Spatial modulation of nonlinear flexural vibrations of membrane resonators

TL;DR

This study investigates the nonlinear vibrational behavior of membrane resonators under strong drive, revealing unique concentric ring patterns and localized overtones, modeled through coupled nonlinear oscillators.

Contribution

It introduces a phenomenological model capturing complex nonlinear vibrational patterns in membrane resonators, highlighting localized overtones and circular symmetry.

Findings

Membranes exhibit concentric ring deflection patterns at high drive.

Different membrane regions oscillate at multiples of the drive frequency.

A coupled oscillator model explains the observed nonlinear phenomena.

Abstract

We study the vibrational motion of membrane resonators upon strong drive in the strongly nonlinear regime. By imaging the vibrational state of rectangular siliconnitride membrane resonators and by analyzing the frequency response using optical interferometry, we show that upon increasing the driving strength, the membrane adopts a peculiar deflection pattern formed by concentric rings superimposed onto the drum head shape of the fundamental mode. Such a circular symmetry cannot be described as a superposition of a small number of excited linear eigenmodes. Furthermore, the different parts of the membrane oscillate at different multiples of the drive frequency, an observation that we denominate as 'localization of overtones'. We introduce a phenomenological model that is based on the coupling of a very small number of effective nonlinear oscillators, representing the different parts of the membrane, and that describes the experimental observations.

Equations

$$u(r,t)\cong \sum _{n\ge 1}{q}_n(t)h_n\left(r\right)e^{i2n\pi f_{\mathrm{d}}t}+\text{mbox}c.c.,$$

$$u(r,t)\cong q_2(t)h_2\left(r\right)e^{i4\pi f_{\mathrm{d}}t}+q_1(t)h_1\left(r\right)e^{i2\pi f_{\mathrm{d}}t}+\text{mbox}c.c.,$$

$${\ddot{q}}_2(t)$$

$${\ddot{q}}_1(t)$$

$$-{\gamma }_1(q_1{)}^3(t)+F_{\mathrm{d}}\cos \left(2\pi f_{\mathrm{d}}t\right)$$

Full text

Micro- and nanoscale mechanical resonators bear rich potential to expand their applications in a broad range of areas, such as noise sensing Weber et al. (2016), optical quantum metrology Kampel et al. (2017); Aspelmeyer et al. (2014), nanoelectromechanical logic gates Mahboob and Yamaguchi (2008); Guerra et al. (2010) and parametric oscillators Rugar and Grütter (1991); Seitner et al. (2017). Mechanical resonators can have different size scales, and may be even as small as single carbon nanotubes or patterned graphene sheets Eichler et al. (2011); Zhang et al. (2015). Among larger devices, quasi-two dimensional membrane oscillators are interesting for several reasons, e.g. they have resonant behavior in a broad range of frequencies (e.g. from 100 kHz to several MHz) which make them attractive as broad-band transducers for vibratory energy harvesting Jia et al. (2016). They are also important in quantum engineered systems, since flexural modes can be easily coupled to other degrees of freedom such as photons Sankey et al. (2010); Purdy et al. (2013); Andrews et al. (2014); Xu et al. (2016) and cold atoms in hybrid optomechanical architectures Camerer et al. (2011); Jöckel et al. (2015). Suspended membrane resonators operating in the single-mode regime have been studied in the weakly nonlinear limit and modeled as a Duffing oscillator Antoni et al. (2013). Increasing the membrane oscillation amplitudes enforces the nonlinearity, thereby giving rise to nonlinear coupling between eigenmodes Nayfeh and Mook (2008); Manevich and Manevitch (2005); Schuster (2009). So far, experiments reported nonlinear interaction involving only very two interacting modes Van der Avoort et al. (2010); Westra et al. (2010); Antonio et al. (2012); Güttinger et al. (2017); Chen et al. (2017), and only when matching an internal resonant condition between eigenfrequencies Shoshani et al. (2017).

