The shift in resonant frequency of a cantilever that is vibrating in a field is given by⁸:

To get the change in vibration amplitude due to a field at the surface, the first derivative of the force with respect to the distance from the surface must be calculated.

Images of surface fields are typically measured by making two scans of the surface. In the first scan, the topography of the surface is established. In the second scan, the probe is scanned at a fixed distance from the surface (see Figure 4-12).

FIGURE 4-12 Topography is measured when the probe is vibrated near the surface, When the probe is vibrated at a fixed distance from the surface, surface fields are measured.

There are several more sophisticated techniques for measuring surface fields such as oscillating the force setpoint, or actually moving the probe up and down as it is scanned across the surface, illustrated in Figure 4-13.

FIGURE 4-13 Methods for separating the topography information from the field information in the field modes. Left: Hosaka Method. Right: Set-point Method.

The field modes require a tip that is coated with either an electrically conductive or magnetic material. Because the probes are coated, the resolution of the field modes are typically much less than the resolution of the topography modes. The decrease in resolution is correlated to the thickness of the coating. The success of making field measurements ultimately depends on getting a high quality coating on a probe. The coatings can be damaged while scanning if the probe contacts the surface.