DFRT revisited: from feedback optimization to full DAQ acquisition

The Data Acquisition module (DAQ) in LabOne has been addressed in 2 recent blog posts: A deep dive into data acquisition with the DAQ tool & Choose the right tool to acquire lock-in data. One other great application area for this acquisition tool is related to imaging and scanning microscopy. Let’s illustrate this by revisiting the concept of Dual Frequency Resonance Tracking (DFRT), which had been described in an earlier blog post using the discontinued ziControl user interface, and apply this to all the new LabOne tools. The benefits of LabOne are many-fold:

  1. A more systematic and quantitative optimization process
  2. Digital recording of internal signals, thus removing additional AD/DA stages
  3. No limit on the number of channels, which would otherwise be limited to 4 with Aux Outputs
  4. Real-time visualization of any of the recorded channels within LabOne

 

Contact resonance and choice of parameters

For a cantilever beam, the contact resonance (beam clamped at both ends) is typically around 6.2 times higher than the free-space resonance (beam clamped only at one end). Such resonance can be found with the Sweeper module, along with the Resonance tool within the Math sub-tab that calculates all resonator parameters such as the Q-factor, the resonance frequency f0, the resonance phase, and the full width at half maximum (FWHM). Half of the FWHM is usually a good value for the modulation frequency in the MOD module. This module generates an amplitude-modulated (AM) signal centered around the carrier frequency f0. The corresponding sweep for the R1, R2 and R3 demodulator amplitudes as functions of the carrier frequency typically looks like the one depicted in Figure 1.

Figure 1: Sweep of the contact resonance (blue) along with the 2-sideband amplitude from demodulators 2 and 3 (red and green).

Such sweep is obtained while the MOD module, as shown on Figure 2, is enabled with an electric drive on both carrier and modulation sent to the same Signal Output 1. The actuation is generally routed via the tip bias voltage while the sample is on ground (or vice-versa, depending on the microscope setup).

Figure 2: Modulation tab in AM mode for bimodal excitation. Click to see the corresponding generated signal.

Feedback optimization

For DFRT, the PID input corresponds to the difference of sideband amplitudes R3-R2, whereas the PID output acts upon the center frequency of the resonance. In its steady state at resonance, the value for R3-R2 is zero, which is used as PID setpoint given that the aim of such feedback is to match the driving frequency with the resonance frequency of the sensor. The PID output corresponds to Oscillator Frequency 1, where the resonance frequency is entered as center frequency with an appropriate range (at least 1/10 of the center frequency).

To optimize such PID, let’s select the first-order low-pass DUT model (LP 1st). This requires 2 external parameters: gain and bandwidth. These values can be measured from an open-loop transfer function, which in our case corresponds to the previously recorded sweep shown in Figure 1. Ideally, it is the difference R3-R2 as a function of Oscillator Frequency 1 which should be represented (instead of R2 and R3). In general terms, this corresponds to the slope of the PID output versus the PID input around the setpoint in open loop. For the bandwidth, we can choose the FWHM measured at resonance. With LabOne 20.01 and higher, there is a Linear Fit function that calculates automatically the gain from the slope. This can be done for the slope of R2 near resonance, multiplied by 2 since the actual gain should be measured on the slope of R3-R2 directly (which can only be displayed with the Arithmetic Unit on the UHFLI Lock-in Amplifier, an option not available on other instruments). Once these parameters are determined experimentally, we can feed them into the PID Advisor and run a closed-loop simulation response or Bode plot. The choice of the target closed-loop bandwidth should be aligned with the expected pixel dwell time, that is, how long the probe will remain at that pixel position before moving to the next pixel. This is all summarized in Figure 3.

Figure 3: Use the right DUT model with experimentally validated parameters. Follow the 6-step process to ensure that everything is optimized.

In addition, the PID Advisor provides a Filter BW for the demodulator settings: this corresponds to the lock-in time constant of the matching demodulator. In the case of DFRT, however, the PID input corresponds to the difference of sideband amplitudes coming from the MOD tab (and not from a single demodulator) and so it is necessary to copy this number manually, which is typically 2 to 5 times higher than the Target BW of the PID, into the MOD tab. After clicking ‘To PID’ the proper P and I values are entered in the PID Settings on the left-hand side and the feedback is ready to be closed. By default, the slope is positive if the PID input increases for an increase of PID output; if the slope is reversed in the Sweeper, a negative sign needs to be entered manually for the P gain.

Integration with third-party controller: example with Park Systems

Once the resonance tracking feedback is running, it is possible to record all relevant parameters such as the amplitude and phase of carrier and sidebands, the frequency shift and the error signal from the PID. An End of Line (EOL) trigger from the Scan Generator is used to trigger the DAQ, fed as Trigger signal setting ‘Demod 1 DIO’ (digital). The number of lines and pixels are entered in the Grid sub-tab as Rows and Columns. Only the number of lines needs to be triggered as the number of pixels is automatically adjusted to the grid with the corresponding data transfer rate and eventually averaged, depending on the user’s choice (see this blog post for more details).

Figure 4: Real-time visualisation of acquired image while scanning.

Image Visualization in LabOne and in SmartScan Software

To illustrate how this method works under experimental conditions, we validated the above procedure on a ferroelectric BFO sample on a Park NX20 Microscope. A Pt/Ir coated cantilever was used to ensure good electric contacts with a contact resonance frequency around 340 kHz and a stiffness of 2.8 N/m in free space. DFRT was performed on the HF2LI Lock-in Amplifier with the following options enabled: HF2LI-MF, HF2LI-MOD, HF2LI-PID. Figure 5 shows the corresponding data capture from the DAQ module, including some of the parameter settings (click to enlarge with more parameters). The image acquired by the SmartScan software through the analog Aux Output further validates the equivalence of the two data acquisition methods.

Figure 5: Simultaneous image acquisition with SPM Control Software from Park Systems using analog Aux Output along with LabOne DAQ recording internal signals.

Figure 6: some exemple data collection acquired simultaneously, including frequency shift, PFM phase of demodulator 2, PID error signal, PFM amplitude of demodulator 2 (from top left to bottom right).

Even if only one time-domain image can be displayed at once for one DAQ module, all data are recorded to be analyzed off-line. Figure 6 illustrates such simultaneously acquired images, saved in HDF5 format and later processed in Python or Matlab:

Figure 6: some exemple data collection acquired simultaneously, including frequency shift, PFM phase of demodulator 2, PID error signal, PFM amplitude of demodulator 2 (from top left to bottom right).

Conclusion

The feedback optimization procedure described here and validated experimentally can be applied to any type of linear system; only the choice of DUT model (All Pass, first- or second-order filter, Resonator Frequency) may differ depending on the system response.

Furthermore, the alignment of all internal signals and the image acquisition can be performed digitally thanks to the DAQ module and can be synchronized with any third-party equipment, thus removing the need for additional AD/DA conversion stages.

Acknowledgements

I am grateful to Ilka Hermes from Park Systems Europe (PSE) for providing us with great measuring conditions in their demo lab on a Park NX20 Atomic Force Microscope, and to Claudius Riek for on-site support and data capture.

References

Rodriguez, B.J. et al. Dual-frequency resonance-tracking atomic force microscopy. Nanotechnology, 18, 475504 (2007).

Tracking Resonance Frequency without Phase: the DFRT Method