In this Letter we analyze a novel, nonlinear vibrational state of a membrane resonator, achieved under strong drive. We observe the appearance of spatially modulated overtones. Upon increasing the driving strength, the membrane adopts a characteristic ring-shaped deflection pattern in which different parts of the membrane oscillate at different subharmonically excited multiples of the driving frequency. Such a ring-shaped pattern is distinctly different from the waveform and the symmetry of the flexural eigenmodes of the rectangular membrane. In other words, this regular pattern can only be reconstructed by using a superposition of a multitude of eigenmodes of the membrane. Moreover, the presence of subharmonically excited tones points out that the system is in a strongly nonlinear vibrational regime. Part of this phenomenon was reported before Zhou et al. (2017), but remained unexplained. Here we analyze it in detail and present a phenomenological model that captures the main features and semi-quantitatively describes the experimental findings. The starting point of this model is that, under driving, the profile is well described by

[TABLE]

where $r$ is the radial coordinate of the 2D membrane, $\textit{h}_{n}(r)$ are arbitrary spatial profile functions, and the amplitudes of oscillation are $\textit{q}_{n}(t)$, respectively. The amplitudes $\textit{q}_{n}(t)$ play the role of effective oscillators with different eigenfrequencies $f_{n}$ and having sizeable amplitudes at different positions on the membrane, described by the functions $\textit{h}_{n}(r)$, i.e. the localized overtones. We assume a nonlinear interaction $V=\sum_{n\geq 2}\lambda_{n}q_{1}^{n}q_{n}$ between a mode $q_{1}$ with eigenfrequency $f_{1}$ and its overtones. This model is referred to as $1:n$ nonlinear coupling model.

We will show that, to describe our experimental observations, a small number of modes is sufficient, representing a significant simplification compared to the classical model based on linear eigenmodes of a rectangular membrane. Our conclusions are thus that spatially modulated overtones represent, indeed, a new paradigm in the nonlinear dynamics of membranes which goes beyond previous theoretical pictures, based on the approach of nonlinear interaction of few linear eigenmodes Antonio et al. (2012); Güttinger et al. (2017); Chen et al. (2017).

The sample fabrication and measurement principles of the siliconnitride (SiN) membranes have been described in detail elsewhere Petitgrand et al. (2001); Hernandez et al. (2002); Waitz et al. (2012, 2015); Yang et al. (2017) and are summarized in the Supplementary Material (SM). Flexural modes of the membrane are excited by applying an AC voltage $V_{\textrm{exc}}$ $\cdot$ sin(2$\pi\textit{f}_{\textrm{d}}$t) to the piezo ring that causes a uniform thickness change of the piezo, resulting in an inertial excitation of the membrane, see Fig. 1(a). The vibrational state of the membrane installed in a vacuum chamber is observed by two different types of optical interferometry. The imaging white light interferometer (IWLI) is able to spatially resolve the deflection profile and to obtain the average amplitude response by integrating the deflection profile over a selected area on the membrane surface Petitgrand et al. (2001). The Michelson interferometer (MI) is focusing on one particular position of the membrane with a spot diameter of $\mathrm{\sim}$2 $\mu$m. Further experimental details are given in the SM.

In the linear response regime, we find resonances corresponding to the eigenfrequencies $\textit{f}_{(\textit{m,n})}$ of the flexural eigenmodes which are characterized by the integers m and n indicating the number of deflection maxima in the two spatial directions of the membrane plane Yang et al. (2017). For example, we find $\textit{f}_{\textrm{(1,1)}}$ = 321 kHz and $\textit{f}_{\textrm{(2,2)}}$ = 646 kHz for the eigenfrequencies of (1,1) and (2,2) mode, respectively, both with quality factors in the order of 20000. The eigenfrequencies of all modes discussed in this article as well as further mechanical parameters are listed in Table 1 in the SM.

