detectors are governed by the same basic principles (see, for example, [22]). As described in detail in appendix A, the dynamic properties of such a system are described by the quality factor Q , by a driving amplitude a$_{0}$ and by its natural angular frequency ω$_{0}$ = 2 πν$_{0}$ = $^{√}$c/m$_{eff}$ , with ν$_{0}$ natural frequency and m$_{eff}$ effective mass of the cantilever[23]. In the steady state regime, the response of the system to a harmonic excitation a$_{0}$ω 2 0 cos(2 πνt ) can be described by a (complex) amplitude A ( ν ), or by two components X ( ν ) and Y ( ν ), the in-phase and out of phase components of the oscillation, corresponding to the real and complex parts of the complex amplitude A ( ν ). The time response of the system is then a ( t ) = X ( ν ) cos(2 πνt ) + Y ( ν ) sin(2 πνt ) = | A ( ν ) | cos (2 πνt + ϕ ( ν )) where ϕ ( ν ) is the phase between the driving force and the response. At its natural frequency ν$_{0}$ the phase is − π/ 2, the (complex) oscillation amplitude is A ( ν$_{0}$ ) = − iY ( ν$_{0}$ ) = − ia$_{0}$Q and the in-phase component X ( ν$_{0}$ ) vanishes. Figure 1 shows a schematic set-up of the main components used in a DSFM detection electronics. The multiplication stages together with the filters essentially calculate the two components X ( ν ) and Y ( ν ) -relations 18 and 19 in appendix A- from the measured oscillation of the cantilever a ( t ). When enabled, the PI-controller of the DSFM detection electronics adjusts the driving frequency ν$_{0}$ of the excitation signal in order to have X ( ν ) = 0. In this way, the system is locked to the natural frequency of the cantilever, tracks this frequency if it varies due to tip-sample interaction and generates an output proportional to the shift of the resonance frequency. In addition to the main DSFM components figure 1 shows the signals (in the frequency domain) along the different points in the DSFM detection path. A key component of any DSFM detection unit is the voltage (VCO) or numerically (NCO) con- trolled oscillator that generates the excitation and reference signals for the lock-in detection scheme. In most applications this oscillator drives the piezo that induces motion of the can- tilever. The corresponding deflection a ( t ) is analyzed by multiplication with two reference signals in quadrature. We will first assume the most general case where the frequency ν of the cantilever motion and that of the reference oscillator are different. Then, multiplication of the deflection signal a ( t ) = a$_{0}$ cos(2 πνt ) with the reference signal a$_{ref}$ ( t ) = cos(2 πν$_{ref}$ t ) results in two quadrature signals x$_{q}$ ( t ) = a ( t ) cos (2 πν$_{ref}$ t ) and y$_{q}$ ( t ) = a ( t ) sin (2 πν$_{ref}$ t ) with frequency components at ν$_{Δ}$ = ν − ν$_{ref}$ , and ν$_{Σ}$ = ν + ν$_{ref}$ . As shown in appendix A, x$_{q}$ ( t ) and y$_{q}$ ( t ) can be obtained from X ( ν ) and Y ( ν ) by multiplication with ( M$_{Δ}$ ( t ) + M$_{Σ}$ ( t )), where M$_{Δ}$ ( t ) is a rotation matrix generating a clockwise rotation with frequency ν$_{Δ}$ , and 4