Schematic description of a typical lock-in type DSFM detection unit. The signal to be analyzed by the DSFM detection is assumed to be centered around some frequency . It enters the detection unit at the input “in” is amplified and usually high-pass filtered (for simplicity the corresponding components are not shown) before being multiplied with two reference signals in quadrature at a frequency . After this multiplication, the signal is shifted to the frequencies and . The resulting signals are then low-pass filtered to remove the higher frequency component , resulting in two averaged signals and . For sufficiently small interaction is proportional to the frequency shift and can be used to readjust the driving frequency of the VCO (or NCO) by means of an appropriate feedback loop (PI-controller). The output of the PI-controller used to adjust the excitation frequency is then proportional to the frequency shift .
Simple approximation of the noise density (black) for a thermally excited cantilever as discussed in the main text together with the correct noise density (dashed). For large quality factors, most of the noise is within the main peak at the resonance frequency.
Main graph: thermal noise error of the phase as a function of the (relative) oscillation amplitude . The black, solid, thin line ending at corresponds to the relation obtained from the relation known in the literature, which diverges for small oscillation amplitude. The black, dotted line corresponds to the relation [Eq. (12)], which is not correct at the singular point . The thick, solid line (red) shows the correct relation calculated from the probability distribution discussed in the Appendix B [see inset and relation (B1)]. Finally, the thin, dotted line (red) corresponds to the approximation [relation (14)], which has the correct low and large amplitude limits. Inset: probability distributions for different (relative) oscillation amplitude . The probability distributions have been calculated for the range of oscillation amplitudes . The probability distribution for is flat while that for is essentially Gaussian and has the highest peak at .
Main graph: spectral noise density of a 0.4 N/m cantilever measured with a digital lock-in amplifier. For this noise measurement, no external excitation was applied to the cantilever and the motion of the cantilever was measured using the beam-deflection technique. The larger (red) points correspond to experimental noise data, the solid line to a fit assuming a constant offset and a Lorenz function (see main text) and the smaller (pink) points show the error between this fit and the measured data points. Inset: log-log plot of the total noise as a function of bandwidth for a noise measurement centered at the peak of the main noise curve. For small bandwidth, the frequency noise shows the typical behavior (slope −1 in the log-log plot). However, for high bandwidth (, with quality factor) the total noise saturates.
Frequency noise of the DSFM detection electronics measured as a function of oscillation amplitude for two different bandwidths (50 Hz and 100 Hz). For this measurement, the same cantilever as that used for the previous experiment was utilized (force constant of 0.4 N/m). The cantilever was excited by the DSFM electronics with the phase-locked loop enabled, and the frequency output was fed into a digital lock-in amplifier in order to determine the total noise of the frequency measurement of the DSFM detection unit. For small oscillation amplitude of the cantilever (, see main text), the frequency noise is independent of oscillation amplitude. For large amplitude the noise decreases linearly (slope 1 in the log-log plot).
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