We utilize the IWLI signal integrated over the entire membrane area to record the nonlinear vibration behavior under intermediate and strong sinusoidal excitation and with a driving frequency $\textit{f}_{\textrm{d}}$ around $\textit{f}_{\textrm{(1,1)}}$. With $\textit{V}_{\textrm{exc}}$ = 2.5 mV to 5.0 mV, the resonance curves exhibit a Duffing-type nonlinearity, as shown in Fig. 1(b), but when exceeding $\textit{V}_{\textrm{exc}}\mathrm{>}$ 5.0 mV, the amplitude rises weaker than expected for the Duffing model but persists over a larger frequency range. This observation signals the onset of the spatial modulation phenomenon. The spatial deflection profiles captured by IWLI for $\textit{V}_{\textrm{exc}}$ = 15.0 mV are shown in Fig. 2(a). When continuously sweeping the driving frequency $\textit{f}_{\textrm{d}}$, the spatial appearance of the deflection pattern changes from single drum head type at $\textit{f}_{\textrm{d}}$ = $\textit{f}_{\textrm{1}}$ (see marker in Fig. 1(b)) to a crater type at $\textit{f}_{\textrm{d}}$ = $\textit{f}_{\textrm{2}}$, where the amplitude at the membrane center decreases and the central part of the deflection pattern adopts a flat shape. When further increasing $\textit{f}_{\textrm{d}}$ to $\textit{f}_{\textrm{3}}$, the flat area evolves into a separate drum head shape. Finally, at $\textit{f}_{\textrm{4}}$ again a minimum occurs in the center surrounded by two rings. The observed deflection patterns reveal a frequency-dependent spatial modulation of the (1,1) mode, which cannot be described as a simple superposition of few (m,n) modes, since it has a circular symmetry contrary to the axial symmetry of the higher (m,n) modes in a rectangular membrane.

With the help of sweep-up and ring-down experiments performed in the MI at different positions on the membrane, one can reveal that the modulation is caused by frequencies which are identified as overtones of the (1,1) mode with 2$\textit{f}_{\textrm{(1,1)}}$ and 3$\textit{f}_{\textrm{(1,1)}}$, as we will show in the following.

We perform a slow sweep-up of $\textit{f}_{\textrm{d}}$ at constant $\textit{V}_{\textrm{exc}}$ from a starting value below the resonance frequency of a particular mode up to a maximum frequency above the resonance frequency but within the frequency range with large amplitude. At this frequency we switch off the drive. The measurement spot is close to one edge of the membrane and chosen such that the maximum deflection stays smaller than a quarter of the laser wavelength (see SM for details). A Fast Fourier Transform (FFT) algorithm is performed on the time domain oscillation displacement data in narrow time windows to get a time-resolved power spectrum of the sweep-up and ring-down processes. Figure 2(b) shows the time evolution during the sweep-up of the power spectrum for a moderate excitation for $\textit{f}_{\textrm{d}}$ = 321.88 kHz. The subsequent ring-down part is displayed in Fig. SLABEL:fig:image11 in the SM section 6. In contrast to the linear response regime shown in Fig. SLABEL:fig:image10 in the SM, we observe response at multiple frequencies which are setting in one after another at different $\textit{f}_{\textrm{d}}$. Strong signals are observed at $f_{\textrm{d}}$, $2f_{\textrm{d}}$, and $3f_{\textrm{d}}$, respectively. These responses evolve during ring-down towards the overtones of the first mode, namely to $f_{(1,1)}$, $2f_{(1,1)}$, and $3f_{(1,1)}$. We note that 2$\textit{f}_{\textrm{(1,1)}}$ = 642 kHz can clearly be distinguished from the eigenfrequency $\textit{f}_{\textrm{(2,2)}}$ = 646 kHz of the (2,2) mode, given the frequency resolution of the FFT algorithm $\mathrm{\sim}$ 1 kHz. Similar multiple frequency response has also been found for SiC and Ge membrane systems Zhou et al. (2017). The multiple frequency response complements the observations from the IWLI and gives a strong hint that the spatial modulation state corresponds to spatially varying superpositions of the ground mode and its overtones. In addition, stroboscopic IWLI measurements show (see Fig. SLABEL:fig:image08(d) in the SM), that at $\textit{f}_{\textrm{d}}$ = $\textit{f}_{\textrm{3}}$ the central part of the membrane (inner disc with a drum head shape) oscillates at different frequency than the outer annulus which oscillates at $\textit{f}_{\textrm{d}}$.

