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Imaging the vibrations of nanomechanical resonators means measuring their flexural mode shapes from the dependence of their frequency response on in-plane position. Applied to two-dimensional resonators, this technique provides a wealth of information on the mechanical properties of atomically-thin membranes. We present a simple and robust system to image the vibrations of few layer graphene (FLG) resonators at room temperature and in vacuum with an in-plane displacement precision of $\approx0.20$ $\mu$m. It consists of a sturdy vacuum enclosure mounted on a three-axis micropositioning stage and designed for free space optical measurements of vibrations. The system is equipped with ultra-flexible radio frequency waveguides to electrically actuate resonators. With it we characterize the lowest frequency mode of a FLG resonator by measuring its frequency response as a function of position on the membrane. The resonator is suspended over a nanofabricated local gate electrode acting both as a mirror and as a capacitor plate to actuate vibrations at radio frequencies. From these measurements, we estimate the ratio of thermal expansion coefficient to thermal conductivity of the membrane, and we measure the effective mass of the lowest frequency mode. We complement our study with a globally gated resonator and image its first three vibration modes. There, we find that folds in the membrane locally suppress vibrations.

Optimization process

For simulating the laws of intensity distribution in image plane, the theory of Fresnel diffraction had been used based on Talbot effect of grating. Under the approximation condition of Fresnel diffraction, the space distribution of complex amplitude in the diffraction field in image plane had been expressed for transitivity function in object plane. The calculation results had shown three parts. The first one was that the width in the flat-top range of maximum intensity was increasing and the border was very clear when the scratch number was increasing. The Talbot effect was more and more obvious as the scratch number of the grating was larger and larger. The second one was that the central positions of the maximum intensity in image plane were in x=±kd on image plane, where k was integral number and d was grating constant. The Talbot effect will be loss with the distance between grating plane and image plane larger and larger. The third one was very important. When the distance between the grating plane and image plane was half-odd multiple of Talbot distance, the central positions of the maximum intensity in image plane were in x=±(2k+1)d on image plane. The positions were the centers of two neighbor-hood positions of maximum intensity in image plane when the distance between the grating plane and image plane was integral number times of Talbot distance. The important phenomenon was found. The phenomenon was the maximum light intensity appeared in the same time in x=±kd and x=±(2k+1)d on image plane.