Real time cantilever signal frequency determination using digital signal processing

Abstract

We describe a digital signal processing method for high precision frequency evaluation of approximately sinusoidal signals based on a computationally efficient method. We demonstrate frequency measurement enabling sensitive measurement of the oscillatory force exerted on a micromechanical cantilever. We apply this technique to detection of the force signal arising in a micromechanically detected magnetic resonance force microscopy electron spin resonance signal. Our frequency detection measurements agree well with the theoretical noise analysis presented here, and we find that due to the excellent sensitivity of optical displacement detection, our sensitivity is limited only by the thermal displacement noise of the cantilever.

Dielectric fluctuations in force microscopy: Noncontact friction and frequency jitter

“…Detector noise is also assumed to be white. The power spectrum of coordinate fluctuations P ␦x det is independent of f at low frequencies, but the conversion from coordinate fluctuations to frequency fluctuations 46 introduces the f 2 dependence of Eq. ͑6.7͒.…”

“…The analysis of Fig. 2 indicates that b Յ f therm , so that the minimum detectable mean-squared jitter is given by 13,46 ͗͑␦f c ͒ 2 ͘ min Ϸ bP ␦f c therm = k B Tb 2 2 x rms 2 k c .…”

“…͑6.6͒ approximately accounts for the dielectric in the present configuration. The smallest P ␦f c ͑f͒ that can be detected experimentally is set by the magnitude of this quantity in the absence of the sample, which has been predicted to have the low-frequency limit 13,46 …”

“…Section V describes the laboratory procedures used in Sec. VI to show that the predicted jitter from a PMMA film exceeds the minimum detectable frequency jitter 13,46 for the apparatus of Refs. 17 and 18.…”

Advances in mechanical detection of magnetic resonance

“…As in atomic force microscopy, feedback can be used to shorten the time required for the cantilever's amplitude to reach steady state in response to a resonant force. 37,54 Optimal control theory has been applied to design high-performance MRFM cantilever controllers, 55 implemented at audio frequency using analog 56 and digital electronics [57][58][59][60] and at radio frequency using digital heterodyning downconversion in conjunction with a field programmable gate array. 61 Another approach to mitigating the slow natural response time of a cantilever is to detect the spins mechanically using force gradients instead of forces.…”