial role so this long-range interaction is not included inour simulations.With the harmonic spring constants of the model chosen,its linear response can be determined and the dispersioncurve obtained. The system is driven with a uniform distri-bution of acceleration noise, with � ranging between�10�3 m/s2 �compared to the � �10�4 m/s2 that will beneeded for ILM production�. The beam displacements of theentire array are recorded as a function of time. A Fouriertransform �FT� of a single oscillator displacement as a func-tion of time yields the frequencies that the particular cantile-ver experiences during the given time interval. Likewise, aFT �taken over a ﬁnite spatial interval� of the displacementsof all the cantilevers at any instant of time will yield theexcited wave vector modes that are participating in the col-lective motion. Performing FTs in both space and timechanges the real-space data (x,t) representation to reciprocalspace (�,k). Such a two-dimensional FT of the beam dis-placement data is shown in Fig. 7�a�. The time interval isequivalent to 1000 periods of oscillation. A one-dimensionalFT for the oscillations of a single cantilever from the middleof the sample is shown in Fig. 7�b�. In both of these framesit is clear that the simulated bandwidth is almost the same forthe acoustic and optic branches.The nonlinear onsite quartic spring constant in Eqs. �2�and �3� is determined in the following manner: �1� k4a andk4b are set equal to each other and �2� the value is deter-mined so that the upper-branch frequency in a simulationshows the same frequency shift as for experiment at a powerlevel below the threshold for the uniform mode instability.This power still needs to be sufﬁciently high so that thenonlinear frequency shift of the resonant frequency can bedetermined accurately. A complete list of simulation param-eter values can be found in Table II.
III. RESULTS AND DISCUSSION
A. Trapping and locking
1. Experiment
Two sequential time-dependent response measurementsof the 152-cantilever array �sample C� for a large amplitude
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driver are shown in Fig. 8. The cantilever positions can beidentiﬁed by the white lines to the left of the time�0 marker.The high-power, PZT driver (�20 V) is chirped from the topof the optic band, f a , to 1.011 f a between time�0 and 14.2ms as indicated by the dotted vertical line in the ﬁgure. TheCCD camera images a portion of the sample, which is 100cantilevers in this case. The dark tracks versus time identifylarge amplitude localized excitations. Although the startingconditions are the same for the experiments shown in Figs.8�a� and 8�b� different results are observed as would be ex-pected for a process initiated by random noise. Note thatsome moving ILMs form during the chirp phase. At longertimes between 14.2 ms and 72.7 ms, with a cw driver someILMs continue to receive energy while the others die out. Atsome point the large amplitude ILMs become trapped at alattice site. This can be seen around lattice site 60 in Fig. 8�a�and at two locations, namely, 38 and 80 in Fig. 8�b�. Afterthe driver is turned off at time�72.7 ms the trapped ILMsdecay.
Figure 9 shows cantilever responses versus time with thehigh-power, chirped driver for the 248-element array, sampleD in Table I. Before the pulse is turned on (time�0), eachstripe corresponds to a stationary cantilever. The PZT voltageis again about 20 V. The viewing size is now 230 cantilevers.Figures 9�a� and 9�b� show different results for identicalstarting conditions. Here, the driver frequency is chirpedfrom 0.9986 f a to 1.016f a . Figure 9�c� shows the results fora slightly different starting condition, where the driver fre-quency is chirped from 0.9986 f a to 1.034 f a . As long as thechirping starts near the top of the band and ends 2–3 %higher than f a , the different chirping schemes give similarresults. The pulse duration is 48.9 ms and the chirp time is16.2 ms �dotted line�. The pulse interval is highlighted inthese ﬁgures. At early times ILMs are observed to form,move, oscillate and hop until the chirp ends; then a fewILMs continue to pick up energy and become trapped atlattice sites. Using an optical probe coupled to a spectrumanalyzer to monitor the center of such a pinned ILM con-ﬁrms that the ﬁxed mode is frequency-locked to the driver.After the pulse, these stationary ILMs lose energy, becomebroader, break free from the trapping site, and sometimesoscillate about it. Note that these oscillations range over10–20 lattice sites and hence they have a much larger am-
FIG. 7. Simulated linear response for a di-element cantilever array. Thephysical parameters of the di-element array are given in Table II. Applyinglow noise amounting to ��0�10�3 m/s2, the displacements are measuredas a function of time. �a� The time-space Fourier transform of the shortercantilever displacements gives the excitation pattern in k space only wherenormal modes exist. The three experimental frequencies, 147.0, 143.2, and127.7 kHz at the top and bottom of the upper band, and top of the lowerband of sample D are reproduced. The bottom of the lower band frequency�123.3 kHz� in simulation is much higher than the experimental one �60.8kHz�. �b� The time Fourier spectrum of a single cantilever displacementpattern.