To further test this hypothesis of localized overtones we perform MI measurements at different positions on the membrane. Figure 2(c) shows the relative amplitudes of the overtones observed in the power spectrum for a line profile from one edge to the other, crossing the center of the membrane, see inset of Fig. 2(b). At each spot a sweep-up is performed and the intensities of the overtones are evaluated at fixed detuning at the various spots by separately integrating the power of each time window within a corresponding frequency range. The total intensity reveals a large plateau that extends over a width of roughly 200 $\mu$m in the center of the membrane. The fundamental frequency $f_{\mathrm{d}}$ has the highest amplitude at the outer rim of this plateau and strongly decreases towards the center. In the central part the intensity of the 2${}^{\textrm{nd}}$ overtone exceeds the one of the ground mode. The 3${}^{\textrm{rd}}$ and 4${}^{\textrm{th}}$ mode have weaker intensity and are even more centered. This observation evidences the existence of spatially localized overtones, confirming the interpretation of the IWLI data.

The detailed microscopic analysis of the observed spatial overtones and their appearance is a challenging theoretical task and beyond the aim of this work. The main physical aspects can be captured by the phenomenological ansatz given in Eq. (1) that describes that the membrane is vibrating at distinct sections at different frequencies. We note that any profile function can be obtained as a linear combination of the eigenmodes (m,n) since the latter form a complete orthonormal basis. An example is given in the SM in which we project the ring-shape of the deflection onto the linear eigenmodes. This yields a very slowly convergent series: as expected intuitively, a superposition of many modes is needed to reproduce a circular ring-shape (N $\mathrm{\geq}$ 100) of the rectangular membrane (see SM). By contrast, the spatially modulated overtones $\textit{u}_{n}$ can be viewed as renormalized eigenmodes appearing only for strong driving. They can be mathematically defined on the concept of an invariant manifold in an elastic nonlinear medium Pesheck et al. (2001). For simplicity we describe here the situation with only two oscillators, i.e. the 1:2 model. The general $1:n$ model is detailed in the SM. We thus consider the equation

[TABLE]

$\textit{h}_{\textrm{1}}(t)$ is localized at a ring of radius R from the center and has the eigenfrequency $f_{\textrm{1}}\simeq\textit{f}_{\textrm{(1,1)}}$, the second one, and $\textit{h}_{\textrm{2}}(t)$, is localized at the center, with eigenfrequency $2f_{\textrm{1}}$, as shown in Fig. 2(a) for $\textit{f}_{\textrm{d}}$ = $\textit{f}_{\textrm{3}}$.

Finally, in our model, we assume a nonlinear coupling between the two effective oscillators and we set the following coupled equations for the amplitudes $\textit{q}_{\textrm{1}}$ and $\textit{q}_{\textrm{2}}$:

[TABLE]

Equations (3) and (4) mean that the membrane is effectively composed of an inner disc coupled to an outer ring. The ring oscillator is characterized by a Duffing nonlinearity with strength $\alpha_{\textrm{1}}$, and both are damped with damping constants $\Gamma_{\textrm{1}}$ and $\Gamma_{\textrm{2}}$. Most importantly, we also postulate a nonlinear resonant interaction between the two oscillators, given by the last terms in Eqs. (2) and (3) with the interaction strength $\lambda$, which can be derived from the nonlinear potential $V=\lambda{q_{\mathrm{1}}}^{2}q_{\mathrm{2}}/2$. This interaction is important since it implies a correlated behavior of the two amplitudes which we actually observe in the system, as we discuss next.

