A Kalman Filter for Amplitude Estimation in High-Speed Dynamic Mode Atomic Force Microscopy

Abstract

A fundamental challenge in dynamic mode atomic force microscopy (AFM) is the estimation of the cantilever oscillation amplitude from the deflection signal, which might be distorted by noise and/or high-frequency components. When the cantilever is excited at resonance, its deflection is typically obtained via narrow-band demodulation using a lock-in amplifier (LIA). However, the bandwidth of this measurement technique is ultimately bounded by the low-pass filter, which must be employed after demodulation to attenuate the component at twice the carrier frequency. Furthermore, to measure the amplitude of multiple frequency components, such as higher eigenmodes and/or higher harmonics in multifrequency AFM, multiple LIAs must be employed. In this paper, the authors propose the estimation of amplitude and phase using a linear time-varying Kalman filter that is easily extended to multiple frequencies. Experimental results are obtained using square-modulated sine waves and closed-loop AFM scans, verifying the performance of the proposed Kalman filter. IndexTerms-Amplitude estimation, atomic force microscopy (AFM), field-programmable gate array (FPGA) implementation, high bandwidth, Kalman filter, state estimation.

A review of demodulation techniques for multifrequency atomic force microscopy

“…Also, the Kalman filter equations have a complexity of for n modeled frequencies resulting in significant computational requirements beyond three modeled frequencies. This system representation has seen success in tracking power system voltage phasors [55] and more recently high-speed AFM [31,32].…”

“…As a result, MF-AFM experiments usually employ multiple lock-in amplifiers in parallel. However, this introduces an inherent bandwidth limitation as high-frequency mixing products must be low-pass filtered [28,31].…”

Lyapunov Estimator for High-Speed Demodulation in Dynamic Mode Atomic Force Microscopy

“…Under certain conditions the Kalman filter for amplitude estimation [12] is equivalent to the Lyapunov estimator, as will be demonstrated in the following. First, it is shown that the two methods are equivalent when the covariance matrix of the Kalman filter, P, is constant.…”

“…However, this method is prone to noise and disturbances from unwanted harmonics. Recent developments include the low-latency coherent demodulator [9], the high-bandwidth lock-in amplifier [10], and Kalman filter [11], [12]. The latter has also been extended for multiharmonic imaging [13], [14].…”

A review of demodulation techniques for amplitude-modulation atomic force microscopy

“…The demand for a high tracking bandwidth while maintaining insensitivity to additional frequencies in the signal has motivated the development of filters such as the time-varying Kalman filter [ 44 ] and Lyapunov filter [ 45 – 46 ]. These methods are based on a linear parametric model of the cantilever deflection signal and were shown to be extendable for the estimation of multiple frequencies for multifrequency AFM [ 47 – 49 ].…”