TABLE II. List of parameters used in the simulations.
ParametersCantilever aCantilever b
Mass5.46�10�13 kg4.96�10�13 kgDamping constant, �8.75 ms8.75 msHarmonic, k20.303 kg/s20.353 kg/s2
Anharmonic, k45.0�108 kg/s2 m25.0�108 kg/s2 m2
Harmonic intersite, kI0.0241 kg/s20.0241 kg/s2
FIG. 8. Cantilever excitation versus time showing the production, interac-tion and decay of ILMs. These experimental results for sample C are takenwith the 1D CCD camera. Frames �a� and �b� have identical starting condi-tions. The PZT frequency is chirped from f a�136.3 kHz to 1.011 f a . Thechirp ends at 14.2 ms �dotted line�. The dark regions identify localizedexcitations. Highlighted region corresponds to the time where the pulse ison. Some localized excitations become trapped during this cw phase. Pulseduration is 72.7 ms.
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plitude than the one unit cell oscillations described in Refs.38 and 43. The moving ILMs have speeds ranging from 2–6lattice sites per millisecond. This should be compared to themaximum possible speed of an optic branch linear modewave packet of about 13 lattice sites per millisecond. Theoval identiﬁers in Figs. 9�a� and 9�c� indicate regions wherefairly uniform background excitations can be seen. Such pat-terns exist even after the pulse. The amplitude of such exci-tations in Fig. 9�c� is less than in frames �a� and �b� andprobably accounts for the absence of any motion of thetrapped ILM after the pulse is turned off. A brief descriptionof some of these results has already been given.68
Close inspection of the large-amplitude trapped ILMs inFig. 9 reveals an experimental artifact in the central region ofthe ILM image. The experiments indicate a smaller ampli-tude at the center of the mode than actually exists there. Asystematic investigation shows this feature is false. By takinga 2D image of two trapped ILMs, which are shown in Fig.10, it is possible to identify the single peaked nature of theexcitation. The two arrows in the ﬁgure identify the center ofthe two localized excitations. In each case the large ampli-tude peak is centered on the short cantilever.
2. Simulations
To understand in more detail a number of the featuresobserved in Figs. 8 and 9 it is necessary to turn to numericalsimulations, which have parameters very similar to those inthe experiment. Figure 11 summarizes some of the results ofsuch an investigation. The chirping scheme used in thissimulation is shown in Fig. 11�a�. The system is initializedwith a small amount of random noise. The system is thenallowed to settle for 500 periods of the resonant frequency, atwhich point the driver is turned on (time�0). The driverfrequency starts at the top of the upper band, then continu-ously increases linearly to frequency 1.027 f a over a timeinterval of 2500 periods. When the uniform mode becomeshighly excited the noise perturbation triggers the modula-tional instability. In the cw mode the ﬁxed driver frequencyis 1.027 f a until the driver is turned off at 7500 periods.Figure 11�b� shows a density-plot of the energy of eachlattice site as a function of time for a typical simulation re-sult. The energy of a particular lattice site in the di-elementarray, at a given instant of time, is
Eai�12 ma�ddt xai�
2�12 k2axai2 �14 k4axai4
�14 kI�xai�xbi�2�14 kI�xai�xbi�1�2,
�4�Ebi�12 mb�ddt xbi�
2�12 k2bxbi2 �14 k4bxbi4
�14 kI�xbi�xai�1�2�14 kI�xbi�xai�2,
where i indexes the unit cell number, and the actual latticesite number is given by j�2i�1 and j�2i for the long�cantilever a) and short �cantilever b) sites, respectively. Theenergy consists of the kinetic energy and onsite potentialterms, as well as the potential energy stored in the couplingsprings. The parameter set used here for sample D is given inTable II. The total number of cantilevers is 250 with ﬁxed-
FIG. 9. Cantilever excitation versus time showing the production and decayof ILMs. Experimental results for sample D taken with the 1D CCD camera.Frames �a� and �b� have identical starting conditions. The frequency of thePZT is chirped from 0.9986 f a to 1.016 f a . For frame �c� the chirp extendsfrom 0.9986 f a to 1.034 f a . Chirp ends at 16.2 ms �dotted line�. Dark pat-terns identify localized excitations. The pulse is on over the highlighted timeinterval and its duration is 48.9 ms. Ovals indicate some of the many regionswhere fairly uniform background excitations can be seen. The amplitude ofthis normal mode noise is larger in �a� and �b� than it is in �c�. After Ref. 68.
