Figure 1

(a) Experimental setup for the linear response measurement of the autoresonant (AR) ILM state with a uniform probe perturbation. The array is composed of alternating 50 and $55\mu \mathrm{m}$ length by 300 nm thick cantilevers. The pump driver and probe signals are added and used to excite the array uniformly with a PZT. The ILM position is monitored by a combination of a line focused laser beam and a 1D CCD camera (not shown). A laser diode (LD) illuminates a short cantilever to the side of the ILM and the reflected beam is detected by a position-sensitive detector (PSD). The displacement signal is analyzed with a lock-in amplifier. A typical pump amplitude is 14 V, while the probe amplitude is 12 mV. (b) Spatial pattern of the odd symmetry ILM. (c) Pattern of an even linear local mode (LLM). The asymmetric LD heating of the localized vibration region permits the even LLM to be excited even though the PZT drives the array uniformly.Reuse & Permissions

Figure 2

(a) Experimentally observed AR ILM amplitude as a function of the pump frequency, showing two transitions. Upper abscissa: Pump frequency normalized to the top of the optic branch. Lower abscissa: Difference frequency between the pump and the top of the optic branch ($f_T=140.0\mathrm{kHz}$) normalized by the bandwidth (3.1 kHz). The stable AR region is 140.46–144.85 kHz, or 0.148–1.57 by the normalized difference frequency. (b) Simulated nonlinear response of the AR state in the hard nonlinear lattice with driver appropriate to the experimental level. Top trace: AR region, identifying its two transitions. Middle trace: $F$ denotes fast frequency rate required to reach the AR state. Bottom two traces: $S$ denotes slow up and down scanning; no AR state occurs. The stable frequency region is 137.56–146.54 kHz, or 0.102–2.29 by the normalized difference frequency. The top of the band frequency is 137.14 kHz and the bandwidth is 4.1 kHz. Curves are shifted up or down for clarity.Reuse & Permissions

Figure 3

Experimental response spectra for the AR ILM state produced by a uniform driver. The two strong sidebands identify the soft-phase mode associated with this state. Spectra are aligned from (0.161–1.55 by the normalized difference frequency). The upper and lower frequency limits are near the two bifurcation frequencies. The gap frequency of these sidebands decreases and the response grows as the pump frequency approaches the upper bifurcation point. The additional very weak sidebands approaching the lower bifurcation transition are the even LLM and its partner and lower frequency band modes, as described in the text. Both frequency axes are normalized by the optical bandwidth.Reuse & Permissions

Figure 4

Simulated linear response spectra of the AR state produced by a uniform driver. The entire stable frequency region of the pump frequency is shown, i.e., 0.112–2.26 by the normalized difference frequency. At the upper bifurcation point, the phase mode gap frequency softens and response diverges. At the lower bifurcation point, the phase mode overlaps with the top band mode, shown as small peaks near the bottom of the figure. This AR transition occurs when the soft-phase mode intersects these linear modes via four-wave mixing. The band mode maxima near the lower bifurcation region are magnified 20 times.Reuse & Permissions

Figure 5

Sine component of the experimental probe response near by the lower bifurcation frequency. Spectra are ordered by the pump frequency from 0.355 to 0.161 in 0.016 step. The large peak at the center is the pump signal. The lower bifurcation takes place at 0.148. The large positive amplitude identifies the soft-phase-mode oscillation of the ILM, the corresponding negative peak is its four-wave-mixing partner. The even LLM, activated by asymmetric heating of the ILM, appears as an amplitude modulation response. As the driver frequency is decreased, the highest frequency band mode of the array, identified with an arrow, approaches the partner of the soft-phase mode. (Lower frequency band modes appear as a sequence of positive and negative peaks.) Near the transition the even LLM frequency crosses the soft-phase-mode partner, dashed curve, second from the bottom. The transition takes place when the soft-mode partner intersects the lower frequency odd symmetry band mode.Reuse & Permissions