In Fig. 3 we show more curves from the same data set as depicted in Fig. 1(b). The orange lines are the same fits to the duffing model as in Fig. 1(b), the green curves are calculated with the 1:2 coupling model and the blue curves using 1:2 and 1:3 coupling. The fitting parameters and their analyses are given in the SM. The model fits the experimental curves well. The small deviations can be explained by the simplicity of the model, e.g. neglecting the contributions of the higher overtones.

The analysis of the development of the spatial overtones is completed by studying the deflection amplitude in a small area. Figure 4(a) shows examples of response curves integrated over a square with size 80 x 80 $\mu$m in the central area (see dashed line in inset) of the membrane for increasing $\textit{V}_{\textrm{exc}}$. A zoom into the maxima is given in panel (b). It reveals a serious of maxima and minima, before the amplitude eventually breaks down. The number of observable maxima increases with $\textit{V}_{\textrm{exc}}$ in agreement with the increasing plateau length when integrating over the whole membrane area. For small $\textit{V}_{\textrm{exc}}$ = 10 mV only the first maximum and minimum are accessible, for $\textit{V}_{\textrm{exc}}$ = 100 mV four maxima and minima are obtained. We interpret this observation as follows: Upon increasing $f_{\textrm{d}}$ the diameter of the circular shaped overtones increase. The first maximum signals that $u_{1}$ reaches the edges of the integration window, thereby reducing the averaged amplitude. When further increasing $f_{\textrm{d}}$, more overtones are excited, increasing the amplitude and subsequently leave the central area, thereby reducing the averaged amplitude again. As an example we fit the curve recorded for $\textit{V}_{\textrm{exc}}$ = 10 mV with our theoretical model, comprising two tones, $\textit{q}_{\textrm{1}}$ and $\textit{q}_{\textrm{2}}$. To account for the fact that upon increasing $\textit{f}_{\textrm{d}}$, the radius of the ring increases and partially leaves the integration area, we use a linearly decreasing weight factor for the mode $\textit{q}_{\textrm{1}}$. The solid line in Fig. 4(c) is the result of the fitting with the theoretical model which describes the experimental observation qualitatively correctly. The individual contributions and the total fitted amplitude and further details are given in the SM. This concept can straightforwardly be extended to the situation with three and more modes. However, the number of fitting parameters increases rapidly. Summarizing, the oscillating amplitude in the central area of the membrane can be interpreted as the gradual transfer of vibrational energy from $q_{1}$ to higher overtones $q_{n}$. In conclusion, by combination of two complementary experimental approaches and the nonlinear $1:n$ coupling model we revealed a novel deflection state caused by a nonlinear coupling mechanism between several overtones of a single flexural mode of a membrane oscillator. This deflection state is hallmarked by a spatial modulation pattern, in which different parts of the membrane oscillate at different frequencies. For rectangular membranes the spatial modulation state can be distinguished from the linear deflection pattern by a distinctly different symmetry. The coupling mechanism is mediated by nonlinear interaction and activated by strong driving.

Acknowledgments

The authors thank R. Waitz for help in sample fabrication. We are indebted to W. Belzig, T. Dekorsy, M. Dykman, V. Gusev, M. Hettich, P. Leiderer, S. Shaw, I. Wilson-Rae, R. S. Edwards, L. Q. Zhou and the SFB767 Nanomechanics Discussion Group for fruitful discussion and comments about the work. The authors gratefully acknowledge financial support from the China Scholarship Council, the European Union’s Horizon 2020 program for Research and Innovation under grant agreement No. 732894 (FET Proactive HOT) and the Deutsche Forschungsgemeinschaft via the Collaborative Research Center SFB767.