FIG. 10. Two-dimensional image of the amplitude pattern for sample Dshowing two trapped ILMs. Note that each ILM is centered on a shortcantilever site. The two stationary ILM patterns are obtained by scanning themirror in front of the CCD shown in Fig. 6. A scanner, synchronized withthe CCD, is used to vibrate the mirror. The cantilevers and the overhangregion are illuminated by the laser and the small amplitude ones appearbright in this image. Strongly vibrating cantilevers appear darker. The shapeof each pinned ILM is single peaked.
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end boundary conditions. As expected there is no qualitativedifference between simulations with the periodic- and ﬁxed-end boundary conditions except near the boundaries.During the chirping phase of the driver many movingILMs appear which then coalesce into several trapped ILMs.Most of the ILMs appear to die out after only a few hundredperiods of oscillation. In this particular simulation, fourmodes can be identiﬁed that persist much longer than theothers. These are identiﬁed in Fig. 11�b� as LS1, LS2, LS3,and NL1. The ﬁrst three of these are trapped at particularlattice sites and in addition, have their frequency locked tothat of the driver. Because of this synchronization with thedriver there is a continual transfer of energy to the ILMs,which keeps them pinned at the particular lattice sites andprevents them from decaying. The fourth large-amplitudeILM, NL1, fails to frequency lock to the driver in the cwregion, and hence ultimately decays after a time lapse of�2600 periods. Once the driver has been turned off, thethree locked states decay. Mode LS3 is seen to decay peace-fully, maintaining its symmetry without further motion in thelattice. In contrast, LS1 and LS2 are close enough togetherso that once the decay begins and they become unpinned,they repel one another, and move in the lattice.
Figure 11�c� shows the average energy for a particularlocked ILM �LS3� as a function of time as determined by
ELS3� �j�43
45E j ,�5�
where the E j are deﬁned by Eq. �4�. The low frequencyoscillationoftheaveragemodeenergyatfrequency0.0056 f a decays slowly over a long time interval. It requiresthe presence of the driver and disappears immediately afterthe driver is turned off. The ﬁgure shows that during thechirp, the energy of this mode increases with time to anamplitude value compatible with the ﬁxed synchronizationfrequency of the driver. Thus, the amplitude of such a lockedILM is rigidly connected to the driver frequency.To explore the development of locked ILMs from an-other view, the double Fourier transform of the cantilevermotion over a speciﬁc time interval is taken. The resultingtime development of excitations both in real space and in kspace is shown in Fig. 12. The energy density as a functionof time for this particular simulation is presented in Fig.12�a�. The same parameters are used as those for Fig. 11;only the initial noise is different. The frequency chirp (f a→1.027 f a) lasts for 2500 periods �left solid vertical line in�a��. At long times only one locked ILM appears. An exami-nation of the energy versus time for the center cantilever andtwo neighboring sites shows a monochromatic strength os-cillation frequency at 0.0056 f a the same as was describedabove for LS3. The pulse is turned off at time�7500 periods�right solid vertical line in �a��. By time–space Fourier trans-forming the displacement over the different time intervalsidentiﬁed by the dotted lines: b, c, d, e, f, and g, in Fig.12�a�, the (�,k) excitation plots shown in frames �b�–�g� areobtained. The dotted curves in these six frames identify thelinear dispersion curve of the optic branch for the di-elementcantilever array. Figures 12�b� and 12�c� show time cuts dur-ing the chirp. The position of the nonlinear dispersion curveand the interference between different ILMs can be seen inFig. 12�c�. The early and late stages of the single locked stateappear in Figs. 12�d� and 12�e�. Figures 12�f� and 12�g� iden-tify the decaying state. A brief description of these ﬁndingshas been presented earlier.68
