Nanomechanical membrane resonators are extensively used in a variety of applications due to their high quality factor and sensitive response in the linear working range. The quantitative characterization of mechanical properties of nanomechanical resonators paves the way to understand the vibrational dynamics of nanomechanical systems and provides the opportunity to extend the application into nonlinear regime. In the nonlinear regime, the nanomechanical systems present tremendous vibrational and coupling behavior which can provide massive information beyond the linear regime. Thus extending the investigation into nonlinear is significantly important. In this thesis, we present experimental studies of the bending waves of freestanding silicon nitride (SiN) nanomembrane resonators in different dynamic regimes by using optical profilometry, such as Imaging White Light Interferometer (IWLI) and Michelson Interferometer (MI), in varying environments such as pressure and temperature.
To quantitatively characterize the basic mechanical properties of nanomembrane resonators, we introduce a method, named Vibrometry in Continuous Light (VICL) that enables us to disentangle the response of the membrane from that of the excitation system, thereby giving access to the eigenfrequency and the quality (Q) factor of the membrane by fitting a damped driven harmonic oscillator model to the experimental data. We verify the performance of the method by studying two modes of a 478 nm thick SiN freestanding membrane and find Q factors of 2 \times 10444 for both modes at room temperature. Finally, we observe a linear increase of the resonance frequency of the ground mode with temperature which makes nanomembrane resonators suitable for high-sensitive temperature sensors.
In the second phase of experiments, we study the vibrational motion of mechanical resonators in the strong nonlinear regime. By imaging the vibrational state of rectangular SiN membrane resonators and by analyzing its frequency response using optical interferometry, we show that upon increasing the driving strength, the membrane adopts a peculiar deflection pattern of concentric rings superimposed onto the drum head shape of its fundamental mode. Such a circular symmetry cannot be described as a superposition of a small number of excited linear eigenmodes. Furthermore, different parts of the membrane vibrate at different multiples of the drive frequency, an observation that we term "localization of overtones". We introduce a phenomenological model based on the coupling between effective nonlinear resonators to represent different parts of the membrane, which agrees well with the experimental observations.
Further more, we investigated the phenomenon of persistent response in ultra-strongly driven membrane oscillators. The term "persistent response" denotes the development of a vibrating state with nearly constant amplitude over a very wide frequency range (up to 50% of the drive frequency). We reveal its underlying mechanism by directly imaging the vibrational state of membrane resonators. Beyond the spatial modulation, at even larger driving strength nonlinear interaction and sub-harmonically driven parametric resonance between different exural modes and their localized overtones govern the persistent response. Our result is important for understanding nonlinear resonance behavior appearing in different fields of physics as well as for the development of amplitude-stable mechanical resonators with broadband tunable frequency.