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TopIntroduction
In the nanoscale, the surface of a sample can be imaged by many different methods, however, atomic force microscope (AFM) introduced in 1986 (Binnig & Quate, 1986) has appeared to be currently one of the most widely used, mainly because of its high spatial resolution, high surface sensitivity and ease of application. AFM relies on the interaction between a sample and a sharp probe attached to a flexible cantilever placed over the sample surface. The most widely used mode of AFM operation is the tapping mode also known as intermittent contact mode (IC-AFM) or amplitude modulated AFM (Zhong, Inniss, Kjoller, & Elings, 1993). In the tapping mode, the cantilever is excited to vibrate at (or close to) its resonant frequency and is brought close to the studied sample surface so that the tip makes intermittent contacts (tapping) with the surface once in every oscillation period. The contact with the surface shifts the resonant frequency and alters the amplitude and the phase of the cantilever vibrations. These vibrations are detected with an optical system where a laser beam is reflected from the back of the cantilever and then it falls onto a position-sensitive photodiode (PSPD), as shown in Figure 1. When the cantilever is scanned over the sample, the oscillation amplitude is maintained at a set-point value through a feedback loop that adjusts the distance between the cantilever and the surface. Therefore, the feedback signal reveals the topography of the sample surface. The non-linear character of the tip–sample interaction results also in the appearance of higher harmonics of the fundamental oscillation of the cantilever (Bhushan, 2010; Sahin et al., 2004).
Figure 1. Setup of an atomic force microscope operated in intermittent contact mode
Conventional detection systems need at least a few oscillation periods of the studied data to determine the signal amplitude. This disadvantage is also a common feature of lock-in amplifiers. In this case, to estimate the amplitude, it is necessary to apply low-pass filter causing significant delay. As a result, RMS to DC converters and lock-in amplifiers offer rather limited measurement bandwidth and thus, they are not suitable for high-speed imaging (Karvinen & Moheimani, 2014). For use in high-speed tapping-mode AFM, the peak-hold method was developed, which in turn allowed to measure the amplitude of the cantilever within only one oscillation cycle (T. Ando et al., 2001). Nevertheless, the drawback of the peak-hold method is that it is strongly affected by the noise, because only two extremal values of the signal are recorded. On the other hand, the synchronous detection method (Pawłowski, Dobiński, Szmaja, Majcher, & Smolny, 2015) implementing Fourier method (Kokavecz, Tóth, Horváth, Heszler, & Mechler, 2006), could offer even greater bandwidth than the peak-hold method but has relatively complex hardware requirements. The proposed detection system based on the Goertzel algorithm (Goertzel, 1958), provides a high speed of recording information about the amplitude of the cantilever, with good signal-to-noise ratio and requires simple hardware implementation.
The remainder of this paper is organized as follows. Firstly, Goertzel algorithm is described. Then results of computer simulations are analyzed. In the next section, the hardware implementation is presented. The paper concludes with an examination of the experimental results.
TopTheoretical Background
The Goertzel algorithm allows to compute a single Fourier component of the signal x(n) of the length N (Goertzel, 1958; Ifeachor & Jervis, 2002; Oppenheim & Schafer, 2010; Rao, Kim, & Hwang, 2010). The DFT equation for k-th spectral point is:
(1)