The time evolution of Fig. 12 shows that initially theregion above the linear dispersion curve is almost uniformlyexcited �Fig. 12�b�� in k space, while in real space there aremany different levels of excitations and localized excitationspeeds. By the middle of the chirp, several horizontal-linetracks, characteristic of stationary ILMs, emerge in thek-space representation �Fig. 12�c��. In the initial stage afterthe chirp one strong horizontal excitation is accompanied byﬂat, relatively weak side bands as shown in Fig. 12�d�. Theside bands are caused by the strength modulation of thelocked ILM. Both the background and these side bands to thelocked mode are nearly gone by Fig. 12�e�. The blue line atthe 151 kHz, k�0 point, in Fig. 12�e�, is a real feature whichoccurs at the driver frequency. It comes about because thecenter of the locked ILM is in phase with the driver, whilethe rest of the lattice is out of phase. In k space, the lockedILM is represented by a near uniform distribution while the
FIG. 11. Computer simulations of ILM production, dynamics, and decay.Cantilever parameters are listed in Table II. �a� Driver frequency as a func-tion of time for this simulation. Pulse is on from 0 to 7500 periods. Driverfrequency starts at f a , increases up to 1.027 f a linearly until 2500 periods,and then, remains at that frequency. The magnitude of the acceleration iskept constant, ��1.0�104 m/s2. �b� A density plot of the energy versustime. Dark regions identify localized modes. During the time that the cwdriver is on three locked stationary ILMs �LS1, LS2, LS3� are found. NL1identiﬁes an ILM, which is not locked to the driver. Hopping motion ofILMs can be seen in the oval markers. �c� Energy as a function of time forthe ILM labeled LS3 averaged over three lattice sites. The characteristicoscillation frequency of the slowly decaying feature is 0.0056 f a with a Q�13. Horizontal arrow shows a linear energy lifetime �8.75 ms� used in thissimulation.
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out of phase component of the lattice is represented by a neardelta function. These two contributions cancel at k�0 pro-ducing the blue spot. After the pulse is turned off, the ILMloses energy, and its frequency decreases as shown in Figs.12�f� and 12�g�.
3. Discussion
Due to the lack of dynamic range and linearity of theamplitude measurement only limited experimental informa-tion about ILM sizes has been obtained. Even though thesizes of ILMs shown in Figs. 8 and 9 appear broader than inthe simulation shown in Fig. 11, this may be a feature of theexperiment, since the measured darkness is not proportionalto the amplitude of the cantilever due to the nonlinearity inthe optical response of the cantilever motion and the electri-cal response of the CCD.
The general picture from observations and simulations isthat a number of moving localized modes are created ini-tially, but that there are only very few modes, which grow inamplitude and survive until the end of the chirp and evenfewer survive until the end of the cw pulse. From our simu-lations, these ILMs are frequency locked to the driver. In thisway sufﬁcient energy can be transferred to maintain theirstrongly localized state and ﬁxed vibrational amplitude.Other ILMs that fail to lock their frequency to the driver areseen to decay. In theoretical work Ro¨ssler and Page34 havereported a locked in-phase driven single-peaked ILM without-of-phase background for a realistic anharmonic potential.Their description is somewhat similar to that observed herein Figs. 12�d� and 12�e�.Simulations demonstrate that the mobility of localizedmodes decreases with increasing amplitude,37 and if the am-plitude becomes large enough, the localized mode is trappedat a site.42 Although the nonlinear KG lattice does not sup-port a Peierls–Nabarro potential43,44 due to the presence ofinternal degrees of freedom,40,69 a pinning potential of somesort still appears to be a valid concept.The two trapped ILMs shown in Fig. 10 have maximumamplitude at the short cantilever site, as do the locked ILMsin the simulations shown in Fig. 11�b�. Thus, the single-peaked mode is more stable than the double peaked mode, inagreement with the early studies of stationary and movingILMs in antiferromagnets14 which have both anharmonic in-trasite and intersite potentials.
