Rapid reconstruction of a strong nonlinear property by a multiple lock-in technique

We propose and discuss a rapid reconstruction procedure for strongly nonlinear signals and validate it for dynamic force microscopy. Harmonics of the cantilever resonance frequency shift, generated by a low-frequency modulation of the tip-sample distance, are detected by a phase-locked loop followed by 12 lock-in amplifiers. The distance dependence of the frequency shift can be reconstructed by summing up the sampled Fourier components with judiciously assigned phase shifts. Following a successful test with a model potential, we report a measurement of the frequency shifts induced by the force field above a KBr(001) surface at room temperature in ultrahigh vacuum. Experimental spectra justify the neglect of harmonics beyond tenth order in the range where clear atomic-scale contrast appears in images of the lower harmonic intensities. A high-resolution three-dimensional frequency shift dataset was measured in 400 s. The method can in general be applied to any single-valued physical quantity with a smooth nonlinear dependence on a control variable.