B. Interactions
1. Experiment
Various types of interactions have been identiﬁed inwhich a locked/pinned ILM participates. Figure 13 addressesthe interaction of a pinned ILM with traveling ILMs. Thedata, a magniﬁed image of a section of Fig. 9�a�, is presentedin Fig. 13�b�, and a schematic representation of the timeevolution identifying important features is given in Fig.13�a�. After the pulse is turned on at time�0, the initialexcitation stage followed by modulational breakup into sev-eral small amplitude localized excitations is marked as re-gion �A� in Fig. 13�a�. Compare with the data in Fig. 13�b�.At longer time, three main excitation sequences appear asoutlined in Fig. 13�a�. The chirp interval extends to 16.2 ms.Within the center sequence �region B�, hopping of an ILMacross the lattice can be observed. The random hopping mo-tion of an otherwise pinned ILM is a consequence of itsinteraction with traveling ILMs and/or background excita-tions associated with the other normal modes of the array.Markers C-1 and C-2 identify collisions between traveling-ILMs and/or other excitations and a strongly pinned ILM. Ineach case the traveling excitation appears to be reﬂectedfrom the stationary one.In addition to pinned ILMs, an underlying normal modeexcitation pattern is another characteristic feature. The ovalsshown in Fig. 9 mark three of the many regions where ﬁnegrained, small amplitude background excitation patterns canbe seen, even after the end of the pulse. These excitations are
FIG. 12. �Color� Simulated formation, locking and decay of an ILM in thek space. �a� Energy density as a function of time. All parameters used are thesame as for Fig. 11, except for the initial noise. The pulse is turned on attime�0 and off at 7500 periods. Chirping starts from the top of the bandf a , and ends at 1.027 f a at time�2500 periods. The solid lines show whenthe chirp ends and when the pulse is turned off. Only one ILM is trappedduring the cw part of the pulse. A low frequency oscillation of 0.0056 f a isobserved at the center ILM site, similar to that shown in Fig. 11�c�. TheQ�15. �b�–�g�: Time–space Fourier transform of the displacement in sev-eral time windows. The time windows for these frames are indicated by theletters b–g above frame �a�. The time development during the initialchirped excitation are shown in frames �b� and �c� locked state in frames �d�and �e�, and the decaying ILM state in frames �f� and �g�. Dotted curves inframes �b�–�g� identify the linear dispersion curve for the upper band. Notethe side bands in frame �d� and the blue line at k�0 in frame �e�. After Ref.68.
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maintained by the strong driver via nonlinear effects. Theexact mechanism of this normal mode excitation is still notcompletely under experimental control.The fact that an identical experimental starting conditionin Figs. 9�a� and 9�b� give different end results demonstratesthe importance of the underlying excitation spectrum of thearray. Even when some level of impurities is present �un-avoidable for a fabricated system�, random ILM creation ispossible if the normal mode excitation effect is larger thanthe impurity effect, which is the case in these experiments.Also the observation of random ﬁnal conditions during thecw driver interval is experimental evidence that the intrinsicpinning effect dominates any underlying impurity effects.Hence the nonrepeatability of the pinning location stronglysupports our claim that the localized excitations in the can-tilever array are intrinsic localized modes, as opposed to im-purity modes.
2. Simulations
Numerical studies show behavior similar to experimentregarding the interaction of a pinned ILM with travelingILMs. Examination of Fig. 11�b� in the vicinity of the pinnedILMs LS2 and LS3 in the time interval 2000–4000 periods,shows the reﬂection of localized wavepackets. The reﬂectionof a moving ILM wave packet from a locked ILM comesabout because of the effective rigidity of the latter. The am-plitude and frequency of a locked ILM are ﬁxed by the fre-quency and strength of the driver. Since the amplitude of the
locked ILM is ﬁxed and there is no remaining freedom, col-liding ILMs are reﬂected, as though from a boundary.Most ILMs created during the initial breakup stage ofthe uniform mode do not lock to the driver, instead theydecompose into small amplitude traveling ILM/wavepacketswhich continually collide with locked ILMs. Near the middleof the chirping stage some locked ILMs are still easilymoved by collisions with such traveling modes, since thepinning effect is still weak. Due to the stability differencebetween the single- and double-peaked modes, ILMs remainlonger as single-peaked modes, and when the pinning effectis weak, an ILM appears to hop among its preferred sites asshown by the oval near LS1 in Fig. 11�b�. Collisions of anunlocked ILM with small amplitude traveling ILM/wavepackets can also be seen in the oval associated with NL1.The hopping process is evident.The repulsive interaction between two strong ILMs canbe seen in Fig. 11�b� to start at around 9000 periods. Afterthe pulse is turned off at 7500 periods, the two ILMs, LS1,and LS2 lose energy, broaden and unpin. Once they are mo-bile the repulsion between them can be observed.
C. Excitation of a pinned ILM
1. Experiment
Since a locked ILM appears to be a stable, somewhatrigid excitation state with a ﬁxed amplitude, there is value inexamining its time dependence at higher resolution. Figure14�a� shows an expanded image of the pinned ILM from Fig.8�a�. Clearly the amplitude near the center of the ILM isoscillating with time. Since the experimental signal for thecentral site, 58, is saturated, only the signal for the perimeterregion can be used in any analysis. Let the total signal fromsite 55 to site 61, excluding 58, be called
FIG. 13. Experimental demonstration of pinning and hopping of ILMs.Magniﬁed section of part of Fig. 9�a�. Enhanced contrast to emphasize onlystrongly excited modes. Even sites are the shorter cantilever sites. Chirpends at 16.2 ms. �b� Experimental data: The initial excitation stage �A� isfollowed by breaking up into several localized excitations, the �B� regionshows the hopping of an ILM, region �C-1� and �C-2� illustrate collision andrepulsive interaction, and region �D� identiﬁes the ﬁnal stationary lockedstate. Note that long-lived strongly excited modes are only at even �short�cantilever sites. The bright center of these ILMs is an experimental artifact.
FIG. 14. Low frequency excitations of a pinned ILM. �a� Magniﬁed imageof Fig. 8�a�. Only cantilever sites shown; the darker the image, the strongerthe signal. There is one stable, pinned ILM centered at site 58. �b� The totalnormalized signal �exp obtained from the signals at site 55 through site 61 asdescribed in the text. �c� Normalized lateral difference signal �exp , as de-scribed in the text. The signal at site 58 is almost saturated and cannot beused in the analysis.
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S� �n�55n�58
61Sn ,�6�
where Sn is the signal at the nth site. With the time averageof Eq. �6� represented by �S�, then the dimensionless quan-tity of interest becomes
�exp�S/�S�.�7�
The lateral difference signal can be deﬁned in a similar man-ner. Let
A� �n�55
57Sn� �n�59
61Sn�8�
�exp�A/�S�.�9�
Figures 14�b� and 14�c� show the measured total ampli-tude, �exp, and the lateral difference, �exp, versus time, re-spectively. An oscillating �exp indicates that there are lowfrequency amplitude modes of the ILM. Figure 14�c� showsan oscillation in �exp, which identiﬁes slow lateral motion ofthe pinned ILM relative to its center. Both of these oscilla-tions represent a slow transient response of the pinned ILMover the entire time that the cw driver is on. The completetime dependence is shown in Fig. 15�a�. The relaxation timeis very long compared to the linear energy lifetime �8.75 ms�of a cantilever.The Fourier transform of the data in Fig. 15�a� gives thefrequencies associated with these small oscillations. The �exp
spectrum, shown in Fig. 15�b� has one broad peak centeredaround 0.011 f a . The �exp spectrum displays several peaks atsomewhat lower frequencies.Another experimental property of these small oscilla-tions is shown in Fig. 16 where the dependence of theirfrequency versus pinning strength can be seen. This ﬁgure isa magniﬁed image of part of Fig. 9�b�. The initial travelinglocalized modes at short times are pinned by the end of thechirp. At longer times there is the ﬁne-grained oscillationpattern that appears on all three stationary localized modes.In some regions these patterns suggest that the center ofgravity of the pinned mode is moving side to side while inother regions it appears that the center of gravity is ﬁxed andthat the excitation takes a variable intensity character. Notethat the period of this oscillation at site 162 �region denotedby the arrow� is larger than for the other two modes shown inthe ﬁgure. Since the mode at 162 is losing energy and be-coming less strongly pinned �see Fig. 9�b� for longer times�,the period of this ﬁne structure pattern increases. It is a sig-nature of the pinning strength.
2. Simulations
The oscillatory behavior found for the locked ILM statein the experiment provides the motivation to compare theresults with simulations. The simulation of the energy versustime for the three central sites of LS3 during the cw regionshown in Fig. 11�c� illustrate that a low frequency mono-chromatic oscillation in the ILM strength is an importantfeature. A similar strong oscillatory feature in the ILMstrength has been observed for the single locked ILM at site74 shown in Fig. 12�a�. Besides this prominent oscillation,which has most of the amplitude on the center site, there areother weaker oscillatory properties of the ILM that can beidentiﬁed.A magniﬁed density plot of the energy for an ILMpinned at site 74 in Fig. 12�a� is presented in Fig. 17�a�. Inthis time window where the cw driver has been on for sometime the energy at sites nearby the center of the locked ILMoscillates with time. These time dependent data are now ana-lyzed using similar notation in the same way as the experi-
FIG. 15. �a� Transient response of the low frequency excitations of thepinned ILM. The complete time interval of the pinned ILM is shown. �b�The Fourier transform spectrum of the low frequency excitations of the �exptype. A broad spectrum centered near 0.011 f a is shown by the arrow. �c�The Fourier transform spectrum of the low frequency excitations of the �exptype. Several peaks at low frequency region around 0.001→0.009 f a indi-cated by the arrow are observed.
FIG. 16. Another experimental example of small amplitude excitation abouta stable ILM state. Magniﬁed section of part of Fig. 9�b�. Three ILMs areseen and the lower two become locked. Initially all three show a ﬁne-grained oscillation but the top ILM at around time�15–22 ms, in the regionof the arrow, loses energy, the period of the small oscillation increases andﬁnally the ILM becomes unpinned.
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ments were analyzed with Eqs. �7� and �9�. The two quanti-ties of interest are �1� the total energy over a restricted regionaround the pinned mode, excluding the central site, deﬁnedas
E� �n�72n�74
76En ,�10�
where En is the energy at nth site. With the time average ofEq. �10� identiﬁed as �E�, the normalized value is
��E/�E�.�11�
The second quantity �2�, the lateral difference energy, is de-ﬁned as
�� �n�72
73En� �n�75
76En ,�12�
to obtain the normalized value
���/�E�.�13�
Equations �11� and �13� now can be used to characterize thesmall scale oscillatory behavior shown in Fig. 17�a�. Figure17�b� gives the oscillations in the total energy, �, the centralsite, while Fig. 17�c� shows oscillations in �, indicating lat-eral motion of the pinned ILM relative to its center.To identify the signiﬁcant spectral regions, the Fouriertransforms of the two curves in Figs. 17�b� and 17�c� aretaken and displayed in Fig. 18. As with the monochromaticlow frequency oscillation of the central peak at 0.0056 f a
discussed earlier, the characteristic frequencies for the twoquantities presented here are very small compared to thehighest frequency normal mode, f a . There are two broadpeaks in the � spectrum, 0→0.015 f a and 0.02→0.04 f a asshown in Fig. 18�a�. The lateral mode spectrum � also hastwo broad peaks, 0.003→0.011 f a and 0.02→0.04 f a asshown in Fig. 18�b�.
3. Discussion
With regard to the stability and rigidity of the lockedILM, several types of deformations from that stable shapehave been observed experimentally. One broad frequency re-gion of response is seen for each of the excitation types,namely, �exp and �exp. Since the experimental signal for thecentral site of such an ILM is saturated, the identities for thedifferent kinds of modes cannot be made as yet. Using thesame kind of analysis in the simulations, two broad fre-quency responses are found for the two kinds of dynamicalsignatures. In addition a single large amplitude, low fre-quency mode is found for the central cantilever of the ILM,which appears to represent oscillations in the ILM strength.Given the basic model used to describe the experimental sys-tem and the absence of experimental data for the centralcantilever, the fact that the small oscillation spectra do notmatch in detail is perhaps not too surprising. Still a key resulthas been obtained: both experiment and simulations showthat low frequency oscillations are a natural feature of lockedILMs. The strength oscillation found in simulations indicatesthat the central cantilever response of the ILM must be mea-sured in experiment in order to develop a more completeunderstanding of these low frequency oscillatory modes andto determine whether or not a more detailed simulationmodel is warranted.
D. Trapped ILM oscillations
1. Experiment
After the ﬁnish of the cw pulse, oscillatory features canstill be detected. The excitation region around site 80 in Fig.8�b� shows interesting behavior as the pinned ILM decays.
FIG. 17. Simulated demonstration of low frequency excitations about thepinned locked ILM state. �a� Magniﬁed section of Fig. 12�a� which showsone stable ILM pinned at site�74. Oscillation of energy between sites 72and 76 can be seen. �b� Normalized total energy � over sites 72–76 whilethe cw mode driver is on to bring out low frequency oscillations, as de-scribed in the text. �c� Normalized lateral difference energy � to bring outother types of low frequency oscillations, as described in the text.
FIG. 18. Fourier transform spectra showing low frequency modes. The cwdriver time interval time�2500–7500 periods in Fig. 12�a� is used. �a� The� spectrum shows response in two different frequency regions, 0→0.015 f a and 0.02 →0.04 f a , illustrated by the arrows. �b� The two broadpeaks in the � spectrum, 0.003 →0.011 f a and 0.02→0.04 f a are due tolateral oscillations.
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Figure 19 can be characterized as an oscillating wavepackettrapped by the pinned ILM. Only pixels corresponding tocantilever sites are shown in this ﬁgure in order to enhancethe excitation pattern. The main feature is the pinned ILM,which has evolved from a previously locked ILM at the endof the pulse. As time passes, the amplitude of the centerregion decreases, and the ILM broadens. Additionally, sev-eral running wavepackets trapped within the center regioncan be identiﬁed.An example of another kind of trapped oscillation abouta pinned ILM is shown in Fig. 20. Figure 20�a� presents themagniﬁed image of a portion of Fig. 9�a�, which shows astationary ILM just after the end of the pulse. The brightcenter in Fig. 20�a� is an optical artifact. The change in the
darkness of the center region with time indicates that theamplitude of the pinned mode is decreasing. Superimposedon this pinned excitation is an oscillating mode. Its period ofoscillation increases with time, as shown schematically inFig. 20�b�.
2. Simulations
Figure 21 shows the evolution of ILM-trapped wavepackets in simulation. This is a magniﬁed picture of a regionof Fig. 12�a�, after the end of the pulse �dotted line�. Initiallya variable intensity mode pattern can be seen. Later, a run-ning wavepacket pattern becomes prominent.Since there is no driver, all the energy for these trappedmodes comes from the locked state before the pulse is turnedoff. Part of this locked mode is converted into the trappedrunning wave packets, still conﬁned to the original spatialregion of the ILM. When the amplitude in the central regionbecomes sufﬁciently small so that the pinning effect weak-ens, and if enough energy is converted to the trapped runningmodes, then the pinned mode can move. Thus, the trappedrunning modes and the lateral oscillation of the perimeter arethe precursors to the ILM launching from the pinned loca-tion.
IV. CONCLUSIONS
Experiments have been carried out in which the creation,interaction, and relaxation of intrinsic localized modes pro-duced in di-element cantilever arrays have been imaged andrecorded. The experimental method permits examination ofthe vibration envelope of a micromechanical oscillator array.By using this experimental setup together with a chirpeddriver, the evolution of the unstable uniform mode into ILMshas been examined in some detail. Initially traveling ILMsare created but ﬁnally locked ILMs are pinned at speciﬁcsites in the array. The ILM hopping motion and subsequentstationary behavior can be explained by an amplitude depen-dent pinning effect plus the collision with traveling ILMs/wave packets which stems both from unlocked ILM excita-tions and normal-mode excitations created by the strongdriver. A repulsive interaction between a stable, locked ILMand a moving unlocked ILM has been observed. Small am-
FIG. 19. Trapped excitations oscillating around a pinned ILM. The data arefrom a section of Fig. 8�a�. Only cantilever sites are shown and the darkerthe image, the stronger the signal. The pulse is turned off at time�72 ms.Decay of the pinned ILM is shown. From time�92 ms, traverse patternsdue to running wave packets trapped in the ILM envelope appear as obliquelines.
FIG. 20. A trapped excitation oscillating around a pinned but decaying ILM.�a� Experimental image. The ILM becomes unpinned at time �56 ms duringwhich time the period of the lateral oscillation increases. �b� Schematicimage. This ILM starts to oscillate laterally from time �50 ms, which isshown as the dotted line.
FIG. 21. Simulation of trapped modes in a decaying but pinned ILM. This isa magniﬁed section of Fig. 12�a�. The pulse is turned off at time�7500periods. The decay of the pinned ILM is evident from its broadening. After8700 periods, running wave packets trapped in the ILM envelope appear asoblique patterns.
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plitude excitations of the pinned ILM from the stable lockedstate, and trapped excitations by a pinned ILM also havebeen identiﬁed. The stability and rigidity of locked ILMsdepend on the existence of the cw driver plus damping. Bymodeling the array with a nonlinear Klein–Gordon lattice,many of the experimental observations can be identiﬁed andexamined in more detail, such as the synchronization andpinning, the rigidity and dynamics of the locked ILM mode,and the repulsive interaction.Three possible extensions to these experiments, whichwould provide more dynamical information, should be men-tioned. �1� The combination of single cantilever optics ofFig. 2 with the experimental setup of Fig. 6 to identify thelow frequency modulation of the ILM center. �2� Modulatethe laser beam close to the driver frequency to observe thedisplacement amplitude and phase using the same CCD. Thecantilever would again modulate the laser beam, and theslow CCD would then pick up the difference frequency sig-nal as long as the difference is smaller than the speed of thecamera. This method could be used to observe both the sta-tionary and traveling locked mode. �3� Another extensionwould be to use optimal control65 in this ﬁnite array to excitethe individual eigenvector of a speciﬁc ILM.
ACKNOWLEDGMENTS
Conversations with J. P. Sethna, R. H. Silsbee, and J. B.Page have been very helpful. Supported by NSF–DMR andCCMR. The fabrication of samples was performed at theCornell Nanofabrication Facility, which is also supported bythe NSF.
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715Chaos, Vol. 13, No. 2, 2003Study of ILMs in MEMS cantilever arrays
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